## 2 Heat Exchanger Model
### Dynamic Characteristic Analysis  
The purpose of studying the dynamic characteristics of heat exchangers is to understand the change of the operating parameters at some key positions in the system, the response characteristics of the physical parameters at a section through which the fluid flows to the control variables of the system, and the impact of the operating parameters and structural parameters of the system on the response characteristics. The dynamic equation of a heat exchanger is a partial differential equation, which usually requires a variety of assumptions before column writing. For complex heat transfer equipment, the writing of dynamic equations is much more complicated, and it will be more difficult to solve the dynamic equation. Therefore, to explain the basic laws of the dynamic characteristics of heat transfer objects, some empirical formulas can be used to describe them. For example, the dynamic characteristics of heat exchangers can be expressed by the following approximate relation [7, 25, 26]. (1). The influence of inlet temperature on the outlet temperature of the thermal fluid, that is, the channel characteristics of \(\Delta T_{1i}\rightarrow\Delta T_{1o}\). If described by a transfer function, it can be expressed as:  
\[G(s)=\frac{K_{2}}{\frac{W_{1}}{G_{1}}+1}=\frac{K_{2}}{Ts+1} \tag{11}\]  
where, \(K_{21}\) is the magnification of the channel, \(W_{1}\) refers to the amount of hot fluid stored in the heat exchanger, \(G_{1}\) refers to the flow rate of thermal fluid. (2). The influence of inlet temperature \(T_{2i}\) and flow rate \(G_{2}\) of refrigerant and flow rate \(G_{1}\) of hot fluid on outlet temperature \(T_{1o}\) of hot fluid, namely the characteristics of \(T_{2i}\to T_{1o}\), \(G_{1}\to T_{1o}\) and \(G_{2}\to T_{1o}\) channels. If expressed in terms of the transfer function, can be approximated as:  
\[G(s)=\frac{K}{(T_{1}s+1)(T_{2}s+1)}e^{-T_{2}s} \tag{12}\]  
where:  
\[ \begin{cases}T_{1}=\frac{W_{1}/G_{1}+W_{2}/G_{2}}{2}\\ T_{2}=\frac{W_{1}/G_{1}+W_{2}/G_{2}}{8}\end{cases}  \tag{13}\]  
where, \(K\) refers to the static magnification of each channel, \(W_{1}\) is the storage volume of the hot fluid, \(W_{2}\) is the storage volume of the cold fluid. As can be seen from Equation (12), the dynamic characteristics of these three channels can be approximated as second-order inertia links with pure hysteresis. This is because when the refrigerant and the hot fluid exchange heat, the heat needs to be transferred first to the intermediate wall, and then to the hot fluid from the intermediate wall to form a second-order inertia link. In addition, the pure lag due to residence time is considered. In this paper, the outlet temperature of the tube heat exchanger in a pharmaceutical factory was studied. The refrigerant is ethylene glycol, and the hydrothermal fluid is toluene. By controlling the flow of refrigerant to stabilize the outlet temperature of the hydrothermal solution. The actual data from the production process is shown in Table 2. By plugging the data into Equations (7) and (12), the transfer function controlling channel 1 is obtained, as shown in Equation (14). The transfer function represents the control effect of refrigerant flow rate on thermal fluid outlet temperature. \[G_{1}(s)=\frac{K_{1}}{(T_{1}s+1)(T_{2}s+1)}e^{-T_{2}s}=\frac{1.9}{105.7s^{2}+25.7 s+1}e^{-5.14s} \tag{14}\]  
It can be seen from Equation (14) that the heat exchanger outlet temperature control system is a system with a large inertia and a large lag. At the same time, the transfer function of the input temperature of hot fluid, the inlet temperature of refrigerant, and the flow rate of hot fluid acting on the exit temperature is given. By substituting the data into Equations (8) and (11), the transfer function controlling channel 2 is obtained, as shown in Equation (15). The transfer function represents the control effect of the inlet temperature of the hot fluid on the outlet temperature of the hot fluid.