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Proceedings of the ASME 2017 Fluids Engineering Division Summer Meeting
FEDSM2017
July 30-August 3, 2017, Waikoloa, Hawaii, USA
FEDSM2017-69582
NUMERICAL STUDY OF HEAT TRANSFER IN TURBULENT
CROSS-FLOW OVER POROUS-COATED CYLINDER
N. Rahmati
Department of Mechanical Engineering, Amirkabir
University of Technology, Tehran, Iran
Z. Mansoori
Energy Research Center, Amirkabir University of
Technology, Tehran, Iran
M. Saffar-Avval
Department of Mechanical Engineering, Amirkabir
University of Technology, Tehran, Iran
G. Ahmadi
Department of Mechanical and Aeronautical
Engineering, Clarkson University, Potsdam, NY,
USA
engineering applications [1,2]. The properties of porous media
such as high thermal conductivity, high effective surface area,
and increased fluid mixing has led to the use of porous
wrapping as an effective tool to enhance heat transfer [3].
In recent years, several researchers have studied porouscoated cylinders for enhancing heat transfer. However, these
efforts have been limited mostly to laminar flows over porous
layer. For example, Odabaee et al. [4] studied forced
convection heat transfer from a cylinder embedded inside a
metal foam layer numerically, where the laminar flow over and
inside the porous medium was assumed. They evaluated the
optimum porous layer thickness for maximum heat transfer
enhancement, and showed that the porous wrapped cylinder has
higher heat transfer rate compared to the fin-tubed heat
exchanger. Saada et al. [5] investigated the natural convection
heat transfer around a horizontal porous-covered cylinder and
assumed that the fluid flow over and within the porous layer is
laminar. Bhattacharyya and Singh [1] conducted a numerical
study of the laminar flow around a porous layer wrapped
cylinder, where the mixed convection is studied for different
values of flow parameters. Al-Sumaily et al. [3] investigate
numerically the forced convection heat transfer from a porous
wrapped cylinder.
In many cases the cross-flow over porous-covered cylinder
is in turbulent regime and the fluid flow in the porous layer
could be laminar or turbulent depending on porosity properties.
In [6,7] the turbulent fluid flow in porous layer was modeled
and the corresponding flow configuration and heat transfer
ABSTRACT
In the present paper, a numerical study has been conducted
to investigate the heat transfer from a constant temperature
cylinder covered with metal foam. The cylinder is placed
horizontally and is subjected to a constant mean cross-flow in
turbulent regime. The Reynolds Averaged Navier-Stokes
(RANS) and Darcy-Brinkman-Forchheimer equations are
combined and used for flow analysis. The energy equation used
assumes local thermal equilibrium between fluid and solid
phases in porous media. The k-ω SST turbulence model is used
to evaluate the eddy viscosity that is implemented in the
momentum and energy equations. The flow in the metal foam
(porous media) is in laminar regime. Governing equations are
solved using the finite volume SIMPLEC algorithm. The effect
of thermophysical properties of metal foam such as porosity
and permeability on the Nusselt number is investigated. The
results showed that using a metal porous layer with low
porosity and high Darcy number in high Reynolds number
turbulent flows markedly increases heat transfer rates. The
corresponding increase in the Nusselt number is as high as 10
times that of a bare tube without the metal foam.
INTRODUCTION
Heat transfer from a horizontal cylinder covered with a
porous layer in cross flows has received considerable attention
over the last several decades.
Design of compact heat
exchanger, effective cooling of electronic systems, geothermal
energy system, catalytic reactors, and refrigeration are a few
1
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Reynolds number, 2  f U  Rs  f
Re
t
Time, (s)
Temperature,
(K)
T
u
Radial velocity component
U
Free stream velocity, (m/s)
v
Tangential velocity component
Greek symbols

Porosity
were investigated. In most situations, a composite porous-fluid
system is used where the flow in the clear fluid region is
turbulent while in the porous layer the flow remains in laminar
regime. For example, Sobera et al. [8] conducted a numerical
study for heat and mass transfer in turbulent flow from a
circular cylinder sheathed by a porous layer with applications
in human clothing. Kuznetsov [9] investigated the turbulent
flow in a composite porous-fluid duct and reported that the
roughness of the porous-fluid interface significantly impacts
the turbulent flow regime in the clear fluid region as well as the
overall heat transfer in the duct. In this analysis, a roughness
related to the pore diameters of the porous layer was applied at
the interface.
In the present study, a computational model was used and
the convective heat transfer under turbulent flow regime from a
porous-coated cylinder was studied. Here it is assumed that the
flow in the porous region remains laminar, and the surface
roughness at the porous-fluid interface was accounted for in the
boundary condition of the clear fluid turbulent flow equations.
The SST k-ω turbulence model, which is suitable for predicting
the separation of flow over the porous-fluid interface, was used
in the simulations. The simulations results showed that the
porous surface wrapping significantly increase the heat
transfer.
p
Porous layer thickness
f
Dynamic viscosity, (kg/ms)
t
t , n

Turbulent viscosity, (kg/ms)
Dimensionless turbulent viscosity, (kg/ms)
 c 
p eff


1
Density, (kg/m3)
Effective thermal capacity of the porous

  1     c  ,(J/m3K)
Specific heat ratio,   c 
 c 
medium,   c p
s
f
p eff
p
f
Tangential coordinate
Tangential angle measured from the forward
stagnation point
Subscripts

f
p
s
NOMENCLATURE
c
CF
cp
Forchheimer coefficient
Cp
Pressure coefficient
Da
e
F
kf
Darcy number
Binary parameter for porosity
Binary parameter
Thermal conductivity of fluid, (W/mK)
Effective thermal conductivity of porous
material,  k f  1    k s , (W/mK)
keff
K
k
Nu
Nuave
Nu0
P
Pr
R
Solid specific heat, J/kg K
Free stream
Fluid
Porous medium
Solid
Specific heat, (J/kgK)
GOVERNING EQUATIONS
In this section, the governing equations for turbulent flow
over a porous-coated cylinder are described. Fig. 1 shows the
computational domain for the flow over a porous-coated
cylinder that is is placed horizontally in a uniform cross flow.
The outer boundary condition is set at R∞=30Rp. The tube
surface temperature is kept constant at a temperature that is
higher than the ambient temperature (T∞). It is assumed that the
porous media is isotropic, and the fluid and solid phases in the
porous media are locally at thermal equilibrium.
Permeability of the metal foam, (m2)
Turbulent kinetic energy, (m2/s2)
Local Nusselt number
Total average Nusselt number
Total average Nusselt number of a solid
cylinder
pressure, (N/m2)
Prandtl number, c p  f k f
Rs
Radial coordinate
Radius of cylinder, (m)
Rc
Thermal conductivity ratio, keff k f
Fig. 1 A schematic of computational domain and fluid flow
in the cylindrical coordinate.
2
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In this paper, we use the single-domain approach to model
the composite system of the clear fluid zone and porous
medium. Hence, a single set of equations is used in the
computations to model the fluid flow in the porous medium as
well as the fluid region. The momentum equations are
generated by combining the Reynolds Averaged Navier-Stokes
(RANS) and Darcy-Brinkman-Forchheimer equations [10].
Also, other equations are obtained by combining the governing
equations in the porous medium and fluid region. The final
non-dimensional equations can be expressed as:
  ru 
r

v
0

T 1   ruT  1   vT 


t r r
r 
  T 
 1 T 

 
Pr
Pr
Rc  t ,n    (4)

   r  Rc  t ,n  



t
t
2  1  r 
  1  r  
 




r

r
RePr r






where  t =0.85. The non-dimensional parameters in equations
(1)-(4) are:
2  f U  Rs
cp  f
K
Re 
, Da  2 , Pr 
,
f
kf
Rs
(5)
  c eff
keff
t

,R 
, 
  c p  c k f t ,n  f

(1)
  ru 2 

 uv 
 



u  1  e  1  e  v2 



t  r r
r 
er 




f
The switching parameters in these equations are given in
Table1.
The SST k-ω turbulence model developed by Menter [11] is
used for evaluation of turbulence properties including the nondimensional eddy viscosity in the fluid region. The transport
equations for k and ω are not shown here for the sake of
brevity.

 1  t ,n  u  
   r 1   u 




t
n
,


  
r
P 2  1 
r  1 

 



r Re  r
r

r






v 

( )  
1  
u  1  
r

 r 1  t ,n  r   r   r 1  t ,n  r 

2  r r 





e Re 


 2 1  t ,n   1 v u 



r  r


r
 



C e
 2e

F
u F
u 2  v2 u 
Da
 Re Da

Table 1. The switching parameters in clear fluid region and fluid
flow in porous medium
Fluid flow in porous
Fluid region
F
1
0
1
e
(2)
  ruv 

 v2 
 
 


v  1  e  1  e  uv 



t  r r
r 
er 





( cp )eff
   c p  f  1    cs
 c 
keff
 k f  1   ks
kf
t
0
t
p f
NUMERICAL METHOD
The set of governing equations is discretized on the
staggered grid by using finite volume method. A pressure
correction-based iterative algorithm, SIMPLEC (Van Doormal
and Raithby) is used for solving the governing equations. The
convective and diffusion terms are discretized through a power
law differencing scheme. For convergence criteria, the
maximum relative errors on the dependent variables are set to
10-5.
A non-uniform grid distribution along the radial direction
and uniform grid distribution along the tangential direction is
generated. The grid independence study is examined by
considering four sets of non-uniform grids, namely, 197×100
(grid 1), 259×140 (grid 2), 293×200 (grid 3), and 350×220
(grid 4). Here the first and second numbers being the number
of grid points, respectively, in r- and θ-directions. The
minimum cell size in each grid in r-direction is 0.3mm, 0.1mm,
 
v 
v  
1
 r 1  t ,n  
 1  t ,n   
1 P 2  1 
1  r
 


r





r

r  Re  r
r





 

 u v  
 
   r 1  t ,n  
 1  t ,n   v



r
r
1
1





 

 2u  

r
r
r  
r
 
 


2 



 v

e Re 


1  t ,n    r  u 



r




r
r 
 r








 2e

C e
F
v F
(u 2  v2 v 
Da
Re
Da


Parameter
(3)
3
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ratio for the cylinder with metal foams is in the order of 4-10,
which represents significant augmentation of the heat transfer
by using a porous layer over the cylinder.
0.05 mm and 0.025 mm. The difference between the average
Nusselt number in grid 3 and grid 4 is less than 0.3%. Hence,
grid 3 with the minimum cell size of 0.05 mm in r-direction is
selected and used for the computation.
To check the validity of the numerical methods, the
predicted local Nusselt number for a bare tube for laminar flow
at Reynolds number of Re=20 are compared with the earlier
results of Denis and Chang [12] in Fig. 2. This figure shows
that good agreement of the present predictions of the local
Nusselt number with the earlier results. The predicted average
Nusselt number of a bare tube for different Reynolds numbers
in turbulent flow regime are compared with the empirical
correlations presented by Incropera et al. [13] in Table 2. This
table shows that the model predictions are in reasonable
agreement with the data.
Fig. 3 Pressure coefficient distributions at Re=106.
Fig. 2 Local Nusselt numbers for a bare cylinder with those
at Re=20.
Fig. 4 Average Nusselt number ratio (Nu/Nu0) versus Re
number.
Table 2. Comparison of the predicted average Nusselt
numbers for a bare cylinder with the empirical models of
Incropera et al. [13], at different Reynolds number.
Re
N u a v e (our results)
N u a v e [1 3 ]
104
105
5×105
Relative error
(%)
58.35
229.1
670.8
53.84
216.19
701.88
8.38
5.97
4.43
The effect of variations of Darcy number on the average
Nusselt number are also studied and the results for Re=104 are
shown in Fig. 5. It is seen that the Nusselt number increases
with increasing the Darcy number. Foams with low Darcy
numbers behave the same as a solid layer because low Darcy
number represents low permeability. For Re=104, Fig. 5 shows
that there no change in the heat transfer by reducing the Darcy
number from 10-6 to 10-8.
One key parameters is the porosity of porous layer. The
porosity affects the Darcy resistance in the momentum
equations and also the effective thermal conductivity of porous
material (keff). The effect of porosity on the velocity is small,
but the effective thermal conductivity significantly changes
with the porosity. Fig. 6 shows that increasing the porosity
reduces the effective thermal conductivity, and decreasing the
porosity enhances the heat transfer.
The predicted pressure coefficient distributions for Re=106
are plotted in Fig. 3 and are compared with the earlier
numerical simulations of Ong et al. [14]. It is seen that the
present results are in conformity with the earlier works.
RESULTS AND DISCUSSION
In this section, the effects of key parameters of fluid and
metal foam layer such as Reynolds number, Darcy number, and
porosity of porous layer on the heat transfer are evaluated and
discussed. Fig. 4 shows the variations of the average Nusselt
number ratio (Nu/Nu0) with flow Reynolds number for a
porous wrapped cylinder with Da=10-4. Here metal foam with
the following properties is used: porosity=0.9, dimensionless
thickness=1.0, CF=0.167. It is seen that the Nusselt number
CONCLUSIONS
In this study, we investigated the heat transfer from a
horizontal cylinder covered with the metal foam layer under
4
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turbulent flow regime with constant temperature. The RANS
equations with the SST k-ω turbulence model in conjunction
with the single-domain approach were used in the simulations.
It was assumed that the fluid flow in porous medium remained
in laminar regime. The finite volume method with SIMPLEC
algorithm was used for numerical integration of the governing
equations, and the computation model was validated by
comparison of the model predictions with the earlier results and
empirical equations. It was shown that the presence of porous
wrapping significantly augmented the heat transfer from the
cylinder. The corresponding Nusselt number increase to almost
10 times that of a bare tube. Effects of thermophysical
properties of foam layer including the Darcy number and
porosity on the Nusselt number were also studied.
[3]
[4]
[5]
[6]
[7]
[8]
Fig. 5 Average Nusselt number versus log(Da) at Re=104.
[9]
[10]
[11]
[12]
[13]
Fig. 6 Average Nusselt number versus porosity at Re=104,
Da=10-4.
[14]
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[1]
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