Details

The mathematical problem involved is:

for

subject to the following initial conditions (IC) and boundary conditions (BC)

IC: ,

BC 1: for ,

BC 2: .

The material parameters of the sphere that affect heat transfer are , the thermal diffusivity and , the thermal conductivity .

The heat transfer coefficient for the sphere/bath system accounts for the resistance associated with heat loss from the sphere to the fluid and is a function of the local mixing properties of the bath as well as the fluid thermal properties.

To solve this problem it is convenient to introduce the following dimensionless variables:

, , and .

Thus the PDE and IC/BCs become:

for ,

IC : ,

BC1: for ,

BC2: ,

where is a dimensionless quantity called the Biot number, a measure of the relative importance of resistance, heat conduction within the sphere, and resistance of heat loss to the surrounding fluid.

Using the method of separation of variables, you obtain the following solution:

,

where are the roots of and .