# 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
In the presented example, 11 Beta lines were used to model the turbine characteristics, while 7 Beta lines were used for the compressor because its \"modeling area\" was smaller. Nevertheless, it should be considered that for more complex efficiency contour shapes, it might not be possible to model all areas of the map with complete accuracy using this approach. In such cases, some local refinement of the Beta lines and reduced speed lines will be required. However, this issue was not addressed in this study. The turboprop engine model for performance calculations can be divided into two somewhat independent blocks: the core engine and the free power turbine. In the first stage, the operating conditions for the core engine (compressor, burner, and high-pressure turbine) are determined. To evaluate a compressor--turbine working line, the calculating process is provided in the proposed approach. It is done by solving a system of equations for the compressor rotational speed (Nc) variable, where the input parameters were the Beta line for the compressor and the turbine inlet temperature (TIT) Tt4. The solution criteria are that the minimum turbine flow area from the iterating process matched the minimum turbine flow area at DP (A\({}_{\text{HPTmin}}=\text{A}_{\text{HPTmin}}\text{DP}  
In the presented algorithm, the variable parameters contain the parameters for a given engine control section, such as pressure, temperature, and mass flow rate. This function is then solved for Y using the fsolve function defined in Matlab 2024b. The core engine operating parameters determined in this way are used to calculate engine parameters for varying ambient conditions and compressor rotor speeds. The physical parameters are derived from the corrected mass flow and rotational speed relationships, respectively (6) and (8), by appropriately rearranging them. Additionally, Tt4 is calculated by:  
\[Tt4=Tt0\times Tt4/Tt0(N_{C~{}cor}), \tag{10}\]  
After performing all these operations, the parameters are prepared to calculate the engine according to the calculation scheme for DP. These calculations make it possible to determine engine parameters at control cross-sections up to the 45 th section, as well as fuel consumption and mass flow rate as a function of compressor rotor speed. The free power turbine can operate over a wide range of rotational speeds, independently of the rotational speed of the HP spool. However, its power is determined by the gas parameters at the inlet and outlet of it, as well as its performance map. In the calculation process for a given rotational speed of a free power turbine (N\({}_{FPT}\)), the corrected speed is  
Figure 5: Algorithm for determining the parameters of the turbine–compressor working line. determined according to Equation (9). Subsequently, the efficiency is obtained from the free power turbine characteristic curve for the free power turbine pressure ratio (FPTPR) that satisfies the condition:  
\[FPTPR=Pt45/(P0/Pt5P9\_DP), \tag{11}\]  
where Pt5P9\({}_{\text{DP}}=\) Pt5\({}_{\text{DP}}\)/P9 \(\_\)DP is the ratio of the total pressure in the FPT outlet to the static pressure in the engine exit diffuser at DP condition. All the parameters in the engine cross sections for the new of-design conditions are evaluated in this way. Some of them are used for performance calculation according to Formulas (1)-(3).