32

Basics of Heat
exchangers
Dr. Abdullah Al-Faruk
Dr. Abdullah Al-Faruk
◊ Heat exchangers are devices that facilitate the exchange of heat between two fluids that
33
are at different temperatures while keeping them from mixing with each other.
 Heat exchangers differ from mixing chambers in that they do not allow the two
fluids involved to mix.
Heat ◊ Heat exchangers are commonly used in practice in a wide range of applications:
 Heating and air-conditioning systems in a household
exchangers  Chemical processing
 Waste heat recovery
 Power production in large plants.

ME
◊ Heat transfer in a heat exchanger usually involves convection in each fluid and
conduction through the wall separating the two fluids.
◊ In heat exchanger analysis, it is convenient to work with an overall heat transfer
coefficient U that accounts for the contribution of all these effects on heat transfer.
◊ The rate of heat transfer between the two fluids at a location in a heat exchanger
depends on the magnitude of the temperature difference at that location, which varies
along the heat exchanger.
 ◊ Heat exchangers are typically classified according to
Dr. Abdullah Al-Faruk
flow arrangement and type of construction.
34

◊ The simplest heat exchanger is one for which the hot

and cold fluids move in the same or opposite directions
Types of in a concentric tube (or double-pipe) construction.
Heat  In the parallel-flow arrangement, the hot and cold
Exchanger fluids enter at the same end, flow in the same
direction, and leave at the same end.
 In the counter-flow arrangement, the fluids enter at
opposite ends, flow in opposite directions, and
leave at opposite ends.
ME

Double-pipe heat exchanger and associated temperature profiles

Dr. Abdullah Al-Faruk
◊ Inspecting the temperature profile of parallel flow heat exchanger shows that no matter
how long the exchanger is, the final temperature of the colder fluid can never reach the
35
exit temperature of the hotter fluid in parallel flow.
◊ For counter-flow, on the other hand, the final temperature of the cooler fluid may
exceed the outlet temperature of the hotter fluid, since a favorable temperature gradient
Types of exists all along the heat exchanger.
Heat ◊ An additional advantage of the counter-flow arrangement is that for a given rate of heat
flow, less surface area is required than in parallel flow.
Exchanger ◊ In fact, the counter-flow arrangement is the most effective of all heat exchanger
arrangements.

ME
 ◊ In compact heat exchangers, the two fluids usually move perpendicular to each other,
Dr. Abdullah Al-Faruk
and such flow configuration is called cross-flow.
36
◊ The cross-flow is further classified as unmixed flow and mixed flow, depending on the
flow configuration.
 The cross-flow is said to be unmixed since the plate fins force the fluid to flow
Types of through a particular inter-fin spacing and prevent it from moving in the transverse
Heat direction (i.e., parallel to the tubes).
 The cross-flow said to be mixed since the fluid now is free to move in the
Exchanger transverse direction. Both fluids are unmixed in a car radiator.
◊ The presence of mixing in the fluid can have a significant effect on the heat transfer
characteristics of the heat exchanger.

ME

Cross-flow heat exchanger, both fluids unmixed. Cross-flow heat exchanger, one fluid mixed and one unmixed.
 Dr. Abdullah Al-Faruk ◊ The shell-and-tube heat exchangers are the most common type of heat exchanger in
37
industrial applications.
◊ Shell-and-tube heat exchangers contain a large number of tubes (sometimes several
hundred) packed in a shell with their axes parallel to that of the shell.
Types of ◊ Heat transfer takes place as one fluid flows inside the tubes while the other fluid flows
outside the tubes through the shell.
Heat ◊ Baffles are commonly placed in the shell to force the shell-side fluid to flow across the
Exchanger shell to enhance heat transfer and to maintain uniform spacing between the tubes.
◊ Note that the tubes are open to some large flow areas called headers at both ends of the
shell, where the tube-side fluid accumulates before entering and after leaving the tubes.

ME

Shell-and-tube heat exchanger

with one tube pass Shell-and-tube heat exchanger with one shell pass and one tube pass
 ◊ Shell-and-tube heat exchangers are further classified according to the number of shell
Dr. Abdullah Al-Faruk
and tube passes involved.
38
◊ Heat exchangers in which all the tubes make one U-turn in the shell, for example, are
called one-shell-pass and two-tube-passes heat exchangers.
◊ Likewise, a heat exchanger that involves two
Types of passes in the shell and four passes in the tubes
Heat is called a two-shell-passes and four-tube-
passes heat exchanger.
Exchanger

ME

Shell-and-tube heat exchanger

with two tube passes
Multipass flow arrangements in shell-and-tube heat exchangers.
 Dr. Abdullah Al-Faruk
◊ An essential part of any heat exchanger analysis is the
determination of overall heat transfer coefficient.
39
◊ It is defined in terms of the total thermal resistance to heat
transfer between the two fluids.
◊ It is determined by accounting for conduction and
Overall heat convection resistances between the fluids separated by
composite plane and cylindrical walls, respectively.
transfer ◊ The thermal resistance network associated with this heat
coefficient transfer process involves two convection and one
conduction resistances:
1 ln 𝐷𝑜 /𝐷𝑖 1
𝑅𝑡𝑜𝑡𝑎𝑙 = + +
ℎ𝑖 𝜋𝐷𝑖 𝐿 2𝜋𝐿𝑘 ℎ𝑜 𝜋𝐷𝑜 𝐿
◊ The rate of heat transfer between the two fluids is
∆𝑇
ME 𝑞𝑟 = = 𝑈𝐴𝑠 ∆𝑇 = 𝑈𝑖 𝐴𝑖 ∆𝑇 = 𝑈𝑜 𝐴𝑜 ∆𝑇
𝑅𝑡𝑜𝑡𝑎𝑙
 where 𝐴𝑠 is the surface area and U is the overall heat
transfer coefficient based on 𝐴𝑠 . Similarly, 𝑈𝑖 is based
on the inner wall surface area 𝐴𝑖 .
◊ Canceling ∆𝑇, the above equation reduces to
1 1 1
Two heat transfer surface areas (for thin 𝑅𝑡𝑜𝑡𝑎𝑙 = = =
tubes, Di < Do and thus Ai < Ao)
𝑈𝐴𝑠 𝑈𝑖 𝐴𝑖 𝑈𝑜 𝐴𝑜
 Dr. Abdullah Al-Faruk
◊ Substituting 𝑅total , we obtain
40
1 1 1 1 ln 𝐷𝑜 /𝐷𝑖 1
= = = + +
𝑈𝐴𝑠 𝑈𝑖 𝐴𝑖 𝑈𝑜 𝐴𝑜 ℎ𝑖 𝜋𝐷𝑖 𝐿 2𝜋𝐿𝑘 ℎ0 𝜋𝐷𝑜 𝐿
 Note that 𝑈𝑖 𝐴𝑖 = 𝑈𝑜 𝐴𝑜 , but 𝑈𝑖 ≠ 𝑈𝑜 unless 𝐴𝑖 = 𝐴𝑜 .
Overall heat  Therefore, the overall heat transfer coefficient U of a heat exchanger is meaningless
transfer unless the area on which it is based is specified.
coefficient  This is especially the case when one side of the tube wall is finned and the other
side is not, since the surface area of the finned side is several times that of the un-
finned side.
◊ When the wall thickness of the tube is small and the thermal conductivity of the tube
material is high, as is usually the case, the thermal resistance of the tube is negligible
ME
(𝑅𝑤𝑎𝑙𝑙 < 0) and the inner and outer surfaces of the tube are almost identical (𝐴𝑖 ≈
𝐴𝑜 ≈ 𝐴𝑠 ).
◊ Then the overall heat transfer coefficient simplifies to
1 1 1
≈ + where 𝑈 ≈ 𝑈𝑖 ≈ 𝑈𝑜
𝑈 ℎ𝑖 ℎ𝑜
 The individual convection heat transfer coefficients inside and outside of the tube,
ℎ𝑖 and ℎ𝑜 , are determined using the convection relations.
 Dr. Abdullah Al-Faruk
◊ Representative values of the overall heat transfer coefficient U are given in Table.
41
◊ Note that the overall heat transfer coefficient ranges from about 10 W/m2 ·K for gas-to-
gas heat exchangers to about 10,000 W/m2 ·K for heat exchangers that involve phase
changes.
◊ This is not surprising, since gases have very low thermal conductivities, and phase-
Overall heat change processes involve very high heat transfer coefficients.
transfer
coefficient

ME
Dr. Abdullah Al-Faruk
◊ After a period of operation the heat-transfer surfaces may become coated with various
deposits present in the flow systems, or the surfaces may become corroded as a result
42
of the interaction between the fluids and the material used for construction of the heat
exchanger.
◊ In either event, this coating represents an additional resistance to the heat flow, and
thus results in decreased performance. Also, the fluid flow passage is reduced.
Fouling ◊ Therefore, the performance of heat exchangers usually deteriorates with time as a result
factors of accumulation of deposits on heat transfer surfaces.
◊ The overall effect is usually represented by a fouling factor, or fouling resistance, 𝑅𝑓 ,
which must be included along with the other thermal resistances making up the overall
heat-transfer coefficient.
◊ The fouling factor is obviously zero for a new heat exchanger and increases with time
ME
as the solid deposits build up on the heat exchanger surface.
◊ The fouling factor depends on the operating temperature and the velocity of the fluids,
as well as the length of service. Fouling increases with increasing temperature and
decreasing velocity.
◊ In applications where it is likely to occur, fouling should be considered in the design
and selection of heat exchangers. It may be necessary to select a larger and thus more
Deposits on the heat expensive heat exchanger to ensure that it meets the design heat transfer requirements
exchanger tubes even after fouling occurs.
Dr. Abdullah Al-Faruk
◊ Fouling factors must be obtained experimentally by
43
determining the values of U for both clean and dirty
conditions in the heat exchanger.
◊ The fouling factor is thus defined as
1 1
Fouling 𝑅𝑓 =
𝑈dirty
−
𝑈clea𝑛
factors ◊ The overall heat transfer coefficient relation for a double-
pipe heat exchanger needs to be modified to account for the
effects of fouling on both the inner and the outer surfaces
of the tube.
◊ For an double-pipe heat exchanger, it can be expressed as
ME
1 1 1
= = = 𝑅total
𝑈𝐴𝑠 𝑈𝑖 𝐴𝑖 𝑈0 𝐴0
1 𝑅𝑓,𝑖 ln 𝐷𝑜 /𝐷𝑖 𝑅𝑓,𝑜 1
= + + + +
ℎ𝑖 𝜋𝐷𝑖 𝐿 𝐴𝑖 2𝜋𝐿𝑘 𝐴𝑜 ℎ0 𝜋𝐷𝑜 𝐿
 where 𝑅𝑓,𝑖 and 𝑅𝑓,𝑜 are the fouling factors at those
surfaces.
◊ Representative values of fouling factors are given in Table
 ◊ To design or to predict the performance of a heat exchanger, it is essential to relate the
Dr. Abdullah Al-Faruk total heat transfer rate to inlet and outlet fluid temperatures, the overall heat transfer
44 coefficient, and the total surface area for heat transfer.
◊ An engineer have to select a heat exchanger that will achieve a specified temperature
change in a fluid stream of known mass flow rate, or to predict the outlet temperatures
of the hot and cold fluid streams in a specified heat exchanger.
Heat ◊ There are two methods used in the analysis of heat exchangers.
exchanger  The log mean temperature difference (or LMTD) method
analysis  The effectiveness–NTU (the number of transfer units) method.
◊ The above relations may readily be obtained by applying overall energy balances to the
hot and cold fluids.
◊ The energy balances and the subsequent analysis are
ME subject to the following assumptions
 Heat exchanger is insulated from its surroundings, the
only heat exchange is between the hot and cold fluids.
 Axial conduction along the tubes is negligible.
 Potential and kinetic energy changes are negligible.
Overall energy balances for the hot and cold fluids of a  The fluid specific heats are constant.
two-fluid heat exchanger. Here 𝑖 is the fluid enthalpy  The overall heat transfer coefficient is constant.
Dr. Abdullah Al-Faruk ◊ The temperatures of fluids in a heat
45
exchanger are generally not
constant but vary from point to
point as heat flows from the hotter
Log mean to the colder fluid.
temperature ◊ Even for a constant thermal
difference resistance, the rate of heat flow will
(LMTD) therefore vary along the path of the
exchangers because its value
method depends on the temperature
difference between the hot and the
cold fluid in that section.
◊ The changes in temperature for
ME both fluids, necessitate the heat
transfer rate equation of the form
with the overall heat transfer
coefficient 𝑈 as
𝑞 = 𝑈𝐴∆𝑇𝑚
 where ∆𝑇𝑚 is an appropriate Temperature distributions for a parallel-flow heat exchanger
mean temperature difference.
Dr. Abdullah Al-Faruk ◊ Applying an energy balance to differential elements in the hot and cold fluids, ∆𝑇𝑚 may be
46
determined. Each element is of length 𝑑𝑥 and heat transfer surface area 𝑑𝐴.
𝑑𝑞 = −𝑚ሶ ℎ 𝑐𝑝,ℎ 𝑑𝑇ℎ = −𝐶ℎ 𝑑𝑇ℎ and 𝑑𝑞 = 𝑚ሶ 𝑐 𝑐𝑝,𝑐 𝑑𝑇𝑐 = 𝐶𝑐 𝑑𝑇𝑐
Log mean  where 𝐶ℎ and 𝐶𝑐 are the hot and cold fluid heat capacity rates, respectively.
◊ The rate of heat transfer across the surface area 𝑑𝐴 may also be expressed as
temperature 𝑑𝑞 = 𝑈∆𝑇𝑑𝐴
difference  where ∆𝑇 = 𝑇ℎ − 𝑇𝑐 is the local temperature difference between hot and cold fluids.
(LMTD) ◊ To determine the integrated form, substitute 𝑑𝑇ℎ and 𝑑𝑇𝑐 into the differential form of
method 𝑑 ∆𝑇 = 𝑑𝑇ℎ − 𝑑𝑇𝑐 ↪ 𝑑 ∆𝑇 = −𝑑𝑞
1
+
1
𝐶ℎ 𝐶𝑐

ME

◊
Dr. Abdullah Al-Faruk ◊ Substituting for 𝑑𝑞 and integrating across the heat exchanger
2 2
47 1 1 𝑑 ∆𝑇 1 1
𝑑 ∆𝑇 = −𝑈∆𝑇𝑑𝐴 + ↪ න = −𝑈 + න 𝑑𝐴
𝐶ℎ 𝐶𝑐 1 ∆𝑇 𝐶ℎ 𝐶𝑐 1
Log mean ◊ Substituting for 𝐶ℎ and 𝐶𝑐
temperature Δ𝑇2 1 1 Δ𝑇2 𝑇ℎ,𝑖 − 𝑇ℎ,𝑜 𝑇𝑐,𝑜 − 𝑇𝑐,𝑖
ln = −𝑈𝐴 + ↪ ln = −𝑈𝐴 +
difference Δ𝑇1 𝐶ℎ 𝐶𝑐 Δ𝑇1 𝑞 𝑞
(LMTD) ↪ ln
Δ𝑇2
=−
𝑈𝐴
𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 − 𝑇ℎ,𝑜 − 𝑇𝑐,𝑜
method Δ𝑇1 𝑞
◊ For the parallel-flow heat exchanger, Δ𝑇1 = 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 and Δ𝑇2 = 𝑇ℎ,𝑜 − 𝑇𝑐,𝑜
◊ Rearranging after substitution yields the total heat transfer rate
Δ𝑇2 − Δ𝑇1
ME 𝑞 = 𝑈𝐴
ln Δ𝑇2 /Δ𝑇1
 Subscripts 1 and 2 designate opposite ends of heat exchanger
◊ Comparing the above expression with 𝑞 = 𝑈𝐴∆𝑇𝑚 , it concludes
that the appropriate average temperature difference is a log mean
temperature difference,
Δ𝑇2 − Δ𝑇1
∆𝑇𝑚 =
ln Δ𝑇2 /Δ𝑇1
Dr. Abdullah Al-Faruk ◊ The counter-flow heat exchanger configuration provides for heat transfer between the
48 hotter portions of the two fluids at one end, as well as between the colder portions at the
other, in contrast to the parallel-flow exchanger.
Log mean ◊ The change in the temperature difference, ∆𝑇 = 𝑇ℎ − 𝑇𝑐 , with respect to x is nowhere as
large as it is for the inlet region of the parallel-flow exchanger. However, the outlet
temperature temperature of cold fluid may now exceed the outlet temperature of hot fluid.
difference ◊ Note that for the same inlet and outlet temperatures, the log mean temperature difference
(LMTD) for counter-flow exceeds that for parallel flow, ∆𝑇1𝑚,𝐶𝐹 > ∆𝑇1𝑚,𝑃𝐹 .
method ◊ Hence, the surface area required to effect a prescribed heat transfer rate 𝑞 is smaller for
the counter-flow than for the parallel-flow arrangement, assuming the same value of 𝑈.

ME

Temperature distributions for a

counter-flow heat exchanger
 ◊ For more complex heat exchangers such as the shell-and-tube arrangements with
Dr. Abdullah Al-Faruk
several tube or shell passes and with cross-flow exchangers having mixed and unmixed
49 flow, the mathematical derivation of an expression for the mean temperature difference
becomes quite complex.
Log mean ◊ The usual procedure is to modify the simple LMTD by correction factors, which have
temperature been published in chart. The ordinate of each chart is the correction factor F.
difference ◊ To obtain the true mean temperature for any of these arrangements, the LMTD
calculated for counter-flow must be multiplied by the appropriate correction factor,
(LMTD) Δ𝑇2 − Δ𝑇1
method ∆𝑇𝑚 = 𝐹 LMTD = 𝐹
ln Δ𝑇2 /Δ𝑇1

ME

Correction factor to counter-

flow LMTD for heat exchanger
with one shell pass and two (or
a multiple of two) tube passes
 Dr. Abdullah Al-Faruk
◊ Thermal analysis and design of heat exchangers to given specifications by 𝑞 = 𝑈𝐴∆𝑇𝑚
50
relation is convenient when all the terminal temperatures necessary for the evaluation
of the appropriate mean temperature (LMTD) are known.
◊ However, numerous occasions when the performance of a heat exchanger (i.e., 𝑈) is
Effectiveness known or can at least be estimated but the temperatures of the fluids leaving the heat
–NTU exchanger are not known.
◊ This type of problem is encountered in the selection of a heat exchanger or when the
Method unit has been tested at one flow rate, but service conditions require different flow rates
for one or both fluids.

◊ In heat exchanger design, the problem to determine the outlet temperatures or the total
heat load, given the size (𝐴) and the convective performance (𝑈) of the unit is referred
ME
as a rating problem.
◊ The outlet temperatures and the rate of heat flow can be found only by a rather tedious
trial-and-error procedure if the charts presented in the preceding section are used.
◊ In such cases, it is desirable to circumvent entirely any reference to the logarithmic or
any other mean temperature difference.
◊ The effectiveness–NTU method is preferable to employ an alternative approach termed.
 ◊ To obtain an equation for the rate of heat transfer that does not involve any of the outlet
Dr. Abdullah Al-Faruk temperatures, let introduce the heat exchanger effectiveness 𝜀.
51
◊ The effectiveness is defined as the ratio of the actual rate of heat transfer in a given
heat exchanger to the maximum possible rate of heat exchange.
𝜀 = 𝑞/𝑞max
Effectiveness ◊ The maximum possible rate of heat transfer would be obtained in a counter-flow heat
–NTU exchanger of infinite heat transfer area.
Method ◊ To determine 𝑞max , we first recognize that the maximum temperature difference is the
difference between the inlet temperatures of the hot and cold fluids. That is, 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 .
◊ To illustrate this point, consider a situation for which 𝐶𝑐 < 𝐶ℎ , in which
case, 𝑑𝑇𝑐 > 𝑑𝑇ℎ according to 𝑑𝑞 = −𝐶ℎ 𝑑𝑇ℎ = 𝐶𝑐 𝑑𝑇𝑐 .
◊ The cold fluid would then experience the larger temperature change, and
ME since 𝐿 → ∞, it would be heated to the inlet temperature of the hot fluid
(𝑇𝑐,𝑜 = 𝑇ℎ,𝑖 ). Accordingly,
𝑞 = 𝐶𝑐 𝑇𝑐,𝑜 − 𝑇𝑐,𝑖 ↪ 𝑞𝑚𝑎𝑥 = 𝐶𝑐 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 if 𝐶𝑐 < 𝐶ℎ
◊ Similarly, if 𝐶ℎ < 𝐶𝑐 , the hot fluid would experience the larger
temperature change and would be cooled to the inlet temperature of the
cold fluid (𝑇ℎ,𝑜 = 𝑇𝑐,𝑖 ).
𝑞 = 𝐶ℎ 𝑇ℎ,𝑖 − 𝑇ℎ,𝑜 ↪ 𝑞𝑚𝑎𝑥 = 𝐶ℎ 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 if 𝐶ℎ < 𝐶𝑐
 Dr. Abdullah Al-Faruk
◊ Depending on which of the heat capacity rates is smaller, the general expression of
𝑞𝑚𝑎𝑥 is
52
𝑞𝑚𝑎𝑥 = 𝐶𝑚𝑖𝑛 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 𝐶𝑚𝑖𝑛 is equal to 𝐶𝑐 or 𝐶ℎ , whichever is smaller
◊ For prescribed hot and cold fluid inlet temperatures, it provides the maximum heat
transfer rate that could possibly be delivered by an exchanger.
Effectiveness ◊ The effectiveness is as follows, depending on the smaller heat capacity rates
–NTU 𝐶ℎ 𝑇ℎ,𝑖 − 𝑇ℎ,𝑜 𝐶𝑐 𝑇𝑐,𝑜 − 𝑇𝑐,𝑖
𝜀= or 𝜀=
Method 𝐶𝑚𝑖𝑛 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 𝐶𝑚𝑖𝑛 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖
◊ Once the effectiveness, 𝜀 of a heat exchanger is known, the rate of heat transfer can be
determined directly from
𝑞 = 𝜀𝐶𝑚𝑖𝑛 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖
◊ This relation expresses the rate of heat transfer in terms of the effectiveness, the smaller
ME heat capacity rate, and the difference between the inlet temperatures. It does not require
the outlet temperatures. It is suitable for analysis and design purposes.
◊ For any heat exchanger it can be shown that
𝜀 = 𝑓 NTU, 𝐶𝑚𝑖𝑛 /𝐶𝑚𝑎𝑥 where 𝐶𝑚𝑖𝑛 /𝐶𝑚𝑎𝑥 is equal to 𝐶𝑐 /𝐶ℎ or 𝐶ℎ /𝐶𝑐
◊ The number of transfer units (NTU) is a dimensionless parameter that is widely used
for heat exchanger analysis and is defined as
NTU = 𝑈𝐴/𝐶𝑚𝑖𝑛
 Dr. Abdullah Al-Faruk ◊ The method of deriving an expression for the effectiveness of a heat exchanger,
53 consider a parallel-flow heat exchanger for which 𝐶𝑚𝑖𝑛 = 𝐶ℎ . We then obtain
𝐶ℎ 𝑇ℎ,𝑖 − 𝑇ℎ,𝑜 𝑇ℎ,𝑖 − 𝑇ℎ,𝑜
𝜀= ↪ 𝜀=
𝐶𝑚𝑖𝑛 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖
Effectiveness ◊ Now consider the expression form LMTD analysis
–NTU Δ𝑇2 1 1 𝑇ℎ,𝑜 − 𝑇𝑐,𝑜 𝑈𝐴 𝐶𝑚𝑖𝑛
ln = −𝑈𝐴 + ↪ ln =− 1+
Method Δ𝑇1 𝐶ℎ 𝐶𝑐 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 𝐶𝑚𝑖𝑛 𝐶𝑚𝑎𝑥
𝑇ℎ,𝑜 − 𝑇𝑐,𝑜 𝐶𝑚𝑖𝑛
↪ = exp −NTU 1 + 𝐶𝑟 let 𝐶𝑟 =
𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 𝐶𝑚𝑎𝑥

◊ Rearranging the left-hand side of this expression as

ME 𝑇ℎ,𝑜 − 𝑇𝑐,𝑜 𝑇ℎ,𝑜 − 𝑇ℎ,𝑖 + 𝑇ℎ,𝑖 − 𝑇𝑐,𝑜
=
𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖
◊ Equating 𝑞 = 𝐶ℎ 𝑇ℎ,𝑖 − 𝑇ℎ,𝑜 and 𝑞 = 𝐶𝑐 𝑇𝑐,𝑜 − 𝑇𝑐,𝑖 to get the
expression for 𝑇𝑐,𝑜 as follows,
𝐶𝑚𝑖𝑛
𝑇𝑐,𝑜 − 𝑇𝑐,𝑖 = 𝑇ℎ,𝑖 − 𝑇ℎ,𝑜 ↪ 𝑇𝑐,𝑜 = 𝑇𝑐,𝑖 + 𝐶𝑟 𝑇ℎ,𝑖 − 𝑇ℎ,𝑜
𝐶𝑚𝑎𝑥
 Dr. Abdullah Al-Faruk
◊ It follows that by substituting 𝑇𝑐,𝑜 ,
54
𝑇ℎ,𝑜 − 𝑇𝑐,𝑜 𝑇ℎ,𝑜 − 𝑇ℎ,𝑖 + 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 − 𝐶𝑟 𝑇ℎ,𝑖 − 𝑇ℎ,𝑜
=
𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖
Effectiveness ◊ Using the expression of effectiveness , we obtain
𝑇ℎ,𝑜 − 𝑇𝑐,𝑜 𝑇ℎ,𝑜 − 𝑇𝑐,𝑜
–NTU 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖
= −𝜀 + 1 − 𝜀𝐶𝑟 ↪
𝑇ℎ,𝑖 − 𝑇𝑐,𝑖
= 1 − 𝜀 1 + 𝐶𝑟
Method ◊ Substituting the above expression and solving for 𝜀, we obtain for the parallel-flow
heat exchanger
1 − 𝜀 1 + 𝐶𝑟 = exp −NTU 1 + 𝐶𝑟 ↪ 𝜀 1 + 𝐶𝑟 = 1 − exp −NTU 1 + 𝐶𝑟
1 − exp −NTU 1 + 𝐶𝑟
𝜀=
ME 1 + 𝐶𝑟
◊ The above 𝜀–NTU relation applies for any parallel-flow heat exchanger, irrespective of
whether the minimum heat capacity rate is associated with the hot or cold fluid, since
the same result may be obtained for 𝐶𝑚𝑖𝑛 = 𝐶𝑐 .
◊ Similar expressions can be developed for a variety of heat exchangers, and
𝑇ℎ,𝑖 − 𝑇ℎ,𝑜 representative results are summarized in Table, where 𝐶𝑟 is the heat capacity ratio
𝜀= 𝐶𝑟 = 𝐶𝑚𝑖𝑛 /𝐶𝑚𝑎𝑥 .
𝑇ℎ,𝑖 − 𝑇𝑐,𝑖
 Dr. Abdullah Al-Faruk

55

Effectiveness
–NTU
Method

ME
 Dr. Abdullah Al-Faruk

56 ◊ The foregoing expressions are represented graphically, the abscissa corresponds to the
total number of transfer units,

Effectiveness
–NTU
Method

ME

Effectiveness of a parallel-flow heat exchanger Effectiveness of a counter-flow heat exchanger

 Dr. Abdullah Al-Faruk

57

◊ Two general types of heat exchanger problems are commonly encountered.

◊ In the heat exchanger design problem, the fluid inlet temperatures and flow rates, as
Design and well as a desired hot or cold fluid outlet temperature, are prescribed.
Performance  The design problem is then one of specifying a specific heat exchanger type and
determining its size—that is, the heat transfer surface area A—required to achieve
Calculations the desired outlet temperature.
 The design problem is commonly encountered when a heat exchanger is to be
custom-built for a specific application.

ME ◊ Alternatively, in a heat exchanger performance calculation, an existing heat exchanger

is analyzed to determine the heat transfer rate and the fluid outlet temperatures for
prescribed flow rates and inlet temperatures.
 The performance calculation is commonly associated with the use of off-the-shelf
heat exchanger types and sizes available from a vendor.
 Dr. Abdullah Al-Faruk

58
◊ For heat exchanger design problems,
 The NTU method may be used by first calculating 𝜀 and (𝐶𝑚𝑖𝑛 /𝐶𝑚𝑎𝑥 ).
 The appropriate equation (or chart) may then be used to obtain the NTU value,
Design and which in turn may be used to determine 𝐴.
Performance
Calculations ◊ For a performance calculation,
 The NTU and (𝐶𝑚𝑖𝑛 /𝐶𝑚𝑎𝑥 ) values may be computed.
 The effectiveness, 𝜀 may then be determined from the appropriate equation (or
chart) for a particular exchanger type.
 Since 𝑞𝑚𝑎𝑥 may also be computed from 𝑞𝑚𝑎𝑥 = 𝐶𝑚𝑖𝑛 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 , it is a simple
ME
matter to determine the actual heat transfer rate from the requirement that 𝑞 =
𝜀𝑞𝑚𝑎𝑥 .
 Both fluid outlet temperatures may then be determined from 𝑞 = 𝐶𝑐 𝑇𝑐,𝑜 − 𝑇𝑐,𝑖
and 𝑞 = 𝐶ℎ 𝑇ℎ,𝑖 − 𝑇ℎ,𝑜 .