# Research on the Dynamic Characteristics Analysis and Power Control Method of Heat Pipe Reactors
## 2 Dynamic Modelling of HPR
### Thermal Resistance Network for HPR  
The thermal resistance network for the heat transfer process in the core of the HPR is shown in Figure 6. The evaporation section of the heat pipe in the core does not have a heat pipe wall, and the monolith is used as the nominal heat pipe wall for the evaporation section. The helium gap between the fuel and the monolith is very thin, so its heat capacity is neglected and only the thermal conductivity is calculated. In Figure 6, \(P_{f}\) is the heat generated by equivalent fuel in W, \(R_{g}\) is the thermal resistance of the air gap, \(R_{m}\) is the thermal resistance of the matrix, \(R_{1}\) is the radial thermal resistance of the suction core in the evaporation section, \(R_{2}\) is the radial thermal resistance of the heat pipe wall in the condensation section, \(R_{3}\) is the radial thermal resistance of the heat tube wall in the condensation section, \(R_{c}\) is the thermal resistance of the convective heat transfer between the coolant and the heat tube wall in the condensation section, and the unit of the thermal resistance is K/W. \(T_{a}\) is the average temperature of SCO\({}_{2}\) in the cooling medium in K.  
The general form of heat transfer equations of fuel and each thermal resistance is shown in Equation (10). \[ \begin{array}{l}(mc)_{f}\frac{dT_{f}}{dt}=P_{f}-P_{g}\\ (mc)_{f}\frac{dT_{m}}{dt}=P_{m,in}-P_{m,out}\\ (mc)_{i}\frac{dT_{i}}{dt}=P_{i,in}-P_{i,out},i=1,2,3\\ (mc)_{c}\frac{dT_{a}}{dt}=P_{c,in}-P_{c,out}\end{array}  \tag{10}\]  
where \(\left(mc\right)_{f}\) is the heat capacity of the equivalent fuel, \(\left(mc\right)_{m}\) is the heat capacity of the equivalent monolith, \(\left(mc\right)_{i}\) is the heat capacity of each part of the heat pipe, and \(\left(mc\right)_{c}\) is the heat capacity of SCO\({}_{2}\) in a single channel, both in units of \(J/K\). The thermal calculations for the fuel, monolith, heat pipe, and cooling fluid are based on the collective parameter method. The fuel temperature is \(T_{f}\), the monolith temperature is \(T_{m}\), the temperature of each part of the heat pipe is \(T_{i}\), and their units are all \(K\). \(P_{f}\) is the heat generated by equivalent fuel, \(P_{g}\) is the heat input to the air gap. \(P_{m,in}\) and \(P_{m,out}\) are the heat input to and output from the monolith, respectively. \(P_{i,in}\) and \(P_{i,out}\  
\[ \begin{array}{l}P_{f}=P_{full}\cdot n_{r}\\ P_{g}=\frac{T_{f}-T_{m,1}}{R_{g}}=P_{m,1}=\frac{T_{m,1}-T_{m}}{R_{m}/2}=\frac{T_{ f}-T_{m}}{R_{g}+R_{m}/2}\\ P_{m,2}=\frac{T_{m}-T_{m,2}}{R_{m}/2}\\ P_{i,1}=\frac{T_{i1}-T_{i}}{R_{f}/2},P_{i,2}=\frac{T_{i}-T_{i2}}{R_{i}/2},i=1,2,3\\ P_{c,in}=\frac{T_{i2}-T_{c}}{R_{f}3/2+R_{c2}},P_{cout}=mc_{p}(T_{out}-T_{in}) \end{array}  \tag{11}\]  
where \(P_{0}\) is the nuclear power at the initial steady state, and \(c_{p}\) is the specific heat capacity of the coolant at a constant pressure. In the thermal resistance network, in order to calculate the input and output heat for the monolith and each part of heat pipe, the front-end temperatures of the monolith and heat pipe thermal resistance in Figure 5 are defined as \(T_{m,1}\) and \(T_{i,1}\) and the back-end temperatures are defined as \(T_{m,2}\) and \(T_{i,2}\), respectively; \(i\) refers to thermal resistances 1 to 3 in Figure 5.  
The heat transfer equation for the HPR thermal resistance network model is obtained through substituting Equation (11) into Equation (10) and is shown in Equation (12). \[