ChemE 109 - Numerical and Mathematical Methods
in Chemical and Biological Engineering
Fall 2007
Solution to Homework Set 6
1. The following conservation equation describes the steady-state temperature distribution in a
metal square:
∂
2
T
∂x
2
+
∂
2
T
∂y
2
= 0
subject to the following boundary conditions:
∂T
∂y
(
x,
0) = 0
,
∂T
∂y
(
x,
1) = 0
T
(0
,y
) = 0
,
T
(1
,y
) =
f
(
y
)
(a) The centered ﬁnite diﬀerence method will be employed to compute the temperature proﬁle
in the rectangle. Set up the grid in (
x,y
) dimensions, with (
i,j
) serving as local indices,
as shown below:
(i-1, j)
(i, j)
(i+1, j)
(i, j-1)
(i, j+1)
Use three equidistant interior nodes and the same discretization step in both
x
and
y
dimensions (i.e.
i
= 1
,...,
5,
j
= 1
,...,
5 and Δ
x
= Δ
y
). Show the exact form of the
equations needed to compute
T
ij
, for
i
= 2
,
3
,
4 and
j
= 2
,
3
,
4.
(b) Use the method of separation of variables to compute an analytic series solution for the
above problem.
Solution:
(a) In the ﬁgure below, the shadowed area is the metal square.
j
= 0
,
6 are ﬁctitious nodes
due to the boundary conditions on x-sides. The unknown variable is
T
j
i
, where the upper
1