Download Numerical study of non-Dracy forced convection in a rectangular

Numerical study of non-Dracy forced convection in a rectangular duct
saturated with a power-law fluid.
R. NEBBALI and K. BOUHADEF
Laboratoire des Transports Polyphasiques et Milieux Poreux
University Houari Boumedien of Science and Technology (USTHB)
B.P.32, El-Alia 16111, Alger
ALGERIA
Abstract: - Forced convection heat transfer to hydrodynamically and thermally fully developed laminar flow of
power-law non-Newtonian fluid in a porous rectangular duct has been investigated for the T thermal boundary
condition. The solutions for the velocity and temperature fields were obtained numerically using the finite volume
method. The averaged Nusselt number and the Fanning factor, given by these solutions were found to be in good
agreement with the literature results. Computations were performed over a range of Darcy number, power-law
indices, duct aspect ratio and dimensionless inertia factor.
Key words: rectangular ducts, power-law fluid, porous medium, heat transfer.
1 Introduction
The non-Newtonian fluids flowing through porous
media are extensively used in industry such as oil
recovery, packed bed reactors, filtration processes,
ceramic processing and many others.
The literature review shows that several studies
have been devoted to fluid flow and heat transfer in
porous media. Nevertheless, the majority of these
studies are relative to the Newtonian fluids. The
effects of channeling phenomena and thermal
dispersion in a two-dimensional porous channel were
investigated in [1]. The mixed convection problem in a
partially porous channel was studied numerically in
[2], where the effects of the porous layer thickness on
the Nusselt number as well as the pressure losses were
studied. The available studies concerning fluid flow
and heat transfer in rectangular ducts, are divided into
two types. The first one deals with non-Newtonian
fluids without the presence of a porous medium. In the
study published in [3], the author has presented a
numerical investigation of heat transfer enhancement
in laminar flow of viscoelastic fluids in rectangular
ducts. It was found that the enhancement of the heat
transfer is due to a secondary flow set up by the
second normal stress tensor. The analysis reported in
[4] deals with heat transfer mechanism in rectangular
ducts for non-Newtonian fluids. The effects of
temperature dependence viscosity and buoyancyinduced secondary flow on heat transfer have been
outlined. The second type of works takes into account
the porous medium, but with the flowing of a
Newtonian fluid. In reference [5], a numerical
investigation of laminar forced convection in a threedimensional square duct packed with an isotropic
granular material and saturated with a Newtonian fluid
was performed. It was found that channeling
phenomena and thermal dispersion effects are reduced
in a three-dimensional duct compared with previously
reported results for a two-dimensional channel.
Thus, It appears from the literature review that the
effect of the presence of a porous matrix on the flow
and heat transfer in rectangular ducts, with non
Newtonian fluids is not deeply analyzed.
In the present work, a numerical investigation of
laminar forced convection flow in a three dimensional
rectangular duct filled with a porous medium and
saturated with a power-law fluid is achieved. The duct
configuration and coordinates system are depicted in
Fig.1. Parametric studies were conducted to examine
the effects of the Darcy number, the power law index,
local inertial term and the aspect ratio of the duct on
the heat transfer as well as on the friction loss. In this
study, the modified Brinkmann-Forchheimer extended
Darcy model is used to describe the hydrodynamic
behavior of the non-Newtonian fluid.
y
H
z
B
x
Figure 1 Configuration and coordinates system
2 Mathematical Formulation
A steady laminar forced non-Newtonian fluid flow
into a rectangular duct filled with a porous medium is
considered. The porous medium is assumed to be
isotropic, homogeneous and in thermal equilibrium
with the fluid, which obeys to the power-law model.
The flow is hydrodynamically and thermally
developed. As a result of the continuity equation, the
flow is unidirectional and it is expressed in terms of
the axial velocity w, alone which does not depend on
the axial z position. All physical properties are
considered constant, and the viscous dissipation is
neglected. Consequently, the equations of momentum
and energy that govern the fluid flow and the heat
transfer in the present problem are as follows:
n
1  n    50 K 

 
K* 

2.Ct  3n  1   3 
n 1
2
(4)
where K and  are respectively the intrinsic
permeability and the porosity of the porous medium.
The turtuosity factor Ct has been defined differently
by various authors. In this study the expression given
in [7] is adopted.
2  8n  

Ct  
3  9n   3 
n
 10 n 3  75 

 
 6n 1  16 
310n3 10n11
(5)
where
Momentum:
n   n  0.31  n 
1   w  1   w  p




 n x  x   n y  y  z

 * ( n 1)
C
w
.w 
w.w  0
*
K
K
(1)
  2T  2T 
T
   2  2 
z
y 
 x

(2)
In the above equations, x, y are the transversal
coordinates and z is the axial coordinate, w is the
velocity component in the axial direction,
 p  z represents the axial pressure gradient that is
constant under the assumed conditions, T is the
temperature,  is the viscosity, and  the thermal
diffusivity. For a power-law non-Newtonian fluid, the
expression for the viscosity is given by
2
2

 w  
*  w 
  
 
   
  x 
  y  
As a result of the hydrodynamically and thermally
fully developed flow and for the T type boundary
condition [8] which is considered in the present work,
the term  T z in the energy equation (2) can be
expressed as

T  T Tw dTm
z Tm Tw dz
Energy:
w
(6)
. Here, Tm designates the mean (bulk) temperature, and
Tw is the wall temperature of the duct.
The momentum and the energy equations were
converted into dimensionless form. This was
accomplished by introducing the following
dimensionless variables:
X
W
n 1
2
(3)
where  * is the consistency factor and n the power-law
index.
In the momentum equation, K * denotes the modified
permeability [6] that depends on the structure of the
porous medium and the power-law index of the fluid
and is given by
(7)
x
Dh
w
wm
Y
P
y
Dh
p
 w 2m
Z

z
Dh
T  Tw 
dT
w m D 2h m
dz
(8)
With these definitions and by making use of the
relation (7), the resulting partial differential equations
are as follows:
  W    W  P




Re
X  X  Y  Y  Z

W n 1
n 1
n 1 2 W  A  Re W W  0
Da
(9)
 2  2
W 


0
2
2
Wm  m
X
Y
(10)
In the above equations,  is the dimensionless
viscosity, while Da, Re and  denotes respectively the
Darcy and the Reynolds numbers and the
dimensionless inertia factor. They are given by:
Re 
w 2m n D nh
(11)
*


K 
Da 
 p 
  D h
z 
f 
1
 w 2m
2
(19)
where k is the thermal conductivity of the fluid, and
the peripherally averaged heat transfer coefficient h
can be written in the form
h
k w m D h / 4 dTm
(Tw  Tm ) d z
(20)
* 2 n 1
D 2h

C
(12)
Accordingly, the expression for the peripherally
averaged Nusselt number can be reduced to
(13)
Nu  
Da
The boundary conditions for the present configuration
are such that a no-slip condition occurs at the
impermeable walls that are considered to be
isothermal at a temperature Tw. Then, the relevant
dimensionless boundary conditions along the solid
boundaries are:
W0
and
0
(14)
and along the symmetry planes X  0 and Y  0 :
W
0
X
and

0
X
(15)
W
0
Y
and

0
Y
(16)
An additional constraint, which is used to deduce the
axial pressure gradient, is that global mass
conservation must be satisfied. This is expressed as:
Wm  1
(17)
The parameters of main interest in this study are the
peripherally averaged Nusselt number and the Fanning
factor, which are respectively given by
Nu 
hDh
k
(18)
1
4 m
(21)
3 Numerical procedure
The forgoing equations together with the given
boundary conditions were solved numerically using
the control volume formulation outlined in [9], which
ensures conservation of momentum and energy over
each control volume. The nonlinear terms, involved by
the presence of the porous medium, in the momentum
equation, were treated as source terms that were
linearized using the source term linearization scheme
described in [9]. In the problem under consideration
the velocity field is independent of the temperature
field and is then solved in first. In order to obtain the
mesh independence of the present solutions, numerical
computations were performed for different values of
the power-law index and the Darcy and Reynolds
numbers. It was found that a 50*50 grid yielded results
with less than 2% error from those obtained using a
100*100 grid.
The present results were compared with the
existing solutions reported in the literature for various
cases. The validity and accuracy of the numerical
model was verified for the pressure drop results (fRe).
As shown in table 1, a comparison was made with the
results given in [8] with the correlation:
 1.733

7.4942 
 5.8606 
 n

f Re 
4
n
(22)
n
0.5
1
correlation
5.7217
14.227
Present study
6.0092
14.203
Table 1 Comparison of fRe results with [8]
1000000
1.5
34.82
33.24
n=0.5
n=1
n=1.5
100000
The numerical predictions of the Nusselt number for
the case of forced convection in non porous duct
( Da   ) and the results reported by [10] are given in
table 1 for different values of aspect ratio and powerlaw index. It appears, from table 1, that the agreement
is very good for n=1 and n=1.5, while the difference is
not significant for n=0.5.
fRe
10000
1000
100
10
1
1E-5

n
0.1
0.2
0.5
numerical
5.8706 4.8028 3.5131
0.5
Ref [10]
6.0312 4.9255 3.6001
numerical
5.9280 4.8378 3.3954
1
Ref [10]
5.9078 4.8283 3.3923
numerical
5.9131 4.8640 3.3732
1.5
Ref [10]
5.8378 4.8031 3.3328
Table 1 Comparison of Nu results with [10]
10
1
0,1
0,01
1E-3
1E-4
Darcy number Da
1
3.1252
3.2079
2.9796
2.9775
2.9202
2.8863
4 Results and discussion
Parametric studies were carried out to analyze the
effects of Darcy number, power-law index, aspect
ration and the dimensionless inertia factor on the
hydrodynamic and heat transfer results. Each of these
parameters was varied over a wide range.
Figure 2 Variation of the dimensionless pressure drop
with Darcy number for Re=1 and C=0.15
For high Darcy numbers, the pressure drop remains
constant, in agreement with non porous duct behavior.
It is shown in figure 2 that the pressure losses are more
significant with shear-thickening fluids (n1). As the
power law index is decreased, the friction factor
decreases significantly, effects of the power-law index
diminish and become negligible at high values of the
Darcy number.
On the figure 3, the velocity profiles at the midline
of the duct are shown for various Darcy numbers. As
the Darcy number decreases, the boundary layer
thickness decreases and the flat portion of the velocity
profile extends gradually toward the solid surfaces.
4.1 Hydrodynamic results
2,2
-6
Da=10
Da=10
-2
Da=0.1
Da=1
Da=10
2,0
Da=10
-4
-3
1,8
1,6
Midline velocity W
Fig.2 shows the variation of friction factor with
Darcy number. It reports that the presence of porous
medium affects considerably the pressure drop. For
small Darcy number, i.e. low permeability, the porous
material presents a high resistance to the flow; hence
the pressure drop is maximal. As the Darcy number
increases this resistance to the flow decreases, and
then the pressure drop decreases rapidly.
1,4
1,2
1,0
0,8
0,6
0,4
0,2
0,0
0,0
0,2
0,4
0,6
0,8
1,0
X
Figure 3 Midline velocity for n=0.5 and several values
of Da.
4.2 Heat transfer results
8,5
7,5
6,5
6,0
5,5
4,5
0,2
0,4
0,6
0,8
1,0
Aspect ratio r
Figure 5 Averaged Nusselt number with r for Da=10-4
and C=0.15
The effects of Darcy and Reynolds numbers on the
averaged Nusselt number evolution with the powerlaw index are presented in Fig.6. For Da=100 the
Nusselt number decreases with the increasing of the
power-law index. It appears also that the Reynolds
number has no effects on the Nusselt number. This is
due to the fact that the non porous duct behavior is
reached. For Da=0.01 the Nusselt number increases
with the increasing of the power-law index and the
Reynolds number (inertia effects). It is found that the
effect of the Reynolds number on the Nusselt number
is more significant for shear-thinning fluids (n<1).
4,25
n=1.5
n=1
n=0.5
Da=0.01
4,00
Nusselt number Nu
Nusselt number Nu
7,0
5,0
5,0
4,5
n=0.5
n=1
n=1.5
8,0
Nusselt number Nu
The effects of the Darcy number and the power-law
index on the averaged Nusselt number are displayed in
Fig.4. As it is shown in this figure, when the Darcy
number increases, the Nusselt number diminishes; this
is explained by the fact that when the permeability
decreases, the fluid velocity increases near the walls,
what leads to an increase in the heat transfer at the
walls. The shear-thickening fluids exhibit a highest
heat transfer as long as the Darcy number is less than a
critical value approximately equal to 0.04. For Darcy
numbers greater than this value, the heat transfer for
shear-thinning fluids (n<1) becomes higher. This
behavior was reported by [11] for the case of porous
parallel plates channel. For low values of the
permeability, which basically correspond to the
Darcian regime, the averaged Nusselt number reaches
an asymptotic value for all fluids. While, for high
permeability, the Nusselt number approaches another
asymptotic value of the non porous duct, which is
different for each fluid.
The averaged Nusselt number is plotted against the
aspect ratio in Fig.5. It is shown that the Nusselt
number decreases with the increasing of the aspect
ratio value, while the augmentation of the power-law
index involves an increase in the heat transfer
coefficient.
4,0
3,5
Re=10
Re=20
Re=50
3,75
3,50
3,25
Da=100
3,00
3,0
0,6
1E-6
1E-5
1E-4
1E-3
0,01
0,1
1
10
0,8
1,0
1,2
1,4
Power-law index n
Darcy number Da
Figure 4 Averaged Nusselt number evolution for Re=1
and C=0.15
Figure 6 Effects of Reynolds number on the evolution
of Nu with the power-law index for C=0.15
5 Conclusion
A numerical study of laminar forced convection in a
porous rectangular duct saturated with a nonNewtonian power-law fluid was carried out. The flow
and heat transfer problems were solved numerically by
the finite volume method. The flow was assumed to be
hydrodynamically and thermally developed. The
effects of the Darcy number, the power-law index, the
aspect ratio and the dimensionless inertial coefficient
on flow and heat transfer in the duct were investigated.
Among the most important results, it was found that
the peripherally averaged Nusselt number increases
with a decreasing Darcy number, and this heat transfer
enhancement becomes more significant as the powerlaw index increases as long as the Darcy number is
less than a critical value approximately equal to 0.04.
The inverse effect was observed for Darcy numbers
greater than this critical value. On the other hand the
pressure losses augments drastically with the
decreasing of the Darcy number. It was also shown
that shear-thinning fluids exhibit the lowest pressure
losses comparatively to the Newtonian and the shearthickening fluids.
Nomenclature
A
Constant
B, H major and minor sides of the duct
C
inertial factor
Da
Modified Darcy number
Dh
hydraulic diameter of the duct
f
Fanning factor
H
peripherally averaged heat transfer coefficient
K
permeability
K*
modified permeability
k
thermal conductivity
n
power-law index
Nu
peripherally averaged Nusselt number
P
dimensionless pressure
p
pressure
r
aspect ration of the duct (=H/B)
Re
Reynolds number
T
temperature
Tm
bulk mean temperature
Tw
Wall temperature
W
dimensionless axial velocity
Wm
dimensionless mean axial velocity
w
axial velocity
wm
mean velocity
X, Y,Y dimensionless coordinate system.
x, y, z coordinate system

thermal diffusivity

porosity

dimensionless viscosity

viscosity

dimensionless inertia factor
*
consistency index

dimensionless temperature
m
dimensionless bulk mean temperature

density
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