Simulation of low-frequency PWR neutron flux fluctuations

Progress in Nuclear Energy 117 (2019) 103039

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Simulation of low-frequency PWR neutron flux fluctuations a,∗

a

a

b

a

a

M. Viebach , C. Lange , N. Bernt , M. Seidl , D. Hennig , A. Hurtado a b

T

Chair of Hydrogen and Nuclear Energy, Institute of Power Engineering, Technische Universität Dresden, George-Bähr-Straße 3b, 01069, Dresden, Germany PreussenElektra GmbH, Tresckowstraße 5, 30457, Hannover, Germany

ARTICLE INFO

ABSTRACT

Keywords: Neutron noise KWU PWR

Several KWU type PWRs have experienced an unexplained cycle-by-cycle change of neutron flux fluctuation amplitudes. The phenomenon is a matter of ongoing research. It has drawn attention to long-known but also not entirely understood long-range correlations in the time-dependent neutron flux signals. A sufficient understanding of these correlations and the corresponding phase relations is needed to explain the cycle-by-cycle change of the neutron flux fluctuation amplitudes. As previous research has shown, coherent deflection of a large number of fuel assemblies could contribute to the observed fluctuation patterns. Therefore, the contribution at hand investigates the effect of a quasi-coherent deflection of all fuel assemblies on the neutron flux fluctuations. The considered model assumes that this behavior has an impact in the reflector only. This impact is quantified via CASMO5 calculations and subsequent DYN3D simulations. In addition to the reflector perturbation, fluctuations of the inlet temperature distribution are superimposed. The simulation results are discussed along with selected data of measured neutron flux fluctuations. On the one hand, it is verified that coherent fuel assembly deflections can contribute to the measured neutron flux patterns as the simulations reproduce the main characteristics of the relevant neutron flux fluctuation phenomena. On the other hand, it turns out that the perturbations in the reflector may not be the only path of action relevant for the observed correlations. Approaches for future research are given.

1. Introduction Neutron flux fluctuations are a natural phenomenon present in nuclear reactors. Commonly referred to as neutron noise or reactor noise, they have been studied since the early days of nuclear reactors (cf. Thie, 1979). Neutron flux power spectrum monitoring is one part of reactor vessel surveillance activities as deviations in the neutron flux fluctuation patterns from the corresponding references may indicate fatigue of certain mechanical components of the reactor. An observation recently attracting attention is the cycle-by-cycle change of the neutron-flux fluctuation amplitudes experienced since 2001 by several German KWU (Kraftwerk Union AG) PWRs ((German) Reactor Safety Commission (RSK, 2013)). The changed amplitudes are empirically related to the change of the mechanical type of fuel assemblies (FA) (Seidl et al., 2015; (German) Reactor Safety Commission (RSK, 2015)), but the actual mechanisms leading to those details of the neutron noise effects relevant for higher or lower amplitudes are still not found. In order to explain the changed amplitudes, one has to have a sufficient general understanding of the neutron noise phenomena ∗

occurring in the considered reactors. In case of the concerned KWU PWRs, this especially involves the fluctuation patterns in the range of , which are recently referred to by many authors (cf. Seidl et al., 2015; Rouchon et al., 2017; Girardin et al., 2017; Viebach et al., 2018a; Chionis et al., 2018b) and which lack a generally accepted explanation. These patterns are the following: Signals from axially aligned detectors are in-phase, signals from detectors within the same radial core half are in-phase, and signals from detectors of different core halves are out-ofphase. One may add that the main contribution to the noise signals lies in the range of developing a maximum at about 0.8 Hz. At this frequency, all signals are highly coherent. Moreover, signals from axially aligned detectors have high coherence in the entire range of . The paper at hand aims to better understand these properties, while the observed effect of changed amplitudes is not directly tackled. It was pointed out by Viebach et al. (2018a) that coherent fuel assembly (FA) deflection may be an essential part of the mechanisms causing the observations given above. Picking up this result, the current paper investigates possible effects of a quasi-coherent time-dependent deflection of all FAs on the neutron flux fluctuations. The applied modelling strategy rests upon the model hypothesis

Corresponding author. E-mail address: [email protected] (M. Viebach). URL: http://tu-dresden.de/ing/maschinenwesen/iet/wket (M. Viebach).

https://doi.org/10.1016/j.pnucene.2019.103039 Received 6 November 2018; Received in revised form 18 March 2019; Accepted 22 April 2019 0149-1970/ © 2019 Elsevier Ltd. All rights reserved.

Progress in Nuclear Energy 117 (2019) 103039

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of the “moderator” path and that of the “displacement” path perturb the time and space dependent neutron kinetics, which can be monitored by neutron flux detectors. The deformation of certain mechanical components can, in principle, also cause changes in the distance between source and detector. But this “geometry” effect is out of scope of this paper. It is considered negligible here because the detectors are comoving within the FAs and the FAs, being the main source of neutrons, do virtually not change their mutual distances as they perform quasicoherent deflections. The postulated coherent deflection of all fuel assemblies is a realization of the structure kinematics module in the “displacement” path. For this mode of deflection, only the outer fuel assemblies see a change of their surroundings (the entire core deflects as a whole). Using a comoving frame of reference, this picture can be reinterpreted as a timedependent change of the radial reflector, i. e. the reflector changes its local thickness with time. Consequently, XS data variation caused by fuel assembly displacements is exclusively located in the reflector. This dependence was quantified by CASMO5 (Studsvik Scandpower, Inc., a) calculations. The FA deflections themselves were simulated by a simple mechanical model of the FAs and the CB/RPV system in MATLAB (The MathWorks Inc., 2016). For simplicity, the driving forces are considered as white Gaussian noise being independent of any thermal-hydraulics perturbations. It should be emphasized that the mechanical model essentially aims on providing quasi-coherent FA deflections rather than modelling the exact conditions as precisely as possible. Thermal-hydraulics processes, the “moderator” path and neutron kinetics are simulated via the reactor dynamics code DYN3D (Rohde et al., 2016; Kliem et al., 2016). The source code was modified by changing the reflector material constants (XS data) to be dependent on the local reflector thickness. This change couples the “displacement” path, which is simulated prior to the DYN3D run, to the neutron kinetics. Preliminary results of corresponding analyses were presented by Viebach et al. (2018b). It has to be emphasized that a dedicated validation of DYN3D for neutron fluctuations simulations is still ongoing work (cf. the work of Demazière et al. (2017) for an example how to verify the results of neutron flux fluctuation simulations performed by a certain code). Moreover, the applicability of time-domain codes in general for noise simulations is a recent question in reactor dynamics (cf. the work of Olmo-Juan et al. (2019)). For the time being, the authors rely on the validation cases summarized by Rohde et al. (2016). Especially low-frequency fluctuations may be attributed to the range of applications covered by these cases. The independently performed DYN3D neutron noise investigations of Rohde et al. (2018) may also be consulted to create additional confidence in the applicability of the code. In the current paper, the full XS data set of the radial reflector is considered for the time-dependent perturbations. Additionally, temperature fluctuations at the core inlet are taken into account to study their influence to the neutron flux fluctuations. As the added temperature fluctuations are correlated essentially only within each core quadrant and travel vertically up the core, the corresponding noise patterns compete with those effected by quasi-coherent FA deflections. The simulated data is compared to corresponding measured data. It should be mentioned that Chionis et al. (2018a) and more recently Torres et al. (2019) also study “synchronous” fuel assembly vibrations, i. e. for a central 5 x 5 fuel assembly cluster, using CASMO5/SIMULATE/S3K (Studsvik Scandpower, Inc. (a, c, b)) and employing a 3-LoopPWR model. The paper is structured as follows: The main part starts with a description of the model (overview, relevant DYN3D details, mechanical model, time-dependent variations in the reflector, fluctuations of the inlet temperature). After briefly summarizing the technical details of simulation preparation and data post-processing, results are presented for a DYN3D run with only reflector perturbations and for a DYN3D run with both reflector perturbations and inlet temperature perturbations. The simulated data is accompanied by measured data as a reference. A

Fig. 1. Illustration of the origin of the detector signals. Thermal-hydraulics (TH) fluctuations of the coolant flow are generated, transported and modulated by thermal-hydraulics processes and directly give variations in the interaction cross section (XS) values via the “moderator” path (temperature and density effects). The thermal-hydraulics fluctuations also drive time-dependent deflections of mechanical components by virtue of fluid-structure interaction. The deflections are transmitted throughout the mechanical system by structure kinematics processes causing local displacements of most of the RPV internals. These displacements go along with another variation of the XS values giving rise to the “displacement” path. The contributions of both paths cause neutron flux fluctuations and thus fluctuations of the detector signals. The geometry effect of fluctuating distances between radiation sources and the detectors is not considered in this paper.

proposed by Viebach et al. (2018a). The hypothesis was slightly enhanced by generalizing the source of the neutron flux fluctuations to be thermal-hydraulics (TH) fluctuations of the coolant flow in general (rather than TH fluctuations at the core inlet). For the sake of clarity, the entire model hypothesis is summarized in the following (cf. Fig. 1): On its way from the reactor pressure vessel (RPV) inlet nozzles to the outlet nozzles, the coolant flow develops, modulates and transports TH fluctuations (inhomogeneities, vortices, etc.), which cause time-dependent modulations of the local material parameters relevant for the fission chain reaction. On the one hand, these fluctuations perturb processes like cooling of the fuel rods and moderation of neutrons. The corresponding impact on the macroscopic cross sections (XS) is called “moderator” path. On the other hand, the coolant flow fluctuations trigger time-dependent deformations and vibration of RPV internals such as the core barrel (CB) or the FAs. The impact of their local displacements on the local macroscopic XS is called “displacement” path. Modelling the latter involves the definition of forces acting on the mechanical components by the coolant, the definition of a dynamical model for the considered mechanical components, and the definition of the XS dependency on the displacements. Finally, both the contribution 2

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Fig. 2. Performed calculations. “External” forces drive deflections of the considered core internals (CB and FAs). The corresponding FA deflections are translated to reflector thickness values which are finally provided to the neutron kinetics simulation. The impact of the reflector thickness variations is quantified with CASMO5 and provided in form of coefficients.

discussion of the simulation results and a comparison of them with the measured data closes the main part of the paper. The paper is finished by drawing conclusions.

thermal-hydraulics are coupled in an implicit scheme. Each thermalhydraulics time-step tTH is subdivided into neutron-kinetics time-steps tTH ). The equations are integrated from t = n tTH tNK (with tNK to t = (n + 1) tTH iteratively. Within each outer thermal-hydraulicsneutron-kinetics iteration step, thermal hydraulics and neutron kinetics are separately solved (operator splitting): The neutron-kinetics timesteps are (iteratively) integrated, the power is calculated, the thermalhydraulics time step is (iteratively) integrated, and the XS data is updated. This outer iteration is repeated until convergence (the operator splitting is relaxed). Afterwards, the next thermal-hydraulics time-step is processed. A more detailed description of the coupling in DYN3D is given by Grundmann et al. (2005, see Fig. 7.4). Appendix A may be consulted for a brief introduction to general thermal-hydraulics-neutron-kinetics coupling schemes. In order to take into account reflector-thickness fluctuations and corresponding changes of the reflector group constants (XS), the DYN3D source code had to be modified. Fig. 4 shows the modified DYN3D feedback loop (i. e. one thermal-hydraulics-neutron-kinetics iteration). The pre-calculated time- and node-dependent values drefl of the water layer thickness in the reflector (cf. Sec. 2.1.3) enter the program via an additional input file. In each thermal-hydraulics (TH) iteration, the nodal reflector group data XS is updated by a simple additionally introduced feedback (perturbation)

2. Simulation of neutron noise 2.1. Description of the model The simulation presented here models a quasi-coherent time-dependent deflection of all fuel assemblies being responsible for the observed neutron flux fluctuation patterns. It is assumed that the deflections of the outermost fuel assemblies perturb the radial neutron reflector of the reactor, i. e. by altering the thickness of the water layer between the outer fuel assemblies and the core shroud. In the current investigation, the impact of the fuel assemblies’ elongation relative to one another is not considered in order to focus on the reflector effect. Fig. 2 illustrates the chain of calculations performed. The neutron flux fluctuations are simulated with the nodal, deterministic, time-domain reactor dynamics code DYN3D. In order to cover low-amplitude neutron flux fluctuations, the accuracy of the DYN3D fission source iterations was increased, accepting an increase in computation time. In the simulation, the fluctuations are triggered by time-dependent changes of the node-averaged group constants in the reflector, while these changes result from the changing deflection state of the respective adjacent FAs. At every time step, these FAs' current deflection states correspond to a certain local water layer thickness in the reflector. This thickness values are translated to the current set of reflector group constants, using the results of corresponding calculations with the lattice code CASMO5 performed beforehand the DYN3D runs. The FAs’ time-dependent deflection states themselves are calculated within a simple mechanical model and are provided as reflector thickness values to the DYN3D simulation via an input file. In the mechanical model, the FA deflections are driven by stochastic forces, which represent generic coolant fluctuation forces. One fraction of the forces directly acts on the individual FAs, and the other one acts on the CB/RPV-system. The CB/RPV-system is coupled to all FAs and therefore acts as an additional, collective drive for the dynamics of all FAs. In addition to the mechanical perturbation of the system, thermalhydraulics perturbations are considered, i. e. by assuming stochastically fluctuating cold-leg temperatures. According to a correlation law experimentally determined at the ROCOM facility (Grunwald et al., 2002; Kliem et al., 2008), the four cold-leg flows carrying the temperature fluctuations are distributed to the inlet temperatures of the 193 coolant channels.

XS(TH,d refl ) = XS0 (TH) (1 + d refl CXS).

(1)

The variable TH stands for the group constants’ dependencies on moderator temperature Tmod , moderator density mod , and boron concentration c bor . And the set XS0 (TH) is the group constants previously updated w. r. t. Tmod , mod , and c bor by the standard DYN3D routines. The value drefl quantifies the current deviation of the thickness from the initial (non-perturbed) state. This deviation is coupled linearly to the group constants by respective coefficients CXS . In principle, CXS depends on the variables Tmod , mod , c bor , but the current investigation is limited to the linear reflector thickness feedback. Therefore, the coefficients CXS do not carry thermal-hydraulics dependencies. Future stages of the implementation will cover this dependency as well. For Eq. (1), note that all variables carry space-dependence and, except for CXS , also timedependence. The introduction of the reflector perturbation does not change the coupling scheme of thermal-hydraulics and neutron-kinetics. In the paper at hand, the full set of group constants available for the reflector in DYN3D is taken into account for the feedback: the diffusion constants D1 and D2 , the macroscopic absorption cross sections, abs,1 and abs,2 , the macroscopic scattering cross section 1 2 , the assembly discontinuity factors, ADF1 and ADF2 , and the inverse neutron velocities, invvel1 and invvel2 . As described, the group constants only of the reflector are time dependent. This assumption follows from the idea of a quasi-coherent deflection of all FAs. For such behavior, at arbitrary considered heights, the corresponding axial section of the FA ensemble basically performs global lateral shifts, only. Therefore, the coherent deflection of all FAs does not change the mutual distance of any pair of FAs. Thus, the intralattice conditions are unchanged (the effect of the shear between the

2.1.1. Neutron kinetics and thermal hydraulics Fig. 3 illustrates the nodal structure of the DYN3D simulations along with the locations of the incore detectors in the actual reactor. The model used for the simulations represents a KWU Vor-Konvoi PWR at 100 % power at end of cycle. After a steady-state calculation, the neutron noise is simulated via a transient calculation. In the transient calculation, the equations of neutron kinetics and 3

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Fig. 3. Nodal structure in DYN3D. The reflector nodes are filled gray (side with dark and corner with light shading). In the x-y layout (Fig. 3a), channel numbers are directly given within the nodes. In the x-y-z layout (Fig. 3b), axial node indices are given on the right-hand side. For comparison with measurement signals, the core-map coordinate system is added and channels with fuel assemblies carrying detector lances are marked with dots (Fig. 3a). The axial positions of the six incore detectors at each lance are indicated by bars along with their axial index (Fig. 3b).

Fig. 5. Beams representing the FAs performing forced vibration triggered by the forces Fi,x and Fiy , resp. The displacements wi,x and wi,y are indicated by gray dashed curves.

Fig. 4. Scheme of the modelled perturbations being coupled to DYN3D (one thermal-hydraulics-neutron-kinetics iteration). Changes relative to the original (unmodified) code are given in gray color.

approach is applied for the y-direction. Assuming only one active deflection mode, the beams’ time dependent deflection wi can be written as

axial layers is neglected). Consequently, only the volume around the FA lattice, i. e. the lateral reflector, is affected by the shifts. This fact becomes more clear for changing the frame of reference from a stationary one to a comoving one. Furthermore, the perturbation introduced by this has a long-range correlation.

(2)

wi (t , z ) = fi (z ) Ai (t ). The shape function fi is approximated by a cosine function

fi (z ) = cos

L

L L , 2 2

z , z

(3)

with the length L of the beam. The amplitude function Ai (t ) is determined by the ordinary differential equation of the harmonic oscillator

2.1.2. Mechanical model For simplicity, each individual FA i of the 193 FAs is mechanically represented by two independent homogeneous beams with a simple support at top and bottom (s. Fig. 5): one beam for the x- and one beam for the y-direction. The two beams are distinguished by the subscripts x and y. In the following, only the x-direction's equations are presented (suppressing the subscript x at the corresponding variables). The same

Fi d2 (t ) = 2 Ai (t ) + 2 mi dt

i

0, i

d Ai (t ) + dt

2 0, i Ai (t )

(4)

with the effective mass mi and the natural frequency 0, i of the undamped system. In order to perturb the oscillation, the driving force Fi 4

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and a linear damping term with the damping coefficient i were added to the actual equation of the harmonic oscillator. The force Fi is considered as being composed of an “external” contribution Fi,ext and an internal contribution Fi,int (5)

Fi (t ) = Fi,ext (t ) + Fi,int (t ).

The “external” contribution represents the FA drive by coolant flow fluctuations. It is modeled by white Gaussian noise for simplicity. The internal contribution represents the FA drive by vibrating mechanical components that the FAs are connected to. Here, the FAs are coupled to the vibration of the CB/RPV-system. Hence,

Fi,int (t ) =

2 0, i mi ACB (t )

(6)

with the function ACB representing the amplitude of the vibrating CB/ RPV-system. Its dynamics is analogously modeled by a harmonic oscillator

FCB,int (t ) + FCB,ext (t ) d2 (t ) = 2 ACB (t ) + 2 mCB dt +

CB

0,CB

d ACB (t ) dt

2 0,CB ACB (t )

(7)

with added driving and damping. The driving force is composed, as well, of an “external” contribution FCB,ext , which is modeled by white Gaussian noise, and an internal contribution FCB,int , which is the reaction force from the FAs:

FCB,int (t ) =

193

2 0, i mi (Ai (t )

ACB (t )).

Fig. 6. Amplitude functions Ai (t ) for the x-direction of all 193 fuel assemblies. Periods of coherent motion alternate with those of incoherent motion.

It has to be emphasized that the values of stochastic external forces correspond, in principle, to a certain thermal-hydraulics state. Therefore, a change of the thermal-hydraulics state would go along with changes of the magnitudes and probably further characteristics of these forces. Here, the magnitude and the nature of the forces, i.e. white Gaussian noise, are fixed to enable an independent investigation of the reflector effect. Anyway, the investigations presented here base one single thermal-hydraulics state, only. Fig. 6 illustrates the solution of the run for the x-direction as a waterfall plot. (Note that the spatial arrangement of the FAs is not a part of the mechanical model. Thus, for simplicity, touch between FAs is also disregarded.) It can be seen that the desired coherent deflection takes place most of the time. The maximum amplitudes are in the range of millimeters.

(8)

i=1

The coupling of the FAs to the CB/RPV-system aims on achieving a quasi coherent deflection of all FAs. Note that the model used here results from strong simplifications of the actual coupling and the actual set of involved mechanical components. Starting from zero-deflection state at rest,

{A1 , A2 , …, A193 , ACB } = {0,0, …, 0,0}, (9)

{A1 , A2 , …, A193 , ACB } = {0,0, …, 0,0},

and driving via the stochastic forces F1,ext , F2,ext , …, F193,ext , FCB,ext , the system of ordinary differential equations, Eqs. (4)–(9), is solved using MATLAB's ODE45. The forces are quantified by their standard deviations (F1,ext ) , (F2,ext ) , …, (F193,ext ) , (FCB,ext ) and change rates t (F1,ext ) , t (F2,ext ) , …, t (F193,ext ) , t (FCB,ext ) . Two independent runs are necessary in order to obtain both the time dependent deflection for the x-direction and that for the y-direction. For the paper at hand, the mechanical parameters are set equal for all FAs. Note that only the outermost FAs are relevant for the mechanical perturbation (cf. Sec. 2.1.3); because of their similar burnup, these FAs are similar also from the mechanical point of view. Table 1 shows the parameter values used for the simulations, here. The chosen harmonic frequencies, damping coefficients and masses are in the respective ranges of realistic values. The magnitude of the force acting on the core barrel is of the same range as the one used by Altstadt and Weiss (1999) for a VVER-440. As mentioned, the mechanical model aims on providing a quasi-coherent deflection of all fuel assemblies. Therefore, the magnitude of the forces individually driving the FAs was chosen sufficiently small. The perturbation rate was set to match with the thermal-hydraulics time step.

2.1.3. Perturbation of the radial reflector The group constants used in the performed DYN3D simulations are calculated with the lattice code CASMO5. Fig. 7 illustrates the modeled layers of the reflector: moderator, baffle, and again moderator. The corner nodes, cf. Fig. 7b, have an extra layer for the core barrel (CB). In the time-dependent DYN3D simulations, the first moderator layer is assumed to have varying width d wl . In case the adjacent FAs are deflected in the direction of the baffle, each considered axial FA section (cf. Fig. 3b) is dislocated in this direction, with the dislocation magnitude determined by the deflection magnitude and the considered FA bow shape (cf. Eq. (3)). Correspondingly, these dislocations have opposite sign for the adjacent FAs deflecting away from the baffle. From the section-wise dislocations of adjacent FAs, the respective values d wl follow for each reflector node. Fig. 8 shows which FAs are considered for the respective values d wl of each reflector channels' nodes. In case of one considered adjacent FA, this FA's nodal dislocation wi (t , z ) is directly taken as the variation d wl . In case of three considered adjacent FAs, the average of their dislocations is taken as the thickness variation. The translation from FA deflections to reflector thickness variations is done via a PYTHON script. The impact of variations in d wl on the group constants used by DYN3D was determined by a series of CASMO5 runs with their inputs differing in d wl . The dependence of the group constants on the value d wl enters DYN3D via a linear approximation (cf. Eq. (1)) utilizing the coefficients CXS . The coefficients, obtained from CASMO5 runs with d wl {0 mm, 20 mm} , are shown in Table 2. The value d wl = 10 mm is

Table 1 Parameters used for the simulation of the time-dependent FA deflection (values in brackets used in sensitivity analysis only). k (=i|CB)

1,2, …, 193 CB

f0, k =

0, k /(2

)

k

mk

(Fk,ext )

t (Fk,ext )

in Hz

in %

in kg

in N

in s

1.0 (1.5) 8.0

2.0 (3.0) 2.0

800 33,640

5 50,000

0.1 0.1

5

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Fig. 7. Illustration of the change of the water layer between fuel assembly and baffle shown for both types of considered radial reflector setups.

2008)), the channel inlet temperatures are calculated as

TCh =

l Ch l T.

(11)

l

As known from earlier studies (Viebach et al., 2018a), the timedependent variation of the inlet temperature introduces perturbations travelling up the thermal-hydraulics channels. Consequently, these perturbations qualitatively differ from those defined for the reflector in Sec. 2.1.3 because those act for all nodes of a reflector channel simultaneously. 2.2. Results This section presents the results of a DYN3D run with perturbations in the reflector, the results of another DYN3D run with both perturbation in the reflector and perturbation of the inlet temperature distribution, and results of measurements at a KWU built PWR (operating at full power at end of cycle). For spectral representation, the following functions are used: Information about a signal's power at certain frequencies is obtained by studying the auto power spectral density (APSD):

APSDN , S (f ) =

Fig. 8. Nodal structure in DYN3D (only 1 quarter is shown (cf. Fig. 3a)). Reflector nodes are filled gray (side with dark and corner with light shading). For each reflector node, arrows indicate whose FAs' deflections are considered to influence the reflector thickness.

(12)

N

with S˜ (f , T ) being the (discrete) Fourier transform of a signal sequence S of (temporal) length T. In the paper at hand, signals are always fluctuations about the respective mean value normalized to that mean value. The notation … N indicates averaging over N signal sequences and the asterisk ⋆ denotes complex conjugation. The corresponding integral quantity is the standard deviation

considered as the value for the unperturbed reflector. Thus, this value is used in the DYN3D steady-state calculation, which is prior to the actual noise calculation.

S

2.1.4. Perturbation of the inlet temperature The inlet temperatures TCh, C h= 1, …, 193, of the thermal-hydraulics channels are perturbed by random fluctuations. These have been generated of assumed fluctuations Tl (t ), l = 1, 2, 3, 4 of the four cold-leg temperatures

S (t ) 2

=

(13)

t

with … t denoting the temporal average. In order to correlate two signals S1 and S2 , the cross power spectral density (CPSD):

CPSDN , S1 S2 (f ) =

(10)

Tl (t ) = Tl,mean + Tl (t ).

1˜ S (f , T ) S˜ (f , T ) T

1 ˜ S1 (f , T ) S˜2 (f , T ) T

(14)

N

can be used for calculating the two signals' coherence (COH )

with Tl,mean representing the mean temperature of loop l. All loops have equal Tl,mean . Each perturbation time series Tl (t ) is a set of Gaussian white noise random numbers. Via an experimentally determined mixing matrix ch l (cf. ROCOM test facility (Grunwald et al., 2002; Kliem et al.,

COH2N , S1 S2 (f ) =

|CPSDN , S1 S2 (f )|2 APSDN , S1 (f )APSDN , S2 (f )

(15)

and their phase (PHA )

Table 2 Coefficients of the reflector thickness feedback. Values are given in %/mm. reflector type

CD1

C D2

C

side corner

0.332 0.398

0.0566 −0.00690

0.606 1.78

abs,1

C

C

abs,2

−0.982 −0.978

1

2

0.586 −0.474

6

CADF1

CADF2

Cinvvel1

Cinvvel2

−0.596 −0.489

10.3 11.1

−0.0136 −0.0193

0.000165 −0.00316

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PHAN , S1 S2 (f ) = arg (CPSDN , S1 S2 (f ) ) .

values for higher frequencies. The values taken from positions in the center (Ch144 and Ch225) have a magnitude about one order lower than the value taken from outer regions (Ch33 and Ch114). In addition to a dominant peak at f = 1 Hz a minor peak at f = 2 Hz and at about f = 7.5 Hz can be seen. All signals have high coherence around f = 1 Hz and also at f = 2 Hz . At higher frequencies, only the position near to the reference position has high coherence. Above f = 9 Hz , the coherence values remain quasi constant at relatively high values. The phase is zero for the position next to the reference position. For the positions being in the opposite core half w. r. t. the reference position, the phase is π for most of the frequencies. Only in a narrow band around f = 2 Hz , the opposite core half has zero phase w. r. t. the reference position. Fig. 9c and d compare the simulated signals axially. For each signal, the APSD s have similar shapes, again. The values from the bottom are slightly lower than those at the top of the core. The same peaks as in the radial comparison can be found. All signals have very high coherence values and their phase is zero at all frequencies.

(16)

The coherence takes values 0 COH2 1, only. A coherence COH2 = 0 for some considered frequency fcon means that the two considered signals' contents at this frequency are uncorrelated. A value COH2 > 0 for some considered frequency fcon means that the signals’ contents at this frequency have each a correlated fraction or are correlated in case of COH2 = 1. The phase PHA is the phase angle between the correlated fractions. The sequences S, also known as sweeps, used for the data analyses consist of 1024 values with an overlap of 512 values. A Hamming window was used when cutting the raw data time series into the sequences. In case of the DYN3D simulations, the raw data has a length of 100 s and a sample frequency of 100 Hz . This corresponds to DYN3D runs each simulating 100 s with a thermal-hydraulics time-step of 0.01 s . From measurement, raw data was extracted also with a length of 100 s but with a sampling frequency of 125 Hz . For the noise simulations, the time-dependend inputs, i. e. of the thickness values drefl (t ) and of the channel inlet temperature values Tch (t ) , were prepared with a time-step that eqauls the thermal-hydraulics time-step. The time-dependent output of the simulations is given by the nodal power density P (t ). Its fluctuations are used as a measure of the neutron flux fluctuations. Note that the actual timedependent simulation follows a time-independent run in order to provide a steady-state solution, which is used for initialization of the timedependent simulation. As slight transients of the mean values were observed for the first simulated seconds of neutron fluctuations, the analyses of the simulated data were performed for 9 sweeps starting at t = 7.85 s , only.

2.2.2. DYN3D run with both reflector and inlet temperature perturbation Fig. 10 shows the results for the simulation with both the perturbations in the reflector and the perturbations on the inlet temperature distribution (standard deviation of the temperature T l = 0.1 K , l = 1,2,3,4 ; perturbation every t = 0.1 s ) The APSDs (Fig. 10a and c) resemble those of the DYN3D run with only reflector perturbations (cf. Fig. 9a and c) but with the main peaks at 1 Hz and 2 Hz less pronounced. In general, the APSD values are increased in the full frequency range. Fig. 10b shows coherence and phase for a radial comparison of the results. At frequencies below 2.5 Hz and above 6 Hz , coherence is lower than for the simulation with only reflector perturbations. At other frequencies, i. e. in the range between 2.5 Hz and 6.0 Hz, coherence is the higher the closer the detector locations of the correlated signals. Between 0.5 Hz and 1.5 Hz and above 8.5 Hz, signals from different core

2.2.1. DYN3D run with reflector perturbation Fig. 9 shows the results for the simulation with perturbations in the reflector. Fig. 9a and b compare the simulated signals radially at mid layer. For each signal, the APSDs have a similar shape to one another, i. e. with a global maximum in the low-frequency range and decreasing

Fig. 9. Spectral representation of nodal power density fluctuations simulated with DYN3D for only reflector perturbations. 7

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Fig. 10. Spectral representation of nodal power density fluctuations simulated with DYN3D for both reflector and thermal-hydraulics perturbations (

halves are out-of-phase, and elsewhere, those signals are in-phase. Fig. 10d shows coherence and phase for an axial comparison of the results. The picture is qualitatively different from that of the simulations with only reflector perturbations (Fig. 9d). For all frequencies, the coherence is lowered. At about 1.5 Hz and about 3.5 Hz , it has sink frequencies (i. e. around these frequencies, coherence sinks to values close to zero). But at about 1 Hz and 2 Hz, coherence is also close to one. Below 3.5 Hz , the phase falls quasi linearly (with increasing slope for increasing spatial distances between the considered detector combinations). Only in a narrow band around 1 Hz , the phase is zero.

T

= 0.1 K )

axial and radial shapes of the fluctuation magnitude distribution do not depend on the mechanical FA parameters varied here. 2.2.4. Measurement data As a reference, some measured in-core data is presented here. The data was measured in a KWU Vor-Konvoi PWR at end of cycle (EOC) with approx. 10 ppm boron-acid concentration, operating at approximately full power (consult Viebach et al. (2018a) for more details on the raw data). It has to be emphasized that the time series used here were cut out with a length of 100 s and a sample frequency of 125 Hz from a given full data set in order to provide a reference data set with its basic properties similar to those of the simulated data. As shown by Viebach et al. (2018a), the measured fluctuation-pattern phenomena listed in the introduction of the paper at hand can be well observed for the full data set. Nevertheless, those 100 s picked for the paper at hand were chosen as (on a subjective basis) they exhibit these phenomena particularly pronounced: For the peripheral fuel assemblies' lances LB11 and L-O05, the APSDs exhibit relative peak heights of roughly one order of magnitude at 0.8 Hz (cf. Fig. 11a and c); these lances’ radial comparison exhibits a coherence of almost 0.9 at this frequency (cf. Fig. 11b); for the shown axial comparison (cf. Fig. 11d), the coherence is almost 1.0 in the range .

2.2.3. Comparison between various DYN3D perturbations Table 3 lists the σ values given next to the APSDs shown here and the σ values for DYN3D runs with weaker temperature fluctuations ( T = 0.05 K ) and with the reflector perturbation switched off, resp. It can be seen that the effects of the applied perturbations are all of the same order and do not cancel out each other (one can rather see 2 2 2 Case I + Case IV Case III ). Table 4 gives an overview about the sensitivity of the induced fluctuations on the mechanical parameters of the fuel assemblies. The corresponding DYN3D runs were performed with the temperature fluctuations at the inlet switched off. Both an increase of the natural frequency of the FAs and an increase of the damping lead to a decrease of the magnitude of the induced fluctuations of the power density. The

Table 3 Standard deviation of power density fluctuations normalized to mean for various perturbation cases. Case No. I) II) III) IV)

P

Refl.Pert. yes yes yes no

T in K 0.00 0.05 0.10 0.10

in % at Positions

Ch33 0.350 0.369 0.414 0.208

18

Ch114 0.106 0.139 0.206 0.174

18

Ch144 0.117 0.148 0.216 0.183

18

8

Ch225 0.363 0.378 0.424 0.223

18

Ch33 0.281 0.322 0.405 0.265

6

Ch33 0.341 0.368 0.422 0.220

14

Ch33 0.347 0.358 0.400 0.220

25

Ch33 0.300 0.314 0.371 0.245

31

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Table 4 Standard deviation of power density fluctuations normalized to mean for varying combinations of the mechanical FA parameters. Combination No.

I) II) III)

k

P

in %

2.0 3.0 2.0

f0, k in Hz

1.0 1.0 1.5

in % at Positions

Ch33

18

0.350 0.269 0.313

Ch114 0.106 0.082 0.091

18

Ch144

18

0.117 0.091 0.091

Ch225 0.363 0.282 0.311

2.3. Discussion

18

Ch33

6

0.281 0.217 0.256

Ch33 0.341 0.262 0.307

14

Ch33 0.347 0.266 0.308

25

Ch33

31

0.300 0.231 0.267

the outer FAs are similar to one another, their actual harmonic frequencies may slightly differ from one another. Additionally, the peaks’ sharpness may indicate that the damping coefficient δ was chosen too low. The simulated data exhibits an additional peak at 2 Hz. It is a result of slight oscillations of the reactivity, which can be explained as follows: The case with all FAs non-deflected represents the state of optimum neutron-leakage conditions (symmetry). And the asymmetric case with all fuel assemblies deflected in one direction makes up a state with worse conditions for neutron economy (negative reactivity). Within one period of coherent oscillation of all FAs, the system starts being symmetric, changes to be asymmetric with all FAs deflected in one direction, changes again to be symmetric, and changes again to be asymmetric but with all FAs deflected in the other direction. Finally, the system returns to the intitial configuration. Thus, two periods of alternation between the symmetric and an asymmetric state have taken place giving rise to an effect having a frequency double of that of the FA oscillation. This explanation is in line with the high coherence and phase zero of all signals in a narrow band around 2 Hz . Another explanation for the peak at 2 Hz may be a non-linear effect (cf. Appendix B). Anyway, the peak at 2 Hz is not a direct result of the perturbation, which does not exhibit such peak (cf. Appendix C). As can be seen in Fig. 11a and c, pronounced peaks at 2 Hz are absent for the measured data. But note that Fig. 11a exhibits some tiny elevation of the APSDs at 2 Hz. Here, further data analyses is necessary.

The simulation with perturbations exclusively in the reflector (cf. Fig. 9) reproduces main features of the measured neutron fluctuation phenomena as can be seen for a comparison with the measured data (Fig. 11): Out-of-phase behavior at opposite azimuthal core halves, inphase behavior within the core halves, in-phase behavior along the FAs, high coherence of all signals for frequencies below 2 Hz , APSD-peak at about 1 Hz . As set in the mechanical model, the peak at 1 Hz corresponds to the resonance frequency of the fuel assemblies. Additionally, the APSDs show an asymptotic behavior similar to the measured reference (this also applies for the simulation with both kinds of perturbations active). The simulated fluctuation amplitudes are in the range of tenths of percents. They vary under variation of the mechanical FA parameters (cf. Table 4) as expected: (a) An increase of the damping coefficient causes intensified restriction of the FA deflection, leading to a decrease of the elongations of the FAs; (b) An increase of the FA natural frequency shifts the frequency range with largest mechanical response (peak, cf. Fig. Appendix C) into the direction of lower response of neutron kinetics an thermal hydraulics (reactor transfer function). Consequently, the magnitude of the induced neutron flux fluctuations is decreased w. r. t. those obtained with the reference FA values. Further elaborating on the APSDs, one notices the simulated peaks to be relatively sharp compared to the measured ones. This could be explained by the used values of the mechanical parameters: Although

Figure 11. Spectral representation of measured neutron flux fluctuations. 9

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For the simulation with both reflector and temperature perturbations, the APSD-peaks at 1 Hz and 2 Hz are less pronounced (Fig. 10a and c) than for the exclusive reflector perturbation (Fig. 9a and c), such that relative height of the peak at 1 Hz is of the same range as in the measurements (Fig. 11a and c). Temperature perturbations travelling up the channels (consult Viebach et al. (2018a) for spectral representations of DYN3D results with temperature fluctuations at the inlet being the only perturbation) contribute to the APSDs also. The interplay between the effects of temperature perturbations and mechanical perturbations can be observed because both effects are of the same magnitude and do not cancel out (cf. Table 3). The temperature fluctuations are correlated mainly within but not between the four core quadrants, because each coolant loop mainly feeds that quadrant of coolant channels that is closest to the respective loop (cf. the data given by Kliem et al. (2008)) and cross flow is not taken into account. Consequently, the radial comparison of the simulations with both kinds of perturbations, Fig. 10b, shows increased values for neighboring channels and decreased values for spatially separated channels (relatively to the results with only reflector perturbations, Fig. 9b). Correspondingly, the radial and axial phase relations obtained in the case with only reflector perturbations get destroyed by the temperature fluctuations to a great extend (provided the temperature fluctuations are strong enough). Only around 1 Hz , different core halves are still out-of-phase, and different heights are still in phase. In Fig. 10d, the temperature fluctuations travelling up the core cause the phase to decrease linearly. The latter phenomenon cannot be seen in the considered measured data, although it is known also for KWU type reactors as shown by Grondey et al. (1992). For comparing both sets of simulation results with the considered measured data, one finds better agreement for the pure reflector perturbations. Although the APSDs look more natural with added temperature fluctuations, relevant correlations and phase dependencies vanish. Therefore, temperature fluctuations seem to contribute to the measured signals less than suggested. A feature of both sets of simulations are small fluctuation amplitudes, which becomes obvious when comparing the absolute APSDvalues of the simulations with those of the measurement. This means that the considered perturbation mechanism in the reflector is too weak to cause fluctuations as high as the measured ones. Furthermore, evaluation of the peak heights of the APSDs for different radial positions (Figs. 9a, 10a and 11a) shows that attenuation with decreasing radius is stronger for the simulation results (about factor 10) than for the measured data (about factor 3–6). This may mean that the coherent perturbation sources are not located solely in the reflector or that certain perturbation sources are missing for the center of the core. (Note that the integral quantity σ has a weaker radial dependence: Magnitude of inner and outer positions differ by factor 3 for the simulation with exclusive reflector perturbation, by factor 2 for the simulation with reflector and inlet-temperature perturbation, and by about factor 1.5 for the measurement.) Both suggestions point into promising directions of future studies: In the present work, quasi-coherent deflections of all fuel assemblies as well as the corresponding impact on the reflector are considered. Although mechanically modelled (cf. Fig. 6), the impact of inter-FA effects is out-of-scope of this paper. Nevertheless, such effects may contribute to the fluctuation amplitudes on the one hand, and on the other hand, they may also reduce the strong radial dependency of the fluctuation amplitudes. Note that axial phase and coherence properties would not remarkably change w. r. t. the results from the exclusive reflector perturbations (Fig. 9d). Regarding inter-FA gaps, consequences of static fuel assembly bow may be addressed. Considering results of Wanninger et al. (2018), the

outer fuel assemblies may not be able to perform time-dependent deflections, as these FAs may touch the core shroud. Consequently, correlated time-dependent water-gap variations may take place between FA and FA rather than between FA and shroud, which would substantially affect larger regions of the reactor core. We have to emphasize that the model chain implemented for the contribution at hand represents a minimalist attempt to investigate the suggested case of coherently deflecting FAs with respect to the arising neutron flux fluctuation patterns. Note that this kinematic behaviour of the FA ensemble is enabled by considering a mainly coherent drive of all fuel assemblies by the core barrel. This assumption is motivated by the contribution of Laggiard et al. (1995), which reports that a mechanical drive is needed to explain measured FA motion. Forces actually exciting the CB/RPV-system may arise from vortices travelling down the downcomer as considered by Altstadt and Weiss (1999) using the Kármán-vortex-street model. Examples for other mechanisms causing such behavior are direct mechanical coupling of neighboring assemblies or large-scale cross-flow fluctuations. Coherent fuel assembly deflections provided, the determination of their impact on the XS data is the crucial detail of the setup. In addition to the proposed modifications mentioned above, the used approach via CASMO5 has to be checked. Preliminary results of corresponding work with Serpent (cf. Leppänen et al. (2016)) was presented by Bernt et al. (2018). 3. Conclusions The work presented here contributes to explain the cycle-by-cycle change of neutron noise amplitudes empirically observed in several KWU PWRs. It elaborates on the correlations and phase relations of measured neutron flux signals as an understanding of these is required for an explanation of the changed fluctuation amplitudes. As suggested in previous research (Viebach et al., 2018a), a coherent time-dependent deflection of all FAs is assumed to trigger a timedependent change of the reflector thickness via the deflection states of the outermost FAs. The simulations with DYN3D and CASMO5 show that main features of the neutron noise phenomena (APSD, coherence, phase) can be qualitatively reproduced with this assumption. Furthermore, the FA properties (damping and natural frequency) influence the noise level (quantified by the standard deviation of the neutron flux fluctuations). The results confirm that fuel assembly deflection, provided coherent deflections, is a promising approach to understand the phenomena found by measurements. Further investigation is needed to determine the actual spatial properties of the FAs’ collective deflection and the actual mechanisms triggering this behavior. In order to overcome discrepancies pointed out here, possible modifications of the shown model are given: consideration of the effect of time dependently changing inter-FA gaps, fixation of the outer FAs, and further investigation on the XS-data dependency on FA deflection. Along with efforts to further confirm the applicability of DYN3D for neutron noise simulations, these modifications are matter of subsequent work. Acknowledgments This work was supported by the German Federal Ministry for Economic Affairs and Energy (Project DURATEA, grant number 1501490). The responsibility for the content of this publication lies with the authors. We thank Ulrich Rohde, Sören Kliem, and Denis Janin for discussion and technical support. We also thank the reviewers of this work for their time and helpful comments.

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Appendix A. Thermal-hydraulics/neutron-kinetics coupling schemes The coupling of thermal-hydraulics and neutron-kinetics is an important component of corresponding reactor-physics calculations. As both disciplines carry their own specific features, coupling is not straight-forward. In general, several methods for coupling thermal-hydraulics and neutron-kinetics can be distinguished: such as operator splitting, Picard, and Jacobian-free Newton-Krylov. In the operator splitting, the thermalhydraulics equations are solved separately from the neutron-kinetics equations with the coupling being realized via the boundary conditions of the respective equations of the two disciplines. The Picard method overcomes certain issues of operator splitting by iterating the equations of both disciplines together, i. e. not separately. In Jacobian-free Newton-Krylov, the coupled equations for thermal-hydraulics and neutron-kinetics are iterated simultaneously, rather than subsequently, which leads to improved accuracy and numerical properties. As this coupling method requires a larger extent of implementation efforts, code couplings are rather performed with operator splitting methods than with Jacobian-free Newton-Krylov methods. The contributions of Zerkak et al. (2015) and Zhang et al. (2018) and as well as references given there might be consulted for a deeper understanding on these methods. Appendix B. Higher Harmonics of Sinusoidal Perturbation

m

A sinusoidal perturbation with a given angular frequency ω can cause oscillations of the neutron flux density with multiples of this frequency = m , as can be derived from point-kinetics: For simplicity, assume point kinetics without delayed neutrons,

(t )

d n (t ) = dt

n (t ), n (0) = n 0 .

For a perturbation (t ) =

n (t ) = n 0exp

{

0

(1

(B.1) 0 sin(

t ) , the solution can be found by separation of variables,

}

cos( t )) .

(B.2)

The normalized deviation from the initial state takes the form

n (t ) =

n (t ) n 0 n0

= exp

{

(1

cos( t ))

}

1=

m=1

1 m!

( )

m

cos( t ))m ,

(1

(B.3)

using the power series representation of the exponential function. Examination of the contributions for given values m reveals the mentioned oscillations with frequencies m : (B.4)

m = 1: (1

cos( t )),

m = 2: (1

cos( t )) 2 = 1

2cos( t ) + cos2 ( t ) = 1

m = 3: (1

cos( t ))3 = 1

3cos( t ) + 3cos2 ( t )

2cos( t ) + cos3 ( t ) = 1

1

cos(2 t ) , 2

3cos( t ) + 3cos2 ( t )

(B.5) 3cos( t ) + cos(3 t ) , 4

(B.6) (B.7)

m > 3: contributes cos(m t ). Appendix C. Auto Power Spectral Density of Deflection

Fig. 12. Exemplary APSD of the reflector layer variation d wl (t ) (at Ch10 ).

One may think that the peak in APSDs of the power densities at two times of the fuel assemblies’ harmonic frequency (cf. Fig. 9) resulted from higher harmonics of the time-dependent fuel assembly deflection. But this is not the case as can be seen from the APSDs of the time-dependent reflector thickness values d wl (t ) . Fig. 12 gives an example. Only two peaks are present: one corresponding to the harmonic frequency of the FAs, the other one to that of the CB/RPV-system (note that the latter one is developed slightly below the actually given harmonic frequency).

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