Heat transfer by natural convection in a vertical porous layer

=

~/q>

=

0

(2.11)

which imply that @, p ahd ~ are periodic functions. To obtain the equations for these quantities, we take horizontal mean of equations (2.7) ahd (2.8) which yield 0

= O~-Vp - q

(2.12)

"~ ~ = ~--~<"~,,k4 ~ ~-~"

(2.13)

Equation (2.8) and its horizontal mean yield

.

+

(214)

To simplify this equation, we neglect the left hand side of equation (2.13) which implies that heat transfer by conduction and convection reach an equilibrium in response to a time change of W and @.

Under this approxima-

tion and using the fact that at a distance far away in the z-direction the temperature distribution is isothermal, we get

~zBT=

R( < w e >

-

) .

(2.15)

N. Rudraiah, T, Masuoka and M.S. Malashetty

64

By substituting this m

Vol. i0, No. 1

into equation (2.14) we have the

three equations (2.9), (2.12) and (2.14) for the three unknowns @, p and q. To determine the condition for the onset of convection and the corresponding heat transfer, w e follow the viscous flow analysis of Platzman

[_10]. In this method @, p and

q can be expanded into a series and each term is expanded as an eigen function expansion.

To determine the conditlorm

for the onset of convection the llnearlzed version of equations (2.9), (2.12) and (2.14) are enough.

In that

case, we look for the solutions of the form E),p,~o<( e~, pa¢, ~ ) Here @~,p~ , ~

exp [ ( ° ' ~ / ~ ) t ]

(2.16)

which are functions of x,y,z, are called

normal modes and the wave numbers are denoted by subscripto~. They satisfy the equations

v.~ o

~-~9~

0

=

= %k

=

we

(2.1~) -v

p~ -

%

(2.18)

+ -~ V 2 @~

(2.19)

and appropriate boundary conditions. If o~*, @* , p* ~,

0~ , Poc and ~

and ~ *

are the complex conjugate of

respectively then w e define the inner

product as

(w~:, gec)= w~ gac

(2.20)

Vol. i0, No. i

where

(...)

CONVECTION IN A VERTICAL POROUS LAYER

( = <

65

>) represents an average over a

volume bounded by the cell when the vertical periodicity is assumed. ~.(~

Then from (2.18), using the condition q~V P; =

p;) = 0, we have

(w~, @js) = (@<, w/~ )

(2.21)

Similarly, making use of coditionv .(@V

@* ) = 0 for each

mode, v~ get

( V20~ , %) = (06, V2Oj~).

(2.22)

Then#om (2.20) to (2.22) f o r rea#l R > 0, we have o-~e~, eoc) = (e~, ~-,@~) = ~-~@~, 8•) = o~*(@~ , @~). if @~c is different from zero, ~ =

o-~* .

(2.23)

This implies that

the eigenvalue o~ocis real and hence @oc , Poc and q~'~are real. This mea~s that any oscillatory motion does not arise when equilibrium breaks down.

(~-o-~) (e¢, @9) = ( ~ @ ~ ,

From (2.20) to (2.22) we also have

@~) - ( @ ~ , 6~9@j~ ) = o.

This implies that any two normal modes @oc, @~ O"ocand 0"~ respectively are orthogonal.

(2.24)

with eigenvalues

In other words

(2.25) which converts the sequence @~, @~ sequence.

.... into an orthonormal

This take care of the secular and resonance terms

that arises in the process of solving non-linear 3.

equations.

Normal Mode Analysis

The linear theory gives only the condition for the onset of

66

N. Rudraiah, T. MasuokaandM.S. Malashetty

Vol. I0, No. 1

convection and is silent about the determimation of heat transport for the amplitudes cannot be determined. case we have to resort to the non-llnear theory.

In that These

aspects are considered in this section using the normal mode analysis. Normal Modes In the discussion of onset of convection, ~ is assumed to be zero.

This does not mean that the motion under considera-

tion is necessarily two-dimensional in the usual sense of the word for u and w may still depend on y. define the stream function u~

=

~,

w~

=-

If ~ is zero, we

as ~x

(3.1)

Then equating (2.18) and (2.19) after eliminating the pressure, take the form

o

÷

(32)

In the normal mode analysis we look for the solutions of (3.2) and (3.3) in the form

~ ( x , , ,~, ~

~ f~Cz) ~

~:~)

(x,z) = f~ (z) ¢~(x)

(3.4)

where f ~ satisfies the following periodic condition (hereafter the subscript ~

-!--2f BZ ~

is omitted)

-a2f

(3.5)

Vol. I0, NO. I

CONVECTION IN A VERTICAL POROUS LAYER

where a is dimensionless wave number. Then equations (D2-a2)g + D@

(3.2) and (3.3) take the form =

Lo--

0

(3.6)

÷

= o

(3.7)

at x = + I

(3.8)

The boundary conditions are g = O, D@ = 0 where D =

~.

Eliminating @ between (3.6) and (3.7), we get

and the boundary conditions are g = (D2-a2)g = 0 The

at x = ~ 1

(3.10)

~ l u t i o n of (3.9), satisfying (3.10), is g = K sin ( n ~ x )

(3.11)

where K is a constant and n is an integer (n = 1,2,3, .... ). At the same time the following relat ion should hold O-- =

R(n2~2

(3.12)

+ a2 )

By making use of (3.7), (3o11) and (3.12) we obtain the temperature @ @ = -

~(n2y~2 + a2) nrc

cos ( n ~ x ) .

(3.13)

Similarly making use of (3.1) and (3.11), we obtain the axial velocity w in the form w =

- Knucos

(n~x)

(3.14)

67

68

N. Rudraiah, T. Masuoka and M.S. Malashetty

Here keeping the generality, we can choose f2 orthonormality condition Hence

K=

I.

The

< w 2 > = Io

~ .

For marginal equation

(2.25) gives f 2 w 2 =

Vol. i0, No. I

stability6 ~ = 0 and hence setting G - = 0 in

(3.12), we obtain

R

=

~.~2 + a2) 2 /C2

for n = I.

(3.15)

Convection can occur when R is raised to a minimum value Rc =

Tf.2

(3.16)

with a critical wave number a = O. Rayleigh number is based

to infinite wave length in

We use this with the understanding

term is only a convenient Physically

that the

on half of the spacing of the plates.

The zero wave number corresponds the z-direction.

It is remembered

that the

expression for "long wave lengths".

this implies uni-cellular

critical wave number a = 0 equation

pattern.

For this

(3.12), using

(3.16) with

n = I, becomes

,~--= Equation

Rc

(1 --IT.-).

(3.'v7)

(3.12) is numerically

evaluated for different

values of g-and a and the results are depicted in figure 2. When R is raised to a minimum value R c =

7~2 in the neutral

curve (o~= 0), convection with a wave number a = 0 can occur. V~en R < Rc, o--is negative for all the disturbances disturbances

will decay.

and the

The question how the cell size

changes with R, when ~ exceeds R c, can be answered by determining

the amplitude which is the relam of non-linear

theory discussed

in the next section.

Vol. i0, No. I

CONVECTION IN A VERTICAL POROUS LAYER

69

Amplitude and Heat Transfer We expand O as e

= Z

A~(t)@~ (x,y,z)

(3.18)

where A~ using the orthonormality property (2.25) is

A~= (O,@~) = e e l .

(3.19)

The Fourier co-efficient A ~ ( ~ = ~ , amplitudes.

,...) are called

Similarly, we expand the velocity ~ and the

pressure p by

= ~A.~., p = Z A . ~ Equations

dA~

(3.20)

(2.14) and (3.19), then give

= (@~,w)+ ~(@oc, V2O) -R Oatw (-<~>)

--O~(~.V )0

+ @<(i~. v )@> .

(3.21)

Using (2.19) to (2.22) and (3.18) to (3.20) we can write

(,~,~,w)

+ ~ (o,~, ~,20) = (,,,,~,o) +

~(

=

(G'o,~, o)

=

O--Ao~ .

And due to vertical periodicity

V2Ooc 90)

(3.22)

of O~,the fifth term on the

right hand side of (3.21) vanishes.

As w and @ are expressed

by (3.18) and (3.20) the amplitude equation (3.21) takes the form

=~-Aoc- R Ooc w(- ) - @oc(~.V )O.

(3.23)

Here, let us adequately approximate @ and w, with only one mode.

Even with this first approximation,

a considerable

70

N. Rudraiah, T. Masuoka and M.S. Malashetty

Vol. i0, No. 1

insight is obtained, so far as the neighbourhood of the onset of convection is concerned. By substituting @ = A~@~c (3.23) and using dA~

¢ ~-

and w = A~w~ into equation

@~(~. V ) @ = O, we obtain ~

= ~A~- R( - < w ~ > ) A i .

(3.24)

Here, A~ represents the intensity of convection %4hen A ~ is small, equatlon (3.24) shows that convection may develop exponentially with time which is usually the case in linear theory.

When A ~ b e c o m e s

large, the second term on the right

hand side arising due to momentum and heat advection terms cannot be neglected so that convection is suppressed by these advection terms and finally reaches a stable steady state. Therefore in steady state equation (3.24) becomes

A~

=

~( )

(3.25)

Using the structure of convection given by equations (3.13) and (3.14) we obtain after using equation (3.12)

- -~ C ~ % ~ )

~L

~

J

(3.26)

<~e> The heat transfer is usually expressed in terms of the Nusselt number Nu, a dimensionless number defined as the ratio of total heat transfer to the heat transfer by oonduetion only.

~T ~'E

Considering the temperature gradient

~e ~zv

= -I + g-E +

Vol. i0, No. 1

and

CONVECTION IN A VERTICAL POROUS LAYER

71

~ z T g i v e n by equation (2.15), we have ~'u

=-

>

=

I + R

(3.2s)

Z=0 For the onset of convection R must exceed the critical value F.c and since our analysis is valid only for R slightly greater than Rc, we may take only the first term of the expanded series where a value n=1 is used in equation (3.26). Then equation (3.28), using (3.27) and (3°26), becomes

Nu

= 1 +

2x4

(~2,+a2.), 2

[ (2+a2)2]RJ 1 -

7t2

(3.29)

"f~'e have shown earlier that in the case of vertical flow,

the motion is unicellular because the critical wave number at which convection sets in is a = 0, independent of R. wave number maximizes Nu for a given R. using a = 0, we have R Nu= 1 + 2 ( I - ~ £ )

This

Then from (3.29),

= 1 + 20--.

(3.30)

This equation is applicable atleast in the vicinity of the critical point of the occurence of convection. The Nu given by equation (3.30) is numerically evaluated for different values of R and the figure 3.

ruesults are shown in

From this figure it is clear that R > ~ 2, Nu

increases and it becomes almost independent of R for R > 3 ~ 2. This uniform nature of Nu for R > 3 ~T2 is mainly due to the limitation that our results are valid only for values of R slightly greater than Rco

To obtain improved r e s u l t s we

have to use the large amplitude analysis°

72

N. Rudraiah, T. Masuoka and M.S. Malashetty

Vol. I0, No. 1

Conclusions The analysis employed in this paper concerns with the study of determining the condition for the onset of convection and the mean heat transfer due to convection in a vertical porous layer of large extent heated from below and cooled from above°

From this it can be concluded that:

(I) Convection is uni-cellular and is independent of time at neutral stability. (2) The critical Raylelgh number occurs at zero wave number and the detailed relationship between the Rayleigh number, wave number and growth rate are given by equation (3.12) and the results are depicted in figure 2. clear that when

R<

From these figure it is R c (Rc is the value of R at a=0)

o--is negative, the disturbances will decay and the cell size changes with R when R > R c.

In particular'•

is independent of "a" for values of a ~ 0.5 and increases with an increase i n o - a n d a. (3) The mean heat transfer is maximum at the zero wave number°

The relationship between the Nusselt number

and the Rayleigh number are given by (3.30) and the results are depicted in figure 3.

This figure shows

that our analysis is valid only for values of R ~ 3 R c.

Large amplitude analysis is needed to

predict the mean heat transfer for R >

3 RcO

Vol.

I0, NO.

1

CONVECTION

IN A V E R T I C A L P O R O U S l A Y E R

160 ,0~ =0.3

40-,

0

1 FIG. 2 - R E L A T I O N

2

fl

BETWEEN

3 6" . a

AND

R

f Nu

O

i

0

i

;~2

!

2;K2

3X2

R

FIG,3-HEAT TRANSFER BY FREE CONVECTION

73

74

N. Rudraiah, T. Masuoka and M.S. Malashetty Nomenclature

A

amplitude ;

a

were number ;

C

P

specific heat at constant pressure;

g

acceleration due to gravity;

k

unit vector in the z-direction;

k

perneability

of the porous medium;

half of the spacing between ]'Tu

Nusselt nu~ber ;

~?

pressttre in fluid;

the plates;

~.~ean filter velocity of the fluid ~

Lapwood Rayleigh ntu~.ber (Ra.k/~2);

lla

~ayleigh number

T

t emperat tire ;

t

t i11e ;

(u,v,w);

(o~#]z;x).~

x,y,z space co-ordinates. Greek Symbols coefficient

of thermal expansion;

temperature gradient; heat capacity ratio @

temperature i

~n

coefficient

~*

visccsity~

of effective thermai conductivity;

kinematic viscosity P

density~

~(~cD) m / ( p C p ) f ]

Vol. I0, No. 1

Vol. i0, No. 1

CONVECTION IN A VERTICAL POROUS LAYER

grov~h r ate; 7

average temperature;

6

porosity of the medium;

~u

stream function.

Subscripts b

value ~% equilibrium state;

f

value of fluid;

m

value of mixture of solid and fluid;

o

value at origin;

s

value of solid;

~.~

corresponds to a wave n~unber vector of normal mode; >

horizontal average~ ) vol~e

average. Acknowledgment

This work was supported by the UGC under DSA programme. The work of one of us (T.~) was sponsored by INSA-JSPS senior scholars exchange programme.

The research by one

of us (~[SM) was supported by the UGC, India under its Faculty Improvement Programme.

References I.

N.Rudraiah, and P.K.Srimani,

'Finite amplitude cellular

convection in a fluid saturated porous layer', Proc.Roy.SocoLondon A373, 199 (1980). 2.

C.G.Bankvall,

'Natural Convection in a vertical permeable

space', Warme-und Stoffubertragung, 3.

7_, 22 (1974).

J.E.Weber, The Boundary-layer regime for convection in a vertical porous layer, Int.J.Heat ~!ass Transfer 18, 569, (1975).

75

76

N. Rudraiah, T. Masuoka and M.S. Malashetty

4. P.J. Burns,L.O.Ohow and O.L.Tien,

Vol. i0, No. 1

'Convection in a Vertical

slot filled with porous insulation,'

Int.J.Heat

~,.~ass Transfer 20, 919 (1977). 5.

N.Rudraiah and S.T.Nagaraj,

'~,[atural convection through

a vertical porous strattun', Int.J.Eng6.Sci, I_~.~, 589 (1977). 6.

N.Rudraiah,

M.Venkatacha~appa

and l:l.S.Ealashetty, 'Oberbeck

convection through a vertical porous stratum' Proceedirgs

of the Indian Academy of Sciences

(To appear)

(19[]2).

7. T.L;asuoka, Y.Yokote and Toliatsuhara,

'Heat transfer by

natural convection in a vertical porous layer' Bulletin of the JSIfE, 24, 995 (1981). 8. A.Bejan and R.Anderson,

'Heat transfer across a vertical

impermeable partition imbeded in porous medium' Int.J.Heat Liass TransfeT,

24, 1237 (1981).

9. S.A.Bories and Iv1.A.Combarn2us, 'Natural convection ina slopping porous layer', J.Fluid lJech., 57, 63(1973). 10. G.W.Platzman,

'The spectral dynamics of laminar convection',

J.Pluid ~Tech., 23, 481-510 11. T.Easuoka,

(1965).

'Heat transfer by free convection in a porous

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'Asymptotic analysis of natural

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Int.