# A Finite Element Method for Nonlinear and Fully Coupled Transient Hydrothermal Transfer
## 2 Governing Equations of Moisture and Temperature Coupling  
The phenomenon of moisture and heat transport in composite materials and porous media has been studied for many decades in the science and engineering research community (see review papers [1,20]). A comprehensive approach in modeling the moisture mass and heat transport has been developed, and the proposed governing equation was manipulated and separated into two components defining the mechanisms for isothermal moisture diffusion and heat-induced moisture diffusion. This approach yields a system of coupled partial differential equations consisting of a moisture transfer equation and a heat balance equation. Suppose in a coordinate system \(\mathbf{x}\) (whose components are \(x_{i}\), where \(i\) = 1, 2, 3), there is a body occupying a space \(\Omega\), which is surrounded by a surface \(S\). The moisture and temperature inside the body may change from point to point and vary with time variable \(t\). Assume \(H(\mathbf{x}\), \(t)\) is the moisture content (which is the mass of water per mass of wet solid, in kg/m\({}^{3}\)) and \(T(\mathbf{x},t)\) is the temperature (K). Both of them are changed continuously with space coordinates \(x_{i}\) and time \(t\). The basic law of the diffusion of moisture and heat conduction may be stated as [21],  
\[f_{i}=-D_{m}H_{i}-\gamma D_{h}T_{j} \tag{1a}\] \[\frac{1}{\rho c}q_{i}=-\lambda D_{m}H_{i}-D_{h}T_{j} \tag{1b}\]  
where the comma \",\" means the partial derivative (e.g., \(H_{j}\) means \(\partial H/\partial x_{i}\)), \(q_{i}\) is the components of the heat flux vector \(\mathbf{q}_{j}\),\(f_{i}\) is the rate of moisture transfer per unit area along the \(x_{i}\) direction, \(\rho c\) is the volumetric heat capacity (J/(m\({}^{3}\)K)), and the notations \(D_{m}\) and \(D_{h}\) are the moisture diffusivity (m\({}^{2}\)/s) and thermal diffusivity ((W/(m\(\cdot\)K))/(kg/m\({}^{3}\cdot\)J/(kg\(\cdot\)K)) = m\({}^{2}\)/s), respectively. The paramaters \(\gamma\) (in kg/(m\({}^{3}\cdot\)K)) and \(\lambda\) (in m\({}^{3}\cdot\)K/kg) are coupling coefficients between the temperature and moisture transfer. The conservation of mass of moisture (neglecting the generation of internal moisture inside the material) requires that [21]  
\[-f_{i,i}=\frac{\partial H}{\partial t}\] (2a) while the conservation of energy (neglecting the generation of internal heat) yields [21] \[-q_{i,i}=\rho c\frac{\partial T}{\partial t} \tag{2b}\]  
Substituting Equation (1a,b) into Equation (2a,b), a system of simultaneous equations is obtained  
\[\
abla\cdot(D_{m}\
abla H)+\
abla\cdot(\gamma D_{h}\
abla T)=\frac{\partial H }{\partial t} \tag{3a}\]\[\
abla\cdot(\lambda D_{m}\
abla H)+\
abla\cdot(D_{h}\
abla T)=\frac{\partial T}{ \partial t} \tag{3b}\]  
where \(\
abla\) is the Nabla operator. On the left-hand side of Equation (3a), the first term represents the moisture movement due to moisture gradient, and the second term describes the moisture movement due to temperature gradient. While on the left-hand side of Equation (3b), the first term reflects the heat conduction due to moisture gradient, the second term is the heat conduction due to temperature gradient. The coefficient of moisture diffusion \(D_{m}\) and the thermal diffusivity \(D_{h}\) can strongly depend on moisture concentration and temperature [21]. In this situation, Equation (3) is highly nonlinear, and the exact solution is impossible even for the simplest loading and boundary conditions. Considering the highly nonlinearly coupled nature of the problem, a finite element code will be developed. For a more systemic analysis, below material properties are introduced: \(c_{0}=\frac{c\lambda D_{h}}{\gamma D_{h}}\), \(D_{11}=c_{0}D_{m}\), \(D_{12}=D_{21}=\rho c\lambda D_{m}\), \(D_{22}=\rho cD_{h}\), and a generalized moisture transfer vector \(\mathbf{J}=c_{0}\mathbf{f}\), where \(D_{22}\) is the thermal conductivity (W/(mK)) and \(D_{11}\), by analogy, can be understood as the moisture conductivity.