# Heat Transfer in a Bar with Rectangular Cross Section

Heat Transfer in a Bar with Rectangular Cross Section

Consider a bar with a rectangular cross section subject to both temperature and heat flux boundary conditions.

The governing equation and boundary conditions in dimensionless form are:

2

∂

∂

2

x

2

∂

∂

2

y

0≤x≤L

0≤y≤W

∂θ

∂x

x=0

q

w

k

x=0

θ(x=L)=0

x=L

θ(y=0)=0

y=0

θ(y=W)=

θ

b

y=W

with , , and .

W=1

L=1

k=0.01

The Chebyshev orthogonal collocation method implemented in Mathematica with =31 collocation points gives the temperature distribution in the bar represented in the snapshots for user-set values of and .

N

p

q

w

θ

b

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

θ=sin( L)sinh( y)+1-cos( x)

∞

∑

n=0

2

L

λ

n

θ

b

λ

n

λ

n

sinh( W)

λ

n

q

w

k

sinh( y)+sinh( (W-y))

λ

n

λ

n

sinh( W)

λ

n

λ

n

where =.

λ

n

(2 n+1) π

2L

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