C
HAPTER
5:
T
RANSIENT
C
ONDUCTION
In practical terms, very few processes
operate at a true steady-state.
Variations in operation may arise from
“noise” caused by controller accuracy
and/or mechanical variation inherent
to the different pieces of equipment
that make up the process
Processes such as this may be
considered to be at steady state
For other processes, disturbances
consistently arise, causing the
controller to continuously adjust to
search for the specified set-point
Processes such as these virtually
always operate in transient mode.
dT/dt
0

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D
EALING
W
ITH
N
ON
-S
TEADY
S
TATE
C
ONDUCTION
Transient conduction may be simple
or extremely difficult to consider,
depending on the assumptions
applied.
Consider the case of
suddenly changing the temperature of
one surface of an object.
If temperature gradients within the
solid may be neglected (small, high
k
)
lumped capacitance method
If the only significant temperature
gradient is 1D
approximations of
heat diffusion equation solutions
If significant temperature gradients
exist in 2-3 dimensions
finite
element/finite difference method

L
UMPED
C
APACITANCE
M
ETHOD
The lumped capacitance method can
be used to find
dT/dt
if it is assumed
that the temperature
T
(
x,y,z
) is
identical
throughout the
entire object
at
any instantaneous time point
.
With no spatial variation in
T
,
d
2
T/dx
2
,
d
2
T/dy
2
,
and
d
2
T/dz
2
are all
zero
.
This assumption is reasonable when
R
(conduction through body) (
x
/
kA
)
<<
R
(heat loss at surface) (1/
hA
)
if the
body is very small
if the
thermal conductivity (k) is
very large
(
k
)
if
h, h
r
are small
q
z
T
y
T
x
T
k
2
2
2
2
2
2
t
T
c
p

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