A long cylinder is initially at zero temperature and the boundary at is maintained at zero temperature. Find the temperature in the cylinder resulting from spatially uniform internal energy generation, (W/m).

The temperature satisfies the following equations:

(18) | |||

This case is number R01B0T0G1. The GF solution equation for the temperature contains only the internal heating term:

(19) |

(20) | |||

(21) |

where and are Bessel functions and the eigenvalues given by . The two integrals in the above solution will be considered one at a time. The spatial integral over acts on one term, and it may be simplified by the substitution and then evaluated as shown below:

The time integral acts only on the exponential term of the series for the GF, given by

(22) |

(23) |

(24) |

**Steady Solution.** Next the steady solution for the solid sphere
with internal heating will be derived. The steady temperature satisfies
the following differential equation:

(25) | |||

The general solution may be found by integrating the differential equation twice:

The constants of integration and may be determined by applying the boundary condtions. For to be bounded at requires . At the boundary condition then determines . Then the steady temperature is given by