# A Practical Approach to Modeling and Performance Analysis of a Turboshaft Engine Using Matlab
## 2 Modeling the Performance of a Turboshaft Engine with a Free Power Turbine
### Off Design Model of Turboshaft Engine  
It is assumed that the model will be applicable for calculations relevant to the steady-state calculation of the turboshaft engine at higher rotational speeds of the high-pressure shaft HPS. According to [15], under these conditions, critical flow can be assumed in the minimum cross-section of the turbine, and the expansion ratio across the HPT, as well as its efficiency, remain practically constant. Additionally, it is assumed that the process indices in engine components, apart from the compressor and the power turbine, remain constant. However, if reliable characteristics of these parameter variations are available, they can be implemented into the model. In practice, this means that the compressor map and the power turbine map are additional required. To prepare a numerical representation of the turbine and compressor maps, an approach using an additional variable, the Beta line, was employed, as presented in studies by [13; 31]. This approach is illustrated in Figure 3, where the Beta line is depicted on the compressor and turbine maps as an additional parameter. This allows for the representation of the compressor parameters: corrected compressor mass flow \(m_{C~{}cor}\), compressor pressure ratio CPR, and compressor efficiency, in the form of two-dimensional tables, as follows:  
\[ \begin{array}{l}\mathrm{m}_{\mathrm{C~{}cor}}=\mathrm{f}(\mathrm{N}_{\mathrm{ cor}},\mathrm{Beta}_{\mathrm{c}})\\ \mathrm{CPR}=\mathrm{f}(\mathrm{N}_{\mathrm{cor}},\mathrm{Beta}_{\mathrm{c}}) \\ \mathrm{C~{}EFF}=\mathrm{f}(\mathrm{N}_{\mathrm{cor}},\mathrm{Beta}_{\mathrm{c}}),\end{array}  \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline
**Calculated Engine Section** & **cp [J/kg/K]** & **k**[29] \\ \hline Compressor & 1016 & 1.4 \\ Burner & 1200 & - \\ HPT & 1172 & 1.321 \\ FPT & 1142 & 1.335 \\ Engine Diffuser & 1127 & 1.342 \\ \hline \end{tabular}
\end{table} 
Table 2: Gas average specific heat value cp, isentropic index k.  
and for the free power turbine the product of the corrected rotational speed and the corrected mass flow rate \(\mathrm{N_{FFR~{}cor}}\) m\({}_{\mathrm{FFR~{}cor}}\), free power turbine pressure ratio FPTPR and free power turbine efficiency PT EFF, as follows:  
\[ \begin{array}{c}\mathrm{N_{FFR~{}cor}}\ \mathrm{m_{FFR~{}cor}}=\mathrm{f(N_{FFT~{}cor},Beta_{FFT})}\\ \mathrm{FPTPR}=\mathrm{f(N_{FFT~{}cor},Beta_{FFT})}\\ \mathrm{FPT~{}EFF}=\mathrm{f(N_{FFT~{}cor},Beta_{FFT})},\end{array}  \tag{5}\]  
where: corrected compressor mass flow  
\[\mathrm{m_{C~{}cor}}=\mathrm{m_{2}}\frac{\mathrm{Pt_{0}}}{\mathrm{Pt_{2}}} \sqrt{\mathrm{Tr_{2}}/\mathrm{Tr_{0}}}, \tag{6}\]  
corrected free power turbine mass flow  
\[\mathrm{m_{FFR~{}cor}}=\mathrm{m_{45}}\frac{\mathrm{Pt_{0}}}{\mathrm{Pt_{45}}} \sqrt{\mathrm{Tr_{0}}}, \tag{7}\]  
corrected compressor rotor speed  
\[\mathrm{N_{C~{}cor}}=\mathrm{N_{C}}\sqrt{\mathrm{Tr_{0}}/\mathrm{Tr_{2}}}, \tag{8}\]  
corrected free power turbine rotor speed  
\[\mathrm{N_{FFR~{}cor}}=\mathrm{N_{FFT}}\sqrt{\mathrm{Tr_{0}}/\mathrm{Tr_{0}}}, \tag{9}\]  
The proposed methods for Beta lines construction are slightly modified in this study to adapt them to the specific characteristics observed, primarily for turbines. In such cases, it is difficult to draw straight lines connecting the ends of the characteristics, as these often have a significantly shorter range at lower rotational speeds. Therefore, the Beta lines are constructed as polylines, which evenly divide the lines of constant reduced rotational speed. This is illustrated, for instance, in the turbine characteristic shown in Figure 4. A similar methodology is applied to compressor characteristics modeling. Figure 3: Compressor map and turbine map with Beta lines. This approach allows for an accurate representation of the limitations and extreme points of the modeled component characteristics. To enhance accuracy, the number of Beta lines can be increased.