## 2 Materials and Methods  
### Description of the Multi-Phase Model  
Summarizing the multi-Fickian model presented in [11] for wood below the _FSP_, the variables for transient moisture transport are the concentration of bound water in the cell walls \(c_{b}\), the concentration of water vapour in the cell lumens \(c_{v}\), and the temperature \(T\). Denoting by \(\mathbf{D}_{b}\) and \(\mathbf{D}_{v}\) the diffusion tensors for bound water and water vapour phases and by \(\mathbf{K}\) the thermal conductivity tensor, the governing equations of the problem are:  
\[\frac{\partial c_{b}}{\partial t}=\ -\
abla\cdot\mathbf{J}_{b}+\dot{c}_{bv} \tag{1}\]  
\[\frac{\partial c_{v}}{\partial t}=\ -\ \
abla\cdot\mathbf{J}_{v}-\ \dot{c}_{bv} \tag{2}\]  
\[c_{w}\partial\frac{T}{\partial t}=\ -\ \
abla\cdot\mathbf{J}_{H}-\ \
abla\cdot \mathbf{J}_{b}h_{b}-\ \
abla\cdot\mathbf{J}_{v}h_{v}+\ \dot{c}_{bv}h_{bv} \tag{3}\]  
where \(\
abla\) is the nabla operator, \(\mathbf{J}_{b}\) and \(\mathbf{J}_{v}\) are the fluxes of bound water and water vapour, and \(\mathbf{J}_{H}\) represents the thermal flux:  
\[\mathbf{J}_{b}=-\mathbf{D}_{b}\
abla c_{b}\,\mathbf{J}_{v}=-\mathbf{D}_{v} \
abla c_{v},\ \mathbf{J}_{H}=-\mathbf{K}\
abla T \tag{4}\]  
In Equation (3), \(c_{w}\) represents the specific heat and \(\varrho\) the wood density, the coupling term \(\dot{c}_{bv}\) is the sorption rate between the two water phases (see Figure 1), \(h_{b}\) and \(h_{v}\) are the specific enthalpies and \(h_{bv}=h_{b}-h_{v}\) is the specific enthalpy of the transition from the bound water to the water vapour. The moisture content \(MC\) is defined as \(c_{b}/\varrho_{0}\) where \(\varrho_{0}\) is the dry wood density. The sorption rate in Equations (1)-(3) is defined as:  
\[\dot{c}_{bv}=\ H_{c}(\varrho_{0}MC_{bl}-c_{b}) \tag{5}\]  
where \(H_{c}\) represents the moisture dependent reaction rate and \(MC_{bl}\) is the moisture content in equilibrium with the relative humidity. In Equation (5), the \(MC_{bl}\) has the meaning of temperature-dependent sorption isotherms. These are defined by using the Anderson-McCarthy model (see Appendix A). In the present work, according to [11], an average between the temperature dependent adsorption and desorption isotherms is used, while a model for sorption hysteresis is not included. Since the bound water cannot pass the external surfaces and it is restricted in the cell walls, the model includes only exchanges of vapour and heat with the ambient air. Therefore, the first boundary condition of Equation (6) holds on all the external surfaces in relation to variable \(c_{b}\). For the other variables, the second and third boundary conditions in Equation (6) apply for the external surfaces exposed to the variable \(RH\) and \(T\):  
\[\mathbf{n}\cdot\mathbf{J}_{b}=\mathbf{0},\ \mathbf{n}\cdot\mathbf{J}_{v}=\ k_{v} ^{w}\ c_{v}^{\prime}-\ k_{v}^{a}c_{v}^{d},\ \mathbf{n}\cdot\mathbf{J}_{v}=\ k_{T}(T-\ T^{a}) \tag{6}\]  
where \(\mathbf{n}\) represents the outward normal direction to the surface, \(c_{v}^{a}\) and \(T^{a}\) are the water vapour concentration and temperature of the air, \(k_{v}^{w}\) and \(k_{v}^{a}\) the surface permeances corresponding to wood temperature and air temperature, and \(k_{T}\) is the thermal emission coefficient. The expressions of the permeances are reported in Appendix A. In Equation (6), \(c_{v}^{\prime}=c_{v}/\varphi\) represents the concentration of water vapour divided by the wood porosity \(\varphi\). The concentration \(c_{v}\) is related to the partial vapour pressure \(p_{v}\) through the ideal gas law:  
\[c_{v}=\ \varphi\ p_{v}M_{H2O}/RT \tag{7}\]where \(R\) is the gas constant and \(M_{H2O}\) the molecular mass of water.