SIMULTANEOUS MOISTURE AND HEAT TRANSFER IN POROUS SYSTEMS WITH PARTICULAR REFERENCE TO DRYING

T . 2 . H A R M A T H Y Dioision of Building Research, National Research Council of Canada, Ottawa, Canada

Based on an examination of the liquid-vapor equilibria and of the mass and energy transfer processes in porous systems, a theory has been developed for simultaneous mass and heat transfer during the pendular state of such systems. A set of differential equations is presented which, when solved with the appropriate initial and boundary conditions, yields the complete moisture content, temperature, and pressure history of the system. By theoretically simulating various drying processes it is shown that during the falling rate period the evaporation-condensation mechanism is the governing mechanism of drying. The application of the theory to transient heat conduction in the presence of moisture and to transient vapor transfer through porous media is also illustrated.

ROM among the several theories so far suggested to explain migration of moisture in porous media, three have won

general recognition : the diffusion theory, the capillary flow theory, and the “evaporation-condensation” theory.

The movement of moisture by diffusion was explicitly proposed as the principal flow mechanism by Lewis (1921), Tuttle (1925), Sherwood (1929, 1930, 1932), Sherwood and Comings (1933), Newman (1931), Childs (1936), Kamei (1937), and many other research workers.

The fundamentals of the capillary flow theory were laid down by Buckingham (1907), who introduced the concept of “capillary potential” and postulated the mechanism of “un- saturated capillary flow.” The latest version of this theory is presented by the basic assumption that the moisture flux is proportional to the gradient of the chemical potential of the moisture (Edlefsen and Anderson, 1943). With this and other assumptions the fundamental equation of moisture migration turns out to be identical with, or similar to, Fick’s second law with concentration-dependent diffusion coefficient (Miller and Miller, 1955; Remson and Randolph, 1962).

The evaporation-condensation theory assumes that migra- tion of moisture takes place entirely in the gaseous phase (in the pores). Gurr et al. (1952), Hutcheon (1958), and Kuzmak and Sereda (1957) proved that when the system is subjected to a temperature gradient this assumption is essen- tially correct, even at relatively high pore saturation. The evaporation-condensation mechanism was utilized by Henry (1939), Cassie et al. (1940), Walker (1961), and others in de- scribing the movement of moisture in beds of textile ma- terials.

There are three comprehensive theories of simultaneous moisture and heat transfer processes not directly related to the above described three basic theories : those advanced by Philip (1957a, b, 1958a, b), Krischer (1942), Lykov and Mykhaylov (1961), and Lebedev (1961).

Characteristics of Drying Processes Early work by Keen (1914), Fisher (1923, 1927), Sherwood

(1929, 1930, 1932), and Sherwood and Comings (1933) in- dicated that under constant environmental conditions the process of drying can be divided into a “constant rate” and one or two “falling rate” periods. When the initial moisture

content is high enough, a considerable amount of moisture leaves the porous medium at a very high, approximately constant rate, which is roughly equal to the rate of evaporation from a continuous water surface under identical environmental conditions. During this initial period the temperature of the system also remains constant (Figure 1, a) and, as a rule, equal to the wet-bulb temperature of the environment.

At a point before the end of the constant rate period the temperature begins to rise slowly, and a few, apparently “dry” patches show up on the surface of the system (taken here as of the form of an infinite slab). This point, C, in Figure 1, a, referred to as the “upper critical point” of drying, indicates the beginning of the breakdown of the “funicular state” and the unfolding of the “pendular state” of the system. [In the “funicular state” there are continuous threads of moisture in the pores; in the “pendular state” there are not. For further discussion see the work of Haines (1927, 1930)].

U p to this point the moisture distribution is fairly uniform throughout the system (Figure 1, b ) , because of the high mobility of moisture in the funicular state. With the break- down of the continuous moisture threads, however, the mobility decreases sharply. The upper critical point thus also marks the inception of a gradual changeover to the migration of moisture by the evaporation-condensation mechanism.

The point at which the departure from the constant rate drying is first noticed in a plot of average moisture content us. time is commonly referred to as “the critical point” of drying (C*, Figure 1, a) . That a t which the last “wet” patches dis- appear from the surface (point c,) is called in this study the “lower critical point.” Here the transition of the system from the funicular to the pendular state is essentially completed, and henceforth moisture movement takes place predominantly in the gaseous phase-i.e., by the evaporation-condensation mechanism.

In this study the interval tcu < t < tcl is referred to as the “transient” period, and the region t > tCz as the falling rate period of drying. Following point Cz, the temperature of the system first rises very rapidly, then tends to level off at the so- called “pseudo-wet-bulb temperature” (Bell and Nissan, 1959; Nissan et al., 1959). Shortly before the end of the drying process it rises again a t an increased rate to the level of final temperature.

92 I & E C F U N D A M E N T A L S

t D t E f F f G

T I M E

c z W I- z 0 V W CLi. 3 I- v,

0 - z

t = O

f C ”

f G I

0 L

S E C T I O N A C R O S S S L A B Figure 1 . Characteristic curves of a drying porous slab

a. Average moisture content and temperature b. Moisture distribution

The moisture distribution curves shown in Figure 1, 6 , are based on information published by Kamei (1937), Johnson (1961), and Corben and Newitt (1955).

Outline of Present Theory

The present theory is based on two fundamental assump- tions. The first is implied by the statement that it is an evaporation-condensation theory.

Assumption 1. All movement of moisture in a porous system takes place in the gaseous phase.

This assumption is not an inherent feature of the theory, but because of the lack of knowledge concerning the mechanism of liquid-phase flow it is a t present an unfortunate necessity. Even though moisture migration in the liquid phase is never completely extinct, there is ample evidence (Christensen, 1944; Osobaet al., 1951; Philip, 1957a; Richards, 1931; Richardson, 1961; Richardson et al., 1952) that in the pendular state the mobility of moisture in the liquid phase is so small that in most practical problems it need not be considered.

Because of Assumption 1 the applicability of the present theory must be limited to the pendular state of porous systems in general, and to the falling rate period in connection with drying problems.

In contrast with Assumption 1, Assumption 2 is concerned with the modeling of the texture of the porous system, and is therefore an inherent feature of the theory. Before introducing Assumption 2, however, it is necessary to describe briefly the “constituents” of a “porous system.” In general, this system can be modeled as three phases, solid, “liquid” (adsorbed), and gaseous; and three components, the material of the solid,

water, and “air.” is regarded as yielding two “constituents.”)

In this theory the solid phase is looked upon as a single- component material that exhibits isotropic characteristics in macroscopic dimensions. I t contains a network of inter- connected pores, the surfaces of which are the source of a force field. This field manifests itself in the depression of equilibrium vapor pressure of the liquid contained in the pores.

The liquid phase is also considered to be a single-component phase. In the presence of the above-mentioned force field the physical and thermodynamic properties of this phase are more or less dependent on the amount in the pores-Le., on the relative pore saturation.

The gaseous phase is regarded as an ideal binary mixture of water vapor and of fictitious air molecules.

Assumption 2. The phases of a porous system are so finely distributed that from the macroscopic standpoint the system is a quasi-one-phase system. (It is not implied that the phases are distributed into macroscopically small units. In fact, each phase may form a macroscopically large, continuous network.)

An important limitation of the analogy between porous and truly one-phase systems must not be overlooked. In a truly one-phase system the components are dispersed on a molecular scale, so that the mass and energy fluxes are associated with molecular transport phenomena (diffusion and conduction, respectively). In a quasi-one-phase system, on the other hand, there are separate phases in any macroscopically small volume element. As the relative movement of the phases takes place by convection, the convective mass and energy transfers are primary transport phenomena. In addition,

(A component that appears in two phase

V O L 8 NO. 1 F E B R U A R Y 1 9 6 9 93

Table I. Phase Constituent Vol. Concn., Cc. /Cc. Mass Concn., G./Cc.

Expressions for local Volume and Mass Concentrations of Phases and Constituents of the Porous System

6, = 6 g u + 6aa v@ Gaseous Air a =

Vapor

Liquid Solid

conduction of heat occurs in all phases and mass transfer by diffusion in those that contain more than one component.

Assumption 2 implies three additional assumptions, the first two of which simply represent the extension to quasi-one-phase systems of two basic assumptions of nonequilibrium thermo- dynamics (Fitts, 1962). These “subassumptions” are as follows :

I t is possible to assign local values to the mass concentrations of the constituents and to the specific internal energy-in other words, to specify aO,( r, t ) , aQa( r, t ) , a2(r, t ) , 8*(r, t ) : and E(r, t ) . These quantities are averages over microscopically large but macroscopically small periods of time and cells in space. Some expressions for the mass (and also volume) concentrations of the various phases and constituents of the porous system are summarized in Table I.

As the slight dependence of the density of liquid phase on the relative pore saturation can be disregarded, as far as the mass concentrations are concerned the porous system is an ideal system.

Even if the system as a whole is a non- equilibrium system, there is a local equilibrium within each macroscopically small cell of it. For a truly one-phase system this assumption takes care of the extensibility of all thermo- dynamic relations to nonequilibrium systems. For a quasi- one-phase system it has another implication:

ASSUMPTIOS 2C. In a quasi-one-phase system the local values of the chemical potential for each component are the same in all phases. For ordinary porous systems the practical meaning of this assumption is that the partial pressure of water vapor in the gaseous phase is equal to the equilibrium vapor pressure of the liquid (adsorbed) phase.

The validity of Assumption 2 and all its implications depends primarily on how closely the actual texture of the porous system approximates a quasi-one-phase texture. As for Assumption 2C, calculations indicate that in drying processes the equilibrium between the liquid and gaseous phases can be taken for granted in most practical problems if the adsorption of moisture is essentially a surface phenomenon.

ASSUMPTION 2A.

ASSUMPTION 2B.

liquid-Vapor Equilibrium

The relation expressing the liquid-vapor equilibrium, the so-called sorption relation, is characteristic of the particular porous system. I t is of the form

m = m(P,, T )

The most common representation of this relation is by the sorption isotherms (Brunauer et al., 1938; Kiselev, 1958)- Le., by the family of curves m = m(P,) or m = m(P,/P,O) for T = const.

A logical way of including all three variables, m, P,, and T, in a single relation was shown by Vassiliou and White (1948), who presented the sorption relation as a relation be- tween the moisture content and the mean normal curvature of the liquid surfaces in the pores--Le., as rn = m(r) . An equiva- lent relation is

e(= m/e) = e ( r ) (2)

If r is expressed from the Kelvin equation

(3)

and Equation 3 is substituted in the B ( r ) relation, an expression is obtained that correctly represents the dependence of m on P , and T.

In a strict sense the Kelvin equation is applicable only to the capillary condensation region (P,/P,” > 0.4), where sorption is brought about by the development of curved liquid surfaces in the pores. In the region of true adsorption (P,/P,” < 0.3) the retention of water molecules is caused by inter- molecular forces, and the surface of the liquid (adsorbed) phase follows more or less the geometry of the pore surfaces. However, an m us. r relation, with r from Equation 3, correctly describes the dependence of m on P , and T in almost the entire 0 < P,/P,” < 1 sorption domain. Thus the validity of Equa- tion 2 can be extended to a major portion of the region of true sorption, provided that the original physical meaning of r is overlooked and it is regarded simply as a convenient param- eter.

The generalized liquid-vapor equilibrium relation (as presented by Equation 2) for a particular porous system can be calculated from its sorption isotherms with the aid of Equation 3. The main difficulty is the hysteresis phenomenon; be- cause of it m is not a single-valued function of P,/P,” in the region of capillary condensation. An “effective” sorption isotherm has to be selected, the shape of which depends on the nature of the process to be examined. For example, for most technical drying problems the effective sorption isotherm consists of a desorption scanning curve (Rao, 1941) and a section of the main desorption isotherm. I t seems, however, that drying processes are not overly sensitive to minor changes in the effective equilibrium isotherm, so that it is usually per- missible to use the main desorption branch as representing the effective m us. P,/P,” relation.

Because of some experimental advantages the 0.9 < P , / P t < 1 (or r > 100 A,) section of the equilibrium relation in desorption can be more conveniently studied with the aid of the so-called mercury-injection technique (Purcell, 1949 ; Richardson, 1961). Some experimental data concernidg the liquid-vapor equilibrium, in both desorption and adsorption, for a certain brown clay brick are plotted in Figure 2. The curves, referred to hereafter simply as the B ( r ) functions, have been tabulated and will be used later as the effective liquid- vapor equilibrium relations for this kind of brick.

As the gaseous phase is regarded here as an ideal binary mixture

(4) p , ( 3 = - P

With the aid of this and the following two empirical equations

( 5 )

( 6 ) and

u = uo - PT

94 l & E C F U N D A M E N T A L S

0 P o i n t s O b l a i n e d b y m e r c u r y - i n j e c t i o n t e c h n i q u e

3 0 0.8

0. 7

P o i n t s C a l c u l a l e d f r o m s o r p t i o n c u r v e s The e l r l r e l a t i o n f o r d e s o r p t i o n

- - - -The g i r l r e t a t i o n f o r a d s o r p t i o n

f 1:: 1 0.4

1

1 IO 10’ 104 105

Figure 2. Generalized liquid-vapor equilibrium relation for a certain clay brick

r A.

Equation 3 becomes

where

(7)

or may be regarded as an empirical constant. With Equation 7 the basic effective equilibrium relation now can be presented in the following more explicit form:

0 = s[r(P,p,T)l (9)

The derivatives of this relation, with respect to p, P, and T,

will also be used later as part of a series of background informa- tion. The (dO/dr)(r) function is obtainable by numerical differentiation of the appropriate O(r) function and is, in general, also presented in tabulated form.

Molecular Transport Processes

According to the model of the porous system presented, simultaneous molecular mass and heat transfer can take place only in the gaseous phase. I t is known from Onsager’s work (1 931) that in such simultaneous processes coupling occurs between the flux-force pairs, giving rise to various secondary phenomena. [The words “flux” and “force” (or “driving force”) are used according to the terminology of nonequilib- rium thermodynamics.] A numerical study of these phenom- ena (Harmathy, 1967) revealed that in drying processes the mass fluxes induced by driving forces other than concentration gradients are negligible. Admittedly, the conduction heat flux and the Dufour energy flux may assume values of the same order of magnitude; yet because of the insignificant magnitude of the total molecular energy flux in the gaseous phase, neglecting the Dufour energy flux is justifiable.

By virtue of these arguments the molecular fluxes occurring in the gaseous phase of a porous system may be satisfactorily described by the following two equations:

Energy flux (excluding the fluxes associated with mass fluxes) :

go = -@k,vT

Mass fluxes (Bird et al., 1960) :

Here, because of the isotropy of the porous medium, the effec- tive porosity, @, could be used for the fractional free area. Because of the tortuosity of the pores D is some fraction of the coefficient of diffusion of water vapor in air outside the pores.

Equations 13 and 14 can be recognized as Fourier’s law of heat conduction and Fick’s law of diffusion, respectively.

Convective Mass Transport

The mass transport described by Equation 14 takes place on a molecular level in relation to the local center of mass of the gaseous phase. In general, however, the local center of mass is also in motion (convection) in relation to the stationary phases of the system.

The equation of motion (in a macroscopic sense) of the gaseous phase is obtained by the application of Darcy’s law:

Here vp is the (local) mass average velocity of the gaseous phase :

Darcy’s “law” is an experimentally obtained expression. I t can also be derived from the fundamental Navier-Stokes equations of motion under certain simplifying assumptions (Harmathy, 1967). Because of these assumptions the applica- bility of Darcy’s law must be limited to the Poiseuille flow of liquids occurring a t very small pressure gradients, but it is permissible to extend it formally to gases if the permeability, K ~ , is not regarded as a property of the porous medium alone but as a factor that depends also on some characteristics of the gas (Creutz and Zumwalt, 1962).

When the porous medium is partially saturated with mois- ture, K~ in Equation 15 must be interpreted as the “effective” permeability, which is related to the “limiting” permeability- Le., to the permeability in the absence of moisture--K,O, as

K g = T K g o

where I, the relative permeability, is a function of the relative pore saturation, 0. Experimental data concerning this func- tion for the brown clay brick mentioned above (with helium as the gaseous phase) are plotted in Figure 3. In the prac- tically important range of 0 < 0 < 0.4, where the moisture (or the “wetting phase,” in general) is practically immobile (Osoba et al., 1951; Richardson, 1961; Richardson et al., 1952), the experimental finding may be satisfactorily described by the following empirical equation (shown as a continuous line in the figure) :

{ = exp(-5.W) (18)

Such a function is expected to be practically independent of the properties of the gas, and probably also of the pore struc- ture of the medium (Osoba et al., 1951 ; Richardson, 1961).

VOL. 8 NO. 1 F E B R U A R Y 1 9 6 9 95

0 0.2 0.4 0.6 o. a 1.0 R E L A T I V E S A T U R A T I O N m i & z 6

Figure 3. clay brick to gases in presence of moisture

Relative permeability of a certain

Equations of Continuity of Matter

By examining the expressions listed in Table I and taking into account the following relation between the relative pore saturation and effective porosity

it may be seen that a porous system is unambiguously defined by the e(r, t ) , y(r, t ) , P ( r , t ) , and T(r, t ) functions. Because of the assumed local equilibrium 6 can be expressed in terms of 'p, P, and T (see Equation 9), so that the model system actually possesses three degrees of freedom. For practical reasons Q,

P, and T a r e used here as the three independent state variables.

To determine the p(r, t ) , P(r, t ) , and T ( r , t ) functions, three independent equations are needed. Two equations can be obtained by applying the equation of continuity to the two mobile constituents of the system, water vapor and air (Bird et al., 196')):

where no, = ~ p o v o u = j u u + 8ouv,

n o a = 8paVga = j,, + beavo

(22)

(23)

and as the j ' s are to be referred to the mass average velocity, vo, their sum adds up to zero.

As the gaseous phase is regarded as an ideal mixture

p o = M,P/RT (24)

p a = M a P / R T ( 2 5 )

(26) and

P~ = P / R T [ M W p + M,(1 - Q)I By substituting jgU and j u a from Equation 14 and vo from

Equation 15 (neglecting the pQg term which, for gaseous flow, is always very small) in Equations 22 and 23 and making use of Equation 17 and some expressions in Table I one obtains

After substituting from Equations 24 to 28 and from Table I into Equations 20 and 21, two second-order, nonlinear, partial differential equations are obtained:

AiV2p + BiV2P + DI(VV)~ + G ( V P ) 2 + G1Vp V P f

96 l & E C F U N D A M E N T A L S

The coefficients are, in general, functions of p, P, and T. Their expressions are listed in Table 11.

Energy Transport Equation

To obtain the third equation one can use the law of conserva- Here the following form of this law seems to tion of energy.

be the most convenient starting point (Bird et al., 1960) :

v . q - A - - - v . v P = 0 (31) bt

The applicability of this equation rests on Assumption 2. I t has also been assumed that only gravitational forces are effective.

The mass average velocity for the entire porous system is defined as

According to Assumption 1 the liquid and solid phases are stationary phases-Le., nz = n, = v 2 = v, = 0. By com- bining Equations 16 and 32 and using plausible values for the 6's it is easy to show that, in general, IvI < 0.0005 IvuI. I t will be assumed that v zz 0; thus the term v vP disappears from Equation 31.

The enthalpy change can be written as

dH = ( g)T,* dP + ( + ( $ ) p , , (33)

For the gaseous constituents and for the material of the For the liquid phase ( b H 2 / d 0 ) P , T = solid (bH/a0) , , , = 0.

- (dQ/b0),,, because of the relation

Q = H , - H2 (34)

Furthermore, ps = const., p z z pw z const., and ( c ~ ) ~ = ( c ~ ) ~ = const. (Berezin et al., 1963). Thus, with the aid of well known thermodynamic relations, the following expressions are obtained for the four constituents of the system:

dH, = ( C , ) ~ ~ T (35)

dHu = (cp),dT (36)

dH, = A dP + (c,),dT (37) Ps

The (bQ/&),,, term in Equation 38 is generally negligible outside the domain of true adsorption.

q in Equation 31 is the conductive energy flux in a macro- scopically small but microscopically large volume element. (Since v z 0, it is approximately equal to the heat flux in a stationary coordinate system.) By making use of Assumption 2 one can extend the validity of Fourier's law (see Equation 13 for the gaseous phase) to macroscopically small cells of the system by writing

q = - k V T ( 3 9 )

Because of its simplicity, the following empirical equation has been selected to describe k for a porous system (Kingery, 1960) :

k = [kU'%(l - 0) + kind + kan( l - €)]'in (40)

where - 1 2 n 5 1 and is to be determined experimentally.

as -avo and 17 as

A in Equation 31 can be expressed, on a macroscopic scale, V P ; consequently, with the aid of Equations 15

K O A = { ( v P ) 2

'Io

After expanding the first two terms in Equation 31 and com- bining them with the equations of continuity (Equations 20 and 21), substituting from Equations 34 to 39 and 41 and from Table I in Equation 31, and taking into account that v k 0, one obtains:

C3V'T + E d V P ) ' + Fa(VT) ' + H3Vp . V T + J3VP * V T =

a p b P bT K3 - + La - + Ma -

at at at (42)

The coefficients C3. . . M3 are also listed in Table 11. All three equations of the simultaneous moisture and heat

transfer in porous systems can be written in the following general form:

AkV'p + BkV'P + CkV'T + Dk(Vrp)' + Ek(VP)'

Fk(VT)' + G ~ V P * V P + H k V p . V T + JkVP * V T =

where k = 1, 2, 3. Some of the coefficients Ak . . . M k are equal to zero; a number of others are negligible in many practical applications.

Initial Conditions

The initial conditions are presented by the following three expressions:

rpo = p ( r , 0 ) , PO = P ( r , 0 ) , TO = T ( r , 0); r in [VI (44)

Because of Assumption 1 the present theory is applicable only if the porous system is in a pendular state. This seems to be the case whenever 0." is less than its value at r 2000 A. according to the effective equilibrium 0(r ) relation. This statement implies that the finer the pore structure of the porous medium, the wider the applicability of the theory.

The fact that in studying drying processes the theory is limited to the falling rate period is of little practical significance, since the history of average moisture content during the con- stant rate period is amenable to simple calculations (Bagnoli, 1963).

In general, mav a t the lower critical point-Le., at t = t c t is the only available information for the theoretical simulation of the falling rate period of drying. As the time t = tCz be- comes the t = 0 level for the simulation, one may write for msv a t t = tc l . The distribution of moisture at the lower critical point is, unfortunately, far from uniform (Figure 1, b ) . To eliminate the use of estimated moisture distribution it is advisable to start the theoretical simulation at some higher uniform moisture content-i.e., at some m-0 > ( r n ~ ) ~ , that occurs at some (undefined) t-0 < 0 time-and then disregard the results obtained for the period t - 0 < t < 0.

As rp has been selected as one of the three state variables of the system, actually qpo(r) [or rp(t-o)], not mo(r ) [or m(t-o)], should be specified in the initial conditions. The conversion from m to rp is done with the aid of the following equation,

VOL. 8 NO. 1 F E B R U A R Y 1 9 6 9 97

which is obtained by combining Equation 7 with the expres- sion of liquid vapor equilibrium, Equation 2 [r (e) is the inverse of the B(r) relation].

The selection of T o ( ? ) (within reasonable limits) seems to have very little effect on the moisture history of the system. The errors associated with its inaccurate selection fade away in relatively short time.

Boundary Conditions

The boundary conditions are, in general, presented by the following equations: 3 (46)

9 = d r , t ) or VV = vdr, t ) P = P(r, t ) or VP = v P ( r , t ) r in [SI T = T(r , t ) or V T = V T ( r , t ) t 2 0

When the first set of conditions does not apply, expressions for V(qr, t ) and V T ( r , t ) can be obtained by examining nu, and q a t the boundary of the porous system.

Only one set of boundary conditions is discussed in detail here: that applicable to a system completely surrounded by air of controlled (not necessarily constant) water vapor content, pressure, and temperature. The expression for P a t the bound- ary is immediately obvious

P ( r , t ) = P,(t), r in [SI, t 2 0 (47) According to the film concept of mass transfer, the diffusive

flux of water vapor at the pore openings through a laminar boundary layer of air (referred to unit area of the system) can be expressed as

(48)

where v takes care of the possible extension of the actual transfer area due to convective mass transport. The heat flux through the same boundary layer may be expressed analogously as

q = -h[T(r , t ) - T,( t ) ] ; r in [SI, t 2 0 (49)

The terms in the square brackets in Equations 48 and 49 are vector quantities oriented (for positive values) perpendicularly toward the boundary surface. [In a strict sense the transfer co- efficients defined by these two equations are applicable only ifn,, + 0 and npa + 0 (Bird et al., 1960).]

By combining Equations 48 and 49 with Equations 14 and 39 the following two boundary conditions result:

RT PD

Vp = va- [ d r , t ) - pm(t)]; I in [SI, t 2 0 (50)

h k V T = - [T(r , t ) - T m ( t ) ] ; r in [SI, t 2 0 (51)

For transfer processes taking place between a vertical slab (with x = 0 and x = L boundary planes) and air a t atmospheric pressure and a temperature Tm (within, say, 0' and 50' C.) contained in an enclosure of a temperature T,, the transfer coefficients can be expressed as follows :

a = a* + 1.629 X 10-61 ( T , - T ) / B p 4 x = 0 (52)

h = 2.75 X 108 a + u,e(T,4 - T4)/(T, - T ) i x = L (53)

where a* and e are empirical constants and can be determined from simple drying experiments (Harmathy, 1967). I n general T, = T,.

Table 111. Expressions Used in Computer Calculations To Describe Expressions Used

Liquid-vapor equilibrium

Relative permeability us. Equation 18

Tabulated 6 = O(r) functions (Figure 2) with r from Equation 7

relative saturation rela- tion

Heat of sorption Q = x, aQ/ae = o Thermal conductivity Equation 40

Simultaneous heat and moisture transfer For 0 < x < L t > t-0 or t > 0 For 0 < x < L t = t-0 or t = 0

Equations 54, coefficients listed in

m-0 or mo = const. (uniform) T-0 or TO = const. (uniform) P-0 or Po = const. (uniform) 9-0 or 90 to be obtained by sub-

Table I1

stituting values of 8-0, T-0, and P-0 or 60 , To, and PO in Equation 45

For x = 0 or x = L t 2 t-0 or t 2 0

P = Pstm Equation 55 with (Y from Equation 52 Equation 56 with h from Equation 53

and k from Equation 40

Values of Constants Appearing in the Expressions listed in Table 111

Value Appearing in Material Properties

1.0063 X l o 7 Table I1 Table I1 0.815 X l o 7 Table I1 1.8646 X l o 7

4.1793 X l o 7 Table I1 0,128 Table I1 0.8 Eq. 53 0.02613 X 106 Eq. 40 2.207 X 106 Eq. 40 0.616 X 106 Eq. 40

Table I1 28.952 Table I1 18.016

0 26 Table I1 i:83 x 10-4 Table I1

Table I1 Table I11 2.4418 X 1010

2.68 Table I1

2.5 X 10-10

Table I1 0,99707

Physical Constants 1.01325 X 106 Table I11 8.3149 X lo' 5.67 X 10-5 Eq. 53

Table I1

Geometric Constants 18.39 Eq. 52 1 .o Table I11

Empirical Constants 1.209658 X Eqs. 7, 45 5080 Eqs. 7, 45 0.25 Eq. 40

b Fm. 52 >

-1- - - Eqs. 7, 4! 0.167

1 . o Eq. 55 121.2 Eqs. 7, 45

Source

a

Exptl.

Est. Est.

Exptl.

(1

a

0

(I

a D

Exptl.

Exptl.

Exptl.

(L

a

a

a

D

.l

Exptl. Exptl.

Exptl. Exptl. Est. Exptl. Est. Est. Est.

a Values listed in several well known handbooks. b Given in captions of jigures.

Experimental Work and Computer Calculations

A number of drying experiments and computer calculations have been performed. I n the experiments the variation of the weight arid the temperature (at the center) of clay brick slabs (of the material mentioned) of 1-cm. thickness, 18.39 cm. square, were recorded under controlled environmental conditions. Some characteristic curves concerning three runs (cases 1, 2, and 3) are plotted in Figures 4, 5, and 6, together with the corresponding curves of theoretical simulation.

98 I & E C F U N D A M E N T A L S

- i x ) e r i m e n t a l __ T h e P r e t i c a i

0.05 -M

0 1 2 3 1 5 6 7 8 9 1 0 1 1 1 2 1 3 T I M E , h r .

Figure 4. Drying experiment 1 and its theoretical simulation

0 ?a

0 015

0.010

L

0.005

0

Initial Conditions Boundary Conditions m-0 = 0.1 05 ccJcc. (uniform) qm = prnl = q,z = 0.010539 mole/mole

r, = rml = rmz = 318.90 K. P, = P,1 = P,z = P.tm a* = ax* = az* = 2.478 X 10-6 mole/

T-0 = 306.2' K. (uniform) P-0 = Patrn (uniform)

sq. cm. set.

Equilibrium represented by O(r) relation for desorption (Figure 2)

0.15

mav

0 10

0 05

- E x p e r i m e n t a l --- E s t i m a t e d T he0 re t I t i I __

-

0.015

0.010

p-

0. w5

0 0 1 2 3 0 5 6 1 8 9 1 0 1 1 1 2 1 3 l I

T I M E . h r .

Figure 5. Drying experiment 2 and its theoretical simulation

Initial Conditions m q = 0.1 05 cc./cc. (uniform)

T-0 = 302.7'' K. (uniform) P-o = Pam (uniform)

Boundary Conditions 'p, = 'p-1 = q m z = 0.009009 mole/

r, = rml = rmz = 308.8' K. P, = P,1 = P,z = Patrn a* = a ~ * = az* = 2.478 X

mole

mole/ sq. cm. set.

Equilibrium represented by O(r) relation for desorption (Figure 2)

m = 1 a t x = o (55) T o one-dimensional transfer problems the following forms 3p - ( - l )m+l ya

b X - -

of Equation 43 are applicable

d T h m = 2 at x = L (56) ax k - = (-l)n'+l - ( T - Tmm) A k b20 - f BI, b2P - + c k b2T - f Dk (gy f Ek (gy f

ax2 3x2 b X 2

- The various equations and conditions used in the computer bx bx bx calculations are summarized in Table 111. The values of some bP . a p a. b~ ap b~

Fk r;)' 4- GI,- - + HI, 0 f Jk g * - - bP bT

Krc - + LI, - f MI, - constants appearing in these equations and in the expressions of coefficients in Table I1 are given in Table IV. [Any material property could be regarded as a function of T, P,

The boundary conditions are now and p (or 0) without unduly complicating the numerical

(k = 1, 2 ~ 3 ) (54) bo bt bt at

VOL. 8 NO. 1 F E B R U A R Y 1 9 6 9 99

h :o;Yai1inp\ r a t e p e r i o d

I

- - E x p e r i m e n t a l - - - E r t i m a t e d -- T h e o r e t i c a l

I \ \ I

1 0 1 I I j I I j 1 j 1 1 1 0 1 2 3 4 5 6 7 8 9 1 0 1 1 12 1 3 1 4

T I M E , h i .

Figure 6. Initial Conditions Boundary Conditions m-o = 0.1 05 cc./cc. (uniform)

Drying experiment 3 and its theoretical simulation

9, = ~ , 1 = V,Z = 0.003548 mole/ mole

1-0 = 294.8' K. (uniform) P-o = Patm (uniform)

r, = rml = T , ~ = 298.50 K. P, = P,1 = P,z = P a m Ly* = cy1* = cy2* 0

Equilibrium represented by O(r) relation for desorption (Figure 2)

o hr .

0.2 0.4 0.6 0.8 x crn.

600-

451 2.5

40 - - 2 - 1.5

I / - 0.5 - / -

d+ I

0 0.2 0.4 0.6 0.8 1.0 35

0 0.2 0.4 0.6 0.8 1.0 x crn. x cm.

Figure 7. Theoretical distribution curves in simulation of drying experiment 1 a. Moisture. b. Temperature. c. Pressure.

Initial Conditions Boundary Conditions m-o = 0.105 cc/cc. (uniform) 9, = $0-1 = 9,z = 0.010539

1-0 = 306.2' K. (uniform) P-o = Patrn (uniform)

mole/mole r, = rmt = rmz = 31 8.90 K. P, = P,1 = P-2 = Pstm

CY* = CYI* = a2* = 2.478 X 10- mole/sq. cm. scc.

Equilibrium represented by O(r) relation for desorption (Figure 2)

100 l & E C F U N D A M E N T A L S

0 0.2 0.4 0.6 0.8 1.0 x cm.

25C v

22 - -

x crn x cm. Figure 8. Theoretical distribution curves for heat transfer through a moist slab with moisture desorption

a. Moisture. b. Temperature. c. Pressure.

Initial Conditions Boundary Conditions mo = 0.0303 cc./cc. (uniform) (0, = (0,l = p,z = 0.0134 mole/

rml = 31 8.90 K., rm2 = 294.80 K. P, = Pml = Pm2 = Patrn cy* = cy1* = cyz* = 0

mole TO = 294.8' K, (uniform) Po = Patrn (uniform)

Equilibrium represented b y 8(r ) relation for desorption (Figure 2 )

procedure.] The values of the constants related to the initial and boundary conditions have been selected as representative of the experimental processes (Figures 4 to 6).

Details of the numerical procedure have been given (Har- mathy, 1967). A backward difference or implicit scheme (Douglas, 1961) was selected that is known to yield stable solutions for any selection of the space and time grid spacings.

The moisture content history-Le., the m ( x , t ) function, which is undoubtedly the principal information sought-is not available explicitly from the solution of Equation 54. It has to be obtained by additional calculations, using the p(x, t ) , P(x, t ) , and T(x, t ) solution functions and the liquid-vapor equilibrium relation Equations 2 and 7.

The agreement between the experimental and theoretical curves is, in general, satisfactory (Figures 4 to 6). Some discrepancies are attributable to the insufficient constancy of T , and p, during the experiments (due to certain experimental difficulties), the not strictly one-dimensional character of the actual transfer processes, and certain compromises made in the selection of some material properties (especially the Q = X = const. assumption).

In Figure 7 the computed moisture content, temperature, and pressure histories of the brick slab (for the falling rate period) are plotted for case 1. The moisture distribution curves are similar to those plotted on the basis of experimental data in Figure 1, 6.

0

The drying processes so far discussed come to an end with the development of equilibrium between the system and the surroundings. To show the general validity of the present theory, two entirely different processes (cases 4 and 5 ) were also simulated on the computer. The information in Tables I11 and IV is applicable to these computations also.

In case 4 a desorption process is examined that eventually reaches a steady state characterized by the absence of mass fluxes and by a constant energy flux. The steady-state mois- ture content distribution will be governed by the desorption liquid-vapor equilibrium corresponding to a constant water vapor pressure and an approximately linear temperature dis- tribution (Figure 8), in agreement with the observations of Vassiliou and White (1948) and Paxton and Hutcheon (1952).

The steady state is now characterized by the absence of conductive heat flux and by a constant water vapor flux. The steady-state mois- ture content distribution is determined by the (local) adsorp- tion equilibrium, corresponding to a constant temperature and a linear vapor pressure distribution.

Discussion

A large number of other computer calculations indicate that the effective liquid-vapor equilibrium is one of the major factors that determine the shape of the ma" us. t relation in drying. Also, for any realistic selection of the initial moisture

Case 5 represents an adsorption process.

VOL. 8 NO. 1 F E B R U A R Y 1969 101

0.0°5 m 0.004 -

0.003 m € e

m

$ E

0.002

0.001 w -

I O 22.6

22.2

g+ + 22.0

i 2.5

0

N c; -10

E‘ W 0

0

2 E G -20 a

I a

-30

1 1 1 1

0, oo hr

0.5

u 0 0.2 0.4 0.6 0.8 1.0

21.4 -40

x cm. x cm. x cm. Figure 9. Theoretical distribution curves for vapor transmission through a moist slab with moisture adsorption

a. Moisture. b. Temperature. c. Pressure.

Initial Conditions Boundary Conditions ma = 0.00077 cc./cc. (uniform) TO = 294.8’ K. (uniform)

(D,I = 0.00364 mole/mole 9 - 2 = 0.01339 mole/mole

P, = P-1 = Pm2 = Patrn Ly* = Lyl* =

Po = P,b (uniform) T, = rml = rmz = 294.80 K.

= 0

Equilibrium represented by the B(r) relation for adsorption (Figure 2)

distribution at the lower critical point, double S-shaped moisture distributions will arise for some time (Figure 7, u) . Bagnoli (1 963) believed that such distributions are brought about by capillary flow mechanism only.

A plot of log mav us. t is approximately a straight line, a t least for some time after the onset of the pendular state condi- tions. At a given slab thickness the slope of this line appears to be proportional to some power of the rate of drying during the constant rate period.

If the “apparent thermal conductivity,” k,,,, of moist ma- terials is defined by replacing the conductive heat flux by the total energy flux in Equation 39, the following equation can be obtained:

k,, = k =k Q %; - when n g e ~ ~ V T , + whenn,, 11 V T

(57) ILI

I t has long been believed that the presence of moisture always increases the value of k,,,. Cammerer (1939) ex- pressed the percentage increase in the value of k,,, as a unique function of the moisture content. Krischer and Rohnalter (1940) developed a more elaborate expression, which is applicable to the steady state of systems at very high relative saturation.

Figures 7 to 9 and Equation 57 reveal that very often, and in drying processes inevitably, k,, < k. In processes charac-

terized by the nBo + 0 condition in steady state, k can differ from k,,, only under transient conditions, if the system is in its pendular state. Ac knowledgmenl

In selecting this area for study the author was influenced by N. B. Hutcheon. The author thanks R. F. Legget for per- mission to use this work as a doctoral dissertation and A. F. Orlicek and Kurt Peters, University of Technology, Vienna, Austria, for consenting to its publication. In the preparation of the computer program S. D. Baxter, J. D. Taylor, and T. A. Hills yielded invaluable help.

Nomenclature Some symbols used as the coefficients of Equation 43 or

as expressions in these coefficients (see Table 11) are not listed. u = empirical constant, cm. sec2/gram b = empirical constant = M,X/R, K. B = height of vertical surface, cm. c p = specific heat at constant pressure, sq. cm./sec.2 K. D = effective coefficient of diffusion of water vapor in air

e = empirical constant, dimensionless E = specific internal energy, sq. cm./sec.2 g = gravitational acceleration, cm./sec.2 h = heat transfer coefficient, gram/secaa K. H = specific enthalpy, sq. cm./sec2 J = mass flux relative to mass average velocity, gram/sq.

inside a porous medium, sq. cm./sec.

cm. sec.

1 0 2 I h E C F U N D A M E N T A L S

k = thermal conductivity, gram ~ m . / s e c . ~ O K. L = thickness, cm. m = volumetric fraction of liquid-phase moisture, “mois-

M = molecular weight; without subscript: molar mean

n = empirical constant, dimensionless n = mass flux with respect to stationary coordinates,

p = partial pressure; without subscript: total (macro-

P,o = equilibrium vapor pressure of free water, gram/cm.

q = conductive energy flux relative to mass average veloc-

Q = latent heat of sorption of water, sq. cm./sec.2 r = parameter; radius or mean normal curvature of liquid

r = position vector, cm. R = gas constant, gram sq. cm./sec.2 O K. mole [SI = surface of porous system t = time, sec. T = temperature, O K. (unless otherwise stated) v = velocity; without subscript or with subscript g: mass

[VI = volume bounded by surface [SI x = coordinate, cm.

GREEK LETTERS a a* P 6 = mass concentration, gram/cc. A E = true porosity, cc./cc. { = relative permeability, dimensionless 7 0 K~

K ~ O

X Y = empirical factor, dimensionless p u BO = empirical constant, gram/sec.2 u, @ = effective porosity: volume fraction occupied by

= mole fraction of water vapor in gaseous phase, moles/

ture content,” cc./cc.

weight of gaseous phase, gram/mole

gram/sq. cm. sec.

scopic) pressure, gram/cm. sec.2

sec2

ity, gram/sec3

surface, cm.

average velocity, cm./sec.

= coefficient of mass transfer, moles/sq. cm. sec. = empirical constant, moles/sq. cm. sec. = empirical constant, gram/sec2 O K.

= viscous dissipation per unit volume, gramcm./ sece3

= coefficient of shear viscosity, gram/cm. sec. = relative saturation = m / ~ , dimensionless = effective permeability of condensed phases of porous

= limiting permeability of porous medium to gases in

= latent heat of vaporization of free water, sq. cm./sec.Z

= density; true density, gram/cc. = surface tension of water, gram/sec2

= Stefan-Boltzmann constant, g rarn /~ec .~ O K.4

system to gases, sq. cm.

absence of liquid phase, sq. cm.

gaseous phase, cc./cc.

mole MATHEMATICAL OPERATIONS

v = nabla vector v2 = Laplacian operator

SUBSCRIPTS a = air app = apparent atm = atmospheric av = average c C = at critical point e = enclosure g = gaseous phase ga g u z = in interior of slab k = 1, 2, 3 1 = liquid phase; lower 0 = t = 0 (at lower critical point) -0 = beginning of theoretical simulation s = solid phase u = upper u = water vapor w = water a) = in bulk of surrounding air 1 = on side x = 0 2 = on side x = L

= during constant rate period

= air in gaseous phase of porous system = water vapor in gaseous phase of porous system

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RECEIVED for review March 4, 1968 ACCEPTED September 19, 1968

Contribution from the Division of Building Research, National Research Council of Canada, published with the approval of the Director of the Division.

VOL. 8 NO. 1 F E B R U A R Y 1 9 6 9 103