Solution Methodology for OneDimensional Transient Conduction
There are several different methods for solving 1D, transient conduction problems that
begin as a uniform temperature and are suddenly exposed to convection.
Lumped Capacitance Method (LCM)
A solid body changes temperature with time in a spatially uniform manner
p
bt
i
VC
hA
b
e
T
T
T
t
T
ρ
=
=
−
−
−
∞
∞
)
(
The density,
ρ
, and the specific heat, C
p
, are of the solid body.
The Biot number is a
nondimensional number that expresses the ratio of convection away from a solid body
to the conduction within the solid body.
Criteria for
LCM
:
Biot Number
based on volume to surface area ratio must be
less
than 0.1
A
V
L
k
hL
Bi
c
c
=
≤
=
1
.
0
Looking a little more closely at the exponent of the solution:
Number
Fourier
,
where
2
c
p
c
c
c
p
L
t
Fo
C
k
BiFo
t
L
L
k
L
h
t
VC
hA
bt
α
=
=
=
=
=
The Fourier number is a nondimensional time.
If LCM is not valid
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This note was uploaded on 07/25/2008 for the course ME 410 taught by Professor Benard during the Spring '08 term at Michigan State University.
 Spring '08
 BENARD
 Heat Transfer

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