Analysis of Heat and Mass Transfer in a Porous Cavity Containing Water near Its Density Maximum

doi:10.4236/am.2011.28127

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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

E-mail: [email protected]

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



1Tc

 



 







)



(8.010





 (˚C)

,3.010mkg







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





 (1)



1/2

3/2

11.75

150

uu v

puu

xK K





 

















 



(2)



1/2

3/ 2

11.75

150

vuv

pvv

yK K





























 



(3)













 (4)

uv D

 c





 (5)

The appropriate initial and boundary conditions are:

0:0,,, 01,01

tuvccxy





 

0 :0,0,01,1

tuv xy





 



0,,,0, 01

uvccy x





 

0,,,0 &1,01

uv ccxy





  

Dimensionless variables are defined as follows:









,,,,

, with,and

hchc

xy uv

XYUVT

cc pL

Ccc

cc v





 













The non-dimensional form of the Equations (1)-(5) is

obtained as:







 (6)

221/

3/ 2

PrPr1.75 ()

150

XDa



















 



(7)



1/ 2

3/2

Pr Pr1.75

150

VU V

PVV

YDa

Ra TR NC





















 









(8)

 T





 (9)

1CC

XYSc

 C





 (10)

The initial and boundary conditions in dimensionaless

form are:

0:0,0, 01,01UV CTXY





 

M. MUTHTAMILSELVAN

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0:0,0,01, 1

UVX Y





 



0,1,1,0, 01UVC TYX

0,0,0,0&1, 01UVC TXY

The dimensionless variables are defined as:

Darcy number, Pr





, Pr andtl number,

Sc D



, Schmidt number,









L

Solutal Rayleigh number, 1c





, density inversion

parameter, ,

NRa

 Buoyancy ratio number.

The average Nusselt number and Sherwood number is

d, d

avg avg

NuX ScX











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

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

M. MUTHTAMILSELVAN

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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,”

Journal of Computational Physics, Vol. 98, No. 1, 1992,

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



coefficient of thermal expansion, K–1



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