# 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
### On Design Model of Turboshaft Engine  
The engine calculation model for the Design Point (DP) is developed in the MATLAB 2024b environment based on functional blocks representing individual engine modules. The block diagram of the engine is presented in Figure 2 and its structure is consistent with the engine structure presented in Figure 1. In each engine block, the mainline averaged parameters of total temperature (Tt), total pressure (Pt), and mass flow rate (m) of the gas stream are calculated. Additionally, power is determined for the compressor and turbines, while fuel consumption is calculated for the burner. Calculations within the blocks are performed in accordance with the computational standards for these elements, as presented in studies by [25; 26]. For this reason, additionally required parameters are the efficiencies (eff) and pressure loss coefficients (PL) of the individual components at the Design Point. The HP shaft is balanced to satisfy the compressor and high-pressure turbine power balance equation, while the burner is balanced to meet the energy balance equation. The determination of the operating parameters of the free power turbine (FPT) is done iteratively by adjusting the expansion ratio across the turbine (free power turbine pressure ratio FPTPR) until the desired gas velocity at the diffuser outlet is achieved, while simultaneously ensuring that the static pressure at the diffuser outlet matches the ambient pressure. An additional parameter determined is the minimum cross-section area of the high-pressure turbine (A_HPT_min), which is calculated based on the critical flow condition in the HPT first stage guide vane. It is assumed that the total parameters of the gas flow in this section are equal to those at the turbine inlet (parameters in section 4). Supporting calculation blocks included a block simulating ambient conditions according to the ISA standard and a performance calculation block, where the following parameters are computed:  
Output shaft power:  
\[\mathrm{P}=\mathrm{G_{eff}}\;\mathrm{P_{FTP}}=\mathrm{G_{eff}m_{45}}\;\mathrm{ cp}(\mathrm{Tt_{45}}-\mathrm{Tt_{5}}), \tag{1}\]  
Output torque:  
\[\mathrm{T}=\frac{\mathrm{P}}{\omega}, \tag{2}\]  
Specific fuel consumption SFC  
\[SFC=\frac{m_{f}}{\mathrm{P}}, \tag{3}\]  
Figure 2: Turboshaft engine on-design model. where: \(\mathrm{P},\mathrm{P}_{\mathrm{FTP}}[\mathrm{W}]\)--output shaft power and power of the free power turbine, \(m_{45}\left[\frac{\mathrm{kg}}{\mathrm{s}}\right]\)--gas mass flow rate in section 45, \(\mathrm{cp}\left[\frac{\mathrm{J}}{\mathrm{kg}\mathrm{K}}\right]\)--gas specific heat values at constant pressure, \(\mathrm{T}\mathrm{t}\mathrm{t}\mathrm{{}_{45}},\mathrm{T}\mathrm{t}\mathrm{{}_ {5}}[\mathrm{K}]\)--total temperature in section 45 and 5, \(\mathrm{T}[\mathrm{N}\mathrm{m}]\)--torque, \(\omega\left[\frac{1}{\mathrm{s}}\right]\)--angular velocity of output shaft, \(\mathrm{SFC}\left[\frac{\mathrm{kg}}{\mathrm{W}\mathrm{s}}\right]\)--specific fuel consumption, \(\mathrm{m}_{\mathrm{f}}\left[\frac{\mathrm{kg}}{\mathrm{s}}\right]\)--fuel mass flow, and \(\mathrm{G}_{\mathrm{eff}}[-]\)--gear efficiency. All calculations are performed considering irreversible processes and the semi-perfect gas model, with parameters determined according to the relations presented in [29] for pure air and hydrocarbon fumes. The presented model has been verified and applied in calculations conducted in [8; 25; 30]. For the average specific heat of the combustion process in the burner, the model described in [7] was used. The fuel used in the calculations is represented by its chemical formula \(\mathrm{C}_{12.5}\mathrm{H}_{23.5}\), and its calorific value amounted to 43 MJ/kg. The computed average specific heat values for air in the compressor, the combustion process in the burner and fumes in the high-pressure turbine, and the free turbine, based on these models, are summarized in Table 2 for DP calculations.