## 2 Heat Exchanger Model
### Static Characteristic Analysis  
Static characteristics in a process control system are defined as the relationship between controlled parameters and control variables in a steady state, which can be described by the corresponding algebraic equations. The static characteristic equation of the heat exchanger temperature control system can be derived from the two basic equations of the heat transfer process, namely, Equations (1) and (2). \[q=G_{2}c_{2}(T_{2i}-T_{2o})=G_{1}c_{1}(T_{1o}-T_{1i}) \tag{1}\]  
where, \(q\) refers to the heat transfer rate, the unit is w; \(G_{1}\) and \(G_{2}\) refer to the mass flow of hot and cold liquids, kg/s; \(c_{1}\) and \(c_{2}\) are the average specific heat capacities of hot and cold liquids, the unit is J/ (kg.\({}^{\circ}\)C); \(T_{1i}\) and \(T_{1o}\) are the hot liquid inlet and outlet temperatures, \(T_{2i}\) and \(T_{2o}\) are the cold liquid inlet and outlet temperatures, the unit is \({}^{\circ}\)C.  
The heat transfer rate \(q\) from a hot liquid to a cold liquid is given by the heat transfer theorem:  
\[q=KF\Delta T \tag{2}\]  
where, \(K\) refers to the heat transfer coefficient, w/(m\({}^{2.\circ}\)C); \(F\) refers to the heat transfer area, m\({}^{2}\); \(\Delta T\) means average temperature difference, \({}^{\circ}\)C.  
The average temperature difference \(\Delta T\) is represented by the logarithmic average:  
\[\Delta T=\frac{(T_{2i}-T_{1o})-(T_{2o}-T_{1i})}{\ln\frac{T_{2i}-T_{1i}}{T_{2o }-T_{1i}}}=\frac{\Delta t_{1}-\Delta t_{2}}{\ln\frac{\Delta t_{1}}{\Delta t_{ 2}}} \tag{3}\]  
where,  
\[ \begin{cases}\Delta t_{1}=T_{2i}-T_{1o}\\ \Delta t_{2}=T_{2o}-T_{1i}\end{cases}  \tag{4}\]  
when \(\Delta t_{1}/\Delta t_{2}\leq 2\), or between 0.33 and 3, the logarithmic mean can be replaced by the arithmetic mean, and the deviation between the two is within 5%. The arithmetic mean is:  
\[\Delta T=\frac{\Delta t_{1}+\Delta t_{2}}{2}=\frac{(T_{2i}-T_{1o})+(T_{2o}-T_ {1i})}{2} \tag{5}\]  
After finishing Equations (1), (2) and (5), the static characteristics of the heat exchanger can be expressed as:  
\[\frac{T_{1o}-T_{1i}}{T_{2i}-T_{1i}}=\frac{1}{\frac{G_{1}C_{1}}{KF}+\frac{1}{2 }\Big{(}1+\frac{G_{1}C_{1}}{G_{2}C_{2}}\Big{)}} \tag{6}\]  
The output flow of the heat exchanger is the heat flow outlet temperature \(T_{1o}\), and the input variables have four groups, respectively, refrigerant flow rate \(G_{2}\), hot fluid inlet temperature \(T_{1i}\), refrigerant inlet temperature \(T_{2i}\), and hot fluid flow rate \(G_{1}\). The effect  
Figure 1: Structure diagram of the heat exchanger. of these four groups of input variables on the thermal fluid outlet temperature can be expressed as four control channels. Since the above equation is not linearized, choosing to linearize it will result in the static magnification of the four channels. By taking the derivative of Equation (6), the influence of the refrigerant flow rate \(G_{2}\) on the outlet temperature of the thermal fluid can be obtained. The static amplification factor of control channel 1, namely channel \(\Delta G_{2}\rightarrow\Delta T_{1o}\), is:  
\[K_{1}=\frac{dT_{1o}}{dG_{2}}=\frac{G_{1}c_{1}(T_{2i}-T_{1i})}{2G_{2}^{2}c_{2} \Big{[}\frac{G_{1}c_{1}}{KF}+\frac{1}{2}\Big{(}1+\frac{G_{1}c_{1}}{G_{2}c_{2}} \Big{)}\Big{]}^{2}} \tag{7}\]  
By taking the derivative of Equation (6), the influence of the hot fluid's inlet temperature \(T_{1i}\) on the outlet temperature of the thermal fluid can be obtained. The static amplification factor of control channel 2, namely channel \(\Delta T_{1i}\rightarrow\Delta T_{1o}\) is:  
\[K_{2}=\frac{dT_{1o}}{dT_{1i}}=1-\frac{1}{\frac{G_{1}c_{1}}{KF}+\frac{1}{2} \Big{(}1+\frac{G_{1}c_{1}}{G_{2}c_{2}}\Big{)}} \tag{8}\]  
By taking the derivative of Equation (6), the influence of the cold fluid inlet temperature \(T_{2i}\) on the outlet temperature of the thermal fluid can be obtained.