1
MECE 4364 Heat Transfer
MECE 4364 Heat Transfer
Prof. Dong Liu
Prof. Dong Liu
Department of Mechanical Engineering
Department of Mechanical Engineering
University of Houston
University of Houston
1
Lecture 12 – Oct 5, 2010
Transient Conduction:
Transient Conduction:
Spatial Effects and the Role of
Spatial Effects and the Role of
Analytical Solutions
Analytical Solutions

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2
Plane Wall
Solution to the Heat Equation for a Plane Wall with
Symmetrical Convection Conditions
•
If the lumped capacitance approximation can not be made, consideration must
be given to spatial, as well as temporal, variations in temperature during the
transient process.
•
For a plane wall with symmetrical convection
conditions and constant properties, the
heat
equation
and
initial/boundary
conditions are:
2
2
1
T
T
x
t
α
∂
∂
=
∂
∂
(5.26)
(
)
,0
i
T
x
T
=
(5.27)
0
0
x
T
x
=
∂
=
∂
(5.28)
(
)
,
x
L
T
k
h T L t
T
x
∞
=
∂
⎡
⎤
−
=
−
⎣
⎦
∂
(5.29)
•
Existence of
seven independent variables:
(
)
, ,
,
, ,
,
i
T
T
x t T
T
k
h
α
∞
=
(5.30)
How may the functional dependence be simplified?
The Semi-Infinite Solid
•
A solid that is initially of uniform temperature
T
i
and is assumed to extend
to infinity from a surface at which thermal conditions are altered.
(
)
,
i
T
x
t
T
→ ∞
=
Second boundary condition

3
Semi
Semi-
Infinite Solid
The Semi-Infinite Solid
•
A solid that is initially of uniform temperature
T
i
and is assumed to extend
to infinity from a surface at which thermal conditions are altered.
•
Special Cases
:
Case 1:
Change in
Surface Temperature
(T
s
)
(
)
(
)
0,
,0
s
i
T
t
T
T
x
T
=
≠
=
(
)
,
x
erf
2
t
s
i
s
T
x t
T
T
T
α
−
⎛
⎞
=
⎜
⎟
−
⎝
⎠
(5.57)
(
)
s
i
s
k T
T
q
t
πα
−
′′ =
(5.58)
Semi
Semi-
Infinite Solid (cont.)
Infinite Solid (cont.)
(
)
(
)
1
2
2
2
/
,
exp
4
erfc
2
o
i
o
q
t
x
T
x t
T
k
t
q x
x
k
t
α
π
α
α
′′
⎛
⎞
−
=
−
⎜
⎟
⎝
⎠
′′
⎛
⎞
−
⎜
⎟
⎝
⎠
(5.59)
Case 2:
Uniform Heat Flux
(
)
s
o
q
q
′′
′′
=
(
)
0
0,
x
T
k
h T
T
t
x
∞
=
∂
⎡
⎤
−
=
−
⎣
⎦
∂
(
)
2
2
,
2
2
i
i
T
x t
T
x
erfc
T
T
t
hx
h
t
x
h
t
exp
erfc
k
k
k
t
α
α
α
α
∞
−
⎛
⎞
=
⎜
⎟
−
⎝
⎠
⎡
⎤
⎛
⎞
⎡
⎤
⎛
−
+
+
⎢
⎥
⎜
⎟
⎜
⎢
⎥
⎜
⎝
⎣
⎦ ⎢
⎥
⎠
⎝
⎣
⎦
(5.60)
Case 3:
Convection Heat Transfer
(
)
,
h T
∞

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