# Cooling by a Cylindrical Pin Fin

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:

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:

T

w

T

∞

h

W/K

2

m

ϕ=-

T-

T

∞

T

w

T

∞

1

r

∂

∂r

∂ϕ

∂r

2

∂

∂

2

z

with and .

0≤r≤

r

0

0≤z≤L

The associated boundary conditions are then:

∂ϕ

∂z

z=0

∂ϕ

∂r

r=0

∂ϕ

∂r

h

k

r=

r

0

ϕ(z=L)==1

ϕ

w

where is the thermal conductivity of the cylindrical pin fin and =1.

k (W/m K)

r

0

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.

N=21

The analytical solution of the differential equation obtained by separation of variables [1] is given by:

ϕ=

∞

∑

n=1

2 ( r)cosh( z)

B

i

J

0

λ

n

λ

n

cosh( L)()+

λ

n

J

0

λ

n

r

0

2

()

λ

n

r

0

2

B

i

B

i

h

r

0

k

λ

n

f(λ)=(λ )(λ )-(λ )

r

0

J

1

r

0

J

0

r

0

B

i

We have found excellent agreement between our numerical solution and the analytical solution.