Building and Environment 37 (2002) 313–318www.elsevier.com/locate/buildenv

Vapour condensation and moisture accumulation in porous buildingwall

Jerzy Wyrwa l∗, Andrzej MarynowiczTechnical University of Opole, Faculty of Civil Engineering, Katowicka 48, 45-061 Opole, Poland

Received 29 March 2000; received in revised form 2 August 2000; accepted 23 November 2000

Abstract

Simultaneous one-dimensional heat and vapour transfer with condensation in a porous wall is analytically investigated. Spatiallysteady-state distribution of accumulated moisture, less than the critical content, is described. Closed-form analytical expressions for thetemperature, condensation rate and moisture content are obtained. The presented model requires material properties which are relativelysimple and easy to determine. The results of the paper are illustrated with an example of multilayer building wall under climatic conditions.c© 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Heat; Vapour; Condensation; Porous material; Critical moisture content

1. Introduction

Simultaneous heat and vapour transfer with condensationin porous materials is of practical importance in applicationsin civil engineering. The transport of vapour across buildingwalls and its possible condensation increases the thermalconductivity of the building porous materials and may causestructural damage.

The theoretical and experimental study of the heat andmoisture transfer in building walls has been the target of alot of important research work. We can refer to the works ofAndersson [1], Budaiwi et al. [2], De Freitas et al. [3], Kiessl[4], Kohonen [5], KAunzel and Kiessl [6], as well as Pedersen[7]. However, very little work exists on the particular subjectof vapour condensation in porous materials.

Water-vapour condensation within a porous wall has beenobserved, particularly when the wall is exposed to large tem-perature diCerences and high humidity environments. Thisphenomenon was Drst rigorously studied by Ogniewicz andTien [8] where the coupling between temperature and con-centration of condensing vapour was taken into account. Ina reference by Motekef and El-Masri [9], one-dimensionaltransport of heat and mass with phase change in a porousslab was studied, and analytical solutions for the cases ofimmobile and mobile condensate were obtained. Recently

∗ Corresponding author. Tel.=fax: +48-077-456-50-84.E-mail address: wyrwal@ss5.po.opole.pl (J. Wyrwa l).

Shapiro and Motakef [10] proposed an analytical solutionfor a large class of transient problems, and compared theirresults with experimental data.

The present paper uses the approach of Motakef andEl-Masri [9] for the problem of one-dimensional Jow ofheat and diCusion of vapour in a porous wall. It is assumedthat the moisture content in the wet zone is lower than thecritical one and, as a consequence of this, the liquid wateris practically immobile. This leads to very simple relationsdescribing the temperature and moisture proDles as well asthe condensation rate in the wet zone. The major advantageof the proposed solution is its relative ease in determiningthe moisture accumulation due to vapour condensation fordiCerent materials in diCerent conditions.

2. Formulation of the problem

Consider the one-dimensional transfer of heat and vapourin a homogeneous porous wall (Fig. 1), the boundaries ofwhich are exposed to two diCerent environments: an indoorenvironment with temperature Ti and humidity ’i , and anoutdoor environment with temperature Te and humidity ’e.Under winter conditions the indoor temperature is higherthan the outdoor one, and vapour diCuses towards the colderboundary.

For a constant pressure system, the concentration of satu-ration vapour is a unique function of temperature. Therefore,

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314 J. Wyrwa l, A. Marynowicz / Building and Environment 37 (2002) 313–318

Nomenclature

Csat concentration of saturation vapour (kg=kg)D vapour diCusion coeMcient in the air (m2=s)Dp vapour permeability (kg=m s Pa)K dimensionless coeMcient (dimensionless)l width of the condensation zone (m)L latent heat of condensation (J=kg)psat saturation vapour pressure (Pa)R condensation rate (kg=(m3 s))R thermal resistance (m2 K=W)Rmax maximum condensation rate (kg=(m3 s))Rv gas constant of vapour (J=(kg K))T temperature (K)Te exterior temperature (K)Ti interior temperature (K)Tl temperature of exterior surface of the wet

zone (K)T0 temperature of interior surface of the wet

zone (K)W moisture content (m3=m3)Wcap capillary saturation (m3=m3)

Wcr critical moisture content (m3=m3)Wmax maximum moisture content (m3=m3)W0 initial moisture distribution (m3=m3)t time (s)tcr time at which critical value is reached (s)x co-ordinate (m)Z thermal resistance (m2 s Pa=kg)

Greek letters

�e exterior surface heat transfer coeMcient(W=(m2 K))

�i interior surface heat transfer coeMcient(W=(m2 K))

� porosity of the material (m3=m3)� thermal conductivity (W=(m K))� material density (kg=m3)�a air density (kg=m3)�w liquid density (kg=m3)’e exterior relative humidity (dimensionless)’i interior relative humidity (dimensionless)

the vapour saturation–concentration curve in the wall isdeDned by the temperature distribution. Depending onthe values of the prescribed boundary conditions, thevapour concentration proDle may touch the saturation–concentration curve in the wall. The diCusing vapour wouldthen undergo a phase change and condense in some regionof the wall. With the relative humidity at the boundaries lessthan 100%, condensation occurs over the wet zone, sepa-rated from the boundaries by two dry zones as illustrated inFig. 1.

The condensation of vapour in the wet zone can be con-sidered to be simultaneously a vapour sink, water sourceand heat source. Hence, three processes of vapour diCu-sion, vapour condensation and heat conduction are coupledthrough the condensation rate. The vapour concentration,moisture content and temperature proDles in the wet zone areobtained by the simultaneous solution of the three coupledconservative equations for heat, vapour and liquid water.

At moisture contents less than the critical one, conden-sate is in a pendular state [11] and does not exhibit any ten-dency to migrate. Beyond the critical moisture content, asthe pendular drops coalesce and the capillary pores are wet-ted, condensate is propelled by surface tension forces fromwetter to drier regions.

In the absence of moisture migration in the wet zone, thesystem of diCerential equations for heat and vapour transfermay be written as

ddx

(�

dTdx

)+ LR= 0; x ∈ (0; l); (1)

ddx

[�a�D

dCsat(T )dx

]− R= 0; x ∈ (0; l); (2)

where x is the co-ordinate (m), � the coeMcient of heat con-duction (W=(m K)), T the temperature (K), L the latent heatof condensation (J=kg), R the condensation rate (kg=(m3 s)),l the width of the condensation zone (m), �a the air den-sity (kg=m3), � the porosity of the material (m3=m3), D thevapour diCusion coeMcient in the air (m2=s), and Csat theconcentration of saturation vapour (kg=kg).

Eq. (1) is subjected to the following boundary conditions:

T =

{T0 at x = 0;

Tl at x = l;(3)

where T0 is the temperature of interior surface of the wetzone (K), and Tl the temperature of exterior surface of thewet zone (K).

The problem of calculation of the location of wet zone,and the temperatures of its surfaces has been solved byMotakef and El-Masri [9].

In the process considered, there Drst occurs a relativelyshort initial transient stage in which the temperature andvapour concentration Delds are developing within the porousslab. During this phase a very small quantity of liquid wateris accumulated in the porous material (Fig. 2). The initialtransient stage is of little signiDcance due to its relativelyshort duration and, therefore, is not studied here. Condensa-tion of vapour is deDned here as the accumulation of liquidwater beyond the phase described above.

Beyond the initial transient time, the temperature andvapour concentration proDles remain invariant with time,vapour condenses continuously in the wet zone, the conden-sate accumulates with time, and for moisture contents lessthan critical the transport of water in liquid phase within the

J. Wyrwa l, A. Marynowicz / Building and Environment 37 (2002) 313–318 315

Fig. 1. Porous slab exposed to two humid environments.

Fig. 2. Theoretical evaluation of moisture content.

capillaries can be ignored. Therefore, conservation equationfor liquid water simpliDes to (Ogniewicz and Tien [8])

�wdWdt

= R; x ∈ (0; l); W ¡Wcr ; (4)

where �w is the liquid density (kg=m3), W the moisturecontent (m3=m3), t the time (s), andWcr the critical moisturecontent (m3=m3), with the initial condition

W =W0(x) at t = 0; (5)

where W0 is the initial moisture distribution in the wet zone(m3=m3).

The critical moisture content for selected building mate-rials is given in Table 1.

Motakef and El-Masri [9] deDned the solution which satis-Des the above conditions as the Drst spatially steady regime.In such a regime the temperature and vapour concentrationproDles are at steady state. There is no condensate motion.The moisture content in the wet zone increases linearly withtime, and the location of the wet region is spatially Dxed anddetermined by the continuity of heat and vapour Juxes atthe wet–dry boundaries. When the local value of moisturecontent reaches its critical level, the capillary forces leadto the migration of condensate into the dry regions and thesubsequent expansion of the wet zone.

Table 1Suctional parameters for selected building materials [7]

Material Density � Critical Capillary(kg=m3) moisture saturation Wcap

content Wcr (m3=m3)(m3=m3)

Ordinary concrete 2400 0.095 0.15Common brick 1700 0.034 0.30Gypsum 950 0.095 0.19Aerated concrete 500 0.200 0.35Glass wool 50 0.950 0.01Exp. polystyrene 30 0.500 0.09

3. Solution method

Consider a wet zone of some thickness and boundarytemperatures as presented in Fig. 1. The energy and vapourcontinuity equations are coupled through the condensationrate term. By eliminating this term, Eqs. (1) and (2) arereduced as follows:

ddx

[�

dTdx

+ �a�DLdCsat(T )

dx

]= 0: (6)

Condensation occurs throughout the width of the wet regionand, hence, the vapour concentration is a unique functionof the temperature distribution. Therefore, Eq. (6) is thesecond-order non-linear diCerential equation in terms of onlytemperature.

Let us take into account the following relation:

Csat(T ) =psat(T )�aRvT

; (7)

where psat is the saturation vapour pressure (Pa), and Rv thegas constant of vapour (J=(kg K)).

Making use of the Clausius–Clapeyron equation to ex-press the vapour pressure derivativedpsat

dT=Lpsat

RvT 2 ; (8)

it can be easily veriDed that the energy equation (6) takesthe form of heat transfer equation

ddx

[k(T )

dTdx

]= 0: (9)

The apparent thermal conductivity [12] used in Eq. (9) isdeDned in the following way:

k(T ) = �+DpL2

Rv

psat(T )T 2

(1 − Rv

LT); (10)

where the vapour permeability is of the form

Dp =�DRvT

: (11)

The apparent thermal conductivity includes the heatconducted across the wet zone in the absence of condensa-tion and heat released by condensation. Fig. 3 shows appar-ent thermal conductivity curves vs. temperature for selectedbuilding materials (with properties shown in Table 2).

316 J. Wyrwa l, A. Marynowicz / Building and Environment 37 (2002) 313–318

Fig. 3. Apparent thermal conductivity vs. temperature for selected buildingmaterials.

Table 2Transport coeMcients for selected building materials [13]

Material Density Heat Vapour� (kg=m3) conductivity � permeability Dp

(W=(m K)) (kg=(m s Pa))

Common brick 1800 0.910 29 × 10−12

Cellular concrete 500 0.220 63 × 10−12

Cement–particle board 450 0.160 104 × 10−12

Mineral wool 100 0.055 133 × 10−12

Glass Dbre mat 80 0.050 167 × 10−12

As can be seen in Fig. 3 for insulating materials (glass-Dbermat and mineral wool) apparent thermal conductivitystrongly depends on temperature while for such materialsas common brick and cellular concrete this dependence isweak.

Obtaining an analytical solution for the non-linear dif-ferential equation (9), governing heat transfer through wetzone, is not possible. In order to get this, we shall replacethe apparent thermal conductivity (10) by an exponentialfunction deDned as follows

k(T ) ≈ k(T0)K (T0−T )=(T0−Tl); (12)

which maintains the proper character of the temperature de-pendence. It can be easily seen that the dimensionless ma-terial coeMcient K is given by

K =k(Tl)k(T0)

; 0¡K¡ 1: (13)

Introducing function (12) into Eq. (9) and integratingthe result, the following Drst-order non-linear diCerentialequation with separated variables is obtained:

K (T0−T )=(T0−Tl) dT = C1 dx; (14)

where C1 is a constant. Solving the above equation in con-junction with boundary conditions (3) yields

−T0 − TllnK

K (T0−T )=(T0−Tl) = C1x + C2; (15)

where coeMcients C1 and C2 are expressed by the relations

C1 =K − 1l

C2 ;

C2 = − T0 − TllnK

: (16)

The solution of the temperature Deld will, therefore, takethe form

T (x) = T0 − T0 − TllnK

ln[(K − 1)

xl

+ 1]

(17)

and is logarithmically dependent on x. The obtained analyt-ical result (17) is easy to evaluate, and shows in the simpleequation how the solution changes in response to the changein all physical variables.

The condensation rate per unit volume can be obtainedfrom Eqs. (1) and (17) as follows:

R(x) =K2Rmax

{[(K − 1)x=l+ 1]}2 ; (18)

where the maximum of condensation rate can be written as

Rmax ≡ R(l) = − �(T0 − Tl)l2L

(K − 1)2

K2 lnK: (19)

The condensation rate exhibits a strong dependence on theratio of the temperature drop across the wet zone to thesquare of that zone width. Therefore, if that ratio increases,the condensation rate will increase as well.

The condensate distribution in the wet zone is obtainedby integration of Eq. (4) with the initial condition (5)

W (x; t) =R(x)�w

t + C; (20)

where the constant C is equal to

C =W0(x): (21)

It is easy to check that the moisture content Deld is Dnallyobtained as

W (x; t) =K2Rmax

�w

t{[(K − 1)x=l+ 1]}2 +W0(x);

W ¡Wcr (22)

and loses its validity beyond the critical value. The con-densate content increases linearly with time and follows thesame proDle as the condensation rate. The maximum of themoisture curve (22) can be found as

Wmax(t) =W (t; l) =Rmax

�wt +W0(l); W ¡Wcr : (23)

The condensate content reaches its critical value when

Wmax(tcr) =Wcr ; (24)

where tcr is the time at which that value is reached (s).The critical time indicates how long the Drst spatially

steady period of condensation lasts.

J. Wyrwa l, A. Marynowicz / Building and Environment 37 (2002) 313–318 317

Table 3Properties of the materials in the wall used as an example for calculation [13]

Material Heat conductivity � Thermal resistance R Vapour permeability Dp Vapour resistance Z(W=(m K)) (m2 (K=W)) (kg=(m s Pa)) (m2 s Pa=kg)

Cement–lime plaster 0.900 0.01 13 × 10−12 7:7 × 108

Glass wool 0.050 1.50 167 × 10−12 4:5 × 108

Light concrete 0.790 0.19 8 × 10−12 187:5 × 108

Fig. 4. Cross section of the wall used as an example for calculation.

4. Practical example

In order to demonstrate the eCects of the various param-eters as well as the initial and boundary conditions on con-densation and condensate accumulation, a practical exampleis presented here. The properties of materials (Table 3) andthe external conditions were chosen to simulate a porouswall. A multilayered building wall is considered as an ex-ample (Fig. 4). The values of parameters employed in thecalculations are as follows:

• latent heat of condensation, L= 2:5 × 106 J=kg,• gas constant of vapour, Rv = 461:9 J=(kg K), and• water density, �w = 1000 kg=m3.

Before analysing condensation it must be establishedwhether, for the conditions considered, condensation takesplace. If at any point in the porous slab the vapour pres-sure is higher than the saturation value corresponding tothe temperature at that point, condensation occurs and theanalysis which includes phenomenon of condensation mustbe followed. The conditions considered above indicate thepresence of condensation in the layers of glass wool andlight concrete (Fig. 4).

In order to calculate the location of wet zone within theanalysed wall and temperatures of its surfaces the computerprogram HUMIDITY is used [14]. This program is designedfor numerical steady-state analysis of the thermal and mois-ture conditions in multilayered walls with diCerent thermaland moisture characteristics. With the program HUMIDITY,for the layer of glass wool, one obtains

Fig. 5. Moisture accumulation in the wet zone of glass wool due to vapourcondensation.

• interior temperature of the wet zone, T0=282:96 K (9:8◦C),• exterior temperature of the wet zone, Tl=273:66 K (0:5◦C),

and• width of the wet zone, l= 0:035 m.

Eqs. (13) and (19) give the following results:

• dimensionless coeMcient, K = 0:8277, and• maximum condensation rate, Rmax=1:53×104 kg=(m3 s).

Because of the large vapour permeability of glass wooland temperature diCerence in the wet zone, the condensationrate is relatively large as well.

The condensate accumulation in wet region of glass woolcan be obtained from the moisture content Deld (22), andfor such parameters as [7]

• initial moisture content (at’=60%); W0=0:0006 m3=m3,and

• capillary saturation, Wcap = 0:01 m3=m3,

is illustrated in Fig. 5. It shows that the condensate contentis larger near the cool surface of the wet zone than near thewarm one and the point of its maximum, equal to capillarysaturation, lies closer to the colder boundary.

Due to large condensation rate occurring during the anal-ysed spatially steady regime, this regime is maintained overa short period of time. Consequently, the time at which cap-illary saturation of glass wool is reached is short as well(Fig. 5). It can be easily veriDed that for light concrete thistime is very long.

318 J. Wyrwa l, A. Marynowicz / Building and Environment 37 (2002) 313–318

5. Conclusions

In civil engineering there is an increasing demand forcalculation methods to estimate the moisture behaviourof building components. Current tasks, such as preservinghistorical buildings or restoring existing constructions, areclosely related to questions concerning present and futuremoisture conditions in a building structure. Due to expen-sive and time-consuming experimental investigations ofthe moisture behaviour of porous building materials, cal-culation studies are becoming increasingly important. Thevalidation of a calculation model requires reliable exper-imental investigations with well-documented initial andboundary conditions, as well as accurate material propertiesand measurements results.

In this paper, one-dimensional simultaneous heat andvapour transfer with phase change in a porous slab has beeninvestigated. A steady analytical solution has been givento the heat transfer equation, where the apparent thermalconductivity is non-constant. Closed-form solutions for thecondensation rate and condensate content are obtained. Fur-thermore, as the resulting equations are solved analytically,the obtained results are universal. The usefulness of anymoisture transport model is very much dependent on itsapplicability, which is determined by the availability of thecorresponding transfer coeMcients. The model presentedrequires material properties which are comparatively simpleand easy to determine. Therefore, a lot of building materialsand constructions can be examined.

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