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Rod and plate
Choose the material, the length, the plate temperature and what happens at the far end.
Air and surface
Convection (h) and radiation (emissivity) take heat sideways out of the rod and the block.
Cooling block
A separate lumped block of the same material cools in the room: Newton law against Stefan-Boltzmann.
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The science
Conduction (Fourier's law)
Heat flows from hot to cold with flux q = -k dT/dx. In a rod this gives the heat equation dT/dt = alpha d2T/dx2 - (h P / rho c A)(T - Ta) - (eps sigma P / rho c A)(T^4 - Ta^4), with thermal diffusivity alpha = k / (rho c), perimeter P, cross-section A. It is solved with explicit finite differences (101 nodes, time step at 40 % of the stability limit alpha dt / dx^2 < 1/2). The hot plate is a fixed-temperature end; the far end is either insulated or a heat sink at room temperature.
Exact solution used as a check
With no side losses and a long rod, a sudden plate temperature gives T(x,t) = Ta + (Tp - Ta) erfc(x / (2 sqrt(alpha t))). The dashed yellow curve in the first graph is this formula, and a model check compares it with the finite-difference numbers. The heat front depth grows like 2 sqrt(alpha t), not linearly: doubling the rod needs four times the time.
Convection and Newton cooling
Air carries heat from a surface at the rate h A (T - Ta) (Newton's law of cooling). A block that is uniform in temperature (lumped body, valid when the Biot number Bi = h Lc / k is below 0.1) cools as T = Ta + (T0 - Ta) e^(-t/tau) with tau = m c / (h A). The coefficient h depends on air speed and geometry; here it is a slider (natural convection is roughly 5-10, a fan 30-100 W/m2K).
Radiation (Stefan-Boltzmann)
A surface at absolute temperature T emits eps sigma T^4 with sigma = 5.67e-8 W/m2K4; with the room absorbing, the net loss is eps sigma A (T^4 - Ta^4). Because of the fourth power, radiation dominates for hot objects and almost vanishes near room temperature. A shiny surface (eps near 0.05) holds its heat much longer than a black one. With ambient at 0 K the cooling law has the closed form T = (T0^-3 + 3 k t)^(-1/3), used as a model check.
Thermal equilibrium and steady state
The dotted grey curve is the steady state of a fin with losses, T - Ta = (Tp - Ta) cosh(m(L - x)) / cosh(mL) with m = sqrt(h_eff P / (k A)); radiation is linearised as h_r = 4 eps sigma Tm^3. The rod approaches it after a few diffusion times L^2 / alpha. At equilibrium the plate power equals the power lost sideways (and into the sink).
Simplifications
One-dimensional rod (no temperature variation across the 2 cm diameter), constant material properties, uniform h and emissivity, no heat exchange between rod and block, a perfectly thermostatted plate. Real h is not a constant; real air also moves the heat in plumes.
What to try
- Copper rod, pure conduction: the yellow erfc curve lies exactly on the red rod profile. Then switch to glass: the same plate heats it a thousand times more slowly.
- Check the diffusion time in the readouts: double the rod length and watch the time to heat the far end grow fourfold.
- Windy day: raise h and see the profile drop towards the steady curve much sooner; the convection bar in the power graph grows.
- Bright vs matte: set emissivity to 0.05 then 0.95 and compare the yellow radiation-only curve of the hot block.
- Heat sink at the far end: the profile becomes a straight line and the plate power settles at k A (Tp - Ta) / L.