drying granular porous media - theory and experiment

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DRYING GRANULAR POROUSMEDIA - THEORY ANDEXPERIMENTStephen Whitaker a & William T-H. Chou aa Department of Chemical EngineeringUniversity of California, Davis, California,95616, U.S.AVersion of record first published: 19 Oct 2007.

To cite this article: Stephen Whitaker & William T-H. Chou (1983): DRYINGGRANULAR POROUS MEDIA - THEORY AND EXPERIMENT, Drying Technology: AnInternational Journal, 1:1, 3-33

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DRYING TECHNOLOGY, '1(1), 3-33 (1983-84)

DRYING GRANULAR POROUS MEDIA - THEORY AND EXPERIMENT

Stephen Whitaker and William T-H. Chou Department of Chemical Engineering

Univers i ty of C a l i f o r n i a Davis , C a l i f o r n i a 95616

U.S.A.

-: mass t r a n s f e r , hea t t r a n s f e r , c a p i l l a r y

a c t i o n , evapora t ion f r o n t .

ABSTRACT

From a gene ra l theory of d ry ing g r anu l a r porous media, we have c ons t ruc t ed a s i m p l i f i e d theory t h a t c o n s i s t s of a s e t of coupled, volume-averaged t r a n s p o r t equa t ions f o r t h e temperature and t h e s a t u r a t i o n . The theory i nco rpo ra t e s t h e l i q u i d and vapor phase con- t i n u i t y equa t i ons , combines t h e l i q u i d , s o l i d and vapor phase thermal energy equa t ions i n t o a s i n g l e temperature equa t ion and makes u se of Darcy 's law f o r t h e l i q u i d phase t o account f o r moisture t r an s - p o r t owing t o c a p i l l a r y ac t i on . By pure ly q u a l i t a t i v e reasoning , one can show t h a t combined hea t and mass t r a n s f e r t h e o r i e s of d ry ing can- no t provide a complete t h e o r e t i c a l explana t ion of d ry ing phenomena and a d e t a i l e d comparison between theory and experiment suppor t s t h i s po in t of view. Specula t ion concerning t h e l o g i c a l course of subsequent t h e o r e t i c a l s t u d i e s is provided.

INTRODUCTION

The motion of bo th l i q u i d and vapor phase moisture through a

r i g i d , i n e r t , porous s o l i d is descr ibed by t h e governing equa t ions

and boundary cond i t i ons f o r mass, momentum and energy. A d e t a i l e d

l i s t i n g of t h e s e governing equa t ions and boundary cond i t i ons is

Copyright 0 1983 by Marcel Dekker, Inc. 0737-3937/83/01010003$3.50/0

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4 WHITAKER AND CHOU

given elsewhere (Whitaker, Sec. 11, 1977), and as a point of refer-

ence we list the transport equations as

api +V.(p v ) = 0, i = 1,2 - at i-i

where i = 1 refers to water and i = 2 refers to air. The species

velocity vector is denoted by v. and the mass average velocity vector -1

by x. The fluid motion is assumed to be in the Stokes flow regime

and quasi-steady so that the Navier-Stokes equations reduce to Eq. 2,

and we have neglected viscous dissipation and compressional work so

that the thermal energy equation (Whitaker, Sec. 5.5, 1983) reduces

to Eq. 3 provided variations in the thermal conductivity can be

neglected. The drying process under investigation is illustrated in

Fig. 1 and we would like to be able to predict the saturation and

temperature profiles in such a system in the absence of adjustable

parameters. The details of this multiphase system are shown in Fig.

2 where the rigid, inert solid phase is identified as the o-phase,

the @-phase represents the liquid which may be continuous or discon-

tinuous, and the y-phase consists of the vapor plus air which is con-

sidered to be a single inert species.

When Eqs. 1 through 3 are volume-averaged, and the boundary

conditions at the three phase interfaces applied, the following

transport equations are obtained (Whitaker, Sec. 5, 1980) - Dry Air - . .

Wet Sand . .. , ' . . - . . . . ' . . . . . . . . . . . . .

-1.0

Figure 1. One-Dimensional Drying of Wet Sand

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DRYING GRANULAR POROUS MEDIA

T o t a l Thermal Energy Equat ion

Liqu id Phase Equat ion of Motion

Liquid Phase C o n t i n u i t y Equat ion

Gas Phase Equat ion of Motion

-Y

Gas Phase C o n t i n u i t y Equation

Gas Phase D i f f u s i o n Equat ion

I n d e r i v i n g Eq. 4 , t h e p r i n c i p l e of l o c a l thermal e q u i l i b r i u m

(Whitaker, Sec. 3 . 3 , 1980) h a s been invoked s o t h a t <T> and 

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WHITAKER AND CHOU

F igure 2 . Wet Granular Porous Medium

r e p re sen t s p a t i a l average q u a n t i t i e s whi le C r e p r e s e n t s a mass P

f r a c t i o n weighted cons t an t p r e s s u r e hea t c apac i t y . Equations 5 and

7 a r e no th ing more than Darcy ' s law f o r t h e l i q u i d and gas phases

r e s p e c t i v e l y . The l i q u i d phase c o n t i n u i t y equa t ion con t a in s a l o c a l

time r a t e of change of mass owing t o t h e v a r i a t i o n of t h e l i q u i d

phase volume f r a c t i o n wi th t ime, a convect ive t r a n s p o r t term which

r e s u l t s from c a p i l l a r y a c t i o n , and a source term <m> which accounts

f o r t he evapora t ion o r condensat ion of t he l i q u i d . The gas phase

c o n t i n u i t y equa t ion i s of a s i m i l a r form, bu t s i n c e t h e gas phase

d e n s i t y p i s not cons t an t t h e i n t r i n s i c phase average dens i t y Y Y Y

appears i n t h i s volwne-averaged c o n t i n u i t y equa t ion . I n w r i t i n g

Eq. 8 we have neglec ted t he d i s p e r s i v e term (Whitaker, Sec. 5, 1980)

Y s i n c e i t w i l l be smal l compared t o t h e convec t ive t r a n s p o r t , <v >. Y "Y

The gas phase d i f f u s i o n equa t ion given by Eq. 9 ha s been w r i t t e n f o r

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DRYING GRANULAR POROUS MEDIA 7

t h e evapora t ing s p e c i e s and i t has t h e same form a s Eq. 8 wi th t h e

exception of t h e d i f f u s i v e term on t h e r i g h t hand s i d e .

I n a d d i t i o n t o t h e t r an spo r t equa t ions given by Eqs. 4 and 9 ,

t h e r e is a volume f r a c t i o n c o n s t r a i n t given by

and s e v e r a l thermodynamic r e l a t i o n s t h a t can be expressed a s

Th i s £ o m of t h e Clausius-Clapeyron equa t ion con t a in s t he Kelvin

e f f e c t c o r r e c t i o n ; however, su r f ace t en s ion h a s a n e g l i g i b l e e f f e c t

on vapor p r e s su re f o r t h e system we wish t o s tudy and t h e unknown

l e ng th parameter r does not p r e sen t a problem.

The c r u c i a l assumption t h a t is made i n o rde r t o s impl i fy t h i s

s e t of equa t ions is l a r g e l y based on t h e t r a d i t i o n a l a n a l y s i s of t h e

S t e f a n d i f f u s i o n tube experiment (Lee and Wilke, 1954). Th i s evapora-

t i o n process is i l l u s t r a t e d i n Fig. 3 and i n t he a n a l y s i s of t h e f l u x

of t h e evapora t ing spec i e s , it i s t r a d i t i o n a l t o assume t h a t t he gas

phase p r e s su re i s cons t an t and ignore t h e gas phase momentum euqat ion .

Th i s would appear t o c r e a t e a problem wi th t h e de te rmina t ion of <v >; "Y

however, i t is easy t o show t h a t t h e mass average v e l o c i t y can be

represen ted i n terms of t h e d i f f u s i o n v e l o c i t y of t h e evaporat ing

s p e c i e s , and t h i s a l lows t h e momentum equat ion t o be neglec ted . To

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8 WHITAKER AND CHOU

show this we make use of the jump condition (Slattery, 1981) at the

6-y interface illustrated in Fig. 3.

Herey is the velocity of the interface, 2 is a unit normal vector, and the form of Eq. 16 assumes that there is no accumulation or con-

sumption of ith species in the interfacial region. When species 2 is

insoluble in the liquid Eq. 16 can be used to obtain

where p v and p v represent mass fluxes in the gas phase. Equations 1-1 2-2

17 can be used to express the velocity of the inert as

Since the mass average velocity is defined by

v = WIXl + W2X2 *

we can use Eq. 18 to estimate !:as*

If conditions are such that

P2 << P6. P1 << P6

*The symbol: indicates an order of magnitude.

one can see that Eq. 20 simplifies to

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dry a i r *

Y - phase

Figure 3. Evaporat ion from a S t e f a n D i f f u s i o n Tube

Under t h e s e c i rcumstances i t i s easy t o show (Bi rd , S tewar t and Ligh t -

f o o t , Sec. 16 .1 , 1960) t h a t t h e mass average v e l o c i t y can be expressed

i n terms of t h e mass d i f f u s i o n v e l o c i t y accord ing t o

and t h e gas-phase momentum e q u a t i o n has been bypassed.

The t r a d i t i o n of ignor ing t h e gas-phase momentum equa t ion i s w e l l

e s t a b l i s h e d i n t h e s t u d y of what a r e commonly r e f e r r e d t o a s " d i f f u s i o n

problems", and t h e a n a l y s i s l e a d i n g t o Eq. 23 , a long w i t h t h e exper i -

menta l r e s u l t s ob ta ined wi th t h e S t e f a n d i f f u s i o n tube (Lee and Wilke,

1954) , c e r t a i n l y f o r t i f i e s t h i s t r a d i t i o n . However, one can f i n d a

f l aw i n t h i s p o i n t of view by a l lowing t h e i n e q u a l i t y p << pB t o be

s a t i s f i e d by t h e s p e c i a l c o n d i t i o n p2 = 0 . Under t h e s e c i rcumstances

we would have a s i n g l e component evapora t ion process i n t h e S t e f a n d i f -

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WHITAKER AND CHOU I f u s ion tube shown i n F ig . 3 and t h e p r e s su re drop-flow r a t e r e l a t i o n i n

t h e y-phase would be governed by t h e Hagen-Poiseui l le law. Under t he se I

circumstances, i t would obviously be unacceptable t o i gno re t h e gas-phase

momentum equat ion . It seems t h a t neg l ec t i ng momentum cons ide r a t i ons i s a

process t h a t is not completely understood, and an i n t e r e s t i n g example of

t h i s is t h e Graham r e l a t i o n (Jackson, 1977) t h a t de sc r i be s t h e r e l a t i v e ' f l u x e s of two s p e c i e s i n a porous medium under i s o b a r i c cond i t i ons . Given1

a c ons t an t p r e s su re system, t h e r a t i o of f l u x e s i s e n t i r e l y governed by

t h e momentum exchange process between t h e d i f f u s i n g s p e c i e s and t h e r i g i d

porous medium.

While unanswered ques t i ons remain concerning t h e S t e f an d i f f u s i o n

tube , we r e l y on t h e a n a l y s i s l e ad ing t o Eq. 23 and t h e good agreement

between theory and experiment and proceed a long t h e same l i n e s i n our

a n a l y s i s of t h e dry ing process i l l u s t r a t e d i n Fig. 1. Under t he se

circumstances t h e d i f f u s i o n equa t ion given by Eq. 9 is reduced t o

(Whitaker, Sec. 5, 1980)

Here we a r e forced t o d i s c a r d Eq. 7 from our l is t of governing equa-

t i o n s wi th t h e thought t h a t any computed gas-phase p r e s s u r e g r ad i en t

i s l i k e l y t o be e r roneous .

From Eq. 22 we know t h a t t h e gas-phase mass f l u x i s j u s t equa l

t o t h e mass f l u x of t he evapora t ing s p e c i e s

pv_ = P1xl, i n t h e y-phase (25)

and from Eq. 24 we s e e t h a t t h i s f l u x i s represen ted a s

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I n our a n a l y s i s of bo th t h e gas phase and l i q u i d phase mois ture t r an s -

p o r t , i t i s convenient t o work i n terms of a s a t u r a t i o n def ined a s

When even a smal l amount of l i q u i d is p r e sen t t h e s a t u r a t i o n is re-

l a t e d t o t h e 0-phase volume f r a c t i o n

S = E / ( l - E ) , when E >> C P ~ > ~ / P ~ e o B (28)

t hus , any func t i ona l dependence on E can be represen ted by a func- 0 t i o n a l dependence on S provided t h e i n e q u a l i t y i n Eq. 28 i s s a t i s f i e d .

This means t h a t Eq. 1 5 a l lows u s t o express Y a s a f unc t i on of S 1

and <T> when E~ is non-zero and t h e mass f r a c t i o n appear ing i n Eq. 26

can be represen ted a s

Here we have used nomenclature given elsewhere (Whitaker, Sec. 5 ,

1980) and we cont inue t h a t po l i cy t o w r i t e

This a l lows us t o exp re s s t he gas-phase mois ture f l u x a s

This express ion i s of t he same form a s t h a t given by Luikov (Eq.

6.33, 1966); however, Luikov's r ep r e sen t a t i on i nc ludes on ly a

d i f f u s i v e f l u x s i n c e t h e r e l a t i o n given by Eq. 23 ha s no t been used

and Darcy 's law f o r t h e gas phase i s r e t a i n e d . Use of Eqs. 26 and

31 i n Eq. 24 l e a d s t o

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12 WHITAKER AND CHOU

In our analysis of the liquid phase moisture transport, we

define the capillary pressure as

and neglect gradients in the ges phase pressure so that Eq. 5 takes

the form

The capillary pressure will depend on the surface tension, the contact

angle, the structure of the porous medium, the history of the moisture

transport process, and the volume fracticn of the liquid EB. Since

the surface tension depends on temperature, we can express the func-

tional dependence of as C

 = F(S, <T>, other parameters) c ( 3 5 )

If one is willing to assume that the "other parameters" in Eq. 35 are

independent of the spatial coordinates, the gradient of <pc> is given

by

Use of this relation in Eq. 34 allows us t o express the liquid phase

moisture flux as

This result has some of the same characteristics as the flux relation

proposed by Luikov (Eq. 6 . 4 2 , 1 9 6 6 ) ; however, Luikov and a host of

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subsequent authors have ignored the effect of gravity. That gravity

is important in drying processes was made clear by the work of Ceaglske

and Hougen (1937), and in a more recent study (Chen, 1982) it has been

made clear that the gravitational effect can never be neglected when the liquid phase is continuous. If the gravitaticnal term in Eq. 37

is omitted, any comparison between theory and ~xperiment must rely on

adjustable parameters which have no fundamental physical significance.

Substitution of Eq. 37 into the liquid phase continuity equation leads

to a form analogous to Eq. 32 given by

where the transport coefficients are defined by

In terms of the nomenclature used here the liquid phase moisture flux

is given explicitly as

It is important to keep in mind that the transport coefficients

appearing in Eq. 42 can be determined independently and should not be

thought of as adjustable parameters to be determined by comparing

theory with drying experiments.

Because the saturation S is the dependent variable of choice in

an analysis of this type, it is convenient to add Eqs. 29 and 34 to

obtain a moisture transport equation of the form

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- a c as = 4- $(%I) -;s + (k<T>+ g(m$ - ;g.k] at

t t t t t (43) capillary Kelvin surface Clausius- gravity action effect tension Clapeyron effect

gradient effect

Once again we note that this moisture transport equation is similar to

that given by Luikov (Eq. 6.48, 1966; Eq. 11, 1975) or more recently

by Le Pourhiet and Borier (1982); however, in those studies the gravi-

tational term was neglected.

We now turn our attention to the thermal energy equation given

by Eq. 4 and make use of Eqs. 31 and 42 to obtain

In general the convective transport of thermal energy can be neglected

(Luikov, Eq. 6.25, 1966); however, there is no difficulty in taking

this mechanism into account and we have done so in our calculations.

When the second inequality in Eq. 21 is valid, Eq. 32 is quasi-

steady and we can use that result to express the source term in

Eq. 44 as

When this result is used in Eq. 44 and the convective transport

neglected, the form is identical to that given by Luikov (Eq. 6.49,

1966). For the special case where the liquid phase moisture flux is

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z e r o , pB<v > = 0 , one can use Eq. 6 a long w i t h t h e approximation -B given by Eq. 28 t o e x p r e s s t h e source term a s

When t h i s r e s u l t is used i n Eq. 41 and t h e convec t ive t r a n s p o r t i s

neg lec ted one o b t a i n s a form s i m i l a r t o t h a t given e a r l i e r b y Luikov

(Eq. 6.77, 1966); however, t h e form involv ing Eq. 46 appears o f t e n

i n t h e l i t e r a t u r e (Husain et a l . , 1973; Mikhailov and S h i s h e d j i e v ,

1975; Szen tgyargyi and Molnar, 1978; Le P o u r h i e t and B o r i e s , 1982)

and i t i s not always c l e a r t h a t t h e p rocesses under i n v e s t i g a t i o n a r e

r e s t r i c t e d by p <v > = 0. For t h e p a r t i c u l a r c a s e under c o n s i d e r a t i o n B *B t h e l i q u i d phase mois ture f l u x is n o t z e r o and t h e i n e q u a l i t i e s g iven

by Eq. 21 a r e s a t i s f i e d . This encourages u s t o use Eq. 45 i n Eq. 44

and e x p r e s s t h e thermal energy equa t ion a s

Equat ions 43 and 47 apply i n r e g i o n s where t h e l i q u i d phase volume

f r a c t i o n E is no t z e r o and t h e gas phase d e n s i t y of t h e evapora t ing B

s p e c i e s is r e l a t e d t o t h e s a t u r a t i o n and tempera ture through t h e

Clausius-Clapeyron equa t ion . I n a "dry" reg ion where E = 0 t h e B s a t u r a t i o n t r a n s p o r t equa t ion reduces t o

where aG/as i s given by

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16 WHITAKER AND CHOW

I n t h e dry r e g i o n t h e thermal energy is g r e a t l y s i m p l i f i e d and t a k e s

t h e form

= 1. s t f f - i < ~ > ) , d r y r e g i o n

While t h e a n a l y s i s l e a d i n g t o Eqs. 4 through 9 is f a i r l y r i g o r o u s and

t h e approximations used i n t h e development of Eqs. 43, 4 7 , 48, and 50

a r e reasonable , t h e c u r r e n t form of t h e theory cannot p rov ide an

a c c u r a t e d e s c r i p t i o n of t h e d r y i n g process excep t under c e r t a i n c i r -

cumstances.

THE PHYSICS OF DRYING

I n t h e process d e p i c t e d i n F ig . 1, wet sand is d r i e d b y - p a s s i n g

warm, d ry a i r over t h e top s u r f a c e . The bottom s u r f a c e is impermeable.

This is t h e type of system s t u d i e d by Ceaglske and Hougen (1937) and

more r e c e n t l y by Cunningham and Kel ley (1980), and i n F ig . 4 we have

shown s a t u r a t i o n and tempera ture p r o f i l e s measured by Ceaglske and

Hougen. On t h e b a s i s of t h e s a t u r a t i o n p r o f i l e t h e porous medium has

been s e p a r a t e d i n t o t h r e e r e g i o n s which can be d e s c r i b e d a s f o l l o w s :

Region I - I n t h i s reg ion t h e s a t u r a t i o n i s g r e a t e r than t h e

i r r e d u c i b l e s a t u r a t i o n S and t h e l i q u i d phase is cont inuous.

C a p i l l a r y a c t i o n causes t h e l i q u i d t o move toward t h e d r y i n g

s u r f a c e and t h e r e l a t i v e l y high thermal c o n d u c t i v i t y a long w i t h

t h e "sink" of thermal energy i n Region I1 g i v e s r i s e t o a

n e a r l y uniform tempera ture i n Region I.

Region I1 - When t h e s a t u r a t i o n f a l l s below S t h e l i q u i d phase

becomes d i scont inuous and l i q u i d phase t r a n s p o r t becomes n e g l i -

g i b l e . The g a s phase t r a n s p o r t of mois tu re i s from t h e r e g i o n

of h igh tempera ture t o t h e r e g i o n of low tempera ture , t h u s t h e

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Figure 4. Experimental S a t u r a t i o n and Temperature P r o f i l e s f o r One-Dimensional Drying

m o i s t u r e t r a n s p o r t i n Region I1 is d i r e c t e d away from t h e dry ing

s u r f a c e .

Region I11 - I n t h i s r e g i o n t h e p a r t i a l p r e s s u r e of water vapor

is no longer t i e d t o t h e temperature by t h e Clausius-Clapeyron

equa t ion and t h e g r a d i e n t i n t h e p a r t i a l p r e s s u r e causes a gas

phase mois ture t r a n s p o r t toward t h e d r y i n g s u r f a c e .

A t t h i s p o i n t i t becomes c l e a r t h a t Eq. 43 cannot c o r r e c t l y p r e d i c t

t h e mois ture t r a n s p o r t when t h e s a t u r a t i o n i s l e s s than t h e i r r e d u c i b l e

s a t u r a t i o n and g r e a t e r than t h e s a t u r a t i o n i n t h e "dry" reg ion , i . e . ,

t h e theory f a i l s i n t h e reg ion S >Y/pB. I f one were w i l l i n g 1

t o a d j u s t t h e parameters i n Eq. 43, agreement between theory and

experiment could s u r e l y b e ob ta ined ; however, t h e parameters i n Eq. 43

a r e a l l known i n terms of other experiments o r o t h e r t h e o r i e s and a r e

not f r e e t o be a d j u s t e d .

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The failure of the theory in Region I1 appears to be associated

with our treatment of the gas phase momentum equation, and the pressure

gradient in Region I1 must be sufficient to generate a convective trans-

port toward the drying surface that is greater than the gas phase dif-

fusive transport away from that surface. The magnitude of the gas phase

pressure gradient that is required to produce the desired moisture flux

can be estimated from the data of Ceaglske and Hougen. The mass flux

is given by Ceaglske (1936) as

and Chou (1981) has estimated the Darcy's law permeability as

We can use Darcy's law to obtain

and it is convenient to express this result in terms of the liquid

head, h. This leads to

The thickness of the sand layers studied by Ceaglske and.Hougen

ranged from 2.5 to 5.0 cm, thus the pressure difference required to

produce the observed mass flux was on the order of cm of water.

Here it seems clear that a complete theory requires a return to the

original formulation (Whitaker, 1977) and the incorporation of Darcy's

law for the gas phase; however, there is something to be learned by

carrying the current theory to completion and comparing its predic-

tions with experimental data. In the following paragraphs we indicate

a method by which this can be accomplished.

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EVAPORATION FRONT DRYING MODEL

I n o r d e r t o u t i l i z e Eq. 4 3 w e must e l i m i n a t e Region I1 s o t h a t

t h e d r y i n g process i s modgled i n terms of a " c a p i l l a r y wet" r e g i o n

and a "dry" reg ion . T h i s model i s s i m i l a r i n some r e s p e c t s t o t h a t

used by Brown e t a l . (1980) and we can proceed t o i t i n a formal manner

by c o n s t r u c t i n g a jump c o n d i t i o n t o e l i m i n a t e Region 11. We begin by

w r i t i n g t h e s a t u r a t i o n t r a n s p o r t equa t ion a s

where t h i s r e s u l t reduces t o Eq. 4 3 i n Region I and Eq. 48 i n Region

111. Use of t h e L e i b n i t z r u l e a l lows u s t o i n t e g r a t e Eq. 55 from

x = L ( t ) - t o x = l ( t ) + E~~~ t o o b t a i n

Here & is t h e u n i t v e c t o r p o i n t i n g toward t h e d r y i n g s u r f a c e and w ..A

is t h e v e l o c i t y of t h e jump g iven by

The f i r s t term i n Eq. 56 r e p r e s e n t s an accumulat ion of mois tu re a t a

s i n g u l a r s u r f a c e and t h e r e i s no way t o e v a l u a t e t h i s term wi thout

knowing t h e s a t u r a t i o n d i s t r i b u t i o n i n Region I1 and how i t changes

w i t h t ime. I f we assume t h a t t h e s a t u r a t i o n d i s t r i b u t i o n i s indepen-

den t of t ime, t h e f i r s t term i n Eq. 56 i s zero and we have

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WHITAKER AND CHOU

While dropping t h e f i r s t term i n Eq. 56 would be u n s a t i s f a c t o r y a t a

t ime when Region I1 f i r s t appears , i t might b e q u i t e r e a s o n a b l e f o r

longer t imes when Region I1 would assume a quasi-s teady c o n d i t i o n .

The s a t u r a t i o n S i s s p e c i f i e d by Eq. 27 i n t h e form 111

where Y is determined by Eqs. 11 and 1 5 and t h e tempera ture a t 1

t h e boundary between Region I1 and Region 111. I t might seem p l a u s i b l e

t o t a k e S e q u a l t o t h e c r i t i c a l s a t u r a t i o n S s i n c e t h e c r i t i c a l I

s a t u r a t i o n was used t o l o c a t e t h e boundary between ~ e ~ i o n s I and 11.

However, t h i s would l e a d t o a s t e a d i l y i n c r e a s i n g s a t u r a t i o n i n Region

I because t h e r e would be an i n f l u x of m o i s t u r e owing t o gas-phase d i f -

f u s i o n and t h e volume of t h e r e g i o n i s d e c r e a s i n g , and w e chose i n s t e a d

t o s p e c i f y SI a s t h e s a t u r a t i o n a t t h e p o i n t where t h e Region I f l u x

is zero . This means t h a t t h e average s a t u r a t i o n i n Region I i s a

c o n s t a n t a f t e r t h e appearance of Region I1 and Eq. 58 s i m p l i f i e s t o

(SI - S I I I ) s ~ = [&) .", x = "t) (60)

I n deve lop ing t h e jump c o n d i t i o n f o r thermal energy e q u a t i o n ;e f o l l o w

t h e same procedure; however, i n t h a t c a s e i t i s r e a s o n a b l e t o n e g l e c t

s e n s i b l e h e a t t r a n s p o r t i n t h e gas phase s o t h a t t h e jump c o n d i t i o n

t a k e s t h e fonn

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DRYING GRANULAR POROUS MEDIA 2 1

I n a d d i t i o n t o t h e t r a n s p o r t equa t ions and the jump c o n d i t i o n s one

needs t h e boundary c o n d i t i o n s f o r mass and energy a t t h e d r y i n g

s u r f a c e . These have been d i scussed elsewhere (Whitaker, Sec. 4 , 1980).

and w i l l no t be presen ted h e r e . The two-region model d i scussed h e r e

has been so lved numer ica l ly by Chou (1981) f o r t h e c o n d i t i o n s used by

Ceaglske and Hougen (1937), and f o r t h o s e c o n d i t i o n s t h e Kelvin e f f e c t

is n e g l i g i b l e and t h e l i q u i d phase t r a n s p o r t owing t o s u r f a c e t e n s i o n

g r a d i e n t s i s i n s i g n i f i c a n t . Under t h e s e circumstances t h e s a t u r a t i o n

t r a n s p o r t e q u a t i o n s i m p l i f i e s t o

and t h e thermal energy equa t ion t a k e s t h e form

I n Region I11 t h e s e two t r a n s p o r t e q u a t i o n s a r e rep laced w i t h Eqs.

48 and 50.

The dry ing process i l l u s t r a t e d i n F ig . 1 is assumed t o be one-

dimensional , thus t h e one-dimensional form of Eqs. 62 and 63 and Eqs.

48 and 50 need t o b e solved numerical ly i n o r d e r t o o b t a i n a compari-

son between theory and experiment. During t h e e a r l y s t a g e s of d ry ing

when on ly Region I is p r e s e n t , t h e s o l u t i o n scheme fol lowed t h a t

p rev ious ly used by Whitaker (1980) except i n t h i s c a s e bo th t h e s a t u -

r a t i o n t r a n s p o r t equa t ion and t h e thermal energy equa t ion were so lved

s imul taneous ly . Both equa t ions were solved by f u l l y i m p l i c i t schemes

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that required inversion of tri-diagonal matrices. Linearization was

always accomplished by evaluating the coefficients at the previous

time step.

The boundary conditions associated with Eqs. 62 and 6 3 are given

by

Here is the outwardly directed unit normal for the wet porous medium,

and Eqs. 6 4 through 67 represent the appropriate form of the boundary

conditions when only Region I is present. In Eq. 65 k represents a g

mass transfer coefficient while h in Eq. 67 represents a heat transfer

coefficient. The initial condition was taken to be

so that a zero flux condition existed at t = 0.

Two important characteristics of the implicit solution scheme

should be noted. To begin with, we found that only the fully impli-

cit method was stable and any modifications of this method, such as

the Crank-Nicolson method, were found to be unstable. Secondly, we

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found t h a t i t was e s s e n t i a l t o r ep r e sen t t h e f l u x boundary cond i t i ons

i n terms of c e n t r a l d i f f e r e n c e formulas appl ied a t a po in t l oca t ed

a t a d i s t a n c e Ax12 i n t e r i o r t o t h e a c t u a l boundary. For example, i f

t h e g r i d numbering s i t u a t i o n is such t h a t

t h e c a p i l l a r y f l u x term i n Eq. 64 i s expressed a s

Use of t h e f a l s e po in t method (Ames, 1977) o r h igher o rde r expres-

s i o n s f o r t h e f i r s t d e r i v a t i v e always l e d t o an uns t ab l e mode. This

is a l s o t r u e of t h e f l u x terms t h a t appear i n t he jump cond i t i ons

given by Eqs. 60 and 61.

When Region 111 appears , t h e f l u x cond i t i ons given by Eqs. 65

and 67 a r e rep laced wi th

where aG/as is given by Eq. 4 9 . I n t h e formula t ion of t he numerical

s o l u t i o n f o r t h e combined Region I- Region I11 model i t is convenient

t o s c a l e t h e d i s t a n c e i n each reg ion by t h e time dependent l eng th of

each region. To i l l u s t r a t e , we cons ider a t r a n s p o r t equat ion of t h e

form

I n Region I t h e dimensionless d i s t a n c e is def ined by

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while in Region 111 we use

WHITAKER AND CHOU

(75)

The transport equation given by Eq. 73 can be expressed as

Here 8 and U are dimensionless quantities given by

where D is some reference diffusivity. Equations 76 and 77 are 0

connected by a jump condition at x = L(t) and one needs to compute

L(t) in order to obtain a numerical solution. For this particular

model, Chou (1981) has shown that the jump condition is located by

the expression

The numerical solution of Eqs. 48, 50, 62 and 63, subject to the appro-

priate boundary conditions, represents a difficult computational effort

and the interested reader is referred to Chou (1981) for the details.

COMPARISON BETWEEN THEORY AND EXPERIMENT

In Fig. 5 the calculated and experimental values of the satura-

tion profiles are compared for various values of the average satura-

tion. When S >S everywhere in the system excellent agreement is

obtained; however, this is not a severe test of the theory since the

saturation profiles for S > Sosatisfy the quasi-steady form of Eq.

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Curve SaVg Experimental Data 1 77% 2 54% A

3 33% v 4 22% 5 6.3% 0 6 3 1%

Figure 5. Saturation Profiles of Ceaglske and Hougen--Theory and Experiment

62 and Ceaglske (1936) provided the profiles only as a function of

the average saturation and not as a function of time. When the

saturation at the drying surface drops below the critical value the

physical system enters the Region I - Region I1 domain while the mathematical model enters the Region I - Region 111 domain with the two regions connected by the jump conditions. The agreement between

theory and experiment in this case is very poor with the theory pre-

dicting an excessively large dry region. In Fig. 6 the calculated

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WHITAKER AND CHOU

Curve Savg Experimental Data

Figure 6. Temperature Profiles of Ceaglske and Hougen--Theory and Experiment

and measured temperatures are compared and there we see that the pre-

dicted temperatures rise too rapidly when Region I1 appears in the

physical system. Since the model generates a dry region under these

circumstances we expect the calculated temperatures to be too high;

however, during the latter stages of drying when most of the system

is dry we find good agreement between theory and experiment. The

most dramatic effect of the removal of Region I1 from the theoretical

framework is illustrated in Fig. 7 where we have shown the dimension-

less drying rate as a function of the average saturation. There we

see that the predicted rates are much lower than the measured rates

during the early part of the falling rate period. Since the elimi-

nation of Region I1 by the jump conditions lengthens the diffusion

path through the dry porous medium, the results are exactly what one

would expect.

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Figure 7. 'Drying Rates of Ceaglske and Hougen--Theory and Experiment

I

1 I I. . D m

....a . -0.. .* .

PARAMETERS IN THE THEORETICAL CALCULATIONS

The comparison between theory and experiment shown in Figs. 5

through 7 has been made in the absence of any adjustable parameters.

However, some of the parameters have been obtained directly from

Ceaglske (1936) and Ceaglske and Hougen (1937) thus assuring reason-

ably good agreement between certain aspects of the theory and expe-

riments. For example, the capillary pressure-saturation curve was

available from Ceaglske and liougen (1937) as was the mass transfer

coefficient at the drying surface. The particle size distribution

was also provided so that a reasonable estimate of the Darcy's Law

permeability could be made (Bear, 1972; Wyckoff and Botstet, 1936).

The effective diffusivity of water vapor was determined from a

correlation for the molecular ,diffusivity in air (Reid and Sherwood,

1977); and the theoretical relation of Ryan et al. (1981) for the

- s . 2

A .* A A A A ~ A ~ .. A A

y 3

A m

A

Temperature. " C Curve Dry -Bulb Wet-Bulb Experimental -

Data 1 77 8 35.8 2 64.0 32.5 . 3 55.0 30 1 A

.

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effective diffusivity as a function of porosity. The film heat

transfer coefficient was obtained by analogy from the mass transfer

coefficient and the effective thermal conductivity was predicted on

the basis of sand-air and sand-water experimental data (Birch and

Clark, 1940; Niven, 1940; Somerton, 1958); Touloukian, 1970). This

involved a linear extrapolation on the basis of volume fraction in

order to obtain a value for the three-phase system.

In addition to these physical parameters, there were several

parameters associated with the numerical solution of the coupled,

non-linear, partial differential equations describing the saturation

and temperature. Care was taken to insure that the results presented

in Figs. 5 through 7 were independent of the parameters associated

with the numerical method.

CONCLUSIONS

In this work we have shown that diffusion-like theories of

drying cannot describe the complete spectrum of moisture transport

mechanisms that occur during the drying of a granular porous medium.

In particular, when the saturation falls below the critical satura-

tion a gas-phase convective transport, caused by a pressure gradient,

must exist. To force the issue and compare theory and experiment we

have eliminated this region, identified as Region 11, with a jump

condition. This has led to the following results: (1) The calcu-

lated saturation profiles do not agree well with the experimental

data when the saturation falls below the critical saturation since

the model replaces Region I1 with a dry region; (2) The calculated

temperatures are higher than the measured temperatures during the

intermediate stages of drying. This is obviously caused by the

prediction of an excessively large dry region; (3) The predicted

drying rates are much lower than the measured drying rates during

the first part of the falling rate period owing to the excessively

large dry region.

It is clear from our comparison between theory and experiment

that the gas-phase momentum equation must be included in any com-

prehensive theory of drying granular material. On the other hand,

if one is interested only in the saturation profiles for S > So the

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c u r r e n t theory can be used w i th confidence. I n a d d i t i o n , i f one i s

concerned w i th a system i n which t h e l eng th of t h e dry reg ion i s

l a r g e compared t o t h e l eng th of Region 11, t h e c u r r e n t theory should

be s a t i s f a c t o r y .

ACKNOWLEDGEMENT

Th i s work was supported by NSF Grants CME 7911341 and CPE-

8116528.

NOMENCLATURE

Roman L e t t e r s

c cons tan t p r e s su re h e a t c apac i t y (kcal/kgK) P

C mass f r a c t i o n weighted h e a t c apac i t y ( kca l l kg K) P

D *(') e f e c t i v e d i f f u s i v i t y t e n s o r f o r d i f f u s i o n of s p e c i e s ~ e f f ... (m 5 IS)

Ah vap

J e

k

k g

K~ c e f f

K 22s

2 g r a v i t y v e c t o r (m I s )

Y/ Y , mass f r a c t i o n 1 Y

p y / B ( l - c ) 1 - an e f f e c t i v e d i f f u s i v i t y

t en so r (m / s )

2 f i l m hea t t r a n s f e r c o e f f i c i e n t (kcal/m sK)

entholpy of vapo r i za t i on per u n i t mass (kca l lkg)

mois ture f l u x vec to r (m/s)

thermal conduc t i v i t y (kcal/msK)

mass t r a n s f e r c o e f f i c i e n t (m/2)

e f f e c t i v e thermal conduc t i v i t y (kcal/msK)

-[K a / a S ] / [ ~ ( 1 - E ~ ) ] , a c a p i l l a r y flow permea- * 2 B b i l i t y (m I s )

p K /uB(l-co) , a g r av i t y f low permeabi l i ty ( s ) BzB 2

Darcy's law permeabi l i ty f o r t h e B-phase (m )

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WHITAKER AND CHOU

2 Darcy's l a w permeabi l i ty f o r t h e y-phase (m )

permeabi l i ty ( m L / s ~ )

t h i cknes s of t h e porous medium (m)

p o s i t i o n of t h e jump cond i t i on between Region I and Region I11 (m)

3 mass rate of evapora t ion p e r u n i t volume (kglm s )

u n i t normal v e c t o r

2 p re s su re (Nlm )

2 r e f e r ence vapor p r e s s u r e (N/m )

gas cons t an t f o r t h e ith s p e c i e s (NmIkgK)

a r ad iu s of cu rva tu r e a s soc i a t ed wi th t h e 0-y i n t e r f a c e (m)

(cop0 + E ~ ~ ) /pB(l-co) , s a t u r a t i o n

t h e i r r e d u c i b l e s a t u r a t i o n of t h e we t t i ng l i q u i d r e f e r r e d t o a s t h e c r i t i c a l s a t u r a t i o n

t i m e ( s )

temperature (K)

temperature f a r removed from t h e porous medium (K)

r e f e r ence temperature i n t h e Clausius-Clapeyron equa t i on (K)

mass d i f f u s i o n v e l o c i t y of t h e ith s p e c i e s ( r / s )

v e l o c i t y of t h e ith s p e c i e s (m/s)

mass average v e l o c i t y (mls)

v e l o c i t y of t h e 0-y i n t e r f a c e and t h e v e l o c i t y of t h e s i n g u l a r s u r f a c e s e p a r a t i n g Regions I and 111 (mls)

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Greek Letters

e h

volume fraction of the n-phase

2 IJ viscosity (Ns/m )

v 2 kinematic viscosity (m Is)

P 3 density (kg/m )

th 3 density of the i species (kglm )

Pl- density of the evapgrating species far from the porous medium (kglm )

u surface tension (N/m)

Wi mass fraction of the ith species

Mathematical Symbols

<$ >' intrinsic phase average of $ n n <$,,> phase average of $ n <$> spatial average of $

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Brown, L.F., Kashiwa, B.A., Vanderborgh, N.E., and Corlett, R.C. 1980. The kinetic behavior of subituminous coal drying: effects of confining pressure. In: Mujumdar, A.S. (ed). Drying '80, Vol. 2, Hemisphere Publishing Corp., New York, NY.

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32 WHITAKER AND CLOU

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drying granular porous media - theory and experiment

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