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.