CFD for non-newtonian CMC fluid flow through concentric annuli with centre body rotation ,...

International Journal on Science and Technology (IJSAT)

Vol. 2, Issue 3, PP121-126, July-September, 2011

CFD for non-newtonian CMC fluid flow through concentric annuli with centre

body rotation

Md Mamunur Rashid

Faculty, Bangladesh Institute of Management (www.bim.org.bd)

4, Sobhanbag, Mirpur Road, Dhaka-1207

E-mail: [email protected]

Abstract The laminar flow of non-Newtonian fluid through

concentric annuli with centre body rotation has

been studied numerically. The scope of this study is

limited to numerical prediction of axial velocity

profiles and tangential velocity profiles at steady

state condition. A general computer program

“TEACH-T” has been modified for this purpose.

The program was used after sufficient justification.

In the present study, confined flow through

concentric annuli with centre body rotation is

examined numerically by solving the modified

Navier-Stokes Equations.

Keywords Annuli, Flux, Hydraulic diameter, iso-viscous,

Peclect number

Nomenclature and list of symbols

m Mass flow rate , Kg/sec

Non-dimensional temperature profiles

R Half diameter of the pipe or tube

µ Laminar viscosity, N-S/m2

Q Volumetric flow rate, m3/s

Gz Graetz number

Pr Prandtl number

U Bulk axial velocity

ρ Fluid density, Kg/m3

X Axial distance, m Re bulk flow Reynolds number, 2ρU(Ro-Ri)/µ

Ri radius of inner wall of annulus

Ro radius of outer wall of annulus

u, v, w axial, radial & tangential velocities

x, r, coordinate directions

ζ (r-Ri)/( Ro-Ri)

ξ x/( Ro-Ri)

K non-Newtonian fluid consistency

index

n power –law exponent

1. INTRODUCTION

Generally, developing countries, like Bangladesh has

no other alternative but to approach the foreign

consults to solve problems arising due to fluid flow of

non-Newtonian nature. A computer package, which

can run on available personal computers, will be of

much help for Bangladesh. In the present study, a

detailed computational investigation on the non-

Newtonian fluid flow through concentric annuli with

centre body rotation with CMC

(carboxymethylcellulose) as a working fluid will be

carried out. The geometry and dimensions of the non-

Newtonian fluid flow is based on the experimental

studies Escudier et al. (1995) [1-2]. Pls see the author

was presented some works in this arena [4-8]. The

specific objectives of this study are to develop a

computer program for theoretical investigation of

combined axial and tangential laminar velocity of

concentric annular with centre body rotation, to study

the constant rotational speed with different Reynolds

number of concentric annuli with centre body rotation

flow and final attempt would be made to establish

reliabilities, suitability and assessment of the quality

of this program through comparing the results

obtained with those available in the literature.

2. GOVERNING DIFFERENTIAL EQUATIONS

This work is concerned with steady laminar flow in

concentric annuli with center body rotation. The

rheological equation used in this work is well-known

power law, viz.

121

http://www.bim.org.bd/



n

rzz

v

r

uK

(1)

Where, rz in shear stress, n is a temperature

independent exponent, which is equal to unity in the

present work and consistency index K, which is also

temperature independent.

RiRo

Schematic

diagram of

the numerical

model

Figure 1: Schematic diagram of the numerical model

The fluid flow in concentric annuli with center body

rotation (shown in figure1) is considered under the

following conditions: a) the fluid flows in laminar

and steady. b) The fluid density ρ, consistency index

K, thermal conductivity k, and heat capacity Cp are

temperature independent. Under the assumptions

stated above, the continuity and momentum equations

for an incompressible fluid in cylinder co-ordinate

(r, , z) system are:

Continuity:

0

r

V

z

v

r

v rzr

(2)

Momentum:

rzrz

VV

r

VV rrzrrrr

zr

r

rzrz

VV

r

VV rzr

zr 2

rzrz

VV

r

VV zrzzzrz

zz

r

(3)

Where the stress tensors are given by

n

rr

V

r

VK

n

zz

VK

n

zzz

VKP

2

n

zrzr

r

V

z

VK

n

r

r

VKP

2

n

rrr

r

VKP

2

3. DISCRETIZED GOVERNING

DIFFERENTIAL

EQUATIONS In the present study the finite volume approach, as

described by Gosman et al. [1989], is adopted [3].

Typical however, the Newtonian term, which is

included in the present study, is introduced through

the source terms. In his approach, the governing

differential equations are discretized by integrating

them over a finite number of control volumes or

computational cells, into which the solution domain is

divided [10-12]. Discretized transport will take the

following quasi-linear form:

(ap-b) фp= anbфnb + C

Where the anb are coefficients multiplying the values

of ф at the neighbouring nodes surrounding the

central node P. The umber of neighbour depends on

the interpolation practice or differencing schemes

used. Here ap is the co-efficient of фp given by

ap=n

anb;

n

Summation over neighbours (N, S, E, W)

For the present study, the hybrid Scheme is used.

The name Hybrid indicates a combination of the

Central Difference Scheme (CDS) and Upwind

Difference Scheme (UDS). For the range of peclect

number (ρuL/Γ) -2< Pe < 2, both the diffusion and

convective term are evaluated by the CDS. Outside

this range convective terms are evaluated using the

UDS and the diffusion terms are evaluation using

CDS. Boundary conditions of the present study are at

inlet boundary, flat profile of axial velocity is

122



specified, at outlet boundary, the gradients of all

variables are set to zero in the axial direction and at

wall boundaries; outer wall velocity is set to a

constant value.

4. RESULT AND DISCUSSION

The entire investigation domain is divided into a non-

uniform grid arrangement of 52 X32 with multiple

repetition is used. Fine grid spacing was used near the

solid walls and a relative course grid was used in the

flow region. For the present study the following

values of parameters are chosen:

Power law index, n=0.75

Consistency index, K=0.04 N-s/m2

Density ρ=1000 kg/m3,

Outer radius Ro=0.0502 m

Inner radius Ri=0.0254 m

Length, X=5.775 m and

Rotational speed of inner pipe, N=126 rpm.

Figure 2. Axial velocity profiles for CMC fluid at

Re=110


Re=350


Re=4400

Figures 2, 3 and 4 represent the developing axial

velocity profiles i.e. dimensionless radius. The

profiles at different non-dimensional axial distances

are shown in such a way, that the gradual change in

profiles from flat to developed parabolic type can be

easily inspected. The last curve (at length to hydraulic

diameter ratio, X/Dh=104) in each case shows the

developed axial velocity profile and compared with

experimental results of Escudier et al. (1995). The

prediction is for the iso-viscous laminar flow of

fluids. From figure 2 the last curve of the graph

shows the developed velocity profile and points of

experimental result from Escudier et al. (1995). The

maximum velocity for experimental result is 1.7 and

for numerical solutions it is 1.5. Hence a percentage

deviation of 12% in maximum velocity is observed at

the last station. In the figure 4 at the last station the

maximum velocity for experimental result is 1.4 and

numerical solution it is 1.65. Hence percentage

deviations of 17% in maximum velocity observed.

For both figures 2 and 3, it is seen for laminar flow

(low Reynolds number respectively 110 and 350) the

maximum velocity occurs near the inner wall. Higher

rotation of inner pipe gives rise to higher shear stress

adjacent to the inner wall resulting lower viscosity,

consequently maximum velocity near the inner wall

123



occurs due to lower shear stress. The flow becomes

turbulent at Re =4400, the present prediction is

carried out under the laminar flow assumption. This

is because the study of turbulent flow is beyond the

scope of present study. The results obtained for Re =

4400 is presented in figure 4. This figure shows that

the maximum velocity for experimental result is 1.22

and numerical solution it is 1.423. Hence a

percentage deviation of 16% in maximum velocity

observed at the last station. Due to existence of

turbulence the experimental velocity profile appears

to be flatter than that by the present numerical

prediction.

Figure 5. Tangential velocity profiles for CMC fluid

at Re=110


at Re=350


at Re=4400

Figures 5, 6 and 7 represent the tangential velocity

profiles. The tangential velocity levels within the

annular gap are progressively increased close to the

centre body. It is seen that the tangential velocity

gradually transforms to developed profiles and the

curve for the length to hydraulic diameter ratio,

X/Dh=104 of the figure are compared with

experimental results of Escudier et al. (1995). In

figure 5 it is observed that curves for length to

hydraulic ratio of 50 and 104 superimpose.

Experimental tangential velocity curves of Escudier

et al. (1995) are not in agreement with our numerical

prediction. The reason is that Escudier et al. (1995)

mentioned that in their showed experiments turbulent

diffusion was present. Due to this turbulent diffusion

the fluid particles move from higher velocity region

to lower velocity region and hence uniform velocity

occur at the central region of the annuli. But in our

numerical scheme turbulent diffusion was not

considered. In figure 8, Newtonian laminar tangential

flow for Glucose at Reynolds number 800 is

compared with analytical data for following equation:

rr

V rrr

rr

2

111

2

2

2

1

2

1

2

2

1

124



Figure 8. Tangential velocity profiles for Glucose at

Re=800

and it is found to be in excellent agreement. This

shows the validity of present numerical predictions.

In the experiments of Escudier et al. (1995), the

centre body is slightly distorted, and it is impossible

to achieve a concentric geometry over the entire

length of the test section. In our numerical prediction

this concentricity was not considered. In contrast to

the negligible influence of friction factor, rotating has

a strong influence on the tangential mean velocities,

which generally reveal a triple layer structure of

speed; Nouri and law (1994) also reported very

similar observation for CMC. As the Reynolds

number is increased the tangential velocity levels

within the annular gap are progressively reduced

except for Re 110, which is shown in figure 9.

Figure 9. Effects of Re of tangential velocity profiles at X/Dh=104

Therefore that the tangential velocity gradient in the

inner layer must be substantially higher than in the

outer layer. From figure 7 that the tangential velocity

profile is gradually developed. Escudier et al. (1995)

has shown that the tangential velocity reveal three

distinct regions across the radial for Reynolds

number. Flow regions have been categorized as

region adjacent to inner pipe, central region of almost

constant velocity and region adjacent to outer pipe.

Present predictions reveal a pattern of exponential

decay of tangential velocity towards the outer wall.

This is due to laminar assumption in the present

study.

5. COCLUSION

The main findings are summarized below: a) the axial

velocity profile at inlet is flat and gradually

transforms to parabolic shape. b) Maximum axial

velocity occurs at a region close to the inner wall.

This tendency is particularly noticeable at a Reynolds

number of 110. c) The tangential velocity decreases

with the increase of radius; while near the outer wall

it changes slowly and d) increasing the Reynolds

number for constant rotational speed produces a

progressively reduced level of the tangential velocity,

except for anomalous behavior at low Reynolds

number for CMC.

6. RECOMMENDATIONS

The same prediction can be carried out for the

turbulent cases by incorporating the turbulent

transport equations Similar study can be made for

eccentric annular with centre body rotation. (E.g.

LUDS, Quick Scheme) can be used to have better

accuracy in this type of prediction [9-12]. Similar

study can be made for different size, length, diameter

and rotational speed. Similar prediction can be made

giving the inner body rotation with vibration.

REFERENCES

[1 ] Escudier, M.P. and Gouldson, I.W., Concentric annular flow

with centre body rotation of a Newtonian and shear thinning

liquid. Int. Journal of Heat and Fluid Flow. Vol. 16. No.3

(1995).

125



[2 ] Nouri, J.M. and Whitelaw, J.H., Flow of Newtonian and non-

Newtonian Fluids in a concentric annulus with rotation of the

inner cylinder, J. Fluid Engineering, vol.116, pp821-827,

(1994).

[3 ] Gosman AD, and Iderials FJK, TEACH-T: A general

computer program for two dimensional turbulent re-

circulating flows, Department of Mechanical Engineering,

Imperial College, London, SW7,1976

[4 ] Patankar SV, and Spalding DB, Heat and Mass transfer in

Boundary layers, 2nd Edn. Intertext Books, London, 1970

[5 ] Rashid MM, and Naser J.A., Computational Fluid Dynamics

For Newtonian Fluid Flow Through Concentric Annuli With

Center Body Rotation, Proceedings of the Fourth

International Conference on Mechanical Engineering,

December,26-28, 2001, Dhaka, Bangladesh, Volume-II,

Section –IV (Fluid Mechanics), pp. 119-123

[6 ] Rashid MM, and Naser J.A., Non-Newtonian Fluid Flow

Through Concentric Annuli, Proceedings of the First BSME-

ASME International Conference on Thermal Engineering, 31

December, 2001-2 January 2002, Dhaka, Bangladesh; pp.

S46-S51

[7 ] Rashid MM, , Numerical Simulation - A Modern Concept of

Chemical Industries, In the Proceedings of the 1st Annual

Paper Meet and International Conference on Chemical

Engineering, February 12, 2002, IEB Chandpur, Bangladesh ,

Paper No-7, pp.72-77

[8 ] Rashid MM, Numerical simulation of non-Newtonian fluid

flow through concentric annuli with center body rotation,

M.Sc in Mechanical Engineering Thesis, BUET, 1996.

[9 ] Yuan SW, Foundations of Fluid mechanics, Prentice-Hall of

India Private limited, New Delhi, 1969.

[10 ] Anderson DA, Tannehill JC, and Pletcher RH, Computational

Fluid Mechanics and Heat Transfer, Hemisphere Publishing

Corporation, Washington DC, 1984

[11 ] Popovska F, and Wilkinson WL, Laminar heat mass transfer

to Newtonian and non-Newtonian fluids in tubes, Chemical

engineering Science, 32, 1154-1164,1977

[12 ] Nouri, J.M. Umur, J.M., and Whitelaw, J.H., Flow of

Newtonian and non-Newtonian Fluids in concentric and

eccentric Annuli, Journal of Fluid Mechanics,253,617-641,

(1993).

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CFD for non-newtonian CMC fluid flow through concentric annuli with centre body rotation ,...

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