## 2 Materials and Methods
### Combustion Chamber
#### 2.2.1 Conventional Combustion  
To predict the NOx emissions of the conventional combustion chamber at flight conditions, a methodology presented in [4] is used. This methodology enables the calculation of emissions at altitude conditions based on ground level emission data. The reference emissions are published by the ICAO [5]. To use the p3-T3-method, reference functions of the emission index (EI), which is the mass flow of emissions per fuel flow, the pressure, and the fuel-to-air ratio (FAR) have to be created:  
\[EI_{NO_{x}}=f(T_{3})\qquad p_{3,Ref}=f(T_{3})\qquad FAR_{Ref}=f(T_{3}) \tag{1}\]  
The index 3 stands for the combustion chamber entry. To receive information regarding \(T_{3}\), the reference engine model of the CF6 is used and the reference functions are approximated with quadratic functions. The equation below is used to determine emissions at every point in the flight envelope in terms of the emission index.  
\[EI_{NO_{x}}=EI_{NO_{x},Ref}\cdot\left(\frac{p_{3}}{p_{3,Ref}}\right)^{\alpha} \cdot\left(\frac{FAR}{FAR_{3,Ref}}\right)^{\beta} \tag{2}\]  
Appropriate values for \(\alpha\) and \(\beta\) can be found in [6].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Unit** & **Jet A-1** & **Drop-In** & **PtL** & **LH2** \\ \hline LHV & MJ/kg & 43.138 & 43.639 & 44.184 & 119.96 \\ Density & kg/m\({}^{3}\) & 814.7 & 777.2 & 752.7 & 70.2 \\ EI CO\({}_{2}\) & kgCO\({}_{2}\)/kgFuel & 3.164 & 3.132 & 3.101 & 0.0 \\ EI H\({}_{2}\)O & kgH\({}_{2}\)O/kgFuel & 1.220 & 1.297 & 1.374 & 8.937 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Key characteristics and emission properties of applied energy carriers.  
These emissions are appropriate for the original CF6 engine, but not for the engines used for the D261+ concept in this paper. As the same p3-T3-method is applied for the entry into service (EIS) 2040 case, the reference functions have to be adapted. This adaptation is accomplished by the DLR/Stoppler method [6; 7; 8].  
The DLR/Stoppler method is based on a simplified description of the actual physical and chemical processes for the nitrogen oxide formation. Therefore, many influences on this formation can be considered and analysed as follows:  
\[\frac{EI_{NO_{x}}}{EI_{NO_{x},Ref}}=\frac{1+\frac{1}{FAR}}{1+\frac{1}{FAR_{Ref}} }\cdot\frac{\textit{in}_{Air,Ref}}{\textit{in}_{Air}}\cdot\left(\frac{p_{3}}{ p_{3,Ref}}\cdot\frac{T_{PZ,Ref}}{T_{PZ}}\right)^{c}\cdot e^{\frac{E}{R}}\cdot \frac{1}{\textit{in}_{T}}-\frac{1}{\textit{in}_{T,Ref}}\right) \tag{3}\]  
Similar to the p3-T3-method, the reference functions are approximated using combustion chamber entry temperature and values from the CF6 engine. Further constants are \(E=-312\) kJ/Mol and the universal gas constant \(R=8.3143\) J/(Mol K). The temperature in the primary zone \(T_{PZ}\) can be calculated with the entry temperature \(T_{3}\) and the equilibrium temperature \(T_{\Phi}\) as follows:  
\[T_{PZ}=\frac{T_{3}+T_{\Phi}}{2} \tag{4}\]  
Furthermore, the flame temperature \(T_{\text{FL}}\) is calculated with the equilibrium temperature \(T_{\Phi}\), the stoichiometry temperature \(T_{\Phi=1}\), and the weighting factor \(r_{\text{FL}}\):  
\[T_{FL}=r_{FL}\cdot T_{\Phi}+(1-r_{FL})\cdot T_{\Phi=1} \tag{5}\]  
The constant values for \(c\) (Equation (3)) and \(r_{\text{FL}}\) (Equation (5)) are selected so that the emission indices \(EI_{NO_{x}}\) and \(EI_{NO_{x},Ref}\) in Equation (3) match at 30% and 100% ground thrust.  
The emissions adapted by the DLR/Stoppler method are still the emissions for the ground level case. For the D261+, the adapted ground emissions are used to determine emissions at flight conditions with the previously presented p3-T3-method.