International Communications in Heat and Mass Transfer 33 (2006) 39–48

www.elsevier.com/locate/ichmt

An analytical method to calculate the coupled heat and moisture

transfer in building materialsB

Menghao Qin, Rafik Belarbi *, Abdelkarim Aıt-Mokhtar, Alain Seigneurin

LEPTAB, University of La Rochelle, La Rochelle, France

Available online 8 September 2005

Abstract

A dynamic model for evaluating the transient thermal and moisture transfer behavior in porous building materials was

presented. Both heat and moisture transfer were simultaneously considered and their interactions were modeled. An analytical

method has been proposed to calculate the coupled heat and moisture transfer process in building materials. The coupled system

was first subjected to Laplace transformation, and then the equations were solved by introducing the Transfer Function Method.

The transient temperature and moisture content distribution across the material can thus be easily obtained form the solutions. The

results were compared with the experimental data and other analytical solutions available in the literature; a good agreement was

obtained.

D 2005 Elsevier Ltd. All rights reserved.

Keywords: Heat and moisture transfer; Porous building materials; Analytical modelling; Transfer function method

1. Introduction

Moisture accumulation within the material of a building envelop can lead to poor thermal performance of the

envelope, degradation of organic materials, metal corrosion and structure deterioration. In addition to the building’s

construction damage, moisture migrating through building envelopes can also lead to poor interior air quality as high

ambient moisture levels result in microbial growth, which may seriously affect human health and be a cause of allergy

and respiratory symptoms. Therefore, the investigation of heat and moisture transfer in porous building materials is

important not only for the characterization of behavior in connection with durability, waterproofing and thermal

performance but also for avoiding health risk due to the growth of microorganisms.

Moisture problems in building materials are results of simultaneous heat and moisture transfer in building

envelopes and indoor air. Luikov [1] has firstly proposed a mathematical model for simultaneous heat and mass

(moisture) transfer in building porous materials. The conservation equations include the mass, the momentum and the

energy conservation equations. The constitutive equations are described by Darcy’s law, Fick’s law and Fourier’s law.

But the solutions are either numerical or complicated involving complex eigenvalues.

Generally, the coupled system for temperature and moisture potential can be handled by both analytical and

numerical approaches, depending on the specific problem considered. Mikhailov and Ozisik [2] have got the

0735-1933/$ - s

doi:10.1016/j.ich

B Communica

* Correspondi

E-mail addr

ee front matter D 2005 Elsevier Ltd. All rights reserved.

eatmasstransfer.2005.08.001

ted by J. Taine and A. Soufiani.

ng author.

ess: rafik.belarbi@univ-lr.fr (R. Belarbi).

M. Qin et al. / International Communications in Heat and Mass Transfer 33 (2006) 39–4840

analytical solutions for linear problems, based on the classical integral transform approach. However, it was later

discovered by the same authors [3] that the existence of complex eigenvalues, not accounted for in the solutions

previously reported, could significantly alter the temperature and moisture distributions. Recently, they again found

another complex eignvalue [4]. 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. Cheroto et al. [6]

presented a modified lumped system analysis method to yield approximate solutions, avoiding difficulties experi-

enced by Mikhailov et al., but sacrificing accuracy. Recently, Fudym et al. [7] proposed some analytical solutions of

the one-dimensional transient coupled heat and mass transfer linear equations in a semi-infinite medium, based on an

extension of the thermal quadrupole formalism.

In this paper, the Transfer Function Method (TFM) is used to solve the coupled heat and moisture equations in

building material. The coupled system is first subjected to Laplace transformation. Thus the coupled partial

differential equations are reduced to the coupled Ordinary Differential Equations (ODE). Then, the TFM is applied

to solve the fourth-order ODE. The resolution process is purely analytical and does not involve the calculation of

complex eigenvalues. Finally, the analytical results have been compared with experimental results obtained from

literature in order to highlight the pertinence of our approach.

2. Problem formulation

2.1. Governing equations

A typical heat and mass transfer problem in porous building materials can be governed by Luikov’s equations [1].

The phase-change occurring within the slab acts as a heat source or sink, which results in the coupled relationship

between mass transfer and heat transfer. In a coupled problem, the heat of adsorption or desorption is generally one of

the sources or sinks as well. In the present study, one-dimensional governing equations with coupled temperature and

moisture for a porous wall are considered, and the effect of the adsorption or desorption heat is added. Material

properties and pressure are considered to be constant throughout the material. A local thermodynamic equilibrium

between the fluid and the porous matrix is assumed, and the equations are as follows:

qCp

BT

Bt¼ k

B2T

Bx2þ qCm ehlv þ cð Þ Bm

Btð1Þ

qCm

Bm

Bt¼ Dm

B2m

Bx2þ Dmd

B2T

Bx2: ð2Þ

Eq. (1) expresses the balance of thermal energy within the body; the last term in this equation represents the heat

sources or heat sinks due to liquid-to-vapor phase-change and to the adsorption or desorption process. Eq. (2)

expresses the balance of moisture within the medium; the last term in this equation represents the moisture source or

moisture sink related to the temperature gradient. All the material properties mentioned above are effective properties.

The relationship between m and moisture content w is as follows: m =w/Cm.

2.2. Boundary conditions

Let us consider the classical case of a porous building wall, with thickness l, as shown in Fig. 1. At the boundaries

of building materials, the latent heat of vaporization becomes part of the energy balance, and the mass diffusion

caused by the temperature and moisture gradients affects the mass balance [8]. Therefore, the boundary conditions are

given as follows:

kBT x;tð Þ

Bx

����x¼0

¼ a1 T 0; tð Þ � T1½ � þ 1� eð Þhlvb1 m 0;tð Þ � m1½ � ð3Þ

� kBT x;tð Þ

Bx

����x¼l

¼ a2 T l;tð Þ � T2½ � þ 1� eð Þhlvb2 m l;tð Þ � m2ð Þ½ � ð4Þ

Fig. 1. The schematic view of the studied configuration.

M. Qin et al. / International Communications in Heat and Mass Transfer 33 (2006) 39–48 41

Dm

Bm x;tð ÞBx

����x¼0

þ kmdBT x;tð Þ

Bx

����x¼0

¼ b1 m 0;tð Þ � m1½ � ð5Þ

� Dm

Bm x;tð ÞBx

����x¼l

� kmdBT x;tð Þ

Bx

����x¼l

¼ b2 m l;tð Þ � m2�½ ð6Þ

Eqs. (3) and (4) express the heat flux in terms of convection heat transfer and the phase-change energy transfer.

Eqs. (5) and (6) 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 flux under the influence of a temperature gradient and a moisture gradient,

respectively. The terms to the right side of the equal sign describe the amount of moisture drawn off from or into the

surfaces. The initial temperature and moisture values in building material are defined:

T x;0ð Þ ¼ Tb ð7Þ

m x;0ð Þ ¼ mb ð8Þ

2.3. The dimensionless representation

By introducing dimensionless quantities to Eqs. (1)–(8), we obtain the system in the dimensionless form as:

B2h1 X ;sð ÞBX 2

� Bh1 X ;sð ÞBs

þ eKoBh2 X ;sð Þ

Bs¼ 0 0bXb1; sN0ð Þ ð9Þ

LuB2h2 X ;sð ÞBX 2

� Bh2 X ;sð ÞBs

þ LuPnB2h1 X ;sð ÞBX 2

¼ 0 0bXb1; sN0ð Þ: ð10Þ

And the boundary conditions and initial conditions are given by:

Bh1 0;sð ÞBX

� Biq h1 0;sð Þ þM1ð Þ � 1� eð Þd Kod Lud Bim h2 0;sð Þ þ N1ð Þ ¼ 0 ð11Þ

� Bh1 1;sð ÞBX

� Biq h1 1;sð Þ þM2ð Þ � 1� eð Þd Kod Lud Bim h2 1;sð Þ þ N2ð Þ ¼ 0 ð12Þ

Bh2 0;sð ÞBX

þ PnBh1 0;sð Þ

BX� Bim h2 0;sð Þ þ N1½ � ¼ 0 ð13Þ

� Bh2 1;sð ÞBX

� PnBh1 1;sð Þ

BX� Bim h2 1;sð Þ þ N2½ � ¼ 0 ð14Þ

h1 X ;0ð Þ ¼ 0; h2 X ;0ð Þ ¼ 0; ð15Þ

M. Qin et al. / International Communications in Heat and Mass Transfer 33 (2006) 39–4842

Where various dimensionless groups are defined as

h1 X ;sð Þ ¼ T x;tð Þ�TbTb�Tref

Dimensionless temperature h2 X ;sð Þ ¼ m x;tð Þ�mb

mb�mrefDimensionless moisture

s ¼ Dtl2; D ¼ k

qCp

��Dimensionless time X ¼ x

lDimensionless coordinate

Pn ¼ d Tb�Trefmb�mref

Possnov number Lu ¼ Dme

D; Dme ¼ Dm

qCm

��Luikov number

Ko ¼ Cm

Cphlv

mb�mref

Tb�TrefKossovitch number Biq ¼ al

k; Bim ¼ bl

DmDimensionless heat and

mass transfer coefficient

M1 ¼ Tb�T1Tb�Tref

; M2 ¼ Tb�T2Tb�Tref

; N1 ¼ mb�m1

mb�mref; N2 ¼ mb�m2

mb�mrefFe ¼ edPndKo ¼ eddd Cm

Cpdhlv

¼ 0:3d2d 0:10:92 d2450 ¼ 159:78

Dimensionless constants

*Tref and mref are the reference temperature and moisture potential, respectively.

3. Resolution

By applying Laplace transformation to Eqs. (9)–(15), the governing equations become:

d2Ph1

dX 2¼ s

Ph1 � seKo

Ph2 ð16Þ

Lud2Ph2

dX 2¼ s

Ph2 � sLuPnh1

P ð17Þ

and the boundary conditions become:

dPh1 0;sð ÞdX

� BiqPh1 0;sð Þ þ M1

s

� �� 1� eð Þd Kod Lud Bim

Ph2 0;sð Þ þ N1

s

� �¼ 0 ð18Þ

� dPh1 1;sð ÞdX

� BiqPh1 1;sð Þ þ M2

s

� �� 1� eð Þd Kod Lud Bim

Ph2 1;sð Þ þ N2

s

� �¼ 0 ð19Þ

dPh2 0;sð ÞdX

þ PndPh1 0;sð ÞdX

� BimPh2 0;sð Þ þ N1

s

� �¼ 0 ð20Þ

� dPh2 1;sð ÞdX

� PndPh1 1;sð ÞdX

� BimPh2 1;sð Þ þ N2

s

� �¼ 0 ð21Þ

where, s is the Laplace transformation parameter.

Then, the Transfer Function Method (TFM) is applied. Introducing transformation function u(x,s) such that:

Ph1 X ; sð Þ ¼ eKou X ; sð Þ ð22ÞPh2 X ; sð Þ ¼ u X ; sð Þ � 1

sdd2u X ; sð Þ

dX 2ð23Þ

Eq. (16) is automatically satisfied. And Eq. (17) becomes:

d4udx4

� s

Lu1þ Luð Þ d

2udx2

þ s2 1� eLuPnKoð ÞLu

u ¼ 0 ð24Þ

Thus, the coupled ODE (16) and (17) is reduced to a single fourth-order ODE (24). Assume the solution of u(x,s)

in the following form,

u X ; sð Þ ¼X4i¼1

fi sð Þe piX ð25Þ

M. Qin et al. / International Communications in Heat and Mass Transfer 33 (2006) 39–48 43

where, pi ¼ffiffiffiffiffiffisqi

p; qi ¼ aiffiffiffiffiffiffi

2Lup 1þ Luþ bi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Luð Þ2þ 4eLu2PnKo

q� �1=2; ai ¼

� 1 i¼ 1;21 i¼ 3;4

; bi ¼�1 i¼ 1;31 i¼ 2;4

��.

Obviously, q3=�q1, q4=�q2.

The coefficient ni(s) (i=1,2,3,4) is determined by using Eqs. (11)–(14). Results are written as the following form:

K½ � f sð Þf g ¼ Qf g

When h1(0, s)=0, h2(0, s)=0, h1(1, s)=0, h2(1, s)=0, matrix [K] equals to

K½ � ¼

eKo eKo eKo eKoeKoexp p1ð Þ eKoexp � p1ð Þ eKoexp p1ð Þ eKoexp � p1ð Þ

1� q21 1� q22 1� q21 1� q221� q21�

exp p1ð Þ 1� q22�

exp � p1ð Þ 1� q21�

exp p1ð Þ 1� q22�

exp � p1ð Þ

2664

3775

Qf g ¼ �M1

s;

�M2

s;

� N1

s;

� N2

s

� �TThe solution of Eq. (25) is determined by Cramer’s rule.

fk sð Þ ¼AkþBke

�qkffis

p

s eqkffis

p�e�qk

ffis

pð Þ ; k ¼ 1;2

� AjþBjeqjffis

p

s eqjffis

p�e

�qjffis

pð Þ ; k ¼ 3;4; j ¼ k � 2

8><>: ð26Þ

where:

A1 ¼�N2eKo�M2 1�q2

2ð ÞeKo q2

2�q2

1ð Þ ; A2 ¼�M2 1�q2

1ð ÞþN2eKo

eKo q22�q2

1ð Þ ;

B1 ¼�N1eKo�M1 1�q2

2ð ÞeKo q2

2�q2

1ð Þ ; B2 ¼�M1 1�Lq21ð ÞþN1eKo

eKo q22�q2

1ð Þ :

Then, we can get the solutions in Laplace domain, which are expressed as follows:

Ph1 X ; sð Þ ¼ eKo

X2i¼1

Aish qiXffiffis

pð Þ þ Bish qi X � 1ð Þ ffiffis

pð Þsd sh qi

ffiffis

pð Þ ð27Þ

Ph2 X ; sð Þ ¼

X2i¼1

1� q2i� Aish qiX

ffiffis

pð Þ þ Bish qi X � 1ð Þ ffiffis

pð Þsd sh qi

ffiffis

pð Þ : ð28Þ

Eqs. (27) and (28) are the temperature and moisture distribution in the transformation domain, respectively.

Applying the inversion theorem for the Laplace transformation, the results listed above can be transformed into time

domain. The results are listed as the following equations:

h1 X ;sð Þ ¼X4i¼1

X2j¼1

Qij

2qj2

ffiffiffiffisp

rexp �

n2ij

4s

!� nijerfc

nij

2ffiffiffis

p� �" #

ð29Þ

h2 X ;sð Þ ¼X4i¼j

X2i¼j

1� q2i� Qij

2qj2

ffiffiffiffisp

rexp �

n2ij

4s

!� nijerfc

nij

2ffiffiffis

p� �" #

ð30Þ

where nij ¼qi j�Xð Þ; i¼1; 2qi�2 j�1ð ÞþXð Þ; i¼3; 4

; j¼1;2ð Þ; Qij

�is represented by the following matrix:

A1 B1

A2 B2

� B1 � A1

� B2 A2

2664

3775.

Eqs. (29) and (30) give the temperature and moisture distribution inside the building material.

4. Discussion of resolution method

In order to compare the present method with previous analytical approaches, a test case was applied in the

following analysis. The physical problem involves a one-dimensional porous slab, initially at uniform temperature

Method [9]Method [3]Present method

ττ = 1.6

τ = 0.2

τ = 0.05

θ1 (

X,τ

)

X

1.40

1.00

1.20

0.80

0.60

0.40

0.20

0.00

-0.201.000.800.600.400.200.00

Fig. 2. Comparison of dimensionless temperature distributions with previous analytical solutions [3] and [9].

M. Qin et al. / International Communications in Heat and Mass Transfer 33 (2006) 39–4844

and uniform moisture content. One of the boundaries, which is impervious to moisture transfer, is in direct contact

with a heater. The other boundary is in contact with the dry surrounding air, thus resulting in a convective boundary

condition for both the temperature and the moisture content. For comparison purposes, the same geometry and

boundary conditions used in [9] is here implemented. The parameters assume the following numerical values:

Lu =0.4, Pn =0.6, e=0.2, Ko =5.0, Bim=Biq=2.5, and Q =0.9. The calculated results of dimensionless temperature

and moisture distributions at different dimensionless time, s =0.05, 0.2, 1.6 are presented in Figs. 2 and 3.

Method [3]Method [9]

X

Present method

0.00 0.20 0.40 0.60 0.80 1.00-0.20

0.00

0.20

0.40

0.60

0.80

1.20

1.00

1.40

τ = 1.6

τ = 0.2

τ = 0.05

θ 2 (

X,τ

)

Fig. 3. Comparison of dimensionless moisture distributions with previous analytical solutions [3] and [9].

INSULATION LAYERS

THERMOCOUPLES

SUPPORT

CONDITIONED AIR ' OUT '

ENVIRONMENTAL CHAMBER

HEAT FLUXMETER (THE COLD PLATE)

SUPPORTING MESH

CONDITIONED AIR ' IN '

Fig. 4. The schematic diagram of the experimental apparatus of Wijeysundera and Hawlader [10].

M. Qin et al. / International Communications in Heat and Mass Transfer 33 (2006) 39–48 45

Furthermore, the results from an integral transform solution method [9] and the results from [3], which includes one

pair of complex conjugate roots, are also plotted. To the current graph scale, it can be seen that the results from the

current approach and [9] agree well. The small discrepancies between the present method and [3] might be caused by

not including enough complex roots when using the eigenvalue method. More complex roots might be required for

the small values of s [9].

5. Experimental validation

In order to validate the achievement of the present new analytical method in predicting the coupled heat and

moisture transfer in building materials, it was checked against experimental results of Wijeysundera et al. [10,11]. For

this purpose, the temperature distribution, heat flux, total moisture gain and the liquid distribution are calculated. The

building material used in the experiment is fibrous insulation. One side of the slab (side 1) is exposed to an ambient

that is warm and humid, and the other side (side 2) is impermeable and subjected to a cold temperature. A schematic

diagram of the experimental arrangement is shown in Fig. 4. The cold plate is located on the top face of the test

section. A heat flow meter is sandwiched between the top surface of the insulation layers and the cold plate. The dry

insulation slab is made up of five layers. Copper–constantan thermocouples are located at the interfaces of insulation

layers [11].

Initially, the slab is assumed to be fully dry with a uniform temperature equal to the ambient temperature and vapor

density equal to the ambient density. It is suddenly subjected to a temperature drop at the cold side. The total test times

X

Present method

0.00

0.20

0.20 0.40

RH: 70%Exp.

(T-T

2)/(

T1-

T2)

0.40

0.60

0.80

1.00

RH: 98%Exp.

0.800.60 1.00

oT = 33 CT = 6.8 C2

1o

Present method

Fig. 5. Comparison of simulated temperature profiles of present method with experimental data after the quasi-steady state (Lu =2.03�10�3,

Biq=26.5, Bim=0.435, Fe =159.8).

Hea

t Flu

x (W

m-2

)

0.00

26

30

Time (h)400200100 300

Present method

600500 700

Exp.

34

38

42

46T1 = 40.0ºC, T2 = 20.4ºC, l = 76.1 mm,

α = 12 W m-2K-1RH = 96%, 1

Fig. 6. The comparison of computed heat flux with the experimental data (Lu =2.03�10�3, Biq=26.5, Bim=0.435, Fe =159.8).

M. Qin et al. / International Communications in Heat and Mass Transfer 33 (2006) 39–4846

were up to 600 h (25 days). The initial and physical parameters in the calculation are determined according to

experimental conditions. Furthermore, compared with the heat of phase-change, the heat of adsorption or desorption

is very small in building application, and is neglected in the following calculation. The experimental data, solutions

by present method and the other simulation results were shown in the following figures. The dimensionless variable

X =x/l is introduced.

Fig. 5 shows the dimensionless temperature distributions in the insulation slab after the quasi-steady state under

different relative humidity at the warm side. The calculated results of the present method fit the experimental data well.

Experimental and simulated heat flux at the cold side are given as a function of time in Fig. 6. The test conditions

are also listed in the figure. There is a good agreement between the computed and measured data. The fluctuation of

the experimental heat flux is probably due to the slight variations in the experimental conditions [10,12]. Fig. 7 shows

the comparison of computed moisture gain with the experimental data. The computed moisture gain agrees well with

the measured data.

The comparison of the computed and measured spatial distribution of the average liquid concentration is

shown in Fig. 8. In the experimental measurements, the values of liquid concentration were determined as an

Moi

stur

e G

ain

(kg

m-3

)

150

Present method

Time (h)0.00

50

100

100

200 300 400 500

Exp.

600 700

350

250

200

300α = 12 W m-2K-1RH = 96%, 1

T1 = 40.0ºC, T2 = 20.4ºC, l = 76.1 mm,

Fig. 7. The comparison of computed moisture gain with the experimental data (Lu =2.03�10�3, Biq=26.5, Bim=0.435, Fe =159.8).

0.040.030.00 0.020.01

x (m)

Ave

rage

Liq

uid

Con

tent

(kg

m-3

)

50

100

150

200

250

69.5H

48H

Present Method

24H

Time

300

0.05 0.06

Exp.

α = 12 W m-2K-1RH = 97%, 1

T1 = 40.5ºC, T2 = 20.8ºC, l = 62 mm,

Fig. 8. Comparison of computed average liquid content with the experimental data (Lu =2.03�10�3, Biq=21.59, Bim=0.354, Fe =159.8).

M. Qin et al. / International Communications in Heat and Mass Transfer 33 (2006) 39–48 47

average water gain for each part of the wall along the x-axis by gravimetric method. But, in current analytical

model, along the x-axis inside the wall, there are 30 spatial nodes. The moisture flux at each node was calculated

by integrating the moisture content at the interface over the time interval. The moisture accumulation can thus be

obtained. Close agreement between experimental and current calculated average liquid concentration is evident in

the figure. The moisture accumulation increases with depth and a sharp increase near the cold side. The liquid

concentration was expected to increase with x because the local temperature decreased as x increased, causing the

local relative humidity to be higher, which results in more liquid accumulation due to adsorption.

6. Conclusion

This paper proposes an analytical approach to calculate the coupled heat and moisture transfer in the porous

building materials, which consists of applying the Laplace transform technique and a Transfer Function Method to

solve the problem. The results have been compared with experimental data and the other exact solutions available in

the literature. A good agreement is obtained. Our method can serve to evaluate the accuracy of approximate or

numerical solutions. The main advantage of the suggested method in comparison with others available in the

literatures consists in the fact that it is easier to use and gives an accurate result.

Nomenclature

Cm Specific moisture (kg kg�1 8M �1)

Cp Specific heat (J kg�1 K�1)

Dm Conductivity coefficient of moisture content (kg m�1 s�1 8M �1)

h Latent heat (kJ kg�1)

hlv Heat of phase change (kJ kg�1)

k Thermal conductivity (W m�1 K�1)

l Thickness of the specimen (m)

m Moisture potential (8M)

mp Moisture flux due to phase change (kg m�3)

s Laplace transformation parameter

t Time (s)

T Temperature (K)

w Moisture content (kg kg�1)

M. Qin et al. / International Communications in Heat and Mass Transfer 33 (2006) 39–4848

Greek letters

a Convective heat transfer coefficient (W m�2 K�1)

b Convective moisture transfer coefficient (m s�1)

c Heat of adsorption or desorption (kJ kg�1)

e Ratio of vapor diffusion coefficient to total moisture diffusion coefficient

n Density (kg m�3)

d Thermogradient coefficient (8M K�1)

u Transformation function

Subscript

b Initial condition

References

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[2] M.D. Mikhailov, M.N. Ozisik, Unified Analysis and Solutions of Heat and Mass Diffusion, Wiley, New York, 1984.

[3] P.D.C. Lobo, M.D. Mikhailov, M.N. Ozisik, On the complex eigenvalues of Luikov system of equations, Dry. Technol. 5 (1987) 273–296.

[4] R.N. Pandey, S.K. Srivastava, Solutions of Luikov equations of heat and mass transfer in capillary porous bodies through matrix calculus: a

new approach, Int. J. Heat Mass Transfer 42 (1999) 2649–2660.

[5] W.J. Chang, T.C. Chen, C.I. Weng, Transient hygrothermal stress in an infinitely long annular cylinder: coupling of heat and moisture, J.

Therm. Stress 14 (1991) 439–454.

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[10] N.E. Wijeysundera, M.N.A. Hawlader, Effect of condensation and liquid transport on the thermal performance of fibrous insulation, Int. J.

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1865–1878.