Rect Inclined

8/6/2019 Rect Inclined

1/4

-~~- ~-~ ~ ~- - ~-~~~- -~~

The Seventh Asian Congress of Fluid Mechanics

Dec 8 -12, 1997, Chennai (Madras) ,

NATURAL CONVECTION IN A RECTANGULAR POROUS CAVITY

M.Venkatachalappa 1.5.Shivakumara M.Sankar B.M.R.Prasanna

UGC-DSACentre in FluidMechanics,Departmentof Mathematics,BangaloreUniversity,Bangalore560001,India.

ABSTRACT The present numerical study based on ADI (Alternate Direction Implicit)

method combined with upwind differencing scheme in the presence of non-Darcy effects

on natural convection in a cavity with high porosity porous media show that the inertia

effects, viscosity and aspect ratios have significant influence on the streamline pattern

and temperature distributions. These effects reduce the heat transfer rate from the

rectangular cavity with side walls maintained at different temperatures and the horizontal

walls kept adiabatic and become more noticeable near the walls. The results also show

that as resistance due to solid matrix increases the temperature profile show channellingeffects.

1. Introduction

Because of its relevance in variety of situations like geothermal system, thermal

insulation, coal and grain storage, solid matrix heat exchangers, nuclear waste disposal and so on,

natural convection in a porous medium is a well developed field of investigation. Most works on

convection in porous media are concerned with the channel ( either vertical or horizontal) of

infinite extent and limited to low or moderate Darcy Numbers and effective Brinkman viscosity

equal to fluid viscosity[l]. Vafai and Kim [2] have studied convection in a-vertical channel for

the case of fully developed flow by considering either constant heatflux or temperatures at theboundaries. Their analysis was limited to the case of effective Brinkman viscosity equal to the

fluid viscosity. Most of the practical problems cited above, particularly the industrial problems

involve sparsely packed porous cavities with high permeability having effective Brikman

viscosity much larger than the fluid viscosity. In spite of its importance much work has not been

done on natural convection in cavities involving boundary and inertia effects with. effective

viscosity different from fluid viscosity. In this paper we consider a rectangular porous cavity with

side walls maintained at uniform but different temperatures with top and bottom walls closed andinsulated.

2. Mathematical Formulation

We consider a two dimensional rectangular porous cavity of height H and width L filled

with homogeneous, isotropic, sparsely packed porous material of high permeability K. The topand bottom walls are closed and insulated and side walls are maintained at constant but different

561


2/4

temperatures. Based on the Boussinesq approximation, the governing equations using th

vorticity - stream function formulation in nondimensional form are

s aT + U aT + V aT = 1- V2 Ti7t ax ay Pr

1-

[

E ~ + U ~ + V E;

]

= Gr aT+ A

(

~ + ~J

- Da-lt,E2 i7t ax ay 2 ay ax2 ay2

2V'=-t"

(1)

(2)

(3)

where

u = a'l'ay ,

v=_a'ax '

(4

r - av - au x - x y - .r"'-ax BY' -L' -L'

co \jI 8 - 80t, = - , ' = - , T =. ., 80 =

alL2 ex. 8h - 8c

tv 't=~,

Uv

_, of,u= -, -aiL a/L-.2

aiL a2 ~8h + 8c V2 = ~+ aT:1-, ax r2

The dimensionlessparametersare:

P = ~ D = K A = J G = gP(8h- 8c)L3 A = Hr ,a 2' ,r 2' L .ex. L ~ v

Initially the fluid is quiescent and the temperature is uniform throughout. The initial an

boundary conditions in the non-dimensional form are:

't = 0 : U = v.. = ' = 0, T = t, = 0, 0 ~ X ~ A , 0 ~ y ~ 1't>o: '=~=O T=-lO at y= O

ay" .

a'' = - = 0 T = + 1.0ay ,

a' aT '=-=O -=0

ax ' ax

at Y = 1

at x= 0 andX=A

3. Numerical Method

The modified two step ADI technique is employed to advance the fields of temperature

and vorticity at the interior grid points across a time step n l:1't to the new level ( n + 1 ) l:1'to I

order to improve the stability of the numerical scheme, the nonlinear convective terms ar

evaluated using second order upwind difference method, the diffusion terms are approximated

using central difference and the time derivative is by forward difference. The method of SLOR i

then employed to solve the elliptic stream function equation for new stream function field

Finally the velocities are computed from the stream function solutions using equation (4). Th

whole process described above is repeated for each time steps until steady state ~lution i

obtained. Prior to the calculations, as a partial verification of the computational procedure,. th

results are compared with the solutions given by Wilkes and Churchill [3] in the absence o

562


3/4

--

porousmediumand are foundto be in good agreement.

4. Results and Discussion

Nonlinear natural convection in a rectangular porous cavity is investigated numerically

for the values of Gr = 50000, Pr = 7, A = 0.1,1 & 3, S = 1 and porosity E = 0.9 and the results

are shown in Figs. 1 to 4.

Figures 1a and 1b show that effect of Darcy number on streamlines for the different

values of viscosity ratio. We found that for Da = 0.01 the flow structure is symmetric to the axes

(see Fig. la). But for Da = 0.001 the streamlines distorted slightly and oriented along the

principal diagonal of the cavity and the central stream lines are almost in elliptic shape. (see Fig.

Ib). From these figures it is also evident that the effect of viscosity ratio on the streamline pattern

is to change the position of maximun value of stream function.

Figures 2a and 2b illustrate the isotherm contours. From Fig. 2a, we find that thetemperature stratification is dominant near the top and bottom of the cold and hot walls

respectively. As Da decreases the temperature stratification also decreases (see Fig. 2b). This is

because when Da decreases the Darcy resistance is dominant and the convection reduces. It is

observed that as the viscosity ratio increases, the shift of isotherms towards the hot wall is less

pronounced and also their effect is to reduce the thermal boundary layer growth.

To know the influence of aspect ratio, the numerical results for A = 3 are reported in

Figs. (3a) and (3b) in terms of streamlines and temperature contours. We note that increase in

the value of A distorts the pattern of streamlines considerably compared to those for smaller

values of A. In the case of isotherms, there exists large temperature gradients which indicates aboundary layer structure in the regions adjacent to the thermally active ( hot and cold) walls.

The rate of heat transfer across the cavity is obtained by evaluating the average Nusseltnumber NUat the hot wall and the variation of mean Nusselt number with dimensionless time is

depicted in Figs. 4a and 4b. From this it is evident that NU decreases with the increase in

viscosity ratio and decrease in Da. The effect of A on the heat transfer rate is noticeable for Da =

0.01 but not so significantfor Da = 0.001 (see Figs. 4a and 4b). We also found that the steadystate can be obtained faster with the decrease in the value of Da.

Acknowledgement

The authors are very thankful to Prof. N. Rudraiah, INSA Senior Scientist, for his valuable

discussions. This work was supported by UGC under DSA and COSIST Programmes.

References

[1] Hong,J.T., Tien, C.L, and Kaviany,M, Int. J. Heat Mass Transfer,28, 11,2149 (1985).

563


4/4

{:J/J

[2] Kim, SJ and Vafai, K, Int. 1. Heat Mass Transfer, 32, 665 (1989).

[3] Wilkes, J.O and Churchill, S.W, AI.Ch.E. J, 12,161 (1966).

Nomenclature

A aspect ratio

Gr Grashof numberH height of the cavity

L width of the cavityPr Prandtl number

S ratio of specific heatt time

T dimensionless temperature

u vertical velocity

U dimensionless vertical velocity

v horizontal velocity

V dimensionless horizontal velocityx vertical co-ordinate

X dimensionless vertical co-ordinate

y horizontal co-ordinateY dimensionless horizontal co-ordinate

00

y

0.5 00,

y0.5 0

0

y0.5 0

0

y

0.5

X 0.5

-,0..0.1-- -1-3

x

:\'

- ,0.-0.1-- -1

-3

x

.--~--

~:-::\\

lHt:

O'/---

.

C:\:'

l, ': ::

\: : I ~. ' :

\', ,':

\' \. .. - -: .: '. ./ :1

3.1 , ~=-~,-.:.-~

.~

"~I(bJ

Fig. 1 STREAMLINES (e) D. . 0.01 &(b) D. = 0.001 :..i~'J~bI

y

0.5 00

y

0.5Fig. 3 STREAMLINES & ISOTHERMS De = 0.01. A = :

X 0.5

10 10

-,0.-0.1- - -1

----- - 3

"'.","'.a.on'

NuNu 5

-"".

Documents

Rect Inclined