Theoretical Background
To permit heat transfer analysis of heat exchanger, it is necessary to define a mean
temperature difference between cold and hot fluids, which change as the fluids pass
through the heat exchanger. The following analysis is referred to cocurrent flow
arrangement and the result is also applicable to countercurrent arrangement.
The rate of heat transfer dq from the hot to the cold fluid through an elemental area
dA about the location A is given by:
dq=U dA ∆T
(1)
This heat transfer dq should be equal to the heat given up by the hot fluid flowing
from position A to A+dA
h
ph
h
dT
C
m
dq

=
(2)
Similarly for the heat gain by the cold fluid
c
h
dT
dT
T
d

=
∆
)
(
(3)
The temperature difference between the hot and cold fluid is
c
h
T
T
T

=
∆
(4)
Differentiating equation(4) gives
c
h
dT
dT
T
d

=
∆
)
(
(5)
Combining equations (2),(3) and (5)
)
1
1
(
)
(
pc
c
pc
h
C
m
C
m
dq
T
d
+

=
∆
(6)
Substituting equation (1) into (6) leads to
dA
C
m
C
m
U
T
T
d
pc
c
ph
h
)
1
1
(
)
(
+

=
∆
∆
(7)
Integrating equation (6) over the whole heat exchanger gives
pc
c
ph
h
C
m
C
m
T
T
q
1
1
2
1
+
∆

∆
=
(8)
Similarly, by integrating equation (7) over the whole heat exchanger leads to
t
m
pc
c
ph
h
A
U
C
m
C
m
T
T
)
1
1
(
)
ln(
2
1
+
=
∆
∆
(9)
where Um is the mean heat transfer coefficient
∫
=
A
t
m
UdA
T
U
0
1
(10)
combining new equations (9) and (11) leads to
m
l
m
t
m
t
T
U
A
T
T
T
T
U
A
q
,
2
1
2
1
)
/
ln(
∆
=
∆
∆
∆

∆
=
(11)
m
l
T
,.
∆
is called the logmean temperature difference(LMTD)
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ln(
2
1
2
1
T
T
T
T
T
lm
∆
∆
∆

∆
=
∆
(12)
in case of countercurrent floe when m
ph
C
=
c
m
pc
C
then Δ
0
T
=Δ
l
T
in such case L’Hopitals rule may be applied to show that
L
m
l
T
T
T
∆
=
∆
=
∆
0
,
(13)
if Δ
o
T
is not more than 50% grater than Δ
L
T
the LMTD given by Eq 14 can be
approximated by the arithmetic mean temperature difference to within about 1%. It is
to be noted that LMTD is always less than the arithmetic mean. It is also to be noted
that LMTD has been derived for singlepass heat exchangers. To use LMTD for
multipass heat exchangers a correction factor must be applied . Correction factors are
commonly given in graphs applying to a particular heatexchanger geometry.
The overall heat transfer coefficient in Eq1 takes into account thermal resistance in
the inside and outside films, resistance due to fouling and resistance due to conduction
through the tube wall.
∑
=
thermal
R
A
U
1
(14)
i
o
o
cu
i
o
o
i
i
o
R
R
h
L
k
D
D
A
h
A
A
U
+
+
+
+
=
1
2
)]
/
[ln(
1
π
(15)
Fouling factor resistance occurs due to the builtup of a scale on the heat transfer
surface. As a result, the heat transfer coefficient of the heat exchanger which has been
in service for a while, will be lower than that of the clean heat exchanger. The thermal
resistance of scale is determined from
cl
f
o
i
sc
U
U
R
R
R
1
1

=
+
=
(16)
Where U is the overall heat transfer coefficient of the clean heat exchanger and U is
the overall heat transfer coefficient of fouled heat exchange.
The film coefficients in Eq17 can be computed from dimensionless correlation based
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 Fall '09
 M.Elgaily
 Heat Transfer, Log mean temperature difference, LMTD, overall heat transfer

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