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Generalized Power and Drag

Generalized Power and Drag - Generalized Power and Drag...

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Generalized Power and Drag Eugene M. Cliff March 14, 2001 1 Introduction The purpose of these notes is to supplement the text with additional material related to level-flight performance. We consider several extensions: inclusion of weight variations in power-required and in drag; and speed-drag results includ- ing Mach dependence. 2 Generalized Speed and Power 2.1 Classical Power Required The term power required ( P r ) refers to the product of level-flight drag and speed. Since equilibrium flight requires that L = W and T = D , it follows that the propulsive system must supply power in the amount P r or the airplane will be losing energy, that is slowing down or descending. Our text (p 133) analysis leads to the result P r = 1 / 2 C Do ρ S V 3 + 2 K W 2 ρ S V . (1) It should be clear that P r depends on true-airspeed ( V ), on altitude (through the air-density ρ ) and on the weight ( W ). Graphs of P r vs V are a two-parameter family depending on weight and altitude. To simplify the altitude dependence we use equivalent airspeed V e , where V e = V · σ , σ being the density ratio ( ρ/ρ sl ). Using this we are led to P r σ = 1 / 2 C Do ρ sl S V 3 e + 2 K W 2 ρ sl S V e . (2) Thus, if we plot P r σ vs V e then for a given weight, we get one plot that works for all altitudes. To find the actual power-required at a given altitude and true airpseed (and, of course, at the given weight) we would 1. Compute V e using the given value of V and ρ 2. Enter the graph with V e and read-off the P r σ value 1
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3. Compute P r by dividing by σ Note that we still need a one-parameter family of curves - one for each weight. 2.2 Incorporating Changes in Weight The result (2) is an improvement over (1) in that the altitude dependence has been supressed. It turns out that by suitably re-defining the speed and power variables we can also ‘absorb’ the weight dependence. We shall do this in two steps. We begin by simply dividing all the terms in (2) by W 3 / 2 . The first term on the right will then read: 1 / 2 C Do ρ sl S V 3 e /W 3 / 2 , while the second term will be: 2 K W 1 / 2 ρ sl S V e .
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