Point and linear heat sources

If the generation of heat is concentrated in a small, limited space we speak of local heat sources, which can be idealised as point, line or sheet singularities. For example, an electrically heated thin wire can be treated as a linear heat source. Alongside such technical applications these singularities have significant theoretical importance in the calculation of temperature fields, cf. [2.1]. 

The calculation of the temperature field produced when a point heat source exists will now be explained. We will consider a body which extends infinitely in all directions, with a spherical hollow space of radius R inside it, Fig. 2.41. At the surface of the sphere the boundary condition of a spatially constant heat flux % = % (t) is stipulated. This corresponds to a heat source within the hollow space with a thermal power (yield) of 

The limit R — 0 at a given time /, keeping Q constant, leads to a point heat source of strength Q(t), which is located at the centre of the sphere which is taken to be the origin r = 0 of the radial (spherical) coordinate. 

The temperature field outside the spherical hollow satisfies the differential equation 

according to the convolution theorem, No. 6 in Table 2.2 (page 143), is 

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