Aircraft Performance: Method

Aircraft Performance

What the tool computes

Classical point and mission performance of an airplane described by a parabolic drag polar, C_D = C_D0 + K C_L², its weight, wing area and maximum lift coefficients, and a simple model of its propulsion: a propeller driven by a shaft power at an efficiency η (constant or tabulated against speed), or a jet with a thrust that does not vary with speed. The air is the 1976 U.S. Standard Atmosphere at the entered geometric altitude. From these the tool derives the drag and power required against speed, the characteristic speeds, the level-flight speed range, the rate and angle of climb with the best-climb speeds and the hodograph, the ceilings, the Breguet range and endurance, and the take-off and landing distances by segments. Every quantity is printed with the equation it comes from and the substituted numbers; the library behind the page is checked against closed forms and an independent numerical integration (test-scripts/test_aircraft_performance.php, 192 checks).

Atmosphere

Geopotential altitude H = r₀Z/(r₀ + Z), a lapse of 6.5 K/km to 11 km, isothermal to 20 km and the warmer layers above, pressure from the hydrostatic equation and density from the perfect-gas law with R = 287.053 J/(kg·K). The density ratio σ = ρ/ρ₀ is referred to the model's own sea-level density (1.224999 kg/m³, printed as 1.2250 in the standard) so that σ = 1 exactly at sea level. The implementation reproduces the printed Table IV rows to every digit.

Drag polar and the characteristic speeds

With K = 1/(π e AR) (or entered) the polar's optima follow from calculus. The lift-to-drag ratio is largest where the parasite and induced drag are equal: C_L = √(C_D0/K), (L/D)_max = 1/(2√(C_D0K)). The power required D·V is least where the induced drag is three times the parasite drag: C_L = √(3C_D0/K), which is also where C_L^3/2/C_D is largest, with the value (1/4)(3/(K C_D0^1/3))^3/4 and L/D = (√3/2)(L/D)_max. The jet-range parameter C_L^1/2/C_D is largest at C_L = √(C_D0/(3K)). In level flight L = W gives V = √(2W/(ρ S C_L)) at each of these lift coefficients: the stall speed at C_Lmax, the minimum-drag speed V_md = √(2W/(ρS)) (K/C_D0)^1/4, the minimum-power speed V_mp = V_md/3^1/4. The drag at any speed is ½ρV²S C_D0 + 2KW²/(ρV²S), the two terms being the parasite and induced parts the tool tabulates and draws.

Thrust and power available

A propeller airplane delivers the power P_A = ηP and the thrust T_A = ηP/V; the thrust is largest at low speed and, with a constant η, would be infinite at rest. The tool therefore says which efficiency it used wherever a low-speed thrust matters (the take-off ground roll at 0.7 V_LO) and accepts a table η(V) for more realistic static and climb thrust. A normally aspirated piston engine loses power with altitude as P/P₀ = σ − (1 − σ)/7.55 (Gagg and Ferrar); a turbocharged or flat-rated engine can be held constant. A jet delivers a thrust independent of speed that falls with altitude as T₀σ^m (m = 1 for the turbojet approximation, about 0.7 for high-bypass turbofans in the troposphere) or is held constant; its power available T·V grows linearly with speed.

Level flight, climb and ceilings

Level flight is possible where P_A ≥ P_R. The lower limit is the stall speed when the excess power is positive there, otherwise the lower crossing of the curves (the back side of the power curve); the upper limit is the top speed where the curves cross again. The quasi-steady rate of climb is R/C = (P_A − P_R)/W and the climb angle sin γ = (T − D)/W, the acceleration along the flight path neglected (the kinetic-energy correction matters for jets above about Mach 0.5 and is not applied). The best rate of climb is searched between the stall and top speeds; for constant power available it sits at the minimum-power speed with (R/C)_max = ηP/W − P_R,min/W, which Anderson writes as ηP/W − 0.8776√((W/S)/(ρC_D0))/(L/D)_max^3/2; for constant thrust the closed form uses Z = 1 + √(1 + 3/((L/D)_max²(T/W)²)). The tool prints both the search and the closed form so that an efficiency table or a speed-dependent thrust shows its effect. The best angle of climb is at the largest excess thrust: the stall speed for a propeller airplane with constant η, the minimum-drag speed for a jet with constant thrust. The hodograph plots vertical against horizontal speed; the best angle is the tangent from the origin, the best rate the highest point. The best rate of climb is recomputed at each altitude: the absolute ceiling is where it falls to zero, the service ceiling where it falls to 100 ft/min (0.508 m/s).

Range and endurance

For a propeller airplane the fuel weight flow is c·P with c the BSFC converted to weight per unit of shaft work, so that dR = (η/c)(L/D) dW/W. Integrated at constant lift coefficient and efficiency this is the Breguet range R = (η/c)(L/D)_max ln(W₀/W₁), flown at V_md, and the endurance E = (η/c)(C_L^3/2/C_D)_max √(2ρS)(W₁^−1/2 − W₀^−1/2), flown at V_mp at constant altitude. For a jet the fuel flow is c_t·T, so dR = (V/c_t)(L/D) dW/W: at constant altitude and lift coefficient R = 2√(2/(ρS)) (1/c_t)(C_L^1/2/C_D)_max(√W₀ − √W₁) and E = (1/c_t)(L/D)_max ln(W₀/W₁). The tool prints the speeds at the start and end of cruise for each optimum (they fall with √W). The values are still-air cruise values with no reserves, climb or descent; the check script integrates the same differential equations numerically over the fuel burn and finds the closed forms exact.

Take-off

The ground roll is the integral of the equation of motion along the runway, NACA TN 1258 eq. (9): s = ∫ (W/g) V dV / [T − C_DqS − μ_r(W − C_LqS)]. With a constant thrust it has the closed form s_g = (1/(2gK_A)) ln[(K_T + K_AV_LO²)/K_T], K_T = T/W − μ_r, K_A = (ρ/(2W/S))(μ_rC_L − C_D) (Raymer, Anderson). The thrust is taken at 0.7 V_LO, Anderson's rule for the propeller whose thrust falls as ηP/V across the run (TN 1258 found that Diehl's linear net-thrust approximation reproduces measured ground runs within about 7 %); the tool also prints Anderson's simpler estimate s ≈ V_LO²W/(2g[T − D − μ_r(W − L)]_0.7V). The lift and drag during the roll use the ground-attitude C_L (the roll is shortest at C_L = μ_r/(2K)) and the parasite increment of the take-off configuration. With wind the integral is taken in the ground frame, the forces at airspeed, from airspeed V_w to V_LO; the Pilot's Handbook's rules of thumb (a 10 % headwind saves about 19 %, a 10 % tailwind costs about 21 %) follow from it. Rotation is t_R seconds at V_LO; the transition is a circular arc at V_TR = 1.15 V_s with a load factor of 1.2, radius V_TR²/(0.2g), to the climb angle sin γ = (T − D)/W in the take-off configuration; if the arc reaches the obstacle height the distance is √(R² − (R − h_obs)²), otherwise a straight climb (h_obs − h_TR)/tan γ follows. Ground effect on the induced drag, runway slope and wheel spin-up are not included; the distances are unfactored.

Landing

Approach at 1.3 V_s on the glide-slope angle from the obstacle height; flare on an arc at 1.23 V_s with a load factor of 1.2 (h_f = R(1 − cos γ_a), s_f = R sin γ_a; if h_f exceeds the obstacle height the flare starts there); touchdown at k_TDV_s; a free roll of t_fr seconds; then braking with μ_b, the braking-roll lift coefficient (zero with spoilers), the landing-configuration drag and an optional reverse thrust taken as a fraction of the thrust available at 0.7 V_TD: s_b = (1/(2gK_A)) ln[K_T/(K_T + K_AV_TD²)] with K_T = −T_rev/W − μ_b. The Handbook's observations are reproduced: with braking predominant the roll grows in proportion to the weight and, at the same indicated speed, as 1/σ with altitude (16 % longer at 5,000 ft). Certification factors (the 1.667 of 14 CFR 25 on the landing distance) are not applied.

Units and conventions

Speeds are true airspeeds in knots in both unit systems, with ft/s or m/s where a formula needs them; distances, heights and ceilings, forces and the fuel weight go through the unit dictionary; powers are printed in hp or kW, rates of climb in ft/min or m/s, range in nautical miles or kilometres, endurance in hours. The BSFC (lb of fuel per hp-hour) and the TSFC (per hour) are plain numbers that mean the same in both systems. The headwind is positive; a tailwind is entered negative. No factor of safety or load level enters this tool: it produces performance, not strength checks.

Limits

Parabolic polar only (no drag rise, no Mach limits), quasi-steady climb, propulsion without a Mach or temperature dependence, standard-day atmosphere, constant efficiency and specific fuel consumption with altitude, no turning performance, no ground effect, no runway slope, no engine-failure cases. For a jet transport the 0.7 V_LO thrust rule is unnecessary (the thrust is constant) and the quasi-steady climb is 5–10 % optimistic at cruise Mach numbers.