ISO 18075:2018

INTERNATIONAL ISO
STANDARD 18075
First edition
2018-03
Steady-state neutronics methods for
power-reactor analysis
Méthodes stationnaires en neutronique pour l'analyse des réacteurs
de puissance
Reference number
©
ISO 2018
© ISO 2018
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting
on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address
below or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Fax: +41 22 749 09 47
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2018 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
3.1 Terms . 1
3.2 Abbreviations . 4
4 Relation to other standards . 4
5 Methods of calculation . 5
5.1 General . 5
5.2 Conditions to be considered . 5
5.3 Fine-group cross-sections . 6
5.3.1 Basic data . 6
5.3.2 Preparation of fine-group cross-sections . 6
5.3.3 System dependent spectrum calculations . 6
5.3.4 Weighting function . 6
5.4 Preparation of broad-group libraries . 6
5.4.1 General. 6
5.4.2 Choice of cell and supercell . 7
5.4.3 Cell environment. 7
5.4.4 Calculation model . 7
5.5 Collapse to few-groups . 8
5.6 Calculation of reactivity, reaction rate, and neutron flux distributions . 8
5.6.1 Models . 8
5.6.2 Uncertainties and assumptions . 9
5.7 Calculation of reaction rates in reactor components . 9
5.8 Depletion calculations .10
5.9 Common practices .11
5.9.1 Pressurized water reactor (PWR) core physics method .11
5.9.2 Boiling water reactor (BWR) core physics methods .12
5.9.3 Liquid metal reactor (LMR) core physics methods .13
5.9.4 Heavy water reactor HWR core physics methods .15
5.9.5 High temperature gas cooled reactor (HTGR) core physics methods .17
6 Verification and validation of the calculation system .18
6.1 Overview .18
6.2 Verification .18
6.2.1 General.18
6.2.2 Unit testing .18
6.2.3 Integral testing.19
6.3 Validation .19
6.3.1 Unit testing .19
6.3.2 Integral testing.20
6.4 Biases and uncertainties .21
7 Documentation .21
8 Summary of requirements .22
Bibliography .23
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).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see the following
URL: www .iso .org/ iso/ foreword .html.
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-
[2]
19.3-2011 (R2017) .
iv © ISO 2018 – All rights reserved

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 become available and
more refined calculation methods are developed, the reliability of the results of the steady-state power
reactors will be considerably enhanced.
INTERNATIONAL STANDARD ISO 18075:2018(E)
Steady-state neutronics methods for power-reactor
analysis
1 Scope
This document provides guidance for performing and validating the sequence of steady-state
calculations leading to prediction, in all types of operating UO -fuel commercial nuclear reactors, of:
— reaction-rate spatial distributions;
— reactivity;
— change of nuclide compositions with time.
The document provides:
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.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
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 http:// www .electropedia .org/
3.1 Terms
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
cell
one or more reactor sections with associated coolant (and possibly additional moderator and
structural material) which, 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 which 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.
3.1.4
data set
collection of microscopic cross-sections and nuclear constants encompassing the range of materials
and reaction processes needed for the application area of interest
3.1.4.1
averaged data set
data set prepared by averaging an evaluated data set or a processed continuous 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.
Application-independent averaged data sets for a wide range of reactor analysis, e.g. light water reactors, are
dealt with in American National Standard Nuclear Data Sets for Reactor Design Calculations, ANSI/ANS-19.1-
[1]
2002 (R2011) .
3.1.4.2
evaluated data set
data set which 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 process.
Note 2 to entry: An evaluated data set is intended to be independent of reactor composition, geometries, energy
group structures, and spectra.
3.1.4.3
processed continuous 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 compositions, geometries, energy-group
structures, and spectra.
3.1.5
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.
2 © ISO 2018 – All rights reserved

3.1.6
few-group
energy-group (typically 2-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 temperature.
3.1.7
lattice
lattice cell
normally refers to a fuel-assembly cell with its associated immediate environment, such as the volume
of moderator associated with it
3.1.8
calculation method
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”.
3.1.9
reactivity
property of the whole reactor, not just of a given material composition, is the ratio of the net production
rate of neutrons (excess of neutrons produced by fission over those absorbed) to the production rate
due to fissions
Note 1 to entry: Quantitatively, the core reactivity, ρ, can be represented as:
ρ = (λ−1)/λ = 1 – (1/k )
eff
where
λ is the eigenvalue of the steady-state neutron balance equation;
k is the effective neutron multiplication constant.
eff
Note 2 to entry: quantity (1 minus the eigenvalue of the steady-state neutron balance equation, written as:
MΦ = λ FΦ
where
Φ is the neutron flux;
F is the neutron yield operator;
M is the scattering, absorption, and leakage operator.
Note 3 to entry: The effective multiplication factor k is the inverse of λ. Reactivity is a unitless, pure number. It
eff
−5
is, however, often written in terms of smaller “units”, such as milli-k = 0,001, pcm = 0,000 01 = 10 or “dollars”
(and cents), where 1 dollar is taken as the value of the delayed-neutron fraction in the system of interest.
3.1.10
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
3.1.11
verification calculation
process of determining that a model implementation accurately represents the developer’s conceptual
description of the model and the solution to the model
3.2 Abbreviations
BWR boiling water reactor
HTGR high temperature gas cooled reactor
HWR heavy water reactor
LMR liquid metal reactor
PWR pressurized water reactor
4 Relation to other standards
The following American National Standards are related to this document:
[1]
— Nuclear Data Sets for Reactor Design Calculations, ANSI/ANS-19.1-2002 (R2011) , 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 which 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.
— Guide for Acquisition and Documentation of Reference Power Reactor Physics Measurements for
[3]
Nuclear Analysis Verification, ANSI/ANS-19.4−2017 ; and Requirements for Reference Reactor Physics
[4]
Measurements, ANSI/ANS 19.5-1995; W2005 .
Validation of calculation systems requires comparison with available integral 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
[5]
R2017 , 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−19.3−2011; 2017.
[6]
— Quality Assurance Program Requirements for Nuclear Facility Applications, ANSI/ASME-NQA 1 2015 .
This standard deals with quality assurance, including that for computer programs.
[8]
— Guidelines for the Documentation of Computer Software, ANSI/ANS 10.3 1995 W2005 . This standard
includes requirements for computer programs.
— Verification and Validation of Non-Safety Related Scientific and Engineering Computer Programs for
[9]
the Nuclear Industry, ANSI/ANS-10.4−2008; W2016 . This standard deals with requirements or
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;
[10]
R2016 . This standard deals with methods to respond to users’ requirements in computer
programs.
4 © ISO 2018 – All rights reserved

5 Methods of calculation
5.1 General
Calculations within the scope of this document would typically be performed in a sequence of steps. A
typical sequence might be:
a) Spectrum Calculation. Averaged-data set cross-sections, nuclide number densities, and geometrical
information (usually repeating cells or supercells) are used to calculate an application-dependent
neutron spectrum for each different reactor region or composition.
b) Cross-Section Collapsing. Collapsing averaged-data set cross-sections to few-group form, using
spectrum calculated in a) above.
c) Cross-Section Homogenization. The spectra obtained above are used to homogenize cross-sections
and number densities over pin cells and assemblies.
d) Flux Distributions and Reactivity. The broad-group cross-sections and geometrical information
about the reactor obtained above is used to calculate reactivity and few-group flux spatial
distributions in the reactor.
e) Reaction Rates Calculations. The preceding information is used to compute reaction rates in physical
reactor components.
f) Exposure. Calculation of changes in nuclide composition of fuel and possibly other reactor
components with exposure are obtained based on the above data.
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.
5.2 Conditions to be considered
Consideration shall be given to all conditions which 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:
a) presence of control elements (rods, cruciforms, or other forms), and degradation 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 BWR;
h) presence of sources, detectors, structural materials, and experimental devices;
i) spatial variations in temperatures;
j) fuel temperature;
k) spatial and temporal variations of important nuclides, e.g. xenon, samarium, and actinides.
5.3 Fine-group cross-sections
5.3.1 Basic data
The primary sources of basic nuclear data that are used for the generation of fine-group constants
are evaluated data sets. Examples of these are the ENDF, JENDL, BROND, JEFF, and CENDL evaluated
data sets. The properties and criteria for selecting these sources of basic nuclear data are specified in
[1]
ANSI/ANS 19.1-2002 R2011 .
5.3.2 Preparation of fine-group cross-sections
The preparation of application-dependent fine-group constants from existing application-independent
fine-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.
5.3.3 System dependent spectrum calculations
The fine-group cross-section set (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 the 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.
5.3.4 Weighting function
The fine-group 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. It should be noted that results may be sensitive to the modelling of the spectra and
the choice of group structure.
5.4 Preparation of broad-group libraries
5.4.1 General
There are three distinct steps for generating broad-group libraries:
a) processing of continuous or point-wise cross-sections, accounting for self-shielding and Doppler
broadening effects and collapsing these data into a fine-group library using an appropriate
spectrum;
b) performance of fine-group transport calculation for a simplified model of the reactor to obtain a
fine-group spectrum;
c) utilization of the fine-group spectrum to obtain broad-group libraries.
6 © ISO 2018 – All rights reserved

5.4.2 Choice of cell and supercell
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 (e.g. sometimes this formalism is 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 non-cell materials such as water channels,
control-rods, and structural materials. Other non-cell 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 consistent with symmetry assumptions.
5.4.3 Cell environment
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,
in moderator or coolant, or in both (e.g. Doppler broadening), and variations in density of coolant or
moderator, or both, shall be included in the calculation. Corrections for a non-uniform temperature
distribution within the cell and supercell should be made, or the temperature distribution should be
included in the calculation.
5.4.4 Calculation model
The calculation model often can be considered to consist of two parts, the geometric model and the
neutronic transport model, though the two parts may not be clearly separable.
5.4.4.1 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 term which stands as
the surrogate of the missing leakage.
5.4.4.2 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. There are two basic approaches in modelling neutron transport
in power reactors: deterministic and Monte Carlo. Various deterministic methods are used, such as
collision probability, discreet ordinates and nodal diffusion methods. 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 served primarily as a guide to the validity of other
(deterministic procedures) but is now used in mainstream applications as well.
Deterministic transport models generate numerical solutions (by collision probability methods, for
instance) of the integral transport equation. Approximations may also be 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, in using discrete ordinate methods,
the analyst shall show 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.
5.5 Collapse to few-groups
When performing full reactor calculations it is usually adequate and desirable to collapse the cross-
sections from the fine-group structure into a broad-group set. The actual group structure chosen
should depend on the type of calculation and the sensitivity of that calculation to the group structure.
When collapsing cross-sections to a broad-group, important system characteristics – such as reaction
rates in a unit-cell, reactivity of the cell and core, or reaction-rate ratios – should be preserved. This
preservation is an attempt to maintain some of the detailed representation of the fine-group calculation
in the coarser broad-group 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 account for all important effects of space and energy
that cannot be adequately modelled 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 constitute the starting (reference) points for calculations of change in nuclide composition with
time (depletion calculations).
5.6 Calculation of reactivity, reaction rate, and neutron flux distributions
5.6.1 Models
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 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 ). This is
eff
appropriate, for example, in describing the closeness to criticality of a reactor in its shutdown 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. Under
these circumstances, a value of k different from unity represents an artificial device. In addition, code-
eff
system bias and uncertainties may lead to a non-unity k as a critical reference point. The definition of
eff
reactivity 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 fine-group or broad-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 synthesis methods;
b) solving the transport equations by discrete-ordinates or collision-probability methods or by the
method of characteristic;
c) solving the reactor neutron balance equations by nodal diffusion methods;
d) the continuous-energy Monte Carlo method with spatially detailed representation and large
number of neutron histories, provides a superior alternative to deterministic methods.
8 © ISO 2018 – All rights reserved

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.
5.6.2 Uncertainties and assumptions
In setting up a computer model of the reactor core intended to simulate steady-state neutronic
behaviour (core-follow or predictive analysis), certain assumptions and simplifications are usually
made. Depending on the number and extent of simplifications and the number of assumptions, each of
them can result in errors which are cumulative. Therefore, the results of the calculations can only be
considered as approximate.
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 which 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 which separately are a function of only one or two
dimensions (spatial separability);
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 which 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 of flux between the spatial points within a spatial mesh structure, the
dimensions of which may be specified somewhat arbitrarily;
g) the use of bucklings to simulate leakage effects in the directions are not explicitly represented;
h) the assumption that dissimilar media may be homogenized;
i) the use of pre-calculated region-homogenized (typically lattice-homogenized) cross-sections at a
predefined power history;
j) the use of interpolation or curve fitting technique 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 stepping outside this range, caution
should be exercised to try to ensure that there are no fortuitously cancelling error, and to be in a better
position to judge the reasonableness of the behaviour of the method and of the results obtained.
5.7 Calculation of reaction rates in reactor components
When a model which 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. 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 reaction rates.
5.8 Depletion calculations
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, and decay of the nuclide. Yields of fission
products, including lumped pseudo fission products, and decay constants of the nuclides of interest are
basic nuclear data which shall be available to 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 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 may be under
the control of the analyst. If so, the choice should be made 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.
...