Cooling by a Cylindrical Pin Fin

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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:

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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 [1] 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.

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Contributed by: Housam Binous and Ahmed Bellagi  (August 2015)


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Reference

[1] T. Bennett, Transport by Advection and Diffusion, New York: Wiley, 2012.



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