ISO/DIS 18075:2025(en)

ISO TC 85/SC 6

Secretariat: DIN

Date: 2025-05-22

Reactor technology — Power reactor analysis - Steady-state neutronics methods

Technologie du réacteur — Méthodes stationnaires en neutronique pour l’analyse des réacteurs de puissance

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Content

Foreword v

Introduction vi

1 Scope 1

2 Normative references 1

3 Terms, definitions and abbreviations 1

3.1 Terms 1

3.2 Abbreviations 5

4 Relation to other standards 5

5 Methods of calculation 6

5.1 General 6

5.2 Conditions to be considered 7

5.3 Multi-group cross sections 7

5.3.1 Basic data 7

5.3.2 Preparation of multi-group constants 7

5.3.3 System dependent spectrum calculations 8

5.3.4 Choice of cell and supercell 8

5.3.5 Cell environment 8

5.3.6 Calculation model 9

5.4 Collapse to few-groups and spatial homogenization 9

5.5 Calculation of reactivity, reaction rates, and neutron flux distributions 10

5.5.1 Models 10

5.5.2 Uncertainties and assumptions 10

5.6 Calculation of reaction rates in reactor components 11

5.7 Depletion calculations 12

5.8 Common practices 13

5.8.1 General 13

5.8.2 PWR core physics methods 13

5.8.3 BWR core physics methods 14

5.8.4 LMR core physics methods 15

5.8.5 PHWR core physics methods 17

5.8.6 HTGR core physics methods 18

6 Verification and validation of the calculation system 19

6.1 Overview 19

6.2 Verification 20

6.2.1 General 20

6.2.2 Unit testing 20

6.2.3 Integral testing 21

6.3 Validation 21

6.3.1 Overview 21

6.3.2 Unit Testing 21

6.3.3 Integral testing 22

6.3.4 Code to code comparisons 22

6.4 Biases and uncertainties 23

7 Documentation 23

8 Summary of requirements 23

Annex A (informative) Computer Codes in Common Use 25

Bibliography 32

Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.

The procedures used to develop this document and those intended for its further maintenance are described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the different types of ISO documents should be noted. This document was drafted in accordance with the editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of any patent rights identified during the development of the document will be in the Introduction and/or on the ISO list of patent declarations received (see www.iso.org/patents).

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This document was prepared by Technical Committee ISO/TC 85, Nuclear Energy, Nuclear Technologies, and Radiological Protection, Subcommittee SC 6, Reactor Technology. This document is based on a standard developed by the American Nuclear Society (ANS) of which the current version is ANSI/ANS-19.3-2022 ^[2].

The main changes compared to the previous edition are as follows:

— addition of new terms in Clause 3;

— figures slightly revised;

— 5.8.5 was amended so that CANDU is only an example;

— Correction of internal references;

— Table A.1 corrected;

— editorial amendments in the whole document;

Any feedback or questions on this document should be directed to the user’s national standards body. A complete listing of these bodies can be found at www.iso.org/members.html.

Introduction

The design and operation of nuclear reactors require knowledge of the conditions under which a reactor will be critical, as well as the degree of subcriticality or supercriticality when these conditions change. In addition, knowledge is required of the spatial distribution of neutron reaction rates in reactor components as a prerequisite, for example, for inferring proper power and temperature distributions to ensure the satisfaction of thermal-limit and safety-limit requirements. Both reaction-rate spatial distributions and reactivity can be and have been measured by suitable experimental techniques, either in mock-ups or in the operating reactors themselves. These quantities can also be calculated by various techniques. Available reactor experimental data have been used to validate the steady-state neutronic calculations within reasonable margins. As more accurate nuclear cross sections and improved calculation methods have become available, steady-state neutronic calculations have been utilized extensively for nuclear fuel and core designs and analyses and thus have become increasingly important.

Steady-state neutronics methods for power reactor analysis

This document provides guidance for performing and validating the sequence of steady-state calculations leading to prediction in all types of operating commercial nuclear reactors, of the following:

— reaction-rate spatial distributions;

— reactivity;

— change of nuclide compositions with time.

The document provides the following:

a) guidance for the selection of computational methods;

b) criteria for verification and validation of calculation methods used by reactor core analysts;

c) criteria for evaluation of accuracy and range of applicability of data and methods;

d) requirements for documentation of the preceding.

There are no normative references in this document.

ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https://www.iso.org/obp

— IEC Electropedia: available at https://www.electropedia.org/

3.1.1

application-dependent multigroup

discrete energy-group structure that is intermediate between the application-independent multigroup structure and a few-group structure

Note 1 to entry: The application-dependent multigroup structure can be such that the group constants are dependent on reactor composition through an estimated neutron energy spectrum. An application dependent multigroup data set is one type of averaged data set.

3.1.2

application-independent multigroup

discrete energy-group structure that is sufficiently detailed that the group constants may be considered as being independent of reactor composition, geometry, or spectrum for a wide range of reactor analysis

Note 1 to entry: The application-independent multigroup structure can be employed directly in reactor-design spectrum calculations, or it can be employed to generate group constants in an application-dependent multigroup structure. An application-independent multigroup data set is one type of averaged data set.

3.1.3

averaged data set

data set prepared by averaging an evaluated data set or a processed continuous-energy data set with a specified weighting function over a specified energy group structure

Note 1 to entry: The group structure and weighting functions may be selected to be application dependent or application independent, e.g, light water reactors (LWRs), are dealt with in ANSI/ANS 19.1 2019^[1]

3.1.4

cell

supercell

one or more reactor components with associated coolant (and possibly additional moderator and structural material) that, for computational purposes, are assumed to form a spatially repeating array in the reactor

Note 1 to entry: The simplest example of a cell is the “pin cell” in which a single fuel rod or pin is surrounded by coolant (e.g. light water, heavy water, or sodium). Another example is a bundle of fuel rods cooled by heavy water within a housing, surrounded by a heavy water moderator space.

Note 2 to entry: More complex geometric configurations are also used for some applications. These are often referred to as “supercells”, or sometimes “(fuel) assembly cells”, although the exact definition of the term varies greatly between reactor types and is even somewhat subjectively defined for a particular reactor type. Supercells, in the context of this document, represent more complex “cell” configurations that involve a collection of contiguous cells forming an assumed repeating array within the reactor, or augmented cells incorporating additional regions to serve as a computational artifice, e.g. to account for significant spectrum effects due to compositions outside the cell, or cell configurations including a reactivity device in addition to fuel, coolant, moderator and poison.

3.1.5

collapse

method by which a many energy group set of cross sections is reduced to fewer energy groups

3.1.6

data set (or dataset)

collection of microscopic cross sections and nuclear constants encompassing the range of materials and reaction processes needed for the application area of interest.

3.1.7

dilution cross section

cross section, which refers to a background cross section that represents the effects of other isotopes in a material, particularly in the context of self-shielding in nuclear reactions

3.1.8

evaluated data set

data set that is completely and uniquely specified over the ranges of energy and angles important to reactor calculations

Note 1 to entry: Such a data set is based upon available information (experimental measurement results and nuclear theories) and employs a judgment as to the best physical description of the interaction processes. An evaluated data set is intended to be independent of reactor composition, geometries, energy group structures, and spectra.

3.1.9

experimental data

any experimentally measured quantity or quantities

Note 1 to entry: As such it is applied herein to both differential cross-section measurements and integral measurements (e.g. control-rod worth) obtained from reactor experiments or operations.

3.1.10

few-group

energy-group (typically two-group) structure that is adopted for a particular application

Note 1 to entry: The few-group constants for a region are dependent on a specific reactor composition and geometry through a calculated energy spectrum, and are also dependent on other conditions such as pressure, temperature, Xenon concentration, void fraction, etc.

3.1.11

fine group

energy group structure (typically hundreds of groups for LWR, thousands for LFMR) that is adopted for a particular application.

Note 1 to entry: The fine-group constants are dependent on temperature.

3.1.12

lattice

normally refers to an array of cells

3.1.13

lumped pseudo fission product

fictitious fission product that is used in depletion calculations to represent a combination of certain minor fission products that are not tracked in an explicit fission product model

Note 1 to entry: A lumped pseudo fission product is characterized by effective neutron cross sections and decay constants.

3.1.14

methods of calculation

mathematical equations, approximations, assumptions, associated numerical parameters, and calculational procedures that yield the calculated results

Note 1 to entry: When more than one step is involved in the calculation, the entire sequence of steps comprises the “calculation method”.

3.1.15

multigroup

more than a few groups, can be hundreds or thousands of energy groups

3.1.16

probability table parameters

parameters for a probability table that is used to process cross sections in the unresolved resonance energy range

3.1.17

processed continuous-energy data set

data set prepared by expansion or compaction of an evaluated data set using specified algorithms

Note 1 to entry: Such a data set is intended to be independent of reactor composition, geometries, and spectra.

3.1.18

quasi-steady state condition

evolution of reactivity and flux is sufficiently low between two time intervals such that we can consider that the core is under a succession of static states.

3.1.19

reaction rate

rate at which neutrons interact with nuclei at a point in the reactor, a volume of the phase space or as an integral over the reactor, according to the context

EXAMPLE The reaction rate for absorption, scattering, and fission.

3.1.20

reactivity

reactivity means the quantity (1 minus the eigenvalue, λ, of the steady-state neutron balance equation), written as

where

Φ is the neutron flux;

F is the neutron fission yield operator; and

M is the scattering, absorption, and leakage operator.

Note 1 to entry: The effective multiplication factor k_eff is the inverse of λ. Reactivity is a unitless, pure number. It is, however, often written in terms of smaller “units”, such as milli-k = 0,001, pcm = 0,000 01 = 10⁻⁵ or “dollars” (and cents), where 1 dollar is taken as the value of the delayed neutron fraction in the system of interest.

3.1.21

time-average model (CANDU)

time-average cell-homogenized few-group macroscopic cross sections at each bundle location in the core are calculated as flux-weighted quantities over the fluence range experienced by the fuel during its residence at that specific location

Note 1 to entry: Consequently, the flux, fluence range, and macroscopic cross sections at each location are linked together and depend on the individual refueling rate of the channel in which the fuel bundle resides. The resulting self-consistency problem is solved iteratively to determine all the results of the time-average model.

Note 2 to entry: The CANada Deuterium Uranium (CANDU) reactor time-average model is also known as the equilibrium-core model.

3.1.22

ultrafine group

energy group structure (typically ten thousand energy groups) that may allow to explicitly describe the resolved resonances

Note 1 to entry: The ultrafine-group constants can be dependent on temperature when they include narrow-resonance reactions.

3.1.23

experimental validation

process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model

3.1.24

numerical validation

process of determining the degree to which a model is accurate by comparison to a previously validated code system

3.1.25

verification

process of determining that a model implementation accurately represents the developer’s conceptual description of the model and the solution to the model

The following American National Standards are related to this document:

— Nuclear Data Sets for Reactor Design Calculations, ANSI/ANS-19.1-2019,^[1] defines the criteria to be employed in the preparation of application-independent cross-section data files from experimental data and theoretical models. This document covers subsequent space and energy averaging processes that may be employed to prepare cross sections for use in the representation of the core and its environment, and the subsequent calculation of the spatial distribution of neutron reaction rates in the core and of the core reactivity. There may be many ways of carrying out the space and energy averaging to obtain few-group cross sections, and no unique path for the preparation or use of cross sections employed in design calculations is defined, required, or recommended by this standard.

— A Guide for Acquisition and Documentation of Reference Power Reactor Physics Measurements for Nuclear Analysis Verification, ANSI/ANS-19.4−2017 (R2022) ^[3]; and Requirements for Reference Reactor Physics Measurements, ANSI/ANS 19.5-1995; W2005^[4].

Validation of calculation systems requires comparison with available experimental results. The preceding standards contain criteria for performing and documenting such experiments, in order to be most useful for this purpose.

— Determination of Thermal Energy Deposition Rates in Nuclear Reactors, ANSI/ANS 19.3.4−2002 R2017,^[5] provides criteria for the establishment of the thermal energy deposition rate distribution within a nuclear reactor core. Since the accuracy with which this can be done is dominated by the accuracy with which neutron reaction rates can be calculated, ANSI/ANS-19.3.4–2002; R2017 is closely related to ANSI/ANS−19.3−2022.

— Quality Assurance Program Requirements for Nuclear Facility Applications, ANSI/ASME-NQA 1 2022.^[6] This standard deals with quality assurance, including that for computer programs.

— Verification and Validation of Non-Safety Related Scientific and Engineering Computer Programs for the Nuclear Industry, ANSI/ANS-10.4−2008 (R2021).^[9] This standard deals with requirements for verifying and validating computer codes, such as those used for neutronics calculations.

— Accommodating User Needs in Scientific and Engineering Computer Software, ANSI/ANS-10.5−2006; R2021.^[10] This standard deals with methods to respond to users’ requirements in computer programs.

Calculations within the scope of this document would typically be performed in a sequence of steps. A typical sequence might be:

a) Utilization of averaged-data set cross sections, nuclide number densities, and geometrical information (usually repeating cells or supercells) to calculate an application-dependent many-group neutron spectrum and within-cell spatial distribution for each different reactor region and composition.

b) Utilization of the spectra and spatial distributions from step (a) to collapse and homogenize averaged-data set cross sections to few-group form.

c) Utilization of the few-group cross sections from step (b) and geometrical information about the reactor to calculate reactivity and few-group flux spatial distributions in the reactor.

d) Utilization of the preceding information to compute reaction rates in physical reactor components (including the reactor vessel if needed).

e) Utilization of the preceding information to calculate changes in nuclide composition of fuel and possibly other reactor components with exposure.

In some applications, the collapse to few group form is not used, and the multigroup cross sections and geometrical information about the reactor are used to calculate reactivity and multigroup spatial distributions in the reactor.

Not all steps in the sequence would normally be executed for a given problem. It is not a requirement of this document that a particular sequence of calculations, such as the one previously listed, be used. Similarly, the use of the preceding sequence does not, in itself, demonstrate compliance with this document. The use of a specific calculation procedure shall be justified by the procedure presented in Clause 6. However, the preceding sequence does provide an adequate framework within which most of the problems in steady-state reactor physics calculations can be discussed. Therefore, each of the aforementioned steps will be discussed in later passages of this subclause.

A summary of the requirements of this document is given in Clause 8.

Consideration shall be given to all conditions that significantly affect the calculated quantities. The method of calculation shall be capable of treating the reactor composition or configuration under the conditions being studied.

Important conditions that may be significant include, but are not limited to the following:

a) presence of control elements (rods, cruciforms, or other forms), and degradation (or depletion) of the effectiveness of control elements;

b) presence and spatial distribution of burnable or soluble absorbers;

c) presence of adjacent, unlike fuel assemblies;

d) composition and geometric layout of fuel in an assembly;

e) dependence of coolant or moderator density upon conditions, or their spatial dependence;

f) depletion dependent conditions, including previous power history, coolant-density history, control-element history, and soluble-absorber history of fuel assemblies;

g) presence of materials or conditions, or both, outside the core, such as the core shroud in a boiling water reactor (BWR);

h) presence of sources, detectors, structural materials, and experimental devices;

i) spatial variations in temperatures (e.g., fuel temperature, coolant temperature, and moderator temperature);

k) spatial and temporal variations of important nuclides (e.g., xenon, samarium, and actinides).

The primary sources of basic nuclear data that are used for the generation of multi-group constants are evaluated data sets. Examples of these are the Evaluated Nuclear Data File Version B (ENDF, /B, see[11],[12] and[17]), Japanese Evaluated Nuclear Data Library (JENDL, see[15] and[16]), Russian Library of Evaluated Neutron Reaction Data (BROND, see[14] and[18]), Joint Evaluated Fission and Fusion Library (JEFF, and see[13]), and Chinese Evaluated Nuclear Data Library (CENDL, see[19]) evaluated data sets. The properties and criteria for selecting these sources of basic nuclear data are specified in ANSI/ANS 19.1-2019^[1].

Processing evaluated data sets

When preparing multigroup constants directly from evaluated data sets, the procedures for the preparation of averaged data sets described in^[1] should be followed. The multigroup constants can be sensitive to the selection of an energy-dependent weighting spectrum and to the choice of group structure. The smaller the number of energy groups, the greater the sensitivity will be. Therefore, an estimate of the reactor spectrum is needed and should be obtained from measurements in identical or similar reactors or from analytical models of neutron slowing-down or source spectra. Results may be sensitive to the modeling of the spectra.

Collapsing application-independent averaged (or multigroup) data sets

The preparation of application-dependent multi-group constants from existing application-independent multi-group constants shall entail use of an application-dependent energy spectrum estimate (see 5.3.3 and 5.4). This procedure employs a weighting spectrum that is selected to preserve important system-dependent characteristics during the averaging process. These characteristics usually include reaction rates, and may include other quantities.

The multi-group cross-section set (see 5.3) should be used in the calculation of the neutron energy spectra in the system under investigation. The energy spectra are established by the geometry, material composition, and operating conditions of the reactor in an interplay of neutron leakage with reactions such as absorptions and scattering. The neutron energy spectrum may vary from one region of the core to another and it may be necessary to compute the spectra for several representative regions of the reactor core.

Many reactor cores can be thought of as composed of repeating units called cells, such as a single fuel pin cell or a fuel assembly cell (this formalism can be extended to absorber pins or water holes), with its associated structures, coolant, and moderator (where this is distinct from the coolant).

Once a cell is selected, one approach is to compute the spectrum representative of this cell. It is necessary to inspect the cell and its surroundings to determine if the spectrum in the cell is generated by the cell and its similar surroundings alone, or if the spectrum in the cell is influenced by parts of the reactor not made of similar cells. When the spectrum is influenced by regions of the core external to the cell, a supercell may be defined, and the spectrum characteristic of the supercell is computed. The supercell may be a repeating unit of the reactor containing noncell materials such as water channels, control-rods, and structural materials. Other noncell regions such as absorber pins, when present, should be included in the supercell if they significantly influence the spectrum. For either a cell or a supercell, outer boundary conditions are specified to be consistent with symmetry assumptions.

The assumption that a reactor is made of an array of similar cells or supercells is, at best, an approximation, and if the spectrum in the cell is influenced by external regions, these effects should be included in the spectrum calculations. These effects may be caused by leakage across the cell or supercell boundaries and thus may be energy and direction dependent. Temperature effects in fuel (e.g., Doppler broadening), temperature effects in the moderator and/or coolant, and variations in density or composition of coolant and/or moderator shall be included in the calculation. Corrections for a nonuniform temperature distribution within the cell should be made, or the temperature distribution should be included in the calculation.

The environment can be explicitly modeled in the case of lattice calculations.

General

The calculation model of the cell or supercell often can be considered to have two aspects - the geometric model and the neutronic transport model, though the two aspects may not be clearly separable.

Geometric model

The geometric model refers to the manner in which the physical configuration of the cell is represented in the mathematical solution. Geometric approximations may be employed when all aspects of the physical configuration are not of comparable importance, the primary objective being to reduce the number of dimensions employed in the solution of the problem or to transform to a more convenient or simplified geometry. Different geometric approximations may be made concerning the same physical configuration for different purposes. The choice of geometric models appropriate to the analyses shall be justified and documented.

In some calculations, one geometric dimension of the model may be dropped, as long as the leakage in the missing direction is taken into account by the judicious inclusion of a buckling or leakage term that stands as the surrogate of the missing leakage.

Neutronic transport model

Various calculation procedures may be utilized to describe neutron transport phenomena in cell studies. Different degrees of approximation may be made depending on the nature of the problem and the objectives of the calculation. A very detailed type of calculation is continuous-energy Monte Carlo. This statistical procedure follows “histories” of large numbers of individual neutrons. Initially, this technique has served primarily as a guide to the accuracy of other procedures but may be used in mainstream applications as well.

Other transport models generate numerical solutions (by collision probability methods, for instance) of the transport equation. Approximations also are introduced in representing energy-transfer kernels. The analyst shall demonstrate and document that the transport model used is appropriate to the problem under consideration. For example, the analyst shall demonstrate that the spatial mesh, the order of scattering (P1, P3, etc.), and the order of quadrature (in Sn methods) are adequate to achieve stated accuracy levels for the calculated reactivity and reaction rates.

When performing full reactor calculations, it is usually adequate and desirable to collapse the cross sections from the multi-group structure into a few groups. The actual group structure chosen should depend on the type of calculation that is to use the few-group data and the sensitivity of that calculation to the group structure.

It is usually desirable to also convert the heterogeneous unit cells into equivalent spatially homogeneous cells for use in full reactor calculations.

When collapsing cross sections to few groups and performing spatial homogenization, important system characteristics – such as reaction rates in a unit-cell, reactivity of the cell and core, or reaction-rate ratios – should be preserved to the extent practical. This preservation is an attempt to maintain an equivalence between the many group heterogeneous calculation and the coarser few group homogeneous calculation. The actual quantity or quantities preserved and the method of doing this should depend on the intended use of the few-group data.

The calculation used in the collapse shall include or approximately account for all important effects of space and energy that cannot be adequately modeled in the calculations to follow, such as self-shielding and spectrum dependence on surrounding materials.

The cross sections of each nuclide present, to a significant degree, shall be retained individually whenever calculations of individual reaction rates are to be carried out. These cross sections should also be the starting points for depletion calculations (e.g., calculations of changes in nuclide composition with time).

The calculations being considered in this section have as their objective the computation of a measure of closeness to criticality of a specified reactor-core configuration, and the reaction rates as a function of position in the core under a steady-state or quasi steady-state condition. A number of models may be used for this purpose.

A frequently used measure of closeness to criticality is the effective multiplication factor (k_eff). This is appropriate, for example, in describing the closeness to criticality of a reactor in its shut-down condition. However, most steady-state reactor calculations are intended to represent conditions at critical or an artificial steady-state for the purpose of calculating reactivity margins or reactivity coefficients. In addition, code-system bias and uncertainties may lead to a non-unity k_eff as a reference point. The definition of reactivity, ρ, in this document is then (1-1/k_eff).

For the purpose of discussion in this subclause, it is assumed that cross sections for all regions of the reactor have been generated in multi-group or few-group homogenized form by the techniques described previously. A number of models are in common use for performing neutron-flux calculations. Some of these are:

a) solving the diffusion equations by finite-difference, finite-element, or other methods;

b) solving the transport or simplified transport equations by discrete-ordinates or collision-probability methods or by the method of characteristics;

c) solving the reactor neutron balance equations and reaction rate distributions by nodal or other methods.

Thermohydraulic and thermal models linked with neutronic simulation are needed for core calculation.

The preceding examples of models are by no means exhaustive of models that may be used. However, they are sufficient to illustrate the variety of methods being used, each of which may have characteristic types of uncertainties and assumptions.

Usually, the model used to describe neutron transport in the reactor calculation is an approximation to a more accurate model. For example, diffusion theory is an approximation to transport theory. Thus, there will be some error because of the model per se. In the implementation of a model via a computer program, it is common for additional approximations to be made. For example, it may be assumed that the neutron flux or current remains constant over small areas or along small line segments. Thus, the solution produced by the computer program will be an approximation to the solution of the model equations.

The following are examples of many modelling assumptions or approximations that are commonly made, and that may contribute to a calculation bias and/or to uncertainties:

a) the assumption that neutron flux in the core as a function of all three spatial dimensions may be represented as the product of functions that separately are a function of only one or two dimensions (spatial separability) although this assumption is used less extensively today;

b) geometrical transformations used to model the physical situation;

c) the use of artificial boundary conditions within the core (e.g., at the boundaries of heavily-absorbing control slabs or cruciforms);

d) assumptions of symmetry for configurations that are not precisely symmetric;

e) the choice of a small number of energy groups to represent the neutron energy variation in the core;

f) the assumption of linearity or simplified variation of flux between the spatial points within a spatial mesh structure, the dimensions of which may be specified somewhat arbitrarily;

g) the choice of a limited number of directions or spherical harmonics moments to represent the angular variation of the flux in transport or approximate transport solvers;

h) the use of bucklings to simulate leakage effects in the directions not explicitly represented;

i) the assumption that dissimilar media may be homogenized;

j) the use of pre-calculated region-homogenized (typically lattice-homogenized) cross sections at a predefined power history;

k) the use of interpolation or curve fitting techniques for the calculation of cross sections at local conditions.

All of the preceding assumptions or approximations are, in principle, amenable to numerical studies aimed at establishing the deviation of the normally used procedures from more precise solutions of the model equations. Numerical methods should normally be used only within the range of parameters for which the biases or uncertainties of the methods are known.

When a model that simplifies the physical description is used, means shall be provided to convert the results of the model calculation into reaction rates in the physical components as required by the application. For the reaction rate calculations to be valid, the cross sections shall be a function of temperature, density, and other core conditions. If the calculation procedures make use of simplifying assumptions, such as separability of local and overall flux variations, the procedure shall specify how local reaction rates are obtained, and the basis or justification for the technique employed shall be described.

The reaction rates thus calculated are used for a variety of purposes. Some examples are:

— computation of heat generation rates for heat transfer and thermal hydraulic calculations, which, in turn, are used to verify thermal limits;

— computation of change in nuclide composition of fissile-nuclide-bearing materials as a function of position in the core in order to predict fuel inventory;

— computation of shutdown margins, control-rod worths, and reactivity worths of other components;

— computation of the relationship between detector response and in-core or ex-core reaction rates.

In a critical reactor, the rate of change in the concentration of a nuclide is the difference between the rate of production of that nuclide and the rate of destruction of that nuclide. The most significant production mechanisms are neutron capture in the transmutation precursor, the decay of another nuclide, and direct fission yield (in the case of fission products). The most significant destruction mechanisms are fission, neutron capture in the nuclide, (n, xn) reactions, decay of the nuclide, and fission gas release to the plenum. Yields of fission products, including lumped pseudo fission products, and decay constants of the nuclides of interest are basic nuclear data that are used by any such calculation.

One common procedure is to assume that the supercell representation (or lattice calculation) and associated neutron flux spectrum have adequate accuracy to allow depletion calculations. The accuracy also depends on the number of rings modelled in the rod. The neutron spectrum is used in summing the product of the cross sections and fluxes into total fission, capture, and (n, xn) reaction rates. These, together with the fission yields and decay constants, provide the production and loss terms for each nuclide. The result is a set of coupled differential equations for the concentration of all nuclides of interest. These equations are solved simultaneously by numerical methods, through the use of discrete time steps. The equations may be reduced to linear equations, since this is appropriate to the solution technique. The time steps shall be set sufficiently small to ensure numerical stability of the solution technique, and accuracy appropriate to the application. It is also necessary to ensure that the flux level in depleting regions does not rise so rapidly that the required time step length becomes exceedingly short. In such cases the numerical solution method shall include estimates of the variation of the flux level within the chosen time step length.

In order to lengthen the permissible time step, it is common practice to assume that some nuclides with very large decay constants are in equilibrium at all time steps. The choice of these nuclides should be validated to be consistent with the intended applicability of the results.

Fission, capture, and (n, xn) reaction rates near the beginning of exposure are normally obtained from the zero exposure cross sections of all nuclides of interest. As the depletion calculation proceeds from zero exposure, concentrations change and group averaged cross sections may change as a result. Both spectrum changes and changes in resonance self-shielding factors contribute to these cross-section changes. Thus, it may be necessary to recalculate the supercell or lattice spectra with updated concentrations at intervals in order to obtain group averaged cross sections and other parameters such as diffusion coefficients as a function of exposure, number density, etc.

The interval between supercell or lattice spectrum calculations is normally under the control of the analyst. The form in which this can be specified may vary; e.g. fuel exposure interval, or percent change in concentration of a specified nuclide (usually ²³⁵U). The procedure for selecting a satisfactory interval is straightforward. The interval is reduced in successive calculations until the differences in concentrations are less than the expected uncertainty for the intended application. This procedure shall be used in the absence of established criteria that can be justified for the application.

The few-group cross sections obtained from the supercell or lattice calculation may be used to perform a spatial reactor depletion calculation. The cross sections may be calculated as functions of exposure, nuclide concentrations, and/or other variables (such as moderator density and temperature, fuel temperature, void fraction, soluble boron concentration in LWR, and presence of a control device) or may be retained as functional fits or as tabular data with interpolation. Extrapolation of data is not recommended. If there is no alternative, extrapolation should be used very carefully and a posteriori validated.

The preceding discussion is applicable if it has been judged that the supercell or lattice representation and spectrum have adequate accuracy to allow proceeding to depletion calculations through the use of macroscopic cross sections.

This assumes that the number-density ratios will not change with time as the core depletes with locally and possibly globally varying power level, moderator temperature, void distribution, etc. If this cannot be justified, a more accurate approach shall be used.

The supercell or lattice calculation is used to generate few-group cross sections as a function of exposure, nuclide concentrations, and/or other variables, which are retained for use in the few-group reactor core calculation. The depletion equations may then be solved as before, or individually for each explicit nuclide. In the first case the macroscopic cross sections will include dependence on separately defined history variables, such as power level, moderator temperature and density, fuel temperature, ¹³⁵Xe concentration, etc.

This procedure will produce different results to the extent that the few-group spectra from the reactor calculation differ from the few-group spectra in the supercell calculations.

Experience with pressurized water reactors (PWRs), BWRs, pressurized heavy water reactors (PHWRs), high-temperature gas-cooled reactors (HTGRs), and liquid metal reactors (LMRs) is sufficiently extensive that a set of common practices has been developed for each. General descriptions of these practices are given in the following passages^[1]. The fact that a particular reactor or reactor type is not included in these passages does not imply that the methods used in its design or operation do not fulfill the requirements of this document. The requirements of this document are delineated in Clause 8.

The accuracy of the various models can be checked by experimental and numerical validation.

There are in general three distinct steps in steady-state PWR reactor physics analysis: library generation, assembly lattice calculations, and reactor core calculations. The purpose of these calculations is to provide the basis for reactor core design, including safety and economic evaluations. Figure 1 illustrates the sequence of such analyses and their relationships to each other.

Figure 1 — General flow of data for PWR physics calculations

Reactor physics analysis begins with continuous-energy or multi-group cross-section libraries and other nuclide data that have been generated by the processing of data from a version of an evaluated data set. These multigroup libraries typically contain 40 to 200 or more energy groups with tables of thermal cross sections and resonance integrals as a function of temperature and dilution cross sections. Libraries may also contain the probability table parameters. Thermal cross sections are generated through the use of a “generic” PWR spectrum. The cross sections in such libraries may or may not have been modified subsequent to the processing from evaluated data in order to improve agreement between downstream models (lattice and nodal models) and measurements. These libraries, as shown in Figure 1, provide the basis for multigroup lattice physics calculations.

PWR fine-group lattice physics calculations typically are two-dimensional transport theory calculations that employ a detailed energy and spatial mesh. They usually are performed for a single lattice, although they may involve several lattices for special-purpose applications, such as for calculating reflector properties. These calculations have three separate but complementary components: fine-group pin cell calculations, intermediate-group lattice calculations, and nuclide depletion calculations. Initially, ultrafine-group thermal and epithermal calculations are performed for the individual fuel pins, or fuel pin types and water rods or water holes, and structural components within the lattice. Fuel rods containing integral burnable poisons are usually modelled in greater detail and/or with more precise methods.

The pin cells may use either a heterogeneous detailed spatial model or a homogenized cell model, and fine-group cross sections for the cells are edited from the results of the ultrafine-group calculations. Next, fine-group calculations are performed for the lattice as a whole, with each cell or detailed region retaining its own identity. Usually, both the ultrafine-group calculations and the fine-group calculations are based on transport theory. The number of energy groups in the fine-group calculation may range from as few as six to as many as are present in the library. Few-group cross sections then are edited from the fine-group calculation for a homogeneous representation of the lattice. Finally, depletion calculations are performed for the nuclides whose concentrations change with time, including lumped pseudo fission products. More sophisticated methods are required to calculate burnable absorber depletion properly.

The spatially homogenized and energy-collapsed cross sections edited from the fine-group calculations are used as input parameters for nodal or detailed transport calculations. Consequently, different sets of fine-group calculations are performed for different combinations of thermal-hydraulic variables, fuel exposure, soluble boron concentration, xenon concentration, nuclide decay (e.g., shutdown time), and presence of control rods. The values of individual variations generally are chosen so that they span the range of conditions expected to be encountered during core operation.

Nodal core calculations are performed for three-dimensional homogenized core analysis. Individual three-dimensional fuel assembly cells are divided axially into stacks of homogenized nodes, each of which is typically 10 cm to 30 cm high. Nodal calculations are largely based on diffusion theory with two or more energy groups. The nodal codes account for significant thermal-hydraulic feedback and presence of control rods in their cross sections and typically include explicit nuclide depletion for actinides, major fission products, and burnable absorbers within each node.

There are in general three distinct steps of BWR reactor physics analysis: library generation, lattice calculations, and nodal calculations. Figure 2 below illustrates the sequence of such analyses and their relationship to each other.

Figure 2 — General flow of data for BWR physics calculations

The general sequence of BWR calculations is the same as for the PWR counterparts, although the calculations differ in some particulars. Most of the application independent fine-group libraries for BWR applications are generated by processing data from a recent version of an evaluated data set. These libraries typically contain 40 to 200 or more energy groups. The nuclide cross sections within a specific group structure are processed through the use of a “generic” BWR energy spectrum. These libraries, as shown in Figure 2, provide the cross-section data for fine-group lattice physics calculations.

BWR fine-group lattice physics calculations typically are two-dimensional transport theory calculations that employ a fine spatial mesh. They usually are performed for a single lattice, although they may involve several lattices for special-purpose applications, such as for calculating reflector properties. These calculations have three separate but complementary components: fine-group pin cell calculations, intermediate-group lattice calculations, and nuclide depletion calculations. Initially, fine-group thermal and epithermal calculations are performed for the individual fuel pins, or fuel pin types and water rods or water holes, and structural components within the lattice. Fuel rods containing integral burnable poisons are usually modeled in greater detail and/or with more precise methods. These fine-group calculations typically employ the same energy group structure as the cross-section library. The pin cells then are homogenized, and intermediate-group cross sections for the homogenized cells are edited from the results of the pin cell calculations. Next, intermediate-group calculations are performed for the lattice as a whole, with each homogenized pin cell retaining its own identity. Usually, both the pin cell calculations and the intermediate-group calculations are based on transport theory. The number of energy groups in the intermediate-group calculation may range from as few as six to as many as are present in the library. Few-group cross sections then are edited from the intermediate-group calculation for a homogenized representation of the lattice. Finally, nuclide depletion calculations are performed for those nuclides whose concentrations change with time, including lumped pseudo fission products. More sophisticated methods are required to calculate burnable absorber depletion properly. The few-group cross sections edited from the intermediate-group calculations are used as input to nodal calculations. Consequently, different sets of intermediate-group calculations are performed for different combinations of thermal-hydraulic variables, fuel exposure, nuclide decay (e.g., shutdown time), and control rod state. The values of individual variations generally are chosen so that they span the range of conditions expected to be encountered during core operation.

Nodal core calculations are performed for three-dimensional homogenized core analysis. Individual bundles are divided into stacks of homogenized nodes, each of which typically is 10 to 30 cm high. Nodal calculations are based on diffusion theory. Although effective one-group nodal models that preserve the fundamental-mode multiplication factor (k_eff) of two- or three-group nodal models are still in use, many now employ two or three-group advanced nodal models for this purpose. The nodal codes account for significant thermal-hydraulic feedback into their cross sections and may include explicit nuclide depletion for actinides, major fission products, and burnable absorbers within the node. However, most current codes use “macroscopic” depletion models (macroscopic cross sections parameterized as functions of exposure, instantaneous void fraction, void fraction history, fuel temperature, control rod presence, and control history) and treat nuclide behavior of only ¹³⁵I, ¹³⁵Xe, and (perhaps) ¹⁴⁹Pm and ¹⁴⁹Sm explicitly.

Feedback effects between local power and moderator density (void) are prominent in BWRs. Consequently, the flow distribution for the reactor core calculation, whether input or internally calculated, should correspond to a condition where the core pressure drop is the same for all parallel flow channels. It is important that all pressure drop effects and local void formation be accurately modeled. These variables strongly influence the reaction rate distributions and reactivity of the core.

The steps for carrying out the reactor physics analysis of an LMR are illustrated by the flow diagram shown in Figure 3. This diagram resembles those for LWRs. However, greater detail needs to be included in the energy structure used for the reactor calculations, as the essential reactions occur in a wider energy range and as the energy spectrum changes significantly with design detail and spatial location in the core.

The generation of fine-group cross sections from evaluated nuclear data files also serves as the starting point for LMR analysis. Fine-group constants are commonly generated for a fine-energy-group structure (50 to 250 energy groups) through the use of one of two different approaches.

In the ultrafine-group approach, “smooth” cross sections for a very fine energy structure (2 000 or more energy groups) are first processed from the evaluated data by averaging the cross sections in each small energy interval. The broad structural resonances are explicitly represented in the ultrafine-group constants, while the narrow resonances are excluded and treated separately. Infinite-medium ultrafine-group spectra are then computed for the ultrafine energy structure through the use of representative fast reactor compositions; the resonance reaction contributions are evaluated using the narrow resonance approximation. In the unresolved resonance range, another pathway is to use probability tables owing to the neutron and gamma width fluctuations. This is important to account for self-shielding in this energy range. Fine-group cross sections are then generated through the use of the detailed smooth reaction rates and resonance reaction rates, along with the ultrafine energy spectrum.

Figure 3 — General flow of data for LMR physics calculations

In the second approach, fine-group cross sections for a particular application are generated through the use of the shielding factor methodology. This method utilizes an application-independent master cross-section library, as described in ANSI/ANS 19.1 2019.^[1] Such a cross-section library contains both group-averaged cross sections (averaged over a representative fast reactor spectrum) and tables of group-¬dependent resonance self-shielding factors, which together are used to prepare cross sections that are properly resonance self-shielded, for specific conditions of interest. The data are normally provided in a group structure of 50 to 250 energy groups. The resonance self-shielding factors for each resonant isotope and each energy group are normally provided in table form as a function of the concentration of the resonant isotope (represented by the background cross section) and its temperature. These shielding factor data are combined with the “unshielded” (infinitely dilute) cross sections by a cross-section preparation code that calculates the appropriate background cross sections for each reactor region and performs the table interpolation to obtain the appropriate resonance self-shielding factors.

Because the neutron mean free path is large compared to the small pin diameter in most fast reactor systems, heterogeneity effects are of relatively minor importance. However, heterogeneity corrections are commonly applied to the multigroup cross sections for the appropriate assembly design. In the ultrafine group approach, for a given pin-cell geometry, heterogeneity effects are generally evaluated through the use of an integral transport calculation on a fine energy mesh. In the shielding-factor method, heterogeneity effects on resonance shielded cross sections are commonly incorporated based on equivalence theory during the preparation of application dependent cross sections for a specific composition and temperature.

Few-group cross sections appropriate to the various reactor regions are ordinarily generated by collapsing the fine-group cross sections (prepared by either the ultra-fine group approach or the shielding factor method) through the use of neutron energy spectra obtained from one or more spatial calculations. The collapsing calculation utilizes a neutronics model in which representative reactor materials are placed in a simplified configuration. Few-group cross sections are generated for multiple spatial regions where the material compositions and neutron energy spectrum remain roughly constant.

Reactor calculations of reactivity effects, neutron reaction rates, and flux distributions are performed with three-dimensional nodal (or finite difference) diffusion theory methods; assemblies are radially homogenized; and corrections for transport effects are sometimes obtained from two dimensional transport codes in r z, x y, or triangular geometries. Three-dimensional transport codes (deterministic and Monte Carlo) are also being utilized with increased frequency for LMR analysis.

The fuel depletion calculations are carried out through subdivision of the burn cycle into one or more subintervals, and performance of an explicit depletion calculation in each region of the reactor over each subinterval through the use of average reaction rates over the subinterval. These average reaction rates are based on three-dimensional fluxes calculated at both the beginning and end of the subinterval. The isotopes to be considered in the depletion equations, as well as their transmutation reactions, are specified by the analyst. The depletion calculations are performed for radially homogenized models of the assemblies; pinwise power densities and nuclides are usually “reconstructed” by interpolation of assembly wise quantities.

A generic model for PHWR reactor physics analysis is shown in Figure 4. CANDU is an example of a PHWR. Reactor physics analysis begins with multigroup, microscopic cross sections or with cross sections based on analytical spectra and with other nuclide data that have been collected into one or more cross-section libraries. These libraries usually have been processed from extensive tabulations. The cross sections in such libraries may have been modified subsequent to processing from evaluated data in order to improve agreement between measurements and results computed by downstream lattice models. The general methodology for PHWR reactor physics calculations is a 3-step process.

Figure 4 — General flow of data for PHWR physics calculations

a) The first step consists of lattice calculations for the bare lattice, i.e., for basic unit-cells containing fuel, coolant, pressure and other tubes if present and the surrounding moderator volume, but excluding the representation of any interstitial reactivity devices. The lattice calculation is usually performed for a single cell (or sometimes for multiple lattice cells) in two (or sometimes three) dimensions with a transport theory code. It is usually carried out in a detailed geometrical model, with a detailed representation of the flux spectrum, and for all relevant lattice conditions, such as: fuel depletion; fuel, coolant, and moderator temperatures; densities; etc. In performing lattice calculations it is normally necessary to repeat the lattice spectrum calculation at different burnups in order to account for the effect of changes in nuclide inventory. The lattice calculation provides homogenized properties for the cell (or for each cell in a multiple-cell configuration) in a few (usually two) energy groups, for input to the finite-core calculation.

b) The second step consists of supercell calculations in three dimensions, to determine the effect of interstitial reactivity devices on homogenized properties in their vicinity. This effect is cast in the form of few-energy-group (usually two-energy group) “incremental” cross sections, which are to be added to the bare-lattice cross sections (calculated in step a) of cells traversed by a reactivity device. Supercell calculations are performed with a transport theory code, and usually in three dimensions, because PHWR reactivity devices may be perpendicular to the fuel channels. The device incremental cross sections are determined by performing two supercell calculations, one with the device included in the model, and another with the device excluded, and subtracting the homogenized cross sections obtained in the two cases.

c) The third step is the three-dimensional-diffusion-theory calculation for the entire core, using two or more energy groups. In this calculation, the finite-reactor model is obtained by superimposing the lattice cross sections obtained in step (a) and the device incremental cross sections obtained in step (b). The finite-reactor code calculates the core eigenvalue and the global distribution of flux and power. The fuel at different positions in the core has a different value of fluence, and its properties are evaluated as a function of its fluence in a snapshot or core-follow calculation, or are fluence-averaged in the time-average core model.

For an accurate calculation of the power distribution in the PHWR, it is important to generate cross sections for each fuel bundle in the core, accounting for local conditions (fuel temperature, coolant density, moderator purity, etc.) and the burnup history of the fuel bundle, as well as the effects of the environment if necessary. Different approximations can be used to treat the local parameters and their history in the PHWR reactor core and to provide lattice properties corresponding to an instantaneous state of the core.

The simplest technique is the uniform-parameter method. In this method, the lattice properties are functions of fuel burnup only, by assuming effective core-average conditions for each fuel bundle. An improved approach is the local-parameter method (also called the macro-depletion method in LWR applications). In this approach, the lattice properties are functions of fuel burnup and other local parameters. Since the macroscopic cross sections used in the interpolation are generated beforehand at various assumed operating conditions, this method does not take into account the dependence of the cross sections on a bundle’s history. A more accurate approach, the micro-depletion method, has been developed and used for the CANDU reactor. This method is as simple as the macro-depletion method but it tracks both the microscopic cross section of a nuclide and its number density, which depends on the depletion history.

A more rigorous method is the history-based local-parameter method developed for PHWRs. This is recognized as the most accurate way to calculate the evolution of nuclear properties, but it is not always performed, depending on the lattice code used.

In this method, the lattice code is coupled directly with the core-analysis code, and the lattice calculations are performed for each bundle at each time step, so as to treat local parameters and the history of each bundle individually. This method relies, in practice, on the assumption that the lattice properties associated with a given reactor state can be calculated (perhaps directly within the core-analysis code) with a simplified lattice cell code with only a few energy groups. While it would be desirable to use lattice properties from a modern lattice cell code directly for history-based local-parameter calculations in the core-analysis codes, this is still impractical for routine calculations at this time because of the computational effort required.

The homogenized lattice properties are usually calculated for a single lattice cell (considered as the heterogeneous medium) with reflective or periodic boundary conditions, without considering the effect of the environment. In some cases, multiple lattice cells (multicell) can be used to correct for the effect of the neighborhood on a single lattice cell or the effect of the presence of the reflector. The multicell methodology, developed for the PHWR reactor, maintains intact the basic structure of the single-lattice-cell-based micro-depletion method.

The accuracy of the PHWR lattice and core calculations may be checked by comparison with measurements or against results of other fundamental methods, such as Monte Carlo methods.

The HTGR model outlined in Figure 5 illustrates, in summary fashion, the general steps in the sequence of design codes used in the physics analysis for HTGRs. The primary result being sought is the evolution of the core power and flux distribution throughout the life of the reactor core. Data are also produced on nuclide concentrations as a function of depletion, control-rod reactivity worth, core shutdown margins, core reactivity, reactivity coefficients, and core kinetics parameters.

Figure 5 — General flow of data for HTGR physics calculations

Reactor physics analysis begins with cross sections collected into fine-group libraries (see Reference [2]). These libraries have been processed from extensive tabulations of basic, evaluated cross-section data. Generally, the most recent evaluation is used.

The calculation sequence is similar to the models used for LWRs. Unit-cell calculations or lattice (assembly) codes, which read the fine-group libraries, are used to generate few-group (also known as broad-group) cross-section data sets. The few-group cross sections form the starting point for multidimensional reactor core calculations. Because the fuel is in the form of coated particles in HTGRs, particle self-shielding is important and is usually accounted for in the unit-cell calculations.

The few-group cross-section data are used in combination with geometric, fuel loading, and operational data to perform the physics analysis. Typically, this analysis may start with simplified, often point reactor, calculations and then proceed to two- and three-dimensional finite- difference or nodal methods for solution of the diffusion equation in the reactor core geometry. Design problems for which transport theory is required include determination of flux disadvantage factors, boundary conditions for control-rods and burnable absorbers, and other situations involving anisotropic scattering or strong neutron absorption or leakage.

Depletion calculations of fuel, fertile material, burnable absorbers, and fission products are performed throughout the reactor core lifetime and with sufficient spatial resolution. Typically, basic point reactor fuel-cycle studies provide rapid determination of fuel loadings followed by two- and three-dimensional depletion studies. Core leakage in the depletion calculations is modelled by a group dependent leakage correction to the fission source term to reduce errors due to spectrum changes with depletion. Cross-section variations with exposure and temperature are accounted for as necessary.

Verification and validation are two complementary aspects in determining the range of applicability of the calculation system. As defined in 3.1, verification assesses the fidelity of the calculation system to the theoretical models upon which it is based, numerical validation assesses the accuracy by comparison to a previously validated Monte Carlo method, while experimental validation assesses the accuracy with which the calculation system predicts real-world behavior. A simple example of verification is the determination of the accuracy with which a diffusion theory code produces solutions to the neutron diffusion equation, irrespective of whether diffusion theory is an appropriate representation for the behaviour of interest. An example of validation is the comparison of the calculation system’s predicted value for, say, the reaction rate distribution with the reaction rate distribution that was actually measured.

The components of the calculation system shall be tested both individually and collectively. Testing of a single component is referred to as unit testing, while testing of the system as a whole is referred to as integral testing. The components of the calculation systems employed for core reactivity and power distributions for most power reactors generally fall within one of four general categories:

a) nuclear data;

b) lattice codes (and supercell codes, if applicable);

c) linkage codes; or

d) reactor core codes.

Although these categories are not universal, extension to those types of reactors that have additional components is expected to be straightforward.

Unit testing of nuclear data will not be discussed herein, as it falls outside the scope of this document. The interested reader is referred instead to Reference [1].

Verification typically involves comparisons with closed-form analytic solutions or reference results from a previously verified code. Verification is typically performed by the code developer, with the results included within the documentation distributed with the code.

Overview

Analytic and/or simplified artificial benchmarks are the primary tool employed for unit testing within the context of verification. In general, verification is a binary process: the code either replicates the reference result to an acceptably fine level of precision or it does not. If it does not, the coding shall be examined until the reasons for the discrepancies are identified and understood. If these reasons are found to be errors in the code, the code shall be revised, and the verification process shall be repeated.

Lattice codes

Unit testing of lattice codes typically poses a number of challenges, foremost of which is the construction of an artificial nuclear data library in the format required by the code. In such cases, it is acceptable to extract the “solver” portion of the code and use it with a front-end driver that produces the cross sections, nuclides, and geometry information required by the solver.

Several benchmark specifications exist to test the ability of lattice codes to predict reactivity and power distributions.^[20] In most such cases, the benchmark documentation includes results from a number of lattice codes, and the results from the lattice code of interest can be compared both to the reference results and to the results obtained by other codes based on the same methodology. In addition, with the advent of modern computers with large amounts of memory, lattice codes sometimes may be applied to benchmarks that formerly were employed only for reactor core codes. The most likely of these benchmarks are those that involve a two-dimensional representation of a symmetric fraction of a reactor core.

Although unit testing of lattice codes is usually thought of in terms of reactivity and power distributions, it also shall include testing for nuclide production and depletion. No matter how accurately a lattice code calculates the flux distribution, its predictions for reactivity and power distributions will be degraded with burnup if it does not also calculate nuclide production and depletion correctly.

Linkage codes

The purpose of the linkage code is to transform the output from the lattice-code calculations (homogenized cross sections, etc.) into input for the reactor core code. The accuracy of this transformation can be assessed by comparing the results from the lattice-code calculations with the input as reconstructed by the reactor-core code. Given criteria for acceptable differences, such comparisons not only satisfy the requirements for unit testing of the linkage code but also provide information for guidelines about the length of burnup steps, etc., for the lattice-code calculations.

Reactor core codes

A number of benchmarks exist for unit testing of reactor-core codes, complete with macroscopic cross sections. These benchmarks encompass all of the reactor types considered in 5.8. Unfortunately, few of them include isotopic production and depletion.

Integral testing for verification purposes is a difficult task, because there are few, if any, appropriate benchmarks for nuclear systems that encompass the full range of the type of analyses that are performed by the calculation systems of interest in this document.

Experimental validation usually involves comparisons with real-world measurements, such as those from critical experiments or operating reactors. Typically, the code developer could perform much of the validation, with the results included within the documentation distributed with the code. The activity of validation can also be separated from development. Further validation against results from the operating reactor of interest or a very similar reactor should also be performed before the calculation system is employed to model that reactor on a routine basis. Example resources of criticality benchmarks and reactor physics benchmarks available for validation include, respectively, the International Criticality Safety Benchmark Evaluation Project Handbook^[21] and the International Handbook of Evaluated Reactor Physics Benchmark Experiments^[22].

General

Unit testing for validation purposes may be performed in either of two ways. The most direct, of course, is to compare code predictions directly to measured results. However, measured results typically are integral rather than unit parameters. A second approach is to compare predictions from the code of interest with results from a code of comparable or better accuracy that already has undergone verification and validation.

Lattice codes

Typically, lattice codes perform calculations in two dimensions rather than three dimensions. Consequently, comparisons with measured data are usually made for zero-power rather than at-power conditions, so that three-dimensional effects may be treated in some accurate but relatively simple manner (e.g., assigning an input value for axial buckling). Critical experiments often are good candidates for unit testing of lattice codes, so long as geometric distortions are minimal.

Unit testing for lattice codes may also be performed by comparing their predictions with results from another code of comparable or better accuracy that already has undergone verification and validation. Monte Carlo codes often serve this function. The nuclear data libraries of the two codes shall be sufficiently similar that the differences observed are due to the two codes themselves and not to their libraries. In practice, this restriction typically means that the two nuclear data libraries are derived from the same version of the same evaluated data set.

Linkage codes

The only purpose of the linkage codes is to transform output from lattice-code calculations into input for reactor-core codes. Consequently, there is no need for unit testing for validation of linkage codes.

Reactor core codes

Reactor core codes typically rely upon input from lattice-code calculations and cannot perform calculations for real-world applications without that input. Consequently, unit testing of a reactor core code for validation cannot be performed against measured data unless that code represents each fuel pin, absorber pin, water hole, etc., explicitly. In that case, unit testing and integral testing become identical.

Integral testing for validation purposes usually involves comparisons with measurements from operating reactors. In these comparisons, due consideration shall be given to uncertainties or tolerances in enrichment, composition, densities, as-built dimensions, and other engineering state variables. In addition, results from critical experiments or from more rigorous codes may be used to supplement these comparisons when appropriate.

In most cases, prior to the release of the code, the code developer or other entity performs calculations for a number of reactors over several cycles and compares the predicted and measured critical conditions, control device reactivity worth and neutron flux distributions under different conditions. Such comparisons establish general guidelines for the accuracy that may be expected when the code is applied to similar calculations for reactors of the same general type.

If the calculation system is to be used for licensing purposes or for in-core fuel management, similar comparisons should be made with measurements for the specific reactor of interest. If the reactor has not yet begun operation, such comparisons should be made with measurements from other reactors that are very similar to the one of interest. Furthermore, it is imperative that the model employed during integral testing be exactly the same as that which will be used for routine applications.

The advantage of using results from critical experiments is that the uncertainties associated with such measurements are not always well known but usually are smaller and more readily quantifiable than those from operating reactors, even when the reactors are at zero-power conditions. On the other hand, critical experiments typically are performed at room temperature and have little if any thermal-hydraulic feedback. In addition, critical experiments are usually limited to zero-burnup fuel, and they often feature different geometries from those in power reactors, for both the fuel design and the core arrangement.

Comparisons and benchmarking may be made with results from other codes, of comparable or greater accuracy, that have already undergone verification and validation. At the present time, most reactor-core codes are based on some variation of neutron diffusion theory. Consequently, codes based on transport theory (e.g. Monte Carlo or nodal or detailed transport codes) may satisfy this objective. However, two restrictions apply. First, there shall be no overlap in the components of the two systems, except for nuclear data libraries. Second, the nuclear data libraries employed by the two systems shall be sufficiently similar that the differences observed are due to the codes themselves and not to their libraries. In practice, this restriction means that the two nuclear data libraries are derived from the same version of the same evaluated data set and therefore, should be equivalent.

A typical approach to a thorough code-to-code comparison is to first prepare a detailed three-dimensional model of the reactor with surrounding facility components of neutronic importance to properly capture the details of the reactor physics conditions being simulated. Often the detailed models are prepared using high-fidelity Monte Carlo codes. Subsequently, simplifications to the model are introduced and the effects quantified to evaluate biases introduced into the model for the reactor physics parameters being assessed. Simplifications can include, but are not limited to, homogenization, removal of impurities, removal of reactor surroundings, introduction of average or standardized material properties, simplification of geometries, voiding of insignificant materials/components, preparation of a two-dimensional model, temperature corrections, etc. The same nuclear data libraries are utilized with both detailed and simplified models such that the final simplification bias can be appropriately quantified.

Preparation of simplified models facilitates the development of near-identical models, within the respective restrictions of varying neutronics codes, for a code-to-code comparison. Code-to-code comparisons are investigated with the neutronics codes using the same nuclear data libraries, thus providing code-to-code biases. The resultant simplification and code-to-code biases are effectively quantified for the evaluated reactor physics measurements being simulated. Appropriate bias corrections can now be applied when using the validated codes, with the nuclear data library utilized in the code-to-code analysis, to simulate the evaluated reactor system. Additional nuclear data library comparisons can be investigated using the detailed or simplified models with the same code but by instead changing which library is utilized to simulate given measurements. In some instances, this may be necessary to account for bias contributions from nuclear data when different codes do not use comparable nuclear data libraries or when different processing measures were performed for the same nuclear data library.

The results from the validation process, particularly integral testing, can be used to establish biases and uncertainties for the predictions from the calculation system over its demonstrated range of applicability. In addition, confidence levels shall be specified for the established uncertainties.

If the calculation system is applied beyond its demonstrated range of applicability, the assigned uncertainties shall be increased by an amount that reflects the extent of the extrapolation and the possibility that other factors or phenomena can influence the results obtained. However, be aware that for licensing applications, extending the use of the code system to a new range of applicability will require additional validation of the system in the new area of applicability.

A written report or reports of the verification and validation of the calculation system shall be prepared. This report:

a) shall describe the selection of models and methods and the calculation system — including methods, cross-section data, and validation procedure — with sufficient detail, and clarity to demonstrate that the verification and validation conform to this document;

b) shall identify experimental data and results of more accurate calculation methods, if applicable, and state explicitly which parameters are compared;

c) shall state the area or areas of applicability;

d) shall state the biases and the uncertainties in the calculation system over each area of applicability, together with the means for developing composite error estimates and confidence level.

The documentation shall be sufficiently complete to provide an auditable path. In those instances where the foregoing documentation is proprietary in nature, documentation edited through the exclusion of the proprietary information shall be prepared and areas omitted because of proprietary considerations shall be noted.

In documentation of calculations of more limited scope (e.g., reload fuel design, core following), some of the aforementioned requirements may be satisfied by referencing appropriate existing documentation. However, the material described in items c) and d) of this subclause shall always be included in documentation of calculated results.

Over and above the requirements of this document, additional design validation documentation requirements exist and may be applied such as those of ANSI/ASME NQA 2:2022^[7].

Compliance with the intent of this document can be demonstrated for an intended area of applicability of the calculation system used by meeting the following requirements:

a) Selection of models and methods:

1) consideration of all conditions of reactor composition, temperature, and configuration that significantly affect the calculated quantities and justification of the resulting model approximations (see 5.2 and 5.5.2);

2) preparation of multi-group constants, if employed, in conformance with ANSI/ANS-19.1-2002,^[1] through the use of an application dependent energy spectrum calculation (see 5.3.2);

3) justification of geometric and neutronic transport approximations utilized in the spectrum calculation (see 5.3.3);

4) inclusions of all important space and energy effects in the calculation utilized for the generation of broad-group cross sections, if broad-group cross sections are employed (see 5.4);

5) demonstration of capability, as required by the application, to retrieve required neutron reaction rates in the physical reactor components from the computations, and to justify any assumptions that need to be made in order to perform this retrieval (see 5.6);

6) justification of the spectrum calculation interval used in depletion calculations, and justification that the numerical integration time step is sufficiently small to ensure numerical stability and accuracy appropriate to the application (see 5.7).

b) Calculation system validation. Establishment of the degree of agreement over a limited area of applicability by correlating experimental results or results of calculations from a validated more accurate model with results obtained from the system being validated (see Clause 6).

c) Evaluation of accuracy. Evaluation of accuracy and range of applicability of data and methods by establishment of biases and uncertainties, with confidence levels, for the calculations that include allowance for uncertainties in the comparison data (see 6.3).

d) Documentation. Documentation of details of the preceding procedures (see Clause 7).

1. 
   (informative)
   
   Computer Codes in Common Use

The list of computer codes given in this annex is selected for illustrative purposes, and other codes are in common use. Inclusion in this list does not imply endorsement. The codes as listed in Table A.1 are organized under categories previously discussed in 4.9 of this document. It should be mentioned that there are often restrictions on access to codes. Many/most codes may not be publicly available.

Table A.1 — Computer codes in common use<Tbl_--></Tbl_-->

A.1   References

[Α.1]   ANSI/ANS-19.1-2019, Nuclear Data Sets for Reactor Design Calculations, American Nuclear Society, La Grange Park, Illinois.

[Α.2]   LA‑12740‑M, R. E. MacFARLANE and D. W. MUIR, The NJOY Nuclear Data Processing System: Version 91, Los Alamos National Laboratory (Oct. 1994).

[Α.3]   ORNL/TM-2016/43, D. WIARDA, M. E. DUNN, N. M. GREENE, M. L. WILLIAMS, C. CELIK, and L. M. PETRIE, AMPX-6: A Modular Code System for Processing ENDF/B, Oak Ridge National Laboratory (2016); available from Radiation Safety Information Computational Center as CCC-834; https://www.ornl.gov/sites/default/files/AMPX-6.pdf (accessed February 1, 2023).

[Α.4]   B. J. TOPPEL, H. HENRYSON II, and C. G. STENBERG, “ETOE‑2/MC²‑2/SDX Multiple Cross-Section Processing,” presented at RSIC Seminar-Workshop Multigroup Cross Sections, Oak Ridge, Tennessee, March 14, 1978.

[Α.5]   ANL/NE-11-41, Rev. 2, C. H. LEE and W. S. YANG, MC²-3: Multigroup Cross Section Generation Code for Fast Reactor Analysis, Argonne National Laboratory (2013); https://doi.org/10.2172/1414284.

[Α.6]   EGG‑2589, R. A. GRIMESEY, D. W. NIGG, and R. L. CURTIS, COMBINE/PC—A Portable ENDF/B Version 5 Neutron Spectrum and Cross‑Section Generation Program, Idaho National Engineering Laboratory (1990).

[Α.7]   G. MARLEAU, “Fine Mesh 3-D Collision Probability Calculations Using the Lattice Code DRAGON,” Proc. Int. Conf. Physics of Nuclear Science and Technology, Long Island, New York, October 5–8, 1998, American Nuclear Society (1998).

[Α.8]   A. HÉBERT, G. MARLEAU, and R. ROY, “Application of the Lattice Code DRAGON to CANDU Analysis,” Trans. Am. Nucl. Soc., 72, 335 (1995).

[Α.9]   EIR-Bericht Nr. 539, D. MATHEWS, J. STEPANEK, S. PELLONI, and C. E. HIGGS, The NJOY Nuclear Data Processing System: The MICROR Module, EIR-Bericht (1984).

[Α.10]   P. KOCH and D. WALTI, “MICROX. Two‑Region Flux Spectrum Code for Calculating Broad Group Cross Sections,” General Atomics (1992).

[Α.11]   J. L. VUJIC and W. R. MARTIN, “Two‑Dimensional Collision Probability Method with Anisotropic Scattering for Vector and Parallel Processing,” Proc. Int. Conf. Physics of Reactors: Operation, Design, and Computation (PHYSOR‑90), Marseilles, France, April 23–27, 1990, vol. 3, p. IV‑104 (1990).

[Α.12]   M. EDENIUS, B. H. FORSSEN, and C. GRAGG, “The Physics Model of CASMO‑4,” Proc. Int. Topl. Mtg. Advances in Mathematics, Computations, and Reactor Physics, Pittsburgh, Pennsylvania, April 28–May 2, 1991, vol. 2, p. 10.1 1-1, American Nuclear Society (1991).

[Α.13]   WCAP‑11597‑A, T. Q. NGUYEN et al., Qualification of the PHOENIX‑P/ANC Nuclear Design System for Pressurized Water Reactor Cores, Westinghouse Electric Corporation (1988).

[Α.14]   J. R. ASKEW, F. J. FAYERS, and P. B. KEMSHELL, “General Description of the Lattice Code WIMS,” J. Brit. Nucl. Energy Soc., 5, 564 (1966); see also AEEW-M 1785, M. J. HALSALL, The Use of WIMSD4 and LWRWIMS, READWT and FILSIX, to Generate Two‑Group Data for Reactor Calculations, Atomic Energy Establishment (July 1980).

[Α.15]   J. J. CASAL, R. J. J. STAMM’LER, E. A. VILLARINO, and A. A. FERRI, “HELIOS: Geometric Capabilities of a New Fuel‑Assembly Program,” Proc. Int. Topl. Mtg. Advances in Mathematics, Computations, and Reactor Physics, Pittsburgh, Pennsylvania, April 28–May 2, 1991, vol. 2, p. 10.2 1‑1, American Nuclear Society (1991).

[Α.16]   LA-UR-03-1987, X-5 MONTE CARLO TEAM, MCNP—A General Monte Carlo N-Particle Transport Code, Version 5, Los Alamos National Laboratory (Apr. 2003).

[Α.17]   LA-UR-07-6632, J. S. HENDRICKS et al., “MCNPX, Version 26E,” Los Alamos National Laboratory (Oct. 2007).

[Α.18]   S. LOUBIERE, R. SANCHEZ, M. COSTE, A. HÉBERT, Z. STANKOVSKI, C. VAN DER GUCHT, and I. ZMIJAREVIC, “Apollo2 Twelve Years Later,” Proc. Int. Conf. Mathematics and Computation, Reactor Physics and Environmental Analysis in Nuclear Applications, Madrid, Spain, September 27–30, 1999, vol. 2, p. 1298 (1999).

[Α.19]   M. A. JESSEE, D. WIARDA, K. T. CLARNO, U. MERTYUREK, and K. BEKAR, “TRITON: A Multipurpose Transport, Depletion, and Sensitivity and Uncertainty Analysis Module,” Sec. 3.1 in ORNL/TM-2005/39, SCALE Code System, Version 6.2.4, W. A. WIESELQUIST, R. A. LEFEBVRE, and M. A. JESSEE, Eds., Oak Ridge National Laboratory (2020); available from Radiation Safety Information Computational Center as CCC-834.

[Α.20]   L. M. PETRIE, K. B. BEKAR, D. F. HOLLENBACH, and S. GOLUOGLU, “CSAS6: Control Module for Enhanced Criticality Safety Analysis Sequences With KENO VI,” Sec. 2.2 in ORNL/TM-2005/39, SCALE Code System, Version 6.2.4, W. A. WIESELQUIST, R. A. LEFEBVRE, and M. A. JESSEE, Eds., Oak Ridge National Laboratory (2020); available from Radiation Safety Information Computational Center as CCC-834.

[Α.21]   J. LEPPÄNEN, M. PUSA, T. VIITANEN, V. VALTAVIRTA, and T. KALTIAISENAHO, “The Serpent Monte Carlo Code: Status, Development and Applications in 2013,” Ann. Nucl. Energy, 82, 142 (2015); https://doi.org/10.1016/j.anucene.2014.08.024.

[Α.22]   PM RCA 77‑12, R. STAMM’LER, S. OLSSON, and O. BERNANDER, The PHOENIX Computer Program for Fuel Assembly and Pin Cell Calculations, ASEA‑Atom (1977).

[Α.23]   S. BORRESEN and T. SKARDHAMAR, “Recent Developments in the Lattice Physics Code RECORD,” Proc. Int. Topl. Mtg. Advances in Reactor Physics, Mathematics, and Computations, Paris, France, April 27–30, 1987, vol. 3, p. 1513 (1987).

[Α.24]   M. YAMAMOTO, H. MIZUTA, K. MAKINO, and R.-T. CHIANG, “Development and Validation of TGBLA BWR Lattice Physics Methods,” Proc. Topl. Mtg. Reactor Physics and Shielding, Chicago, Illinois, September 17–19, 1984, vol. 1, p. 364, American Nuclear Society (1984).

[Α.25]   M. YAMAMOTO, Y. ANDO, T. UNAMO, H. MIZUTA, and H. SEINO, “Recent Developments in TGBLA Lattice Physics Code,” Proc. Int. Topl. Mtg. Advances in Reactor Physics, Mathematics, and Computations, Paris, France, April 27–30, 1987, vol. 3, p. 1549 (1987).

[Α.26]   R.-T. CHIANG and S. P. CONGDON, “Resonance Improvements for Lattice Analysis,” Proc. Int. Conf. Physics of Reactors: Operation, Design, and Computation (PHYSOR-90), Marseilles, France, April 23–27, 1990, vol. 2, p. IX-34 (Apr. 1990).

[Α.27]   R.-T. CHIANG, S. SITARAMAN, A. L. JENKINS, K. ASANO, and K. KOYABU, “Development, Validation and Applications of Innovative MCNP-Based BWR Fluence Computational Package MF3D,” Proc. 9th Int. Conf. Nuclear Engineering, Nice, France, April 8–12, 2001.

[Α.28]   D. KNOTT and E. WEHLAGE, “Description of the LANCER02 Lattice Physics Code for Single-Assembly and Multibundle Analysis,” Nucl. Sci. Eng., 155, 331 (2007); https://doi.org/10.13182/NSE155-331.

[Α.29]   J. D. IRISH and S. R. DOUGLAS, “Validation of WIMS-IST,” Proc. 23rd Annual Conf. Canadian Nuclear Society, Toronto, Ontario, Canada, June 2–5, 2002, Canadian Nuclear Society (2002).

[Α.30]   LA‑10049‑M, R. E. ALCOUFFE, F. W. BRINKLEY, D. R. MARR, and R. D. O’DELL, User’s Guide for TWODANT: A Code Package for Two‑Dimensional, Diffusion-Accelerated, Neutral Particle Transport,” Los Alamos National Laboratory (1984).

[Α.31]   P. HUTT, N. GAINES, M. McELLIN, R. J. WHITE, and M. J. HALSALL, “The UK Core Performance Code Package,” J. Brit. Nucl. Energy Soc., 30, 291 (1991).

[Α.32]    “NESTLE Version 5.2.1: Few-Group Neutron Diffusion Equation Solver Utilizing the Nodal Expansion Method for Eigenvalue, Adjoint, Fixed-Source Steady-State and Transient Problems,” North Carolina State University, Electric Power Research Center, RSICC Computer Code Number C00641, Radiation Safety Information Computational Center (July 2003).

[Α.33]   ANL‑82‑64, K. L. DERSTINE, DIF3D: A Code to Solve One‑, Two‑, and Three‑Dimensional Finite‑Difference Diffusion Theory Problems, Argonne National Laboratory (1984).

[Α.34]   ANL‑83‑2, B. J. TOPPEL, A User’s Guide for the REBUS‑3 Fuel-Cycle-Analysis Capability, Argonne National Laboratory (1983).

[Α.35]   B. ROUBEN, “RFSP-IST, The Industry Standard Tool Computer Program for CANDU Reactor Core Design and Analysis,” Proc. 13th Pacific Basin Nuclear Conf. (PBNC 2002), Shenzhen, China, October 21–25, 2002.

[Α.36]   AECL-11047, B. ROUBEN, Overview of Current RFSP-Code Capabilities for CANDU Core Analysis, Atomic Energy of Canada Limited (Jan. 1996).

[Α.37]   B. ROUBEN, “Technology Transfer: CANDU Fuel-Management Code RFSP,” Proc. 11th Pacific Basin Nuclear Conf. (PBNC ’98), Banff, Alberta, Canada, May 3–7, 1998.

[Α.38]   B. ROUBEN, “Recent Trends in Methodologies for CANDU Finite-Core Analysis at AECL,” Proc. 5th Int. Canadian Nuclear Society Conf. Simulation Methods in Nuclear Engineering, Montréal, Quebec, Canada, September 8–11, 1996, Canadian Nuclear Society (1996).

[Α.39]   DOE‑HTGR‑86‑109, Rev. 1, G. SHIRLEY and C. FOUSE, Users Manual—MHTGR Core Decay Heat, U.S. Department of Energy (July 1987).

[Α.40]   GA‑A16657, Rev., R. J. ARCHIBALD and P. K. KOCH, A Users Guide and Programmers Guide to the GAUGE Two‑Dimensional Neutron Diffusion Program, General Atomics (June 1990).

[Α.41]   WCAP‑10966‑A, Y. S. LIU et al., ANC: A Westinghouse Advanced Nodal Computer Code, Westinghouse Electric Corporation (1985).

[Α.42]   SOA‑89/04, K. SMITH et al., “SIMULATE‑3 Methodology,” Studsvik of America (1989); see also W. R. COBB, R. S. BORLAND, B. L. DARNELL, P. L. VERSTEEGEN, and D. M. VERPLANCK, “SIMULATE‑E (Mod. 3) Computer Code Manual,” in EPRI‑NP‑4574‑CCM, ARMP‑2 System Documentation, Part II, Chap. 8, Electric Power Research Institute (1987).

[Α.43]   U. LINDELOW, C.‑A. JONSSON, and S.‑O. LINDAHL, “The ABB Atom PWR Core Design Code Package,” Proc. Jahrestagung Kerntechnik ’90, Nuremberg, Germany, May 1980, p. 15; see also RCA 77‑113, O. BERNANDER et al., Reactor Core Design and Performance Evaluation Methods, ASEA‑Atom (1977).

[Α.44]   BAW‑10180‑A, Rev. 1, G. H. HOBSON et al., NEMO—Nodal Expansion Method Optimized, Babcock & Wilcox (Mar. 1993).

[Α.45]   EMF-96-029, Vol. 1, S. K. MERK et al., Reactor Analysis System for PWRs, Volume 1: Methodology Description, Siemens Power Corporation (Jan. 1997).

[Α.46]   EMF-96-029, Vol. 2, N. A. ANGUIANO et al., Reactor Analysis System for PWRs, Volume 2: Benchmarking Results, Siemens Power Corporation (Jan. 1997).

[Α.47]   G. HOBSON et al., “ARTEMIS^TM Core Simulator: Latest Developments,” Proc. Joint Int. Conf. on Supercomputing in Nuclear Applications and Monte Carlo (SNA + MC 2013), Paris, France, October 27–31, 2013.

[Α.48]   UM-NERS-09-0001, T. DOWNAR, Y. XU, V. SEKER, and N. HUDSON, “PARCS v3.0 U.S. NRC Core Neutronics Simulator,” University of Michigan (Mar. 2010).

[Α.49]   C. W. LINDENMEIER, D. H. TIMMONS, G. R. CORRELL, and P. URBAN, “Advanced Methodology for LWR Core Design,” Proc. Int. Reactor Physics Conf., Jackson Hole, Wyoming, September 18–22, 1988, Vol. I, p. I-101, American Nuclear Society (1988).

[Α.50]   NEDO‑20953A, J. A. WOOLLEY, Three‑Dimensional BWR Core Simulator, General Electric Nuclear Energy (Jan. 1977).

[Α.51]   R. L. CROWTHER et al., “Three Dimensional Simulation of Large Power Reactors,” Proc. Int. Topl. Mtg. Advances in Mathematical Methods for the Solution of Nuclear Engineering Problems, Munich, Germany, April 27–29, 1981.

[Α.52]   S. BORRESEN, “A Simplified, Coarse Mesh, Three‑Dimensional Diffusion Scheme for Calculating the Gross Power Distribution in a Boiling Water Reactor,” Nucl. Sci. Eng., 44, 37 (1971); https://doi.org/10.13182/NSE71-A18903.

[Α.53]   S. PALMTAG, J. LAMAS, J. FINCH, A. GODFREY, and B. R. MOORE, “The Advanced BWR Core Simulator AETNA02,” Proc. Int. Conf. Physics of Reactors (PHYSOR 2008), Interlaken, Switzerland, September 14–19, 2008.

[Α.54]   ANL‑83‑1, R. D. LAWRENCE, “The DIF3D Nodal Neutronics Option for Two‑ and Three‑Dimensional Diffusion-Theory Calculations in Hexagonal Geometry,” Argonne National Laboratory (1983).

[Α.55]   W. S. YANG, P. J. FINCK, and H. S. KHALIL, “Reconstruction of Pin Burnup Characteristics from Nodal Calculations in Hexagonal Geometry,” Proc. Int. Conf. Physics of Reactors: Operation, Design, and Computation (PHYSOR-90), Marseilles, France, April 23–27, 1990, vol. 2, p. VIII‑22 (1990).

[Α.56]   IGE-208, Rev. 4, E. VARIN, A. HÉBERT, R. ROY, and J. KOCLAS, A User Guide for DONJON—Version 3.01, École Polytechnique de Montréal (Aug. 2005).

[Α.57]   ORNL/TM-13221, W. A. RHOADES and R. L. CHILDS, The TORT Three-Dimensional Discrete Ordinates Neutron Transport Code (TORT Version 3), Oak Ridge National Laboratory (Oct. 1997).

[Α.58]   W. A. RHOADES and R. L. CHILDS, “The DORT Two-Dimensional Discrete Ordinates Transport Code,” Nucl. Sci. Eng., 99, 88 (1988); https://doi.org/10.13182/NSE88-A23547.

[Α.59]   C. B. CARRICO, G. PALMIOTTI, and E. E. LEWIS, “Nodal Variational Transport Methods in Three Dimensions: Cartesian, Triangular, and Hexagonal Geometries,” Trans. Am. Nucl. Soc., 64, 273 (1991).

[Α.60]   LA-UR-05-3925, Rev., R. E. ALCOUFFE, R. S. BAKER, J. A. DAHL, S. A. TURNER, and R. C. WARD, PARTISN: A Time-Dependent, Parallel, Neutral Particle Transport Code System, Los Alamos National Laboratory (May 2005).

[Α.61]   AECL‑6878, H. C. CHOW and M. H. M. ROSHD, SHETAN—A Three‑Dimensional Integral Transport Code for Reactor Analysis, Atomic Energy of Canada Limited (1980).

[Α.62]   R. N. BLOMQUIST, “VIM,” Proc. Int. Topl. Mtg. Advances in Mathematics, Computations, and Reactor Physics, Pittsburgh, Pennsylvania, April 28–May 2, 1991, Vol. 5, p. 30.4 2‑1, American Nuclear Society (1991).

[Α.63]   M. A. JESSEE et al., “Polaris—2D Light Water Reactor Lattice Physics Module,” Sec. 3.2 in ORNL/TM-2005/39, SCALE Code System, Version 6.2.4, W. A. WIESELQUIST, R. A. LEFEBVRE, and M. A. JESSEE, Eds., Oak Ridge National Laboratory (2020); available from Radiation Safety Information Computational Center as CCC-834.

[Α.64]   ANL/NE-18/4, Y. S. JUNG, C. LEE, and M. A. SMITH, PROTEUS-NODAL User Manual, Argonne National Laboratory (2018).

[Α.65]   B. KOCHUNAS, B. COLLINS, D. JABAAY, T. J. DOWNAR, and W. R. MARTIN, “Overview of Development and Design of MPACT: Michigan Parallel Characteristics Transport Code,” Proc. Int. Conf. Mathematics and Computational Methods Applied to Nuclear Science & Engineering (M&C 2013), Sun Valley, Idaho, May 5–9, 2013, p. 42, American Nuclear Society (2013).

[Α.66]   NLR-03, NFI Licensing Topical Report (1994) [in Japanese].

[Α.67]   Y. KANAYAMA, et al., “Applicability of the Advanced Fuel Assembly Design Code NEUPHYS for LWR Next Generation Fuels,” GENES4/ANP2003, Kyoto, Japan, September 15-19 (2003).

Bibliography

The user is advised to review each of the following references to determine whether it, a more recent version, or a replacement document is the most pertinent for each application. When alternate documents are used, the user is advised to document this decision and its basis.

[1] ANSI/ANS-19.1-2019, Nuclear Data Sets for Reactor Design Calculations, American Nuclear Society, La Grange Park, Illinois, USA

[2] ANSI/ANS-19.3-2022, Steady-State Neutronics Methods for the Analysis of Power Reactors, American Nuclear Society, La Grange Park Illinois, USA

[3] ANSI/ANS-19.4-2017 (R2022), A Guide for Acquisition and Documentation of Reference Power Reactor Physics Measurements for Nuclear Analysis Verification, American Nuclear Society, La Grange Park, Illinois, USA

[4] ANSI/ANS-19.5-1995; W2005, Requirements for Reference Reactor Physics Measurements, American Nuclear Society, La Grange Park, Illinois, USA

[5] ANSI/ANS-19.3.4-2022, The Determination of Thermal Energy Deposition Rates in Nuclear Reactors, American Nuclear Society, La Grange Park, Illinois, USA

[6] ANSI/ASME NQA-1-2022, Quality Assurance Requirements for Nuclear Facility Applications, published by ASME International, New York, New York, USA

[7] ANSI/ASME NQA 2:1989, Quality Assurance Requirements for Nuclear Facility Applications, published by ASME International, New York, New York, USA

[8] ANSI/ANS-10.3-1995 (withdrawn), Documentation of Computer Software, American Society of Mechanical Engineers, New York, New York, USA

[9] ANSI/ANS-10.4-2008; (R2021), Verification and Validation of Non-Safety Related Scientific and Engineering Computer Programs for the Nuclear Industry, American Nuclear Society, La Grange Park, Illinois., USA

[10] ANSI/ANS-10.5-2006; (R2021), Accommodating User Needs in Scientific and Engineering Computer Software Development, American Nuclear Society, LaGrange Park, Illinois, USA

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[12] M.B. Chadwick, et al., “ENDF/B-VII.0: Nuclear Data Library for Science and Technology Cross Sections, Covariances, Fission Product Yields and Decay Data”, Nuclear Data Sheets, Vol. 112, No. 12 (December 2011), p. 2887. https://doi.org/10.1016/j.nds.2011.11.002

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[21] International Criticality Safety Benchmark Evaluation Project Handbook, OECD Nuclear Energy Agency (2021).

[22] International Handbook of Evaluated Reactor Physics Benchmark Experiments, OECD Nuclear Energy Agency. (2021); https://www.oecd-nea.org/jcms/pl_20279 (accessed February 1, 2023).

1. Some computer codes in common use are listed in Annex A ↑