Cooling by a Cylindrical Pin Fin
In electronic systems, a fin is a heat sink or a passive heat exchanger that cools a device by dissipating heat into the surrounding medium (e.g. air). This diagram shows a cylindrical pin fin, used to maximize heat transfer to a fluid between two walls:[more]
The walls are at a high temperature . The fluid flowing over the pin has a free stream temperature . The heat transfer coefficient between the pin wall and the surrounding medium is labeled (in ). If one introduces the dimensionless temperature , the governing equation is:
with and .
The associated boundary conditions are then:
(axial symmetry condition),
(radial symmetry condition),
(continuity of heat flux at the boundary fin/surrounding air), and
(constant temperature at the wall),
where is the thermal conductivity of the cylindrical pin fin and .
The Demonstration plots the contours of the dimensionless temperature for a user-set value of the Biot number. This solution is based on Chebyshev orthogonal collocation with collocation points.
The analytical solution of the differential equation obtained by separation of variables  is given by:
, where is the Biot number and are the zeros of the nonlinear function: .
We have found excellent agreement between our numerical solution and the analytical solution.[less]
 T. Bennett, Transport by Advection and Diffusion, New York: Wiley, 2012.