ML20036C004
| ML20036C004 | |
| Person / Time | |
|---|---|
| Site: | Haddam Neck File:Connecticut Yankee Atomic Power Co icon.png |
| Issue date: | 06/02/1993 |
| From: | Office of Nuclear Reactor Regulation |
| To: | |
| Shared Package | |
| ML20036C003 | List: |
| References | |
| IEB-88-002, IEB-88-2, NUDOCS 9306080267 | |
| Download: ML20036C004 (10) | |
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UNITED STATES 5
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WASHINGTON, D.C. 20555-0001 ENCLOSURE SAFETY EVALUATION BY THE OFFICE OF NUCLEAR REACTOR REGULATION RESPONSE TO THE NRC BULLETIN 88-02 (RAPIDLY-PROPAGATING CRACKS IN STEAM GENERATOR TUBES)
HADDAM NECK PLANT CONNECTICUT YANKEE ATOMIC POWER COMPANY DOCKET NO. 50-213
1.0 INTRODUCTION
A steam generator (SG) tube ruptured at North Anna Unit 1 in July 1987. The cause of the tube rupture was determined to be high-cycle fatigue due to flow-induced vibration.
It was shown that of the possible flow-induced vibration mechanisms, fluidelastic instability was the most-likely cause.
Fluidelastic instabilities can result from coupling between flow-induced dynamic forces and SG tube vibrations.
Instability occurs when flow velocity is sufficiently high so the energy absorbed from the fluid forces exceeds the energy dissipated by damping. Motions developed by a tube in the fluidelastically-unstable mode are quite large in comparison to the other known mechanisms.
Thus, it is possible for an unstable fixed-boundary condition tube to deflect to an amount, which will produce fatigue-inducing stresses. As a result of the North Anna Unit 1 incident, there has been renewed interest in the possibility of a fluidelastic instability in units of similar design. The more recent incident in February 1991 at Mihama-2 at the Kansai Electric nuclear power plant in Japan, was another example of a similar type of failure.
In February 1988, NRC issued Bulletin 88-02 " Rapidly Propagating Fatigue Cracks in Steam Generator Tubes," (Reference 1) requesting licensees with Westinghouse SGs to assess the potential of such tube failures in its plants.
Connecticut Yankee Atomic Power Company (CYAPC0 or licensee) and its consultants performed a fluidelastic instability analysis in response to Bulletin 88-02, for the Westinghouse Model 27 SGs at Haddam Neck, and determined that no corrective action was required at that plant (Reference 2).
During the review of the licensee's analysis, the staff requested additional information on the analytical model used by the licensee and its consultants.
In its response, (Reference 3) the licensee qualified its earlier analytical model by means of a similar evaluation of fluidelastic instability for the known failed SG tube at North Anna Unit 1.
The licensee also responded to other staff concerns regarding establishment of a quantitative measure for the onset of instability and justification for the selection of damping ratios employed in the fluidelastic instability analyses.
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. aA 2.0 DISCUSSION In its response to NRC Bulletin 88-02, (References 2 and 3) the licensee attempted to quantify the likelihood of a fluidelastic instability in Westinghouse Model 27 SGs at Haddam Neck, and to identify tubes which might lead to failure due to flow-induced vibration. The SG tubes of special interest are the U-tubes in the inner rows, which are not supported by anti-vibration bars (AVBs).
The= nonuniform AVB elevations above the tube sheet gives rise to a secondary flow peaking effect.
It is believed that this plant-specific characteristic might have an adverse effect on fluidelastic instability of U-tubes immediately below the lowest AVB. Consequently, a detailed geometric model of the AVBs, including unequal elevations, was employed in the present analysis.
Since the vibratory response is limited in the plane of U-tubes, it was assumed that the significant tube displacements are in a direction normal-to the plane of the tube. Modal analyses have shown that the dominant mode of vibration is an out-of-plane flexure.
Predictions by three-dimensional two-phase flow codes indicate considerable cross-flow in the U-bend region for typical recirculating-type SGs.
This cross-flow can cause damaging flow-induced vibration of the tubes with minimal supports.
I Denting, which is due to the accumulation of deposits such as magnetite, in the annular openings between the outer tube surface and the support plate, can change the plate-to-tube interaction from a limited-displacement-type to a fully-or partially-clamped type. The dominant frequency for the out-of-plane response depends on the type of boundary condition assumed at the intersections of a U-tube with the uppermost support plate.
For this reason, analyses were performed assuming both fully-clamped and flexible supports.
Modal analyses with flexible end supports yield dominant frequencies which are significantly lower.
In general, such tubes would be more susceptible to fluidelastic instabilities.
The licensee's flow-induced vibration analysis consists of three parts:
(1) a:
thermal-hydraulic simulation of the flow field on the secondary side of the SG using the ATHOS3 three-dimensional two-phase flow code, (2) a modal analysis by the ADINA finite-element code which computes the natural frequencies of.the-dominant vibration modes and mode shapes of the susceptible ' tubes, and (3) a fluidelastic instability analysis which utilizes predicted data from thermal-hydraulic and modal analyses in a post-processing procedure, and determines effective flow velocities between adjacent tubes and critical gap velocities which would introduce fluidelastic instability.
2.I THERMAL-HYDRAULIC SIMULATION OF WESTINGHOUSE MODELS 27 AND 51 STEAM GENERATORS Thermal-hydraulic analyses of the Westinghouse Models 27 and 51 SGs were performed using the ATHOS3 computer code (Reference 4). The analyses predicted the three-dimensional distribution of two-phase velocities and mixture densities in the secondary flow field of the SGs.
These parameters
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. are needed to calculate effective fluid velocities at various locations in the tube bundle and compare them to critical velocities which may cause damaging tube vibrations. A finite-difference grid with over 10,000 active cells was used in the analysis. Since the velocity components computed by ATHOS3 are defined on the surfaces of a flow cell, the tube gap velocity and density i
distributions along a particular tube, required for tube vibration evaluation, are determined by a post-processor from the ATHOS3 output. The post-processor generates a data file which contains this information for all the tubes of interest in the model and serves as part of the input data required for tube vibration analyses. Because the majority of the flow cells contain more than one tube within a cell, the tube gap velocity and density surrounding a tube are obtained by interpolation of the ATHOS3 calculated velocities (defined on the cell surfaces) and densities (defined at the center of the cell). The post-processor performs the necessary interpolations to determine in-plane and out-of-plane velocities at specific intervals along the length of the tubes.
Besides these parameters, the code calculates the circulation ratio (CR)
(defined as the ratio between the mass flow rate through the tube bundle to the feedwater mass flow rate). This is an important parameter since it establishes the total bundle flowrate and average loading on the tubes. The licensee's analysis predicted a value of CR which is about 20% higher than the assumed value for Model 27 SGs. The higher predicted value of CR may be due to an underestimation of empirical loss coefficients for the tube support plates, downcomer and steam separators.
If mineral deposits accumulated in the flow holes and annular gaps around tubes in the support plates, the associated loss coefficients would be higher than those assumed for a unit with nominal design openings. A higher value of CR than anticipated for actual plant operation would result in higher axial velocities in the wrapper region, which in turn implies higher predicted cross-flow velocities. Thus, the net effect would be to introduce some conservatism in the evaluation of fluidelastic instability. The analysis underpredicted the inlet and outlet temperatures by about 5'F, which is considered acceptable. ATHOS3 predictions differ from accepted plant values by about 1-2* for similar evaluations of a CE System 80, and Electricite de France Model 51 SGs.
A possible explanation for the slightly higher underprediction of primary fluid temperatures is the fouling of the tube bundle, which has not been accounted for in the Haddam Neck ATHOS3 simulation.
Fouling introduces an additional resistance to heat transfer, and an accompanying increase in the primary temperature in order to achieve the same overall heat load with a lower effective value of heat transfer coefficient. The predicted values of primary inlet and outlet temperatures are also affected by the particular heat transfer correlation selected.
The ATHOS3 code iterates on the absolute values of primary inlet and outlet temperatures, such that the primary temperature drop is maintained at a level known a priori. That level is determined during an initialization phase of the calculation from a global enthalpy balance employing input data for the operating parameters.
There was good agreement between the measured and predicted values of primary temperature drop, which is a direct indication of power level.
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- The predicted flow patterns indicate that the flow in the SG is quite i
unsymmetrical. The heat imbalance between the hot and cold sides causes much higher velocities on the hot side, while the contraction of the wrapper causes a convergent flow pattern near the top of the U-bend.
Before the flow reaches the top, it acquires an outward radial component, seeking to take the path of least resistance through the bundle.
This latter effect has been observed in many SG code predictions, and is attributable to the higher solid-to-fluid friction for cross-flow against the apex of the tube bundle, as opr ed to axial flow through the sides of the bundle. The outward. radial f' 3 rings about a cross-flow component in a horizontal plane, which is of s-ficant interest with regard to the potential for fluidelastic instability.
.The tubes in the apex region are of primary interest from fluidelastic instability considerations.
In this region, the cross-flew velocity in a horizontal plane has an upper bound of about 1.0 ft/sec. The corresponding tube gap velocity is about 2.1 ft/sec.
Since the cross-flow velocity has a finite circumferential component, the velocity component normal to the plane would be lower. Thus, the effective velocity normal to the U-bends can be expected to be less than 2 ft/sec for all tubes in the apex region.
A comparison of cross-flow velocities in the apex region for the Models 51 and 27 SGs revealed that the cross-flow tube gap velocities predicted by the ATHOS3 code for the Model 51 SG are substantially higher than those predicted for Model 27.
In the Model 51 simulation, the AVBs and biccked tubes were not considered, but the large difference in the cross-flow velocities cannot be explained on this basis alone. Most likely, the higher cross-flow velocities are due to different thermal-hydraulic conditions in the U-bend region, and to some extent, due to the lower solid-to-fluid friction of a less tightly packed tube bundle (the pitch-to-diameter ratio is 1.464 for the Model 51 compared to 1.375 for the Model 27).
2.2 MODAL ANALYSES The modal analyses were performed by using the ADINA (automatic Qynamic Incremental Nonlinear analysis) computer code.
The U-tubes of five rows in the region of interest and having appropriate geometric parameters were each considered separately. The flexural and material properties of the Inconel 600 tubes were modelled with 40 two-noded beam-pipe elements. The combined inertial effects of the tube metal, the primary fluid and the secondary mixture were represented as an effective beam density.
Denting at the i
uppermost support plate was approximated as a fully-clamped support condition in the initial calculations.
The complete finite element model consisted of a linear system with 234 degrees of freedom. The lowest four eigenvalues of the system were calculated by means of a determinant search algorithm. The resulting fundamental mode of each tube was determined to be an out-of-plane flexure for which all points move in the same direction. Higher mode shapes were determined to be in-plane side sways, and out-of-plane flexures in which the two quadrants of the U-tube move opposite to one another.
These mode shapes, along with the natural frequencies of dominant modes, are provided in the licensee's submittal. As anticipated, the U-tube frequencies are significantly influenced by the length of the straight sections and support
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. conditions. Since U-tube frequencies can have an important Learing'on the critical cross-flow velocities, the support conditions and the straight sections of the tube had to be accurately modelled.
Previous studies in similar SGs suggest that partially fixed, rather than completely fixed
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conditions, more accurately represent tube supports (Reference 5). An additional modal analysis was conducted to examine this effect. This analysis, which was qualitative in nature, indicated that mode shapes retain their general shapes, but motions occur at lower natural frequencies as fixities are relaxed.
Since it is not possible to assign precise moduli for tube support flexibility based on available data for the Model 27 SGs, this effect was considered qualitatively in evaluating fluidelastic stability.
2.3 FLUIDELASTIC INSTABILITY ANALYSIS Several rows of tubes in the U-bend region which are not supported by anti-vibration bars were analyzed for fluidelastic instability. Those rows immediately below the lower set of AVBs may be subject to local flow peaking effects due to an off-design characteristic of unequal AVB elevations._ The fluidelastic instability analysis utilized data from both thermal-hydraulic and modal analyses in a post-processing procedure, for the determination of-effective and critical velocities.
In a conventional analysis, a quantitative measure of instability is taken as the ratio of the effective velocity, V,,,,
ThisratioissometimesreferredtoastEe to the critical velocity, V,kt.
" susceptibility ratio," S =
/V, greater than 1.0 implies an on,s,et 'oY. A susceptibility ratio equal to or -fluidelastic ins displacements can grow quite large and cause tube rupture.
In this ratio, the effective velocity depends on the-span-wise distributions of flow velocity and density, and on the mode shape of vibration. The critical velocity is based on experimental data and has been shown to be dependent upon the tube's natural frequency, damping, geometry,' pattern and fluid density, along with the appropriate correlation coefficients. The detailed calculation of this ratio using velocity and density distributions, requires three-dimensional thermal / hydraulic and tube vibration calculations which are very complex and time consuming. Alternately, a simplified, one-dimensional version of this ratio has been used to provide a more rapid, relative assessment of tubes to fluidelastic instability. The critical velocity is defined by Conners' equation, where 2xm,f Ve =K!Do Peo:
i K - Conncrs' coefficient, taken as 3.3 f = natural frequency of tube, Hz D. - tube outer diameter i
m* - effective mass per unit tube length
( = total damping ratio p,
= effective secondary fluid density.
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. The value'of Conners' Coefficient K - 3.3 is considered to be conservative.
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Larger values leading to proportionately higher critical velocities are consistent with some experimental data.
The value of the total damping ratio of 0.03 was inferred from the experimental data by Pettigrew, et al (Refernce 6). This comprehensive collection of damping data for single and two-phase flow conditions includes.
data obtained by Axisa, et al. (Reference 7). Although considerable work has-been done in the last decade, apparently, damping data on steam-water mixtures is still scarce, it appears that damping ratios obtained in steam-water are.
generally lower than in air-water, the latter configuration being the source of much of the database.
Pettigrew and Axisa have suggested that this may be due to a lower surface tension in steam-water mixtures.
Damping in two-phase flow primarily depends on void fractions, flow regimes, fluid and structural densities.
It has been shown to have a finite, but weak dependence on frequency, confinement (pitch-to-diameter ratio) and mass flux.
Since damping in two-phase flow appears to have the strongest dependence on void fraction, the licensee has chosen to account for this effect in all of its fluidelastic instability analyses. He most recently published data (Reference 6) was utilized in damping-void fraction correlations for both the Models 27 and 51 analyses. The measured data for damping in steam-water was derived from experiments conducted in France during the early 1990s.
A staff concern relative to the ATHOS3 Code was that unequal phase velocities (or slip) occurred in the axial direction only., The licensee's explanation for this relates to i.he early development and testing of the ATHOS3 Code, when it was determined that it was difficult to achieve convergence with the
" general slip" option, which allows slip. in e.11 coordinate directions. ' This was due to discontinui+ies in interphase friction correlations in certain regions of the flow ra '.mes.
An " algebraic slip" option, where slip is allowed only in the predominant flow direction, was adcpted to overcome this difficulty.
With this model, the relative velocity between the vapor phase and the liquid is deduced from a drift-flux formulation, while the mixture velocity is calculated from a mixture-momentum equation.
The staff had raised concerns regarding the effective fluid velocity in the U-bend region of the SG tube bundle.
The selection of a suitable, effective velocity in this region is complex, because the general rules for determining effective velocities which apply to the straight tube section do not necessarily apply to the curved tubes in the U-bend.
For example, the method of taking the weighted averages of the square of the velocity with the square of the displacement may not be completely valid for the U-bend region. A further complication is that there are separate phase velocities in the axial direction available from the thermal-hydraulic calculation. The usual practice is to employ.a mixture velocity, which again lacks rigorous justification. The fluid velocity obtained from the Code (actually, an
" approach" velocity into a cell) must be converted into a tube gap velocity and the velocity at a node along the tube _must be interpolated from values at cell's edge. The same method has been used for the determination of effective velocity for the Model 27 at.Haddam Neck and the Model 51 at North Anna. The
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staff recognizes the empirical nature of certain assumptions in the analysis.
In some cases, supporting data is not available. However, the licensee has attempted to perform bounding calculations in those instances.
The staff finds the overall analytical approach acceptable.
2.4 RESULTS OF FLVIDELASTIC INSTABILITY ANALYSES FOR MODELS 27 AND 51 SG'S For fluidelastic evaluation, susceptibility ratio (S - V measure of the potential for unstable flow-induced vibraN, o/Vg )he data for represents a T
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the Model 27 SG shows that the maximum value of S for all tube rows of interest lies between 0.80 and 0.88 and occurs consistently at the largest column number in each row (Figure 1). The location of the maximum value is predicted to occur near the periphery of the bundle, where velocities tend to be the highest. The latter effect is due to the reduced resistance of flow parallel to the tubes as compared w' h normal flow. The secondary flow tends to " skirt" around the tube bundle.
Some of the higher values of S are also found at the lowest column numbers (e.g., column 51), near the vertical plane of symmetry.
Due to the limitations in the spatial resolution of the ATHOS3 calculation, and the inability of any existing thermal-hydraulic code to predict the exact profiles of voi<i fraction and velocities in the U-bend region,.it was.not.
expected that the model would be able to single out the exact location of the failed tube at North Anna Unit 1.
The resolution is of the order of three tube pitches even with 11,000 cells, and interpolation had to be employed to evaluate individual tube locations. With regard to the accuracy of prediction at a given computational node, comparisons with experimental data have shown that ATHOS3 can capture only the general shape of the void fraction or velocity profile alung a tube row but local differences may be as high as 10%
at any node. Other available thermal-hydraulic codes can predict these void fraction and veloci+v parameters with about the same degree of accuracy.
Hence, it is impor u to interpret the results in the proper perspective.
The susceptibility. e
, S, for the failed tube, R9C51, at North Anna was determined to be 0.852, a value in the range of the maximum predicted for the Model 27. However, a higher value of S - 0.956 was calculated only three columns away which is within the spatial resolution of the 'model. A preat ted value of S as large as 1.210 prevails at the outermost column of row 9, column 93.
At column 51 of row 10, only a. single pitch away from the failed North Anna tube, the value of S is 0.998, which is essentially the conventionally-accepted value for the onset of instability.
Larger values of S are also predicted for the innermost and outermost tube columns of row 10, e.g.,
columns 48 through 50 and 90 through 93. The maximum value of susceptibility ratio for row 10 is S - 1.464, which occurs at outermost tube column 93. The average value of S for all columns of row 10 is 0.946.
Several tubes of row 10 of the Model 51 at North Anna Unit 1 appear to w close to instability if they are indeed unsupported by AVBs and are also dented at the uppermost support plate. As the bend radir increases, the utural frequency and the critical velocity decrease resuit rg in higher values of S for a given cros; fl ow. The data for row 11, shows higher values of S, with the average and
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maximum vhlues b'eing 1.098 and 1.756, respectively. A comparison of critical velocities for the Model 27 and Model 51 for a tube of the same bend radius shows that the values of V for the Model 51 are generally higher. This is dueprimarilytothelarge,r,,lubediameterfortheModel 51 (0.875 in. vs.
0.750 in.), as noted earlier. However, V in Model 51 are also predicted to be much higher than Model 27. Asaresull,,,thesusceptibilityratiosinthe Model 51 are generally higher than in the Model 27 SGs.
In its review, the staff has taken into consideration a number of conservatisms in the licensee.'s analyses. These include' the selection of empirical coefficients such as the Conners' coefficient and darping ratios, which were selected so that the resulting critical velocity V 'he empirical would tend to beinthelowrangewithrespecttothealternatechoicesofla factors. This would lead to higt.or values of S, and a conservative prediction of fluidelastic instability.
Other assumptions in the analysis which introduce similar conservatisms include tube support flevability considerations. The analysis is based on the assumption of rigid tube supports. The effect of assuming flexible supports is to decrease the natural frequency for the dominant out-of-plane modes and critical velocities, thus increasing the predicted values of S.
The analysis includes the effect of the AVBs on overall secondary flow distribution, but prediction of local flow effects is highly ccmplex and remains elusive with the currently-available techniques.
These flow peaking effects for the Model 27 SG are likely to be smaller than those for the Model 51 SG, since the AVBs in the Model 27 have a circular cross-section, as opposed to a rectangular shape. The rod-like AVBs of Model 27 provide less of a perturbation of the secondary flow field.
In addition, the licensee's analysis indicates that flow peaking does not seem likely in the region of the most susceptible tubes for the Model 27 SG.
The circulation ratios, CR, have been selected at the high end of the design range.
The design value of CR is usually determined by the vendor using a one-dimensional model, based on mea::ured or est ' =ated pressure drop for devises such as the steam separator, tube support plates, and downcomer. The licensee's database on the Westinghouse Model 27 includes design values of CR in the range 3.8 < CR < 4.8.
The value of CR = 4.54 predicted by ATHOS3 was used, although the CR might have a value as low as 3.8 in practice.
It seems likely that due to the accumulation of deposits in the secondary flow passages through tube support plate gaps and flow holes over 24 years of operation at Haddam Neck, the CR might indeed be at the low end of the design range. A lower value of CR would result in lower axial velocities throughout the wrapper region, hence, a value of CR - 4.54, based on assumed values of pressure drop coefficients, provides some conservatism in the analysis.
The natural frequency predicted by ADINA for the failed North Anna Unit 1 tube R9C51 was 65.6 Hz, which is higher than the Westinghouse computed value of 60 Hz. This implies that the value assumed for the lenath of the U-tube straight stub might not be representative of the actual North Anna geometry. However, the effect of an overprediction of natural frequency would be an increase in the calculated value for critical velocity and a decrease in susceptibility ratios predicted for North Anna Unit 1.
Thus, if the natural frequency of
1Li22:**122 60 Hz for R9C51 is accepted, all susceptibility ratios predicted for row 9 of North Anna Unit I would be increased by about 10%.
In light of the licensee's conservatively-based approach, as discussed above, the staff concludes that fluidelastic instability is unlikely to occur in the Model 27 SGs at Haddam Neck.
3.0 CONCLUSION
S The likelihood of failure of "9 tubes at Haddam Neck due to fluidelastic instability has been ascertained in response to NRC Bulletin 88-02. The licensee's calculated results indicate that the parameter, S, which is an indicator of fluidelastic instability is below the critical limit of 1.0 for the most susceptible tubes.
The licensee has verified the accuracy and reliability of its analytical model by means of a similar evaluation of fluidelastic instability for the known failed tube, R9C51, at North Anna Unit 1.
The result of the benchmark comparison showed that for North Anna Unit 1, the licensee's model did indeed predict susceptibility ratio, S, exceeding 1.0, the conventionally accepted value above which fluidelastic instability can occur. The model predictions also indicated that values of S > 1.0 occurred at locations within the spatial resolution applicable to the thermal-hydraulic calculation (three tube pitches), of the failed North Anna Unit I tube, R9C51.
Susceptibility ratios predicted for the Model 27 were typically smaller than 0.65, with a maximum value of S = 0.88 near the bundle periphery, where flow peaking is not expected to play a significant role.
Based on its review of the fluidelastic instability analysis for Model 27 SGs at Haddam Neck, the staff concludes that the licensee has generally adopted a conservative approach, and fluidelastic instability is not likely to occur at these SGs. Therefore, corrective actions such as downcomer flow restriction, or tube plugging are not considered necessary for the Haddam Neck SGs.
Principal Contributor: J. Rajan Date:
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-REFERENCES 1.
NRC Bull.atin 88-02, " Rapidly Propagating Fatigue Cracks in' Steam Generator Tubes," dated February 1988.
2.
JAYCOR Report J5439-89-00R1 "Fluidelastic ' Instability Analysis of the U-Bend Region of a Westinghouse Model 27 Steam Generator," dated July 1989.
JAYCOR Report Jk681-90-00RO " Response to NRC request for Additional 3.
Information - Haddam Neck," dated June 1991.
4.
L. W. Keeton, A. K. Singha1, and G. Srikantiah, "ATHOS3 - A Computer Program for Therm 6 Hydraulic Analysis of Steam Generators," EPRI Report NP-4604-CCM, Volunie 1:
" Mathematical and Physical Models and Method of Solution," July 1985.
5.
C. C. Schoof, T P. Khatua, L. M.;Shusto, S. P.' Winder, and J. M.#
Thomas, "Millst..se II Degraded Support Analysis:, Analysis of Tube -
Vibrations and Tube-Support Interaction Forces," Report FaAA-R-85-04-02, Failure Analysis Associates, Palo Alto, CA, December.1986.
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6.
M. J. Pettigrew, C. E. Taylor and B. S. Kim, " Vibration of Tube Bundles in Two-Phase Cross-Flow:
Part 1 - Hydrodynamic. Mass and Damping," ASME-PVT Vol. Ill, November 1989.
Presented at 1988 International Symposium on Flow-Induced Vibration and Noise, ASME WAM, PVP Division,-Volume 2, Chicago, Nov. 27-Dec. 2, 1988.
7.
F. Axisa, M. Wullschleger, B. Villard and C. Taylor,'"Two-Phase Cross-Flow Damping in Tube Arrays, ASME-PVT Vol.133.
Presented at ASME-PVP Conference, Pittsburgh, June 1988.
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