# 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-λcot(λ)-Bi=0

n

n

Bi

τ

Bi

τ