## 2 First Principle Model of System
### First Principle Modeling  
For studying the influence of air flow rate and aviation kerosene flow rate on gas temperature in the combustor, when establishing the mathematical model of gas temperature in combustor, firstly, the gas temperature in combustor of the test system was taken as output variable; then, the aviation kerosene flow-rate was taken as input variable, and finally, a mathematical model from the aviation kerosene flow rate to gas temperature was established. To simplify the problem, it is assumed that the temperature of high-speed hot air flow is equal everywhere in the combustor. Therefore, the whole system is regarded as a lumped parameter link, and the heat balance equation in the combustor of the HHHTS can be obtained by the law of conservation of energy. \[V\rho_{p}c_{p}\frac{\mathrm{d}T}{\mathrm{d}t}=Q_{fuel}+Q_{air-in}-Q_{air-ex}-Q_ {wall}-Q_{water} \tag{1}\]  
where \(V\) is the volume of the system combustor; \(\rho_{p}\) is the density of gas in the system combustor; \(c_{p}\) is the specific heat capacity of gas in the system combustor; \(T\) is the gas temperature of the system combustor; \(Q_{fuel}\) is the heat produced by the combustion of aviation kerosene which entering the combustor in a unit time; \(Q_{air-in}\) is the heat of the air entering the combustor in a unit time; \(Q_{air-ex}\) is the heat of the high temperature gas leaving the combustor in a unit time; \(Q_{wall}\) is the heat caused by cooling heat conduction of the combustor wall in a unit time; \(Q_{water}\) is the heat taken away with the circulating cooling water leaving the combustor in a unit time; \(Q_{fuel}\), \(Q_{air-in}\), \(Q_{air-ex}\), \(Q_{wall}\), and \(Q_{water}\) satisfy the following equations. \[Q_{fuel}=H\rho_{fuel}w_{s} \tag{2}\]  
\[Q_{air-in}=\rho_{air-in}w_{air-in}c_{p1}T_{in} \tag{3}\]  
\[Q_{air-ex}=\rho_{air-ex}w_{air-ex}c_{p2}T \tag{4}\]  
\[Q_{wall}=KA_{1}(T_{f}-T_{\infty}) \tag{5}\]  
\[Q_{water}=\rho_{water}w_{water}c_{water}(T_{w}-T_{w0}) \tag{6}\]  
where \(H\) is the calorific value of aviation kerosene; \(\rho_{fuel}\) is fuel density; \(w_{s}\) is fuel flow-rate; \(\rho_{air-in}\) is the density of input air; \(w_{air-in}\) is the flow-rate of input air; \(c_{p1}\) is specific heat capacity of input air; \(T_{in}\) is temperature of input air; \(\rho_{air-ex}\) is the density of output air; \(w_{air-ex}\) is the flow-rate of output air; \(c_{p2}\) is specific heat capacity of output air; \(K\) is the heat transfer coefficient of the combustor wall; \(A_{1}\) Heat transfer area of the combustor wall; \(T_{f}\) is the inner temperature of the combustor wall; \(T_{\infty}\) is cooling environmental temperature;  
Figure 1: The main structure of the system. \(\rho_{water}\) is the density of cooling water; \(w_{water}\) is the flow-rate of cooling water; \(c_{water}\) is specific heat capacity of cooling water; \(T_{w}\) is the outlet temperature of cooling water; \(T_{w0}\) is the inlet temperature of cooling water. It is assumed that the gas mass in the combustor of the system is an invariant parameter; the following equation can be obtained from the mass conservation law. \[\rho_{air-ext}w_{air-ex}=\rho_{fuel}w_{s}+\rho_{air-in}w_{air-in} \tag{7}\]  
Because \(\rho_{fuel}w_{s}\) is much less than \(\rho_{air-in}w_{air-in}\) in an actual working condition, it can be approximated that \(\rho_{air-ex}w_{air-ex}\approx\rho_{air-in}w_{air-in}\). Similarly, it also can be approximated that \(c_{p1}\approx c_{p2}\approx c_{p}\). It is assumed that \(T_{w}=\beta T\) and \(T_{f}=\alpha T\), so the following equation can be obtained from Equations (2)-(7)  
\[ \begin{array}{l}V\rho_{p}c_{p}\frac{\mathrm{d}T}{\mathrm{d}t}+\rho_{air-in }w_{air-in}c_{p}T+KA_{1}\alpha T+\rho_{water}w_{water}c_{water}\beta T\\ =H\rho_{fuel}w_{s}+\rho_{air-in}w_{air-in}c_{p}T_{in}+KA_{1}T_{\infty}+\rho_{ water}w_{water}c_{water}T_{w0}\end{array}  \tag{8}\]  
where \(\alpha\) and \(\beta\) are proportional coefficients.