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BWR instrument tube vibrations: Interpretation of measurements and simulation
Ann. Nucl. Energy, Vol. 21, No. 12, pp. 759-786, 1994
Pergamon 0306-4549(93)E0004-J
Copyright © 1994ElsevierScienceLtd Printed in Great Britain. All rights reserved 0306-4549/94$7.00+ 0.00
BWR INSTRUMENT TUBE VIBRATIONS: INTERPRETATION OF MEASUREMENTS A N D SIMULATION I. PAZSIT I A N D O. G L O C K L E R 2 ~Department of Reactor Physics, Chalmers University of Technology, S-412 96 G6teborg, Sweden 2Fuel and Physics Department, Ontario Hydro, 595 Bay St. A7-BI2, Toronto, Ontario, Canada M5G 2C2 (Received 11 August 1993; in revised form 23 October 1993)
Abstract--In 1986 and 1987, neutron noise measurements were made in two Swedish BWRs on a number of detector strings, in order to detect and quantify detector guide tube vibrations, and possibly impacting. Vibrations of various amplitude were detected in several strings, but quantification of the strength of vibrations, including impacting, was not trivial. For quantification of the vibrations, there appeared a need for spectral descriptors other than just the amplitude of the vibration peak in the detector auto power spectral densities (APSDs). To find such descriptors, and investigate their sensitivity, numerical simulation of the signal of a vibrating detector both with and without impacting was performed. Two possible descriptors of impacting, namely the widening of the vibration peak and the distortion of the amplitude probability distribution (APD) function were investigated. The first of these two was found to be a relatively reliable indicator of impacting, in accordance with the experimental findings. The use of the APD is much less effective, and reasons for this are discussed in the paper.
!. INTRODUCTION The fact that excessive instrument tube vibrations may occur in BWRs has been known from the mid1970s. Extensive review of the measurements and diagnostic analysis can be found in Thie (1979) and Kosfily (1980). The significance of this phenomenon arises since excessive vibrations can lead to impacting and channel box corner wear, as well as detector damage. It is therefore important to be able to monitor the vibrations in order to avoid impacting. However, there is no absolute quantitative method available for determining impacting or even judging the probability of impacting. The most effective information carrier is the signal of the L P R M detectors (local power range monitors) within the instrument tube concerned. It is easy to observe the presence of the vibrations through the peak in the detector signal APSDs in the frequency range 2-5 Hz. However, it is not easy to use this information to determine whether or not impacting occurs. One possibility that had been investigated in detail is to relate the magnitude of the vibration peak in the normalized A P S D (NAPSD) of the detector signals to absolute displacement. One problem is that the magnitude of the resulting neutron signal fluctuations, and thus that of 759
the peak in the APSD, is determined by both the local neutron flux gradient and the actual two-dimensional trajectory of the vibration. These are usually unknown, different for each tube and reactor, and can even change in time during the cycle independently of the amplitude of vibrations. Nonetheless, criteria for impacting, based on a threshold in detector signal NCPSDs have been elaborated and used (Mott, 1976; Fry et al., 1977). Such threshold criteria are rather individual to each core type, and are not easy to transfer from one core to another. The lack of generally applicable quantification methods was clearly felt when in 1986 and 1987 the Studsvik Noise G r o u p (currently within EuroSim AB), of which one of the present authors was a member, evaluated in-core noise measurements in two Swedish BWRs. The main purpose of the measurements was to quantify possible detector tube vibrations. It was observed in the measurements that several strings vibrated, but the threshold principle above could not be applied due to differing core and detector parameters. As reported in/~kerhielm et al. (1986) and P~izsit et al. (1987), during the evaluation of the measurements, new descriptors were also invoked. Of the traditional ones, the magnitude of the fundamental resonance as well as the first higher harmonics were used. In
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at various elevations. From independent investigations, it had been suspected that some of the detector tubes may vibrate, and the aim of the investigations • the phase of the cross spectrum between detectors was to quantify the vibrations. in the same string; The results of these investigations were reported in • widening of the vibration peak; more detail by ,~kerhielm et al. (1986), here we only • distortion of the APD function. give a brief summary. In some strings, no or little Out of these three, the first one is based on the fact vibration was found. Auto- and cross-spectra in those that the linear phase of the CPSD between two strings follow a pattern that was described by the detectors in the same string, observed earlier in BWR simple model of Wach and Kositly (1974). For such measurements and elaborated in Section 2, gets dis- strings the phase curve of the CPSD between two torted, eventually becomes zero, within the frequency detectors shows a linear dependence on frequency up range of vibrations if the latter are strong. Its use was to some 10-15 Hz, whereas the absolute value of the reported in ~kerhielm et al. (1986) and Pfizsit et al. CPSD, and even more the coherence, show a charac(1987, 1988). The second descriptor, the widening of teristic sink structure. At integer multiples of the the vibration peak, was noticed first in numerical frequency equal to the inverse of the bubble transit experiments by G16ckler and Frei (1986). It was used time z, that is at in interpreting both measurement series mentioned f = (n/r); n = 0, 1, 2, 3.... (1) above. The third one was suggested to the present authors by Thie 0985). It was not used in the the phase curve crosses zero, and the coherence and Barsebfick measurements, but attempts were made to the CPSD have a local maximum. At half-integer use this principle in the Oskarshamn measurements, multiples of 1/r, i.e. at without success. To find out the applicability of the f = (n + 1/2)/z; n = 0, 1, 2, 3 .... (2) APD distortion concept, further numerical simulations were made by Gl6ckler and Frei that will be the phase curve crosses ± 7r, and the coherence and reported in this paper. CPSD have a local minimum, that is a sink. The content of this paper is as follows. In Sections 2 Such a case is demonstrated by the noise spectra of and 3, the measurements made in Barsebfick 1 and the detectors in string No. 18, LPRMs 181-184, see Oskarshamn 2 will be described, respectively. In Figs 1 and 2. Here LPRM 181 stands for the detector Section 4 the numerical simulation model will be in string No. 18 at level l, which is the highest axial described, and results of the investigations presented. level. The same convention is used in numbering the Finally, the results of the measurements and the detectors throughout the paper. Corresponding to the simulation results will be compared. The conclusion is fact that this tube does not vibrate, the APSDs of the that the peak widening, although not an absolute and detector signals show no peaks, only a smooth struceasily quantifiable method, may be effectively used for ture (Fig. 1). The phase and coherence curves follow detection of impacting. On the other hand, the distor- the pattern described above very closely (Fig. 2). The tion of the APD function is not a useful indicator of phase is linear up to between 10 and 15 Hz, above impacting. The main reason for this is that usually, which frequency the reactor transfer breaks down and only a small portion of the power content of the signal the background and instrumentation noise starts to is contained in the vibration peak, thus the APD is dominate. Accordingly, the coherence vanishes above dominated by the noise from sources other than the this frequency, as seen from the figures. From the vibration itself. In addition, in the presence of a slope of the linear part of the phase curves it can be saddle-point type gradient in the neutron flux at the seen that the bubble transit time is smaller between detector position, the distortion of the APD due to the detectors on the highest axial level. impacting is reduced quite significantly. In strings No. 02 and 03, vibrations can be observed. We take first string No. 02, where moderate vibrations can be seen. This can be demonstrated by the APSD on level 4 (Fig. 3b). A large peak is seen at 2. THE MEASUREMENTS IN BARSEB,~CK-I 6 Hz. However, there are no peaks in the APSDs of The first series of measurements were made in the LPRM detectors on the higher axial level (see e.g. Barseb~ick-l, a 440 MW Swedish BWR, late spring Fig. 3a), indicating that the largest amplitude of 1986 by Studsvik. Neutron noise signals were taken vibrations may be found on level 4. from 6 detector strings, each containing 4 LPRMs. In A further support for the assumption of vibrations addition, measurements were made by a TIP detector in string No. 02 can be found in the phase behaviour addition, three new parameters or descriptors were used. These were:
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and sink structure of CPSD and coherence (Fig. 4). On levels 1-2 and 2-3 (not shown here for brevity), the phase and sink structure show a vibration-free behaviour, described in and around equations (1) and (2). However, between levels 3 and 4 a distortion of both the phase curve and the coherence can be seen. The linear phase behaviour disappears above 6 Hz, which is the frequency of the vibrations in this case, and the sink structure is also distorted. By the latter it is meant that the minima and maxima of the coherence function do not coincide with the "n-crossings" and zero-crossings of the phase curve, respectively. The most interesting string in this analysis is string No. 03, where the strongest vibrations were detected. However, looking at the APSDs on levels 1-4 (Fig. 5), this may not appear so obvious. On level'l, no explicit vibration peak can be seen (Fig. 5a). On level 2, a small thin peak can be seen (Fig. 5b), which looks much less significant than the peak in Fig. 3b, belonging to LPRM 024. On level 3 the peak at 6 Hz is much broader, and another broad peak can be suspected at around 13 Hz (Fig. 5c). On level 4 these two peaks can only be surmised, they are broadened so much that only an increased value of APSD can be seen when compared to LPRM 031 (Fig. 5d). Compared to the higher axial levels, an overall increase of the APSD magnitude at all frequencies can be observed. One indication of the fact that in this case one has strong vibrations is found in the phase of the CPSDs (Fig. 6). Between levels 1 and 2 the phase is (piecewise) linear (Fig. 6a), but deviates from the normal (vibration-free) linear dependence. Between levels 2 and 3, the phase is practically zero around 6 Hz, that is around the vibration frequency (Fig. 6b), and between levels 3 and 4 (Fig. 6c) it is zero everywhere above 3 Hz (that is even over the first harmonics around 13 Hz). The coherence functions show the same trend of distortion, that is a deviation from the vibration-free sink structure. The conclusion is that strong vibrations influence the signals of LPRMs 031-034, and that the vibration amplitude is the largest at level 4. However, the fact that the phase is zero instead of linearly increasing, is a good indication of vibrations, but cannot be used to find out if impacting occurs or not. For this one needs to return to the APSDs. There are two main reasons which suggest that impacting may occur, both taken from LPRM 033. One is the peak around 12 Hz. Here, one possibility is that this peak is due to flux curvature effects, which become more pronounced with larger vibration amplitudes. If this is the case, the presence of this peak makes it also likely that impacting occurs, but gives no sure indication. Another possibility for the occurrence of this
higher frequency peak can be that it is the eigenfrequency of the fuel box, whose vibration is induced by impacting. The second reason to suspect impacting is the peak widening of the APSDs on the lower axial levels. The peaks in LPRMs 033 and 034 (if any in the latter) are much broader than that of LPRM 024, or other peaks found in the measurement (not shown in this paper). Taking all the above facts into account, it was judged that impacting occurred in string 03 with a very high probability. In view of the fact that from the measured spectra, it is not possible to decide if the higher frequency peak corresponds to flux curvature effects or fuel box frequency, further that the fuel box frequency was not known at the time of the measurements, the only direct indication for impacting came from the observed peak widening. APD functions were not calculated at the time of the original evaluation. As it was mentioned, the occurrence of impacting was confirmed after shut-down by detector tube and fuel box wear. Although APD functions were not calculated at the time of the measurements, it is interesting to see how these look for the vibrations in string No. 03 especially when now we know that impacting did occur. These functions were thus calculated in retrospective from the measured data. The results are shown in Fig. 7. Since the data series available for this analysis were shorter than the original ones, the APD functions have an uneven quality. At this point it is also necessary to mention what type of distortion of the Gaussian APD can be expected in case of impacting. From the theory of one-dimensional bounded processes with reflecting boundaries, see e.g. (Papoulis, 1965), the shape of the APD is given as a cut bellshaped distribution with sharp boundaries (Fig. 8). This form is due to the fact that the tails of the original (non-bounded) distribution are bent back to the accessible domain of the independent variable. However, from the results shown in Fig. 7, it is quite obvious that there is no sign of distortion which would indicate impacting. This is in line with the findings of the Oskarshamn measurements below. The reasons will be discussed later.
3. THE MEASUREMENTS IN OSKARSHAMN-2
In summer 1987, a series of measurements were made in Oskarshamn-2, with similar goals as in the Barsebfick (B-I) case. These measurements were made by the plant personnel and recorded to analog tape. The data processing was done in Studsvik. In general, the signal quality was not as good as in the B-1
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Fig. 8. APD function of a one-dimensional Gaussian bounded process. measurements, and this is reflected on the calculated spectra too. The analysis and the interpretation of the measurement was also performed by using the same principles as at B-1 (P~tzsit et al., 1987). In these measurements the APD functions were calculated as well. A general difference from the B-1 case was that these measurements were taken at lower power and higher recirculation flow, which led to a smaller portion of local boiling noise in the detector signals. This had the consequence that the linear phase between adjacent detectors was distorted already by very slight vibrations. In particular, a linear phase behaviour up to 15 Hz, as observed in the B-1 measurements (see Fig. 2a--c) was not found in any of the 0-2 detector tubes. Due to the above reason and due to the relatively poor signal quality, the confidence level of the statements on vibration amplitude and impacting was lower than in the B-1 case. Despite the fact that the APDs were also calculated in this case, no direct indications for impacting were found. From these measurements we show results from one detector string only, from string No. 4. This was one out of three strings, where high vibration amplitudes and a high probability of impacting was found. The APSDs, APDs, phase and coherence are shown in Figs 9 and 10. In the APSDs, peaks are seen at 3 Hz, and some peaks may be incidentally suspected even at 6 Hz in Figs 9b and d. It is difficult to say whether these peaks are already broadened (indicating impacting) or just normal. The phase is deviating from linear over 1 Hz and is close to zero. The sink structure of the coherence is also deviating from the regular pattern. All these facts indicate a high probability of impacting. The APD functions are nevertheless normal (Gaussian) for all 4 detectors (Figs 9e-h).
Although in the case of the Oskarshamn measurements, there is no direct proof of the fact that impacting occurred, altogether it can be stated that according to experimental evidence in both measurements, the APD functions do not show the expected distortion in case of impacting. This latter fact is somewhat puzzling and it is interesting to find an explanation. A first idea was that the explanation lies in the fact that the type of expected distortion of the APD function was taken from a one-dimensional model, describing a bounded Gaussian process with reflecting boundary conditions, as explained in Section 3. The case of detector signals from a vibration differs from this model in two ways. First, the vibration itself is only bounded in r(t) with the condition r2(t) = x2(t) + y2(t) <~R 2
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and not in x or y separately. Second, the detector signal &i(t) is a linear combination of linear and quadratic functions of x(t) and y(t) in the form [see equation (8)]
~i(t)~Ax(t)+ By(t)+Cx2(t)+ Dy2(t)
(4)
It is not easy to see the effect of condition (3) on the APD of 6i(t), or to relate the APD of the latter to the one in Fig. 8. At any rate, we have not found an analytical expression for the APD of equation (4). Since neither this APD function, nor the widening of the vibration peak in the APSDs with impacting is easy to study theoretically, numerical simulations were made to quantify them. These simulations will be reported in the next section. 4. THEORY
AND
NUMERICAL
SIMULATIONS
The general question is in what way the impacting influences spectral characteristics, in particular the shape of the peak in the APSD as well as the APD. These questions are not easily answered through analytical investigations. Regarding the first, there is no direct way to study APSD with impacting from analytical models. The distortion of the APDs can be easily treated in case of a one-dimensional bounded process with reflective boundaries; however, in the case of two-dimensional motions and the constraint equation (3), there are no results available. In addition, when considering the motion of the detector tube in a curved flux, terms proportional to x 2 and yZ will also appear in the neutron detector signals, which means a further significant complication for analytical description. The effect of impacting was thus investigated through a numerical model where detector tube
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f~(t) andf.(t). These forces were described as a regular sequence of pulses with random, uncorrelated coefficients. This model was first suggested to the authors by Antonopoulos-Domis (1982), and was reported in detail by Pfizsit et al. (1984). The essence of the tube motion model is as follows. The two-dimensional motion of the rod around its equilibrium position, which is chosen as the origin of the coordinate system, is described by the equations 3~(t) + 2~5c(t) + co02x(t) = £ (t) (5) y (t) + 2~j~(t) + co~v(t) =f. (t)
f~(t) and f.(t) are then simulated as the discrete processes
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Here a, and b, are independent Gaussian sequences with mean zero and variance unity. Thus, the force components are assumed uncorrelated, and no anisotropy of the motion is assumed. This latter constraint could easily be removed, but this was found unnecessary, since only single detector signals will be considered in this paper. The strength of the force is described by the scalar constant F. Since the motion constraint is represented by a parameter R [see equation (3)] which was kept constant throughout the calculations, in this model the probability or frequency of impacting will be a unique, monotonously increasing function of F. In other words, impacting can be induced by choosing F large enough. Using equations (6) in (5), for a given sequence a, and b, the latter can be integrated, and thus x(t) and y(t) are obtained. Although the force train in this model is specified at discrete times nat, the motion itself is calculated with a much finer time mesh. This is the same procedure that has already been used by Pfizsit et al. (1984). The present work contains extensions in two respects. First, as mentioned above, the
BWR instrument tube vibrations motion is constrained here, and for F large enough, frequent impacting will occur. The impacting is described such that when during the simulation of the motion, x(t) and y(t) fulfil equation (3) with an equality sign, that is the tube hits the border, the radial velocity component will be reversed with a user specified elasticity factor which lies between zero and unity. Second, when calculating the detector signal from the displacement components, allowance is made for the flux curvature, so that the relationship reads as
6~(to)~Ax(to) + By(to) + Cx2(to) + Dy'(to) (7) Here, the coefficients A, B, C and D are related to the first and second derivatives of the static flux and are thus constants (Thie, 1979; Pfizsit et al.,-1988), and x2(to) and y2(to) are symbolic notations for the Fourier transforms of x2 (t) and y 2(t). The parameters A-D are treated here only formally and not given dimensions or a physically realistic absolute value. It is only the different mutual ratios of these parameters that have an importance for the present calculations. Since the coefficients A-D do not depend on the frequency, only on the rod and (equilibrium or average) detector position, equation (7) can be equivalently written as
6 ~ ( t ) ~ A x ( t ) + By(t) + Cx2(t) + Dy:(t)
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It is also assumed here that the prompt part of the detector signal t~i(t) is proportional to the flux fluctuation 6~(t) in the frequency range of interest. In a simulation, x(t) and y(t) are determined, from which 6~(t) is calculated via (8). Then, spectral and A P D analysis of the data is performed just as with measured detector signals. A series of calculations were made where the two new features were systematically varied over a range of values. The first one was the impacting parameter, or force coefficient F. Starting with a low value of F, with no impacting, it was then increased such that moderate, strong and excessive impacting was induced. The strength of impacting was determined from the frequency of impacting compared to the vibration frequency. This parameter can be determined qualitatively, visually from the trajectory pattern (Fig. 1 la), and can be also quantitatively determined from the simulation itself. Then, the induced neutron noise and its spectral descriptors were calculated through equation (7) for all four vibration amplitudes. In this part of the study, the parameters A-D were varied such as to describe different contributions from the flux curvature effects relative to the effect of the linear flux gradient. We started with
773
C = D = 0, corresponding to no flux curvature, then the values of C and D were increased in a few steps such that they represented a strong flux curvature. A high flux curvature was said to have been simulated when the first higher harmonics (double frequency peak) in the neutron noise APSD became significant. Cases with C and D having opposite sign were considered too. This latter case corresponds to a saddle point in flux curvature. A rather large number of calculations were made by different parameter combinations, but for practical reasons only a few interesting cases are shown. The four different types of mechanical motion, that is the displacement trajectories with different force coefficients F a r e shown in Fig. 1 la. It is seen that with the lowest value of the force coefficient F, the vibration is completely free of impacting, whereas with the highest value of F, quite frequent impacting occurs. The corresponding A P D functions and spectra are shown in Figs 1 lb and 1 lc. For simplicity, these functions are only shown for the x component, because the statistics of the x and y components are the same. In all calculations of the neutron noise that will be shown here, only these trajectories are used. The elasticity factor in these cases is equal to unity. Calculations were made with various values of the elasticity factor, but the impact of varying this parameter on trajectory and neutron noise characteristics was insignificant. The APSDs of x clearly show in Fig. l l b the widening of the vibration peak with increasing impacting. Inspection of the A P D functions of x(t) (Fig. l l c ) show, on the other hand, that as soon as impacting occurs ( F = 20 and upwards), the displacement APDs get distorted. That is, although the process is only bounded in r = x / r ~ T y 2, the APD of x (and y) is still very similar to that of the onedimensional process in Fig. 8. In other words, the 2-D character of the motion does not affect the distortion of the A P D of x and y. In Figs 12-14 spectra and APD functions of the neutron noise are seen. In one figure, a family of calculations is shown, with fixed A, B, C and D, belonging to the four different trajectory types with different values of impacting. In Fig. 12, C = D = 0, that is the neutron flux around the detector tube is assumed to be linear. This also means that there is no double frequency effect. The power spectra (Fig. 12a) show the same peak widening with increasing impacting as the vibration displacement spectra. The APD functions also follow the same trend as the displacement components (Fig. 12b). As soon as impacting starts, the APD functions of the detector signals get distorted just as with a 1-D linear process.
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785
A P D will also be Gaussian. Since in the simulations, reported in this paper, all noise is due to vibrations, this effect could not be modelled. This explains the difference between the measurements and the simulation. The disadvantage of the A P D is thus that it is affected by all frequency components of the signal. The vibration peak, on the other hand, is only due to the vibrations (except for a usually separable background), thus the distortion (widening) of the vibration peak can be much easier detected. The agreement between the experiments and simulations was thus also good. 5. CONCLUSIONS The measurements and the numerical simulations showed that the distortion of the phase of the CPSD and the widening of the vibration peak in the APSD of the neutron detector signals is a significant indication of detector tube vibrations and even impacting. The distortion of the A P D functions of the detector signals on the other hand does not appear to be a useful tool for detecting impacting. The statistics of the detector signals is not dominated by the vibrations but by other sources and the background, thus on the unprocessed A P D the distortions due to impacting cannot be observed. A narrow-band filtering of the signal, centred on the vibration peak, could presumably help this problem. Acknowledgement--The measurements and interpretation
were originally made by the Studsvik Noise Group, recently within EuroSim AB. The noise data for the recent reevaluation of the measurements were obtained from them and used with their kind permission. Thanks are also due to the respective power plants for their permission to publish measurement data. We thank Lasse Urholm and Lennart Norberg for their expert help in producing, editing and importing graphics with unlimited patience.
REFERENCES
.~kerhielm F., Pfizsit I., Bergdahl B-G., Oguma R., Sandell S. and Lorenzen J. (1986) Noise measurements on a few LPRM detectors (SPND and fission chambers) in Barseb~ick I. Studsvik Report NI-86/9. Antonopoulos-Domis M. (1982) Personal communication. Fry D. N. et al. (1977) Summary of ORNL Investigations of In-Core Instrument Tube Vibrations in BWR-4. ORNL/ NUREG/TM-101, Oak Ridge National Laboratory. G16ckler O. and Frei Z. (1986) Personal communication. Kosfily G. (1980) Prog. Nucl. Energy 5, 145. Mott J. E. (1976) Trans. Am. Nucl. Soc. 23, 465. Papoulis A. (1965) Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York. Pfizsit I., Antonopoulos-Domis M. and Gl6ckler O. (1984) Prog. Nucl. Energy 14, 165.
786
I. Pfizsit and O. Gl6ckler
P~tzsit I., Akerhielm F., Bergdahl B-G., Lorenzen J. and Oguma R. (1987) Noise analysis of certain LPRM strings in Oskarshamn II with regard to detector tube vibrations. Studsvik Report NI-87/34.
P~.zsit I., Akerhielm F., Bergdahl B-G. and Oguma R. (1988) Prog. Nucl. Energy 21, 107. Thie J. A. (1979) Nucl. Technology 45, 5. Thie J. A. (1985) Personal communication. Wach D. and Kos~ly G. (1974) Atomkernenergie 23, 244.
Pergamon 0306-4549(93)E0004-J
Copyright © 1994ElsevierScienceLtd Printed in Great Britain. All rights reserved 0306-4549/94$7.00+ 0.00
BWR INSTRUMENT TUBE VIBRATIONS: INTERPRETATION OF MEASUREMENTS A N D SIMULATION I. PAZSIT I A N D O. G L O C K L E R 2 ~Department of Reactor Physics, Chalmers University of Technology, S-412 96 G6teborg, Sweden 2Fuel and Physics Department, Ontario Hydro, 595 Bay St. A7-BI2, Toronto, Ontario, Canada M5G 2C2 (Received 11 August 1993; in revised form 23 October 1993)
Abstract--In 1986 and 1987, neutron noise measurements were made in two Swedish BWRs on a number of detector strings, in order to detect and quantify detector guide tube vibrations, and possibly impacting. Vibrations of various amplitude were detected in several strings, but quantification of the strength of vibrations, including impacting, was not trivial. For quantification of the vibrations, there appeared a need for spectral descriptors other than just the amplitude of the vibration peak in the detector auto power spectral densities (APSDs). To find such descriptors, and investigate their sensitivity, numerical simulation of the signal of a vibrating detector both with and without impacting was performed. Two possible descriptors of impacting, namely the widening of the vibration peak and the distortion of the amplitude probability distribution (APD) function were investigated. The first of these two was found to be a relatively reliable indicator of impacting, in accordance with the experimental findings. The use of the APD is much less effective, and reasons for this are discussed in the paper.
!. INTRODUCTION The fact that excessive instrument tube vibrations may occur in BWRs has been known from the mid1970s. Extensive review of the measurements and diagnostic analysis can be found in Thie (1979) and Kosfily (1980). The significance of this phenomenon arises since excessive vibrations can lead to impacting and channel box corner wear, as well as detector damage. It is therefore important to be able to monitor the vibrations in order to avoid impacting. However, there is no absolute quantitative method available for determining impacting or even judging the probability of impacting. The most effective information carrier is the signal of the L P R M detectors (local power range monitors) within the instrument tube concerned. It is easy to observe the presence of the vibrations through the peak in the detector signal APSDs in the frequency range 2-5 Hz. However, it is not easy to use this information to determine whether or not impacting occurs. One possibility that had been investigated in detail is to relate the magnitude of the vibration peak in the normalized A P S D (NAPSD) of the detector signals to absolute displacement. One problem is that the magnitude of the resulting neutron signal fluctuations, and thus that of 759
the peak in the APSD, is determined by both the local neutron flux gradient and the actual two-dimensional trajectory of the vibration. These are usually unknown, different for each tube and reactor, and can even change in time during the cycle independently of the amplitude of vibrations. Nonetheless, criteria for impacting, based on a threshold in detector signal NCPSDs have been elaborated and used (Mott, 1976; Fry et al., 1977). Such threshold criteria are rather individual to each core type, and are not easy to transfer from one core to another. The lack of generally applicable quantification methods was clearly felt when in 1986 and 1987 the Studsvik Noise G r o u p (currently within EuroSim AB), of which one of the present authors was a member, evaluated in-core noise measurements in two Swedish BWRs. The main purpose of the measurements was to quantify possible detector tube vibrations. It was observed in the measurements that several strings vibrated, but the threshold principle above could not be applied due to differing core and detector parameters. As reported in/~kerhielm et al. (1986) and P~izsit et al. (1987), during the evaluation of the measurements, new descriptors were also invoked. Of the traditional ones, the magnitude of the fundamental resonance as well as the first higher harmonics were used. In
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at various elevations. From independent investigations, it had been suspected that some of the detector tubes may vibrate, and the aim of the investigations • the phase of the cross spectrum between detectors was to quantify the vibrations. in the same string; The results of these investigations were reported in • widening of the vibration peak; more detail by ,~kerhielm et al. (1986), here we only • distortion of the APD function. give a brief summary. In some strings, no or little Out of these three, the first one is based on the fact vibration was found. Auto- and cross-spectra in those that the linear phase of the CPSD between two strings follow a pattern that was described by the detectors in the same string, observed earlier in BWR simple model of Wach and Kositly (1974). For such measurements and elaborated in Section 2, gets dis- strings the phase curve of the CPSD between two torted, eventually becomes zero, within the frequency detectors shows a linear dependence on frequency up range of vibrations if the latter are strong. Its use was to some 10-15 Hz, whereas the absolute value of the reported in ~kerhielm et al. (1986) and Pfizsit et al. CPSD, and even more the coherence, show a charac(1987, 1988). The second descriptor, the widening of teristic sink structure. At integer multiples of the the vibration peak, was noticed first in numerical frequency equal to the inverse of the bubble transit experiments by G16ckler and Frei (1986). It was used time z, that is at in interpreting both measurement series mentioned f = (n/r); n = 0, 1, 2, 3.... (1) above. The third one was suggested to the present authors by Thie 0985). It was not used in the the phase curve crosses zero, and the coherence and Barsebfick measurements, but attempts were made to the CPSD have a local maximum. At half-integer use this principle in the Oskarshamn measurements, multiples of 1/r, i.e. at without success. To find out the applicability of the f = (n + 1/2)/z; n = 0, 1, 2, 3 .... (2) APD distortion concept, further numerical simulations were made by Gl6ckler and Frei that will be the phase curve crosses ± 7r, and the coherence and reported in this paper. CPSD have a local minimum, that is a sink. The content of this paper is as follows. In Sections 2 Such a case is demonstrated by the noise spectra of and 3, the measurements made in Barsebfick 1 and the detectors in string No. 18, LPRMs 181-184, see Oskarshamn 2 will be described, respectively. In Figs 1 and 2. Here LPRM 181 stands for the detector Section 4 the numerical simulation model will be in string No. 18 at level l, which is the highest axial described, and results of the investigations presented. level. The same convention is used in numbering the Finally, the results of the measurements and the detectors throughout the paper. Corresponding to the simulation results will be compared. The conclusion is fact that this tube does not vibrate, the APSDs of the that the peak widening, although not an absolute and detector signals show no peaks, only a smooth struceasily quantifiable method, may be effectively used for ture (Fig. 1). The phase and coherence curves follow detection of impacting. On the other hand, the distor- the pattern described above very closely (Fig. 2). The tion of the APD function is not a useful indicator of phase is linear up to between 10 and 15 Hz, above impacting. The main reason for this is that usually, which frequency the reactor transfer breaks down and only a small portion of the power content of the signal the background and instrumentation noise starts to is contained in the vibration peak, thus the APD is dominate. Accordingly, the coherence vanishes above dominated by the noise from sources other than the this frequency, as seen from the figures. From the vibration itself. In addition, in the presence of a slope of the linear part of the phase curves it can be saddle-point type gradient in the neutron flux at the seen that the bubble transit time is smaller between detector position, the distortion of the APD due to the detectors on the highest axial level. impacting is reduced quite significantly. In strings No. 02 and 03, vibrations can be observed. We take first string No. 02, where moderate vibrations can be seen. This can be demonstrated by the APSD on level 4 (Fig. 3b). A large peak is seen at 2. THE MEASUREMENTS IN BARSEB,~CK-I 6 Hz. However, there are no peaks in the APSDs of The first series of measurements were made in the LPRM detectors on the higher axial level (see e.g. Barseb~ick-l, a 440 MW Swedish BWR, late spring Fig. 3a), indicating that the largest amplitude of 1986 by Studsvik. Neutron noise signals were taken vibrations may be found on level 4. from 6 detector strings, each containing 4 LPRMs. In A further support for the assumption of vibrations addition, measurements were made by a TIP detector in string No. 02 can be found in the phase behaviour addition, three new parameters or descriptors were used. These were:
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I. P~sit and O. Gl6ckler
and sink structure of CPSD and coherence (Fig. 4). On levels 1-2 and 2-3 (not shown here for brevity), the phase and sink structure show a vibration-free behaviour, described in and around equations (1) and (2). However, between levels 3 and 4 a distortion of both the phase curve and the coherence can be seen. The linear phase behaviour disappears above 6 Hz, which is the frequency of the vibrations in this case, and the sink structure is also distorted. By the latter it is meant that the minima and maxima of the coherence function do not coincide with the "n-crossings" and zero-crossings of the phase curve, respectively. The most interesting string in this analysis is string No. 03, where the strongest vibrations were detected. However, looking at the APSDs on levels 1-4 (Fig. 5), this may not appear so obvious. On level'l, no explicit vibration peak can be seen (Fig. 5a). On level 2, a small thin peak can be seen (Fig. 5b), which looks much less significant than the peak in Fig. 3b, belonging to LPRM 024. On level 3 the peak at 6 Hz is much broader, and another broad peak can be suspected at around 13 Hz (Fig. 5c). On level 4 these two peaks can only be surmised, they are broadened so much that only an increased value of APSD can be seen when compared to LPRM 031 (Fig. 5d). Compared to the higher axial levels, an overall increase of the APSD magnitude at all frequencies can be observed. One indication of the fact that in this case one has strong vibrations is found in the phase of the CPSDs (Fig. 6). Between levels 1 and 2 the phase is (piecewise) linear (Fig. 6a), but deviates from the normal (vibration-free) linear dependence. Between levels 2 and 3, the phase is practically zero around 6 Hz, that is around the vibration frequency (Fig. 6b), and between levels 3 and 4 (Fig. 6c) it is zero everywhere above 3 Hz (that is even over the first harmonics around 13 Hz). The coherence functions show the same trend of distortion, that is a deviation from the vibration-free sink structure. The conclusion is that strong vibrations influence the signals of LPRMs 031-034, and that the vibration amplitude is the largest at level 4. However, the fact that the phase is zero instead of linearly increasing, is a good indication of vibrations, but cannot be used to find out if impacting occurs or not. For this one needs to return to the APSDs. There are two main reasons which suggest that impacting may occur, both taken from LPRM 033. One is the peak around 12 Hz. Here, one possibility is that this peak is due to flux curvature effects, which become more pronounced with larger vibration amplitudes. If this is the case, the presence of this peak makes it also likely that impacting occurs, but gives no sure indication. Another possibility for the occurrence of this
higher frequency peak can be that it is the eigenfrequency of the fuel box, whose vibration is induced by impacting. The second reason to suspect impacting is the peak widening of the APSDs on the lower axial levels. The peaks in LPRMs 033 and 034 (if any in the latter) are much broader than that of LPRM 024, or other peaks found in the measurement (not shown in this paper). Taking all the above facts into account, it was judged that impacting occurred in string 03 with a very high probability. In view of the fact that from the measured spectra, it is not possible to decide if the higher frequency peak corresponds to flux curvature effects or fuel box frequency, further that the fuel box frequency was not known at the time of the measurements, the only direct indication for impacting came from the observed peak widening. APD functions were not calculated at the time of the original evaluation. As it was mentioned, the occurrence of impacting was confirmed after shut-down by detector tube and fuel box wear. Although APD functions were not calculated at the time of the measurements, it is interesting to see how these look for the vibrations in string No. 03 especially when now we know that impacting did occur. These functions were thus calculated in retrospective from the measured data. The results are shown in Fig. 7. Since the data series available for this analysis were shorter than the original ones, the APD functions have an uneven quality. At this point it is also necessary to mention what type of distortion of the Gaussian APD can be expected in case of impacting. From the theory of one-dimensional bounded processes with reflecting boundaries, see e.g. (Papoulis, 1965), the shape of the APD is given as a cut bellshaped distribution with sharp boundaries (Fig. 8). This form is due to the fact that the tails of the original (non-bounded) distribution are bent back to the accessible domain of the independent variable. However, from the results shown in Fig. 7, it is quite obvious that there is no sign of distortion which would indicate impacting. This is in line with the findings of the Oskarshamn measurements below. The reasons will be discussed later.
3. THE MEASUREMENTS IN OSKARSHAMN-2
In summer 1987, a series of measurements were made in Oskarshamn-2, with similar goals as in the Barsebfick (B-I) case. These measurements were made by the plant personnel and recorded to analog tape. The data processing was done in Studsvik. In general, the signal quality was not as good as in the B-1
769
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Although in the case of the Oskarshamn measurements, there is no direct proof of the fact that impacting occurred, altogether it can be stated that according to experimental evidence in both measurements, the APD functions do not show the expected distortion in case of impacting. This latter fact is somewhat puzzling and it is interesting to find an explanation. A first idea was that the explanation lies in the fact that the type of expected distortion of the APD function was taken from a one-dimensional model, describing a bounded Gaussian process with reflecting boundary conditions, as explained in Section 3. The case of detector signals from a vibration differs from this model in two ways. First, the vibration itself is only bounded in r(t) with the condition r2(t) = x2(t) + y2(t) <~R 2
(3)
and not in x or y separately. Second, the detector signal &i(t) is a linear combination of linear and quadratic functions of x(t) and y(t) in the form [see equation (8)]
~i(t)~Ax(t)+ By(t)+Cx2(t)+ Dy2(t)
(4)
It is not easy to see the effect of condition (3) on the APD of 6i(t), or to relate the APD of the latter to the one in Fig. 8. At any rate, we have not found an analytical expression for the APD of equation (4). Since neither this APD function, nor the widening of the vibration peak in the APSDs with impacting is easy to study theoretically, numerical simulations were made to quantify them. These simulations will be reported in the next section. 4. THEORY
AND
NUMERICAL
SIMULATIONS
The general question is in what way the impacting influences spectral characteristics, in particular the shape of the peak in the APSD as well as the APD. These questions are not easily answered through analytical investigations. Regarding the first, there is no direct way to study APSD with impacting from analytical models. The distortion of the APDs can be easily treated in case of a one-dimensional bounded process with reflective boundaries; however, in the case of two-dimensional motions and the constraint equation (3), there are no results available. In addition, when considering the motion of the detector tube in a curved flux, terms proportional to x 2 and yZ will also appear in the neutron detector signals, which means a further significant complication for analytical description. The effect of impacting was thus investigated through a numerical model where detector tube
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f~(t) andf.(t). These forces were described as a regular sequence of pulses with random, uncorrelated coefficients. This model was first suggested to the authors by Antonopoulos-Domis (1982), and was reported in detail by Pfizsit et al. (1984). The essence of the tube motion model is as follows. The two-dimensional motion of the rod around its equilibrium position, which is chosen as the origin of the coordinate system, is described by the equations 3~(t) + 2~5c(t) + co02x(t) = £ (t) (5) y (t) + 2~j~(t) + co~v(t) =f. (t)
f~(t) and f.(t) are then simulated as the discrete processes
fx(t) = F
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Here a, and b, are independent Gaussian sequences with mean zero and variance unity. Thus, the force components are assumed uncorrelated, and no anisotropy of the motion is assumed. This latter constraint could easily be removed, but this was found unnecessary, since only single detector signals will be considered in this paper. The strength of the force is described by the scalar constant F. Since the motion constraint is represented by a parameter R [see equation (3)] which was kept constant throughout the calculations, in this model the probability or frequency of impacting will be a unique, monotonously increasing function of F. In other words, impacting can be induced by choosing F large enough. Using equations (6) in (5), for a given sequence a, and b, the latter can be integrated, and thus x(t) and y(t) are obtained. Although the force train in this model is specified at discrete times nat, the motion itself is calculated with a much finer time mesh. This is the same procedure that has already been used by Pfizsit et al. (1984). The present work contains extensions in two respects. First, as mentioned above, the
BWR instrument tube vibrations motion is constrained here, and for F large enough, frequent impacting will occur. The impacting is described such that when during the simulation of the motion, x(t) and y(t) fulfil equation (3) with an equality sign, that is the tube hits the border, the radial velocity component will be reversed with a user specified elasticity factor which lies between zero and unity. Second, when calculating the detector signal from the displacement components, allowance is made for the flux curvature, so that the relationship reads as
6~(to)~Ax(to) + By(to) + Cx2(to) + Dy'(to) (7) Here, the coefficients A, B, C and D are related to the first and second derivatives of the static flux and are thus constants (Thie, 1979; Pfizsit et al.,-1988), and x2(to) and y2(to) are symbolic notations for the Fourier transforms of x2 (t) and y 2(t). The parameters A-D are treated here only formally and not given dimensions or a physically realistic absolute value. It is only the different mutual ratios of these parameters that have an importance for the present calculations. Since the coefficients A-D do not depend on the frequency, only on the rod and (equilibrium or average) detector position, equation (7) can be equivalently written as
6 ~ ( t ) ~ A x ( t ) + By(t) + Cx2(t) + Dy:(t)
(8)
It is also assumed here that the prompt part of the detector signal t~i(t) is proportional to the flux fluctuation 6~(t) in the frequency range of interest. In a simulation, x(t) and y(t) are determined, from which 6~(t) is calculated via (8). Then, spectral and A P D analysis of the data is performed just as with measured detector signals. A series of calculations were made where the two new features were systematically varied over a range of values. The first one was the impacting parameter, or force coefficient F. Starting with a low value of F, with no impacting, it was then increased such that moderate, strong and excessive impacting was induced. The strength of impacting was determined from the frequency of impacting compared to the vibration frequency. This parameter can be determined qualitatively, visually from the trajectory pattern (Fig. 1 la), and can be also quantitatively determined from the simulation itself. Then, the induced neutron noise and its spectral descriptors were calculated through equation (7) for all four vibration amplitudes. In this part of the study, the parameters A-D were varied such as to describe different contributions from the flux curvature effects relative to the effect of the linear flux gradient. We started with
773
C = D = 0, corresponding to no flux curvature, then the values of C and D were increased in a few steps such that they represented a strong flux curvature. A high flux curvature was said to have been simulated when the first higher harmonics (double frequency peak) in the neutron noise APSD became significant. Cases with C and D having opposite sign were considered too. This latter case corresponds to a saddle point in flux curvature. A rather large number of calculations were made by different parameter combinations, but for practical reasons only a few interesting cases are shown. The four different types of mechanical motion, that is the displacement trajectories with different force coefficients F a r e shown in Fig. 1 la. It is seen that with the lowest value of the force coefficient F, the vibration is completely free of impacting, whereas with the highest value of F, quite frequent impacting occurs. The corresponding A P D functions and spectra are shown in Figs 1 lb and 1 lc. For simplicity, these functions are only shown for the x component, because the statistics of the x and y components are the same. In all calculations of the neutron noise that will be shown here, only these trajectories are used. The elasticity factor in these cases is equal to unity. Calculations were made with various values of the elasticity factor, but the impact of varying this parameter on trajectory and neutron noise characteristics was insignificant. The APSDs of x clearly show in Fig. l l b the widening of the vibration peak with increasing impacting. Inspection of the A P D functions of x(t) (Fig. l l c ) show, on the other hand, that as soon as impacting occurs ( F = 20 and upwards), the displacement APDs get distorted. That is, although the process is only bounded in r = x / r ~ T y 2, the APD of x (and y) is still very similar to that of the onedimensional process in Fig. 8. In other words, the 2-D character of the motion does not affect the distortion of the A P D of x and y. In Figs 12-14 spectra and APD functions of the neutron noise are seen. In one figure, a family of calculations is shown, with fixed A, B, C and D, belonging to the four different trajectory types with different values of impacting. In Fig. 12, C = D = 0, that is the neutron flux around the detector tube is assumed to be linear. This also means that there is no double frequency effect. The power spectra (Fig. 12a) show the same peak widening with increasing impacting as the vibration displacement spectra. The APD functions also follow the same trend as the displacement components (Fig. 12b). As soon as impacting starts, the APD functions of the detector signals get distorted just as with a 1-D linear process.
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T
T
BWR instrument tube vibrations Basically the same can be said on the case with moderate flux curvature (Fig. 13), with one exception. Figure 13a shows that in the power spectra there is a peak at twice the fundamental resonance frequency, due to the flux curvature. With increasing impacting, this peak also broadens. At F = 60, the broadening of the fundamental and first harmonic peak is so large that they largely overlap; actually, the double frequency peak is hard to see at all. The same effect, even more pronounced, can be seen in Fig. 14a, which corresponds to a case with even larger flux curvature. This is also in a very good agreement with the measurements in Barseb~ick (cf. Figs 5c and d). Regarding the APDs, it is seen that despite the functional difference between 8 ~ (t) and a 1-D linear bounded process, the A P D function of the former gets distorted the same way as the latter when becoming bounded. This means that the conjecture of why the A P D functions in the measurements did not show distortion was incorrect. The functional differences between ~ ( t ) and a linear bounded process do not lead to a difference in the behaviour of the APDs. However, it is interesting to note that there exist certain circumstances when the A P D of the detector signal may not be distorted despite heavy impacting. This is the case shown in Fig. 14b. Here, C = - 6 0 and D = 60 correspond to a strong flux curvature with a saddle point. The different sign in C and D leads to the fact that the A P D function looks relatively Gaussian even with heavy impacting. That is, under certain circumstances, the APDs may look Gaussian even with impacting. However, this is a relatively special case, and although we do not have a flux map from the measurements discussed in this paper, we do not believe that this can be the main reason for the fact that A P D distortion was never seen. We have though some indications that strong flux curvatures may occur close to detector tubes, because in the Barseb/ick measurements, in a few cases opposite phase was found between an SPND and a TIP detector in the same string and at the same axial elevation, indicating that they were vibrating in opposite flux gradients. The most likely reason for the fact that A P D distortion is not found in the measured detector signals even with strong impacting can be that the vibration peak represents only a small portion of the spectral power content of the signal. As seen in e.g. Figs 3b, 5c, 9a~l, the spectra are largest at low frequencies, and the vibration peak is at least one decade lower than the PSD value around zero frequency. Thus, the latter frequency components will dominate the signal, and since their characteristics can be assumed to be undistorted Gaussian, the signal
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A P D will also be Gaussian. Since in the simulations, reported in this paper, all noise is due to vibrations, this effect could not be modelled. This explains the difference between the measurements and the simulation. The disadvantage of the A P D is thus that it is affected by all frequency components of the signal. The vibration peak, on the other hand, is only due to the vibrations (except for a usually separable background), thus the distortion (widening) of the vibration peak can be much easier detected. The agreement between the experiments and simulations was thus also good. 5. CONCLUSIONS The measurements and the numerical simulations showed that the distortion of the phase of the CPSD and the widening of the vibration peak in the APSD of the neutron detector signals is a significant indication of detector tube vibrations and even impacting. The distortion of the A P D functions of the detector signals on the other hand does not appear to be a useful tool for detecting impacting. The statistics of the detector signals is not dominated by the vibrations but by other sources and the background, thus on the unprocessed A P D the distortions due to impacting cannot be observed. A narrow-band filtering of the signal, centred on the vibration peak, could presumably help this problem. Acknowledgement--The measurements and interpretation
were originally made by the Studsvik Noise Group, recently within EuroSim AB. The noise data for the recent reevaluation of the measurements were obtained from them and used with their kind permission. Thanks are also due to the respective power plants for their permission to publish measurement data. We thank Lasse Urholm and Lennart Norberg for their expert help in producing, editing and importing graphics with unlimited patience.
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