Hold a mug of hot cocoa and three things happen at once. The handle slowly warms in your fingers — that is conduction, heat creeping through solid material. Steam curls upward and carries warmth away — that is convection, heat riding along a moving fluid. And your face feels the glow even without touching the mug — that is radiation, heat crossing open space as invisible light. Every heating or cooling problem in nature is some blend of these three.
The easiest one to pin down is conduction. Heat flows faster when the two sides are at very different temperatures, when the path is wide, and when the material is happy to pass heat along; it flows slower when the material is thick. Put that sentence into symbols and you get Fourier's Law:
Here $\dot{Q}$ is the heat flowing per second (in watts), $\Delta T$ is the temperature gap, $A$ the area, $L$ the thickness, and $k$ the material's thermal conductivity. Try a brick wall with $k = 0.9$, $A = 1\ \text{m}^2$, $\Delta T = 20\ \text{K}$, and $L = 0.2\ \text{m}$: that gives $\dot{Q} = 0.9 \times 1 \times 20 / 0.2 = 90\ \text{W}$ leaking outward — roughly one light bulb's worth of heat, lost continuously.
It is often cleaner to think in thermal resistance $R_{\text{th}} = L/(kA)$, so that $\dot{Q} = \Delta T / R_{\text{th}}$ — exactly like Ohm's law, with temperature playing the role of voltage and heat flow the role of current. Wall layers add their resistance in series, $R_{\text{total}} = \sum_i R_i$, which is why a thin sheet of foam ($k \approx 0.04$) can out-insulate a thick brick. Convection follows $\dot{Q} = hA\,\Delta T$, and radiation obeys the steep fourth-power law $P = \varepsilon\sigma A T^4$ — double an object's absolute temperature and it radiates $2^4 = 16$ times as hard. The sliders map straight onto these: $L$, $A$, $k$, and $\Delta T$ drive conduction, $h$ drives convection, and $\varepsilon$ drives radiation.
Try this in the sim above. In Conduction, slide $k$ down from a copper-like 400 toward 0.04 and watch $\dot{Q}$ collapse while $R_{\text{th}}$ shoots up. Push the thickness $L$ higher and see the heat flow roughly halve as $L$ doubles. Then switch to Radiation and raise $\Delta T$: notice the heat output climb far faster than the temperature itself — the unmistakable signature of that $T^4$ law.
| Symbol | Quantity | SI Unit |
|---|---|---|
| $k$ | Thermal conductivity | W m⁻¹ K⁻¹ |
| $h$ | Convection coefficient | W m⁻² K⁻¹ |
| $\varepsilon$ | Emissivity (0=mirror, 1=blackbody) | dimensionless |
| $R_{\text{th}}$ | Thermal resistance | K W⁻¹ |
| $\sigma$ | Stefan-Boltzmann constant = 5.67×10⁻⁸ | W m⁻² K⁻⁴ |