International Communications in Heat and Mass Transfer 32 (2005) 454–463

www.elsevier.com/locate/ichmt

Natural convection in porous cavity with sinusoidal

bottom wall temperature variationB

Nawaf H. Saeid*

School of Mechanical Engineering, University of Science Malaysia, 14300 Nibong Tebal, Pulau Pinang, Malaysia

Available online 20 December 2004

Abstract

Numerical study of natural convection in a porous cavity is carried out in the present paper. Natural convection

is induced when the bottom wall is heated and the top wall is cooled while the vertical walls are adiabatic. The

heated wall is assumed to have spatial sinusoidal temperature variation about a constant mean value which is

higher than the cold top wall temperature. The non-dimensional governing equations are derived based on the

Darcy model. The effects of the amplitude of the bottom wall temperature variation and the heat source length on

the natural convection in the cavity are investigated for Rayleigh number range 20–500. It is found that the average

Nusselt number increases when the length of the heat source or the amplitude of the temperature variation

increases. It is observed that the heat transfer per unit area of the heat source decreases by increasing the length of

the heated segment.

D 2004 Elsevier Ltd. All rights reserved.

Keywords: Natural convection; Porous cavity; Non-uniform wall temperature; Numerical study

1. Introduction

Convective heat transfer in fluid-saturated porous media is a research topic of practical importance

due to the wide range of geophysical and engineering applications. These include high performance

insulation for buildings, grain storage, energy efficient drying processes, solar collectors, etc.

Representative reviews of these applications and other convective heat transfer applications in porous

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cheatmasstransfer.2004.02.018

ated by J.P. Hartnett and W.J. Minkowycz.

593 7788; fax: +60 4 594 1025.

ress: n_h_saeid@yahoo.com.

N.H. Saeid / Int. Commun. Heat and Mass Transf. 32 (2005) 454–463 455

media may be found in the recent books by Ingham and Pop [1], Nield and Bejan [2], Vafai [3] and

Bejan and Kraus [4].

The problem of natural convection in a porous cavity whose four walls are maintained at different

temperatures or heat fluxes is one of the classical problems in porous media. Much research work, both

theoretical and experimental, has been done on this type of convective heat transfer problems. The

natural convection can be induced by either heating from side with horizontal walls adiabatic or heating

from below with vertical walls adiabatic. A good deal of research work on the heating from side problem

has been presented by Walker and Homsy [5], Bejan [6], Goyeau et al [7], Mohamad [8] and in the

recent paper by Saeid and Pop [9], while the natural convection induced by heating from below has been

studied by Horne and O’Sullivan [10], Prasad and Kulacki [11], Kazmierczak and Muley [12] and Nield

[13] among others.

The literature shows that the flow and heat transfer characteristics for the constant boundary

temperature condition is generally studied for this type of cavity. However, very little work has been

done for the natural convection in porous cavities with boundary walls having non-uniform

temperatures. The problem of free convection in a vertical porous layer with walls at non-uniform

temperatures has been studied by Storesletten and Pop [14], Bradean et al. [15] and Yoo [16]. The effect

of non-uniform temperature on the convection in a fluid-saturated porous medium between two infinite

horizontal walls has been studied by Yoo and Schultz [17]. They obtained an analytical solution for both

vertical and horizontal porous layers at low Rayleigh number conditions.

The aim of this paper is to study numerically the natural convection in porous cavity with non-

uniform hot wall temperature and uniform cold wall temperature. The hot and cold walls are the

horizontal walls while the vertical walls are adiabatic. The heated wall is the bottom wall and it has

spatial sinusoidal temperature variation about a constant mean value which is higher than the cold top

wall temperature. This spatial sinusoidal temperature variation occurs in the applications when a

cylindrical heater is placed on a flat wall. There will be one contact point between the circular cross-

section of the heater and the wall, which gives maximum temperature at the contact region. The

temperatures before and after the contact region is less than the maximum value because the heater

surface is relatively far from the wall at these regions.

2. Basic equations

A schematic diagram of the two-dimensional cavity of length 2L and height L filled with a porous

media, under the present investigation, is shown in Fig. 1. All the cavity walls are impermeable and the

vertical walls are adiabatic. A finite heat source of length 2D is located on the bottom surface which is

otherwise adiabatic. The heat source is affected by the presence of sinusoidal temperature variation about

a constant mean value which is higher than the upper wall constant temperature.

In the porous media, the following assumptions are made:

1. The convective fluid and the porous media are in local thermal equilibrium.

2. The properties of the fluid and the porous media are constants.

3. The viscous drag and inertia terms of the momentum equations are negligible.

4. The Boussinesq approximation is valid.

5. Darcy law is applicable.

Fig. 1. Schematic diagram of the physical model and coordinate system.

N.H. Saeid / Int. Commun. Heat and Mass Transf. 32 (2005) 454–463456

Under these assumptions, the conservation equations for mass, momentum and energy for the two-

dimensional steady natural convection in the porous cavity are:

Bu

Bxþ Bv

By¼ 0 ð1Þ

Bu

By� Bv

Bx¼ � gbK

y

BT

Bxð2Þ

uBT

Bxþ v

BT

By¼ a

B2T

Bx2þ B

2T

By2

� �ð3Þ

where u, v are the velocity components along x- and y-axes, T is the fluid temperature and the physical

meaning of other quantities are mentioned in the nomenclature. It is assumed that the temperature of the

hot wall has a sinusoidal variation about a mean value of T̄h in the form:

Th xð Þ ¼ T¯ h þ e T¯ h � TcÞcos px=2Dð Þ�

ð4Þ

where e denotes the amplitude of the hot wall temperature variation. Eqs. (1)–(3) are subject to the

following boundary conditions:

u � L; yð Þ ¼ 0; BT � L; 0ð Þ=Bx ¼ 0 ð5aÞ

u L; yð Þ ¼ 0; BT L; yð Þ=Bx ¼ 0 ð5bÞ

v x; Lð Þ ¼ 0; T x;Lð Þ ¼ Tc ð5cÞ

v x; 0ð Þ ¼ 0; and T x; 0ð Þ ¼ Th xð Þ at � DVxVD; BT x; 0ð Þ=By ¼ 0 at � DNxND ð5dÞ

N.H. Saeid / Int. Commun. Heat and Mass Transf. 32 (2005) 454–463 457

Eqs. (1)–(3) may be written in terms of the stream function defined as u=Bw/By and v=�Bw/Bx.

Subsequent non-dimensionalisation using

X ¼ x

L; Y ¼ y

L; h ¼ T � T0

T¯ h � Tc; W ¼ w

að6Þ

where T0=(T̄h+Tc)/2, leads to the following dimensionless forms of the governing equations:

B2W

BX 2þ B

2WBY 2

¼ � RaBhBX

ð7Þ

BWBY

BhBX

� BWBX

BhBY

¼ B2h

BX 2þ B

2hBY 2

ð8Þ

where the Rayleigh number is defined as Ra=(gbK(T̄h�Tc)L)/ya, and the boundary conditions (5a) (5b)

(5c) (5d) become

W � 1; Yð Þ ¼ 0; Bh � 1; Yð Þ=BX ¼ 0 ð9aÞ

W 1;Yð Þ ¼ 0; Bh 1;Yð Þ=BX ¼ 0 ð9bÞ

W X ; 0ð Þ ¼ 0; h X ; 0ð Þ ¼ 0:5þ ecos pX=2Hð Þ at � HVXVH ;Bh X ; 0ð Þ

BY¼ 0 at � HNXNH

ð9cÞ

W X ; 1ð Þ ¼ 0; h X ; 1ð Þ ¼ � 0:5 ð9dÞ

where H=D/L. The physical quantities of interest in the present investigation are the local and the

average Nusselt numbers along the hot wall which are defined respectively as:

Nu ¼ � BhBY

� �Y¼0

; and NuP ¼

Z H

0

NudX ð10Þ

3. Numerical method

The flow and heat transfer characteristics are symmetrical around X-axis (Fig. 1). Due to this

symmetry, only one half of the cavity has been considered for the computational purpose. The coupled

system of Eqs. (7) and (8) subjected to the boundary conditions (9a) (9b) (9c) (9d) is integrated

numerically using the finite volume method as described by Patankar [18]. The quadratic upwind

N.H. Saeid / Int. Commun. Heat and Mass Transf. 32 (2005) 454–463458

differencing QUICK scheme by Hayase et al. [19] is used for the convection terms formulation, whereas

the central difference scheme is used for the diffusive terms. The QUICK scheme uses three-point

quadratic interpolation for the control volume face values of the dependent variable and it has a third-

order accurate approximation for the uniform grid spacing. The discretisation equation for the general

control volume is derived for the uniform grid spacing. Implementation of the boundary conditions

requires a separate integration for the boundary and near boundary control volumes as well as the corners

control volumes. The linear extrapolation, known as mirror node approach, has been used for the

implementation of the boundary conditions. The number of grid points in both X- and Y-directions is

taken as 32�32 with uniform spaced mesh. The resulting algebraic equations are solved by line-by-line

using the Tri-Diagonal Matrix Algorithm iteration. The iteration process is terminated under the

following condition:

Xi; j

/ni; j � /n�1

i; j =Xi; j

����������/n

i; j

����������V10�5 ð11Þ

where / stands for either h or W and n denotes the iteration step.

4. Results and discussion

The results for the isothermal heat source temperature (e=0 in the present formulation), H=0.5 and for

the Rayleigh number of Ra=100 are presented in the form of isotherms and streamlines as shown in Fig.

2. These contours are almost same to those given by Prasad and Kulacki [11] for the same particular

case. The local Nusselt number defined in Eq. (10) is calculated at Y=0 using the boundary and the next

two grid values of the non-dimensional temperature in the Y-direction, which has a third-order accuracy

also. The average Nusselt number is compared with that given by Prasad and Kulacki [11] for the

Fig. 2. Isotherms (left) and streamlines (right) for Ra=100, H=0.5 with e=0, |wmax|=5.333.

Table 1

Comparison of average Nusselt number (NuP)

Ra H=0.2 H=0.5 H=0.8

Ref. [11] Present Ref. [11] Present Ref. [11] Present

20 0.620 0.578 0.855 0.839 0.970 0.966

50 0.823 0.821 1.360 1.405 1.368 1.515

100 1.345 1.354 2.290 2.317 2.631 2.650

200 1.970 2.074 3.296 3.322 3.810 3.798

500 3.132 3.032 4.640 4.744 5.556 6.082

N.H. Saeid / Int. Commun. Heat and Mass Transf. 32 (2005) 454–463 459

isothermal heat source temperature and for different values of H and Ra as shown in Table 1. The

differences between the present values and the values given in Ref. [11] are in the calculation of the

Nusselt number since there is no difference in the thermal and flow fields shown in Fig. 2. These results

provide confidence to the accuracy of the present numerical method to study the effect of the bottom

wall temperature variation on the natural convection in the porous cavity.

The effect of the amplitude of the bottom wall temperature variation on the average Nusselt number

(NuP) for different values of H is shown in Fig. 3 for Ra=100. The variation of Nu

Pwith H for the

isothermal heat source (e=0) is also presented in the same figure as a reference. It is observed that for all

the values of the heat source length, the average Nusselt number increases with increasing amplitude of

the bottom wall temperature variation. Next, the effect of the Rayleigh number on the average Nusselt

number for e=1 is shown in Fig. 4. The average Nusselt number increases with increasing length of the

heat source for the whole Rayleigh number range 20–500. Fig. 5 shows the variation of the average

Nusselt number per unit length of the heat source for different values of Rayleigh number for e=1.0. Itcan be observed from this figure that for a given Ra, the ratio Nu

P=H (representing the heat transfer per

unit area of the heat source) decreases as the length of the heated segment increases.

The isotherms and the streamlines for Ra=100, e=0.5 and for different heat source length are shown inFig. 6. It can be seen from this figure that |wmax|=5.266 when H=0.2 (Fig. 6a) and it increases to

|wmax|=6.732 when H is increased to 0.5 (Fig. 6b). Increasing the heat source length H from 0.5 to 0.8

Fig. 3. Variation of NuP

with H for Ra=100.

Fig. 4. Variation of NuP

with H for e=1.0.

N.H. Saeid / Int. Commun. Heat and Mass Transf. 32 (2005) 454–463460

leads to further increase in |wmax| from 6.732 to 7.162. This conforms to the results presented in Fig. 5

that the heat transfer per unit area of the heat source decreased by increasing the length of the heated part

of the bottom wall.

5. Conclusions

The natural convection in a two-dimensional cavity filled with a porous medium is analysed

numerically in the present investigation. The natural convection is induced by heating the bottom wall

and cooling the top wall of the cavity while the sidewalls are thermally insulated. The heated wall is

assumed to have sinusoidal temperature variation about a constant mean value. The numerical results are

presented for the Rayleigh number range of Ra=20–500 for different heat source length (H=0.1–0.8 of

the cavity length) and for different amplitude of the bottom wall temperature variation (e=0.1–1.0). It isfound that the average Nusselt number (Nu

P) increases with increasing amplitude of the hot wall

temperature variation for all the values of Ra and H considered in the analysis. It is noticed that the

Fig. 5. Variation ofPNu=H with H for e=1.0.

Fig. 6. Isotherms (left) and streamlines (right) for Ra=100, e=0.5 with (a) H=0.2, (b) H=0.5 and (c) H=0.8.

N.H. Saeid / Int. Commun. Heat and Mass Transf. 32 (2005) 454–463 461

N.H. Saeid / Int. Commun. Heat and Mass Transf. 32 (2005) 454–463462

average Nusselt number increases with increasing H for a given Ra but the ratio NuP=H(representing the

heat transfer per unit area of the heat source) decreases by increasing the length of the heated segment H.

Nomenclature

D half of the heat source length

g gravitational acceleration

H non-dimensional length of the heat source D/L

K permeability of the porous medium

L cavity height

Nu local Nusselt number

NuP

average Nusselt number

Ra Rayleigh number for porous medium

T fluid temperature

Tc temperature of the cold wall

Th temperature of the hot wall

u, v velocity components along x- and y-axes, respectively

U, V non-dimensional velocity components along X- and Y-axes, respectively

x, y Cartesian coordinates

X, Y non-dimensional Cartesian coordinates

Greek letters

a effective thermal diffusivity

b coefficient of thermal expansion

e non-dimensional amplitude of the hot wall temperature variation

h non-dimensional temperature

y kinematic viscosity

w stream function

W non-dimensional stream function

References

[1] D.B. Ingham, I. Pop (eds.), Transport Phenomena in Porous Media, Pergamon, Oxford (1998), Vol. II (2002).

[2] D.A. Nield, A. Bejan, Convection in Porous Media, 2nd ed., Springer, New York, 1999.

[3] K. Vafai (Ed.), Handbook of Porous Media, Marcel Dekker, New York, 2000.

[4] A. Bejan, A.D. Kraus (Eds.), Heat Transfer Handbook, Wiley, New York, 2003.

[5] K.L. Walker, G.M. Homsy, J. Fluid Mech. 87 (1978) 449.

[6] A. Bejan, Lett. Heat Mass Transf. 6 (1979) 93.

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[12] M. Kazmierczak, A. Muley, Int. J. Heat Fluid Flow 15 (1994) 30.

[13] D.A. Nield, Int. J. Heat Fluid Flow 15 (1994) 157.

[14] L. Storesletten, I. Pop, Fluid Dyn. Res. 17 (1996) 107.

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[15] R. Bradean, D.B. Ingham, P.J. Heggs, I. Pop, Int. J. Heat Mass Transfer 39 (1996) 2545.

[16] J.S. Yoo, Int. J. Heat Mass Transfer 46 (2003) 381.

[17] J.S. Yoo, W.W. Schultz, Int. J. Heat Mass Transfer 46 (2003) 4747.

[18] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Washington, 1980.

[19] T. Hayase, J.A.C. Humphrey, R. Greif, J. Comput. Phys. 98 (1992) 108.