Lagged and unlagged bars

On this page we will consider the flow of heat down a bar, assuming that it is perfectly lagged so that no heat escapes from the sides. All the heat energy entering one end of the bar eventually leaves the other end; this is known as parallel heat flow. The drop in temperature is linear.

With an unlagged bar, however, heat is lost and so the rate of fall of temperature varies down the bar.
The temperature variation and the heat flow are shown for both bars in Figure 2(a) and (b).

Note that this is the flow of heat down a bar when steady state has been reached. Initially the temperature distribution will depend on the rate at which a pulse of heat energy can travel down the bar, and this depends on the specific heat capacity of the material and not the thermal conductivity.

Clearly this is a very large value: the heat loss from a normal house in winter would seem to be enormous. However, when heat is conducted through the glass it warms the layer of air just outside the window, thus raising the temperature of the outer glass surface (Figure 3). This reduces the temperature gradient and hence the loss of heat is also reduced. It also explains why rooms loose more heat on windy days – the layer of warm air is being continually blown away.

Conduction – metals, non-metals, molecules and free electrons!

(i) If one end of a metal rod is heated the atoms gain energy and their vibrations increase. This energy is then passed on to the other atoms and to the numerous free electrons in the metal. Since the electrons are very small they can travel rapidly around within the specimen, transferring their energy by collision to other electrons and other atoms. In metals, therefore, heat is carried mainly by the motion of these free electrons, although some energy is of course transferred by interatomic vibration.
The conductivity of metals does vary with temperature and at very low temperatures the lattice vibrations carry the heat energy more easily than the free electrons do.

(ii) Non-metals

In non-metallic solids there are virtually no free electrons, and therefore the only way that heat can travel through the specimen is by the direct transfer of energy from one atom to another. Debye suggested that the energy was transferred by some sort of elastic wave propagated through the specimen. These waves are sometimes called phonons, and travel with the speed of sound through the solid. Wiedmann and Franz showed that the ratio of thermal conductivity to electrical conductivity is the same for all metals, and this supports the view that a good conductor of electricity is also a good conductor of heat.

The composite bar

Consider a bar made of two materials both of the same cross-sectional area A, one a good conductor and the other a poor conductor, and assume that the bar is perfectly lagged, so that no heat leaves its sides (Figure 4).

The heat flowing into one end of the bar is equal to that flowing out the other end, and so the rate of flow of heat in one material is equal to that in the other material.
Therefore it the temperature of one end of the bar is q₁ and that of the other end q₂, and if the temperature of the join is q, then:

For the good conductor: dH/dt = [-k₁A (q₁- q)]/L₁

For the poor conductor: dH/dt = [-k₂A (q - q)]/L₂

where k₁ and k₂ are the thermal conductivities of the two materials and L₁ and L₂ are the lengths of the two sections of the bar.
Since the rates bf heat flow are equal, we have:

[-k₁A (q₁- q)]/L₁ = [-k₂A (q - q)]/L₂

So if the lengths are equal, the temperature drop through the good conductor will be much less than that through the poor conductor.