Assessing the “sufficient ventilation” requirement for Austrian buildings: development of a Monte Carlo based spreadsheet calculation to estimate airing intervals and mould risk in window ventilated buildings

Abstract In Austria the lack of guidelines or standards has caused many discussions and disputes on the question if “sufficient ventilation” can be ensured with window airing only, in particular in newly constructed, airtight residential buildings. This work presents the development of a calculation method aiming to provide a simple-to-use tool to estimate the risk of mould growth and the window airing interval required to ensure good indoor air quality assuming a range of different boundary conditions and occupant behaviours. The method implements a Monte Carlo approach calculating 1000 single zone mass balances for carbon dioxide (on a room level) and water vapor (on a housing level). Air infiltration through the building envelope is accounted using the so-called LBL-model. The time interval between window airing required to comply with CO2 limit value is estimated by calculating the time evolution of the CO2 concentration for 1000 different parameter combinations. The mould risk is estimated by a 1000-fold calculation of the daily averaged indoor air humidity and the resulting water activity on critical wall surfaces. The results are displayed as probability distributions providing information on the risk that the queried situation can or cannot ensure “sufficient ventilation”. Exemplary calculations for bedrooms of new multifamily buildings estimate that intervals between window airing events (to keep time-averaged CO2-concentration below 1000 ppm), will vary between 23 and 190 minutes (representing the 5th and the 95th percentile). This is clearly below an acceptable intervention interval for bedrooms. For living rooms, the assessment shows a strong sensitivity on the “accessible” air volume. The humidity assessment for this type of housing suggests that mould growth could occur in about 17% of the cases even though air exchange corresponding to two airing events per day were assumed. An additional outdoor air exchange of up to 40 m³/h would be required to reduce the mould risk fraction to <1%, suggesting the need for mechanical ventilation concepts in residential housing to enable healthy indoor environment independently of occupant behaviour.


Introduction
Increasing energy efficiency and building airtightness requirements have led to an ongoing debate about suitable measures to ensure proper indoor air quality and humidity levels in residential buildings.In Austria "sufficient ventilation" is required by building regulations (OIB, 2019), without further specifying how this should be accomplished or to what extent active intervention by building occupants, for example, by window airing, is reasonable to assume.The lack of guidelines or standards has caused many discussions and disputes on the question if "sufficient ventilation" can be ensured with window airing only, in particular in newly constructed residential buildings.This work presents the development of a calculation method commissioned by the Austrian ministry of climate action, environment, energy, mobility, innovation and technology.It estimates the risk of mould growth and the time intervals between window airing events required to maintain carbon dioxide (CO 2 ) concentration below the Austrian guideline limit value (Tappler et al., 2017).This calculation method has been implemented in MS Excel and is freely available for download (BMK, 2021) with the aim of providing an objective decision basis for stakeholders.Further details are documented in the project report (Greml et al., 2021), the following two chapters are a translated summary thereof.

Method
As required by the contracting authority the calculation method should be easy to implement.It should not require a special software, for example, building performance simulation tool, and its application should allow for different levels of user knowledge and engagement.That means that it should not be necessary to enter all calculation parameters as this could overwhelm the user.However, the use of many static default parameters could create a substantial bias in the calculation results.Therefore, the authors opted for a Monte Carlo (MC) based approach where the results of one query are based on 1000 calculations with varying input parameters (except for those parameters fixed by the user).The extend of variation of the parameters not specified by the users are defined in probability density functions (PDFs).In that way, the result is not a single value but a value range providing also information about the uncertainty.The calculations use a single zone mass balance model for estimating CO 2 and water vapour concentration (H 2 O) in the zone's air volume.

Single zone mass balance
For this calculation the air density can be considered constant in good approximation.Therefore, a mass balance or volume balance model can be used interchangeably.In the mass balance models CO 2 serves as an indicator for the indoor air quality and H 2 O as a relevant quantity for the evaluation of the mould risk.It is assumed that the respective component (CO 2 or H 2 O) can be introduced either from the outside via natural infiltration, window ventilation or via a source located in the room or building.The removal is represented in the model via exfiltration or window ventilation.Water vapour buffering in walls and interiors is not considered.Since H 2 O mass balance is calculated for daily averages this simplification is considered reasonable.It is assumed that the component under consideration (CO 2 or H2O) is ideally mixed in the zone being evaluated (Barp et al., 2009;Schnieders, 2003).For the CO 2 assessment a certain room of the building is evaluated.The H 2 O assessment is applied to the entire dwelling zone.Since mould risk is only considered an issue in residential settings, the H 2 O assessment is only applied to residential buildings, while CO 2 assessment can also be performed for schools and office buildings.

Estimation of intervals between window airing
CO 2 concentration is often used as an indicator of indoor air quality.It is important to note that in general it should only be considered an indicator for occupancy related ventilation demand (Persily, 1997(Persily, , 2015)).Nevertheless, previous studies suggest that for typical Austrian residential dwellings, fresh air supply needed to maintain decent CO 2 concentration is greater than air supply needed to control other pollutants like VOC's from building materials (Rojas et al., 2015;Tappler et al., 2014).CO 2 is inert and sorption effects can be neglected in indoor environments.Therefore, the difference in mass balance between inflow and outflow can be represented as an increase or decrease in concentration, resulting in the following differential equation: Here, _ V in and _ V out are the incoming and outgoing air flow rates, respectively, due to in-/exfiltration (or window ventilation).Neglecting density differences, these can be equated and _ Where AER is the air exchange rate in (1/h) and V is the room volume in (m 3 )._ V CO2 is the emission rate of CO 2 by occupants in (m 3 /h), c a , c out and c are the respective CO 2 concentrations in volume fractions (-).In the model, CO 2 is assumed to be emitted only by people present in the room through respiration.The emission rate depends on the activity level and the age of the persons.The outdoor air concentration c a is assumed to be 450 ppm.The above differential equation can be easily solved to determine the concentration profile as a function of time t as follows: Here, c 0 is the initial concentration and c 1 is the steady-state equilibrium concentration, which can be determined as follows: As stated above, the question of whether the "sufficient ventilation" requirement can be fulfilled via in-/exfiltration and window airing is answered by assessing if the time intervals between window airing events needed to stay below a CO 2 limit value c LV is reasonable.In this case AER is solely determined by natural in-/exfiltration.Starting from an initial concentration c 0 , the time period until the limit value c LV is reached can be determined as follows: In general, it cannot be assumed that a window airing event (after the limit value has been reached) will bring indoor air concentration to ambient air level.Therefore, the initial concentration c 0 has to be determined by estimating the concentration after a typical window airing event.If the air exchange rate during the window airing event AER WA and the duration of the ventilation process t WA are given, c 0 can be determined as follows: The stationary concentration for the window airing event c 1WA can be calculated in analogy to Equation (3).Note that this concentration will typically not be reached during t WA by far.
In the Austrian IAQ guideline different air quality classes are defined (Tappler et al., 2017).For the context of this work, Class 2 ( 1000 ppm) or Class 3 ( 1400 ppm), depending on the room type, are relevant.However, the guideline does not specify limit values for instantaneous CO 2 values, but for the arithmetic mean of the CO 2 concentration over a defined assessment period t AP : Therefore, Equation ( 4) must be adjusted accordingly to give the time until the limit value as defined in the Austrian guideline c LV , is reached.The concentration mean c t ð Þ can be written as a function of time as: Equation ( 7) can no longer be solved for t analytically.Therefore, a numerical approach is implemented.Equation ( 7) is solved for 192 timesteps, from 0 to 2t AP : The time t LV when the limit value is reached or exceeded, is determined when c t ð Þ !c LV : To obtain the initial value c 0 an iterative calculation must be applied.To reduce computational demand, an approximative calculation for c 0 was developed, showing reasonable accuracy and little influence on the final results (Greml et al., 2021).
Additionally, the window airing interval required when assuming ideal airing events t LV, id is also calculated for informational purpose.This corresponds to the assumption that initial concentration after airing is equal to ambient concentration, i.e. c 0 ¼ c a :

Estimation of mould risk
In order to assess the mould risk, the indoor air humidity must be determined.In contrast to CO 2 , it depends not only on internal sources and outdoor air exchange.It is also significantly influenced by the buffer effect of building materials, furnishings and objects of daily use.Therefore, a time-dependent calculation on a room level was not considered sensible.Instead, the humidity mass balance is calculated over an entire dwelling as daily average.The mould risk calculation was only implemented for residential housing.The vapor density q a of the ambient air at indoor temperature T i (K) is calculated for a given ambient air temperature T a (K) and ambient air humidity u a (-) as follows.Herein, the saturation vapor pressure E is calculated according to Magnus' formula, for example (WMO, 2018): Assuming that humidity is distributed evenly throughout the home and that temperature differences between rooms might exist, for example, cooler bedroom, the vapour density in the coolest room with temperature T i, min (K) can be calculated as follows.
Here, _ m H2O is the humidity source strength and _ V is the average outdoor air flow rate for the entire home (referenced to the spatially averaged indoor temperature T i ).Note that the vapor density, which has the unit mass per unit volume, for example (kg/m 3 ), is converted via a linear approach (ideal gas law) to the reference temperature T i, min : On the basis of this vapor density, the so-called water activity a w of the mould-critical wall surfaces can be calculated as follows.
It corresponds to the relative humidity of the bulk (room) air as it takes the temperature of the wall surface T si : The risk of mould growth is high at a w values >0.8, provided this condition is present over a certain period of time (few days) (see, e.g., € ONORM B 8110-2,2 2003; Sedlbauer, 2001;UBA, 2016).This calculation method assumes substantial risk of mould growth if the calculated daily average is >0.8.For Austrian climates the mould-critical wall surfaces are the ones with the lowest temperature, that is, at thermal bridge locations.The interior surface temperature T si of a thermal bridge, can be estimated using the so-called f RSI factor.It is a dimensionless quantity for assessing thermal bridges, see for example, DIN 4108-2.
Depending on the building age and thermal standard, assumptions can be made for f RSI , for example, old building: 0.5, new construction: 0.7, passive house: 0.9.Since the temporal variation of the ambient air temperature T a is strongly damped on the inner surface (due to the heat capacity of the wall structure), the 24 h moving average of the ambient temperature b T a is used in Equation ( 11).This has shown to be a good assumption for typical massive construction in previous studies (Rojas et al., 2015).

Estimation of natural in-/exfiltration
Both, the estimation of the time interval between airing and the estimation of the mould risk, require input regarding the air exchange rate due to in-and exfiltration.It depends on the airtightness of the building and is driven by wind and temperature differences between inside and outside (chimney effect), resulting in a high temporal variability.Various approaches to determine natural in-/exfiltration are known from the literature, they can be roughly divided into the following categories (see, for example, Liddament, 1996): Category 1: "Rules of thumb"/empirical factors, for example, DIN 1946-6 Category 2: Simplified theoretical models, for example, LBL model (Sherman, 1980) Category 3: Detailed physical multi-zone model/simulation, for example, CONTAM (NIST n.d.) The contracted assignment for this work precluded the use of simulation software with a detailed physical model according to category 3. The input effort would be too extensive and the use would ultimately be reserved for experts only.Thus, a choice had to be made between the first two categories.While the first category has the advantage of simplicity, it has the disadvantage that it cannot directly consider influences of weather, i.e. the temporal variability of natural in-/exfiltration which is critical when evaluating indoor air quality.Therefore, a well-known and well documented model from category 2, the Lawrence Berkeley Laboratory (LBL) model (Sherman, 1980) was chosen.Numerous studies have validated this model or used it for their investigations (e.g., Berge, 2011;Binamu & Lindberg, 2002;Hayati et al., 2014).In the LBL model, the in/exfiltrated air flow rate is calculated as follows: Herein, A 0 is the effective leakage area, a measure of the air leakage of the building envelope.It can be derived from a building air tightness measurement, i.e. from the n 50 value.The temperature difference DT between indoor and outdoor air and the meteorological wind speed v are weighted (non-linearly) by a stack effect factor f s and a wind factor f w , respectively.Further details on the LBL model and its parameters can be found in general summary literature (e.g., Liddament, 1996) or in the original literature (Sherman, 1980).

Monte Carlo approach
The calculations described above require a number of input parameters with respect to building type, geometry, tightness, terrain, etc. and information about the building occupants and the resulting source strengths (of CO 2 and H 2 O).Furthermore, the highly variable ambient conditions (temperature, humidity and wind strength) enter into the calculations.The question arises how to deal with the uncertainty or variability of these input parameters.Most input parameters can be assigned a value range rather than a single value.In a deterministic calculation approach, one would have to select a single representative value for each input parameter.The big disadvantage of this is, that the information about the variance of the input parameters and their influence on the variance of the result is lost.Especially for the evaluation of indoor air quality and mould risk, this loss of information is problematic, because issues might occur for non-representative conditions.The methodology should therefore be able to identify and quantify not only the "worst case" but also the "intermediate" and "best case" conditions and their probability of occurrence.For this purpose, all input parameters must be varied within their value range.This quickly leads to an unmanageable number of combinations.The so-called Monte-Carlo (MC) approach offers a remedy (see, e.g., Rubinstein & Kroese, 2017).Instead of computing the entire parameter space (every possible combination), this approach computes only a random sample from the entire pool of value combinations.In the method presented here, 1000 parameter combinations are computed for each query.The value of each parameter is randomly chosen according to a stored probability distribution function (PDF).In total 170 PDFs are defined, see Table 1.Beta-distributions are used to generate the PDFs as their shape can be defined with two shape parameters, resulting a useful choice to model random variables with finite limits.Most of them were defined based on "estimated guesses" by the authors, some of them based on literature values.It is the goal to continually update the PDFs as soon as the necessary empirical data is available.Nevertheless, for any given parameter, users can alter the stored PDFs as needed or specify a single value depending on the uncertainty of the query.Figure 1 shows three examples of the stored PDFs based beta distributions with different shapes.
As can be seen in Table 1 one or more PDFs have been defined depending on the parameter, for example, the PDF of the room area should depend on the selected room type (bedroom, living room, etc.).For many parameters, the PDF must be selected depending on the building type (single-family house, multi-family house, office building, etc.) or building standard (old building, standard new construction, low-energy construction, etc.).Further details about all the assumed PDFs can be found in (Greml et al., 2021).
It should be noted that this approach requires statistical independence of the different parameters, for example, the number of children in the classroom, should not depend on the parameter "room size".In reality, there may be dependencies between some of the parameters.However, this was considered when defining the PDFs and their value range.For example, in reality, in rare cases, class occupancy with only 5 or 40 students will occur, but it can be assumed that a correspondingly smaller or larger classroom will be selected.Thus, the range of values for the number of students was restricted to 15 to 25 students.Altogether, the authors assume that the effect of these dependencies on the final results are small.Nevertheless, this should be investigated further.An obvious exception regarding statistical independence are the parameters describing the weather conditions, i.e. temperature, humidity and wind speed.Therefore, those value ranges are not defined via PDFs, but with metrological data sets.These contain hourly values for representative years for the stored locations (15 Austrian cities).The weather data sets were generated using Meteonorm software (Meteotest n.d.).A random number is used to select any hour between January 1 and February 28.Only core winter months were chosen to focus on critical periods within which insufficient window ventilation by occupants may occur.

Assessment metric
The results of the presented calculation method are used to assess the "sufficient ventilation" requirement in terms of CO 2 concentration and mould risk in the following manner.
CO 2 concentration: The median of the 1000 calculated time intervals between window airing events, required to comply with the Austrian IAQ guideline, is compared with a time interval defined as reasonable for occupant intervention depending on the room type.This definition is summarised in Table 2.If the calculated time interval is smaller than the reasonable intervention time, than ventilation solely via manual window airing (and in-/exfiltration) is not considered acceptable.Note that the median is used for comparison, i.e. 50% of the calculated cases are expected to require shorter window airing intervals.
Mould risk: The calculation tool estimates the mould risk for two separate scenarios.One considers regular occupancy.Here, the corresponding humidity sources (persons, cooking, showering, plants, aquarium, … ) and a reasonable minimal occupant intervention (window airing twice a day) are accounted for.The other scenario considers absence, for example, during a winter vacation.Here, reduced humidity sources (plants, aquarium, … ), no window airing and reduced indoor temperatures are assumed.The mould risk is reported as fraction of cases (of the 1000 calculations) where the calculated water activity a w is >0.8.Based on the authors judgement, a mould risk fraction <1% is considered uncritical, fractions >5% are highlighted as critical.

Exemplary results
This section presents selected calculation examples to show the applicability of the calculation tool and the plausibility of the results.The only three obligatory inputs are the location, the type of building and the building standard.The following exemplary queries show the results for a

Window airing requirement based on CO 2 concentration
In an exemplary query the window airing requirement for the bedroom was assessed.Note that for each query the results vary slightly.Figure 2 shows the results of the timely evolution of the CO 2 concentration and the histogram of the required window airing interval.The query results estimate that for 50% of the cases (median) the limit value of the Austrian guideline (average concentration in assessment period of 8 hours <1000 ppm) would be exceeded every 68 minutes or sooner.In 5% (95%) of the cases the window airing interval would need to be 23 (190) minutes or shorter.This result shows that for newly constructed multi-family buildings a ventilation strategy based on manual window airing is unacceptable for the bedroom if one wants to comply with the guideline and assumes that airing within an 8 h rest period is unacceptable.
In another exemplary query the window airing requirement for the living room was assessed.For living rooms, the Austrian guideline suggests that the average CO 2 concentration should not exceed 1400 ppm.This work assumes that a time interval between manual interventions by the occupants of greater two hours is acceptable for the living room.Therefore, the results for the living room are not as clear as for the bedroom.When the variable input parameters are based on the defined PDFs, the 5 th percentile, the median and the 95 th percentile of the airing interval are 24, 64 and 146 minutes, respectively.So only in a few cases (5-10%) the airing interval would be greater two hours.However, when the floor area of the living room is specified to be 50 m 2 instead of being variable between 16 and 46 m 2 (P5 and P95), the median airing interval is 130 minutes and therefore assessed as acceptable (see Figure 3).This sensitivity to air volume underlines the importance of using dynamic models when assessing bio-effluents.

Mould risk based on daily averaged indoor air humidity
In an exemplary assessment of the mould risk around 17% of the cases show a risk of mould growth (water activity a w > 0.8) during occupancy if the variable input parameters are based on the defined PDFs (Figure 4a).The tool also calculates the additional air exchange required to reduce the mould risk fraction to 1%.For this example, up to around 40 m 3 /h would be needed.If a construction with minimal thermal bridges (f RSI ¼ 0.9 þ/-0.03corresponding to Passive House quality instead of 0.7 þ/-0.06 corresponding standard new construction) is assumed, the fraction exhibiting mould risk is reduced to about 4.5%.In this case an additional air flow rate of up to $10 m 3 /h would be needed to reduce the fraction to 1%.If the humidity source strength is assumed to be "low" (resulting in a P5-P95 range of 2.0 À 6.7 L/day, instead of 2.3 À 8.4 L/day) the mould risk fraction is at around 9%.Only the combination of both assumptions (minimal thermal bridges and low humidity source strength) would reduce the fraction with mould risk to about 2% (Figure 4b).Note that, although not shown here, results indicate that for certain cases (e.g., high airtightness values) the absence scenario will be more critical in terms of mould risk.

Conclusions
The presented calculation method, and its implementation in a spreadsheet software, can be used to asses if natural in-/exfiltration and window airing can provide sufficient ventilation to comply with given CO 2 concentration limits and ensure low mould risk.The novelty is the "simple-to-use" stochastic approach based on simplified physical models.It accounts for uncertainty and variability of the input parameters providing uncertainty information in the outputs.The current implementation is geared towards Austrian buildings and circumstances, reflected in the stored PDFs defining the variance of input parameters and in the applied assessment metrics.An adaption or amendment for other countries or regions is easily possible.Exemplary results indicate that for current Austrian construction practices, it is not reasonable to rely solely on window airing by the

Figure 2 .
Figure 2. Window airing requirement for the bedroom of a new multifamily building (see text).Left: Evolution of time average of CO 2 concentration without and with airing for the median case, and for selected percentiles without airing.Right:Histogram of window airing interval if Austrian limit value is not to be exceeded.

Figure 3 .
Figure 3. Window airing requirement for a 50 m 2 living room of a new multifamily building (see text).Left: Evolution of time average of CO 2 concentration without and with airing for the median case, and for selected percentiles without airing.Right:Histogram of window airing interval if Austrian limit value is not to be exceeded.

Figure 4 .
Figure 4. Histogram for number of cases with different water activity results.Values >0.8 correspond to cases with substantial mould risk.Left: Result for standard newly constructed multifamily building in Vienna.The remaining parameters are defined by the stored PDFs.Right: Results for newly constructed multifamily building in Vienna with minimal thermal bridges (Passive House standard) and low humidity load assumption.

Table 1 .
Summary of all relevant input parameters and the corresponding number of defined probability density functions (PDFs) and what the selection of the PDF depends on fin curly bracketsg.

Table 2 .
Assumptions of reasonable time interval between occupant intervention.multifamily building in Vienna.Unless otherwise noted all the other parameters were not further specified.Their values were chosen based on the stored probability density functions generating random variations for the 1000 Monte-Carlo calculations.