Rotor Blade Icing, Power Penalty and Electrothermal Anti-Icing

At 241 radial stations, momentum theory supplies hover/axial inflow and $U=\sqrt{(\Omega r)^2+V_n^2}$. The stagnation collection efficiency uses Langmuir–Blodgett’s modified inertia parameter and the droplet-range fit in NASA CR-2005-213852, PDF pp. 6–7, equations (3)–(7). An equivalent cylinder based on projected airfoil thickness supplies the overall collection factor used by the drag fits; this geometry approximation is separate from stagnation collection.

The local Messinger balance is $\dot m_c=\dot m_i+\dot m_e+\dot m_r$. At 0 °C, convection, evaporative cooling, supercooled-water sensible heating and droplet kinetic energy determine the freezing rate. If all remaining water freezes, 26 bisection iterations solve the subfreezing surface temperature; if no water freezes, they solve the wet equilibrium temperature. The dry branch uses ice saturation pressure. Ice thickness is $\delta=\dot m_i t/\rho_i$. Stagnation heat transfer follows $Nu=1.14Pr^{0.4}Re^{0.5}$ (NASA report, equation 16), with the entered multiplier. Heat/mass transfer uses the Chilton–Colburn analogy and Buck (1996) saturation pressures.

Flemming rime drag is equation (14) as reproduced in NASA CR-175092, PDF p. 40; forward-flight correction is unity for the axial-flow model. Actual roughness is limited to the smaller of the entered maximum and local ice thickness; negative empirical increments are rejected at onset. The Gray option implements equation (2) in Gray (1964), NASA TN D-2166, PDF p. 9, with equal icing and evaluation incidences. Its nose-radius parameter is 100 times the radius/chord ratio (percent chord). Both water-content radicals are retained: the prefactor uses √(LWC·βₘ), and the phase uses √LWC. The base sine phase is set to zero outside 0–180°. Historical mph, chord in inches, minutes, g/m³ and °F arguments are explicitly converted. The Flemming glaze coefficient functions are not defined in the accessible reproduction; that branch is not claimed here. Blade elements give $dP=\tfrac12 B\rho c\Delta C_d U(\Omega r)^2dr$, integrated by trapezoids.

Running-wet heat includes convection relative to recovery temperature, droplet sensible/kinetic energy, mass-transfer-limited evaporation, radiation and backing losses. The evaporative solution uses 24 bisection iterations for the lowest temperature at or above the wet setpoint that evaporates the full incoming flux, then applies the corresponding energy balance. Neither anti-icing mode adds a fusion load: incoming water remains liquid or becomes vapour. Electrical power is the surface duty integrated over the entered wrap width and all blades, divided by efficiency.

Steady, spatially uniform cloud; monodisperse spherical droplets represented by MVD; straight blades with linearly varying chord; equal blade conditions. Cloud humidity is referenced to supercooled liquid water. Dry-air properties use ideal-gas density, Sutherland viscosity and a temperature-dependent conductivity fit; $c_{p,a}=1006$, $c_{p,w}=4180$, $c_{p,i}=2100$ J/(kg·K), $L_f=333700$ and $L_v(0)=2501000$ J/kg. Radiation emissivity is 0.85. Accretion neglects conduction into the blade and radiation, as in the stagnation reference model; heater sizing includes entered backing losses and radiation.

Ice fluxes and thickness are stagnation-line quantities, not spanwise ice mass or horn geometry. Both heat and droplet flux at stagnation are conservatively applied over the entire heater strip; enlarging the strip therefore increases duty linearly. The rotor holds thrust and speed; icing changes profile drag only. Clean lift and inflow are retained, so induced-power changes from lift degradation are excluded. The power baseline is shaft power, excluding propulsion electrical losses.

For preliminary studies of classical supercooled liquid-cloud icing, MVD 5–50 µm, ambient −40 to −0.1 °C, pressure 60–105 kPa, hover or axial climb, and tip Mach number ≤0.55. Incompressible heat/mass transfer and an equivalent-cylinder collection approximation are used. Applicability to small drone airfoils and low Reynolds numbers requires separate experimental validation; the numerical input bounds are not a validation envelope. The classic correlations can have large errors outside their originating airfoil and tunnel datasets.

Flemming is restricted to dry rime with negligible liquid runback. Evaporation can make its incoming-water freezing fraction slightly below unity even in rime. Gray additionally requires subfreezing total temperature and a positive radius/incidence factor. The original fit can predict negative drag changes for thin, sharp-nosed airfoils; this power-penalty tool flags those cases instead of treating them as a benefit. It is a separate published empirical fit, not Flemming’s glaze model. At very small ice thicknesses the roughness cap is an onset extrapolation. Neither correlation models stall, lift loss, tip-loss changes, rough ice feathers, asymmetric loads, ice shedding or rotor control margins; the estimated profile penalty may substantially understate the total degradation.

Fixed clean geometry is retained throughout an exposure. A warning checks the entire plotted cloud exposure, including times later than the current animated readout, and appears when ice exceeds 5% of local chord anywhere on the blade. Shorten the cloud exposure duration to keep the complete timeline within this fixed-geometry range. The default 90 s timeline stays below the limit. Runback is a mass outflow at the station, not a downstream re-freezing prediction. No radial or chordwise film transport, evolving horn shapes, SLD splash/rebound, compressible icing CFD, cyclic de-icing or transient heater/skin conduction is solved. Running-wet clears the modeled heated area only; downstream icing remains possible. The evaporative search is limited to 80 °C.

Spanwise calculations and time curves use 241 points each; readouts use shape-preserving cubic interpolation. No coarse-grid picture interpolation or reduced-physics runtime shortcut is used. The exposure animation reuses the solved cloud/thermal state. NASA reference cases test collection, freezing fraction and measured stagnation thickness, not complete rotor torque or an installed heater.