# Potentials for Pressure Gain Combustion in Advanced Gas Turbine Cycles
## Materials and Methods
### Combustor Model  
The modeling of pressure gain combustion was crucial for this study. However, the single correct model for PGC did not exist. That is all the more especially true as many different approaches aim to increase pressure during combustion. For this steady-state zero-dimensional simulation (also the lumped volume approach), each component was represented by a transfer function or tabulated values in order to link inlet and outlet states. Functions for mass flow, total temperature, and total pressure were necessary. For PGC, the mass flow was simply the sum of inlet air flow and fuel flow. The outlet temperature was calculated as for conventional combustion from the energy balance. In GTlab-Performance, lookup tables created in the software Cantera were used to determine the outlet temperature for the equilibrium state. The difficulty, as indicated in the Introduction, was associated with the outlet pressure. This steady-state approach required a single value representative for the intermittent process. Therefore, only a few models were applicable, from which the following were implemented. For completeness, they are repeated here: the models provided by [PERSON] [28] and [PERSON] [18] calculate the pressure ratio (PR) from an enthalpy ratio (HR), which is given by the fuel flow. First, a non-dimensional heat is calculated giving the enthalpy ratio from an energy balance around the combustor (Equations (3) and (4)). The pressure ratio was then derived in Equation (5) using a tuning factor \(C\), which was 0.12 according to [PERSON] and 0.105 according to [PERSON]. \[q_{0}=\frac{LHV}{\left(1+AFR\right)\gamma R_{g}T_{in}} \tag{3}\]  
\[HR=1+q_{0}\left(\gamma-1\right)\left(1-pf\right) \tag{4}\]  
\[PR=HR^{\frac{C_{T}}{\gamma-1}} \tag{5}\]  
The purge fraction \(pf\) will not be used in this analysis and is set zero throughout this paper. However, this could represent a handle for off-design operation. A different model was defined by [PERSON] [15]. Outlet temperature \(T_{out}\) and isochoric combustion temperature \(T_{B}\) were calculated from the energy balance around the combustor (see Equation (6)). From this temperature, the isochoric pressure \(p_{B}\) was found using an isochoric change of state (Equation (7)). The isochoric combustion was then followed by an isentropic expansion to the outlet condition. The outlet pressure \(p_{out}\) was calculated using the outlet temperature \(T_{out}\) and the isentropic relation in Equation (8). \[Q=\dot{m}c_{v}\left(T_{B}-T_{in}\right)=\dot{m}c_{p}\left(T_{out}-T_{in}\right) \tag{6}\]  
\[\frac{T_{B}}{T_{in}}=\frac{p_{B}}{p_{in}} \tag{7}\] \begin{table}
\begin{tabular}{l c} \hline \hline
**Pressure Losses for Recuperator** \\ \hline Inlet cold side & 4.5\% \\ Outlet cold side & 1.75\% \\ Inlet hot side & 4\% \\ Outlet hot side & 2\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Assumptions for recuperator pressure losses from [4]. \[\frac{p_{out}}{p_{B}}=\frac{T_{out}}{T_{B}}^{\frac{\gamma}{\gamma-1}} \tag{8}\]  
Additionally, these authors inferred a PGC model that matched the CFD data provided by Paxson [9], which will be denoted as Mix. The results of CFD simulations by Paxson are also plotted in Figure 2 and were regarded as more accurate than the simple combustor models above. The complete setup of the experimentally-validated, one-dimensional, time-accurate, reactive, computational fluid-dynamics Euler solver was presented in [12]. Two different CFD datasets are available. The difference was associated with the fuel distribution within the combustion tube. For the data denoted Paxson CFD1, the tube was completely filled with the air fuel mixture, whereas an air buffer was present at the tube outlet in the dataset Paxson CFD2. The developed model from [19] was inferred from simplified pulse detonation combustion. A typical pulse detonation tube can be split into two combustion regimes. First, some sort of ignition source initiated a subsonic deflagration.