International Journal of Engineering & Technology IJET Vol: 9 No: 9
- 201 -
An analysis and comparison of tube natural
T
frequency modes with fluctuating force
frequency from thermal cross-flow fluid in
300 MWE PWR
Tasneem M. Shah, Zafar U. Koreshi, Sadaf Siddiq1
Abstract— Reactor components including fuel assemblies
and heat exchanger tubes are beam-type structures with
natural or forced boundary conditions or mixed type
boundary conditions. It is difficult to model such systems
accurately as it consists of both thermal fluid flow behavior
and structure of hollow fuel rods. An attempt is made to decouple the problem into thermal fluid flow and structures.
First, a vibration analysis of the fuel rod is carried out
resulting in the form of natural frequencies and
corresponding mode shapes. Then, three-dimensional
transient thermal fluid flow transport equations are solved
across a hollow cylinder with specified boundary conditions.
This gives a transient temperature profile or the vorticity of
flow around the fuel rod in the form of fluctuating force
having a certain frequency. A comparison is made between
the fluctuating force frequency of the coolant fluid and the
hollow rod natural frequency; a phenomenon called lock-in
is obtained in both directions, resulting in much larger
amplitude of resonance vibration of the rod. The natural
frequencies of the fuel rod for a 300 Mwe PWR Nuclear
Power Plant are calculated and compared with numerically
calculated Turbulence Induced Vibration (TIV) fluctuating
force frequencies. The present paper is very useful to
evaluate the fuel rod vibration with various boundary
conditions. An important result is obtained for the variation
in the vorticity along the axis; it is found that in the range 0700 mm, the vorticity is in the range 0-5 Hz i.e. it approaches
the first-mode natural frequency. A similar trend is seen
towards the end of the flow channel when the vorticity is 25
Hz corresponding to the third mode of natural frequency.
Further numerical investigation is needed with variable
coolant velocity and temperature.
I. INTRODUCTION
HE well-known problem of fuel degradation and potential
damage to instrumentation or fuel rods [1-3] in Pressurized
Water Reactors (PWR’s) is investigated by studying the flow
field in the core of a 300 MWe reactor. Of the various flow
induced vibration mechanisms we have focused on turbulence
induced vibration as it is the prime excitation source in axial
flow and can lead to fretting wear damage in the core
internals.
The flow-field analysis is centered on the vorticity as it is
important to investigate how close it is from the natural
frequencies. In the first part, a modal analysis is carried out
resulting in natural frequencies of the cylinder. In the second
part, hydrodynamic analysis is carried out which gives the
vorticities (TIV). The objective is to compare the frequencies
to get the lock-in (resonance) conditions. An empirical
approach for cross-flow induced vibrations has been taken by
Khushnood et al [5] while Kang et al [6] has carried out a
similar analysis for a 5×5 bundle.
This work is useful for understanding the vulnerable domains
in a coolant flow channel in a PWR which has yet to undergo
detailed theoretical analysis.
We consider the fuel as a single body, and neglect the
cladding or internal structure of a rod. The coolant flow
channel is modeled as shown in Fig. 1. Due to symmetry, we
have considered one-fourth of the flow channel as shown in
Fig. 2.
Index Terms— Computational Fluid Dynamics, Finite Element
Method, Modal Analysis, Pressurized Water Reactor.
1
Dr. Tasneem M. Shah is a Professor at Air University, Islamabad,
Pakistan ; ph: 0092(300)5269610; fax: 0092(51)9260158; e-mail:
dr.tasneem@ mail.au.edu.pk.
Dr. Zafar Ullah Koreshi is a Professor at Air University, Islamabad,
Pakistan ; e-mail: zafar@ mail.au.edu.pk.
Engr. Sadaf Siddiq is an Assistant Professor at Air University, Islamabad,
Pakistan ; e-mail: sadaf@ mail.au.edu.pk.
1936091 IJET-IJENS @ International Journals of Engineering and Sciences IJENS
International Journal of Engineering & Technology IJET Vol: 9 No: 9
- 202 -
with a frequency 5 (Hz). Figs. (5-6) show the second and third
modes with corresponding frequencies 13 and 27 (Hz). Since
the present analysis is done for lock-in condition at the bottom
baffle plate so we are only interested in the first mode of the
clamped rod (5 Hz).
Fig. 1. Coolant flow domain between 4-Fuel Rods
Fig. 3. Elements in FE Model
Fig. 2. One-fourth of Fluid Domain for Computation
II. FEM MODAL ANALYSIS OF FUEL ROD
The modal analysis of the bundle of fuel rod is carried out
by the standard finite element package ANSYS [7]. To find
frequency and mode shape, we have considered a single fuel
rod supported by the baffle plates. The rod is modeled by 8node SHELL63 element. For building 1 FE model of a rod,
14,800 elements are needed (Fig. 3). For the boundary
condition of the FE analysis, both ends of the rod are fixed.
Material properties for the components of the rod for a typical
300 Mwe PWR used here for the reference values in the
present analysis are listed in Table (1).
Fig. 4. First Mode Shape of Single Fuel Rod
A. Finite Element Analysis Results:
Natural frequencies obtained from the FE analysis for a rod
clamped at both ends are obtained by solving appropriate
second order differential equations using ANSYS for
calculating mode shapes and frequencies together with
material properties given in Table 1. Obtained frequencies are
summarized in Table (2), and corresponding mode shapes are
depicted in Figs. (4-6). Fig. 4 shows the first mode appeared
1936091 IJET-IJENS @ International Journals of Engineering and Sciences IJENS
International Journal of Engineering & Technology IJET Vol: 9 No: 9
- 203 III. FLUID FLOW ANALYSIS
A. Final Stage
The 3-D hydrodynamic model is solved on single fuel rod
supported at the bottom baffle plate where the inlet velocity,
temperature and pressure are specified given in Table (2). The
velocities and temperature are calculated by the code and
vorticites are obtained using velocity-vorticity relations [9].
TABLE 3
INPUT PARAMETERS FOR FLOW ANALYSIS
Fig. 5. Second Mode Shape of Single Fuel Rod
S#
1
Quantity
ρ
Units
2
Vin
3
4
P
5
Tin
K
6
Tout
K
Ts
Description
Water Density
Value
1000.0
m/s
Velocity at
Entrance
8.0
bar
K
Pressure
Temperature at
fuel rod surface
Inlet
Temperature
Outlet
Temperature
155.0
618
kg / m
3
561
575
B. GOVERNING EQUATIONS:
The conservation of mass, momentum and energy in three
dimensions for a turbulent, incompressible, Newtonian fluid
are given by [9]:
∂u
∂x
+ ∂∂yv + ∂∂wz = 0,
(1)
ρ (u ∂∂ux + v ∂∂uy + w ∂∂uz ) + ∂∂px =
Fig. 6. Third Mode Shape of Single Fuel Rod
∂
∂y
TABLE 1
GEOMETRY AND MATERIAL PROPERTIES OF FUEL ROD
S#
1
2
3
4
Quantity
Diameter
Height
Thickness
Density
5
E
6
ν
Units
(mm)
(mm)
(mm)
Description
Fuel rod
Fuel Rod
Fuel Rod
Fuel Rod
Value
10.0
3210.0
0.78
12.4
N / m2
Young Modulus
1.88 × 10
-
Poisson Ration
0.29
g / cm 3
Frequency (Hz)
5
13
27
Mode Shape
Fig. 1
Fig. 2
Fig. 3
∂p
∂y
=
∂
∂y
11
∂
∂x
(2)
( μ ∂∂vx − ρu ′v ′) +
[ μ ∂∂yv − ρv ′v ′] + ∂∂z [ μ ∂∂vz − ρw′v ′]
ρ (u ∂∂wx + v ∂∂wy + w ∂∂wz ) + ∂∂pz =
TABLE 2
NATURAL FREQUENCY AND MODE SHAPES OF SINGLE FUEL ROD
S. No.
1
2
3
( μ ∂∂ux − ρu ′u ′) +
[ μ ∂∂uy − ρu ′v ′] + ∂∂z [ μ ∂∂uz − ρu ′w′]
ρ (u ∂∂vx + v ∂∂yv + w ∂∂vz ) +
∂
∂y
∂
∂x
∂
∂x
(3)
( μ ∂∂wx − ρu ′w′) +
[ μ ∂∂wy − ρv ′w′] + ∂∂z [ μ ∂∂wz − ρw′w′]
( 4)
ρC p (u ∂∂Tx + v ∂∂Ty + w ∂∂Tz ) = ∂∂x [k ∂∂Tx ] + ∂∂y [k ∂∂Tz ]
+ ∂∂z [ k ∂∂Tz ]
(5)
where u , v, w are mean velocities and u ′, v ′, w′ are turbulent
fluctuations. A Mac-type staggered grid system is used to
locate the flow variables. The velocities are stored on the cell
faces and the pressure and temperature are stored at the cell
centers. Due to the staggering of the mesh three different
types of control volume are required for the momentum
equations and the continuity equation in the interior region,
with straightforward modifications near the boundaries. A
detailed description is given in [10].
1936091 IJET-IJENS @ International Journals of Engineering and Sciences IJENS
International Journal of Engineering & Technology IJET Vol: 9 No: 9
- 204 -
These equations are solved using standard 3-D Fluent [8]
hydrodynamic code using Finite Volume method. 0.81 Million
fluid elements with hexahedral unstructured mesh are taken in
the computational domain Fig. (2). Results for velocity,
vorticity and temperatures are obtained and depicted in Figs.
(6-13).
No slip boundary conditions for velocities on the solid surface
are given whereas temperature is kept constant. Symmetric
conditions for all variables are used on other boundaries.
Fig. 8. Temperature Contours on Cross-Sectional Area at z=1600 mm
Fig. 6. Velocity Contours on Cross-Sectional Area at z=1600 mm
Fig. 9. Velocity Across Radial Direction
Fig. 7. Vorticity Contours on Cross-Sectional Area at z=1600 mm
Fig. 10. Vorticity across Radial Direction
1936091 IJET-IJENS @ International Journals of Engineering and Sciences IJENS
International Journal of Engineering & Technology IJET Vol: 9 No: 9
- 205 IV. RESULTS AND DISCUSSIONS
Fig. 11. Temperature across Radial Direction
The modal analysis of a single rod were carried out and the
results are given for natural frequencies for first, second and
third modes which are in the range of 5 to 27 Hz. [Table (3)].
The hydrodynamics analysis was performed and results were
obtained for vorticity (TIV) along the fuel rod. From Fig. 12,
it is observed that vorticity at axial distance 0-700 mm from
the bottom of fuel rod is in the range of 0-5 Hz which
increased exponentially from 5-60 Hz up to 1200 (mm) and
then suddenly is fallen to 25 Hz at the middle of the rod (1500
mm). From here onward along the rod, it remained 25 Hz till
the top of the rod (3210 mm).
These results have shown that lock-in (resonance) condition
has occurred at both ends of the fuel rod which may be
damaged the assembly at the bottom baffle plate in the power
reactor. To avoid such accident, the coolant velocity may be
reduced or controlled by some phenomena not to occur lockin condition. Further numerical investigations will be carried
out by varying coolant flow conditions to avoid lock-in
condition.
V. CONCLUSION
A comparison was done between fluctuating frequency
of the coolant fluid flow and the hollow rod material
frequency of a typical 300 Mwe PWR under operation.
This has been observed that the resonance (lock-in)
condition occurred at both ends of the fuel rod resulting
damage the assembly at the bottom baffle plate in the
power reactor. Numerical experiments have shown that
these phenomena could be controlled by lowering the
coolant flow rate (velocity). Further numerical
investigations will be carried out by considering the flow
rate of the coolant through pressure difference
Fig. 12. Vorticity along Axial Direction
REFERENCES
[1]
[2]
Fig. 13. Velocity along Axial Direction
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Kazimi, M . S, , “High Performance Fuel Design for Next Generation
PWRs”: Final Report Center for Advanced Nuclear Energy Systems
(CANES), MIT-NFC- PR-082. 2006.
[3] King, S. J., Young, M. Y., Seel, D. D, Conner, M. E., Lu, R. Y.and
Paramonov, D. V., “Flow Induced Vibration and Fretting Wear in
PWR”, International Conference on Nuclear Engineering, American
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[4] Mian, Z., and Nayyar, A. H., 1999, “Pakistan’s Chashma Nuclear
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[5] S. Khushnood, Z.M.Khan, M.A.Malik, Z.U.Koreshi and M.A.Khan,
2003, “Cross-Flow Induced Vibrations in Tube Bundles: A Review”,
Proc. of the Eleventh International Conference on Nuclear Engineering,
(ICONE-11-36261), April 20-23, 2003, Tokyo, Japan.
[6] Kang, H. S., Choi, M. H., Yoon, K. H., Song, K. N. and Jung, Y. H.
2004, “A Vibration Analysis and Test of a 5×5 Rod Bundle”, ICONE-12
Arlington, Virginia, April 25-29.
[7] ANSYS 10.0 Inc. Pittsburg, USA
[8] Fluent 6.1 Inc, Lebanon, New Hampshire, USA
[9] Fletcher, C. A., Computational Techniques for Fluid Dynamics,
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[10] S. Sivaloganathan and G. J. Shaw, A multigrid method for Recirculating
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1936091 IJET-IJENS @ International Journals of Engineering and Sciences IJENS