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International Journal of Heat and Mass Transfer 53 (2010) 3035–3044

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Transient model for coupled heat, air and moisture transfer throughmultilayered porous media

Fitsum Tariku a,*, Kumar Kumaran b, Paul Fazio c

a British Columbia Institute of Technology, 3700 Willingdon Ave., Burnaby, BC, Canada V5G 3H2b National Research Council, Institute for Research in Construction, 1200 Montreal rd., Ottawa, Ont., Canada K1A 0R6c Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Que., Canada H3G 1M8

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

Article history:Received 26 February 2009Received in revised form 30 September2009Accepted 1 October 2009Available online 13 April 2010

Keywords:Transient HAMMoistureHygrothermalBuilding envelope

0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.03.024

* Corresponding author.E-mail address: [email protected] (F. Tariku).

Most building materials are porous, composed of solid matrix and pores. The time varying indoor andoutdoor climatic conditions result heat, air and moisture (HAM) transfer across building enclosures. Inthis paper, a transient model that solves the coupled heat, air and moisture transfer through multilayeredporous media is developed and benchmarked using internationally published analytical, numerical andexperimental test cases. The good agreements obtained with the respective test cases suggest that themodel can be used to assess the hygrothermal performance of building envelope components as wellas to simulate the dynamic moisture absorption and release of moisture buffering materials.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The three aspects of building design: durability, indoor humid-ity level, and energy performance are interrelated and have to beconsidered simultaneously as part of an optimized building design.The thermal and moisture dynamic response of building enclosureshave strong impact on the humidity condition of the indoor spaceand energy consumption of the building. To accurately capture theinfluence of the building enclosure on the indoor environment andHVAC systems, a transient heat, air and moisture (HAM) transfermodel that handles the coupled heat, air and moisture transferphenomena through building enclosure is essential. The model en-ables dealing with the three important aspects of whole buildinghygrothermal analysis.

The first aspect relates to assessing the degree of moisture buf-fering effects of interior lining materials. El Diasty et al. [1] andJones [2] suggested that as much as one third of the moisture re-lease into the indoor air could be absorbed by interior moisturebuffering materials. These materials have a potential of modulatingthe indoor humidity level [3,4], especially in cases where the ven-tilation rate is low [1,5]. Thus, obtaining a detailed account of thedynamic moisture absorption and release of moisture bufferingmaterials is a crucial step in more accurately predicting the indoorhumidity level and fluctuation over time.

ll rights reserved.

Another advantage of having a detailed transient HAM model asone of the basic building blocks of whole building hygrothermalmodel is that it enables more accurately capturing the potentialmoisture release from the building enclosure to the indoor space.In fact, Christian [6] stated that moisture sources from construction(e.g. initial moisture content of concrete), and from wet soilthrough foundation walls and floor slab could dominate all internalmoisture sources. Similarly, TenWolde [7] recently emphasized theimportance of quantification of the moisture release from founda-tion slabs when calculating indoor humidity levels. Christian [6]estimated a total of 200 l moisture release by an average houseconstructed with lumber having an average of 19% moisture con-tent; and 90 l of water release per cubic meter of concrete usedduring the construction. The effect of these significant moisture re-leases to the indoor space and the moisture exchange between out-door and indoor environments, including the wind-driven rainload, on the overall hygrothermal performance of a building canbe more effectively estimated and understood using a transientcoupled HAM model [8,9].

A third advantage of the use of a transient HAM model whenconducting whole building performance analysis is that it yieldsa better estimation of energy demand for heating or cooling of abuilding. This is possible due to the fact a transient HAM modeltakes into account the effect of moisture in the heat transferthrough building enclosures. Usually energy simulation modelsignore the moisture effect when conducting the thermal analysis[10], and use constant thermal storage and transport (thermal

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2010.03.024mailto:[email protected]://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmt

Nomenclature

Cpa specific heat capacity of air (J/K kg)Cpv specific heat capacity of water vapor (J/K kg)Cpl specific heat capacity of liquid water (J/K kg)Dl liquid conductivity (s)g acceleration due to gravity (m/s2)hfg latent heat of condensation/evaporation (J/kg)jv vapor diffusion flux (kg/m2 s)jl liquid conduction flux (kg/m2 s)ka airflow coefficient (s)_mc moisture condensation/evaporation rate (kg/s)

M molar mass of water (0.01806 kg/mol)Pv vapor pressures (Pa)Ps suction pressures (Pa)

P_

saturated vapor pressure (Pa)Patm atmospheric pressure (Pa)R universal gas constant (8.314 J/K mol)

T temperature (�C)V air velocity (m/s)Yv mass fraction of water vapor (dimensionless)Yl mass fraction of liquid water (dimensionless)w moisture content (kg/m3)

Greek symbolsdv vapor permeability (s)H sorption capacity (kg/m3)l air dynamic viscosity (kg/m s)qa density of air (kg/m3)qw density of water (kg/m3)qm density of material (kg/m3)/ relative humidity (dimensionless)x absolute humidity (kg/kg-air)

0 25 50 75 100RH (%)

Moi

stur

e co

nten

t

~95

Capillary Water region

Hygroscopic sorption region

Fig. 1. Equilibrium moisture content profile of a typical material.

Water vapor Liquid water

3036 F. Tariku et al. / International Journal of Heat and Mass Transfer 53 (2010) 3035–3044

conductivity and heat capacity, respectively) property values de-spite the fact that these properties can be strongly dependent onmoisture content. For example, as the moisture content of woodincreases to 10% [11] its corresponding thermal storage capacityincreases by 30% as compared to its dry state; likewise, the thermalconductivity of lime silica brick increases more than twice as themoisture content increase to full saturation [12]. This implies thatarbitrary choices of values for the thermal transport and storageproperty of materials may result in an incorrect prediction of heatflux through building enclosure as demonstrated in Hagentoft’s[13] simple calculation of heat fluxes with and without moisturein a structure. Other important effects of moisture on energy calcu-lations, quite often omitted in whole building energy analysis tools,are the latent heat transfer across the building enclosure and thelocal heating and cooling effects that are generated within thestructure due to moisture phase changes (condensation and evap-oration, respectively). In this paper, details on the developmentand validation of a heat and moisture transfer model that takesinto account critical issues such as moisture buffering effects,moisture sources and the effects of moisture on heat transfer ispresented.

(c)(a) (b)

Fig. 2. Moisture in idealized pores [14]: (a) pores in hygroscopic region, (b) pores incapillary water region and (c) pores in high end of hygroscopic region.

2. Mathematical models for coupled heat, air and moisturetransfer through porous media

Most building materials are porous, and composed of solidmatrices and pores. In the pores, moisture can exist in any of thethree thermodynamic states of matter, i.e. the gas (vapor), liquidand solid (ice) states. However, moisture movement is possibleonly in the vapor and liquid states. The main mechanisms of mois-ture transfer can be by vapor diffusion, capillary suction or a com-bination of both, depending up on the moisture content of thematerial. Materials have unique equilibrium moisture contentcharacteristic curve that covers the hygroscopic and capillarywater regions. These regions are commonly referred to as sorptionisotherm and water retention curves, respectively, for which a typ-ical equilibrium moisture content characteristic curve is shown inFig. 1. In the hygroscopic region, the pores are mainly filled withwater vapor (Fig. 2(a)) and consequently, the moisture transportis mainly by vapor diffusion. Liquid water transport is possiblefor the case where the pores are filled with liquid water(Fig. 2(b)). This flow mechanism is very active in the capillarywater region, where the relative humidity is over 95%. Both vaporand liquid transport can co-exist in the higher end of the hygro-scopic region, as illustrated in Fig. 2(c). In this region, both vapor

diffusion and capillary suction are active in large and small pores,respectively. Vapor diffuses in the open pores and condenses onthe capillary meniscus, whereas on the other end of the meniscus,water evaporates into the next open pore space. This implies thatthe diffusion path is reduced, resulting in an increase in the rateof moisture.

2.1. Moisture transfer

The basic governing equation for moisture flow through a por-ous medium, given by Eq. (3), can be derived by adding the speciesconservation equations of water vapor (Eq. (1)) and liquid water(Eq. (2)).

qm@Yv@tþ qm divðVYvÞ þ divðjvÞ ¼ � _mc ð1Þ

qm@Yl@tþ divðjlÞ ¼ _mc ð2Þ

InterfaceMaterials

Material 1 Material 2

Moisture content

Relative humidity

Materials

Discontinuity in Moisture content

Fig. 4. Relative humidity and moisture content profiles at the interface of twodissimilar materials.

F. Tariku et al. / International Journal of Heat and Mass Transfer 53 (2010) 3035–3044 3037

gives

qm@Yv@tþ @Yl@t

� �þ qm divðVYvÞ þ divðjvÞ þ divðjlÞ ¼ 0 ð3Þ

Expressing the transient term in terms of moisture content

i:e:; qm@Yv@t þ

@Yl@t

� �¼ @w

@t

� �, and rewriting the vapor mass fraction

in the second term in terms of the humidity ratio, and substitutingthe vapor diffusion and liquid conduction fluxes (in the third andfourth terms, respectively) with Fick’s and Darcy’s law, respec-tively, yields moisture balance equation (4).

@w@tþ divðqaVxÞ þ div �dv

@Pv@xi

� �þ div Dl

@Ps@xiþ qwg

� �� ¼ 0

ð4Þ

The moisture balance equation (Eq. (4)) is comprised of variousmoisture driving potentials, specifically: w, x, Pv and Ps. These driv-ing potentials, and the associated gradients, can be expressed interms of a single flow potential. The chosen flow potential in thiswork is relative humidity (/) since it is continuous at the interfaceof two layers of materials having different moisture storage proper-ties (sorption and moisture retention), in contrary to moisture con-tent, which is discontinuous. This is illustrated in Figs. 3 and 4 asfollow: the relative humidity at the contact surfaces of material 1and material 2 are equal since the vapor pressure and temperatureare continuous at the interface. However, as shown in Fig. 3, theequilibrium moisture contents of the respective contacting surfacesare different (W1 and W2). Consequently, as illustrated in Fig. 4 atthe interface the moisture content profile becomes discontinuous asit jumps from W1 to W2. Relative humidity, on the other hand, iscontinuous throughout the computational domain. Consequently,all terms in the moisture balance equation (Eq. (4)) are mathemat-ically transformed using relative humidity as a driving potential asfollow:

(a) Transient term: @w@t

� �@w@t¼ @w@/� @/@t; H ¼ @w

@/where H ¼ @w

@/is the sorption

capacity ðSlop of sorption—moisture retention curveÞ@w@t¼ H @/

@tð4AÞ

(b) x in the vapor convection term

x ¼ 0:622 � PvPatm � Pv

; Patm � Pv � Patm; Pv ¼ P_

ðTÞ � /

x ¼ 0:622 � P_

ðTÞ � /Patm

; Cc ¼0:622Patm

where Patm is atmospheric pressure

x ¼ Cc P_

�/ ð4BÞ

Moi

stur

e co

nten

t

Relative humidity

Material 2

Material 1W_2

W_1

RH

Fig. 3. Sorption isotherm of two dissimilar materials showing different level ofequilibrium moisture content at given relative humidity.

(c) Vapor pressure gradient in the vapor diffusion term: ð@Pv@xiÞ

@Pv@xi¼@ P

_

ðTÞ � /� �

@xi¼ / @ P

_

ðTÞ@xi

þ P_ @/@xi

;@ P_

ðTÞ@xi

¼ @ P_

ðTÞ@T

� @T@xi

where P_

ðTÞ is the saturation vapor pressure, which is a function oftemperature T

@Pv@xi¼ / @ P

_

@T� @T@xiþ P

_ @/@xi

ð4CÞ

(d) Suction pressure gradient in liquid conduction term: @Ps@xi

Suction pressure can be expressed as a function of temperatureand relative humidity using Keleve’s equation: PsðT;/Þ ¼� qwRTM lnð/Þ, where R is the universal gas constant and M is themolecular weight of water molecule. Thus, an expression for suc-tion pressure gradient as function of temperature and relativehumidity (Eq. (4D)) can be obtained by making use of the partialdifferentiation of Keleve’s equation as follow:

@Ps@xi¼ @Ps@T� @T@xiþ @Ps@/

@/@xi

;@Ps@T¼ �qwR

Mlnð/Þ; @Ps

@/¼ �qwRT

M� 1/

@Ps@xi¼ �qwR

Mlnð/Þ @T

@xiþ T

/@/@xi

� �ð4DÞ

Finally, substituting Eqs. (4A), (4B), (4C), (4D) into Eq. (4) gives

H@/@t|fflffl{zfflffl}

@w@t

¼ @@xi

dv /@ P_

@T� @T@xiþ P

_ @/@xi

24

35

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}@Pv@xi

�qaVi0:622Patm

P_

�/

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

x

0BBBBBBB@

1CCCCCCCA

þ @@xi

DlqwRM

lnð/Þ @T@xiþ T

/@/@xi

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

@Ps@xi

�Dlqw~g

0BBBBB@

1CCCCCA

This expression can be simplified to Eq. (5), which represents themathematical model implemented in this paper for the general caseof non-isothermal moisture transfer through multilayered porousmedia.

H@/@t¼ @@xi

D/@/@xiþ DT

@T@xi

� �� @@xi

Dlqw~g þ qaViCc P_

�/� �

ð5Þ

where

D/ ¼ dv P_

þDlqwRM

T/

� �; DT ¼ dv/

@l P_

@Tþ Dl

qwRM

lnð/Þ

0@

1A

Eq. (5) can be reduced for a simpler case where moisture transfer ina porous media is considered an isothermal process, and not consid-ering either airflow or gravity effects as: H @/

@t ¼ @@xi ðD/@/@xiÞ. If


moisture content is used as a flow variable, the moisture transferequation can be written as: @w

@t ¼ @@xi ðDm@w@xiÞ.

Combining these equations provides a relationship between themoisture conduction coefficient, D/, and moisture diffusivity, Dm.This relation permits deducing both the liquid conduction coeffi-cient and liquid conductivity from measurable quantities of mois-ture capacity, vapor permeability and moisture diffusivity.

D/ ¼ Dm �H ¼ dv P_

þDlqwRM

T/

� �; Dl ¼

MqwR

� /T

Dm �H� dv P_

� �

1 HAMSTAD stands for Heat, Air and Moisture Standard Development.

2.2. Heat transfer

The conservation equation for internal energy and enthalpy arederived from the conservation equation of total stored energy, asgiven in Eq. (6). The total stored energy (E) of a system is thesum of internal energy (U), kinetic energy (KE) and potential en-ergy (PE), E = U + KE + PE. The conservation equation for the totalstored energy can be derived by considering a control volume,and accounting for the rate of change of stored energy in the con-trol volume (term I), transport of energy in and out of the controlvolume by convection (term II) and diffusion (term III) as well asthe work done by external forces at the surface of the control vol-ume including viscous forces (term IV) and by gravity (body) force(term V) and heat source (or sink) (term VI) [15].

@ðqeÞ@t|fflffl{zfflffl}

I

þdivðqVeÞ|fflfflfflfflfflffl{zfflfflfflfflfflffl}II

¼ �divðjqÞ|fflfflfflfflffl{zfflfflfflfflffl}III

þ divðbsVÞ|fflfflfflfflffl{zfflfflfflfflffl}IV

þqð g!� VÞ|fflfflfflfflffl{zfflfflfflfflffl}V

þ _Q|{z}VI

ð6Þ

where e is energy per unit mass ðe ¼ Em ¼ uþ 12 jV2j þ g � xiÞ and qe is

the energy per unit volume. After rearranging some mathematicalexpressions, the conservation equation for the total energy can beexpressed in terms of enthalpy, h, as provided in Eq. (7) [15].

@ðqhÞ@tþ divðqVhÞ ¼ �divðjqÞ þ _Q s ð7Þ

where jq is a diffusion term, which comprises heat transfer by con-duction and enthalpy transport due to moisture transfer and Qs is aheat source (or sink) term. Rewriting the transient, convection anddiffusion terms using mixture enthalpy (moisture, air and solid ma-trix), and subsequent simplification of Eq. (7) yields the mathemat-ical model, implemented in this paper, for transient heat transferthrough porous media, Eq. (8).

qmCpeff@T@tþ qaðCpa þxCpvÞdivðVTÞ þ divð�keff gradðTÞÞ

¼ _mchfg þ _mcTðCpv � CplÞ þ _Q s ð8Þ

where Cpeff and keff are the effective specific heat capacity and ther-mal conductivity (which take moisture effect into account), respec-

tively, and _mc ¼ div dv @Pv@xi� �

� qa divðVxÞ is the amount of moisturecondensation/evaporation in kg/s.

2.3. Airflow through porous media

Airflow through a porous media can be expressed by usingPoiseuille’s law of proportionality [16], which relates the pressuregradient to flow velocity (Eq. (9)).

V ¼ � kal

divðPÞ ð9Þ

In building physics applications, air is considered incompressibledue to the very low airflow speeds, and low pressure and tempera-ture changes that are encountered in practice. Consequently, theconservation equation for air mass balance is given by Eq. (10).

divðqaVÞ ¼ 0 ð10Þ

Combining the mass balance, given in Eq. (10), and momentumbalance, provided in Eq. (9), gives Eq. (11), which is implementedin this paper to compute airflow velocities through buildingenclosures.�divðdadivðPÞÞ ¼ 0 ð11Þ

where da ¼ qa kal (air permeability).

3. Numerical tool for transient HAM analysis

The mathematical models implemented in this paper are Eqs.(5), (8) and (11) for moisture, heat and air transport through mul-tilayered porous media, respectively. The solution to the air bal-ance equation, flow through a porous media with perfect contactbetween adjacent layers, is relatively straightforward if the air per-meability of the medium is assumed to be constant, which is a gen-erally the case in building physics applications. In this instance, Eq.(11) is solved independently for the pressure distribution in themedium of a given boundary pressure condition. Subsequently,Eq. (9) is used to calculate the airflow velocity field. The knownvelocity field will then be used in the convection transport termsof moisture and energy equations, Eqs. (5) and (8), respectively.

Fig. 5 shows a graphical representation of the interdependencyof heat, air and moisture transfer in a porous media. The heat andmoisture balance equations (Eqs. (8) and (5), respectively) arehighly coupled in a way that the heat transfer solution dependson the moisture balance solution and vice versa. Temperatureand moisture content can affect the air density and permeabilityof the porous media, and in turn the mechanisms of convectiveheat and moisture transfer. In the heat balance equation, Eq. (8),the thermal storage and transfer properties of materials (effectiveheat capacity, Cpeff, and apparent thermal conductivity, keff ) as wellas the local heat source/sink (associated with moisture phasechange, _mc) depend on the moisture state of the domain. On theother hand, the temperature fields affect the moisture transfer pro-cess since the temperature gradient is one of the moisture drivingforces as indicated in the moisture balance equation (Eq. (5)).Moreover, the vapor permeability, moisture transfer coefficients(D/ and DT) and saturated vapor pressure, which are importantparameters in the moisture balance equation, are temperaturedependent. In addition to the strong coupling between the heatand moisture balance equations, the equations themselves arehighly nonlinear due to the fact that neither the transfer nor thestorage coefficient of the respective balance equations are con-stants but are functions of the corresponding driving potentials.

For example, the moisture and heat transfer properties of a loadbearing material [17], which is used in one of the HAMSTAD1 bench-mark exercises, are presented in Figs. 6 and 7, respectively. Fig. 6shows the nonlinearity of the relationship of the sorption capacityand vapor permeability with relative humidity as well as the liquiddiffusivity with moisture content. As moisture content (or relativehumidity) increases these properties exhibit more nonlinear behaviordepicting a high increase in sorption capacity, decreases in vaportransport and significant increase in liquid water transport. Likewise,Fig. 7 shows the moisture dependency of the thermal properties forthis same material; the heat storage capacity and thermal conductiv-ity of the material increases with moisture content.

3.1. Numerical tool

To obtain the temperature and relative humidity field acrossthe computational domain (multilayered building envelop

Latent heat, Enthalpy, Heat transfer and storage

coefficient

Heat

Air

Convective Moisture transfer

Convective Heat transfer

Saturated vapor pressure, Moisture transport and

storage coefficients

Moisture

DensityPermeability

Fig. 5. Graphical representation of the interdependency of heat, air and moisture transfer. The arrows show the influence of an entity on the correspondingly linked entitythrough its effect on the parameters described in the accompanying text box.

0

200

400

600

20% 40% 60% 80%Relative Humidity

Sorp

tion

Cap

acity

(kg/

m3 )

0.E+00

4.E-13

8.E-13

1.E-12

0% 20% 40% 60% 80% 100%Relative Humidity

Vapo

r Pe

rmea

bilit

y (s

)

0.E+00

2.E-09

4.E-09

6.E-09

0 50 100 150Moisture content (kg/m3)

Liqu

id

Diff

usiv

ity (m

2 /s)

Fig. 6. Typical moisture transport properties curves. As moisture content increasesthe transport properties exhibit nonlinear behavior.

700800900

10001100

Moisture Content (kg/m

Hea

t Cap

acity

(J

/kgK

)

700800900

10001100

0 50 100 150Moisture Content (kg/m3 )

Hea

t Cap

acity

(J

/kgK

)

01234

0 50 100 150M oisture content (kg/m 3)

Ther

mal

C

ondu

ctiv

ity

(W/K

m)

Fig. 7. Typical thermal properties as a function of moisture content.


component), the coupled and nonlinear partial differential equa-tions (PDEs) need to be solved simultaneously. Here, a finite-ele-ment based computational tool called COMSOL Multiphysics2 andMatLab3 were used to solve these equations. In addition to a solver,

2 COMSOL Multiphysics: http://www.comsol.com/.3 Mathworks http://www.mathworks.com.

COMSOL Multiphysics has a graphical user interface (GUI) to createcomputational domain geometry, an automated and user controlledmesh generator, and it also has an integrated post processing capa-bility for plotting, interpolating and integrating simulation results.The COMSOL Multiphysics computational tool has a library of prede-fined models to solve familiar engineering problems such as convec-tion diffusion, fluid dynamics, heat transfer and other problems. Italso has a provision to apply equation based modeling techniques,referred as ‘‘PDE Modes”, for solving problems that may not besolved by the standard modules. Using this numerical technique,the developer formulates the PDEs that govern the physical phenom-ena, and solves them using the built-in solver.

In this paper, the three-coupled transient HAM equations weresimultaneously solved using the COMSOL Multiphysics time-dependent solver. The solver is based on an explicit scheme withvariable time stepping. The user can predefine the maximum timestep so that it matches with the boundary conditions changeperiods.

http://www.comsol.com/http://www.mathworks.com

0

20

40

60

80

100

0 5 10 15 20x (cm)

Moi

stur

e co

nten

t (kg

/m3 )

300 h100 h

1000 h

HAMFitAnalytical

Fig. 9. Moisture profiles of the structure at 100, 300 and 1000 h from the initialmoisture content of 80.8 kg/m3.


4. Benchmarking of the transient coupled HAM model

In this section, the newly developed transient coupled HAMmodel is benchmarked against published test cases. The test casescomprise an analytical verification, comparisons with other mod-els, and validation of simulation results with experimental data.Judkoff and Neymark [18] recommend these three classes of modelevaluation methods to test whether the mathematic models thatare incorporated in the numerical tool describe the physical pro-cesses of interest adequately. Here, two of the five benchmarkingexercises that were designed under the European HAMSTAD pro-ject and a drying experiment carried out by Maref et al. [19] arepresented as they cover the three categories of model test cases.The HAMSTAD project was initiated to develop standard test cases,by which the accuracy of the existing and newly developed hygro-thermal models should be evaluated [17]. The complete bench-marking exercises that are undertaken to test the model arereported in Tariku [20].

1000 Hours--Top section

20

30

40

50

60

Moi

stur

e C

onte

nt

(kg/

m3 )

Analytical HAMFit

4.1. Analytical verification – HAMSTAD Benchmark Exercise #2

In this benchmark exercise, a schematic of which is given inFig. 8, the isothermal drying process of a relatively wet 200 mmthick homogeneous layer structure is considered. The initial hygro-thermal conditions of the structure are 20 �C and 95% relativehumidity. The level of relative humidity of the surrounding envi-ronment is changed so that the structure dries out by moistureredistribution and release to the surroundings. The top (exterior)and bottom (interior) surfaces of the structure are exposed to45% and 65% relative humidity, respectively, while the tempera-ture is kept constant at 20 �C. The heat and mass transfercoefficients for both surfaces are 25 W/m2 K and 1E � 3 s/m,respectively. The material properties of the structure are given inTable 1 below. This benchmark exercise is a test case that has ananalytical solution. This is possible due to the fact that the dryingprocess is isothermal, and the boundary conditions and hygrother-mal properties of the material are assumed to be constant. The fulldescription of this benchmark exercise is given in Hagentoft [17].

The accuracy of the numerical model in simulating the dryingprocess of the structure is verified by comparing the model resultswith the analytical solutions, which are provided in the HAMSTADproject report. The transient moisture profiles (moisture content inkg/m3) across the structure, which result due to the continuous re-lease of moisture from the structure to the surrounding through its

200 mm

Top Surface

Bottom Surface

X

Fig. 8. A structure with initial moisture content of 81 kg/m3 (95% relativehumidity), and boundary conditions of 45% and 65% relative humidity at the topand bottom surfaces, respectively.

Table 1Hygrothermal properties of the monolithic struacture.

Sorption isotherm w ¼ 1161� 10:118 lnð/Þð Þ

0:869 kg=m3

Vapor diffusion 10�15 sMoisture diffusivity 6 � 10�10 m2/sThermal conductivity 0.15 W/m KHeat capacity 4.2 � 105 J/m3 K

boundary surfaces, are used as verification parameters. Fig. 9shows the initial moisture content and the moisture distributionacross the structure at 100, 300 and 1000 h. In this and the follow-ing figures, the simulation results of the model are designated as‘‘HAMFit”. The moisture distributions at the top, middle and bot-tom sections of the structure at 1000 h are presented in Figs. 10,11 and 12, respectively. As can be seen in these figures, the newlydeveloped model produced excellent results that clearly show highdegree of agreement with the analytical solutions.

4.2. Comparative analysis —HAMSTAD Benchmark Exercise #4

In this benchmark exercise, the dynamic responses of the HAMmodel for a well-defined heat and moisture transfer problem are

0 1 2 3 4 5x (cm)

Fig. 10. Moisture distribution of the top section of the structure at 1000 h.

1000 Hours--Middle section

55

58

61

64

67

70

5 6 7 8 9 10 11 12 13 14 15

x (cm)

Mo

istu

re C

on

ten

t (k

g/m

3 )

Analytical HAMFit

Fig. 11. Moisture distribution of the middle section of the structure at 1000 h.

1000 Hours--Bottom section

30

35

40

45

50

55

60

15 16 17 18 19 20x (cm)

Moi

stur

e C

onte

nt

(kg/

m3 )

Analytical HAMFit

Fig. 12. Moisture distribution of the bottom section of the structure at 1000 h.


compared with other models’ simulation results. The prerequisitefor such type of comparative analysis is that all model inputparameters including geometrical representation, dimensions, ini-tial conditions, internal and external boundary conditions, andmaterial properties of the building envelope systems have to beprescribed and consistently used by all participating models. Thefull description of the benchmark exercise can be found in Hagent-oft [17]. Here, a brief description of the problem, input parametersand results are presented.

The test case deals with heat and moisture transfer in a two-layer wall system exposed to realistic internal and external bound-

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Fig. 13. Boundary conditions imposed on the indoor and outdoor surfa

ary conditions. The wall system is composed of a load-bearinglayer on the exterior, and finishing layer on the interior of the wallsystem. The load-bearing layer is 100 mm thick and has a densityof 2050 kg/m3 and specific heat capacity of 840 J/K kg; the finishinglayer has a thickness of 20 mm, density 790 kg/m3 and specificheat capacity of 870 J/K kg. Realistic time dependent boundaryconditions that are applied at the external and internal surfacesof the wall are shown in Fig. 13. The variable heat and moistureloads on the exterior surface due to solar radiation and rain arerepresented by equivalent outdoor temperature (Fig. 13a) andwind-driven rain flux (Fig. 13(c)), respectively. The time dependentindoor moisture load that may be related to occupant activity, isrepresented by variable indoor vapor pressure (Fig. 13(b)). The out-door air temperature and vapor pressure, as well as the indoor airtemperature are held constant with values of 10 �C, 1150 Pa and20 �C, respectively. This test case is more challenging [21] as it in-volves severe climatic load that causes surface condensation on theexterior surface due to nighttime cooling (low equivalent temper-ature), and frequent occurrences of wetting and drying of the walldue to the alternating rain and solar radiation loads. Moreover, theproblem involves rapid rainwater absorption at the interfaces andhigh rate of moisture movement within the layers due to the extre-mely high liquid water absorption property of the load-bearinglayer.

The initial hygrothermal conditions of the two layers are 20 �Cand 40% temperature and relative humidity, respectively. The masstransfer coefficients of the interior and exterior surfaces are 3E � 8and 2E � 7 s/m, respectively. The heat transfer coefficients for thecorresponding surfaces are 8 and 25 W/m2 K, respectively. For

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ces: (a) thermal load, (b) vapor pressure and (c) wind-driven rain.


comparison purposes, the model simulation results (designated as‘‘HAMFit”) are superimposed on the corresponding six HAMSTADproject participants’ solutions. Fig. 14 shows the transient surfacemoisture contents and temperatures of the outer and inner sur-faces of the wall for the entire simulation period. The moisturecontent and temperature profiles of the wall system after 96 hare presented in Fig. 15. As can be seen in these figures, the simu-lation results of the model are in very good agreement with theother six models’ solutions. In whole building hygrothermal mod-eling, the coupling of building enclosure and indoor environment isthrough interior surfaces, and therefore, it is important to accu-rately predict the hygrothermal states of these surfaces to obtainuseful results.

4.3. Experimental validation—Laboratory controlled experiment

In this section, a drying experiment carried out by Maref et al.[19] is used for validation and testing of the model. The model’sprediction is compared with this laboratory controlled measureddata. In fact, the main objective of Maref’s experiment was to

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Fig. 14. The transient surface moisture content and temperature of the o

provide measured data by which building envelope models couldbe tested and validated. The experiment was done on full-scale sizewall having equal height and width of 2.4 m. The wall system iscomprised of a wood frame, sheathing board (11.5 mm thickOSB) and vapor barrier (polyethylene sheet) that are installed onthe outside and interior surfaces of the frame, respectively. Thecavity between the vertical wood studs is filled with glass fiberinsulation. The experiment was designed to measure the dryingrate of a wetted sheathing board (OSB) as it is exposed to con-trolled indoor and outdoor boundary conditions. At the beginningof the experiment, the equilibrium moisture content of the wettedOSB was 330 kg/m3, which is equivalent to 99.6% relative humid-ity. This initial moisture condition was attained by carrying out apreconditioning process that involved: soaking the OSB in a waterbath, and thereafter, wrapping it up with polyethylene sheet toallow moisture redistribution across the thickness of the panel.As part of the experimental setup, all surfaces of the wood framewere coated with vapor tight paint to restrict moisture exchangewith the surroundings including the OSB. Furthermore, the edgesof the OSB were sealed to prevent moisture loss through these

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uter and inner surfaces of the wall for the entire simulation period.

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Fig. 15. The moisture content and temperature profiles of the wall system at 96 h.


surfaces. These preliminary actions suggest that the drying processis one-dimensional and takes place through the OSB planer sur-faces. During the experiment, any weight loss recorded by theweighing system was interpreted as moisture loss (drying) of theOSB to the outdoor environment. The basis for this assumptionare the following: (1) the weight of the wood-frame remains thesame since its moisture exchange with the surrounding is re-stricted by the paint; (2) moisture accumulation in the insulationis insignificant due to its non-hygroscopic nature; (3) condensationon the exterior surface of the polyethylene sheet is insignificantsince the indoor and outdoor temperature conditions are nearlythe same.

The OSB used in this experiment had a density of 650 kg/m3,thermal conductivity of 9.41E � 02 W/m K and specific heat capac-ity of 1880 J/kg K. Its moisture storage and transport propertiesthat include the sorption isotherm, vapor permeability and liquiddiffusivity are known from the experimental study. The density,thermal conductivity, heat capacity and vapor permeability ofthe glass fiber insulation are 11 kg/m3, 3.66E � 02 W/m K, 1256 J/kg K and 1.30E � 10 kg/s Pa m, respectively. Since the insulationis non-hygroscopic its moisture storage capacity is very low, andtherefore neglected in the modeling. Moreover, due to its capillarynon-active nature, the liquid water transport property was set tozero. The vapor permeability of polyethylene sheet is

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2.29E � 15 kg/s Pa m. As far as hygrothermal modeling is con-cerned, vapor permeability is the most important hygrothermalproperty of the polyethylene sheet; the remaining propertiesincluding moisture storage, thermal storage, liquid permeabilityand thermal resistance values were set to zero. Since the polyeth-ylene sheet was directly exposed to the indoor boundary condition,it was modeled as a surface vapor resistance rather than as a layer.

The two remaining and essential input data for benchmarkingof hygrothermal models using laboratory-controlled experimentsare: initial and boundary conditions. The initial equilibrium mois-ture content of the OSB was 330 kg/m3, and the corresponding rel-ative humidity (from sorption isotherm curve) was 99.6%. In thesimulation, the initial moisture content was assumed to be uni-form across the OSB thickness. This is based on the step taken dur-ing the preconditioning process, more specifically, wrapping thewetted OSB with a polyethylene sheet to allow moisture redistri-bution. The initial temperature condition of the wall system wasassumed to be 25 �C and uniform across the thickness. The bound-ary conditions to which the wall system was exposed were con-trolled and measured over the course of the experiment. Thetemperature and relative humidity conditions of the outdoor envi-ronment were fairly constant at 25 �C and 25%, respectively. Formost of the time during the experiment, the temperature differ-ence across the wall was between 1 and 2 �C. This small tempera-

20 25 30 35Days

ExperimentHAMFit

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sured and simulated drying curves of OSB.


ture difference coupled with the presence of insulation in the cav-ity makes the drying process nearly an isothermal process. The in-door relative humidity was generally higher and more variablethan the outdoor relative humidity (with a mean value of 40%).However, its effect on the drying process was very limited due tothe presence of the polyethylene sheet, which essentially createsan interior adiabatic boundary condition for moisture transfer.

In the simulation, the mass transfer coefficients are determinedfrom the heat transfer coefficients using Lewis relation [11]. Sincethe experiment is carried out in the indoor environment, a heattransfer coefficient that is recommended in the IEA/Annex 24[22] for interior flat surface 8 W/m2 K is adopted for both interiorand exterior surfaces. Based on Lewis relation, the mass transfercoefficients for the exterior surface is determined to be 5.80E � 8,1.53E � 11 and 5.80E � 8 s/m, respectively. The effective masstransfer coefficient of the interior surface was 1.53E � 11 s/m,which is calculated by superimposing the vapor resistance of poly-ethylene sheet on the vapor flow resistance created by the moist-air boundary layer.

In Fig. 16 the simulation result for the OSB drying curve derivedfrom the model (HAMFit) is superimposed on the laboratory re-sults, which depict the transient moisture content of the OSB at dif-ferent times over the drying period. As can be seen in the figure,the model prediction is in excellent agreement with the experi-mental results for the entire drying period. During this period,the OSB lost 2.5 kg of moisture per square meter of OSB. The aver-age moisture content of the OSB by weight was reduced from 51%(initial state) to 16% (end of experiment).

5. Conlusion

The thermal and moisture dynamic responses of building enclo-sures, essential inputs for whole building hygrothermal models,have strong impact on the overall performance of the building. Thisis due to the fact that the moisture stored in the structure affectsthe indoor humidity and energy flow across the structure, andthereby HVAC equipment size. Moreover, building enclosures canhave significant influence on the indoor humidity level dependingon the moisture buffering capacity of the interior lining materials.The dynamic influences of building enclosures on the indoor envi-ronment and HVAC systems can be captured by using a transientmodel that handles coupled heat, air and moisture transfer throughmultilayered porous media.

In this paper, a transient heat, air and moisture transfer model isdeveloped based on basic conservation of mass and energy equa-tions. The governing partial-differential equations (PDEs) of thethree transport phenomena are coupled and solved simultaneouslyfor temperature, relative humidity and pressure. The modelaccommodates nonlinear transfer and storage properties of mate-rials, moisture transfer by vapor diffusion, capillary liquid watertransport and convective heat and moisture transfer through mul-ti-layered porous media. The PDEs are derived in such a way thateach PDE is described with a single driving potential, which is con-tinuous across the interfaces of adjoining materials. Consequently,an equation-based modeling technique, which requires less time ofimplementation and provides high degree of transparency andflexibility of modeling, is used for solving the coupled PDEs. Thetransient HAM model is successfully benchmarked against threepublished test cases. The test cases are comprised of an analyticalverification, comparisons with other models and validation of sim-ulation results with experimental data. The good agreement ob-tained with the respective test cases suggest that the model

development and implementation are satisfactory, and therefore,can be further coupled with an indoor model to create a wholebuilding hygrothermal model. The development and benchmark-ing of a holistic model that utilizes the transient model developedin this paper as one of its building block will be presented in sub-sequent paper.

Acknowledgement

The authors thank Dr. Wahid Maref for providing the experi-mental data used in the third benchmark exercise.

References

[1] R. El Diasty, P. Fazio, I. Budaiwi, Modelling of indoor air humidity: the dynamicbehavior within an enclosure, Energy Build. 19 (1992) 61–73.

[2] R. Jones, Indoor humidity calculation procedures, Build. Services Eng. Res.Technol. 16 (3) (1995) 119–126.

[3] C. Simonson, M. Salonvaara, T. Ojanen, Heat and mass transfer between indoorair and a permeable and hygroscopic building envelope: part 1 – fieldmeasurements, J. Therm. Envelope Build. Sci. 28 (1) (2004) 63–101.

[4] C. Simonson, M. Salonvaara, T. Ojanen, Heat and mass transfer between indoorair and a permeable and hygroscopic building envelope: part II – verificationand numerical studies, J. Therm. Envelope Build. Sci. 28 (2) (2004) 161–185.

[5] A. TenWolde, Mathematical model for indoor humidity in houses duringwinter, in: Proceedings of Symposium on Air Infiltration, Ventilation andMoisture Transfer, Building Thermal Envelope Coordinating Council,Washington, DC, 1988.

[6] J.E. Christian, Moisture Sources, Moisture Control in Buildings, ASTM ManualSeries: MNL 18, 1994, pp. 176–182.

[7] A. TenWolde, C.L. Pilon, The effect of indoor humidity on water vapor release inhomes, in: Proceedings of Thermal Performance of the Exterior Envelopes ofWhole Buildings X International Conference, Dec. 2–7, Clearwater, FL, 2007.

[8] F. Tariku, M.K. Kumaran, Hygrothermal modeling of aerated concrete wall andcomparison with field experiment, in: Proceeding of the 3rd InternationalBuilding Physics/Engineering Conference, August 26–31, Montreal, Canada,2006, pp. 321–328.

[9] F. Tariku, S. Cornick, M. Lacasse, Simulation of wind-driven rain effects on theperformance of a Stucco–Clad wall, in: Proceedings of Thermal Performance ofthe Exterior Envelopes of Whole Buildings X International Conference, Dec. 2–7, Clearwater, FL, 2007.

[10] N. Mendes, F.C. Winkelmann, R. Lamberts, P.C. Philippi, Moisture effects onconduction loads, Energy Build. 35 (2003) 631–644.

[11] ASHRAE Handbook of Fundamentals, American Society of Heating,Refrigeration, and Air-Conditioning Engineers, Atlanta, 2005.

[12] H. Kuenzel, A. Karagiozis, A. Holm, A hygrothermal design tool for architectsand engineers, in: Moisture Analysis and Condensation Control in BuildingEnvelopes, ASTM Manual Series 50: Chapter 9, 2001.

[13] C.-E. Hagentoft, Heat, air and moisture transfer through new and retrofittedinsulated envelope parts, IEA Annex 24 HAMTIE, Final Report, vol. 5, Task 5:Performances and Practice, ISBN 90-75741-04-9, 1996.

[14] C.-E. Hagentoft, Building Physics Fundamentals, Report R-97:1, Department ofBuilding Physics, Chalmers University of Technology, Sweden, 1997.

[15] K. Kuo, Principles of Combustion, Wiley, New York, ISBN 0-471-09852-3, 1986.[16] H. Hens, Building Physics – Heat, Air and Moisture, Fundamentals and

Engineering Methods with Examples and Exercises, Ernst & Sohn A Wiley,ISBN 978-3-433-01841-5, 2007.

[17] C.-E. Hagentoft, HAMSTAD – Final report: methodology of HAM-modeling,Report R-02:8, Gothenburg, Department of Building Physics, ChalmersUniversity of Technology, 2002.

[18] R. Judkoff, J. Neymark, Building energy simulation test (BESTEST) anddiagnostic method, NREL/TP-472-6231, National Renewable Energy Lab,Golden, CO, 1995.

[19] W. Maref, M. Lacasse, K. Kumaran, M.C. Swinton, Benchmarking of theadvanced hygrothermal model – hygIRC with mid-scale experiments, eSim2002 Proceedings, University of Concordia, Montreal, 2002, pp. 171–176.

[20] F. Tariku, Whole building heat and moisture analysis, Ph.D. Thesis, ConcordiaUniversity, Montreal, Canada, 2008.

[21] C.-E. Hagentoft, A. Kalagasidis, B. Adl-Zarrabi, S. Roels, J. Carmeliet, H. Hens,J. Grunewald, M. Funk, R. Becker, D. Shamir, O. Adan, H. Brocken, K.Kumaran, R. Djebbar, Assessment method of numerical prediction models forcombined heat, air and moisture transfer in building components:benchmarks for one-dimensional cases, J. Therm. Envelope Build. Sci. 27(4) (2004) 327–352.

[22] C. Sanders, Heat, air and moisture transfer through new and retrofittedinsulated envelope parts, IEA Annex 24 HAMTIE, Final Report, vol. 2, Task 2:Environmental Conditions, K.U. Leuven, Belgium, 1996.

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Transient model for coupled heat, air and moisture transfer through multilayered porous mediaIntroductionMathematical models for coupled heat, air and moisture transfer through porous mediaMoisture transferHeat transferAirflow through porous media

Numerical tool for transient HAM analysisNumerical tool

Benchmarking of the transient coupled HAM modelAnalytical verification – HAMSTAD Benchmark Exercise #2Comparative analysis —HAMSTAD Benchmark Exercise #4Experimental validation—Laboratory controlled experiment

ConlusionAcknowledgementReferences

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