Heat Equation: Boundary Value Example
Consider a rod of length Lwhose ends are controlled to temperature 0 and
to which is applied a point heat source at x = L/4, raising the
temperature there to , and a point heat sink at x = 3L/4 , reducing
the temperature there to . We state without proof that the steady-
state solution to the heat equation in this case is as shown below.
1
T
+
1
T
−
Steady-State Temperature
Heat Equation: Boundary Value Example (2)
At time t = 0 the heat source and sink are removed while the ends of
the rod continue to be controlled to a temperature of 0. Determine
the temperature distribution across the bar as a function of time.
Temperature at t = 0
1
T+
From Equations 18 and 24,
Now , the temperature distribution at time t = 0 , is an odd
function about the midpoint of the rod. Since
are even functions about the midpoint,
=
−
=
t
L
n
n
n
L
xn
ectxT
1
sin),(
2
Heat Equation: Boundary Value Example (3)
)(
0xT
Heat Equation: Boundary Value Example (4)
)(
0xT
1
T
+
0
1
0=x
Lx =
4/Lx =
4/3Lx =
2/Lx =
L
x
sin
1
0=x
Lx =
4/Lx =
4/3Lx =
2/Lx =
Heat Equation: Boundary Value Example (5)
To compute for n even:
)/4()(
10
=
LxTxT
Lx
4/0
( )
+
=
=
LL
n
Ldx
L
xn
LxTdx
L
xn
xT
L
c
4/
0
1
00
/sin/42sin)(
2
We find that for nany multiple of four. The only nonzero
n
c
0=
n
c
Heat Equation: Boundary Value Example (6)
The results are
−=
=
1
2
6
1
2
2
9
18
8
Tc
Tc
Heat Equation: Boundary Value Example (7)
How well does the series (Equation 26) at t = 0 represent the initial
temperature distribution (Equation 25)? Let us examine the
accuracy at the peak, x = L/4, where we should find .
From Equation 26
so Equation 27 becomes
1
)0,4/( TLT =
Heat Equation: Boundary Value Example (8)
However, the sum of the inverse squares of the odd integers is
which is yet another important result attributed to Leonhard Euler,
and so, as desired,
It takes quite a few terms of the infinite series to achieve an accurate
approximation when t = 0. However, for even a relatively small
lapse of time the first term in the series dominates all others.
Consider just the first three terms at x = L/4:
Heat Equation: Boundary Value Example (9)
If only enough time has elapsed to reduce the first exponential term to
then
The spatial distribution across the bar is then well represented by
Clearly, the diffusion represented by the heat equation quickly smoothes
out peaks.
8.0
22 /4 =
−Lt
e
Heat Equation: Boundary Value Example (10)
Takeaways (What’s important here?):
In the mechanical engineering discipline of heat transfer, solving the
heat equation often involves separation of variables and expansion
in Fourier series.
Homework Assignment 23
Read:
Chapter 9, Section 9.1
Work:
Short Quiz #3