On the conditions for absolute minimum fuel burn for turbofan powered, civil transport aircraft and a simple model for wave drag

Abstract

In a recent series of papers, Poll and Schumann have been developing a simple model for estimating fuel burn for turbofan powered, civil transport aircraft for a given mass, Mach number and flight level and in a specified ambient temperature profile for all phases of flight. This paper focuses upon the combination of Mach number and flight level at which an aircraft cruises with the absolute minimum fuel burn. For a given aircraft type, the information necessary to determine these conditions must be specified and this poses a challenge. An initial attempt to obtain these data has been described previously by the first author. In this paper, the optimum conditions are found using a completely different approach. Starting from first principles and using established theory, the equations governing the situation where engine overall efficiency and airframe lift-to-drag ratio both have local maxima at the same flight condition are developed. This special case is termed the “design optimum” condition and, for a specified aircraft mass and a specified atmospheric temperature versus pressure profile, it gives the lowest possible fuel burn for any aircraft and engine combination. The design optimum occurs at a particular Mach number and Reynolds number, and it is a fixed characteristic of the aircraft. The analysis reveals the significance of Reynolds number variations, wave drag, including its derivatives with respect to both lift coefficient and Mach number, and the atmospheric properties. Whilst wave drag is notoriously difficult to determine accurately, it is found that solutions to the equations are not particularly sensitive to the accuracy of this quantity. Consequently, a simple, physically realistic model can give good results. An appropriate model is developed and a complete, approximate solution is obtained. Taking the International Standard Atmosphere as the design atmosphere, results are presented for the 53 aircraft types previously considered by Poll and Schumann. Relative to the design optimum conditions, when Reynolds number is constant and wave drag is zero, compressibility alone reduces L/D by about 5%, reduces lift coefficient by about 1.5% and increases drag coefficient by about 3.5%. Reynolds number variation has little effect upon L/D, but it reduces lift coefficient and drag coefficient by a further 7% and 8% respectively. The reduction in lift coefficient has a significant impact on the optimum cruise flight level. In general, an aircraft’s operating optimum will not coincide with its design optimum, but deviations are expected to be small. Therefore, using the design optimum solution as a reference point, an improved version of the operating optimum estimation method described by Poll and Schumann in previous work is developed. This allows the estimation of the conditions for absolute minimum fuel burn for an aircraft of given mass flying thorough any atmosphere. Updated coefficients for the 53 aircraft types are given.