An analytical solution to coupled heat and moisture diffusion transfer in porous materials

International Journal of Heat and Mass Transfer 43 (2000) 3621±3632

www.elsevier.com/locate/ijhmt

An analytical solution to coupled heat and moisture di€usion transfer in porous materials Win-Jin Chang a,*, Cheng-I Weng b a

Department of Mechanical Engineering, Kung Shan Institute of Technology, Tainan, Taiwan b Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan Received 16 September 1999; received in revised form 7 December 1999

Abstract An analytical method is proposed for solving the problem of coupled heat and moisture transfer in porous materials. The coupled partial di€erential equations and boundary conditions are ®rst subjected to Laplace transformation, the equations are reduced to ordinary di€erential equations, then the equations are converted into a single fourth-order ordinary di€erential equation by introducing a transformation function. The solution of the equation can be easily obtained, and thus, the temperature and moisture distributions in the transform domain can be determined. Finally, the transformed values are analytically or numerically inverted to obtain the time domain results. Therefore, the transient solution at any given time can be evaluated. The results are identical with published analytical solutions for a special case using decoupling technique, and they agree with a published analytical solution for wood slab. The method is compact enough to be generally applied to problems of heat and mass transfer in porous media. # 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction Simultaneous heat and moisture transfer accompanied by phase change and/or absorption heat in some porous and composite materials is a process which occurs frequently in various engineering applications. The moisture movement contributes to heat transfer, while phase-change and/or heat of absorption within the material act as heat sources or sinks. The coupled system for temperature and moisture potential can be handled through both analytical and numerical approaches, depending on complexities of the speci®c problem considered. For linear problems, the analytical solutions, based on classical integral transform approach, have been obtained by Mikhailov

* Corresponding author. E-mail address: [email protected] (W.-J. Chang).

et al. [1]. It was later discovered by the same group [2] that the existence of complex eigenvalues, not accounted for in the solutions previously reported, could signi®cantly alter the temperature and moisture distributions. Liu et al. [3] later recon®rmed such ®ndings by discovering a pair of complex eigenvalue. Recently, Mikhailov et al. [4] again found another complex eigenvalue. In the cited papers, there exist numerical diculties in computing the complex conjugate eigenvalues that lead to an incorrect solution containing high frequency oscillation of limited usefulness. Chang et al. [5] applied a decoupling technique to coupled governing equations, but failed to address the case of simultaneous coupling of governing equations and boundary conditions. Ribeiro et al. [6] treated the coupled system by applying the generalized integral transform technique with complicated procedures. Recently, Cheroto et al. [7] presented a modi®ed lumped system analysis method to yield approximate

0017-9310/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 7 - 9 3 1 0 ( 0 0 ) 0 0 0 0 3 - X

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W.-J. Chang, C.-I. Weng / Int. J. Heat Mass Transfer 43 (2000) 3621±3632

Nomenclature cm cp C D Dm hC hm hLV I0 I1 J0 J1 k K0 K1 ` L m m0

moisture capacity (kg/kg 8M) heat capacity (J/kg K) moisture content (kg/kg) equivalent di€usion coecient of moisture content (m2/s) conductivity coecient of moisture content (kg/m s 8M) convective heat transfer coecient (W/m2 K) convective moisture transfer coecient (kg/m2 s 8M) heat of phase change (kJ/kg) modify Bessel function of the ®rst kind of order zero modify Bessel function of the ®rst kind of order ®rst Bessel function of the ®rst kind of order zero Bessel function of the ®rst kind of order ®rst thermal conductivity (W/m K) modify Bessel function of the second kind of order zero modify Bessel function of the second kind of order ®rst half thickness of specimen (m) equivalent di€usion coecient of temperature (m2/s) moisture potential (8M) initial moisture potential (8M)

solutions, avoiding diculties experienced by Mikhailov et al., Liu et al., and Ribeiro et al., but sacri®cing accuracy. In this paper, a di€erent analytical approach is developed. The coupled system is ®rst subjected to Laplace transformation. The coupled partial di€erential equations thus are reduced to the coupled ordinary di€erential equations. Then, the temperature and moisture in the transformed domain are expressed in terms of a transformation function. Consequently, the system of ordinary di€erential equations can be reduced to a single fourth-order ordinary di€erential equation. The solution of the equation can be obtained and the coecients are determined from the boundary conditions. Finally, the transformed solution can be analytically or numerically inverted to yield the time domain results. 2. Problem formulation A typical heat and mass transfer problem is governed by Luikov's equations [8], which relate to drying a porous moist slab under constant pressure. The phase-change occurring within the slab act as heat

m S t T T0 T Y0 Y1

Laplace transformation Laplace transformation time (s) temperature (K) initial temperature (K) Laplace transformation Bessel function of the zero Bessel function of the ®rst

of m parameter

of T second kind of order second kind of order

Greek symbols r material density (kg/m3) n coupling coecient due to moisture migration (K/8M) l coupling coecient due to heat conduction (8M/K) f transformation function g heat of absorption or desorption (kJ/kg) e ratio of vapor di€usion coecient to coecient of total moisture di€usion Subscripts 0 initial condition m moisture 1 ambient atmosphere

source or sink resulting in the coupled relationship between mass transfer and heat transfer. In a coupled problem, the heat of absorption or desorption is generally one of the sources or sinks as well. This heat is not negligible for some hygrothermal materials [9,10]. In the present study, one-dimensional governing equations with coupled temperature and moisture for a porous slab are considered, and the e€ect of the absorption or desorption heat is added. Material properties and pressure are considered to be constant throughout the material. A local thermodynamic equilibrium between the ¯uid and the porous matrix is assumed. Moreover, the coupled equations can be generalized to apply to cases of hollow cylindrical and hollow spherical geometries. The equations are rcp x 1ÿ2n

  @T @ @T @m ˆk x 1ÿ2n ‡ rcm …ehLV ‡ g†x 1ÿ2n @t @x @x @t …1a†

rcm x 1ÿ2n

    @m @ @m @ @T ˆ Dm x 1ÿ2n ‡ Dm d x 1ÿ2n @t @x @x @x @x …1b†

W.-J. Chang, C.-I. Weng / Int. J. Heat Mass Transfer 43 (2000) 3621±3632

where n ˆ 1=2 for slab, n ˆ 0 for hollow cylinder, and n ˆ ÿ1=2 for hollow sphere. T is the temperature, and m is the moisture potential, k and Dm are the thermal and moisture conductivity coecients, respectively, cp and cm are the heat and moisture capacities of the medium, respectively, r is the material density, hLV is the heat of evaporative phase-change, g represents the heat of absorption or desorption, d is the thermogradient coecient, and e is the ratio of the vapor di€usion coecient to the coecient of total moisture di€usion. All the material properties mentioned above are e€ective properties. The moisture potential m is related to the moisture content C, and C ˆ cm m

…2†

The coupling di€usion system represented by Eqs. (1a) and (1b) contains not only general di€usion equations, but also some source or sink terms. The governing equation (1a) expresses the balance of thermal energy within the body; the last term in this equation represent the heat sources or heat sinks due to liquid-tovapor phase-change and to the heat of absorption or desorption. Similarly, Eq. (1b) expresses the balance of moisture within the medium; the last term in this equation represents the moisture source or moisture sink with respect to the temperature gradient. To simplify the notation, dividing Eq. (1a) by rcp x 1ÿ2n and using the expression for @ =@ x…x 1ÿ2n @ T=@ x† obtained from Eq. (1a) into Eq. (1b), then rearranging the two new equations, yields   @ @T @T @m x 1ÿ2n ˆ ÿn …3a† Lx 2nÿ1 @x @x @t @t   @ @m @m @T Dx 2nÿ1 x 1ÿ2n ˆ ÿl @x @x @t @t

…3b†

Dˆ

k rcp

…4a†

kDm   rcm k ‡ Dm d…ehLV ‡ g†

…4b†

cm …ehLV ‡ g† nˆ cp

lˆ

cp Dm d  cm k ‡ Dm d…ehLV ‡ g† 

duction, respectively; L and D are also always positive expressing the equivalent temperature di€usion coef®cient and the equivalent moisture di€usion coecient, respectively. The moisture in Eq. (3a) will play the role of a heat source for the temperature distribution, if the moisture rate is positive (i.e. @ m=@ t > 0), and act as a heat sink if the moisture rate is negative (i.e. @ m=@ t < 0). Similarly, the temperature may play the role of a moisture source or a moisture sink, depending on the temperature rate being positive or negative. Therefore, the coupling di€usion system rewritten as Eqs. (3a) and (3b) more compactly and clearly represents the same physical process modeled by Eqs. (1a) and (1b). At the boundaries of the domain, the latent heat of vaporization becomes part of the energy balance, and the mass di€usion caused by the temperature and moisture gradients a€ects the mass balance. The associated hygrothermal boundary and initial conditions are k

  @ T…x 1 , t† ˆ hC1 T…x 1 , t† ÿ T11 @x   ‡ …1 ÿ e †hLV hm1 m…x 1 , t† ÿ m11

…5a†

  @ T…x 2 , t† ˆ hC2 T…x 2 , t† ÿ T12 @x   ‡ …1 ÿ e †hLV hm2 m…x 2 , t† ÿ m12

…5b†

ÿk

@ m…x 1 , t† @ T…x 1 , t† ‡ Dm d @x @x   ˆ hm1 m…x 1 , t† ÿ m11

…5c†

@ m…x 2 , t† @ T…x 2 , t† ÿ Dm d @x @x   ˆ hm2 m…x 2 , t† ÿ m12

…5d†

Dm

ÿDm

where Lˆ

3623

…4c†

…4d†

In Eqs. (3a) and (3b), n and l are positive coupling coecients due to moisture migration and heat con-

T…x, 0† ˆ T0

…6a†

m…x, 0† ˆ m0

…6b†

Eqs. (5a)±(5d) constitute the natural boundary conditions for temperature and moisture, respectively. Eqs. (5a) and (5b) represent the heat balance at x ˆ x 1 and x ˆ x 2 : The two equations express the heat ¯ux in terms of convection heat transfer and the phase-change energy transfer. Eqs. (5c) and (5d) represent the moisture balance at the two surfaces; the two terms on the left-hand side of the equal sign describe the supply of moisture ¯ux under the in¯uence of a temperature gradient and a moisture gradient, respectively. The

3624

W.-J. Chang, C.-I. Weng / Int. J. Heat Mass Transfer 43 (2000) 3621±3632

terms to the right side of the equal sign describes the amount of moisture drawn o€ from or into the surfaces. Eqs. (6a) and (6b) represent initial temperature and moisture value within the domain, respectively.

3. Method of solution Applying the Laplace transformation to Eqs. (3a), (3b), (5a)±(5d) and (6a), (6b) with respect to t, they become    2nÿ1 d 1ÿ2n dT Lx x ˆ ST ÿ T0 ÿ nSm ‡ nm0 …7a† dx dx Dx 2nÿ1

  d dm x 1ÿ2n ˆ Sm ÿ m0 ÿ lST ‡ lT0 dx dx

   1 , S† dT…x  1 , S† ÿ T11 k ˆ hC1 T…x dx S   m11  1 , S† ÿ ‡ …1 ÿ e †hLV hm1 m…x S 

n

ji

Slab Hollow cylinder

1/2 0

Hollow sphere

ÿ1/2

e pi x I0 …pi x† for i ˆ 1, 2 K0 …pi x† for i ˆ 3, 4 e pi x =x

 S† ˆ n f…x, S† ‡ T0 T…x, L S

…10a†

  1 1 d df m0 f…x, S† ÿ x 2nÿ1 x 1ÿ2n ‡ L S dx dx S

 m…x, S† ˆ

…10b† we ®nd that Eq. (7a) is automatically satis®ed and Eq. (7b) becomes

…8a†

ÿk

 1 , S†  1 , S† dm…x dT…x ‡ Dm d dx dx   m11  1 , S† ÿ ˆ hm1 m…x S

Geometry

…7b†



 2 , S† dT…x  2 , S† ÿ T12 ˆ hC2 T…x dx S   m12  2 , S† ÿ ‡ …1 ÿ e †hLV hm2 m…x S

Table 1 The functions ji …x, S †, shown in Eq. (12), for slab, hollow cylinder and hollow sphere

…8b†

    d 1ÿ2n d 2nÿ1 d 1ÿ2n df x x x x dx dx dx dx     S D d df ÿ x 2nÿ1 x 1ÿ2n 1‡ D L dx dx 2nÿ1

‡

…11†

S 2 …1 ÿ nl † fˆ0 LD

Therefore, the system of coupled ODE equations (7a) and (7b) is reduced to a single Eq. (11). Eq. (11) is a fourth-order ODE. Assume the solution of f…x, S † in the following form

Dm

…8c†

fˆ

4 X xi …S †ji …x, S†

…12†

iˆ1

 2 , S†  2 , S† dm…x dT…x ÿ Dm d dx dx   m12  2 , S† ÿ ˆ hm2 m…x S

ÿDm

…8d†

where ji represents di€erent functions, as shown in Table 1, for slab, hollow cylindrical and hollow spherical geometries. p In Table 1, pi ˆ Sqi , in which qi is de®ned as 2

 0† ˆ T0 T…x, S

…9a†

 31=2 s  2 ai 4 D D 4lnD 5 qi ˆ p 1 ‡ ‡ bi 1ÿ ‡ L L L 2D

m0 S

…9b†

and where

 m…x, 0† ˆ

 † and m…S  † are the Laplace transformation where T…S of T…t† and m…t†, respectively; and S is Laplace transformation parameter. Introducing a transformation function f…x, S † such that

8 <1 ai ˆ 1 : ÿ1 and

for i ˆ 1, 2 for i ˆ 1, 2, 3, 4 for i ˆ 3, 4

…13†

if slab and hollow sphere if hollow cylinder if slab and hollow sphere

W.-J. Chang, C.-I. Weng / Int. J. Heat Mass Transfer 43 (2000) 3621±3632

 bi ˆ

ÿ1 for i ˆ 1, 3 : 1 for i ˆ 2, 4

Q1 ˆ

The coecients xi …S † …i ˆ 1, 2, 3, 4† can be determined by using Eqs. (8a)±(8d). The results are then written in the matrix form  ‰K Š x…S † ˆ fQg

…14†

in which 0

K11 B K21 ‰K Š ˆ B @ K31 K41

K12 K22 K32 K42

1

K13 K23 K33 K43

K14 K24 C C K34 A K44

 T  x…S † ˆ x1 …S †, x2 …S †, x3 …S †, x4 …S †

…15a†

…15b† …15c†

T

where fx…S †g and fQg stand for the transpose of fx…S †g and fQg, respectively, and where K1j ˆ

  …1 ÿ e †hLV hm1 2nÿ1 d djj kn djj 1ÿ2n x x ‡ dx L dx S dx  1 ÿ hC1 n ‡ …1 ÿ e †hLV hm1 jj L at x ˆ x 1 , j ˆ 1, 2, 3, 4 …16a†

  …1 ÿ e †hLV hm2 2nÿ1 d djj kn djj K2j ˆ x 1ÿ2n x ÿ dx L dx S dx  1 ÿ hC2 n ‡ …1 ÿ e †hLV hm2 jj L at x ˆ x 2 , j ˆ 1, 2, 3, 4 …16b†    djj Dm d d x 2nÿ1 x 1ÿ2n S dx dx dx   djj hm1 2nÿ1 d 1ÿ2n ‡ x x dx S dx djj hm1 Dm …dn ‡ 1 † ÿ j ‡ L dx L j at x ˆ x 1 , j ˆ 1, 2, 3, 4

K3j ˆ ÿ

K4j ˆ

 1 hC1 …T0 ÿ T11 † ‡ …1 ÿ e †hLV hm1 …m0 ÿ m11 † S …16e†  1 hC2 …T0 ÿ T12 † ‡ …1 ÿ e †hLV hm2 …m0 ÿ m12 † S …16f†

Q3 ˆ

hm1 …m0 ÿ m11 † S

…16g†

Q4 ˆ

hm2 …m0 ÿ m12 †: S

…16h†

The solution of Eq. (14) can be determined by Cramer's rule. xi …S † ˆ

 T  Q ˆ Q1 , Q2 , Q3 , Q4 T

Q2 ˆ

   djj Dm d d x 2nÿ1 x 1ÿ2n dx S dx dx   djj hm2 2nÿ1 d 1ÿ2n x ‡ x dx S dx djj hm2 Dm …dn ‡ 1 † ÿ j ÿ L dx L j at x ˆ x 2 , j ˆ 1, 2, 3, 4

3625

jZi j , jKj

i ˆ 1, 2, 3, 4

…17†

where jZi j is the determinant of the matrix resulting from ‰KŠ in which the ith column is replaced by the column vector fQg: Substituting Eq. (17) into Eq. (12), Eqs. (10a) and (10b) can be rewritten as 4 X T0  S† ˆ n T…x, x …S †ji …x, S† ‡ L iˆ1 i S

 m…x, S† ˆ

4  X 1 iˆ1

 m0 ÿ q2i xi …S †ji …x, S† ‡ : L S

…18a†

…18b†

As it is dicult in general to ®nd the inverse Laplace  S † and m…x,  transformation of the functions T…x, S† analytically, a numerical inversion method [11] may be used. This method has been proven to yield accurate results in previous research [12]. However, for some special cases, the functions at any given time can be evaluated using the inversion theorem for the Laplace transformation. 4. Analysis of a special case

…16c†

…16d†

As a special case of the foregoing analysis, we now consider an in®nitely long hollow cylinder, having inner and outer radii x 1 and x 2 , respectively, subjected to symmetrical hygrothermal loadings. The mathematical formulation is expressed as follows:   1 @ @T @T @m L x ˆ ÿn …19a† x @x @x @t @t D

  1 @ @m @m @T x ˆ ÿl x @x @x @t @t

…19b†

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W.-J. Chang, C.-I. Weng / Int. J. Heat Mass Transfer 43 (2000) 3621±3632

T…x 1 , t† ˆ T1 ,

T…x 2 , t† ˆ T1

m…x 1 , t† ˆ m0 ,

m…x 2 , t† ˆ m0

T…x, 0† ˆ T0 ,

m…x, 0† ˆ m0 :

…20†

In this problem, the moisture contains vapor phase only, then the phase change within the cylinder is excluded. A similar problem have been solved in previous research [5] by using decoupling techniques. Comparing the present problem with the previous general problem, we ®nd that hC1 ˆ hC2 ˆ hC 4 1,

hm1 ˆ hm2 ˆ hm 4 1,

p1 ˆ p3 ,

p2 ˆ p4 , m11 ˆ m12 ˆ m1 ˆ m0 , Lˆ Fig. 1. Schematic representation of the wood slab.

k , rcp

Dˆ

q1 ˆ q3 ,

kDm , rcm …k ‡ Dm dg†

cp Dm d lˆ : cm …k ‡ Dm dg†

Fig. 2. Temperature at the center and surface of wood specimen for case 1.

q2 ˆ q4 ,

nˆ

cm g , cp

hLV ˆ 0,

…21†

W.-J. Chang, C.-I. Weng / Int. J. Heat Mass Transfer 43 (2000) 3621±3632

3627

This problem is a special case of the coupled system discussed previously. The corresponding coecients are now reduced to 2

n 6 L I0 …p1 x 1 † 6 6n 6 I0 …p1 x 2 † 6L 6  ‰K Š ˆ 6 6 1 2 6 ÿ q1 I0 …p1 x 1 † 6 L 6  6 4 1 ÿ q21 I0 …p1 x 2 † L

fQgT ˆ



T1 ÿ T0 , S

n I0 …p2 x 1 † L n I0 …p2 x 2 † L   1 2 ÿ q2 I0 …p2 x 1 † L   1 ÿ q22 I0 …p2 x 2 † L

T1 ÿ T0 , S

n K0 …p1 x 1 † L n K0 …p1 x 2 † L   1 2 ÿ q1 K0 …p1 x 1 † L   1 ÿ q21 K0 …p1 x 2 † L

3 n K0 …p2 x 1 † 7 L 7 7 n 7 K0 …p2 x 2 † 7 L 7   7 1 7 2 ÿ q2 K0 …p2 x 1 † 7 7 L 7   7 5 1 ÿ q22 K0 …p2 x 2 † L

…22†

 0,

…23†

0

Then xk …S † in Eq. (17) can be simpli®ed as xk …S † ˆ

     L…T1 ÿ T0 † 1 1 2 2 ÿ 2
ÿ q ÿ q g Zk ‡ k 2 1 L L Sn q2 ÿ q21  ÿp  ÿp
ÿp

ÿp
 K0 Sqk x 2 ÿ K0 Sqk x 1 …d1k ‡ d2k † ‡ I0 Sqk x 1 ÿ I0 Sqk x 2 …d3k ‡ d4k † ÿp
ÿp
ÿp
ÿp

, I0 Sqk x 1 K0 Sqk x 2 ÿ I0 Sqk x 2 K0 Sqk x 1

k ˆ 1, 2, 3, 4

…24†

where  gk ˆ

ÿ1 for k ˆ 1, 3 0 for k ˆ 2, 4

and  Zk ˆ

0 for k ˆ 1, 3 1 for k ˆ 2, 4

and dij is the Kronecker delta which is de®ned as  dij ˆ

1 when i ˆ j 0 when i 6ˆ j

Consequently, Eqs. (18a) and (18b) become      2  X 1 1 2 2  S† ˆ Tÿ 1 ÿ T0
T…x, ÿ q ÿ q ÿ d ‡ 1n 2 1 d2n L L S q22 ÿ q21 nˆ1
 ÿp
 ÿp
ÿp
 ÿp
ÿp
 ÿp T0 K0 Sqn x 2 ÿ K0 Sqn x 1 ‰I Š0 Sqn x ‡ I0 Sqn x 1 ÿ I0 Sqn x 2 K0 Sqn x ÿp
ÿp
ÿp
ÿp

‡ S I0 Sqn x 1 K0 Sqn x 2 ÿ I0 Sqn x 2 K0 Sqn x 1

 m…x, S† ˆ


ÿ
2 ÿ L…T1 ÿ T0 † L1 ÿ q21 L1 ÿ q22 X ÿ 2
‰ ÿ d1n ‡ d2n Š Sn q2 ÿ q21 nˆ1
 ÿp
 ÿp
ÿp
 ÿp
ÿp
 ÿp m0 K0 Sqn x 2 ÿ K0 Sqn x 1 I0 Sqn x ‡ I0 Sqn x 1 ÿ I0 Sqn x 2 K0 Sqn x ÿ
ÿ
ÿ
ÿ

‡ p p p p S I0 Sqn x 1 K0 Sqn x 2 ÿ I0 Sqn x 2 K0 Sqn x 1

…25a†

…25b†

The temperature and moisture distributions in the transformation domain, Eqs. (25a) and (25b) can be transformed to time domain by the inversion theorem for the Laplace transformation [13]. From the theorem we then have

3628

W.-J. Chang, C.-I. Weng / Int. J. Heat Mass Transfer 43 (2000) 3621±3632

Fig. 3. Moisture content at the center and surface of wood specimen for case 1.

Wk …x, t† ˆ

T1 ÿ T0 ÿ
2pi q22 ÿ q21    … g‡i1 1 1 2

ÿ ÿ q2 L gÿi1 z     L 1 U1  d1k ‡ ÿ q21 d2k n L V1       1 L 1 U2 2 2 ÿ q1 d1k ‡ ÿ q2 d2k ‡ L n L V2 ezt dz ‡ T0 d1k ‡ m0 d2k …26†

where k ˆ 1, 2; W1 and W2 represent T and m, respectively; and
ÿp
 ÿp  ÿp
Uk ˆ K0 zqk x 2 ÿ K0 zqk x 1 I0 zqk x
ÿp
 ÿp
 ÿp ‡ I0 zqk x 1 ÿ I0 zqk x 2 K0 zqk x ezt , k ˆ 1, 2

…27a†

ÿp
ÿp
ÿp
ÿp
Vk ˆ I0 zqk x 1 K0 zqk x 2 ÿ I0 zqk x 2 K0 zqk x 1 , k ˆ 1, 2 …27b† The singularities of the integrand are z ˆ 0, p p zq1 ˆ ian , and zq2 ˆ ian , which correspond to the simple poles z ˆ 0, z ˆ ÿa2n =q21 , z ˆ ÿa2n =q22 , respect-

ively, where an satis®es J0 …an x 1 †Y0 …an x 2 † ÿ J0 …an x 2 †Y0 …an x 1 † ˆ 0

…28†

Note that the relationships I0 …ian x† ˆ J0 …an x† and K0 …ian x† ˆ ÿpi 2 ‰J0 …an x† ÿ iY0 …an x†Š are used to obtain the above equation. Using the residue theorem to evaluate Eq. (26), yields Wk …x, t† ˆ …T1 ÿ T0 †d1k
Res…0 †
ÿ ÿ L1 ÿ q22 ‡ …T1 ÿ T0 † 2 q ÿ q21  2    L 1 2 ÿ q1 d2k  d1k ‡ n L ! 1 1 X ÿ q21 a2
Res ÿ n2 ‡ …T1 ÿ T0 † L2 q1 q2 ÿ q21 nˆ1     L 1  d1k ‡ ÿ q22 d2k n L ! 1 X a2n
Res ÿ 2 ‡ T0 d1k ‡ m0 d2k q2 nˆ1 k ˆ 1, 2:

…29†

W.-J. Chang, C.-I. Weng / Int. J. Heat Mass Transfer 43 (2000) 3621±3632

The residue at z ˆ 0 can be evaluated for small arguments I0 …z†11, I1 …z†1z=2, K0 …z†1 ÿ ln z, and K1 …z†11=z; then the limit can be taken by letting z 4 0, that is Res…0 † ˆ 1:

…30†

The residue at z ˆ ÿa2n =q21 , and z ˆ ÿa2n =q22 may be obtained from ! ÿ
Uk ÿ a2n =q2k a2n Res ÿ 2 ˆ , ÿa2n =q2k …dVk =dz†zˆÿa2n =q2 qk …31† k k ˆ 1, 2: Di€erentiating Eq. (27b), we have

ÿ x 2 I1

ÿp
ÿp
zqk x 2 K0 zqk x 1

ÿp
ÿp
 ‡ x 1 I0 zqk x 2 K1 zqk x 1

p In Eq. (32), inserting ian in place of zqk , and using the relationships I0 …z†K1 …z† ‡ I1 …z†K0 …z† ˆ 1=z, I0 …ian x† ˆ J0 …an x†, and Vk ˆ 0 in Eq. (27b), the following form is obtained J 2 …an x 2 † ÿ J 20 …an x 1 † a2n dVk ÿ 2 ˆ 0 , 2J0 …an x 1 †J0 …an x 2 † qk dz zˆÿa2n =q2k

…32†

…33†

k ˆ 1, 2: Similarly,

Uk

ÿp
ÿp
dVk qk  ˆ 1=2 x 1 I1 zqk x 1 K0 zqk x 2 dz 2z ÿp
ÿp
ÿ x 2 I0 zqk x 1 K1 zqk x 2

3629

a2 ÿ 2n qk

!

 p  Y0 …an x 2 † ÿ Y0 …an x 1 † J0 …an x † 2   ‡ J0 …an x 1 † ÿ J0 …an x 2 † Y0 …an x † 2 2  eÿ…an =qk †t , k ˆ 1, 2 …34†

ˆÿ

Finally, substituting Eqs. (33) and (34) into Eq. (31), and then substituting Eqs. (31) and (30) into Eq. (29), the following results are obtained:

Fig. 4. Temperature at the center and surface of wood specimen for case 2.

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W.-J. Chang, C.-I. Weng / Int. J. Heat Mass Transfer 43 (2000) 3621±3632

Fig. 5. Moisture content at the center and surface of wood specimen for case 2.

Wk …x, t† ˆ …T1 ÿ T0 †d1k ÿ p…T1 ÿ T0 †

   1 J a x X 0 … n 1 † ÿ J0 …an x 2 † J0 …an x †Y0 …an x 2 † ÿ J0 …an x 2 †Y0 …an x † J 20 …an x 1 † ÿ J 20 …an x 2 †

nˆ1


J0 …an x 1 †

( ÿ )
       1 2 ÿ L1 ÿ q22 L 1 L 1 2 ÿ…a2n =q21 †t 2 ÿ…a2n =q22 †t L ÿ q1
ÿ q1 d2k e ‡ T0 d1k d1k ‡ ‡ 2 ÿ q2 d2k e d1k ‡ n L n L q2 ÿ q21 q22 ÿ q21

…35†

‡ m0 d2k k ˆ 1, 2

In order to compare the results written in Eq. (35) with the results of [5], Eq. (35) can be rewritten as follows: T…x, t† ÿ T0 p ˆ 1ÿ T1 ÿ T0 B1 ‡ B2   1 X J0 …an x 1 † ÿ J0 …an x 2 † J0 …an x 1 †U0 …an x † J 20 …an x 1 † ÿ J 20 …an x 2 †
ÿ 2 2 2 2  B1 eÿan t=q1 ‡ B2 eÿan t=q2

m…x, t† ÿ m0 pLB1 B2 ˆ l…T1 ÿ T0 † D…B1 ‡ B2 †   1 J a x X 0 … n 1 † ÿ J0 …an x 2 † J0 …an x 1 †U0 …an x † J 20 …an x 1 † ÿ J 20 …an x 2 † nˆ1 ÿ
2 2 2 2  ÿ eÿan t=q1 ‡ eÿan t=q2

…36b†

where

nˆ1

…36a†

U0 …an x † ˆ J0 …an x †Y0 …an x 2 † ÿ J0 …an x 2 †Y0 …an x †

…37a†

W.-J. Chang, C.-I. Weng / Int. J. Heat Mass Transfer 43 (2000) 3621±3632

B1 ˆ

1 ÿ Dq21 nl

…37b†

B2 ˆ

Dq22 ÿ 1 nl

…37c†

and where …T ÿ T0 †=…T1 ÿ T0 † and …m ÿ m0 †=l…T1 ÿ T0 † are dimensionless forms of T and m, respectively. The temperature and moisture distributions, as written in Eqs. (36a) and (36b), are identical to those obtained by previous research [5].

5. Numerical results and discussions Now we consider a wood slab, subjected to symmetrical hygrothermal loadings; the heat and moisture transfer are coupled within the slab and its boundaries. Two cases are discussed here. For case 1, the coupled system is modeled by Eqs. (3a), (3b) to (6a), (6b). The geometry and the material properties of the wood slab shown in Fig. 1, which were used by Liu et al. [3], are used in the numerical calculations for comparison. Therefore, absorption heat in Eqs. (3a) and (3b) is assumed to be negligible and the geometry and boundary conditions in Eqs. (5a)±(5d) are assumed to be symmetric. The results can be obtained from Eqs. (18a) and (18b), and the data are as follows: n ˆ 1=2,

T0 ˆ 108C,

T1 ˆ 1108C,

m1 ˆ 48M, r ˆ 370 kg=m3 ,

m0 ˆ 868M,

k ˆ 0:65 W=…m K †,

Dm ˆ 2:2  10ÿ8 kg=…m s 8M †, hLV ˆ 2500 kJ=kg, e ˆ 0:3,

cp ˆ 2500 J=…kg K†,

d ˆ 2:08M=K, s 8M †,

hC1 ˆ hC2 ˆ hC ˆ 2:25 W=…m2 K †, cm ˆ 0:01 kg=…kg 8M†,

hm1 ˆ hm2 ˆ hm ˆ 2:5  10ÿ6 kg=…m2

` ˆ 0:012 m,

g ˆ 0:

For case 1, the temperature and moisture evolution at the surface and at the middle of the slab are shown in Figs. 2 and 3. It can be seen that the results from the two di€erent methods agree well, even though the initial moisture content deviates a little, as shown in Fig. 3. The discrepancies might be caused by not including enough complex eigenvalues when using the eigenvalue method [3,4]. For case 2, to evaluate the ambient temperature e€ects on the moisture of the medium during drying, we make T0 ˆ T1 ˆ 258C, and the other data are kept the same as for case 1. Fig. 4 shows the temperature evolution at the surface and at the center of the slab.

3631

The temperature of the slab at initial time is equal to that of the atmosphere, while the moisture of the slab is higher than that of the atmosphere, so that moisture is transferred from the surface of the slab as vapor to the atmosphere. Initially, this evaporation causes cooling of the slab at the surface. Subsequently, the temperature within the slab also decreases due to heat conduction. After the temperature decreases to a value of about 98C, it begins to increase due to heat convection from the surfaces. Simultaneously, the slab gains heat from the ambient atmosphere and loses heat through latent heat of evaporation. According to our mathematical analysis, the slab should reach an equilibrium temperature at T ˆ 258C after 30 h. The moisture content evolution is shown in Fig. 5. As expected, the moisture content of the slab decreases with increases in time, until it reaches an equilibrium value of 0.04 kg/kg after 30 h. The moisture content at the surfaces is lower than in the interior. Comparison of Figs. 3 and 5 indicates that the temperature has some e€ects on moisture migration during drying. Generally, the coupling e€ect between the temperature and moisture content becomes signi®cant as the value D=L approaches unity, while it will diminish as n
l approaches zero. In this study, according to the data given by Liu et al., they are D=L11=12 and n
l11=20:

6. Conclusion This study proposes a new analytical approach that consists of applying the Laplace transform technique and a transformation function to solve the problem of coupled temperature and moisture transport. The results, shown to be the same as published analytical solutions obtained using a decoupled technique, compare very well against published analytical solution for a wood slab con®guration, and can serve to evaluate the accuracy of approximate or numerical solutions. Therefore, the method is recommended for analytically solving problems involving coupled temperature and moisture transport in porous materials.

References [1] M.D. Mikhailov, M.N. Ozisik, Uni®ed Analysis and Solutions of Heat and Mass Di€usion, Wiley, New York, 1984. [2] P.D.C. Lobo, M.D. Mikhailov, M.N. Ozisik, On the complex eigenvalues of Luikov system of equations, Drying Technol 5 (1987) 273±286. [3] J.Y. Liu, S. Cheng, Solution of Luikov equations of heat and mass transfer problems in porous bodies, I. J. Heat Mass Transfer 34 (1991) 1747±1754.

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[4] R.N. Pandey, S.K. Srivastava, M.D. Mikhailov, Solutions of Luikov equations of heat and mass transfer in capillary porous bodies through matrix calculus: a new approach, I. J. Heat Mass Transfer 42 (1999) 2649± 2660. [5] W.J. Chang, T.C. Chen, C.I. Weng, Transient hygrothermal stress in an in®nitely long annular cylinder: coupling of heat and moisture, J. Thermal Stress 14 (1991) 439±454. [6] J.W. Ribeiro, R.M. Cotta, M.D. Mikhailov, Integral transform solution of Luikov's equations for heat and mass transfer in capillary porous media, I. J. Heat Mass Transfer 36 (1993) 4467±4475. [7] S. Cheroto, S.M.S. Guigon, J.W. Ribeiro, R.M. Cotta, Lumped-di€erential formulations for drying in capillary porous media, Drying Technol 15 (1997) 811±835.

[8] A.V. Luikov, Heat and Mass Transfer in Capillary-porous Bodies, Pergamon Press, Oxford, 1966 (Chap. 6). [9] R. Li, J.F. Bond, W.J.M. Douglas, J.H. Vera, Heat of desorption of water from cellulosic materials at high temperature, Drying Technol 10 (1995) 999±1012. [10] S.J. Gregg, K.S.W. Sing, Adsorption, Surface Area and Porosity, 2nd ed., Academic Press, London, 1982. [11] T.C. Chen, C.I. Weng, Generalized coupled transient thermoelastic plane problems by Laplace transform ®nite element method, ASME J. Appl. Mech 59 (1990) 510±517. [12] T.C. Chen, C.I. Weng, W.J. Chang, Transient hygrothermal stresses induced in general plane problems by theory of coupled heat and moisture, ASME J. Appl. Mech 59 (1992) 10±16. [13] H.S. Caralaw, J.C. Jaege, Conduction of Heat in Solids, 2nd ed., Oxford University Press, London, 1959.