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WOLFRAM|DEMONSTRATIONS PROJECT

Heat Transfer in a Bar with Rectangular Cross Section

​
q
w
1
θ
b
20
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
∂
θ
∂
2
x
+
2
∂
θ
∂
2
y
=0
, with
0≤x≤L
and
0≤y≤W
,
∂θ
∂x
x=0
=-
q
w
k
(continuity of heat flux at the
x=0
boundary),
θ(x=L)=0
(constant temperature at the
x=L
boundary),
θ(y=0)=0
(same constant temperature at the
y=0
boundary), and
θ(y=W)=
θ
b
(different constant temperature at the
y=W
boundary),
with
W=1
,
L=1
, and
k=0.01
.
The Chebyshev orthogonal collocation method implemented in Mathematica with
N
p
=31
collocation points gives the temperature distribution in the bar represented in the snapshots for user-set values of
q
w
and
θ
b
.
The analytical solution of the differential equation obtained using the separation of variables technique [1] is given by:
θ=
∞
∑
n=0
2
L 
λ
n
θ
b
sin(
λ
n
 L)sinh(
λ
n
 y)
sinh(
λ
n
 W)
+
q
w
k
1-
sinh(
λ
n
 y)+sinh(
λ
n
 (W-y))
sinh(
λ
n
 W)
cos(
λ
n
 x)
,
where
λ
n
=
(2 n+1) π
2L
.
We have found excellent agreement between our numerical solution and the analytical solution.
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