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Conceptual Design Methods Hybrid-Electric Transport Aircraft

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Conceptual Design Methods Hybrid-Electric Transport Aircraft ( conceptual-design-methods-hybrid-electric-transport-aircraft )

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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 𝑃𝑆𝑒𝑝𝑝𝑙𝑦,𝐸𝑙𝑒𝑐 remains constant. Considering now 𝐻𝐸 as a function of the flight conditions, the energies supplied can be written as: and 𝐸𝑆𝑒𝑝𝑝𝑙𝑦,𝐸𝑙𝑒𝑐 = 𝑃𝑆𝑒𝑝𝑝𝑙𝑦,𝐸𝑙𝑒𝑐 Β· 𝑑 (4.22) (4.23) (4.24) 𝐸𝑆𝑒𝑝𝑝𝑙𝑦,𝐹 𝑒𝑒𝑙 = 𝐸𝑆𝑒𝑝𝑝𝑙𝑦,𝑇 π‘œπ‘‘π‘Žπ‘™ βˆ’ 𝐸𝑆𝑒𝑝𝑝𝑙𝑦,𝐸𝑙𝑒𝑐 = 𝐸𝑆𝑒𝑝𝑝𝑙𝑦,𝑇 π‘œπ‘‘π‘Žπ‘™ βˆ’ 𝑃𝑆𝑒𝑝𝑝𝑙𝑦,𝐸𝑙𝑒𝑐 Β· 𝑑 Equations 4.22 and 4.23 can be combined into Equation 4.17. d𝐢 =π‘π‘‡π‘–π‘šπ‘’ +𝑐𝐹𝑒𝑒𝑙 Β·(d𝐸𝑆𝑒𝑝𝑝𝑙𝑦,π‘‡π‘œπ‘‘π‘Žπ‘™ βˆ’π‘ƒπ‘†π‘’π‘π‘π‘™π‘¦,𝐸𝑙𝑒𝑐)+𝑐𝐸𝑙𝑒𝑐 ·𝑃𝑆𝑒𝑝𝑝𝑙𝑦,𝐸𝑙𝑒𝑐 =0 d𝑑 d𝑑 Equation 4.24 yields to the definition of the CI for constant electric power as expressed in Equation 4.25. πΆπΌπ‘ƒπΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘,π‘π‘œπ‘›π‘ π‘‘ = π‘π‘‡π‘–π‘šπ‘’ + 𝑃𝑆𝑒𝑝𝑝𝑙𝑦,𝐸𝑙𝑒𝑐 Β· (𝑐𝐸𝑙𝑒𝑐 βˆ’ 𝑐𝐹𝑒𝑒𝑙) = βˆ’d𝐸𝑆𝑒𝑝𝑝𝑙𝑦,π‘‡π‘œπ‘‘π‘Žπ‘™ (4.25) 𝑐𝐹 𝑒𝑒𝑙 d𝑑 As the cost for electricity, 𝑐𝐸𝑙𝑒𝑐, is in the numerator of Equation 4.25, an increase in 𝑐𝐸𝑙𝑒𝑐 leads to a larger value of πΆπΌπ‘ƒπΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘,π‘π‘œπ‘›π‘ π‘‘. As the result, the optimum economical flight speed increases with increasing electricity price. Moreover, πΆπΌπ‘ƒπΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘,π‘π‘œπ‘›π‘ π‘‘, becomes zero under the specific price relation given by Equation 4.26. 𝑐𝐹𝑒𝑒𝑙 = 𝑐𝐸𝑙𝑒𝑐 + π‘π‘‡π‘–π‘šπ‘’ (4.26) 𝑃𝑆𝑒𝑝𝑝𝑙𝑦,𝐸𝑙𝑒𝑐 This condition results in an economical Mach number which coincides with the ESAR op- timum cruise speed as the term, 𝑑𝐸𝑆𝑒𝑝𝑝𝑙𝑦,π‘‡π‘œπ‘‘π‘Žπ‘™/𝑑𝑑, equals zero. Under the condition of a fuel price 𝑐𝐹𝑒𝑒𝑙 much greater than the electricity price 𝑐𝐸𝑙𝑒𝑐, Equation 4.25 yields the formulation: πΆπΌπ‘ƒπΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘,π‘π‘œπ‘›π‘ π‘‘ = 𝑐𝑇 π‘–π‘šπ‘’ βˆ’ 𝑃𝑆𝑒𝑝𝑝𝑙𝑦,𝐸𝑙𝑒𝑐 (4.27) (𝑐𝐸𝑙𝑒𝑐<<𝑐𝐹𝑒𝑒𝑙) 𝑐𝐹𝑒𝑒𝑙 According to Equation 4.27, different scenarios can be discussed with respect to the relation between 𝑃𝑆𝑒𝑝𝑝𝑙𝑦𝐸𝑙𝑒𝑐 and the ratio 𝑐𝑇 π‘–π‘šπ‘’/𝑐𝐹 𝑒𝑒𝑙 which reflects the conventional CI expressed in Equation 4.19.

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