International Journal on Science and Technology (IJSAT)

Vol. 2, Issue 5, June, 2012, pp180-186

CFD for Newtonian Glucose 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: mamun87245@gmail.com

Abstract Computational fluid dynamics (CFD) has been very

successful in modeling turbulent gas and liquid flows in

many situations. It can provide answers to complex flow,

heat transfer and chemical reaction (usually combustion)

problems that are simply not solvable by any other means

except testing. Newtonian fluid flow phenomenon is very

important in all the pharmaceutical industries and in

many chemical industries. A computer programme can be

capable of predicting Newtonian behavior. This is because

of experimental investigation of Newtonian fluid flow is

not only expensive, laborious and time consuming; it is

impossible in many cases. The power of prediction

enables us to operate existing equipment more safely and

efficiently. Predictions of the relevant process help us in

forecasting and even controlling potential dangers. These

predictions offer economic benefits and contribute to

human well being. The existing analytical and numerical

techniques can only deal with very specific ideal cases.

Under this situation, a reliable computer programme,

which can run on a personal computer, is very much

desirable. For this reason, the laminar flow of Newtonian

fluids 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. The computer program is used for

the prediction of the axial and tangential velocities. In the

present study, confined flow through concentric annuli

with centre body rotation is examined numerically by

solving the modified Navier-Stokes equations.

Measurement of the axial and tangential components of

velocity is presented in non-dimensional form for a

Newtonian fluid. The annular geometry consists of a

rotating Centre body with angular speed of 126 rpm and a

radius ratio of 0.506. The solution of governing set of

partial differential equations is done by finite difference

computation. A non-uniform grid arrangement of 52 X 32

with multiple repetitions is used. The governing equations

have been integrated numerically with the aid of a finite –

volume method. The Hybrid scheme and central

differencing were adopted to properly account for

convection-diffusion effects, and the coupling of

continuity with the momentum equations was treated with

SIMPLE algorithm. The numerical predictions have been

confirmed by comparing them with experimentally derived

axial and tangential velocity profiles obtained for a

Newtonian. For the Newtonian (Glucose) fluid, the study

was carried out for Reynold’s number of 800 and 1200.

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

180

International Journal on Science and Technology (IJSAT)

Vol. 2, Issue 5, June, 2012, pp180-186

1. INTRODUCTION

In the present study, a detailed computational

investigation on the Newtonian fluid flow through

concentric annuli with centre body rotation with

glucose as the working fluid will be carried out. The

geometry and dimensions of the Newtonian fluid flow

is based on the experimental studies Escudier et al.

(1995) [1]. The present study deals with numerical

investigation of Newtonian fluid flow through

concentric annuli with centre body rotation. The fluids

are dilute solution of Glucose. The glucose solution is

a 1:1 w/w mixture of a glucose syrup (Crestar) and

water. 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 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.

RiRo

Schematic

diagram of

the numerical

model

Figure 1: Schematic diagram of the numerical model

2. PREVIOUS WORK

Nouri et al. (1994) [2] presented experimental results

for flow of Newtonian and non-Newtonian fluids in a

concentric annulus with rotation of the inner cylinder.

Experiment was conducted for annular passage flow

with an outer brass pipe of nominal inside diameter D°

of 40.3 mm, length of 2.0 m and an inner stainless

steel rod of 20 mm diameter, Din. They pointed out

that the mean velocity and the corresponding

Reynolds shear stresses of Newtonian and non-

Newtonian fluids were measured in a fully developed

concentric flow with a diameter ratio of 0.5 and inner

cylinder rotational speed of 300 rpm. With the

Newtonian fluid in laminar flow, the effects of the

inner shaft rotation were a uniform increase in the

drag co-efficient by about 22 percent, a flatter and less

skewed axial mean velocity and swirl profile with a

narrow boundary close to the inner wall with a narrow

boundary close to the inner wall with a thickness of

about 22 percent of the gap between the pipes.

Escudier et. al. (1995) [1]performed experiments with

test section consists of five modules each of 1.027 m

length and one of 0.64 m which gives an overall

length of 5.775. It was length to hydraulic diameter

ratio of 116. The experimental flow geometry was that

of a concentric smooth walled annulus with rotating

centre body of radius ratio 0.506. The centre body

rotated at a speed not exceeding 126 rpm. They

pointed out increasing the bulk velocity (for constant

rotational speed) produces a progressive reduction in

the level of the tangential velocity that is similar for

the Glucose and carboxymethylcellulose (CMC)

fluids, except anomalous behavior for CMC at low

Reynolds number. Moreover in this area author is

presented some works [5-8].

3.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.

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. The fluid flow in concentric

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International Journal on Science and Technology (IJSAT)

Vol. 2, Issue 5, June, 2012, pp180-186

annuli with center body rotation 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

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

4.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 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 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.

5. SOLUTION ALGORITHM

The procedure developed for the calculation of the

flow field has been given the name SIMPLE, which

stands for Semi-Implicit Method for Pressure-Linked

Equations. The procedure has been described in

Patankar and Spalding (1970) .Operations in the order

of their execution is as follows:

a) to guess the pressure field p*.

b) to solve the momentum equations to obtain u*,v*,

and w*

c) to solve the p* equation

d) to calculate p by adding P/ to P

*

e) to calculate u,v and w from their starred values

using the

velocities –correction formulas.

(3)

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International Journal on Science and Technology (IJSAT)

Vol. 2, Issue 5, June, 2012, pp180-186

f) Treat the corrected pressure p as a new guessed

pressure p*,

return to step b, and repeat the whole procedure

until a

converged solution is obtained.

Numerical solution of the governing equation for

equation for transport of momentum is obtained by

using the SIMPLE algorithm [10-12]. The fluid is

considered Newtonian. The flow geometry is

concentric with the inner pipe rotating. More corrector

stages may be added following procedure used for the

second corrector stage. For the flow concentric annuli

with centre body rotation in the calculation domain

can be done by inserting ‘internal’ boundary

condition.

6. RESULT AND DISCUSSION

The results of numerical simulation of Newtonian

fluid flow through concentric annuli with centre body

rotation are presented and compared with the

experiments of Escudier et al. (1995) [1]. The results

are obtained by the numerical method described in

above solution algorithm section. The solution

domain was bounded by the inlet plane, exit plane,

outside solid wall, inside wall with constant rotational

speed and axis symmetry. 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=1.00

Consistency index, K=0.01 N-s/m2

Density ρ=1134 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. Grid Independence Test for Glucose of

Re=800

at X/Dh=104

Figure 3. Axial velocity profiles for Glucose at

Re=800

Grid independence test is necessary to test whether

the predicted results are independent of grid. At a

constant Reynolds number, Re= 800 with 32 X22,

42X32 and 52X32 grids. The 52 x 32 grid gave

reasonably grid predictions when compared to the

theoretical predicted of Yuan et al. [1969] as shown

in figure 2. Figures 3 and 4 represent the developing

axial velocity profiles. The profiles at different non-

dimensional axial distance is shown in such a way ,

thus the gradual changes in profiles from flat to

developed parabolic type can be easily inspected.

The last curve (at length to hydraulic diameter ratio,

X/Dh=104) of Figure 3 shows the developed velocity

profile compared with experimental result of Escudier

et al. (1995) [1]. The last curve (at length to hydraulic

diameter ratio, X/Dh=104) of figure 3 is compared

with laminar Newtonian profile. From figure 4, the

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International Journal on Science and Technology (IJSAT)

Vol. 2, Issue 5, June, 2012, pp180-186

maximum velocity for experimental result is 1.22 and

numerical solution 1.38. Hence again the difference

may have occurred due to developed of turbulence by

inner rotating pipe. From figure 3, the maximum

velocity for laminar Newtonian profile is 1.484 and

numerical solution 1.49.

Figure 4. Axial velocity profiles for Glucose at

Re=1200

Figure 5. Tangential velocity profiles for Glucose at

Re=800

Hence percentage of deviation 0.5% in maximum

velocity profile observed. This indicates the validity

of the present methodology. For both figures 3 and 4

the maximum velocity occurs near center of the

annuli for Newtonian fluid. Figures 5 and 6 represent

the tangential velocity profiles for Newtonian fluids.

The gradual change of tangential velocity profile is

shown in concentric annuli with centre body rotation.

The last curve (at length to hydraulic diameter ratio,

X/Dh=104) of figure 5 is in excellent agreement with

the theoretical data. The last curve (at length to

hydraulic diameter ratio, X/Dh=104) of figure 6 is

compared with the experimental data of Escudier et

al. (1995). Also in the case of Newtonian fluid it is

shown that as the Reynolds number is increased, the

tangential velocity levels within annular gap are

progressively reduced. The same qualified behavior

was found by Escudier el al. (1995) and Nouri and

law (1994) [1-2].

Figure 6. Tangential velocity profiles for Glucose at

Re=1200

Figure 7. Effects of Re of tangential velocity profiles at X/Dh=104

Due to the turbulent diffusion of fluid at Reynolds

number=1200 this predicted results shown in figure 6

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International Journal on Science and Technology (IJSAT)

Vol. 2, Issue 5, June, 2012, pp180-186

are not in good 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 was not considered. Thus

that for the Newtonian fluid flow the tangential

velocity gradient in the inner layer must be

substantially higher than in the outer layer. This

expression has a consequence of the torque being

constant within the annular gap and the assumption of

laminar sub-layers at each surface. This situation for

a non-Newtonian fluid is more complex, although

qualitatively, the same trend evidently exists. The

present prediction failed to reproduce this behavior.

As the Reynolds number is increased the tangential

velocity levels within the annular gap are

progressively reduced except for non-Newtonian

fluid (CMC) at Re 110, which is shown in figure 6. In

figure 5 Newtonian laminar tangential flow for

Glucose at Reynolds number 800 is compared with

rr

V rrr

rr

2

111

2

2

2

1

2

1

2

2

1

The analytical data for following equation (Yuan S.W

1969) [9] and it is found to be in excellent agreement.

This shows the validity of present numerical

predictions.

7.COCLUSION

Laminar axial and tangential flows through

concentric annuli with centre body rotation have been

studied simulated for Newtonian (glucose). The main

findings are summarized below: a) For Newtonian

fluids, 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. c) The tangential velocity decreases with the

increase of radius. Near the inner all it changes

sharply, 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.

8.RECOMMENDATIONS

The same prediction can be carried out for the

turbulent cases by incorporating the turbulent

transport equations for both Newtonian fluid and non-

Newtonian fluid. 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 [10-12]. 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).

[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 ,

185

International Journal on Science and Technology (IJSAT)

Vol. 2, Issue 5, June, 2012, pp180-186

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).

He also worked as adjunct faculty at DIU, BOU. IBAISU, BUBT,

BUET, IPM, DIPTI and Planning Academy. Prior this job he

worked as a Mechanical Engineer of Jamuna Fertilizer Company, Bangladesh for seven years. He obtained Bachelor of Science in

Mechanical Engineering degree from RUET (erstwhile BITR) in

1993, Master of Science in Mechanical Engineering degree from BUET in 1996 and Master of Business Administration degree from

BOU in 2004. He completed a Diploma in Computer Science and

Applications, a Post Graduate Diploma in Human Resource Management and a Post Graduate Diploma in Marketing

Management. He also completed Management Accounting, Project

Management, Safety and Maintenance Management, Information Technology in Business and Artificial Neural System courses in

Graduate level study at Industrial and Production Engineering

Department of BUET, Bangladesh. He has 34 publications in renowned Journals and International Conferences. He participated

at training programs in Singapore for 3 weeks on Mechatronics

System Technology and 8 weeks on TQM and ISO in Hyderabad, India. He has also attended some training program in Bangladesh

like Project Management, Vibration Monitoring and Maintenance

Management etc. Now, he has been conducting doctoral research at Kitami Institute of Technology, of Japan since January 2010. His

main research theme is customer needs analysis for product

development. He is member of the following professional organization/body: IEB, JSPE, BSME, BSTD, IPM and BCS. He

can be reached by e-mail: mamun87245@gmail.com

186

Md Mamunur Rashid is

Bangladeshi National and was born in 1970. He has been serving as a

Faculty, Production Management

Division at Bangladesh Institute of Management (BIM), Dhaka since

February 2004. As a faculty he has

been facilitating for the Graduate and Professional training program

in the areas of Product

Development, TQM, Quality Management System

(ISO9001:2008), Productivity and

Competitiveness, Project Management with MS Project-2007

and Industrial Safety Management .