Applied Thermal Engineering 25 (2005) 61–71www.elsevier.com/locate/apthermeng

Study of heat and moisture migration propertiesin porous building materials

Z.Q. Chen *, M.H. Shi

Department of Power Engineering, Southeast University, No. 2 Si Pai Lou, Nanjing 210096, Jiangsu Province, PR China

Received 25 December 2003; accepted 6 May 2004

Available online 13 August 2004

Abstract

Based on the non-equilibrium thermodynamic theory, the thermal driving forces and the fluxes in heat

and moisture migration process for unsaturated porous building materials are analyzed. The mechanisms of

heat and moisture migration in unsaturated porous building materials are discussed and the pheno-

menological equations to describe the immigrating process in unsaturated porous building materials areestablished. By means of the diffusion law and the equation of state for ideal gas, the expressions of

coefficients in the phenomenological equations are deduced. The effects of temperature, water content or

partial vapour pressure on the phenomenological coefficients are also discussed.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Heat and moisture migration; Phenomenological coefficients; Non-equilibrium thermodynamics; Porous

building materials

1. Introduction

Problems involving heat and moisture migration in porous building materials arise in a numberof engineering interests, such as wall drying, the solar house designing, cooling load calculating ofair conditioning, etc. Affected by porous structure, temperature gradients, moisture gradients andenvironmental characteristic, heat and moisture transport in the porous building materials is quitecomplex. Richards [1] firstly established the equation of unsaturated flow in porous materials,which is on the basis of Darcy’s law and the principle of continuous motion. Philip and Vries [2]

* Corresponding author. Tel.: +86-25-8620-5393; fax: +86-25-5771-4489.

E-mail address: zqchen@seu.edu.cn (Z.Q. Chen).

1359-4311/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.applthermaleng.2004.05.001

62 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) 61–71

developed a moisture migration model at inhomogeneous temperature profiles, in which moisturemigration affected by temperature gradient was taken over. In the process of heat and moisturemigration, the gradients of temperature, moisture and pressure, the main driving forces of heatand moisture migration in unsaturated porous media, influence one another [3]. Due to thecomplicated structure of porous building materials, it is difficult to consider micro-phenomena. Inorder to describe the continuity of porosity and other parameters in unsaturated porous mediaapproximately, unsaturated models were developed by using the method of representative ele-mentary volume [4,5]. Because of water evaporation, phase-change is taking place in unsaturatedporous materials. To decide the transport properties, migration coefficients vs. temperature andvolume moisture content were studied by Shah et al. [6]. The steady-flux measurements ofmoisture diffusivity in unsaturated porous media were studied by Richards [7]. In a paper seriesWilson, Hall and coworkers [8–11] studied the water movement in porous building materials byuse of unsaturated flow theory and obtained some experimental verifications of capillaryabsorption of water for different building materials and structures. The different moisturetransport mechanisms and some interfacial phenomena in porous materials were investigated byFreitas et al. [12] using the theory of Luikov [3] and Philip and Vries [2]. Based on a one-dimensional steady model, the dynamic and diffusive behavior of a three-layer building envelopeelement was investigated by Taylor et al. [13]. More recently, real-time thermal and moistureparameters in buildings were studied theoretically and experimentally [14,15]. All these previousworks are very instructive, but the migration phenomenological coefficients are still not to bedecided owing to the complexity of heat and moisture migration processes in porous materials.This paper will present a method to predict the phenomenological coefficients with considerationof heat and moisture migration mechanisms, in which a non-equilibrium approach is employed toexplain combined heat transfer and water, including liquid and vapour, movement.

2. Linear phenomenological equations of heat and moisture migration

Heat and moisture migration in porous building materials is the process affected by theinteractions of temperature field, moisture field, and partial vapour pressure field. Generally, thecoupled heat and moisture migration can be described in linear non-equilibrium thermodynamictheory [16]. According to the Curie’s principle, the thermal driving forces and the fluxes in heatand mass transfer process are coupled. So the linear phenomenological equations of heat andmoisture migrations in porous building materials can be described as

~Jq ¼ � Lqq

T 2rT � LqL

Trh � Lqv

TrPV ð1Þ

~JL ¼ � LLq

T 2rT � LLL

Trh þ qLK~g ð2Þ

~JV ¼ � LVq

T 2rT � LVV

TrPV ð3Þ

where,~Jq,~JL and~JV are heat flux, liquid mass flux and vapour mass flux, respectively. T , h and PVare the temperature, water content and partial vapour pressure. qL and K are liquid water density

Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) 61–71 63

and unsaturated hydraulic conductivity. We assume, k ¼ LqqT 2 , apparent thermal conductivity;

k1P ¼ LqV

T , migration coefficient; kP ¼ LVVT , infiltration coefficient; k11

P ¼ LVqT 2 , thermal infiltration

coefficient; km ¼ LLLT , mass diffusivity; k1

m ¼ LqLT , mass thermal diffusivity; k11

m ¼ LLqT 2 , thermomass

diffusivity.So, Eqs. (1)–(3) can be written as

~Jq ¼ �krT � k1mrh � k1

PrPV ð4Þ

~JL ¼ �k11m rT � kmrh þ qLK~g ð5Þ

~JV ¼ �k11P rT � kPrPV ð6Þ

3. Analysis of thermodynamics flux

In order to describe the linear phenomenological coefficients, the mechanisms of heat, moisturemigration and thermodynamic flux are analyzed below.

3.1. Mechanism of heat migration

Heat transfer in unsaturated building materials under temperature gradient covers heat con-duction, infiltration convection heat transfer, radiation heat transfer and phase-change heattransfer. They are interactive. For there is no large temperature difference in unsaturated buildingmaterials, so radiation heat transfer is ignored.

According to the analysis above, the heat flux is consisted of heat conduction heat flux ~Jqd

and convection heat flux ~Jqc caused by infiltration fluid flow.

~Jq ¼~Jqd þ~Jqc ¼ �kerT þ ð~JLhL þ~JVhV þ~J ahaÞ ð7Þ

3.2. Mechanism of moisture migration

Moisture migration in unsaturated porous building materials includes diffusive migration andinfiltration fluid flow. The diffusive migration has the forms of molecular diffusion, Knudsendiffusion and surface diffusion.

3.2.1. Liquid water mass fluxLiquid flux in porous building materials consists of infiltration flow and surface diffusion

caused by the gradients of moisture and temperature. According to Darcy law [17], the infiltrationliquid flow ~JLC can be described as

~JLC ¼ �qLðDhLrh þ DTLrT � K~gÞ ð8Þ

where, DhL ¼ K o/

oh

�� ��, DTL ¼ K o/oT

�� ��, / is hydraulic potential.In non-isothermal conditions, surface diffusion fluid flow ~JLD caused by adsorption and

desorption is [18]

64 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) 61–71

~JLD ¼ �qLDTDrT ð9Þ

where, DTD ¼ C

qLTis the coefficient of adsorption-diffusion.

So, the liquid flux ~JL is

~JL ¼~JLC þ~JLD ¼ �qL½DhLrh þ ðDTL þ DTDÞrT � K~g� ð10Þ

3.2.2. Vapour fluxThe total vapour flux includes the vapour convection flux, general molecular diffusion and

Knudsen diffusion flux under the gradients of temperature and vapour partial pressure. Assumingthat the vapour convection velocity is ~V , so vapour flux ~JV is [19]

~JV ¼ qV~V � DerqV ð11Þ

where, De and qV are effective diffusion coefficient and vapour density, respectively.

3.2.3. Air fluxThe total air flux also includes air convection flux and diffusion flux. For the convection

velocity of air is the same as that of vapour [19], so the air flux is

~J a ¼ qa~V � Derqa ð12Þ

where, qV is air density.

4. Phenomenological coefficients

According to the analysis of thermodynamic flux and ideal gas state equation, the pheno-menological coefficients are deduced below.

4.1. Liquid water

The phenomenological coefficients include mass diffusivity km and thermomass diffusivity k11m .

Comparing Eqs. (5) and (10), they are respectively described as

km ¼ qLDhL ¼ qLKo/oh

�������� ð13Þ

k11m ¼ qLðDTL þ DTDÞ ¼ qL K

o/oT

��������

�þ DTD

�ð14Þ

4.2. Vapour

In the unsaturated porous building materials, comparing to vapour flux, air flux is very small[19], so the net transfer of air is zero

~J a ¼~0 ð15Þ

Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) 61–71 65

Using Eq. (15) to (12) the vapour convection velocity ~V is described as

~V ¼ Derqa=qa ð16Þ

Combined Eqs. (11) and (16), the vapour flux is

~JV ¼ DeqV

rqa

qa

��rqV

qV

�ð17Þ

Assuming that the air and vapour obey the ideal gas law in the unsaturated porous buildingmaterials, therefore the vapour flux is described as [19]

~JV ¼ � DePRVT ðP � PVÞ

rPV ð18Þ

Therefore, by using Eqs. (12) and (18), the phenomenological coefficients of vapour flux is de-scribed as

kP ¼ DePRVT ðP � PVÞ

ð19Þ

k11P ¼ 0 ð20Þ

where, De is the effective diffusion coefficient, RV is general gas constant, P and PV are total gaspressure and partial vapour pressure, respectively.

4.3. Heat

The enthalpies of liquid water, vapour and air in unsaturated porous building materials are

hL ¼ CLT ð21Þ

hV ¼ CPVT þ HC ð22Þ

ha ¼ CPaT ð23Þ

Combining Eq. (7) and Eqs. (21)–(23) the heat flux is

~Jq ¼ �ðke þ k11m CLT ÞrT � kmCLTrh � kPðHC þ CPVT ÞrPV þ qLCLTK~g ð24Þ

Compared Eqs. (4) and (24), the phenomenological coefficients of heat flux––apparent thermalconductivity k, migration coefficient k1

P and mass thermal diffusivity k1m are

k ¼ ke þ k11m CLT ð25Þ

k1P ¼ kPðCPVT þ HCÞ ð26Þ

k1m ¼ kmCLT ð27Þ

66 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) 61–71

5. Effects of temperature and moisture on the phenomenological coefficients

According to the analysis above, the phenomenological coefficients of heat and moisturemigrations are functions of temperature and moisture or partial vapour pressure. For sandybuilding block, the effects of temperature and moisture on the phenomenological coefficients areanalyzed below.

5.1. Thermodynamic and thermal physical parameters

5.1.1. Hydraulic potential / and conductivity KThe hydraulic potential and conductivity are functions of temperature and water content. They

can be described by Eq. (28) [20] and Eq. (29) respectively [23].

/ ¼ /S

he

� ��4

expð�cT Þ ð28Þ

K ¼ KS

/S

/

� �2:75

ð29Þ

where, e, /S, KS and c are porosity, saturated hydraulic potential, saturated hydraulic conductivityand surface extended coefficient, respectively. For sandy block with mean pore diameter of 0.3mm, e, /S, KS and c are 0.39, )0.0315 m, 1.76· 10�5 m/s and 2.189· 10�3 C�1 [20].

5.1.2. Partial vapour pressure PVIn the unsaturated porous building materials, vapour partial pressure obeys the following

thermodynamic relation [19]:

PV ¼ PS expg/RVT

� �ð30Þ

where, PS is the saturated vapour partial pressure and g is the acceleration to gravity.

5.1.3. Effective thermal conductivity ke

The effective thermal conductivity in unsaturated porous building materials can be describedapproximately as

ke ¼ kSð1� eÞ þ kLh þ kgðe � hÞ ð31Þ

where, kS, kL and kg are the thermal conductivity of solid, liquid and gas (vapour and air),respectively.

5.1.4. Effective diffusion coefficient De

The effective diffusion is the combination of general modular diffusion and Knudsen diffusion.So, its coefficient is

De ¼DatmDKn

Datm þ DKn

ð32Þ

Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) 61–71 67

General modular diffusion coefficient [21]

Datm ¼ 4:942 10�4eT 1:5=ðPf0Þ ð33Þ

Knudsen diffusion coefficient [22]

DKn ¼8e2

3f0Sg

2RVTpMV

� �0:5

ð34Þ

where s, S and M are tortuosity factor, specific area for BET and molecular weight of watervapour. For sandy block with mean particle diameter of 0.3 cm, the coefficient of adsorption-diffusion is very small and it can be neglected.

5.2. Effects of temperature and moisture on phenomenological coefficients

For sandy block with mean particle diameter of 0.3 mm and porosity of 0.39, the effectivecurves of temperature and moisture on phenomenological coefficients––apparent thermal con-ductivity k, mass thermal diffusivity k1

m, mass diffusivity km and thermomass diffusivity k11m are

showed in Figs. 1–4 respectively. As shown, the phenomenological coefficients increase with theincreasing of temperature and water content. But the mass thermal diffusivity k1

m, mass diffusivitykm and thermomass diffusivity k11

m are closed to zero when the water content is very small. So whenthe water content is below 0.1, the liquid water immigration, the effect of heat on liquid watermigration and effect of liquid water migration on heat transfer are very small. The relations ofmigration coefficient and infiltration coefficient with temperature and vapour partial pressure arein Figs. 5 and 6. As shown, the effects of temperature on migration coefficient and infiltrationcoefficient are larger than the effects of vapour partial pressure.

3.5

7.0

10.5

14.0

0.05 0.25 0.45

100

70

40

T=10

a

Water content θ

Appa

rent

ther

mal

con

duct

ivity

λ x

104 (k

W/m

°C)

Appa

rent

ther

mal

con

duct

ivity

λ x

104 (k

W/m

°C)

0

3.5

7.0

10.5

14.0

0 50 100 150

b0.39

0.3

θ=0.1

Temperature T (°C)

Fig. 1. Apparent thermal conductivity curve.

-1

0

1

2

3

4

0.05 0.25 0.45

a

100

70

40

T=10

Water content θ

Mas

s th

erm

al d

iffus

ivity

λ1 m

(kW

/m)

-1

0

1

2

3

4

0 50 100 150

b

0.39

0.3

θ=0.1

Temperature T (°C)M

ass

ther

mal

diff

usiv

ity λ

1 m (k

W/m

)

Fig. 2. Mass thermal diffusivity curve.

0

2.5

5.0

7.5

10.0

0.05 0.25 0.45

a

1007040

T=10

-1.0

1.4

3.8

6.2

8.6

11.0

0 50 100 150

b

0.39

0.3

θ=0.1

Temperature T (°C)

Mas

s di

ffudi

vity

λ

x103 (k

g/m

s)m

Water content θ

Mas

s di

ffusi

ty λ

mx1

03 (kg/

ms)

Fig. 3. Mass diffusivity curve.

68 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) 61–71

These calculated values are used in the numerical simulation of heat and moisture transfer inunsaturated soil. The predicted results are compared with the experimental data obtained fromone-dimensional column of soil and the agreement is satisfactory [18].

-1

0

1

2

0.05 0.25 0.45

1007040

T=10

a

Watercontent θ

Ther

mo-

mas

s di

ffusi

vity

λ x

106 (k

g/m

s°C

)m

-1

0

1

2

0 50 100 150

0.39

0.3θ=0.1

b

TemperatureT(°C)Th

erm

o-m

ass

diffu

sivi

ty λ

x 1

06 (kg/

ms°

C)

m" "

Fig. 4. Thermomass diffusivity curve.

6.0

6.5

7.0

7.5

8.0

0 600 1200 1800

100

70

40

T=10

a

Partialvapor pressure Pv (Pa)

Mig

ratio

n co

effic

ient

λ1 x

108 (k

W/m

°C)

p

6.0

6.5

7.0

7.5

8.0

8.5

0 50 100 150

0.390.3

θ=0.1

b

Temparature T (°C)

Mig

ratio

n co

effic

ient

λ1 p x

108 (k

W/m

°C)

Fig. 5. Migration coefficient curve.

Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) 61–71 69

6. Conclusions

The phenomenological equation to describe the immigration process of heat and mass inunsaturated porous building material is established. The phenomenological coefficients of heatand moisture are deduced. The phenomenological coefficients of heat and moisture migration are

2.4

2.6

2.8

3.0

0 600 1200 1800

100

70

40

T=10

a

Partrial vapor pressure Pv (Pa)

Infil

tratio

n co

effic

ient

λp

x 1

011(k

g/m

Pa s

)

Infil

tratio

n co

effic

ient

λp

x 1

011(k

g/m

Pa s

)2.30

2.63

2.96

3.29

0 50 100 150

0.390.30.2

θ=0.1

b

Temperature T (°C)

Fig. 6. Infiltration coefficient curve.

70 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) 61–71

very useful for simulation of the heat and moisture migration process in porous building mate-rials. The effects of temperature, water content or partial vapour pressure on the phenomeno-logical coefficients are also discussed. For sandy block with mean particle diameter of 0.3 mm andporosity of 0.39, it is concluded that the liquid water immigration, the effect of heat on liquidwater migration and effect of liquid water migration on heat transfer in the porous buildingmaterials are very small when the water content is below 0.1.

Acknowledgements

The authors thank the support fromNational BasicResearch Project of China, no.G2000263-03.

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