Transient Cooling of a Sphere

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This Demonstration shows transient heat conduction in a sphere of radius . At time , the sphere is held at a uniform temperature . At time , the sphere is immersed in a well-mixed cooling bath at temperature . The sphere loses heat from its surface according to Newton's law of cooling: , where is a heat transfer coefficient. Assume that at any time in the cooling process, the temperature distribution within the sphere depends solely on the radial coordinate; in a spherical coordinate system, the temperature is symmetric with respect to the azimuthal and polar angles.

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The Demonstration finds the 15 first roots of and displays the density plot of the sphere's temperature for user-set values of the Biot number, , and the dimensionless time . Larger values of or correspond to cooler temperatures of the sphere.

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Contributed by: Housam Binous and Brian G. Higgins (January 2013)
Open content licensed under CC BY-NC-SA


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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 .



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