Water vapor transport to a semi-infinite material with simultaneous varying surface relative humidity and temperature

. The water vapor transfer between the indoor air and hygroscopic finishing materials is of importance for the moisture balance of the room. Most protocols for determining the effect are based on isothermal conditions and cycling relative humidity in the form of square wave or sinusoidal functions. A new analytical solution for a material exposed to a both time varying surface relative humidity and temperature is presented in the paper. The time varying temperature inside the material is assumed to follow the surface temperature throughout the material layer since the reaction time for temperature changes in a reasonable thin surface material is rather short compared with the one for moisture changes. The semi-infinite approach is justified by the fact that the penetration depth for moisture variations are very limited for diurnal variations. The analytical approach and solution are presented in the paper


Introduction
Moisture uptake in the interior surface materials due to varying humidity in the indoor air is of importance both for the humidity levels of the room itself and for the moisture conditions in the surface materials.There have been research on this topic [1][2][3] and standards [4] and some ongoing measurements [5].
However, there has been less focus on cases with both a varying RH and temperature.Some recent results on this are presented in [6][7].This study gives new handy analytical expressions for the effect of combined variations in RH and temperature.In parallel to this paper, an experimental validation of the model is under way.

Governing equations
In this study the surface vapor resistance is neglected which means that the surface RH is always equal to the room RH.This will represent the case of maximum interaction.

Equations
The moisture balance equation reads: Here, v (kg/m 3 ) is the humidity by volume and w (kg/m 3 ) is the moisture content.
The analysis in this paper assumes that the temperature of the surface material always follows the interior temperature without any delay.This is a reasonable assumption since temperature changes are much more rapid than moisture changes and that it is only the thin interior surface layer that is affected by variations in indoor cyclic moisture variations.

Simplified equations
Two simplifications will be introduced.The first one is that the vapor diffusion coefficient v  (m 2 /s) is constant: The second simplification is that the slope of the sorption curve is constant.Furthermore, hysteresis is neglected.
The moisture balance equation then becomes: Here, vs is the humidity by volume at saturation.Introducing the vapor moisture diffusivity av (m 2 /s): We have a RH step change at the boundary, x=0, at time zero.The surface RH is: The analytical solution [1] for a semi-infinite domain 0, 0 Here, erfc is the complimentary error function.The penetration depth, i.e. the depth were we approximately find half the disturbance of what happened at the boundary has propagated is: Example: For wood this depth is around 0.0005 m after 1h, 0.0007 m after 2h, 0.003 m after 1 day and 0.007 m after a week.

Periodic change
If we have the following variation of the RH at the material surface, x=0: Here, tp (s) is the time period of the cosinusoidal variation.For the steady periodic case, the following well-known expression [1] for a semi-infinite domain The periodic penetration depth pv d (m), i.e. the depth were the amplitude of the RH has diminished with a factor 1 0.37 Example: For wood this depth is around 0.001-0.002m for a diurnal variation (tp =24h) and 0.03 m for a yearly one.
These numbers show that the penetration depth of moisture in to the material is very small for diurnal variations.
This periodic variation is of particular interest in this paper.The moisture flow g (kg/m 2 /s) into the material at x=0 becomes: Principle sketch of the diurnal boundary RH variation (9) and the moisture flow into the material (12).The total moisture uptake during a half period, mA, is indicated by the hatched area.The time t0 is also marked.
We are interested in the total uptake of moisture by the surface during a half period.The magnitude of the moisture uptake mA (kg/m 2 ) from time t0 to time t0+tp/2 is: The time t0 is chosen so that the net uptake of moisture is zero at this time, see Fig 1 .For the periodic RH variation according to (9) and ( 12) it is equal to tp/8.For steady periodic variations the net accumulation of moisture in the material is on average zero over time.The amplitude of the integrated mass flow (kg/m 2 ), (13), into and out from the material surface during a half period, [1]: Here, the material temperature is equal to the boundary temperature.It is purely a function of time: A variable substitution is introduced: The moisture balance equation is transformed to: Here, we have an equation that is similar to the onedimensional moisture balance equation with the diffusivity av equal to 1.The equation is linear when using this transformed time variable and superposition techniques can be used.We therefore only need the basic solution for a unit-step change or sinusoidal variations to handle complex time-wise changes in the RH.For a unit step change we for instance have: And for a cosinusoidal variation (steady-periodic) in the  regime with the time period p  : 5 Wave train , and low RH value 0 A    and temperature T2 for: From the definition of  (17):   Figure 1 shows the boundary values as functions of t and  .The fraction, α, of the time period p  when the RH has its highest value (and Using Fourier series in regime, the wave train can be written as:

Moisture uptake
The moisture flow g (kg/m 2 /s) into the material is: For the isothermal case ( The integrated m (kg/m 2 ) moisture uptake from time zero to time t is: The formula can be reformulated using a variable substitution, ( ) Figure 4 shows fm, (32), for a case.As seen from the figure the moisture is leaving the material at time t0 until time t0+tp/2 when the moisture uptake is increasing again.The total moisture uptake/release during a half cycle is equal mA, which is illustrated in Fig 4.  The moisture uptake during a half period determined by ( 13) and (31-32) becomes: The time lag t0 and the amplitude function fA is practically found by integrating the moisture flow over time m(t), (31).The amplitude mA is then obtained from the difference between mmax and mmin.See Fig Due to symmetry we will get the same amplitude of the integrated mass flow in to the material if let av1 occur at the high boundary RH and av2 at the low one: ( ) E3S Web of Conferences 1 0 (2020) 72, 4005 NSB 2020 ttp://doi.org/10.1051/e3sconf/20201720h 4005

Example
The material surface is made of spruce: The RH will be high when the temperature is low and low when the temperature is high.The following boundary values are assumed.The average RH in the material will be lower than the average of the boundary due to changes of the temperature over time.
The magnitude of the integrated mass flow into and out from the material surface is:

Conclusions
Explicit analytical formulas for the moisture distribution inside a material that is exposed to cyclic variations in RH and temperature are derived.The boundary RH and temperature have different values during each half period.The formula can be used to estimate the uptake and release of moisture to an exposed material surface as long as the periodic penetration depth is much smaller than the thickness of the material.
change A material surface layer with initial relative humidity of 0  and temperature of 0 T is considered.

Fig 2 .
Fig 2. Boundary RH and temperature as functions of t and .
26) When α=0.5, the high and low values of v a are equal and the second term will vanish.This represent the isothermal For other values of α there will be a change in the average value of RH in the material from 0  .Since we know the solution for the heat conduction equation for cosinusoidal boundary conditions (20) we get:

Figure 3 Fig 3 .
Figure3shows the relative amplitude of RH and the average RH as a function of depth from the surface using (27) for a specific case.

Fig 4 .
Fig 4. Parameter fm, (32), for the integrated moisture uptake as a function of time t during a day.Diurnal variations (tp=24h).

Table 1 .
4. For the studied case the time t0 is approximately equal to tp/4, i.e the time when RH goes from a high to a low value.The amplitude parameter A The amplitude parameter A f for the determination of the total moisture uptake during a half cycle.
f of the integrated mass flow into and out from the material surface (33) is given in Table1.