PhD Thesis Khairy Elsayed

FACULTY OF ENGINEERING

Department of Mechanical Engineering

Analysis and Optimization of

Cyclone Separators Geometry

Using RANS and LES

Methodologies

Thesis submitted in fulfillment of the requirements for the

award of the degree of Doctor in de Ingenieurswetenschappen

(Doctor in Engineering) by

Khairy Elsayed

Brussels, October 2011

Advisor: Prof. Dr. Ir. Chris Lacor

Analysis and Optimization of Cyclone Separators Geometry Using RANS and LES Methodologies

by

Khairy Elsayed

Submitted to the Department of Mechanical Engineering, in partial fulfillment of the requirements

for the degree ofDoctor in Engineering

Vrije Universiteit BrusselOctober 2011

Advisor: Prof. Dr. Ir. Chris Lacor

Analysis and Optimization of Cyclone Separators Geometry Using RANS and LES Methodologies

Khairy Elsayed

Department of Mechanical Engineering, Vrije Universiteit BrusselPleinlaan 2, B-1050 Brussels, Belgium

Thesis submitted in partial fulfillment of the requirements for the academic degree of Doctor in Engineering

Promoter:Prof. dr. ir. Chris LacorJury: Prof. dr. ir. Johan Deconinck, voorzitterProf. dr. ir. Rik Pintelon, vice-voorzitterProf. dr. ir. Gunther Steenackers, secretarisProf. dr. ir. Gert DesmetProf. dr. ir. Harry van den Akker (Delft University of Technology, Netherlands)Prof. dr. ir. Herman Deconinck (Von Karman Institute, Belgium)

© 2011 Khairy Elsayed

2011 Uitgeverij University PressLeegstraat 15 B-9060 Zelzate Tel +32 9 342 72 25E-mail: [email protected] www.universitypress.be

Vrije Universiteit Brussel – Faculteit Ingenieurswetenschappen Pleinlaan 2 – 1050 Brussel Contact: +32 (0)2 629 39 10 http://www.vub.ac.be/IR – [email protected]

ISBN 978-94-9069-594-1

All rights reserved. No parts of this book may be reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the author.

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Abstract

The gas-solids cyclone separator is an industrial equipment that has been

widely used for more than a century. Due to its industrial relevance, a

large number of experimental, theoretical and computational studies have

been reported in the literature aimed at understanding and predicting the

performance of cyclones in terms of pressure loss and collection efficiency

(cut-off diameter). The currently used mathematical models for the predic-

tion of cyclone performance, however, exhibit limited accuracy and gener-

ality. Moreover, the cyclone performance can be calculated using the artifi-

cial neural networks approach. An alternative approach is to simulate the

gas-particle flow field in a cyclone by computational fluid dynamics (CFD).

As a result of the recent progress of computational power and numerical

techniques, CFD has been widely applied to industrial flow problems. The

cyclone performance parameters are governed by many operational pa-

rameters (e.g., the gas flow rate and temperature) and geometrical param-

eters. This study focuses only on the effect of the geometrical parameters

on the flow field pattern and performance of the tangential inlet cyclone

separators using three different approaches, the most robust mathemat-

ical models, artificial neural networks and CFD approaches. The study

was limited to reverse-flow gas-solids cyclone separators operating at low

solids loading.

The objective of this study is four-fold. First, to determine the most sig-

nificant factors affecting the cyclone performance based on the previous

studies and statistical analysis of data using response surface methodol-

ogy. Second, to study each (significant) parameter separately to obtain

more details about its effect on the flow field pattern and the performance.

Third, to obtain the most efficient cyclone design for minimum pressure

drop (using the most robust mathematical models, artificial neural net-

works and CFD dataset). Finally, to obtain the most efficient cyclone de-

sign for best performance (minimum pressure drop and minimum cut-off

diameter) using multi-objective optimization techniques with two different

optimization techniques (both the Nelder-Mead with desirability function

and the genetic algorithms (NSGA-II)).

The response surface methodology has been performed using dataset ob-

tained from the Muschelknautz method of modeling (MM) to determine

the most significant parameters. Four geometrical factors have significant

effects on the cyclone performance viz., the vortex finder diameter, the in-

i

let width, the inlet height and the cyclone total height. There are strong

interactions between the effect of inlet dimensions and the vortex finder

diameter on the cyclone performance. The same investigation has been re-

peated using artificial neural network approach based on the experimental

pressure drop. A radial basis neural network (RBFNN) is developed and

employed to model the pressure drop for cyclone separators. The neural

network has been trained and tested by experimental data available in

literature. The result demonstrates that artificial neural networks can of-

fer an alternative and powerful approach to model the cyclone pressure

drop. The analysis indicates the significant effect of the vortex finder di-

ameter and the vortex finder length, the inlet width and the total height.

Furthermore, Four mathematical models (Muschelknautz method “MM”,

Stairmand, Ramachandran and Shepherd and Lapple) have been tested

against the experimental values. The residual error (the difference be-

tween the experimental value and the model value) of the MM model is

the lowest.

The numerical simulations of cyclone flow were carried out by solving

the unsteady-state, three-dimensional Reynolds averaged Navier-Stokes

(RANS) equations combined with a closure model for the turbulent stresses

and the large eddy simulation approach. The modeling of the cyclonic flow

by computational fluid dynamics (CFD) simulation has been reported be-

fore in the literature. Using the experimental data available in literature,

a generic assessment was carried out on a number of turbulence closure

models. Only the Reynolds stress turbulence model (RSM) and large eddy

simulation (LES) captured the cyclone flow field best compared to the ex-

perimental measurements.

The CFD model also predicted the collection efficiency, where the partic-

ulate phase was treated in a Lagrangian framework by tracking a large

number of particles of different size classes through the computational do-

main. The stochastic nature of the particle motion due to the fluid turbu-

lence was taken into account by a particle dispersion model. There was a

reasonable agreement between the calculated and measured cut-off diam-

eter for both the RSM and LES results.

The effect of the cone tip-diameter on the flow field and performance of

cyclone separators was investigated because of the discrepancies and un-

certainties in the literature about its influence. Three cyclones with dif-

ferent cone tip diameters were studied using large eddy simulation (LES).

The flow field pattern has been simulated and analyzed with the aid of ve-

locity components and static pressure contour plots. The obtained results

demonstrate that the cone tip-diameter has an insignificant effect on the

collection efficiency (the cut-off diameter) and the pressure drop. The sim-

ii

ulation results agree well with the published experimental results and the

mathematical models’ trend.

The effect of the cyclone inlet dimensions on the performance and the flow

field pattern has been investigated computationally using the Reynolds

stress turbulence model (RSM) for five cyclone separators. The maximum

tangential velocity in the cyclone decreases with increasing the cyclone in-

let dimensions. Increasing the cyclone inlet dimensions decreases the pres-

sure drop. The cyclone cut-off diameter increases with increasing cyclone

inlet dimension. Consequently, the cyclone overall efficiency decreases due

to weakness of the vortex strength. The effect of changing the inlet width

b is more significant than the inlet height a, especially for the cut-off di-

ameter. The optimum ratio of inlet width to inlet height b/a is from 0.5 to

0.7.

The effect of the vortex finder dimensions (both the diameter and length)

on the performance and flow field pattern has been investigated computa-

tionally using the large eddy simulation (LES) for nine cyclone separators.

The maximum tangential velocity in the cyclone decreases with increasing

the vortex finder diameter. Whereas, a negligible change is noticed with

increasing the vortex finder length. Increasing the vortex finder length

makes a small change in both the static pressure, axial and tangential ve-

locity profiles. However, decreasing the vortex finder diameter gradually

changes the axial velocity profile from the inverted W to the inverted V pro-

file. Decreasing the cyclone vortex finder diameter increases the maximum

tangential velocity. The maximum tangential velocity approaches asymp-

totically 1.589 times the inlet velocity when decreasing the vortex finder

diameter. The Euler number (dimensionless pressure drop) decreases with

increasing the vortex finder diameter. Increasing the vortex finder length

slightly increases the Euler number. The Stokes number increases with

increasing the vortex finder diameter and slightly increases as the vortex

finder length is increased.

The effect of the cyclone height (both the barrel and cone) on the perfor-

mance and flow field pattern has been investigated computationally for six

cyclone separators. The maximum tangential velocity in the cyclone de-

creases with increasing the cyclone (barrel or cone) height. Increasing the

barrel height, makes a small change in the axial velocity, whereas increas-

ing the cone height changes it considerably. Increasing the cyclone (barrel

or cone) height decreases both the pressure drop and the cut-off diameter.

The changes in the performance beyond h/D = 1.8 are small at constant

cone height, whereas the performance improvement stops after hc/D = 4.0(Ht/D = 5.5) at constant barrel height where h is the barrel height, hc is

the cone height, Ht is the total cyclone height and D is the barrel diameter.

iii

The effect of changing the cone height on the flow pattern and performance

is more significant than that of the barrel height.

The CFD model was used to predict the pressure drop and the collection

efficiency of a range of cyclone geometries based on Stairmand’s high-

efficiency design. These predictions were used to obtain an algebraic equa-

tion that relates the performance of a cyclone to its design and a limited

set of dimensionless quantities (Euler number and Stokes number). This

approach towards predicting cyclone performance by varying many geo-

metrical parameters has not been reported before.

To obtain new optimized cyclone separators, several optimization studies

have been conducted in this thesis. Both the response surface methodology

(RSM) and the radial basis function neural network (RBFNN) have been

used as meta-models. Three different sources of data have been used to fit

the second order polynomial in case of RSM and for training the RBFNN.

These data come from analytical models, experimental measurements and

CFD simulations. Two optimization techniques have been used to optimize

the cyclone geometry for minimum pressure drop, namely, the Nelder-

Mead and the genetic algorithms techniques. To handle the bi-objective

optimization problem (both the pressure drop and the cut-off diameter),

two approaches have been applied, the desirability function and NSGA-II

techniques.

All the new optimized cyclones obtained either for single objective or for

bi-objective problems exhibit better performance than the Stairmand de-

sign. Moreover, a new correlation between the Stokes number and the

Euler number is obtained. The new correlation can be used to estimate

the Stokes number if the Euler number is known.

iv

Acknowledgments

First and foremost, I would like to gratefully acknowledge the enthusiastic

supervision of my promoter Prof. Chris Lacor who gave me the opportunity

to do a PhD under his guidance. I particularity thank him for our weekly

technical discussions, which had a major influence on this thesis. I am in-

debted to him for showing great confidence in me and always pushing me

to achieve greater heights, as well as for granting me sufficient freedom to

pursue my own ideas. I can say for sure that the past years at VUB have

been the most productive days of my learning.

I also thank all the members of my thesis committee: Prof. Johan Decon-

inck, Prof. Rik Pintelon, Prof. Gunther Steenackers, Prof. Gert Desmet,

Prof. Harry van den Akker and Prof. Herman Deconinck, whose construc-

tive criticism and valuable suggestions improved the quality of this disser-

tation.

I warmly thank the IT support of our system administrator Alain Wery.

His support is invaluable for the research at our department. I greatly

appreciate him for his good mood and everlasting patience through the

perpetual stream of requests and computer problems coming towards him.

I am yet to meet someone who is so patient and always ready to help oth-

ers. Thank you very much, Alain!

The support of our secretary Jenny D’haes started even before I arrived in

Belgium. She was there for help, starting from filling down my admission

forms in Dutch, to organizing my PhD defense. Thanks a lot Jenny. A

word of thanks should also go to Birgit Buys and my Egyptian colleague

Mahmoud El-kafafy who helped me in printing the draft version.

I am pleased to acknowledge my colleagues, Ghader Ghorbaniasl. The

many discussions on mathematics and physics I have had with Ghader

were always fruitful. Santhosh Jayaraju and Kris van den Abeele gave

v

me the template that was used for this thesis, and in doing so, saved a lot

of much needed time for me. I also enjoyed the scientific discussion with

them. In this regard, I should mention Willem Deconinck as well. I taught

the students a basic techniques in computer simulation course with him.

He made it fun to do so with his pleasant mood and sense of humor, even

though I was under the pressure of writing my thesis at that time. I would

like to thank Willem once more for proofreading of some part of my the-

sis. At my first days at VUB during which we were office mates, I have

shared many laughs and a lot of joy with Mahdi Zakyani. I am pleased

to acknowledge my present and former colleagues Dean Vucinic, Matteo

Parsani, Patryk Widera, Xiadong Wang, Vivek Agantori, Florian Krause

and Dinesh Kumar. We have nice discussions and good fun at the coffee

corner.

In addition, I am sincerely thankful to Prof. Momtaz F. Sedrak, Prof.

Ahmed F. Helal, Prof. Mohammed M. Abdelrahman, Prof. Mohammed

Fatouh and Prof. Samira Elshereef who played a major role in my scien-

tific career. I have learned a lot from them during my Master and Bachelor

studies. I consider them as good examples for Egyptian professors. All my

thanks are given to the Egyptian community at Belgium for advices, sup-

port and continuous encouragement. Special thanks are given to Omar

Ellabban, Sameh Sorror, Romany Abskharon, Ehab Khatab and Wael Mo-

hammed for the help and advice they gave me during my stay here at

Brussel, especially at my first days at Belgium.

I cannot forget to give all thanks to the spirit of my late parents who I am

indebted with all my life. Lastly, and most importantly, my utmost grat-

itude is reserved to my dear wife and my two sons Omar and Ahmed for

their patience and encouragement.

vi

Jury Members

President Prof. dr. ir. Johan Deconinck

Vrije Universiteit Brussel

Vice-President Prof. dr. ir. Rik Pintelon


Secretary Prof. dr. ir. Gunther Steenackers


Internal Member Prof. dr. ir. Gert Desmet


External members Prof. dr. ir. Harry van den Akker

Delft University of Technology

Prof. dr. ir. Herman Deconinck

Von Karman Institute

Promoter Prof. dr. ir. Chris Lacor


vii

Contents

1 Introduction 1

1.1 Overview of dust collectors . . . . . . . . . . . . . . . . . . . . 1

1.2 Cyclone separators: types and principals . . . . . . . . . . . 2

1.2.1 Advantages and disadvantages of cyclones . . . . . . 5

1.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Principals of cyclonic separation . . . . . . . . . . . . 7

1.2.4 Factors affecting the cyclone performance . . . . . . . 10

1.3 Motivation of this work . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . 13

2 Literature Review 15

2.1 Classification of study approaches . . . . . . . . . . . . . . . 15

2.2 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Experimental methods . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Computational fluid dynamics (CFD) . . . . . . . . . . . . . . 17

2.5 Discrepancy in the previous studies . . . . . . . . . . . . . . 18

2.5.1 The cone tip diameter . . . . . . . . . . . . . . . . . . . 18

2.5.2 The dust outlet geometry . . . . . . . . . . . . . . . . . 19

2.5.3 The inlet dimensions . . . . . . . . . . . . . . . . . . . 19

2.5.4 The vortex finder dimensions . . . . . . . . . . . . . . 20

2.5.5 The cyclone heights . . . . . . . . . . . . . . . . . . . . 21

2.5.6 Previous optimization studies . . . . . . . . . . . . . . 21

2.6 Summary and research plan . . . . . . . . . . . . . . . . . . . 23

3 Governing Equations 25

3.1 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 The governing equations for the gas phase . . . . . . . . . . 28

3.2.1 Reynolds averaged Navier Stokes (RANS) . . . . . . . 29

3.2.2 Reynolds stress model (RSM) . . . . . . . . . . . . . . 31

3.2.3 Large eddy simulation (LES) . . . . . . . . . . . . . . 32

3.3 Discrete phase modeling . . . . . . . . . . . . . . . . . . . . . 37

ix

3.3.1 Governing equations for the particles . . . . . . . . . 37

3.3.2 Modeling the particle phase . . . . . . . . . . . . . . . 40

3.3.3 Stochastic trajectory approach . . . . . . . . . . . . . 41

4 Sensitivity Analysis of Geometrical Parameters 45

4.1 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.1 Response surface methodology (RSM) . . . . . . . . . 46

4.1.2 Design of experiment (DOE) . . . . . . . . . . . . . . . 47

4.1.3 Analysis of response surfaces . . . . . . . . . . . . . . 48

4.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 The cone-tip diameter . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.1 Numerical simulation . . . . . . . . . . . . . . . . . . 54

4.2.2 Results and discussion . . . . . . . . . . . . . . . . . . 63

4.2.3 The flow pattern in the three cyclones . . . . . . . . . 64

4.2.4 Comparison with mathematical models . . . . . . . . 70

4.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 The dust outlet geometry . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 Numerical simulation . . . . . . . . . . . . . . . . . . 73

4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 The Vortex Finder Dimensions 89

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2 Numerical settings . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2.1 Configuration of the tested cyclones . . . . . . . . . . 92

5.2.2 Solver settings . . . . . . . . . . . . . . . . . . . . . . . 92

5.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . 93

5.2.4 Grid independency study . . . . . . . . . . . . . . . . . 93

5.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . 95

5.3.1 The axial variation . . . . . . . . . . . . . . . . . . . . 95

5.3.2 The flow pattern . . . . . . . . . . . . . . . . . . . . . . 96

5.3.3 The radial variation . . . . . . . . . . . . . . . . . . . . 99

5.3.4 The cyclone performance . . . . . . . . . . . . . . . . . 102

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 The Inlet Dimensions 109

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109


6.2.1 Configuration of the five cyclones . . . . . . . . . . . . 111


6.2.3 Selection of the time step . . . . . . . . . . . . . . . . . 111

6.2.4 CFD grid . . . . . . . . . . . . . . . . . . . . . . . . . . 112

x

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3.1 The axial variation of the flow properties . . . . . . . 113

6.3.2 The flow pattern . . . . . . . . . . . . . . . . . . . . . . 115

6.3.3 The cyclone performance . . . . . . . . . . . . . . . . . 120

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7 The Cyclone Height 127

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127


7.2.1 Configuration of the tested cyclones . . . . . . . . . . 128


7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.3.1 The axial variation . . . . . . . . . . . . . . . . . . . . 130

7.3.2 The radial variation . . . . . . . . . . . . . . . . . . . . 131

7.3.3 The flow pattern . . . . . . . . . . . . . . . . . . . . . . 135

7.3.4 The performance . . . . . . . . . . . . . . . . . . . . . 137

7.3.5 The cone height versus the barrel height . . . . . . . 141

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8 Optimization 147

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.2 Single-objective using MM model . . . . . . . . . . . . . . . . 149

8.2.1 CFD comparison between the two designs . . . . . . . 150

8.2.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 154

8.3 Single-objective using RBFNN . . . . . . . . . . . . . . . . . 158

8.3.1 Radial basis function neural networks (RBFNN) . . . 158

8.3.2 Evaluation of different mathematical models . . . . . 164

8.3.3 Design of experiment (DOE) . . . . . . . . . . . . . . . 169

8.3.4 CFD Comparison between the two designs . . . . . . 179

8.3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 185

8.4 Multi-objective optimization using GA . . . . . . . . . . . . . 186

8.4.1 Artificial neural network (ANN) approach . . . . . . . 187

8.4.2 Single objective optimization . . . . . . . . . . . . . . 194

8.4.3 Optimal cyclone design for best performance . . . . . 202

8.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 204

8.5 Multi-objective optimization using CFD data . . . . . . . . . 215

8.5.1 Design variables and approaches . . . . . . . . . . . . 215

8.5.2 The desirability function . . . . . . . . . . . . . . . . . 220

8.5.3 Artificial neural network (ANN) approach . . . . . . . 231

8.5.4 Optimization Using Genetic Algorithms . . . . . . . . 235

8.5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 239

xi

9 Conclusions and Future Directions 243

9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

9.1.1 The most significant geometrical factors . . . . . . . . 243

9.1.2 The impact of geometry . . . . . . . . . . . . . . . . . 244

9.1.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . 245

9.1.4 Multi-objective optimization . . . . . . . . . . . . . . . 246

9.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 248

A Mathematical models 251

A.1 General assumptions . . . . . . . . . . . . . . . . . . . . . . . 251

A.2 Barth model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

A.3 The Muschelknautz method of modeling (MM) . . . . . . . . 256

A.4 Stairmand model for pressure drop . . . . . . . . . . . . . . . 259

A.5 Purely empirical models for pressure drop . . . . . . . . . . . 260

A.6 Iozia and Leith model for the cut-off diameter . . . . . . . . 261

A.7 Rietema model for cut-off diameter . . . . . . . . . . . . . . . 262

B Optimization Techniques 263

B.1 Nelder-Mead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

B.2 Genetic algorithms (GA) . . . . . . . . . . . . . . . . . . . . . 266

B.2.1 Description of the genetic algorithm process . . . . . 268

B.2.2 Genetic operators . . . . . . . . . . . . . . . . . . . . . 268

B.3 Multi-objective optimization . . . . . . . . . . . . . . . . . . . 272

Bibliography 281

xii

Nomenclature

Upper-Case Roman

Ai inlet cross sectional area [m]

AR total inside area of the cyclone [m2]

Bc cyclone cone-tip diameter [m]

CD particle drag coefficient

Cin inlet dust concentration, [kg/m3]

D cyclone barrel diameter [m]

Dx cyclone vortex finder diameter [m]

Eu Euler number [-]

Fdrag drag force [N]

Frx Froude number (inertia force / gravitational force) [-]

HCS length of the control surface [m]

Ht cyclone total height [m]

K fluctuating kinetic energy [m2/s2]

K vortex finder entrance factor

Li distance between the inlet section and cyclone center [m]

Le distance between the outlet section and the barrel top [m]

Ln cyclone natural length [m]

P fluctuating kinetic energy production [m2/s3]

P mean pressure [N/m2]

Pr Prandtl number [-]

Qin gas volume flow rate [m3/s]

R cyclone radius [m]

R2 coefficient of multiple determination

Rb dust outlet radius [m]

Rij Reynolds stress tensor [m2/s2]

Rep Reynolds number based on the relative particle velocity [-]

ReR cyclone body Reynolds number [-]

Rm geometric mean radius [m]

Rx vortex finder radius [m]

xiii

S vortex finder length [m]

Sg geometrical swirl number [-]

Stk50 Stokes number at the cut-off diameter [-]

V cyclone volume [m3]

Lower-Case Roman

a cyclone inlet height [m]

acceleration [m/s2]

b cyclone inlet width [m]

d50 cut-off diameter [µm]

dp particle diameter [m]

f total friction factor [-]

fair gas friction factor [-]

fr friction factor due to wall roughness [-]

fsm friction factor for smooth wall [-]

gi acceleration due to gravity in i direction [m/s2]

h barrel height [m]

hc cone height [m]

k turbulent kinetic energy [m2/s2]

ks wall relative roughness of the cyclone wall [m]

m mass [kg]

mp dust mass flow rate [kg/s]

p static pressure [N/m2]

t flow physical time [s]

tres flow average residence time [s]

ui flow velocity component in i direction [m/s]

ui mean velocity [m/s]

u′i fluctuating velocity component in i direction [m/s]

upi particle velocity in i direction [m/s]

vx mean gas velocity through the vortex finder [m/s]

vzw wall axial velocity [m/s]

x particle diameter [µ m]

x50 cut-off diameter [µm]

xi position [m]

Upper-Case Greek

∆ filter width [m]

∆P pressure drop in the cyclone [N/m2]

∆Pbody pressure drop in the cyclone body [N/m2]

∆Px pressure drop in the vortex finder [N/m2]

xiv

Lower-Case Greek

α gas moment-of-momentum ratio at inlet [-]

β ratio of inlet width to the cyclone radius [-]

δij Kronecker delta [-]

ε turbulence dissipation rate [m2/s3]

εν viscous dissipation [m2/s3]

µ dynamic viscosity [kg/(m s)]

µt turbulent (eddy) viscosity [kg/(m s)]

ν kinematic viscosity (ν = µ/ρ) [m2/s]

νt turbulent (eddy) kinematic viscosity (νt = µt/ρ) [m2/s]

ξ spatial separation [m]

ρ gas density [kg/m3]

ρbulk bulk density of the solid [kg/m3]

ρp particle density [kg/m3]

ρstr bulk density of the strand layer at the walls [kg/m3]

σij viscous stress tensor [N/m2]

τij subgrid scale stress tensor [N/m2]

Subscripts

CS at the control surface

g gas

in at the inlet surface

p particle properties

θ angular (tangential component)

w close to the wall

Abbreviations

ANOVA Analysis Of Variance

ANN Artificial Neural Networks

CFD Computational Fluid Dynamics

CFL Courant-Friedrichs-Lewy number

DNS Direct Numerical Simulation

DOE Design Of Experiment

DPM Discrete Phase Modeling

GA Genetic Algorithm

xv

GCI Grid Convergence Index

GEC Grade Efficiency Curve

LDA Laser Doppler Anemometry

LES Large Eddy Simulation

MM Muschelknautz Method of modeling

PIV Particle Image Velocimetry

RANS Reynolds Averaged Navier-Stokes

RBFNN Radial Basis Function Neural Network

RNG Renormalization Group

RSM Response Surface Methodology

RSM Reynolds Stress turbulence Model

SGS Subgrid Scale Model

xvi

Chapter 1

Introduction

1.1 Overview of dust collectors

There are four principal types of industrial dust collectors [77] namely,

inertial separators, fabric collectors, wet scrubbers and electrostatic pre-

cipitators. The selection of one type depends mainly on the particle size as

shown in Fig. 1.1. The classification of different dust collectors is shown in

Fig. 1.2.

The inertial separators separate dust from gas streams using a combina-

tion of forces, such as centrifugal, gravitational, and inertial. These forces

move the dust to an area where the forces exerted by the gas stream are

minimal. The separated dust is moved by gravity into a hopper, where

it is temporarily stored. The three primary types of inertial separators

are settling chambers, baffle chambers, and centrifugal collectors (e.g., cy-

clone separator). A settling chamber consists of a large box installed in the

ductwork. The sudden expansion at the chamber reduces the speed of the

dust-filled airstream and heavier particles settle down. Settling chambers

are simple in design and can be manufactured from almost any material.

However, they are seldom used as primary dust collectors because of their

large space requirements and low efficiency. A practical use is as preclean-

ers for more efficient collectors. Baffle chambers use a fixed baffle plate

that causes the conveying gas stream to make a sudden change of direc-

tion. Large-diameter particles do not follow the gas stream but continue

into a dead air space and settle. Baffle chambers are used as precleaners

for more efficient collectors.

Fabric collectors are commonly known as baghouses. Fabric collectors use

filtration to separate dust particulates from dusty gases. They are one

1

Chapter 1. Introduction

Particle size (micron)

10-3

10-2

10-1

100

101

102

103

Settling chambers

Cyclone separator

Liquid scrubbers

Filters

Electrostatic precipitators

Figure 1.1: Suitable methods for removing particles from a gas stream [128]

of the most efficient types of dust collectors available and can achieve a

collection efficiency of more than 99% for very fine particulates.

Dust collectors that use liquid are commonly known as wet scrubbers. In

these systems, the scrubbing liquid (usually water) comes into contact with

a gas stream containing dust particles. The greater the contact of the gas

and liquid streams, the higher the dust removal efficiency.

The electrostatic Precipitators use electrostatic forces to separate dust par-

ticles from exhaust gases. A number of high-voltage, direct-current dis-

charge electrodes are placed between grounded collecting electrodes. The

contaminated gases flow through the passage formed by the discharge and

collecting electrodes. The airborne particles receive a negative charge as

they pass through the ionized field between the electrodes. These charged

particles are then attracted to a grounded or positively charged electrode

and adhere to it [77].

1.2 Cyclone separators: types and principals

The gas cyclones belong to the type of centrifugal separators. A gas cyclone

is a stationary mechanical device that utilizes centrifugal force to separate

solid or liquid particles from a carrier gas. The flow enters near the top

through the tangential inlet, which gives rise to an axially descending spi-

ral of gas and a centrifugal force field that causes the incoming particles

to concentrate along, and spiral down, the inner walls of the cyclone sep-

arator. The collected particulates are allowed to exit out an underflow

2

1.2. Cyclone separators: types and principals

Figure 1.2: Classification of dust collectors

3


Figure 1.3: Typical cyclone separator

pipe while the gas phase reverses its axial direction of flow and exits out

through the vortex finder (gas outlet tube) [77]. Figure 1.3 shows a typical

cyclone separator.

The cyclone separator is one of the most efficient and robust dust sepa-

rators. Its robustness results from lack of moving parts and the ability

to withstand harsh operating environments. Moreover, cyclones are well

suited for high pressure and temperature applications.

Centrifugal collectors use cyclonic action to separate dust particles from

the gas stream. In a typical cyclone, the dust gas stream enters tangen-

tially forcing the flow into a spiral movement. The centrifugal force cre-

ated by the circular flow throws the dust particles toward the wall of the

cyclone. After striking the wall, the particles fall into a hopper located

underneath. The most common types of centrifugal, or inertial, collectors

in use today are single-cyclone separators and multiple-cyclone separators

(multiclone). Single-cyclone separators create a dual vortex to separate the

dust from the gas. The main vortex spirals downward and carries most of

the heavier particles. The inner vortex, created near the bottom of the

4


cyclone, spirals upward and carries finer dust particles. Multiple-cyclone

separators consist of a number of small-diameter cyclones, operating in

parallel and having a common gas inlet and outlet. Multiclones operate on

the same principle as cyclones–creating a main downward vortex and an

ascending inner vortex.

1.2.1 Advantages and disadvantages of cyclones

Compared with the other separation devices, the cyclone separators ad-

vantages are [77]:

• the collected product remains dry and, normally useful.

• low capital investment and maintenance costs in most applications.

• very compact in most applications.

• can be used under extreme processing conditions, in particular at

high temperatures and pressures and with chemically aggressive feeds.

• no moving parts.

• very robust.

• can be constructed from most any material suitable for the intended

service including plate steel, casting metals, alloys, aluminum, plas-

tics, ceramics, etc.

• can be equipped with erosion or corrosion resistant or ‘particle re-

pelling type liners, such as Teflon. Internal surfaces may be electro

polished to help combat fouling.

• can be fabricated from plate metal or, in the case of smaller units,

cast in molds.

• can, in some processes, handle sticky or tacky solids with proper liq-

uid irrigation.

• can separate either solids or liquid particulates; sometimes both in

combination with proper design.

Some disadvantages of cyclones are [77]:

• low efficiency for particle sizes below their ‘cut-off diameter when op-

erated under low solids-loading conditions.

• usually higher pressure loss than other separator types, including

bag filters and low pressure drop scrubbers.

• subject to erosive wear and fouling if solids being processed are abra-

sive or ‘sticky.

• can operate below expectations if not designed and operated properly.

Although this problem, as well as the erosion and fouling problem

mentioned above, is not unique to cyclones.

5


Figure 1.4: Applications of cyclone separator

1.2.2 Applications

Due to the mentioned advantages, cyclones have found application in vir-

tually every industry where there is a need to remove particles from a gas

stream. Figure 1.4 presents some examples of cyclones industrial applica-

tions with wide range of sizes, locations and applications. Today, cyclone

separators are found in:

• ship unloading installations

• power stations

• spray dryers

• fluidized bed and reactor riser systems (such as catalytic crackers

and cockers)

• synthetic detergent production units

• food processing plants

• crushing, separation, grinding and calcining operations in the min-

eral and chemical industries

• fossil and wood-waste fired combustion units (normally upstream of

6


a wet scrubber, electrostatic precipitator or fabric filter)

• vacuum cleaning machines

• dust sampling equipment

Cyclones have also been used to classify solids on the basis of their charac-

teristic such as their mass, density, size, or shape. Because of their simple

construction and high reliability, cyclones are also used very effectively

to separate two-phase gas-liquid mixtures, such as the entrained droplets

exiting a venturi scrubber or other types of scrubber. Other examples in-

clude the removal of water droplets from steam generators and coolers and

oil-mist from the discharge of air compressors. Likewise, they have been

widely applied in process machinery to remove entrained oil and hydrocar-

bon droplets generated from spraying, injection, distillation, or most any

process that results in the production of entrained droplets or a two-phase

mixture. They have even been used as inlet devices to prevent foaming in

gravity separation drums [77].

1.2.3 Principals of cyclonic separation

In centrifugal devices, the dust-laden gas is initially brought into a swirling

motion. The dust particles are slung outward to the wall, and transported

downward to the dust outlet by the downwardly directed gas flow near the

wall. A sketch of a standard reverse-flow, cylinder-on-cone cyclone with a

tangential, slot-type inlet is shown in Fig. 1.5.

For the standard, reverse-flow cyclone, (with a so-called slot type of entry)

the swirling motion is brought about by designing the inlet in such a man-

ner that it forces the gas to enter the unit on a tangent to the inner body

wall. The inlet is normally of rectangular cross section. As the gas swirls,

it moves axially downwards in the outer part of the separation space. In

the conical part of the cyclone, the gas is slowly forced into the inner re-

gion of the cyclone, where the axial movement is upwardly directed. This

flow pattern is often referred to as a double vortex: an outer vortex with

a downwardly directed axial flow and an inner one with an upwardly di-

rected flow. The gas exits the cyclone through the so-called vortex finder,

which extends downward from the center of the roof. This outlet pipe goes

by many different names, with vortex tube and dip-tube being the most

common, aside from the vortex finder [77]. The particles in the inlet gas

are slung outwards to the wall in the centrifugal field, and are transported

to the dust exit by the downwardly directed gas flow near the wall. Below

more details of the flow pattern in the separation space will be given.

The geometry of a cyclone with a slot type inlet is determined by the fol-

lowing eight dimensions as shown in Fig. 1.5:

7


aS

h

D

b

Ht

Dx

Bc

hc

Figure 1.5: Sketches of a reverse-flow, cylinder-on-cone cyclone with a tangential

inlet. The geometrical notation is indicated in the right sketch

1. the body diameter (barrel diameter) D2. the total height of the cyclone (from roof to dust exit) Ht

3. the vortex finder diameter Dx

4. the vortex finder length (from the roof of the separation space) S5. the inlet height a6. the inlet width b7. the height of the conical section hc or the height of the cylindrical

section h8. the cone-tip diameter (dust exit diameter) Bc

1.2.3.1 Real vortex flow

Swirling flow, or vortex flow, occurs in different types of equipment, such as

cyclones, hydrocyclones, spray dryers and vortex burners [77]. Two basic

types of swirling flows can be distinguished:

1. forced vortex flow, which is a swirling flow with the same tangential

velocity distribution as a rotating solid body

2. free vortex flow, which is the way a frictionless fluid would swirl.

8


Tangential velocity

Distance from center

Forced vortex

Free vortexRankine vortex

Figure 1.6: The tangential velocity distribution in a real vortex [77]

The tangential velocity in such a swirl is such that the moment-of-

momentum of fluid elements is the same at all radii.

The tangential velocity distribution in a real swirling flow is intermediate

between these two extremes. Now imagine first that the swirling fluid has

an infinite viscosity (behaves like a solid body). Hence, no shearing motion

exists between fluid layers at different radii. In this case, the fluid ele-

ments at all radial positions are forced to have the same angular velocity

Ω which equals vθ/r where vθ is the tangential velocity. This is the forced

vortex flow or solid-body rotation:

vθ = Ωr (1.1)

In the other extreme, if the swirling fluid has no viscosity, the motion of

a given fluid element is not influenced by the neighboring elements at

smaller and larger radii. If in such a fluid, we bring an element to a

smaller radius, its tangential velocity will increase, since its moment-of-

momentum (mvθr) will be conserved. Such a vortex is called a free or

frictionless vortex. In such a flow, we have rvθ = C, with C a constant,

so that:

vθ =C

r(1.2)

This is the second basic swirl flow. A real swirling flow normally has a

core of near solid-body rotation surrounded by a region of near loss-free

rotation as sketched in Fig. 1.6. This is called a Rankine vortex.

The flow and pressure distribution within cyclones is more easily under-

stood if we make clear the relation between static and dynamic pressures;

p and 1/2ρv2, respectively, with ρ the density. The well-known Bernoulli

9


equation for steady flow of a frictionless, constant density fluid, which can

be derived from the Navier-Stokes equations, states that:

p

ρ+ gh+

1

2v2 = constant along a streamline (1.3)

In this equation, we recognize the static and dynamic pressures (the lat-

ter is often called the velocity head) as the first and third terms on the

left-hand side. They have been divided by the fluid density. This equa-

tion shows that static and dynamic pressures can be interchanged in the

flow field [77]. In areas where the velocity is high, the static pressure will

be low and vice versa. It is especially important to appreciate this interde-

pendence between static and dynamic pressure when dealing with swirling

flows [77]. The left-hand side of Eq. 1.3 is sometimes called Bernoulli’s tri-

nomial. The second term is unimportant relative to the two others when

discussing gas cyclones, since the fluid density is relatively low, and height

differences not very large. In an actual flow situation, the fluid is not fric-

tionless. Frictional dissipation of mechanical energy will therefore cause

Bernoullis trinomial to decrease in the flow direction, i.e. the trinomial

is no longer constant, but decreases along a streamline. Frictionless flow

is, nevertheless, a reasonably good approximation in the outer part of the

swirl in a cyclone; Bernoulli’s trinomial does not change very much there

[77].

1.2.4 Factors affecting the cyclone performance

Figure 1.7 indicates the possible factors affecting the cyclone performance

and flow pattern. These factors can be sub-classified as follows:

1. Cyclone dimensions

• Cyclone diameter

• Inlet height

• Inlet width

• Vortex finder diameter

• Vortex finder length

• Cylinder height

• Cyclone total height

• Cone tip diameter

2. Particle properties

• Density

• Shape

• Diameter and distribution

10

1.3. Motivation of this work

• Mass loading

3. Gas properties

• Velocity

• Density

• Viscosity

• Temperature

• Pressure

4. Other factors

• Wall roughness

• Shape of vortex finder

• Eccentricity of vortex finder

1.3 Motivation of this work

In spite of the fact that the use of cyclone separators is common in many

industrial applications, an accurate prediction tool for their behaviors is

still not available. The challenge of this work is therefore, a detailed study

of the flow phenomena in cyclones and the design of an optimum cyclone

separator (minimum pressure drop and maximum collection efficiency).

The cyclone performance parameters are governed by many operational

parameters (e.g. the gas flow rate and temperature) and geometrical pa-

rameters. This study focuses only on the effect of the geometrical param-

eters on the flow field pattern and on the performance of the tangential

inlet cyclone separators.

For that the specific goals of the work are the following:

• Determine the most significant factors affecting the cyclone perfor-

mance based on previous studies and statistical analysis of data.

• Study each (significant) parameter separately to obtain more details

on its effect on the flow field pattern.

• To obtain the most efficient cyclone design for minimum pressure

drop (using the most robust mathematical models, artificial neural

networks and CFD data sets).

• To obtain the most efficient cyclone design for best performance (min-

imum pressure drop and minimum cut-off diameter) using multi-

objective optimization techniques.

11

Ch

ap

ter

1.

Intro

du

ction

Figure 1.7: Cause and effect plot for cyclone separator

12

1.4. Outline of the thesis

The study is limited to reverse flow gas cyclone separators operating at low

mass loading.

1.4 Outline of the thesis

The thesis is organized in ten chapters. After an introduction and overview

given in the previous sections, chapter 2 deals with the different study

approaches of cyclone separators. In addition, an overview is presented of

work reported in the literature on the application of CFD in cyclone mod-

eling. Moreover, the discrepancy of the results from the previous studies

has been discussed in details. Furthermore, a summary of the previous op-

timization studies is given. Chapter 3 presents the governing equations

for the carrier (gas) and the discrete phase (solids) in detail. The applica-

tion of response surface methodology and design of experiment statistical

techniques to estimate the most significant geometrical parameters are

given in chapter 4. Furthermore, chapter 4 deals with the uncertainty

of the significant effect of cone tip diameter and the necessity of including

the dust outlet geometry in the simulation domain. Chapter 5 presents

the study of the effect of the vortex finder dimensions, whereas chapter

6 deals with the effect of the inlet dimensions. The effects of both the

cone height and the barrel height on the flow field and performance are

discussed in chapter 7. The new optimized cyclones for minimum pres-

sure drop and best performance using different techniques are analyzed in

chapter 8. The used dataset are collected from different sources; math-

ematical models calculations, artificial neural network models and CFD

simulations data. Chapter 9 summarizes the main conclusions and some

future directions. The thesis contains two appendices. Appendix A, deals

with the details of eight mathematical models used to predict the cyclone

performance parameters. The details of the two optimization techniques

used in this thesis are given in appendix B.

13


14

Chapter 2

Literature Review

The most important parameter that affects the cyclone performance and

flow pattern is the cyclone geometry. For reversed flow cyclones, there are

seven geometrical parameters, viz. the inlet height a, the inlet width b,the vortex finder diameter (gas outlet tube diameter) Dx, the vortex finder

length S, the cylindrical part height h, the cyclone total height Ht, and the

cone-tip diameter Bc. All these dimensions are expressed in terms of bar-

rel diameter D as shown in Fig. 2.1. The two performance indicators used

are the pressure drop and the particle separation (collection) efficiency.

The latter is normally expressed as a “grade efficiency curve” a graph of

the collection efficiency against the particle diameter. For low mass load-

ing cyclone separators, the cut-off diameter x50 is usually given instead of

grade efficiency curves.

2.1 Classification of study approaches

There is a widespread literature on the effect of cyclone geometry on per-

formance, using one or more of the four main approaches of study, which

are:

1. Analytical methods (mathematical models), which can be classified

into [194]:

(a) theoretical and semi-empirical models

(b) statistical models

2. Experimental measurements

3. Computational fluid dynamics (CFD) simulations

4. Artificial neural networks (ANN) approach

Recently, optimization studies based on data available from one of the

main four approaches have been performed. Also artificial neural net-

15

Chapter 2. Literature Review

Figure 2.1: The cyclone separator dimensions

works become a tool to study the effect of cyclone geometry on performance.

2.2 Mathematical models

The theoretical models were developed by many researchers e.g., Shep-

herd and Lapple [157], Alexander [1], First [58], Stairmand [166], Barth

[9], Avci and Karagoz [5], Zhao [192], Karagoz and Avci [90] and Chen and

Shi [22]. These models were derived from physical descriptions and math-

ematical equations. They require a very detailed understanding of gas

flow pattern and energy dissipation mechanisms in cyclones. In addition,

due to using different assumptions and simplifying conditions, different

theoretical or semi-empirical models can lead to significant differences be-

tween predicted and measured results. Predictions by some models are

sometimes twice the experimental values [172].

Since the first application of aerocyclones in 1886 [3], theories for the es-

timation of both particle collection efficiency and pressure drop of cyclone

have been developed by many contributors using different methods with

various simplifying assumptions. During the past 50 years, interest in

particle collection and pressure theories has steadily increased [196]. The

most widely used mathematical models for the cut-off diameter and pres-

sure drop estimation are:

• Barth model [9]

• The Muschelknautz method of modeling (MM) [29, 77, 114–116, 174,

175]

16

2.3. Experimental methods

• Stairmand model [165]

• Shepherd and Lapple model [157]

• Casal and Martinez-Bent model [21]

• Ramachandran model [139]

• Iozia and Leith model [84]

• Rietema model [142]

The interested reader can refer to appendix A for more details about these

eight models.

2.3 Experimental methods

There are numerous experimental investigations performed on the cyclone

separators. The majority of these studies used either laser doppler anemom-

etry (LDA) or particle image velocimetry (PIV) to obtain the flow field pat-

tern. Some of the studies only measured the pressure drop and collec-

tion efficiency without any details of the flow fields. For instance, Dirgo

and Leith [43] measured the collection efficiency and pressure drop for the

Stairmand high efficiency cyclone at different flow rates. Hoekstra et al.

[75] measured the mean and fluctuating velocity components for gas cy-

clones with different geometric swirl numbers (SG = πDxD4ab ) by means of

the laser doppler anemometry technique. The experimental data shows

the strong effect of the geometric swirl number on the mean flow charac-

teristics, in particular with respect to vortex core size and the magnitude of

the maximum tangential velocity. It is shown that the forced vortex region

of the flow is dominated by the so-called precessing vortex core.

Hoffmann et al. [76] investigated the effect of the cyclone length on the

separation efficiency and the pressure drop experimentally and theoret-

ically by varying the length of the cylindrical segment of a cylinder-on-

cone cyclone. They found for cyclone lengths from 2.65 to 6.15 cyclone

diameters, a marked improvement in cyclone performance with increas-

ing length up to 5.5 cyclone diameters; beyond this length the separation

efficiency was dramatically reduced. For the interested reader, other ex-

perimental results on cyclones can be found in [36, 43, 66, 74–76, 93, 102,

105, 121–124, 127, 138, 152, 163, 184].

2.4 Computational fluid dynamics (CFD)

Boysan et al. [14] presented the first CFD investigation in the field of cy-

clone separators. From that time, the CFD technique becomes a widely

used approach for the flow simulation and performance estimation for cy-

clone separators. For example, Griffiths and Boysan [68] computationally

17


investigated three cyclone samplers. They reported that the CFD predicted

pressure drops are in excellent agreement with the measured data. The

CFD modeling approach is also able to predict the features of the cyclone

flow field in great details, which providing a better understanding of the

fluid dynamics in cyclone separators [68]. Consequently, CFD approach is

a reliable and relatively inexpensive method of examining the effects of a

number of design changes. Moreover, this makes the CFD methods rep-

resent a cost-effective route for geometry optimization in comparison with

the experimental approach. Another example, Gimbun et al. [64] success-

fully applied CFD to predict and to evaluate the effects of temperature

and inlet velocity on the pressure drop of gas cyclones [194]. The success-

ful application of CFD technique in different studies in cyclone separators

has been reported by many researchers [e.g., 6, 11, 23, 49, 50, 52, 62–

64, 146, 147, 186, 198]. Nevertheless, CFD is still more expansive in com-

parison with the mathematical models approach. The main reasons behind

the cost of the CFD approach with respect to the mathematical methods

are:

1. In essence, the CFD process requires expert intervention by an ex-

pert researcher at every stage (mesh generation, solver settings and

post processing).

2. The license cost of the grid generator, solver and post processor.

3. The running cost especially for unsteady simulations which need also

parallel processing.

4. CFD results always need (i) validation with experimental results (ii)

perform the same simulation on different grids to be sure that the

obtained results are grid independent.

2.5 Discrepancy in the previous studies

In this section, the effect of the geometrical parameters on the performance

in terms of the two indicators; the pressure drop and the separation effi-

ciency (cut-off diameter) will be discussed briefly based on the available

literatures. More details will be presented in the subsequent chapters.

2.5.1 The cone tip diameter

Very little information is available on the effects of changing the cone bot-

tom (tip) diameter, which determines the cone shape if other cyclone di-

mensions are fixed [184]. Regarding this effect, discrepancies and uncer-

tainties exist in the literature. Bryant et al. [17] observed that if the vor-

tex touched the cone wall, particle re-entrainment occurred and efficiency

decreased, so collection efficiency will be lower for cyclones with a small

18

2.5. Discrepancy in the previous studies

cone opening (cone tip diameter). While according to Xiang et al. [184], a

cone is not an essential part for cyclone operation, although it serves the

practical purpose of delivering collected particles to the central discharge

point. However, Zhu and Lee [200] stated that the cone provides greater

tangential velocities near the bottom for removing smaller particles.

2.5.2 The dust outlet geometry

Conventional cyclones always have a dustbin attached to the cone to collect

the separated solid particles. When a gas flow stream enters the dustbin

(closed at bottom), some of the flow will return into the cone and distribute

some of the separated particles. This phenomena called “re-entrainment”

and it will affect the separation efficiency of the cyclone [138]. There has

been little work concerning the dust outlet geometries [e.g., 40, 47, 78,

123].

Regarding this influence, discrepancies and uncertainties exist in the lit-

erature. Xiang and Lee [186] reported that the dustbin connected to the

cyclone should be incorporated in the flow domain as it affects the results

obtained. On the other hand, numerous studies were performed with-

out dustbin [e.g., 159, 178] with good matching with experimental results.

Obermair et al. [123] performed cyclone tests with five different dust outlet

geometries to find the influence of the dust outlet geometry on the sepa-

ration process. They showed that separation efficiency can be improved

significantly by changing the dust outlet geometry, and they reported that

further research is needed to clarify precise effects of dust outlet geometry.

The effect of a dipleg (a vertical tube between the cyclone and the dustbin)

was posed and investigated by several researchers [e.g., 78, 92].

2.5.3 The inlet dimensions

The effects of the inlet dimensions on the cyclone performance (pressure

drop and cut-off diameter) have been reported in many articles. Casal

and Martinez-Benet [21] proposed the following empirical formula for the

dimensionless pressure drop (Euler number),

Eu = 11.3

(ab

Dx

)2

+ 2.33 (2.1)

implying proportionality with the square of the inlet area. On the other

hand, Ramachandran et al. [139] proposed,

Eu = 20

a b

D2x

SD

HD

hD

Bc

D

1/3

(2.2)

19


i.e., a linear relation with the inlet area. Iozia and Leith [84, 85] presented

a correlation to estimate the cut-off diameter x50 and found proportionality

to (ab)0.61. The importance of the inlet dimensions becomes clear after the

study of the natural length (or vortex length) by several researchers, e.g.,

Alexander [1]. The cyclone has two spiral motions, outer and inner. In the

reverse flow cyclone, the outer vortex weakens and changes its direction

at a certain axial distance Ln from the vortex finder [29]. This distance

is usually called the turning length, natural length or vortex length of the

cyclone. The inlet area is one of the relevant parameters influencing the

natural length. Alexander [1] found that Ln decreased proportionally to

the inlet area but the opposite trend has been also reported [29].

Numerous studies have been performed for the effect of geometrical pa-

rameters on the flow pattern and performance [e.g., 15, 62, 102, 140, 184]

while the effect of cyclone inlet dimensions remained largely unexplored.

The articles investigating the effect of cyclone geometry report only briefly

on the effect of inlet section dimensions without sufficient details about the

effects on the flow pattern and velocity profiles. The new trend is to study

the multi-inlet cyclone [e.g., 103, 187, 195].

2.5.4 The vortex finder dimensions

The vortex finder size is an especially important dimension, which sig-

nificantly affects the cyclone performance as its size plays a critical role

in defining the flow field inside the cyclone, including the pattern of the

outer and inner spiral flows. Saltzman and Hochstrasser [151] studied

the design and performance of miniature cyclones for repairable aerosol

sampling, each with a different combination of three cyclone cone lengths

and three gas outlet diameters. Iozia and Leith [84] optimized the cyclone

design parameters, including the gas outlet diameter, to improve the cy-

clone performance using their optimization program. Kim and Lee [95]

described how the ratio of the diameters of cyclone body D and the vortex

finder Dx affected the collection efficiency and pressure drop of cyclones,

and proposed an energy-effective cyclone design. Moore and Mcfarland

[111] also tested cyclones, with six different vortex finders, and concluded

that the variation in the gas outlet diameter under the constraint of a

constant cyclone Reynolds number produced a change in the aerodynamic

particle cut-off diameter. Recently, Hoekstra [74] investigated the effect of

gas outlet diameter on the velocity profile using 2-D axisymmetric simu-

lations. Lim et al. [102] examined experimentally the effect of the vortex

finder shape on the collection efficiency at different flow rates but without

any explanation on its effect of the flow field pattern and velocity profiles.

Raoufi et al. [140] duplicated numerically the same study of Lim et al.

20

2.5. Discrepancy in the previous studies

[102] with limited details about the effect of the gas outlet diameter on the

flow field pattern and velocity profile.

2.5.5 The cyclone heights

Limited literatures are available for the effect of cyclone height. Zhu and

Lee [200] have conducted detailed experiments on cyclones of different

height and found that, the cyclone height can influence considerably the

separation efficiency of the cyclones. However, they did not provide any

information about the flow pattern or even explanation for the efficiency

results. Hoffmann et al. [76] investigated the effect of the cyclone length

on the separation efficiency and the pressure drop experimentally and the-

oretically. The cyclone performance improves with increasing length up

to 5.5 cyclone diameters beyond this length the separation efficiency was

dramatically reduced. However, they did not present any contour plot or

velocity profile to assist the explanation for the effect of cyclone height

on performance. Recently, Xiang and Lee [186] have repeated the same

study of Zhu and Lee [200] for the effect of cyclone height computationally

via steady three-dimensional simulation using Reynolds stress turbulence

model (RSM). They found that the tangential velocity decreases with in-

creasing cyclone height, which is responsible for the lower separation effi-

ciency observed in long cyclones. The explanation of this behavior was not

adequate. Moreover, no particle tracking study was presented.

2.5.6 Previous optimization studies

Due to the wide range of industrial applications of the cyclone separator,

it was a matter of study for decades. However, the optimization studies on

it is quite limited in literature. Moreover, many of these studies are not

coherent studies. Ravi et al. [141] carried out a multi-objective optimiza-

tion of a set of N identical reverse-flow cyclone separators in parallel by

using the non-dominated sorting genetic algorithm (NSGA). Two objective

functions were used: the maximization of the overall collection efficiency

and the minimization of the pressure drop. Non-dominated Pareto optimal

solutions were obtained for an industrial problem in which 165 m3/s of

air was treated. In addition, optimal values of several decision variables,

such as the number of cyclones and eight geometrical parameters of the cy-

clone, are obtained. Their study shows that the barrel diameter, the vortex

finder diameter, and the number of cyclones used in parallel, are the impor-

tant decision variables influencing the optimal solutions. Moreover, their

study illustrates the applicability of NSGA in solving multi-objective opti-

mization problems involving gas-solid separations. The main drawbacks of

21


their study are: (1) They used the model of Shepherd and Lapple [157] for

predicting the dimensionless pressure drop (Euler number). In the Shep-

herd and Lapple model, the Euler number depends on only three factors

(Eu = 16ab/D2x) and they used it to optimize the seven geometrical parame-

ters. (2) The barrel diameter, number of parallel cyclones and the gas veloc-

ity have been included into the optimization design space. Consequently, it

is not devoted to the geometrical ratio. (3) They used many side constraints

on the geometrical values (0.4 ≤ a/D ≤ S/D, 0.15 ≤ b/D ≤ (1 − Dx/D)/2if 0.5 ≤ Dx/D ≤ 0.6) these constraints prevent searching for the global op-

timization geometrical ratios for the seven geometrical parameters. (4) No

table for the non-dominated Pareto front points are presented from which

the designer can select certain geometrical ratio set (optimal solution).

Swamee et al. [172] investigated the optimum values of the number of

cyclones to be used in parallel, the diameter of cyclone barrel D and exit

pipe Dx, when a specified flow rate of gas is to be separated from solid

particles, and the cut diameter is already specified. They used Stairmand

model for calculation of pressure drop and Gerrard and Liddle formula

for the cut-off diameter [172] which is not a widely used model. Instead

of handling two objective functions, they blended the two objective into a

single objective problem which is not the suitable method to considering

two conflicting objectives (the pressure drop and cut-off diameter).

Safikhani et al. [148] performed a multi-objective optimization of cyclone

separators. First, they simulated many cyclones to obtain the pressure

drop and the cut-off diameter and used artificial neural network approach

to obtain the objective function values. Finally, a multi-objective genetic

algorithms are used for Pareto based optimization of cyclone separators

considering two conflicting objectives. However, the design variables are

only four (instead of seven): the barrel height, the cone height, the vortex

finder diameter and length. So they ignored the effect of inlet dimensions,

which has been acknowledged by other researchers as significant geometri-

cal parameters for the cyclone flow field and performance (cf. Elsayed and

Lacor [50, 52–54]). Moreover, they did not explain why they selected these

particular parameters. Furthermore, they applied four side constraints on

the four tested variables, which prevent searching for the global optimiza-

tion.

Pishbin and Moghiman [130] applied genetic algorithm for optimum cy-

clone design. They studied the seven geometrical parameters. The data

used for optimization was obtained from 2-D axisymmetric simulations.

However, the flow in the cyclone separator is 3-D unsteady. Instead of

using multi-objective genetic algorithm (e.g., non-dominated sorting ge-

netic algorithm II (NSGA-II) [34]) they used the weighted-sum genetic al-

gorithm. In this technique, a weighting factor is assigned for each objective

22

2.6. Summary and research plan

function based on the user preference. The main shortages of the Pishbin

and Moghiman [130] study are: (1) How to select the weighting factor,

in scientific and engineering problems, it is a non-trivial task to find the

one solution of interest to the decision maker [26]. The decision maker’s

weight (no matter how defined) could be greater than necessary as more

acceptable solutions are missed. Optimizing mostly profit could lead to

poor quality or reliability, not a good compromise [26]. The weighted-sum

genetic algorithm usually does not find all Pareto front points of inter-

est. But this approach is a simple approach for handling multi-objective

optimization problem. Another simple but better result can be obtained

using the desirability function approach [54, 126]. (2) No table for the non-

dominated Pareto front points is presented from which the designer can

select a certain geometrical ratio.

Safikhani et al. [149] carried out a multi-objective optimization using the

genetic algorithm technique to obtain the best vortex finder dimension (di-

ameter and length) and shape (convergent and divergent). Four design

variables have been investigated; the vortex finder diameter, angle, upper-

part length and lower-part length of the vortex finder. They applied neural

networks to obtain a meta-model for the pressure drop and collection effi-

ciency from CFD dataset. The main shortages of the Safikhani et al. [149]

study are: (1) They used dimensional values instead of dimensionless, and

applied side constraints, which prevent the optimization procedure from

obtaining global optimization. (2) The selection of only the vortex finder

dimension as the design variables and neglecting the interaction with the

vortex finder diameter with the other dimensions, especially the inlet di-

mensions [52, 53].

2.6 Summary and research plan

After studying the existing literature on cyclones, the following conclusion

can be drawn:

• The separation mechanism inside cyclone separators is not well un-

derstood yet, and needs more investigations.

• Nearly all published articles have no systematic and complete study

for the effect of geometrical parameters on the flow field and perfor-

mance. In more details: the geometry parameters are not given as

dimensionless numbers; the effect of a certain parameter is obtained

with no knowledge about the effect of others or possible interactions;

the results of different articles are sometimes in contradiction.

• In some cases, there are a lot of results about the effect of a certain

parameter on the performance but on different dimensions (and not

23


on the same dimensionless ratio) and also sometimes at different op-

erating conditions

• Some parameters have less interest compared with others like the

effect of vortex finder shape and number of inlet sections.

• The particle tracking in almost all computational investigations are

not well reported (or missing), and need more investigations

• Some articles give only the flow pattern results without a study of

the effect of any parameter on the performance, or with limited dis-

cussion.

From the remarks above, it is clear that more investigation is still needed

about the flow patterns inside cyclone separators and how the geometri-

cal parameters affect the performance. This guided us to construct the

following research plan:

1. Undertake a systematic study for the effect of seven geometrical pa-

rameters to obtain the most significant factors on the cyclone perfor-

mance. This step can be subdivided into:

(a) Application of design of experiment to obtain a table of runs, the

performance parameters will be estimated using the most robust

mathematical models.

(b) Application of the response surface methodology to determine

the most significant factors and any possible interactions.

2. Once, the most significant factors are selected, CFD simulations for

each particular parameter will be performed to investigate in detail

the effect of these parameters on the flow field pattern and perfor-

mance (pressure drop and cut-off diameter).

3. Optimization of the cyclone geometry, first single-objective optimiza-

tion for minimum pressure drop and then for best performance (multi-

objective optimization for minimum pressure drop and minimum cut-

off diameter). The required data for optimization can be obtained us-

ing mathematical models, artificial neural networks data (based on

experimental measurements available in literature) or CFD simula-

tions.

24

Chapter 3

Governing Equations

3.1 Turbulence

Hinze [73] described turbulence as follows: “Turbulence fluid motion is an

irregular condition of flow in which various quantities show a random vari-

ation with time and space coordinates so that statistically distinct average

values can be discerned”. Wilcox [183] explains that turbulence consists

of a continuous spectrum of scales that vary from smallest to largest over

several orders of magnitude. The idea of a series turbulent eddies is often

used. He also stated that: “A turbulent eddy can be thought of as a lo-

cal swirling motion whose characteristic dimension is the local turbulence

scale.” These eddies overlap in space and the large ones carry the smaller

ones. The conversion of energy in a turbulent flow follows a cascading pro-

cess where the kinetic energy is transferred from the larger eddies to the

smaller ones.

The energy cascade

The concept of the energy cascade has been introduced in 1922 by Richard-

son [132]. The idea is that the kinetic energy enters the turbulence at the

largest scales of motion. This energy is then transferred by inviscid pro-

cesses to smaller and smaller scales until the smallest scales. The energy

is dissipated by viscous processes [35], Fig. 3.1. In 1941, Kolmogorov iden-

tified the smallest scales of turbulence to be those that now bear his name

[35, 132]. In this concept, the turbulence can be considered to be composed

of eddies of different sizes that overlap in space. The largest eddies are

characterized by the length scale l which is comparable to the flow scale

25

Chapter 3. Governing Equations

Figure 3.1: Schematic representation of the energy cascade [32, 35]. η is the Kol-

mogorov length scale, lDI is the length scale dividing the dissipation and the iner-

tial subrange, lEI is the length scale dividing the energy-containing range and the

inertial subrange, l0 is the length scale and L is the characteristic length [35].

(the large scales are of the order of the flow geometry), and a characteristic

velocity u which is on the order of the turbulence intensity.

If l and u are the length and velocity scales of the largest eddy, the time

scale is derived as,

τ =l

u(3.1)

The large energy containing eddies give away their kinetic energy to slightly

smaller-scale eddies with which the large scales interact. The process of

kinetic energy transfer continues in a similar fashion until the smallest

scale eddies are reached, where the frictional forces become so large that

the kinetic energy is converted into internal energy. This process of energy

transfer and dissipation is referred to as the energy cascade process. The

scales at which the dissipation (ε) takes place are the smallest scales, and

are also referred to as the Kolmogorov scales. They can be estimated from

the large-scale properties as follows,

ε =u2

τ=

u3

l(3.2)

26

3.1. Turbulence

Figure 3.2: Energy spectrum for a turbulent flow [183]

Since the processes of dissipation in the smallest scales are due to viscous

forces, the properties of the smallest eddies can be estimated using the flow

kinematic viscosity (ν) and the dissipation (ε) itself. The length, velocity

and time scales are given by:

lη =

(ν3

ε

)1/4

(3.3)

uη = (νε)1/4

(3.4)

τη =(νε

)1/2(3.5)

The turbulent length scale l is related to the wave number κ as κ = 2π/l.The energy spectrum E(κ) for a turbulent flow is as shown in Fig. 3.2.

From dimensional analysis, the Kolmogorov -5/3 law characterizes the in-

ertial subrange which is given by,

E(κ) = Cκε2/3κ−5/3 (3.6)

Cκ is the Kolmogorov constant.

27


3.2 The governing equations for the gas phase

All fluid motions (laminar or turbulent) are governed by a set of dynami-

cal equations namely the continuity, momentum and the energy equation

(Navier-Stokes equations),

∂ρ

∂t+

∂

∂xi(ρui) = 0 (3.7)

∂

∂t(ρui) +

∂

∂xj(ρuiuj) = − ∂p

∂xi+

∂τij∂xj

(3.8)

∂

∂t(ρE) +

∂

∂xi(ρHui) =

∂

∂xi(τjiuj − qi) (3.9)

ui(~x, t) represents the i-th component of the fluid velocity at a point in

space ~x and time t.

p(~x, t) is the static pressure.

τij(~x, t) are the viscous stresses.

ρ(~x, t) is the fluid density.

E and H are the total energy and total enthalpy per unit mass.

qi in Eq. 3.9 is the heat flux which is proportional to the temperature

gradient.

qi = −κ∂T

∂xi(3.10)

where κ is the thermal conductivity.

The Mach numbers associated with air flow in cyclone separators are very

nominal, which allows the flow to be treated as incompressible. Further-

more, the air behaves as a Newtonian fluid, in which case the viscous

stresses are related to the incompressible fluid motion using a property

of fluid, viscosity [86].

τij = 2µ

(sij −

1

3skkδij

)(3.11)

sij is the instantaneous strain rate tensor given by,

28

3.2. The governing equations for the gas phase

sij =1

2

(∂ui

∂xj+

∂uj

∂xi

)(3.12)

For incompressible flows, Eqs. 3.7 and 3.8 are simplified to the following

form,

∂uj

∂xj= 0 (3.13)

∂ui

∂t+ uj

∂ui

∂xj= −1

ρ

∂p

∂xi+ ν

∂2ui

∂xj∂xj(3.14)

In this thesis, the temperature effects are ignored and hence Eq. 3.9 is

uncoupled from the continuity and momentum equations.

The four main numerical procedures for solving the Navier-Stokes equa-

tions are the direct numerical simulation (DNS), the large eddy simula-

tion (LES), the detached eddy simulation (DES) and the reynolds aver-

aged navier stokes (RANS) approach. The most accurate approach is DNS

where the whole range of spatial and temporal scales of turbulence are

resolved. Since all the spatial scales, from the smallest dissipative Kol-

mogorov scales (lη) up to the energy containing integral length scales (l),are needed to be resolved by the computational mesh, the number of points

required in one direction is of the order,

N =l

lη(3.15)

The number of points required for a resolved DNS in three dimensions can

be estimated as,

N =

(l

lη

)3

∼(ul

ν

)9/4

= Re9/4 (3.16)

The number of grid points required for fully resolved DNS is enormously

large, especially for high Reynolds number flows, and hence DNS is re-

stricted to relatively low Reynolds number flows. DNS is generally used

as a research tool for analyzing the mechanics of turbulence, such as tur-

bulence production, energy cascade, energy dissipation, noise production,

drag reduction, etc [86].

3.2.1 Reynolds averaged Navier Stokes (RANS)

When the flow is turbulent, it is convenient to analyze the flow in two

parts, a mean (time-averaged) component and a fluctuating component

29


[86],

Ui = U i + u′

i

P = P + p′

Tij = T ij + τ′

ij

Overline is a shorthand for the time average and in case of RANS, Ui ≡ Ui

and u′

i=0. The above technique of decomposing is referred to as Reynolds

Decomposition. Inserting this decomposition into the instantaneous equa-

tions and time averaging results in the Reynolds averaged Navier-Stokes

equations (RANS).

∂U j

∂xj= 0 (3.17)

∂U i

∂t+ U j

∂U i

∂xj= −1

ρ

∂P

∂xi+ ν

∂2U i

∂xj∂xj− ∂

∂xj

(u

′

iu′

j

)(3.18)

u′

iu′

j in the last term of Eq. 3.18 represents the correlation between fluctu-

ating velocities and is called the Reynolds stress tensor. All the effects of

turbulent fluid motion on the mean flow are lumped into this single term

by the process of averaging [86]. This will enable great savings in terms

of computational requirements. On the other hand, the process of aver-

aging generates six new unknown variables. Now, in total there are ten

unknowns (3-velocity, 1-pressure, 6-Reynolds stresses) and only four equa-

tions (1-continuity, 3 components of momentum equation). Hence, we need

six equations to close this problem. This is referred to as the Closure prob-

lem. Based on the way we close the Reynolds stress tensor, there are two

main categories, namely the eddy viscosity models and the Reynolds stress

model.

The Reynolds stress tensor resulting from time averaging of the Navier-

Stokes equations is closed by replacing it with an eddy viscosity multiplied

by velocity gradients. This is referred to as the Boussinesq assumption.

u′

iu′

j = −νt

(∂U i

∂xj+

∂U j

∂xi

)(3.19)

where νt is the turbulent (eddy) kinematic viscosity. In order to make Eq.

3.19 valid upon contraction because of Eq. 3.17, it should be rewritten as,

30


u′

iu′

j = −νt

(∂U i

∂xj+

∂U j

∂xi

)+

2

3ρδijk (3.20)

where δij is the Kronecker delta, δij = 1 if i = j and δij = 0 if i 6= j. k is the

turbulent kinetic energy given by,

k =1

2u

′

iu′

i (3.21)

The eddy viscosity is treated as a scalar quantity and is determined using

a turbulent velocity scale v and a length scale l, based on the dimensional

analysis.

νt ∼ vl (3.22)

There are different types of eddy viscosity models (EVM) based on the way

we close the eddy viscosity. Algebraic or zero equation EVM’s normally

use a geometric relation to compute the eddy viscosity. In one equation

EVM’s, one turbulence quantity is solved and a second turbulent quantity

is obtained from algebraic expression. These two quantities are used to

describe the eddy viscosity. In two equation EVM models the two turbulent

quantities are solved to describe the eddy viscosity. The interested reader

can refer to [24, 177, 183] for more details.

In the Reynolds stress models (RSM), an equation is solved for each Reynolds

stress component as well as one length scale determining equation. How-

ever, RSM’s are computationally much more demanding when compared

to EVM’s.

3.2.2 Reynolds stress model (RSM)

RSM is regarded as the most appropriate RANS turbulence model for cy-

clone flows [178]. Equation 3.18 can be written as [147],

∂Ui

∂t+ Uj

∂Ui

∂xj= −1

ρ

∂P

∂xi+ ν

∂2Ui

∂xj∂xj− ∂

∂xjRij (3.23)

where Rij = u′iu

′j is the Reynolds stress tensor. The RSM turbulence model

provides differential transport equations for evaluation of the turbulence

31


stress components (Eq. 3.24).

∂

∂tRij + Uk

∂

∂xkRij =

∂

∂xk

(νtσk

∂

∂xkRij

)−[Rik

∂Uj

∂xk+Rjk

∂Ui

∂xk

]

− C1ε

K

[Rij −

2

3δijK

]− C2

[Pij −

2

3δijP

]− 2

3δijε

(3.24)

where the turbulence production terms Pij are defined as [147]:

Pij = −[Rik

∂Uj

∂xk+Rjk

∂Ui

∂xk

], P =

1

2Pij (3.25)

With P being the fluctuating kinetic energy production. νt is the turbulent

(eddy) kinematic viscosity; and σk = 1, C1 = 1.8, C2 = 0.6 are empirical

constants The transport equation for the turbulence dissipation rate, ε, is

given as [98]:

∂ε

∂t+ Uj

∂ε

∂xj=

∂

∂xj

[(ν +

νtσε

) ∂ε

∂xj

]− Cε1 ε

KRij

∂Ui

∂xj− Cε2 ε

2

K(3.26)

In Eq. (3.26), K =1

2u′iu

′i is the fluctuating kinetic energy, and ε is the

turbulence dissipation rate. The values of constants are σε = 1.3, Cε1 =1.44 and Cε2 = 1.92.

3.2.3 Large eddy simulation (LES)

The large eddy simulation technique was developed based on an implica-

tion from Kolmogorov’s theory of self-similarity that the large eddies of the

flow are dependent on the geometry while the smaller scales are more uni-

versal [86]. Hence, the big three-dimensional eddies which are dictated by

the geometry and boundary conditions of the flow involved are directly cal-

culated (resolved) whereas the small eddies which tend to be more isotropic

are modeled.

Large eddy simulation (LES) treats the large eddies more exactly than

the small ones. In LES, the large scales in space are calculated directly as

illustrated in Fig. 3.3. The LES technique is based on a separation between

large and small scales. A grid size first has to be determined. Those scales

that are of a characteristic size greater than the grid size are called large or

resolved scales, and others are called small or subgrid scales. The subgrid

scales are included by way of a model called the subgrid model [154]. An

32


elaborate explanation on LES can be found in several text books such as

[132, 150, 183].

Figure 3.3: Decomposition of the energy spectrum [154]

3.2.3.1 LES equations

In case of RANS, the instantaneous continuity and momentum equations

(Eq. 3.7 and 3.8) are time averaged to obtain a steady form of the averaged

equations (Eq. 3.17 and 3.18). In case of LES, instead of time-averaging,

the instantaneous time-dependent equations are filtered. Filtering is a

method that separates the resolvable scales from the subgrid scales. Fil-

tering can be performed in either wave number space or the physical space.

The filter cut-off should lie somewhere in the inertial range of the spectrum

(Fig. 3.2).

In finite volume methods, box filters are always used because the finite

volume discretization itself implicitly provides the filtering operation. One

of the earliest volume average box filters was given by Deardorff [33].

φ(X, t) =1

∆3

∫ x−0.5∆x

x−0.5∆x

∫ y−0.5∆y

y−0.5∆y

∫ z−0.5∆z

z−0.5∆z

φ(ξ, t)dξdηdζ (3.27)

φ = φ+ φs (3.28)

In the above equation, φ denotes the resolvable scale filtered variable and

φs denotes the sub-grid scale fluctuation. ∆ is the filter width given by

∆ = (∆x∆y∆z)1/3.

Leonard [100] defined a generalized filter as a convolution integral which

is given by,

φ(X, t) =

∫ ∫ ∫G(X − ξ; ∆)φ(ξ, t) d3ξ (3.29)

33


G is the filter function that determines the scale of resolved eddies. The

filter function is normalized by requiring that,

∫ ∫ ∫G(X − ξ; ∆) d3ξ = 1 (3.30)

The filter function in terms of the volume average box filter (Eq. 3.27) can

be written as,

G(X − ξ; ∆) =

1/∆3, |x− ξ| < ∆x/2

0, otherwise

Finally, the decomposition of the flow into a filtered part and a sub-grid

part looks like,

Ui = Ui + usi

P = P + ps

Tij = Tij + τsij

The hat operator . in the above equations represents the filtering oper-

ation as opposed to the time-averaging in case of RANS. Moreover, con-

trary to RANS, where the average of fluctuations is zero, in LES,Ui 6= Ui

[150] and us 6= 0. Further details on the filtering methods can be found in

[32, 97, 132, 183].

Inserting the above decomposition into the instantaneous equations re-

sults in the following filtered Navier-Stokes equations,

∂Uj

∂xj= 0 (3.31)

[∂Ui

∂t+ Uj

∂Ui

∂xj

]= −1

ρ

∂P

∂xi+ ν

∂2Ui

∂xj∂xj−

∂τsgsij

∂xj(3.32)

where τsgsij are the sub-grid scale stresses.

3.2.3.2 SGS modeling

From the energy cascade, explained in the beginning of this chapter, it

is apparent that the energy transfer occurs from the bigger scales to the

34


smaller scales. Hence, the main purpose of an SGS model is to represent

the energy sink [86]. The representation of the energy cascade is an aver-

age process. However, locally and instantaneously the transfer of energy

can be much larger or much smaller than the average. Moreover, there is

also the phenomenon of energy backscatter in the opposite direction [129].

Ideally speaking, SGS models should actually account for all these phe-

nomena. However, if the grid scale is much finer than the dominant scales

of the flow, even a crude SGS model will result in good predictions of the

behavior of the dominant scales [86].

The sub-grid scale stresses τsgsij in Eq. 3.32 are given by,

τsgsij = UiUj − UiUj (3.33)

By using the definition of filtering as given by Eq. 3.28 we can further work

out τsgsij as,

τsgsij = −(UiUj −

[

(Ui + usi )(Uj + us

j)

])(3.34)

τsgsij = −usiu

sj︸︷︷︸

Reynolds

+(−Uius

j −Ujus

i )︸︷︷︸Cross−term

+ UiUj − UiUj︸︷︷︸

Leonard

(3.35)

Leonard [100] shows that the Leonard stresses can significantly drain en-

ergy from the resolvable scales and they can be directly computed. On the

other hand, Wilcox [183] mentions that Leonard stresses are of the same

order of magnitude as the truncation error when a finite-difference scheme

of second-order accuracy is used, and thus it is implicitly represented. The

cross-term stresses are dispersive in nature and largely account for the

backscatter effects. Modeling them with a purely dissipative model such as

Smagorinsky would be in conflict because of its dispersive nature [97]. In

many applications, it is assumed that the Leonard and cross-term stresses

can be neglected, and only the Reynolds stresses remain to be modeled. It

is the same case in the present work [86]. The interested reader is referred

to Sagaut [150] for the detailed review of various SGS models available in

literature.

Smagorinsky model

One of the simplest SGS model is the Smagorinsky model [161]. The un-

known subgrid-scale stresses are modeled employing the Boussinesq as-

sumption as in the case of RANS. The subgrid-scale stress are related to

35


the eddy viscosity as follows,

τij −1

3τkkδij = −νt

(∂Ui

∂xj+

∂Uj

∂xi

)(3.36)

The eddy-viscosity is modeled as,

νt = L2s

√2SijSij (3.37)

where Ls is the length-scale for the sub-grid scale and is given by CsV1/3,

where V is the computational cell volume. It is interesting to note that the

length scale is now the filter width rather than the distance to the closest

wall as in RANS. Cs is a constant which is taken to be 0.17. The only

disadvantage of the Smagorinsky model is the constant Cs, which is not

really a constant, but is flow dependent. It is found to vary between 0.065

[110] and 0.3 [89]. In the dynamic version, which was first proposed by

Germano et al. [60], Cs is dynamically computed based on the information

provided by the resolved scales of motion. The specification of Ls as CsV1/3

is not justifiable in the viscous wall region as it incorrectly leads to a non-

zero turbulent shear-stress at the wall. In order to rectify this, Moin and

Kim [110] use a Van Driest damping function to specify the length scale

as,

Ls = CsV1/3

[1− exp

(y+

A+

)](3.38)

where y+ = uτd/ν is the non-dimensional distance from wall, uτ is the wall

shear stress velocity, d is the distance to the nearest wall and A=25 is the

Van Driest constant.

The above-described SGS model is a standard version as defined in Smagorin-

sky [161]. The LES simulations in the present thesis are performed em-

ploying the Fluent flow solver. The Smagorinsky model implemented in

Fluent deviates slightly from the standard version in the following ways

[86],

• The length-scale for the sub-grid scale is computed as min(κd,CsV1/3).

κ is the von Karman constant (typically a value of 0.41 is used), d is

the distance to the closest wall. κd is indeed one of the first mixing

length models in the literature to handle the turbulent viscosity and

was proposed by Prandtl [133]. Van Driest damping is basically an

improved version of Prandtl’s mixing length model. Both the Prandtl

and the Van Driest model are algebraic and from the zero-equation

models category.

36

3.3. Discrete phase modeling

• The constant Cs in Fluent is taken to be 0.1 instead of 0.17 as was

originally proposed. The value of 0.17 for Cs was originally derived

for homogeneous isotropic turbulence in the inertial subrange. How-

ever, this value was found to cause excessive damping of large-scale

fluctuations in transitional flows near solid boundaries, and has to be

reduced in such regions [59]. A Cs value of around 0.1 has been found

to yield the best results for a wide range of flows, and is the default

value in Fluent.

Dynamic Smagorinsky-Lilly model

Germano et al. [60] and subsequently Lilly [101] conceived a procedure in

which the Smagorinsky model constant Cs is dynamically computed based

on the information provided by the resolved scales of motion [59]. The

dynamic procedure thus obviates the need for users to specify the model

constant Cs in advance. The Smagorinsky model constant is dynamically

computed instead of given as an input to the solver, but clipped to zero or

0.23 if the calculated model constant is outside this range to avoid numer-

ical instabilities [59]. The second advantage of the dynamic Smagorinsky-

Lilly model over the Smagorinsky model is the treatment near the wall.

In the dynamic Smagorinsky-Lilly model, a damping function for the eddy

viscosity near the wall is not required, since the model constant goes to

zero in the laminar region just near the wall [55, 113].

3.3 Discrete phase modeling

3.3.1 Governing equations for the particles

Based partly on the physical properties of dust particles and partly on

the mathematical modeling effort required, there are certain reasonable

assumptions made to describe the particles transport in a fluid medium

[86]. The major simplifying assumptions are as follows,

• The particles are assumed to be spherical.

• The ratio of particle to fluid density is very large: The density of the

dust particle is much higher when compared to the fluid medium

which is air.

• Drag force is the dominant force: This is a direct result of the previous

assumption. Since the density of the particles are much higher than

the density of the fluid medium, several forces such as the lift force,

Basset force and buoyancy force can be readily discarded as they have

negligible effect on the particles transport [86].

37


102

100

10-2

10-4

10-7 10-5 10-110-3

One-Way

Coupling

Two-Way

Coupling

Four-Way

Coupling

Negligable

effect on

turbulence

Particles

enhance

production

Particles

enhance

dissipation

Fluid Particles Fluid Particles Fluid Particles Particles

Φp

τp /τe

Dilute suspension Dense suspension

Figure 3.4: Map for particle-turbulence modulation [44]. φp is the ratio of particles

volume to the volume occupied by particles and fluids. τp is the particle response

time, τp = ρpd2/(18µ), where ρp is the particle density, d is the particle diameter, ρ

is the fluid viscosity. τe is the turnover time of large eddy (time scale= l/u) [44].

• One-way coupling: The phenomenon of mutual mass, momentum

and energy transfer between the phases is termed as coupling. El-

ghobashi [44] proposed a map of regimes of interactions between

particles and fluid turbulence as shown in Fig. 3.4. For values of

dispersed-phase volume fraction less than 10−6, particles have negli-

gible effects on turbulence and this is termed as one-way coupling.

The volume fraction of dust particles we are dealing with in the

present thesis is much less than 10−6 and hence one-way coupling

is assumed. In the second regime which lies between 10−6 − 10−3,

the existence of particles can augment the turbulence if the ratio of

the particle response time to the turnover time of a large eddy is

greater than unity, or can attenuate turbulence if the ratio is less

than unity. This interaction is called two-way coupling. In the third

regime where the volume fractions are greater than 10−3, in addi-

tion to two-way coupling between particles and turbulence, particle

collisions take place and hence this regime is termed as four-way cou-

pling.

38


Incorporating all the above assumptions, the Lagrangian equations gov-

erning the particle motion can be written as [86]:

dxp

dt= up (3.39)

dup

dt= Fd (u− up) + gx

(ρp − ρ)

ρp(3.40)

xp is the particle position, gx is the gravitational force, ρ and ρp are the

density of the fluid and the particle respectively.

Generally, the particle moves with a different velocity than the fluid at any

given point. The difference in fluid velocity (u) and the particle velocity

(up), termed as the slip velocity (u − up), leads to an unbalanced pressure

distribution as well as viscous stresses on the particle surface which yields

a resulting force called drag force. In Eq. 3.40, the term Fd (u− up) is the

drag force per unit particle mass. Fd is given by [86]:

Fd =1

τp

CdRep24

(3.41)

where τp is the particle relaxation time given by,

τp =ρpd

2p

18µ(3.42)

Laws of drag coefficient

The drag coefficient Cd is a function of particle Reynolds number (Rep).

Various experimentally based empirical correlations for the drag coeffi-

cient based on Rep are available in the literature. The Reynolds number of

the particle is defined as:

Rep = ρ dp|u− up|

µ(3.43)

In Fluent, the drag coefficient for spherical particles is calculated by using

the correlations developed by Morsi and Alexander [112]. It is given by,

Cd = a1 +a2Rep

+a3Re2p

(3.44)

where a1, a2 and a3 are constants that apply to smooth spherical particles

in a stipulated range of Rep as given in Table 3.1.

39


Table 3.1: Drag coefficient parameters used for Morsi and Alexander [112]

Rep a1 a2 a3< 0.1 0 24 0

0.1 < 1.0 3.69 22.73 0.0903

1 < 10.0 1.222 29.1667 -3.8889

10.0 < 100.0 0.6167 46.5 -116.67

100.0 < 1000.0 0.3644 98.33 -2778

1000.0 < 5000.0 0.357 148.62 -4.75

5000.0 < 10000.0 0.46 -490.546 57.87

10000.0 < 50000.0 0.5191 -1662.5 5.4167

3.3.2 Modeling the particle phase

Coming to the fundamental mathematical modeling of two-phase flow, the

two most widely used approaches are the Eulerian continuum approach

and the Lagrangian trajectory approach.

Eulerian continuum approach

In an Eulerian approach, the particles are treated as a second fluid which

behaves like a continuum and the equations are developed for average

properties of the particles. For example, the particle velocity is the average

velocity over an averaging volume. This approach is most suitable when

one requires a macroscopic field description of dispersed phase properties

such as pressure, mass flux, concentration, velocity and temperature. Eu-

lerian approach is more suitable for simulating large-scale particle flow

processes. However, this approach requires sophisticated modeling in or-

der to describe the key effects and phenomena found in industrial pro-

cesses [31, 86].

Lagrangian trajectory approach

A Lagrangian approach is useful when the particle phase is so diluted that

the description of particle behavior by continuum models is not feasible.

The motion of a particle is expressed by ordinary differential equations

in Lagrangian coordinates and are directly integrated to obtain individ-

ual tracks of particles [86]. To solve the Lagrangian-equation for a par-

ticular moving particle, the dynamic behavior of the gas phase (generally

obtained by an Eulerian approach) and other particles surrounding this

moving particle should be pre-determined. Since the particle velocity and

the corresponding particle trajectory are calculated for each particle, this

approach is more suitable to obtain the discrete nature of motion of par-

ticles. However, to obtain statistical averages with reasonable accuracy, a

40


large number of particles will have to be tracked. An advantage of using

the Lagrangian approach is the ability to vary easily the physical prop-

erties associated with individual particles such as diameter, density, etc.

Moreover, local physical phenomena related to the particle flow behavior

can be easily probed. Hence, the Lagrangian models can also be used for

validation, testing and development of continuum models [31].

The Lagrangian approach is classified into two types namely, determinis-

tic trajectory methods and Stochastic trajectory methods based on the effect

of turbulence. In a deterministic method, all the turbulent transport pro-

cesses of the particle phase are neglected where as the stochastic method

takes into account the effect of fluid turbulence on the particle motion by

considering instantaneous fluid velocity in the formulation of the equation

of particle motion. In the present thesis, the dust particles are modeled

with a stochastic Lagrangian approach [86].

3.3.3 Stochastic trajectory approach

One of the most frequently used models is the eddy interaction model

(EIM) first introduced by Hutchinson et al. [83] and further developed

by Gosman and Ioannides [67].

The instantaneous motion of particles governed by Equations 3.39 and

3.40 can be written in a general form as given below [86].

dx

dt= up (3.45)

dup

dt=

1

τp(u− up) + g (3.46)

The instantaneous fluid velocity u in the above equation is represented as

the sum of the mean and fluctuating velocity,

u = U + u′

(3.47)

Assuming isotropic turbulence, we have,

u′2 = v′2 = w′2 =2

3k (3.48)

where k is the turbulent kinetic energy. Furthermore, it is assumed that

the local velocity fluctuations of the fluid phase obey a Gaussian probabil-

ity density distribution. Most stochastic models in practical use are de-

41


rived from the formulation of Gosman and Ioannides [67], which is given

by,

u′

=

√2

3k ∗ ζ (3.49)

where ζ is a random number drawn from a normal probability distribu-

tion with zero mean and unit standard deviation. The minimal random

number generator of Park and Miller with Bays-Durham shuffle [135] is

implemented [86]. The random number generator returns a uniform ran-

dom derivative with zero mean and unit standard deviation.

The chosen fluctuation is referred to a turbulent eddy whose size (length

scale) and life-time (time scale) is known. Sommerfeld et al. [164] proposed

the following relations for eddy parameters,

te = ctk

ε(3.50)

le = te

√2

3k (3.51)

where ct was taken to be 0.3.

Figure 3.5 shows a 2-D schematic representation of an eddy inside a rect-

angular domain. At any given particle position (xp, yp), the eddy param-

eters are first evaluated based on the local fluid kinetic energy and dissi-

pation rate. The particle position (xp, yp) is assumed to be located at the

center of this hypothetical eddy. It is accepted that each eddy has its own

fluctuation u′

, which remains constant until the particle leaves this eddy.

The particle leaving an eddy is based on a certain interaction time of the

particle with the eddy. Once this interaction time is reached while time

integration of particle equations, the particle is assumed to have left the

present eddy. Now, based on the new position of the particle, new eddy pa-

rameters are calculated and a new fluctuation u′

is assigned to this eddy.

This procedure may be repeated for as many interaction times as required

for the particle to traverse the required distance. If a statistically signif-

icant number of particles are tracked in this way, the ensemble averaged

behavior should represent the turbulent dispersion induced by the prevail-

ing fluid field [67]. The interaction time is the minimum of two time scales,

one being a typical turbulent eddy lifetime and the other the crossing-time

of the particle in the eddy [67].

tint = min(te, tc) (3.52)

42


Figure 3.5: 2-D illustration of a particle within an eddy [86]

The crossing-time is defined as,

tc = −τp ln

[1−

(le

τp|u− up|

)](3.53)

where τp is the particle relaxation time, le the eddy length scale and |u−up|the magnitude of slip velocity. In circumstances where le/(τp|u − up|) > 1,

Eq. 3.53 has no solution. This can be interpreted as the particle trapped

by an eddy, in which case tint = te [67].

The mentioned eddy interaction model is needed only for RANS simula-

tion to take into account the effect of turbulence on the particle. In LES

simulations, the effect of the resolved velocity fluctuations on the parti-

cles is accounted for and there is no need for an eddy interaction model

like in RANS [86]. In this thesis, the effect of the subgrid scale velocity

fluctuations on the particle dispersion is assumed negligible and hence not

modeled. Figure 3.6 represents a flow chart demonstrating the steps in-

volved in tracking one injected particle.

43


Start with the location of one particle injected from the inlet surface at a certain point

Compute the surrounding cells of the current control volume where the particle lies

Interpolate the flow variables at the particle position

Determine the eddy parameters (and fluctuating velocity using interpolated flow variables and the

random number generator in case of RANS)

Compute the forces acting on the particle

Determine the integration time-step based on the cell size & eddy parameters

Perform a time-step integration to obtain updated particle position and velocity

Check the distance between particle position & the nearest boundary cell (wall or outlet)

Particle is stuck on wall (trapped) or reached outlet (escaped)

Is the particle still in the current cell?

Is the current cell a boundary cell?

No

No

Is the distance to wall/outlet <= particle radius?

No

Figure 3.6: Flow chart demonstrating the steps involved in tracking one injected

particle [86]

44

Chapter 4

Sensitivity Analysis of

Geometrical Parameters

4.1 Sensitivity analysis

The cyclone separator performance and the flow field are affected mainly

by the cyclone geometry where there are seven geometrical parameters,

namely, the inlet section height a and width b, the vortex finder diameter

Dx and length S, the barrel height h, the cyclone total height Ht and the

cone-tip diameter Bc. all of these parameters are always expressed as a

ratio of cyclone diameter D, as shown in Fig. 4.1 and Table 4.1.

In this study, only the effect of geometry was taken into account. Neverthe-

less, what about the effect of flow rate on the performance. Overcamp and

Scarlett [125, p 369, Fig. 6] studied the effect of changing Reynolds num-

ber on the cut-off diameter (Stokes number) and found that for Reynolds

number values beyond 1E4, the effect of increasing the Reynolds number

on the cut-off diameter is very limited. Furthermore, Karagoz and Avci

[90, p 863, Fig. 7] studied the effect of increasing the Reynolds number

on the pressure drop and found that beyond Reynolds number of 2E4 any

increase in the Reynolds number has nearly no effect on the pressure drop.

The values of the Reynolds number for all tested cases in this thesis have

Reynolds number higher than 2E4. Consequently, the effect of flow rate

Table 4.1: The Stairmand high-efficiency design

a/D b/D Dx/D Ht/D h/D S/D Bc/D0.5 0.2 0.5 4.0 1.5 0.5 0.375

45

Chapter 4. Sensitivity Analysis of Geometrical Parameters

Figure 4.1: The cyclone separator dimensions

can be safely neglected.

4.1.1 Response surface methodology (RSM)

The usual method of optimizing any experimental set-up is to adjust one

parameter at a time, keeping all others constant, until the optimum work-

ing conditions are found. Adjusting one parameter at a time is necessar-

ily time consuming, and may not reveal all interactions between the pa-

rameters. In order to fully describe the response and interactions of any

complex system a multivariate parametric study must be conducted [30].

Since there are seven geometrical parameters to be investigated, the best

technique is to perform this study using the response surface methodology

(RSM).

RSM is a powerful statistical analysis technique which is well suited to

model complex multivariate processes, in applications where a response

is influenced by several variables, and the objective is to optimize this re-

sponse. Box and Wilson first introduced the theory of RSM in 1951 [13].

RSM today is the most commonly used method of process optimization. Us-

ing RSM one may model and predict the effect of individual experimental

parameters on a defined response output, as well as locating any inter-

actions between the experimental parameters which otherwise may have

been overlooked. RSM has been employed extensively in the field of engi-

neering and manufacturing, where many parameters are involved in the

process [70, 106, 118, 167–170].

46

4.1. Sensitivity analysis

In order to conduct a RSM analysis, one must first design the experiment,

identify the experimental parameters to adjust, and define the process re-

sponse to be optimized. Once the experiment has been conducted and the

recorded data tabulated, the RSM analysis software models the data and

attempts to fit a second-order polynomial to this data [30]. The generalized

second-order polynomial model used in the response surface analysis was

as follows:

Y = β0 +

7∑

i=1

βiXi +

7∑

i=1

βiiX2i +

∑∑

i<j

βijXiXj (4.1)

where β0, βi, βii, and βij are the regression coefficients for intercept, linear,

quadratic and interaction terms, respectively. Xi and Xj are the indepen-

dent variables, and Y is the response variable (Euler number).

4.1.2 Design of experiment (DOE)

The statistical analysis is performed through three main steps. Firstly,

construct a table of runs with combinations of values of the independent

variables via the commercial statistical software Statgraphics centurion

XV by giving the minimum and maximum values of the seven geometrical

factors under investigation as input. Secondly, perform the runs by esti-

mating the pressure drop (Euler number) using the MM model (cf. Sec. A.3,

page 256). Thirdly, fill in the values of pressure drop in the Statgraphics

worksheet and obtain the response surface equation with main effect plot,

interaction plots, Pareto chart and response surface plots beside the opti-

mum settings for the new cyclone design.

Table 4.2 depicts the parameters ranges selected for the seven geometrical

parameters. The study was planned using Box–Behnken design, with 64

combinations. A significant level of P < 0.05 (95% confidence) was used in

all tests. Analysis of variance (ANOVA) was followed by an F-test of the

individual factors and interactions.

Table 4.2: The values of the independent variables

Variables minimum center maximum

Inlet height, a/D =X1 0.4 0.55 0.7

Inlet width, b/D =X2 0.14 0.27 0.4

Vortex finder diameter, Dx/D =X3 0.2 0.475 0.75

Total cyclone height, Ht/D =X4 3.0 5.0 7.0

Cylinder height, h/D =X5 1.0 1.5 2.0

Vortex finder length, S/D =X6 0.4 1.2 2.0

cone-tip diameter, Bc/D =X7 0.2 0.3 0.4

47


Fitting the model

The analysis of variance (ANOVA) shows that the resultant quadratic poly-

nomial models adequately represented the input data with the coefficient

of multiple determination R2 being 0.92848. This indicates that the ob-

tained quadratic polynomial model was adequate to describe the influence

of the independent variables studied [189]. Analysis of variance (ANOVA)

was used to evaluate the significance of the coefficients of the quadratic

polynomial models (see Table 4.3). For any of the terms in the models, a

large F-value (small P-value) would indicate a more significant effect on

the respective response variables.

Based on the ANOVA results presented in Table 4.3, the variable with the

largest effect on the pressure drop (Euler number) was the linear term of

vortex finder diameter, followed by the linear term of inlet width and inlet

height (P < 0.05); the other four linear terms (barrel height, vortex finder

length, cyclone total height and cone-tip diameter) did not show a signif-

icant effect (P > 0.05). The quadratic term of vortex finder diameter also

had a significant effect (P < 0.05) on the pressure drop; however, the effect

of the other six quadratic terms was insignificant (P > 0.05). Furthermore,

the interaction between the inlet dimensions and vortex finder diameters

(P < 0.05) also had a significant effect on the pressure drop, while the effect

of the remaining terms was insignificant (P > 0.05).

4.1.3 Analysis of response surfaces

For visualization of the calculated factor, main effects plot, Pareto chart

and response surface plots were drawn. The slope of the main effect curve

is proportional to the size of the effect, and the direction of the curve spec-

ifies a positive or negative influence of the effect [61](Fig.4.2(a)). Based on

the main effect plot, the most significant factor on the Euler number are

(1) the vortex finder diameter, with a second-order curve with a wide range

of inverse relation and a narrow range of direct relation, (2) direct relation

with inlet dimensions, (3) inverse relation with cyclone total height and

insignificant effects for the other factors.

Pareto charts were used to summarize graphically and display the rela-

tive importance of each parameter with respect to the Euler number. The

Pareto chart shows all the linear and second-order effects of the param-

eters within the model and estimates the significance of each with re-

spect to maximizing the Euler number response. A Pareto chart displays

a frequency histogram with the length of each bar proportional to each

estimated standardized effect [30]. The vertical line on the Pareto chart

judges, whether each effect is statistically significant within the generated

48


response surface model; bars that extend beyond this line represent ef-

fects that are statistically significant at a 95% confidence level. Based on

the Pareto chart (Fig. 4.2(b)) and ANOVA table (Table 4.3) there are four

significant parameters (six terms in the ANOVA table ) at a 95% confidence

level: the negative linear vortex finder diameter; the linear inlet width; the

linear total cyclone height; a second-order vortex finder diameter; negative

interaction between vortex finder diameter and inlet dimensions. These

are the major terms in a polynomial fit to the data. Therefore, the pareto

chart is a perfect supplementation to the main effects plot.

To visualize the effect of the independent variables on the dependent ones,

surface response of the quadratic polynomial models were generated by

varying two of the independent variables within the experimental range

while holding the other factors at their central values [189]. Thus, Fig. 4.2(c)

was generated by varying the inlet height and the inlet width while hold-

ing the other five factors fixed at their central value. The trend of the curve

is linear, with more significant effect for inlet width, with no interaction

between the inlet height and width. The response surface plots given by

Figs. 4.2(d), 4.2(e) and 4.2(f) show that there are interactions between both

inlet width and inlet height with the vortex finder diameter. The effect of

cyclone total height is less significant with respect to the vortex finder di-

ameter, but its effect is higher than that of the vortex finder length, the

barrel height and the cone-tip diameter.

4.1.4 Conclusions

Mathematical modeling (the Muschelknautz method of modeling (MM))

has been used to understand the effect of the cyclone geometrical parame-

ters on the cyclone performance. The most significant geometrical param-

eters are:

1. the vortex finder diameter

2. the inlet section width

3. the inlet section height

4. the cyclone total height.

The effect of both the barrel height and the vortex finder length on the cy-

clone separator performance are small in comparison with these most sig-

nificant geometrical parameters. There are strong interactions between

the effects of inlet dimensions and the vortex finder diameter on the cy-

clone performance.

This study confirms the insignificant effect of the cone-tip diameter on the

cyclone performance. However, the discrepancy exists in literature for this

49


Table 4.3: Analysis of variance and the regression coefficients∗

Variable Regression coefficient F-Ratio P-Value

β0 -43.0742

Linearβ1 178.176 8.11 0.0075β2 372.26 19.79 0.0001β3 -161.452 232.04 0.0000β4 -1.55344 0.6 0.446β5 8.5875 0 0.9691β6 -7.23112 0.1 0.757β7 19.5663 0 0.9537

quadraticβ11 1.08238 0 0.9931β22 -12.2111 0 0.9446β33 403.419 107.8 0.0000β44 -0.223597 0.09 0.7641β55 -2.67108 0.05 0.8223β66 1.81257 0.15 0.6994β77 -62.1739 0.04 0.8364

Interactionβ12 91.0488 0.22 0.6427β13 -355.892 14.75 0.0005β14 0.459314 0 0.9726β15 -3.27883 0 0.9514β16 2.19997 0 0.9465β17 26.2787 0.01 0.9191β23 -720.685 42.42 0.0000β24 1.03571 0 0.9467β25 -2.53478 0 0.9675β26 4.2616 0.01 0.9112β27 -5.28466 0 0.9862β34 5.2034 0.51 0.4799β35 2.77536 0.01 0.9249β36 0.985086 0 0.9568β37 32.579 0.05 0.8221β45 -0.0452174 0 0.9911β46 0.345301 0.02 0.8902β47 -1.5016 0.01 0.9404β56 -0.422227 0 0.9667β57 3.82354 0 0.9622β67 -6.40134 0.02 0.8945

R2 0.92848

∗ Bold numbers indicate significant factors as identified by the analysis of variance (ANOVA) at the 95%confidence level.

50


(a) Main effect plot (b) Pareto chart

(c) X1 versus X2 (d) X1 versus X3

(e) X2 versus X3 (f) X3 versus X4

Figure 4.2: Analysis of design of experiment (cf. Table 4.2)

51


geometrical factor motivates us to study this individual parameter in more

details using large eddy simulation methodology as given in Sec. 4.2. Be-

fore proceed to apply CFD technique in the optimization of the cyclone ge-

ometry and to study the effect of each significant factor factor, a question

appears. Should the dust outlet geometry be included in the simulation

domain? The answer of this question is given in Sec. 4.3.

4.2 The cone-tip diameter

Until now, a considerable number of investigations has been performed ei-

ther on small sampling cyclones or larger industrial cyclone separators, for

example; Buttner [19], Iozia and Leith [84], Kim and Lee [95], Zhu et al.

[199], Elsayed and Lacor [50, 52, 53] and Safikhani et al. [147]. In these

studies, almost all of the cyclone dimensions listed in Table 4.4, were var-

ied and the changes in cyclone performance characteristics brought about

by these variations were studied. However, very little information is avail-

able on the effect of changing the cone bottom (tip) diameter (which deter-

mines the cone shape if other cyclone dimensions are fixed [184]) on the

flow pattern and performance. Regarding this effect, discrepancies and

uncertainties exist in the literature. Bryant et al. [17] observed that if the

vortex touched the cone wall, particle re-entrainment occurred and effi-

ciency decreased, so the collection efficiency will be lower for cyclones with

a small cone opening (cone-tip diameter). However, according to Stern et

al. [171] (cited in Xiang et al. [184]), a cone is not an essential part for

cyclone operation, whereas it serves the practical purpose of delivering

collected particles to the central discharge point. However, Zhu and Lee

[200] stated that the cone provides greater tangential velocities near the

bottom for removing smaller particles. Furthermore, the sensitivity anal-

ysis presented in Sec. 4.1 indicates the insignificant effect of the cone-tip

diameter on the cyclone performance.

However, the understanding and knowledge of the flow field inside a cy-

clone has been developed rapidly over the last few years, the exact mecha-

nisms of removing particles are still not fully understood. Therefore, most

existing cyclone theories are based on simplified models or depend upon

empirical correlations [23]. Xiang et al. [184] carried out experiments with

cyclones of different cone dimensions and evaluated a few models, namely

Barth [9], Leith and Licht [99] and Iozia and Leith [85]. All these models

could simulate correctly the trend of Xiang’s experimental data. However,

the quantitative agreement was not satisfactory. CFD has a great poten-

tial to predict the flow field characteristics and particle trajectories as well

as the pressure drop inside the cyclone [68]. Chuah et al.[23] carried out a

numerical investigation on the same cyclone dimensions used by Xiang et

52

4.2. The cone-tip diameter

al. [184] with the commercial finite volume code Fluent. Using different

turbulence models they proved that Fluent with Reynolds stress turbu-

lence model (RSM) predicts well the cyclone collection efficiency and the

pressure drop. The CFD simulation results from Chuah et al. [23] agree

well with Xiang’s experimental results in that cyclones with a smaller cone

diameter result in a slightly higher collection efficiency compared to cy-

clones with a bigger cone-tip diameter (only if the cone-tip diameter is not

smaller than the gas exit tube diameter). Moreover, the change in the

pressure drop will not be significant when the cone size is varied. Both

Xiang and Chuah did not give any results about the effect of the cone-tip

diameter on the flow field inside the cyclone separator, except some plots

for axial and tangential velocity profiles at two stations in the flow field for

Chuah et al. [23]. Xiang and Lee [185] computationally investigated the

effect of the cone-tip diameter on the flow field using the Reynolds stress

turbulence model. They did not present any contour plots for either the

static pressure, tangential and axial velocity. However, the comparisons

between the tangential and axial velocity profiles at different sections in-

dicating no valuable difference between the three cyclones [185, Fig. 8, p.

216 and Fig. 9, p. 217 ], they mentioned that the cone-tip diameter has

a significant effect on the flow field. No particle tracking study has been

performed in the study of Xiang and Lee [185].

Currently a better understanding of the flow field inside cyclone separators

is an important concern, especially with the application of large eddy sim-

ulation (LES). The present study was undertaken in an effort to carry out

a numerical study on the effect of the cone-tip diameter on the flow field

and the cyclone performance using LES available in Fluent commercial

finite volume solver.

Table 4.4: The geometrical dimensions of the three cyclones§

DimensionLength Dimension ratio(mm) (dimension/D)

Body diameter, D 31 1Gas outlet diameter, Dx 15.5 0.5Inlet height, a 12.5 0.4Inlet width, b 5 0.16Cyclone height, Ht 77 2.5Cylinder height, h 31 1Gas outlet duct length, S 15.5 0.5

Cone-tip diameter, Bc

Cyclone I 19.4 0.625Cyclone II 15.5 0.5Cyclone III 11.6 0.375

§ The outlet section is above the cylindrical barrel surface by Le = 0.5D. The inlet section located at adistance Li = 0.75D from the cyclone center (cf. Fig. 4.3).

53


!

"

#

"

Sa

h

D

b

H t

D x

L i

L e

B c

Figure 4.3: Schematic diagram for the cyclone geometry and coordinate definition

4.2.1 Numerical simulation

4.2.1.1 Configuration of the three cyclones

The cyclones used in this study had a reversed flow tangential inlet. The

geometry and dimensions are shown in Fig. 8.1 and Table 4.4. Three cy-

clones with different cone-tip diameters are used viz., Bc/D= 0.625, 0.5

and 0.375. The three cyclones are identical to those used by both Xiang

et al. [184, 186] and Chuah et al. [23]. Four plotting sections are used to

investigate the effect of the cone-tip diameter Bc on the velocity profiles as

given by Table 4.5.

4.2.1.2 Selection of the turbulence model (RANS versus LES)

For the turbulent flow in cyclones, the key to the success of CFD lies with

the accurate description of the turbulent behavior of the flow [68]. To

model the swirling turbulent flow in a cyclone separator, there are dif-

ferent turbulence models available in Fluent. These range from the stan-

dard k − ε model to the more complicated Reynolds stress model (RSM)

Table 4.5: The position of different plotting sections

Section S1 S2 S3 S4

z (mm) 5 15 30 50z/D 0.16 0.48 0.97 1.61

54


and large eddy simulation (LES) methodology as an alternative for RANS

models. The standard k−ε, RNG k−ε and Realizable k−ε models were not

optimized for strongly swirling flows found in cyclones [23]. The Reynolds

stress turbulence model (RSM) requires the solution of transport equations

for each of the Reynolds stress components and yields an accurate predic-

tion on swirl flow pattern, axial velocity, tangential velocity and pressure

drop on cyclone simulation [159].

Large eddy simulation (LES) has been widely accepted as a promising nu-

merical tool for solving the large-scale unsteady behavior of complex tur-

bulent flows. Encouraging results have been reported in recent literature

and demonstrate the ability of LES to capture the swirling flow instability

and the energy containing coherent motion of such highly swirling flows

[41]. LES methodology has been used in many articles to study the highly

swirling flow in cyclone separators, [e.g., 38, 39, 48, 155, 156, 159, 191].

It will be used in this study to reveal the effect of changing the cone-tip

diameter on the turbulent flow in the cyclone separator. The simulation

will start with the Reynolds stress turbulence model for flow initialization,

then the large eddy simulation methodology will be applied.

4.2.1.3 Solver settings

Selection of the discretization schemes

The choice of the discretization schemes has a tremendous influence on the

simulation results and the Fluent solver offers many different schemes

for pressure-velocity coupling, pressure, momentum, kinetic energy, rate

of kinetic energy dissipation discretization [59]. Both Kaya and Karagoz

[91] and Shukla et al. [158] investigated the performance of different dis-

cretization schemes in the steady and unsteady simulation of cyclone sep-

arators. The schemes used in this study are given in the following para-

graphs together with an explanation of the reasons behind their selection.

Kaya and Karagoz [91] reported the advantages of SIMPLEC (semi im-

plicit method for pressure linked equations consistent) scheme for pressure-

velocity coupling in terms of convergence. For the pressure discretization

they stated that only the PRESTO (pressure staggering option) pressure

interpolation scheme only can predict precisely the mean velocity profiles

static pressure distribution and the pressure drop in the cyclone separator

with good agreement with the experimental values. This scheme is also

recommended by the Fluent manual [59] for highly swirling flows.

For momentum, the QUICK (quadratic upwind differencing) scheme has

been recommended by both Kaya and Karagoz [91] and Fluent manual

[59] for the flow in cyclone separators. For the discretization of kinetic

55


energy and its dissipation rate equation, the second-order upwind scheme

has been used [158]. The first-order upwind scheme has been used for

the discretization of the Reynolds stress equations [91, 158]. For the LES

simulations the bounded central difference scheme is the default and the

recommended convection scheme by the Fluent manual [59].

The time step

The unsteady simulation started with the Reynolds stress model for ini-

tialization with a time step of 1E-4s using implicit coupled solution al-

gorithm. For the LES simulation, the time step is 1E-5 s. The selected

time step results in an average inlet Courant number of 2.3 for the three

cyclones. But as the solver is a segregated implicit solver, there is no sta-

bility criterion that needs to be met in determining the time step (and

consequently the Courant number) [59, 137]. However, to model transient

phenomena properly the Fluent manual [59] suggested using a time step

of at least one order of magnitude smaller than the smallest time constant

in the system. In the cyclone separator studies, the average residence time

(cyclone volume/ gas volume flow rate) is widely used to estimate the time

step [23, 43, 52]. In this study, the cyclone volumes (calculated by the Flu-

ent solver) are 5.045387E-5 m3, 4.738813E-5 m3 and 4.468306E-5 m3 for

cyclones I-III respectively. For a flow rate of 30E-3 m3/s, the corresponding

average residence time values are 0.00168s, 0.00158s and 0.001489s for

cyclones I-III respectively, i.e., the used time step is just a small fraction

of the average residence time. This confirms that the used time step can

reveal the transient phenomena properly. However, the interest in this

study lies in the simulation of averaged scalars and vectors (the average

velocities and pressures in order to estimate the cyclone performance). To

resolve the high-frequency phenomena in the time domain may be a time

step smaller by two orders of magnitude (tiny fraction) than the average

residence time would be required which is not the case in this study. Fur-

thermore, to verify that the choice for the time step was appropriate after

the calculation is complete; Fluent manual [59] suggests to check the max-

imum value of Courant number at the most sensitive transient regions of

the domain (in this study, it is the central region) should not exceed a value

of 20-40 [59]. For the three cyclones the maximum values of Courant num-

ber are 9.78, 12.1, 16.92 for cyclones I-III respectively. This verifies again

that the choice of the time step was proper.

Convergence criteria

With regard to the convergence criteria, two aspects should be considered.

Firstly, the scaled residuals should be below 1E-5 (The default convergence

56


criterion of Fluent is that scaled residuals of all equations fall below 1E-

3). Secondly, some representative quantities such as velocity and pressure

should be monitored until they are constant [137]. Although the present

simulations were converged at about (t=1.5-1.6 s), they were only termi-

nated at t=2s to get more accurate time averaged values. After achieving

constant tangential velocity with time at a certain point in the middle of

the cyclone domain, Fluent begins the data sampling for time statistics for

the whole domain (the velocity components and the static pressure) each

time step for sufficiently long time (t=1.5s until t=2.0s). From this step,

the following time averaged values are available; the mean and the root

mean square values of the static pressure p, the velocity magnitude v, the

x-velocity vx, the y-velocity vy and the z-velocity vz. The mean z-velocity vz

is identical to the mean axial velocity vaxial. The mean tangential velocity

vθ can be obtained using Fluent custom field function [cf. 59] according to

Eq. (4.2).

vθ = vx cos θ + vy sin θ (4.2)

where vx is the time averaged x-velocity, vy is the time averaged y-velocity

and θ is the angular coordinate.

Boundary conditions and other settings

Velocity inlet boundary condition is applied at inlet, outflow at gas outlet

and wall (no-slip) boundary condition at all other boundaries. The air in-

let velocity Uin equals 8 m/s, corresponding to air inlet volume flow rate

Qin=30 l/s, air density 1.0 kg/m3 and dynamic viscosity of 2.11E-5 Pa s,

leading to a Reynolds number of 1.18E4 based on the cyclone diameter and

the area averaged inlet velocity. The turbulence intensity I equals 5% and

the turbulence characteristic length equals 0.07 times the inlet width [74].

At the cyclone inlet, the Reynolds stress specific method in Fluent solver is

the Reynolds stress components. The diagonal components of the Reynolds

stress tensor (normal stresses) are assigned to 2kin/3, kin = 32

(I U2

in

),

where kin is the kinetic energy at the inlet [59, 81]. The shear stresses

(non-diagonal components) at the inlet are set to zero. To take into ac-

count the stochastic component of the turbulent flow at the inlet for the

LES simulation, artificial perturbations have been generated using the

spectral synthesizer method available in the Fluent solver [59, 82, 162],

where the fluctuation velocity components are computed by synthesizing

a divergence-free velocity-vector field from the summation of 100 Fourier

harmonics [59]. The fluctuations are added to the mean inlet velocity.

The reason for introducing these artificial perturbations instead of select-

ing the no-perturbation option in the Fluent solver, is that the unpertur-

57


bated flat turbulent profiles at inlet generates unrealistic turbulent eddies

[59]. For the near-wall treatment, the enhanced wall function [59] has

been used in the RSM simulation. For subgrid scale model, the dynamic

Smagorinsky-Lilly model [59, 60, 101] has been used. The Smagorinsky

model constant is dynamically computed instead of given as an input to

the solver, but clipped to zero or 0.23 if the calculated model constant is

outside this range to avoid numerical instabilities [59]. The second advan-

tage of the dynamic Smagorinsky-Lilly model over the Smagorinsky model

is the treatment near the wall. In the dynamic Smagorinsky-Lilly model, a

damping function for the eddy viscosity near the wall is not required, since

the model constant goes to zero in the laminar region just near the wall

[113].

The boundary condition at the gas outlet is the outflow boundary condition

provided by Fluent [59], where all transport variables have a zero normal

gradient. This boundary condition is valid for fully developed flow, and

this is the reason why the vortex finder is extended 1/2 cyclone diameter

above the top of the cyclone in the present study to allow the exit flow

to be a fully developed flow [185]. The effect of this distance on the flow

field has been the subject of different investigations [e.g., 153, 178], where

this distance was varied between zero and four cyclone diameters. Wang

et al. [178] investigated the effect of outflow length on the flow field and

velocity profile and suggested to have the gas outlet boundary condition at

a distance longer than the cyclone radius.

The grid independency study

The grid independence study has been performed for the three tested cy-

clones. Three levels of grid for each cyclone have been tested, to be sure

that the obtained results are grid independent. The hexahedral computa-

tional grids were generated using GAMBIT grid generator and the simula-

tions were performed using Fluent 6.3.26 commercial finite volume solver

on a 8 nodes CPU Opteron 64 Linux cluster.

The computational results of the three grid types are presented in Ta-

ble 4.6. As seen the maximum difference between the results obtained

from the fine and medium meshes is 1% for the calculation of the cut-off

diameter and the Euler number which is in the range of experimental error

[52, 137]. It has been observed that even medium grids provide a sufficient

grid independency. However, for excluding any uncertainty, computations

have been performed using the fine grid, where the total number of grid

points was not that critical with respect to the computation overhead [10].

Consequently, the used grid produces grid independent results (the author

only checked the mean values, so for future studies with unsteady phe-

58


Table 4.6: The details of the grid independence study for cyclones I-IIIa

Cyclone I II III

N Eu x50 N Eu x50 N Eu x50

Coarse 632153 2.48 1.396 513021 2.36 1.335 513991 2.869 1.24Medium 861077 2.405 1.355 863852 2.28 1.257 712576 2.712 1.21Fine 1021616 2.39 1.35 1025778 2.27 1.25 1027982 2.687 1.2

% differenceb 3.76 3.4 3.96 6.8 6.77 3.33

% differencec 0.63 0.37 0.44 0.56 0.93 0.83

a N is the number of hexahedral cells, Eu is the Euler number (dimensionless pressure drop = pressuredrop / average kinetic energy at inlet) and x50 is the cut-off diameter; the particle diameter that willproduce 50% collection efficiency (cf. Sec. 3.3).b The percentage absolute difference between the coarse and fine grid values for Euler number andcut-off diameter.c The percentage absolute difference between the medium and fine grid values for Euler number andcut-off diameter.

nomena like vortex-core precession (cf. Ref. [38]), the effect of the grid on

the Strouhal number associated with the simulated vortex-core precession

[38] should be included in the grid independency study probably requiring

finer grids, but this is not part of the present study). Moreover, to evaluate

accurately the numerical uncertainties in the computational results (espe-

cially because of the large difference between the results obtained on the

coarse and the fine mesh which is about 7%), the concept of grid conver-

gence index (GCI) was adopted using three grid levels per cyclone.

Grid convergence index (GCI)

Roache [143–145] suggested a quantitative measure for the grid conver-

gence; the grid convergence index (GCI). The GCI can be computed using

two levels of grid; however, three levels are recommended in order to es-

timate accurately the order of convergence and check that the solution is

within the asymptotic range of convergence [160]. For a consistent numer-

ical analysis the discretized equations will approach the solution of the

actual equations as the grid resolution approaches zero [160]. The appro-

priate level of grid resolution is a significant issue in numerical investiga-

tions. It is a function of many variables including the flow condition, type

of analysis, geometry and many other variables.

The GCI is based upon a grid refinement error estimator derived from

the theory of the generalized Richardson extrapolation [160]. The GCI

is a measure of how far the computed value is away from the value of

the asymptotic numerical value. Consequently, it indicates how much the

solution would change with a further refinement of the grid. A small value

of GCI indicates that the computation is within the asymptotic range.

59


The GCI on the fine grid is defined as:

GCIfine =

Fs|ε|(rp − 1)

(4.3)

where Fs is a factor of safety. Fs = 3 for comparison of two grids and 1.25

for comparison over three grids or more.

For the coarse grid:

GCIcoarse =

Fs|ε|rp(rp − 1)

(4.4)

ε is a relative error measure of the key variable f between the coarse and

fine solutions,

ε =f2 − f1

f1(4.5)

where f2 is the coarse-grid numerical solution obtained with grid spacing

h2. f1 is the fine-grid numerical solution obtained with grid spacing h1. ris the grid refinement ratio (r = h2/h1 > 1). For complicated geometries

r is replaced by the ratio of the number of control volumes in the fine and

coarse mesh [107] which is the case in this study,

r12 =

(N1

N2

) 1D

(4.6)

where D = 2 and 3 for two-dimensional and three-dimensional geometries

respectively [143, pp. 410]. N1 and N2 are the number of control volumes

in the fine and coarse mesh respectively.

p is the order of the discretization method. p equals two if the second order

discretization is used for all terms in space [107] (However, Slater [160]

stated that if all discretization in space was of second-order, p will be less

than 2. The difference is due to grid stretching, grid quality, non linearity

in the solution, presence of shocks, turbulence modeling and perhaps other

factors). For the grid refinement study, three meshes have been used with

N1, N2 and N3 cells for the fine, medium and coarse three-dimensional

mesh respectively.

r12 =(

N1

N2

) 13

, r23 =(

N2

N3

) 13

, e12 = f2 − f1, e23 = f3 − f2, where ei,i+1 =

fi+1 − fi is the difference in the key variable f resulting from the use of

different grids. If r12 = r23 then,

p = ln

(e23e12

)/ln(r) (4.7)

60


If r12 6= r23 which is the case in this study, Roache [145] proposed to solve

Eq. 4.8

e23(rp23 − 1)

= rp12

[e12

(rp12 − 1)

](4.8)

Equation 4.8 is transcendental in p. Using the iterative technique with

relaxation factor introduced in Roache [144, 145]

p = ω ρ+ (1− ω)ln(β)

ln(r12)(4.9)

where β =(rp12−1)e23(rp23−1)e12

, ω = 0.5 and ρ is the previous iteration of p. The

author suggest to use ρ = ln(e23e12

)/ln(r12) as a first guess. The iteration

will stop if |p−ρp | < 1E − 5.

Now one can calculate, ε12 = f2−f1f1

, ε23 = f3−f2f2

, GCIfine12 = 1.25|ε12|

(rp12−1)and

GCIfine23 = 1.25|ε23|

(rp23−1). GCI

fine12 should be smaller than GCI

fine23

To check if the solution is in the asymptotic range, α ≈ 1 (cf. Eq. 4.10)

α =rp12GCI

fine12

GCIfine23

(4.10)

The Richardson extrapolation can be used to obtain the value of f when

the grid spacing h vanishes (h → 0) [2, 143].

fexact = f1 + (f1 − f2)/ (rp12 − 1) (4.11)

Table 4.7 presents the grid convergency calculations using GCI method

and three grid levels for cyclones I -III. The following conclusions have

been obtained from the GCI analysis:

• The results are in the asymptotic range for the three cyclones, be-

cause the obtained values for α are close to unity.

• The ratio R is less than unity this means monotonic convergence [2]

(Ali et al. [2] classified the possible convergence conditions into three

groups, namely (1) monotonic convergence; 0 < R < 1 (2) oscillatory

convergence; R < 0 (3) divergence; R > 1.) .

• There is a reduction in the GCI value for the successive grid refine-

ments (GCIfine12 < GCI

fine23 ) for the two variables (Eu and x50). This in-

dicates that the dependency of the numerical results on the cell size

has been reduced. Moreover, a grid independent solution has been

achieved. Further refinement of the grid will not give much change

61


in the simulation results. For the two variables (Eu and x50), the ex-

trapolated value is only slightly lower than the finest grid solution.

Therefore, the solution has converged with the refinement from the

coarser grid to the finer grid [2]. Figure 4.4 presents a qualitative

proof that the obtained results are in the asymptotic range, i.e., the

obtained results are mesh independent.

Table 4.7: Grid convergency calculations using GCI method and three grid levels

for cyclones I - III

i Ni fi ri,i+1 ei,i+1 εi,i+1 GCIfinei,i+1% Ra αb

I

Eu

0c 2.37501 1021616 2.3900

1.0586 0.0150 0.0063 0.78492 861077 2.4050 0.2013 1.0063

1.1085 0.0750 0.0312 1.55973 632153 2.4800

x50

0 1.34711 1021616 1.3500

1.0586 0.0050 0.0037 0.26492 861077 1.3550 0.1224 1.0037

1.1085 0.0410 0.0303 0.72523 632153 1.3960

II

Eu

0 2.25231 1025778 2.2700

1.0589 0.0100 0.0044 0.97542 863852 2.2800 0.1256 1.0044

1.1897 0.0800 0.0351 1.51943 513021 2.3600

x50

0 1.24131 1025778 1.2500

1.0589 0.0070 0.0056 0.87402 863852 1.2570 0.0902 1.0056

1.1897 0.0780 0.0621 1.56523 513021 1.3350

III

Eu

0 2.68351 1027982 2.6870

1.1299 0.0250 0.0093 0.16192 712576 2.7120 0.1607 1.0093

1.1150 0.1570 0.0579 1.31273 513991 2.8690

x50

0 1.19621 1027982 1.2000

1.1299 0.0100 0.0083 0.39132 712576 1.2100 0.3361 1.0083

1.1150 0.0300 0.0248 1.42113 513991 1.2400

a R=ε12/ε23.b α=

(

rp12 GCI12)

/GCI23.c The value at zero grid space (h → 0). i=1, 2 and 3 denote the calculations at the fine, medium andcoarse mesh respectively.

62


N -1

Eul

ernu

mbe

r

Cut

-off

diam

eter

0 5E-07 1E-06 1.5E-062

2.2

2.4

2.6

2.8

3

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5Cyclone I (Eu)Cyclone I (X50)Cyclone II (Eu)Cyclone II (X50)Cyclone III (Eu)Cyclone III (X50)

(h --> 0)

Figure 4.4: Qualitative representation of the grid independency study. The Euler

number and the cut-off diameter for each cyclone at the three grid levels. N−1 is

the reciprocal of the number of cells, h → 0 means the value at zero grid size (cf.

Table 4.7). To obtain a smooth curve; the spline curve fitting has been applied in

Tecplot post-processing software.

4.2.2 Results and discussion

Validation of results

The obtained numerical results are compared with the LDA velocity mea-

surements of Hoekstra [74] measured using laser doppler anemometry

(LDA) system. Figure 4.5 shows the comparisons between the LES simula-

tion and the measured axial and tangential velocity profiles at axial station

Z=94.25 cm from the cyclone bottom [74]. The LES simulation predicts

a similar trend as observed experimentally. The non exact matching be-

tween experimental and LES simulation has been reported in some other

literatures [e.g., 38]. Considering the complexity of the turbulent swirling

flow in the cyclones, the agreement between the simulations and measure-

ments is considered to be quite acceptable. Another comparison between

the current LES results and the Reynolds stress turbulence model (RSM)

results of Xiang and Lee [186] for cyclone III is given in Fig. 4.6 which in-

dicates LES can also depict the main flow features of cyclonic flow as the

Reynolds stress turbulence model can do.

63


Radial position / Cyclone radius

Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

LDASimulation

Radial position / Cyclone radiusA

xial

velo

city

/Inl

etve

loci

ty-0.5 0 0.5

-0.4

-0.2

0

0.2

0.4

0.6

LDASimulation

Figure 4.5: Comparison of the time averaged axial and tangential velocity between

LDA measurements [74] and the LES simulations at section Z=94.25 cm from the

cyclone bottom (cf. Hoekstra [74] for more details about the used cyclone geometry

in this validation).

4.2.3 The flow pattern in the three cyclones

The pressure field

Figure 4.7 shows the time-averaged static pressure contours plots. In the

three cyclones, the static pressure decreases radially from the wall to the

center. A negative pressure zone appears in the forced vortex region (cen-

tral region) due to high swirling velocity. The pressure gradient is largest

along the radial direction, while the gradient in the axial direction is very

limited. The cyclonic flow is not symmetrical as is clear from the shape of

the low-pressure zone at the cyclone center (twisted cylinder). The static

pressure contour plots for the three cyclones are almost the same.

The velocity field

Based on the contours plots of the time averaged tangential velocity (Fig. 4.7)

and the radial profiles at sections S1, S2, S3 and S4 shown in Fig. 4.8 (cf.

Table 4.5), the following comments can be drawn. The maximum tangen-

tial velocity equals around 1.25 times the average inlet velocity and occurs

in the annulus cylindrical part. The tangential velocity distribution for

the three cyclones are nearly identical at the corresponding sections. The

tangential velocity profile at any section is composed from two regions, in-

ner and outer. In the inner region the flow rotates approximately like a

solid body rotation (forced vortex), where the tangential velocity increases

with radius. After reaching its peak the tangential velocity decreases with

radius in the outer part of the profile (free vortex). This profile is so-called

64


Dimensionless distance

Dim

en

sio

nle

ss

tan

ge

nti

al

ve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

1.25Present studyXiang and Lee (2005)


Dim

en

sio

nle

ss

axia

lve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1-0.5

-0.25

0

0.25

0.5Present studyXiang and Lee (2005)

Figure 4.6: Comparison of the time averaged tangential and axial velocity between

RSM results of Xiang and Lee [186] and the current LES results at Z/D=1.29 for

Cyclone III (Bc/D = 0.375). Note: the dimensionless radial distance = the distance

/ the cyclone radius, the dimensionless velocity = the velocity / inlet velocity.

Rankine type vortex which include a quasi-forced vortex in the central re-

gion and a quasi free vortex in the outer region.

The radial profiles given in Fig. 4.8 represent the time averaged tangential

velocity in the lower part of the cyclone. The tangential velocity distribu-

tions at the bottom sections show good axis-symmetrical distribution. The

tangential velocity profiles for the three cyclones are identical in the inner

region, where the maximum tangential velocity nearly equals the inlet ve-

locity and occurs at a position 0.25 - 0.45 of the cyclone radius as given in

Table 4.8.

The axial velocity contours (Fig. 4.7) indicate the existence of two flow

streams. Downward flow directed to the cyclone bottom (negative axial

velocity), and upward flow directed to the vortex finder exit. The axial ve-

locity plots for the three cyclones are nearly identical to the corresponding

sections in the conical part. The axial velocity equals zero at the walls and

maximum close to the position of maximum tangential velocity. The axial

velocity profiles shown in Fig. 4.8 exhibit a severe asymmetrical feature.

Table 4.8: Comparison between the maximum tangential velocity value and its

position at different sections.

Section S1 S2 S3 S4

Cyclone I II III I II III I II III I II III

vθmax/ v∗in 1.08 0.97 1.02 1.02 0.95 1.05 0.96 0.89 1.04 0.94 0.88 1.035

x/R† 0.3 0.28 0.26 0.3 0.33 0.33 0.32 0.37 0.41 0.35 0.43 0.425

∗ The ratio between the maximum tangential velocity and the area average inlet velocity.† The dimensionless distance between the centerline and the point of maximum velocity, R is the cycloneradius.

65


Comparison of the velocity profiles in the three cyclones

However, from the previous discussions it is clear that, the effect of the

cone-tip diameter on the flow field in the conical section is insignificant, in

comparison with other geometrical parameters such as the vortex finder

diameter. Nevertheless, in this section a comparison between the axial and

tangential velocity profiles at four sections (Table 4.5) will be analyzed as

presented in Fig. 4.9.

The tangential velocity profiles in the forced vortex region are nearly iden-

tical in the three cyclones at each sections. The tangential velocity in the

free vortex region increases as the cone-tip diameter is reduced. The tan-

gential velocity profiles for the three cyclones are almost the same.

The axial velocity profile has the shape of an inverted W for all cyclones.

The highest axial velocity occurs at 0.25 - 0.5 of the cyclone radius down

the vortex finder until the cyclone bottom, and between 0.25 and 0.5 of the

cyclone radius in the annulus and through the vortex finder. No consid-

erable difference exists in the axial velocity profiles for the three cyclones.

Since the axial velocity profiles is almost the same for the three cyclones,

the average residence time of particles is nearly the same. Furthermore,

the position of the highest axial velocity moves inward in the conical part

as the cone-tip diameter reduced.

From the previous analysis, the region of downward flow is nearly the

same, for the three cyclones, while the tangential velocity slightly increases

as the cone-tip diameter reduced, so the particles will experience a higher

tangential velocity for cyclone III than in other cyclones for the same time

(as the region of downward axial velocity is nearly equal). This results in

a slightly higher collection efficiency. This is consistent with the measured

results reported by Xiang et al. [184] and simulation by Xiang and Lee

[186] and Chuah et al. [23]. The change of the cone-tip diameter affects

the flow field in the cyclone separator but this change is so limited, i.e.,

the reduction of cone-tip diameter enhances the collection efficiency but

with a small percentage, as the flow field pattern is so closed for the three

cyclones.

The DPM results

In order to calculate the effect of the cone-tip diameter on the cut-off diam-

eter, 104 particles were injected from the inlet surface with particle veloc-

ity equals the gas inlet velocity. The particle density is 860 kg/m3 and the

maximum number of time steps for each injection was 9E5 steps. The DPM

analysis results for the three cyclones are shown in Table 4.9 and Fig. 4.10.

It is found that the cut-off diameter decreases slightly with decreasing the

66


Figure 4.7: The contours plots for the time averaged flow variables at Y=0 . From

top to bottom: the static pressure N/m2, the tangential velocity and axial velocity

m/s. From left to right cyclone I, II and III respectively.

67



Dim

en

sio

nle

ss

tan

ge

nti

al

ve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

1.25S1S2S3S4


Dim

en

sio

nle

ss

axia

lve

locit

y-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

-0.5

-0.25

0

0.25

0.5S1S2S3S4


Dim

en

sio

nle

ss

tan

ge

nti

al

ve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

1.25S1S2S3S4


Dim

en

sio

nle

ss

axia

lve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1-0.5

-0.25

0

0.25

0.5S1S2S3S4


Dim

en

sio

nle

ss

tan

ge

nti

al

ve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

1.25S1S2S3S4


Dim

en

sio

nle

ss

axia

lve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1-0.5

-0.25

0

0.25

0.5S1S2S3S4

Figure 4.8: The radial profile for the time averaged tangential and axial velocity

at different sections on the X-Z plane (Y=0) for each cyclone. From top to bottom:

Cyclone I, II and III respectively.

68



Dim

en

sio

nle

ss

tan

ge

nti

al

ve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

1.25Cyclone ICyclone IICyclone III


Dim

en

sio

nle

ss

axia

lve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1-0.5

-0.25

0

0.25



Dim

en

sio

nle

ss

tan

ge

nti

al

ve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10

0.25

0.5

0.75

1



Dim

en

sio

nle

ss

axia

lve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1-0.5

-0.25

0

0.25



Dim

en

sio

nle

ss

tan

ge

nti

al

ve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10

0.25

0.5

0.75

1



Dim

en

sio

nle

ss

axia

lve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1-0.5

-0.25

0

0.25



Dim

en

sio

nle

ss

tan

ge

nti

al

ve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10

0.25

0.5

0.75

1



Dim

en

sio

nle

ss

axia

lve

locit

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1-0.25

0

0.25


Figure 4.9: Comparison between the radial profile for the time averaged tangential

and axial velocity at different sections on the X-Z plane (Y=0). From top to bottom:

section S4 - S1 respectively.

69


cone-tip diameter while the pressure drop is increasing slightly. Conse-

quently, the effect of the cone-tip diameter on the cyclone performance is

insignificant.

The trend of changing the cut-off diameter with the cone-tip diameter

given by Chuah et al. [23] (Qin = 60l/min) supports the conclusion of

the slightly decrease of the cut-off diameter by decreasing the cone-tip di-

ameter (insignificant effect), Table 4.9.

4.2.4 Comparison with mathematical models

Table 4.10 presents a comparison between the Euler number (dimension-

less pressure drop) and the cut-off diameter obtained from CFD, experi-

mental investigation [184] and seven mathematical models, viz. (the Barth

model [9], the Muschelknautz method of modeling (MM) [116, 174], the

Stairmand model [165], the Casal and Martnez-Benet model [21], the Shep-

herd and Lapple model [157], the Iozia and Leith model [85] and the Ritema

model [142] (cf. Sec. 2.2)

The Euler numbers obtained from the models of Shepherd and Lapple,

Casal and Martnez-Benet are constant, because these models do not in-

clude the effect of the cone-tip diameter Bc in their formulas. The three

other models (Barth, MM and Stairmand) indicate less effect on both the

Euler number and the cut-off diameter by changing the cone-tip diameter.

The models of Iozia and Leith in addition to that of Rietma indicate no

change in the cut-off diameter with changing the cone-tip diameter. The

results of mathematical models and the experimental investigation sup-

port the CFD results that the cone-tip diameter has an insignificant effect

on the cyclone separator performance.

Table 4.9: The cut-off diameter and pressure drop for the three cyclones

Cyclone I II III

Bc/D 0.625 0.5 0.375

Cut-off diameter [µm] 1.35 1.25 1.2

Cut-off diameter [µm] (Chuah et al. [23]∗) 1.65 1.45 1.1

Pressure drop [N/m2] 76.5 72.7 86

Euler number Eu 2.39 2.27 2.687

∗ Qin = 60 l/min.

70


Bc /D

Eul

ernu

mbe

r[-]

Cut

-off

diam

eter

[mic

ron]

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70

1

2

3

4

5

1

1.2

1.4

1.6

1.8

Euler number [-] (Exp.)

Euler number [-] (LES)

Cut-off diameter [micron] (LES)

Figure 4.10: The effect of cone-tip diameter on the pressure drop (Euler number)

and the cut-off diameter (with spline curve fitting to get a smooth curve).

4.2.5 Conclusions

Large eddy simulation has been used to study the effect of the cone-tip

diameter on the cyclone flow field and performance. Three cyclones with

different values of Bc/D viz. 0.625, 0.5 and 0.375 (at constant vortex finder

diameter Dx/D = 0.5) have been investigated. The following conclusion

can be drawn.

• The cone-tip diameter has an insignificant effect on the flow pattern

and performance.

• As the cone-tip diameter decreases, the maximum tangential velocity

increases slightly, while its position is almost the same.

Table 4.10: The cyclone performance parameters using CFD, Experimental [184]

and different seven mathematical models

Cyclone CFD Barth MM Stairmand∗ Sphered∗ Casal∗ Iozia† Ritema† Exp.§

I 2.39 6.94 4.88 6.68 4.1 4.07 - - 2.8

Euler number Eu [-] II 2.27 7.40 4.95 6.69 4.1 4.07 - - 2.8

III 2.687 7.43 4.95 6.69 4.1 4.07 - - 3.25

I 1.35 1.22 2.13 - - - 1.44 1.4 3.01

Cut-off diameter x50 [µm] II 1.25 1.22 2.089 - - - 1.44 1.4 2.60

III 1.2 1.28 2.089 - - - 1.44 1.4 2.36

∗ The mathematical model used for estimation of the pressure drop only.† The mathematical model used for estimation of the cut-off diameter only.§ Different particle density.

71


• The flow pattern and performance parameters of the three cyclones

are almost the same.

• Decreasing the cone-tip diameter increases the pressure drop slightly.

The reverse trend is obtained for the cut-off diameter.

• Seven mathematical models used for estimation of the effect of cone-

tip diameter on the cyclone performance, and all of them support the

CFD results for the insignificance of varying the cone-tip diameter on

the cyclone performance.

• However, the main finding of the current study is the insignificant

effect of the cone-tip diameter on the cyclone performance, in com-

parison to the other geometrical parameters like the vortex finder

diameter Dx or the inlet dimensions [cf. 50, 52], the cone is an essen-

tial part for cyclone operation. If the cone would be removed from the

cyclone, it will be only cylindrical part and that one will have another

geometry (not the conventional cyclone). Moreover, the particles are

collected over the cone wall and then moved to the cyclone bottom.

Consequently, a very low collection efficiency will be expected for a

cyclone separator without a cone.

4.3 The dust outlet geometry

Conventional cyclones always have a dustbin attached to the cone to collect

the separated solid particles. When a gas flow stream enters the dustbin

(closed at bottom), some of the flow will return the cone and carry some of

the separated particles. This phenomena is called “re-entrainment” and it

will affect the separation efficiency of the cyclone [138].

However, many works have been carried out to investigate the influence of

different geometrical parameters such as cyclone length, inlet and outlet

pipe geometries on the performance of cyclones [15, 62, 102, 140, 184],

there has been little work concerning the dust outlet geometries [40, 47,

78, 123].

Regarding this influence, discrepancies and uncertainties exist in the liter-

ature. Xiang and Lee [186] reported that the dustbin connected to the cy-

clone should be incorporated in the flow domain as it affects the results. On

the other hand, numerous studies were performed without dustbin [e.g.,

159, 178] with good matching with experimental results.

Obermair et al. [123] performed cyclone tests with five different dust outlet

geometries to find the influence of the dust outlet geometry on the sepa-

ration process. They showed that separation efficiency can be improved

significantly by changing the dust outlet geometry, and they reported that

further research is needed to clarify precise effects of dust outlet geometry.

72

4.3. The dust outlet geometry

The effect of a dipleg (a vertical tube between the cyclone and the dustbin)

was posed and investigated by several researchers [78, 92].

The previous studies on the effect of dust outlet geometry on the cyclonic

flow can be classified into the following categories:

1. Comparison between cyclones with dustbin and that with dustbin

plus dipleg [78, 123, 138]. Obermair et al. [123] investigated exper-

imentally the effect of different dust outlet geometries on the flow

pattern. Nevertheless, the given flow pattern was limited to the cy-

clone bottom. The effects of the dust outlet geometry on the velocity

profiles are not given.

2. Comparison between the cyclone with and without dipleg (prolonged

cyclones). Kaya and Karagoz [92] numerically investigated the flow

characteristics and particle collection efficiencies of conventional (with-

out dipleg) and prolonged cyclones.

3. Comparison between cyclone with and without dustbin [47]. Elsayed

and Lacor [47] numerically investigated two cyclones with and with-

out dustbin. They reported a negligible effect of the dustbin on the

performance. They reported further research is needed with the cy-

clone with dipleg and dust hopper (dustbin plus dipleg).

All studies above do not contain enough information about the effect of

the dust outlet geometry on the flow pattern, velocity profiles and sepa-

ration performance. Furthermore, there is no clear comparison between

the four cases: separation space only (cylinder on cone), cyclone with cylin-

drical shaped dustbin, cyclone with vertical tube (dipleg) and cyclone with

dustbin plus dipleg. The present numerical investigation aims at exam-

ining the influence of the dust outlet geometry on the flow pattern and

the cyclone performance using the Reynolds stress turbulence model. This

serves two purposes: (1) to demonstrate whether simulation of the separa-

tion space only without including the dustbin or the dipleg is sufficient to

estimate the cyclone performance. (2) to give information for the designer

about the effect of the configuration under the separation space (the dust-

bin or the dipleg) on the flow pattern and performance.

4.3.1 Numerical simulation

4.3.1.1 Configurations of the four tested cyclones

The numerical simulations were performed on four cyclones. Cyclone I has

only the cylinder on cone shape (separation space only), cyclone II has a

dustbin, cyclone III has a dipleg while cyclone IV has a dust hopper (dust-

bin plus dipleg). Figure 4.11 and Table 4.11 give the cyclones dimensions.

73


Table 4.12 gives more details for the used cyclones, including the number

of cells, cyclone volume and the flow residence time for each cyclone. Nine

sections are used to plot the velocity profiles as shown in Table 4.13.

(a) I (b) II (c) III (d) IV

(e) The surface mesh

Figure 4.11: Schematic diagrams and surface meshes for the four tested cyclone

separators

4.3.1.2 Solver settings

Based on the study of Kaya and Karagoz [91] for the best selection of nu-

merical schemes to be used with RSM model, the following discretization

schemes have been used. The PRESTO scheme has been used for the pres-

74


Table 4.11: The geometrical dimensions of the four tested cyclones∗

Dimension Length Dimension ratio

(m) (Dimension/D)

Body diameter, D 0.205 1

Inlet height, a 0.105 0.5

Inlet width, b 0.041 0.2

Gas outlet diameter, Dx 0.105 0.5

Gas outlet duct length, S 0.105 0.5

Cone-tip diameter, Bc 0.076875 0.375

Cylinder height, h 0.3075 1.5

Cyclone height, Ht 0.82 4

∗ The outlet section is above the cyclone surface by Le = 0.5D. The inlet section located at a distanceLi = 0.75D from the cyclone center, the height of the dustbin and the dipleg, LD = 2D.

sure interpolation, the SIMPLEC algorithm for pressure velocity coupling,

the QUICK scheme for momentum equations, the second-order upwind

for the turbulent kinetic energy and the first-order upwind discretization

scheme for the Reynolds stresses [52], cf. Sec. 4.2.1.3 for more details.

Chuah et al. [23] stated that the time step should be selected as a tiny

fraction of the residence time tres. From Table 4.12, the value of tres varies

between 0.237 and 0.399 s. Therefore a time step of 1E-4 is an accept-

able value for the current simulations [52]. The simulations have been

performed using FLUENT 6.3.26.

4.3.1.3 CFD grid and boundary conditions

The mesh sensitivity study has been performed for the four tested cyclones

with three levels for each cyclone, to be sure that the obtained results

are grid independent. For example, three different meshes with respec-

tively 130596, 260230 and 478980 cells have been used for cyclone I. The

computational results on the three grids are presented in Table 4.14. As

the maximum difference between the results is less than 5%, so the grid

template 130596 produces the grid independent results [52, 146]. It has

been observed that even 130596 grid provides a sufficient grid indepen-

Table 4.12: The details of the four tested cyclones

Cyclone I II III IV

Number of cellsa 260230 593125 322286 441062

Cyclone volume x102 [m3] 1.99 3.35 2.18 2.76

tres [s] b 0.237 0.399 0.259 0.328

a The total number of hexahedral cells after the mesh sensitivity study

b The average residence time, tres = V/Qin where V is the cyclone volume and Qin is the gas flow rate.

75


Table 4.13: The position of different plotting sections∗

Section S1 S2 S3 S4 S5 S6 S7 S8 S9

z`/D 2.75 2.5 2.25 2 1.75 1.5 1.25 1.0 0.75∗ z` is measured from the top of the inlet section (cf. Fig. 4.11(a)).

dency. However, for excluding any uncertainty, computations have been

performed using the 260230 cells grid, where the total number of grid

points was not that critical with respect to the computation overhead [10].

Figure 4.11(e) shows the surface mesh of the four cyclones. The hexahedral

computational grids were generated using the GAMBIT grid generator.

The boundary condition at the inlet section is the velocity inlet. An out-

flow boundary condition is used at the outlet. The no-slip (wall) boundary

condition is used at the other boundaries [52]. The air volume flow rate

Qin=0.08405 m3/s for all cyclones, air density 1.225 kg/m3 and dynamic

viscosity of 178.940E-6 Pa s. The turbulent intensity equals 5% and char-

acteristic length equals 0.07 times the inlet width [74].

DPM settings

A discrete phase modeling (DPM) study has been performed by injecting

104 particles from the inlet surface with a particle density of 860 kg/m3

and with a particle size ranging from 0.025 until 5 µm at a velocity equals

to the gas velocity.

4.3.2 Results

Validation of the numerical model

In order to validate the obtained results, it is necessary to compare the pre-

diction with experimental data. The comparison performed with the mea-

surements of Hoekstra [74] of the Stairmand cyclone using Laser doppler

anomemetry (LDA). The present simulation are compared with the mea-

sured axial and tangential velocity profiles at an axial station located at

94.25 cm from the cyclone bottom (Dx/D = 0.5) as shown in Fig. 4.12 (cf.

Table 4.14: The details of the grid independency study for cyclone I

Total number of cellsStatic pressure drop Cut-off diameter

N/m2 µm

130596 955.51 1.48260230 960.25 1.5478980 961.12 1.51% difference∗ 0.584 1.987

∗ The percentage difference between the coarsest and finest grid

76


Radial position /cyclone radius

Tan

gint

ialv

eloc

ity/i

nlet

velo

city

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

LDARSM

Radial position /cyclone radius

Axi

alve

loci

ty/i

nlet

velo

city

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

-0.25

0

0.25

0.5

LDARSM

Figure 4.12: Comparison of the time averaged tangential and axial velocity be-

tween the LDA measurements, Hoekstra [74] and the current Reynolds stress tur-

bulence model (RSM) results at 94.25 cm from the cyclone bottom.

Hoekstra [74] for more details about the used cyclone in this validation).

The RSM simulation matches the experimental velocity profile with un-

derestimation of the maximum tangential velocity, and overestimation of

the axial velocity at the central region. Considering the complexity of the

turbulent swirling flow in the cyclones, the agreement between the simu-

lations and measurements is considered to be quite acceptable.

The cyclone pressure drop is calculated as the pressure difference between

the inlet and the average pressure across the vortex finder exit [74]. The

experimental pressure drop of the cyclone can be calculated by the differ-

ence between the static pressures at the inlet and outlet [137]. A com-

parison of the pressure drop, the cut-off diameter (at particle density of

2740 kg/m3) obtained from the experimental data [74], CFD prediction is

shown in Table 4.15. Table 4.15 indicates a very small deviations from

the experimental values in both the calculated pressure drop and cut-off

diameter. As the errors are less than 4%, so it is in the same magnitude

as the experimental error [137]. The above comparison results show that

the numerical model employed in this study can be used to analyze the gas

flow field and performance of the cyclone separator.

Table 4.15: Validation of the computational pressure drop and cut-off diameter

Static pressure drop [N/m2] Cut-off diameter [micron]

Experimental [74] 300 1

CFD 309 0.965

% error 3 3.5

77


The dominant velocity component of the gas flow in cyclones is the tan-

gential velocity, which results in the centrifugal force for particle separa-

tion [186]. The axial velocity is responsible more than the gravity for the

transport of particles to the collection devices [29, 104]. These velocity

components will be discussed in details in order to investigate the effect

of the dust outlet geometry on the flow properties. Moreover, the pressure

distribution in the swirling flow in these four cyclone separators will be

discussed in details.

The axial variation of the flow properties

Figures 4.13 and 4.14 present the radial profiles of the time-averaged tan-

gential and axial velocity and static pressure at nine axial stations. As

expected, the tangential velocity profiles exhibit the so-called Rankine vor-

tex, which consists of two parts, an outer free vortex and an inner solid

rotation in the center (Fig. 4.13). The tangential velocity distribution in

the inner region is rather similar in different sections for the same cy-

clone. In the outer region, due to the sharp drop in velocity magnitude in

the near wall region, the distribution is different but the maximum tan-

gential velocity is similar in all sections. Generally, the tangential velocity

distribution varies only slightly with axial positions for the same cyclone,

which is also reported in other articles [e.g., 66, 127, 163, 186]. This means

that, if the tangential velocity increases at one section of the cyclone, it will

increase at all other sections. The axial velocity profiles at nine different

stations are shown in Fig. 4.13. Two types of axial velocity profiles are

observed. Cyclone I, III and IV show an inverted W profile. Only cyclone

II (with dustbin) has an inverted V profile. The reason for the two differ-

ent axial velocity profiles can be explained by the change in the flow field

pattern caused by the dustbin.

The radial profiles of the time averaged static pressure are given in Fig. 4.14

for the four cyclones. Like for the tangential velocity, the axial variations

of static pressure are very small for the same cyclone. An exception is the

cyclone I which shows some variations in the central part. Figure 4.15

compares the static pressure profiles for the four cyclones at sections S7-

S9 (located at the cylindrical part of the cyclone, which is the most effective

part of the cyclone in the separation process, the location of the highest area

average tangential velocity). The plots at sections S7-S9 are also represen-

tative for the other sections, as the axial variations in the flow variables

are small. From the comparison between the radial profiles of the four

cyclones, the minimum pressure at the cyclone center is almost the same

for all cyclones. The static pressure radial profiles of cyclones I, III and IV

78

4.3

.T

he

du

stou

tlet

geom

etry

Radial position/ Cyclone radius

Ta

ng

en

tia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 10

5

10

15

20

25

30

S1S2S3S4S5S6S7S8S9


Ta

ng

en

tia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 10

5

10

15

20

25

30

S1S2S3S4S5S6S7S8S9


Ta

ng

en

tia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 10

5

10

15

20

25

30

S1S2S3S4S5S6S7S8S9


Ta

ng

en

tia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 10

5

10

15

20

25

30

S1S2S3S4S5S6S7S8S9


Axia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 1-5

0

5

10

15

20S1S2S3S4S5S6S7S8S9


Axia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 1-5

0

5

10

15



Axia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 1-5

0

5

10

15



Axia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 1-5

0

5

10

15


Figure 4.13: The radial profile for the time averaged tangential and axial velocity at different sections.

79


almost coincide. Cyclone II (with dustbin) depicts fewer gradients in the

radial direction. It has the lowest maximum pressure value. Referring to

Fig. 4.15, there is no difference between the radial pressure profiles at the

three sections for the same cyclone.


Sta

tic

pre

ssu

re(N

/m2)

-1 -0.5 0 0.5 1

-200

0

200

400

600

800

1000

1200

1400

S1S2S3S4S5S6S7S8S9

(a) Cyclone I


Sta

tic

pre

ssu

re(N

/m2)

-1 -0.5 0 0.5 1

0

200

400

600

800

1000

1200

S1S2S3S4S5S6S7S8S9

(b) Cyclone II (dustbin)


Sta

tic

pre

ssu

re(N

/m2)

-1 -0.5 0 0.5 1

-200

0

200

400

600

800

1000

1200

1400

S1S2S3S4S5S6S7S8S9

(c) Cyclone III (dipleg)


Sta

tic

pre

ssu

re(N

/m2)

-1 -0.5 0 0.5 1

0

200

400

600

800

1000

1200

1400

S1S2S3S4S5S6S7S8S9

(d) Cyclone IV (dustbin plus dipleg)

Figure 4.14: The radial profiles for the time-averaged static pressure at different

sections for the four cyclones.

The velocity profiles

Figure 4.16 compares the tangential and axial velocity profiles at sections

S7-S9. The variation of the velocity profiles (both the axial and the tan-

80


gential) from cyclone to cyclone is mainly located at the central region for

both the tangential and axial velocity. Since the effect of the dust outlet ge-

ometry on the tangential velocity (centrifugal force) is minor, it is expected

that the collection efficiency (cut-off diameter) of the four cyclones will be

comparable.


Sta

tic

pre

ssu

re(N

/m2)

-1 -0.5 0 0.5 10

200

400

600

800

1000

1200

1400

1600With dust binWithout dust binWith diplegWith dipleg & dust bin

(a) The four cyclones at section S7


Sta

tic

pre

ssu

re(N

/m2)

-1 -0.5 0 0.5 10

200

400

600

800

1000

1200

1400


(b) The four cyclones at section S8


Sta

tic

pre

ssu

re(N

/m2)

-1 -0.5 0 0.5 10

200

400

600

800

1000

1200

1400


(c) The four cyclones at section S9

Figure 4.15: The radial profile for the time-averaged static pressure at different

sections for the four cyclones.

The axial velocity in the inner vortex is either reported as an inverted V

81


or inverted W-shaped profile,i.e., with a maximum (V-shaped) or a dip (W-

shaped) at the symmetry axis [29]. Hoekstra et al. [75] stated that the

shape of the axial velocity profile is affected by the cyclone geometry. They

referred the dip in the inverted W profile to the loss of swirl in the vortex

finder (the friction force of the vortex finder wall attenuates the swirling

flow), which results in an adverse pressure gradient at the centerline [78].

Hence, fluid with fewer swirls is drawn back from the exit pipe into the cy-

clone. This core flow prevails throughout the entire separation space of the

cyclone in spite of the attenuation of swirl in the conical part of the cyclone

[78]. This explains the reason behind the inverted W-shaped profile ex-

hibited by cyclones I (cylinder on cone), cyclone III (dipleg) and cyclone IV

(dustbin plus dipleg). However, why cyclone II (with dustbin) exhibit the

inverted V-shaped profile? In cyclone II, the dustbin has an equal diame-

ter to the cyclone barrel, which means sudden expansion to the downward

flow and sudden contraction to the upward flow (the flow inside the dustbin

also has two streams due to the absence of a vortex stabilizer [75] which

can prevent the re-entrainment of the collected particles). The upward gas

flow (directed from the end of dustbin) has a higher kinetic energy espe-

cially at the cone-tip diameter (sudden contraction) which can overcome

the adverse pressure gradient at the centerline (caused by the swirl at-

tenuation in the vortex finder) and results in the inverted V axial velocity

profile. One more question may appear now, why cyclones III (dipleg) and

IV (dustbin plus dipleg) did not exhibit the inverted V-shaped profile? The

reason can be referred to the diameter of the dustbin directly connected to

the cyclone. For cyclones III and IV there is no change in the flow area at

the connection, consequently no flow acceleration happens. The effect of

dustbin dimensions (diameter, height) still need more investigations. The

author believe, if a cone is inserted at the entrance of the cone-tip inside

the dustbin of cyclone II, the axial velocity will become inverted W-shaped.

However, the dimensions and location of this cone still need more investi-

gations (cf. Obermair et al. [123] for more details).

The flow pattern

Regarding the effect of neglecting the dust outlet geometry (dustbin or

dipleg or dustbin plus dipleg) in the simulating domain on the flow field

pattern, the following comments can be drawn (Fig. 4.17).

1. From the comparison between the static pressure contour plots of

each cyclone and that of cyclone I, the highest value of static pres-

sure is obtained in case of cyclone III (dipleg). The lowest value is

given by cyclone II (dustbin). The highest value of the static pressure

82



Ta

ng

en

tia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 10

5

10

15

20

25

30



Axia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 1-10

-5

0

5

10

15

20



Ta

ng

en

tia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 10

5

10

15

20

25

30



Axia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 1-10

-5

0

5

10

15

20



Ta

ng

en

tia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 10

5

10

15

20

25

30



Axia

lve

locit

y(m

/s)

-1 -0.5 0 0.5 1-10

-5

0

5

10

15

20



at different sections on the X-Z plane (Y=0) at the inlet region (sections S7, S8 and

S9). From top to bottom: section S7-S9.

83


in cyclone IV (dustbin plus dipleg) is in between that for cyclones III

and II. This indicates a slight underestimation of the pressure drop

by neglecting the effect of dipleg or dustbin plus dipleg (cyclone III, IV

versus cyclone I). Furthermore, a slight overestimation of the pres-

sure drop is observed by neglecting the effect of the dustbin (cyclone

II versus cyclone I).

2. The tangential velocity pattern is very similar in the four cyclones

(Rankine profile). The highest value is almost the same for the four

cyclones. Consequently, the collection efficiency (cut-off diameter) of

the four cyclones will be almost the same. Moreover, the contour plots

for the tangential velocity of the four cyclones are quite similar in the

main separation space (cylinder and cone).

3. The axial velocity patterns for cyclones I, III and IV have the shape

of an inverted W profile while that of cyclone II has an inverted V

profile, indicating different flow behavior. Close to the cone bottom

there are different flow patterns as a consequence of different dust

outlet geometry.

Figure 4.18 shows a qualitative view of the complex flow in the four tested

cyclones with the streamtraces plots of the time averaged velocities colored

by the time-average axial velocity. The swirling, downward flow at the

outer region of the cyclone is clearly visible. Near the bottom of the cyclone,

it is diverted into an upward flow near the cyclone center. In cyclone II

(with dustbin) and IV (with dustbin and dipleg), the flow behavior in the

dustbin is quite different because of the different length of the dustbin.

The performance

Figure 4.19 presents the grade efficiency curves (GECs) for the four tested

cyclones. As expected, the frictional efficiencies of all the cyclones are seen

to increase with the increase in particle size. The shapes of the grade

collection efficiency curves of all models have a so-called S shape [195].

It is clear from Fig. 4.19 that the effect of the cyclone dustbin or dipleg,

on the cut-off diameter (particle diameter of 50% collection efficiency) is

small. Neglecting the effect of dust outlet geometry in the cyclone simu-

lation slightly overestimates the cut-off diameter (when compare it with

cyclone I). A deeper look to the GECs indicates the variation of the collec-

tion efficiency for particles diameters larger than 1.5 µm, with a higher

efficiency for cyclone II and the lower efficiency for cyclone I. This behavior

is due to the increase in the separation space and the change in the max-

imum tangential velocity. For particles with diameters less than 0.8 µm,

84


Figure 4.17: The contour plots for the time averaged flow variables at sections Y=0

and throughout the inlet section. From left to right: the static pressure (N/m2), the

tangential and the axial velocity (m/s).

85


Figure 4.18: The streamtraces plots for the time averaged flow variables, colored

by the average axial velocity (m/s).

86


the collection efficiency tends to zero except in case of cyclone IV which

depict a non zero collection efficiency.

In order to estimate the effect of dust outlet geometry on the performance

parameters, the pressure drop and cut-off diameter have been calculated.

Table 4.16 shows the effect of dust outlet geometry absence from the simu-

lation domain on the pressure drop and cut-off diameter. From the compar-

ison between the estimated pressure drop of cyclone I and the other three

cyclones, the following comments are obtained. A slight underestimation

is obtained when omitting the dipleg or dustbin plus dipleg from the sim-

ulation space. A slight overestimation result from omitting the dustbin.

The pressure drop values given in Table 4.16 support the results obtained

from the analysis of the static pressure contour plots.

Table 4.16: The effect of dust outlet geometry on the cyclone performance

Cyclone I II (dustbin) III (dipleg) IV (dustbin plus dipleg)

Pressure drop (N/m2) 960 890 1017 1008

Cut-off diameter (µm) 1.5 1.0 1.25 1.2

Particle diameter [micron]

Col

lect

ion

effic

ienc

y

1 2 3 4 5

0.2

0.4

0.6

0.8

1Without dustbinWith dustbinWith diplegWith dustbin plus dipleg

Figure 4.19: The grade efficiency curves for the four cyclones.

4.3.3 Conclusions

Four cyclones have been simulated using the Reynolds stress model (RSM),

to study the effect of the dust outlet geometry on the cyclone separator

performance, flow pattern and velocity profiles. The following conclusions

have been obtained.

87


• The maximum tangential velocity in the four cyclones is very similar.

• No radial acceleration occurs in the cyclone space (the maximum tan-

gential velocity is nearly constant throughout the cyclone).

• The cyclone without dustbin slightly overestimates both the pressure

drop and cut-off diameter. So the simulations with and without dust-

bin will produce nearly the same performance parameters.

• The cyclone without dipleg slightly underestimates the pressure drop

and overestimates the cut-off diameter. Consequently, the simula-

tions with and without dipleg will produce nearly the same perfor-

mance parameters.

• The axial velocity patterns obtained by the four cyclones are differ-

ent.

• If the main target of the CFD investigation is the performance pa-

rameters, one can safely simulate only the main separation space

(cylinder on cone). However, if the aim is to investigate the flow field

pattern, the dust outlet geometry should be included in the simula-

tion domain.

4.4 Closure

From the previous sections, it becomes clear that the most significant fac-

tors are four (the vortex finder diameter, the inlet width, the inlet height

and the total cyclone height (cone height)). Consequently, any minor mod-

ification in these particular factors will result in a considerable change in

the cyclone performance. The effect of the cone-tip diameter on the cyclone

performance is insignificant. Since the main target of this thesis is to op-

timize the cyclone performance, it is accepted to exclude the dust hopper

(dustbin or dipleg) from the simulation domain and use the obtained CFD

data to obtain the optimized cyclone design.

88

Chapter 5

The Vortex Finder Dimensions

5.1 Introduction

The vortex finder size is an especially important dimension, which sig-

nificantly affects the cyclone performance as its size plays a critical role

in defining the flow field inside the cyclone, including the pattern of the

outer and inner spiral flows. Numerous studies have been performed for

the effect of geometrical parameters on the flow pattern and performance

[15, 52, 62, 102, 140, 184] whereas only limited number of studies have

been devoted to the effect of the vortex finder dimensions.

Iozia and Leith [84] optimized the cyclone design parameters, including

the vortex finder diameter, to improve the cyclone performance using an

optimization code. Kim and Lee [95] described how the ratio of the di-

ameters of cyclone body, and the vortex finder diameter Dx/D affected the

collection efficiency and pressure drop of cyclones, and proposed an energy-

effective cyclone design. Moore and Mcfarland [111] also tested cyclones,

with six different vortex finders, and concluded that the variation in the

gas outlet diameter under the constraint of a constant cyclone Reynolds

number produced a change in the aerodynamic particle cut-off diameter.

Bakari and Hamdullahpur [7] investigated experimentally the effects of

both the inlet gas velocity, the cyclone inlet width, the vortex finder length

and the vortex finder diameter on the cyclone performance. They reported

that the vortex finder length and the vortex finder diameter have a strong

effect on the cyclone performance parameters. Moreover, the results in-

dicated that the vortex finder length has a direct effect on the cyclone

performance. The longer vortex finder minimizes the short circuiting of

incoming gases, preventing the dusty gas from flowing directly from the

inlet to the outlet. Zhu and Lee [200] carried out a set of experimental

89

Chapter 5. The Vortex Finder Dimensions

(a) The cyclone geometry (b) The surface mesh for cyclone S1 (D5)

Figure 5.1: Schematic diagram for the cyclone separator

investigations on the particle collection efficiency of small cyclones oper-

ating at high-flow rates. Special emphasis was given to the effects of the

barrel height and the vortex finder length on the particle collection effi-

ciency. The length ratios of the barrel height h and vortex finder length Sto the cyclone body diameter D were varied from 0.75 to 4.5 and from 0.5 to

1.5, respectively. Pressure drop decreased substantially either as the bar-

rel height h became longer or as the vortex finder length S became shorter.

It was also found that the difference between the cyclone barrel height hand the vortex finder length S affects the particle collection characteristics

significantly. The optimum performance will be obtained if (h−S)/D = 1.0.

The performance of a cyclone, with different vortex finders, was evaluated

by Lim et al. [102] to examine the effect of the vortex finder shape on

the characteristics of the collection efficiency. Four cylinder-shaped and

six cone-shaped vortex finders were designed and employed to compare

the collection efficiencies of the cyclone, at flow rates of 30 and 50 l/min.

The cylinder-shaped-vortex finders had different diameters and the cone-

shaped vortex finders had different cone lengths. The result indicates that

two cone-shaped vortex finders, with different diameters, had the collec-

tion efficiencies between those of the cylinder-shaped vortex finders with

the same diameter, and that a smaller pressure drop per flow rate unit

could be achieved for the cone-shaped design, but the cone length did not

affect the collection efficiency and pressure drop of the cyclone. Raoufi et

al. [140] duplicated numerically the same study of Lim et al. [102] with

limited details about the effect of the gas outlet diameter on the flow field

90

5.1. Introduction

pattern and velocity profile.

You-hai et al. [188] simulated the three-dimensional gas-phase flow field

in the cyclone separator with different vortex finder diameters. The results

show that when the diameter of the vortex finder decreases, the downward

flow decreases and the tangential velocities of the whole cyclone separa-

tor increase, but at the cost of the pressure drop increase. Horvath et al.

[79] refereed the reason behind the two classes of axial velocity in cyclone

separator (class V and class W) to the influence of the vortex finder di-

ameter. The pressure drop and collection efficiency of a swirl tube with

different vortex finder geometries were studied numerically by Jian and

You-hai [87]. The gas flow fields were simulated by the Reynolds stress

model (RSM) and the stochastic tracking approach in discrete phase model

(DPM). The results indicate that the decrease of the vortex finder diameter

leads to higher tangential velocity, which helps to improve the separation

efficiency. The back flow can be observed in converging coned-shaped vor-

tex finder, meanwhile the diverging coned-shape vortex finder can make

the flow move smoothly to reduce pressure drop.

Ficici et al. [57] performed an experimental study using three cylinder-

shaped vortex finders with diameters of 80, 120 and 160 mm. They inves-

tigated the effects of gas inlet velocity, the vortex finder diameter Dx and

length S on the cyclone performance. They reported a linear relationship

between the length of the vortex finder and the pressure loss. Khalkhalia

and Safikhania [94] performed a multi-objective optimization of a cyclone

vortex finder shape using CFD simulations data set. Two meta-models

based on the evolved group method of data handling (GMDH) type neural

networks are used as fitness functions for Pareto-based optimization.

In summary, all previous studies reported the significant effect of the vor-

tex finder dimensions on the cyclone performance and flow pattern. Nev-

ertheless, the previous studies are not coherent, and did not present suf-

ficient details about the effect of these two geometrical parameters on the

pressure drop and the cut-off diameter. Moreover, detailed studies about

the effect of the vortex finder dimensions on the velocity profiles are scarce

in the literature.

The present study is intended to computationally investigate the effect of

increasing the vortex finder diameter Dx and length S on the pressure drop

and cut-off diameter and to obtain more details about the flow field pattern

and velocity profiles using the large eddy simulation (LES) methodology.

91


5.2 Numerical settings

5.2.1 Configuration of the tested cyclones

The numerical simulations were performed on five cyclones with different

vortex finder diameters (at constant S) and five cyclones with different

vortex finder length (at constant Dx). Figure 5.1 and Table 5.1 give the

cyclones geometrical dimensions.

5.2.2 Solver settings

The simulations started with unsteady simulation using the Reynolds stress

turbulence model with a time step of 1E-4 for initialization of the flow field.

Afterwards, the turbulence model switched to the large eddy simulation

with a time step of 1E-5s using implicit coupled solution algorithm. The

selected time step results in an average inlet Courant number of 0.0288

for the tested cyclones. The cyclones volumes’ and the corresponding res-

idence times for the tested cyclones are given in Table 5.2. The minimum

value of residence time is 0.0812s i.e., the used time step is just a small

fraction of the average residence time. This confirms that the used time

step can reveal the transient phenomena properly. For the tested cyclones

the maximum values of the courant number are less than 0.2. This verifies

again that the choice of the time step was proper (cf. Sec. 4.2.1.3, page 56

for more details).

Table 5.1: The geometrical dimensions of the tested cyclonesa

Dimension Cycloneb dimension/D (h − S)/D Dx/SInlet height, a 0.375Inlet width, b 0.2625Barrel height, h 1.5c

Total cyclone height, Ht 4.0Cone tip-diameter, Bc 0.375

Vortex finder diameter, Dx S/D = 0.5

D1 0.30 1.0 0.6D2 0.35 1.0 0.7D3 0.40 1.0 0.8D4 0.45 1.0 0.9D5 0.50 1.0 1.0

Vortex finder length, S Dx/D = 0.5

S1 0.5 1.0 1.0S2 0.625 0.875 0.8S3 0.875 0.625 0.625S4 1.0 0.5 0.5S5 1.0 1.0c 1.0

a Body diameter, D = 31 mm. The outlet section is above the cylindrical barrel surface by Le = 0.5D.The inlet section located at a distance Li = D from the cyclone center, cf. Fig. 5.1(a).

b Cyclone D5 and S1 are identical.

c cyclone S6 has different barrel height h/D = 2.0 .

92

5.2. Numerical settings

Table 5.2: The details of the tested cyclones

Cyclone D1 D2 D3 D4 D5 =S1 S2 S3 S4 S5

Number of cellsa 601482 620088 657841 666336 714029 795321 798143 798256 888612

Cyclone volume x 105 [m3] 6.765 6.802 6.845 6.894 6.949 6.948677 6.947 6.946 8.116

tresb [s] 0.0812 0.0816 0.0821 0.0827 0.0834 0.0834 0.0834 0.0834 0.0974

a The total number of hexahedral cells after the grid independency study, cf. Sec. 5.2.4

b The residence time tres = V/Qin where V is the cyclone volume and Qin is the gas flow rate (50 l/min).

5.2.3 Boundary conditions

Velocity inlet boundary condition is applied at inlet, outflow at the gas

outlet and wall (no-slip) boundary condition at all other boundaries. The

air inlet volume flow rate Qin=50 l/min, air density 1.0 kg/m3 and dynamic

viscosity of 2.11E-5 Pa s. The turbulence intensity I equals 5% and the

turbulence characteristic length equals 0.07 times the inlet width [52, 55,

74]. All other settings are identical to that given in Sec. 4.2.1.3.

5.2.4 Grid independency study

The grid independence study has been performed for the tested cyclones.

Three levels of grid for each cyclone have been tested, to be sure that

the obtained results are grid independent. The hexahedral computational

grids were generated using GAMBIT grid generator and the simulations

were performed using Fluent 6.3.26 commercial finite volume solver on

an eight nodes CPU Opteron 64 Linux cluster. To evaluate accurately the

numerical uncertainties in the computational results, the concept of grid

convergence index (GCI) was adopted using three grid levels per cyclone.


The grid convergence index (GCI) proposed by Roache [143–145] was em-

ployed to test the grid independence of the simulations. The GCI are com-

Table 5.3: The details of the grid independence study for cyclone S1 (D5)

Number of cells Eua Cut-off diameter

714029 3.375 1.9851174029 3.18 1.4361793459 3.145 1.318

% differenceb -6.8148 -33.602% differencec -1.1006 -8.2173

a Euler number is the dimensionless pressure drop Eu = ∆P/( 12ρV 2

in) where ∆P is the static pressure

drop, ρ is the gas density, Vin is the gas inlet velocity. b The percentage difference between the coarsestand finest grid. c The percentage difference between the fine and finest grid.

93


puted using three levels of grids in order to estimate accurately the order of

convergence and check that the solution is within the asymptotic range of

convergence [160]. The GCI is based upon a grid refinement error estima-

tor derived from the theory of the generalized Richardson extrapolation

[55, 160]. The GCI is a measure of how far the computed value is away

from the value of the asymptotic numerical value. Consequently, it indi-

cates how much the solution would change with a further refinement of

the grid. A small value of GCI indicates that the computation is within the

asymptotic range (cf. Sec. 4.2.1.3 for more details).

Table 5.4 presents the grid convergency calculations using GCI method for

three grid levels for cyclone S1 as an example for the tested cyclones. The

following conclusions have been obtained from the GCI analysis [55]:

• The results are in the asymptotic range, because the obtained values

for α are close to unity.

• The ratio R is less than unity this means monotonic convergence [2].


ments (GCIfine12 < GCI

fine23 ) for the two variables (Eu and x50). This in-

dicates that the dependency of the numerical results on the cell size

has been reduced and a grid independent solution has been achieved.

Further refinement of the grid will not give much change in the sim-

ulation results. For the two variables (Eu and x50), the extrapolated

value is only slightly lower than the finest grid solution. Therefore,

the solution has converged with the refinement from the coarser grid

to the finer grid [2]. Figure 5.2 presents a qualitative proof that the

obtained results are in the asymptotic range.

Table 5.4: Grid convergency calculations using GCI method and three grid levels

for cyclone S1

i Ni fi ri,i+1 ei,i+1 εi,i+1 GCIfinei,i+1

% Ra αb

Eu

0c 3.1336

1 1793459 3.1450

1.1517 0.0350 0.0111 0.45202 1174029 3.1800 0.1815 1.0111

1.1803 0.1950 0.0613 1.8228

3 714029 3.3750

x50

0 1.2703

1 1793459 1.3180

1.1517 0.1180 0.0895 4.5223

2 1174029 1.4360 0.2342 1.08951.1803 0.5490 0.3823 14.4222

3 714029 1.9850

a R=ε12/ε23 .

b α=(

rp12

GCI12

)

/GCI23 .

c The value at zero grid space (h → 0). i=1, 2 and 3 denote the calculations at the fine, medium and coarse mesh respectively.

94

5.3. Results and discussions

N -1

Eul

ernu

mbe

r

Cut

-off

diam

eter

0 5E-07 1E-063

3.1

3.2

3.3

3.4

1.2

1.4

1.6

1.8

2

2.2Euler number

Cut-off diameter

(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h -- > 0)(h --> 0)


number and the cut-off diameter for cyclone S1 at the three grid levels. N−1 is

the reciprocal of the number of cells, h → 0 means the value at zero grid size (cf.

Table 5.4). To obtain a smooth curve; the spline curve fitting has been applied in

Tecplot post-processing software.



z`/Da 2.75 2.5 2.25 2 1.75 1.5 1.25 1.0 0.75

a z`measured from the inlet section top (cf. Fig. 5.1(a)).

5.3 Results and discussions

The flow velocity can be decomposed into three components. The tangen-

tial and the axial velocity components are the major velocity components

in comparison with the radial velocity component. Xiang and Lee [186]

stated that the tangential velocity is the dominant gas velocity in gas cy-

clones, which results in the centrifugal force for particle separation. The

axial component is responsible for the two flow streams (downward and

upward).

5.3.1 The axial variation

Figures 5.3 and 5.4 present the radial profiles of the time-averaged tan-

gential and axial velocity and static pressure at nine axial stations (cf. Ta-

ble 5.5). As expected, the tangential velocity profiles exhibit the so-called

Rankine vortex, which consists of two parts, an outer free vortex and an

inner solid rotation in the center. The tangential velocity distribution in

the inner region is rather similar at different sections for the same cyclone.

95


In the outer region, due to the sharp drop in the velocity magnitude in the

near wall region, the distribution is different but the maximum tangen-

tial velocity is similar at all sections for the same cyclone (S1-S5). The

maximum tangential velocity increases with decreasing the vortex finder

diameter.

The cyclones S1 to S5 show the inverted W profile but cyclones D1 to D3

show the inverted V profile. The radial profiles of the time averaged static

pressure are given in Fig. 5.3. Like for the tangential velocity, the ax-

ial variations of static pressure are very small for the same cyclone. The

maximum value of the static pressure decreases when the vortex finder

diameter is increased for cyclones D1 to D5. However, the maximum value

of the static pressure slightly increases when the vortex finder length is

decreased for cyclones S1 to S4. Cyclone S5 (Dx/D = 0.5, S/D = 0.5,

h/D = 2.0) differs than cyclone S4 in only the barrel height (h/D = 1.5).

From Fig. 5.4, a slight difference in the maximum tangential velocity be-

tween the two cyclones, which predict close values for the collection effi-

ciency. Moreover, the axial velocity profiles for the two cyclones are close

but less central dip in the axial velocity is exhibited in cyclone S5. The dis-

tinct difference between cyclones S4 and S5 is that in the maximum static

pressure, where cyclone S5 indicates a reduction in the maximum static

pressure. This indicates that the effect of changing the barrel height is

more significant than that of the vortex finder length. The sharp changes

in the radial profiles appear in Fig. 5.4 are due to crossing the vortex finder.

5.3.2 The flow pattern

Figure 5.5 shows the contour plots of the time-averaged static pressure,

tangential and axial velocity for cyclones D1-D5. The time-averaged static

pressure decreases radially from the wall to the center. A negative pres-

sure zone appears in the forced vortex region (central region) due to high

swirling velocity. The pressure gradient is largest along the radial direc-

tion, whereas the gradient in axial direction is very limited. The cyclonic

flow is not symmetrical as is clear from the shape of the low-pressure zone

at the cyclone center (twisted cylinder). Two vortical motions are exist one

moving down (outer vortex) and the other moving up (inner vortex). The

highest value of the static pressure decreases with increasing the vortex

finder diameter. Consequently, a smaller pressure drop can be expected

when increasing the vortex finder diameter. The tangential velocity pat-

tern is very similar in all cyclones (Rankine profile). The highest value

decreases with increasing the vortex finder diameter. Consequently, a bet-

ter collection efficiency can be expected when decreasing the vortex finder

96

5.3

.R

esu

ltsan

dd

iscussio

ns

D1 D2 D3 D4 D5

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

0

4

8

12

16

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

0

4

8

12

16

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

0

4

8

12

16

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

14

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-5

0

5

10


Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

0

4

8


Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6

8S1S2S3S4S5S6S7S8S9

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6S1S2S3S4S5S6S7S8S9

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-5

-2.5

0

2.5

5

7.5


Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

100

200

300

400

500

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

0

100

200

300

400

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

0

80

160

240

320

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

-40

0

40

80

120

160

200

240

280

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

Figure 5.3: The radial profile for the time-averaged tangential and axial velocity and static pressure at different sections for

cyclones D1 - D5.

97

Ch

ap

ter

5.

Th

eV

orte

xF

ind

er

Dim

en

sion

s

S1 S2 S3 S4 S5

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-5

-2.5

0

2.5

5

7.5


Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-5

0

5


Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-5

0

5


Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

-4

-2

0

2

4

6

8


Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

-4

-2

0

2

4

6

8


Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

Figure 5.4: The radial profile for the time-averaged tangential and axial velocity and static pressure at different sections for

cyclones S1 - S5.

98


diameter.


tangential and axial velocity for cyclones S1-S5. The highest value of the

static pressure slightly decreases with increasing the vortex finder length.

The tangential velocity pattern is very similar in all cyclones (Rankine

profile). The highest value is very closed for cyclones S1 to S4 but there is

a small difference between cyclones S4 and S5. The axial velocity patterns

for the five cyclones (S1 to S5) have the shape of an inverted W profile.

Cyclone S5 exhibit a less central dip in the axial velocity.

5.3.3 The radial variation

The tangential and axial velocity profiles at section S6 (as a representative

for the other sections, because the axial variations in the flow variables are

small) for the tested cyclones are compared in Fig. 5.7. From the compar-

ison between the radial profiles of the five D cyclones, the minimum pres-

sure at the cyclone center is almost the same for cyclones (D3 - D5). The

slope of the static pressure radial profile becomes flatter with increasing

the vortex finder diameter. Contrarily, the variation of the static pressure

for cyclones S1 to S5 is very small.

Decreasing the cyclone vortex finder diameter, increases the maximum

tangential velocity. The maximum tangential velocity approaches asymp-

totically 1.589 times the inlet velocity when decreasing the vortex finder

diameter. The effect of increasing the vortex finder length on the maximum

tangential velocity is limited and small reduction in the maximum tan-

gential velocity by increasing the barrel height for the same vortex finder

dimensions (cyclone S5).

The variation of axial velocity with changing the vortex finder length is

limited close to the wall, especially in the outer part. In the central region,

the change in axial velocity profile is more pronounced with the dip in

axial velocity decreasing with increasing the vortex finder length. Cyclone

S5 exhibit the smallest dip among cyclones S1 to S5.

The variation of axial velocity in the five D cyclones is limited close to the

wall with a big change in the central part. The axial velocity profile grad-

ually changes from the inverted W to be inverted V with decreasing vortex

finder diameter. Again, limited effect on the axial velocity profile (inverted

W) for the vortex finder length is recorded. Moreover, increasing the vortex

finder length has only small influence on the central part. Furthermore,

The influence of the barrel height is more pronounced than that of the vor-

tex finder length. Generally, the effect of changing the vortex finder length

99

Ch

ap

ter

5.

Th

eV

orte

xF

ind

er

Dim

en

sion

s

D1 D2 D3 D4 D5

Figure 5.5: The contour plots for the time averaged flow variables at sections Y=0 and throughout the inlet section for

cyclones D1 - D5. From top to bottom: the static pressure (N/m2), the tangential velocity (m/s) and the axial velocity (m/s).

From left to right cyclone D1 - D5.

100

5.3

.R

esu

ltsan

dd

iscussio

ns

S1 S2 S3 S4 S5

Figure 5.6: The contour plots for the time averaged flow variables at sections Y=0 and throughout the inlet section for

cyclones S1 - S5. From top to bottom: the static pressure (N/m2), the tangential velocity (m/s) and the axial velocity (m/s).

From left to right cyclone S1 - S5.

101


on the axial velocity profile is very small in comparison with the vortex

finder diameter.

The shape of the axial velocity profile is affected by the cyclone geometry

[75]. Hoekstra et al. [75] referred the dip in the inverted W profile to the

loss of swirl in the vortex finder (the friction force of the vortex finder wall

attenuates the swirling flow), which results in an adverse pressure gradi-

ent at the centerline [78]. Hence, fluid with fewer swirls is drawn back

from the exit pipe into the cyclone. This core flow prevails throughout the

entire separation space of the cyclone in spite of the attenuation of swirl

in the conical part of the cyclone [78]. This explains the reason behind the

inverted W-shaped profile exhibited by cyclones S1 to S5. However, why

decreasing the vortex finder diameter gradually change the axial velocity

profile from inverted W to exhibit the inverted V-shaped profile? The an-

swer is, when the vortex finder diameter decreases, the swirl in the cyclone

increases (as is clear from the tangential velocity profiles for cyclones D1

to D5). Consequently, the flow can overcome the adverse pressure gradient

and exhibit the inverted V profile for cyclone D1.

5.3.4 The cyclone performance

In order to estimate the effect of the vortex finder dimensions on the per-

formance parameters, the Euler number (the dimensionless pressure drop)

have been calculated. A discrete phase modeling (DPM) study has been

performed by injecting 104 particles from the inlet surface with a particle

density of 860 kg/m3 and with a particle size ranging from 0.025 until 5

micron.

Figure 5.8 shows a sharp decrease of the Euler number (dimensionless

pressure drop) with increasing the vortex finder diameter Dx and a small

increase with increasing the vortex finder length S. This behavior can

be explained as follows. The pressure drop in the cyclone is composed of

three main contributions [52]: (1) the pressure drop at the inlet section.

(2) the pressure drop in the cyclone body due to swirling motion and due

to wall friction, this contribution may increase with increasing the cyclone

barrel height (cyclone S5) or with increasing the vortex finder length (cy-

clones S1 - S4) as the wall friction will increase due to friction with a larger

wall surface, or decreases as the vortex strength will decrease because the

maximum tangential velocity decreases. (3) the main contribution to the

cyclone pressure drop is the energy loss in the vortex finder, which mainly

depends on the maximum tangential velocity in the cyclone. As is clear

from Fig. 5.7 the maximum tangential velocity decreases with increasing

the vortex finder diameter. As the inlet section is the same in the all tested

102


Radial position (m)

Sta

ticpr

essu

re(N

/m2)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

0

100

200

300

400

500

Dx/D=0.30Dx/D=0.35Dx/D=0.40Dx/D=0.45Dx/D=0.50

Radial position (m)

Sta

ticpr

essu

re(N

/m2)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

S/D=0.5

S/D=0.625

S/D=0.875

S/D=1.0 (h/D=1.5)

S/D=1.0 (h/D=2.0)

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-2

0

2

4

6

8

10

12

14

Dx/D=0.30Dx/D=0.35Dx/D=0.40Dx/D=0.45Dx/D=0.50

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-2

0

2

4

6

8

10

12

S/D=0.5

S/D=0.625

S/D=0.875

S/D=1.0 (h/D=1.5)

S/D=1.0 (h/D=2.0)

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-2

0

2

4

6

8Dx/D=0.30Dx/D=0.35Dx/D=0.40Dx/D=0.45Dx/D=0.50

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-5

-2.5

0

2.5

5

S/D=0.5

S/D=0.625

S/D=0.875

S/D=1.0 (h/D=1.5)

S/D=1.0 (h/D=2.0)

Figure 5.7: Comparison between the radial profiles for the time averaged static

pressure, tangential and axial velocity at section S6.

103


cyclones, the pressure drop in the inlet section does not vary with increas-

ing the vortex finder dimensions. The sharp decrease of the Euler number

between the D cyclones with increasing the Dx is mainly due to the de-

crease in the pressure drop as a result of the decrease in the maximum

tangential velocity. The Euler number in cyclones (S1 - S4) increases as

the vortex finder length increases. Since, the maximum tangential veloc-

ity in the four cyclones is almost the same. Consequently, this trend is due

to the small increase in the energy loss at the wall surface with increasing

S. Cyclone S exhibits a sudden (small) drop in the Euler number. There

are two competing contributions: increase of the pressure drop due to fric-

tion (the barrel height in cyclone S5 is larger than that for cyclone S4)

and decrease of the pressure drop due to the small reduction in the vortex

strength (maximum tangential velocity, cf. Fig. 5.7).

Dx

S

Eul

ernu

mbe

r

Stk

50x

10

3

0.3 0.35 0.4 0.45 0.5

0.6 0.8 1

3

4

5

6

7

8

9

10

11

12

0

0.5

1

1.5

2

2.5

Euler number (Dx)

Stk50 x 103 (Dx)

Euler number (S)

Stk50 x 103 (S)

Figure 5.8: The variation of the Euler number and the Stokes number with the

vortex finder dimensions.

The trend of increasing the Stokes number with increasing the vortex

finder diameter is quite reasonable, as the centrifugal force affecting par-

ticles attenuates when the swirl intensity (maximum tangential velocity)

decreases (Fig. 5.8). The Stokes number slightly increases as the vortex

finder length is increased (cyclone S1 - S4). The insignificant change of

Stokes number is quite reasonable because of very limited changes in the

flow pattern (cf. Fig. 5.7). The small increase in the Stokes number can be

explained with the aid of the contours plots given in Fig. 5.6. The zone of

peak axial velocity increases with increasing S. Consequently, the possi-

bility of carrying bigger particles to escape with the upward flow slightly

increased. Cyclone S5 depicts a sudden reduction in the Stokes number

104


values in comparison with cyclone S4. This drop is due to separation space

increase in cyclone S5, and the possibility of particles to be captured in-

creases. Although, the vortex strength decreased with a small amount due

to the reduction in the maximum tangential velocity.

Figure 5.9 presents the variation of the cyclone performance parameters

with the ratio Dx/S. It is clear that the effect of changing Dx/S depends

on the variables (Dx or S). Consequently, there are two curves per perfor-

mance parameter (Euler number and Stokes number). The common point

in Fig. 5.9 is cyclone D5 (=S1).

Dx /S

Eul

ernu

mbe

r

Stk

50x

103

0.6 0.8 13

4

5

6

7

8

9

10

11

12

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6Euler number (Dx)

Stk50 x 103 (Dx)

Euler number (S)

Stk50 x 103 (S)


ratio of DxS

(cyclone S5 is excluded).

The variation of the performance parameters with the ratio of (h−S)/D is

presented in Fig. 5.10 for cyclones S1 - S5. Increasing (h− S)/D decreases

both the Euler number and Stokes number. However, the values of the

performance parameters depend on the value of the barrel height as is

clear from the values for cyclone S5.

In order to obtain the Euler number-Stokes number relationship, Fig. 5.11

has been drawn. It indicates a general relationship (trend) between the

two dimensionless numbers irrespective to the geometrical parameters

values. Two second-order polynomials have been proposed by Elsayed and

Lacor [54, 56], Eqs. 5.1 and 5.2.

Equation 5.2 presents a good matching for the performance parameters

105


(h-S)/ D

Eul

ernu

mbe

r

Stk

50x

103

0.5 0.625 0.75 0.875 13

3.2

3.4

3.6

3.8

4

2

2.1

2.2

2.3

2.4Euler number (S1-S4)

Stk50 x 103 (S1-S4)

Euler number (S5)

Stk50 x 103 (S5)


ratio of h−SD

.

for cyclones D1 -D5. For smaller values of Euler number, there is un-

derestimation of the Stokes number if the Eq. 5.2 is applied. The differ-

ence between the two correlations can be referred to two reasons. Firstly,

Eq. 5.1 is based on the CFD simulations data for both Euler number and

Stokes number, whereas Elsayed and Lacor correlation [56] (Eq. 5.2) was

obtained from experimental data for Euler number values and Iozia and

Leith model for the Stokes number values. Secondly, Eq. 5.1 is limited

to only four geometrical parameters. The other three factors are fixed,

h = 1.5, S = 0.5 and Bc = 0.375. Consequently, Eq. 5.1 is not suitable

to fit the Euler number - Stokes number relationship for cyclones S1 - S5

because the values of S is away from the range of applicability.

Stk50 = 100.3533(log10(Eu))2−1.1645log10(Eu)−2.3198 (5.1)

Stk50 = 100.3016(log10(Eu))2−0.9479log10(Eu)−2.5154 (5.2)

5.4 Conclusions

Nine cyclones of different vortex finder dimensions (diameter and length)

have been simulated using the large eddy simulation (LES) methodology,

106

5.4. Conclusions

Euler number

Sto

kes

num

berx

10

3

3 4 5 6 7 8 9

1

1.5

2

2.5

3

Simulation

Correlation 1

Correlation 2

Figure 5.11: The variation of the Stokes number with the Euler number for cy-

clones D1–D5.

to study the effect of the vortex finder dimensions on the performance and

flow pattern. The following conclusions have been obtained.

• The maximum tangential velocity in the cyclone decreases with in-

creasing the vortex finder diameter. A negligible change is noticed

with increasing the vortex finder length.

• Very limited axial variations in the flow variables are reported with

changing the vortex finder dimensions for the same cyclone.

• Increasing the vortex finder length, makes a small change in both

the static pressure, axial and tangential velocity profiles. However,

decreasing the vortex finder diameter gradually changes the axial

velocity profile from the inverted W to the inverted V profile.

• Decreasing the cyclone vortex finder diameter, increases the maxi-

mum tangential velocity. The maximum tangential velocity approaches

asymptotically 1.589 times the inlet velocity when decreasing the vor-

tex finder diameter.

• The Euler number (dimensionless pressure drop) decreases with in-

creasing the vortex finder diameter Dx. Increasing the vortex finder

length S slightly increases the Euler number.

• The Stokes number increases with increasing the vortex finder di-

ameter, because the centrifugal force affecting particles attenuates

when the swirl intensity (maximum tangential velocity) decreases.

The Stokes number slightly increases as the vortex finder length is

107


increased (cyclone S1 - S4). The insignificant change of Stokes num-

ber is quite reasonable because of very limited changes in the flow

pattern.

• The effect of changing Dx/S on the performance parameters depends

on the variables (Dx or S).

• Increasing (h − S)/D decreases both the Euler number and Stokes

number. However, the values of the performance parameters depend

on the value of the barrel height.

108

Chapter 6

The Inlet Dimensions

6.1 Introduction

The effects of cyclone inlet section dimensions on the cyclone performance

(pressure drop and cut-off diameter) have been reported in many articles.

Casal and Martinez-Benet [21] proposed the following empirical formula

for the dimensionless pressure drop (Euler number),

Eu = 11.3

(aD

bD

Dx

D

)2

+ 2.33 (6.1)

implying proportionality with the square of the inlet area. Ramachandran

et al. [139] on the other hand proposed,

Eu = 20

a b

D2x

SD

HD

hD

Bc

D

1/3

(6.2)

i.e. a linear relation with the inlet area. Iozia and Leith [84, 85] presented

a correlation to estimate the cut-off diameter d50 and found proportional-

ity to (a b)0.61. The importance of inlet dimensions becomes clearer after

the study of natural length (or vortex length) by several researchers, e.g.,

Alexander [1]. The cyclone has two spiral motions, outer and inner. In the

reverse flow cyclone, the outer vortex weakens and changes its direction

at a certain axial distance Ln from the vortex finder [29]. This distance

is usually called the turning length, natural length or vortex length of the

cyclone. The inlet area is one of the relevant parameters influencing the

natural length. Alexander [1] found that Ln decreased proportionally to

the inlet area (Ai = a b) but the opposite trend has been also reported [29].

109

Chapter 6. The Inlet Dimensions


rameters on the flow pattern and performance [e.g., 15, 62, 102, 140, 184]

while the effect of cyclone inlet dimensions remained largely unexplored.

The articles investigating the effect of cyclone geometry report only briefly

on the effect of inlet section dimensions on the cyclone performance with-

out sufficient details about their effects on the flow pattern and velocity

profiles. A new trend is the use of multi-inlet cyclone [e.g., 103, 187, 195].

The effects of cyclone inlet on the flow field and performance of cyclone

separators have been numerically investigated by Zhao et al. [198]. They

compared the performance of two types of cyclones with the conventional

single inlet and spiral double inlets using the Reynolds stress turbulence

model. The results show that the new type cyclone separator with spiral

double inlet can improve the symmetry of gas flow pattern and enhance

the particle separation efficiency. While their finding is for double inlets

cyclone, it supports the importance of the effect of the inlet section dimen-

sions on the performance of cyclone separator. The significant effects of the

cyclone inlet dimensions on the cyclone performance have been acknowl-

edged in many articles [e.g., 5]. For two inlets cyclone separators, Zhao

[193] reported the possibility of increasing the cyclone efficiency without

significantly increasing the pressure drop by improving the inlet geometry

of the cyclone. The effect of inlet section angle has been tested by many re-

searchers. Qian and Zhang [137] computationally investigated the effect of

the inlet section angle. The pressure drop of the cyclone decreases to a 30%

lower value than that for conventional cyclone, if the inlet section angle θbecomes 45 . However, Qian and Wu [136] reported only 15% reduction in

the pressure drop for θ = 45 .

In summary, all articles mentioned above did not study the effect of the

inlet height or width dimensions on the performance and flow pattern but

they studied the effect of the inlet configurations (inclined instead of tan-

gential), or the effect of the number of inlets (single or double) or the shape

of the inlet section (rectangular duct or nozzle). The present study is in-

tended to computationally investigate the effect of increasing the cyclone

inlet width and height on the pressure drop and cut-off diameter and ob-

taining more details about the flow field pattern and velocity profiles. The

study will be done using RANS and the Reynolds stress turbulence model

(RSM) will be used as a closure turbulence model.

110



6.2.1 Configuration of the five cyclones

The numerical simulations were performed on five cyclones with different

inlet dimensions. Figure 8.1 and Table 6.1 give the cyclones dimensions.

Table 6.2 gives more details for the used cyclones including the number of

cells, geometric swirl number, cyclone volume, flow residence time and the

inlet velocity for each cyclone.

In swirling flow, the swirl number usually characterizes the degree of swirl.

In cyclone separators, the swirling flow is characterized by the geometric

swirl number. The geometric swirl number Sg is a measure for the ratio of

tangential to axial momentum [75, 154], defined by [74],

Sg =π Dx D

4 Ain(6.3)

where Dx is the vortex finder diameter, D is the cyclone body diameter, and

Ain is the inlet cross-sectional area. For industrial cyclones, the geomet-

rical swirl number usually varies between 1 and 5 [74]. Table 6.2 shows

that, the tested cyclones cover this range. Implying that the obtained re-

sults can be applied to the industrial cyclones.


A velocity inlet boundary condition is used at the cyclone inlet, i.e., a veloc-

ity normal to the inlet is specified. An outflow boundary condition is used

at the outlet. The no-slip boundary condition is used at the other bound-

aries. The air volume flow rate Qin=50 l/min for all cyclones, air density

1.0 kg/m3 and dynamic viscosity 2.11E-5 Pa s. The turbulent intensity

equals 5% and the characteristic length equals 0.07 times the inlet width

[74].

6.2.3 Selection of the time step

The average residence time in the cyclone is determined from the cyclone

dimensions and gas flow rate [43]. The residence time tres = V/Qin where

V is the cyclone volume and Qin is the gas flow rate. This value is used

to select the time step. The time step for the unsteady simulation should

be a tiny fraction of the average residence time [23]. The tres ≈ 0.08s(for all tested cyclones) as shown in Table 6.2. So a time step of 5E-4 is

111


Table 6.1: The geometrical dimensions of the tested cyclones§

Dimension Cyclone∗ dimension/D b/a

Gas outlet diameter, Dx 0.5Vortex finder insertion length, S 0.5Cone tip-diameter, Bc 0.375Cylinder height, h 1.5Cyclone height, Ht 4.0

Inlet height, a b/D=0.2625A1 0.25 1.05A2 0.375 0.7A3 0.50 0.525

Inlet Width, b a/D=0.375B1 0.15 0.4B2 0.2625 0.7B3 0.375 1.0

§ Body diameter, D = 31 mm. The outlet section is above the cylindrical barrel surface by Le = 0.5D.The inlet section located at a distance Li = D from the cyclone center.

∗ Cyclones A2 and B2 are identical.

an acceptable value for the current simulation for accurate results and

achieve scaled residuals less than 1e-5 for all variables. The selected time

step results in an average inlet Courant number of 29.25, 28.88, 21.67,

30.40, 21.45 for cyclones A1, A2, A3, B1 and B3 respectively. However,

as the solver is a segregated implicit solver, there is no limitations on the

Courant number for stability.

6.2.4 CFD grid

Figure 6.1(b) shows the surface grid of cyclone A2 used in this study. The

hexahedral computational grids were generated using GAMBIT grid gen-

erator.

A grid independency study has been performed for the five tested cyclones.

Three grid levels for each cyclone have been tested, to be sure that the

obtained results are grid independent. For example, for cyclone A2 (B2)

three levels of meshes with respectively 490164, 714029 and 1174029 cells

have been used. The computational results are presented in Table 6.3. As

seen, the maximum difference between the results is less than 5%, so the

grid with 490164 cells can be considered as adequate [146]. It has been ob-

Table 6.2: The details of the five tested cyclones

Cyclone A1 A2=B2 A3 B1 B3

Number of cells∗ 705088 714029 820362 706370 816714

Sg 5.984 3.989 2.992 6.981 2.793

Cyclone volume x105 [m3] 6.878 6.95 7.017 6.875 7.012

tres [s] 0.0825 0.0834 0.0842 0.0825 0.0841

Inlet velocity [m/s] 13.214 8.809 6.607 15.416 6.166

∗ The total number of hexahedral cells after the grid independence study

112

6.3. Results

!

"

#

"

Sa

h

D

b

Ht

Dx

L i

Le

Bc

(a) The cyclone geometry (b) The surface mesh for cyclone A2 (B2)

Figure 6.1: Schematic diagram and surface mesh for the cyclone separator

served that even 490164 grid provides a sufficient grid independency. How-

ever, for excluding any uncertainty, computations have been performed us-

ing the 714029 cells grid, where the total number of grid points was not

that critical with respect to the computation overhead [10].

Table 6.3: The details of the grid independence study for cyclone A2

Number of cells Static pressure drop [N/m2] Cut-off diameter [µm]

490164 95.99 0.99714029 100.922 1.01174029 98.68 1.02% difference∗ 2.73 2.94

∗ The percentage difference between the coarsest and finest grid

6.3 Results

6.3.1 The axial variation of the flow properties

The tangential velocity is the dominant component of the gas flow in cy-

clones, which results in the centrifugal force for particle separation [186].

113


Moreover, the development of axial velocity profile in axial direction will

be analyzed for the five cyclones. Nine sections are used to plot the velocity

profiles as shown in Table 6.4.

Figures 6.2, 6.3 and 6.4 present the radial profiles of the time-averaged

static pressure, tangential and axial velocity at 9 axial stations. As ex-

pected, the tangential velocity profiles exhibit the so-called Rankine vor-

tex, which consists of two parts, an outer free vortex and an inner solid

rotation in the center (Fig. 6.3). The tangential velocity distribution in the

inner region is rather similar at different sections for the same cyclone. In

the outer region, due to the sharp drop in velocity magnitude in the near

wall region, the distribution is different and the change in the value of

maximum tangential velocity is rather limited. Generally, the tangential

velocity distribution varies only slightly with axial positions for the same

cyclone, which is also reported in other articles [e.g., 66, 127, 163, 186].

This means that, if the tangential velocity increases at one section of the

cyclone, it will increase at all other sections. The same conclusion can be

drawn from the radial profile of static pressure with higher values of pres-

sure drop expected for cyclone A1 in comparison with cyclones A2 and A3.

Cyclone B1 also depicts higher values of pressure in comparison with cy-

clones B2 and B3 (Fig. 6.2). The axial velocity profile has the shape of an

inverted W for all cyclones except B3 with the shape of an inverted V, as a

result of the change in the flow field pattern due to very wide inlet section.

Table 6.4: The position of different plotting sections†


z`/D 2.75 2.5 2.25 2 1.75 1.5 1.25 1.0 0.75

† z`measured from the inlet section top

To evaluate the effect of increasing the cyclone inlet width on the tangen-

tial and axial velocity profiles, the tangential and axial velocity profiles at

section S9 (close to the inlet section) for the three cyclones (both for A cy-

clones and B cyclones) are compared in Fig. 6.5. As is clear from Fig. 6.5

the variation of axial velocity close to the wall is limited when changing

the inlet width or height. The axial velocity profiles for the three cyclones

are very similar except at the central region. The most important is the ef-

fect of cyclone inlet height or width on the tangential velocity (proportional

to the centrifugal force, which is the main force in the separation process).

Increasing the cyclone inlet width or height decreases the maximum tan-

gential velocity. Cyclone A1 and B1 have the maximum tangential velocity

in comparison with other cyclones. This means that decreasing the cyclone

inlet dimension will enhance the collection efficiency.

114

6.3. Results



tangential and axial velocity for cyclones A1, A2 and A3. It is observed

that, the highest value of the static pressure decreases with increasing the

inlet height. The tangential velocity pattern is very similar in all cyclones

(Rankine profile). The highest value decreases with increasing the inlet

height, so that a better collection efficiency can be expected when decreas-

ing the inlet height. The axial velocity patterns for the three cyclones have

the shape of an inverted W profile.


tangential and axial velocity for cyclones B1, B2 and B3. It can be seen

that, the non-symmetry of the flow increases with increasing the inlet

width. The highest value of the static pressure decreases with increas-

ing the inlet width. The tangential velocity pattern is very similar for all

cyclones (Rankine profile). The highest value decreases with increasing

the inlet width, so that a better collection efficiency can be expected when

decreasing the inlet width. The axial velocity patterns for cyclones B1 and

B2 have the shape of an inverted W profile while that of cyclone B3 has an

inverted V profile. In cyclone B3, the inlet width (b/D=0.375) is wider than

the gap between the cyclone barrel and the vortex finder wall. As a result,

some part of the incoming flow will impact the vortex finder and hence will

not experience any swirling motion around the vortex finder. This results

in different axial velocity profile in cyclone B3 in comparison with the other

two cyclones. Furthermore, this will cause excessive stresses on the vortex

finder, vibrations and noise.

From the inspection of Fig. 6.7 (also Fig. 8.2 page 152, Fig. 8.15 page 181 &

Fig. 8.24 page 199), two observations can be made: the flow is asymmetric

near the cone bottom and there is a large variation in the time-averaged

axial velocity in this region. These observations are also found in the liter-

ature; for example, Gronald and Derksen [69, Fig. 5], Kaya and Karagoz

[92, Fig. 8] and Wang et al. [178, 179, Fig. 6]. However, no explana-

tion for this flow pattern is given. The author thinks this flow pattern can

be explained as follows: (1) The cyclone has only one inlet; consequently,

the flow should be asymmetric. However, this asymmetry should diminish

far away from the inlet section. (2) Another effect comes from the cyclone

bottom, which is closed in these simulations (no flow from the cyclone bot-

tom). As a consequence of this boundary condition, the descending flow

reverses its direction at the cone bottom resulting in a large asymmetry

115


A1 B1

Radial position [-]

Sta

ticpr

essu

re[N

/m2 ]

-1 -0.5 0 0.5 1-50

0

50

100

150

200

250

300

S1S2S3S4S5S6S7S8S9

Radial position [-]

Sta

ticpr

essu

re[N

/m2 ]

-1 -0.5 0 0.5 1-50

0

50

100

150

200

250

300

350

S1S2S3S4S5S6S7S8S9

A2 B2

Radial position [-]

Sta

ticpr

essu

re[N

/m2 ]

-1 -0.5 0 0.5 1-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

Radial position [-]

Sta

ticpr

essu

re[N

/m2 ]

-1 -0.5 0 0.5 1-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

A3 B3

Radial position [-]

Sta

ticpr

essu

re[N

/m2 ]

-1 -0.5 0 0.5 1-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

Radial position [-]

Sta

ticpr

essu

re[N

/m2 ]

-1 -0.5 0 0.5 1-25

0

25

50

75

100

125

S1S2S3S4S5S6S7S8S9


sections. Note: A2=B2.116

6.3. Results

A1 B1

A2 B2

A3 B3

Figure 6.3: The radial profile for the time-averaged tangential velocity at different

sections. Note: A2=B2.117


A1 B1

A2 B2

A3 B3

Figure 6.4: The radial profile for the time-averaged axial velocity at different sec-

tions. Note: A2=B2.118

6.3. Results

Radial position [-]

Sta

ticpr

essu

re[N

/m2 ]

-1 -0.5 0 0.5 1-50

0

50

100

150

200

250

300a/D=0.25a/D=0.375a/D=0.5

Radial position [-]

Sta

ticpr

essu

re[N

/m2 ]

-1 -0.5 0 0.5 1-50

0

50

100

150

200

250

300

350b/D=0.15b/D=0.2625b/D=0.375



119


at this region. The author believes that the extent of asymmetry is not

just a function of the cyclone geometry and boundary condition at the cone

bottom, but also the operating condition like the gas flow rate. This asym-

metry causes the obtained axial velocity variations (spot-like distribution

close to the cone bottom). (3) Another reason can be the boundary condi-

tion at the gas outlet (vortex finder exit). A fully developed flow (outflow

boundary condition) is assumed. Actually, there is no guarantee that the

flow is fully developed. May be the flow is still developing throughout a

long distance after the vortex finder exit. However, Wang et al. [178] re-

ported that the effect of the gas outlet length on the simulated velocity

becomes insignificant after a distance of 1/2 the cyclone diameter.

In spite of the above-mentioned interpretations, the following parameters

need to be investigated to accurately explain the reason of this flow behav-

ior: (i) The effect of including the dustbin in these simulations. (ii) The ef-

fect of the gas outlet tube length. (iii) The effect of the operating condition;

e.g., the gas Reynolds number (it includes the effect of inlet velocity, gas

density, and viscosity), the operating pressure, and temperature. More-

over, it may be worthwhile to perform these investigations experimentally

(using the particle image velocimetry (PIV) technique or any other flow

visualizations technique) to check also if these phenomena (observations)

come only from the numerical solution (schemes, settings, boundary con-

ditions, etc.).

6.3.3 The cyclone performance

In order to estimate the effect of cyclone inlet dimensions on the perfor-

mance parameters, the pressure drops have been calculated and presented

in Fig. 6.8. A discrete phase modeling (DPM) study has been performed by

injecting 104 particles from the inlet surface with a particle density of 860

kg/m3 and with a particle size ranging from 0.025 until 5 micron.

6.3.3.1 The effect of the inlet height

Figure 6.8 shows a rapid decrease in the pressure drop when increasing

the inlet height for 0.25 ≤ a/D ≤ 0.4 and a smaller decrease for a/D ≥ 0.4.

This behavior can be explained as follows. The pressure drop in the cyclone

is composed of three main contributions: (1) the pressure drop at the inlet

section (decreased by increasing the inlet dimensions). (2) the pressure

drop in the cyclone body due to swirling motion and due to wall friction,

this contribution decreases with increasing the cyclone inlet height (as the

vortex strength will decrease). (3) the main contribution to the cyclone

120

6.3. Results

A1 A2 A3

Th

est

ati

cp

ress

ure

N/m

2T

he

tan

gen

tia

lvelo

city

m/s

Th

ea

xia

lvelo

city

m/s


and throughout the inlet section. From top to bottom : the static pressure N/m2,

the tangential velocity m/s and the axial velocity m/s. From left to right cyclone A1

through cyclone A3.

121


B1 B2 B3

Th

est

ati

cp

ress

ure

N/m

2T

he

tan

gen

tia

lvelo

city

m/s

Th

ea

xia

lvelo

city

m/s


and throughout the inlet section. From top to bottom : the static pressure N/m2,

the tangential velocity m/s and the axial velocity m/s. From left to right cyclone B1

through cyclone B3.

122

6.3. Results

pressure drop is the energy loss in the exit tube, which mainly depends on

the maximum tangential velocity in the cyclone. As is clear from Fig. 6.5

the maximum tangential velocity decreases with increasing cyclone inlet

height. Consequently, in general the total cyclone pressure drop will de-

crease with increasing the cyclone inlet height. Figure 6.8 also depicts

the pressure drop for the three cyclones using four different mathemati-

cal models; Muschelknautz method of modeling (MM) [29, 175], Casal and

Martinez-Benet [21], Shepherd and Lapple [157], Stairmand [166] indicat-

ing the same overall trend obtained using RSM simulations for the pres-

sure drop with better matching between the MM model and the current

RSM simulation.

The effect of the cyclone inlet height on the cut-off diameter (particle di-

ameter of 50% collection efficiency) is shown in Fig. 6.8. The general trend

is an increase of the cut-off diameter with increasing cyclone inlet height,

due to weakness of the vortex. Figure 6.8 also depicts the cut-off diameter

for the three cyclones using two mathematical models; Iozia and Leith [85]

and Rietema [142] indicating the same overall trend obtained using RSM

simulations for the cut-off diameter with exact matching between the cur-

rent RSM results and Rietema model.

6.3.3.2 The effect of the inlet width

Figure 6.8 shows a rapid decrease in the pressure drop when increasing the

inlet width for 0.15 ≤ b/D ≤ 0.27 and a smaller decrease for b/D ≥ 0.27.

Generally, both the cyclone pressure drop and the cut-off diameter de-

crease with increasing the cyclone inlet width. The used mathematical

models again indicate the same overall trend obtained using RSM simula-

tions for cut-off diameter and pressure drop.

Figure 6.8 indicates the need of applying a multi-objective optimization

procedure to get the optimum value for the inlet dimensions. Both the col-

lection efficiency (cut-off diameter) and the pressure drop in cyclone sep-

arator are important objective functions to be optimized simultaneously

[146]. The effects of changing the cyclone inlet dimensions on pressure

drop and collection efficiency are opposite. Increasing the inlet width will

save more driving power but leads to reduced collection efficiency (larger

cut-off diameter). From the graph, the optimum value will be close to b/D= 0.25 with large reduction in pressure drop (energy losses) and small in-

crease in cut-off diameter. Elsayed and Lacor [50] estimated the optimum

value for inlet width (b/D) equals 0.236.

123


6.3.3.3 The inlet height versus the inlet width

Figure 6.9 shows that, the impact of changing either the inlet height or

width on the pressure drop is almost the same. However, the effect of

changing the inlet width on the cut-off diameter is more significant in

comparison with that of the inlet height. When plotting the pressure drop

against the ratio of inlet width to inlet height, it becomes clear the opti-

mum range of b/a is from 0.5 until 0.7.

6.4 Conclusions

Five cyclones of different inlet width and height have been simulated, us-

ing the Reynolds stress model (RSM), to study the effect of cyclone inlet

dimensions on the cyclone separator performance and flow pattern. The

following conclusions have been obtained.


creasing both the cyclone inlet width and height.

• No acceleration occurs in the cyclone space (the maximum tangential

velocity nearly constant throughout the cyclone). The axial variation

of both the static pressure and axial velocity is very limited.

• Increasing the cyclone inlet width or height decreases the pressure

drop at the cost of increasing the cut-off diameter. So an optimiza-

tion procedure is needed to estimate the optimum value of inlet di-

mensions.

• Wider inlet cyclones (b/D > gap between the cyclone barrel and the

vortex finder are not preferred.

• The effect of changing the inlet width on the cut-off diameter is more

significant in comparison with that of the inlet height.

• The optimum ratio of the inlet width to the inlet height b/a is from

0.5 to 0.7.

124

6.4. Conclusions

a/D

Pre

ssu

red

rop

[N/m

2]

Cu

t-o

ffd

iam

ete

r[m

icro

n]

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

100

200

300

400

500

600

700

800

900

1000

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2RSMMMCasalShepherdStairmandRSMIoziaRietma

Pressure drop

Cut-off diameter

b/D

Pre

ssu

red

rop

[N/m

2]

Cu

t-o

ffd

iam

ete

r[m

icro

n]

0.1 0.15 0.2 0.25 0.3 0.35 0.40

100

200

300

400

500

600

700

800

900

1000

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2RSMMMCasalShepherdStairmandRSMIoziaRietma

Pressure drop

Cut-off diameter

Figure 6.8: The effect of inlet dimensions on the pressure drop and cut-off diameter

using CFD simulations and different mathematical models.

125


Inlet height a/D

Inlet width b/DP

ressu

red

rop

[N/m

2]

Cu

t-o

ffd

iam

ete

r[m

icro

n]

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

0.15 0.2 0.25 0.3 0.35 0.4

0

100

200

300

400

500

600

700

800

900

1000

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Effect of inlet heightEffect of inlet width

Pressure drop

Cut-off diameter

b/a

Pre

ssu

red

rop

[N/m

2]

Cu

t-o

ffd

iam

ete

r[m

icro

n]

0.2 0.4 0.6 0.8 1 1.20

100

200

300

400

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6Pressure drop [N/m 2]

Cut-off diameter [micron]

Figure 6.9: Comparison between the effect of inlet height and width on the pres-

sure drop and cut-off diameter using CFD simulations.

126

Chapter 7

The Cyclone Height

7.1 Introduction


rameters on the flow pattern and performance [15, 52, 62, 102, 140, 184]

whereas only limited number of studies have been devoted to the effect

of the cyclone height. Zhu and Lee [200] have conducted detailed exper-

iments on cyclones of different height and found that the cyclone height

can influence considerably the separation efficiency of the cyclones. How-

ever, they did not provide any information about the flow pattern nor an

explanation for the efficiency results. Hoffmann et al. [76] investigated the

effect of the cyclone height on the separation efficiency and the pressure

drop experimentally and theoretically. They found improvement in cyclone

performance with increasing the total height Ht up to 5.5 times cyclone

diameters beyond this length the separation efficiency was dramatically

reduced. However, they did not present any contour plot or velocity profile

to support the explanation for the effect of the cyclone height on the perfor-

mance. Recently, Xiang and Lee [186] have repeated the study of Zhu and

Lee [200] for the effect of cyclone height via steady three-dimensional sim-

ulations using the Reynolds stress turbulence model (RSM). They found

that the tangential velocity decreases with increasing the cyclone height,

which is responsible for the lower separation efficiency observed in long

cyclones. The reason for this behavior however, was not analyzed in detail.

Moreover, no particle tracking study was presented.

The present study is intended to computationally investigate the effect of

increasing cyclone (barrel and cone) height on the pressure drop and cut-

off diameter and to obtain more details about the flow field pattern and

velocity profiles. A RANS approach with the Reynolds stress turbulence

127

Chapter 7. The Cyclone Height

(a) The cyclone geometry (b) The surface mesh for cyclone

C1 (B2)

Figure 7.1: Schematic diagram for the cyclone separator

model (RSM) is used.


7.2.1 Configuration of the tested cyclones

The numerical simulations were performed on four cyclones with different

barrel heights (at constant cone height) and three cyclones with different

cone heights (at constant barrel height). Figure 7.1 and Table 7.1 give the

cyclones dimensions.


Table 7.2 shows the details of the boundary conditions. The air volume

flow rate Qin=50 L/min for all cyclones, air density 1.0 kg/m3 and dynamic

viscosity of 2.11E-5 Pa s. The turbulent intensity equals 5% and charac-

teristic length equals 0.07 times the inlet width [52, 74].

128


Table 7.1: The geometrical dimensions of the tested cyclonesa

Dimension Cyclone b dimension/D hc/h Htc

Inlet height, a 0.375Inlet width, b 0.2625Gas outlet diameter, Dx 0.5Vortex finder insertion length, S 0.5Cone tip-diameter, Bc 0.375

Cone height, hc h/D = 1.5C1 2.5 1.666 4.0C2 3.5 2.333 5.0C3 4.5 3.0 6.0

Barrel height, h hc/D = 2.5

B1 1.0 2.5 3.5B2 1.5 1.666 4.0B3 2.0 1.25 4.5B4 2.5 1.0 5.5

a Body diameter, D = 31 mm. The outlet section is above the cylindrical barrel surface by Le = 0.5D.The inlet section located at a distance Li = D from the cyclone center.

b The cone height is hc = Ht − h, where Ht is the total cyclone height. Cyclone C1 and B2 areidentical.

c Cyclone C2 and cyclone B4 are equal in Ht/D but they are different in both h/D and hc/D.

Grid independency study

A grid independency study has been performed for the tested cyclones.

Three different grids have been tested for each cyclone, to be sure that the

obtained results are mesh independent. For example, meshes with respec-

tively 490164, 714029 and 1174029 cells have been used for cyclone B2.

The computational results on the three grids are presented in Table 7.4.

As it can be seen the maximum difference between the results is less than

5%, so the results on the 490164 cells grid can already be considered as suf-

ficiently accurate. However, for excluding any uncertainty, computations

have been performed using the 714029 cells grid, where the total number

of grid points was not that critical with respect to the computation over-

head [10, 52]. Figure 7.1(b) shows the surface grid of cyclone C1(B2) used

in this study. Table 7.3 gives the total number of cells used for each cyclone

after the grid independency study. The hexahedral grids were generated

using the GAMBIT grid generator.

Table 7.2: The boundary conditions

Boundary Inlet Outlet Cone tip Other surfaces

Condition Velocity inlet Outflow Wall (no-slip) Wall (no-slip)

129


Table 7.3: The details of the tested cyclonesa

Cyclone B1 B2 (C1) B3 B4 C2 C3

Number of cells 688170 714029 712183 786865 770556 820362

a The total number of hexahedral cells after the grid independency study

7.3 Results

The flow velocity can be decomposed into three components. The tangen-

tial and the axial velocity components are the major velocity components

in comparison with the radial velocity component. Xiang and Lee [186]

stated that the tangential velocity is the dominant gas velocity in gas cy-

clones, which results in the centrifugal force for particle separation. The

axial component is responsible for the two flow streams (downward and

upward).

7.3.1 The axial variation

Figures 7.2 - 7.4 present the radial profiles of the time-averaged tangential

and axial velocity and static pressure at nine axial stations (cf. Table 7.5).

As expected, the tangential velocity profiles exhibit the so-called Rankine

vortex, which consists of two parts, an outer free vortex and an inner solid

rotation in the center (Fig. 7.2). The tangential velocity distribution in the

inner region is rather similar at different sections for the same cyclone.

In the outer region, due to the sharp drop in velocity magnitude in the

near wall region, the distribution is different but the maximum tangential

velocity is similar at all sections for the same cyclone. The axial velocity

profiles at nine different stations are shown in Fig. 7.3. All cyclones show

the inverted W profile but the central dip decreases with increasing the

Table 7.4: The details of the grid independence study for cyclone C1 (B2)

Number of cells Eua Stk50x103b

490164 3.475 1.188714029 3.654 1.21174029 3.573 1.224% differencec 2.74 2.94

a Euler number is the dimensionless pressure drop Eu = ∆P/( 12ρV 2

in) where ∆P is the static pressuredrop, ρ is the gas density, Vin is the gas inlet velocity.

b The Stokes number based on the cut-off diameter; Stk50 = ρpx250Vin/(18µD) [37]. It is the ratio

between the particle relaxation time; ρpx250/(18µ) and the gas flow integral time scale; D/Vin where ρp

is the particle density =860 kg/m3, µ is the gas viscosity.

c The percentage difference between the coarsest and finest grid

130

7.3. Results



z`/Da 2.75 2.5 2.25 2 1.75 1.5 1.25 1.0 0.75

a z`measured from the inlet section top (cf. Fig. 7.1(a)).

barrel (or cone) height (Except at the cyclone bottom for cyclones C2 and

C3, where the axial velocity shows an inverted V profile). The radial pro-

files of the time averaged static pressure are given in Fig. 7.4. Like for

the tangential velocity, the axial variations of the static pressure are very

small for the same cyclone. The variations become negligible with increas-

ing barrel (or cone) height. Furthermore, the maximum value of the static

pressure decreases when the barrel height is increased.

7.3.2 The radial variation

The tangential and axial velocity profiles at section S6 (as a representative

for the other sections, because the axial variations in the flow variables are

small) for the six cyclones are compared in Fig. 7.5. The variation of axial

velocity with changing barrel height is limited close to the wall, especially

in the cylindrical part. In the central region, the change in axial velocity

profile is more pronounced with the dip in the axial velocity decreasing

with increasing the barrel height. This is the result of the flatting of the

pressure distribution results in a smaller pressure force. This may explain

also the change of the axial velocity from cyclones B1 to B4. Increasing the

cyclone barrel height decreases the maximum tangential velocity. Cyclone

B1 has the maximum tangential velocity in comparison with the other cy-

clones. The effect of increasing the barrel height on the maximum tan-

gential velocity is limited. The variation of axial velocity in the three C

cyclones is limited close to the wall with changing the cone height. The

axial velocity profiles are very similar except at the central region due to

change in the axial velocity profile. Increasing the cyclone cone height de-

creases the maximum tangential velocity. Cyclone C1 has the maximum

tangential velocity in comparison with the two other cyclones (C2 and C3).

From the comparison between the radial profiles of the four B cyclones,

the minimum pressure at the cyclone center is almost the same for all cy-

clones (B1 - B4). The slope of the static pressure radial profile becomes

flatter with increasing the barrel height. The minimum pressure at the

cyclone center is almost the same for all C cyclones. The static pressure

radial profiles of cyclones C2 and C3 are very close. Increasing the cyclone

131


C1 (B2) B1

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-2

0

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

C2 B3

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-2

0

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

C3 B4

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-2

0

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

S1S2S3S4S5S6S7S8S9

Figure 7.2: The radial profile for the time-averaged tangential velocity at different

sections. Note: C1=B2.

132

7.3. Results

C1 (B2) B1

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6

8S1S2S3S4S5S6S7S8S9

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6

8S1S2S3S4S5S6S7S8S9

C2 B3

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6

8S1S2S3S4S5S6S7S8S9

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6

8S1S2S3S4S5S6S7S8S9

C3 B4

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6

8S1S2S3S4S5S6S7S8S9

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6

8S1S2S3S4S5S6S7S8S9

Figure 7.3: The radial profile for the time-averaged axial velocity at different sec-

tions. Note: C1=B2.

133


C1 (B2) B1

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

250

300

S1S2S3S4S5S6S7S8S9

C2 B3

Radial position (m)

Sta

ticpr

essu

re(N

/m2)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-50

0

50

100

150

200

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

250

300

S1S2S3S4S5S6S7S8S9

C3 B4

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-50

0

50

100

150

S1S2S3S4S5S6S7S8S9

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

250

300

S1S2S3S4S5S6S7S8S9


sections. Note: C1=B2.

134

7.3. Results

height (either barrel or cone) decreases the pressure drop, the maximum

tangential velocity (vortex strength) and the dip in the axial velocity pro-

file. The effect of increasing the cone height on the axial velocity profile is

predominant with respect to the barrel height.

The swirling motion of the gas generates a strong radial pressure gradient,

the pressure being low in the centre of the vortex and high at the periph-

ery. As the strongly swirling gas enters the confines of the vortex finder

on its way out of the cyclone, the swirl is attenuated through friction with

the wall. This means that further up the vortex finder the pressure in the

centre is higher than at the exit of the separation space: a reverse pres-

sure gradient is present [78] as is clear from Fig.7.5. This drives an axial

flow with dip in the centre of the vortex finder (inverted W profile); this

core flow prevails throughout the entire separation space of the cyclone in

spite of the attenuation of swirl in the conical part of the cyclone. With

increasing the cone height the pressure distribution becomes flatter conse-

quently the pressure force causes the dip in the axial velocity at the center

line becomes less and less. That may explain also the change of the axial

velocity from cyclones C1 to C3, Figs. 7.5 and 7.6.



tangential and axial velocity for cyclones C1-C3. The time-averaged static

pressure decreases radially from the wall to the center. A negative pres-



tion, whereas the gradient in axial direction is very limited. The cyclonic

flow is not symmetrical as is clear from the shape of the low-pressure

zone at the cyclone center (twisted cylinder). Two vortical motions are

exist one moving down (outer vortex) and the other moving up (inner vor-

tex). The highest value of the static pressure decreases with increasing

the cone height. The tangential velocity pattern is very similar in all cy-

clones (Rankine profile). The highest value decreases with increasing the

cone height but the differences between cyclones C2 and C3 are small, so

that a better collection efficiency can be expected when decreasing the cone

height.


tangential and axial velocity for cyclones B1-B4. The highest value of the

static pressure decreases with increasing the barrel height. The tangen-

tial velocity pattern is very similar in all cyclones (Rankine profile). The

highest value decreases with increasing the barrel height, but the differ-

135


Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-50

0

50

100

150

200

250hc/D=2.5hc/D=3.5hc/D=4.5

Radial position (m)S

tatic

pres

sure

(N/m

2 )-0.015 -0.01 -0.005 0 0.005 0.01 0.015

-100

-50

0

50

100

150

200

250

300

h/D=1.0

h/D=1.5

h/D=2.0

h/D=2.5

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-2

0

2

4

6

8

10

12

14hc/D=2.5hc/D=3.5hc/D=4.5

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

2

4

6

8

10

12

h/D=1.0

h/D=1.5

h/D=2.0

h/D=2.5

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6

8hc/D=2.5hc/D=3.5hc/D=4.5

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6

8h/D=1.0

h/D=1.5

h/D=2.0

h/D=2.5



136

7.3. Results

ences between the four cyclones are small. The axial velocity patterns for

the four cyclones have the shape of an inverted W profile.

7.3.4 The performance

In order to estimate the effect of cyclone height on the performance pa-

rameters, the Euler number (the dimensionless pressure drop) have been

calculated. A discrete phase modeling (DPM) study has been performed by

injecting 104 particles from the inlet surface with a particle density of 860

kg/m3 and with a particle size ranging from 0.025 until 5 micron.

Figure 7.8 and Table 7.6 show a sharp decrease of the Euler number with

increasing the barrel height until h/D > 1.8 (Ht/D > 4.3) and a gradual

decrease beyond. This behavior can be explained as follows. The pres-

sure drop in the cyclone is composed of three main contributions [52]: (1)

the pressure drop at the inlet section. (2) the pressure drop in the cy-

clone body due to swirling motion and due to wall friction, this contribu-

tion may increase with increasing the cyclone height as the wall friction

will increase due to friction with a larger wall surface, or decreases as

the vortex strength will decrease because the maximum tangential veloc-

ity decreases. (3) the main contribution to the cyclone pressure drop is

the energy loss in the exit tube, which mainly depends on the maximum

tangential velocity in the cyclone. As is clear from Fig. 7.5 the maximum

tangential velocity decreases with increasing cyclone barrel height. As the

inlet section is the same in all cyclones, the pressure drop in the inlet sec-

tion does not vary with increasing barrel height. The sharp decrease of

the Euler number between cyclones B1 and B2 is due to the decrease in

the maximum tangential velocity. There are two competing contributions:

increase of the pressure drop due to friction and decrease of the pressure

drop due to the reduction in the vortex strength. At the beginning, the wall

friction effect is small in comparison with the effect of vortex strength. For

longer cyclones, this effect becomes larger (but still less than that of vortex

strength decay). This explains the small variation of the Euler number

with the barrel height for h/D > 1.8, which is clear from Fig. 7.5 where the

maximum tangential velocity of cyclones B3 and B4 are very close.

The behavior of the Stokes number curve as a function of barrel height

is quite reasonable with increasing barrel height (separation space), the

possibility of particles to be captured increases due to the increased cy-

clone space. However, the vortex strength decreased with a small amount

due to the reduction of the maximum tangential velocity, the main contri-

bution here is the collecting surface. The Stokes number curve becomes

137



and throughout the inlet section. From left to right : the static pressure (N/m2), the

tangential velocity (m/s) and the axial velocity (m/s). From top to bottom cyclone

C1-C3.

138

7.3

.R

esu

lts

Figure 7.7: The contour plots for the time averaged flow variables at sections Y=0 and throughout the inlet section. From

top to bottom: the static pressure (N/m2), the tangential velocity (m/s) and the axial velocity (m/s). From left to right cyclone

B1 - B4.

139


nearly flat between cyclones B3 and B4 due to the small changes in both

the axial and the tangential velocity profile between the two cyclones. Both

Ramachandran [139] and Iozia [84] models agree with the CFD results in

the trend of decreasing both the Euler number and Stokes number with

increasing barrel height, but differ in slope and values.

Ht /D

Eul

ernu

mbe

r

Stk

50x

103

3.5 4 4.5 53

3.5

4

4.5

5

5.5

6

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4Euler number (CFD)Euler number (Ramachandran model)Stk50 x 103 (CFD)Stk50 x 103 (Iozia model)

Figure 7.8: The effect of barrel height on the Euler number and the Stokes number.

Table 7.6: The Euler numbers and Stokes numbers for cyclones B1-B4

Cyclone B1 B2 B3 B4

h/D 1.0 1.5 2.0 2.5

hc/h 2.5 1.666 1.25 1.0

Ht/D 3.5 4 4.5 5.0

Eu 4.39 3.654 3.33 3.09

Eu (Ramachandran model [139]) 5.71 4.77 4.17 3.73

Stk50 x 103 1.32 1.2 1.01 0.95

Stk50 x 103 (Iozia model [84]) 1.94 1.82 1.72 1.64

Figure 7.9 and Table 7.7 show a sharp decrease of both the Euler number

and the Stokes number with increasing the cone height until hc/D=3.3 and

a gradual decrease when 3.3 > hc/h > 4.0. This behavior can be explained

as follows. As the inlet section is the same in all cyclones, the pressure

drop in the inlet section may not vary with increasing the cone height.

140

7.3. Results

The sharp decrease in the Euler number between cyclones C1 and C2 is

due to the huge decrease in the pressure drop in the cyclone body due

to the drop in the maximum tangential velocity (the decay of the vortex

strength). There are two competing contributions, increase of the pressure

drop due to friction and decrease of the pressure drop due to the reduction

in the vortex strength. At the beginning, the decay in the pressure drop

due to the decay of the vortex strength overrides the effect of increasing

the pressure drop due to wall friction for longer cyclones. This explains

the small variation of the Euler number with the cone height for hc > 4.0,

which is clear from Fig. 7.5 where the maximum tangential velocities of

cyclones C2 and C3 are very close.

The trend of decreasing Stokes number with increasing cone height is

quite reasonable, as more separation space exists, and the possibility of

particles to be captured increases. Although, the vortex strength decreased

with a small amount - due to the reduction in the maximum tangential

velocity- we estimate that the main contribution to the collection efficiency

comes from the increased collecting surface with increasing the cone height.

The reason of nearly constant Stokes number after Ht/D =5.5 is the change

in the axial velocity profile. Figure 7.5 shows a higher kinetic energy of the

flow at the cyclone bottom for cyclone C3, that will enhance re-entrainment

of some of the captured particles to escape with the upward flow. Conse-

quently low collection efficiency and higher cut-off diameter (Stokes num-

ber). Because, the differences between the axial and tangential velocity

profile between cyclones C2 and C3 are limited.

Table 7.7: The Euler numbers and Stokes numbers for cyclones C1-C3

Cyclone C1 C2 C3

hc/h 1.666 2.333 3.0

hc/D 2.5 3.5 4.5

Ht/D 4 5 6

Eu 3.654 2.749 2.584

Stk50 x 103 1.2 0.465 0.315

7.3.5 The cone height versus the barrel height

• Increasing the cyclone total height (either by increasing the cone or

the barrel height) will decrease the maximum tangential velocity.

• The effect of cone height on the axial velocity profile is much larger

than that of the barrel height.

• Both the Euler and the Stokes numbers decrease with increasing the

141


hc /D

Ht /D

hc /hE

uler

num

ber

Stk

50x

103

2 2.5 3 3.5 4 4.5 5

3.5 4 4.5 5 5.5 6 6.5

1.5 2 2.5 3 3.5

1

1.5

2

2.5

3

3.5

4

0

0.5

1

1.5

2

2.5

3

3.5

4Euler number

Stk50 x 103

Figure 7.9: The effect of cone height on the pressure drop (Euler number) and

cut-off diameter (Stokes number).

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

-2

0

2

4

6

8hc /D=2.5hc /D=3.5hc /D=4.5



142

7.3. Results

total height either by increasing the barrel or cone height. The effect

of changing the cone height is more important than that of changing

the barrel height (cf. Table 7.6, 7.7 and Fig. 7.11).

• The effect of increasing the ratio of cone to barrel height hc/h on the

cyclone performance depends on the dependent variable. If hc/h in-

creases at constant cone height, the Euler number increases linearly

with decreasing the barrel height. If hc/h increases at constant barrel

height, the Euler number decreases with increasing the cone height.

• The effect of changing hc/h at constant cone height has a negligible

effect on the Stokes number.

• Increasing hc/h at constant barrel height decreases the Stokes num-

ber. This effect becomes negligible for hc/h > 2.75 (Fig. 7.11). This be-

havior can be explained by inspecting the variation of the time aver-

aged static pressure, tangential and axial velocity profiles with cone

and barrel height, Fig. 7.12. As is clear from Fig. 7.12 the differences

between the maximum tangential velocity in cyclones C2 and C3 are

negligible. Furthermore, the effect of the cone height on the flow field

is more significant than that of the barrel height.

143


Ht /D

Eul

ernu

mbe

r

Stk

50x

10

3

3 3.5 4 4.5 5 5.5 6 6.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6Euler number (Barrel height)Euler number (Cone height)Stk50 x 103 (Barrel height)Stk50 x 103 (Cone height)

hc /h

Eul

ernu

mbe

r

Stk

50

x1

03

0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.250

1

2

3

4

5

6

0

1

2

3

4

5

6Euler number (Barrel height)Euler number (Cone height)Stk50 x 103 (Barrel height)Stk50 x 103 (Cone height)


barrel and cone height.

144

7.3. Results

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

250

300

h/D=1.0

h/D=1.5

h/D=2.0

h/D=2.5

hc/D=3.5

hc/D=4.5

Radial position (m)

Sta

ticpr

essu

re(N

/m2 )

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-100

-50

0

50

100

150

200

250

300

h/D=1.0

h/D=1.5

h/D=2.0

h/D=2.5

hc/D=3.5

hc/D=4.5

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-2

0

2

4

6

8

10

12

h/D=1.0

h/D=1.5

h/D=2.0

h/D=2.5

hc/D=3.5

hc/D=4.5

Radial position (m)

Tan

gent

ialv

eloc

ity(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-2

0

2

4

6

8

10

12

h/D=1.0

h/D=1.5

h/D=2.0

h/D=2.5

hc/D=3.5

hc/D=4.5

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

-4

-2

0

2

4

6

8h/D=1.0

h/D=1.5

h/D=2.0

h/D=2.5

hc/D=3.5

hc/D=4.5

Radial position (m)

Axi

alve

loci

ty(m

/s)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

-4

-2

0

2

4

6

8h/D=1.0

h/D=1.5

h/D=2.0

h/D=2.5

hc/D=3.5

hc/D=4.5

Figure 7.12: The radial profile for the time averaged static pressure, tangential and

axial velocity at three different sections for the six cyclones. From top to bottom

: static pressure, tangential and axial velocity. From left to right: S6 - S9. Note:

h/D = 1.5 also represents hc/D = 2.5

145


7.4 Conclusions

Six cyclones of different barrel and cone height have been simulated using

the Reynolds stress model (RSM), to study the effect of cyclone height on

the performance and flow pattern. The following conclusions have been

obtained.


creasing the cyclone (barrel or cone) height.

• No acceleration occurs in the cyclone space (the maximum tangential

velocity nearly constant throughout the same cyclone).

• Increasing the barrel height, makes a small change in the axial ve-

locity.

• Increasing the cyclone barrel height decreases the pressure drop and

the cut-off diameter. The changes in the performance beyond h/D =1.8 are small.

• Increasing the cone height makes a considerable change in the axial

velocity.

• Both the pressure drop and the cut-off diameter decrease with in-

creasing the cyclone cone height. The performance improvement stops

after hc/D = 4.0 (Ht/D = 5.5).

• The effect of changing the barrel height is less significant on the per-

formance and the flow pattern in comparison with the effect of the

cone height.

146

Chapter 8

Optimization

8.1 Introduction

Mathematical optimization refers to the selection of a best element from

some set of available alternatives. In the simplest case, this means solving

problems in which one seeks to minimize or maximize a real function by

systematically choosing the values of real or integer variables from within

an allowed set. Generally, the use of the word optimization implies the

best result under the circumstances [176].

In 1951, Stairmand [166] presented one of the most popular design guide-

lines for the high-efficiency cyclone separators [147]. Stairmand presented

the geometrical ratios for the seven geometrical parameters as: a/D = 0.5,

b/D = 0.2, Dx/D = 0.5, Ht/D = 4.0, h/D = 1.5, S/D = 0.5 and Bc/D =0.375. These values have been obtained based on the Stairmand model

[165] which suffers from many shortages [50].

In the Stairmand model [165], the velocity distribution has been obtained

from a moment-of-momentum balance, estimating the pressure drop as

entrance and exit losses combined with the loss of static pressure in the

swirl. The main drawbacks of the Stairmand model are: (1) neglecting the

entrance loss by assuming no change of the inlet velocity occurs at the inlet

area; (2) assuming a constant friction factor; (3) the effect of the particle

mass loading on the pressure drop is not included. All these drawbacks

are overcome in the Muschelknautz method of modeling (MM) [77] intro-

duced by Muschelknautz and Trefz [116, 117]. The main benefit of MM

over other models is its ability to take the following effects into account: a)

wall roughness due to both the physical roughness of the materials of con-

struction and to the presence of collected solids. b) the effect of the mass

loading and the Reynolds number on cyclone performance. c) the change

147

Chapter 8. Optimization

of flow velocity throughout the cyclone [77]. Consequently, the optimiza-

tion procedure using a data obtained from MM model via response sur-

face methodology will result in better results than the simple Stairmand

model. An alternative approach is using the available experimental data

in literature using the artificial neural networks (ANNs) approach for the

optimization process. The third source of data is the CFD simulations to

train the ANNs.

A general unconstrained optimization problem can be expressed as [182]:

min fi(x), i = 1, 2, . . . , Isubject to

x = [x1, x2, . . . , xn]T ,x ∈ X

(8.1)

where x = [x1, x2, . . . , xn]T is the vector of design variables (cyclone dimen-

sions) with total number of n, X denotes the design space. The fi(x) stands

for objective functions with total number of I. I should be larger than or

at least equal to 1, which suggests the multi-objective or single objective

optimal problem. In Eq. 8.1, f1(x) is the Euler number and f2(x) is the

cut-off diameter.

The design variables are seven geometrical parameters, namely, Dx/D,

a/D, b/D, Ht/D, h/D, S/D and Bc/D. The side constraints are: 0.3 ≤Dx/D ≤ 0.65, 0.2 ≤ a/D ≤ 0.65, 0.15 ≤ b/D ≤ 0.3, 3 ≤ Ht/D ≤ 6, 0.75 ≤h/D ≤ 2, 0.4 ≤ S/D ≤ 0.75 and 0.1 ≤ Bc/D ≤ 0.45.

There are numerous optimization techniques that can be used for the ge-

ometry optimization in the cyclone separator. However, only two tech-

niques have been used in this thesis (Nelder-Mead technique and the ge-

netic algorithms (GA)) for robustness, and availability in the used soft-

ware. In cases of multi-objective optimization studies, two approaches

have been applied, the desirability function and the non-sorted dominated

genetic algorithm (NSGA-II) techniques have been selected due to robust-

ness, and availability. More details about the used optimization techniques

are given in appendix B. A detailed literature review for the previous opti-

mization studies is given in Sec. 2.5.6, page 21.

The present study is an attempt to obtain new optimized cyclone separator

designs based on the MM model, experimental data sets and CFD simu-

lations data sets and to investigate the effect of each cyclone geometrical

parameter on the cyclone performance using response surface methodol-

ogy (RSM) and CFD simulations. Table 8.1 summaries the conducted opti-

mization studies.

148

8.2. Single-objective using MM model

Table 8.1: Summary of optimization studies

No.Design Objective Source Meta-model Optimization

variables functions of data technique

1 7 1 MM model RSM Nelder-Mead2 7 1 Experimental data (Exp.) ANN Nelder-Mead3 7 2 Exp. and Iozia and Leith model ANN GA and NSGA-II

4 4 2 CFD simulations data ANNDesirability function,

GA and NSGA-II

8.2 Optimization of the Euler number using

MM model and Nelder-Mead technique

The present study aims to obtain a new optimized cyclone separator for

minimum pressure drop based on the MM model and to investigate the

effect of each cyclone geometrical parameter on the cyclone performance

using CFD simulations. This section is an extension to the sensitivity

analysis study presented in Sec. 4.1, page 45.

!

"

#

"

Sa

h

D

b

Ht

Dx

L i

Le

Bc

Figure 8.1: Schematic diagram for Stairmand cyclone separator

Table 8.2 gives the optimum values for cyclone geometrical parameters

for minimum pressure drop estimated by MM using the downhill simplex

optimization technique available in Statgraphics XV software.

149


Table 8.2: The geometrical parameters for minimum pressure drop using MM

model

Factor Low High Optimum

a/D 0.5 0.75 0.618b/D 0.14 0.4 0.236Dx/D 0.2 0.75 0.622Ht/D 3 7 4.236h 1.0 2.0 1.618S 0.4 2 0.620Bc 0.2 0.4 0.382

8.2.1 CFD comparison between the two designs

Numerical settings

The air volume flow rate Qin=0.08 m3/s for the two cyclones (inlet velocity

for Stairmand design is 19 m/s and 13.1 m/s for the new design), air density

1.0 kg/m3 and dynamic viscosity of 2.11E-5 Pa s. The turbulent intensity

equals 5% and characteristic length equals 0.07 times the inlet width [75].

Velocity inlet boundary condition is applied at inlet, outflow at the gas

outlet and wall boundary condition at all other boundaries.

The finite volume method has been used to discretize the partial differen-

tial equations of the model using the SIMPLEC (Semi-Implicit Method for

Pressure-Linked Equations-Consistent) method for pressure velocity cou-

pling and QUICK scheme to interpolate the variables on the surface of

the control volume. The implicit coupled solution algorithm was selected.

The unsteady Reynolds stress turbulence model (RSM) was used in this

study with a time step of 0.0001 s. The residence time (cyclone volume/gas

volume flow rate) of the two cyclones are close (≈ 0.25 s).

The grid refinement study shows that a total number of about 134759 hex-

ahedral cells for Stairmand cyclone and 154746 hexahedral cells for the

new design are sufficient to obtain a grid-independent solution, and fur-

ther mesh refinement yields only small, insignificant changes in the nu-

merical solution. These simulations were performed on an eight nodes

CPU Opteron 64 Linux cluster using Fluent commercial software. The

geometrical values are given in Table 8.3 for the two cyclones (cf. Fig. 8.1).

Table 8.3: The values of geometrical parameters for the two designs (D=0.205 m)

Cyclone a/D b/D Dx/D Ht/D h/D S/D Bc/D Li/D Le/DStairmand design 0.5 0.2 0.5 4 1.5 0.5 0.36 1.0 0.618

New design 0.618 0.236 0.622 4.236 1.618 0.620 0.382 1.0 1.618

150


Table 8.4: The position of different sectionsa

Section S1 S2 S3 S4 S5 S6 S7

z’/Db 2.75 2.5 2.25 2.0 1.75 1.5 0.25

a Sections S1–S5 are located in the conical section, section S6 at the cylindrical part and S7 locatedthrough the inlet section.b z’ measured from the inlet section top

Results and discussion

The pressure field

Figure 8.2 shows the contour plot at Y=0 and at section S7 (at the middle of

inlet section, Table 8.4). In the two cyclones, the time-averaged static pres-

sure decreases radially from the wall to center. A negative pressure zone

appears in the forced vortex region (central region) due to high swirling

velocity. The pressure gradient is largest along the radial direction, while

the gradient in the axial direction is very limited. The cyclonic flow is not

symmetrical as is clear from the shape of the low pressure zone at the

cyclone center (twisted cylinder). However, the two cyclones have almost

the same flow pattern, but the highest pressure of the Stairmand design

is nearly twice that of the new design, implying that the new design has a

lower pressure drop.

The pressure distributions presented in Figs. 8.3 and 8.4 of the two cy-

clones at sections S1–S6 depict the two parts pressure profile (for Rankine

vortex). Once again, the highest static pressure for Stairmand design is

more than twice that of the new design at all sections while the central

value is almost the same for the two cyclones irrespective to the section

location. This indicates that, the new design has a lower pressure drop

with respect to the Stairmand design.

The velocity field

Based on the contour plots of the time–averaged tangential velocity, Fig. 8.2,

and the radial profiles at sections S1–S6 shown in Figs. 8.3 and 8.4, the

following comments can be drawn. The tangential velocity profile at any

section is composed of two regions, an inner and an outer one. In the

inner region, the flow rotates approximately like a solid body (forced vor-

tex), where the tangential velocity increases with radius. After reaching

its peak the velocity decreases with radius in the outer part of the profile

(free vortex). This profile is a so-called Rankine type vortex as mentioned

151



and S7. From top to bottom: static pressure [N/m2], tangential velocity [m/s] and

axial velocity [m/s]. From left to right Stairmand design and new design respec-

tively.

152


Table 8.5: The performance parameters for the two cyclones

Method Eu [-] ∆p [N/m2] x50 [µm]

Stairmand designMM 5.79 1045 1.54CFD 6.592 1190 1.0

New designMM 5.24 450 1.77CFD 5.672 487 1.6

before, including a quasi-forced vortex in the central region and a quasi-

free vortex in the outer region. The maximum tangential velocity may

reach twice the average inlet velocity and occurs in the annular cylindrical

part. The tangential velocity distributions for the two cyclones are nearly

identical in pattern and values (dimensionless), with the highest velocity

occurring at 1/4 of the cyclone radius for both cyclones. This implies a

nearly equal collection efficiency for both cyclones, as the centrifugal force

is the main driving force for particle collection in the cyclone separator.

The axial velocity profiles for the two cyclones are also very close, exhibit-

ing a M letter shape (also known as inverted W axial velocity profile in

some other literatures (cf. Horvath et al. [79])). Part of the flow in the

central region moves downward in the two cyclones. This phenomena has

been shown in the axial velocity pattern in other published articles [e.g.,

79, 159].

The DPM results

In order to calculate the cut-off diameters of the two cyclones, 104 particles

were injected from the inlet surface with a velocity equals the inlet gas ve-

locity. The particle density ρp is 860 kg/m3. The grade efficiency curves for

the two designs are plotted in Fig. 8.5. The DPM analysis results and the

pressure drops for the two cyclones are depicted in Table 8.5. An accept-

able agreement between the CFD results and the MM mathematical model

has been obtained. While the difference between the two cyclone cut-off di-

ameters is small, the saving in pressure drop is considerable (nearly half

the value of Stairmand cyclone).

Based on the flow pattern analysis and the DPM results. One can con-

clude that the cyclone collection efficiency for the two cyclones should be

very close, with the advantage of low pressure drop in the new design.

The authors want to emphasis that only small changes in the geometrical

dimensions of the two designs led to this improvement in the performance.

153


8.2.2 Conclusions

Both mathematical modeling (the Muschelknautz method of modeling (MM))

and CFD investigation have been used to understand the effect of the cy-

clone geometrical parameters on the cyclone performance and a new opti-

mized cyclone geometrical ratios based on MM model has been obtained.

The new cyclone design is very close to the Stairmand high efficiency de-

sign in the geometrical parameter ratio, but superior in low pressure drop

at nearly the same cut-off diameter. The new cyclone design results in

nearly one-half the pressure drop obtained by the old Stairmand design at

the same volume flow rate.

154



at different sections on the X-Z plane (Y=0) at sections S1–S3 . From top to bot-

tom: section S1–S3. From left to right: time-averaged static pressure, tangential

velocity and axial velocity respectively.

155


Figure 8.4: The radial profile for the time–averaged tangential and axial velocity

at different sections on the X-Z plane (Y=0) at sections S4–S6 . From top to bot-

tom: section S4–S6. From left to right: time-averaged static pressure, tangential

velocity and axial velocity respectively.

156



Col

lect

ion

effic

ienc

y[-]

10-1 100 101

0.25

0.5

0.75

1Stairmand designNew design

Figure 8.5: The grade efficiency curves for the two designs

157


8.3 Optimization of the Euler number using

RBFNN and Nelder-Mead technique

Recently, Artificial neural networks (ANNs) have been widely applied in

the fields of modeling, prediction, fault detection and process control. In

the field of performance evaluation for cyclone separators, unfortunately,

ANNs have not been paid enough attention on their algorithmic advan-

tages [197]. There are few articles about the application of neural network

in the field of cyclone separator.

There are six main objectives of this study. (1) Application of the artificial

neural network to model the pressure drop using experimental dataset.

(2) Investigation of the effect of the seven geometrical parameters on the

pressure drop based on the trained RBFNN. (3) A detailed comparison be-

tween the experimental pressure drop values and the estimated values

obtained from different mathematical models. In order to recommend the

best mathematical model for future use. (4) Application of the response

surface methodology to study the effect of each geometrical parameter on

the pressure drop and test the interaction between these parameters using

the trained RBFNN. (5) Obtaining the optimum design (geometrical ratios)

for minimum pressure drop. (6) CFD study of the new cyclone separator

and compare its performance and flow pattern with the Stairmand design.

8.3.1 Radial basis function neural networks (RBFNN)

Why RBFNN

Radial basis function neural networks (RBFNNs) are powerful and inter-

esting networks due to their rapid training, generality and simplicity [16].

Girosi and Poggio [65] and Hartman and Keeler [72] proved that RBFNNs

are universal approximators and can approximate any continuous func-

tion with arbitrary accuracy. Training of these networks is very fast, and

they are very good at interpolation [190]. Niros and Tsekouras [120] stated

that radial basis function neural networks (RBFNN) have certain advan-

tages over other types of neural networks including better approximation

capabilities, simple network structure, and faster learning.

Zhao and Su [197] tested three different types of artificial neural network

to model the pressure drop in cyclone separators, viz. the back propa-

gation neural network (BPNN), the radial basis function neural network

(RBFNN) and the generalized regression neural network (GRNN). They

stated that compared with the BPNN and GRNN, the RBFNN provides

superior prediction performance criteria, better capability of approxima-

tion and high robustness.

158

8.3. Single-objective using RBFNN

In this study, the radial basis function neural network (RBFNN) is em-

ployed to model the pressure drop in the cyclone separator.

The structure of RBFNN

The radial basis function neural network (RBFNN) is a kind of 3-layered

forward network with multi inputs and multi outputs. The first layer has

(m) inputs, while the second is a hidden layer with (L) units, and the third

layer has (n) outputs, Fig. 8.6. The transformation function from the in-

puts to the hidden units, varied radial basis functions (RBF) on different

occasions, is nonlinear, whereas the mapping of the hidden units to the

outputs is linear [180].

Theoretically, RBFNN has the ability of approaching nonlinear mapping

arbitrarily, fr : Rm → Rn, as defined by:

y = fr(x) = W0 +

L∑

i=1

Wiφ (‖x− ci‖) (8.2)

where x ∈ Rm is the input vector, y ∈ Rn is the output vector, ci ∈ Rm

(i = 1, 2, · · ·L) is the center vector, ‖ · ‖ is the Euclidian norm, Wi ∈ Rm

(i = 1, 2, · · ·L) is the weight vector, W0 ∈ Rm is the bias vector, and φ(·) is

the Gaussian function, a non negative and nonlinear function with radial

symmetry and attenuation versus center, in the form of:

φ(v) = exp(−v2/2σ2) (8.3)

where σ is a width constant (spread factor ) [180].

In order to comprehensively compare the model performance, the evalua-

tion parameters, normalized mean squared error E2 and correlation coef-

ficient R are employed as follows [197]:

E2 =1

nΣn

i=1(yNi − yNi)2

(8.4)

where n is the number of test cases, y is the actual variable, y is the

RBFNN output variable and Ni is the neurons number of input layer in

the RBFNN.

R =Σn

i=1(yNi − yN )(yNi − yN

)√Σn

i=1(yNi − yN )2Σn

i=1

(yNi − yN

)2 (8.5)

159


1

∑

∑

∑

x1

x2

xm

W0

Wi

y1

y2

yn

Input layer Hidden layer Output layer

Figure 8.6: Schematic diagram for the radial basis function neural network

Modeling procedures

The ANN modeling procedure can be divided into the following steps: se-

lecting the variables, dividing the sample, optimizing parameters, training

and testing simulation and evaluating performance [197]. In this study, all

calculations are carried out on a MacBook pro laptop with the hardware

configurations: processor, Intel Core 2 Duo (2.4 GHz); memory, 4.0 GB

(DDR3-1067 2G 2); hard drive, 320 GB (7200 rpm); with Mac OS X 10.5

system.

The Euler number

The pressure drop across a cyclone separator essentially depends on the

dimensions and operating conditions. Generally, it is proportional to the

average dynamic pressure at the inlet and is often defined as [197]

∆P = Eu

(12ρgV

2in

)(8.6)

where Eu is Euler number (the dimensionless pressure drop also called

the pressure drop coefficient [197]). The Euler number is a complex non-

linear function of the cyclone geometrical dimensions and is not affected

by operating conditions in the high Reynolds number ( Re > 5E4) [50, 77].

The Euler number will be constant for any cyclone configuration regard-

less of size as long as the dimension ratios remain the same, although the

pressure drop varies with different operating conditions (due to the effect

160


of ρg and Vin). Therefore, pressure drop can be established by determin-

ing experimentally or theoretically for a particular cyclone design and also

be modified by the semi-empirical correlations to take the effect of solid

loading [197].

In order to determine the Euler number more accurately, all eight dimen-

sions of the cyclone are selected to establish the ANN models because they

have the effect on the Euler number to different extent [139, 197]. Usu-

ally, these dimensions can be characterized by the barrel diameter D and

expressed as seven dimensionless geometric ratios [197]:

Eu = f

(Dx

D,a

D,b

D,S

D,Ht

D,h

D,Bc

D

)(8.7)

According to Eq. 8.7, seven independent dimensionless geometrical vari-

ables and one dependent variable (the Euler number of the cyclone) are se-

lected as respectively the input and output parameters in the ANN model,

as presented in Table 8.6. For simplicity, the division of each factor by the

barrel diameter D will be dropped.

A dataset of 98 samples obtained from the measurements of pressure drop

for different cyclone designs available in the literature [42, 139, 197] is

used in the present investigation to evaluate the prediction performance

of the ANN models. Table 8.7 presents more details about the used dataset

including the minimum, mean, maximum and range of the seven dimen-

sionless geometrical ratios. Due to the large difference in the order of mag-

nitude of the value (cf. Table 8.7), the available dataset is transformed into

-1 to 1 interval using the Matlab intrinsic function; mapminmax in order

to avoid solution divergence [197]. The ANN calculations have been per-

formed using the neural network toolbox available from Matlab commer-

cial software 2010a.

Descriptive statistical parameters of the input dataset

Table 8.8 shows Pearson product moment correlations between each pair of

variables. These correlation coefficients range between -1 and +1 and mea-

sure the strength of the linear relationship between the variables. More-

over, shown in parentheses is the P-value which tests the statistical signif-

Table 8.6: The input and output variables for ANN model

Input parameters Output diameter

Variables X1 X2 X3 X4 X5 X6 X7 ySpecification Dx a b S Ht h Bc Eu

161


Table 8.7: Descriptive statistical parameters for the training dataset

Variable Dx a b S Ht h Bc

Minimum 0.25 0.113 0.067 0.39 1.158 0.501 0.14Mean 0.429 0.630 0.211 0.891 3.283 1.189 0.342Maximum 0.667 1.0 0.4 3.052 10.97 3.5 1.0Range 0.417 0.887 0.333 2.662 9.812 2.999 0.86

icance of the estimated correlations. P-values below 0.05 indicate statisti-

cally significant non-zero correlations at the 95.0% confidence level. The

following pairs of variables have P-values below 0.05: (1) Dx with a and Bc;

(2) a with b, Ht and Bc; (3) b with Ht and Bc;(4) S with Ht, h and Bc; (5) Ht

with h and Bc; (6) h with Bc. These conclusions can be obtained also from

Fig. 8.7. From this analysis, however there are some correlations between

the input variables, but this dataset is still reliable. The same dataset has

been used successfully by other researchers [e.g., 194, 197]. Furthermore,

it is the only available experimental data set in the literature.

K-fold cross validation

For the calibration of the RBFNN, the spread factor σ plays an important

role in the regression model. To obtain the optimum value for σ a multi-

step search technique is used. In the multi-step search technique, the

seeking for the optimum value is performed in two steps [197]. The first

step is a coarse search to determine the best range of values. The second

search is a fine search in the best range. To avoid overfitting, the K-fold

cross validation has been employed [8, 80, 197]. The original sample of

data is randomly portioned into K subsamples. A single subsample is used

for the validation (testing) and the other remaining K-1 subsamples are

used for the training. The process of training and testing is then repeated

for each of the K possible choices of the subset omitted from the training.

The average performance on the K omitted subsets is then our estimate

of the generalization performance. This procedure has the advantage that

is allows us to use a high proportion of the available data (a fraction 1 -

1/K) for training, while making use of all the data points in estimating the

Table 8.8: Correlations between each pair of input variables∗

Dx a b S Ht h BcDx -0.377 (0.000) -0.121 (0.234) 0.092 ( 0.367) 0.189 (0.061) 0.096 (0.346) 0.199 (0.049)a -0.377 (0.000) 0.442 (0.000) 0.148 (0.145) 0.288 (0.004) -0.052 (0.610) 0.270 (0.007)

b - 0.121 (0.234) 0.442 (0.000) 0.170 (0.093) 0.239 (0.017) 0.194 (0.054) 0.243 (0.015)

S 0.092 (0.367) 0.148 (0.145) 0.170 (0.093) 0.378 (0.000) 0.685 (0.000) 0.526 (0.000)

Ht 0.189 (0.061) 0.288 (0.004) 0.239 (0.017) 0.378 (0.000) 0.393 (0.000) 0.555 (0.000)

h 0.096 (0.346) -0.052 (0.610) 0.194 (0.054) 0.685 (0.000) 0.393 (0.000) 0.470 (0.000)Bc 0.199 (0.049) 0.270 (0.007) 0.243 (0.015) 0.526 (0.000) 0.555 (0.000) 0.470 (0.000)∗ The bold value shown in parentheses is the P-value.

162


Figure 8.7: Qualitative representation of the correlations between each pair of

input variables. For uncorrelated pair of variables, the data will be well distributed

and no linear correlation can be obtained e.g., Dx and b (first row, third column).

The high correlation between Ht and h is clear (fifth row, sixth column), where one

can fit easily fit a straight line.

generalization error. The disadvantage is that we need to train the net-

work K times. Typically K = 10 is considered reasonable and most widely

used [18]. For the radial basis function neural networks (newrb in Matlab

2010a), the learning process is a must to obtain the weights (the width of

the radial basis function units). In order to obtain the optimum value of

the spread factor σ, the multi-step search technique with 10-fold cross val-

idation from the interval of (0 - 1) with the performance goal of 1E-5, the

maximum number of neurons in the hidden layer equals the training sam-

ple size of 98 neurons, the number of neurons to add between displays is

10. Based on the mentioned settings, the optimum value of σ equals 0.191

(E2=1.1321E-06), this value is different than that obtained by Zhao and

Su [197] (σ equals 0.32 (E2=5.84E-04)) the reason can be referred to the

lower goal used in the current study (the goal used in Zhao and Su [197]

was 1E-4, and all other settings are identical).

8.3.1.1 Fitting the ANN

Table 8.9 presents more details about the validation of the used RBFNNs.

Both the average, minimum, maximum and range of the input (Euler num-

ber) and the predicted Euler number are given. It is clear from Table 8.9

that the ANNs preserved the descriptive statistical parameters of the in-

163


Table 8.9: Validation of the used RBFNN∗

Experimental MM Stairmand Ramachandran Shepherd

x y x y x y x y x y

Average 23.268 23.268 15.150 15.150 20.206 20.206 22.543 22.543 17.774 17.774

Minimum 2.3 1.745 1.34 1.164 2.88 2.892 1.85 1.793 0.957 1.197Maximum 155.3 155.985 138.0 137.235 132.0 133.521 153.0 150.854 92.2 92.543

Range 153.0 154.24 136.66 136.071 129.12 130.629 151.15 149.061 91.243 91.346

Correlation Coefficient, R 0.999 0.996 0.999 0.999 0.999

Mean squared error, E2 1.311E-4 1.212E-4 9.185E-5 1.442E-4 5.411E-5

Intercept 0.017 0.012 0.011 0.020 0.005Slope 0.999 0.999 0.999 0.999 1.0

∗ x is the input to the RBFNN and y is the predicted value. Both x and y represent the Euler number.The values of R, E2, intercept and slope are that for the testing stage.

put data. The correlation coefficient between the input and the output and

the mean squared error are given for each RBFNN. The intercept and the

slope of the adjusted line between the input and the predicted value of the

ANN are also given.

The configured RBFNN predictions versus experimental data and four

other models for cyclone Euler number are shown in Fig. 8.8. According

to Fig. 8.8, it can be seen that the ANN models are able to attain the

high training accuracy. The training mean square errors for the exper-

imental values and the four mathematical models (MM, Stairmand, Ra-

machandran and Shepherd and Lapple), have the values 1.311E-4, 1.212E-

4, 9.185E-5, 1.442E-4 and 5.411E-5 respectively (Fig. 8.8). This indicates

that, compared with traditional models of curve fitting, the models based

on artificial intelligence algorithm have a superior capability of nonlinear

fitting. Especially, the RBFNN has its unique and optimal approximation

characteristics in learning process [197].

Figure 8.8 illustrates the agreement between the ANN input and output.

The obtained relation is a typical linear relation with a coefficient of cor-

relation close to 1 (R > 0.999). The agreement between the input and

output of the ANN is also clear from the value of the mean squared error

E2 (< 1.5E − 4). That means, the trained neural network predicts very

well the Euler number values and can be used in cyclone design and per-

formance estimation. Table 8.9 and Fig. 8.8 present different performance

indicators as a validation of the proposed model for experimental values.

8.3.2 Evaluation of different mathematical models

In order to evaluate the performance of the four tested mathematical mod-

els in comparison with the experimental values, the percentage residual

164


Input value

Pre

dict

edva

lue

50 100 150

50

100

150 Data pointLinear fit

(a) Experimental values

Input value

Pre

dict

edva

lue

50 100 150

50

100


(b) MM model

Input value

Pre

dict

edva

lue

50 100 150

50

100


(c) Stairmand

Input value

Pre

dict

edva

lue

50 100 150

50

100


(d) Ramachandran model

Input value

Pre

dict

edva

lue

20 40 60 80 100

20

40

60

80

100Data pointLinear fit

(e) Shepherd and Lapple model

Figure 8.8: Linear regression of the RBFNN.

165


Run number

Res

idua

lerr

or%

0 20 40 60 80 100-200

-100

0

100

200

300

400MMStairmandRamachandranShepherd

Figure 8.9: Percentage residual error for the four tested models based on 98

dataset.

error (Eq. 8.8) for each model has been plotted in Fig. 8.9.

% error =Model value - Experimental value

Experimental value∗ 100 (8.8)

Figure 8.9 depicts that MM model underestimate the pressure drop by

around 50%. Also the percentage errors for other models are between ±50%. The peaks in error are almost the same for all models. The residual

error of MM model is almost the lowest. The reasons of these peaks may be

due to the high values of geometrical swirl number for some of the avail-

able dataset (cf. Ramachandran et al. [139]), where Sg = πDx D/(4a b). Sg

varies between 1 and 5 for industrial cyclones [74]. The Sg values in the 98

dataset used in this study varies from 2.18 to 92.67. As the swirl number is

a measure for the ratio of tangential to axial momentum [74, 75, 154], the

high values of Sg may cause violation of the simplified assumptions used

in the models. Figure 8.9 seems to indicate that the MM model among the

other mathematical models is best suited for estimation of the pressure

drop.

The effect of geometrical parameters on the Euler number

The effects of the geometrical parameters on the Euler number are de-

picted in Fig. 8.10. To study the effect of each parameter, the tested RBFNN

model has been used by varying one parameter at a time from its minimum

to maximum values of the available 98 dataset, while the other parameters

are kept constant at their mean values (cf. Table 8.7). Figure 8.10(a) indi-

cates the significant effect of the vortex finder diameter Dx and the vortex

finder length S, the inlet width b and the total height Ht. Less effect is due

to the cylinder height h (for h > 2.5) and the inlet height a (for a > 0.55).

166


The effect of the geometrical parameters on the Euler number obtained

from the MM model (Fig. 8.10(b)) is very close to that obtained from the

analysis of the experimental dataset but with underestimation of the Eu-

ler number. This supports the use of the MM model in the Euler number

estimation for cyclone separators [29, 50, 77]. The situation for the Stair-

mand model is not the same as that for the experimental data or MM

model. Here, the effect of many geometrical parameters attenuated. The

effect of the inlet width becomes insignificant. That is not realistic, as for

high values of inlet width, a considerable part of the incoming flow will im-

pact directly the vortex finder which increases the entrance loss and conse-

quently, the total pressure drop in the cyclone separator. The Ramachan-

dran model was initially constructed by curve fitting based on the used 98

dataset (cf., Ramachandran et al. [139] for more details). So it depicts a

better agreement with the experimental values of the Euler number than

that of Stairmand model. The Ramachandran model predicts almost the

same Euler number variation with the inlet width. In the Shepherd and

Lapple model only the inlet dimensions (aandb) and vortex finder diameter

Dx affect the pressure drop (Eu = 16ab/D2x) as is clear from Fig. 8.10(e).

Figure 8.11 compares the effect of each individual geometrical parameter

using predictions with the ANN based on respectively the experimental

data and the four tested models. The following conclusions can be drawn

from the analysis of Fig. 8.11:

Dx : All the models (except Shepherd and Lapple) show the same variation

of the Euler number with increasing Dx. For Dx > 0.5 there is an ex-

act matching between the results of MM model and the experimental

values.

a : Both the Ramachandran and the Shepherd and Lapple models predict

a (nearly) linear relation between the inlet height and the Euler num-

ber. The effect of changing a predicted by the Stairmand model is in

accordance with the experimental values for a > 0.8.

b : For small values of b (b < 0.15), the MM model results are in good agree-

ment with the experimental values. Also the trends of all models in

this range are matching the trend of the experimental values. Be-

yond this range, Both the Ramachandran and the Stairmand models

agree well with the experimental values trend.

S : The trend of the MM model results is similar to that of the experi-

mental values, although there is a shift in the values of the Euler

number. The Shepherd and Lapple model does not present any effect

of changing S on the Euler number (for Shepherd and Lapple model,

Eu = 16ab/D2x), while the MM model always underestimates the ef-

167


Dx, a, b, B c

h, S

Ht

Eul

ernu

mbe

r

0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5 3 3.5

2 4 6 8 10

0

10

20

30

40

Dx

abSHt

hBc

(a) Experimental values

Dx, a, b, B c

h, S

Ht

Eul

ernu

mbe

r

0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5 3 3.5

2 4 6 8 10

0

10

20

30

40

Dx

abSHt

hBc

(b) MM model

Dx, a, b, B c

h, S

Ht

Eul

ernu

mbe

r

0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5 3 3.5

2 4 6 8 10

0

10

20

30

40

Dx

abSHt

hBc

(c) Stairmand model

Figure 8.10: The effect of geometrical parameters on the Euler number using the

trained neural networks based on experimental dataset and four different mathe-

matical models.

168


Dx, a, b, B c

h, S

Ht

Eul

ernu

mbe

r

0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5 3 3.5

2 4 6 8 10

0

10

20

30

40 Dx

abSHt

hBc

(d) Ramachandran model

Dx, a, b, B c

h, S

Ht

Eul

ernu

mbe

r

0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5 3 3.5

2 4 6 8 10

0

10

20

30

40 Dx

abSHt

hBc

(e) Shepherd and Lapple model

Figure 8.10: (continued) The effect of geometrical parameters on the Euler number

using the trained neural networks based on experimental dataset and four differ-

ent mathematical models.

fect of S. Both the Stairmand and the Ramachandran models may

over/underestimate its effect on Euler number.

Ht : Both the Stairmand and the Ramachandran models do not show a

significant effect of Ht on the Euler number. The trends of both MM

and Ramachandran model are almost the same as that for the exper-

imental values.

h : For higher values of h all models (except the Shepherd and Lapple

model) give nearly the same value of Euler number.

Bc : Nearly, all the models (except the Shepherd and Lapple model) show

the same trend in the changing the Euler number with the cone tip

diameter.

8.3.3 Design of experiment (DOE)

Table 8.10 depicts the parameters ranges selected for the seven geometri-

cal parameters. The study was planned using Box-Behnken design, with

64 combinations. A significant level of P < 0.05 (95% confidence) was used

in all tests. Analysis of variance (ANOVA) was followed by an F-test of the

169


Table 8.10: The values of the independent variables used in the design of experi-

ment


Vortex finder diameter, Dx=X1 0.2 0.475 0.75

Inlet height, a=X2 0.4 0.55 0.7

Inlet width, b=X3 0.14 0.27 0.4

Vortex finder length, S=X4 0.4 1.2 2.0

Total cyclone height, Ht=X5 3.0 5.0 7.0

Cylinder height, h=X6 1.0 1.5 2.0

Cone tip diameter, Bc=X7 0.2 0.3 0.4

individual factors and interactions.

Fitting the model

Analysis of variance (ANOVA) showed that the resultant quadratic poly-

nomial models adequately represented the experimental data with the co-

efficient of multiple determination R2 being 0.965843. This indicates that

the quadratic polynomial model obtained was adequate to describe the in-

fluence of the independent variables studied [189]. Analysis of variance

(ANOVA) was used to evaluate the significance of the coefficients of the

quadratic polynomial models (see Table 8.12). For any of the terms in the

models, a large F-value (small P-value) would indicate a more significant

effect on the respective response variables.

Based on the ANOVA results presented in Table 8.12, the variable with

the largest effect on the pressure drop (Euler number) was the linear term

of vortex finder diameter, followed by the linear term of inlet width and

vortex finder length (P < 0.05); the other four linear terms (inlet height,

barrel height, cyclone total height and cone tip diameter) did not show a

significant effect (P > 0.05). The quadratic term of vortex finder diameter,

vortex finder length and cyclone total height also had a significant effect (P

< 0.05) on the pressure drop; however, the effect of the other four quadratic

terms was insignificant (P > 0.05). Furthermore, the interaction between

the inlet dimensions and vortex finder diameters (P < 0.05) also had a

significant effect on the pressure drop, while the effect of the remaining

terms was insignificant (P > 0.05).

Analysis of response surfaces



170


Table 8.11: The geometrical dimensions and Euler number for the used cyclones

X1 X2 X3 X4 X5 X6 X7 Y1

Exp. No. a/D b/D Dx/D Ht/D h/D S/D Bc/D Euler Number

1 0.7 0.27 0.75 5 1 1.2 0.3 4.16

2 0.55 0.27 0.475 7 2 2 0.3 6.93

3 0.55 0.27 0.475 3 2 0.4 0.3 9.30

4 0.55 0.27 0.2 7 1.5 1.2 0.2 65.705 0.7 0.4 0.475 7 1.5 1.2 0.3 11.90

6 0.55 0.14 0.475 5 1 1.2 0.4 4.34

7 0.55 0.4 0.475 5 1 1.2 0.4 9.19

8 0.55 0.27 0.475 5 1.5 1.2 0.3 7.94

9 0.55 0.27 0.475 3 1 2 0.3 8.87

10 0.4 0.4 0.475 7 1.5 1.2 0.3 5.8811 0.55 0.27 0.475 5 1.5 1.2 0.3 7.94

12 0.55 0.4 0.75 5 1.5 0.4 0.3 3.24

13 0.55 0.14 0.2 5 1.5 0.4 0.3 24.50

14 0.7 0.4 0.475 3 1.5 1.2 0.3 13.40

15 0.7 0.27 0.2 5 1 1.2 0.3 105.0016 0.7 0.27 0.475 5 1.5 0.4 0.4 10.30

17 0.4 0.4 0.475 3 1.5 1.2 0.3 7.05

18 0.7 0.27 0.75 5 2 1.2 0.3 4.07

19 0.55 0.4 0.475 5 2 1.2 0.2 9.19

20 0.55 0.27 0.475 7 1 0.4 0.3 7.5021 0.55 0.27 0.2 3 1.5 1.2 0.2 77.50

22 0.55 0.14 0.75 5 1.5 0.4 0.3 2.42

23 0.55 0.4 0.2 5 1.5 0.4 0.3 130.00

24 0.55 0.27 0.475 5 1.5 1.2 0.3 7.94

25 0.4 0.14 0.475 7 1.5 1.2 0.3 3.17

26 0.4 0.27 0.2 5 1 1.2 0.3 41.9027 0.4 0.27 0.2 5 2 1.2 0.3 40.70

28 0.55 0.27 0.475 5 1.5 1.2 0.3 7.94

29 0.55 0.27 0.475 5 1.5 1.2 0.3 7.94

30 0.55 0.27 0.75 7 1.5 1.2 0.2 3.35

31 0.55 0.4 0.475 5 1 1.2 0.2 9.4532 0.55 0.4 0.2 5 1.5 2 0.3 128.00

33 0.55 0.14 0.2 5 1.5 2 0.3 23.80

34 0.55 0.27 0.75 3 1.5 1.2 0.4 3.82

35 0.55 0.14 0.475 5 2 1.2 0.4 4.22

36 0.55 0.27 0.475 5 1.5 1.2 0.3 7.9437 0.55 0.14 0.475 5 2 1.2 0.2 4.33

38 0.55 0.27 0.75 3 1.5 1.2 0.2 3.86

39 0.4 0.27 0.75 5 1 1.2 0.3 2.99

40 0.55 0.27 0.475 5 1.5 1.2 0.3 7.94

41 0.55 0.27 0.475 7 1 2 0.3 7.09

42 0.7 0.14 0.475 3 1.5 1.2 0.3 6.0343 0.55 0.14 0.75 5 1.5 2 0.3 2.23

44 0.7 0.27 0.2 5 2 1.2 0.3 103.00

45 0.55 0.4 0.75 5 1.5 2 0.3 3.05

46 0.55 0.27 0.2 7 1.5 1.2 0.4 64.10

47 0.4 0.27 0.75 5 2 1.2 0.3 2.9248 0.4 0.27 0.475 5 1.5 2 0.4 5.70

49 0.7 0.27 0.475 5 1.5 2 0.4 9.64

50 0.4 0.27 0.475 5 1.5 0.4 0.2 6.34

51 0.4 0.27 0.475 5 1.5 0.4 0.4 6.12

52 0.55 0.27 0.475 3 2 2 0.3 8.5153 0.7 0.14 0.475 7 1.5 1.2 0.3 4.70

54 0.4 0.27 0.475 5 1.5 2 0.2 5.88

55 0.55 0.14 0.475 5 1 1.2 0.2 4.51

56 0.7 0.27 0.475 5 1.5 0.4 0.2 10.60

57 0.55 0.27 0.2 3 1.5 1.2 0.4 76.30

58 0.4 0.14 0.475 3 1.5 1.2 0.3 4.0659 0.55 0.27 0.75 7 1.5 1.2 0.4 3.25

60 0.55 0.27 0.475 3 1 0.4 0.3 9.77

61 0.7 0.27 0.475 5 1.5 2 0.2 9.91

62 0.55 0.27 0.475 5 1.5 1.2 0.3 7.94

63 0.55 0.4 0.475 5 2 1.2 0.4 9.0164 0.55 0.27 0.475 7 2 0.4 0.3 7.30

New design∗ 0.618 0.236 0.618 4.236 1.618 0.618 0.3819 5.24

Stairmand design† 0.5 0.2 0.5 4 1.5 0.5 0.36 5.79

∗ The new design based on MM model and downhill simplex optimization scheme (Euler number =5.24, cut-off diameter =1.77µm ).†The standard Stairmand high efficiency cyclone design, (Euler number =5.79, cut-off diameter =1.54µm based on MM model).

171


Table 8.12: Analysis of variance of the regression coefficients of the fitted quadratic

equationa


β0 138.604

Linearβ1 -485.694 96.89 0.0000β2 20.2354 0.06 0.8152β3 325.517 59.28 0.0000β4 -10.7497 6.87 0.0140β5 -8.7097 0.06 0.8161β6 -19.1361 0.57 0.4575β7 -167.261 0.60 0.4464

Quadraticβ11 515.706 489.19 0.0000β22 113.392 2.09 0.1590β33 -13.0018 0.02 0.9017β44 10.9817 15.89 0.0004β55 1.04634 5.63 0.0247β66 6.06181 0.74 0.3974β77 232.346 1.74 0.1983

Interactionβ12 -158.258 7.61 0.0101β13 -345.242 27.22 0.0000β14 -4.21438 0.15 0.6981β15 8.42261 3.83 0.0602β16 31.8848 3.43 0.0744β17 95.7341 1.24 0.2752β23 -143.344 1.40 0.2473β24 20.5571 1.09 0.3060β25 -3.23107 0.17 0.6851β26 -16.3805 0.27 0.6076β27 -6.62067 0.00 0.9668β34 -26.3282 1.34 0.2569β35 0.0205865 0.00 0.9982β36 32.7526 0.81 0.3758β37 -28.156 0.02 0.8782β45 -0.62389 0.18 0.6763β46 -2.06662 0.12 0.7294β47 -18.6467 0.40 0.5334β56 -2.60425 1.21 0.2803β57 2.35394 0.04 0.8437β67 -5.32565 0.01 0.9112

R2 0.965843

a Bold numbers indicate significant factors as identified by the analysis of variance (ANOVA) at the 95%confidence level.

172


is proportional to the size of the effect and the direction of the curve spec-

ifies a positive or negative influence of the effect [50, 61] (Fig.8.12(a)).

Based on the main effect plot, the most significant factors on the Euler

number are: (1) the vortex finder diameter Dx, with a second–order curve

with a wide range of inverse relation and a narrow range of direct relation,

(2) the inlet width b linearly related to the Euler number, (3) the cyclone

total height Ht, (4) the vortex finder length S, whereas the other factors

have an insignificant effect.

Pareto charts were used to graphically summarize and display the rela-

tive importance of each parameter with respect to the Euler number [50].

The Pareto chart shows all the linear and second-order effects of the pa-

rameters within the model and estimates the significance of each with re-

spect to maximizing the Euler number response. A Pareto chart displays

a frequency histogram with the length of each bar proportional to each

estimated standardized effect [30]. The vertical line on the Pareto charts

judges whether each effect is statistically significant within the generated

response surface model; bars that extend beyond this line represent effects

that are statistically significant at a 95% confidence level. Based on the

Pareto chart (Fig. 8.12(b)) and ANOVA table (Table 8.12) there are five sig-

nificant parameters (eight terms in the ANOVA table; Table 8.12) at a 95%

confidence level: the vortex finder diameter Dx, the inlet width b, the total

cyclone height Ht, the vortex finder length S and the inlet height a (due to

interaction with Dx). Therefore, the Pareto chart is a perfect supplement

to the main effects plot.




while holding the other factors at their central values [189] as shown in

Fig. 8.13. Thus, Fig. 8.13(b) was generated by varying the inlet height aand the inlet width b while keeping the other five factors constant. The

trend of the curve is linear, with a more significant effect for the inlet

width b, and a weak interaction between the inlet height a and width b.The response surface plots of Figs. 8.13(a), 8.13(c) and 8.13(d) show that

there are strong interactions between the vortex finder diameter Dx and

respectively the inlet height a, the cyclone total height Ht and the vortex

finder length S. The effect of vortex finder length S is less significant with

respect to the vortex finder diameter Dx, but its effect is higher than that

of the barrel height h and the cone tip diameter Bc (cf., Fig. 8.12).

173


8.3.3.1 Optimization (Nelder-Mead method)

In this optimization problem, the objective function is the Euler number

(f1(x) in Eq. 8.1). Table 8.13 gives the optimum values for cyclone geomet-

rical parameters for minimum pressure drop. It is clear from Table 8.13

that the new optimized design is very close to the Stairmand design in

many geometrical parameters, whereas the new ratios will lead to the min-

imum pressure drop. To understand the effect of this small change in the

geometrical ratios on the flow field pattern and performance, a CFD study

for the two designs is needed.

Table 8.13: The geometrical parameters for minimum pressure drop using RBFNN

Factor Low High Stairmand design Optimum

Dx 0.2 0.75 0.5 0.487

a 0.5 0.75 0.5 0.629

b 0.14 0.4 0.2 0.203

S 0.4 2.0 0.5 0.733

Ht 3.0 7.0 4.0 4.852

h 1.0 2.0 1.5 1.633

Bc 0.2 0.4 0.375 0.383

174


Dx

Eul

ernu

mbe

r

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70

10

20

30

40

50Exp.MMStairmandRamchandranShepherd

(a) Dx

a

Eul

ernu

mbe

r

0 0.2 0.4 0.6 0.8 10

5

10

15

20


(b) a

b

Eul

ernu

mbe

r

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

5

10

15

20

25


(c) b

S

Eul

ernu

mbe

r

0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

35

40

45

50

55

60


(d) S

Figure 8.11: Comparison between the effect of each geometrical parameters on the

Euler number using different models and experimental values based on RBFNN.

175


Ht

Eul

ernu

mbe

r

2 4 6 8 10 120

10

20

30

Exp.MMStairmandRamchandranShepherd

(e) Ht

h

Eul

ernu

mbe

r

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15


(f) h

Bc

Eul

ernu

mbe

r

0.2 0.4 0.6 0.8 10

10

20

30

40

Exp.MMStairmandRamchandranShepherd

(g) Bc

Figure 8.11: (continued) Comparison between the effect of each geometrical param-

eters on the Euler number using different models and experimental values based

on RBFNN.

176


(a) Main effects plot

(b) Pareto chart

Figure 8.12: Analysis of design of experiment

177


(a) Dx versus a

(b) a versus b

(c) Dx versus Ht

(d) Dx versus S

Figure 8.13: The response surface plots.178


8.3.4 CFD Comparison between the two designs

Numerical settings


for Stairmand design is 19 m/s and 14.9 m/s for the new design), air density

1.0 kg/m3 and dynamic viscosity of 2.11E-5 Pa s. the turbulent intensity



and wall boundary condition at all other boundaries. The Reynolds stress

turbulence model has been used to reveal the turbulence characteristics in

the cyclone separators.




pling and QUICK scheme to interpolate the variables on the surface of the

control volume. The implicit coupled solution algorithm was selected. The

unsteady Reynolds stress turbulence model (RSM) was used in this study

with a time step of 0.0001 s.

The grid refinement study using different levels of grid shows that a total

number of 134759 hexahedral cells for the Stairmand cyclone and 381709

hexahedral cells for the new design are sufficient to obtain a grid-independent

solution, and further mesh refinement yields only small, insignificant changes

in the numerical solution. The hexahedral meshes have been obtained us-

ing the GAMBIT commercial software. These simulations were performed

on an 8 nodes CPU Opteron 64 Linux cluster using Fluent 6.3.26 commer-

cial software. The geometrical values for the two cyclones (cf. Fig. 8.1) are

given in Table 8.14. The surface mesh for Stairmand cyclone is given in

Fig. 8.14(a).

Table 8.14: The values of geometrical parameters for the two designs (D=0.205 m)∗

Cyclone a/D b/D Dx/D Ht/D h/D S/D Bc/DStairmand design 0.5 0.2 0.5 4 1.5 0.5 0.375

New design 0.628 0.203 0.487 4.852 1.633 0.732 0.382

∗ The outlet section is above the cyclone surface by Le = 0.618D. The inlet section located at adistance Li = D from the cyclone center.

179


(a) The Stairmand design (b) The new design

Figure 8.14: The surface meshes for the two designs


The pressure field

Figure 8.15 shows the contour plot at Y=0. In the two cyclones, the time-

averaged static pressure decreases radially from the wall to center. A neg-

ative pressure zone appears in the forced vortex region (central region)

due to high swirling velocity. The pressure gradient is largest along the

radial direction, while the gradient in the axial direction is very limited.

The cyclonic flow is not symmetrical as is clear from the shape of the low

pressure zone at the cyclone center (twisted cylinder). However, the two

cyclones have almost the same flow pattern, but the highest pressure of

the Stairmand design is nearly 1.5 times that of the new design, implying

that the new design has a lower pressure drop.

The pressure distribution presented in Figs. 8.16 and 8.17 of the two cy-


vortex). Again, the highest static pressure for Stairmand design is more

than 1.5 times that of the new design at all sections while the central value

is almost the same for the two cyclones irrespective of the section location.

This indicates that, the new design has a lower pressure drop with respect

to the Stairmand design.

180


Figure 8.15: The contour plots for the time averaged flow variables at sections

Y=0 (cf. Fig.8.1). From top to bottom: Stairmand design and the new design re-

spectively. From left to right: the static pressure (N/m2), the tangential and axial

velocity (m/s). Note: both cyclones have the same barrel diameter and air volume

flow rate.

181


Distance from center (m)

Sta

ticpr

essu

re(N

/m2)

-0.1 -0.05 0 0.05 0.1

0

500

1000

1500

2000

2500New designStairmand design


Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1

0

0.5

1

1.5

2

2.5New designStairmand design


Axi

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-0.1 -0.05 0 0.05 0.1

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Sta

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/m2)

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/m2)

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2000



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nlet

velo

city

-0.1 -0.05 0 0.05 0.1-0.4

-0.2

0

0.2

0.4

0.6

0.8



at different sections on the X-Z plane (Y=0) at sections S1–S3 . From top to bottom:

section S1–S3. From left to right: time-averaged static pressure, tangential and

axial velocity respectively.

182



Sta

ticpr

essu

re(N

/m2)

-0.1 -0.05 0 0.05 0.1

0

500

1000

1500

2000



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velo

city

-0.1 -0.05 0 0.05 0.1

0

0.5

1

1.5

2



Axi

alve

loci

ty/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1-0.4

-0.2

0

0.2

0.4

0.6

0.8



Sta

ticpr

essu

re(N

/m2)

-0.1 -0.05 0 0.05 0.10

500

1000

1500

2000



Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1

0

0.5

1

1.5

2



Axi

alve

loci

ty/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1-0.4

-0.2

0

0.2

0.4

0.6

0.8



Sta

ticpr

essu

re(N

/m2)

-0.1 -0.05 0 0.05 0.10

500

1000

1500

2000



Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1

0

0.5

1

1.5

2



Axi

alve

loci

ty/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1

-0.2

0

0.2

0.4

0.6

0.8



at different sections on the X-Z plane (Y=0) at sections S4–S6 . From top to bottom:

section S4–S6. From left to right: time-averaged static pressure, tangential and

axial velocity respectively.

183



Section S1 S2 S3 S4 S5 S6

z`/D b 2.75 2.5 2.25 2.0 1.75 1.5

a Sections S1–S5 are located in the conical section, section S6 at the cylindrical part.b z` is measured from the top of the inlet section (cf. Fig. 8.1).

The velocity field

Based on the contour plots of the time–averaged tangential velocity, Fig. 8.15,

and the radial profiles at sections S1–S6 shown in Figs. 8.16 and 8.17, the

following conclusions can be drawn. The tangential velocity profile at any

section is composed of two regions, an inner and an outer one. In the

inner region, the flow rotates approximately like a solid body (forced vor-

tex), where the tangential velocity increases with radius. After reaching

its peak the velocity decreases with radius in the outer part of the profile

(free vortex). This profile is a so-called Rankine type vortex as mentioned

before, including a quasi-forced vortex in the central region and a quasi-

free vortex in the outer region. The maximum tangential velocity may

reach twice the average inlet velocity and occurs in the annular cylindrical

part. The tangential velocity distribution for the two cyclones are approx-

imately nearly identical in pattern and values (dimensionless), with the

highest velocity occurring at 1/4 of the cyclone radius for both cyclones.

This implies a nearly equal collection efficiency for both cyclones, as the

centrifugal force is the main driving force for particle collection in the cy-

clone separator. The axial velocity profiles for the two cyclones are also

very close, exhibiting the inverted W axial velocity profile [79]. Part of the

flow in the central region moves downward in the two cyclones. This phe-

nomena has been shown in the axial velocity pattern in other published

articles [e.g., 79, 159].

The DPM results


were injected from the inlet surface with a velocity equals the air inlet ve-

locity. The particle density ρp is 860 kg/m3 and the maximum number of

time steps for each injection was 200000 steps. The DPM analysis results

and the pressure drops for the two cyclones are depicted in Table 8.16. Al-

though, the difference between the two cyclone cut-off diameters is small,

the saving in the pressure drop is considerable (nearly 25% the value of

Stairmand cyclone).

Based on the flow pattern analysis and the DPM results, one can conclude

that the cyclone collection efficiency for the two cyclones is very close, with

184


Table 8.16: The pressure drop and the cut-off diameter for the two cyclones

∆p (N/m2) x50 (µm)

Stairmand design 1190 1.0New design 864 0.8

the advantage of low pressure drop in the new design. The authors want

to emphasis that only small changes in the geometrical dimensions of the

two designs lead to this improvement in the performance.

8.3.5 Conclusions

In order to predict accurately the complexly non linear relationships be-

tween pressure drop and geometrical dimensions, a radial basis neural

network (RBFNN) is developed and employed to model the pressure drop

for cyclone separators. The neural network has been trained and tested by

the experimental data available in literatures. The following conclusions

can be drawn from analysis of the obtained results:

• The result demonstrates that artificial neural networks can offer an

alternative and powerful approach to model the cyclone pressure drop.

• Four mathematical models (Muschelknautz method “MM”, Stairmand,

Ramachandran and Shepherd and Lapple) have been tested against

the experimental values. The residual error of MM model is the low-

est. Also, one can multiply the calculated value using MM by 1.5 to

get the experimental value (as a rough approximation)

• The analysis indicates the significant effect of the vortex finder diam-

eter Dx and the vortex finder length S, the inlet width b and the total

height Ht.

• The response surface methodology has been used to fit a second order

polynomial to the RBFNN.

• The second-order polynomial has been used to get a new optimized

cyclone for minimum pressure drop using Nelder-Mead technique.

• A comparison between the new design and the standard Stairmand

design has been performed using CFD simulation with the Reynolds

stress model to get a clear vision of the flow field pattern in the new

design.

• CFD results show that, the new cyclone design is very close to the

Stairmand high efficiency design in the geometrical parameter ratio,

and superior for low pressure drop at nearly the same cut-off diame-

ter.

• The new cyclone design results in nearly 75% of the pressure drop

obtained by the old Stairmand design at the same volume flow rate.

185


8.4 Single and multi-objective optimization us-

ing RBFNN and GA

Both the pressure drop and the cut-off diameter in a cyclone separator

can be decreased or increased by varying the cyclone dimensions. For an

accurate optimal design of a cyclone, it is quite necessary to use a reli-

able model for its performance parameters. Optimization of gas cyclone is,

indeed, a multi-objective optimization problem rather than a single objec-

tive optimization problem that has been considered so far in the literature

[53, 146]. Both the pressure drop and the collection efficiency in gas cy-

clones are important objective functions to be optimized simultaneously

in such a real-world complex multi-objective optimization problem [54].

These objective functions are either obtained from experiments, empirical

models or computed using very timely and high-cost computational fluid

dynamic (CFD) approaches. Modeling and optimization of the parameters

are investigated in the present study, by using radial basis function arti-

ficial neural networks and multi-objective genetic algorithm optimization

technique in order to maximize the collection efficiency (minimize the cut-

off diameter) and minimize the pressure drop.

The optimization studies conducted in the previous sections used the Nelder-

Mead technique [119] which suffer from two drawbacks, (1) the final solu-

tion may be affected by the start point, (2) the obtained optimum may be

a local minimum. To avoid this situation, the application of evolutionary

method like the genetic algorithm must.

Study objectives

There are four objectives of this study. (1) Investigation of the effect of the

seven geometrical parameters on the cyclone separator performance (the

pressure drop and cut-off diameter) based on the experimental data for

the pressure drop and the most robust mathematical models for the cut-off

diameter. (2) Study the possible interaction between the seven geometri-

cal parameters affecting the cyclone performance using response surface

methodology. (3) Multi-objective optimization to obtain new geometrical

ratios for optimum performance (minimum pressure drop and minimum

cut-off diameter). (4) Obtaining the optimum design (geometrical ratios)

of the cyclone separator for minimum pressure drop using the genetic al-

gorithm optimization technique, followed by a comparison of the numeri-

cal simulations of the optimal design and the Stairmand design using the

Reynolds stress turbulence model.

186

8.4. Multi-objective optimization using GA

8.4.1 Artificial neural network (ANN) approach

In this study, the radial basis function neural network (RBFNN) has been

used to model the effect of cyclone dimensions on the Euler number (pres-

sure drop). The details of this step have been explained in Sec. 8.3.1,

page 160.

The cut-off diameter (Stokes number)

The source of the training data for the cut-off diameter has been obtained

from the application of Iozia and Leith model [85]. This model has been

approved as an acceptable approach for calculating the cyclone cut-off di-

ameter [84, 85]. The cut-off diameter x50 is the particle diameter which

produces 50% collection efficiency.

A question may appear here, why the authors employed the cut-off diame-

ter instead of the collection efficiency, like in the study of Ravi et al. [141].

Firstly, for low mass loading cyclone separator, the cut-off diameter can

replace the collection efficiency, since one can fit the grade efficiency curve

using the cut-off diameter via some correlations, cf. Hoffmann and Stein

[77, page 91] for more details. Moreover, many models can predict well the

cut-off diameter but exhibit different grade efficiency curves [77, page 97].

Secondly, the cut-off diameter (instead of the collection efficiency) has been

used as an objective function in many recent publications [e.g., 50, 148].

Moreover, the selection of the cut-off diameter or the collection efficiency

for low mass loading cyclones can be considered as a researcher choice.

The Iozia and Leith model [84] is similar to the model of Barth [9] both

based on the equilibrium-orbit theory (Force balance). The cut-off diameter

can be calculated as [84]:

x50 =(9 µ C) /

(π HCS ρp V 2

θmax

)1/2(8.9)

where µ is the gas viscosity, Q is the gas volume flow rate, ρp is the particle

density. HCS is the core height (height of the control surface, Fig. 8.18),

Vθmax is the maximum tangential velocity, that occur at the edge of the

control surface CS. In this model, the value of core diameter dc, and the

tangential velocity at the core edge; Vθmax are calculated from regression

of experimental data using Eq. A.34.

Vθmax = 6.1Vin

(a b/D2

)−0.61(Dx/D)

−0.74(Ht/D)

−0.33

(8.10)

dc = 0.52D(a b/D2

)−0.25(Dx/D)

1.53(8.11)

187


Figure 8.18: The control surface (core edge) used in the Iozia and Leith model [85].

where Vin is the area-average inlet velocity. The following expression ob-

tained from trigonometry relations and can be used to calculate HCS .

HCS =(R− Dx

2 )(Ht − h)

R− (Bc2 )

+ (h− S) if Bc > Dx

= (Ht − S) if Bc ≤ Dx (8.12)

It is clear from this model that the most important geometrical parameters

that affect the cyclone cut-off diameter are the vortex finder diameter,the

ratio of inlet area to exit area, cyclone height.

Based on this model, the cut-off diameter is a function of the inlet gas ve-

locity (i.e., a function of both gas volume flow rate, Barrel diameter, inlet

section height and width), gas viscosity and particle density. For this par-

ticular study presented in Fig.8.20. The following values have been used:

Barrel diameter D =0.1 m, air flow rate =0.8333 l/s, air viscosity 1.0E-5 Pa

s and particle density 860 kg/m3. This means that the obtained results will

be valid for this particular case (used for just demonstration). However,

the authors believe the variation of the cut-off diameter due to variations

of cyclone geometrical dimensions is superior to the effect of these operat-

ing parameters, which is quite difficult to cover their range of operating

conditions.

The cut-off diameter x50 for cyclone separator is always given in units

of µm. Another way to represent x50 is using a dimensionless number;

Stokes number. The Stokes number based on the cut-off diameter; Stk50 =ρpx

250Vin/(18µD) [37]. It is the ratio between the particle relaxation time;

ρpx250/(18µ) and the gas flow integral time scale; D/Vin.

188


Table 8.17: Validation of the used RBFNN to model the Euler number∗

x y

Average 23.268 23.268Minimum 2.3 1.745

Maximum 155.3 155.985

Range 153.0 154.24

Correlation Coefficient, R 0.999

Mean squared error, E2 1.311E-4Intercept 0.017

Slope 0.999∗ x is the input to the RBFNN and y is the predicted value.

Both x and y represent the Euler number.

The values of R, E2, intercept and slope are that for the testing stage.

Fitting the ANNs

Tables 8.17 and 8.18 present more details about the validation of the used

RBFNNs. Both the average, standard deviation, minimum, maximum and

range of the input and the predicted values are given. It is clear from

Tables 8.17 and 8.18 that the ANNs preserved the descriptive statistical

parameters of the input data. The correlation coefficient between the input

and the output and the mean squared error are given for each RBFNN.

The intercept and the slope of the adjusted line between the input and the

predicted value of the ANN are also given.

The configured RBFNN predictions versus experimental data for the Euler

number and the Iozia and Leith mathematical model for the Euler number

and cut-off diameter are shown in Fig. 8.19. According to Fig. 8.19, it can

be seen that the ANN models are able to attain the high training accu-

racy. The training mean square errors for the experimental values and the

Iozia and Leith model have the values 1.311E-4 and 3.258E-4 respectively

(Fig. 8.19). This indicates that, compared with traditional models of curve

fitting, the models based on artificial intelligence algorithm have a supe-

rior capability of nonlinear fitting. Especially, the RBFNN has its unique

and optimal approximation characteristics in learning process [197].

Figure 8.19 illustrates the agreement between the ANNs input and out-

put. The obtained relation is a typical linear relation with a coefficient of

correlation close to 1 (R > 0.999). The agreement between the input and

output of the ANN is also clear from the value of the mean squared error

E2. That means, the trained neural networks predict very well both the

Euler number and cut-off diameter values and can be used in cyclone de-

sign and performance estimation. Tables 8.17, 8.18 and Fig. 8.19 present

different performance indicators as a validation of the proposed model for

experimental values.

189


Table 8.18: Validation of the used RBFNN to model the cut-off diameter∗

x y

Average 8.219 8.219Minimum 3.64 3.716

Maximum 15.3 15.405

Range 11.66 11.689

Correlation Coefficient, R 0.99915

Mean squared error, E2 3.258E-4Intercept 0.014

Slope 0.999∗ x is the input to the RBFNN and y is the predicted value.

Both x and y represent the cut-off diameter.

The values of R, E2 , intercept and slope are that for the testing stage.

Input value (x)

Pre

dict

edva

lue

(y)

40 80 120

40

80

120

Data pointLinear fit

y=0.999 x + 0.0167

R=0.99964

E2= 1.311E-4

(a) The Euler number

Input value (x)

Pre

dict

edva

lue

(y)

4 6 8 10 12 14

4

6

8

10

12

14

Data pointLinear fit

y=0.998 x + 0.014

R=0.99915

E2= 3.258E-4

(b) The cut-off diameter

Figure 8.19: Linear regression of the RBFNNs for the Euler number and the cut-off

diameter.

8.4.1.1 The effect of geometrical parameters on the cut-off diam-

eter based on RBFANN

The effect of the geometrical parameters on the Euler number based on

the trained RBFNN has been presented in Sec. 8.3.2, page 166. The study

acknowledge the significant effect of the vortex finder diameter Dx and

the vortex finder length S, the inlet width b and the total height Ht. Less

effect is due to the cylinder height h (for h > 2.5) and the inlet height a (for

a > 0.55).

The effects of the geometrical parameters on the cut-off diameter are de-

picted in Fig. 8.20. To study the effect of each parameter, the tested RBFNN

model has been used by varying one parameter at a time from its minimum

to maximum values of the available 98 dataset, while the other parameters

190


Dx, a, b, B c

h, S

Ht

X50

[mic

ron]

0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5 3 3.5

2 4 6 8 10

0

2

4

6

8

10

12

14

Dx

abSHt

hBc

Figure 8.20: The effect of geometrical parameters on the cut-off diameter based on

the Iozia and Leith model [85]. Note: The plotted curves are obtained for a test

case with the following settings, Barrel diameter=0.1 m, air flow rate=0.8333l/s,

air viscosity=1.0E-5 Pa s, particle density=860 kg/m3.

are kept constant at their mean values (cf. Table 8.7). Figure 8.20 indicates

the significant effect of the vortex finder diameter Dx and the vortex finder

length S, the inlet width b, the inlet height a and the total height Ht. Less

effect is due to the cylinder height h and the cone tip diameter Bc. More

analysis is given in Table 8.19.

The significant geometrical parameters on the cut-off diameter

(Stokes number) using the response surface methodology (RSM)

Table 8.20 represents the parameters ranges selected for the seven geo-

metrical parameters. The study was planned using Box–Behnken design,

with 64 combinations. A significant level of P < 0.05 (95% confidence) was

used in all tests. Analysis of variance (ANOVA) was followed by an F-test

of the individual factors and interactions [53].

191


Table 8.19: The variation of the cut-off diameter with cyclone dimensions using the

RBFNN model (cf., Fig. 8.20)

Factor Analysis

Dx

The vortex finder diameter has the most significant effect on

the cut-off diameter x50 (the highest slope in Fig. 8.20). The

slope is very high until Dx = 0.5 and any further increase in

Dx produces a small change in x50. In general, increasing Dx

increases x50 (decreasing the collection efficiency), this is one

of the main reasons of the trade-off between the Euler number

and the cut-off diameter objectives. This makes the

optimization of cyclone geometry a multi-objective procedure.

b

The variation of x50 with the inlet width is similar in trend

and significance to that for Dx but here the slope changes at

b = 0.25.

S and a

The effect of the vortex finder length and the inlet section

height on the cut-off diameter is almost paralleled up to

S = 1.5 and a = 0.6 afterwards they lose their significance and

become nearly constant.

h

Increasing the barrel height slightly decreases the cut-off

diameter with nearly linear relation. This trend has been

reported by other researchers using CFD simulations, e.g.,

Elsayed and Lacor [51].

Ht∗

The effect of the cyclone total height is basically due to two

effects the cone height and barrel height. The curve can be

subdivided into four main regions. Sharp decrease in x50 up to

Ht = 3.25, no valuable difference between 3.25 and 5.25, sharp

increase between 5.25 and 8, and insignificant effect beyond 8.

Bc

The effect of the cone-tip diameter on the cut-off diameter is

quite small. First, increasing the cone-tip diameter slightly

decreases the cut-off diameter up to Bc = 0.55 and any further

increment increases the cut-off diameter. This trend has been

reported by other researchers, e.g., Elsayed and Lacor [46, 55].

∗ Due to the interaction between the geometrical parameters, especially between Ht with S and Dx (cf.,Fig. 8.22 and Table 8.21), the obtained conclusions may not be applicable generally and the application ofresponse surface methodology to analysis the effect of each particular parameter must.

192


Table 8.20: The values of the independent variables used in the design of experi-

ment


Inlet height, a=X1 0.4 0.55 0.7Inlet width, b=X2 0.14 0.27 0.4Cone tip diameter, Bc=X3 0.2 0.3 0.4Vortex finder diameter, Dx=X4 0.2 0.475 0.75Barrel height, h=X5 1.0 1.5 2.0Total cyclone height, Ht=X6 3.0 5.0 7.0Vortex finder length, S=X7 0.4 1.2 2.0

Analysis of variance (ANOVA)

Analysis of variance (ANOVA) showed that the resultant quadratic polyno-

mial models adequately represented the experimental data with the coef-

ficient of multiple determination R2 being 0.984099 (cf., Table 8.21). This

indicates that the quadratic polynomial model obtained was adequate to

describe the influence of the independent variables studied [189]. Analy-

sis of variance (ANOVA) was used to evaluate the significance of the coef-

ficients of the quadratic polynomial models (see Table 8.21). For any of the

terms in the models, a large F-value (small P-value) would indicate a more

significant effect on the respective response variables [50, 53].

Based on the ANOVA results presented in Table 8.21, the variable with the

largest effect on the Stokes number (cut-off diameter) was the linear term

of vortex finder diameter, followed by the linear term of the cyclone total

height, the vortex finder length and the inlet width (P < 0.05); the other

three linear terms (inlet height, barrel height, and cone tip diameter) did

not show a significant effect (P > 0.05). The quadratic term of cyclone

total height, vortex finder diameter and vortex finder length also had a

significant effect on the pressure drop; however, the effect of the other four

quadratic terms was insignificant. Furthermore, the interaction between

Dx with (Ht, S, b) and between S with (h, Ht) also had a significant ef-

fect on the Stokes number, while the effect of the remaining terms was

insignificant.

Analysis of response surfaces



is proportional to the size of the effect and the direction of the curve speci-

fies a positive or negative influence of the effect [50, 61], Fig.8.21(a). Based

on the main effect plot, the most significant factors on the Stokes number

are: (1) the vortex finder diameter Dx, with a second–order curve of di-

rect relation. (2) the cyclone total height Ht inversely related to the Stokes

number. (3) the vortex finder length S with direct relationship. (4) the inlet

193


dimensions width b and height a inversely related to the Stokes number.

Whereas the other factors have an insignificant effect. The main effect plot

supports the analysis given in Table 8.19, except for Ht where the strong

interaction between the cyclone total height and the vortex finder length

affected the trend given in Fig. 8.20.

Pareto charts were used to summarize graphically and display the relative

importance of each parameter with respect to the Stokes number [50]. The

Pareto chart shows all the linear and second-order effects of the parame-

ters within the model and estimates the significance of each with respect

to maximizing the Stokes number response. A Pareto chart displays a

frequency histogram with the length of each bar proportional to each es-

timated standardized effect [30]. The vertical line on the Pareto charts

judges whether each effect is statistically significant within the generated

response surface model; bars that extend beyond this line represent ef-

fects that are statistically significant at a 95% confidence level. Based on

the Pareto chart (Fig. 8.21(b)) and ANOVA table (Table 8.21) there are five

significant parameters at a 95% confidence level: the vortex finder diame-

ter Dx, the total cyclone height Ht, the vortex finder length S and the inlet

dimensions a and b. Therefore, the Pareto chart is a perfect supplement to

the main effect plot.




while holding the other factors at their central values (cf., Table 8.20) [189]

as shown in Fig. 8.22. Thus, Fig. 8.22(a) was generated by varying the

total height Ht and the vortex finder length S while keeping the other

five factors constant. The response surface plots presented in Fig. 8.22

illustrate the strong interactions between Ht with (S and Dx) and Dx with

(S and b).

8.4.2 Single objective optimization using GA

The genetic algorithm optimization technique has been applied to obtain

the geometrical ratios for minimum pressure drop (Euler number). The

objective function is the Euler number (using the trained radial basis func-

tion neural network presented in Sec. 8.3.1.1, page 163). The design vari-

ables are the seven geometrical dimensions of the cyclone separator.

194



(b) Pareto chart. A=a, B=b, C=Bc, D=Dx, E=h, F=Ht, G=S,

FG=Ht*S, etc.

Figure 8.21: Analysis of design of experiment for the Stokes number

195


(a) Ht versus S

(b) Dx versus Ht

(c) Dx versus S

(d) b versus Dx

Figure 8.22: The response surface plots for the Stokes number. Note: the stokes

number values are multiplied by 1000.

196


GA settings

Table 8.22 presents the settings used to obtain the optimum design for

minimum pressure using global optimization Matlab toolbox (Matlab 2010a

commercial package). The evolution of the cost function for the best indi-

vidual is given in Fig. 8.23. After 70 iterations (14400 function counts), the

Matlab code stops generating a new population since the average change

in the fitness value becomes less than 1E-6. The total calculation time for

this optimization problem was 102 seconds. Table 8.23 gives the optimum

values for cyclone geometrical parameters for minimum pressure drop es-

timated by the artificial neural network using the genetic algorithm op-

timization technique. It is clear from Table 8.23 that the new optimized

design is very close to the Stairmand design in many geometrical param-

eters, whereas the new ratios will result in minimum pressure drop. To

understand the effect of this small change in the geometrical ratios on the

flow field pattern and performance, a CFD study for the two designs is

needed [53].

Generation

Fitn

ess

valu

e

10 20 30 40 50 60 704

4.5

5

5.5

6

6.5

7

7.5

8

Figure 8.23: Evolution of the cost function for the best individual

8.4.2.1 Comparison between the two cyclone designs using CFD

Numerical settings


for Stairmand design is 19 m/s and 16 m/s for the new design), air density

197


1.0 kg/m3 and dynamic viscosity of 2.11E-5 Pa s. the turbulent intensity



and wall boundary condition at all other boundaries [53].







with a time step of 0.0001 s.

The grid refinement study using different levels of grid shows that a total

number of 134759 hexahedral cells for the Stairmand cyclone and 378963

hexahedral cells for the new design are sufficient to obtain a grid indepen-

dent solution, and further mesh refinement yields insignificant changes in

the numerical solution. The hexahedral meshes have been obtained using

the GAMBIT commercial software. These simulations were performed on

an 8 nodes CPU Opteron 64 Linux cluster using Fluent 6.3.26 commer-

cial software. The geometrical values for the two cyclones are given in

Table 8.24.


The pressure field

Figure 8.24 shows the contour plot at Y=0. In the two cyclones, the time-

averaged static pressure decreases radially from the wall to center. A neg-

ative pressure zone appears in the forced vortex region (central region)

due to high swirling velocity. The pressure gradient is largest along the

radial direction, while the gradient in the axial direction is very limited.

The cyclonic flow is not symmetrical as is clear from the shape of the low-

pressure zone at the cyclone center (twisted cylinder). However, the two

cyclones have almost the same flow pattern, but the highest pressure of

the Stairmand design is nearly 1.5 times that of the new design, implying

that the new design has a lower pressure drop.

The pressure distributions presented in Figs. 8.25 and 8.26 of the two cy-


vortex). Again, the highest static pressure for Stairmand design is more

than 1.5 times that of the new design at all sections while the central value

is almost the same for the two cyclones irrespective of the section location.

198


Figure 8.24: The contour plots for the time averaged flow variables at sections Y=0.

From top to bottom: Stairmand design and the new design respectively. From left

to right: the static pressure (N/m2), the tangential and axial velocity (m/s). Note:

both cyclones have the same barrel diameter and air volume flow rate.

199



Sta

ticpr

essu

re(N

/m2 )

-0.1 -0.05 0 0.05 0.10

500

1000

1500

2000



Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1

0

0.5

1

1.5

2



Axi

alve

loci

ty/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5



Sta

ticpr

essu

re(N

/m2 )

-0.1 -0.05 0 0.05 0.10

500

1000

1500

2000



Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1

0

0.5

1

1.5

2



Axi

alve

loci

ty/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5



Sta

ticpr

essu

re(N

/m2 )

-0.1 -0.05 0 0.05 0.10

500

1000

1500

2000



Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1

0

0.5

1

1.5

2



Axi

alve

loci

ty/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5



at different sections on the X-Z plane (Y=0) at sections S1–S3 (cf., Table 8.25). From

top to bottom: section S1–S3. From left to right: time-averaged static pressure,

tangential and axial velocity respectively.

This indicates that, the new design has a lower pressure drop with respect

to the Stairmand design.

The velocity field

Based on the contour plots of the time–averaged tangential velocity pre-

sented in Fig. 8.24, and the radial profiles at sections S1–S6 shown in

Figs. 8.25 and 8.26, the following conclusions can be drawn. The tangen-

tial velocity profile at any section is composed of two regions, an inner and

an outer one. In the inner region, the flow rotates approximately like a

solid body (forced vortex), where the tangential velocity increases with ra-

dius. After reaching its peak the velocity decreases with radius in the outer

200



Sta

ticpr

essu

re(N

/m2 )

-0.1 -0.05 0 0.05 0.10

500

1000

1500

2000



Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1

0

0.5

1

1.5

2



Axi

alve

loci

ty/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5



Sta

ticpr

essu

re(N

/m2 )

-0.1 -0.05 0 0.05 0.10

500

1000

1500

2000



Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1

0

0.5

1

1.5

2



Axi

alve

loci

ty/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5



Sta

ticpr

essu

re(N

/m2 )

-0.1 -0.05 0 0.05 0.10

500

1000

1500

2000



Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1

0

0.5

1

1.5

2



Axi

alve

loci

ty/I

nlet

velo

city

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5



at different sections on the X-Z plane (Y=0) at sections S4–S6 (cf., Table 8.25). From

top to bottom: section S4–S6. From left to right: time-averaged static pressure,

tangential and axial velocity respectively.

part of the profile (free vortex). This profile is a so-called Rankine type vor-

tex. The maximum tangential velocity may reach twice the average inlet

velocity and occurs in the annular cylindrical part. The tangential veloc-

ity distributions for the two cyclones are approximately nearly identical in

pattern and values (dimensionless), with the highest velocity occurring at

1/4 of the cyclone radius for both cyclones. This implies a nearly equal col-

lection efficiency for both cyclones, as the centrifugal force is the main driv-

ing force for particle collection in the cyclone separator. The axial velocity

profiles for the two cyclones are also very close, exhibiting the inverted W

axial velocity profile.

201



Col

lect

ion

effic

ienc

y[-]

10-1 100 101

0.25

0.5

0.75

1Stairmand designNew design

Figure 8.27: The grade efficiency curves for the two designs

The DPM results


were injected from the inlet surface with a velocity equals the air inlet ve-

locity and particle density ρp = 860kg/m3. The grade efficiency curves for

the two designs are plotted in Fig. 8.27. The DPM analysis results and the

pressure drops for the two cyclones are depicted in Table 8.26. Although,

the difference between the two cyclone cut-off diameters is small, the sav-

ing in pressure drop is considerable (nearly 32.5% the value of Stairmand

cyclone).

Based on the flow pattern analysis and the DPM results, one can conclude

that the cyclone collection efficiency for the two cyclones is very close, with

the advantage of low pressure drop in the new design. The authors want

to emphasis that only small changes in the geometrical dimensions of the

two designs lead to this improvement in the performance.

8.4.3 Optimal cyclone design for best performance

NSGA-II settings

Table 8.27 presents the genetic operators and parameters for multi-objective

optimization. The Euler number values have been obtained from the arti-

ficial neural network trained by experimental values. The Stokes number

values are obtained from Iozia and Leith model [85]. In order to investi-

gate the effect of different geometrical and operational parameters on the

Pareto front, sixteen test cases with different barrel diameter, gas flow rate

and particle density have been tested, cf. Table 8.28. The sixteen test cases

covers: 1) Two barrel diameters, 31 mm and 205 mm. 2) Four levels of air

flow rates, 50, 60, 70 and 80 l/min. 3) Five values of particle density, 860,

202


1000, 1500, 1750 and 2000 kg/m3.

Pareto front

The Pareto front (non dominated points) for the sixteen test cases are pre-

sented in Fig. 8.28(a). Figure 8.28(a) clearly demonstrate tradeoffs in ob-

jective functions Euler number and Stokes number from which an appro-

priate design can be compromisingly chosen by the designer [148]. All the

optimum design points in the Pareto front are non-dominated and could be

chosen by a designer as optimum cyclone separator [148]. The correspond-

ing geometrical ratios of the Pareto front shown in Fig. 8.28(a) are given

in Table 8.29 for test case 1 and Table 8.30 for test case 9. Three points

A, B and C are indicated in Figs. 8.28(c) and 8.28(c) and Tables 8.29 and

8.30. Point A indicates the point of minimum Euler number and maximum

Stokes number. Point B indicates the point of maximum Euler number and

minimum Stokes number. Point C indicates an optimal point for the multi-

objective optimization problem.

In order to obtain the Euler number- Stokes number relationship, Fig. 8.28(b)


two dimensionless numbers irrespective to the barrel diameter, gas flow

rate, particle density. A second-order polynomial has been fitted between

the logarithms of Euler number and Stokes number, Eq. 8.13. The ob-

tained correlation can fit the data with a coefficient of correlation R2 =0.98643 as shown in Fig. 8.28(b).

Stk50 = 100.3016(log10(Eu))2−0.9479log10(Eu)−2.5154 (8.13)

Bubble plots for Pareto front

For visual inspection of the effect of the seven geometrical parameters on

the two conflicting performance parameters, the bubble plots on Pareto

front points have been drawn for each geometrical parameter. However,

only figures for test case 1 (Fig. 8.29) and 9 (Fig. 8.30) are presented, but

all other cases depict the same results (trend).

Figure 8.29 indicates that: a) Decreasing the vortex finder diameter Dx de-

creases the Stokes number and increases the Euler number, Fig. 8.29(b).

b) Generally speaking, increasing the inlet height a increases the Euler

number and decreases the Stokes number. c) A similar trend is exhibited

203


by the inlet width b but due to interaction with other geometrical and op-

erational variables, one could see a range of bubble sizes in the region of

best performance (lower values for both the Euler and Stokes numbers).

d) The higher values of total cyclone height Ht will produce less Stokes

number, intermediate values could produce less Euler number, smaller-

intermediate values could produce the optimum performance due to inter-

action with other variables. e) Short barrels will produce better collection

efficiency (low Stokes number) and higher Euler numbers. Intermediate

values results in low Euler number values. Long barrels can produce the

best performance. f) Short vortex finder may produce higher values of Eu-

ler numbers or higher values of Stokes number due to strong interaction

with other variables. Long vortex finder can produce the optimum perfor-

mance. g) Generally speaking, the variation of the cone-tip diameter Bc

has no effect on the performance parameter. The above comments is re-

stricted to the range of each geometrical variables located on the Pareto

front and not for the whole range of values (cf., Fig. 8.29 for the range of

each geometrical parameters).

8.4.4 Conclusions

To predict the complex non-linear relationships between the performance

parameters and the geometrical dimensions, two radial basis neural net-

works (RBFNNs) are developed and employed to model the Euler number

and Stokes number for cyclone separators. The neural networks have been

trained and tested by the experimental data available in literature for Eu-

ler number (pressure drop) and Iozia and Leith model [85] for the Stokes

number (cut-off diameter). The effects of the seven geometrical parame-

ters on the Stokes number have been investigated using the trained ANN.

To declare any interaction between the geometrical parameters affecting

the Stokes number, the response surface methodology has been applied.

The trained ANN has been used as an objective function to obtain the

cyclone geometrical ratios for minimum Euler number using the genetic

algorithms optimization technique. A CFD comparison between the new

optimal design and the Stairmand design using the Reynolds stress turbu-

lence model has been performed. A multi-objective optimization technique

using NSGA-II technique has been applied to determine the Pareto front

for the best performance cyclone separator.

The following conclusions can be drawn from analysis of the obtained re-

sults:

204



alternative and powerful approach to model the cyclone performance.

• The analysis indicates the significant effect of the vortex finder di-

ameter Dx and the vortex finder length S, the inlet width b, the inlet

height a and the total height Ht on the cyclone performance.

• The response surface methodology has been used to fit a second-order

polynomial to the RBFNN for the cut-off diameter. The analysis of

variance of the cut-off diameter indicates a strong interaction be-

tween Dx with (Ht, S, b) and between S with (h, Ht).

• The trained RBFNN has been used to get a new optimized cyclone for

minimum pressure drop (Euler number) using the genetic algorithm

optimization technique.


design has been performed using CFD simulation with the Reynolds

stress turbulence model to get a clear vision of the flow field pattern

and performance in the new design.

• CFD results shows that, the new cyclone design are very close to the

Stairmand high efficiency design in the geometrical parameter ratio,

and superior in low pressure drop at nearly the same cut-off diameter.


obtained by the old Stairmand design at the same volume flow rate.

This confirms that the obtained design using the genetic algorithm is

better than that obtained using Nelder-Mead technique which results

in 75% of the Stairmand pressure drop [53].

• The two trained RBFNNs have been used in a multi-objective opti-

mization process using NSGA-II technique. Sixteen test cases with

different barrel diameter, gas flow rate and particle density have

been tested and plotted. The Pareto fronts for the 16 test cases are

very close. A second-order polynomial has been fitted between the

logarithms of Euler number and Stokes number to obtain a general

formula, Stk50 = 100.3016(log10(Eu))2−0.9479 log10(Eu)−2.5154 with a coeffi-

cient of correlation R2 = 0.98643. This formula can be used to obtain

the Stokes number if the Euler number is known.

205


Table 8.21: Analysis of variance and the regression coefficients of the fitted

quadratic equation for the Stokes numbera

Source Regression coefficient F-Ratio P-Value

β0 -0.0470554Linearβ1 0.942933 6.65 0.0154β2 -1.3178 24.12 0.0000β3 2.10188 0 1β4 8.3493 1145.05 0.0000β5 -0.843633 0 1β6 -0.527695 243.98 0.0000β7 1.46453 137.75 0.0000Quadraticβ11 -1.17696 0.21 0.6489β22 1.44904 0.18 0.6737β33 -3.50314 0.37 0.5476β44 2.33086 9.38 0.0048β55 0.281211 1.49 0.232β66 0 39.06 0.0000β77 0.243505 7.33 0.0114Interactionβ12 0.726218 0.03 0.8558β13 0 0 1β14 -2.09219 1.25 0.2732β15 0 0 1β16 0.121646 0.22 0.6401β17 -0.187708 0.09 0.7726β23 0 0 1β24 -4.84457 5.03 0.0330β25 0 0 1β26 0.262495 0.78 0.3842β27 -0.466053 0.39 0.5352β34 0 0 1β35 0 0 1β36 0 0 1β37 0 0 1β45 0 0 1β46 -0.869946 38.41 0.0000β47 1.44156 16.87 0.0003β56 0 0 1β57 0 39.06 0.0000β67 -0.370128 58.83 0.0000

R2 0.984099

a Bold numbers indicate significant factors as identified by the analysis of variance (ANOVA) at the 95%confidence level.

Table 8.22: Genetic operators and parameters for single objective optimization

Population type: Double vectorInitial range: [0.2 0.1 0.1 0.3 2.0 0.65 0.05 ; 0.8 0.8 0.6 0.8 8.0 2.5 0.75]Fitness scaling: RankSelection operation: Tournament (tournament size equals 4)Elite count: 2Crossover fraction: 0.8Crossover operation: Intermediate crossover with the default value of 1.0Mutation operation: The constraint dependent defaultMaximum number of generations (iterations): 1400Population size: 200

206


Table 8.23: The optimized cyclone separator design for minimum pressure drop

Factor Low High Stairmand design Optimum design

Dx 0.2 0.75 0.5 0.549a 0.5 0.75 0.5 0.595b 0.14 0.4 0.2 0.201S 0.4 2.0 0.5 0.595Ht 3.0 7.0 4.0 4.549h 1.0 2.0 1.5 1.411Bc 0.2 0.4 0.375 0.275

Table 8.24: The geometrical parameters for the two designs (D=0.205 m)∗

Cyclone a/D b/D Dx/D Ht/D h/D S/D Bc/DStairmand design 0.5 0.2 0.5 4 1.5 0.5 0.375New design 0.595 0.201 0.549 4.549 1.411 0.595 0.275

∗ The outlet section is above the cyclone surface by Le = 0.618D. The inlet section located at adistance Li = D from the cyclone center.


Section S1 S2 S3 S4 S5 S6

z`/D b 2.75 2.5 2.25 2.0 1.75 1.5

a Sections S1–S5 are located in the conical section, section S6 at the cylindrical part.b z` is measured from the top of the inlet section.

Table 8.26: The performance parameters for the two cyclones

Design Method ∆p (N/m2) Euler number x50 (µm) Stokes number x 103

CFD 803 6.338 0.804 0.114

New design ANN 584.4 4.613 2.938 1.815

Ramachandran model [139] 877.98 5.523

CFD 1190 6.567 1.0 0.209

Stairmand design ANN 1015.8 5.606 3.314 1.931Ramachandran model [139] 699.66 4.846

Table 8.27: Genetic operators and parameters for multi-objective optimization

Population type: Double vectorPopulation size: 105 (i.e., 15* number of variables which is the default in Matlab)Initial range: [0.2 0.1 0.1 0.3 2.0 0.65 0.05 ; 0.8 0.8 0.6 0.8 8.0 2.5 0.75]Selection operation: tournament (tournament size equals 2)Crossover fraction: 0.8Crowding distance fraction 0.35Crossover operation: Intermediate crossover with the default value of 1.0Number of generations (iterations): 1400 (i.e., 200* number of variables which is the default in Matlab)

207


Table 8.28: The diameters, air flow rates and the particle densities for the sixteen

test cases

Case D [mm] Q [l/min] ρp [kg/m3]

1 205 50 860

2 205 60 860

3 205 70 860

4 205 80 860

5 205 50 1000

6 205 50 1500

7 205 50 1750

8 205 50 2000

9 31 50 860

10 31 60 860

11 31 70 860

12 31 80 860

13 31 50 1000

14 31 50 1500

15 31 50 1750

16 31 50 2000

208


Table 8.29: The seven geometrical parameters and the obtained Euler number and

Stokes number for the nondominated points (Pareto-front) for test case 1 (cf. Table

8.28)

point Dx a b S Ht h Bc Euler number Stokes number x 103

1 0.306 0.659 0.385 0.410 6.957 1.779 0.387 23.843 0.552

2B 0.306 0.688 0.398 0.404 6.993 1.779 0.317 27.322 0.540

3 0.618 0.229 0.213 0.411 6.774 1.885 0.495 1.026 2.785

4 0.326 0.331 0.240 0.451 6.668 1.885 0.444 6.485 0.798

5C 0.360 0.295 0.253 0.443 6.664 1.910 0.459 4.892 0.937

6 0.670 0.230 0.213 0.410 6.777 1.859 0.496 0.879 3.478

7 0.585 0.226 0.217 0.419 6.763 1.901 0.492 1.159 2.089

8 0.306 0.685 0.299 0.401 6.985 1.789 0.425 18.474 0.576

9 0.306 0.666 0.361 0.407 6.965 1.782 0.398 22.300 0.55810 0.305 0.516 0.271 0.439 6.840 1.868 0.428 12.977 0.633

11 0.303 0.286 0.318 0.449 6.611 1.930 0.462 8.404 0.699

12 0.476 0.240 0.217 0.434 6.743 1.901 0.471 1.915 1.524

13 0.312 0.622 0.277 0.419 6.943 1.909 0.429 14.816 0.619

14 0.308 0.686 0.379 0.427 6.974 1.781 0.459 23.145 0.55415 0.592 0.229 0.213 0.413 6.831 1.895 0.494 1.117 2.491

16 0.320 0.570 0.273 0.423 6.898 1.909 0.434 12.767 0.658

17 0.598 0.229 0.213 0.412 6.770 1.893 0.494 1.095 2.565

18 0.309 0.441 0.266 0.450 6.769 1.881 0.435 10.629 0.674

19 0.430 0.307 0.231 0.438 6.735 1.898 0.465 3.211 1.22920 0.306 0.667 0.389 0.417 6.950 1.779 0.367 24.941 0.550

21 0.306 0.392 0.277 0.441 6.708 1.882 0.443 9.963 0.676

22 0.326 0.331 0.209 0.451 6.668 1.900 0.471 5.515 0.823

23 0.308 0.686 0.348 0.427 6.974 1.807 0.459 21.132 0.564

24 0.680 0.229 0.213 0.409 6.787 1.857 0.495 0.849 3.640

25 0.307 0.653 0.296 0.421 6.972 1.816 0.420 17.606 0.58826 0.307 0.368 0.338 0.439 6.708 1.895 0.430 11.399 0.659

27 0.514 0.235 0.217 0.429 6.709 1.901 0.479 1.596 1.718

28 0.310 0.596 0.275 0.424 6.918 1.899 0.429 14.387 0.622

29 0.559 0.246 0.221 0.453 6.785 1.898 0.455 1.486 1.915

30 0.399 0.308 0.229 0.438 6.707 1.891 0.460 3.732 1.10231 0.516 0.244 0.241 0.422 6.728 1.904 0.486 1.797 1.674

32 0.413 0.256 0.280 0.444 6.667 1.927 0.472 3.510 1.162

33 0.306 0.507 0.352 0.437 6.811 1.848 0.460 16.110 0.602

34 0.307 0.644 0.333 0.423 6.960 1.807 0.401 19.929 0.575

35 0.646 0.229 0.219 0.409 6.797 1.863 0.497 0.963 3.113

36A 0.692 0.228 0.213 0.408 6.819 1.855 0.498 0.815 3.805

37 0.558 0.216 0.230 0.439 6.802 1.922 0.486 1.311 1.945

Minimum 0.303 0.216 0.209 0.401 6.611 1.779 0.317 0.815 0.540

Maximum 0.692 0.688 0.398 0.453 6.993 1.930 0.498 27.322 3.805

A indicates the point of minimum Euler number and maximum Stokes number. B indicates the point ofmaximum Euler number and minimum Stokes number. C indicates an optimal point for themulti-objective optimization problem. (cf. Fig. 8.28(c)).

209



Stokes number for the nondominated points (Pareto-front) for test case 9 (cf. Table

8.28)

point Dx a b S Ht h Bc Euler number Stokes number x 103

1A 0.686 0.236 0.236 0.478 6.910 1.915 0.473 1.008 3.743

2 0.308 0.655 0.390 0.423 6.902 1.995 0.471 21.688 0.5583 0.308 0.692 0.390 0.423 6.902 1.995 0.472 22.898 0.551

4 0.585 0.235 0.235 0.418 6.925 1.727 0.471 1.355 2.436

5 0.314 0.330 0.236 0.431 6.837 1.864 0.470 6.568 0.750

6 0.309 0.486 0.335 0.426 6.869 1.931 0.471 13.947 0.621

7 0.578 0.237 0.235 0.421 6.684 1.975 0.471 1.356 2.0088 0.309 0.272 0.347 0.424 6.897 1.967 0.472 8.025 0.697

9 0.310 0.538 0.369 0.429 6.882 1.984 0.467 16.799 0.597

10 0.473 0.238 0.235 0.478 6.923 1.971 0.471 2.105 1.481

11 0.308 0.375 0.382 0.423 6.901 1.989 0.471 12.168 0.634

12 0.432 0.238 0.236 0.445 6.785 1.947 0.472 2.494 1.30013 0.321 0.327 0.236 0.431 6.839 1.862 0.470 6.223 0.777

14 0.308 0.606 0.383 0.424 6.896 1.990 0.471 19.736 0.571

15 0.308 0.643 0.381 0.424 6.895 1.986 0.472 20.775 0.565

16 0.407 0.255 0.240 0.470 6.899 1.907 0.472 3.119 1.165

17 0.309 0.333 0.353 0.424 6.882 1.958 0.471 10.021 0.665

18 0.452 0.236 0.236 0.448 6.793 1.945 0.472 2.255 1.39419 0.678 0.236 0.236 0.473 6.893 1.919 0.473 1.029 3.615

20 0.625 0.237 0.236 0.449 6.800 1.944 0.472 1.187 2.891

21 0.309 0.596 0.379 0.425 6.894 1.988 0.471 19.039 0.578

22 0.313 0.325 0.262 0.430 6.847 1.884 0.471 7.197 0.730

23 0.664 0.237 0.236 0.476 6.911 1.914 0.473 1.076 3.40424 0.309 0.365 0.345 0.425 6.876 1.946 0.471 10.740 0.656

25 0.608 0.235 0.235 0.438 6.921 1.797 0.472 1.261 2.691

26B 0.308 0.692 0.390 0.423 6.902 1.995 0.472 22.898 0.551

27 0.309 0.436 0.381 0.424 6.899 1.989 0.464 14.120 0.616

28 0.308 0.427 0.360 0.424 6.886 1.962 0.471 13.112 0.626

29C 0.330 0.290 0.252 0.465 6.890 1.955 0.471 5.639 0.818

30 0.658 0.245 0.239 0.477 6.910 1.924 0.473 1.146 3.290

31 0.308 0.362 0.365 0.424 6.883 1.668 0.467 11.924 0.64832 0.520 0.251 0.244 0.470 6.906 1.684 0.472 1.996 1.671

33 0.631 0.236 0.236 0.449 6.796 1.945 0.472 1.164 2.964

34 0.549 0.242 0.240 0.430 6.909 1.783 0.471 1.613 1.823

35 0.316 0.270 0.341 0.425 6.891 1.966 0.472 7.482 0.726

36 0.686 0.236 0.236 0.478 6.910 1.915 0.473 1.008 3.74337 0.380 0.239 0.237 0.477 6.890 1.922 0.473 3.327 1.071

Minimum 0.308 0.235 0.235 0.418 6.684 1.668 0.464 1.008 0.551Maximum 0.686 0.692 0.390 0.478 6.925 1.995 0.473 22.898 3.743

A indicates the point of minimum Euler number and maximum Stokes number. B indicates the point ofmaximum Euler number and minimum Stokes number. C indicates an optimal point for themulti-objective optimization problem. (cf. Fig. 8.28(d)).

210


Euler number

Sto

kes

num

berx

10

3

0 5 10 15 20 25

0.5

1

1.5

2

2.5

3

3.5

4

Case 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9Case 10Case 11Case 12Case 13Case 14Case 15Case 16

(a) Pareto fronts for 16 test cases,

linear scale

Euler number

Sto

kes

num

berx

10

3

5 10 15 20 25 30350.5

1

1.5

2

2.5

3

3.5

44.5

55.5

66.5

7

Pareto pointsStk50=10 0.3016[LOG10(Eu)] 2-0.9479 LOG10(Eu)-2.5154

(b) Pareto fronts for 16 test cases, log

scale

Euler number

Sto

kes

num

berx

103

5 10 15 20 25

0.5

1

1.5

2

2.5

3

3.5

4A

BC

(c) Pareto front for test case 1

Euler number

Sto

kes

num

berx

103

5 10 15 20 25

0.5

1

1.5

2

2.5

3

3.5

4A

BC

(d) Pareto chart for test case 9

Figure 8.28: Pareto fronts for different test cases.

211

Ch

ap

ter

8.

Op

timiz

atio

n

(a) Dx, range: 0.303 - 0.692 (b) a, range: 0.216 - 0.688 (c) b, range: 0.209 - 0.398

(d) Ht, range: 6.611 - 6.993 (e) h, range: 1.779 - 1.930 (f) S, range: 0.401 - 0.453

Figure 8.29: Bubble plots for different geometrical parameters for test case 1 (cf. Fig. 8.28(c) and Table 8.29).

212

8.4

.M

ulti-o

bje

ctive

op

timiz

atio

nu

sing

GA

(a) Dx, range: 0.308 - 0.686 (b) a, range: 0.235 - 0.692 (c) b, range: 0.235 - 0.390

(d) Ht, range: 6.684 - 6.925 (e) h, range: 1.668 - 1.995 (f) S, range: 0.418 - 0.478

Figure 8.30: Bubble plots for different geometrical parameters for test case 9 (cf. Fig. 8.28(d) and Table 8.30).

213



Stokes number for the three points, A, B and C for the 16 test cases (cf. Table 8.28)

Point Dx a b S Ht h Bc Euler number Stokes number x 103

A1 0.692 0.228 0.213 0.408 6.819 1.855 0.498 0.815 3.805

B1 0.306 0.688 0.398 0.404 6.993 1.779 0.317 27.322 0.540

C1 0.360 0.295 0.253 0.443 6.664 1.910 0.459 4.892 0.937

A2 0.700 0.204 0.227 0.446 6.422 1.811 0.470 0.820 4.260

B2 0.301 0.697 0.399 0.561 6.989 1.840 0.449 28.222 0.538

C2 0.415 0.414 0.287 0.449 6.803 1.776 0.443 6.052 1.037A3 0.676 0.209 0.204 0.431 4.378 1.980 0.496 0.868 4.108

B3 0.304 0.624 0.398 0.581 6.983 1.116 0.344 32.296 0.561

C3 0.363 0.278 0.230 0.459 6.404 1.354 0.445 4.791 1.000

A4 0.671 0.205 0.212 0.427 6.524 1.824 0.467 0.819 3.796

B2 0.303 0.622 0.399 0.467 6.767 1.975 0.479 22.621 0.556C4 0.347 0.237 0.271 0.582 6.769 1.913 0.476 4.873 0.931

A5 0.669 0.224 0.204 0.537 6.946 1.895 0.479 0.900 3.654

B5 0.301 0.672 0.387 0.416 6.851 1.905 0.463 23.579 0.538

C5 0.343 0.228 0.282 0.475 6.925 1.822 0.487 4.647 0.893

A6 0.697 0.207 0.207 0.400 6.734 1.974 0.465 0.706 4.142B6 0.301 0.671 0.400 0.500 6.982 1.228 0.477 29.367 0.538

C6 0.380 0.241 0.303 0.486 6.715 1.845 0.427 4.557 1.026

A7 0.660 0.214 0.205 0.432 6.962 1.934 0.495 0.810 3.391

B7 0.301 0.690 0.322 0.459 6.988 1.918 0.450 20.739 0.557

C7 0.401 0.350 0.258 0.451 6.928 1.920 0.454 4.735 1.039

A8 0.660 0.214 0.205 0.432 6.962 1.934 0.495 0.810 3.391B8 0.301 0.690 0.322 0.459 6.988 1.918 0.450 20.739 0.557

C8 0.333 0.278 0.263 0.457 6.869 1.940 0.450 5.582 0.832

A9 0.686 0.236 0.236 0.478 6.910 1.915 0.473 1.008 3.743

B9 0.308 0.692 0.390 0.423 6.902 1.995 0.472 22.898 0.551

C9 0.330 0.290 0.252 0.465 6.890 1.955 0.471 5.639 0.818A10 0.699 0.211 0.205 0.409 6.499 1.937 0.444 0.741 4.328

B10 0.302 0.636 0.399 0.424 6.854 1.614 0.488 23.886 0.543

C10 0.375 0.298 0.250 0.465 6.726 1.875 0.445 4.619 0.997

A11 0.687 0.212 0.232 0.568 6.702 1.900 0.465 0.955 4.035

B11 0.300 0.670 0.399 0.481 6.988 1.693 0.484 25.971 0.534C11 0.367 0.259 0.282 0.461 6.688 1.857 0.424 4.801 0.972

A12 0.694 0.202 0.202 0.512 6.999 1.794 0.499 0.736 4.071

B12 0.301 0.686 0.400 0.431 6.840 1.191 0.439 29.948 0.533

C12 0.392 0.368 0.281 0.455 6.874 1.731 0.472 5.822 0.980

A13 0.700 0.235 0.201 0.602 6.752 1.810 0.481 0.901 4.286

B13 0.300 0.574 0.365 0.408 6.743 1.907 0.454 19.174 0.565C13 0.348 0.308 0.239 0.426 6.577 1.925 0.477 5.047 0.895

A14 0.692 0.202 0.203 0.464 6.963 1.901 0.463 0.722 4.161

B14 0.305 0.695 0.398 0.434 6.881 1.785 0.498 24.604 0.543

C14 0.398 0.389 0.272 0.440 6.899 1.849 0.467 5.614 0.992

A15 0.676 0.237 0.205 0.437 6.779 1.879 0.469 0.890 3.717B15 0.301 0.688 0.367 0.562 6.975 1.704 0.480 25.765 0.549

C15 0.332 0.224 0.278 0.450 6.645 1.799 0.456 4.912 0.869

A16 0.684 0.217 0.202 0.424 6.725 1.962 0.380 0.823 4.299

B16 0.314 0.687 0.388 0.400 6.943 1.792 0.495 21.743 0.565

C16 0.348 0.292 0.245 0.457 6.850 1.887 0.477 4.969 0.891

A indicates the point of minimum Euler number and maximum Stokes number. B indicates the point ofmaximum Euler number and minimum Stokes number. C indicates an optimal point for themulti-objective optimization problem, cf. Fig. 8.28(c).

214

8.5. Multi-objective optimization using CFD data

8.5 Single and multi-objective optimization us-

ing CFD data

8.5.1 Design variables and approaches

The sensitivity analysis presented in Sec. 4.1 reported that the most sig-

nificant factors affecting the cyclone performance are, the vortex finder di-

ameter Dx, the inlet width b, the inlet height a and the total cyclone height

Ht. The effects of the barrel height h, the vortex finder length S and the

cone-tip diameter Bc are insignificant. Therefore, only four significant ge-

ometrical factors have been used in this study. The three other factors are

fixed based on the Stairmand high efficiency design, i.e. Bc/D = 0.375,

h/D = 1.5 and S/D = 0.5. The selection of the values for both h and S are

also based on the conclusion of Zhu and Lee [200] (when both the pressure

drop and the particle collection efficiency are considered, a cyclone which

has (h-S)/D of 1.0 would be an optimum design).

Optimization of a gas cyclone is, indeed, a multi-objective optimization

problem rather than a single objective optimization problem that has been

considered so far in the literature [146]. Both the pressure drop and the

cut-off diameter in gas cyclones are important objective functions to be op-

timized simultaneously in a multi-objective optimization problem. These

objective functions values are either obtained from experiments, empirical

models or computed using computational fluid dynamic (CFD) approaches.

CFD modeling and optimization of the parameters are investigated in the

present study and multi-objective Nedler-Mead optimization algorithms

are used in order to maximize the collection efficiency (minimize the cut-off

diameter) and minimize the Euler number (dimensionless pressure drop).

The desirability function approach has been used to handle the two objec-

tive function.

The application of the Nelder-Mead technique suffers from one big dis-

advantage. The optimal point may be local minimum because the tech-

nique depends on the starting point. This was the motivation to use also

the genetic algorithm technique. Furthermore, the radial basis function

neural networks can attain high accuracy as curve fitting approach than

the response surface [53]. Consequently, the application of the radial ba-

sis function neural networks to model the relationship between the per-

formance parameters and the four geometrical parameters will result in

more accurate results. The optimization studies given in the previous sec-

tions were applied for the seven geometrical parameters but the current

study is focusing only on the most significant four geometrical parameters.

The above-mentioned studies used meta-models using mathematical mod-

els and experimental measurements but this study is based only on CFD

215


simulations performed on sampling cyclone with a diameter of 31E-3 m.

Study objectives

There are four objectives of this study. (1) Investigation of the effect of the

four geometrical parameters on the cyclone separator performance based

on CFD simulations results. (2) Study the possible interaction between

the four geometrical parameters using the response surface methodology.

(3) Application of the multi-objective optimization technique to obtain new

geometrical ratios for minimum pressure drop and minimum cut-off diam-

eter, followed by a comparison of the numerical simulations for the new

design and the Stairmand design using the Reynolds stress turbulence

model. (4) Application of the genetic algorithm technique instead of the

desirability function approach using the two trained radial basis function

neural networks as fitness functions.

Study outline

This study is performed in six stages. The first stage is the application

of the response surface methodology (Box-Behnken design) to design an

experiment to study the effect of four geometrical parameters (the inlet

height and width, the vortex finder diameter, and the cyclone total height)

on the cyclone performance using Statgraphics statistical software. Sec-

ondly, the obtained 27 test cases (designs) have been computationally sim-

ulated using the Reynolds stress turbulence model and discrete phase

modeling with the Fluent solver. In the third stage, the Euler numbers

and the cut-off diameters obtained are used to fit a second order poly-

nomial (response surface) for each response (the Euler number and the

cut-off diameter). The obtained polynomials have been used to study the

variation of the two responses with the four geometrical parameters. Fur-

thermore, the obtained polynomials have been used to obtain new cyclone

geometrical ratios using the multi-objective optimization between the two

conflicting objectives (the Euler number and the cut-off diameter) using

the desirability function approach. The fourth stage is a computational

investigation: a numerical comparison between the new design and the

Stairmand design has been performed. Fifthly, replace the response sur-

face methodology with the artificial neural networks approach and study

the effect of each geometrical parameter on the cyclone separator perfor-

mance. The last stage is an optimization study using the genetic algorithm

instead of desirability function.

216


Design of experiment

In order to model a complex multivariate process where the responses are

influenced by several variables, the response surface statistical technique

seems the best approach [50]. The steps are as follows: (1) construct the

design of experiment by identifying the four tested geometrical parameters

(minimum and maximum values) and also to decide upon the dependent

variables (the Euler number and the cut-off diameter). Statgraphics com-

mercial statistical software gives 27 runs to be performed (cf. Table 8.32)

using the Box-Behnken design of experiment method [118]. (2) Once the

runs have been conducted (using CFD simulations) and recorded data in-

serted in the table; Statgraphics software fits a second order polynomial to

this data (one response surface per dependent variable) [30]. The second-

order polynomial (response surface) has the form [50]:

Yk = β0 +4∑

i=1

βiXi +4∑

i=1

βiiX2i +

∑∑

i<j

βijXiXj (8.14)

where β0, βi, βii, and βij are the regression coefficients for intercept, linear,

quadratic and interaction terms, respectively. Xi and Xj are the indepen-

dent variables, and Yk is k the response variable (k = 1 for the Euler num-

ber and k = 2 for the cut-off diameter). (3) The third step, is the analysis

of the response surface plot, main effect plots, Pareto chart and interaction

plots.

Analysis of response surface plots

Analysis of variance (ANOVA) showed that the resultant quadratic poly-

nomial models adequately represent the used data with the coefficient of

determination R2, being 0.94284 and 0.973468 for the Euler number and

cut-off diameter respectively. This indicates that the quadratic polynomial

models obtained were adequate to describe the influence of the indepen-

dent variables studied [189]. Analysis of variance (ANOVA) was used to

evaluate the significance of the coefficients of the quadratic polynomial

models (cf. Tables 8.33 and 8.34). For any of the terms in the models, a

large F-value (small P-value) would indicate a more significant effect on

the respective response variables.

Based on the ANOVA results presented in Table 8.33, the variable with

the largest effect on the pressure drop (Euler number) was the linear term

of vortex finder diameter (β3), the linear terms of inlet height and width

(P < 0.05) (β1 and β2 respectively), whereas the linear term of cyclone

total height did not show a significant effect (P > 0.05). The quadratic

217


Table 8.32: The Box-Behnken’s design matrix and the responses of the Euler num-

ber and cut-off diametera

Run X1 X2 X3 X4 Euler number Cut-off diameterb

No. a/D b/D Dx/D Ht/D (-) (µm)

1 0.250 0.2625 0.50 3 3.500 1.546

2 0.500 0.2625 0.75 4 2.827 2.541

3 0.375 0.3750 0.25 4 27.257 1.1584 0.375 0.2625 0.50 4 3.475 1.683

5 0.375 0.2625 0.75 3 2.333 2.444

6 0.250 0.2625 0.50 5 2.952 1.364

7 0.375 0.1500 0.50 4 2.726 1.353

8 0.375 0.1500 0.50 5 2.530 1.2849 0.375 0.2625 0.25 5 17.712 0.956

10 0.500 0.3750 0.50 4 9.086 2.163

11 0.250 0.2625 0.75 4 1.413 1.939

12 0.375 0.1500 0.50 3 3.000 1.455

13 0.375 0.1500 0.75 4 1.211 1.826

14 0.375 0.3750 0.50 3 7.500 2.08115 0.500 0.2625 0.50 5 5.904 1.787

16 0.375 0.3750 0.50 5 6.326 1.836

17 0.250 0.2625 0.25 4 12.720 0.860

18 0.375 0.2625 0.25 3 21.000 1.084

19 0.500 0.2625 0.25 4 25.440 1.12720 0.250 0.3750 0.50 4 4.543 1.651

21 0.375 0.2625 0.50 4 4.770 1.683

22 0.375 0.3750 0.75 4 3.029 2.610

23 0.500 0.1500 0.50 4 3.634 1.513

24 0.250 0.1500 0.50 4 1.817 1.15525 0.500 0.2625 0.50 3 7.000 2.025

26 0.375 0.2625 0.50 4 4.770 1.683

27 0.375 0.2625 0.75 5 1.968 2.156

a The values of h/D = 1.5, S/D = 0.5 and Bc/D = 0.375 are identical to that of Stairmand highefficiency design, so the variation in the total cyclone height is due to the variations of the cone height.

b The value of cut-off diameter depends on the gas velocity and density, particle density. In this study,D = 31mm, gas volume flow rate Qin = 50 l/min, gas density ρ = 1.0, gas viscosity µ = 2.11E − 5 Pa sand particle density ρp = 860 kg/m3.

term of vortex finder diameter also had a significant effect on the pressure

drop; however, the effect of the other three quadratic terms was insignifi-

cant. Furthermore, the interaction between the inlet dimensions and vor-

tex finder diameter also had a significant effect on the pressure drop (β13

and β23), whereas the effect of the remaining terms was insignificant. Ta-

ble 8.34 confirms the significant effect of all linear terms of the inlet di-

mensions, vortex finder diameter and total cyclone height on the cut-off

diameter. Moreover, the interaction between the inlet width and the vor-

tex finder diameter also had a significant effect on the cut-off diameter,

whereas the effect of the remaining terms was insignificant.

For the visualization of the results of the analysis, main effects plot, Pareto

chart and response surface plots were drawn. The slope of the main effect

curve is proportional to the size of the effect, and the direction of the curve

specifies a positive or negative influence of the effect [50, 61] (Fig.8.31(a)).

Based on the main effect plot and the Pareto chart shown in Fig. 8.31,

the most significant factors on the Euler number are: (1) the vortex finder

diameter Dx, with a non-linear relation with a wide range where it is in-

versely proportional to Euler number and a narrow range with a direct

218


Table 8.33: Analysis of variance and regression coefficients for the Euler number∗


β0 1.88508Linearβ1 41.3522 14.21 0.0012β2 134.36 33.78 0.0000β3 -49.1776 209.03 0.0000β4 -4.14082 0.94 0.3430Quadraticβ11 16.9991 0.1 0.7509β22 -44.8511 0.5 0.4878β33 87.9024 44.97 0.0000β44 0.368311 0.2 0.6602Interactionβ12 48.4622 0.41 0.5306β13 -90.448 7.01 0.0155β14 -1.096 0.02 0.8992β23 -185.69 28.5 0.0000β24 -1.56444 0.03 0.8708β34 2.923 0.47 0.5017

R2 0.94284


proportional relation. (2) the inlet width b and the inlet height a almost

linearly related to the Euler number, (3) the cyclone total height Ht has

an insignificant effect. Pareto charts were used to summarize graphically

and display the relative importance of each parameter with respect to the

Euler number [50]. The Pareto chart shows all the linear and second-

order effects of the parameters within the model and estimates the signif-

icance of each with respect to maximizing the Euler number response. A

Pareto chart displays a frequency histogram with the length of each bar

proportional to each estimated standardized effect [30]. The vertical line

on the Pareto charts judges, whether each effect is statistically significant

within the generated response surface model; bars that extend beyond this

line represent effects that are statistically significant at a 95% confidence

level. Based on the Pareto chart (Fig. 8.31(b)) there are six significant pa-

rameters at a 95% confidence level: the vortex finder diameter Dx; the

inlet width b; the inlet height a and the combinations a Dx, b Dx and D2x.

Therefore, the Pareto chart is a perfect supplement to the main effects

plot. To visualize the effect of the independent variables on the dependent

ones,response surfaces of the quadratic polynomial models were generated

by varying two of the independent variables within the experimental range

whereas holding the other factors at their central values [189] as shown in

Fig. 8.31. Thus, Fig. 8.31(e) was generated by varying the inlet height aand the inlet width b whereas keeping the other factors fixed at their cen-

tral values. The trend of the curve is linear, with a more significant effect

for the inlet width b, and a weak interaction between the inlet height a

219


Table 8.34: Analysis of variance and regression coefficients for the cut-off

diameter∗


β0 0.365539Linearβ1 1.84206 67.21 0.0000β2 -1.23737 89.76 0.0000β3 0.494019 524.79 0.0000β4 -0.00682157 15.14 0.0009Quadraticβ11 -2.25164 0.82 0.3763β22 -0.708116 0.06 0.8152β33 -0.307782 0.25 0.6237β44 0.00841817 0.05 0.8308Interactionβ12 2.73778 0.59 0.4531β13 2.68 2.77 0.1116β14 -0.112 0.08 0.7837β23 7.07071 18.61 0.0003β24 -0.164444 0.14 0.7170β34 -0.16 0.63 0.4360

R2 0.973468


and width b. The response surface plots of Figs. 8.31(c), 8.31(d) and 8.31(f)

show that there are strong interactions between the vortex finder diameter

Dx and the inlet dimensions a and b.

From the analysis of the design of experiment for the cut-off diameter, the

effect of variation of the vortex finder diameter Dx on the cut-off diameter

is opposite to that on the Euler number (cf. Fig. 8.32(a)). The Pareto chart

given in Fig. 8.32(b) indicates the significance effect of the cyclone total

height Ht on the cut-off diameter, but its effect is minor in comparison with

the three other factors. The significant interaction exists only between the

vortex finder diameter and the inlet dimensions.

8.5.2 Multi-objectives optimization using the desirabil-ity function

The desirability function approach

From the previous analysis, it is observed that the optimal values for the

geometrical parameters that minimize the pressure drop are different from

the values that minimize the cut-off diameter (cf. Figs. 8.31 and 8.32).

As a result, a multi-objective optimization procedure is needed. The uti-

220



(b) Pareto chart

Figure 8.31: Analysis of design of experiment for the Euler number

lization of desirability function proposed by Harrington [71] is the most

popular and strongly suggested method for multiple response optimiza-

tion problems [20] to convert the problem into single-objective. The Stat-

graphics statistical package uses this desirability function approach for

optimization of multiple response problems. The desirability function is

first defined for each response. The desirability function d(y) expresses the

desirability of a response value equal to y on a scale of 0 (if the response

value is in an unacceptable range) to 1 (for the optimum value), for mini-

mization of response variable. The procedure will then find the settings of

the experimental factors that maximize a combined desirability function,

which is a function that expresses the desirability of a solution involving

m, where m here equals 2 (one for the Euler number and the other for the

221


(c) Dx versus a

(d) Dx versus b

(e) a versus b

(f) Dx versus Ht

Figure 8.31: (continued) Analysis of design of experiment for the Euler number.

Note: In Figs 8.31(c) - 8.31(f), for each plot of two independent variable, all other

variables are hold at their central values.

222



(b) Pareto chart

Figure 8.32: Analysis of design of experiment for the cut-off diameter

cut-off diameter), responses through the function of the form [131],

D =dI11 dI22 . . . dImm

1/(∑m

j=1 Ij)(8.15)

where dj is the calculated desirability of the jth response and Ij is an im-

pact coefficient that ranges between 1 and 5 [131]. It represents the im-

portance relative of each response over the other responses [88]. Where

the default value is 3. In this study, more importance is given to the Eu-

ler number (I1 = 5 and I2 = 3). When a response is to be minimized, the

223


(c) Dx versus a

(d) Dx versus b

(e) a versus b

(f) Dx versus Ht

Figure 8.32: (continued) Analysis of design of experiment for the cut-off diameter.

Note: In Figs 8.32(c) - 8.32(f), for each plot of two independent variable, all other

variables are hold at their central values.

224


Table 8.35: The optimized cyclone separator design for best performance using the

desirability function

Factor Minimum Center Maximum Stairmand design Optimum

a 0.25 0.375 0.5 0.5 0.256b 0.15 0.2625 0.375 0.2 0.151Dx 0.25 0.5 0.75 0.5 0.415Ht 3 4 5 4.0 4.56

desirability of a predicted response equal to yj is defined as,

dj =

1 yj < lowj(yj−highj

lowj−highj

)lowj ≤ yj ≤ highj

0 yj > highj

(8.16)

where lowj and highj are the minimum and maximum values of jth re-

sponse. In this study, the minimum and maximum values obtained in the

data sheet have been used (cf. Table 8.32). The desirability plots are given

in Fig. 8.33. For more information about statistical model used for mul-

tiple response optimization refer to Ref. [131]. Table 8.35 presents the

optimum values of geometry parameters that minimize the values of the

Euler number and the cut-off diameter, which gives optimum desirability

D = 0.83 to minimize the Euler number and cut-off diameter. It is clear

from Table 8.35 that the new optimized design differs from the Stairmand

design in many geometrical parameters, whereas the new ratios will re-

sult in minimum pressure drop and minimum cut-off diameter. The ratio

of inlet width to height b/a = 0.589 lays in the optimal cyclone lies in the

recommended range of b/a from 0.5 to 0.7 proposed by Elsayed and Lacor

[52]. To understand the effect of these changes in the geometrical ratios on

the flow field pattern and performance, a CFD study for the two designs is

needed.

8.5.2.1 CFD comparison between the Stairmand and optimal de-

sign

The Fluent solver has many turbulence models available for simulating

turbulent flow. It is generally recognized that only the Reynolds stress

model (RSM) and large eddy simulation (LES) can capture the main fea-

tures of the highly complicated swirling flow in cyclone separators [11, 23,

50, 52, 62–64, 74, 91, 159, 186, 198]. The Reynolds stress turbulence model

has been used in this study to reveal the turbulent flow in the two cyclone

separators. For the detailed governing equation for both the Reynolds av-

eraged Navier-Stokes equation (RANS) and the discrete phase modeling

225


(a) Dx versus a

(b) Dx versus b

(c) a versus b

(d) Dx versus Ht

Figure 8.33: The desirability plots. Note: for each plot of two independent variable,

all other variables are hold at their optimal values.

226


(DPM) the reader can refer to Elsayed and Lacor [52]. The geometrical

values are given in Table 8.36 for the two cyclones (cf. Fig. 8.34).

Numerical settings

The air volume flow rate Qin=50 l/min for the two cyclones, air density is

1.0 kg/m3 and dynamic viscosity 2.11E-5 Pa s. The turbulent intensity


A velocity inlet boundary condition is applied at inlet, outflow at gas outlet

and wall boundary conditions at all other boundaries.







with a time step of 0.0001s.

(a) The cyclone geometry (b) The surface mesh for the new design

Figure 8.34: The cyclone geometry and the surface mesh for the new design.

Table 8.36: The values of geometrical parameters for the two designs (D=31E-3 m)

Cyclone a/D b/D Dx/D Ht/D h/D S/D Bc/D Li/D Le/DStairmand design 0.5 0.2 0.5 4 1.5 0.5 0.375 1.0 0.5

New design 0.256 0.151 0.415 4.56 1.5 0.5 0.375 1.0 0.5

227


Grid independency study

The grid independence study has been performed for the tested cyclones.

Three levels of grid for each cyclone have been tested, to be sure that

the obtained results are grid independent. The hexahedral computational

grids were generated using Gambit grid generator and the simulations

were performed using Fluent 6.3.26 commercial finite volume solver on a

8 nodes CPU Opteron 64 Linux cluster. To evaluate accuratelyto estimate

accurately the numerical uncertainties in the computational results, the

concept of grid convergence index (GCI) was adopted using three grid lev-

els per cyclone.


Table 8.37 presents the grid convergency calculations using GCI method

for three grid levels for each cyclone. The following conclusions have been

obtained from the GCI analysis [55]:

• The results are in the asymptotic range, because the obtained values

for α are close to unity.

• The ratio R is less than unity this means monotonic convergence [2].


ments

(GCIfine12 < GCI

fine23 ) for the two variables (Eu and x50). This indicates

that the dependency of the numerical results on the cell size has been

reduced. Moreover, a grid independent solution has been achieved.

Further refinement of the grid will not give much change in the sim-

ulation results. For the two variables (Eu and x50), the extrapolated

value is only slightly lower than the finest grid solution. Therefore,

the solution has converged with the refinement from the coarser grid

to the finer grid [2]. Figure 8.35 presents a qualitative proof that the

obtained results are in the asymptotic range.

• The value of εi,i+1 represent the relative change in each value from

coarse to medium and from medium to fine mesh. For example, ε1,2 =0.0256 for the Euler number in the new design means the percentage

change in the Euler number when the mesh becomes 986748 cells

instead of 717353 cells equals 2.56%. This means no need to use

the fine mesh and the usage of the medium mesh of 717353 cells is

sufficient. Another example, ε1,2 = 0.0052 for the cut-off diameter

in the Stairmand design means the percentage change in the cut-

off diameter when the mesh becomes 848783 cells instead of 622253

228


cells equals 0.52%. This means no need to use the fine mesh and the

usage of the medium mesh of 622253 cells is sufficient.

In summary, the grid refinement study shows that a total number of about

622253 hexahedral cells for Stairmand cyclone and 717353 hexahedral

cells for the new design are sufficient to obtain a grid-independent solu-

tion, and further mesh refinement yields only small, insignificant changes

in the numerical solution.

Table 8.37: Grid convergency calculations using GCI method using three grid levels

for each cyclone.

i Ni fi ri,i+1 ei,i+1 εi,i+1 GCIi,i+1% Ra αb

New

desi

gn

Eu

ler

nu

mber 0c 2.5195

1 986748 2.5380

1.1121 0.0650 0.0256 0.9112

2 717353 2.6030 0.0331 1.0256

1.2553 2.0140 0.7737 4.00993 362679 4.6170

Cu

t-off

dia

mete

r 0 0.6621

1 986748 0.6710

1.1121 0.0190 0.0283 1.6647

2 717353 0.6900 0.0669 1.02831.2553 0.2920 0.4232 5.0609

3 362679 0.9820

Sta

irm

an

dd

esi

gn

Eu

ler

nu

mber 0 5.4310

1 848783 5.4860

1.1090 0.1200 0.0219 1.2521

2 622253 5.6060 0.0999 1.0219

1.2044 1.2270 0.2189 3.90103 356181 6.8330

Cu

t-off

dia

mete

r 0 1.7187

1 848783 1.7210

1.1090 0.0090 0.0052 0.1678

2 622253 1.7300 0.0489 1.00521.2044 0.1850 0.1069 0.8173

3 356181 1.9150

a R=ε12/ε23 .

b α=(

rp12

GCI12

)

/GCI23 .

c The value at zero grid space (h → 0). i=1, 2 and 3 denote the calculations at the fine, medium and coarse mesh respectively.


Flow field pattern

Figure 8.36 shows the contour plot at Y=0 and throughout the inlet sec-

tion. In the two cyclones, the time-averaged dimensionless static pressure

(Euler number) decreases radially from the wall to center. A negative pres-



tion, whereas the gradient in the axial direction is very limited. The cy-

clonic flow is not symmetrical as is clear from the shape of the low-pressure

zone at the cyclone center (twisted cylinder). The flow asymmetry is more

pronounced in the new cyclone. However, the two cyclones have almost the

229


N -1

Eul

ernu

mbe

r

Cut

-off

diam

eter

0 1E-06 2E-062

3

4

5

6

7

1

1.5

New design (Eu)

New design (X50)

Stairmand design (Eu)

Stairmand design (X50)

(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)(h --> 0)


number and the cut-off diameter for the two cyclones using the three grid levels.

N−1 is the reciprocal of the number of cells, h → 0 means the value at zero grid

size (cf. Table 8.37). To obtain a smooth curve; the spline curve fitting has been

applied in Tecplot post-processing software.

same flow pattern, but the Euler number of the Stairmand design is nearly

twice that of the new design.

The dimensionless static pressure distribution presented in Fig. 8.37 for

the two cyclones indicates that the highest dimensionless static pressure

for the Stairmand design is more than twice that of the new design at all

sections whereas the central value is almost the same for the two cyclones.

This indicates that, the new design has a lower dimensionless pressure

drop than the Stairmand design. However, these results are obtained at

different inlet velocity for the two cyclones (to have the same air flow rate).

The same Euler number values would be obtained if the two cyclones work

at the same inlet velocity because the Euler number is not a function of

flow velocity if the Reynolds number is higher than 2E4 [50].

The tangential velocity profile is composed of two regions. In the inner re-

gion, the flow rotates approximately like a solid body (forced vortex), where

the tangential velocity increases with radius. After reaching its peak the

velocity decreases with radius in the outer part of the profile (free vortex).

The tangential velocity distributions for the two cyclones are nearly iden-

tical in pattern (Rankine profile). The inner part of the tangential velocity

distribution of the two cyclones is very similar. The outer part for the new

design is flatter in comparison with the Stairmand cyclone. This implies

that there is more space in the optimal cyclone where the particles are

subjected to high centrifugal force. Whereas, the maximum dimension-

less tangential velocity for Stairmand cyclone is higher than that for the

230


new design, the cyclone performance is not only affected by the maximum

tangential velocity but also with the separation space (the new design is

longer than the Stairmand design)

The axial velocity profiles for the two cyclones are different in values and

shape. Stairmand cyclone exhibit the inverted W axial velocity profile. The

new design exhibit the inverted W axial velocity profile away from the inlet

section and the inverted V elsewhere. This is due to the very high swirl

exist at the inlet section in case of the new design.

Performance parameters

To calculate the cut-off diameters of the two cyclones, 104 particles were

injected from the inlet surface with a particles density ρp = 860kg/m3 and

the maximum number of time steps for each injection was 2E9 steps. The

DPM analysis results and the Euler number for the two cyclones are given

in Table 8.38. Table 8.38 introduces also a comparison between the CFD

results and four different mathematical models viz., the Ramachandran

model [139], the Muschelknautz method of modeling (MM) [50, 116, 174],

the Iozia model [85], the Ritema model [142](cf. Hoffmann and Stein[77]

for more details about these mathematical models). However, no-good

matching between the two approaches (CFD and mathematical models)

is obtained, they agree in the trend of superior performance of the new

design. The Euler number and cut-off diameter for the new design is ap-

proximately half that of Stairmand design.

8.5.3 Artificial neural network (ANN) approach

Artificial neural networks (ANNs) have become an attractive approach for

modeling highly complicated and nonlinear system [53, 180, 197]. In this

study, the radial basis function neural network (RBFNN) has been used to

model the effect of cyclone dimensions on both the pressure drop and the

cut-off diameter. For more details about the radial basis function neural

networks, the reader can refer Sec. 8.3.1, page 158. Two RBFNNs have

been trained using the 27 data set obtained from the CFD simulations for

both the Euler number (the dimensionless pressure drop) and the cut-off

diameter.

In this study, the performance parameters are assumed as functions of

only the four geometrical parameters, whereas all other parameters kept

constant, Eq. 8.17.

231


New design Stairmand design


and throughout the inlet section. From top to bottom: the dimensionless static

pressure (divided by the dynamic pressure at inlet), the dimensionless tangential

velocity, axial velocity. Note: both cyclones have the same air volume flow rate.

232


Table 8.38: The cyclone performance parameters using CFD simulations and four

mathematical models

Parameter Cyclone Statgraphics CFD Ramachandran MM Iozia Ritema

Euler numberStairmand design 5.673 5.606 4.85 5.33 - -New design 2.221 2.603 2.71 2.99 - -

Cut-off diameter, µmStairmand design 1.706 1.73 - - 1.69 1.55New design 0.865 0.69 - - 0.95 0.71


Sta

ticpr

essu

re/D

ynam

icpr

essu

reat

inle

t

-1 -0.5 0 0.5 1-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4



Tan

gent

ialv

eloc

ity/I

nlet

velo

city

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2 New designStairmand design

Figure 8.37: The radial profiles of the time averaged static pressure and tangential

velocity at z’/D=1.5.

Eu = f1

(a

D,b

D,Dx

D,Ht

D

)

x50 = f2

(a

D,b

D,Dx

D,Ht

D

)(8.17)

Due to the large difference in the order of magnitude of the value (cf. Ta-

ble 8.35), the available dataset is transformed into -1 to 1 interval using

the Matlab intrinsic function; mapminmax in order to avoid solution diver-

gence [53, 197]. The RBFNN calculations have been performed using the

neural network toolbox available from Matlab commercial software 2010a.

The cut-off diameter x50 for cyclone separator is always given in units of

µm. Another way to represent x50 is using a dimensionless number; Stokes

number Stk50 = ρpx250Vin/(18µD) [37]. It is the ratio between the particle

relaxation time; ρpx250/(18µ) and the gas flow integral time scale; D/Vin.

Fitting the RBFNNs

The configured RBFNNs predictions versus the CFD data for the Euler

number and cut-off diameter are shown in Fig. 8.38. It can be seen that

233


the RBFNN models are able to attain the high training accuracy. The

training mean square errors are zeros (i.e., identical matching between

the input and output, the reason behind that may refer to the consistency

between the used data in the simulation using the design of experiment,

which is not the case for the study of Elsayed and Lacor [53] using ex-

perimental data set), Fig. 8.38. This indicates that, in comparison with

traditional models of curve fitting, the models based on an artificial intelli-

gence algorithm have a superior capability of nonlinear fitting. Especially,

the RBFNN has its unique and optimal approximation characteristics in

learning process [53, 197].

Figure 8.38 illustrates the agreement between the RBFNNs input and out-

put. The obtained relation is a typical linear relation with a coefficient of

correlation close to 1 (R > 0.999). The agreement between the input and

output of the RBFNN is also clear from the value of the mean squared er-

ror E2. Consequently, the trained neural networks predict very well both

the Euler number and cut-off diameter values and can be used in cyclone

design and performance estimation. Figure 8.38 present different perfor-

mance indicators as a validation of the proposed models for the trained

data.

Input value (x)

Pre

dict

edva

lue

(y)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

20

22

24


y=0.999 x -2.2E-15

R=0.99999

E2= 0.0

(a) The Euler number

Input value (x)

Pre

dict

edva

lue

(y)

1 1.5 2 2.5

1

1.5

2

2.5 Data pointLinear fit

y=0.999 x +4.03E-16

R=0.99999

E2= 0.0

(b) The cut-off diameter

Figure 8.38: Linear regression of the RBFNNs for the Euler number and the cut-off

diameter.

234


Dx, a, b

HtE

uler

num

ber

0.2 0.3 0.4 0.5 0.6 0.7

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

5

10

15

Dx

abHt

(a) Euler number

Dx, a, b

Ht

X5

0[m

icro

n]

0.2 0.4 0.6

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Dx

abHt

(b) Cut-off diameter

Figure 8.39: The effect of geometrical parameters on the cyclone performance.

The effect of the four geometrical parameters on the cyclone per-

formance based on RBFNNs

The effects of the geometrical parameters on both the Euler number and

the cut-off diameter are depicted in Figs. 8.39(a) and 8.39(b). To study the

effect of each parameter, the tested RBFNNs models have been used by

varying one parameter at a time from its minimum to maximum values

of the available CFD dataset, whereas the other parameters are kept con-

stant at their mean values (cf. Table 8.35). Figures 8.39(a) and 8.39(b)

indicate the significant effect of the vortex finder diameter Dx, the inlet

width b, the inlet height a. Less effect is assigned to the total cyclone

height Ht. More analysis is given in Tables 8.39 and 8.40.

8.5.4 Optimization Using Genetic Algorithms

Optimal cyclone design for minimum pressure drop

The genetic algorithm optimization technique has been applied to obtain

the geometrical ratios for minimum pressure drop (Euler number). The ob-

jective function is the Euler number (using the trained radial basis func-

tion neural network). The design variables are four geometrical dimen-

sions, the inlet height a, the inlet width b, the vortex finder diameter Dx

and the total cyclone height Ht. These four variables are the most signifi-

cant factors which affect the cyclone performance [50].

235


Table 8.39: The variation of the Euler number with cyclone dimensions using the

RBFNN model (cf., Fig. 8.39(a))

Factor Analysis

Dx

The most significant effect is that of the vortex finder diameter Dx

with inverse relationship when increasing Dx up to ≈ 0.65 after

which the relation becomes direct, Fig. 8.39(a). This can be explained

as follows: Although the pressure loss in the vortex finder decreases

with increasing the vortex finder diameter like the case of viscous

flow in a pipe, the Euler number in the cyclone body instead will

increase due to the decrease of the flow area just after the flow

entrance from the inlet region (the annular space between the barrel

and the vortex finder). This analysis indicates the large contribution

of the pressure loss in the vortex finder to the total Euler number (the

pressure loss at the entrance, the pressure loss in the cyclone body,

and the pressure loss in the vortex finder [50]).

a and b

The inlet height a and width b is almost linearly related to the Euler

number. But, why the relation is direct (at the same flow rate

increasing the inlet dimensions decreases the inlet velocity.

Consequently,the loss in the vortex finder will decrease)? The reason

is the increase in the pressure drop at the inlet section, due to

deviation of the inlet flow when it mixes with the swirling flow.

Ht

The effect of changing the total cyclone height Ht on the Euler

number is very small.

Table 8.41 presents the settings used to obtain the optimum design for

minimum pressure using global optimization Matlab toolbox in Matlab

2010a commercial package. Table 8.42 gives the optimum values for cy-

clone geometrical parameters for minimum pressure drop estimated by the

artificial neural network using the genetic algorithm optimization tech-

nique. It is clear from Table 8.42 that the new optimized design is very

close to the Stairmand design in many geometrical parameters, whereas

the new ratios will result in the minimum pressure drop.

Optimal cyclone design for best performance using NSGA-II

In case of cyclone separator geometry optimization for minimum Euler

number and minimum cut-off diameter, the objectives are conflicting with

each other. There is no best solution for which all objectives are optimal

simultaneously [181]. The increase of one objective will lead to the de-

236


Table 8.40: The variation of the cut-off diameter with cyclone dimensions using the

RBFNN model (cf., Fig. 8.39(b))

Factor Analysis

Dx

The vortex finder diameter has the most significant effect on the

cut-off diameter x50 (the highest slope in Fig. 8.39(b)). The slope is

very high for Dx = 0.4 − 0.55 and any further increase or decrease in

Dx beyond the above range produces a small change in x50.

Increasing the vortex finder diameter decreases the swirling intensity

in the cyclone (i.e. Reduction in the centrifugal force). Consequently,

low collection efficiency (higher x50) is obtained. In the meantime, the

increase of Dx decreases the pressure drop. This is one of the main

reasons of the trade-off between the Euler number and the cut-off

diameter objectives. This makes the optimization of cyclone geometry

a multi-objective procedure.

a and b

The variation of x50 with the inlet width and height are similar in

trend and significance to that for Dx but here the slope is high in the

range of a = 0.3− 0.4 b = 0.2− 0.275.

Ht

The effect of the cyclone total height on the cut-off diameter is due to

the effect of the cone height as the barrel height is fixed in this study.

The slope is very small, with a general trend of inverse relation

(increasing the separation space, enhances the collection efficiency).

crease of the other objective. Then, there should be a set of solutions, the

so-called Pareto optimal set or Pareto front, in which one solution cannot

be dominated by any other member of this set [56].

Recently, a number of multi-objective genetic algorithms (MOGAs) based

on the Pareto optimal concept have been proposed. The well known non-

dominated sorting genetic algorithm II (NSGA-II) proposed by Deb et al.

[34] is one of the most widely used MOGAs since it provides excellent re-

sults as compared with other multi-objective genetic algorithms proposed

[25].

Table 8.43 presents the genetic operators and parameters for multi-objective

optimization. The Euler number and the Stokes number values have been

obtained from the artificial neural network trained by the CFD data set.

The Pareto front (non dominated points) is presented in Fig. 8.40(a) and

Table 8.44. Figure 8.40(a) clearly demonstrate tradeoffs in objective func-

tions (Euler number and Stokes number). All the optimum design points

in the Pareto front are non-dominated and could be chosen by a designer

as optimum cyclone separator [148]. This set of designs makes the Pareto

237


Table 8.41: Genetic operators and parameters for single objective optimization

Population type Double vectorInitial range [0.25 0.15 0.25 3 ; 0.5 0.375 0.75 5 ] for a, b, Dx and Ht respectivelyFitness scaling RankSelection operation: Tournament (tournament size equals 4)Elite count 2Crossover fraction 0.8Crossover operation Intermediate crossover with the default value of 1.0Mutation operation The constraint dependent defaultMaximum number of generations: 800Population size 200

front approach more preferred than the desirability function approach which

gives only one design point. Three points A, B and C are indicated in

Fig. 8.40(a). Point A indicates the point of minimum Euler number (max-

imum Stokes number). Point B indicates the point of maximum Euler

number (minimum Stokes number). Point C indicates an optimal point for

the multi-objective optimization problem.

In order to obtain the Euler number-Stokes number relationship, Fig. 8.40(b)


two dimensionless numbers irrespective to the four geometrical parame-

ters values. A second-order polynomial has been fitted between the loga-

rithms of Euler number and Stokes number, Eq. 8.18. The obtained corre-

lation can fit the data with a coefficient of determination R2 = 0.99613 as

shown in Fig. 8.40(b). Elsayed and Lacor [56] presented another correla-

tion between the Euler number and the Stokes number, Eq. 8.19. Equa-

tion 8.19 presents a good matching for only high values of Euler numbers.

For smaller values of Euler number, there is underestimation of the Stokes

number. The difference between the two correlations can be referred to two

reasons. Firstly, the new correlation is based on the CFD simulations data

for both Euler number and Stokes number, whereas Elsayed and Lacor

Table 8.42: The optimized cyclone separator design for minimum pressure drop

using genetic algorithm

Factor Low High Stairmand design∗ Optimum design†

a 0.25 0.5 0.5 0.492

b 0.15 0.375 0.2 0.158

Dx 0.25 0.75 0.5 0.617

Ht 3.0 5.0 4.0 4.535

Euler number 5.606 2.369

Cut-off diameter 1.706 1.704

∗ The values for the Euler number and the cut-off diameter for the Stairmand design have been obtainedfrom CFD simulations, cf. Table 8.38.

† The values for the Euler number and the cut-off diameter for the new optimal design have beenobtained from the trained RBFNN.

238


Table 8.43: Genetic operators and parameters for multi-objective optimization

Population type Double vectorPopulation size 60 (i.e., 15* number of variables which is the default in Matlab)Initial range [0.25 0.15 0.25 3 ; 0.5 0.375 0.75 5 ] for a, b, Dx and Ht respectivelySelection operation tournament (tournament size equals 2)Crossover fraction 0.8Crowding distance fraction 0.35Crossover operation Intermediate crossover with the default value of 1.0Maximum number of generations 800 (i.e., 200* number of variables which is the default in Matlab)

correlation was obtained from experimental data for Euler number values

and Iozia and Leith model for the Stokes number values. Secondly, the

new correlation is limited to only four geometrical parameters. The other

three factors are fixed, h = 1.5, S = 0.5 and Bc = 0.375. Consequently, the

new correlation (Eq. 8.18) is valid only for these values.

Stk50 = 100.3533(log10(Eu))2−1.1645log10(Eu)−2.3198 (8.18)

Stk50 = 100.3016(log10(Eu))2−0.9479log10(Eu)−2.5154 (8.19)

8.5.5 Conclusions

CFD simulations data have been used to understand the effect of four

geometrical parameters on the cyclone performance and to optimize the

cyclone geometry. Two meta-models have been used viz., the response

surface and the radial basis function neural network approaches. Two

optimization techniques have been applied, the desirability function with

Nelder-Mead technique and the non-sorted dominated genetic algorithm

NSGA-II.

• The response surface methodology has been used to fit two second-

order polynomials to the Euler number and cut-off diameter obtained

from CFD simulations. The analysis of variance of the Euler number

indicates a strong interaction between Dx with (a, b) and between Dx

with b only for the cut-off diameter.

• The bi-objective functions have been converted to single-objective func-

tion using the desirability function approach. A new optimal de-

sign has been obtained using the Nelder-Mead technique available in

Statgraphics commercial software. The ratio of inlet width to height

b/a = 0.589 lays in the optimal cyclone lies in the recommended range

of b/a from 0.5 to 0.7 proposed by Elsayed and Lacor [52]. The new de-

sign and the Stairmand design have been computationally compared

239


Euler number

Sto

kes

num

berx

103

5 10 15 20 25 30 35

1

2

3

4

5

6

7 A

BC

(a) Pareto front in linear scale

Euler number

Sto

kes

num

berx

10

3

5 10 15 20 25 30

2

4

6

8

10

Non-dominated pointsElsayed & Lacor correlationCurve-fitting

Stk50=10 0.3533 (LOG10(Eu))^2- 1.1645 (LOG10(Eu) -2.31 98

R2=0.99613

(b) Pareto front in log scale. Note: The correlation

(Eq. 8.19) given by Elsayed and Lacor [45] are based on

seven geometrical parameters.

Figure 8.40: Pareto front plots obtained from the variation of four geometrical

ratios.

240



Stokes number for the nondominated points (Pareto-front).

point a b Dx Ht h S Bc Euler number Stokes number x 103

1 0.500 0.375 0.250 4.994 1.5 0.5 0.375 33.750 0.511

2 0.391 0.296 0.250 4.979 1.5 0.5 0.375 20.806 0.569

3 0.374 0.270 0.252 4.961 1.5 0.5 0.375 17.883 0.595

4A 0.250 0.150 0.750 4.899 1.5 0.5 0.375 0.756 6.958

5 0.252 0.152 0.362 4.893 1.5 0.5 0.375 3.326 1.262

6 0.250 0.150 0.732 4.846 1.5 0.5 0.375 0.799 6.4467 0.320 0.214 0.300 4.929 1.5 0.5 0.375 8.617 0.840

8 0.363 0.268 0.300 4.945 1.5 0.5 0.375 12.160 0.778

9 0.259 0.159 0.472 4.960 1.5 0.5 0.375 2.086 2.080

10 0.391 0.284 0.252 4.966 1.5 0.5 0.375 19.737 0.581

11 0.419 0.327 0.256 4.983 1.5 0.5 0.375 23.514 0.56812 0.251 0.151 0.711 4.853 1.5 0.5 0.375 0.854 5.903

13 0.455 0.365 0.251 4.976 1.5 0.5 0.375 29.693 0.528

14 0.251 0.151 0.695 4.858 1.5 0.5 0.375 0.895 5.504

15 0.298 0.189 0.441 4.910 1.5 0.5 0.375 3.277 1.554

16 0.376 0.270 0.262 4.960 1.5 0.5 0.375 16.657 0.63017 0.472 0.358 0.251 4.986 1.5 0.5 0.375 30.324 0.525

18B 0.500 0.375 0.250 4.994 1.5 0.5 0.375 33.750 0.51119 0.255 0.155 0.608 4.810 1.5 0.5 0.375 1.214 3.820

20 0.252 0.152 0.497 4.879 1.5 0.5 0.375 1.759 2.380

21C 0.286 0.183 0.346 4.907 1.5 0.5 0.375 4.954 1.103

22 0.351 0.233 0.295 4.949 1.5 0.5 0.375 10.568 0.789

23 0.251 0.151 0.638 4.897 1.5 0.5 0.375 1.050 4.354

24 0.265 0.154 0.460 4.881 1.5 0.5 0.375 2.193 1.780

25 0.493 0.321 0.250 4.991 1.5 0.5 0.375 28.441 0.531

26 0.312 0.206 0.320 4.869 1.5 0.5 0.375 7.136 0.94127 0.352 0.234 0.295 4.950 1.5 0.5 0.375 10.728 0.784

28 0.486 0.365 0.250 4.992 1.5 0.5 0.375 31.833 0.518

29 0.252 0.152 0.730 4.840 1.5 0.5 0.375 0.817 6.387

30 0.252 0.153 0.676 4.861 1.5 0.5 0.375 0.957 5.071

31 0.251 0.151 0.650 4.895 1.5 0.5 0.375 1.013 4.56532 0.426 0.333 0.252 4.977 1.5 0.5 0.375 25.126 0.551

33 0.261 0.150 0.747 4.838 1.5 0.5 0.375 0.798 6.822

34 0.272 0.161 0.553 4.880 1.5 0.5 0.375 1.630 2.945

35 0.253 0.154 0.553 4.880 1.5 0.5 0.375 1.451 3.022

36 0.251 0.151 0.711 4.853 1.5 0.5 0.375 0.854 5.90337 0.266 0.284 0.252 4.966 1.5 0.5 0.375 13.428 0.633

Minimum 0.25 0.150 0.250 4.810 1.5 0.5 0.375 0.7555 0.511Maximum 0.50 0.375 0.750 4.995 1.5 0.5 0.375 33.7497 6.958

A indicates the point of minimum Euler number and maximum Stokes number. B indicates the point of maximum Euler number and minimum

Stokes number. C indicates an optimal point for the multi-objective optimization problem. (cf. Fig. 8.40(a)).

to get a clear vision for the differences between the flow field pattern

and performance in the two designs. The CFD simulations results

and four mathematical models confirmed the better performance of

the new design in comparison with the Stairmand design.


alternative and powerful approach to model the cyclone performance

better than the response surface methodology. The used RBFNN pre-

sented zero mean squared error and almost unity coefficient of deter-

mination.


eter Dx and the inlet dimensions a and b on the cyclone performance.

Moreover, the range of high influence is given for each geometrical

parameters using the trained RBFNNs.

• The trained RBFNN for the Euler number has been used to get a

new optimized cyclone for minimum pressure drop (Euler number)

241


using the genetic algorithm optimization technique. The new cyclone

design is very close to the Stairmand high efficiency design in the

geometrical parameter ratio, and superior in low pressure drop at

nearly the same cut-off diameter. But, the optimal design obtained

from the desirability function results in a better collection efficiency

(smaller cut-off diameter) as is clear from Table 8.38 because there

the cut-off diameter is included in the desirability function, whereas,

the obtained result are for single objective (Euler number).


mization process using NSGA-II technique. The Pareto front is pre-

sented for the designer with a wide choice for selection.

• A second-order polynomial has been fitted between the logarithms

of Euler number and Stokes number to obtain a general formula,

Stk50 = 100.3533(log10(Eu))2−1.1645 log10(Eu)−2.3198 with a coefficient of de-

termination R2 = 0.98643. This formula can be used to obtain the

Stokes number if the Euler number is known at h = 1.5, S = 0.5 and

Bc = 0.375.

242

Chapter 9

Conclusions and Future

Directions

9.1 Conclusions

9.1.1 The most significant geometrical factors

The geometrical parameters in cyclone separators affect significantly the

flow field and performance parameters. There are seven geometrical pa-

rameters, which can be classified into four classes, namely, the inlet di-

mensions (height and width), the vortex finder dimensions (diameter and

length), the cyclone height (cone and barrel) and the cone-tip diameter.

These dimensions do not have the same influence on the cyclone perfor-

mance. Consequently, the first step was to identify the most significant

parameters and the possible interaction between them.

The Muschelknautz method of modeling (MM)) has been used to under-

stand the effect of the cyclone geometrical parameters on the cyclone per-

formance. The most significant geometrical parameters are:

1. The vortex finder diameter

2. The inlet section width

3. The inlet section height

4. The cyclone total height (cone height).

The effect of both the barrel height and the vortex finder length on the cy-

clone separator performance are small in comparison with these most sig-

nificant geometrical parameters. There are strong interactions between

243

Chapter 9. Conclusions and Future Directions

the effects of inlet dimensions and the vortex finder diameter on the cy-

clone performance.

Large eddy simulation methodology has been used to study the effect of the

cone-tip diameter on the cyclone flow field and performance. The analysis

of results indicates the insignificant effect of the cone-tip diameter on the

flow pattern and performance.

Four cyclones have been simulated using the Reynolds stress model (RSM),

to study the effect of the dust outlet geometry on the cyclone separator

performance, flow pattern and velocity profiles. The results approved that

if the main target of the CFD investigation is the performance parameters,

one can safely simulate only the main separation space (cylinder on cone).

However, if the aim is to investigate the flow field pattern, the dust outlet

geometry should be included in the simulation domain.

9.1.2 The impact of geometry

The vortex finder dimensions: The maximum tangential velocity in the

cyclone decreases with increasing the vortex finder diameter. A neg-

ligible change is noticed with increasing the vortex finder length. In-

creasing the vortex finder length, makes a small change in both the

static pressure, axial and tangential velocity profiles. However, de-

creasing the vortex finder diameter gradually changes the axial ve-

locity profile from the inverted W to the inverted V class. Decreasing

the cyclone vortex finder diameter, increases the maximum tangen-

tial velocity. The Euler number (dimensionless pressure drop) de-

creases with increasing the vortex finder diameter. Increasing the

vortex finder length slightly increases the Euler number. The Stokes

number increases with increasing the vortex finder diameter, because

the centrifugal force affecting particles attenuates when the swirl in-

tensity (maximum tangential velocity) decreases. The Stokes number

slightly increases as the vortex finder length is increased.

The inlet dimensions: The maximum tangential velocity in the cyclone

decreases with increasing both the cyclone inlet width and height. In-

creasing the cyclone inlet width or height decreases the pressure drop

at the cost of increasing the cut-off diameter. The effect of changing

the inlet width on the cut-off diameter is more significant in compari-

son with that of the inlet height. The optimum ratio of the inlet width

to the inlet height is from 0.5 to 0.7.

The cyclone heights: The maximum tangential velocity in the cyclone

decreases with increasing the cyclone (barrel or cone) height. In-

creasing the barrel height, makes a small change in the axial veloc-

244

9.1. Conclusions

ity. Increasing the cyclone barrel height decreases the pressure drop

and the cut-off diameter. The changes in the performance beyond

h/D = 1.8 are small. Increasing the cone height makes a consider-

able change in the axial velocity. Both the pressure drop and the cut-

off diameter decrease with increasing the cyclone cone height. The

performance improvement stops after hc/D = 4.0 (Ht/D = 5.5). The

effect of changing the barrel height is less significant on the perfor-

mance and the flow pattern in comparison with the effect of the cone

height.

9.1.3 Optimization

Several new optimized cyclone geometrical ratios have been obtained and

presented in this thesis. All the new geometrical ratios result in better

performance than the Stairmand design. For example, The new cyclone

design ratios obtained using MM model and Nelder-Mead technique for

minimum pressure drop, are very close to the Stairmand high efficiency

design in the geometrical parameter ratios, and superior for low pressure

drop at nearly the same cut-off diameter. The new cyclone design results

in nearly one-half the pressure drop obtained by the Stairmand design at

the same volume flow rate.

Artificial neural networks

In order to accurately predict the complexly non linear relationships be-

tween pressure drop and geometrical dimensions, a radial basis neural

network (RBFNN) is developed and employed to model the pressure drop

for cyclone separators. The neural network has been trained and tested by

the experimental data available in literatures.


alternative and powerful approach to model the cyclone performance.

• Four mathematical models (Muschelknautz method “MM”, Stairmand,

Ramachandran and Shepherd and Lapple) have been tested against

the experimental values. The residual error of MM model is the low-

est. Moreover, one can multiply the calculated value using MM by

1.5 to get the experimental value (as a rough approximation).

• The response surface methodology has been used to fit a second-order

polynomial to the RBFNN.

• The second-order polynomial has been used to get a new optimized

cyclone for minimum pressure drop using the Nelder-Mead technique.

245



design has been performed using CFD simulation to obtain a clear

vision of the flow field pattern in the new design.


obtained by the Stairmand high efficiency design at the same volume

flow rate.

9.1.4 Multi-objective optimization

Two multi-objective optimization approaches have been applied in this

thesis, namely the desirability function and the non-dominated sorted ge-

netic algorithm (NSGA-II). Moreover, two sources of data have been used,

namely: (1) The experimental data for the pressure drop (Euler number)

and the Iozia and Leith model for the cut-off diameter (Stokes number) for

the optimization of the seven geometrical parameters. (2) CFD simulations

for the optimization of only four geometrical parameters.

Seven geometrical parameters

Two radial basis neural networks (RBFNNs) are developed and employed

to model the Euler number and the Stokes number for cyclone separators.

The neural networks have been trained and tested by the experimental

data available in literature for Euler number (pressure drop) and the Iozia

and Leith model [85] for the Stokes number (cut-off diameter). The effects

of the seven geometrical parameters on the Stokes number have been in-

vestigated using the trained ANN. To declare any interaction between the

geometrical parameters affecting the Stokes number, the response surface

methodology has been applied. The trained ANN has been used as an ob-

jective function to obtain the cyclone geometrical ratios for minimum Euler

number using the genetic algorithms optimization technique. A CFD com-

parison between the new optimal design and the Stairmand design has

been performed. A multi-objective optimization technique using NSGA-

II technique has been applied to determine the Pareto front for the best

performance cyclone separator.

• The trained RBFNN has been used to get a new optimized cyclone for

minimum pressure drop (Euler number) using the genetic algorithm

optimization technique.


obtained by the Stairmand design at the same volume flow rate. This

246

9.1. Conclusions

confirms that the obtained design using the genetic algorithm is bet-

ter than that obtained using Nelder-Mead technique which results in

75% of the Stairmand pressure drop [53].


mization process using NSGA-II technique. Sixteen test cases with

different barrel diameter, gas flow rate and particle density have

been tested. The Pareto fronts for these test cases are very close.

A second-order polynomial has been fitted between the logarithms

of the Euler number and the Stokes number to obtain a general for-

mula, Stk50 = 100.3016(log10(Eu))2−0.9479 log10(Eu)−2.5154. This formula

can be used to obtain the Stokes number if the Euler number is

known.

Four geometrical parameters

CFD simulations data have been used to understand the effect of four

geometrical parameters on the cyclone performance and to optimize the

cyclone geometry. Two meta-models have been used viz., the response

surface and the radial basis function neural network approaches. Two

optimization techniques have been applied, the desirability function with

Nelder-Mead technique and the non-sorted dominated genetic algorithm

NSGA-II.

• The response surface methodology has been used to fit two second-

order polynomials to the Euler number and the cut-off diameter ob-

tained from CFD simulations. The analysis of variance of the Euler

number indicates a strong interaction between Dx with (a, b) and be-

tween Dx with b only for the cut-off diameter.

• The bi-objective functions have been converted to single-objective func-

tion using the desirability function approach. A new optimal design

has been obtained using the Nelder-Mead technique. The ratio of

inlet width to height b/a = 0.589 i.e., in the optimal cyclone lies in

the recommended range of b/a from 0.5 to 0.7 proposed by Elsayed

and Lacor [52]. The new design and the Stairmand design have been

computationally compared to get a clear vision for the differences be-

tween the flow field pattern and performance in the two designs. The

CFD simulations results confirmed the superior performance of the

new design in comparison with the Stairmand design.

• The result demonstrates once more that artificial neural networks

can offer an alternative and powerful approach to model the cyclone

performance better than the response surface methodology. The used

247


RBFNN presented zero mean squared error and almost unity coeffi-

cient of determination.


eter Dx and the inlet dimensions a and b on the cyclone performance.

Moreover, the range of high influence is given for each geometrical

parameters using the trained RBFNNs.

• The trained RBFNN for the Euler number has been used to get a

new optimized cyclone for minimum pressure drop (Euler number)

using the genetic algorithm optimization technique. The new cyclone

design is very close to the Stairmand high efficiency design in the

geometrical parameter ratio, and superior in low pressure drop at

nearly the same cut-off diameter. But, the optimal design obtained

from the desirability function results in a better collection efficiency

(smaller cut-off diameter) because there the cut-off diameter is in-

cluded in the desirability function, whereas, the obtained result are

for single objective (Euler number).


mization process using NSGA-II technique. The Pareto front is pre-

sented for the designer with a wide choice for selection.

• A second-order polynomial has been fitted between the logarithms

of the Euler number and the Stokes number to obtain a general for-

mula, Stk50 = 100.3533(log10(Eu))2−1.1645 log10(Eu)−2.3198. This formula

can be used to obtain the Stokes number if the Euler number is

known at h = 1.5, S = 0.5 and Bc = 0.375.

9.2 Future Directions

As a recommendation of future work, the same study is to be performed but

at different flow rates and different particle densities. Furthermore, the

effect of the cyclone dimensions on natural vortex length and precessing

vortex core needs more investigation using large eddy simulation method-

ology.

The present study on the dust outlet geometry is to be enlarged to include

the following. (1) The effect of dustbin and dipleg shape, length, diameter,

and interior details (like the inner cone at the dustbin inlet) and optimiza-

tion of these dimensions. (2) The effect of air and dust flow rates, and

particle properties. (3) perform the same study but using large eddy simu-

lation (LES) to check the effect of dust outlet geometry on the vortex core

precession.

248

9.2. Future Directions

The following issues still need more investigation. (1) test more mathe-

matical models against the experimental value. (2) comparison between

the support vector machine approach and neural networks approach for

cyclone separator performance estimation. (3) create a neural network

model to design the cyclone separator and estimate its performance pa-

rameters. (4) generate performance curves for each geometrical and oper-

ating parameters that affect the cyclone performance to help the designer

in predicting the change of the performance due to change in the cyclone

loading and operating conditions.

Furthermore, many parameters can be taken into consideration in the

optimization process, e.g., erosion rate for harsh environments, surface

roughness, interaction between particles using two-phase flow modeling.

Another idea is to perform an geometry optimization (aeroacoustic) study

to reduce the cyclone-noise especially for domestic applications.

249


250

Appendix A

Mathematical models

Since the first application of aerocyclones in 1886 [3], theories for the es-

timation of both particle collection efficiency and pressure drop of cyclone

have been developed by many contributors using different methods with

various simplifying assumptions. During the past 50 years, interest in

particle collection and pressure theories has steadily increased [196]. The

most widely used mathematical models for the cut-off diameter and pres-

sure drop estimation are:

• Barth model [9]

• The Muschelknautz method of modeling (MM) [29, 77, 114–116, 174,

175]

• Stairmand model [165]

• Shepherd and Lapple model [157]

• Casal and Martinez-Bent model [21]

• Ramachandran model [139]

• Iozia and Leith model [84]

• Rietema model [142]

A.1 General assumptions

Some simplifying assumptions are common to all these models. They can

be considered as offering a good compromise between accurate prediction

and simplification of the equations [3]. They are:

• The particles are spherical.

• The particle motion is not influenced by the presence of neighboring

particles.

• The radial velocity of the gas equals zero.

251

APPENDIX A. Mathematical models

!"#$%&'(

)*+,#*&(

(-.#/"%'()0(

Figure A.1: The control surface concept in the equilibrium-orbit model.

• The radial force on the particle is given by Stokes’ law.

A.2 Barth model

In 1956, Barth [9] proposed a simple model based on force balance (clas-

sified as one of the equilibrium-orbit models [77]). This model enables to

obtain the cut-off diameter and the pressure drop values. Barth proposed

calculating the wall velocity (the tangential velocity near the wall of the

cyclone surface) and the tangential velocity at the control surface CS (

Fig. A.1) through two steps:

1. calculate the wall velocity vθw (the velocity outside CS) from the inlet

velocity vin.

2. use its value to calculate the tangential velocity, vθCS at CS.

In a cyclone with a slot type of rectangular inlet, the inlet jet is compressed

against the wall, resulting in a decrease in the area available for the in-

coming flow and an increase in the velocity. Barth accounted for this by

introducing α, which is defined as the ratio of the moment-of-momentum

of the gas in the inlet and the gas flowing along the wall.

α =vinRin

vθwR(A.1)

252

A.2. Barth model

Figure A.2: Inlet flow pattern for tangential inlet cyclone [77]

where Rin is the radial position of the center of the inlet (Fig. A.2), For a

slot inlet Rin = R− b2 , where b is the inlet width and R = D/2 is the cyclone

radius.

Hoffmann and Stein [77] gave an algebraic relations for α, the simplest of

which is

α = 1− 0.4

(b

R

)0.5

(A.2)

So vθw can be calculated as vθw = vinRin

αR , as the inlet velocity is given or

calculated from vin = Qa b where Q is the gas volume flow rate, a is the inlet

height.

To get from vθw to the tangential velocity at CS vθCS , Barth gave the fol-

lowing relation

vθCS =vθw(

RRx

)

1 + HCSRπfvθwQ

(A.3)

Introducing Eq. (A.1), one obtains

vθCS =πRinRxvx

a bα+HCSπfRin(A.4)

Where Rx is the vortex finder radius Rx = Dx

2 , vx is the mean axial velocity

in the vortex finder vx = QπR2

x, and HCS is the height of the control surface

extending from the bottom of the vortex finder to the cyclone bottom.

253


For the friction factor f , the following relation can be used for hydraulically

smooth cyclone surface according to Hoffmann and Stein [77],

f = 0.05(1 + 3√co) (A.5)

Where co is the mass ratio of dust feeding the cyclone to the gas flow rate

(dimensionless).

Estimation of the pressure drop

Barth subdivided the pressure drop into three contributions:

1. the inlet losses (Barth assumed that this loss could be effectively

avoided by good design).

2. the losses in the cyclone body.

3. the losses in the vortex finder.

The pressure drop in the cyclone body can be estimated from,

∆Pbody =1

2ρv2x

(Dx

D

) 1(

vxvθCS

− H−S0.5Dx

f)2 −

(vθCs

vx

)2

(A.6)

where f is the friction factor (calculated from Eq. A.5). This model ac-

counts for the effect of solid loading upon pressure loss via the total friction

factor f .

The pressure drop in the vortex finder can be estimated using a semi- em-

pirical approach as,

∆Px =1

2ρv2x

((vθCS

vx

)2

+K

(vθCS

vx

) 43

)(A.7)

where K is the vortex finder entrance factor (K = 3.41 for rounded edge

and K = 4.4 for sharp edge)

The total pressure drop ∆P = ∆Pbody + ∆Px can be made dimensionless

using the average inlet velocity vin = Qab leading to the so called Euler

number Eu

Eu =∆P

0.5ρv2in(A.8)

254

A.2. Barth model

Estimation of the cut-off diameter

As mentioned above, the Barth’s model [9] is based on an equilibrium-orbit

model [77]. This model considers the imaginary cylindrical surface CS that

is formed by continuing the vortex finder wall to the bottom of the cyclone

Fig. A.1. It is based on the force balance of a particle that is rotating in CSat radius Rx = Dx

2 . The outwardly directed centrifugal force is balanced

against the inward drag caused by the gas flowing through surface CS into

the inner part of the vortex.

Fcentrifugal = ma = mpv2θCS

Rx= ρp

(π6x3) v2θCS

Rx(A.9)

where the subscript p is for particle properties and x is the particle diame-

ter. Since, for a creep flow, Rep ≪ 1 where Rep =ρgxvrp

µg, the drag coefficient

for sphere CD = 24Rep

. Hence, the Stokesian drag force can be estimated as:

Fdrag = 3πµgvrCS x (A.10)

where vrCS is the uniform radial gas velocity in the surface of CS given by:

vrCS =Q

πDxHCS(A.11)

The following expression obtained from trigonometry relations can be used

to calculate HCS

HCS =(R−Rx)(H − h)

R− (Bc2 )

+ (h− S) if Bc > Dx

= (H − S) if Bc ≤ Dx (A.12)

Large particles are therefore centrifuged out to the cyclone wall (because

the centrifugal force is larger than the drag force. Whereas, small particles

are dragged in and escape out through the vortex finder. The particle size

for which the two forces balance the particles that orbit in equilibrium in

CS is taken as the cyclone’s cut-off diameter x50; it is the particle size that

stands a 50 - 50 chance of being captured. This particle size is called of

fundamental importance and is a measure of the intrinsic separation ca-

pability of the cyclone. Here, all the gas velocity components are assumed

constant over CS for the computation of the equilibrium-orbit size. Equat-

ing the forces, in Eqs. A.9 and A.10 gives the cut-off diameter x50 as:

x50 =

√9vrCS µg Dx

ρp v2θCS

(A.13)

255


A.3 The Muschelknautz method of modeling

(MM)

Hoffmann and Stein [77] stated that the most practical method for mod-

eling cyclone separators at the present time is the Muschelknautz method

(MM) [29, 77, 114–116, 174, 175]. The roots of the Muschelknautz method

(MM) extend back to an early work performed by Barth [9] as it is based

on the equilibrium orbit model [77].

The pressure loss in cyclone

According to the MM model, the pressure loss across a cyclone occurs,

primarily, as a result of friction with the walls and irreversible losses

within the vortex core, the latter often dominating the overall pressure

loss, ∆p = ∆pbody+∆px. In a dimensionless form, it is defined as the Euler

number.

Eu =1

12ρv

2in

[∆pbody +∆px] (A.14)

The wall loss, or the loss in the cyclone body is given by,

∆pbody = fAR

(0.9 Q)

ρ

2(vθwvθCS)

1.5(A.15)

where ρ is the gas density, Q is the gas volume flow rate, AR is the total

inside area of the cyclone contributing to frictional drag. It encompasses

the inside area of the roof, the barrel cylinder, the cone, and the external

surface of the vortex finder.

AR = Aroof +Abarrel +Acone +Avortex finder

= π[R2 −R2

x + 2Rh+ (R+Rb)√(Ht − h)2 + (R −Rb)2

]+2πRxS

(A.16)

where R = D/2, Rx = Dx/2, Rb = Bc/2.

As in the Barth’s model [9, 77], the tangential gas velocity in the entire

space between the wall and the vortex finder can be significantly higher

than the inlet velocity due to constriction of the inlet jet (see Fig. A.2). For

tangential inlet, the inlet jet is compressed against the wall, resulting in

a decrease in the area available for the incoming flow, and an increase in

the velocity. Barth accounts for this by introducing α (see Eq. A.1).

256

A.3. The Muschelknautz method of modeling (MM)

Muschelknautz computes the entrance constriction coefficient α for a con-

ventional slot-type inlet from the following empirical formula, Eq.A.17

[77].

α =1

β

1−

√√√√1 + 4

[(β

2

)2

−(β

2

)]√1− (1− β2)(2β − β2)

1 + c0

(A.17)

where β = b/R, c0 is the ratio of the mass of the incoming solids to the

mass of the incoming gas in the stream feeding the cyclone.

Knowing α, along with vin, Rin and R, one can compute the wall veloc-

ity, vθw (velocity in the vicinity of the wall), vθw = (vinRin)/(αR). Now

compute the geometric mean radius, Rm =√RxR which is needed in the

computation of a wall axial velocity vzw .

vzw =0.9Q

π(R2 −R2m)

(A.18)

Trefz and Muschelknautz [174] found that, approximately 10% of the in-

coming gas “short-circuits” the cyclone and flows radially inwards in a spi-

ral like manner along the roof and down the outside of the vortex finder.

As a consequence, approximately 90% of the incoming flow Q directly par-

ticipates in the flow along the walls and in the formulation of the inner

vortex [77].

To calculate the friction factor f , Muschelknautz and Trefz [77] defined the

cyclone body Reynolds number ReR as:

ReR =RinRmvzwρ

Htµ(A.19)

with ρ and µ are the gas phase density and absolute viscosity, respectively.

Ht is the cyclone total height. The friction factor of the clean gas fair =

f

(ksR,ReR

)where

ksR

is the wall relative roughness of the cyclone wall

(ks = 0.046 [mm] for commercial steel pipe,ksR

6< 6E − 4 for non-negative

logarithm in Eq. A.22). The gas friction factor can be expressed as the sum

of two components, one for smooth wall, fsm, plus an added contribution

due to wall roughness fr.

fair = fsm + fr (A.20)

fsm = 0.323Re−0.623R (A.21)

257


fr =

log

1.6

ksR

− 0.0005999

2.38

−21 +

2.25E5

Re2R

(ksR

− 0.000599

)0.213

(A.22)

The total frictional drag f within a cyclone consists of two components in

the MM, that due to drag on the (pure) gas phase fair and that due to an

additional drag imposed by the moving strand of solids, which is present

at the walls. The total friction factor f becomes,

f = fair + 0.25

(R

Rx

)−0.625√

η c0 Frx ρ

ρstr(A.23)

The second term in this equation is the frictional contribution due to the

solids, where η is the overall efficiency, that is the fraction of incoming

solids collected by the cyclone (Hoffmann and Stein [77] suggested to as-

sume a value of 0.9 to 0.99). Frx is the Froude number (Frx = vx√2 Rx g

)

where vx is the average axial velocity through the vortex finder vx (vx =Q

π R2x

), ρstr term represents the bulk density of the dust or strand layer at

the walls and can be taken as 0.4ρbulk where ρbulk is the bulk density of the

solid [77].

The tangential velocity of the gas at the inner core radius Rcs (see Fig. A.3)

is given by,

vθcs = vθw

(RRx

)

[1 +

f AR

√

RRx

2 Q

] (A.24)

The second contribution to pressure drop is the loss in the core and in the

vortex finder and is given by,

∆px =

[2 +

(vθcsvx

)2

+ 3

(vθcsvx

)4/3]1

2ρv2x (A.25)

Cut-off diameter

A very fundamental characteristic of any lightly loaded cyclone is its cut-off

diameter x50 produced by the spin of the inner vortex. This is the practical

diameter that has a 50% probability of capture. The cut-off diameter is

258

A.4. Stairmand model for pressure drop

Q

0.1Q

Rc

Rcs

Vz

Vx

Vөw

Vөcs

Figure A.3: Geometric parameters and velocities used in MM model [77]

analogous to the screen openings of an ordinary sieve or screen [77]. In

lightly loading cyclones, x50 exercises a controlling influencing on the cy-

clone’s separation performance. It is the parameter that determines the

horizontal position of the cyclone grade-efficiency curve (fraction collected

versus particle size). For low mass loading, the cut-off diameter can be

estimated in MM using Eq. A.26 [77].

x50 =

√9 µ (0.9 Q)

π (ρp − ρ)v2θcs(Ht − S)(A.26)

A.4 Stairmand model for pressure drop

Stairmand [165] estimated the pressure drop as entrance and exit losses

combined with the static pressure loss in the swirl.

Eu = 1 + 2q2(2 (D − b)

Dx− 1

)+ 2

(4 a b

πD2x

)2

(A.27)

259


where q is given by,

q =−(

Dx

2(D−b)

)0.5+(

Dx

2(D−b) +4ARGa b

)0.5(2ARGa b

) (A.28)

where AR is the total wall area of the cyclone body, including the inner

walls of the lid, the cylindrical and the conical sections and the outer wall

of the vortex finder, given by Eq. A.29. G(= f/2 where f is the friction factor)is a wall friction factor, which Stairmand set equal to 0.005.

AR =π(D2 −D2

x

)

4+πDh+πDxS+

π (D +Bc)

2

((H − h)

2+

(D −Bc

2

)2)0.5

(A.29)

A.5 Purely empirical models for pressure drop

1. Sphered and Lapple model [157]

Eu =16 a b

D2x

(A.30)

2. Casal and Martinez-Bent model [21]

Eu = 3.33 + 11.3

(a b

D2x

)2

(A.31)

3. Ramachandran model

The Ramachandran et al. [139] model was developed through a sta-

tistical analysis of pressure drop data for ninety-eight cyclone de-

signs. The model is shown to perform better than the pressure drop

models of Shepherd and Lapple [157], and Barth [9] in comparison

with experimental results.

Eu = 20

a b

D2x

SD

HD

hD

Bc

D

1/3

(A.32)

260

A.6. Iozia and Leith model for the cut-off diameter

!!

"!!

!!

!!!

!!!

!!!

!

!!!

!!!

!!!

!!!!!!!!

!!!!

!!!!

Figure A.4: The control surface used in Iozia and Leith model [85].

A.6 Iozia and Leith model for the cut-off di-

ameter

The Iozia and Leith model [84] is similar to the model of Barth [9] as it is

also based on the equilibrium-orbit theory (Force balance). Iozia and Leith

[84] gave the following expression for the cut-off diameter,

x50 =(9 µ Q) /

(π HCS ρp V 2

θmax

)1/2(A.33)

where HCS is the core height (height of the control surface of Barth’s

model) Vθmax is the maximum tangential velocity, that occurs at the edge

of the control surface CS, Fig. A.4. In this model however the value of

the core diameter dc and the tangential velocity at the core edge Vθmax are

calculated from regression of experimental data using the following equa-

tions.

Vθmax = 6.1vin

(ab/D2

)−0.61(Dx/D)

−0.74(Ht/D)

−0.33

(A.34)

dc = 0.52D(ab/D2

)−0.25(Dx/D)1.53 (A.35)

It is clear from this model that the most important geometry parameters

that affect the cyclone collection efficiency are the vortex finder diameter,

the ratio of inlet area to exit area and the cyclone height.

261


A.7 Rietema model for cut-off diameter

The Rietema model [142] estimates the cut-off diameter x50 using Eq. A.36.

x50 =

√µ ρ Q

(ρp − ρ) Ht ∆P(A.36)

This model relates, the separation cut-off diameter x50 to the pressure

drop. Hence, the pressure drop needs to be predicted to use the model.

A good pressure drop model for this purpose is that of Shepherd and Lap-

ple. The interested reader can refer to Hoffmann and Stein [77] for more

details.

262

Appendix B

Optimization Techniques

Two optimization techniques have been used throughout this thesis, namely

the Nelder-Mead technique and the genetic algorithms. In the following

sections, the details of these techniques will be explained.

B.1 Nelder-Mead

The Nelder-Mead method, also known as downhill simplex method is a

commonly used nonlinear optimization technique, The technique was pro-

posed by Nelder and Mead [119] and is a technique for minimizing an

objective function in a many-dimensional space. It requires only function

evaluations, and no calculation of derivatives [134]. The downhill simplex

nonlinear optimization technique has been used by many researchers [e.g.,

4, 12]. According to Bernon et al. [12] Powell’s algorithm and the down-

hill simplex one are ones of the most used minimization algorithms; the

downhill-simplex algorithm became the most performant. Further more,

the Statgraphics XV package has been used for design of experiment and

optimization, with the only available optimization technique is the Nelder-

Mead technique. In this study, the target is to obtain the global optimum

values. Consequently, no linear constrains applied.

The idea is to employ a moving simplex in the design space to surround

the optimal point and then shrink the simplex until its dimensions reach a

specified error tolerance [96]. In n-dimensional space, a simplex is a figure

of n+1 vertices connected by straight lines and bounded by polygonal faces.

If n = 2, a simplex is a triangle; if n = 3, it is a tetrahedron.

For two variables, the simplex is a triangle and the method is a pattern

search that compares function values at the three vertices of a triangle.

The worst vertex where f(x, y) is largest, is rejected and replaced with a

263

APPENDIX B. Optimization Techniques

new vertex. A new triangle is formed, and the search is continued. The pro-

cess generates a sequence of triangles (which might have different shapes),

for which the function values at the vertices get smaller and smaller. The

size of the triangles is reduced, and the coordinates of the minimum point

are found [108]. The algorithm is stated using the term simplex (a general-

ized triangle in N dimensions) and will find the minimum of a function of

N variables. It is effective and computationally compact. In the following

paragraphs, a brief explanation is given for this technique. The interested

reader can refer to Mathews and Fink [108].

!

R

G

W

Md

d

Reflection

B

GW

M C2

Contraction

R

M

C1

B

GW

M

Shrinkage

S

B

GW

Md

d

Expansion

d

E

R

B

GW

Original Simplex

Figure B.1: Basic operations in the downhill simplex method for two dimensions

space [108]

The initial triangle BGW

Let f(x, y) be the function that is to be minimized. To start, we are given

three vertices of a triangle: V k = (xk, yk), k = 1, 2, 3. The function f(x, y) is

then evaluated at each of the three points zk = f(xk, yk) for k = 1, 2, 3. The

subscripts are reordered so that z1 ≤ z2 ≤ z3. We use the notation [108],

B = (x1, y1), G = (x2, y2), and W = (x3, y3) (B.1)

to help remember that B is the best vertex, G is good (next to best), and W

is the worst vertex.

264

B.1. Nelder-Mead

Midpoint of the good side

The construction process uses the midpoint of the line segment joining B

and G. It is found by averaging the coordinates:

M =B +G

2=

(x1 + x2

2,y1 + y2

2

)(B.2)

Reflection using the point R

The function decreases as we move along the side of the triangle from W

to B, and it decreases as we move along the side from W to G. Hence it is

feasible that f(x, y) takes on smaller values at points that lie away from

W on the opposite side of the line between B and G. We choose a test point

R that is obtained by ‘Reflecting’ the triangle through the side BG. To

determine R, we first find the midpoint M of the side BG. Then draw

the line segment from W to M and call its length d. This last segment is

extended a distance d through M to locate the point R (see Fig. B.1). The

vector formula for R is [108]

R = M + (M −W ) = 2M −W (B.3)

Expansion using the point E

If the function value at R is smaller than the function value at W, then

we have moved in the correct direction toward the minimum. Perhaps the

minimum is just a bit further than the point R. So we extend the line

segment through M and R to the point E. This forms an expanded triangle

BGE. The point E is found by moving an additional distance d along the

line joining M and R (see Fig. B.1). If the function value at E is less than

the function value at R, then we have found a better vertex than R. The

vector formula for E is [108]

E = E + (R−M ) = 2R−M (B.4)

Contraction using the point C

If the function values at R and W are the same, another point must be

tested. Perhaps the function is smaller at M, but we cannot replace W

with M because we must have a triangle (for 2-D space). Consider the two

midpoints C1 and C2 of the line segments WM and MR, respectively

(see Fig. B.1). The point with the smaller function value is called C, and

the new triangle is BGC.

265


Shrink toward B

If the function value at C is not less than the value at W, the point G and

W must be shrunk towards B (see Fig. B.1). The point G is replaced with

M, and W is replaced with S, which is the midpoint of the line segment

joining B with W [108].

Logical decisions for each step

A computationally efficient algorithm should perform function evaluation

only is needed. In each step, a new vertex is found, which replace W. As

soon as it is found, further investigation is not needed, and the iteration

step is completed. The logical details for two-dimensional cases are ex-

plained in Fig. B.2.

B.2 Genetic algorithms (GA)

The genetic algorithm is an optimization technique for solving both con-

strained and unconstrained optimization problems that is based on natu-

ral selection, the process that drives biological evolution [109]. The genetic

algorithm repeatedly modifies a population of individual solutions. At each

step, the genetic algorithm selects individuals at random from the current

population to be parents and uses them to produce the children for the

next generation. Over successive generations, the population evolves to-

ward an optimal solution. The genetic algorithm can be used to solve a

variety of optimization problems that are not well suited for standard opti-

mization algorithms, including problems in which the objective function is

discontinuous, no differentiable, stochastic, or highly nonlinear [109]. The

genetic algorithm uses three main types of rules at each step to create the

next generation from the current population [109]:

• Selection rules select the individuals, called parents, that contribute

to the population at the next generation.

• Crossover rules combine two parents to form children for the next

generation.

• Mutation rules apply random changes to individual parents to form

children.

The genetic algorithm differs from a classical, derivative-based, optimiza-

tion algorithm in two main ways, as summarized in Table B.1.

266

B.2

.G

en

etic

alg

orith

ms

(GA

)

If f (R) < f (G) es No

Case (i)either reflect or extend

Case (ii)either contract or shrink

If f (B) < f (R)Yes No

Replace W with R Compute E and f (E)

If f (E) < f (B)Yes No

Replace W with E Replace W with R

If f (R) < f (W)Yes No

Replace W with R

Compute C-(W+M)/2or C=(M+R)/2and f (C)

Compute E and f (E)

If f (C) < f (W)Yes No

Replace W with C Compute S and f (S)

Replace W with SReplace G with M

Start next cycle ...


Start next cycle ... Start next cycle ...



Start a new cycle

Figure B.2: The logical decisions for the Nelder-Mead algorithm

267


Table B.1: Comparison between the classical algorithms and the genetic algorithm

Classical Algorithm Genetic Algorithm

• It generates a single point at each itera-tion.

• The sequence of points approaches anoptimal solution.

• It selects the next point in the sequenceby a deterministic computation.

• In most cases, it find a relative (local)optimum that is closest to the startingpoint.

• It generates a population of points ateach iteration.

• The best point in the population ap-proaches an optimal solution.

• It selects the next population by compu-tation, which uses random number gen-erators.

• It can find the global optimum solutionwith a high probability.

B.2.1 Description of the genetic algorithm process

Genetic algorithms use a population of configurations, called individual, to

evolve over a number of generations. Each individual is represented by its

genetic material, called a chromosome and every variable in each individ-

ual is termed as the gene. For optimization purpose, the chromosome is

described by the design variables.

The process starts with an initial population of n individuals. The perfor-

mance of each individual is then evaluated in regard to the fitness function

and the handling of constraints (if some are considered) [28]. A selection

is done in the population to identify valuable parents. Higher is the per-

formance of an individual, higher is its probability to become parent [28].

Two parents are match randomly to exchange their genetic materials to

form the offspring for the next generation. This exchange process is called

crossover. If this process does not happen, the parents are directly trans-

ferred to the next generation meaning the cloning of these individuals. Af-

ter the crossover operator and before forming the next generation, all the

individuals are forced to undergo a mutation process [28]. The evolution

procedure is repeated until the population converges to a certain level or

simply if the maximum number of generation is reached, Fig. B.3.

B.2.2 Genetic operators

Population

Population type specifies the type of the input to the fitness function [109].

There are two options in Matlab, double vector and Bit string (binary chro-

mosome). The double vector option has been used in this study. A double

vector chromosome is simply a row vector of double values. These values

can be thought of as genes. Thus, 7 genes/chromosome means a double

268

B.2. Genetic algorithms (GA)

Start

Create initial random population

Evaluate fitness for each individual

Store best individuals (chromosomes)

Creating the mating pool (selection of parents)

Create the next generation by applying crossover & mutation

(new generation: offspring)

Optimal solution found?Stooping criteria met?

Yes

Stop

Evaluate fitness for each individual

No

Figure B.3: Flow chart for the genetic algorithms process

vector of 7 elements [27]. The population size n specifies how many indi-

viduals there are in each generation.

Fitness scaling

The scaling function converts raw fitness scores returned by the fitness

function to values in a range that is suitable for the selection function.

Scaling function specifies the function that performs the scaling. One can

choose from the following functions [109]: Rank, proportional, Top or Shift

linear. The rank option has been used in this study. Rank scales the raw

scores based on the rank of each individual, rather than its score. The

rank of an individual is its position in the sorted scores. The rank of the

fittest individual is 1, the next fittest is 2, and so on. Rank fitness scaling

removes the effect of the spread of the raw scores.

269


Selection

Different types of selections are implanted in the optimization toolbox in

Matlab 2010a [109], but only the tournament selection is used in this study

(it is the only available selection type for multi-objective optimization in

the Matlab toolbox). The tournament selection randomly identifies some

competitors from the population to compete against each other. The one

with the highest performance win a parent status. The tournament selec-

tion permits to control the selective pressure put on the population. The

population diversity is adjusted by modification of the competitor num-

ber. Greater competitor numbers in the tournament increase the chances

to focus the search over the best individuals meaning a greater selective

pressure. On the other hand, with only two competitors, the possibility of

becoming a parent remains open to a larger band of the population mean-

ing a lesser selective pressure [28]. In other words, the tournament size

can take values between 1 and n (population size). Larger values give

more chances to the best samples to be selected and to create offsprings.

It favors a rapid, although perhaps premature, convergence to a local op-

timum. Very small values result in a more random selection of parents

[173]. The default value of tournament size of four has been used.

Reproduction

Reproduction options determine how the genetic algorithm creates chil-

dren at each new generation. Elite count specifies the number of individ-

uals who are guaranteed to survive to the next generation. the Matlab

manual [109] suggested to use Elite count to be two (a positive integer

less than or equal to population size). Crossover fraction specifies the frac-

tion of the next generation that crossover produces. Mutation produces

the remaining individuals in the next generation. Matlab manual [109]

suggested to use a crossover fraction of 0.8 (a fraction between 0 and 1).

Crossover

Crossover combines two individuals, or parents, to form a new individual,

or child (offspring), for the next generation [109]. the Matlab optimiza-

tion toolbox offers six functions: Scattered, single point, two point, inter-

mediate, heuristic and arithmetic. The intermediate function has been

used in this study. Intermediate (also, called weighted crossover [28]) cre-

ates children by a random weighted average of the parents. Intermedi-

ate crossover is controlled by a single parameter ratio: child1 = parent1+

rand*Ratio*(parent2 - parent1). If the ratio is in the range [0,1], the chil-

dren produced are within the hypercube defined by the parents locations

270

B.2. Genetic algorithms (GA)

at the opposite vertices. If Ratio is in a larger range, say 1.1, children can

be generated outside the hypercube. Ratio can be a scalar or a vector of

length equals the number of variables. If the ratio is a scalar, all the chil-

dren lie on the line between the parents. If the ratio is a vector, children

can be any point within the hypercube.

Mutation

Mutation functions make small random changes in the individuals in the

population, which provide genetic diversity and enable the genetic algo-

rithm to search for a broader space. To specify the function that performs

the mutation in the mutation function field, one can choose from the follow-

ing functions: constraint dependent default, gaussian, uniform and adap-

tive feasible. The constraint dependent default chooses: Gaussian if there

are no constraints or adaptive feasible otherwise. 1) Gaussian adds a ran-

dom number to each vector entry of an individual. This random number

is taken from a gaussian distribution centered on zero. The standard de-

viation of this distribution can be controlled with two parameters. The

scale parameter (default value of 1.0) determines the standard deviation

at the first generation. The shrink parameter controls how the standard

deviation shrinks as generations go by. If the shrink parameter is 0, the

standard deviation is constant. If the Shrink parameter is 1 (the default

value), the standard deviation shrinks to 0 linearly as the last generation

is reached. 2) Uniform is a two-step process. First, the algorithm selects

a fraction of the vector entries of an individual for mutation, where each

entry has the same probability as the mutation rate of being mutated. In

the second step, the algorithm replaces each selected entry by a random

number selected uniformly from the range for that entry. 3) Adaptive fea-

sible randomly generates directions that are adaptive with respect to the

last successful or unsuccessful generation. A step length is chosen along

each direction so that linear constraints and bounds are satisfied. The

constraint dependent default has been used in this study.

Stopping criteria

The stopping criteria determines what causes the algorithm to terminate.

Matlab optimization toolbox has the following stopping criteria, the default

values are given in parentheses.

• Generations specific the maximum number of iterations the genetic

algorithm performs (100).

• Time limit specifies the maximum time in seconds the genetic algo-

rithm runs before stopping (∞).

271


• Fitness limit - If the best fitness value is less than or equal to the

value of fitness limit, the algorithm stops (−∞).

• Stall generations - If the weighted average change in the fitness func-

tion value over stall generations is less than Function tolerance, the

algorithm stops (50).

• Stall time limit - If there is no improvement in the best fitness value

for an interval of time in seconds specified by Stall time limit, the

algorithm stops (∞).

• Function tolerance - If the cumulative change in the fitness func-

tion value over stall generations is less than Function tolerance, the

algorithm stops (1E-6).

B.3 Multi-objective optimization

In case of cyclone separator geometry optimization for minimum Euler

number and minimum cut-off diameter, the objectives are conflicting with

each other. There is no best solution for which all objectives are optimal si-

multaneously [181]. The increase of one objective will lead to the decrease

of the other objective. Then, there should be a set of solutions, the so-called

Pareto optimal set or Pareto front, in which one solution cannot be domi-

nated by any other member of this set. The definition of domination is as

given in Wang et al. [181]

For minimal problem, a solution a ∈ X dominates a solution

b ∈ X(a ≻ b) if and only if it is superior or equal in all objectives

and at least superior in one objective. This can be expressed as

follows [181]:

a ≻ b, if

∀i ∈ 1, 2, . . . ,m : fi(a) ≤ fi(b)∧∃j ∈ 1, 2, . . . ,m : fj(a) < fj(b)

where m is the number of objective functions.

Each solution of Pareto optimal set is called a non-inferior solution, which

is corresponding to one point on the Pareto front. A general example of the

Pareto front with two objectives is illustrated in Fig. B.4. In this example,

the Pareto front is composed of six points, which are indifferent to each

other (denoted by filled circles • in Fig. B.4). While, points with hollow

circles are not belonging to the Pareto front, since they are dominated by

the Pareto front points.

272

B.3. Multi-objective optimization

Figure B.4: Pareto front for two objective functions.

Nondominated sorting genetic algorithm (NSGA-II)

Recently, a number of multi-objective genetic algorithms based on the Pareto

optimal concept has been proposed. The well known nondominated sort-

ing genetic algorithm (NSGA-II) proposed by Deb et al. [34] is one of the

most widely used multi-objective genetic algorithm since it provides ex-

cellent results as compared with other multi-objective genetic algorithms

proposed [25]. A brief description of processes of NSGA-II is presented

below, Fig. B.5.

Initially, a parent population of size n is generated randomly. All individ-

uals in this population are sorted into different front levels based on the

domination of pair comparison. Each front level is assigned a fitness (or

a rank) which equals its non-domination level. Level 1 is the top level in

which the individual is dominated by none of the other individuals; level 2

is the secondary level in which the individual is dominated by some indi-

viduals only in level 1, and so on. In the same front level, the location of the

finite number of solutions is expected to be distributed uniformly. In other

words, a large diversity of the individuals can prevent the results from

sticking into a local optimum. Therefore, another feature, called crowding

distance, is adopted to evaluate the local aggregation of individuals. The

273


definition of crowding distance is [34, 181]:

Cj =

Nobj∑

i=1

F j+1i − F j−1

i

(F ji )max − (F j

i )min

(B.5)

where Cj is the crowding distance of point j on the Pareto front, (F ji ) is the

value of the fitness function i at point j (i = 1, 2 for two objective functions).

For boundary points, the crowding distance is set to the maximum value

of the system in order to ensure that these points can survive to the next

generation. In the same rank level, individuals who have larger crowding

distance also have more opportunities to be selected [182].

Simulated binary crossover

In NSGA-II, a simulated binary crossover operator is used, which simu-

lates the working principle of the single point crossover operator on binary

strings [182]. Let x1,ti and x2,t

i denote two parent individuals for ith selec-

tion in generation t. The procedure of computing the children individuals

x1,t+1i and x2,t+1

i from parent individuals x1,ti and x2,t

i is described below. A

spread factor βi is defined as the ratio of the absolute difference in children

values to that of parent values:

βi =

∣∣∣∣∣x2,t+1i − x1,t+1

i

x2,ti − x1,t

i

∣∣∣∣∣ (B.6)

Firstly, a random number ui ∈ [0, 1] is generated, whereafter, from a speci-

fied probability distribution function, the ordinate βqi is found so that the

area under the probability curve from 0 to βqi is equal to the chosen ran-

dom number ui. The probability distribution used to create a child indi-

vidual is derived to have a similar search power as that in a single-point

crossover in binary-coded GAs and is given as follows [182]:

P (βi) =

0.5(η + 1)βiη, if βi ≤ 1

0.5(η + 1)/βiη+2, otherwise

(B.7)

where, η is the distribution index which can be any nonnegative real num-

ber. A lager value of η gives a higher probability for creating near parent

individuals and a small value of η allows distant individuals to be selected

as children individuals. Using Eq.B.7, the βqi can be calculated as follows

[182]:

βqi =

(2ui)1

η+1 , if ui ≤ 0.5(

12(1−ui)

) 1η+1

, otherwise(B.8)

274

B.3. Multi-objective optimization

Thereafter, the children individuals can be obtained by [182]:

x1,t+1i = 0.5

[(1 + βqi)x

1,ti + (1− βqi)x

2,ti

]

x2,t+1i = 0.5

[(1 + βqi)x

1,ti + (1− βqi)x

2,ti

] (B.9)

Note that two children individuals are symmetric about the parent indi-

viduals. This is deliberately used to avoid any bias towards any particular

parent individual in a single crossover operation.

Mutation Operator

Let xk be the component of an individual xi, which is going to be mu-

tated. xuk and xl

k stand for the maximum and minimum value of this com-

ponent in all individuals, respectively. The mutated individual yk can be

calculated as follows [182]:

yk = xk + δq(xuk − xl

k

)(B.10)

Here δq is a mutation parameter, which stands for:

δq =

(2r + (1− 2r)(1− δ1)

ηm+1)1/(ηm+1) − 1, if r ≤ 0.5

1−(2r(1 − r) + 2(r − 0.5)(1− δ2)

ηm+1)1/(ηm+1)

otherwise

(B.11)

where ηm is the mutation index which is set to 20 (1- crossover fraction) in

general, and r is a random number. The intermediate variables δ1 and δ2stand for:

δ1 =xk − xl

k

xuk − xl

k

(B.12)

δ2 =xuk − xk

xuk − xl

k

(B.13)

275


Figure B.5: Flow chart for the NSGA-II process

276

List of Publications

Journal Articles

Published

1. K. Elsayed, C. Lacor. Modeling and Pareto Optimization of Gas Cy-clone Separator Performance Using RBF Type Artificial Neural Net-works and Genetic Algorithms. Powder Technology. In Press, Ac-cepted Manuscript, Available online 17 October 2011doi: http://dx.doi.org/10.1016/j.powtec.2011.10.015

2. K. Elsayed, C. Lacor. Numerical and Empirical Modeling of the FlowField and Performance in Cyclones of Different Cone-Tip Diameters.Computers & Fluids, Vol. 51, No. 1, pp. 48 - 59, 2011.doi: http://dx.doi.org/10.1016/j.compfluid.2011.07.010

3. K. Elsayed, C. Lacor. Modeling, Analysis and Optimization of Air-cyclones Using Artificial Neural Network, Response Surface Method-ology and CFD Simulation Approaches. Powder Technology, Vol.212, No. 1, pp. 115 - 133, 2011.doi: http://dx.doi.org/10.1016/j.powtec.2011.05.002

4. K. Elsayed, C. Lacor. The Effect of Cyclone Inlet Dimensions onthe Flow Pattern and Performance. Applied Mathematical Mod-elling, Vol. 35, No. 4, pp. 1952 - 1968, 2011.doi: http://dx.doi.org/10.1016/j.apm.2010.11.007

5. K. Elsayed, C. Lacor. Optimization of the Cyclone Separator Geom-etry for Minimum Pressure Drop Using Mathematical Models andCFD Simulations.Chemical Engineering Science, Vol. 65, No. 22,pp. 6048 - 6058, 2010.doi: http://dx.doi.org/10.1016/j.ces.2010.08.042 (listed inthe Top 25 Hottest Articles Chemical Engineering > Chemical Engi-neering Science, October to December 2010).

Accepted with revision

1. K. Elsayed, C. Lacor. The Effect of Cyclone Vortex Finder Dimen-sions on the Flow Pattern and Performance using LES.Computers & Fluids, 2011.

277

http://dx.doi.org/10.1016/j.powtec.2011.10.015

http://dx.doi.org/10.1016/j.compfluid.2011.07.010


http://dx.doi.org/10.1016/j.apm.2010.11.007

http://dx.doi.org/10.1016/j.ces.2010.08.042

With reviewers

1. K. Elsayed, C. Lacor. CFD Modeling and Multi-Objective Optimiza-tion of Cyclone Geometry Using Desirability Function, Artificial Neu-ral Networks and Genetic Algorithms.Applied Mathematical Modelling. Submitted, 2011.

2. K. Elsayed, C. Lacor. A CFD Study of the Effect of the Dust OutletGeometry on the Performance and Hydrodynamics of Gas Cyclones.Computers & Fluids. Submitted, 2011.

3. K. Elsayed, C. Lacor. Numerical Study of the Effect of Cyclone Cone& Barrel Height On The Flow Pattern And Performance.Applied Mathematical Modelling. Submitted, 2011.

International conference proceedings

1. K. Elsayed, C. Lacor. Single and Multi-Objective Optimization of theCyclone Separator Geometry Using Artificial Neural Network andGenetic Algorithm, EUROGEN 2011, Evolutionary and Determinis-tic Methods for Design, Optimization and Control with Applicationsto Industrial and Societal Problems, Capua, Italy, 14-16 September2011.

2. K. Elsayed, C. Lacor. The Effect of Dust Outlet Shape on the FlowPattern and Performance of Cyclone Separators, 10th InternationalSymposium on Experimental and Computational Aerothermodynam-ics of Internal Flows (ISAIF10), Brussel, Belgium, 4-7 July 2011.

3. K. Elsayed, C. Lacor. The Effect of the Cyclone Separator ConeHeight on the Performance Using Artificial Neural Network Modeland CFD Simulations, 10th International Symposium on Experimen-tal and Computational Aerothermodynamics of Internal Flows (ISAIF10),Brussel, Belgium, 4-7 July 2011.

4. K. Elsayed, C. Lacor. A CFD Study of the Effect of Cyclone BarrelHeight on Its Performance Parameters, 8th International Conferenceon Computational Fluid Dynamics in the Oil & Gas, Metallurgicaland Process Industries, Trondheim, Norway, 21-23 June 2011.

5. K. Elsayed, C. Lacor. Numerical Investigations of the Effect of Dif-ferent Dust Outlet Designs on the Cyclone Performance and FlowPattern, 14th International Conference on Aerospace Sciences andAviation Technology (ASAT-14), Cairo, Egypt, 24-26 May 2011.

6. K. Elsayed, C. Lacor. Optimization of the Cyclone Separator Geom-etry for Minimum Pressure Drop Based on Artificial Neural NetworkModel And CFD Simulation, ECCOMAS thematic conference, CFD &Optimization, Antalya, Turkey, 23-25 May 2011.

278

7. K. Elsayed, C. Lacor. Multi-Objective Optimization of Gas CycloneBased On CFD Simulation, ECCOMAS thematic conference, CFD &Optimization, Antalya, Turkey, 23-25 May 2011.

8. K. Elsayed, C. Lacor. The effect of cyclone height on the flow patternand performance using LES, Tenth International Congress of FluidDynamics (ICFD10), ASME, Egypt, ICFD10-EG-3003, Ain Soukhna,Red Sea, Egypt, 16-19 December 2010.

9. K. Elsayed, C. Lacor. The effect of cyclone inlet width on the flowpattern and performance, Tenth International Congress of Fluid Dy-namics (ICFD10), ASME, Egypt, ICFD10-EG-3085, Ain Soukhna,Red Sea, Egypt, 16-19 December 2010.

10. K. Elsayed, C. Lacor. Numerical study on the effect of cyclone in-let height on the flow pattern and performance, Tenth InternationalCongress of Fluid Dynamics (ICFD10), ASME, Egypt, ICFD10-EG-3068, Ain Soukhna, Red Sea, Egypt, 16-19 December 2010.

11. K. Elsayed, C. Lacor. The effect of cyclone dustbin on the flow pat-tern and performance, Tenth International Congress of Fluid Dynam-ics (ICFD10), ASME, Egypt, ICFD10-EG-3092, Ain Soukhna, RedSea, Egypt, 16-19 December 2010.

12. K. Elsayed, C. Lacor. Application of Response Surface Methodologyfor Modeling and Optimization of the Cyclone Separator for Mini-mum Pressure Drop, Fifth European Conference on ComputationalFluid Dynamics (ECCOMAS CFD10), Lisbon, Portugal, 14-17 June2010.

13. K. Elsayed, C. Lacor. The Effect of Vortex Finder Diameter on Cy-clone Separator Performance and Flow Field, Fifth European Confer-ence on Computational Fluid Dynamics (ECCOMAS CFD10), Lisbon,Portugal, 14-17 June 2010.

14. V. Agnihotri, K. Elsayed, C. Lacor, S. Verbanc. Numerical Studyof Particle Deposition in the Human Upper Airways With Emphasison Hot Spot Formation and Comparison Of LES and RANS Models,Fifth European Conference on Computational Fluid Dynamics (EC-COMAS CFD10), Lisbon, Portugal, 14-17 June 2010.

15. K. Elsayed, C. Lacor. Modeling of the Gas and Particle Flow in theCyclone Separator Using LES, RANS and Mathematical Models, 14th

International Conference on Applied Mechanics and Mechanical En-gineering (AMME-14), Military Technical College, Cairo, Egypt, 25-27 May 2010.

16. K. Elsayed, C. Lacor. Optimization of the Cyclone Separator Geom-etry Based On CFD Simulation, ERCOFTAC day, Vrije UniversiteitBrussel, Brussels, Belgium, 3rd December 2009.

279

17. K. Elsayed, C. Lacor. A CFD Study of the Effects of Cone Dimen-sions on the Flow Field of Cyclone Separators Using LES, 13th In-ternational Conference on Aerospace Sciences & Aviation Technology(ASAT-13), Military Technical College, Cairo, Egypt, 26-28 May 2009.

18. K. Elsayed, C. Lacor. Investigation of the Geometrical ParametersEffects on the Performance and the Flow-Field of Cyclone Separa-tors Using Mathematical Models and Large Eddy Simulation, 13th

International Conference on Aerospace Sciences & Aviation Technol-ogy (ASAT-13), Military Technical College, Cairo, Egypt, 26-28 May2009.

National Conference proceedings

1. K. Elsayed, C. Lacor. Study Of the Effects of Geometrical Parame-

ters on the Performance of Cyclone Separators, 8th National Congress

on Theoretical and Applied Mechanics (NCTAM2009), Vrije Univer-

siteit Brussel, Brussels, Belgium, 28-29 May 2009.

2. K. Elsayed, C. Lacor. Effects of Geometry Parameters on the Flow

Field of Cyclone Separator, Poster day, the GRAduate School in ME-

CHanics (GRASMECH 2008), Royal Military Academy, Brussels, Bel-

gium, 3rd October 2008.

Technical reports

1. K. Elsayed, S. Jayaraju, C. Lacor. Wood Pellet Transport in the

Biomass Burner of the Rodenhuize Power Plant, Research report, La-

borelec Company, Belgium, April 2009.

2. K. Elsayed, C. Lacor. The State of the Art for Flow in Cyclone Sepa-

rator, Internal report, Vrije Universiteit Brussel, Brussels, Belgium,

April 2008.

280

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