Heat Transfer: Conduction and Thermal Conductivity

Heat Transfer: Conduction and Thermal Conductivity

Heat, wherever a temperature difference exists, always moves from the hotter region towards the colder

one, and it does so by one (or more) of three physically distinct mechanisms. Conduction is the

mechanism most relevant to heat moving through solids (and, much less effectively, through fluids at

rest).

The mechanism of conduction

In conduction, heat is passed along through a material by direct contact between neighbouring atoms or

molecules, without the material as a whole undergoing any bulk (large-scale) motion. Atoms/molecules in a

hotter region are vibrating (in a solid) or moving (in a fluid) more vigorously, on average, than those in

a cooler, adjacent region; through repeated collisions and interactions with their immediate neighbours,

some of this extra kinetic energy is passed on, region by region, from the hot end towards the cold end,

even though no single atom travels any macroscopic distance. In metals, this process is greatly assisted

by the "sea" of free (conduction) electrons that are not bound to any particular atom and can carry

kinetic energy rapidly across relatively large distances within the metal -- which is exactly why metals

that are good electrical conductors (their free electrons carrying electric current) also tend to be good

thermal conductors (the same free electrons carrying heat energy).

The law of steady-state conduction

Consider a rod (or a slab) of some conducting material, of length (or thickness) $L$ and uniform

cross-sectional area $A$, whose curved side surface is perfectly lagged (insulated) so that heat can only

flow along its length, and whose two ends are maintained at steady, fixed temperatures, with the hot end

at a higher temperature than the cold end by $\Delta T$. Once a steady state has been reached (every

point along the rod has settled to its own fixed temperature, no longer changing with time), the rate at

which heat flows through the rod, $\dfrac{Q}{t}$, is found experimentally to be:

• directly proportional to the cross-sectional area $A$ (a thicker rod conducts more heat per second),
• directly proportional to the temperature difference $\Delta T$ between the two ends (a bigger temperature difference drives a faster heat flow), and
• inversely proportional to the length $L$ (a longer rod, all else equal, conducts less heat per second, since the same temperature difference is now spread over a longer path).

Combined into a single relation,

where the constant of proportionality $k$, characteristic of the particular material, is called its

thermal conductivity, with SI unit $\text{W m}^{-1}\text{K}^{-1}$.

Conductors, insulators, and conduction in series

Materials with a large value of $k$ -- copper, silver, aluminium, and metals generally -- are called good

(thermal) conductors, and are chosen wherever heat must be moved efficiently: cooking vessels, heat

sinks for electronic components, and radiators. Materials with a very small value of $k$ -- wood, glass

wool, cork, thermocol (expanded polystyrene), and, notably, still (unconvecting) air -- are called

insulators or poor conductors, and are chosen wherever heat loss must be minimised: the walls of a

thermos flask, winter clothing (which works mainly by trapping a layer of still air), and building

insulation. …