Propeller Thrust Calculator and Paramotor Takeoff

For an open actuator disk in still air, $A=\pi D^2/4$, $T=2\rho A v_i^2$ and $P_{ideal}=Tv_i$. The static figure of merit is $FM=P_{ideal}/P_{shaft}$, giving

NASA TM-20210021872, attached page 17 (printed p. 11), equations (1)–(3), supplies the axial momentum equations and the static definition of figure of merit. This is the classical Rankine–Froude/Glauert actuator-disk approach. The lossless ceiling uses FM = 1. Disk pressure rise is T/A; mass flow is ρAvᵢ; the ideal static far wake is 2vᵢ and has area A/2.

For the estimated ground run, the tool holds the effective air-power fraction equal to the entered static FM as an explicit low-speed approximation. It solves $FM P_{shaft}=T(V+v)$ and $T=2\rho A v(V+v)$ by 42 bisection iterations. This continuation is a modelling assumption, not a measured forward-flight figure of merit.

At each station $V=V_g+V_{wind}$, drag is $\tfrac12\rho(C_D A)V^2$. Assumed inflated-wing lift is $L=W(V/V_{to})^2$. A trike has ground resistance $\mu(W-L)$; foot launch omits it. NASA SP-367, attached page 152 (printed p. 144), explains the ground-roll force balance; the chosen lift buildup and drag area are this tool's simplifications. Then $F=T-D-\mu(W-L)$ and $a=F/m$.

The force chart uses 161 speed stations, independently of the integration grid. Bounded minimisation checks the continuous force curve, including both endpoints; integration starts with a split at an interior minimum when present.

Distance and time use adaptive Simpson quadrature on normalised ground speed. Each panel is compared with its two half-panels; the estimated error is the absolute difference divided by 15. Both dimensionless integrals must meet a local relative tolerance of 10⁻⁶ plus an absolute allowance of 10⁻⁹ times the panel width. Accepted panels use the Richardson-corrected estimate. Refinement is limited to 26 levels and 2048 processed panels. A numerical conditioning guard rejects a positive minimum force smaller than 10⁻⁸ times the larger of static thrust and total weight. Nonpositive force, failed convergence or an unresolved force margin leaves both run and time blank, with an explanatory message. These limits control numerical accuracy, not uncertainty in the physical inputs.

UIUC checks use the measured APC Slow Flyer 10×4.7 coefficients at 4029 and 5960 RPM. Independent expected thrust is $C_T\rho n^2D^4$ and input shaft power is $C_P\rho n^3D^5$, with n in revolutions per second. FM = 0.690801 is derived only from the separate 4880 RPM row, $FM=\sqrt{2/\pi}\,C_T^{3/2}/C_P$, and is held fixed for both tests. Density 1.225 kg/m³ is a dimensional reconstruction convention, not the reported tunnel density; these tests compare dimensionless performance. Their 2% thrust tolerance allows the measured change in figure of merit with RPM.

For a measured static thrust and shaft-power pair, calculate $FM=T^{3/2}/(P_{shaft}\sqrt{2\rho A})$ using coherent SI units. Additional checks reconstruct induced velocity from the momentum equation to verify axial power balance and confirm that the axial solver recovers the static result at zero airspeed. An inverse-power reference reuses the 4029 RPM measurement with the same separately calibrated FM; it is not an additional experiment.

Numerical regressions include the independent reviewer's near-limit example (15 m/s, 3.897 m² drag area: 225.2405 m and 17.43317 s) and a case with a small positive force minimum between chart stations. Their expected values were independently computed with SciPy QUADPACK and Brent root finding. They verify the integration of these equations, not measured takeoff performance.

• One isolated, unshrouded propeller in uniform, incompressible air, away from walls and ground-effect augmentation.
• Shaft power is mechanical power at the propeller in the entered atmospheric conditions. Power is constant during the modelled run; no automatic engine derating is applied.
• FM includes static aerodynamic and installation losses only to the extent represented by the user's measurement or estimate. The default FM, drag area and rolling coefficient are illustrative, not catalogue data.
• Total mass is constant, the ground is level, thrust is horizontal, and headwind is steady and aligned with travel.
• The wing is already inflated and its effective lift coefficient stays constant through the run. Lift reaches weight at the user-supplied true airspeed. Effective drag area is held constant.
• Foot launch has no wheel resistance and no force from the runner. Pilot ability, canopy inflation, line dynamics and the transition into flight are outside the model.
• Standard gravity is 9.80665 m/s². Static thrust-to-weight uses total weight, not mass.

Use this tool for preliminary estimates and comparisons. It cannot establish a safe launch distance, obstacle clearance or flight capability. The default run is an inflated-wing acceleration segment and can be much shorter than a real launch. Use the aircraft and wing manufacturer's information and flight-test data for operational decisions.

The static actuator-disk result does not resolve pitch, RPM, tip Mach number, stall, blade stress, swirl, cage blockage or motor–propeller matching. Without a measured figure of merit, no universal accuracy percentage is justified. For example, FM = 0.65 gives only about 75% of the lossless static thrust at the same shaft power; using FM = 1 would overpredict that result by about one third.

Holding the static loss factor constant during axial flight is a low-speed approximation. A warning is shown when lift-off airspeed exceeds static induced velocity, an author-selected screening limit rather than a universal aerodynamic boundary. Real fixed-pitch thrust curves can differ substantially. No takeoff measurement has been used to validate the run model. The UIUC cases test static thrust for one small propeller; they do not validate paramotor launch predictions or a universal FM.

Only nonnegative headwind below lift-off airspeed is supported. Near zero net accelerating force the run becomes highly sensitive to input errors and numerical resolution; use a detailed model and measured thrust data. The minimum of the continuous force curve is checked between the chart stations. Run and time are displayed only after both adaptive integration error tests pass and the minimum force clears the numerical conditioning guard; otherwise they remain blank. A high far-wake-speed warning is a screening check only; low wake speed does not guarantee low blade-tip Mach number.