an assessment of corewide coherency effects in the multichannel ...

AN ASSESSMENT OF COREWIDE COHERENCY EFFECTS IN THE MULTICHANNEL MODELING OF THE INITIATING PHASE OF A SEVERE ACCIDENT IN A SODIUM FAST REACTOR

FISSION REACTORS KEYWORDS: sodium fast reactor, core disruptive accident, multichannel approach

M. GUYOT,a* P. GUBERNATIS,a C. SUTEAU,a R. LE TELLIER,a and J. LECERF b Commissariat a` l’E´nergie Atomique, DEN/DTN/STRI/LMA, CE de Cadarache, 13115 St. Paul-lez-Durance, France b Commissariat a` l’E´nergie Atomique, DEN/DER/SESI/LSMR, CE de Cadarache, 13115 St. Paul-lez-Durance, France a

Received August 2, 2012 Accepted for Publication July 17, 2013 doi:10.13182/NT12-123

To consolidate the safety assessment for liquid-metal fast breeder reactors (LMFBRs), hypothetical core disruptive accident (HCDA) sequences have been extensively studied over the past decades. Numerous analyses of the socalled initiating phase (or primary phase) of a HCDA have been made with the safety analysis system code SAS4A. The SAS4A accident analysis code requires that subassemblies or groups of subassemblies be represented together as independent channels. For simulating a severe accident sequence, a subassembly-to-channel assignment procedure has to be implemented to produce the consistent SAS4A input decks. Generally, one uses imposed criteria over relevant reactor parameters to determine the subassemblyto-channel arrangement. The multiple-assembly-per-channel approach introduces corewide coherency effects, which can affect the reactivity balance and therefore the overall accident development. In this paper, a subassembly-tochannel assignment procedure based on the subassembly power-to-flow ratio is presented and implemented to

I. INTRODUCTION

To consolidate the safety assessment for liquidmetal fast breeder reactors (LMFBRs), hypothetical core

*E-mail: [email protected] NUCLEAR TECHNOLOGY

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generate the SAS4A input decks over a range of parameter values. The corresponding SAS4A calculations have been performed on a large LMFBR. The purpose of the present series of calculations is to investigate the magnitude of errors encountered in the analysis of the initiating phase related to the subassembly-to-channel arrangement selection, by comparison with a one-subassembly-per-channel reference solution. It appears that a refinement in the channel arrangement substantially reduces corewide coherency effects. Analysis of the calculations also suggests that an accurate representation of the scenario requires the number of channels to be on approximately the same order of magnitude as the total number of subassemblies. Numerical results are examined to provide the reader with quantitative measurements of bias related to subassemblyto-channel arrangement. Note: Some figures in this paper may be in color only in the electronic version.

disruptive accident (HCDA) sequences have been extensively studied over the past decades.1 A severe accident scenario is generally broken down into subphases, such as the initiating phase, the transition phase, the expansion phase, and the decay heat–removal phase, for applying specific code to each phase.2 During the early part of the accident, which is called the initiating phase, the wrapper 21

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COREWIDE COHERENCY EFFECTS IN MULTICHANNEL MODELING OF ACCIDENT IN SFR

tubes keep their mechanical integrity. Material disruption and dispersal are primarily one-dimensional. For this reason, evaluation methodology for the initiating phase relies on a multichannel approach. Typically, a channel represents an average pin in a subassembly or a group of similar subassemblies. In this context, channel-to-channel thermal and momentum exchanges are not considered. For simulating the accident sequence, a subassembly-tochannel assignment procedure has to be implemented. Generally, one uses imposed criteria over relevant reactor parameters to determine the subassembly-to-channel arrangement. The multichannel approach introduces corewide coherency effects, which can affect the reactivity balance and therefore the overall accident development. The SAS4A computational tool has been developed for the analysis of the initiating phase of HCDAs in LMFBRs (Ref. 3). SAS4A belongs to the fourth generation of the SAS (safety analysis system) series of codes. While previous methods of initiating-phase analysis were based on a SAS-type code with up to 10 channels,4,5 SAS4A was extended to model 200 channels. Such a modeling is believed to be required for reactor applications. Whereas the multichannel-related corewide coherency phenomenon is well known among the scientific community, quantitative measurements of the magnitude of errors related to the channel arrangement selection are scarce in the specialized literature. A previous attempt was made in Ref. 5, where a special procedure was implemented to overcome the ten-channel limitation of the SAS code. It was shown that coherency effects could be reduced by this treatment. However, the restriction on the number of channels prevented determination of the deviation to a detailed simulation. The treatment of uncertainties related to the lumping of subassemblies has also been investigated in the framework of the PARSEC computer program.6 In this approach, different subassembly-to-channel assignments with up to 100 channels are used to model a 205subassembly, half-symmetric core. The results presented in Ref. 6 provide useful information on the corewide coherency effects. For example, it is shown that without a detailed incoherency treatment, the calculations may yield artificially high reactivity ramp rates. As a consequence, prompt criticality is avoided in the 100-channel calculation. The general goal of the PARSEC-1 code is to develop a fast-running accident analysis code for the initiating-phase simulations. To this aim, simplified models are built based on more accurate codes. In particular, the PARSEC-1 code is based on a crude fuel and cladding motion treatment with potential important associated uncertainties, whereas prefailure cladding motion, fuel pin failure, and subsequent material dispersal are of primary importance regarding the consequences of a severe accident scenario. Fuel and cladding motion and relocation are closely coupled with the subassembly hydraulics and the neutronics. As a consequence, corewide coherency effects are likely to be impacted by this crude modeling. The main purpose for 22

developing a new approach in this paper is to increase the accuracy of the calculations in accident situations by decreasing unnecessary conservatisms of the modeling. The innovative features introduced by the next generation of nuclear reactors require a more predictive tool to investigate corewide coherency effects in severe accident conditions. This paper aims to investigate this particular point in providing measurements of this effect for the whole range of partitioning using the most up-to-date computational models. In the remainder of this paper, a subassembly-tochannel assignment procedure based on the subassembly power-to-flow ratio is presented and implemented to generate SAS4A input decks over a range of parameter values. The corresponding SAS4A transient calculations have been performed on a large LMFBR. An overview of the computational tools and models required for the study is provided. A loss-of-coolant-flow accident is studied assuming the failure of the reactor shutdown system. This scenario is believed to cover the key phenomena encountered during core disruption. For example, such accidents could result from an unprotected loss of electrical power to the primary sodium pumps. The purpose of the present series of calculations is to investigate the magnitude of errors encountered in the analysis of the initiating phase related to the subassemblyto-channel arrangement selection, by comparison with a one-subassembly-per-channel reference solution. The goal is to overcome the limitations of previous studies.4–6 In this paper, the analysis is made on a 3600-MW(thermal) sodium fast reactor (SFR) with fresh fuel. Analysis of the numerical results as well as conclusions and perspectives are discussed.

II. SAS4A CODE CAPABILITY II.A. General Overview

The SAS4A code is designed to perform deterministic analysis of severe accidents in LMFBRs. For a loss-offlow (LOF) transient, the code calculates coolant heating and boiling, cladding and fuel heating and melting, and relocation of the different materials. The reactivity feedbacks from fuel heating (axial expansion and Doppler), coolant heating and boiling, and fuel and cladding relocation are computed using first-order perturbation theory. The feedback model is coupled to a point kinetics module to provide an estimate of the reactor power level. SAS4A uses a multichannel treatment. Each channel consists of a fuel pin and its associated coolant and structure and represents an average pin in a subassembly or a group of similar subassemblies. The SAS4A channels are connected at their inlet to the lower sodium plenum and at their outlet to the upper sodium plenum, where a NUCLEAR TECHNOLOGY

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uniform pressure is prescribed. The neutronics behaviors of the different channels are coupled via a point kinetics model of the core.3 The particular features of SAS4A point kinetics and feedback models are presented in Sec. II.B. Heat transfer within each pin is computed with a two-dimensional (r-z) heat conduction equation. Singleand two-phase coolant flows are modeled with a onedimensional multiple-bubble boiling model. Also included is a fuel pin module that calculates the fuel pin characterization in response to both the preirradiation phase and transient conditions.7 In SAS4A, post–fuel pin failure phenomena such as fuel relocation and migration of all moving components in the channel are computed with the aid of a multicomponent, multiphase nonequilibrium hydrodynamic model.8 II.B. Point Kinetics and Feedback Models

In response to both thermal and thermal hydraulics changes, the reactivity effects in each channel have to be calculated and integrated to provide an estimate of the reactor power level. A detailed description of the methods and approximations used for the neutronics calculations can be found in Ref. 3 and is out of the scope of this paper. Our aim, rather, is to emphasize the particular features of the point kinetics and feedback models within the multichannel approach framework. Our starting point is the writing of the point kinetics equation as well as the differential equations for the delayed neutron precursors (for k~1, . . . , K, where K is the number of precursor groups): X dw r(1) (t){b (t)~ w(t)z lk ck (t) dt L k

ð1Þ

and

reactivity. The net reactivity is the sum of different components: r(1) ~rdop zrcool zrclad zrfuel ,

ð3Þ

where rdop 5 fuel Doppler feedback reactivity rcool 5 coolant density feedback reactivity rclad 5 cladding axial expansion and relocation feedback reactivity rfuel 5 fuel axial expansion and relocation feedback reactivity. Other components such as scram or core radial expansion reactivities are not modeled in our approach. The SAS4A feedback model employs a set of precalculated regional reactivity feedback coefficients. The coolant, cladding, and fuel feedback effects are conveniently described Lr by a mass coefficient Lm , where c stands for coolant, c cladding, or fuel. The reactivity change is then computed from rc ~

X X Lr (Ci , j)Dmc (Ci , j) , Lmc j C

ð4Þ

i

where Ci 5 channel number j 5 axial region number of channel Ci Dmc (Ci , j) 5 mass variation in channel Ci and in axial region j from initial steady state to current state. The numerical values of the feedback coefficients have to be provided to the SAS4A model by an external neutronics computation. A description of the method and the hypothesis used for the evaluation of the feedback coefficients is presented in Sec. III. Since the Doppler effect is related to the broadening of cross-section resonances, a special procedure has to be used to compute the Doppler coefficient. The Doppler feedback effect is conveniently described by a temperature coefficient. It is generally obtained from successive criticality calculations with effective cross sections at different temperatures.9 For fast neutron reactors, the Doppler coefficient shows an approximately 1/T behavior10: Lr Lmc (Ci , j)

dck b (t)~{lk ck (t)z k w(t) , dt L

ð2Þ

where w 5 neutron flux amplitude r(1) 5 net reactivity P b ~ bk 5 total effective delayed neutron fraction k

lk 5 decay constant for the delayed neutron precursor k L 5 effective prompt neutron generation time bk 5 effective delayed neutron fraction for precursor k. Superscript (1) in the expression of the net reactivity is denoted to indicate that first-order theory is assumed to predict reactivity feedback effects, except for the Doppler effect, which requires an exact decomposition of the NUCLEAR TECHNOLOGY

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Lr LT

~ Doppler

Kd , T

ð5Þ

where Kd is called the Doppler constant and is considered to be insensitive to temperature changes. The Doppler feedback reactivity evaluation is based on a summation 23

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over all fuel axial nodes and channels of mass-weighted fuel temperature contributions. This leads to the following formulation of the Doppler effect: rdop ~

XX Ci

j

mf (Ci , j) Tf (Ci , j) ln , Kd (Ci , j) mf , ss (Ci , j) Tf , ss (Ci , j)

ð6Þ

where Kd (Ci , j ) 5 Doppler coefficient for channel Ci including weighting factor for axial node j mf (Ci , j ) 5 current fuel mass

Fig. 1. Three-lozenge-based discretization of the hexagon in SNATCH.

mf , ss (Ci , j ) 5 initial steady-state fuel mass Tf (Ci , j ) 5 current average fuel temperature Tf , ss (Ci , j ) 5 initial steady-state average fuel temperature. To account for variation in fuel temperature and moderator void simultaneously, SAS4A models the voided and unvoided Doppler effects. Thus, the quantity Kd (Ci , j ) is evaluated assuming a linear interpolation between the two values: Kd (Ci , j ) ~ Ad (Ci , j ) ð1{v (Ci , j )Þ zBd (Ci , j ) v (Ci , j )

,

ð7Þ

SNATCH perturbation tools have been used in the present work to compute first-order coolant, fuel, and cladding reactivity worths and exact Doppler reactivity worth. The SNATCH first-order perturbation engine provides the derivative of the reactivity over any isotopic density, and thus, the reactivity coefficients are found by summing the different contributions of all the involved isotopes in all the regions of the core. As each assembly d is horizontally partitioned into three lozenges l, the SAS-consistent firstorder feedback coefficients for coolant, fuel, and cladding for a given channel Ci can be written as ! Lr 1 X 1 X Lr (Ci )~ (l ) , Lm Nass (Ci ) d[C 3 l[d Lm

where v (Ci , j ) 5 void fraction in axial node j of channel Ci Ad (Ci , j ) 5 unvoided Doppler coefficient for channel Ci

together with the expression of

Bd (Ci , j ) 5 voided Doppler coefficient for channel Ci . As well as for coolant, cladding, and fuel density coefficients, Doppler coefficient calculations are performed by an external neutronics code. The details of the computations are provided hereafter.

ð8Þ

i

Lr (l): Lm

X Lr Lr (l )~ (iso, l ) , pm (iso, l ) Lm Lm iso[l

ð9Þ

where Nass (Ci ) 5 number of subassemblies in channel Ci

III. ERANOS CODE SYSTEM DESCRIPTION

The deterministic ERANOS neutronics system11 consists of data libraries, codes, and calculation procedures that have been developed mainly for fast reactor simulations. A high-order discrete ordinates transport solver has been implemented recently to improve the ERANOS code system capabilities. Such a solver named SNATCH (Ref. 12) has the ability to model core geometries based on hexagonal assemblies. The three-lozenge-based discretization of the hexagon is depicted in Fig. 1. Perturbation tools consistent for the SNATCH solver have also been developed in order to compute first-order derivatives or exact decomposition of a given quantity of interest with respect to some system parameters. The 24

pm (iso, l ) 5 mass proportion of isotope iso in a region l Lr (iso, l ) 5 derivative of the reactivity over the mass Lm of isotope iso in a region l as computed by the SNATCH perturbation engine. In Eqs. (8) and (9), axial meshing summation is omitted for convenience; an extension of the formula to account for the axial summation is straightforward. A similar formulation can be found for the Doppler effect. In this particular case, exact decomposition rather than derivatives is used to account for the microscopiccross-section modifications. Self-shielded microscopic cross sections are prepared using the ECCO cell code of the ERANOS distribution.13 The voided and unvoided Doppler coefficients are computed using a uniform change NUCLEAR TECHNOLOGY

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in fuel temperature from 1500 K to 2200 K. For this reason, four sets of effective cross sections are calculated.

procedure can now be presented. Let (Qi )i and (Ei )i be the different flow rate and enrichment zones and (SA) be the set of fueled subassemblies. (Qi )i and (Ei )i can be written as Qi ~fd [ (SA) ; qd ~qi g

IV. TEST MODEL

and

IV.A. Geometry

The model used in the present study pertains to a large, sodium-cooled, oxide-fueled reactor. Main design parameters of the core are listed in Table I. This core is made of three different coolant flow rate zones and two different enrichment zones (inner core and outer core). A beginning-of-life state (fresh fuel) is selected as the core state of this evaluation (i.e., no preirradiation phase). The use of a one-third symmetry leads to the use of 151 hexagonal fuel assemblies. IV.B. Subassembly-to-Channel Arrangement Method Description

A channel lumping procedure intends to account for the nonsynchronicity of phenomena between the multiple channels. The channel lumping can be produced by using several reactor parameters such as power-to-flow ratio, distribution profile of reactivity coefficients, burnup, etc., leading to different subassembly-to-channel assignments. The procedure presented here does not aim to find an optimum arrangement regarding the number of channels; the objective, rather, is to provide a support to parametric analysis. A subassembly-to-channel arrangement procedure based on the subassembly power-to-flow ratio has been successfully implemented to produce the consistent SAS4A input decks. The presence of different enrichment and flow rate zones imposes the corresponding subassemblies to be automatically spread into different channels. This constraint leads to a coarse three-channel representation of the core (one in the inner enrichment zone and two in the outer enrichment zone). The power-to-flow–ratio criterion e must then be adjusted in order to refine the SAS4A channel arrangement. A general formulation of the

Core Design Parameters Specification

Core power Inlet/outlet temperature Core height Number of fuel subassemblies Number of pins per subassembly Total effective delayed neutron fraction

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where q and e represent the subassembly flow rate (in kilograms per second) and the subassembly plutonium enrichment (in percent), respectively. Using these definitions, let us define the set of subassemblies (Ck,l i )i,k,l for all k and l: C k,l i (e)~ (P=q)d vie , d [ Qk \ El ; (i{1)eƒ1{ maxQ \E (P=q) k

l

ð10Þ where P represents the subassembly total power (in watts). Within a given flow rate zone k and enrichment aggregates subassemblies sharing similar zone l, C k,l i power-to-flow ratios P/q. In this context, a subassemblyto-channel arrangement based on the power-to-flow ratio C~(Ci )i can be written: C~ C k,l i =1 :

ð11Þ

The cardinal of C corresponds to the number of channels. For the selected core with 453 subassemblies arranged in a one-third symmetry, a complete description requires a 151-channel SAS4A input file. This corresponds to the finest possible description and stands for the reference arrangement. The power-to-flow–ratio criterion e is then varied over a range from 10% to 0.0001%. The different values and the corresponding number of channels are summarized in Table II. The subassembly-to-channel assignment layouts over a one-third section of the core are plotted in the Appendix. IV.C. Calculation of Physics Parameters

TABLE I

Item

Ei ~fd [ (SA) ; ed ~ei g ,

3600 MW(thermal) 650 K/800 K 1m 453 271 3.69|1023

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Coolant, fuel, and cladding initial temperature maps as well as coolant initial pressures are provided by SAS4A steady-state computations. The initial power distribution calculation has been performed using the SNATCH solver. Six energy groups are employed in the multigroup treatment. Microscopic cross sections are based on JEFF2.2 data and processed using the ECCO cell code of the ERANOS distribution.13 Figure 2a shows the spatial distribution of the initial power, and Fig. 2b shows the power-to-flow ratio at the core median plan. 25

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TABLE II Subassembly-to-Channel Arrangements Criterion (%)

—

10

5

1

0.5

0.1

10{2

10{3

10{4

Number of channels

3

13

21

43

50

75

106

142

151

Fig. 2. Spatial maps of (a) initial power (in kW) and (b) power-to-flow ratio (in kW/kg?s21) .

Reactivity worths are summarized in Table III. The sodium and cladding reactivity coefficients of the core present negative values at the upper and lower ends of the active core region and in the outer core where a leakage of neutrons prevails. Otherwise, the coefficients show positive values. The axial distribution of the sodium reactivity coefficient for the three-channel case is represented in

TABLE III Total Reactivity Worths First-order coolant reactivity worth First-order cladding reactivity worth First-order fuel reactivity worth Fuel Doppler reactivity worth With sodium voiding Without sodium voiding 26

2.2 $ 10.1 $ 277.9 $ 218.3 $ 224.2 $

Fig. 3; the corresponding channel arrangement can be found in the Appendix.

V. NUMERICAL RESULTS

The test accident is initiated by a flow coastdown from nominal flow and full power. Failure of the diverse shutdown systems is assumed. Such a scenario is often called unprotected LOF (ULOF). Numerical solutions for the different subassembly-to-channel arrangements were analyzed. V.A. Accident Scenario

During the initiating phase of an ULOF, an imbalance between energy production and removal results from the flow rate reduction. The initiating phase sequence can be split into two parts. Before sodium boiling begins, the NUCLEAR TECHNOLOGY

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Fig. 3. Sodium reactivity coefficient for the three-channel case.

compensation of the sodium density effect by the Doppler effect and the fuel axial expansion determines the net reactivity. The reactor power remains close to its nominal value. In the second part, positive sodium voiding reactivity insertion can occur and induce a power increase. The axial relocation of the fuel and the cladding takes place due to loss of cooling and the overheating of the materials. The subsequent accident development is determined by the cladding and fuel dispersal, which could lead to LOFdriven transient of power conditions if sufficient positive reactivity is introduced to produce significant energetics. V.B. Results

Figure 4 provides information concerning the overall accident development modification while refining the core representation. The onset-of-boiling time, the onset-ofdryout time, the first-occurrence-of-pin-failure time, and the end-of-initiating-phase time are represented. In Table IV, the relative deviations of the different quantities with respect to the reference one-subassembly-per-channel case

are shown. The results show that coolant boiling onset occurs later in the 3-channel case than in the 151-channel case and to a lesser extent in the 13-channel case [,4 s (11%) later and 1 s (4%) later, respectively]. The justification of this deviation lies in the fact that hot spots are reduced when a large number of subassemblies are gathered together as channels. Figure 5 shows the steadystate coolant temperatures for the hottest channel in the different cases. The steady-state variations observed in Fig. 5 are the consequence of coherency effects in the core representation. These steady-state variations are the reasons the event initiation times decrease when the level of incoherency increases. It should be noted that this difference is significantly reduced when considering the 21-channel case (,2%) and becomes insignificant when 43 channels are used. In Figs. 6 through 11, the different feedback reactivity coefficients are plotted as functions of time for the different subassembly-to-channel arrangements. The comparison of the results shows that the accident development is substantially modified when a coarse representation

TABLE IV Relative Deviations* with Respect to the Reference Case Number of channels Peak power Peak reactivity Time of boiling onset Time of dryout Time of pin failure End-of-simulation time

3

13

21

43

50

75

106

142

j2100 76.6 211.0 211.8 212.9 212.2

250.7 250.4 24 24.4 21.3 24.9

214.7 233.5 21.7 21.9 20.5 22.1

280.4 241.1 20.03 20.1 0.02 21.3

20.5 223.2 20.03 20.2 20.01 21.5

24.1 217.9 20.03 20.03 0.01 20.8

9.8 23.99 0.001 20.14 0.02 21.3

21.3 20.7 0.001 20.2 0.1 0.4

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*In percent. NUCLEAR TECHNOLOGY

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Fig. 4. Different sequence event times versus number of channels.

Fig. 5. Steady-state coolant temperatures (hottest channel) for different subassembly-to-channel arrangements.

of the core is used. In our case, it is seen that a refined channel arrangement yields milder accident consequences. Intuitively, it seems obvious that a more refined representation of the reactor would yield a smoother reactivity insertion. In Fig. 12, the value of the maximal normalized power peak during the initiating phase as well as the corresponding net reactivity peak (in dollars) are plotted as functions of the number of channels. 28

As expected, the three-channel case presents the steepest coolant voiding reactivity ramp due to the important corewide coherency added by the modeling of the core (see Fig. 6). In the three-channel case, the coolant voiding contribution of one channel corresponds to a 0.9-$ reactivity insertion. This leads subsequently to an important amount of positive reactivity insertion due to cladding dispersal and a premature loss of hex-can NUCLEAR TECHNOLOGY

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Fig. 6. Coolant feedback reactivity versus time.

Fig. 7. Cladding motion feedback reactivity versus time.

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Fig. 8. Fuel axial expansion feedback reactivity versus time.

Fig. 9. Fuel Doppler feedback reactivity versus time.

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Fig. 10. Fuel motion feedback reactivity versus time.

Fig. 11. Net feedback reactivity versus time.

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Fig. 12. Net reactivity and power peaks versus number of channels.

integrity. As a consequence, prompt criticality is reached in this case. While the first part of the sequence is almost identical in every case but the three-channel simulation, the second part shows more differences between the cases. Fuel and cladding motion reactivity insertion are stronger than sodium voiding reactivity insertion (see Table III). The net feedback reactivity is thus more sensitive to fuel and cladding dispersal than to sodium boiling. As a consequence, on the second part of the transient, more channels are required to provide an accurate simulation. This point is well illustrated by the 13-channel case, where most of the discrepancy from the reference case is coming from the cladding motion reactivity evaluation (see Fig. 7). From these observations, it seems that the number of channels employed in the simulation should be increased if cladding and fuel dispersal occur during the transient. It should be noted that the selected subassembly-tochannel arrangement refinement is based on the powerto-flow–ratio criterion, which directly improves the prediction of the thermal behavior during the ULOF. Coolant boiling and voiding as well as cladding and fuelmelting coherency effects are directly reduced. A more accurate treatment would be to account for the neutronics behavior in the same way as for the thermal behavior. To this aim, the channel lumping procedure can include an additional criterion to aggregate subassemblies sharing a similar profile of reactivity coefficients. Such a procedure 32

is expected to provide more accurate results with a smaller number of channels. It is found that the 106-channel case yields an accurate prediction of the accident scenario. Relative discrepancies to the reference one-subassembly-perchannel case are , 4% for the peak reactivity prediction. This analysis suggests that an accurate representation of the scenario requires the number of channels to be of the same order of magnitude as the total number of subassemblies. For evaluating the consequences of the initiating phase, it is relevant to study the fuel-melting fraction spatial maps at the onset of the transition phase. Results for the reference case are displayed in Fig. 13. Within point kinetics approximation, no flux shape calculations are carried out. This implies that the hottest channels at the ULOF onset are unchanged during the transient. Fuelmelting fraction at final time is thus expected to be higher for the channels having a higher initial power-to-flow ratio. This point is confirmed by comparing Fig. 2 and Fig. 13. Absolute deviations of the averaged fuel-melting fraction with respect to the one-subassembly-per-channel calculation are shown in Fig. 14. This provides a good illustration of the sequence convergence while refining the channel arrangement. Once again, it is seen that the 106channel case presents very close results in comparison to the reference calculation. This set of calculations not only shows the importance of treating incoherency effects in NUCLEAR TECHNOLOGY

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crucial for whole-core subchannel simulations where the number of channels can be increased by one or two orders of magnitude.

VI. CONCLUSIONS AND PERSPECTIVES

Fig. 13. Averaged fuel-melting fraction, in percent, at onset of transition phase.

severe accident analysis but also indicates the minimal number of channels required to obtain the same level of accuracy as a one-to-one correspondence. The results obtained here can be used to speed up the calculations in keeping the same degree of accuracy. V.C. Computational Cost

Figure 15 shows the computational cost associated with the different subassembly-to-channel arrangements. We can notice that the computing time is quite reasonable even for the 151-channel case (, 4 h). The computing time per channel is roughly constant between the different calculations. As a consequence, the gain we get in using 106 channels in place of 151 channels is ,30% of the overall computing time in keeping the same degree of accuracy in our results. Elimination of corewide coherency effects by this method will be fully realized when dealing with a more accurate modeling of the fuel bundle. For example, the subchannel model of SAS4A (Ref. 14) can benefit from this approach by drastically reducing both computing time and memory storage. This point is NUCLEAR TECHNOLOGY

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A technique for producing subassembly-to-channel arrangements based on power-to-flow ratio has been presented. Corresponding SAS4A calculations have been performed on a typical ULOF scenario. It appears that an increased number of channels yields substantially milder corewide incoherency effects. Numerical results have been provided to give quantitative aspects. A onesubassembly-per-channel reference case is computed for comparison. It is found that a too-large criterion leads to substantial differences in the overall accident development, while a refined subassembly-to-channel arrangement increases the accuracy of the calculation. From the results, it is observed that the number of channels employed in the simulation should be increased if cladding and fuel dispersal occur during the transient. In this case, coherency effects are higher because cladding and fuel dispersal reactivity insertion are stronger than sodium voiding reactivity insertion. Results from the 106channel case proved to be in very good agreement with the reference 151-channel case. Thus, a power-to-flow criterion of 0.01% seems a reasonable choice for reducing corewide coherency effects and for obtaining an accurate prediction of the sequence. This choice provides the same level of accuracy as a one-subassembly-per-channel correspondence with a 30% reduction of the overall computing time. These results are useful to save both computing time and memory storage. These two points are crucial for running whole-core subchannel calculations. Given the accuracy needed, the interested reader can find here quantitative elements for selecting a proper subassembly-to-channel arrangement for his or her own calculations. This paper analyzes a beginning-of-cycle case with fresh fuel. It should be pointed out that other cases with higher burnups, different designs, or different accident initiators need to be analyzed in the future to confirm that effects of the channel selection on the accident outcome are similar. While the technique presented here uses a monocriterion procedure, a multicriterion-based subassembly-tochannel arrangement technique can also be investigated. For example, the channel lumping procedure could include a neutronics parameter such as the subassembly reactivity worth or the axial profile of the different feedback reactivity coefficients. A multicriterion parametric analysis similar to the one presented here can thus be performed to provide more insight into the corewide incoherency effects related to the multichannel modeling. Such an analysis could be led to investigate the possibility of finding an optimum arrangement regarding the number of channels. 33

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Fig. 14. Absolute differences of the averaged fuel-melting fraction with respect to the one-subassembly-per-channel calculation.

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Fig. 15. Computational time versus number of channels.

APPENDIX CHANNEL ARRANGEMENT REPRESENTATION

Fig. A.1. The three-channel-arrangement representation. NUCLEAR TECHNOLOGY

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Fig. A.2. The 13-channel-arrangement representation. 35

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Fig. A.3. The 21-channel-arrangement representation.

Fig. A.4. The 43-channel-arrangement representation. 36


Fig. A.6. The 75-channel-arrangement representation. NUCLEAR TECHNOLOGY

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