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WOLFRAM|DEMONSTRATIONS PROJECT

Transient Cooling of a Sphere

​
Biot number
1.5
dimensionless time
0.05
This Demonstration shows transient heat conduction in a sphere of radius
r
0
. At time
t<0
, the sphere is held at a uniform temperature
T
i
. At time
t=0
, the sphere is immersed in a well-mixed cooling bath at temperature
T
∞
. The sphere loses heat from its surface according to Newton's law of cooling:
qn=h(T-
T
∞
)
, where
h
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.
The Demonstration finds the 15 first roots of
1-
λ
n
cot(
λ
n
)-Bi=0
and displays the density plot of the sphere's temperature for user-set values of the Biot number,
Bi
, and the dimensionless time
τ
. Larger values of
Bi
or
τ
correspond to cooler temperatures of the sphere.
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