Transport in Porous Media 3 (1988), 95-106 9 1988 by Kluwer Academic Publishersl

95

Free Convection from a Vertical Plate with Nonuniform Surface Heat Flux and E m b e d d e d in a Porous Medium R A M A SUBBA R E D D Y G O R L A and R O B E R T T O R N A B E N E Department of Mechanical Engineering, Cleveland State University, Cleveland, OH 44115, U.S.A. (Received: 17 March 1987; revised 8 September 1987)

Abstract. An analysis is presented for the calculation of heat transfer due to free convective flow along a vertical plate embedded in a porous medium with an arbitrarily varying surface heat flux. By applying the appropriate coordinate transformations and the Merk series, the governing energy equation is expressed as a set of ordinary differential equations. Numerical solutions are presented for these equations which represent universal functions and several computational examples are provided. Key words. Free convection, heat transfer, Porous media, natural convection. 1. N o m e n c l a t u r e f g h k K L m q Ra* T u v x y a /3 h p tp 0

dimensionless stream f u n c t i o n acceleration due to gravity locat heat transfer coefficient thermal c o n d u c t i v i t y permeability length of the plate constant heat transfer rate modified R a y l e i g h n u m b e r temperature velocity c o m p o n e n t in x-direction velocity c o m p o n e n t in y-direction c o o r d i n a t e along plate m e a s u r e d f r o m leading edge c o o r d i n a t e m e a s u r e d n o r m a l to plate thermal diffusivity coefficient of v o l u m e t r i c expansion dimensionless n o r m a l c o o r d i n a t e wedge parameter dimensionless streamwise c o o r d i n a t e density stream function dimensionless t e m p e r a t u r e

96

RAMA SUBBA REDDY GORLA AND ROBERT TORNABENE

Subscripts w condition at the surface condition at the boundary layer edge

2. Introduction Natural convective heat transfer in porous media has become a subject of increased recent investigations. Some of the applications of this subject include groundwater systems for industrial and agricultural uses, buried electrical cable and transformer coils and in the storage of radioactive nuclear waste materials. A similarity solution was obtained by Cheng and Minkowycz [1] for the steady free convection along a vertical plate adjacent to a porous medium. Kaviany and Mittal [2] performed experiments to determine the local heat transfer rate from an isothermal vertical plate placed next to a saturated porous medium. Merk [3] derived the momentum and energy boundary layer equations and used the 'wedge' variable as one of the independent coordinates. Following a series solution approach, Merk evaluated only the first term in the series and presented the governing differential equation for the second term. Chao and Fagbenle [4] presented an investigation of a reappraisal of Merk's procedure by providing the corrected sequence of the differential equations governing the universal functions associated with the method. The present work was undertaken in order to study the problem of free convective flow about a vertical wall with arbitrarily prescribed surface heat flux and embedded in a porous medium. A Merk-type series was applied to obtain the nonsimilar solutions for heat transfer. The strength of the method is that, by utilizing the solutions for the universal functions provided in this paper, the heat transfer rates may be readily evaluated in a straightforward manner for any specified surface heat flux. Several numerical examples are presented in this paper. Recently, Gorla and Zinalabedini [5] investigated the problem of free convection from a vertical late with nonuniform surface temperature.

3. Governing Equations and Transtormations Consideration is given here to the steady laminar boundary layer free convec~tive flow over a vertical impermeable surface with a prescribed continuous surface heat flux. The surface is assumed to be embedded in an unbounded region of saturated porous medium with constant permeability, K. The flow through the porous medium is assumed to follow Darcy's law. The properties of the fluid and the porous medium are taken as constant. Employing the Boussinesq approximation, we may write the governing equations as follows: cgu Ov --+= O, Ox Oy

(1)

FREE CONVECTION FROM A VERTICAL PLATE

aT + OX

u --

v

=

3T

v --

Oy

-

32T

= a ---

(2)

0Y2 '

-3y'

(4)

fl(T-

P = P~[ 1 -

97

T~)].

(5)

In the above equations, u and v are the Darcy's velocity components in x and y directions, p the density, /x the viscosity, fl the thermal expansion coefficient of the fluid, K the permeability of the saturated porous medium, o~ the thermal diltusivity, g the gravitational acceleration, p the pressure and T the temperature. Figure 1 shows the flow development and coordinate system. The appropriate boundary conditions are given by y =0:

v =0,

or

q~,(x)

0y

k

y---+oo: u = 0 ,

T=T~.

(6)

The continuity equation is automatically satisfied by introducing a stream function 41 defined by 04,

04,

U= ~y,

v-

Ox"

(7)

Then (x, y) coordinate system is transformed into the (~, ~) system by introducing x

(8)

: ~, (y/L)Ra .1/3 1(/ --

(9)

~1/3

where Ra* - P~g~qwKL2

alxk We now introduce a stream function and temperature function as follows to = ~ . R a , l/3 ~2/3 f(~, rl ) T-T~

O- i, qw(O " r'~ \~-ff2a,-l;~ !

(10)

1

g,/3.

(11)

RAMA SUBBA REDDY GORLA AND ROBERT TORNABENE

98

1.0

gravity field 0.8

i,, y~v coordinate system 0.6

0.4

0.2

1.0

I

I

I

1.5

2.0

2.5

fs

Fig. 1. a versus/6(0). Substituting the expressions in Equations (8)-(1 l), the conservation equations are transformed into ['= 0

(12) ~-~-

~-~]

(13)

with the boundary conditions = O; f ( f , O) = O, rl----~:

f'(~:, oo) = 0

f"(G O) = - 1 , (14)

where the prime denotes difterentiation with respect to 7, and A is defined by

;~ -

00X0 o,(0 0~

(15)

FREE CONVECTION FROM A VERTICAL PLATE

99

where q~(~)- L Or(t ) -- V G ~ I

~ .

4. S o l u t i o n

Due to the presence of the Jacobian on the right-hand side of Equation (13), the following Merk-type series solution is proposed: oA f(t,~l)=fo()t, rl)+4~'fl()`,'q)+

16 2 02)` ~ 0 ~ . f2()`, r~) + . 9 9

(16)

After substituting Equation (16) into Equation (13) and collecting terms free of 3)`/)`t and then terms common to 4~(O)`/O~), 16~2(oa)`/ota), etc, a set of ordinary differential equations is obtained. Zeroth order:

f~,+ 2 + ~ fofo-

+ x fro) , 2 =0.

(17)

First order: f,~, + [ 2 )`, ,, (5 ) k3 + ~ ) [ f o f l + flfg] - ~ + 2), 9 fief{ + f'dfl

=

1~ of,, f~',ofq o;t

472,

Second order: ,

2

f~'--2(~-~ /~)for2-}-' t 2[f~f2--f0f2]-1-(3 -}-2-) )`\ (f0f2~'q- f2f~) =

1

(19)

~ - fgf~).

The appropriate boundary conditions are given by

r/=0:

fo= f l = f2 = 0, f{; = - 1 ,

f~ = f~ = 0,

r~--+oo: f[~= f ~ = f ; = O .

(20)

In the above equations, for any streamwise location ~, )` is fixed and may be regarded as a parameter. The above set of ordinary differential equations was integrated by means of a fourth-order R u n g e - K u t t a method. The solution for fo01) is first obtained and then f~(9) and f2(rJ) are evaluated by using the known values of fo, f~ and f~. The initiM functions fo, f~, and f{~ must be determined

100

RAMA SUBBA REDDY GORLA AND ROBERT TORNABENE

with high accuracy since the higher-order functions f~ and f2 and their derivatives are expected to be sensitive to these. The computations start with the conditions at the wall and proceed in a forward direction to the edge of the boundary layer in a stepwise fashion. T h e boundary condition fa(1, 0) must be specified b e f o r e the integration over the boundary layer starts. We then search for a value of fg(a,0) that will generate a solution that yields f~--*0 as rl--->o~ within an error margin of 10 -.2. This is accomplished by the half-interval method. In practice, rl = oo must be replaced by "q = r/=. rl= is chosen large enough so that the solution shows little further change for r~ larger than rt=. The integration of fl requires the derivatives of Ofo/0t and Of~/01. For this purpose we use a second-order central difference scheme as follows:

Ofo _ -fo(1 + 21, -q) + 8fo(1 + 1 1 , ,/) - 8fo(i - 11, rt) + fo(1 - 21, rt) OA

12AA

(21)

Similar expression was used for Of~/Oa. A choice of ~Xt = 0.0001 was made to insure accuracy. The final correct solution for fl(~) was generated by providing

1.0

0.8

0.6

0.4

0.2

|

0

J

!

I

0.5

1.0

1.5

f~' (0) Fig. 2.

A versus f;(O).

101

FREE CONVECTION FROM A VERTICAL PLATE

1,0 1

0.8

0.6

0.4

0.2

0

I

I

I

1

I

0.05

0.10

0.15

0.20

0.25

-G(o) Fig. 3.

A versus f~(0).

guesses for f'((A, 0) and searching for a value of f'((A, 0) which will generate a solution that satisfies f;(A, ~=)~< 10 -a2. The numerical integration of Equation (19) was similar to the integration of Equations (17) and (18). The solution of Equation (19) required less computational time than that of Equatiofi (18) since Equation (19) does not contain any partial derivatives with respect to A. Satisfactory solutions were assumed to have been obtained when f:i(A, n~) ~ lO-~2Figures 1-3 contain the surface values of the derivatives of the universal functions. These values will be used in the examples worked out in the next section.

5. Results and Discussion When the surface heat flux is prescribed, the dimensionless heat transfer rate may be written as

102

RAMA SUBBA REDDY GORLA AND ROBERT TORNABENE

Nu. g a *1/3 - f ' ( sr 0)

(22)

where Nux = (hx/R) the local Nusselt number. Four numerical examples for specified surface heat flux will be presented here.

Case 1. Linear Surface Heat Flux Distribution Consider the linear case where qw(~) has the specific form qw(s~) = 1 + m~.

(23)

We see from (15) that Or(~) = C(1%" m~) 2/3

(24)

and consequently

O0, )t = - -

2ms~ -

or(0 0~

30 + m~:)"

(25)

The corresponding Merk series coefficients are given by

Oh Om~ 4~: 0~ - 3(1 + m02 02a 16s~2 0~:2 -

64

(26) m2~:2

3 (1 + ms~)3

(27)

At any streamwise location x (or so), the Merk series coefficients may be calculated from Equations (26) and (27) while m is specified. Then, by means of Equations (26) and (22) and Figures 1-3 for the numerical solution of the surface first derivative of f, the dimensionless heat transfer rate may be calculated by simple arithmetic. Comparisons are made in Figure 4 for m = 1.0. It may be observed that the Nusselt number increases with streamwise distance.

Case 2. Sinusoidal Heat Flux Distribution Consider the sinusoidal case where qw(~) has the form q.,,,(~) = sin(~).

(28)

Here it is shown that )t-

2~: 3 tan ~:

(29)

and

16s~a 02~ _ 0~2

64sr 3 sin a ~ [1 - ~: cot s~].

(31)

103

FREE CONVECTION FROM A VERTICAL PLATE

qw(F~) = 1 + m ~

1.0

m=l

0.5 Z

Y I

2

Fig. 4.

I 4

I 6

,

Nusselt number versus streamwise distance (% = 1 + m~).

Figure 5 shows the Nusselt number variation with streamwise distance.

Case 3.

Power Law Type Surface Heat Flux Distribution.

If qw(~) is of the form to produce nonsimilar flow, qw(~) =

(1 + U)

(32)

It can be shown that

21

1

a =3t1+C~

(33)

and 0A

8n2~ n

4 sr 0-~ = 3(1 + ~,)2,

(34)

02A 16~ z ~-~ = 32nZ~"[(n - 1) - (n + 1)~"].

(35)

The results for the variation of Nusselt number with streamwise distance are displayed in Figures 6 and 7.

Case 5.

Exponential Surface Heat Flux Distribution

Consider an exponential heat flux of the form qw(~) = e "e.

(36)

104

RAMA SUBBA REDDY GORLA AND ROBERT TORNABENE

/

q w ( ~ ) = s i n (n~) n = 1.0

7

I . . . . . . . . . . .

0

0.5

Fig. 5.

I 1.0

.

!

1.5

Nusselt n u m b e r versus streamwise distance (qw = sin(n~)).

qw(~) = 1+~ n

1.0

n-1

0.5 z

f .....

Fig. 6.

I 2

...................

I ................... 4

I 6

Nusselt n u m b e r versus streamwise distance (qw = 1 + ~", n ~ 1).

I 8

FREE CONVECTION FROM A VERTICAL PLATE

qw(~;) = I + U '

1.0

x -1

105

0.5

z

Fig. 7.

I

I

1.0

1.5

Nusselt n u m b e r versus streamwise distance (q~ = l + ~n, n = 2).

qw(~) = e n~,

1.0

_.=

I

0.5

0.5

z

] 1

0

Fig. 8.

I 2

Nusselt n u m b e r versus streamwise distance (qw = e"~).

106

R A M A SUBBA REDDY G O R L A AND R O B E R T T O R N A B E N E

We now have Or(~) =

C(e-r 2/3

(37)

and se 0 g(~:) 2 A = O,(sc) O~ - 3~n

(38)

with the Merk coefficients OA 8 4~ ~-~ = ~ n~:

(39)

16 2 32A 0-~ = 0.

(40)

and

The results for the Nusselt number variation with the streamwise distance are shown in Figure 8.

6. Concluding Remarks A Merk-type series has been applied to obtain the local nonsimilarity solution for the heat transfer from a vertical plate embedded in a saturated porous medium with an arbitrarily varying surface heat flux. Utilizing the numerical solutions of the universal functions provided in this paper, the heat transfer rates may be calculated in a relatively straightforward manner for any specified surface beat flux distribution. As no iteration procedure is involved in the final calculations, the method presented in this paper is easy for practical applications.

References 1. Cheng, P. and Minkowycz, W. J., 1977, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geophys. Res. 82, 2040-2044. 2. Kaviany, M. and Mittal, M., 1985, An experimental study of vertical plate natural convection in porous media, ASME Bound Volume, Heat Transfer in Porous Media and Particulate Flows, HTS Vol. 46, pp. 175-179. 3. Merk, H. J., 1959, Rapid calculations for boundary layer transfer using wedge solutions and asymptotic expansions, J. Fluid Mech. 5, 460-480. 4. Chao, B. T. and Fagbenle, R. O., 1974, On Merk's method of calculating boundary layer transfer, Int. J. Heat Mass Trans. 17, 223-240. 5. Gorla, R. S. R. and Zinalabedini, A. H., 1987, Free convection from a vertical plate with nonuniform surface temperature and embedded in a porous medium, Trans. A S M E , J. Energy Resour. Tech. 109, 26-30.