# A Finite Element Method for Nonlinear and Fully Coupled Transient Hydrothermal Transfer
## 5 Illustrative Examples
### Transient 1D Moisture and Heat Transfer in a Solid Disk with Constant Material Properties  
The exact solution for this problem is also available. Therefore, the developed FEM model will be used to study the problem, and the results will be compared with the exact solutions. In order to obtain an exact solution, it is assumed that the problem is axisymmetric and the moisture content and heat only vary with the radial coordinate \(r\) and time. The disk is initially at 0 moisture content and 0 temperature. The boundary condition is such that the moisture content and temperature on the outer surface of the disk (at \(r=b\)) are suddenly changed to \(H_{0}\) and \(T_{0}\), respectively. Since \(D_{m}\), \(D_{h}\), \(\gamma\), and \(\lambda\) are all constants, the 1D form of the equations governing the moisture content and temperature along the radial direction is as follows:  
\[[D]\bigg{(}\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{ \partial r}\bigg{)}\Bigg{\{} \begin{matrix}H\\ T\end{matrix} \Bigg{\}}=[E]\frac{\partial}{\partial t}\Bigg{\{} \begin{matrix}H \\ T\end{matrix} \Bigg{\}} \tag{26}\]  
\[\text{where}\ [D]= \begin{bmatrix}D_{m}&\gamma D_{h}\\ \lambda D_{m}&D_{h}\end{bmatrix} \ \text{and}\ [E]= \begin{bmatrix}1&0\\ 0&1\end{bmatrix} .\]  
The conduction Equation (26) can be solved by a standard separation-of-variables technique. The result is  
\[\Bigg{\{} \begin{matrix}V\\ T\end{matrix} \Bigg{\}}=\Bigg{\{} \begin{matrix}H_{0}\\ T_{0}\end{matrix} \Bigg{\}}-\sum_{n=1}^{\infty}\frac{2J_{0}(\gamma_{n}\xi_{p} ^{t})}{\gamma_{n}J_{1}(\gamma_{n})}\bigg{(}\exp(-\Big{(}\frac{\gamma_{n}}{b} \Big{)}^{2}\xi_{1}t)A_{1}\{a_{1}\}+\exp(-\Big{(}\frac{\gamma_{n}}{b}\Big{)}^{2 }\xi_{2}t)A_{2}\{a_{2}\}\bigg{)} \tag{27}\]  
in which \(J_{0}\) is the zero-order Bessel function of the first kind, \(J_{1}\) is the first-order Bessel function of the second kind, \(\gamma_{n}\) are the roots of the equation \(J_{0}(\gamma_{n})=0\), \(\xi_{1}\) and \(\xi_{2}\) are eigenvalues, and \(\{a_{1}\}\) and \(\{a_{2}\}\) are corresponding eigenvectors of Equation (20). The constants \(A_{1}\) and \(A_{2}\) are determined by the initial conditions of the problem, and they are the same as Equation (23). Once again, the material properties are taken as \(\lambda=0.122\), \(\gamma=2.053\), \(D_{m}=\frac{D_{0}}{1-\lambda\gamma}\), and \(D_{h}=10D_{m}\)[21]. Note that the 2D numerical model has been developed. Of course, the 2D numerical model can be used to solve the 1D problem (which has a known analytical solution) and to verify the numerical model. As such, the problem is also studied by the two-dimensional finite element modeling. Due to axisymmetric, only one-fourth of the disk needs to model. A finite element mesh configuration with 20 divisions along the radial direction and 10 divisions along the circumferential direction is shown in Figure 9.  
Figure 8: Distribution of temperature at different times. Analytical solutions and the finite element solutions are given in Tables 3 and 4. Both solutions agree with each other very well. If finer mesh configurations are used, the finite element results and the analytical solutions are identical. In both cases, the relative difference between the proposed finite element model and the analytical solution is less than 1%.