In Chapter 2 we solved various heat conduction problems in various geometries
in a systematic but highly mathematical manner by (1) deriving the
governing differential equation by performing an energy balance on a differential
volume element, (2) expressing the boundary conditions in the proper
mathematical form, and (3) solving the differential equation and applying the
boundary conditions to determine the integration constants. This resulted in a
solution function for the temperature distribution in the medium, and the solution
obtained in this manner is called the analytical solution of the problem.
For example, the mathematical formulation of one-dimensional steady heat
conduction in a sphere of radius r0 whose outer surface is maintained at a uniform
temperature of T1 with uniform heat generation at a rate of g ·
0 was expressed
as (Fig. 5–1)