nestle - researchgate · nestle version 5.2.1 few-group neutron diffusion equation solver utilizing...

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/236368235

NESTLE: Few-group neutron diffusion equation solver utilizing the nodal

expansion method for eigenvalue, adjoint, fixed-source steady-state and

transient problems

Article · June 1994

DOI: 10.2172/10191160

CITATIONS

65

READS

390

6 authors, including:

Some of the authors of this publication are also working on these related projects:

Reduced Order Modeling Based Uncertainty Quantification, Sensitivity Analysis and Model Calibration View project

Paul J. Turinsky

North Carolina State University

117 PUBLICATIONS 666 CITATIONS

SEE PROFILE

All content following this page was uploaded by Paul J. Turinsky on 27 June 2015.

The user has requested enhancement of the downloaded file.

https://www.researchgate.net/publication/236368235_NESTLE_Few-group_neutron_diffusion_equation_solver_utilizing_the_nodal_expansion_method_for_eigenvalue_adjoint_fixed-source_steady-state_and_transient_problems?enrichId=rgreq-6dfd0c656dae6cb8d7d2c40864464f29-XXX&enrichSource=Y292ZXJQYWdlOzIzNjM2ODIzNTtBUzoyNDUwNDk2ODY0OTExMzhAMTQzNTQzNTgwNjIzNQ%3D%3D&el=1_x_2&_esc=publicationCoverPdf

https://www.researchgate.net/publication/236368235_NESTLE_Few-group_neutron_diffusion_equation_solver_utilizing_the_nodal_expansion_method_for_eigenvalue_adjoint_fixed-source_steady-state_and_transient_problems?enrichId=rgreq-6dfd0c656dae6cb8d7d2c40864464f29-XXX&enrichSource=Y292ZXJQYWdlOzIzNjM2ODIzNTtBUzoyNDUwNDk2ODY0OTExMzhAMTQzNTQzNTgwNjIzNQ%3D%3D&el=1_x_3&_esc=publicationCoverPdf

https://www.researchgate.net/project/Reduced-Order-Modeling-Based-Uncertainty-Quantification-Sensitivity-Analysis-and-Model-Calibration?enrichId=rgreq-6dfd0c656dae6cb8d7d2c40864464f29-XXX&enrichSource=Y292ZXJQYWdlOzIzNjM2ODIzNTtBUzoyNDUwNDk2ODY0OTExMzhAMTQzNTQzNTgwNjIzNQ%3D%3D&el=1_x_9&_esc=publicationCoverPdf

https://www.researchgate.net/?enrichId=rgreq-6dfd0c656dae6cb8d7d2c40864464f29-XXX&enrichSource=Y292ZXJQYWdlOzIzNjM2ODIzNTtBUzoyNDUwNDk2ODY0OTExMzhAMTQzNTQzNTgwNjIzNQ%3D%3D&el=1_x_1&_esc=publicationCoverPdf

https://www.researchgate.net/profile/Paul_Turinsky?enrichId=rgreq-6dfd0c656dae6cb8d7d2c40864464f29-XXX&enrichSource=Y292ZXJQYWdlOzIzNjM2ODIzNTtBUzoyNDUwNDk2ODY0OTExMzhAMTQzNTQzNTgwNjIzNQ%3D%3D&el=1_x_4&_esc=publicationCoverPdf


https://www.researchgate.net/institution/North_Carolina_State_University?enrichId=rgreq-6dfd0c656dae6cb8d7d2c40864464f29-XXX&enrichSource=Y292ZXJQYWdlOzIzNjM2ODIzNTtBUzoyNDUwNDk2ODY0OTExMzhAMTQzNTQzNTgwNjIzNQ%3D%3D&el=1_x_6&_esc=publicationCoverPdf



NESTLE

Version 5.2.1

Few-Group Neutron Diffusion Equation Solver Utilizing TheNodal Expansion Method for Eigenvalue, Adjoint, Fixed-

Source Steady-State and Transient Problems

Revised July 2003

Electric Power Research CenterNorth Carolina State University

Raleigh, NC 27695-7909

Copyright 1994, 2001,2003 by

NORTH CAROLINA STATE UNIVERSITY

P.O. Box 7909

Raleigh, NC 27695-7909

All Right Reserved

No part of this report may be reproduced in any form

without the written permission of North Carolina State University

2

tion

ty);

nvalue

ution

ource

model

dent.

ups

Three,

rative

egy is

ation

upon

ased

mial

ture

ergy

sumed

fuel

Abstract

NESTLE is a FORTRAN77 code that solves the few-group neutron diffusion equa

utilizing the Nodal Expansion Method (NEM). NESTLE can solve the eigenvalue (criticali

eigenvalue adjoint; external fixed-source steady-state; or external fixed-source or eige

initiated transient problems. The code name NESTLE originates from the multi-problem sol

capability, abbreviatingNodal Eigenvalue,Steady-state,Transient,Le core Evaluator. The

eigenvalue problem allows criticality searches to be completed, and the external fixed-s

steady-state problem can search to achieve a specified power level. Transient problems

delayed neutrons via precursor groups. Several core properties can be input as time depen

Two or four energy groups can be utilized, with all energy groups being thermal gro

(i.e.upscatter exits) if desired. Core geometries modeled include Cartesian and Hexagonal.

two and one dimensional models can be utilized with various symmetries. The non-linear ite

strategy associated with the NEM is employed. An advantage of the non-linear iterative strat

that NESTLE can be utilized to solve either the nodal or Finite Difference Method represent

of the few-group neutron diffusion equation. For Cartesian geometry, the NEM is based

quartic polynomial expansion functions; whereas, for Hexagonal geometry the NEM is b

upon the semi-analytic nodal method utilizing trigometric, hyperbolic trigometric and polyno

expansion functions and the conformal mapping technique.

Thermal-hydraulic feedback is modeled employing a Homogenous Equilibrium Mix

(HEM) model, allowing two-phase flow to be treated. However, only the continuity and en

equations for the coolant are solved, implying a constant pressure treatment. The slip is as

to be one in the HEM model. A lumped parameter model is employed to determine the

3

ed to

ized by

r a

rms of

e, and

ection

r this

ith

nergy

ient

be

s are

define

temperature. Decay heat groups are used to model decay heat.

The thermal conditions predicted by the thermal-hydraulic model of the core are us

correct cross-sections for temperature and density effects. Cross-sections are parameter

color, control rod state (i.e. in or out) and burnup, allowing fuel depletion to be modeled. Eithe

macroscopic or microscopic model may be employed. All cross-sections are expressed in te

a Taylor’s series expansion in coolant density, coolant temperature, effective fuel temperatur

soluble poison number density. For the Hexagonal-Z geometry, an intra-nodal cross-s

treatment is used to improve the accuracy due to the coarse nodalization required fo

geometry.

Pin-power reconstruction capability is provided for Hexagonal-Z geometry, w

reconstruction always based upon two energy groups independent of whether two or four e

groups are utilized in the diffusion equation solution.

Memory management is accomplished utilizing a container array to facilitate effic

memory allocation. In this manner various problems with different dimensionality can

executed without code re-compilation. To facilitate the understanding of coding, procedure

used extensively and an electronic dictionary program, NESTLE.DICT has been created to

the meaning of code variables.

4

Table of Contents

I Introduction............................................................................................. 1

II Theoretical Foundations....................................................................... 5

II.1 Nodal Model - Cartesian Geometry .................................................... 5II.1.a Eigenvalue Problem........................................................................ 5II.1.b Non-Linear Iterative Strategy....................................................... 11

II.2 Outer-Inner Solution Method for FDM Equations ........................... 17II.2.a Inner Iteration Acceleration.......................................................... 19II.2.b Outer Iteration Acceleration......................................................... 23

II.3 Steady-State Fixed-Source Problem ................................................. 32II.3.a Fixed-Source Scaling Factor Method........................................... 33

II.4 Nodal Model - Hexagonal Geometry................................................ 37II.4.a Eigenvalue Problem...................................................................... 37II.4.b Intranodal Burnup Gradient Treatment ........................................ 44II.4.c Non-Linear Iterative Strategy....................................................... 48II.4.d Pin-power Reconstruction Method............................................... 54

II.5 Transient Problem............................................................................. 61

II.6 Adjoint Problem................................................................................ 66

II.7 Cross-Section Model......................................................................... 68II.7.a Macroscopic Model ...................................................................... 68II.7.b Microscopic Model ...................................................................... 72

II.8 Control Option Searches ................................................................... 75

II.9 Hydrodynamic Model ....................................................................... 76II.9.a Field Equations............................................................................. 76II.9.b Equation Discretization................................................................ 77II.9.c Fuel Temperature Model .............................................................. 84II.9.d Steady-State Model ...................................................................... 87II.9.e Effective Heat Transfer Coefficient Evaluation ........................... 88II.9.f Decay Heat Model ........................................................................ 89

III References........................................................................................... 91

IV User’s Guide ....................................................................................... 93

IV.1 Code Control Parameter Data File .................................................. 95

v

IV.2 Geometry Data File ....................................................................... 102IV.2.a Geometry Input ......................................................................... 107

IV.3 Cross-Section Data File................................................................. 112

IV.4 Kinetic Data File ........................................................................... 131

IV.5 Solution Method Control Data File ............................................... 133

IV.6 Initial Exposure Data File.............................................................. 137

IV.7 Initial Isotopic Number Densities Data File.................................. 139

IV.8 Pin-Power Data File ...................................................................... 140

V Programmer’s Guide ........................................................................ 142

V.1 Dependence Diagram...................................................................... 142

V.2 Summary of Procedures.................................................................. 142

V.3 Variables’ Definitions..................................................................... 143

V.4 Variables’ Storage........................................................................... 143

V.5 Machine Specific Instructions ........................................................ 144

V.6 Geometry Treatment ....................................................................... 145

V.7 Installation ...................................................................................... 145

vi

vii

Listing of Tables

Table 1 :Non zero entries in the 16 by 16 two-node NEM problem................. 14

Table 2 :Values of g2(u) as a function of u....................................................... 41

Table 3 :Values of g(u,0) as a function of u...................................................... 43

Table 4 :List of required file names. ................................................................. 94

Table 5 :Listing of procedures and their functions. ........................................ 156

Table 6 :Listing of fcb files containing named COMMON blocks................. 168

Table 7 :Sample interactive session with NESTLE.DICT.............................. 169

viii

Listing of Figures

Figure 1: Overview of NESTLE nested iterative solution strategy. ................. 29

Figure 2: Hex geometry dimensions and axis orientation................................. 37

Figure 3: Conformal mapping of a hexagon to a rectangle............................... 38

Figure 4: v- vs. y-coordinates at x=R(30.5/2). .................................................. 47

Figure 5: Mapping scale function at u=a/2........................................................ 47

Figure 6: Corner Flux Configuration. ............................................................... 58

Figure 7: Thermal-hydraulic mesh notation...................................................... 79

Figure 8: Radial material geometry figures for different core geometries...... 109

Figure 9: Dependence diagram of the NESTLE code..................................... 147

tion

ty);

nvalue

ution

bles:

ernal

gard to

(

Three,

ilable,

core

dary

the

d by a

nodal

etric,

axial

I. Introduction

NESTLE is a FORTRAN77 code that solves the few-group neutron diffusion equa

utilizing the Nodal Expansion Method (NEM). NESTLE can solve the eigenvalue (criticali

eigenvalue adjoint; external fixed-source steady-state; or external fixed-source or eige

initiated transient problems. The code name NESTLE originates from the multi-problem sol

capability, abbreviatingNodal Eigenvalue,Steady-state,Transient,Le core Evaluator. The

eigenvalue problem allows criticality searches to be completed on one of the following varia

soluble boron, coolant inlet temperature, control rod position or core power level. The ext

fixed-source steady-state problem can also search on these same parameters, now in re

achieving a specified power level.

Two or four energy groups can be utilized, with all groups being thermal groupsi.e.

upscatter exists) if desired. Core geometries modelled include Cartesian and Hexagonal.

two and one dimensional models can be utilized. Various core symmetry options are ava

including quarter, half and full core for Cartesian geometry and one-sixth, one-third and full

for Hexagonal geometry. Zero flux, non-reentrant current, reflective and cyclic boun

conditions are treated

The few-group neutron diffusion equation is spatially discretized utilizing theNodal

Expansion Method (NEM). For Cartesian geometry, quartic polynomial expansion for

transverse integrated fluxes are employed. Transverse leakage terms are represente

quadratic polynomial. For Hexagonal geometry, a conformal mapping based hexagonal

method is employed. The transverse integrated flux expansion consists of trigonom

hyperbolic trigonometric, and polynomial functions. The transverse leakage term in the

1

ed in

xagon.

oups.

ation,

g a

tically

e the

Shift

lerate

ndent

ter and

od is

ments

lation.

tegy,

up

ither

ergy

direction is represented by a quadratic polynomial while the radial contribution is express

terms of the mapping scale function and the physical currents on the surfaces of the he

DiscontinuityFactors (DFs) are utilized to correct for homogenization errors.

Transient problems utilize a user specified number of delayed neutron precursor gr

Time dependent inputs include coolant inlet temperature and flow; soluble poison concentr

and control banks’ positions. Time discretization is done in a fully implicit manner utilizin

first-order difference operator for the diffusion equation. The precursor equations are analy

solved assuming the fission rate behaves linearly over a time-step.

Independent of problem type, an outer-inner iterative strategy is employed to solv

resulting matrix system. Outer iterations can employ Chebyshev acceleration, Weilandt

acceleration with flux extrapolation, and the Fixed Source Scaling Technique to acce

convergence. Inner iterations employ either color line or point SOR iteration schemes, depe

upon problem geometry. Values of the energy group dependent optimum relaxation parame

the number of inner iterations per outer iteration to achieve a specified L2 relative error reduction

are determined a priori. The non-linear iterative strategy associated with the NEM meth

utilized. This has advantages in regard to reducing FLOP count and memory size require

versus the more conventional linear iterative strategy utilized in the surface response formu

In addition, by electing to not update the coupling coefficients in the nonlinear iterative stra

the Finite DifferenceMethod (FDM) representation, utilizing the box scheme, of the few-gro

neutron diffusion equation results. The implication is that NESTLE can be utilized to solve e

the nodal or FDM representation of the few-group neutron diffusion equation.

Thermal-hydraulic feedback is modelled employing aHomogenousEquilibrium M ixture

(HEM) model, allowing two-phase flow to be treated. However, only the continuity and en

2

sumed

eter

ourant

ially

in a

y heat.

ation

d that

on to

e code

ed to

ized by

s

ictor-

the

ed and

t the

ptions

or

olant

ensity.

equations for the coolant are solved, implying a constant pressure treatment. The slip is as

to be one in the HEM model. The fuel temperature is determined utilizing a lumped param

model. The SETS method is used for the temporal treatment to overcome the material C

limit on numerical stability. A conventional staggered mesh formulation is used in spat

discretizing the fluid’s equations. Flow is assumed to be parallel to the axial direction with

closed channel. A user specified number of decay heat groups are used to model deca

Direct deposition in the coolant of fission energy is accounted for. Equation of State inform

is provided via polynomials, whose coefficients are provided as input. It should be recognize

the thermal-hydraulic model was developed with a pin-cell geometry as its basis. Adopti

other geometries, such as appear in gas-cooled reactors, would likely require some sourc

modifications.

The thermal conditions predicted by the thermal-hydraulic model of the core are us

correct cross-sections for temperature and density effects. Cross-sections are parameter

color, control rod state (i.e. in or out) and burnup, implying fuel burnup modelling capabilitie

exist. Either a macroscopic or microscopic fuel depletion model may be employed. A Pred

Corrector formulation is used to solve the depletion equations. With the election of

microscopic option, depletion equations for the U234 through U236 and U238 through Pu242

depletion chains, two lumped fission product groups, and a simple burnable poison are solv

used in conjunction with burnup dependent microscopic cross-sections to construc

macroscopic cross-sections. The I-Xe and Pm-Sm chains are also modelled, with various o

to determine their number densities (i.e. equilibrium, transient, peak Sm-no Xe, no Sm nor Xe,

frozen). All cross-sections are characterized in terms of a Taylor’s series expansion in co

density, coolant temperature, effective fuel temperature, and soluble poison number d

3

ut.

uracy.

n is

oups

flux,

n to

tion is

art file

n of

tainer

Taylor’s series terms utilized (e.g. linear or quadratic in coolant density) are specified via inp

An intranodal cross section treatment can be used with Hexagonal geometry to improve acc

Pin power reconstruction is available for Hexagonal geometry. A two-group formulatio

utilized to complete pin-power reconstruction independent of whether two or four energy gr

are utilized in solving the neutron diffusion equation.

Output edits include predicted values of the key core attributes, such as power,

temperatures, isotopic number densities and burnup spatial distributions, in additio

documenting key input options and convergence behavior parameters. The output informa

biased towards the sort of information a nuclear designer of a power reactor requires. A rest

is written, allowing restart for branch cases, re-initiation of core depletion, continuatio

iterations towards a tighter convergence, or re-initiation of a transient.

Memory management is accomplished via a container array. Code determined con

array pointers are used to facilitate problem specific memory allocation (e.g.trading off of spatial

and energy detail within a fixed total memory size).

4

the

[1,2].

f the

sverse

tion,

d

and,

tron

II. Theoretical Foundations

II.1. Nodal Model - Cartesian Geometry

II.1.a. Eigenvalue Problem

The following section describes the standard NEM formulation for the solution of

three-dimensional, Cartesian geometry, multi-group, eigenvalue neutron diffusion equation

The principal characteristics of the polynomial nodal method are its quartic expansions o

one-dimensional transverse-integrated flux and quadratic leakage model for the tran

leakage.

Consider the general form of the steady-state multi-group neutron diffusion equa

written in standard form and with the group constants (i.e. properly weighted cross-sections an

discontinuity factors) already available from a lattice physics calculation forg = 1, 2,..., G

(1)

where the dependence of each quantity on the spatial coordinate has been suppressed,

Dg= diffusion coefficient [cm]

φg= neutron flux [cm-2sec-1]

Σtg = total macroscopic cross section [cm-1]

Σsgg’= group-to-group scattering cross section [cm-1]

χg= fission neutrons yield

k= multiplication factor (i.e. critical eigenvalue)

vg= average number of neutrons created per fission

Σfg= macroscopic fission cross section [cm-1]

As with most modern nodal methods, we begin by intergrating the multi-group neu

∇– Dg∇φg Σtgφg+⋅ Σsgg′φg′χg

k----- vg′Σ fg′φg′

g′ 1=

G

∑+g′ 1=

G

∑=

r

5

s. For

, the

x-

in

diffusion equation over a material-centered spatial node which has homogenized propertie

Cartesian geometry we rewrite Eqn. (1) for the arbitrary spatial nodel,

(2)

where, , and

For simplicity, in cases where redundant equations exist in all three directions

illustrating equations will be only given in the x-direction. Using Fick’s Law, which in the

direction can be expressed as,

(3)

where,

allows Eq. (2) to be rewritten as:

(4)

Integration of Eq. (4) over the volume of nodel generates a local neutron balance equation

terms of the face-averaged net currents and the node volume average flux.

(5)

D– gl

x2

2

∂∂ φg

l r( ) Dgl

y2

2

∂∂ φg

l r( )– Dgl

z2

2

∂∂ φg

l r( ) Agl φg

l r( )+– Qgl r( )=

g 1 G,( )∈

r( ) x y z, ,( ) Vl∈≡ x y∆ z∆∆ Volume of nodel≡=

Agl Σtg

l Σsggl

–xg

l

k-----vgΣ fg

l–=

Qgl

r( ) Qgg′l φg′

lr( )

g′ g≠

G

∑ Σsgg′l φg′

lr( )

xgl

k----- vg′Σ fg′

l φg′l

r( )g′ g≠

G

∑+g′ g≠

G

∑==

jgxl

r( ) Dgl

x∂∂ φg

lr( )–=

jgxl

r( ) x-component of the net neutron current≡

x∂∂

jgxl

r( )y∂

∂jgyl

r( )z∂

∂jgzl

r( ) Agl φg

lr( )++ + Qg

lr( )=

1

xl∆

-------- Lgxl

( ) 1

yl∆

------- Lgyl

( ) 1

zl∆

------- Lgzl

( ) Agl φg

l+ + + Qg

l=

6

ted

en in

ome

It is the

lation

three-

. This

, of the

where, assuming nodel is centered around the coordinate’s origin, the volume integra

quantities are defined below:

and,

where,

Eq. (5) is known as the nodal balance equation. Now for the neutron diffusion equation writt

this form, in order to obtain the spatial neutron flux distribution, one must devise s

relationship between the node average flux and the face-averaged net (surface) currents.

equations used to compute the surface currents in Eq. (5) which distinguish one nodal formu

from another. In NEM, the widely used method of transverse-integration is used, where the

dimensional diffusion equation is integrated over the two directions transverse to each axis

generates three one-dimensional equations, one for each direction in Cartesian coordinates

following form,

φgl 1

Vl

----- φgl

r( ) x y z Node volume average flux≡ddd

z∆ l

2-------–

z∆ l

2-------

∫y∆ l

2-------–

y∆ l

2-------

∫x∆ l

2-------–

x∆ l

2-------

∫=

Qgl 1

Vl

----- Qgl

r( ) x y z Node volume average source≡ddd

z∆ l

2-------–

z∆ l

2-------

∫y∆ l

2-------–

y∆ l

2-------

∫x∆ l

2-------–

x∆ l

2-------

∫=

1

xl∆

--------Lgxl 1

xl∆

-------- Jgx+l

Jgx-l

–( ) 1

Vl

-----x∂

∂jgxl

r( ) x y zddd

z∆ l

2-------–

z∆ l

2-------

∫y∆ l

2-------–

y∆ l

2-------

∫x∆ l

2-------–

x∆ l

2-------

∫= =

Jgx ±l

Average x-directed net current on node facesx∆ l

2--------±≡

7

as a

such

(6)

where,

and,

In NEM, the one-dimensional averaged flux that appears in Eq. (6), is expanded

general polynomial,

(7)

where is the node average flux, implying for Eq. (7) to be true that must be chosen

that the basis functions satisfy

(8)

Note that for quartic NEM, the method used in NESTLE, the summation extends toN = 4. The

first four basis functions in NEM can be expressed as follows [1],

(9)

which can be shown to also satisfy the following,

xdd

jgxl

x( ) Agl φgx

l+ x( ) Qgx

lx( ) 1

yl∆

-------Lgyl

x( )–1

zl∆

-------Lgzl

x( )–=

Lgyl

x( ) 1

zl∆

-------y∂

∂jgyl

r( ) y z Average y-direction transverse leakage≡dd

yl∆2

-------–

yl∆2

-------

∫zl∆2

-------–

zl∆2

-------

∫=

Lgzl

x( ) 1

yl∆

-------z∂

∂jgzl

r( ) z y Average z-direction transverse leakage≡dd

zl∆2

-------–

zl∆2

-------

∫yl∆2

-------–

yl∆2

-------

∫=

φgxl

x( ) φgl

agxnl

f n x( )n 1=

N

∑+=

φgl

f n x( )

f n x( ) xd

x∆ l

2-------–

x∆ l

2-------

∫ 0 for n 1 ...,N,= =

x

xl∆

-------- ; 3

x

xl∆

-------- 2 1

4---;–

x

xl∆

-------- 3 1

4--- x

xl∆

-------- ;–

x

xl∆

-------- 4 3

10------ x

xl∆

-------- 2

– 180------+

f1 f2 f3 f4

8

total

e node

roup,

ides

ts from

nergy

any

h

me [3]

oment

ctions,

, and

nsion

(10)

At this point it is appropriate to consider the elementary concept of accounting for the

number of equations and that of unknowns. For a three-dimensional Cartesian geometry, th

average andN expansion coefficients in each direction appear per node per energy g

implying a total of 3N+1 equations are required. The nodal balance equation, Eq. (5), prov

one equation, where now Eqs. (3) and (7) are used to eliminate face-averaged net curren

this equation. Surface current and flux continuity provide 6 more equations per node per e

group. So forN=2, there would be an equal number of equations and unknowns without

further development. However, forN= 4, two additional unknowns are introduced for eac

direction per node per energy group. This is addressed by using a weighted residual sche

applied to Eq. (6), which in essence provides the additional equations (referred to as the m

equations) needed,

(11)

where the two weighting functions for n = 1,2 are chosen to be the same as the basis fun

namelyωn(x) = fn(x), as those used in the one-dimensional flux expansion1. Here, the first and

second (actually linear combination of zeroth and second) moments of the flux, source

leakage for each groupg are defined by,

The first term in Eq. (11) is evaluated by using Eqs. (3) and (7) and the definition of the expa

1. This constitutes amoments weightingscheme; if one usesωn(x) = fn+2(z) for n = 1,2 it is known asGaler-kin weighting. Numerical experiments favormoments weighting.

f nx

l∆2

--------± 0 for n 3 4,==

ω< n x( )xd

djgxl

x( ) Agl φgxn

l+>, Qgxn

l 1

yl∆

-------Lgyxnl

–1

zl∆

-------Lgzxnl

–=

ωn x( ) φgxl

x( ), >< ωn x( ) Qgxl

x( ), >< ωn x( ) Lgyl

x( ), >< ωn x( ) Lgzl

x( ),< >

φ lgxn Ql

gxn Llgyxn Llgzxn

9

re the

known,

dratic

the y-

e

g the x-

odes.

coefficients, and completing the integration (i.e. inner product) analytically.

One last point which needs to be addressed before Eq. (11) can be solved a

transverse leakage terms appearing on the right hand side. Their spatial dependency is un

so their “shape” must be approximated. The most popular approximation in NEM is the qua

transverse leakage approximation. For example, the x-direction spatial dependence of

direction transverse leakage is approximated by,

(12)

where is the average y-directed leakage in nodel, and the coefficients and can b

expressed in terms of average y-directed leakages of the two nearest-neighbor nodes alon

direction (i.e. nodesl-1 andl+1) so as to preserve the node average leakages of these three n

The quadratic expansion coefficients can be shown to be given by,

(13)

(14)

where,

(15)

Lgyl

x( ) Lgyl

ρgy1l

f 1 x( ) ρgy2l

f 2 x( )+ +≅

Lgyl

ρgy1l ρgy2

l

ρgy1l

gl

xl∆( ) Lg

l 1+Lg

l–( ) x

l∆ 2 xl 1–∆+( ) x

l∆ xl 1–∆+( ) Lgy

lLgy

l 1––( ) x

l∆ 2 xl 1+∆+( ) x

l∆ xl 1+∆+( )+[ ]=

ρgy2l

gl

xl∆( )

2Lgy

l 1+Lgy

l–( ) x

l∆ xl 1–∆+( ) Lgy

l 1–Lgy

l–( ) x

l∆ xl 1+∆+( )+[ ]=

gl

xl∆ x

l 1+∆+( ) xl∆ x

l 1–∆+( ) xl 1–∆ x

l∆ xl 1+∆+ +( )[ ]

1–=

10

the

nd to

on,

mith

de

rather

(

oup

tegy,

alled

ing

oved

mated

dated

This

orces

rents

rage

forms;

usable

e base

actor

II.1.b. Non-Linear Iterative Strategy

The most common manner of solving the matrix system associated with NEM is

response-matrix formulation. To minimize computer run time and memory requirements, a

facilitate the capability to solve either the NEM or Finite Difference Method (FDM) formulati

the non-linear iterative strategy is employed in NESTLE. This technique was developed by S

[4,5,6] and successfully implemented into the Studsvik QPANDA and SIMULATE co

packages. The documentation available on this technique is scarce, but it turns out to be

simplistic and almost trivial to implement in a FDM code which utilizes the box-schemei.e.

material-centered).

The basic idea is applicable to the standard FDM solution algorithm of the multi-gr

diffusion equation. Solving the FDM based equation utilizing an outer-inner iterative stra

every outer iterations (where is somewhat arbitrary but can be optimized) the so-c

“two-node problem” calculation (a spatially-decoupled NEM calculation spanning two adjoin

nodes) is performed for every interface (for all nodes and in all directions) to provide an impr

estimate of the net surface current at that particular interface. Subsequently, the NEM esti

net surface currents are used to update (i.e. change) the original FDM diffusion coupling

coefficients. Outer iterations of the FDM based equation are then continued utilizing the up

FDM coupling coefficients for outer iterations. The entire process is then repeated.

procedure of updating the FDM couplings is a convergent technique which progressively f

the FDM equation to yield the higher-order NEM predicted values of the net surface cur

while satisfying the nodal balance Eq. (5), thus yielding the NEM results for the node-ave

flux and fundamental mode eigenvalue. The advantages of this technique come in many

the storage requirements are minimal because the two-node problem arrays are re-

(disposable) at each interface, the rate of convergence is nearly comparable to that of th

FDM algorithm being used, the number of iteratively determined unknowns is reduced by a f

N∆ 0 N∆ 0

N0∆

11

e of

n be

icity,

ions

terms

he x-

/node

with

ment

cond

of 6 (node flux vs. partial surface current), and the simplicity of the algorithm and eas

implementation, compared to any other nodal technique, is far superior.

The two-node problem produces an 8G X 8G linear system of equations which ca

constructed by applying the standard NEM relations to two adjoining nodes. For simpl

consider two arbitrary adjoining nodes in the x-direction. Denote these notes asl andl+1:

Substitution of the one-dimensional expansion, Eq. (7), into Fick’s law yields express

for the average x-direction net surface currents at the left(-) and right(+) interfaces of nodel,

(16)

Now, assume the node average flux, criticality constant, and all transverse direction

are known from a previous iteration; then, the total number of unknowns associated with t

direction two node problem is 8G, which corresponds to the 4 expansion coefficients/group

(x) G groups (x) two nodes. The 8G constraint equations are obtained as follows. We begin

the substitution of Eq. (16) into the nodal balance equation for node l, to yield the zeroth mo

constraints (G equations/node),

(17)

A similar substitution into the moment-weighted equation, Eq. (11), yields the first and se

moment constraints (2G equations/node),

(18)

(19)

Nodel

Nodel+1x- x+

jgx±l D– g

l

xl∆

---------- agx1l

3agx2l 1

2---agx3

l 15---agx4

l±+±≡

D– gl

xl∆ x

l∆---------------- 6agx2

l 25---agx4

l+

1

yl∆

-------Lgyl

–1

zl∆

-------Lgzl

Agl φg

lQgg′

l

g′ g≠

G

∑ φg′l

+––=

60

xl∆

--------Dg

l

xl∆

-------- Agl

+ agx3l

Qgg′l

ag′x3l

g′ g≠

G

∑ 10Aglagx1

l– 10 Qgg′

lag′x1

l

g′ g≠

G

∑+– 101yl∆

--------ρgy1l 1

zl∆-------ρgz1

l+

=

140

xl∆

---------Dg

l

xl∆

-------- Agl

+ agx4l

Qgg′l

ag′x4l

g′ g≠

G

∑ 35Aglagx2

l– 35 Qgg′

lag′x2

l

g′ g≠

G

∑+– 351yl∆

--------ρgy2l 1

zl∆-------ρgz2

l+

=

12

The

using

ons)

q. (7),

sics

of the

tially

t each

t the

t of a

ly [7].

=2.

Similar equations can be written for node l+1, producing a total of 6G equations.

continuity of net surface current constraints at the interface (G equations) are obtained by

Eq. (16) at the adjoining interface of the two nodes,

(20)

Last, the continuity (or discontinuity) of surface-averaged flux constraints (G equati

are obtained by equating the surface-averaged fluxes of the two adjoining nodes by using E

(21)

where and are the Discontinuity Factors (DFs) obtained from lattice phy

calculations. Do note that continuity conditions are never imposed on the outside surfaces

two-node problem, since the two-node problem is deliberately formulated to be spa

decoupled. Continuity is assured in the formulation of the FDM based equations.

Eqs.(7) through (21) constitute the 8G system of equations needed to be solved a

interface. This matrix system, after taking advantage of its reducability and by noting tha

even-moment expansion coefficients don’t change whether the node is on the left or righ

two-node problem, can be reduced to smaller systems which can be solved quite efficient

The following table illustrates this more efficient arrangement of unknowns for the case of G

D– gl

xl∆

---------- agx1l

3agx2l agx3

l

2----------

agx4l

5----------+ + +

D– gl

xl 1+∆

--------------- agx1l 1+

3agx2l 1+ agx3

l 1+

2-----------

agx4l 1+

5-----------–+–=

dgx +l φg

l agx1l

2----------

agx2l

2----------+ + dgx –

l 1+ φgl 1+ agx1

l 1+

2-----------

agx2l 1+

2-----------+–=

dgx ±l

dgx ±l 1+

13

8G

duce

cient

*Refers to order of polynomial that transverse integrated

flux expansion coefficient is associated with.

In NESTLE, the two-node problems are solved by utilizing the analytic solution to the 8G X

matrix system. This was accomplished by employing symbolic manipulator software to pro

the FORTRAN code segment used in NESTLE. This approach is computationally more effi

Table 1: Non zero entries in the 16 by 16 two-node NEM problem.

Eqn Grp Nod a b c d e f g h i j k l m n o p

0th Moment 1 l x x

0th Moment 2 l x x

2nd Moment 1 l x x x x

2nd Moment 2 l x x x x

0th Moment 1 l+1 x x

0th Moment 2 l+1 x x

2nd Moment 1 l+1 x x x x

2nd Moment 2 l+1 x x x x

1st Moment 1 l x x x x

1st Moment 2 l x x x x

1st Moment 1 l+1 x x x x

1st Moment 2 l+1 x x x x

Cur Con 1 x x x x x x x x

Cur Con 2 x x x x x x x x

Flx Dis 1 x x x x

Flx Dis 2 x x x x

UNKNOWN NODE GROUP EXP. COEF.*

a l 1 2b l 2 2c l 1 4d l 2 4e l+1 1 2f l+1 2 2g l+1 1 4h l+1 2 4i l 1 1j l 2 1k l 1 3l l 2 3m l+1 1 1n l+1 2 1o l+1 1 3p l+1 2 3

14

to

-node

-node

e

n all

wn in

nt, the

be

is

FDM

ere to

is

dard

LE,

d. The

rface

than utilizing a direct matrix solver (e.g. LU decomposition); however, it limits the values of G

those directly programmed for. Also note that on boundaries special treatments of the two

problems are required. Depending upon the specified boundary condition (BC), one

problems may originate (e.g. zero flux BC), or on interior axis geometry unfolding may b

required to create a two-node problem (e.g. cyclic BC).

Solutions of the two-node problems provide NEM evaluated values of the currents o

surfaces for specified values of the node average fluxes [recall they were assumed kno

solving the two-node problems]. To correct the FDM based expression for the surface curre

following approach is utilized. The coupling coefficient update to the FDM equation can

implemented by simply expressing the FDM net surface current at thex+ face of nodel as

follows,

(22)

The first term on the RHS is the normal FDM approximation for a box scheme, where

the actual FDM diffusion coupling coefficient between nodes l and l+1,

(23)

The second term on the RHS represents the nonlinear NEM correction applied to the

scheme. The (+) sign between the flux values in the second term of Eq. (22) is purposely th

improve the convergence behavior of the nonlinear iterative method [8]. Note that if

zero, which it initially is in NESTLE’s implementation, then Eq. (22) corresponds to the stan

FDM definition of the net surface current. This is the basis for the FDM option within NEST

where now two-node problem solves and coupling coefficients updates are never complete

value of is determined by setting Eq. (22) equal to the NEM two-node predicted su

Jgx +l FDM, Dgx +

l FDM,

xl∆ x

l 1+∆+2

-----------------------------

-----------------------------– φgl 1+

φgl

–[ ]Dgx +

l NEM,

xl∆ x

l 1+∆+2

-----------------------------

----------------------------- φgl 1+

φgl

+[ ]–=

Dgx +l FDM,

Dgx +l FDM, Dg

lDg

l 1+x

l∆ xl 1+∆+( )

Dgl

xl∆ Dg

l 1+x

l 1+∆+------------------------------------------------------=

Dgx +l NEM,

Dgx +l NEM,

15

r this

one

n for

inally,

n the

current value, using the associated node average flux values in Eq. (22) and solving fo

quantity.

Summarizing, to apply a NEM update after outer iterations of the FDM routine,

solves the two-node problem at a given interface, then (with the expansion coefficients know

that interface) one calculates the NEM estimate of the net surface current using Eqn.(16) F

one equates this result to Eq. (22), and solves for the value of which will be used i

subsequent set of FDM iterations.

N0∆

Dgx +l NEM,

16

DM

ting,

stem.

tion,

and

tions

lems

the

TLE’s

the

onal

(23)

. (8)

II.2. Outer-Inner Solution Method for FDM Equations

The only large matrix that requires solution for the non-linear iterative method is the F

representation of the multi-group diffusion equation. Much work has been done on formula

understanding and implementing the iterative solution of this large, sparse matrix sy

NESTLE takes advantage of this wealth of knowledge in its iterative solution implementa

utilizing an outer-inner iterative strategy.

The “Outer-Inner Method” refers to outer iterations to update the fission source term

inner iteration to approximately solve the resulting fixed source problem. The outer itera

correspond to a “Power Method.” This method can be applied to both Fixed Source Prob

[FSP] and the Associated Eigenvalue Problem [AEVP]. Shortly it will be shown that both

fixed source steady-state and transient problems are representable as FSP in NES

formulation. Although the AEVP involves additional calculations for the eigenvalue, basically

iteration schemes for both problems are similar. We will discuss the AEVP first.

Returning to Eq. (5), the FDM representation of this equation in three-dimensi

Cartesian geometry within homogenous nodel can be expressed as follows:

(24)

where the non-zero values of the coupling coefficients are obtained via Eqs.(22) and

and L denotes the total number of nodes. Substituting in the definitions for and into Eq

and rearranging terms we obtain

(25)

This equation can be written in terms of matrix notation spanning the spatial domain as

(26)

Cgl l ′, φg

l ′

l ′ 1=

L

∑ Agl φg

l+ Qg

l=

Cgl l ′,

Agl

Qgl

Cgl l ′,

l ′ l≠

L

∑ φgl ′

Σtg

l Σsgg

lCg

l l,+–( )+ φg

lΣsgg′

l

g′ g≠

G

∑ φg′l

–xg

l

k----- υg′

g′ 1=

G

∑ Σ fg′l φg′

l=

Agφg Σsgg′φg′g′ g≠

G

∑–1k---xg υg′Σ fg′

l φg′g′ 1=

G

∑=

17

has a

X L)

GL)

ps.

y Eq.

es for

ive

e rate

ups,

plies

where the “bar” over the node average flux value now denotes a column vector. Matrix

seven-banded matrix structure for three-dimensional Cartesian geometry. In turn, the G (L

matrix systems expressed by Eq. (26) can be collected to write the following single (GL X

matrix system.

(27)

The matrix is block lower triangular in structure for that portion applicable to the fast grou

The outer-inner iteration process is summarized as follows: For the AEVP specified b

(27), given an arbitrary initial vector , the outer iterations generate successive estimat

the flux vector by the process

(28)

where how the criticality constant (i.e.eigenvalue) is updated will be discussed later. The iterat

matrix associated with the outer iterations is

(29)

The properties of the iterative matrix has a significant role in determining the convergenc

of the power iterations [9,10].

In solving Eq. (28), advantage is taken of the structure of the matrix. For the fast gro

solving from low to high energy group number results in energy group decoupling. This im

that we may solve a system of linear equations of the form

(30)

where,

(31)

Ag

Aφ 1k---Fφ=

A

φ0( )

φ

φq( ) 1

kq 1–( )---------------A

1–Fφ

q 1–( )=

Q A1–F=

Q

A

Agφgq( )

Sgq( )

=

Sgq( ) Σsgg′φg′

q( ) 1

kq 1–( )---------------+

g′ g≠

G

∑ χg νg′Σ f g′φg′q 1–( )

g′ 1=

G

∑=

18

oups

g Eqn.

ups’

attering

lor

ions.

ree-

direct

of

llows.

For the thermal groups, NESTLE assumes the group fluxes for all other thermal gr

except the one being updated are known. This produces energy group decoupling, allowin

(30) to be utilized. So called “scattering” iterations are then completed after all thermal gro

fluxes are updated. Stationary acceleration is employed to accelerate convergence of the sc

iterations.

II.2.a. Inner Iteration Acceleration

To solve Eq. (30) we introduce the inner iterations. In this work we employ a Multi-Co

Point or Line SOR Method, depending upon problem geometry, for the inner iterat

Specifically, a Red-Black Point or Line SOR method is used in NESTLE for two or th

dimensional Cartesian geometry, respectively. For one-dimensional Cartesian geometry, a

matrix solve is utilized since the group-wise A matrix is triangular allowing employment

Gaussian elimination.

Mathematically, this approach is a multi-splitting method and can be expressed as fo

(32)

where,

(33)

and

(34)

(35)

φ φp where vectorφp spans nodes of color ''p''⊕=

φpm 1+( )

Bp1–

S Cpp′φp′m 1+( )

Cpp′φp′m( )

p′ p 1+=

P

∑+p′ 1=

p 1–

∑+= for p 1,2,...,P=

A Ap and non-square matrixAp equals rows ofA that span nodes of color ''p''⊗=

Ap Bp Cpp′

p′ p≠

P

∑–= for p = 1,2,...,P

φpm 1+( )

φpm( )

ω φpm 1+( ) φp

m( )–( )+=

19

in the

cheme

that

per

error

that the

since

ese

ptable.

ss-

he

arizes

lue of

nergy

Note that the group g and outer iteration count (q) indices have been suppressed for clarity

above equations. The matrix is square and has either a diagonal structure for the point s

or block diagonal structure composed of tridiagonal blocks for the line scheme. This implies

the action of indicated in Eqs. (32) is simple to evaluate. A total of inner iterations

outer iterations are completed, this value determined such that the specified relative

reduction from the 0th iterative error for the inner iterations is achieved.

To a priori determine the value of the optimum relaxation parameter,ω and [which

are energy group dependent but dependence notation has been surpressed], it is assumed

iterative matrix associated with this inner iterative method is symmetrizable. This is not true

the NEM corrections to the FDM coupling coefficients invalidate symmetry; however, th

corrections have been found to be relatively small so the symmetrizable assumption is acce

Making this assumption, we can expressω in terms of the spectral radius of the associated Gua

Seidel iteration matrix, , as follows,

(36)

Clearly . Therefore, calculation of the spectral radius of t

associated Gauss-Seidel iterative matrix is the heart of this procedure. The following summ

the details of the computational procedure used in NESTLE to obtain an estimate of the va

ω, which is based upon the DIF3D methodology [10]. These steps are completed for each e

group.

Bp

Bp1–

NI∆

NI∆

ρ LG-S

( )

ω 2

1 1 ρ LG-S( )–[ ]

1/2+

------------------------------------------------=

LG-S

LSOR

ω( )with ω 1= =

20

st

d

0.1 Step 1.Starting with an arbitrary non-negative initial guess vector , complete at lea

ten Gauss-Seidel iterations in solving the following equation.

0.2 Step 2.Following each iteration withm >10, estimate the upper and lower bounds of the

spectral radii using the following equations.

Compute the corresponding relaxation factors given by

0.3 Step 3.Terminate iteration when either

or mequals a specified upper limit [10,11]. The optimum factorω is then set toω(m). This

test forces tighter convergence ofω when is close to unity to ensure the require

numerical accuracy is achieved.

x0( )

Ax 0=

λ m( ) xm( )

xm( )

,⟨ ⟩

xm( )

xm 1–( )

,⟨ ⟩-----------------------------------≡

λm( )

MAXixi

m( )

xim 1–( )----------------≡

λ m( )MINi

xim( )

xim 1–( )----------------≡

ω m( ) 2

1 1 λ m( )–[ ]

1/2+

---------------------------------------≡

ω m( ) 2

1 1 λm( )

–[ ]1/2

+---------------------------------------≡

ω m( ) 2

1 1 λ m( )–[ ]

1/2+

---------------------------------------≡

ω m( ) ω m( )–

2 ω m( )–

5--------------------<

ρ LG-S( )

21

uch

d of

may

ation of

ner of

antial

ns has

roup.

tio of

s are

0.4 Step 4.Determine the number of inner iterations required for each outer iteration , s

that the value of satisfies the following equation:

where

and denotes the desired relative error reduction from the initial iteration to the en

-th iteration. It is suggested that a very small number for not be used since it

force excessive inner iterations [10].

The advantages of these accelerations strategies are clear. The automated determin

the optimum overrelaxation factors relieves users of the burden of the trial and error man

specifying optimum parameters for a large class of reactor models. In addition, subst

computational time can be saved since the need to check the convergence of inner iteratio

been removed by using a fixed number of predetermined inner iterations for each energy g

The outer iterations defined by Eq. (28) are slow to converge, since the dominance ra

the iterative matrix, Eq. (29), is close to one. Two complementary acceleration technique

utilized in NESTLE to accelerate the outer iterations of the AEVP.

NI∆

NI∆

LSOR

ω( )( )NI 1–∆

LG S–

⋅ t2 NI 1–∆2

t2 NI∆2

+[ ]1/2

εin≤=

t NI∆ ω 1–[ ]NI 1–∆

2------------------

ρ LG S–

( )[ ]1/2

1 NI 1–∆( )+ 1 ρ LG-S

( )–( )1/2

[ ]=

εin

NI∆ εin

22

ased

. No

s the

yshev

ctors

od [9,

ressed

hod.

ction

yshev

in the

,

lied to

h to

II.2.b. Outer Iteration Acceleration

The outer iterations for the AEVP are accelerated by using either a polynomial b

acceleration method or an eigenvalue shift acceleration method with flux extrapolation

knowledge of higher eigenvalues are required to utilize either method. We first discus

polynomial based acceleration method, which utilizes Chebyshev polynomials. Cheb

polynomials [12] are used to obtain the best linear combinations of the previous iterative ve

so as to minimize the error. The method implemented is the Chebyshev Semi-Iterative meth

10, 11, 13]. In this method, the error vector associated with the acceleration method is exp

in terms of a linear combination of the error vectors of the underlying interactive met

Acceleration of the iteration is achieved by minimizing the error vector by appropriate sele

of the expansion coefficients, which is determined to be those associated with Cheb

polynomials. Further details of the mathematical background of this method can be found

related references [9, 10].

Since the rate of convergence in the AEVP is dependent on the dominance ratio

the Chebyshev acceleration method detailed in Refs. [9, 10, 11, 13] can therefore be app

iterations,

(37)

provided that a suitable estimate of is obtained. NESTLE follows the DIF3D approac

solve the AEVP in which we accelerate the fission sourceΨ [13], whereΨ is defined as

(38)

The accelerated iterative procedure can then be expressed as follows:

(39)

σ Q

φq( ) 1

kq 1–( )---------------Qφ

q 1–( )=

σ Q

Ψ νg′Σ fg′φg′g′ 1=

G

∑ νΣ f φ= =

Ψn* p+( ) 1

kn* p 1–+( )

------------------------QΨn* p 1–+( )

=

23

yshev

to be

ented

trix,

are

ower

tone

are

where

(40)

(41)

and

and p denotes the successive fission source iterations employed within a Cheb

cycle (i.e.since last updating the estimate of ). Note the dominance ratio needs

estimated in order for the scheme to work. This is accomplished using the procedure implem

in DIF3D [10] as now outlined. Do note that versus is the relevant outer iterative ma

since fission source versus flux extrapolation is employed.

Since an accurate estimate of is not known when the outer iterations

commenced, a “boot-strap” process is required. By performing a limited number of p

iterations, a reasonable initial estimate of is obtained. Only when all but the first over

mode are essentially damped out, high-order cycles based on accurate estimates of

Q νΣ f A1–

χ=

Ψn

*p+( )

Ψn

*p 1–+( )

αp Ψn

*p+( )

Ψ–n

*p 1–+( )

βp Ψn

*p 1–+( )

Ψ–n

*p 2–+( )

++=

kn* p+( )

kn* p 1–+( ) Ψ

n* p+( )22

Ψn* p+( )

Ψn* p 1–+( )

( , )

----------------------------------------------------

=

α12

2 σ Q( )–----------------------=

β1 0=

αp4

σ Q( )------------- p 1–( )γ[ ]cosh

pγ[ ]cosh-------------------------------------

=

βp 1σ Q( )

2-------------–

αp 1–=

γ 1–cosh 2

σ Q( )------------- 1–

=

n*

outer iteration index where acceleration begins=

p 1≥( )

σ Q( ) σ Q( )

Q Q

σ Q( )

σ Q( )

σ Q( )

24

:

ev

llest

ing

ence

w

being

utilized [10, 14]. More precisely, the algorithm can be described in terms of four basic steps

Step 1. A minimum of three power iterations are performed initially. The first Chebysh

acceleration cycle is begun on outer iteration (n* + 1), where (n* + 1) is the sma

integer such that for which the dominance ratio estimate, satisfies the follow

criterion:

where

Step 2. Using as the dominance ratio estimate for , the accelerated iterative sequ

given by Eqns. (39) and (41) is carried out for iterations with . At first lo

degree polynomials are applied repeatedly with estimates of the dominance ratio

updated continuously according to

where

n*

3≥ σ

0.4 σ 1.0≤ ≤

σ Rn*( )

Rn*( )

,⟨ ⟩

Rn* 1–( )

Rn* 1–( )

,⟨ ⟩---------------------------------------------

1/2

=

Rn*( )

Ψ n*( ) Ψn* 1–( )

–≡

σ σ Q( )

n*

p+( ) p 1≥

σ′ σ2---

1– γ( )coshp 1–

--------------------------- 1+ cosh=

γ Cp 1–2 σ–

σ------------

En*,p-1=

En*,p-1Ψ

n* p+( )Ψ

n* p 1–+( )– 2

Ψn* 1+( )

Ψn*( )

– 2

-----------------------------------------------------------=

Cp 1– y( ) Chebyshev polynomial of degreep 1–( )=

p 1–( ) 1– ycosh[ ] y 1>,cosh=

25

ction

een

han

cycle

mial

d up

are

The polynomials are at least of degree 3 and are terminated when the error redu

factor is greater than the theoretical error reduction factor:

The theoretical error reduction factor is the error reduction which would have b

achieved if were equal to , the true dominance ratio. If is greater t

this, the acceleration cycle has not been as effective as it should have been, so a new

is started using the updated dominance ratio estimate, . Alternatively, the polyno

degree will be terminated if the reduction in theL2 relative residual of the diffusion

equation, defined as

falls below a specified value.

Step 3. After the estimates for have converged higher degree polynomials are applie

to the maximum degree specified.

Step 4.The outer iterations are terminated at outer iteration n if the following four criteria

met:

En* p, 1–

En*,p-1 Cp 1–2 σ–

σ------------

1–>

σ σ Q( ) En* p 1–,

σ′

Aφn* p+( ) 1

kn( )--------Fφ

n* p+( )–

2

1

kn* p+( )

-----------------Fφn* p+( )

2

-------------------------------------------------------------------

Aφn*( ) 1

kn( )--------Fφ

n*( )–

2

1

kn*( )

----------Fφn*( )

2

-----------------------------------------------------

--------------------------------------------------------------------

σ Q( )

26

the

ed

, the

ed to

f the

to be

effects

and are

pend

rmine

where are input parameters. The following meanings of

“normed” stopping criteria should be noted:

= L2 norm of the relative residual of the outer iterative equation

= norm of the true error of the fission source

= L2 norm of the relative residual of the diffusion equation. [Note that for a fix

source problem, whether associated with an external source or transient problem

normalization shown in the denominator of the expression bounded by is chang

theL2 norm of the source.]

Modification to this basic scheme is made in the actual implementation in NESTLE o

Chebyshev polynomial acceleration. Due to various thermal-hydraulic feedback effects,

discussed later, the coefficient matrices and in Eq. (27) are changed whenever such

are accounted for in the system. That is, since feedback effects change cross sections

dependent upon the flux solution, our matrix problem is truly non-linear since and de

upon the flux solution. Since the non-linearity is weak, one can guess a flux solution, dete

kn( )

kn 1–( )

– εk≤

Ψn( )

Ψn 1–( )

– 2

Ψn( )

Ψn 1–( )

,⟨ ⟩1/2

------------------------------------------- εΨ2≤

11 σ–------------

maxiΨi

n( ) Ψin 1–( )

–

Ψin( )---------------------------------- εΨ∞

<

Aφn( ) 1

kn( )--------Fφ

n( )–

2

1

kn( )--------Fφ

n( )

2

------------------------------------------------- εφ<

εk ε, Ψ2εΨ∞

andεφ,

εΨ2

εΨ∞L∞

εφ

εφ

A F

A F

27

flux

tions

plete

that it

. An

yshev

d. The

small

entire

in our

omial

ctions

the

rmal-

NEM

s is

, or a

hed,

latest

the feedback effects and appropriately modify and , and solve for the flux. This updated

solution can then be used to re-initiate the cycle until both the feedback and flux solu

converge. One way to handle these effects is to update the and matrices after com

termination of the outer iteration process. This approach has a clear disadvantage in

requires large computational time to obtain converged solutions for feedbacks and flux

alternate approach is to update the coefficient matrix for feedback effects during the Cheb

acceleration process. In doing so, a substantial reduction in computation time can be realize

latter approach can be justified by observing that the feedback effects are relatively

perturbations to the original system from a reactor physics point of view and hence, the

Chebyshev acceleration scheme is not jeopardized. This modified scheme is incorporated

work in such a manner that the matrices are updated just before a new Chebyshev polyn

acceleration cycle begins. The same approach is taken in regard to updating the NEM corre

to the coupling coefficients.

Figure 1 summarizes the overall nested iterative solution strategy used within

NESTLE code. This strategy has been demonstrated to be efficient and robust. A the

hydraulic feedback iteration is completed each time a new Chebyshev cycle is started. A

non-linear iteration is completed when either after a sufficient number of outer iteration

completed so that a specified relative L2 error reduction in the fission source is achieved

specified maximum number of outer iterations counting from the past NEM iteration is reac

which ever occurs first. Mathematically this is expressed as follows, where denotes the

outer iteration where a NEM non-linear iteration has been completed andm denotes the number

of outer iterations completed since this update.

A F

A F

n

28

value

, of the

ative

ient

Figure 1: Overview of NESTLE nested iterative solution strategy.

The alternative outer iterative method to Chebyshev acceleration employs an eigen

shift approach to decrease the dominance ratio or spectral radius, which ever is applicable

outer iterative matrix. Specifically, the Weilandt shift method is employed. Now the outer iter

equation is given by the following

(42)

where denotes a diagonal matrix defined as follows for the diagonal coeffic

associated with energy groupg and nodem.

Ψn m+( )

Ψn m 1–+( )

– 2

Ψn m+( )

Ψn m 1–+( )

,⟨ ⟩1/2

-----------------------------------------------------------

Ψn( )

Ψn 1–( )

– 2

Ψn( )

Ψn 1–( )

,⟨ ⟩1/2

-------------------------------------------

---------------------------------------------------------------- εΨNEM≤

OR

m MNEM=

NEM Non-Linear Iterations

Thermal-Hydraulic Feedback Iterations

FDM Outer Iterations

FDM Scattering Iterations

FDM Inner Iterations

φn* p+( )

AλW

n* 1–( )

kn* 1–( )

-----------------

FSn* 1–( )

–

1–1

kn* p 1–+( )

------------------------

FλW

n* 1–( )

kn* 1–( )

-----------------

FSn* 1–( )

–

φn* p 1–+( )

=

FSn* 1–( )

29

the

,

t shift

r

for

es not

uter

, one

ates of

or a

hev

r NEM

tions

shev

(43)

In this manner energy group de-coupling is retained when solving Eq. (42), facilitating

implementation of the two and four energy group options within NESTLE. In Eq. (42), (n*-1)

denotes the last outer iteration where the shift has been updated,p the number of outer iterations

i.e. cycle, since the last time the shift has been updated, and the relative Weiland

with reference to the outer iterative estimate ofk. The upper limit on is provided as use

input, which must be less than or equal to 1.0. Internal to NESTLE, a fixed schedule

approaching the user specified value from below is implemented to assure that the shift do

exceed the true value ofk-1 during the early iterative estimates. To further accelerate the o

iterations, a stationary extrapolation of the flux is employed, mathematically expressed as

(44)

Note that be setting the relative Weilandt shift to zero and using stationary acceleration

obtains an outer iteration strategy based solely upon stationary acceleration. Updated estim

the eigenvalue are obtain by employing Galerkin weighting.

(45)

Termination of a Weilandt cycle is determined by either a maximum specified cycle

reduction in theL2 relative residual of the diffusion equation as defined earlier for the Chebys

acceleration method. As with Chebyshev acceleration, the various matrices are updated fo

coupling coefficients, thermal-hydraulic feedback, fission product and criticality search varia

only at the conclusion of a Weilandt shift cycle.

Termination of the outer iterations is done using the same criteria as for Cheby

FSn* 1–( )

gmχgm νg′mΣ fg′m

φg'mn* 1–( )

φgmn* 1–( )

-----------------

g′ 1=

G

∑=

λWn* 1–( )

λWn* 1–( )

φn* p+( )

φn* p 1–+( )

Ω φn* p+( )

φn* p 1–+( )

– +=

kn( ) φ

n( )Fφ

n( ),⟨ ⟩

φn( )

Aφn( )

,⟨ ⟩-------------------------------=

30

e used

tion.

ative

acceleration, except that the norm of the true error of the fission source can no longer b

since it requires an estimate for the value of , which is not available from Weilandt accelera

This stopping criteria is changed to the norm of the relative residual of the outer iter

equation.

L∞

σ

L∞

maxiΨi

n( ) Ψin 1–( )

–

Ψin( )---------------------------------- εΨ∞

<

31

32

II.3. Steady-State Fixed-Source Problem

Real reactors utilize fixed neutron sources to facilitate start-ups and assure high enough

count rates for nuclear instrumentation used for control and protection. We refer to the analysis of

this situation as a Fixed Source Problem [FSP]. Mathematically, the multi-group diffusion

equation for a steady-state FSP is as follows,

(46)

where dependence has been surpressed and denotes the external neutron source.

This equation can be solved utilizing nearly exactly the same method as utilized for the

AEVP, except now appears on the RHS in the NEM equations associated with the AEVP.

This applies to both the FDM equation and two-node problem equations. The biggest difference

in the solution of the FSP versus AEVP originates because the FSP does not involve determining

the fundamental eigenvector. This impacts the outer iterations of the FDM equations in the

following manner. For the AEVP, the rate of convergence of the Power Method is determined by

the dominance ratio of the outer iterative matrix, ; by contrast, for the FSP the rate of

convergence is determined by the spectral radius, , where note that, . The

implication for the Chebyshev Semi-Iterative method is whenever appeared in the

governing equations, it should be replaced by . The other implication for the Chebyshev

Semi-Iterative method is that the FSP versus AEVP outer iterations will converge much slower

since for problems of interest. When Weilandt Shift is employed,

neither of these issues appear. A special implementation of the Coarse Mesh Rebalance method,

as now described, is utilized for the FSP to accelerate convergence.

∇ Dg∇φg⋅– Σtgφg+ Σsgg′φg′g′ 1=

G

∑ χg νg′Σ fg′φg′ Sextg+

g′ 1=

G

∑+=

r Sextg

Sextg

σ Q( )

ρ Q( ) ρ Q( ) keff=

σ Q( )

ρ Q( )

σ Q( ) ρ Q( )< keff 1≈=

ith

erative

flux

e used

ingle

antly

imate

y in

c(q) is

hting

II.3.a. Fixed-Source Scaling Factor Method

When the FSP is near-critical (i.e. approaches unity), convergence rates w

Chebyshev acceleration are unacceptably slow. This convergence is slow even when the it

flux shape is correct but the magnitude is in error. To accelerate convergence of the

magnitude, a global coarse mesh rebalance [12] is proposed. This acceleration option can b

with when either Chebyshev or Weilandt Shift acceleration are employed. Application of a s

scaling prior to the start of a new Chebyshev acceleration cycle sometimes can signific

reduce the required number of outer iterations. This reduction is achieved by an approx

procedure that attempts to scale the current iterative flux vector to the exact flux vector.

For steady-state the FDM based matrix equation analogous to Eq. (27) is

(47)

Now assume that theqth outer iterative estimate of the flux has the correct shape but is off onl

magnitude by a factor of c(q) from the exact solution,i.e.

(48)

Then it follows that an improvedqth iterate is given by

(49)

or in terms of the fission source

(50)

where is the Chebyshev accelerated fission source. The fixed source scaling factor,

defined so as to preserve neutron balance in an integral sense. Utilizing Galarkin weig

defines c(q) as follows

(51)

ρ Q( )

A F–( )φ Sext=

φ cq( )φ

q( )=

φq( )

cq( )φ

q( )=

Ψq( )

cq( )Ψ

q( )=

Ψq( )

cq( ) φ

q( )Sext,⟨ ⟩

φq( )

A cq( )( ) F c

q( )( )–( )φq( )

,⟨ ⟩--------------------------------------------------------------------------=

33

sed in

inates

as a

l, the

be

er

ated as

51)

r

(51)

e.

noted

This method does differ from the fundamental mode contamination adjustment approach u

DIF3D [10].

The dependence on the scale factor of the matrix operators indicated in Eq. (51) orig

because of thermal-hydraulic (T-H) feedback, implying the solution of Eq. (51) for c(q) involves a

non-linear root search. Difficulty in this search originates because the Eq. (51) RHS h

singularity, which is addressed as follows. It is known that when the reactor is close to critica

flux can be approximated by the AEVP flux. This implies that the Eq. (51) RHS can

approximated as

(52)

where is the eigenvalue (i.e. ). Since the second bracketed term varies much slow

than the first bracketed term as c(q) varies for a near critical system, the second term is tre

constant. We next assume that varies linearly with c(q).

(53)

The values of and are obtained by explicitly evaluating the Eq. (

RHS for the two scale factor values and using the resulting values in Eqn. (52) to solve fo

values. Substituting Eq. (53) into Eq. (52), and using this equation as the RHS of Eq.

produces a quadratic equation in terms of c(q), with one root denoted being the desired valu

For a steady-state problem the following steps are completed to implement the just

procedure:

Step 1: Calculate 0th outer iterative operator estimates and , based upon

flux used in T-H feedback calculations and accounting for external parameters (e.g.control

rod position).

λ0 c q( )( )1 λ0 c q( )( )–----------------------------

φq( )

Sext,⟨ ⟩

φq( )

F cq( )( )φ

q( ),⟨ ⟩

---------------------------------------------≅

λ0 λ0 keff=

λ0 cq( )( )

λ0 cq( )( ) λ0 c1

q( )( )λ0 c2

q( )( ) λ0 c1q( )( )–

c2q( )

c1q( )

–( )--------------------------------------------- c

q( )c1

q( )–( )+=

λ0 c1q( )( ) λ0 c2

q( )( )

λ0

c3q( )

A0 F0 flux φ0( )

=

34

ting

and

.

by

r the

actors

ately

nd the

lues of

and

ction

Step 2: Solve the FSP iteratively for a fixed number of outer iterations .

Step 3:Set and calculate the following: operator estimates and by repea

Step 1 using flux = (Step 2) flux, Eqn. (51) RHS, and .

Step 4:Set = (Step 3 Eq. (51) RHS) and calculate the following: operator estimates

by repeating Step 1 using flux, Eq. (51) RHS, and

Step 5: Solve quadratic equation for and calculate operator estimates and

repeating Step 1 using flux.

This basic process is repeated every so many outer iterations as specified by user input.

The just noted method scales energy groups equally, thus it does not account fo

energy spectrum shift that occurs as a result of T-H feedback. This is important in water re

due to the dependance of moderating power on water density. This effect can be approxim

accounted for as follows: Assume that leakage can be approximated by a treatment a

Prompt Jump approximation can be used to estimate the flux energy spectrum shift. The va

are spatially dependent and obtained from the current estimate of the flux distribution

prior to the scale factor impact on cross-sections via T-H feedback (i.e. after Step 2). Specifically

for a two-group problem, suppressing spatial dependance notation, we obtain

(54)

Now an improved estimate for the flux ratio can be obtained as follows, where cross-se

values now reflect the scale factor via T-H feedback (i.e.during Steps 3-5).

(55)

q( )

c1q( )

1= A1 F1

λ0 c1q( )( )

c2q( )

A2

F2 flux c2q( )

Step 2×= λ0 c2q( )( )

c3q( )

A3 F3

flux c3q( )

Step 2×=

DgBg2

Bg2

B22 1

D2------

Σr1

φ1q( )

φ2q( )---------

Σa2–=

φ1

φ2-----

i

D2 ciq( )( )B2

2 Σa2 ciq( )( )+

Σr1 ciq( )( )

--------------------------------------------------------=

35

r the

thod

ation

This

when

.

his is

r than

f the

We are now free to set either to the Step 2 flux value, and using Eq. (55) solve fo

other group flux. NESTLE selects in its implementation and solves for . The above me

can be generalized to a multi-group formulation and is done so in the NESTLE implement

for the case G=4.

The scaling process can be very effective in obtaining the correct flux magnitude.

avoids a serious problem associated with FSP type problems, which is particularly troubling

the initial guess of the flux is higher than the converged value (i.e. approaches from above)

However, when the reactor is very close to critical, the scaling process may break down. T

because with high neutron multiplication, and are nearly equal and are much large

which implies that to get an accurate estimate of a very accurate estimate o

shape of is required.

φ1 or φ2

φ1 φ2

Aφ Fφ

Sext A F–( )φ

φ

36

not

ll align

15].

hin

delity

nd

le, to

under

caling

ous to

ed. The

II.4. Nodal Model - Hexagonal Geometry

II.4.a. Eigenvalue Problem

Utilization of NEM for Hexagonal (Hex) geometry introduces several complications

encountered for Cartesian geometry, originating because the surfaces of the Hex do not a

with the Cartesian axis. This can be seen in Figure 2.

R.D. Lawrence addressed these difficulties in implementing the Hex NEM option in DIF3D [

NESTLE through version 5.0.2 utilized this earlier work, adapting it for implementation wit

the context of the non-linear iterative method. However, this treatment has questionable fi

due to the coarseness of hexagonal nodes,e.g. cross section of a fuel assembly. Chao a

Tsoulfanidis [16] applied a conformal mapping, which transforms a hexagon into a rectang

the diffusion equation before the transverse integration. The Laplacian operator is invariant

the conformal mapping. Therefore, the diffusion equation remains unchanged except for s

factors which multiply the diffusion operator. The resulting transverse equations are analog

the ones for rectangular nodes when intranodal cross section spatial dependence is treat

1

3-------h

2

3-------h

v u

x

y

h

Figure 2: Hex geometry dimensions and axis orientation.

37

ping

the

ode is

conformal mapping approach has been verified by Chao and Shattila [17] and Knight,et.al. [18].

Starting with Version 5.0.3, Version 5 series of NESTLE have adapted the conformal map

approach for implementation within the non-linear iterative method. The derivation of

governing equations in the Hexagonal-Z geometry is presented in this section.

Consider the following three-dimensional, multigroup diffusion equation:

. (56)

Assume that the hexagonal node is in the complex plane and the rectangular n

in the complex plane as shown in Figure 3.

The transformed diffusion equation for nodel and energy groupg has the following form:

(57)

Dgx

2

2

∂

∂

y2

2

∂

∂

z2

2

∂

∂+ +

– ΣRg

χg

k------νΣ f g

–+

φg x y z, ,( )

Σsg' g→

χg

k------νΣ f g'

+ φg' x y z, ,( )

g' g≠

G

∑=

W u iv+=

Z x iy+=

F A(0,0) B(a/2,0)

F

A(0,-R)

B

E

D (0,R)

CE(-a/2,b) D C(a/2,b)

v

u

Figure 3: Conformal mapping of a hexagon to a rectangle.

Dgl

u2

2

∂

∂

v2

2

∂

∂

z2

2

∂

∂+ +

– ΣRg

l χg

k------νΣ f g

l–

g2

u v,( )+ φgl

u v z, ,( )

Σsg' g→

l χg

k------νΣ f g'

l+

g2

u v,( )φg'l

u v z, ,( )g' g≠

G

∑=

38

must

size

up

effect,

fusion

:

where is a mapping function associated with the mapping from (x,y) to (u,v) coordinates.

To preserve the total area under the mapping, the side length of the hexagon ( )

be related to the base (a) and height (b) of the rectangle by

and . (58)

Selecting thev-coordinate to span from 0 tob, the u-coordinate to span from-a/2 to a/2 (see

Figure 3), and thez-coordinate (note that the lower casez denotes the normal axial direction) to

span from -c/2 to c/2, we apply the transverse integration on Eq. (57) over thev- andz-coordinates

on nodel and define:

(59)

. (60)

to obtain the following transverse integrated equation:

. (61)

Since the radial node size,i.e. one hexagonal assembly, is larger than the radial node

normally used for western LWRs,i.e. one fourth of a fuel assembly, the treatment of the burn

gradient effect within a hexagonal assembly is necessary. To capture the burnup gradient

spatially dependent cross sections are introduced. The burnup gradient effect on the dif

coefficients is assumed to be negligible. The spatially dependent cross section is written as

g2

u v,( )

R h 3⁄=

a 1.82308R≅ b 1.42510R≅

φgul

u( ) 1bc------ φg

lu v z, ,( ) vd zd

c– 2⁄

c 2⁄

∫0

b

∫=

g2

u( )

1bc------ g

2u v,( )φg

lu v z, ,( ) vd zd

c– 2⁄

c 2⁄

∫0

b

∫

φgul

u( )----------------------------------------------------------------------------- 1

b--- g

2u v,( ) vd

0

b

∫≅=

Dgl

u2

2

∂

∂– ΣRg

l χg

k------νΣ f g

l–

g2

u( )+ φgul

u( )

Σsg' g→

l χg

k------νΣ f g'

l+

g2

u( )φg'ul

u( )g' g≠

G

∑– L– gul

u( )=

39

. (62)

btain

of Eq.

, (62)

where is the node flux-volume weighted average cross section. By substituting Eq

into Eq. (61) and collecting the spatially varying part on the right hand side of Eq. (61), we o

the following equation:

, (63)

where

. (64)

Details on how to calculate and are presented in Section II.4.b.

The one-dimensional transverse integrated flux that appears on the left hand side

(63) is approximated using the following expansion:

, (65)

where

. (66)

denote polynomials of ordern which are orthonormal on the interval with

weight , implying that

Σxguu( ) Σxg

⟨ ⟩ δΣxguu( )+=

Σxg⟨ ⟩

Dgl

u2

2

∂

∂– ΣRg

l⟨ ⟩

χg

k------ νΣ f g

l⟨ ⟩–

g2

u( )+ φgul

u( )

Σsg' g→

l⟨ ⟩

χg

k------ νΣ f g'

l⟨ ⟩+

g2

u( )φg'ul

u( )g' g≠

G

∑– Qgul

u( ) Lgul

u( )–=

Qgul

u( ) δΣRgu

lu( )

χg

k------δνΣ f gu

lu( )–

g2

u( )φgul

u( )=

δΣsg' gu→

lu( )

χg

k------δνΣ f g'u

lu( )+

g2

u( )φg'ul

u( )g' g≠

G

∑–

Σxg⟨ ⟩ δΣxgu

u( )

φgl

u( ) Agl

kglu( )cosh Bg

lkg

lu( )sinh ang

lWn u( )

n 0=

2

∑+ +=

kgl

g2

a 2⁄( ) ΣRg

l⟨ ⟩

χg

k------ νΣ f g

l⟨ ⟩– Dg

l⁄=

Wn u( ) a 2⁄– a 2⁄,[ ]

g2

u( )

40

l

es of

ients

n the

. (67)

The polynomials that satisfy these conditions are determined to be as follows:

, (68)

, (69)

and , (70)

where

. (71)

The mapping scale function, , is symmetric inu and is approximated by a polynomia

function:

. (72)

The polynomial coefficients are selected such that as required. The valu

taken from Ref. [16] and given in Table 2 are used to determine the expansion coeffic

of Eq. (72).

The leakage terms are directly expressible in terms of the surface currents i

1a--- g

2u( )Wm u( )Wn u( ) ud

a 2⁄–

a 2⁄

∫ δmn=

W0 u( ) κ0=

W1 u( ) κ1u=

W2 u( ) κ1

u2 κ0 κ1⁄( )2

–

κ1 κ2⁄( )2 κ0 κ1⁄( )2–

---------------------------------------------------------=

κn1

2a--- u

2ng

2u( ) ud

0

a 2⁄

∫

---------------------------------------------=

g2

u( )

g2

u( ) cnu

a 2⁄----------

n

n 0=

6

∑=

g2

u( ) ud0

a 2⁄∫ a

2---=

g2

u( )

cn

Table 2: Values ofg2(u) as a function ofu.

u/a/2 0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

g2(u) 2.279 1.594 1.353 1.173 1.025 0.897 0.786 0.690 0.609 0.544 0.507

Lug

lu( )

41

e

e two

or each

gon is

transverse directions, which in terms of thez- andv-directed leakage is given by:

. (73)

The axial leakage component, , is assumed to be given by a simple parabola in :

, (74)

where thez-direction node-averaged leakage in nodel can be expressed in terms of surfac

average currents by

, (75)

and the expansion coefficients of Eq.(74) can be solved for in terms of and

so as to preserve the node-averaged leakages in nodel-1, l, andl+1 :

and . (76)

Thev-directed transverse leakage component is given by:

, (77)

where and are the profiles of the physical currents on the two top surfaces and th

bottom surfaces of the hexagon, respectively. The physical currents are known on average f

face of the hexagon. Assuming that the physical current on each surface of the hexa

constant, we can write

for and for . (78)

Lug

lu( ) Lug

z l,u( ) Lug

v l,u( )+=

Lug

zu

Lug

z l,u( ) Lug

z l,ρ1g

lf 1 u( ) ρ2g

lf 2 u( )+ +=

Lug

z l, 1∆z------ Jg z+,

z l,Jg z-,

z l,–( )=

ρig

l

Lug

z l 1–,Lug

z l 1+,

u

-3a/2 -a/2 a/2 3a/2

l-1 l l+1

ρ1g

lLug

z l 1+,Lug

z l 1–,–

4---------------------------------------------= ρ2g

lLug

z l 1+,Lug

z l 1–,+

12----------------------------------------------

Lug

z l,

6---------–=

Luv l,

u( ) 1b---g u 0,( ) JT u( ) JB u( )–[ ]=

JT JB

JT u( ) JTM= a 2⁄– u 0≤ ≤ JT u( ) JTP= 0 u a 2⁄≤ ≤

42

is

es of

iated

he

usly for

where the meaning of the surface-averaged current subscript notation is depicted below.

The surface current on the bottom surfaces of hexagon is defined in a similar manner.

The profile of the linear mapping scale function appearing in Eq. (77)

approximated using the following function:

. (79)

The values of taken from Ref. [16] and given in Table 3 are used to obtain the valu

and appearing in Eq. (72).

For the v-direction transverse integrated diffusion equation, the derivation and assoc

transverse leakage term treatment are nearly the same as done for theu-direction transverse

integrated diffusion equation. For thez-direction transverse integrated diffusion equation, t

derivation and associated transverse leakage term treatment are the same as done previo

Cartesian geometry.

JBPJBM

JTMJTP

g u 0,( )

g u 0,( ) 1 u–u

------------ 1 3⁄

γ1x γ0+( )≅

g u 0,( )

γo γ1

Table 3: Values ofg(u,0) as a function ofu.

u/a/2 0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

g(u,0) 8.078 1.739 1.372 1.183 1.053 0.947 0.851 0.756 0.650 0.510 0.000

43

the

e

two-

omial

II.4.b. Intranodal Burnup Gradient Treatment

The two-dimensional intranodal cross section distribution is approximated using

following polynomial function:

, (80)

where:

.

Clearly the expansion coefficients of the polynomial have been selected to make th

coefficients equal to surface and node average cross-section values. Similarly, the

dimensional flux distribution within an assembly can also be represented by the same polyn

expansion:

, (81)

so it follows that the coefficients take on the surface and node average flux values.

Σxgx y,( ) Sng

Pn x y,( )n 1=

7

∑=

S1gΣxg

west... west surface average cross section,=

S2gΣxg

east... east surface average cross section,=

S3gΣxg

northwest... northwest surface average cross section,=

S4gΣxg

southeast... southeast surface average cross section,=

S5gΣxg

northeast... northeast surface average cross section,=

S6gΣxg

southwest... southwest surface average cross section,=

S7gΣxg

⟨ ⟩ ... node average cross section,=

Pn x y,( ) cmnx( )

xm

cmny( )

ym

+( )m 0=

2

∑ c3nxy c4nxy2

+ +=

Pn x y,( ) Sng

φg x y,( ) FngPn x y,( )

n 1=

7

∑=

Fng

44

ross

two-

n

t

flux-

an

Utilizing these polynomial expansions, the node flux-volume average value of the c

section, which is defined to preserve reaction rates, defined as

(82)

can be evaluated. Next, the following steps are performed in order to transform the

dimensional distribution in theZ-plane domain ( ) into a one-dimensional distributio

in one of the coordinates of theW-plane ( ). At the west ( ) and eas

( ) surfaces of the conformally-mapped rectangle, the transversely-integrated,

volume weighted cross section can be calculated as:

, (83)

where denotes a function which transforms a point at in theW-plane toy-

coordinate at in the Z-plane. The relationship between they- andv-coordinates is

shown in Figure 4. Note that the function is approximated by a polynomial function:

. (84)

Utilizing , , and values obtained from Eqs. (82) and (83), we c

approximate the profile of in Eq. (62) with a second-order polynomial function:

. (85)

Σxg⟨ ⟩

xdR 3–2

--------------

0

∫ ydx–

3------- R–

x

3------- R+

∫ Σxgx y,( )φg x y,( ) xd

0

R 32

-----------

∫ ydx

3------- R–

x–

3------- R+

∫ Σxgx y,( )φg x y,( )+

3 32

----------R2

φg⟨ ⟩

------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

Z x iy+=

W u iv+= u a 2⁄–=

u a 2⁄=

Σxg

a2---±

Σxg

R 32

-----------± y, φg

R 32

-----------± y, g

a2---± f v y( ),

ydR 2⁄–

R 2⁄∫

bφga2---±

g2 a

2---±

--------------------------------------------------------------------------------------------------------------------------=

f v y( ) a 2⁄± v,( )

x R 3± 2⁄=

f v y( )

f v y( ) R cnyR---

n

n 0=

3

∑=

Σxga 2⁄–( ) Σxg

a 2⁄( ) Σxg⟨ ⟩

Σxgu( )

Σxgu( ) Σxg

⟨ ⟩ 12--- Σxg

a2---

Σxg

a2---–

– u

a 2⁄----------

+=

12--- Σxg

a2---

Σxg

a2---–

+ Σxg

⟨ ⟩–12---

3u

a 2⁄----------

21–

+

45

e

on:

into

).

Similarly, the one-dimensional flux distribution in theu-direction appearing in Eq. (64) can b

written as:

, (86)

where

. (87)

The mapping scale function at can also be approximated by a polynomial functi

. (88)

The fitted versus exact values of are shown in Figure 5. By substituting Eq. (84)

Eq. (88), we can determine the function which appears in Eqs. (83) and (87

φg u( ) φg⟨ ⟩ 12--- φg

a2---

φga2---–

– u

a 2⁄----------

+=

12--- φg

a2---

φga2---–

+ φg⟨ ⟩–

12---

3u

a 2⁄----------

21–

+

φga2---±

φg

R 32

-----------± y, g

a2---± f v y( ),

yd

R 2⁄–

R 2⁄

∫

bg2 a

2---±

-----------------------------------------------------------------------------------=

u a 2⁄±=

ga2---± v,

cnvb---

n

n 0=

6

∑=

g a 2⁄± v,( )

g a 2⁄± f v y( ),( )

46

Figure 4: v- vs.y-coordinates atx=R(30.5/2).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

v/R

y/R

data

fv(x,y)

Figure 5: Mapping scale function atu=a/2.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

g(a

/2,v

)

v/b

data

fitted __

47

anner

ain the

cted

at the

23) is

e

plies

urface

oduces

hich is

nd to

er

II.4.c. Non-Linear Iterative Strategy

For Hexagonal-Z geometry, the non-linear iterative strategy is applied in the same m

as for Cartesian geometry. For each surface of a node, a two-node problem is solved to obt

NEM predicted surface-averaged current based upon the FDM flux solution utilizing corre

coupling coefficients. The corrected coupling coefficients are determined demanding th

FDM and NEM predicted currents agree. In the radial plane for Hexagonal geometry, Eq. (

modified to read as follows:

. (89)

The conformal mapping is used when formulating the two-node problem, with thu-

direction in theW-plane always selected as the direction transversing the two nodes. This im

that three two-node problems are solved for each node to obtain the three radial plane s

average currents associated with half of the hexagonal surfaces. The two-node problem pr

a 10G x 10G linear system of equations, where G denotes the number of energy groups, w

constructed by applying NEM relations to two adjoining nodes. The 10G unknowns correspo

the five coefficients in Eq. (65) x two nodes x G neutron energy groups. For simplicity, consid

two arbitrary conformally-mapped nodes in theu-direction and denote these nodes as nodel and

nodel+1 .

In order to solve the 10G unknowns, the following constraints are applied:

• Surface flux discontinuity on the interior surface.

• Surface current continuity on the interior surface.

Dgx +l FDM, 2Dg

lDg

l 1+

Dgl

Dgl 1+

+---------------------------=

nodel nodel+1u- u+

48

• The matching of the zeroth moment projection of the two sides of Eq. (63).

• The matching of the first moment projection of the two sides of Eq. (63).

• The matching of the second moment projection of the two sides of Eq. (63).

• Volume-averaged flux determined by the FDM solution.

Radial Directions (Example -u-direction):

Flux Discontinuity

(90)

Current Continuity

(91)

dgx+l

Agl kg

la

2---------

cosh Bgl kg

la

2---------

sinh a0g

l κ0

a1g

l κ1a

2------------------- a2g

l κ1

a2

4------

κ0

κ1------

2

–

κ1κ2------

2 κ0

κ1------

2

–

-----------------------------------------+ + + + =

dgx-l 1+

Agl 1+ kg

l 1+a

2-----------------

Bgl 1+ kg

l 1+a

2-----------------–

sinh a0g

l 1+ κ0 a1g

l 1+ κ1a2---– a2g

l 1+ κ1

a2

4------

κ0

κ1------

2

–

κ1κ2------

2 κ0

κ1------

2

–

-----------------------------------------+ +–cosh

Dgl

Agl

kgl kg

la

2---------

sinh Bgl

kgl kg

la

2---------

cosh a1g

l κ1

a2g

laκ1

κ1κ2------

2 κ0

κ1------

2

–

-----------------------------------------+ + +– =

Dgl 1+

A– gl 1+

kgl 1+ kg

l 1+a

2-----------------

sinh Bgl 1+

kgl 1+ kg

l 1+a

2-----------------

cosh a1gl 1+ κ1

a2g

laκ1

κ1κ2------

2 κ0

κ1------

2

–

-----------------------------------------–+ +

49

Zeroth Moment Balance

, (92)

where

, (93)

, (94)

, (95)

and . (96)

First Moment Balance

, (97)

where

, (98)

, (99)

, (100)

α0g

lAg

l ξ0la2g

l+ =

Q0g

lL0g

l– ΣRg

l⟨ ⟩

χg

k------ νΣ f g

l⟨ ⟩– φg

l⟨ ⟩– Σsg' g→

l⟨ ⟩

χg

k------ νΣ f g'

l⟨ ⟩+

φg'l⟨ ⟩

g' g≠

G

∑+

α0g

l2

κ0Dglkg

l

a------------------- kg

l a2---

sinh–=

ξ0g

l 2Dgl κ0κ1–

κ1 κ2⁄( )2 κ0 κ1⁄( )2–

---------------------------------------------------------=

Q0g

l 1a--- W0 u( )g2

u( )Qugu( ) ud

a 2⁄–

a 2⁄

∫=

L0g

lκ0 Lug

z l, 975817911---------------

JTP JTM JBP– JBM–+

b-----------------------------------------------------------+=

β1g

lBg

l β1gg'

lB

g'

l

g' g≠

G

∑ ζ1g

la1g

l ζ1gg'

la1g'

l

g' g≠

G

∑+ + + Q1g

lL1g

l–=

β1g

l κ1

2a------ 4Dg

lkg

l a2---

sinh 2Dglkg

la kg

l a2---

cosh– ΣR0g

l χg

k------νΣ f 0g

l–

snh1g

l+=

β1gg'

l Σsg' g→

l⟨ ⟩

χg

k------ νΣ f g'

l⟨ ⟩+

– snh1g'

l=

ζ1g

l ΣRg

l⟨ ⟩

χg

k------ νΣ f g

l⟨ ⟩–=

50

, (101)

, (102)

, (103)

and . (104)

Second Moment Balance

, (105)

where

, (106)

, (107)

, (108)

ζ1gg'

l Σsg' g→

l⟨ ⟩

χg

k------ νΣ f g'

l⟨ ⟩+

–=

Q1g

l 1a--- W1 u( )g2

u( )Qugu( ) ud

a 2⁄–

a 2⁄

∫=

L1g

lκ1a

ρ1g

l

6--------

67730697803------------------

JTP JTM– JBP– JBM+

b-----------------------------------------------------------+=

snh1g

l 1a--- W1 u( )g2

u( ) kglu( )sinh ud

a 2⁄–

a 2⁄

∫=

α2g

lAg

l α2gg'

lA

g'

l

g' g≠

G

∑ γ2g

la0g

l ξ0la2g

l ξ2gg'

la2g'

l

g' g≠

G

∑+ + + + Q2g

lL2g

l–=

α2g

l 1

4aκ1

κ1

κ2------

2 κ0

κ1------

2

–

---------------------------------------------------- 16Dgl κ1

2

kgl

------ kgl a2---

sinh– 8κ02Dg

lkg

l kgla

2--------

sinh+

=

aκ12

8Dgl

kgl a2---

cosh 2aDglkg

l kgla

2--------

sinh–+

ΣRg

l⟨ ⟩

χg

k------ νΣ f g

l⟨ ⟩–

csh2g

l+

α2gg'

l Σsg' g→

l⟨ ⟩

χg

k------ νΣ f g'

l⟨ ⟩+

csh2g'

l

g' g≠

G

∑–=

γ2g

l 1

4aκ1

κ1

κ2------

2 κ0

κ1------

2

–

---------------------------------------------------- 8Dgl κ1

2κ0a– Dgl 8aκ0

2κ1

κ1

κ2------

2 κ0

κ1------

2

–

--------------------------------------- aκ128Dg

l κ0+ +

=

51

, (109)

, (110)

, (111)

, (112)

and . (113)

Volume-averaged flux

(114)

where

(115)

ξ2g

l 1

4aκ1 κ1 κ2⁄( )2 κ0 κ1⁄( )2–

----------------------------------------------------------------------

8Dgl κ1

3a

a2

12------

κ0

κ1------

2

–

– Dgl

8aκ02κ1( )+

κ1 κ2⁄( )2 κ0 κ1⁄( )2–

-----------------------------------------------------------------------------------------------

=

aκ12 8Dg

l κ1 a 2⁄( )2 κ0 κ1⁄( )2–[ ] 2Dg

la

2κ1–

κ1 κ2⁄( )2 κ0 κ1⁄( )2–

--------------------------------------------------------------------------------------------------+

ΣRg

l⟨ ⟩

χg

k------ νΣ f g

l⟨ ⟩–

+

ξ2gg'

l Σsg' g→

l⟨ ⟩

χg

k------ νΣ f g'

l⟨ ⟩+

g' g≠

G

∑–=

Q2g

l 1a--- W2 u( )g2

u( )Qugu( ) ud

a 2⁄–

a 2⁄

∫=

L2g

l κ1a2

κ1 κ2⁄( )2 κ0 κ1⁄( )2–

---------------------------------------------------------Lug

z l,

12---------

ρ2g

l

30--------

4169149796------------------


b-----------------------------------------------------------+ +

=

κ1 κ0 κ1⁄( )2

κ1 κ2⁄( )2 κ0 κ1⁄( )2–

--------------------------------------------------------- Lug

z l, 975817911---------------


b-----------------------------------------------------------+

–

csh2g

l 1a--- W2 u( )g2

u( ) kglu( )cosh ud

a 2⁄–

a 2⁄

∫=

α3g

lAg

la0g

l+ φg⟨ ⟩ l FDM,

=

α3g

l 1a--- kg

lu( )cosh g

2u( ) ud

a 2⁄–a 2⁄∫=

52

wing

y. For

atrix

atrix

. For

ming

atrix

olved

lems,

22).

Axial Direction:

Flux Discontinuity

(116)

Current Continuity

(117)

Nodal Balance

(118)

First Moment

(119)

Second Moment

(120)

For the z-direction, the same matrix structure as for Cartesian geometry results allo

rearrangement of the associated two-node problem coefficient matrix to achieve reducibilit

the u-direction, the two-node problem for two groups can be reduced from a 20 X 20 m

problem to a 16 X 16 matrix problem by performing a Gaussian elimination. The 16 X 16 m

system can be further reduced to two 4 X 4 matrix problems and one 8 X 8 matrix problem

four groups the 40 X 40 matrix problem can be reduced to 32 X 32 matrix system by perfor

Gaussian elimination. The 32 X 32 matrix system can be further reduced to two 8 X 8 m

problems and one 16 X 16 matrix problem. The associated matrix problems are s

analytically to reduce floating point operations required. Having solved the two-node prob

the corrections to the coupling coefficients can be obtained as previously indicated in Eq. (

dgz+l

agz1l

agz2l

+[ ] dgz–l 1+

agz1l 1+

agz2l 1+

–[ ]+ 2 dgz–l 1+ φg

l 1+dgz+

l φgl

–[ ]=

Dgl

zl∆

------- agz1l

3agz2l 1

2---agz3

l 15---agz4

l+ + +

Dgl 1+

zl 1+∆

-------------- agz1l 1+

3agz2l 1+

–12---agz3

l 1+ 15---agz4

l 1+–+– 0=

Dgl

zl∆

-------

–1

zl∆

------- 6agz2

l 25---

agz4l

+2

3h------

Lgxl

Lgul

Lgvl

+ +( )– Agl φg

l– Qgg′

l φg′l

g′ g≠

G

∑+=

10 Aglagz1

lQgg′

lag′z1

l

g′ g≠

G

∑+– 60

zl∆

------- Dg

l

zl∆

-------

Agl

+ agz3l

Qgg′l

ag′z3l

g′ g≠

G

∑–+ 102

3h------

ρgxy1l

=

35 Aglagz2

lQgg′

lag′z2

l

g′ g≠

G

∑+– 140

zl∆

--------- Dg

l

zl∆

-------

Agl

+ agz4l

Qgg′l

ag′z4l

g′ g≠

G

∑–+ 352

3h------

ρgxy2l

=

53

limits

ction

ction

tion

arying

ithin

ttice

flux

the

erage

ik has

orage

where

n the

as for

II.4.d. Pin-power Reconstruction Method

Pin-power reconstruction is necessary when utilizing nodal methods so that thermal

can be verified. For NESTLE Hexagonal geometry, we have utilized a pin-power reconstru

method similar to the method developed by Chao and Shatilla [17]. The pin-power reconstru

method implemented in NESTLE is described below.

A common assumption of pin-power reconstruction methods is that the flux distribu

within a node can be represented as the product of a smooth intra-nodal flux and a highly v

flux based upon the lattice physics solution. The intra-nodal flux represents the smooth w

node spatial flux variation of the homogenized problem. It corrects for the fact that the la

physics calculations are completed with reflective boundary conditions. The highly varying

obtained from the lattice physics solution accounts for the pin cell heterogeneities within

lattice, under the condition of reflective boundary conditions. When normalized to a node av

flux of one, the highly varying flux function is referred to as a flux form factor.

We have also adopted the Studsvik approach with regards to the form factors. Studsv

elected to utilize a power versus flux form factor. This results in some computation and st

reductions. We can begin to mathematically state their method as follows for nodel, where node

dependence have been suppressed for clarity: The power at cylindrical coordinates ,

r=0 denotes the hexagonal node center, is determined using

(121)

where the power form factor is defined by

. (122)

The power form factors, on a unit cell average basis, are input into NESTLE based upo

lattice physic code’s predictions. Control rod insertion dependence is treated the same

r θ,( )

p r θ,( ) pIntra

r θ,( ) FF r θ,( )×=

FF r θ,( ) pLattice

r θ,( )

p⟨ ⟩Lattice---------------------------------=

54

ing as

the

intra-

to a

oup-

er

alues

tions is

ion

cross-sections; that is, the unrodded and rodded power form factors are combined us

weights the axial fraction of the node unrodded and rodded, respectively.

The intra-nodal power is obtained from the intra-nodal group fluxes via

(123)

As used in Eq. (121), the value of is evaluated at a pin’s center and multiplied by

unit cell average basis power form factor for that pin to obtain the pin power. To evaluate the

nodal power using Eq. (123) one must evaluate the intra-nodal flux and cross-section.

The pin-power reconstruction option in the current version of NESTLE is restricted

two neutron energy group problem (NG=2) due to the complication in determining the gr

wise intra-nodal fluxes for NG > 2. For a NESTLE problem with NG > 2, the pin-pow

reconstruction is performed by group collapsing the flux and macroscopic cross section v

into two-group fluxes and cross sections as shown in Eqs. 124 through 126.

(124)

(125)

(126)

The pin-power reconstruction methodology based upon the two-group fluxes and cross sec

detailed below.

To obtain the intra-nodal flux, we start by writing the two-group neutron diffus

equation over a homogeneous hexagonal node in the following form:

, (127)

pIntra

r θ,( ) κgΣ fgIntra

r θ,( )g 1=

G

∑= φgIntra

r θ,( )

pIntra

r θ,( )

φGcollapsed φg

g G∈∑=

ΣxG

collapsed Σxgφg

g G∈∑ φg

g G∈∑⁄=

ΣsG G''→

collapsed Σxg g'→φg

g G∈∑

g' G'∈∑

φgg G∈∑⁄=

∇2φ Bφ– 0=

55

to

Eq.

to

where and . We can cast this matrix equation in

diagonal form by first defining

, (128)

where are the eigenvectors of . Now, let which allows us to rewrite

(127) as:

. (129)

Substitute and into Eq. (129) and denote and

obtain:

. (130)

The solution to Eq. (130) in cylindrical coordinates can be shown to be given by

(131)

and

B

ΣR1νΣ f 1

–( )

D1--------------------------------

ν– Σ f 2

D1---------------

Σs12

D2-----------–

Σa2

D2--------

= φφ1

φ2

=

P ψ1 ψ2≡

ψ11 ψ12

ψ21 ψ22

=

ψi Bψi λiψi= φ PΦ=

∇2Φ

λ1 0

0 λ2

Φ– 0=

µ2 λ1–= ν2 λ2= Φ1 Φµ= Φ1 Φν=

∇2 Φµ

Φν

µ20

0 ν–2

Φµ

Φν+ 0=

Φµ r θ,( ) Aµ α( ) µr θ α–( )cos( )sin Bµ α( ) µr θ α–( )cos( )cos+[ ] αd

0

2π

∫=

Cµ α( ) µr θ α–( )sin( )cos Dµ α( ) µr θ α–( )sin( )sin+[ ] αd

0

2π

∫+

56

We

need

e 12

d

of the

nomial

e

t flux

each

t three

orner

d are

by a

e the

d the

s shown

, (132)

where , , , and are direction-dependent superposition coefficients.

numerically perform the integration operations shown in Eqs. (131) and (132), implying we

to determine the values of , , , and at discrete values of . There ar

known values that can be utilized as constraints,i.e. six homogeneous corner point flux values an

six homogeneous surface average flux values. For simplicity, we assumed that the profile

surface average flux on a hexagonal surface can be approximated by a quadratic poly

where . Two corner point flux values and th

surface average flux value are imposed as constraints to determinec0, c1, andc2 coefficients. We

can then determine the flux value at the center of the hexagonal surface ( ). The poin

values at the center of the six hexagonal along with the six corner point flux values for

energy group can be imposed as constraints to evaluate the superposition coefficients a

values of .

The surface average fluxes are readily available from the nodal solution while the c

point flux values are not. The corner point flux values for the pin-power reconstruction metho

obtained in the following manner. First the flux in the vicinity of a corner point is represented

polynomial function in (x,y), with the associated expansion coefficients fitted so as to preserv

node volume-averaged fluxes of the three nodes surrounding the corner point an

heterogeneous surface average fluxes on the three interfaces meeting at the corner point, a

in Figure 6.

Φν r θ,( ) Aν α( ) νr θ α–( )cos( )sinh Bν α( ) νr θ α–( )cos( )cosh+[ ] αd

0

2π

∫=

Cν α( ) νr θ α–( )sin( )cosh Dν α( ) νr θ α–( )sin( )sinh+[ ] αd

0

2π

∫+

A α( ) B α( ) C α( ) D α( )

A α( ) B α( ) C α( ) D α( ) α

φgsurface

s( ) c0 c1s c2s2

+ += s R– 2⁄ R 2⁄,[ ]∈

s 0=

α

57

ven

r

e

int flux

ysics

cross

de and

ective

n as a

The polynomial function used to approximate the flux in the vicinity of the corner point is gi

by

. (133)

Selecting a coordinate system such that (x,y)= corresponds to the location of the corne

point of interest, the corner point flux is simply given by . Th

resulting heterogeneous flux is continuous at a corner point. The homogeneous corner po

is then defined as:

(134)

The corner point ADF values that appear on the RHS are obtained from the lattice ph

calculation. This process is repeated for the other five corner points of nodel, providing all the

constraints required to solve for the expansion coefficients of the intra-nodal flux.

To evaluate the intra-nodal power distribution, the cross-section spatial variation a

the node must also be determined. This variation is associated with the homogenous no

corrects for the burnup distribution error made in the lattice physics code because refl

boundary conditions are assumed. This is accomplished by expanding the cross-sectio

Figure 6: Corner Flux Configuration.

node 1

node 2 node 3

s1 s2

s3

y

x

φg x y,( ) c0gc1g

x c2gy c3g

x2

c4gy

2c5g

xy+ + + + +=

0 0,( )

φgcorner( )Het φg 0 0,( ) c0g

= =

φgcorner( )Hom

φgcorner( )Het

ADFgcorner

----------------------------=

58

is

the

f the

seven

cross-

model

he node

s. The

uate

flux is

at even

s, the

s since

In all

surface

ers,

lue of

35)

tain

s are

polynomial inr andθ according to

, (135)

where and . The base of the polar coordinates

from the center of the hexagon to the center of the east surface.

Eq. (135) has eight unknown coefficients. Note that when solving for , one of

coefficients can be chosen arbitrarily because of a lack of linear independence o

expansion defined by Eq. (135). This implies that only seven constraints are required. The

constraints are obtained by demanding that the node and six surface average values of the

section be preserved. The cross-section values are obtained for the macroscopic depletion

based upon the node and surface average burnups. For the microscopic depletion model, t

and surface average burnups and isotopics are used to obtain the cross-section value

implication is that with pin-power reconstruction, the depletion calculation must now eval

surface average quantities based upon the homogenous flux. Since the homogenous

discontinuous across surfaces, the burnup and isotopics will also be discontinuous. Note th

though the user may elect to only do pin-power reconstruction on the highest power node

depletion calculation for surface average quantities must be completed on all node surface

the location of the highest power nodes under various core conditions is not known a priori.

cases, the node average thermal-hydraulic conditions are used to correct the node and

average cross-sections.

The last calculation that remains to be completed is the determination of pin-pow

which proceeds as follows. Eqs. (131) and (132) are evaluated at the fuel pin center va

for each fuel pin within the node to obtain the axially dependent fuel pin fluxes. Eq. (1

is also evaluated at the fuel pin center value of for each fuel pin within the node to ob

the axially dependent fuel pin cross-sections. When the group dependent flux value

κΣ f g

lr θ,( ) C0g

lC1ng

l γnn 1=

3

∑ C2ng

l γn2

n 1=

3

∑ C3gl γ1

3– γ2

3 γ33

–+( )+ + +=

γn r θ αn–( )cos= αn n 1–( ) π 3⁄( )= r θ,( )

C1n

C1n

r θ,( )

r θ,( )

59

xially

l pin

alize

at the

m

er

ting

in the

dial

multiplied by the cross-section values and summed over energy groups, the intra-nodal a

dependent fuel pin power is obtained. This quantity is multiplied by the axially dependent fue

power form factor to obtain the unnormalized axially dependent fuel pin power. To norm

these results, it is demanded that the axially dependent fuel pin relative powers be such th

associated node power is preserved, implying

(136)

In this expression,l denotes the node dependence andn denotes the fuel pin dependence. Fro

the values, the total peaking factor FQ value is determined assuming a flat axial pow

distribution within the node and determining the maximum value over alln. Having the axially

dependent fuel pin power distribution, the total fuel pin power is obtained by axially integra

those values associated with a specific pin and dividing by the active fuel pin length to obta

pin-wise radial power distribution. This information allows the determination of the ra

peaking factor , which completes the pin-power reconstruction calculations.

pnl

rn θn,( )∆vnl

n 1=

total number of fuel pins in a node

∑ plV

l=

pnl

rn θn,( )

F∆H

60

tron

source

up g

) and

. The

f the

utrons,

poral

II.5. Transient Problem

Under transient conditions, both the multi-group diffusion equation and delayed neu

precursor equations must be solved. These equations, accounting for an external neutron

and utilizing six precursor groups, are given by (suppressing andt dependences for clarity):

(137)

and

(138)

where the notation is identical as before except as now noted.

= neutron speed for energy group g

= fraction of prompt neutrons born into energy group g

= fraction of delayed neutrons for precursor group i born into energy gro

= neutron precursor concentration in precursor group i

= decay constant for precursor group i

= fraction of all fission neutrons emitted per fission in precursor group i

= total fraction of fission neutrons which are delayed

Alternately, the ‘eigenvalue initiated’ transient equations can be obtained from Eqs. (137

(138) by setting and by replacing with everywhere.

The neutron kinetics equations, Eqs. (137) and (138), involve differentials in space and time

time dependence is a difficult problem to treat in neutronics modeling due to the stiffness o

associated equations. The time constants range from very small, associated with prompt ne

to very long, associated with the longer lived precursors. NESTLE numerically treats the tem

r

1vg-----

t∂∂φg Σsgg′

φg′ 1 β–( )χgP( ) νg′Σ fg′φg′ χgi

D( )λiCi ∇ Dg⋅ ∇φg Σtgφg– Sextg++

i 1=

ID( )

∑+g′ 1=

G

∑+g′ 1=

G

∑=

t∂∂Ci βi νgΣ fgφg λiCi–

g 1=

G

∑= for i 1,...,ID( )

=

vg

χgP( )

χgiD( )

Ci

λi

βi

β

Sextg0= νgΣ fg νgΣ fg( ) k⁄

61

ology

crete

nce.

nd will

f the

[tn,

must

rator

early

dependence in a manner that results in a FSP, which can be solved utilizing the method

developed for the steady-state FSP.

The first step in this conversion to a FSP is to discretize the time domain into dis

times tnand to approximate the time derivative of the flux at time tn+l by a backward differe

(139)

This assures unconditional stability. Do note that spatial dependance notation has been a

continue to be suppressed in the equations.

To develop an expression for the precursors’ concentrations at time tn+1 in terms o

flux , Eq. (138) is solved utilizing the Integrating Factor method over time span

tn+1] to obtain

. (140)

To solve the above integral, a functional form for the time dependent neutron fission source

be assumed over time span [tn, tn+1]. Consistent with the backward difference ope

approximation for the time derivative of the flux, the fission source is assumed to vary lin

between time-steps,

(141)

Incorporating this approximation into Eq. (140) and rearranging terms, we obtain

, (142)

where

t∂∂φg

tn 1+

φg tn 1+( ) φg tn( )–

tn∆-------------------------------------------=

φg tn 1+( )

Ci tn 1+( ) Ci tn( )eλi tn∆–

βi υgΣ fgφg t′( )eλi tn 1+ t′–( )–

dt′g 1=

G

∑tn

tn 1+

∫+=

vgΣ fgφg t( ) vgΣ fgφg tn( )vgΣ fgφg tn 1+( ) vgΣ fgφg tn( )–

tn∆----------------------------------------------------------------------- t tn–( )+=

Ci tn 1+( ) Ci tn( )eλi tn∆–

Fin

0 νgΣ fgφg tn( ) Fin

1 νgΣ fgφg tn 1+( )g 1=

G

∑+g 1=

G

∑+=

62

) as the

of the

ould be

outer

.

fluxes,

e

dent

ents.

o-

(143)

and

(144)

Now substituting Eqs. (139) and (142) into Eq. (137) one obtains

(145)

where

(146)

In Eq. (145) all cross-sections are evaluated at time tn+1. Inspection of Eq. (145) indicates it to be

a FSP, with modified operators and source from the steady-state FSP. We refer to Eq. (145

transient FSP. Hence the application of NEM to the transient FSP and the iterative solution

resulting coupled equations can proceed exactly the same as for the steady-state FSP. As w

expected, the values of flux and adjusted FDM coupling coefficients at time for the 0th

iterative step are based upon their values at time tn.

If we proceed in this manner, in the two-node problems spatial moments of

would appear. As Eq. (146) indicates, is dependent upon and

The implication is that the expansion coefficients associated with the transverse integrated

obtained from solution of the two-node problems, at the previous time tn must be saved. The sam

is true for the precursor concentrations, which are treated like the flux for time depen

problems solved by NEM. This would substantially increase the computer memory requirem

To overcome this difficulty, further approximations are required in formulating the tw

Fin

1 βi

λi tn∆------------- tn∆ 1

λi---- 1 e

λi tn∆––( )–=

Fin

0F– i n

1 βi

λi---- 1 e

λi tn∆––( )+=

1tnνg∆

--------------φg tn 1+( ) ∇ D⋅ g∇φg tn 1+( ) Σtgφg tn 1+( )+– Σsgg′φg′ tn 1+( )

g′ 1=

G

∑=

1 β–( )χgP( ) χgi

D( )λiFin

1

i 1=

ID( )

∑+

+ vg′Σ fg′φg′ tn 1+( ) Sef fgtn 1+( )+

g′ 1=

G

∑

Sef fgtn 1+( ) χgi

D( )λiFin

0vg′Σ fg′φg′ tn( ) χgi

D( )λiCi tn( )eλi tn∆–

i 1=

ID( )

∑+g′ 1=

G

∑i 1=

ID( )

∑ 1tnvg∆

-------------φg tn( ) Sextgtn 1+( )+ +=

tn 1+

Sef fgtn 1+( )

Sef fgtn 1+( ) φg tn( ) Ci tn( )

63

is

hat the

ions of

pact

TLE

that is,

e flux

ues are

uted.

fferent

nd the

is that

sient

tate

5) and

node problems. Recognizing that the within node spatial dependence of

associated with the contributions from the delay neutrons and the external neutron source, t

external neutron source is assumed to be constant within a node, and that these contribut

neutrons are small, one would expect within node spatial shape to have little im

on the solution. This justifies treating spatial dependence approximately. In NES

this approximate treatment is done in the same manner as for the transverse leakages;

using a quadratic polynomial as indicated in Eq. (12). Now only node average values of th

and precursor concentrations at time must be saved. The node average precursor val

solved for via back substitution using Eq. (142) after the node average flux has been comp

This implementation does create one problem that must be addressed. Since di

spatial treatments are used at times and , the solution of the steady-state FSP a

transient FSP, now for steady-state conditions, will not agree. The practical consequence

when one utilizes the steady-state FSP solution to determine initial conditions for the tran

FSP, the flux will undergo a very mild transient with time even when the initial steady-s

conditions are preserved. This annoyance can be avoided by regrouping terms in Eqns. (14

(146) as follows.

(147)

where

(148)

(149)

Sef fgtn 1+( )

Sef fgtn 1+( )

Sef fgtn 1+( )

tn

tn tn 1+

∇ Dg∇φg tn 1+( ) Σtgφg tn 1+( )+⋅– =

Σsgg′φg′ tn 1+( ) χg νg′Σ fg′φg′ tn 1+( ) Sef fgtn 1+( )+

g′ 1=

G

∑+g′ 1=

G

∑

χg 1 β–( )χgP( ) βiχgi

D( )

i 1=

ID( )

∑+=

Sef fgtn 1+( ) χgi

D( ) λiFin

1 βi–( )i 1=

I D( )

∑

νg′Σ fg′φg′ tn 1+( )g′ 1=

G

∑=

1tnvg∆

-------------φg tn 1+( )– Sef fgtn 1+( )+

64

f

e all

that

eated

verse

ce the

nted in

e

in the

g the

iterative

node

ents

sure

sient

eby

ence,

flux.

Eq. (147) is recognized to be identical to the steady-state FSP except for the replacement o

with the modified effective source . Under steady-state conditions, equals sinc

the additional terms in Eq. (149) cancel. This is true even for the moments of

appear in the two-node problems provided the within node spatial dependence is tr

consistently for the variables appearing in Eq. (149) at times tn and tn+1; in particular, when they

are all treated by a quadratic polynomial as previously indicated by Eq. (12) for the trans

leakages. The implication is that the transient FSP under steady-state conditions will produ

same solution as produced by the steady-state FSP. This approach has been impleme

NESTLE.

Since the values of node average flux at time tn+1, the unknowns, now appear in th

modified effective source, an iterative approach is required. This is easily accomplished with

context of the non-linear iterative method of solving the NEM equations. Recall when solvin

two-node problems, node average fluxes are assumed known based upon the latest outer

values available for the FDM equation’s solution. For transient problems, this corresponds to

averaged flux values at time tn+1, which are precisely the values requires to evaluate the mom

of that appear in the two-node problems. This approach is utilized within NESTLE to as

the steady-state FSP solution does not “drift” when used as an initial condition in the tran

FSP.

To improve upon the initial iterative estimate of the fission source and flux, ther

hopefully minimizing the number of outer and inner iterations required to achieve converg

the 0th outer iterative estimates of both are determined utilizing a time extrapolated

Specifically, assuming an exponential time dependence, the0th iterative flux at time stepn+1 is

given by

(150)

Sextg

Sef fgSef fg

Sextg

Sef fgtn 1+( )

Sef fg

φg tn 1+( ) φg tn( )eωgn

∆tn

= whereωgn

1∆tn 1–--------------- ln

φg tn( )φg tn 1–( )---------------------

=

65

lue

n the

n be

ns, the

the

their

ux is

flux.

metric

n. The

sulting

attering

s (

rmal

oint

ients

ng not

ed flux

ethod

. To

is

EM

II.6. Adjoint Problem

The adjoint solution to the few-group neutron diffusion equation for the eigenva

problem is of interest. This follows since the adjoint flux can be used to estimate the effect o

reactivity of perturbations via the Raleigh quotient, derived from perturbation theory; and, ca

used to estimate kinetics parameters utilizing point-reactor kinetics theory. For these reaso

ability to solve for the adjoint flux of the eigenvalue problem has been incorporated into

NESTLE code.

For the FDM solution, the development of the equations that need to be solved and

solution are straight forward. The matrix system that needs to be solved for the adjoint fl

obtained by transposing the matrices of the matrix system that is solved for the ‘forward’

Since the matrices on a per energy group basis are symmetric in space, the only non-sym

components occur because of energy group coupling originating from scattering and fissio

transpose of the matrices associated with fission and scattering are easily taken and the re

matrix system solved. Since down-scatter now becomes up-scatter for the transposed sc

operator, one solves the few-group diffusion equation by sweeping from low to high energiei.e.

from high energy group number to low energy group number) in the outer iterations. The

scattering iterations are completed if up-scatter exists in the ‘forward’ problem.

For the NEM solution, the situation is more complicated. Mathematically the adj

solution we seek should not only include the group fluxes but also all the expansion coeffic

for the transverse integrated fluxes. This implies that the matrices we should be transposi

only correspond to the nodal balance equations, but also include all the transverse integrat

constraint equations. These matrices never really appear in the nonlinear NEM iterative m

like they would in a more traditional surface current response NEM solution methodology

overcome this incompatibility and to greatly simplify the adjoint flux solution for the NEM, it

assumed that the diffusion coupling correction coefficients that originate in the nonlinear N

66

ient to

ations

Since

sion

lving

ing the

flux

r the

ction

hese

non-

ses no

ndent

the

rmal-

ined

eeds

atically

the

roup.

e, the

iterative method do not change when the core is perturbed. When using the Raleigh quot

estimate reactivity changes due to core perturbations, this implies that the resulting perturb

to the matrix operators do not include perturbations to the coupling correction coefficients.

the coupling correction coefficients can be thought of as corrections to the normal FDM diffu

coupling coefficients, this approximation should be acceptable for many applications invo

estimating core reactivity.

Having made the above assumption, one need no longer be concerned with evaluat

adjoint, transverse integrated flux expansion coefficients, since they only couple to the

solution through the coupling correction coefficients. Therefore, to obtain the adjoint flux fo

NEM one follows exactly the same procedure as for the FDM, except that the coupling corre

coefficients determine by the ‘forward’ solution appear in the matrix operator. Since t

coupling correction coefficients make the spatial originated component of the matrices

symmetric, this must be treated in addition to the non-symmetry in energy groups. This cau

practical problems within NESTLE, where now the transpose of the energy group depe

matrices that appear in the inner iterations are utilized.

A final point needing discussion in regard to solving for the adjoint flux concerns

treatment of cross-section feedback corrections. In NESTLE all feedback effects due to the

hydraulics and the transient fission products are frozen at their ‘forward’ solution determ

values. This implies that before an adjoint solution can be completed, a ‘forward’ solution n

to be completed to obtain the feedback corrected cross-section values. This is done autom

by NESTLE when the adjoint solution option is selected.

To facilitate subsequent utilization of the adjoint flux solution, NESTLE contains

option to write out to a user specified file its values as a function of spatial node and energy g

In this manner other computer codes can utilize this file as input to evaluate, for exampl

core’s reactivity response to perturbations employing the Raleigh quotient.

67

del.

uation

olor

sity,

only

When

irectly

nsities

single

educed

odels,

f the

color,

olant

llows

p g

II.7. Cross-Section Model

NESTLE has the option of utilizing a macroscopic or microscopic cross-section mo

The macroscopic model determines the macroscopic cross-sections used in the diffusion eq

solver by characterizing them as a function of the following: node burnup; color (where c

refers to the initial composition of the material within the node), rod in or out, coolant den

coolant temperature, effective fuel temperature, and soluble poison concentration. The

isotopic number densities calculated are for the I-Xe and Pm-Sm transient fission products.

used in conjunction with microscopic cross-sections, these isotopes’ effects can be d

accounted for. The microscopic cross-section model also determines the isotopic number de

for the U234-U236 and U238-Pu242 fuel chains, two lumped fission product groups, and a

isotope depletable burnable poison. The relative advantages of both approaches are r

computational resources and increased accuracy for the macroscopic and microscopic m

respectively. We now discuss the macroscopic model, to be followed by a discussion o

microscopic model.

II.7.a. Macroscopic Model

The macroscopic model represents macroscopic cross-section for a given fuel

burnup and rod insertion as a Taylor series expansion in terms of coolant density, co

temperature, effective fuel temperature and soluble poison number density as fo

(suppressing fuel color, burnup and rod insertion notation):

(151)

where

= macroscopic cross-section for reaction type x and energy grou

without transient fission products corrected to local conditions

Σxg a1xga n 1+( )xg

n 1=

2

∑ ρ∆ c( )na4xg

Tc∆ a5xgTFeff

∆ a n 5+( )xg

n 1=

3

∑ N∆ sp( )n+ + + +=

Σxg

68

m

m

(PPM)

or of

atial

, one

cally

uces

tion

= expansion coefficients

= change in coolant density (g/cm3) from reference condition

= change in coolant temperature (˚F) from reference condition

= charge in square root of effective fuel temperature (˚F) fro

reference condition

= change in soluble poison number density (cm-3 x 10-24) fro

reference condition.

The soluble poison number density change accounts for both soluble poison concentration

and coolant density (ρc) changes. The effective fuel temperature is evaluated by

(152)

where

Wp = pellet weighting factor, which accounts for resonance flux depression in the interi

the pellet

Wc = core statistical weighting factor, that compensates for the lack of detail in the sp

description of the core

= volume average fuel pellet temperature (oF)

= surface average fuel pellet temperature (oF).

To obtain the macroscopic cross-section for node l free of transient fission products

employs Eq. (151) using the expansion coefficients for the fuel color of node l quadrati

interpolated to the node l burnup, accounting for control rod effects as follows

(153)

where is the fraction of node l rodded. This treatment, for coarse axial meshing, prod

the artificial behavior of control rod “cusping” when plotting integral rod worth versus inser

ajxg

ρc∆ ρc ρc0( )

–=

TC∆ TC TC0( )

–=

TFeff∆ TFeff

TFeff

0( )–=

Nsp∆ Nsp Nsp0( )

–=

TFeffTC WC WpTF 1 Wp–( )TFSrf

TC–+[ ]+=

TF

TFSrf

Σxgl

1 f Roddedl

–( ) Σxgl

( )Unrodded f Roddedl Σxg

l( )Rodded+=

f Roddedl

69

ction

d Sm

manner

ions for

ium

sient

es are

time-

spatial

depth. Finally correcting for the transient fission products’ effect on the absorption cross-se

we obtain

(154)

where,

with and denoting the Xe135 and Sm135 number densities for node l. The Xe an

microscopic absorption cross-sections are represented and evaluated in exactly the same

as the macroscopic cross-sections. As implied by the above equations, the reference condit

the macroscopic cross-section input are xenon and samarium free.

The following options exist in NESTLE in regard to establishing xenon and samar

number densities: equilibrium, transient, no xenon nor samarium, no xenon and tran

samarium, and frozen at restart values. When the option requires, the number densiti

determined by solving the I135-Xe135 and Pm149-Sm149 chain depletion equations. The

dependent depletion equations for the iodine-xenon chain are given by (again suppressing

dependence)

(155)

(156)

where subscripts I and Xe denote I135 and Xe135, respectively, and

= nuclei number density of isotope i

Σagl Σag

l ΣXeag

l∆ ΣSmag

l∆+ +=

ΣXeag

l∆ σXeag

lNXe

l=

ΣSmag

l∆ σSmag

lNSm

l=

NXel

NSml

tdd

NIl

t( ) γ Il Σ f g

lt( )φg

lt( )

g 1=

G

∑ λIlNI

lt( )–=

tdd

NXel

t( ) λI NIl

t( ) γ Xel Σ f g

lt( )φg

lt( )

g 1=

G

∑ λXeNXel

t( )–+=

σXeag

lt( )φg

lt( )NXe

lt( )

g 1=

G

∑–

Nil

t( )

70

dy-state

= microscopic absorption cross section of isotope i

= macroscopic fission cross section

= node average flux

= effective yield (atoms/fission) of isotope i

= decay constant of isotope i

Likewise, the depletion equations for the Promethium-Samarium chain are given by

(157)

(158)

where subscripts Pm and Sm denote Pm149 and Sm149, respectively. The pseudo stea

solutions of Eq. (155) through (158) are given by

(159)

(160)

for the Iodine-Xenon depletion chain and,

(161)

(162)

σi ag

lt( )

Σ f gt( )

φgl

t( )

γ il

λi

tdd

NPml

t( ) γPml Σ f g

lt( )φg

lt( )

g 1=

G

∑ λPmNPml

t( )–=

tdd

NSml

t( ) λPmNPml

t( ) σSmag

lt( )φg

lt( )NSm

lt( )

g 1=

G

∑–=

NI ∞

l

γ Il Σ f g

l

g 1=

G

∑ φgl

λI-----------------------------=

NXe∞

l

λI NI ∞

l γ Xel Σ f g

l

g 1=

G

∑ φgl

+

λXe σXeag

l

g 1=

G

∑ φgl

+

------------------------------------------------------=

NPm∞l

γPml Σ fg

l

g 1=

G

∑ φgl

λPm----------------------------------=

NSm∞

l λPmNPm∞l

σSmag

l

g 1=

G

∑ φgl

-----------------------------=

71

ber

s and

multi-

l, it is

inetics

ber

s, and

cross-

for the Promethium-Samarium depletion chain. Note that the equilibrium isotopic num

densities and flux are coupled. This is addressed by updating the number densitie

subsequently cross-sections during the outer iteration process associated with solving the

group diffusion equation for either the eigenvalue problem or steady-state FSP.

For the transient solutions, forward differencing over the time-step produces

(163)

(164)

(165)

(166)

where is the time-step associated with transient fission product conditions. In genera

selected smaller and larger than the time-steps used in the core depletion and neutron k

solutions, respectively.

II.7.b. Microscopic Model

The microscopic model only differs from the macroscopic model in that the num

densities of the U234-U236 and U238-Pu242 fuel chains, two lumped fission product group

a simple burnable poison are explicitly calculated and used to determine the macroscopic

sections. This implies that the macroscopic cross-section for node l is determined from

NIl

t t∆+( ) NIl

t( ) t γ Il Σ f g

l

g 1=

G

∑ t( )φgl

t( ) λI NIl

t( )–∆+=

NXel

t t∆+( ) NXel

t( ) t γ Xel Σ f g

l

g 1=

G

∑ t( )φgl

t( ) λI NIl

t( )+∆+=

λXeNXel

t( ) σXeag

lt( )φg

lt( )NXe

lt( )

g 1=

G

∑––

NPml

t t∆+( ) NPml

t( ) t∆ γPml Σ f g

lt( )φgl

t( ) λPmNPml

t( )–g 1=

G

∑+=

NSml

t t∆+( ) NSml

t( ) t∆ λPmNPml

t( ) σSmag

lt( )φg

lt( )NSm

lt( )

g 1=

G

∑–+=

t∆

72

n (

mple

67) are

rnup

l is

otopic

s

upon

g the

(167)

where

= macroscopic cross-section for the background (Bk) isotopes (i.e. without isotopes in set

)

=set of fissile and fertile isotopes, lumped fission products and simple burnable poisoi.e.

U234-U236 and U238-Pu242 chains, two lumped fission product groups, and si

burnable poison)

As before, the macroscopic and microscopic cross-sections appearing on the RHS of Eq.(1

determined by interpolating expansion coefficients for the fuel color of node l to the node l bu

and using the results in Eq. (151).

The computational effort associated with the microscopic cross-section mode

associated with evaluating node dependent microscopic cross-sections and solving the is

depletion equations, which can be expressed for the fuel isotopes in general form as follow

(168)

where

Do note that Plutonium isotopic production is assumed to occur instantaneously

neutron capture. Eq. (168) is solved utilizing the Integrating Factor Technique, assumin

Σxg

lΣxg

Bk( )l

σi xg

lNi

l

i ςF∈∑+=

Σxg

Bk( )l

ςF

ςF

tdd

Nil

t( ) pil

t( )Ni 1–l

t( ) dil

t( )Nil

t( )–=

pil

t( )

λi 1–

or

σi 1– cg

lt( )φg

lt( )

g 1=

G

∑

production coefficient= =

dil

t( ) λi σi ag

lt( )φg

lt( )

g 1=

G

∑+ destruction coefficient= =

73

her the

OR-

given

g the

uently

code

f Eq.

inear

d used

Pm-

pletion

nt is

t

cross-sections constant at the end of time-step values and flux constant, set equal to eit

beginning of time-step value (PREDICTOR option) or time-step averaged value (PREDICT

CORRECTOR option) producing

(169)

When using the PREDICTOR-CORRECTOR option, the time-step averaged flux value is

by

(1)

The end of time-step value of the flux appearing in this equation is periodically updated durin

flux iterations, resolving the depletion equations for new number densities and subseq

updating cross-sections each time this occurs. The periodicity of updates is specified via

input. For the fissile and fertile chains and simple burnable poison, the integral on the RHS o

(169) can be analytically evaluated, since the solutions are composed of l

combinations of exponentials. The analytic solutions of Eq. (169) have been determined an

in NESTLE. Ref. [16] provides further details on the analytic solutions.

Two lumped fission products are used to model all fission products except I-Xe and

Sm. Their pseudo number densities are determined by solving the associated pseudo de

equations. In terms of Eq. (168) notation, the lumped fission products production coefficie

given by

(170)

where is the fission yield for Lumped Fission (LF) productj. Destruction is assumed to no

exist.

Nil

tn 1+( ) Nil

tn( )ed– i

ltn 1+( ) tn 1+ tn–( )

pil

tn 1+( ) Ni 1–l

t′( )ed– i

ltn 1+( ) tn 1+ t ′–( )

t′d

tn

tn 1+

∫+=

φgl⟨ ⟩n 1+

12--- φg

ltn( ) φg

ltn 1+( )+( )=

Nil

t( )

pLF j

lt( ) γ LF j

l Σ fgl

t( )φgl

t( )g 1=

G

∑=

γ LF j

l

74

ore is

-state

n can

lso be

nvalue

ta is

of the

ontrol

epeated

d core

this

is not

II.8. Control Option Searches

Various options exist in NESTLE to adjust a certain control parameter such that the c

made critical for the AEVP or achieves the desired specified core power level for the steady

FSP. Either control rods’ position, coolant inlet temperature or soluble poison concentratio

be selected as the control parameter to adjust. For the AEVP, core power level can a

selected. NESTLE adjusts the selected control parameter by contrasting the desired eige

(i.e. keff = 1.0) or core power level with the current outer iterative predicted value. This da

used to develop a linear expression for the value of the desired core attribute as a function

selected control parameter, from which a new estimate of the value of the selected c

parameter to achieve the desired core attribute value can be estimated. This process is r

every so many outer iterations until both the convergence criteria on achieving the desire

attribute value and neutronic solution are mutually satisfied. For slowly convergent FSP

approach may fail if the predicted core power level used in developing the linear expression

adequately converged.

75

w up

ed,

of

m the

nt. The

ddition,

l node

II.9. Hydrodynamic Model

II.9.a. Field Equations

The hydrodynamic model used in NESTLE models single and two phase coolant flo

closed coolant channels. A Homogenous Equilibrium Mixture (HEM) model is employ

limiting model applicability to low quality fluids where slip does not occur. The system

equations which describe the average conditions within the flow channel are obtained fro

mass continuity and energy conservation equations, assuming pressure to be consta

constant pressure assumption removes the need to consider the momentum equation. In a

an Equation of State is used to provide closure.

The one-dimensional, mass continuity equation along a specified channel for a radia

ij is

(171)

Similarly, the energy conservation equation assuming constant pressure is given by

(172)

where

= coolant density

= coolant mass velocity

= coolant internal energy

= volumetric power density from heat deposited directly in the coolant

= fuel rod surface heat flux into the coolant

= total cross-sectional area for coolant flow within the node

ACij

z( )∂ρC

ijz t,( )

∂t---------------------- ∂

∂z-----– GC

ijz t,( )AC

ijz( )=

ACij

z( )t∂

∂ ρCij

z t,( )UCij

z t,( )( )z∂

∂GC

ijz t,( )AC

ijz( )UC

ijz t,( )( )–=

Pz∂

∂ GCij

z t,( )ACij

z( )

ρCij

z t,( )-----------------------------------

– qSij

z t,( )SFij

qCij

z t,( )ACij

z( )+ +

ρCj

GCij

UCij

qCij

qSij

ACij

76

en

ure T

ocity,

tem of

ty as a

used

uid

= total fuel rod surface area per unit axial length within the node

= coolant pressure

Note that the field equations contain only the single spatial variablez due to the assumption of a

homogenous, closed channel.

The heat flux qS is obtained using Newton’s Law of Cooling as follows

(173)

where

= coolant temperature

= lumped (i.e. radially averaged) fuel temperature

=effective heat transfer coefficient

The effective heat transfer coefficient,heff is defined so as to provide the correct heat flux wh

the lumped versus surface fuel temperature is used in Newton’s Law. The coolant temperatc

is evaluated in terms of coolant internal energyUc using an Equation of State.

These two partial differential equations contain three unknowns: coolant mass vel

coolant internal energy and coolant density; a third equation is required to form a closed sys

equations. This third equation is provided by an Equation of State expressing coolant densi

function of coolant internal energy. For the steady-state analysis, the field equations are

setting the temporal derivative terms equal to zero.

II.9.b. Equation Discretization

In general the field equations are integrated over the flow stream fromzk-1/2to zk+1/2.

This axial spatial mesh defines the volume of a radial nodeij , as shown in Figure 7, equivalent to

the neutronic nodel. This spatial discretization results in node centered values for the fl

SFij

P

qSij

z t,( ) heffij

z t,( ) TFij

z t,( ) TCij

z t,( )–( )=

TCij

TFij

heffij

77

duces

been

e

properties and UC, and node boundary values for the mass velocityGC.

Discretizing the mass continuity equation by integrating over the staggered mesh pro

(174)

Now the Mean Value Theorem is used to approximate the time derivative term to obtain

(175)

where thebar over a variable denotes a node average value. Note that the axial mesh has

selected such thatAc is constant for and . The integral of th

spatial derivative term yields

ρC

ACij

z( )∂ρC

ijz t,( )

∂t----------------------dz

zk 1/2–

zk 1/2+

∫ z∂∂

GCij

z t,( )ACij

z( )( )dz

zk 1/2–

zk 1/2+

∫–=

ACij

z( )∂ρC

ijz t,( )

∂t----------------------dz

zk 1/2–

zk 1/2+

∫ VCl

dρCl

t( )dt

-----------------=

z zk 1 2⁄– zk 1 2⁄+,[ ]∈ VCl

ACij

zk∆=

78

zk 3/2+

zl 3/2+

=

zk 1/2+

zl 1/2+

=

zk 1/2–

zl 1/2–

=

Nodel+1

Nodel

ACl 3/2+ ρC

l 3/2+GC

l 3/2+, ,

ρCl 1+

UCl 1+

,

ACl 1/2+ ρC

l 1/2+GC

l 1/2+, ,

ρCl

UCl

,

ACl 1/2– ρC

l 1/2–GC

l 1/2–, ,

Radial Nodeij

Figure 7: Thermal-hydraulic mesh notation.

79

r by

time

mass

sity is

is

cement

alues

to be

t node

(176)

Finally a backward difference time advancement scheme is employed producing

(177)

The energy conservation equation, Eq. (172) is discretized in a similar manne

integrating along the flow stream to eliminate the spatial derivative, and a semi-implicit

treatment is employed. The convective term is linearized by using new time-step level

velocity and past time-step level for the other parameters. Furthermore, the coolant den

assumed to be constant over the time-step interval. The result of this discretization scheme

(178)

where the time averaged terms appear as a result of the central difference time advan

treatment and are defined as

(179)

and

(180)

Note the convective and work terms in the energy equation contains fluid property v

at the node boundaries. However, as stated previously, the fluid property values are

calculated as node averages. An intuitive approach would be to spatially average adjacen

z∂∂

GCij

z t,( )ACij

z( )( )dz

zk 1/2–

zk 1/2+

∫ GCl 1/2+

t( )ACl 1/2+

GCl 1/2–

t( )ACl 1/2–

–=

VCl ρ

l n 1+,ρ

l n,–

tn∆------------------------------- GC

l 1/2,n 1+ +AC

l 1/2+GC

l 1/2,n– 1+AC

l 1/2––+ 0=

VCl ρC

l n, Ul n 1+,

Ul n,

–tn∆

---------------------------------- GCl 1 2 n 1+,⁄+

ACl 1 2⁄+

UCl 1 2 n,⁄+

GCl 1 2 n 1+,⁄–

ACl 1 2⁄–

– UCl 1 2 n,⁄+( )–=

PGC

l 1 2 n,⁄+AC

l 1 2⁄+

ρCl 1 2 n,⁄+

----------------------------------------GC

l 1 2 n,⁄–AC

l 1 2⁄–

ρCl 1 2 n,⁄–

----------------------------------------–

qSl n 1+,

SFl

zl∆ qC

l n 1+,VF

l+ +–

qSl n 1+, qS

l n,qS

l n 1+,+

2-------------------------------=

qCl n 1+, qC

l n,qC

l n 1+,+

2-------------------------------=

80

ver, it

]. The

nsity

s of

ms of

lant

bove

average property values along the flow direction to obtain the node boundary values. Howe

has been found that a donor cell averaging technique is necessary for numerical stability [17

donor cell average is defined for the coolant internal energy as

(181)

and for the coolant density as

(182)

for flow in the positive direction, the only modelling capability within NESTLE.

To solve the two coupled discretized equations, the Equation of State for coolant de

in terms of coolant internal energy is linearized as follows

(183)

This equation is then substituted into Eq. (177) to solve for the coolant mass velocity in term

coolant internal energy to produce

(184)

Finally, the above equation is substituted into Eq.(139) to produce an equation only in ter

coolant internal energy

(185)

This equation is solved for at each radial node by sweeping in the direction of coo

flow. The following auxiliary relationships are used to evaluate terms appearing in the a

UCl 1 2 n,⁄+

UCl n,

= and UCl 1 2 n,⁄–

UCl 1– n,

=

ρCl 1 2 n,⁄+ ρC

l n,= and ρC

l 1 2 n,⁄– ρCl 1– n,

=

ρCl n 1+,

ρCl n,

UCl

∂

∂ ρCl

tn

UCl n 1+,

UCl n,

–( )+≅

GCl 1 2 n 1+,⁄+ 1

ACl 1 2⁄+

----------------- GCl 1 2 n 1+,⁄–

ACl 1 2⁄– VC

l

tn∆-------

–UC

l∂

∂ ρCl

tn

UCl n 1+,

UCl n,

–( )=

VCl

tn∆-------

ρCl n, ∂ρC

l

∂UCl

-----------

tn

ACl 1 2⁄+

UCl n,

–

UCl n 1+,

GCl 1 2 n 1+,⁄–

ACl 1 2⁄–

UCl 1 n,–

UCl n,

–( )=

PGC

l 1 2 n,⁄+AC

l 1 2⁄+

ρCl n,----------------------------------------

GCl 1 2 n,⁄–

ACl 1 2⁄–

ρCl 1 n,–

----------------------------------------–

qSl n 1+,

SFl

z∆ lqC

l n 1+,VC

l VCl

tn∆-------

ρCl n,

UCl n,

+ + +–

UCl n 1+,

81

olant

olant

void

aving

equation:

Coolant Volume:

Total Fuel Rod Surface Area Per Unit Axial Length:

Coolant Volumetric Heat Source:

where

= wet fraction for the node when rodded

= fuel rod radius

= number of fuel pins within the node

= distribution fraction of energy directly deposited within the fuel

= total volumetric power density for heat deposited within the node.

Having obtained values for the coolant internal energy at the new time-step, the co

densities are evaluated at the new time-step using an Equation of State for the liquid if co

internal energy indicates sub-cooled conditions for the node. If bulk boiling is indicated, the

fraction is first determined as follows (suppressing superscripts)

(186)

where

= saturated coolant liquid internal energy

= saturated coolant vapor internal energy

= saturated coolant liquid density

= saturated coolant vapor density

from which the coolant density is determined as now indicated

(187)

These same equations are utilized in determining introduced back in Eq. (183). H

VCl

f Roddedl

Vl⋅=

SFl

2πrFNFuelPinsl

=

qCl n,

VCl

1 dF–( )qTl n,

Vl

=

f Roddedl

r F

NFuelPinsl

dF

qTl n,

αρCL

UC UCL–( )

ρCVUCV

UC–( ) ρCLUCL

UC–( )–---------------------------------------------------------------------------------=

UCL

UCV

ρCL

ρCV

ρC αρCV1 α–( )ρCL

+=

UCd

dρC

82

e new

terial

bility

ontrol

8] is

ally

ight

e-step

ilizing

sity as

nergy,

fluid.

icated

ed in

havior,

ed

evaluated the coolant density, Eq. (177) is used to solve for the coolant mass velocity at th

time-step.

The above approach is not unconditionally stable and must satisfy the Courant ma

limit due to the degree of semi-implicitness introduced in linearizing the equations. This sta

limit for certain transients restricts the time-step sizes to values smaller than required to c

truncation errors. The Stability-Enhancing Two-Step (SETS) Method developed at LANL [1

used to allow a Courant material limit violating treatment. Since this method was origin

utilized within the context of the six-equation model used within the TRAC code, a sl

modification of the SETS method is required for the current application.

The stabilizing energy conservation equation used to solve for is given by

(188)

Do note the increased degree of implicitness of this equation. Estimates of the current tim

values are determined solving the previous introduced set of equations. Having stab

predicted values of coolant internal energy, they are then utilized to update the coolant den

noted before. Also, the coolant temperature is determined based upon coolant internal e

using an Equation of State for sub-cooled fluid and the saturation temperature for saturated

This approach has been shown to allow large time-steps without stability problems, as ind

by solving the transient equations at steady-state conditions and observing no drift.

To initiate the process, the volumetric power density at the new time-step is estimat

the same manner as the flux; that is, it is time extrapolated, assuming exponential time be

using power density values atn-1 andn. The past time value of the surface heat flux is employ

UCl n 1+,

VCl ρC

l n 1+,UC

l n 1+, ρCl n,

UCl n,

–tn∆

------------------------------------------------------------

G( Cl 1 2⁄+ n 1+,

ACl 1 2⁄+

UCl n 1+,

–=

G– Cl 1 2⁄– n 1+,

ACl 1 2⁄–

UCl 1 n 1+,–

) PGC

l 1 2⁄+ n 1+,AC

l 1 2⁄+

ρCl n 1+,-----------------------------------------------

GCl 1 2 n 1+,⁄–

ACl 1 2⁄–

ρCl 1 n 1+,–

-----------------------------------------------–

–

qSl n 1+,

SFl

z∆ lqC

l n 1+,VC

l+ +

83

tions,

ed to

aulic

ith the

of the

on for

olant

values.

alues.

so all

lance

ies. In

value.

based

aging

ed

de is

le of

n each

as the initial estimate for the new time step. Using these values in the thermal-hydraulic equa

they provide initial estimates of the coolant and fuel conditions at the new time-step, us

correct cross-sections and commence the flux iterations. With the explicit thermal-hydr

option selected, these values are never updated and used throughout the flux iteration. W

implicit thermal-hydraulic option selected, the SETS process is repeated as new estimates

volumetric power density and heat flux become available, associated with the iterative soluti

the flux.

For the feedback correction of cross-sections with respect to coolant density, co

temperature, and effective fuel temperature, all must be evaluated as node average

Likewise, the coolant temperature appearing in Newton’s Law requires node average v

Previously we stated that we are solving for node average values of coolant properties,

would seem in order. However, thinking about the steady-state solution of the energy ba

equation it becomes clear that coolant internal energy is really evaluated at node boundar

this sense donoring is done counter-flow from the node boundary value to the node average

Therefore the node average coolant temperature is determined using an Equation of State

upon the average of the coolant internal energy at axial elevationsk-1 andk. The node average

coolant density used to correct cross-sections follows a similar approach, now directly aver

densities. This subtlety becomes important for large axial meshing (e.g. 2D radial geometry).

II.9.c. Fuel Temperature Model

The lumped (i.e. radially averaged) fuel temperature is obtained by utilizing a lump

parameter heat conduction model, in which a simple energy balance for each radial no

performed. This approach should be valid for transients where the fuel pin-wise radial profi

the fuel temperature stays close to the steady-state profile. The rate of energy change i

84

de heat

ces

sity is

91) to

e

ge fuel

node, ignoring axial heat conduction, can be expressed as the difference between the no

source and the energy lost due to heat transported radially:

(189)

where r denotes the radial coordinate for an average pin in node ij and

= fuel internal energy

= fuel density

= volumetric power density from heat deposited directly in the fuel

= heat flux within the fuel

Using Fourier’s Law of Thermal Conductivity expresses the heat flux within the fuel as

(190)

wherekF denotes the fuel’s thermal conductivity. Substituting Eq. (190) into Eq. (189) produ

(191)

The enthalpy is now expressed in terms of the fuel specific heat ( ) and temperature, den

assumed constant, and fuel specific heat is assumed slowly varying in time allowing Eq. (1

be rewritten as

(192)

Integrating over the nodel volume occupied by fuel, denoted,

(193)

where indicates the fraction of nodel occupied by fuel, applying the central difference tim

advancement scheme, and rearranging terms yields the final expression for the node avera

temperature.

qFij

r z t, ,( )

t∂∂ ρF

ijr z t, ,( )UF

ijr z t, ,( )( ) qF

ijr z t, ,( ) ∇r qhf

ijr z t, ,( )⋅–=

UFij

ρFij

qFij

qhfij

qhfij

r z t, ,( ) k– Fij

r z t, ,( )∇rTFij

r z t, ,( )=

t∂∂ ρF

ijr z t, ,( )UF

ijr z t, ,( )( ) qF

ijr z t, ,( ) ∇r kF

ijr z t, ,( )∇rTF

ijr z t, ,( )⋅+=

cpF

ρFcPF

ijr z t, ,( )

∂TFij

r z t, ,( )∂t

---------------------------- qFij

r z t, ,( ) ∇r kFij

r z t, ,( )∇rTFij

r z t, ,( )⋅+=

VFl

f Fl

Vl×=

f Fl

85

rface

luated

sfer

Also

ing the

93)),

r to

tion

ion.

luate

icated

linear

tially

(194)

In obtaining Eq. (194) the volume integral over the heat conduction term is converted to a su

integral via Green’s Theorem, and the resulting expression for the surface heat flux is eva

using Newton’s Law of Cooling given by Eq. (173). Do note that the effective heat tran

coefficient is treated explicitly in both the coolant and fuel energy conservation equations.

the fuel volumetric heat source that appears in Eq. (192) has been replaced in Eq. (194) us

following expression.

(195)

Since only appears in the ‘heat sink’ term in Eq. (194) and is a function of (see Eq. (1

the value of may be varied by fuel color to account for fuel density variations by colo

overcome the input limitation of only inputting the core average fuel density. The deple

equations will also correctly reflect fuel density variations captured by the fuel volume fract

In addition to lumped fuel temperature, the surface fuel temperature is required to eva

the effective fuel temperature used to correct cross-sections for Doppler broadening, as ind

in Eq. (152). This is obtained by characterizing surface fuel temperature as a function of

power density for a reference coolant temperature, in terms of a polynomial. The spa

dependent linear power density is given by

(196)

The surface fuel temperature is then determined using

(197)

ρFVFl

tn∆--------------cPF

l n 1+, heffl n,

2--------

SFl

zl∆+ TF

l n 1+, heffl n,

2--------

SFl

zl∆ TC

l n,TC

l n 1+,+( )=

ρFVFl

tn∆--------------cPF

l n 1+, heffl n,

2--------

– SFl

zl∆ TF

l n,dFqT

l n,V

l+ +

qFl n,

VFl

dFqTl n,

Vl

=

VFl

f Fl

f Fl

TCRef

qLl n, SF

l

NFuelPinsl

---------------------- qS

l n,

dF--------

=

TFSrf

l n,f TSrf

qLl n,

( ) TCl n,

TCRef–( )+=

86

d fuel

n the

locity

noring

mporal

ons are

of the

ansfer

ficient

where denotes the polynomial function.

II.9.d. Steady-State Model

For steady-state conditions, the governing equations used to solve for the coolant an

conditions are obtained by setting the temporal derivative to zero. When this is done i

coolant’s mass continuity equation, Eq. (171), it is seen that the product of coolant mass ve

and cross-sectional flow area must be constant up the flow channel. Also the concept of do

no longer enters since node average coolant values only appeared because of the te

derivative terms. These implications lead to the following discretized equations:

Coolant Mass Continuity

(198)

where subscript In denotes the inlet to radial node ij associated with node l.

Coolant Energy Conservation

(199)

Fuel Energy Conservation

(200)

Surface fuel temperature is evaluated as indicated for the transient conditions. These equati

iteratively solved as new estimates of the flux become available, providing new estimates

surface heat flux and volumetric heat densities. During these iterations the effective heat tr

coefficient is also updated, producing consistent values for the effective heat transfer coef

f TSrfqL

l n,( )

GCl 1 2⁄+ AC

l 1 2⁄–

ACl 1 2⁄+

-----------------

GCl 1 2⁄– ACIn

ij

ACl 1 2⁄+

-----------------

GCIn

ij= =

UCl 1 2⁄+

UCl 1 2⁄–

PACIn

ijGCIn

ij 1

ρCl 1 2⁄+

----------------- 1

ρCl 1 2⁄–

----------------–

qCl

VCl

qFlVF

l+ +–=

TFl

TCl VF

l

heffl

SFl

zl∆

-----------------------

qFl

+=

87

cient

fficient

rmined

ws.

and

d by

itial

. Now

e node

olution

raulic

n, but

heat

e fuel

and lumped fuel temperature as now described.

II.9.e. Effective Heat Transfer Coefficient Evaluation

For the lumped fuel temperature model to be utilized, the effective heat transfer coeffi

must be evaluated. For steady-state conditions we can select the effective heat transfer coe

such that the correct values of the lumped fuel temperature result, these temperatures dete

utilizing a more detailed fuel pellet model. This implies the following:

(201)

One can now solve forheff given values of and for a fixed coolant temperature as follo

(202)

Note thatheff has been characterized as a function of since the fuel thermal conductivity

gap closure, both functions of fuel temperature, are the main reasons whyheff changes. This

characterization is captured using a polynomial representation.

For steady-state calculations, an initial estimate of fuel temperature is obtaine

characterizing it as a function of linear power density in terms of a polynomial. Given this in

lumped fuel temperature estimate, the effective heat transfer coefficient can be evaluated

Eq. (200) can be used to calculate a new estimate of the lumped fuel temperature once th

average coolant temperature and volumetric heat density have been evaluated. As the flux s

is iterated, this sequence of calculations is repeated. The iteration of the thermal-hyd

equations not only addresses feedback between its solution and the neutronic solutio

addresses the non-linearities in calculating the lumped fuel temperature due to effective

transfer coefficient dependency on fuel temperature.

For transient calculations, the same iterative sequence is employed; however, now th

qF AF heffSF TF TCRef–( )=

TF qF

heff

qF AF

SF TF TCRef–( )

------------------------------------=

TF

88

at flux,

ng the

ture in

o but

the

ting in

at

heat.

fit a

us, the

ential

specific heat is also updated due to fuel temperature dependency. In addition the surface he

which appears in the transient coolant energy conservation equation, is also updated utilizi

updated effective heat transfer coefficient, coolant temperature and lumped fuel tempera

Newton’s Law.

II.9.f. Decay Heat Model

When the reactor shuts down, the reactor power does not immediately drop to zer

falls off rapidly according to a negative period, eventually determined by the half-life of

longest-lived delayed neutron group. Even then, the transuranics and fission products exis

the fuel continue to decay (β and γ) at decreasing rates for long periods of time. The he

generated from isotopic decay of these isotopes following reactor shutdown is called decay

Although there are many isotopes involved in the complex decay chain, it is customary to

measured decay heat curve for a high burnup reactor with a series of decay heat groups. Th

model is analogous to the handling of delayed neutrons.

Accounting for decay heat, the total volumetric heat density is given by

(203)

where

= concentration of decay heat group i

= disintegration rate (decay constant) [sec-1]

= fraction of the total fission energy appearing as decay heat for decay heat group i

= = total fraction of the fission energy appearing as decay heat

The concentration of decay heat precursors can be expressed by the following differ

equation.

qT r t,( ) 1 αT–( ) κgΣ fg r t,( )g 1=

G

∑ φg r t,( ) ςiDi r t,( )i 1=

I DH( )

∑+=

Di r t,( ) Mev

cm3

-----------

ςi

αi

αT αii 1=

I DH( )

∑

89

grated

ource

nterval

sired

rium.

4) and

.

(204)

To develop an expression for the decay heat precursor concentrations, a time-inte

expression is derived by integrating Eq. (204) from tn to tn+1. This integration results in

(suppressing dependence for clarity)

(205)

To solve the above integral, a functional form for the time dependent neutron fission s

density must be developed. Assume the fission source density is constant over the time i

at the past time-step value,i.e.

(206)

Incorporating this approximation into Eq. (205) and rearranging terms we obtain the de

expression.

(207)

In steady-state it is generally assumed that even the longest-lived group is in equilib

The steady-state concentration is calculated by setting the time derivative to zero in Eq. (20

solving for the precursor concentration producing,

(208)

This equation is utilized to determine the initial conditions required for the transient solution

∂Di r t,( )∂t

--------------------- αi κgΣ fg r t,( )φg r t,( ) ςiDi r t,( )–g 1=

G

∑= for i 1,...,IDH( )

=

r

Di tn 1+( ) Di tn( )eςi tn∆–

αi κgΣ fg t′( )φg t′( )eςi– tn 1+ t′–( )

dt′g 1=

G

∑tn

tn 1+

∫+=

t′ tn tn 1+,[ ]∈

κgΣ fg t′( )φg t′( )g 1=

G

∑ κgΣ fg tn( )φg tn( )g 1=

G

∑=

Di tn 1+( ) Di tn( )eςi tn∆– αi

ςi----- 1 e

ςi tn∆––[ ] κgΣ fg tn( )φg tn( )

g 1=

G

∑+=

Di∞

αi

ςi----- κgΣ fgφg

g 1=

G

∑=

90

ctor

and

d,”

tive

te-

y

III. References

[1] B. R. Bandini,A Three-Dimensional Transient Neutronics Routine for the TRAC-PF1 Rea

Thermal Hydraulic Computer Code, PhD Dissertation, Pennsylvania State University (1990).

[2] R. D. Lawrence, “Progress in Nodal Methods for the Solution of the Neutron Diffusion

Transport Equations,”Progress in Nuclear Energy, 17, No. 3, 271 (1986).

[3] B. A. Finlayson and L. E. Scrivin, “The Method of Weighted Residuals - A Review,”Applied

Mechanics Review, 19, No. 9, 735 (1966).

[4] K. S. Smith, “Nodal Method Storage Reduction by Non-Linear Iteration,”Trans. Am. Nucl.

Soc., 44, 265 (1983).

[5] K. S. Smith, “QPANDA: An Advanced Nodal Method for LWR Analyses,”Trans. Am. Nucl.

Soc., 50, 265 (1985).

[6] K. S. Smith and K. R. Rempe, “Testing and Applications of the QPANDA Nodal Metho

Proc. Intl. Topl. Mtg. Advances in Reactor Physics, Mathematics and Computation, 2, 861 (1987).

[7] P. R. Engrand, G. I. Maldonado, R. Al-Chalabi and P. J. Turinsky, “Non-Linear Itera

Strategy for NEM: Refinement and Extension,”Trans. Am. Nucl. Soc., 65, 221 (1992).

[8] G. H. Hobson,Private Communication(1991).

[9] L. A. Hageman and D. M. Young,Applied Iterative Methods, Computer Science and Applied

Mathematics, Academic Press, Inc., Orlando (1981).

[10] K. I. Derstine, “DIF3D: A Code to Solve One- Two- and Three-Dimensional Fini

Difference Diffusion Theory Problems,” ANL-82-84, Argonne National Laboratory (1984).

[11] R. S. Varga,Matrix Iterative Analysis, Prentice Hall, Inc., Englewood Cliffs, New Jerse

(1962).

[12] S. Nakamura,Computational Methods in Engineering and Science, Wiley and Sons, Inc.,

New York (1977).

91

ith

-5,”

nal

al

- I:

-II:

al-

M-

of

,”

[13] S. K. Zee,Numerical Algorithms for Parallel Processors Computer Architectures w

Applications to the Few-Group Neutron Diffusion Equations, PhD thesis, North Carolina State

University (1987).

[14] L. A. Hageman and C. J. Pfeifer, “The Utilization of the Neutron Diffusion Program PDQ

WAPD-TM-385, Bettis Atomic Power Laboratory, Westinghouse Power Corporation (1965).

[15] R. D. Lawrence, “The DIF3D Nodal Neutronics Option for Two- and Three-Dimensio

Diffusion-Theory Calculations in Hexagonal Geometry,” ANL-83-1, Argonne Nation

Laboratory (1983).

[16] Y. Chao and N. Tsoulfanidis, “Conformal Mapping and Hexagonal Nodal Methods

Mathematical Foundation,”Nucl. Sci. Eng., 121, 202-209 (1995).

[17] Y. Chao and Y.A. Shattila, “Conformal Mapping and Hexagonal Nodal Methods

Implementation in the ANC-H Code,” Nucl. Sci. Eng., 121, 210-225 (1995).

[18] M. Knight , P. Hutt, and I. Lewis, “Comparison of PANTHER Nodal Solutions in Hexagon

z Geometry,”Nucl. Sci. Eng., 121, 254-263 (1995).

[19] R. F. Barry, “LEOPARD - A Spectrum Dependent Non-Spatial Depletion Code for the IB

7094,” WCAP-3269-26, Atomic Power Division, Westinghouse Electric Corporation (1963).

[20] T. A. Porsching, J. H. Murphy and J. A. Redfield, “Stable Numerical Integration

Conservation Equations for Hydraulic Networks,”Nuclear Science and Engineering, 43, 218

(1971).

[21] J. H. Mahaffy, “A Stability-Enhancing Two-Step Method for Fluid Flow Calculations

Journal of Computational Physics, 46, 239 (1982).

92

ffort

given

g the

is

cept

quick

with a

ames

quick

s are

wn in

cution

e

eter

the

nly

IV. User’s Guide

The NESTLE code has been written with a view to minimizing the input preparation e

as much as possible. A brief description of the function of each subprogram in NESTLE is

in the Programmer’s Guide section along with the flow diagram of the code. By separatin

type of input (e.g. cross section, geometry, or program control) into distinct input files, it

possible to setup widely varied problems with little input preparation effort. All input data ex

for restart files are on disk and in free format (except the alphanumeric strings). Thus

editing is possible and comments to identify each input data can be attached to each data

double blank between the input and comment. The alphanumeric string variables for file n

generally have enough length (A40) so that file names can be assigned to them for later

identification of the files. Due to the large amount of data written in the restart files, these file

written unformatted to save on storage and facilitate fast retrieval of the data by the code.

The logical units assigned to each file, file name, and the contents of the files are sho

Table 4. The control parameter data file name is specified as a parameter on the exe

statement, i.e.nestle File Name. If File Name is not specified, it defaults to the nam

NESTLE.CNTL . All other “Free to Select” file names are specified in the code control param

date file, the exception being Unit 33 files which are specified in the Unit 3 file.

The following sections describe the input data required for each of the ASCI input files.If a fixed

formatted read is involved, immediately after the variable name in parenthesis is indicated

applicable format. In addition, for lines of input that are contained within a DO loop or are o

read if certain other input values have certain values, aFORTRAN style DO or IF notation with

indentation is employed.

93

Table 4: List of required file names.

LogicalUnit

I/O File Name Contents

1 I Free to Select Code Control Parameter Data

2 I Free to Select Geometry Data

3 I Free to Select Cross Section Data

4 I Free to Select Solution Method Control Data

5 I Free to Select Restart Data-Read (Binary)

10 O Free to Select Restart Data-Write (Binary)

33 I Free to Select Cross Section Data (Optional)

55 O Free to Select Hardcopy Output

77 I Free to Select Node-wise Initial Isotopic Number Densities(Optional)

79 I Free to Select Node-wise Initial Exposure (Optional)

81 I Free to Select Corner Point Discontinuity Factors and Pin-wiseForm Factors for Pin Power Reconstruction(Optional)

82 O Free to Select Pin Power Output

94

ation

IV.1. Code Control Parameter Data File

(Unit 1 => “CNTRL”)

TITLE (A80)

Title line to identify run in hardcopy output

NBUSTEP

Maximum of the number of burnup mask values input for the soluble poison concentr

or (burnup steps+1).

NBMAX

Maximum number of control rod banks (groups) to be input.

GEOM (A40)

Geometry data file name (Unit 2)

XSECT (A40)

Cross section data file name (Unit 3)

KINET (A40)

Kinetic data file name (Unit 4)

PERFM (A40)

Solution method control file name (Unit 5)

RESTRT (A40)

Restart file name [to read] (Unit 9)

OUTPUT (A40)

Hardcopy output file name (Unit 55)

OUTADJ (A40)

Unformatted output file name for adjoint solution (Unit 10)

IADJ (A5)

Adjoint option (“Y”/ “N”)

95

ISAVEADJ (A5)

Write adjoint solution to file OUTADJ (“Y”/ “N”)

ITRAN (A5)

Transient option (“Y”/ “N”)

IRSTRT (A5)

Restart option (“Y”/ “N”)

ITYPE (A5)

Problem type

Eigenvalue (“EVP”)

Fixed-source (“FSP”)

NXSEC (A5)

Cross-section corrections to be applied (“Y”/ “N”)

IF (NXSEC.EQ. “Y”) THEN READ

ASRCH (A5)

Criticality/Power level search option (“Y”/ “N”)

IF (ASRCH.EQ. “Y”) THEN READ

IWHICH

Parameter to search upon

Power level (relative)=1

Soluble poison concentration (PPM)=2

Coolant inlet temperature (oF)=3

Lead (I.D.=highest value) control bank withdrawn (inches)=4

IF(ITYPE.EQ.”EVP’) THEN READ

CKE_TARGET

Target keff value to match in search

96

ENDIF EVP

ENDIF ASRCH

ANFDBK (A5)

Thermal-hydraulic feedback on (“Y”/ “N”)

ENDIF NXSEC

ABUCKL (A5)

Buckling correction option to be applied (“Y”/ “N”)

AVEBU(1)

Cycle average burnup at first burnup step (MWD/MTM)

IBURN (A5)

Depletion case (“Y”/ “N”)

IF (IBURN.EQ. “Y”) THEN READ

NBU

Number of burnup steps

(DELBU(I),I=2,NBU+1)

Burnup steps’ sizes (MWD/MTM)

ENDIF IBURN

(IXE(I), I=1,NBU)

Xenon-samarium conditions for each burnup step

No Xe and Sm = 1

Freeze Xe and Sm = 2

Equilibrium Xe and Sm = 3

No Xe and transient Sm = 4

Transient Xe and Sm = 5

No Xe and freeze Sm = 6

97

t

,

BP,


NPPMX

Number of burnup steps that letdown soluble poison concentration provided a

DO IBU=1,NPPMX

BUPM(IPM),PPM(IPM)

Cycle burnup (MWD/MTM) and soluble poison concentration (PPM)

ENDDO IBU ...Burnup step loop

(ZB(IBK),IBK=1,NBMAX)

Control banks (group) axial withdrawal position (in)

PRCNT

Power level (% of full rated)

ENDIF NXSEC

AL3 (A5)

Long input echo options: Yes/No (“Y”/ “N”)

NPC (A5)

Long hardcopy output option (“Y”/ “N”)

IF (NPC.EQ. “Y”)THEN READ

NOUTLONG

Number of variables whose node values are to be output

(AOUTLONG(N),N=1,NOUTLONG) (10(A5,1X))

Names of variables whose node values are to be output

PREL, FLUX, DCOOL, TCOOL, TFUEL, BU, I135, XE135, PM149, SM149

U234, U235, U236, U238, PU239, PU240, PU241, PU242, LFPG1, LFPG2,

ADJFL

ENDIF NPC

98

by

ut

save

CRTON (A5)

CRT output option (“Y”/ “N”)

DO IBU=1,NBU+1...For each burnup step

ISAVE(IBU) (A5)

Write steady-state restart file (“Y”/”N”) at burnup step IBU

IF (ISAVE(IBU).EQ. “Y”) THEN READ

OUT(IBU) (A40)

Steady-state restart file name [to write] at burnup step IBU

ENDIF ISAVE(IBU)

ENDDO IBU ...Burnup step loop

IF (ITRAN.EQ. “Y”) THEN READ

ISAVETR (A5)

Write transient restart file (“Y”/”N”)

ENDIF ITRAN

IF (ISAVETR.EQ. “Y”) THEN READ

OUTTR (A40)

Transient restart file name [to write]. This file is written at every time indicated

the kinetic file input variable TIMEPR(IT), that indicates times when outp

should be produced. Note transient restart file is overwritten each time to

space.

ENDIF ISAVETR

IEXP (A5)

Initial node-wise exposure map available (“Y”/ “N”)

IF (IEXP.EQ. “Y”) THEN READ

FINITEXP (A40)

99

-at the

nal

)

rm

Name of the file containing the initial node-wise exposures (Unit=79)

ENDIF IEXP

OEOC (A5)

End of depletion ASCII restart file option (“Y” / “N”)

If OEOC = “Y”, NESTLE will write new code control parameter file (CNTRL file), nodewise exposure map, and node-wise number density map (for microscopic depletion)end of the depletion. This option is valid for a steady-state problem only.

Note: the following pin power reconstruction option is currently available for hexago

geometry only.

PPR (A5)

Pin power reconstruction option (“Y”/ “N”)

IF (PPR.EQ. “Y”) THEN READ

FPIN (A5)

Corner point discontinuity factors and pin-wise form factors available (“Y”/ “N”

IF (FPIN.EQ. “Y”) THEN READ

INPPIN (A40)

File name containing the corner point discontinuity factors and pin-wise fo

factors (Unit=81)

ELSEIF (FPIN.EQ. “N”) THEN READ

NHRPIN

Number of radial rings of pins surrounding center pin.

Number of pins within the assembly = NPIN

When PPR = “Y” and FPIN = “N”, NESTLE will set all corner point

NPIN 6 ii 1=

NHRPIN 1–

∑ 1+=

100

discontinuity factors and form factor values to 1.0.

ENDIF FPIN

OUTPPR (A5)

Output pin-wise power values (“Y” / “N”)

IF (OUTPPR.EQ. “Y”) THEN READ

OUTPIN (A40)

Pin power output file name (Unit=82)

ENDIF OUTPPR

ENDIF PPR

101

ions.

IV.2. Geometry Data File

(Unit 2 => “GEOM”)

NSHAP (A5)

Node shape in radial plane

Hexagonal (“HEXA”)

Cartesian (“CART”)

IDRUN (A5)

Core symmetry

IF (NSHAP .EQ. “HEXA”)

Full core (“FCORE”)

One-third core (“TCORE”) 5 to 9 o’clock

One-sixth core (“SCORE”) 5 to 7 o’clock

One-dimensional axial core (“AXIAL”)

IF (NSHAP. EQ. “CART”)

Full core (“FCORE”)

Half core (“HCORE”) 3 to 9 o’clock

Quarter core (“QCORE”) 3 to 6 o’clock

One-dimensional axial core (“AXIAL”)

ENDIF NSHAP

IF (NSHAP.EQ."CART") THEN READ

NX,NY

The x and y total mesh numbers applicable to initial homogenous material reg

NMULXY

Number of mesh to create from each input x and y material mesh.

ELSEIF (NSHAP.EQ."HEXA") THEN READ

102

all

n.

al

e

NHR

Number of radial rings of bundles (assemblies) surrounding center bundle.

ENDIF NSHAP

NZ

The z total mesh number applicable to initial material regions.

NMULZ

Number of mesh to create from each input z material mesh.

NFIGURE

Number of different radial configurations (basic figures) of core materials over

elevation.

LIHO,LIPS,LIZU,LIZD

Boundary conditions: Radial exterior, radial interior, z-up, z-down

(reflective=0, zero flux=1, non-reentrant=2, cyclic=3, not applicable=4)

For radial exterior, z-up and z-down boundaries, LIHO cannotequal 3.

IF(NSHAP.EQ."CART") THEN READ

BPITCHX,BPITCHY

Pitch for bundles (assemblies) in the x and y directions (in).

NSUBX,NSUBY

Number of different x and y material mesh sizes to utilize in numerical solutio

(NSPACX(I),DDX(I),I=1,NSUBX)

Number of consecutive x material mesh (NSPACX(I)) of constant x mesh size

(DDX(I)) running from west to east. [Note the sum of NSPACX(I) must equ

NX.]

(NSPACY(J),DDY(J),J=1,NSUBY)

Number of consecutive y material mesh (NSPACY(J)) of constant y mesh siz

103

ual

al

(DDY(J)) running from north to south. [Note the sum of NSPACY(J) must eq

NY.]

ELSEIF (NSHAP.EQ."HEXA") THEN READ

DELH

Pitch of bundles (assemblies) (in)

ENDIF NSHAP

(See section IV.2.a entitled Geometry Input for detailed description of geometry input.)

DO IB=1,NFIGURE ...For each color figure

NBASIC(IB)

Basic figure I.D. (=1,2,...)

DO IY=1,NY...For each y mesh

(NCOL2DT(IX,IY,IB), IX=NXSTART(IY),NXEND(IY),NXSKIP)

Core material colors defined for initial radial material mesh.

ENDDO...Y mesh loop

ENNDO...Color figure loop

Blank Line


(NROT2DT(IX,IY),IX=NXSTART(IY),NXEND(IY),NXSKIP)

Mesh clockwise rotation of core material used to define surface ADFs for initi

radial material mesh.

(Cartesian: 0o = 0, 90o = 1, 180o = 2, 270o = 3)

(Hexagonal: 0o = 0, 60o = 1, 120o = 2, 180o = 3, 240o = 4, 300o = 5)

ENDDO IY

Blank Line

DO IY=1,NY

104

(K))

(NREF2DT(IX,IY),IX=NXSTART(IY),NXEND(IY),NXSKIP)

Mesh diagonal axis reflection of core material used to define surface ADFs

for initial radial material mesh. (No reflection = 0,

Reflection about NW to SE node diagonal of original orientation = 1)

ENDDO IY

NSUBZ

Number of different z material mesh sizes to utilize in numerical solution.

(NSPACZ(K),DDZ(K),K=1,NSUBZ)

Number of consecutive z material mesh (NSPACZ(K)) of constant z mesh size (DDZ

running from down to up. [Note the sum of NSPACZ(K) must equal NZ.]

IZCOLS,IZCOLE

Starting and ending axial material mesh numbers for fuel.

(NCOLZT(IZ),IZ=1,NZ)

Basic figure I.D. assigned to initial axial material mesh.

Blank Line

(NTOPZT(IZ),IZ=1,NZ)

Mesh axial reflection of core material used to define surface ADFs for initial

axial material mesh. (Up and down surfaces’ ADFs reversed)

IF (NXSEC.EQ."Y") THEN READ

Blank Line


(LROD2DT(IX,IY),IX=NXSTART(IY),NXEND(IY),NXSKIP)

Control bank (group) I.D. defined for radial material mesh.

(Bank Present=1, 2,...,NBACU, Bank Not Present=0)

ENDDO...Y mesh loop

105

RODOFFSET

Elevation above bottom of fuel when control bank fully inserted (in).

ENDDO NXSEC


(LSEXT2DT(IX,IY),IX=NXSTART(IY),NXEND(IY),NXSKIP)

External source locations defined for radial material mesh

(Source Present=1, Source Not Present=0)

ENDDO IY

Blank Line

(LSEXTZT(IZ),IZ=1,NZ)

External source locations defined for axial material mesh.

(Source Present=1, Source Not Present=0)

106

et

ape

ian-Z

ne-

full

axial

etries.

ed on

the

ted for

mesh

X,NY)

. For

undle

nds to

This

ndary

core

in the

ctions

IV.2.a. Geometry Input

To simplify the input of geometric information, NESTLE utilizes a highly flexible and y

automated input approach. The required input is all provided on Unit 2 => “GEOM”. Core sh

(NSHAP) can be either Cartesian-Z or Hexagonal-Z. Core symmetries (IDRUN) for Cartes

include the following: full core, half core (3 to 9 o’clock), quarter core (3 to 6 o’clock), and o

dimensional axial core. Core symmetries (IDRUN) for Hexagonal-Z include the following:

core, one-third core (5 to 9 o’clock), one-sixth core (5 to 7 o’clock), and one-dimensional

core. Figure (8) shows examples of radial material geometry figures for each of these geom

If the number of axial mesh points is set to one and reflective boundary conditions are utiliz

z-up and z-down surfaces, a two-dimensional model results.

The number of homogenous material region mesh points input in conjunction with

above core shape and symmetry input is used to automatically determine the input expec

the geometry figures. For Cartesian-Z core shape the number of x (NX) and y (NY) material

dictates the Cartesian core layout and hence geometry figures. Figure (8) corresponds to (N

= (36,36), (36,18) and (18,18) for the full, half and quarter core problems, respectively

Hexagonal-Z core shape the number of radial rings of bundles surrounding the central b

(NHR) dictates the hexagonal core layout and hence geometry figures. Figure (8) correspo

NHR = 10. Note in both these figures the ‘0’ entries in certain material mesh locations.

indicates to the NESTLE code the edge of the geometry to be analyzed where bou

conditions will be applied.It is required that every material mesh that is created from the

core shape, symmetry and material mesh number input have a value input for all geometry

figure inputs.

The material mesh sizes are determined based upon input. For the Hexagonal-Z

shape, the bundle pitch (DELH) uniquely specifies radial mesh size. Variable mesh sizes

axial direction are allowed. For the Cartesian-Z core shape, variable mesh sizes in all dire

107

each

Y and

BX,

irection

k and

input

iables

d z-

core

r core

xterior

entrant

ly be

ial

are

gure

ndle

ed to

pe and

the

(i.e.x, y and z) are allowed. Rather than requiring mesh sizes in each direction to be input for

mesh, mesh sizes are input separately for each direction with the span (NSPACX, NSPAC

NSPACZ) and size (DDX, DDY and DDZ) of a fixed mesh size input over all spans (NSU

NSUBY and NSUBZ).

To facilitate mesh refinementfor Cartesian-Z core shape only, from every initial

homogenous material mesh, refined numerical solution meshes can be generated in each d

(i.e. x, y and z) each with the same properties as the original material mesh. Feedbac

depletion effects will than be applied to the refined numerical solution mesh. This simplifies

and is very convenient when running fine-mesh benchmarks using the FDM. The input var

NMULXY and NMULZ provide the required information to complete the mesh refinement.

Boundary conditions are specified in terms of radial exterior, radial interior, z-up an

down boundaries. The radial exterior and interior boundaries locations depend upon the

shape and symmetry specified. For example, with Cartesian-Z core shape and quarte

symmetry the interior boundaries correspond to the north and west surfaces and the e

boundaries correspond to the south and east surfaces. Reflective, zero current, non-re

current and cyclic boundary conditions are treated. The cyclic boundary condition may on

used on the radial interior boundaries.

Cross-section (e.g. fuel) colors are input via geometry figures for each unique rad

configuration (NCOL2DT) as shown in Figure (8). These radial material geometry figures

than assigned to the axial material mesh via an axial mask (NCOLZT). A similar geometry fi

input style is used for all input quantities that are spatially dependent.

The final point to note with regard to geometry input concerns NESTLE’s usage of bu

pitch (i.e.BPITCHX and BPITCHY) for Cartesian-Z core shape. These variables are only us

determine the fuel bundle boundaries used in the control of output edits. Given the core sha

symmetry, mesh size and layout, and bundle pitch, NESTLE will automatically determine

108

n C-E

0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0

bundle boundaries, including the capability to recognize off-set bundles as encountered i

cores and quarter and half assemblies when symmetry does not correspond to full core.

0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 1 7 7 7 7 7 7 1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 1 1 1 1 1 7 7 7 7 7 7 1 1 1 1 1 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 1 7 7 7 7 9 9 3 3 9 9 7 7 7 7 1 2 2 2 2 0 0 0 0 0 0 0 0 0 2 2 2 2 1 1 1 7 7 7 7 9 9 3 3 9 9 7 7 7 7 1 1 1 2 2 2 2 0 0 0 0 0 0 0 2 2 2 2 1 7 7 8 8 5 5 3 3 4 4 3 3 5 5 8 8 7 7 1 2 2 2 2 0 0 0 0 0 2 2 2 2 1 1 1 7 7 8 8 5 5 3 3 4 4 3 3 5 5 8 8 7 7 1 1 1 2 2 2 2 0 0 0 2 2 2 2 1 7 7 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 7 7 1 2 2 2 2 0 0 0 2 2 1 1 1 7 7 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 7 7 1 1 1 2 2 0 0 0 2 2 1 7 7 8 8 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 8 8 7 7 1 2 2 0 2 2 2 2 1 7 7 8 8 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 8 8 7 7 1 2 2 2 2 2 2 2 1 7 7 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 7 7 1 2 2 2 2 2 1 1 1 7 7 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 7 7 1 1 1 2 2 2 1 7 7 9 9 3 3 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 3 3 9 9 7 7 1 2 2 2 1 7 7 9 9 3 3 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 3 3 9 9 7 7 1 2 2 2 1 7 7 3 3 4 4 3 3 6 6 3 3 6 6 3 3 6 6 3 3 6 6 3 3 4 4 3 3 7 7 1 2 2 2 1 7 7 3 3 4 4 3 3 6 6 3 3 6 6 3 3 6 6 3 3 6 6 3 3 4 4 3 3 7 7 1 2 2 2 1 7 7 9 9 3 3 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 3 3 9 9 7 7 1 2 2 2 1 7 7 9 9 3 3 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 3 3 9 9 7 7 1 2 2 2 1 1 1 7 7 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 7 7 1 1 1 2 2 2 2 2 1 7 7 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 7 7 1 2 2 2 2 2 2 2 1 7 7 8 8 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 8 8 7 7 1 2 2 2 0 0 2 2 1 7 7 8 8 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 8 8 7 7 1 2 2 0 0 0 2 2 1 1 1 7 7 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 7 7 1 1 1 2 2 0 0 0 2 2 2 2 1 7 7 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 7 7 1 2 2 2 2 0 0 0 2 2 2 2 1 1 1 7 7 8 8 5 5 3 3 4 4 3 3 5 5 8 8 7 7 1 1 1 2 2 2 2 0 0 0 0 0 2 2 2 2 1 7 7 8 8 5 5 3 3 4 4 3 3 5 5 8 8 7 7 1 2 2 2 2 0 0 0 0 0 0 0 2 2 2 2 1 1 1 7 7 7 7 9 9 3 3 9 9 7 7 7 7 1 1 1 2 2 2 2 0 0 0 0 0 0 0 0 0 2 2 2 2 1 7 7 7 7 9 9 3 3 9 9 7 7 7 7 1 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 1 1 1 1 1 7 7 7 7 7 7 1 1 1 1 1 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 1 7 7 7 7 7 7 1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0

Cartesian-Z Full Core

Figure 8: Radial material geometry figures for different core geometries.

109

2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0

2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0

2 2 1 7 7 3 3 4 4 3 3 6 6 3 3 6 6 3 3 6 6 3 3 6 6 3 3 4 4 3 3 7 7 1 2 2 2 1 7 7 9 9 3 3 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 3 3 9 9 7 7 1 2 2 2 1 7 7 9 9 3 3 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 3 3 9 9 7 7 1 2 2 2 1 1 1 7 7 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 7 7 1 1 1 2 2 2 2 2 1 7 7 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 7 7 1 2 2 2 2 2 2 2 1 7 7 8 8 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 8 8 7 7 1 2 2 2 0 0 2 2 1 7 7 8 8 5 5 3 3 5 5 3 3 6 6 3 3 5 5 3 3 5 5 8 8 7 7 1 2 2 0 0 0 2 2 1 1 1 7 7 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 7 7 1 1 1 2 2 0 0 0 2 2 2 2 1 7 7 3 3 5 5 3 3 5 5 3 3 5 5 3 3 5 5 3 3 7 7 1 2 2 2 2 0 0 0 2 2 2 2 1 1 1 7 7 8 8 5 5 3 3 4 4 3 3 5 5 8 8 7 7 1 1 1 2 2 2 2 0 0 0 0 0 2 2 2 2 1 7 7 8 8 5 5 3 3 4 4 3 3 5 5 8 8 7 7 1 2 2 2 2 0 0 0 0 0 0 0 2 2 2 2 1 1 1 7 7 7 7 9 9 3 3 9 9 7 7 7 7 1 1 1 2 2 2 2 0 0 0 0 0 0 0 0 0 2 2 2 2 1 7 7 7 7 9 9 3 3 9 9 7 7 7 7 1 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 1 1 1 1 1 7 7 7 7 7 7 1 1 1 1 1 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 1 7 7 7 7 7 7 1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0

C a r t e s i a n - Z H a l f C o re

3 6 6 3 3 6 6 3 3 4 4 3 3 7 7 1 2 6 3 3 5 5 3 3 5 5 3 3 9 9 7 7 1 2 6 3 3 5 5 3 3 5 5 3 3 9 9 7 7 1 2 3 5 5 3 3 5 5 3 3 5 5 7 7 1 1 1 2 3 5 5 3 3 5 5 3 3 5 5 7 7 1 2 2 2 6 3 3 5 5 3 3 5 5 8 8 7 7 1 2 2 2 6 3 3 5 5 3 3 5 5 8 8 7 7 1 2 2 0 3 5 5 3 3 5 5 3 3 7 7 1 1 1 2 2 0 3 5 5 3 3 5 5 3 3 7 7 1 2 2 2 2 0 4 3 3 5 5 8 8 7 7 1 1 1 2 2 2 2 0 4 3 3 5 5 8 8 7 7 1 2 2 2 2 0 0 0 3 9 9 7 7 7 7 1 1 1 2 2 2 2 0 0 0 3 9 9 7 7 7 7 1 2 2 2 2 0 0 0 0 0 7 7 7 1 1 1 1 1 2 2 2 2 0 0 0 0 0 7 7 7 1 2 2 2 2 2 2 0 0 0 0 0 0 0 1 1 1 1 2 2 2 2 2 2 0 0 0 0 0 0 0 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0

C a r t e s i a n - Z Q u a r t e r C o re

Figure 8 (cont): Radial material geometry figures for different core geometries.

110

0 0 0 0 5 5 5 0 0 0 0 0 5 5 5 5 5 5 5 5 5 5 0 0 5 5 5 5 4 4 4 5 5 5 5 0 0 5 5 5 4 4 4 4 4 4 5 5 5 0 5 5 5 4 4 4 1 4 1 4 4 4 5 5 5 5 5 4 4 4 4 3 3 3 3 4 4 4 4 5 5 5 5 4 4 1 3 3 3 3 3 3 3 1 4 4 5 5 0 5 4 4 4 3 3 2 3 3 2 3 3 4 4 4 5 0 0 5 5 4 1 3 3 3 3 3 3 3 3 3 1 4 5 5 0 0 5 5 4 4 3 3 3 3 3 3 3 3 3 3 4 4 5 5 00 5 5 5 4 4 3 2 3 3 3 3 3 2 3 4 4 5 5 5 0 0 5 5 4 4 3 3 3 3 3 3 3 3 3 3 4 4 5 5 0 0 5 5 4 1 3 3 3 3 3 3 3 3 3 1 4 5 5 0 0 5 4 4 4 3 3 2 3 3 2 3 3 4 4 4 5 0 5 5 4 4 1 3 3 3 3 3 3 3 1 4 4 5 5 5 5 4 4 4 4 3 3 3 3 4 4 4 4 5 5 5 5 5 4 4 4 1 4 1 4 4 4 5 5 5 0 5 5 5 4 4 4 4 4 4 5 5 5 0 0 5 5 5 5 4 4 4 5 5 5 5 0 0 5 5 5 5 5 5 5 5 5 5 0 0 0 0 0 5 5 5 0 0 0 0

Hexagonal-Z Full Core

0 5 5 5 4 4 3 2 3 3 3 0 5 5 4 4 3 3 3 3 3 3 0 5 5 4 1 3 3 3 3 3 3 0 5 4 4 4 3 3 2 3 3 2 5 5 4 4 1 3 3 3 3 3 3 5 5 4 4 4 4 3 3 3 3 4 5 5 5 4 4 4 1 4 1 4 4 0 5 5 5 4 4 4 4 4 4 5 0 5 5 5 5 4 4 4 5 5 5 0 5 5 5 5 5 5 5 5 5 5 0 0 0 0 5 5 5 0 0 0 0

Hexagonal-Z Third Core

33 3

3 3 32 3 3 2

3 3 3 3 34 3 3 3 3 4

4 4 1 4 1 4 45 4 4 4 4 4 4 5

5 5 5 4 4 4 5 5 55 5 5 5 5 5 5 5 5 5

0 0 0 0 5 5 5 0 0 0 0

Hexagonal-Z Sixth Core

Figure 8 (cont): Radial material geometry figures for different core geometries andsymmetries

111

IV.3. Cross-Section Data File

(Unit 3 => “XSECT” & Unit 33 => “AXSCIN”)

NG

Total number of energy groups.

NGT

Number of thermal energy groups.

ICOLXY

Total number of material colors (i.e. cross section sets).

NBUMAX

Maximum of the number of burnup mask values input for the cross-sections.

IF(NXSEC.EQ. “Y”) THEN READ

NPREC

Number of delayed neutron precursor groups.

NDECAY

Number of decay heat precursor groups.

NTERMMACRO,NTERMMACRI

Number of cross-section coefficients input for macroscopic cross-sections

w/o and w/ rods in.

NTERMCSCATRO,NTERMCSCATRI

Number of cross-section coefficients input for macroscopic scattering kernels

w/o and w/ rods in.

NTERMFPRO,NTERMFPRI

Number of cross-section coefficients input for transient fission products

microscopic absorption cross-sections w/o and w/ rods in.

[In following, IXSxxx values imply the following:

112

Base = 1,

Linear-Coolant Density (g/cm3) = 2,

Quadratic-Coolant Density (g/cm3) = 3,

Linear-Coolant Temperature (oF) = 4,

Linear in Square Root-Effect. Fuel Temperature (oF) = 5,

Linear-Soluble Poison Number Density (1/b) = 6,

Quadratic-Soluble Poison Number Density (1/b) = 7),

Cubic-Soluble Poison Number Density (1/b) = 8]

IF(NTERMMACRO.GT.0) READ

(IXSMACRO(ITERM),ITERM=1,NTERMMACRO)

Cross-section coefficients to be input for macroscopic cross-sections

w/o rods in.

IF(NTERMMACRI.GT.0) READ

(IXSMACRI(ITERM),ITERM=1,NTERMMACRI)

Cross-section coefficients to be input for macroscopic cross-sections

w/ rods in.

IF(NTERMCSCATRO.GT.0) READ

(IXSCSCATRO(ITERM),ITERM=1,NTERMCSCATRO)

Cross-section coefficients to be input for macroscopic scattering kernels

w/o rods in.

IF(NTERMCSCATRI.GT.0) READ

(IXSCSCATRI(ITERM),ITERM=1,NTERMCSCATRI)

Cross-section coefficients to be input for macroscopic scattering kernels

w/ rods in.

113

.

IF(NTERMFPRO.GT.0) READ

(IXSFPRO(ITERM),ITERM=1,NTERMFPRO)

Cross-section coefficients to be input for transient fission products

microscopic absorption cross-sections w/o rods in.

IF(NTERMFPRI.GT.0) READ

(IXSFPRI(ITERM),ITERM=1,NTERMFPRI)

Cross-section coefficients to be input for transient fission products

microscopic absorption cross-sections w/ rods in.

IMICRO (A5)

Microscopic cross-section option (“Y”/”N”).

IF(IMICRO.EQ. “Y”) THEN READ

INUMDEN (A5)

Node-wise isotopic number densities available (“Y” / “N”).

IF (INUMDEN.EQ. “Y”) THEN READ

FNUMDEN (A40)

Name of the file containing the node-wise initial isotopic number densities

NTERMMICRO

Number of cross-section coefficients input for microscopic cross-sections

w/o rods in.

NTERMMICRI

Number of cross-section coefficients input for microscopic cross-sections

w/ rods in.

IF(NTERMMICRO.GT.0) READ

(IXSMICRO(ITERM),ITERM=1,NTERMMICRO)

114

functionutronnctioned

cursor

utron

Cross-section coefficients to be input for microscopic cross-sections

w/o rods in.

IF(NTERMMICRI.GT.0) READ

(IXSMICRI(ITERM),ITERM=1,NTERMMICRI)

Cross-section coefficients to be input for microscopic cross-sections

w/ rods in.

ENDIF IMICRO

ENDIF NXSEC

AXSEC (A5)

Cross-section colors are read from different input files (“Y”/ “N”).


RLI,RLX,RLPM

Decay constants (I-135, Xe-135, Pm-149) (1/sec)

The next data, the delayed neutron precursor decay constants, can be entered as aof both isotope and delayed neutron precursor group or as a function of delayed neprecursor group only. To enter the delayed neutron precursor decay constants as a fuof both isotope and precursor group, write “ALAMDAMI” in a line preceding the delayneutron precursor decay constants input (see example below).

Delayed neutron precursor decay constants as a function of delayed neutron pregroup only:

(ALAMDA(I),I=1,NPREC)

Delayed neutron precursor decay constants (1/sec).

Delayed neutron precursor decay constants as a function of both delayed neprecursor group and isotope:

ALAMDAMI (mandatory card of format A8)

DO ISOT=1,8...For each isotope

(ALAMDAMIC(ISOT,IPREC), IPREC=1,NPREC)

115

42.

Isotopic dependent delayed neutron precursor decay constants (1/sec).

ENDDO ISOT

Note: ISOT value corresponds to the following isotope:

1=U-234, 2=U-235, 3=U-236, 4=U-238, 5=Pu-239, 6=Pu-240, 7=Pu-241, 8=Pu-2

DO IG=1,NG...For each energy group

(XHIDMI(IG,IPREC),IPREC=1,NPREC)

Delayed neutron fission spectrum.

ENDDO IG ...Energy loop

(ZETA(I),I=1,NDECAY)

Decay heat precursors decay rates (1/sec).

IF(IMICRO.EQ. “Y”) THEN

DO ISOT=1,NISOT... For each isotope (U-234, U-235, U-236, U-238,

Pu-239, Pu-240, Pu-241, Pu-242)

(BETAMI(ISOT,IPREC),IPREC=1,NPREC)

Delayed neutron yields for isotope.

ENDDO ISOT...Isotope loop


(XHIPMI(ISOT,IG),ISOT=1,9)

Prompt neutron spectra for each isotope ISOT

(1=U-234, 2=U-235, 3=U-236, 4=U-238,

5=Pu-239, 6=Pu-240, 7=Pu-241, 8=Pu-242, 9=Am-241)

ENDDO IG...Energy loop


Pu-239, Pu-240, Pu-241, Pu-242)

116

).

Am-

(ALPHAI(ISOT,IDECH),IDECH=1,NDECAY)

Decay heat group yields for isotope.



Pu-239, Pu-240, Pu-241, Pu-242)

GINMI(ISOT),GXNMI(ISOT),GPNMI(ISOT)

Transient fission product yields (I135, Xe135, Pm149) for isotope.



RNUU4(IG),RNUU5(IG),RNUU6(IG)

Nu values for isotopes (U-234, U-235, U-236).

RNUU8(IG),RNUP9(IG),RNUP0(IG),RNUP1(IG),RNUP2(IG),RNUA1(IG)

Nu values for isotopes (U-238, Pu-239, Pu-240, Pu-241, Pu-242, Am-241

ENDDO IG...Energy group loop

RKU34, RKU35, RKU36

Kappa values (Mev) for isotopes (U-234, U-235, U-236).

RKU38, RKPU39, RKPU40, RKPU41, RKPU42, RKAM41

Kappa values (Mev) for isotopes (U-238, Pu-239, Pu-240, Pu-241, Pu-242,

241).

GLFP1,GLFP2

Lumped fission products 1 and 2 yields.

ENDIF IMICRO

ENDIF NXSEC

NFUEXY

Number of non-fuel material colors.

117

L

Cross section table sets input follow.

Units: Macroscopic cross section (1/cm), microscopic cross sections (barns),

burnup (MWD/MTM)

DO ICOL=1,ICOLXY ...For each material color

Note ‘I’ used below is determined from ICOL and denotes internal storage index of ICOth

material color entry.

IF(AXSEC.EQ. “Y”) READ

AXSCIN (A40)

Name of file containing cross-sections for a specific color.

Following input on file name “XSECT” if AXSEC = “N” and on file name “AXSCIN” if AXSEC =

“Y”.

Header Line (A80)

NESTLE does not utilize-to help with data file preparation.

Header Line (A80)

NESTLE does not utilize-to help with data file preparation.

AFUEL (A5)

Material cross-sections now to be entered corresponds to a fuel

(i.e. fissionable) (“Y”/”N”).

NCOLOR(I)

Material color number of cross-sections to be entered.

118

n

IF(AFUEL.EQ. “Y”) READ

NMAX(I)

Number of burnup steps for fuel material color that cross sections

provided at. [For non-fuels set equal to 1.]

WTFRRO(I), WTFRRI(I), FUFR(I)

Volume fractions for coolant w/o rods, coolant w/ rods, and fuel.

RHOWREF(I), TCOLREF(I), TFREF(I), REFB(I)

Reference conditions cross-sections evaluated at: coolant density (lbm/ft3),

coolant temperature (oF), fuel effective temperature (oF), and reference soluble boro

concentration (ppm).

(BUBOS(N,I),N=1,NMAX(I))

Burnup values the cross-sections are input at.

DO ITERM=1,NTERMMACRO ...For each cross-section coefficient

DO N=1,NMAX(I) ...For each burnup step


(XSECRO(N,I,IRX,IG,ITERM),IRX=1,NRXMAX)

Macroscopic cross sections coefficients w/o rods:

IMICRO= “Y”: transport, absorption.

IMICRO= “N”: transport, absorption, nu-fission, kappa-fission, nu.


ENDDO N...Burnup step loop

ENDDO ITERM ...Cross-section coefficient loop

DO ITERM=1,NTERMCSCATRO ...For each cross-section coefficient



119

(XSECSCRO(N,I,IG,IGP,ITERM),IGP=1,NG-1)

Scattering transfer macroscopic cross sections coefficients w/o rods;

from group igp (or igp+1 when upscatter) to group ig.

(Note within group (ig to ig) scattering is not entered, hence the

non-square matrix.





DO ITERM=1,NTERMCSCATRI ...For each cross-section coefficient

DO N=1,NMAX(I). ..For each burnup step


(XSECRI(N,I,IRX,IG,ITERM),IRX=1,NRXMAX)

Macroscopic cross sections coefficients w/ rods:

IMICRO= “Y”: transport, absorption.

IMICRO= “N”: transport, absorption, nu-fission, kappa-fission, nu.




DO ITERM=1,NTERMMACRI ...For each cross-section coefficient



(XSECSCRI(N,I,IG,IGP,ITERM),IGP=1,NG-1)

Scattering transfer macroscopic cross sections coefficients

w/ rods; from group igp (or igp+1 when upscatter) to group ig.

120

(Note within group (ig to ig) scattering is not entered, hence the

non-square matrix.




ENDIF NXSEC



B2COL(N,I,IG)

Buckling (1/cm2) Normally input as 0’s for 3D geometries.



IF (AFUEL.EQ. “Y”) THEN READ


(XHIPN(N,I,IG),IG=1,NG)

Prompt fission neutron spectrum


ENDIF AFUEL

DO N=1,NMAX(I) ...For each burnup step.


(ADFS(I,N,IADF,IG),IADF=1,NADFS)

Assembly Discontinuity Factors w/o rods:

(Cartesian (NADFS=6): N, E, S,W, UP, DOWN)

(Hexagonal (NADFS=8): NE, E, SE, SW, W, NW, UP, DOWN)


121

tive

unit





(ADFSRD(I,N,IADF,IG),IADF=1,NADFS)

Assembly Discontinuity Factors w/ rods:

(Cartesian (NADFS=6): N, E, S, W, UP, DOWN)

(Hexagonal (NADFS=8): NE, E, SE, SW, W, NW, UP, DOWN)



IF(IMICRO.EQ. “N”) THEN READ


(VELOCN(N,I,IG),IG=1,NG)

Neutron velocity (cm/sec).


ELSEIF (IMICRO.EQ. “Y”) THEN READ


(VELOCN(N,I,IG),IG=1,NG), (BFACT(N,I,IPREC),IPREC=1,NPREC)

Neutron velocity (cm/sec), the delayed neutron importance.

The delayed neutron importance is defined as the ratio of the effec

delayed neutron yields over the delayed neutron yields from lattice

assembly depletion: ,

where


βeff β BFACT×=

β βisot νΣ f g

isotφgg 1=

G

∑isot=1

# fiss. isot.

∑

νΣ f g

isotφgg 1=

G

∑isot=1

# fiss. isot.

∑

⁄=

122

ENDIF ... IMICRO


DO ITERM=1,NTERMFPRO ...For each cross-section coefficient



(XFPNRO(N,I,IG,IFP,ITERM),IFP=1,2)

Microscopic absorption cross-sections coefficients w/o

rods: (Sm149, Xe135)

ENDDO IG ...Energy group loop



DO ITERM=1,NTERMFPRI ...For each cross-section coefficient



(XFPNRI(N,I,IG,IFP,ITERM),IFP=1,2)

Microscopic absorption cross-sections coefficients

w/ rods: (Sm-149, Xe-135)

ENDDO IG ...Energy group loop



IF(IMICRO.EQ. “N”) THEN READ


GIN(N,I),GXN(N,I),GPN(N,I)

Transient fission products yields (I135, Xe135, Pm149).


123


(BETAN(N,I,IPREC),IPREC=1,NPREC)

Delayed neutron yields.



(ALPHAN(N,I,IDECH),IDECH=1,NDECAY)

Decay heat group yields.


ENDIF IMICRO

ENDIF AFUEL

ENDIF NXSEC



IF(IMICRO.EQ. “Y”) THEN READ

Number densities of isotopes in following order (nuclei/cm3x10-24):

(UPUDEN(I,IRX),IRX=1,3)

(U234, U235, U236)


U238, Pu239, Pu240, Pu241, Pu242)


(Lumped FP#1, Lumped FP#2)


(Burnable Poison, Am-241)

DO N=2,NMAX(I) ... For each burnup step

Number densities of isotopes at other burnup steps (nuclei/cm3x10-24):

124

(INITND(I,N,IRX),IRX=1,4)

(U234, U235, U236, U238)


(Pu239, Pu240, Pu241, Pu242)


(Burnable Poison, Am-241)

ENDDO N

DO ITERM=1,NTERMMICRO ...For each cross-section coefficient



(UCHAINRO(N,I,IRX,IG,ITERM),IRX=1,6)


(UCHAINRO(N,I,IRX,IG,ITERM),IRX=13,16 and IRX=20,21)

Microscopic cross sections coefficients w/o rods:

(absorption, fission) for each isotope:

1st line (U-234, U-235, U-236)

2nd line (U-238, Pu-239, Pu-240)

3rd line (Pu-241, Pu-242, Am-241)


Microscopic absorption cross sections coefficients w/o

rods: Lumped FP#1, Lumped FP#2

UCHAINRO(N,I,19,IG,ITERM)

Microscopic absorption cross sections coefficients w/o

rods: Burnable Poison


125



DO ITERM=1,NTERMMICRI ...For each cross-section coefficient



(UCHAINRI(N,I,IRX,IG,ITERM),IRX=1,6)


(UCHAINRI(N,I,IRX,IG,ITERM),IRX=13,16 and IRX=20,21)

Microscopic cross sections coefficients w/ rods:

(absorption, fission) for each isotope:

1st line (U-234, U-235, U-236)

2nd line (U-238, Pu-239, Pu-240)

3rd line (Pu-241, Pu-242, Am-241)


Microscopic absorption cross sections coefficients w/

rods: Lumped FP#1, Lumped FP#2

UCHAINRI(N,I,19,IG,ITERM)

Microscopic absorption cross sections coefficients w/

rods: Burnable Poison




ENDIF IMICRO

ENDIF AFUEL

ENDIF NXSEC

126

ure.

End of input on file name “AXSCIN” if AXSEC = “Y” & returning to file name “XSECT”.

ENDDO I...Material color loop

Remaining input on file name “XSECT”.

Blank Line


NFRODS

Number of fuel rods per bundle (assembly)

FUELRAD

Fuel rod radius (in)

WC,WP

Fuel temperature rise and average fuel temperature Doppler effective fuel

temperature weight factors.

NPOLTAF

Order of polynomial fitting average fuel temperature versus linear power density.

(COF_TAF(I),I=0,NPOLTAF-1)

Fitting coefficients for average fuel temperature (oF) versus

linear power density (kw/ft).

NPOLHFF

Order of polynomial fitting effective heat transfer coefficient versus fuel temperat

(COF_HFF(I),I=0,NPOLHFF-1)

Fitting coefficients for effective heat transfer coefficient (kw/ft2-oF) versus

fuel temperature (oF).

NPOLTSF

Order of polynomial fitting surface fuel temperature versus linear power density.

(COF_TSF(I),I=0,NPOLTSF-1)

127

Fitting coefficients for surface fuel temperature (oF) versus

linear power density (kw/ft).

MOLW_COL

Molecular weight of coolant.

ATMW_SOLP

Atomic weight of soluble poison.

ABUN_SOLP

a/o of absorbing isotope of soluble poison (e.g.B-10).

PIN

Coolant pressure (psia). Only use within NESTLE is for editing.

TCOLSAT,UCOLG,RHOG

Coolant saturation temperature (oF), saturated vapor internal energy (BTU/lbm)

and saturated vapor density (lbm/ft3).

DEPF

Fraction of fission energy deposited directly within fuel.

TCOLIN

Coolant inlet temperature (oF). Only used for non-transient problems.

TCOLMIN,TCOLMAX

Coolant inlet temperature (oF) search range for criticality or power level search.

GMASS

Coolant mass velocity (lb/hr-ft2). Only used for non-transient problems.

GMASS > 0 => Inlet mass velocity entering bottom fuel node, implying

cross-sectional flow rate = fuel node wet fraction X node x-y plane area.

GMASS < 0 => Inlet mass velocity below bottom axial node, implying

cross-sectional flow area = node x-y plane area.

128

NPOLUCOL

Order of polynomial fitting coolant liquid internal energy versus temperature.

(COF_UCOL(I),I=0,NPOLUCOL-1)

Fitting coefficients for coolant internal energy (BTU/lbm) versus temperature (oF).

NPOLTCOL

Order of polynomial fitting coolant liquid temperature versus internal energy.

(COF_TCOL(I),I=0,NPOLTCOL-1)

Fitting coefficients for coolant liquid temperature (oF) versus internal energy

(BTU/lbm).

NPOLRHOWC

Order of polynomial fitting coolant liquid density versus internal energy.

(COF_RHOWC(I),I=0,NPOLRHOWC-1)

Fitting coefficients for coolant liquid density (lbm/ft3) versus internal energy

(BTU/lbm).

NPOLCPF

Order of polynomial fitting fuel specific heat versus temperature.

(COF_CPF(I),I=0,NPOLCPF-1)

Fitting coefficients for fuel specific heat (BTU/lbm-oF) versus temperature (oF).

RHOWTF

Fuel density (lbm/ft3).

RATIOHMFUEL

Ratio of heavy metal density to fuel material density.

(Used to convert MTM to MT fuel material)

QQT

Core power density at rated full power (kw/liter).

129

ENDIF NXSEC

(RT(IG),IG=1,NG)

Initial estimates of relative group flux values.

(SEXT(IG),IG=1,NG)

External source strengths (1/cm3-sec).

130

IV.4. Kinetic Data File

(Unit 4 => “KINET”)


NPERT

Number of times that time-dependent input parameters are provided.

NPRINT

Number of times that time-dependent output is to be printed.

NTSPN

Total number of time spans, each utilizing a constant time step size over the

time span.

ENDDO ITRAN

IF (IXE.EQ.4 OR IXE.EQ.5) THEN READ

NFP

Number of time-steps for transient fission product case.

ENDIF IXE

IF(ITRAN.EQ. “Y”)THEN READ

DO IT=1,NPERT...For each transient time perturbation

TIMETR(IT),PMT(IT),TINT(IT),GMINT(IT)

Time (sec), soluble poison concentration (PPM), coolant inlet temperature (oF),

coolant mass flow rate (lb/ft2-sec) (4 values per line)

TIMETR(IT),(ZPT(IBK,IT), IBK=1,NBACU)

Time (sec), control banks steps withdrawn (1+NBACU values per line)

[Note TIMETR(1) must equal 0.]

ENDDO...Transient time perturbation loop

(TIMEPR(IT),IT=1,NPRINT)

131

Time (sec) output is to be printed.

(TIMESP(IT),DELT(IT),IT=1,NTSPN)

Use time step size DELT(IT) (sec) from time TIMESP(IT-1) to TIMESP(IT) (sec)

where TIMESP(0) = 0.

ENDDO ITRAN

IF (IXE.EQ.4 ORIXE.EQ. 5) THEN READ

DO IFP=1,NFP...For each fission product transient time perturbation

TIMEFP(IFP),PRFP(IFP),PMFP(IFP),TINFP(IFP)

Time (hrs), core power (relative), soluble poison (PPM),

coolant inlet temperature (oF).

[Note that if IBOR=”Y” the parameter being searched upon will overwrite the

input value of this parameter.]

TIMEFP(IFP),(ZBFP(IBK,IFP),IBK=1,NBACU)

Time (hrs), Control banks steps withdrawn (1+NBACU values per line)

[Note that if IBOR=”Y” the parameter being searched upon will overwrite the

input value of this parameter.]

ENDDO...Fission product transient time perturbation loop

ENDIF IXE

132

ion

IV.5. Solution Method Control Data File

(Unit 5 => “PERFM”)

N_THRMITR

Maximum number of thermal scattering iterations.

KITR

Maximum number of outer iterations.

EPSK

Outer iteration stopping criteria on eigenvalue

EPSOT

Outer iteration stopping criteria on L2 norm of relative residual of the outer iterative

equation

EPSRESID

Outer iteration stopping criteria on L2 norm of relative residual of the diffusion equat

EPSINF

Outer iteration stopping criteria on Linf norm of relative fission source error (Chebyshev

acceleration) or outer iterative equation (Weilandt Shift)

EPSDET

Inner iteration convergence criteria on L2 relative error reduction

ACONV (A5)

Stopping criteria required to be satisfied after a coefficient matrix update (“Y”/ “N”)

AMETHOD (A5)

Solution method (“FDM”/ “NEM”)

IF (AMETHOD.EQ. “NEM”) THEN READ

NNEM

Maximum number of iterations between NEM coupling coefficients’ updates

133

T-H Off: Number of outer iterations/update

T-H On: Number of T-H calls/update

EPSNEM

Outer iteration L2 relative error reduction criteria used for determining when NEM

coupling coefficients’ updates are to be completed

ENDIF AMETHOD

IF (IBURN.EQ. “Y”) THEN READ

ADEPL

Depletion method: Predictor or Predictor-Corrector (“PRED”/”CORR”)

IF(ADEPL.EQ. “CORR”) THEN READ

NDELPC

Frequency of Corrector updates for Predictor-Corrector method

T-H Off: Number of outer iteration/update

T-H On: Number of T-H calls/update

ENDIF ADEPL

ENDIF IBURN

AOUTER (A5)

Chebyshev acceleration on (“CHEB”)

Weilandt Shift acceleration on (“WEIL”)

IUPCHE,EPREDRESID

Upper limit on Chebyshev order or Weilandt Shift cycle, reduction of L2 relative

residual error for the diffusion equation from starting new Chebyshev order or

Weilandt Shift cycle

IF(AOUTER.EQ. “WEIL”) THEN READ

IF(ITRAN.EQ. “Y”.OR.IADJ.EQ. “Y”) THEN READ

134

WEILANDTEI,WEILANDTFS

Forward eigenvalue or steady-state and transient or adjoint upper limit on

Weilandt shift with respect tok-1

ELSE READ

WEILANDTEI

Forward eigenvalue or steady-state upper limit on Weilandt shift with respect tok-1

ENDIF ITRAN

OMEGAEXT

Stationary extrapolation parameter for the flux

ENDIF AOUTER

AISC (A5)

Steady-state scaling factor acceleration on (“Y”/”N”)

IF (AISC.EQ. “Y”) THEN READ

ISC

Outer iteration when steady-state scaling factor acceleration first applied.

ASPSH (A5)

Steady-state spectral shift correction on (“Y”/”N”)

ENDIF AISC


IONE

Number of outer iterations per time step control logic.

Convergence criteria controled=1

Fixed number=2

IF (IONE.EQ.2) THEN READ

IOK

135

Number of outer iterations per time step.

ENDIF IONE

ANFDBKIMP (A5)

Implicit (versus explicit) thermal-hydraulics to be used for transients (“Y”/”N”)

AISCTR (A5)

Transient scaling factor acceleration on (“Y”/ “N”)

IF (AISCTR.EQ. “Y”) THEN READ

ISCTR

Outer iteration when transient scaling factor acceleration first applied.

ASPSHTR

Transient spectral shift correction on (“Y”/”N”)

ENDIF AISCTR

ENDIF ITRAN

136

be

IV.6. Initial Exposure Data File

(Unit 79=> “FINITEXP”)

Line 1: TITLE

Line 2:Blank or comment line

Line 3: NNODE

Number of nodes whose axially-dependent exposure values are provided.

To minimize the length of the input file, input for fresh nodes (exposure=0.0) can

skipped.



DO IY=1, NY ... for each y mesh

(NODEID(IY,IX),IX=NXSTART(IY),NXEND(IY),NXSKIP)

Node ID number as a function of x- and y-mesh number.

See Section IV.2 for the meaning of NY, NXSTART(IY), NXEND(IY), and

NXSKIP.

ENDDO

Line 6+NY: Blank or comment line

DO I=1,NNODE ... for each node

INODE

Node ID number

(EXP(INODE,IZ),IZ=IZCOLS,IZCOLE)

Node exposure (MWd/MTU).

See Section IV.2 for the meaning of IZCOLS and IZCOLE.

ENDDO

137

gonal

(see

The following input data are needed for hexagonal assembly calculation only:

INITSUREXP (A5)

Option to provide the initial surface exposure values on the six surfaces of each hexa

assembly (“Y”/ “N”).

If the initial surface exposure input is not available, then the burnup gradient treatment

Section II.4.b) will not be completed.

The following is the format for the initial surface exposure input (if desired).

IF (INITSUREXP.EQ. “Y”) THEN READ

DO I=1,NNODE

((BUSURF(INODE,IZ),IZ=IZCOLS,IZCOLE),ISURF=1,6)

Surface exposure value (MWd/MTU).

ISURF corresponds to the following hexagonal surface:

West =1 , East =2, Northwest=3, Southeast=4, Northeast=5, Northwest=6


ENDDO

ENDIF

138

139

IV.7. Initial Isotopic Number Densities Data File

(Unit 77=> “FNUMDEN”)

Line 1: TITLE


Line 3: NNODE

Number of nodes whose axially-dependent exposure values are provided.

To minimize the length of the input file, input for fresh nodes (exposure=0.0) can be

skipped.



DO IY=1, NY ... for each y mesh

(NODEID(IY,IX),IX=NXSTART(IY),NXEND(IY),NXSKIP)

Node ID number as a function of x- and y-mesh number.

See Section IV.2 for the meaning of NY, NXSTART(IY), NXEND(IY), and

NXSKIP.

ENDDO

Line 6+NY: Blank or comment line

DO I=1,NNODE... for each node

INODE

Node ID number

DO IZ=IZCOLS,IZCOLE ... for each z mesh in the fuel region

(ANUMDEN(INODE,IZ,ISOT),ISOT=1,13)

Isotopic number densities.

ISOT=1 -> U234, ISOT=2 -> U235, ISOT=3 -> U236, ISOT=4 -> U238,

ISOT=5 -> Pu239, ISOT=6 -> Pu240, ISOT=7 -> Pu241, ISOT=8 -> Pu242,

ISOT=9 -> LFP1, ISOT=10 -> LFP2, ISOT=11 -> DBP, ISOT=12 -> Am241,

ISOT=13 -> Sm


ENDDO

ed.

find

IV.8. Pin-Power Data File

(Unit 81=> “INPPIN”)

Note: pin power reconstruction option is available for Hexagonal geometry only.

IF (PPR.EQ. “Y”.and. FPIN.EQ. “Y”) THEN READ

NINP

Number of corner point & form factor data set to be input

NPIN

Number of form factor data provided within a hexagonal fuel assembly.

DO IPIN=1,NPIN ... For each pin

PINLOCX(IPIN), PINLOCY(IPIN)

The x- and y-coordinate of the center of the fuel pin where the form factor is provid

ENDDO IPIN

DO IDATA=1,NINP ...For each node

NCOLTEMP

Material number color of pin-power data to be entered. NESTLE code will then

the index I where NCOLOR(I)1 = NCOLTEMP

DO N=1,NMAX(I) 2...For each burnup

DO IG=1,NG...For each neutron energy group

(CPADF(I,N,IADF,IG),IADF=1,6)

Corner Point Assembly Discontinuity Factors w/o rods

ENDDO IG

1. See Section IV.32. See Section IV.3

1

3

26

5

4Location of Corner Point Discontinuity Factor

140

ENDDO N

IF (NXSEC1.EQ. “Y”) THEN

DO N=1,NMAX(I) ... For each burnup step

(CPADFRD(I,N,IADF,IG),IADF=1,6)

Corner Point Assembly Discontinuity Factors with rods

ENDDO N

ENDIF NXSEC


(FF(I,N,IPIN),IPIN=1,NPIN)

Pin-wise form factor without rods.

ENDDO N

IF (NXSEC.EQ. “Y”) then

DO N=1,NMAX(I)

(FFRD(I,N,IPIN),IPIN=1,NPIN)

Pin-wise form factor with rods.

ENDDO N

ENDIF NXSEC

ENDDO IDATA

ENDIF PPR

1. See Section IV.1

141

that

sed

ures

g to

king.

ption

the

d with

nsient

of the

ction, a

e the

here

each

long

cific

he

V. Programmer’s Guide

This section presents information about the organization and coding of NESTLE

should prove useful to individuals who wish to understand and modify the officially relea

version of NESTLE. Since NESTLE contains over 26,000 lines of FORTRAN, 132 proced

(i.e. subroutines, main program and function subprograms) and 69 fcb files (i.e. files containing

one or more named COMMON blocks referred to in INCLUDE statements), those wishin

understand the code structure should be prepared to commit significant time to this underta

V.1. Dependence Diagram

The dependence diagram for the NESTLE code is shown in Figure (9). With the exce

of the main program, in Figure (9) the subroutines are listed alphabetically indicating

subroutines called one-level deep. Four main control paths exist within the code, associate

the solution of the eigenvalue and steady-state external source, eigenvalue adjoint, tra

fission product, and transient neutronics problems. These main control paths utilize many

same procedures. Where different subroutines are required to perform nearly the same fun

suffix at the end of the subroutine name is utilized to distinguish the function. For exampl

suffixes ‘k’ (or ‘tr’ ), ‘ss’ and ‘ad’ refer to respectivelykinetic (or transient),steady-state, and

adjoint; and, the suffixes ‘c’ and ‘h’ refer to respectivelyCartesian andHexagonal geometries.

V.2. Summary of Procedures

As noted above, there are a total of 132 procedures incorporated into NESTLE. T

names and a brief description of their functions are given in Table (5). The header of

procedure contains a brief description via COMMENT lines. Most procedures have very

calling argument lists which are required to allow variables to have problem spe

dimensionality without recompilation. This point will be further explained below in t

142

ronic

code

s. To

that

able’s

bles’

dated

ilizing

ere

buted

ocks

s

In this

iring

for

if an

ear for

by a

he

subsection entitle Variables’ Storage.

V.3. Variables’ Definitions

Rather than define variables throughout the source code of NESTLE, an elect

dictionary has been created. The electronic dictionary consists of the FORTRAN

NESTLE.DICT and the associated data base containing variables’ names and definition

determine the definition of a variable, the user initiates execution of NESTLE.DICT, indicates

a definition is sought, and enters the variable name. NESTLE.DICT then outputs the vari

definition. This is all done in an interactive manner. Options also exist to add and delete varia

names and definitions. At the conclusion of the NESTLE.DICT execution the data base is up

reflecting any changes made, if so desired. Table (7) shows an interactive session ut

NESTLE.DICT. It is most conveniently used by opening a separate window wh

NESTLE.DICT will be controlled from. This code and the associated data base are distri

with the NESTLE code.

V.4. Variables’ Storage

Variables with dimensionality are stored either in a container array or in common bl

of fixed dimensionality. The container array (i.e. A Array) stores all variables whose dimension

are problem specific, such as number of energy groups and number of spatial nodes.

manner the dimensionality of variables can be specified at run-time without requ

recompilation for problems with different dimensionality. Also memory is only allocated

variables that will be employed for the specific problem as specified via input. For example,

eigenvalue problem is to be solved no memory space is allocated for variables that only app

a neutronics transient problem. The A array is specified (including its size indicated

PARAMETER statement) in named COMMON block ‘array’ stored in file ‘array.fcb’. T

143

INE

format

or.

b’

r (

MON

clude

N

ment

e fcb

de’

ming

the

d to

ction

E’ is

call

the

rent

starting address of all variables stored in the A array are determined in SUBROUT

POINTER. The variable names associated with the starting addresses all have the

‘LXXXN’ where ‘XXX’ is the name of the variable that the starting address is being given f

The named COMMON block ‘varlen’ stored in file ‘varlen.fcb’ and varlens in file ‘varlens.fc

contain all the pointer variables.

As noted above, COMMON blocks are utilized to store variables that are either scalai.e.

have no dimensionality) or have problem independent dimensionality. These named COM

blocks are incorporated into the source code via the ‘include’ statement using the format ‘in

yyy.fcb’ where ‘yyy.fcb’ is the file name of a file containing one or more named COMMO

blocks. A listing of the fcb files is presented in Table (6). The advantage of the ‘include’ state

is that modifications to a COMMON block need only be made in one place, the appropriat

file. The danger when using the UNIX ‘make’ feature is that not all procedures which ‘inclu

the altered fcb file will be recompiled since they have not been ‘touched’. A sound program

practice when altering fcb files is to ‘grep’ on *.f files the fcb file names altered and ‘touch’ all

procedures containing the altered fcb files.

V.5. Machine Specific Instructions

The only machine specific instruction that is utilized by NESTE is the clock call use

provide timing information that is output. Rather than have the machine specific clock instru

scattered throughout the source code, a function subroutine named ‘FUNCTION GTIM

called. Within this subroutine are the machine specific clock calls. Currently the clock

instructions for DEC ULTRIX, IBM AIX and SUN OS operating systems are included, with

clock calls not utilized commented out. In this manner NESTLE can be ported to diffe

platforms and only the file gtime.f needs to be modified.

144

, the

ted in

dial

a

-Z

when

h the

urface

n

dition

index,

de is

tside

the

tive

just

aries

and y

. The

l.

s by

V.6. Geometry Treatment

Since both Cartesian-Z and Hexagonal-Z geometry can be treated by NESTLE

geometry treatment needs to be fairly flexible. Most of the setup of the geometry is comple

SUBROUTINE GEOMETRY. For most variables the indices associated with the two ra

directions (IX,IY) are rolled into a single index (IXY) indirectly indexed vi

IXY=NDNUM(IX,IY). This greatly facilitates handling both Cartesian-Z and Hexagonal

geometries. In addition, no memory is wasted to store variables outside the solution domain

a ‘raggedy edge’ solution domain is employed. In addition to the nodes associated wit

solution domain, an extra layer of nodes is added radially about the core such that every s

associated with either an interior boundary (i.e. symmetry boundary) or exterior boundary has a

extra node located across the surface. This is done to facilitate the various boundary con

treatments. Since these extra nodes IXY index starts after the solution domain nodes IXY

they can be easily recognized by the value of IXY. In this manner whether a neighbor no

within the solution domain can be determined. If it is determined that the neighbor node is ou

the solution domain, than a vector NBC(IXY) contains the information required to apply

specified boundary condition. For cyclic boundary conditions, NBC(IXY) stores the nega

value of the IXY of the neighboring node associated with cyclic symmetry. The treatment

noted greatly simplifies code logic in regard to geometry.

The SUBROUTINE GEOMETRY also determines for Cartesian geometry the bound

of all fuel bundles (assemblies) by using the mesh layout and the bundle pitches in the x

directions. The capability to treat ‘offset’ bundles as in C-E cores has been incorporated

resulting information is utilized to determine bundle average attributes and in output contro

V.7. Installation

NESTLE code may be installed on most platforms without the need to edit makefile

145

the

the

s, it

ilation

using the included build script. Simply issue the command:

% build <platform>

where <platform> is the name of the Unix platform (e.g.hp, ibm, sun, etc.) that NESTLE is to be

installed on. The build script will set compilation options for the desired platform and call

included makefile to compile NESTLE. Thus, to install NESTLE on the IBM workstation,

command is:

% build ibm

Currently, the build script supports HP, IBM, DEC, and Sun Workstations. For other platform

may be necessary to edit the included generic makefile in order to set the proper comp

options. To execute NESTLE:

% nestle.exe<CNTL file name>

Note that any CNTL input file name may be used.

146

C GASCATH INITAL DEPLETEOLLAP PINPOWER OUTPUTSS

CRT

147

MAIN|

INPDATA GTIME POINTER FILE_NUMDEN GEOMETRY INPEDIT CONVER GASCATDEPLETES XSFDBK STARTER STEADYN PINLOC XSECBUC CORNERFLX PINC

SLOWTRAN ADJOINT OUTPUTAD TRANSIT OUTCYC

ADJOINT|

XSFDADJ WSHIFT TRIDIA0 LSORB0 OUTINADJ OUTP

AVGEDIT|

RSTRING

AVGEDIT-PIN|

RSTRING

BURNNODE|

CALACV CHAIN

Figure 9: Dependence diagram of the NESTLE code.

NTWOC

EARCH NONNEMC NONNEMHTRIDIA0 LSORB

.

151

NONNEMH|

NONNETH NONPLMH NONONEH NONTWOH NONONEC NO

NONONEC|

DIRECT4 DIRECT8B DIR4FULL DIR2FULL

NONONEH|

ONENODE4 ONENODE8

NONTWOC|

DIRECT4 DIRECT8B DIRECT8 DIRECT16

NONTWOH|

ONENODE4 ONENODE8 TWONODE8 TWONODE16

OUTIN|

GTIME LSORB CHEBY1 WEILANDT1 RELPOWER SXENON KSEARCH CNTROD PSTHFDBKS DEPLETE MFST RELPOWER SFST XSFDBK WSHIFT


RIDIA0 LSORB0 OUTIN PEAK

SHIFT TRIDIA0 LSORB0 OUTIN

.

153

PINPOWER|

PINFLX PINFLXMATA DIR12FULL

POINTER|

INPUTCK

SETUP0|

SORCE0 TRIDIA

SFST|

AMF SCALEXCT SCALING RELPOWER

SLOWTRAN|

PINTER XSFDBK OUTPCRT RELPOWER THFDBK TXENON CNTROD WSHIFT TOUTPCRT OUTPUTSS

STEADYN|

PINTER XSFDBK NORM NORMFSP SXENON RELPOWER THFDBKS CNTROD W


.

155

XSECBU|

PINTER

XSECBUS|

PINTER


ellet

rrently

ations.

linear

hecks

and

essage

SUBROUTINEADJOINTThis subroutine controls the calculation of the adjoint flux.

SUBROUTINEAMFThis subroutine determines (A-F)*flux for the fixed-source scale factor method.

SUBROUTINEANMERGEThis subroutine merges alphanumeric string arrays together to form single string.

SUBROUTINEAVGEDITThis subroutine edits out radial and axial core averaged properties.

SUBROUTINEAVGEDIT-PINThis subroutine edits out maximum value of axially-averaged pin power and maximum ppower.

SUBROUTINEBURNNODEThis subroutine solves the isotopic depletion equations.

SUBROUTINEBURNNODESThis subroutine solves the isotopic depletion equations on the six surfaces of a hexagon (cuworks for NSHAP= “HEXA” only).

SUBROUTINECALACVThis subroutine determines the interaction rates required to solve the isotopic depletion equ

SUBROUTNECHAINThis subroutine uses the integrating factor technique to analytically solve a coupleddepletion chain.

SUBROUTINECHEBY1This subroutine applies the semi-implicit Chebyshev polynomial acceleration method and cfor convergence for the steady state problem.

SUBROUTINECHEBYTRThis subroutine applies the semi-implicit CHEBYCHEV polynomial acceleration methodchecks for convergence for the transient problem.

SUBROUTINECHECKThis subroutines recognizes the different non-permissible runs and flags back an error malong with suggestion for the alternative run case.

Table 5: Listing of procedures and their functions.

156

uction

odel

odelf each

ector

d. All

ll

d. All

oyed..

SUBROUTINECNTRODThis subroutine determines the fraction of control group inserted.

SUBROUTINECONVERThis subroutine converts the input data units to NESTLE’s internal working units.

SUBROUTINECORNERFLXThis subroutines determines the corner point flux values needed for the pin power reconstr(currently works for NSHAP = “HEXA” only).

SUBROUTINEDECAYHNThis subroutine solves the decay heat precursor equations.

SUBROUTINEDEPLETEThis subroutine controls the depletion, determining number densities for the microscopic mand returning the macroscopic cross section expansion coefficients.

SUBROUTINEDEPLETESThis subroutine controls the depletion, determining number densities for the microscopic mand returning the macroscopic cross section expansion coefficients at the six surfaces ohexagonal assembly (currently works for NSHAP = “HEXA” only)

SUBROUTINEDIR12FULLThis subroutine solves a 12 x 12 matrix system needed for the pin power reconstruction. Varguments are employed. All unknowns are evaluated. Full matrix structure is assumed.

SUBROUTINEDIR2FULLThis subroutine analytically solves a 2 x 2 matrix system. Vector arguments are employeunknowns are evaluated. Full matrix structure is assumed in analytic solution.

SUBROUTINEDIR4FULLThis subroutine analytically solves a 4 x4matrix system. Vector arguments are employed. Aunknowns are evaluated. Full matrix structure is assumed in analytic solution.

SUBROUTINEDIRECT2X2This subroutine analytically solves a 2 x 2 matrix system. No vector arguments are employeunknowns are evaluated. Full matrix structure is assumed in analytic solution.

SUBROUTINEDIRECT16This subroutine analytically solves a 16 x 16 matrix system. Vector arguments are emplOnly half of unknowns are evaluated. Matrix sparsity is taken advantage in analytic solution

Table 5 (cont): Listing of procedures and their functions.

157

ll

d. All

d. All

ed for

rovides

ternal

gather

SUBROUTINEDIRECT4This subroutine analytically solves a 4 x4matrix system. Vector arguments are employed. Aunknowns are evaluated. Matrix sparsity is taken advantage in analytic solution.

SUBROUTINEDIRECT8This subroutine analytically solves a 8 x 8 matrix system. Vector arguments are employeunknowns are evaluated. Matrix sparsity is taken advantage in analytic solution.

SUBROUTINEDIRECT8BThis subroutine analytically solves a 8 x 8 matrix system. Vector arguments are employeunknowns are evaluated. Matrix sparsity is taken advantage in analytic solution.

SUBROUTINEECHOINPThis subroutine echoes out the radial input figures.

SUBROUTINEFILE_CNTThis subroutine reads the general control input parameters.

SUBROUTINEFILE_GEOThis subroutine reads the geometrical parameters.

SUBROUTINEFILE_KINThis subroutine reads the kinetic parameters required for the transient runs.

SUBROUTINEFILE_PPRThis subroutine reads the corner point discontinuity factors and pin-wise form factors needthe pin power reconstruction (currently works for NSHAP = “HEXA” only).

SUBROUTINEFILE_PRFThis subroutine reads the parameters used to control the solution methods employed and pthe convergence criteria.

SUBROUTINEFILE_XSCThis subroutine reads the cross sections and T-H input parameters.

SUBROUTINEFLUIDCONThis subroutine calculates coolant temperature, density, and void fraction based upon inenergy.

SUBROUTINEGASCATCThis subroutine is used for the cart geometry only. It calculates the needed parameters tothe nodes with the same color together (i.e. blacks with blacks...etc). for Cartesian geometry, two


158

.

gatheree.

ndary

AIX

ulateve the

start

rol

ntrol,

rything

inlet

different colors are used. this will be utilized in the solution of the finite difference equations

SUBROUTINE GASCATHThis subroutine is used for the hex geometry only. It calculates the needed parameters tothe nodes with same color together (i.e. blacks with blacks...etc). For Hexagonal geometry, thrdifferent colors are used. this will be utilized in the solution of the finite difference equations

SUBROUTINEGEOMETRYThis subroutine sets up the geometry including B.C. mesh expansion and bundle bouidentification for output control.

FUNCTION GTIMEThis subroutine returns back the elapsed time (in seconds) applicable to ULTRIX oroperating system.

SUBROUTINEHYPINTThis subroutine contains three functions: COSH0, SINH1, COSH2. The three functions calcthe zeroth, first, and second moment integration of the hyperbolic functions needed to solconformal mapping based hexagonal nodal equations. (For NSHAP = “HEXA” only).

SUBROUTINEINITALThis subroutine initializes the flux, fission source, and T-H conditions guesses, or for reoption on reads in the restart file.

SUBROUTINEINIVALThis subroutine edits out the initial input data.i.e. cross section data, geometry data, and contoptions, if users choose long edits for the inputs (AL3 = “Y”).

SUBROUTINEINPDATAThis subroutine controls the overall input reading.

SUBROUTINEINPEDITThis subroutine edits out input associated with the transient case, method of solution cocontrol options, I/O file names, and if elected cross sections and T-H parameters.

SUBROUTINEINPUTCKThis subroutine checks whether alphanumeric input was correctly entered and converts eveto upper case letters.

SUBROUTINEKSEARCHThis subroutine performs criticality search on four different parameters: soluble poison,coolant temperature, power level, and control bank insertion.


159

laroutine

for the

letable

letable

every

SUBROUTINELAMDASUBThis subroutine determines the scale factor for the fixed-source scale factor method.

SUBROUTINELINEARThis subroutine completes the linear interpolation or extrapolation.

SUBROUTINELSORBThis subroutine calls the following two subroutines:SORCE - which calculates the RHS of the finite difference equations for a specific color.TRIDIA - which solves for the flux by solving a tridiagonal system of equations for a particucolor. The flux is also accelerated using the omegas which were precalculated in the subrLSORB0.

SUBROUTINELSORB0This subroutine calculates the number of inner per outer and the acceleration parameterscolor line SOR equations.

PROGRAMMAINThis is the main routine for NESTLE.

SUBROUTINEMFSTThis subroutine controls the multi fixed source scale factor method.

SUBROUTINEMICROXNTThis subroutine determines the microscopic cross section expansion coefficients for the depisotopes.

SUBROUTINEMICROXNTSThis subroutine determines the microscopic cross section expansion coefficients for the depisotopes on the six surface of the hexagonal assembly (NSHAP = “HEXA” only).

FUNCTION NEWPAGEStarts a new page of output.

SUBROUTINENONNEMCThis subroutine solves the NEM equations.********* For Cartesian Geometry **************The nodal method used here is nonlinear NEM, We solve a one or two node problem forinterface in the core to update the coupling coefficients.

SUBROUTINENONNEMHThis subroutine solves the NEM equations.


160

ometry.pling

urrent

urrent

nsion

nsion

.

********* For Hexagonal Geometry **************The nodal method used here is a conformal mapping based nodal method for hexagonal geWe solve a one or two node problem for every interface in the core to update the coucoefficients.

SUBROUTINENONNETCThis subroutine determines the currents for NEM.********* For Cartesian Geometry **************

SUBROUTINENONNETHThis subroutine determines the currents for NEM.********* For Hexagonal Geometry **************

SUBROUTINENONONECThis subroutine solves a one-node problem. It is used only for edge nodes without zero cboundary condition when updating the net currents by NONNEM.********* For Cartesian Geometry **************

SUBROUTINENONONEHThis subroutine solves a one-node problem. It is used only for edge nodes without zero cboundary condition when updating the net currents by NONNEM.********* For Hexagonal Geometry **************

SUBROUTINENONPLMCThis subroutine calculates the leakages for every node in every direction and the expacoefficients for the quadratic leakage approximation.********* For Cartesian Geometry **************

SUBROUTINENONPLMHThis subroutine calculates the leakages for every node in every direction and the expacoefficients for the quadratic leakage approximation.********* For Hexagonal Geometry **************

SUBROUTINENONTWOCThis subroutine solves the two node problem and returns the NEM current JNEM.********* For Cartesian Geometry **************

SUBROUTINENONTWOHThis subroutine solves the two node problem and returns the NEM current JNEM.********* For Hexagonal Geometry **************

SUBROUTINENORMThis subroutine normalizes the flux and fission source to a core relative average power = 1


161

d. All

d. All

tate

int

ient

st.

SUBROUTINENORMFSPThis subroutine normalizes the flux and fission source to an input specified scaling.

SUBROUTINEONENODE4This subroutine analytically solves a 4 x 4 matrix system. Vector arguments are employeunknowns are evaluated. Matrix sparsity is taken advantage in analytic solution.

SUBROUTINEONENODE8This subroutine analytically solves a 8 x 8 matrix system. Vector arguments are employeunknowns are evaluated. Matrix sparsity is taken advantage in analytic solution.

SUBROUTINEOUTCYCThis subroutine writes the node-wise isotopic number densities.

SUBROUTINEOUTINThis subroutine performs outer-inner iterations utilizing FDM or NEM option for the steady ssolution.

SUBROUTINEOUTINADJThis subroutine performs outer-inner iterations utilizing FDM or NEM option for the adjosolution.

SUBROUTINEOUTINTRThis subroutine performs outer-inner iterations utilizing FDM or NEM option for the transsolution.

SUBROUTINEOUTPCRTThis subroutine outputs values to the screen (i.e.CRT).

SUBROUTINEOUTPOINTThis subroutine outputs point-wise values of parameter passed through calling argument li

SUBROUTINEOUTPUTADThis subroutine outputs the adjoint problem’s solution.

SUBROUTINEOUTPUTSSThis subroutine outputs the steady state problem’s solution.

SUBROUTINEOUTPUTTRThis subroutine outputs the transient problem’s solution.


162

.

group>2

ower

e two-

gian

, inlet

les the

lerkin

SUBROUTINEPEAKThis subroutine determines the total peaking factor and location.

SUBROUTINEPERTURBThis subroutine interpolates the time dependent input parameters for the transient problem

SUBROUTINEPINCOLLAPThis subroutine collapses kappa-sigma-fission cross sections and fluxes for NG>2 into two-cross sections and fluxes for the pin power reconstruction (NSHAP = “HEXA” only). NGintroduces complexities in determining pin-wise fluxes.

SUBROUTINEPINLOCThis subroutine determines the fuel pin locations within the hexagonal assembly for the pin preconstruction (NSHAP = “HEXA” only).

SUBROUTINEPINPOWERThis subroutine performs a pin power reconstruction in hexagonal geometry based upon thgroup cross sections and fluxes (NSHAP = “HEXA” only).

SUBROUTINEPINTERThis subroutine completes quadratic interpolation (or extrapolation) using Lagranpolynomial.

SUBROUTINEPOINTERThis subroutine determines the A array pointers (i.e.starting locations of arrays).

SUBROUTINEPRECRThis subroutine solves the delayed neutron precursor equations.

FUNCTION PROPPOLYThis subroutine calculates specified property as function of a stated dependence.

SUBROUTINEPSEARCHThis subroutine performs power level search on three different parameters: soluble poisoncoolant temperature, and control bank insertion.

SUBROUTINERELPOWERThis subroutine determines the total core power level accounting for decay heat and scapower density to a relative core power level = 1.

SUBROUTINERESIDDetermines the relative residual of the diffusion equation and the eigenvalue using Ga


163

ed by

cale

ource

ining

steady

factor

IDIA.e the, and

IDIA.Theed inouter

weighting of the diffusion operators.

SUBROUTINERSTRINGThis subroutine loads a numeric value into an alphanumeric variable.

SUBROUTINESCALAPRXThis subroutine calculates the ratio of the effective external source to the fission source usthe fixed-source scale factor method.

SUBROUTINESCALEXCTThis subroutine determines the ratio of the effective external source to (A-F)*FLUX to get sfactor update.

SUBROUTINESCALINGThis subroutine scales the flux and fission source using the scale factor from the fixed-sscale factor method.

SUBROUTINESETUP0This subroutine solves the homogeneous problem using color line G-S in support of determoptimum relaxation parameters and number of iterations per outer iteration.

SUBROUTINESFSTThis subroutine performs the single fixed-source scaling technique procedure for the FSPstate case.

SUBROUTINESHAPECORThis subroutine adjusts the flux for coolant spectral effects within the fixed-source scalemethod.

SUBROUTINESLOWTRANThis subroutine provides overall control of the transient fission product problem.

SUBROUTINESORCEThis subroutine calculates the RHS for the tridiagonal system to be solved by subroutine TRThe tridiagonal system results from using the color line SOR method which is used to solvfinite difference form of the diffusion equations. The RHS here includes fission, scatteringdiffusion terms.

SUBROUTINESORCE0This subroutine calculates the RHS for the tridiagonal system to be solved by subroutine TRThe equation to be solved is A*PHI=0 where A is the coefficient matrix and PHI is the flux.RHS here does not include any fission or scattering. It only includes diffusion terms. It is usobtaining the color line SOR extrapolation parameters and number of inner iterations per


164

ion in

ts.

for the

for the

r line

ys for

ctorge in

ctorge in

iteration.

SUBROUTINESPECSHFTThis subroutine determines the B**2 values used in making the coolant spectral shift correctthe fixed-source scale factor method.

SUBROUTINESTARTERThis subroutine sets initial values to initiate the transient from.

SUBROUTINESTEADYNThis subroutine provides overall control of the steady state solution.

SUBROUTINESXENONThis subroutine solves for the steady state number densities of the transient fission produc

SUBROUTINETHFDBKKThis subroutine calculates coolant internal energy, coolant density, and fuel temperaturestransient problem.

SUBROUTINETHFDBKSThis subroutine calculates coolant internal energy, coolant density, and fuel temperaturessteady state problem.

SUBROUTINETRANSITThis subroutine provides overall control of the transient problem.

SUBROUTINETRIDIAThis subroutine solves a tridiagonal system of equation which results from using the coloSOR method. It solves for the flux of a particular color each time it is called.

SUBROUTINETRIDIA0This subroutine factors the tridiagonal matrices associated with color line and assigns to arraeach color.

SUBROUTINETWONODE8This subroutine analytically solves a 8 x 8 matrix system used in NONTWOH routine. Vearguments are employed. All unknowns are evaluated. Matrix sparsity is taken advantaanalytic solution.

SUBROUTINETWONODE16This subroutine analytically solves a 16 x 16 matrix system used in NONTWOH routine. Vearguments are employed. All unknowns are evaluated. Matrix sparsity is taken advantaanalytic solution.


165

e step

for

for

roup

eption.

sion

ity, fuel

ity, fuelnsion

ity, fuel

SUBROUTINETXENONThis subroutine solves for the transient number densities of the transient fission products.

SUBROUTINEUPDATEThis subroutine saves time-step values of various parameters for usage in the next timsolution.

SUBROUTINEWEILANDT1This subroutine is applicable to Wielandt shift with stationary acceleration and checksconvergence for the steady state problem.

SUBROUTINEWEILANDTRThis subroutine is applicable to Wielandt shift with stationary acceleration and checksconvergence for the transient problem.

SUBROUTINEWSHIFTThis subroutine determines Wielandt shift utilizing the Sutton method to retain energy gdecoupling at inner iterative level.

SUBROUTINEXSECBUThis subroutine determines the nuclear properties (e.g.cross section expansion coefficients) at thnode color and burnup except for the depletable isotopes modeled using the microscopic o

SUBROUTINEXSECBUCThis subroutine determines the corner point discontinuity factor and form factor expancoefficients (NSHAP = “HEXA” only).

SUBROUTINEXSECBUSThis subroutine determines the surfaces’ nuclear properties,e.g. cross section expansioncoefficients (NSHAP = “HEXA” only).

FUNCTION XSECPOLYThis subroutine calculates cross sections accounting for coolant temperature, coolant denstemperature and soluble poison feedback corrections.

FUNCTION XSECPOLY2This subroutine calculates cross sections accounting for coolant temperature, coolant denstemperature and soluble poison feedback corrections. Difference with XSECPOLY is dimeof calling arguments arrays.

FUNCTION XSECPOLYSThis subroutine calculates cross sections accounting for coolant temperature, coolant dens


166

agonal

djoint

utronic

utronic

rnup

temperature and soluble poison feedback corrections on the six surfaces of the hexassembly (NSHAP = “HEXA” only).

SUBROUTINEXSFDADJThis subroutine determines the matrix transpose of the coefficient matrix required for the aflux solution.

SUBROUTINEXSFDBKThis subroutine determines the macroscopic and microscopic cross sections and other nenode values and uses them in determining the coefficient matrix.

SUBROUTINEXSFDBKSThis subroutine determines the macroscopic and microscopic cross sections and other nevalues on the six surfaces of a hexagonal assembly (NSHAP = “HEXA” only).

SUBROUTINEXSMODThis subroutine calculates the flux-volume weighted cross sections for the within-node bugradient treatment in the hexagonal nodal method (NSHAP = “HEXA” only).


167

adf.fcb cntl2.fcb hexdim.fcb perttr.fcb thermo.fcbadj.fcb confmap.fcb hgeo.fcb pertv.fcb thmargin.fcbarray.fcb conv.fcb multit.fcb pinpow.fcb tim.fcbbasic.fcb convfact.fcb nemcnt.fcb power.fcb time.fcbbcs.fcb crhs.fcb nemtime.fcb restinp.fcb time1.fcbbcshex.fcb crit.fcb nline.fcb restotp.fcb timetr.fcbbpitch.fcb crod.fcb nonfue.fcb rodkin.fcb varlen.fcbbuckl.fcb dataf.fcb nterm.fcb soln2.fcb varlens.fcbbundle.fcb depletepc.fcb numsurf.fcb spectral.fcb veloc.fcbburn.fcb extsor.fcb only.fcb srf.fcb xeopt.fcbbypass.fcb flamdold.fcb opti.fcb start.fcb xsec1.fcbche.fcb fpxs.fcb outlong.fcb th_cof.fcb xsec2.fcbcheby.fcb gasch.fcb param.fcb thcoef.fcb xspolycom.fcbcntl.fcb geom.fcb pardrwm.fcb thermk.fcb

Table 6: Listing of fcb files containing named COMMON blocks

168

View publication statsView publication stats

NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE VERSION 1.0 NESTLE NESTLE ELECTRIC POWER RESEARCH CENTER NESTLE NESTLE NORTH CAROLINA STATE UNIVERSITY NESTLE NESTLE COPYRIGHT 1994 - BY NCSU NESTLE NESTLE NESTLE NESTLE NESTLE DICTIONARY PROGRAM NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE NESTLE

WHICH OPTION DO YOU WISH TO ENABLE (E=EXIT, F=FIND, D=DELETE, A=ADD)? F

WHAT IS THE VARIABLE NAME THAT YOU WISH THE DEFINITION FOUND FOR?SW

DEFINITION: MESH DEPENDENT FLUX [SOLUTION TO THE PROBLEM] / EIGENVECTOR FOR THE EIGENVALUE PROBLEM

WHICH OPTION DO YOU WISH TO ENABLE (E=EXIT, F=FIND, D=DELETE, A=ADD)? A

WHAT IS THE VARIABLE NAME YOU WISH TO ADD? TEST1

INPUT DEFINITION FOR THE VARIABLE:JUST A TEST VARIABLE FOR THE DICTIONARY PROGRAM

WHICH OPTION DO YOU WISH TO ENABLE(E=EXIT, F=FIND, D=DELETE, A=ADD)?F

WHAT IS THE VARIABLE NAME THAT YOU WISH THEDEFINITION FOUND FOR? TEST1

DEFINITION:JUST A TEST VARIABLE FOR THE DICTIONARY PROGRAM

WHICH OPTION DO YOU WISH TO ENABLE (E=EXIT, F=FIND, D=DELETE, A=ADD)? D

WHAT IS THE VARIABLE NAME THAT YOU WISH THE DEFINITION DELETED FOR?TEST1

ARE YOU CERTAIN THAT YOU WANT THE VARIABLE DEFINITION DELETED (Y,N)?Y

VARIABLE DEFINITION DELETED.

WHICH OPTION DO YOU WISH TO ENABLE (E=EXIT, F=FIND, D=DELETE, A=ADD)?E

IF VARIABLE NAMES AND DEFINITIONS HAVE BEEN ADDED OR DELETED, DO YOU WANT TO SAVE CHANGES IN DATA BASE (Y,N)?Y

Table 7: Sample interactive session with NESTLE.DICT

169

https://www.researchgate.net/publication/236368235

nestle - researchgate · nestle version 5.2.1 few-group neutron diffusion equation solver utilizing...

Documents