Kuppan T. Heat Exchanger Design Handbook
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464
Chapter
10
10.2
Determination
of
Added
Mass
Coefficient,
Cm,
for
SinglePhase
Flow
The added mass coefficient is estimated either by the analytical method of Blevins
[8]
or from
the experimental database of Moretti et al.
[95].
Blevins Correlation
Blevins
[8]
gave an analytical formula to determine added mass coefficient of a single flexible
tube surrounded by an array of rigid tubes as shown in Fig. 34a. This is a reasonable approxi-
mation
of
the more complex case of all flexible cylinders. The expression
for
added mass
coefficient
Cm
is given as
(D,/D)l
+
1
c,
=
(D,/D)2
-
1
where
D,/D
=
(1
+
0.5
p/D)p/D
in which
D,
is the equivalent diameter of
a
tube array. The
equivalent diameter
0,
is used to represent the confinement due to surrounding tubes.
Experimental Data of Moretti et al.
[95]
The experimental results of Moretti et al.
[95]
for a single flexible tube surrounded by rigid
tubes in a hexangular array
or
in a square array as shown in Fig. 34b with a pitch-to-diameter
ratio from
1,25
to 1.50 is given in Fig. 35. Their test results for
Cm
for four pitch-to-diameter
ratios are given in Table
7.
TEMA included this figure (Fig.
35)
for determination of added
mass coefficient
[
191.
10.3
Natural Frequencies
of
Tube Bundles
The tubes in a heat exchanger
are
slender
beams
and the most flexible members. They vibrate
at discrete frequencies when they are excited. The natural frequency depends on their geometry,
material properties like Young's modulus, density and material damping, tube-to-support inter-
Q
0
U
0
0
Q
Figure
34
Tube
arrangement for determining added mass coefficient, (a) Blevins model
[8];
and
(b)
Morretti model. (From Ref.
95.)
465
Flow-Induced Vibration
t
pitch
imntr
1.0
1
I
I
I
I
I
1
.Z
1.3
1.4
1.5
1.6
1.7
1.8
Pitch-to-diameter
rCrtl0
Figure
35
Added mass coefficient. (From Ref.
95.)
actions, etc. The lowest frequency at which the tubes vibrate is known as the natural frequency.
The coincidence of the exciting frequency with the natural frequency is called resonance. The
conventional shell and tube heat exchangers have either straight tubes or U-tubes with baffle
supports at intermediate points and fixed at their ends. The determination of the natural fre-
quencies of the straight tubes and U-tubes is dealt with next.
Estimation of Natural Frequencies of Straight Tubes
Hand Calculation Method.
Though a rigorous analysis considering the multispan
(Fig.
36)
modeling is possible, the state of the
art
on vibration prediction method recommends a simpler
but a slightly conservative approach treating the individual spans separately
[
1
11.
For
example,
the simpler hand calculations method of Timoshenko and Young
[96]
basically consider a tube
Table
7
Added Mass Coefficient,
C,
cnl
Pitch- to-
diameter ratio Triangular pitch Square pitch
1.25 1.756 1.5 19
1.33 1.429 1.38 1
1.42 1.347 1.286
1
SO
1.274 1.272
466
Chapter
10
Figure
36
Multispan uniform
beam
with fixed ends.
as a continuous elastic beam on multiple supports with fixed end conditions and simply
sup-
ported at the intermediate points (baffle locations). The individual spans (Fig.
37)
are each
treated as a separate beam with its own end conditions (fixed-pinned, pinned-pinned, etc.).
Natural frequencies
of
the individual spans are calculated and the lowest frequency
of
the
spans
is taken as the representative figure for the whole tube and hence for the tube bundle.
Baffles
Tubesheet
Tubesheets
Figure
37
Individual tube span.
(a)
Pinned-pinned;
(b)
fixed-pinned;
(c)
fixed-fixed.
467
Flow- Induced Vibration
Table
8
Frequency Constant,
h,,
and Euler Buckling Load,
F,,
Pinned-pinned Fixed-pinned Fixed-fixed
k,
nn
0.25(4n
+
1)n 0.5(2n
+
1)n
Fcr
K’EIIL’
2x’EIIL’
4n’EIlL
The Timoshenko and Young method of treating the individual spans separately to find the
natural frequency
is
recommended by Singh and Soler
[
111,
Zukaskus
[
131, TEMA, and others.
The frequency equation in a generalized form including the effect of axial stress is given by:
n
=
1,2,3,
.
.
.
The expression for axial stress factor
xB
in terms of tube axial load
F
and Euler’s critical
buckling load
F,,
is given by
where
F,
=
A,
0,
(54)
In Eq.
54,
F,
is tube axial load,
A,
is tube metal cross sectional area, and
0,
is tube longitudinal
stress. In Eq. 53, the plus sign is for tensile load and the negative sign is for compressive load
included in the tubes. The expression for frequency constant
h,,
for the nth mode and for
Euler’s buckling load for various end conditions are given
in
Table
8,
and the values
of
h,,
up
to fifth mode are given in Table
9.
Rigorous Approaches to Calculate Straight Tube Natural Frequencies.
These include the
methods
of
Timoshenko and Young
[96]
and
ABAQUS
[97].
Natural Frequencies
of
Low
Finned Tubes.
These may be calculated by the TEMA method.
U-Tube Natural Frequency
The expression for the lower bound
of
the natural frequency of U-tubes included in TEMA for
various forms of tube symmetry and U-bend support configuration (Fig.
38)
is of the form
Table
9
Numerical Values
for
Frequency Constant
A,
up to Fifth Mode
First Second Third Fourth
~ ~ ~~ ~~~~~
Fifth
End conditions mode mode mode mode mode
Pinned-pinned
3.14 16
6.2832
9.4248 12.5664
15.7080
Fixed-pinned
3.9269
7.0686 10.2102
13.3518 16.4934
Fixed-fixed
4.7 124 7.8540 10.9956
14.1372 17.2788
468
Chapter
10
b
a
f
-:
-.
/J
L
Detailed charts are given in TEMA
[19]
to determine
C,
for various U-bend support geome-
tries.
10.4
Damping
Damping plays an important role on the stability
of
the tube array. Damping limits the tube
response when the tube is excited by any one
of
the excitation mechanisms. The higher the
damping, the lower will be the tube response. Damping determines the critical flow velocity
for FEI. The critical velocity increases with increasing damping. Damping in a vibrating
system
is due to several possible energy dissipation mechanisms. According to Pettigrew et al.
[98],
various energy dissipation mechanisms are
Internal or material damping
Viscous damping
Squeeze film damping in the baffle hole clearance
Flow dependent damping
Frictional damping
Joint damping at the tube and tube-sheet interface.
Energy dissipated by tube impact on the support and by the traveling waves
Determination
of
Damping
Determination
of
damping by experimental methods is very difficult. Empirical correlations
for determining damping for gases and liquids are given next.
Gases.
From the review and analysis
of
the available data on damping of multispan heat
exchanger tubes in gases, Pettigrew, Goyder, Quio, and Axisa
[98]
recommended empirical
correlations to calculate damping ratio
cn.
The damping ratio is defined as the ratio
of
actual
469
Flow-Induced Vibration
damping over critical damping. The logarithmic decrement of damping
6
is equal to
2x6,.
According to these authors for gas flow, frictional damping and impacting are the dominant
energy dissipation mechanisms, with baffle thickness a dominant parameter, whereas diametral
clearance between the tube and baffle support is much less significant.
Liquids.
According to Pettigrew, Rogers, and Axisa [99] the most important damping mecha-
nisms in liquid flow are viscous damping (tube to fluid), squeeze film damping, and friction
damping. They fitted experimental data with
an
empirical model considering viscous and
squeeze film damping resulting in the following empirical correlation. The semi-empirical
expressions recommended to estimate damping for design purposes in single-phase fluids,
gases, and liquids are given.
Gas Flow on the Shell Side [98].
0.7
N
-
1
fh
e,=-----
for
fh
<
12.7
100
N
12.7
N-
1
.*.
6
=
3.463
x
10-’
N
th
(57)
~
where
tb
is the tube support thickness,
L,
the characteristic span length, and
N
the number of
tube supports for the tube under consideration. These expressions are valid for
D
=
12-15 mm,
tb
=
6-25 mm,
f=
20-600
Hz,
and baffle hole clearances 0.4-0.8 mm.
TEMA included
Eq.
58
for all baffle thicknesses.
Liquid
Flow
on the Shell Side. The determination of
6
involves either dynamic (absolute)
viscosity
p
in the expression suggested in TEMA or kinematic viscosity
v
in the expression of
Pettigrew et al. [99]. The relation between them is:
absolute viscosity
p
Kinematic viscosity
v
=
specific gravity
:.
p
=
v
x
specific gravity
The units are centistokes for
v
and centipoise for
p.
Note: TEMA uses the symbol
v
instead of
p.
Pettigrew, Rogers, and Axisa [99] give the equation
where
De
_-
-
1.7
-
P
for triangular arrays
D
D
P
=
1.9
-
for square arrays
(6
1
b)
D
Note: If
cn
is less than
0.006,
assume
&
=
0.006
[9] to account for friction damping.
TEMA included the following expressions for liquid flow, but the reference source is not
given. The damping parameter
6
is the greater of
6,
and
&:
4
70
Chapter
I0
Table
10
Hydrodynamic Mass Coefficient,
Mass
Per Unit Length, and Damping Parameter
1.
Hydrodynamic mass coefficient,
C,,,
Blevins correlation
Moretti et al.
Read from Fig.
35
or from Table
7
d
DJD)?
+
1
C,
=
(D,,/D)?
-
I
2.
Mass per unit length
lID’C,,pn,
lD2p
+
n(D2
-
DZ)p
m=p+2
4 4 4
3.
Tube natural frequency
i. Straight tube
L=
A,=
0,
=
F
=
o,A,
F,,=
l8
=
I=
ill
=
ii. U-tube
llJr
=
C“
=
s/r
=
(use TEMA dimensional units)
R=
4.
Damping parameter
TEMA
Others
3.41
D
Gases
6,
=
___
N-
1
m
fn
6
=
3.463
X
10-’
-
th
for
tl,
<
12.7
N
N-1
t
5
6
=
0.314
-
ifr
for
t,,
2
12.7
N
6
=
max{6,,
&,}(use TEMA dimensional units)
Liquid
D
i.
-
=
(1.7
p/D)-’
for triangular
array
D,
=(
1.9
p/~)-’
for
square array
Note:
If
6
is less than
0.0377,
then assume
6
=
0.0377.
Flow-
Induced
Vibration
4
71
3.41 D
6,
=-
m
fn
(62b)
m
:.
6
=
ma~{6~,6,}
(63)
Note: use
TEMA
dimensional units,
D
=
in.,
p5
=
lbm/ft3,
p
=
centipoise, and
rn
=
lbm/ft.
10.5
Other Values
Fill the values of added mass coefficient, natural frequency, and damping parameter into the
format given in Table
10.
10.6
Two-Phase
Flow
Determination of damping for two-phase flow is detailed in Welding Research Bulletin No.
389 [62].
NOMENCLATURE
=
area
of
cross section of opening in the Helmhotz cavity resonator
=
tube cross sectional area, m2(in.*)
=
speed of sound in the shellside medium
=
effective speed of sound in the shellside fluid
=
lift coefficient for vortex shedding
=
added mass coefficient
=
reduced damping in nth mode
=
turbulence force coefficient
=
constant to determine U-tube natural frequency
=
tube outside diameter
=
baffle diameter
=
equivalent confinement diameter in a tube array
=
(1
+
O.Sp/D)p
to calculate
C,,,
=
1.7p/D
for triangular array to calculate
tn
=
1.9p/D
for square array to calculate
tn
=
tube inside diameter
=
modulus of elasticity of the tube material
=
axial load on tubes
=
critical buckling load
of
the tube span analyzed
=
Helmhotz cavity resonator resonance frequency,
s-'
=
acoustic resonance frequency,
s-l
=
tube natural frequency,
s-'
=
frequency of U-bend region
=
vortex shedding frequency,
s-l
=
turbulent buffeting frequency in a tube array,
s-l
=
proportionality constant,
=
9.81
mls'
=
1.0
in
SI
units and dimensionless
=
32.174
lbm ftAbf
s
Chapter
10
=
moment of inertia of the cross section
=
vibration mode number;
j
=
n
=
1
is fundamental mode
=
acoustic modal indices
=
fluid elastic instability constant
=
characteristic dimension for acoustic standing wave
=
unsupported tube span length for flow-induced vibration
=
unsupported tube span for flow-induced vibration or characteristic tubespan
length, m (in.)
=
central baffle spacing
=
baffle spacing inlet
=
baffle spacing outlet
=
characteristic tube length
=
longitudinal tube pitch, mm (in.)
=
total tube length
=
acoustic chamber dimensions
=
length of Helmhotz cavity resonator neck
=
modal mass
=
molecular weight
of
shell-side gas or vapor
=
mass per unit length
of
tube,
=
m,
+
m,
+
ms
=
mass per unit length of
tube
at a distance
x
along the tube axis
=
fluid added mass per unit length
=
contained fluid mass per unit length
=
structural mass per unit length
=
number of tube spans for the tube being analyzed
=
number of solid baffles
=
shell-side pressure, Pa(psi)
=
rms value of shellside pressure Pa(psi)
=
tube pitch, m(in.)
=
volume flow rate of
gas
phase in two-phase flow
=
volume flow rate of liquid phase
in
two-phase flow
=
U-tube bend radius, shell radius, m(in.)
=
universal gas constant
=
Reynolds number based on the tube outside diameter, dimensionless,
ULNv
=
U-bend dimensions
=
Helrnhtoz cavity resonator neck radius, m(ft)
=
sound pressure level, dB
=
Strouhal number, dimensionless
=
absolute temperature
of
shellside gas,
('R/"K)
=
transverse tube pitch
=
time
=
baffle support plate thickness
U,
UBell,
UTmker,
Uothen
=
reference crossflow velocity
ucr
=
critical fluid velocity for fluid elastic excitation
U(X)
=
crossflow velocity distribution
U,
=
upstream velocity
vh
=
volume of Helmhtoz cavity resonator
=
kinematic viscosity of the fluid, centistokes
V
473
Flow-Induced Vibration
=
longitudinal pitch ratio,
&/D
=
pitch ratio
=p/D
=
transverse pitch ratio,
T,/D
=
axial distance
=
amplitude of vibration
=
rms
amplitude of tube vibration
=
mean square amplitude of vibration
=
maximum tube response due to resonance due to vortex shedding
=
shell-side gas or vapour compressibility factor
=
frequency constant
=
tube bank solidity, dimensionless
=
logarithmic decrement of damping
=
27&
=
Chen number
=
nondimensional parameter
=
factor to account for tube axial load on tube natural frequency
=
mode constant for tube natural frequency, dimensionless
=
critical damping ratio
=
density of tube material
=
density of tube-side fluid
=
density
of
shell-side fluid
=
shell longitudinal stress,
+
for tensile stress,
-
for compressive stress
=
Amold’s slenderness ratio
=
ratio of specific heat of gas at constant pressure to constant volume
=
absolute viscocity, centipoise
REFERENCES
1.
Wambsganss, M. W., Tube vibration and flow distribution in shell and tube heat exchangers,
Heat
Transfer Eng.,
8(3) 62-71 (1987).
2.
Paidoussis, M. P., Flow induced vibrations in nuclear reactors and heat exchangers: Practical experi-
ences and state of knowledge, in
Practical Experiences with Flow Induced Vibration
(Nandascher
and
D.
Rockwell, eds.); Springer-Verlag, Berlin,
1980,
pp.
1-18.
3.
Paidoussis, M.
P.,
Fluid elastic vibration of cylinder arrays in axial and crossflow: State of
art,
J.
Sound Vibration,
76,
329-359 (1 98 1).
4.
Au-Yang, M.
K.,
Blevins, R.
D.,
and Mulcahy, T.
M.,
Flow induced vibration analysis of tube
bundles-A proposed Section
I11
Appendix N nonmandatory code,
Trans. ASME,
J.
Pressure Vessel
Technol., 113,
257-267 (1991).
5.
Chen,
S.
S.,
Guidelines for the instability
flow
velocity tube arrays in crossflow,
J.
Sound Vibration,
93(3), 439-455 (1984).
6.
Weaver,
D.
S.,
and Fitzpatrick,
J.
A., A review of flow induced vibration in heat exchangers,
Int.
Conf. Flow Induced Vibration,
sponsored by BHRA, Bowness-on-Windermere, England,
12-1 4
May
1987.
7.
Au-Yang,
M.
K.,
Flow
induced vibration: Guidelines for design, diagnosis and trouble shooting of
common power plant components,
Trans. ASME,
J.
Pressure Vessel Technol.,
207,
326-334 (1985).
8.
Blevins, R.
D.,
Flow Induced Vibration,
2nd ed., Van Nostrand Reinhold, New York,
1990.
9.
Sandifer,
J.
B., Guidelines for flow induced vibration prevention in heat exchangers, Welding Re-
search Council Bulletin
372,
May
1992.
10.
Chen,
S. S.,
Flow
Induced Vibration
of
Circular Cylindrical Structures,
Hemisphere; New York,
1987.