Kuppan T. Heat Exchanger Design Handbook
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Ch
n
p
t
Y
r
.?
t
.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Figure
1
1
(a)
Crossflow: unmixed-mixed-the stronger/C,,,, fluid mixed, E-C*-NTU
chart:
(b)
F-R-P-
NTU chart;
F
as
a
function
of
P
for
constant
R
(solid lines) and constant NTU (dashed lines).
By
substituting
N
=
1.2.3.
. . .
in
Eqs.
59
and
61,
equations for thermal relations are ob-
tained for the specific arrangements by Nicole
[28],
and this is given
in
Table
5
(Eqs.
T9-Tl2)
for one row. two rows, three rows, and four rows.
For
a larger number of tube
rows
(for all
practical purposes, when
N
exceeds
5),
the solution approaches that of unmixed-unmixed
cross-
flow arrangement. Values of
F
for
N
=
1,2,3,4
are shown
in
Figs.
13-16
and are always less
than the basic case of unmixed-unmixed crossflow (Fig.
8).
Mitltiyu.s.s
Tube
Rows
Cross-Coirnte$o~* Arrarigements, Both
Fluids
Unmi.\-ed.
and
Midtiple
Tube
Rort*s
in
Mirltipass
Tirbu
Ro~vs. Cross-Counturjlo~* Arrangements.
This would apply to
a manifold-type air cooler
in
which the tubes
in
one row are connected to the next by U-bends.
The solutions are based on Ref.
28.
Solutions for the
2
rows-:!
pass and
3
rows-3
pass cases
are based on Stevens et al.
[29].
The general formula for thermal effectiveness referred to the
air side
(fin
side) is given by
[28]:
P”R(I
11
-,i
(62)
55
Heat Exchanger Thermohydraulic Fundamentals
56
Chapter
2
Figure
12
Unmixed-unmixed
crossflow
arrangement.
The expressions for for various cases are
as
follows:
1.
Two tube rows, two passes, as shown in Fig.
17a
[29]:
;+
=
6
(1
-
y
j,
?KR
K
=
I
-
exp[--Y
2.
Three tube rows, three passes
as
shown in Fig. 17b
[29]:
K=
1
-expi-?)
3.
Four tube rows, four passes as shown
in
Fig.
17c
[28]:
K
=
1
-
exp[-y
4.
Five tube rows, five passes
[28]:
K=
1
-exp(-y
j
57
Heat Exchanger Thermohydraulic Fundamentals
5.
Six
tube
rows,
six
passes
[28]:
Table
5
Thermal Effectiveness Relations for Tube Rows with Single Pass Arrangement
General formula, Ref. (28)
Flow arrangement
Eq.
no.
Note-These formulas are valid for
R
=
1
One tube row.
1
4-
P~
=-(I
-
e-KR)
R
K=
1
-exp(-NTU)
Two tube rows. T10 1281
1
r-5
P,=-[l
+RK?)]
R
K=
1
-exp(-y]
j@3-
2
Three tube rows.
T11 [28]
1
e
3KR
4
PI+
R
-[
1
+
RK~
(3
-
K)
+
(3/2)
R'K'
K=
1 -exp(-y)
2
Four tube rows.
T12 [28]
JKR
[1+
RK2
(6
-
4K+
K')
+
4R2K4
(2
-
K)
+
(813)
R'K'
K
=
1
-
exp(-NTU/4)
c
58
Chapter
2
8.9
0.6
8,5
Figure
59
Heat Exchanger Thennohydraulic Fundamentals
1.0
8.9
&
8,8
&
C
0
.*
c.,
0
2
0"
8.7
iL
0.6
0.5
Figure
60
Chapter
2
1.0
8.9
c
.C(
U
0
0.s
8.1
0.2
0.3
8.4
6.5
0.6
8.7
0.8
0.8
p1,
Thermal effectiveness
Figure
15
Three
tube
rows;
F
as a
function
of
P
for
constant
R
(solid lines) and constant
NTU
(dashed lines).
61
Heat Exchanger Thermohydraulic Fundamentals
I
.e
9.8
8.
S
Figure
62
Chapter
2
(a)
1--
@---
Figure
17
Multipass tube rows.
(a)
Two rows-two pass;
(b)
three rows-three pass;
(c)
four rows-four
pass.
NTU
]
K
=
1
-
exp(-T
(67b)
Values
of
F
referred to the fluid outside the tube side (air side) for
N
=
2,
3,
4,
5,
and
6
are shown
in
Figs.
18, 19, 20, 21,
and
22,
respectively. The
F
values are always higher than
the basic case (Fig.
8).
When
N
becomes greater than
6, F
approaches
1,
that is, pure counter-
current flow.
6.
Multiple Tube Rows in Multipass Cross-Counterflow Arrangements; One Fluid Unrnixed
Throughout; Other Fluid (Tube Fluid) Unmixed in Each Pass but Mixed Between Passes).
1.
Four rows, two passes, with two rows per pass. Tube-side fluid mixed at the header and
the other fluid is unmixed throughout, coupling in inverted order [Fig. 23(a)] [28]:
K
K'
1
1--+-
[1-e
]
[
2
8
]
4KR
+RK')'
K
=
1
-
exp[-T)
NTU
2.
Six rows, two passes, with three rows per pass. Tube-side fluid is mixed between passes,
that is, at the header, and the other fluid is unmixed throughout, coupling in inverted order
[Fig. 23(b)] [26]:
2A-RA'-6
P,
=
1-R6
A
=
a,,C,,(3KR)
+
alC1(3KR)
+
alC2(3KR)
a,,=
1
-(1
-a3
al
=
3RK'(3
-
2K)
6
=
a;C,,(6KR)
+
2a,,a,C1(6KR)
+
(a:
+
2a,,al)C2(6KR)
+
2ala,C7(6KR)
+
aiC4(6KR)
K
=
I
-
expl-7NTU
j
Heat Exchanger Thermohydraulic Fundamentals
63
1
.ta
0.8
8.8
0.7
0.0
8.5
8.2
0.3
6.4
0.6
0.e
8.7
e.
a
1
.e
pl.
Thermal
effectiveness
Figure
18
Two tube rows, two passes.
Both
fluids unmixed throughout;
fluid
I
inverted coupling between passes.
F
as a function
of
P
for
constant
R
(solid lines) and constant NTU (dashed lines).