Applied Mathematics, 2011, 2, 927-934
doi:10.4236/am.2011.28127 Published Online August 2011 (http://www.SciRP.org/journal/am)
Analysis of Heat and Mass Transfer in a Poro us Cavity
Containing Water near Its Density Maximum
Murugan Muthtamilselvan
Department of Applied Mathematics, Bharathiar University, Coimbatore, India
Received September 23, 2010; revised May 31, 2011; accepted June 7, 2011
Abstract
A numerical study is performed on steady natural convection inside a porous cavity with cooling from the
side walls. The governing equations are solved by finite volume method. Representative results illustrating
the effects of the thermal Rayleigh number, density inversion parameter, buoyancy ratio and Schmidt num-
ber on the contour maps of the fluid flow, temperature and concentration are reported. It is found that the
number of cells which form in the cavity varies primarily with the density inversion parameter and is always
even due to the symmetry imposed by the cold sidewalls. In addition that the flow becomes weaker as the
Darcy number decreases from the pure fluid limit towards the Darcy-flow limit.
Keywords: Convection, Double Diffusive, Porous Medium, Finite Volume Method
1. Introduction
Natural convection in enclosures containing a fluid-
saturated porous medium has received much attention
recently because the data concerning buoyant enclosure
flow are in great demand for many traditional and con-
temporary applications such as insulating systems for
buildings and heat exchanger devices, energy storage
systems, material processing and geothermal systems
[1,2]. Another reason leading to study of the convection
in porous medium enclosures is the interest in funda-
mental phenomena of double diffusive and density of the
convective flow. A comprehensive review of the litera-
ture on double diffusion, natural convection in saturated
porous media may be found in [3-6].
Convection in water behaves differently around the
temperature region of 4˚C due to the anomalous behavior
of density around this temperature and the density of
water at both side walls of the cavity various linearly
with the temperature. A number of studies have investi-
gated the effect of the density extremum for water in
detail. Nansteel et al. [7] studied the natural convection
of cold water in the vicinity of its density maximum in a
rectangular enclosure in the limit of small Rayleigh
number. They observed that the strength of the counter
rotating flow decreases with decreasing aspect ratio. Ma-
hidjiba et al. [8] investigated onset of convection in a
horizontal anisotropic porous layer saturated with water
near 4˚C. It is found that the onset of motion dependent
permeability ratio and inversion parameter. The Brink-
man’s extension of Darcy’s law has been used in a study
by Bannacer et al. [9] to investigate double diffusive
convection in anisotropic porous media with high poros-
ity. It is demonstrated that the anisotropic properties of
the porous medium considerably modify the heat and
mass transfer rates from that expected under isotropic
conditions.
Very recently, Muthtamilselvan et al. [10] studied
numerically convection in a two-sided lid-driven heat
generating porous cavity with alternative thermal bound-
ary conditions. They found that the variation of the av-
erage Nusselt number is non-linear for increasing values
of the Darcy number with uniform and non-uniform
heating condition. The aim of the present study a double
diffusive natural convection flow in a square cavity filled
with a fluid saturated porous medium when the bottom
wall is heated.
2. Mathematical Analysis
Consider a steady-state two-dimensional square cavity
filled with a porous medium of length L as shown in
Figure 1. Different temperature and concentrations are
imposed between the bottom (h

, ch) and vertical side
walls (c

, cc), where the hc

  and hc . The top
wall is considered to be adiabatic. The physical proper-
cc

928 M. MUTHTAMILSELVAN
Figure 1. Flow configuration and coor dinate syste m.
ties are considered to be constant except the density
variation in the body force term of the momentum equa-
tion which is satisfied by the Boussinesq’s approxima-
tion. The density of the cold water is assumed to vary
with temperature according to the following equation
 
2
01
1Tc cc  

 

0


31
)
 6
(8.010
T


 (˚C)
23
,3.010mkg
c
K 


where T

and c

are the coefficients for thermal and
concentration expansions. In the present investigation the
porous medium is assumed to be hydrodynamically and
thermally isotropic and saturated with a fluid that is in
local thermal equilibrium (LTE) with the solid matrix. A
general Brinkman-Forchheimer extended Darcy model is
used to account for the flow in the porous medium. Us-
ing the above assumptions, the governing equations for
mass, momentum and energy can be written in the di-
mensional form as
0
uv
xy


 (1)

2
1/2
22
2
3/2
0
1
11.75
150
uu
uv
xy
uu v
puu
xK K


 








 

(2)

2
1/2
22
2
3/ 2
00
1
11.75
150
vv
uv
xy
vuv
pvv
yK K

g















 

(3)
2
uv
xy






 (4)
2
cc
uv D
xy
 c


 (5)
The appropriate initial and boundary conditions are:
0:0,,, 01,01
cc
tuvccxy



 
0 :0,0,01,1
c
tuv xy
yy
 
 

0,,,0, 01
hh
uvccy x



 
0,,,0 &1,01
cc
uv ccxy



  
Dimensionless variables are defined as follows:


 
2
2
0
,,
,,,,
, with,and
c
hc
c
hchc
hc
xy uv
XYUVT
LL
cc pL
Ccc
cc v


 






,
P 
The non-dimensional form of the Equations (1)-(5) is
obtained as:
0
UV
XY



 (6)
2
221/
2
3/ 2
1
PrPr1.75 ()
150
UU
UV
XY
PU
UU
XDa
Da









 

2
UV
(7)


2
1/ 2
22
2
3/2
2
1
Pr Pr1.75
150
Pr
T
VV
UV
XY
VU V
PVV
YDa
Da
Ra TR NC










 




(8)
2
1
Pr
TT
UV
XY
 T


 (9)
2
1CC
UV
XYSc
 C


 (10)
The initial and boundary conditions in dimensionaless
form are:
0:0,0, 01,01UV CTXY


 

M. MUTHTAMILSELVAN
929
0:0,0,01, 1
TC
UVX Y
YX


 

0,1,1,0, 01UVC TYX
0,0,0,0&1, 01UVC TXY
The dimensionless variables are defined as:
2,
K
Da
H
Darcy number, Pr


, Pr andtl number,
Sc D

, Schmidt number,

23
2
hc
T
g
T
Ra



L
,
Solutal Rayleigh number, 1c
hc
R





, density inversion
parameter, ,
c
T
Ra
NRa  Buoyancy ratio number.
The average Nusselt number and Sherwood number is
11
00
0
d, d
avg avg
YY
TC
NuX ScX
YY





0
3. Method of Solution
The governing equations along with the boundary condi-
tions are solved numerically employing finite volume
method using staggered grid arrangement. The semi-
implicit method for pressure linked equation (SIMPLE)
is used to couple momentum and continuity equations as
given by Patankar [11]. The third order accurate deferred
QUICK scheme of Hayase et al. [12] is employed to
minimize the numerical diffusion for the convective
terms for both the momentum equations and energy
equation. The solution of the discretized momentum and
pressure correction equations is obtained by TDMA
line-by-line method. For complete details and code vali-
dation, the author is referred to the prior publication [10,
13]. The grid independence test is performed using suc-
cessively sized uniform grids, 21 × 21 to 91 × 91. After
grid independence check considering the accuracy and
the computational time, all the computations are per-
formed with a 41 × 41 grid.
4. Results and Discussion
Computations have been carried out for various values of
the Darcy number (Da = 10–4 - 10–1), the thermal
Rayleigh number (RaT = 102 - 105), the buoyancy ratio
number (N = 0.2 - 0.8), the density inversion parameter
(R = 0.1 - 0.3), the Schmidt number (Sc = 5 - 50) and
fixed value of Pr = 11.573 and porosity (ε = 0.4). The
results are presented as streamlines, isotherms and iso-
concentrations. The rate of heat and mass transfer in the
enclosure is measured in terms of the average Nusselt
and Sherwood numbers.
Figure 2 illustrates the streamlines, isotherms and iso-
concentration of the numerical results for various thermal
Rayleigh number. When RaT = 102 the flow is seen to be
very weak as observed from streamlines. Therefore, the
temperature and isoconcentration distributions are simi-
lar to that with stationary fluid and the heat transfer is
due to purely conduction. When RaT = 103, streamline
show the major cells occupied entire cavity and the mi-
nor cells are visible near bottom corners of the cavity.
During conduction dominant mass transfer, the isocon-
centration contours with c = 0.06 occur symmetrically
near the side walls of the cavity. The other isoconcentra-
tion contours with c ≥ 0.13 are smooth curves which
span the entire cavity and they are generally symmetric
with respect to the vertical symmetric line. As thermal
Rayleigh number increases from 104 to 105 streamline
shows that the bottom corner minor cells are reduced its
strength and size. The corresponding temperature con-
tour illustrates the smooth cure bended towards bottom
of the cavity. It should be noted in Figure 2, with in-
creasing RaT from 102 to 104 the minor cells grow its size
and reduced when RaT = 105. It is observed that the con-
duction heat transfer switches to convection heat transfer
with increasing RaT from 102 to 104. Again the conduc-
tion heat transfer dominates when RaT = 105.
Streamline, isotherm and isoconcentration contours
are displayed in Figure 3, for different values of Schmidt
number. The streamline shows the minor cells are occu-
pied top corners of the cavity and it increases its size and
strength when Sc increases. The circulations are greater
near the center and least at the wall due to no slip
boundary conditions. The greater circulation in each half
of the box follows a progressive wrapping around the
centers of rotation and a more pronounced compression
of the isotherms toward the boundary surfaces of the
cavity occurs. Due to greater circulations near the central
core at the top half of the cavity, there are small gradi-
ents in temperature whereas a large stratification zone of
temperature is observed at the vertical symmetry line due
to stagnation of flow.
Figure 4 shows the average Nusselt and Sherwood
numbers for different density inversion parameter with
different Darcy number. Decreasing of Darcy number
decreased the heat and mass transfer rate. In such situa-
tion the heat transfer is dominated by conduction. Figure
5 illustrate that the average Nusselt and Sherwood num-
ber for thermal Rayleigh number Vs density inversion
parameter. Decreasing of thermal Rayleigh number de-
creased the heat and mass transfer rate. Figure 6 shows
the behavior of the average Nusselt and Sherwood num-
ber for different N Vs Sc. It is found that the average
Nusselt and Sherwood number gets minimum in the den-

M. MUTHTAMILSELVAN
930
Figure 2. Streamlines, isotherms, iso-concentr ation for differe nt RaT, Da = 10–2, N = 0.5, Sc = 5, R = 0.2.

M. MUTHTAMILSELVAN
931
Figure 3. Streamlines, isotherms, iso-concentr ation for differe nt Sc, Da = 10–2, N = 0.2, RaT = 105.
Figure 4. Average Nusselt numbe r and Sherwood number for Da Vs R.

M. MUTHTAMILSELVAN
932
Figure 5. Average Nusselt numbe r and Sherwood number for RaT Vs R.
Figure 6. Average Nusselt numbe r and Sherwood number for Sc Vs N.
sity maximum region. For such a situation, the dual cell
structure inhibits the direct convective transfer of energy
from one cell to another. This phenomenon results essen-
tially from the inversion of the fluid density at 4˚C and is
one of its most significant effects on the mechanism of
heat transfer by convection of water within the cavity.
5. Conclusions
Numerical computations are performed to study natural
convection heat and mass transfer in a porous enclosure
containing water near its density maximum. It is ob-
served that the density inversion leaves strong effects on
fluid flow, heat and mass transfer due to the formation of
bi-cellular stature. The heat and mass transfer rate be-
haves nonlinearly with density inversion parameter and
Schmidt number. The heat and mass transfer rates are
found to decrease with decreasing thermal Rayleigh
number. At the onset of convection dominant mode, the
temperature contour lines get compressed toward the side
walls and they tend to get deformed towards the upward
direction.
6. References
[1] D. A. Nield and A. Bejan, “Convection in Porous Media
(Third Edition),” Springer-Verlag, New York, 2006.
[2] K. Vafai, “Handbook of Porous Media,” 2nd Edition,
Marcel Dekker Inc., New York, 2005.
doi:10.1201/9780415876384
[3] F. Alavyoon, “On Natural Convection in Vertical Porous
Enclosures Due to Prescribed Fluxes of Heat and Mass
Transfer at the Vertical Boundaries,” International Jour-
nal of Heat and Mass Transfer, Vol. 36, No. 10, 1993, pp.
2479-2498. doi:10.1016/S0017-9310(05)80188-7
[4] M. Mamou, P. Vasseur and E. Bilgen, “Multiple Solution
for Double-Diffusive Convection on a Vertical Porous
Enclosure,” International Journal of Heat and Mass
Transfer, Vol. 38, No. 10, 1995, pp. 1787-1798.

M. MUTHTAMILSELVAN
933
doi:10.1016/0017-9310(94)00301-B
[5] M. Nishimura, M. Wakamatsu and A. M. Morega, “Os-
cillatory Double-Diffusive Convection in a Rectangular
Enclosure with Combined Horizontal Temperature and
Concentration Gradients,” International Journal of Heat
and Mass Transfer, Vol. 41, No. 11, 1998, pp. 1601-1611.
doi:10.1016/S0017-9310(97)00271-8
[6] I. Sezai and A. A. Mohamad, “Three-Dimensional Dou-
ble-Diffusive Convection in a Porous Cubic Enclosure
Due to Opposing Gradients of Temperature and Concen-
tration,” The Journal of Fluid Mechanics, Vol. 400, 1999,
pp.333-353. doi:10.1017/S0022112099006540
[7] M. W. Nansteel, K. Medjani and D. S. Lin, “Natural
convection of Water near Its Density Maximum in Rac-
tangular Enclosure: Low Rayleigh Number Calculations,”
Physics of Fluids, Vol. 30, No. 2, 1987, pp. 312-317.
doi:10.1063/1.866379
[8] A. Mahidjiba, L. Robillard and P. Vasseur, “On Set of
Convection in a Horizontal Anisotropic,” International
Communications in Heat and Mass Transfer, Vol. 27, No.
6, 2000, pp. 765-774.
doi:10.1016/S0735-1933(00)00157-3
[9] R. Bennacer, A. Tobbal, H. Beji and P. Vasseur, “Double
Diffusive Convection in a Vertical Enclosure Filled with
Anisotropic Porous Media,” International Journal of
Thermal Sciences, Vol. 40, No. 1, 2001, pp. 30-41.
doi:10.1016/S1290-0729(00)01185-6
[10] M. Muthtamilselvan, M. K. Das and P. Kandaswamy,
“Convection in a Lid-Driven Heat Generating Porous
Cavity with Alternative Thermal Boundary Conditions,”
Transport in Porous Media, Vol. 82, No. 2, 2010, pp.
337-346. doi:10.1007/s11242-009-9429-7
[11] S. V. Patankar, “Numerical Heat Transfer and Fluid
Flow,” Hemisphere, Washington DC, 1980.
[12] T. Hayase, J. A. C. Humphrey and R. Grief, “A Consis-
tently Formulated QUICK Scheme for Fast and Stable
Convergence Using Finite-Volume Iterative Procedures,”
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pp. 108-118. doi:10.1016/0021-9991(92)90177-Z
[13] P. Kandaswamy, M. Muthtamilselvan and J. Lee,
“Prandtl Number Effects on Mixed Convection in a Lid-
Driven Porous Cavity,” Journal of Porous Media, Vol. 11,
2008, pp. 791-801. doi:10.1615/JPorMedia.v11.i8.70

934 M. MUTHTAMILSELVAN
Nomenclature
C dimensional concentration
Da Darcy number
G gravitational acceleration
Gr Grashof number
L enclosure length
KT thermal diffusion ratio
K effective thermal conductivity of the porous medium
Nu Nusslet number
Nuavg average Nusslet number
P Pressure
Pr Prandtl number
R dimensional density inversion parameter
Rac solute Rayleigh number
RaT thermal Rayleigh number
Sc Schmidt number
Sh Sherwood number
Shavg average Sherwood number
T dimensionless temperature
U, V dimensionless velocities in X- and Y-direction
u, v velocities in x- and y-direction
X, Y dimensionless Cartesian coordinates
x, y Cartesian coordinates
Greek symbols

effective thermal diffusivity of porous medium, m2·s–1
T

coefficient of thermal expansion, K–1
c

coefficient of solute expansion

 temperature difference

temperature, ˚C
K permeability of porous medium, m2

effective dynamic viscosity
v effective kinematic viscosity

fluid density, kg·m–3

Porosity

dimensionless time
Subscripts
Avg Average
C cold wall
H hot wall