Modes of heat transfer with rate equations

Heat transfer is the transfer of thermal energy because of a temperature difference. It occurs by conduction, convection and radiation, and in most practical problems two or three modes act together.

Conduction

Energy moves from a hotter to a colder part of a body (or between bodies in contact) by molecular vibration, free-electron motion and collisions, without bulk movement of matter. It is the only mode in an opaque solid. The rate is given by Fourier's law:

$$Q = -kA\frac{dT}{dx}$$

where $k$ is thermal conductivity (W/m K), $A$ the area normal to heat flow and $dT/dx$ the temperature gradient. The minus sign shows that heat flows toward falling temperature. Example: heat flow through a furnace wall.

Convection

Energy is carried by the motion of a fluid over a surface. It is conduction plus bulk fluid motion. If the motion is caused by a fan or pump it is forced convection; if it is caused by density differences it is natural convection. Newton's law of cooling gives the rate:

$$Q = hA\,(T_s - T_\infty)$$

where $h$ is the convective heat transfer coefficient (W/m$^2$ K). Example: cooling of an engine radiator by air, or a hot cup of tea cooling in a room.

Radiation

Energy is emitted by every body above absolute zero as electromagnetic waves. It needs no medium and also works in vacuum. For a real surface:

$$Q = \varepsilon\sigma A\,(T_s^4 - T_{sur}^4)$$

with $\sigma = 5.67\times10^{-8}\ \text{W/m}^2\text{K}^4$, $\varepsilon$ the emissivity and temperatures in kelvin. Example: heat received from the sun, heat loss from a boiler furnace to its surroundings.