UNCERTAINTY QUANTIFICATION IN HIGH DENSITY FLUID … · Computational Fluid Dynamics Sensitivity...

UNCERTAINTY QUANTIFICATION IN

HIGH-DENSITY FLUID RADIAL-INFLOW

TURBO-EXPANDERS AND DIFFUSERS FOR

RENEWABLE LOW-GRADE

TEMPERATURE CYCLES

AIHONG ZOU

Master of Thermophysics Engineering

Submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Chemistry, Physics and Mechanical Engineering

Science and Engineering Faculty

Queensland University of Technology

2019

Dedicated to

My most beloved parents.

iv Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for Renewable

Low-grade Temperature Cycles

QUT Verified Signature

vi Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for Renewable


Keywords

Renewable Energy

Organic Rankine Cycle (ORC)

Radial-inflow Turbine

Radial-inflow Turbo-expander

Conical Diffuser

Annular-radial Diffuser

High-density Fluid

Uncertainty Quantifications (UQ)

Equation of State (EOS)

Computational Fluid Dynamics

Sensitivity Analysis

Total-to-static Efficiency

Pressure Recovery

vii Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for Renewable


Abstract

In order to reduce greenhouse gas emissions, the use of renewable energy for the

generation of electricity is considered to be an attractive alternative to fossil fuels.

Employing low-grade temperature resources from renewable energy sources (such as

biomass energy, waste energy, ocean thermal energy, and geothermal energy) is

considered to be a potential solution to the generation of electricity, rather than using

fossil fuels. With the purpose of producing electricity from low-grade temperature

resources, Organic Rankine Cycle (ORC) is becoming a leading thermodynamic cycle

that is capable of extracting more energy, thanks to the use of high-density fluids

compared to other conventional cycles.

Turbines are key components for electricity generation, and thus play a

significant role in ORC. Radial-inflow turbines are commonly used, as studies have

shown their overall suitability and performance for low-grade temperature, high-

density fluid power cycles. In order to develop an advanced and robust optimised

design for radial-inflow turbines, it is critical to develop numerical techniques and

consider uncertainties in the optimisation process. Thus, an advanced and robust

framework coupling a Computational Fluid Dynamics (CFD) solver with an

Uncertainty Quantification (UQ) approach is proposed in this study as an effective

way to implement radial-inflow turbine sensitive analysis.

The first part of this study validates the CFD solver, and builds a robust

framework for connecting the CFD solver with the UQ approach. The UQ analysis of

the high pressure ratio, single stage, ideal gas radial-inflow turbine performance used

in the Sundstrand Power Systems T-100 Multi-purpose Small Power Unit is

investigated. A deterministic, three-dimensional (3D) volume-averaged CFD solver is

coupled with a UQ approach that employs a non-statistical generalised Polynomial

Chaos (gPC) representation, based on a pseudo-spectral projection method. One of the

advantages of this approach is that it does not require any modification of the CFD

code for the propagation of random disturbances in the aerodynamic and geometric

fields. The stochastic results highlight the importance of the blade thickness and

trailing edge tip radius for the total-to-static efficiency of the turbine, compared to the

rotational speed and trailing edge tip length. From a theoretical point of view, the use

viii Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for

Renewable Low-grade Temperature Cycles

of the gPC representation on an arbitrary grid also allows investigation of the

sensitivity of the blade thickness profiles in terms of turbine efficiency. The gPC

approach is also applied to couple random parameters. The results show that the most

influential coupled random variables are trailing edge tip radius coupled with

rotational speed. More importantly, the proposed framework has been validated, and

the 3D CFD results for total-to-static efficiency are compared against the experimental

data at the rig conditions. The results are in really good agreement with the

experiments, with a maximum difference less than 1%. This framework validation

builds the foundation for further work. In the following stages, UQ analysis of a high-

density fluid radial-inflow turbine needs to be conducted to improve the overall ORC

efficiency.

The inclusion of uncertainties in the design of ORC radial-inflow turbines for

renewable low-grade temperature power cycles is becoming a crucial aspect in the

development of robust and reliable power blocks that are capable of handling a better

range of efficiencies over a wider range of operational conditions. Modelling high-

density fluids using existing Equations of State (EOS) adds complexity to improving

the system’s efficiency, and little is known about the effect that the uncertainties of

EOS parameters may have on turbine efficiency. The purpose of this study is to

quantify the influence of coupled uncertain variables on the total-to-static efficiency

of a radial-inflow ORC turbine using high-density fluid R143a in a low-grade

temperature renewable power block. To this end, a stochastic solution is obtained by

combining a multi-dimensional gPC approach with a full 3D viscous turbulent CFD

solver for high-density radial-inflow turbo-expander. Both operational conditions

(inlet total temperature, rotational speed and mass flow rate) and EOS parameters

(critical pressure and critical temperature) are investigated, highlighting their

importance for turbine efficiency, based on considering three EOSs, namely, Peng-

Robinson (PR), Soave-Redlich-Kwong (SRK), and HHEOS. This study, which is

performed for both nominal and off-design operational conditions, highlights that the

inlet temperature as the most influential operational uncertain parameter, while critical

pressure is the most sensitive parameter for the three EOSs tested. More importantly,

it demonstrates that the SRK EOS has a higher level of sensitivity, in particular under

off-design operational conditions. This is a crucial aspect to be taken into account for

robust designs of ORC turbines for low-grade temperature renewable power cycles

ix Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for Renewable


working at various conditions. The proposed stochastic approach may consequently

positively support the renewable energy sector to develop more robust and viable

systems.

As the connecting component to the ORC turbo-expander outlet, high-density

fluid diffusers are also key components designed to improve the efficiency of ORC.

However, investigations into the robust optimal design of high-density fluid diffusers

are lacking, which hampers the improvement of overall ORC efficiency. The validated

robust framework developed in this study is utilised to effectively implement

sensitivity analysis of the high-density fluid conical diffuser. R143a, a potential high-

density fluid, is employed in this analysis. Both operating and geometric parameters

have significant impact on the performance of conical diffusers, and thus a

performance analysis is conducted using the proposed framework. This study

quantifies the influence of coupled and multiple uncertain parameters on a high-

density fluid conical diffuser. It is shown that swirl velocity has more impact than inlet

axial velocity on pressure recovery under various geometric conditions such as length

and angle of the high-density fluid conical diffuser. This study highlights the need to

achieve a robust optimal high-density fluid diffuser design in order to improve overall

ORC efficiency.

After investigating the radial-inflow turbo-expander and the conical diffuser

using high-density working fluid, in order to design a more appropriate diffuser to

match current existing R143a turbo-expanders, a performance comparison between the

preliminary design of a conical diffuser and annular-radial diffuser is conducted. The

numerical results show that the conical diffuser geometry using R143a has difficulty

in achieving optimal static pressure recovery. For the same conditions, the annular-

radial diffuser has higher performance than the conical diffuser on pressure recovery.

This study further highlights the need to achieve a high performance, high-density

fluid diffuser design in order to improve overall ORC efficiency, which is a critical to

further development of renewable power solutions.

For the purpose of analysing the uncertainty quantification for the whole radial-

inflow turbine, including both a radial-inflow turbo-expander and a newly designed

annular-radial diffuser, the operational conditions (inlet total temperature, rotational

speed, and mass flow rate) are considered as uncertain parameters. Importantly, the

mean value efficiency of the Improved-Complete-Turbine is 4.4% higher than the

x Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for Renewable


Preliminary-Turbo-expander. The Improved-Complete-Turbine also shows a more

robust performance than the Preliminary-Turbo-expander, with lower Coefficient of

Variation regarding turbine efficiency.

Uncertainty Quantification analysis has been conducted for the Improved-

Complete-Turbine. It will contribute to the design of robust turbomachinery, capable

of working well under uncertain operational conditions. The full three-dimensional

turbine simulations that have been carried out using high-density fluid in this project

are of high significance. This study will pave the way to reliable and robust ORC

turbines for low-grade temperature renewable power cycles, and will consequently

have a positive impact on the renewable energy sector. This developed framework can

also be applied in designing many other engineering applications for uncertain

conditions.

xi Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for Renewable


Table of Contents

Statement of Original Authorship .............................................................................................v

Keywords ................................................................................................................................ vi

Abstract .................................................................................................................................. vii

Table of Contents .................................................................................................................... xi

List of Figures ....................................................................................................................... xiii

List of Tables ........................................................................................................................ xix

List of Abbreviations ............................................................................................................ xxi

List of Publication ............................................................................................................... xxiii

Acknowledgements ...............................................................................................................xxv

Chapter 1: Introduction ...................................................................................... 1

1.1 Background .....................................................................................................................1

1.2 Research Problems..........................................................................................................7

1.3 Research Objectives........................................................................................................7

1.4 Research Significance .....................................................................................................8

1.5 Research Innovation .......................................................................................................9

1.6 Thesis Outline ...............................................................................................................10

Chapter 2: Literature Review ........................................................................... 15

2.1 ORC applications for Low-grade Temperature Renewable Energies ...........................15

2.2 ORC Turbo-expanders ..................................................................................................20

2.3 Diffusers .......................................................................................................................34

2.4 Uncertainty Quantification ...........................................................................................37

2.5 Summary .......................................................................................................................43

Chapter 3: Coupled Uncertainty Quantification – Deterministic Flow Solver

Methodology 45

3.1 Computational Fluid Dynamics Solver.........................................................................45

3.2 generalised Polynomial Chaos Approach .....................................................................51

3.3 Robust Coupled UQ-CFD Framework .........................................................................56

Chapter 4: Validation and Application of the UQ-CFD Framework to the

Ideal Gas Turbo-expander ...................................................................................... 59

4.1 Numerical Validation of the Deterministic CFD solver ...............................................59

4.2 Parametric Study of Operational and Geometric Conditions .......................................67

4.3 Validation of the UQ-CFD Framework for Ideal Gas Turbo-expander .......................75

4.4 Conclusion ....................................................................................................................78

xii Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for Renewable


Chapter 5: Application of the UQ-CFD Framework to an ORC Radial

Turbo-expander ........................................................................................................ 81

5.1 Computational Fluid Dynamics Characteristics ........................................................... 82

5.2 Uncertainty Quantification Parameters ........................................................................ 87

5.3 Deterministic and Stochastic Results at Nominal Conditions ...................................... 89

5.4 Stochastic Analysis at Off-design Conditions ............................................................ 100

5.5 Conclusion ................................................................................................................. 111

Chapter 6: Sensitivity Analysis of High-density Conical Diffusers using UQ-

CFD Framework .................................................................................................... 115

6.1 Computational Model ................................................................................................ 116

6.2 Deterministic Flow Characteristics Analysis ............................................................. 120

6.3 Sensitivity High-density Conical Diffuser to Axial and Swirling Velocities ............ 129

6.4 Conclusion ................................................................................................................. 139

Chapter 7: Development and Analysis of a More Robust ORC Radial

Turbine 141

7.1 Introduction ................................................................................................................ 142

7.2 Preliminary design of diffusers .................................................................................. 142

7.3 Numerical Modelling of Complete ORC Radial Turbine .......................................... 146

7.4 Performance Analysis of ORC Radial Turbo-expander Fitted with Two Different

Diffusers ............................................................................................................................... 147

7.5 Comparison between Preliminary ORC Radial Turbine and Improved ORC Radial

Turbo-expander Fitted with Annular-radial Diffuser ........................................................... 151

7.6 Stochastic Analysis of Preliminary and Improved ORC Turbines under Operational

Uncertainties ......................................................................................................................... 154

7.7 Stochastic Analysis at Off-design Conditions ............................................................ 163

7.8 Conclusions ................................................................................................................ 167

Chapter 8: Conclusions and Suggestions ....................................................... 169

8.1 Research Summary and Concluding Remarks ........................................................... 169

8.2 Research Limitations .................................................................................................. 173

8.3 Directions for Future Research .................................................................................. 174

Bibliography ........................................................................................................... 176

xiii Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


List of Figures

Figure 1-1: Schematic diagram of geothermal energy with ORC binary power

cycle. .............................................................................................................. 2

Figure 1-2: Thesis outline. ......................................................................................... 13

Figure 2-1: Schematic of a biomass ORC system (Rahbar, et al., 2017). .................. 16

Figure 2-2: Schematic of an OTEC ORC system (Rahbar, et al., 2017). .................. 17

Figure 2-3: (a) The sketch of an axial turbo-expander. (b) The sketch of a

radial-inflow turbo-expander (Baskharone, 2006). ...................................... 20

Figure 3-1: Computational UQ-CFD framework of gPC application processing

in radial-inflow turbine. ............................................................................... 56

Figure 3-2: The detailed computational UQ-CFD framework with relative

equations and steps. ..................................................................................... 57

Figure 3-3: Convergence rate of variance of efficiency using Peng-Robinson vs

P-order regarding 𝑃𝑐 − 𝑇𝑐. .......................................................................... 58

Figure 4-1: The whole geometry of the Jones’s radial-inflow turbine....................... 60

Figure 4-2: Geometry of periodic one blade passage of the Jones’ radial-inflow

turbine. ......................................................................................................... 61

Figure 4-3: Grid Refinement independent investigations. ......................................... 62

Figure 4-4: Mesh of rotor. .......................................................................................... 62

Figure 4-5: Three-dimensional closer view of the O–H grid of rotor blade

passage: (a) at hub; (b) at shroud. ................................................................ 63

Figure 4-6: Mesh of Stator: (a) all of the nozzles of stator; (b) one periodic

nozzle passage of stator. .............................................................................. 63

Figure 4-7: Residual convergence at rig condition. ................................................... 64

Figure 4-8: Static pressure distribution at mid-span along turbine. ........................... 65

Figure 4-9: Relative Mach number at mid-span along the turbine. ........................... 65

Figure 4-10: Variation of total-to-static efficiency with rotational speed.................. 67

Figure 4-11: Parametric evaluation of inlet temperature’s effect on the turbine’s

efficiency...................................................................................................... 69

Figure 4-12: Parametric evaluation of outlet pressure’s effect on the turbine’s

efficiency...................................................................................................... 69

Figure 4-13: Parametric evaluation of mass flow rate’s effect on the turbine

efficiency...................................................................................................... 70

Figure 4-14: Parametric evaluation of the turbine efficiency on Rotational

Speed (RPM). ............................................................................................... 71

Figure 4-15: Geometric variation on TE Meridional Length and TE Tip Radius. .... 72

Figure 4-16: Parametric evaluation of the turbine efficiency on TE Tip Radius. ...... 72

xiv Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


Figure 4-17: Parametric evaluation of the turbine efficiency on TE Meridional

Tip Length. ................................................................................................... 73

Figure 4-18: Blade thickness profile geometric study. .............................................. 74

Figure 4-19: Parametric evaluation of the blade thickness profile’s effect on the

turbine efficiency. ........................................................................................ 74

Figure 4-20: Legendre quadrature points and arbitrary support points for

rotational speed for P = 1, 3, 5, 7, 9, 11. ...................................................... 75

Figure 4-21: Convergence rates of the variance of the rotational speed in

respect to total-to-static efficiency. .............................................................. 76

Figure 4-22: The 1st order of Sobol’s indices of each uncertain parameter’s

contribution to 𝜂𝑇 − 𝑆 (a) R – RPM (b) L – RPM (c) L– R ; The 2nd

order Sobol’s indices of each uncertain parameter’s contribution to

𝜂𝑇 − 𝑆 (d) R – RPM, L – RPM, R – L. ......................................................... 77

Figure 5-1: 3D view of the O-H grid around the stator. ............................................. 84

Figure 5-2: 3D view of the O-H grid (a) Rotor blade at the hub. (b) Rotor blade

at the shroud. ................................................................................................ 84

Figure 5-3: Legendre quadrature points for 𝑃𝑐 − 𝑇𝑐 for 𝑃 = 5, 7, 9. ........................ 88

Figure 5-4: Probability Density Functions (PDF) of the total-to-static efficiency

coefficient 𝜂𝑇 − 𝑆 obtained using the PR EOS in the presence of

uncertainties for 𝑃𝑐 − 𝑇𝑐. ........................................................................... 88


coefficient 𝜂𝑇 − 𝑆 obtained using the PR EOS in the presence of

uncertainties for for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚. ................................................... 89

Figure 5-6: Deterministic Isentropic Mach number profile at middle span of

rotor blade for the nominal case using three EOSs. ..................................... 90

Figure 5-7: Deterministic T-h curve along the blade for the nominal case using

three EOSs. ................................................................................................... 91

Figure 5-8: PDF for 𝜂𝑇 − 𝑆 for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚, uncertain parameters

using three EOSs with P = 7. ....................................................................... 93

Figure 5-9: The mean and standard deviation of Isentropic Mach number at

middle span of rotor blade for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚 uncertain

parameters using three EOS with P = 7. (a) The whole blade

Isentropic Mach number profile. (b) Streamwise 0.015-0.045 blade

Isentropic Mach number profile at Suction Side. ........................................ 94

Figure 5-10: The skin friction coefficient (𝐶𝑓) profile at middle span of rotor

blade with 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚 using three EOS with P = 7. (a) The

whole blade skin friction coefficient (𝐶𝑓) profile. (b) Streamwise

0.015-0.045 skin friction coefficient (𝐶𝑓) profile at Suction Side............... 95

Figure 5-11: (a) 1st order and (b) 2nd order of Sobol’s indices of each uncertain

parameter contribution of 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚 for 𝜂𝑇 − 𝑆 using three

EOSs with P = 7. .......................................................................................... 96

Figure 5-12: PDF for 𝜂𝑇 − 𝑆 of the 3rd set 𝑃𝑐 − 𝑇𝑐 uncertain parameters using

three EOSs with P = 7. ................................................................................. 97

xv Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for Renewable



parameter’s contribution to 𝑃𝑐 − 𝑇𝑐 by using three EOSs for 𝜂𝑇 − 𝑆

with P = 7. .................................................................................................... 98

Figure 5-14; The mean and standard deviation of Isentropic Mach number at

middle span of rotor blade for 𝑃𝑐 − 𝑇𝑐 uncertain parameters using

three EOS with P = 7. (a) The whole blade Isentropic Mach number

profile. (b) Streamwise 0.015-0.045 blade Isentropic Mach number

profile at Suction Side. ................................................................................. 99

Figure 5-15: The mean and standard deviation of skin friction coefficient (𝐶𝑓)

profile at middle span of rotor blade for 𝑃𝑐 − 𝑇𝑐 uncertain parameters

using three EOS with P = 7. (a) The whole blade skin friction

coefficient (𝐶𝑓) profile. (b) Streamwise 0.015-0.045 skin friction

coefficient (𝐶𝑓) profile at Suction Side. ...................................................... 99

Figure 5-16: Total-to-static efficiency map charts for off-design conditions

with 𝑃𝑐 − 𝑇𝑐 uncertain parameters using the three EOSs with P = 7

regarding three 𝑇𝑇𝑖𝑛 varying three 𝑅𝑃𝑀. ................................................. 103


with 𝑃𝑐 − 𝑇𝑐 uncertain parameters using the three EOS with P = 7

regarding three 𝑅𝑃𝑀 varying three 𝑇𝑇𝑖𝑛. ................................................. 103

Figure 5-18: Pressure versus temperature for R143a based on REFPROP NIST

with different thermodynamic states. ......................................................... 105

Figure 5-19: The 1st order Sobol’s indices at off-design conditions with 𝑃𝑐 −𝑇𝑐uncertain parameters with P = 7. (a) TTin = 400K and 80%

nominal RPM (b) TTin = 400K and120% nominal RPM (c) TTin =

450K and 80% nominal RPM (d) TTin = 450K and 120% nominal

RPM. .......................................................................................................... 106

Figure 5-20: The 2nd order Sobol’s indices at off-design conditions with 𝑃𝑐 −𝑇𝑐 uncertain parameters with P = 7. (a) 𝑇𝑇𝑖𝑛 = 400K and 80%

nominal RPM (b) 𝑇𝑇𝑖𝑛 = 400K and 120% nominal RPM (c) 𝑇𝑇𝑖𝑛 =

450K and 80% nominal RPM (d) 𝑇𝑇𝑖𝑛 = 450K and 120% nominal

RPM. .......................................................................................................... 107

Figure 5-21: PDF for 𝜂𝑇 − 𝑆 with 𝑃𝑐 − 𝑇𝑐 uncertain parameters using three

EOS with P = 7. (a) 𝑇𝑇𝑖𝑛 = 400K and 80% nominal RPM (b) 𝑇𝑇𝑖𝑛 =

400K and 120% nominal RPM (c) 𝑇𝑇𝑖𝑛 = 450K and 80% nominal

RPM (d) 𝑇𝑇𝑖𝑛 = 450K and 120% nominal RPM. ..................................... 108

Figure 5-22: Skin friction coefficient (𝐶𝑓) profile along the rotor blade with

𝑃𝑐 − 𝑇𝑐 at 𝑇𝑇𝑖𝑛 = 400K and 80% nominal RPM using three EOS

with P = 7. (a) PR (b) HHEOS (c) SRK. The velocity vectors

including leading edge at pressure and suction sides of rotor blade

using three EOS with P = 7. (d) PR (e) HHEOS (f) SRK. ........................ 110

Figure 6-1: Sketch of Conical Diffuser adapted from Clausen’s experiment

(Clausen, et al., 1993). ............................................................................... 116

Figure 6-2: Computational mesh of longitudinal view. ........................................... 118

Figure 6-3: Computational mesh of inlet circumferential view. .............................. 118

xvi Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


Figure 6-4: Sketch of the main flow characteristics with changing U, W and A. .... 120

Figure 6-5: (a), Contour plot of velocity u of case R in the middle plane of

diffuser. (b), velocity u on S0/S4/S8 profiles of the case R. ...................... 121

Figure 6-6: Contour Plot Velocity U regarding (a) case A, Umin Wmin. (b)

case B, Umin Wmax. (c) case C, Umax Wmin. (d) case D, Umax

Wmax. ......................................................................................................... 123

Figure 6-7: Deterministic S8 profiles of U-W for cases A-D. .................................. 124

Figure 6-8: (a) Contour Plot Velocity_U regarding: (a), case (E) Lmin and

Amin. (b), case (F) Lmin and Amax. (c), case (G) Lmax and Amin.

(d), case (H) Lmax and Amax. .................................................................... 127

Figure 6-9: Deterministic S8 profiles of U-W for case E-H. .................................... 128

Figure 6-10: Legendre quadrature points for U-W for 𝑃 = 5, 7, 9. .......................... 129

Figure 6-11: PDF Cp for P = 5, 7, 9 gPC for U-W at L = 510mm, A = 10°. ........... 130

Figure 6-12: UQ analysis of velocity u regarding U-W for Case E at L =

410mm, A = 8°. (a) Mean value of velocity u. (b) Standard Deviation

of velocity u. (c) Variance of velocity u. (d) CoV of velocity u. ............. 132

Figure 6-13: UQ analysis of velocity u regarding U-W for Case F at L =

410mm, A = 12°. (e) Mean value. (f) Standard Deviation. (g)

Variance. (h) CoV. .................................................................................... 133

Figure 6-14: UQ analysis of velocity u regarding U-W for Case R at L =

510mm, A = 10°. (a) Mean value. (b) Standard Deviation. (c)

Variance. (d) CoV. .................................................................................... 134

Figure 6-15: UQ analysis of velocity u regarding U-W for Case G at L =

610mm, A = 8°. (a) Mean value. (b) Standard Deviation. (c) Variance.

(d) CoV. ...................................................................................................... 134

Figure 6-16: UQ analysis of velocity u regarding U-W for Case H at L =

610mm, A = 12°. (e) Mean value. (f) Standard Deviation. (g)

Variance. (h) CoV. .................................................................................... 135

Figure 6-17: Standard deviation of velocity u on S8 profile under different

geometric conditions. ................................................................................. 136

Figure 6-18: Close view of Figure 6-17 under different geometric conditions.

(a) L410-A12. (b) L510-A10. (c) L610-A8. (d) L610-A12. ........................ 137

Figure 6-19: Sobol's indices of Cp for U-W. (a) First order. (b) Second order. ....... 138

Figure 6-20: PDF for U-W under different geometric conditions. ........................... 139

Figure 7-1: Sketch of conical diffuser. ..................................................................... 143

Figure 7-2: Sketch of annular-radial diffuser. .......................................................... 145

Figure 7-3: Geometry of one blade passage of two full turbines with two

different diffusers respectively. .................................................................. 146

Figure 7-4: The velocity streamline of axial velocity of the conical diffuser. ......... 148

Figure 7-5: The velocity streamline of the annular-radial diffuser. (a) Overall

view. (b) Closer view for the dash box for vector. .................................... 148

xvii Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


Figure 7-6: Skin friction coefficient of conical diffuser. ......................................... 149

Figure 7-7: Skin friction coefficient of annular-radial diffuser hub. ....................... 150

Figure 7-8: Skin friction coefficient of annular-radial diffuser shroud.................... 151

Figure 7-9: Mach number at mid-span for the Preliminary-Turbo-expander at

nominal conditions. .................................................................................... 152

Figure 7-10: Mach number at mid-span for the Improved-Complete-Turbine at


Figure 7-11: Isentropic Mach number profile at middle span of rotor blade for

Preliminary-Turbo-expander and the Improved-Complete-Turbine at


Figure 7-12: Closer view of Figure 7-11 for Streamwise from 0-0.07. ................... 154

Figure 7-13: Legendre quadrature points for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚 for 𝑃 = 5, 7,

9.................................................................................................................. 155

Figure 7-14: Probability Density Functions (PDF) of the total-to-static

efficiency coefficient 𝜂𝑇 − 𝑆 obtained for Improved-Complete-

Turbine in the presence of uncertainties for for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚

for P = 5, 7, 9. ............................................................................................ 156

Figure 7-15: (a) The mean and standard deviation of Isentropic Mach number

at middle span of rotor blade for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚 for both turbines

with P = 7. (b) Closer View of (a) for Streamwise 0.015-0.06.................. 157

Figure 7-16: (a) The skin friction coefficient (𝐶𝑓) profile at middle span of

rotor blade with 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚 for both turbines with P = 7. (b)

Closer view of (a) for Streamwise 0.015-0.075. ........................................ 158

Figure 7-17: The 1st order of Sobol’s indices of each uncertain parameter

contribution of 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚 for 𝜂𝑇 − 𝑆 with P = 7 for both

turbines. ...................................................................................................... 159

Figure 7-18: The 2nd order of Sobol’s indices of each uncertain parameter

contribution of 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚 for 𝜂𝑇 − 𝑆 with P = 7 for both

turbines. ...................................................................................................... 160

Figure 7-19: PDF for 𝜂𝑇 − 𝑆 for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛 − 𝑄𝑚uncertain parameters with

P = 7 for both turbines. .............................................................................. 161

Figure 7-20: UQ analysis of Mach number for Standard Deviation (a) – (b), for

variance (c) – (d), and for CoV (e) – (f) for Preliminary-Turbo-

expander and Improved-Complete-Turbine respectively. ......................... 162

Figure 7-21: The 1st order Sobol’s indices at off-design conditions with 𝑃𝑐 −𝑇𝑐uncertain parameters with P = 7 for two turbines. ................................ 165

Figure 7-22: The 2nd order Sobol’s indices at off-design conditions with 𝑃𝑐 −𝑇𝑐 uncertain parameters with P = 7 for two turbines. ............................... 165

Figure 7-23: PDF for 𝜂𝑇 − 𝑆 with 𝑃𝑐 − 𝑇𝑐 uncertain parameters at 𝑇𝑇𝑖𝑛=

400K and 80% nominal RPM off-design conditions with P = 7 for two

turbines. ...................................................................................................... 166

xviii Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for



𝑃𝑐 − 𝑇𝑐 at 𝑇𝑇𝑖𝑛 = 400K and 80% nominal RPM for two turbines. ......... 167

xix Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


List of Tables

Table 2-1: Brief summary of tested cubic EOSs in literature review. ....................... 24

Table 2-2: Latest radial-inflow turbine numerical studies using various types of

EOS. ............................................................................................................. 25

Table 2-3: Typical UQ methods (Le Maître & Knio, 2010). ..................................... 38

Table 3-1: The coefficients for Equations (3-9) and (3-10). ...................................... 48

Table 3-2: gPC type and underlying random variable (Xiu, 2010). .......................... 52

Table 3-3: Weight and density function for gPC distribution (Xiu, 2010). ............... 53

Table 4-1: Boundary Condition of the Study Case. ................................................... 64

Table 4-2: Results Comparison for the Rig Conditions. ............................................ 67

Table 4-3: Characteristics of the uncertain parameters studied. ................................ 68

Table 4-4: Mean, standard deviation and CoV of the total-to-static efficiency

for each individual uncertain parameter for P = 11 and coupled

parameters for P = 5. .................................................................................... 77

Table 5-1: R143a turbine design parameters at nominal conditions (Sauret &

Gu, 2014). .................................................................................................... 82

Table 5-2: PR, SRK, and HHEOS models. ................................................................ 86

Table 5-3: Characteristics (mean and support values) of the studied uniformly

distributed uncertain parameters. ................................................................. 87

Table 5-4: Deterministic total-to-static efficiency 𝜂𝑇 − 𝑆 for the nominal case

with three different EOSs............................................................................. 90

Table 5-5: Mean, standard deviation, and CoV of the 𝜂𝑇 − 𝑆 for (𝑅𝑃𝑀 −𝑇𝑇𝑖𝑛 − 𝑄𝑚) using three EOSs with P = 7. ................................................. 92

Table 5-6: Mean and standard deviation and CoV of the 𝜂𝑇 − 𝑆 for the 2nd set

case obtained with gPC (𝑃𝑐 − 𝑇𝑐) coupled by three EOS with P = 7. ....... 97

Table 5-7: The mean values of 𝜂𝑇 − 𝑆 for off-design conditions with 𝑃𝑐 −𝑇𝑐 uncertain parameters by the three EOSs with P = 7 (based on

uniform distribution laws).......................................................................... 101

Table 5-8: The 𝐶𝑜𝑉 × 10 − 3of 𝜂𝑇 − 𝑆 at off-design conditions with 𝑃𝑐 −𝑇𝑐uncertain parameters for the three EOSs with P = 7. ............................ 104

Table 6-1: Boundary conditions of the reference case R for high-density fluid

conical diffuser, L = 510mm, A = 10........................................................ 117

Table 6-2: The range of uncertain parameters: inlet velocity, swirl velocity. ......... 120

Table 6-3: Deterministic results regarding extreme U-W cases at L and A

constant (L = 510mm, A = 10°).................................................................. 122

xx Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for Renewable


Table 6-4: Deterministic results regarding L and A cases at U-W constant (U =

11.6 m/s, W = 56.1 rad/s). .......................................................................... 126

Table 6-5: Mean and CoV of Cp for coupled U-W uncertain parameters by gPC

under different geometric conditions. ........................................................ 131

Table 7-1: Geometric parameters of the conical diffuser. ........................................ 143

Table 7-2: Geometric parameters of the annular-radial diffuser. ............................. 145

Table 7-3: Grid study of pressure recovery for different diffusers. ......................... 146

Table 7-4: Total-to-static efficiency 𝜂𝑇 − 𝑆 for the whole turbines using

different diffusers. ...................................................................................... 147

Table 7-5: The pressure recovery coefficient 𝐶𝑝 for both diffusers. ....................... 147

Table 7-6: Deterministic total-to-static efficiency 𝜂𝑇 − 𝑆 for Preliminary-

Turbo-expander and the Improved-Complete-Turbine at nominal

conditions. .................................................................................................. 152

Table 7-7: Mean and support values of the uniformly distributed uncertain

parameters. ................................................................................................. 155

Table 7-8: Mean, standard deviation, and CoV of the 𝜂𝑇 − 𝑆 for (𝑅𝑃𝑀 −𝑇𝑇𝑖𝑛 − 𝑄𝑚) for Preliminary-Turbo-expander and Improved-

Complete-Turbine with P = 7. ................................................................... 157

Table 7-9: The mean values of 𝜂𝑇 − 𝑆 under 80% nominal RPM and 𝑇𝑇𝑖𝑛=

400K with 𝑃𝑐 − 𝑇𝑐 uncertain parameters by PR with P = 7 (based on

uniform distribution laws). ......................................................................... 164

xxi Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


List of Abbreviations

Nomenclature

𝑃 𝑀𝑃𝑎 pressure 𝑇 𝐾 temperature

�� 𝑘𝑔. 𝑠−1 mass flow rate 𝑐𝑝0 𝐽. 𝑚𝑜𝑙−1. 𝐾−1 zero pressure ideal gas heat capacity

𝑊 rad/s Swirl Velocity 𝐴 ° Half Cone Angle

𝐿 mm Length 𝐶𝑝 - Pressure Recovery Coefficient

𝑈 m/s Inlet Velocity 𝑇u inlet turbulence intensities

𝑆𝑛 - Swirl Number 𝑃2 Pa Static pressure of outlet

𝑃1 Pa Static pressure of inlet 𝑃01 Pa Total pressure of inlet

Greek symbols

Ω RPM rotational speed 𝜂 % efficiency 𝜎 - standard deviation

𝜔 - acentric factor 𝜇 - mean value

Subscripts Abbreviation

in Turbine inlet UQ Uncertainty Quantification SS Suction Side

is Isentropic CFD Computational Fluid Dynamics PS Pressure Side

out Turbine outlet PDF Probability Density Function TE Trailing Edge

𝑇 Total CoV Coefficient of Variation LE Leading Edge

𝑇 − 𝑆 Total-to-static VR Velocity Reduction EOS Equation of State

𝑇 − 𝑇 Total-to-total gPC generalised Polynomial Chaos PR Peng-Robinson

𝑆 Static SRK Soave-Redlich-Kwong

𝑐 critical point

xxii Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


xxiii Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


List of Publication

Publications:

Zou, A., Chassaing, J.-C., Persky, R., Gu, Y., & Sauret, E. (2019). Uncertainty

Quantification in high-density fluid radial-inflow turbines for renewable low-grade

temperature cycles. Applied Energy, 241, 313-330. (IF = 7.9)

Zou, A., Sauret, E., Chassaing, J. C., Li, W., & Gu, Y. Quantified high-density fluid

conical diffuser performance with uncertain parameters by flow characteristic

analysis. Applied Thermal Engineering. Under review.

Zou, A., Sauret, E., & Gu, Y. Uncertainty Quantification in high-density radial-inflow

turbo-expander comparison with and without connecting designed diffuser for

renewable low-grade temperature cycles. Energy. In preparation.

Zou, A., Sauret, E., & Gu, Y. A review of Uncertainty Quantification for non-ideal

gas turbomachinery in Organic Rankine Cycle application. Applied Energy. In

preparation.

xxiv Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


Conferences:

Zou, A., Sauret, E., Chassaing, J. C., Saha, S. C., & Gu, Y. (2015). Stochastic analysis

of a radial-inflow turbine in the presence of parametric uncertainties. Full paper has

published in Proceedings of the 6th International Conference on Computational

Methods (Vol. 2). ScienTech.

Zou, A., Sauret, E., Chassaing, J. C., Li, W., & Gu, Y. Quantified high-density fluid

conical diffuser performance with uncertain parameters by flow characteristic

analysis. The 9th International Conference on Computational Methods, Rome, Italy,

2018. Oral Presentation.

Zou, A., Sauret, E., & Gu, Y. Numerical comparisons of conical and annular-radial

diffusers’ performance for high-density radial-inflow turbines. The 21st Australasian

Fluid Mechanics Conference, Adelaide, Australia, 2018.

Zou, A., Sauret, E., & Gu, Y. (2018). Numerical comparisons of conical and annular-

radial diffusers’ performance for high-density radial-inflow turbines. Full paper has

published in Proceedings of the 21st Australasian Fluid Mechanics Conference.

ScienTech. Oral Presentation.

xxv Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


Acknowledgements

Firstly, I would like to express my profound gratitude and grateful thanks to my

principal supervisor Dr. Emilie Sauret for her patient guidance and great support

during my PhD journey. Her vast experience and systematic guidance have been

invaluable. I really appreciate her help through every meeting and by email feedback,

which has enabled me to adequately address an array of complexities in the research

engineering problems. I also deeply appreciate her moral support for the hardships I

faced during my PhD journey.

I would like to express my sincere thanks to my associate supervisor, Prof.

YuanTong Gu for his great support. I also would like to thank my associate supervisor,

Dr Wei Li, who gave me a lot of constructive advice during this journey. I extend my

thanks to our collaborator, A/Prof. Jean-Camille Chassaing in Sorbonne Université in

Paris, France, for his advice and extensive support for our cooperative research.

I gratefully acknowledge the scholarship award provided by the Science and

Engineering Faculty, Queensland University of Technology. I also would like to

acknowledge the Australian Research Council (ARC) for its financial support

(DE130101183) for my scholarship. In addition, I would like to thank the “John and

Gay Hull Top-up scholarship” for their financial support during my research journey.

These awards allow me to conduct my PhD research to completion. I would like to

acknowledge the cutting-edge resources supplied by the High Performance Computing

(HPC) centre of Queensland University of Technology. These resources greatly

assisted in generating all the research data efficiently.

Thanks to Dr. Christina Houen of Perfect Words Editing for editing my thesis

according to university guidelines and those of the Institute of Professional Editors

(IPEd).

Importantly, I would like to express my sincere thanks to my friends, and all

members of the Laboratory of Advanced Modelling and Simulation in Engineering

and Science (LAMSES), which is a fantastic research platform in Australia; and

especially, thanks to the Computational Fluid Dynamics (CFD) subgroup members in

LAMSES.

xxvi Uncertainty Quantification in High-density Fluid Radial-inflow Turbo-Expanders and Diffusers for


Chapter 1: Introduction 1

Chapter 1: Introduction

This chapter presents the background of the project in Section 1.1, the research

problem in Section 1.2, and research objectives in Section 1.3. The research

significance is outlined in Section 1.4. The research’s innovation is presented in

Section 1.5. Section 1.6 outlines the succeeding chapters of the thesis.

1.1 BACKGROUND

Attention to renewable energy as a favourable source of electricity generation is

increasing, given the rise in fossil fuel consumption, and hence greenhouse gas

emissions (Chen, Xu, & Chen, 2012); Costall, Hernandez, Newton, and Martinez-

Botas (2015). Renewable energy sources, including biomass (Qiu, Shao, Li, Liu, &

Riffat, 2012), waste (Lecompte, Huisseune, Van Den Broek, Vanslambrouck, & De

Paepe, 2015), ocean thermal energy (Sun, Ikegami, Jia, & Arima, 2012) and

geothermal energy (Heberle & Brüggemann, 2010; Lentz & Almanza, 2006), can

transfer their energy into electricity (Vaja & Gambarotta, 2010). The Organic Rankine

Cycle (ORC) is considered to be a leading technology for these renewable energy

conversion processes (Fiaschi, Manfrida, & Maraschiello, 2012), as it can extract more

energy from renewable sources.

In general, high-temperature renewable resources (220-400°C) are the most

appropriate for commercial production of electricity generated by dry steam and flash

steam systems employing an ORC system. A low-to-medium grade temperature

resource, typically around 150-220°C, is strongly recommended for utilisation in local

district heat supplements using ORC systems (Rahbar, Mahmoud, Al-Dadah,

Moazami, & Mirhadizadeh, 2017). In recent decades, ORC is considered to be an

efficient technique for converting a low-grade temperature (<150°C) resource to

generate electricity. The advantages of employing ORC compared to traditional steam

power systems include effective operation of energy resources, smarter systems, and

economical performance (Yamamoto, Furuhata, Arai, & Mori, 2001).

At low-grade temperature conditions, the utilisation of an organic vapour in

place of steam is of interest for medium size power plants (50-500 kW), as the real

organic fluids can release more energy (Al-Sulaiman, Dincer, & Hamdullahpur, 2010;


Chacartegui, Sánchez, Muñoz, & Sánchez, 2009; Quoilin, Lemort, & Lebrun, 2010).

In those ORC cycles, the turbo-expander and diffuser are critical components for

transferring the thermal energy into mechanical energy and then into electricity energy.

In particular, radial-inflow turbines have been shown to be suitable for a 50-500

kW power-generation range (Badr, O'callaghan, Hussein, & Probert, 1984) and well-

suited for ORCs (Pini, Persico, Casati, & Dossena, 2013). However, they are also

sensitive to operational conditions while using high-density fluid (Sauret & Gu, 2014),

thus requiring more robust design, which is partly addressed in this thesis. Currently,

ORC is proven to be reliable and adaptable (Fiaschi, et al., 2012).

A typical ORC cycle including a turbo-expander and a diffuser is presented in

Figure 1-1, which shows a thermodynamic model for a binary power cycle (Fiaschi, et

al., 2012). As depicted in Figure 1-1, an ORC system includes preheater, evaporator,

turbo expander, diffuser, condenser, and pump.

The heat from the low-grade temperature source is pumped into the evaporator.

Then the high-density working fluid flows into the ORC radial-inflow turbine to

generate power. The exhaust vapor flows into the condenser, where it is condensed by

the cooling water. The condensed working fluid is pumped back to the evaporator and

another new cycle begins (Wang, Liu, & Zhang, 2013).

Figure 1-1: Schematic diagram of geothermal energy with ORC binary power cycle.


Radial-inflow turbines, including turbo-expanders and diffusers, play a critical

role in ORC, and are key components for converting the heat energy into electricity

(Badr, et al., 1984). The main components of a radial-inflow turbine are turbo-

expander (a stator and a rotor) and a diffuser. The gas goes into the stator and then

rotor, and finally through the diffuser, and out of the radial turbine. The combined

components, the volute, stator, and rotor of the turbine, can be called the turbo-

expander. The study of turbo-expanders and diffusers is of great importance in

improving the efficiency of the cycle to extract more energy from low-grade

temperature heat sources.

In order to achieve optimum performance from low-grade temperature

renewable power blocks, one of the main requirements is to have robust designs,

capable of handling fluctuations and operating well under adverse conditions. This is

especially important for turbines in Organic Rankine Cycles (ORC), as the power

delivered must be as high as possible for those low-grade temperature renewable

cycles. Recently, extensive investigations (Al Jubori, Al-Dadah, Mahmoud, & Daabo,

2017; Da Lio, Manente, & Lazzaretto, 2017; Fiaschi, et al., 2012; Fiaschi, Manfrida,

& Maraschiello, 2015; Kim & Kim, 2017b; Pan & Wang, 2013; Sauret & Rowlands,

2011; Song, Gu, & Ren, 2016a; Xia, Wang, Wang, & Dai, 2018; Zhu, Deng, & Liu,

2015) focused on meanline models of the ORC radial-inflow turbines to maximise

their isentropic efficiency. The main consideration of the meanline model is to design

the velocity triangles and inlet/outlet dimensions of the rotor. However, the three-

dimensional effects within the rotor passage must be modelled robustly to ensure the

effective expansion of the fluid (White, 2015). Furthermore, most of those studies did

not account for uncertainties in their meanline model, which again prevents robust

design of radial-inflow turbines. In order to develop robust designs of high-density

radial-inflow turbines, uncertainties must be accounted for at the design level. These

uncertainties arise from the variability of the normal operational conditions of the

system, e.g. inlet temperature, the geometric parameters due to manufacturing

tolerances, and the numerical representation of the physical system, including

mathematical models and boundary conditions. In this work, the combined effects of

several parametric uncertainties (Faragher, 2004) on the performance of a radial

turbine operating with high-density fluids will be investigated.


Considering the key components for converting energy into electricity, not only

radial-inflow turbo-expanders, but also diffusers, play pivotal roles in ORC

applications (Badr, et al., 1984). Diffusers are positioned at the outlet of radial-inflow

turbo-expanders, with the aim of recovering pressure and thereby increasing the whole

turbine efficiency. While numerical studies of radial-inflow turbines have been

extensively performed (Boncinelli, Rubechini, Arnone, Cecconi, & Carlo Cortese,

2004; Cho, Cho, Ahn, & Lee, 2014; Harinck, Pasquale, Pecnik, van Buijtenen, &

Colonna, 2013; Pini, et al., 2013; Sauret & Gu, 2014; Sauret & Rowlands, 2011;

Wheeler & Ong, 2013; Zou, Sauret, Chassaing, Saha, & Gu, 2015), the exhaust flow

through the high-density fluid diffusers has not received much consideration so far,

and needs to be investigated. The turbine components are quite crucial to achieving

high ORC efficiency (Pini, et al., 2013). However, the performance of diffusers is

easily neglected when the whole ORC system performance is considered. In order to

develop robust designs of high-density diffusers, uncertainties must also be accounted

for at the design level. These uncertainties arise from the variability of the normal

operational conditions of the system, e.g., inlet velocity and swirling velocity.

Investigations into the uncertainty quantification analysis of high-density fluid

diffusers are lacking, which hinders the improvement of overall ORC efficiency.

Importantly, comprehensive understanding of the influence of input uncertainties will

enhance the reliability of a risk-based design, increase design confidence, reduce risks,

improve safety, and refine the system’s operating range (Cinnella & Hercus, 2010).

A suitable diffuser to fit the radial-inflow turbo expander is significant for

maximizing the whole turbine’s efficiency by improving pressure recovery. Conical

diffusers are widely employed to connect the downstream of turbo expanders (Klein,

1981), as they have a simple geometry. However, based on an experimental

investigation by Abir and Whitfield (1987), the flow characteristics of ideal gas conical

diffusers are unstable, while curved annular diffusers and radial diffusers present more

stable flow conditions. Due to the lack of experiments with high-density fluids to

compare the performance of these diffusers, it is of interest to conduct numerical

studies to compare different diffusers’ geometries, in particular, conical and annular-

radial ones using high-density fluids. Recently, Keep, Head, and Jahn (2017) had a

constrained preliminary design for an annular-radial diffuser to fit their existing

supercritical CO2 turbine. However, in their study, the inlet swirl angle was set to zero,


which does not accurately represent the real flow direction out of the turbo expander

rotor, due to the uncertain flow characteristics through the radial-inflow machine. In

other words, the uncertainty quantification was not considered. In previous studies,

diffusers were investigated independently from the whole turbine, and did not include

the turbo expander. So far, in current numerical tools for conducting numerical studies

on diffusers, it is quite difficult to accurately set the diffusers’ inlet boundary

conditions, which correspond to the outlet flow of the turbo expanders. Inlet boundary

conditions are known to affect the flow in diffusers, and as such, in this study, our

proposed R143a radial-inflow turbo expander (Sauret & Gu, 2014) is built as the inlet

part of the diffuser. Limited understanding has been established regarding the way the

flow characteristics of these two typical diffusers employing high-density fluid affect

the efficiency of ORC turbines, and thus further influence the overall efficiency of the

low-grade temperature ORC. It is necessary to numerically compare the performance

of the preliminary design of a conical diffuser with an annular-radial diffuser, fitting

the conditions from the current existing 400kW R143a radial-inflow turbine to form a

whole radial-inflow turbine. Then Uncertainty Quantification (UQ) methods will be

applied to analyse the optimal whole radial-inflow turbine, including radial-inflow

turbo-expander and suitable diffuser, in order to evaluate which individual or coupled

parameters affect its performance.

In order to have confidence in the numerical simulations and include

uncertainties in the design process and optimisation of robust radial-inflow turbines,

UQ methods appear to be a powerful solution. UQ is a mathematical approach

employed to determine the likely certain outcomes in an uncertain system (Faragher,

2004). Any engineering system is subject to uncertainties, which can come from the

random variation of geometric parameters and operating conditions. These

uncertainties cannot be removed from the system and are called “aleatory”

uncertainties (Faragher, 2004). In addition, the numerical representation of this system

introduces uncertainties through the mathematical models and boundary conditions

used. These “epistemic” uncertainties (Faragher, 2004), however, can be reduced, as

they are due to modelling errors, and make it impossible to isolate their influence from

these parameters without using uncertainty quantification methods. Some

uncertainties, namely aleatory uncertainties, cannot be removed, while others, the

epistemic uncertainties, can be reduced (Faragher, 2004). The UQ method is


considered a powerful solution for robust engineering design, specifically for

evaluating the influence of uncertain input variables on the outcome (Daroczy, Janiga,

& Thevenin, 2016).

A systematic review of UQ approaches (Walters & Huyse, 2002) shows that the

Monte Carlo (MC) and Polynomial Chaos (PC) approaches approximate the

probability distribution of objective functions; however, MC is considered an

extremely expensive computation and a poor convergence technique (Walters &

Huyse, 2002), especially for complex geometries (Sankaran & Marsden, 2011). In

comparison, the PC approach has recently become more attractive, as it delivers an

exact means of propagating uncertainty, provides high order results, and dramatically

reduces computational costs (Modgil, Crossley, & Xiu, 2013). The gPC approach is a

non-statistical representation of random processes based on a pseudo-spectral finite

element approach (Ghanem & Spanos, 2003; Spanos & Ghanem, 1989) as extended

by Xiu, Lucor, Su, and Karniadakis (2002). The gPC approach also provides

exponential convergence (Walters & Huyse, 2002). Xiu et al. (Xiu & Karniadakis,

2003) presented incompressible flow simulations modelled by input uncertainty with

gPC. The gPC method results demonstrate that computational efficiency and accuracy

are dramatically increased compared to the MC approach (Cacuci, Ionescu-Bujor, &

Navon, 2004; Huyse, 2001). Zou, et al. (2015) presented a UQ study on an ideal gas

radial-inflow turbine including a stator and a rotor, showing that the most sensitive

random variables are the trailing edge tip radius combined with the rotational speed.

This study shows that geometric uncertainties due to manufacturing tolerances are one

of the critical aspects that can influence turbine efficiency. However, ideal gas, not

high-density fluid, was employed as a working fluid. Due to the importance of high-

density fluid in accurately predicting turbine performance, understanding the effect of

Equation of State (EOS) uncertainties is also critical to better design of turbines.

To sum-up, there are few Uncertainty Quantification studies on radial-inflow

turbines employing high-density fluid. This work executes performance analysis and

further identifies the parameters that are most influential on the performance of both

radial-inflow turbo-expanders and diffusers employing R143a as a working fluid. The

objective of this work is to highlight the significance of multiple uncertainties, with

possible random distributions, on the stochastic response of the performance for a

whole radial-inflow turbine that includes a radial-inflow turbo-expander and a suitable


diffuser. An advanced robust framework is established connecting a CFD solver and a

UQ approach for the ORC radial-inflow turbine UQ analysis.

In the following literature review Section, renewable energy, Organic Rankine

Cycle, radial-inflow turbines, diffusers, and uncertainty quantification will be

reviewed.

1.2 RESEARCH PROBLEMS

Due to the complexity of applications involving radial-inflow turbines and high-

density fluids, an advanced and robust framework needs to be established. The main

research problem is to propose a new design for an ORC radial-inflow turbine which

is more robust to on- and off-design conditions under uncertainties. Another research

problem is how to develop a robust framework to couple a CFD solver and UQ, which

involves multiple uncertain parameters. The parameters, such as operational

conditions and geometric parameters, have significant impacts on the ORC radial

turbine design in low-grade temperature power cycles.

Therefore, the major research problems that will be addressed in this project are:

How to develop an advanced and robust framework coupling a CFD solver and

a UQ approach that can be efficiently used for the optimal design of ORC radial

inflow turbines?

How will different parameters, individually or coupled, influence efficiency of

the ORC radial-inflow turbine in low-grade temperature power cycles, and

which one has the dominant impact?

What type of diffuser is more suitable for the current existing R143a radial-

inflow turbine? What is the performance of the whole turbine including turbo-

expander and diffuser?

1.3 RESEARCH OBJECTIVES

This study aims to build a robust design for a whole radial-inflow turbine for

low-grade temperature power cycles, using an advanced and robust UQ-CFD

framework. An advanced and robust framework needs to be built for the ORC radial-

inflow turbine UQ analysis. This numerical framework will be established based on

the concept of coupling a CFD solver and a UQ approach. It will give accurate results


for the complex working conditions of radial-inflow turbines that use high-density

fluid. The developed model will assist the optimised design of an ORC turbo-expander.

The newly established robust framework will be able to identify the most sensitive

individual or coupled uncertain parameters that affect the radial-inflow turbine

efficiency, so that a robust design can be established.

Specifically, major objectives of this research work include:

To develop a robust framework that couples a CFD solver and a UQ

approach to ORC radial-inflow turbine optimal design in a low-grade

temperature power cycle.

Using the newly established framework, to obtain a comprehensive

understanding of parameters (such as operational conditions like

temperature, pressure; or geometrically uncertain parameters) that affect the

efficiency of ORC radial-inflow turbines in low-grade temperature power

cycles.

To investigate the most suitable diffuser for an ORC turbo-expander

working with high-density fluids.

1.4 RESEARCH SIGNIFICANCE

This research significance of this study is as follows:

The research is of great significance to climate change. Over the next several

decades, with the purpose of preventing global warming of more than 2°C,

it is recommended that developed countries must aggressively reduce

emissions of greenhouse gas to a point of near-zero by 2050 (Metz, 2007).

Development of mixed pattern renewable energy to take the place of

traditional fossil fuel energy would require energy supplies with the target

of zero carbon dioxide emission. In order to achieve this target, ORC is a

promising technique for utilising low-grade temperature resources for

renewable energy.

The research is significant to energy demands. Global population and

urbanisation are continuing to grow. For example, developed countries are

estimated to have the highest average urban growth rate of 3.3% per annum

between 2010 and 2050 (Madlener & Sunak, 2011). Services to improve the


living standards of the growing population have put enormous pressure on

energy demands, which are estimated to increase by 25% by 2050 (Bohi,

2013). To satisfy energy demands and limit greenhouse gas emissions

simultaneously, the use of renewable energies must be extended, and more

efficient systems developed.

The research is quite significant to economic development. The Conference

of Mayors Climate Protection Agreement (CPA) points out that a significant

rationale will stimulate economic development through creating renewable

energy jobs (Yi, 2013). Developing renewable energy through more

efficient systems will reduce reliance on fossil fuels and create new jobs in

the renewable energy sector. This is an opportunity to supplement the loss

of jobs in the traditional fossil fuel energy sector.

This work contributes to design of more efficient ORC turbines for low-

grade temperature renewable resources. Robust ORC turbine designs must

be established. This requires designers to take into account the

quantification of uncertainties at an early stage in the design process. In

particular, the sensitivity of the ORC turbines to high-density fluids in both

on- and off-design conditions must be considered in order to develop

reliable and robust turbine designs capable of maintaining high efficiencies

over a wide range of conditions.

1.5 RESEARCH INNOVATION

The research innovations of this study are as follow:

For the first time, both nominal and off-design operational conditions are

thoroughly investigated using a coupled UQ-CFD approach, highlighting

the most influential operational uncertain parameters (both individually and

coupled) that affect high-density radial-inflow turbine performance.

This study is the first to conduct a comprehensive sensitivity analysis of

different EOS under various uncertain parameters that affect radial-inflow

turbine performance.

This is the first study to quantify the influence of coupled and multiple

uncertain parameters on a high-density fluid conical diffuser.


This study presents a UQ analysis for a whole turbo-expander with an

adapted diffuser working with a high-density fluid, and proposes a new

design of an ORC radial-inflow turbine that is more robust to on- and off-

design conditions under uncertainties.

1.6 THESIS OUTLINE

This section describes the structure of this dissertation. The thesis comprises

eight chapters based on a series of finished manuscript publications with original

contributions to the literature. Following this chapter, there are seven chapters, with

their synopsis presented below.

In Chapter 1, an overarching review of the background of renewable energy is

introduced. The Organic Rankine Cycle for renewable energy blocks, high-density

fluid radial-inflow turbines, typical types of diffusers, and the uncertainty

quantification approach, are briefly intoduced. The research problems, research

objectives, research significance and innovations are also defined.

In Chapter 2, there are three main sections of the literature review. The first

section reviews the renewable energy sector. The second section is about the power

blocks, which are divided into Organic Rankine Cycles, radial-inflow turbines, and

diffusers, including conical diffusers and annular-radial diffusers. The literature

review of radial-inflow turbines includes both ideal gas turbines and high-density fluid

turbines. In this section, the computational and experimental investigations of radial-

inflow turbines and various types of diffusers using both ideal gas and high-density

fluid are discussed. The third section reviews the UQ approach, including its

definition, and focuses on the applications of UQ related to radial-inflow turbines and

diffusers, and high-density fluid. Finally, the existing findings and limitations are

discussed.

Chapter 3 is on Methodology. This chapter introduces the computational

framework developed in this work. First, computational fluid dynamics solvers for

rotating machineries are discussed, then the generalised polynomial chaos method as

the UQ tool is detailed. The chapter covers polynomial theory basics and generalised

polynomial chaos (gPC) used for this study. In addition, the whole automatic robust

framework connecting the CFD solver with the UQ approach is introduced.


In Chapters 4–7, advances in the field pertaining to uncertainty quantification

analysis for ORC radial-inflow turbine are presented, based on finalised and in-

preparation papers for this resarch.

Chapter 4 is based on a published conference paper. The development of an

advanced robust numerical framework to examine the uncertainty quantification

technique connecting with the radial-inflow turbine CFD workflows using ideal gas is

discussed in this chapter. In particular, the CFD solver validation is presented as the

foundation for the following works. This study demonstrates the reliability and

robustness of the framework to conduct the uncertainty quantification analysis for

radial-inflow turbines.

Chapter 5 presents the work for the UQ analysis on high-density fluid radial-

inflow turbine. It provides in-depth on- and off-design conditions for the uncertainty

quantification analysis of the high-density fluid radial-inflow turbo-expander using

R143a fluid for low-grade temperature ORC renewable energy applications.

Chapter 6 applies the developed framework to the investigation of the sensitivity

of a high-density fluid conical diffuser.

Chapter 7 considers the numerical comparison between a conical diffuser and

annular-radial diffuser. In this chapter, the performance of the preliminary design of a

conical diffuser and an annular-radial diffuser is compared, matching the conditions

from our existing R143a radial-inflow turbo-expander. Then, an extensive uncertainty

quantification analysis of the whole turbine is conducted to verify the robustness of

the developed turbine.

In Chapter 8, the overarching conclusions of the project are drawn. Results from

all the previous chapters are summarised and interpreted to draw conclusions,

including the optimal whole turbine performance and the most important individual

and/or coupled uncertain parameters. The last section of this chapter proposes future

work utilising the current research as a foundation. Moreover, this section presents the

research limitations, the practical implications, and the future research developments.


A schematic of the thesis outline is shown in Figure 1-2.

CHAPTER

Methodology

CFD Model Uncertainty Quantification (UQ) Framework Benchmark

Research Schedule

1. Validation of the UQ-CFD framework.

2. Evaluation of the sensitive parameters in high-density fluid ORC

radial-inflow turbine using UQ-CFD framework.

3. Application of the UQ-CFD framework to high-density fluid conical

diffusers.

4. Robust Design of a complete turbo-expander with optimum diffuser

geometry.

Introduction

Background Research Objectives Significance Thesis Research

Problem & Aims &Innovation Outline Framework

1

3

2

Literature Review

Renewable Energy Turbo-expander Diffuser Uncertainty Quantification

High-density Fluid Meanline Numerical Off-design Experiments

8

4-7

Conclusions & Future Work

Conclusions Limitations Future Work

Figure 1-2: Thesis outline.

Chapter 2: Literature Review 15

Chapter 2: Literature Review

This chapter begins with an overview of an Organic Rankine Cycle (ORC)

application for low-grade temperature renewable energies in Section 2.1. A concise

introduction to ORC turbo-expanders is presented in Section 2.2. A contemporary

review of diffusers is introduced in Section 2.3. Then the literature regarding Uncertain

Quantification (UQ) methods, including the applications employing UQ, is reviewed

in Section 2.4. Section 2.5 highlights the conclusions and implications from the

literature.

2.1 ORC APPLICATIONS FOR LOW-GRADE TEMPERATURE

RENEWABLE ENERGIES

Energy is a critical topic for global economic development, and is treated as a

key factor for all industries and production processes (Rahbar, et al., 2017). With the

development of the worldwide economy, the shortage of energy and the pollution of

environments are becoming more and more serious (Dolz, Novella, García, &

Sánchez, 2012). The demand for energy is significantly increasing, with a 56% world

growth rate predicted from 2010 to 2040 (Bohi, 2013). According to a report from the

International Energy Agency (Agency, 2012), the present tendency extended to 2050

indicates that energy consumption will grow by 70% and pollution emissions will

increase by 60% compared to 2011. By 2050, pollution emissions will lead to the

global average temperature rising by 6°C, resulting in radical climate change as well

as unsustainable energy development. Thus, energy consumption and energy

efficiency have extraordinary influences on environmental development.

The energy sector is currently facing many challenges: fossil fuel limitations,

environment pollution due to greenhouse gases, and rapidly increasing demand

accompanying fast-growing urbanisation. Nowadays, in order to solve these issues and

balance energy supply and demand and environmental deterioration due to fossil fuel

combustion, renewable energy resources and energy usage efficiency are the focus of

attention (Abdmouleh, Alammari, & Gastli, 2015).

In this thesis, among many kinds of renewable energy, biomass (Qiu, et al.,

2012), ocean thermal energy conversion (Sun, et al., 2012) and geothermal energy


(Heberle & Brüggemann, 2010; Lentz & Almanza, 2006), are focused on rather than

other types of renewable energy, as these three renewable energies can employ the

promising ORC technique to extract energy from low-grade temperature resources.

These three kinds of renewable energies contribute to multiple and/or mixed electricity

generation for government targets. Thus, these three types of renewable energy in ORC

applications will be introduced in brief in this section to give an overall picture.

Figure 2-1: Schematic of a biomass ORC system (Rahbar, et al., 2017).

As the fourth largest energy resource, biomass can supply about 10% of the

world’s energy demands (Dolz, et al., 2012). Biomass from agricultural processing is

widely utilised; for instance, from the furniture industry, agriculture, or forest residues.

The schematic of a biomass ORC system is shown in Figure 2-1. Biomass transforms

its combustion heat energy into electricity energy by employing the ORC technique.

The heat is generated from the biomass feed burner, and then transfers the heat via flue

gases to the heat-transfer fluid. The heat-transfer fluid goes to the evaporator in the

ORC to vaporise the high-density fluid (Tchanche, Lambrinos, Frangoudakis, &

Papadakis, 2011); in this study, the working fluid is R143a. A thermodynamic model

of a 2kW biomass Combined Heated and Power system employing ORC with HFE

7000, n-pentane, and HFE7100 as working fluid is investigated by Liu, Shao, and Li

(2011). In their results, the thermodynamic efficiency of ORC is determined not only


by sub-cooling but also by its superheating efficiency. The ORC’s efficiency relies on

some additional aspects, such as the operational conditions, selection of the working

fluid, the fluid temperature in the boiler, and the heating amount supported by the

Combined Heated and Power system. In Europe, more than 120 power plants

employing the ORC technique are in commercial operation, utilising biomass

combustion heat (Bini, Guercio, & Duvia, 2009), which is part of the energy mix. This

evidences the need to further develop those technologies and make their contribution

higher. In most of these power plants, radial-inflow turbines play quite a significant

role in converting energy and generating electric power, and are the focus of the

present research.

Figure 2-2: Schematic of an OTEC ORC system (Rahbar, et al., 2017).

Ocean thermal energy conversion (OTEC) is another promising renewable

energy technique that can transfer thermal power from the ocean’s natural thermal

gradient into electricity. The schematic of an OTEC ORC system is illustrated in

Figure 2-2. ORC is an operational solution to converting low-grade temperature heat

into electricity power. The warm water from the ocean’s surface is employed as the

heat medium to vaporise the high-density fluid in the evaporators of ORC. At the same

time, the cold water from the deep layer of the ocean is utilised as the cooling medium

to condense the high-density fluid. Sun, et al. (2012) investigated and optimised the

performance and energy efficiency of ORC in OTEC by employing different working

fluids with both ammonia and R134a. From their results, ammonia is indicated as the


most suitable working fluid for ORC application in OTEC in terms of the output

power. Wang et al. (Wang & Hung, 2010) conducted the investigation employing

OTEC coupled with solar energy to generate the electricity. The results showed that

the pressure, inlet temperature, and exit temperature of the turbine condenser influence

the turbine’s efficiency, employing R113, R114, and R123 as dry working fluids and

R11, R152a, and R500 as wet working fluids. Compared to the dry working fluids, the

wet working fluids indicated better turbine efficiency in terms of steep saturated

vapour in the 𝑇 − 𝑆 diagram. Nithesh, Chatterjee, Oh, and Lee (2016) designed an

ORC radial-inflow turbo-expander for a 2 kWe capacity OTEC energy application,

with a rotational speed of 34,000 RPM and inlet and outlet temperatures of 24.5 °C

and 14 °C respectively, employing R-22 as the working fluid. The rotor tip and shroud

radii were 24 mm and 19 mm respectively; the blade widths at rotor inlet and outlet

were 6 mm and 11 mm respectively; the axial length was 17.5 mm; and the length of

the diffuser was 62 mm. In addition, the significance of the number of blades, blade

filleting and stagger angle in terms of turbine efficiency were presented as well. In

another study, Nithesh and Chatterjee (2016) designed another ORC radial-inflow

turbine for OTEC energy with 2 kWe power output, employing R134a as the working

fluid. The inlet and outlet temperatures of the turbine were 24.5 °C and 14 °C

respectively; the rotational speed of the turbine was 22,000 RPM; the rotor tip and

shroud radii were 35.5 mm and 22 mm respectively; the blade widths of rotor inlet and

outlet were 6 mm and 13 mm respectively.

Geothermal energy is another type of low-grade temperature renewable energy

using the ORC technique. The schematic of a geothermal ORC binary power cycle is

expressed in Figure 1-1. At approximately three kilometres depth from the earth’s

surface, the estimated geothermal energy resources are 43,000,000 × 1018 J (DiPippo,

2012). The heat source temperature of geothermal energy varies from 50-350°C. The

most commonly available temperature of the resource is normally from 100-220°C for

the medium-temperature geothermal resources. However, it is for important for

potential heat resources with a low-grade temperature from 70-110°C to be utilised

(Liu, Chien, & Wang, 2004). Such heat resources are the most widespread technique

employed to generate electricity in binary cycle power plants. The heat is transferred

to the organic working fluid in ORC evaporators and goes back through the brine to


the injection lines at a low-grade temperature (Quoilin, Van Den Broek, Declaye,

Dewallef, & Lemort, 2013).

Lots of investigations have focused on fluid selection related to geothermal

energy (Liu, Duan, & Yang, 2013; Liu, Wei, Yang, & Wang, 2017; Madhawa

Hettiarachchi, Golubovic, Worek, & Ikegami, 2007; Zhai, Shi, & An, 2014). However,

as shown by (Persky & Sauret, 2018), fluid selection cannot be decoupled from the

cycle design and turbine design. (Sauret & Gu, 2014) study is one of the few that look

at radial-inflow turbine design for geothermal energy. Sauret and Gu (2014) designed

an ORC radial-inflow turbine for geothermal energy application using a one-

dimensional meanline model employing R143a as a working fluid. This study is a key

study in ORC radial-inflow turbine investigation; it not only built the foundation for

later many similar investigations into one-dimensional meanline design, but also it

supplies guidance for three-dimensional numerical studies under off-design

conditions, such as the one presented in this thesis. However, as Sauret and Gu (2014)

mentioned, in their future work, Uncertainty Quantification analysis is necessary to

achieve a more robust and comprehensive understanding of radial-inflow turbine

design. This thesis, accordingly, presents an uncertainty quantification study of a

radial-inflow turbine.

It is evident that ORC is a critical technology in these renewable energy systems.

In terms of the key components for converting energy into electricity, radial-inflow

turbo-expanders and diffusers play pivotal roles in ORC applications (Badr, et al.,

1984). By improving the efficiency of radial-inflow turbines, the current low

efficiency of the ORC cycle can be dramatically enhanced. In this regard, radial-inflow

turbines have been extensively used (Boncinelli, et al., 2004; Cho, et al., 2014; Harinck,

et al., 2013; Pini, et al., 2013; Sauret & Gu, 2014; Sauret & Rowlands, 2011; Wheeler

& Ong, 2013; Zou, et al., 2015). Diffusers are positioned at the rotor outlet of the

radial-inflow turbines, aiming to recover the pressure and thereby increase the

turbine’s efficiency. The studies that investigate ORC high-density fluid turbines and

diffusers will be reviewed in the following sections.


2.2 ORC TURBO-EXPANDERS

In section 2.1, renewable energy in ORC applications, including biomass energy,

OTEC energy, and geothermal energy was discussed. In this section, investigations

into the turbo-expanders are reviewed.

Turbo-expanders are the most significant components in an ORC system, as they

generate mechanical energy. Thus, analysis of ORC turbo-expanders, especially for

low-grade temperature renewable power blocks, has received wide attention from

resarchers. Generally, the turbo-expander accelerates the fluid and changes the fluid

direction, and thus results in reducing the stagnation enthalpy through the rotation of

the rotor. These changes produce mechanical energy through energy transfer processes

(Hall & Dixon, 2013). Upstream of the rotor, the stator accelerates the working fluid

and delivers it to the rotor with a large, absolute, tangential velocity.

The absolute tangential velocity then decreases through the rotor, and the

decrease decides the amount of useful mechanical energy generated (Hall & Dixon,

2013).

In general, the two common types of turbo-expanders are radial and axial turbo-

expanders. In radial-inflow turbines, the flows enter the rotor in the radial direction.

With a large tangential velocity, the flow turns 90° throughout the rotor and leaves the

rotor following an axial direction. On the other hand, in an axial turbo-expander, the

flow always remains in the axial direction without any radial direction of velocity.

Sketches of an axial turbo-expander and radial-inflow turbo-expander are presented in

Figure 2-3 (a) and (b) respectively.

Figure 2-3: (a) The sketch of an axial turbo-expander. (b) The sketch of a radial-

inflow turbo-expander (Baskharone, 2006).


Radial-inflow turbo-expanders can be effectively utilised in single shaft turbines

in power ranges from as low as 1 kW up to approximately 2 MW (small-scale capacity)

electricity generation applications (Li, Pei, Li, & Ji, 2013). Consequently, radial-

inflow turbo-expanders have advantages over the axial turbo-expanders in industry,

which usually employs radial-inflow turbo-expanders (Bao & Zhao, 2013) below 2

MW usually employing radial-inflow turbo-. In the past decades, NASA conducted

extensive investigations into small turbo-expander design, and concluded that radial-

inflow turbo-expanders are favoured over axial turbo-expanders in small-scale power

generation (Kofskey & Nusbaum, 1969; Kofskey & Wasserbauer, 1966; Rohlik, 1968;

Wood, 1962). In addition, it is noteworthy that axial turbo-expanders need a

tremendously thin blade at the trailing edge to obtain high efficiency for a small scale

system (Dunham & Panton, 1973). The advantages of radial-inflow turbo-expanders

are a compact system, easy manufacture, and light-weight construction, which provide

high efficiency and robust design. Radial-inflow turbo-expanders are employed for

widespread applications, managing a wide range of net power output, rotating speed,

and mass flow rate (Hall & Dixon, 2013). High-density fluid radial-inflow turbo-

expanders are the focus of the main study objective of this thesis. High-density fluid

significantly affects radial-inflow turbine performance in ORC applications, and

accordingly, investigations into this will be introduced in the next section.

Equations of State for High-density Fluids

The analysis of the fluid dynamics of turbo-expanders using high-density fluids

(non-ideal gases) requires comprehensive consideration. This is relevant in regard to

low-grade temperature ORC power cycles where more energy can be extracted from

the high-density fluids. Hence, research into high-density fluid for turbo-expanders is

gaining a lot of attention.

However, high-density fluids do not follow the classical ideal gas laws. Redlich

and Kwong (1949) were among the first to develop an Equation of State (EOS),

Redlich-Kwong (RK), for high-density fluids. The equation includes two individual

coefficients, giving satisfactory results above the critical temperature for any

pressures. The coefficients are dependent on the gas composition, as discussed by

Redlich and Kwong (1949). Equations (2-1)-(2-3) are the primary equations of the

next several EOS methods listed below, which are modifications from the RK

equation. The RK EOS equations are as follows:


𝑝 =𝑅𝑇

𝑉𝑚 − 𝑏−

𝑎

√𝑇𝑉𝑚(𝑉𝑚 + 𝑏)

(2-1)

𝑎 =0.42748𝑅2𝑇𝑐

2.5

𝑝𝑐

(2-2)

𝑏 =0.08662𝑅𝑇𝑐

𝑝𝑐

(2-3)

Where 𝑝 is the gas pressure; 𝑅 is the gas constant; 𝑇 is temperature; 𝑉𝑚 is the

molar volume; 𝑇𝑐 is the temperature at the critical point; 𝑝𝑐 is the pressure at the

critical point.

Peng and Robinson (1976) developed a new two-constant equation, Peng-

Robinson (PR) EOS, which is known for its good balance between simplicity and

accuracy, especially close to the critical point (Agrawal, Cornelio, & Limperich,

2012); it is a further extension of the RK model, and is thus expected to behave like

the SRK model. Its advantage is the prediction of liquid phase densities as well. Span

and Wagner (1996) used Span-Wagner (S-W) EOS to study the critical region and to

extrapolate the behavior of carbon dioxide thermodynamic properties. Carbon dioxide

covering the fluid region, using this EOS, is up to 1100K for temperature and up to

800MPa for pressure. Aungier (1995) modified the Redlich-Kwong (RK) two-

parameter EOS for high-density fluids. The prediction accuracy of the thermodynamic

parameters such as enthalpy and entropy for several compounds was demonstrated

over an increased application range. The Soave-Redlich-Kwong (SRK) model (Soave,

1972) is an extension of the original Redlich-Kwong (RK) model. Nagel and Bier

(1996) investigated the validity of SRK EOS for mixtures of R125, R143a and R134a,

which was confirmed by comparison with experimental results. Lemmon (2004)

evaluated mixture models in Helmholtz energy thermodynamic properties of

refrigerant mixtures containing R-32, R-125, R-134a, R143a, and R-152a. The

independent variables are the density, temperature, and composition. The model could

be used to calculate the thermodynamic properties of mixtures, including dew and

bubble point properties. Park et al. (Park, Lim, & Lee, 2002) investigated the vapor–

liquid equilibrium data for six binary mixtures of isobutene + HFC-32, +HFC-125,

+HFC-134a, +HFC-143a, +HFC-152a, and +HFC-227ea. They were correlated with

the Peng–Robinson–Stryjek–Vera (P-R-S-V) EOS. Their results had good agreements


with experiments. The HHEOS is a 17-term Lemmon-Jacobsen equation based on

Helmholtz energy (Lemmon & Tillner-Roth, 1999), with the capacity to predict

accurately the properties for multicomponent mixtures, and is valid for temperatures

up to 450K and pressures up to 50MPa (Lemmon & Tillner-Roth, 1999). The National

Institute of Standards and Technology (NIST) Reference Fluid Thermodynamic and

Transport Properties (REFPROP) database takes account of high-density fluid

behaviour. The NIST REFPROP model is considered the international standard for

fluid properties (Lemmon, Huber, & McLinden, 2013). This model employs high-

fidelity empirical correlations in the Helmholtz energy Equation of State (HHEOS) to

calculate the transport properties of high-density fluid (Lemmon, Huber, & McLinden,

2007). A brief summary of tested cubic EOSs is given in Table 2-1.

According to Table 2-1, it is noteworthy that Peng-Robinson and Soave-Redlich-

Kwong’s equation of state for R143a as a working fluid has been validated

experimentally.


Table 2-1: Brief summary of tested cubic EOSs in literature review.

Due to the importance of EOSs in the accurate prediction of turbine performance,

understanding the effect of Equation of State uncertainties becomes critical to better

design of turbines.

EoS Name Fluid Temp. and Pressure Range Experiment

Validation

Redlich-Kwong (Barrick,

Anderson, & Robinson Jr,

1986)

Carbon

Dioxide 323-423K; 1.6-10.7 MPa Yes

Soave-Redlich-Kwong

(SRK) (Nagel & Bier,

1996)

R125/R143a/13

4a

204.557-363.760K;0.01675-

3.963 MPa Yes

Peng-Robinson (PR)

(Nagel & Bier, 1996) R143a/R134a

205.023-360.730K;0.02143-

3.940MPa Yes

Peng-Robinson-Stryjek-

Vera (Park, et al., 2002)

R600a 293.15-323.15K;0.3045-

0.6832 MPa Yes

HFC-32 283.15-303.15 K;1.1090-

1.9240 MPa Yes

HFC-125 293.15-313.15 K;1.2036-

2.0030 MPa Yes

HFC-134a 303.15-323.15 K;0.7700-

1.3200 MPa Yes

HFC-143a 323.15-333.15 K;2.3030-

2.8690 MPa Yes

HFC-152a 303.15-323.15 K;0.9050-

1.5000 MPa Yes

HFC-227ea 293.15-313.15 K;0.2760-

0.9222 MPa Yes

HFC-32/R600a 301.75-321.75 K;1.885-

3.057 MPa Yes

HFC-

134a/R600a

303.15-323.15 K; 0.877-

1.451 MPa Yes

HFC-

152a/R600a

293.15-333.15 K; 0.578-

1.599 MPa Yes

HFC-

227ea/R600a

303.15-323.15 K; 0.646-

1.081 MPa Yes

R-401a 208.15-518.15K; 0.01-3.8

MPa Yes

Span-Wagner (Span &

Wagner, 1996)

Carbon

Dioxide

216.6K-1100K; 0.3MP-

800MPa Yes


Table 2-2: Latest radial-inflow turbine numerical studies using various types of EOS.

Author Working

Fluid

Geometry EOS

Sauret and Gu (2014) R143a Radial turbine Peng-Robinson (Peng

& Robinson, 1976)

Li and Ren (2016) R123 Radial turbine Aungier-Redlich-

Kwong (Aungier, 1995)

Kim and Kim (2017a) R152a Radial turbine Aungier-Redlich-


Kim and Kim (2017b) R143a Radial turbine Aungier-Redlich-


Zheng, Hu, Cao, and

Dai (2017)

R134a Radial turbine Peng-Robinson (Peng

& Robinson, 1976)

Wheeler and Ong

(2013)

Pentane,

R245fa

Radial turbine Helmholtz (Lemmon &

Span, 2006; Span &

Wagner, 2003)

Dong et al. (2018) R245fa Radial turbine

nozzle

Soave–Redlich–Kwong

(Pedersen, Thomassen,

& Fredenslund, 1984)

Nithesh, et al. (2016) R22 Radial-inflow

turbo-expander

Soave–Redlich–Kwong

(Pedersen, et al., 1984)

Xia, et al. (2018) R245fa Radial-inflow

turbine

Helmholtz (Lemmon &

Span, 2006; Span &

Wagner, 2003)

Wu and Pan (2018) R134a Radial-inflow

turbine

Peng-Robinson (Peng

& Robinson, 1976)

Deligant, Sauret, Danel,

and Bakir (2018)

SES36,

R245fa

Radial-inflow

turbine

Peng-Robinson (Peng

& Robinson, 1976)

Various EOSs for these high-density gases are available in the literature (Poling,

Prausnitz, & O'connell, 2001), as presented in


Table 2-2. According to these studies, the EOSs were considered the most

accurate for their study. However, the reasons for choosing a suitable EOS for high-

density in ORC applications have not been thoroughly investigated. Furthermore, most

EOSs derive from the classical meanfield theory, which explains critical fluctuations

by illustrating the critical regions of a fluid. As such, there are uncertainties in the

mathematical model itself, but also in the errors in measurement of the EOS data

around the fluid’s critical point. This explains why most EOSs are expected to result

in uncertain property predictions in the close vicinity of the critical region (Poling,

Prausnitz, John Paul, & Reid, 2001).

Reliability of critical pressure, critical temperature and acentric factors, among

others, is questionable (Cinnella, Marco Congedo, Pediroda, & Parussini, 2011), and

thermodynamic model uncertainties must be included in the turbine design. The

application of uncertainty quantification techniques to high-density fluid has mainly

been investigated for shock tubes (Congedo, Colonna, Corre, Witteveen, & Iaccarino,

2010), 2D airfoil geometries (Congedo, Corre, & Martinez, 2011; Merle & Cinnella,

2015), and 2D nozzle blades (Bufi, Cinnella, & Merle, 2015; Colonna, Rebay, Harinck,

& Guardone, 2006). It is now critical to characterise the deviations in efficiency from

the idealised deterministic conditions for various equations of states (Panizza, Iurisci,

Sassanelli, & Sivasubramaniyan, 2012). Very recently, Zhao, Mecheri, Neveux,

Privat, and Jaubert (2017) investigated Uncertainty Quantification analysis for EOS,

and identified HHEOS as most appropriate to CO2 as a working fluid for a high-

temperature supercritical turbine, which is different from the subcritical low-grade

temperature turbine in this study. In this thesis, R143a is utilised as the high-density

working fluid for numerical studies of ORC subcritical turbo-expanders and diffusers.

ORC Radial-inflow Turbine Design

Although extensive investigations have looked at the working fluid selection in

ORC cycles (Branchini, De Pascale, & Peretto, 2013; Hung, 2001; Larsen, Pierobon,

Wronski, & Haglind, 2014; Saleh, Koglbauer, Wendland, & Fischer, 2007; Toffolo,

Lazzaretto, Manente, & Paci, 2014), the design of ORC turbines is quite significant

for the improvement of ORC cycle performance. The literature review has revealed

that the meanline design of a radial-inflow turbo-expander using ideal gas is well

researched, and the design processes have been summarised in many key studies

(Aungier, 2005; Ebaid, Bhinder, & Khdairi, 2003; Hall & Dixon, 2013; Marcuccilli &


Zouaghi, 2007; Moustapha, Zelesky, Baines, & Japikse, 2003; Rodgers & Geiser,

1987; Whitfield, 1990; Whitfield & Baines, 1976; Whitfield & Baines, 1990; Yang,

1991). However, these analyses are based on ideal gas, not organic high-density fluids.

Higher efficiency of radial-inflow turbines can be attained by operating non-ideal

fluids or high-density fluids, as real organic high-density working fluids can release

more energy (Al-Sulaiman, et al., 2010; Chacartegui, et al., 2009; Quoilin, et al.,

2010). Thus, it is necessary to develop robust meanline designs of radial-inflow

turbines that are appropriate for these high-density fluids. This requires accurate

modelling of high-density fluid properties using appropriate EOS, as discussed in

Section 2.2.1.

A widespread radial-inflow turbine design methodology is usually called

meanline design, which considers the velocity triangles and fluid properties through

the turbine. Meanline modelling is based on a one-dimensional assumption that there

is a mean streamline through the stages, so that conditions on the mean streamline are

an average of the passage conditions (Moustapha, et al., 2003). An early study was

conducted by Badr, et al. (1984) on an ORC turbo-expander employing a specific

rotational speed and diameters. In a typical study, Sauret and Rowlands (Sauret &

Rowlands, 2011) compared different fluids that will generate different turbine designs

with different characteristics, such as size and RPM. Their results indicated that

optimal designs were obtained for R134a and n-pentane.

Zhu, et al. (2015) built a meanline design model for an ORC radial-inflow

turbine, following the work proposed by Aungier (2005). The results showed that

velocity ratio and velocity speed, expansion ratio, and turbine size, are key parameters

of the radial-inflow turbine efficiency in ORC system applications. Their study

presented a 50kW rotor with an inlet temperature of 147°C using REFPROP for

calculating working fluid properties. Ventura, Jacobs, Rowlands, Petrie-Repar, and

Sauret (2012) developed a similar methodology; however, it also coupled a meanline

model with loss models to evaluate the turbine efficiency. This methodology was

developed to automatically select feasible machines according to pre-defined

performance or geometric features for a given problem. Later, similar studies that

coupled meanline design with loss models were published (Erbaş & Bıyıkoğlu, 2015;

Fiaschi, et al., 2015; Hu, Li, Zheng, Wang, & Dai, 2015; Rahbar, Mahmoud, Al-

Dadah, & Moazami, 2015a; Wang, et al., 2013). The meanline model was coupled


with the ORC thermodynamic model by Wang, et al. (2013) and Hu, Li, et al. (2015),

allowing the effect of varying radial-inflow turbine efficiency to be considered in cycle

analysis investigations. The purpose of these was to diverge from the decoupled

method, enabling a more optimal framework to be investigated. Comparably, both

Rahbar, Mahmoud, Al-Dadah, and Moazami (2015b) and Erbaş and Bıyıkoğlu (2015)

coupled the meanline model with optimisation methodology, which optimised the

various input variables to achieve the highest turbine performance. Rahbar, et al.

(2015b) conducted radial-inflow turbine optimisation, employing a genetic algorithm

to maximise turbine performance using different high-density working fluids. The

main purpose of their study was to obtain the highest design point efficiency of

turbines. In contrast to Rahbar, et al. (2015b), Erbaş and Bıyıkoğlu (2015) also used

genetic algorithm, but optimised the part-load efficiency rather than the design point

efficiency of turbines. Although meanline design models coupled with other

approaches (such as loss model, thermodynamic model, and optimisation model) show

improvements, the Uncertainty Quantification aspects have not been taken into

account.

In summary, one-dimensional meanline design models were developed by many

studies for the ORC radial-inflow turbine’s preliminary design, and a considerable

focus has been on the inclusion of loss models. However, the loss models are based on

empirical data, which is gathered from experiments on radial-inflow turbines

employing ideal gas, not high-density fluid. Furthermore, as remarked by Uusitalo,

Turunen-Saaresti, Honkatukia, Colonna, and Larjola (2013), accurate evaluation for

turbine performance needed experimental data, which was lacking; thus it is

noteworthy that most of the studies have not been totally validated by experimental

information using high-density fluid. The meanline design model aims to design the

velocity triangles and inlet/outlet dimensions of the rotor. However, the three-

dimensional effect with the rotor passage must be modelled robustly to ensure the

effective expansion of the fluid (White, 2015). Normally, this is not fully considered

in meanline design modelling; thus, there is a need for a more advanced technique,

like the CFD solver, to solve it. Numerical studies on ORC radial-inflow turbines

employing CFD technique will be introduced in the following section.


Numerical Study

The Computational Fluid Dynamics (CFD) technique is a powerful solution for

conducting numerical studies of ORC turbines (Klonowicz, Borsukiewicz-Gozdur,

Hanausek, Kryłłowicz, & Brüggemann, 2014). CFD numerical study is not only used

to forecast performance, but also to optimise ORC radial-inflow turbine design. A CFD

solver capable of handling high-density fluid has been developed (Cinnella &

Congedo, 2005; Hoffren, 1997). The numerical study of ORC turbines must consider

high-density fluid complex properties for higher accuracy.

Usually, three-dimensional geometry designs used in CFD studies are

constructed based on one-dimensional meanline designs. Zheng, et al. (2017) used

ANSYS-CFX to assess the preliminary steady-state three-dimensional design of a

radial-inflow turbine employing R134a as the candidate working fluid. CFD was also

employed to decide the number of rotor blades. In CFD technique development,

Hoffren, Talonpoika, Larjola, and Siikonen (2002) tailor-made an existing Navier-

Stokes solver to simulate a supersonic turbine stator using high-density fluid. Their

results were consistent with the calculated results of one-dimensional meanline design.

Colonna, et al. (2006) demonstrated CFD numerical studies employing Euler solver,

and presented the flow characteristics in the stator by using different equations of state.

Their study indicated that the Peng-Robinson-Stryjek-Vera and Span-Wagner

techniques had quite similar results; however, results were significantly different when

employing the ideal gas law.

Harinck, Turunen-Saaresti, Colonna, Rebay, and van Buijtenen (2010) simulated

a supersonic ORC turbine stator, and compared the 𝑘 − 휀 and standard 𝑘 − 𝜔

turbulence models. The results showed that the choice between 𝑘 − 휀 and standard

𝑘 − 𝜔 turbulence models had minor influence on the flow field, in particular the Mach

number and the overall flow structure, with a difference in isentropic efficiency of less

than 2%. The main difference appears for the prediction of the shock wave interacting

with the boundary layer. Moreover, Sauret and Gu (2014) also successfully applied

𝑘 − 휀 model and validated their CFD model with meanline design. For further CFD

numerical set-up, the first order Upwind numerical scheme were employed for steady-

state radial turbines simulations in (Dong, et al., 2018; Dong, Xu, Luo, Zhuang, &

Quan, 2017; Fiaschi, Innocenti, Manfrida, & Maraschiello, 2016; Sauret & Gu, 2014).

However, the second order Upwind numerical scheme were used for unsteady-state


simulations such as (RAI, 1987)and transient state simulations in (Setoguchi,

Santhakumar, Takao, Kim, & Kaneko, 2002). Harinck, et al. (2013) presented a

steady-state three-dimensional viscous CFD of Tri-O-Gen ORC radial-inflow turbine,

including the radial-inflow stator, rotor, and diffuser. This CFD numerical study by

ANSYS CFX for an ORC turbine employed REFPROP to generate fluid property

tables. The three-dimensional simulation results show that stator geometry was

improved, manufactured and experimentally tested. However, as the high pressure

ratio and radial arrangement of nozzles presented a straight mean line, dense vapour

expansion through the turbine stator was influenced by strong oblique shock waves.

Based on the demonstrated three-dimensional RANS simulations, the shock waves,

together with the viscous wake of the blade, generated a huge variation in flow outlet

angle and velocity along the circumference of the rotor inlet. In addition, as there was

a lack of measurement of flows working in the high-density fluids region, their CFD

codes were not appropriately validated by comparison against high-density fluid flow

experiments. Wheeler and Ong (2014) conducted a similar numerical investigation, in

which steady and unsteady state simulations were compared for ORC turbines. The

CFD simulation results showed that unsteady simulations presented the same tendency

as the steady ones; nevertheless, there was an obvious interaction between the stator

trailing edge shocks and the rotor leading edge, which resulted in a big drop in turbine

efficiency.

Sauret and Gu (2014) conducted a three-dimensional steady state RANS

numerical study on an ORC radial-inflow turbine, employing R143a as a working fluid

for a geothermal energy application. The geometry of the radial-inflow turbine in the

CFD model was based on their meanline design. The CFD technique was utilised using

an ANSYS-CFX module in their study, and a 𝑘 − 휀 turbulence model was employed

for the CFD solver, in conjunction with the Peng-Robinson EOS. They mentioned that

in their future work, the REFPROP database would replace the Peng-Robinson EOS.

After Sauret and Gu (2014) study, similar investigations were conducted (Kim & Kim,

2017b; Rahbar, Mahmoud, & Al-Dadah, 2016; Zheng, et al., 2017). Kim and Kim

(2017b) designed the same size turbine as Sauret and Gu (2014) using the same

working fluid, R143a, and also conducted a similar CFD numerical study. However,

the NIST REFPROP was applied to their radial-inflow turbine performance analysis,


and the CFD calculation results in terms of ORC turbines performance analysis were

argued to be more reliable than Sauret and Gu (2014) study.

Most of these CFD numerical investigations were verified against one-

dimensional meanline designs, as internal measurement of small radial turbines is

difficult to conduct, and published experimental data, as outlined by Sauret (2012),

lack essential information for reproduction of the geometry and validation of

numerical codes. This lack of experimental data for code validation is even more

critical for high-density radial-inflow turbines. As outlined by Congedo et al.

(Congedo, Corre, & Cinnella, 2011), the optimised design of ORC turbines relies on

the availability of accurate and efficient CFD tools. Very recently, Alshammari,

Pesyridis, Karvountzis-Kontakiotis, Franchetti, and Pesmazoglou (2018) conducted a

CFD study on an ORC radial-inflow turbine using an ANSYS Turbogrid for a meshing

tool and ANSYS CFX for the solver. The CFD results had good agreement, with a

maximum deviation of 1.15% in total efficiency in their experimental results. This

study is good for building confidence in terms of CFD solver validation work

compared against experimental investigation. However, this study did not mention

EOS, and did not consider Uncertainty Quantification in the numerical investigations.

The CFD technique can accurately predict ORC radial-inflow turbine design.

Investigations of radial-inflow turbines under off-design conditions will be introduced

in Section 2.2.4.

Off-design Modelling

In order to achieve optimum performance from advanced renewable power

blocks, one of the main requirements is to have robust designs that are capable of

handling fluctuations and operating well under adverse conditions. This is especially

important for turbines in Organic Rankine Cycles (ORC), as the delivered power must

be as high as possible for those low-grade temperature renewable cycles. It has been

shown that radial-inflow turbines are well-suited expanders for ORCs (Pini, et al.,

2013) but are also sensitive to operational conditions while using high-density fluids

(Sauret & Gu, 2014).

In addition, the thermodynamic parameters of the ORC heat sources may not be

as stable and controllable (Delgado-Torres & García-Rodríguez, 2010; Shokati,

Ranjbar, & Yari, 2015; Tchanche, et al., 2011) as other power systems. The mass flow


rate and the input temperature from the heat sources will change with the quantity of

power plant production and the operation process (Campana et al., 2013). In addition,

the operational conditions of the ORC systems may vary distinctly when the

supplement of the heat source and/or cooling source is unstable (Song, Gu, & Ren,

2016b). In this case, the optimal cycle conditions may move the radial-inflow turbine

away from the nominal design conditions to off-design conditions. The performance

of the ORC system at off-design conditions is a significant consideration in the ORC

radial-inflow design and optimisation phase. Some investigations (Calise, Capuozzo,

Carotenuto, & Vanoli, 2014; Han, Chen, Lin, & Jin, 2015; Hu, Li, et al., 2015; Hu,

Zheng, Wu, Li, & Dai, 2015; Mazzi, Rech, & Lazzaretto, 2015; Song, et al., 2016b)

focused on the off-design within the cycles. As described before, the radial-inflow

turbine is the key component for the organic vapour expansion of high-density fluid

and the power output in the ORC system. Thus, off-design performance evaluation of

the radial-inflow turbine is quite essential and meaningful for the whole ORC system

analysis. However, off-design radial-inflow turbine investigations are not extensive so

far. In this thesis, the radial-inflow turbine is the main research target. The present

investigations of radial-inflow turbines under off-design conditions are introduced

below.

Sauret and Gu (2014) conducted an R143a-400kW radial-inflow turbine

performance under off-design conditions. The results showed the best efficiency point

was gained at the fixed 400kW power output. In addition, compared to off-design

conditions, efficiency of the nominal conditions was quite close to the best efficiency,

with a 1.4% variation. At nominal turbine power conditions, the highest efficiency was

achieved at the nominal mass flow rate with a decreased pressure ratio, obtained by

changing the static pressure at the outlet. Later, Kim and Kim (2017b), following the

research concept of Sauret and Gu (2014), investigated the radial-inflow turbine

performance map under off-design conditions, employing R143a as a working fluid.

Their results indicated that the rotational speed and the incidence angle towards the

rotor blade greatly affect the radial-inflow turbine efficiency and net power output.

Zheng, et al. (2017) also conducted a similar off-design performance evaluation for a

radial-inflow turbine, employing R134a as working fluid. The performance of nominal

design point was compared to off-design points. In these investigations (Kim & Kim,

2017b; Sauret & Gu, 2014; Zheng, et al., 2017), the off-design input variables were:


the rotational speed RPM, the inlet temperature 𝑇𝑇𝑖𝑛 , and the pressure ratio. The

nominal design point is the most capable of handling a variation of these three

variables (rotational speed, inlet temperature, and pressure ratio), where the radial-

inflow turbine can maintain a relatively high performance over the range of variation.

These numerical off-design investigations did not consider uncertainties under

off-design conditions. The off-design performances need to be conducted

experimentally as well, so that numerical models can accurately predict over a wide

range of operational conditions to generate the turbine performance map. This is useful

to guide an optimised and robust design process for a high-density fluid radial-inflow

turbine.

Experimental Studies of ORC Radial-inflow Turbines

The experimental investigations of ORC turbines not only validate the CFD

solvers for numerical studies of ORC turbines, but also build comprehensive

understanding of high-density fluid behaviour in ORC turbines.

In the early stages, the experimental investigation of radial-inflow turbines

focused on air as the working fluid (Borges, 1990; Dambach, Hodson, & Huntsman,

1999; Simpson, 2013; Spence & Artt, 1997). However, the ideal gas cannot reflect the

characteristics of high-density fluid flow through the turbines.

In order to demonstrate how high-density fluid affects the radial-inflow

performance, experimental studies employing high-density fluid need to be conducted.

However, three-dimensional radial-inflow turbine experiments using high-density

fluid as the working fluid are currently very limited (Kang & Chung, 2011). Kang

(2012) designed a radial-inflow turbine directly connected to the high-speed

synchronous generator using R245fa. The average total-to-total turbine efficiencies

were 76.0, 77.5 and 82.2%, and the average cycle efficiencies were 5.05, 5.24 and

5.66% when the average evaporator temperatures were 77.1, 79.5 and 82.3 °C,

respectively. The cycle efficiency is low. With increased evaporator temperature, both

turbine and cycle efficiency increase. However, no uncertainties were considered in

their experiments. Recently, Shao et al. (Shao, Zhu, Meng, Wei, & Ma, 2017)

experimentally investigated the operational characteristics and performance of the

ORC radial-inflow turbine, using R123 as working fluid. According to their

experimental results, with the heat source temperature increasing, both isentropic


efficiency and thermal efficiency increased, with different corresponding relationships

(between inlet temperature and turbine efficiency) from the numerical studies (Kim &

Kim, 2017b; Sauret & Gu, 2014), as introduced in section 2.2.4. In a higher net power

output experimental study, Alshammari, et al. (2018) conducted a radial-inflow turbine

experiment. Due to the limitation of the experimental environment, the experimental

results compared with CFD numerical studies under off-design operational conditions

were not at design conditions.

These experiments were limited by a fixed power output, and only used one

particular working fluid, such as R123. Experiments also lacked an evaluation of

uncertainties, and in particular, of the temperature fluctuations of the renewable heat

resource supply.

The above literature review regarding the turbo-expander includes high-density

fluids and equations of state, meanline design, numerical studies, off-design, and

experimental studies. As the downstream of turbo-expanders, diffusers are also

significant in their effect on the overall ORC efficiency.

2.3 DIFFUSERS

As previously mentioned, an ORC system includes turbine and diffuser, which

are critical components, and evaporator, condenser and working fluid pump. Many

investigations have been conducted for ORC turbo expanders; a typical study is that

of Fiaschi, et al. (2012); but ORC diffusers have been neglected in the research.

Diffusers are positioned at the downstream of turbo expanders, aiming to recover

exhaust kinetic energy as static pressure, and thereby increase the whole ORC

efficiency.

Improving the efficiency of radial-inflow turbines can dramatically enhance

ORC cycle efficiency, which is not high, at approximately 8-12% in commercial

industries for low-grade temperature resources (Kutlu, Li, Su, Pei, & Riffat, 2018).

Based on the literature, the overall ORC cycle efficiency will be increased by

approximately 1%, when the efficiency of the turbomachinery improve 3-5%. The 1%

increase is a massive improvements for cycles whose commercial efficiencies are

around 10%. In this regard, there have been extensive studies of radial-inflow turbines,

as already described in Section 2.2.


As described previously, a suitable diffuser to fit the radial-inflow turbo

expander is significant to maximise the whole turbine’s efficiency. Conical diffusers

are widely employed to connect the downstream of turbo expanders (Klein, 1981) as

they have simple geometries and an easy manufacturing process. In the the early

stages, most studies which investigated the flow characteristics of diffusers were based

on experiments (Azad, 1996; Azad & Kassab, 1989; Baghdadi & McDonald, 1975;

Clausen, Koh, & Wood, 1993; Fox, McDonald, & Va, 1971; Klein, 1981; Senoo,

Kawaguchi, & Nagata, 1978). As reviewed by Klein (Klein, 1981), experiment-based

results show that the inlet conditions of turbulence and swirl collectively affect the

performance of the conical diffuser, and that swirl is the most effective way to prevent

flow separation. The relationship between the flow regime and the swirl number was

investigated in detail by Baghdadi et al. (Baghdadi & McDonald, 1975), while the

turbulent swirling flow of conical diffusers has been extensively reported, particularly

in experimental swirling studies (Azad, 1996; Azad & Kassab, 1989; Clausen, et al.,

1993; Fox, et al., 1971; Klein, 1981; Senoo, et al., 1978) as summarised by Azad

(1996). An extensive investigation on recirculation and the separation near diffuser

walls integrated in the ‘ERCOFTAC conical diffuser’ database by Clausen, et al.

(1993) has become a popular test case for the validation of numerical codes,

However, investigations into the robust optimal design of high-density fluid

diffusers are lacking, which hinders the improvement of overall ORC efficiency. In

addition, limited understanding has been established regarding the way flow

characteristics of diffusers affect the efficiency of the low-grade temperature ORC. In

terms of the performance analysis of conical diffusers, numerical modelling appears

to be an effective way to guide the optimisation process. With the development of

numerical modelling, it is evident that CFD has the capability to compute complex

engineering applications by employing high-density fluid as working fluid (Cinnella

& Congedo, 2004). Various computational studies on the ERCOFTAC diffuser have

been conducted to capture the complex flow phenomenon (Armfield, Cho, & Fletcher,

1990; Bounous, 2008; Olivier & Balarac, 2010; Page, Giroux, & Massé, 1996; Sauret,

Persky, Chassaing, & Lucor, 2014)), and to explore various turbulence models which

best capture this flow. Armfield, et al. (1990) discussed various inlet swirl profiles

which have impact on flow and predicted turbulence in swirling flow conical diffusers

using ideal gas.


However, these investigations consider ideal gas systems, which fail to

comprehensively represent high-density fluid in low-temperature ORCs. Recently,

From, Sauret, Armfield, Saha, and Gu (2017) demonstrated the Algebraic Reynolds

Stress Model (EARSM) as the most suitable turbulence model for conical diffuser

flows. They established high-density fluid (R143a) flow regimes, and compared the

fluid behaviour characteristics in detail between ideal gas and high-density fluid in the

ERCOFTAC conical diffuser configuration. However, the overall performance of

high-density diffuser was not investigated by From, et al. (2017). In addition, based on

the experimental investigation by Abir and Whitfield (1987), the flow characteristics

of conical diffusers are unstable, while the curved annular diffuser and the radial

diffuser present more stable flow conditions. Due to the lack of experiments with high-

density fluids to compare the performance of these diffusers, it is of interest to conduct

a numerical study to compare the conical diffuser and annular-radial diffuser using

high-density fluid. Recently, Keep, et al. (2017) developed a constrained preliminary

design for an annular-radial diffuser to fit their existing supercritical CO2 turbine,

based on their numerical analysis. However, no comparison with experiments was

performed for validation. In previous studies, diffusers were investigated

independently from the whole turbine, and did not include the turbo expander. Inlet

boundary conditions are known to affect the flow in diffusers, and as such, in this

study, the proposed R143a radial-inflow turbo expander (Sauret & Gu, 2014) was built

as the inlet part of the diffuser.

To sum up, these experimental and numerical studies of conical diffusers and

annular-radial diffusers cover the flow under an ideal gas regime. Limited

understanding has been reached on how the flow characteristics of these typical

diffusers, such as conical and annular-radial, which employ high-density fluid, affect

the efficiency of ORC turbines and thus further influence the overall efficiency of the

low-grade temperature ORC.

Thus, this thesis also aims to compare the performance of the preliminary design

of a conical diffuser and an annular-radial diffuser, fitting the conditions from the

current existing 400kW R143a radial-inflow turbine.


2.4 UNCERTAINTY QUANTIFICATION

In order to develop robust ORC radial-inflow turbine design, uncertainties need

to be accounted for. Uncertainty Quantification (UQ) is the science of quantitative

characterization and reduction of uncertainties in applications. The UQ method assigns

a probability distribution to each uncertain variable around its mean value, and then

propagates this uncertainty through the mathematical model to the output (Faragher,

2004). Various uncertainty quantification techniques have been developed in the

literature and are discussed in the following section.

Uncertainties can be classified into Aleatory Uncertainty and Epistemic

Uncertainty. Aleatory Uncertainty is physical variability in the system or in its existing

environment (Iaccarino, 2009). It is not strictly due to a lack of knowledge and cannot

be reduced. For instance, operational conditions and/or material properties result in

aleatory uncertainties. Aleatory uncertainty is usually characterised by employing

probabilistic methodology. Epistemic uncertainty is a potential shortage that is caused

by lack of knowledge. It may be generated from assumptions in the derivation of the

employed mathematical model, and/or simplifications in terms of the correlation in

physical processing. It is clear that those epistemic uncertainties can be reduced by,

for instance, improving the physical models and/or by implications from experimental

observations. Typical examples of epistemic uncertainties are chemical kinetics model

assumptions and turbulence model suppositions.

Typical UQ Methods

There are several typical and widely used UQ methods in engineering

applications, especially in fluid dynamic investigation in turbomachinery and/or airfoil

fields; their brief description can be found in Table 2-3.


Table 2-3: Typical UQ methods (Le Maître & Knio, 2010).

Name Description Characteristics

Monte Carlo

Method

(MC)

Uses a large number of values

of the input variables to

calculate values repeatedly for

the output variables. Statistics,

such as the mean and variance

can be calculated.

Very expensive computational

cost. Sampling distribution can

be large resulting in an increase

of simulation time or failure.

Don’t supply accurate

information about the tails of

the output probability density

distribution (PDF). Chooses

random sample of every input

variable according to PDF, and

calculates output results from

the response surface equation.

Polynomial Chaos

(PC)

Employs a polynomial-based

stochastic space to represent

and propagate uncertainty in the

form of probability density

distribution. PC supplies a

mathematical framework to

separate the stochastic results of

a system response from the

deterministic ones.

PC allows high-order

representation and supplies

higher computational efficiency

compared to MC.

However, chaos expansion

converges slowly for turbulent

flow fields.

Generalised

Polynomial Chaos

Method (gPC)

Spectral representation of the

uncertainty with the

decomposition into separate

deterministic and random

components.

The gPC is extended from PC,

and it also referred as the

Askey-chaos, employing the

orthogonal polynomials from

the Askey scheme in random

space.

Speed-up factors from 1000 to

100,000 compared to MC

depending on problem. The

gPC converges much faster

than PC and MC for turbulent

flow fields. It also can supply

accurate results at the same

time.

Stochastic

Collocation

Method (SC)

Performs calculation at specific

collocation points in the

stochastic domain. Highly

efficient to deal with

uncertainty propagation and

nonlinear responses.

Reduces the curse of

dimensionality. It depends on a

large number of random

variables, while keeping a high

level of accuracy.

Monte Carlo (MC) approach is a well-known and widely-used UQ technique in

engineering applications (Fishman, 2013). In it, all deterministic solutions are

employed to compute the statistical characterisation of interest (Le Maître & Knio,

2010). Monte Carlo can supply the entire probability density function of any system

variable; however it suffers a high computational cost, as a great number of samples


are needed for reasonable accuracy. Polynomial Chaos (PC) is presented as a more

efficient probabilistic technique for uncertainty propagation (Xiu & Karniadakis,

2002). The Polynomial Chaos method was pioneered (Ghanem & Spanos, 1991;

Spanos & Ghanem, 1989) based on the homogeneous chaos theory of Wiener (1938),

which was a spectral expansion of the random variables. The typical polynomial chaos

expansion depends on the Hermite polynomials associated with Gaussian random

variables (Xiu & Em Karniadakis, 2002). However, the Polynomial Chaos method is

limited to non-linear dynamic systems and has slow convergence for turbulent flows

(Xiu, et al., 2002). Because of these limitations, Xiu, et al. (2002) further developed

the Polynomial Chaos method (Ghanem & Spanos, 1991; Spanos & Ghanem, 1989)

into generalized Polynomial Chaos (gPC), which was chosen from the hypergeometric

polynomials of the Askey scheme (Askey & Wilson, 1985). The underlying random

variables in gPC are not restricted to Gaussian random variables. Instead, the random

variables were selected based on the stochastic input and the weight function of these

random variables, determining the type of orthogonal polynomials to be employed as

the basis in stochastic space (Xiu & Em Karniadakis, 2002). In order to investigate the

performance of gPC, its accuracy and efficiency was examined by Xiu et al. (Xiu,

Lucor, Su, & Karniadakis, 2003). They employed first-order and second-order

ordinary differential equations with random parameters. Their results showed gPC was

much more efficient than the Monte Carlo method and Polynomial Chaos method. The

gPC presents a distinct advantage over the traditional Monte Carlo method, yielding

exponentially fast convergence of the errors instead of algebraically convergence of

the errors (Xiu, et al., 2003).

The generalised Polynomial Chaos method has been developed both in the global

context, which uses spectral expansions spanning all of stochastic space, and in a local

context, which employs localised spectral representations. The former is termed an

intrusive technique; it needs new established codes and/or solvers for the reformulated

systems (Nechak, Berger, & Aubry, 2011). The latter approach is denoted as a non-

intrusive technique, since the original code can be treated as a black box. The non-

intrusive method is becoming more popular, as it does not require the modification of

the deterministic solver. A comparison between intrusive and non-intrusive methods

is presented by Onorato et al. (Onorato, Loeven, Ghorbaniasl, Bijl, & Lacor, 2010)

while non-intrusive approaches are detailed in (Loeven, Witteveen, & Bijl, 2007). The


advantage of the non-intrusive approach compared to the intrusive approach is that it

uses the original model code, and it is evidenced to be more beneficial for stochastic

dynamic systems, as they need no modifications of the system model while the

intrusive method does. A non-intrusive method, referred to as Stochastic Collocation

(SC), is another UQ approach, which is easily implemented and leads to the solution

of deterministic problems, like the Monte Carlo, Polynomial Chaos, and generalised

Polynomial Chaos methods (Mathelin, Hussaini, & Zang, 2005; Xiu & Hesthaven,

2005). The Stochastic Collocation method depends non-linearly on the driving random

variables. In addition, the Stochastic Collocation method reduces considerably the

curse of dimensionality and thus saves computational cost, and allows the designer to

deal with a moderate number of random variables, while keeping a high level of

accuracy (Eldred & Burkardt, 2009).

The work presented in this study focuses on establishing a reliable and robust

framework for the design of high-density radial-inflow turbines using the uncertainty

quantification method. Once established and validated, this framework can be

modified to accommodate any type of UQ methods. However, for the sake of

implementation simplicity, and because the cutting-edge High Performance Computer

resource at QUT can be used to handle the overall computation task, a non-intrusive

generalised Polynomial Chaos method is chosen, allowing a good compromise

between accuracy and computational cost in the context of turbomachinery. In the

following section, the applications of generalised Polynomial Chaos methods will be

introduced.

Examples of UQ for High-density Fluids and Turbomachinery

Applications

In order to develop robust designs of high-density radial-inflow turbines,

uncertainties must be accounted for at the design level. These uncertainties arise from

the variability of the normal operational conditions of the system, such as inlet

temperature, the geometric parameters due to manufacturing tolerances, and the

numerical representation of the physical system, including mathematical models and

boundary conditions. Some uncertainties, namely aleatory ones, cannot be removed,

while others, the epistemic uncertainties, can be reduced (Faragher, 2004). More

uncertainties need to be taken into account at the beginning of design. Meeting

performance requirements in various circumstances, including uncertain operation


conditions and manufacturing tolerances as encountered in practical engineering

applications, is also necessary. Importantly, comprehensive understanding of the

influence of input uncertainties will enhance the reliability of a risk-based design,

increase design confidence, reduce risks, improve safety, and refine the systematic

operating range (Cinnella & Hercus, 2010). As outlined by Congedo et al. (Congedo,

Corre, & Cinnella, 2011), the optimised design of ORC turbines relies on the

availability of accurate and efficient CFD tools, because of the scarcity or lack of

available experimental data. In order to have confidence in the numerical simulations

and include uncertainties in the design process and optimisation of robust radial

turbines, uncertainty quantification methods appear to be a powerful solution for

robust engineering design, specifically for evaluating the influence of uncertain input

variables on the outcome (Daroczy, et al., 2016).

In addition, all these parameters in the numerical simulations are fixed, which

makes it impossible to isolate the influence of individual parameters as well as to

evaluate their coupled effects. However, by coupling sensitivity analysis and robust

design, it is possible to accurately quantify these uncertainties. It is thus critical to

include the effects of the different interacting sources of uncertainty to achieve better

designs by minimising the deviations from the optimal operational conditions

(Panizza, et al., 2012). For probabilistic analysis, the Monte Carlo method is the most

popular method. However, even if statistical approaches are straightforward to

implement, they suffer from prohibitive computational costs and poor convergence

rates, especially for complex geometries (Sankaran & Marsden, 2011). The Monte

Carlo technique has been applied to the performance analysis of turbomachines.

Panizza, et al. (2012) used a Monte Carlo approach to propagate uncertainties in a

meanline 1D compressor code. Despite the overall results for the quantification of

impeller geometrical uncertainties, they suggested the use of CFD for more accurate

results. (Javed, Pecnik, & van Buijtenen, 2013) successfully applied a surrogate-based

Monte Carlo approach to quantify the manufacturing uncertainties of a centrifugal

compressor and optimise its design. However, due to the large number of uncertainties

in turbomachine design, (Panizza, et al., 2012) also suggested the use of non-statistical

uncertainty quantification methods with sparse grid techniques to overcome the curse

of dimensionality of these methods. Both stochastic collocation and gPC methods have

been applied to 2D airfoils using either ideal gas (Chassaing & Lucor, 2010) or high-


density fluid (Cinnella, et al., 2011; Merle & Cinnella, 2015). Modgil, et al. (2013)

claimed to be the first to apply a gPC method using sparse grid to the design

optimisation of a high-pressure ideal gas turbine blade in regard to three geometrical

uncertainties. Unlike the sensitivity study carried out by Javed, et al. (2013), neither

Bufi, et al. (2015) nor Modgil, et al. (2013) considered the tip clearance uncertainties.

The use of high-density gas to extract more energy in renewable power cycles

adds new uncertainties in the numerical model. As highlighted by Harinck, Colonna,

Guardone, and Rebay (2010) for 2D expanders, and Sauret and Gu (2014) for 3D

radial-inflow turbines, the high-density fluid model dramatically affects the turbine

performance predictions. Various equations of state (EOS) for these high-density gases

are available in the literature (Poling, Prausnitz, & O'connell, 2001), as discussed in

Section 2.2.1. However, they suffer uncertainties in the mathematical model itself, but

also from the errors in measurements of the EOS data around the fluid’s critical point.

Reliability of critical pressure, critical temperature and acentric factors, among others,

are questionable (Cinnella, et al., 2011), and thermodynamic model uncertainties must

be included in the turbine design. Application of uncertainty quantification techniques

to high-density fluid has been investigated mainly for shock tubes (Congedo, et al.,

2010), 2D airfoil geometries (Congedo, Corre, & Martinez, 2011; Merle & Cinnella,

2015), and 2D nozzle blades (Bufi, et al., 2015; Colonna, et al., 2006). In particular,

high-density fluid flows are adapted to stochastic analyses, as they are sensitive to

variations in upstream thermodynamic properties. Zou, et al. (2015) presented a UQ

study on an ideal gas radial-inflow turbine including a stator and a rotor, showing that

the most sensitive random variables are the trailing edge tip radius combined with the

rotational speed.

Overall, however, very few UQ investigations on entire radial-inflow turbines in

low-grade temperature ORC applications have been reported by Zou, et al. (2015). For

diffusers, Sauret, et al. (2014) conducted a UQ investigation on conical diffusers using

ideal gas. According to their stochastic results, the inlet velocity was the most

important random variation, compared to swirling velocity, inlet turbulent kinetic

energy, diffuser length, and cone half angle on the pressure recovery, both individually

and coupled with a second uncertain variable. However, very limited studies have been

reported on UQ investigation of conical diffusers employing high-density fluid as a

working fluid, especially in low-to-medium temperature ORC applications.


2.5 SUMMARY

Four main aspects have been reviewed in this literature review chapter:

renewable energy in ORC application, turbo-expanders, diffusers, and uncertainty

quantification for ORCs. Regarding these four areas, some important conclusions have

been drawn as below.

From this review of the current research status of ORC techniques, it is

clear that ORC is a powerful solution to conversion of low-medium

temperature renewable energies, including biomass, OTEC, and

geothermal energy, into electricity in a cleaner way in power plants.

Many investigations have evidenced the capability of the ORC technique

to convert these renewable energies into electric power. However, further

advancements in efficiency are needed to make this technology highly

competitive in the energy market.

R143a high-density fluid has been evidenced as a potential fluid for ORC

system applications, and is the candidate for the high-density working

fluid used in this research. EOS have been established for R143a, and

those equations have been validated experimentally.

Many studies using one-dimensional meanline design have investigated

the preliminary design of ORC radial-inflow turbines. However, most of

them lack full validation by experiments employing high-density fluids.

Furthermore, the meanline design model investigates velocity triangles

and inlet/outlet dimensions of the rotor, but does not consider the three-

dimensional effect of the rotor passage. This must be accounted for to

make sure of the effective expansion of the high-density fluid through

the rotor.

Extensive CFD numerical investigations have been conducted for ORC

radial-inflow turbines. However, as the experiments are very difficult to

conduct, most of them have not been completely validated using high-

density fluid experimental data. At this stage, most CFD investigations

are verified against the meanline design and play a guidance role for

experimental studies.


Many investigations into radial-inflow turbine performance under off-

design conditions have been conducted over a wide range of operational

conditions to illustrate the performance map. These studies offer relevant

guidance for the optimisation and robust design of ORC radial-inflow

turbines.

Most meanline designs and CFD studies have not accounted for

uncertainties in their models, which prevents robust design of ORC

radial-inflow turbines.

Due to both high expenses and complex process issues of the ORC radial-

inflow experiments, it is quite difficult to conduct experiments using

various sizes of turbines employing various working fluids. Thus, at this

stage, experimental results for a wide range of net power outputs and

various working fluids are still lacking. In addition, most of the high-

density fluid radial-inflow turbines have not considered uncertainty

quantification of their systems under off-design operational conditions.

Most investigations of diffusers focus on experimental studies using

ideal gas. Very few high-density fluid diffusers, including conical

diffusers and annular-radial diffusers, have been conducted to fit high-

density fluid radial-inflow turbines in ORC applications.

Compared with the Monte Carlo method, the generalised Polynomial

Chaos method has speed-up factors from 1000 to 100,000 depending on

different problems, and also can support accurate results at the same time.

Very limited Uncertainty Quantification analysis has been conducted

regarding the whole of ORC radial-inflow turbines and ORC diffusers.

Some Uncertainty Quantification investigations have been carried out on

parts of ORC radial-inflow turbines, such as the stator or rotor. Very few

Uncertainty Quantification investigations of radial-inflow turbines and

diffusers have been proposed using ideal gas but not high-density fluids.

Chapter 3: Coupled Uncertainty Quantification – Deterministic Flow Solver Methodology 45

Chapter 3: Coupled Uncertainty

Quantification – Deterministic

Flow Solver Methodology

In this chapter, the implementation techniques of Computational Fluid Dynamics

and Uncertainty Quantification will be documented concisely, including the

Computational Fluid Dynamics Solver and the generalised Polynomial Chaos method

as an Uncertainty Quantification technique in Section 3.1 and Section 3.2 respectively.

Additionally, a robust framework coupling Computational Fluid Dynamics with

Uncertainty Quantification will be introduced in Section 0.

3.1 COMPUTATIONAL FLUID DYNAMICS SOLVER

The CFD solver ANSYS-CFX v18.0 has been used to perform steady-state 3D

viscous simulations of radial-inflow turbines. In this thesis, for the radial-inflow turbo-

expanders and diffusers study, Reynolds-Averaged Navier Stokes (RANS) equations

for viscous compressible flows were applied in a finite volume solver adapted to

accommodate dense gas simulations. The CFX solver is pressure-based coupled solver

(ANSYS 18.0 CFX-Solver Theory guide, 2017).

The Reynolds-Averaged Navier Stokes (RANS) equations for viscous

compressible flows are presented below, and solved using a finite volume solver

adjusted for high-density fluid numerical simulations (ANSYS 18.0 CFX-Solver

Theory guide, 2017). 𝑈𝑖 = 𝑈�� + 𝑢𝑖, 𝑈�� =1

∆𝑡∫ 𝑈𝑖𝑑𝑡

𝑡+∆𝑡

𝑡. 𝑈𝑖 is divided into an average

component 𝑈��, and a time varying component 𝑢𝑖.

𝜕��

𝜕𝑡+

𝜕

𝜕𝑥𝑗

(𝜌𝑈𝑖 ) = 0

(3-1)

𝜕𝜌𝑈𝑖

𝜕𝑡+

𝜕

𝜕𝑥𝑗(𝜌𝑈𝑖𝑈𝑗

) = −𝜕��

𝜕𝑥𝑗+

𝜕

𝜕𝑥𝑗(𝜏𝑖𝑗 − 𝜌𝑢𝑖

′𝑢𝑗′ ) + 𝑆𝑀

(3-2)

Where 𝜏 is the molecular stress tensor including both normal and shear

components of the stress 𝜌𝑢𝑖′𝑢𝑗

′ is the Reynolds stresses.


Following the results from Sauret and Gu (2014), the standard 𝑘 − 𝜖 turbulence

model with scalable wall function was chosen for the turbo-expander simulations,

associated with a first order numerical scheme for the turbulence variables for

robustness considerations. Convergence is achieved once the Root Mean Squared

(RMS) for mass, momentum, and turbulence variables approaches the residual target

of 1×10-6. Two-equation turbulence models are very widely employed, as they offer a

good compromise between numerical effort and computational accuracy. Both the

velocity and length scale are solved employing separate transport equations (ANSYS

18.0 CFX-Solver Theory guide, 2017). Based on previous work in Sauret and Gu

(2014), the two-equation turbulence model 𝑘 − 휀 is used in this study. The 𝑘 − 휀

model utilises the gradient diffusion hypothesis to relate the Reynolds stresses to the

mean velocity gradients and the turbulent viscosity. The turbulent viscosity is

modelled as the product of a turbulent velocity and turbulent length scale. The

turbulent velocity scale is calculated from the turbulent kinetic energy, which is

derived from the solution of its transport equation. The turbulent length scale is

estimated from two properties of the turbulence field, the turbulent kinetic energy and

the dissipation rate. The dissipation rate of the turbulent kinetic energy is supplied

from the transport equation solution.

𝑘 is the turbulence kinetic energy and is defined as the variance of the

fluctuations in velocity. 휀 is the turbulence eddy dissipation.

The 𝑘 − 휀 model presents two new variables in the system of equations. The

continuity equation is as follows:

Where 𝑆𝑀 is the sum of boby forces, 𝜇𝑒𝑓𝑓 is the effective viscosity accounting

for turbulence, and 𝑝′ is the modified pressure, defined by:

𝑝′ = 𝑝 +2

3𝜌𝑘 +

2

3𝜇𝑒𝑓𝑓

𝜕𝑈𝑘

𝜕𝑥𝑘

(3-3)

2

3𝜇𝑒𝑓𝑓

𝜕𝑈𝑘

𝜕𝑥𝑘 is the divergence of velocity. It is neglected in ANSYS CFX (ANSYS

18.0 CFX-Solver Theory guide, 2017).

The 𝑘 − 휀 model is based on the eddy viscosity concept, thus:

𝜇𝑒𝑓𝑓 = 𝜇 + 𝜇𝑡 (3-4)


Where 𝜇𝑡 is the turbulence viscosity. The 𝑘 − 휀 model for the turbulence

viscosity in connection with the turbulence kinetic energy and dissipation by the

relation is:

𝜇𝑡 = 𝐶𝜇𝜌𝑘2

휀

(3-5)

Where 𝐶𝜇 is 0.9 as a constant.

The values of 𝑘 and 휀 come directly from the differential transport equations for

the turbulence kinetic energy and turbulence dissipation rate:

𝜕(𝜌𝑘)

𝜕𝑡+

𝜕

𝜕𝑥𝑗(𝜌𝑈𝑗𝑘) =

𝜕

𝜕𝑥𝑗[(𝜇 +

𝜇𝑡

𝜎𝑘)

𝜕𝑘

𝜕𝑥𝑗] + 𝑃𝑘 − 𝜌휀 + 𝑃𝑘𝑏

(3-6)

𝜕(𝜌휀)

𝜕𝑡+

𝜕

𝜕𝑥𝑗(𝜌𝑈𝑗휀)

=𝜕

𝜕𝑥𝑗[(𝜇 +

𝜇𝑡

𝜎𝜀)

𝜕휀

𝜕𝑥𝑗] +

휀

𝑘(𝐶𝜀1𝑃𝑘 − 𝐶𝜀2𝜌휀 + 𝐶𝜀1𝑃𝜀𝑏)

(3-7)

Where 𝐶𝜀1, 𝐶𝜀2, 𝜎𝑘 are constants.

𝑃𝑘𝑏 and 𝑃𝜀𝑏 represent the influence of the buoyancy forces, which are described

below. 𝑃𝑘 is the turbulence production due to viscous forces, which is modelled

employing:

𝑃𝑘 = 𝜇𝑡 (𝜕𝑈𝑖

𝜕𝑥𝑗+

𝜕𝑈𝑗

𝜕𝑥𝑖)

𝜕𝑈𝑖

𝜕𝑥𝑗−

2

3

𝜕𝑈𝑘

𝜕𝑥𝑘(3𝜇𝑡

𝜕𝑈𝑘

𝜕𝑥𝑘+ 𝜌𝑘)

(3-8)

For the diffuser cases, the turbulence model is the 𝐵𝑆𝐿𝑘𝜔𝐸𝐴𝑅𝑆𝑀 model, based

on 𝐵𝑆𝐿 − 𝑘𝜔 added to 𝐸𝐴𝑅𝑆𝑀. The computational modelling of fluid dynamics in

the high-density fluid conical diffuser extends from previous work by From, et al.

(2017). The detailed governing equations, turbulence model, and complete validation

are presented in (From, et al., 2017).

For the rotating conical diffuser simulation, because of the rapid variation in

viscosity and density in high-density fluid, which changes spatially throughout the

domain, a completely viscous compressible solver adapted for low Mach numbers was

selected to constitute any of these characteristic variations.


The RANS equations for viscous compressible flows are carried out in the finite

volume solver, which is modified for low Mach numbers to accommodate high-density

fluid simulations. According to Gatski and Speziale (2006) investigations, an

extension of the Explicit Algebraic Reynolds Stress Model (EARSM) model to extend

the initial work from Wallin and Johansson (2000) is conducted as the turbulence

model so as to accurately capture the streamline curvature and rotational effects

through the explicit addition of the rotation Ω𝑖𝑗 and anisotropy 𝑎𝑖𝑗 tensors.

In this thesis, the 𝐵𝑆𝐿 − 𝑘𝜔 model proposed by Menter (1994) is coupled with

the EARSM. The 𝐵𝑆𝐿𝑘𝜔𝐸𝐴𝑅𝑆𝑀 is presented in Equation (3-10) and (3-11). This

model couples the turbulent kinetic energy dissipation rate (𝜖) by Menter, Garbaruk,

and Egorov (2012) with the specific rate of dissipation 𝜔 equation by Wilcox (1998);

thus the near-wall turbulence is modelled employing 𝜔 for the better accuracy and the

far-wall region is modelled employing 𝜖.

𝜕(𝜌𝑘)

𝜕𝑡+

𝜕

𝜕𝑥𝑗(𝜌𝑈𝑗𝑘) =

𝜕

𝜕𝑥𝑗[(𝜇 +

𝜇𝑡

𝜎𝑘3)

𝜕𝑘

𝜕𝑥𝑗] + 𝑃𝑘 − 𝛽′𝜌𝑘𝜔,

(3-9)

𝜕(𝜌𝜔)

𝜕𝑡+

𝜕

𝜕𝑥𝑗(𝜌𝑈𝑗𝜔)

=𝜕

𝜕𝑥𝑗[(𝜇 +

𝜇𝑡

𝜎𝜔3)

𝜕𝜔

𝜕𝑥𝑗] + (1 − 𝐹1)2𝜌

1

𝜎𝜔2𝜔

𝜕𝑘

𝜕𝑥𝑗

𝜕𝜔

𝜕𝑥𝑗

+ 𝛼3

𝜔

𝑘𝑃𝑘 − 𝛽3𝜌𝜔2.

(3-10)

The coefficients in Equation (3-9) and (3-10) are shown in Table 3-1.

Table 3-1: The coefficients for Equations (3-9) and (3-10).

𝛽′ 𝛼1 𝛽1 𝜎𝑘1 𝜎𝜔1 𝛼2 𝛽2 𝜎𝑘2 𝜎𝜔2

0.09 5/9 0.075 2 2 0.44 0.0828 1 1

0.856

The viscous production for the turbulence kinetic energy, 𝑃𝑘 is demonstrated as:

𝑃𝑘 = 𝜇𝑡

𝜕𝑈𝑖

𝜕𝑥𝑗(

𝜕𝑈𝑖

𝜕𝑥𝑗+

𝜕𝑈𝑗

𝜕𝑥𝑖) −

2

3

𝜕𝑈𝑘

𝜕𝑥𝑘(3𝜇𝑡

𝜕𝑈𝑘

𝜕𝑥𝑘+ 𝜌𝑘).

(3-11)


The near-wall treatment as a blended formulation is presented as Equation (3-12)-

(3-14) in the solver. The blending function in the transport Equation (3-10) is described

as:

F1 = tanh(arg14). (3-12)

Where

arg1 = min (max (√𝑘

𝛽′𝜔y,500𝜈

y2𝜔) ,

4𝜌k

CDk𝜔𝜎𝜔2y2),

(3-13)

With

CDk𝜔 = max (2𝜌1

𝜎𝜔2𝜔

𝜕𝑘

𝜕𝑥𝑗

𝜕𝜔

𝜕𝑥𝑗, 1 × 10−10).

(3-14)

Thus, the Reynolds stress tensor is demonstrated as:

𝑢𝑖′𝑢𝑗

′ =2

3𝑘𝛿𝑖𝑗 − 𝟐𝜐𝒆S𝒊𝒋 + 𝑘𝑎𝑖𝑗

𝑒𝑥𝑡𝑟𝑎. (3-15)

The full details are given in (From, et al., 2017).

The 𝜐𝑒 effective viscosity is presented as:

𝜐𝑒 = −1

2𝛽1𝑘𝜏,

(3-16)

𝑎𝑖𝑗𝑒𝑥𝑡𝑟𝑎 = 𝛽2(𝐒𝛀 − 𝛀𝐒) (3-17)

𝛀 =𝟏

𝟐𝜏 (

𝜕��𝑖

𝜕𝑥𝑗−

𝜕��𝑗

𝜕𝑥𝑖) − 𝜏𝜖𝑖𝑗𝛀𝑟𝑜𝑡. 𝐒 =

𝟏

𝟐𝜏 (

𝜕��𝑖

𝜕𝑥𝑗+

𝜕��𝑗

𝜕𝑥𝑖). (3-18)

This formulation automatically changes from wall functions to the low-Reynolds

near-wall formulation when the mesh is refined.

In addition, with the purpose of evaluating the transport properties due to

compressible effects of the high-density fluid, the energy within the fluid is computed

by the Total Energy Equation (3-19). According to the enthalpy, ℎ𝑡𝑜𝑡𝑎𝑙:


𝜕(𝜌ℎ𝑡𝑜𝑡𝑎𝑙)

𝜕𝑡−

𝜕𝑃

𝜕𝑡+

𝜕

𝜕𝑥𝑖(𝜌 𝑈𝑥,𝑦,𝑧 ℎ𝑡𝑜𝑡𝑎𝑙) =

𝜕

𝜕𝑥𝑖(𝜆

𝜕𝑇

𝜕𝑥𝑖) +

𝜕

𝜕𝑥𝑖(𝑈𝑥,𝑦,𝑧 𝜏𝑖𝑗).

(3-19)

𝜌 is the density, 𝑃 the pressure, 𝜆 the fluid thermal conductivity, 𝑇 the static

temperature, 𝜏𝑖𝑗 the stress tensor, and 𝑈𝑥,𝑦,𝑧 are the three velocity components of the

velocity vector.

A compressible solver coupled with a Pressure-Velocity, cell-centred and non-

staggered, was employed, addressing the three momentum equations with pressure at

every integration point. The solver calculates compressible flows at any Mach number

by implicit discretisation of the product of the density and mass-carrying adverting

velocity. The implicit discretisation is achieved by Newton-Raphson linearisation

between the new and the current iteration, which demonstrates the current iteration of

density is with linearisation associated with pressure. The pressure is interpolated

employing the Rhie and Chow (1983) method. Based on their method, the mass flow

items are discretised to circumvent the decoupling of pressure and velocity at adjacent

cells. This method contains linearisation of nonlinear equations, conducted in a matrix

solution as well. The discrete system of linear equations is addressed employing an

Algebraic Multi-Grid method.

According to the boundedness principles presented by Barth and Jespersen

(1989), the variable ‘β’ was calculated in a nonlinear manner so that it is close enough

to β ≈ 1 for every mesh node; a second-order advection scheme is employed. The

second-order accuracy is for the spatial discretisation. The Courant Friedrichs Lewy

(CFL) number is according to the time step. This is in terms of the estimation of the

fluid domain time scale according to the length scale and peak velocity scale in the

dynamic scale, density, total mass, mass flow, and impact of compressive time scale.

The CFL number does not exceed 5. The CFL number was measured at 4.25 for high-

density fluid. For simplicity, the time scale is considered with a set constant value of

one.

The main numerical set-up in CFX for these numerical investigations is

introduced below.

The numerical scheme is Upwind, and turbulence numeric option is the First

Order for stability reasons. The turbulence intensity is set up as 5%. The interface

between the stationary and rotational frame was set as the mixing plane interface


boundary condition. The periodic boundary condition was set, and thus only one blade

passage was modelled. The calculation convergence is attained, when the Root Mean

Squared (RMS) for mass, momentum, and turbulence variables achieve the residual

target 1×10-6. The frozen rotor interface was employed between the rotational frame

and the diffuser. The frozen rotor interface model was selected because it agrees well

with experimental data (Jones, 1994) with a 1% maximum deviation as detailed in

Chapter 4.

3.2 GENERALISED POLYNOMIAL CHAOS APPROACH

The succeeding discussion explains some of the theory underlying the gPC

algorithm and Uncertainty Quantification applications for real world problems.

generalized Polynomial Chaos method:

In this work, the generalized Polynomial Chaos method (gPC) (Spanos &

Ghanem, 1989) is the stochastic solver used for the propagation of parametric

uncertainties in high-density fluid ORC turbine configurations. This approach is being

increasingly employed for CFD-based uncertainty quantifications (Xiu & Karniadakis,

2003). The spectral representation of any aerodynamic random variables 𝑢(𝑦, 𝜽), such

as efficiency in the present study, is based on the following approximation:

𝑢(𝒚, 𝜽)𝑛 = ∑ ��𝑚(𝑦)ɸ𝑚(𝜽)

𝑀−1

𝑚=0

(3-20)

Where 𝑀 denotes the number of modes in the spectral decomposition, 𝑦

represents the deterministic variables, and 𝜽 = [𝜽1, ⋯ , 𝜽𝑁]𝑇 is an N-dimensional

vector of random variables with independent components and prescribed distributions.

The polynomials’ basis {ɸ𝑚(𝜽)} must be chosen in order to satisfy the orthogonality

condition with respect to input uncertainty distributions. The total number of modes,

𝑀, is a function of the number of random variables 𝑁 and the order 𝑃 of the expansion,

as 𝑀 =(𝑃+𝑁)!

𝑃!𝑁!− 1.

A Galerkin projection of the stochastic solution onto each member of the local

orthogonal basis is used to compute the spectral coefficients of the Polynomials Chaos

expansion ��𝑚(𝒚) as:


��𝑚(𝒚) = ⟨𝑢(𝒚, 𝜽), ɸ𝑚(𝜽) ⟩

⟨ɸ𝑚(𝜽), ɸ𝑚(𝜽) ⟩ (3-21)

=1

⟨ɸ𝑚(𝜽), ɸ𝑚(𝜽) ⟩ ∫ 𝑢(𝒚, 𝜽)ɸ𝑚(𝜽)

𝛺

𝒫(𝜽)𝑑𝜽 (3-22)

Where ⟨. , . ⟩ is the scalar product and 𝛺 denotes the support of the random space

described by the probability density function 𝒫(𝜽) of the random variable 𝜽. In this

work, the integral in Equation (3-21) is evaluated using tensor product quadrature

(Trefethen, 2008).

A general polynomial is:

𝑄𝑛(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ ⋯ + 𝑎1𝑥 + 𝑎0, 𝑎𝑛 ≠ 0 (3-23)

Where 𝑎𝑛 is the polynomial coefficient.

The leading coefficient of one is the monic form of the polynomial.

𝑃𝑛(𝑥) =𝑄𝑛(𝑥)

𝑎𝑛= 𝑥𝑛 +

𝑎𝑛−1

𝑎𝑛𝑥𝑛−1 + ⋯ + ⋯

𝑎1

𝑎𝑛𝑥 +

𝑎0

𝑎𝑛

For the gPC polynomials, a set of polynomials exists that supplies an optimal

basis for different types of probability distribution. They are derived from the family

of hypergeometric orthogonal polynomials in the Askey scheme (Xiu, 2010).

Correspondence between different types of gPC and the underlying random variables

is presented in Table 3-2.

Table 3-2: gPC type and underlying random variable (Xiu, 2010).

Type Distribution of Z gPC Basis Polynomials Support

Continuous

Gaussian Hermite (-∞,+∞)

Gamma Laguerre [0, +∞)

Beta Jacobi [a,b]

Uniform Legendre [a,b]

Discrete

Poisson Charlier {0,1,2…}

Binomial Krawtchouck {0,1, …N}

Negative Binomial Meixner {0,1,2…}

Hypergeometric Hahn {0,1, …N}

The choice of a specific basis function depends on orthogonality with respect to

the weight function (𝑤 ). Thus, the weight function and the density function for

different distributions are shown in Table 3-3.


Table 3-3: Weight and density function for gPC distribution (Xiu, 2010).

Distribution Density Function Polynomial Weight

Function

Gaussian 1

√2𝜋𝑒

−𝑥2

2⁄ Hermite 𝑒−𝑥2

2⁄

Gamma 𝑥𝛼𝑒−𝑥

Γ(𝛼 + 1) Laguerre 𝑥𝛼𝑒−𝑥

Beta (1 − 𝑥)𝛼(1 + 𝑥)𝛽

2𝛼+𝛽𝐵(𝛼 + 1, 𝛽 + 1) Jacobi

(1 − 𝑥)𝛼(1+ 𝑥)𝛽

Uniform 1

2 Legendre 1

The Askey scheme of continuous hypergeometric polynomials is used in this

study. Legendre is a special case for Jacobi for 𝛼 = 𝛽 = 0. Setting 𝛼 = 0 in the Gamma

function will result in an exponential distribution. In this study, as widely used

polynomials, the Legendre polynomials are described briefly as follows:

Legendre Polynomials:

A class of orthogonal polynomials that satisfies:

𝑃𝑛+1 =2𝑛 + 1

𝑛 − 1𝑥𝑃𝑛(𝑥) −

𝑛

𝑛 + 1𝑃𝑛−1(𝑥), 𝑛 > 0

(3-24)

∫ 𝑃𝑛

+1

−1

(𝑥)𝑃𝑚(𝑥)𝑑𝑥 =2

2𝑛 + 1𝛿𝑚𝑛

(3-25)

For probabilistic analysis, a wide range of probabilistic variations for some

engineering problems can be expressed as uniform distributions. The orthogonal

families of polynomials form a basis of the function space 𝐿𝑤2 = {𝐹: ⟨𝑓𝑓⟩ < ∞}.

In this study, as bounded supports for the input random parameters were defined,

a uniform distribution law was adopted to propagate these parametric uncertainties.

Thus, the Legendre polynomials are quite advantageous in gPC settings. Legendre

polynomials are defined on the interval [-1,+1].

The Legendre polynomials, {𝐿𝑒𝑛(𝑥), 𝑛 = 0,1, … }, are an orthogonal basis of

𝐿𝑤2 [−1,1] with respect to the weight function 𝑤(𝑥) = 1/2 for all 𝑥 ∈ [−1,1]. They

are typically normalised so that 𝐿𝑒𝑛(1) = 1 , in which case they are supplied by

(Canuto, Hussaini, Quarteroni, & Thomas Jr, 2012):


𝐿𝑒𝑛(𝑥) =1

2𝑛∑ (−1)𝑙 (

𝑛𝑙

) (2𝑛 − 2𝑙

𝑛) 𝑥𝑛−2𝑙

[𝑛 2⁄ ]

𝑙=0

.

(3-26)

Here [𝑛 2]⁄ denotes the integral part of 𝑛 2⁄ . The polynomials are even when 𝑛

is even and odd when 𝑛 is odd.

The Legendre polynomials satisfy the recurrence relations:

𝐿𝑒𝑛+1(𝑥) =2𝑛 + 1

𝑛 + 1𝑥𝐿𝑒𝑛(𝑥) −

𝑛

𝑛 + 1𝐿𝑒𝑛−1(𝑥)

(3-27)

With 𝐿𝑒0(𝑥) = 1 and 𝐿𝑒1(𝑥) = 𝑥 . The first seven Legendre polynomials are

supplied by (Abramowitz & Stegun, 1966; Canuto, et al., 2012). 𝐿𝑒0(𝑥) = 1 ,

𝐿𝑒1(𝑥) = 𝑥 , 𝐿𝑒2(𝑥) =1

2(3𝑥2 − 1) , 𝐿𝑒3(𝑥) =

1

2(5𝑥3 − 3𝑥) , 𝐿𝑒4(𝑥) =

1

8(35𝑥4 −

30𝑥2 + 3) , 𝐿𝑒5(𝑥) =1

8(63𝑥5 − 70𝑥3 + 15𝑥) , 𝐿𝑒6(𝑥) =

1

16(2313𝑥6 − 315𝑥4 +

105𝑥2).

The Gauss-Legendre quadrature is according to the following formula (Le

Maître & Knio, 2010):

∫ 𝑓(𝑥)𝑑𝑥1

−1

= ∑ 𝜔𝑖𝑓(𝑥𝑖)

𝑛𝑞

𝑖=1

+ 𝑅𝑛𝑞 , (3-28)

Where

𝑅𝑛𝑞 =22𝑛𝑞+1(𝑛𝑞!)4

(2𝑛𝑞 + 1)[(2𝑛𝑞)!]3𝑓(2𝑛𝑞)(𝜉) , − 1 < 𝜉 < 1

(3-29)

Where 𝑛𝑞 is the number of collocation points, 𝑥𝑖 is the coordinate of the 𝑖-th

collocation point, and 𝜔𝑖 is the corresponding weights.

The coordinates 𝑥𝑖 are the zero of 𝐿𝑒𝑛𝑞(𝑥), and the weights are supplied by:

𝜔𝑖 =2

(1 − 𝑥𝑖2)[𝐿𝑒𝑛𝑞

′ (𝑥𝑖)]2

(3-30)

For gPC post-processing, it must be noted that once the spectral coefficients are

determined, the mean 𝜇𝑢 and the variance 𝜎𝑢2 of 𝑢(𝒚, 𝜽) can be directly obtained by:


𝜇𝑢 = ��0(𝒚, 𝑡), 𝜎𝑢2 = ∑ ��𝑚(𝒚)

𝑀−1

𝑚=1

(3-31)

The standard deviation is 𝜎 = √𝜎𝑢2 and Coefficient of Variation CoV = 𝜎𝑢/ 𝜇𝑢.

Moreover, the distribution of the stochastic parameter of interest 𝑢(𝒚, 𝜽) can be

computed using Monte-Carlo sampling of the Polynomial Chaos Expansion (3-20) at

low computational cost. Finally, the sensitivity analysis of the stochastic solution to

the input parametric uncertainties is carried out by analysing the Sobol’s coefficients

(Sobol, 1993) whose evaluation is straightforward once the decomposition of the

stochastic solution on the Polynomial Chaos basis is performed.

The Sobol’s coefficient is introduced by Sobol (1993) for the sensitivity estimate.

It is a mathematical model with a function 𝑓(𝑥), where 𝑥 = (𝑥1, … , 𝑥𝑛), and is

defined as an n-dimensional cube:

𝐾𝑛 = {𝑥|0 ≤ 𝑥𝑖 ≤ 1; 𝑖 = 1, … , 𝑛}, (3-32)

The sensitivity of 𝑓(𝑥) will be estimated with respect to different variables or

their groups.

If 𝑓(𝑥) ∈ 𝐿2, then all 𝑓𝑖1…𝑖𝑛∈ 𝐿2. Thus,

𝐷 = ∫ 𝑓2(𝑥)𝑑𝑥 − 𝑓02

𝐾𝑛

(3-33)

and 𝐷𝑖1…𝑖𝑛= ∫ ⋯ ∫ 𝑓𝑖1…𝑖𝑛,

21

0

1

0𝑑𝑥𝑖1

… 𝑑𝑥𝑖𝑛 (3-34)

𝐷 = ∑ 𝐷𝑖1…𝑖𝑛

⋀

(3-35)

𝑓(𝑥) and 𝑓𝑖1…𝑖𝑛(𝑥𝑖1

, … , 𝑥𝑖𝑛) are random. 𝐷 and 𝐷𝑖1…𝑖𝑛

are their variances.

These variances characterise how the corresponding functions change. Then, the

sensitivity estimative is presented as:

𝑆𝑖1…𝑖𝑛= 𝐷𝑖1…𝑖𝑛

/𝐷 (3-36)

∑ 𝑆𝑖1…𝑖𝑛= 1

⋀

(3-37)


3.3 ROBUST COUPLED UQ-CFD FRAMEWORK

The proposed framework relies on automatic coupling between the uncertainty

quantification and the CFD solver. To achieve this coupling, the methodology is as

follows:

Literature review is a necessary and crucial part of the methodology to

investigate for this framework.

CFD solver validation is the foundation for implementing this framework.

The automatic loop regarding UQ-CFD had been developed by

programming scripts, which is a critical and significant step to achieving

accurate results, and reducing the labour and/or computational cost.

A UQ-CFD framework is applied to conduct uncertainty quantification

analysis on ORC radial-inflow turbines.

In this study, an automatic computational framework is developed, including

geometry creation, mesh generation, and CFD solver calculation with UQ calculation

and post-processing. The connections between parts ①-④ are compiled using Matlab

and Python codes, as illustrated in Figure 3-1.

Figure 3-1: Computational UQ-CFD framework of gPC application processing in

radial-inflow turbine.


Figure 3-2: The detailed computational UQ-CFD framework with relative equations

and steps.

The automatic work-flow of this framework eventually reduces manual errors

and dramatically improves work efficiency. It is noteworthy that this framework can

be a benchmark and be applied broadly in related engineering applications. In real

power plant applications, the coupled uncertain parameters (variables) are rather

common than single uncertain variables. Figure 3-1 and Figure 3-2 further demonstrate

the steps outlined above to show a real-world application employing the generalised

Polynomial Chaos method to obtain the post-processing of the quantity of interest. The

CFD calculations are carried out based on the quadrature points and the post-

processing provision of mean, variance, standard deviation, CoV, and Sobol indices,

as shown in Figure 3-1 and Figure 3-2.

In addition, the method of coupled uncertain parameters can identify which

single parameter plays more important role in the coupled uncertain parameters group.

Thus, the Sobol’s indices are very useful to conduct the Uncertainty Quantification

analysis employing coupled uncertain parameters. Using single uncertain parameter

and coupled uncertain parameters is to identify the turbine efficiency affected by single

uncertain parameter and coupled uncertain parameters respectively. In addition, when

using coupled uncertain parameters, the Sobol’s indices is applied to the turbine

efficiency from each contribution of coupled uncertain parameters in order to identify

the main contributors to the 1st order and the 2nd order variance estimated with the gPC

approach (Tang, Eldred, & Swiler, 2010).


Figure 3-3: Convergence rate of variance of efficiency using Peng-Robinson vs P-

order regarding 𝑃𝑐 − 𝑇𝑐.

According to Figure 3-3, the residuals value of the variance 1×10-7 is less than

the standard deviation n×10-4 of any uncertain parameter in the next following

Chapters. Thus, the numerical method does not influence the Uncertainty

Quantification analysis results.

Chapter 4: Validation and Application of the UQ-CFD Framework to the Ideal Gas Turbo-expander 59

Chapter 4: Validation and Application of

the UQ-CFD Framework to the

Ideal Gas Turbo-expander

This chapter details the validation work of the UQ-CFD approach to the

performance analysis of the high pressure ratio, single stage radial-inflow turbine used

in the Sundstrand Power Systems T-100 Multi-purpose Small Power Unit. A

deterministic three-dimensional volume-averaged CFD solver is coupled with a non-

statistical generalised Polynomial Chaos (gPC) representation, based on a pseudo-

spectral projection method applied to investigate this complete, three-dimensional,

high-pressure ratio radial-inflow turbine.

In detail, Section 4.1 will present the validation work, which employs an open

data Jones’s radial-inflow turbine, using ideal gas as a working fluid; the validation

will cover the three-dimensional geometry, grid refinement independence study, and

numerical deterministic simulations. Section 4.2 will discuss the parametric study of

the validated Jones’s radial-inflow turbine. Then, Uncertainty Quantification analysis

is applied to the Jones’s turbine in Section 4.3. A summary of the results will be

presented in the conclusion of this chapter in Section 0.

4.1 NUMERICAL VALIDATION OF THE DETERMINISTIC CFD

SOLVER

The radial-inflow turbine employed in this work has been developed by

Sundstrand and experimentally tested by Jones (1996). In this thesis, the radial-inflow

turbine is denoted as Jones’s turbine. This geometry has become an open benchmark

after the work of Sauret (2012), who reproduced the geometry and provided initial

CFD results.

The test case at nominal conditions is a 120 kW, 5.7 pressure ratio turbine used

in the Sundstrand Power Systems T-100 Multi-purpose Small Power Unit. The rig

conditions are employed in this chapter for the validation and application of the

coupled generalised Polynomial Chaos method – CFD approach.


Three-dimensional Geometry

To reproduce the three-dimensional geometry of the turbine, ANSYS-BladeGen

is employed to build the three-dimensional nozzle and rotor blades. The geometry is

presented below in Figure 4-1, and the full details are presented in Sauret (2012),

including the blade profiles, rotor hub and shroud, rotor blade angle, and thickness

distributions of hub and shroud respectively.

Figure 4-1: The whole geometry of the Jones’s radial-inflow turbine.

The periodic one blade passage of the whole geometry is presented in Figure 4-2,

which is the geometric model in the CFD calculation.


Figure 4-2: Geometry of periodic one blade passage of the Jones’ radial-inflow

turbine.

This radial-inflow turbine contains 19 stator nozzles and 16 rotor blades. The tip

clearance is from 0.4mm at the leading edge to 0.23mm at the trailing edge.

Grid Refinement Independence Study

Three-dimensional geometry and mesh of one blade passage, including stator,

rotor and part of the diffuser, are reproduced in the ANSYS turbomachinery package.

The mesh is generated using ANSYS-TurboGrid for the flow passage for both rotor and

stator. The non-dimensional grid spacing at the wall 𝑦𝑤+ ranges from 20 to 140, which

is the recommended range, as the log-law wall function is valid for 𝑦𝑤+ values above

15 and under 100 for machine Reynolds numbers of 1×105; the transition affects the

boundary layer formation and skin friction, and up to 500 for Reynolds numbers of

2×106 when the boundary layer is mainly turbulent throughout (Manual, 2000). The

boundary layer refinement control is 4×106, with Near Wall Element Size Specification

to reach the 𝑦𝑤+ (non-dimensional wall element size) requirement.

There are seven cases investigated for the mesh, as shown in Figure 4-3. Seven

meshes from 300,000 nodes to 900,000 nodes were investigated for the mesh

refinement study. It can be seen that when the mesh reaches 700,000 nodes, mesh

convergence is achieved.


Figure 4-3: Grid Refinement independent investigations.

After the grid refinement study, the total mesh number is 712,082, including

stator, rotor and part of the diffuser. The grid quality was checked using indicators

such as orthogonality of the cells and aspect ratios. All of the computations were

performed until full convergences of the flow variables were achieved. The residuals

were dropped down below 10-6.

The converged mesh of the rotor is presented in Figure 4-4. A closer view of the

rotor mesh at hub and shroud is shown in Figure 4-5.

Figure 4-4: Mesh of rotor.


Figure 4-5: Three-dimensional closer view of the O–H grid of rotor blade passage:

(a) at hub; (b) at shroud.

The converged mesh of the stator is shown in Figure 4-6(a) and (b).

Figure 4-6: Mesh of Stator: (a) all of the nozzles of stator; (b) one periodic nozzle

passage of stator.

Validation

The three-dimensional viscous flow simulation is conducted in ANSYS CFX.

Reynolds-Averaged Navier-Stokes equations are solved in this simulation using

ANSYS-CFX. The turbulence model (detailed in Section 3.1) is employed for the

simulations. The boundary conditions are summarised in Figure 4-1. Three-

(a) (b)

(a) (b)


dimensional geometry and the mesh of the periodic one blade passage, which includes

stator, rotor and part of the diffuser, are reproduced in the ANSYS turbomachinery

package. The working fluid is air. The mass flow of the stator inlet is 0.0173 kg/s,

reduced to one blade passage.

Table 4-1: Boundary Condition of the Study Case.

𝑃𝑖𝑛

(kPa)

𝑃𝑜𝑢𝑡

(kPa)

𝑇𝑇𝑖𝑛

(K)

𝑅𝑃𝑀

(RPM)

𝑄𝑚

(Kg/s)

Blade

Number

Nozzle

Number

Experiment

Rig 413.6 72.4 477.6 71700 0.0173 16 19

The numerical calculations were carried out until full convergences were

achieved. The residual convergence in terms of the momentum and mass were below

10-6, as presented in Figure 4-7.

Figure 4-7: Residual convergence at rig condition.

The static pressure distribution of the radial-inflow turbine is shown in Figure

4-8.


Figure 4-8: Static pressure distribution at mid-span along turbine.

The highest pressure region is in the inlet of the stator. The lowest pressure

distribution region is in the outlet of the diffuser. The pressure reduces from 413 kPa

at the stator entrance to 67.3 kPa at the rotor exit.

Figure 4-9: Relative Mach number at mid-span along the turbine.


This radial-inflow turbine was designed to work with a high subsonic flow.

According to Figure 4-9, the maximum Mach number, approximately 1.2, occurs at

the nozzle throat of the stator passage outlet, and the nozzle is choked. Choked flow is

a phenomenon that limits the mass flow rate of a compressible fluid flow through

sudden expansions of turbine nozzles. Generally, it is the mass flux, after which a

further reduction in downstream pressure will not lead to an increase in mass flow rate.

As a compressible fluid achieves the speed of sound (i.e. Mach number of 1), pressure

changes can no longer be communicated upstream, as the speed of which these

pressure changes are propagated is limited by the speed of sound. The turbine nozzle

has the effect of isolating the upstream side from the downstream side at the throat.

Due to this effect, any reduction in downstream pressure will have no effect on the

flow rate, as the increased pressure differential is not 'felt' upstream of the restriction.

Thus, the choked flow will significantly influence the passage flow of the rotor, and

thus affect the turbine’s efficiency (Baines, 2003). A large low-speed area is on the

pressure side of rotor blades, as Figure 4-9 shows.

The isentropic efficiencies defined in Equation (4-1) are a function of the

enthalpy drop, as follows:

𝜂𝑇−𝑆 =ℎ𝑇𝑖𝑛

− ℎ𝑇𝑜𝑢𝑡

ℎ𝑇𝑖𝑛− ℎ𝑆𝑖𝑠𝑜𝑢𝑡

(4-1)

Based on Figure 4-10, three-dimensional CFD total-to-static and total-to-total

efficiencies are compared against the experimental data at the rig conditions. The

results show very close agreement, with a maximum difference of less than 1% for the

total-to-static efficiency.


Figure 4-10: Variation of total-to-static efficiency with rotational speed.

It is significant to note that it was not possible to rebuild the geometry exactly,

due to the shortcomings of the software. The CFD analysis results are compared

against experimental data at the rig conditions, Sauret’s results (Sauret, 2012), and

Odabaee’s results (Odabaee, Shanechi, & Hooman, 2014), as presented in Table 4-2.

Table 4-2: Results Comparison for the Rig Conditions.

Experiment

Rig Condition

(Jones, 1996)

Sauret’s

Results

(Sauret,

2012)

Odabaee’s Results

(Odabaee, et al., 2014)

This Study’s

Results

𝜂𝑡−𝑠(%) 86.4 86.6 84 86.6

Power (kW) 36.7 - 36.4 36.6

The results are in really good agreement with data for the experimental rig

conditions, with a maximum difference of less than 1% for the total-to-static efficiency,

and less than 0.3% for the power, which validates the CFD solver for radial-inflow

turbine using ideal gas.

4.2 PARAMETRIC STUDY OF OPERATIONAL AND GEOMETRIC

CONDITIONS

The robust framework introduced in Section 0 has been utilised in this section.

In order to carry out the Uncertainty Quantification analysis for this radial-inflow


turbine, a parametric study is necessary to find out which parameter(s) will present a

non-linear surface response of interest for the Uncertainty Quantification approach.

In order to investigate the uncertain parameters’ effects on the efficiency of the

turbine, the parametric study range is listed in Table 4-3, which will also apply for the

Uncertainty Quantification employing the gPC algorithm in the following section. In

Table 4.3, the mean value represents the experimental data, and the support of the

uncertain parameters is given. Trailing Edge is denoted as TE.

In Table 4-3, the wider support range of the uncertain parameters derived from

Odabaee, et al. (2014) is noteworthy.

Table 4-3: Characteristics of the uncertain parameters studied.

According to Equation (4-1), it is clear that efficiency depends on enthalpy,

which is influenced by temperature, pressure, mass flow, and rotational speed of the

radial-inflow turbine. Hence, the temperature, pressure, mass flow rate, and rotational

speed are potentially influential parameters that affect turbine efficiency. The

parametric study of the temperature, pressure, mass flow rate, and rotational speed to

affect turbine efficiency is presented in Figure 4-11 to Figure 4-14 respectively.

Uncertain Parameter Experiment Rig

Value Support

Temperature of Stator Inlet 𝑇𝑇𝑖𝑛 (K) 477.6 [447.6, 507.6 ]

Pressure of Diffuser Outlet 𝑃𝑜𝑢𝑡 (kPa) 72.4 [60, 84.8 ]

Mass Flow Rate of Stator Inlet 𝑄𝑚 (Kg/s) 0.0173 [0.0147, 0.0199]

Rotational speed 𝑅𝑃𝑀 (RPM) 71700 [57360, 86040]

TE Meridional Length 𝐿 (mm) 35.0012 [33.1, 42 ]

TE Tip Radius 𝑅 (mm) 36.83 [31.1, 37]

Blade Thickness peak position along the

meridional length (%) 41 [21,71]


Figure 4-11: Parametric evaluation of inlet temperature’s effect on the turbine’s

efficiency.

Figure 4-12: Parametric evaluation of outlet pressure’s effect on the turbine’s

efficiency.


Figure 4-13: Parametric evaluation of mass flow rate’s effect on the turbine

efficiency.

From Figure 4-11 to Figure 4-13, it can be seen that the trends for parametric

investigations in terms of temperature and pressure are linear. Thus, temperature and

pressure are of no particular interest for the application of Uncertainty Quantification

anlaysis for a radial-inflow turbine using ideal gas, not high-density fluid. According

to Figure 4-11, when the inlet temperature increases, the total-to-static efficiency of

the turbine increases accordingly. Similarly, based on Figure 4-13, when the mass flow

rate of the stator inlet increases, the total-to-static efficiency of the turbine increases

as well. These results can be explained by the fact that the inlet temperature increases

the enthalpy drop, and then the efficiency will be increased, based on Equation (4-1).

When the outlet pressure decreases, the total-to-static efficiency of turbine declines, as

demonstrated in Figure 4-12.

According to Figure 4-14, the parametric study of rotational speed trend is non-

linear. When the rotational speed is approximately 71700 RPM, the total-to-static

efficiency of the turbine has a maximum value of around 86.66%. Then, as the

rotational speed increases, the total-to-static efficiency declines after the peak value.


Figure 4-14: Parametric evaluation of the turbine efficiency on Rotational Speed

(RPM).

The Tip Length and Tip Radius of the Trailing Edge are significant parameters

that affect the ideal gas radial-inflow turbine (Odabaee, et al., 2014). The TE Tip

Length and TE Tip Radius are defined in Figure 4-15. The red point “A” in Figure

4-15 is the geometry varying point, corresponding to the TE position at the shroud.

The arrows’ direction is the geometry varying direction. When “A” point is moving in

a horizontal direction, the TE tip length will change. It is important to note that when

“A” point is moving in the vertical direction (TE Tip Radius), the blade height will be

modified, but the tip clearance will be kept at the initial value simultaneously.


Figure 4-15: Geometric variation on TE Meridional Length and TE Tip Radius.

Figure 4-16: Parametric evaluation of the turbine efficiency on TE Tip Radius.

Figure 4-16 shows that the response of the turbine’s total-to-static efficiency to

the variation of the TE meridional tip length is non-linear. The initial value of the TE

meridional tip length is 35 mm. From this study, it can be seen that the TE meridional


tip length in terms of maximum total-to-static efficiency is at approximately 33.5 mm,

not at the initial value point, which leads to an increase efficiency of 0.6%.

Figure 4-17: Parametric evaluation of the turbine efficiency on TE Meridional Tip

Length.

Based on Figure 4-17, it can be seen that the efficiency of the turbine is sensitive

to the TE tip radius. The response of the total-to-static efficiency to the variation of the

TE tip radius is non-linear. The initial value of the TE tip radius is 36.83 mm. It is

obvious that the maximum total-to-static efficiency is approximately 86.72% when the

TE tip radius is 34.0 mm, not 36.83 mm, which results in an increased efficiency of

2.4% over the initial turbine efficiency.

As indicated, these six different rotor blade thickness profiles affect the ideal gas

radial-inflow turbine efficiency (Odabaee, et al., 2014); they were established for the

parametric study, as demonstrated in Figure 4-18. The maximum value of the blade

thickness is kept constant while its location is moved along the tip length, thus

modifying the profile curve’s shape.


Figure 4-18: Blade thickness profile geometric study.

Figure 4-19: Parametric evaluation of the blade thickness profile’s effect on the

turbine efficiency.

As shown in Figure 4-16, Figure 4-17, and Figure 4-19, based on these

parametric investigations in terms of rotational speed, TE tip radius, TE tip length and

blade thickness have non-linear response surfaces. One can also note that maximum

efficiency is obtained at values of the TE tip radius, TE tip length and blade thickness,


in contrast to the initial Jones’s geometry, indicating that optimisation of this turbine

can be achieved. Rotational speed, TE tip radius, TE tip length, and blade thickness,

are thus used as random inputs for the application of the gPC method.

4.3 VALIDATION OF THE UQ-CFD FRAMEWORK FOR IDEAL GAS

TURBO-EXPANDER

The framework detailed in Section 0 has been employed to couple the CFD

calculation with Uncertainty Quantification analysis processing for the ideal gas

Jones’s turbine.

The mean and support for the four random parameters (rotational speed, TE tip

radius, TE tip length and blade thickness) are summarised in Table 4-3.

For the convergence study, Figure 4-20 shows the gPC Legendre quadrature

points for polynomial orders’ P range of 1, 3, 5, 7, 9, 11, respectively, when rotational

speed is the random variable.

Figure 4-20: Legendre quadrature points and arbitrary support points for rotational

speed for P = 1, 3, 5, 7, 9, 11.


Figure 4-21: Convergence rates of the variance of the rotational speed in respect to

total-to-static efficiency.

In Figure 4-21, symbols represent simulations, while lines are the corresponding

linearly fitted decay rates. It can be seen that the error line trend decreases when the

P-order increases, showing good convergence rates. Similar trends are observed for all

uncertain variables.

In Table 4-4, ω, L and R refer to the rotational speed, TE tip length, and TE tip

radius, respectively. The comparison of the results using single and coupled uncertain

parameters is presented.

Regarding each uncertain parameter, the mean value μ, the standard deviation σ,

variance σv2, and the coefficient of variation CoV = σ/μ of the total-to-static efficiency

with the gPC at P = 11 are presented in Table 4-4. It can be seen that the blade thickness

profile has the most influential effect on the turbine total-to-static efficiency, closely

followed by the TE tip radius R, while L doesn’t appear to be a critical geometric

parameter in regard to efficiency. The gPC method was also applied for coupled

uncertain parameters with a lower polynomial order P = 5 in order to minimise the

computational cost. When parameters are coupled, the most influential coupled

random variables on the total-to-static efficiency are R-ω.


Table 4-4: Mean, standard deviation and CoV of the total-to-static efficiency for each

individual uncertain parameter for P = 11 and coupled parameters for P = 5.

gPC 1D (P = 11) gPC 2D (P = 5)

Variable 𝑹𝑷𝑴 𝑳 𝑹 Blade

Thickness 𝑹 −𝑹𝑷𝑴

𝑳− 𝑹𝑷𝑴

𝑹 − 𝑳

μ 85.09 86.72 85.65 85.5 83.27 85.34 81.68

σ×10-3 13.409 1.720 16.611 17.400 25.360 14.971 13.102

σ ʋ2×10-3 0.180 0.003 0.276 0.303 0.0643 0.0224 0.0172

CoV×10-3 15.759 1.983 19.393 20.400 30.454 17.543 16.040

Figure 4-22: The 1st order of Sobol’s indices of each uncertain parameter’s

contribution to 𝜂𝑇−𝑆 (a) R – RPM (b) L – RPM (c) L– R ; The 2nd order Sobol’s

indices of each uncertain parameter’s contribution to 𝜂𝑇−𝑆 (d) R – RPM, L – RPM, R

– L.

The Sobol’s indices, representing the contribution to the variability of the turbine

performances by means of overall efficiency. In order to identify the individual

(a) (b)

(c) (d)


contribution of coupled uncertain parameters affecting the total-to-static efficiency of

the radial-inflow turbine, Sobol’s variance indices must be computed in this Chapter.

According to Figure 4-22(a) and (c), the R (TE tip radius) plays more important

role compared to RPM and L (TE tip length) respectively, while the L (TE tip length)

has the smallest impact to turbine efficiency compared to RPM and R (TE tip radius)

as shown in Figure 4-22(b)-(c). Figure 4-22(d) shows that combined effects of

uncertainties R - RPM are the most important on the turbine efficiency compared to

the other two L - RPM and R – L. All of these Sobol’s indices results have a good

agreement with the results regarding the CoV in Table 4-4.

4.4 CONCLUSION

A deterministic three-dimensional CFD solver is coupled with a gPC approach

and successfully applied to investigate a complete three-dimensional high-pressure

ratio radial-inflow turbine. The uncertainty quantification has been applied to the

performance analysis of the radial turbine for the propagation of various aerodynamic

and geometric uncertainties. The main conclusions are as follows:

The grid refinement was implemented, which can satisfy the requirement of

calculation accuracy. The validation work is the foundation for employing

a CFD solver to conduct further radial-inflow turbine numerical

investigations. Three-dimensional CFD calculations for total-to-static

efficiencies of the radial-inflow turbine are compared with the open data

experimental data, including the rig conditions. The results are in close

agreement with experimental data, with a maximum deviation of less than

1% for total-to-static efficiency.

The parametric study for the Jones’s turbine is produced by the previously

introduced robust framework. The parametric study can supply guidance for

choosing uncertain parameter(s) which can dramatically non-linearly affect

the turbine’s efficiency and for further application of Uncertainty

Quantification analysis. Eight parameters were investigated: inlet

temperature, outlet pressure, mass flow rate, tip clearance, rotational speed,

TE meridional length, TE tip radius, and blade thickness. However, only

rotational speed, TE meridional tip length, TE tip radius, and blade thickness


present a non-linear surface response. Thus, these four uncertain parameters

were employed for application of further Uncertainty Quantification

investigations.

gPC has been applied to the performance of radial turbines for the

propagation of various aerodynamic and geometric uncertainties. The

convergence rate for a single uncertain parameter has been carefully

checked, showing that the stochastic spectral projection decreases

dramatically with the increase of polynomial order.

The initial deterministic study highlighted the non-linear response of the

total-to-static turbine efficiency in regard to the variations of the rotational

speed, TE tip radius, TE tip length and blade thickness. From the

preliminary study, for the CoV of the total-to-static efficiency, the most

influential uncertainty is the blade thickness, closely followed by the TE tip

radius. When the gPC approach is applied to couple random parameters, the

most influential coupled random variables are the trailing edge tip radius

with the rotational speed.

The CFD solver coupled with an Uncertainty Quantification framework has

been demonstrated to work well, properly implemented and validated, so

that it can be utilised to any engineering applications, in particular to

investigating a high-density radial-inflow turbine in this thesis.

Chapter 5: Application of the UQ-CFD Framework to an ORC Radial Turbo-expander 81

Chapter 5: Application of the UQ-CFD

Framework to an ORC Radial

Turbo-expander

The inclusion of uncertainties in the design of turbines for renewable low-grade

temperature power cycles is becoming a crucial aspect in the development of robust

and reliable power blocks capable of handling a better range of efficiencies over a

wider range of operational conditions. Modelling high-density fluids using existing

Equations of State add complexity to improving the system efficiency and little is

known on the effect that the uncertainties of Equations of State (EOS) parameters may

have on the turbine efficiency. The purpose of this chapter is to quantify the influence

of coupled uncertain variables on the total-to-static efficiency of a radial-inflow

Organic Rankine Cycle turbine with a high-density fluid R143a in a low-grade

temperature renewable power block. To this end, a stochastic solution is obtained by

combining a multi-dimensional generalized Polynomial Chaos approach with a full

three-dimensional viscous turbulent Computational Fluid Dynamics solver for high-

density radial-inflow turbines. Both operational conditions (inlet total temperature,

rotational speed and mass flow rate) and EOS parameters (critical pressure and critical

temperature) are investigated, highlighting their importance for turbine efficiency

based on the consideration of three Equations of State, namely, Peng-Robinson (PR),

Soave-Redlich-Kwong (SRK), and HHEOS. The study in this chapter, which is

performed for both nominal and off-design operational conditions, highlights the inlet

temperature as the most influential operational uncertain parameters, while the critical

pressure is the most sensitive parameter for the three Equations of State tested. More

importantly, it demonstrates a higher level of sensitivity of the SRK EOS, in particular

at off-design operational conditions. This is a crucial aspect to take into account for

the robust designs of Organic Rankine Cycle turbines for low-grade temperature

renewable power cycles working at various conditions. It is expected that the proposed

stochastic approach may consequently positively support the renewable energy sector

to develop more robust and viable systems.


The aim of this study is to thoroughly characterise, from a statistical point of

view, the sensitivity of the efficiency of a high-density radial-inflow turbine to

parametric uncertainties. The considered EOSs and the deterministic 3D-CFD solver

are presented in Section 5.1. The considered EOSs and the deterministic three-

dimensional CFD introductions and the effects of uncertainties in nominal and off-

design operating conditions using different EOSs are described in Section 5.3.

Stochastic analysis at off-design conditions is conducted in Section 5.4, and

concluding remarks are drawn in Section 5.5.

5.1 COMPUTATIONAL FLUID DYNAMICS CHARACTERISTICS

The 400kW-R143a ORC radial-inflow turbine designed by Sauret and Gu

(Sauret & Gu, 2014) is used to apply the gPC technique and analyse the sensitivity of

uncertain parameters on the performance of this low-grade temperature high-density

gas turbine. All geometric and nominal point conditions are provided in (Sauret &

Gu, 2014) and not repeated in full here. Table 5-1 briefly summarises the main

parameters at nominal conditions, denoted here as the nominal case.

Table 5-1: R143a turbine design parameters at nominal conditions (Sauret &

Gu, 2014).

Global

Variables Value Unit

Geometric

Parameters Value Unit

𝑅𝑃𝑀 24250 RPM 𝑁𝑠 19 -

𝑇𝑇𝑖𝑛 413 K 𝑁𝑟 16 -

��𝑚 17.24 kg/s 𝐷𝑅𝑖𝑛 127.17 mm

𝑃𝑇𝑖𝑛 5 MPa

𝑃𝑆𝑜𝑢𝑡 1.835 MPa

W 400 kW

Computational Model

The CFD solver ANSYS-CFX v18.0 has been used to perform steady-state 3D

viscous simulations. Following the results from Sauret and Gu (2014), the standard

𝑘 − 휀 turbulence model with scalable wall function was chosen, associated with a first


order numerical scheme for the turbulence variables for considerations of robustness.

Harinck, Turunen-Saaresti, et al. (2010) simulated a supersonic ORC turbine stator,

and compared the 𝑘 − 휀 and standard 𝑘 − 𝜔 turbulence models. The results showed

that the choice between 𝑘 − 휀 and standard 𝑘 − 𝜔 turbulence models had minor

influence on the flow field, in particular the Mach number and the overall flow

structure, with a difference in isentropic efficiency of less than 2%. The main

difference appears for the prediction of the shock wave interacting with the boundary

layer. However, in our case, such strong interactions are not present in the turbine and

both turbulence models give similar results for the global parameters investigated in

this study, in particular the efficiency of the turbine with 0.01% difference; results are

not presented in this paper for sake of conciseness. Moreover, Sauret and Gu (2014)

also successfully applied 𝑘 − 휀 model and validated their CFD model with meanline

design. This study follows the work of Sauret and Gu (2014). Their validation work

gives us confidence in the suitability of this model for turbomachinery and allows us

to maintain a good balance between accuracy and computational cost.

Convergence is achieved once the Root Mean Squared (RMS) for mass,

momentum, and turbulence variables approaches the residual target of 1×10-6. An O-

H grid for both the stator and the rotor is built with a total grid number of 1,359,907

nodes, including the extension of the domain at the front of the stator. The three-

dimensional computational mesh for the numerical study of one blade passage is

presented in Figure 5-1 for the stator and in Figure 5-2 for the rotor hub and shroud.

The average non-dimensional grid space at the wall is yw+=703, which is close to the

recommended value of 500 for RANS simulations using wall function-based

turbulence models at approximately Reynolds number = 107, corresponding to the

nominal conditions (Sauret & Gu, 2014). Based on the original study from (Sauret &

Gu, 2014), this value is sufficient to conduct pioneering numerical investigations of

the R143a radial-inflow turbine for the uncertainty quantification analysis.


Figure 5-1: 3D view of the O-H grid around the stator.

Figure 5-2: 3D view of the O-H grid (a) Rotor blade at the hub. (b) Rotor blade

at the shroud.

The generalized Polynomial Chaos (gPC) approach used in this work had been

validated against Monte Carlo calculation for five uncertain parameters affecting the

performance of a conical diffuser (Sauret, et al., 2014).The maximum difference

between the gPC and the MC approach for both the mean and standard deviation was

less than 1%. The coupled gPC-CFD robust framework had been verified in (Zou, et

al., 2015), demonstrating the convergence in P (polynomial order) of our gPC

approach, i.e. evaluating the polynomial order from which the results do not vary

anymore. Furthermore, the CFD solver validation and grid independence of the CFD

results associated with the nominal case (refer to Table 5-1) have been explored by

Sauret and Gu (Sauret & Gu, 2014) against the meanline analysis with results showing

good agreement between both the CFD results and the meanline analysis. Thus, the

robust framework including gPC approach, coupled gPC-CFD, and CFD solver have

(a) (b)


been validated by our previous studies. These validation and verification studies give

confidence in conducting the pioneering three-dimensional simulations using our

robust framework.

The boundary conditions are set based on the meanline design proposed by

Sauret and Gu (Sauret & Gu, 2014) with an inlet mass flow rate of 17.24 kg.s-1 and a

total inlet temperature of 413K. The outlet pressure is fixed at the outlet of the diffuser

to 1.835 MPa, the total-to-static pressure ratio being ∏ = 2.72𝑇−𝑆 , while the rotational

speed of the rotor is 24,250 RPM.

The interface between the stationary and rotational frame was set as the mixing

plane condition and the frozen rotor interface was applied between the rotational frame

and the diffuser. The periodic boundary condition was built so that one blade passage

is modelled.

Equations of State (EOS)

In order to investigate the effect of the high-density properties on the

performance of the radial-inflow turbine, three commonly used EOSs have been

selected: the Soave-Redlich-Kwong (SRK) model (Soave, 1972), which is an

extension of the original Redlich-Kwong (RK) model; the cubic EOS of Peng-

Robinson (PR) (Peng & Robinson, 1976), which is known for its good balance

between simplicity and accuracy, especially close to the critical point (Agrawal, et al.,

2012) and is a further extension of the RK model, and is thus expected to behave

similarly to the SRK model; and the HHEOS, which is 17-term Lemmon-Jacobsen

equation based on Helmholtz energy (Lemmon & Tillner-Roth, 1999), with the

capacity to predict accurately the properties for multicomponent mixtures, and is valid

for temperatures up to 450K and pressures up to 50MPa (Lemmon & Tillner-Roth,

1999). These three EOSs are summarised in Table 5-2.


Table 5-2: PR, SRK, and HHEOS models.

PR SRK HHEOS

𝑝 =𝑅𝑇


𝑎𝛼

𝑉𝑚2 + 2𝑏𝑉𝑚 − 𝑏2

𝑝 =𝑅𝑇


𝑎𝛼

𝑉𝑚(𝑉𝑚 + 𝑏)

𝛼(𝜏, 𝛿

= 𝛼0(𝜏, 𝛿)

+ 𝑎𝑟(𝜏, 𝛿)

𝑎 =0.457235𝑅2𝑇𝑐

2

𝑝𝑐 𝑎 =

0.42747𝑅2𝑇𝑐2

𝑝𝑐 𝜏 =

𝑇𝑟

𝑇

𝑏 =0.077796𝑅𝑇𝑐

𝑝𝑐 𝑏 =

0.08664𝑅𝑇𝑐

𝑝𝑐 𝛿 =

𝜌

𝜌𝑟

𝛼 = (1 + 𝑘(1 − 𝑇𝑟0.5))

2

𝑤𝑖𝑡ℎ 𝑇𝑟 =𝑇

𝑇𝑐

𝛼 = (1 + 𝑘(1 − 𝑇𝑟0.5))

2

𝑤𝑖𝑡ℎ 𝑇𝑟 =𝑇

𝑇𝑐

𝛼 =𝑎

𝑅𝑇

𝑘 = 0.37464 + 1.54226𝜔

− 0.26992𝜔2

𝑘 = 0.480 + 1.574𝜔

− 0.176𝜔2

The acentric factor of refrigerant R143a is 𝜔 = 0.2615; 𝑇𝑐 = 345.86𝐾 and

𝑃𝑐 = 3761𝑘𝑃𝑎 are respectively the critical temperature and pressure of R143a

(Outcalt & McLinden, 1997). 𝑉𝑚 is the molar volume and 𝑅 is the universal gas

constant.

In order to describe the high-density properties, the CFD solver calculates the

enthalpy and entropy employing relationship, as presented in ( ANSYS® Academic

Research CFX Guidance, Release 18, 2017) in detail. These relationships are based on

the zero pressure ideal gas specific heat capacity 𝑐𝑝0, and the derivatives of the EOSs.

Coefficient 𝑐𝑝0 is obtained by a fourth-order polynomial defined by Poling et al.

(Poling, Prausnitz, & O'connell, 2001) for both PR and SRK EOSs. The same

coefficients are used for these two EOSs. Both equations in their dimensional forms

(Table 5-2) are thus function of five material-dependent parameters (𝑇𝑐, 𝑃𝑐, 𝜔, 𝑅, 𝑐𝑝0).

For the HHEOS, the zero pressure ideal gas specific heat capacity is obtained by fitting

the values reported by Yokoseki et al. (Lemmon & Jacobsen, 2000). In this study, only

𝑃𝑐 and 𝑇𝑐 will be considered as uncertain parameters and investigated in detail, as


they have been identified as some of the most questionable EOS parameters (Cinnella,

et al., 2011).

5.2 UNCERTAINTY QUANTIFICATION PARAMETERS

In this work, the generalised Polynomial Chaos method is the stochastic solver

used for the propagation of parametric uncertainties in high-density fluid ORC turbine

configurations. This approach is being increasingly employed for CFD-based

uncertainty quantifications (Xiu & Karniadakis, 2003). The robust framework

introduced in Section 0 has also been employed in this section.

Uncertain Parameters

The support range for the first set of uncertain parameters (𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚)

has been established, based on the acceptable deviation between the test and design

conditions, given in the ASME PTC22 standard (ASME PTC 22 2014) as 2% for each

parameter, and listed in Table 5-3. For the second set of uncertain parameters (𝑃𝑐 −

𝑇𝑐), in order to be more realistic and to closely follow the experimental data established

and/or summarised by several authors (Barret & Candau, 1992; Outcalt & McLinden,

1997; Yaws, 1999), an uncertainty of ±4.5% for 𝑃𝑐 , ±0.5% for 𝑇𝑐 (Table 5-3) is

selected. As bounded supports for the input random parameters were defined, a

uniform distribution law was adopted to propagate these parametric uncertainties.

Table 5-3: Characteristics (mean and support values) of the studied uniformly

distributed uncertain parameters.

Set

Name

Uncertain

Parameter

Symbol Unit Mean

Value

Supports

Rotational speed 𝑅𝑃𝑀 𝑟𝑝𝑚 24250 [23765 -24735]

1st set Total inlet

temperature

𝑇𝑇𝑖𝑛 𝐾 413 [404-421]

Mass flow rate ��𝑚 𝑘𝑔. 𝑠−1 17.24 [16.375-18.1]

2nd set

Critical pressure 𝑃𝑐 𝑀𝑃𝑎 3.761 [3.592-3.930]

Critical temperature 𝑇𝑐 𝐾 345.86 [344-347.6]


Order of the Polynomial Representation

As a direct consequence of the choice of uniform input distributions, the

Legendre polynomials are selected as optimal basis in the spectral representation of

the Polynomial Chaos expansion (Equation (3-20)). In order to illustrate the UQ

framework, Figure 5-3 shows the grid of quadrature points used for P = 5, 7, and 9,

where the deterministic CFD solver must be employed to compute the flow solution

for each Legendre quadrature point.

Figure 5-3: Legendre quadrature points for 𝑃𝑐 − 𝑇𝑐 for 𝑃 = 5, 7, 9.


coefficient 𝜂𝑇−𝑆 obtained using the PR EOS in the presence of uncertainties for 𝑃𝑐 −

𝑇𝑐.



coefficient 𝜂𝑇−𝑆 obtained using the PR EOS in the presence of uncertainties for

for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚.

Figure 5-4 and Figure 5-5 present the distribution of the total-to-static efficiency

𝜂𝑇−𝑆 (𝜂𝑇−𝑆 =ℎ𝑇𝑖𝑛

−ℎ𝑇𝑜𝑢𝑡

ℎ𝑇𝑖𝑛−ℎ𝑆𝑖𝑠 𝑜𝑢𝑡

). The size of the support represents all expected values of

𝜂𝑇−𝑆, and the values with highest probability are indicated by the plateau (or highest

peak) in the shape of the PDF. As observed in Figure 5-4 and Figure 5-5, no major

differences in the distributions of 𝜂𝑇−𝑆 are visible when the polynomial order P used

in gPC expansion is greater than P = 7, which is an appropriate trade-off to balance

accuracy and computational burden, and will thus be used for all the stochastic

computations performed in this work.

5.3 DETERMINISTIC AND STOCHASTIC RESULTS AT NOMINAL

CONDITIONS

Deterministic Results at Nominal Conditions

The efficiency and flow fields of the R143a radial-inflow turbine at nominal

conditions (as presented in Table 5-1) are computed using the CFD solver. The total-

to-static efficiencies of the nominal case using the three EOSs are presented in Table


5-4. Based on Table 5-4, the efficiencies obtained with the three EOSs are very close

to each other, with a maximum difference of only 0.2% between PR and SRK.

Table 5-4: Deterministic total-to-static efficiency 𝜂𝑇−𝑆 for the nominal case

with three different EOSs.

Case Name

EOS

PR SRK HHEOS

𝜂𝑇−𝑆 𝜂𝑇−𝑆 𝜂𝑇−𝑆

Nominal Case 0.8655 0.8637 0.8650

In addition to the total-to-static efficiency, an important flow characteristic of

the radial-inflow turbine is the Isentropic Mach number, in particular in the critical

region near the leading edge (LE) and trailing edge (TE). The Isentropic Mach number

profiles at middle span along the rotor blade, obtained using the three EOSs at nominal

conditions, are presented in Figure 5-6, and plotted at both the suction side (SS) and

pressure side (PS) along the non-dimensional meridional coordinate.

Figure 5-6: Deterministic Isentropic Mach number profile at middle span of

rotor blade for the nominal case using three EOSs.


In Figure 5-6, it is observed that the trend of the Isentropic Mach number

distributions is quite similar for all three EOSs, especially at Pressure Side. Overall,

the deterministic CFD results using the three EOSs at nominal conditions are close.

HHEOS and PR give almost identical Isentropic Mach number profiles, while there’s

a slight difference with SRK compared to the two other EOSs around the blade

Leading Edge at the Suction Side. All the three EOSs are in good agreement, with

previous results from Sauret and Gu (Sauret & Gu, 2014) validating the present CFD

code with all three different EOSs.

The T (Temperature)-h (Enthalpy) curve demonstrate the thermodynamics

phenomenon along the blade as presented in Figure 5-7. The trend of T-h curves using

these three EOSs are similar. The lowest total enthalpy at Edges of blade using SRK

compared to the other two EOSs not presenting much difference as shown in black

box, while the highest total enthalpy along the blade passage profiles as shown along

the middle profiles in Figure 5-7.

Figure 5-7: Deterministic T-h curve along the blade for the nominal case using

three EOSs.


Stochastic Analysis of Uncertain Operational Conditions

In this section, the gPC algorithm previously described in Section 3.2 is

employed in order to quantify the variability of the turbine performance under

uncertain operational conditions of the R143a radial-inflow turbine. In the first set of

uncertain parameters, the coupled uncertainties of rotational speed, inlet total

temperature, and mass flow rate (𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚) are considered (Table 5-3).

The mean, standard deviation and CoV (CoV = 𝜎/𝜇, standard deviation of the

stochastic solution is 𝜎 and its mean value is 𝜇) of the total-to-static efficiency for this

first set of uncertain parameters are presented in Table 5-5. As mentioned earlier, the

mean total-to-static efficiencies are very close using all EOSs, similar to what was

observed for the deterministic calculations (Table 5-4). The CoV of the efficiency

using SRK is the highest, and approximately 6.5% higher than PR, which has the

lowest CoV.

Table 5-5: Mean, standard deviation, and CoV of the 𝜂𝑇−𝑆 for (𝑅𝑃𝑀 −

𝑇𝑇𝑖𝑛− ��𝑚) using three EOSs with P = 7.

PR SRK HHEOS

𝜇 𝜎

× 10−3

𝐶𝑜𝑉

× 10−3 𝜇

𝜎

× 10−3

𝐶𝑜𝑉

× 10−3 𝜇

𝜎

× 10−3

𝐶𝑜𝑉

× 10−3

1st set

case 0.866 3.291 3.801 0.863 3.494 4.047 0.865 3.386 3.914

The Probability Density function (PDF), corresponding to the first set of

uncertainties ( 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 ) , which illustrates the variability of the global

interest parameter, efficiency, using the three EOSs, is shown in Figure 5-8. Figure 5-8

shows that the distribution for 𝜂𝑇−𝑆 differs slightly between the three EOSs. The range

of support of the PDF for PR and HHEOS is identical. However, a small difference is

observed for the most probable values, where the peak of the PDF for PR is around

𝜂𝑇−𝑆 = 0.87, while high probability values for HHEOS are obtained for 0.865. Note

that the shape of the PDF for SRK is similar to those of HHEOS, but its support is

slightly shifted to lower values of 𝜂𝑇−𝑆.


Figure 5-8: PDF for 𝜂𝑇−𝑆 for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚, uncertain parameters using

three EOSs with P = 7.

The efficiencies obtained from Figure 5-8 are more or less identical and centred

around the nominal efficiency from the deterministic case (Table 5-4). The nominal

conditions are not sensitive to 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 for these three EOS. Thus, the PDF

is similar between these three EOS and the width (probability range) of the PDF is

reduced as not much sensitivity is observed.

Figure 5-9 presents the mean value and the standard deviation of the Isentropic

Mach number associated with −𝑇𝑇𝑖𝑛− ��𝑚 , employing the three EOSs.


Figure 5-9: The mean and standard deviation of Isentropic Mach number at

middle span of rotor blade for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 uncertain parameters using three

EOS with P = 7. (a) The whole blade Isentropic Mach number profile. (b)

Streamwise 0.015-0.045 blade Isentropic Mach number profile at Suction Side.

The variations of the Isentropic Mach number along the rotor blade show no

significant difference between PR and HHEOS. It can also be seen that the largest

standard deviation appears at Suction Side around Leading Edge using all EOSs, which

indicates that the most sensitive region in terms of Isentropic Mach number happens

in this region, which may be highly affected by the upstream flow coming from the

stator. Moreover, at approximately 0.03 streamwise location, the maximum standard

deviation for SRK is about 44% higher than PR and HHEOS, as shown in Figure 5-9

(b). SRK shows the largest standard deviation at Suction Side from 0.0 to 0.3 along

the streamwise direction compared to the other two EOS. This result shows that the

Isentropic Mach number may be more sensitive to this first set of uncertain parameters

when obtained using SRK.


Figure 5-10: The skin friction coefficient (𝐶𝑓) profile at middle span of rotor blade

with 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 using three EOS with P = 7. (a) The whole blade skin

friction coefficient (𝐶𝑓) profile. (b) Streamwise 0.015-0.045 skin friction coefficient

(𝐶𝑓) profile at Suction Side.

Figure 5-10 shows the profiles of the mean and standard deviation of the skin

friction coefficient 𝐶𝑓 along the rotor blades using the three EOSs. There is very little

difference between these three EOS tested as shown for the standard deviation of 𝐶𝑓.

As for the Isentropic Mach number distribution along the rotor blades, the overall

results are close for all three EOS for the skin friction coefficient. The inset figure

(Figure 5-10 (b)) shows that the mean value of the 𝐶𝑓 using SRK is the lowest and has

the largest standard deviation compared to the two other EOS. The 𝐶𝑓 for SRK at

approximately 0.032 streamwise is approaching zero highlighting a higher risk to

generate the recirculation or separation bubble.

The sensitivity analysis using Sobol’s indices is applied to the efficiency from

each source of uncertain parameters in order to identify the main contributors to the 1st

order and the 2nd order variance estimated with the gPC approach (Tang, et al., 2010).

(a) (b)



parameter contribution of 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 for 𝜂𝑇−𝑆 using three EOSs with P = 7.

The Sobol’s indices, indicating the contribution to the variability of the turbine

performances by means of efficiency, are shown in Figure 5-11. It is clear that 𝑇𝑇𝑖𝑛

has the strongest influence on the efficiency using all EOSs. Small variations of 𝑇𝑇𝑖𝑛

cause large enthalpy changes of high density fluids, and as such influence dramatically

the efficiency of the turbine (Poling, Prausnitz, John Paul, et al., 2001). In the 2nd

order Sobol’s indices, 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛are the most important coupled parameters. SRK

shows the strongest contribution for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛, while HHEOS presents the smallest

one. It can also be seen that SRK estimates a much larger effect on efficiency for the

couple 𝑇𝑇𝑖𝑛− ��𝑚 compared to PR and HHEOS.

In summary, the UQ analysis for the uncertain operational conditions showed

that the most important single parameter is 𝑇𝑇𝑖𝑛, and the dominating coupled parameter

is 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛. SRK shows the strongest contribution to coupled uncertain

parameters. In order to further evaluate different EOSs’ influence on R143a radial-

inflow turbine performance, the 2nd set of uncertain parameters (𝑃𝑐 − 𝑇𝑐) is calculated

in the following section.

Stochastic Analysis at Nominal Conditions of Pc-Tc

The quantification analysis associated with total-to-static efficiencies is

presented in Table 5-6. The smallest CoV of efficiency is obtained using SRK, and is

approximately 54.2% higher than the one obtained using PR, which gives the largest

CoV.

(a) (b)


Table 5-6: Mean and standard deviation and CoV of the 𝜂𝑇−𝑆 for the 2nd set

case obtained with gPC (𝑃𝑐 − 𝑇𝑐) coupled by three EOS with P = 7.

PR SRK HHEOS

𝜇 𝜎

× 10−3

𝐶𝑜𝑉

× 10−3 𝜇

𝜎

× 10−3

𝐶𝑜𝑉

× 10−3 𝜇

𝜎

× 10−3

𝐶𝑜𝑉

× 10−3

0.8669 0.7859 0.8166 0.8645 1.0885 1.2593 0.8654 0.7926 0.9159

The PDF for total-to-static efficiency in Figure 5-12 shows some differences in

the shape of the PDF obtained using different EOS. However, differences in the mean

efficiency are minimal.

Figure 5-12: PDF for 𝜂𝑇−𝑆 of the 3rd set 𝑃𝑐 − 𝑇𝑐 uncertain parameters using

three EOSs with P = 7.

Contrary to the propagation of uncertainties in 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚, the range of

support of the PDF, the range of possible values for 𝜂𝑇−𝑆, given uncertainties in 𝑃𝑐 −

𝑇𝑐, differs depending on the considered EOSs. For 𝑃𝑐 − 𝑇𝑐, the PDF presents slightly

more difference between these three EOS (Figure 5-12) with the width (probability

range) of the PDF for SRK slightly larger than for the other two EOS. This is probably

because of the slightly stronger sensitivity of SRK compared to the other two EOS


(See Figure 5-13 (b)).Best efficiencies are obtained using PR, whose distribution

clearly follows a uniform distribution law. The size of the support obtained for PR and

HHEOS is nearly identical, but those of SRK results are much larger, meaning that the

variability of the efficiency due to uncertain parameters is more pronounced. From this

point of view, the stochastic performances of the turbine obtained using SRK should

be considered as less robust (or more sensitive) to uncertainties than when using PR

and HHEOS. The PDF is slightly dissymmetric using SRK and HHEOS, while it is

symmetric using PR.

The 1st order and the 2nd order Sobol’s indices results for the 2nd set of uncertain

parameters are presented in Figure 5-13. According to Figure 5-13 (a),

compared to 𝑇c , 𝑃𝑐 is the largest contributor affecting the total-to-static efficiency for

all three EOSs. 𝑇c shows little impact for SRK compared to the other two EOSs.

Figure 5-13 (b) shows that combined effects of uncertainties in 𝑃𝑐 and 𝑇c are

negligible, confirming the linear behaviour of this stochastic configuration on the

shape of the PDF, as already observed. It is thus obvious that the highest (but limited)

2nd order of Sobol’s indices is obtained for SRK.


parameter’s contribution to 𝑃𝑐 − 𝑇𝑐 by using three EOSs for 𝜂𝑇−𝑆 with P = 7.

The stochastic distribution of the Isentropic Mach number and skin friction

coefficient (𝐶𝑓) using three EOSs is presented in Figure 5-14 and Figure 5-15. Based

on Figure 5-14 and Figure 5-15 , the trends of the Isentropic Mach number and skin

friction coefficient profiles are similar. Both the Isentropic Mach number and the skin

friction coefficient (𝐶𝑓) profiles along the rotor blade at nominal conditions using SRK

are lower than PR and HHEOS at Suction Side close to the Leading Edge.

(a) (b)

(a) (b)


Figure 5-14; The mean and standard deviation of Isentropic Mach number at

middle span of rotor blade for 𝑃𝑐 − 𝑇𝑐 uncertain parameters using three EOS with P =

7. (a) The whole blade Isentropic Mach number profile. (b) Streamwise 0.015-0.045

blade Isentropic Mach number profile at Suction Side.

Figure 5-15: The mean and standard deviation of skin friction coefficient (𝐶𝑓) profile

at middle span of rotor blade for 𝑃𝑐 − 𝑇𝑐 uncertain parameters using three EOS with

P = 7. (a) The whole blade skin friction coefficient (𝐶𝑓) profile. (b) Streamwise

0.015-0.045 skin friction coefficient (𝐶𝑓) profile at Suction Side.

For the Isentropic Mach number and skin friction coefficients, it can be

concluded that the influence of the EOSs choice is rather limited as there is little

difference between all EOSs. However, the standard deviation of both the Isentropic

Mach number and 𝐶𝑓 based on the insertion figures using SRK is larger than when

using PR and HHEOS as demonstrated in Figure 5-14(b) and Figure 5-15(b). These

results are consistent with the Sobol’s 2nd results, presented in Figure 5-13 (b).

(a) (b)


There is not much difference on Mach number and skin friction coefficient (𝐶𝑓)

at nominal conditions, which explains the PDF results. In addition, there is no

recirculation in the turbine for all of the stochastic cases at nominal conditions, as all

𝐶𝑓 values remain above zero. This explains that the PDF remains with high efficiency

value, as the results are not very sensitive to the uncertainties at the nominal conditions.

In other words, at nominal conditions, these three EOS are not very sensitive to these

uncertaintes. Thus, there is not much variation of losses and then not much variation

of turbine efficiencies. Based on Figure 5-9(b), Figure 5-10(b), Figure 5-14(b), and

Figure 5-15(b), Mach number and skin friction coefficient (𝐶𝑓) of SRK are slightly

lower than for the other two EOS. These results have good agreement with the lower

efficiency using SRK at the nominal conditions obtained in Table 5-4 for the

deterministic results. The SRK is not influenced much by the uncertainties and the

turbine efficiencies remain lower for the two sets of uncertain parameters using SRK

compared to PR and HHEOS.

In summary, from the nominal condition analysis, the total inlet temperature in

the first set of operational uncertain parameters is observed to have the most important

influence on the efficiency for all three EOSs. Critical pressure in the second set of

uncertain parameters is the paramount uncertain parameter for the efficiency for all

three EOSs. Furthermore, according to all these investigations at nominal conditions,

similar results are obtained with the three EOSs. However, SRK gives the largest CoV,

and shows the largest influence of 𝑃𝑐 − 𝑇𝑐 coupled uncertain parameters compared to

the two other EOSs. Thus, SRK may be more sensitive to uncertainties, and may not

be the most appropriate choice for robust optimisation of high-density fluid radial

inflow turbines.

5.4 STOCHASTIC ANALYSIS AT OFF-DESIGN CONDITIONS

Under operation, ORC turbines may not operate at full nominal capacity

conditions, but instead, may operate at partial capacity under off-design conditions, in

the event geothermal resources are not continuously available (Fiaschi, et al., 2015).

For instance, the geothermal binary cycles as demonstrated in (DiPippo, 2012), have

a variable amount of available geothermal heat, which results in a variable mass flow

rate or thermodynamic conditions of the produced organic stream at the radial turbine

inlet. An in-depth comprehension of the characteristics of the designed R143a radial-


inflow turbine under off-design conditions, using stochastic approaches such as

uncertainty quantification, is an important step towards robust optimisation of those

turbines, allowing them to efficiently work under variable conditions that deviate from

the design conditions.

Based on off-design study (Kim & Kim, 2017b; Sauret & Gu, 2014), it has been

shown that rotational speed and inlet total temperature are two parameters that can

vary under operations and can dramatically affect efficiency. In order to evaluate the

sensitivity of the three EOSs through 𝑃𝑐 − 𝑇𝑐 at extreme off-design conditions, a low

and high rotational speed, corresponding to 80% and 120% of the nominal rotational

speed, respectively, and low (𝑇𝑇𝑖𝑛= 380K) and high (𝑇𝑇𝑖𝑛

= 450K) inlet temperatures

are selected, according to Sauret and Gu (2014), to perform uncertainty quantification

analyses. 𝑇𝑇𝑖𝑛= 400K is employed as the low inlet temperature in this study, and the

reason of choosing this temperature will be further explained in the following section.

The uniformly distributed 𝑃𝑐 − 𝑇𝑐 uncertain parameters’ range is the same as for the

range used in the previous study (see Section 5.3.3).

Mean, Standard Deviation and CoV of Efficiency

The mean values of the total-to-static efficiency, 𝜂𝑇−𝑆 for the nominal and off-

design conditions, are presented in Table 5-7 for the three EOSs. The mean values of

efficiency at off-design using PR are the same as the deterministic solutions of Sauret

and Gu (2014), and the three EOSs provide similar results, except for HHEOS at the

extreme inlet temperatures. As discussed in Sauret and Gu (2014), the efficiencies

increase with the rotational speed up to the nominal value before decreasing at all inlet

temperatures (Figure 5-16). At the nominal rotational speed, the efficiency increases,

then decreases to reach a maximum around the nominal temperature (Figure 5-17). It

can also be noted that at 𝑇𝑇𝑖𝑛 = 380K, the efficiencies obtained using HHEOS

equations are extremely low, around 0.36 - 0.38, and more than 50% lower than the

values obtained with the two other EOSs. The reason for these low values is explained

in detail below.

Table 5-7: The mean values of 𝜂𝑇−𝑆 for off-design conditions with 𝑃𝑐 −

𝑇𝑐 uncertain parameters by the three EOSs with P = 7 (based on uniform distribution

laws).


𝑇𝑇𝑖𝑛

RPM

380K 400K 413K 450K

80%

Nominal

120%

PR SRK HHEOS PR SRK HHEOS PR SRK HHEOS PR SRK HHEOS

0.8588 0.8584 0.3622 0.8223 0.8126 0.8190 0.8085 0.7999 0.8065 0.7498 0.7426 0.7557

0.8618 0.8668 0.3840 0.8722 0.8708 0.8715 0.8655 0.8645 0.8654 0.8432 0.8388 0.8570

0.6905 0.7259 0.3605 0.8662 0.8385 0.8295 0.8492 0.8570 0.8489 0.8692 0.8682 0.8989

Table 5-7 makes clear that the efficiency obtained using the HHEOS is well

below the two other EOSs for all RPM at the lowest temperature 𝑇𝑇𝑖𝑛= 380K, at around

0.38 versus 0.7.

The map charts showing the mean efficiencies and standard deviations at

nominal and off-design conditions are presented in Figure 5-16 and Figure 5-17 for

the three EOSs. According to Figure 5-16, at 𝑇𝑇𝑖𝑛= 400K and 𝑇𝑇𝑖𝑛

= 413K at 80%

nominal RPM and 120% nominal RPM, the efficiencies are lower than those at

nominal RPM, and have much larger standard deviations for all three EOS. At off-

design RPM, as presented in Figure 5-17, the standard deviation of efficiencies is

getting smaller when the inlet total temperature is increasing. Because the lower

temperature is closer to the critical point, it is more sensitive to small temperature

variations, which cause large enthalpy variations in the near-critical point region,

which directly affects efficiency (Kim & Kim, 2017b). These results may indicate that

a larger standard deviation of efficiency is obtained at lower temperatures, showing

that it is mandatory to study the sensitivity of the efficiency to uncertainties at off-

design conditions in order to fully characterise the stochastic response of the system.



with 𝑃𝑐 − 𝑇𝑐 uncertain parameters using the three EOSs with P = 7 regarding three

𝑇𝑇𝑖𝑛 varying three 𝑅𝑃𝑀.


with 𝑃𝑐 − 𝑇𝑐 uncertain parameters using the three EOS with P = 7 regarding three

𝑅𝑃𝑀 varying three 𝑇𝑇𝑖𝑛.


The CoV of efficiencies at off-design conditions with 𝑃𝑐 − 𝑇𝑐 uncertain

parameters is listed in Table 5-8. According this table, the CoV is quite high at 𝑇𝑇𝑖𝑛 =

380K and 120% using all EOSs; for instance, it is approximately 44, 21 and 64 times

higher than nominal conditions using PR, SRK, and HHEOS respectively. It can also

be noted that the CoV is high for HHEOS compared to the two other EOSs for the

lowest inlet temperature for all RPM. At 𝑇𝑇𝑖𝑛 = 380K, increasing the RPM

dramatically increases the CoV for the three EOSs. These results are attributed to this

low inlet temperature being close to the critical temperature point (𝑇𝑐= 345.86K) of

the R143a fluid. The low inlet temperature leads to temperatures within the turbine

that are even closer to the critical or even below for the HHEOS, leading the

deterministic solver to deal with liquid phases, which will be detailed below, based on

Figure 5-18. Close to the critical point, the EOSs are also known (Poling, Prausnitz,

John Paul, et al., 2001) to be more sensitive to small temperature variations through

large enthalpy variations, directly affecting efficiency. When the inlet total

temperature is higher, HHEOS gives high efficiency and small CoV. However, PR

supplies high efficiency and small CoV at lower inlet total temperature. When the

temperature is higher, the fluid property is close to supercritical, as presented in Figure

5-18; as (Zhao, et al., 2017) presented, HHEOS is more appropriate to supercritical

high-density fluid than cubic EOS.

Table 5-8: The 𝐶𝑜𝑉 × 10−3of 𝜂𝑇−𝑆 at off-design conditions with 𝑃𝑐 −

𝑇𝑐uncertain parameters for the three EOSs with P = 7.

𝑇𝑇𝑖𝑛

RPM

380K 400K 413K 450K

80%

Nominal

120%

PR SRK HHEOS PR SRK HHEOS PR SRK HHEOS PR SRK HHEOS

2.502 2.291 27.605 4.163 4.385 4.188 2.951 3.481 3.321 2.599 2.481 0.717

5.736 3.229 26.592 0.283 0.706 0.561 0.817 1.259 0.916 1.197 1.744 0.516

35.93 25.95 58.905 3.865 4.231 4.044 2.556 3.949 3.603 0.543 1.306 0.348

In order to better understand these results, the fluid properties of R143a are

investigated and the pressure-temperature curve for R143a from REFPROP presented

in Figure 5-18.


Figure 5-18: Pressure versus temperature for R143a based on REFPROP NIST

with different thermodynamic states.

As shown in Figure 5-18, the fluids are classified in the following manner:

subcritical vapour, supercritical vapour, subcritical liquid, supercritical liquid, and

mixed subcritical vapour/subcritical liquid. These different colour codes are divided

by the critical point where the critical temperature and critical pressure and the liquid

line are obtained, using the REFPROP database (Outcalt & McLinden, 1997). It is

noticed that there is a region comprising mixed subcritical vapour and subcritical

liquid, and this temperature range is between 300K and 345.86K for the R143a fluid.

While investigating the temperature field in the turbine obtained with the HHEOS for

all the gPC cases at 𝑇𝑇𝑖𝑛= 380K, it was observed that the temperature in the rotor was

below the 345.86K threshold, and thus the CFD solver had to solve in the mixed

subcritical vapour and subcritical liquid region ( ANSYS® Academic Research CFX

Guidance, Release 18, 2017); this condition cannot be handled properly in CFX unless

code adaption is performed, which is not within the scope of this study. The two other

EOSs, however, do not fall under the threshold temperature, and thus the calculations

remain accurate.

In succeeding sensitivity analysis, the lowest inlet total temperature of the off-

design conditions was increased to 400K to avoid any mixed thermodynamic state for

the HHEOS simulations. For fair comparison with the other two EOSs, the aim of this

chapter is to evaluate the sensitivity of the EOSs at nominal and off-design conditions.

However, it is important to highlight here that results can be strongly affected by


temperature variations from different EOSs. While this is not the purpose of this

chapter, it still good to remember that as a consequence, the CFD solver needs to be

adapted to handle mixed flow regions.

Detailed Sensitivity Analysis at Extreme Off-design Conditions

A comparison based on the off-design conditions of inlet total temperature and

rotational speed is conducted hereafter with the use of Sobol’s indices. For the first

order Sobol’s indices of coupled 𝑃𝑐 − 𝑇𝑐, as presented in Figure 5-19(a) – (d), 𝑃𝑐 is the

dominating factor compared to 𝑇𝑐 for all of the conditions, using all EOSs. The three

EOSs show no distinct change when the rotational speed changes at 𝑇𝑇𝑖𝑛 =400K. At

high 120% nominal RPM, SRK shows variation when the inlet total temperature

varies; a stronger effect of 𝑃𝑐 over 𝑇𝑐 is observed for SRK compared to the two other

EOSs.

Figure 5-19: The 1st order Sobol’s indices at off-design conditions with 𝑃𝑐 −

𝑇𝑐uncertain parameters with P = 7. (a) TTin = 400K and 80% nominal RPM (b)

TTin = 400K and120% nominal RPM (c) TTin = 450K and 80% nominal RPM (d)

TTin = 450K and 120% nominal RPM.

(a) (b)

(c) (d)


Based on Figure 5-20(d) with 𝑇𝑇𝑖𝑛 = 400K and 120% nominal RPM, combined

effects in the uncertain parameters only have an influence on efficiency for SRK. It is

interesting to note that the effect of coupled uncertainties may strongly change for a

given EOS, depending on the operating conditions; as, for instance, PR and HHEOS

for 120% nominal RPM at 𝑇𝑇𝑖𝑛 = 400K (Figure 5-20c) and 𝑇𝑇𝑖𝑛 = 450K (Figure

5-20d).

Figure 5-20: The 2nd order Sobol’s indices at off-design conditions with 𝑃𝑐 −

𝑇𝑐 uncertain parameters with P = 7. (a) 𝑇𝑇𝑖𝑛 = 400K and 80% nominal RPM (b)

𝑇𝑇𝑖𝑛 = 400K and 120% nominal RPM (c) 𝑇𝑇𝑖𝑛 = 450K and 80% nominal RPM (d)

𝑇𝑇𝑖𝑛 = 450K and 120% nominal RPM.

Figure 5-21 presents the PDF based on efficiency, with 𝑃𝑐 − 𝑇𝑐 uncertain

parameters at extreme off-design conditions using three EOSs.

(a) (b)

(c) (d)


Figure 5-21: PDF for 𝜂𝑇−𝑆 with 𝑃𝑐 − 𝑇𝑐 uncertain parameters using three EOS

with P = 7. (a) 𝑇𝑇𝑖𝑛 = 400K and 80% nominal RPM (b) 𝑇𝑇𝑖𝑛 = 400K and 120%

nominal RPM (c) 𝑇𝑇𝑖𝑛 = 450K and 80% nominal RPM (d) 𝑇𝑇𝑖𝑛 = 450K and 120%

nominal RPM.

It is noticed that the shape of the probability density functions of a given EOS

varies depending to off-design conditions. It is clear from Figure 5-21(c) at 𝑇𝑇𝑖𝑛=

450K and 80% nominal RPM that the resulting PDF of the efficiency with uniform

distribution of random variables in 𝑃𝑐 − 𝑇𝑐 also gives a uniform distribution. However,

due to its narrower support than those of PR and SRK, HHEOS shows a prediction of

efficiency that is less sensitive to uncertainties compared to its counterparts, while

given the highest probable values of 𝜂𝑇−𝑆.

However, the stochastic response of the efficiency is different at 𝑇𝑇𝑖𝑛= 400K and

80% nominal RPM in Figure 5-21(a). The plateau in the uniform shape of the PDF is

less visible, and we remark that the PDF of 𝜂𝑇−𝑆 obtained for HHEOS presents two

small peaks, meaning that two sets of the most probable values of the efficiency may

be obtained. It can be deduced that aerodynamic nonlinearities are more pronounced

for this operating condition, compared to Figure 5-21(c).

(a) (b)

(c) (d)


Contrary to the previous operating point, stochastic results obtained at 𝑇𝑇𝑖𝑛=

400K and 120% nominal RPM show that best efficiency is obtained for SRK, where

the PDF of 𝜂𝑇−𝑆 shows a more pronounced peak than HHEOS and PR, thus reducing

the range of more probable values of the efficiency in the presence of uncertain 𝑃𝑐 −

𝑇𝑐.

It is noticed that the shape of the probability density functions vary with the EOS,

and that the responses are sensitive to the 𝑃𝑐 − 𝑇𝑐 parameters. It is noteworthy that at

𝑇𝑇𝑖𝑛= 400K and 80% nominal RPM in Figure 5-21(a), the PDF response using HHEOS

is clearly bimodal; while, using SRK, a bimodal may also be identified, based on

Figure 5-21(a). These bimodal responses exhibit bifurcations in flow characteristics

(Chassaing & Lucor, 2010). In other words, for efficiencies at 𝑇𝑇𝑖𝑛= 400K and 80%

nominal RPM, the PDF response is the signature of the system’s nonlinearity using

HHEOS and SRK. However, according to Figure 5-21 (a) - (b), at 𝑇𝑇𝑖𝑛= 400K for both

low and high rotational speed, the density functions are unimodal using PR, with

distributions close to uniform. Compared to the level of randomness in the stochastic

response, the efficiencies using HHEOS at 𝑇𝑇𝑖𝑛 = 450K and 120% nominal RPM, as

shown in Figure 5-21(d), are significantly higher than PR and SRK.

In summary, at low inlet total temperature and low rotational speed using all

EOSs, the PDF of 𝜂𝑇−𝑆 due to uniform distributions of the input random variables 𝑃𝑐 −

𝑇𝑐 are not uniform anymore, due to the presence of aerodynamic nonlinearities which

are reported in the random space. Moreover, the efficiency map chart (Figure 5-16 and

Figure 5-17) illustrates large standard deviations of efficiencies at these conditions,

resulting in potentially important variations in the performance of the ORC turbine

compared to the design operating point.

Thus, to further understand this behaviour, it is interesting to investigate the

stochastic variation of the 𝐶𝑓 along the rotor blade and the velocity vectors at 𝑇𝑇𝑖𝑛 =

400K and 80% nominal RPM, using all EOSs. Figure 5-22 presents both the skin

friction coefficient 𝐶𝑓 and the velocity vectors along the rotor blades using the three

EOSs. It can be clearly seen that for all three EOSs, the 𝐶𝑓 is close to zero, indicating

that the flow field is near separation.



𝑃𝑐 − 𝑇𝑐 at 𝑇𝑇𝑖𝑛 = 400K and 80% nominal RPM using three EOS with P = 7. (a) PR

(b) HHEOS (c) SRK. The velocity vectors including leading edge at pressure and

suction sides of rotor blade using three EOS with P = 7. (d) PR (e) HHEOS (f) SRK.

A closer look shows that the 𝐶𝑓 at the pressure side of the TE for PR (Figure

5-22(a)) remains positive, even including the standard deviation, and thus no

separation is observed on the corresponding velocity vectors (Figure 5-22(d)) for all

possible values of the input uncertain parameters. While moving to HHEOS, it can be


observed that the standard deviation of some 𝐶𝑓 values is extremely close to zero and

that some small recirculation regions are present on the velocity vectors (Figure

5-22(e)). Finally, the SRK plots evidence of a negative 𝐶𝑓 (Figure 5-22(c)) and clear

separation and recirculation region (Figure 5-22 (f)). As presented in the dash magenta

square, which corresponds to 0.3 streamwise location, standard deviations using SRK

are 24.7% and 5.3% higher than PR and HHEOS respectively. At the same time, in the

black square corresponding to 0.42 streamwise location, standard deviations

employing SRK are 72.9% and 34.7% larger than PR and HHEOS respectively. These

results highlight the nonlinear response observed for the PDF of 𝜂𝑇−𝑆, using HHEOS

and SRK at 𝑇𝑇𝑖𝑛 = 400K and 80% nominal RPM, as shown in Figure 5-21(a).

Separation or recirculation in the rotor passages is still obtained in the stochastic case,

thus greatly increasing the unwanted losses and reducing the efficiency of the turbine

(Kim & Kim, 2017b) in the presence of uncertain parameters of the EOS. These results

are consistent with the efficiencies map chart (Figure 5-16 and Figure 5-17), in which

the highest and lowest efficiencies use PR and SRK respectively, as presented in

Figure 5-22(a) and (d). At low inlet total temperature and low rotational speed (𝑇𝑇𝑖𝑛 =

400K and 80% RPM), SRK shows the highest sensitivity to 𝑃𝑐 − 𝑇𝑐 through negative

flow characteristics; in particular, recirculation and separation affect efficiency.

5.5 CONCLUSION

In the present work, Uncertainty Quantification analyses of both nominal and

off-design operational conditions using three different Equations of State (Peng-

Robinson, Soave-Redlich-Kwong, and Helmholtz energy) are investigated for use in

design of an R143a high-density radial-inflow turbine. The stochastic analysis of both

operational uncertain parameters, (rotational speed, inlet total temperature, and mass

flow rate combined) and Equation of State uncertain parameters (critical pressure and

critical temperature) is presented employing different Equations of State. The uniform

distribution law is used to propagate these uncertain parameters. The Uncertainty

Quantification results for three quantities of interest including Isentropic Mach

number, skin friction coefficient, and total-to-static efficiency, are evaluated. In terms

of radial inflow turbine total-to-static efficiency, it has been observed that the effect of

EOS choice is not negligible. Among the three Equations of State, the Soave-Redlich-

Kwong is the most sensitive prediction for high-density in the close critical region.

The following conclusions are drawn:


The results of Uncertainty Quantification analysis showed that uncertainty

in 𝑇𝑇𝑖𝑛 has the strongest influence on the stochastic response of the total-to-

static efficiency.

According to these uncertain parameters (rotational speed, inlet total

temperature, and mass flow rate combined) in Uncertainty Quantification

analysis, the Coefficient of Variation of the efficiency using Peng-Robinson

is approximately 6.5% lower than Soave-Redlich-Kwong, as Peng-

Robinson and Soave-Redlich-Kwong have the smallest and highest

Coefficient of Variation respectively. Consequently, for these sets of

uncertain parameters, Peng-Robinson is shown to be the most robust

Equation of State and Soave-Redlich-Kwong is shown to be the most

sensitive.

The Uncertainty Quantification analysis for all sets of uncertain parameters

for the Isentropic Mach number and skin friction coefficient indicates that

the most sensitive region is located around the leading edge of the rotor

blade.

The sensitivity to randomness is more pronounced at off-design conditions

than at nominal conditions. Based on the results, Soave-Redlich-Kwong

may be more sensitive to uncertainties and thus it may not be the most

suitable choice for the robust optimization of high-density radial-inflow

turbines working over a wider range of operational conditions.

The proposed technique demonstrated its capabilities for investigating

random input parameters, which is crucial to the development of robust

designs at both nominal and off-design conditions. For low-grade

temperature Organic Rankine Cycle configurations in particular, it will be

crucial to investigate input random parameters that will strongly affect the

performance of the high-density fluid radial-inflow turbines at both nominal

and off-design conditions. Furthermore, this study will greatly benefit the

robust design of Organic Rankine Cycle turbines for low-grade temperature

renewable energy generation, and will therefore have a progressive impact

on renewable energy applications.


Future work should be devoted to the use of these stochastic tools to perform

robust optimization of the Organic Rankine Cycle turbines in the sense that the design

of the turbine should give, in the presence of uncertainty, robust levels of aerodynamic

performance. In particular for low-grade temperature Organic Rankine Cycle

applications, the most robust Equation of State would be critical for the numerical

investigation of high-density radial-inflow turbines.

Chapter 6: Sensitivity Analysis of High-density Conical Diffusers using UQ-CFD Framework 115

Chapter 6: Sensitivity Analysis of High-

density Conical Diffusers using

UQ-CFD Framework

As the connecting component to the ORC turbine outlet, high-density fluid

diffusers are key components designed to improve the efficiency of ORC. However,

investigations into the robust optimal design of high-density fluid diffusers are lacking,

which restricts the improvement of overall ORC efficiency. A robust framework

coupling the UQ approach with CFD and NIST REFPROP is used to effectively

implement sensitivity analysis of high-density fluid conical diffusers, as described in

Section 3.2 and Section 0. R143a, a potential high-density fluid, is employed in this

analysis. Both operating and geometric parameters have significant impact on the

performance of conical diffusers, and thus a performance analysis is conducted using

the proposed framework. It is shown that the swirl velocity has more impact than the

inlet axial velocity on pressure recovery under various geometric conditions of the

length and angle of the high-density fluid conical diffuser. It is also shown that high-

density fluid flows in diffusers are adapted to stochastic analyses, as they are sensitive

to variations in upstream thermodynamic properties. However, to the best of the

author’s knowledge, very few studies have reported on UQ investigation of conical

diffusers employing high-density fluid as working fluid, especially in low-to-medium

temperature ORC applications.

This work is the first attempt to execute a performance analysis, and to further

identify the most influential parameters on conical diffuser efficiencies employing

R143a as a working fluid. This study highlights the need to achieve a robust, optimal,

high-density fluid diffuser design in order to improve overall ORC efficiency. The

objective of this work is to highlight the influence of multiple uncertainties, with

possible random distributions, on the stochastic response of diffusers. The content is

outlined as follows: the computational model is described in Section 6.1. The

deterministic results are analysed in Section 6.2. The sensitivity analysis is detailed in

Section 6.3. Finally, the conclusion is presented in Section 6.4.


6.1 COMPUTATIONAL MODEL

The methodology of the study and its robust framework are described in Section

0. The detail of the computational model will be introduced in the following sections.

Geometry

The geometric illustration of the ERCOFTAC swirling turbulent conical

diffuser, as applied in this work, is provided by Clausen et al. (Clausen, et al., 1993).

A honeycomb directing inlet is placed at the inlet of the conical diffuser to generate

the turbulence and swirl velocity, depicted as the rotating domain in Figure 2.

According to Bounous (Bounous, 2008), a length of 1.02 m for the tailpipe was found

to be the best for a more accurate prediction of the profiles at the exit of the diffuser.

The Cartesian coordinate xw is the axis along the diffuser axis, and yn is the axis normal

to the diffuser wall. All the dimensions, including diffuser length (L), half cone angle

(A), and the typical line profiles (S0/S4/S7/S8), are provided in Figure 6-1.

Figure 6-1: Sketch of Conical Diffuser adapted from Clausen’s experiment

(Clausen, et al., 1993).

It is noteworthy that the line profile S8 is used to evaluate the velocity variation

from wall to centreline, as detailed in Section 6.3 below. The S distance as shown in

Figure 6-1 can be calculated as:

𝑆 =𝐿

cos 𝐴− [130 + 𝐿 × tan 𝐴] × sin 𝐴 (6-1)

This geometry is employed for the high-density fluid diffuser simulations, with

adapted boundary conditions to match the ORC turbine conditions (see Section 6.2).


The pressure recovery coefficient Cp of a diffuser is most frequently defined as

the static pressure rise through the diffuser divided by the inlet dynamic head (Japikse

& Baines, 1998):

𝐶𝑝 = 𝑃2 − 𝑃1

𝑃01 − 𝑃1 .

(6-2)

Where 𝑃2 is the static pressure of the diffuser outlet; 𝑃1 is the static pressure of

the diffuser inlet; and 𝑃01 is the total pressure of the diffuser inlet.

The Cp value is used here as an indicator of the diffuser’s performance, as the

more pressure is recovered, the higher the ORC turbine efficiency.

Boundary Conditions of the Reference Case

The boundary conditions of the reference case for the high-density fluid conical

diffuser are presented in Table 6-1. This reproduces a realistic operating state of the

fluid at the rotor exit of an ORC radial-inflow turbine in terms of temperature and

pressure. No-slip boundary conditions are enforced on the wall of the diffuser. The

reference case establishes a preliminary proposal for an RG diffuser that operates based

on Clausen’s experimental geometry (Clausen, et al., 1993), which is introduced in

Section 3.1. We denote this reference case as case R.

Table 6-1: Boundary conditions of the reference case R for high-density fluid conical

diffuser, L = 510mm, A = 10.

Validation and Mesh Refinement

Previously, the model for the ideal gas conical diffuser has been successfully

validated by From, et al. (2017) and was experimentally investigated by Clausen, et

Boundary Conditions Value Unit

swirl velocity W 56.1 rad/s

inlet mean velocity U 11.6 m/s

inlet turbulence intensities Τu 5 %

inlet temperature T 367.23 K

outlet pressure

swirling number

P

Sn

1.845

0.278

MPa

-


al. (1993). The grid is used with three-dimensional hexahedron O-H grid elements,

comprising 920,000 nodes as presented in Figure 6-2 and Figure 6-3.

Figure 6-2: Computational mesh of longitudinal view.

Figure 6-3: Computational mesh of inlet circumferential view.

In the grid independence study carried out by From, et al. (2017), the nearest

grid point from the wall is 7×10-6m, which satisfies the requirement of y+ < 2.

Convergence is achieved once the Root Mean Squared (RMS) for mass, momentum,

and turbulence variables approaches the residual target of 1×10-6.


The overarching objective of this study is to quantify the influence of

uncertainties on the non-linear aerodynamics performance of high-density fluid

conical diffusers. Uncertainty Quantification is employed in order to quantitatively


predict the influence of random input variables on stochastic outputs. The choice of

UQ parameters should be motivated by the fact that some input variables are inherently

random or uncertain (aleatory uncertainty). These random variables may have low or

high effects on the stochastic results. In order to avoid the curse of dimensionality,

only the random parameters which strongly affect the stochastic response are usually

retained. Uncertainty parameters exhibiting high impact on diffuser efficiencies were

chosen, following previous parametric numerical studies of diffusers (Armfield, et al.,

1990; Bounous, 2008; Olivier & Balarac, 2010; Page, et al., 1996; Sauret, et al., 2014),

which have identified them as having the highest impact on the diffuser efficiencies.

These two uncertain parameters are: operation conditions’ inlet axial velocity U, and

swirl velocity W. In this work, the two parameters are propagated simultaneously. The

coupled uncertain parameter is U-W. Hereinafter, we denote two-coupled uncertain

parameters as 2D (two-dimensional).

When applying UQ, a critical point is the way to model the uncertain parameters,

which involves careful selection of their random distribution range. The swirling flows

in diffusers are categorised by Armfield and Fletcher (1989) as weak swirl, moderate

swirl, and strong swirl, based on the ‘Swirl Number’ (Sn) which is defined by Cho

(Cho, 1990) and calculated as:

S𝑛 =∫ 𝑈𝑊𝑟2𝑅𝑖𝑛

0𝑑𝑟

𝑅𝑖𝑛 ∫ 𝑈2𝑟𝑅𝑖𝑛

0𝑑𝑟

. (6-3)

The Sn is calculated using the set values of U and W, and is obtained from the

CFD calculations. However, U and W restrain each other to prevent both near-wall

separation and centreline recirculation of diffusers. A moderate Sn value is defined

between 0.18 and 0.3 (From, et al., 2017). The U and W maintain an acceptable range

for ±3% and ±2.6% respectively, leading to variations of Sn between ±0.8% and

±1.3%, the maximum Sn being kept below the 0.3 limit established by (From, et al.,

2017). However, From et al. (From, et al., 2017) demonstrated that their real gas case

still presents slight recirculation at the centreline for Sn = 0.278. Thus, we maintained

Sn below 0.278, but close enough to see any influence of U and W due to separation

and/or recirculation. As bounded supports for the input random parameters were

defined, a uniform distribution law was adopted to propagate these parametric

uncertainties.


Table 6-2: The range of uncertain parameters: inlet velocity, swirl velocity.

We define this range of uncertain parameters through Eq. (6-3) for a moderate

Sn number, based on case R geometry. Case R establishes a preliminary proposal for

0.18 < Sn < 0.3 (From, et al., 2017). Hence, as presented in Table 6-2, the ranges of

uncertain parameters are listed, and the Sn ranges lie in the moderate swirl range

(From, et al., 2017).

6.2 DETERMINISTIC FLOW CHARACTERISTICS ANALYSIS

Flow Characteristics Description

Cone half-angle (A) is an important geometric condition that impacts on the near-

wall separation and centerline recirculation of diffusers (Armfield, et al., 1990). The

main flow characteristics associated with the variation of U, W and the diffuser angle

A at the S8 profile of the conical diffuser are demonstrated in Figure 6-4.

Figure 6-4: Sketch of the main flow characteristics with changing U, W and A.

Uncertain Parameter Range

inlet velocity U [m.s−1] [11.3, 11.9]

swirl velocity W [rad.s −1 ] [54.4, 57.8]


The near-wall, centerline and VR (velocity reduced) positions are demonstrated

at ①, ②, and ④ respectively. ③, the green rectangle in Figure 6-4, demonstrates

the centerline region. The region enclosed by blue shaded ④ area corresponds to a

velocity u below 1m/s. As 1m/s is reduced at least 92% compared to the maximum

velocity about 12m/s, it can be considered a risky limit where near-wall separation and

centerline recirculation can occur. Hereinafter, this region is denoted as the velocity

reduction (VR) region. If W increases, or U decreases, or A increases, recirculation may

happen at the centerline at position ③, where the velocity u on the S8 profile is

negative. Increased W and decreased U lead to density-induced rotational momentum

towards the wall and promote velocity reduction at the centerline, which can form

recirculation (From, et al., 2017). Swirl momentum becomes weak as the cross-section

area rapidly grows, so that the tangential component of the velocity does not reach the

centerline region, and thus the flow velocity in the axial direction becomes weak

(Azad, 1996). The generation of centerline recirculation and near-wall separation

occurs readily in VR regions, and subsequently affects Cp. Hence, analysis of VR

regions is of significance for evaluating Cp. This is also important while designing

robust diffusers that avoid recirculation and separation at off-design conditions.

Deterministic Results of Reference Case

Deterministic CFD computations are conducted based on the presented model

described in section 6.1 above. The boundary conditions of case R are described in

section 6.1.2, with U = 11.6 m/s, W = 56.1 rad/s, L = 510 mm, A = 10°, Cp = 0.7275.

Figure 6-5: (a), Contour plot of velocity u of case R in the middle plane of

diffuser. (b), velocity u on S0/S4/S8 profiles of the case R.

(a) (b)


The velocity contour plot (Figure 6-5(a)) and axial velocity profiles (Figure

6-5(b)) for case R were obtained at the three main locations within the diffuser: S0, S4,

and S8. On the S8 profile of case R, as shown in Figure 6-5(b), velocity u becomes

negative from Yn/Y0 > 0.75, and recirculation appears in the centreline region, as shown

in Figure 6-5(a). The Sn of case R is 0.278, close to the 0.3 limit stated by (From, et

al., 2017). As the flows generated by high-density gas R143a form a large reduction at

the centreline, slight recirculating flows of axial velocities occur at the exit of the

diffuser. For the case R, the detailed recirculation reasons at the centreline are

explained in (From, et al., 2017).

Deterministic Results of Extreme U-W Conditions

In order to have a better understanding of the flow characteristics within the

support range, four deterministic simulations are set up to correspond to the minimum

and maximum values of the two uncertain parameters coupled in a 2D manner, as

introduced in Table 6-3.

The Lratio is defined as the ratio of the length from the inlet to the onset of the VR

region (L0) and the total length (L); Lratio = L0/L, L0 and L are demonstrated in Figure

6-6(a), with an increased Lratio leading to a smaller VR region and thus better

performance for the diffuser. The black dash lines in Figure 6-6 (a, b, c, and d)

represent L0. The Lratio for cases A, B, C and D are listed in Table 6-3.

Table 6-3: Deterministic results regarding extreme U-W cases at L and A

constant (L = 510mm, A = 10°).

Case Name Case Description U (m/s) W(rad/s) Cp Sn Lratio

A Umin Wmin 11.3 54.4 0.7219 0.2784 0.7712

B Umin Wmax 11.3 57.8 0.7078 0.2941 0.7205

C Umax Wmin 11.9 54.4 0.7346 0.2775 0.8329

D Umax Wmax 11.9 57.8 0.7243 0.2807 0.7756

It is noteworthy that the Sn value for all the cases is higher than 0.277, as

presented in Table 6-3, which is very close to the limit of the moderate value, Sn =

0.278, as indicated in (From, et al., 2017). According to Table 6-3, when W increases

by 6%, comparing case A to case B (or case C to case D), the Lratio decreases about 7%,


with fixed minimum U. The value of Cp for case B decreases by approximately 3%

compared to case A, which decrease is impacted by the variation of Lratio. These

observations are also linked with a decrease of the Sn value from case D to C and from

B to A, where W reduces. If U increases by 6% in case C compared to case A, the Lratio

for case C increases by approximately 8%. Consequently, the Cp of case C improves

approximately 2% compared to case A.

The deterministic CFD contour plots of the velocity u of cases A, B, C, and D

are shown in Figure 6-6. The region enclosed by the Magenta Line in Figure 6-6 is the

VR region, which is the velocity u < 1m/s.

Figure 6-6: Contour Plot Velocity U regarding (a) case A, Umin Wmin. (b)

case B, Umin Wmax. (c) case C, Umax Wmin. (d) case D, Umax Wmax.

(c) (d)

VR

(a) (b) Case A Case B

Case C Case D

L0

L

S8


According to Figure 6-6, comparing case A to case B, or case C to case D, when

the swirling velocity is increasing, the recirculation region becomes larger. When

comparing case A to case C, or case B to case D, the recirculation region will increase

as well when the axial inlet velocity increases. The explanation for these results is that

when the velocity is higher at the centerline than in the near-wall region, based on the

mass conversion analysis, a weak velocity region will appear in the near-wall region

rather than at the centerline (Armfield, et al., 1990).

The variations of velocity u on S8 profiles among cases A-D and case R are

presented in Figure 6-7. Based on Figure 6-7, the peak of velocity u near the wall for

case B and D is higher than for case A, C and R. The results can be explained by the

increased W, which affects induced high centrifugal forces to generate the tangential

component in the near-wall region (Armfield, et al., 1990). Thus, increased W resists

boundary layer separation by enhancing the momentum of transport (From, et al.,

2017). As there is negative velocity u appearing on the S8 profile for all cases,

recirculation happens for cases A-D.

Figure 6-7: Deterministic S8 profiles of U-W for cases A-D.


Deterministic Results of L-A Conditions

In order to better understand the flow characteristics of high-density fluid

through different geometric conditions, four combinations of the diffuser length (L)

and half angle (A) are studied here, as listed in Table 6-4.

From the existing literature (Hah, 1983; Okhio, Horton, & Langer, 1986; Senoo,

et al., 1978), the divergence angle has been estimated to range from 8-12 degrees, and

not many investigations of the ‘ideal’ length of the conical diffusers have been

proposed in the literature. As such, we are considering those 8-10-12 degrees angles

and ±20% length (410mm-510mm-610mm) in order to investigate the sensitivity of

these acceptable diffuser geometries to the uncertain parameters U-W.


Table 6-4: Deterministic results regarding L and A cases at U-W constant (U =

11.6 m/s, W = 56.1 rad/s).

Case Name Case Description L (mm) A (°) Cp Sn Lratio

E Lmin Amin 410 8 0.7338 0.2753 -

F Lmin Amax 410 12 0.7299 0.2781 0.7351

G Lmax Amin 610 8 0.7207 0.2783 0.7849

H Lmax Amax 610 12 0.6909 0.2799 0.5241

From Figure 6-8, it is clear that L0 decreases as the length of the diffuser

increases. However, according to Table 6-4, when A increases approximately 50%

comparing cases E to F (or cases G to H), the Lratio declines by about 26%. The Cp

value is only slightly affected by a decrease of 0.5% for the shortest diffuser between

case E and F. However, for the longest diffuser, the increase of angle leads to a large

Cp drop of approximately 4.4%. One possible explanation lies in the almost constant

value of Sn between E and F, while an increase is observed between G and H, leading

to the conclusion that a combination of both high length and high angle is damaging

the performance of the diffuser. We can also note that while L increases by about 50%,

comparing cases E to G (or cases F to H), the Lratio decreases 21%. This again

highlights the impact of the length at a constant diffuser angle.

Based on Figure 6-8(a), there is no recirculation for case E, which consequently

presents the highest Cp value. When the VR region increases comparing case F to case

E (or comparing case H to case G), as presented in Figure 6-8, the swirl increase by

the increase of the half-angle may become more effective in generating recirculation

in the centreline.


Figure 6-8: (a) Contour Plot Velocity_U regarding: (a), case (E) Lmin and

Amin. (b), case (F) Lmin and Amax. (c), case (G) Lmax and Amin. (d), case (H)

Lmax and Amax.

(a) (b)

(b) (d)

VR

VR VR

Case E Case F

Case G Case H

S8


Figure 6-9: Deterministic S8 profiles of U-W for case E-H.

The velocity u for case E-H and case R is presented in Figure 6-9. The velocity

u for case E is positive, and for case F, it is close to zero, from about Yn/Y0 = 0.75. The

negative velocity u of case R occurs at approximately Yn/Y0 = 0.8. However, the

velocity u for case G and case H starts to be negative from Yn/Y0 = 0.8 and 0.5

respectively. These results indicate that recirculation increases in the centerline of

diffusers and is linked to the lowest Cp and highest Sn values for case H.

In summary, the swirl momentum is the reason for the centrifugal force and

momentum of transport, which can push the fluid toward the near wall and prevent the

boundary layer separating. Nevertheless, swirl momentum produces an undesirable VR

region in the centerline, which is prone to recirculation. Moreover, varying the half-

cone angle dramatically changes the area ratio of the fluid domain, thereby attenuating

the swirl and axial momentum, which causes formation of recirculation at the

centerline of the diffuser. Deterministic results show that U-W have an effect on the

performance of the diffuser through changes in velocity profiles both near the wall and

at the centerline. However, how the U-W uncertainties will impact the diffuser


performance for different fixed geometries is not clear. Thus, uncertainty

quantification analysis for U-W will be conducted in the following section. Hence, it

is a critical challenge to quantify the coupled U-W to convert kinetic energy behind the

turbines into high static pressure recovery.

6.3 SENSITIVITY HIGH-DENSITY CONICAL DIFFUSER TO AXIAL

AND SWIRLING VELOCITIES

The gPC approach is applied to quantify the variation of performance (Cp) of

the diffuser under uncertain U-W parameters at four different geometric conditions and

for the reference case. The convergence of gPC and sensitivity analysis of uncertain

parameters are presented in this section.

Convergence Analysis


Legendre polynomials are selected as the optimal basis in the spectral representation

of the Polynomial Chaos expansion (Equation (3-20)). With the purpose of

demonstrating the UQ framework, Figure 6-10 shows the grid of quadrature points for

P = 5, 7, and 9 as used in the deterministic CFD solver to compute the flow solution.

Figure 6-10: Legendre quadrature points for U-W for 𝑃 = 5, 7, 9.


The effect of the polynomial order P on the accuracy of the gPC formulation is

investigated with the coupled U-W parameters, as presented in Figure 6-11.

Figure 6-11: PDF Cp for P = 5, 7, 9 gPC for U-W at L = 510mm, A = 10°.

The convergence among three approximation orders (P = 5, P = 7 and P = 9) is

shown in Figure 6-11. In Figure 6-11, the PDFs show very small differences among

all three P orders. For the 2D gPC (U-W) stochastic analysis, the polynomial order, P,

is then set to P = 7, which is an appropriate trade-off between accuracy and

computational burden.

Stochastic Results

The Coefficient of Variation (CoV), expresses the sensitivity of different coupled

variables. According to Table 6-5, when two parameters U-W are coupled under

different geometric conditions, the highest and lowest mean values of CoV are for L =

610mm, A = 8° and L = 410mm, A = 8°, respectively. These results have good

agreement with Section 6.2.4, as L = 410mm, A = 8° is the only case far from

recirculation, and thus shows the least sensitivity of Cp to U-W. The four other cases

are all close recirculation or have clear recirculation, and because of that seem more

sensitive to U-W on the Cp evaluation. The case L = 410mm, A = 8° also presents the


highest Cp mean value, and with the lowest CoV, it demonstrates that high

performance with low sensitivity can be reached at the same time. Thus, the group case

having the highest Cp but also being the least sensitive would be more appropriate in

a robust design perspective.

Table 6-5: Mean and CoV of Cp for coupled U-W uncertain parameters by gPC

under different geometric conditions.

gPC 2D U-W (P = 7)

Case

Name gPC (Case E) gPC (Case F) gPC (Case R) gPC (Case G) gPC (Case H)

Geometry L = 410mm,

A = 8°

L = 410mm,

A = 12°

L = 510mm,

A = 10°

L = 610mm,

A = 8°

L =610mm,

A = 12°

μ 0.7335 0.7116 0.7254 0.7215 0.6926

σ×10-3 4.601 13.038 13.340 13.374 12.954

CoV×10-3 6.633 17.775 18.090 18.535 18.204

The standard deviation of velocity u on S8 profile for U-W under different

geometric conditions is presented in Table 6-5. The standard deviation of velocity u

under L410A12 is higher at the near-wall region than for other geometric conditions.

On the contrary, when the half-cone angles of the diffuser are smaller, such as 8° and

10°, L410A8, L510A10 and L610A8, the standard deviation velocity u of the S8 profile

is large. For the two larger angle group cases, a clear recirculation at the centreline is

present, and the stochastic results show that having a well-developed recirculation

reduces the sensitivity of the velocity profile to U-W. However, the sensitivity is higher

in group case L610A8, which is consistent with previous analysis of results, and also

highlights a greater sensitivity of the velocity profile to U-W when the flow is very

close to recirculation.

With the purpose of having better visualisation of the stochastic distribution of

the velocity u, the mean value, standard deviation, variance, and CoV contour plots are

presented for the different diffusers’ geometries in Figure 6-12 to Figure 6-16.


Figure 6-12: UQ analysis of velocity u regarding U-W for Case E at L =

410mm, A = 8°. (a) Mean value of velocity u. (b) Standard Deviation of velocity u.

(c) Variance of velocity u. (d) CoV of velocity u.

Based on Figure 6-12(a), no negative velocity u in x direction region can be

observed. According to Figure 6-12(b) and (c), both higher standard deviation region

and high variance region occur at the centreline. It is noteworthy that the CoV shows

very light variation at the centreline of the diffuser near the outlet. This result is

consistent with the lowest CoV obtained for L = 410mm, A = 8° geometry condition,

as presented in Table 6-5. These results show that the coupled uncertain parameter U-

W is not sensitive under L = 410mm, A = 8° geometry condition, probably due to the

absence of separation and recirculation in this case.


Figure 6-13: UQ analysis of velocity u regarding U-W for Case F at L = 410mm,

A = 12°. (e) Mean value. (f) Standard Deviation. (g) Variance. (h) CoV.

When the cone half-angle increases to 12° at the shortest length (L = 410mm) in

Figure 6-13, compared to Figure 6-12, negative values of the mean velocity u occur at

the centreline near the outlet of the diffuser, as shown in Figure 6-13(e). According to

Figure 6-13(f) and (g), a region of high standard deviation and high variance appears

at the fringe of the centreline recirculation near x = 0.3-0.35m.

Figure 6-13(h) similarly shows that a high CoV zone is generated around the

recirculation. The high sensitivity of this case at the fringe of the recirculation region

is explained by the change of velocity u’s direction. Just before the recirculation, the

velocity is positive, and the uncertain parameters U-W do not affect flow behaviour.

This is similar to what happens within the recirculation region, where the negative

velocity is not affected by the U-W uncertain parameters. However, the interface

between positive-negative velocities is where the highest sensitivity is observed, due

to the very low positive-negative velocity values. This case also shows a bit more

sensitivity near the walls close to the outlet (Figure 6-13(f)), as the flow velocity is

weakening. The obvious difference is the CoV. The highest CoV region as presented

in Figure 6-14(d) shows that the coefficient of variation region at the centreline under

L = 510mm, A = 10° is big and strengthened.


Figure 6-14: UQ analysis of velocity u regarding U-W for Case R at L =

510mm, A = 10°. (a) Mean value. (b) Standard Deviation. (c) Variance. (d) CoV.

Figure 6-15: UQ analysis of velocity u regarding U-W for Case G at L = 610mm, A =

8°. (a) Mean value. (b) Standard Deviation. (c) Variance. (d) CoV.

With the same cone half-angle, compared to Figure 6-12 (b)-(c), the standard

deviation and variance of velocity u under L = 610mm, A = 8° (Figure 6-15) is larger


at the beginning of the recirculation region compared to L = 410mm, A = 8°. This

means the transition region from positive to negative under L = 610mm, A = 8° is

longer than L = 410mm, A = 8°. Thus, the velocity u regarding U-W under L = 610mm,

A = 8° is more sensitive than under L = 410mm, A = 8°. This result has good agreement

with the finding that the CoV of Cp under L = 610mm, A = 8° is much higher than

under L = 410mm, A = 8°. When the diffuser length increases, a larger recirculation

region will be generated. In addition, the interface of positive-negative velocities will

be more sensitive with increases in the diffuser length.

Figure 6-16: UQ analysis of velocity u regarding U-W for Case H at L =

610mm, A = 12°. (e) Mean value. (f) Standard Deviation. (g) Variance. (h) CoV.

Similarly, Figure 6-16(e) for L = 610mm, A = 12° illustrates that the increased

cone half-angle results in a large kinetic energy generating a larger recirculation region

compared to Figure 6-15(a) (L = 610mm, A = 8°). Figure 6-16(f)-(g) indicates that the

standard deviation and variation are smaller than in Figure 6-15(b)-(c). Based on

Figure 6-16(h), the highest CoV region is thinner compared to Figure 6-15(d). This

result can be explained by the recirculation under L = 610mm, A = 12° being large and

fully established, and having less variation between positive and negative velocity

compared to the A = 8° case (Figure 6-15). This result is also consistent with Table


6-5, which shows the CoV of Cp under L = 610mm, A = 12° being smaller than for L

= 610mm, A = 8°.

Based on Figure 6-17, the deterministic and mean profiles of S8 for Case E are

extremely close to each other. This means that there is negligible variation between

the deterministic and the mean profile of S8 for Case E. According to Figure 6-17, it

is clear that variations happen between deterministic and mean values regarding S8

velocity u for Case F, R, G, and H. In particular, for Case R, the mean value from Yn/Y0

= 0.95 to the end of the S8 velocity u profile is higher than the standard deviation range

as marked in the green circle. It is noteworthy that for case F, R, and G, the variation

between deterministic and mean values of S8 velocity u happens close to the zero

(velocity u = 0) line. This explains that the velocity u will more sensitive close to zero,

which is the transition from positive to negative value.

Figure 6-17: Standard deviation of velocity u on S8 profile under different geometric

conditions.


Figure 6-18: Close view of Figure 6-17 under different geometric conditions. (a)

L410-A12. (b) L510-A10. (c) L610-A8. (d) L610-A12.

According to Figure 6-18(a), from Yn/Y0 = 0.65 to 0.95, there exists a discrepancy

in the deterministic velocity u and the mean velocity u. As shown in Figure 6-13(h),

the highest CoV of velocity u is located from Yn/Y0 = 0.65 to 0.95, and the interface

between the positive-negative velocities occurs in this region. This region is highly

susceptible to recirculation, where negative velocity u is observed starting from Yn/Y0

= 0.80. Based on Figure 6-18(b), it is found that the deterministic velocity u is higher

than the mean velocity u commencing at Yn/Y0 = 0.65. Moreover, Figure 6-14(d) shows

regions of high CoV of velocity u from 0.65 to 1.0.

The deterministic velocity u is lower than the mean velocity u from Yn/Y0 = 0.65

to 0.80, as shown in Figure 6-18(c). As shown in Figure 6-15(d), the highest CoV of

velocity u is also located from Yn/Y0 = 0.65 to 0.80. In addition, the deterministic

velocity u at approximately Yn/Y0 = 0.75 is negative (less than 0m/s), which infers the

generation of recirculation in this region. According to Figure 6-18(d), the difference

between the deterministic and mean velocity u increases from approximately Yn/Y0 =

(a) (b)

(c) (d)


0.4 to 0.5; afterwards this difference gradually decreases from Yn/Y0 = 0.5 to 0.9. It is

noteworthy that at approximately Yn/Y0 = 0.5, the maximum discrepancy between the

deterministic and mean velocity u is observed; additionally, the velocity u is 0 m/s.

This observation reflects what is depicted in Figure 6-16(h), where the highest CoV of

velocity u also covers Yn/Y0 = 0.5, and it is here that recirculation is prevalent. It is

noteworthy that for L410-A12, L510-A10, L610-A8, and L610-A12, the variation

between deterministic and mean value of S8 velocity u happens close to zero (velocity

u = 0) line. This explains that the velocity u will be more sensitive close to zero, which

is the transition from positive to negative value.

The CoV analysis shows the effect of multiple and combined uncertainties. In

order to identify the individual contribution of uncertain parameters affecting the

stochastic value Cp, Sobol’s variance indices must be computed. In Figure 6-19, both

the first order (Figure 6-19(a)) and second order (Figure 6-19(b)) Sobol indices are

computed under different L-A geometric parameters and in the presence of random

conditions in both U and W.

Figure 6-19: Sobol's indices of Cp for U-W. (a) First order. (b) Second order.

According to Figure 6-19(a), the first order for U and W does not show

significant difference, and the first order Sobol’s indices for all cases shows that W

contribution is higher than U. Those results indicate that the stochastic output

parameter Cp is more sensitive to W than to U at any L/A geometric parameter

conditions for the high-density fluid conical diffuser. Based on Figure 6-19(b), the

second-order Sobol’s indices (Figure 6-19(b)) are very small, showing that U-W

coupled have negligible non-linear effects on the stochastic response. We can note that

L410A8, for which the deterministic results did not present recirculation (Figure

6-8(a)) at the nominal velocities, is the least influenced geometry.

(a) (b)


Figure 6-20: PDF for U-W under different geometric conditions.

According to Figure 6-20, the probability density function (PDF) exhibits

characteristics of a multivariate uniform distribution under L410A8 geometric

conditions, which means that the Cp is a linear function of these two-coupled uncertain

parameters, U-W. Because the cross-section area grows when the half-cone angle

increases, the tangential component of the velocity avoids entering the centreline

region to generate recirculation and lead to a non-linear influence on the performance.

6.4 CONCLUSION

In this study, uncertainty quantification analysis of the high-density fluid conical

diffuser is conducted with uncertain operating conditions, including nominal and

varied geometric conditions.

A robust framework is established to couple Uncertainty Quantification with a

Computational Fluid Dynamics solver coupled with the NIST REFPROP database.

More importantly, this framework can be considered as a benchmark to be utilised in

related engineering applications. The deterministic and stochastic analyses for the

high-density fluid conical diffuser are investigated comprehensively by density-

induced swirl flow.

The difference between deterministic results and the stochastic mean values of

the velocity u (under diffuser length = 410mm - diffuser half cone angle = 12°; diffuser

length = 510mm - diffuser half cone angle = 10°; diffuser length = 610 mm - diffuser

half cone angle = 8°; diffuser length = 610mm - diffuser half cone angle = 12°) is due


to local non-linear effects in the recirculation regions. Therefore it is significant in

these cases to refer to statistical results to identify the most probable values of the

velocity u due to the uncertain parameters.

Results associated with Coefficient of Variance and Sobol’s indices against

pressure recovery show that swirling velocity has more effect than inlet velocity for

the high-density fluid conical diffuser. It was observed that uncertainties in coupled

swirling velocity and inlet velocity play a more important role under a longer and

smaller half-cone angle diffuser (diffuser length = 610mm, diffuser half cone angle =

8°) than do the other geometric conditions.

When a large centreline recirculation region occurs, it is interesting to note that

coupled swirling velocity and inlet velocity uncertainties do not have much effect on

performance, probably because recirculation is already too large to be influenced by

small uncertainties. On the other hand, the case near recirculation shows the highest

sensitivity to inlet velocity and swirling velocity. This is an essential point to consider

while designing a high-density fluid conical diffuser, as working with close

recirculation may result in a more sensitive design. In future work, this proposed

framework can be easily applied to the rational design of robust diffusers for ORC

turbines.

Chapter 7: Development and Analysis of a More Robust ORC Radial Turbine 141

Chapter 7: Development and Analysis of a

More Robust ORC Radial

Turbine

The use of high-density fluids leads to small compact radial-inflow turbines with

a high velocity at the exit of the rotor. Such high velocities then enter the diffuser, and

can negatively impact the recovery process due to losses associated with separation

and recirculation. It is thus of paramount importance to improve design diffusers for

high-density radial-inflow turbines. However, while many studies have focused on

conical diffusers, as introduced in Section 2.3, they mainly considered ideal gas. A

previous studies (From, et al., 2017), highlighted the different behaviour of high-

density fluids in diffusers, thus showing the need for a more detailed understanding of

the flow and design optimisation of diffusers for high-density turbines. A suitable

diffuser to fit the radial-inflow turbo expander is needed to maximise the whole

turbine’s efficiency. However, so far, in current numerical tools, it is quite difficult to

accurately set the diffusers’ inlet boundary conditions, which correspond to the outlet

flow of the turbo expanders when conducting numerical studies on diffusers alone.

The influence of inlet boundary conditions is known to affect the flow in diffusers, and

as such, in this study, our proposed R143a radial-inflow turbo expander (Sauret & Gu,

2014) is built as the inlet part of the diffuser to form a complete radial-inflow turbine.

Fitting these two typical ORC diffusers to an existing turbo-expander will affect the

complete turbine’s performance; this phenomenon lacks comprehensive understanding

in the literature.

The numerical model is briefly introduced in Section 7.1. Then, the content is

divided into two independent parts. The first part is the numerical comparison of two

preliminary diffuser geometries, a conical diffuser and an annular-radial diffuser,

matching the conditions from our existing 400kW R143a radial-inflow turbine, as

described in Section 7.2. The performance analysis of the ORC radial-inflow turbo-

expander fitted with two different types of diffusers is carried out in Section 7.4. Then

the deterministic study, a comparison between a preliminary ORC radial-inflow turbo-

expander and an improved ORC radial-inflow turbo-expander fitted with an annular-

radial diffuser at nominal operation conditions, is presented in Section 7.5. Then the


stochastic analysis of the improved ORC radial-inflow turbine under uncertain

operational parameters and off-design conditions are carried out in Section 7.6 and

Section 7.7 respectively. A summary is presented in Section 7.8.

7.1 INTRODUCTION

In this chapter, a Computational Fluid Dynamics (CFD) technique and the robust

framework introduced in Section 0 is employed. The numerical simulations have been

conducted by employing the ANSYS CFX package based on the Finite Volume

Method to perform steady-state 3D viscous simulations ( ANSYS® Academic

Research CFX Guidance, Release 18, 2017). Reynolds-Averaged Navier Stokes

(RANS) equations for viscous compressible flows were applied. Convergence is

achieved once the Root Mean Squared (RMS) residuals for mass, momentum, and

turbulence variables approach the residual target of 1×10-6. The CFD turbulence model

is detailed in Section 3.1. Following the work on the radial-inflow turbine carried out

in Section 5.3, and the validation of the R143a radial-inflow turbine by Sauret and Gu

(Sauret & Gu, 2014), the 𝑘 − 𝜖 model is chosen in this section to model the complete

turbine, including, stator, rotor, and diffuser.

In order to investigate the effect of the high-density fluid properties on the whole

turbine performance, the cubic EOS of Peng-Robinson (PR) (Peng & Robinson, 1976)

which is known for its good balance between simplicity and accuracy, especially near

the critical point, is chosen. PR is investigated and evidenced in Chapter 5: to be the

most robust and most appropriate Equation of State for the R143a radial-inflow

turbine.

The investigated R143a turbo-expander presented in Section 5.3 is denoted

‘Preliminary-Turbo-expander’. The Preliminary-Turbo-expander fitted with the new

designed radial-annular diffuser is denoted ‘Improved-Complete-Turbine’. A more

robust Improved-Complete-Turbine will be investigated through deterministic

analysis and Uncertainty Quantification analysis and will be compared against the

Preliminary-Turbo-expander as well.

7.2 PRELIMINARY DESIGN OF DIFFUSERS

According to the experimental investigation by Abir and Whitfield (1987), the

flow characteristic of conical diffusers is unstable, while the curved annular diffuser


and the radial diffuser demonstrate more stable flow characteristics. Recently, Keep,

et al. (2017) designed an annular-radial diffuser to fit their existing supercritical CO2

radial-inflow turbine. However, they modelled an independent diffuser not connected

to the turbo-expander. In this study, the preliminary annular-radial diffuser is designed

to fit a current existing radial-inflow turbo-expander. The inlet dimensions of the two

diffusers are constrained by the outlet of the upstream R143a turbo expander, as

detailed in (Sauret & Gu, 2014).

The preliminary conical diffuser geometry is built based on the geometric

similarity of From et al.’s design (From, et al., 2017) to fit an existing R143a radial-

inflow turbo expander. The sketch of the conical diffuser is presented in Figure 7-1.

The streamwise location is non-dimensional. It is X/Xw, as presented in Figure 7-1.

We denote the red point as 𝑆𝑐, which corresponds to a streamwise location = 0.7 along

the diffusing wall, which will be used for the skin friction coefficient analysis in

section 7.4 to evaluate the near-wall flow recirculation and separation.

Figure 7-1: Sketch of conical diffuser.

Using the geometric similarity principle, the original ERCOFTAC diffuser

(From, et al., 2017) was scaled down. The geometric parameters for the preliminary

conical diffuser design are presented in Table 7-1.

Table 7-1: Geometric parameters of the conical diffuser.

Name Symbol Value

Inlet outer radius R0 46.1 mm

Inlet inner radius R1 14.1mm

Outlet Radius R2 77.6 mm

Diffuser Length L0 180 mm

Extension Length L1 360.1 mm

Half Cone Angle A 10°


The annular-radial diffuser is created according to theoretical descriptions

(Japikse & Baines, 1998; Keep, et al., 2017). Moller (Moller, 1966) demonstrated an

annular-radial diffuser design method for the bend with no change in passage region

along the flow path, which is a key study for combining the axial and radial diffuser.

In Moller’s experiment using ideal gas, the non-dimensional width ℎ/𝑑 is 0.1-0.2, and

the optimal deterministic analysis value is 0.143. Due to lack of experimental studies

using high-density fluid, in this study, the ℎ/𝑑 is also set at 0.143 for the preliminary

design of the annular-radial diffuser. The ℎ is demonstrated in Figure 7-2. The 𝑑 is

calculated by Equation (7-1), and 𝛼1is presented in Table 7-2.

ℎ = 𝛼1 ∗ 2√𝑟𝑠2 − 𝑟ℎ

2 (7-1)

The sketch of the annular-radial diffuser is presented in Figure 7-2. The

streamwise locstion is non-dimensional. The 0 is set at the beginning of the shroud and

the hub of the diffuser. The streamwise location on the shroud is X/Xs and on the hub

is X/Xh, as presented in Figure 7-2. In Figure 7-2, the magenta and blue points, denoted

as 𝑆𝑎−𝑠1 and 𝑆𝑎−𝑠2 , correspond to the streamwise shroud locations 0.22 and 0.5

respectively. The green point in Figure 7-2 is denoted as 𝑆𝑎−ℎ corresponding to the

streamwise 0.13 along the hub. The typical points 𝑆𝑎−𝑠1 and 𝑆𝑎−𝑠2 are utilised for the

numerical analysis in Section 7.4. Based on the inlet dimension constraints from the

existing R143a radial-inflow turbo expander (Section 5.3), the optimal diffuser

dimensions for rs, rh, L, r, r0, and h are constrained by the equations given in (Moller,

1966).


Figure 7-2: Sketch of annular-radial diffuser.

The geometric parameters of the preliminary design of the annular-radial

diffuser are presented in Table 7-2.

Table 7-2: Geometric parameters of the annular-radial diffuser.

Name Symbol Value(mm)

Shroud radius rs 46.1

Hub radius rh 14.1

Inlet axial length L 15

Transition radius r 26.9

Outlet radius r0 120

Radial passage scaling factor α1 0.143

Radial Passage width h 13.1

The performance of both diffusers is described by the pressure rise coefficient,

𝐶𝑝, as defined in Equation (6-2).

0

Xs

0 Xh


7.3 NUMERICAL MODELLING OF COMPLETE ORC RADIAL

TURBINE

The pressure recovery coefficient 𝐶𝑝 is monitored for both diffusers, and the

nominal mesh is shown to be converged, with less than 0.01% difference between the

nominal and coarse mesh and almost no changes between the nominal and coarse. The

nominal mesh sizing is employed in this study for both diffusers.

Table 7-3: Grid study of pressure recovery for different diffusers.

Conical diffuser Annular-radial diffuser

Mesh 𝐶𝑝 𝐶𝑝

Coarse 0.7266 0.7533

Nominal 0.7269 0.7536

Fine 0.7268 0.7536

The investigations were conducted using different turbulent models, such as 𝑘 −

휀, 𝑘 − 𝜔, Shear Stress Transport (SST), and Explicit Algebraic Reynolds Stress Model

(EARSM), for the Improved-Complete-Turbine. All of the results employing different

turbulent models show very minor different influences (less than 0.5%) on the total-

to-static efficiencies of complete turbines with conical diffuser and annular-radial

diffuser. Thus in this chapter, based on Chapter 5, the two-equation turbulence model

𝑘 − 휀 is employed for the investigation of complete turbines.

Due to modelling the whole turbine, the periodic boundary condition is built so

that only one passage is modelled for these two full turbines, including stator, rotor,

and two types of diffusers, as presented in Figure 7-3.

Figure 7-3: Geometry of one blade passage of two full turbines with two different

diffusers respectively.


In this study, as the whole turbine is modelled, the main boundary conditions of

the 400kW-R143a ORC radial-inflow turbo expander designed by Sauret and Gu

(Sauret & Gu, 2014) are used: inlet mass flow rate of 17.24 kg.s-1, total inlet

temperature of 413K, outlet pressure of diffuser 1.835 MPa, rotational speed of the

rotor 24,250 RPM, total blade number of stator 19, and total blade number of rotor 16.

Walls are set to a no-slip condition. All detailed geometric and design conditions are

provided in (Sauret & Gu, 2014) and not repeated in full here.

7.4 PERFORMANCE ANALYSIS OF ORC RADIAL TURBO-EXPANDER

FITTED WITH TWO DIFFERENT DIFFUSERS

The total-to-static efficiency and flow fields of the whole R143a radial-inflow

turbine including turbo expander and diffusers are calculated. In the total-to-static

equation (𝜂𝑇−𝑆 =ℎ𝑇𝑖𝑛

−ℎ𝑇𝑜𝑢𝑡

ℎ𝑇𝑖𝑛−ℎ𝑆𝑖𝑠 𝑜𝑢𝑡

), the outlet is the diffuser outlet and the inlet is the inlet

of the stator. The total-to-static efficiencies are presented in Table 7-4.

Table 7-4: Total-to-static efficiency 𝜂𝑇−𝑆 for the whole turbines using different

diffusers.

Diffuser type 𝜂𝑇−𝑆

Conical diffuser 0.8738

Annular-radial diffuser 0.9039

Thus, overall efficiency is improved by approximately 3.4% using the annular-

radial diffuser compared to the conical diffuser.

Table 7-5: The pressure recovery coefficient 𝐶𝑝 for both diffusers.

Diffuser type 𝐶𝑝

Conical diffuser 0.7269

Annular-radial diffuser 0.7536

Overall pressure recovery coefficient has approximately 3.7% improvement

employing the annular-radial diffuser compared to the conical diffuser, as presented in

Table 7-5, which explains the increase in the overall whole ORC turbine efficiency.

Further Uncertainty Quantification analysis for the turbo-expander’s

performance using different EOSs, as investigated in Section 5.3, showed that PR EOS

is not the most sensitive to uncertain parameters, and as such, the superiority of the


annular-radial diffuser is expected to hold under uncertainties. Further design

optimisation may be needed to improve those results even further.

In order to better understand the flow characteristics of the two diffusers, the

velocity streamlines at the periodic plane for both diffusers are presented in Figure 7-4

and Figure 7-5.

Based on Figure 7-4, a small near-wall separation marked in the Red Rectangle

happens, which is located near the outlet of the diffuser. Furthermore, there is a

recirculation at the inlet centreline of the conical diffuser (Purple Rectangle).

Figure 7-4: The velocity streamline of axial velocity of the conical diffuser.

Compared to the conical diffuser, neither obvious separation nor recirculation is

observed in the annular-radial diffuser, as shown in Figure 7-5(a), and in a closer view

of the dash box for the vector in Figure 7-5(b).

Figure 7-5: The velocity streamline of the annular-radial diffuser. (a) Overall

view. (b) Closer view for the dash box for vector.

Sc


In order to investigate the flow phenomenon regarding the near-wall separation,

the characteristics of the boundary layer along the wall of the diffusers are evaluated

by the skin friction coefficient (𝐶𝑓), as shown in Figure 7-6 for the conical diffuser,

and in Figure 7-7 and Figure 7-8 for the annular-radial diffuser.

Figure 7-6: Skin friction coefficient of conical diffuser.

Based on Figure 7-6, from approximately streamwise 0.7 (point 𝑆𝑐 in Figure

7-1), the skin friction coefficient is zero for the conical diffuser. These results have

good agreement with Figure 7-4 in the Red Rectangle region, which means it is starting

to generate the near-wall separation in this region.


Figure 7-7: Skin friction coefficient of annular-radial diffuser hub.

As presented in Figure 7-7, there is an obvious drop of the skin friction

coefficient at the annular-radial diffuser hub at approximately streamwise 0.13, at the

beginning of the bending (point 𝑆𝑎−ℎ in Figure 7-2). The geometry changes from

straight to bending, which results in a change of the flow direction, and thus the

velocity reducing at the turning region.

As demonstrated in Figure 7-8, a skin friction coefficient peak occurs at

streamwise 0.22 ( 𝑆𝑎−𝑠1 point in Figure 7-2), which indicates that the near-wall

separation is difficult to generate in this region. However, from approximately

streamwise 0.5 located from the annular-radial section to the radial-radial section

(𝑆𝑎−𝑠2 point in Figure 7-2), the skin friction coefficient is low, but higher than zero.

These results show that the near-wall separation more easily happens in the radial-

radial section than in the annular and annular-radial sections for the annular-radial

diffuser. No obvious near-wall separation occurs in this annular-radial diffuser.


Figure 7-8: Skin friction coefficient of annular-radial diffuser shroud.

In summary, these results agree well with the experimental results in (Abir &

Whitfield, 1987). The diffuser performance improves when curving the passage from

the flow cone angle to the radial direction. The curved radial direction passage better

converts the high kinetic energy from the upstream rotor into static pressure than the

straight direction does. Moreover, the conical diffuser needs a very long extension to

achieve static pressure recovery, which may be a space limitation for the whole turbine

layout. The annular-radial diffuser effectively avoids recirculation at the centerline that

is generated in the conical diffuser. Thus, the annular-radial diffuser is selected to fit

the current existing preliminary turbo-expander in order to form an improved complete

turbine, which will be further investigated in the following sections.

7.5 COMPARISON BETWEEN PRELIMINARY ORC RADIAL TURBINE

AND IMPROVED ORC RADIAL TURBO-EXPANDER FITTED WITH

ANNULAR-RADIAL DIFFUSER

The total-to-static efficiency and flow characteristics of the R143a Improved-

Complete-Turbine at nominal conditions (as presented in Table 5-1) are calculated

employing the CFD technique. The total-to-static efficiencies of the Preliminary-

Turbo-expander and the Improved-Complete-Turbine at nominal conditions are


presented in Table 7-6. According to Table 7-6, the difference between the two turbine

geometries in terms of total-to-static efficiencies obtained is 4.4%, with a clear benefit

from the new annular-radial diffuser.

Table 7-6: Deterministic total-to-static efficiency 𝜂𝑇−𝑆 for Preliminary-Turbo-

expander and the Improved-Complete-Turbine at nominal conditions.

Preliminary-Turbo-expander Improved-Complete-Turbine

𝜂𝑇−𝑆 0.8655 0.9039

The Mach numbers at mid-span for the Preliminary-Turbo-expander and

Improved-Complete-Turbine at nominal conditions are presented in Figure 7-9 and

Figure 7-10 respectively. As shown in Figure 7-9 and Figure 7-10, the peak value of

Mach numbers for the Preliminary-Turbo-expander is 1.05, which is approximately

2% higher than for the Improved-Complete-Turbine (Mach number = 1.03). It is

noteworthy that the highest Mach number region is the exit of the stator, and the

Preliminary-Turbo-expander will present a higher risk of choked conditions compared

to the Improved-Complete-Turbine. In addition, at the exit of the Improved-Complete-

Turbine, the Mach number is low, which indicates this newly designed annular-radial

diffuser has the capability to convert high velocity speed (out of rotor) into low

velocity speed. The suitable diffuser shows a significant improvement in the

Improved-Complete-Turbine performance.

Figure 7-9: Mach number at mid-span for the Preliminary-Turbo-expander at

nominal conditions.


Figure 7-10: Mach number at mid-span for the Improved-Complete-Turbine at

nominal conditions.

The rotor blade is the key component to be investigated in terms of the flow

characteristics. Two significant flow characteristics of the rotor blade are the Isentropic

Mach number and the skin friction coefficient, in particular in the crucial region close

the leading edge (LE). The Isentropic Mach number profiles and the skin friction

coefficient (𝐶𝑓) at middle span along the rotor blade obtained from the two different

geometries, Preliminary-Turbo-expander and the Improved-Complete-Turbine at

nominal conditions, are plotted at both the Suction Side (SS) and Pressure Side (PS)

along the non-dimensional meridional coordinate in Figure 7-11. Closer views of the

black dash box in Figure 7-11 are shown in Figure 7-12.

Figure 7-11: Isentropic Mach number profile at middle span of rotor blade for

Preliminary-Turbo-expander and the Improved-Complete-Turbine at nominal

conditions.


Figure 7-12: Closer view of Figure 7-11 for Streamwise from 0-0.07.

Figure 7-11 shows that the trends of the Isentropic Mach number and skin

friction coefficient distributions are quite close for both rotor blades, especially at the

PS. However, we can see in Figure 7-12, at a closer view of the black dash box of

Figure 7-11, that the Mach number and skin friction coefficient around the blade LE

at the suction side (SS) show slightly larger fluctuations, which may lead to lower

efficiency and decrease the robustness of the turbine while operating at off-design

conditions.

According to the deterministic study for these two turbines, the Improved-

Complete-Turbine shows a more uniform distribution of the Mach numbers than the

Preliminary-Turbo-expander does. Further Uncertainty Quantification analysis for

these two turbines will be investigated in the following sections.

7.6 STOCHASTIC ANALYSIS OF PRELIMINARY AND IMPROVED ORC

TURBINES UNDER OPERATIONAL UNCERTAINTIES

In this section, the gPC algorithm previously described in Section 3.2 is

employed in order to quantify the variability of the turbine performance under

uncertain operational conditions for both the Preliminary-Turbo-expander and the

Improved-Complete-Turbine.


The coupled uncertainties of rotational speed, inlet total temperature, and mass

flow rate (𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚), and the support range for these uncertain parameters,

are shown in Table 7-7. They correspond to the first set of uncertain parameters

presented in Table 5-3. The input random parameters of bounded supports were


outlined, and a uniform distribution law was utilised to propagate these uncertain

parameters.

Table 7-7: Mean and support values of the uniformly distributed uncertain

parameters.

Uncertain Parameter Symbol Unit Mean Value Supports

Rotational speed 𝑅𝑃𝑀 𝑟𝑝𝑚 24250 [23765 -24735]

Total inlet temperature 𝑇𝑇𝑖𝑛 𝐾 413 [404-421]

Mass flow rate ��𝑚 𝑘𝑔. 𝑠−1 17.24 [16.375-18.1]

Order of the Polynomial Representation


Legendre polynomials are selected as an optimal basis in the spectral representation of

the Polynomial Chaos expansion (Equation (3-20)). In order to illustrate the UQ

framework, Figure 7-13 demonstrates the grid of quadrature points for these three

coupled uncertain parameters employed for P = 5, 7, and 9, where the deterministic

CFD solver must be used to calculate the flow solution for each Legendre quadrature

point.

Figure 7-13: Legendre quadrature points for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 for 𝑃 = 5, 7, 9.



coefficient 𝜂𝑇−𝑆 obtained for Improved-Complete-Turbine in the presence of

uncertainties for for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 for P = 5, 7, 9.

Figure 7-14 presents the probability density function distribution of the total-to-

static efficiency 𝜂𝑇−𝑆 . As observed in Figure 7-14, no major differences in the

distributions of 𝜂𝑇−𝑆 are visible when the polynomial order P employed in gPC

expansion is greater than P = 7, which is a suitable trade-off to balance accuracy and

computational burden, and will be thus employed for all the stochastic calculations in

this analysis.

Stochastic Analysis of Two Turbines

The mean (𝜇), standard deviation (𝜎), and CoV of the total-to-static efficiency

for the uncertain parameters are presented in Table 7-8. The mean of the efficiency for

the Improved-Complete-Turbine is approximately 4.4% higher than for the

Preliminary-Turbo-expander. Furthermore, the CoV for the Improved-Complete-

Turbine is approximately 21.8% lower than for the Preliminary-Turbo-expander. The

performance of the Improved-Complete-Turbine shows a large improvement and thus

can contribute to the overall robustness of the full turbine.


Table 7-8: Mean, standard deviation, and CoV of the 𝜂𝑇−𝑆 for (𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛−

��𝑚) for Preliminary-Turbo-expander and Improved-Complete-Turbine with P = 7.

𝜇 𝜎 × 10−3 𝐶𝑜𝑉 × 10−3

Preliminary-Turbo-expander 0.86571 3.2906 3.8011

Improved-Complete-Turbine 0.90393 2.6872 2.9729

Figure 7-15 presents the mean value and the standard deviation of the Isentropic

Mach number associated with 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 for Preliminary-Turbo-expander

and Improved-Complete-Turbine.

Figure 7-15: (a) The mean and standard deviation of Isentropic Mach number at

middle span of rotor blade for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 for both turbines with P = 7. (b)

Closer View of (a) for Streamwise 0.015-0.06.

According to Figure 7-15(a), the variations of Isentropic Mach number along the

rotor blade show no significant difference between these two rotor blade profiles. It

can also be observed that the largest standard deviation occurs at SS around LE for

both rotor blades. This shows that the most sensitive region for the Isentropic Mach

number happens in this region, as demonstrated in Figure 7-15 (b), which may be

affected by the Stator upstream flow. Furthermore, at approximately 0.032 streamwise

location, the maximum standard deviation for the Preliminary-Turbo-expander is

about 11% higher than for the Improved-Complete-Turbine, as shown in Figure

7-15(b). This result shows that the Isentropic Mach number may be more sensitive to

these coupled uncertain parameters when obtained for the Preliminary-Turbo-

expander.


Figure 7-16 presents the profiles of the mean and standard deviation of the skin

friction coefficient 𝐶𝑓 along the rotor blades for these two turbines. The closer view,

as shown in Figure 7-16(b), demonstrates that the 𝐶𝑓 at approximately 0.032

streamwise location, the maximum standard deviation for the Preliminary-Turbo-

expander, is about 12% higher than for the Improved-Complete-Turbine. These results

have good agreements with the Mach number results in Figure 7-15.

Figure 7-16: (a) The skin friction coefficient (𝐶𝑓) profile at middle span of

rotor blade with 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 for both turbines with P = 7. (b) Closer view of

(a) for Streamwise 0.015-0.075.

In addition to the CoV for uncertainty quantification analysis, the sensitivity

analysis employing Sobol’s indices is applied to the turbine efficiency from each

source of these uncertain parameters, with the purpose of identifying the main

contributors to the first order and the second order variance with the gPC technique

(Tang, et al., 2010). The first order and the second order Sobol’s indices, indicating

the contribution of uncertain variables to the variability of the turbine performances

by means of total-to-static efficiencies, are presented in Figure 7-17 and Figure 7-18

respectively.

Based on Figure 7-17, it is clear that 𝑇𝑇𝑖𝑛is the most important uncertain

parameter for these two turbines. The first order Sobol’s indices regarding 𝑇𝑇𝑖𝑛 are

more important for the Preliminary-Turbo-expander than for the Improved-Complete-

Turbine.


Figure 7-17: The 1st order of Sobol’s indices of each uncertain parameter

contribution of 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 for 𝜂𝑇−𝑆 with P = 7 for both turbines.

Based on Figure 7-17, it is clear that 𝑇𝑇𝑖𝑛 has a much stronger influence for the

Preliminary-Turbo-expander than for the Improved-Complete-Turbine. Small

variations of 𝑇𝑇𝑖𝑛 lead to large enthalpy variance of high-density fluids (Poling,

Prausnitz, John Paul, et al., 2001) and so significantly affect the turbine’s efficiency.

By decreasing the 𝑇𝑇𝑖𝑛 contribution, ��𝑚 and RPM contributions from the Improved-

Complete-Turbine have slightly increased. As 𝑇𝑇𝑖𝑛 is the most critical parameter to

affect enthalpy and turbine efficiency, based on the investigation in Chapter 5, then

the Improved-Complete-Turbine promises to reduce the influence of this critical

parameter while the other two parameters do not affect the efficiency significantly.


Figure 7-18: The 2nd order of Sobol’s indices of each uncertain parameter

contribution of 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 for 𝜂𝑇−𝑆 with P = 7 for both turbines.

For the second order Sobol’s indices, as shown in Figure 7-18, 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛are

more important coupled parameters than the other two. The Preliminary-Turbo-

expander illustrates the strongest influence on turbine efficiency, while 𝑅𝑃𝑀 −

𝑇𝑇𝑖𝑛has a smaller contribution to the efficiency of the Improved-Complete-Turbine.

While the contributions of 𝑇𝑇𝑖𝑛− ��𝑚 and 𝑃𝑀 − ��𝑚 to the Improved-Complete-

Turbine increase in comparison to the Preliminary-Turbo-expander, the scale of the

second order Sobol indices is small enough to not significantly affect the turbine’s

efficiency. As such, the respective contributions of the coupled uncertain parameters

are not of critical importance for the turbine’s performance and sensitivity.

The Probability Density function (PDF), corresponding to the uncertain

parameters (𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚 ) which show the variability of the global interest

parameter, efficiency, for the Preliminary-Turbo-expander and Improved-Complete-

Turbine, is presented in Figure 7-19. A difference is observed for the most probable

values where the peak of the Preliminary-Turbo-expander is around 𝜂𝑇−𝑆 = 0.87,

while the high probability values for the Improved-Complete-Turbine are

approximately obtained as 0.905. It is noteworthy that the shape of the PDF for the


Preliminary-Turbo-expander is similar to the Improved-Complete-Turbine; however,

its support is shifted to the lower values of 𝜂𝑇−𝑆.

Figure 7-19: PDF for 𝜂𝑇−𝑆 for 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚uncertain parameters with P

= 7 for both turbines.

In order to better visualise the stochastic distribution of the Mach number along

the rotor blade, the mean value, standard deviation, variance, and CoV contour plots

for the Preliminary-Turbo-expander and the Improved-Complete-Turbine are

presented in Figure 7-20.


Figure 7-20: UQ analysis of Mach number for Standard Deviation (a) – (b), for

variance (c) – (d), and for CoV (e) – (f) for Preliminary-Turbo-expander and

Improved-Complete-Turbine respectively.

Based on Figure 7-20(a) and (b), the standard deviation distribution in the

magenta dash box for the rotor blade for the Improved-Complete-Turbine is much

lower than for the Preliminary-Turbo-expander, in particular on the SS along the rotor

blade. The standard deviation at the Trailing Edge is also smaller for the Improved-

Complete-Turbine than for the Preliminary-Turbo-expander. These results show good


agreement with Table 7-8, which also indicates that the Improved-Complete-Turbine

obtained a lower standard deviation in terms of the total-to-static turbine efficiency

than did the Preliminary-Turbo-expander. Similarly, the variance distribution along

the rotor blade in the purple dash box for the Improved-Complete-Turbine is lower

than for the Preliminary-Turbo-expander, as demonstrated in Figure 7-20(c) and (d)

respectively. It is noteworthy that the CoV distribution shows an obvious difference in

the brown dash box between Figure 7-20 (e) and (f). There is a larger region with a

higher CoV value for the Preliminary-Turbo-expander (Figure 7-20(e)) compared to

the Improved-Complete-Turbine in Figure 7-20(f). This result is consistent with the

result in the Table 7-8 for the CoV regarding the total-to-static turbine efficiency. The

Improved-Complete-Turbine shows less sensitivity to the coupled uncertain

parameters 𝑅𝑃𝑀 − 𝑇𝑇𝑖𝑛− ��𝑚, and a more robust response than the Preliminary-

Turbo-expander.

7.7 STOCHASTIC ANALYSIS AT OFF-DESIGN CONDITIONS

As described in Section 5.4, ORC turbines may not always work at full nominal

condition capacity, but instead, may work under off-design conditions because of

discontinuity in the availability of heat resources (Fiaschi, et al., 2015). Uncertainty

quantification analysis under off-design conditions is an important step towards robust

optimisation of turbines, allowing them to efficiently work under variable conditions

that deviate from the design conditions (nominal conditions).

In Section 5.4, it has been shown that a lower rotational speed (80% nominal

RPM) and lower inlet total temperature (𝑇𝑇𝑖𝑛= 400K) can dramatically affect the

turbine’s efficiency. As concluded in Section 5.5, Peng-Robinson is the most robust

Equation of State for these uncertain parameters tested in Chapter 5 and Chapter 7.

Thus, for the purpose of evaluating the sensitivity of Peng-Robinson through 𝑃𝑐 − 𝑇𝑐

under this off-design condition (80% nominal RPM; 𝑇𝑇𝑖𝑛= 400K), Uncertainty

Quantification analysis is conducted for the Improved-Complete-Turbine, and

compared against the Uncertainty Quantification analysis of the Preliminary-Turbo-

expander employed in Chapter 5. The uniformly distributed 𝑃𝑐 − 𝑇𝑐 uncertain

parameters range is the same as for the range utilised in the previous study (see Section

5.3.3).


The mean values of the total-to-static efficiency, 𝜂𝑇−𝑆 for the Improved-

Complete-Turbine and Preliminary-Turbo-expander under 80% nominal RPM, 𝑇𝑇𝑖𝑛=

400K are presented in Table 7-9 using the Peng-Robinson (PR) Equation of State

(EOS).

Table 7-9: The mean values of 𝜂𝑇−𝑆 under 80% nominal RPM and 𝑇𝑇𝑖𝑛=

400K with 𝑃𝑐 − 𝑇𝑐 uncertain parameters by PR with P = 7 (based on uniform

distribution laws).

𝝁 𝝈 × 𝟏𝟎−𝟑 𝑪𝒐𝑽 × 𝟏𝟎−𝟑

Preliminary-Turbo-expander 0.8224 3.4443 4.1882

Improved-Complete-Turbine 0.8668 2.8729 3.3143

According to Table 7-9, the mean of the efficiency for the Improved-Complete-

Turbine is approximately 5.4% higher than for the Preliminary-Turbo-expander,

thanks to the improvements in pressure recovery of the radial-annular diffuser. In

addition, the CoV for the Improved-Complete-Turbine is about 20.9% lower than for

the Preliminary-Turbo-expander. The Improved-Complete-Turbine presents a better

and more robust performance at lower rotational speed and lower inlet temperature.

The influences of 𝑃𝑐 − 𝑇𝑐 at off-design conditions for both Preliminary-Turbo-

expander and Improved-Complete-Turbine are investigated. For the first order Sobol’s

indices of the coupled parameters 𝑃𝑐 − 𝑇𝑐, as presented in Figure 7-21, 𝑃𝑐 is the

dominating factor compared to 𝑇𝑐 for both turbines. The two turbines show no distinct

change for the first order Sobol’s indices.


Figure 7-21: The 1st order Sobol’s indices at off-design conditions with 𝑃𝑐 −

𝑇𝑐uncertain parameters with P = 7 for two turbines.

Figure 7-22: The 2nd order Sobol’s indices at off-design conditions with 𝑃𝑐 −

𝑇𝑐 uncertain parameters with P = 7 for two turbines.

Based on Figure 7-22, for the second order Sobol’s indices, the Preliminary-

Turbo-expander demonstrates a stronger influence on efficiency, while 𝑅𝑃𝑀 −

𝑇𝑇𝑖𝑛has a weaker contribution to the efficiency of the Improved-Complete-Turbine.


Figure 7-23 presents the PDF based on efficiency with 𝑃𝑐 − 𝑇𝑐 uncertain

parameters at 𝑇𝑇𝑖𝑛 = 400K and 80% nominal RPM off-design conditions for the

Preliminary-Turbo-expander and Improved-Complete-Turbine.

Figure 7-23: PDF for 𝜂𝑇−𝑆 with 𝑃𝑐 − 𝑇𝑐 uncertain parameters at 𝑇𝑇𝑖𝑛= 400K

and 80% nominal RPM off-design conditions with P = 7 for two turbines.

It is noteworthy that at 𝑇𝑇𝑖𝑛= 400K and 80% nominal RPM in Figure 7-23, the

PDF response for the Preliminary-Turbo-expander shows slightly more oscillation

compared to the Improved-Complete-Turbine, which shows there is a bit more non-

linear response for the Preliminary-Turbo-expander than for the Improved-Complete-

Turbine. A difference is clear for the most probable values where the peak of

Preliminary-Turbo-expander is around 𝜂𝑇−𝑆 = 0.825, while the high probability

values for the Improved-Complete-Turbine are about 0.865. The support for

Improved-Complete-Turbine is shifted to higher values of 𝜂𝑇−𝑆.

In order to further understand the uncertain behaviour, it is interesting to

investigate the stochastic variation of the 𝐶𝑓 along the rotor blade and the velocity

vectors at 𝑇𝑇𝑖𝑛 = 400K and 80% nominal RPM for these two turbines.



𝑃𝑐 − 𝑇𝑐 at 𝑇𝑇𝑖𝑛 = 400K and 80% nominal RPM for two turbines.

According to Figure 7-24, at approximately 0.3 streamwise location, the

standard deviations for the Preliminary-Turbo-expander are 38.81% higher than for

the Improved-Complete-Turbine. At the same time, at approximately 0.42 streamwise

location, the standard deviation for the Preliminary-Turbo-expander is much closer to

zero compared to the Improved-Complete-Turbine. The risk of generating separation

in this region for the Preliminary-Turbo-expander is higher than for the Improved-

Complete-Turbine. At low inlet total temperature and low rotational speed (𝑇𝑇𝑖𝑛 =

400K and 80% RPM), the Preliminary-Turbo-expander shows higher sensitivity

to𝑃𝑐 − 𝑇𝑐, while the Improved-Complete-Turbine seems slightly more robust to these

coupled parameters.

7.8 CONCLUSIONS

Comparing the total-to-static efficiency of the whole turbine fitted with two

diffusers, the annular-radial diffuser design produces favourable results compared to

the conical diffuser. The total-to-static efficiency and pressure recovery obtained with

the annular-radial diffuser are approximately 3.4% and 3.7% higher than the conical

diffuser, respectively.


Based on the preliminary designs of these two types of diffusers, the annular-

radial diffuser seems slightly more favourable than the conical diffuser to fit our

current R143a radial-inflow turbo expander. The appropriate choice of diffuser is

critical to improving the whole turbine’s efficiency.

The Uncertainty Quantification and sensitivity analysis for the uncertain

operational conditions have been conducted for the Improved-Complete-Turbine,

which includes the newly designed annular-radial diffuser fitted to the Preliminary-

Turbo-expander. The Uncertainty Quantification analysis results show that the inlet

temperature for the Preliminary-Turbo-expander play a more dominant role than for

the Improved-Complete-Turbine. The coupled uncertain inlet temperature and

rotational speed is more significant for the Preliminary-Turbo-expander than for the

Improved-Complete-Turbine in their effect on turbine efficiency. Importantly, the

mean value efficiency of the Improved-Complete-Turbine shows 4.4% improvement

compared to the Preliminary-Turbo-expander. In addition, the coupled uncertain

parameters show less coefficient of variation in terms not only of turbine efficiency

but also of the Mach number distribution along the rotor blade profiles for the

Improved-Complete-Turbine compared to the Preliminary-Turbo-expander, including

under off-design conditions (low inlet temperature and low rotational speed). Overall,

the Improved-Complete-Turbine presents a slightly more robust performance

compared to the Preliminary-Turbo-expander, and is a promising avenue to further

improve the robustness and performance of the ORC turbine.

Improving the efficiency of the Improved-Complete-Turbine will increase the

overall ORC efficiency, which is significant to further development of the renewable

energy sector.

Chapter 8: Conclusions and Suggestions 169

Chapter 8: Conclusions and Suggestions

This thesis presents a holistic, comprehensive Uncertainty Quantification

analysis of a complete radial-inflow turbine, including a radial-inflow turbo-expander

and a diffuser. The overarching objective of this research is to establish a

comprehensive understanding of uncertain parameters’ influence on the efficiency of

the radial-inflow turbine. In this chapter, an overall summary and concluding remarks

are provided in Section 8.1. In the following section, the limitations of this research

will be detailed. The chapter will conclude with suggestions and recommendations for

future research.

The conclusions pertaining to the main research studies presented in Chapters 3-

8 are summarised as follows.

8.1 RESEARCH SUMMARY AND CONCLUDING REMARKS

Overall, a systematic Uncertainty Quantification analysis for a radial-inflow

turbo-expander and diffuser is conducted. A complete radial-inflow turbine, including

a turbo-expander and a newly designed annular-radial diffuser, is developed, and

Uncertainty Quantification is applied to this improved turbine. The conclusions to the

chapters of this research are summarised hereafter.

Chapter 3: This chapter details the methodology used to couple the

deterministic computational flow solver to the generalised Polynomial Chaos

approach. The Reynolds-Averaged Navier Stokes (RANS) equations for viscous

compressible flows implemented in the finite volume solver ANSYS-CFX v18.0 were

employed to carry out steady-state three-dimensional simulations for a radial-inflow

turbo-expander, diffuser, and a complete radial-inflow turbine. The solver was

adapted to accommodate for high-density fluid simulations using the REFPROP NIST

database. The generalised Polynomial Chaos is employed as the stochastic method for

the Uncertainty Quantification analysis. A robust framework has been established to

couple the CFD solver with the Uncertainty Quantification method to apply to radial-

inflow turbines for low-grade temperature Organic Rankine Cycle applications.

Chapter 4: The deterministic three-dimensional CFD solver coupled with the

gPC approach is successfully validated and applied to investigate a complete three-


dimensional high-pressure ratio, single stage radial-inflow turbine used in the

Sundstrand Power Systems T-100 Multi-purpose Small Power Unit.

The main conclusions are as follows:

A grid refinement study was performed to satisfy the requirements of

calculation accuracy. The three-dimensional CFD calculations for total-to-

static efficiencies of the ideal gas radial-inflow turbine are validated against

the open data experimental data at the rig conditions, with a maximum

difference of less than 1%.

Eight parameters were studied for the parametric study: inlet temperature,

outlet pressure, mass flow rate, tip clearance, rotational speed, TE

meridional length, TE tip radius, and blade thickness. Only the rotational

speed, TE meridional tip length, TE tip radius, and blade thickness

demonstrate a non-linear surface response, and so were further investigated

using the stochastic approach.

The convergence investigation using generalised Polynomial Chaos has

been carefully checked, showing that the stochastic spectral projection

decreases dramatically with the increase of polynomial order.

The radial-inflow turbine efficiency shows a non-linear response to

rotational speed, TE tip radius, TE tip length, and blade thickness, and has

been evaluated using Uncertainty Quantification analysis. The most

important individual uncertain parameter is blade thickness, followed by the

trailing edge tip radius. The most important two coupled random variables

are the trailing edge tip radius with the rotational speed.

Chapter 5: The Uncertainty Quantification analysis under on- and off-design

operational conditions is conducted using different Equations of State (Peng-

Robinson, Soave-Redlich-Kwong, and Helmholtz Energy Equation) for the R143a

turbine. The stochastic analysis of the operational uncertain parameters — rotational

speed, inlet total temperature, and mass flow rate combined — and the uncertain

parameters from the Equations of State — critical pressure and critical temperature —

is presented employing these different Equations of State. The following conclusions

are drawn:


Based on the Uncertainty Quantification analysis in terms of operational

uncertain parameters (rotational speed, inlet total temperature, and mass

flow rate combined), 𝑇𝑇𝑖𝑛most importantly affects the stochastic response

of the total-to-static efficiency of the radial-inflow turbine.

The Peng-Robinson Equation of State presents the most robust behaviour in

regard to the ORC radial-inflow turbine’s efficiency, while the Soave-

Redlich-Kwong EOS shows the most sensitive characteristics for two

groups of uncertain parameters (rotational speed-inlet total temperature-

mass flow rate; critical pressure-critical temperature). Thus, Soave-Redlich-

Kwong may not be the most appropriate choice for the robust optimisation

of ORC radial-inflow turbines working over a wider range of operational

conditions.

The sensitivity of the radial-inflow turbine’s efficiency to randomness is

more prominent at off-design conditions than at nominal conditions for

coupled critical pressure-critical temperature uncertain parameters.

Chapter 6: Uncertainty Quantification analysis of the high-density fluid conical

diffuser is carried out with uncertain operating conditions, including under nominal

and varied geometric conditions.

The uncertain parameter couple, swirling velocity and inlet velocity, shows

more prominent effect on the pressure recovery under diffuser length =

610mm and diffuser half-cone angle = 8° than for the other geometric

conditions.

Differences between deterministic results and the stochastic mean values of

the velocity u (under diffuser length = 410mm – diffuser half-cone angle =

12°; diffuser length = 510mm – diffuser half-cone angle = 10°; diffuser

length = 610 mm – diffuser half-cone angle = 8°; diffuser length = 610mm

– diffuser half-cone angle = 12°) were owing to local non-linear effects in

the recirculation regions. Thus, it is critical in these cases to denote

statistical results to identify the most probable values of the velocity u as a

result of the uncertain parameters.

If there is large centreline recirculation region, the coupled uncertain

parameters of swirling velocity with inlet velocity do not much affect the


conical diffuser pressure recovery, which is explained by the fact that the

recirculation is already too large to be affected by small variations from

these uncertainties.

The highest sensitivity from coupled uncertain parameters of inlet velocity

with swirling velocity for the cases are near the recirculation region, which

is the interface of positive-negative velocity u. This highlights that when the

diffusers are close to recirculation, the high-density conical diffuser design

may be more sensitive, which may be a problem for the robustness of a

complete ORC turbine.

Chapter 7: In order to establish the most appropriate diffuser design for an ORC

turbine, a numerical comparison of the total-to-static efficiency for a whole turbine

fitted with two different diffusers has been carried out. The Uncertainty Quantification

analysis with coupled uncertainty parameters (rotational speed, inlet total temperature,

and mass flow rate combined) for the completed radial-inflow turbine, including

radial-inflow turbo-expander and annular-radial diffuser, was then performed, and the

results compared to the preliminary turbine to highlight the performance and

robustness improvements.

The results show that the high-density annular-radial diffuser design

performs better than the conical diffuser. The pressure recovery of the

annular-radial diffuser is 3.7% higher than the conical diffuser.

According to the preliminary designs of these two types of diffusers, the

annular-radial diffuser is more suitable than the conical diffuser to fit our

current R143a radial-inflow turbo-expander.

The Uncertainty Quantification analysis for the coupled uncertain

operational conditions have been carried out regarding the Improved-

Complete-Turbine, which includes the newly designed annular-radial

diffuser connected with the Preliminary-Turbo-expander.

The mean value efficiency of the Improved-Complete-Turbine is 4.4%

higher than for the Preliminary-Turbo-expander. This increase could

improve the overall efficiency of the ORC cycle by approximately 2%,

referring to the experimental investigation employing n-Hexane as working

fluid (Pethurajan & Sivan, 2018).


The coupled uncertain parameters present a lower coefficient of variation

for both the turbine efficiency and the Mach number distribution along the

rotor blade profiles for the Improved-Complete-Turbine compared to the

Preliminary-Turbo-expander for both nominal and off-design conditions.

In summary, the Improved-Complete-Turbine shows a more robust

performance than the Preliminary-Turbo-expander, which is critical to

improving the overall Organic Rankine Cycle efficiency.

8.2 RESEARCH LIMITATIONS

Although this study has provided a comprehensive characterisation of the radial-

inflow turbine based on Uncertainty Quantification analysis, there are still several

limitations that need further work.

Lack of experiments: The Improved-Complete-Turbine should be validated

against experimental investigations. A completed turbine needs to be built,

including the turbo-expander and the annular-radial diffuser. Both the

nominal and off-design conditions should be compared between the

experimental data and the computational results.

Computational cost: The case studies presented in this thesis are based on

numerical Computational Fluid Dynamics calculation and Uncertainty

Quantification analysis as a mathematical method. More advanced

Uncertainty Quantification approach such as the Stochastic Collocation

method using sparse grids will improve work efficiency and save the

computational costs.

Working fluid selection: In this study, one high-density working fluid,

R143a, is investigated. It will be interesting to investigate other high-density

working fluids such as R134a and CO2, which are commonly encountered

in renewable energy applications, and check that the sensitivity analysis

outcomes obtained for R143a still hold for those various high-density fluids.

Structural considerations: This work has addressed the important point of

the aerodynamic performance of the ORC radial-inflow turbine. However,

this work should be coupled with the investigation of the structural

components to make sure the turbine is mechanically robust as well. Finite


Element Analysis (FEA) could be coupled to the current UQ-CFD approach,

and a full uncertainty quantification could be carried out on both fluid and

structural parameters.

All of the numerical study were considering for subsonic regime, however

shocks appeared when the turbine was not working at nominal conditions

but at off-design conditions for some Uncertainty Quantification cases, in

particular, at the outlet of nozzles with the Mach number higher than 1.1.

However, we focused the study on the blades in the rotor in this thesis and

have not investigated in details the effect of these shocks onto the turbine

performance. However, it will be needed consideration in future work.

8.3 DIRECTIONS FOR FUTURE RESEARCH

This study presents a comprehensive Uncertainty Quantification analysis for an

Organic Rankine Cycle turbine in renewable energy applications. This would serve as

a steppingstone for conducting additional research in the foreseeable future in order to

produce more robust Organic Rankine Cycle turbines.

A more advanced stochastic approach, such as stochastic collocation with

sparse grids, will be needed for the radial-inflow turbine Uncertainty

Analysis to save computational cost. These stochastic tools would support

the performance of robust optimisation of the ORC turbines, in the sense

that the design of the turbine should give, in the presence of uncertainty,

robust levels of aerodynamic performance.

Structural analysis using, for example, Finite Element Analysis, should be

considered and coupled with Computational Fluid Dynamics

simultaneously, so that extensive Uncertainty Quantification analysis for

radial-inflow turbines can be conducted, and reliable and robust turbine

design established.

It would be interesting to investigate the tested Equations of State for other

high-density fluids commonly suggested in ORC applications, such as

R134a and CO2. The sensitivity analysis for different high-density fluids

employing different Equations of State will help to better understand how

the compressible fluid properties affect the turbine’s efficiency.


Considering the gPC method needs to couple with the CFD solver, different

turbulence models need to be considered, however this research problem is

out of the scope in this thesis. Uncertainty Quantification analysis of

different turbulence models would further improve our understanding for

robust design of turbomachinery and will be conducted in future work. In

this thesis, the computational work is based on the steady state assumption.

The transient state study will be considered under off-design condition in

future work.

The proposed UQ-CFD technique is crucial to developing more robust complete

radial-inflow turbines. It will greatly benefit low-grade temperature Organic Rankine

Cycle application, and will therefore have progressive influence on the renewable

energy sectors.


Bibliography

Abdmouleh, Z., Alammari, R. A., & Gastli, A. (2015). Review of policies encouraging

renewable energy integration & best practices. Renewable and Sustainable

Energy Reviews, 45, 249-262.

Abir, A., & Whitfield, A. (1987). Stability of Conical and Curved Annular Diffusers

for Mixed-Flow Compressors. In ASME 1987 International Gas Turbine

Conference and Exhibition (pp. V001T001A066-V001T001A066): American

Society of Mechanical Engineers.

Abramowitz, M., & Stegun, I. (1966). Handbook of mathematical functions. American

Journal of Physics, 34(2), 177-177.

Agency, I. E. (2012). Energy Technology Perspectives 2014: Harnessing Electricity's

Potential: OECD Publishing.

Agrawal, A., Cornelio, A. A., & Limperich, D. (2012). Investigation of Cubic EOS

Models for HFO-1234yf Refrigerant Used In Automotive application.

Al-Sulaiman, F. A., Dincer, I., & Hamdullahpur, F. (2010). Exergy analysis of an

integrated solid oxide fuel cell and organic Rankine cycle for cooling, heating

and power production. Journal of Power Sources, 195(8), 2346-2354.

Al Jubori, A. M., Al-Dadah, R. K., Mahmoud, S., & Daabo, A. (2017). Modelling and

parametric analysis of small-scale axial and radial-outflow turbines for Organic

Rankine Cycle applications. Applied energy, 190, 981-996.

Alshammari, F., Pesyridis, A., Karvountzis-Kontakiotis, A., Franchetti, B., &

Pesmazoglou, Y. (2018). Experimental study of a small scale organic Rankine

cycle waste heat recovery system for a heavy duty diesel engine with focus on

the radial inflow turbine expander performance. Applied Energy, 215, 543-555.

ANSYS 18.0 CFX-Solver Theory guide. On. (2017).

ANSYS® Academic Research CFX Guidance, Release 18. On. (2017).

Armfield, S., Cho, N., & Fletcher, C. A. (1990). Prediction of turbulence quantities for

swirling flow in conical diffusers. AIAA journal, 28(3), 453-460.

Armfield, S., & Fletcher, C. (1989). Comparison of k–ϵ and algebraic Reynolds stress

models for swirling diffuser flow. International Journal for Numerical

Methods in Fluids, 9(8), 987-1009.

Askey, R., & Wilson, J. (1985). Some basic hypergeometric polynomials that

generalize Jacobi polynomials Memoirs Amer. Math. Soc. AMS Providence RI,

319.


ASME PTC 22 (2014). Elsevier.

Aungier, R. (1995). A fast, accurate real gas equation of state for fluid dynamic

analysis applications. Journal of fluids engineering, 117(2), 277-281.

Aungier, R. H. (2005). Turbine Aerodynamics. New York: ASME.

Azad, R. S. (1996). Turbulent flow in a conical diffuser: A review. Experimental

Thermal and Fluid Science, 13(4), 318-337.

Azad, R. S., & Kassab, S. Z. (1989). Turbulent flow in a conical diffuser: Overview

and implications. Physics of Fluids A: Fluid Dynamics, 1(3), 564-564.

Badr, O., O'callaghan, P., Hussein, M., & Probert, S. (1984). Multi-vane expanders as

prime movers for low-grade energy organic Rankine-cycle engines. Applied

Energy, 16(2), 129-146.

Baghdadi, S., & McDonald, A. (1975). Performance of three vaned radial diffusers

with swirling transonic flow. Journal of Fluids Engineering, 97(2), 155-160.

Baines, N. C. (2003). Radial Turbine Design. In H. Moustapha (Ed.), (pp. 199-327).

White River Junction, VT: Concepts NREC

Bao, J., & Zhao, L. (2013). A review of working fluid and expander selections for

organic Rankine cycle. Renewable and Sustainable Energy Reviews, 24, 325-

342.

Barret, M., & Candau, Y. (1992). Thermodynamic Properties Computation of Two

Possible Substitute Refrigerants: R143a (1, 1, 1-trifluoroethane) and R125

(pentafluoroethane).

Barrick, M. W., Anderson, J. M., & Robinson Jr, R. L. (1986). Solubilities of carbon

dioxide in cyclohexane and trans-decalin at pressures to 10.7 MPa and

temperatues from 323 to 423 K. Journal of Chemical and Engineering Data,

31(2), 172-175.

Barth, T., & Jespersen, D. (1989). The design and application of upwind schemes on

unstructured meshes. In 27th Aerospace sciences meeting (pp. 366).

Baskharone, E. A. (2006). Principles of turbomachinery in air-breathing engines:

Cambridge University Press.

Bini, R., Guercio, A., & Duvia, A. (2009). Organic Rankine Cycle (ORC) in biomass

applications for cogenerative systems in association with adsorption chillers.

In Proceedings of 5th Dubrovnik Conference on Sustainable Development of

Energy, Water and Environment Systems, Dubrovnik, Croatia, Sept.

Bohi, D. R. (2013). Analyzing demand behavior: a study of energy elasticities: RFF

Press.


Boncinelli, P., Rubechini, F., Arnone, A., Cecconi, M., & Carlo Cortese. (2004). Real

gas effects in turbomachinery flows: A computational fluid dynamics model

for fast computations. Journal of Turbomachinery, 126, 268-276.

Borges, J. E. (1990). A Three-Dimensional Inverse Method for Turbomachinery: Part

II—Experimental Verification. Journal of Turbomachinery, 112(3), 355-361.

Bounous, O. (2008). Studies of the ERCOFTAC Conical Diffuser with OpenFOAM.

Branchini, L., De Pascale, A., & Peretto, A. (2013). Systematic comparison of ORC

configurations by means of comprehensive performance indexes. Applied

Thermal Engineering, 61(2), 129-140.

Bufi, E. A., Cinnella, P., & Merle, X. (2015). Sensitivity of supersonic orc turbine

injector designs to fluctuating operating conditions. In Proceedings of the

ASME 2015 Turbo Expo Turbine Technical Conference, ASME TURBO EXPO

2015 June 15-19, 2015, Montreal, Canada.

Cacuci, D. G., Ionescu-Bujor, M., & Navon, I. M. (2004). Sensitivity and Uncertainty

Analysis: Applications to large-scale systems (Vol. 2): CRC Press.

Calise, F., Capuozzo, C., Carotenuto, A., & Vanoli, L. (2014). Thermoeconomic

analysis and off-design performance of an organic Rankine cycle powered by

medium-temperature heat sources. Solar Energy, 103, 595-609.

Campana, F., Bianchi, M., Branchini, L., De Pascale, A., Peretto, A., Baresi, M.,

Fermi, A., Rossetti, N., & Vescovo, R. (2013). ORC waste heat recovery in

European energy intensive industries: Energy and GHG savings. Energy

Conversion and Management, 76, 244-252.

Canuto, C., Hussaini, M. Y., Quarteroni, A., & Thomas Jr, A. (2012). Spectral methods

in fluid dynamics: Springer Science & Business Media.

Chacartegui, R., Sánchez, D., Muñoz, J. M., & Sánchez, T. (2009). Alternative ORC

bottoming cycles FOR combined cycle power plants. Applied Energy, 86(10),

2162-2170.

Chassaing, J.-C., & Lucor, D. (2010). Stochastic investigation of flows about airfoils

at transonic speeds. AIAA journal, 48(5), 938-950.

Chen, Q., Xu, J., & Chen, H. (2012). A new design method for Organic Rankine Cycles

with constraint of inlet and outlet heat carrier fluid temperatures coupling with

the heat source. Applied Energy, 98, 562-573.

Cho, N.-H. (1990). Computation of turbulent swirling flows in conical diffusers with

tailpipes.

Cho, S.-Y., Cho, C.-H., Ahn, K.-Y., & Lee, Y. D. (2014). A study of the optimal

operating conditions in the organic Rankine cycle using a turbo-expander for

fluctuations of the available thermal energy. Energy, 64, 900-911.


Cinnella, P., & Congedo, P. (2004). A Numerical Solver for Dense Gas Flows. Paper

presented at 34th AIAA Fluid Dynamics Conference and Exhibit,

Cinnella, P., & Congedo, P. M. (2005). Numerical solver for dense gas flows. AIAA

journal, 43(11), 2458-2461.

Cinnella, P., & Hercus, S. (2010). Robust optimization of dense gas flows under

uncertain operating conditions. Computers & Fluids, 39(10), 1893-1908.

Cinnella, P., Marco Congedo, P., Pediroda, V., & Parussini, L. (2011). Sensitivity

analysis of dense gas flow simulations to thermodynamic uncertainties.

Physics of Fluids, 23(11), 116101.

Clausen, P. D., Koh, S. G., & Wood, D. H. (1993). Measurements of a swirling

turbulent boundary layer developing in a conical diffuser. Experimental

Thermal and Fluid Science, 6(1), 39-48.

Colonna, P., Rebay, S., Harinck, J., & Guardone, A. (2006). Real-gas effects in ORC

turbine flow simulations: influence of thermodynamic models on flow fields

and performance parameters. In ECCOMAS CFD 2006: Proceedings of the

European Conference on Computational Fluid Dynamics, Egmond aan Zee,

The Netherlands, September 5-8, 2006: Delft University of Technology;

European Community on Computational Methods in Applied Sciences

(ECCOMAS).

Congedo, P. M., Colonna, P., Corre, C., Witteveen, J., & Iaccarino, G. (2010). Robust

simulation of nonclassical gas-dynamics phenomena. In Proceedings of the

Summer Program (pp. 27).

Congedo, P. M., Corre, C., & Cinnella, P. (2011). Numerical investigation of dense-

gas effects in turbomachinery. Computers & Fluids, 49(1), 290-301.

Congedo, P. M., Corre, C., & Martinez, J.-M. (2011). Shape optimization of an airfoil

in a BZT flow with multiple-source uncertainties. Computer Methods in

Applied Mechanics and Engineering, 200(1), 216-232.

Costall, A., Hernandez, A. G., Newton, P., & Martinez-Botas, R. (2015). Design

methodology for radial turbo expanders in mobile organic Rankine cycle

applications. Applied Energy, 157, 729-743.

Da Lio, L., Manente, G., & Lazzaretto, A. (2017). A mean-line model to predict the

design efficiency of radial inflow turbines in organic Rankine cycle (ORC)

systems. Applied Energy, 205, 187-209.

Dambach, R., Hodson, H. P., & Huntsman, I. (1999). An Experimental Study of Tip

Clearance Flow in a Radial Inflow Turbine. Journal of Turbomachinery,

121(4), 644-650.


Daroczy, L., Janiga, G., & Thevenin, D. (2016). Analysis of the performance of a H-

Darrieus rotor under uncertainty using Polynomial Chaos Expansion. Energy,

113, 399-412.

Delgado-Torres, A. M., & García-Rodríguez, L. (2010). Analysis and optimization of

the low-temperature solar organic Rankine cycle (ORC). Energy Conversion

and Management, 51(12), 2846-2856.

Deligant, M., Sauret, E., Danel, Q., & Bakir, F. (2018). Performance assessment of a

standard radial turbine as turbo expander for an adapted solar concentration

ORC. Renewable Energy.

DiPippo, R. (2012). Geothermal power plants: principles, applications, case studies

and environmental impact: Butterworth-Heinemann.

Dolz, V., Novella, R., García, A., & Sánchez, J. (2012). HD Diesel engine equipped

with a bottoming Rankine cycle as a waste heat recovery system. Part 1: Study

and analysis of the waste heat energy. Applied Thermal Engineering, 36, 269-

278.

Dong, B., Xu, G., Li, T., Quan, Y., Zhai, L., & Wen, J. (2018). Numerical prediction

of velocity coefficient for a radial-inflow turbine stator using R123 as working

fluid. Applied Thermal Engineering, 130, 1256-1265.

Dong, B., Xu, G., Luo, X., Zhuang, L., & Quan, Y. (2017). Analysis of the supercritical

organic Rankine cycle and the radial turbine design for high temperature

applications. Applied Thermal Engineering, 123, 1523-1530.

Dunham, J., & Panton, J. (1973). Experiments on the design of a small axial turbine.

In Conference on Heat and Fluid Flow in Steam and Gas Turbine Plant,

Coventry, England (pp. 56-65).

Ebaid, M. S. Y., Bhinder, F. S., & Khdairi, G. H. (2003). A Unified Approach for

Designing a Radial Flow Gas Turbine. Journal of Turbomachinery, 125(3),

598-606.

Eldred, M., & Burkardt, J. (2009). Comparison of non-intrusive polynomial chaos and

stochastic collocation methods for uncertainty quantification. AIAA paper,

976(2009), 1-20.

Erbaş, M., & Bıyıkoğlu, A. (2015). Design and multi-objective optimization of organic

Rankine turbine. International Journal of Hydrogen Energy, 40(44), 15343-

15351.

Faragher, J. (2004). Probabilistic Methods for the Quantification of Uncertainty and

Error in Computational Fluid Dynamic Simulations.

Fiaschi, D., Innocenti, G., Manfrida, G., & Maraschiello, F. (2016). Design of micro

radial turboexpanders for ORC power cycles: From 0D to 3D. Applied Thermal

Engineering, 99, 402-410.


Fiaschi, D., Manfrida, G., & Maraschiello, F. (2012). Thermo-fluid dynamics

preliminary design of turbo-expanders for ORC cycles. Applied Energy, 97,

601-608.

Fiaschi, D., Manfrida, G., & Maraschiello, F. (2015). Design and performance

prediction of radial ORC turboexpanders. Applied Energy, 138, 517-532.

Fishman, G. (2013). Monte Carlo: concepts, algorithms, and applications: Springer

Science & Business Media.

Fox, R. W., McDonald, A. T., & Va, R. R. V. (1971). Effects of swirling inlet flow on

pressure recovery in conical diffusers. AIAA Journal, 9(10), 2014-2018.

From, C., Sauret, E., Armfield, S., Saha, S., & Gu, Y. (2017). Turbulent dense gas

flow characteristics in swirling conical diffuser. Computers & Fluids, 149,

100-118.

Gatski, T. B., & Speziale, C. G. (2006). On explicit algebraic stress models for

complex turbulent flows. Journal of Fluid Mechanics, 254, 59-78.

Ghanem, R. G., & Spanos, P. D. (1991). Stochastic finite elements: a spectral

approach (Vol. 387974563): Springer.

Ghanem, R. G., & Spanos, P. D. (2003). Stochastic finite elements: a spectral

approach: Courier Corporation.

Hah, C. (1983). Calculation of various diffuser flows with inlet swirl and inlet

distortion effects. AIAA Journal, 21(8), 1127-1133.

Hall, C., & Dixon, S. L. (2013). Fluid mechanics and thermodynamics of

turbomachinery: Butterworth-Heinemann.

Han, W., Chen, Q., Lin, R.-m., & Jin, H.-g. (2015). Assessment of off-design

performance of a small-scale combined cooling and power system using an

alternative operating strategy for gas turbine. Applied Energy, 138, 160-168.

Harinck, J., Colonna, P., Guardone, A., & Rebay, S. (2010). Influence of

thermodynamic models in two-dimensional flow simulations of

turboexpanders. Journal of turbomachinery, 132(1), 011001.

Harinck, J., Pasquale, D., Pecnik, R., van Buijtenen, J., & Colonna, P. (2013).

Performance improvement of a radial organic Rankine cycle turbine by means

of automated computational fluid dynamic design. Proceedings of the

Institution of Mechanical Engineers, Part A: Journal of Power and Energy,

227(6), 637-645.

Harinck, J., Turunen-Saaresti, T., Colonna, P., Rebay, S., & van Buijtenen, J. (2010).

Computational study of a high-expansion ratio radial organic rankine cycle


turbine stator. Journal of Engineering for Gas Turbines and Power, 132(5),

054501.

Heberle, F., & Brüggemann, D. (2010). Exergy based fluid selection for a geothermal

Organic Rankine Cycle for combined heat and power generation. Applied

Thermal Engineering, 30(11-12), 1326-1332.

Hoffren, J. (1997). Adaption of FINFLO for real gases. Helsinki University of

Technology, Laboratory of Applied Thermodynamics, Helsinky, Finnland.

Hoffren, J., Talonpoika, T., Larjola, J., & Siikonen, T. (2002). Numerical simulation

of real-gas flow in a supersonic turbine nozzle ring. Journal of engineering for

gas turbines and power, 124(2), 395-403.

Hu, D., Li, S., Zheng, Y., Wang, J., & Dai, Y. (2015). Preliminary design and off-

design performance analysis of an Organic Rankine Cycle for geothermal

sources. Energy Conversion and Management, 96, 175-187.

Hu, D., Zheng, Y., Wu, Y., Li, S., & Dai, Y. (2015). Off-design performance

comparison of an organic Rankine cycle under different control strategies.

Applied Energy, 156, 268-279.

Hung, T.-C. (2001). Waste heat recovery of organic Rankine cycle using dry fluids.

Energy Conversion and Management, 42(5), 539-553.

Huyse, L. (2001). Free-form Airfoil Shape Optimization Under Uncertainty Using

Maximum Expected Value and Second-order Second-moment Strategies.

Iaccarino, G. (2009). Quantification of uncertainty in flow simulations using

probabilistic methods.

Japikse, D., & Baines, N. C. (1998). Diffuser design technology(Book). White River

Junction, VT: Concepts ETI, Inc, 1998.

Javed, A., Pecnik, R., & van Buijtenen, J. (2013). Optimization of a Centrifugal

Compressor Impeller Design for Robustness to Manufacturing Uncertainties.

In ASME Turbo Expo 2013: Turbine Technical Conference and Exposition (pp.

V05AT23A023-V005AT023A023): American Society of Mechanical

Engineers.

Jones, A. (1996). Design and test of a small, high pressure ratio radial turbine. Journal

of turbomachinery, 118(2), 362-370.

Jones, A. C. (1994). Design and test of a small, high pressure ratio radial turbine. In

ASME 1994 International Gas Turbine and Aeroengine Congress and

Exposition (pp. V001T001A045-V001T001A045): American Society of

Mechanical Engineers.

Kang, S. H. (2012). Design and experimental study of ORC (organic Rankine cycle)

and radial turbine using R245fa working fluid. Energy, 41(1), 514-524.


Kang, S. H., & Chung, D. H. (2011). Design and Experimental Study of Organic

Rankine Cycle (ORC) and Radial Turbine. In ASME 2011 Turbo Expo:

Turbine Technical Conference and Exposition (pp. 1037-1043): American

Society of Mechanical Engineers.

Keep, J. A., Head, A. J., & Jahn, I. H. (2017). Design of an efficient space constrained

diffuser for supercritical CO2 turbines. In Journal of Physics: Conference

Series (Vol. 821, pp. 012026): IOP Publishing.

Kim, D.-Y., & Kim, Y.-T. (2017a). Preliminary design and performance analysis of a

radial inflow turbine for ocean thermal energy conversion. Renewable Energy,

106, 255-263.

Kim, D.-Y., & Kim, Y.-T. (2017b). Preliminary design and performance analysis of a

radial inflow turbine for organic Rankine cycles. Applied Thermal

Engineering, 120, 549-559.

Klein, A. (1981). Review: Effects of Inlet Conditions on Conical-Diffuser

Performance. Journal of Fluids Engineering, 103(2), 250-250.

Klonowicz, P., Borsukiewicz-Gozdur, A., Hanausek, P., Kryłłowicz, W., &

Brüggemann, D. (2014). Design and performance measurements of an organic

vapour turbine. Applied Thermal Engineering, 63(1), 297-303.

Kofskey, M., & Nusbaum, W. (1969). Cold performance evaluation of 4.97-inch

radial-inflow turbine design for single-shaft Brayton cycle space-power

system.

Kofskey, M. G., & Wasserbauer, C. A. (1966). Experimental performance evaluation

of a radial-inflow turbine over a range of specific speeds.

Kutlu, C., Li, J., Su, Y., Pei, G., & Riffat, S. (2018). Off-design performance modelling

of a solar organic Rankine cycle integrated with pressurized hot water storage

unit for community level application. Energy Conversion and Management,

166, 132-145.

Larsen, U., Pierobon, L., Wronski, J., & Haglind, F. (2014). Multiple regression

models for the prediction of the maximum obtainable thermal efficiency of

organic Rankine cycles. Energy, 65, 503-510.

Le Maître, O., & Knio, O. M. (2010). Spectral methods for uncertainty quantification:

with applications to computational fluid dynamics: Springer Science &

Business Media.

Lecompte, S., Huisseune, H., Van Den Broek, M., Vanslambrouck, B., & De Paepe,

M. (2015). Review of organic Rankine cycle (ORC) architectures for waste

heat recovery. Renewable and Sustainable Energy Reviews, 47, 448-461.


Lemmon, E., Huber, M., & McLinden, M. (2013). NIST Standard Reference Database

23: Reference Fluid Thermodynamic and Transport Properties-REFPROP,

Version 9.1, National Institute of Standards and Technology, Standard

Reference Data Program, Gaithersburg.

Lemmon, E. W. (2004). Equations of State for Mixtures of R-32, R-125, R-134a, R-

143a, and R-152a. Journal of Physical and Chemical Reference Data, 33(2),

593-620.

Lemmon, E. W., Huber, M. L., & McLinden, M. O. (2007). REFPROP: Reference

fluid thermodynamic and transport properties. NIST standard reference

database, 23(8.0).

Lemmon, E. W., & Jacobsen, R. T. (2000). An international standard formulation for

the thermodynamic properties of 1, 1, 1-trifluoroethane (HFC-143a) for

temperatures from 161 to 450 K and pressures to 50 MPa. Journal of Physical

and Chemical Reference Data, 29(4), 521-552.

Lemmon, E. W., & Span, R. (2006). Short fundamental equations of state for 20

industrial fluids. Journal of Chemical & Engineering Data, 51(3), 785-850.

Lemmon, E. W., & Tillner-Roth, R. (1999). A Helmholtz energy equation of state for

calculating the thermodynamic properties of fluid mixtures. Fluid phase

equilibria, 165(1), 1-21.

Lentz, Á., & Almanza, R. (2006). Solar–geothermal hybrid system. Applied Thermal

Engineering, 26(14-15), 1537-1544.

Li, J., Pei, G., Li, Y., & Ji, J. (2013). Analysis of a novel gravity driven organic

Rankine cycle for small-scale cogeneration applications. Applied energy, 108,

34-44.

Li, Y., & Ren, X.-d. (2016). Investigation of the organic Rankine cycle (ORC) system

and the radial-inflow turbine design. Applied Thermal Engineering, 96, 547-

554.

Liu, B.-T., Chien, K.-H., & Wang, C.-C. (2004). Effect of working fluids on organic

Rankine cycle for waste heat recovery. Energy, 29(8), 1207-1217.

Liu, H., Shao, Y., & Li, J. (2011). A biomass-fired micro-scale CHP system with

organic Rankine cycle (ORC) – Thermodynamic modelling studies. Biomass

and Bioenergy, 35(9), 3985-3994.

Liu, Q., Duan, Y., & Yang, Z. (2013). Performance analyses of geothermal organic

Rankine cycles with selected hydrocarbon working fluids. Energy, 63, 123-

132.

Liu, X., Wei, M., Yang, L., & Wang, X. (2017). Thermo-economic analysis and

optimization selection of ORC system configurations for low temperature

binary-cycle geothermal plant. Applied Thermal Engineering, 125, 153-164.


Loeven, G., Witteveen, J., & Bijl, H. (2007). Probabilistic collocation: an efficient

non-intrusive approach for arbitrarily distributed parametric uncertainties. In

Proceedings of the 45th AIAA Aerospace Sciences Meeting and Exhibit, AIAA

paper (Vol. 317).

Madhawa Hettiarachchi, H. D., Golubovic, M., Worek, W. M., & Ikegami, Y. (2007).

Optimum design criteria for an Organic Rankine cycle using low-temperature

geothermal heat sources. Energy, 32(9), 1698-1706.

Madlener, R., & Sunak, Y. (2011). Impacts of urbanization on urban structures and

energy demand: What can we learn for urban energy planning and urbanization

management? Sustainable Cities and Society, 1(1), 45-53.

Manual, A. U. s. (2000). Ansys Inc.

Marcuccilli, F., & Zouaghi, S. (2007). Radial inflow turbines for Kalina and organic

Rankine cycles. Paper presented at Proceedings European Geothermal

Congress, Unterhaching, Germany

Mathelin, L., Hussaini, M. Y., & Zang, T. A. (2005). Stochastic approaches to

uncertainty quantification in CFD simulations. Numerical Algorithms, 38(1-3),

209-236.

Mazzi, N., Rech, S., & Lazzaretto, A. (2015). Off-design dynamic model of a real

Organic Rankine Cycle system fuelled by exhaust gases from industrial

processes. Energy, 90, 537-551.

Menter, F., Garbaruk, A., & Egorov, Y. (2012). Explicit algebraic Reynolds stress

models for anisotropic wall-bounded flows. Progress in Flight Physics, 3, 89-

104.

Menter, F. R. (1994). Two-equation eddy-viscosity turbulence models for engineering

applications. AIAA Journal, 32(8), 1598-1605.

Merle, X., & Cinnella, P. (2015). Bayesian quantification of thermodynamic

uncertainties in dense gas flows. Reliability Engineering & System Safety, 134,

305-323.

Metz, B. (2007). Climate change 2007: mitigation: contribution of working group III

to the fourth assessment report of the intergovernmental panel on climate

change: Intergovernmental Panel on Climate Change.

Modgil, G., Crossley, W. A., & Xiu, D. (2013). Design optimization of a high-pressure

turbine blade using generalized polynomial chaos (gPC). In 10th World

Congress on Structural and Multidisciplinary Optimization (Vol. 1, pp. 1-16).

Moller, P. (1966). A radial diffuser using incompressible flow between narrowly

spaced disks. Journal of Basic Engineering, 88(1), 155-162.


Moustapha, H., Zelesky, M. F., Baines, N. C., & Japikse, D. (2003). Axial and radial

turbines (Vol. 2): Concepts NREC White River Junction, VT.

Nagel, M., & Bier, K. (1996). Vapour-liquid equilibrium of ternary mixtures of the

refrigerants R125, R143a and R134a. International Journal of Refrigeration,

19(4), 264-271.

Nechak, L., Berger, S., & Aubry, E. (2011). A polynomial chaos approach to the robust

analysis of the dynamic behaviour of friction systems. European Journal of

Mechanics-A/Solids, 30(4), 594-607.

Nithesh, K., Chatterjee, D., Oh, C., & Lee, Y.-H. (2016). Design and performance

analysis of radial-inflow turboexpander for OTEC application. Renewable

Energy, 85, 834-843.

Nithesh, K. G., & Chatterjee, D. (2016). Numerical prediction of the performance of

radial inflow turbine designed for ocean thermal energy conversion system.

Applied Energy, 167, 1-16.

Odabaee, M., Shanechi, M. M., & Hooman, K. (2014). CFD simulation and FE

analysis of a high pressure ratio radial inflow turbine. In 19AFMC: 19th

Australasian Fluid Mechanics Conference (pp. 1-4): Australasian Fluid

Mechanics Society.

Okhio, C., Horton, H., & Langer, G. (1986). The calculation of turbulent swirling flow

through wide angle conical diffusers and the associated dissipative losses.

International journal of heat and fluid flow, 7(1), 37-48.

Olivier, M., & Balarac, G. (2010). Large Eddy Simulation of a High Reynolds Number

Swirling Flow in a Conical Diffuser.

Onorato, G., Loeven, G., Ghorbaniasl, G., Bijl, H., & Lacor, C. (2010). Comparison

of intrusive and non-intrusive polynomial chaos methods for CFD applications

in aeronautics. In V European Conference on Computational Fluid Dynamics

ECCOMAS, Lisbon, Portugal (pp. 14-17).

Outcalt, S. L., & McLinden, M. O. (1997). An equation of state for the thermodynamic

properties of R143a (1, 1, 1-trifluoroethane). International journal of

thermophysics, 18(6), 1445-1463.

Page, M., Giroux, A., & Massé, B. (1996). Turbulent swirling flow computation in a

conical diffuser with two commercial codes. In 4th annual conference of the

CFD Society of Canada, Ottawa.

Pan, L., & Wang, H. (2013). Improved analysis of Organic Rankine Cycle based on

radial flow turbine. Applied thermal engineering, 61(2), 606-615.

Panizza, A., Iurisci, G., Sassanelli, G., & Sivasubramaniyan, S. (2012). Performance

uncertainty quantification for centrifugal compressors: 1. stage performance

variation. In Proceedings of ASME Turbo Expo.


Park, J. Y., Lim, J. S., & Lee, B. G. (2002). High pressure vapor-liquid equilibria of

binary mixtures composed of HFC-32, 125, 134a, 143a, 152a, 227ea and

R600a (isobutane). Fluid Phase Equilibria, 194, 981-993.

Pedersen, K. S., Thomassen, P., & Fredenslund, A. (1984). Thermodynamics of

petroleum mixtures containing heavy hydrocarbons. 1. Phase envelope

calculations by use of the Soave-Redlich-Kwong equation of state. Industrial

& Engineering Chemistry Process Design and Development, 23(1), 163-170.

Peng, D.-Y., & Robinson, D. B. (1976). A new two-constant equation of state.

Industrial & Engineering Chemistry Fundamentals, 15(1), 59-64.

Persky, R., & Sauret, E. (2018). Assessment of turbine performance variability in

response to power block design decisions for SF6 and CO2 solar thermal power

plants. Energy Conversion and Management, 169, 255-265.

Pethurajan, V., & Sivan, S. (2018). Experimental Study of an Organic Rankine Cycle

Using n-Hexane as the Working Fluid and a Radial Turbine Expander.

Inventions, 3(2), 31.

Pini, M., Persico, G., Casati, E., & Dossena, V. (2013). Preliminary design of a

centrifugal turbine for organic rankine cycle applications. Journal of

Engineering for Gas turbines and power, 135(4), 042312.

Poling, B. E., Prausnitz, J. M., John Paul, O. C., & Reid, R. C. (2001). The properties

of gases and liquids (Vol. 5): Mcgraw-hill New York.

Poling, B. E., Prausnitz, J. M., & O'connell, J. P. (2001). The properties of gases and

liquids (Vol. 5): Mcgraw-hill New York.

Qiu, G., Shao, Y., Li, J., Liu, H., & Riffat, S. B. (2012). Experimental investigation of

a biomass-fired ORC-based micro-CHP for domestic applications. Fuel, 96,

374-382.

Quoilin, S., Lemort, V., & Lebrun, J. (2010). Experimental study and modeling of an

Organic Rankine Cycle using scroll expander. Applied Energy, 87(4), 1260-

1268.

Quoilin, S., Van Den Broek, M., Declaye, S., Dewallef, P., & Lemort, V. (2013).

Techno-economic survey of Organic Rankine Cycle (ORC) systems.

Renewable and Sustainable Energy Reviews, 22, 168-186.

Rahbar, K., Mahmoud, S., & Al-Dadah, R. K. (2016). Mean-line modeling and CFD

analysis of a miniature radial turbine for distributed power generation systems.

International Journal of Low-Carbon Technologies, 11(2), 157-168.

Rahbar, K., Mahmoud, S., Al-Dadah, R. K., & Moazami, N. (2015a). Modelling and

optimization of organic Rankine cycle based on a small-scale radial inflow

turbine. Energy Conversion and Management, 91, 186-198.


Rahbar, K., Mahmoud, S., Al-Dadah, R. K., & Moazami, N. (2015b). Parametric

analysis and optimization of a small-scale radial turbine for Organic Rankine

Cycle. Energy, 83, 696-711.

Rahbar, K., Mahmoud, S., Al-Dadah, R. K., Moazami, N., & Mirhadizadeh, S. A.

(2017). Review of organic Rankine cycle for small-scale applications. Energy

conversion and management, 134, 135-155.

RAI, M. (1987). Unsteady three-dimensional Navier-Stokes simulations of turbine

rotor-stator interaction. In 23rd Joint Propulsion Conference (pp. 2058).

Redlich, O., & Kwong, J. (1949). On the Thermodynamics of Solutions. V. An

Equation of State. Fugacities of Gaseous Solutions. Chemical Reviews, 44(1),

233-244.

Rhie, C., & Chow, W. L. (1983). Numerical study of the turbulent flow past an airfoil

with trailing edge separation. AIAA journal, 21(11), 1525-1532.

Rodgers, C., & Geiser, R. (1987). Performance of a high-efficiency radial/axial

turbine. Journal of turbomachinery, 109(2), 151-154.

Rohlik, H. E. (1968). Analytical determination of radial inflow turbine design

geometry for maximum efficiency.

Saleh, B., Koglbauer, G., Wendland, M., & Fischer, J. (2007). Working fluids for low-

temperature organic Rankine cycles. Energy, 32(7), 1210-1221.

Sankaran, S., & Marsden, A. L. (2011). A stochastic collocation method for

uncertainty quantification and propagation in cardiovascular simulations. J

Biomech Eng, 133(3), 031001-031001-031001-031012.

Sauret, E. (2012). Open design of high pressure ratio radial-inflow turbine for

academic validation. In ASME 2012 International Mechanical Engineering

Congress and Exposition (pp. 3183-3197): American Society of Mechanical

Engineers.

Sauret, E., & Gu, Y. (2014). Three-dimensional off-design numerical analysis of an

organic Rankine cycle radial-inflow turbine. Applied Energy, 135, 202-211.

Sauret, E., Persky, R., Chassaing, J.-C., & Lucor, D. (2014). Uncertainty

quantification applied to the performance analysis of a conical diffuser. Paper

presented at 19th Australasian Fluid Mechanics Conference, RMIT University,

Melbourne, Australia

Sauret, E., & Rowlands, A. S. (2011). Candidate radial-inflow turbines and high-

density working fluids for geothermal power systems. Energy, 36(7), 4460-

4467.


Senoo, Y., Kawaguchi, N., & Nagata, T. (1978). Swirl Flow in Conical Diffusers.

Bulletin of JSME, 21(151), 112-119.

Setoguchi, T., Santhakumar, S., Takao, M., Kim, T., & Kaneko, K. (2002). A

performance study of a radial turbine for wave energy conversion. Proceedings

of the Institution of Mechanical Engineers, Part A: Journal of Power and

Energy, 216(1), 15-22.

Shao, L., Zhu, J., Meng, X., Wei, X., & Ma, X. (2017). Experimental study of an

organic Rankine cycle system with radial inflow turbine and R123. Applied

Thermal Engineering, 124, 940-947.

Shokati, N., Ranjbar, F., & Yari, M. (2015). Exergoeconomic analysis and

optimization of basic, dual-pressure and dual-fluid ORCs and Kalina

geothermal power plants: A comparative study. Renewable Energy, 83, 527-

542.

Simpson, A. T. (2013). Numerical and Experimental Study of the Performance Effects

of Varying Vaneless Space and Vane Solidity in Radial Turbine Stators.

Journal of Turbomachinery, 135(3), 031001-031001-031001-031012.

Soave, G. (1972). Equilibrium constants from a modified Redlich-Kwong equation of

state. Chemical Engineering Science, 27(6), 1197-1203.

Sobol, I. M. (1993). Sensitivity estimates for nonlinear mathematical models.

Mathematical modelling and computational experiments, 1(4), 407-414.

Song, J., Gu, C.-w., & Ren, X. (2016a). Influence of the radial-inflow turbine

efficiency prediction on the design and analysis of the Organic Rankine Cycle

(ORC) system. Energy Conversion and Management, 123, 308-316.

Song, J., Gu, C.-w., & Ren, X. (2016b). Parametric design and off-design analysis of

organic Rankine cycle (ORC) system. Energy Conversion and Management,

112, 157-165.

Span, R., & Wagner, W. (1996). A new equation of state for carbon dioxide covering

the fluid region from the triple‐point temperature to 1100 K at pressures up

to 800 MPa. Journal of physical and chemical reference data, 25(6), 1509-

1596.

Span, R., & Wagner, W. (2003). Equations of state for technical applications. II.

Results for nonpolar fluids. International Journal of Thermophysics, 24(1), 41-

109.

Spanos, P. D., & Ghanem, R. (1989). Stochastic finite element expansion for random

media. Journal of engineering mechanics, 115(5), 1035-1053.

Spence, S. W. T., & Artt, D. W. (1997). Experimental performance evaluation of a

99.0 mm radial inflow nozzled turbine with different stator throat areas.


Proceedings of the Institution of Mechanical Engineers, Part A: Journal of

Power and Energy, 211(6), 477-488.

Sun, F., Ikegami, Y., Jia, B., & Arima, H. (2012). Optimization design and exergy

analysis of organic rankine cycle in ocean thermal energy conversion. Applied

Ocean Research, 35, 38-46.

Tang, G., Eldred, M., & Swiler, L. P. (2010). Global sensitivity analysis for stochastic

collocation expansion. CSRI Summer Proceedings, 2009, 100.

Tchanche, B. F., Lambrinos, G., Frangoudakis, A., & Papadakis, G. (2011). Low-grade

heat conversion into power using organic Rankine cycles – A review of various

applications. Renewable and Sustainable Energy Reviews, 15(8), 3963-3979.

Toffolo, A., Lazzaretto, A., Manente, G., & Paci, M. (2014). A multi-criteria approach

for the optimal selection of working fluid and design parameters in Organic

Rankine Cycle systems. Applied Energy, 121, 219-232.

Trefethen, L. N. (2008). Is Gauss quadrature better than Clenshaw–Curtis? SIAM

review, 50(1), 67-87.

Uusitalo, A., Turunen-Saaresti, T., Honkatukia, J., Colonna, P., & Larjola, J. (2013).

Siloxanes as Working Fluids for Mini-ORC Systems Based on High-Speed

Turbogenerator Technology. Journal of Engineering for Gas Turbines and

Power, 135(4), 042305-042301-042305-042309.

Vaja, I., & Gambarotta, A. (2010). Internal Combustion Engine (ICE) bottoming with

Organic Rankine Cycles (ORCs). Energy, 35(2), 1084-1093.

Ventura, C. A. M., Jacobs, P. A., Rowlands, A. S., Petrie-Repar, P., & Sauret, E.

(2012). Preliminary Design and Performance Estimation of Radial Inflow

Turbines: An Automated Approach. Journal of Fluids Engineering, 134(3),

031102-031101-031102-031113.

Wallin, S., & Johansson, A. V. (2000). An explicit algebraic Reynolds stress model

for incompressible and compressible turbulent flows. Journal of Fluid

Mechanics, 403, 89-132.

Walters, R. W., & Huyse, L. (2002). Uncertainty analysis for fluid mechanics with

applications.

Wang, S., & Hung, T. (2010). Renewable energy from the sea-organic Rankine Cycle

using ocean thermal energy conversion. In Energy and Sustainable

Development: Issues and Strategies (ESD), 2010 Proceedings of the

International Conference on (pp. 1-8): IEEE.

Wang, X., Liu, X., & Zhang, C. (2013). Performance Analysis of Organic Rankine

Cycle With Preliminary Design of Radial Turbo Expander for Binary-Cycle

Geothermal Plants. Journal of Engineering for Gas Turbines and Power,

135(11), 111402-111401-111402-111408.


Wheeler, A. P., & Ong, J. (2014). A study of the three-dimensional unsteady real-gas

flows within a transonic ORC turbine. In ASME Turbo Expo 2014: Turbine

Technical Conference and Exposition (pp. V03BT26A003-V003BT026A003):

American Society of Mechanical Engineers.

Wheeler, A. P. S., & Ong, J. (2013). The Role of Dense Gas Dynamics on Organic

Rankine Cycle Turbine Performance. Journal of Engineering for Gas Turbines

and Power, 135(10), 102603-102601-102603-102609.

White, M. (2015). The design and analysis of radial inflow turbines implemented

within low temperature organic Rankine cycles. City University London

Whitfield, A. (1990). The preliminary design of radial inflow turbines. Journal of

Turbomachinery, 112(1), 50-57.

Whitfield, A., & Baines, N. C. (1976). A general computer solution for radial and

mixed flow turbomachine performance prediction. International Journal of

Mechanical Sciences, 18(4), 179-184.

Whitfield, A., & Baines, N. C. (1990). Design of radial turbomachines.

Wiener, N. (1938). The homogeneous chaos. American Journal of Mathematics, 60(4),

897-936.

Wilcox, D. C. (1998). Turbulence modeling for CFD (Vol. 2): DCW industries La

Canada, CA.

Wood, H. J. (1962). Current technology of radial-inflow turbines for compressible

fluids. In ASME 1962 Gas Turbine Power Conference and Exhibit (pp.

V001T001A009-V001T001A009): American Society of Mechanical

Engineers.

Wu, H.-Y., & Pan, K.-L. (2018). Optimum design and simulation of a radial-inflow

turbine for geothermal power generation. Applied Thermal Engineering, 130,

1299-1309.

Xia, J., Wang, J., Wang, H., & Dai, Y. (2018). Three-dimensional performance

analysis of a radial-inflow turbine for an organic Rankine cycle driven by low

grade heat source. Energy Conversion and Management, 169, 22-33.

Xiu, D. (2010). Numerical methods for stochastic computations: a spectral method

approach: Princeton University Press.

Xiu, D., & Em Karniadakis, G. (2002). Modeling uncertainty in steady state diffusion

problems via generalized polynomial chaos. Computer Methods in Applied

Mechanics and Engineering, 191(43), 4927-4948.


Xiu, D., & Hesthaven, J. S. (2005). High-order collocation methods for differential

equations with random inputs. SIAM Journal on Scientific Computing, 27(3),

1118-1139.

Xiu, D., & Karniadakis, G. E. (2002). The Wiener--Askey polynomial chaos for

stochastic differential equations. SIAM Journal on Scientific Computing, 24(2),

619-644.

Xiu, D., & Karniadakis, G. E. (2003). Modeling uncertainty in flow simulations via

generalized polynomial chaos. Journal of computational physics, 187(1), 137-

167.

Xiu, D., Lucor, D., Su, C.-H., & Karniadakis, G. E. (2002). Stochastic modeling of

flow-structure interactions using generalized polynomial chaos. Journal of

Fluids Engineering, 124(1), 51-59.

Xiu, D., Lucor, D., Su, C.-H., & Karniadakis, G. E. (2003). Performance evaluation of

generalized polynomial chaos. In Computational Science—ICCS 2003 (pp.

346-354): Springer

Yamamoto, T., Furuhata, T., Arai, N., & Mori, K. (2001). Design and testing of the

organic Rankine cycle. Energy, 26(3), 239-251.

Yang, Y. L. (1991). A Design Study of Radial Inflow Turbines in Three-Dimensional

Flow Ph.D Thesis. Massachusetts Institute of Technology, The United States

Yaws, C. L. (1999). Chemical properties handbook: McGraw-Hill.

Yi, H. (2013). Clean energy policies and green jobs: An evaluation of green jobs in

US metropolitan areas. Energy Policy, 56, 644-652.

Zhai, H., Shi, L., & An, Q. (2014). Influence of working fluid properties on system

performance and screen evaluation indicators for geothermal ORC (organic

Rankine cycle) system. Energy, 74, 2-11.

Zhao, Q., Mecheri, M., Neveux, T., Privat, R., & Jaubert, J.-N. (2017). Selection of a

Proper Equation of State for the Modeling of a Supercritical CO2 Brayton

Cycle: Consequences on the Process Design. Industrial & Engineering

Chemistry Research, 56(23), 6841-6853.

Zheng, Y., Hu, D., Cao, Y., & Dai, Y. (2017). Preliminary design and off-design

performance analysis of an Organic Rankine Cycle radial-inflow turbine based

on mathematic method and CFD method. Applied Thermal Engineering, 112,

25-37.

Zhu, S., Deng, K., & Liu, S. (2015). Modeling and extrapolating mass flow

characteristics of a radial turbocharger turbine. Energy, 87, 628-637.

Zou, A., Sauret, E., Chassaing, J.-C., Saha, S. C., & Gu, Y. (2015). Stochastic analysis

of a radial-inflow turbine in the presence of parametric uncertainties. In


Proceedings of the 6th International Conference on Computational Methods

(Vol. 2): ScienTech.

UNCERTAINTY QUANTIFICATION IN HIGH DENSITY FLUID … · Computational Fluid Dynamics Sensitivity...

Documents

Transcript of UNCERTAINTY QUANTIFICATION IN HIGH DENSITY FLUID … · Computational Fluid Dynamics Sensitivity...