Natural convection in porous cavity with sinusoidal bottom wall temperature variation

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 B

Communicated by J.P. Hartnett and W.J. Minkowycz. * Tel.: +60 4 593 7788; fax: +60 4 594 1025. E-mail address: [email protected].

0735-1933/$ - see front matter D 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2004.02.018

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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 nonuniform 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 crosssection 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. 2. 3. 4. 5.

The convective fluid and the porous media are in local thermal equilibrium. The properties of the fluid and the porous media are constants. The viscous drag and inertia terms of the momentum equations are negligible. The Boussinesq approximation is valid. Darcy law is applicable.

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

Under these assumptions, the conservation equations for mass, momentum and energy for the twodimensional steady natural convection in the porous cavity are: Bu Bv þ ¼0 Bx By

ð1Þ

Bu Bv gbK BT  ¼  By Bx y Bx

ð2Þ


2  BT BT BT B2 T u þ 2 þv ¼a Bx By Bx2 By

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 T¯ h in the form:  Th ð xÞ ¼ T¯ h þ e T¯ h  Tc Þcosð px=2DÞ ð4Þ where e denotes the amplitude of the hot wall temperature variation. Eqs. (1)–(3) are subject to the following boundary conditions: uð  L; yÞ ¼ 0;

BT ð  L; 0Þ=Bx ¼ 0

ð5aÞ

uð L; yÞ ¼ 0;

BT ð L; yÞ=Bx ¼ 0

ð5bÞ

vð x; LÞ ¼ 0;

T ð x; LÞ ¼ Tc

ð5cÞ

vð x; 0Þ ¼ 0; and T ð x; 0Þ ¼ Th ð xÞ at  DVxVD;

BT ð x; 0Þ=By ¼ 0 at  DNxND

ð5dÞ

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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

Y¼

y ; L

h¼

T  T0 ; T¯ h  Tc

W¼

w a

ð6Þ

where T 0=(T¯ h+Tc)/2, leads to the following dimensionless forms of the governing equations: B2 W B2 W Bh þ ¼  Ra 2 2 BX BY BX

ð7Þ

BW Bh BW Bh B2 h B2 h  ¼ þ BY BX BX BY BX 2 BY 2

ð8Þ

where the Rayleigh number is defined as Ra=(gbK(T¯ hTc)L)/ya, and the boundary conditions (5a) (5b) (5c) (5d) become Bhð  1; Y Þ=BX ¼ 0

Wð  1; Y Þ ¼ 0; Wð1; Y Þ ¼ 0;

ð9aÞ

Bhð1; Y Þ=BX ¼ 0

ð9bÞ

Wð X ; 0Þ ¼ 0; hð X ; 0Þ ¼ 0:5 þ ecosð pX =2H Þ at  HVX VH;

Wð X ; 1Þ ¼ 0;

hð X ; 1Þ ¼  0:5

Bhð X ; 0Þ ¼ 0 at  HNX NH BY ð9cÞ 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

Bh  BY

 ; Y ¼0

P

and Nu ¼

Z

H

N udX

ð10Þ

0

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

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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 thirdorder 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 3232 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:    X  /ni; j  /n1 i; j    i; j

=

  X  /ni; j V105   i; j

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, |w max|=5.333.

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Table 1 P Comparison of average Nusselt number (Nu) Ra 20 50 100 200 500

H = 0.2

H = 0.5

H = 0.8

Ref. [11]

Present

Ref. [11]

Present

Ref. [11]

Present

0.620 0.823 1.345 1.970 3.132

0.578 0.821 1.354 2.074 3.032

0.855 1.360 2.290 3.296 4.640

0.839 1.405 2.317 3.322 4.744

0.970 1.368 2.631 3.810 5.556

0.966 1.515 2.650 3.798 6.082

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 P P (Nu) for different values of H is shown in Fig. 3 for Ra=100. The variation of Nu with 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. It P can be observed from this figure that for a given Ra, the ratio Nu=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 in Fig. 6. It can be seen from this figure that |w max|=5.266 when H=0.2 (Fig. 6a) and it increases to |w max|=6.732 when H is increased to 0.5 (Fig. 6b). Increasing the heat source length H from 0.5 to 0.8

P Fig. 3. Variation of Nu with H for Ra=100.

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P Fig. 4. Variation of Nu with H for e=1.0.

leads to further increase in |w max| 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 is P found that the average Nusselt number (Nu) 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

P Fig. 5. Variation of Nu=H with H for e=1.0.

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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.

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P average Nusselt number increases with increasing H for a given Ra but the ratio Nu=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 P Nu average Nusselt number Ra Rayleigh number for porous medium T fluid temperature temperature of the cold wall Tc temperature of the hot wall Th 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 a b e h y w W

letters effective thermal diffusivity coefficient of thermal expansion non-dimensional amplitude of the hot wall temperature variation non-dimensional temperature kinematic viscosity stream function non-dimensional stream function

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