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ChE 120B
Lumped Parameter Models for Heat Transfer and the Blot Number
Imagine a slab that has one dimension, of thickness 2d, that is much smaller than the other two
dimensions; we also assume that the slab is homogeneous with temperature and positionindependent physical
parameters. At time
t
= 0, the slab is placed into contact with a liquid of constant temperature,
T
c
. Heat transfer
from the slab to the fluid is governed by Newton's law of cooling:
s
c
Qh
A
TT
(
1
)
in which
Q
is the heat flux.
A
is the surface area for heat transfer,
T
s
is the surface temperature of the specimen,
and
21
hW cm
K
is the heat transfer coefficient. Within the specimen itself, heat transfer is by conduction
T
Qk
A
x
(2)
11
kW cm
K
is the thermal conductivity of the specimen. At the boundary of the sample, the heat flux
given by the two expressions must be equal:
s
sc
T
hT T
k
x
(
3
)
s
T
x
is the temperature gradient at the specimen surface. Equation 3 is used as a one boundary condition in
solving for the transient temperature distribution within the specimen,
,
Tx
t
:
(
4
)
2
2
;
s
T
Tk
tx
C
p
3
gc
m
is the density and
CJg K
the heat capacity of the specimen.
is known as the
thermal diffusivity and has units of cm
2
sec
1
. If the sample is cooled simultaneously from both sides, as is
typically the case, the symmetry of the problem gives a second boundary condition:
0,
0
T
t
x
(
5
)
The specimen is initially assumed to be at a uniform temperature
T
0
:
0
,0
T
(
6
)
To simplify the problems and to help in identifying important parameters, we define the following
dimensionless variables:
81
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2
0
;
c
c
TT
x
t
t
dT
Td
which reduces the problem and the initial and boundary conditions to the following:
2
2
,0
1,
Initial Condition
(7a)
0,
0,
Boundary Condition
@0
x
(7b)
1,
,
hd
k
Boundary Condition
x
(7c)
The dimensionless group
hd k
is known as the Biot modulus, Bi. This modulus is a measure of the relative
rates of convective to conductive heat transfer. With this choice of boundary conditions, the problem can be
solved by a separation of variables technique. The exact solution is an infinite series expansion and is available
in texts such as that of Carslaw and Jaeger (1959). A better insight into the physics of the problem can be
obtained by examining certain limiting cases of this solution. First, let:
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 Winter '10
 Zasadinski

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