A pin fin has an elliptical external surface, as shown in the "pin fin image".

The walls are at a high temperature . The fluid flowing over the pin has a free stream temperature . The coefficient of heat transfer between the pin wall and the surrounding medium is labeled (in ). If one introduces the dimensionless temperature

, the governing equation is:

.

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 .

This Demonstration plots the contours of the dimensionless temperature for a user-set value of the Biot number and the shape factor .