ORNL-TM-3359.txt

ORNL-TM-3259

Contract No. W-ThC5-eng-20

Reactor Division

ROD: A NUCLEAR AND FUEL-CYCLE ANALYSIS CODE
FOR CIRCULATING-FUEL REACTORS

H. F. Baumsn

G. W. Cunninghsam,JIT
, J. L. Lucius

H. T. Kerr

C. W. Craven, Jr.

 

This report was preparad =5 an sccocunt of work
sponam:cd by the United States Government, Neither
the Un.ne‘d States nor the United States Atomic Enesgy

D » BOr any of the employees, nor any of
their conmactors, subcontractors, of their employees,
makes 30y warranty, express ot implied, or assumes any
legal lLiability er responsibility for the accuracy, com.
pleteness or usefulress of any information, apparatus,
product or process Jisclosed, or represents that its nse
would not infringe privately owned rights,

 

 

SEPTEMBER 1971

OAK RIDGE NATIONAL LABORATORY
Oak Ridge, Tennessee
operated by
UNTON CARBIDE CORPORATION
for the
U. S. ATOMIC ENERGY COMMISSION

 
iii

CONTENTS
Page
FOTEWOTE soanccecnocatneecosossnnoness sescereassraseaiesascssencs v
ADSETECE vevniatvesraccasansascrncnsnnanas cessvrereecrsearaneasaaa vii
Acknowledgments ...ccieercnconnrnionaanaaen ctieaseantaneavanans e ix
Computer Code Abstract ....... ................................. xi
1. Introduction ... iiieiereririrrneroveens cematesmsmnancnoncsnss 1.1
2. The History of ROD ........... bt evessesaanas Ferecessneasaane 2.1
%. The Functions of the ROD Program «-«...cccvveevennses ceaeanns . 3.1
L. Tnput Description ............ feeeatotsssanacansasoacanacnnns 4.1
Section A. MODRIC ..ovvece Ceeccaccane tereserateeceeeaanan k.3
Sectionn B. ERC eevvccvosassocasnanonns Ceetreetsearoan eee 4,18
Section C. Fission Product and Delayed Neutron Data ..... 4.29
Section D. OPTL ....... Ceaacsecnecnnana teveracncasaseran . 431
5. Discussion of Tnput ....cevccuciane cecessenvesensanaaeanuenna 5.1
6. User Informetion ..e...... feeenieanes feterreeeieeanas ceererss 6.1
CONETOL CATAS .evvceeciscnnnenavasnannes Crereeene feeeeenes 6.1
Cross-Section Tape ee-v:sss. G eteedececceatancacaenas vecaes 6.2
ROD SUDYOULINES aevsvvseseennnencrversnnanns een N -
7. Theory ...... cecescescsreansaacasennrans T eeee Tl
MODRIC=ERC tevevovverccsacacosannennns cesanonacas esacssnee Tol
Two-Dimensional Synthesis eveveeveseereioereracavares cesse Teb
OPTT vevvvvavacsanenessessesnosnnannnnsns Ceteecanacacaaan . 7.7
HISTRY oc-vvcreoncornnans Ceetasecsssteteresaacnnaasennnne . T7.16
8. Sample PrODLEM «eeveceeereoneennneenns e eretaeene e 8.1
G. References -...cccvevaencan ctmenresnareranes cheecoecscansonns 9.1
Appendix A. The ERC Equations ................ cvesersrsescsenaoas Al
Appendix B. Basic MODRIC Equations seveececeancnaaccacenaa.. eeee B.l
Appendix C. TFission-Product Treatment ...... ceeeen vereeereens Cil

Appendix D. The Processing Study Option ...... camnes cecsscacassse Dol
FOREWORD

The evolubtionary nature of the ROD program (see Seetion 2, The History
of ROD) has led to certain practical limitations on the information pre-
sented in this report. We have described the uses for which the program
was intended, the theory and methodology employed, and rather completely
the information required for spplying the program. We have not attempted
a ccomprehensive description of the programming itself.
ABSTRACT

ROD (Reactor Optimum Design) is a computer code for simultaneously
optimizing the core design and performing the fuel-cycle analysis for
cireulating-fuel reactors. It comsists of & multigroup diffusion calcu-
lation, including multiple thermel groups with neutron upscatter, in one-
dimension or in two-dimensional synthesis. combined with an eguilibrium
fuel-cycle caleulation. Cross sections in the CITATION format are required.
The egquilibrium caleulation is a detailed model of the fuel cycle, in-
cluding the effects of processing and of nuclear transmutation and decay.
Fuel-cycle costs and fission-product concentrations are calculated, the
fission products by an independent calculation from internally stored two-
Zroup cross sections. Special features of ROD are an optimization routine
based on the gradient-projection method, a flux-plotting cption, and a
subprogram for simple time-dependent calculations based on reacticn rates

from the main program.

Keywords: %breeding performance, compuber codes, fluid-fueled
resctors, fuel-cycle costs, nuclear analysis, optimizations, conceptual
design, cores, delayed neutrons, equilibrium, fission products, neutron
£lux, parametric studies, processing, time dependent.
ACKNOV.LEDGMENTS

A number of perscns, not excluéing the suthors of this report, have
made significant contributiors <o the development of the ROD code.
H. F. Baumen and H. T. Kerr have been the principle users ¢f RCD and have
guided its development frcm the user’s standpoint. The theoretical de-
velopment has been the work of L. G. Alexander, T. W. Kerlin, and
o. W. Craven, Jr. The programming was done by J. L. Lucius and
G. W. Cunningham. Sections of pre-existing programs which have been in-
corporated into ROD (see Chapter 2, The History of ROD) were written by
J. Replogle, W. 7. Kephart, M. J. Bell, and R. S. Carlsmith.
COMPUTER COLE ABSTRACT

Neme: ROD

Computer for Which Code is Designed: The code is designed for
computers in the IBM-360 series which have directly addressable
storage of 300 thousand words or more.

Problems Solved: For nuclear reactors the code solves the eigenvalune
problem with or without a critical concertration search for one
dimension or for a two-dimensional synthesis, giving flux and fission-
density distributions. It performs a fuel-cycle analysis, including
costs, for circulating-fuel reactors, either at equilibrium, for
continuous processing, or time-dependent (by assuming & separable
time-dependence) for batch or continuous processing. The equilibrium
problem may be solved for up to three independent fuel or fertile
streaems. The equilibrium concentrations for the fuel-chain nuclides
and up to 200 fission-product nuclides are obtained. The optimum
values of selected core-design and fuel-cycle paremeters may be ob-
tained, based on the maximizatior of a selected function related to
the reactor performance.

Method of Solution: The neutromics calcenlation is performed by &
multigroup, one-dimensional or synthetic two-dimensional diffusion
calculation, including multiple thermal groups with neutron upscatter.
The equilitrium calculation uses the reaction rates from the diffusion
calculaticn to determine the equilibrium concentrations of the fuel-
chain nuclides. The concentrations of the fission products are ob-
tained from & self-conteined two-group calculation. An iterative
process is continued until the diffusion and equilibrium ceslculations
converge on a common set of nuclide concentrations. The optimization
is based on the gradienteprojection method. The time~dependent calcu-
lation (optional) uses average reaction rates from the main calcu-
lation to ecmlculate the concentrations of the principle fuel nuclides
as a function of time.

Restrictions on Complexity of the Problem: The major limiting values
which restrict the complexity of a problem are 15 energy groups,

 
14.

including four thermal Yé;roups, 30 nuclides per region, 10 regions

per dimension, 2 dimensions, 50 nuclides in the equilibrium calcu-
lation, 200 fission products, and 20 optimization variables.

Typical Machine Time: The running time on the IBM 360/75 varies widely
depending on the type of problem. Single cases require from about
one minute for a one~dimensional problem with six groups (5 fast,

1 thermal) to gbout 5 minutes for a two-dimensional synthesis problem
with 9 groups (5 fast, 4 thermsl). Cases run as part of & series (as
in an optimization) run in a half to a third the time required for a
single case (because they are started with the flux distribution from
the previous case). An optimization with five variebles, nine groups,
in one dimension runs in about an hour.

Unusual Features of the Program: The ability to closely model the
behavior of a ei:rculating-fuel reactor, including such factors as the
loss of delayed neutrons from fuel circulating outside the core, is an
unusual feature of the program. Others are the availability of rou-
tines for optimization and for flux plotiing.

Related and Auxiliary Programs: The code is designed to use a micro-
scopic cross-section tape generated by the code XSDRN.

Status: ROD is in production use at ORNL on the IBM 360/75 and 360/91.
Machine Requirements: About 300 thousand words of core storage and
three I/O devices excluding input-output and system requirements are
needed by the progrem. Two additional I/0 devices, for awxiliary
output and flux-plotting, are optional. Plotting requires a CALCOMP
CRT plotter.

Programming Language Used: FORTRAN IV.

Operating System: IBM 05/360 with FORTRAN H compiler.

Programming Informastion: The program ccnsists of about €000 FORTRAN
statements. Arrays of fixed dimensions are provided for all data
within the program, which requires about 300 thousand L-byte words of
core storage.

Users Information: The code and report may be cbtained through the
Argonne Code Center at Argonne National Laboratory.
xiii

15. Reflerences:

1. H. F. Bamman et al., ROD: A Nuclear and Fuel-Cycle Analysis Code
for Cireulating-Fuel Reactors, USAEC Report ORNL-IM-3359, Cak
Ridge National Lsboratory.

2, N. M. Greene and C. W. Craven, Jr., XSDRN: A Discrete Ordinates
Spectral Averaging Code, USAEC Report ORNL-IM-2500, Oak Ridge
National Laboratory, July 1969.

H, F. Bauman

G. W. Cunnipgham, ITI
J. L. Lucius

H, T. Kerr

C. W. Craven, Jr.

Oak Ridge National Laboratory
P, 0. Box Y

Oak Ridge, Tennessee 37830
1.1

CHAPTER 1

INTRODUCTION

The Rod (Reactor Optimum Design) code is unigue among reactor anai-
ysis codes in two respects; it was developed for the core design and fuel-
cycle analysis of circulating-fuel reactors; and it incorporates a package
for the optimization of certain design parameters. It is limited to one-
dimensional, or a synthesis of two one-dimensional, neutron diffusion cal-
culations, and is therefore suited for conceptual design studies of re-
actors rather than the detailed calculation of a given core geometry.

Circulating-fuel reactors, which include the aqueous homogeneous and
the molten-salt reactor types, are different in several characteristics
Prom fixed-fuel reactors. The fuel is perfectly mixed so that its com-
position at a given time is the same everywhere in the system. However,
there may De more than one {iuid stream, &s vwhen & fertile stream is used
as a blanket, or the fuel may be circulated through a fixed moderator, as
in the molten-salt reactor, so that the overall core composition may be
different in different core zomes. Circulating-fuel reactors are usually
designed for continuous processing of the fuel to remove fission products
and to adjust the fissile concentration. In many designs, the fuel
reaches an equilibriuvm composition in a relatively short time. The calcu-
lation of the reactor performance at equilibrium is then 2 most important
consideration. Finally, in circulating-fuel reactors, delayed neutrons
emitted from the fuel circulating outside the core, as in a heat exchanger,
are largely lost to the chain reaction and must be accounted for in the
reactcr neutron balance. The ROD code has been designed to take all of
these factors into account.

ROD also includes a subprogram for nonequilidbrium calculations,
‘designed to celcilate an average performance for & reactor over some
interval of time (e.g., a reactor lifetime), either with batch fuel pro-
cessing or with continuous processing. This option may aiso be used to
calculate the aversge performance c¢f a reactor in the intervel from start-
uwp until equilibrium is established. The method uses average reaction
rates from & space-energy dependent calculation to calculate the time-
dependent concentrations of the most important nuclides. This 1is, of
1.2

course, an approximation, in that space-energy and time-dependent effects
are not considered simultaneously.

The optimization package in ROD is based on the gradient-projection
method, or the method of steepest ascent. It will very a given set of
parameters (e.g., core dimensions, processing cycle tiwes) within limits,
in & series of cases to sesarch out the values of the parameters which
maximize the value of a given objective function (e.g., the breeding
ratio, the inverse of the fuel-cycle cost). Optimization may be used in
combination with a parameter survey; for example, the effect of a given
parameter can be determined wito other parameters adjusted to their

optimum values for each case, rather than merely held fixed af some arbi-
trary value.
2.1

CHAPTER 2

THE HISTORY OF RCD

The ROD code was not "written” but rather "evolved". Parts of ROD
were written, put together, taken apart, and revised by a number of people
over a long period of time. A code with the scope and versatility of ROD
could probably not have been attained without this long periocd of develop-
ment. However, the evolutionary cheracter of ROD has resulted almost in-
evitebly in some disadvantages, chiefly that the input and output lack a
consistent format, that many sections of the code have not been described
by their au‘thofs except in the original FORTRAN, that parts of it are
seldom used or cbsolete, and that the program has become complex and dif-
ficult to change. Fortunately, the code was considersbly unified in
adapting it for the IBM System 360 computers in 1968.

In the beginning (in the 1950's) fluid-fuel reactor caleulations were
made with the one-dimensional diffusion-theory code GNU, written in machine
languege at General Motors Corporation, and ERC (for equilibrium reactor
calculation), an equilibrium code writter in FORTRAN at ORNL. In 1960,
GNU was replaced by MODRIC,l also a one-dimensional diffusion-theory code,
but written in FORTRAN, which made modification of the program more prac-
tical. At this time, of course, the neutronic and equilibrium calculations
were performed separately.

The moment of conception for ROD came in 1961 when J. L. Lucius,
under the direetion of L. G. Alexander and T. W. Kerlin, joined MODRIC
and ERC into a single code called MERC.Z In this cambination, & neutron
diffusion calculation altermated with an equilibrium calculation, so that
reaction rates were sqpp]ied from the diffusion calculation to the equi-
librium cz_a.lculation, and new equilibrium concentrations were supplied to
the succeeding diffusion calculation until both converged on a single set
of concentrations. This process is still the heart of the ROD calculation.

Over the next few years the code was expanded and improved. The
fission-product treatment in ERC was expanded to treat first 75 and later
125 nuclides. The number of fuel streams for whicn equilibrium could be
caleulated was increased from two to three. A provision was added for
calculating the withdrawal of fuel at a final concentration beyond the
2.2

equilibrium burnup — useful in caleculating certain solid-fuel cores such
as a pebble-hed. A two-dimensional synthesis was developed, by

C. W. Craven, Jr., in which & two-dimensional calculation was synthesized
from two one-dimensional calculations; for example, a cylindrical core
from an axial and a radial calculatiocn.

In 1964, T. W. Kerlin suggested that the most efficient method for
finding the set of parameters which gave the best performance in a parti-
cular core design was to use a computerized optimization technique. 4&bout
this same time, W. L. Kephart at the Oak Ridge Geseous Diffusion Plant
had developed an optimization code (unpublished) based on the gradient-
projection method.” We decided to link MERC with the optimization pack-
age; the resuits was a combined code called OPTIMERC. Development of
OPTIMERC continued through 1965 and it was used extensively for molten-
salt reactor caleulations through mid-1968.

The OPTIMERC code had one major operational fault, which was that
the entire program would not fit into the core of the IBM 7090 computer
in use at that time. During a calculation, therefore, information was
continually stored and retrieved from magnetic tazpe. This resulted in
long running times — up to 2 or 3 hr for complex optimization prob-
lems — and frequent job failures because of tape input-output errors.

The TBM 360 Model 75, which became availsble at ORNL in 1967, seemed
ideal for a large program like OPTIMERC because of its large core ca-
pecity. To take full advantage of the new computer, however, it was
necessary to reprogram OPTIMERC to eliminate much of the information
handling. We decided not only to reprogram OPTIMERC for the Model 75,
but to take this opportunity to integrate better the various parts of
the program, and to enlarge the capacity of the code to handle larger
prob_}ems.

The prograrming of the new code, which was named ROD (for Reactor
Optimum Design), was undertaken by G. W. Cunninghasm, urder the direction
of J. L. Lucius and the guidance of C. W. Craven, Jr., H. T. Kerr,and
H. F. Bauman. The important new features of the ROD code were:

i. A1l operations, after reading of the cross-section information,
are contained in core.
2.3

2. Multiple thermal groups (with upscatter) are permitted.

3. Two-dimensional synthesis is performed by energy groups.

4. ERC was expanded:

a) up to four materials are permitted, of which three may bde
treated as fuel streams,

b) the meximum number of nuclides in the equilibrium calcu-
lation was expanded from 25 to 50,

¢) the maximum number of fission-product nuclides was ex-
panded fram 125 to 200.

5. Standard optimization variables and objective function were
built-in (no programming required).

6. A standerd CITATION cross-section tape is read.

T. Cross-section sets are assigned by region; up to five cross-
section sets are permitted.

At sbout this time, M. J. Bell, of the CORNL Chemical Technoleogy Divie-
sion, needed a method for calculating in detail the effects of the pro-
cessing removal of various fission products from a molten-salt reactor. He
used the basic ROD calculation, bubt substituted his own subroutine for the
calculation of the fission-product absorptions. This treatment is now
an option in ROD. Its use is limited to single cases (i.e., optimization
is precluded).

In 1970, an option was added to ROD to permit the calculetion of the
average performesnce of & reactor over one or more batch processing cycles.
It is based on a zero-dimension, one-group, time-dependent code (un-
published) written by R. S. Carlsmith in 1966, which we expanded and re-
vised for inelusion in ROD. Called HISTRY, it takes reaction rates from
the diffusion calculation to caleulate the concentrations of the important
Tuel nuclides as a2 functionof time. The time-weighted average concen-
trations are then supplied to the next diffusion calculation, and this
iterative process continued until there is no further change in the
average concentratians.

This brief history of ROD shows how it bas grown and develcped over
a period of years, and we can only assume that further additions and im-
provements will be made. Some fubture developments that have already been
suggested are:
2.l+

1. Replacing MODRIC with ORNL's latest diffusion code, CITATION,
which would permit finite-difference two-dimensional calculations.

2. Reprogramming of the "solid-fuel" option (which was not included
in the conversion of OPTIMERC to ROD).

3. The standardization of the input format and the elimination of
dual nuclide-identifications.

3ince some of these changes, and others not yet conceived, may be
made in the future, this report has been organized so far as possible
into independent chapters.
5.1

CHAPTER 3

THE FUNCTIONS OF THE ROD PROGRAM

The ROD program consists or four principal parts called MODRIC, ERC,
OPTI, and HISTRY. The functions of these parts, as well as several
important options, are described in this section. A flow diagram of the
program is given in Fig. 3.1.

MODRIC is the neutronics section of the code. It is a multigroup,
one-dimensional or synthesis two-dimensional, diffusion-theory calculation.
It can perform a criticality search, or simply solve the eigenvalue prob-
lem. It can be run independently, without preparing input data for the
other sections of the code. The main output of MODRIC is the critical
concentrations and the flux and fission density distributions.

ERC calculztes the equilibrium composition of the reactor and per-
forms the economies caleulations. It requires reaction rates from MODRTC
and in turn supplies nuclide concentrations for the next MODRIC iteration.
Tterations between MODRIC and ERC proceed until the calculations converge
on & common set of pueclide concentrations.

ERC calculates the equilibrium concentrations of the principle fuel
nuclides, for which reaction rates are cbtained from MCDRIC, and also
the concentrations of up to 200 fission-product nuclides, for each of
which it mskes a two-group calculation based on a thermal cross section
and a resonance integral. The two-group cross-section data, along with
fission yields, are stored in a separate section of the data deck known
as "permenent data."

The 250 ERC nuclides may be divided into as many as ten groups for
processing, each with its own processing equation in each stream. Cost
data mey be supplied for any of the principle nuclides from which the
various components of the fuel-cycle cost are calculated. Most of the
ROD output is assembled in the ERC section, including the neutron balance,
fissile inventory, breeding ratio, fuel yield, and feed and production
rates for the prineiple nuclides.

OPIT is an optimization package, based on the gradient-projection
method. It controls the running of a series of cases, in which certain
reactor parameters may be varied systematically within limits, in order
3.2

ORNL-DWG 71-1884

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
I |
l |
| [
{
| |
VARS TO MERC | I
LINKAGE '
“ | |
| |
| VARS ERC TO MODRIC MODRIC TC ERC |
(VARIABLES SPECIFIED) l L INKAGE LINKAGE i
| | |
| e |
l l
e | | |
* l o l
' |
| l
! 2 conversen? |
{ |
i
o ves__ 3
] OPTICNS 0

 

 

Fig. 3.1. Flow

 

 

 

OPTI TO MERC
LINKAGE

 

|

 

 

OPTI

 

 

 

MERC TO OPTI
LINKAGE

 

l

 

 

of the ROD Program.

 

 

 
3.3

to find, within some tolerance, the set of parameters which gives a
maximm value of 2 prescribed objective function. Parameters which may
be varied by OPTI include region thicknesses and fuel volume fractions,
the boundary position between adjoining regions, processing cycle times,
and, in HISTRY, the time in a batch cycle that the feed is switched from
one fissile fuel to anocther. The objectiwve function is & sum of the
following components, each of which may be weighted with an optional
constant coefficient, including zero: breeding ratio, fuel yield, recip-
rocal fueli-cycle cost, reciprocal discounted fuel cost, specific power,
an inverse function of the meximwm fast flux, and a function of the breed-
ing gain and the specific power called the conservetion coefficient.

HISTRY is a subprogram which can calculate the concentrations of the
principle fuel nuclides as a function of time, rather than just at egni-
librium. It uses reaction rates obtained from MODRIC-ERC (MERC), and
supplies time<weighted average nuclide ccncentrations to the next iterw
ation of MERC. The iteration process proceeds until MERC and HISIRY con-
verge on a common set of concentrations. The HISTRY option is designed
to calculate the average performance of a reactor during a batch proces-
sing cycle, or over a reactor lifetime of several processing cycles. It
calculates the discounted fuel cost, based on a present-worth calculation
of all purchases and saleg of fissile fuel and carrier materials, over
the lifetime of the reactor for & specified discount rate. It alsc pro-
vides useful information on the feed and production rates and inventories
of the principle fuel nuclides as a function of time. Its main limitation
is that the space-~energy calculation is considered independent of time;
which resulis in some lack of rigor in the burnup calculstion. ROD-HISTRY,
therefore, should not be considered as a substitute for more sophisticated
‘burnup codes.

Among the main options available in ROD, perhaps the most useful is
the "variables specified” option. This is a provision in the OPTI section
of the code to control the running of a series of cases in which any of
the standard optimization parsmeters ("varisbles") are preset to any de-
sired values. After the base case, essentially the only data reguired
are the values of those parameters which are to be changed from the
3.4

preceding case, since all other data are held fixed. Furthermore each
case is started with the fiux and fission density distribution from a
previous case, which gives, typically, a factor of 3 or 4 saving in
computation time over the running of an identical series of cases in-
dependently.

The output options are very important. So much output is potentially
avaeilable from the (typically) 60 or 70O cases of an optimization problem
that we have devised &n elsborate system for selecting output. The ceases
are divided into four categories as follows:

1. A base case [the first case run).

2. The final or cptimum case from an optimizaetion; or the "variables
specified” cases when this option is selected.

%. The final case from each optimization cycle (gradient projection).

L. Intermediate cases in an optimization.

The output itself is divided into 29 segments, any or none of which
may be gelected for each type of case above. In addition, an input edit
of 15 segments is aveilable. Two independent sets of output may be
specified for any problem. The first, or "detailed" output, can include
any or all of the edit options. The second, or "short" output, can in-
clode most of the edit options except scme of the iInfrequently required
tables. Either set may be omitted.

Finally, an opticn is available for flux rlotting. It is included
in the output options by type of case as deseribed gbove. The plotting
information is recorded on magnetic tape for use on a CRT (cathode ray
tube) plotter. The fluxes for any or all energy groups may be selected
for plotting on either & linear or logaritimic scale.
k.1

CEAPTER 4 -

INPUT DESCRIPTION

The input for ROD is divided into four main sections:
A. MODRIC, including HISTRY

B. ERC
C. Fission product and delayed neutron data
D. OPTT

The input description is given as concisely as possible. Items marked
by an asterisk are discussed in greater detail in the following section,
"Discussion of Input.” The numbers following the varieble name, where
glven, are suggested values for the data. These are intended tc be helpful
to users with little or no experience with ROD, and may not be appropriate
for every case.

The input instructions are intended to be self-explanatory; hcwever,
the following introduction may be helpful.

The diffusion calculation is one-dimensional, along a line such as
& reactor axis or radius. The reactor composition must be specified' for
the various regions along each line of computation. The results of the
calculation are applied to a three-dimensional volume, ususlly in spher-
ical or cylindrical geometry. For 2-D synthesis caleulasions s in partic-
ular, it is necessary to specify the composition of the subregions of the
reactor that are not along an axis or radius of calculation. Usually a
number of subregions are of the same composition; for example, axial,
radial, and "corner” blanket subregions may all have the same composition.
A superregion is a set of subregions all of the same camposition. The |
superregion composition 1s specified by assigning a volume fraction for
each material to that superregion. |

A material is defined by its nuclide composition. There are two
classes of materials; those of fixed camposition, such as a graphite
moderator or a Hastelloy-N structural material, and those of camposition
determined by the feed, processing, and nuclear reaction rates of the
system. The latter materials are referred to as streams; a fuel stream
for a molten-salt reactor, for example, typically consists of the carrier
salt nuclides, the fissile and fertile nuclides, and the fission-product

nuclides.
4,2

The program limjitations on the number of materials, regions, and

other input parameters are summarized in Table Lk.1.

Teble 4.1l. Program Limits for RCD Imput Parameters

 

Limits in MODRIC

Number of materials L

Note: The first three materials may
be treated as streams (with

processing ).
Number of nuclides, per region 50
Number of search nuclides 30
Nunber of energy groups 15
Number of space dimensions 2
Number of regions, per dimension 10
Nurber of superregions 20

Limits in ERC

Number of principal nuclides 50
Number of fission-product nuclides 200

Iimits in OPTL

Number of wvariables 20

 
4.3

Section A. MODRIC

 

Card Number, Descripticn, and Format
A-1 Title (18ak).

A-2 Comment (18Ak).

A-3 Program control (¥2.0,I2,I1).

 

 

 

Columm Name Su%i;sl?d Description
1.2 FITF Dimension option
>0 Two=dimensional synthesis
<0 One dimensicn
3.0 MAYXEP 5 Maximm nurber of MODRIC-ERC iteratlons per
case |
5 MERC MODRIC cnly opticn

>0 MODRIC and ERC
=0 MODRIC only
Cards A-% to 7. Output optians for detailed printout. See Teble
L.2. The detailed output may be omitted by means of a dummy control
card (refer to Chapter €).
A-lt Base case (case zero), (50I1).
A-5 Final case in an optimization; variables specified cases {50I1).
A.5 Final case in each optimizatior cycle (5011).
A-T Intermediate cases in an optimization (50I1).
Cards A-8 to 11. Output options for short printout. (Same as cards
AL to T.) To omit short printout, leave cards A-8 to 11 blank.
A-12 Convergence information.* (2I3,6E10.4,I10)

1-3 NRFLX Not usezd

4.6 NRFLXN i Flux pormalization
= 1 Normalize dimension 1 true flux to
dimension 2

= 0 Do not normalize as above

 

¥*
Asterisks refer to Chapter 5, Discussion of Input.
"

Table 4.2. Output Options

Enter 1 where output is desired. Ctherwise enter
0 or leave blank.

 

 

Column
on Output Table Controlled
Carxrd

1® ¥ODRIC data by group, region, ard dimension. Usually omitted.

2% Macroscopic cross sections and homogenized atom densities by
region after each criticality search. Usuzlly amitted.

38 MODRIC data by region and dimension. Usually cmitted.

La Normalized 2-D synthesis MODRIC data by nuclide, region and
dimension after each MODRIC pass. Usually omitted.

5a Data supplied as input to ERC. Atom densities and reaction
rates by auclide and material.

6 k-effective and upscattering date by iteration. Usually
cmitted.

7 The main ERC output table end neutron balance.® Atom den-
sities, inventories, and feed and production rates by nuclide
and materizal.

g Fission product atom densities and ebsorptions by nuclide.

9 Atom densities supplied to MODRIC each MERC iteration.
Usually amitted.

10 Region thicknesses and other region information. (This table
is also obtained in option 21.)

11 Volumes, total and by material, by super region.

12 Processing information.

13 MODRIC neutron balance by group and dimension.

14 Neutron absorptions and productions by region and dimension.

152 Meacroscoplc cross sections by group, region, and dimension.

16 Homogenized atom densities by region and dimension.

17 MODRIC fluxes and figsion densities. Specify for "MODRIC
only" runms.

18 Normalixed point fluxes and fission density distribution.

198 Exercise option to plot fluxes.

20P Table of optimization data. Usually cmitbed.

210 Region thicknesses and other region information. Super-

region volume fractions.
4.5

Table 4.2 (contd)

 

 

Column
on Output Table Controlled
Card
Eéb Objective function output summary and optimization variables
used.
23 Edit of de2%ta supplied to the HISTRY subprogramn.

Note: Options 24 to 38 control the edit of input informetion. These
options are ignored excepi when specified for the base case (case
zero).

2k Initial atom densities by material.
25 MODRIC control and search information.
26 Cross-section listing. Enter the specified integer to obtain

one of the following fouvr options:

O No output

1 Title of each cross-secticn set

2 Title and list of nueclides in each cross-section set

32 Complete listing of each cross-section set

27 Energy group boundary table.

26 Dimension information.

202 Initial fission density distribution.

30 Initial homogenized atom densities by region.

Z1 ERC input card edit.

32 Nuelide correspondence table.

33 Subregion--super region correspondence. (The "picture" of
the reactor.)

3k Super region volume fractions.

352 Permanent data. (Atomic mass, beta decay constant, two-
group cross sections, fission yield, by nuclide.)

36 Delayed neutron data.

378 Scurce and recycle-fraction data by material.

33 HISTRY input edit.
1.6

Table 4.2 (contd)

 

Columm
on Output Table Controlled
Card

 

Note: Options 39 to 43 are oubtput options for the HISTRY subprogram.
Option 39 must be specified tc obtain any of the HISTRY output.

HISTRY k_.. by iteration may be specified-on card A-18.

39 Atom densities, inventories, eigenvalues, and conversion
ratios.

40 Cumulative purchases.

k3 Incremental purchases.

Lg Neutron absorptions and productions.

L3 Costs.

502 ERC output for non-converged nuclides by iteration. Used

only for study of ERC convergence.

 

Soption not available for short printout.
bOption not available for base case.
“ERC output every pass may be specified on card B-l.
7-16

17-26

27-36

3746

4756

5T7-66

67=76

FIGVPL

cgNCSo 2.0
CONIT1 2.0

CDELT 1.0E-05

CTL  1.0E-OL
CP1CEN 1.2

NITEXT 3

k.7

Factor for true flux and fission power

dersity caleculation. Enter:

Cne-~dimension:

Sphere ' 1.0

Cylinder Overall height

Slab Product of overall lengths

of second and third sides

Two-dimension:

Slak ~cylinder 2.0

Slab-slab Overall length of third

side
Factor by which the convergence criteria
are tightened for base case.
Factor by which the convergence criteria
are loosened for the first MERC iteration.
The convergence requirement on the change
cf upscatter acceleration treatment from
iteration to iteration.
The tolerance on the upscatter acceleration
treatment approach to wnity.
Factor which limits the change in de/dk in
the criticality search.
The minimum number of MODRIC iterations
required after the upscatter treatment has
converged.

Cards A-15 to 17 control the flux plotting option. If no flux plots
are required, these cards may be left blank. The symbols used to
designate the neutron erergy groups are given in Table 4.3.

A<13 Flux plot control (4I5,4E10.4).

1-5
6-10

IWADFP
IGRID

Not used

Grid optiomns:

= 1 Linear

2 Semilog, space-coordinate logrithmic
5 Semilog, flux logrithmic

= 4 Iog-log

H

i
4.8

Table L.3. Symbols Used in Flux Plots

 

Syl Neutron Energy
ol Group

 

3, 13

h, 1k

2, 15

10

DK < O X + B G

 
5.9

11.15 TPTLIN Point-line options:
= 0 Points only
= 1 ILine only
2 Points and line
5 Histogram
16-20 NPRNG Range options:
= 0 Range determined by data extrema

= ]. Range specified in next four fields

21=30 XMINN Minimm of space-coordinate range
3140 XMAXX Maximum of space-coordinate range
41-50 YMIWN Minimum of flux range
51-60 YMAYX Maximum of flux range

A-1k,1-36 Plot title (9Ak).
A-15,1-36 Space-coordinate axis label (9AL).
A-16,1-36 Flux axis label (QA4).
A-1T7 Groups to be plotted (16I2).
1-2 NgGPS Total number of groups to be plotted.
3 NPLTGP(I) Remsining fields identical. Enter group
number of each group to be plotted.
Cards A-18 to 34 are for input to the HISTRY subprogram. If HISTRY
is not used, these cards may be left blank.
A-18 HISTRY conmtrol information (6I5,30X,E10.4k).

1-5 ISTRY Activate HISTRY subprogram.
>0 Yes
= 0 No
6-10 KIIM 30 limit on X iterations.
11-15 KBUG 0 Printout, K by iteration.
> 0. Yes
= 0 No
16-20 NV Activate converter-breeder option.”

= 0 Not activated
When keff is greater than 1.0:
21-25
26-30

61-70

NCY
NF'S

SWCH

A-10 HISTRY data (8E10.k).

1-10
11-20
21-30

3140
41.50
51-€0
61-T70

71380

TMAX

DT 0.05
XPR 240
PUR

TAU

THMX

SHIFT 7.0
TPLK 0.0001

4.10

1 Bypass the criticality search
2 Withdraw feed in criticality search
= % Withdraw uranium mixture in fuel in

criticality search; return as needed before

i

resuming normal feed

Number of batch processing cycles.

Second feed key nuclide number.* (Ignored
when SWCH is zero. )

Time, full-power months, of switch to
second feed. Zero for no second Teed.

Time, full-power months per cycle.

Time step, months.

Number of time steps per normal data
period.®

Feed options:

= 0 5 feed, Th fertile

> 0 Pu feed, Th fertile

< 0 Low enriched U feed

Time constant for -Pa removal, if any;
fraction removed per second.

Maximm atom density of fertile nuclide
if fertile-buildup cption is used.”
Otherwise zero.

The number of data printouts before the
frequency of output shifts from twelve
times normal to normal.*

Tolerance on Kepe.

A-19.1 Carrier cost control information™ (6I5).

1-5

NS

The number of carrier nuclides for which
costs are specified in ERC. Zero to five
permitted.
L.n

610 Nsc(z) Remaining fields identical. Enter the
ERC number of each carrier nuclide for
which & cost is specified.

A-20 Control of the restarting atom densities (after the first cycle)
(1475.1)

Enter z restart factor for each nuclide in order.
1.0 If the nuclide is recovered and recycled.
0.0 Otherwise.
The above entries apply to the fertile nuclide if its stom density
is held constant throughout the cycle. If it is allowed to deplete
during the cycle, enter:

2.0 If it is recovered and recycled.
-1.0 Otherwise.
A-21 to 34 HISTRY nuclide information.* One card for each nuclide
(4+E10.L,2%,48).
1-10 CSP(T1) Tnitial stom density for nuclide T.
1120 FRM(I) Feed fractions for feed nuclides. Other-
wise 0.0.
21-30 FRS(I) Feed fractions for second feed, if any.
3140 SL(I) Removal fraction for sale, if any.
4350 DENT(T) Nuclide identification.

This ends tke HISIRY date section.

\A-35 designates a series of cards of identical format on which the

initial atom density of each nuclide ic each materizal is entered.
A-35 Atom densities by material (I2,5{¥3,E1l.Lk)).

1-2 MX Material number.*®
3-5 TH(I) MODRIC nuclide number (NPET).
6-16 TEMP{I) Atom density (atom/tn-cm) in material

{(RHPT). Must be non-zeroc.
Remaining pairs of fields identicel. Use as-many cards as reguired
for each material. Enter the material number in the I2 field on
each card.
A-36 Blank card
4.12

A-3T Material names (BaAkL).
1-8 HPLMAT Name of material 1.

Remaining fields identical. Enter names of materials in order.
4-38 designates a series of cards, cne feor each region level in
dimersion 1, in order, on which the super region number is entered
for each subresgion of the reactor. These cards create a 2D
"sicture" of the super-region distribution in the reactor. For 2-D
problems in cylindrical geometry, by convention, the axial dimension
is 1 and the radisl dimension is 2.

A-38 Super-region subregion corresporndence (picture).* (I3,3X,10I3).

1-3 L Fegion level in dimension 1.
L6 Blank
7-9 NTEMP(K) Super-region number assigned to subregion

defined by region levels (I,K), where K

is the region level in dimensicn 2.
Other fields identical. Eanter in order of K up to number of region
levels in dimension 2.

A-3Q Blank card
A-40 designates a series of cards, one for each super region, in

order, on which the volume fraction of each material is entered.
A-LC Volume fractions of materials in each super region (12,284 4E7.4).

1-2 J Super-region number.
3-10 HELVAL, Super-regicn name.
11-17 vFS(M,T) Volume fraction of material M.

Remaining fields identical. Enter volume fraciions in order by
material. Nobte: The volume fraction of the last materiai X is set by
the code so that the volume fractions sum to 1.0 in each super region.

A1 Group structure.* (3I2)
1-2 NG Tetel number of energy groups
3k NETH Group number of the last epitherms] group.
5.6 NTH Group number of the last thermal group.

 
4.13

A-42 MOLRIC convergence information.™ (I10,2E10.4,5I10).

1-10

11-20
21-30
3140
41-50

5160
61-70

71-80

TCPN

EPS1

EPS2

I™MAX

MAXPPT

=

-

3.0E~05

3.0E-0Oh

300

100

Convergence options:

= 1on keff

2 on fission density

= 5 on both

Convergence requirement on keff in suc-
cessive iterations.

Convergence requirement on fission den-
sity in successive iteratioms.

Masdimum number of MODRIC iterations ailowed.
Not used

Not used

Perform criticality sezrch:

= 1 Yes

= 0 Ko

Maximum number of optimization cases (may
not exceed 200).

When no criticality search is specified, omit cards A-43 and k.
A-L3 MODRIC search information.* (I10,3E10.4,I5).

1-10

11-20
21-30

3140
L1hs

ICH

RMD
XX

EPSL
MS

3

1.0
5.0

1.0E=0h

= 5 Search on atom density.

Note: No other MODRIC search cptians are
used in ROD.

Desired ke e

Tnitial estimate of dc/dk, fractional
change in atom density of search nuclides
per unit change in k.*

Tolerance for ke e

The material altered by the criticality

search.

Al Criticality search nuclides.” (2LI3).

1-3
46

NSE

NPC(L)

Total number of s=arcn nuoclides.
Reraining fields identical. Enter the
MODRIC number of each search nuclide.
Uze an additional card (same formet) if
needed.
Cards A-45 to 54 specify informstion. for dimension 1.

A-lt5 Two-dimensional synthesis information.* (L0I2). May be left blank
for a 1-D case.

1-2 KPRE(LD) Core region number.
3.k IR3C(T, Ten identical fields. Enter numbers of
LD)

regions, in order, to which core region

buckiing from other dimension is to be

applied.

23-2k MTAB(1I, Ten identicel fields. Enter numbers of
Lp) regions, in order, to which transverse

leakage is to be distributed.

A-L6 designates a series of cards, one for each region, in order,

on which the region information is entered.

A-L6 Region information (2X,284,E10.3,3I5).

3-10 AME Region name. |
11-20 THICK Region thickness, cm. i
2125 MESH Number cf mesh spaces.™ |
26-30 RXS ROD order number of croses section set to |

be applied in region. {(Limited to integers
1 through 5).
31-3% NF Region contains fissilz material.™
= 0 No
= 1 Yes

ALT Blank card
AX8 chell thickness.* (7(I3,E7.4)). If no shells are gpecified, use a

blank card.
1-3 IB(I) Region number to the inside of the shell.
4.0 TEMP (I ) " Shell thickness, cn.

Remaining pairs of fields identical. Enter region numbers and shell
thicknesses in order. Erd data with a blank field. Use an additional
card if needed.
L.15

4-kS designates a series of cards on which the boundary conditions
are specified ty energy group, one card for each set of boundary
conditions, in order by groups. <f the same boundary conditions
apply to all groups, ounly one card is required. The conmstants a,
b, ard ¢ apply to the origin, or imner boundery, and 4, e, and f to
the outer boundary. Appropriate values for the constants may be
selected from Table L.k.

Teble 4.k. Boundary Condition Constants

 

Constant

 

Boundary Condition
acrad DbDore cor T

 

Reflected, current zero 0.0 1.0 0.0
No return current, extrapo-

lation distance = 2.13D 0.94 1.0 0.0
No return current, extrapo-

lation distance = 2.0D 1.0 1.0 0.0
Flux zero 1.0 0.0 - 0.0

 

A-L9 Boundery conditions (272,I6,6E10.4).

1-2 IL Group number of higlest energy group for
which boundary conditions apply.
3.k IT Group number of lowest energy group Tfor
which boundary conditions apply.
5«10 XCK Not used
11-20 TEMP{1) a (refer to Table 4+.2).
2130 TEMP(2) b
31-40 TEMP(3) ¢
h1-50 TEMP (L) d
51-60 TEMP(5) e
61-70 TEMP(6) f
L.,1%6

A-50 designates & series of cards on which are entered the attenu-
ation coefficients for shells between regions, if any. If all

coefficients are 1o be 1.0, or if no shells are specified, use a2

blank card.
A-50 Shell attenuation coefficients by energy groun (3(212,2E8.L4)).
1-2 IS(L) Group
3L TH(L) Region inside of shell.
5.12 TEMP(L) g (refer to Eq. B.9, Appendix B)
13-20 TEMH(L) h (refer to Eq. B.10, Appendix B)

Remsining groups of four fields identical. Enter coefficients in
order by group and region. End data with a blank field.

A-51 Ceametry, buckling, and initial fission density distributior options

(312).

1-2 IPEPT(ID) Geometry option:
= 1 Slab
= 2 Cylinder
= 5 Sphere

%l IBGPT(ID) 2 Buckling option:
= 1 Group and region dependent
= 2 Independent of group and region
= 3 Input by group and regiocn
Note: The buckling option is ignored in
spherical geometry, or when the 2-D
synthesis is specified.

5.5 IFgPT(ILD) 2 Initial fission density distribution:
= 1 Flat
= 2 Cosinpe

Omit card A-52 unless buckling option 1 wes selected. Omit for sphere
and 2-D synthesis. '
A-52 Buckling optior 1.* (3E10.k).
1-10 TWD(1D) Gamne,
13-20 YH(LD) yorh, am
21-30 Z(1LD) Z, em
4.17

Card A-55. Omit for sphere. Blank for 2-D synthesis.
A-5% Buckling option 2 (El0.Lk). Omit unless option 2 was selected.
1-10 BSQ{1D) Value for buckling.
Card A-54%. Omit for sphere, 2-D synthesis.
A-5k Buckling option 3 (7E10.4). Omit unless option 3 was selected.
Enter bucklings in order by group and region.
For a two-dimensional synthesis, repeat cards A-45 to 54 with infor-
mation for dimension 2. Otherwise, this ends the dimension infor-
mation.
A-55 designates a series of cards on which the ERC nuclide number is
indicated for each MODRIC nuclide, by material. Requires at least
one card per material. Use as many cards as needed. Nuclides mzay
be entered in any order. Fields may be left blank where convenient.
Oomit A-55 for "MCDRIC only” rums.
A-55 MODRIC-ERC nuclide number correspondence (I2,10(I3,1X,I3)).

12 J Material nmumber.
3.5 NTEMP(I) ERC nuclide number.*
T=10 IDTEMP(I) MODRIC nuclide number.

Remaining pairs of fields identical. Fields may be left blank.
A-55 Blank card. Omit for "MODRIC only".
A.5T7 Blank card. Omit for "MODRIC only”.
A58 Cross section set assigmment.™ (5I3).
1-3 KX@DR(I) Set number of cross gection set on CITATION
tape to be stored in core.

Remaining fields identical. Up to five cross section sets may be
stored for use in the ROD calculation. The order in which they are
specified esteblishes the ROD order mumber for card A-46.

- For MODREC-only, end the data here.
4,18

Section B. ERC

The card mumber in the ERC section not only identifiies the input,
but is alsc an integer variable, entered in columms 1 to 23 on each card,
which defines the format for the data on that card. This allows the data
cards in this section to be assembled in any order, and, although they

are usually arranged in seguence, no error 1s introduced by misplacing a
card within the section.

B-1 ERC data (3X,9I5).

Column

 

1-3
L8

9-13
14-18
19-23
228

29-33

3438
3943

Suggested
Neme Value
KARD ool
1gD 0
NEW
MAX Lo
189
IFPg 2
TRE
NRy1
NgPP

Deseription

Output by MERC iteration option.*

= 0 Normal

> 0 Output options 7, 9, and 38-43, if
specified, printout for each MERC iteration.
Not used

Maximum mumber of ERC iterations.

Kot uwsed

Fission product option (refer to Appendix
3).

O Omit fission product alculation

1 A11 fission product nuclides ecalcunlated
in ERC.

= 2 Normal. Reaction rates for selected
fission products may be calculated in
MODRIC.”

ERC number of fission product reference
element.*

W

u

‘Not used

ROD options:

= O Optimization

1 Variables specified

2 Base case only.”™

—1 Processing study option.™

n

0
L".. 19

L8 NWECV Processing cost option.™
> C Volume basis
< 0 Qther
B-2 ERC convergence and other data (3X,6E10.4).
1-3 KARD Qo2
4-13 C¢NVEG 1.0E-Ok Convergence, ERC atom densities.
1423 B26 0.5 Atom density damping/forcing coefficiemt.*
2h .33 B27 2.0 Limit on change of atom density per
iteration.”*
3LA3 P Reactor power, Mw(thermal).
Lh 5% F36 Plant factor.*
5h 63 E36 Thermal efficiency.
B-3 ERC residence times and other data (3X,6E1C.Y).
1.3 KARD 0035
I GES3 Fission-product resonance integral.
123 AK Not used
24 33 B5% Fuel residence time, in core, sec.
(For loss of delayed neutrons calculation.)
3h 43 B56 Fuel residence time, out of core, sec.
Ll 53 BST o.L Scaling factor for processing plant capital
cost.™
54«63 WINT Interest rate, fraction per annum.

B-4 Tolerance for MERC convergence and other data (3X,3E10.k4).
1-3 KARD o0

4-13 W7 Fission-product thermsl spectrum factor.*
1423 SEPS 1.0E-03 Tolerance for MERC convergence.
2433 UPLTJ 1.01 Limiting factor for change of the recycle

fraction per iteration.*
Cards B-5 to 7 are currently no® activated.
B-8 Transverse dimension factor (3X,E10.4). Qmit for sphere or 2-D

synthesis.
1.3 KARD 008
L.13 FUDGE Transverse dimension factor:

Cylinder - oversall height, em
k.20

Slab - product of second and third over-
all lengths, cm®.
Cards B-9 to 13 specify the processing information. In the B-9 ard
B=10 series, one card is required for each material to be processed
as a stream.
B-O Processing data (3X,315).

1-3 KARD GOS

4.8 J Material number.

913 NTIME Number of processing cycle times to be
defined for msterial J (limit 10).

%-18 NPGEQ Number of processing equations to be de-

fined for material J (limit 10).
Card B-10 may be cmitted if all times are in days.
B-10 Time wnits (3X,I5,1084%).

1-3 XARD 010G
L8 J Material number.
9.12 TUNITS(1) DAYS The first time is the master cycle time

which must be given in days.
Other fields identical. Enter units for processing times i. order.
Onlv the following entries are permitted:
SECS
MINS
HPUR
DAYS
YEAR
Card B-1ll designates & series of cards, one for each processing time
for each material stream on which time and cost information are
entered. Cards may be amitted for times not being used, without
changing the "number of times" or card B-9.
B-11 Processing cycle times (3X,21%,2E10.4).
1-3 KARD 01l
4.6 J Meterial number.
T=9 NT Time number.
L.21

10=19 GPTIME Processing cycle time, in units specified

(J,NT) on card B-10.* FEnter full-cepacity
operating time, not calendar time.

20-29 WPCV(J, Processing cost factor, volume basis.™
NT) (Zero permitted.)

B-12 designates a series of cards, one for each processing equation,

on which the times used in each equation are indicated by entering

1.0 in the proper positions.

B-12 Processing equations.* (3X,2I3,10F6.0).

1-3 KARD 012
L6 J Material number.
7=9 K Processing equation number.

Remaining ten fields identical. Enber 1.0 in each field correspomding
to a cycle time to be used in the processing equation.

(Time)
10-15 (1)
16.21 (2)
22-27 (3)
28-33 (%)
34-39 (5)
%0-45 (6)
4651 (1
52-57 (8)
58-63 (9)
6k -59 {10)

Card B-13 designates a series of cards, one for each processing
equation, on which the group of auclides to be treated by each
equation may be given a keyword identification.

B-13 Processing nuclide group names (3X,2I3,3A%).

1-3 KARD 013

L6 J Materiel number.

7-9 X Number of the processing equation.
10-21 HPLPGE Keyword name of the nuclide group to be

treated by the equation.
L.22

Cards B-1L4 to 19 are currently not activated.
B-20 designates a series of cards, one for each material to be
treated as a stream.

B-20 Stream data (3X,I3,LE10.4,I10,E10.%).

1-3 KARD 020

L6 J Meterial number.

7-16 SPIVEV External volume, ft>.%*

17-26 (3) Not used
27-36 sTP(J) Holdup ime in the processing plant, days.®
3746 STR(J) Operating time on reserve fuel, days.™
47-56 NW1(J) 0 Withdrawal option:

0 Fluid fuel

1 Solid fuel. (Note: The solid fuel

option is currently not activated.)

5766 XE21(J) A fixed poison fraction (used for xenon).*
B-2]1 designates a series of cards, one for each group of contiguouis
nuclides for which information is identical, for which nonstandard
values of the following data are to be entered. (Cmit for nuclides
for which standard values apply.)

Neame Std. Vaiue
Fraction processed per cycle B(1,T) 1.0
Fraction removed in processing SCE(I,J) 1.0
Fraction recycle to same
streem {5=JD) RCF(1,J,JD) 1.0
Fraction recycled to othner
streams (JAJD) RCF(I,J,JD) 0.0

B-21 Removal and rzcycle data* (3X,I2,213,11,288.4,I2,4E7.4).
1.3 KARD 021

L5 J Material number.

6-8 I First nuclide for which data applies.
9-11 Il Last nuclide for which deta applies.
12 NESCE E and SCE data coantrol:

O Use standard values.
1l Read values fram following two fields.
4.23%

13-20 DuML Enter E(I,J), fraction processed per cycle.

21-28 DUM2 ™ter SCE(I,J), fraction removed, or lost
in recycling.

29-30 NRCF Pecycle fraction contrcol:

0 Use standard values.
1l Read from next four fields.

31-37 DUM(1) Recycle fraction to material 1.
3844 DM(2) Recycle fraction to material 2.
L5-51 DUM(3) Recycle fraction to materiel 3.
5253 DuM(4 ) Recycle fraction to materimel L.

B-22 designates a series of cards, one for each principel nuclide
in each meterial treated as a stream, on which the feed, ataom
density, and recycle options are entered. The options are listed
in Tabie 4.5. Cards may be omitted for nuclides for which all data

are zero.
B-22 Feed, atom density, and recycle optioms.* (3X,213,77.3,13,5%,3I5).
1-3 KARD 033
b6 J Material number.
7-9 I ERC nuclide number.
10-16 Q(I,J) Special feed rate or atom density
17-15 TUMp specifications. See Teble 4.5
2520 M??fi(l, Feed option (Q).
30-34 N(I,J) Atam density optiom (N).
35-39 J§?5(I, Recycle option (J).

B-23 designates a series of cards, one for each principsal nuclide,
on which are eniered data required for the material balance.
B-23 ERC material balance data (3X,715,2E10.3,28:%,I3).

1-3 KARD 025

46 I ERC nuclide number.

T7-9 Ie(1,1) Processing source nuclide nmumber.™
10-12 IP(1.2) Processing source nuclide number.
13.15 I7(1,1) Transmutation source nuclide number.

16.18 IT(1,2) Transmutation source nuclide number.
i .2k

Tsble 4+.5. Feed, Atom Density, end Recycle
Options, Card B-22

 

Option

 

Number Description
Feed Options:

0 No feed.

1l Feed rate calculated by ERC to maintain the equilibrium or
critical concentration.

2 Feed rate specified as input. Enter (kg/dey) in the form
#0.7yyixx in colunns 11-19, where xx is the exponent for the
data yyy.

3 Feed rate proportional to the feed rate of another nuclide.

Enter in the form ixx.yyyfzz in columns 10-13, where:

¥ = the reference feed nuclide
t.yyyizz = the ratic of the feed rate of nuclide I to that
of the reference feed nuclide.

Atom Density Options:

o
1
2

Atam density held constant.
Equilibrium atom density calculated by ERC.

Critical atom density calculated by ERC. This option
permitted for one nuclide only.

Atom density adjusted by ERC to keep reaction rate constant.

Special &% option. Celculates the average atom density of
238y over the reactor lifetime for startup with 25U feed.
Enter in the form #xx.yyy:zz in columms 10-13, where:

xx = the core lifetime, calendar years.
+1'Vy+zz = the ratio of <38y to &5y in the feed.

Befer to Eq. (4.32), Appendix A. Specific for 235U as
ERC nuclide 5.

Specifies a8 pseudo-ntclide representing the loss of delayed
reutrons.

Specifies a pseudo-nuclide for the fixed poison fraction.

‘Specifies & psewdonuslide representing -the lumped fission
products.

 
34'.25

Table 4.5 (contd)

 

Option
Number

Description

 

recycle Fraction Options:

0

Recyele fractions held constant.

The following options allow ERC to calculate the recycle fractions for
the fuel nuclides for a breeder reactor in which excess fuel may be
produced for sale. In each case, any remaining fuel from materials 1,
2, ard 3 is recycled to material 1.

1

2

\J‘!

pSl

D

Specifies the key nuclide for the sale of excess fuel,
any, based on the composition of material 1.

Specifies the key nuclide for the sale of excess fuel,
any, based on the mixed composition of materials 1, 2,
3‘

Sverifies the key nuclide for the sale of excess fuel,
any, based or the camposition of material 3.

Specifies the key nuclide for the sale of excess fuel, if
any, vased on the mixed composition of materials 2 and 3.

Specifies a nuclide to be sold in proportion to the key
naclide in option 1.
Specifies a auclide to be sold in proportion to the key
nuclide in option 2.

Specifies & nuclide to be so0lid in proportion to the key
nuclide in option 3.

K

K BB

Specifies s nuciide to be sold in proportion to the key
nuclide in cption %.

 
19-21
202k
25-34
35k
e 52
2335

ID(I,1)
ID(I,2)
AMASS(T)
AMBA(T)
HYLL
NPGFNI(I)

b 26

Decay source nuclide number.
Decay source nuclide number.
Atomic mass of nuclide I.

Beta decay constant for nuclide I.
Nuclide name.

Number of processing group.*

B-2L designates a series of cards, one for each nuclide, in each

m:terial, for which cost data are assigned.
B-2i Value of materials (3X,213,3E10.L).

1-3

L-6

7-9
16-19

20-29

50-39

4049

KARD

J

I
wl(I,J)

W3(I,J)

Wi (I,J)

Ww(I,J)

o2k

Material number.

ERC nuclide number.

Value of nuclide I in materials 1, 2, ani
3 in the reactor system. The value
assigned to nuclide I in materisl 1 is
autamatically assigned in materials 2 and
3 also.*

Unit processing cost, $/kg (non-zero for
weight-basis calculation omly).*
Processing unit capital cost factor {(non-
zero for capital cost basis calculation
only).*

Value of nuclide I in material J as feed
material. Assign for each material.

B.25 designates & series of cards, one for each nuclide for which
an interest rate different from that entered on card B-35 is speci-
field for the calenlstion of an inventory charge. Omit if none are
different.
B-25 Interest rates {3X,I3,E1C.L).

1-3
) -6

7-16

KARD
I
w2(1)

025

ERC nuclide number.
Interest rate, fraction per year.

Caxds B-26 tc 28 are currently not activated.
k.27

R-29 List fertile nueclides (3X,23I3).
1-3 ¥ARD 029
26 122N1

- Remaining fields identical. Znier the ERC nuclide numbers designating
the fertiie nuclides for the breeding ratio calculation. Refer to
Eq. (A.37), Appendix A.

B-30 Iist fissile precursors (3X,2313).
1-3 KARD 030
L6 122N2
Remaining fields identical. ZEnter the ERC nuclide numbers of fissile
precursors, for example, = Pa.

B-31 List fissile nuclides for breeding ratio (3X,23I%..
1-3 KARD 031
k-6 122D
Remaining fields identical. Znter the ERC nuclide numbers designating
the fissile nuclides for the breeding ratio eczlculation.

B-32 IList fissile nuclide for mean eta (3X,231I3).
1-3 KARD 032
L6 123D
Remaining fields identiezl. ZEnter the ERC nuclide numbers designating
the fissile nuclides for the mesn eta caleulation. Refer to Eg. (A.38],
Appendix A.

B-33 List fissile nuclides and precursors for inventory (3X,23I3).
1.3 KARD 033
L& 129N
Remaining fields identical. Enter the ERC nuclide numbers designating
the fissile nuclides ari precursors for the fissile inventory calcu-
lztion. Refer to Eg. (A.45), Apperdix A.

B-3k List fissile nuclides and precursors for processing loss (3X,23I3).
1-3 KARD 0%
L6 L30X
Remeiring fields identicai. ZEnter the ERC nuclide numbers designating
the fissile nuclides and precursors for the processing loss calcu-
lation {numerator). Refer to Eq. {A.46), Appendix A.
L.23

B-35 List fissile nuclides for processing loss (3X,25I3).
1-3 “{ARD 035
L_§g L300

Remaining fields identical. Enter the ERC nuclide numbers desig-

nating the fissile nuclides (only) for the processing loss caleu-
lation {denominator).
Cerd B-36 is currently not activated.
B-3T7 List fissile nuclides for fixed poison fraction (3X,23I3).
-3 .ED o337
L5 ¥21L

b

Remaining fields identicel. Enter the ERC nuclide mmbers desig-
nating tne fissile nuclides (only) for the fixed poison-fraction
celculation. Refer to Eq. (A.34), Appendix A.

B-3C Blank card. This ends ERC datz Sectiorn B.
k.2c

-~

Seection C. Fission Product and Delayed Neutron Deta

This seetion cortains the fission yields, two-group cross sections,
and traznsmutation and decay chain data for up to 200 fission-product
nuclides. It is referred to as permasnent data, because, once set up, it
may be used for the calculation of any thermal reactor. However, the
processing group number has been superimposed on the permanent data in
this section, and this would be expected tc change with the processing
method employed. Finally, the last six cards in this section contain
the data for six groups of delayed neutron precursors.

C-1 Fissionable nuclide correspondence (5I3).
Fields may be left blank for mnclides not being used. Specific for
ZRC numbers in thke range of 1 to 1.

1-3 NFPYS(1) ERC nuclide number for Z52Th.
L6 KFPYS(2) ERC nuclide number for <=7U.
7-5 NPYS(3) ERC nuclide number for 22°U.
10-12 NFPYS(L) ERC nuelide number for Z3CU.
13-15 NFPYS(5) ERC nuclide number for Z>°pu.
1515 NFPYS{6) FRC nuclide number for <%ipu.

£-2 designates a group of up to 200 cards, one for sach fission-
product nuciide. End the fission-product datea with a blank card.
* Fissicn-product permanent data (5I3,E9.1,8E6.1,2A%,T60,I1).

1-3 I ERC nuclide nuxber (starting with 51).
46 ITl Transmutation source nuclide number.
7«3 ITe Transmmtation source nuclide number.
10-12 ID1 Decgy source nuclide number.

13-1%5 IDz Decey source nuclide number.

16.-2L BDECAY Beta decay constant.
25230 SA2200 Absorption eross section (2200 m/sec).
31-35 RESINT Resonance integral.

3742 YT(1) Fission yield from <>=Th.
L3048 YT(2) Fission yield fram 23U,
195k ¥YT(3) Fission yield fram 35U.

55-60 YT(%) Fission yield from 23%y.
L.30

61-66 YT(5} Fission yield from Z>°Pu.
67-72 YT(&) Fission yield from 2%pyu.
3-79 BAL1 Nuclide name.
8o NPG Processing group number.
C-3 designates a group of six cards, one for each delayed neutron
group.
C-3 Delayed neutron data (I2,7E10.L).
1.2 I Delayed neutron group.
z-12 AMAZ{T) Decay constant.
13-22 YT(1) Delayed neutron fraction for =*2Th.
23-32 YT(2) Delayed neutron fraction for 2371,
33.L2 YT(3) Delayed neutron fraction for 23%y.
L3-52 YT(%) Delayed neutron fraction for =38y.
5262 YT(5) Delayed neutron fraction for 2>°pu.
63-72 YT(6) Delayed neutrom fraction for 2%ipu,

This ends data Section €. It is the end ¢f the data for running
a single case; that is, when the optimization or variables specified
options are not required.
N

§g:tion D. 0OPTI

The data for optimization or for the varisbles specified option are

entered in this section.
section may be omitted.

When these options are not required, this

The coefficients for the standard terms of the objective function

are entered on card D-i.

The valu2 of each coeffiyient determines

the weight of each objective in the optimization. Terms that need
not be considered in a particular optimization may be given zero

O=1 Objective function coefficients.* (TE10.L).

coefficieants.

1-10 #ECL
11-20 ¢BC2
21-30 #BC3
3140 gnck
L1.50 #BCS
51-60 @BCS
61-70 ¢BCT

D=2 Allowable flux (E10.4).

1-1C

FLXALW

Coefficient for breeding ratio.
Coefficient for fuel yleld.
Coefficient for reciprocal of fuel-cycle
cost.

Coefficient for reciprocal of fuel
specific inventory.

Ccefficient for the group one (damage)
fast flux factor.

Coefficient for the conservation coef-
ficient.

Coefficient for the reciprocal of dis-
counted fuel cost (from HISTRY).

The allowable group 1 (fast) flux. Must
be none-zero if the coefficient of the
flux factor (card D-1) is non-zero.

Cards D3 to S5 form a set; one such set is required for each

variablas~specified case.

The indevendent varisbles which may be

specified (that is, assigned fixed wvalues) or optimized are given
in Table 4.6. They include region thicknesses, the volume fractions
of each material in each superregion, the locations of region
beundaries, processing cycle times, and the time at which feeds may
be switched in a HISTRY cycle.
4,32

Table L.6. Types of Variables

 

 

 

Indices
Description 111£e
er 7 5 3
Region thickness 1 Region Dimension None
Volune fraction 2 Material Super None
region

Bound.a.rya' 9 Region Dimension None
Processing cycle

time 10 Material Time number None
Feed switch time 11 None

 

8omit cards D-3 to 5 for optimization.
D-3 Number of variables (2I5).

1-5 NVSPC
6-10 NVAR

Case numbar.

Rumber of variables specified for given

case.

D-4 Case information (I5,E1C.L,IS).

1-5 MAX3S
6-15 HELP

16-20 NSET

D-5 designates a group of cards, one for each variable whose value
is t2 be changed from the preceding case.

MERC iteration l1limit f~r case.

Factor by which all convergence criteria

are multiplied for case.

Control of initial flux, fission density
distributica, and atom densities:

= O Taken from previous case
> 0 Taken from base case.

this group must equal NVAR cn card b-3.

Note: A card "D-9, optimization variable” is compatible in format
and may be used as a D-5 card. The fields containing optimization

information are not read at this point.

The number of cards in
4.33

D-5 Specified variable (LI3,10X,E10.%,L0X,24ak).

1-3
L6
7-9

10.12

23-32

7580

ITYPE(T)
INDX1(T)
INDX2(I)
INDX3(I)
XB(1)
HPLYPT

Varisble type number (see Table %.6).
First subscript.

Second subscript.

Third subscript, if any.

Specified value for varisble.

Nume of variable.

End the varisbles-specified date with a blenk card. This ends the
data deck for a variables-specified run.

D=t Number of OPTI cycles (I3).

1.5

MAXCYC 10

Maximum rumber of optimization cycles
(gradient projections).*

DT OPTI control (3I5,E5.0,5810.%4).

1.5
6-10

11.1%

16-20

21-30

3140

L1-50

51-60

N

KNTVEC -l
NH@I1-D 0
CM¥; +1.0
ALPHA 0.05
BETA .10
SFl 1.1
EFS 0.G003

Numbe:r of OPTI variebles.™

Vectsr count for parallel.tar-eat
acceleration methcd.

= —«1 New caze. (No other options
activated)

Fumber of cycles & variable is held at a
limit (zero permitted).*

Ascent/descent control.

+1.0 to maximize objective function
=1.0 to minimize objective function
Fracticn of range eacnh variable is moved
to calculate derivatives.

Fraction of range that the controlling
varisble is moved in the initial step along
a vector.

Step factor by which BETA is multiplied
af'ter each successful step.

Step tolerance; a lower limit or the
fractional improvement ir the objective
finciion required for a step to be con-

sidered successful.
61-T0

PER

0.003

L.3k

Cycle tolerance; & lower limit on the
fractional improvement in the objective
function per cycle required Ior continu-
ation of the optimization.

D-8 OPTI control (I5,2E10.%,I5).

1-5

6-15

16-25
26-30

51-35

NORED

ALPLIM
KAFLSM

2.0

0.02

Bypass the reduced-step option:

= 1 Yes

= 0 No

Factor by which ALPHA and BETA are reduced
in the reduced-step option.

Minimum ALFHA permitted.™

Activate end-effect option 1. Ridge-
analysis factor reduced when derivative

is negative.*

= 1 Yes

= 0 %o

Activate end-effect option 2. Variables
with negative derivatives restrained after
an unsuccessful first step. Interval scan
anitted.

= 1 No

= O Yes

D-9 designates a group of cards, cne for each variable, on which
the starting value and range are entered. The number of cards in
this group must equal N on card D-7.

D-9 Optimization variable® {LI3,3E10.4,25X,FS5.C,2ahk).

1-3

4.6
7-9
10-12
13-22
23-32
3342

ITYPE(Z)

INDXL(I)

INDX2(1)

INDX3(I)
(1)
¥B(T)
XH(I)

Verisble type number. {Refer to Table
.6).

First subzcript.

Second subseript.

Third subscript, if any.

Minimum value {lower limit of range).
Initial value.

Maxcimum value {upper limit of range).
k.35

e8.T2 SLEFAC 1.0 Initial value of ridge-analysis factor.
(1) Mzst be non-zero and nct greater than
1.0. Usually 1.0.
This ends the ROD data for an optimization runm.
This ends the input descripticn.

 
5.1

CHAPTER 5

DISCUSSION OF INPUT

Many of the features of ROD require mwore exposition than is appro-
priate for the preceding section, "Description of Imput”. Such items,
which have been arked with asterisks in the cdescription, are discussed
ir this section. 3Iach discussion is keyed to the appropriate irput card
number.

L-12 Convergence Information
Refer to card A-L2 for a discussion of the convargence criteria.
=13 HISTRY Control Information

Convertertreeder optian. At the beginning of a bateh processing
cycle, a converter reactor may have a temporary excess of fissile material,
because fission-product poisoning has been reduced to zero and fissiie
material {e.g., =-U) mey be aveilsble from prezursors (e.g., °>>Pa’ pro-
duced in the previous cycle. HESTRY normally "sells” any excess fissile
material. ‘'(he converter-breeder ontion provides the following alterna-
tives, which apply when kefi‘ exceeds 1.0:

1. Shut off the eriticality search. This allows the excess fissile
to be retained in the system, thus deigying the point at which fissile
Teed must again be resumed. This expedient introduces a small error in
the fissile balanze, (vecause K pe is greater then 1.0) but may be the
vest alternative waen k off only slightly exceeds 1.0 for a short time,

2. Witndrew feed in the criticality search. Note, however, that
it is usuelly not practical to withdraw feed from an actuwal reactor.

5. WVithéraw, in the criticality search, uranium nuclides in the
oroportions present in the fuel stream. This simuiastes withholding some
of the uranium separated from the fuel at the end of a cycle, and feeding
it back as needed during the following cycle, before resuming ncrmal feed.
The simulation is imperfect in that the instantaneous fuel composition is
used rather than the composition at tiz erd of the c¢ycle. This alterns-
tive is recummended when considerable excess fuel is available at the

beginning of a cycle.

 
5.2

Second feed. For reactors with plutcaium feed, where uranium is
recovered at the end of a cycle but plutonium is not, there is an ad-
vantage in switching to a uranium feed near the end of the cycle. To
activate this option, enter the time in the cycle at which the feed is
to be switched and specify the ERC number of the key nuclide of the
second feed (e.g., 5, for 235U, for enriched urenium as second feed). Also
specify the feed fractions for the second feed on cards A-21 to 3k.

A-19 HISTRY Data

Frequency of output. The frequency of output is determined by
specifying the number of timesteps ir a data storage interval. A fre-
quency of ance or twice & year is adequate for most of a typical cycle.
However, the fuel caomposition usually changes rapidly at the beginning
of a cycle, and the program provides for 12 times the normal {regquincy
of output at the begirning of a cycle, that is, data ance or twice a
month. SHIFT specifies the total number of printouts at the higher fre-
quency {starting with the first at time zero).

Fertile buildup option. Certain fuel cycles are characterized Dy
a fissile inventory which starts at e high level and decreases. (me
then has the option of starting with a lower {issile and fortile in-
ventory and adding fertiie materisl with time instead of removing fissile.
To exercise this option, specify the final fertile atem-density desired
as TEMX, and enter a smaller atum-density for the initial velue.

A-19.)1 HISTRY Carrier Cost Control Infcrmalion

Fuel carrier cost data sre usually part of the ERC imput. The
carrier cost in HISTRY is calculeted from the ratio, carrier cost per
kilogram of thorium purchased, whkich is calculated frem ERC data as
follows:

), {sT(d,1)%a(M))

1=I,NS
M=NSC(T)

sI(1,1)

SCR = + WL(1,1)

where
SCR = the carrier cost ratio,
5.3

STI(M,1) = the inventory of carrier nuclide M in naterial 1,

SI(1,1)} = the inventory of thorium in material 1,

WL(I,1) = the cost per kilogram assigned to each nuclide,

NSC(I) = the ERC numbers of the carrier nuclides specified on card
4-15.1,

£-21 AISTRY Nuclide Information

The HISTRY subprogram is set up for a specific canfiguration of the
EZRC data as follows:

Nuclides 1 to 12 in order are: =3Th, =3py, 3y, B4y, 235;, 236y
257y, 238y, 39y, 240py, 4lpy, ang 242py.

HISTRY nuclide 13, fixed sbsorlers, corresponés +o the summation of
ERC nuclides 13 to 50, except 25.

HISTRY nuclide 14, fission products, ccrresponds to the summation
of ERC nuelides 25 (lumped fission products}, 229 (1%Sam), and 2351 (15iam).

A-35 Atom Dens.ties by Materiai

By conventior, the fuel stream is material 1, the fertile stream,
if any; is material 2, and the moderator, if any, is the last material
specified.

A-30 Super Region Subregion Correspondence

The "super region” wes conceived as a convenient method of indicatin
The distribution of materials (that is, the volume fractions) iz regions
of the 2-D reactor that do not lie on one of the calculational axes. It
is convenient to assign subregions of identical campositiorn the same
super region number. (The same form of data is followed for a 1-D reactor,
although the form then has no special utility.)

A<l Croup Structure

When only one therms) group is indicated, the upscatter treatment is
shut off. We have noted that the use of multiple thermal groups with
upscatier, although valuable in certain cases, results in increased
running time by & factoa: € 2 or more.
5.4

A-42 L3 MODRIC Convergence and Seerch Information

A certain balance in the convergence requirements of the wvarious
sections of the code 1s essential to the efficient functioning of ROD.
It is convenient wo use EPSL, the tolerance for keff’ as a reference.
Experience has shown that the ratio of the value of a given cenvergence
criterion to that selected £~ EPSL SHbuld be about as indicsated in
Teble 5.1. The recommended absolute value of EPSL of 1.0 X 10~¢ is
adequate for most single or "variables specified” cases, and gives results
for the breeding ratio and fissile inventory consistent to about one part
per thousand. It is sometimes necessary to tighten the convergence cri-
teria for difficult optimization problems, because the directiorn of the
optimization iepends on the ratio of the results of two very similar
cases, and therefore may be very sensitive to slight imprecision in the

individual casés.

Tabls 5.1. Recommended Balance Among Convergence Criteria

 

 

Infiizbgird Namz Descriphion Rfiii?:eiged
EPSL
A-43 EPSL Tolerance for K pp 1.0
AL2 EPSl Convergence on k_ap 0.3
AL2 EPS2 Convergence on fission density 3.0
A-12 CTPL Tolerance, upscatter acceleration 1.0
A-12 CDELT Convergence, upscatter acceleration 0.1
B-2 CYNVEG Convergence, ZRC atom densities 1.0
B-4 SEPS Tolerance, MERC balance 10.0

 

The balance between tightness of convergence and the running time is
critical for optimizations. When the convergence criteria are set tight
enough (say, 107> for EPSL) to get very precise derivative calculations,
the running times become long. If set too loose (say, 10" for EPSL) the
derivative calculations may became so imprecise as to direct an optimizaticon
55

vector in z false directioan. Optimizations usually run most efficiently

at moderately tight convergance (near 10™% for EPSL).

Two "tricks' are employed in ROD to save running time (refer to card
A-12). We have found that optimization runs can be made efficiently with
relatively looser convergence wren the convergence is tighter for the
tace case. This gives a firr starting flux distribution for the first
optimization case, whereas otherwise The flux distribtution may tend to
cnange over the first few cases even though the convergence criteria are
satisfied. The other trick is tc loosen the convergconce slightly for the
first MERC iteratlon, since it is wasteful for the diifusion calculation
to be tightly converged until it has received a set ol altered concen-
trations from ERC. To accommodate this provision, =nd-rto prevent MERC
from stopping with a fortuitous balance while MODRIZ and ERC are not con-

verged, the program requires a minimumm of two MERC iterations per case.

A-i3 MODRIC Search Information

 

The value of dC/dk, the ratio of change in composition to change in
keff’ is highly dependent on the reactor composition. The best guide
Tor selecting an initial value is tc look &t the final value calculated

in a similar case (given as CPl in the output).
ALl Criticality Search Nuclides

It is usually more efficient to specify as search nuclides all the
nuclides in whe fissile chain rather than just the main fuel nuclide.
Specifically, those nuclides whose concentrations tend to vary with the
concentration of the fuel nuclide should be included as search nuclides,
while those nuclides which tend to reach an equilibrium concentration
independent of the fuel nuclide should not be included. If such "in-
dependent” nuclides are treated as search nuclides, they tend to cause
the ecaleulation to oscillate between MCDRIC and ERC. Suspect this effect

if more than three MERC iterations are required for convergence.
A5 Two-Dimensional Symthesis Information

The *two-dimensional synthesis is described in Chapter 7, p. T.6.
By convention, for a 2-D synthesis in cylindrical gecmetry, dimension
1l is axial and dimension 2 is radisal.
5.6

In the 2-D synthesis, there is a provision for adjusting the flux
in the core to take into account neutrons which leave the core in the
transverse direction. The region specified as "core" can only be the
center region of the reactor. The computed net flow of neutrons out of
this region (by group) determines a buckling for the calculation of
transverse 'leakage" in the other dimension. The leakage neutrons may
be distributed in proportion tc ths abscorptiorns in as many regions as
mey be desired. These are the "transverse leakage distribution"” regions

and should always include the core regicn.

A-46 Region Information

 

A reascnable number of mesh spaces per region might range from 5 for
a small region to 50 for a large region. Avoid large differences in the
size of a mesh interval fram one region to the next. The running time
is not very sensitive to the number of mesh spaces, and is moderately
affected by the number of regions.

The fission-density distribution is calculated over 111 regions

specified as containing fissile material.
A48 and 50 Shell Thicknesses and Attenuation Coefficients

If desired, regions may be separated by "shells" in which the neutron
current may be attenuated (refer to Appendix B).

A-52 Buckling Option 1

This cption calculates the buckling by group and region from the

equations:
Slab:
2 _ T 2 T 2
B™ = (y+ 7D) * (z+ 7D)
Cylinder:
2 _ P 2
B™ = (h + D) ?
where

¥,2,h = perpendicular dimensions of the reactor,

 
5eT

¥ = constant for calculating the extrapclation distance,

D = diffusion coefficient, a function of group and region.
A-55 MODRIC-ERC NWuclide Number Correcpondence

The fissionatle nuclides (for which fission yields are given on

caré C-2) must be assigned ERC nuclide mumbers in the range 1 to 15.
A-58 Cross-Section Set Assigrment

For many calculaticns the use of a single cross-section set weighted
for the average flux-spectrum is adequate. However, Jor cases in which
the flux-spectrum effects are different in different regions of the re-
actor, cross sections appropriate to the various spectra can be prepared
and assigned to the different regicns.

B-1 ERC Data

A "MERC iteration" is one pass through MODRIC and ERC. Typically
two or three MFRC iterations are reguired for convergence. Normally,
output is obtained for only the fingl iteration. For the purpose of code
development, some of the ERC and HTSTRY output may be obtained each ~
iteration.

The fission-product reference element must éorrespond to an artifi-
ciel element in MODRIC which has cross sections for a 1/v absorber with
0_2200 = 1.0. ~

Some of the important fission products (e.g., ~*Sm) may be included
explicitly in the multigroup diffusion czlculation, if desired. To 4o
this, select fission product option 2, 1list the nuclides in the MODRIC-
ERC correspondence table, and include them on the cross-section tape.

Such nuclides are edited separately by material in the second part of the
ERC output table {output option 7), but are included in the lumped fission
rrocducts in the preceding summary neutron balance.

Base-case only‘qption. When this option is specified the progrem
will stop after running the base case (case zero) even when the data deck
is otherwsie set up for the cptimization or variablesespecified options.
Tt may be used to check tThe base case before proceeding with, say, a long
optimization run.
5.8

The processing study option substitutes a more sophisticated fission-
product treatment for that normally used in ERC. It is described in
Appendix D.

Three processing cost options are availeble (refer to card B-24).

For molten-salt reactors, the processing cost depends mainly on the volume
of salt streem processed, and is, therefore, usually caleculated on the

"volume basis.”
B-2 ERC Convergence and Other Data

The atom-density damping/forcing coefficient is a factor by which the
calculated change in atom density per iteration is multiplied. It is
usually set less than 1.0 to dampen cyecling of the atomr densities from
iteration to iteration.

The 1limit cn the change in atom density permitted per iteration is
used Lo help prevent cycling and to prevent atom densities from becoming
negative. The limit is in effect for each ERC iteration after the first.

The plant factor is defined as the anticipated energy production as
a fraction of the energy that would be produced if the plant were operated
continuously at full power.

B~-3 ERC Residence Times and (Other Dsata

The capital ¢o3T of a processing plant is assumed to be proportional
to its capacity raised to a fractional power called the scaling factor.

B-k Tolerance for MERC Convergence and Other Data

The fission-product thermal spectrum factor is defined as follows:

1
W= ————,
7 298
& (75 273
where
Anf = lethargy width of fast energy groups,
T = temperature, °C.

The limiting factor for change of the recycle fraction is used to
dampen oscillations in ERC, and is applied in each iteration after the

first.

 
5.9

B-11 Processing Cycle Times

A processing cycle time is the time reguired to process one system
volume of a material. Processing may consist o several steps, and each
step can have its own cycle time. Usuaily the cost of processing can be
related to one or two main steps, and these may have a processing cost

facter, volume basis, assigned to them, as follows:

PCV . =1U .t(l'SF)
o, n,j 1,Jj ’

where
PCV = processing cost factor, volume basis,

n = number of processing c¢ycle time associated with a given

processing step,
j = material number,

U = unit prceessing cost, dollars per cubic foot of meterial j,
for processing step n in a reference plaut,

t = throughput, cubic feet of material j processed per dey in
step n in the reference plant,

SF = scaling factor. The capital cost of & processing plant is
assumed to be proporticnal to its capacity raised to a
fractional power, the scaling factor.

Processing cycle time 1 is used as a reference cycle time for calculabing
the material holdup in the processing plant. Refer to Eq. (4.40),
Appendix A.

B-12 Processing Eguations

The removal rate of any nuclide in ERC is calculated for each material
stream by the processing equation to which it is assigued. The removal
rate is calculated from the cycle times for the processing steps in which
it is removed. The cycle times that apply to each group eguation are in-
dicated on the processing equation card by entering a 1.0 in the position
corresponding to the number of each cycle time. For example, if the nu-
clides in processing group one are removed from material one in processing

steps 2 and L, the card for material one processing equation one should
5.10

have 1.0 entered in fields 2 (col. 16-21) and & (col. 28-33), which
correspond to cycle times 2 and 4; +he other fields are left blacnk.

B-20 Strezm Data

 

The code calculates the volume of each material in the reactor core.
However, in a circulating-fuel reactor, a considersable volume of the fuel
stream is outside the core in piping and Leat exchangers. To allow the
code to calculate the true reactor inventoriss and inventory costs, the
emount of such extermal volume for each stream may be entered here.

In addition, if the stream is processed, the holdup time in the
processing plant may be entered. The volume of the holdup is then calcu-~
lated as a function of the reactor volume and the ratio of the holdup time
to the reference processing cycle time (processing cycle time 1).

Reserve fuel. If the reactor requires a fissile feed, a fuel reserve
sufficient to feed the reactor for some period of time mey be included in
the inventory. This is calculated from the net burnup (burnup less pro-
duction) of the feed nuclide (see Eg. A.43, Appendix A.

Fixed poison fraction. The strong fission-product poison 1°SXe may
be treated as a special case in molien-salt reactors. It is insclublie in
the salt, and is either removed by gas stripping or is absorbed by the
graphite moderator. Its true removal efficiency is not easily determined.
The problem has been side-stepped by assigning a removal efficiency of
1.0 to the gas-stripping process, and adding a fixed poison-fraction to
allow for the holdup of xenon in the moderator.

B-21 Removal and Recycle Data

The "removal efficiency” in processing may be defined as the produet
of E, the fraction processed per cycle, and SCE, the fraction removed (or
lost, for nuclides which are recycled). Normally the removal efficiency
is 1.0, and E and SCE are automaticelly assigned the wvalue of 1.0. This
value is not appropriate for gll nuclides, however, and other values may
be assigned on cards B-2l. Some examples are:

() Nuclides which are recycled (either back to the stresm fram

which they were removed, or to another stream) such as the uranium nuclides.
5.11

They may be assigned an SCE of 0.0, or a small non-zero value representing
the fraction lost per cycle in processing.

(b) DNuclides which are only partially removed by processing. An
appropriate removal fraction may be assigned.

(¢) DNuclides which are removed in smaller side streams taken from
another processing step. They may be assigned the appropriate processing
fraction. (Alternatively, such nuclides may be placed in separate proces-
sing groups, to which independent processing cycle times. can be assigned.)

(&) To simunlate batch processing. E values greater than 1.0 may be
assigned to groups of fission procducts to simulate the greater removal in
batch processing compared to continuous processing for the same cycle
time. E values less than 1.0 may be assigned to nuclides such as the
plutorniums, when, because of decreasing concentrations during a cycle,
their removal at the end of a bateh cycle is less thar would be obtained
by continuous removal (based on their average corcentrations and the av-
erage carrier discard rate).

Recycle fraction. Any nueclide which, wholly or partially, is not
removed in processing, presumebly remains in its original stream and is
returned to the reactor. In the calculation this is considered recycle
to the same stream, and the recycle fraetion for a stream to itself is
automatically set at 1.0. Similarly, the reecyele fraction from one stream
vo another stream is automatically set at 0.0.

Sometimes it is essential to recycle certain nuclides from one stream
to another, as in the case of a reactor with a separate fertile stream,
cr bianket. The key to the high performance of such reactors is in the
trinsfer of the fissile material bred in the fertile stream to a fuel
stream. DNuclides to be so transferred must be assigned ERC nuclide numbers
In the range 1 to 25, and their recycle fractions set appropriately on
cards B-21. The permitted wvalues for the recycle fractions are 1.0 and
0.0. The lumped fission products may not be transferred.

B-22 Feed, Atom-Density, and Recycle Options

ERC works best when one nuclide, usually the most important fuel
nuclide, is selected as the "key nuclide" for these options. It is

normaliy assigned feed option 1, atom-density option 2, and one of the
5.12

recycle key-nuclide options 1 to 4. It is alsc usually a criticality-
search nuclide ir MCDRIC. This selectlion allows MERC tc either feed cr
sell the key nuclide, as required, as MERC converges on a solution.

For a reactor with a breeding ratio near 1.0, we might wish to sell
one nuclide if the reactor were a breeder, and feed another nuclide if it
were not. In theory, MERC could do this, but in practice, because of
imbalance in ERC, it may sell cne fuel nuclide while feeding the other.
We recommend that a preliminary case be run to determine the breeding
ratio, an! the key nuclide then be selected accordingly.

When the EISTRY option is used, atom-density option 2 (criticality)
may be specified for the key nuclide, but option O should be specified
for the notker nuclides in the fertile-fissile chain. This will hold the
atom densities fixed in ERC at the time-averaged values obtained from
HISTRY.

B-2% ERC Material-Balance Dzata

 

Any nuclide for which one of the non-zero recycle options was speci-
fied (card B-22) must be specified as its own processing source. In
addition, =3>Pa should be specified as a processing source for =77,
assuming that =33U formed by the decay of 237Pa in the processing plant
is returned to the reactor. The processing group number determines the
processing equation applied to a2 given nuclide in each material. Note
that the processing equations are material-dependent while the processing
group assigmments are not. The number of processing groups 1s limited to
ten. The reference processing cycle time is applied to all nuclides for
which no processing group number is specified.

B-24 Value of Materials

 

The value of each material may be specified (for the purpose of cam-
puting inventory and replacement costs) by assigning a value to one or
more nuclides in that material. A different value may be assigned to a
nuclide in the system from the value assigned to the same nuclide as feed
material. This allows, for example, that Z>°U be assigned a higher value
as feed material than as a part of the fuel stream, where it is contami-
nated with =°U.

 
5.15

Three copticns are availsble for computing processing costs. One,
based on the volume of material processed, has already been discussed.
The others are the welght-basis option, based on the weight of some
nuelide or nuclides processed, and the capital-cost-basis option, based
en the unit capital cost of the processing plant. (Refer to Eq. A.60,
Appendix A). The options are selected merely by making the appropriate
coefficient non-zero.

The preccessing unit-capital-cost factor is defined, for a reference

processing plant, as follows:

(€)(1)
U=—m——,
Qs
where
U = unit capital cost factor,
C = capital investment, $,
i = interest rate,
Q = throughput of nueclide I, kg/day,
s = scaling factor (refer to card B-3).

D=1 Objective~Function Coefficients

The standard equation for the objective function is:

C=aB+bY+ce/M+da/IT+~ eF+ fC+ g/D,

where

O
n

objective function,
a,b,c,d,e,f,g = coefficients,

B = breeding ratio,
Y = fuel yield, percert of fissile inventory per year.
M = fuel-cycle cost, mills/kwhr,
T = specific inventory, kg fissile/MW(thermsl),
F = flux factor,

F —-F

-tz
1014

F_ = allowable group-l f£lux; input on caxrd D-2,

 
5.1k

Fm = @ maximum group-l point flux in the core, calculated
in MERC.

When (Fa - Fm) >0, F is set to zero.

C = conservation ccefficient,
100(B-1)
- 2
T2
D = discounted fuel cost (from HISTRY), in mills/kwhr.

D-2 Allowable Flux

 

The allowable group-l flux may be specified as required to limit the
fast-neutron damage to the materials in the Lore. Whenever the allowsble
flux is exceeded, the flux factor, a negative term, reduces the value of
the objective funetion. The net effect is to srift the optimum to that
set of conditions which gives the highest performance consistent with a
peak flux exceeding the allowable flux by same margin which depends on
the relative weight given the flux factor.

D-6 Number of OPTT Cycles

The number of OPTI cycles (gradient projections) should be set to
stop the run before the rumning-time limit is exceeded. An OPTI c¢ycle,
or gradient projection, consists of the calculation of derivatives to
determine a gradient wvector, the taking of steps along the vector, and
the calculation of the maximmm of the objective function along the wvector.
When & run is stopped in the middle of a cycle, the information generated
for that cycle is wasted. Specifying zero as the number of cycles will
helt the calculation at the end of the base zase.

D=7 OPTI Control

Although 20 OPTTI variables are allowed, there are good reasons for
holding the number of variables as small as possible. The most obvious
is to save running time. lLess obvious but perhaps more important is a
certain decrease in precision as the number of variables ic increased.
This may be explained as follows: As steps are teken along a vector,

we can think of the varisbles one by cne reaching their optimum values
5-15

and continuing on beyond, until the variables beyond their optima are
balanced by those that have not reached their optima, and the objective
function reaches its maximm. The greater the number of variables, the
more likely that some of them will stop at some distance from their true
optima. This effect can be countered by tightening the convergence
criterie and reducing the step size ;, but, of course, at the expense of
increasing the running time.

When a varisble reaches one of its limiting values, it may be held
there for any specified number of optimization cycles. During these
cycles, no derivative is calculated for the held variable and it does not

affect the size of step of the other variables.
D8 OPTT Control

The minimum ATPHA is the mechanism which halts the step-reduction
procedure. If a step reduction would result in an ATPHA less than the
minimum, the reduction is not permitted and the optimization is terminated.
The minimum ATPHA determines the "fineness” to which the optimum is lo-
cated. In reactor calculations there is little incentive to locate the
optimum with great precision, and the step reduction is cften bypassed,
or held to one reduction by the choice of minimum ALPHA.

End-effect options. The end-effect is described in Chapter T, p.T7.15.
End-effect option 1 applies a reduction in the ridge-analysis factor when-
ever the derivative is negative, as well as whenever it changes sign.

When this option is used, it may be advantageous to select initial wvalues
for the variables such that the initial derivatives are likely to be
positive.

End-effect option 2 is applied only after a normal first step has
been unsuccessful. This must occur at least once at the end of every
optimization. The variables with negative derivatives are then restrained
(by a factor of 0.01) and steps are started along e vector determined
essentially by the positive derivatives. The interval scan is omitted.
Often, successful steps can be taker along the new vector. In general, we

recommend the activation of end-effect option 2.
5.16

D-G Optimization Variablie

The sensitivity of an optimization to a given variable is influenced
by the range assigned that veriable. The step factors ALFHA and BETA are
defined as fractional factors of the range: therefore, when the range is
large, the steps will be large and the resulting change in the objective
function is likely to be large. The optimization works best when thle
change in the objective function is about the same for each variable —
that is, when the "derivatives” are of roughly the same magnitude. Cften
this is not the case. When the optimization is dominated by one or two
variables, the "ridge effect” may result; that is, the path of the
optimization jumps back and forth across a "ridge” as the valus of the
dominant variable is alternately too high and toc low. The "parallel
tangent" and "ridge-analysis"” techniques are used in OPTI to get around
the ridge effect (see Chapter 7), but the optimization must run several
cycles to accumulate thke irformation required for these techniques. When
+the user knows from experience that a given variable will tend to dominate,
the sensitivity of the optimization to this variable can be reduced by
giving it a small range ccmpared to the range of other veriables. However,
a small range is not always practical. The range should be large enough
to include, with high probebility, the optimum value of the variable
(otherwise the entire run could bz wasted). When the range camnot prac-
ticelly be reduced, the semnsitivity to a given variable may be reduced by
assigning it a fractional initial ridge-analysis factor. The ridge-
analysis factor is applied to both derivative and vector-step calculations.

Frozen variable. If the minimum and maximum vaiues of a variable
are set egual to its initiel velue, it becomes & 'frozen" variable. No
derivatives or other caleulations are made for frozen variables. The
frozen variable has two uses:

(1) To remove a variable from the cptimization, perhaps just for a
particular run, without removing it from the edit ol the variables.

(2) To satisfy the requirement that the region thicknesses on either
side of a boundary variable be variables, without increasing the number
of active wvariables.
6.1

CHAPTER 6

USER INFORMATION

Control Cards

Typical control cards for running ROD on the ORNL IBM 360/75 or
360/91% computers are:

//HFBR JOB (12073)4'Y~12 9104~-2 H BAUMAN', MSGLEVEL=],
// CLASS=FTYPRUN=HCLD4REGICKN=1536K
//R0O0 EXEC LINKNGO sPARNMGLINK=*LIST!,REGION.GO=1536K

//LIMKLFT33F001 DD UNIT=TAPEQ,H?BEL=(,NL),VDLUME=SER=33}> X
// BISP=(CLD,PASS), b X
// DCR=(RECFM=FBS,LRECL=BC,BLKSIZE=3200)

F/7LIMNK GSYSIN DD =

INCLUDE FT33F001
/=
//G0.FTOSFO01 DD SYSOUT=A,DCB=(RECFM=FRAy LRECL=133,BLKSIZE=3458),C
// SPACE= (3458, (200D)4RLSE)C .
//60.FTOSFO01 DO SYSOUT=A,DCB=(RECFM=FBA, LRECL=133,BLKSIZE=3458),
// SPACE=(3458, (600) 4RLSE) |
//GOLETCLFO01 DD UNIT=SYSDA:SPACE={TRK,{27)),DCB={RECFN=F 4BLKSIZE=50}
//GO.FTO2F001 DD UNIT=TAPEQy LABEL=(yNL)s VOLUFKE=SER=2,D1$P=0LD, X
/7 BCB={RECS=VB, LRECL=360CyBLKSIZE=2604)
//GC.CRTTAPE OO UNIT=TAPESsLABEL=(sNL)},DISP=0LDyVOLUME=SER=4S
J/GOLFTIOFOC1 DD =%
/=
/7

 

*When running on the ORNL/91, only two nine-track tapes are avail-

gble; therefore, when plotting is required, the program must reside on
disk.

bFor running with the program on tape. If the progrem resides on
disk, the following cards should be used instead:

//LINKL.FT33F001 DD VOLUME=REF=2272Z17,DISP=SHR, X
// bCB={RECFNM=FBSs LRECL=8L,BLKSIZE=3200)» X
/7 DSNAIPE=AZ 4G4 .P35614.C12073,BAUMANG A

°To omit detailed output, replace these two cards with the following:

//GOFTOSFO01 DD DURNY

d'For plotting only. Otherwise, omit this card so that no plotting
tape will be mounted.
6.2
The input/output devices are listed in Table 6.1. ROD requires a
minimum core region allocetion of 120CK bytes (300K words)-

Table 6.1. Input/Output Devices Used
for the ROD Program

 

 

mzil Stepneme - ddname Use
1 Gfp.FTOLF001 Disk-scratch
2 Gp.FTO2F001 Tepe or disk — cross section
1ibrary
4 G . CRITAPE Tape, for plotting, O track
8 G .FTOSF0O01 Auxiliary output
9 G . FTO9FO01 Standard output
10 G@.FT10F001 Stendard input
33 LINK.FT33F001 Tape or disk for ROD

 

Cross-8ection Tape

 

The cross-section tape used by ROD is in the same format as that
used by the code CITATION.® This binary tape is normally prepared by the
code XSDRN* but may be prepared from cards using the CITATION auxiliary
program.

The format for a cross-section set on the tape is given in Table
6.2.

The n,2n reaction, if any, is included in the fission cross section

 

as follows:
* true
Op = O, + °h,2n s
¥ true
r = Voo + 2.0X Gn’2n s
¥
> ¢
el
“f
where 0" and " are reported on the cross-section tape.

f
6.3

Table 6.2. Format of the ROD Cross-Section Tape

 

Record

 

Number Vexriable Name Description
1 (TPTIT(I),I=1,18)
TPTIT T2 character title for set
2 NTYFE,NTAPE,NGT,
NDS,NUS
NTIYPE Not used
NTAPE Number of nuclides in set
NGT Number of energy groups in set
NDS Not used
NUS Not used
3 (FUS(I),I=L,HGT),

(ET(1),I=1,NGT),
(EM(I),I=1,NGT)

FUS |

ET

EM

Fission source distribution function by

- group

Upper energy of each discrete group
(ev)

Mean energy of each discrete group (eV),
that is, the energy corresponding to
the midpoint lethargy of the group

Repeat records 4 and 5 for each nucligde.

L

NID,N2,N3,N:,
NE:(AME(I);I=176):
(TAB(I),I=1,60)

5 x5 2SR E

Nuclide number

Not used

Not used

Not used

Not used

2L cheracter nuclide name
Not used
6.4

Teble 6.2 (contd)

 

 

Record . . 1=
Nuzber Variable Name Description
5 (ALA(IG),FISS(IG),
ALTR(IG),FNUS(IG),
DUMMY, IG=1,NGT),
((m™w(IG,L),1=1,NGT,
I1G=1,NGT)
ATA(IG) Absorption cross section for group IG
FIss(IG) - Fission cross section for grouwp IG
ALTR(IG) Transport cross section for group IG
FNUS(IG) Neutrons/fission for group IG
UMMy Not used
PMU(IG,L) Total scattering from group
IG to group L
€ NEND
NEND Closure record {==1)

As many sets as desired may be included on the tape, but only five may
be used by the ccde for any given case.

 
€.5

The “Be(n,d) cross section is treated specifically by MODRIC nu-
clides number L and 7O so that the buildup of ©1i from ®Be may be treated
explicitly. For ®Be (MODRIC nuclide number L) the group 1 absorption

cross section on the tape is given by

ot(1) = o (1) + of _ (1) .

n,zn

We say that

and create a MODRIC nuclide number 70O such that

679(1) = o (1) .

n,o
The cross-section tape must have nuclides number 4 and 70 on it, in

that order, where the cross sections listed for nuelide 7O are all zero.

ROD Subroutines

The subroutines in ROD, with their fumction and the location from
which they are called, are given in Table 6.3.
6.6

 

 

Table 6.3. The ROD Subroutines
Subroutine Called From Description
A. Tnput and Caleulation Subroutines

MATN Control calculation.

ZER@ MATN Zeroes cammon.

SPLIT General input.

PART2 SPLIT Diffusion caleulation input.

RPCS1 MATN Processes microscopic cross section tape.

CIN MAIN Equilibrium caleunlation input.

BTMZD ERCUT Converts the equilibrium material densi-
ties to the form used by the diffusion
calculation. (Entry point in CIN)

MgD3cC MATIV Diffusion calculation.

CRSCH MFD8C Criticality search.

RCSHN DPALL,MZDEC Performs & geometry-dependent region
boundary correction on flux snd fission
density integrations.

SAMMY MZD3C Calculates k-effective for two-dimensional
synthesis,

ERCP MAIN,MfiDSC Argument = 0: Calculates ebsorptions,
fissions, and neutrons produced by nuclide
and region.

Argument = 9: Two.dimensional synthesis;
normalizes dimensior one results to
dimension two, and sets up the linkage be-
tween the diffusion caleculation and equi-
librium calculation.

DPALIL MATW,MODSC Argument = 2: Computes region densities

PART2 fror material densities.

Argument = 3: Modifies densities after
criticality search.

Argument = -%: Corrects macroscopic cross
sections after criticality search.

Argument = 5: Calculates a complete set of
macroscopic c¢ross sections.

Argument = 6: Calculates mesh end region
boundaries.
6.7

Table 6.3 {contd)

 

Subroutine Called From

Deseription

 

ERCM
CYCIB

ERCJUT

HISTRY

FLUX
@UTSET
EPSILN
TIGHT
SLACK
BELL

SET

ERCM
ERCM

ERCEUT

ERCM, MATN

Argument = 7: Fission density normalization.
Equilibrium calculation.

Recyele calculation.

Fuel-cycle and economics calculations.

Time ~deperdent calculation.

Argument = 1: Performs the linkage to the
optimization routine. Calculates the
objective function.

Argument = 2: Performs the linkage from
the optimization routine back to the dif-
fusion-equilibrium calculation. Assures
that the super region volume fractions
sum to wnity by modifying the final ma-
terial (moderator) volume fraction. Re-
calculates the diffusion volume fractions.
Performs the boundary wvariable manipu-
lation.

Argmment = 3: Calculates the region vol-
umes and the equilibrium material volumes.
Treats the processing equations by calcu-
lating removal efficiencies.

Optimization routine.

Calculates true flux arnd aversge fission
power density.

Controls the output options in optimization.

Multiplies all convergence epsilons by an
input constant (varisbles-specified cases

only).

Divides all convergence epsilons by an
input constant.

Muitiplies all convergence epsilons by an
input constant.

Interfaces a special fission-product
treatment.

Stores a set of material densities, fluxes,
and -fission densities for use at a later
time.
6.8

Table 6.3 (contd)

 

 

Subroutine Called From Deseription

RESET MATIY Restores the previously saved material
densities, fluxes, and fission densities.
(Entry point in SET)

BEUND ALLDg Changes region thicknesses when the bound-
ary variable option hes been specified for
an optimization or varigbles-specified
case.

FIXQ CIN Input processing routine.

BLPCK itializes CPMMPN/PRESET/.

DATA

B. Edit Subroutines

EDOL SPLIT Edits nuclide number correspondence table.

EDO2 SPLIT Edits super region subregion correspondence.

EDO3 ALID@, SPLIT Edits super region volume fractions.

EDO4 ALIDS, MAIN Edits region thickness, mesh spaces, cross-
section set, whether fissionable, distance
to start of region, and distance to end of
region.

ZD05 MATN its super region volumes and meterial
volumes.

EDOT MATIN,MZDSC Edits macroscopic cross sections.

EDOS MATN Edits the material densities after the
equilibrium calculaticn.

EDO9 MATN,M@D3C Edits homogenized region densities.

ED1C MATN Edits processing information.

ED12 MATN Edits group neutron balance.

ED13 ERCP,MAIN Edits absorptions, neutrons produced, and
fissions by nuclide and region.

ED1h MATN Edits optimization summary table.

ED15 MATN Edits total region absorptions, neutrons
produced., fissions, and ratio of pro-
duccions to absorptions.

ED16 MATN Edits forward leskage, transverse leakage,

absorption, fission, and neutrons produced
by region.
6.9

Table 6.3 (contd)

 

 

Subroutine Called From Description

EDL7 SPLIT ¥dits plotting input cptions.

ED18 SPLIT Edits diffusion calculation convergence
and econtrol information.

ED20 MAIN Edits true flux, Tission density, and
aversge fission power density.

ED21 ED22, ERCAUT Edits short summary table of the eguilibe
rium caienlation results. The table in-
cludes breeding ratio, eta, vield, fuedl-
cycle cost, fissile inventory and proces-
sing losses.

ED22 ERC@UT Fdits equilibrium calculation summary
table.

ED22E ERC@UT Edits equilibrium calculation summary
table. (Entry point in ED22)

ED23 ERCfiUT Edits fission-product densities and absorp-
tions for material one only.

ED2k ERCP Edits input to equilibrium calculation

' from diffusion calculation. The table in-
cludes densities, absorptions, neutrons
produced, and fissions by nuclide and
meterial.

D25 WLKEP Edits input for an optimization case.

ED26 MATN Edits input for a varisbles-specified case.

ED27 ERCHUT Edits input to HISTRY from MERC.

cguT CIN Edits input for ERC.

GONE WLKEP Bdits debugging information for opti-
mization.

DIVID MATN Controls case heading printout.

FHLPR DIVID Prints case heading.

BIFCK Initializes C@MMPN/LETTER/. Stores

DATA characters to be printed.

HT SPUT HISTRY Edits results from HISTRY.

 
6.10

Teble 6.3 (contd)

 

Subroutine Called From Deseription

 

C. Plotting Subroutines

SETPLT MATN Initializes plotting packsage.
ELAPLT MATW Sets up the fluxes t¢ be plotted.
ADDTIY FLXPLT Puts additional heading on plots.
PLTTRM MATN Terminates plotting.

GEGPLT Never called Dummy subroutine.

@PTPLT MATN Dummy subroutine

BPLATS FLXPLT

BLAR BPLT

XY BPLYT

L3 BPL@T

TITLE BPLYT

LINE BPLYT

LINSCL BPLYT

HIST BPLET

CRTGRD RPLAT

RANGE BPLYT

D. Additional System Subroutines and Functions Required

ALGG BPL#T,RPCS1 Natural logarithm function.
ALPG1O CRTGRD Commor. Logarithm function.
cgs PART2 Cosine function.
CRT™ BPL#T,CRTGRD,

HIST, LINE,

PLTTRM, SETPLT,

TITLE
CRTNUM™ ADDTIT,1G,XY
CRTSYM ™ ADDTIT,BPLAT,

1G,LINE, TLTLE

EXP ERCM,MATN Exponential function.
6.11

Table 6.3 (contd)

 

 

Subroutine Called From Descripction
ICcLfcK CLN,ERCM, Returns CPU time in hundredths of seconds.
ERC@UT ,MATN,
MPD3C , WLKEP
IDAY DIVID Returns the date as 8 EBCDIC characters.
SLITE MATN,PART2 Alters status of sense lights.
SLITET MATN Tests and records status of sense lights.
SQRT ELISTRY Square root functiorn.

 

®Plotting routines for a CALCEMP model 835 cathode-ray tube plotter.
T.1

CHAPTER T

THEORY
MODRIC~-ERC™

MODRIC is a typical neutron-diffusion-theory code. It allows 50
neutrcn enexgy groups with downscattering from a group to any of the
following ten groups. It has been medified in ROD, where ornly 15 energy
groups are permitted, but where a multiple-thermal group treatment with
upscatter has been added.

It will perform concentration searches on specified elements. The
output consists of critical concentrations, group macroscopic cross sec-
tions, normalized nuclear events (absorptions, fission, leakage, etc.) by
region and group, absorptions and fissions by material and region, group
flux distributions, and fission density distributions. The basic MODRIC
equations are given in Appendix B.

Basically, ERC solves two main equations. They are:

Wiy 2

= Qij + R,,+ F,.+ T,.+ D,

v . .
i] iJ i ij

j at

Equation (7.2) is just the conservetion requirement, saying that enough
fissiie material must be added (or removed) in iteration s to overcome
the neutron production deficiency (or excess) in iteration (s-1). These

are inner iterations in ERC. The terms are defined as:

 

*This section has been largely excerpted from Ref. 2.
Subscripts:
i =
j=
k =

7.2

volume of stream j, cm”,

atams of nueclide 1 per barn cm of stream ],

time, sec,

feed rate of nuclide i into stream j, atoms/sec,

rate of production of nuclide i in stream j due to recycle
from other streams, atoms/sec,

rate of production of fission fragment i in stream J,
atoms/sec,

rate of production of nuclide i in stream J due to neutron
sbsorptions in other nuclides, atoms/sec,

rate of production of nuclide i in stream J due to radio-
active decay of other nuclides, atmms/sec,

rate coefficient for loss of nuclide i in stream Jj because
of neutron capture, atoms/sec/atam/tarn cm,

rate coefficient for loss of nuclide i in stream j because
of radioactive decay, atoms/sec/atom/barn cm,

rate coefficient for loss of nuclide i in stream j because
of processing removal, atoms/sec/atcm/barn cm,

rate coefficient for production of nuclide i in stream j
because of recycle from stream j, atoms/sec/atom/barn om,
neutrons produced per fission in nuclide i,

reaction rate coefficient, number of fissions in nuclide 1
per atom/barn cm in stresm j in region k per fission neutron
born in reactor,

reaction rate coefficient, number of absorptions in nuclide
i per atom/barn cm in stream j in region k per fission

neutron born in re=ctor.

nuclide,
stream, or material,

region.

The use of stream and region indexes allows reactors with two or more

streams in the same region to be analyzed.
7.3

The sequence of ccmputations goes es follows. MODRIC, with trial
velues of the Nij’ calculates absorption rates which are comverted (in
ERC) to the reaction rate coefficients, Cijk' These are used in Eq. (7.1),
with specified feed and removal rates, to calculate new values of the
Nia' Equaticn (7.2) then indicates the adjustment to be made in the con-
centration of & key fissile nuclide in order to restore the balance be-
tween neutron productions and losses. Since this new concentration is no
longer consistent with the solution of Eq. (7.1l), an adjustment in feed
rate, Q,l , or removal rate coefficient, 940 must be made and Eg. (7.1)
solved again for a new set of N'j s, which are then tested sgain in Eg.
(7.2). This sequence of computations is indicated in Fig. 7.1. Note that
an essential step in the ERC inmer iteration, not shown on Fig. 7.1, is
the adjustment of Q1 or qia by application of & single equation from the
coupled set, Eq. (7. 1) with the adjusted 11"" obtained from Eg. (7.2),
prior to the next complete solution of the system of equations, Eq. (7.1),
with the revised feed or removal rate.

The equilibrium concentration calculations in ERC use reaction rate
coefficients (C ) obtained from an earlier MODRIC calculation. How=-
ever, the 1n1t1a1 concentrations used in the MODRIC calculetion will not,
in general, sgree with the equilibrium concentrations computed by ERC.
‘This new set of concertrations will alter the neutron spectrum and flux
distribution, thereby changing the reaction rate coefficieats. Therefore,
it is necessary to repeat the MODRIC criticality calculation with the
latest value for the estimated concentrations to get new reaction rate
coefficients. This process is repeated until the MODRIC and ERC con-
centraticns converge.

The reaction rate coefficients (Cijk) used in ERC are spectrum-
averaged cross sections which are available directly from MODRIC. The
MODRIC calculation gives Aik and VFik’ the absorptions and neutron pro-
ductions in nuclide i in region k, normalized to 1.0 total neutron pro-
duced. The distribution of nuclear events between multiple streams in a
region is accomplished by introducing the stream volume fractions, fjk’
in this manrer:
T4
ORNL ING 64-3815

First guess at concentrations

 

 

 

 

 

 

 

 

 

 

 

    
  
  
  
  

§ 1 wome
I |
| |
l Calculate keffv] I
| Is |
A | Ne Yes '
l [ Chenge Conc. |

L T e e e

ERC

’r-HCalculate equ:.]ibrnm doncentrations | :
l {Calculate concentrat;on to make keps=1.0 I
l Do these two concentrations l
# agree? l
| No Yes |

 

 

o —— —

Do MODRIC and ERC agree on all
concentrations?

Yes

Is maximum number of
iterations exceeded?

 

 

et~

Fig. 7.1. MERC Flow Diagram.
-5

_ ,r8boms of 1 in stream J in region Kk .
A sk = %k’ (T otoms of T In remion & ) s (7.3

atoms of i in stream j in region k) (7.14)

= VyFo atoms of i in regiom k

Vi Fige = ViFax

The multiplying factor in each term is

 

atoms of i in stream j in region k Nij Jk
( ) = (7.5)
atoms of & in region k ?

where the units omn these factors are

N . = atoms of i in stream j (7.6)
ij barn om of stream j :

 

£, - cm> og st.ream.j in region k (7.7)
cn- of region k

The nuclide, stream, and region dependent absorption and production terms

are automatically transferred from the MODRIC link to the ERC link of

the MERC calculation. ERC cobtains the reaction rate coefficients (in-

tensive quantities) from the absorption znd production terms (extensive

quantities) by dividing by Nij , the stream concentration

A,
c;'jk = fi:.g.}' ’ (7.8)
ij
v, F,
i Tijk ;
Yitisk = _‘N:]L . (7.9)

‘The absolute reaction rate coefficient (Cy jk) is obtained in the ERC
calculation using the total neutron production rate as determined by the
reactor power

'c'f‘jk = C?.;jk X 3.1 X 10% p ¥ X 10*2% | (7.10)

where
7.6

3.1 X 10%*©
P

v

number of fissions/sec/megawatt,

power level in megawatts,

average numter of neutrons produced per fission =

N,

iklj%flvi
) S (7.11)

T
N.. C.
T T 1K
A similar argument applies for the fission reaction rate.

The ERC equations, including those for calculation of inventories

and fuel-cycle costs, are given in Appendix A.

Two-Dimensional Synthesis

The two-dimensional synthesis in ROD is based on the assumption that
for a symmetrical cylindrical reactor, the shape of the flux distribution
in the axisal direction everywhere is well represented by the flux distri-
bution along the axis, end similarly that the shape of the flux distri-
bution in the radial direction everywhere 1s well represented by a radial
flux distribution at the midplane. The complete spatial flux distri-
bution may then be inferred from two one-dimensional calculations, one
along the axis and the other along & radius at the midplane.

The two-dimensional synthesis is implemented in the MODRIC section
of ROD as follows:

Starting from an assigned initial flux distribution (e.g., cosice),
a new set of fluxes and ncutron reaction rates is calculated, for the
first neutron energy group, along the reactor axis. At each region
boundary, the net transport of neutrons (for the group) across the bound-
ary is calculated. The user may select an appropriate boundary (e.g.,
that of the core region) where a buckling is caleulated for use in the
second dimension.

Still for the first ensrgy group, a new set of fluxes and reaction
rates is calculated for the radial dimension. The radial fluxes are
normalized to the axial dimensiorn. The buckling cbtained from the first
dimension may be applied to specified regiors to calculate the transveise
leakage or net transport of neutrons transverse to the line of calculation.
TeT

The reactions of these neutrons are assigned to the various nueclides and
regions in proportion to the reaction rates obtained for specified regions
selected from the first dimension (the transverse leakage distribution
regions). A%t the same time the net forward transport of neutrons across
each boundary is caleulated, from which an appropriate boundary may be
selected for the caleulation of a buckling to be used in the first
dimension.

MODRIC then performs the same calculations for each of the remaining
energy groups in twn. The reactions from the axial and radial calcu-
lations including the transverse distribution reactions are summed and
normalized to give the required fission rate. MODRIC iterations are cone

tinued until the flux, fission density, and k__ . convergence criteria

£
are met.

OFTI

 

The optimization routine (OPTI) is based on the method of steepest
ascent. It permits up to 20 reactor parameters to be varied, within
limits, in order to find a maximum (or minimum) in a specified objectifie
funetion, calculated from the MERC output (see Chapter 5, paragrapk D-1),
It requires two links with MERC; one to transmit the values of the vari-
ables to MERC at the start of the celeulation, and another to calculate
and return to OPTI the resulting value of the objective fumction.

Starting values, and ranges, defined by upper and Xcwer limits, are
supplied for each variable. OPTI transmits the startirg values to MERC,
which runs a base-point case and returns the base-point result (value of
the objective function). OPTI then changes the first variable by a small
increment (ALPHA, a specified fraction of the range) and obtains a new
objective function result from MERC to approximate the (partial) derivetive
of the objective function with respect tc the Pirst variable:

2.2 2
i i

where
7.8

0 = objective function,

Xi = variable i.

By this method, derivatives are calculated for each varisble in turn.
The derivatives determine the direction of a vector (in hyperspace) origi-
nating at the base point and extending in the direction of the most rapid
increase in the objective function, that is, the path of steepest ascent.
OPTI now changes th= values of all the varigbles simultaneously in order
to teke steps along this vector. The variable with the largest derivative
is changed by a unit, (BETA, & specified fraction of its range), while
each of the other variables is changed by a fraction of a unit, determined
by the ratio of its derivative to the largest derivative. At the end of
each step, a new value for the objective function is obtained from MERC.
OPTL compares this velue with the previous one; if it is greater, by a
speciried increment, OPTI proceeds with another step; if not, a parabola
is fitted to the last three points along the vector, and the maximum of
the parsbola is taken as a new base point. If the value of the objective
function a2t the maxdimum of the parabola does not exceed the value for the
last successful step, the parabolic fit is rejected and the last successful
step is taken as the new base point.

Additional optimization cyecles, like the one just described, are
started from each new base point, until the increase in the objective
function in the last cyecle is less than a specified inecrement, or until
no successful step 1s found in the cycle. OPTT can then decia:re the last
base point to be the optimum, and terminate the search, or can reduce the
size of the increments (ALPHA and BETA) and carry the search further.

Reducing the size of the increments increases the precision of the
optimization; in particular, the finite-difference approximation to the
deriva.tivé approaches the true value as the increment ALPHA approaches
zerc. In practice, the precision of the optimization is limited by the
precision of the calculation of the cbjective function; e.g., in ROD, byv
the precision of the MERC calculation. The increments must not be reduced
to the point that the vector calculations founder in the imprecision of
the reactor calculations.
1.9

The steepest ascent method works very well for regular response
surfaces; however, in many practical problems the response.surface forms
a sharp ridge; the steepest ascent method is notoriously inefficient for
such cases. beczuse the path of the search tends to zig-zag back and
forth across the ridge. However, accelerztion methods have been developed
which handle such problems fairly well. Two acceleration methods avail-
able in OPTI are the "psrallel tangent' method and & method invented in
the development of OPTIMERC called the "ridge analysis" method.

Consider, for simrlicity, a two-variable problem, so that the response
surface can be represented b, & contour map in which the x and y coordi-
nates are the two variables, expressed as fractions of their range, and
the contour height is the wvalue of the objective function. For the case
of a regular response surface, the maximum, or peak, is surrounded by
nearly circular contours as shown in Fig. 7.2, so that a vector indicating
the path of steepest ascent points toward the meximum. But suppose that
the contcurs are elliptically elongated, fcrming a sharp ridge, as shown
in Fig. T7.3. A vector indicating the direction ol steepest ascent from
any random point {except on the ridge) now points across the crest of the
ridge rather than toward the peak. Consequently, the search path tends
to zig-zag back and forth across the ridge, gradually working up toward
the peak (vectors Vi to Vg, Fig. T.3).

The "ridge analysis" method is based on the observation that the
derivative, d0/dY, with respect to = dominating variable will change sign
every time a vectsr crosses a ridge. If the Y component of the succeeding
vector is decreased by & suitable factor every time d.O/d.Y changes sign,
the X componert beccmes relatively more important and the vectors {Rz to
Ry, Fig. 7.k) tend to follow the ridge.

The "parallel tangent” method is based on the observation that alter-
nate base points tend to line up along a ridge. When this is so, a vector
determined by two alternate base points will point toward the peak (vector
P, determined by base points Vz and V., Fig. 7.5).

Both methods may be used similtaneously (veetor Kfs, determined by
base points Rp and Ry, Fig. 7.6).

The ridge-analysis method is "free", that is, it does not require
that additional cases be calculated. The parallel-tangent method, since
 

T.10

ORNL-DWG 71-1887

Vv,

END

START

 

Fig. T.2.
Surface.

Path of Hypothetical Optimization for a Regular Response
 

Fig. 7.3.
Surface.

T.11

ORNL-DWG 71-1888

 

Path of Hypothetical Optimization for a Ridged Response
7.12

ORNL-DWG 71-1885

   

 

R, - / / START
RZ/ V:,, Vz

 

 

X

Fig. T.4. Path of Hypothetical Optimization Fmploying the Ridge-

Analysis Technique.
7.13

 

 

<
=
-
v,
>
-
w
-
B
o
~
T -
- w b et
N v
o
=
L= o
“~}
-
o L
= 2
— o

Fig. 7.5. Path of Hypothetical Optimization Employing the Parrslel-
Tangent Technigue.
ORNL-DWG 71-2566

 

 

 

X

Fig. 7-6. Path of Hypothetical Optimization Employing the Ridge-
Analysis and Parallel-Tangent Techniques Simultaneously.
7.15

it employs an "extra" (P) vector not in the normal series, requires that
at least one additional casre be calculated tc determine whether any
successful steps can be taker along the P vector.

In ROD, the ridge-analysis method is used in calculating every vector
after the first, and the parallel-tangent method is used to calculate
"RP" vectors after every normal vector following the third. Experieace
has shown that the RP vectors, although much of their work may have al-
ready been done by the ridge-analysis method, are successful often enough
to justify using both methods together.

Another phenomenon, the "end effect”, occurs as the optimum is
approached. It occurs because the epproximetion to the derivative is
one-sided; that is, the increment is taken in only cme (the positive)
direction. If a2 variable is at about its optimum value, irnuireasing or
decreasing its value can only give a poorer result; therefore, the one-
sided approximation to the derivative becomes negative as the optimum is
approached, whereas the trne derivetive would approach zero. Toward the
end of the optimization, oftten all the derivatives but one have beccme
negative. The presence of one positive derivative means that one vari-
eble, at least, must have a more favorable value; it mey be, however,
that no successful step is fourd. This can occur when the variables with
negative derivatives are actually at asbout their optima, so that stepping
in either the positive or negative direction would be unfavorable. As
the varisble with the positive derivative is scepped toward its optimum,
the others .tep off {heir optima, and the net :fect is an unsuccaessful
step. There are several theoreticel solutions to this problem; the most
obvious is tc use a two-sided approximation to the derivative. However,
this would - ouble the number of cases required to caleculate derivatives.
We have elected to provide two options that require no additional cases.
The first is to apply the reduction in the ridge-analysis factor whenever
the derivative is negative (in addition to whenever it changes sign).
Thus variables with consistently negative derivatives are restrained more
and more as the optimization proceeds. The other is to limit the move-
ment of variables with negative derivatives (by a factor of 0.01) after
a normal first step along a vector has been unsuccessful. This is done
7.16

instead of the interval scan described below. The procedure is bypassed
when the derivatives are either all negative or all positive.

Another procedure in OPTI, called the interval scan,is used when the
first step along & vector is unsuccessful. It runs one or more cases
along the vector in the interval between the base point arnd the first
step, starting a factor of 0.6 from the base, to test whether the optimum
might lie in this interval. If one or more successful steps are taken in
this intervel, the usval parsbolic fit is applied; if not, the cycle is

considered unsuccessf:il.

HISTRY

 

The premise of the HISTRY code is that a great amount of useful
information can be obtained from a simple time-dependent reactor calcu-
lation when a good set of average reaction rates are available for the
principle nuclides in the fertile-fissile chain. HISTRY can give an
approximation to the time-dependent fuel.cycle analysis of a reactor,
adequate for many purposes (e.g., conceptual design studies) where the
effort of a full space-energy-time dependent calculation meyr not be
Justified.

HISTRY solves the time-dependent material.balance equations for the
chain of 12 nuclides beginning with ®°2Th and ending with ®#%pu. A1l
other neutron absorbers are lumped together into two groups; the fixed
absorbers, mainly moderator, core structure, and fuel carrier nuclides,
and the fission products. The fission products may be treated either as
& fixed poison, or as a poison which builds-in as a parabolic function
during the cycle.

The first step in HISTRY is a criticality search. Thne initial ke
is adjusted so that keff
nizclides or mixtures of nuclides may be designated for feed or for sale
‘depending on whether or not the reactor is breeding at the given time-
step). After the critical concentrations are determined, the concen-
trations of the other nuclides at the end of the time-step are calculated
from the material-baslance eguations. The above calculations are repeated

ff
will average 1.0 over each time step, Various

for as many time-steps as specified over the reactor cycle. For the case
.17

of batch processing, the reactor lifetime may be divided into several
batch processing cycles. HISTRY celculates atom densities, inventories,
neutron absorptions and productions, and purchases or sales for each
nuclide as & function of time. The time-averaged atom densities are
calculated and returned to MERC for the next iteration.

The fuel and carrier costs are calculated by the discounted cash
fiow method.
9.1

CHAPTER O

REFERENCES

J. Replogle, MODRIC — A& One-Dimensional Diffusion Code, USAEC Report
K-1520, Oak Ridge Gaseous Diffusion Plant (Sept. 6, 1962).

L. G. Alexander et al., The MERC-1 Equilibrium Reactor Code, USAEC
Report ORNL-TM-847, Oak Ridge National Laboratory (Apr. 22, 196L).

J. R. Copper and W. L. Kephat, Some Improvements in the Gradient
Search Method of Optimization, USAEC Report K-DP-1063, Oak Ridge
Gaseous Diffusion Plant (December 1965).

T. B. Fowler and D. R. Vondy, Nuclear Reactor Core Analysis (Code:
CITATION, USAEC Report ORNL-TM-2496, Osk Ridge Natiomal Laboratory
(July 1969).

N. M. Greene and C. W. Craven, Jr., XSDRN: A Discrete Ordinates

Spectral Averaging Code, USAEC Report ORNL-IM-250C, Oak Ridge
National Laboratory (July 1949).

R. Q. Wright, BPL@T, A Generalized Fortran Plot Package for the
IBM 3éC, TIDBITS, Vol. 4, No. 6, June 1968.
LPPENDICES
Subscrigts

Name

TguEes

Al

Appendix A

THE ERC EQUATIORS

Description
Nuclide number
Material number
Number of material fram which recycled
Nuclide number, processing source
Nuclide number, transmutation source
Nuclide number, decay source
Nuelide number of reference feed nuclide

Delayed neutron group number

Input Variabies (in order of appearance in equations)

Name

 

P
CIMI
MATS

SCE
GPTIME

M655

AMASS

BETA
AMA3
B35
B56
XE21

Description
Power, MW(th)
Initial atom density in material stream, atoms/barn cm
Number of materials
Fraction processed
Fraction removed or lost in processing
Processing cycle time
Fission product yield
Feed option
Feed rate, kg/day
Atomic weight
Decay constant
Atom-density option
Delayed neutron fraction
Delayed neutron decay constant
Residence time, in core, sec
Residence time, out of core, sec

Fixed poison fraction
A.2

 

Name Description

L22N1 Array, nuclide numbers of fertile nuclides

12282 Array, nuclide numbers of fissile precursor nuclides

122D Array, nuclide numbers of fissile nuclides

123D Array, nuclide numbers of fissile nuclides

120N Array, nuclide mumbers of fissile nuclides

LZ0N Array, nuclide numbers of lissile and fissile precursor
nuclides

L30D Array, nuclide nwmbers of fissile nuclides

N2lL Array, nuclide numbers of fissile nuclides

™ Holdup time in processing plant, days

TR Operating time on reserve fuel, days

F36 Plant factor

E36 Thermal efficiency

Value, in system, $/kg

Interest rate, fraction per year

Unit processing cost, weight basis, $/ke
Processing plant unit cost, $

gfi&ififi

Processing cost, reference, volume basis
B5T Scaling factor
Value of feed, $/kg

S

Camputed Variables

ffffi geiigizi Description

F 1 Fission rate, fissions/sec X 10%¢

CIM 2,25,26, Atom density in material stream, atams/barn-em
29-35

CAP 3 Neutron capture rate

CA Neutron absorption rate

CF Neutron fission rate

B Total neutron absorptions

W™ Mean nu

CN Neutron production rate
 

oo efiziog

FVB 6

S16 T

R 8

v

SDC

RCF

TS

FI3S 9

CRoL 10,12,13,
2

FRATE 11,1416

TRSK 24

BDS 17

PRS 18

D 19

TRAS 20-23

FFFS 36

BR 37

ETA 3

ST 29

FI Lo,k1

RSI 42,43

T@TT Lh

FISSI 145

PL 46

YIELD 47

FRTE 48

FRTEI 49

PDRTE 50

PDRAT 51

PCRTE 52

A.3

Description

Neutron balance, neutrons produced per absorption
Total neutrons produced

External recycle rate (recycle to material from
other materials)

Material volume, cm®

Discharge concentration (SDC = CIM for fluid-fusl
reactors)

Recycle fraction
= GPTIME(1l). Reference processing cycle time, days
Fission product source

Processing source (internal recycle)

Feed rate, atoms/sec

Capture sink

Decay sink

Processing sink

Decay source

Transmutation source

Fraction of fissions in material 1
Breeding ratio

Mean eta

Inventory in reactor system, kg
Inventory in processing plant, kg
Inventory in reserve, kg

Inventory total, kg

Totel fissile inventory, kg

Fissile loss in processing as fraction of burnup
Fissile yield, percent of fissile inventory per year
Feed rate, by material, kg/dey

Feed rate, total, kg/day

Production rate, by material, kg/day
Procduction rate, total, kg/day
Processing rate, by material, kg/day
 

Nme e
FCRIEI 53
PE36 54
FCIC 55
FCICI 56

RC 51,58
RCI 59

PC 60
PCI 61

SV

STS

PCR 62
PCRI 63
FCC 64
ANDX 65
ANDI 66
RPIV 67
CPIV 68
TETAIT 69
RSIV T0
CALC T1-7k
BCD I
Al 6

A2 T

A3 8

Al TS

A 80

Equations in ERCM

Fission rate

Ak

Description

Processing rate, total, kg/day

Power, electrical, MW

Inventory charges, by material, mills/kwhr(e)
Inventory charges, total, mills/kwhr(e)
Replacement charges, by material, mills/kwar{e)
Replacement charges, total, mills/kwhr(e)
Processing charges, by material, mills/kwhr(e)
Processing charges. total, mills/kwhr(e)
Material volume, £t~

Processing cycle time, days

Production credit, by material, mills/kwhr(e)
Production credit, total, mills/kwhr(e)
Fuel-cycle cost, mills/kwhr(e)

Neutron absorptions, by material

Neutron absorptions, total, normalized to ETA
Neutron captures, total, normalized to ETA
Fissions, total, normalized to ETA

Total inventory, kg

Processing removal rate, kg/day

Calculated value for recycle fraction
Processing source (external recycle)

Decay source

Decay and removal sink

Burnup

Transmutation source

Net sink

F = P*3,1E.8 (A.1)
A.S

Atom density
CIM(I,J) = ciMi(I,J) ell I,J (A.2)
Capture rate
CAP(I,J) = CA(I,J) - CF(I,J) I <50 (4.3)
Total neutron gbsorptions

MATS N200

B=y ), CIM(I,J)*CA(I,J) (A.b)
=1 =1

Mean nu

MATS 20
Y, CIM(I,J)*CN(I,J)
J=1 I=1
= MATS 20 (4.5)

Y ), CIM(T,T)*CF(I,J)
J=1 I-1

Reutron balance
FVB = F*VM/B (A.6)

Total neutrons produced

Is 5

IZl CIM(I,J)*CN(I,J) (A.7)

O

S16 =

OIS

1
External recycle rate
MATS Ir(1,2)

R[1,J] = V(JPp)* SJP)* *(1.0-
;;,::l IPCZ_',IP(glg )*SDC(IPC,JP)*E(IPC,JP)*(1.0-SCE(IPC,JP) )
IPCLO
*RCF(IPC,JP,J))/TS(IP)) (4.8)
for IPCAI

IPC=T end JPAT
A.6

Fission product source
13
FISS(I,J) = FVB* p, CIM(LZ,d)*CP(1J,J)*¥(I,J)
IJ=1
I > 21 except 2L
Feed rate and processing source equations:

For M655(I,J) = 0O

il

CRG4(I,J) = 0.0

FRATE(I,J) = 0.0

For M655(1,J) = 1
If IP(1+1,1) =TI

I

or
If IP(I+1,2) = I  CRSL(I,d) = 0.0

If TIP(I+1,1)#1 and IP(I+1,2){I

CROL (T,d) = V(JI)*E(I,J)*(1.0-SCE(I,J))*RCF(I,J,J)/Ts(J)

(4.9)

(A.10)

(4.11)

(A.12)

(4.13)

FRATE(I,J) = CTM(I,J)*(TRSK(I,J +BDS(I,J)+PRS(I,J) - CRO4(I,J))

- (r(z,J) + FISS(I,J) + TRAS(I,J) + D(I,J))

For M655(I,J) = 2
FRATE(I,J) = (Q(I,J)/AMASS(I))*6.9710648 E-3
For M655(I,J) = 3
FRATE(I,J) = Q(I,J)*FRATE(ISTAR(I,J),J)
Beta decay sink coefficient
"BDS(I,J) = V{J)*AMBA(I)
Processing removal sink coefficient

PRS(I,J) = V{(J)*E(I,J)/Ts(J)

(A.1h)

(A.25)

(A.16)

—~
»
|

-]

s

(A.18)
Beta decay source rate

ID(I,2)
D(I,T) = V(J) * ),  CIM(IDL,J)*AMBA(ID1) (2.19)
ID1=ID(I,1l)
ID1£0
Transmutation source rate
I7(1,2)
TRAS(I,J) = FVB * ), CIM(IT1,J)*CAP(ITL,T) (4.20)
IT1=IT(I,1)
IT10 I <50
IT(1,2)
TRAS(I,J) = FVB * ), CIM(ITL,J)*CA(IT1,T)  (A.21)
IT1=IT(I,1)
IT1£0 I > SC
TRAS(223,J) = FVB*CIM(222,J)*CA(222,T)*0.47 (A.22)
TRAS(224,J) = FVB*CIM(222,J)*CA(222,J)*0.53 (A.23)
Transmutation sink coefficient
TRSK(I,J) = FVB*CA(I,J) (A.24)
Atom-density equations:
For N(I,J; =0
CIM(I,J) = CIMI(I,J) (4.25)
For N(I,J) =1

If M55(1,F) =1

FRATE(T,J +R(T,J+FISS(I,J)+TRAS(I,J)+D(I,J)

cmM(1,3) = TRSK(I,J)+BDS(I,J5+PRS?1}37:E§§Eff,J5 === (a.26)

If M655(I,J7) £1
If IP(I+1,1) =1I
cr
If IP(1+1,2) =1 CRO4(I,J) = 0.0 (A.27)
A.8

If IP(I+1,1) # I and IP(T+1,2) # I

CROL(T,J) = V(J)*E(I,J)*(l.gSEJ§CE(I,J))*RCF(I;QJJ)

FRATE(T,J)+R(I,J )+ FLSS(T,J +TRAS(Z,T)+D(I,J)

CIM(I,T) = =kl T+ 505 (T 3)Ens T,J )=CRO%(L,d

For N(I,J) = 2

 

MATS N200 MATS 50
CIM(I,T) = CIM(I,T) + Z=L =2 JI=1 II=1
CN(I,T)

For N(I,J) = 3
CIM(I,J) = B*CIMI(I,J)

For N(I,J) = &
(Option for =°5U only)

A3l = CIN(3,J)*CA(3,J)
A51 = CIM(5,J)*CA(5,d)
211 = CIM(1,J)*CA(1,T)
Al = cIM{%:,T)*CA(L,d)
A2 = CIM(2,J)*CA(2,J)
F31 = CIM(3,J)*CF(3,J)

51 = CIM(5,J)*CF(5,J)

_A31L + AS1 - A1l - AW] + A21 + A21
FALE = ST - L)

 

UAL = 3.452E-3*P*F36%Q(I,J)*FAf16

0.863EL*P*F36*CA(T,J)

B2 = 73)

 

TAR = ISTAR(I,J)

BT1 = UB2*TAR

(4.28)

(A.29)

(A.30)

(A.31)
A.9

_ %0 - EXP(~BT1)

ET1
UAl
UAB = Gp5
EIB = UAB - (Q(I,J)*(cMx(3,J) + CIMI(5,d))

CIM(I,J) = UAB - (EX1¥EIB) (4.32)

For N(I,J) =5
(delayed neutron nuclide)

MAS [15
N 1 >
CIM(I,J) = B¥CA(T,T) * JVZI:=1 sz-lcm{K’M) CN(X,M)

x 62 BETA(X, L) *(1.0-EXP(-AMA3 (L.)*B55) )*(1.0 - EXP(-AMA3(L)*B56))
AMA3 (L) *B55% (1.0-EXP{ ~-AMA5(L)* (B55 + B56) -

 

I=1
(4.33)
For N{(1,J) = €
(fixed poison fraction)
N21L(N21K)
XE21L(T) * ), o CTM(X, 3 )*CA(K, T)
K=N21L(21
- A.
CIM(I,J) = SRCA(T T} (A.34)
For N(I,J) =7
(1umped fission preoduct)
25:30
/,  CIM(II,J)*CA(II,J)
II=1

CIH(I,J) = B%CA(T,T

INE(II,J)#0 for fission products treated specifically in MCDRIC.
A.10

Equations in ERCPUT

 

Freetion of fission in material 1 (fuel)

 

 

 

NS5O
;l CIM(I,1)*CF(1,1)
FFFS = gima 5 (A.36)
Z Pf CIM(I,J)*CF(I,J)
J=1 1I=1
Breeding ratio
MaTS | L22NL (K22 ) 122N2 (K22n2)
. CIM(X,J)*CAP(K,J) - 2 CIM(K,J)*CA(X,J)
sr - =L K=122N1(1) K=122N2(1)
- MATS 122D(K22D)
CIM(X,J)*CA(K,J) (A.37)
J=1 K=122D(1)
Mean eta
4 = > (4.38)
MATS 123D(K23D)
CIM(K,J)*CA(X,d)

J=1 K=L23D{(1)

Material inventory in kg

 

 

CIM(I,J)*V(J)*AMASS(I) I = AL
SI(I1,J) = L502,5 J = 1,MATS (8.39)
Processing inventory in kg
T =1,16
SDC(T,I)*V(T)*E(T,I)*TR(J)*AMASS(T) T _ 'y
pr(1,5) = 2L TS(}7*602-% = LA
(4.40)
Exception:

If I =2 and IP(3,1) =2 or IP(3,2) = 2

J)*E(2,J)*AMASS(2
PI(2,d) = —J—Fy——(-y-g“——'i—lsnig;%*zés); ¢ 02?3 (A.%1)
A-ll

it

PI(I,J) = 0.0y J = 1,MATS

Reserve inventory in kg

SI(I,d) = o.o} T = 17,N200

 

RSI(I,J) = 0.0 I = 1,N200 if N(I,J; £ 2 (a.42)
J = 1,MATS
If N(I,J) =2
0
Bil,2)
FXVM * |CIM(I,J)*CA(T,T) - ), CIM{IX,J)*CAP(IX,J)
oL B¥6C2.3

* AMAS*1.1*TR(J) (A.43)

Total inventory in kg
TPTI(I,T) = SI(I,J) + PI(I,J) + RSI(I,J) (A.LL)

Fissicnable inventory in kg

_ MATS L2ON(K29N)

FISSI = Z‘, E SI(K,d) + PI(K,J) + RSI(K,J) (A.L5)
J=1 K=I29N{1)

Processing losses

MATS 30N (K30N)
Z Tg(g) * E SDC(K,J)*E(K,J)*SCE(K,J)
I, = J=1 X=130N(1) (A.46)
MATS L30D{K30D)
CIM{K,J)*CA(K,J)
J=1 K=L30D(1)

FXVM
B
Yield

_ 37.8%VM*P*F36%(BR-1.0-PL)

YIELD FTAYFISST (A.LT)

Feed rate ir kg/day

FRTE{I,J) = FRATE(I,J)*AMASS(I)*143.4501 TT - tfis (A.48)
£.12

MATS

FRTEI(I) = ), FRTE(I,J) I = 1,N50 (A.49)
J=1

Production rate in kg/day

PIRTZ(I,J) = SDC(I,J)*V(J)*E(I,J%"S*gio - SCE(I,J))* AMASS(I)*143.4501

3
* ’J!;r (1.0 - RCF(Z,J,JP)) § - i’ggs (A.50)
=1 7

MATS

PDRAT(I) = ), PDRTE(I,J) I=1,8%0 (A.51)
F=1

Processing rate in kg/day

 

PCRTE(T, ) = SDC(I,J)*V(J);g%fiJ)*AMASS(I)')fll"3',"501 (4.52)
I = 1,850
J = 1,MATS
MATS
FCRTEI(I) = J, PCRTE(I,J) I=1,50 (4.53)
J=1
Power, electrical
PE36 = P*E36 (A.54)

Inventory charges in mills/kwhr(e)

0. L4 E-3*PTT(T,J)*WL(I,T)*W2(I)

 

FCIC(I,J) = R (4.55)
I = 1,50
J = 1,MATS
MATS
FCICI(I) = ), FCIC(I,T) I= 1,850 (4.56)

Jd=1
AolB
Replacement charges in mills/kwhr(e)

0.0417*FRTE(I,JT)*W5(I,J)

 

RC(I,J) = P50 if M655(I,F) £ 0
RC(I,J) = 0.0 if M655(1,J) =0 I =1,N0
J = 1,MATS
MATS
RCI(I) = ), Ro(I,J) I =1,K50
J=1

Processing costs in mills/kwhr

PCRTE(I,J)*W3(I,J)*0.0k1T

 

 

 

 

 

P(I,9) = PE36
+ W (I,J)*(PCRTE(I,J )*F36 )**R57*0,114E~3
PESG*F36
WINEW(I ,J)*0.0417 [S‘V(J)*E I J)]
* T555 L= STs(Jg BT
I = 1,850
J = 1,MATS
MATS
PCI(I) = ), PC(I,d) T = 1,550
Production credits in mills/kwhr(e)
0.0417*PDRTE(T, J)*W(I,J) I =-1,150
PCR(I,J) = PESS T = LMATS

MAT'S
PCRI(I) = ), PCR(I,J) I = 1,N50
71

(A.57)

(A.58)

(A.59)

(4.60)

(A.61)

(A.62)

(A.63)
A1k

Fuel-cycle costs in mills/kwhr(e)

MATS NSO
FCC = ), y, FCIC(I,J) + RC(I,J) + PC(I,J) - PCR(I,J)
F=1 T=1

Neutron absorptions

ANDX(I,J) = CIM(I,JT)*CA(I,J)

MATS
ANDI(I) = ETA * ), ANDX(I,J)
J=1
Neutron captures
MATS
RPTV(I) = ETA * ), CIM(I,d)*CAR(I,J)
J=1
Fissions
MATS
CPIV(I) = ETA * ), CTM(I,J)*CF(I,J)
=1

Total inventory

MATS
TPTALI(I) = », SI(I,J) + PI(I,J) + RSI(I,J)

Processing removal rate, kg/day

MATS

(A.64)

(A.65)

(A.66)

(A.67)

(A.68)

(4.69)

RSIV(I) = ;:1 SDC(IL,J)*V(J)*E(I,J)*SCE(T,T)*AMASS(T)*143.4501/T5(J)

Equations in CYCI8
Recycle fractions

If CAIC = 1.0

RCF(1,1,1) = 1.0
RCF(I,1,2) = 0.0
RCF(1,1,3) = 0.0

(A.70)
A5

RCF(I,2,1) = 1.0
RCF(I,2,2) = 0.0
RCF(I,2,3) = 0.0
RCF(I,3,1) = 1.0

RCF(1,3,2) = 0.0

RCF(I,3,3) = 0.0
If CAIC < 1.0

M655(I,3) = O

If J655(1,3) = 1
RCF(I,1,1) = CAIC

A(I,1) - BCD(I,2) - BCD(I,3)

 

 

CALC = BCD(T, 1) (8.71)
If J655(I,J3) = 2
RCF(I,1,1) = CAIC
RCF(I,2,1) = CALC
RCF(I1,3,1) = CALC
A(T,1)
CALC = THTSTE T BN T RN (72
If J655(1,d) = 3
RCF(I,3,1) = CAIC
_ A(1,1) - BCD(I,1) -~ BCD(I,2)
C.AI!C = Bmf,j) (A'F{B)
If J655(1,F) = 4
RCF(I,2,1) = CALC
RCF(I,3,1) = CALC
_ A(T,1) - BCD(I,1) '
CALC = 5D(1,2) - BOD(E,3) (h70)
Processing source
IP(I,2)
BCD(I,J) = SDC(LAL,J)*V(J)*E(LAL,J)*(1.0 - SCE(IAL,J)) (A.75)

 

1AT=TP(1,1) TS(J)
LALAD
A.16

Decay source
I0{(1,2)
AU(I,1) = ), CIM(ID1,1)*V(1)*AMBA(TDL)

ID1=ID(I,1)
ID1{0

Decay and removal sink

SDC(I,1)*E(I 1)]

A2(I,1) = V(l)*[CIM(I,l)*W(I) + TS(1

Burnup

A3(I,1) = CA(I,1)*CIM(I,1)

Transmutation source

IT(1,2)
A(I,L) = ), CIM(ITL,1)*CAP(IT1,1)
ITI=IT(I,1)
IT1£0

Net sink

A(I,1) = FVB*(A3 - A4) - AL + A2

(a.76)

(A.TT)

(A.78)

(A.79)

(A.80)
Appencix B

BASIC MODRTC EQUATIORNS

The following description of the basic MODRIC equations is taken
fram Ref. 1.

The diffusion eguation which describes the neutron flux o9(r,g) at
a point r in the reactor within the energy group g has the form:

 

Vefb(r,g) + R(I,S) ¢(r,g) + S(r,g) =0, (B-l)
where
. g+10
R(r,g) = _]JS -5, (B.2)
g-1
XP(@)+ 2 MU (r,i)
g s = + g
S(r,g) = g0 . ( .3)
D(Aug)

For the geometries here Vo9 = ¢” + -19_- ¢’ , where

O fer a slab,
1l for a cylinder,
2 for a sphere.

In order to state the problem in its entirety, it is convenient to
change the notation so that the dependence of the variables and parameters
on & particular energy group is omitted. Thus, the diffusion equation can

P

]

be written:
¢7(x) + £ ¢/(x) + R(r) ¢(r) + 5(r) = 0 . (B.4)

The boundary conditicns at the origin (r = 0) and the outer bowndary
(r = RN) are:

 

*
May include the extrapolated distance.
B'E

a$(0) + 2bD¢’ (0) = ¢ , (B.3)

ap(Ry) + 2eD6’ (Ry) = £, (8.6)

vhere &, b, ¢, 4, e, and £ are parameters which can vary with the energy
group.

Another type of boundary condition occurs between regions at radius
RI. This takes the following form for an interface with no shell:

[6(R;) = 208" (R)]_ = [(R,) — 20" ()], (B.7)

[6(R) + 2D¢" (R)]_ = [$(Ry) + 2D¢’ (R)], , (8.8)
where the - and + signs refer to values at the left and right of the
interface respectively. An interface with a shell is a stubregion from
RI to RJ with attenuation of the neutron currents given by the following:

GROL#(R,) — 2p¢’ (R)]_ = RO[6(R;) — 208" (R )], (B.9)

RLo(R,) + 209 (R)]_ = EEOL(R,) + 208" (R,)], (.10)

Again, G and H can vary with energy group. The - and + signs refer to
values to the left of RI and the right of RJ respectively. We can use
Egs. (B.9) and (B.10) for the special case of no shell by setting

BI = RJ. ¥or continuity of the flux and current across the interface,

G = H =1, but thic is not necessary if desired otherwise.
3
é(r,g)

B.3

Ncmenclature

Radial distance for a cylinder or sphere, longitudinal
distance through a slab

lethargy group index
Neutron flux per unit lethargy at distance r in group g
Macroscopic absorption cross section” for group g

Macroscopic scattering cross section” for heavy elements
from group g to group 1

Macroscople fission cross section* in group g multiplied
by the neutrons/fission

Transverse buckling

Iiffusion coefficient™

lethargy width of grouwp g

Fraction of fission neutrons emitted in group g
Fission density at distance r

Removal term in diffusion equation

Source term in diffusion equation

 

*
Assumed to be constant over & region, but can vary with g.
Cll

Appendix C

FISSION PRODUCT TREATMENT®

The fission-product reaction rate coefficient is obtained by reference

to a specified standard absorber:
CE?=CR£, (c.l)
GB

where
CFP = fission-product reaction rate coefficient,
CR = reference material reaction rate coefficient,
GFP = effective fission-product absorption cross section,
EB = effective reference material absorption cross section.
Trhe effective cross section ratio is obtained from a two-group
formulation:

=FP  (0191/¢p + GE)FP

 

 

;B - R (c.2)
(01¢1/¢2 + Op)
where
, Uth
J oé(u) du (RT)
0, = fast absorption cross section = ° - 2 y
Uth Uth
82 = absorption cross section averaged over the thermal flux,
¢, = Fast flux,

¢> = average thermal flux.
For a2 two-group treatment, all neutrons removed from the fast group must
either be absorbed or leak from the reactor while thermal:

Eg 1 =%, 62+ DB% . (c.3)

Ignoring leskage,

 

*
This section has been revised from Ref. 2.
3z - zae/le . (C.%)

Also,
Oz =1 a.izoo ’ (C.5)

where

f = thermal spectrum factor = 4(;}) (¥-_%9—.—-—;73) for & Maxwell-~

Boltzmann distribution,
cr§2°° = 2200 m/s ebsorption cross section.
Substituting Zgs. (C.2), (C.4), and (C.5) in%to (C.1l) gives

(K(RD) + 622090)FF

c* - P , (C.6)

x

 

where
=
aps
LRJ_ f o
a = (K(RI) + c§2°°)3.

K is calculated as foliows:
K = WS,

1 <
W-f—E—:.nput

5 z,a_a/z:Rl = value actomatically calculated by MODRIC for the
camposition being studied.

The nuclear constants for a 1/v absorber with a 2200 m/s cross section

of 1.0 barns are built into the code. Therefore, the reference element

mest correspond to an artificial element in MODRIC which has cross sectiors

for a 1/v ebsorber with 0§3°° = 1.0.

Certain important fission products may be calrulated explicitly by
the mltigroup diffusion calculation in MODRIC, using fissiom product
option 2. Cross sections for such nuclides must be provided in the usual
format on the cross section tape.
Appendix D

THE PROCESSING STUDY OPTION

An Alternative Calculation of Fission-Product Poisoning

For some applications it is necessary to compute the fission-product
poisoning using a model other than the first-order removal process con-
sidered in the ERC calculations. An option has been provided in RCD with
which it is possible tc call a subroutine, named BELL, to ve supplied by
the user, to perform an alternstive calculation of the lumped fission-
product poisoning (refer to card B-l). If this option is designated,
equilibrium concentrations will be computed in the ERC calculation for
only those nuclides trested explicitly in the MODRIC diffusion calculation.
¥When both the diffusion and equilibrium calculations have converged to
within the specified limits for a given value of the limped fission-product
concentration, the program will call subroutine RELL to obtain a new esti-
mate of the lumped fission-product concentration. This subroutine is
called with the following argument list: SUBRPUTINE BELL (C@NC, ABSRC,
PR, Y, BAL, P8, SV, RIFAC, ALPHA, FCC, C@NCFP, ETA). The varisbles C@NC
and ABSRC are each dimensioned for fifty values in the calling program
end contain, using the ERC nimbering system, the atam densities, in
atoms /fbarn-cm, and the relative sbsorption rates per unit atom density,
gbsorption/(fissile absorption-atom/barn-cm), for the ERC principle nu-
clides. The definitions of the remaining variables are:

BR = breeding ratio,
Y = fuel yield, percent per annum,

BAL = ratio of two previous iterations of the converged
equilibrium calculation,
P8 = reactor thermal power, MW,
SV = fuel salt volume, £t>,

RIFAC = Tescpnance integral factor, flux per unit lethargy/thermal
flux,
ALPHA = spectrum-averaged neutron cross section for a 1/v absorber
whose 2200 m/sec cross section is 1.0 bern, barus,
FCC = fuel-cycle cost, mills/kwhr(e),
D.2

C@NCFP

H

new value of lumped fission-product concentration computed
by subroutine BELL, atoms/barn-cm,
ETA = effective value of n€, neutrons produced/fissile sbsorption.

Witk the exception of the variable CENCFP, the values of a1l the
arguzents are computed by ROD for use by subroutine BELL. BRELL uses the
information in the argument list, and an independent set of fission yields,
decay schemes, thermal neutron cross sections, and resonance integrals to
campute the fission-product inventories and poisoning for a given set of
processing conditions. The individual fission-product poisonings are
sumned, and this lumped fission-product poisoning is used to compute the
concentration of t'ie reference 1/v absorber which would produce the same
poisoning. This concentration is the value of the variable C@NCFP which
is returned to the calling program in ROD. The diffusion and equilibrium
calculations are repeated using the new value of the lumped fission-product
concentration, and the process is continued until the lumped fission-
product concentratlion from two successive iterations converges to within
a8 predtermined relative error. At this point the entire process may be
repeated for another set of processing conditions, or the calculations
may be terminated. When using this option to calculate the poisoning by
the lumped fission products it is necessary to remove the permanent fission
product data from the ERC input and to specify atom-density option O on
ERC input card B-22 for the lumped fission-product nuclide. This specifies
that the value of the lumped fission-product concentration is not to be
changed in the equilibrium calculation.