# Stochastic Simulation of Mould Growth Performance of Wood-Frame Building Envelopes under Climate Change: Risk Assessment and Error Estimation

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Hygrothermal Model

#### 2.1.1. Wall Configuration and Material Properties

#### 2.1.2. Boundary Conditions and Climatic Realisations

- h
_{ce}—exterior heat transfer coefficient (W/m^{2}·K) - β
_{ve}—exterior vapour transfer coefficient (s/m) - v—wind speed (m/s)

- r
_{bv}—wind-driven rain deposited on the exterior wall surface - F
_{E}—rain exposure factor, reflects different exposure types, for buildings lower than 10 m; the rain exposure factor can be assumed to have a value of 1.4 for the severe exposure category, 1.0 for the medium exposure category, and 0.7 for the sheltered category - F
_{D}—rain deposition factor, reflects different roof designs; it can be assumed to have a value of 0.35 for a steep-slope roof, 0.5 for a low-slope roof and 1.0 for a wall subject to rain runoff - F
_{L}—empirical constant, 0.2 kg·s/(m^{3}·mm) - θ—angle between wind direction and normal to the wall
- U—hourly average wind speed at 10 m height, m/s
- r
_{h}—rainfall intensity, horizontal surface, mm/h

_{E}) was considered as 1, and the rain deposition factor (F

_{D}) was considered as a stochastic variable. To simulate rain leakage, a moisture source was deposited on the exterior surface of the spun-bonded polyolefin (SBPO), and the quantity of this moisture source was considered as a stochastic variable as well. The effect of cladding ventilation was simulated using a source/sink approach, which considers the air in the drainage cavity as a heat and moisture source by imposing a cladding ventilation rate (ACH) that was considered as a stochastic variable. The range of these three stochastic variables and the stochastic brick properties will be discussed in Section 2.2.

- MI—moisture index of a specific year
- DI—normalised drying index of a specific year
- WI—normalised wetting index of a specific year

- U—wind speed in a specific hour, m/s
- R
_{h}—horizontal rain in a specific hour, mm

- w
_{sat}—humidity ratio at saturation in a specific hour - w—humidity ratio of ambient condition in a specific hour

#### 2.1.3. Wall Orientations

#### 2.2. Literature Review of Stochastic Variables

#### 2.2.1. Rain Deposition Factor

_{D}= 0.5 under historical and future periods. It can be seen that the maximum value for catch ratio, with F

_{D}= 0.5 and WS = 10 m/s, is around 1. This value for the historical period is slightly lower than the values obtained from field measurement and numerical simulation for a mid-rise building [8,9], or low-rise building [43]. The value for catch ratio is higher for the future projected climate period as compared to the historical period, because of the higher projected wind speed in the future. The randomness of the scatter plot is due to the randomness of the wind direction, although most of the time the wind direction is normal to the facade. Therefore, the wind-driven rain calculated using the ASHRAE semi-empirical model, and with F

_{D}= 0.5, can be considered as the amount of rain deposited on the façade at the worst position (top or top edge). Similarly, the hourly catch ratio values were also calculated for F

_{D}= 0.35 and F

_{D}= 1, the maximum and 2-years’ averaged catch ratio at these two levels of F

_{D}were presented in Table 4. Since catch ratio with F

_{D}= 0.35, is 30% lower than that with F

_{D}= 0.5, the scenario of F

_{D}= 0.35 can be considered as a moderate protection level, i.e., a 0.1-m overhang [9]. Whereas for the scenario with F

_{D}= 1.0, this can be considered as the wall subject to rainwater runoff or the façade on a high-rise building, since the average catch ratio is around 0.38, which is very close to the average catch ratio of a tall building reported in the literature [44]. To consider the uncertainty of the building geometry, a small range of values for F

_{D}was assigned for each level of F

_{D}, whereby, the stochastic F

_{D}value was categorised into three ranges: 0.35 to 0.56, to represent the wall with some protection; 0.56 to 0.78, to represent the wall without any protection, and; 0.78 to 1, to represent the wall subject to rainwater runoff or a wall in a high-rise building.

#### 2.2.2. Moisture Source from Rain Leakage

- WDRPI—wind-driven rain pressure index
- WDR—wind-driven rain, calculated based on ASHRAE wind-driven rain model
- DRWP—driving-rain wind pressure, calculated by hourly wind velocity through Bernoulli’s principle
- α, β—for different configurations of wall assemblies based on their response to the WDR intensity and DRWP respectively during water tightness tests. For the brick wall, the value of α is 0.9506 and β is 1.0442.

- Moisture source—the amount of water that reaches the sheathing membrane per unit time (ml/min)
- a, b—adjustment coefficients derived from fitting the measured moisture source to corresponding values of WDRPIs. The details of the measurements and derivation of adjustment coefficients have been demonstrated based on a vinyl-clad wall by Xiao et al. [47]. The same procedure was also applied to the brick wall to obtain the two adjustment coefficients where: a = 7.998 × 10
^{−6}and b = 0.6737.

#### 2.2.3. Cladding Ventilation Rate

^{2}[49,50]. Generally, an insect screen is installed at the top openings, and the cavity depths are either 25 mm or 38 mm. This type of opening design leads to a variation of ACH between 0 and 10. For example, Finch and Straube reported the ACH of a brick wall with a 38 mm ventilated cavity, which was installed in the BEGHut, a test facility at the University of Waterloo. The ACH value was in the range of 0 to 9.6 with an average value of 2.2 for an entire year [49]. Simpson monitored the hygrothermal performance of a brick wall with a 25 mm ventilated cavity, which was located in the Burnaby campus of the British Columbia Institute of Technology (BCIT). The derived ACH value ranged between 1 and 11 with an average of 6 from February to June [50]. It is noted that the ACH can be increased up to 90 for a 20 mm deep cavity when there is no insect screen; however, this type of wall was only used for the experiment, walls 20 mm deep cavity without an insect screen is not a common construction practice in Canada [48].

#### 2.2.4. Brick Properties

_{eff}) of brick were considered as stochastic material properties. The material properties of red matt clay brick were used from a baseline model [32]. To quantify the range of these two brick properties, a literature review was conducted to determine the A-value and θ

_{eff}of similar types of brick. A summary of these values is given in Table 7. The effective saturation water content, θ

_{eff}, is that of the maximum degree of saturation as may occur over the long-term immersion. However, the effective saturation water content is subject to a large uncertainty depending on different test methods used to assess this value, be this using the long-term with partial immersion in water, 5 h boiling saturation, or vacuum saturation test methods. According to Mensinga [53], the 5 h boiling saturation water content is about 20% lower than that obtained using vacuum saturation. Even using the same test method, the measurement uncertainty is quite high. For example, the vacuum saturation is largely dependent on the vacuum pressure applied to the specimen, and the uncertainty in applied vacuum pressure can result in ca. ±20% of uncertainty for the θ

_{eff}from vacuum saturation [53]. Furthermore, the properties amongst different specimens in the same brick category have large uncertainties due to the manufacturing process. According to Kumaran [33], the uncertainty in the vacuum saturation water content amongst 9 brick specimens is about ±40%. Zhao [54] measured the effective saturation water content of 23 brick specimens within the same brick category through 2 weeks of partial immersion; it was found that the uncertainty can be as high as ±37%. By considering the values for the effective saturation water content of clay brick and the uncertainties reported from different literature, the range for the effective saturation water content within one type of brick was set as ±40%. A normal probability distribution was generated by assuming the mean value as 0.217 m

^{3}/m

^{3}, with a standard deviation of 0.043 m

^{3}/m

^{3}.

^{2}s

^{0.5}, and standard deviation as 0.005 kg/m

^{2}s

^{0.5}; as such, the uncertainty also falls within ±40%. According to the statistical analysis of 23 brick specimens by Zhao et al. [57], the effective saturation water content is positively correlated with the A-value with a correlation coefficient of 0.6. For this paper, then, the probability distributions of effective saturation water content and A-value were assumed as positively correlated with a coefficient of 0.6. Figure 8 shows the probability density functions of the brick properties, the correlation between A-value and θ

_{eff}, and the stochastic moisture storage curves, liquid diffusivity curves derived from stochastic A-value and θ

_{eff}.

_{eff}was used to scale the moisture storage curve, and the liquid diffusivity curve was calculated based the stochastic θ

_{eff}and A-value through Equation (9) [58,59]:

- D
_{w}—liquid diffusivity at unsaturated water content, m^{2}/s - θ
_{eff}—effective saturation water content, kg/m^{3}, 10^{3}∙ m^{3}/m^{3} - θ—unsaturated water content, kg/m
^{3}, 10^{3}∙ m^{3}/m^{3} - A—water absorption coefficient, kg/m
^{2}s^{0.5} - b—shape factor, determines the slope of the liquid diffusivity curve; value can be between 5 and 10; In this paper, b was assumed as 7.5.

#### 2.3. Literature Review of Sampling Methods

#### 2.4. Implementation of Sobol Sequence-Based Sampling

#### 2.5. Error Estimation and Risk Assessment

- ${Q}_{n}^{\left(i\right)}\left(f\right)$—the estimator of mean or standard deviation of the ith randomised sequence
- ${\overline{Q}}_{n,r}\left(f\right)$—the average of the estimators of r sets of randomised sequences
- r—the number of randomised sequences; in this paper, r is 10

^{3}× 10, 2

^{4}× 10 up to 2

^{8}× 10. For the sub-spaces, the error estimation started from a sample size of 2

^{5}× 10 ended up with 2

^{8}× 10, as the sample size of each sub-space is approximately the total sample size divided by the number of intervals. The maximum sample size was determined by considering the balance between the computational cost and the accuracy of the results. For the whole sample space, the sample size of 2

^{8}× 10 gave a standard error in the magnitude of 10

^{−3}, whilst, for each sub-space, the sample size of 2

^{8}× 10 gave a standard error in the magnitude of 10

^{−2}. According to Hou et al. [29], the standard errors calculated from Equations (10) and (11) is roughly in the same order of magnitude as the absolute error that was obtained by comparing the results at different sample sizes (2

^{3}× 10, 2

^{4}× 10 up to 2

^{7}× 10) with a very large reference sample size—40,960. The results of standard error were discussed in Section 3.1 and Section 3.2. Then the mould growth risk for different climatic realisations and orientations were assessed based on the results from the highest sample size.

## 3. Results and Discussion

#### 3.1. The Whole Sample Space

^{−3}for both historical and future periods. As shown in Table 10, the future period has a higher mould growth index than the historical period, but the standard deviation in the future is very similar to that in the historical period. The standard error in the future period is slightly lower than that in the historical period. The mould growth index listed in Table 10 reflects the overall mould growth risk in the city of Ottawa, with all the climatic realisations, orientations and other stochastic variables taken into account.

#### 3.2. Different Climatic Realisations

^{−2}after 1280 runs, except for the mean values of the future period, which needs 2560 runs to ensure all the standard errors fall into the magnitude of 10

^{−2}.

^{−2}. It can be seen that most of the climatic realisations have a higher mould growth index in the future than in the historical period. The future period also has a higher variability in the mould growth index than the historical period given the higher standard deviation. The variation in mould growth index among different climatic realisations is more significant than the variation in moisture index as shown in Figure 2. As shown in Figure 14a,b, there is no clear relationship between mould growth index and moisture index based on the stochastic simulation results when all the orientations were taken into account. However, the significance of the relationship was increased when the analysis only focused on the orientation that receives the highest amount of WDR (The sample size was locally expanded for the worst orientation to guarantee the standard errors were in an order of magnitude of 10

^{−2}).

#### 3.3. Different Orientations

^{−2}after 1280 runs, which means for each orientation 80 runs can achieve a standard error lower than 0.1.

#### 3.4. Different Mould Growth Risk Mitigation Strategies

_{D}represents the low level of rain deposition factor (0.35~0.56) with all the other stochastic variables (climatic realisations, rain leakage moisture source, cladding ventilation rate, and brick properties) varying in their full range of values. As such, if the mould growth index can nonetheless be reduced by controlling one specific variable in consideration of all the uncertainties, the influence of this variable can be considered as a robust control strategy. The 640-run stochastic simulation based on RQMC gave the standard error of mean value and standard deviation of mould index for each level of each risk control variable of ca. 0.05.

#### 3.4.1. The Influence of Rain Deposition Factor and Cladding Ventilation Rate

#### 3.4.2. The Influence of Rain Leakage Moisture Source

_{D}increased from 1.5 to 1.8, and that at the high level of ACH increased from 2.0 to 2.7. (e.g., compare results given in Figure 17b to those in Figure 18a).

_{D}and ACH were reduced (Figure 18b). The 75 percentile mould growth index at a low level of FD decreased from 1.5 to 1, and that for a high level of ACH decreased from 2 to 1.5 (Figure 17b compare to Figure 18b). Therefore, it can be said that improvements in water tightness can reduce mould growth risk overall.

#### 3.5. The Influence of Brick Properties

^{2}value for the water penetration coefficient is higher than that for A-value. Although the A-value and effective saturation water content are the two properties that are measured from laboratory tests, the water penetration coefficient, i.e., the ratio of A-value to effective saturation content, is a more useful parameter in reflecting the water penetration capability of brick and assessing the influence on mould growth performance of the wood sheathing in wood-frame wall assemblies.

## 4. Conclusions

^{−2}after 1280-runs for all 15 climatic realisations and 16 wall orientations, which indicates for each climatic realisation or wall orientation, the standard error can be controlled at the magnitude of 10

^{−2}after 80-runs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 9.**Categorisation of different sampling methods. Note: The sampling points generated by the Quasi-Monte Carlo methods are deterministic points. To allow error estimation, the points from Quasi-Monte Carlo methods have to be randomised. Thus, it is referred to as Randomized Quasi-Monte Carlo (RQMC).

**Figure 12.**Convergence rate for different climatic realisations. Note: there are 15 climatic realisations, therefore, the sample size for each realisation is roughly equal to the total sample size divided by 15. The marks of “⧫” are outliers.

**Figure 15.**Convergence rate for different orientations. Note: there are 16 orientations, therefore, the sample size for each orientation is roughly equal to the total sample size divided by 16. The marks of “⧫” are outliers.

**Figure 17.**Boxplots of 2-year’s averaged mould growth index at different levels of mould growth risk control variables. FD: Rain deposition factor; MS: Rain leakage moisture source; ACH: Cladding ventilation rate. For ACH, the high level is the 4 opening ventilated design, the middle level is the 2 opening ventilated design and the low level is the vented cavity which has no opening on the top. The marks of “⧫” are outliers.

**Figure 18.**Boxplots of 2-year’s averaged mould growth index with scenarios of rain leakage moisture source in the future period. Note: The marks of “⧫” are outliers.

Bulk Density (kg/m ^{3}) | Porosity (m ^{3}/m^{3}) | Effective Saturation Water Content (m ^{3}/m^{3}) | Capillary Water Content (m ^{3}/m^{3}) | Vapour Resistance Factor_Dry (−) | Water Absorption Coefficient (kg/m ^{2}s^{0.5}) | Heat Capacity (J/kg·K) | Heat Conductivity (W/m·K) | |
---|---|---|---|---|---|---|---|---|

Red matt clay brick | 1935 | 0.265 | 0.217 | 0.162 | 129 | 0.0268 | 800 | 0.5 |

Air space | 1.2 | 0.99 | - | - | 1 | - | 1006 | 0.025 |

SBPO * | 464 | 0.012 | 0.012 | 0.01 | 305 | 0.00031 | 1250 | 0.248 |

OSB | 650 | 0.4 | 0.38 | 0.27 | 753 | 0.0022 | 1880 | 0.094 |

Mineral wool | 37 | 0.92 | 0.9 | 0.9 | 1.09 | - | 840 | 0.032 |

Polyethylene | 1256 | - | - | - | 1 × 10^{6} | - | 2100 | 0.16 |

Gypsum board | 700 | 0.65 | 0.42 | 0.4 | 138 | 0.0019 | 870 | 0.16 |

Interior Heat Transfer Coefficient (W/m^{2}·K) | Interior Vapour Transfer Coefficient (s/m) | Short Wave Absorptivity (−) | Long Wave Emissivity (−) |
---|---|---|---|

8 | 1.52 × 10^{−8} | 0.6 | 0.9 |

Authors | Year | Building Geometry ^{1} | Approach | Catch Ratio | Notes |
---|---|---|---|---|---|

Choi [41] | 1993 | Bldg1, 4: 1: 1 Bldg2, 4: 8: 2 | CFD simulation | Bldg1, 0.05–0.47, WS ^{2}, 10 m/s; 0.34–1.17, WS, 20 m/s; Bldg2, 0–0.4, WS, 10 m/s; 0.04–0.82, WS, 20 m/s | The variation of catch ratio at each wind speed depends on the positions on the façade. |

Blocken and Carmeliet [42] | 2000 | Bldg1, 4: 25: 7 Bldg2, 8: 25: 7 | CFD simulation | Bldg1, 0–0.5 Bldg2, 0–0.4 WS, 0–6 m/s | Bldg 1 flat roof, Bldg 2 steep slope roof. The catch ratio is in a fixed position on the façade at middle height. |

Blocken and Carmeliet [43] | 2001 | Same as above | CFD simulation & field measurement | Bldg1, 1.58 Bldg2, 1.26 WS, 10 m/s | The catch ratio is a specific catch ratio of 1 mm raindrop at the worst position on the façade-top edge. |

Kubilay et al. [44] | 2013 | Tower building 35: 5: 4 | CFD simulation & field measurement | Maximum specific catch ratio of 0.5 and 1 mm raindrop at 10 m/s WS is around 2.8 | Averaged catch ratio at the worst position after two rain events are 0.3 and 0.5, respectively |

Foroushani et al. [7] | 2014 | Cubic building 10: 10: 10 | CFD simulation | 0.6 at the worst position at WS 5 m/s | 0.6 m overhang helps protect the upper half of the façade up to 80% |

Kubilay et al. [9] | 2017 | 19: 16: 8 | CFD simulation | 1.2 at the worst position at WS 10 m/s | A window sill with a size of 0.1 m reduces the catch ratio by 37% |

Ge et al. [8] | 2017 | 20: 39: 15 | Field measurement | 1.0 at the worst position at WS 8 m/s. Averaged catch ratio at worst position 0.213 | 0.6 m overhang reduces the catch ratio by 30% to 90% depending on different positions |

^{1}The dimensions listed in building geometry are height: length: width

^{2}WS: Wind Speed.

Rain Deposition Factor | Maximum Catch Ratio | Average Catch Ratio |
---|---|---|

0.35 | Historical, 0.8; Future, 1.1 | Historical, 0.129; Future, 0.135 |

1 | Historical, 2.4; Future, 3.1 | Historical, 0.371; Future, 0.384 |

**Table 5.**Water entry rate of different types of wall assemblies (Data from reference [10]).

Wall Assembly | Drainage Cavity | Water Entry Rate (L/min) | Spray Rate (L/min) | ^{1} Pressure(Pa) | Highest Water Entry Ratio | Moisture Source on Sheathing Membrane (SBPO) |
---|---|---|---|---|---|---|

Brick veneer | Yes | 0.042 | 0.85 | 75 | 4.9% | 0.7% |

Stucco | Yes | 0.191 | 1.7 | 0 | 11.2% | 1.7% |

Stucco | No | |||||

Fibre Board | Yes | 0.15 | 0.85 | 150 | 17.6% | 2.6% |

Fibre Board | No | 0.014 | 0.85 | 300 | 1.65% | 0.25% |

EIFS ^{2} | No | 0.218 | 3.4 | 300 | 6.41% | 0.96% |

Vinyl | No | 0.059 | 0.85 | 300 | 6.94% | 1% |

^{1}The pressure listed above was the pressure at which the highest water entry rate was obtained. In principle, a higher pressure will likely lead to a higher water entry rate, however, the water-tightness tests are subject to measurement uncertainties and the water entry rate for some deficiency types may not be sensitive to the applied pressure, therefore, the highest water entry ratio does not necessarily occur at the highest pressure.

^{2}Exterior Insulation and Finish Systems.

Authors | Year | Cladding | Cavity Depth (mm) | Opening | ACH |
---|---|---|---|---|---|

VanStraaten and Straube [48] | 2004 | Brick | 20 | 1600 mm^{2} at top and bottom (2 of 10 × 80 at each position, clear no screen) | 0–90 |

Finch and Straube [49] | 2007 | Brick | 38 | 1300 mm^{2} at top & bottom (2 of 10 × 65 at each position, with bug screen) | 0–9.6, Average 2.2 |

Ying Simpson [50] | 2010 | Brick1: 2.44 m height; Brick2: 4.88 m height | 25 | 1560 mm^{2} on top (2 of 12 × 65 with bug screen); 1872 mm ^{2} (2 of 12 × 78) on bottom | Brick 1: 1–11, Average 6; Brick 2: 1–6, Average 4. |

Langmans et al. [14] | 2016 | Brick | 40 | 1050 mm^{2} at top and bottom (2 of 15 × 35 at each position) | 1 opening, 1–10, 85% of the time below 6; 2 openings, ACH doubled |

Vanpachtenbeke et al. [51] | 2020 | Brick | 40 | 1050 mm^{2} at top and bottom (2 of 15 × 35 at each position) | In between 5 and 10 |

Authors | Years | Name of Brick | Density (kg/m^{3}) | A-Value (kg/m^{2}s^{0.5}) | θ_{eff} (m^{3}/m^{3}) |
---|---|---|---|---|---|

Kumaran et al. [32] | 2002 | Red matt clay brick | 1935 | 0.0268 | 0.217 (Vacuum) |

Textured coated clay brick | 1821 | 0.0322 | 0.333 (Vacuum) | ||

Mensinga [53] | 2009 | Clay brick 1 | 2212 | 0.032 | 0.192 (5 h boiling) 0.228 (Vacuum) |

Clay brick 2 | 2223 | 0.028 | 0.182 (5 h boiling) 0.219 (Vacuum) | ||

Zhao [54] | 2012 | Old building brick Dresden1 | 1948 | 0.0219 | 0.179 (Partially immersed in water for 2 weeks) |

Old building brick Dresden2 | 1736 | 0.034 | 0.32 (Partially immersed in water for 2 weeks) | ||

Yousefi [55] | 2019 | Clay brick | 2080 | 0.012 | 0.116 (Partially immersed in water for 8 days) |

Aldabibi [56] | 2020 | Reclaimed exterior brick | 1968 | N/A | 0.242 (5 h boiling) |

New exterior brick | 1904 | N/A | 0.198 (5 h boiling) |

Authors | Year | Simulation Objects | Sampling Methods | Number of Stochastic Variables | Sample Size | Convergence Size |
---|---|---|---|---|---|---|

Lomas and Eppel [64] | 1992 | Whole building energy model | Random | 70 | 100 | 100 |

Salonvaara et al. [17] | 2001 | Hygrothermal model | Random | 16 | 100 | N/A |

Hyun et al. [65] | 2007 | A building ventilation model | Latin Hypercube | 13 | 30 | N/A |

Macdonald [66] | 2009 | Infiltration rate as a function of temperature & wind speed | Random; Stratified; Latin Hypercube | 2 | 100; repeated 100 times | 100 for all three sampling methods |

Zhao et al. [18] | 2011 | Hygrothermal model | Random | 36 | 400 | N/A |

Burhenne et al. [62] | 2011 | Whole building energy model | Random; Latin Hypercube; Stratified sampling; Sobol sequence-based sampling | 4 | 16, 32, 64, 128, 256, 512; each size repeated 100 times | Random sampling: 256; Other: 64 |

Defraeye et al. [67] | 2013 | Hygrothermal model | Random | 6 | 2000 | N/A |

Janssen [63] | 2013 | Hygrothermal model | Random; Optimized Latin Hypercube (OLHS) | 4 | 10, 20, 50, 100, 250, 500; each size repeated 10 times | OLHS converged faster than random |

Goffart et al. [68] | 2015 | Whole building energy model | Latin Hypercube | 14 | 600 | 400 |

Hou et al. [29] | 2019 | Hygrothermal model | Random; OLHS; Sobol; Neiderreiter-Xing; lattice sequence | 7 | 80, 160, 320, 640, 1280 | QMC converged faster than MC for smooth objective functions |

Bui et al. [69] | 2020 | Hygrothermal model | Latin Hypercube | 5 | 1000 | 600 |

Variables | Distribution | Range | Intervals |
---|---|---|---|

Climatic realisation | Discrete | R01–R15 | - |

Orientation | Uniform | 0–360 | 16 Orientations, interval of N 348.75~11.25, NNE 11.25~33.75 and so on… |

Rain deposition factor | Uniform | 0.35–1; | Low 0.35~0.56; Middle 0.56~0.78; High 0.78~1. |

Rain leakage moisture source (% of wind-driven rain) | Normal | 0–2.0 | Low 0.3 (0.35); Middle 0.56 (0.35); High 0.8 (0.35) |

Cladding ventilation rate (ACH) | Normal | 1–20 | Low 3 (0.67); Middle 5.5 (1.4); High 10.5 (3.5) |

Water absorption coefficient (A-value) of brick (kg/m^{2}·s^{0.5}) | Normal | 0.0161–0.0389 | 0.0268 (0.005) |

Effective saturation water content of brick (m^{3}/m^{3}) | Normal | 0.108–0.325 | 0.217 (0.043) |

Historical Period | Future Period | |
---|---|---|

Mean value | 1.06 ^{1} SE 0.007 | 1.44 SE 0.004 |

Standard deviation | 0.91 SE 0.007 | 1.08 SE 0.004 |

^{1}SE: standard error.

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**MDPI and ACS Style**

Wang, L.; Defo, M.; Xiao, Z.; Ge, H.; Lacasse, M.A.
Stochastic Simulation of Mould Growth Performance of Wood-Frame Building Envelopes under Climate Change: Risk Assessment and Error Estimation. *Buildings* **2021**, *11*, 333.
https://doi.org/10.3390/buildings11080333

**AMA Style**

Wang L, Defo M, Xiao Z, Ge H, Lacasse MA.
Stochastic Simulation of Mould Growth Performance of Wood-Frame Building Envelopes under Climate Change: Risk Assessment and Error Estimation. *Buildings*. 2021; 11(8):333.
https://doi.org/10.3390/buildings11080333

**Chicago/Turabian Style**

Wang, Lin, Maurice Defo, Zhe Xiao, Hua Ge, and Michael A. Lacasse.
2021. "Stochastic Simulation of Mould Growth Performance of Wood-Frame Building Envelopes under Climate Change: Risk Assessment and Error Estimation" *Buildings* 11, no. 8: 333.
https://doi.org/10.3390/buildings11080333