The Three Modes of Heat Transfer

Thermodynamics predicts the end state: the cup of tea will reach room temperature. It says nothing about whether that takes one minute or one hour. Heat transfer is the study of rate, and rate is what every piece of equipment is actually sized by.

A radiator, a heat sink, a kettle element and a heat exchanger are all specified in watts. None of them can be designed from an equilibrium argument.

Conduction is energy transfer by molecular interaction with no bulk motion of the material. Hot atoms vibrate more vigorously and jostle their neighbours; in metals, free electrons carry most of the energy — which is exactly why good electrical conductors are good thermal conductors.

Fourier's law: Q = −k·A·(dT/dx). Rate is proportional to the temperature *gradient*, not to the temperature difference alone — a thick wall with the same ΔT conducts less. The minus sign encodes the second law: heat flows down the gradient, never up it unaided.

The range spans a factor of 15 000. Note also that most insulation works by *trapping air* — the fibres are there to stop the air convecting, not because they insulate well themselves.

Convection is conduction into a fluid layer, followed by that fluid moving away and taking the energy with it. It is therefore never purely a heat transfer problem — it is a fluid mechanics problem with a temperature attached, which is why the boundary layer reappears here.

Newton's law of cooling: Q = h·A·(T_s − T_∞). The equation is trivial and all the difficulty is inside h.

h is not a material property. It depends on velocity, geometry, orientation and fluid properties, and finding it is what the dimensionless-number correlations of the convection module are for. This is also why a fan cools you: it raises h without changing the air at all.

• Free (natural) convection — the fluid moves because heating changed its density. Slow, and the reason a radiator warms a whole room.
• Forced convection — a fan or pump drives the flow. Much larger h.
• Boiling and condensation — a phase change absorbs or releases latent heat at nearly constant temperature, giving enormous h. This is why steam is used to move heat around a power station.

Radiation is electromagnetic energy emitted by every surface above absolute zero. It needs no medium — it crosses a vacuum, which is how the sun heats the earth.

Stefan-Boltzmann: Q = ε·σ·A·(T⁴ − T_surr⁴), with σ = 5.67 × 10⁻⁸ W/m²K⁴ and emissivity ε from 0 (perfect reflector) to 1 (black body).

The fourth power is what makes radiation behave unlike the other two. Doubling the absolute temperature multiplies the radiated power by sixteen, so radiation is a modest contributor at room temperature and utterly dominant in a furnace. And because it is absolute temperature, kelvin is not optional — using celsius here is a silent, catastrophic error.

Emissivity is a surface property, not a bulk one, and it can differ enormously from the visual appearance: polished aluminium has ε ≈ 0.05 while white paint has ε ≈ 0.9 in the infrared, despite both looking bright. This is why a vacuum flask is silvered and why spacecraft thermal control is entirely a question of surface finish.

A hot bar standing in a room conducts along its length, convects from its surface into the air, and radiates to the walls — simultaneously, and in different directions. Its temperature profile is the result of the competition.

This is why fin analysis has two terms in one differential equation: a fin conducts along itself while convecting from its surface, and fin efficiency is essentially the ratio between what the two mechanisms can carry. It is also why a vacuum flask attacks all three separately — vacuum kills conduction and convection, silvering kills radiation, and a narrow neck limits what is left.

Because conduction and convection are both linear in ΔT, resistances add in series and parallel exactly as electrical ones do. A composite wall with an air film on each side collapses to a single sum, which is the workhorse calculation of the whole subject.

Radiation resists the analogy, because T⁴ is not linear. It can be linearised about an operating point to give a radiation heat transfer coefficient, but that is an approximation valid only near the temperature it was derived at.