Chapter 5 Heat Exchangers
Chapter 5 Heat Exchangers
Chapter 5 Heat Exchangers
Heat Exchangers
5.1 Introduction
Heat exchangers are devices used to transfer heat between two or more fluid streams
at different temperatures. Heat exchangers find widespread use in power generation,
chemical processing, electronics cooling, air-conditioning, refrigeration, and automo-
tive applications. In this chapter we will examine the basic theory of heat exchangers
and consider many applications. In addition, we will examine various aspects of heat
exchanger design and analysis.
71
72 Mechanical Equipment and Systems
Q = U A∆Tm (5.5)
where A is the total surface area for heat exchange that U is based upon. Later we
shall show that
1 1 1 1
= + + (5.7)
UA (ηo hA)i Skw (ηo hA)o
where η0 is the surface efficiency of inner and outer surfaces, h is the heat transfer
coefficients for the inner and outer surfaces, and S is a shape factor for the wall
separating the two fluids.
The surface efficiency accounts for the effects of any extended surface which is
present on either side of the parting wall. It is related to the fin efficiency of an
extended surface in the following manner:
µ ¶
Af
ηo = 1 − (1 − ηf ) (5.8)
A
The thermal resistances include: the inner and outer film resistances, inner and
outer extended surface efficiencies, and conduction through a dividing wall which
keeps the two fluid streams from mixing. The shape factors for a number of useful
wall configurations are given below in Table 1. Additional results will be presented
for some complex doubly connected regions.
Equation (5.7) is for clean or unfouled heat exchanger surfaces. The effects of
fouling on heat exchanger performance is discussed in a later section. Finally, we
should note that
U A = Uo Ao = Ui Ai (5.9)
however,
Uo 6= Ui (5.10)
Finally, the order of magnitude of the thermal resistances in the defintion of the
overall heat transfer coefficient can have a significant influence on the calculation of
the overall heat transfer coefficient. Depending upon the nature of the fluids, one
or more resistances may dominate making additional resistances unimportant. For
example, in Table 2 if one of the two fluids is a gas and the other a liquid, then it is
easy to see that the controlling resistance will be that of the gas, assuming that the
surface area on each side is equal.
74 Mechanical Equipment and Systems
Geometry S
A
Plane Wall
t
2πL
Cylindrical Wall µ ¶
ro
ln
ri
4πri ro
Spherical Wall
r o − ri
Fluid h [W/m2 K]
∆T2 − ∆T1
∆TLM T D = (5.11)
∆T2
ln
∆T1
where ∆T1 and ∆T2 represent the temperature difference at each end of the heat
exchanger, whether parallel flow or counterflow. The LMTD expression assumes that
Heat Exchangers 75
the overall heat transfer coefficient is constant along the entire flow length of the heat
exchanger. If it is not, then an incremental analysis of the heat exchanger is required.
The LMTD method is also applicable to crossflow arrangements when used with the
crossflow correction factor. The heat transfer rate for a crossflow heat exchanger may
be written as:
Q = F U A∆TLM T D (5.12)
where the factor F is a correction factor, and the log mean temperature difference is
based upon the counterflow heat exchanger arrangement.
The LMTD method assumes that both inlet and outlet temperatures are known.
When this is not the case, the solution to a heat exchanger problem becomes some-
what tedious. An alternate method based upon heat exchanger effectiveness is more
appropriate for this type of analysis. If ∆T1 = ∆T2 = ∆T , then the expression for
the LMTD reduces simply to ∆T .
UA
NTU = (5.16)
Cmin
It is now a simple matter to solve a heat exchanger problem when
76 Mechanical Equipment and Systems
ǫ = f (N T U, Cr ) (5.17)
where Cr = Cmin /Cmax .
Numerous expressions have been obtained which relate the heat exchanger effec-
tiveness to the number of transfer units. The handout summarizes a number of these
solutions and the special cases which may be derived from them.
For convenience the ǫ − N T U relationships are given for a simple double pipe heat
exchanger for parallel flow and counter flow:
Parallel Flow
1 − exp[−N T U (1 + Cr )]
ǫ= (5.18)
1 + Cr
or
− ln[1 − ǫ(1 + Cr )]
NTU = (5.19)
1 + Cr
Counter Flow
1 − exp[−N T U (1 − Cr )]
ǫ= , Cr < 1 (5.20)
1 + Cr exp[−N T U (1 − Cr )]
and
NTU
ǫ= , Cr = 1 (5.21)
1 + NTU
or µ ¶
1 ǫ−1
NTU = ln , Cr < 1 (5.22)
Cr − 1 ǫCr − 1
and
ǫ
NTU = , Cr = 1 (5.23)
1−ǫ
For other configurations, the student is referred to the Heat Transfer course text, or
the handout. Often manufacturer’s choose to present heat exchanger performance in
terms of the inlet temperature difference IT D = (Th,i − Tc,i ). This is usually achieved
by plotting the normalized parameter Q/IT D = Q/(Th,i − Tc,i ). This is a direct
consequence of the ǫ − N T U method.
First, a fluid experiences an entrance loss as it enters the heat exchanger core due to
a sudden reduction in flow area, then the core itself contributes a loss due to friction
and other internal losses, and finally as the fluid exits the core it experiences a loss
due to a sudden expansion. In addition, if the density changes through the core as
a result of heating or cooling an accelaration or decelaration in flow is experienced.
This also contributes to the overall pressure drop (or gain). All of these effects are
discussed below.
Entrance Loss
The entrance loss for an abrupt contraction may be obtained by considering Bernoulli’s
equation with a loss coefficient combined with mass conservation to obtain:
¢ 1 G2
∆pi = 1 − σi2 + Kc
¡
(5.24)
2 ρi
where σ is the passage contraction ratio and G = ṁ/A, the mass flux of fluid. In
general,
Core Loss
In the core, we may write the pressure drop in terms of the Fanning friction factor:
4f L 1 G2
∆pc = (5.26)
Dh 2 ρ m
Since the fluid density may change appreciably in gas flows, acceleration or decel-
eration may occur. We consider a momentum balance across the core
¢ 1 G2
σe2
¡
∆pe = − 1 − − Ke (5.29)
2 ρe
78 Mechanical Equipment and Systems
where we have assumed that pressure drop (rise) is from left to right. Once again σ
is the area contraction ratio and G is the mass flux of fluid.
G2 ¡
· µ ¶ µ ¶ µ ¶¸
2
¢ 4L ρi ρi ¡ 2
¢ ρi
∆p = 1 − σi + Kc + f +2 − 1 − 1 − σe − Ke
2ρi Dh ρ m ρe ρe
(5.31)
Now the fluid pumping power is related to the overall pressure drop through appli-
cation of conservation of energy
1 ṁ
Ẇp = ∆P (5.32)
ηp ρ
where ηp is the pump efficiency. The efficiency accounts for the irreversibilities in the
pump, i.e. friction losses.
Entrance and exit loss coefficients have been discussed earlier in Chapter 3. The
handout provides some additional information useful in the design of heat exchangers.
It is clear that a Reynolds number dependency exists for the expansion and contrac-
tion loss coefficients. However, this dependency is small. For design purposes we may
approximate the behaviour of these losses by merely considering the Re = ∞ curves.
These curves have the following approximate equations:
Ke = (1 − σ)2 (5.33)
and
Kc ≈ 0.42(1 − σ 2 )2 (5.34)
d2 θ dA dθ hP ds
A + − θ=0 (5.41)
du2 du du k du
80 Mechanical Equipment and Systems
The governing equation is valid for both axial and radial systems having varying
cross-sectional area and profile. The term ds/du is the ratio of lateral surface area to
the projected area. It is related to the profile function y(u) through
s µ ¶2
ds dy
= 1+ (5.42)
du du
The above equation may be taken as unity, i.e. (dy/du)2 ≈ 0 for slender fin profiles,
without incurring large errors. Thus, for slender fins having varying cross-sectional
area and profile the governing equation becomes
d2 θ dA dθ hP
A + − θ=0 (5.43)
du2 du du k
The governing equation for one dimensional conduction with convection is appli-
cable to systems in which the lateral conduction resistance is small relative to the
convection reistance. Under these conditions the temperature profile is one dimen-
sional. The conditions for which Eq. (5.37) is valid are determined from the following
criterion:
hb
Bi = < 0.1 (5.44)
k
where Bi is the Biot number based upon the maximum half thickness of the fin profile.
The fin Biot number is simply the ratio of the lateral conduction to lateral convection
resistance
b
Rconduction
Bi = = kA (5.45)
Rconvection 1
hA
dθ(ue ) he
+ θ(ue ) = 0 (5.46)
du k
for truncated fins where he is the convection heat transfer coefficient for the edge
surfaces, or
dθ(ue )
=0 (5.47)
du
the adiabatic tip condiction. At the fin base (u = uo )
θ(uo ) = θo (5.48)
Heat Exchangers 81
is generally prescribed.
For axial fins it will be convenieint to take ue = 0 and uo = L, while for radial fins
ue = ro and uo = ri . In subsequent sections, analytic results will be obtained for each
class of fin for various profile shapes. Once the solution for the temperature excess
for a particular case has been found, the solution for the heat flow at the fin base
may be obtained from the Fourier rate equation
dθ
Qb = −kA (5.49)
du
applied to the base of the fin.
Qb Qb
ηf = = (5.50)
Qmax hAs θb
where Qmax is the maximum heat transfer rate if the temperature at every point
within the fin were at the base temperature θb . The fin effectiveness may be defined
as
Qb,f in Qb
ǫ= = (5.51)
Qb,bare hAb θb
where Qb,bare is the heat transfer from the base of the fin when the fin is not present,
i.e. L → 0.
In many heat sink design applications, it is often more convenient to consider the
fin resistance defined as
θb
Rf in = (5.52)
Qb
The use of the fin resistance is more appropriate for modelling heat sink systems,
since additional resistive paths may be considered.
Jakob (1949) and Eckert and Drake (1972). Analytical methods have been success-
fully applied to a number of applications of extended surfaces such as longitudinal
fins, pin fins, and circular annular fins. A table of widely used solutions is provided
in the class notes.
1 1 ln(ro /ri ) 1
= + + (5.53)
UA hi (2πri L) 2πkw L ho (2πro L)
where ri and ro denote the radii of the inner pipe. The heat transfer coefficient hi is
computed for a pipe while the heat transfer coefficient ho is computed for the annulus.
If both fluids are in turbulent flow, the heat transfer coefficients may be computed
using the same correlation with D = Dh , otherwise, special attention must be given
to the annular region.
The pressure drop for each fluid may be determined from:
· ¸
4f L 1
∆p = Σ + ΣK ρV 2 (5.54)
Dh 2
However, care must be taken to understand the nature of the flow, i.e. series,
parallel, or series-parallel.
Example 5.1
Examine the following double pipe heat exchanger. Water flowing at 5000 kg/hr
is to be heated from 20 C to 35 C by using hot water from another source at 100
C. If the temperature drop of the hot water is not to exceed 15 C, how much tube
length is needed in a parallel flow arrangement if a nominal 3 inch outer pipe and
nominal 2 inch inner pipe are used. Assume teh inner pipe wall thickness is 1/8 inch
Heat Exchangers 83
1 1 1
= + Rw + (5.55)
UA hi Ai ho Ao
where Ai and Ao denote the inner and outer areas of the tubes. The heat transfer
coefficient hi is computed for a tube while the heat transfer coefficient ho is computed
for tube bundles in either parallel or cross flow depending on whether baffling is
used. Special attention must be given to the internal tube arrangement, i.e. baffled,
single pass, multi-pass, tube pitch and arrangement, etc., to properly predict the heat
transfer coefficient. Often, unless drastic changes occur in the tube count, the shell
side heat transfer coefficient will not vary much from an initial prediction. Often a
value of ho = 5000 W/m2 K is used for preliminary sizing.
The heat transfer surface area is calculated from:
Ao = πdo Nt L (5.56)
84 Mechanical Equipment and Systems
where do is the outer diameter of the tubes, Nt is the number of tubes, and L is the
length of the tubes.
The number of tubes that can fit in a cylindrical shell is calculated from:
πDs2
Nt = CT P (5.57)
4CLPt2
The factor CT P is a constant that accounts for the incomplete covereage of circular
tubes in a cylindrical shell, i.e. one tube pass CT P = 0.93, two tube passes CT P =
0.9, and three tube passes CT P = 0.85. The factor CL is the tube layout constant
given by CL = 1 for 45 and 90 degree layouts, and CL = 0.87 for 30 and 60 degree
layouts. Finally, Pt is the tube pitch and Ds is the shell diameter.
The shell diameter may be solved for using the above two equations, to give:
r ¸1/2
CL Ao Pt2
·
Ds = 0.637 (5.58)
CT P do L
The shell side heat transfer coefficient is most often computed from the following
experimental correlation:
N uDe = 0.36Re0.55
De P r
1/3
(5.59)
for 2 × 103 < ReDe < 1 × 106 . The effective diameter De is obtained from
and
ṁPt
Gs = (5.64)
Ds CB
where Nb is the number of baffles, C is the clearance between adjacent tubes, B is
the baffle spacing, while f is determined from
1 1 t 1
= + + (5.66)
UA (ηo hA)i kw Aw (ηo hA)o
where ηo is the overall surface efficiency. In most compact heat exchanger design
problems, the heat transfer and friction coefficients are determined from experimental
performance charts or models for enhanced heat transfer surfaces. The pressure drop
is also computed using the general method discussed section 5.4.
Example 5.5
Air enters a heat exchanger at 300 C at a rate of ṁ = 0.5 kg/s. and exits with
a temperature of 100 C. If a surface similar to 1/9 − 22.68 (see handout) is used,
calculate the pressure drop in the core, heat transfer coefficient for the core, and the
surface efficiency for the core. The heat exchanger core has the following dimensions:
W = 10 cm, L = 30 cm, kf = 200 W/mK, tw = 1.5 mm.
Example 5.6
You wish to cool an electronic package having dimensions of 5 cm by 5 cm which
produces 100 W . The package and heat sink are to be mounted on a circuit board
which forms a channel with another circuit board. The spacing between the two
circuit boards is 25 mm and each board is approximately 40 cm x 40 cm. The air
speed in the channel is 3 m/s and has a temperature of 25 C. An aluminum, (k = 200
W/mK, ǫ = 0.8), finned surface similar to that used in compact heat exchangers is to
be considered, i.e 3/32 − 12.22 (see handout). It is to be attached using a conductive
adhesive tape having a thermal conductivity (ka = 50 W/mK) and thickness of 1
mm. As the chief packaging engineer you must determine:
temperature, however you do not need to re-iterate the solution for the heat transfer,
merely comment on the effect of the mean bulk temperature change. In the case
of the pressure drop calculations, use the properties given below to interpolate the
outlet air densities for each stream.
Assume properties for air at 500 C are: ρ = 0.456 kg/m3 , ν = 78.5 × 10−6 m2 /s,
k = 0.056 W/mK, Cp = 1.093 KJ/kgK, P r = 0.70, and at 20 C are: ρ = 1.205
kg/m3 , ν = 15.0 × 10−6 m2 /s, k = 0.025 W/mK, Cp = 1.006 KJ/kgK, P r = 0.72.
up. However, depending upon the nature of the fluid other factors may contribute to
fouling.
1 1 1 1
= + Rf,i + + Rf,o + (5.67)
UA ηi hi Ai Skw ηo ho Ao
The fouling resistance may be computed from
tf
Rf = (5.68)
kf Aw
for a plane wall, and
ln(df /dc )
Rf = (5.69)
2πkf L
for a tube.
Unfortunately, fouling in heat exchangers has not been modelled adequately for
predictive purposes. Some typical values of fouling resistances are given in Table 3
for a number of fluids.
Heat Exchangers 89
5.8 References
Bejan, A., Heat Transfer, 1993, Wiley, New York, NY.
Kakac, S. (ed.), Boilers, Evaporators, and Condensers, 1991, Wiley, New York,
NY.
Kakac, S. and Liu, H., Heat Exchangers: Selection, Rating, and Thermal Perfor-
mance, 1998, CRC Press, Boca Raton, FL.
Kays, W.M. and London, A.L., Compact Heat Exchangers, 1984, McGraw-Hill,
New York, NY.
Kern, D.Q. and Kraus, A.D., Extended Surface Heat Transfer, 1972, McGraw-
Hill, New York, NY.
McQuiston, F.C. and Parker, J.D., Heating, Ventilation, and Air Conditioning:
Analysis and Design, 1988, Wiley, New York, NY.
Rohsenow, W.M., Hartnett, J.P, Cho, Y.I., Handbook of Heat Transfer, 1998,
McGraw-Hill, New York, NY.
Shah, R.K. and Sekulic, D., Fundamentals of Heat Exchanger Design, 2003, Wiley,
New York, NY.
Smith, E.M., Thermal Design of Heat Exchangers, 1995, Wiley, New York, NY.