# 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 Table 1, the maximum relative difference is about 0.05%, which is quite small. [PERSON] et al. [16] argued that the effect of moisture content on the heat conduction is negligible (i.e., \(\lambda\) in Equation (3b) is taken as 0). They ignored the influence of \(D_{m}\) on the temperature and used the following uncoupling system of moisture and temperature relations:  
\[\
abla\cdot(D_{m}\
abla H)+\
abla\cdot(\gamma D_{h}\
abla T)=\frac{\partial H }{\partial t} \tag{24a}\] \[\
abla\cdot(D_{h}\
abla T)=\frac{\partial T}{\partial t} \tag{24b}\]  
In Figures 1-4, results for fully coupling (Equation (3)) and uncoupling (Equation (24)) are shown for sudden moisture and sudden temperature changes on the surfaces of the bar. The results of uncoupling and fully coupling are small when time is small or at the steady state. In fact, at the steady state, Equations (3) and (24) show that the moisture and temperature are governed, respectively, by two independent equations, \(\
abla\cdot(D_{m}\
abla H)=0\) and \(\
abla\cdot(D_{h}\
abla T)=0\). They are, therefore, totally uncoupling at the steady state. Generally, at the transient state, the differences between the uncoupling and fully coupling are very significant. Therefore, it is important that when studying the transient solutions of the moisture and temperature, fully coupling between the moisture and heat conductions should be considered. Otherwise, the results will be considerably different from the exact values or even completely wrong. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(t\)/\(t_{0}\),\(t_{0}\)=\(\frac{b^{2}}{D_{0}}\) & \(x\)/\(b\) & \begin{tabular}{c} _T/(\(\lambda H_{0}\))_ \\ **Finite Element** \\ \end{tabular}  & \begin{tabular}{c} _T/(\(\lambda H_{0}\))_ \\ **(Analytic)** \\ \end{tabular}  & \ \begin{tabular}{c} _H/H\({}_{0}\)_ \\ **Finite Element** \\ \end{tabular}  & \begin{tabular}{c} _H/H\({}_{0}\)** \\ **(Analytic)** \\ \end{tabular}  \\ \hline \multirow{6}{*}{0.125} & 0.0 & 0.000 & 0.000 & 1.000 & 1.000 \\ \cline{2-6}  & 0.1 & 0.01241 & 0.01241 & 0.8850 & 0.8850 \\ \cline{2-6}  & 0.2 & 0.02360 & 0.02360 & 0.7812 & 0.7812 \\ \cline{2-6}  & 0.3 & 0.03249 & 0.03248 & 0.6989 & 0.6989 \\ \cline{2-6}  & 0.4 & 0.03819 & 0.03818 & 0.6460 & 0.6460 \\ \cline{2-6}  & 0.5 & 0.04015 & 0.04015 & 0.6278 & 0.6278 \\ \hline \multirow{6}{*}{0.25} & 0.0 & 0.0000 & 0.0000 & 1.0000 & 1.0000 \\ \cline{2-6}  & 0.1 & 0.003726 & 0.003732 & 0.9655 & 0.9654 \\ \cline{2-6}  & 0.2 & 0.007087 & 0.007099 & 0.9343 & 0.9342 \\ \cline{2-6}  & 0.3 & 0.009754  
Figure 1: Variation of moisture content through bar for a sudden moisture content \(H_{0}\) on both sides of the bar. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(t\)/\(t_{0}\),\(t_{0}\)=\(\frac{b^{2}}{D_{0}}\)} & \multirow{2}{*}{\(x\)/\(b\)} & \(T\) & \multirow{2}{*}{\(T\) (Analytic)} & \multirow{2}{*}{\( \begin{array}{c}Relative \\ \text{\bf Difference (\%)}\end{array} \)} & \multirow{2}{*}{\( \begin{array}{c}H\)/(\(\gamma T_{0}\)) \\ \text{\bf Finite}\end{array} \)} & \multirow{2}{*}{\( \begin{array}{c}H\)/(\(\gamma T_{0}\)) \\ \text{\bf Finite}\end{array} \)} & \multirow{2}{*}{\( \begin{array}{c}H\)/(\(\gamma T_{0}\)) \\ \text{\bf(Analytic)}\end{array} \)} & \multirow{2}{*}{\( \begin{array}{c}Relative \\ Difference (\%)\end{array} \)} \\ \hline \multirow{7}{*}{0.01} & 0.0 & 1.0000 & 1.0000 & 0 & 0.0000 & 0.0000 & 0 \\ \cline{2-7}  & 0.1 & 0.8862 & 0.8862 & 0 & 0.4451 & 0.4440 & 0.25 \\ \cline{2-7}  & 0.2 & 0.7878 & 0.7879 & 0 & 0.6886 & 0.6862 & 0.35 \\ \cline{2-7}  & 0.3 & 0.7136 & 0.7137 & 0.01 & 0.7394 & 0.7358 & 0.49 \\ \cline{2-7}  & 0.4 & 0.6675 & 0.6675 & 0 & 0.7172 & 0.7155 & 0.24 \\ \cline{2-7}  & 0.5 & 0.6517 & 0.6517 & 0  
Figure 4: Variation of temperature through bar for a sudden temperature change \(T_{0}\) on both sides of the bar. Figure 3: Variation of moisture content through bar for a sudden temperature change \(T_{0}\) on both sides of the bar. Figure 2: Variation of temperature through bar for a sudden moisture content \(H_{0}\) on both sides of the bar.