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MODULE 7
HEAT EXCHANGERS
7.1 What are heat exchangers?
Heat exchangers are devices used to transfer heat energy from one fluid to
another. Typical heat exchangers experienced by us in our daily lives include
condensers and evaporators used in air conditioning units and refrigerators.
Boilers and condensers in thermal power plants are examples of large industrial
heat exchangers. There are heat exchangers in our automobiles in the form of
radiators and oil coolers. Heat exchangers are also abundant in chemical and
process industries.
There is a wide variety of heat exchangers for diverse kinds of uses, hence the
construction also would differ widely. However, in spite of the variety, most
heat exchangers can be classified into some common types based on some
fundamental design concepts. We will consider only the more common types
here for discussing some analysis and design methodologies.
7.2 Heat Transfer Considerations
The energy flow between hot and cold streams, with hot stream in the
bigger diameter tube, is as shown in Figure 7.1. Heat transfer mode is by
convection on the inside as well as outside of the inner tube and by conduction
across the tube. Since the heat transfer occurs across the smaller tube, it is this
internal surface which controls the heat transfer process. By convention, it is
the outer surface, termed Ao, of this central tube which is referred to in
describing heat exchanger area. Applying the principles of thermal resistance,
t
T
di
do
Figure 7.1: End view of a tubular heat exchanger
r
ln o r 
 i
1
1
R


ho Ao
2  kl
hi Ai
If we define overall the heat transfer coefficient, Uc, as:
Uc 
1
RAo
Substituting the value of the thermal resistance R yields:
1
1


U c ho
r
ro ln o r 
 i
A
 o
k
hi Ai
Standard convective correlations are available in text books and handbooks for
the convective coefficients, ho and hi. The thermal conductivity, k, corresponds
to that for the material of the internal tube. To evaluate the thermal resistances,
geometrical quantities (areas and radii) are determined from the internal tube
dimensions available.
7.3 Fouling
Material deposits on the surfaces of the heat exchanger tubes may add
more thermal resistances to heat transfer. Such deposits, which are detrimental
to the heat exchange process, are known as fouling. Fouling can be caused by a
variety of reasons and may significantly affect heat exchanger performance.
With the addition of fouling resistance, the overall heat transfer coefficient, Uc,
may be modified as:
1
1

 R"
Ud Uc
where R” is the fouling resistance.
Fouling can be caused by the following sources:
1) Scaling is the most common form of fouling and is associated with
inverse solubility salts. Examples of such salts are CaCO3, CaSO4,
Ca3(PO4)2, CaSiO3, Ca(OH)2, Mg(OH)2, MgSiO3, Na2SO4, LiSO4, and
Li2CO3.
2) Corrosion fouling is caused by chemical reaction of some fluid
constituents with the heat exchanger tube material.
3) Chemical reaction fouling involves chemical reactions in the process
stream which results in deposition of material on the heat exchanger
tubes. This commonly occurs in food processing industries.
4) Freezing fouling is occurs when a portion of the hot stream is cooled to
near the freezing point for one of its components. This commonly occurs
in refineries where paraffin frequently solidifies from petroleum products
at various stages in the refining process. , obstructing both flow and heat
transfer.
5) Biological fouling is common where untreated water from natural
resources such as rivers and lakes is used as a coolant. Biological microorganisms such as algae or other microbes can grow inside the heat
exchanger and hinder heat transfer.
6) Particulate fouling results from the presence of microscale sized particles
in solution. When such particles accumulate on a heat exchanger surface
they sometimes fuse and harden. Like scale these deposits are difficult to
remove.
With fouling, the expression for overall heat transfer coefficient
becomes:
1

Ud
r
ln( o r ) 1
1
i

  R"
k
ho
r 
hi   i r 
o
7.4 Basic Heat Exchanger Flow Arrangements
Two basic flow arrangements are as shown in Figure 7.2. Parallel and
counter flow provide alternative arrangements for certain specialized
applications. In parallel flow both the hot and cold streams enter the heat
exchanger at the same end and travel to the opposite end in parallel streams.
Energy is transferred along the length from the hot to the cold fluid so the outlet
temperatures asymptotically approach each other.
In a counter flow
arrangement, the two streams enter at opposite ends of the heat exchanger and
flow in parallel but opposite directions. Temperatures within the two streams
tend to approach one another in a nearly linearly fashion resulting in a much
more uniform heating pattern. Shown below the heat exchangers are
representations of the axial temperature profiles for each. Parallel flow results
in rapid initial rates of heat exchange near the entrance, but heat transfer rates
rapidly decrease as the temperatures of the two streams approach one another.
This leads to higher exergy loss during heat exchange. Counter flow provides
for relatively uniform temperature differences and, consequently, lead toward
relatively uniform heat rates throughout the length of the unit.
t1
t2
t2
T1
T1
T1
T2
Parallel Flow
t1
T1
Temperature
T2
t1
t2
T2
Counter Flow
Temperature
T2
t2
t1
Position
Position
Fig. 7.2 Basic Flow Arrangements for Tube in Tube Heat Exchangers.
7.5 Log Mean Temperature Differences
Heat flows between the hot and cold streams due to the temperature
difference across the tube acting as a driving force. As seen in the Figure
below, the difference will vary with axial position within the HX so that one
must speak in terms of the effective or integrated average temperature
differences.
Counter Flow
T1
Parallel Flow
1
T1
t1
t2
2
t1
2
t2
T2
1
T2
Position
Position
Temperature Differences Between Hot and Cold Process Streams
Working from the three heat exchanger equations shown above, after
some development it if found that the integrated average temperature difference
for either parallel or counter flow may be written as:
  LMTD 
1   2
 
ln  1 
 2 
The effective temperature difference calculated from this equation is known as
the log mean temperature difference, frequently abbreviated as LMTD, based on
the type of mathematical average that it describes. While the equation applies
to either parallel or counter flow, it can be shown that eff will always be
greater in the counter flow arrangement. This can be shown theoretically from
Second Law considerations but, for the undergraduate student, it is generally
more satisfying to arbitrarily choose a set of temperatures and check the results
from the two equations. The only restrictions that we place on the case is that it
be physically possible for parallel flow, i.e. 1 and 2 must both be positive.
Another interesting observation from the above Figure is that counter
flow is more appropriate for maximum energy recovery. In a number of
industrial applications there will be considerable energy available within a hot
waste stream which may be recovered before the stream is discharged. This is
done by recovering energy into a fresh cold stream. Note in the Figures shown
above that the hot stream may be cooled to t1 for counter flow, but may only be
cooled to t2 for parallel flow. Counter flow allows for a greater degree of
energy recovery. Similar arguments may be made to show the advantage of
counter flow for energy recovery from refrigerated cold streams.
7.6 Applications for Counter and Parallel Flows
We have seen two advantages for counter flow, (a) larger effective
LMTD and (b) greater potential energy recovery. The advantage of the larger
LMTD, as seen from the heat exchanger equation, is that a larger LMTD
permits a smaller heat exchanger area, Ao, for a given thermal duty, Q. This
would normally be expected to result in smaller, less expensive equipment for a
given application.
This should not lead to the assumption that counter flow is always a
superior. Parallel flows are advantageous (a) where the high initial heating rate
may be used to advantage and (b) where the more moderate temperatures
developed at the tube walls are required. In heating very viscous fluids, parallel
flow provides for rapid initial heating. The quick decrease in viscosity which
results may significantly reduce pumping requirements through the heat
exchanger. The decrease in viscosity also serves to shorten the distance
required for flow to transition from laminar to turbulent, enhancing heat transfer
rates. Where the improvements in heat transfer rates compensate for the lower
LMTD parallel flow may be used to advantage.
A second feature of parallel flow may occur due to the moderation of
tube wall temperatures. As an example, consider a case where convective
coefficients are approximately equal on both sides of the heat exchanger tube.
This will result in the tube wall temperatures being about the average of the two
stream temperatures. In the case of counter flow the two extreme hot
temperatures are at one end, the two extreme cold temperatures at the other.
This produces relatively hot tube wall temperatures at one end and relatively
cold temperatures at the other. Temperature sensitive fluids, notably food
products, pharmaceuticals and biological products, are less likely to be
“scorched” or “thermally damaged” in a parallel flow heat exchanger.
Chemical reaction fouling may be considered as leading to a thermally damaged
process stream. In such cases, counter flow may result in greater fouling rates
and, ultimately, lower thermal performance. Other types of fouling are also
thermally sensitive. Most notable are scaling, corrosion fouling and freezing
fouling. Where control of temperature sensitive fouling is a major concern,
parallel flow may be used to advantage.
7.7 Multipass Flow Arrangements
In order to increase the surface area for convection relative to the fluid volume,
it is common to design for multiple tubes within a single heat exchanger. With
multiple tubes it is possible to arrange to flow so that one region will be in
parallel and another portion in counter flow. An arrangement where the tube
side fluid passes through once in parallel and once in counter flow is shown in
the Figure below. Normal terminology would refer to this arrangement as a 1-2
pass heat exchanger, indicating
that the shell side fluid passes
through the unit once, the tube
side twice. By convention the
number of shell side passes is
always listed first.
The primary reason for using
multipass designs is to increase
the average tube side fluid
velocity in a given arrangement.
In a two pass arrangement the
fluid flows through only half the
tubes and any one point, so that
the Reynold’s number is
effectively doubled. Increasing the Reynolds’s number results in increased
turbulence, increased Nusselt numbers and, finally, in increased convection
coefficients. Even though the parallel portion of the flow results in a lower
effective T, the increase in overall heat transfer coefficient will frequently
compensate so that the overall heat exchanger size will be smaller for a specific
service. The improvements achievable with multipass heat exchangers is
sufficiently large that they have become much more common in industry than
the true parallel or counter flow designs.
The LMTD formulas developed earlier are no longer adequate for
multipass heat exchangers. Normal practice is to calculate the LMTD for
counter flow, LMTDcf, and to apply a correction factor, FT, such that
 eff  FT  LMTDCF
The correction factors, FT, can be found theoretically and presented in analytical
form. The equation given below has been shown to be accurate for any
arrangement having 2, 4, 6, .....,2n tube passes per shell pass to within 2%.
 1 P 
R 2  1 ln 
1  R  P 
FT 
2  P R  1  R2  1 

 R  1 ln 

2
 2  P R  1  R  1 




where the capacity ratio, R, is defined as:
T1  T2
t 2  t1
R
The effectiveness may be given by the equation:
P
1  X 1/ N shell
R  X 1/ N shell
provided that R1. In the case that R=1, the effectiveness is given by:
P
N shell
Po
 Po   N shell  1
where
Po 
and
X 
t2  t1
T1  t1
Po  R  1
Po  1
As an alternative to using the formulas for the correction factors, which can
become tedious for non-computerized calculations, charts are available. Several
are included in standard texts. Experience has shown that, due to variability in
reading charts, considerable error can be introduced into the calculations and the
equations are recommended. When charts are used, they should be reproduced
at a sufficiently large scale, and considerable care should be used in making
interpolations.
7.8 Limitations of Multipass Arrangements
Since the 1-2 heat
exchanger uses one parallel pass
and one counter current, it
follows that the maximum heat
recovery for these units should
be between that of parallel and
counter flow. As a practical
limit it is important that nowhere
Figure . Temperature Profiles for a 1-2 HX with a
in the unit should the cold fluid Temperature Cross.
temperature exceed that of the
hot fluid. If so, then heat transfer is obviously in the wrong direction. Such a
situation can arise in a multipass heat exchanger as seen in Figure 6. This unit
represents a cold fluid, located on the tube side of the heat exchanger, making
two passes through the unit, the hot fluid, on the shell side, traveling across the
unit only once. Here the cold fluid is heated to a temperature slightly above that
of the hot fluid near the exit for the two streams. At this axial location, near the
left end of the unit, the temperature of the cold fluid in the first pass remains
well below that of the hot fluid so that considerable heat transfer occurs. The
cold fluid in the second pass is slightly above that of the hot fluid at the same
location. The small temperature difference between the second pass cold fluid
and the hot stream, indicates that only a small amount of heat will be transferred
between these streams. Overall heat will flow from hot to cold fluid, but a
portion of the heat transfer surface is being used in a counter productive way.
This condition is termed as a temperature cross.
In the limit the hot fluid exit temperature could be cooled to the average
of the cold fluid inlet and exit temperature. This would, however, be highly
inefficient and would require an excessively large surface area. Some engineers
advocate that good design should not permit a temperature cross, indicating that
the 1-2 should operate with the same heat recovery limit as a true parallel flow.
The preferred method of attaining additional heat recovery is to stage heat
exchangers in series so that no temperature cross occurs in any unit. An
equivalent solution is to put multiple 1–n arrangements within a single shell. A
2–4 unit is the equivalent of 2 1–2 units provided that the total heat transfer area
is equal. Similarly a 3–6 unit is the equivalent of 3 1–2 units with equal overall
area.
Other engineers suggest that a small temperature cross may be acceptable
and may provide a less expensive design than the more complex alternatives. If
one were to plot the locus of points where the temperature cross occurs for the
1-2 heat exchanger on the temperature correction chart, it would be found to
correspond to a relatively narrow range of FT values ranging from about 0.78 to
0.82. Lower values of FT may be taken as an indication that a temperature cross
will occur.
A second consideration is that at lower FT values the slope, dFT/dP,
becomes extremely steep. This is an indication that the temperature efficiency,
P, is asymptotically approaching its upper limit and the design has no margin to
accommodate uncertainties. A good rule of thumb is that the minimum slope of
dFT/dP, which is negative, should not fall below -1.5. Instead a 2-4 or even a 36 should be selected to provide the needed operational design margin.. Similar
restrictions exist for these designs as well. In the limit a counter flow design
may be the only suitable selection for high heat recovery applications.
7.9 Effectiveness-NTU Method:
In our previous discussions, we have been looking at practical HX
designs using the LMTDCF with a Ft correction factor to account for
the mixed flow conditions. Now we wish to consider an alternate,
more recent approach that is in common use today. This is the
effectiveness-NTU method.
Effectiveness, 
Consider two counter-flow heat exchangers, one in which the cold
fluid has the larger T (smaller mcp) and a second in which the cold
fluid has the smaller T (larger mcp):
t > T
MCp > mcp
T2
T1
T > t
mcp > MCp
t2
T2
t1
T1
t2
t1
We may see in the first case that, because the cold fluid heat capacity
is small, its temperature changes rapidly. If we seek efficient energy
recovery, we see that in the limit a HX could be designed in which the
cold fluid exit temperature would reach that of the hot fluid inlet. In
the second case, the hot fluid temperature changes more rapidly, so
that in the limit the hot fluid exit temperature would reach that of the
cold fluid inlet.
The effectiveness is the ratio of the energy recovered in a HX to that
recoverable in an ideal HX.

  cp  t2  t1 
m
  cp  T1  t1 
m
  C  T  T 
M
p
1
2

  C  T  t 
M
t > T
p
1
T > t
1
Canceling identical terms from the numerator and denominator of
both terms:

t  t 
T  t 
2
1
1
1

t > T
T  T 
T  t 
1
2
1
1
T > t
We see that the numerator, in the two cases, is the temperature change
for the stream having the larger temperature change.
The
denominator is the same in either case:

 Tmax
In the LMTD-Ft method an
effectiveness was defined:
T  t 
1
1
P
t 2  t1
T1  t1
Note that the use of the upper case T in the numerator, in contrast to
our normal terminology, does not indicate that the hot fluid
temperature change is used here. The max subscript over-rides the
normal terminology and indicates that this refers to the side having
the larger temperature change.
Number of Transfer Units (NTU)
Recall that the energy flow in any HX is described by three equations:
Q = UAeff
Q = -MCpT
Q = mcpt
HX equation
1st Law Equation
1st Law Equation
We may generalize the latter two expressions, using -NTU
terminology as follows:
Q = (MCp)minTmax
Q = (MCp)maxTmin
Again the use of the upper case letters is over-ridden by the use of the
subscripts.
If we eliminate Q between the HX equation and one of the 1 st Law
equations,
UAeff = (MCp)minTmax
This expression may be made non-dimensional by taking the
temperatures to one side and the other terms to the other side:
NTU 

UA
 C
M
p


 Tmax
  eff
min
Physically we see that a HX with a large product UA and a small
(MCp)min should result in a high degree of energy recovery, i.e.
should result in a large effectiveness, .
Capacity Ratio, CR
The final non-dimensional ratio needed here is the capacity ratio,
defined as follows:
CR 
( M  Cp ) min
( M  Cp ) max

 Tmin
 Tmax
In the LMTD-Ft method a
capacity ratio was defined:
R
T1  T2
t 2  t1
-NTU Relationships
In the LMTD-Ft method, we found a general equation which
described Ft for all 1-2N, 2-4N, 3-6N, etc. heat exchangers. Another
relationship, not given here, is required for cross flow arrangements.
In a similar fashion, we may develop a number of functional
relationships showing
 =  (NTU, CR)
or, alternatively:
NTU = NTU(,CR)
These relationships are shown in tables in standard text books. For
example, we find that the relationship for a parallel flow exchanger is:

1 e
1  CR  e NTU (1 CR )
 NTU  (1 CR )
Note: These correlations are
not general. Specific
correlations will be given for
different kind of HX.
The -NTU method offers a number of advantages to the designer
over the traditional LMTD-Ft method. One type of calculation where
the -NTU method may be used to clear advantage would be cases in
which neither fluid outlet temperature is known.
Th1
Hot
fluid
∆T1
Cold
fluid
Th1
Th2
∆T2
Tc2
Hot
fluid
∆T1
dTh
Tc1
dQ
Cold
fluid
Tc1
Th2
∆T2
dTc
Tc2
End 1
End 2
End 1
End 2