# Research on Real-Time Model of Turboshaft Engine with Surge Process
## 2 Turboshaft Engine Component Level Model  
The structure and section definition of the turboshaft engine model established in this paper are shown in Figure 1 and the numbers such as 1,2,3, etc. represent the inlet or outlet sections of different parts of the engine. The main components include airscrew, inlet, compressor, combustion chamber, gas turbine, power turbine, and nozzle. Through the aero-dynamic and thermo-dynamic calculation of each component, the parameters such as temperature, pressure, and flow of each section can be obtained. In addition, each component also needs to meet the balance equation of the mass flow or power. Thus, establishing the mathematical model of the engine with the component method includes three aspects: (1) Completing the aero-thermal calculation of each component along the engine intake process; (2) Establishing and solving the balance equation reflecting the common working relationship between the components in the steady-state operation of the engine; (3) Establishing and solving the balance equation reflecting the common working relationship between the components in the transient operation of the engine. When the engine is working in a steady-state, the power of the rotating parts should be balanced, and the mass flow through each section should meet the continuous condition so that the following five balance equations can be obtained:  
(1) Power balance between compressor and gas turbine:  
\[f_{1}=\frac{P_{GT}}{P_{C}/\eta_{mc}+P_{ext}}-1 \tag{1}\]  
where \(P_{GT}\) is the gas turbine power, \(P_{C}\) is the compressor power, \(P_{ext}\) is the extracted compressor shaft power and \(\eta_{mc}\) is the mechanical efficiency of the compressor shaft. (2) Power balance between airscrew and power turbine:  
\[f_{2}=\frac{P_{PT}}{P_{L}/\eta_{mL}}-1 \tag{2}\]  
where \(P_{PT}\) is the power turbine power, \(P_{L}\) is the power required by airscrew and \(\eta_{mL}\) is the mechanical efficiency of the power turbine shaft. Figure 1: The layout of turboshaft engine. 3. The flow continuity equation of the gas turbine: \[f_{3}=\frac{W_{g41C}-W_{g41 CX}}{W_{g41 CX}}\] (3) where \(W_{g41C}\) is the real gas flow of combustor exit, \(W_{g41X}\) is the real gas flow calculated according to \(W_{g41C}\) at the gas turbine inlet. (4) The flow continuity equation of the power turbine: \[f_{4}=\frac{W_{g45C}-W_{g45 CX}}{W_{g45 CX}}\] (4) where \(W_{g45C}\) is the real gas flow of gas turbine exit, \(W_{g45 CX}\) is the real gas flow calculated according to \(W_{g45C}\) at the power turbine inlet. (5) The flow continuity equation of the nozzle: \[f_{5}=\frac{W_{g8}-W_{g8C}}{W_{g8}}\] (5) where \(W_{g8}\) is the gas flow at nozzle inlet, \(W_{g8C}\) is the nozzle gas flow calculated by the dense flow function. In steady-state calculation, these five equations are expected to be equal to zero, and the Newton-Raphson (N-R) method shown in Equation (6) is used to solve these five equations. In addition, the relative speed of the compressor \(N_{g}\), the relative speed of power turbine \(N_{P}\), the coefficient of compressor pressure ratio \(Z_{C}\), the coefficient of gas turbine pressure ratio \(Z_{GT}\), the coefficient of power turbine pressure ratio \(Z_{PT}\) are chosen as guesses for the five operating equations which will be continuously updated in iterations until \(f_{1}\) to \(f_{5}\) are less than \(10^{-5}\). \[\left\{ \begin{array}{l}X=X-\lambda J^{-1}E\\ J=\
abla_{X}E\end{array} \right. \tag{6}\]  
where \(X=\left[ \begin{matrix}-\bar{N}_{g}\bar{N}_{P}Z_{C}Z_{GT}Z_{PT}\end{matrix}  \right]^{T}\) is the vector composed of five guesses, \(E=\left[f_{1}f_{2}f_{3}f_{4}f_{5}\right]^{T}\) is the vector composed of the five errors, \(\lambda\) is the step size, \(J\) is the Jacobian matrix of \(E\) to \(X\).