Baseline (every case starts here). A lumped body of heat capacity C exchanges heat with its surroundings through a conductance UA: C dT/dt = −UA (T − Tₐ), so T(t) = Tₐ + (T₀ − Tₐ) e^(−UA·t/C) and the time constant is τ = C/UA.

What this lab adds. C dT/dt = P − UA·(|ΔT|/|ΔT₀|)ⁿ·ΔT − ε σ A (T⁴ − Tₐ⁴) with temperatures in kelvin inside the radiation term, σ = 5.670×10⁻⁸ W m⁻² K⁻⁴. The exponent n lets the convective part grow with ΔT (n ≈ 0.25 for laminar natural convection; n = 0 is Newton). The heater P moves the equilibrium to Tₐ + P/UA in the Newton limit.

Honest simplifications. One uniform temperature (no internal gradients), constant ambient, grey-body surface radiating to surroundings at Tₐ. The conductance, area and emissivity sliders are illustrative values, not measured data. Cases defined only by a rate constant k are given a nominal C = 1000 J/K so that UA = 1000·k; only the ratio UA/C is physical there.

Independent checks. The page compares its RK4 integration with the closed form in the Newton limit, with a 8× smaller step for your current settings, and verifies that the net power vanishes at the computed equilibrium. A separate Python test repeats these with its own integrator.

Lumped Cooling Case Studies