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4.4 Operational Economics Optimization 61 4.4.2 Constant Electric Power Supply In this section, the CI is derived for hybrid-electric propulsion system whose operations result in a constant electric power supply. The formulation of the CI for constant electric power supply is relevant for the hybrid-electric aircraft concept investigated in Section 5.3. In this context, the hybrid-electric propulsion system is composed of an electric motor power running at constant power during a given segment of the mission. Under the assumption that the efficiency chain of the electrical system from the electric energy and power device to the electric motor is invariable during the operation, the electric power supplied πππ’ππππ¦,πΈπππ remains constant. Considering now π»πΈ as a function of the flight conditions, the energies supplied can be written as: and πΈππ’ππππ¦,πΈπππ = πππ’ππππ¦,πΈπππ Β· π‘ (4.22) (4.23) (4.24) πΈππ’ππππ¦,πΉ π’ππ = πΈππ’ππππ¦,π ππ‘ππ β πΈππ’ππππ¦,πΈπππ = πΈππ’ππππ¦,π ππ‘ππ β πππ’ππππ¦,πΈπππ Β· π‘ Equations 4.22 and 4.23 can be combined into Equation 4.17. dπΆ =πππππ +ππΉπ’ππ Β·(dπΈππ’ππππ¦,πππ‘ππ βπππ’ππππ¦,πΈπππ)+ππΈπππ Β·πππ’ππππ¦,πΈπππ =0 dπ‘ dπ‘ Equation 4.24 yields to the definition of the CI for constant electric power as expressed in Equation 4.25. πΆπΌππΈππππ‘πππ,ππππ π‘ = πππππ + πππ’ππππ¦,πΈπππ Β· (ππΈπππ β ππΉπ’ππ) = βdπΈππ’ππππ¦,πππ‘ππ (4.25) ππΉ π’ππ dπ‘ As the cost for electricity, ππΈπππ, is in the numerator of Equation 4.25, an increase in ππΈπππ leads to a larger value of πΆπΌππΈππππ‘πππ,ππππ π‘. As the result, the optimum economical flight speed increases with increasing electricity price. Moreover, πΆπΌππΈππππ‘πππ,ππππ π‘, becomes zero under the specific price relation given by Equation 4.26. ππΉπ’ππ = ππΈπππ + πππππ (4.26) πππ’ππππ¦,πΈπππ This condition results in an economical Mach number which coincides with the ESAR op- timum cruise speed as the term, ππΈππ’ππππ¦,πππ‘ππ/ππ‘, equals zero. Under the condition of a fuel price ππΉπ’ππ much greater than the electricity price ππΈπππ, Equation 4.25 yields the formulation: πΆπΌππΈππππ‘πππ,ππππ π‘ = ππ πππ β πππ’ππππ¦,πΈπππ (4.27) (ππΈπππ<<ππΉπ’ππ) ππΉπ’ππ According to Equation 4.27, different scenarios can be discussed with respect to the relation between πππ’ππππ¦πΈπππ and the ratio ππ πππ/ππΉ π’ππ which reflects the conventional CI expressed in Equation 4.19.PDF Image | Conceptual Design Methods Hybrid-Electric Transport Aircraft
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