ACM 95/100c
April 9, 2004
Problem Set II
1. (20 pts) The heat transfer in a onedimensional homogeneous bar in 0
< x < L
is described by
the equation
∂T
∂t

κ
∂
2
T
∂x
2
= 0
,
where
κ
is a constant. The temperatures of the ends of the bar are fixed (by attaching each end to
a large reservoir) such that
T
(0
, t
) =
T
0
,
T
(
L, t
) =
T
1
where
T
0
and
T
1
are constant temperatures.
(a) Show that using the transformation
T
=
w
(
x, t
) +
1
L
(
T
1

T
0
)
x
+
T
0
an equation for
w
(
x, t
) can be obtained with homogeneous end conditions.
(b) Solve the equation for
w
by a seriessolution method with initial temperature distribution
given by
T
(
x,
0) =
f
(
x
)
,
0
< x < L.
Note: This is an initial condition for
T
, not
w
.
(c) What is the temperature distribution in the bar when
t
→ ∞
?
2. (40 pts) A rod with insulated sides occupies the portion 1
< x <
2 of the
x
axis. The thermal
conductivity of the rod material is not constant but depends on
x
in such a way that the rod
temperature
T
(
x, t
) satisfies the heat equation
∂T
∂t

A
2
∂
∂x
x
2
∂T
∂x
= 0
where
A
is a constant. The boundary and initial conditions are
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 NilesA.Pierce
 Heat, Heat Transfer, 2 j, 20 pts, temperature distribution

Click to edit the document details