# Early Fault Detection and Operator-Based MIMO Fault-Tolerant Temperature Control of Microreactor
## 4 Modeling
### Thermal Models of Heat Spreader  
In this section, heat spreaders are modeled based on the laws of heat transfer. The three heat transfer laws used are [PERSON]'s law, [PERSON]'s cooling law, and [PERSON]'s law. The relational equation for heat conduction in Part A\({}_{n}\)(\(n=1,2,3\)) is obtained using the heat transfer laws and Tables 1 and 2 as follows. For the left side of the model equation, it is the time derivative of the thermal energy of the Part. The left-hand side shows the total amount of heat energy exchanged by external factors, indicating the heat energy transfer per unit time. Note that since the defined Parts are adjacent to each other, there is interference due to heat transfer. Note that \(S^{\prime}=2S_{1}+2S_{2}+S_{3}-S_{5}\). In addition, \(u_{d_{1}}\) and \(u_{d_{3}}\) are the heat absorbed from the Peltier device and are discussed in Section 4.3.  
\[\text{Part A}_{1}:\frac{d(T_{0}-T_{a_{1}})m_{a}c_{a}}{dt}= 2u_{d_{1}}-\alpha S^{\prime}(T_{0}-T_{a_{1}})+\alpha_{\omega}S_{6}(T _{a_{1}}-T_{\omega_{1}})+\frac{\lambda_{a}S_{3}(T_{a_{1}}-T_{a2})}{dx}+\epsilon _{a}\sigma S^{\prime}(T_{a_{1}}^{4}-T_{0}^{4}) \tag{3}\] \[\text{Part A}_{2}:\frac{d(T_{0}-T_{a_{2}})m_{a}c_{a}}{dt}= -\alpha(T_{0}-T_{a_{2}})(2S_{7}+2S_{8})+\alpha_{\omega}S_{9}(T_{a _{2}}-T_{\omega_{2}})-\frac{\lambda_{a}S_{3}(T_{a_{1}}-T_{a_{2}})}{dx}\] \[-\frac{\lambda_{a}S_{3}(T_{a_{3}}-T_{a_{2}})}{dx}+\epsilon_{a} \sigma(T_{a_{2}}^{4}-T_{0}^{4})(2S_{7}+2S_{8})\] (4) \[\text{Part A}_{3}:\frac{d(T_{0}-T_{a_{3}})m_{a}c_{a}}{dt}= 2u_{d_{3}}-\alpha S^{\prime}(T_{0}-T_{a_{3}})+\alpha_{\omega}S_{6}( T_{a_{3}}-T_{\omega_{3}})+\frac{\lambda_{a}S_{3}(T_{a_{3}}-T_{a_{2}})}{dx}+ \epsilon_{a}\sigma S^{\prime}(T_{a_{3}}^{4}-T_{0}^{4}) \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Symbol** & **Description** & **Value/Unit** \\ \hline \(T_{0}\) & Initial temperature (outside temperature) & [K] \\ \(T_{\text{c}}\) & Heat-absorbing surface temperature of Peltier device & [K] \\ \(T_{\text{h}}\) & Heat dissipation surface temperature of Peltier device & [K] \\ \(T_{a_{n}}\) & Temperature of Part A\({}_{n}\) & [K] \\ \(T_{\text{m}_{n}}\) & Temperature of Part W\({}_{\text{k}}\) & [K] \\ \(i\) & Input current of Peltier device & [A] \\ \(S\) & Seebeck coefficient of Peltier device & 0.053 V/K \\ \(K\) & Thermal conductivity of Peltier device & 0.63 W/K \\ \(R\) & Resistance of Peltier device & 2.01 \\ \(\epsilon_{a}\) & Emissivity of Aluminum & 0.2 \\ \(\epsilon_{w}\) & Emissivity of water & 0.93 \\ \(\sigma\) & Stefan–Boltzmann constant & \(5.67\times 10^{-8}\,\text{W}/(\text{m}^{2}\cdot\text{K}^{4})\) \\ \(\alpha\) & Heat transfer coefficient of air & \(180\,\text{W}/(\text{m}^{2}\cdot\text{K})\) \\ \(\alpha_{w}\) & Heat transfer coefficient of water & \(\text{W}/(\text{m}^{2}\cdot\text{K})\) \\ \(\lambda_{a}\) & Thermal conductivity of aluminum & \(135\,\text{W}/(\text{m}\cdot\text{K})\) \\ \(\lambda_{w}\) & Thermal conductivity of water & \(0.602\,\text{W}/(\text{m}\cdot\text{K})\) \\ \(c_{a}\) & Specific heat of aluminum & \(900\,\text{J}/(\text{kg}\cdot\text{K})\) \\ \(c_{w}\) & Specific heat of water & \(4180\,\text{J}/(\text{kg}\cdot\text{K})\) \\ \(m_{a}\) & Mass of aluminum & [kg] \\ \(m_{w}\) & Mass of water & [kg] \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of modeling.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Symbol** & **Value** & **Unit** & **Symbol** & **Value** & **Unit** \\ \hline \(S_{1}\) & \(2.6\times 10^{-3}\) & \(\text{m}^{2}\) & \(S_{2}\) & \(7.0\times 10^{-4}\) & \(\text{m}^{2}\) \\ \(S_{3}\) & \(9.8\times 10^{-3}\) & \(\text{m}^{2}\) & \(S_{4}\) & \(9.0\times 10^{-4}\) & \(\text{m}^{2}\) \\ \(S_{5}\) & \(9.0\pi\times 10^{-6}\) & \(\text{m}^{2}\) & \(S_{6}\) & \(3.0\pi\times 10^{-4}\) & \(\text{m}^{2}\) \\ \(S_{7}\) & \(1.4\times 10^{-3