Natural convection with coupled mass transfer in porous media

I~T. O3MM. HEAT MASS TRANSFER Vol. i0, pp. 465-476, 1983

0735-1933/83 $3.00 + .00

@P~n

Press Ltd~ Printed in the I/f/ted ;tates

NATURAL CONVECTION WITH COUPLED MASS TRANSFER IN POROUS MEDIA

D. J, Close D i v i s i o n of Energy Technology Commonwealth S c i e n t i f i c I n d u s t r i a l and Research O r g a n i z a t i o n Highett, Victoria 3190, Australia

(C~L,~nicated by J.P. Hartnett and W.J. M i n ~ z )

ABSTRACT For c e r t a i n t y p e s of packed beds used f o r t h e r m a l energy s t o r a g e , h i g h d i s c h a r g e r a t e s Can be a c h i e v e d i f t~e e f f e c t i v e conductance i s i n c r e a s e d over t h a t a c h i e v a b l e with s t a g n a n t gases and l i q u i d s . A packed bed c o n t a i n i n g a f l u i d m i x t u r e w i t h one c o n d e n s i n g component and w i t h i t h e f i u l d m i x t u r e s i m u l t a a e o u s l y c o n v e c t i n g may a c h i e v e the r e q u i r e d high conductanCes. An a n a l o g y has been developed w i t h a s i m i l a r system w i t h no coupled mass t r a n s f e r . I t p r o v i d e s some u s e f u l i n s i g h t s i n t o t h e r o l e of g a s / v a p o u r m i x t u r e p r o p e r t i e s on the e f f e c t i v e c o n d u c t i v i t y of such s y s t e m s , and s u g g e s t s t h a t v e r y l a r g e i n c r e a s e s i n e f f e c t i v e c o n d u c t i v i t y are a c h i e v a b l e . 1.

Introduction

A proposed method f o r s t o r i n g t h e r m a l energy i s

illustrated

in Fig. I.

The a r r a n g e m e n t of t h e r m a l e n e r g y d e l i v e r y and d i s c h a r g e l i n e s a r i s e s the need t o s e p a r a t e high p r e s s u r e f l u i d ,

465

from

f o r example steam, from t h e s t o r a g e

466

D.J. Close

median

container.

This

tainer

walls.

packing within the

The

avoids

Vol. i0, No. 6

material

thermochemical

the use of expensive

and high strength

bed may be sensible,

and the interstitia]

The vapour is supplied from a liquid pool

fluid

phase

con-

change or

is a gas/vapour

mixture.

Ln the base of the container,

and a

pump with spray may be required to ensure that the packing remains wet. system

resembles

employed

a

controlled

that

vapour

proposed

only.

addition

The

by gas

or withdrawal

Nemecek vapour

et

a].

mixture

[]],

except

that

is suggested

of the non-condensing

gas,

so

The they

that

by

the system can

be maintained at one atmosphere pressure.

it is expected of natural

that the use of the gas/vapour mixture and the promotion

convection

in the effective

within

the packing will

conductivity

lead to substantial

increases

of the packing, an outcome required for satis-

factory store operation.

This paper

is concerned

convection

in packed

with

transfer

heat

beds only.

with

with Its

storage requires further work. parameters

mass

extension

Combarnous and Bories

numerically

bounding

problem

ol

oi natural

on an analogy thermal

energy

important

The System Model

[2], survey models of natural convection

the simplest model

gives

and with no mass transfer. reasonable

time and also with more recent measurements,

has

to the

based

However it is useful in identifying

under Steady state conditions

The

of an analysis

transfer,

and the role of gas/vapour mixtures.

2.

media

the development

coupled

two

dimensional

surfaces

system

is

agreement

dimension

uniform spheres.

The assumptions a r e ,

[2]:

dp,

When solved

with data

to that

13].

visualised

as

at top and bottom are isothermal

a characteristic

in porous

for example

shown

in Fig.

2.

The

with T2>T I.

The packing

sphere diameter

in a bed of

Vol. i0, No. 6

Mat

NA%l~:~AL CENVEL-TIQg IN POROUS M]~DIA

~

oxtrlctlon

•

-,

[

I

~

)I

ZI

L Delivery llne

467

/-~-.,

~,

to spray

llO0(f I---,--I \__.__"

! of

~

FIG. I Thermal energy storage system with packing and heat transfer fluid separated

m T,

FIG. 2 Model of f l u i d s a t u r a t e d porous medium with natural c o n v e c t i o n

(i)

the B o u s s i n e s q a p p r o x i m a t i o n h o l d s ;

(ii)

f l u i d and s o l i d p r o p e r t i e s are c o n s t a n t , and an i s o t r o p i c e f f e c t i v e bed conductance A* a c c o u n t s f o r h e a t t r a n s f e r by c o n d u c t i o n w i t h i n t h e medium;

(iii)

(v.grad)v terms are neglected;

(iv)

Darcy's Law holds;

(v)

rate

coefficients

between f l u i d and p a c k i n g are high enough to

assume l o c a l t e m p e r a t u r e e q u a l i t y . The

resulting

dimensionless equations for

steady s t a t e

w i t h a stream

f u n c t i o n ~ i n t r o d u c e d , [ 2 ] , are aZT ' a2T ' H . ,aT' ,ST'. ( )2 ax_f~ + az_f~ . ~ tu _~T + w a_ETj = o

z + (~)z Nu*

=

f~

~

H _ ~aT' =

z + 5 Ka-~-~T

the

fluid

comprises

p e r f e c t gas p r o p e r t i e s

' .-

({) C2)

0

aT' H 1 az' "dx' + ~ fo w'T'dx'

For coupled mass t r a n s f e r , (vi)

~,

(3)

a d d i t i o n a l assumptions are: an

inert

gas

and a

c o n d e n s i n g vapour w i t h

( f o r example a i r and water Vapour), and the

p r o p e r t i e s of the m i x t u r e and i t s components are assumed c o n s t a n t ;

468

D.J. Close

Vol. I0, ~b. 6

the surfaces of the packing material are covered with liquid which

(vii)

flows vertically downwards through the packing and has no influence on the applicability of Darcy's Law;

(viii)

following equality

(v) above a n d between

pressure which

the

Eckert

liquid

is assumed

and

vapour

Faghri pressure

to be equal

[4], there

is

and vapour

local

partial

to the vapour pressure of

pure liquid at that temperature.

3. 3.1

Derivation of Equations

Conservation of enersy An energy balance with m the potential for vapour diffusion yields

A*

~2m. 8 8(PdWhm ) + 8z-~) = 8-x(PdUhm ) +

(82T 82T~ ......82m ~ + 8z--~; + Pm~nv£~x--~

+ ~8

(pgu~h~) + 8~

(4),

(p~wgh~)

Conservation of the condensing and non-condensing components yield 8 8 8 8-x (Pd um) + 8zz (Pd wm) + ~ 8 +2 8-x (Pd u) ~

8 (p~u~) + ~

"~ /82m 82m) (O~w~) - Pm D^ ~x-~~ + ~

= 0

(Pdw) = 0

(5)

(6).

Expanding derivatives in (4) and (5) and substituting from (5) and (6) in (I), A .82T 82T. *{~x-~ + ~Z-~) + pm D* (hv - h~) 8h£ + p~w~ ~ A

.82m 82m~ 8h m 8h m 8h~ £~x--~+ az--~Z; = pd u ~ + pd w ~ + p£u~

8m 8m h~(PdU ~-~ + pd w ~ )

(7)

vapour mass balance on the whole cell in Fig. 2 yields

Pd d.

:

0

It will be assumed that this relationship holds locally, and from (vii) Section 2, u£ = O. The first assumption can also be justified on the basis that

the enthalpy

comparison

with

flows

those

in the liquid phase will

in the vapour

phase.

This

in general be small assumption

in

was made by

Vol. i0, No. 6

~O3WVECrION

Eckert

and Faghri

better

in their case since diffusion

horizontal cells.

[4]

as well

for diffusion

as vertical

IN POROUS MEDIA only,

469

although the justif~catio~

is

only does not give rise to the large

fluxes,

encountered

in natural

convection

In general hm = hm(T,m ) and h£ = h i ( T ) . However it follows from (viii) Section 2, that m = m(T). Then h m = hm(T) dm

X~

+ PmD#(hv - h i ) ~ tSZT

A~

and an effective conductance A "~ =

p o s t u l a t e d as in [4] and [8].

a2T.

m d m d S T ,h

,Sx-~ + 8z-~)= PdU (d-~---h£ ~

)" ~ x

dh

Then from (7), _

PdW (d~

dm _ db£ 8T h£ ~ m d--#--)~

(8).

dh dm dh£ Values of d-~' h£ ~ and m d-~ for air/water vapour at one atmosphere dh£ and various temperatures suggest that m ~ can be neglected, and an e

"effective" specific heat C

~,SZT

8ZT~

can be postulated.

8T

Then equation (8) reduces to

8T

~Sx-~ + 8z-~j = P d U C * ~ + PdWL'#~

With dimensionless

quantities

T', x' , z' as used in equations

(9).

(I)-(3)

and dimensionless velocities u" and w", (7) becomes

Hz

3.2

82T '

02T '

H (u" ST'

~x-~ + ~z-~ = ~

,,ST'

~-~v + w a-~v )

(10).

Conservation of momentum For the gas mixture 8P 8x

Pu=o K

8z

pmg -

(11)

w = 0

(12).

If the mixture is assumed to be a perfect gas, then

Pm

p

(l+m) McM d

= R-T

(M c + mM d)

(13).

From the assumptions, Pc = P c (T), and P - -c= m

P

or

M

/(~d

+ m)

m = (MclMd)(Pc/P)/(1-Pc/P).

(14).

470

D.J. Close

From (II) and (12), and

dPm - Md (Mc

dT

_p

Pm = Pm (T)

dPc

Pm

RT Md - I ) - ~

Cross d i f f e r e n t i a t i n g au

Vol. I0, No. 6

(15).

- -~

(9) and (10) and u s i n g (13),

dPm

aT

~+g-d~

~+~

aw

(16).

~=0

From the Clausius-Clapeyron equation, dPc _ Pchfg d ~ - R T z , and for a perfect gas ~ = I/T. C

dP m Putting

dT

= -pm~' and using (15),

then~' =611

-

and from (9),

~(M - ~z

-M d) RThfgl

(17),

a_uu _ g6'Pm aT az ~ + ~

aw ~ = 0.

u,,=O~' aZ'

With a stream function @' defined by, 8z~ '

H

aT'

Hz

Oz '2 + ERa** ~-~+ ~ 3.3

~-~

'

w. = _ ~_~i ax'

0

(18)

ax, ~ =

Determination of Nusselt number The

effective

conductance

of the packed

bed

for unit width

in the y

direction is defined as ~ = ~- ~ AT L " At steady state, and assuming no lateral transfer between cells [2], then the energy flux across any horizontal plane must also be Q. aT area I by Ax, AQ = - h ~-* azAX-- + PdWhmAX + p~w£h£Ax. assumption will be made that locally, PdWm + p£w£ = 0. Hence

AQ = - k**aT az Ax + PdWhmAx

With

hm --

dh

PdWmh~ Ax"

dh (r - % )

and

a constant, dh

~T L = - f~ k** ~-~ dx + fo Pdw d - ~ (T - T1)dx

as in For an

As in Section 3.1, the

Vol. i0, No. 6

NATJRAL ~ I O N f~

f~

But

IN POROUS MEDIA

-

dm L dh PdW ~ (T - TI) h£dx + fo PdW d ~

-

Pdw ~

dh

dm

471

(TI - To)dX

(T 1 - T O ) h£dx .

,?

- .

L ~dWdX.

dh

Pd~ (d-~ - h~ H~ )(TI - To)dX : (d-~ - h~ ~)(TI - T o) fo

f~ PdWdX = O, from the conservation of non condensing gas, hence L

~T

L

dhm

Q = - fo A'~ ~z dx + fo Pdw [dT

dan - h£ dT ] (T

or

Nu"~" = - f~ aT' ~ dx' + H ~

3.4

S i m i l a r i t y with h e a t t r a n s f e r o n l y s o l u t i o n s E q u a t i o n s (1O),

and

(3).

Nu~

= f(Ra~'k),

- Tl)dX ,

foI w"T'dx'.

(19)

(18) and (19) form a s e t d i r e c t l y analogous to ( I ) ,

Consequently s o l u t i o n s providing

that

and data

the

yielding

assumptions

are

Nu* = f(Ra*) satisfied.

(2)

lead to

An obvious

example where t h i s would n o t be t r u e r e l a t e s to the p o s s i b l e v a r i a t i o n of ~' with t e m p e r a t u r e .

For p e r f e c t g a s e s , ~ w i l l always d e c r e a s e with t e m p e r a t u r e

whereas (17) shows t h a t ~' may i n c r e a s e or d e c r e a s e .

Hence i n one system, ~'

may be d e c r e a s i n g with T i n c r e a s i n g i n some l o c a t i o n s b u t i n c r e a s i n g w i t h T at others, producing significant

flow and e n e r g y t r a n s f e r s d i f f e r e n t

from the

heat t r a n s f e r o n l y case.

4. From the d e f i n i t i o n are

critical.

E f f e c t of Gas Mixture P r o p e r t i e s of ~'

i n (17), t h e r e l a t i v e magnitudes of Mc and Md

For enhancement of

convection,

the m o l e c u l a r weight of t h e

n o n - c o n d e n s i n g component should be h i g h e r whereas f o r s u p p r e s s i o n i t be lower.

should

I f ~' i s n e g a t i v e t h e n f o r the geometry shown i n F i g . 2 c o n v e c t i o n

would n o t o c c u r , b u t would do so i f t h e p o s i t i o n s of t h e hot and cold p l a t e s were r e v e r s e d .

472

D.J. Close From (17)

if

Mc
then

can be d e d u c e d from t h e

fact

Vol, i0, No. 6

~'>0. that

I f Mc>Md, t h e n a t some T, ~' = O. as T t e n d s

to the freezing

temperature,

becomes v e r y s m a l l a s d o e s m / ( l + m ) .

When T t e n d s t o t h e b o i l i n g

m-~,

mixtures

and m / ( l ' + m ) ~ l .

helium or methane,

For

87.9°C r e s p e c t i v e l y . appropriate

beds. general

and

Bories

Further

data

[2]

summarise

measurements

hydrogen, 14.3°C and

temperatures.

considerable by

Buretta

Nu~' v s .

and

Ra* d a t a

Berman

[3]

for

confirm

t r e n d and e x t e n d t h e Ra* r a n g e .

[3] for water saturated beds of glass spheres can be

represented by Nu* = 0.4 (Ha**) ½ for 500 < Ra** is assumed

for 500 < 10,000.

to follow

< Ra ^ < I0,000.

the effective

Hence Nu ~

= 0.4

Between Ra* and Ra**' of 40 and 500 the

the data

in

[3].

With the reservation that

constant properties are used in the calculations, Table showing

1.5°C,

Effective Conductance of Packed Beds

Data from [2] and

relation

are

normal c o n v e c t i o n would o c c u r a t

v a l u e s o f Ra ~k, b u t n o t a t h i g h e r

Combarnous

their

to ~'=0

m

temperature,

of water vapour with

corresponding

At l o w e r t e m p e r a t u r e s ,

5.

packed

saturated

temperatures

This

I has been prepared

conductances of beds of 8 mm diameter glass spheres,

1 m high, for four saturating fluids at one atmosphere.

The permeability was

calculated from the Kozeny-Carman relationship d23 K =

P ~

36k(1-~) z

with a value of k of 4.8 for spheres from [7].

The void fraction is assumed

to be 0.4 and the diameter chosen ensures that no significanL extrapolation of Nu* vs Ra* data is required.

In one case T 2 is held constant at I00°C and

T I at 0, 40 and 80°C corresponding 90°C.

to average

temperatures

of 50,

In the other, T I is held constant at 0°C, with T 2 at 20, 60 and I00°C

corresponding to average temperatures of 10, 30 and 50°C.

Property data and

methods for calculating D, Pm and A m were obtained from steam tables, [6].

70 and

[5] and

D* was taken as 0.2 D, from [4]. Three

important features emerge.

Firstly very large increases in heat

fluxes are achieved with coupled mass convection.

Secondly higher conduct-

Vol. I0, NO. 6

NATORAL ~ I O N

IN POROUS MEDIA

473

TABLE 1 P a r a m e t e r s f o r a 1 m high bed of 8 mm diameter glass spheres Saturating Gas or Gas Nixture

T2 = lO0OC

Air

Ra* Nu* A*

T~I = OoC

.

T1=80°C

TI=40°C

TI=0°C

T2=20°C

T2=60°C

AT=20K 3.25

AT=60K 13.4

AT=lOOK 31.2

AT=20K 13.8

AT=60K 27.1

1

1

1

I

1

0.483 0.483

0.434 0.434

0.388 0.388

0.307 0.307

0.346 0.346

Freon 12

Ra* Nu* A*

54.8 1.45 0.438 0.635

226 5.74 0.387 2.22

530 9.21 0.339 3.12

225 5.72 0.254 1.45

458 8.74 0.295 2.58

Air/Water Vapour

Ra ~'~ Nu ~-~ A~'# A

487 8.29 2.42 21.5

790 11.2 0.671 7.54

519 9.11 0.453 4.13

32.7 I 0.314 0.314

159 4.27 0.369 1.58

Freon 12/ Water Vapour

Ra~'# Nu~ A~-# A

10600 41.2 0.943 38.9

9980 40.0 0.480 19.2

6500 32.2 0.367 11.8

467 8.79 0.257 2.26

1990 17.8 0.305 5.43

ances are obtained with the bed containing the Freon 12/water vapour mixture. This can be attributed to its higher (pd C*) and ~', and lower vm more than offsetting its lower A~'#. Thirdly relatively constant heat flux is achieved with the condensing mixtures when T 2 is held constant. This is attributable dm m 6' to the parameters ~ in C~ and ~ in increasing while AT decreases. The opposite effect occurs when T 1 is held constant but AT varies as in this case AT,

and ~

°

decrease together. 6.

Conclusions and Further Work

The original impetus for this study arose from the need to achieve very large increases in the effective conductance of packed beds used for thermal energy

storage.

They

coupled mass transfer.

should

be

obtainable

with

natural

convection

and

474

D.J. Close

The

model

presented

in this

analogy with heat transfer alone variability bility

in properties

of the buoyancy

Vol. I0, No. 6

paper

relies

in a similar

on

the

system.

development

of

an

Due to the greater

in the coupled mass transfer case, and the possiaiding,

hindering

or preventing

convective

flows,

further study of the effects of property variations and the various gas and vapour mixtures that might be used is necessary. models noted in, for example tional

factors

[2], require study.

such as convection

inside

In addition, more complex For energy storages, addi-

containers

of height

to diameter

ratios near I, non isothermal heating and cooling surfaces and operation at values of Ra** well beyond the existing Ra* range must be tackled.

Acknowledgement

The

author wishes

to thank his colleagues

at the CSIRO

Division of Energy Technology for stimulating discussions.

Nomenclature

C~

=

dh m d--T-- - h£

[J.kg.-1K-11;

Cp,

specific heat of gas [J.kg.-IK-l};

dp,

packing dimension [m];

D,

diffusivity of gas/vapour mixture [mZ.s-l];

D~ ,

effective diffusivity of stagnant packed bed [mZ.s-l];

g,

gravitational acceleration [ m . s - 2 ] ;

H,

height of packing [m];

hm, h~,

specific enthalpy of gas mixture [J.kg-l];

hfg,

latent heat of vaporisation of condensing component [J.kg-l];

hv , k,

enthalpy of vapour [J.kgl];

L,

convection ceil dimension in horizontal direction [m];

specific enthalpy of liquid [J.kg-l];

constant in Kozeny-Carman equation;

Mc ,

molecular weight of condensing component [kg/kg mole];

Md , m,

molecular weight of non-condensing component

[kg/kg mole];

mass fraction of condensing component in gas mixture. Ratio of condensing to non-condensing component [kg/kg];

P,

gas mixture pressure [Pa];

Vol. 10, No. 6

NAqURALOCR'A/ECPION IN PONCYJSMEDIA

Pc ~

partial pressure of condensing component [ P a ] ;

Rc , R,

gas constant of condensing component [ J . k g . ' l K - 1 ] ; universal gas constant [J.kg mole.-1K-1]; oC

Ra*

= g~(~)~.

AT H.

m

Ra~W~

Rayleigh Number for heat transfer only;

--

(PdC*z

=

'

= X/A*.

Nu**

= k/X**. Nusselt Number for coupled heat and mass transfer; heat flow rate per unit area [W.m-2];

T,

temperature [K];

To,

reference temperature [K];

T1,T2,

temperatures of upper and lower surfaces of packed bed [K];

Nusselt Number for heat transfer only;

T-T 1 T t

= T2_TI ; U~

gas velocity in x direction [m.s-l];

U~

= PmCpUH" k* '

u~, w£

mean liquid velocities in x and z directions [m.s'l];

V~

velocity [m.s-1];

W~

gas velocity in z direction [m.s-l];

u"

Z,

-

PdC*UH A**

PmCp wL W t

x,z,

.

g~"-v---'X** AT H. Rayleigh Number for'coupled heat and mass m transfer;

Nu *

x' = x/L;

475

wit

•

'

PdC*WL

coordinates as shown in Fig. 2 [m]; z' = z/H; gas compressibility.

Greek symbols 6,

coefficient of volumetric expansion [K'l];

~'

= 611

m -/-~(M c

-

Md) ~ ,

]

[K-l];

476

D.J. Close

Vol. i0, No. 6

6,

void fraction of packed bed;

@,

stream function, u' = ~@/Oz', w' = -~@/8x';

~' ,

stream function, u" = ~'/~z',

A,

effective thermal conductivity of packed bed [W.m-I.K-I];

X*,

w" = -~'/~x';

effective thermal conductivity of stagnant packed bed with no coupled mass transfer [W.m-I.K-I];

A~ ,

effective thermal conductivity of stagnant packed bed with coupled mass transfer [W.m-l.K-l];

Vm,

kinematic viscosity of gas or gas mixture [m2.s'l];

Pd'

density of non-condensing gas; mass per unit volume of mixture

[kg.m-3]; 0~,

density of liquid [kg.m-3];

Pm'

density of gas or gas mixture Ikg.m-3];

K,

permeability of packed bed [m2];

~,

dynamic viscosity of gas or gas mixture [N.s.m-2];

AT

= T 2 - T 1 [K]. References

I.

J.J. Nemecek, D.E. Simmons and T.A. Chubb, Demand sensitive storage in molten salts, Solar Energy 20, 213-217 (1978).

2.

M.A. Combarnous and S.A. Bories, Hydrothermal convection in saturated porous media, Advances in Hydroscience, I0, 231-307 (1975).

3.

R.J. Buretta and A.S. Berman, Convective heat transfer in a liquid saturated porous layer, Journal of Applied Mechanics, 43, 249-253 (1976).

4.

E.R.G. Eckert and M. Faghri, A general analysis of moisture migration caused by temperature differences in an unsaturated porous medium, Int. J. Heat Mass Transfer, 23, 1613-1623 (1980).

5.

Warren M. Rohsenow McGraw-Hill (1973).

6.

R. Byron Bird, Warren E. Phenomena. Wiley (1960).

7.

M.J.R. Wyllie and A.R. Gregory, Fluid flow through unconsolidated porous aggregates, Ind. and Ensng. Chem., 47, 1379-1388 (1955).

8.

E.R.G. Eckert and E. Pfender, Heat and mass transfer in porous media with phase change, in Proc. 6th International Heat Transfer Conference, Toronto, 1978, Vol. 6, pp. 1-12. Hemisphere, Washington, D.C. (1978).

and James

P.

Stewart

Hartnett,

and

Edwin

energy

Handbook of Heat Transfer.

L.

Lightfoot,

Transport