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60 4 Methods for Flight Technique Optimization of Hybrid-Electric Aircraft tiating Equation 4.13 with respect to the block time yields the cost increase per change in time as defined in Equation 4.17. dπΆ = ππ πππ + ππΉ π’ππ Β· dπΈππ’ππππ¦,πΉ π’ππ + ππΈπππ Β· dπΈππ’ππππ¦,πΈπππ = 0 (4.17) dπ‘ dπ‘ dπ‘ A local minimum leads to establish the condition of having Equation 4.17 equals to zero. Rewriting Equation 4.17 as a function of the design parameter π»πΈ in Equation 4.18 results in the general definition of the CI for hybrid-electric aircraft in Equation 4.19. dπΆ =πππππ+ dπ‘ (ππΉπ’ππ Β· (1 β π»πΈ) + ππΈπππ Β· π»πΈ) Β· dπΈππ’ππππ¦,πππ‘ππ + πΈππ’ππππ¦,πππ‘ππ Β· (ππΈπππ β ππΉπ’ππ) Β· dπ»πΈ (4.18) (4.19) ππ πππ + πΈππ’ππππ¦,π ππ‘ππ Β· (ππΈπππ β ππΉ π’ππ) Β· dπ»πΈ πΆπΌ= dπ‘ =β dπΈππ’ππππ¦,π ππ‘ππ dπ‘ dπ‘ dπ‘ ππΉπ’ππ Β·(1βπ»πΈ)+ππΈπππ Β·π»πΈ The application of the CI is demonstrated in the following for three different hybrid-electric propulsion system characterized namely by a constant degree of hybridization for energy, a constant electric power supply and a constant amount of electric energy consumption. 4.4.1 Constant Degree of Hybridization for Energy For a hybrid-electric aircraft configuration equipped with a propulsion system whose op- erational utilization results in a constant π»πΈ, the fractional change of π»πΈ versus time in Equation 4.19 equals zero. dπ»πΈ =0 (4.20) dπ‘ Combining Equation 4.19 and Equation 4.20, the expression of CI for constant π»πΈ is obtained in Equation 4.21. πΆπΌπ»πΈππππ π‘ = πππππ (4.21) ππΉπ’ππ Β·(1βπ»πΈ)+ππΈπππ Β·π»πΈ πΆπΌπ»πΈππππ π‘ becomes the ratio of πππππ and the specific cost of the energy weighted by π»πΈ. A value of π»πΈ tending to zero leads to the traditional cost index definition considering in this case the total energy supplied. For a fixed πππππ and a ππΈπππ far less than ππΉπ’ππ, the πΆπΌπ»πΈππππ π‘ optimization would lead to higher economical Mach number compared to the one obtained by using the traditional cost index. This effect will be even more pronounced for higher π»πΈ values. An example of a hybrid-electric concept for the application of πΆπΌπ»πΈππππ π‘ is presented in Section 5.4. For this hybrid-electric aircraft concept, π»π π’π π is fixed at 50% during cruise. Under the assumption that the efficiency of the hybrid-electric propulsion system does not vary significantly during cruise, π»πΈ remains constant.PDF Image | Conceptual Design Methods Hybrid-Electric Transport Aircraft
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