HEAT EXCHANGERS-2
Prabal Talukdar
Associate Professor
Department of Mechanical Engineering
IIT Delhi
E-mail: prabal@mech.iitd.ac.in
p
P.Talukdar/ Mech-IITD
Multipass and Crossflow
The subscripts 1 and 2 represent the inlet and
outlet, respectively..
T and t represent the shell- and tube-side
temperatures respectively
temperatures,
P.Talukdar/ Mech-IITD
T and t represent the shelland tube
tube-side
side temperatures
temperatures,
respectively
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P.Talukdar/ Mech-IITD
Design with LMTD
LMTD method is very suitable for determining the size of a heat
exchanger
h
tto realize
li prescribed
ib d outlet
tl t ttemperatures
t
when
h th
the mass fl
flow
rates and the inlet and outlet temperatures of the hot and cold fluids are
specified.
With the LMTD method, the task is to select a heat exchanger that will
meet the prescribed heat transfer requirements. The procedure to be
followed by the selection process is:
Select-type
off Heat
H t
Exchanger
Determine- Inlet, Oulet
t
temp,
Heat
H t transfer
t
f rate
t
Using energy balance
Calculate ΔTlm
and F if neccessary
Obtain U
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Calculate As
Alternative of LMTD
A second kind of problem encountered in heat exchanger analysis is the
determination of the heat transfer rate and the outlet temperatures of the
hot and cold fluids for prescribed fluid mass flow rates and inlet temperatures
when the type and size of the heat exchanger are specified.
specified
The heat transfer surface area A of the heat exchanger in this case is known,
but the outlet temperatures
p
are not. Here the task is to determine the heat
transfer performance of a specified heat exchanger or to determine if a heat
exchanger available in storage will do the job.
The LMTD method could still be used for this alternative problem, but the
procedure would require tedious iterations, and thus it is not practical. In an
attempt to eliminate the iterations from the solution of such problems, Kays
andd L
London
d came up with
i h a method
h d in
i 1955 called
ll d the
h effectiveness–NTU
ff ti
NTU
method, which greatly simplified heat exchanger analysis
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Effectiveness-NTU
Effectiveness
NTU method
Effectiveness
.
ε=
Q
Q max
=
Actual heat transfer rate
Maximum possible heat transfer rate
Actual heat transfer rate
.
Q = Cc (Tc,outt − Tc,in
i ) = C h (Th ,in
i − Th , outt )
Maximum temperature difference that can occurs
ΔTmax = Th ,in
i − Tc,in
i
Maximum possible heat transfer
.
Q max = C min (Th ,in − Tc,in )
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Maximum heat transfer
The heat transfer in a heat exchanger will reach its maximum value
when
(1) the cold fluid is heated to the inlet temperature of the hot fluid or
(2) The hot fluid is cooled to the inlet temperature of the cold fluid.
These two limiting conditions will not be reached simultaneously
unless the heat capacity rates of the hot and cold fluids are
identical (i
(i.e.,
e Cc = Ch).
) When Cc ≠ Ch, which is usually the case
case,
the fluid with the smaller heat capacity rate will experience a larger
temperature change, and thus it will be the first to experience the
maximum temperature,
p
, at which point
p
the heat transfer will come to
a halt.
Maximum possible heat transfer
.
Q max = Cmin ( Th ,in − Tc ,in )
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Example
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ExampleC
Contd.
d
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Example
-Contd.
C td
The temperature rise of the cold fluid in a
heat exchanger will be equal to the
temperature drop of the hot fluid when the
mass flow rates and the specific heats of
the hot and cold fluids are identical.
P.Talukdar/ Mech-IITD
Effectiveness relation
parallel-flow double-pipe heat exchanger
.
C
T h , out = Th ,in − c (Tc,out − Tc,in )
Ch
Q = Cc (Tc, out − Tc,in ) = C h (Th ,in − Th , out )
⎛ 1
1 ⎞⎟
⎜
= − UA s
+
ln
⎜
&
& c C pc ⎟⎠
Th ,in − Tc,in
⎝ m h C ph m
Th ,out − Tc,out
ln
Th ,out − Tc,out
Th ,in − Tc,in
=−
UA s ⎛ Cc
⎜⎜1 +
Cc ⎝ C h
and after adding and subtractingTc, in gives
C
Th ,in − Tc,in + Tc,in − Tc,out − c (Tc,out − Tc,in )
UA s ⎛ Cc
Ch
⎜1 +
ln
=−
Th ,in − Tc,in
Cc ⎜⎝ C h
simplifies to
P.Talukdar/ Mech-IITD
⎡ ⎛ C
ln ⎢1 − ⎜⎜1 + c
⎢⎣ ⎝ C h
⎞
⎟⎟
⎠
⎞ Tc, out − Tc,in ⎤
UA s ⎛ Cc ⎞
⎟⎟
⎜⎜1 +
⎟⎟
=
−
⎥
Cc ⎝ C h ⎠
⎠ Th ,in − Tc,in ⎥⎦
⎞
⎟⎟
⎠
Effectiveness relation
We now manipulate the definition of effectiveness to obtain
.
ε=
.
Q
Q max
=
Cc (Tc, out − Tc,in )
Tc, out − Tc,in
C
⎯
⎯→
= ε min
C min (Th ,in − Tc,in )
Th ,in − Tc,in
Cc
Taking either Cc or Ch to be Cmin (both
approaches give the same result), the
relation above can be expressed more
conveniently as
⎡ UA s ⎛ C min ⎞⎤
⎜⎜1 +
⎟⎟⎥
1 − exp ⎢−
⎣ C min ⎝ C max ⎠⎦
ε pparallel _ flow =
C i
1+ min
C max
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⎡ ⎛ C
ln ⎢1 − ⎜⎜1 + c
⎢⎣ ⎝ C h
⎞ Tc,out − Tc,in ⎤
UA s ⎛ Cc
⎟⎟
⎜⎜1 +
⎥=−
Cc ⎝ C h
⎠ Th ,in − Tc,in ⎥⎦
⎞
⎟⎟
⎠
results
⎡ UA s ⎛ Cc
⎜⎜1 +
1 − exp ⎢−
Cc ⎝ C h
⎣
ε parallel _ flow =
C min ⎛ Cc ⎞
⎜⎜1 +
⎟⎟
Cc ⎝ C h ⎠
⎞⎤
⎟⎟⎥
⎠⎦
NTU
UAs
=
C min
Number of transfer units
NTU =
Capacity ratio
C min
c=
C max
Note that NTU is proportional to A
Note
that NTU is proportional to As . Therefore, for specified values of U and Cmin, the value of NTU is a measure of the heat transfer surface area As . Thus, the larger the NTU, the larger the heat exchanger.
.
(m C p ) min
⎡ UA s ⎛ C min ⎞⎤
⎜⎜1 +
⎟⎟⎥
1 − exp ⎢−
⎣ C min ⎝ C max ⎠⎦
ε parallel _ flow =
C
1 + min
C max
ε = function ( UAs / C min , C min / C max ) = function( NTU, c)
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UAs
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Effectiveness for heat
exchangers (from Kays and London, Ref. 5).
P.Talukdar/ Mech-IITD
Effectiveness for heat
exchangers (from Kays and London, Ref. 5).
P.Talukdar/ Mech-IITD
Effectiveness for heat
exchangers (from Kays and London, Ref. 5).
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Discussions
1. The value of the effectiveness ranges from 0 to 1.
It increases rapidly with NTU for small values (up
to about NTU 1.5) but rather slowly for larger
values. Therefore, the use of a heat exchanger with
a large NTU (usually larger than 3) and thus a
l
large
size
i cannot be
b justified
j ifi d economically,
i ll since
i
a
large increase in NTU in this case corresponds to a
small increase in effectiveness. Thus, a heat
exchanger
g with a veryy high
g effectiveness mayy be
highly desirable from a heat transfer point of view
but rather undesirable from an economical point of
view.
2 For a given
2.
gi en NTU and capacit
capacity ratio c = Cmin
/Cmax, the counter-flow heat exchanger has the
highest effectiveness, followed closely by the
cross-flow heat exchangers with both fluids
unmixed.
P.Talukdar/ Mech-IITD
Discussions
3. The effectiveness of a heat exchanger is
independent of the capacity ratio c for
NTU values of less than about 0.3.
4. The value of the capacity ratio c ranges between
0 and 11. For a given NTU,
NTU the effectiveness
becomes a maximum for c = 0 and a minimum
for c = 1. The case c = Cmin /Cmax → 0
corresponds to Cmax →∞ , which is realized
d i a phase-change
during
h
h
process iin a condenser
d
or
boiler. All effectiveness relations in this case
reduce to ε = εmax = 1 - exp(NTU) regardless of
yp of heat exchanger.
g Note that the
the type
temperature of the condensing or boiling fluid
remains constant in this case. The effectiveness
is the lowest in the other limiting case of c =
Cmin/Cmax = 1,
1 which
hich is realized
reali ed when
hen the heat
capacity rates of the two fluids are equal.
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Conclusions
Note that the analysis
y of heat exchangers
g with unknown outlet temperatures
p
is
a straight forward matter with the effectiveness–NTU method but requires
rather tedious iterations with the LMTD method.
When all
Wh
ll the
th inlet
i l t andd outlet
tl t ttemperatures
t
are specified,
ifi d the
th size
i off the
th heat
h t
exchanger can easily be determined using the LMTD method.
Alternatively, it can also be determined from the effectiveness–NTU method
Alternatively
by first evaluating the effectiveness from its definition and then the NTU
from the appropriate NTU relation given in tabular form.
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Compact heat exchangers
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