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## Solid sphere with convection.

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 Where . The temperature is given by the boundary-heating term of the GF solution equation in the form (27)

The large-time GF for this case is given by:   (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.   Next: Steady Temperature in a Up: EXAMPLES, TEMPERATURE FROM GF Previous: Solid cylinder with internal
2004-01-31