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ChE 120B
4  1
Transient Heat Conduction and Combining Variables (Set #4)
Examples of the general equations:
oneD transient heating of a semiinfinite solid bar
Boundary conditions
T
=
T
s
for all
T
(0,
t
)
At time
t
= 0, one end of the bar is raised to temp
T
s
and kept there.
0
x
TT
→∞
→
( )
0
,0
at
0
Tx
T
t
=
=
Can simplify the general equations for a solid:
2
2
tx
α
∂
∂
=
∂
∂
Define new variable
0
0
s
θ
−
=
−
so as to make
( )
()
0
0
2
2
0
0,
1 and
s
s
x
t
t
θθ
=
∞=
−
==
−
∂∂
=
∂
∂
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4  2
Combination of Variables
When no obvious length or time scale exists in the problem, it is often necessary to create one
by combining variables.
Define a new variable
4
x
n
t
α
⎫
=
⎬
⎭
Note that
n
is dimensionless.
Applying the chain rule for derivatives:
2
n
tt
n
θ
∂
−∂
=
∂
∂
22
1
4
x
tn
∂
∂
=
∂
∂
so that our original partial differential equation becomes ordinary.
2
2
20
.
dd
n
dn
dn
θθ
+
=
Let
p
dn
∂
=
integrating this expression gives
dp
np
dn
+
=
2
dp
ndn
p
=−
2
np
n
C
=
−+
A
2
1
n
p
Ce
−
=
2
1
n
d
dn
−
=
ChE 120B
4  3
and a second integration gives
()
2
12
0
.
n
n
nC
ed
n
C
θ
−
=+
∫
Examine the boundary conditions. Note that the first two conditions:
,0
0
0
4
x
x
n
t
α
⎫
=⎫
⎪⎪
=
⎬⎬
∞=
⎪
⎪
⎭
⎭
combine into one boundary condition
when
0 or
,
t
xn
→
⎛⎞
⎜⎟
→∞
⎝⎠
( )
0
n
→∞ =
This is an essential condition for combination of variables to work. For an ordinary second
order differential equation, we can have only 2 boundary conditions.
An inside tip:
Combination of variables — The general approach
1)
To reduce the partial differential equation into an ordinary differential equation, we need
to combine 2 variables into one.
A.
Check B.C.'s — if two of the B.C.'s cannot be made equivalent, forget it, cannot be
done.
B.
Often used if no length or timescale available.
Start by assuming that
,
bc
ax t
η
=
and use chain rule for derivatives:
dd
d
dt
d
dt
θη
=
2
22
2
2
d
d
dx
d
x
d
dx
ηη
∂
∂
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4  4
and substitute into original equation:
()
2
11
2
2
1
bc
b c
dd
d
ax ct
abx
t
ab b
x
t
d
θθ
θ
αα
ηη
η
−−
−
=+
−
reorganize to get:
( )
22
1
2
1
01
.
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This note was uploaded on 03/22/2011 for the course CHE 120B taught by Professor Zasadinski during the Winter '10 term at UCSB.
 Winter '10
 Zasadinski

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