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Transcript of CFD for Newtonian Glucose Fluid Flow through Concentric Annuli with Centrebody Rotation
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7/31/2019 CFD for Newtonian Glucose Fluid Flow through Concentric Annuli with Centrebody Rotation
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International Journal on Science and Technology (IJSAT)
Volume II, Issue V, pp.180-186
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180
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
AbstractComputational 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 isnot 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 Reynolds number of 800 and1200.
KeywordsAnnuli, Flux, Hydraulic diameter, iso-viscous,
Peclect number
Nomenclature and list of symbolsm Mass flow rate , Kg/sec
Non-dimensional temperature profiles
R Half diameter of the pipe or tube
Laminar viscosity, N-S/m2
Q Volumetric flow rate, m3/s
Gz Graetz numberPr Prandtl number
http://www.bim.org.bd/http://www.bim.org.bd/http://www.bim.org.bd/mailto:[email protected]:[email protected]:[email protected]://www.bim.org.bd/ -
7/31/2019 CFD for Newtonian Glucose Fluid Flow through Concentric Annuli with Centrebody Rotation
2/7
International Journal on Science and Technology (IJSAT)
Volume II, Issue V, pp.180-186
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181
U Bulk axial velocity
Fluid density, Kg/m3
X Axial distance, mRe bulk flow Reynolds number, 2U(Ro-Ri)/
Ri radius of inner wall of annulusRo radius of outer wall of annulus
u, v, w axial, radial & tangential velocitiesx, r, coordinate directions
(r-Ri)/( Ro-Ri)
x/( Ro-Ri)
K non-Newtonian fluid consistency
index
n powerlaw exponent
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.
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 thatof a concentric smooth walled annulus with rotating
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International Journal on Science and Technology (IJSAT)
Volume II, Issue V, pp.180-186
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182
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
annuli with center body rotation is considered under
the following conditions: a) the fluid flows in laminar
and steady. B) The fluid density , 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
rrzrrrrz
rr
rzrz
VV
r
VV
rzrzr
2
rzrz
VV
r
VV
zrzzzrz
z
z
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) p= anbnb + CWhere the anb are coefficients multiplying the
values of 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 p 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
(3)
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4/7
International Journal on Science and Technology (IJSAT)
Volume II, Issue V, pp.180-186
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183
Difference Scheme (CDS) and Upwind Difference
Scheme (UDS). For the range of peclect number
(uL/) -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
velocitiescorrection formulas.
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 thesecond corrector stage. For the flow concentric annuli
with centre body rotation in the calculation domain
can be done by inserting internal boundary
condition.
6. RESULTANDDISCUSSION
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 =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.
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International Journal on Science and Technology (IJSAT)
Volume II, Issue V, pp.180-186
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184
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
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 tohydraulic diameter ratio, X/Dh=104) of figure 6 is
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International Journal on Science and Technology (IJSAT)
Volume II, Issue V, pp.180-186
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185
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
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
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7/7
International Journal on Science and Technology (IJSAT)
Volume II, Issue V, pp.180-186
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186
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 flowwith centre body rotation of a Newtonian and shear thinning
liquid. Int. Journal of Heat and Fluid Flow. Vol. 16. No.3
(1995).
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Md Mamunur Rashid is
Bangladeshi National and was born
in 1970. He has been serving as a
Faculty at Bangladesh Institute of
Management (BIM), Dhaka since February 2004. 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 has around 34 publications in renowned
Journals (17) and International Conferences (17). Now, he
has been conducting doctoral research at Kitami Institute of
Technology, of Japan since January 2010. He is member
of the following professional organization/body: IEB,
JSPE, BSME, BSTD, IPM and BCS.
