MONTE CARLO LARGE SCALE REACTOR PHYSICS …

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MONTE CARLO LARGE SCALE REACTOR PHYSICS CALCULATIONS W. Bernnat Universitaet Stuttgart, IKE Pfaffenwaldring 31, 70550 Stuttgart, Germany [email protected] S. Langenbuch and W. Zwermann Gesellschaft fuer Anlagen- und Reaktorsicherheit (GRS) mbH Forschungsgelaende, 85748 Garching, Germany [email protected]; [email protected] Abstract Monte Carlo criticality calculations for large scale LWR reactors based on both point-wise and multigroup nuclear data are presented and discussed. The applied methods and nuclear data had been validated previously by re-calculating numerous LWR related benchmark experiments. As applications, several states of a large BWR at BOL with cold and hot operating conditions and different control rod configurations were regarded and calculated by the MCNP-4C and KENO-Va programs, based on JEF-2.2 point-wise and multigroup cross sections, respectively. Corresponding calculations were also performed for PWR and RBMK systems. Details of these calculations are analysed, like the convergence of power distributions, reaction rates and neutron spectra in small spatial regions. There is a strong need to control the source convergence of such calculations since in large LWR systems regions with very weak coupling exist, especially for cold conditions with high moderator density. Therefore, criticality calculations with neutron generation sizes up to 100,000 neutrons and total number of histories of up to 200 millions were performed, and the results were compared with deterministic transport calculations to analyse the source distribution convergence.

Transcript of MONTE CARLO LARGE SCALE REACTOR PHYSICS …

MONTE CARLO LARGE SCALE REACTOR PHYSICS CALCULATIONS

W. Bernnat Universitaet Stuttgart, IKE

Pfaffenwaldring 31, 70550 Stuttgart, Germany [email protected]

S. Langenbuch and W. Zwermann

Gesellschaft fuer Anlagen- und Reaktorsicherheit (GRS) mbH Forschungsgelaende, 85748 Garching, Germany

[email protected]; [email protected]

Abstract

Monte Carlo criticality calculations for large scale LWR reactors based on both point-wise and multigroup nuclear data are presented and discussed. The applied methods and nuclear data had been validated previously by re-calculating numerous LWR related benchmark experiments. As applications, several states of a large BWR at BOL with cold and hot operating conditions and different control rod configurations were regarded and calculated by the MCNP-4C and KENO-Va programs, based on JEF-2.2 point-wise and multigroup cross sections, respectively. Corresponding calculations were also performed for PWR and RBMK systems. Details of these calculations are analysed, like the convergence of power distributions, reaction rates and neutron spectra in small spatial regions. There is a strong need to control the source convergence of such calculations since in large LWR systems regions with very weak coupling exist, especially for cold conditions with high moderator density. Therefore, criticality calculations with neutron generation sizes up to 100,000 neutrons and total number of histories of up to 200 millions were performed, and the results were compared with deterministic transport calculations to analyse the source distribution convergence.

Introduction Important reactor design and safety parameters must be determined with a high degree of

accuracy and reliability. Corresponding reactor physics calculations therefore should be performed by methods which are based on first principles and validated on representative experiments and operational data. There are extensive good experiences with the application of continuous and multigroup Monte Carlo programs based on established evaluated nuclear data for compact systems represented by a large number of benchmark experiments and research reactor operational data. Therefore, one can conclude that these methods are also suitable for large power reactors. Most design calculation methods for large power reactors use simplified methods for spectral and cell or assembly calculations and few group diffusion theory for the final 2D or 3D calculations for the determination of the power and flux density distribution. These methods mainly are validated by comparisons with experimental data from operational conditions. An extension to ranges not covered by experiments or operational data may cause large uncertainties. For several problems in reactor safety and reactor design, an independent and validated method compared to the design methods should be available for the verification of the most important design parameters and safety related values. Examples for such problems are the verification of the reactor design and the analysis of severe accidents like reactivity accidents caused by control rods, boron dilution, or by movement of fissionable, absorbing or moderating material in damaged cores, but also criticality problems of spent fuel storage taking into account burn-up credit.

Applications of the Monte Carlo Method for 3D full core calculations are in principle possible,

since the geometric description of core and reflector in detail is relatively easy due to the repeated structure options of modern programs. These options enable a pin by pin description of all assemblies in the core together with a detailed description of the control rods. They can be realised, e.g., in the continuous Monte Carlo code MCNP [1] or the multigroup codes KENO-Va and KENO-VI [2]. For cores at begin of life without burn-up and build-up of fission products, detailed 3D calculations even for large LWRs can be performed with comparable small computer resources and input effort.

As examples, full core calculations were performed for a typical large BWR with 1,300 MWel, at

begin of life for different states and control rod configurations. Further calculations were performed for PWR and RBMK-systems. For the interpretation of the results, additionally cell calculations were performed and partly compared with deterministic SN methods to analyse the convergence behaviour of the source distribution and reaction rates. The calculations were performed on both high performance workstations and massive parallel computers like Cray T3E/512 and Hitachi SR 8000.

Monte Carlo Calculations for a Large Boiling Water Reactor The analysed BWR core consists of 840 Uranium fuel assemblies of two different types of 8 x 8

arrays of fuel pins with different enrichments, with a central water pin, and 1 or 2 Gadolinium pins. The core was calculated with MCNP-4C and KENO-Va. The model is very detailed, with explicit representations of each fuel pin, and each absorber pin for the controlled states. Some details are shown in Fig 1. The MCNP calculations were performed with JEF-2.2 [3] point data, and the KENO-Va calculations with JEF-2.2 292 group data in AMPX format used in the SCALE code package [2], generated with the 1D spectral code RESMOD [4] (solving the slowing down equation for resolved resonance range based on point-wise cross sections) for the various fuel pin cells. The applicability of both methods was demonstrated, among many other benchmark activities, by re-calculation of the KRITZ-2 [5] and VENUS-2 [6] experiments [7,8].

Fig. 1. Horizontal cut through the MCNP model of a 1,300 MW BWR Top: The whole reactor core. Bottom: A four-bundle of fuel assemblies in its surrounding.

The results for the multiplication constants for various operational and accidental conditions of

the described BWR are shown in Table 1 and additionally in Fig. 2. (These calculations were performed in the framework of investigations on the criticality conditions in the course of severe accidents; a part of these results was already given in [9].) Generally, the keff results showed sufficiently good convergence and comparable small statistical errors in reasonable computer time. The results matched the usual statistical checks, provided that a reasonable neutron starting source was chosen. Not only the total reactivity worth of control rod banks and single control rods, but also the reactivity as a function of bank position could be determined with sufficient statistical accuracy. For all calculated cases, the MCNP and KENO results were in excellent agreement, with relative differ-ences not exceeding 0.3 % as it was also found for several benchmark experiments (e. g. the KRITZ-2

assemblies). This indicates that the multigroup method can also be successfully applied even for situations where the pre-condition of regular lattices is not strictly fulfilled (perturbations by water gaps and absorber rods between the fuel assemblies), and that it is appropriate as an alternative to Monte Carlo calculations with point data, when independent calculation methods are desired.

Table 1. Multiplication constants for a BWR core in different states calculated from MCNP-4C and KENO-Va calculations with JEF-2.2 nuclear data, along with the corresponding reactivity differences. Statistical uncertainties are ~ 0.0005 in MCNP and ~ 0.0003 in KENO (1 σ).

Case description MCNP KENO ∆ρ

Cold, uncontrolled 1.1337 1.1323 0.0011

Hot, uncontrolled, 0 % void 1.1228 1.1189 0.0031

Hot, uncontrolled, 40 % void 1.0995 1.0960 0.0029

Cold, controlled 0.9476 0.9457 0.0021

Cold, ~1 of 4 control rods withdrawn 1.0109 1.0091 0.0018

Cold, controlled, 1 central control rod withdrawn 0.9765 0.9744 0.0022

Cold, controlled, 5 central control rods withdrawn 1.0518 1.0510 0.0007

Cold, controlled, 6 central control rods withdrawn by 53 cm 0.9983 0.9988 -0.0005

Hot, controlled, 0 % void 0.8786 0.8771 0.0020

Although we did not experience problems with the convergence of the multiplication factor, as observed for weakly coupled systems and discussed in a number of publications (for recent investigations cf. [10]), we found some difficulties for the axial and azimuthal power distribution for large cores due to the weak coupling of core regions. Here, the power distributions converged very slowly with a different characteristic compared to the effective multiplication constant. This was the case for all Monte Carlo programs applied (MCNP, KENO-V and MORSE). An example is the axial power distribution from a full core calculation at BOL, cold conditions of the large BWR (with a number of control rods inserted such that the core is approximately critical, denoted as “Cold, ~1 of 4 control rods withdrawn” in Table 1). To study the convergence of the power distribution without the knowledge of the exact solution, a system with four identical azimuthal quadrants and exact axial symmetry was chosen.

The axial power distribution turned out not converge according to the evaluated statistical errors

given for axial intervals. The estimated relative statistical error after 500 generations with 20,000 neutrons per generation was less than 0.20 % (one standard deviation), whereas the corresponding power rates in symmetric intervals differed by up to 9 % An improvement was seen by substantially increasing the total number of neutron histories. In Fig. 3, the evolution of the deviations of axial symmetry with increasing number of histories is displayed, where the calculation was performed with 100,000 neutrons per generation. With 140 million histories, the deviation of the distribution from symmetry is reduced to less than 1 % for all axial intervals. Nevertheless, also in this case the deviation from symmetry (for an idealised symmetric system) was significantly larger than the corresponding estimated statistical error of 0.05 % or less (this applies even if 3 σ is used as statistical uncertainty).

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Fig 2. Multiplication constants for a BWR core in different states calculated from MCNP-4C and KENO-Va calculations with JEF-2.2 nuclear data (top), along with the corresponding reactivity differences (bottom). Statistical uncertainties for the multiplication constants are ~ 0.0005 in MCNP and ~ 0.0003 in KENO (1 σ).

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Fig. 3. Deviation from symmetry of the radially and azimuthally integrated power in three axial sections for a BWR core as a function of the number of neutron histories, calculated with MCNP. Cold state; approximately critical control rod pattern. Relative uncertainties are given as 0.02-0.05 % for the last batch after 140 million neutrons run.

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Fig. 4. MCNP results of the axially integrated power in four quadrants for a BWR core as a function of the number of neutron histories. Cold state; approximately critical control rod pattern. The error bars indicate three times the uncertainties given for MCNP.

A similar situation was found for the power rates of the four quadrants which should be identical due to the core symmetry. The calculated values are displayed in Fig. 4 in dependence of the neutron histories. Also for these calculations, we observed deviations from symmetry much larger than the estimated statistical error. The deviations are even larger than for the axial distribution. After 140 million neutron histories, The deviations from symmetry for two of the quadrants are almost 10 %, whereas the estimated statistical uncertainties are approximate 0.02 % on the 1 σ level.

To exclude a bias due to the initial conditions, the initial source distribution was chosen carefully

(homogeneous over the whole core such that the symmetries of the core are preserved for the average distribution of the starting neutrons), and additionally, the first 4 million neutron histories were ignored for tallying the power distribution. Generally, we found that the estimated statistical errors of power rates were much lower than the deviations from the corresponding symmetrical position. Since there is an exact symmetry of the problem, the deviation from symmetry does not influence the effective multiplication factor. Therefore, if symmetry is applicable, it should be taken into account, of course. If the symmetry is slightly perturbed the influence on power distribution may be also difficult to calculate by Monte Carlo for weakly coupled systems and deviations from the true solution may be comparable large as the deviations from the symmetrical solution found for the investigated LWR. Here more comparisons with deterministic converged solutions should be made to verify the power distributions calculated by Monte Carlo.

Calculation of the Power Distribution for a Pressurized Water Reactor After the investigations on a large BWR, we also analysed a large PWR. To study the source

convergence for symmetrical conditions, we analysed a simplified PWR at BOL in the hot full power and cold states. The fuel geometry and material data were taken from the Rowlands pin cell UOX benchmark [11]. For studying axial distributions, also calculations were performed for pin cells with the height of a PWR core, in an infinite lattice in the radial directions.

The convergence of the power density in seven equally large axial sections of a pin cell calculated

by MCNP-4C and KENO-Va are shown in Fig. 5 together with a deterministic 2D SN solution calculated by the program TWODANT [12] with 18 group cross sections, collapsed from 292 group data prepared by the 1D cell code RESMOD. The form of the power density distribution has its origin in the axial decreasing moderator density with the maximum shifted into the lower part of the core. The agreement of the two Monte Carlo solutions with the TWODANT solution is here quite sufficient. Nevertheless, there are deviations from the TWODANT solution which are outside the estimated 3 σ uncertainties of the Monte Carlo results. Part of the deviations may also originate from systematic differences of the solutions of the Monte Carlo and deterministic calculations, in addition to the not fully reached convergence of the Monte Carlo distributions. Qualitatively the same is observed for the corresponding Monte Carlo and TWODANT calculations on the full size simplified core, with slightly larger deviations.

However, if we consider the axially integrated power densities in the four quadrants (see Fig. 6)

similar deviations from symmetry were found as in the BWR case. Also for this case, the deviations from symmetry were substantially larger than the estimated statistical errors, and only a very slow convergence of the power in the four quadrants to their average value is observed, in contrast to the much more satisfactory axial power distributions.

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Fig. 5. Axial power distribution for a pin cell with the length of the active zone of a typical PWR (active length 371 cm). Hot state; an axial coolant density distribution is used to simulate PWR full power conditions. Geometry and materials are from the Rowlands UO2 pin cell benchmark. MCNP and KENO Monte Carlo results in comparison with the TWODANT reference solution. 20.000,000 active particle histories (1,000 generations with 20,000 neutrons/generation). Relative uncertainties are given as 0.06-0.15 % for MCNP and 0.11-0.42 % for KENO.

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Fig. 6. MCNP results of the axially integrated power in four quadrants for a simplified PWR core (360 cm diameter of the active zone) as a function of the number of neutron generations (with 20,000 neutrons/generation). Hot state; an axial coolant density distribution is used to simulate PWR full power conditions. Geometry and materials of the fuel cells are from the Rowlands UO2 pin cell benchmark. The error bars indicate three times the uncertainties given for MCNP.

It is interesting to note that the problem of convergence of the axial power distribution also was found for a radially infinite lattice of fuel pins with PWR core height in a fully axially symmetric condition including the moderator density (cold state), see Fig. 7. A corresponding two dimensional deterministic multigroup transport calculation with TWODANT showed correct symmetry and served as reference solution. The axial region was again subdivided into seven identical sections. The maximum of the power density is in zone 4 of the 7 zones. The zones 1 and 7, 2 and 6, as well as 3 and 5 should be identical. Fig. 8 shows that this is not yet the case even after the simulation of 1,000 batches with 20,000 neutrons each, with MCNP. The deviations of the corresponding zone power densities from the symmetric values also here remarkably exceed the statistical errors estimated by the Monte Carlo program. Even after 1,000 generations with 20,000 neutrons, deviations from reference solution of up to 4 % were found, whereas the evaluated statistical errors did not exceed 0.12 % (1 σ) for each axial interval. After substantially increasing the number of neutrons per generation to 100,000, a significant decrease of the axial asymmetry is observed. This can be seen from Fig. 9, where the deviations from symmetry for the axial intervals are plotted for 20,000 and 100,000 neutrons per generation.

Sensitivity studies with axially variable cell importances showed partly an improvement for the

axial distribution; for the deviations from quadrant symmetry inside a large core, however, there is no improvement possible, since every quadrant has the same importance. Here, more detailed investigations are necessary to find an improved procedure to obtain reliable reaction rate distributions for large systems.

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Fig. 7. Axial power distribution for a pin cell with the length of the active zone of a typical PWR (active length 371 cm). Cold state. Geometry and materials are from the Rowlands UO2 pin cell benchmark. MCNP and KENO Monte Carlo results in comparison with the TWODANT reference solution. 20.000,000 active particle histories (1,000 generations with 20,000 neutrons/generation). Relative uncertainties are given as 0.06-0.12 % for MCNP and 0.12-0.53 % for KENO.

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Fig. 8. MCNP results of the power in seven axial sections for a pin cell with the length of the active zone of a typical PWR (371 cm) as a function of the number of neutron generations (with 20,000 neutrons/generation). Cold state. Geometry and materials are from the Rowlands UO2 pin cell benchmark. The error bars indicate three times the uncertainties given for MCNP.

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Fig. 9. MCNP results for the axial asymmetry of the power in a pin cell with a length of 371 cm after 1,000 generations with 20,000 and 100,000 neutrons/generation. Cold state. Geometry and materials are from the Rowlands UO2 pin cell benchmark.

Monte Carlo Calculations for an RBMK System Another application was an RBMK fuel channel. This consists of 18 fuel pins inside a tube with

an inner diameter of approx. 8 cm filled with boiling water as coolant, surrounded by a graphite block with a side length of 25 cm serving as moderator. Compared with a typical light water reactor, the RBMK is much larger, with a height of the active zone of 700 cm.

The axial power distribution for this assembly was calculated in order to check the convergence

behaviour in comparison to the LWR systems. For this, the coolant density was taken as uniform over the whole length of the fuel channel. In Fig. 10, the deviation from symmetry of the axial power from an MCNP calculation after 20 million neutron histories is displayed. In fact, it can be seen again that in this case, the deviations are much larger (up to 2 %) than the estimated relative uncertainties of 0.09-0.18 % (1 σ). Also for this system, a weak coupling of the lower and upper regions is given. Nevertheless, it turned out that the deviations from symmetry are comparable in size to those observed in LWR systems with only approximate the half axial dimension for the same number of neutron histories (cf. Figs. 3 and 7): The corresponding maximum values were ~2.5 % for the BWR core and ~4 % for the PWR cell. These comparably moderate convergence problems for the RBMK system are due to the fact that in the graphite moderator, the neutrons can cover much longer distances than in a light water moderated system, increasing the coupling of the lower und upper regions of the reactor and partially compensating the effect of the large axial dimension of the RBMK reactor.

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Fig. 10. Deviation from symmetry of the axial power distribution for an RBMK fuel channel cell (700 cm active height), calculated with MCNP. Hot state. 20.000,000 active particle histories (1,000 generations with 20,000 neutrons/generation). Relative uncertainties are given as 0.09-0.18 %. Pin-wise Power Distribution Inside a Large Reactor Core

As a particular application of the Monte Carlo method to large scale reactor calculations, we

attempted to determine the pin-wise power distribution inside the 1,300 MWel BWR described earlier.

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Fig. 11. MCNP results for the pin-wise power distribution in axial fuel assembly sections of 53 cm length located in the central part (upper figure) and near the periphery (lower part) of a large BWR core in cold state; approximately critical control rod pattern. 200.000,000 active particle histories (4,000 generations with 50,000 neutrons/generation). Relative uncertainties are given as 1.3-1.8 % for the central part and 2.6-3.9 % for the interval located near the periphery. (The gadolinium fuel pins are displayed in red, the water pins in green.)

Again, each fuel pin was subdivided axially in seven intervals to account for the axial variation of the power distribution. The calculations were performed with MCNP. Two representative fuel assembly sections inside the reactor were evaluated, one of them in the central region of the reactor, and the other in a peripheral part, to consider regions with both high and low neutron fluxes, and therefore with low and high statistical uncertainties. The x-y positions of these assemblies are marked in blue in the upper part of Fig. 1. For the central fuel assembly, the central axial section was selected, whereas for the peripheral fuel assembly, the second axial section from the bottom was taken for evaluating the pin power distribution. Again, the reactor is in a near critical state, with approximately three out of four absorber elements inserted. The pin power distributions were calculated in fuel assemblies located in uncontrolled four-bundles.

Due to the extremely small spatial regions as compared to the volume of the whole reactor core, a

very large number of neutron histories had to be evaluated to obtain statistical uncertainties which are small compared to the overall variation of the pin power in the considered volumes. In total, 200 million histories were taken for tallying.

In Fig. 11, the calculated pin power distribution is displayed for both of the fuel assembly

intervals. The pronounced tilt in the distribution between two corners of the fuel assemblies, which is particularly visible for the central assembly, is due to the positions of these corners, the one being located closer to a control rod than the other. Low power values are at positions of fuel pins with gadolinium.

The pin powers could be determined with satisfactorily small estimated relative statistical

uncertainties, not exceeding 2 % for the central pin sections, and not exceeding 4 % for the peripheral pin sections, on the 1 σ level. From our earlier findings, one should keep in mind, however, that these estimated statistical uncertainties do not properly account for uncertainties in the power distribution on the large scale, with significantly larger deviations from axial and azimuthal symmetry.

Computational Aspects Concerning the computer requirements for Monte Carlo calculations of differential quantities in

large reactor cores, the following observations were made. Using 128 nodes on a T3E, the total elapsed time for the simulation of 200 millions of neutrons in the large BWR, as it was done for calculating the pin power distributions at selected positions, was 12 hours. A single processor high performance workstation would need approximately two weeks for this task. Parallel processing therefore is necessary to keep the turnaround time of jobs in reasonable time limits. However, the use of MCNP-4C (with PVM as multiprocessor support) on the T3E of the High Performance Computing Centre HLRS in Stuttgart is limited due to the comparably small memory size of 128 MB per node. Larger problems cause difficulties with dynamic memory management. Therefore, for such problems massive parallel systems with substantially more memory are necessary, as it is available at HLRS on the Hitachi RS 8000 with 1GB/processor. The routine application of the PVM version of MCNP on this platform, however, is difficult due to the very limited support for PVM. Here an improvement is expected when an MPI version of MCNP will be available. With appropriate massive parallel computers, the use of Monte Carlo codes, especially MCNP, for large cores and detailed tallying are possible, and a new field of applications can be opened with the direct simulation of reactor problems under operational and accidental conditions.

Conclusions The experience with the calculation of power distributions (and other reaction rate distributions)

for large reactors showed that the calculated values have to be investigated carefully since the errors in the results may be underestimated. This situation should be improved, especially the estimation of statistical errors due to slow source convergence. Of course, this may also require the simulation of much more neutron histories. A use of massive parallel computers is necessary to calculate such problems in reasonable time. Therefore, a part of the MCNP calculations was performed on a CRAY T3E/512 using 64 or 128 nodes. Further investigations with non-symmetric configurations should be performed and compared with converged deterministic transport solutions. Nevertheless, it could be demonstrated that for important design and safety related reactor physics problems 3D full core calculations can be performed for very detailed core models with excellent integral results and promising differential results if large numbers of neutron histories are tracked.

Acknowledgement

This work was supported by the German Federal Ministry of Economics and Labour (BMWA).

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