Unsteady-state temperature field in a semi-infinite body heated by a disk surface heat source

The paper deals with a three-dimensional axisymmetric problem on temperature field in a semi-infinite body heated by a time-constant heat flux distributed uniformly within a circular region on a free surface. Exact formulas are obtained for the unsteady-state temperature on the edge of the heating zone and for the average temperature within the region being heated. A non-obvious property of radial inversion of the value of average temperature is revealed. A new expression for unsteady-state temperature field, which is convenient for use in numerical calculations, is derived in an integral form on a circular cylindrical surface which is a continuation of the boundary of the heating zone. A general analytical solution of the problem is found in a closed form for the steady-state mode. The steady-state temperature field is expressed in terms of a combination of elliptic integrals. Particular cases are treated, in which the general expression derived for temperature assumes a simpler form. The decrease in the degree of concentration of heat flux with increasing depth is calculated. Fields of isotherms are constructed for both steady-state and unsteady-state cases, and the pattern of their distribution is analyzed. The asymptotics of the obtained dependences at low and high values of dimensionless time and in the vicinity of and away from the region being heated are studied. The range of validity of the adopted model of linear heat conduction is identified.