In this study the flow field and heat transfer properties of a steady, two-dimensional flow field in a porous domain between two parallel plates is investigated numerically by using a discretized numeric code. Analysis has been carried for Reynolds number based on particle sizes ranging from 60 to 1000. Numerical results are compared with different numerical methods used for predicting this kind of flow. Results are obtained for different regime, various p Re numbers and the effect of Particles size is also investigated. Solutions indicate that by increasing the p Re , the flow in the porous media remains laminar where the flow has turbulence characteristics for p Re <50. Moreover, by increasing p Re , the value of average Nusselt number increases. Also, reducing the particle size affects the Nusselt number and it increases while the porosity remains the same.

1. 1 Copyright © 2009 by ASME
Proceedings of the Seventh International ASME Conference on Nanochannels, Microchannels and Minichannels
ICNMM2009
June 22-24, 2009, Pohang, South Korea
ICNMM2009-82287
PREDICTION OF FORCED CONVECTION FLOW IN A PARALLEL PLATE CHANNEL FILLED
WITH POROUS MEDIA
Dana Ehyaei
1
,Hamed Honari
2
, M. Rahimian
3
1
Mechanical Engineering Department, University of Tehran, dana.ehyaee@me.ut.ac.ir
2
Mechanical Engineering Department, University of Tehran, hhonari@me.ut.ac.ir
3
Mechanical Engineering Department, University of Tehran, rahimyan@ut.ac.ir
ABSTRACT
In this study the flow field and heat transfer properties of a
steady, two-dimensional flow field in a porous domain between
two parallel plates is investigated numerically by using a
discretized numeric code. Analysis has been carried for
Reynolds number based on particle sizes ranging from 60 to
1000. Numerical results are compared with different numerical
methods used for predicting this kind of flow. Results are
obtained for different regime, various pRe numbers and the
effect of Particles size is also investigated. Solutions indicate
that by increasing the pRe , the flow in the porous media
remains laminar where the flow has turbulence characteristics
for pRe <50. Moreover, by increasing pRe , the value of
average Nusselt number increases. Also, reducing the particle
size affects the Nusselt number and it increases while the
porosity remains the same.
INTRODUCTION
Force convection heat transfer from a porous media of
isothermal particles suspended in a channel has been
encountered in various applications. Simulation of motion of
underground waters, pressure drop and flow field in filters and
increasing heat transfer in heat transfer media is some of its
applications. In Figure (1) a sketch of the model geometry is
shown. The duct is made of three sections, initially fluid enters
the duct from section I and after passing section II (Porous
media) it exits the duct by section III. The porous media in
section II is modeled directly by putting rods with square
sections. The plates are adiabatic and the particles are
isothermal.
Darci [1] model was the first model which was used in
solution of porous media. In this model pressure drop in each
direction was correlated with average velocity in the same
direction. This model was not suitable for high Reynolds
number and was corrected by using inertial terms in momentum
equations. Some investigators use more complication forms of
this model, which have additional nonlinear terms and
empirical constants. Poulikakos and Renken [2], [3], Chou and
Lien [4], Calmidi and Mahajan [5], Cheng and Hsu [6] used
this approach. For facilitating analyses in the beginning the
porosity was usually assumed as a constant which is
conveniently called a constant porosity model in this study.
However Roblee et al and Benenati and Brosilow based on
their experimental results observed that porosity varied
significantly in the near wall region. Schwartz and co-workers
conducted experimental studies and measured the maximum
velocity in the near wall region which is normally called the
channeling effect These phenomena directly validated that the
porosity which was regarded as a variable was more realistic.
Furthermore Cheng et al pointed out that in much of the
literature the porosity was simulated as a damped oscillatory
function of the distance from the wall and the damped
oscillatory phenomenon was insignificant as the distance was
larger than five particle diameters for packed beds. Therefore in
concerning both of the practical use and convenient theoretic
model the variation of the porosity is assumed as an
exponential function of the distance from the solid wall and is
called a variable porosity model for comparing with the
constant porosity model. Based on the above experience two

2. 2 Copyright © 2009 by ASME
different models have been adopted to derive the individual
equations of fluid flow and heat transfer for the porous
medium. However, no-slip condition does not satisfy near any
solid particle in theoretical works with Darci category model.
Several authors solved details of flow inside of a microscopic
periodicstructure. Eidsath et al [7], Coulaud et al [8], Fowler
and Bejan [9] carried out two dimensional numerical
simulations for flow across banks of circular cylinders. Larson
and Higdon [10] analyzed Stokes flow through lattice of
spheres. Kuwahara et al [11], [12] assumed a macroscopic
uniform flow to pass through a lattice of square rods placed
regularly in an infinite space and solved flow in a block
constructed from four neighboring square cylinders. Rahimian
and Pourshaghaghi [13] calculated the fluid flow
microscopically in porous media without volumetric averaging.
Porous media was generated by random distribution of square
cylinders. Navier-Stokes and energy equations were solved
considering diffusion and convection terms in fluid region by a
point collocated method. Rahimian and Abasiun [14] solved
Navier-Stokes and energy equations in a staggered grid domain
by SIMPLE method. In the staggered grids the velocities at the
surface of the filled cells, i.e. porous materials, fixed zero while
in point collocated methods the velocity at the center of the
filled cells were fixed. In a staggered grid the points with
known velocities are four times greater than the point
collocated methods. There were no considerable changes in the
results. The objective of this research is to find flow field and
heat transfer properties of a porous domain between two
parallels using structured grids. SIMPLE method is the method
of solution.
I II III
Figure 1 – model geometry
GOVERNING EQUATIONS AND NUMERICAL
SOLUTIONS
For numerical solution the flow is assumed to be
turbulent, two-dimensional and incompressible. All fluid
properties are assumed to be constant. The turbulence model
used in this study is the standard k model with the
standard wall function. So, the governing equations used are as
follows:
The continuity equation can be written as follows:
0).( 




t
(1)
Conservation of momentum in an inertial reference is described
by
gp
t
 


).().()( (2)
which for the Turbulent flow the transport equations for
turbulence energy and dissipation are:
    


 

























bk
jk
t
j
i
i
GG
x
k
x
ku
xt
k (3)
and
   
  k
CGCG
k
C
xx
u
xt
bk
j
t
j
i
i
2
231



































(4)
In these equations, kG represents the generation of turbulence
kinetic energy due to the mean velocity gradients, bG is the
generation of turbulence kinetic energy due to buoyancy, and
 21 ,CC ,and 3C are constants. k and  are the turbulent
Prandtl numbers for k and  , respectively.
And the energy equation is:
 ).(.))(.()(  effeff JhTkpEE
t


 (5)
where effk is the conductivity ( tkk  , where tk is the
turbulent thermal conductivity, defined according to the
turbulence model being used), and J is the diffusion flux. The
first three terms on the right-hand side of equation (5) represent
energy transfer due to conduction, species diffusion, and
viscous dissipation, respectively. In this equation
2
2



p
hE .
The definition for porosity, the most intuitive porous media
parameter used in this article is as in equation (6):
SolidsofVolumeVoidsofVolume
VoidsofVolume~

 (6)
RESULTS AND DISCUSSION
The channel dimensions were chosen 0.51.5 m (height
length) see Figure 1.
The length of each three sections is equal to 0.5m. Numerical
method was performed using a structured grid which the
adapted typical is shown in Figure 2. First of all to validate the
solution the particle sizes are 0.5mm and the effect of particle
sizes is then investigated.
The ~ which is the porosity remains the same for all the
answers and is equal to 0.9 and Pr=0.7.

3. 3 Copyright © 2009 by ASME
Figure 2 – structured grid
In Figure 3, temperature contours are shown typically. The
condition used is described in Table 1.
pRe = 310 Pr=0.7
Temperature Inlet = 300 K Temperature Particles=390 K
 =0.9 Particle size= 10mm
Table 1
305 315 325 335 345 355 365 375 385
305 315 325 335 345 355 365 375 385
Figure 3 – temperature contours
In figure 4 and 5, the solution is validated with other numerical
solutions in which it can be seen that they are in good
agreement.
+
+ +
++
+
+
++
+
++++
Re
AveNu
101
102
103100
101
102
Galloway et al
Satterfield et al
Wilke and Hougen
De Acetis and Thodos
Present Study
+
p
Figure 4 – Ave Nu versus pRe , ~ =0.9, particle size 0.5 mm
+
+
+
+
+
+ + + + + + + + + +
Re
-dp/dx
100
101
102
10310-1
100
101
102
Darcy Forchheimer 1
Darcy Forchheimer 2
Darcy Law 1
Darcy Law 2
Present Study
+
p
Figure 5 – Pressure drop versus pRe , ~ =0.9 and particle
size 0.5 mm
As it can be seen from figure 4, the average Nu number varies
approximately linearly with pRe .
The bulk temperature in the duct is shown in figures 6
and 7 for pRe = 20, pRe =50 for different interval size of
meshing (See figure 8), as it can be seen the bulk temperature
does not fluctuate in the Porous media (Section II) but for
pRe =50 it fluctuates in the Third section. When pRe is 50
the Re number is 5555 so the flow would be turbulent in the
duct. It seems that the flow fields in section I, IIII are turbulent

4. 4 Copyright © 2009 by ASME
for pRe under investigation in this paper ( pRe >60) and flow
in porous media has laminar characteristics.
x
t
0.2 0.4 0.6 0.8 1 1.2 1.4
300
302
304
306
308
310
Cell=1
Cell=2
Cell=3
Cell=4
Cell=5
Figure 6 - bulk temperature for pRe = 20
x
Tbulk
0.2 0.4 0.6 0.8 1 1.2 1.4
300
302
304
306
Cell=1
Cell=2
Cell=3
Figure 7- bulk temperature for pRe = 50
Figure 8 – different interval size of meshing
A particles size is a parameter that influences the flow and
heat transfer parameters. It is shown that by increasing the
parameter size the average Nusselt number will decreases (See
figure 9). Also in figure 10 the effect of particle size on
pressure drop is investigated.

+
+
+
+
+
+
++
+
++
+
+
+
+
Re
AveNu
101
102
103
20
40
60
80
100
120 Present Study - Pasrticle size 5 mm
Present Study - Particle size 10 mm
Present Study - Particle size 15 mm
Present Study - Particle size 20 mm
Present Study - Particle size 25 mm
Wilke and Hougen
De Acetis and Thodos
Galloway et al
Satterfield et al

+
p
Figure 9 – Effect of particle size on Ave Nu versus pRe ,
~ =0.9
+
Re
-dp/dx
100
101
102
103
100
101
102
103
Darcy Forchheimer 1
Darcy Forchheimer 2
Darsy Law 1
Darcy Law 2
Present Study - Particle size 0.5 mm
Present Study - Particle size 10 mm
Present Study - Particle size 15 mm
Present Study - Particle size 20 mm
Present Study - Particle size 25 mm
+
p
Figure 10 -Effect of particle size on pressure drop versus
pRe , ~ =0.9
Conclusion
A direct numerical method for simulating of flow in porous
media was discussed. A random distribution of solid obstacles
was performed. The effect of channel walls and porous media
on velocity and temperature field were taken into account. The
result were validated with other reported data and showed great
agreement. It was observed that the average Nu number varies
approximately linearly with pRe . The nature of the flow
regimes were also investigated in the porous media. At the end
the effect of particle size was also reported that by increasing
the particles size the average Nu number will decrease like the
pressure drop.

5. 5 Copyright © 2009 by ASME
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