Christmas Pyramid: Candle Heat, Chimney Flow and Fan Torque Lab

Presets

Live readouts

Graphs

Rotor speed vs time (green = theoretical steady speed)

Drive torque and friction vs speed (dot = now)

Steady rpm vs number of candles

Stall torque vs blade angle (dashed = analytic best)

Numerical checks

running...

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. Space pauses, R resets.

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 Q_c = n q f_c, the rest is radiation.

Q_c = n q f_c, wax rate = n q / 43 MJ/kg

2. Chimney flow. The warm column of height H has a buoyancy pressure Δp_b = ρ g H ΔT / T₀. The temperature rise follows from energy conservation, ΔT = Q_c / (ρ w A c_p), where A = πR² is the fan disc and w the air speed. The fan costs a pressure drop ½ ρ w² K, with K = K_b φ tan²β (K_b = 2 is an assumed blockage constant: steeper blades block more of the flow). Balancing the two gives a closed form:

w = [ 2 g H Q_c / ( ρ c_p T₀ A (1 + K) ) ]^1/3

Doubling the candles raises w only by 2^1/3 = 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 τ_stall = ηφρA w² r tanβ. Because w itself falls with K, τ_stall peaks where tan²β = 3 / (K_b φ), about 56 degrees for φ = 0.7.

4. The rotor. The carousel has moment of inertia I = (tiers) m_f r_f² and obeys

I dω/dt = τ − τ₀ tanh(ω/0.05) − b ω − c_d ω|ω|

With c_d = 0 the steady speed is ω_ss = (τ_stall − τ₀) / (ηφρAwr² + b) and the spin-up time constant is I divided by that denominator. If τ_stall < 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 c_p ΔT = Q_c; (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 K_b, 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.