Aircraft Sizing and Constraints: Method

Aircraft Sizing and Constraints

What the tool computes

The first sizing of a fixed-wing aircraft in three steps. A constraint analysis finds the wing loading W₀/S and the thrust-to-weight ratio T₀/W₀ (jet) or the power-to-weight ratio P₀/W₀ (propeller) that meet every performance requirement entered. A mission analysis turns the range, the loiter time and the fixed segment fractions into the fuel fraction, and the weight equation gives the take-off weight; the wing area and the thrust or power follow from the design point. A range-payload diagram then shows what the sized aircraft can carry how far. T₀ and P₀ are sea-level static values; W₀ is the take-off weight. The air is the 1976 U.S. Standard Atmosphere.

Drag polar

C_D = C_D0 + K C_L², K = 1/(π e AR). C_D0 is entered or estimated as C_fe S_wet/S from an equivalent skin-friction coefficient on the wetted area. The maximum lift-to-drag ratio is 1/(2√(C_D0K)) at C_L = √(C_D0/K). With flaps down the profile-drag increments ΔC_D are added and a lower airplane efficiency factor is used; NASA RP-1060 takes e = 0.7 with flaps against 0.85 clean, flap increments of 0.01, 0.02 and 0.03 at 15°, 25° and 35°, and 0.015 for the landing gear. Compressibility drag is not modelled: enter a C_D0 that holds at the cruise Mach number.

Constraint lines

Along the flight path, with the weight fraction β = W/W₀, the load factor n and the rate of climb dh/dt, the thrust required is

T/W₀ = β [ q C_D0/(β W₀/S) + K n² β (W₀/S)/q + (1/V) dh/dt ], q = ρV²/2.

For a jet the line is T₀/W₀ = (T/W₀)/α with the thrust lapse α = T/T₀ = σ^m, taken independent of speed; in cruise α is multiplied by the cruise rating. For a propeller the line is P₀/W₀ = (T/W₀) V/(η α) with the shaft-power lapse α = σ − (1 − σ)/7.55 of a normally aspirated engine (Gagg and Ferrar) or α = 1, and a propeller efficiency η for cruise, for climb and for take-off. These lapse models are deliberate simplifications; the exponent, the rating and the efficiencies are the places to fit a particular engine.

• Stall speed: W₀/S ≤ ρ V_s² C_Lmax/2 in the landing configuration, a vertical limit.
• Cruise, maximum speed, sustained turn: the equation above with dh/dt = 0 at the entered speed and altitude; the turn with its load factor n.
• Rate of climb and service ceiling: the equation with n = 1 at the speed that needs the least thrust (jet: the root of ρC_D0V⁴/(β W/S) − (dh/dt) V − 4Kβ(W/S)/ρ = 0) or the least power (propeller: the minimum-power speed, C_L = √(3C_D0/K)), but not below 1.2 V_s clean.
• Climb gradients: T/W = [N/(N − 1)] (1/(L/D) + G) at C_L = C_Lmax/k², the small-angle balance of NASA RP-1060 equations (3.3) and (3.4); k = 1.2 in the take-off configuration with the gear up (second segment) and 1.3 in the approach configuration at landing weight. The factor N/(N − 1) applies with one engine inoperative. 14 CFR 25.121(b) asks 2.4, 2.7 and 3.0 percent for two, three and four engines in the second segment and 25.121(d) 2.1, 2.4 and 2.7 percent in the approach climb; 14 CFR 23.2120 asks 8.3 percent after take-off and 3 percent in a balked landing with all engines for low-speed level 1 and 2 airplanes.

Field lengths: the jet correlations of NASA RP-1060

Loftin correlated the FAR balanced take-off field length of 19 jet aircraft with the take-off parameter (W₀/S)/(σ C_L,T T₀/W₀), which is the ground run without drag and friction, equation (3.11), up to a constant. His Table 4.II reads 159, 185 and 212 lb/ft² for 6,000, 7,000 and 8,000 ft, that is 37.7 ft per lb/ft²: the tool's k_TO. The line is T₀/W₀ = k_TO (W₀/S)/(σ C_L,T × length). The FAR landing field length (the measured distance from 50 ft divided by 0.6) correlates with the square of the approach speed V_A = 1.3 V_s; Table 4.I reads 129 knots for 5,000 ft, so k_L = 0.3 ft/kt², and the landing wing loading is ρ V_A² C_Lmax/(2 × 1.69), divided by the landing weight fraction. Both constants are inputs. The correlations describe jet transports and business jets of the 1960s and 1970s on a standard day.

Field lengths: segments

The same segments as the Aircraft Performance tool. Take-off: ground roll to 1.1 V_s by the integrated equation of motion of NACA TN 1258, s = ln((K_T + K_AV²)/K_T)/(2gK_A) with K_T = T/W − μ_r and K_A = ρ(μ_rC_L − C_D)/(2W/S), the thrust taken at 0.7 of the lift-off speed; rotation for t_R seconds; a transition arc at 1.15 V_s and a load factor of 1.2; a straight climb to the obstacle. Landing: approach at 3° from the obstacle, flare arc at 1.23 V_s, free roll at the touchdown speed 1.15 V_s, braking roll with C_L = 0 and no reverse thrust. The take-off line is the thrust or power, found by bisection, at which distance × factor equals the required length; the landing limit is the wing loading at which it does. The factors turn distances into field lengths: 1.15 on the all-engine take-off distance to 35 ft (14 CFR 25.113(a)(2)) and 1/0.6 on the landing distance (14 CFR 121.195(b)).

Design point

The upper envelope of the lines is the least thrust or power that meets every requirement at a wing loading; the feasible region lies above it and to the left of the stall and landing limits. The automatic design point is the lowest point of the envelope inside the limits; where the envelope is level there (a climb-gradient line governs) the highest wing loading is taken, which gives the smallest wing. An entered point is checked against each requirement.

Mission fuel fractions

Each segment multiplies the weight by a fraction. Warm-up, taxi, take-off, climb, descent and landing are entered fractions. Cruise and loiter follow the Breguet equations: for a jet W_end/W_start = exp(−R c/(V L/D)) and exp(−E c/(L/D)) with the thrust specific fuel consumption c; for a propeller exp(−R c/(η L/D)) and exp(−E V c/(η L/D)) with the brake specific fuel consumption c. A blank L/D is the best of the polar for the segment: 0.866 (L/D)_max for jet cruise and propeller loiter, (L/D)_max for jet loiter and propeller cruise. The fuel fraction is W_f/W₀ = (1 + f_res)(1 − product of the fractions). NASA RP-1060 writes the single-segment case as W_f/W_g = 1 − exp(−R/B) with the Breguet factor B = V(L/D)/c, equation (3.13).

Take-off weight

W₀ = (W_payload + W_crew)/(1 − W_f/W₀ − W_e/W₀) with the empty-weight fraction W_e/W₀ = A W₀^C (W₀ in lb). With C = 0 this is equation (3.41) of NASA RP-1060, where 1 − A is the useful-load fraction. The equation is solved for its smallest root; the growth factor dW₀/dW_payload = 1/(1 − W_f/W₀ − (1 + C) W_e/W₀) says how many pounds the aircraft grows per pound of payload. The trade table resizes the aircraft for a 10 percent change of range, payload, L/D and fuel consumption and 1 percent of the empty-weight fraction. The propeller default A = 0.5795, C = 0.0037 is a least-squares fit to the gross and empty weights of six general-aviation aircraft in NASA SP-468 Table III (2,300 to 12,500 lb, all within 5.5 percent of the fit); for other classes enter constants fitted to comparable aircraft.

Range-payload diagram

With the take-off weight, the empty weight and the crew fixed, the cruise legs are scaled until the fuel carried is used (less the reserve fraction): at maximum payload with fuel up to the take-off weight, with full tanks and the payload reduced to hold the take-off weight, and with full tanks and no payload. L/D and fuel consumption are held at the mission values, so the diagram is a first estimate.

The default case and what was checked

The page opens on the ten-passenger executive jet of NASA RP-1060 chapter 4, variant 1: Mach 0.8, 2,174 n. mi. plus 500 n. mi. of reserve, C_D0 = 0.0220, aspect ratio 7, FAR field lengths of 6,000 and 5,000 ft. The report's match point 1 is 118.0 lb/ft² and T₀/W_g = 0.315, read from its charts and engine curves, with W_g = 19,906 lb, W_e = 11,147 lb and W_f = 5,594 lb (Table 4.V). The tool gives 116.5 lb/ft², 0.318, 19,961 lb, 11,178 lb and 5,618 lb; the weights differ by 0.3 percent because the report rounds the fuel fraction to 0.281 before dividing. The report lists a speed of sound of 576 knots at 39,200 ft where the 1976 atmosphere gives 573.6, so its 460.8 knots is Mach 0.803 here. The propeller option uses the Cessna Skyhawk row of NASA SP-468 Table III as requirements (stall 49 mph, cruise 138 mph at 8,000 ft, C_D0 = 0.0319, aspect ratio 7.32) with illustrative climb and field requirements and returns 13.1 lb/ft², 14.6 lb/hp and 2,262 lb against the airplane's 13.1, 15.3 and 2,300. The test script reproduces Tables 4.I to 4.V of the report row by row, compares 19 parameter sets with an independent scipy implementation and flies the sized aircraft through the Aircraft Performance library.

Limits

Parabolic polar without compressibility; thrust independent of speed and constant propeller efficiencies; quasi-steady climb; no wind, slope or temperature offset; the segment method gives all-engine distances, not a balanced field length; one cruise L/D and fuel consumption per segment; the empty-weight law is statistical. The result is a starting point for the Aircraft Performance, Drag Build-Up and Vortex Lattice Analysis tools, not a design.