Longitudinal heat conduction in a rod

When a rod shaped object, for example a bolt or a pillar, is heated at one end heat flows along the axial direction and is transferred to the environment through the outer surface of the object. The heat release from fins is a similar heat conduction problem, which we will look at in the next section. Finally there are a number of measuring procedures for the determination of the thermal conductivity which are based on the comparison of temperature drops in rods made of different materials, see [2.2], 

We will consider a rod of length L with constant cross sectional area Aq and constant circumference U. One end of the rod is kept at constant temperature i)u by the input of heat. Heat flows along the direction of the rod axis, and will be transferred to the surroundings via the outer surface of the rod, Fig. 2.7. The cross sectional area of the rod is so small that the temperature across the whole cross section can be assumed to be constant, it only changes in the x-coordinate direction. In a rod element with volume AH = AqAx, the heat flow through the outer surface of the rod released to the surroundings which have a constant temperature t s is 

This release of heat has the same effect as a heat sink in the rod material with the power density 

Putting this into the differential equation (2.38) for linear heat flow with a heat source (n = 0) we obtain, with x instead of r 

Assuming a constant heat transfer coefficient a and putting as an abbreviation 

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