# Analytical Calculation Model of the TV3-117 Turboshaft Working Regimes Based on Experimental Data
## 4 Analytical Calculation Model  
As mentioned above, the analytical calculation model is applied to determine the regimes of the TV3-117 MT turboshaft engine. It should be noted that only some main parameters and performance from the takeoff regime are known, as presented in Table 13.  
The first step of calculation of the TV3-117 MT engine regimes is to determine the main and similarity parameters from the takeoff regime, because the variation for the other regimes is reported to the similarity parameter values from the takeoff regime. In this way, to obtain accurate values is essential to use polynomials, for air and gas enthalpy and entropy, which are functions of working temperature [21]. In the case of air and gases (for ideal combustion), enthalpy and entropy are thus defined:  
\[h_{a}(T_{a}),s_{a}(T_{a})^{p0}=f(T_{a})\quad h_{g:id}\big{(}T_{g}\big{)},s_{g: id}\big{(}T_{g}\big{)}^{p0}=f\big{(}T_{g}\big{)} \tag{10}\]  
where the polynomial function is the form:  
\[f(T)=a_{1.a,g}+a_{2.a,g}\times T+a_{3.a,g}\times T^{2}+a_{4.a,g}\times T^{3}+ a_{5.a,g}\times T^{4}+a_{6.g}\times T^{5}. \tag{11}\]  
In case of gases (for real combustion), the enthalpy and entropy have the next forms:  
\[h_{g}\big{(}T_{g}\big{)}=x_{id}\times i_{g:id}\big{(}T_{g}\big{)}+x_{a,ex} \times h_{a}\big{(}T_{g}\big{)} \tag{12}\]  
\[s_{g}\big{(}T_{g}\big{)}^{p0}=x_{id}\times s_{g:id}\big{(}T_{g}\big{)}^{p0}+x_ {a,ex}\times s_{a}\big{(}T_{g}\big{)}^{p0} \tag{13}\]  
where the mass participation \(x_{id},x_{a.ex}\) depending on air excess \(\lambda_{ex}\) and air-fuel ratio \(min_{L}\)  
\[x_{id},x_{a.ex}=f(\lambda_{ex},min_{L}). \tag{14}\]  
Apart from enthalpy and entropy polynomials, the calculation model requires some additional parameters or coefficients for which regular values are chosen, as shown in Table 14. \begin{table}
\begin{tabular}{c c c c c} \hline
**P [HP]** & **SFC [kg/HP/h]** & \(T_{3.t}\) [\({}^{\circ}\)C] & \(\pi_{Ct}\) [-]** & **N [\%]** \\ \hline
2225 & 0.230 & 975 & 9.6 & 97.6 \\ \hline \end{tabular}
\end{table} 
Table 13: TV3-117 MT certain main data at the takeoff regime. Thus, using specific data from Tables 13 and 14, the values of the similarity parameters for the takeoff regime are calculated as described in the particular calculation below. Since there is not sufficient data to determine certain main engine parameters, a preliminary step was to determine the air excess that is useful for the air and gas mass flow calculation. The calculation method of air excess starts from the enthalpy equality relation between reactants HR and reaction products HP from the general combustion reaction [22]. \[H_{R}\Big{(}T_{2,t},T_{f}\Big{)}=H_{P}(T_{3,t}) \tag{15}\]  
Additionally, the enthalpies are defined as a function of temperature under a polynomial form:  
\[H_{R,P}(T)=R_{i}\times\bigg{(}a_{1}+a_{2}\times\frac{T}{2}+a_{3}\times\frac{T ^{2}}{2}+a_{4}\times\frac{T^{3}}{2}+a_{5}\times\frac{T^{4}}{2}+\frac{b_{1}}{T} \bigg{)} \tag{16}\]  
From the above equality, a second-degree equation in the unknown \(\alpha_{ex}\) is derived. The solution of this equation provides the values of air excess. \[\alpha_{ex}^{2}\times a+\alpha_{ex}\times b+c=0\ \ \ \Rightarrow\ \ \alpha_{ex,1,2}=\frac{-b\pm\sqrt{\Delta}}{2\times a}\ \ \ \Rightarrow\ \ \alpha_{ex} \tag{17}\]  
\[\alpha_{ex}=f(T_{3,t},\pi_{C,t})\ \ \Rightarrow\ \ \alpha_{ex}=3.687\ \ \Rightarrow\ \ f_{C}=\frac{1}{\alpha_{ex}\times min_{L}} \tag{18}\]  
After the air excess was determined, the calculation continues to determine the main engine parameters and performance that are components of the similarity parameters.