Local and Overall Condensation Heat Transfer Behavior in Horizontal Tube Bundles, Heat Transfer Eng., 171, 19–30.. KRAUS University of Akron Akron, Ohio 11.1 Introduction 11.2 Governing
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CHAPTER 11
Heat Exchangers
ALLAN D KRAUS
University of Akron Akron, Ohio
11.1 Introduction 11.2 Governing relationships 11.2.1 Introduction 11.2.2 Exchanger surface area 11.2.3 Overall heat transfer coefficient 11.2.4 Logarithmic mean temperature difference 11.3 Heat exchanger analysis methods
11.3.1 Logarithmic mean temperature difference correction factor method 11.3.2 –Ntumethod
Specific–Nturelationships 11.3.3 P –Ntu,cmethod
11.3.4 ψ–P method
11.3.5 Heat transfer and pressure loss 11.3.6 Summary ofworking relationships 11.4 Shell-and-tube heat exchanger
11.4.1 Construction 11.4.2 Physical data Tube side Shell side 11.4.3 Heat transfer data Tube side Shell side 11.4.4 Pressure loss data Tube side Shell side 11.5 Compact heat exchangers 11.5.1 Introduction 11.5.2 Classification ofcompact heat exchangers 11.5.3 Geometrical factors and physical data 11.5.4 Heat transfer and flow friction data Heat transfer data
Flow friction data 11.6 Longitudinal finned double-pipe exchangers 11.6.1 Introduction
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11.6.2 Physical data for annuli Extruded fins
Welded U-fins 11.6.3 Overall heat transfer coefficient revisited 11.6.4 Heat transfer coefficients in pipes and annuli 11.6.5 Pressure loss in pipes and annuli
11.6.6 Wall temperature and further remarks 11.6.7 Series–parallel arrangements 11.6.8 Multiple finned double-pipe exchangers 11.7 Transverse high-fin exchangers
11.7.1 Introduction 11.7.2 Bond or contact resistance ofhigh-fin tubes 11.7.3 Fin efficiency approximation
11.7.4 Air-fin coolers Physical data Heat transfer correlations 11.7.5 Pressure loss correlations for staggered tubes 11.7.6 Overall heat transfer coefficient
11.8 Plate and frame heat exchanger 11.8.1 Introduction
11.8.2 Physical data 11.8.3 Heat transfer and pressure loss 11.9 Regenerators
11.9.1 Introduction 11.9.2 Heat capacity and related parameters Governing differential equations 11.9.3 –Ntumethod
11.9.4 Heat transfer and pressure loss Heat transfer coefficients Pressure loss
11.10 Fouling 11.10.1 Fouling mechanisms 11.10.2 Fouling factors Nomenclature
References
11.1 INTRODUCTION
A heat exchanger can be defined as any device that transfers heat from one fluid to another or from or to a fluid and the environment Whereas in direct contact heat exchangers, there is no intervening surface between fluids, in indirect contact heat
exchangers, the customary definition pertains to a device that is employed in the trans-fer of heat between two fluids or between a surface and a fluid Heat exchangers may
be classified (Shah, 1981, or Mayinger, 1988) according to (1) transfer processes,
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(2) number offluids, (3) construction, (4) heat transfer mechanisms, (5) surface com-pactness, (6) flow arrangement, (7) number offluid passes, and (8) type ofsurface
Recuperators are direct-transfer heat exchangers in which heat transfer occurs
between two fluid streams at different temperature levels in a space that is separated
by a thin solid wall (a parting sheet or tube wall) Heat is transferred by convection from the hot (hotter) fluid to the wall surface and by convection from the wall surface
to the cold (cooler) fluid The recuperator is a surface heat exchanger
Regenerators are heat exchangers in which a hot fluid and a cold fluid flow
al-ternately through the same surface at prescribed time intervals The surface of the regenerator receives heat by convection from the hot fluid and then releases it by convection to the cold fluid The process is transient; that is, the temperature ofthe surface (and of the fluids themselves) varies with time during the heating and cooling
of the common surface The regenerator is a also surface heat exchanger
In direct-contact heat exchangers, heat is transferred by partial or complete
mix-ing ofthe hot and cold fluid streams Hot and cold fluids that enter this type ofex-changer separately leave together as a single mixed stream The temptation to refer
to the direct-contact heat exchanger as a mixer should be resisted Direct contact
is discussed in Chapter 19 In the present chapter we discuss the shell-and-tube heat exchanger, the compact heat exchanger, the longitudinal high-fin exchanger, the trans-verse high-fin exchanger including the air-fin cooler, the plate-and-frame heat ex-changer, the regenerator, and fouling
11.2 GOVERNING RELATIONSHIPS 11.2.1 Introduction
Assume that there are two process streams in a heat exchanger, a hot stream flowing with a capacity rate C h = ˙m h C ph and a cooler (or cold stream) flowing with a capacity rate C c = ˙m c c ph Then, conservation ofenergy demands that the heat transferred between the streams be described by the enthalpy balance
q = C h (T1 − T2) = Cc (t2 − t1) (11.1) where the subscripts 1 and 2 refer to the inlet and outlet of the exchanger and where
the T ’s and t’s are employed to indicate hot- and cold-fluid temperatures, respectively.
Equation (11.1) represents an ideal that must hold in the absence oflosses, and
while it describes the heat that will be transferred (the duty ofthe heat exchanger)
for the case of prescribed flow and temperature conditions, it does not provide an indication ofthe size ofthe heat exchanger necessary to perform this duty The size
ofthe exchanger derives from a statement ofthe rate equation:
q = UηSθm = U hηov,hShθm = U cηov,cScθm (11.2)
whereShandScare the surface areas on the hot and cold sides of the exchanger,Uh
andU c are the overall heat transfer coefficients referred to the hot and cold sides of
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the exchanger andθm is some driving temperature difference The quantitiesηov,h
andηov,c are the respective overall fin efficiencies and in the case of an unfinned exchanger,ηov,h= ηov,c= 1
The entire heat exchange process can be represented by
q = U hηov,hS hθm = U cηov,cS cθm = C h (T1 − T2) = Cc (t2 − t1) (11.3) which is merely a combination ofeqs (11.1) and (11.2)
11.2.2 Exchanger Surface Area
Consider the unfinned tube oflengthL shown in Fig 11.1a and observe that because
ofthe tube wall thickness δw, the inner diameter will be smaller than the outer diameter and the surface areas will be different:
In the case ofthe finned tube, shown with one fin on the inside and outside ofthe tube
wall in Fig 11.1b, the fin surface areas will be
S f o = 2n o b o L (11.5b) wheren i andn o are the number offins on the inside and outside ofthe tube wall, respectively, and it is presumed that no heat is transferred through the tip of either of
the inner or outer fins In this case, the prime or base surface areas
Sbi = (πd i − n iδf i)L (11.6a)
Sbo = (πd o − n oδf o)L (11.6b) The total surface will then be
S i = S bi + S f i = (πd i − n iδf i + 2n i b i )L
or
S i=πd i + n i (2b i− δf i )L (11.7a)
S o=πd o + n o (2b o− δf o )L (11.7b) The ratio of the finned surface to the total surface will be
S f i
Si =
2ni b i L
πd i + n i (2b i− δf i )L =
2ni b i
πd i + n i (2bi− δf i) (11.8a)
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Sf o
So =
2noboL
πd o + n o (2b o− δf i )L =
2nobo
πd o + n o(2bo− δf o) (11.8b)
The overall surface efficiencies ηov,h and ηov,c are based on the base surface operating at an efficiency of unity and the finned surface operating at fin efficiencies
ofηf iandηf o Hence
ηov,iS i = S bi+ ηf i S f i
= S i − S f i+ ηf i S f i
or
ηov,i= 1 − Sf i
Si
and in a similar manner,
ηov,o= 1 −S f o
So
Figure 11.1 End view of(a) a bare tube and (b) a small central angle ofa tube with both
internal and external fins oflength,L.
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Notice that when there is no finned surface,S f i = S f o= 0 and eqs (11.9) reduce to
ηov,i= ηov,o= 1
and that with little effort, the subscripts in eqs (11.4) through (11.9) can be changed
to reflect the hot and cold fluids
11.2.3 Overall Heat Transfer Coefficient
In a heat exchanger containing hot and cold streams, the heat must flow, in turn, from the hot fluid to the cold fluid through as many as five thermal resistances:
1 Hot-side convective layer resistance:
Rh= h 1
hηov,hS h (K/W) (11.10)
2 Hot-side fouling resistance due to an accumulation offoreign (and undesirable)
material on the hot-fluid exchanger surface:
R dh= 1
hdhηov,hSh (K/W) (11.11)
Fouling is discussed in a subsequent section
3 Resistance ofthe exchanger material, which has a finite thermal conductivity and which may take on a value that is a function ofthe type ofexchanger:
R m=
δw kmSm (K/W) plane walls
ln(do)(di)
2πk m Ln t (K/W) circular tubes
(11.12)
whereδmis the thickness ofthe metal,S mthe surface area of the metal, andn t the number oftubes
4 Cold-side fouling resistance:
Rdc= 1
hdcηov,cSc (K/W) (11.13)
5 Cold-side convective layer resistance:
R h= 1
hcηov,c Sc (K/W) (11.14) The resistances listed in eqs (11.10)–(11.14) are in series and the total resistance
can be represented by
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1
US =
1
hhηov,hSh +
1
hdhηov,hSh + R m+
1
hdcηov,cSc +
1
hcηov,cSc (11.15)
where, for the moment,U and S on the left side of eq (11.16) are not assigned any
subscript Equation (11.16) is perfectly general and may be put into specific terms depending on the selection of the reference surface, whether or not fouling is present and whether or not the metal resistance needs to be considered Ifeq (11.16) is solved forU, the result is
S
h hηov,h S h+
S
h dhηov,h S h + SR m+
S
h dcηov,c S c +
S
h cηov,c S c
(11.16)
and ifthe thickness ofthe metal is small and thermal conductivity ofthe metal is high, the metal resistance becomes negligible and
S
hhηov,hSh +
S hdhηov,hSh +
S hdcηov,cSc +
S
hcηov,cSc
(11.17)
Several forms of eq (11.17) are:
• For a hot-side reference with fouling,
1
h hηov,h +h 1
dhηov,h +h 1
dcηov,c
S h
S c +
1
h cηov,c
S h
S c
(11.18)
• For a cold-side reference with fouling,
1
hhηov,h
Sc
Sh+
1
hdhηov,h
Sc
Sh +
1
hdcηov,c +
1
hcηov,c
(11.19)
• For a hot-side reference without fouling,
1
h hηov,h + 1
h cηov,c
Sh
S c
(11.20)
• For a cold-side reference without fouling,
1
hhηov,h
S c
Sh +
1
hcηov,c
(11.21)
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• For an unfinned exchanger where ηov,h = ηov,c = 1 and a hot-side reference
without fouling,
U h= 1
1
h h+
1
h c
S h
S c
(11.22)
• For an unfinned exchanger and a cold-side reference without fouling,
U c= 1
1
h h
Sh
S c +
1
h c
(11.23)
11.2.4 Logarithmic Mean Temperature Difference
For the four basic simple arrangements indicated in Fig 11.2,θmin eqs (11.2) and
(11.3) is the logarithmic mean temperature difference, which can be written as
θm= LMTD = ∆T1− ∆T2
ln(∆T1/∆T2)=
∆T2− ∆T1 ln(∆T2/∆T1) (11.24)
T2
T s
T2
T2
T1
t2
t2
t2
t1
t1
t1
t s L
L
L
L
( )a
( )c
( )b
( )d
T1
T2
Figure 11.2 Four basic arrangements for which the logarithmic mean temperature
differ-ence may be determined from eq (11.23): (a) counterflow; (b) co-current or parallel flow;
(c) constant-temperature source and rising-temperature receiver; (d) constant-temperature
re-ceiver and falling-temperature source