Transient Cooling of a Sphere
Transient Cooling of a Sphere
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.
r
0
t<0
T
i
t=0
T
∞
qn=h(T-)
T
∞
h
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.
1-cot()-Bi=0
λ
n
λ
n
Bi
τ
Bi
τ