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Modelling of heat and moisture transfer in buildings
Energy and Buildings 34 (2002) 1033±1043
Modelling of heat and moisture transfer in buildings I. Model program Xiaoshu LuÈ* Laboratory of Structural Engineering and Building Physics, Department of Civil and Environmental Engineering, Helsinki University of Technology, P.O. Box 2100, FIN-02150 Hut, Finland Received 15 November 2001; accepted 2 February 2002
Abstract The overall objective of this work is to develop an accurate model for predicting heat and moisture transfer in buildings including building envelopes and indoor air. The model is based on the fundamental thermodynamic relations. Darcy's law, Fick's law and Fourier's law are used in describing the transfer equations. The resultant nonlinear system of partial differential equations is discretised in space by the ®nite element method. The time marching scheme, Crank±Nicolson scheme, is used to advance the solution in time. The ®nal numerical solution provides transient temperature and moisture distributions in building envelopes as well as temperature and moisture content for building's indoor air subject to outdoor weather conditions described as temperature, relative humidity, solar radiation and wind speed. A series measurements were conducted in order to investigate the model performance. The simulated values were compared against the actual measured values. A good agreement was obtained. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Heat transfer; Moisture transfer; Buildings; Mathematical modelling
1. Introduction Moisture damage is one of the most important factors limiting a building's service life. High moisture level can cause metal corrosion, wood decay and structure deterioration. In addition to the building's construction damage, moisture migrating through building envelopes can lead to poor interior air quality as high ambient moisture levels result in microbial growth. According to Andersen and Korsgaard [1], below 45% indoor relative humidity at 20±22 8C almost no house dust mites are able to survive, but at higher humidity the number of mites increases rapidly up to several thousand mites per gram house dust. In some climates, control of house dust mite allergen loads has been found to be possible by seasonal regulation of indoor relative humidity [2]. Therefore, solving moisture problems is necessary in prolonging a building's service life and avoiding growth microorganisms. Moisture problems in buildings are results of heat and moisture transfer in building envelopes and indoor air. For decades, many researchers have devoted to the work of modelling of heat and moisture transfer in buildings. There are various modelling methods and models. However, there * Tel.: 358-9-451-5305; fax: 358-9-451-3724. E-mail address: [email protected] (X. LuÈ).
are still lacks in the knowledge that de®nes the problem well. The main dif®culty is highlighted by the following facts. Firstly, most of building envelopes are porous media. The structure of a porous medium is complex. Secondly, the moisture transport in a porous medium is a multi-component and multi-phase process. Thirdly, boundary conditions are dif®cult to de®ne. Fourthly, from the experimental point of view, material properties are dif®cult to determine. Material properties are not showing a consistency for different specimens. And the last, from the mathematical point of view, the highly nonlinear governing equations are dif®cult to handle. The complexity, plus the importance, of the modelling work leads to the following fact: even though there exist a great variety of not only model methods, but also numbers of models for heat and moisture transfer in porous media or in buildings, each has its own aim and scale. The feasibility of the model depends primarily on the assumption related to numerous phenomenological auxiliary equations. This fact will be revealed in Section 2. In thispaper, anaccuratemodelofcoupledheat andmoisture transfer in building envelopes and indoor air is developed. The model is then served as an important tool in predicting indoor conditions for further performing control strategies. The modelling and control application will be presented by two parts: paper (I) and paper (II). This paper presents the ®rst part.
0378-7788/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 7 7 8 8 ( 0 2 ) 0 0 0 2 1 - X
X. LuÈ / Energy and Buildings 34 (2002) 1033±1043
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Nomenclature [C] CP D [D] [F] h hm hT m [N] P q qh Q S t T V w
capacitance matrix specific heat (J kg 1 K 1) mass or moisture transfer coefficient (s) stiffness matrix load vector enthalpy (J kg 1) moisture transfer coefficient (m s 1) heat transfer coefficient (W m 2 K 1) moisture flow rate (kg s 1) row vectors containing element shape functions pressure (Pa) moisture flux (kg m 2 s 1) heat flux (W m 2) heat transfer rate (W) saturation time (s) temperature (K) volume (m3) moisture content (kg m 3)
Greek letters D difference of quantities f porosity Z ventilation rate (s 1) l thermal conductivity (W m r density (kg m 3) r differential operator P sum of quantities Subscripts air air c capillary con conductive eff effective g gas i index in indoor j index l liquid out outdoor P pressure rad radiative s solid sol solar radiative sou source surface surface T temperature v vapour Superscripts air air in indoor k index out outdoor
1
K 1)
This paper is organised into the following sections. Following the introduction is a literature review. General governing equations are proposed in Section 3. The test box and real test house for investigating the model performances are described thereafter. The results of simulated against measured values are shown. 2. Literature review The engineering analysis of the heat and moisture transfer in porous media can be dated as early as 1920s in the ®elds of drying science. Drying of solid was studied and postulated as a molecular diffusion process consisting of diffusion from the interior to the surface and the evaporation from the surface [3]. In addition to the drying science, at the same time moisture movement in porous media was explained in terms of surface tension forces or capillary action in other engineering ®elds (e.g. [4,5]). These early basis works have been generalised and extended by the pioneered works [6±8], where a general description of heat and moisture transfer in porous media was presented. The energy equation was incorporating with vapour and liquid transport caused by capillary forces. The capillary force was represented in terms of gradients of the moisture content and temperature. Whitaker [9±12] averaged the transport equations on a representative elementary volume (REV) at the continuum level and obtained the governing equations in a higher level. This modelling method overcomes the modelling dif®culty that porous media are heterogeneous. These early works have been surveyed in [9]. The REV method has been adopted widely by many researchers since then (e.g. [13,14]). Heat and moisture transfer models exist in many engineering areas including drying applications, oil recovery, water resources, soil and building applications, though the underlying applied physical models and mathematical equations are similar. In the following, we shall give a summary of the applications of the model equations. Attention is paid to describing the application scales and simpli®cations of the general model equations, especially of the moisture transfer equations. Many literature works assume a constant gaseous pressure as the atmospheric pressure throughout porous media. This assumption has been adopted in modelling the natural drying processes [6,15]. Heat and moisture transfer for brick and mortar was studied in [15], which focused on the effect of initial moisture content and convective transfer coef®cients on the evaporative cooling of the materials. Some authors have assumed constant material properties for porous media. The assumption leads to the linear property of the system equations, which greatly simpli®es the problems [16,17]. Paper [16] compared the results of drying of a timber section with constant material properties and varied material properties. It was found that there was
X. LuÈ / Energy and Buildings 34 (2002) 1033±1043
virtually no difference in results. One-dimensional model equations were used in [17] to study convective drying for brick and mortar. The simulated results were compared with the calculated and measured results by Kallel et al. [15]. Good agreement was obtained. Many authors used different heat and moisture transfer equations for different regions depending on the state of the presented water. Two regions were determined [18] by de®nitions of the hygroscopic region where only bound water remains and the nonhygroscopic region where free water exists. Different formulation equations were used in [19] for the funicular region where capillary ¯ow of liquid is dominating and for the pendular region where moisture is transported by vapour diffusion. In most of the modelling works, general heat and moisture transfer equations have been used [20,21]. Paper [20] investigated the heat and moisture transport in initially fully saturated concrete for drying process. Wood plank drying was simulated in [21]. In building applications, paper [22] used general heat and moisture model to study drying by forced convection in porous media. The moisture transport in the model was mainly due to the liquid and gaseous ¯ow. The simulations were performed for clay brick and wood. Paper [23] presented a general model in studying heat and moisture transfer in building structures. The simulation was conducted for a building basement garage. In [24], vapour diffusion and capillary ¯ow was considered in the moisture ¯ow. The simulations were carried out for a building roof. The results were compared with the traditional simple calculation methods, the dew point method and Claser's method. These models do not include the mechanism of bound water transport, which is obvious important in modelling drying processes. The above-reviewed models are heat and moisture transfer models for building envelopes. When building indoor air is involved, often the coupled heat and moisture transfer for indoor air is modelled in lumped form, where a well-mixed indoor air distribution is assumed (e.g. [25]). Some authors modelled the turbulent natural convection of indoor air, normally a computational ¯uid dynamics (CFD) software is needed (e.g. [26]). 3. Mathematical model equations This paper adopts the REV method in deriving heat and moisture transfer equations for building envelopes. The approach is to start with conservation equations, constitutive equations and state equations by means of thermodynamics of Gibbs relation. 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. The cross-transport phenomena (e.g. Soret effect and its reciprocal Dufour effect) are neglected.
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3.1. Governing equations for building envelopes The building envelope (porous medium) is assumed to consist of liquid phase and gaseous phase. Gaseous pressure is assumed to be constant as the atmospheric pressure. Like most of the moisture transfer models used in building applications, the mechanism of bound water transport is neglected. The complete differential form of the conservation laws for moisture and energy can be summarised as follows. 3.1.1. Moisture conservation equation For total moisture content w containing liquid and vapour, we have @w r ~ ql ~ qv 0 @t
(1)
where ¯ux ~ ql presents the capillary ¯ow and ¯ux ~ qv presents the vapour diffusion. By Darcy's law and Fick's law, we have ~ ql
Dl r Pc
(2)
~ qv
Dv r Pv
(3)
where Pc and Pv, are capillary and vapour pressure, respectively. 3.1.2. Energy conservation equation Only one equation for energy is required if we assume that the temperature is the same in all the phases. @ rCP T r hv~ qcon 0 qv hl~ ql r ~ @t
(4)
where rCP presents the averaged thermal storage, h presents the enthalpies for vapour and liquid phases and ~ qcon is the thermal conduction term. Here no heat source is assumed. By Fourier's law ~ qcon
leff rT
(5)
where leff is the effective thermal conductivity. Calculations for the averaged rCP and leff are according to the weighted volumetric distribution (e.g. [27]). 3.1.3. Boundary conditions For envelopes' surfaces exposed to outdoor air, we have, for example: out ~ ql ~ qv jsurface;out hout m rv
rv jsurface;out
(6)
hv~ qcon jsurface;out qv hl~ ql ~ hout T Tout
Tjsurface;out qsol
(7)
where qsol presents the solar energy ¯ux received by the surface. Similar boundary conditions can be obtained for envelopes' surfaces exposed to indoor air. ~ ql ~ qv jsurface;in hin m rv jsurface;in
rin v
(8)
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Fig. 1. Sorption isotherm for plywood [28].
hv~ qcon jsurface;in hin qv hl~ ql ~ T Tjsurface;in
Tin qh;rad (9)
where qh,rad presents the radiative heat ¯ux received by the surface. 3.1.4. Initial conditions Initial conditions are given by a measurement or by assuming a steady state condition. 3.1.5. Thermodynamic equilibrium and constraints To close the above system of equations, the phase equilibrium equation for building material has to be given. This is described as the moisture retention function. Traditionally in building applications, the function is divided into two ranges. The ®rst range, hygroscopic range, presents the relative humidity up to 95 or 99%. In this range, the function is often called as sorption isotherm. It gives a function of the amount of water respective to the relative humidity at a constant temperature. The sorption isotherm can be represented by Langmuir equation or BET equation. Fig. 1 shows the example of the sorption isotherm for plywood [28]. It presents the hysteresis. In building applications, the temperature-dependent factor is often neglected. When relative humidity is approaching 100%, in the nonhygroscopic range, water is bound by capillary condensation and the sorption isotherm cannot provide enough information any more. The equilibrium moisture content at a given hydraulic pressure is normally expressed by a function between the capillary pressure and the moisture content, called moisture retention curve by some authors. Fig. 2 gives the schematic picture of a moisture retention curve. The calculation of capillary pressure is usually represented by the empirical Leverret approach (e.g. [29]). Often, sorption isotherms and moisture retention curves are ®tted by empirical functions.
It should be noted that in some capillary region in a hygroscopic range, both the sorption isotherm and the moisture retention curve give the same information and can be formulated with each other by Kelvin's law. 3.2. Governing equations for building indoor air The transition from laminar to turbulent natural convection for building indoor air depends on the Rayleigh number. A signi®cant heating and ventilation may cause a turbulence of the indoor air. Generally, a well-mixed indoor air can be assumed to simplify the calculation. The heat and moisture transfer equations for indoor air can be written as P dTin Qsou i Qi Z Tout Tin (10) dt VCPair rair VCPair rair P drin mi msou out in v Z rv (11) rv i dt V V P where Qi is the heat transfer rate from the building P envelopes which contribute to the indoor air and mi is
Fig. 2. Schematic picture of a moisture retention curve.
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Fig. 3. Schematic picture of the measurement equipment.
the moisture ¯ow rate from the building envelopes which contribute to the indoor air. And Qsou and msou are the building's indoor heat and moisture generation rates, or source rates. Initial condition is normally determined by a measurement. Eqs. (1)±(11) are coupled heat and moisture transfer equations for building envelopes and indoor air which can be solved numerically by the ®nite element method. 3.3. Numerical procedure The Galerkin's weighted residual method is used to discretise the space for Eqs. (1) and (4). The state variable for moisture transfer is chosen as vapour pressure when liquid phase is not generated or capillary pressure, denoted as P. This is mainly due to the consideration that vapour and capillary pressures have continuous properties at the interfaces of contacted materials. Temperature is selected as the state variable for transfer. For an element, the state variables are approximated by linearising two node elements, for example: P Ni Pi Nj Pj and
or
P NfPg ) P NfPg
T NfTg
(12)
where [N] is the row vectors containing element shape functions. Applying Galerkin's weighted residual formula to Eqs. (1) and (4) Z T @w r ~ ql ~ N qv 0 (13) @t Z T @ rCP T r hv~ N qcon 0 qv hl~ qv r ~ @t
where [CP], [CT], [CPP], [CPT], [CTT], [CTP], {fP} and {fT} are corresponding matrices consisting of the coef®cients derived from Eqs. (13) and (14). And P_ denotes the time derivative of P. By assembling the element equations, in association with indoor air equations (Eqs. (10) and (11)), we get the global ordinary differential equations simpli®ed as " # CPP 0 fP CPP CPT P P_ (17) T 0 CTT CTP CTT fT T_ The system of Eq. (17), can be written in a simpler form as _ DfXg fFg CfXg
(18)
where {X} is the state variable vector, [C] the capacitance matrix, [D] the stiffness matrix and {F} is the load vector. A suitable time marching procedure is needed in obtaining the solution in time from a given initial condition. Based on the structure of the matrix [D], Crank±Nicolson time marching scheme is applied. Applying the central difference Crank±Nicolson scheme, we get C 12 DtDfX k1 g C
1 k 2 DtDfX g
12 Dt fFgk1 fFgk (19)
where k presents time step index. The method is unconditional stable. However, the time step is controlled in this work in order to obtain desired accuracy. The resultant algebric equation system Eq. (19) needs to be solved at each time. Newton's iterative method can be used. As matrices [C] and [D] are often dependent on temperature and moisture content, [F] is often a nonlinear
(14) where [N]T presents the transpose of [N]. Inserting boundary conditions, applying the phase equilibrium relations and rearranging Eqs. (13) and (14), the following system of ordinary differential equations is obtained: _ CPP fPg CPT fTg ffP g CP fPg
(15)
_ CTT fTg CTP fPg ffT g CT fTg
(16)
Fig. 4. The wall constructions of the test box.
1038 X. LuÈ / Energy and Buildings 34 (2002) 1033±1043
Fig. 5. Simulated against measured indoor (a) temperatures and (b) moisture contents in the test box (series 1); (c) temperatures and (d) moisture contents in the test box (series 2).
X. LuÈ / Energy and Buildings 34 (2002) 1033±1043
function, the implicit method requires to solve a very large system of nonlinear algebric equations at each time step. The computing load is signi®cant. Simpli®cation is often made in modelling equations. One way to overcome this heavy computation is to apply the implicit method only to parts of the state variables, vapour or capillary pressure for example. For each time step, the assembled system of equations of moisture transfer, similar to Eq. (19) but much smallersized, is solved implicitly. Old time temperature values are used. With the new time step moisture content values, the assembled system of equations of temperature is then solved explicitly. This method reduced the computing time greatly. It is worthwhile noting here that for a complicated problem, the method of combination of explicit and implicit methods considerably limits the time step selection, thus a fully implicit method is recommended.
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4. Model verification and simulation results The testing of the model's computational performance and the veri®cation of its predictions were performed by a test box and a real test house. The test box was 454 mm wide by 454 mm long with height 345 mm. An independent air pump was installed to one side of the box in order to provide required ventilation air. The pump was connected to a climatic chamber to maintain the desired air temperature and relative humidity. Outdoor air was heated or cooled in the climatic chamber when passing through a copper pipe installed inside the climatic chamber. The air ¯ow rate in the pipe was from 4:2 10 4 to 6:1 10 4 m 3 s 1 at the atmospherical pressure and room temperature. There were water traps acted as condensation collector for handling the condensed air. Additionally, an air mixer was contained
Fig. 6. Simulated against measured indoor (a) temperatures and (b) moisture contents in the test box with wood pieces as storage.
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inside the box in order to maintain a well-mixed indoor temperature and moisture content in the box. The schematic picture of the measurement equipment is shown in Fig. 3. The walls forming the test box were composed of an outer layer of 12 mm plywood panel with oil-base paint, an layer of 48 mm insulation and an inside layer of 12 mm plywood panel with oil-base paint (see Fig. 4). The test box was surrounded by the laboratory room. The measuring instruments used in the measurement were sensors and computers. The measurement can be classi®ed into two categories: the measurements with and without internal thermal and moisture storage. In the second category, some small wood pieces were mounted inside the box. The material physics properties as well as their heat and moisture transport properties have been published in [30]. Some local sources have also been considered (e.g. [31]).
Fig. 7. Schematic picture of the constructions of Anjala house.
Fig. 8. Calculated against measured indoor (a) temperatures and (b) moisture contents in Anjala house.
X. LuÈ / Energy and Buildings 34 (2002) 1033±1043
The insulation (mineral wool) has about 30% moisture content of its saturation value at 50% relative humidity. Here the dependence of sorption properties on temperatures was not included. The effective thermal conductivity was estimated from [27] 1=4
1=4 leff l1=4 g Sg f ll Sl f ls 1
f4
(20)
and the averaged thermal mass was given as rCP rg CP;g Sg rl CP;l Sl rs CP;s Ss
(21)
Additional information necessary to complete the model included the surface heat and moisture transfer coef®cients which must be determined experimentally in principle. Very often, a well-known Lewis relation is used: hin m
hin T rair CPair
(22)
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where hin T is the convective heat transfer coef®cient. According to [30], an empirical formula was used: hin T 2jTsurface;in
Tin j0:25
(23)
A value of 0.9 was used the emissivity in calculating the radiative heat exchange. For the surface of plywood panel with oil-base paint, the following empirical relation was employed in our calculation [32]: hin m
1 75 103
(24)
2 1 K 1 and hout Moreover, hout T 6:25 W m m 0:001 ms were used as the average values of the exterior surface transfer coef®cients. The results of experimental tests were compared with model simulations to verify the model predictions. The comparison is summarised in the following two groups.
Fig. 9. Simulated indoor (a) temperatures and (b) relative humidity with different surface heat transfer coefficients.
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The ®rst group was the representative of the base case of the test box displayed in Fig. 5. The air coming from outdoor was heated or cooled in the climatic chamber and then entered and left the box with ¯ow rates from 4:2 10 4 to 6:2 10 4 m3 s 1 at atmosperical pressure and room temperature. The air mixed fan mixed the incoming air in the box. The velocity of the air was from 0.2 to 0.6 ms 1. The temperatures of incoming air, indoor air in the box and outgoing air were then measured and saved to the computer. The second group was the representative of the case with indoor thermal and moisture storage. Fourteen wood pieces were mounted inside the box. Results are demonstrated in Fig. 6. These ®gures demonstrate that the model ®ts the experimental data. Besides the test box, the model has been veri®ed with our real test houses. We take Anjala house as an example. The Anjala house is a museum house situated in Anjala, Finland. The schematic picture of its wall constructions is presented in Fig. 7. The house is a two-storey house with ¯oor area 11:5 m 27 m. There was no heating operation and the house was naturally ventilated during the measured period. A real photo and detailed description of Anjala house was presented in [30]. Fig. 8 shows the validation results. As Anjala house contains crawl space and our program has not been ready for that, the simulated indoor air temperature tended somewhat lower than the measured result especially in winter time. Nevertheless, good agreement was obtained. The difference between simulated and measured data was less than 10% compared to the absolute values. In simulating the Anjala house, the transient solar radiation was calculated from the program developed in our local laboratory. The convective heat transfer coef®cient was calculated from Eq. (23), the surface moisture transfer coef®cients were then obtained by Lewis formula. For the exterior 2 1 walls, hout K 1 and hout were T 25 W m m 0:003 ms used as the average values. Wind information was not available. Therefore, an approximate value 0.3 h 1 was adopted as an averaged ventilation rate throughout the calculations. The above-described parameters are not intrinsic properties of the ¯ow. Their values are dependent on the temperature, moisture distribution on the surfaces and the surface geometry as well as wind situations. Fig. 9 demonstrates the differences of the temperatures and relative humidity for the interior surface of the walls with different heat transfer coef®cient values for the test box. The differences are clearly recognised. The picture demonstrates the importance of the effect of the heat transfer coef®cient on determining heat and moisture transfer in buildings. 5. Conclusions Through an extensive survey work, a heat and moisture transfer model for buildings is proposed in this paper. For
indoor air, a well-mixed model is assumed. The model has been validated with our test box and real test houses. There are certainly many areas requiring further research. As already demonstrated, the surface heat and moisture transfer coef®cients have big effects on predicting heat and moisture transfer behaviours and have to be determined experimentally especially for some extreme cases. Secondly, thermal and moisture behaviours in window are only limited in a simple situation where the window is considered as one component of building envelopes. Effects of window frames etc. are not included yet. Thirdly, the effect of raining wetting is not considered. Even though the model developed in this paper does not cover every heat and moisture aspect, it is still proved to a useful model allowing obtaining high accuracy for building researchers. A companion paper will show how the model is applied to the study of the control of indoor thermal and moisture levels by properly heating the indoor air and ventilating the outdoor air. Acknowledgements The author would like to thank the Finnish Academy for the ®nancial support. Special thanks to professor M. Viljanen who provided many ideas for this study. Thanks also go to M. Suihkonen for her experimental work. References [1] I. Andersen, J. Korsgaard, Asthma and the indoor environment: assessment of the health implications of high indoor air humidity, Environment International 12 (1984) 121±127. [2] S. King, Asthma, house dust mites and indoor climate, Architectural Science Review 40 (1997) 43±47. [3] W.K. Lewis, The rate of drying of solid materials, Industrial Engineering Chemistry 13 (1921) 427±432. [4] W. Gardnen, J.A. Widtsoe, The movement of soil moisture, Soil Science 11 (1920) 215±232. [5] L.A. Richards, Capillary conduction of liquids through porous mediums, Journal of Applied Physics 1 (1931) 318±333. [6] J.R. Philip, D.A. DeVries, Moisture movement in porous materials under temperature gradients, Transactions of American Geophysics Union 38 (1957) 222±232. [7] D.A. De Vries, Simultaneous transfer of heat and moisture in porous media, Transactions of American Geophysics Union 39 (1958) 909± 916. [8] A.W. Luikov, Heat and mass transfer in capillary-porous bodies, Pergamon Press, Oxford, 1966. [9] S. Whitaker, Simultaneous heat, mass and momentum transferÐa theory of drying, Advanced Heat Transfer 13 (1977) 119±203. [10] S. Whitaker, Flow in porous media. I. A theoretical derivation of Darcy's law, Transport in Porous Media 1 (1986) 3±26. [11] S. Whitaker, Flow in porous media. II. The governing equations for immiscible, two-phase flow, Transport in Porous Media 1 (1986) 105±126. [12] S. Whitaker, Flow in porous media. III. Deformable media, Transport in Porous Media 1 (1986) 127±154. [13] Y. Bachmat, J. Bear, Macroscopic modelling of transport phenomena in porous media. 1. The continuum approach, Transport in Porous Media 1 (1986) 213±240.
X. LuÈ / Energy and Buildings 34 (2002) 1033±1043 [14] J. Bear, Y. Bachmat, Macroscopic modelling of transport phenomena in porous media. 2. Applications to mass, momentum and energy transport, Transport in Porous Media 1 (1986) 241±269. [15] F. Kallel, N. Galanis, B. Perrin, R. Javelas, Effects of moisture on temperature during drying of consolidation porous materials, Transactions of ASME 115 (1993) 724±733. [16] H.R. Thomas, K. Morgan, R.W. Lewis, A fully nonlinear analysis of heat and mass transfer problems in porous bodies, International Journal of Numerical Methods Engineering 15 (1980) 1381±1393. [17] K. Murugesan, K.N. Seetharamu, P.A. Aswatha Narayana, A onedimensional analysis of convective drying of porous materials, Heat and Mass Transfer 32 (1996) 81±88. [18] F. Couture, W. Jomaa, J.R. Puiggali, Relative permeability relations: a key factor for a drying model, Transport in Porous Media 23 (1996) 303±335. [19] C.L.C. Huang, Multi-phase moisture transfer in porous media subjected to temperature gradient, International Journal of Heat Mass Transfer 22 (1979) 1295±1307. [20] J. Selih, A.C.M. Sousa, T.W. Bremner, Moisture transport in initially fully saturated concrete during drying, Transport in Porous Media 24 (1996) 81±106. [21] N. Ouelhazi, G. Arnaud, J.P. Fohr, A two-dimensional study of wood plank drying. The effect of gaseous pressure below boiling point, Transport in Porous Media 7 (1992) 39±61. [22] N.S. Ben, P. Perre, Detailed study of a model of heat and mass transfer during convective drying of porous media, International Journal of Heat and Mass Transfer 31 (1988) 957±967.
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[23] P. Haupl, J. Grunewald, H. Fechner, Coupled heat air and moisture transfer in building structures, International Journal of Heat Mass Transfer 40 (1997) 1633±1642. [24] C.R. Pedersen, Prediction of moisture transfer in building constructions, Building and Environment 27 (1992) 387±397. [25] S.P.W. Wong, Simulation of simultaneous heat and moisture transfer by using the finite difference method and verified tests in a test chamber, ASHRAE Transactions 96 (1990) 472±486. [26] G. Gan, H.B. Awbi, Numerical simulation of the indoor environment, Building and Environment 29 (1994) 449±459. [27] W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to Ceramics, 2nd Edition, Wiley/Intersciences, New York, 1976. [28] R.F. Richards, D.M. Burch, W.C. Thomas, Water vapor sorption measurements of common building materials, ASHRAE Transactions 98 (1992) 475±483. [29] K.S. Udell, Heat transfer in porous media considering phase change and capillarityÐthe heat pipe effect, International Journal of Heat Mass Transfer 28 (1985) 485±495. [30] RIL 117, LaÈmmoÈn ja kosteudeneristys, Suomen RakennusinsinoÈoÈrien Liitto, Finland, 1997. [31] M. Viljanen, J. Bergman, S. Grabko, X. Lu, R. YrjoÈlaÈ, Ensuring the long service life of unheated buildings, Helsinki University of Technology, TKK-TRT-100, Espoo, 1999. [32] J. HaÈnninen, A method to analyse the transient moisture physical behaviour of building constructions in hygroscopic moisture region, Master thesis, Helsinki University of Technology, Espoo, 1986 (in Finnish).
Modelling of heat and moisture transfer in buildings I. Model program Xiaoshu LuÈ* Laboratory of Structural Engineering and Building Physics, Department of Civil and Environmental Engineering, Helsinki University of Technology, P.O. Box 2100, FIN-02150 Hut, Finland Received 15 November 2001; accepted 2 February 2002
Abstract The overall objective of this work is to develop an accurate model for predicting heat and moisture transfer in buildings including building envelopes and indoor air. The model is based on the fundamental thermodynamic relations. Darcy's law, Fick's law and Fourier's law are used in describing the transfer equations. The resultant nonlinear system of partial differential equations is discretised in space by the ®nite element method. The time marching scheme, Crank±Nicolson scheme, is used to advance the solution in time. The ®nal numerical solution provides transient temperature and moisture distributions in building envelopes as well as temperature and moisture content for building's indoor air subject to outdoor weather conditions described as temperature, relative humidity, solar radiation and wind speed. A series measurements were conducted in order to investigate the model performance. The simulated values were compared against the actual measured values. A good agreement was obtained. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Heat transfer; Moisture transfer; Buildings; Mathematical modelling
1. Introduction Moisture damage is one of the most important factors limiting a building's service life. High moisture level can cause metal corrosion, wood decay and structure deterioration. In addition to the building's construction damage, moisture migrating through building envelopes can lead to poor interior air quality as high ambient moisture levels result in microbial growth. According to Andersen and Korsgaard [1], below 45% indoor relative humidity at 20±22 8C almost no house dust mites are able to survive, but at higher humidity the number of mites increases rapidly up to several thousand mites per gram house dust. In some climates, control of house dust mite allergen loads has been found to be possible by seasonal regulation of indoor relative humidity [2]. Therefore, solving moisture problems is necessary in prolonging a building's service life and avoiding growth microorganisms. Moisture problems in buildings are results of heat and moisture transfer in building envelopes and indoor air. For decades, many researchers have devoted to the work of modelling of heat and moisture transfer in buildings. There are various modelling methods and models. However, there * Tel.: 358-9-451-5305; fax: 358-9-451-3724. E-mail address: [email protected] (X. LuÈ).
are still lacks in the knowledge that de®nes the problem well. The main dif®culty is highlighted by the following facts. Firstly, most of building envelopes are porous media. The structure of a porous medium is complex. Secondly, the moisture transport in a porous medium is a multi-component and multi-phase process. Thirdly, boundary conditions are dif®cult to de®ne. Fourthly, from the experimental point of view, material properties are dif®cult to determine. Material properties are not showing a consistency for different specimens. And the last, from the mathematical point of view, the highly nonlinear governing equations are dif®cult to handle. The complexity, plus the importance, of the modelling work leads to the following fact: even though there exist a great variety of not only model methods, but also numbers of models for heat and moisture transfer in porous media or in buildings, each has its own aim and scale. The feasibility of the model depends primarily on the assumption related to numerous phenomenological auxiliary equations. This fact will be revealed in Section 2. In thispaper, anaccuratemodelofcoupledheat andmoisture transfer in building envelopes and indoor air is developed. The model is then served as an important tool in predicting indoor conditions for further performing control strategies. The modelling and control application will be presented by two parts: paper (I) and paper (II). This paper presents the ®rst part.
0378-7788/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 7 7 8 8 ( 0 2 ) 0 0 0 2 1 - X
X. LuÈ / Energy and Buildings 34 (2002) 1033±1043
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Nomenclature [C] CP D [D] [F] h hm hT m [N] P q qh Q S t T V w
capacitance matrix specific heat (J kg 1 K 1) mass or moisture transfer coefficient (s) stiffness matrix load vector enthalpy (J kg 1) moisture transfer coefficient (m s 1) heat transfer coefficient (W m 2 K 1) moisture flow rate (kg s 1) row vectors containing element shape functions pressure (Pa) moisture flux (kg m 2 s 1) heat flux (W m 2) heat transfer rate (W) saturation time (s) temperature (K) volume (m3) moisture content (kg m 3)
Greek letters D difference of quantities f porosity Z ventilation rate (s 1) l thermal conductivity (W m r density (kg m 3) r differential operator P sum of quantities Subscripts air air c capillary con conductive eff effective g gas i index in indoor j index l liquid out outdoor P pressure rad radiative s solid sol solar radiative sou source surface surface T temperature v vapour Superscripts air air in indoor k index out outdoor
1
K 1)
This paper is organised into the following sections. Following the introduction is a literature review. General governing equations are proposed in Section 3. The test box and real test house for investigating the model performances are described thereafter. The results of simulated against measured values are shown. 2. Literature review The engineering analysis of the heat and moisture transfer in porous media can be dated as early as 1920s in the ®elds of drying science. Drying of solid was studied and postulated as a molecular diffusion process consisting of diffusion from the interior to the surface and the evaporation from the surface [3]. In addition to the drying science, at the same time moisture movement in porous media was explained in terms of surface tension forces or capillary action in other engineering ®elds (e.g. [4,5]). These early basis works have been generalised and extended by the pioneered works [6±8], where a general description of heat and moisture transfer in porous media was presented. The energy equation was incorporating with vapour and liquid transport caused by capillary forces. The capillary force was represented in terms of gradients of the moisture content and temperature. Whitaker [9±12] averaged the transport equations on a representative elementary volume (REV) at the continuum level and obtained the governing equations in a higher level. This modelling method overcomes the modelling dif®culty that porous media are heterogeneous. These early works have been surveyed in [9]. The REV method has been adopted widely by many researchers since then (e.g. [13,14]). Heat and moisture transfer models exist in many engineering areas including drying applications, oil recovery, water resources, soil and building applications, though the underlying applied physical models and mathematical equations are similar. In the following, we shall give a summary of the applications of the model equations. Attention is paid to describing the application scales and simpli®cations of the general model equations, especially of the moisture transfer equations. Many literature works assume a constant gaseous pressure as the atmospheric pressure throughout porous media. This assumption has been adopted in modelling the natural drying processes [6,15]. Heat and moisture transfer for brick and mortar was studied in [15], which focused on the effect of initial moisture content and convective transfer coef®cients on the evaporative cooling of the materials. Some authors have assumed constant material properties for porous media. The assumption leads to the linear property of the system equations, which greatly simpli®es the problems [16,17]. Paper [16] compared the results of drying of a timber section with constant material properties and varied material properties. It was found that there was
X. LuÈ / Energy and Buildings 34 (2002) 1033±1043
virtually no difference in results. One-dimensional model equations were used in [17] to study convective drying for brick and mortar. The simulated results were compared with the calculated and measured results by Kallel et al. [15]. Good agreement was obtained. Many authors used different heat and moisture transfer equations for different regions depending on the state of the presented water. Two regions were determined [18] by de®nitions of the hygroscopic region where only bound water remains and the nonhygroscopic region where free water exists. Different formulation equations were used in [19] for the funicular region where capillary ¯ow of liquid is dominating and for the pendular region where moisture is transported by vapour diffusion. In most of the modelling works, general heat and moisture transfer equations have been used [20,21]. Paper [20] investigated the heat and moisture transport in initially fully saturated concrete for drying process. Wood plank drying was simulated in [21]. In building applications, paper [22] used general heat and moisture model to study drying by forced convection in porous media. The moisture transport in the model was mainly due to the liquid and gaseous ¯ow. The simulations were performed for clay brick and wood. Paper [23] presented a general model in studying heat and moisture transfer in building structures. The simulation was conducted for a building basement garage. In [24], vapour diffusion and capillary ¯ow was considered in the moisture ¯ow. The simulations were carried out for a building roof. The results were compared with the traditional simple calculation methods, the dew point method and Claser's method. These models do not include the mechanism of bound water transport, which is obvious important in modelling drying processes. The above-reviewed models are heat and moisture transfer models for building envelopes. When building indoor air is involved, often the coupled heat and moisture transfer for indoor air is modelled in lumped form, where a well-mixed indoor air distribution is assumed (e.g. [25]). Some authors modelled the turbulent natural convection of indoor air, normally a computational ¯uid dynamics (CFD) software is needed (e.g. [26]). 3. Mathematical model equations This paper adopts the REV method in deriving heat and moisture transfer equations for building envelopes. The approach is to start with conservation equations, constitutive equations and state equations by means of thermodynamics of Gibbs relation. 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. The cross-transport phenomena (e.g. Soret effect and its reciprocal Dufour effect) are neglected.
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3.1. Governing equations for building envelopes The building envelope (porous medium) is assumed to consist of liquid phase and gaseous phase. Gaseous pressure is assumed to be constant as the atmospheric pressure. Like most of the moisture transfer models used in building applications, the mechanism of bound water transport is neglected. The complete differential form of the conservation laws for moisture and energy can be summarised as follows. 3.1.1. Moisture conservation equation For total moisture content w containing liquid and vapour, we have @w r ~ ql ~ qv 0 @t
(1)
where ¯ux ~ ql presents the capillary ¯ow and ¯ux ~ qv presents the vapour diffusion. By Darcy's law and Fick's law, we have ~ ql
Dl r Pc
(2)
~ qv
Dv r Pv
(3)
where Pc and Pv, are capillary and vapour pressure, respectively. 3.1.2. Energy conservation equation Only one equation for energy is required if we assume that the temperature is the same in all the phases. @ rCP T r hv~ qcon 0 qv hl~ ql r ~ @t
(4)
where rCP presents the averaged thermal storage, h presents the enthalpies for vapour and liquid phases and ~ qcon is the thermal conduction term. Here no heat source is assumed. By Fourier's law ~ qcon
leff rT
(5)
where leff is the effective thermal conductivity. Calculations for the averaged rCP and leff are according to the weighted volumetric distribution (e.g. [27]). 3.1.3. Boundary conditions For envelopes' surfaces exposed to outdoor air, we have, for example: out ~ ql ~ qv jsurface;out hout m rv
rv jsurface;out
(6)
hv~ qcon jsurface;out qv hl~ ql ~ hout T Tout
Tjsurface;out qsol
(7)
where qsol presents the solar energy ¯ux received by the surface. Similar boundary conditions can be obtained for envelopes' surfaces exposed to indoor air. ~ ql ~ qv jsurface;in hin m rv jsurface;in
rin v
(8)
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Fig. 1. Sorption isotherm for plywood [28].
hv~ qcon jsurface;in hin qv hl~ ql ~ T Tjsurface;in
Tin qh;rad (9)
where qh,rad presents the radiative heat ¯ux received by the surface. 3.1.4. Initial conditions Initial conditions are given by a measurement or by assuming a steady state condition. 3.1.5. Thermodynamic equilibrium and constraints To close the above system of equations, the phase equilibrium equation for building material has to be given. This is described as the moisture retention function. Traditionally in building applications, the function is divided into two ranges. The ®rst range, hygroscopic range, presents the relative humidity up to 95 or 99%. In this range, the function is often called as sorption isotherm. It gives a function of the amount of water respective to the relative humidity at a constant temperature. The sorption isotherm can be represented by Langmuir equation or BET equation. Fig. 1 shows the example of the sorption isotherm for plywood [28]. It presents the hysteresis. In building applications, the temperature-dependent factor is often neglected. When relative humidity is approaching 100%, in the nonhygroscopic range, water is bound by capillary condensation and the sorption isotherm cannot provide enough information any more. The equilibrium moisture content at a given hydraulic pressure is normally expressed by a function between the capillary pressure and the moisture content, called moisture retention curve by some authors. Fig. 2 gives the schematic picture of a moisture retention curve. The calculation of capillary pressure is usually represented by the empirical Leverret approach (e.g. [29]). Often, sorption isotherms and moisture retention curves are ®tted by empirical functions.
It should be noted that in some capillary region in a hygroscopic range, both the sorption isotherm and the moisture retention curve give the same information and can be formulated with each other by Kelvin's law. 3.2. Governing equations for building indoor air The transition from laminar to turbulent natural convection for building indoor air depends on the Rayleigh number. A signi®cant heating and ventilation may cause a turbulence of the indoor air. Generally, a well-mixed indoor air can be assumed to simplify the calculation. The heat and moisture transfer equations for indoor air can be written as P dTin Qsou i Qi Z Tout Tin (10) dt VCPair rair VCPair rair P drin mi msou out in v Z rv (11) rv i dt V V P where Qi is the heat transfer rate from the building P envelopes which contribute to the indoor air and mi is
Fig. 2. Schematic picture of a moisture retention curve.
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Fig. 3. Schematic picture of the measurement equipment.
the moisture ¯ow rate from the building envelopes which contribute to the indoor air. And Qsou and msou are the building's indoor heat and moisture generation rates, or source rates. Initial condition is normally determined by a measurement. Eqs. (1)±(11) are coupled heat and moisture transfer equations for building envelopes and indoor air which can be solved numerically by the ®nite element method. 3.3. Numerical procedure The Galerkin's weighted residual method is used to discretise the space for Eqs. (1) and (4). The state variable for moisture transfer is chosen as vapour pressure when liquid phase is not generated or capillary pressure, denoted as P. This is mainly due to the consideration that vapour and capillary pressures have continuous properties at the interfaces of contacted materials. Temperature is selected as the state variable for transfer. For an element, the state variables are approximated by linearising two node elements, for example: P Ni Pi Nj Pj and
or
P NfPg ) P NfPg
T NfTg
(12)
where [N] is the row vectors containing element shape functions. Applying Galerkin's weighted residual formula to Eqs. (1) and (4) Z T @w r ~ ql ~ N qv 0 (13) @t Z T @ rCP T r hv~ N qcon 0 qv hl~ qv r ~ @t
where [CP], [CT], [CPP], [CPT], [CTT], [CTP], {fP} and {fT} are corresponding matrices consisting of the coef®cients derived from Eqs. (13) and (14). And P_ denotes the time derivative of P. By assembling the element equations, in association with indoor air equations (Eqs. (10) and (11)), we get the global ordinary differential equations simpli®ed as " # CPP 0 fP CPP CPT P P_ (17) T 0 CTT CTP CTT fT T_ The system of Eq. (17), can be written in a simpler form as _ DfXg fFg CfXg
(18)
where {X} is the state variable vector, [C] the capacitance matrix, [D] the stiffness matrix and {F} is the load vector. A suitable time marching procedure is needed in obtaining the solution in time from a given initial condition. Based on the structure of the matrix [D], Crank±Nicolson time marching scheme is applied. Applying the central difference Crank±Nicolson scheme, we get C 12 DtDfX k1 g C
1 k 2 DtDfX g
12 Dt fFgk1 fFgk (19)
where k presents time step index. The method is unconditional stable. However, the time step is controlled in this work in order to obtain desired accuracy. The resultant algebric equation system Eq. (19) needs to be solved at each time. Newton's iterative method can be used. As matrices [C] and [D] are often dependent on temperature and moisture content, [F] is often a nonlinear
(14) where [N]T presents the transpose of [N]. Inserting boundary conditions, applying the phase equilibrium relations and rearranging Eqs. (13) and (14), the following system of ordinary differential equations is obtained: _ CPP fPg CPT fTg ffP g CP fPg
(15)
_ CTT fTg CTP fPg ffT g CT fTg
(16)
Fig. 4. The wall constructions of the test box.
1038 X. LuÈ / Energy and Buildings 34 (2002) 1033±1043
Fig. 5. Simulated against measured indoor (a) temperatures and (b) moisture contents in the test box (series 1); (c) temperatures and (d) moisture contents in the test box (series 2).
X. LuÈ / Energy and Buildings 34 (2002) 1033±1043
function, the implicit method requires to solve a very large system of nonlinear algebric equations at each time step. The computing load is signi®cant. Simpli®cation is often made in modelling equations. One way to overcome this heavy computation is to apply the implicit method only to parts of the state variables, vapour or capillary pressure for example. For each time step, the assembled system of equations of moisture transfer, similar to Eq. (19) but much smallersized, is solved implicitly. Old time temperature values are used. With the new time step moisture content values, the assembled system of equations of temperature is then solved explicitly. This method reduced the computing time greatly. It is worthwhile noting here that for a complicated problem, the method of combination of explicit and implicit methods considerably limits the time step selection, thus a fully implicit method is recommended.
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4. Model verification and simulation results The testing of the model's computational performance and the veri®cation of its predictions were performed by a test box and a real test house. The test box was 454 mm wide by 454 mm long with height 345 mm. An independent air pump was installed to one side of the box in order to provide required ventilation air. The pump was connected to a climatic chamber to maintain the desired air temperature and relative humidity. Outdoor air was heated or cooled in the climatic chamber when passing through a copper pipe installed inside the climatic chamber. The air ¯ow rate in the pipe was from 4:2 10 4 to 6:1 10 4 m 3 s 1 at the atmospherical pressure and room temperature. There were water traps acted as condensation collector for handling the condensed air. Additionally, an air mixer was contained
Fig. 6. Simulated against measured indoor (a) temperatures and (b) moisture contents in the test box with wood pieces as storage.
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inside the box in order to maintain a well-mixed indoor temperature and moisture content in the box. The schematic picture of the measurement equipment is shown in Fig. 3. The walls forming the test box were composed of an outer layer of 12 mm plywood panel with oil-base paint, an layer of 48 mm insulation and an inside layer of 12 mm plywood panel with oil-base paint (see Fig. 4). The test box was surrounded by the laboratory room. The measuring instruments used in the measurement were sensors and computers. The measurement can be classi®ed into two categories: the measurements with and without internal thermal and moisture storage. In the second category, some small wood pieces were mounted inside the box. The material physics properties as well as their heat and moisture transport properties have been published in [30]. Some local sources have also been considered (e.g. [31]).
Fig. 7. Schematic picture of the constructions of Anjala house.
Fig. 8. Calculated against measured indoor (a) temperatures and (b) moisture contents in Anjala house.
X. LuÈ / Energy and Buildings 34 (2002) 1033±1043
The insulation (mineral wool) has about 30% moisture content of its saturation value at 50% relative humidity. Here the dependence of sorption properties on temperatures was not included. The effective thermal conductivity was estimated from [27] 1=4
1=4 leff l1=4 g Sg f ll Sl f ls 1
f4
(20)
and the averaged thermal mass was given as rCP rg CP;g Sg rl CP;l Sl rs CP;s Ss
(21)
Additional information necessary to complete the model included the surface heat and moisture transfer coef®cients which must be determined experimentally in principle. Very often, a well-known Lewis relation is used: hin m
hin T rair CPair
(22)
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where hin T is the convective heat transfer coef®cient. According to [30], an empirical formula was used: hin T 2jTsurface;in
Tin j0:25
(23)
A value of 0.9 was used the emissivity in calculating the radiative heat exchange. For the surface of plywood panel with oil-base paint, the following empirical relation was employed in our calculation [32]: hin m
1 75 103
(24)
2 1 K 1 and hout Moreover, hout T 6:25 W m m 0:001 ms were used as the average values of the exterior surface transfer coef®cients. The results of experimental tests were compared with model simulations to verify the model predictions. The comparison is summarised in the following two groups.
Fig. 9. Simulated indoor (a) temperatures and (b) relative humidity with different surface heat transfer coefficients.
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The ®rst group was the representative of the base case of the test box displayed in Fig. 5. The air coming from outdoor was heated or cooled in the climatic chamber and then entered and left the box with ¯ow rates from 4:2 10 4 to 6:2 10 4 m3 s 1 at atmosperical pressure and room temperature. The air mixed fan mixed the incoming air in the box. The velocity of the air was from 0.2 to 0.6 ms 1. The temperatures of incoming air, indoor air in the box and outgoing air were then measured and saved to the computer. The second group was the representative of the case with indoor thermal and moisture storage. Fourteen wood pieces were mounted inside the box. Results are demonstrated in Fig. 6. These ®gures demonstrate that the model ®ts the experimental data. Besides the test box, the model has been veri®ed with our real test houses. We take Anjala house as an example. The Anjala house is a museum house situated in Anjala, Finland. The schematic picture of its wall constructions is presented in Fig. 7. The house is a two-storey house with ¯oor area 11:5 m 27 m. There was no heating operation and the house was naturally ventilated during the measured period. A real photo and detailed description of Anjala house was presented in [30]. Fig. 8 shows the validation results. As Anjala house contains crawl space and our program has not been ready for that, the simulated indoor air temperature tended somewhat lower than the measured result especially in winter time. Nevertheless, good agreement was obtained. The difference between simulated and measured data was less than 10% compared to the absolute values. In simulating the Anjala house, the transient solar radiation was calculated from the program developed in our local laboratory. The convective heat transfer coef®cient was calculated from Eq. (23), the surface moisture transfer coef®cients were then obtained by Lewis formula. For the exterior 2 1 walls, hout K 1 and hout were T 25 W m m 0:003 ms used as the average values. Wind information was not available. Therefore, an approximate value 0.3 h 1 was adopted as an averaged ventilation rate throughout the calculations. The above-described parameters are not intrinsic properties of the ¯ow. Their values are dependent on the temperature, moisture distribution on the surfaces and the surface geometry as well as wind situations. Fig. 9 demonstrates the differences of the temperatures and relative humidity for the interior surface of the walls with different heat transfer coef®cient values for the test box. The differences are clearly recognised. The picture demonstrates the importance of the effect of the heat transfer coef®cient on determining heat and moisture transfer in buildings. 5. Conclusions Through an extensive survey work, a heat and moisture transfer model for buildings is proposed in this paper. For
indoor air, a well-mixed model is assumed. The model has been validated with our test box and real test houses. There are certainly many areas requiring further research. As already demonstrated, the surface heat and moisture transfer coef®cients have big effects on predicting heat and moisture transfer behaviours and have to be determined experimentally especially for some extreme cases. Secondly, thermal and moisture behaviours in window are only limited in a simple situation where the window is considered as one component of building envelopes. Effects of window frames etc. are not included yet. Thirdly, the effect of raining wetting is not considered. Even though the model developed in this paper does not cover every heat and moisture aspect, it is still proved to a useful model allowing obtaining high accuracy for building researchers. A companion paper will show how the model is applied to the study of the control of indoor thermal and moisture levels by properly heating the indoor air and ventilating the outdoor air. Acknowledgements The author would like to thank the Finnish Academy for the ®nancial support. Special thanks to professor M. Viljanen who provided many ideas for this study. Thanks also go to M. Suihkonen for her experimental work. References [1] I. Andersen, J. Korsgaard, Asthma and the indoor environment: assessment of the health implications of high indoor air humidity, Environment International 12 (1984) 121±127. [2] S. King, Asthma, house dust mites and indoor climate, Architectural Science Review 40 (1997) 43±47. [3] W.K. Lewis, The rate of drying of solid materials, Industrial Engineering Chemistry 13 (1921) 427±432. [4] W. Gardnen, J.A. Widtsoe, The movement of soil moisture, Soil Science 11 (1920) 215±232. [5] L.A. Richards, Capillary conduction of liquids through porous mediums, Journal of Applied Physics 1 (1931) 318±333. [6] J.R. Philip, D.A. DeVries, Moisture movement in porous materials under temperature gradients, Transactions of American Geophysics Union 38 (1957) 222±232. [7] D.A. De Vries, Simultaneous transfer of heat and moisture in porous media, Transactions of American Geophysics Union 39 (1958) 909± 916. [8] A.W. Luikov, Heat and mass transfer in capillary-porous bodies, Pergamon Press, Oxford, 1966. [9] S. Whitaker, Simultaneous heat, mass and momentum transferÐa theory of drying, Advanced Heat Transfer 13 (1977) 119±203. [10] S. Whitaker, Flow in porous media. I. A theoretical derivation of Darcy's law, Transport in Porous Media 1 (1986) 3±26. [11] S. Whitaker, Flow in porous media. II. The governing equations for immiscible, two-phase flow, Transport in Porous Media 1 (1986) 105±126. [12] S. Whitaker, Flow in porous media. III. Deformable media, Transport in Porous Media 1 (1986) 127±154. [13] Y. Bachmat, J. Bear, Macroscopic modelling of transport phenomena in porous media. 1. The continuum approach, Transport in Porous Media 1 (1986) 213±240.
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