# A Finite Element Method for Nonlinear and Fully Coupled Transient Hydrothermal Transfer
## 5 Illustrative Examples
### Transient 1D Moisture and Heat Transfer with Constant Material Properties  
In this example, transient moisture and heat conduction in a bar are considered for which the material properties \(D_{m}\), \(D_{h}\), \(\gamma\), and \(\lambda\) are all constants. The exact solution to this problem is available and will be used to validate the finite element method. The 1D form of the equations governing the moisture content and temperature, Equation (3), is  
\[[D]\frac{\partial^{2}}{\partial x^{2}} \begin{cases}H\\ T\end{cases} =[E]\frac{\partial}{\partial t} \begin{cases}H\\ T\end{cases}  \tag{18}\]  
where \([D]= \begin{bmatrix}D_{m}&\gamma D_{h}\\ \lambda D_{m}&D_{h}\end{bmatrix} \) and \([E]= \begin{bmatrix}1&0\\ 0&1\end{bmatrix} \). Assume the bar is along the \(x\)-axis and its length is denoted as \(b\). The left side and the right side of the bar are at \(x=0\) and \(x=b\), respectively. Initially, the moisture content and the temperature of the bar are 0. At \(t=0\), the moisture content and temperature on both sides of the strip are suddenly changed to \(H_{0}\) and \(T_{0}\), respectively. Assume the solution of Equation (18) is  
\[ \begin{cases}H\\ T\end{cases} = \begin{cases}H_{0}\\ T_{0}\end{cases} -\frac{4}{\pi}\sum\limits_{m=1,3,5,\dots}^{\infty}\frac{1}{m} sin\Big{(}\frac{m\pi x}{b}\Big{)}exp\Big{(}-\Big{(}\frac{m\pi}{b}\Big{)}^{2} \xi t\Big{)}A\left\{a\right\} \tag{19}\]  
in which \(\left\{a\right\}\) is an unknown column of two elements, \(A\) and \(\xi\), constants to be determined. Boundary conditions are automatically met by Equation (18). The constants \(A\), \(\xi\), and \(\left\{a\right\}\) are determined by satisfying the equilibrium equations and initial conditions of the problem. Putting Equation (19) into Equation (18) yields  
\[([D]-\xi[E])\left\{a\right\}=0 \tag{20}\]  
Thus, \(\xi\) has two values, \(\xi_{1}\) and \(\xi_{2}\), which are the eigenvalues of the matrix \([D]-\xi[E]\). The corresponding eigenvectors for \(\left\{a\right\}\) are denoted as \(\left\{a_{1}\right\}\) and \(\left\{a_{2}\right\}\), respectively. For real problem, \(\xi_{1}\) and \(\xi_{2}\) should be positive. As a result, the solution can be written as  
\[ \begin{cases}H\\ T\end{cases} = \begin{cases}H_{0}\\ T_{0}\end{cases} -\frac{4}{\pi}\sum\limits_{m=1,3,5,\dots}^{\infty}\frac{1}{m} sin\Big{(}\frac{m\pi x}{b}\Big{)} \tag{21}\] \[\times\Big{(}exp(-\big{(}\frac{m\pi}{b}\big{)}^{2}\xi_{1}t)A_{1} \left\{a_{1}\right\}+exp(-\big{(}\frac{m\pi}{b}\big{)}^{2}\xi_{2}t)A_{2} \left\{a_{2}\right\}\Big{)}\]  
The initial conditions will be used to obtain \(A_{1}\) and \(A_{2}\). Therefore, at \(t=0\), one obtains  
\[\frac{4}{\pi}\sum\limits_{m=1,3,5,\dots}^{\infty}\frac{1}{m}sin\Big{(}\frac{ m\pi x}{b}\Big{)}(A_{1}\left\{a_{1}\right\}+A_{2}\left\{a_{2}\right\})= \begin{cases}H_{0}\\ T_{0}\end{cases}  \tag{22}\]  
Since \(\frac{4}{\pi}\sum\limits_{m=1,3,\dots}^{\infty}\frac{1}{m}sin\Big{(}\frac{m \pi x}{b}\Big{)}=1\), Equation (22) yields  
\[ \begin{cases}A_{1}\\ A_{2}\end{cases} =\Big{[}\left\{a_{1}\right\}&\left\{a_{2}\right\}\Big{]}^{-1 } \begin{cases}H_{0}\\ T_{0}\end{cases}  \tag{23}\]  
The following material properties are used for the numerical calculation: \(\lambda=0.122\), \(\gamma=2.053\), \(D_{m}=\frac{D_{0}}{1-\lambda\gamma}\), and \(D_{h}=10D_{m}\)[21]. In total, 20 quadratic 2D elements and 42 nodal points are used to construct the grid pattern. There are 21 nodal points along the \(x\) direction and 2 nodal points normal to the \(x\) direction. The values of \(H\) and \(T\) as a function are found and are given in Tables 1 and 2 for various values of time while \(x\) varies from 0.0 to 0.5\(b\) in increments of 0.1\(b\). Due to symmetry, the left half and the right half of the bar have the same temperature and moisture distributions. It can be seen that the deviations between the finite element and analytical results are very small. In Table 1, the relative difference between the proposedfinite element model and the analytical solution is almost 0; therefore, it is not given.