Propeller and Jet Thrust: Method

Propeller and Jet Thrust

What the tool computes

Three classical models of thrust production, chosen by the Model select. The momentum theory treats the propeller as a disc that adds momentum to a streamtube and gives the ideal (upper-bound) relation between thrust, power and airspeed. The blade-element momentum theory resolves the blade into sections whose lift and drag, with the induced velocities of the annulus each section sweeps, give the thrust, torque, power and efficiency of a real propeller geometry and their variation with the advance ratio. The gas-turbine cycle follows the air through a turbojet, a separate-exhaust turbofan or a ramjet station by station and gives the fuel-air ratio, the specific thrust, the thrust specific fuel consumption and the thermal, propulsive and overall efficiencies. The air is the 1976 U.S. Standard Atmosphere at the entered altitude. The library behind the page is checked against closed forms, an independent numpy/scipy implementation and the measured curves of NACA Report 640 (test-scripts/test_propeller_jet_thrust.php).

Momentum theory (Rankine, Froude)

An actuator disc of area A = πD²/4 moving at V through air of density ρ accelerates the air that passes it. Far upstream the streamtube carries V; at the disc V + w; far downstream V + 2w, the induced velocity doubling behind the disc because the pressure returns to ambient while the disc has done its work. Mass flow ṁ = ρA(V + w), thrust T = ṁ·2w = 2ρA(V + w)w (the momentum given to the stream, equal to the pressure jump Δp·A across the disc) and ideal power P = T(V + w) (the kinetic energy added per second). The ideal efficiency is therefore η_i = TV/P = V/(V + w), which with the thrust loading T/(qA), q = ½ρV², is η_i = 2/(1 + √(1 + T/(qA))): a lightly loaded disc is efficient, a heavily loaded one is not, and no propeller beats this bound. NACA Report 640 writes the same relation in coefficient form, η_i = 2/(1 + V_s/V) with V_s/V = √(1 + 8C_T/(πJ²)). Given a shaft power the tool solves the cubic 2ρA(V + w)²w = P for w (the thrust an ideal propeller would give), given a thrust it takes w = ½[−V + √(V² + 2T/(ρA))]. At rest the induced velocity is w₀ = √(T/(2ρA)) and the ideal power P₀ = T^3/2/√(2ρA); the ratio P₀/P_shaft is the figure of merit of a static propeller or rotor. The assumed actual efficiency η_p places a real propeller beside the ideal one: T = η_pP_s/V, or P_s = TV/η_p, and η_p/η_i is the fraction of the ideal efficiency that the blades realise.

Blade-element momentum theory

The blade is divided into elements of chord c and blade angle β at radius r. Each element sees the airspeed V plus the axial induced velocity w_a and the rotational speed Ωr minus the swirl w_t (half the slipstream rotation, by the same argument that halves the axial induction at the disc), so the resultant W makes the inflow angle φ = atan[(V + w_a)/(Ωr − w_t)] with the plane of rotation and the section's angle of attack is α = β − φ. Its lift and drag, per unit radius for B blades, are ½ρW²Bc·C_l and ½ρW²Bc·C_d; resolved into the thrust and torque directions, dT/dr = ½ρW²Bc(C_lcos φ − C_dsin φ) and dQ/dr = ½ρW²Bc(C_lsin φ + C_dcos φ)r. The momentum balance of the annulus 2πr dr, with Prandtl's factor F = (2/π)acos[exp(−B(R − r)/(2r sin φ))] (and the same form at the hub) that accounts for the finite number of blades through the tip vortices, gives dT = 4πρrF(V + w_a)w_a dr and dQ = 4πρr²F(V + w_a)w_t dr. Equating the two descriptions, with the local solidity σ′ = Bc/(2πr), gives κ = σ′C_y/(4F sin²φ) = w_a/(V + w_a) and κ′ = σ′C_x/(4F sin φ cos φ) = w_t/(Ωr − w_t), C_y = C_lcos φ − C_dsin φ, C_x = C_lsin φ + C_dcos φ, which with the definition of φ reduce to one equation in φ, V cos φ = (1 − κ)(1 − a′)Ωr sin φ, a′ = κ′/(1 + κ′). Written in velocities rather than in Glauert's induction factors a = w_a/V and a′ = w_t/(Ωr), the equation stays regular at V = 0 (κ = 1 there), so the static case is solved like any other; the root is bracketed between 0 and 90° and found by bisection (the scheme of Ning 2014 for wind turbines, in the propeller sign convention). Forty elements from the hub radius to 0.995R are integrated by the midpoint rule. Thrust, torque and power become the coefficients C_T = T/(ρn²D⁴), C_P = P/(ρn³D⁵), C_Q = C_P/(2π) and the efficiency η = TV/P = JC_T/C_P, J = V/(nD); the map against J at fixed rpm sweeps the airspeed to the zero-thrust point.

The NACA 5868-9 blade and the section polar

NACA Report 640 (Hartman and Biermann 1938) tested seven full-scale 10 ft propellers in the 20-foot tunnel. Its Figure 2 gives the blade-form curves of the Navy 5868-9 blade of Clark Y section: chord b/D, thickness h/b and the geometric pitch p/D at the 15°, 25° and 35° settings. The tool carries those curves digitized at 0.05R (chord from 0.038D at 0.2R to a maximum of 0.076D at 0.5R and 0.040D at 0.95R; thickness from 0.27 at 0.3R to 0.09 at the tip; the twist β(r) − β(0.75R) from +17.1° at 0.3R to −2.9° at 0.95R, taken from the p/D curves through β = atan(p/(2πr)) and consistent between the three settings within 0.1°). Setting the blade angle at 0.75R shifts the whole distribution. NACA measured blade angles from the flat lower surface of the Clark Y, so the zero-lift angle of the stated polar is referred to that face, about −5°; with the lift slope 0.10 per degree, a drag polar C_d = 0.008 + 0.10(h/b)² + 0.010(C_l − 0.35)² and C_l,max = 1.35, the theory reproduces the report's zero-thrust advance ratios (0.82, 1.04, 1.30 and 1.55 at 15°, 20°, 25° and 30°) within 1.2 % and its thrust coefficients within a few percent in the working range; its efficiency sits three to eight percent above the measured propulsive efficiency, which includes the drag the slipstream adds to the engine nacelle. These polar constants are illustrative and editable; a measured section polar at the blade Reynolds number is better. Past C_l,max the lift decays toward the flat-plate value 0.9 sin 2α and the drag rises toward 1.8 sin²α, a rough model that flags the stalled elements and the low-J, high-pitch cases where the comparison with the report shows the theory's limits. The report's curves for the three-blade propeller at 15° to 35° (Figures 9, 11 and 12), read from the scanned figures, are overlaid on the charts whenever that blade is run with three blades at 10 ft and one of those angles.

Gas-turbine cycle

The stations follow the NASA Glenn Beginner's Guide to Propulsion: 0 free stream, 2 compressor face, 3 compressor exit, 4 burner exit, 5 turbine exit, 9 nozzle exit, 13 and 19 the fan stream of a turbofan. Total temperature is constant through the inlet, the total pressure is recovered by π_d (times the Mil-Spec factor 1 − 0.075(M − 1)^1.35 above Mach 1 when asked). The compressor raises the total pressure by PR_c; the isentropic exit temperature follows from the entropy function φ(T) = ∫c_pdT/T through p_t3/p_t2 = exp[(φ_3i − φ₂)/R] and the actual enthalpy rise is the isentropic one divided by the adiabatic efficiency η_c (the same for the fan). The burner holds π_b of the pressure and the energy balance (1 + f)h_t4 = h_t3 + fη_bh_PR, with the enthalpies measured from 298.15 K, gives the fuel-air ratio f. The turbine supplies the compressor and fan work, (1 + f)(h_t4 − h_t5) = (h_t3 − h_t2) + α(h_t13 − h_t2); its isentropic drop is the actual one divided by η_t and its pressure ratio follows from φ. The nozzle keeps the total temperature and π_n of the pressure: a convergent nozzle chokes when p_t9/p₀ exceeds the critical ratio (1.893 for γ = 1.4, computed from the gas model), leaving the exit at M = 1 and p₉ > p₀, the pressure thrust (p₉ − p₀)A₉ being carried as an addition to the effective exhaust velocity; a convergent-divergent nozzle expands to p₀. The kinetic efficiency η_n scales V₉² as on the Glenn nozzle page. Thrust per unit of inlet air is then F/ṁ₀ = [(1 + f)V_9e − V₀ + α(V_19e − V₀)]/(1 + α), the TSFC is ṁ_f/F, the thermal efficiency the kinetic-energy rise of the streams over the fuel energy, the propulsive efficiency FV₀ over that rise, and their product the overall efficiency FV₀/(ṁ_fh_PR). A ramjet is the same cycle with no compressor or turbine (PR_c = 1, h_t5 = h_t4).

Gas properties

The variable-c_p model uses the NASA Glenn polynomials of McBride, Zehe and Gordon (2002) for air (the "Air" record of their thermo.inp, N₂ 78.084 %, O₂ 20.9476 %, Ar 0.9365 %, CO₂ 0.0319 % by mole), c_p/R = a₁T⁻² + a₂T⁻¹ + a₃ + a₄T + a₅T² + a₆T³ + a₇T⁴ with the integrated enthalpy and entropy, in the 200–1000 K and 1000–6000 K intervals; the combustion products are treated as air, the usual first approximation at fuel-air ratios of a few percent (Mattingly's "level 2" analysis). The constant-c_p option is the textbook cold-air cycle with γ = 1.4, for which the ideal turbojet (every efficiency 1, nozzle expanded) has the closed forms of the parametric cycle analysis: τ_r = 1 + (γ − 1)M₀²/2, τ_c = PR_c^(γ−1)/γ, T₉ = T_t4/(τ_rτ_c), V₉/a₀ = √[(2/(γ − 1))(T₉/T₀)(τ_rτ_cτ_t − 1)]; the library reproduces them to machine precision.

Units and conventions

Lengths, speeds, forces and powers go through the site's unit dictionary; the rpm, blade count, angles, coefficients and efficiencies are plain numbers. The turbine inlet temperature is entered in kelvin in SI mode and degrees Rankine in USCS mode, the heating value in MJ/kg or Btu/lb, the air flow in kg/s or lb/s; the specific thrust is given in N per kg/s or lbf per lb/s and the TSFC in mg/(N·s) or lb/(lbf·h), with the other unit beside it. The propeller's rpm is the shaft speed; n = rpm/60 in the coefficients. The static case (V = 0) is allowed in both propeller models.

Limits

Momentum theory ignores swirl, profile drag and tip loss and is a bound, not a prediction. The blade-element theory is a steady, incompressible, two-dimensional-section model: no compressibility correction (the tip Mach number is printed; above about 0.85 the measured thrust falls below the theory, see NACA Report 639), no three-dimensional stall delay, no nacelle or body interference, no Reynolds-number variation of the polar; results at high pitch and low advance ratio, where the inner elements stall, are indicative only. The cycle analysis is parametric (design-point) analysis with constant component efficiencies and no cooling air, bleed, power extraction or mixing; the products are treated as air; the turbofan's streams exhaust separately.