� � � c Nomenclature ρ air density W aircraft weight S reference area (wing area) b wing span Iaverage wing chord λ taper ratio AR wing aspect ra t io o wing root bending inertia E Young’s mo dulus cCPPg gravity elec electric p ower max maximum thrust power ( = (ηPelec)max) η overall propulsive efficiency L lift coefficient d wing profile drag coefficient CDA0 drag area of non-wing components τ airfoil thickness/chord ratio δ tip deflection Assumed Design Variables The primary design variables here are assumed to be AR and S. Other design variables such as τ, λ, etc., may also be considered, although for brevity these will not be shown in the a r gument lists below. The following dependent auxilliary functions and constraint functions are assumed to be known from suitable analyses: c(AR, S) = S/AR (1) b(AR, S) = √S × AR (2) Wwing(AR, S) = 0.6 τ c 2 b ρfoam g (3) W (AR, S) = Wwing + Wfuse (4) � 2 �4 4I0(AR, S) = 0.04 τ3 c (5) δ W1+λ fuse b2 1+2λ (AR, S) = (6) b EI0 96 1+λ The velocity for level flight and corresponding chord R eynolds number are also convenient aux-illiary functions. � �1/22W V (AR, S) = (7) ρ S CL V c Re(AR, S) = (8) ν Minimum Flight Power The battery power needed to sustain level flight is given by the flight power relation. 1 1 C2 LPelec(AR, S) = ρV 3 S CDA0 + cd + (9) η 2 S π e AR Most of the variables in this expression depend implicitly on the design variables. The W (AR, S) and V (AR, S) functions were considered earlier. The overall power conversion efficiency depends 1 Improved Performance Estimates for Optimization 23 Mar 06Lab 7 Lecture Notes� � on the velocity, η(V ) which is evaluated in terms of the design variables by direct substitution of expression (7). η(AR, S) = η(V (AR,S)) (10) Similarly, the profile drag depends on the CL and Reynolds number, cd(CL, Re) which again can likewise be given in terms of the design variables. cd(AR, S) = cd(CL, Re(AR,S)) (11) Maximum Flight Speed The maximum flight speed Vmax is given implicitly by the flight power relation (9) , with Pelec set to the maximum available power. It is convenient to consider Pelec and η together, as the maximum available thrust power Pmax(V ) = (ηPelec)max. CDA0 C2 � Pmax =1 ρ V 3 S + cd(CLmin, Remax) + Lmin (12) 2 max S π e AR 2W/S = (13) CLmin ρ V 2 max Vmax c Remax = (14) ν Although equations (12), (13), (14) cannot be explicitly solved for Vmax(AR, S), it is possible to solve them by a reasonably simple iterative procedure. We note that at maximum speed the induced drag is likely to be small, and the profile cd only weakly dependent on V . Hence, the initial Vmax value can be guessed, and then subsequently improved. The iteration proceeds as follows: 0) Assume some expected V˜max value. 1) Compute corresponding C˜Lmin . 2W˜CLmin = ρ ˜2 (15) V Smax 2) Evaluate all o t her quant ities which depend on V or CL. ˜Vmax c˜Remax = (16) ˜Pν max = Pmax(Vmax) (17) c˜d = cd(˜ ˜CCLmin, Remax) (18) 2 C˜Di = Lmin (19) πeAR 3) Calculate improved Vmax from power relation (12). 2˜�1/3 VPmax max = ρ (CDA0 + Sc˜d + S ˜(20) CDi ) The iteration can be repeated by starting again a t 1). Only one such additional pass is likely to be necessary in most cases.
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