# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 2 Modeling of the Turboshaft Engine Performance
### Turboshaft Engine Performance Simulation Model  
Since this paper focuses on the performance of the turboshaft engine, a component-level model is established to simulate the engine performance. The model is based on the thermodynamic cycle and the match of different components. In the simulation model, each component is considered as a \"black box\" and its performance is represented by the characteristic map or empirical formula. When the component performance is specified, the thermodynamic relations between inlet and outlet can be built to simulate the variation of the gas state [20; 21; 22]. In this situation, if the thermodynamic parameters at the inlet are known, the parameters at the outlet can be derived. The usual parameters of interest are mass flow, pressure (total or static), temperature (total or static), Mach number, fuel-air ratio, etc. Once all the cycle parameters are determined, the engine performance can be obtained. To improve the accuracy of the model, several factors that affect the properties of the gas are involved in this model, such as temperature and humidity. In addition, the effects of bleed air and turbine cooling are also taken into consideration. Various applications have shown that the component-level model is accurate enough to predict the engine performance [23; 24; 25; 26]. Figure 1: The configuration of the turboshaft engine. The model consists of two simulation modules. The first one is used for thermodynamic cycle calculation at the design point of the engine and the other one is applied to calculate the cycles at other operating points. The former is called the design point module and the latter is called the off-design module. For the design point, the operating condition of the engine is fixed and a high level of performance is usually specified for each component. Moreover, the match of cycle parameters and component performance should also be reasonable. In this case, the design point module can determine the key flow areas and other essential thermodynamic parameters of the engine. Thus, the design cycle serves as a cycle reference point to perform off-design cycle calculation at other operating points. In the off-design module, several suitable component maps are required to predict the component performance at the operating point that deviates from the design point, especially for the compressor and turbines. However, in most cases, the genuine maps are not available. Thus, some standard maps are often chosen as substitutes for the genuine maps. To adapt the standard maps to the simulation model, the standard maps are coupled with the design values of component performance in order to adjust the characteristic values of the maps. Taking the crucial compressor map as an example, shown in Figure 2, there are three characteristic parameters to represent the compressor performance, namely, the pressure ratio (\(\pi_{com}\)), the corrected mass flow (\(w_{com,cr}\)), and the adiabatic efficiency (\(\eta_{com}\)). Another important parameter in the map is relative corrected spool speed (\(N_{com,cr}\)) which reflects the operating condition of the compressor. The map indicates the variations of \(\pi_{com}\) and \(\eta_{com}\) over \(w_{com,cr}\) under each \(N_{com,cr}\). To facilitate reading the data from the map, the characteristic parameters are often defined as follows:  
\[\left\{ \begin{array}{l}w_{com,cr}=f_{1}(N_{com,cr},Z_{com})\\ \pi_{com}=f_{2}(N_{com,cr},Z_{com})\\ \eta_{com}=f_{2}(N_{com,cr},Z_{com})\\ \end{array} \right., \tag{1}\]  
where \(Z_{com}\) denotes the location of the compressor operating point in the compressor map. By introducing the parameter \(Z_{com}\), the map can be conveniently integrated into the simulation model. In this situation, two parameters, \(N_{com,cr}\) and \(Z_{com}\), are essential to determine the compressor performance since the compressor performance only depends on \(N_{com,cr}\) and \(Z_{com}\).