Building and Environment 46 (2011) 1354e1360

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Building and Environment

journal homepage: www.elsevier.com/locate/buildenv

Contribution to analytical and numerical study of combined heat

and moisture transfers in porous building materials
K. Abahri*, R. Belarbi, A. Trabelsi
LEPTIAB, University of La Rochelle, Av. Michel Crépeau, 17042 cedex1 La Rochelle, France

a r t i c l e i n f o a b s t r a c t

Article history: In this paper, one-dimensional model for evaluating coupled heat and moisture transfer in porous
Received 20 September 2010 building materials was proposed. The transient partial differential equations system was solved
Received in revised form analytically for Dirichlet boundary conditions. It consists first to introduce the Laplace transformation
1 December 2010
and then to use the potential function technique. This approach allows simplifying the initial mathe-
Accepted 29 December 2010
matical problem to a fourth order ordinary differential equation which can be easily solved. This solution
was used to assess the transient temperature and moisture distribution across materials. A comparison
Keywords:
with numerical models from Luikov [2] and Vafai et al. [12] was performed, a good agreement was
Porous materials
Coupled heat and moisture transfer
obtained.
Partial differential equations Ó 2011 Elsevier Ltd. All rights reserved.
Analytical solution
Potential function

1. Introduction presented an exact solution of Luikov system for a moist spherical

capillary porous body. Their analytical procedure is a combination
The coupled heat and moisture transfer in porous media has of the matrix functions and the Laplace transformation. Chang et al.
been widely studied due to its presence in many fundamental and [9] proposed an analytical solution obtained for the slab under
industrial applications. Particularly in building science, researches natural hygrothermal boundary conditions. Their approach is based
have been carried out in the past few decades to improve building on the use of transformation functions. These authors [9] cited the
energy efficiency and indoor air quality [1]. work of Liu et al. [4] who introduced potential functions corre-
To describe hygrothermal transfer in capillary porous media, sponding to their system of equations, by a change of variables for
Luikov developed a model [2] assuming analogy between moisture temperature and moisture content. They used the boundary
migration and heat transfer. Moreover, he assumed that capillary condition of the third kind and compared their results with those of
transport is proportional to moisture and temperature gradients. Thomas et al. [10] who conducted experimental studies on wood
Also, by analogy with specific heat, he introduced the specific mass (pine). In the cited papers [5], there exist analytical difficulties to
capacity which is defined as the derivative of the water content find the solution when the domain of convergence of the series is
with respect to the mass potential. This model is applicable for both limited.
hygroscopic and non-hygroscopic materials. It was used by several This paper presents a Potential Transfer Function Method applied
researchers [3e7]. to partial differential equations. It concerns one-dimensional heat
In order to solve the coupled system for temperature and and moisture transfer through a plane geometry of porous building
moisture potentials, many authors used both analytical and component. Non-symmetric Dirichlet boundary conditions are
numerical approaches. Generally, the solution of the governing considered. These conditions are more suitable for applications with
partial differential equations depends on the specific problem high value of the dimensionless Biot number (typically, for cases
considered. Qin et al. [8] presented one of these specific cases where the average temperature and relative humidity at the surface
addressed to a multi-layer building material and they obtained of the building component converge very quickly to the ambient
satisfactory numerical and experimental results. Ribeiro [5] temperature and relative humidity as confirmed by Younsi [11]). The
inverse Laplace Transformation was exactly obtained. The numerical
solution was also performed using the same boundary conditions.
* Corresponding author.
Moreover, comparison with other numerical model given by Luikov
E-mail address: kamilia.abahri@univ-lr.fr (K. Abahri). [2] and that of Vafai et al. [12] was achieved.

0360-1323/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.buildenv.2010.12.020
 K. Abahri et al. / Building and Environment 46 (2011) 1354e1360 1355

been introduced by Luikov [2] and recently developed by Qin et al.

[6]. It is defined as a linear function of the moisture content w. It is
expressed by:

w ¼ Cm m
The dimension of specific mass capacity Cm is kg kg1 M,
where M denotes a mass transfer degree. This relation explains
that, in thermodynamic equilibrium condition (i.e. in hygrothermal
equilibrium), there is a definite distribution of moisture in a porous
media. With an increase in the total mass of moisture, its content in
the separate parts of a body increases. As expected, the moisture
transfer takes place from the higher potential toward the lower one.
This moisture transfer inside pores can be attributed to the
mechanism of liquid transfer under consideration: evaporation of
liquid takes place at one meniscus of the pore and vapor consid-
eration at the opposite one. It stands to reason that the quantity of
liquid evaporating from one meniscus must be equal to the quantity
of vapor condensation on the opposite meniscus.
To facilitate the writing and to appear easily, the Laplace
transformation of the following groups were introduced from Eq.
(1) and (2):

k
L ¼ (3)
rCp

kDm
D ¼ (4)
rCm ½k þ Dm dð3hlv þ gÞ

Cm ð3hlv þ gÞ
v ¼ (5)
Fig. 1. Schematic representation of the geometry. Cp

2. Analytical modeling of the problem Cp Dm d

l ¼ (6)
Cm ½k þ Dm dð3hlv þ gÞ
The used heat and mass transfer model is based on Luikov’s [2]
approach which is extensively employed in the prediction of heat Thus the following simplified equations are obtained. These
and moisture migration in porous building materials. However, as equations are similar to those obtained by Chang and Weng [9]:
usually found in many previous studies, the following assumptions
are still adopted in this study: v2 T vT vm
L 2 ¼ v (7)
vx vt vt
- The material is considered homogenous and the thermo-
physical properties are assumed constant.
- A local thermodynamic equilibrium between the fluid and the
porous matrix is assumed. v2 m vm vT
D 2 ¼ l (8)
- The initial moisture content and temperature repartition in the vx vt vt
wall is uniform.

Energy and mass conservation equations can be expressed by [6]

Table 1
and [4]:
Experimental drying conditions and product properties.

vT v2 T vm Property Value
rCp ¼ k 2 þ rCm ð3hlv þ gÞ (1)
vt vx vt Cm(Kg/Kg M) 0.01
Cp(J/(KgK) 2500
d( M/K) 2
vm v2 m v2 T k(W/mK) 0.65
rCm ¼ Dm 2 þ dDm 2 (2)
vt vx vx r(Kg/m3) 370
Dm(Kg/ms M) 2.2  108
where, T is the temperature, m moisture potential, t time, Cp and Cm Tb( C) 10
are heat and moisture capacities of the medium, k and Dm are Ti( C) 60
thermal and moisture diffusion coefficients, respectively, r is the T0( C) 110
l(m) 0.024
dry body density, 3 the ratio of vapor diffusion coefficient to coef-
hltv(J/Kg) 2.5  109
ficient of total moisture diffusion, g the heat of absorption or mb( M) 86
desorption, d the thermogradient coefficient and hlv the heat of mi( M) 45
phase change. m0( M) 4
The moisture potential concept, based on thermodynamic simi- g 0
3 0.3
larities between heat and mass transfer in hygroscopic materials, has
1356 K. Abahri et al. / Building and Environment 46 (2011) 1354e1360

Table 2 linear differential equation and solved using a rigorous matrix

Convergence of the series, for initial moisture content and temperature, function of algebra technique. The solution of Eq. (15) can be written in the
the number of term n.
following form:
n 40 60 95 140 300 400 550 600
Tb( C) 10.40 10.26 10.18 10.108 10.04 10.03 10.01 10.01 X
4
mb( M) 85.05 85.38 85.56 85.80 85.90 85.94 85.03 85.98 4ðx; sÞ ¼ xi ðsÞepi x (16)
i¼1

In which:
pffiffi
2.1. Boundary conditions pi ¼ sqi

The physical system is modeled as a porous building wall with 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2


impermeable and adiabatic horizontal boundaries. The vertical ai 4 D D 2 4lvD5
boundaries are kept at different uniform temperatures and mois-
qi ¼ pffiffiffiffiffiffiffi 1 þ þ bi 1 þ
2D L L L
ture: (Ti, mi) for x ¼ 0 and (T0, m0) for x ¼ l, where l is the thickness
of the wall, as shown in Fig. 1 At the initial time, temperature T(x, 0)  
and moisture m(x, 0) are at initial values Tb and mb, respectively. 1; i ¼ 1; 2 1; i ¼ 1; 3
ai ¼ bi ¼
1; i ¼ 3; 4 1; i ¼ 2; 4
2.2. Method of solution
q3 ¼ q1 and q4 ¼ q2
The coupled system is first subjected to a symmetric Laplace The coefficient xi ðsÞ (i ¼ 1,2,3,4) can be determined by using Eqs.
transformation. The coupled partial differential equations modeled (11) and (12). The results are written as the following homogeneous
by Eqs. (7) and (8) are thus reduced to two ordinary differential algebraic system:
equation as:
½KfxðsÞg ¼ fQ g
d2 T
L 2 ¼ sT  T b  vsm þ vmb (9) Where:
dx

2 3
v v v v
6 v
L
v
L
v
L
v
L 7
6 Lexpðp1 lÞ Lexpðp2 lÞ Lexpðp1 lÞ Lexpðp2 lÞ 7
6 1 1 1 1 7
K ¼ 6 2 2 2 2 7
6
4 L  q1 L  q2 L  q1 L  q2 7
5

1  q2 expðp lÞ
1 2
 1 2
 1 2

L 1 1 L  q2 expðp2 lÞ L  q1 expðp1 lÞ L  q2 expðp2 lÞ

d2 m 2 3
D 2 ¼ sm  mb  lsT þ lTb (10) Ti Tb
dx s
6 T0 T 7
Where TðsÞ and mðsÞ are the Laplace transformation of T(t) and 6 s b 7
Q ¼ 6 mi m 7
m(t), respectively; and s is Laplace transformation parameter. 4 s b 5
m0 mb
Likewise, time invariant boundary conditions during a process s
have similar symmetric transformations.For x ¼ 0:
By using the elimination of Gauss method, the solution of Eq. (16) is
T m given by:
Tð0; sÞ ¼ i ; mð0; sÞ ¼ i (11)
s s
For x ¼ l:

T0 m0
Tðl; sÞ ¼ ; mðl; sÞ ¼ (12)
s s
Then, the temperature and moisture in the transformed domain are
expressed in terms of a transformation function. Thus, by intro-
ducing a potential function 4(x, s) such that:

v T
Tðx; sÞ ¼ 4ðx; sÞ þ b (13)
L s

1 1 v2 4ðx; sÞ mb
mðx; sÞ ¼ 4ðx; sÞ  þ (14)
L s vx2 s
Eq. (9) is automatically satisfied and Eq. (10) becomes:


v4 4ðx; sÞ s D v2 4ðx; sÞ s2
 1 þ þ ð1  lvÞ4ðx; sÞ ¼ 0 (15)
vx4 D L vx2 DL
The coupled system (9) and (10) is converted to the fourth order Fig. 2. Change in temperature with time at the center of the wall (x ¼ 0.012).
 K. Abahri et al. / Building and Environment 46 (2011) 1354e1360 1357

By using the same technique, for the other similar terms and by
substituting these transformations in Eqs. (18) and (19), we obtain
the exact solution of temperature and moisture potential profiles
given by Eqs. (21) and (22):

"  n  n2 p2 t

vX 2
x 2A X N
1 ðq lÞ2 n px bi ðx  lÞ
Tðx; tÞ ¼ Ai þ i e i sin þ
L i¼1 l p n¼1 n l l
 
 #
2B X N
1n n2 p2 t=ðqi lÞ2 n pðx  lÞ
þ i e sin þ Tb (21)
p n¼1 n l

2

X "  n  n2 p2 t
1 x 2A XN
1 ðq lÞ2 n px
mðx; tÞ ¼  q2i Ai þ i e i sin
i¼1
L l p n¼1 n l


Fig. 3. Change in moisture content with time at the center of the wall (x ¼ 0.012). Bi ðx  lÞ
þ
l
 
#
2Bi X 1n n2 p2 t=ðqi lÞ2
N
n pðx  lÞ
8 þ e sin þ mb
>
> ffi p n¼1 n l
>
>
p
eqk l ps ffi
>
<
Ak þBpffi k k ¼ 1; 2
sðeqk l s eqk l s Þ (22)
xi ðsÞ ¼ p ffi
>
> 
Aj þBj eqk l s
ffi ffi k ¼ 3; 4; j ¼ k  2
>
>
p p

: ð Þ
qk l s
> e
qk l s
s e
2.3. Convergence of solutions

So that the coefficient Ai can be expressed that: In order to validate the applicability of the present analytical
  method for heat and moisture migration prediction in building
ðm0  mb Þv  1  Lq22 ðT0  Tb Þ materials, a physical application on a wood slab building envelope
A1 ¼   has been presented using the material properties given from [9]
v q22  q21
which are associated to the boundary conditions cited in Table 1.
This study focuses on the convergence of analytical solutions for
 
ðm0  mb Þv þ 1  Lq21 ðT0  Tb Þ the temperature and moisture content. For this, the initial condi-
A2 ¼  2  tions at the specimen center are checked, first by considering real
v q2  q21
values only and then by increasing the number of terms. The
  results obtained from different n real roots for Tb and mb are shown
ðmi  mb Þv þ 1  Lq22 ðTi  Tb Þ in Table 2.
B1 ¼  
v q22  q21 The input data are Tb ¼ 10  C and mb ¼ 86 M. For n < 140, the
initial conditions can be roughly satisfied (see Table 2). Then, for
  values of n greater than 140, the same input values Tb and mb are
ðmi  mb Þv  1  Lq21 ðTi  Tb Þ
B2 ¼  2  approximately the same.
v q2  q21

With these solutions, we can go back up toward 4(x, s) which is:

4ðx; sÞ ¼ x1 eP1 x þ x2 eP2 x þ x3 eP3 x þ x4 eP4 x (17)

Substituting Eq. (17) into (13) and (14), the steady part leads to the
following solutions:
 pffiffi  pffiffi
vX 2
Ai sh qi x s þ Bi sh qi ðx  lÞ s T
Tðx; sÞ ¼  pffiffi þ b (18)
L i¼1 s sh qi l s s

2
  pffiffi  pffiffi
X 1 Ai sh qi x s þBi sh qi ðxlÞ s mb
mðx;sÞ ¼ q2i  pffiffi þ (19)
i¼1
L s sh qi l s s

The inverse Laplace transformation is directly found by using the

complex variable theory methods given by Spiegel [13]. Eqs. (18)
and (19) are consists of 4 similar terms which takes the global
pffiffi pffiffi
formshðx sÞ=s shðl sÞ. From [13] their inverse Laplace trans-
formation it can be written in the following form:

 pffiffi   n  n2 p2 t
sh x s x 2X N
1 ðq lÞ2 n px
L1  pffiffi ¼ þ e i sin (20) Fig. 4. Comparison of temperature value in summer conditions for coupled and non-
s sh l s l p n¼1 n l
coupled heat and moisture transfer (x ¼ 0.025 m).
1358 K. Abahri et al. / Building and Environment 46 (2011) 1354e1360

Fig. 5. Comparison of moisture content value in summer conditions for coupled and Fig. 7. Comparison of moisture content value in winter conditions for coupled and
non-coupled heat and moisture transfer (x ¼ 0.025 m). non-coupled heat and moisture transfer (x ¼ 0.025 m).

Figs. 2 and 3 show variations with time of temperature and are the most examples targeted by this study. Specially, it concerns
moisture ratio, respectively, at the center of the building component. the environmental material of construction as wood.
Fig. 2 indicates a large decrease in the temperature for short times as The purpose of the following development is to improve the
a result of moisture vaporization. For longer times, the temperature impact of the coupling heat and moisture on the wall building
in the region became positive when vaporization effects were behavior of 0.05 m thickness subjected to summer and wintry
weaker and the temperature distribution is governed by diffusion meteorological weather data.
[14]. The maximum temperature at the center was 83.66  C which To realize this study, the last physical properties of wood given
stabilized from 480 s, while moisture potential stabilized at 6.104 s. by [9] are used and the meteorological data of La Rochelle are
Thus, the stationary state was reached more quickly for temperature introduced too. The initial boundary conditions of temperature and
than for moisture. This result can be illustrated by the intrinsic moisture content are: Tb ¼ 10  C and mb ¼ 50 M.
properties of the material used; more precisely the coefficient of The same configuration (Fig. 1) used in the first study is
thermal diffusivity is bigger than the moisture diffusion coefficient. undertaken where mi and Ti takes respectively the following values:
12.5 M and 25  C. Outdoor ambient relative humidity and
3. Physical application for building materials temperature values (m0, T0) are defined by the meteorological data
for one year period. The transition from relative humidity to the
The present analytical solution serves to predict exactly the moisture content is obtained from the sorption isotherm of wood
coupled heat and moisture transfer phenomena in porous building described by Hameury [15].
component. As application for this aspect, the building materials The simulation results are illustrated by Figs. 4 to 7. Comparison
of time temperature dependence in the two following cases was
achieved. The first one assumes a non-coupled heat and moisture

Fig. 6. Comparison of temperature value in winter conditions for coupled and non-
coupled heat and moisture transfer (x ¼ 0.025 m). Fig. 8. Comparison of analytical temperature distributions with numerical resolution.
 K. Abahri et al. / Building and Environment 46 (2011) 1354e1360 1359

512 respectively. The detailed application of the FEM for solving

heat transfer problems is done in [20].
The numerical code is successfully validated by comparing the
present solutions with the analytical solution. The predicted
temperature and moisture distributions, at different times, inside
the porous slab are compared with numerical simulation and the
results are presented in Figs. 8 and 9, respectively. The migration of
capillary water is significant in the neighborhood of the cold
surface. The moisture potential profile decreases with time reach-
ing a low value and a uniform distribution at the end. While, the
temperature at the center of the slab increases along the time
compared to its initial condition. The maximum differences
between the simulated temperature or moisture and the numerical
temperature or moisture are 0.2  C and 0.1 M respectively. It can be
seen that the numerical model shows a good agreement with the
analytical model. As expected, the moisture of the slab decreases
with increase in time, until it reaches an equilibrium value of
Fig. 9. Comparison of analytical moisture potential distributions with numerical 24.51 M after 16 h. In the other case, temperature increases in time
resolution. to stabilize at 83.66  C after a short time as was indicated before.
Another test problem for validation of numerical models
consists in comparing the numerical data obtained by Wijey-
transfer phenomena where the second takes into account the sundera and Zheng [21] and analytical solution that we have
coupled terms in the mass and energy balances equations. Simu- obtained. Wijeysundera and Zheng used a model with a different
lation results are given in summer and winter conditions in the mathematical formulation than that used in this study. This study
center of the building component x ¼ 0.025 m (Figs. 4 and 5). These uses a different model of heat and moisture transfer based on Vafai
figures indicate that the coupling term in energy balance equation et al. equations [12]. The aim of this following study is to compare
has no effect on the temperature values because the involved mass our analytical resolution with the numerical study of Wijeysundera
transfer quantity and moisture storage capacity of the material that and Zheng. The building material characteristics and their
appears in the coupling term of Eq. (1) are very low. However, the boundary conditions are calculated by [21].
moisture content profiles shown in Figs. 6 and 7 save the same The moisture and temperature distribution are shown in Figs. 10
trends with significant difference. In fact, coupling terms, due to the and 11 at different time levels. These results are compared to those
thermal diffusion, in Eq. (2) are not negligible compared to the mass given by [21], under the same conditions, a good agreement is shown.
transfer terms. This means the moisture transfer in the building The vapor content migration shown in Fig. 10 reaches the steady
construction has a significant influence on the latent cooling load. state for the same period used by the heat transfer. But for the same
The sensible cooling load is not significantly affected. The similar time temperature drop occurs slowly compared to the vapor density.
results can also be found in [16e18]. In this case, the Lewis number described by the ratio of thermal
diffusion to moisture diffusion is close to one and the thermal and
4. Numerical simulation moisture transfer kinetics are of the same range. It can be concluded
that the boundary conditions and the physical properties of materials
In reality, the input parameters of models of heat and mass affect directly the magnitude of heat and moisture transfer.
transfer are variable. In this case, the problem becomes complex The maximum difference between numerical results of the two
and the existence of analytical solutions is usually compromised. models is less than 10% for temperature distribution and less than
Hence, the recourse to the numerical simulation is generally rec- 5% for moisture content.
ommended. In this section, the numerical solution in the case of The small difference between the results given by the both
constant parameters is performed and validated by comparison models is due to their assumptions. In fact, the present analytical
with the proposed analytical solution.
The numerical resolution of the coupled partial differential
equations is undertaken using COMSOL Multiphysics code [19]. It is
a powerful environment for modeling and solving a variety of
research and engineering problems based on the Finite Element
(FE) method. Moreover, it is particularly adapted for the treatment
of Multiphysical problems where several phenomena are simulta-
neously studied. Thus, the assessment of the incidence of heat and
moisture transfer on other phenomena such as the ingress of
aggressive agents (chlorides, carbonation) and the ingress of
pollutants (Volatile Organic Compounds) becomes possible. In
addition to that, different kind of meshes are easily generated
which is more convenient for 3D studies.
COMSOL PDE solving presents great assortment of FE algo-
rithms: (UMFPACK, GMRES, Multi-grid.) [19]. In the present study
it concerns the UMFPACK solver. It solves general systems of the
form Ax ¼ b using the nonsymmetric-pattern multifrontal method
and direct LU factorization of the spares matrix A Solution is
established by applying the Quadratic-Lagrange method. The Fig. 10. Comparison between analytical and numerical Wijeysundera data given for
degrees of freedom and number of elements are about 2050 and moisture content.
1360 K. Abahri et al. / Building and Environment 46 (2011) 1354e1360

m Moisture potential ( M)
mp Moisture flux due to phase change (kg m3)
RH Relative humidity (kgwater kg1
air )
S Laplace transformation parameter
T Time (s)
T Temperature (K)
w Moisture content (kg kg1)

Greek symbols
a Convective heat transfer coefficient (W m2 K1)
b Convective moisture transfer coefficient (m s1)
g Heat of absorption or desorption (kJ kg1)
3 Ratio of vapor diffusion coefficient to total moisture
diffusion coefficient
r Material density (Kg/m3)
d Thermogradient coefficient ( M K1)
4 Transformation function
Fig. 11. Comparison between analytical and numerical Wijeysundera data given for
temperature.
Subscripts
model is more accurate in the hygroscopic domain where the B Initial condition
Wijeysundera and Zheng [21] model considers capillarity region. So i, o Reference condition
the last model is more precised near the saturation domain.
References
5. Conclusion
[1] ANNEX 41: Subtask 2 Experimental analysis of moisture buffering. IEA 2007.
This paper presents analytical solutions for the coupled heat and [2] Luikov AV. Heat and mass transfer in capillary-porous bodies. Oxford: Per-
gamon Press; 1966. chap. 6.
moisture transfer in porous building materials under some
[3] Thomas HR, Morgan K, Lewis RW. Afully monolinear analysis of heat and mass
reasonable assumptions based on local thermodynamic equilib- transfer problems in porous media. International Journal of Numerical
rium between the fluid and the porous matrix. One-dimensional Methods in Engineering 1980;15:1381e93.
partial differential equations are utilized to describe the heat and [4] Liu JY, Cheng S. Solution of Luikov equations of heat and mass transfer in capillary-
porous bodies. International Journal of Heat and Mass Transfer 1991;34:1747e54.
moisture migration processes within a porous slab. In the first step, [5] Ribeiro JW. Complete and satisfactory solutions of Luikov equations of heat
an analytical approach is developed by using combination of Lap- and moisture transport in a spherical capillary-porous body. International
lace transformation functions and potential function according to Communications in Heat and Mass Transfer 2000;27:975e84.
[6] Qin M, Belarbi R. Development of an analytical method for simultaneous heat
Dirichlet boundary conditions. and moisture transfer in building materials utilising transfer function method.
Several numerical simulations using the finite element method Journal of Materials in Civil Engineering ASCE 2005;17(5):492e7.
are conducted to compare the results obtained analytically. A good [7] Belarbi R, Qin M, Aït-Mokhtar A, Nilsson LO. Experimental and theoretical
investigation of nonisothermal transfer in hygroscopic building materials.
agreement is shown. Building and Environment 2008;43:2154e62.
As application, the effect of heat and moisture coupling on [8] Qin M, Belarbi R, Ait-Mokhtar A, Nilson L. Coupled heat and moisture transfer
environmental material building subjected to summer and winter in multi-layer building materials. Construction and Building Materials
2009;23:967e75.
weather conditions are studied. The simulation results show that [9] Chang W, Weng C. An analytical solution to coupled heat and moisture
the thermal diffusion affects strongly the moisture migration in the diffusion transfer in porous materials. International Journal of Heat and Mass
building envelope (wall) by the contribution of the coupled terms Transfer 2000;43:3621e32.
[10] Thomas HR, Lewis RW, Morgan K. An application of the finite element method
in partial differential equations. But the contribution of the mois-
to the drying of timber. Wood Fiber 1980;11(4):237e43.
ture terms in energy balance equation has few incidences on [11] Younsi R, Kocaefe D, Kocaefe Y. Three-dimensional simulation of heat and
temperatures values. moisture transfer in wood. Applied Thermal Engineering 2006;26:1274e85.
As well, other published numerical data of Wijeysundera and [12] Vafai K, Whitaker S. Simultaneous heat and mass transfer accompanied by
phase change in porous insulation. Journal of Heat Transfer-Transactions of
Zheng [21] compare reasonably well with our analytical solutions. the ASME 1986;108:132e40.
These comparisons are carried out to find a good correspondence [13] Spiegel M. Variables complexes (Cours et Problèmes). série Schaum. Mc Graw
between the distribution of temperature and moisture content Hill; 1973.
[14] Dantas LB, Orlande HRB, Cotta RM. An inverse problem of parameter esti-
inside the porous slab for both numerical models that are used for mation for heat and mass transfer in capillary porous media. International
the present work. Our analytical solution can be used to evaluate Journal of Heat and Mass Transfer 2003;46:1587e98.
the accuracy of approximate or other numerical solutions in [15] Hameury S. Moisture buffering capacity of heavy timber structures directly
exposed to an indoor climate: a numerical study. Building and Environment
building materials. 2005;40:1400e12.
[16] Künzel HM, Holm A, Zirkelbach D, Karagiozis AN. Simulation of indoor
Nomenclature temperature and humidity conditions including hygrothermal interactions
with the building envelope. Solar Energy 2005;78:554e61.
[17] Mendes N, Winkelmann FC, Lamberts R, Philippi PC. Moisture effects on
conduction loads. Energy and Building 2003;35(7).
Cm moisture storage capacity (kg kg1 M1) [18] Qin M, Belarbi R, Aït-Mokhtar A, Allard F. Simulation of coupled heat and
Cp Specific heat (J kg1 K1) moisture transfer in air-conditioned buildings. Automation in Construction
2009;18:624e31.
Dm Moisture diffusion coefficient content (kg m1 s1 M1) [19] Femlab AG. Comsol 3.3 multiphysics FEM software package. Göttin-gen; 2007.
H Latent heat (kJ kg1) [20] Minkowycs WJ, Sparrow EM, Murthy JY. Handbook of numerical heat transfer.
hlv Heat of phase change (kJ kg1) 2nd ed. USA: Wiley; 2006.
[21] Wijeysundera NE, Zheng BF. Numerical simulation of the transient moisture
K Thermal conductivity (W m1 K1) transfer through porous insulation. International Journal of Heat and Mass
L Thickness of the specimen (m) Transfer 1995;39:995e1003.