## 3 Principles Demonstration through Application Example
Bearing in mind that, in practice, estimated health factors are always characterized by uncertainty, as a result of the presence of noise in the measured quantities, when the deviation caused by the non-steady operation is well within this uncertainty interval, it could be considered acceptable, as it would have the same effect as measurement uncertainty. This means that when the slope is below a certain value, even though the operating condition is not perfectly steady state, it could be acceptable in terms of uncertainty in the estimated health coefficients. The picture for all health indices is shown in \[\text{Flow factor}:\ SW_{i}=\left(W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)/\left( W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)_{ref}\]1 The health index with the largest deviation should be used to establish the interrelation with the slope  
\[\text{Flow factor}:\ SW_{i}=\left(W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)/\left( W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)_{ref}\]2 Time evolution of the slope of the exhaust gas temperature for the transient of \[\text{Flow factor}:\ SW_{i}=\left(W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)/\left( W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)_{ref}\]3 (**a**) overall picture; (**b**) zoom over a shorter time interval. \[\text{Flow factor}:\ SW_{i}=\left(W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)/\left( W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)_{ref}\]4 Health factor deviation in function of the local slope of the exhaust gas temperature for all health indices. (i.e., SW25). For the data shown in Figure 6, if we take, for example, a value of 0.25% as the maximum acceptable deviation for steady state, then the largest corresponding slope is 1 Kelvin/min. A question that arises now is whether the limit defined through this procedure is the same in different types of transients. What we have seen above is that for speed increase, whether it is steep or more gradual, the transient behavior in terms of the equivalent health factors is similar. Speed decrease is another representative type of transient (the entire operation of the engine consists of speed decrease and increase variations). The equivalent health factors calculated for the ramp down are shown in \[\text{Flow factor}:\ SW_{i}=\left(W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)/\left( W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)_{ref}\]5 We observe a pattern similar in form to the one in Figure 3, but the deviations are negative in this case. This is because the time derivatives disturbing the steady-state equation have an opposite sign (we have now deceleration instead of acceleration). We also observe that the asymptotic approach to the steady-state values happens at longer intervals, which we should expect to behave somewhat differently due to the fact that the operating lines move along different paths on the component maps. If we now observe the interrelation between slopes and the equivalent health factor deviations, the difference observed above is reflected in different lines, as shown in \[\text{Flow factor}:\ SW_{i}=\left(W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)/\left( W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)_{ref}\]6 A comparison with Figure 6 shows that, for the same threshold in virtual factor deviation, a smaller slope threshold has to be applied. For example, for virtual factors deviating less than 0.25%, a slope with a maximum of 0.4 K/min has to be enforced. \[\text{Flow factor}:\ SW_{i}=\left(W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)/\left( W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)_{ref}\]7 Health factor deviation in function of the local slope of the exhaust gas temperature trend: (**a**) overall picture; (**b**) zoom over a shorter window. \[\text{Flow factor}:\ SW_{i}=\left(W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)/\left( W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)_{ref}\]8 Variation in all health factors during the slow-down transient.