Find the temperature in a solid sphere of radius , initially at elevated
temperature , that is suddenly immersed in a fluid at constant temperature
.
The heat transfer coefficient for the process is a constant value, .
The temperature satisfies the following equations:

(26) | |||

As stated, this is case RS03B1T1. Note that the convection boundary condition provides that the heat flux at will be positive for . There are two driving terms, however one of them can be made homogeneous by suitable choice of a normalized temperature. (Generally it is better to zero out boundary conditions in favor of initial conditions but the purpose of this example is to demonstrate the convection boundary term.) Let . Then the differential equation is unchanged, the initial condition is , and the boundary condition may be written in standard form as

(27) |

(28) | |||

where the eigenvalues are found from and where . The time integral is easily evaluated to give

(29) | |||

For numerical evaluation the steady-state term of the series should be replaced by its constant value. As , the sphere takes on the fluid temperature. That is, . Substitute this constant value in place of the steady-state portion of the above series to find the following result:

(30) | |||

Note the sign change.