an unsaturated flow of moisture in porous hygroscopic media at low moisture contents

Pergamon Chemical Em]ineerin~l Science. Vol. 52, No . 19, p p 3383 3392, 1997 t 1997 Elsevier Science Lid. All r ights reserved

Printed in G r e a t Br i la in PIh - - A A ^ ^ --------*----.----------S0009__2fi09(97)00140__ 1 0009 2509~97 $17.00 + ( } 1 ()0

An unsaturated flow of moisture in porous hygroscopic media at low moisture contents

N. H. Kolhapure and K. V. Venkatesh* Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai,

Mumbai 400 076, India

(Received 1 August 1996)

Abstract--A mathematical description of thermal drying process involving various mechanisms of moisture migration such as water sorption, diffusion and solid-moisture interactions in hygroscopic materials is developed for unsaturated flow region. The model has been formulated using microscopic mass balance and volume averaging techniques. The constitutional laws such as Darcy's law are modified to account for solid-moisture interactions. Effective diffusion coefficients were estimated using moisture isotherms for three distinct systems of differing hygroscopicity. Drying curves were simulated using the estimated effective diffusion coefficient for potato, and compared with the experimental data. The analysis demonstrates the strong dependence of the effective diffusion coefficient on the shape of the moisture isotherm. ~ 1997 Elsevier Science Ltd

Keywords: Multiphase transport; drying; moisture sorption isotherm.

I N T R O D U C T I O N

The problem of multiphase flow and transport in porous hygroscopic materials is one of long-standing interest and is of increasing importance. Although this problem has been studied from both experimental and theoretical perspectives since at least the beginning of this century, it continues to receive attention in current literature. Information regarding moisture transfer rates is required for optimal design of dryers which are energy intensive units. Thermal drying of porous hygroscopic materials is one of the processes involving multiphase transport phenomena. In thermal drying of unsaturated porous medium, the external boundary layer of the air is heated by convection while conduction heats the porous medium. Thus, drying of unsaturated porous media is an extremely complex phenomenon involving simultaneous heat and mass transfer processes (Bouraoui et al., 1994). Understanding the mechanism of water sorption and diffusion in drying of porous media is of particular importance in relation to our ability to correctly inter- pret transport related phenomena.

Mathematical modelling is a useful way to describe various processes occurring in multiphase transport during drying of unsaturated porous hygroscopic medium. Here, by hygroscopic medium, we mean a

*Corresponding author. Tel.: (091)(22)5782545 extn 2263; fax: 091-22-578-3480; e-mail: [email protected].

system where there is a strong interaction between moisture and the solid matrix. Most of the current drying models are based on relations such as Darcy's law and Fick's law, which are developed for single- phase flow (Hassanizadeh and Gray, 1986a, b). How- ever, unsaturated flow in porous solids is typically characterized by the presence of air water interface; i.e. by multiphase flow. This means that the matrix permeability, as opposed to molecular diffusion, is a crucial factor affecting drying. To quantify these effects, extensions of Darcy's law and Fick's law are carried out for multiphase flow on the basis of phys- ical and thermodynamic understanding of the system. The other common feature of all these works is that microscopic and macroscopic description of mass balances have been carried out. The two balances are connected through the volume averaging technique (Hassanizadeh and Gray, 1979; Achanta et al., 1994). Though almost all studies consider drying as a multiphase flow problem, these are not valid for hygroscopic materials since the extensions of Darcy's law for multiphase flow do not account for solid-moisture interactions (Stanish et al., 1986). A good review of the existing drying theories is given by Okos (1993).

In light of above arguments, the objective of this paper is: (1) to explore the mass transfer mechanisms that govern thermal drying and to develop a mathematical model to simulate drying curves for porous hygroscopic materials in an unsaturated flow region; (2) to predict effective moisture diffusion coefficient from a given isotherm, which in turn will determine

3383

3384

/ /

S /

> Averaged volume

N. H. Kolhapure and K. V. Venkatesh

/ , / / J ~, ,

/~phase /~

J/ l / / / / / / , ~ / Fpbase

Fig. 1. Schematic diagram of a multiphase flow in porous media.

the drying rates; (3) to compare experimental drying data for spherical potato samples, with those obtained by simulation.

DEVELOPMENT OF MATHEMATICAL MODEL

Drying of unsaturated porous hygroscopic materials typically occurs in the falling rate period. In this region, the effective diffusion coefficient is usually determined through drying experiments from the slope of the drying curve. The effective diffusion coefficient determined by this method has been found to increase with increase in moisture content (Van Ar- sdel and Copley, 1963; Keey, 1975). In this paper, the moisture movement is modelled at very low moisture content. This low moisture content is typically given in terms of water activity less than 75%.

At low moisture contents, the pores mainly consist of bound water and vapour (see Fig. 1). The mechanisms of moisture movement at very low moisture content are significant in mathematical description of drying. Independent moisture movement mechanisms are postulated for each of these phases. Bound water moves by a diffusion process that is driven by a gradient in the chemical potential of the sorbed water molecules. Water vapour and air are transported both by bulk flow driven by a gradient in total gas pressure and by binary diffusion driven by a gradient in the mole fraction of each component. Thus, in unsaturated systems, the effective diffusion coefficient (D~ ff) is a combination of vapour phase diffusion coefficient (D]) and liquid phase diffusion coefficient (D~) (see Fig. 1). Each of these migration mechanisms has to be retained in quantitative mathematical expres- sions to model the drying process.

Model fi~rmulation The typical moisture transport equation during

drying of porous non-shrinking solid for isothermal condition, is usually based on diffusion theory, and is given as

CX - - = V(D]ff VX) (1) Ct

where, X is the moisture content in kg/kg of solid on dry basis and D] ff is the effective diffusion coefficient (Ketelaars et al., 1995; Whitaker, 1977). The effective diffusion coefficient is a combination of vapour and liquid diffusion coefficients, which are both dependent on the moisture content. To model these diffusion coefficients, the following approach is taken (Achanta, 1994; Gray and Hassanizadeh, 1991a, b):

I. Conservation balances for various phases are obtained at microscopic level. Using volume averaging technique these balances are averaged (Hassanizadeh and Gray, 1979).

2. Constitutive laws like Darcy's law are derived in their modified form to account for multiphase flow. These laws are modified using the approach given by Hassanizadeh and Gray (1986b, 1990) in their thermomechanical theory.

3. Using these laws, averaged balances are recast to obtain the individual diffusion coefficient values.

Using the above approach, a multiphase transport model for porous hygroscopic solid is formulated. The main assumptions involved in the formulation are:

1. Isothermal conditions are assumed for drying. Since, the moisture content is very low, the temperature gradient inside the moisture layer is negligible.

2. Equilibrium is assumed between air-water interface inside the solid matrix. Based on this as- sumption, vapour concentrations can be estimated using the moisture isotherm.

3. Since drying is carried out at low moisture contents, shrinkage of the solid is negligible. So, the porosity (e), which is a combination of vapour and liquid volume fractions, remains constant. At low moisture content, volume fraction of liquid is very small compared to volume fraction of vapour. So, volume fraction of vapour can be assumed to be constant (Wang and Brennan, 1995).

The microscopic balances for liquid and vapour phases are given by

- - = - V ( p l v t ) (2) ~t

Cp ~ ~ - = r ip"v, ,) (3)

Using volume averaging technique, the averaged liquid phase mass balance is obtained. The averaging volume used is sum of volumes of/-phase, v-phase and s-phase (see Fig. 1).

c? at (d(P~') + V((P~)'(v~)) + (yh)

+ V((ptvl)) = 0. (4)

An unsaturated flow of moisture

The dispersive flux term, the last term in the above equation, is significant only at higher flow velocities of liquid phase. At low moisture contents, the velocity of liquid is very low and hence, the dispersive flux term becomes negligible compared to convective transport. Further, the density of water can be assumed to be constant. Hence, the balance can be written as

? p~'~ (d) + p 'V( (v , ) ) + (rh> = 0 (5)

Ot

where, e) is the liquid volume fraction, which can be expressed in terms of moisture content and porosity of solid by eq. (6):

s d = (1 - g) ~w X. (6)

Combining the above two equations, we get the averaged liquid-phase balance as below:

~X 1 - - - [pwV. (<v,>) + (rh>]. (7)

at p~(1 -- ~)

Similarly, we can obtain the averaged vapour-phase balance as

? ~ ~'e"< p ">"')

- - v v

t P . . . . . . ,) Ov, freeVf(PV>V/~/,,]

- v . ~ v : ; / _ < m > = o. t8)

<p,>,, /

The traditional Darcy's law, based on capillary pressure gradient (Bird et al., 1960), is modified using the concept of wettability potential, as given by Has- sanizadeh and Gray (1990). Neglecting gravity, the modified law can be given as

vt = - Vp + s----7~ Vs" . (9)

In above eq. (9), the first term on right-hand side represents pressure gradient in water phase. The sec- ond term on right-hand side represents the interaction between liquid phase and solid matrix, fV ~ and f~" are the interaction potentials of water phase and air, respectively. The interaction potential, f~', for phase

is defined by Hassanizadeh and Gray (1990) as

= . , 11o)

where, A" is Helmboltz free energy and s" is volume saturation function, which is defined for the phase • as the ratio of the volume of phase ~ to the total void space. At intermediate unsaturated region, the pressure is a function of only volume saturation of water phase. This implies negligible effect of interfaces on pressure. Hence,

f a p w l ~ w VpW=k~sw)VS . ( t l )

3385

Further simplification of modified Darcy's law can be carried out since vapour density is much less than liquid density. This also implies that the interaction potential of air can be neglected. From the above definition of interaction potential, the solid-moisture interactions can be characterised by following equation:

(n w - n . ) ta w w{aAw'~ -~ s w P ~,-~s"')p,' r" (12) S'a'

The above equation applies to changes between two equilibrium moisture states, when both p" and T are constant. Helmholtz free energy is further defined in terms of the chemical potential as

A" = p"' P~ - - - ( 1 3 ) pW

The general criterion for phase equilibrium in multiphase system is given at constant temperature and is as follows:

d(#~ _ pw) = RTdlna,,. (14)

Differentiation of this equation with respect to saturation gradient will give the general expression for phase equilibrium and is represented by eq. 05):

alnaw \ ~ s ' J , : , r ~s w

Definition of liquid volume fraction given in eq. (6) can be extended to define saturation, in terms of porosity and the moisture content, as

(1 - e ) f s w - - - X. (16) (~) p-'

Combining eq. (11) to eq. (16) and substituting in eq. (9):

K w alnaw v , = ---~(p R T ) ~ VX (17)

Equation (17) is the modified Darcy's law in simpli- fied form. Substituting this in eq. (7), the averaged liquid phase balance is

[ w ° __~X = V" q'K p_ (pwRr) Olnawlvx ~t p~(1 - e) OX J

<a,>] .8)

where (rh) is the mass rate of evaporation of liquid phase. The term can be obtained by assuming negligible changes in vapour density and vapour volume fraction with time and after rearranging the averaged vapour-phase balance, the mass rate of evaporation can be given by eq. (19):

v v V . . . . . . D ' t'<:w> '~'1

V cx\kp 2 / / w L k -~/ J

(19)

3386

Therefore, the overall moisture balance will be

~X [- K pW c~ In aw - O--t = V" [~-~ P S ( ~ e ) ( p W R T ) ~

" ~ v v

(xP /~ V, free ~ ~ J

+ V. VX. (20) pS(1 - e)(1 -~/(PL)~~

This can be further rewritten as

~X ¢?t V" [D~x + D]]VX. (21)

The final drying model is given by the above expression where the liquid and vapour diffusion coefficients are given as

K pW c3 In aw D / qw p~(1 - e) (p~RT) ~X (22)

k Y-Y ;

D] = p~(1 - e) (1 - -~-7fi)(P~)~ (23)

The final moisture transport model can be written to give the radial variation in moisture content in spherical coordinate system, as follows:

The initial and boundary conditions to solve the above model are given as below:

X(r, t = O) = Xo (25)

X(r = R, t) = X~ (26)

In literature, eq. (1) is normally used to define the drying behaviour of porous solids assuming a constant effective diffusion coefficient. The drawback of such models is that the effective diffusion coefficient has to be characterised separately through experiments for different drying conditions such as temperature, pressure, and humidity of external air, etc. Moreover, the effective diffusion coefficient is assumed to be only liquid diffusion coefficient and the contribution of vapour diffusion is neglected, which is significant at low moisture contents. The developed model incorporates both liquid and vapour diffusion terms in the effective diffusion coefficient. The strength of the developed model is the estimation of the effective diffusion coefficient for different drying conditions, using the moisture sorption isotherm at different temperatures.


a nonlinear parabolic equation. It was solved using method of lines. The method of lines generates a set of

VX ODEs which are further solved numerically using the fourth order Multistep Method. A fortran program was developed using the above numerical method for a spherical geometry.

A large number of parameters are involved in the description of the moisture transport model and inter- nal properties of the material. The most important parameters are discussed below.

SIMULATION

The model equation for moisture transport, which gives variation of moisture content with time, is

The isotherm model The water activity mainly represents the state and

mobility of water present inside the solid matrix. The mobility of moisture is a strong function of moisture content. Oswin isotherm is used to represent relationship between water activity and moisture content as given below:

XI/N aw A1/N + X1 m (28)

where, A and N are the constants at a particular temperature and are characteristic of the material being dried.

The diffusivity model In the moisture transport model, the value of per-

meability, K, is required for estimation of the liquid diffusion coefficient. A diffusion model is used for evaluating the permeability and is given by eqs (29) and (30) (Achanta, 1993):

K O e-~/"Rrl (29) ~'~(pWRT)= L, free

DL, free = Do e-E°/RT (30)

It accounts for the activation energy for diffusion of free water through DL, free and adsorbed water through Eb. The model gives the activation energy for diffusion of adsorbed water higher than that of free water and this excess energy is expressed in terms of binding energy (Van Arsdel and Copley,1963). The binding energy is defined as the excess energy required to vaporize the moisture in solid matrix over that of pure water (Cervantes, 1994). The binding energy is calculated using Clausius-Clayperon equation and is given by eq. (31):

[01naw 1 - E~= RT 2[ ~T ]p,x. (31)

The vapour concentrations, required for calcu- lation of vapour diffusion coefficient, are computed in terms of water activity assuming equilibrium between air-water interface (Achanta, 1993):

353.4W (p~')~' 0.61 l (p~)" (32)

T

P~,a~, (p~)V = (33)

RwT


Table 1. Parameter values used in simulation for Oswin isotherm

3387

Parameter A ( T ) N ( T )

Relationship A = A0 + A1 x T N = NO + N I x T Solid-A A =0.018-4.496x10 5×T N = --4.980+ 1.911xl0 2 x T Solid-B* A=0.497 1.236x10 - 3 x T N = -2.251+8.638x10 3 x T Solid-(" A = 1.356 3.372 × 10-3× T N = --2.047 + 7.853 × 10-3 x T

* Obtained from experimental equilibrium data.

Table 2. Different parameter values used in simulation for predicting effective diffusion coefficient

E, (kcal/mol)* n* Dv.jree Do (mZ/s ) * ~:+ p~ (kg/m3) +

8.81 2.1 5x10 -s 1.78 x 10 -5 0.3 1200

* after Van Arsdel and Copley (1963). + after Wang and Brennan (1995).

Table 3. Constants used in simulation for predicting drying curve

X o (kg/kg d.b.) X,, (kg/kg d.b.) P" (N/m 2) r (m)

0.16 0.039 I 0.01

where, P',', is the vapour pressure of water, and is calculated in mm of Hg column using Antoine's constants as

1668.0 log (P~) = 7.967 T - 45.15" (34)

The porosity is obtained from the study carried out by Wang and Brennan (1995). They have shown that porosity remains constant at low moisture content. The other parameters used to simulate the model are obtained from literature and are listed in Table 2. The constants used in the simulation of drying data are listed in Table 3.

EXPERIMENTAL SET-UP

Figure 2 depicts the experimental apparatus used for this work. It consists of: (1) humidity chamber; (2) air supply, heater and cooler; (3) instrumentation: dry bulb and wet bulb temperature controllers, torsion weight balance; (4) data acquisition system.

The humidity chamber maintains a constant humidity in the chamber. The humidity is controlled by dry and wet bulb temperatures. The sample is sus- pended inside the chamber by a wire. The wire is connected to a torsion weight balance which is further connected to a data acquisition system. The humidity inside the chamber is also monitored through a hygrometer which is connected to the data acquisition system.

0 2

1 5

4

I

Fig. 2. Schematic diagram of the experimental set-up. [(1) humidity chamber; (2)potato sample; (3)electronic weight balance; (4) hygrometer; (5) air inlet: (6) air outlet: (7) dry bulb temperature controller; (8) wet bulb temper-

ature controller: (9) data acquisition and control panel.]

Exper imenta l procedure The potato sample to be dried were cut into spheri-

cal shape and then placed in the humidity chamber till it achieves a constant weight. The final moisture content reached is the equilibrium moisture content at the particular humidity maintained inside the chamber and the corresponding humidity is recorded as water activity. Similarly, at various humidity values, one can determine the equilibrium moisture content. These data were then fitted to the Oswin isotherm to represent the equilibrium data. The corresponding Oswin isotherm parameters obtained for potato samples are reported in Table 1.

To get the drying curves, similar experiments were carried out. The sample was placed in the humidity chamber at a particular temperature. Since the model is for low moisture content, to ascertain that the initial moisture content is low, the sample was allowed to

3388

attain equilibrium at low humidity. Then the humidity in the chamber was lowered to a new value corresponding to equilibrium moisture content of 0.039 kg/kg dry solid. The moisture content was determined at different time points. The dimensions of the sample was measured before the drying experiments began. These experiments were carried out at different dry bulb temperatures (30, 40, 50 and 70°C). Each experi- ment was repeated thrice and the average value of moisture content is reported.

R E S U L T S A N D D I S C U S S I O N

Figure 3 represents the sorption isotherm, which is the plot of moisture content versus water activity. Sorption isotherm helps in deciding the state and mobility of the moisture, which plays an important role in deciding solid-moisture interaction. Moisture isotherm shape is useful in predicting drying behaviour.

0.8

'-2.

v

;4

z E.- Z 0 r )

[,r.l

0

0.6

0.4

0.2

0.0 0,0 0.2 0.4 0.6


To illustrate the above point, various shapes of isotherms are considered, which typically represent different levels of hygroscopicity in porous materials. Equilibrium data was obtained for potato sample as per the experimental procedure given earlier and is shown in Fig. 3. The data was fitted using Oswin isotherm (refer Fig. 3) and the corresponding parameters (A,N) are listed in Table 1. This curve is designated as solid-B. Other shapes of isotherm are obtained by varying Oswin isotherm parameters (A,N) and are designated as solid-A and solid-C, respectively. For solid-A (refer to Fig. 3), the water activity is high even at very low moisture contents. It implies that bound water present in the material is negligible. So, it is the representative curve for a material with low hygroscopicity. Whereas, for solid-B (potato) Fig. 3 shows that significant amount of moisture is present at low water activity. This moisture corresponds to bound moisture. So, it is a representative curve for intermediate hygroscopicity or it represents a hygroscopic material. Similarly, we can say that the curve for solid-C is a representative curve for a material with high hygroscopicity.

The state and mobility of water in these materials can be further seen from Fig. 4, which is the plot of binding energy versus moisture content. As discussed earlier, binding energy is defined as the excess energy required to remove the moisture from solid matrix over that of free water. For solid-A, binding energy reaches zero at very low moisture content; i.e. low amount of bound water is present even at low moisture contents. For solid-B, binding energy is higher at the same moisture content; i.e. the water binding has increased or water is more strongly adsorbed onto the solid. The binding energy in solid-B goes to zero at

0.8 1.0 higher moisture content as compared to solid-A. This indicates the increase in the proportion of bound water in solid-B. The same trend continues in solid-C, showing increase in the binding as well as proportion of bound water. This demonstrates that the level of hygroscopicity increases from solid-A to solid-C.

WATER ACTIVITY, a.

Fig. 3. Sorption isotherm curves for different solids (solid-B is potato).

o

z

3 0 0 -

2001 ~

o ] ~ , ~ 7 , 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.00 0.01 0.02 0.03 0.04 0.05

MOISTURE CONTENT, X, (kg/kg d.b.)

t.-: z; ro

0 0

r~

[ - O

r~

1.2E-009

1.0E-O09

8.0E-010 <

v 6.0E-O10

4 . 0 E - 0 1 0

2 . 0 E - 0 7 0

O.OE+O00 0.0

S o l i d - C

S o U d - B

,,'~ ...... r ......... i ......... i ......... i ......... 0.1 0.2 0.3 0.4 0.5

WATER ACTIVITY, a.

Fig. 4. Variation of binding energy with moisture content Fig. 5. Variation of effective diffusion coefficient with water for different solids (solid-B is potato), activity for different solids (solid-B is potato).

An unsaturated flow of moisture 3389

%-

v

C~

U E E.. r.r..1 o ;Z o O3

1.2E-009

1.0E-009

8.0E-010

6.0E-010

4.0E-010

2.0E-010

O.OE+O00 0.000

For Solid-A dt

0.001 0.002 0.003 0.004

-...

v

[-, Z;

t... r..

8 Z; o 03

tr.,

6.0E-011

5.0E-O 11

4.0E-011

3.0E-011

2.0E-011

1.0E-011

O.OE+O00 0.00

For Solid-B

%

0.05 0.10 0.15

MOISTURE CONTENT, X, (Kg/Kg d.b.) MOISTURE CONTENT, X, (kg//kg d.b.)

Fig. 6. Effective diffusion coefficient, liquid diffusion coefficient and vapour diffusion coefficient for solid-A at different

moisture contents.

Fig. 7. Effective diffusion coefficient, liquid diffusion coefficient and vapour diffusion coefficient for solid-B (potato) at

different moisture contents.

Figure 5 depicts the effect of water activity on effective diffusion coefficient values in the above solids. For solid-A because of low bound water, less energy is required to remove this bound water and therefore, it has a high effective diffusion coefficient (~ 10- to m2/s). Whereas, for solid-B and solid-C effective diffusion coefficient decreases at the same water activity due to higher amounts of bound water present in the solid. Solid-B and solid-C have lower effective diffusion coefficients and are of the order of 10- 11 and 10-12 m2/s, respectively.

Figure 6 gives effective diffusion coefficient, liquid diffusion coefficient and vapour diffusion coefficient profiles for solid-A with respect to moisture content. With the decrease in moisture content, solid-moisture interactions become significant. This will increase moisture binding to the solid surface and will result in the decrease of the liquid diffusion coefficient. This moisture, which is loosely adsorbed onto the solid and is removable, is termed as multilayer water content. When moisture content reaches to a point where it is strongly adsorbed and is irremovable as liquid and it is termed as monolayer water content. At this point, liquid diffusion coefficient goes to zero since the chemical potential is same at all points inside solid-A. From the figure, it can be seen clearly that monolayer water content is almost zero. Though liquid diffusion decreases at low moisture content, water activity at this moisture content is very high. So, more of vapour enters into the gaseous phase, increasing the vapour diffusion coefficient, which finally reaches a limiting value of 7.6 × 10-lo m2/s. The effective diffusion coefficient, which is the combination of two, is dominated by liquid diffusion on higher values of moisture content and by vapour diffusion on lower values of moisture content. Therefore, it decreases with decrease in moisture content and increases at very low moisture content. Ketelaars et al. (1995) performed experiments on kaolin clay and obtained data of diffusion coeffi-

cient versus moisture content under isothermal conditions. The data represents a similar trend as given by solid-A at low moisture content. The limiting value of effective diffusion coefficient, as moisture content tends to zero, is of the order of 10 -9 m2/s for kaolin clay, which is close to that of solid-A. Therefore, we can say that kaolin clay is an example of material representing solid-A.

Figure 7 gives effective diffusion coefficient, liquid diffusion coefficient and vapour diffusion coefficient profiles for solid-B with respect to moisture content. Here, liquid diffusion profile shows the same trend as in case of solid-A. But, for solid-B the moisture range over which the change occurs is more. This indicates higher amounts of multilayer moisture content in solid-B than solid-A. Also, liquid diffusion coefficient reaches zero at higher moisture content, indicating higher amounts of monolayer water content. In case of vapour diffusion coefficient, it increases with decrease in moisture content till water activity is high. When water activity starts decreasing, less water vapour enters into the gaseous phase, thereby, decreasing vapour diffusion coefficient, which reaches a limiting value of 1.33 x 10-14 m2/s as moisture content tends to zero. The peak in the vapour diffusion coefficient for solid-B is shifted away from the y-axis, as compared to solid-A.

Figure 8 shows diffusion coefficient effective diffusion coefficient, liquid diffusion coefficient and vapour diffusion coefficient profiles for solid-C with respect to moisture content. The profiles show similar trends as discussed for solid-B. But, for solid-C, multilayer as well as monolayer water content increases and the peak in the vapour diffusion coefficient shifts further away from y-axis. In order to summarize the dis- cussion, we can say that as level of hygroscopicity increases, diffusion coefficient values decreases and multilayer as well as monolayer water content increases. Though the profiles are shown till moisture

3390 N. H. Kolhapure and K. V. Venkatesh

2.0E-011 r~

e~ 1.5E-011

Z

~ 1.0E-011 r~

0

Z 5.0E-012

O.OE+O00 0.0

For Solid-C <~ /

, , , , . . . . , , . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 0.2 0.3 0.4


Fig. 8. Effective diffusion coefficient, liquid diffusion coefficient and vapour diffusion coefficient for solid-C at different

moisture contents.

0.18

,6

0.13

t--; z [-, z 0 0.08

[-,

0.03

For Solid-B (Potato)

Xe=O.039 k g / k g d.b. - - Mode l P r e d i c t i o n s

i ~ E x p t . Data For T=70°C k islam Expt. Data For T=5O°C

l OCxX)OExpt. Data For T=4O°C ~ laAAAa Expt. Dat____aa ForT=3O°C

0 5 10 15 20

TIME, t , ( h r )

Fig. 10. Compar i son of theoretical and experimental drying data of po ta to for different temperatures.

1.2E-010

8.0E-O 11

~ ~ 4.oE-oll

O.OE+O00 , 0.0o

For Solid-B

0.02 0.04 0.06 0.08 0.10


Fig. 9. Change in effective diffusion coefficient for potato (solid-B) at different temperatures.

content reaches zero, in reality it is difficult to achieve it and profiles will shift to the equilibrium moisture content corresponding to the final humidity maintained in the humidity chamber.

Figure 9 gives the effect of increasing temperature on effective diffusion coefficient. As temperature increases, the excess binding energy required to remove bound water from the solid matrix decreases. Thus binding energy of bound water decreases with increase in temperature, giving an apparent increase in the free water content. It is easier to remove water at higher temperature due to decreasing binding energy. Temperature effects the isotherm shape in a way that level of hygroscopicity decreases (Fortes and Okos, 1981; Sun and Woods, 1993). The permeability of free water also increases with temperature [refer eq. (29)]. Both these effects will contribute to increase in effective diffusion coefficient with increase in temperature.

0.25

..~ 0.20

o.15

z r~ ['~ 0.10 z 0 ro

r~ 0.05

[.-,

O o.oo

For Solid-B (Potato)

Xe=0.042 kg/kg d.b. Model Prediction For Potato Sphere

,,t=o U :if: cjO

0 1 2 3

TIME, t, ( h r )

Fig. ll. Comparison of theoretical and literature drying data of potato at 65.5°C for a half-dice (assuming an equivalent radius of 0.5 cm). Literature data is taken from Van

Arsdel and Copley (1963).

Simulation was carried out to give the variation of moisture content with time. Figure 10 shows the comparison of experimental data with predictions at different temperatures. The model is able to predict the drying curve with reasonable accuracy at all temperatures. The deviation in the predictions at higher temperature can be attributed to the changes in the porosity. This clearly demonstrates that the predictions of effective diffusion coefficient from the model are accurate. The model is also simulated to predict the drying curve available for potato in literature (Van Arsdel and Copley, 1963). Figure 11 shows the comparison between literature data and the predicted curve. The simulation was carried out in a spherical coordinate system assuming an equivalent diameter for a half-dice potato sample. The model prediction


shows a reasonable agreement with the literature t data. The predicted drying rates are slightly faster T which can be attributed to the fact that the available v surface area for drying in spherical sample is greater <v> than that of the half-dice sample. X

time, s

temperature, K velocity, m/s averaged velocity, m/s moisture content, kg/kg dry basis

3391

CONCLUSION

The moisture transport model was derived for the drying of porous hygroscopic solids at low moisture content. The model accounts for the solid-moisture interactions through wettability potential. The wettability potential was defined in terms of water activity and was incorporated into Darcy's law. The effective diffusion coefficient was given by the summation of liquid and vapour diffusion coefficient. The diffusion coefficient values were highly dependent on the shape of the isotherm. The binding energy and water activity concept were used to explain the state and mobility of water inside the solid matrix and were used to explain the diffusion coefficient profiles at low moisture content. The model was able to simulate effective diffusion coefficient and drying curves for different temperature values.

The future challenge include extending a similar approach to predict effective diffusion coefficient at higher moisture contents and to incorporate the solid shrinkage due to drying. Such an analysis can be used to predict drying rates at all moisture contents. The estimated drying rates can be used to optimally design industrial dryers.

Acknowledgement

This work was supported by Department of Science and Technology. India, Grant No. SR/SY/E-08/93 under the Young Scientist Scheme.

aw

A A w

DL. free

Dv,free O~x ff

D'x

E~ Eh 1 K

N pW

r R Rw

S'a'

NOTATION water activity, dimensionless Oswin isotherm parameter, K Helmholtz free energy of water phase, kJ/gmol effective diffusion coefficient for free water, m2/s

vapour diffusion coefficient in air, m2/s effective diffusion coefficient, m2/s vapour-phase diffusion coefficient, m2/s liquid-phase diffusion coefficient, mZ/s activation energy of free water, kJ/gmol binding energy, kJ/gmol unit tensor, dimensionless permeability tensor for the solid phase, m 2 mass rate of evaporation, kg/m 2 s Oswin isotherm parameter, K pressure in water phase, N/m 2 radial distance, m universal gas constant, kJ/gmol K universal gas constant based on weight, kJ/g K volume saturation function, dimensionless

Greek letters

P <p>

q

porosity or volume fraction, dimensionless density, kg/m 3 averaged density, kg/m 3 chemical potential, kJ/gmol viscosity, kg/m s interaction potential, kg/m s 2

Superscripts and subscripts ,s phase 0 initial value a or v property of gaseous phase e equilibrium value l or w property of liquid phase o property of phase at standard state s property of solid phase

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an unsaturated flow of moisture in porous hygroscopic media at low moisture contents

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