drying in capillary-porous cylindrical media

Drying in Capillary -Porous Cylindrical Media M. C. ROBBINS and M. N . OZISIK

Lkpament of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, U.S.A.

Analytical solutions are presented for temperature and moisture distributions in a cylindrical capillary-porous body using Luikov’s model and employing a solution technique based on the decoupling of the simultaneous heat and mass diffusion problems. The relative importance of the dimensionless parameters governing drying is presented and a comparison of temperature and moisture profiles for the cylinder and the slab is given.

Des solutions analytiques sont prksenttes pour les distributions de temptrature et d’humiditt dam un corps capillaire- poreux cylindrique en utilisant le modkle de Luikov ainsi qu’une technique de rksolution baske sur le dkcouplage des problkmes de diffusion de chaleur et de matikre simultanks. L’importance relative des paramktres adimensionnels gou- vemant le stkhage est prksentk, et on prksente une comparaison des profils de temp6rature et d’humiditk pour le cylindre et le panneau.

Keywords: dehydration, heat and mass diffusion, mathematical model.

odeling of drying in many biological materials such M as food, wood, and pharmaceutical products has prac- tical and economic importance for industrial processes (Fulford, 1969). A highly desirable approach to internal moisture transfer is to predict moisture and heat transport behavior from physical parameters of the system. An obstacle to such an approach arises since moisture may be transferred by numerous mechanisms within one system. Mechanisms cited in the literature include: liquid movement as a result of capillary action; liquid diffusion as a result of concentration gradients; surface diffusion of liquid at the pore’s surface; vapor diffusion in partially air-filled pores as a result of vapor pressure gradients (or capillary-condensation mechanism); and finally, moisture transfer as result of a temperature gradient (Fulford, 1969). A number of models have been proposed since those of Sherwood (1929a,b; 1930, 1932), who relied simply on diffusion laws to model drying, and Ceaglske and Hougen (1937) who based their theory solely on capillary action. Since various mechanisms of moisture transfer have been identified, it is desirable to com- bine them into a single comprehensive theory of drying. Luikov (1975) in the Soviet Union and Krischer (1956) in Germany attempted to do this. Luikov’s contribution to the theory of heat and mass transfer is based on the phenomeno- logical approach (DeGroot and Mazur, 1962). Krischer developed a set of differential equations for heat and mass transfer in hygroscopic media. His equations include separate transport coefficients for capillary and diffusional mechanisms and his model allows the coefficients to be specific for either liquid or vapor transfer. However, it has been shown that the equations of Krischer are equivalent to those of Luikov and the correspondence between the dimensionless parameters of the two models has been provided (Iwantschewa and Mikhailov, 1968). The advantages of Luikov’s model over Krischer’s model is that the former reduces the number of variable coefficients to be determined

parameters affecting moisture and temperature distributions include the Luikov number, La, which is the coupling parameter, the Posnov, Pn, and Kossovich, KO, numbers as well as the phase change criteria, E , which characterize the relative importance of internal heat and mass transfer, and the dimensionless heat and mass flux numbers, Kiq and Kim, which represent the strengths of the heat source and moisture sink at the surface, respectively (Luikov and Mikhailov, 1965). In this work, the effects of such parameters on heat and mass transfer are examined and the influence of the geometry on temperature and moisture is demonstrated.

Formulation of the problem

We consider capillary-porous material contained within a cylindrically symmetric geometry, initially at a constant temperature to and constant moisture content u, as defined by Luikov (1966). The outer surface of the cylinder is exposed to a medium which permits moisture to be removed from the body thus creating a moisture sink at the surface. The contacting medium may be one through which chemical absorption occurs such as molecular sieves, activated alu- mina, or silica gels (Jones and King, 1977), or one in which osmosis occurs such as a sugar solution (Ponting et al., 1966). The outer surface is also exposed to an externally applied heat flux.

The differential equations governing this process are obtained from Luikov and Mikhailov (1965). In the dimensionless form, these equations are given by

- [ (LU P ~ ) I R ) I aiaR (RaTiaR), (Fulford, 1969). Empirical formulae are available from reports of recent experiments (Sharaf-Eldeen et a]. , 1980); however, only a limited amount of work has been reported on the solutions to mathematical models of drying illustrating (2) O < R < l , F o > O . . . . . . . . . . . . . . . . . . . . . . the effects of physical properties on temperature and moisture distributions (Hayakawa and Rossen, 1977; Mikhailov and The initial conditions are taken as Shishedjiev, 1975). Solutions have been presented for drying

important applications require the analysis in cylindrical in planar media (Robbins and Ozisik, 1988), although many Fo = 0, 0 I R I 1 . . . . . (3a)

geometry (Puiggali and Varichon, 1982). Significant e(R,z%) = 0, for FO = 0, o I R I 1 . . . . . (3b)

T(R,Fo) = 0, for

1262 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 69, DECEMBER, 1991

and the boundary conditions as Equations (5)-(8) are expressed in matrix form as

aT(R,Fo)IaR = 0, for R = 0, Fo > 0 . . . . . (4a)

ae(R,Fo)IaR = 0, for R = 0, Fo > 0 . . . . . (4b)

aT(R,Fo)/aR = Ki, - N Kim,

for R = 1, Fo > 0 . . . . . (4c)

ae(R,Fo)/aR = Pn aT(R,Fo)IaR + Kim,

for R = 1, Fo > 0 . . . . . ( 4 4

where N = (1 - E)LUKO and E is the phase change criteria associated with the definitionof KO* in Equation ( 1 ) . This coupled heat and mass system, Equations (1)-(4), can be decoupled as now described.

Analysis

To decouple Equations (1)-(4) we rearrange them in the form

aTlaFo = [(l + Ko*LuPn)lR] aIaR(RaT1aR)

- [ ( K ~ * L U ) I R ] aiaR(Rae/aR)

O < R < l , F 0 > 0 . ( 5 )

subject to the initial conditions

T(R,Fo) = 0, for Fo = 0, 0 I R I 1 (7a)

B(R,Fo) = 0, for Fo = 0, 0 I R I 1 (7b)

and the boundary conditions

aT(R,Fo)IaR = 0 , for R = 0, Fo > 0 . . . . . (8a)

ae(R,Fo)IaR = 0, for R = 0, Fo > 0 . . . . . (8b)

aT(R, Fo)IaR = Ki, - N Kim,

for R = 1, Fo > 0 . . . . . (8c)

ae(R,Fo)IaR = PnKi, + (1 - N Pn)Ki,,,,

f o r R = 1, Fo > 0 . . . . . . ( 8 4

The above system is now expressed in matrix form and then decoupled following the procedure outlined in Mikhailov and Ozioik (1988).

aiaFo ( 4 ) = [ K I v 2 (41 O < R < l , F o > O (9a)

subject to the initial conditions

(41 = (01, for Fo = 0, 0 5 R I 1 (9b)

and the boundary conditions

ff [ $ I + P (41 = [ g l ,

for R = O , R = l . . . . . . (9c)

where various vectors and matrices are defined as

. . . . . . . . (1 1) 1 1 + Ko*LuPn -Ko*Lu - LuPN Lu

[KI = [ . . . . . . . . a = O , f l = l , ( g ] = O at R = O ( 124

Ki, - NKi, PnKi, + ( 1 - NPn)Ki,

ff = 0, p = 1, [ g ] =

at R = l . . . . . . . ( 12b)

and alan denotes the derivative along the outward drawn normal to the surface.

To decouple the system of Equations (9), a new dependent variable [ Z ] is defined as

. . . . . . . . . . . . . . . . . . . . . . . . . . . [ 4 ] = [ C ] [ Z ) (13)

where [ C ] is the unknown coefficient matrix which must satisfy the condition

. . . . . . . . . . . . . . . . . . . . . [ A ] = [ C ] - ‘ [ K ] [ C ] (14)

such that [ A ] is a diagonal matrix. Clearly, the diagonal elements of [ A ] are the eigenvalues of the matrix [ K ] and the columns of the matrix [ C ] are the eigenvectors. The elements zI and z2 of the vector [ Z) are yet to be determined.

For the present problem, the eigenvalues Xi of [ K ] are determined as

Xi = Lu(1/2)[(1 + Ko*Pn + l /Lu) + ( - l ) i

J ( 1 + Ko*Pn + 1ILu)’ - 4ILu) . . . . . . . . . . (15)

where i = 1 , 2. Then the elements of [C] are determined as

. . . . . . . . . . . . . (16) 1 XI - Lu A2 - Lu -LuPn -LuPn

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 69, DECEMBER, 1991 1263

and its inverse becomes

[c] = (l /(-LuPn(A1 - A,))) -LuPn -(A2 - Lu) . . . . . . . . . . . . . . . . 1 [ Lupn X I - Lu

The application of the transformation Equation (13) to Equa- tions (9) results in the following decoupled system for the determination of the functions (Z) = ( zl 22 I '.

a/aFo (z) = [A] v 2 (21

0 < R c 1, Fo > 0 . . . (18a)

( Z ) = (01, for Fo = 0, 0 I R I 1 . . . . . (18b)

a ( z ] + p a/an {z) = [cI - ' { g ) . . . . . .

which is written in the explicit form as

aZ,/aFo = ( A , / R ) ~ / ~ R ( R at, i a ~ )

O < R < l , F o > O

for Fo = 0, 0 I R I 1

for R = 0, FO > o . . . . .

ZI = 0,

a t , / a R = 0,

az,IaR = [ [Ki , - N Ki,)I(Al - A,)]

+ [(A2 - Lu)/LuPn(X1 - A,)] 1

[PnKi, + (1 - N Pn) Kim]

for R = 1, Fo > 0 . . . . .

and

az2 I ~ F O = (A, I R ) aiaR (R a z2 I a ~ ) 0 c R c 1, Fo > 0 . . . (20a)

22 = 0, for Fo = 0, 0 I R I 1 . . . (20b)

a t 2 / a ~ = 0, for R = 0, FO > o . . . . . . . . (20c)

az21aR = - [ ( K i , - N K i m ) / @ , - h 2 ) ]

- [(A, - Lu)/LuPn(A1 - A 2 ) ]

[PnKi, t (1 - N P n ) K i , I ,

for R = 1, Fo > 0 . . . . . . . . (204

Thus, the problem has been reduced to the solution of two decoupled diffusion Equations (19) for z1 and Equations (20) for z2. These decoupled systems are solved by application of the integral transform technique (Ozigik, 1980).

Once zl and z2 are known, the solution of the original problem defined by Equations (1)-(4) is determined according to the transformation, Equation (13), as

Finally, the solutions for T and 8 become

T(R,Fo) = 2(Ki, - LuKoKi,)Fo + (Ki4 - N Kim) o r 2

(R2/2 - 114) - 2 r n = I C i = l C [J, ( f l m R ) / / 3 ~ J , ( / 3 , ) ] C ~

Fo) . . . . . . . . . . . . . . . . . . . . . . . . . . (22) exP ( - X i

8(R,Fo) = 2LuKimFo + [PnKi, - (1 - N Pn)Kim] o r 2

(R2/2 - 1/4) + 2 m = C I , = I C [J,(pm R ) / P ~ J , ( P m ) ] C y

exp (-Ai p i Fo) ......................... (23)

where N = (1 - e)LuKo, &'s are the positive roots of J1(P,) = 0, and

C! = - [ (Ki, - N Kim)A2

- (Ki, - KoLuKim)] / (Al - X2) . . . . . . . . . . . (24a)

Cj = [ (Ki , - N Kim)AI

- (Ki, - KoLuKi,)] / ( A I - A,) . . . . . . . . . . (24b)

Cy = ( [PnKi, + (1 - N Pn)Kim] A2 - LuKi,] /

(A, - A,) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (24c)

C," = - { [PnKi, + (1 - NPn)Kim]X1 - LuKim) /

(A, - A,) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (244

Results and discussion

The influence of various dimensionless parameters Fo, Kim, Ki,, KO, Lu, Pn, and e on the temperature and moisture distributions within a cylindrical capillary-porous body are examined. Certain restrictions have been reported by Luikov with respect to the relative magnitudes of the Luikov number and the dimensionless product eKoPn; if Lu is small (i.e., Lu < 0.3), then EKoPn < 0.3; con- versely, if Lu is large (Lea, Lu 1 0.3) then &oPn > 0.4 (Luikov and Mikhailov, 1965). Moreover to maintain the physically significant boundary conditions for the above heat and mass transfer problem, restrictions must also be imposed on the relative magnitudes of Ki, and Kim. With these guidelines in mind, we have chosen values for the thermophysical parameters from several sources (Kayakawa and Rossen, 1977; Luikov, 1966) appropriate for biological materials.


Fo = 0.80 -

Fo - 0.40 . - m - 0.24 - Fo - 0.10 Fo - 0.06 .

0 0.5 1

R, Radiur

Figure 1 - Temperature distribution as a function of time for E = 0.2, Kim = 0.5, Ki, = 1.5, KO = 5.0, Lu = 0.4 and Pn = 0.5.

, Po - 1.80 1 t

Fo - 0.80 -

Fo - 0.40

Fo - 0.24

Fo - 0.10 Fo = 0.06

-

0 0.5 1

R, Radius

Figure 2 - Moisture distribution as a function of time for E = 0.2, Kim = 0.5 , Ki, = 1.5, KO = 5.0, La = 0.4 and Pn = 0.5.

The following values of dimensionless parameters are used in the present study:

Fo = 0.05, 0.10, 0.20, 0.40, 0.8, 1.6, 3.2 Kim = 0.1, 0.5 Ki, = 1.5, 2.0, 2 .5 KO = 1.0, 3.0, 5.0, 7.0 LU = 0.01, 0.05, 0.1 Pn = 0.0, 0.4, 0.6, 1.0 E = 0.2, 0.4, 0.8, 1.0. Figures 1 and 2 represent the temperature and moisture

distribution at various Fourier numbers for a group of typical dimensionless parameters: E = 0.2, Ki, = 1.5, Kim = 0.5, KO = 5 .O, Lu = 0.4, and Pn = 0.5. As Fo number increases, temperature increases but moisture decreases (i.e., increasing 0 cortesponds ta decreasing moisture potential u) at every point within the body, The moisture contat remains at its

0 0.6 1

R, Radius

Figure 3 - Effects of Kim number on temperature distribution for E = 0.2, Ki, = 1.5, KO = 5.0, Lu = 0.4 and Pn = 0.5.

Fo - 1.0

-

Fo - 1.0

Fo = 0.Z

m - 0.8

- I . * . * I . . . * I * * I 0 0.5 1

R. Radium

Figure 4 - Effects of Kim number on moisture distribution for E = 0.2, Ki, = 1.5, KO = 5.0, Lu = 0.4 and Pn = 0.5.

initial value near the centerline at (R = 0) for values of the Fo number up to 0.4, but shows pronounced drying near the surface for short times.

Figures 3 and 4 display the effects of the Kim number on temperature and moisture distributions. While the Kim number represents the dimensionless mass flux, it affects the size of both the heat source and moisture sink as seen from boundary conditions Equations (4c) and (4d). Increasing Kim while maintaining other parameters constant produces a significantly drier body at a lower temperature as shown in Figure 3 for temperature and Figure 4 for moisture.

The effects of the dimensionless heat flux, Ki,, on temperature and moisture distributions appear in Figures 5 and 6. Increasing the Kiq number has the straightforward effect of increasing temperatures throughout the medium. However, these higher temperatures subsequently affect the moisture distribution as seen in Equation (2). Higher Ki,


I - ro - 0.84

0.2

t : 3 . 80.1 0 k . e s . 0

0

-0.1

0

- . - KO - 3.0

.... KO-6.0 - KO - 7.0

-

7.-.- .7- -7. .-r. -7.

I . . . . I .

0.6 1

R, Radium

Figure 5 - Effects of Ki, number on temperature distribution for 6 = 0.2, Kim = 0.5, KO = 5.0, Lu = 0.4 and Pn = 0.5.

1 I ' ' . . - ro * 1.0 /,. lo = 1.6

- x+ - 1.6 . ... q - 8 . 0 - - 'g - 8.6

- I . . . . I r

0 0.6 1

R, Radiw

Figure 6 - Effects of Ki, number on moisture distribution for E = 0.2, Kim = 0.5, KO = 5.0, Lu = 0.4 and Pn = 0.5.

numbers result in a drier surface with respect to the interior producing steeper moisture distributions.

Figures 7 and 8 depict the effects of the Kossovich number on temperature and moisture distributions in the medium. Examinations of Figures 7 and 8 along with Figures 5 and 6 show the dramatically opposing effects of the KO number and the Ki, number. The KO number governs the magnitude of the heat sink created by evaporation of liquid within the medium as seen in Equation (1). Increasing KO results in a larger heat sink due to evaporation which lowers the temperature within the medium. As seen in Equation (4c), the KO number also affects the magnitude of the moisture sink at the surface; therefore, the higher KO number creates a smaller moisture sink and thus less drying at the surface.

The Luikov number governs the coupling of heat and moisture transfer within the body (Luikov, 1966). In Figures 9 and 10, the effects of the Luikov number are demonstrated.

r; 0 2 k b E

ro - 1.6

- ro - 1.6

c 4

R, Radium

Figure 7 - Effects of KO number on temperature distribution for E = 0.2, Kim = 0.5, Ki , = 1.5, Lu = 0.4 and Pn = 0.5.

O 3 3

Two ranges of Luikov number and hence the parameter &oPn were selected: (i) Lu < 0.3, EKoPn = 0.1, and (ii) Lu > 0.3, EKoPn = 0.5. The Luikov number dramatically influences the moisture distribution, but has little effect on the temperature distribution. It represents a moisture d i f i - sivity coefficient which combines both diffusional and capillary effects into a single coefficient. Smaller values of the Luikov number result in a temperature distribution which is developed well ahead of the moisture distribution, while for larger values temperature and moisture profiles remain unaffected.

Figure 11 shows the effect of the Posnov number on the moisture distribution. The Pn number characterizes internal mass transfer as seen in Equation (2) and does not affect heat transfer (Luikov, 1966). The temperature profiles remain virtually unchanged with the variation of the P o m n u m b e r over its entire range 0 to 1, but the moisture content gradient


0 0.5 1

R, Radium

Figure 8(b) - Effects of KO number on moisture distribution for E = 0.2, Kim = 0.5, Ki, = 1.5, Lu = 0.4 and Pn = 0.5 for Fo = 1.6.

o'2 t 1 01 ' . ' * ' ' ' ' ' I "

0 0.5 1

R, Radiua

Figure 9(a) - Effects of La number on temperature distribution for E = 0.2, Kim = 0.5, Ki, = 1.5, KO = 1.0 and Pn = 0.5 for Fo = 0.2.

is strongly affected by the variation of the Posnov number (i.e., increasing Posnov number increases moisture gradient). Therefore for a given temperature distribution, it is possible to have a redistribution of the initial moisture content such that at some region, the local moisture content may become larger than the initial distribution. Figure 1 1 illustrates such a case in the region near the center (i.e., in the figure 8 = 0 corresponds to some constant initial moisture content and the negative values correspond to a moisture content higher than the initial value).

In Figures 12 and 13, the effects of the phase change criteria, E , on temperature and moisture distributions are shown. A value of E = 0 implies evaporation occurs only at the surface while a value of e = I implies evaporation occurs inside the body (Luikov, 1966). As E increases toward its maximum value of 1 3 , the magnitudeaf 'the heat source

R, Radium

Figure 9(b) - Effects of Lu number on temperature distribution for E = 0.2, Kim = 0.5, Ki, = 1.5, KO = 1.0 and Pn = 0.5 for Fo = 3.2.

- la -0.01 _.. la-0.06 - - la - 0.10

d fl 0

/ * '

/ : / :'

/ ?o - s.2 Po - 0.8 Po - 0.2

Po - 0.2 -

0 0.5 1

R, Radius

Figure 10 - Effects of Lu number on moisture distribution for E = 0.2, Kim = 0.5, Ki, = 1.5, KO = 1.0 and Pn = 0.5.

and moisture sink reach their maximum values as shown by an examination of the boundary conditions Equations (8c) and (8d); thus, higher temperatures and more drying occur near the surface for higher values of E .

The parameters Ki,, KO, Pn, and E appear to affect the distribution of moisture without changing the total moisture content which results in intersecting moisture distribution curves for a given Fourier number as shown in Figures 6, 8, 1 1 and 13. Furthermore, the phase change criteria, E , has the same effect on the temperature distribution as shown in Figure 12. On the other hand, the Lu and Kim numbers change both the total moisture content as well as the shape of the moisture profile.

The effects of the geometry can be seen in Figure 14 for temperature and Figure 15 for moisture. The cylinder exhibits higher temperatures than the slab for any value of the Fourier number; however, the difference becomes more


1 I . ' " I '

. -m-o.o .... m - 0 . 4

. - - m - o . o - . m - 1.0 , ?O - 3.2.

/

, ?o - 3.2

Fo - 3.2-

- lo - 3.2

lo - 0.2

?O - 0.8 Fo - 0.2.

m - 0.2

-

0 0.6 1

R, Radius

Figure 11 - Effects of Pn number on moisture distribution for E = 0.2, Kim = 0.5, Ki, = 1.5, KO = 1.0 and Lu = 0.2.

" " ' " " " ' 4 - I - 0.2 _... 1-0 .4 - - I - 0.8

0 0 0.6 1

R, Radius Figure 12(a) - Effects of t on temperature distribution for Kim = 0.5, Ki, = 1.5, KO = 5.0, Lu = 0.4 and Pn = 0.5 for Fo = 0.2.

significant at higher Fourier numbers. Figure 15 shows the effect of curvature on the moisture distribution. At low Fo numbers, the cylinder shows slightly increased drying at the surface and a reduction in drying at the center with respect to the slab. As the Fo number increases, the moisture potential of the cylinder is significantly decreased in comparison with the slab.

Acknowledgement

This work was supported in part by the Alcoa Foundation Grant of the Alcoa Technical Center, PA.

Nomenclature

am a9 b Cq

= diffusion coefficient of moisture, m2/s = thermal diffusivity coefficient, m21s = radius of cylinder, m = specific heat capacity, Idlkg; "C

0 0.5 1

R, Radiw

Figure 12(b) - Effects of t on temperature distribution for Kim = 0.5, Ki, = 1.5, KO = 5.0, Lu = 0.4 and Pn = 0.5 for Fo = 0.8, 1.6.

0.9 I " . . I .

I ?o - 0.2

0 0.5 1

R, Radius

Figure 13(a) - Effects of t on moisture distribution for Kim = 0.5, Ki, = 1.5, KO = 5.0, Lu = 0.4 and Pn = 0.5 for Fo = 0.2.

Fo = Fourier number, a, 7 / b 2 = mass flux, kg/m2 s j , = heat flux, H/m2 . s -4

K = diffusivity matrix Kim = dimensionless mass flux, j , b l a, p u,) Ki, = dimensionless heat flux, jq b / A , t, KO = Kossovich number, Lu, l c, r , KO* = modified Kossovich number, €KO L = specific heat of vaporization, Idlkg Lu = Luikov number, a, I a, N = ( 1 - E ) L ~ K o Pn = Posnov number, 6t , l u , r = radial coordinate, m R = dimensionless radial coordinate, rlb t = temperature, "C T = dimensionless temperature, ( t - r , ) /r , , U = moisture content, kg/kg (solid) z , , z2 = elements of vector Z Z = new dependent variable vector


.... 1-0.4

- . - I - 1.0

0 0.5 1

R, Radfua

Figure 13(b) - Effects of E on moisture distribution for Kim = 0.5, Ki, = 1.5, KO = 5.0, Lu = 0.4 and Pn = 0.5 for Fo = 0.8, 1.6.

‘ , ‘ , ‘ , ‘ I ’ - - Cylinder - Slab K1, = 0.6 K$ = 1.6 Pn - 0.6 KO = 6.0 Lu = 0.4 I = 0.2

0.5 1 Fo = 0.2 1

” 0 0.2 0.4 0.6 0.8 1

X or R Figure 14 - Effects of geometry on temperature distribution for E = 0.2, Kim = 0.5, Ki, = 1.5, KO = 5.0, Lu = 0.4 and Pn = 0.5.

Greek letters

a = boundary coefficient p = boundary coefficient 6 = thermal gradient coefficient, 1/”C e = phase change criteria 8 = dimensionless moisture content, (u, - u)/u, Xi = eigenvalues of diffusivity matrix A, = thermal conductivity, M/m s . “C p = density of dry body, kg/m3 7 = time, s

Superscripts and subscripts

m = moisture 0 = initial condition q =heat

l*/ KO I 5.0 Lu = 0.4 € = 0.2

0 0

0 0

0

0 0 .

/ / $0.5 c

- 3 - - _ _ - - 4 -

Po - 0.2

0 0.2 0.4 0.6 0.8 1

X or R

Figure 15 - Effects of geometry on moisture distribution for E = 0.2, Kim = 0.5, Ki, = 1.5, KO = 5.0, Lu = 0.4 and Pn = 0.5.

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drying in capillary-porous cylindrical media

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