Time 0 sHot end 0 °CMid 0 °CFar end 0 °CHeat in 0 WLost by air 0 WRadiated 0 WFPS 0

Drag to orbit, wheel or pinch to zoom. Bar colour = temperature (blue = ambient, red = heater temperature; white glow above about 450 °C). Wavy lines: infrared from the heater. Rising plumes: air heated by the bar. Dots in the bar: heat conducted along it.

Heated Metal Bar: Conduction, Convection and Radiation

One end sits on a hot plate. How far does the heat travel before the air and the glow take it all?

Bar

Material

Surroundings

Far end

Time

Live readouts

Temperature along the bar

Heat ledger over time

Experiments

Each button solves the steady state for several values (headless, same solver) and tabulates it.

Independent numerical checks

Controls

Drag / one finger: orbit; wheel / pinch: zoom. Three camera buttons jump to a front, top or hot-end view.
Presets set material, heater, wind and surface together. Every slider works while the bar heats up; the model reacts at once.
Time scale = simulated seconds per real second. A steel bar needs hours to settle, so use the scale or press "Jump to steady state".
The bar is 0.5 m long with a 40 mm square section; its left end is held at the heater temperature.

The science

Conduction. Fourier's law q = −k dT/dx. The bar is a 1-D chain of 80 cells. Each cell obeys ρ c A dx dT/dt = kA (T_i−1 − 2T_i + T_i+1)/dx − P dx [h (T_i − T_∞) + ε σ (T_i⁴ − T_∞⁴)], A = d², P = 4d, d = 40 mm. Backward Euler with a tridiagonal (Thomas) solve; the temperature-dependent h and radiation coefficient are lagged one Picard iteration.

Convection. Natural convection from a horizontal bar: h_nat = 1.32 (ΔT/d)^1/4 W m⁻² K⁻¹ (laminar correlation for a horizontal cylinder, used here as a typical engineering value). Wind: h_f = 5.7 + 3.8 v (a common flat-plate rule of thumb for v up to a few m/s). Combined as h = (h_nat³ + h_f³)^1/3. Treat all of these as typical, not exact.

Radiation. Grey surface to a large cool room: q = ε σ (T⁴ − T_∞⁴), σ = 5.670×10⁻⁸ W m⁻² K⁻⁴, written as h_r (T − T_∞) with h_r = εσ (T + T_∞)(T² + T_∞²). At 600 °C radiation can exceed convection several times over.

Fin theory (check). With constant h, no radiation and an insulated tip: θ(x) = θ_b cosh(m (L − x)) / cosh(mL), m = √(hP / kA), heat in Q = √(hPkA) θ_b tanh(mL). The numeric solver is compared with this in the check table. Other checks: the erf solution for a semi-infinite solid, the lumped-capacitance decay exp(−t/τ), τ = ρcA/(hP), and a running energy ledger.

Simplifications. 1-D temperature (no variation across the section, valid while the Biot number h·(A/P)/k is below about 0.1), properties of k, ρ, c held constant (they vary with temperature in real metals), no thermal contact resistance at the hot plate, grey surfaces, still room. The finned sink is a single conductance to ambient (illustrative). The plumes, wavy lines and conduction dots are illustrations scaled by the computed quantities, not simulated fluid or photons. Material properties are typical handbook values: spot-check them before quoting. Assets built with Blender; scene by Three.js.