Kuppan T. Heat Exchanger Design Handbook
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194
Chapter
4
Nu
St
=
~
Re(Pr)
(48)
From
Eqs.
47
and
48,
the expression for
j
is given by
.
Nu
J=-
Re( Pr)
"'
(49)
5.2
The
j
and
f
Factors
Bare Tube Bank
Experimental heat-transfer and fluid-friction data for six staggered circular tube patterns and
one in-line arrangement are given in Ref.
36.
The results of Ref.
36
have been presented in
Kays and London
[
171.
The Giedt
[32]
correlation states that the thermal performance of a bare tube bank is giv-
en by
.
0.376
J=-
Re"'
In-Line
Tube
Bank.
For
a
10-row-deep in-line
tube
bank,
the correlation forj is given by
[37]:
j=-
0.33
Re
''.'
This correlation is valid for tube diameter from
in
to
2
in, and Reynolds number range
100-80.000.
Circular Tube-Fin Arrangement
The basic geometry
is
shown in Fig.
27.
Experimental heat-transfer and fluid-friction
data
for
three
circular finned
tube
surfaces
are
given in London et
al.
[38].
The results of Ref.
38
have been presented in Kays and London
[
171.
Figure
27
Tube-fin
details.
Compact Heat Exchangers
I95
Briggs and Young Correlations [39].
For low-fin tube banks, the correlation based on experi-
mental heat-transfer data is
where
[s
=
(I
-
Nftf)lNf]
with a standard deviation of 3.1%. This is applicable for a circular
finned tube with low fin height and high fin density,
Re
=
1000-20,000,
N,
2
6, and equilateral
triangular pitch.
I
For high-fin tube banks the correlation based on experimental heat-transfer data
is
Nu
=
0.1378Re: 'I8Pr
"'
(
f
i"2'6
(53)
with a standard deviation of 5.1%.
For all tube banks, the correlation based
on
regression analysis is
Nu
=
0.134Re:'"'Pr
)O.'"
[
;
[
5
p""
(54)
(55)
with a standard deviation of 5.1%.
Robinson and Briggs Correlations
[40].
For Equilateral Triangular Pitch with High Finned Tubes.
p,
-0917
f
=
9.465ReJ"l6
[---I
with a standard deviation of 7.8%. This is applicable for the following parameter definitions:
Re
=
2000-50,000
df
=
1.562-2.750
PI
=
1.687-4.50
For Isosceles Triangular Layout.
This
is
applicable for
do(mm)
=
18.6-40.9
h/d,
=
0.35-0.56
hlp
=
4.5-5.3
CJ
=
13.3-1 7.4
P,ld
=
1.8-4.6
P,ld
=
1.8-4.6
N,
2
6
Rabas, Eckels, and Sabatino Correlation
[41].
For
tube banks arranged on
an
equilateral
triangular pitch with low-finned tubes,
196
Chapter
4
rn
=
0.4 15
-
0.03461n
(:j
,251
lf
,759
f=
3.805Reiozw
[
'p
[
;
1
where
ol,
and
a,
are a temperature-dependent fluid properties correction factor and a row
correction factor respectively. The row correction factor has been covered earlier. Rabas and
Taborek [7] recommend the following correlations for
a,,:
Equation 60a is from Weierman [42], and
Eq.
60b is from
ESDU
[33]. However, the Prandtl
number format
is
recommended by Rabas and Taborek [7] for liquids and there is no fluid
property correction term for the friction factor
f.
These are applicable for
lf
<
6.35 mm,
loo0
<
Re
<
25,000,
4.76 mm
I
d
131.75
mm,
246
I
Nf
I
1181 finslm, 15.08
mm
I
PI
S
11
1 mm,
10.32 mm
I
PI
I
96.1 1
mm,
PI
5
PI,
dfls
<
40,
and
N,
>
6.
ESDU
Correlation [33].
This is applicable for loo0
<
Re
<
80,000
or 100,OOO and
N,
>
10.
Ganguly et al. [43] Correlation.
f=U
where
3 -
F,
=
2.5
+
-tan
'[0.5(A/Ap
-
5)]
II:
fp
=
0.25P(Re225)
--
197
Compact Heat Exchangers
p
=
2.5
+
1.2(Xl
-
0.85)-'."
+
0.4
-
1
-
0.01
-
1
i'
E
1
For both
j
andffactors,
95%
of the data correlated within
15%
and 100% of the data correlated
within 20%. This is applicable for
If
56.35 mm,
800
5
Re
5
800,000,
20
5
8
5
40,
XI
54, and
Nr
2
4.
Elmahdy and
Biggs
Correlation
[44]. For a multirow staggered circular or continuous fin
tube arrangement:
j
=
C,Ret? (Ma)
where
This correlation is also valid for circular fin-tube arrangement. The experimentally determined
j
factor is 3-5% lower than that predicted analytically.
This correlation is applicable for:
Re=200-2000
DJtf
=
3.00-33.0
tf
/If
=
0.01-0.45
df/Pl=
0.87-1.27
S/tf
=
2.00-25.0
df/Pl
=
0.76-1.40
did,
=
0.37-0.85
CT
=
0.35-0.62
Continuous Fin on Circular Tube
Gray and
Webb
Correlation
14.51.
Four-Row Array.
0.14Rei0.328
--
(
p,
)
-0.502
[
s
)
0.0312
j4
=
For this form,
89%
of
data were correlated within f10%.
Up to Three Rows.
[
[
$,
)-0.031
]0.607(4-Nr)
Jr!
=
0.991
2.24RePw2
j4
This is used for
Nr
=
1-3
(for
Nr
>
4 the row effect is negligible). The standard deviation
is
-4%
to
+8%.
Pressure Drop Correlation. Gray and Webb correlation is based
on
a superposition model
that was initially proposed
by
Rich [46]. The basic model
is
written as
AP
=
APf
+
Apt
(67)
I98
Chapter
4
where
Ap,
is
the pressure-drop component due to the drag force on the tubes, and
Ap,
is
the
pressure-drop component due to the friction factor on the fins. Accordingly, the expressions
for
Ap,
and
Apf
are given by
The mass velocity
G
used in
Eqs.
(68a) and (68b) is evaluated as the same
G
that exists
in
the
finned-tube exchanger. Several correlations are available
for
the friction factor of
flow
normal
to
tube banks [32,36,47]. Parameters are:
N,
=
1
-
8 or more
Re
=
500-24,700
PJd
=
1.97-2.55
Plld
1.70-2.58
s/d
=
0.08-0.64
Re
=
Gd/p
McQuistion Correlation
[48].
Continuous-Fin Tube Heat Exchanger with Four Rows.
j,
=
0.0014
+
0.2618Re;".'
(70)
i
2
i""
where
AIA,
is
the
ratio
of
the total surface area to the area
of
the bone tube. Here, 90%
of
data
were correlated within +10%.
Less
Than Four Rows.
&
-
1
.O
-
1280NrRe,'
'
-
j,
l.O-5120Re,"
for N,
=
1-3 with a standard deviation
of
-9% to +18%.
f=
4.904
x
10-3
+
1.382a2
Compact Heat Exchangers
I99
-
19.1
18.35
-
-
thickness
0.2,
25.4
2.82
zL
All
dimensions
in
mm
Figure
28
Geometrical details
of
a
flat tube-fin
radiator
to
calculate
j
and
f
factors as
per
Eq.
(73).
[Adopted
from
Ref.
(501.1
Parameters are:
Pe,
P,
=
1
-
2 in
a
=
0.08-0.24
35
do=---
in
88
Tube spacing
=
1-2 in
Nf
=4-14 fin/in
tr
=
0.006-0.010 in
Continuous Fin on Flat Tube Array
Performance figures for three geometries are given in Kays and London
[
171. New data are
presented for two new types of rippled
fin
by Maltson et al.
[49].
For these types of layouts,
fin length for calculating fin efficiency is half the spanwise length of fin between tubes. Kroger
[50] presented performance characteristics for a six rows deep (3
x
2)
core geometry given in
Fig. 28, whose
j
and
f
factors are given by
0.174 0.3778
j=----
Re0..?87
’
f=-
(73)
h
These equations were also found to correlate the data for having eight rows deep (4
x
2).
5
CORRELATIONS
FOR
3
AND
FOF
PLATE-FIN
HEAT EXCHANGERS
5.1
Offset Strip
Fin
Heat Exchanger
Wieting correlations [51] consist
of
power-law curve fits
of
the
j
and
f
values
for
22
heat
exchanger surface geometries over two Reynolds number ranges: Re,,
_<
1000, which is primar-
ily lamina, and Reh
2
2000, which is primarily turbulent.
For Re,
I
1000,
j
=
0.483(1,/0,)-()
‘“c”
‘8JRe:s’f‘
(74)
200
Chapter
4
For Reh
2
2000,
j
=
0.242(if/Dh)
-a'322(
ff/Dh)
O
""Re
368
(76)
f
=
l.136(lf/Dh)-0781(tf/Dh)0'5'JRe~19X
(77)
where Reh is the Reynolds number based on hydraulic diameter
Dh
given by
and
a
is the aspect ratio,
slhf
(Fig.
29).
Joshi and Webb
[
161
have developed a more sophisticated theoretical model to predict the
characteristics of
j
and
f
versus Reynolds number for the offset strip fin. This model preferably
includes all geometrical factors
of
the array
(sld,
tf/lf,
tf/s)
and heat transfer to the base surface
area to which the fins are attached.
Manglik and Bergles
[52]
reviewed the available
j
and
f
data and developed improved
correlations for
j
and
f,
with a single predictive equation representing the data continuously
from lamina to turbulent
flow.
Their correlations forj andfare as follows:
and
where
5
=
aspect ratio
=
s/h,;
6
=
f&;
q
=
tf/s
and Reh is based on the hydraulic diameter
Dh
given by the following equation:
These equations correlate the experimental data for the
18
cores within
+20%.
5.2
Louvered
Fin
Based on tests of
32
louvered-fin geometries, Davenport
[
191
developed multiple regression
correlations for
j
and
f
versus Re as follows:
Figure
29
Details of offset
strip
fin.
(From
Ref.
16.)
201
Compact Heat Exchangers
f
=
0.494Re
:.‘9H,
0.46
1000
c
Re
c
4000
where Re,, is the Reynolds number based on louver pitch,
lh
is louver height,
I,
is louver length,
I,
is the louver pitch, and
H
is the fin height. Here 95% of the data could be correlated within
_+6%
for heat transfer, and within
f10%
for the friction factor. The required dimensions are
millimeters. Figure
30
defines the terms in the equations. The characteristic dimension in the
Reynolds number is louver pitch,
&,
and not
Dh.
5.3
Pin
Fin
Heat Exchangers
For the heat-transfer characteristics of pin-finned surfaces, see Kanzaka et al. [29], Kays [53],
Norris
et al. [54], and Bengt Sunden et
al.
[SS], among others. The results of Ref. 53 were
presented in Kays and London
[17].
The works of Kays [53] and
Norris
[54] are limited to the
heat-transfer characteristics of pin fins having their representative length comparable to those
of louvered
or
offset fins currently used [29].
To
materialize
a
heat exchanger that has thermal
performance superior to those
of
offset or louvered fins requires a more finely segmented
surface of less than
1
mm
at maximum. Kanzaka et al.
[29]
studied the heat-transfer and fluid
dynamic characteristics of pin finned surfaces of various profiles, namely, circular, square, and
rectangular, and
both
in-line and staggered arrangements.
I
.,
I
b
Louvre
Ditch
ILouvre
height,Gh
Figure
30
Definitions
of
geometrical parameters
of
a
louvred
fin.
[From
Ref.
19.1
202
Chapter
4
Staggered Arrange men
t
s
For
their test element, the expression for the Nusselt number is
Nu,
=
0.662Re~.’PrPr’’
In-Line Arrangements
The expression for the Nusselt number is
Nu+
=
0.44ORe:
‘Pr
where
(I
is the characteristic length. The expressions for Nusselt number and
$
are
Nu+=4
hd
Re,=-
Gd+
k
CL
Itd
d+=-$
for circular fins
do
=
a~
+
bp
for rectangular fins
2
where
U,
and
6,
are the rectangular pin fin dimensions (i.e., cross section is
a,
x
6J
and
d,
is
the circular fin diameter.
6
FIN EFFICIENCY
Fins are primarily used to increase the surface area and consequently to enhance the total heat-
transfer rate. Both conduction through the fin cross section and convection from the surface
area take place. Hence, the fin surface temperature is generally lower than the base (prime
surface) temperature if the
fin
convects heat to the fluid. Due
to
this, the fin transfers less heat
than if
it
had been at the base temperature. This is described by the
fin
temperature effective-
ness or fin efficiency
qf,
The
fin
temperature effectiveness is defined as the ratio
of
the actual
heat transfer,
q,
through the fin to that which would be obtained,
q,,
if the entire fin were at
the base metal temperature:
q,
=g
(88)
91
The fin efficiency for plate-fin surfaces in heat exchanger design can be determined from
Gardner [56], Kern and Kraus [57], Scott and Goldschmidt
[58],
Schmidt [59], Zabronsky
[60], Lin and Sparrow [61], Shah [19,62], and others. Fin efficiency of a plain rectangular
profile cross section is discussed in this section.
6.1
Fin Length for Some Plate-Fin Heat Exchanger
Fin Configurations
The two most commonly used plate-fin geometries are rectangular and triangular passages,
shown
in
Fig. 31. From a review of Fig. 31, the fin length
Ef
for a rectangular passage
is
Note that no effect of the fin inclination is taken into account in Eq. 89 for the triangular fin.
Numerous fin geometries and corresponding fin lengths for various flow passage configuration
are presented by Shah [24].
203
I
I
Compact Heat Exchangers
'J=@
I
Figure
31
Fin length. (a) Rectangular passage;
(b)
triangular passage.
[24]
6.2
Fin Efficiency
Circular Fin
Kern and Kraus
[571
Formula
for
Rectangular-Profile Circular
Fin.
For circular fins
(Fig.
32),
the fin efficiency is given by
where
m
=
(2h/kft,)
OS,
1,
is the modified Bessel function of the first kind (nth order),
K,,
is the
modified Bessel function of the second kind (nth order),
kf
is
the fin material thermal conduc-
tivity,
re
is the fin radius,
r,
is the tube outer radius, and
tf
is the fin thickness.
The fin efficiency expressed by
Eq.
90
does not lend itself to comparison with the efficien-
cies of fins of other radial profiles but can be adjusted by expressing the efficiency in terms
of
the radius ratio
rJre
and a parameter
+:
where
A,
is the profile area of the fin:
Figure
32
Circular
fin
details.