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How to use
Pick a preset or move any slider. The 3D or 2D pyramid spins up from rest; the green line on the speed graph is the theoretical steady speed. Try: (1) go from 4 to 12 candles and see the rpm only rise by a third; (2) sweep the blade angle and watch the stall-torque graph peak near 56 degrees; (3) raise dry friction until the carousel refuses to start; (4) make the figures heavy and compare spin-up time. Drag to orbit, wheel to zoom. Space pauses, R resets.
The base plate, posts, tier discs, carved figures, candles with holders, fan blades and star were modelled and baked in Blender; Three.js r160 animates flames, the rotor and streamlines from the same model as the 2D lab.
The science
What turns a Christmas pyramid? Candle flames heat the air under the fan. Hot air is lighter, rises through the chimney of height H, and pushes on slanted fan blades. SI units throughout.
1. Heat that reaches the air. A candle burns wax (heat of combustion about 43 MJ/kg) at roughly q = 30 to 50 W; about 70 percent leaves as convective heat Qc = n q fc, the rest is radiation.
Qc = n q fc, wax rate = n q / 43 MJ/kg
2. Chimney flow. The warm column of height H has a buoyancy pressure Δpb = ρ g H ΔT / T0. The temperature rise follows from energy conservation, ΔT = Qc / (ρ w A cp), where A = πR² is the fan disc and w the air speed. The fan costs a pressure drop ½ ρ w² K, with K = Kb φ tan²β (Kb = 2 is an assumed blockage constant: steeper blades block more of the flow). Balancing the two gives a closed form:
w = [ 2 g H Qc / ( ρ cp T0 A (1 + K) ) ]1/3
Doubling the candles raises w only by 21/3 = 1.26. That cube root is why a pyramid with twelve candles is not three times as fast as one with four.
3. Torque on the blades. A blade at angle β to the vertical turns the vertical flow so that, relative to the blade, the air leaves along the blade. The air then has tangential speed w tanβ − ωr (r = 0.7 R, the effective radius). Momentum flow φρAw through the blades gives
τ = η φ ρ A w r ( w tanβ − ω r )
At ω = 0 this is the stall torque τstall = ηφρA w² r tanβ. Because w itself falls with K, τstall peaks where tan²β = 3 / (Kb φ), about 56 degrees for φ = 0.7.
4. The rotor. The carousel has moment of inertia I = (tiers) mf rf² and obeys
I dω/dt = τ − τ0 tanh(ω/0.05) − b ω − cd ω|ω|
With cd = 0 the steady speed is ωss = (τstall − τ0) / (ηφρAwr² + b) and the spin-up time constant is I divided by that denominator. If τstall < τ0 the carousel never starts.
Checks. (1) RK4 integration to steady state equals the closed form; (2) at τ0 = 0 the speed at t = I/k is 1 − 1/e of the final value; (3) energy balance ρ w A cp ΔT = Qc; (4) momentum balance of the chimney; (5) a brute-force scan of blade angle finds the analytic best angle; (6) mechanical efficiency never exceeds the Carnot limit.
Simplifications. One-dimensional plume, uniform velocity, air properties fixed at 20 C, no room draughts, no flame radiation onto the fan, constant blockage coefficient Kb, ideal turning of the flow by the vanes. Real pyramids also lose torque to blade stalling and to the wake of the candles. Absolute rpm should be read as a typical value, not a prediction for one particular pyramid. Heat to work is only about a thousandth of a percent: nearly all the energy leaves as warm air.