# Moisture Redistribution in Full-Scale Wood-Frame Wall Assemblies: Measurements and Engineering Approximation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Methods

#### 2.1. Wall Assemblies and Wetting System

#### 2.2. Moisture Content, Temperature, and Relative Humidity Measurements

_{sat}) was calculated using a slightly modified equation from Buck [25], with output in Pa given temperature θ in °C:

_{sat}, where h is relative humidity (0 ≤ h ≤ 1). To calculate the OSB vapor pressure, a formula from Boardman [9] was used based on a generic OSB sorption isotherm, allowing calculation of the relative humidity h given the moisture content u:

#### 2.3. Load Cell and Counter-Balance System

## 3. Experimental Results

_{init}is the initial value prior to water injection, and S

_{peak}is the maximum value after water injection. The plots start just prior to the water injections. Typically, over 70% of the paper towel area becomes wet after injection, so that pins 1 to 3 inside the field of the paper towel all rise. The MC which represents this wetted area is the average of pins 1 and 3.

## 4. Moisture Transfer Model

#### 4.1. Basic Moisture Transfer Mechanisms and Approximations

^{−1}), A is the surface area (m

^{2}), R is the resistance to water vapor diffusion (m

^{2}h Pa g

^{−1}), and p is water vapor pressure (Pa). The time units are in hours (h) since the process is slow and the data were acquired on an hourly basis. Values of Q were calculated each hour based on the values of p. Each material was represented by a single vapor resistance value; the typical dependence on relative humidity for hygroscopic materials such as lumber and OSB is not included in the model. The model includes additional effective resistance due to the air layers (R

_{air}) on the inside and outside of the OSB. When exterior insulation was added over the outside of the OSB, this further increased the total resistance to moisture flow by R

_{xps}.

_{wet}) coming from the wetted area of the OSB directly under the paper towel (A

_{wet}) and the second (Q

_{nw}) from the rest of the area (A

_{nw}) of the rectangular piece of OSB. In the majority of the OSB area (A

_{nw}), the vapor driving force is the difference between the vapor pressure in the wall cavity (p

_{cav}) and the laboratory (p

_{lab}). The third flow (Q

_{wood}) is constituted by diffusion through the lumber studs, driven by the same pressure difference but with different resistance factors, including wood thickness (R

_{wood}) and, in some experiments, a paint layer (R

_{pnt}).

^{2}to roughly match the rectangular paper towel area. The area of the initially wet OSB is not well defined so the area of the wetted portion was chosen to approximate the paper towel area. Similarly, the moisture content of this area varies by location and is arbitrarily assumed to be represented by the average of moisture pins 1 and 3, which are in the field of the wetted paper towel. The initial water mass used to calculate k also depends on the density of the OSB, which was measured as 540 kg/m

^{3}. The assumption of near-exponential decay to describe the mass of injected water subject to moisture redistribution will be explored later by comparing other possible simple models, including pure exponential decay. The near exponential function f(t) is described in more detail by Whitehead et al. [27] and is shown in Equation (5) below, with k as the scale factor (g), c a dimensionless shape parameter, τ

_{mrd}the time constant (h), and t the time (h).

_{mrd}were used as fit parameters in the moisture balance described below to allow the predicted MC inside the wetted region to track the measured values over time. The c value, which must be above 0, determines this function’s proximity to pure exponential decay.

_{lab}), but the same exponentially weighted average was used:

_{sorp}is a time constant for sorption (set to 100 h) and w is a function that applies exponential weighting over the previous 400 h:

_{sorp}represents the resistance to diffusion in combination with sorption and is another fitting parameter in the moisture balance described in the next section. The full model uses two Q

_{sorp}terms, one for the OSB and the other for the lumber, which have different resistance terms related to paint on the lumber or exterior insulation sometimes covering the OSB, as well as different areas. The time values used for averaging and the sorption time constant are discussed in Appendix B.

#### 4.2. Moisture Transfer Model for the Whole Wall and for the Water Injection Site

_{i}was based on the mass at the previous hour (t

_{i}

_{−1}) and the moisture flows Q from diffusion and sorption (Equation (8)) for a time step Δt set to 1 h:

_{wet}), inward diffusion into the insulated cavity (Q

_{in}), redistribution (Q

_{mrd}) within the OSB or the cavity away from the injection site, and sorption (Q

_{sorp}) scaled by the ratio of wetted area to total area (A

_{wet}/A

_{osb}):

_{d,osb}is the dry density of OSB (540 kg m

^{−3}) and L

_{osb}is its thickness (11 mm).

_{mrd}) is strongly dependent on Equation (5) modeling the mass of water moving out of the wetted area into the surrounding OSB, as well as further net transfer due to redistribution in the air. In those cases where external insulation was added to the OSB, the additional term R

_{xps}was added to Equations (8), (10), and (11).

_{0}), was set to the peak MC measured by the moisture pins. However, the initial value of total mass, m(t

_{0}), was allowed to vary somewhat from initial measured mass. When the peak MC was reached and the simulation started, the mass flow was still far from steady state, so there was little chance of matching the predicted and measured m. It took an additional day or two for the mass flows to stabilize, so the starting value, m(t

_{0}), was adjusted to minimize the difference between predicted and measured values after the system was closer to steady state. Furthermore, it was not possible to use the mass corresponding to the MC reading because an unknown amount of injected water had already moved outside the wetted area before the model even began, given that the injections happened over multiple days.

## 5. Modeling Results and Discussion

#### 5.1. Modeling versus Measured Drying Curves

#### 5.1.1. Results for Poly Walls

_{osb}, with diffusion through the framing assumed to be negligible due to the polyethylene covering, in order to minimize the difference between measurement and model prediction for both the MC within the wetted area and the total mass of injected water remaining in the wall. Model predictions are compared with measurement on the left side of Figure 8. The resulting OSB vapor resistance was equivalent to a vapor permeability of 1.2 ng Pa

^{−1}s

^{−1}m

^{−1}, which corresponds to a diffusion resistance factor of 170. Recall that the water vapor diffusion resistance factor is the ratio of the vapor permeability of still air to that of the material and is thus dimensionless. We assumed that the permeability of air was 198 ng Pa

^{−1}s

^{−1}m

^{−1}based on the calculation methods in ASTM standard E96 [30], using a temperature of 23 °C and pressure of 1 bar. This value for OSB is between the 40% and 70% RH levels for OSB reported in the literature [31,32,33,34].

^{−1}s

^{−1}m

^{−1}, which corresponds to a diffusion resistance factor of 79. This value for XPS is lower than the range of values (94 to 208) reported in the literature [31,32,35]. Furthermore, this value was still too high to model the non-poly wall results discussed next, which used an XPS resistance factor of 55. In all cases, the XPS was caulked to the OSB surface, but in the case of the poly wall, the XPS was also taped to the polyethylene which covered the lumber. We suggest that these lower XPS vapor diffusion resistance factors are realistic for field work, where water vapor can diffuse through alternative paths in addition to directly through the XPS or possibly be carried away by air leakage between the XPS and OSB.

#### 5.1.2. Results for Non-Poly Walls

_{osb}, R

_{xps}, R

_{air}, R

_{sorp}, and R

_{pnt}, yielded a wood permeability of 3 ng Pa

^{−1}s

^{−1}m

^{−1}. This corresponds to a diffusion resistance factor of 63. This value for wood is between the 50% and 75% RH levels for soft woods like spruce, pine, or fir reported in the literature [32,36,37], again showing good agreement with literature values. Table 2 summarizes all the optimized R values in units native to the model (Pa h m

^{2}g

^{−1}). Regarding moisture redistribution away from the injection site, the average values of c and τ

_{mrd}for the near-exponential decay model of the lateral movement were 1.44 and 168 h, respectively. Table 3 summarizes the c and τ

_{mrd}values for each wall, beginning the exploration of the parameter variability depending on insulation levels and water injection amounts.

#### 5.2. Water Vapor Pressure and Component Moisture Flows

_{wet}) is smaller than the flow from the rest of the OSB (Q

_{nw}) because the area is much smaller. More interesting is that the movement of moisture away from the wetting site (Q

_{mrd}) is much higher than flow due to diffusion (Q

_{nw}) very early in the drying, as moisture moves laterally in the OSB and into the cavity air. The magnitude of these flows equalizes as the drying continues. The ability to calculate Q

_{mrd}is the primary fruit of this modeling effort.

#### 5.3. The Shape of the Redistribution Function

_{wet}in the wetted area and p

_{nw}representing the OSB outside of the wetted area. The effective resistance was allowed to vary in order to obtain the best fit. The results in Table 4 show that this is the worst of the models, primarily because it does not have enough driving force, although the shape of the curve is good. This suggests that additional physical mechanisms are at work in moisture redistribution beyond diffusion laterally through the OSB—for example, capillary transport and evaporation. The rest of the models use different simple functional forms to fit the data, each using two fit parameters, a and b, in the first two cases. The equations explored are a linear fit (Q

_{mrd}= at + b), a logarithmic fit (labeled “log”, Q

_{mrd}= a·ln(t) + b), and a pure exponential fit (labeled “exp”) which uses the difference in mass between adjacent hour entries like the near-exponential function to calculate Q

_{mrd}. Figure 13 plots measured MC and the predictions from all the forms for the poly, w/o XPS wall (top), along with the residuals (bottom).

_{mrd}early in the drying and the rapid reduction of this contribution to moisture movement.

_{mrd}parameters allow a good fit but are not as sensitive to obtaining exactly the correct value. Table 5 illustrates this by showing the RMSE when the redistribution function can vary only one fit parameter, τ

_{mrd}. In this case, k is fixed as half of the water mass in the wetted area for both the pure and near-exponential models, and c = 1.44, the average across the non-poly walls when using the near-exponential model. In all but one case, the near-exponential function has a lower RMSE.

#### 5.4. Further Discussion

_{cav}is only slightly less than p

_{wet}and that p

_{cav}is the primary driver of moisture loss from the system. We speculate that water is rapidly distributed within the cavity by multiple mechanisms which include lateral diffusion within the OSB, capillary transport, and evaporation from the injection site into the cavity air. As a result of this moisture redistribution, the MC read by the moisture pins in the field of the paper towel drops rapidly at first, but this does not correspond to rapid moisture loss from the system. This moisture redistribution reduces the risk of local moisture damage at the injection site but does not correspond to what we usually mean by drying—that is, moisture exiting the wall assembly.

_{mrd}and Q

_{in}, to be helpful when creating one-dimensional hygrothermal models of drying after water injection. The work of Boardman, Glass, et al. [23] started exploration of this modeling using field data and assumed a near-exponential shape for the moisture sink. In the case of the “wood w/ XPS” laboratory data measured in this work, the relevant moisture sink to try in capturing field conditions would be a near-exponential approximation with k = 80 (half of the injection), c = 1.17 and τ = 271 h, based on the sum of Q

_{mrd}and Q

_{in}. This is close to the Q

_{mrd}values c = 1.14 and τ = 168 h because Q

_{in}is small, similar to the Q

_{wet}values, which are much smaller than Q

_{mrd}soon after the injection. This earlier work used a larger k value (85% to 55% of injection), with default c = 0.79 and τ = 75 h applied to all tested cases. This default produced adequate results, tracking the MC of the wetted area as it decayed, with a RMSE of 2.6% MC. The present manuscript will inform ongoing work to improve the one-dimensional hygrothermal modeling of the field data.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Load Cell Drift and Temperature Correction

**Figure A1.**Laboratory conditions (

**a**,

**b**) during constant mass experiment, with raw mass and corrected mass (

**c**).

## Appendix B. Calibration of Simple Sorption Model

**Figure A2.**Laboratory conditions (

**a**) and associated mass response (

**b**) and MC response (

**c**) during RH spike, showing sorption model fit.

_{sorp}, set to 100 h. The representative OSB vapor resistance for sorption was 560 Pa h m

^{2}g

^{−1}. Assuming the same OSB permeability used in the full model, this resistance represents 21% of the OSB depth. Figure A2 plots the total mass, measured by load cell, with the model prediction (b) and the measured MC with the predicted MC (c). These predictions were produced using the model described in Equations (9)–(15) but neglecting all the moisture transfer terms except Q

_{sorp}for the OSB (the framing lumber was covered).

## Appendix C. Vapor Pressure Coordination with Room Vapor Pressure

**Figure A3.**Vapor pressure before and after water injection: raw (uncorrected) values (

**a**,

**b**) and corrected values (

**c**,

**d**).

_{lab}= 1214 Pa, p

_{cav}= 1367 Pa, p

_{wet}=1487 Pa, and p

_{nw}= 1511 Pa. The three latter values were lowered by constants of 153, 273, and 297 Pa to produce the corrected values used in subsequent analysis. A close-up of the corrected data is plotted in Figure A3d, along with the full data (c).

Number | Label | Vapor Pressure Correction (Pa) | ||
---|---|---|---|---|

- | - | p_{cav} | p_{wet} | p_{nw} |

1 | Poly, w/o XPS | 201 | 340 | 327 |

2 | Poly, w/ XPS | 153 | 273 | 297 |

3 | Wood, w/o XPS | 145 | 327 | 216 |

4 | Wood, w/ XPS | 253 | 430 | 346 |

5 | Paint, w/o XPS | 203 | 398 | 403 |

6 | Paint, w/ XPS | 144 | 311 | 278 |

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**Figure 1.**Side and front view diagram of wall construction. Points 1 through 7 represent placement of moisture pins.

**Figure 5.**Fractional change in total mass and local OSB moisture content after water injection for poly walls with and without 50 mm XPS covering the OSB. For clarity, only every 50th data point is plotted for each series. The blue vertical line marks the end of the moistening phase, after which drying occurs.

**Figure 6.**Moisture pin data comparing wetted MC with MC outside wetted area (non-wetted) for (

**a**) no XPS covering OSB, and (

**b**) 50 mm XPS covering OSB.

**Figure 7.**Vapor pressures in the wetted OSB region (p

_{wet}), in OSB outside the wetted region (p

_{nw}), and in cavity air (p

_{cav}) compared to laboratory (p

_{lab}).

**Figure 8.**Measurement versus model predictions for both poly walls (0.15 mm polyethylene covering the framing lumber). Subfigures (

**a**) mass results, and (

**c**) MC results, represent the wall without XPS covering the OSB, while (

**b**) mass results, and (

**d**) MC results, represent the wall with 50 mm XPS insulation covering the OSB.

**Figure 9.**Measurement versus model prediction for walls with exposed lumber framing (wood, w/o XPS and wood, w/ XPS). Subfigures (

**a**) mass results, and (

**c**) MC results, represent the wall without XPS covering the OSB, while (

**b**) mass results, and (

**d**) MC results, represent the wall with 25 mm XPS insulation covering the OSB.

**Figure 10.**Measurement versus model prediction for walls with painted framing lumber (paint, w/o XPS and paint, w/ XPS). Variations in room humidity caused the mass fluctuations. Subfigures (

**a**) mass results, and (

**c**) MC results, represent the wall without XPS covering the OSB, while (

**b**) mass results, and (

**d**) MC results, represent the wall with 50 mm XPS insulation covering the OSB.

**Figure 11.**Vapor pressures and flows for walls with exposed framing lumber (wood, w/o XPS and wood, w/ XPS).

**Figure 12.**Vapor pressures and flows for walls with painted framing lumber (paint, w/o XPS and paint, w/ XPS).

**Figure 13.**Different models for moisture movement away from injection site (Q

_{mrd}) in exposed OSB: measured OSB moisture content with model predictions (

**a**) and residuals (

**b**).

Number | Label | Water Injection (g) | XPS Thickness (mm) | Lab RH Stability | Lumber Covering |
---|---|---|---|---|---|

1 | Poly, w/o XPS | 160 | 0 | fair | Poly |

2 | Poly, w/ XPS | 160 | 50.8 | good | Poly |

3 | Wood, w/o XPS | 300 | 0 | good | None |

4 | Wood, w/ XPS | 160 | 25.4 | good | None |

5 | Paint, w/o XPS | 160 | 0 | poor | Paint |

6 | Paint, w/ XPS | 160 | 50.8 | fair | Paint |

R_{osb} | R_{xps} | R_{wood} | R_{sorp} | R_{air} | R_{pnt} |
---|---|---|---|---|---|

2619 | 1954 | 3941 | 550 | 11 | 7 |

Parameter | Wood, w/o XPS | Wood, w/ XPS | Paint, w/o XPS | Paint, w/ XPS | Average |
---|---|---|---|---|---|

c | 0.43 | 1.13 | 1.59 | 2.63 | 1.44 |

τ_{mrd} (h) | 287 | 199 | 89 | 98 | 168 |

Model | Poly, w/o XPS | Poly, w/ XPS | Wood, w/o XPS | Wood, w/ XPS | Paint, w/o XPS | Paint, w/ XPS |
---|---|---|---|---|---|---|

nexp | 0.49 | 0.61 | 0.65 | 0.37 | 0.19 | 0.30 |

exp | 0.15 | 0.21 | 0.54 | 0.40 | 0.22 | 0.21 |

log | 0.72 | 0.76 | - | - | 0.70 | 0.59 |

linear | 1.47 | 1.42 | - | - | 1.22 | 1.11 |

diff | 1.50 | 2.03 | - | - | 1.46 | 1.34 |

**Table 5.**RMSE (% MC) for near-exponential and pure exponential Q

_{mrd}models with only τ

_{mrd}allowed to vary.

Model | Poly, w/o XPS | Poly, w/ XPS | Wood, w/o XPS | Wood, w/ XPS | Paint, w/o XPS | Paint, w/ XPS |
---|---|---|---|---|---|---|

nexp | 0.50 | 1.34 | 1.07 | 0.43 | 0.22 | 0.60 |

exp | 1.38 | 2.17 | 0.81 | 0.98 | 1.16 | 1.50 |

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## Share and Cite

**MDPI and ACS Style**

Boardman, C.R.; Glass, S.V.; Zelinka, S.L.
Moisture Redistribution in Full-Scale Wood-Frame Wall Assemblies: Measurements and Engineering Approximation. *Buildings* **2020**, *10*, 141.
https://doi.org/10.3390/buildings10080141

**AMA Style**

Boardman CR, Glass SV, Zelinka SL.
Moisture Redistribution in Full-Scale Wood-Frame Wall Assemblies: Measurements and Engineering Approximation. *Buildings*. 2020; 10(8):141.
https://doi.org/10.3390/buildings10080141

**Chicago/Turabian Style**

Boardman, Charles R., Samuel V. Glass, and Samuel L. Zelinka.
2020. "Moisture Redistribution in Full-Scale Wood-Frame Wall Assemblies: Measurements and Engineering Approximation" *Buildings* 10, no. 8: 141.
https://doi.org/10.3390/buildings10080141