# Hygro-Thermo-Mechanical Analysis of Brick Masonry Walls Subjected to Environmental Actions
## 3 Hygro-Thermal Simulations and Discussion
### Description of the Numerical Model
The initial and boundary conditions used for the hygro-thermal simulations of the masonry wall are summarized in Table 2. Neumann boundary conditions were imposed on the external and internal faces of the wall. Therefore, heat and moisture fluxes were defined with commonly accepted values for the convective transfer coefficients. In particular, heat transfer coefficients \(h_{T,ext}=25\) W/(m\({}^{2}\cdot\)K) and \(h_{T,int}=8\) W/(m\({}^{2}\cdot\)K) were imposed at the external and internal faces, respectively [61]. It is noted that the heat transfer coefficient selected for the external surface already takes into account the radiative contribution. Furthermore, convective mass transfer coefficients \(h_{m,ext}=2.0\)\(\times\)\(10^{-7}\) s/m and \(h_{m,int}=3.0\)\(\times\)\(10^{-8}\) s/m were considered for the external and the internal faces of the wall, respectively [61]. Two-dimensional quadrilateral elements with quadratic shape functions were used for the discretization of the models. After preliminary sensitivity analyses to determine the optimal mesh size, the average element size was set to 5 mm. The primary variables, namely, temperature and relative humidity, were interpolated based on standard Lagrangian shape \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & **Property** & **Symbol** & **Value** & **Units** & **Source** \\ \hline \multirow{8}{*}{Brick} & Bulk density & \(\rho_{bulk}\) & 1900 & kg/m\({}^{3}\) & \\  & Open porosity & \(\phi_{0}\) & 0.280 & – & \\  & Capillary moisture content & \(w_{cap}\) & 240 & kg/m\({}^{3}\) & \\  & Fitting parameter for sorption isotherm & \(\psi\) & 1.0070 & – & [PERSON] et al. [18] \\  & Water absorption coefficient (\(\parallel\) bed face) & \(A_{w,\parallel}\) & 0.104 & kg/(m\({}^{2}\cdot\)g\({}^{0.5}\)) & \\  & Water absorption coefficient (\(\perp\) bed face) & \(A_{w,\perp}\) & 0.075 & kg/(m\({}^{2}\cdot\)g\({}^{0.5}\)) & \\  & Water vapor resistance factor & \(\mu\) & 34.14 & – & \\  & Specific heat capacity & \(C_{p}\) & 825 & J/(kg\(\cdot\)K) & \\  & Thermal conductivity & \(\lambda_{0}\) & 0.59 & W/(m\(\cdot\)K) & [PERSON] et al. [62] \\  & Thermal conductivity supplement & \(b\) & 12.74 & – & \\ \hline \multirow{8}{*}{Lime mortar (LM)} & Bulk density & \(\rho_{bulk}\) & 2060 & kg/m\({}^{3}\) & \\  & Open porosity & \(\phi_{0}\) & 0.230 & – & \\  & Capillary moisture content & \(w_{cap}\) & 190 & kg/m\({}^{3}\) & \\  & Fitting parameters for adsorption isotherm & \(a_{ads}\) & \(3.80\times 10^{-6}\) & 1/Pa & [PERSON] et al. [18] \\  & & \(m_{ads}\) & 0.352 & – & \\  & Fitting parameters for desorption isotherm & \(a_{ads}\) & \(1.19\times 10^{-5}\) & 1/Pa & \\  & & \(m_{ads}\) & 0.235 & – & \\  & Water absorption coefficient (isotropic) & \(A_{w,ISO}\) & 0.080 & kg/(m\({}^{2}\cdot\)g\({}^{0.5}\)) & \\  & Water vapor resistance factor & \(\mu\) & 15.00 & – & [PERSON] [63] \\  & Specific heat capacity & \(C_{p}\) & 840 & J/(kg\(\cdot\)K) & \\  & Thermal conductivity & \(\lambda_{0}\) & 0.85 & W/(m\(\cdot\)K) & [PERSON] [64] \\  & Thermal conductivity supplement & \(b\) & 9.53 & – & \\ \hline \multirow{8}{*}{Cement mortar (CM)} & Bulk density & \(\rho_{bulk}\) & 2000 & kg/m\({}^{3}\) & \\  & Open porosity & \(\phi_{0}\) & 0.210 & – & \\  & Capillary moisture content & \(w_{cap}\) & 180 & kg/m\({}^{3}\) & \\  & Fitting parameters for adsorption isotherm & \(a_{ads}\) & \(1.43\times 10^{-6}\) & 1/Pa & [PERSON] et al. [18] \\  & & \(m_{ads}\) & 0.285 & – & \\  & Fitting parameters for desorption isotherm & \(a_{ads}\) & \(2.51\times 10^{-5}\) & 1/Pa & \\  & & \(m_{ads}\) & 0.213 & – & \\  & Water absorption coefficient (isotropic) & \(A_{w,ISO}\) & 0.060 & kg/(m\({}^{2}\cdot\)g\({}^{0.5}\)) & \\  & Water vapor resistance factor & \(\mu\) & 25.00 & – & [PERSON] [63] \\  & Specific heat capacity & \(C_{p}\) & 932 & J/(kg\(\cdot\)K) & \\  & Thermal conductivity & \(\lambda_{0}\) & 1.72 & W/(m\(\cdot\)K) & [PERSON] [64] \\  & Thermal conductivity supplement & \(b\) & 9.29 & – & \\ \hline \hline \end{tabular}
\end{table}