## 3 Principles Demonstration through Application Example  
_Application to Typical Transients_  
We will elucidate this behavior by using the example of a mixed-flow turbofan, with a layout as shown in Figure 1. The analysis is carried out using an engine performance model of this configuration [22]. The model is generated in the PROOSIS v6.4.0 simulation environment [23] using its standard library of engine components, which is based on the industry-accepted performance modelling methodology. The off-design performance of all turbomachinery components is obtained from suitable maps, scaled accordingly during the design calculation. The model is adapted to the known engine performance using the procedure described in [22]. The health factors used to characterize the condition of the individual components, as well as the measurements used for monitoring this engine, are shown in the same figure. It is noted here that the health condition of the engine components is represented by the \"health parameters\". Two health parameters per engine component are defined. For a component with entrance at station \(i\) along the engine, these parameters are as follows:  
\[\text{Flow factor}:\ SW_{i}=\left(W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)/\left( W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)_{ref}\]  
\[\text{Efficiency factor}:\ SE_{i}=n_{i}/(n_{i})_{ref}\]  
\(W_{i}\) is the gas mass flow rate at station \(i\), \(Tt_{i}\) is the total temperature at station \(i\), \(Pt_{i}\) is the total pressure at station \(i\), \(n_{i}\) is the efficiency of the component with entrance at station \(i\), while the subscript ref indicates the reference values, i.e., the values of a healthy engine. The use of such factors for describing the health condition of engine components has been discussed by [PERSON] et al. [24]. The deviation of an engine component health parameter from its nominal value indicates the presence of a fault in the corresponding component. Moreover, the more severe the fault, the greater this deviation is. The deviation of a health parameter \(f\) is defined as \(\Delta f(\%)=\frac{f-fo}{fo}\cdot 100\%\), where \(f\) is the value of the health parameter and \(fo\) is its nominal value. To start with, the transient maneuvers of a fuel flow step change and a fuel ramp-up are simulated. The fuel flow and the corresponding variation in some performance parameters are shown in Figure 2. The well-known behavior of the measured quantities that vary significantly over the steep part of the transient and then asymptotically tend towards the steady-state condition is observed. A steady-state diagnostic problem is then solved to determine the values of the 8 health parameters of the turbomachinery components of the engine. The measurements used as input are NH, Wf, Pt13, Pt13, Pt31, Tt31, Tt5 and Pt45. A non-linear square diagnostic problem is solved, where these eight measurements are given and the eight health parameters are the unknowns. The obtained values for the health parameters are shown in Figure 3.  
Figure 1: Mixed-flow turbofan layout and health indices of its components. Measurement location and station numbering. Figure 2: (**a**) Fuel step and fuel ramp simulated; (**b**) Corresponding variation in measurements of a mixed-flow twin-spool turbofan. The following observations can be made: although the engine components' condition has not changed, a significant variation in the health parameters over a part of the transient is observed. Moreover, this variation is not physically consistent, as it indicates that the performance of all the components in terms of efficiency and flow capacity increase. It is well known that such behavior cannot occur when the engine condition changes. Engine deterioration or faults always lead to a reduction in the values of the efficiency factors, while the flow capacity of compressors always reduces also. Additionally, the magnitude of deviations reaches values that would mean catastrophic changes, if they were of the right physical significance.