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Procedía Engineering 56 (2013) 76-81 =

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5th BSME International Conference on Thermal Engineering

MHD natural convection in a rectangular cavity having internal energy sources with non-uniformly heated bottom wall

M. Obayedullah*, M.M.K. Chowdhury

Department of Mathematics, Bangladesh University of Engineering and Technolog, Dhaka-1000, Bangladesh

Abstract

The present study deals with steady natural convection flow in a rectangular cavity containing internally heated and electrically conducting fluid. The bottom wall is non-uniformly heated while the upper wall of the cavity is well insulated. The left and right vertical walls are maintained at constant hot and cold temperature respectively. Numerical results have been obtained for the effect of various internal and external Rayleigh numbers and Hartmann numbers. Results are presented in the form of streamlines, isotherm contours. Average Nusselt numbers are expressed as a function of Rayleigh number. It is found that the temperature, fluid flow and heat transfer strongly depend on internal and external Rayleigh numbers and Hartmann numbers.

© 2013 The Authors. Published by Elsevier Ltd.

Selection and peer review under responsibility of the Bangladesh Society of Mechanical Engineers Keywords: natural convection; heat generation; non-uniform heating; rectangular cavity.

1. Introduction

Natural convection heat transfer induced by internal heat generation has received considerable attention because of its numerous application in geophysics and energy related engineering problems. Oreper and Szekely, 1983 [1] studied the effect of an externally imposed magnetic field on buoyancy driven flow in a rectangular cavity. Ozoe and Maruo, 1987 [2] investigated magnetic and gravitational natural convection of melted silicon and used two dimensional numerical computation for the rate of heat transfer. Garandet et al., 1992 [3] studied natural convection heat transfer in a rectangular enclosure with a transverse magnetic field. Rudraiah et al., 1995 [4] investigated the effect of surface tension on buoyancy driven flow of an electrically conducting fluid in a square cavity in the presence of a transverse magnetic field to see how this force damps hydrodynamic movements.

Gelfgat and Yoseph, 2001 [5] studied the effect of an externally imposed magnetic field on the linear stability of steady convection flow in a horizontally elongated rectangular cavity. Sarris et al., 2005 [6] presented a numerical study of two dimensional natural convection of an electrically conducting fluid in a laterally and volumetrically heated square cavity under the influence of a magnetic field. Mehmet and Elif, 2006 [7] studied the natural convection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls. Kandaswamy et al., 2008 [8] studied MHD natural convection in an enclosure with partially active vertical walls. Nithyadevi et al., 2009 [9] studied magnetoconvection in a

ELSEVIER

* Corresponding author.

Email address: obayed@math.buet.ac.bd

1877-7058 © 2013 The Authors. Published by Elsevier Ltd.

Selection and peer review under responsibility of the Bangladesh Society of Mechanical Engineers doi:10.1016/j.proeng.2013.03.091

cavity with partially active vertical walls having time periodic boundary condition. Kahveci et al., 2009 [10] studied MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure.

As per author's knowledge the literature review revealed that non-uniform temperature profile in the wall was not used in he magnetohydrodynamic natural convection flow in a rectangular cavity with heat generation. In this paper the investigation is carried out on the MHD natural convection flow in a rectangular cavity filled with heat generating fluid having parabolically heated bottom wall.

2. Formulation of the problem

A rectangular cavity filled with viscous incompressible fluid is shown in the Fig. 1. The cavity dimensions are defined by L for width .and H for height. The cavity is isothermally heated from the left vertical wall with a uniform constant temperature Th and the right vertical with temperature Tc (Th > Tc ) .

The bottom horizontal wall is heated with T^ — (T^ —Tc ) — (1--) while the remaining wall is

considered perfectly insulated.

adiabatic

Y//////////////A

Th- (Th-Tc) L (1-L)

Fig. 1 Physical model and coordinate system

Steady two-dimensional laminar magnetohdrodynamic free convective flow of viscous incompressible Boussinesq fluid such as water, air etc. with constant properties is assumed. Under these assumptions, equations of mass, momentum and energy are

du dv „

— + —= 0 (1)

du du 1 dp

u--\-v— =----w

dx dy p dx

rd2u d2u^

Kdx2 cy 2y

dv dv 1 dp

u--h v— =----w

dx dy p dy

d v d2v

—7 +-7

dx dy2

+ gß (T — Tc )-

aB02 v

u--1- v— = a

rd2T d2T^

dx1 dy2

Q p°p

The boundary conditions are:

u(x,0)=u(x,H)=u(0,y) =u(L,y)=0, v(x,0)=v(x,H)=v(0,y) =v(L,y) =0.

x x dT

T(x,0) = Th- (Th-Tc )-(l--) , T(0,y)= Th, T(L,y) = Tc , — (x, H) = 0.

where u and v are the velocity components in the x and y directions respectively; p is the fluid density, P is the coefficient of thermal expansion, cp is the fluid specific heat, Q is the rate of internal heat generation per unit volume, B0 is the magnetic induction, a is the thermal diffusivity and v kinematic viscosity of the fluid. To make the above equations dimensionless we introduce the following non-dimensional variables:

X = x Y = Z- U = — V = — P = pL-,

T~TC n-T

where 0 is the dimensionless temperature.

Substitution of the dimensionless variables into the Eqs.(1)-(4) leads to:

■ + — = 0

dP_ 'dX

f d2U dU

SX1 dYz

TTdv dv dP n

U-+ V-=--+ Pr

dX dY dY

rd2V d2V^

dX2 dY'

+ Rav Pr^-Ha2 PrV

dX dY dX

r d26 d20A

dX2 dY'

RaE Ra

In the above equations RaE is the external Rayleigh number , Raj is the internal Rayleigh number , Pr is the Prandtl number and Ha is the Hartmann number defined respectively by the following equations.

gfi (Th-Tc )L3

g^LL P^.r and Ha2 =

va vak a

The dimensionless boundary conditions are:

U(X,0)=U(X,A) =U(0,Y)=U(1,Y)=0, V(X,0)=V(X,A) =V(0,Y)=V(1,Y)=0,

Q(X.0)=1-X(1-X) Q(0,Y)=1, Q(1,Y)=0 , — (X, A) = 0

where A — H /L is the aspect ratio of the cavity which is taken as 0.75. The heat transfer coefficient in

terms of local Nusselt number is defined by Nu —--where n denotes the normal direction on a plane.

The average Nusselt numbers at the bottom and side walls are computed as

_ 1 _ i *

Nub= JNubdX, Nus= — | NusdY

The discretization process involves a certain amount of error, which can be systematically reduced by a series of grid refinements. To this end, five types of grid densities have been chosen to check for the self-consistency of the present study. The grids chosen are: (a) 22342 nodes, 3453 elements, (b) 28455 nodes,

4421 elements (c) 35796 nodes, 5589 elements (d) 41124 nodes, 6424 elements (e) 47212 nodes, 7387 elements. Type (c) mesh density was found to give sufficient accuracy with modest computational time, and hence selected for the simulation study.

The governing Eqs.(5)-(8) along with the boundary conditions are solved numerically, employing finite element method based on Galerkin weighted residual formulation. To ensure convergence of solutions the following criteria is applied to all dependent variables over the solution domain

< ERMAX

where O represents the dependent variables U, V, P and T ;the indexes i, j refers to space coordinates and the index n is the current iteration.. The value of ERMAX is chosen as 10" 5.

Table 1. Comparison of present numerical results with those obtained by Sathiyamoorty et al., 2007 [11]for the case Pr = 0.71 in the absence of magnetic field.

RaE Nub (Present work) NUb (Sathiyamoorty et al.)

103 3.7196 3.7294

104 4.7520 4.7753

105 6.8042 6.8272

3. Results and discussions

Numerical results are presented to determine the effects of internal heat generation and magnetic field on the natural convection flow of an electrically conducting fluid in a rectangular cavity. The three governing parameters in this problem are the internal Rayleigh number Raj , the external Rayleigh number RaE and the Hartmann number Ha, Values of the internal Rayleigh number Raj range from 10 to 105 and the Hartmann number Ha from 00 to 50 while Pr=0.71 is kept fixed.

Fig. 2. Contour plots for Rge=10\ Ra=102, Ha=20 streamline(top); isotherm(bottom)

Fig. 3. Contour plots for RaE=103 Rai=103, Ha=20 streamline(top); isotherm(bottom)

Fig. 4. Contour plots for RaE=103 Rai=104, Ha=20 streamline(top); isotherm(bottom)

Fig. 5. Contour plots for RaE=103 Rai=105, Ha=20 streamline(top); isotherm(bottom

The effects of internal Rayleigh number Raj has been considered first on the flow and temperature fields. The resulting flows of fluid and its temperature distributions have been shown in Figs. 2-5. where the top row gives the streamlines and the bottom row gives the isotherms for increasing values of internal Rayleigh number Ra=102, 103, 104 and 105 with Pr=0.71 , Raj=103 and Ha=20. Because of parabolic temperature profile in the bottom wall and uniformly heated left vertical wall fluid rises up along the left wall of the cavity and flow down along the right wall forming a single cell called primary cell as shown in the of Fig. 2(top). A single cell is formed because the internal Rayleigh number Rar is smaller than the

external Rayleigh number RaE . With the increase of the internal Rayleigh number a secondary cell has been developed in top left corner of the cavity as shown in the Fig. 4(top). The increasing rate of heat within the cavity due to the increase of the internal Rayleigh Number leads to increase the flow rate in the secondary cell as well as increase in its size until it occupies half of the cavity. This effect of internal Rayleigh number on the flow field is reasonable since internal heat generation assists the buoyancy forces by accelerating the fluid flow. The left cell revolves anticlockwise because of greater internal Rayleigh number and the right cell revolves clockwise as expected. The fluid temperature increases significantly due to increase of the internal Rayleigh number which is shown in the isotherms of Figs. 2-5 (bottom). It is clearly seen that owing to the increase of the internal Rayleigh Number the fluid temperature exceeds the surface temperature.

Figures 6-9 depicts the effects of Hartmann number Ha on the flow and temperature fields. The flow and temperature distributions have been shown in these figures where the top row gives the streamlines and the bottom row gives the isotherms for increasing values of Ha= 00, 10, 20, 50 with the external Rayleigh number RaE=103 , the internal Rayleigh number Ra/=104 and Pr=0.71.

Fig. 6. Contour Fig. 7. Contour Fig. 8. Contour Fig. 9. Contour

plots for RaE=103, plots for RaE=103 plots for RaE=103 plots for RaE=103

Rai=104, Ha=00 Raj=104, Ha=10 Raj=104, Ha=20 Raj=104, Ha=50

streamline(top); streamline(top); streamline(top); streamline(top);

isotherm(bottom) isotherm(bottom) isotherm(bottom) isotherm(bottom

Here two cells are formed as the internal Rayleigh number is greater than the external Rayleigh number. From the figures as shown in the top rows, it can be seen that intensities of the flow decrease owing to increase in the magnetic field. Because of the decrease of the intensity of the flow the left cell is becoming smaller. This is expected since presence of magnetic field usually retards the velocity field.

The corresponding effects of increasing Hartmann number on the isotherms may be viewed from the Figs. 6-9(bottom). It is seen that the isotherms become more curved due to the increase of the magnetic field strength, which is expected; since the magnetic field resists the flow.

Ral= 1.0E2 Ral= 1.0E3 Ral= 1.0E4 Ral= 1.0E5

-20 10

- Ral=1.0E2

- Ral= 1.0E3

- Ral= 1.0E4

- Ral=1.0E5

104 Ra (b)

103 103

- Ha= 00.0

---------- Ha= 10.0 //

----------------- Ha= 20.0 ft ''

--------- Ha= 50.0 // ''

104 Ra

Fig. 10. Nusselt number as a function of external Rayleigh number for the (a) non-uniformly heated bottom wall and (b) heated left wall for varying Raj with Ha=20 (c) non-uniformly heated bottom wall and (d) heated left wall for varying Ha with Ra= 104, all with Pr=0.71.

The overall effects of internal Rayleigh number Rat on the average Nusselt number for non-uniformly heated bottom wall and heated left wall for Pr=0.71, and internal Rayleigh number Ra= 102, 103 , 104 and 105 are displayed in Figs. 10(a) and (b) via average Nusselt number vs external Rayleigh number plot. It is seen that the average Nusselt Number is negative for both the non-uniformly heated bottom wall and heated left wall. In both the cases heat transfer is very high for Raj= 105 .

The effects of Ha on the average Nusselt number for bottom wall and left wall for Pr=0.71 and Ra= 104 and. are shown in Figs. 10(c) and (d). Initially at RaE =103 the heat transfer on the bottom wall is positive and left wall is negative. It is observed that the average Nusselt number smoothly increases as the external Rayleigh number increases. In both the cases average Nusselt numbers are reduced as the strengths of the applied magnetic field are increased.

4. Conclusion

Natural convection flow and heat transfer in a rectangular cavity in the presence of magnetic field is numerically investigated by using finite element method. The investigation is carried out to find the affect of the internal Rayleigh number Ra{ and Hartmann number Ha. Internal Rayleigh number Ra{ affects the flow structure and heat transfer inside the cavity. The heat transfer is enhanced with increasing internal Rayleigh number. The flow characteristics and heat transfer mechanisms inside the cavity also depend strongly upon the strength of the magnetic field. Strong suppression of the convective current can be obtained by applying strong magnetic field. This is why, reductions in the average Nusselt number Nu are produced as the strengths of the applied magnetic field are increased. Thus significant effect of the magnetic field is observed in the heat transfer mechanisms.

References:

[1] Oreper GM, Szekely J., 1983. The effect of externally imposed magnetic field on buoyancy driven flow in a rectangular cavity, J. of Crystal Growth 64, p. 505-515.

[2] Ozoe H, Maruo M., 1987. Magnetic and gravitational natural convection of melted silicon-two dimensional numerical computations for the rate of heat transfer, JSME 30, p. 774.

[3] Garandet JP, Alboussiere T, Moreau R., 1992.. Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field, Int. J. of Heat and Mass Transfer 35, p. 741.

[4] Rudraiah N, Vankatachalappa M, Subraiah CK., 1995. Combined surface tension and buoyancy-driven convection in a rectangular open cavity in the presence of a magnetic field, Internal. J. Non-Linear Mech 30(5), p. 759.

[5] Gelfgat AY, Yoseph PZ., 2001. Effect of an external magnetic field on oscillatory instability of convert^ flows in rectangular cavity, Phys. Fluid 13, p. 2269.

[6] Sarris IE, Kakarantzas SC, Grecos AP, Vlachos NS., 2005. MHD natural convection in a laterally and volumetrica^ heated

square cavity, Int. J. of Heat and Mass Transfer 48, p. 3443.

[7] Mehmet CE, Elif B., 2006. Natural convection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls, Fluid Dynamics Research 38, p. 564.

[8] Kandaswamy P, Sundari SM, Nithyadevi N., 2008. Magneto convection in an enclosure with partially active vertical walls, Int. J. of Heat and Mass Transfer 51, p. 1946.

[9] Nithyadevi N, Kandaswamy P, Sundari SM., 2009. Magneto^me^^ in a square cavity with partially active vertical walls: Time periodic boundary condition, Int. J. of Heat and Mass Transfer 52, p. 1945.

[10] Kahveci K, Oztuna S., 2009. MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure, European Journal of Mechanics 28, p. 744.

[11] Sathiyamoorty M, Tanmay Basak Roy S, Pop I., 2007. Steady Natural convection flow in a square cavity with linearly heated side walls, International Journal of Heat and Mass Transfer 50, p.766.