Transient free convection flow with temperature dependent viscosity in a fluid saturated porous medium

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Itzt.J. &gag Sci. Vol. 30, No. 8, pp. 1083-1087, 1992

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TRANSIENT FREiE CONVECTION FLOW WITH TEMPERATURE DEPENDENT VISCOSITY IN A FLUID SATURATED POROUS MEDIUM K. N. MEWTA and SHOBHA Department of Mathemati~, Indian Institute of Technolo~,

SOOD New ~l~i-110016,

India

Abstract-The effect of temperature dependent viscosity on the heat transfer rate for a transient free convection flow along a non-isothermal vertical surface is studied employing Karman-Pohlhausen integral method. It is observed that a decreasein viscosity increases the heat transferrate and reduces the time to reach steady state.

1. INTRODUCTION Many facets of natural convective

flow have been studied due to its numerous applications in related engineering problems. Various investigators have adopted different methods to analyse the problem. The Key-Pohlha~en integral method has received much attention due to its simplicity and ability to yield results of reasonable accuracy. Cheng [l] applied the method for various convective heat transfer problems for which similarity solution had already been obtained. Later, Cheng and Pop [2] obtained the heat transfer rate for the transient problem using this method. Nakayama and Pop [3] used this method for a Darcian fluid flow over a non-isothermal body of arbitrary shape. Nakayama and Pop [4] studied the same problem by incorporating viscous dissipation effects. Recently Mehta and Sood [5,6] used the method to study the effect of discrete variations in permeability for steady free convective flow and temporal variations in permeability for transient free convective flow about a non-isothe~al vertical flat plate. Although in realistic fluids the viscosity decreases with the increase in temperate, all the above mentioned references assume the viscosity of the fluid to be constant. Blythe and Simpkins [7] have examined thermal convection in fluid saturated porous layered two dimensional rectangular cavity by means of integral relations for the linear dependance of fluidity on temperature. The aim of this paper is to study the effect of variation of viscosity with temperature on the buoyancy induced transient free convection flow across a vertical flat plate embedded in a fluid saturated porous medium. The dependance of viscosity on temperature has been represented by the empirical formula proposed by Poiseuille [S]. The results have been presented for the isothermal as well as non-isothermal wall.

geophysi~l

and energy

2. BASIC

GOVERNING

EQUATIONS

Under Boussinesq approximation equation for conservation of energy in non-dimensional form, can be written as [9]

of mass, Darcy law and equation

K. N. MEHTA

1084

and S. SOOD

T.The initial and

where v(T) is the dimensionless viscosity depending on the local temperature the boundary conditions for the problem under study are T=O

at t=O

(4)

v = 0,

T = T,(x)

aty=O

(5)

U-+0,

T-+0

asy+m

u, u = 0,

Elimination of u from equation (3), on using equations (1) and (2), gives T,? dt

Integeration

dx

T2 i_-)++T)=g v(T)

(6)

of equation (6) with respect to y from 0 to S(x: t) yields d

b

at”I

Tdy+;

a T2

dT Y=a

I 0 v(T)

ay y=.

-dy=-

The temperature profile which satisfies the initial condition (5) is chosen in the form

(7)

(4) and the boundary

conditions

T = TW(x)e-y’G (e _ 1) (elWy’&- 1) The dependance

of viscosity on temperature Y(T) =

is represented

(8) by the Poiseuille formula ]7]

1

(9)

I + klT + k2T2

where kl, kz are positive scalars. Upon substituting equations (8) and (9) in equation (7), we get g

+ (AT,(x)

+ k&T:(x)

+ k,DT:(x)) x g

+ (4A + 6k,CT,(x)

+ 8k,DTZ,(x)

2)

A = 2B

(10)

where A = 6’ and A, B, C and D are constants given by 3e + 1

A =-=0.5613 6e 4e + 2 B = (e = 7.4925

C=

lOe*+4e+ 30e2

1

= 0.3869

D_35e3+15eZ+5e+1 140e3 Equation (10) is a linear first order hyperbolic the condition A(0) = 0.

3. ANALYTIC

AND

= 0.2944 equation in A which is to be solved subject to

NUMERICAL

SOLUTION

For the case of an isothermal wall T,(x) = 1, equation (10) reduces to ~+(A+klC+kzD)~=2B

(11)

1085

Transient free convection Bow

The solution of equation (11) for small time and large time is given, respectively,

by

A=2Bt

(12a)

2Bx A+k,C+k,D

Wb)

and A=

For the fluid with constant viscosity kl = k2 = 0, the results obtained here are in agreement with those obtained by Cheng and Pop [2] except for the slightly different values of the constants A and B on account of the different choice for T made in this study in contrast to the choice for T used in [Z]. To account for the variable wali temperature distribution we consider the case when T,(x) = eh. The resulting equation for A obtained from equation (10) is $

+ eh(A + klCeh -t kzDezb) E

+ ileh(4A + 6klCeh + 8k,Dezh)

A = 2B

(13)

Though the analytic solution for large time can be easily obtained from equation (13) it is not possible to obtain the analytic solution for small time. The analytic solution for large time is given by A=

2Bem4& it(A + klCeAS + k2De*)2

In order to obtain the solution for all time, equation (13) was solved numerically using finite differences. The numerical solution obtained for large time were compared with the analytical results obtained from equation (14) and were found to be in good agreement. The boundary layer thickness 6(x, t) changes from the transient state to the steady state along the limiting characteristic curve obtained by solving the first order linear differential equation dt

-h(A + k,Ceh + kzDe2’*)-’

-_=e

dx

with the initial condition x = 0 at t = 0. Each curve divides the x - t plane into two regions, the upper region representing the steady state flow domain and the lower region representing the transient flow domain. In the case of an isothermal wall, A = 0, the curves are straight lines with slope equal to l/(A + klC + kzD). It is seen from Fig. 1 that decrease in viscosity reduces the time to reach steady state at all location x on the plate. 2.0

1.2

r

1.0 1.5 0.6 t 1.0

0.6

0.4

0.5 0.2

0

.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1.0

0

0 x

Fig. 1. Time to reach the steady state for (a) an isothermal wall A = 0 and (b) a non-isothermal A= 1.

wail

1086

K. N. MEHTA

and S. SOOD

30

30

/ 25

),

=os

k,

=o

h

25

-05

k, -0

20

z g

15

Ii? 10

,k,=15

,kZ=15

5

I 0

01

I

I __.____ 03 04

0.2

3

t k, on average

Fig. 2. Effect of viscosity parameter heat transfer rate.

The non-dimensional expression study can be written as

Fig. 3. Effect of viscosity parameter heat transfer rate.

k,

on average

for the co-efficient of heat transfer for the problem under

--x dT (2e - 1)x 2.5820X Nu/Ra’” = T,(X) dy Y”O= (e - 116 =---X--

(15)

The average heat transfer rate is given by k$Ra”’

= 2.5820

For the case of an isothermal wait (A = 0) for small time 3340(A + k,C + kzD)“‘,

for large time

40

k,=lO k2=

5

30

9 z \

20

1;

10

0

I

I

I

0.1

0.2

0.3

I

I

04

05

t

Fig. 4. Effect of L on average heat transfer rate.

(16)

1087

Transient free convection flow

Equation (16) clearly indicates that an increase in the values of the parameters kl, kz increases the heat transfer rates. The same is also true for the case of a non-isothermal vertical flat plate, the results of which are shown in Figs 2 and 3. The increase in the non-isothermality parameter L, also increases the rate of heat transfer. The effect of the non-isothermality parameter 3Lfor given values of kI and kz is shown in Fig. 4.

4. CONCLUSIONS

The effect of temperature dependent viscosity on the heat transfer rate is found to be quite significant. For larger values of kI and kz the steady state sets in immediately. For the case of a non-isothermal wall the time required to reach the steady state is less than that for an isothermal wall. As expected from physical considerations a decrease in viscosity increases the rate of heat transfer. Although in the present study only one law was taken for the viscosity, the analysis can be done by taking the polynomial approximation for any law of the fluidity v-l(T). In practice the constants occuring in the relations should be so chosen that it represents a satisfactory approximation to the realistic viscosity variations.

REFERENCES [l] [2] [3] [4] [5] [6] [7] [8]

P. CHENG, Len. Heat Mass Transfer 5,243-252 (1978). P. CHENG and I. POP, Int. J. Engng Sci. Z&253-264 (1984). A. NAKAYAMA and H. KOYAMA, 1. Heat Transfer 109,125-130 (1984). A. NAKAYAMA and I. POP, Znt. Commun. Heat Mass Transfer 16,173-180 (1989). K. N. MEHTA and S. SOOD, Int. J. Engng Fhdd Mech. To appear. K. N. MEHTA and S. SOOD, Modell. Simul. Control. To appear. P. A. BLYTHE and P. G. SIMPKINS, Int. J. Heat Mass Transfer 24,4!97-506 (1981). J. R. PARTINGTON, An Advanced Treaties on Physical Chemistry, Vol. 2: Properties of Liquids. Longma, London (1%2). [9] A. BEJAN, Convective Heat Transfer. Wiley, New York (1984). (Received 3 September 1991; accepted 23 September 1991)

NOMENCLATURE P: acceleration due to eravitv -k aT* h=-.-...e. local heat transfer coefficient T: - T,’ i3y* w._,,’ k: thermal condu&ity of the porous medium K: permeability of the porous medium k,, k,: dimensionless viscosity parameters L: length of the vertical flat plate Nu = y: Ra

=

Nusselt number

ii’L(T - T2J: crv_

Rayleigh

number

cvRa t*: dimensionless time UL2 T=T*-T: m : dimensionless temperature

t=-

i _ u=u *: dimensionless velocity in x-direction cyRa L Rain v=u*: dimensionless velocity in y-direction (Y X8 x = t:

dimensionless co-ordinate measuring distance

alo$he Ra

y=

r

plate from the lower stagnation point

y l: dimensionless coordinate normal to the

plate Greek symbols (Y:thermal dilbtsivity /I: co-efficient of thermal expansion 6: dimensionless boundary layer thickness I: constant used in wall temperature V’ v = -:

dimensionless viscosity parameter v, u: heat capacity ratio

Subscript r: reference quantity w: condition at the wall -: ambient condition outside the boundary layer Superscript *: dimensional quantity