{
  "title": "The ideal power cost of hovering",
  "kind": "Deterministic actuator-disk teaching calculation, not measurements",
  "g_m_s2": 9.80665,
  "parameters": {
    "mass_kg": [
      1,
      5,
      20
    ],
    "area_m2_range": [
      0.1,
      10
    ],
    "area_points": 81,
    "density_kg_m3": [
      0.9,
      1.225,
      1.4
    ]
  },
  "rows": 729,
  "formulas": {
    "thrust": "T=m*g",
    "disk_speed": "vi=sqrt(T/(2*rho*A))",
    "far_wake_speed": "vw=2*vi",
    "mass_flow": "mdot=rho*A*vi",
    "ideal_power": "Pi=T*vi=T^(3/2)/sqrt(2*rho*A)"
  },
  "baseline": {
    "mass_kg": 5,
    "disk_area_m2": 1,
    "density_kg_m3": 1.225,
    "thrust_N": 49.033249999999995,
    "disk_loading_Pa": 49.033249999999995,
    "induced_disk_speed_m_s": 4.473653029524241,
    "far_wake_speed_m_s": 8.947306059048483,
    "mass_flow_kg_s": 5.480224961167196,
    "ideal_induced_power_W": 219.35774740991948
  },
  "assumptions": [
    "Steady axial hover in still air, far from ground, ceilings, walls and other rotors.",
    "Open unshrouded ideal disk, uniform loading and inflow, incompressible inviscid flow, no wake swirl.",
    "All supported mass is included and vertical thrust equals weight. Disk area is swept area, not total blade surface area.",
    "For multiple disks, summing area requires noninteracting equal disk loading; overlapping or coaxial disks cannot simply be added."
  ],
  "limitations": [
    "Ideal induced power is a lower-bound reference for this open-disk hover model, not a universal lower bound for every lift mechanism.",
    "Actual rotor shaft power also pays for blade profile drag, nonuniform inflow, swirl and other aerodynamic losses; electrical input adds motor and controller losses.",
    "No empirical efficiency is assumed or fitted. No prediction of battery endurance, motor rating or measured flight performance.",
    "No ground effect, forward flight, wind, transient manoeuvre, acoustics, blade stress or vehicle stability is modelled.",
    "Density values are independent illustrative inputs, not an altitude or weather model."
  ],
  "validation": {
    "momentum_relative_error_max": 4.440892098500626e-16,
    "wake_energy_relative_error_max": 4.440892098500626e-16,
    "mass_power_exponent_1_5": true,
    "quadruple_area_halves_power": true,
    "quadruple_density_halves_power": true,
    "equal_loaded_noninteracting_disk_sum": true,
    "nonpositive_inputs_rejected": true
  },
  "sources": [
    {
      "title": "NASA Glenn Propeller Analysis, momentum and disk velocity",
      "url": "https://www.grc.nasa.gov/WWW/K-12/BGP/propanl.html"
    },
    {
      "title": "NASA Glenn Propeller Thrust, pressure jump and ideal disk assumptions",
      "url": "https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/propeller-thrust/"
    },
    {
      "title": "NASA NDARC Theory Appendix1, induced velocity and power models",
      "url": "https://ntrs.nasa.gov/api/citations/20160012474/downloads/20160012474.pdf"
    }
  ]
}