This Demonstration describes a method for determining the thermal diffusivity of a sphere. The sphere is suddenly immersed in a cooling fluid and the temperature of the fluid is observed as a function of time.
Consider a homogeneous solid sphere of radius
, initially at uniform temperature
that is immersed at time
in a volume
of fluid in a well-stirred, insulated tank at temperature
It is convenient to define the following dimensionless variables:
= dimensionless solid temperature,
= dimensionless fluid temperature,
= dimensionless radial coordinate,
= dimensionless time,
is the radial coordinate,
is the radius of the sphere,
is the thermal diffusivity
is the thermal conductivity,
are the solid and fluid densities, and
are the specific heat capacities of the solid and liquid respectively.
The heat equation, in terms of these dimensionless variables, can be written :
with the boundary conditions:
is finite, with
is the ratio of the heat capacities of the fluid to the solid and
is the volume of the solid.
The following solution is obtained in :
are the nonzero roots of
The only place where the thermal diffusivity of the solid
appears is in the dimensionless time
, so that the temperature rise of the fluid can be used to determine the thermal diffusivity of the solid. Interestingly, the temperature history of the fluid is obtained without requiring the temperature profiles for the solid.