Probabilistic modelling of mould growth in buildings

Abstract

A new probabilistic model for the assessment of mould growth in buildings has been elaborated within interdisciplinary – building physics/structural engineering – cooperation. Both the occurrences of favourable conditions for the growth of mould fungi and their durations are taken into account. The probabilistic approach can be characterised as time dependent based on the theory of stochastic processes. The resulting probability of occurrence of mould growth cycles having deteriorating potential suggests itself as a measure of the mould growth hazard. By the measure, the management of building performance with respect to mould growth hazard may be conceived. The calculations of an illustrative example show that the variability of the outdoor climate conditions can substantially influence the occurrences and durations of favourable conditions for mould growth.

Keywords

Building performance mould growth probabilistic modelling

Introduction

The necessary condition for the growth of mould fungi on a building component is a sufficient duration of favourable temperature and relative humidity (RH) levels. These quantities in turn depend on random variations of material parameters, on external and internal conditions as well as on rather uncertain workmanship. The external conditions include outdoor temperature, RH and solar radiation, also wind-driven rain in some cases. The internal conditions involve indoor air temperature and RH. During the heating season, the indoor air temperature is specified by the applied course of heating. The indoor air RH results from the coupled effect of inhabitant’s behaviour, the hygroscopic properties of indoor surfaces and outdoor climate conditions. As a consequence, the favourable conditions for mould growth occur randomly in time.

The evaluation of hygrothermal performance of building component represents a complex task. The results of simulation are conventionally obtained as hourly time series of RH or moisture content and temperature. The analyses of favourable conditions for mould growth, particularly germination of mould fungi and cycles of mould growth on a given surface, employ these time series.

A review of various deterministic approaches to mould risk evaluation can be found in Vereecken and Roels (2012). Two different approaches are used in the existing models for prediction of mould growth. The basic models indicate only the mould germination conditions. A pioneer study was carried out by Sedlbauer (2002), who developed mould isopleth graphs for prediction of mould fungus formation under different combinations of steady RH and temperature. The advantage of the basic models is their simplicity; however, only a few of them, for example, the Moon’s germination graph method (Moon, 2005) and the time-of-wetness (TOW) method (Adan, 1994; Adan and Samson, 2011), take into account the effect of non-steady humidity and temperature including fluctuation of favourable and unfavourable conditions, which eventually result in fungi extinction. The concept of TOW suggested by Adan (1994) defines the TOW ratio of durations of the wet period (RH ≥ 80%) and the whole (wet + dry) period. The mould growth can occur in periods, in which the inequality TOW > 0.5 is fulfilled. The method was verified for gypsum-based materials and a single fungal species on interior surfaces not considering the effect of temperature variations.

The more advanced approaches, for example, the model developed at VTT Technical Research Centre of Finland (VTT model) by Hukka and Viitanen (1999) or the biohygrothermal model (Krus et al., 2007), also predict time-dependent intensity of mould growth, described usually by a time-dependent mould growth index. The VTT model was later improved and extended for different types of building materials (Viitanen et al., 2010, 2011). However, its verification has shown a significant effect of the uncertainty in mould growth prediction. Therefore, at the current state of knowledge, even the results obtained by more advanced models differ significantly, due to the involved assumptions and simplifications (Vereecken and Roels, 2012; Viitanen et al., 2010, 2011).

An innovative approach to describe the potential of the germination and/or mould growth under general time variation of RH and temperature is suggested in Isaksson et al. (2010). A total daily dose assumed as a product of components of daily averages of RH and temperature is accumulated in consequent days indicating the onset of mould growth by reaching the reference value. The reference dose is adjusted for constant exposure of the studied material to a specified reference climate.

A simplified probabilistic approach to mould growth in buildings is considered in Hagentoft (2011) and Van Gelder et al. (2012). The yearly sums of hours of favourable mould growth conditions are calculated by the Monte Carlo method simulating random variables – material parameters (Hagentoft, 2011; Van Gelder et al., 2012) and climate outdoor conditions out of 30 years available (Hagentoft, 2011). However, the continuous durations of favourable mould growth conditions are not evaluated.

The time-integrated probabilistic approach (Melchers, 1999) has been employed extending the dose concept to doses above the reference value (Thelandersson, 2012). The relative dose referred to the reference dose is related to the Hukka and Viitanen’s mould growth index (Hukka and Viitanen, 1999); the values above 1 indicate the intensity of mould development. Constructing cumulative distribution function (CDF) of yearly maxima of the relative dose, the probability of mould growth intensity, particularly of its onset given by the value 1, has been obtained. However, with respect to experiences in the building sector, too conservative results have been obtained (Thelandersson, 2012). The reason for this may be that the daily growth-favourable doses can attain significantly higher values than those of recovery doses.

The present article focuses on the methodological aspects of the developed probabilistic approach to the mould growth problem. The TOW concept of mould growth modelling is adopted (see Adan and Samson, 2011). The continuous periods of favourable conditions for mould growth are denoted as TOW on cycles given by the original inequality TOW > 0.5. Generally, the bound in the inequality can differ depending on the mould species and surface material (Adan and Samson, 2011). The occurrences of TOW on cycles in time are modelled by the Poisson counting process (Melchers, 1999). The pilot study on the topic (Sadovský et al., 2012a) is extended to practically important durations of mould growth conditions, which are studied as rare events (cf. Sadovský et al., 2012b). The main goal of this study is the determination of the first-passage probability of exceeding a considered duration of TOW on cycle in a given time period. The probability is calculated by the Poisson spike process (Melchers, 1999). An example study concerning probabilistic assessment of the mould growth conditions on the internal surface of an external wall illustrates the suggested approach.

Probabilistic model

Input data of the model

Under the time-invariant input, the continuous random variable material parameters and geometry as well as the discrete random variable workmanship can be considered. The existing material parameter assemblies are relatively small not allowing exact development of distribution functions. The random variability of geometry can be sufficiently described. Studies on workmanship are still scarce facing difficult classification. In the presented probabilistic model, only the vector of continuous random variables of material parameters X _M is included. It is assumed that the components of the vector X _M possess the joint probability density function $f_{X_{M}} (X_{M})$ . Having quantification of the workmanship parameter, its inclusion into the probabilistic model would be a routine task.

The time-variant input is represented by time series of outdoor and indoor climate. The outdoor climate data generally consist of hourly records of temperature, RH, global and diffuse radiation on horizontal surface, wind speed and direction as well as wind-driven rain. The indoor climate involves indoor air temperature and RH resulting from the coupled effect of inhabitant’s behaviour, the hygroscopic properties of indoor surfaces and outdoor climate conditions. In calculations of indoor climate, some simplifying daily, weekly and season-dependent scenarios of inhabitant’s behaviour are commonly utilised.

Probabilistic analysis

Introducing the input data into specially developed codes or commercially available software, the output is provided as time series of RH and temperature values on the considered surface of a building structure, which are then analysed for favourable mould growth conditions. Adopting the TOW concept, the TOW on cycles was studied as events randomly occurring in time. Note that mutual correlations and autocorrelations of outdoor climate components are already included in the output. However, a sufficiently long-lasting record of climate data in a considered location should be available.

The stochastic process of TOW on cycles is designated to a reference period. In relationship to the outdoor climate uncertainty, its natural choice is, for example, 1 year or heating season. The process is characterised by the mean rate occurrence in the reference period ν and stochastic properties of the event, which are the durations D of the TOW on cycles. The random realisations of D are assumed as independent and identically distributed with the CDF F_D(d). The reference period is considered as a time unit.

The main goal of probabilistic analysis is the assessment of the first-passage probability p_f1(a) of exceeding a practically important duration of the mould growth cycle a in the reference period. A conditional scheme of solution is used. First, conditioned by a realisation of input random variables X _M, the conditional probability p_f1c(a| x _M) is calculated. Then, the probability p_f1(a) is obtained as the mean value of the conditional probability with respect to the random variables X _M

p_{f 1} (a) = \int_{X_{M}} p_{f 1 c} (a | x_{M}) f_{X_{M}} (x_{M}) d x_{M}

(1)

Assume one realisation of the material parameters x _M for the corresponding conditional probability p_f1c(a| x _M) the abbreviated notation p_f1c(a) is used. For calculations of p_f1c(a), the occurrences of TOW on cycles are modelled by the Poisson counting process. Consideration of their random durations is implemented by the application of the Poisson spike process (Melchers, 1999). The crossing rate $ν_{a}^{+}$ of the level a by D in the reference period for this process is given as

ν_{a}^{+} = F_{D} (a) [1 - F_{D} (a)] ν

(2)

and the conditional probability p_f1c(a) of the first-level crossing in the reference period is

p_{f 1 c} (a) = 1 - \exp [- ν_{a}^{+}]

(3)

One may also ask about the conditional probability of double occurrence (n = 2) of exceeding the mould growth cycle duration a in the reference period. The model provides the answer by extending equation (3) to

p_{f 1 c} (a | n = 2) = 1 - exp [- ν_{a}^{+}] - ν_{a}^{+} exp [- ν_{a}^{+}]

(4)

A simple way of evaluating the mean value in equation (1) is to apply a suitable kind of Monte Carlo method, for example, the Latin hypercube sampling (Janssen, 2013). This means that the conditional probability p_f1c(a| x _M) is calculated for relatively small number of realisations of the material parameters. In the case of low influence of material parameters on the conditional probability, approximate formulas can be used. Note that in Sadovský et al. (2012a), the total probability p_f1(a) has been calculated by the first-order reliability method (FORM; Melchers, 1999) employing an artificial limit-state function suggested by Wen (1990).

Illustrative example

General description

The considered building structure represented the external wall of a room with a volume of 48.49 m³ that was treated as a one-zone space. The room was a part of a residential house, where all other structural elements of the room were internal ones. The considered external wall was composed of autoclaved aerated concrete (AAC) (300 mm), which was plastered on both sides by mineral plaster (15 mm) (cf. Figure 1). This type of external wall was popular in Slovakia during the second half of the 20th century. The reference period was the first heating season of the wall from its ‘as-built’ state to the quasi-steady state during the period from 1 October to 30 April. The numerical simulation of the hygrothermal performance started on 1 October; however, the initial period of very intensive drying (1–20 October) was excluded from the analysis. The initial moisture contents were 0.17 and 0.19 m³/m³ for AAC and mineral plaster, respectively. These values, obtained from in situ measurements (Mrlik, 1985), correspond to the moisture contents of AAC walls in the early phase after finishing.

The one-dimensional simulation tool NEV3 (Koronthályová and Matiašovský, 1998), which is based on the solution of two coupled equations for heat and moisture transfer, was used for the calculation of the RH and temperature on the internal surface of the external wall. The NEV3 calculation does not take into account the possible air flow through the structure. NEV3 calculated the time-dependent temperature and RH values under the applied initial and boundary conditions, herein represented by the indoor and outdoor climate. The calculated RH and temperature represented a sequence of discrete values, corresponding to the applied hourly time step. This one-dimensional simulation described the hygrothermal state on the internal surface of a flat wall. The analysis of the hygrothermal state in the building elements where two-dimensional (2D) and three-dimensional (3D) effects are of importance (as, for example, room corners) was not involved in this study. However, the applied one-dimensional approach was acceptable taking into account that the main goal of the calculation was to illustrate the suggested probabilistic methodology. In all calculations, standard values of the surface film coefficients for heat transfer h_i = 8 W/(m² K) and h_e = 25 W/(m² K) and consequently of surface film coefficients for vapour diffusion β_vi = 0.0034 m/s and β_ve = 0.01 m/s were applied. The used material parameters are presented in section ‘Material properties’.

Outdoor climate

As the external boundary conditions, the measured hourly values of air temperature, air RH and solar radiation for 13 heating seasons, from 1997 to 2010, from Bratislava (Dfb according to the Kőppen–Geiger climate classification) were used. The actual direct normal radiation on the west-facing wall was calculated using the global and diffusion solar radiation data on a horizontal surface (Markus and Morris, 1980). Considering the case of a well-sheltered wall, the effect of wind-driven rain was not taken into account. The variability of climatic parameters is illustrated in Figures 2 to 4 showing the values for the period from 3 November to 10 December for 2006, 2007 and 2010. The counting of days starts from 1 October.

Indoor climate

The actual values of indoor air RH generally result from simultaneous effect of the following factors: outdoor climate, indoor air temperature, moisture production, air change rate and hygroscopic properties of indoor surfaces (see section ‘Material properties’). During the heating season, the indoor air temperature does not change significantly from the adjusted thermostatic setting, but there is a slight correlation between the outdoor and indoor temperatures (Kalamees et al., 2006).

A single daily scenario was assumed for inhabitant’s behaviour. The considered moisture production in the room is 2.4 kg/day, which corresponded to the standard moisture production of a four-member family, recalculated to one room of a flat (IEA-Annex XIV, 1991). The constant moisture production during the periods when the occupants were not present in the room was 0.025 kg/h; during morning hours, it increased to 0.4 kg/h; and during afternoon and evening hours, it increased to 0.2 kg/h (Table 1). The considered mean air change rate during the winter period (November–15 March) was 0.5 h⁻¹ and during the autumn and spring periods (October, 16 March–April) was 0.7 h⁻¹. The actual hourly mean air change rate values were adjusted to the presence of occupants in the room and their activities. The used daily schedule of air change rate for the winter and autumn/spring periods is given in Table 1. The indoor air temperature during the longer time periods was considered as practically constant, which corresponded to the use of an ideal heating system. The values are presented in Table 2. The higher values during the autumn and spring periods reflect the noticed correlation between the daily mean values of the outdoor and indoor temperatures (Kalamees et al., 2006).

Table 1.

Daily schedule of vapour production and air change rate in the room.

Hours	0–6	6–8	8–16	16–22	22–24
Moisture production (kg/h)	0.025	0.4	0.025	0.2	0.025
Air change rate (winter) (h⁻¹)	0.4	0.83	0.4	0.62	0.4
Air change rate (autumn, spring) (h⁻¹)	0.6	1.03	0.6	0.82	0.6

Table 2.

Indoor air temperatures used in simulation.

Period	October and 16 March–30 April	1–15 November and 1–15 March	16 November–28 February
Indoor air temperature (°C)	22	21	20

For evaluation of indoor air time-dependent RH function, the simulation tool NPI (Koronthályová et al., 2004; cf. also Woloszyn and Rode, 2008) was used. It was based on the solution of the water vapour mass balance equation in the single-zone space, taking into account outdoor climate, indoor air temperature, moisture production, air change rate and hygroscopic properties (water vapour permeability, moisture diffusivity, retention curve; see section ‘Material properties’) of indoor surfaces.

Material properties

Similarly, as in the previous studies (Sadovský et al., 2012a, 2012b), only three selected material parameters, namely, dry thermal conductivity, water vapour resistance factor, and the moisture diffusivity at RH = 100%, were considered as random variables. The mean values of all material parameters including those which were assumed as random variables are given in Table 3. The thermal conductivity dependence on the moisture content was approximated by a linear function. The water vapour permeability was considered as a constant. The retention curve is described by the relation of Hansen type (Hansen, 1986)

u (φ) = u_{cap} \cdot {(1 - \frac{ln φ}{B})}^{- \frac{1}{n}}

(5)

Table 3.

Material parameters.

Material	Porosity (–)	Bulk density (kg/m³)	Specific heat capacity (J/kg K)	B (–)	n (–)	Water vapour resistance factor (–)	D_w₀ (m²/s)	D_w_max (m²/s)	Thermal conductivity (dry) (W/m K)
Mineral plaster	0.24	1900	850	0.01	0.5	25	1E−12	1E−9	0.8
AAC	0.8	500	850	0.0008	0.35	7.6	1E−11	2.5E−8	0.125

AAC: autoclaved aerated concrete.

The moisture diffusivity dependence on the moisture content was approximated by the following exponential function

D_{w} (u) = D_{w 0} \cdot exp (ln (\frac{D_{w c a p}}{D_{w 0}}) \cdot \frac{u}{u_{c a p}})

(6)

Numerical results and discussion

Mould growth cycles

The analysis of the building structure considered started with the calculation of the hourly values of the RH and temperature on the internal surface of the external wall using the material parameter values shown in Table 3. The obtained internal surface temperature showed minor fluctuations around 19°C, thereby the RH of 80% was critical for mould growth (Hukka and Viitanen, 1999). For determination of the TOW on cycles, the modified inequality TOW ≥ 0.5 was adopted. The reason for including the bound by adding equality sign in the TOW ratio inequality was in the discrete, hourly step of RH evaluations. As a consequence, the TOW on cycles were realised at even number of hours.

A total of 13 heating seasons related to the available measurements of climate in Bratislava recorded from the year 1998 to 2011 was considered. In five of the heating seasons, namely, 1998/1999, 1999/2000, 2001/2002, 2005/2006 and 2007/2008, a very favourable outdoor climate resulted in short TOW on cycles lasting at the most for 14 h. The occurrences of TOW on cycles greater than 24 h obtained in the remaining eight heating seasons are summarised in Table 4. The practically important TOW on cycles arose in November and partly also in December, which duly reflects the consideration of the first heating season of an external wall transition from as-built state. However, the occurrences of TOW on cycles greater than 24 h were rare. It is in correspondence to the expectation of well-designed flat wall performance, where 2D and 3D effects as well as the effects of the surface transfer coefficients diversity were negligible.

Table 4.

TOW on cycles greater than 24 h.

	Heating season
	2000/2001	2002/2003	2003/2004	2004/2005	2006/2007	2008/2009	2009/2010	2010/2011
Frequency	2	1	1	1	5	1	2	2
Date of occurrence	14–23 November	14 November–2 December	24 November–1 December	1–8 November	14 November–10 December	1–22 November	17–26 November	4–21 November
Duration (h)	106; 44	436	156	172	46; 190; 176; 34; 54	494	60; 44	62; 140

TOW: time of wetness.

Some parts of the TOW on cycle’s process as obtained for the heating seasons in years 2006/2007, 2007/2008 and 2010/2011 are illustrated in Figures 5 to 7, respectively. The counting of days starts from 1 October. The TOW on bars show durations of favourable mould growth conditions, while the TOW off bars in the negative part of the plot stand for dry intervals. Comparing Figures 5 to 7 and Figures 2 to 4 showing selected outdoor climate data in the corresponding years and months shows extraordinary dependence of mould growth conditions on the outdoor climate. In the selected period, the dominant effect of outdoor temperature can be noticed, and the TOW on periods were coincident with the periods of higher outdoor temperature. In the case of season 2006/2007, the effect of relative high temperature was intensified by the coupled effect of lower sun radiation during the second part of the period. The effect of the outdoor RH was not evident as the differences between RH of particular years during the presented period were insignificant. Results highlighting the importance of outdoor climate for mould growth study have been also presented in Thelandersson (2012).

Note that for application of the Poisson process, the independence of occurrences of rare TOW on cycles is assumed or, at least, a slight dependence is an acceptable alternative. This may not be the case when the cycles arrive in clusters. In this instance, one may reduce the studied sample to that of the cluster maxima (Reiss and Thomas, 2007).

Application of the time-dependent approach

The frequencies of the occurrences of TOW on cycles collected across all 13 heating seasons are drawn in Figure 8. Altogether, 814 occurrences of the cycles were found. The intensity of the process was given by the mean rate occurrence ν per heating season: ν = 814/13 = 62.6.

Referring to the Sedlbauer’s (2002) isopleths, the calculated favourable mould growth conditions of less than 1 day duration are not relevant. Only few practically important TOW on durations were found (Figure 8). This suggested the idea of analysing them as exceedances over a preselected threshold. The exceedances greater than 24, 48 and 96 h were considered. These correspond to N = 15, 11 and 8 occurrences, respectively. As for the threshold of 24 h, clustering of exceedances occurred in the heating season 2006/2007 (Figure 5), and two neighbouring cycles of 46 and 190 h were considered as one cycle of 190 h duration. Similarly was treated the cluster of cycles of 34 and 54 h in Figure 5. Correspondingly, N = 13 was applied. Ordering the exceedances by ascending magnitudes, the upper tail of an empirical distribution function could be created using

F_{D} (d_{i}) = \frac{i}{814 + 1}

(7)

where d_i were the TOW on durations with $i \in 〈 814 - N + 1, \dots, 814 〉$ . F_D(d_i) was approximated by a non-linear regression analysis using the generalised Pareto (GP) probability distribution, which includes both exponential and Pareto CDF representing the so-called light and heavy tail distributions (Reiss and Thomas, 2007). The quality of approximation was evaluated by the correlation coefficient and the variance of residual. An indication for the choice of a probability distribution was also its stability with respect to the threshold shift. The obtained figures are shown in Table 5. As the best approximation, the GP distribution adjusted to N = 11 exceedances was considered (see Figure 9). This theoretical probability distribution was then used for calculations of F_D(a) values in equation (2).

Table 5.

Upper tail approximations by the generalised Pareto CDF.

CDF	Estimations	N = 13	N = 11	N = 8
Generalised Pareto	Variance of residual	8.97E−7	8.32E−7	6.89E−7
Generalised Pareto	Correlation coefficient	0.979	0.972	0.955

CDF: cumulative distribution function.

The conditional first-passage probability of exceeding a considered level a of mould growth cycle was obtained by equations (2) and (3). As shown in the pilot study (Sadovský et al., 2012a), the random variable material parameters had low effect on the resultant conditional probability. Thereby, the conditional probability p_f1c(a) calculated for the mean values of the random variable parameters provides a satisfactory estimate of the total probability p_f1(a). The estimated total probabilities of the first up-crossing of TOW on durations in the first heating season are presented in Figure 10 in daily steps (the symbol-line curve P(D > a) – TOW on process). The lower curve (P(D > a|n = 2) – TOW on process) shows the probability of two exceedances of the considered TOW on duration. It is seen that long durations have negligible chance to occur once again.

Alternative time-integrated approach

The classical approach to time-dependent reliability is the time-integrated approach (Melchers, 1999), in which a stochastic process of adverse events is represented by an extreme value distribution. In our case, the extreme value distribution F_D_max(d_max) of heating season’s maxima d_max of TOW on cycles was approximated by the non-linear regression analysis considering the generalised (unified) extreme value (GEV) distribution (cf. Reiss and Thomas, 2007). Note that because of small sample size of season’s maxima (n = 13), the distribution tests for determination of the underlying distribution function should not be relied on.

In Figure 11, one can distinguish negligible maxima of five favourable heating seasons. In the remaining more or less unfavourable seasons, limitation to maximum value resulted in omission of other relevant TOW on durations, which were taken into account in the suggested time-dependent approach. The corresponding failure (exceedance) probabilities given as 1 −F_D_max(d_max) are shown in Figure 10 (P(D > a) – max TOW on). Comparison with the first-passage probabilities obtained by the suggested approach shows an overall agreement with slightly greater values. Nevertheless, the time-integrated approach does not provide the probabilities of two or several up-crossings.

Conclusion

This article presents a time-dependent probabilistic approach to the mould growth problem in buildings based on the theory of stochastic processes. The occurrence of mould growth cycles was evaluated by the TOW concept and was modelled using the Poisson counting process. Approximating the upper tail of random variable durations of TOW on cycles, the first-passage probability of exceeding a considered duration of mould growth cycle in a reference period was calculated by the Poisson spike process. The approach also allowed calculation of the probabilities of two (or several) up-crossings.

An illustrative numerical study addressed the internal surface of an external wall of a residential building structure. The reference period was the first heating season of the wall transition from ‘as-built’ state to the quasi-steady state. The variability of outdoor climate condition was represented by the data measured in Bratislava since 1998−2011. The practically important TOW on cycles were studied as rare events.

For comparison, the less general time-integrated approach was applied. The stochastic process of the occurrences of TOW on cycles was represented by the extreme value distribution of their heating season’s maximal durations. The probabilities of failure obtained by this approach yielded somewhat more conservative values than the first-passage probabilities being in overall agreement with them. However, it does not provide the probabilities of two (or several) up-crossings.

The numerical results showed that the variability of the outdoor climate conditions substantially influenced occurrences and durations of favourable mould growth conditions. However, this implied that the standard test reference years or design reference years applied in deterministic calculations of the hygrothermal performance of building cannot be used as a surrogate for outdoor climate data in the probabilistic solution of the mould growth problem. A practical implication of this conclusion is that observable deterioration of a building wall by humidity problems may be just the matter of a very unfavourable outdoor climate season.

The assessed probability of occurrence of mould growth cycles having deteriorating potential suggests itself as a measure of the mould growth hazard. In analogy to structural reliability where the probability of failure is represented by the corresponding reliability index, the performance index can be introduced. For the purpose of management of structural reliability for construction works, the Eurocode EN 1990: 2002 “Basis of structural design” recommends defining consequence classes differentiating the consequences of failure or malfunction of the structure. The associated reliability classes may differentiate the structural reliability by the recommended minimum values for the reliability index. Development of a paralleling structure based on the performance index and appropriate consequence classes appears feasible for the management of building performance with respect to the mould growth hazard.

Footnotes

Appendix 1 Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This work was supported by the Slovak Research and Development Agency under the contract no. APVV-0031-10 and by the Scientific Grant Agency VEGA under the contract no. 2/0145/13.

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Woloszyn

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(2008) Tools for performance simulation of heat, air and moisture conditions of whole buildings. Building Simulation 1: 5–24.