Unformatted text preview: cen58933_ch13.qxd 9/9/2002 9:57 AM Page 667 CHAPTER H E AT E X C H A N G E R S
eat exchangers are devices that facilitate the exchange of heat between
two fluids that are at different temperatures while keeping them from
mixing with each other. Heat exchangers are commonly used in practice in a wide range of applications, from heating and airconditioning systems
in a household, to chemical processing and power production in large plants.
Heat exchangers differ from mixing chambers in that they do not allow the
two fluids involved to mix. In a car radiator, for example, heat is transferred
from the hot water flowing through the radiator tubes to the air flowing
through the closely spaced thin plates outside attached to the tubes.
Heat transfer in a heat exchanger usually involves convection in each fluid
and conduction through the wall separating the two fluids. In the analysis of
heat exchangers, 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. In the analysis of heat exchangers, it is usually convenient to work with the logarithmic mean temperature difference LMTD, which is an equivalent mean temperature difference
between the two fluids for the entire heat exchanger.
Heat exchangers are manufactured in a variety of types, and thus we start
this chapter with the classification of heat exchangers. We then discuss the determination of the overall heat transfer coefficient in heat exchangers, and the
LMTD for some configurations. We then introduce the correction factor F to
account for the deviation of the mean temperature difference from the LMTD
in complex configurations. Next we discuss the effectiveness–NTU method,
which enables us to analyze heat exchangers when the outlet temperatures of
the fluids are not known. Finally, we discuss the selection of heat exchangers. H 13
CONTENTS
13–1 Types of Heat
Exchangers 668
13–2 The Overall Heat Transfer
Coefficient 671
13–3 Analysis of Heat
Exchangers 678
13–4 The Log Mean Temperature
Difference Method 680
13–5 The Effectiveness–NTU
Method 690
13–6 Selection of Heat
Exchangers 700 667 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 668 668
HEAT TRANSFER 13–1 I TYPES OF HEAT EXCHANGERS Different heat transfer applications require different types of hardware and
different configurations of heat transfer equipment. The attempt to match the
heat transfer hardware to the heat transfer requirements within the specified
constraints has resulted in numerous types of innovative heat exchanger
designs.
The simplest type of heat exchanger consists of two concentric pipes of different diameters, as shown in Figure 13–1, called the doublepipe heat
exchanger. One fluid in a doublepipe heat exchanger flows through the
smaller pipe while the other fluid flows through the annular space between
the two pipes. Two types of flow arrangement are possible in a doublepipe
heat exchanger: in parallel flow, both the hot and cold fluids enter the heat
exchanger at the same end and move in the same direction. In counter flow,
on the other hand, the hot and cold fluids enter the heat exchanger at opposite
ends and flow in opposite directions.
Another type of heat exchanger, which is specifically designed to realize a
large heat transfer surface area per unit volume, is the compact heat exchanger. The ratio of the heat transfer surface area of a heat exchanger to its
volume is called the area density . A heat exchanger with
700 m2/m3
23
(or 200 ft /ft ) is classified as being compact. Examples of compact heat
exchangers are car radiators (
1000 m2/m3), glass ceramic gas turbine
2
3
heat exchangers (
6000 m /m ), the regenerator of a Stirling engine
20,000 m2/m3). Compact heat
(
15,000 m2/m3), and the human lung (
exchangers enable us to achieve high heat transfer rates between two fluids in
T T Ho t fl Hot flui Cold d Co ld f FIGURE 13–1
Different flow regimes and
associated temperature profiles in
a doublepipe heat exchanger. Cold
in
Hot
out Cold
in
(a) Parallel flow luid fluid Cold
out Hot
in uid Hot
out Hot
in Cold
out
(b) Counter flow cen58933_ch13.qxd 9/9/2002 9:57 AM Page 669 669
CHAPTER 13 a small volume, and they are commonly used in applications with strict limitations on the weight and volume of heat exchangers (Fig. 13–2).
The large surface area in compact heat exchangers is obtained by attaching
closely spaced thin plate or corrugated fins to the walls separating the two fluids. Compact heat exchangers are commonly used in gastogas and gastoliquid (or liquidtogas) heat exchangers to counteract the low heat transfer
coefficient associated with gas flow with increased surface area. In a car radiator, which is a watertoair compact heat exchanger, for example, it is no surprise that fins are attached to the air side of the tube surface.
In compact heat exchangers, the two fluids usually move perpendicular to
each other, and such flow configuration is called crossflow. The crossflow
is further classified as unmixed and mixed flow, depending on the flow configuration, as shown in Figure 13–3. In (a) the crossflow is said to be unmixed since the plate fins force the fluid to flow through a particular interfin
spacing and prevent it from moving in the transverse direction (i.e., parallel to
the tubes). The crossflow in (b) is said to be mixed since the fluid now is free
to move in the 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. FIGURE 13–2
A gastoliquid compact heat
exchanger for a residential airconditioning system. Crossflow
(unmixed) Crossflow
(mixed) Tube flow
(unmixed)
(a) Both fluids unmixed Tube flow
(unmixed)
(b) One fluid mixed, one fluid unmixed FIGURE 13–3
Different flow configurations in
crossflow heat exchangers. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 670 670
HEAT TRANSFER
Tube
outlet FIGURE 13–4
The schematic of
a shellandtube
heat exchanger
(oneshell pass
and onetube
pass). Shell
inlet Baffles Frontend
header
Rearend
header
Tubes Shell
Shell
outlet Shellside fluid
In
Tubeside
fluid
Out
In
Out
(a) Oneshell pass and twotube passes
Shellside fluid
In
Out
Tubeside
fluid
In
Out
(b) Twoshell passes and fourtube passes FIGURE 13–5
Multipass flow arrangements in shellandtube heat exchangers. Tube
inlet Perhaps the most common type of heat exchanger in industrial applications
is the shellandtube heat exchanger, shown in Figure 13–4. Shellandtube
heat exchangers contain a large number of tubes (sometimes several hundred)
packed in a shell with their axes parallel to that of the shell. Heat transfer takes
place as one fluid flows inside the tubes while the other fluid flows outside the
tubes through the shell. Baffles are commonly placed in the shell to force the
shellside fluid to flow across the shell to enhance heat transfer and to maintain uniform spacing between the tubes. Despite their widespread use, shellandtube heat exchangers are not suitable for use in automotive and aircraft
applications because of their relatively large size and weight. Note that the
tubes in a shellandtube heat exchanger open to some large flow areas called
headers at both ends of the shell, where the tubeside fluid accumulates before
entering the tubes and after leaving them.
Shellandtube heat exchangers are further classified according to the number of shell and tube passes involved. Heat exchangers in which all the tubes
make one Uturn in the shell, for example, are called oneshellpass and twotubepasses heat exchangers. Likewise, a heat exchanger that involves two
passes in the shell and four passes in the tubes is called a twoshellpasses and
fourtubepasses heat exchanger (Fig. 13–5).
An innovative type of heat exchanger that has found widespread use is the
plate and frame (or just plate) heat exchanger, which consists of a series of
plates with corrugated flat flow passages (Fig. 13–6). The hot and cold fluids
flow in alternate passages, and thus each cold fluid stream is surrounded by
two hot fluid streams, resulting in very effective heat transfer. Also, plate heat
exchangers can grow with increasing demand for heat transfer by simply
mounting more plates. They are well suited for liquidtoliquid heat exchange
applications, provided that the hot and cold fluid streams are at about the same
pressure.
Another type of heat exchanger that involves the alternate passage of the hot
and cold fluid streams through the same flow area is the regenerative heat exchanger. The statictype regenerative heat exchanger is basically a porous
mass that has a large heat storage capacity, such as a ceramic wire mesh. Hot
and cold fluids flow through this porous mass alternatively. Heat is transferred
from the hot fluid to the matrix of the regenerator during the flow of the hot
fluid, and from the matrix to the cold fluid during the flow of the cold fluid.
Thus, the matrix serves as a temporary heat storage medium. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 671 671
CHAPTER 13 FIGURE 13–6
A plateandframe
liquidtoliquid heat
exchanger (courtesy of
Trante Corp.). The dynamictype regenerator involves a rotating drum and continuous flow
of the hot and cold fluid through different portions of the drum so that any
portion of the drum passes periodically through the hot stream, storing heat,
and then through the cold stream, rejecting this stored heat. Again the drum
serves as the medium to transport the heat from the hot to the cold fluid
stream.
Heat exchangers are often given specific names to reflect the specific application for which they are used. For example, a condenser is a heat exchanger
in which one of the fluids is cooled and condenses as it flows through the heat
exchanger. A boiler is another heat exchanger in which one of the fluids absorbs heat and vaporizes. A space radiator is a heat exchanger that transfers
heat from the hot fluid to the surrounding space by radiation. 13–2 I Cold
fluid Hot
fluid
Heat
transfer
Ti
Hot fluid
Ai
hi THE OVERALL HEAT TRANSFER COEFFICIENT A heat exchanger typically involves two flowing fluids separated by a solid
wall. Heat is first transferred from the hot fluid to the wall by convection,
through the wall by conduction, and from the wall to the cold fluid again by
convection. Any radiation effects are usually included in the convection heat
transfer coefficients.
The thermal resistance network associated with this heat transfer process
involves two convection and one conduction resistances, as shown in Figure
13–7. Here the subscripts i and o represent the inner and outer surfaces of the Cold
fluid
Wall Ao
ho To Ti
Ri = –1
––
hi A i Rwall Ro = –1
––
ho Ao FIGURE 13–7
Thermal resistance network
associated with heat transfer
in a doublepipe heat exchanger. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 672 672
HEAT TRANSFER inner tube. For a doublepipe heat exchanger, we have Ai
DiL and Ao
DoL, and the thermal resistance of the tube wall in this case is
Rwall R
D Di
o
Outer tube
Outer
fluid Inner
fluid (131) where k is the thermal conductivity of the wall material and L is the length of
the tube. Then the total thermal resistance becomes L Heat
transfer ln (Do /Di )
2 kL Inner tube
Ao = π Do L
Ai = π Di L FIGURE 13–8
The two heat transfer surface areas
associated with a doublepipe heat
exchanger (for thin tubes, Di Do
and thus Ai Ao). Rtotal Ri Rwall Ro ln (Do /Di )
2 kL 1
hi Ai 1
ho Ao (132) The Ai is the area of the inner surface of the wall that separates the two fluids,
and Ao is the area of the outer surface of the wall. In other words, Ai and Ao are
surface areas of the separating wall wetted by the inner and the outer fluids,
respectively. When one fluid flows inside a circular tube and the other outside
DiL and Ao
DoL (Fig. 13–8).
of it, we have Ai
In the analysis of heat exchangers, it is convenient to combine all the thermal resistances in the path of heat flow from the hot fluid to the cold one into
a single resistance R, and to express the rate of heat transfer between the two
fluids as
·
Q T
R UA T Ui Ai T Uo Ao T (133) where U is the overall heat transfer coefficient, whose unit is W/m2 · °C,
which is identical to the unit of the ordinary convection coefficient h. Canceling T, Eq. 133 reduces to
1
UAs 1
Ui Ai 1
Uo Ao R 1
hi Ai Rwall 1
ho Ao (134) Perhaps you are wondering why we have two overall heat transfer coefficients
Ui and Uo for a heat exchanger. The reason is that every heat exchanger has
two heat transfer surface areas Ai and Ao, which, in general, are not equal to
each other.
Note that Ui Ai Uo Ao, but Ui Uo unless Ai Ao. Therefore, the overall
heat transfer coefficient U of a heat exchanger is meaningless unless the area
on which it is based is specified. 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 unfinned 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 (Rwall 0) and the inner and outer surfaces of the tube are
almost identical (Ai Ao As). Then Eq. 134 for the overall heat transfer coefficient simplifies to
1
U 1
hi 1
ho (135) where U
Ui
Uo. The individual convection heat transfer coefficients
inside and outside the tube, hi and ho, are determined using the convection
relations discussed in earlier chapters. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 673 673
CHAPTER 13 The overall heat transfer coefficient U in Eq. 135 is dominated by the
smaller convection coefficient, since the inverse of a large number is small.
When one of the convection coefficients is much smaller than the other (say,
hi ho), we have 1/hi 1/ho, and thus U hi. Therefore, the smaller heat
transfer coefficient creates a bottleneck on the path of heat flow and seriously
impedes heat transfer. This situation arises frequently when one of the fluids
is a gas and the other is a liquid. In such cases, fins are commonly used on the
gas side to enhance the product UAs and thus the heat transfer on that side.
Representative values of the overall heat transfer coefficient U are given in
Table 13–1. Note that the overall heat transfer coefficient ranges from about
10 W/m2 · °C for gastogas heat exchangers to about 10,000 W/m2 · °C for
heat exchangers that involve phase changes. This is not surprising, since gases
have very low thermal conductivities, and phasechange processes involve
very high heat transfer coefficients.
When the tube is finned on one side to enhance heat transfer, the total heat
transfer surface area on the finned side becomes
As Atotal Afin Aunfinned (136) where Afin is the surface area of the fins and Aunfinned is the area of the unfinned
portion of the tube surface. For short fins of high thermal conductivity, we can
use this total area in the convection resistance relation Rconv 1/hAs since the
fins in this case will be very nearly isothermal. Otherwise, we should determine the effective surface area A from
As Aunfinned fin Afin (137) TABLE 13–1
Representative values of the overall heat transfer coefficients in
heat exchangers
Type of heat exchanger
Watertowater
Watertooil
Watertogasoline or kerosene
Feedwater heaters
Steamtolight fuel oil
Steamtoheavy fuel oil
Steam condenser
Freon condenser (water cooled)
Ammonia condenser (water cooled)
Alcohol condensers (water cooled)
Gastogas
Watertoair in finned tubes (water in tubes)
Steamtoair in finned tubes (steam in tubes) *Multiply the listed values by 0.176 to convert them to Btu/h · ft2 · °F.
† Based on airside surface area. ‡ Based on water or steamside surface area. U, W/m2 · °C*
850–1700
100–350
300–1000
1000–8500
200–400
50–200
1000–6000
300–1000
800–1400
250–700
10–40
3060†
400–850†
30–300†
400–4000‡ cen58933_ch13.qxd 9/9/2002 9:57 AM Page 674 674
HEAT TRANSFER where fin is the fin efficiency. This way, the temperature drop along the fins
1 for isothermal fins, and thus Eq. 137
is accounted for. Note that fin
reduces to Eq. 136 in that case. Fouling Factor
The performance of heat exchangers usually deteriorates with time as a result
of accumulation of deposits on heat transfer surfaces. The layer of deposits
represents additional resistance to heat transfer and causes the rate of heat
transfer in a heat exchanger to decrease. The net effect of these accumulations
on heat transfer is represented by a fouling factor Rf , which is a measure of
the thermal resistance introduced by fouling.
The most common type of fouling is the precipitation of solid deposits in a
fluid on the heat transfer surfaces. You can observe this type of fouling even
in your house. If you check the inner surfaces of your teapot after prolonged
use, you will probably notice a layer of calciumbased deposits on the surfaces
at which boiling occurs. This is especially the case in areas where the water is
hard. The scales of such deposits come off by scratching, and the surfaces can
be cleaned of such deposits by chemical treatment. Now imagine those mineral deposits forming on the inner surfaces of fine tubes in a heat exchanger
(Fig. 13–9) and the detrimental effect it may have on the flow passage area
and the heat transfer. To avoid this potential problem, water in power and
process plants is extensively treated and its solid contents are removed before
it is allowed to circulate through the system. The solid ash particles in the flue
gases accumulating on the surfaces of air preheaters create similar problems.
Another form of fouling, which is common in the chemical process industry, is corrosion and other chemical fouling. In this case, the surfaces are
fouled by the accumulation of the products of chemical reactions on the surfaces. This form of fouling can be avoided by coating metal pipes with glass
or using plastic pipes instead of metal ones. Heat exchangers may also be
fouled by the growth of algae in warm fluids. This type of fouling is called
biological fouling and can be prevented by chemical treatment.
In applications where it is likely to occur, fouling should be considered in
the design and selection of heat exchangers. In such applications, it may be FIGURE 13–9
Precipitation fouling of
ash particles on superheater tubes
(from Steam, Its Generation, and Use,
Babcock and Wilcox Co., 1978). cen58933_ch13.qxd 9/9/2002 9:57 AM Page 675 675
CHAPTER 13 necessary to select a larger and thus more expensive heat exchanger to ensure
that it meets the design heat transfer requirements even after fouling occurs.
The periodic cleaning of heat exchangers and the resulting down time are additional penalties associated with fouling.
The fouling factor is obviously zero for a new heat exchanger and increases
with time 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.
The overall heat transfer coefficient relation given above is valid for clean
surfaces and needs to be modified to account for the effects of fouling on both
the inner and the outer surfaces of the tube. For an unfinned shellandtube
heat exchanger, it can be expressed as
1
UAs 1
Ui Ai 1
Uo Ao R 1
hi Ai Rf, i
Ai ln (Do /Di )
2 kL Rf, o
Ao 1
ho Ao (138) where Ai
Di L and Ao
Do L are the areas of inner and outer surfaces,
and Rf, i and Rf, o are the fouling factors at those surfaces.
Representative values of fouling factors are given in Table 13–2. More comprehensive tables of fouling factors are available in handbooks. As you would
expect, considerable uncertainty exists in these values, and they should be
used as a guide in the selection and evaluation of heat exchangers to account
for the effects of anticipated fouling on heat transfer. Note that most fouling
factors in the table are of the order of 10 4 m2 · °C/W, which is equivalent to
the thermal resistance of a 0.2mmthick limestone layer (k 2.9 W/m · °C)
per unit surface area. Therefore, in the absence of specific data, we can assume the surfaces to be coated with 0.2 mm of limestone as a starting point to
account for the effects of fouling.
EXAMPLE 13–1 Overall Heat Transfer Coefficient of a
Heat Exchanger Hot oil is to be cooled in a doubletube counterflow heat exchanger. The copper
inner tubes have a diameter of 2 cm and negligible thickness. The inner diameter of the outer tube (the shell) is 3 cm. Water flows through the tube at a rate of
0.5 kg/s, and the oil through the shell at a rate of 0.8 kg/s. Taking the average
temperatures of the water and the oil to be 45°C and 80°C, respectively, determine the overall heat transfer coefficient of this heat exchanger. SOLUTION Hot oil is cooled by water in a doubletube counterflow heat
exchanger. The overall heat transfer coefficient is to be determined.
Assumptions 1 The thermal resistance of the inner tube is negligible since
the tube material is highly conductive and its thickness is negligible. 2 Both
the oil and water flow are fully developed. 3 Properties of the oil and water are
constant.
Properties The properties of water at 45°C are (Table A–9)
k 990 kg/m3
0.637 W/m · °C Pr 3.91
/ 0.602 10 6 m2/s TABLE 13–2
Representative fouling
factors (thermal resistance due
to fouling for a unit surface area)
(Source: Tubular Exchange Manufacturers
Association.) Fluid
Distilled water, sea
water, river water,
boiler feedwater:
Below 50°C
Above 50°C
Fuel oil
Steam (oilfree)
Refrigerants (liquid)
Refrigerants (vapor)
Alcohol vapors
Air Rf , m2 · °C/W 0.0001
0.0002
0.0009
0.0001
0.0002
0.0004
0.0001
0.0004 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 676 676
HEAT TRANSFER The properties of oil at 80°C are (Table A–16). 852 kg/m3
0.138 W/m · °C k
Hot oil
0.8 kg/s 2 cm 1
hi 1
U 3 cm FIGURE 13–10
Schematic for Example 13–1. 490
37.5 10 6 m2/s Analysis The schematic of the heat exchanger is given in Figure 13–10. The
overall heat transfer coefficient U can be determined from Eq. 135: Cold
water
0.5 kg/s Pr 1
ho where hi and ho are the convection heat transfer coefficients inside and outside
the tube, respectively, which are to be determined using the forced convection
relations.
The hydraulic diameter for a circular tube is the diameter of the tube itself,
Dh
D
0.02 m. The mean velocity of water in the tube and the Reynolds
number are ·
m
Ac m ·
m
(1
4 D2) 0.5 kg/s
(990 kg/m3)[1 (0.02 m)2]
4 1.61 m/s and
m Dh Re (1.61 m/s)(0.02 m)
0.602 10 6 m2/s 53,490 which is greater than 4000. Therefore, the flow of water is turbulent. Assuming
the flow to be fully developed, the Nusselt number can be determined from hDh
k Nu 0.023 Re0.8 Pr 0.4 0.023(53,490)0.8(3.91)0.4 240.6 Then, h k
Nu
Dh 0.637 W/m · °C
(240.6)
0.02 m 7663 W/m2 · °C Now we repeat the analysis above for oil. The properties of oil at 80°C are 852 kg/m3
0.138 W/m · °C k 37.5
490 Pr 10 6 m2/s The hydraulic diameter for the annular space is Dh TABLE 13–3 Do Di 0.03 0.02 0.01 m The mean velocity and the Reynolds number in this case are Nusselt number for fully developed
laminar flow in a circular annulus
with one surface insulated and the
other isothermal (Kays and Perkins,
Ref. 8.)
Di /Do Nui Nuo 0.00
0.05
0.10
0.25
0.50
1.00 —
17.46
11.56
7.37
5.74
4.86 3.66
4.06
4.11
4.23
4.43
4.86 m ·
m
Ac [1
4 ·
m
2
(Do 0.8 kg/s
D i2)] (852 kg/m )[1
4 3 (0.032 0.022)] m2 2.39 m/s and Re m Dh (2.39 m/s)(0.01 m)
37.5 10 6 m2/s 637 which is less than 4000. Therefore, the flow of oil is laminar. Assuming fully
developed flow, the Nusselt number on the tube side of the annular space
Nui corresponding to Di /Do
0.02/0.03
0.667 can be determined from
Table 13–3 by interpolation to be Nu 5.45 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 677 677
CHAPTER 13 and k
Nu
Dh ho 0.138 W/m · °C
(5.45)
0.01 m 75.2 W/m2 · °C Then the overall heat transfer coefficient for this heat exchanger becomes U 1
1
hi 1
1
ho 1
7663 W/m2 · °C 1
75.2 W/m2 · °C 74.5 W/m2 · °C Discussion Note that U ho in this case, since hi ho. This confirms our earlier statement that the overall heat transfer coefficient in a heat exchanger is
dominated by the smaller heat transfer coefficient when the difference between
the two values is large.
To improve the overall heat transfer coefficient and thus the heat transfer in
this heat exchanger, we must use some enhancement techniques on the oil
side, such as a finned surface. EXAMPLE 13–2 Effect of Fouling on the Overall Heat
Transfer Coefficient A doublepipe (shellandtube) heat exchanger is constructed of a stainless steel
(k 15.1 W/m · °C) inner tube of inner diameter Di 1.5 cm and outer diameter Do 1.9 cm and an outer shell of inner diameter 3.2 cm. The convection
heat transfer coefficient is given to be hi 800 W/m2 · °C on the inner surface
of the tube and ho 1200 W/m2 · °C on the outer surface. For a fouling factor
of Rf, i 0.0004 m2 · °C/ W on the tube side and Rf, o 0.0001 m2 · °C/ W on
the shell side, determine (a) the thermal resistance of the heat exchanger per
unit length and (b) the overall heat transfer coefficients, Ui and Uo based on the
inner and outer surface areas of the tube, respectively. SOLUTION The heat transfer coefficients and the fouling factors on the tube Cold fluid and shell sides of a heat exchanger are given. The thermal resistance and the
overall heat transfer coefficients based on the inner and outer areas are to be
determined.
Assumptions The heat transfer coefficients and the fouling factors are constant
and uniform.
Analysis (a) The schematic of the heat exchanger is given in Figure 13–11.
The thermal resistance for an unfinned shellandtube heat exchanger with fouling on both heat transfer surfaces is given by Eq. 138 as Outer layer of fouling 1
UAs 1
Ui Ai 1
Uo Ao Ai
Ao R Di L
Do L 1
hi Ai Rf, i
Ai ln (Do /Di )
2 kL Rf, o
Ao where (0.015 m)(1 m)
(0.019 m)(1 m) 0.0471 m2
0.0597 m2 Substituting, the total thermal resistance is determined to be 1
ho Ao Tube wall
Hot
fluid Inner layer of fouling
Cold fluid Hot
fluid Di = 1.5 cm
hi = 800 W/m2·°C
Rf, i = 0.0004 m2·°C/ W Do = 1.9 cm
ho = 1200 W/ m2·°C
Rf, o = 0.0001 m2·°C/ W FIGURE 13–11
Schematic for Example 13–2. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 678 678
HEAT TRANSFER 1
(800 W/m2 · °C)(0.0471 m2)
ln (0.019/0.015)
2 (15.1 W/m · °C)(1 m) R 0.0004 m2 · °C/ W
0.0471 m2 0.0001 m2 · °C/ W
1
0.0597 m2
(1200 W/m2 · °C)(0.0597 m2)
(0.02654 0.00849 0.0025 0.00168 0.01396)°C/ W
0.0532°C/ W
Note that about 19 percent of the total thermal resistance in this case is due to
fouling and about 5 percent of it is due to the steel tube separating the two fluids. The rest (76 percent) is due to the convection resistances on the two sides
of the inner tube.
(b) Knowing the total thermal resistance and the heat transfer surface areas,
the overall heat transfer coefficient based on the inner and outer surfaces of the
tube are determined again from Eq. 138 to be Ui 1
RAi 1
(0.0532 °C/ W)(0.0471 m2) 399 W/m2 · °C Uo 1
RAo 1
(0.0532 °C/ W)(0.0597 m2) 315 W/m2 · °C and Discussion Note that the two overall heat transfer coefficients differ significantly (by 27 percent) in this case because of the considerable difference between the heat transfer surface areas on the inner and the outer sides of the
tube. For tubes of negligible thickness, the difference between the two overall
heat transfer coefficients would be negligible. 13–3 I ANALYSIS OF HEAT EXCHANGERS Heat exchangers are commonly used in practice, and an engineer often finds
himself or herself in a position 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.
In upcoming sections, we will discuss the two methods used in the analysis
of heat exchangers. Of these, the log mean temperature difference (or LMTD)
method is best suited for the first task and the effectiveness–NTU method for
the second task as just stated. But first we present some general considerations.
Heat exchangers usually operate for long periods of time with no change in
their operating conditions. Therefore, they can be modeled as steadyflow devices. As such, the mass flow rate of each fluid remains constant, and the fluid
properties such as temperature and velocity at any inlet or outlet remain the
same. Also, the fluid streams experience little or no change in their velocities
and elevations, and thus the kinetic and potential energy changes are negligible. The specific heat of a fluid, in general, changes with temperature. But, in cen58933_ch13.qxd 9/9/2002 9:57 AM Page 679 679
CHAPTER 13 a specified temperature range, it can be treated as a constant at some average
value with little loss in accuracy. Axial heat conduction along the tube is usually insignificant and can be considered negligible. Finally, the outer surface
of the heat exchanger is assumed to be perfectly insulated, so that there is no
heat loss to the surrounding medium, and any heat transfer occurs between the
two fluids only.
The idealizations stated above are closely approximated in practice, and
they greatly simplify the analysis of a heat exchanger with little sacrifice of
accuracy. Therefore, they are commonly used. Under these assumptions, the
first law of thermodynamics requires that the rate of heat transfer from the hot
fluid be equal to the rate of heat transfer to the cold one. That is,
·
Q ·
mcCpc(Tc, out Tc, in) (139) ·
Q ·
mhCph(Th, in Th, out) (1310) and
where the subscripts c and h stand for cold and hot fluids, respectively, and
··
mc, mh
Cpc, Cph
Tc, out, Th, out
Tc, in, Th, in mass flow rates
specific heats
outlet temperatures
inlet temperatures ·
Note that the heat transfer rate Q is taken to be a positive quantity, and its direction is understood to be from the hot fluid to the cold one in accordance
with the second law of thermodynamics.
In heat exchanger analysis, it is often convenient to combine the product of
the mass flow rate and the specific heat of a fluid into a single quantity. This
quantity is called the heat capacity rate and is defined for the hot and cold
fluid streams as
Ch ·
mhCph and Cc ·
mcCpc (1311) The heat capacity rate of a fluid stream represents the rate of heat transfer
needed to change the temperature of the fluid stream by 1°C as it flows
through a heat exchanger. Note that in a heat exchanger, the fluid with a large
heat capacity rate will experience a small temperature change, and the fluid
with a small heat capacity rate will experience a large temperature change.
Therefore, doubling the mass flow rate of a fluid while leaving everything else
unchanged will halve the temperature change of that fluid.
With the definition of the heat capacity rate above, Eqs. 139 and 1310 can
also be expressed as
·
Q Cc(Tc, out Tc, in) T
Hot fluid
∆T1 Ch ∆T
∆T2
Cold fluid
Cc = Ch
∆T = ∆T1 = ∆T2 = constant (1312) and
·
Q x Ch(Th, in Th, out) (1313) That is, the heat transfer rate in a heat exchanger is equal to the heat capacity
rate of either fluid multiplied by the temperature change of that fluid. Note
that the only time the temperature rise of a cold fluid is equal to the temperature drop of the hot fluid is when the heat capacity rates of the two fluids are
equal to each other (Fig. 13–12). Inlet Outlet FIGURE 13–12
Two fluids that have the same mass
flow rate and the same specific heat
experience the same temperature
change in a wellinsulated heat
exchanger. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 680 680
HEAT TRANSFER Two special types of heat exchangers commonly used in practice are condensers and boilers. One of the fluids in a condenser or a boiler undergoes a
phasechange process, and the rate of heat transfer is expressed as
·
Q T (1314) ·
where m is the rate of evaporation or condensation of the fluid and hfg is the
enthalpy of vaporization of the fluid at the specified temperature or pressure.
An ordinary fluid absorbs or releases a large amount of heat essentially
at constant temperature during a phasechange process, as shown in Figure
13–13. The heat capacity rate of a fluid during a phasechange process must
approach infinity since the temperature change is practically zero. That is,
·
·
·
C mCp → when T → 0, so that the heat transfer rate Q mCp T is a
finite quantity. Therefore, in heat exchanger analysis, a condensing or boiling
fluid is conveniently modeled as a fluid whose heat capacity rate is infinity.
The rate of heat transfer in a heat exchanger can also be expressed in an
analogous manner to Newton’s law of cooling as Condensing fluid . Q Cold fluid ·
Q Inlet ·
mhfg Outlet (a) Condenser (Ch → )
T
Hot fluid . Q UAs Tm (1315) where U is the overall heat transfer coefficient, As is the heat transfer area, and
Tm is an appropriate average temperature difference between the two fluids.
Here the surface area As can be determined precisely using the dimensions of
the heat exchanger. However, the overall heat transfer coefficient U and the
temperature difference T between the hot and cold fluids, in general, are not
constant and vary along the heat exchanger.
The average value of the overall heat transfer coefficient can be determined
as described in the preceding section by using the average convection coefficients for each fluid. It turns out that the appropriate form of the mean temperature difference between the two fluids is logarithmic in nature, and its
determination is presented in Section 13–4. Boiling fluid Inlet Outlet (b) Boiler (Cc → ) FIGURE 13–13
Variation of fluid temperatures in a
heat exchanger when one of the fluids
condenses or boils. 13–4 I THE LOG MEAN TEMPERATURE
DIFFERENCE METHOD Earlier, we mentioned that the temperature difference between the hot and
cold fluids varies along the heat exchanger, and it is convenient to have a
·
mean temperature difference Tm for use in the relation Q UAs Tm.
In order to develop a relation for the equivalent average temperature difference between the two fluids, consider the parallelflow doublepipe heat exchanger shown in Figure 13–14. Note that the temperature difference T
between the hot and cold fluids is large at the inlet of the heat exchanger but
decreases exponentially toward the outlet. As you would expect, the temperature of the hot fluid decreases and the temperature of the cold fluid increases
along the heat exchanger, but the temperature of the cold fluid can never
exceed that of the hot fluid no matter how long the heat exchanger is.
Assuming the outer surface of the heat exchanger to be well insulated so
that any heat transfer occurs between the two fluids, and disregarding any cen58933_ch13.qxd 9/9/2002 9:57 AM Page 681 681
CHAPTER 13 changes in kinetic and potential energy, an energy balance on each fluid in a
differential section of the heat exchanger can be expressed as
·
Q ·
mh Cph dTh T
Th, in .
δ Q = U (Th – Tc ) d As
Th (1316) ∆T ∆T1 and
·
Q ·
mc Cpc dTc ˙
Q
˙
m hCph dTh (1318) ∆T2 Th, out
Tc, out dTc (1317) That is, the rate of heat loss from the hot fluid at any section of a heat exchanger is equal to the rate of heat gain by the cold fluid in that section. The
temperature change of the hot fluid is a negative quantity, and so a negative
·
sign is added to Eq. 1316 to make the heat transfer rate Q a positive quantity.
Solving the equations above for dTh and dTc gives dTh
.
δQ Tc
Tc, in
1 Hot
fluid ∆T1 = Th, in – Tc, in
∆T2 = Th, out – Tc, out
2
dAs
Tc, out
dAs As Th, out Th, in and
˙
Q
˙
mc Cpc dTc (1319) Taking their difference, we get
dTh dTc d(Th ·
1
Q·
m h Cph Tc) 1
·
m c Cpc (1320) The rate of heat transfer in the differential section of the heat exchanger can
also be expressed as
·
Q U(Th Tc) dAs (1321) Substituting this equation into Eq. 1320 and rearranging gives
d(Th
Th Tc)
Tc 1
U dAs ·
m h Cph 1
·
m c Cpc (1322) Integrating from the inlet of the heat exchanger to its outlet, we obtain
ln Th, out
Th, in Tc, out
Tc, in 1
UAs ·
m h Cph 1
·
m c Cpc (1323) ·
·
Finally, solving Eqs. 139 and 1310 for mcCpc and mhCph and substituting into
Eq. 1323 gives, after some rearrangement,
·
Q UAs Tlm (1324) T1
T2
ln ( T1/ T2) (1325) where
Tlm is the log mean temperature difference, which is the suitable form of the
average temperature difference for use in the analysis of heat exchangers.
Here T1 and T2 represent the temperature difference between the two fluids Cold fluid
Tc, in FIGURE 13–14
Variation of the fluid temperatures in a
parallelflow doublepipe heat
exchanger. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 682 682
HEAT TRANSFER
Tc, out
∆T2 Hot
fluid
Th,in
∆T1 Th,out
Cold fluid
Tc, in ∆T1 = Th,in – Tc, in
∆T2 = Th,out – Tc, out (a) Parallelflow heat exchangers
Cold
fluid
Tc, in ∆T2 Hot
fluid
Th,in Th,out ∆T1
Tc, out at the two ends (inlet and outlet) of the heat exchanger. It makes no difference
which end of the heat exchanger is designated as the inlet or the outlet
(Fig. 13–15).
The temperature difference between the two fluids decreases from T1 at
the inlet to T2 at the outlet. Thus, it is tempting to use the arithmetic mean
T2) as the average temperature difference. The
temperature Tam 1 ( T1
2
logarithmic mean temperature difference Tlm is obtained by tracing the actual temperature profile of the fluids along the heat exchanger and is an exact
representation of the average temperature difference between the hot and
cold fluids. It truly reflects the exponential decay of the local temperature
difference.
Note that Tlm is always less than Tam. Therefore, using Tam in calculations instead of Tlm will overestimate the rate of heat transfer in a heat exchanger between the two fluids. When T1 differs from T2 by no more than
40 percent, the error in using the arithmetic mean temperature difference is
less than 1 percent. But the error increases to undesirable levels when T1
differs from T2 by greater amounts. Therefore, we should always use the
logarithmic mean temperature difference when determining the rate of heat
transfer in a heat exchanger. ∆T1 = Th,in – Tc, out
∆T2 = Th,out – Tc, in (b) Counterflow heat exchangers FIGURE 13–15
The T1 and T2 expressions in
parallelflow and counterflow heat
exchangers. CounterFlow Heat Exchangers
The variation of temperatures of hot and cold fluids in a counterflow heat exchanger is given in Figure 13–16. Note that the hot and cold fluids enter the
heat exchanger from opposite ends, and the outlet temperature of the cold
fluid in this case may exceed the outlet temperature of the hot fluid. In the limiting case, the cold fluid will be heated to the inlet temperature of the hot fluid.
However, the outlet temperature of the cold fluid can never exceed the inlet
temperature of the hot fluid, since this would be a violation of the second law
of thermodynamics.
The relation above for the log mean temperature difference is developed using a parallelflow heat exchanger, but we can show by repeating the analysis
above for a counterflow heat exchanger that is also applicable to counterflow heat exchangers. But this time, T1 and T2 are expressed as shown in
Figure 13–15.
For specified inlet and outlet temperatures, the log mean temperature
difference for a counterflow heat exchanger is always greater than that for a
Tlm, PF, and thus a smaller
parallelflow heat exchanger. That is, Tlm, CF
surface area (and thus a smaller heat exchanger) is needed to achieve a specified heat transfer rate in a counterflow heat exchanger. Therefore, it is common practice to use counterflow arrangements in heat exchangers.
In a counterflow heat exchanger, the temperature difference between the
hot and the cold fluids will remain constant along the heat exchanger when
the heat capacity rates of the two fluids are equal (that is, T
constant
·
·
T2, and the last log
when Ch Cc or mhCph mcCpc). Then we have T1
mean temperature difference relation gives Tlm 0 , which is indeterminate.
0
It can be shown by the application of l’Hôpital’s rule that in this case we have
T1
T2, as expected.
Tlm
A condenser or a boiler can be considered to be either a parallel or counterflow heat exchanger since both approaches give the same result. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 683 683
CHAPTER 13 Multipass and CrossFlow Heat Exchangers:
Use of a Correction Factor T
Th, in The log mean temperature difference Tlm relation developed earlier is limited
to parallelflow and counterflow heat exchangers only. Similar relations are
also developed for crossflow and multipass shellandtube heat exchangers,
but the resulting expressions are too complicated because of the complex flow
conditions.
In such cases, it is convenient to relate the equivalent temperature difference to the log mean temperature difference relation for the counterflow
case as
Tlm F Tlm, CF t2
T1 t1
t1 Th
∆T
Cold
fluid Tc, in Th, in T2
t1 Th, out
Tc, out FIGURE 13–16
The variation of the fluid temperatures
in a counterflow doublepipe heat
exchanger.
Cold Tc, in
fluid Th,in T1
t2 (1328) Cold
fluid Hot
fluid Hot
fluid ·
(m Cp)tube side
·
(m C ) Th, out
Tc, in (1327) and
R Tc (1326) where F is the correction factor, which depends on the geometry of the heat
exchanger and the inlet and outlet temperatures of the hot and cold fluid
streams. The Tlm, CF is the log mean temperature difference for the case of
a counterflow heat exchanger with the same inlet and outlet temperatures
and is determined from Eq. 1325 by taking Tl Th, in Tc, out and T2
Th, out Tc, in (Fig. 13–17).
The correction factor is less than unity for a crossflow and multipass shellandtube heat exchanger. That is, F 1. The limiting value of F 1 corresponds to the counterflow heat exchanger. Thus, the correction factor F for a
heat exchanger is a measure of deviation of the Tlm from the corresponding
values for the counterflow case.
The correction factor F for common crossflow and shellandtube heat exchanger configurations is given in Figure 13–18 versus two temperature ratios
P and R defined as
P Hot fluid
Tc, out ∆T1 Crossflow or multipass
shellandtube heat exchanger ∆T2 Th,out Tc, out p shell side where the subscripts 1 and 2 represent the inlet and outlet, respectively. Note
that for a shellandtube heat exchanger, T and t represent the shell and
tubeside temperatures, respectively, as shown in the correction factor
charts. It makes no difference whether the hot or the cold fluid flows
through the shell or the tube. The determination of the correction factor F
requires the availability of the inlet and the outlet temperatures for both the
cold and hot fluids.
Note that the value of P ranges from 0 to 1. The value of R, on the other
hand, ranges from 0 to infinity, with R 0 corresponding to the phasechange
(condensation or boiling) on the shellside and R → to phasechange on the
tube side. The correction factor is F
1 for both of these limiting cases.
Therefore, the correction factor for a condenser or boiler is F 1, regardless
of the configuration of the heat exchanger. Heat transfer rate:
.
Q = UAsF ∆Tlm, CF
where ∆Tlm, CF = ∆T1 – ∆T2
ln(∆T1/∆T2) ∆T1 = Th,in – Tc,out
∆T2 = Th,out – Tc,in
and F = … (Fig. 13–18) FIGURE 13–17
The determination of the heat transfer
rate for crossflow and multipass
shellandtube heat exchangers
using the correction factor. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 684 684
HEAT TRANSFER Correction factor F 1.0 T1
t2
t1 0.9
0.8
R = 4.0 3.0 2.0 1.5 1.0 0.8 0.6 0.4 T2 0.2 0.7
T1 – T2
R = ——–
t2 – t1 0.6
0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (a) Oneshell pass and 2, 4, 6, etc. (any multiple of 2), tube passes t2 – t1
P = ——–
T1 – t1 Correction factor F 1.0 T1 0.9 t2 0.8
R = 4.0 3.0 2.0 1.5 t1 1.0 0.8 0.6 0.4 0.2 0.7 T2 0.6
0.5 T1 – T2
R = ——–
t2 – t1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 t2 – t1
P = ——–
T1 – t1 (b) Twoshell passes and 4, 8, 12, etc. (any multiple of 4), tube passes Correction factor F 1.0 T1 0.9
0.8
R = 4.0 3.0 2.0 1.5 1.0 0.8 0.6 0.4 0.2 t1 0.7
T1 – T2
R = ——–
t2 – t1 0.6
0.5 0 0.1 0.2 T2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (c) Singlepass crossflow with both fluids unmixed Correction factor F 1.0 FIGURE 13–18
Correction factor F charts
for common shellandtube and
crossflow heat exchangers (from
Bowman, Mueller, and Nagle, Ref. 2). t2 t2 – t1
P = ——–
T1 – t1 T1 0.9
0.8
R = 4.0 3.0 2.0 1.5 1.0 0.8 0.6 0.4 t1 0.2 t2 0.7
0.6
0.5
0 T1 – T2
R = ——–
t2 – t1
0.1 0.2 T2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (d ) Singlepass crossflow with one fluid mixed and the other unmixed t2 – t1
P = ——–
T1 – t1 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 685 685
CHAPTER 13 EXAMPLE 13–3 The Condensation of Steam in a Condenser Steam in the condenser of a power plant is to be condensed at a temperature of
30°C with cooling water from a nearby lake, which enters the tubes of the condenser at 14°C and leaves at 22°C. The surface area of the tubes is 45 m2, and
the overall heat transfer coefficient is 2100 W/m2 · °C. Determine the mass flow
rate of the cooling water needed and the rate of condensation of the steam in
the condenser. SOLUTION Steam is condensed by cooling water in the condenser of a power
plant. The mass flow rate of the cooling water and the rate of condensation are
to be determined.
Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well
insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes
in the kinetic and potential energies of fluid streams are negligible. 4 There is
no fouling. 5 Fluid properties are constant.
Properties The heat of vaporization of water at 30°C is hfg
2431 kJ/kg
and the specific heat of cold water at the average temperature of 18°C is
Cp 4184 J/kg · °C (Table A–9).
Analysis The schematic of the condenser is given in Figure 13–19. The condenser can be treated as a counterflow heat exchanger since the temperature
of one of the fluids (the steam) remains constant.
The temperature difference between the steam and the cooling water at the
two ends of the condenser is
T1
T2 Th, in
Th, out Tc, out
Tc, in (30
(30 22)°C
14)°C Steam
30°C Cooling
water
14°C 8°C
16°C That is, the temperature difference between the two fluids varies from 8°C at
one end to 16°C at the other. The proper average temperature difference between the two fluids is the logarithmic mean temperature difference (not the
arithmetic), which is determined from Tlm T2
T1
ln ( T1/ T2) 8 16
ln (8/16) 11.5°C
30°C This is a little less than the arithmetic mean temperature difference of
1 (8
16) 12°C. Then the heat transfer rate in the condenser is determined
2
from ·
Q UAs Tlm (2100 W/m2 · °C)(45 m2)(11.5°C) 1.087 106 W 1087 kW Therefore, the steam will lose heat at a rate of 1,087 kW as it flows through the
condenser, and the cooling water will gain practically all of it, since the condenser is well insulated.
The mass flow rate of the cooling water and the rate of the condensation of the
·
·
·
steam are determined from Q
[m Cp (Tout
Tin)]cooling water
(m hfg)steam to be ·
m cooling water ·
Q
Cp (Tout Tin) 1,087 kJ/s
(4.184 kJ/kg · °C)(22 22°C 14)°C 32.5 kg/s FIGURE 13–19
Schematic for Example 13–3. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 686 686
HEAT TRANSFER and ·
Q
hfg ·
m steam 1,087 kJ/s
2431 kJ/kg 0.45 kg/s Therefore, we need to circulate about 72 kg of cooling water for each 1 kg of
steam condensing to remove the heat released during the condensation process. EXAMPLE 13–4 Heating Water in a CounterFlow Heat Exchanger A counterflow doublepipe heat exchanger is to heat water from 20°C to 80°C
at a rate of 1.2 kg/s. The heating is to be accomplished by geothermal water
available at 160°C at a mass flow rate of 2 kg/s. The inner tube is thinwalled
and has a diameter of 1.5 cm. If the overall heat transfer coefficient of the heat
exchanger is 640 W/m2 · °C, determine the length of the heat exchanger required to achieve the desired heating. Hot
geothermal
160°C
water
2 kg/s
Cold
water
20°C
1.2 kg/s 80°C
D = 1.5 cm SOLUTION Water is heated in a counterflow doublepipe heat exchanger by
geothermal water. The required length of the heat exchanger is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well
insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes
in the kinetic and potential energies of fluid streams are negligible. 4 There is
no fouling. 5 Fluid properties are constant.
Properties We take the specific heats of water and geothermal fluid to be 4.18
and 4.31 kJ/kg · °C, respectively.
Analysis The schematic of the heat exchanger is given in Figure 13–20. The
rate of heat transfer in the heat exchanger can be determined from
·
Q ·
[m Cp(Tout Tin)]water (1.2 kg/s)(4.18 kJ/kg · °C)(80 20)°C 301 kW Noting that all of this heat is supplied by the geothermal water, the outlet
temperature of the geothermal water is determined to be ·
Q ·
[m Cp(Tin Tout)]geothermal → Tout Tin ·
Q
·C
mp 160°C FIGURE 13–20
Schematic for Example 13–4. 301 kW
(2 kg/s)(4.31 kJ/kg · °C) 125°C
Knowing the inlet and outlet temperatures of both fluids, the logarithmic mean
temperature difference for this counterflow heat exchanger becomes T1
T2 Th, in
Th, out Tc, out
Tc, in (160
(125 80)°C
20)°C 80°C
105°C and Tlm T1
T2
ln ( T1/ T2) 80 105
ln (80/105) 92.0°C Then the surface area of the heat exchanger is determined to be ·
Q UAs Tlm → As ·
Q
U Tlm 301,000 W
(640 W/m2 · °C)(92.0°C) 5.11 m2 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 687 687
CHAPTER 13 To provide this much heat transfer surface area, the length of the tube must be As DL → L As
D 5.11 m2
(0.015 m) 108 m Discussion The inner tube of this counterflow heat exchanger (and thus the
heat exchanger itself) needs to be over 100 m long to achieve the desired heat
transfer, which is impractical. In cases like this, we need to use a plate heat
exchanger or a multipass shellandtube heat exchanger with multiple passes of
tube bundles. EXAMPLE 13–5 Heating of Glycerin in a
Multipass Heat Exchanger A 2shell passes and 4tube passes heat exchanger is used to heat glycerin from
20°C to 50°C by hot water, which enters the thinwalled 2cmdiameter tubes
at 80°C and leaves at 40°C (Fig. 13–21). The total length of the tubes in the
heat exchanger is 60 m. The convection heat transfer coefficient is 25 W/m2 ·
°C on the glycerin (shell) side and 160 W/m2 · °C on the water (tube) side. Determine the rate of heat transfer in the heat exchanger (a) before any fouling occurs and (b) after fouling with a fouling factor of 0.0006 m2 · °C/ W occurs on
the outer surfaces of the tubes. SOLUTION Glycerin is heated in a 2shell passes and 4tube passes heat
exchanger by hot water. The rate of heat transfer for the cases of fouling and no
fouling are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well
insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to heat transfer to the cold fluid. 3 Changes in the
kinetic and potential energies of fluid streams are negligible. 4 Heat transfer coefficients and fouling factors are constant and uniform. 5 The thermal resistance of the inner tube is negligible since the tube is thinwalled and highly
conductive.
Analysis The tubes are said to be thinwalled, and thus it is reasonable to
assume the inner and outer surface areas of the tubes to be equal. Then the
heat transfer surface area becomes
As DL (0.02 m)(60 m) 3.77 m2 The rate of heat transfer in this heat exchanger can be determined from ·
Q UAs F Tlm, CF where F is the correction factor and Tlm, CF is the log mean temperature difference for the counterflow arrangement. These two quantities are determined
from T1
T2
Tlm, CF Th, in Tc, out (80 50)°C
Th, out Tc, in (40 20)°C
T1
T2
30 20
ln ( T1/ T2) ln (30/20) 30°C
20°C
24.7°C Cold
glycerin
20°C 40°C
Hot
water
80°C
50°C FIGURE 13–21
Schematic for Example 13–5. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 688 688
HEAT TRANSFER and
P t2
T1 t1
t1 40
20 80
80 0.67 R T1
t2 T2
t1 20
40 50
80 0.75 uF 0.91 (Fig. 13–18b) (a) In the case of no fouling, the overall heat transfer coefficient U is determined from 1 U 1
hi 1
1
160 W/m2 · °C 1
ho 1
25 W/m2 · °C 21.6 W/m2 · °C Then the rate of heat transfer becomes ·
Q UAs F Tlm, CF (21.6 W/m2 · °C)(3.77m2)(0.91)(24.7°C) 1830 W (b) When there is fouling on one of the surfaces, the overall heat transfer coefficient U is U 1
hi 1
1
ho Rf 1
160 W/m2 · °C 1
1
25 W/m2 · °C 0.0006 m2 · °C/ W 21.3 W/m2 · °C
The rate of heat transfer in this case becomes ·
Q UAs F Tlm, CF (21.3 W/m2 · °C)(3.77 m2)(0.91)(24.7°C) 1805 W Discussion Note that the rate of heat transfer decreases as a result of fouling,
as expected. The decrease is not dramatic, however, because of the relatively
low convection heat transfer coefficients involved. EXAMPLE 13–6 Cooling of an Automotive Radiator A test is conducted to determine the overall heat transfer coefficient in an automotive radiator that is a compact crossflow watertoair heat exchanger with
both fluids (air and water) unmixed (Fig. 13–22). The radiator has 40 tubes of
internal diameter 0.5 cm and length 65 cm in a closely spaced platefinned
matrix. Hot water enters the tubes at 90°C at a rate of 0.6 kg/s and leaves at
65°C. Air flows across the radiator through the interfin spaces and is heated
from 20°C to 40°C. Determine the overall heat transfer coefficient Ui of this radiator based on the inner surface area of the tubes. SOLUTION During an experiment involving an automotive radiator, the inlet
and exit temperatures of water and air and the mass flow rate of water are measured. The overall heat transfer coefficient based on the inner surface area is to
be determined.
Assumptions 1 Steady operating conditions exist. 2 Changes in the kinetic
and potential energies of fluid streams are negligible. 3 Fluid properties are
constant. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 689 689
CHAPTER 13
90°C Air flow
(unmixed)
20°C 40°C 65°C
Water flow
(unmixed) Properties The specific heat of water at the average temperature of (90 65)/
2 77.5°C is 4.195 kJ/kg · °C.
Analysis The rate of heat transfer in this radiator from the hot water to the air
is determined from an energy balance on water flow, ·
Q ·
[mCp (Tin Tout)]water (0.6 kg/s)(4.195 kJ/kg · °C)(90 65)°C 62.93 kW The tubeside heat transfer area is the total surface area of the tubes, and is
determined from Ai n Di L (40) (0.005 m)(0.65 m) 0.408 m2 Knowing the rate of heat transfer and the surface area, the overall heat transfer
coefficient can be determined from ·
Q Ui Ai F Tlm, CF → Ui ·
Q
Ai F Tlm, CF where F is the correction factor and Tlm, CF is the log mean temperature difference for the counterflow arrangement. These two quantities are found to be T1
T2
Tlm, CF Th, in Tc, out (90 40)°C
Th, out Tc, in (65 20)°C
T1
T2
50 45
ln ( T1/ T2) ln (50/45) 50°C
45°C
47.6°C and P t2
T1 t1
t1 65
20 90
90 0.36 R T1
t2 T2
t1 20
65 40
90 0.80 uF 0.97 (Fig. 13–18c) Substituting, the overall heat transfer coefficient Ui is determined to be Ui ·
Q
Ai F Tlm, CF 62,930 W
(0.408 m2)(0.97)(47.6°C) 3341 W/m2 · °C Note that the overall heat transfer coefficient on the air side will be much lower
because of the large surface area involved on that side. FIGURE 13–22
Schematic for Example 13–6. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 690 690
HEAT TRANSFER 13–5 I THE EFFECTIVENESS–NTU METHOD The log mean temperature difference (LMTD) method discussed in Section
13–4 is easy to use in heat exchanger analysis when the inlet and the outlet
temperatures of the hot and cold fluids are known or can be determined from
an energy balance. Once Tlm, the mass flow rates, and the overall heat transfer coefficient are available, the heat transfer surface area of the heat exchanger can be determined from
·
Q UAs Tlm Therefore, the LMTD method is very suitable for determining the size
of a heat exchanger to realize prescribed outlet temperatures when the mass
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:
1. Select the type of heat exchanger suitable for the application.
2. Determine any unknown inlet or outlet temperature and the heat transfer
rate using an energy balance.
3. Calculate the log mean temperature difference Tlm and the correction
factor F, if necessary.
4. Obtain (select or calculate) the value of the overall heat transfer coefficient U.
5. Calculate the heat transfer surface area As .
The task is completed by selecting a heat exchanger that has a heat transfer
surface area equal to or larger than As .
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. The heat transfer surface
area A of the heat exchanger in this case is known, but the outlet temperatures
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
and London came up with a method in 1955 called the effectiveness–NTU
method, which greatly simplified heat exchanger analysis.
This method is based on a dimensionless parameter called the heat transfer effectiveness , defined as
·
Q
Qmax Actual heat transfer rate
Maximum possible heat transfer rate (1329) The actual heat transfer rate in a heat exchanger can be determined from an
energy balance on the hot or cold fluids and can be expressed as
·
Q Cc(Tc, out Tc, in) Ch(Th, in Th, out) (1330) cen58933_ch13.qxd 9/9/2002 9:57 AM Page 691 691
CHAPTER 13 ·
·
where Cc mcCpc and Ch mcCph are the heat capacity rates of the cold and
the hot fluids, respectively.
To determine the maximum possible heat transfer rate in a heat exchanger,
we first recognize that the maximum temperature difference in a heat exchanger is the difference between the inlet temperatures of the hot and cold
fluids. That is,
Tmax Th, in Tc, in (1331) 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.e., Cc Ch). When Cc Ch,
which is usually the 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, at which point the heat transfer will come to a halt.
Therefore, the maximum possible heat transfer rate in a heat exchanger is
(Fig. 13–23)
·
Q max Cmin(Th, in Tc, in) ·
where Cmin is the smaller of Ch
mhCph and Cc
clarified by the following example. EXAMPLE 13–7 20°C
25 kg/s Cold
water (1332) ·
mcCpc. This is further Hot
oil
130°C
40 kg/s
. Upper Limit for Heat Transfer in a
Heat Exchanger Cold water enters a counterflow heat exchanger at 10°C at a rate of 8 kg/s,
where it is heated by a hot water stream that enters the heat exchanger at 70°C
at a rate of 2 kg/s. Assuming the specific heat of water to remain constant at
Cp 4.18 kJ/kg · °C, determine the maximum heat transfer rate and the outlet
temperatures of the cold and the hot water streams for this limiting case. SOLUTION Cold and hot water streams enter a heat exchanger at specified
temperatures and flow rates. The maximum rate of heat transfer in the heat exchanger is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well
insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to heat transfer to the cold fluid. 3 Changes in the
kinetic and potential energies of fluid streams are negligible. 4 Heat transfer coefficients and fouling factors are constant and uniform. 5 The thermal resistance of the inner tube is negligible since the tube is thinwalled and highly
conductive.
Properties The specific heat of water is given to be Cp 4.18 kJ/kg · °C.
Analysis A schematic of the heat exchanger is given in Figure 13–24. The heat
capacity rates of the hot and cold fluids are determined from
Ch ·
m hCph (2 kg/s)(4.18 kJ/kg · °C) 8.36 kW/°C Cc ·
m cCpc (8 kg/s)(4.18 kJ/kg · °C) 33.4 kW/°C Cc = mcCpc = 104.5 kW/°C
. Ch = mcCph = 92 kW/°C
Cmin = 92 kW/°C
∆Tmax = Th, in – Tc, in = 110°C
. Qmax = Cmin ∆Tmax = 10,120 kW FIGURE 13–23
The determination of the maximum
rate of heat transfer in a heat
exchanger. 10°C
8 kg/s Cold
water Hot
water
70°C
2 kg/s and FIGURE 13–24
Schematic for Example 13–7. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 692 692
HEAT TRANSFER Therefore Cmin Ch 8.36 kW/°C which is the smaller of the two heat capacity rates. Then the maximum heat
transfer rate is determined from Eq. 1332 to be ·
Q max Cmin(Th, in Tc, in)
(8.36 kW/°C)(70 10)°C 502 kW
That is, the maximum possible heat transfer rate in this heat exchanger is
502 kW. This value would be approached in a counterflow heat exchanger with
a very large heat transfer surface area.
The maximum temperature difference in this heat exchanger is Tmax
Th, in Tc, in (70 10)°C 60°C. Therefore, the hot water cannot be cooled
by more than 60°C (to 10°C) in this heat exchanger, and the cold water cannot
be heated by more than 60°C (to 70°C), no matter what we do. The outlet temperatures of the cold and the hot streams in this limiting case are determined
to be ·
Q Cold
fluid Hot
fluid
.
mh ,Cph
. . Q = mh Cph ∆Th
.
= mc Cpc ∆Tc
If . . mc Cpc = mh Cph Tc, in) → Tc, out Tc, in ·
Q .
mc ,Cpc Cc(Tc, out
Ch(Th, in Th, out) → Th, out Th, in ·
Q
Cc
·
Q
Ch 10°C 502 kW
33.4 kW/°C 25°C 70°C 502 kW
8.38 kW/°C 10°C Discussion Note that the hot water is cooled to the limit of 10°C (the inlet
temperature of the cold water stream), but the cold water is heated to 25°C only
when maximum heat transfer occurs in the heat exchanger. This is not surprising, since the mass flow rate of the hot water is only onefourth that of the cold
water, and, as a result, the temperature of the cold water increases by 0.25°C
for each 1°C drop in the temperature of the hot water.
You may be tempted to think that the cold water should be heated to 70°C in
the limiting case of maximum heat transfer. But this will require the temperature of the hot water to drop to 170°C (below 10°C), which is impossible.
Therefore, heat transfer in a heat exchanger reaches its maximum value when
the fluid with the smaller heat capacity rate (or the smaller mass flow rate when
both fluids have the same specific heat value) experiences the maximum temperature change. This example explains why we use Cmin in the evaluation of
·
Q max instead of Cmax.
We can show that the hot water will leave at the inlet temperature of the cold
water and vice versa in the limiting case of maximum heat transfer when the mass
flow rates of the hot and cold water streams are identical (Fig. 13–25). We can
also show that the outlet temperature of the cold water will reach the 70°C limit
when the mass flow rate of the hot water is greater than that of the cold water. then ∆Th = ∆Tc FIGURE 13–25
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. ·
The determination of Q max requires the availability of the inlet temperature
of the hot and cold fluids and their mass flow rates, which are usually specified. Then, once the effectiveness of the heat exchanger is known, the actual
·
heat transfer rate Q can be determined from
·
Q ·
Q max Cmin(Th, in Tc, in) (1333) cen58933_ch13.qxd 9/9/2002 9:57 AM Page 693 693
CHAPTER 13 Therefore, the effectiveness of a heat exchanger enables us to determine the
heat transfer rate without knowing the outlet temperatures of the fluids.
The effectiveness of a heat exchanger depends on the geometry of the heat
exchanger as well as the flow arrangement. Therefore, different types of heat
exchangers have different effectiveness relations. Below we illustrate the development of the effectiveness relation for the doublepipe parallelflow
heat exchanger.
Equation 1323 developed in Section 13–4 for a parallelflow heat exchanger can be rearranged as
ln Th, out
Th, in Tc, out
Tc, in UAs
1
Cc Cc
Ch (1334) Tc, in) (1335) Also, solving Eq. 1330 for Th, out gives
Th, out Cc
(T
Ch c, out Th, in Substituting this relation into Eq. 1334 after adding and subtracting
Tc, in gives
Th, in Tc, in Tc, in ln Cc
(T
Ch c, out Tc, out Th, in Tc, in) UAs
1
Cc Tc, in Cc
Ch which simplifies to
ln 1 1 Cc Tc, out
Ch Th, in Tc, in
Tc, in UAs
1
Cc Cc
Ch (1336) We now manipulate the definition of effectiveness to obtain
·
Q
·
Q max Cc(Tc, out
Cmin(Th, in Tc, in)
Tc, in) → Tc, out
Th, in Tc, in
Tc, in Cmin
Cc Substituting this result into Eq. 1336 and solving for gives the following
relation for the effectiveness of a parallelflow heat exchanger:
1 UAs
1
Cc exp parallel flow Cc
Ch
(1337) Cc Cmin
Ch Cc 1 Taking either Cc or Ch to be Cmin (both approaches give the same result), the
relation above can be expressed more conveniently as
1 UAs
1
Cmin exp parallel flow 1 Cmin
Cmax Cmin
Cmax
(1338) Again Cmin is the smaller heat capacity ratio and Cmax is the larger one, and it
makes no difference whether Cmin belongs to the hot or cold fluid. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 694 694
HEAT TRANSFER Effectiveness relations of the heat exchangers typically involve the dimensionless group UAs /Cmin. This quantity is called the number of transfer units
NTU and is expressed as
NTU UAs
Cmin UAs
·C )
(m p min (1339) where U is the overall heat transfer coefficient and As is the heat transfer surface
area of the heat exchanger. 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.
In heat exchanger analysis, it is also convenient to define another dimensionless quantity called the capacity ratio c as
Cmin
Cmax c (1340) It can be shown that the effectiveness of a heat exchanger is a function of the
number of transfer units NTU and the capacity ratio c. That is,
function (UAs /Cmin, Cmin /Cmax) function (NTU, c) Effectiveness relations have been developed for a large number of heat exchangers, and the results are given in Table 13–4. The effectivenesses of some
common types of heat exchangers are also plotted in Figure 13–26. More
TABLE 13–4
Effectiveness relations for heat exchangers: NTU UAs /Cmin and
·
·
c Cmin/Cmax (m Cp)min/(m Cp)max (Kays and London, Ref. 5.)
Heat exchanger
type
1 Double pipe:
Parallelflow
Counterflow
2 Shell and tube:
Oneshell pass
2, 4, . . . tube
passes Effectiveness relation
1 exp [ NTU(1 c )]
1c
1 exp [ NTU(1 c )]
1 c exp [ NTU(1 c )] 21 c 1 c2 1
1 exp [ NTU
exp [ NTU 1
1 3 Crossflow
(singlepass)
Both fluids
unmixed 1 Cmax mixed,
Cmin unmixed 1
c (1 Cmin mixed,
Cmax unmixed 1 exp 1 exp( NTU) 4 All heat
exchangers
with c 0 exp NTU0.22
[exp ( c NTU0.78)
c exp {1 c[1
1
c [1 exp ( NTU)]})
exp ( c NTU)] 1] c 2]
c 2] 1 9:57 AM Page 695 695
CHAPTER 13
100 100 5
0.71.00 /
mi
n 5
5
0.2 C Tube
fluid 60 =0 0.5
0 Effectiveness ε, % mi
n ax
Cm 80 0.25
0.50
0.75
1.00 60
40 =0 / ax
Cm C Effectiveness ε, % 80 Shell fluid 40
Tube
fluid
20 20
Shell fluid 0
3
4
5
1
2
Number of transfer units NTU = AsU/Cmin (a) Parallelflow (b) Counterflow 100
n Shell fluid Effectiveness ε, % 40 C 60 80 0.50
0.75
1.00 m /C 100 =0
0.25 x
ma m
i 80 3
4
5
1
2
Number of transfer units NTU = AsU/Cmin C 0 Effectiveness ε, % =0
x
C ma 25
/
0. 0
in
0.5
0.75
1.00 60
Shell fluid 40
20 20
Tube fluid Tube fluid
1 2 3 4 0 5 Number of transfer units NTU = AsU/Cmin
(c) Oneshell pass and 2, 4, 6, … tube passes 60 =0 5
0.2 0
0.5 5
0.7 00
1. 80 5 d
xe
mi
un ,
=0 0.25
4
0.5
2
0.75
1.33
1 Hot
fluid C mi xe Cold fluid
40 4 (d ) Twoshell passes and 4, 8, 12, … tube passes Effectiveness ε, % mi
n / ax
Cm 3 2 Number of transfer units NTU = AsU/Cmin 100 100
80 1 d /C 0 C 9/9/2002 Effectiveness ε, % cen58933_ch13.qxd 60
Mixed
fluid 40 20 20 0 0 Unmixed fluid
1
2
3
4
5
Number of transfer units NTU = AsU/Cmin (e) Crossflow with both fluids unmixed 1 2 3 4 5 Number of transfer units NTU = AsU/Cmin
( f ) Crossflow with one fluid mixed and the
other unmixed FIGURE 13–26
Effectiveness for heat exchangers (from Kays and London, Ref. 5). cen58933_ch13.qxd 9/9/2002 9:57 AM Page 696 696
HEAT TRANSFER extensive effectiveness charts and relations are available in the literature. The
dashed lines in Figure 13–26f are for the case of Cmin unmixed and Cmax mixed
and the solid lines are for the opposite case. The analytic relations for the effectiveness give more accurate results than the charts, since reading errors in
charts are unavoidable, and the relations are very suitable for computerized
analysis of heat exchangers.
We make these following observations from the effectiveness relations and
charts already given:
1. The value of the effectiveness ranges from 0 to 1. It increases rapidly
1
Counterflow ε Crossflow with
both fluids unmixed
0.5
Parallelflow
(for c = 1) 0 0 1 2
3
4
NTU = UAs /Cmin 5 FIGURE 13–27
For a specified NTU and capacity
ratio c, the counterflow heat
exchanger has the highest
effectiveness and the parallelflow the
lowest. ε 0 ε = 1 – e– NTU
(All heat exchangers
with c = 0) 0 1 2
3
4
NTU = UAs /Cmin FIGURE 13–28
The effectiveness relation reduces to
1 exp( NTU) for all
max
heat exchangers when the capacity
ratio c 0. max 1 exp( NTU) (1341) regardless of the type of heat exchanger (Fig. 13–28). Note that the
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, which is realized when the heat capacity rates of
the two fluids are equal. 1 0.5 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 large size cannot be justified
economically, since a large increase in NTU in this case corresponds to
a small increase in effectiveness. Thus, a heat exchanger with a very
high effectiveness may be highly desirable from a heat transfer point of
view but rather undesirable from an economical point of view.
2. For a given NTU and capacity ratio c Cmin /Cmax, the counterflow
heat exchanger has the highest effectiveness, followed closely by the
crossflow heat exchangers with both fluids unmixed. As you might
expect, the lowest effectiveness values are encountered in parallelflow
heat exchangers (Fig. 13–27).
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 1. For a given
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 during a phasechange process in a condenser or boiler. All
effectiveness relations in this case reduce to 5 Once the quantities c Cmin /Cmax and NTU UAs /Cmin have been evaluated, the effectiveness can be determined from either the charts or (preferably) the effectiveness relation for the specified type of heat exchanger. Then
·
the rate of heat transfer Q and the outlet temperatures Th, out and Tc, out can be
determined from Eqs. 1333 and 1330, respectively. Note that the analysis of
heat exchangers with unknown outlet temperatures is a straightforward matter
with the effectiveness–NTU method but requires rather tedious iterations with
the LMTD method.
We mentioned earlier that when all the inlet and outlet temperatures
are specified, the size of the heat exchanger can easily be determined
using the LMTD method. Alternatively, it can also be determined from the
effectiveness–NTU method by first evaluating the effectiveness from its
definition (Eq. 1329) and then the NTU from the appropriate NTU relation in
Table 13–5. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 697 697
CHAPTER 13 TABLE 13–5
NTU relations for heat exchangers NTU UAs /Cmin and c
·
·
(m Cp )min/(m Cp )max (Kays and London, Ref. 5.)
Heat exchanger type NTU relation 1 Doublepipe:
Parallelflow NTU Counterflow NTU Cmin /Cmax 2 Shell and tube:
Oneshell pass
2, 4, . . . tube passes 1
c NTU Cmin mixed,
Cmax unmixed
4 All heat exchangers
with c 0 (1
c 1 NTU 3 Crossflow (singlepass)
Cmax mixed,
Cmin unmixed ln [1 1 ln ln 1 1
1 c 1
1 c )] c 2 ln ln (1 NTU 1
1 c
c 1
1 c2
c2 c)
c ln [c ln (1
c
ln(1
) NTU 2/
2/ ) 1] Note that the relations in Table 13–5 are equivalent to those in Table 13–4.
Both sets of relations are given for convenience. The relations in Table 13–4
give the effectiveness directly when NTU is known, and the relations in
Table 13–5 give the NTU directly when the effectiveness is known. EXAMPLE 13–8 Using the Effectiveness–NTU Method Repeat Example 13–4, which was solved with the LMTD method, using the
effectiveness–NTU method. SOLUTION The schematic of the heat exchanger is redrawn in Figure 13–29,
and the same assumptions are utilized.
Analysis In the effectiveness–NTU method, we first determine the heat capacity rates of the hot and cold fluids and identify the smaller one:
Ch
Cc ·
mhCph
·
mcCpc (2 kg/s)(4.31 kJ/kg · °C) 8.62 kW/°C
(1.2 kg/s)(4.18 kJ/kg · °C) 5.02 kW/°C Cold
water
20°C
1.2 kg/s Hot
geothermal 160°C
brine
2 kg/s 80°C
D = 1.5 cm Therefore, Cmin Cc FIGURE 13–29
Schematic for Example 13–8. 5.02 kW/°C and c Cmin /Cmax 5.02/8.62 0.583 Then the maximum heat transfer rate is determined from Eq. 1332 to be ·
Q max Cmin(Th, in Tc, in)
(5.02 kW/°C)(160
702.8 kW 20)°C cen58933_ch13.qxd 9/9/2002 9:57 AM Page 698 698
HEAT TRANSFER That is, the maximum possible heat transfer rate in this heat exchanger is
702.8 kW. The actual rate of heat transfer in the heat exchanger is ·
Q ·
[mCp(Tout Tin)]water (1.2 kg/s)(4.18 kJ/kg · °C)(80 20)°C 301.0 kW Thus, the effectiveness of the heat exchanger is ·
Q
·
Q max 301.0 kW
702.8 kW 0.428 Knowing the effectiveness, the NTU of this counterflow heat exchanger can be
determined from Figure 13–26b or the appropriate relation from Table 13–5.
We choose the latter approach for greater accuracy: NTU 1
c 1 ln c 1
1 1
0.583 1 ln 0.428 1
0.428 0.583 0.651 1 Then the heat transfer surface area becomes NTU UAs
Cmin → As NTU Cmin
U (0.651)(5020 W/°C)
640 W/m2 · °C 5.11 m2 To provide this much heat transfer surface area, the length of the tube must be As DL → L As
D 5.11 m2
(0.015 m) 108 m Discussion Note that we obtained the same result with the effectiveness–NTU
method in a systematic and straightforward manner. EXAMPLE 13–9 150°C Hot oil is to be cooled by water in a 1shellpass and 8tubepasses heat
exchanger. The tubes are thinwalled and are made of copper with an internal
diameter of 1.4 cm. The length of each tube pass in the heat exchanger is 5 m,
and the overall heat transfer coefficient is 310 W/m2 · °C. Water flows through
the tubes at a rate of 0.2 kg/s, and the oil through the shell at a rate of 0.3 kg/s.
The water and the oil enter at temperatures of 20°C and 150°C, respectively.
Determine the rate of heat transfer in the heat exchanger and the outlet temperatures of the water and the oil. Oil
0.3 kg/s 20°C
Water
0.2 kg/s FIGURE 13–30
Schematic for Example 13–9. Cooling Hot Oil by Water in a Multipass
Heat Exchanger SOLUTION Hot oil is to be cooled by water in a heat exchanger. The mass flow
rates and the inlet temperatures are given. The rate of heat transfer and the outlet temperatures are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well
insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 The thickness of the tube is negligible since it is thinwalled. 4 Changes in the kinetic
and potential energies of fluid streams are negligible. 5 The overall heat transfer coefficient is constant and uniform.
Analysis The schematic of the heat exchanger is given in Figure 13–30. The
outlet temperatures are not specified, and they cannot be determined from an cen58933_ch13.qxd 9/9/2002 9:57 AM Page 699 699
CHAPTER 13 energy balance. The use of the LMTD method in this case will involve tedious
iterations, and thus the –NTU method is indicated. The first step in the –NTU
method is to determine the heat capacity rates of the hot and cold fluids and
identify the smaller one: Ch
Cc ·
mhCph
·
mcCpc (0.3 kg/s)(2.13 kJ/kg · °C)
(0.2 kg/s)(4.18 kJ/kg · °C) 0.639 kW/°C
0.836 kW/°C Therefore, Cmin Ch 0.639 kW/°C and c Cmin
Cmax 0.639
0.836 0.764 Then the maximum heat transfer rate is determined from Eq. 1332 to be ·
Q max Cmin(Th, in Tc, in)
(0.639 kW/°C)(150 20)°C 83.1 kW That is, the maximum possible heat transfer rate in this heat exchanger is 83.1
kW. The heat transfer surface area is As n( DL) 8 (0.014 m)(5 m) 1.76 m2 Then the NTU of this heat exchanger becomes NTU UAs
Cmin (310 W/m2 · °C)(1.76 m2)
639 W/°C 0.853 The effectiveness of this heat exchanger corresponding to c
NTU 0.853 is determined from Figure 13–26c to be 0.764 and 0.47
We could also determine the effectiveness from the third relation in Table 13–4
more accurately but with more labor. Then the actual rate of heat transfer
becomes ·
Q ·
Q max (0.47)(83.1 kW) 39.1 kW Finally, the outlet temperatures of the cold and the hot fluid streams are determined to be
·
Q
·
Q Cc(Tc, out Tc, in) → Tc, out Tc, in
Cc
39.1 kW
66.8°C
20°C
0.836 kW/°C
·
Q
·
Q Ch(Th, in Th, out) → Th, out Th, in
Ch 150°C 39.1 kW
0.639 kW/°C 88.8°C Therefore, the temperature of the cooling water will rise from 20°C to 66.8°C as
it cools the hot oil from 150°C to 88.8°C in this heat exchanger. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 700 700
HEAT TRANSFER 13–6 I SELECTION OF HEAT EXCHANGERS Heat exchangers are complicated devices, and the results obtained with the
simplified approaches presented above should be used with care. For example,
we assumed that the overall heat transfer coefficient U is constant throughout
the heat exchanger and that the convection heat transfer coefficients can be
predicted using the convection correlations. However, it should be kept in
mind that the uncertainty in the predicted value of U can even exceed 30 percent. Thus, it is natural to tend to overdesign the heat exchangers in order to
avoid unpleasant surprises.
Heat transfer enhancement in heat exchangers is usually accompanied by
increased pressure drop, and thus higher pumping power. Therefore, any gain
from the enhancement in heat transfer should be weighed against the cost of
the accompanying pressure drop. Also, some thought should be given to
which fluid should pass through the tube side and which through the shell
side. Usually, the more viscous fluid is more suitable for the shell side (larger
passage area and thus lower pressure drop) and the fluid with the higher pressure for the tube side.
Engineers in industry often find themselves in a position to select heat
exchangers to accomplish certain heat transfer tasks. Usually, the goal is to
heat or cool a certain fluid at a known mass flow rate and temperature to a
desired temperature. Thus, the rate of heat transfer in the prospective heat
exchanger is
·
Q max ·
mCp(Tin Tout) which gives the heat transfer requirement of the heat exchanger before having
any idea about the heat exchanger itself.
An engineer going through catalogs of heat exchanger manufacturers will
be overwhelmed by the type and number of readily available offtheshelf heat
exchangers. The proper selection depends on several factors. Heat Transfer Rate
This is the most important quantity in the selection of a heat exchanger. A heat
exchanger should be capable of transferring heat at the specified rate in order
to achieve the desired temperature change of the fluid at the specified mass
flow rate. Cost
Budgetary limitations usually play an important role in the selection of heat
exchangers, except for some specialized cases where “money is no object.”
An offtheshelf heat exchanger has a definite cost advantage over those made
to order. However, in some cases, none of the existing heat exchangers will
do, and it may be necessary to undertake the expensive and timeconsuming
task of designing and manufacturing a heat exchanger from scratch to suit the
needs. This is often the case when the heat exchanger is an integral part of the
overall device to be manufactured.
The operation and maintenance costs of the heat exchanger are also important considerations in assessing the overall cost. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 701 701
CHAPTER 13 Pumping Power
In a heat exchanger, both fluids are usually forced to flow by pumps or fans
that consume electrical power. The annual cost of electricity associated with
the operation of the pumps and fans can be determined from
Operating cost (Pumping power, kW) (Hours of operation, h)
(Price of electricity, $/kWh) where the pumping power is the total electrical power consumed by the
motors of the pumps and fans. For example, a heat exchanger that involves a
1hp pump and a 1 hp fan (1 hp 0.746 kW) operating 8 h a day and 5 days
3
a week will consume 2017 kWh of electricity per year, which will cost $161.4
at an electricity cost of 8 cents/kWh.
Minimizing the pressure drop and the mass flow rate of the fluids will minimize the operating cost of the heat exchanger, but it will maximize the size of
the heat exchanger and thus the initial cost. As a rule of thumb, doubling the
mass flow rate will reduce the initial cost by half but will increase the pumping power requirements by a factor of roughly eight.
Typically, fluid velocities encountered in heat exchangers range between 0.7
and 7 m/s for liquids and between 3 and 30 m/s for gases. Low velocities are helpful in avoiding erosion, tube vibrations, and noise as well as pressure drop. Size and Weight
Normally, the smaller and the lighter the heat exchanger, the better it is. This
is especially the case in the automotive and aerospace industries, where size
and weight requirements are most stringent. Also, a larger heat exchanger normally carries a higher price tag. The space available for the heat exchanger in
some cases limits the length of the tubes that can be used. Type
The type of heat exchanger to be selected depends primarily on the type of
fluids involved, the size and weight limitations, and the presence of any phasechange processes. For example, a heat exchanger is suitable to cool a liquid
by a gas if the surface area on the gas side is many times that on the liquid
side. On the other hand, a plate or shellandtube heat exchanger is very suitable for cooling a liquid by another liquid. Materials
The materials used in the construction of the heat exchanger may be an important consideration in the selection of heat exchangers. For example, the
thermal and structural stress effects need not be considered at pressures below
15 atm or temperatures below 150°C. But these effects are major considerations above 70 atm or 550°C and seriously limit the acceptable materials of
the heat exchanger.
A temperature difference of 50°C or more between the tubes and the shell
will probably pose differential thermal expansion problems and needs to be
considered. In the case of corrosive fluids, we may have to select expensive cen58933_ch13.qxd 9/9/2002 9:57 AM Page 702 702
HEAT TRANSFER corrosionresistant materials such as stainless steel or even titanium if we are
not willing to replace lowcost heat exchangers frequently. Other Considerations
There are other considerations in the selection of heat exchangers that may
or may not be important, depending on the application. For example, being
leaktight is an important consideration when toxic or expensive fluids are involved. Ease of servicing, low maintenance cost, and safety and reliability are
some other important considerations in the selection process. Quietness is one
of the primary considerations in the selection of liquidtoair heat exchangers
used in heating and airconditioning applications. EXAMPLE 13–10 Installing a Heat Exchanger to Save Energy
and Money In a dairy plant, milk is pasteurized by hot water supplied by a natural gas furnace. The hot water is then discharged to an open floor drain at 80°C at a rate
of 15 kg/min. The plant operates 24 h a day and 365 days a year. The furnace
has an efficiency of 80 percent, and the cost of the natural gas is $0.40 per
therm (1 therm 105,500 kJ). The average temperature of the cold water entering the furnace throughout the year is 15°C. The drained hot water cannot be
returned to the furnace and recirculated, because it is contaminated during the
process.
In order to save energy, installation of a watertowater heat exchanger to preheat the incoming cold water by the drained hot water is proposed. Assuming
that the heat exchanger will recover 75 percent of the available heat in the hot
water, determine the heat transfer rating of the heat exchanger that needs to be
purchased and suggest a suitable type. Also, determine the amount of money
this heat exchanger will save the company per year from natural gas savings. SOLUTION A watertowater heat exchanger is to be installed to transfer energy Hot
80°C water
Cold
water
15°C FIGURE 13–31
Schematic for Example 13–10. from drained hot water to the incoming cold water to preheat it. The rate of heat
transfer in the heat exchanger and the amount of energy and money saved per
year are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The effectiveness of the
heat exchanger remains constant.
Properties We use the specific heat of water at room temperature, Cp 4.18 kJ/
kg · °C (Table A–9), and treat it as a constant.
Analysis A schematic of the prospective heat exchanger is given in Figure
13–31. The heat recovery from the hot water will be a maximum when it leaves
the heat exchanger at the inlet temperature of the cold water. Therefore, ·
Q max ·
mhCp(Th, in Tc, in) 15
kg/s (4.18 kJ/kg · °C)(80
60
67.9 kJ/s 15)°C That is, the existing hot water stream has the potential to supply heat at a rate
of 67.9 kJ/s to the incoming cold water. This value would be approached in a
counterflow heat exchanger with a very large heat transfer surface area. A heat
exchanger of reasonable size and cost can capture 75 percent of this heat cen58933_ch13.qxd 9/9/2002 9:57 AM Page 703 703
CHAPTER 13 transfer potential. Thus, the heat transfer rating of the prospective heat exchanger must be ·
Q ·
Q max (0.75)(67.9 kJ/s) 50.9 kJ/s That is, the heat exchanger should be able to deliver heat at a rate of 50.9 kJ/s
from the hot to the cold water. An ordinary plate or shellandtube heat exchanger
should be adequate for this purpose, since both sides of the heat exchanger involve the same fluid at comparable flow rates and thus comparable heat transfer
coefficients. (Note that if we were heating air with hot water, we would have to
specify a heat exchanger that has a large surface area on the air side.)
The heat exchanger will operate 24 h a day and 365 days a year. Therefore,
the annual operating hours are Operating hours (24 h/day)(365 days/year) 8760 h/year Noting that this heat exchanger saves 50.9 kJ of energy per second, the energy
saved during an entire year will be Energy saved (Heat transfer rate)(Operation time)
(50.9 kJ/s)(8760 h/year)(3600 s/h)
1.605 109 kJ/year The furnace is said to be 80 percent efficient. That is, for each 80 units of heat
supplied by the furnace, natural gas with an energy content of 100 units must
be supplied to the furnace. Therefore, the energy savings determined above result in fuel savings in the amount of Fuel saved Energy saved
Furnace efficiency
19,020 therms/year 1.605 109 kJ/year 1 therm
0.80
105,500 kJ Noting that the price of natural gas is $0.40 per therm, the amount of money
saved becomes Money saved (Fuel saved) (Price of fuel)
(19,020 therms/year)($0.40/therm)
$7607/ year Therefore, the installation of the proposed heat exchanger will save the company $7607 a year, and the installation cost of the heat exchanger will probably be paid from the fuel savings in a short time. SUMMARY
Heat exchangers are devices that allow the exchange of heat
between two fluids without allowing them to mix with each
other. Heat exchangers are manufactured in a variety of
types, the simplest being the doublepipe heat exchanger. In a
parallelflow type, both the hot and cold fluids enter the heat
exchanger at the same end and move in the same direction,
whereas in a counterflow type, the hot and cold fluids enter
the heat exchanger at opposite ends and flow in opposite directions. In compact heat exchangers, the two fluids move
perpendicular to each other, and such a flow configuration is
called crossflow. Other common types of heat exchangers in
industrial applications are the plate and the shellandtube heat
exchangers.
Heat transfer in a heat exchanger usually involves convection
in each fluid and conduction through the wall separating the two
fluids. In the analysis of heat exchangers, it is convenient to cen58933_ch13.qxd 9/9/2002 9:57 AM Page 704 704
HEAT TRANSFER work with an overall heat transfer coefficient U or a total thermal resistance R, expressed as
1
UAs 1
Ui Ai 1
Uo Ao R 1
hi Ai 1
ho Ao Rwall where the subscripts i and o stand for the inner and outer surfaces of the wall that separates the two fluids, respectively.
When the wall thickness of the tube is small and the thermal
conductivity of the tube material is high, the last relation simplifies to
1
U 1
hi 1
Ui Ai
1
hi Ai 1
R
Uo Ao
Rf, i ln (Do /Di )
Ai
2 kL where Rf, o
Ao is the log mean temperature difference, which is the suitable
form of the average temperature difference for use in the analysis of heat exchangers. Here T1 and T2 represent the temperature differences between the two fluids at the two ends (inlet
and outlet) of the heat exchanger. For crossflow and multipass
shellandtube heat exchangers, the logarithmic mean temperature difference is related to the counterflow one Tlm, CF as 1
ho Ao Di L and Ao
Do L are the areas of the inner and
where Ai
outer surfaces and Rf, i and Rf, o are the fouling factors at those
surfaces.
In a wellinsulated heat exchanger, the rate of heat transfer
from the hot fluid is equal to the rate of heat transfer to the cold
one. That is,
·
·
Q mcCpc(Tc, out Tc, in) Cc(Tc, out Tc, in) Tlm ·
mhCph(Th, in Th, out) Ch(Th, in Th, out) where the subscripts c and h stand for the cold and hot fluids,
respectively, and the product of the mass flow rate and the spe·
cific heat of a fluid mCp is called the heat capacity rate.
Of the two methods used in the analysis of heat exchangers,
the log mean temperature difference (or LMTD) method is F Tlm, CF where F is the correction factor, which depends on the geometry of the heat exchanger and the inlet and outlet temperatures
of the hot and cold fluid streams.
The effectiveness of a heat exchanger is defined as
·
Q
Actual heat transfer rate
Qmax Maximum possible heat transfer rate
where
·
Q max and
·
Q T1
T2
ln ( T1/ T2) Tlm 1
ho where U Ui Uo. The effects of fouling on both the inner
and the outer surfaces of the tubes of a heat exchanger can be
accounted for by
1
UAs best suited for determining the size of a heat exchanger
when all the inlet and the outlet temperatures are known. The
effectiveness–NTU method is best suited to predict the outlet
temperatures of the hot and cold fluid streams in a specified
heat exchanger. In the LMTD method, the rate of heat transfer
is determined from
·
Q UAs Tlm Cmin(Th, in Tc, in) ·
·
and Cmin is the smaller of Ch mhCph and Cc mcCpc. The effectiveness of heat exchangers can be determined from effectiveness relations or charts.
The selection or design of a heat exchanger depends on
several factors such as the heat transfer rate, cost, pressure
drop, size, weight, construction type, materials, and operating
environment. REFERENCES AND SUGGESTED READING
1. N. Afgan and E. U. Schlunder. Heat Exchanger: Design
and Theory Sourcebook. Washington D.C.: McGrawHill/Scripta, 1974. 4. K. A. Gardner. “Variable Heat Transfer Rate Correction in
Multipass Exchangers, Shell Side Film Controlling.”
Transactions of the ASME 67 (1945), pp. 31–38. 2. R. A. Bowman, A. C. Mueller, and W. M. Nagle. “Mean
Temperature Difference in Design.” Transactions of the
ASME 62 (1940), p. 283. 5. W. M. Kays and A. L. London. Compact Heat
Exchangers. 3rd ed. New York: McGrawHill, 1984. 3. A. P. Fraas. Heat Exchanger Design. 2d ed. New York:
John Wiley & Sons, 1989. 6. W. M. Kays and H. C. Perkins. In Handbook of Heat
Transfer, ed. W. M. Rohsenow and J. P. Hartnett. New
York: McGrawHill, 1972, Chap. 7. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 705 705
CHAPTER 13 7. A. C. Mueller. “Heat Exchangers.” In Handbook of Heat
Transfer, ed. W. M. Rohsenow and J. P. Hartnett. New
York: McGrawHill, 1972, Chap. 18.
8. M. N. Özisik. Heat Transfer—A Basic Approach. New
,
York: McGrawHill, 1985.
9. E. U. Schlunder. Heat Exchanger Design Handbook.
Washington, D.C.: Hemisphere, 1982.
10. Standards of Tubular Exchanger Manufacturers
Association. New York: Tubular Exchanger
Manufacturers Association, latest ed. 11. R. A. Stevens, J. Fernandes, and J. R. Woolf. “Mean
Temperature Difference in One, Two, and Three Pass
Crossflow Heat Exchangers.” Transactions of the ASME
79 (1957), pp. 287–297.
12. J. Taborek, G. F. Hewitt, and N. Afgan. Heat Exchangers:
Theory and Practice. New York: Hemisphere, 1983.
13. G. Walker. Industrial Heat Exchangers. Washington,
D.C.: Hemisphere, 1982. PROBLEMS*
Types of Heat Exchangers
13–1C Classify heat exchangers according to flow type and
explain the characteristics of each type.
13–2C Classify heat exchangers according to construction
type and explain the characteristics of each type.
13–3C When is a heat exchanger classified as being compact? Do you think a doublepipe heat exchanger can be classified as a compact heat exchanger?
13–4C How does a crossflow heat exchanger differ from a
counterflow one? What is the difference between mixed and
unmixed fluids in crossflow? 13–10C Under what conditions is the thermal resistance of
the tube in a heat exchanger negligible?
13–11C Consider a doublepipe parallelflow heat exchanger
of length L. The inner and outer diameters of the inner tube are
D1 and D2, respectively, and the inner diameter of the outer
tube is D3. Explain how you would determine the two heat
transfer surface areas Ai and Ao. When is it reasonable to
assume Ai Ao As?
13–12C Is the approximation hi ho h for the convection
heat transfer coefficient in a heat exchanger a reasonable one
when the thickness of the tube wall is negligible? 13–5C What is the role of the baffles in a shellandtube heat
exchanger? How does the presence of baffles affect the heat
transfer and the pumping power requirements? Explain. 13–13C Under what conditions can the overall heat transfer
coefficient of a heat exchanger be determined from U (1/hi
1/ho) 1? 13–6C Draw a 1shellpass and 6tubepasses shellandtube
heat exchanger. What are the advantages and disadvantages of
using 6 tube passes instead of just 2 of the same diameter? 13–14C What are the restrictions on the relation UAs Ui Ai
Uo Ao for a heat exchanger? Here As is the heat transfer surface area and U is the overall heat transfer coefficient. 13–7C Draw a 2shellpasses and 8tubepasses shellandtube heat exchanger. What is the primary reason for using so
many tube passes? 13–15C In a thinwalled doublepipe heat exchanger, when
is the approximation U hi a reasonable one? Here U is the
overall heat transfer coefficient and hi is the convection heat
transfer coefficient inside the tube. 13–8C What is a regenerative heat exchanger? How does a
static type of regenerative heat exchanger differ from a dynamic type? The Overall Heat Transfer Coefficient
13–9C What are the heat transfer mechanisms involved during heat transfer from the hot to the cold fluid?
*Problems designated by a “C” are concept questions, and
students are encouraged to answer them all. Problems designated
by an “E” are in English units, and the SI users can ignore them.
Problems with an EESCD icon
are solved using EES, and
complete solutions together with parametric studies are included
on the enclosed CD. Problems with a computerEES icon
are
comprehensive in nature, and are intended to be solved with a
computer, preferably using the EES software that accompanies
this text. 13–16C What are the common causes of fouling in a heat
exchanger? How does fouling affect heat transfer and pressure drop?
13–17C How is the thermal resistance due to fouling in a
heat exchanger accounted for? How do the fluid velocity and
temperature affect fouling?
13–18 A doublepipe heat exchanger is constructed of a copper (k 380 W/m · °C) inner tube of internal diameter Di
1.2 cm and external diameter Do 1.6 cm and an outer tube of
diameter 3.0 cm. The convection heat transfer coefficient is reported to be hi 700 W/m2 · °C on the inner surface of the
tube and ho 1400 W/m2 · °C on its outer surface. For a fouling factor Rf, i 0.0005 m2 · °C/W on the tube side and Rf, o
0.0002 m2 · °C/W on the shell side, determine (a) the thermal
resistance of the heat exchanger per unit length and (b) the cen58933_ch13.qxd 9/9/2002 9:57 AM Page 706 706
HEAT TRANSFER overall heat transfer coefficients Ui and Uo based on the inner
and outer surface areas of the tube, respectively.
13–19
Reconsider Problem 13–18. Using EES (or
other) software, investigate the effects of pipe
conductivity and heat transfer coefficients on the thermal resistance of the heat exchanger. Let the thermal conductivity vary
from 10 W/m · ºC to 400 W/m · ºC, the convection heat transfer coefficient from 500 W/m2 · ºC to 1500 W/m2 · ºC on the inner surface, and from 1000 W/m2 · ºC to 2000 W/m2 · ºC on the
outer surface. Plot the thermal resistance of the heat exchanger
as functions of thermal conductivity and heat transfer coefficients, and discuss the results.
13–20 Water at an average temperature of 107°C and an average velocity of 3.5 m/s flows through a 5mlong stainless
steel tube (k 14.2 W/m · °C) in a boiler. The inner and outer
diameters of the tube are Di 1.0 cm and Do 1.4 cm, respectively. If the convection heat transfer coefficient at the
outer surface of the tube where boiling is taking place is ho
8400 W/m2 · °C, determine the overall heat transfer coefficient
Ui of this boiler based on the inner surface area of the tube.
13–21 Repeat Problem 13–20, assuming a fouling factor
Rf, i 0.0005 m2 · °C/W on the inner surface of the tube.
13–22
Reconsider Problem 13–21. Using EES (or
other) software, plot the overall heat transfer
coefficient based on the inner surface as a function of fouling
factor Fi as it varies from 0.0001 m2 · ºC/W to 0.0008 m2 · ºC/W,
and discuss the results.
13–23 A long thinwalled doublepipe heat exchanger with
tube and shell diameters of 1.0 cm and 2.5 cm, respectively, is
used to condense refrigerant 134a by water at 20°C. The refrigerant flows through the tube, with a convection heat transfer coefficient of hi 5000 W/m2 · °C. Water flows through the
shell at a rate of 0.3 kg/s. Determine the overall heat transfer
Answer: 2020 W/m2 · °C
coefficient of this heat exchanger.
13–24 Repeat Problem 13–23 by assuming a 2mmthick
layer of limestone (k 1.3 W/m · °C) forms on the outer surface of the inner tube.
13–25
Reconsider Problem 13–24. Using EES (or
other) software, plot the overall heat transfer
coefficient as a function of the limestone thickness as it varies
from 1 mm to 3 mm, and discuss the results.
13–26E Water at an average temperature of 140°F and an
average velocity of 8 ft/s flows through a thinwalled 3 in.4
diameter tube. The water is cooled by air that flows across the
12 ft/s at an average temperature
tube with a velocity of
of 80°F. Determine the overall heat transfer coefficient. Analysis of Heat Exchangers
13–27C What are the common approximations made in the
analysis of heat exchangers? 13–28C Under what conditions is the heat transfer relation
·
·
·
Q mcCpc(Tc, out Tc, in) mhCph (Th, in Th, out)
valid for a heat exchanger?
13–29C What is the heat capacity rate? What can you say
about the temperature changes of the hot and cold fluids in a
heat exchanger if both fluids have the same capacity rate?
What does a heat capacity of infinity for a fluid in a heat exchanger mean?
13–30C Consider a condenser in which steam at a specified
temperature is condensed by rejecting heat to the cooling
water. If the heat transfer rate in the condenser and the temperature rise of the cooling water is known, explain how the
rate of condensation of the steam and the mass flow rate of
the cooling water can be determined. Also, explain how the
total thermal resistance R of this condenser can be evaluated
in this case.
13–31C Under what conditions will the temperature rise of
the cold fluid in a heat exchanger be equal to the temperature
drop of the hot fluid? The Log Mean Temperature Difference Method
·
13–32C In the heat transfer relation Q UAs Tlm for a heat
exchanger, what is Tlm called? How is it calculated for a
parallelflow and counterflow heat exchanger?
13–33C How does the log mean temperature difference for a
heat exchanger differ from the arithmetic mean temperature
difference (AMTD)? For specified inlet and outlet temperatures, which one of these two quantities is larger?
13–34C The temperature difference between the hot and cold
fluids in a heat exchanger is given to be T1 at one end and T2
at the other end. Can the logarithmic temperature difference
Tlm of this heat exchanger be greater than both T1 and T2?
Explain.
13–35C Can the logarithmic mean temperature difference
Tlm of a heat exchanger be a negative quantity? Explain.
13–36C Can the outlet temperature of the cold fluid in a heat
exchanger be higher than the outlet temperature of the hot fluid
in a parallelflow heat exchanger? How about in a counterflow
heat exchanger? Explain.
13–37C For specified inlet and outlet temperatures, for what
kind of heat exchanger will the Tlm be greatest: doublepipe
parallelflow, doublepipe counterflow, crossflow, or multipass shellandtube heat exchanger?
·
13–38C In the heat transfer relation Q
UAs F Tlm for a
heat exchanger, what is the quantity F called? What does it represent? Can F be greater than one?
13–39C When the outlet temperatures of the fluids in a heat
exchanger are not known, is it still practical to use the LMTD
method? Explain. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 707 707
CHAPTER 13 13–40C Explain how the LMTD method can be used to determine the heat transfer surface area of a multipass shellandtube heat exchanger when all the necessary information,
including the outlet temperatures, is given.
13–41 Steam in the condenser of a steam power plant is to be
2305 kJ/kg) with
condensed at a temperature of 50°C (hfg
cooling water (Cp 4180 J/kg · °C) from a nearby lake, which
enters the tubes of the condenser at 18°C and leaves at 27°C.
The surface area of the tubes is 58 m2, and the overall heat
transfer coefficient is 2400 W/m2 · °C. Determine the mass flow
rate of the cooling water needed and the rate of condensation of
Answers: 101 kg/s, 1.65 kg/s
the steam in the condenser.
Steam
50°C
27°C 13–45 A test is conducted to determine the overall heat transfer coefficient in a shellandtube oiltowater heat exchanger
that has 24 tubes of internal diameter 1.2 cm and length 2 m
in a single shell. Cold water (Cp 4180 J/kg · °C) enters the
tubes at 20°C at a rate of 5 kg/s and leaves at 55°C. Oil
2150 J/kg · °C) flows through the shell and is cooled
(Cp
from 120°C to 45°C. Determine the overall heat transfer coefficient Ui of this heat exchanger based on the inner surface area
Answer: 13.9 kW/m2 · °C
of the tubes.
13–46 A doublepipe counterflow heat exchanger is to cool
2560 J/kg · °C) flowing at a rate of
ethylene glycol (Cp
3.5 kg/s from 80°C to 40°C by water (Cp 4180 J/kg · °C) that
enters at 20°C and leaves at 55°C. The overall heat transfer coefficient based on the inner surface area of the tube is
250 W/m2 · °C. Determine (a) the rate of heat transfer, (b) the
mass flow rate of water, and (c) the heat transfer surface area
on the inner side of the tube.
Cold water
20°C
Hot glycol 18°C
80°C
3.5 kg/s Water
50°C 40°C FIGURE P13–41
FIGURE P13–46
13–42 A doublepipe parallelflow heat exchanger is to heat
water (Cp 4180 J/kg · °C) from 25°C to 60°C at a rate of 0.2
kg/s. The heating is to be accomplished by geothermal water
(Cp 4310 J/kg · °C) available at 140°C at a mass flow rate of
0.3 kg/s. The inner tube is thinwalled and has a diameter of 0.8
cm. If the overall heat transfer coefficient of the heat exchanger
is 550 W/m2 · °C, determine the length of the heat exchanger
required to achieve the desired heating.
13–43
Reconsider Problem 13–42. Using EES (or
other) software, investigate the effects of temperature and mass flow rate of geothermal water on the length
of the heat exchanger. Let the temperature vary from 100ºC to
200ºC, and the mass flow rate from 0.1 kg/s to 0.5 kg/s. Plot
the length of the heat exchanger as functions of temperature
and mass flow rate, and discuss the results.
13–44E A 1shellpass and 8tubepasses heat exchanger is
used to heat glycerin (Cp 0.60 Btu/lbm · °F) from 65°F to
140°F by hot water (Cp 1.0 Btu/lbm · °F) that enters the thinwalled 0.5in.diameter tubes at 175°F and leaves at 120°F.
The total length of the tubes in the heat exchanger is 500 ft.
The convection heat transfer coefficient is 4 Btu/h · ft2 · °F
on the glycerin (shell) side and 50 Btu/h · ft2 · °F on the water
(tube) side. Determine the rate of heat transfer in the heat exchanger (a) before any fouling occurs and (b) after fouling with
a fouling factor of 0.002 h · ft2 · °F/Btu occurs on the outer surfaces of the tubes. 13–47 Water (Cp
4180 J/kg · °C) enters the 2.5cminternaldiameter tube of a doublepipe counterflow heat exchanger at 17°C at a rate of 3 kg/s. It is heated by steam
2203 kJ/kg) in the shell. If the
condensing at 120°C (hfg
overall heat transfer coefficient of the heat exchanger is 1500
W/m2 · °C, determine the length of the tube required in order to
heat the water to 80°C.
13–48 A thinwalled doublepipe counterflow heat exchanger is to be used to cool oil (Cp 2200 J/kg · °C) from
150°C to 40°C at a rate of 2 kg/s by water (Cp 4180 J/kg ·
°C) that enters at 22°C at a rate of 1.5 kg/s. The diameter of the
tube is 2.5 cm, and its length is 6 m. Determine the overall heat
transfer coefficient of this heat exchanger.
13–49 Reconsider Problem 13–48. Using EES (or
other) software, investigate the effects of oil
exit temperature and water inlet temperature on the overall heat
transfer coefficient of the heat exchanger. Let the oil exit temperature vary from 30ºC to 70ºC and the water inlet temperature from 5ºC to 25ºC. Plot the overall heat transfer coefficient
as functions of the two temperatures, and discuss the results.
13–50 Consider a watertowater doublepipe heat exchanger
whose flow arrangement is not known. The temperature measurements indicate that the cold water enters at 20°C and
leaves at 50°C, while the hot water enters at 80°C and leaves at cen58933_ch13.qxd 9/9/2002 9:57 AM Page 708 708
HEAT TRANSFER 45°C. Do you think this is a parallelflow or counterflow heat
exchanger? Explain.
13–51 Cold water (Cp 4180 J/kg · °C) leading to a shower
enters a thinwalled doublepipe counterflow heat exchanger
at 15°C at a rate of 0.25 kg/s and is heated to 45°C by hot water
(Cp 4190 J/kg · °C) that enters at 100°C at a rate of 3 kg/s. If
the overall heat transfer coefficient is 1210 W/m2 · °C, determine the rate of heat transfer and the heat transfer surface area
of the heat exchanger.
13–52 Engine oil (Cp 2100 J/kg · °C) is to be heated from
20°C to 60°C at a rate of 0.3 kg/s in a 2cmdiameter thinwalled copper tube by condensing steam outside at a temper2174 kJ/kg). For an overall heat
ature of 130°C (hfg
transfer coefficient of 650 W/m2 · °C, determine the rate of
heat transfer and the length of the tube required to achieve it.
Answers: 25.2 kW, 7.0 m
Steam
130°C
Oil
20°C
0.3 kg/s 60°C
55°C FIGURE P13–52
13–53E Geothermal water (Cp 1.03 Btu/lbm · °F) is to be
used as the heat source to supply heat to the hydronic heating
system of a house at a rate of 30 Btu/s in a doublepipe
counterflow heat exchanger. Water (Cp 1.0 Btu/lbm · °F) is
heated from 140°F to 200°F in the heat exchanger as the geothermal water is cooled from 310°F to 180°F. Determine the
mass flow rate of each fluid and the total thermal resistance of
this heat exchanger.
13–54 Glycerin (Cp 2400 J/kg · °C) at 20°C and 0.3 kg/s is
to be heated by ethylene glycol (Cp 2500 J/kg · °C) at 60°C
in a thinwalled doublepipe parallelflow heat exchanger. The
temperature difference between the two fluids is 15°C at
the outlet of the heat exchanger. If the overall heat transfer coefficient is 240 W/m2 · °C and the heat transfer surface area is
3.2 m2, determine (a) the rate of heat transfer, (b) the outlet
temperature of the glycerin, and (c) the mass flow rate of the
ethylene glycol.
13–55 Air (Cp 1005 J/kg · °C) is to be preheated by hot exhaust gases in a crossflow heat exchanger before it enters the
furnace. Air enters the heat exchanger at 95 kPa and 20°C at a
rate of 0.8 m3/s. The combustion gases (Cp 1100 J/kg · °C)
enter at 180°C at a rate of 1.1 kg/s and leave at 95°C. The product of the overall heat transfer coefficient and the heat transfer
surface area is AU 1200 W/°C. Assuming both fluids to be
unmixed, determine the rate of heat transfer and the outlet temperature of the air. Air
95 kPa
20°C
0.8 m3/s Exhaust gases
1.1 kg/s
95°C FIGURE P13–55
13–56 A shellandtube heat exchanger with 2shell passes
and 12tube passes is used to heat water (Cp 4180 J/kg · °C)
in the tubes from 20°C to 70°C at a rate of 4.5 kg/s. Heat is
supplied by hot oil (Cp 2300 J/kg · °C) that enters the shell
side at 170°C at a rate of 10 kg/s. For a tubeside overall heat
transfer coefficient of 600 W/m2 · °C, determine the heat transAnswer: 15 m2
fer surface area on the tube side.
13–57 Repeat Problem 13–56 for a mass flow rate of 2 kg/s
for water.
13–58 A shellandtube heat exchanger with 2shell passes and
8tube passes is used to heat ethyl alcohol (Cp 2670 J/kg · °C)
in the tubes from 25°C to 70°C at a rate of 2.1 kg/s. The heating
is to be done by water (Cp 4190 J/kg · °C) that enters the shell
side at 95°C and leaves at 45°C. If the overall heat transfer coefficient is 950 W/m2 · °C, determine the heat transfer surface area
of the heat exchanger.
Water
95°C
70°C
Ethyl
alcohol
25°C
2.1 kg/s
45°C (8tube passes) FIGURE P13–58
13–59 A shellandtube heat exchanger with 2shell passes and
12tube passes is used to heat water (Cp 4180 J/kg · °C) with
ethylene glycol (Cp 2680 J/kg · °C). Water enters the tubes at
22°C at a rate of 0.8 kg/s and leaves at 70°C. Ethylene glycol enters the shell at 110°C and leaves at 60°C. If the overall heat
transfer coefficient based on the tube side is 280 W/m2 · °C,
determine the rate of heat transfer and the heat transfer surface
area on the tube side. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 709 709
CHAPTER 13 13–60 Reconsider Problem 13–59. Using EES (or other)
software, investigate the effect of the mass flow
rate of water on the rate of heat transfer and the tubeside surface
area. Let the mass flow rate vary from 0.4 kg/s to 2.2 kg/s. Plot
the rate of heat transfer and the surface area as a function of the
mass flow rate, and discuss the results.
13–61E Steam is to be condensed on the shell side of a
1shellpass and 8tubepasses condenser, with 50 tubes in
each pass at 90°F (hfg 1043 Btu/lbm). Cooling water (Cp
1.0 Btu/lbm · °F) enters the tubes at 60°F and leaves at 73°F.
The tubes are thinwalled and have a diameter of 3/4 in. and
length of 5 ft per pass. If the overall heat transfer coefficient
is 600 Btu/h · ft2 · °F, determine (a) the rate of heat transfer,
(b) the rate of condensation of steam, and (c) the mass flow
rate of cold water.
Steam
90°F
20 lbm/s
73°F 60°F
Water
90°F FIGURE P13–61E 13–62E Reconsider Problem 13–61E. Using EES (or
other) software, investigate the effect of the
condensing steam temperature on the rate of heat transfer, the
rate of condensation of steam, and the mass flow rate of cold
water. Let the steam temperature vary from 80ºF to 120ºF. Plot
the rate of heat transfer, the condensation rate of steam, and the
mass flow rate of cold water as a function of steam temperature, and discuss the results.
13–63 A shellandtube heat exchanger with 1shell pass and
20–tube passes is used to heat glycerin (Cp 2480 J/kg · °C)
in the shell, with hot water in the tubes. The tubes are thinwalled and have a diameter of 1.5 cm and length of 2 m per
pass. The water enters the tubes at 100°C at a rate of 5 kg/s and
leaves at 55°C. The glycerin enters the shell at 15°C and leaves
at 55°C. Determine the mass flow rate of the glycerin and the
overall heat transfer coefficient of the heat exchanger.
13–64 In a binary geothermal power plant, the working fluid
isobutane is to be condensed by air in a condenser at 75°C
255.7 kJ/kg) at a rate of 2.7 kg/s. Air enters the con(hfg
denser at 21ºC and leaves at 28ºC. The heat transfer surface Air
28°C
Isobutane 75°C
2.7 kg/s
Air
21°C FIGURE P13–64 area based on the isobutane side is 24 m2. Determine the mass
flow rate of air and the overall heat transfer coefficient.
13–65 Hot exhaust gases of a stationary diesel engine are to
be used to generate steam in an evaporator. Exhaust gases
(Cp 1051 J/kg · ºC) enter the heat exchanger at 550ºC at a
rate of 0.25 kg/s while water enters as saturated liquid and
evaporates at 200ºC (hfg 1941 kJ/kg). The heat transfer surface area of the heat exchanger based on water side is 0.5 m2
and overall heat transfer coefficient is 1780 W/m2 · ºC. Determine the rate of heat transfer, the exit temperature of exhaust
gases, and the rate of evaporation of water.
13–66 Reconsider Problem 13–65. Using EES (or
other) software, investigate the effect of the exhaust gas inlet temperature on the rate of heat transfer, the exit
temperature of exhaust gases, and the rate of evaporation of
water. Let the temperature of exhaust gases vary from 300ºC to
600ºC. Plot the rate of heat transfer, the exit temperature of exhaust gases, and the rate of evaporation of water as a function
of the temperature of the exhaust gases, and discuss the results.
13–67 In a textile manufacturing plant, the waste dyeing water (Cp 4295 J/g · ºC) at 75°C is to be used to preheat fresh
water (Cp 4180 J/kg · ºC) at 15ºC at the same flow rate in a
doublepipe counterflow heat exchanger. The heat transfer
surface area of the heat exchanger is 1.65 m2 and the overall
heat transfer coefficient is 625 W/m2 · ºC. If the rate of heat
transfer in the heat exchanger is 35 kW, determine the outlet
temperature and the mass flow rate of each fluid stream. Fresh
water
15°C Dyeing
water
75°C Th, out
Tc, out FIGURE P13–67 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 710 710
HEAT TRANSFER The Effectiveness–NTU Method
13–68C Under what conditions is the effectiveness–NTU
method definitely preferred over the LMTD method in heat exchanger analysis?
13–69C What does the effectiveness of a heat exchanger represent? Can effectiveness be greater than one? On what factors
does the effectiveness of a heat exchanger depend?
13–70C For a specified fluid pair, inlet temperatures, and
mass flow rates, what kind of heat exchanger will have the
highest effectiveness: doublepipe parallelflow, doublepipe
counterflow, crossflow, or multipass shellandtube heat
exchanger?
13–71C Explain how you can evaluate the outlet temperatures of the cold and hot fluids in a heat exchanger after its effectiveness is determined.
13–72C Can the temperature of the hot fluid drop below the
inlet temperature of the cold fluid at any location in a heat exchanger? Explain.
13–73C Can the temperature of the cold fluid rise above the
inlet temperature of the hot fluid at any location in a heat exchanger? Explain.
13–74C Consider a heat exchanger in which both fluids have
the same specific heats but different mass flow rates. Which
fluid will experience a larger temperature change: the one with
the lower or higher mass flow rate?
13–75C Explain how the maximum possible heat transfer
·
rate Q max in a heat exchanger can be determined when the mass
flow rates, specific heats, and the inlet temperatures of the two
·
fluids are specified. Does the value of Q max depend on the type
of the heat exchanger?
13–76C Consider two doublepipe counterflow heat exchangers that are identical except that one is twice as long as
the other one. Which heat exchanger is more likely to have a
higher effectiveness?
13–77C Consider a doublepipe counterflow heat exchanger. In order to enhance heat transfer, the length of the heat
exchanger is now doubled. Do you think its effectiveness will
also double?
13–78C Consider a shellandtube watertowater heat exchanger with identical mass flow rates for both the hot and cold
water streams. Now the mass flow rate of the cold water is reduced by half. Will the effectiveness of this heat exchanger increase, decrease, or remain the same as a result of this
modification? Explain. Assume the overall heat transfer coefficient and the inlet temperatures remain the same.
13–79C Under what conditions can a counterflow heat exchanger have an effectiveness of one? What would your answer be for a parallelflow heat exchanger?
13–80C How is the NTU of a heat exchanger defined? What
does it represent? Is a heat exchanger with a very large NTU
(say, 10) necessarily a good one to buy? 13–81C Consider a heat exchanger that has an NTU of 4.
Someone proposes to double the size of the heat exchanger and
thus double the NTU to 8 in order to increase the effectiveness
of the heat exchanger and thus save energy. Would you support
this proposal?
13–82C Consider a heat exchanger that has an NTU of 0.1.
Someone proposes to triple the size of the heat exchanger and
thus triple the NTU to 0.3 in order to increase the effectiveness
of the heat exchanger and thus save energy. Would you support
this proposal?
13–83 Air (Cp 1005 J/kg · °C) enters a crossflow heat exchanger at 10°C at a rate of 3 kg/s, where it is heated by a hot
4190 J/kg · °C) that enters the heat exwater stream (Cp
changer at 95°C at a rate of 1 kg/s. Determine the maximum
heat transfer rate and the outlet temperatures of the cold and
the hot water streams for that case.
13–84 Hot oil (Cp 2200 J/kg · °C) is to be cooled by water
(Cp 4180 J/kg · °C) in a 2shellpass and 12tubepass heat
exchanger. The tubes are thinwalled and are made of copper
with a diameter of 1.8 cm. The length of each tube pass in the
heat exchanger is 3 m, and the overall heat transfer coefficient
is 340 W/m2 · °C. Water flows through the tubes at a total rate
of 0.1 kg/s, and the oil through the shell at a rate of 0.2 kg/s.
The water and the oil enter at temperatures 18°C and 160°C,
respectively. Determine the rate of heat transfer in the heat exchanger and the outlet temperatures of the water and the oil.
Answers: 36.2 kW, 104.6°C, 77.7°C
Oil
160°C
0.2 kg/s Water
18°C
0.1 kg/s (12tube passes) FIGURE P13–84
13–85 Consider an oiltooil doublepipe heat exchanger
whose flow arrangement is not known. The temperature measurements indicate that the cold oil enters at 20°C and leaves at
55°C, while the hot oil enters at 80°C and leaves at 45°C. Do
you think this is a parallelflow or counterflow heat exchanger? Why? Assuming the mass flow rates of both fluids to
be identical, determine the effectiveness of this heat exchanger.
13–86E Hot water enters a doublepipe counterflow watertooil heat exchanger at 220°F and leaves at 100°F. Oil enters
at 70°F and leaves at 150°F. Determine which fluid has the
smaller heat capacity rate and calculate the effectiveness of this
heat exchanger.
13–87 A thinwalled doublepipe parallelflow heat exchanger is used to heat a chemical whose specific heat is 1800 cen58933_ch13.qxd 9/9/2002 9:57 AM Page 711 711
CHAPTER 13 J/kg · °C with hot water (Cp 4180 J/kg · °C). The chemical
enters at 20°C at a rate of 3 kg/s, while the water enters at
110°C at a rate of 2 kg/s. The heat transfer surface area
of the heat exchanger is 7 m2 and the overall heat transfer coefficient is 1200 W/m2 · °C. Determine the outlet temperatures
of the chemical and the water. Chemical
20°C
3 kg/s Hot water
110°C
2 kg/s by hot oil (Cp 2200 J/kg · °C) that enters at 120°C. If the heat
transfer surface area and the overall heat transfer coefficients
are 6.2 m2 and 320 W/m2 · °C, respectively, determine the outlet temperature and the mass flow rate of oil using (a) the
LMTD method and (b) the –NTU method.
13–92 Water (Cp 4180 J/kg · °C) is to be heated by solarheated hot air (Cp 1010 J/kg · °C) in a doublepipe counterflow heat exchanger. Air enters the heat exchanger at 90°C at a
rate of 0.3 kg/s, while water enters at 22°C at a rate of 0.1 kg/s.
The overall heat transfer coefficient based on the inner side of
the tube is given to be 80 W/m2 · °C. The length of the tube is
12 m and the internal diameter of the tube is 1.2 cm. Determine
the outlet temperatures of the water and the air.
13–93 FIGURE P13–87
13–88 Reconsider Problem 13–87. Using EES (or
other) software, investigate the effects of the inlet temperatures of the chemical and the water on their outlet
temperatures. Let the inlet temperature vary from 10ºC to 50ºC
for the chemical and from 80ºC to 150ºC for water. Plot the
outlet temperature of each fluid as a function of the inlet temperature of that fluid, and discuss the results.
13–89 A crossflow airtowater heat exchanger with an effectiveness of 0.65 is used to heat water (Cp 4180 J/kg · °C)
with hot air (Cp 1010 J/kg · °C). Water enters the heat exchanger at 20°C at a rate of 4 kg/s, while air enters at 100°C at
a rate of 9 kg/s. If the overall heat transfer coefficient based on
the water side is 260 W/m2 · °C, determine the heat transfer
surface area of the heat exchanger on the water side. Assume
Answer: 52.4 m2
both fluids are unmixed. 13–90 Water (Cp
4180 J/kg · °C) enters the 2.5cminternaldiameter tube of a doublepipe counterflow heat exchanger at 17°C at a rate of 3 kg/s. Water is heated by steam
2203 kJ/kg) in the shell. If the
condensing at 120°C (hfg
overall heat transfer coefficient of the heat exchanger is 900
W/m2 · °C, determine the length of the tube required in order to
heat the water to 80°C using (a) the LMTD method and (b) the
–NTU method. Reconsider Problem 13–92. Using EES (or
other) software, investigate the effects of the
mass flow rate of water and the tube length on the outlet temperatures of water and air. Let the mass flow rate vary from
0.05 kg/s to 1.0 kg/s and the tube length from 5 m to 25 m. Plot
the outlet temperatures of the water and the air as the functions of the mass flow rate and the tube length, and discuss the
results. 13–94E A thinwalled doublepipe heat exchanger is to be
used to cool oil (Cp 0.525 Btu/lbm · °F) from 300°F to 105°F
at a rate of 5 lbm/s by water (Cp 1.0 Btu/lbm · °F) that enters
at 70°F at a rate of 3 lbm/s. The diameter of the tube is 1 in. and
its length is 20 ft. Determine the overall heat transfer coefficient of this heat exchanger using (a) the LMTD method and
(b) the –NTU method.
13–95 Cold water (Cp
4180 J/kg · °C) leading to a
shower enters a thinwalled doublepipe counterflow heat
exchanger at 15°C at a rate of 0.25 kg/s and is heated to 45°C
by hot water (Cp 4190 J/kg · °C) that enters at 100°C at a
rate of 3 kg/s. If the overall heat transfer coefficient is 950
W/m2 · °C, determine the rate of heat transfer and the heat
transfer surface area of the heat exchanger using the –NTU
Answers: 31.35 kW, 0.482 m2
method.
Cold water
15°C
0.25 kg/s
Hot water 13–91 Ethanol is vaporized at 78°C (hfg
846 kJ/kg) in a
doublepipe parallelflow heat exchanger at a rate of 0.03 kg/s 100°C
3 kg/s Oil
120°C 45°C FIGURE P13–95 Ethanol
78°C
0.03 kg/s FIGURE P13–91 13–96 Reconsider Problem 13–95. Using EES (or
other) software, investigate the effects of the
inlet temperature of hot water and the heat transfer coefficient on the rate of heat transfer and surface area. Let the inlet
temperature vary from 60ºC to 120ºC and the overall heat cen58933_ch13.qxd 9/9/2002 9:57 AM Page 712 712
HEAT TRANSFER transfer coefficient from 750 W/m2 · °C to 1250 W/m2 · °C.
Plot the rate of heat transfer and surface area as functions of
inlet temperature and the heat transfer coefficient, and discuss
the results. Steam
30°C 13–97 Glycerin (Cp 2400 J/kg · °C) at 20°C and 0.3 kg/s
2500 J/kg · °C) at
is to be heated by ethylene glycol (Cp
60°C and the same mass flow rate in a thinwalled doublepipe parallelflow heat exchanger. If the overall heat transfer
coefficient is 380 W/m2 · °C and the heat transfer surface area
is 5.3 m2, determine (a) the rate of heat transfer and (b) the
outlet temperatures of the glycerin and the glycol.
13–98 A crossflow heat exchanger consists of 40 thinwalled tubes of 1cm diameter located in a duct of 1 m 1 m
crosssection. There are no fins attached to the tubes. Cold
water (Cp 4180 J/kg · °C) enters the tubes at 18°C with an
average velocity of 3 m/s, while hot air (Cp 1010 J/kg · °C)
enters the channel at 130°C and 105 kPa at an average velocity of 12 m/s. If the overall heat transfer coefficient is 130
W/m2 · °C, determine the outlet temperatures of both fluids
and the rate of heat transfer. 1m
Hot air
130°C
105 kPa
12 m/s 1m Water
18°C
3 m/s FIGURE P13–98 13–99 A shellandtube heat exchanger with 2shell
passes and 8tube passes is used to heat ethyl
alcohol (Cp 2670 J/kg · °C) in the tubes from 25°C to 70°C
at a rate of 2.1 kg/s. The heating is to be done by water (Cp
4190 J/kg · °C) that enters the shell at 95°C and leaves at
60°C. If the overall heat transfer coefficient is 800 W/m2 · °C,
determine the heat transfer surface area of the heat exchanger
using (a) the LMTD method and (b) the –NTU method.
Answer (a): 11.4 m2 13–100 Steam is to be condensed on the shell side of a
1shellpass and 8tubepasses condenser, with 50 tubes in
each pass, at 30°C (hfg 2430 kJ/kg). Cooling water (Cp
4180 J/kg · °C) enters the tubes at 15°C at a rate of 1800 kg/h.
The tubes are thinwalled, and have a diameter of 1.5 cm and
length of 2 m per pass. If the overall heat transfer coefficient
is 3000 W/m2 · °C, determine (a) the rate of heat transfer and
(b) the rate of condensation of steam. 15°C
Water
1800 kg/h
30°C FIGURE P13–100 13–101 Reconsider Problem 13–100. Using EES (or
other) software, investigate the effects of the
condensing steam temperature and the tube diameters on the
rate of heat transfer and the rate of condensation of steam. Let
the steam temperature vary from 20ºC to 70ºC and the tube
diameter from 1.0 cm to 2.0 cm. Plot the rate of heat transfer
and the rate of condensation as functions of steam temperature
and tube diameter, and discuss the results.
13–102 Cold water (Cp 4180 J/kg · °C) enters the tubes of
a heat exchanger with 2shellpasses and 13–tubepasses at
20°C at a rate of 3 kg/s, while hot oil (Cp 2200 J/kg · °C) enters the shell at 130°C at the same mass flow rate. The overall
heat transfer coefficient based on the outer surface of the tube
is 300 W/m2 · °C and the heat transfer surface area on that side
is 20 m2. Determine the rate of heat transfer using (a) the
LMTD method and (b) the –NTU method. Selection of Heat Exchangers
13–103C A heat exchanger is to be selected to cool a hot liquid chemical at a specified rate to a specified temperature. Explain the steps involved in the selection process.
13–104C There are two heat exchangers that can meet the
heat transfer requirements of a facility. One is smaller and
cheaper but requires a larger pump, while the other is larger
and more expensive but has a smaller pressure drop and thus
requires a smaller pump. Both heat exchangers have the same
life expectancy and meet all other requirements. Explain which
heat exchanger you would choose under what conditions.
13–105C There are two heat exchangers that can meet the
heat transfer requirements of a facility. Both have the same
pumping power requirements, the same useful life, and the
same price tag. But one is heavier and larger in size. Under
what conditions would you choose the smaller one?
13–106 A heat exchanger is to cool oil (Cp 2200 J/kg · °C)
at a rate of 13 kg/s from 120°C to 50°C by air. Determine the
heat transfer rating of the heat exchanger and propose a suitable type. cen58933_ch13.qxd 9/9/2002 9:57 AM Page 713 713
CHAPTER 13 13–107 A shellandtube process heater is to be selected to
heat water (Cp 4190 J/kg · °C) from 20°C to 90°C by steam
flowing on the shell side. The heat transfer load of the heater is
600 kW. If the inner diameter of the tubes is 1 cm and the velocity of water is not to exceed 3 m/s, determine how many
tubes need to be used in the heat exchanger. 60°C. If the overall heat transfer coefficient based on the outer
surface of the tube is 300 W/m2 · °C, determine (a) the rate of
heat transfer and (b) the heat transfer surface area on the outer
Answers: (a) 462 kW, (b) 29.2 m2
side of the tube.
Hot oil
130°C
3 kg/s Steam
90°C
Cold water
20°C
3 kg/s
(20tube passes)
60°C
20°C
Water FIGURE P13–107
13–108 Reconsider Problem 13–107. Using EES (or
other) software, plot the number of tube passes
as a function of water velocity as it varies from 1 m/s to 8 m/s,
and discuss the results.
13–109 The condenser of a large power plant is to remove
500 MW of heat from steam condensing at 30°C (hfg 2430
kJ/kg). The cooling is to be accomplished by cooling water
4180 J/kg · °C) from a nearby river, which enters the
(Cp
tubes at 18°C and leaves at 26°C. The tubes of the heat exchanger have an internal diameter of 2 cm, and the overall heat
transfer coefficient is 3500 W/m2 · °C. Determine the total
length of the tubes required in the condenser. What type of heat
Answer: 312.3 km
exchanger is suitable for this task?
13–110 Repeat Problem 13–109 for a heat transfer load of
300 MW. FIGURE P13–113
13–114E Water (Cp 1.0 Btu/lbm · °F) is to be heated by
solarheated hot air (Cp 0.24 Btu/lbm · °F) in a doublepipe
counterflow heat exchanger. Air enters the heat exchanger at
190°F at a rate of 0.7 lbm/s and leaves at 135°F. Water enters at
70°F at a rate of 0.35 lbm/s. The overall heat transfer coefficient based on the inner side of the tube is given to be 20 Btu/h
· ft2 · °F. Determine the length of the tube required for a tube internal diameter of 0.5 in.
13–115 By taking the limit as T2 → T1, show that when
T2 for a heat exchanger, the Tlm relation reduces to
T1
T1
T2.
Tlm
13–116 The condenser of a room air conditioner is designed
to reject heat at a rate of 15,000 kJ/h from Refrigerant134a
as the refrigerant is condensed at a temperature of 40°C. Air
(Cp
1005 J/kg · °C) flows across the finned condenser
coils, entering at 25°C and leaving at 35°C. If the overall heat
transfer coefficient based on the refrigerant side is 150 W/m2
· °C, determine the heat transfer area on the refrigerant side.
Answer: 3.05 m2
R134a
40°C Review Problems
13–111 Hot oil is to be cooled in a multipass shellandtube
heat exchanger by water. The oil flows through the shell, with
a heat transfer coefficient of ho 35 W/m2 · °C, and the water
flows through the tube with an average velocity of 3 m/s. The
tube is made of brass (k 110 W/m · °C) with internal and external diameters of 1.3 cm and 1.5 cm, respectively. Using water properties at 25°C, determine the overall heat transfer
coefficient of this heat exchanger based on the inner surface. 35°C
Air
25°C 13–112 Repeat Problem 13–111 by assuming a fouling factor
Rf, o 0.0004 m2 · °C/W on the outer surface of the tube.
13–113 Cold water (Cp 4180 J/kg · °C) enters the tubes of
a heat exchanger with 2shell passes and 20–tube passes at
20°C at a rate of 3 kg/s, while hot oil (Cp 2200 J/kg · °C) enters the shell at 130°C at the same mass flow rate and leaves at 40°C FIGURE P13–116
13–117 Air (Cp 1005 J/kg · °C) is to be preheated by hot
exhaust gases in a crossflow heat exchanger before it enters cen58933_ch13.qxd 9/9/2002 9:57 AM Page 714 714
HEAT TRANSFER the furnace. Air enters the heat exchanger at 95 kPa and 20°C
at a rate of 0.8 m3/s. The combustion gases (Cp 1100 J/kg ·
°C) enter at 180°C at a rate of 1.1 kg/s and leave at 95°C. The
product of the overall heat transfer coefficient and the heat
transfer surface area is UAs 1620 W/°C. Assuming both fluids to be unmixed, determine the rate of heat transfer.
13–118 In a chemical plant, a certain chemical is heated by
hot water supplied by a natural gas furnace. The hot water
(Cp 4180 J/kg · °C) is then discharged at 60°C at a rate of
8 kg/min. The plant operates 8 h a day, 5 days a week, 52
weeks a year. The furnace has an efficiency of 78 percent, and
the cost of the natural gas is $0.54 per therm (1 therm
100,000 Btu
105,500 kJ). The average temperature of the
cold water entering the furnace throughout the year is 14°C. In
order to save energy, it is proposed to install a watertowater
heat exchanger to preheat the incoming cold water by the
drained hot water. Assuming that the heat exchanger will recover 72 percent of the available heat in the hot water, determine the heat transfer rating of the heat exchanger that needs to
be purchased and suggest a suitable type. Also, determine the
amount of money this heat exchanger will save the company
per year from natural gas savings.
13–119 A shellandtube heat exchanger with 1shell pass
and 14tube passes is used to heat water in the tubes with geothermal steam condensing at 120ºC (hfg 2203 kJ/kg) on the
shell side. The tubes are thinwalled and have a diameter of 2.4
cm and length of 3.2 m per pass. Water (Cp 4180 J/kg · ºC)
enters the tubes at 22ºC at a rate of 3.9 kg/s. If the temperature
difference between the two fluids at the exit is 46ºC, determine
(a) the rate of heat transfer, (b) the rate of condensation of
steam, and (c) the overall heat transfer coefficient.
Steam
120°C 13–121 Air at 18ºC (Cp 1006 J/kg · ºC) is to be heated to
70ºC by hot oil at 80ºC (Cp 2150 J/kg · ºC) in a crossflow
heat exchanger with air mixed and oil unmixed. The product of
heat transfer surface area and the overall heat transfer coefficient is 750 W/m2 · ºC and the mass flow rate of air is twice
that of oil. Determine (a) the effectiveness of the heat exchanger, (b) the mass flow rate of air, and (c) the rate of heat
transfer.
13–122 Consider a watertowater counterflow heat exchanger with these specifications. Hot water enters at 95ºC
while cold water enters at 20ºC. The exit temperature of hot
water is 15ºC greater than that of cold water, and the mass flow
rate of hot water is 50 percent greater than that of cold water.
The product of heat transfer surface area and the overall heat
transfer coefficient is 1400 W/m2 · ºC. Taking the specific heat
of both cold and hot water to be Cp 4180 J/kg · ºC, determine
(a) the outlet temperature of the cold water, (b) the effectiveness of the heat exchanger, (c) the mass flow rate of the cold
water, and (d) the heat transfer rate.
Cold water
20°C
Hot water
95°C FIGURE P13–122
Computer, Design, and Essay Problems 22°C
14 tubes mass flow rate of geothermal water and the outlet temperatures
of both fluids. 120°C Water
3.9 kg/s FIGURE P13–119
13–120 Geothermal water (Cp 4250 J/kg · ºC) at 95ºC is to
be used to heat fresh water (Cp 4180 J/kg · ºC) at 12ºC at a
rate of 1.2 kg/s in a doublepipe counterflow heat exchanger.
The heat transfer surface area is 25 m2, the overall heat transfer
coefficient is 480 W/m2 · ºC, and the mass flow rate of geothermal water is larger than that of fresh water. If the effectiveness of the heat exchanger is desired to be 0.823, determine the 13–123 Write an interactive computer program that will give
the effectiveness of a heat exchanger and the outlet temperatures of both the hot and cold fluids when the type of fluids, the
inlet temperatures, the mass flow rates, the heat transfer surface area, the overall heat transfer coefficient, and the type of
heat exchanger are specified. The program should allow the
user to select from the fluids water, engine oil, glycerin, ethyl
alcohol, and ammonia. Assume constant specific heats at about
room temperature.
13–124 Water flows through a shower head steadily at a rate
of 8 kg/min. The water is heated in an electric water heater
from 15°C to 45°C. In an attempt to conserve energy, it is proposed to pass the drained warm water at a temperature of 38°C
through a heat exchanger to preheat the incoming cold water.
Design a heat exchanger that is suitable for this task, and discuss the potential savings in energy and money for your area.
13–125 Open the engine compartment of your car and search
for heat exchangers. How many do you have? What type are
they? Why do you think those specific types are selected? If cen58933_ch13.qxd 9/9/2002 9:58 AM Page 715 715
CHAPTER 13 you were redesigning the car, would you use different kinds?
Explain.
13–126 Write an essay on the static and dynamic types of regenerative heat exchangers and compile information about the
manufacturers of such heat exchangers. Choose a few models
by different manufacturers and compare their costs and performance.
13–127 Design a hydrocooling unit that can cool fruits and
vegetables from 30°C to 5°C at a rate of 20,000 kg/h under the
following conditions:
The unit will be of flood type that will cool the products as
they are conveyed into the channel filled with water. The products will be dropped into the channel filled with water at one end
and picked up at the other end. The channel can be as wide as 3
m and as high as 90 cm. The water is to be circulated and cooled
by the evaporator section of a refrigeration system. The refrigerant temperature inside the coils is to be –2°C, and the water temperature is not to drop below 1°C and not to exceed 6°C.
Assuming reasonable values for the average product density,
specific heat, and porosity (the fraction of air volume in a box),
recommend reasonable values for the quantities related to the
thermal aspects of the hydrocooler, including (a) how long the
fruits and vegetables need to remain in the channel, (b) the
length of the channel, (c) the water velocity through the channel, (d) the velocity of the conveyor and thus the fruits and
vegetables through the channel, (e) the refrigeration capacity of
the refrigeration system, and (f) the type of heat exchanger for
the evaporator and the surface area on the water side.
13–128 Design a scalding unit for slaughtered chicken to
loosen their feathers before they are routed to featherpicking
machines with a capacity of 1200 chickens per hour under the
following conditions:
The unit will be of immersion type filled with hot water at
an average temperature of 53°C at all times. Chickens with an
average mass of 2.2 kg and an average temperature of 36°C
will be dipped into the tank, held in the water for 1.5 min, and
taken out by a slowmoving conveyor. Each chicken is expected to leave the tank 15 percent heavier as a result of the
water that sticks to its surface. The centertocenter distance
between chickens in any direction will be at least 30 cm. The
tank can be as wide as 3 m and as high as 60 cm. The water is to be circulated through and heated by a natural gas furnace,
but the temperature rise of water will not exceed 5°C as it
passes through the furnace. The water loss is to be made up by
the city water at an average temperature of 16°C. The ambient
air temperature can be taken to be 20°C. The walls and the
floor of the tank are to be insulated with a 2.5cmthick urethane layer. The unit operates 24 h a day and 6 days a week.
Assuming reasonable values for the average properties, recommend reasonable values for the quantities related to the
thermal aspects of the scalding tank, including (a) the mass
flow rate of the makeup water that must be supplied to the
tank; (b) the length of the tank; (c) the rate of heat transfer from
the water to the chicken, in kW; (d) the velocity of the conveyor and thus the chickens through the tank; (e) the rate of
heat loss from the exposed surfaces of the tank and if it is significant; ( f ) the size of the heating system in kJ/h; (g) the type
of heat exchanger for heating the water with flue gases of the
furnace and the surface area on the water side; and (h) the operating cost of the scalding unit per month for a unit cost of
$0.56 therm of natural gas (1 therm 105,000 kJ).
13–129 A company owns a refrigeration system whose refrigeration capacity is 200 tons (1 ton of refrigeration 211
kJ/min), and you are to design a forcedair cooling system for
fruits whose diameters do not exceed 7 cm under the following
conditions:
The fruits are to be cooled from 28°C to an average temperature of 8°C. The air temperature is to remain above –2°C and
below 10°C at all times, and the velocity of air approaching the
fruits must remain under 2 m/s. The cooling section can be as
wide as 3.5 m and as high as 2 m.
Assuming reasonable values for the average fruit density,
specific heat, and porosity (the fraction of air volume in a box),
recommend reasonable values for the quantities related to the
thermal aspects of the forcedair cooling, including (a) how
long the fruits need to remain in the cooling section; (b) the
length of the cooling section; (c) the air velocity approaching
the cooling section; (d) the product cooling capacity of the system, in kg · fruit/h; (e) the volume flow rate of air; and ( f ) the
type of heat exchanger for the evaporator and the surface area
on the air side. cen58933_ch13.qxd 9/9/2002 9:58 AM Page 716 ...
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