#U.S.NRC

United States Nuclear Regulatory Commission

Protecting People and the Environment

NUREG/CR-6958

ORNL/TM-2007/233

LAPUR 6.0 User's Manual

Office of Nuclear Reactor Regulation
AVAILABILITY OF REFERENCE MATERIALS

IN NRC PUBLICATIONS

NRC Reference Material

As of November 1999, you may electronically access

NUREG-sedes publications and other NRC records at

NRC's Public Electronic Reading Room at

http://www.nrc.cgov/readincg-rm.html. Publicly released

records include, to name a few, NUREG-senes.

publications; Federal Register notices; applicant,

licensee, and vendor documents and correspondence;

NRC correspondence and internal memoranda;

bulletins and information notices; inspection and

investigative reports; licensee event reports; and

Commission papers and their attachments.

NRC publications in the NUREG series, NRC

regulations, and Title 10, Energy, in the Code of

Federal Regulations may also be purchased from one

of these two sources.

1. The Superintendent of Documents

U.S. Government Printing Office

Mail Stop SSOP

Washington, DC 20402-0001

Internet: bookstore.gpo.gov

-Telephone: 202-512-1800

Fax: 202-512-2250

2. The National Technical Information Service

Springfield, VA 22161-0002

www.ntis.gov

1-800-553-6847 or, locally, 703-605-6000

A single copy of each NRC draft report for comment is

available free, to the extent of supply, upon written

request as follows:

Address: U.S. Nuclear Regulatory Commission

Office of Administration

Mail, Distribution and Messenger Team

Washington, DC 20555-0001

E-mail: DISTRIBUTION@nrc.gov

Facsimile: 301-415-2289

Some publications in the NUREG series that are

posted at NRC's Web site address

http://www.nrc.gov/reading-rm/doc-collections/nuregs

are updated periodically and may differ from the last

printed version. Although references to material found

on a Web site bear the date the material was accessed,

the material available on the date cited may

subsequently be removed from the site.

Non-NRC Reference Material

Documents available from public and special technical

libraries include all open literature items, such as

books, journal articles, and transactions, Federal

Register notices, Federal and State legislation, and

congressional reports. Such documents as theses,

dissertations, foreign reports and translations, and

non-NRC conference proceedings may be purchased

from their sponsoring organization.

Copies of industry codes and standards used in a

substantive manner in the NRC regulatory process are

maintained atThe NRC Technical Library

Two White Flint North

11545 Rockville Pike

Rockville, MD 20852-2738

These standards are available in the library for

reference use by the public. Codes and standards are

* usually copyrighted and may be purchased from the

originating organization or, if they are American

National Standards, fromAmerican National Standards Institute

11 West 4 211d Street

New York, NY 10036-8002

www.ansi.org

212-642-4900

Legally binding regulatory requirements are stated

only in laws; NRC regulations; licenses, including

technical specifications; or orders, not in

NUREG-series publications. The views expressed

in contractor-prepared publications in this series are

not necessarily those of the NRC.

The NUREG series comprises (1) technical and

administrative reports and books prepared by the

staff (NUREG-XXXX) or agency contractors

(NUREG/CR-XXXX), (2) proceedings of

conferences (NUREG/CP-XXXX), (3) reports

resulting from international agreements

(NUREG/IA-XXXX), (4) brochures

(NUREG/BR-XXXX), and (5) compilations of legal

decisions and orders of the Commission and Atomic

and Safety Licensing Boards and of Directors'

decisions under Section 2.206 of NRC's regulations

(NUREG-0750).

DISCLAIMER: This report was prepared as an account of work sponsored by an agency of the U.S. Government.

Neither the U.S. Government nor any agency thereof, nor any employee, makes any warranty, expressed or

implied, or assumes any legal liability or responsibility for any third party's use, or the results of such use, of any

information, apparatus, product, or process disclosed in this publication, or represents that its use by such third

party would not infringe privately owned rights.
*U.S.NRC

United States Nuclear Regulatory Commission

Protecting People and the Environment

NUREG/CR.-6958

ORNL[TM-2007/233

LAPUR 6.0 User's Manual

Manuscript Completed: December 2007

Date Published: October 2008

Prepared by

Alberto Escrivdi

Jos6 Luis Mufioz Cobo

Jos6 Melara San Roman

Manuel Albendea Darriba

Jos6 March-Leuba

Oak Ridge National Laboratory

P.O. Box 2008

Oak Ridge, TN 37831

T. Huang, NRC Project Manager

NRC Job Code J3319

Office of Nuclear Reactor Regulation

ABSTRACT

This report documents a series of programming upgrades to Version 6.0 of the LAPUR

code. LAPUR, a computer program in FORTRAN, is a mathematical description of the core of a

boiling water reactor. Its two linked modules, LAPURX and LAPURW, respectively solve the

steady-state governing equations for the coolant and fuel and the dynamic equations for the

coolant, fuel, and neutron field in the frequency domain. The main upgrade in LAPUR 6 is the

ability to model part-length fuel rods and fuel spacers explicitly. General implementation

descriptions are followed by a detailed description of input and output parameters of LAPURX

and LAPURW. Sample inputs are included and stability benchmarks are noted.

Paperwork Reduction Act Statement

7

The information collections contained in this NUREG are covered by the requirements of

10 CFR Parts 50, 52, and 110, which were approved by the Office of Management and Budget,

approval number 3150-0011, -0151 and -0036.

This NUREG also contains additional information collections that are subject to the Paperwork

Reduction Act of 1995 (44 U.S.C. 3501 et seq.). The telephone survey of people living in the

Emergency Planning Zone has been submitted to the Office of Management and Budget for

review and approval of the information collection.

Public Protection Notification

The NRC may not conduct or sponsor, and a person is not required to respond to, a request for

information or an information collection requirement unless the requesting document displays a

currently valid OMB control number.

iii

CONTENT

ABSTRACT .................................................................................................................................. i

CONTENT ...................................................................................................................................... v

1.0 INTRODUCTION ................................................................................................................ 1

2.0 LAPUR GENERA L DESCRIPTION ............................................................................... 3

2.1. General Description of the Input and Output ................................................................ 3

2.2. Succinct Description of the Program .......................................................................... 4

2.2.1. LAPURX, The Steady-State Module .................................................................... 4

2.2.2. LAPURW , The Dynamics Module ..................................................................... 4

2.2.3. Intermediate Storage Devices .............................................................................. 7

3.0 LAPUR 6 IMPLEMENTATION ...................................................................................... 9

3.1. Differences Between LAPUR 5.2 and LAPUR 5.1 ...................................................... 9

3.2. Differences Between LAPUR 6.0 R.0 and LAPUR 5.2 ............................................ 10

3.3. LAPUR 6 Input and Output Files ................................................................................ 10

3.4. Operating System Requirements .................................... 10

3.5. Installing and Executing LAPUR-6 ............................................................................. 10

4.0 LAPUR INPUT PARAM ETERS ................................................................................... 13

4.1. LAPUR Dimension Limits ........................................................................................... 13

4.2. Alphabetical Description of the LAPURX Subroutines ............................................. 14

4.3. Preparation of Input for LAPURX ............................................................................. 24

4.3.1. Introduction ........................................................................................................ 24

4.3.2. Input Description ............................................................................................... 25

4.4. Output Description of LAPURX ................................................................................. 34

4.5. Alphabetical Description of the LAPURW Subroutines ............................................. 39

4.6. Preparation of Input for LAPURW ............................................................................. 51

4.6.1. Introduction ........................................................................................................ 51

4.6.2. Input Description ................................................................................................... 52

4.7. Output Description of LAPURW ............................................................................... 58

REFERENCES ............................................................................................................................ 65

Appendix A - Sample LAPURX and LAPURW Inputs For A Typical BWR ........... A-1

LAPURX V6.0 Sample Input Deck: ......................................................................................... A-2

LAPURW V6.0 Sample Input Deck: .............................................................................. I ...... A-26

Appendix B - Sample LAPURX and LAPURW Output For A Typical BWR ............ I ........... B-1

Output File LAPURX.CSV: ...................................................................................................... B-2

v
O utput File LA P U R W .C SV : ..................................................................................................... B-2

Appendix C - Detailed Description of LAPUR 6 Modifications ...................................... C-1

vi
1.0 INTRODUCTION

This report contains an updated User Manual for the new code version, LAPUR 6.0.

LAPUR, a computer program in FORTRAN, is a mathematical description of the core of a boiling

water reactor (BWR). Its two linked modules, LAPURX and LAPURW, respectively solve the

steady-state governing equations for the coolant and fuel and the dynamic equations for the

coolant, fuel, and neutron field in the frequency domain. General implementation descriptions

are followed by a detailed description of input and output parameters of LAPURX and LAPURW.

Sample inputs and outputs are included and stability benchmarks are noted.

The LAPUR code has been validated in the past against a reasonable array of BWR test data.

The validation of the LAPUR 6 version is the topic of a separate report. The old validation data

include:

1. Peach Bottom tests (see Ref. 3).

2. Vermont Yankee stability tests (see Ref. 3).

3. Dresden local stability test (see Ref. 4).

4. Browns Ferry single-loop stability test (see Refs. 5and 6).

5. Susquehanna-2 stability tests (see Refs.7 and 8).

6. Grand Gulf-1 stability tests (see Refs 7 and 8).

a

2.0 LAPUR GENERAL DESCRIPTION

This section presents the computer programLAPUR, a version coded in the FORTRAN

language of the mathematical description of a BWR. This program consists of two autonomous

modules, LAPURX and LAPURW, which are linked by means of an intermediate storing device.

The first module, LAPURX, solves the coolant and the fuel steady-state governing equations as

described in Sect. 3.2.1 of Ref. 1. Maps of core steady-state parameters are generated by

LAPURX and stored into two data files for subsequent use by the dynamics module LAPURW.

The second module, LAPURW, solves the dynamic equations for the coolant, fuel, and neutron

field in the frequency domain according to the formulations developed in Sect. 3 of Ref. 1.

2.1. GENERAL DESCRIPTION OF THE INPUT AND OUTPUT

General input requirements to run a problem are as outlined:

For steady-state calculation (LAPURX), system operating status:

1. State parameters: reactor pressure, thermal power generated, core flow rate, and coolant

temperature at the inlet plenum.

2. Power map: Vertical power shape of representative channels and either steam exit

quality or amount of power generated in each channel.

3. Fraction of power deposited into the coolant by neutron moderation and y ray absorption.

For steady-state calculation (LAPURX), system design parameters:

1. Mechanical: channel box and fuel pin dimensions.

2. Physical: hydraulic diameters, friction multipliers, orifice flow coefficients, fuel and clad

densities, and gap thermal conductance.

For steady-state calculation (LAPURX), user options:

1. Error criteria for the iterative calculations.

2. Adjustable parameters for two-phase correlations.

3. Number of nodes desired in the boiling part of a channel.

4. Output options.

For dynamic calculation (LAPURW), systems parameters:

1. Recirculation loop pressure to flow rate gain and time constant.

For dynamic calculation (LAPURW), neutronic parameters:

1. Effective neutron lifetime table.

2. Steady-state core reactivity table.

3. Delayed neutron fractions and their time constants.

4. Doppler reactivity coefficient.

5. Table of density reactivity coefficients.

3
For dynamic calculation (LAPURW), output options:

1. Frequencies of interest for the dynamic analysis.

2. Edit and plotting options.

3. Stability calculation options.

Note that each subcore must be assigned a set of neutronic parameters.

Steady-state calculation LAPURX generates a map of the thermohydraulic parameters in the

core, that is, of the following:

1. Channel flow rate, pressure drops, and exit quality (or power) for each channel.

2. Nodal coolant density, void fraction, enthalpy, quality, velocities, and friction components

at each node along each channel.

3. Set of coefficients for the dynamics calculation.

The dynamic calculation determines:

1. For the fuel, the response of the heat flux to the coolant and of the average temperature

to driving perturbations of power generation, coolant temperature, and coolant flow rate at

each node in every channel.

2. For the coolant flow in a channel box:

* The response of the coolant parameters at every node to driving perturbations of power

generation at each subcore level, coolant flow rate, and coolant inlet temperature.

* The open-loop transfer function (TF) for the channel hydrodynamics, its natural frequency,

and decay ratio.

* The reactivity feedback induced in each node by the driving perturbations. (This is

accomplished by properly weighting the coolant density and fuel temperature perturbations

with reactivity coefficients.)

3. For each nuclear subcore:

* Feedback reactivity TFs for driving perturbations in power at each subcore, inlet

temperature, and core flow rate.

* Open-loop TF matrix; total core natural frequency and decay ratio.

* Closed-loop TF matrix of reactivity to power.

4. Nyquist and magnitude phase plots for all open-loop TFs; Bode plots for closed-loop TFs.

2.2. SUCCINCT DESCRIPTION OF THE PROGRAM

2.2.1. LAPURX, The Steady-State Module

This module follows the procedure described in Sect 3.2.1 of Ref. 1. Table 1 shows a structural

description of the LAPURX subroutine calling sequence. Descriptions of the subroutines, input,

and output of this program module are presented in Sects. 4.1, 4.2, and 4.3 of this report.

2.2.2. LAPURW, The Dynamics Module

This program is the coded version of the dynamic equations in the frequency domain of Sect. 3

of Ref. 1 and its corresponding appendices.

A structural listing of the subroutine calling sequence is shown in Table 2. Descriptions of the

subroutine, input, and output are presented in Sects. 4.4, 4.5, and 4.6 of this report.

4
Table 1. LAPURX structural listing

MAIN: INPUT: INP1

INP2

INP3

OUTIN

READY: THCL:

FLOW:

HCL:

TSA TP:

HFGP:

VOLUME

VGP:

FN

FPMU

FPK

FPRAN

CHECK

LINEAR

LINEAR

LINEAR

DoX= 1, N channels

SETUP

STEADY:

BOUNDY:

NODE:

SUBOIL:

BULK:

FRICT:

PREPAR

POWNW

DCUMP

TD:

HNBP:

TJLP

VOLUME

FN

DCUMP

FN

VOID

TD:

HNBP:

FPK

VOID

FPMU

AF

FF

FN

FPK

FPRAN

FPMU

FN

DELTP: PREPHI:

End of IXa

DRUMW2

DO IX 1, N channels

SETUP

STEADY:

COEFR: VOLUME

HNBP:

TD:

HBBP

FFF1

OMEGAP

FN

FUEL:

FFF2

DEFNRR

OMEGAP

BESSI

FTK:

FTRC:

FTDKDT:

LINEAR

LINEAR

FTK: LINEAR

OUTST

OUTFUX

OUTCOR

DRUMW1

DRUMW3

End of IX

END

aA convergence test is made at this point to ensure the pressure boundary condition is satisfied.

To avoid repetition, the symbol - substitutes a subroutine structure already described in the table.

5
MAIN: INPUTW:

OUTINW

DRUMR2

TRANS:

Table 2. LAPURW structural listing

INPW1, INPW2, INPW3, INPW4, INPW5, INPW6

INXS2G

INCROD

WEIGHT: Do IX = 1, N channels

DRUMR1

NODOS

COUPXS:

End of IX

Do IW=I, N frequencies

Do IX = 1, N channels

DRUMR1

NODOS

NODREA

REACFA: COE

JOSELE: CON

COND

DF

D.

FUEL:

DIF:

BESSI

FTK:

FTRC:

FTDKDT:

COND

LINEAR

LINEAR

FTK: LINEAR

OUTFUX

DRUMW3

DRUMR3

FUELW

COEFIW

FREQ

End of IX

RECIRC

ONECOR

End of IW

OUTFUW: PRINTA:

ANGLE

WDSKPR: PRINTA

WDSKGH

STABGH: DBDEG

SPLIN

SEVAL

BREAL: EXPIO

BIMAG:

RESPON: ANGLE

EXPIO

MARGIN

PRINTB:

WDSKPR:

WDSKPR:

WDSKGH

STABGH:

MARGIN

CONSIS

LIMIT

ANGLE

SENS: STABGH:

MARGIN

MODE2:STABGH:

MARGIN

LIMIT

STABGH: ~

MARGIN

TRANS: -

STABGH:

CONSIS

SEARCH:

STOP

END

Note: To avoid repetition, the symbol - substitutes a subroutine structure already described in the table.

6
2.2.3. Intermediate Storage Devices

Intermediate storage of data is used for two different purposes: (1) to provide a link between the

steady state and the dynamics calculation and (2) to allow the orderly printing of the output TFs

as a function of frequency while avoiding the need for excessive computer core memory space.

For the first purpose, the storage device can be tape, disk, or drum. The files are addressed as

logical units 1 and 2. Their input/output operations are unformatted and sequential.

For the second purpose, a direct access on line disk or drum is required. Three files are created

and assigned to logical units 10, 11, and 12. Data are transferred to and from the on-line device

unformatted in 8 byte segments per direct access operation.

7

3.0 LAPUR 6 IMPLEMENTATION

3.1. DIFFERENCES BETWEEN LAPUR 5.2 AND LAPUR 5.1

The major improvements of LAPUR 5.2 with respect to LAPUR 5.1 are:

1. The maximum number of thermal-hydraulic regions was parameterized. The default

value has a maximum number of 200 channels. Note: LAPUR calculations with up to 200

channels were verified against results from the old LAPUR 5.1 with no significant loss of

precision. Calculations with more than 200 channels are typically not necessary due to

symmetry considerations, but are possible; however, those calculations have not been

verified here.

2. The maximum number of axial power shapes was parameterized so that each thermalhydraulic region has its own axial and radial power shape. In the default compilation, the

maximum number of axial power shape regions is 200.

3. The maximum number of bundle and fuel rod types was parameterized. In the default

compilation, the maximum number of bundle types is 10. This allows modelling of mixed

cores with up to 10 different fuel types.

4. The maximum number of frequencies was parameterized. The default value is 100. This

allowed for better definition of the TFs and a more robust evaluation of the decay ratio. In

the old version, it was often required to perform two LAPURW calculations, one to

identify the frequency of oscillation using a coarse frequency mesh, and a second

calculation with a fine mesh around the oscillation frequency. With 100 frequencies, a

single calculation works for most cases.

5. All relevant dimensions of internal LAPUR variables were parameterized, so they can be

modified in the future. File PARAM.FI contains the definitions for all these parameters.

Note: these parameters have been left at their original values, and value changes have

not been tested thoroughly; it is the responsibility of the user to verify that any future

changes to these parameters does not result in significant loss of precision.

6. A well known behavior of LAPUR was that it would report very-high channel decay ratios

if the channel was very stable. This required the use to manually check the phase and

gain margins. This behavior was corrected by improving the decay-ratio estimation

algorithm. The problem was related to an internal "spline fit" that LAPUR performs on the

calculated TFs. For a very stable channel, the phase may cross the zero-degree level;

since LAPUR maintains the phases negative, the phase jumps from -5 deg to -355 deg.

The spline fit under those conditions cannot reproduce the discontinuity accurately, and

as a result a very-high decay ratio is reported. This behavior was corrected in

LAPUR5.2, which should calculate and report accurately low-channel-decay-ratio

conditions.

7. The calculation of the Average Saturated Boiling Boundary has been added at the end of

the LAPURX execution. This calculation is performed for output purposes only and does

not affect the LAPURW input files; thus it does not affect the calculated decay ratios.

8. Two new output files are generated by LAPUR. These files have an ASCII "Comma

Separated Value" (CSV) format and contain a summary of the most relevant output

information. The file names are LAPURX.CSV and LAPURW.CSV and can be opened

with most spreadsheet programs for further manipulation of the LAPUR output data.

LAPURX.CSV contains the relative power and flow, the pressure drop, exit quality, and

9
the saturated boiling boundary of each channel. LAPURW.CSV contains the decay ratio

and frequency for each channel and for the core, as well as the open-loop and closedloop TFs as function of frequency.

9. During the detailed code review, the authors found and corrected one miscellaneous

programming errbr.

10. The screen-input routines were removed. Batch is the only available mode for LAPUR

5.2. The executable files have been renamed LAPURX.EXE and LAPURW.EXE.

3.2. DIFFERENCES BETWEEN LAPUR 6.0 R.0 AND LAPUR 5.2

Appendix C contains a detailed description of all the improvements incorporated in

LAPUR 6.0. A short summary of these improvements follows:

1. The two-phase friction multipliers for frictional forces and for local losses, have been

improved, using updated relationships that are more realistic for the newest fuel designs.

2. The capability to model channels with variable areas, in order to represent more

adequately the partial -length rods of the new fuel designs.

3. Several cells in the single-phase region have been defined, and the cell variables

(densities, liquid velocities, temperatures) of the nonboiling region are calculated.

4. Local pressure losses (spacers) have been included.

5. Irreversible pressure losses due to contractions and expansions have been included.

6. During the detailed code review, the authors found and corrected a miscellaneous

programming error.

3.3. LAPUR 6 INPUT AND OUTPUT FILES

The current implementation of LAPUR 6 expects the input data described in Sect. 4 of this report

in two files named LAPURX.DAT and LAPURW.DAT. It generates the main (verbose) output to

two files named LAPURX.OUT and LAPURW.OUT. A summary of the most relevant information

is generated in ASCII Comma Separated Values (CSV) format in files LAPURX.CSV and

LAPURW.CSV. LAPUR 6 also generates intermediate storage files that can be deleted after

successful completion of a run. These files are named LAPUR.D1, LAPUR.D2, LAPUR.D3,

LAPURW.T1, and LAPURW.T2.

3.4. OPERATING SYSTEM REQUIREMENTS

LAPUR5 does not require any special operating system. The current version is compiled as a

Windows Console Application using Microsoft FORTRAN PowerStation.

3.5. INSTALLING AND EXECUTING LAPUR 6

LAPUR 6 is composed of two executable files: LAPURX.EXE, and LAPURW.EXE. LAPURX

performs steady-state and initialization calculations. LAPURW computes the frequency domain

or dynamic part of the calculations.To install LAPUR 6, simply copy the executable files to any

directory.To run LAPUR 6, follow these steps:

1. Generate input files for the steady state (LAPURX) and dynamic (LAPURW) modules.

2. Rename these input files to LAPURX.DAT and LAPURW.DAT.

10
3. Execute LAPURX.EXE

4. Execute LAPURW.EXE

5. Rename the output files LAPURX.OUT, LAPURX.CSV, LAPURW.OUT, and

LAPURW.CSV so that future runs will not overwrite them.

11

4.0 LAPUR INPUT PARAMETERS

4.1. LAPUR DIMENSION LIMITS

All arrays in LAPUR 6 are dimensioned using parameters instead of hard constants. These

parameters are defined in file PARAM.FI, which is included in both LAPURX and LAPURW

during the compilation step. These parameters are summarized in Table 3, and they can be

changed and the program recompiled to generate a version with larger dimension limits;

however, any changes would have to be verified to guarantee that programming errors do not

affect the results.

Table 3. Dimension Limits for LAPUR Arrays

PARAMETER (NPSMAX =200)! Max # of power shapes (>= 6)

PARAMETER (NPSNMAX = 50)! Max # of nodes of power shapes (= 50)

PARAMETER (NCHMAX =200)! Max # of TH channels (>= 7)

PARAMETER (NFBMAX = 10)! Max # of channel types (>= 5)

PARAMETER (NVAMAX = 10)! Max # of variable area channels (>=10)

PARAMETER (NOVAMAX = 90)! Max # of axial intervals channel (= g0)

PARAMETER (NLLMAX = 10)! Max # of local loss channel types (>=10)

PARAMETER (NOLLMAX = 90)! Max # of axial intervals Il-channel (=90)

PARAMETER (NFRMAX = 10)! Max # of fuel rod types (>= 5)

PARAMETER (NFMMAX = 10)! Max # of friction multipliers (>=10)

PARAMETER (NFMNMAX = 90)! Max # of axial intervals fri-mul (= 90)

PARAMETER (NCIMAX = 10)! Max # of channel inlet types (>=10)

PARAMETER (NCINMAX = 10)! Max # of pipes in series chan-in (= 10)

PARAMETER (NCEMAX = 10)! Max # of channel exit types (>=10)

PARAMETER (NCENMAX = 10) ! Max # of pipes in series chan-ex (= 10)

PARAMETER (NAPMAX = 10)! Max # of axial points edit rod (>=10)

PARAMETER (NOBMAX = 90)! Max # of axial nodes boiling (>=90)

PARAMETER (NOTMAX =140)! Max # of axial nodes total (>=140)

PARAMETER (NNBMAX = 50)! Max # of axial nodes nonboiling (>=50)

PARAMETER (NNJMAX = 1)! Max # of horizontal subcores (>= 1)

PARAMETER (NNKMAX = 1)! Max # of vertical subcores (>= 1)

PARAMETER (NNDMAX = 40)! Max # of sets of delay-n characterist (= 1)

PARAMETER (NROMAX = 40)! Max # of initial reactivities (= 1)

PARAMETER (NNLMAX = 40)! Max # of neutron lifetime in subcore (= 1)

PARAMETER (NNCMAX =200)! Max # of neutron coupling coefficien (= 1)

PARAMETER (NDGMAX = 6) ! Max # of delayed n energy groups (>= 6)

PARAMETER (NBTMAX = 3)! Max # of fuel bundle types (4f-b&l r) (>= 3)

PARAMETER (NFTMAX = 8)! Max # of different fuel types (>= 8)

PARAMETER (NXSEC = 7) ! # of Xsect D1,D2,Sal(inc scatt),Sa2,Sfl,Sf2,S1-2 (= 7)

PARAMETER (NECMAX = 3) ! Max # of order power expansion coeff (>= 3)

PARAMETER (NAFMAX = 6)! Max # of axial fuel intervales (>= 6)

PARAMETER (NACMAX =11)! Max # of axial control bundles inter. (>=1 1)

PARAMETER (NDRMAX = 7) ! Max # of tables of dens. react. coeffs (ID 18) (>= 7)

PARAMETER (NDRNMAX = 10)! Max # of values table react. coeffs (>=10)

PARAMETER (NFPMAX = 100)! Max # of frequency points (>=25)

PARAMETER (NSRMAX = 20)! Max # of subcritical reactivity o-pha (>=20)

PARAMETER (NFUMAX = 10) ! Max # of fuel radial nodes (>10)

13
4.2. ALPHABETICAL DESCRIPTION OF THE LAPURX SUBROUTINES

Notes on the format used in these descriptions:

1. "Input" indicates the means by which the subroutine receives the parameters on which it

operates. The parameters may come from an outside source defined by a logical unit,

from the calling program through shared COMMON areas, or as arguments.

2. "Output" indicates the means by which the parameters calculated or modified by the

subroutine are disposed of. These can be transferred to the calling subroutine, a

COMMON area, or an outside device.

Main

Subroutine BOUNDY

Called from: STEADY.

Input: (i) Argument: IX -= channel type.

(ii) COMMONS: DATA, INPT, REDY, REDY2, SETP, SETP2, PRPR,

PRPR2.

Performs: (i) If IOP(3) # 2, where the dot indicates "multiple," the position of the

boiling boundary Znb is read as input; hence, the enthalpy and temperature at the boiling

boundary are calculated accordingly.

(ii) If IOP(3) = 2, the Jens Lottes correlation [Eq. (3.2.4)]* is used to

determine the temperature of inception of subcooled boiling. Subsequently, the boiling

boundary position and physical parameters for the nonboiling part of the flow in the channel

are calculated.

Output: COMMONS: TBOUND, BNDY.

Subroutine BULK

Called from: STEADY.

Input: (i) Argument: IX.

(ii) COMMONS: TVOID, INPT, REDY, BNDY, NOHD, BOIL, BOIL2, BALK.

Performs: Calculation of the distributions of liquid and steam mass flow rates, void

fraction, steam quality, and other parameters of interest at each node along the bulk boiling

region. It also determines the position at which boiling would begin if subcooled boiling did

not occur and the fraction of channel power deposited in its nonboiling length.

Output: COMMONS: BOIL, BOIL2, BALK.

Subroutine COEFR

Called from: FLOW.

Input: (i) Argument: IX.

(ii) COMMONS: DATA, INPT, REDY2, REDY, SETP2, SETP, PRPR2,

BNDY, NOHD, NOHD2, BOIL, BOIL2, FRCT2, FRCT, PRP12, OMGP2, OMGP.

(iii) Output of the subroutine: OMEGAP.

Performs: Calculation of those coefficients in the set of dynamic equations in App. B 10

and B1 1, which are frequency independent. Since some of the D coefficients of the

• Citations of equations, appendices, and paragraphs throughout Sect. 3 refer to Reference 1

unless otherwise specified.

14
subcooled boiling region in App. B10 are frequency dependent, a set of coefficients B, which

contain their frequency independent part, is generated here.

Output: COMMONS: TCOEFR, CEFR, CEFR2, CEFR3.

Subroutine DCUMP

Called from: BOUNDY, NODE

Input: Arguments I, ZM

Performs:

Output: DELPC

Subroutine DEFNRR

Called from: COEFR.

Input: Arguments: NR, DZ, DZR.

Performs: It defines the integration mesh for the coolant equations in the non-heated

boiling sections of the channel. This is a provision to account for the possibility of having a

series of different pipes connected at the exit of the heated channel. The length of the mesh

is taken as DZ, the mesh length in the heated boiling section. The length of a non-heated

section is stored in DZR(I), and the maximum number of nodes allowed in any of these

sections is NR _ 10, that is, the value NRR(I) _• 10 where I is the non-heated section index.

Output: Argument: NRR =_ number of nodes in the non-heated boiling section under

consideration.

Subroutine DEL TP

Called from: STEADY.

Input: COMMONS: DATA, INPT, REDY, SETP, SETP2, PRPR2, BNDY, NOHD,

NOHD2, BOIL, BOIL2, FRCT, PRP12, OMGP, OMEGP2.

Performs: Calculation of the flow pressure losses at the inlet and outlet orifices and at

every node along the length of the channel. It uses the subroutine PREPHI for those nodes

in boiling sections of the channel.

output: COMMON: DELTP.

Subroutine DRUMWI

Called from: FLOW.

Activated by: IOP(1) < 2.

Input: (i) Argument: IX.

(ii) COMMONS: SETP2, SETP, PRPR2, PRPR, BNDY, NOHD, NOHD2,

BOIL, BOIL2, BALK, FRCT2, FRCT, DLTP, PRPI2, PRPI, OMGP2, OMGP, CEFR, CEFR3,

CEFR2, REACOE.

Performs: Saving out of core the steady state thermohydraulic parameters of the coolant

for each node along the length of channel IX. The map thus created will be used to define the

initial conditions for subsequent dynamics calculations.

Output: Logical unit 1: Sequential unformatted writing of the content of the COMMON

statements listed above.

Subroutine DRUMW2

Called from: FLOW.

Activated by: IOP(1) < 2.

Input: COMMONS: IIOP1, DATA, INPT, INPT2, REDY2, REDY, ADD1.

Performs: Saving out of core of the input data and the system parameters determined by

the subroutine READY.

Output: Logical unit 2: Sequential writing of the content of the COMMON statements

15
listed above.

Subroutine DRUMW3

Called from: FLOW.

Activated by: IOP(1) < 2.

Input: (i) Argument: IX.

(ii) COMMONS: ADIN, FUL, FULIX.

Performs: Saving out of core of the thermal parameters of the fuel rods within a channel

for subsequent use in the dynamics calculations.

Outpu: Logical unit 1: Sequential unformatted writing of the COMMON statements

listed above.

Subroutine FLOW

Called from: MAIN.

Input: (i) Argument: NPAGE.

(ii) COMMONS: DATA, INPT, REDY, REDY2, IIOPI, PREBC, ADD1.

Performs: (i) Determination of the flow rate distribution among the fuel channels

subject to uniform inlet and outlet pressure boundary conditions. To this effect an iterative

procedure is followed. First, the thermohydraulic conservation equations are solved along the

length of a channel by means of the subroutines SETUP(IX) and STEADY(IX). The mass

flow rate, exit steam quality, and total pressure drop along the channel are retained prior to

repeating the process with another channel. Once the pressure drops for all channels are

known, they are compared to see if they are equal. If they are not equal, a new flow rate is

estimated for each channel and a new iteration is started.

The results of the first iteration are used to determine the partial derivative of the total

pressure drop with respect to the mass flow rate for each of the channels. These partial

derivatives are used to determine the tentative pressure drop to which all the channels will

converge and the corresponding flow rates. After the second iteration, a parabolic

interpolation scheme is used to estimate the new flow rates, using the results of the two

previous iterations. The maximum number of iterations provided for is 20.

(ii) Generation of the parameters needed as initial conditions for the

subsequent dynamic calculations. To this effect, once that pressure drop convergence has

been achieved, the subroutines SETUP(IX) and STEADY(IX) are called again to repeat the

calculations of the last iteration. The subroutines COEFR(IX) and FUEL(IX) are called this

time. Prior to starting with another channel, all parameters generated for the channel are

transferred to a disk by means of the subroutines DRUMW1 (IX) and DRUMW3(IX) and

printed, if desired, by calling the subroutines OUTST(IX), OUTCOR (IX) and OUTFUX(IX).

Output: (i) COMMON: TFLOW.

(ii) Logical unit 1: The output of the subroutines DRUMW1 (IX),

DRUMW3(IX).

(iii) Logical unit 2: The output of the subroutine DRUMW2.

(iv) Logical unit 6: Under the heading: ***FLOW***, the flow characteristics

of each channel type and the tentative pressure drop for the next iteration at every iteration

step.

Subroutine FRICT

Called from: STEADY.

Input: COMMONS: INPT, REDY, SETP, SETP2, PRPR, BNDY.

Performs: Calculation of the Reynolds number for each of the flow regions and the

corresponding Moody friction factor f and coefficient a2. It uses the function subroutines FF,

AF, and FPMU.

16
Output: COMMONS: FRCT, FRCT2.

Subroutine FUEL

Called from: FLOW.

Input: (i) Argument: IX.

(ii) COMMONS: DATA, INPT, SETP, SETP2, PRPR, BNDY, NOHD,

CEFR, CEFR3, ADIN, ADD1, IIOPI.

Performs: (i) Calculation of the temperature distribution across the fuel element at

every node as described in the fuel steady state calculation (paragraph 3.2.2.1).

(ii) Calculation of the parameters a, 03, and y for the dynamics analysis as

described in Eq. (3.2.42).

(iii) Saving into common FELi the characteristics of up to 10 vertical nodes

as specified by NFPRT at input time.

Output: (i) COMMONS: TFUEL1, FUL, FELl, FULIX.

(ii) Logical unit 6: Error message 3.

Subroutine INPUT

Called from: MAIN.

Input: Logical unit 5.

Performs: (i) Reading of the data which specify the steady state thermohydraulic

calculations to be performed. The subroutines INP1, INP2, and INP3 are used to input some

of the arrays.

(ii) Consistency check of some of the input data.

Output: (i) COMMONS: INPT, INPT2, DATA, REDY, REDY2, ADD1, ADIN, JJJ,

PREBC, TINPUT, ICH, COFC.

(ii) Logical unit 6: Error message 1.

Subroutine INP1

Called from: INPUT

Input: NO, NNN

Performs: Reads an array of integers of length NO in the standard format 8110 for the

first line and 1 OX,7110 for the continuation lines

Output: NNN

Subroutine INP2

Called from: INPUT

Input: NO, NNN, AAA, ID

Performs: Reads a two-dimensional array of floating point variables in the standard

format 10X,7E10.5. The row index is input in ID.

Output: AAA(ID, *)

Subroutine INP3

Called from: INPUT

Input: NO, AAA

Performs: Reads an array of floating point variables in the standard format 110,7E10.5

for the first line and 10X,7E10.5 for the continuation lines

Output: AAA(*)

Subroutine LINEAR

Called from: FTK, FTRC, TSATP, VGP, HFGP.

Input: Arguments: X, XF, YF.

17
Performs: Linear interpolation within the table XF, YF to determine the value of Y that

corresponds to that of X.

Output: Argument: Y.

Subroutine NODE

Called from: STEADY.

Activated by: BOUNGO 9 0.

Input: (i) Argument: IX.

(ii) COMMONS: DATA, INPT, REDY, SETP, PLRPR, PRPR2, BNDY,

Performs: (i) Definition of a new mesh along the channel so that there are N nodes

in the boiling length of the channel.

(ii) Transformation of the nodal parameters to the new coordinate system.

The parameters affected by the transformation are: nodal boundary ordinates ZCOM(J),

normalized cumulative power generation CCOM(J), homogenized enthalpy HAV(J), mixed

steam quality XMIX(J), and friction multipliers EKFL(J).

Output: (i) COMMONS: NOHD, NOHD2.

(ii) Logical unit 6: Error message 4.

Subroutine OMEGAP

Called from: COEFR and PREPHI.

Input: Arguments: J, GP, H, INO.

COMMONS: DATA, INPT, REDY, SETP, PRPR2, BOIL2.

Performs: Calculation of the Jones flow correction factor 0 for the Martinelli-Nelson twophase friction multiplier and its derivatives. J is the node index number, GP is the mass flow

rate in g/ cm2 s, H is the relative subcooling of the liquid phase, and INO is a flag to indicate

whether node J is in a heated section or not.

Output: COMMONS: OMGP, OMGP2.

Subroutine OUTCOR

Called from: FLOW

Input: IX, NPAGE

Performs: Outputs the steady state core conditions to the OUT file.

Output:

Subroutine OUTFUX

Called from: FLOW.

Input: (i) Arguments: IX, NPAGE, NFU.

(ii) COMMONS: DATA, ADDIN, FELl.

Performs: Listing of the steady state thermal parameters of the fuel in each channel at

selected vertical nodes for all the radial nodes in the fuel.

Output: (i) Argument: NPAGE.

(ii) Logical unit 6: Under the heading ***FUEL(IX = )*** the calculated fuel

parameters are printed for each of the selected NFNO vertical nodes and the 10 radial nodes.

Subroutine OUTIN

Called from: MAIN.

Activated by: INPUGO > 1.

Input: COMMONS; DATA, INPT, INPT2, REDY, ADDI, ADIN, PREBC, COFC, JJJ.

Performs: Printout of the input data.

Output: Logical unit 6: Check print of the data read by the subroutine INPUT.

18
Subroutine OUTST

Called from: FLOW.

Activated by: IOP(4) • 0.

Input: (i) Arguments: IX, MMMM, NPAGE.

(ii) COMMONS: DATA, INPT, REDY, SETP, SETP2, PRPR, PRPR2,

BNDY, NOHD, NOHD2, BOIL, BOIL2, BALK, FRCT, FRCT2, DELTP, OMGP, OMEGP2.

Performs: (i) The calculation of the partial derivatives: pl3/aZ and a3/aX for each

node for printing purposes only.

(ii) Printing of the thermohydraulic nodal parameters as determined by the

subroutines READY and STEADY for each of the channels.

Output: (i) Arguments: NPAGE.

(ii) Logical unit 6: Under the heading: **STEADY***ITERATION MMMM:

tables of the thermohydraulic parameters for each node and channel.

Subroutine POWNW

Called from: BOUNDY

Input: none

Performs:

Output ZF, F (in common /POWNW1/)

Subroutine PREPAR

Called from: STEADY.

Input: (i) Argument: IX.

(ii) COMMONS: DATA, REDY, STEP, SETP2.

Performs: (i) For a total of NP + 1 nodes, the calculation of the normalized

cumulative power distribution CIN(I) at the nodal boundaries along the channel IX. The nodal

mesh Z(I) is the one defined when reading in the power distribution in subroutine INPUT. The

entrance to the first node is represented by the index I = 1.

(ii) Calculation of the inlet mass velocity wo in g/s, the base reference inlet

linear velocity u0 in cm/s, and the average heat flux through the heating surfaces in the

channel qave cal/cm 2 s.

Output: COMMONS: PRPR, PRPR2.

Subroutine PREPHI

Called from: DELTP

Input: COMMONS: INPT, REDY, SETP, SETP2, BOIL, BOIL2.

Performs: Calculation of the Martinelli-Nelson two-phase friction multiplier )2 and its

derivative by means of the set of coefficients COEFF in BLOCK DATA as described in App.

B5.

Output: COMMONS: PRPI, PRPI2.

Subroutine READY

Called from: MAIN.

Input: (i) Argument: NPAGE.

(ii) COMMONS: DATA, INPT.

Performs: (i) Determination of the physical characteristics of the coolant water at the

operating conditions in metric engineering units. This is accomplished by means of the

function subroutines: THCL, TSATP, HFGP, VOLUME, VGP, FN, FPMU, FPK, and FPRAN.

(ii) Calculation of the factors ks and r for the slip correlation, according to

Eq. (B2.2) in App. B2.

(iii) Calculation of the coefficients ai for the Martinelli-Nelson two-phase

19
friction multiplier ý2 according to Eq. (B5.2) in App. B5.

(iv) Estimation of the initial distributions of coolant flow rates and power

generation or exit quality among the channels in the core for the iterative procedure

described in Sect. 3.2 of Reference 1.

Output: (i) Argument: PAGE.

(ii) COMMONS: TREADY, REDY, REDY2.

(iii) Logical unit 6: (a) error message 2; (b) under the heading

***READY***, the fundamental thermohydraulic characteristics of the reactor core.

Subroutine SETUP

Called from: FLOW.

Input: (i) Argument: IX.

(ii) COMMONS: INPT, INPT2, JJJ, REDY.

Performs: (i) The selection of the parameters specific to channel type IX from the

arrays in the subroutines INPUT and READY.

(ii) Calculation of the power fraction or exit quality corresponding to the

new estimate of the channel flow rate during the iterative procedure.

Output: COMMONS: SETP, SETP2.

Subroutine STEADY

Called from: FLOW.

Input: (i) Argument: IX.

(ii) COMMONS: INPT, BNDY, BOIL, BOIL2, BALK, TBOUND, TNODE,

TSUBOI, TBULK.

Performs: (i) Initialization to zero of the content of the COMMONS BOIL, BOIL2 and

BALK.

(ii) Manages the solution of the thermohydraulic equations of the coolant

flow by calling the subroutines PREPAR, BOUNDY, NODE, SUBOIL, BULK, FRICT, and

DELTP in succession.

Output: COMMONS: BNDY, BOIL, BOIL2, BALK, TSTEAD.

Subroutine SUBOIL

Called from: STEADY

Input: (i) Argument: IX.

(ii) COMMONS: DATA, INPT, REDY, REDY2, SETP, SETP2, PRPR,

PRPR2, BNDY, NOHD, NOHD2, ADDI.

Performs: Calculation of the liquid and steam mass flow rates, liquid enthalpy, void

fraction, and other parameters of interest at each node along the subcooled boiling region of

the channel. To this effect the continuity and energy equations are solved as described in

Sect. 3.2 of Reference 1. The basis for the iterative procedure followed is that of Reference

4, p. 31. The function subroutines FN, FPK, TD, and HNBP and the subroutine VOID are

used.

Output: (i) COMMONS: BOIL, BOIL2.

(ii) Logical unit 6: Error message 5.

Subroutine VOID

Called from: SUBOIL and BULK.

Input: (i) Arguments: IX, M, X02, H02.

(ii) COMMONS: DATA, INPT, REDY, SETP, BOIL2, ADD1.

Performs: Calculation of the void fraction as a function of steam quality and relative

subcooling of the liquid phase as described in Sect. 3.2 of Reference 1.

20
Output: (i) Argument: B02.

(ii) COMMON: TVOID.

(iii) Logical unit 6: Error message 6.

BLOCK DATA

Performs: Initialization of the variables in the COMMON statements DATA and FNCT

prior to execution of the first statement in the MAIN program.

The content of this data block is:

SG = g - the gravitational acceleration in cm/s,

SBPO, SBP1 = ci and di respectively table of coefficients for the Jones

two-phase flow friction multiplier,

BBB = bj -= table of fitting coefficients of the Martinelli-Nelson friction factor

(2, as described in App. B5.

A tabulation, based on the ASME steam tables, of the saturation temperature,

the specific volume of saturated steam, and the enthalpy of evaporation of water vs pressure

(from 0.2 to 3206.2 psia) is implemented by means of: ARG-ARG2 as pressure in psia, FCTI

FCTII as Tsat in OF, FCT2 FCT22 as vs in ft3

/lb, and FCT3 FCT33 as hfo in Btu/Ib.

Function Subroutine AF

Arcqument: f the smooth Moody friction factor.

Output: a 2 - the exponent of the Reynolds number in the general expression for the

Moody friction factor as described in Eq. (B7.4a).

Function Subroutine AF2

Argument: fl, the friction factor model II.

Output: a 2,

11 the exponent of the Reynolds number in the general expression for the

generic single-phase friction factor model II (Reference 13).

Function Subroutine BESSI

Arguments: N order of the modified Bessel function,

X argument.

Output: The value of IN(X), the modified Bessel function of order N, for the argument

X. The algorithm used is valid for N < 3000 and X < 87. In the program, the subroutine FUEL

uses the N = 0, that is, lo(X), function to represent the radial distribution of the power

generation within the fuel pellets.

Function Subroutine CHECK

Arguments: P pressure in kg/cm 2,

T temperature in 0C,

N and M = indices.

Output: It fixes the lower and upper bounds for P and T so that if the input value is out

of bounds, the boundary value is returned to the calling program. The index N determines

which of the two variables is being checked; that is, N = 1 specifies pressure, while N = 2

specifies temperature. The bounds are

M = 1: 0.006• P< 600.0; -1 T• 505.7;

M = 2: 0.006•< P•< 250.0; -1 <T < 505.7;

M = 3: 0.006<• P<• 600.0; -1 <T:< 800,0.

Function Subroutine FF

Argument: Re = the Reynolds number.

21
output: The smooth Moody friction factor f corresponding to the value of the

argument. It follows the description in App. B7. A maximum of 50 iterations is programmed

for the solution of the transcendental Eq. (B7.3); if convergence is not achieved, that is, f5o -

f49 > 10-5 then the relation,

f = 0,213 Re-° 214

is used.

Function Subroutine FF2(REO)

Argument: REO - the Reynolds number.

Output: The friction factor f 2 model II corresponding to the value of the argument. The

equations of this model are:

f2 = MAX(FF, FFLAM)

FFEm -64.0

Re,

FF=AN-[ )+ CN]DN 1+ (BN R.RI

Re,]

where AN, BN, CN, DN are input-user data, Rr is the relative roughness, and Re[ is the liquid

Reynolds number

Function Subroutine FFFI(A__,BJ

Internal function called from: COEFR

Function Subroutine FFF2(ABJ

Internal function called from: COEFR

Function Subroutine FN(P, T)

Arguments: P pressure in kg/cm 2,

T temperature in 0C.

Output: Liquid saturation enthalpy in cal/g.

Function Subroutine FPK(P, 7)

Internal function called from: HNBP, READY, SUBOIL

Function Subroutine FPMU(P, 7)

Arguments: P pressure in kg/cm2,

T temperature in 0C.

Output: Liquid dynamic viscosity in g/cm2 s.

Function Subroutine FPRAN(P,T)

Arguments: P pressure in kg/cm 2,

T temperature in °C.

Output: Prandtl number for water, as a function of pressure and temperature. The

correlations used are those of the code STABLE (Reference 10, pp. 73 74).

Function Subroutine FTDKDT

Argument: T - temperature in 0C.

output: The temperature derivative of the heat conductivity of the UO2 fuel pellet. It is

based on the correlation used in the subroutine FTK.

22
Function Subroutine FTK(T)

Internal function called from: FTDKDT, FUEL

Function Subroutine FTRC(RHOF_ 17)

Internal function called from: FUEL

Function Subroutine HBBP(PQ)

Internal function called from: COEFR

Function Subroutine HCL(PD, TD)

Internal function called from: THCL

Function Subroutine HFGP(P)

Ar-gument: P pressure in atm.

Output: Evaporation enthalpy in cal/g of water, as a function of system pressure P.

The subroutine LINEAR is used to interpolate from the tables defined in BLOCK DATA prior

to the application of the system of units transformation factors.

Function Subroutine HNBP(PT,G,D)

Arguments: P pressure kg/cm2

T temperature °C,

G flow rate g/s,

D hydraulic diameter cm.

Output: The convective heat transfer coefficient between the coolant and the fuel clad,

in cal/(cm 2 °C s).

Function Subroutine POLY(N,A,X)

Internal function called from: VSCV.

Performs: Evaluates a polynomial trough succesive multiplications.

Function Subroutine TD(PH)

Arguments: P pressure in atm,

H enthalpy in cal/g.

Output: The liquid temperature in °C as a function of pressure and liquid enthalpy. An

iterative procedure is used based on the function subroutine FN (P,T).

Function Subroutine THCL(HP)

Internal function called from: READY

Function Subroutine TJLP(PQZH, TSA 7)

Arguments: P pressure in kg/cm2,

QZ - surface heat flux cal/cm2 s,

H - convective heat transfer coefficient cal/cm 2 °C,

TSAT =- saturation temperature °C.

Output: The temperature of inception of subcooled boiling in 0C. The Jens Lottes

correlation in Eq. (3.2.4) is used.

Function Subroutine TSATP(P)

Argument: P - pressure in atm.

23
Output: Saturation temperature in 0C.

Function Subroutine VGP(P)

Argument: P =- pressure in atm.

Output: Specific volume of saturated steam in g/cm3 .

Function Subroutine VOLUME(PT)

Arguments: P pressure in kg/cm2

,

T temperature in 0C.

Output: Specific volume of liquid water in cm3 /g.

Function Subroutine VSCV(TROV)

Argument: T =- temperature in 0C.

ROV - vapor density in g/cm 3.

Output: Dynamic viscosity of steam in N-s/m 2.

4.3. PREPARATION OF INPUT FOR LAPURX

4.3.1. Introduction

Data are read in by the subroutine INPUT. The input deck consists of a title card, data set cards,

and a last card. The title card can contain up to 72 characters. These characters will be printed

on each page of output to allow for easy identification of the case.

Each data set is preceded by an ID card that identifies the set. Thus, the sets need not be read

inma specific sequence, although sets 1 and 2 have to be read before sets 7 or larger; set 3

before sets 4 and 5; and sets 35, 36, and 37 as well as the groups (38, 39, 40, 41, 42, 43, 44),

(45, 46, 47, 48, 49, 50, 51, 52), (58, 59, 60, 61, 62), (63, 64, 65, 66), (67, 68, 69, 70, 71, 72) and

(73, 74, 75, 76) must be read in sequence. All ID cards carry their number in format (14). Most of

the time the data cards use an (110) format for integers and an (El 0.3) format for the real values.

All numbers are right justified. The input routine will accept commas to separate fields as long as

each filed is less than 10 characters long. Note, however, that most continuation cards start with

a "l0OX" format, so continuation card data must start in column 11 even if commas are used to

separate the fields.

The last card is equivalent to an ID number equal to zero; thus a 0 in column 4 will terminate the

input.

Once the input process is finished, the program will be executed.

When the calculations are finished, a new title card is sought by the subroutine INPUT. If a new

title is found, a new case will be run.

For consecutive runs, only those data sets that change from one run to the next need be

reentered.

Execution stops when the end of data card is found.

24
4.3.2. Input Description

4.3.2.1. First Card

Title ... columns 1 through 72.

4.3.2.2. Data Cards

Each data set is preceded by a card containing only the ID number of the corresponding data set

in format (14).

Description of the data sets:

Data set FORTRAN symbols Format Comment

ID No. I

1 P,TO,POWT,WT,CWBYP,CPROM P,CPRBYP,UNITS 8E10.3 if UNITS=O

PPSI,ENTBTU,POWT,WLBSHR,CWBYP,CPROMP,CPRBYP,UN ITS 8E10.3 if UNITS=1

2 NXE,N,ICOFC(IOP(J),J=1,7) 1015 NXE:_

NCHMAX

N !5 NOBMAX

P,PPSI

TO

ENTBTU

POWT

WT,WLBSHR

CWBYP

CPROMP

CPRBYP

UNITS

P System pressure P (kg/cm 2),

PPSI (psia).

To Core inlet water temperature (OC).

H Core inlet water enthalpy (Btu/lb).

Pth Total thermal power (MWth).

W Total flow rate (active core + bypass)

WT (kg/s); WLBSHR (lb/h).

Fraction of total flow through bypass.

Ypr Fraction of thermal energy given to water through moderation of neutron

and y-ray attenuation.

Same as CPROMP but in bypass region.

Option for units to be used with the following input variables:

Units 0 1

System pressure kg/cm psia

Inlet water condition 0C Btu/Ib

Core total flow kg/s lb/h

Channel mass velocity g/cm2 s 106 Ib/ft2 h

NXE

N

IOP(1)

Number of regions within which the thermohydraulic behavior of the

channels is uniform (- thermal region: same exit quality, same hydraulic

diameter, same flow rate, and same power shape).

Number of axial nodes desired in the boiling region of the channels.

Option for the calculation of the flow distribution among channels in the

steady state and for the calculation of TFs.

IOP(1) 0 1 2 3

flow distribution 0 X 0 X

transfer function 0 0 X X

0 calculated

X not calculated

25
IOP(2) Input option for GPOW(IX) and FPOW(IX) (see IDs 8 and 9,

respectively).

IOP(2) 0 1 2 3

GPOW(IX) X X 0 0

KFPOW(IX) I Xe I Pf I Xe I Pf I

0 necessary

X unnecessary

Xe read as exit quality

Pf read as relative power

IOP(3) Option to specify ZNBM, EKS, and R by input (see IDs 12 and 6,

respectively).

IOP(3) 0 1 2 3

ZNBM X 0 X 0

EKSandR X X 0 0

0 specified by input

X calculated by the program

IOP(4)

lOP(5)

IOP(6)

IOP(7)

If = 0, it will print the steady state results.

If • 0, it will not do it.

Edit option for the results of the subroutine FUEL1.

If = 0, yes. If = 1, no.

Input option for NFNO and NFPRT(I) (see ID 54).

If = 0, edit FUEL1 at default node numbers.

If • 0, data set 54 is required.

Output option for the results of COEFR.

If = 0, yes. If $ 0, no.

3 NPD,(NPPP(I),I=1,NPD) 8110 NPD M _ NPSMAX

10X,7E10.3,

4 ((DELZM(IP,I),I=1,NPPP(IP)),IP=I,NPD) (10X,7E10.3)

10X,7E10.3,

5 ((POWM(IP,I),I=1,NPPP(IP)),IP=1,NPD) (10X,7E10.3)

NPD Number of regions within which the axial power distributions are uniform

(power shape regions).

NPPP(IP) Number of axial nodes for the specification of the relative nodal power ir

the IP-th power shape region.

DELZM(IP,I) Azi Length of the I-th node in region IP (cm) I = 1:bottom of the core.

POWM(IP,I) f(z) Relative nodal power of the I-th node in region IP.

6 EKS,R,SAO,SA1,SA2,SA3,SA4 7E10.3 7E10.3 See 10P(3) ID @I

EKS

R

SAO, SA1,

SA2, SA3,

SA4

k, Constant for Bankoff's slip correlation.

r Constant to modify Bankoffs slip correlation; App. B2.

Sao Coefficients for the calculation of the power distribution within a fuel rod.

Sa4

26
7 kp.IX \ 8110, NXE•<

NXE,(NPOW(IX),IX=I ,NE (10X,7110) NCHMAX

NPOW(IX) Nuclear region number to which the IX-th thermal region belongs.

8 NXE,(GPOW(IX),IX=1,NXE) I 110x,7E10.3) ID 1

GPOW(IX) Gp Channel mass velocity.

(g/cm2)Ys if UNITS = 0.

(MIb/ft h) if UNITS = 1.

Unnecessary when IOP(2) = 0 or 1.

9 NXE,(FPOW(IX),IX=1,NX, 110,7E10.3, See IOP(2), I X ,FOWI)IIIN E (10X,7E10.3) ID 1

FPOW(IX) If IOP(2) = 1 or 3:

= fi, relative power generated in region IX.

If IOP(2) = 0 or 2:

= Xeix, exit quality for region IX.

10 NXE,(EKCPM(IX),IX=1,NXE) I110,7E10.3,

(10X,7E10.3)

EKCPM(IX) kc, Contraction coefficient at the inlet of the channels.

11 NXE,(EKEPM(IX),IX=1,NXE) I(10X,7E 10'.3

EKEPM(IX) ke Contraction coefficient at the exit of the channels. If the flow is expanded,

it has a negative value.

12 1 NXE,(ZNBM(IX),IX=1,NXE) 110,7E10.3, See IOP(3)

ZNBM(IX) Znb Position of the boiling boundary from the bottom of the channel (cm).

13 NXE,(ZELP(IX),IX=1,NXE) I110J7E10.3, (1ox,7ElO.3)

ZELP(IX) ZLp Height of the channel inlet measured from the point at which pressure is

uniform (cm). Used to calculate the APinlet due to gravity.

14 NXE,(NCH(IX),lX=1,NXE) 8110,

(10X,7110)

NCH(IX) Number of channels in thermal region IX.

15 NXE,(NROD(IX),IX=1 ,NXE) 8110,

(10X,7110)

NROD(IX) Number of fuel rods in a channel from region IX.

27
16 NXE,(NDIM(IX),IX=1,NXE) 8110,

(1 OX,711 0) 171 10,7E10.3, NTD•<

17 NTD,(ELM(ITD),ITD=1,NTD) (10X,7E10.3) NFBMAX

110X,7El0.3, FBA

18 NTD,(WHTM(ITD),ITD=1,NTD) 110,7E10.3,

(1 OX,7E1 0.3)

19 NTD,(AOM(ITD),ITD=1,NTD) 110,7E10.3,

(10X,7E10.3)

20 NTD,(ACELM(ITD),JTD=1,NTD) 110,7E10.3,

10X,7E10.3,

21 NTD,(DEM(ITD),ITD=1 ,NTD) 110,7E10.3,

(10X,7E10.3)

22 NTD,(BETOMM(ITD),ITD=1,NTD) 110,7E10.3,

_____ _______________________________________________(1 OX,7E1 0.3) ______

23 NTD,(BETGMMITD),ITD=1 ,NTD) 110,7E10.3,

(10X,7E10.3)

24 NTD,(FPM(ITD),ITD=I,NTD) 110,7E10.3,

(1 OX,7E1 0.3)

25 NTD,(CLAMPM(ITD),ITD=I,NTD) 110,7E10.3,

(10X,7E10.3)

NDIM(IX)

NTD

ELM(ITD)

WHTM(ITD)

AOM(ITD)

ACELM(ITD)

DEM(ITD)

BETOMM(ITD)

BETGMM(ITD)

FPM(ITD)

CLAMPM

Channel type of the IX-th thermal region.

Number of channel types.

L Height of the ITD-th type channel (cm).

Wht Heat transfer area per unit axial length of the channel (cm).

A0 Channel flow area (cm 2).

Ace], The same as AOM(ITD) (cm2).

This value is used for the calculation of the relative water density in the

reactivity feedback model. If the density reactivity coefficient is defined for

the density of active coolant.

A0 = Acei.

De Hydraulic diameter (cm).

Pa Parameter for the calculation of Q in the subcooled boiling region, App.

B6.

O3y Parameter for the calculation of y in the subcooled boiling region, App.

B2.

fp Adjustable parameter to correlate the calculated and measured void

fraction distribution in the subcooled boiling region. It is inversely related

to the fraction of energy invested in voids in the S.B. region, App. B3.

c' Adjustable parameter for the same purpose as fp. This one is directly

related to the mean lifetime of the voids in the S.B. region, App. B4.

28
8110, 26 NXE,(NFUEL(IX),IX=1,NXE) (18X,7110)

27 (OFM(IFUIFU=INTF 110,7E10.3, NTFU <

27 NTFU,(R HU),,U) (10X,TE10.3) NFRMAX

28 NTFU,(TFM(IFU),IFU=1,NTFU) 110,7E10.3,

(1 10,7E10.3)

29 NTFU,(RHOCCM(IFU),IFU=1,NTFU) 110,7E10.3,

(10,7E10.3)

30 NTFU,(EKCLM(IFU),IFU=1,NTFU) 110,7E10.3,

(1 ,7E1 0.3)

31 NTFU,(TCM(IFU),IFU=1,NTFU) 110,7E10.3,

(10X,7E10.3)

32 NTFU,(HGAPM(IFU),IFU=1 ,NTFU) 110,7E10.3,

(10X,7E10.3)

33 NTFU,(TGAPM(IFU),IFU=1,NTFU) 10,7E10.3,

_________________________________________(IOXE1 0.3) ______

NFUEL(IX) Fuel rod type of the IX-th thermal region.

NTFU Number of fuel rod types.

RHOFM(IFU) pfu Density of the fuel in the IFU-th type of fuel rod (g/cm 3).

TFM(IFU) Tf Fuel pellet diameter (cm).

RHOCCM(IFU) cc Cladding heat capacity (cal/cm 3 C).

EKCLM(IFU) k, Cladding thermal conductivity (cal/cm s OC).

TCM(IFU) tc Cladding thickness (cm).

HGAPM(IFU) hgap Gap heat transfer coefficient (cal/cm2 s °C).

TGAPM(IFU) tgap Gap width (cm).

34 NXE,(NFRC(IX),IX=1 ,NXE) 8110,

(10X,7110)

8110, NTFR•<

35 NTFR,(NFM(IFR),IFR=I ,NTFR) (1OX,7110)) NFMMAX

10X,7E10.3,

36 ((DZFM(IFR,I),I=1,NFM(IFR)),IFR=1,NTFR) (1OX,7E10.3)

10X,7E1 0.3,

37 ((EKFM(IFR,I),I=1,NFM(IFR)),IFR=1,NTFR) (1OX,7E10.3)

NFRC(IX)

NTFR

NFM(IFR)

DZFM(IFR,I)

EKFM(IFR,I)

Friction multiplier type of the IX-th thermal region.

Number of friction multiplier types.

Number of axial intervals for the IFR-th type friction multiplier.

Azfr Length of the I-th axial interval within which the friction multiplier is

uniform (cm). (I = 1 bottom of the core).

kf Friction multiplier for the I-th axial interval.

29
8110,

38 NXE,(NIN(IX),IX=1,NXE) (18X,7110)

S8110, NTIN

39 NTIN,(NCM(IIN),IIN=1,NTIN) (10X,7110) NCIMN!

(10OX,7110).3, MA

40 ((DZINM(IIN,I),I=1,NCM(IIN)),IIN=1,NTIN) 10X,7E10.3,

(10X,7E10.3)

41 ((AINM(IIN,I),l=1,NCM(IIN)),IIN=1 ,NTIN) 1OX,7E10.3,

(10X,7E10.3)

42 ((DEINM(IIN,I),I=1,NCM(IIN),IIN=1 ,NTIN) (1OX,7E10.3)

10X,7E10.3,

43 ((EKCINM(IIN,I),I=I,NCM(IIN)),IIN=1 ,NTIN) (1OX,7E10.3)

410X,7E10.3,

44 ((EKFINM(IIN,I),I=1 ,NCM(IIN)),IIN=1,NTIN) (10X,7E10.3)

NIN(IX) Channel inlet piping type of the IX-th thermal region.

NTIN Number of channel inlet piping types.

NCM(IIN) Number of pipes in series in the IIN-th type of piping.

DZINM(IIN,I) Azin Length of the l-th inlet pipe (cm) I = 1 - the nearest pipe to the channel

inlet.

AINM(IIN,I) Ain Flow area of the l-th inlet pipe (cm 2).

DEINM(IIN,I) Dein Hydraulic diameter of the I-th inlet pipe (cm).

EKCINM(IIN,I) Kcin Contraction coefficient at the boundary between the (1+1)-th and the I-th

pipes.

EKFINM(IIN,I) kf,, Friction multiplier for the l-th pipe.

45 NIV~~8110,

45 NXE,(NEX(IX),IX=1,NXE) 810X,7110

8110, NTEX _ 46 NTEX,(NEXM(IEX),IEX= ,NTEX) (10X,7110) NCEMAX

1 0X,7E10.3,

47 ((DZEXM(IEX,I),I=1,NEXM(IEX)),IEX=1,NTEX) (10X,7E10.3)

48 ((AEXM(IEX,I),I=1,NEXM(IEX)),IEX=1,NTEX) 10X,7E10.3,

(10X,7E10.3)

49 ((DEEXM(IEX,I),I=I,NEXM(IEX)),IEX=I,NTEX) (10X,7E10.3)

10X,7E10.3,

50 ((EKEXM(IEX,I),I=1,NEXM(IEX)),IEX=1,NTEX) (10X,7E10.3)

10X,7E10.3,

51 ((EKFEXM(IEX,1),I=I,NEXM(IEX)),IEX=I,NTEX) (10X,7E10.3)

10X,7E10.3,

52 ((THETEM(IEX,I),I=1,NEXM(IEX)),IEX=1,NTEX) (10X,7E10.3)

NEX(IX)

NTEX

NEXM(IEX)

DZEXM(IEX,I)

AEXM(IEX,I)

DEEXM(IEX,I)

EKEXM(IEX,I)

Zex

Aex

Dex

kex

Exit piping type of the IX-th thermal region.

Number of exit piping types.

Number of pipes in series in the IEX-th type of piping.

Length of the I-th exit pipe (cm) I = 1: the nearest pipe to the exit of the

channel.

Flow area of the l-th exit pipe (cm 2).

Hydraulic diameter of the I-th exit pipe (cm).

Contraction coefficient at the boundary between the (I-1)-th and the I-th

pipes.

Friction multiplier for the I-th pipe.

Angle of the I-th pipe tilt from vertical.

EKFEXM(IEX,I) kfex

THETEM(IEX,I) Oex

30
53 1(EPSINP(I),I=1,8) 8E10.3

EPSINP(I) si Convergence criteria for the iterative calculations.

*Calculation of the flow distribution among the channels:

EPSINP(1) (0.02) pressure drop relative error < El.

EPSINP(2) (0.01) total flow relative error < 82.

EPSINP(3) (0.01) mass velocity relative error < &3.

*Calculation of the flow quality in the subcooled boiling region:

EPSINP(4) (0.00002) flow quality error < 4.

*Calculation of void fraction:

EPSINP(5) (0.001) void fraction relative error < 85.

*Calculation of boundary between the subcooled and the bulk boiling regions:

EPSINP(6) (109) normalized degree of subcooling error < 86.

*Calculation of the steady state temperature distribution within a fuel rod:

EPSINP(7) (0.01) fuel temperature relative error < F.

EPSINP(8) (0.05) fuel heat transfer energy balance relative error E s8.

8110, See IOP(6)

54 NFNO,(NFPRT(I),I=1,NFNO) (1OX,7110) NFNO <

NAPMAX

NFNO Number of axial points to edit the transient response of the fuel rods.

NFPRT(I) Axial node number to edit the transient response of the fuel rods.

57 ZN, SN 1OX,2E10.3

ZN Parameter to vary the axial power shape peak location.

SN Parameter to vary the axial power shape peaking factor.

If ID 57 is input, the axial power shape is calculated as

P(Z) = [sin 7r( JIZN]SN

8110,

58 NXE,(NDIMV(IX),IX=I,NXE) (10X,7110)

59 NTDV,(NODV(NV),NV=I,NTDV) ___(1 OX,711 81X,110, 0) NAA NTVAMA

,NTDV)10X,7E10.3,

60 ((DZVM(NV,I),I=1 ,NODV(NV)),NV=1,NTDV) (10X,7E10.3)

10X,7E1 0.3,

61 ((AVM(NV,I),I=1,NODV(NV)),NV=1 ,NTDV) 1X,TE10.3)

10X,7E1 0.3,

62 ((DEVM(NV,I),I=1,NODV(NV)),NV=1 ,NTDV) (1OX,7E10.3)

NDIMV(IX)

NTDV

NODV(NV)

DZVM(NV,I)

AVM(NV, I)

Variable-area channel type of the IX-th thermal region.

Number of variable-area channel types.

Number of axial intervals for the NV-th type, variable-area channel.

Top height from bottom of the I-th variable-area axial interval of channel

variable-area NV-type (cm).

Boiling cell flow area of the I-th variable-area axial interval of channel

variable-area NV-type (cm 2).

31
DEVM(NV,I) Hydraulic diameter of the I-th variable-area axial interval of channel

variable-area NV-type (cm).

63 NXE,(NDIMLL(IX),IX=1,NXE) 8110,

______(1 OX,711 0)

64 NTDLL,(NODLL(NLL),NLL=1 ,NTDLL) 8110, NTDLLM

(10X,7110) NLLMAX

65 ((DZLLM(NLL,I),I=1,NODLL(NLL)),NLL=1 ,NTDLL) 10X,7E10.3,

10X,7E10.3,

66 ((EKLLM(NLL,I),I=1 ,NODLL(NLL)),NLL=1 ,NTDLL) (1OX,7E10.3)

NDIMLL(IX) Local pressure loss type of the IX-th thermal region.

NTDLL Number of local-loss channel types.

NODLL(NLL) Number of axial intervals for the NLL-th type, local loss channel.

DZLLM(NLL,I) Top height from bottom of the I-th local loss axial interval for channel

local loss type NLL-th (cm).

EKLLM(NLL,I) Local loss constant for the I-th axial interval of channel local loss type

NLL-th.

67 NTFU,(IFFM(1FU),IFU=1,NTFU) 8110, NTFUR X

(10X,7110) NFRMAX

68 NTFU,(ANM(IFU),IFU=1,NTFU) I10X,7E10.3,

(10X,7E10.3)

69 NTFU,(BNM(IFU),IFU=1,NTFU) I10,7E10.3,

110,7E10.3,

70 NTFU,(CNM(IFU),IFU=1,NTFU) (10X,7E10.3)

(110,7E 10.3,

71 NTFU,(DNM(IFU),IFU=1,NTFU) (10X,7E10.3,

(10X,7E10.3)

72 NTFU,(RRM(IFU),IFU=I ,NTFU) (10,7E10.3,

(1 OXE1 0.3)

These cards (67-72) are necessary only if the Friction Factor Model /I is used. The equations of

this model are:

FF = MAX(FF, FFLAM)

FFL = 64.0

Re,

where AN, BN, CN,

Reynolds number

IFFM(IFU)

ANM(IFU) AN

BNM(IFU) BN

CNM(IFU) CN

FF=AN. 1+(BN.Rr)+ CN IeeN

DN are input-user data, Rr is the relative roughness, and Re, is the liquid

Friction factor model II option, for the IFU-th fuel rod type.

If IFFM = 2, the friction factor model II is used.

Coeficient AN for the friction factor model II option, for the IFU-th fuel rod

type.

Coeficient BN for the friction factor model II option, for the IFU-th fuel rod

type.

Coeficient CN for the friction factor model II option, for the IFU-th fuel rod

type.

32
DNM(IFU) DN Coeficient DN for the friction factor model II option, for the IFU-th fuel rod

type.

RRM(IFU) Rr Relative roughness for the friction factor model II option, for the IFU-th

fuel rod type.

73 TF ,(FM (1U)1F=1NTU)8110, NTFU _<

73 TFU(IMM(FUIF=INTU)(10OX,711 0) NFRMAX

11 0,7E1 0.3,

74 NTFU,(AJUSTAA1 M(IFU),IFU=1,NTFU) (11X,7E10.3)

I1O,7E10.3,

75 NTFU,(AJUSTAB1M(IFU),IFU=1,NTFU) (11X,7E10.3)

110,7E10.3,

76 NTFU,(AJUSTAC1M(IFU),IFU=1,NTFU) (11X,7E10.3)

These cards (73-76) are necessary only if the Friction Multiplier Model II (Chisholm-Baroczy) is

used. The equations of this model are:

GMSQ = Ig PL

BGM =GMSQ

B = 55.0 BGM < 9.5

[AJUSTAB, .G

520.0 BGM-[AJUSTAB,-G] 5 9' <BGM<28

15000.0

GMSQ . [AJUSTAB, .GP-5

1= AJUSTAC1•

{1.+ (GMSQ- 1.)- [B. AJUSTAA,- XF° 9 - (1.- AJUSTAA,- XF)°09 + AJUSTAA, XF1 .8

where pg is the steam viscosity; 1ItF is the saturated liquid viscosity; p, is the saturated liquid

density; pg is the steam density; AJUSTAAj, AJUSTAB1 , AJUSTAC, are fit factors for G, XF and

Global (default=1.0, 1.0, 1.0); G is the mass flux (kg/m 2 s); and XF is the flow quality.

IFMM(IFU) Friction multiplier model II option, for the IFU-th fuel rod type.

If IFMM= 2, the friction multiplier model II is used.

AJUSTAA1M(IFU) Coeficient AJUSTAA, for the friction multiplier model II option, for the

IFU-th fuel rod type.

AJUSTAB1M(IFU) Coeficient AJUSTAB, for the friction multiplier model II option, for the

IFU-th fuel rod type.

AJUSTAC1M(IFU) Coeficient AJUSTAC, for the friction multiplier model II option, for the

IFU-th fuel rod type.

33
4.4. OUTPUT DESCRIPTION OF LAPURX

First, the date of the day is printed, followed by a listing generated by OUTIN of the input

parameters using the input mnemonics are described in App. C2 of Reference 1.

Next, a list of those system parameters directly calculated from the input parameters is

printed by READY.

What follows next depends on the selected options in card 2. Provided that they have not

been bypassed by the options, the subroutines produce output in the following sequence:

a. During the Iterations*

*Calculation controlled by IOP(1).

Let M be the number of iterations required to find the steady state flow distribution. Then,

FLOWm

prints its output M - 1 times, once for each iteration (i.e., for all m = 1 to M - 1).

b. When the Steady State Is Found

STEADYM(IX)t FUELX(IX)t COEFR(IX)-*

Print their output NXE times, once for each channel, that is, for all IX = 1 to NXE, followed

by the output of FLOWM. The subindex M indicates the values corresponding to the

converged last flow iteration.

tOutput controlled by IOP(4).

tOutput controlled by IOP(5).

Output controlled by IOP(7).

Description of the Output Generated by These Subroutines

- OUTIN

Prints the INPUT parameters using the same mnemonics found in the input instructions.

- READY

The parameters printed are:

Mnemonics Math & Units

IA-INI I IAL

POWER RATIO

TO=TIN

TSAT

HO

HF

HFG

RHOLIN

RHOL

RHOS

MU(TO)

MU(TSAT)

K(TO)

K(TSAT)

(Dix

ZNCHix (Dix

ix

(OC)

(°C)

(cal/g)

(cal/g)

(cal/g)

(g/cm 3)

(g/cm3)

(g/cm 3)

PTO (kg s/cm2)

l-ITsat (kg s/cm 2

)

kTo ( cal

k "at -

1 calsC" kTsat .scmo0

Description

Fraction of total power that is produced per fuel bundle

pertaining to region IX, it will remain constant unless the

exit quality is the variable fixed by NOPT(5).

Water inlet temperature.

Water saturation temperature.

Enthalpy of inlet water.

Enthalpy of water at TSAT.

Enthalpy of evaporation of water.

Density of inlet water.

Density of water at TSAT.

Density of steam at TSAT.

Dynamic viscosity of water at TO.

Dynamic viscosity of water at TSAT.

Heat conductivity of water at TO.

Heat conductivity of water at TSAT.

34
PRAN(TO)

PRAN(TSAT)

ETA

PrTo

PrTsat

pvs

J hfg

Xl

Prandtl number for water at TO.

Prandtl number for water at TSAT.

= 1 - RHOS/RHOL.

Mechanical to thermal power ratio of the evaporation

process.

Correction coefficient for the Bankoff steam water slip

correlation

KS,R ks, r

- STEADY(IX)

The output of this subroutine is performed by OUTST. Note: those variables whose headings in

the printout are self explanatory are not included in this description.

Under the heading "Subcooled and Bulk Boiling Regions"-

I i .Node number

Z z (cm) Vertical position of the lower node boundary.

DP3/DZ

DP3/DX

DPAC

DPDH

DPFR

LP (cm1) az

a13

az

(kg/cm2)

(kg/cm 2)

(kg/cm 2)

Void fraction gradient.

Slope of void fraction vs quality.

Pressure drop from inlet due to acceleration.

Pressure drop from inlet due to gravitation.

Pressure drop from inlet due to friction.

Under the heading "Subcooled Boiling Region":

Subcool enthalpy hfh EnthalpN hfg

FS01 Fraction

the node

FS02 F_ Fraction

difference to saturation, relative to hfg.

of energy invested in voids at the entrance of

,. See App. B3.

of energy invested in voids at the exit of the

x dependent factor for the calculation of the

fetime in the SBR. See App. B4.

FPH1 F(4)

In the next page:

NNB

NSB

NBB

QAV

WO

UO

qav (cal/cm 2 s)

coo (g/s)

Uo (cm/s)

Number of nodes in the nonboiling region (NBR).

Number of nodes in the subcooled boiling region (SBR).

Number of nodes in the bulk boiling region (BBR).

Average heat flux rate in the channel.

Total mass flow rate through the channel.

Volumetric flow rate per unit area (fluid velocity) in the

channel at saturated liquid density equivalent mass.

Crude friction factor multiplier for the NBR.

Parameter for the calculation of the mean lifetime of the

bubbles in the subcooled boiling region.

Node transient time corresponding to the stream density

equivalent mass fluid velocity.

KFNB

LAMO

TAUK

kfrlb

x 0 (S.1)

tk (S)

35
C17

TZNB

ZNB

ZB

ZBB

IZNB

IZB

IZBB

T.1 (°C)

Znb-Zl (cm)

Zb-Z2 (cm)

Zbb (cm)

Eizbb

hz.b (cal/g)

Ah (cal/g)

Pz,. (g/cm3)

p (g/cm3)

Cp.,, (cal/g IC)

Cp., (cal/g 0C)

c P vg (cal/g °C)

HZNB

DELTAH

RHOZNB

RHOLB

Bulk boiling boundary subcooling to void transformation

factor.

Temperature of the water at the subcooled boiling

boundary (SBB).

Position of the SBB.

Position of the boiling boundary if subcooled boiling did

*not occur.

Position of the bulk boiling boundary (BBB).

Fraction of power deposited in the coolant in the NBR.

Fraction of bundle power invested in heating the water to

saturation temperature.

Fraction of power deposited in the coolant below the

BBB.

Enthalpy of the water at the SBB.

Enthalpy increment in the NBR.

Density of water at the SBB.

Average density of water below the BBB.

Specific heat of water at the average temperature of the

NBR.

Specific heat of water at saturation temperature.

Average specific heat of water in the NBR.

Jones correction factor to the Martinelli-Nelson two -

phase friction multiplier. See App. B6.

Slope of the Jones factor vs flow rate relation.

CPSB

CPSAT

CPAVG

OMEGA

DOMDYO

( -ay aQ) 0

H02LAST H02 Degree of subcooling, relative to hfg, at the exit of the last

subcooled boiling node.

FNB fnb Moody single phase friction coefficient in the NBR.

FSB fsb Moody single phase friction coefficient in the SBR.

FBB fbb Moody single phase friction coefficient in the BBR.

A2NB a 2,, Reynolds number exponent for the nonboiling region.

A2SB a 2., Reynolds number exponent for the SBR.

A2BB a 2bb Reynolds number exponent for the BBR.

RE(TO) ReTO Reynolds number for the fluid at temperature TO.

RE(TSAT) ReT, Reynolds number for the fluid at temperature TSAT.

EKFL kf Surface friction factor multiplier for those nodes in the

boiling region.

In the next output page, the results of the solution of the momentum equation are printed in self

explanatory fashion.

- FLOW

M Flow redistribution iteration number.

36
QPL Average pressure drop in the channels (slope

weighted).

WTP Total core flow rate.

IX Channel type number.

NCH Number of channels of a type in the core.

POWFR Power fraction of the core total that is generated in one

channel.

QG (g/cm 2 s) Channel mass velocity.

QX Channel exit steam quality.

QP (kg/cm2

) Channel pressure drop.

WNO (g/s) Mass flow rate in a channel region.

POWT Nominal power generated in the core.

- FUEL

I iVertical node number.

NFU Cross sectional fuel ring zone number.

P (cal/s cm) Power generated in a ring per unit length.

T (OC) Fuel temperature at the center of the ring.

K (cal/s cm IC) Fuel thermal conductivity.

RHOCF (cal/cm3 OC) Specific heat of the fuel.

CAP (cal/cm °C) Heat capacity per unit length of fuel.

RES (s cm °C/cal) Thermal resistance. The last ring includes the gap and

the inner half of the clad. See Eq. (3.2.17).

ALP a

BET Parameters for the transient analysis. See Eq. (3.2.42).

GAM Y

PTP (cal/s cm) Total power generated in the node that is transmitted to

the coolant by convection.

TC (°C) Temperature at the center of the clad.

TG (OC) Temperature of the gap.

TAV (OC) Fuel average temperature.

RFUEL Average thermal resistance of the fuel.

RGAP Thermal resistance of the gap.

RCLD1 Thermal resistance of the inner half of the clad.

RCLD2 Thermal resistance of the outer half of the clad.

EPSW Effect of the film coefficient on the thermal conductance

of the outer half of the clad and water film. See Eq.

(3.2.34).

EPSY Effect of the flow rate on the convective heat transfer

coefficient. See Eq. (3.2.36).

- COEFR

The output of this subroutine if performed by OUTCOR:

HLO (cal/g) Liquid phase enthalpy.

TLO (OC) Liquid phase temperature.

RHO (g/cm3) Liquid phase density.

RHOA (g/cm3) Average nodal density. See Eq. (3.2.8).

HTC (cal/cm2 °C) Convective heat transfer coefficient. See App. BI.

37
DRS

DRH

DRB

ap- (g/cm3

)

as

cP•a (g/cm3)

aH

p•• (g/cm3)

a13

C1 C4

C4 C12

C13 C16,C21,C22

TAU =

TAUA =

TAUL =

TAUIP

P1 Zl

P2 U0

Azi

2u 0

C18 TNBR

U0 Azi

gL

Partial derivative of nodal density with respect to liquid

temperature factor s in the NBR. See Eq. (B9.8).

Partial derivative of nodal density respect to the degree

of subcooling with H in the SBR. See Eq. (B9.14).

Partial derivative of nodal density with respect to the

void fraction 03 in the boiling region. See Eq. (B9.14).

Coefficients of Eq. (B9.17).

Coefficients of Eq. (B9.4).

Coefficients of Eq. (B10.3).

Nonboiling region transit time constant.

Nodal transit time constant. See Eq. (B9.17).

Pressure differential between inlet plenum and boiling

boundary time constant. See Eq. (B10.2b). Note that

INBR = TAUIP (DZ = z).

Nodal momentum transfer time constant. See Eq.

(B310.2a).

Enthalpy to void TF at the boiling boundary. See Eq.

(B9.22b).

Flow to pressure drop from plenum to the boiling

boundary gain coefficient. See Eq. (B10.2b).

C17

C18

38
4.5. ALPHABETICAL DESCRIPTION OF THE LAPURW SUBROUTINES

MAIN

Subroutine ANGLE

Called from: PRINTA, PRINTB and RESPON.

Input: Arguments: A imaginary number,

B real number,

C angle in the range 90,

IFLG - quadrant flag,

ICOUNT =- counter,

I - number of vectors to be printed in a row.

Performs: Calculation of the true angle in degrees defined by A and B. The vectors IFLG,

ICOUNT, and I are used to track the displacement of the vector pointer defined

by A and B in the plane (R, Im) as the subroutine is called repeatedly. The

purpose is to account for crossings of the imaginary axis by the pointer. It adjusts

C by adding 1800 when the pointer moves clockwise, and by subtracting 1800

when it moves counterclockwise.

Output: Argument: C - corrected angle.

Subroutine COEFIW

Called from: TRANS.

Input: (i) Arguments: IX - channel index number,

CSS - complex driving perturbation.

(ii) COMMONS: INPT, REDY, SETP, PRPR, BNDY, NOHD, NOHD2, BOIL,

BOIL2, CEFR, CEFR2, FEEL.

Performs: Calculation of the frequency dependent coefficients needed to solve the

thermohydrodynamic equations.

Output: COMMON: CEFI.

Subroutine COND

Internal routine called from: COUPXS, DIF, JOSELE, PEPE.

Subroutine CONSIS

Internal routine called from: RESPON, SEARCH

Subroutine COUPXS

Internal routine called from: WEIGHT

Subroutine DBDEG

Called from: STABGH.

Input: Arguments: X real part vector,

Y imaginary part vector,

N length of the vectors X, and Y.

Performs: Determination of the magnitude in decibels and phase in degrees of the complex

number X + jY.

Output: Arguments: AMAG - magnitude in decibels,

PH - phase in degrees.

39
Subroutine DIF

Internal routine called from: JOSELE

Subroutine DRUMR1

Called from:

Input:

Performs:

Output:

WEIGHT and TRANS.

IX =_ channel number index.

Sequential reading of the coolant steady state parameter map of a channel into

the COMMON statements listed below. This subroutine is the read equivalent to

DRUMW1 in LAPURX.

COMMONS: SETP2, SETP, PRPR2, PRPR, BNDY, NOHD, NOHD2, BOIL,

BOIL2, BALK, FRCT2, FRCT, DLTP, PRPI2, PRPI, OMGP2, OMGP, CEFR,

CEFR3, CEFR2, REACOE.

Subroutine DRUMR2

Called from: MAIN.

Input:

Performs: Sequential reading of steady state data into the COMMON areas listed below.

This subroutine is the read equivalent to the write subroutine DRUMW2 in

LAPURX.

OutDut: COMMONS: IIOP1, DATA, INPT, INPT2, REDY2, REDY, ADD.

Subroutine DRUMR3

Called from: TRANS.

Input: IX =_ channel number index.

Performs: Sequential reading of the fuel steady state parameter map into the COMMON

areas listed below. This subroutine is the read equivalent to DRUMW3 in

LAPURX.

Output: COMMONS: ADIN, FUL, FULIX.

Subroutine DRUMW3

Called from: TRANS

Input: (i) Argument: IX.

(ii) COMMONS: ADIN, FUL, FULIX.

Performs: Saving out of core of the thermal parameters of the fuel rods within a channel for

subsequent use in the dynamics calculations.

Output: Logical unit 1: Sequential unformatted writing of the COMMON statements listed

above.

Subroutine FREQ

Called from: TRANS.

Input: IX -- channel type index,

CSS complex driving perturbation,

PSIA reactivity weighting factors array,

DKDRHO - density reactivity coefficients array,

CDYPB - complex driving flow rate perturbation,

CDRAMB - complex driving power generation perturbation,

CDTIN - complex driving coolant temperature perturbation,

IW frequency index number,

JJJ flag,

MX maximum number of channel types,

40
MP maximum number of thermohydraulic nodes,

MJ maximum number of subcores in a horizontal plane.

COMMONS: DOPP, REACOE, NODS, INPT,,REDY, SETP2, SETP, BNDY,

BOIL, BOIL2, CEFR, CEFR2, FEEL, CEFI, ADD1.

Performs: (i) Solves the continuity and energy equations for the coolant at each node along

the channel. As indicated in the response of the nodal mass flow rate, liquid

temperature and void fraction (hence, density) to the driving perturbations are

calculated.

(ii) The nodal density perturbations are multiplied by the reactivity coefficients

and weighting factors to yield reactivity contribution of each subcore.

(iii) The momentum equations are subsequently solved to generate the channel

pressure TFs.

(iv) The response of the average temperature of the fuel is determined by means

of Eq. (3.2.43). Subsequently, multiplication by the Doppler reactivity

coefficient and weighting factors produces the Doppler reactivity TFTFs.

Out.pu: (i) Argument: CDROA - coolant density response map.

(ii) COMMONS: FRIQ, FEL2.

Subroutine FUEL

Called from: TRANS

Input: (i) Argument: IX.

(ii) COMMONS: DATA, INPT, SETP, SETP2, PRPR, BNDY, NOHD, CEFR,

CEFR3, ADIN, ADD1, IIOP1.

Performs: (i) Calculation of the temperature distribution across the fuel element at every

node as described in the fuel steady state calculation (paragraph 3.2.2.1).

(ii) Calculation of the parameters a, P3, and y for the dynamics analysis as

described in Eq. (3.2.42).

(iii) Saving into common FELl the characteristics of up to 10 vertical nodes as

specified by NFPRT at input time.

Output: (i) COMMONS: TFUEL1, FUL, FELl, FULIX.

(ii) Logical unit 6: Error message 3.

Subroutine FUEL W

Called from: TRANS.

Input: (i) Arguments: IX = channel type index number,

CSS - complex driving perturbation's frequency.

(ii) COMMONS: INPT, BNDY, NOHD, CEFR3, ADIN, ADDI, FUL, FULIX.

Performs: Solution of fuel dynamics equations, as described in paragraph (3.2.23), for three

unity driving perturbations of power generation, coolant flow rate, and coolant

temperature, one at a time. This subroutine yields the set of fuel TFs aq, bq, cq

and aT, bT, CT for each node along the channel, as described in paragraph (3.2.3).

Output: COMMONS: FEEL, FEL2.

Subroutine INCROD

Called from: INPUTW

Inout:

Performs: Reading of control-rod data when cross sections are specified

Out_ ut:

Subroutine INPUTW

Called from: MAIN.

41
Input: Logical unit 5.

Performs: Reading of the data that specify the dynamic calculation to be performed. The

subroutines INPW1, INPW2, INPW3, INPW4, INPW5, and INPW6 are called to

input some of the data arrays. Three kinds of information are input: (i) reactor

constitution in terms of thermohydraulic channel types, composition of the nuclear

subcores; (ii) neutronic parameters; and (iii) output options.

Output: (i) COMMONS: TITLEW, INOUT, NNNNN, TESTSU, DOPP, RELOOP,

NNEUTR, XNEUTR, NDENRE, DENRE, WINK, NODS, MPLO.

(ii) Logical unit 6: Error message (self explanatory).

Subroutine INPWI

Called from: INPUTW

Input: NO, NNN

Performs: Reads an array of integers of length NO in the standard format 8110 for the first

line and 1 OX,71 10 for the continuation lines

Output: NNN

Subroutine INPW2

Called from: INPUTW

Input: NO, NNN, AAA, ID

Performs: Reads a two-dimensional array of floating point variables in the standard format

10X,7E10.5. The row index is input in ID.

Output: AAA(ID, *)

Subroutine INPW3

Called from: INPUTW

Input: NO, AAA

Performs: Reads an array of floating point variables in the standard format 110,7E1 0.5 for

the first line and 10X,7E10.5 for the continuation lines.

Output: AAA(*)

Subroutine INPW4

Called from: INPUTW

Input: MA, I1, 12, N1

Performs: Reads a two dimensional integer array in the standard format 1oX,7110.

Output: MA

Subroutine INPW5

Called from: INPUTW

Input: MA, I1, 12,13, N1, N2

Performs: Reads a three dimensional integer array in the standard format 1OX,7110.

Output: MA

Subroutine INPW6

Called from: INPUTW

Input: A, I1, 12

Performs: Reads a two dimensional real array in the standard format 10X,7E10.5.

Output:

Subroutine INXS2G

Called from: INPUTW

42
Input: none

Performs: Reads the cross section data

Output: A0, the cross section coefficients

Subroutine JOSELE

Called from: REACFA

Input: BETAXS_ IFT, ICT, D

Performs: Computes the void reactivity coefficient

Output: D

Subroutine LIMIT

Called from: MODE2, RESPON

Input: GM, HO

Performs: Estimates the limit cycle amplitude and average power increase given an

unstable reactor configuration

Output: PMAX, P0, RO, IER

Subroutine LINEAR

Called from: FTK, FTRC

Input: X,Y,XF,YF

Performs: Perform linear interpolation

output: Y

Subroutine MARGIN

Called from: MODE2, RESPON, SEARCH, SENS

Input: GH, W, NW

Performs: This subroutine estimates the phase and gain margins of an open-loop TF

Output: PM, WPM, GM, WGM

Subroutine MODE2

Called from: RESPON

Input: NPAGE

Performs: This subroutine estimates and prints the stability of the first subcritical mode as a

function of subcritical reactivity

Output: none

Subroutine NODOS

Called from: WEIGHT and TRANS.

Input: (i) Arguments: IX - channel index number,

X - vector containing the quantity to be integrated over the

core volumes,

MX maximum length of the vector X,

WE vector with the integrand weighting factors,

MW maximum length of the vector WE,

XAVKJ - zeroed 4 by 10 array,

L, LL, and LLL flags.

(ii) COMMONS: REDY, BNDY, NOHD, NODS, INPT.

Performs: (i) Calculation of the cores boundaries in terms of the thermohydrodynamic node

index numbers along a channel.

(ii) Calculation of the power generated in each nuclear core.

(iii) Calculation of the integrals

43
1i, LWdV; I.W V t XWdV; W1k, .,k , jk,

and the averaging ratio

.fo, X WdV

SW dV

where

Vix,k volume of channel ix within the nuclear core level k,

Vkj-- volume of the nuclear core of coordinates kj.

The effect of the flags is the following: -

If L<0 W(l)=WEforallI=1 toMX;

L > 0 W(l) = WE(I) for all I = 1 to MX;

If LL < 0 calculate the core's boundaries and power [steps (i) and (ii)];

LL _ 0 perform averaging ratio;

LL = N = number of nodes in the boilina reaion of channel, then the

If LLL < 0

LLL > 0

(i) Argument:

(ii) COMMONS: NO

X array needs to be padded with zeroes and shifted;

the vectors X and WE contain center of node values;

the vectors X and WE contain node boundary values;

DS.

Output:

Subroutine

Called from:

Input:

Performs:

Output:

NODREA

TRANS

NBN, IX

Gets the node numbers where there is a boundary of cross section types

Subroutine ONECOR

Called from: TRANS.

Input: (i) Arguments: 1W = frequency index,

1011 pointer index for direct access file 11,

CSS complex driving perturbation,

NDG number of delayed neutron groups,

ELSTR =- effective neutron lifetime,

BTF - vector of delayed neutron fractions,

RAMDA =- vector of delayed neutron lifetimes.

(ii) COMMONS: RCLOOP, INPT, TCHAN.

Performs: (i) Calculation of the forward neutronic reactivity to power TFTFs according to

the point kinetics model.

(ii) Calculation of the Doppler and density reactivity TFTFs.

Output: (i) Logical unit 11: Content of the COMMONS: TFCHAN and TFREAC.

(ii) COMMONS: NKINET, TFREAC.

Subroutine OUTFUW

Called from: RESPON.

Input: (i) Arguments: IX - channel index number,

NPAGE - printed page number,

NFU =-number of radial nodes in the fuel pin.

(ii) COMMONS: TITLEW, WINP, OUTF.

44
Performs: Printing of the fuel TFs aq, bq, cq and aT, bT, CT at selected elevations, with the

help of the subroutine PRINTA.

Output: Logical unit 6: Fuel TFTFs under the headings CAQ, CAT, CBQ, CBT, CCQ, and

CCT as a function of frequency.

Subroutine OUTFUX

Called from: TRANS

Input: IX, NPAGE, NFU_

Performs: Prints fuel information

Output:

Subroutine OUTINW

Called from: MAIN.

Activated by: INPWGO > 1.

Input: COMMON: DATA, TITLEW, INOUT, NNNNN, DOPP, RELOOP, NNEUTR,

XNEUTR, NDENRE, DENRE, NODS, WINP, MPLO.

Performs: Print out the input data and transformation of the frequency units from those read

(Hz) to the working units (rad/s).

Output: Logical unit 6: Check print of the data read by the subroutine INPUT.

Subroutine PRINTA

Called from: OUTFUW, WDSKPR.

Input: Arguments: IW- frequency index number,

W -= frequency vector in rad/s,

CA complex vector,

CB complex vector,

IFLG - quadrant flag,

COUNT = counter.

Performs: Calculation of the magnitude and phase in decibels and degrees, respectively, of

the elements of the two vectors CA and CB prior to printing them as a function of

frequency in hertz. Flags IFLG and ICOUNT track the quadrant to which an

element of the vector belongs so that the subroutine ANGLE can decide the

angle that corresponds to the next element of the vector.

Output: Logical unit 6: Frequency, real part, imaginary part, magnitude (in dB), and phase

(in deg) of the three vectors W, CA, and CB.

Subroutine PRINTB

Called from: RESPON

Input: Arguments: IW- frequency index number,

W - frequency vector in rad/s,

CA complex vector,

CB complex vector,

CC complex vector.

Produces: Printing of the four vectors W, CA, CB, and CB similarly to PRINTA.

Output: Logical unit 6: Magnitude (in dB) and phase (in deg) of vectors CA, CB, and CC

as a function of frequency W in hertz.

Subroutine REACFA

Called from: TRANS.

Input: (i) Arguments: IX- channel type index,

45
NBN - total number of thermohydraulic nodes in the channel,

RHOA = Pa - map of average nodal coolant density in the

channel,

RHOL = P2 - coolant density at saturation temperature.

(ii) COMMONS: NNNNN, NDENRE, DENRE, NODS.

Performs: Determination of the core map of density reactivity coefficients by means of the

function COEF and subroutine JOSELE operating on the input data tables.

Output: Argument: DKDRHO (IX, I, IJ) - map of density reactivity coefficients for

channel type IX in horizontal nuclear core region IJ.

Subroutine RECIRC

Called from: TRANS.

Input: (i) Argument: CSS - complex driving perturbation.

(ii) COMMONS: RCLOOP, TFCHAN.

Performs: Calculation of the recirculation loop TFs as indicated in Eq. (3.3.22).

Output: COMMONS: RCLOOP

Subroutine RESPON

Called from: MAIN.

Input: (i) Arguments: NPACE =- output printed page number,

NK number of cores in the vertical direction,

NJ number of cores in the horizontal plane,

NFU - number of radial nodes within the fuel pin,

NXE -=number of channel types.

(ii) COMMONS: WINP, ADIN, OUTF, TITLEW, NKINET, MPLO.

(iii) Direct access logical units 10, 11, and 12.

Performs: Management of the output of the program.

(i) It takes the information stored on the on line direct access disk and prints it in

an orderly manner.

(ii) Calls the stability analysis routine STABGH to calculate the natural frequency

and decay ratio of the flow channels and the core.

Subroutine SEARCH

Called from: MAIN

Input: NPAGE

Performs: Searches for the position of the complex pole in the full complex plane by

iteration and prints the results.

Output: none

Subroutine SENS

Called from: RESPON

Input: NPAGE

Performs: This subroutine recomputes the reactivity to power TF and estimates and prints

the decay ratio for various values of density reactivity coefficient, and recirculation

loop parameters

Outpu_: none

Subroutine SPLIN

Called from: STABGH

Input:

46
Performs: Spline interpolation

Output:

Subroutine STABGH

Called from: MODE2, RESPON, SEARCH, SENS.

Input: (i) Arguments: IOPT- flag,

IXGHF =- channel type number,

NPAGE - print out page number,

XGH - open-loop TFTF

(ii) COMMONS: WINP, MPLO.

Performs: The stability analysis of the system represented by the open-loop TF XGH.

Smith's method is used to infer the decay ratio and natural frequency as

described in Reference 11, pp. 120 128. A third order Lagrange interpolation

scheme is used to calculate the distance to the (-1, 0) point and subsequently to

generate smooth plots. IOPT = 1 indicates that the stability analysis corresponds

to that of a channel. IOPT = 0 corresponds to the whole core power stability.

Output: (i) Logical unit 6: Under the heading ***STABGH*** the results of the stability

assessment are printed.

(ii) COMMONS: MAGPH, ZXYW, GHDBDG.

Subroutine TRANS

Called from: MAIN, SEARCH.

Input: (i) Argument: NPAGE - printed output page number.

(ii) COMMONS: NNNNN, DOPP, NNEUTR, NDENRE, DENRE, WINP,

RCLOOP, NKINET, NODS, INPT, INPT2, REDY2, REDY, SETP.

(iii) Subroutines: DRUMR1, DRUMR3.

Performs: (i) Definition of direct access files 10, 11, and 12, and zeroing of some arrays.

(ii) Management of the frequency domain dynamic calculations.

First, the reactivity weighting factors for each of the cores are calculated by

the auxiliary subroutine WEIGHT. Second, a big loop is opened to

sequentially solve all the dynamic equations for one particular frequency at a

time. After storing the calculated TFTF in the three direct access files, the loop

is repeated for a new frequency.

The solution of the dynamic equations for a particular frequency proceeds as

follows:

(i) The steady state coolant and fuel parameters corresponding to a

particular channel are read from the intermediate storage device by

means of the subroutines DRUMR1 and DRUMR3.

(ii) Based on the steady state coolant density map and the input reactivity

coefficients, the weighted reactivity factor map is generated by subroutine

REACFA.

(iii) Based on the steady state fuel parameters, the fuel dynamic equations

are solved by means of FUELW for the current channel type and

frequency. This yields the fuel TFs aq,T bq,T, and Cq,T described in

paragraph (3.2.2).

(iv) Prior to the solution of the coolant dynamic equations, a set of frequency

dependent coefficients based on the steady state calculation is generated

by means of the subroutine COEFIW.

(v) The thermohydrodynamic equations of the coolant are solved

sequentially at each of the nodes along the channel by means of the

subroutine FREQ. This subroutine determines a map of the response of

47
coolant density, density reactivity, Doppler reactivity, and pressure drop

to driving perturbations of (1) channel inlet flow rate, (2) inlet coolant

temperature, and (3) power generation at any specific point along the

channel. To this effect, the solution of the thermohydraulic equation is

repeated several times. Unity driving perturbations of inlet flow rate,

power generation in the segment of channel length within the boundaries

of each nuclear core, and coolant inlet temperature are considered one at

a time while the rest of the independent parameters are kept equal to

zero. This process yields the set of channel TFTFs in paragraph 3.2.5.

Note that steps (i) through (vi) are repeated for each one of the channel

types.

(vi) Taking into consideration the lower plenum flow boundary condition, the

flow redistribution impedances in paragraph 3.3.4 are calculated, that is,

the particular driving perturbation to particular channel flow rate TFTF, for

each of the following perturbations: total core flow rate, inlet plenum

temperature, and power perturbation in any of the nuclear cores.

(vii) The reactivity feedback TFTFs [Eqs. (3.3.11) and (3.3.12)] are next

determined for each of the nuclear cores.

(viii) By means of the subroutine RECIRC and based on the input dynamic

characteristics, the recirculation loop TFTFs are determined. They

provide the link between the three driving perturbations mentioned.

(ix) By means of the subroutine ONECOR the forward neutronic TFTF is

determined.

(x) By combining the neutronic and feedback TFTFs the openand closedloop TFTFs are obtained.

Output: Logical unit 10 direct access: Channels fuel and coolant TFTFs.

Logical unit 11 direct access: Point kinetics TFTF

Logical unit 12 direct access: Multicore TFTF

Subroutine WDSKGH

Called from: RESPON.

Input: Arguments: NW- number of frequencies,

NF - direct access file number,

NOR - position of the pointer,

NRECL - number of records separating the values of a TFTF for

two consecutive frequencies.

Performs: Picks from the specified file all of the values pertaining to the TFTF of interest.

Output: Argument: X vector with the TFTF of interest.

Subroutine WDSKPR

Called from: RESPON.

Input: Arguments: W vector of frequencies in rad/s,

NW = number of frequencies,

KMOD2 - flag,

NF file number,

NR pointer position,

NRECF =- number of records separating two consecutive

frequencies of the same TFTF in the disk file,

NDEL - number of records separating the two TFTFs to be printed

on the same line.

48
Performs: Selectively picks the data stored in the direct access file and prints two TFTFs as

a function of frequency by means of PRINTA. Since PRINTA requires two TFTFs

when only one TFTF needs to be printed out, KMOD2 - 0, and thus the second

TFTF is printed as zero.

Output: Through PRINTA the content of the direct access file address by the pointer.

Subroutine WEIGHT

Called from: TRANS.

Input: (i) Argument: ALPKJ =- ratio of total core to individual nuclear core.

Performs: (i) Prior normalization of the input power shapes; the map of power squared

nodal weighting factors is determined.

(ii) By means of the subroutine NODOS, the power and power squared quantities

corresponding to each nuclear core are calculated.

(iii) The ratio,

to ./2dv

ALPKJ = akkj -

where VT and Vkj are the total and core "k,j" volumes, respectively, is

determined. Note, k and j are the vertical and horizontal subcore position

indices.

Output: (i) Argument: ALPKJ.

(ii) COMMONS: NODS, REACWT.

Function Subroutine BESSI(N,X)

Arguments: N order of the modified Bessel function,

X argument.

Output: The value of IN(X), the modified Bessel function of order N, for the argument X.

The algorithm used is valid for N < 3000 and X < 87. In the program, the

subroutine FUEL uses the N = 0, that is, 10(X), function to represent the radial

distribution of the power generation within the fuel pellets.

Function Subroutine BIMAG

Arguments: X magnitude in decibels,

Y phase in degrees.

Output: Imaginary component of the vector defined by X and Y.

Function Subroutines BREAL

Arguments: X magnitude in decibels,

Y phase in degrees.

Output: Real component of the vector defined by X and Y.

Function Subroutine COEF

Called from: REACFA.

Input: Arguments: RA ordinate;

NR number of points in the table, i.e., length of DR and RT;

DR vector with the ordinates of the data points;

RT vector with the abscissas of the data points.

Performs: Lagrange interpolation of the table defined by the vectors (DR, RT) to generate

the value that corresponds to the ordinate RA.

49
output: COEF - ordinate corresponding to RA in the table (DR, RT).

Function Subroutine EXPIO

Argument: X - a real number.

Outt: The value EXP10 = 1Ox.

Function Subroutine FTDKDT(T)

Argument: T - temperature in 0C.

Outpu: The temperature derivative of the heat conductivity of the U0 2 fuel pellet. It is

based on the correlation used in the subroutine FTK.

Function Subroutine FTK(T)

Internal routine called from: FTDKDT, FUEL

Function FTRC(RHOF_, T)

Internal routine called from: FUEL

Double Precision Function SEVAL(N, U, X, Y, B, C, D)

Internal routine called from: STABGH

50
4.6. PREPARATION OF INPUT FOR LAPURW

4.6.1. Introduction

Data are read in from logical unit 5 by the subroutine INPUTW. The input deck consists of a title

card, data set cards, and a last card. The title card can contain up to 72 characters. These

characters will be printed on each page of output for identification convenience.

Each data set is preceded by an ID card that identifies the set. The data sets need not be read in

sequence, but the following should be considered: set 1 must be read before any other set

except for sets 4, 17, 21, and 26; the sets within the groups (5, 6, 7, 8), (9, 10), (11, 12), (15, 16),

and (18, 19, 20, 21) must be read in order.

A 11 ID cards carry their number in format 14. All of the numbers are right justified.

The last card is equivalent to an ID number equal to zero, thus a 0 in column 4 will terminate the

input and initialize the execution of the program.

When the run is finished, a new title card is sought by the subroutine INPUTW. If the end of data

is found in logical unit 5, then the execution stops; otherwise, a new case will be input and run.

Note that for consecutive runs, only those data sets that change need to be entered.

Because the dynamic equations solved by LAPURW are based on linearization about the steady

state conditions calculated by LAPURX, the compatibility with the input of LAPURX must be kept

in mind when preparing input for LAPURW.

The link to LAPURX consists of sequential unformatted reading from the intermediate storage

device into the appropriate COMMON areas. Provisions for the correct files to be accessible

through logical units 1 and 2 must be made.

51
4.6.2. Input Description

4.6.2.1. First Card

Title ... columns 1 through 72.

4.6.2.2. Data Cards

Each data set is preceded by a card containing only the ID in format (14).

Description of the data sets:

Data set FORTRAN symbols Format Comment ID No._______________________________

NX <

1 NX, NJ, NK 315 NCHMAX

NJ = 1

______ ____________________________________________ NK= =1

NX Number of regions within which the fuel channels can be assumed

thermohydraulically equal. Hence, only NX representative channel types

need to be analyzed in detail.

Note that NX has the same value as NXE in the LAPURX steady state

run.

Number of nuclear subcores in a horizontal plane.

Number of nuclear subcores in the vertical direction.

Note that NJ x NK total number of nuclear subcores in which the reactor

core assembly is mathematically partitioned for analysis purposes.

NJ

NK

2 ((NCHXJ (IX,IJ)IX=1,NX),IJ=1,NJ) 1 1OX,7110 1 1

NCHXJ(IX,IF) Number of channels of type IX present in the horizontal nuclear mesh of

index IJ.

Note that SUM(NCHXJ(IX,IJ) for IJ = 1, NJ) = NCH(IX) in LAPURX.

1 3 (ZKIK),IK=1,NK) 10X,4E10.5

ZK(IK) (cm) Elevation, from the bottom of the core, of the upper boundary plane of the

vertical nuclear mesh of index IK.

1 4 1 TAUPY, GAINPY I 10X,2E10.5 I I

TAUPY

GAINPY

(s-) Time constant of the pressure to flow recirculation loop TFTF

Gain of the TFTF mentioned above.

52
NNEUT:<

NNDMAX

5 NNEUT, NDG 215 NDGA

NDG <ý

NDGMAX

6 ((IDGKJ(IK,IJ),IK=1 ,NK),IJ= ,NJ) 10X,7110 7 10X,7E10.5,

7 ((BTFR(IG,IN),IG=1,NDG),IN=1,NNEUT) (10X,7E10.5)

(10X,7E10.5,

8 ((XLAMD(IG,IN),IG=1,NDG),IN=1,NNEUT) 110X,7E10.5,

_____________________________________________(10X7E 10.5) _____

NNEUT Number of different sets of delayed neutron characteristics to be input.

NDG Number of delayed neutron energy groups to be read.

IDGKJ(IK,IJ) Identification index number to locate in the tables the delayed neutron

characteristics of a subcore (IK,IJ) (i.e., IN below).

BTFR(IG,IN) Table of delayed neutron fractions as a function of energy group IG and

subcore index IN.

XLAND(IG,IN) Table of delayed neutron decay constants as a function of energy group

IG and subcore index IN.

9 ((IRHOKJ(IK,IJ),IK=1,NK),IJ=1,NJ) 1OX,7110

10 NNERO,(RHOO(IR),IR=1,NNERO) 110,7E10.5, NNERO <

(10X,7E10.5) NROMAX

IRHOKJ(IK,IJ) Table of index numbers identifying a value of the table RHOO with the

initial reactivity of the subcore of indices (IK,IJ).

NNERO Length of the table RHOO.

RHOO Table of initial reactivity values.

11 ((IELSKJ(IK,IJ),IK=1,NK),IJ=1,NJ) 1OX,7110

12 NNELS,(ELST(IE),IE=INNELS) 110,7E10.5, NNELS _ S(10X,7E10.5) NNLMAX

IELSKJ(IK,IJ) Table of index numbers identifying a value of the table ELST with the

neutron lifetime in the subcore of indices (IK,IJ).

NNELS Length of the table ELST.

ELST Table of neutron lifetime.

NFT, NCOPOL 1415

((((A(IXST,IABC,ICT,IFT),IXST=1,7),IABC=1 ,NCOPOL),ICT=1,2),IFT= 10X,7E10.4,

1,NFT) (10X,7E10.4)

RHOLXS, RHOSXS 10X,2E10.4

NBT 1oX, 15

Input for I = 1, NBT {

13 IBT, NZXS(IBT) 1OX,1415

(ZXS(IZXS, IBT), IZXS = 1, NZXS(IBT)) 1OX, 7E10.4

(JFT(IZXS, IBT), IZXS = 1, NZXS(IBT)) 1OX, 7E10.4

Input for I = 1, NXE {

IX,(NCHXBXJ(IBT, IX), IBT = 1, NBT) 1OX, 7E10.4

A Two group cross section expansion coefficients. IXST refers to the cross

section type (1 - fast diffusion coefficient, 2 - thermal diffusion coefficient,

3 - fast absorption cross section, 4 - thermal absorption cross section, 5 -

53
JFT

NBT

NCHXBJ

NCOPOL

NFT

NZXS

RHOLXS

RHOSXS

ZXS

fast fission cross section, 6 - thermal fission cross section, 7 remonition

from fast group = absorption + scattering). IABC refers to the power

expansion coefficients. ICT = 1 is an uncontrolled bundle and ICT = 2 is a

controlled bundle. IFT refers to the fuel type.

Fuel type of axial interval IZXS for bundle type IBT.

Number of bundle types.

Number of channels of type IX that correspond to bundle type IBT.

Order of the cross section expansion in power series of the density.

Number of different fuel types.

Number of different axial intervals with constant fuel.

Liquid density at pressure used in calculation of A.

Steam density at pressure used in calculation of A.

Upper height of axial fuel interval IZXS for bundle type IBT.

Input for I = 1, NBT {

IBT, NZCR(IBT) 1OX, 1415

(ZCR(IZCR, IBT), IZCR =1, NZC) 10X,7E10.4 1

14 NNBC 1OX, 15

Input for I = 1, NNBC{

IBT, IX 1OX, 1415

(XNBC(IZCR, IBT, IX), IZCR = 1, NZCR(IBT)) 1OX, 7E10.4

NNBC Number of bundle types with nonzero number of controlled bundles.

Bundles not input are taken as uncontrolled.

NZCR Number of regions in which the number of bundles controlled can be

considered constant.

XNBC Number of bundles controlled in axial region IZCR of bundles type IBT

and thermohydraulic region (channel) IX.

ZCR Upper boundary of region IZCR.

Note: ID 14 must be read after ID 13. If IDs 13 and 14 are read, and IDs

number 11, 12, LAPUR will compute neutron lifetime. If IDS 18 through

21 are omitted, LAPUR will compute density reactivity coefficients.

15 ((ICOUPL(KJ1,KJ2),KJI=INKJ), KJ2=1,NKJ) 1 OX,7110 NKJ = NK*NJ

110,7E10.5, NNCOU _ 16 NNCOU,(COUPL(IC),IC=I,NNCOU) (10X,7E10.5) NNCMAX

ICOUPL(KJ1,KJ2) Table of index numbers identifying a value of the table COUPL with the

coupling coefficients between the subcores defined by KJ1 and KJ2,

where KJ = K1 + (J1 - 1)x NK for subcore (K1,J1).

NNCOU Length of the table COUPL.

COUPL Table of neutron coupling coefficients.

17 1CDOPP 1oX,E10.5

CDOPP ak

aT [% °C] Doppler reactivity coefficient of the fuel.

54
18 NOCOE,(LENG(IN0),IN0=1,N0COE) 8110, NOCDERA

(1 OX,711 0) NDRMAX_

19 (((IDNCOE(IX,IK,IJ),IX=1 ,NX),IK=1 ,NK),IJ=1,NJ) (10X,7110)

10X,7E10.5, LENG(I)

20 ((DENSR1 (I,J),J=I ,LENG(I)),I=I1,NOCOE) 10X,7E10.5) NDRNMAX

10X,7E10.5, LENG(I) _

21 ((REACTI (I,J),J=1,LENG(I),I=1,NOCOE) (10X,7E10.5) NDRNMAX

INOCOE0.) DR M

NOCOE

LENG(INO)

IDNCOE

DENSR1

REACT1

pf

ak

ap

Length of the table LENG.

Index table containing the lengths of each of the sets of reactivity tables

to be read into DENSR1 REACT1.

Identification index numbers to locate in the sets of tables the

correspondent to the channel IX and the subcore of indices (IK,IJ).

Set of tabulated values of relative water density.

[% g-1 cm3

] Set of tabulated density reactivity coefficients.

These two matrices define a set of tables of reactivity coefficients in units

of (%AK)/K per g/cm 3 as function of the relative water density. The set

that applies for a particular channel and subcore is specified by IDNCOE.

22 NW,(W(IW),IW=I ,NW) 22 N,(W(W~j=1,N) _F'10,7E1.5) I110'7E10"5' 10X,7E10.5) NFPMAX NW_< W

NW Number of frequency points at which the dynamic response is to be

calculated.

W co (Hz) Frequency.

(1415)

IOPTW(M)

IOPTW(1)

IOPTW(2)

IOPTW(3)

IOPTW(4)

IOPTW(5)

IOPTW(6)

IOPTW(7)

IOPTW(8)

IOPTW(9)

Output options.

If IOPTW(M) = 0, no output of type M is produced by the subroutine

RESPON.

If IOPTW(M) = 1 yes, output is produced.

The following outputs can be controlled:

Fuel TFs aq, aT, bq, bT, Cq, cT. See paragraph 3.2.2.

Channel nodal TFs = unity power perturbation response along a channel.

Channel TFs.

Core flow impedances zy, Zq, ZT.

Recirculation loop TFs.

Core feedback, point kinetics open-loop, and closed-loop TFTFs.

Multicore power perturbation to pressure at the boiling boundary TFTFs

of a channel, CCXK - Cxk. See Eq. (3.2.88c) and matrix C at App. B1 1.

Multicore inlet flow rate to moderator density reactivity contribution to a

core TFTFs CEKJ = Ek. See Eqs. (3.3.1 a) and matrix E at App. B11.

Multicore inlet temperature to moderator density reactivity TFTFs CHKJ

hk. See Eq. (3.3.1c) and vector h at App. B11.

55
IOPTW(10)

IOPTW(1 1)

IOPTW(12)

IOPTW(13)

IOPTW(14)

IOPTW(15)

IOPTW(16)

IOPTW(17)

IOPTW(18)

IOPTW(19)

IOPTW(20)

IOPTW(21)

IOPTW(22)

Multicore power to core flow TFTFs CZQKJ -= fq at Eq. (B12.2).

Multicore power to moderator density reactivity TFTFs, CFKPJK fk,k'.

See Eq. (3.3.1 b) and matrix F at App. B11.

Multicore power to channel flow rate TFTFs CMXJK = matrix M, in Eqs.

(3.3.10) and (B12.9).

Multicore power to total moderator reactivity feedback TF matrix CQ Q

in Eqs. (3.3.23a) and (B12.14).

Multicore total flow rate to total moderator reactivity feedback TF matrix

CR R in Eqs. (3.3.23b) and (B12.16).

Multicore inlet temperature to total moderator reactivity feedback TF

matrix CS = S in Eqs. (3.3.23c) and (B12.18).

Multicore core to core power TF matrix CHQ = matrix H in Eq. (3.3.17)

with Q as in Eq. (B12.14).

Multicore reactor core flow rate to individual core power TF ZR in Eq.

(3.3.20).

Multicore inlet temperature to individual core power TF vector Zs in Eq.

(3.3.20).

Multicore closed-loop reactivity to power TF matrix Z in Eq. (3.3.16).

Sum of the column elements of the feedback TF matrix H, that is,

CDENFBRow = SUM [Hrowcol] for all columns.

Sum of the column elements of the closed-loop TF matrix Z, that is,

CLOSEKRow = SUM [Zrow .ol] for all columns.

CRFBAC and CLTFPK, the, total core reactivity feedback and power

response to a unity reactivity vector perturbation.

CRFBAC = SUM[CDEFBROW] for all rows.

CLTFPK = SUM[CLOSEKRow] for all rows.

24 1 (ISTAB(IX),IX=1,NX+1) 1415,

(1415)

ISTAB(IX) Stability calculation options: = 0 - no; 1 - yes.

IX indicates channel type for which the hydrodynamic stability is to

be calculated. When IX = NX + 1 the total core open-loop TF is

analyzed if ISTAB(NX + 1) = 1.

1415 :J 25 1 (IPLOT(IX)'IX=INX+2) (45

IPLOT(IX) Plotting options: = 0 =_ no; = 1 yes.

IX refers to the open-loop TF of channel type IX.

IX = NX + 1 indicates total core open-loop TF Magnitude phase and

Nyquist plots are produced for the open-loops.

26 REAMUL 7E10.5

REAMUL Density reactivity coefficient multiplier. If REAMUL is negative, its absolute

value is taken, but a sensitivity calculation is performed on density

reactivity coefficient and recirculation loop parameters.

56
1 29 1NRHOSC,(RHOSC(I),1=1, NRHOSC) I10,7E 10.5 1 7

NRHOSC

RHOSC

Number of subcritical reactivities at which out of phase mode stability is

to be determined.

Array of values of the reactivity of the first subcritical neutronic mode.

Note: If ID 29 is read, LAPUR5 will estimate the stability of the first

subcritical neutronic mode with a parallel channel thermohydraulic

feedback. If the subcritical reactivity is known, use that value and

NRHOSC=1. NRHOSC>1 allows for a sensitivity calculation with little

cost.

30 1NSRCH 1 110

NSRCH Number of iterations in the full Laplace domain (with real part of s not

equal to 0., i.e., s = r + jw, r # 0). This is done to improve the estimate of

the decay ratio. It essentially multiplies the computation time by NSRCH

+1.

31 1 WPCNT E 10.4

WPCNT Flow rate (percent of nominal).

Note: If ID 31 is read after ID 4 (ID 4 may not be present), LAPUR5

computes a crude estimate of the gain and time constant of the

recirculation loop based on the flow rate. This is useful for scoping

calculations, but the correct value is recommended for important

calculations.

32 1 GAINPY, TAUPY, RL P2, RLP3, RL Z1, RL Z2 I8F10.0

Extended Recirculation Loop Transfer Function. If this card is present, the pressure-to-core flow

T.F used is:

RLPY - 6Y _ GAINPY[1+RL Zlxs+RL Z2xs21

6n [1+TAUPYxs+RLP2xs 2 +RLP3xs ]

GAINPY

TAUPY

RL P2

RL P3

RL Z1

RLZ2

Gain of the pressure to flow recirculation loop TF

(s-1) Time constant of the pressure to flow recirculation loop TF

Pole constant for the pressure to flow recirculation loop TF

Pole constant for the pressure to flow recirculation loop TF

Zero constant for the pressure to flow recirculation loop TF

Zero constant for the pressure to flow recirculation loop TF

57
4.7. OUTPUT DESCRIPTION OF LAPURW

First, the date is printed, followed by a listing produced by OUTINW of the input

parameters with the same mnemonics and self explanatory headings. What follows next is the

output generated by the subroutine NODOS. This subroutine produces the power in MW,

average coolant density, and void fraction in each of the nuclear subcores in which the reactor is

divided.

Next, the output of the TFs generated by the different subroutines is produced in the

order described below. The normalization factors for the TFs are the following:

For pressure: RHOL xLx L103 (kg/cm2), with RHOL in (g/cm 3) and L in cm.

RHOL is the density of water at saturation temperature.

L is the height of the channel.

For flow: WO is the channel mass flow rate at steady state conditions.

WT is the total core flow rate.

For power: POW is the total power of the channel in cal/s.

POWKJ is the total subcore power.

For temperature: 1 0C.

Description of the Output Generated by the Subroutines

***FUELW

The output of this subroutine is performed by OUTFUW, when invoked by RESPON, if

IOPTW(1) = 1. For each channel IX and selected vertical nodes I, the following nodal TFs are

printed as a function of frequency (o:

Mnemonics Math & Units Description

CAQ aq Coolant flow rate at the node level-to-heat flux from fuel

to coolant TF

CAT aT Coolant flow rate at the node level-to-average fuel pellet

temperature TF (of interest for Doppler reactivity).

CBQ bq Power generation rate in the fuel-to-heat flux to coolant

TF

CBT bT Power generation rate in the fuel-to-average fuel pellet

temperature TF

CCQ cq Liquid phase coolant temperature at the node level-toheat flux to coolant TF

CCT cT Liquid phase coolant temperature-to-average fuel

temperature TF

Note that CAQ aq and CAT aT are equal to (0. , 0. ) at those nodes within the boiling

length of the channel because of the assumption of independence of the convective heat

transfer coefficient.

***FREQ - TRANS

The output of this subroutine is performed from RESPON as follows:

Unity Power Perturbations (when IOPTW(2) = 1)

Under this heading the frequency response along the length of a channel to pressure,

power, and inlet temperature unit perturbation is printed.

The following prefixes are used with the variables printed:

M meaning magnitude in absolute units.

TH meaning phase in degrees.

58
R meaning real part.

I meaning imaginary part.

The variables whose responses are printed for all of the frequencies cO(Hz) are:

DKNB (%AK) Density reactivity response in the NBR.

DKSB (%AK) Density reactivity response in the SBR.

DKBB (%AK) Density reactivity response in the BBR.

DKN (%AK) Density reactivity response in the channel.

Y(NSB+I) Flow rate response at the entrance of the BBR.

Y(NBB+I) Flow rate response at the exit of the BBR.

Y(NR+I) Flow rate response at the exit of the exit.

H(NSB+I) Degree of subcooling response at the entrance of the

BBR.

Y(1) Flow rate response at the entrance of the SBR.

B(NSB+I) Void fraction response at the exit of the SBR.

B(NBB+I) Void fraction response at the exit of the BBR.

B(NR+1). Void fraction response at the exit of the exit pipe.

CDR Doppler reactivity response of the channel.

P(NR+I) Pressure response at the exit of the exit pipe.

P(NBB+I) Pressure response at the exit of the BBR.

P(NSB+I) Pressure response at the exit of the SBR.

P(1) Pressure response at the exit of the NBR.

The various TFS OF A CHANNI

the underlined heading. Note that the u

absolute units Ak.

CAX aix

CABX aix x bix

CBX bix

CCX Cix

CDX dix

CDYX

CDQX

CDTX

CEX

CFX

CHX

CLX

CMX

CNX

EL are printed (if IOPW(3) = 1) for all the channels under

nits of reactivity used in this section are the percent of

Flow/pressure TF for channel.

Open-loop TF of channel.

Channel flow rate to pressure at the boiling boundary TF

Channel power to pressure at the boiling boundary TF

Channel inlet temperature to pressure at the boiling

boundary TF

Channel flow rate to Doppler reactivity TF

Channel power to Doppler reactivity TF

Channel inlet temperature to Doppler reactivity TF

Channel flow rate to density reactivity TF

Channel power to density reactivity TF

Channel inlet temperature to density reactivity TF

Total flow rate to individual channel flow rate TF

Total core power to individual channel flow rate TF

Core inlet temperature to individual channel flow rate TF

dqix

dti,

eix

fix

hix

Lix

mix

Nix

***STABGH

This subroutine when invoked by ISTAB(IX) = 1 will produce the output of the stability

analysis of channel IX in a self explanatory format in accordance with the theory in Reference

12, pp. 29 37. The TF object of the stability analysis is CABXix.

***Core Flow Impedances

Output is done by RESPON provided IOPTW(4) = 1.

CZY Zy Total flow rate to channel inlet pressure TF

59
1. Zy= izx aix x gi.

ix 1. +aix x bix

CZQ Zq Total core power to channel inlet pressure TF

zq=,x aix x gix x Cix Zq Zy /,ix 1. +aix x bix

CZT ZT Inlet temperature to channel inlet pressure TF ZT Zy = X Iaix x gix x dix

ZT 1ZYX + ix xbix

***RECIRC

Output produced by RESPON provided IOPTW(5) = 1. Under the heading,

RECIRCULATION LOOP TFs:

RLPY Pressure to core flow TF:

RLPY = GAINPY

6n 1+TAUPYxs

CQY Total core power to core flow TF:

CQY- 8 -

Y qq x RLPY

6q 1-RLPYxzy

Note that CLYT-- - and CQT - T CLYT x CQY are not printed since T is taken

6Y 8q 8Y

as (0., 0.) in this program.

***ONECOR

Output produced by RESPON provided IOPTW(6) = 1. Under the heading CORE

REACTIVITY FEEDBACK TFs:

CAR Ar Total flow rate to density reactivity TF

Ar = Zeix xLix

ix

CAD Ad Total flow rate to Doppler reactivity TF

Ad-= Zdy x Lix

ix

CBR Br Total core power to density reactivity TF

Br = E fix + eix x Mix

ix

CBD Bd Total core power to Doppler reactivity TF

Bd= I ddy x Mix

ix

CCR Cr Inlet temperature to density reactivity TF

Cr = -hix +eix xNix

ix

CCD Cd Inlet temperature to Doppler reactivity TF

Cd = Z dt, + dyix x Nix

ix

CHAR Har Total core power to density reactivity through flow.

60
Har : Ar X CQY

CHAD Had Total core power to Doppler reactivity TF through the

perturbation of flow rate.

Had -Adx CQY

CHCR Hor Total core power to density reactivity TF through the

perturbation of inlet temperature.

Hcr = Cr x CQT

CHCD Hcd Total core power to Doppler reactivity TF through the

perturbation of inlet temperature.

H-d = Cdx CQT

CHR Hr Total core power to total density reactivity TF This is the

hydrodynamic feedback loop.

Hr = Har + Hcr + Br

CHD Hd Total core power to total Doppler reactivity TF This is the

fuel temperature feedback loop.

Hd =Had +Hcd +Bd

Under the heading NEUTRON KINETICS LOOP:

CG G Total reactivity to total core power TF This is the neutron

kinetics or forward loop.

G = ±qq (point kinetics) 6k

Under the heading FEEDBACK LOOP:

CH H Total core power to total reactivity TF This is the overall

feedback loop.

H = Hr +Hd

Under the heading OPEN-LOOP RESPONSE:

CGH G x H Open-loop reactivity TF of reactivity.

Under the heading CLOSED-LOOP RESPONSE:

GIGH G

G1GH G Closed-loop reactivity to power TF I+GxH

***STABGH

Provided that ISTAB(NXE + 1), the results of the stability analysis of the core open-loop

TF GH are printed by means of RESPON.

***FREQ TRANS

Provided that IOPTW(7 to 22) = 1, the following multicore TFs are printed from RESPON:

CCXK c,,k Subcore k power to pressure at the boiling boundary of

channel X TF Refer to Eq. (3.2.88c) and matrix C at App.

B11.

CEKJ ekj Inlet flow rate into horizontal zone j to moderator density

reactivity feedback of the subcore at level k [i.e., subcore

(k,j)]. According to Eq. (3.3.1a) and App. B11,

ek,j = lexkj x Nchx,j

X

61
CHKJ hk,j Inlet coolant temperature into horizontal zone j to

moderator density reactivity of the subcore at level k [i.e.,

subcore (k,j)]. According to Eq. (3.3.1c) and App. B11,

hk,j = L hx,k,j x Nch×,j X

CZQKJ Zqk.j Subcore (k,j) power to total reactor core flow TF vector zq

at Eq. (B12.2).

CFKJK fkl.j,k Subcore (k1,j) power to moderator density reactivity of

subcore (k,j) TF See Eq. (3.3.1b) and matrix F of App. B.

CMXJK MXJ 1,kJ2 Subcore (k,j 2) power to inlet flow rate to the channel X

placed in zone ji, TF See Eq. (3.3.10) and matrix M in

App. B12.

CQ Q Matrix of subcore (kl,j 1) power to moderator density

reactivity of subcore (k2,j2) TFs. See Eqs. (3.3.11) and

(B15.14).

CR R Vector of total core flow rate to moderator density

reactivity of subcore (k,j) TFs. See Eqs. (3.3.11) and

(B12.16).

CS S Vector of inlet coolant temperature to moderator density

reactivity of subcore (k,j) TFs. See Eqs. (3.3.11) and

(B 12.18).

CHQ Feedback matrix of power at subcore (kl,j1) to reactivity

at subcore (k2,j2), including flow redistribution effects. It

corresponds to Eq. (3.3.23a).

CHRY R* Feedback vector containing the TFs of total core inlet

flow rate to reactivity at subcore (k,j). It corresponds to

Eq. (3.3.23b).

CHST S* Feedback vector containing the TFs of inlet coolant

temperature to reactivity at subcore (k,j). It corresponds

to Eq. (3.3.23c).

CG1MGH Zk2 ,j2,kl,j1 Closed-loop [external reactivity at subcore (k1 ,jl) to

power at subcore (k2,j2)] TF matrix. It corresponds to Eq.

(3.3.16) using Q* as indicated by Eq. (3.3.23a).

CDENFB Vector containing the total reactivity effect on each

subcore due to a simultaneous unity perturbation of

reactivity error in every subcore; that is,

CDENFBjik = Z Q *j,k,j',k'

j' k'

CLOSEK Vector containing the closed-loop power response of

each subcore (k,j) due to a simultaneous unity

perturbation of external reactivity in every subcore; that

is,

CLOSEK j,k = Z I Zj,k,j',k' ALPKJj,,k,

j' k'

CRFBAC Sum of the point kinetics reactivity feedback TFs of each

nuclear subcore.

62
CLTFPK Sum of the closed-loop TFs of each subcore to yield the

total reactor core power response to a simultaneous

unity perturbation of reactivity in each subcore; that is,

CLTFPK = Y Y CLOSEKjk POWKJjk/POWT

j- k'

63

REFERENCES

1. Otaduy-Bengoa, P. J. "Modeling of the Dynamic Behavior of Large Boiling Water

Reactor Cores," Ph.D. dissertation, University of Florida, 1979.

2. March-Leuba, J. A. and E. D. Blakeman, "A Study of Out of Phase Power Instabilities in

Boiling Water Reactor," International Reactor Physics Conference, Jackson Hole, WY,

September 18-22, 1988.

3. March-Leuba, J. A. and P. J. Otaduy, "A Comparison of BWR Stability Measurements

with Calculations Using the Code LAPUR IV," NUREG/CR2998, ORNL/TM 8546, Oak

Ridge National Laboratory, January 1983.

4. March-Leuba, J. A., D. N. Fry, M. E. Buchanan, and C. 0. McNew, Local Stability Tests

in Dresden 2, ORNLITM-9054, Oak Ridge National Laboratory, March 1984.

5. March-Leuba, J. A., R. T. Wood, P. J. Otaduy, and C. 0. McNew, "Stability Tests at

Browns Ferry Unit 1 Under Single-Loop Operating Conditions," NucL. Techno. 74 38-52

(May 1986).

6. March-Leuba, J. A. and P. J. Otaduy, "The Importance of Momentum Dynamics is BWR

Neutronic Stability: Experimental Evidence," Trans. Am. Nuci. Soc. 50 563-564,

November 1985.

7. March-Leuba, J. A. and D. N. Fry, Grand Gulf-I and Susquehanna-2 Stability Test,

ORNLINRC/LTR-87/01, Oak Ridge National Laboratory, April 1987.

8. March-Leuba, J. A. Stability Calculations for the Grand Gulf-i and Susquehanna-2

Boiling Water Reactor, ORNL/NRC/LTR-87/08, Oak Ridge National Laboratory,

September 1987.

9. Jones, A. B. Hydrodynamic Stability of a Boiling Channel, Part 4, KAPL 3070, 1964.

10. Jones, A. B. and D. G. Dight, Hydynamic Stability of a Boiling Channel, Part 2, KAPL

2208, Knolls Atomic Power Laboratory, Schenectady, NY, 1962.

11. Flinta, J. The Hydrodynamic Instability of Boiling Channels, AERTL 1014, AB Atomenergi

Studsvik, Sweden, 1968.

12. Smith, 0. J. M. Feedback Control Systems, McGraw Hill, New York, 1958.

13. Waggener, J. P. "Friction factors for pressure drop calculations". NUCLEONICS, Vol. 19,

No. 11. 1961.

65

APPENDIX A

SAMPLE LAPURX AND LAPURW INPUTS FOR A TYPICAL BWR

A-1
The following input decks represent a quarter-core configuration for a typical BWR

LAPURX V6.0 SAMPLE INPUT DECK:

LAPURX 6 SAMPLE INPUT DECK

982.55, 507.77,1609.40,.2958E+08, 0.0879, 0.031f 0.487, 1.0

2

156, 45, 0, 1, 0, 0, 0, 0, 0 1 3

156, 27,27,27,27, 27,27,27,

27, 27, 27, 27, 27, 27, 27,

27,27,27,27,27,27,27,

27,27,27,27,27,27,27,

27r27, 27, 27, 27,27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27,27r27,27, 27, 27, 27,

27,27,27,27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27,r27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27, 27, 27, 27, 27,

27, 27, 27,r27, 27, 27, 27,

27, 27, 27, 27, 27, 27,r27,

27, 27,

4

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24,- 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 35.9700

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 35.9700

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 35.9700

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24,. 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 35.9700

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15..24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

A-2
33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24r 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 159.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 115.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, .15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24 , 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

A-3
15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.2.4, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, .15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

3.3.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24;, 15.24, 15.24, 15.24;' 15.24, 15.24,

ý15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24,. 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24,. 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

A44
15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15..24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24,. 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24,, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, ý15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24,. 1:5.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

1S.24, .15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, .15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

A-5
.15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

.52, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 152, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 1.5.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24,. 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290r 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

A-6
33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24,. 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15:24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, .15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24,. 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24r 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 151'24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

A-7
15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24,. 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290r 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.2.4, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

A-8
15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, .15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, .5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, .15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, . 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300.

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

A-9
15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.-24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24r, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

.15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, .15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

33.290, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 15.24, 15.24, 15.24,

15.24, 15.24, 15.24, 15.24, 5.08, 46.1300

A-1
5

33.290,

15.24,

15.24,

15.24,

33. 290,

15.24,

15.24,

15.24,

33. 290,

15.24,

15.24,

15.24,

33.290,

15.24,

15.24,

15.24,

33.290,

15.24,

15.24,

.15.24,

0.00000

1.07996

1.23191

1.02922

0.00000

0.97360

1.25721

1.14210

0.00000

1.01270

1.26063

1.09467

0.00000

0.90480

1. 28 616

1.21137

0.00000

1.20689

1.15885

0.85674

0.00000

1.20049

1. 16026

0.86364

0.00000

1. 20555

1.15837

0.85627

0.00000

1.19572

1.16206

0.87001

0.00000

1.20039

1. 16122

0.86427

0.00000

1.20405

1.15850

0.85623

0.00000

1.20291

1.15972

0.85880

0.00000

1. 21959

1.15448

0.83773

0. 00000

1. 21951

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

,0.25122

, 1. 118 60

, 1.21891

,0.91694

,0.23525

, 1. 025 61

,1.25681

,1.03303

,0.24196

,1.06214

,1.25136

,0. 98326

,0.21416

,0.96199

,1.29526

,1. 09966

,0.20353

,1.21625

,1.15396

,0.74505

, 0. 202 52

,1.21118

,1.15602

,0.75260

,0.20421

,1. 21510

, 1. 153 53

,0.74426

,0.20129

,1.20723

r1 .15843

, 0.75 922

,0.20226

,1.21128

,1.15697

,0.75294

,0.20502

,1.21402

,1.15373

,0.74386

,0.20399

,1.21302

,1.15503

,0.74647

,0.20799

,1.22630

,1.14793

,0.72328

,0.20936

,1.22598

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

,0.73510

,1.15825

,1.19724

,0.75246

,0. 67 839

,1.08696

,1.24679

,0.85442

,0.68 366

,1.11966

,1.23302

,0.81102

, 0. 624 03

,1. 03472

,1. 29 629

,0.91166

,0.96285

,1. 21015

,1.12509

,0. 592 66

,0. 95798

, 1.20 625

,1. 12763

,0.60000

,0.96587

,1. 20 937

,1.12479

,0.59189

,0. 95272

,1. 20364

,1. 13050

,0.60633

,0. 95717

,1.20680

,1. 12861

,0 .60014

,0. 96913

, 1. 208 97

,1.12511

,0.59148

,0.96405

,1.20816

,1.12614

,0. 59365

,0. 982 97

,1.21742

,1.11744

,0. 57 069

,0. 98779

,1.21753

15.24, 1

15.24, 1

15.24, 1

15.24,

15.24, 1

15.24,

15.24,

151.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

,0.86341

,1.18343

,1.17836

,0.35709

,0.78220

,1.13727

,1.24076

,0.39265

,0.79596

,1.16475

,1.22100

,0. 37926

,0.7 1655

,1.10295

,1.29987

,0.41830

,1.20857

,1.19129

,1.00854

,0.20017

,1.20275

,1.18884

,1.01129

,0.20156

,1.21159

,1.19059

,1.00842

,0.20013

,1.19751

,1.18751

,1.01427

,0.20280

,1.20234

,1.18961

,1.01224

,0.20175

,1.21466

,1.19052

,1.00884

,0.20026

,1.20997

,1.19055

, 1.00 932

,0.19822

,1.,23157

,1.19588

,0. 99993

,0.19187

,1.23644

,1.19614

.5.24, 15

.5.24, 15

.5.24, 15

5.08, 46

5S.24, 15

15.24, 15

15.24, 15

5.08, 46

15.24, 15

15.24, 15

15.24, 15

5.08, 46

15.24, 15

15.24, 15

15.24, 15

15.24, 35

15.24, 15

15.24, 15

15.24, 15

5.04, 46

,0. 92 565

,1.20006

,1. 16380

,0.17877

,0.82543

,1.17428

,1. 23654

,0.19014

,0.85005

,1. 19726

,1. 21042

,0.18365

,0. 75551

, 1. 158 13

,1.29651

,0.19984

,1.27041

,1. 19579

,1.02431

,0.12869

,1.26348

,1. 19451

,1. 02781

,0.12976

,1. 27142

f1 .19514

,1.02400

,0.12876

,1. 25741

1 .19415

1 .03132

,0.13082

,1.26273

,1.19542

,1.02867

, 0. 12995

f1 .27167

,1.19513

,1.02433

,0.12900

,1.26872

,1. 19571

,1. 02511

,0.12773

,1.29215

,1. 19791

,1. 01392

,0.12271

,1.29450

,1.19797

.24,

.24,

.24,

.1300

.24,

.24,

.24,

.1300

.24,

. 1300

.9700

.24,

.24,

.24,

,.1700

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

15.24,

,0.97400

,1.21160

,1.12902

,0.00000

,0.8 6079

,1.20313

1.21440

'0.00000

,0.8 9553

,1.21806

,1.18097

,0.00000

,0.78894

,1.20187

,1.27498

,0.00000

,1.27076

,1.19656

,0.99067

'0.00000

,1.2 6324

,1.19635

,0.99515

'0.00000

,1.27036

,1.19613

,0.99052

,0.00000

,1.25700

,1.19687

,0.99960

'0.00000

,1.26253

,1.19741

,0.99612

'0.00000

,1.26912

, 1.19641

,0.99095

,0.00000

,1.26697

1.19714

,0.99213

,0.00000

,1.28965

,1.19635

,0.97812

,0.00000

,1.2 9058

,1.19635

,1.02801

,1.22187

, 1.08 672

,0.91220

,1.23092

,1.18812

,0.95206

,1.23959

,1.14693

0.84072

,1.24503

,1.25272

,1.24456

,1.18390

,0.93003

,1.23734

,1.18459

,0. 93588

,1.24387

,1.18356

,0.92980

,1.23167

,1.18589

,0.94148

,1.23704

,1.18569

,0. 93 672

,1. 24276

,1. 18393

,0. 93009

,1.24068

,1.18489

,0.93231

,1.26056

,1.18150

, 0. 914 69

,1.26132

,1. 18134

A-1l1
1.15377

0.83568

0.00000

1.21884

1.15585

0.83780

0.00000

1.21884

1.15217

0.83481

0.00000

1.21023

1.15792

0. 85021

0.00000

1.19419

1.16768

0.87570

0.00000

1.20363

1.16369

0.86095

0.00000

1.19642

1.16702

0.87206

0.00000

1.20224

1.16285

0.86243

0.00000

1.20162

1.16548

0.86622

0.00000

1.19863

1.16532

0.86750

0.00000

1.20360

1.16379

0.86198

0.00000

1.21162

1.16094

0.85124

0.00000

1.21920

1.15565

0.83955

0.00000

1.23087

1.15194

0.82367

0.00000

1.22620

1.15347

0.82995

0.00000

1.22136

1.15454

0.83550

0.00000

1.21914

1.15543

0.84024

0.00000

1.21816

1.15565

0.84063

0.00000

,1.14705

0.72078

,0.20727

,1.22624

,1.14921

,0.72269

,0.21031

,1.22535

,1.14560

,0.72000

,0.20454

,1.21967

,1.15279

,0.73707

,0.19721

,1.20875

,1.16459

,0.76548

,0.20057

,1.21579

,1.15928

,0.74846

,0.19780

,1.21068

,1.16372

,0.76086

,0.20119

,1.21406

,1.15903

0. 74974

,0. 19849

,1.21458

,1.16188

,0.75386

,0.19935

,1.21167

,1.16180

0. 75541

,0.20059

,1.21554

,1.16006

,0. 74865

,0. 20270

,1.22182

,1.15595

,0.73711

,0.20670

,1.22685

,1.14993

,0.72369

,0.20960

,1.23623

,1.14388

,0. 70790

,0.20854

,1.23235

,1.14651

,0.71325

,0.20771

,1. 22831

,1.14866

,0.71785

,0.20583

,1.22698

,1.15028

,0.72356

,0. 20598

,1.22632

,1.15065

,0.72383

,0.20808

,1.11624

,0.56848

,0.97959

,1.21835

,1.11843

,0.57002

,0.99252

,1.21675

1.11501

,0.56786

,0.96716

,1.21297

,1.12349

,0.58439

,0.93158

,1.20596

1 .13705

,0.61251

,0.94756

,1.21140

r1.13036

,0.59534

r 0. 93413

,1.20764

,1.13605

,0.60739

,0.94989

r1.20946

,1.13085

,0.59605

,0.93782

,1.21043

,1.13373

,0.60000

,0.94134

,1.20820

,1.13385

,0.60177

,0.94748

,1.21083

,1.13177

,0.59448

,0.95771

,1.21568

,1.12637

,0.58373

,0.97615

,1.21811

,1.11947

,0.56970

,0.98992

,1.22546

,1.11144

,0.55612

,0.98515

,1.22256

,1.11501

,0.55983

,0.98138

,1.21941

,1.11806

,0.56282

,0.97308

,1.21788

,1.12019

,0.56878

,0.97378

,1.21730

,1.12058

,0.56891

,0.98506

,0.99858

,0.19014

,1.22875

,1.19739

,1.00060

,0.19063

,1.24098

,1.19492

,0.99790

,0.19135

,1.21401

,1.19324

,1.00696

,0.19855

.1.17440

1.19036

,1.02098

,0.20924

,1.19291

,1.19387

,1.01348

00.20133

1.17734

,1.19148

,1.01979

,0.20688

,1.19423

,1.19185

,1.01466

,0.20264

1 .18245

,1.19320

,1.01731

,0.20382

,1.18514

1 .19155

1 .01763

,0.20489

,1.19241

,1.19301

,1.01541

,0.20175

,1.20478

,1.19618

,1.00951

,0.19802

,1.22459

,1.19656

,1.00233

.0.19110

,1.24124

,1.20171

,0.99344

,0.18785

,1.23568

,1.19963

,0.99741

,0.18766

,1.23117

,1.19735

.1.00090

,0.18723

,1.22254

,1.19627

,1.00367

0 .19179

1.22335

,1.19590

1 .00408

.0.19181

,1.23838

,1.01214

,0.12191

,1.28908

,1.19959

,1.01430

,0.12221

,1.29679

,1.19653

,1.01148

,0.12256

,1.27493

,1.19683

,1. 02221

,0.12719

,1.24002

,1.19754

,1.03859

,0.13470

,1.25686

,1.19911

,1.02963

,0.12945

,1.24263

,1.19805

,1.03724

,0.13306

,1.25676

,1.19709

,1.03130

,0.12996

,1.24878

,1.19887

,1.03441

,0.13075

,1.24875

,1.19768

,1.03465

,0.13162

,1.25621

,1.19822

,1.03215

,0.12926

,1.26841

,1.19991

,1.02502

,0.12681

,1.28652

,1.19858

,1.01724

,0.12179

,1.30324

,1.20167

,1.00614

,0.11979

,1.29722

,1.20040

f1.01113

,0.11951

,1.29187

.1.19890

,1.01562

,0.11906

,1.28554

,1.19820

,1.01919

,0.12164

1.28593

,1.19801

,1.01969

,0.12161

,1.30127

,0.97640

*0.00000

11.28727

,1.19818

,0.97861

.0.00000

,1.29120

,1.19502

,0.97582

,0.00000

,1.27490

,1.19705

,0.98773

,0.00000

,1.24689

,1.20087

,1.00635

.0.00000

,1.26132

,1.20066

,0.99596

10.00000

,1.24964

,1.20098

,1.00456

'0.00000

,1.26016

,1.19886

,0.99785

,0.00000

,1.25603

,1.20095

,1.00106

.0.00000

,1.25381

,1.20023

,1.00162

.0.00000

,1.26084

,1.19995

,0.99847

'0.00000

,1.27201

.1.20020

,0.99033

.0.00000

,1.28656

,1.19737

,0.98162

.0.00000

,1.30221

,1.19816

,0.96871

.0.00000

,1.29632

,1.19799

,0.97455

.0.00000

,1.29067

i.119750

,0.97980

.0.00000

,1.28648

,1.19694

,0.98355

.0.00000

,1.28629

r1.19693

,0.98404

.0.00000

,1.30010

,0.91302

,1.25940

,1.18331

,0.91521

,1.26144

,1.17997

,0.91217

1.24881

,1.18377

,0.92541

,1.22647

,1.19057

,0.94695

,1.23880

,1.18854

,0.93484

,1.22918

,1.19029

,0.94432

,1.23712

,1.18713

,0.93648

,1.23511

,1.18948

,0.93985

,1.23218

,1.18913

,0.94076

1.23831

,1.18815

,0.93664

,1.24819

,1.18694

,0.92730

,1.25910

,1.18277

,0.91757

,1.27321

,1.18133

,0.90279

,1. 26781

,1.18209

,0.90921

,1. 26243

,1.18246

,0.91497

,1.25893

,1.18246

,0. 91886

,1.25831

,1.18259

,0.91930

,1.26976

f

F

f

f

J

t

t

r

¥

A-12
1.22663

1.15117

0.83055

0.00000

1.24546

1.14892

0.80955

0.00000

1.24504

1.14619

0.80935

0.00000

1.24517

1.14382

0.80905

0.00000

1.26313

1.13910

0.78465

0.00000

1.26300

1.13904

0.78425

0.00000

1.26172

1.13651

0.78467

0.00000

1.25308

1.13993

0.79823

0.00000

1.25258

1.13972

0.79780

0.00000

1.24184

1.14390

0.81238

0.00000

1.25593

1.13957

0.79206

0.00000

1.16660

1.18719

0.91311

0.00000

1.25664

1.13893

0. 79038

0.00000

1.24571

1.14676

0.80821

0.00000

1.29124

1.12283

0.74713

0.00000

1.23671

1.15312

0.82402

0.00000

1.24667

1.14598

0.80625

0.00000

1.28334

1.12780

0.76198

,1.23208

1.14521

,0.71232

,0.20856

,1.24953

,1.14016

,0.69037

,0.21037

,1.24760

,1.13771

,0.68972

,0.21099

,1. 24672

1.13570

,0. 68897

,0.21531

,1. 26139

,1.12795

,0.66479

,0.21512

,1.26124

,1.12779

,0.66418

,0.21655

,1.25905

,1.12579

,0.66440

,0.21259

,1.25243

,1.13122

,0.67797

,0.21317

,1.25193

,1.13092

,0.67750

,0.20989

,1.24365

,1.13709

,0.69169

,0.22569

,1.25321

,1.12751

,0.67767

,0.22580

,1.18490

,1.18131

,0.81390

,0.22708

,1.25306

,1.12664

,0.67627

,0.22056

,1.24672

,1.13622

,0.69376

,0.22029

,1.28160

,1.10899

,0.62506

,0.21624

,1.24106

,1.14414

,0.70933

,0.22170

,1.24729

,1.13505

.0.69202

,0.21406

,1.27627

,1.11577

,0.63936

,1.22106

,1.11455

,0.55758

,0.98646

,1.23528

,1.10631

,0.53836

,0.99582

,1.23274

,1.10437

,0.53720

,1.00057

,1.23109

,1.10269

,0.53581

,1.02078

,1.24251

,1.09163

,0.51588

,1.02056

,1.24260

,1.09151

,0.51540

,1.02821

,1.23972

,1.08992

,0.51523

,1.01076

,1.23443

,1.09688

,0.52654

,1.01300

,1.23407

,1.09655

,0.52619

,0.99823

,1.22816

,1.10446

,0.53806

,1.02905

.1.23443

,1.09255

,0.53696

,0.92041

,1.18892

,1.15836

0.67587

,1.03455

,1.23415

1.09140

,0.53621

,1.00221

,1.23058

,1.10295

.0.55074

,1.05503

,1.25426

,1.06964

,0.48096

,0.97957

,1.22765

,1.11263

0.56383

,1.00647

,1.23113

,1.10152

,0.54957

,1.02811

,1.25077

,1.07840

,0.49280

,1.19730

,0.99811

,0.18765

,1.24446

,1.20848

,0.98801

,0.18053

,1.25450

,1.20526

,0.98669

,0.18004

,1.26082

,1.20291

,0.98554

,0.17934

,1.28454

,1.21109

,0.97274

,0.17282

,1.28475

,1.21113

,0.97276

,0.17279

,1.29328

,1.20786

,0.97173

,0.17253

,1.27514

,1.20443

,0. 97940

,0.17606

,1.27733

,1.20409

,0.97911

,0.17600

,1.26113

,1.20031

,0.98807

,0. 18004

,1.27916

1 .20531

0 .97313

,0.18208

f 1. 10288

,1.18348

,1.04209

,0.24062

,1.28482

,1.20494

,0.97192

,0.18204

,1.24966

,1.20419

, 0. 98439

,0.18776

,1.33687

,1.21567

,0.94952

,0.15840

,1.22426

,1.20359

.0.99518

,0.19295

,1.25425

,1.20452

,0.98282

.0.18742

,1.31087

,1.21391

,0.96033

,0.16422

,1.19759

,1.01313

,0.11864

,1.31416

,1.20570

,1.00099

,0.11392

r1.32168

,1.20230

,1.00006

,0.11342

,1.32806

,1.19950

,0.99944

,0.11271

,1.35196

,1.20477

,0.98359

,0.10862

,1.35174

,1.20476

,0.98363

,0.10863

,1.35834

,1.20149

,0.98305

,0.10821

,1.34337

,1.19976

,0.99259

,0.11032

,1.34423

,1.19946

,0.99224

,0.11027

,1.32823

,1.19761

,1.00294

,0.11285

,1.34256

,1.20020

,0.98173

,0.11396

,1.15908

,1.19586

,1.05227

,0.15201

,1.34630

r1.19978

,0.98028

,0.11397

,1.31706

,1.20126

,0.99460

,0.11746

,1.40824

,1.20377

,0.95893

,0.09836

,1.29446

,1.20235

,1.00702

,0.12082

,1.32055

,1.20126

0.99277

,0.11736

,1.38819

,1.20344

,0.97189

,0.10189

,1.19532

,0.97701

.0.00000

,1.31830

,1.20000

,0.96209

0.00000

,1.32249

,1.19682

,0.96168

.0.00000

1.32723

,1.19411

,0.96140

.0.00000

,1.34958

,1.19610

,0.94272

.0.00000

,1.34921

,1.19616

,0.94279

.0.00000

,1.35285

,1.19310

,0.94273

.0.00000

,1.34068

,1.19303

,0.95351

.0.00000

,1.34068

,1.19281

,0.95321

.0.00000

11.32601

,1.19306

,0.96535

.0.00000

,1.33792

,1.19147

,0.94175

'0.00000

,1.17640

,1.19986

,1.02113

'0.00000

1.33994

.1.19091

.0.94002

.0.00000

,1.31872

,1.19436

,0.95556

.0.00000

,1.39989

,1.19044

,0.91553

.0.00000

,1.30088

,1.19733

,0.96939

.0.00000

1.32072

,1.19435

0.95383

.0.00000

,1.38577

.1.19180

,0.92967

0.00000

,1.17956

.0.91097

,1.28946

,1.18066

,0.89295

,1.29110

,1.17766

,0.89279

,1.29334

,1.17510

,0.89268

,1.31373

,1.17379

,0.87104

,1.31357

,1.17384

,0.87094

,1.31444

,1.17109

,0.87124

1.30379

,1.17278

,0.88338

,1.30361

,1.17261

,0.88304

,1.29065

,1.17488

,0.89649

,1.30268

,1.17109

,0.87415

,1.17505

,1.19680

,0.97167

,1.30385

,1.17033

,0.87223

,1.28841

,1.17598

,0.88880

,1.35335

,1.16310

,0.83947

,1.27547

,1.18062

,0.90352

,1.29001

,1.17559

,0.88689

,1.34257

,1.16616

,0.85381

#

f

#

l

t

#

t

f

#

#

A-13
0.00000

1.26528

1.13270

0.77911

0.00000

1.21295

1.16605

0.85257

0.00000

1.30723

1.10941

0.72339

0.00000

1.18343

1.18052

0.88913

0.00000

1.19241

1.17500

0.87459

0.00000

1.24136

1.15180

0.81409

0.00000

1.18861

1.18055

0.88423

0.00000

1.23319

1.15404

0.82288

0.00000

1.18585

1.18257

0.88802

0.00000

1.17931

1.18639

0.89794

0.00000

1.18540

1.18238

0.88847

0.00000

1.17534

1.19085

0.90609

0.00000

1.17773

1.18624

0.89796

0.00000

1.18555

1.18391

0.88870

0.00000

1.17463

1.19099

0.90574

0.00000

1.17956

1.19050

0.90295

0.00000

1.17728

1.19032

0.90647

0.00000

1.18037

1.18740

,0.22520

,1.25901

,1.12039

,0.66274

,0.22354

,1.22199

,1.15800

,0.74256

,0.22478

,1.29020

,1.09381

,0.60120

,0.22994

,1.19873

,1.17371

,0.78516

,0.23670

,1.20350

,1.16568

,0.77220

,0.23652

1.24108

,1. 13742

,0.70660

,0.23374

,1.20151

,1. 17117

,0.78321

,0.23275

,1.23482

,1.14189

,0.71410

,0.23316

,1.19984

,1.17335

,0.78694

,0.23039

,1.19606

,1.17911

,0.79515

,0.23433

,1.19889

,1.17316

,0.78725

,0.22849

,1.19279

,1.18251

,0.80658

,0.23154

,1.19498

,1.17906

,0.79520

,0.23526

,1.19961

,1.17371

,0.78882

,0.22876

,1.19264

,1.18263

,0.80632

,0.23102

,1.19651

,1.18091

,0.80433

,0.22973

,1.19413

,1 .18066

,0. 80951

,0.22958

,1. 19597

,1.17673

1.03834

,1.23710

1. 08539

0.52206

,0. 96576

,1. 21525

,1.12954

,0.59933

,1.08932

1. 25737

,1.05292

,0.46116

,0.94509

,1.19873

,1.14945

,0.64453

,0.96696

,1.20065

,1.13991

.0.63491

.1.00757

.1.22550

,1.10495

,0.57130

,0.94637

,1.20067

,1.14570

,0.64646

,1.00338

,1.22144

,1.11064

,0.57646

,0.94084

,1.19997

,1.14797

,0.64987

,0.93044

,1.19821

,1.15504

,0.65521

0.94221

1.19894

,1.14803

,0.65010

,0.91312

,1.19661

,1.15815

,0.66937

,0.93276

,1.19678

,1.15522

,0.65531

,0.93722

,1.19946

,1.14821

,0.65327

,0.91376

,1.19612

,1.15831

,0.66926

,0.91475

,1.19920

,1.15594

,0.66822

,0.91014

,1.19733

,1.15578

,0.67424

,0.91496

,1.19849

,1.15122

,1.29917

,1.20511

,0.96722

,0.17865

,1.18256

,1.19768

,1.01152

,0.20945

1.38533

,1.21410

,0.93453

,0.15396

,1.13181

, 1.18948

,1.03146

,0.22893

1 .15813

,1.18991

,1.02134

,0.22837

1.23563

,1.20196

,0.98583

,0.20546

,1.13439

,1.19130

,1.02693

,0.23615

,1.23009

,1.19952

,0.99197

0.20438

,1.12670

,1.19102

,1.02931

,0.23782

,1.11125

,1.19046

,1.03692

,0.23684

,1.12710

,1.19044

,.1.02961

,0.23780

,1.09230

,1.19083

,1.03956

,0.24848

,1.11297

,1.18963

,1.03709

,0.23702

1.12070

1.19145

,1.02938

,0.24230

,1.09300

,1.19066

,1.03956

,0.24862

,1.09250

,1.19283

,1.03694

,0.25028

,1.08858

,1.19123

,1.03657

,0.25479

,1.09591

,1.19148

,1.03175

,1.36568

,1.19758

,0.97610

,0.11094

, 1.24443

,1.20080

,1.02197

,0.13071

,1.45551

,1.19846

.0.94301

.0.09530

.1.18680

.1.19804

,1.04067

,0.14307

,1.21264

,1.19676

,1.02709

,0.14293

,1.30192

,1.19982

,0.98909

,0.12826

,1.19320

1.19931

,1.03265

,0.14817

,1.29272

,1.19872

,0.99793

,0.12751

,1.18497

,1.19985

,1.03539

,0.14923

, 1.16791

,1.20055

,1. 04577

,0. 14844

,i1 18531

,1. 19924

,1.03565

,0. 14928

,1. 15319

,1. 20198

,1.04730

,0. 15649

,1.16920

,1. 19992

1. 04568

,0.14868

1. 18078

1. 20042

,1.03432

,0.15245

1. 15397

1. 20197

,1. 04702

,0.15667

,1.15498

1.20333

,1.04346

,0.15773

,1.15267

,1.20211

,1.04272

,0.16081

,1.15964

,1.20136

,1.03715

,1.35815

,1.18730

,0.93525

•0.00000

,1.25507

,1.19849

,0.98592

'0.00000

,1.43703

,1.18251

.0.89850

0.00000

,1.20105

,1.19926

,1.00738

0.00000

,1.21982

,1.19593

,0.99206

'0.00000

,1.30531

,1.19160

,0.94889

0.00000

,1.20831

,1.19921

,0.99806

'0.00000

,1.29411

r1.19235

,0.95903

,0.00000

,1.20248

,1.20007

,1.00078

0.00000

,1.18868

,1.20207

,1.01233

,0.00000

1.20202

,1.19949

,1.00124

'0.00000

,1.17946

,1.20395

,1.01402

'0.00000

,1.18796

,1.20175

,1.01243

'0.00000

,1.19970

,1.20028

,0.99936

.0.00000

,1.17926

,1.20415

,1.01383

.0.00000

,1.18291

,1.20431

,1.00963

.0.00000

,1.18153

,1.20360

,1.00985

.0.00000

,1.18759

,1.20213

,1.00444

,1.31747

,1.16533

,0.86430

1.23955

,1.18661

,0.92562

.1.37902

.1.15241

.0.81866

,1.19624

,1.19324

,0.95396

,1.20934

,1.18837

,0.93846

,1.27852

,1.17514

,0.88678

,1.20250

,1.19266

,0.94595

,1.26839

,1.17707

,0.89674

,1.19854

,1.19401

,0.94913

,1.18869

,1.19717

,0.96042

,1.19775

,1.19360

,0.94966

,1.18245

,1.20012

,0.96449

,1.18714

,1.19696

,0.96053

,1.19695

,1.19446

,0.94839

,1.18171

,1.20035

,0.96423

,1.18669

,1.19973

,0.96064

,1.18486

,1.19935

,0.96249

,1.18939

,1.19705

,0.95809

f

f

A-14
0.90336

0.00000

1.21447

1.16457

0.85645

0.00000

1.20420

1.17241

0.87137

0.00000

1.20453

1.17074

0.86933

0.00000

1.20405

1.17135

0.86991

0.00000

1.20453

1.17274

0.87231

0.00000

1.23628

1.15560

0.82627

0.00000

1.25728

1.13822

0.79675

0.00000

1.21422

1.17257

0.85342

0.00000

1.21419

1.17406

0.85455

0.00000

1.21508

1.17857

0.85862

0.00000

1.21640

1.17544

0.85398

0.00000

1.25865

1.14578

0.80551

0.00000

1.28150

1.12985

0.77864

0.00000

1.28195

1.12819

0.77575

0.00000

1.23747

1.15323

0.82684

0.00000

1.23586

1.15651

0.83053

0.00000

1.26089

1.14195

0.79773

0.00000

1.23987

,0.80769

,0.22505

,1.22164

,1.15394

,0.75132

,0.21904

,1.21396

,1.16406

,0.76536

,0.22015

.1.21464

,1.16246

,0.76345

,0.21990

,1.21428

,1.16304

,0.76378

,0.21899

,1.21476

,1.16443

,0.76605

,0.23088

,1.23782

,1.14320

,0.71757

,0.22269

1.25302

,1.12722

,0.68019

,0.24137

,1.22143

,1. 15853

,0.75258

,0.24050

,1.22213

,1.16009

,0.75280

,0.23924

,1.22581

,1.16522

,0. 75445

,0.24087

,1.22595

,1.16167

,0.75042

,0.21642

,1.26026

,1.13485

,0.68893

,0.20304

11.27805

,1.12006

0. 65559

,0.20520

,1.27727

,1.11817

,0.65306

,0.23934

,1.23590

,I .13946

,0.72211

,0.23724

,1. 23604

,1.14300

,0. 72522

,0.23307

,1.25639

,1.12725

,0.68894

,0.24166

,1. 24000

,0.67419

,0.95830

,1.21348

,1.12484

,0.61296

,0.93631

,1.20985

,1.13612

,0.62425

,0. 94117

,1.20960

,1.13470

,0. 62270

0. 93892

,1.20952

,1.13547

,0.62286

,0.93302

1 .21044

,1.13691

,0.62465

,0.98827

,1.22420

,1.11243

10.57888

,1.01328

,1.23192

,1.09377

,0.53677

,0.97257

,1.21479

,1.12999

0.61947

,0.96726

,1.21624

,1.13112

,0.61858

,0.95419

,1.22191

,1.13583

,0.61718

,0.96389

,1.22061

,1.13188

,0.61418

,0.97606

,1.24067

,1.10103

,0.54313

,0.97992

,1.25106

1.08381

,0.50559

,0.98983

,1.24993

,1.08157

,0.50402

,1.00967

,1.22246

1.10874

,0.58593

,0.99774

,1.22366

,1.11261

,0.58808

,1.01816

,1.23599

,1.09338

,0.55236

,1.00221

,1.22663

,0.25984

,1.17706

,1.19579

.1.00599

,0.22741

,1.15080

,1.19493

,1.01780

,0.22590

,1.15597

,1.19408

,1.01638

,0.22530

,1.15321

,1.19424

,1.01737

,0.22543

,1.14725

,1.19530

.1.01898

,0.22600

,1.21527

,1.20095

,0.99438

,0.20467

,1.27504

,1.20141

,0.97638

,0.18323

,1.16565

,1.19967

,1.01107

,0.22615

,1.15877

,1.20145

,1.01160

,0.22462

,1.13938

,1.20758

1.01535

,0.22092

,1.15166

,1.20588

,1.01118

,0.22034

,1.23474

,1.21014

,0.98383

,0.18525

,1.27924

,1.21244

,0.96834

,0.16727

,1.28901

,1.21145

.0.96605

,0.16700

,1.22408

,1.19874

,0.99039

,0.19949

,1.21021

,1.20061

,0.99419

,0.20022

,1.25766

,1.20576

,0.97496

,0.18696

,1.21145

1 .20327

,0.16410

,1.24553

,1.19872

,1.01410

,0.14277

,1.21939

,1.19981

,1.02857

,0.14125

,1.22393

.1.19893

,1.02667

,0.14108

,1.22161

,1.19919

,1.02764

,0.14115

,1.21756

,1.20019

,1.02969

,0.14138

,1.28326

,1.19971

,1.00061

,0.12680

,1.34715

,1.19592

,0.98667

,0.11324

,1.22541

1.20293

,1.01246

,0.14065

,1.21942

,1.20489

,1.01369

,0.13957

1.20235

,1.21096

,1.01937

,0.13706

,1.21300

,1.20891

,1.01440

,0.13679

,1.31483

,1.20497

,0.99601

,0.11378

,1.37033

,1.20260

,0.98480

,0.10195

,1.37807

,1.20142

,0.98205

,0.10190

,1.28624

,1.19611

,0.99496

,0.12200

,1.27475

,1.19852

,0.99898

,0.12243

,1.32866

,1.19911

,0.98055

,0.11431

,1.27638

,1.20078

.0.00000

,1.26012

,1.19558

,0.97922

.0.00000

,1.23880

,1.19878

,0.99380

.0.00000

1.24148

,1.19809

,0.99243

,0.00000

,1.23998

,1.19837

,0.99326

,0.00000

,1.23861

,1.19924

,0.99523

.0.00000

,1.29374

,1.19288

,0.96135

0.00000

,1.34624

,1.18752

,0.94721

.0.00000

,1.24199

,1.19771

,0.97317

.0.00000

,1.23852

,1.19968

,0.97468

.0.00000

,1.22868

,1.20581

,0.98133

'0.00000

,1.23501

1.20351

,0.97626

0.00000

,1.33064

,1.19654

,0.95686

'0.00000

,1.38240

1 .19265

,0.94485

.0.00000

,1.38565

1.19140

,0.94200

.0.00000

,1.29377

1 .18825

,0.95641

.0.00000

.1.28618

.1.19110

,0.96062

.0.00000

,1.33455

,1.18858

.0.94008

.0.00000

,1.28877

,1.19228

,1.24312

,1.18375

,0.92374

,1.22794

,1.18944

,0.93837

,1.22913

,1.18847

,0.93676

1.22822

,1.18891

,0.93749

,1.22814

,1.18991

,0.93969

,1.27124

,1.17793

,0.89936

,1.30819

,1.16806

,0.87921

,1.23346

,1.18730

,0.91694

,1.23190

,1.18913

,0.91870

,1.22841

,1.19484

,0.92520

,1.23159

,1.19220

,0.92029

,1.30309

,1.17661

,0.88811

,1.34201

,1.16817

,0.86954

1.34341

,1.16685

,0.86663

,1.27002

,1.17368

,0.89620

,1.26612

,1.17681

,0.90031

,1.30364

,1.16929

,0.87421

,1.26958

,1.17739

F

f

F

A-15
1:15697

0.82523

0.00000

1.29462

1.13377

0.75915

0.00000

1.29160

1.13059

0.76263

0.00000

1.24754

1.15408

0.81959

0.00000

1.26332

1.14407

0.80062

0.00000

1.23103

1.16510

0.83894

0.00000

1.30250

1.12387

0.75702

0.00000

1.26640

1.14377

0.79905

0.00000

1.26645

1.14564

0.80178

0.00000

1.23309

1.16121

0.84137

0.00000

1.27838

1.13909

0.78250

0.00000

1.27675

1.13753

0.78909

0.00000

1.25583

1.15194

0.81208

0.00000

1.31905

1.11115

0.72627

0.00000

1.22086

1.17111

0.85691

0.00000

1.30909

1.11848

0. 74416

0.00000

1.25353

1.15240

0. 81222

0.00000

1.17144

1.20196

0.91883

0.00000

,1.14181

,0.72105

0. 23444

,1.28579

,1.11436

,0.64840

,0.23383

,1.28145

,1.11237

,0.65124

,0.24277

,1.24419

,1.13830

,0.71304

,0.23979

,1.25744

,1.12731

,0.69183

,0.24624

,1.23083

,1.14996

,0.73452

,0.21568

,1.29187

,1.10855

,0.63903

,0.23284

,1.26088

,1.12872

,0.68762

,0.23033

,1.26085

,1.13080

,0.68953

,0.23265

,1.23298

r1.14969

S0.73090

,0.23419

,1.26971

,1.12203

,0.67124

,0.21626

,1.27022

,1.12434

,0.67387

,0.24255

,1.25055

,1.13469

,0.70523

,0.22526

,1.30193

1.09265

,0.60939

,0.23930

,1.22470

,1.15777

,0.75321

,0.22087

,1.29626

,1.10163

,0.62614

,0.23393

,1.24983

,1.13783

,0.70256

,0.25839

1.18098

,1.19078

,0.81840

,0.25920

,1.11054

,0.58605

,1.02822

,1.25893

,1.07477

,0.51516

,1.03179

,1.25411

,1.07385

,0.51675

,0.99299

,1.22918

,1.10593

,0.57735

,1.01030

,1.23731

,1.09316

,0.55562

,0.97256

,1.22153

,1.11940

,0.59961

1.00158

,1.26136

,1.07000

,0.49881

,0.98895

,1.24074

,1.09417

,0.54899

,0.98038

,1.24142

,1.09648

,0.55022

,0.95081

,1.22249

,1.11973

,0.58957

,1.00081

,1.24681

,1.08573

,0.53569

,0. 97361

,1.24721

,1.08866

,0.53330

'0.99100

,1.23409

,1.10126

,0. 57018

,1.04950

,1.26563

,1.05105

,0.47445

,0.94507

,1.21719

,1.12853

,0.61649

,1.02215

,1.26256

,1.06174

,0.48777

,0.97499

,1.23392

,1.10435

,0.56629

,0.91180

,1.18881

,1.16849

,0.67912

,0.91305

,0.99134

,0.20208

,1.28337

,1.22217

,0.95415

,0.17798

1.29064

,1.21741

,0.95414

,0.17908

,1.21298

,1.20540

,0.98625

,0.21131

, 1.24218

,1.20819

,0.97397

.0.19859

,1.18077

,1.20338

,0.99930

,0.22523

,1.29536

,1.21918

.,0.95272

,0.17061

,1.23990

,1.21051

,0.97554

,0.19627

,1.23233

,1.21159

,0.97805

,0.19739

,1.18357

,1.20187

,1.00256

,0.21607

,1.25790

,1.21423

,0.96646

,0.19288

,1.25197

,1.21337

,0.97161

,0.18927

,1.21680

,1.20915

,0.98127

,0.21128

,1.35213

,1.22003

,0.93285

,0.16154

,1.15135

,1.20214

,1.00841

,0.23502

,1.32112

,1.21922

,0.94440

,0.16654

,1.21770

,1.20955

,0.98466

,0.21003

,1.06207

,1.19263

,1.04738

,0.26636

,1.06272

,0.99385

,0.12386

,1.36380

,1.21006

,0.95707

,0.10908

,1.37023

,1.20539

,0.95848

,0.10954

,1.28448

f1.20124

,0.98973

,0.12988

,1.31691

,1.20111

,0.97818

,0.12112

,1.24987

,1.20216

,1.00154

,0.13953

,1.38901

,1.20528

,0.96369

,0 .10372

,1.32033

,1. 20323

,0.98243

,0.12055

,1.31469

,1.20437

,0.98551

,0.12102

,1.25944

,1.20043

,1.01106

,0.13366

,1.34095

,1.20461

,0.97156

,0.11851

,1.34217

,1.20394

,0.98315

,0.11645

,1.29220

,1.20347

,0.98375

,0.12990

,1.44107

,1.20296

,0.94036

,0.09848

,1.22337

1.20330

,1.01333

,0.14579

,1.41281

,1. 20390

.0.95351

,0.10133

,1.29580

,1.20427

,0.99159

,0.12992

,1.11702

,1.20233

,1.05013

,0.16509

.,1.11711

,0.95440

*0.00000

,1.37540

,1.19392

,0.91207

.0.00000

1.37819

1.18995

,0.91497

.0.00000

1.30057

,1.19150

,0.95055

.0.00000

,1.33067

f1.18901

,0.93792

'0.00000

,1.26766

,1.19507

,0.96358

.0.00000

,1.40424

,1.19023

,0.92108

.0.00000

,1.33680

,1.19210

,0.94245

'0.00000

,1.33448

,1.19307

,0.94548

,0.00000

,1.28047

,1.19449

,0.97434

,0.00000

,1.35498

,1.19136

,0.92980

,0.00000

,1.36029

,1.19272

,0.94307

,0.00000

,1.31084

,1.19195

,0.94387

,0.00000

,1.44018

,1.18466

,0.89485

,0.00000

,1.24814

,1.19788

,0.97714

.0.00000

,1.42061

,1.18737

,0.90980

.0.00000

,1.31116

,1.19487

,0.95248

.0.00000

,1.14652

,1.20223

,1.01815

'0.00000

,1.14584

,0.89426

,1.34410

,1.16822

,0.84074

,1.34348

,1.16477

,0.84437

,1.28079

,1.17560

,0.89042

,1.30426

,1.16975

,0.87491

,1.25568

,1.18252

,0.90625

,1.36395

,1.16285

,0.84597

,1.31016

r1.17212

,0.87684

,1.30928

,1.17334

.0.87985

,1.26435

,1.18146

,0.91402

,1.32509

,1.16935

,0.86200

,1.32835

,1.17058

,0.87284

,1.29054

,1.17467

,0.88354

,1.38867

,1.15377

,0.81696

,1.24146

,1.18695

r0.92233

1.37411

,1.15871

,0.83387

1.28954

,1.17752

,0.88838

,1.16347

,1.20234

,0.97415

,1.16247

P

F

F

F

t

t

t

t

#

#

#

A-16
1.17048

1.20195

0.91915

0.00000

1.15540

1.21733

0.94149

0.00000

1.19175

1.19711

0.90043

0.00000

1.17073

1.21188

0.92755

0.00000

1.19527

1.19699

0.89720

0.00000

1.19157

1.19580

0.90072

0.00000

1.16649

1.21523

0.93418

0.00000

1.16921

1.20992

0.93133

0.00000

1.14290

1.22913

0.96208

0.00000

1.14566

1.22851

0.95818

0.00000

1.17169

1.20872

0.92649

0.00000

1.20691

1.18746

0.88053

0.00000

1.17725

1.22091

0.91511

0.00000

1.17356

1.22218

0.92037

0.00000

1.23334

1.17890

0.84378

0.00000

1.27321

1.14665

0.79866

0.00000

1.17676

1.21672

0.92018

0.00000

1.23246

1.17877

0.85020

,1.18002

,1.19064

,0.81871

,0.25034

,1.17229

,1.20608

,0.84477

,0.23962

,1.20347

r1. 18502

,0.79953

,0.24176

r1.18776

,1.20050

,0.82887

,0.23935

,1.20773

1.18445

,0.79591

,0.24027

,1.20444

1.18307

,0.80118

,0.23969

,1.18556

,1.20374

,0.83696

,0.24066

,1.18648

,1.19869

,0.83407

,0.24318

,1.16679

,1.21811

,0.86905

,0.24370

,1.16941

,1.21725

,0.86487

,0.24258

,1.18735

,1.19741

,0.82793

,0.24362

,1.21473

,1.17343

,0.77839

,0.25470

,1.19468

,1.20508

,0.81834

,0.25331

,1.19103

,1.20737

,0.82254

,0.25250

,1.23457

,1.15988

,0.74152

,0.23761

,1.26496

,1.12853

,0.69045

,0.25272

,1.19315

,1.20303

,0.82165

,0.24577

,1.23559

,1.16258

,0.74502

,1.18829

,1.16842

,0.67951

,086393

1. 18752

,1.18414

,0.70829

,0.89898

,1.20927

,1.15934

,0.66093

,0.86820

,1.19945

,1.17704

,0.69074

,0.89756

,1.21234

,1.15868

,0.65748

'0.90106

,1.20935

,1.15719

,0.66415

,0.86099

,1.19784

,1.17987

,0.69893

,0.87072

,1.19592

,1.17509

,0.69622

,0.83420

,1.18612

,1.19580

,0.73272

,0.83839

,1.18858

,1.19481

,0.72858

,0.87380

,1.19720

,1.17394

,0.68980

,0.92391

,1.21483

,1.14636

0.64139

,0.86661

,1.20749

,1.17778

,0.68341

,0.86054

,1.20488

,1.18145

,0. 68638

,0. 95267

1. 22937

,1.12806

,0.60816

,0.98469

,1.24637

,1.09374

,0. 55539

,0.87412

,1.20527

,1.17861

,0.68384

,0. 92782

,1.23113

,1. 13201

,0.60794

,1.19187

,1.04735

,0.26714

,1.00628

,1.19755

,1.06168

,0.28830

,1.06551

,1.20563

,1.03884

,0.24777

,1.02016

,1.20438

,1.05564

,0.26866

.1.06481

,1.20770

,1.03829

,0.24569

,1.06811

,1.20546

,1.03642

,0.25192

,1.01013

,1.20464

,1.05747

,0.27291

,1.02347

,1.20129

,1.05318

,0.27222

,0.96626

,1.20177

1.07150

,0.29943

,0.97045

,1.20272

,1.07073

.0.2 9562

,1.02777

,1.20189

,1.05232

,0.26940

,1.09939

r1.20705

r1.02588

,0.24039

,1.00805

,1.21587

,1.04999

,0.28036

,1.00200

,1.21406

,1.05468

,0.28016

,1.13920

,1.21661

,1.00388

,0.23463

,1.22271

,1.21702

,0.97419

,0.19825

,1.01422

,1.21221

,1.05543

,0.26341

,1.12601

,1.21797

,1.00997

,0.22821

,1.20175

,1.04999

,0.16560

,1.06613

,1.21089

,1.06421

,0.18029

,1.13018

,1.21201

,1.04395

,0.15279

,1.08333

.1.21475

,1.05991

,0.16678

,1.13196

,1.21330

,1.04274

,0.15129

,1.13287

,1.21172

,1.04125

,0.15598

,1.07167

,1.21655

,1.06220

,0.16946

,1.08475

,1.21268

,1.05860

,0.16922

,1.02347

,1.21806

,1.07502

,0.18732

,1.02836

,1.21860

,1.07378

,0.18490

1.09057

,1.21201

,1.05700

,0.16731

,1.16574

,1.21040

,1.02958

,0.14811

,1.07205

,1.22679

,1.04720

,0.17427

,1.06679

,1.22518

,1.05263

,0.17406

,1.20972

,1.21536

,1.00141

,0.14393

,1.30460

,1.20702

,0.97836

,0.12026

,1.07470

,1.22169

,1.05586

,0.16237

,1.20049

,1.21647

,1.01287

,0.13993

,1.20183

,1.01803

,0.00000

,1.10782

,1.21299

,1.03316

'0.00000

,1.17115

,1.20924

,1.00956

'0.00000

,1.12816

,1.21464

,1.02739

.0.00000

,1.17399

,1.21029

,1.00800

.0.00000

,1.17256

,1.20938

,1.00785

.0.00000

,1.11692

,1.21722

,1.03024

.0.00000

,1.12643

,1.21297

,1.02735

.0.00000

,1.07225

,1.22174

,1.04536

,0.00000

,1.07726

,1.22218

,1.04368

'0.00000

,1.13272

,1.21179

,1.02518

'0.00000

,1.20041

,1.20510

,0.99397

.0.00000

,1.12207

,1.22495

,1.01157

.0.00000

,1.11712

,1.22392

,1.01755

'0.00000

,1.24227

,1.20568,

,0.96162

.0.00000

,1.33047

,1.19294

,0.93729

0.00000

,1.12143

,1.21939

,1.02076

,0.00000

,1.23735

,1.20765

,0.97473

'0.00000

,1.20207

,0.97422

,1.13743

i1.21534

,0.99201

,1.18948

,1.20445

,0.95963

,1.15673

,1.21384

,0.98194

,1.19278

,1.20487

,0.95706

,1.18971

,1.20397

,0.95886

,1.14907

,1.21682

,0.98646

,1.15470

,1.21190

,0.98398

,1.11457

,1.22570

,1.00761

,1.11828

,1.22574

,1.00475

1.15925

,1.21087

,0.98067

1.21089

,1.19748

,0.94214

,1.15760

,1.22319

,0.96731

,1.15294

,1.22320

,0.97328

,1.24530

,1.19350

,0.90745

,1.31132

,1.17281

,0.87331

,1.15637

,1.21766

,0.97410

,1.24378

,1.19514

,0.91746

A-17
0.00000

1.24313

1.17127

0.83649

0.00000

1.23477

1.17553

0.84591

0.00000

1.23237

1.17786

0.85057

0.00000

1.23451

1.17722

0.84827

0.00000

1.23426

1.17717

0.84781

0.00000

1.23309

1.19128

0.85736

0.00000

1.20799

1.20651

0.88490

0.00000

1.19297

1.21610

0.90647

0.00000

1.18810

1.21600

0.91247

0.00000

1.24543

1.17843

0.84142

0.00000

1.30356

1.14054

0. 76905

0.00000

1.24858

1.17590

0.83623

0.00000

1.19257

1.21253

0.90593

0.00000

1.22377

1.18263

0.86923

0.00000

1. 25149

1.16110

0.84006

0.00000

1.24828

1.16535

0. 84313

0.00000

1. 25319

1. 16035

0. 84545

0.00000

1.27502

1.16096

,0.24977

,1.24230

,1.15401

,0.72987

,0.24730

,1.23638

,1.15856

,0.74102

0.24604

,1.23490

,1.16193

,0.74512

,0.24624

,1.23637

,1.16046

,0. 74297

,0. 24646

,1.23611

1 .16034

,0. 74240

,0. 24228

1.23932

,1.17361

,0.75431

,0.26280

,1.21465

,1.18738

,0.78696

,0.25727

,1.20598

,1.19934

,0.80762

,0.25535

,1.20309

,1.20003

,0.81182

,0.25215

,1.24571

,1.15823

,0.73957

.0.24849

.1.28945

,1.11565

.0.66640

,0.25263

,1.24786

,1.15514

,0.73531

,0.25545

.1.20559

,1.19627

,0.80641

,0.24445

,1.22644

,1.16649

,0.76635

,0.23837

,1.24535

,1.14446

,0.73426

,0.24805

,1.24208

,1.14763

,0.73850

,0.22599

,1.25097

,1.14518

,0.73981

,0.21597

,1.27360

,1.14219

,0.94831

,1.23341

,1.12241

,0.59287

,0.93826

,1 .23072

,1.12701

,0. 60422

,0.93189

,1.22959

,1.13147

,0.60777

,0.93325

,1.23103

,1.12892

,0.60567

,0.93270

.1.23125

,1.12887

.0.60530

,0.89572

,1.23725

,1.14137

,0.61835

,0.90245

,1.22053

,1.15790

,0.65266

,0.87316

,1.21678

,1.17210

,0.67028

,0.87176

,1.21261

,1.17344

,0.67214

,0.94909

,1.23932

,1.12534

,0.60476

,1.02876

,1.26571

,1.07588

,0.53683

,0.95815

,1.23982

,1.12180

,0.60140

,0.88044

,1.21397

,1.16979

.0.66756

,0.93257

,1.22357

,1.13761

,0.62681

,0.96663

,1.23299

,1.11372

,0.59410

,0.96311

,1.23101

,1.11734

,0.59986

,0.95070

,1.23776

,1.11409

,0.59598

,0.92595

,1.25662

,1.10861

,1.15305

.1.21790

.1.00031

,0.22264

,1.13964

,1.21730

,1.00532

,0.22576

,1.13215

,1.21680

,1.00997

,0.22724

,1.13482

,1.21809

,1.00715

,0.22630

,1.13420

,1.21775

,1.00712

,0.22647

,1.08731

,1.22814

,1 01660

'0.23833

1.05001

,1.22179

1.03075

,0 .26024

,1.01397

,1.22213

,1.04611

,0.26469

,1.01455

,1.21867

1.04882

,0.26268

,1.12976

,1.22537

,1.00185

,0.22402

,1.24846

,1.23098

,0.95325

,0.18497

,1.14084

,1.22433

,0.99797

,0.22324

,1.02526

,1.21824

,1.04491

,0.26198

.1.11101

,1.21495

,1.01647

,0.23157

,1.16914

,1.21466

,0.99518

,0.20838

,1.15671

,1.21439

0.99763

,0.21646

,1.16393

,1.21611

,0.99529

,0.20390

,1.14776

,1.23113

,0.98846

,1.22729

,1.21437

,1.00283

,0.13604

,1.21261

,1.21549

.1.00808

,0.13809

,1.20550

,1.21534

,1.01338

,0.13912

,1.20842

,1.21612

.1.01032

.0.13829

.1.20839

,1.21597

,1.01009

,0.13854

,1.16672

,1.22805

,1.01817

,0.14619

,1.11552

,1.22620

,1.02632

,0.15996

,1.07946

,1.22931

,1.04454

,0.16249

,1.07713

,1.22847

,1.04956

,0.16052

,1.19928

,1.22137

,1.00049

,0.13559

.,1.32355

,1.21499

,0.94972

,0.11023

,1.20867

,1.22092

,0.99611

,0.13446

,1.08776

,1.22725

,1.04511

,0.15899

,1.17619

,1.21521

,1.01961

,0.13916

,1.23994

r1.20824

,0.99960

,0.12387

,1.22816

,1.20914

,0.99967

,0.12929

,1.23498

,1.21182

f1.00398

,0.12030

,1.22980

,1.22452

,0.99517

,1.25921

,1.20356

,0.96385

,0.00000

,1.24515

,1.20602

,0.97002

'0.00000

,1.23947

,1.20655

0.97527

'0.00000

,1.24349

,1.20667

,0.97219

,0.00000

,1. 24368

,1.20671

,0.97196

.0.00000

,1.21516

,1.21881

,0.97837

,0.00000

,1.16283

,1.21927

,0.98866

'0.00000

1.13258

,1.22409

,1.00859

'0.00000

,1.12616

,1.22342

,1.01439

.0.00000

,1.24016

.,1.20833

,0.96047

.0.00000

,1.35378

,1.19389

,0.90430

,0.00000

.1.24616

,1.20747

,0.95515

.0.00000

,1.13524

,1.22170

,1.00895

,0.00000

,1.21466

,1.20569

,0.98320

,0.00000

,1.27433

,1.19488

,0.96267

'0.00000

,1.26339

,1.19591

,0.96270

'0.00000

,1.26976

,1.19920

,0.96853

'0.00000

,1.27851

,1.20813

,0.95722

,1.2.5966

,1.18919

,0.90558

,1.24830

,1.19266

,0.91296

,1.24447

,1.19392

,0.91792

,1.24776

.1.19386

,0.91523

,1.24785

,1.19397

,0.91491

,1.23534

,1.20640

,0.92214

,1.19444

,1.21279

,0.94092

,1.17222

,1.21969

,0.96187

,1.16555

,1.21868

,0.96823

,1.25315

,1.19404

,0.90512

,1.34049

,1.16921

,0.84005

,1.25739

,1.19207

,0.89939

1.17251

,1.21595

,0.96165

,1.22878

,1.19473

,0.93127

,1.27361

,1.17927

,0.90702

1.26575

1. 18123

,0.90832

,1.27318

,1.18185

,0.91239

,1.29316

,1.18665

,0.89992

F

A-18
0.83296

0.00000

1.28908

1.16242

0.82059

0.00000

1. 26757

1.17529

0.84794

0.00000

1.24379

1.19190

0.87214

0.00000

1.26184

1.19121

0.85340

0.00000

1.32821

1.16252

0.78371

0.00000

1.32911

1.15801

0.78231

0.00000

1.26465

1.19851

0.84480

0.00000

1.25355

1.20281

0.86546

0.00000

1.26589

1.19757

0.84485

0.00000

1.21348

1.22226

0.89486

0.00000

1.21266

1.23002

0.89600

0.00000

1.16493

1.24179

0.94668

0.00000

1.12291

1.26975

0.98146

0.00000

1.11157

1.28008

0.98713

0.00000

1.15965

1.27964

0.95014

0.00000

1.13221

1.28731

0.98215

0.00000

1.15833

1.28049

0.95332

0.00000

1.24848

,0.72790

,0.21343

,1.28742

,1.14157

,0.71385

,0.20721

,1.27488

,1.15708

,0.74079

,0.22925

,1.25261

,1.17143

,0.76828

,0.24265

,1.26517

,1.16545

,0.75174

r0.22050

,1.32455

,1.13178

,0.68176

,0.21676

,1.32460

,1.12857

,0.67863

,0.23128

,1.27636

,1.17243

,0.74563

,0.20271

,1.27594

,1. 18113

,0.76265

,0. 22103

1.28027

,1.17242

,0.74372

,0. 22055

,1. 24384

,1.19958

,0.79226

,0.22122

,1.24593

,1.20562

,0.79282

,0.21319

,1.20645

,1.22327

,0.84384

0. 20126

11.17570

,1.25426

,0.87894

,0.19677

,1.16941

r1.26442

,0.88480

,0.21946

f1.20868

,1.25499

,0.84948

,0.21315

,1.18490

,1.26622

,0.88059

,0.21796

,1.20716

,1.25491

,0.85001

,0.21642

,1.25134

,0.58655

,0.91975

,1.26827

,1.10573

,0.57305

,0.89231

,1.26268

,1.12294

,0.59428

,0.88665

,1.24565

,1.13919

,0.62366

,0.89685

1.25362

1.13001

r0.61384

,0.93035

,1.29930

,1.08657

,0.54854

,0.93074

,1.29802

,1.08464

,0.54426

.0.90274

,1.27114

,1.13216

.0.60666

,0.84687

,1.27501

,1.14313

r0.61804

,0.88647

,1.27554

1.13227

,0.60374

,0.83661

,1.25526

,1.16565

,0.64604

,0.82121

,1.25966

,1.17145

,0.64701

,0.80219

,1.23364

,1.19402

,0.68956

,0.76389

,1.21546

,1.22546

,0.72029

,0.74985

,1.21221

,1.23397

,0.72571

r0.74344

,1.23757

,1.21937

r0.69693

.0.72320

,1.22022

,1.23478

,0.72306

,0.73807

,1.23826

,1.21988

,0.69523

,0.86874

,1.24763

,0.20429

1.14482

1.24051

,0.98515

,0.19165

,1.10473

,1.23972

.1.00290

,0.19611

,1.06865

,1.23335

,1.01768

,0.22137

,1.07692

,1.23950

,1.00556

,0.22870

1.15155

1.26471

.0.95937

,0.19126

,1.16017

1.26142

,0.95923

,0.18669

,1.08200

,1.25402

'0.99955

,0.21810

,1.03780

,1.25728

,1.01642

,0.21221

,1.07508

,1.25693

,1.00163

,0.21382

,1.00539

,1.25261

,1.03638

,0.23747

,0.98588

,1.25912

,1.04165

,0.24236

,0.95182

,1.24366

,1.06786

,0.24609

,0.90093

,1.24176

.1.09579

,0.25814

,0.88403

,1.24340

,1.10123

,0.26206

,0.87683

,1.25969

,1.08720

,0.25866

,0.85227

,1.24794

,1.10584

,0.26704

,0.87325

,1.26115

,1.09024

,0.26076

,1.07060

,1.23630

,0.11942

11.22884

,1.23271

,0.99247

,0.11195

,1. 18615

,1.23572

,1.01395

,0.11391

,1.14274

,1.23429

,1.02349

,0.12863

,1.15646

,1.23769

,1.00450

,0.13319

,1.24506

,1.25056

,0.95817

,0.11068

,1.25554

,1.24663

0.96034

,0.10804

,1.15356

,1.24645

,0.99724

,0.12515

,1.12163

,1.25413

,1.02331

,0.12096

,1.15353

,1.24987

,1.00243

,0.12245

11.07796

,1.25467

,1.03901

,0.13449

,1.06149

,1.26191

,1.04184

,0.13707

,1.01365

,1.25511

,1.07659

,0.13843

,0.95579

,1.26228

,1.10540

,0.14343

*0.93775

,1.26779

1.11089

,0.14490

,0.94859

,1.27790

i,.08603

.0.14193

,0.92134

,1.27159

,1.10869

,0.14567

,0.94659

,1.28102

,1.09151

,0.14291

,1.15529

,1.22841

.0.00000

,1.28124

,1.21583

,0.95265

,0.00000

,1.24110

,1.22250

,0.97663

,0.00000

,1.19311

,1.22367

,0.98742

.0.00000

,1.20929

r1.22368

,0.96591

'0.00000

,1.30725

,1.22609

,0.91361

.0.00000

,1.31513

,1.22211

,0.91617

.0.00000

,1.20256

,1.23288

,0.95747

.0.00000

,1.18514

,1.24311

,0.98432

.0.00000

,1.20555

r1.23657

,0.96244

.0.00000

,1.13183

,1.24437

,1.00214

.0.00000

,1.12156

,1.25082

,1.00334

.0.00000

,1.06454

,1.25130

,1.04424

.0.00000

,1.00541

,1.26711

,1.07380

,0.00000

,0.98898

,1.27484

,1.07834

*0.00000

,1.02004

,1.27987

,1.04766

'0.00000

,0.99016

,1.27702

,1.07357

.0.00000

,1.01835

,1.28083

,1.05244

.0.00000

,1.20261

,1.22122

,1.30349

,1.19176

,0.89152

,1.27122

,1.20214

,0.91725

,1.23099

,1.20932

,0.93435

1.24806

,1.20803

,0.91365

,1.33832

,1.19656

,0.85195

,1.34240

,1.19280

,0.85294

,1.24153

,1.21649

,0.90461

,1.23028

,1.22518

,0.92839

.1.24440

,1.21865

,0.90746

1.17855

,1.23449

,0.95302

.,1.17378

,1.24160

,0.95413

,1.11826

,1.24773

,1.00080

,1.06455

,1.26950

,1.03240

1.05036

,1.27775

,1.03673

,1.09558

,1.27957

,1.00268

,1.06563

,1.28267

,1.03247

,1.09434

,1.28144

,1.00712

,1.23584

1.20527

A-19
1.18309 ,1.16436

0.86563 ,0.75902

0.00000 ,0.23963

1.22346 ,1.21636

1.15298 ,1.13460

0.88649 ,0.76992

,1.13464

,0. 60975

,0. 92194

,1.21058

,1. 10413

,0.59800

,1.00277 ,1.01656

,0.22902 10.11769

,1.13863 ,1.20200

,1.19953 ,1.19321

,1.07312 ,1.05171

,0.23849 ,0.13926

,0.98333 ,0.93177,

,0.00000

,1.22806 ,1.23135,

,1.18087 ,1.17044

,1.01543 ,0.96029

,0.00000

7

156, 1 ,

8,

15,

22,

29,

36,

43,'

50,

57,

64,

71,

78,

85,

92,

99,

106,

113,

120,

127,

134,

141,

148,

155,

2,

9,

16,

23,

30,

37,

44,

51,

58,

65,

72,

79,

86,

93,

100,

107,

114,

121,

128,

135,

142,

149,

156,

0.00710, 0.00

0686, 0.00684,

3,

10,

17,

24,

31,

38,

45,

52,

59,

66,

73,

80,

87,

94,

101,

108,

115,

122,

129,

136,

143,

150,

4,

11,

18,

25,

32,

39,

46,

53,

60,

67,

74,

81,

88,

95,

102,

109,

116,

123,

130,

137,

144,

151,

5,

12,

19,

26,

33,

40,

47,

54,

61,

68,

75,

82,

89,

96,

103,

110,

117,

124,

131,

138,

145,

152,

6,

13,

20,

27,

34,

41,

48,

55,

62,

69,

76,

83,

90,

97,

104,

111 ,

118,

125,

132,

139,

146,

153,

7,

14,

21,

28,

35,

42,

49,

56,

63,

70,

77,

84,

91,

98,

105,

112,

119,

126,

133,

140 , 147,

154,

9

156, 0.00689, 0.00718, 303, 0.00690, 0.

0.00687, 0.00684, 0.0068

0.00681,

0. 00675,

0.00663,

0.00638,

0.00611,

0.00601f

0.00597,

0.00592,

0. 00582,

0.00563,

0.00726,

0.00712f

0.00704,

0. 00680,

0. 00675,

0.00656,

0. 00641,

0.00610,

0.00729,

0.00717,

0.00504,

0.0O

0.00680,

0. 00674,

0.00661,

0.00636,

0.0060 9,

0. 00 601,

0.00596,

0. 00590,

0.00580,

0.00556,

0.00723,

0.00710,

0. 00 699,

0. 0067 9,

0.00674,

0.00656,

0.00635,

0.00608,

0. 00719,

0. 00700,

0. 00534,

0.00679,

0. 00 672,

0.00654,

0. 00 635,

0.00608,

0.00600,

0.00595,

0. 00588,

0.00578,

0.00555,

0.00719,

0. 00710,

0.00693,

0.0067 9,

0.00673,

0.00655r

0.00630,

0.00710,

0.00704,

0.00705,

0. 00 678,

0. 00671,

0.00645,

0.00633,

0.00608,

0.00599,

0.00595,

0.0058 6,

0. 0057 6,

0 .00551.,

0.00717,

0.00709,

0.00689,

0.00678,

0.00671,

0.00653,

0.00627,

0.00664,

0.00649,

0.00075,

0.0067

0.0067

0.0064

0.0063

0.0060

0.0059

0.0059

0.0058

0.0057

0.0054

0.0071

0.0070

0.0068

0.0067

0.0066

0.0065

0.0062

0.0064

0.0064

0.0070

00689, 0.00687,

3, 0.00682, 0.00682,

7, 0.00676, 0.00676,

0, 0.00667, 0.00665,

4, 0.00643, 0.00640,

1, 0.00613, 0.00611,

7, 0.00606, 0.00603,

9, 0.00598, 0.00597,

4, 0.00594, 0.00593,

4, 0.00583, 0.00583f

4, 0.00571, 0.00570,

8, 0.00729, 0.00728,

4, 0.00713, 0.00713,

8, 0.00706, 0.00705,

6, 0.00685, 0.00682,

7, 0.00677, 0.00676,

4, 0.00659, 0.00658,

0, 0.00648, 0.00647,

2, 0.00615, 0.00613,

4, 0.00632, 0.00617,

1, 0.00636, 0.00650,

0, 0.00679, 0.00348,

10

156, 191.19, 194.00, 194.98,

47.55, 47.55, 55.00,

52.51, 48.30, 45.77,

49.94, 48.23, 50.71,

45.69, 45.68, 50.57,

49.72, 49.70, 45.53,

47.03, 45.31, 47.83,

45.98, 50.23, 45.16,

45.14, 45.12, 49.21,

47.50, 49.15, 47.47,

45.05, 43.41, 51.59,

45.71, 46.60, 51.40,

46.52, 48.85, 44.49,

42.69, 48.13, 45.16,

50.51, 46.31, 43.73,

46.68, 49.66, 43.66,

200.85, 42.46, 44.15, 47.57,

50.79, 42.43,

48.27, 47.47,

46.53, 47.48,

45.60, 48.05,

47.98, 52.08,

49.40, 50.17,

49.26, 43.55,

45.11, .51.61,

49.18, 41.74,

49.08, 47.40,

41.55, 44.81,

48.20, 49.94,

45.16, 45.12,

46.70, 48.44,

42.44, 46.62,

47.56,

48.25,

49.93,

47.27,

45.36,

47.78,

47.58,

51.61,

49.14,

44 .97,

47.05,

43.96,

51.79,

43.71,

49.60,

47.55,

45.76,

52.37,

46.36,

43.55,

45.30,

46.88,

49.17,

49.12,

49.02,

50.04,

46.40,

50.42,

44.90,

47.85,

A-20
11

156,

13

156,

14

156,

"49.65,

44.82,

49.41,

45.09,

45.05,

44.57,

42.99,

-0.483,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.378,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

11,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

4,

4,

4,

46.63,

46.57,

43.45,

46.28,

42.79,

44.85,

45.26,

-0.483,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.499,

0.,

0.,

0.,

0.,*

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

14, 15

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

4,

4,

4,

48.37,

48.33,

48.16,

45.98,

43.53,

45.50,

-0.483,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

-0.408,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

48.41,

43.57,

46.43,

47.14,

42.64,

45.07,

44.88,

49.51,

45.17,

44.35,

44.53,

179.87,

45.32,

43.55,

45.77,

46.06,

43.73,

180.33,

47.80,

47.70,

49.25,

46.55,

43.86,

180.82,

-0.483, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408f

-0.408, -0.408,

-0.408, -0.408f

-0.408, -0.408,

-0.408, -0.408,

-0.408f -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

-0.408, -0.408,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

0.,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

4,

4,

4,

8,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

4,

4,

4,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

4,

4,

4,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

4,

4,

4, 4,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

3,

4,

4,

4,

4,

A-21
4, 4, 4, 4, 4, 4, 4,

4, 4, 4, 4, 4, 4, 4,.

4, 4, 4, 4, 4, 4, 4,

4, 4, 4, 4, 4, 4,i 4,

4, 4, 5, 5, 5, 5, 5,

6, 6, 6, 6, 6, 6, 7,

8, 8, 9, 1, 11 11 6

4, 4,

15

156, 60, 60, 60, 60, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74; 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, .74,

74, 74, 74, 74, . 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, .74,

74, 74, 74, 74, 74, 74, 74,.

74, . 74, 74, 74, 74, 74, 74,

74, 74, 74, . 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

74, 74, 74, 74, 74, 74, 74,

92, 96,

16

156, 1, 1, 1, 1, 2, 2, 2,.

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, *2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2,. 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

3, 4,

17

4, 450.260, 450.260, 450.260, 450.260

18

4, 231.250, 259.910, 296.590, 290.130.

19

4, 105.910, 102.420, 98'.980, 95.080

20

4, 105.910, 102.420, 98.980, 95.080

21

4, 1.473, 1.276, 1.107, 0.969

22

4, 0.100, 0.100, 0.100, 0.100

23

4, 0.100, 0.100, 0.100, 0.100

24

A-22
4, 1.300, 1.300, 1.300, 1.300

25

4, 0.125, 0.125, 0.125, 0.125

26

156, 1, 1, 1, 1, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, *2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, .2,

2, 2, 2, 2, 2, 2, 2,

3, 4,

27

4, 10.465, 10.465, 10.500, 10.520

28

4, 1.044, 0.954, 0.876, 0.819

29

4, 0.5125, 0.5125, 0.5125, 0.5125

30

4, 0.0393, 0.0393, 0.0393, 0.0393

31

4, 0.08100, 0.07112, 0.06600, 0.06300

32

4, 0.1690, 0.1690, 0.1690, 0.1690

33

4, 0.01000, 0.01016, 0.00900, 0.00850

34

156, 1, 1, 1, 1, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, . 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2,

3, 4,

35

4, 1, 1, 1, 1

36

450.260,

450.260,

450.260,

A-23
37

53

1E-2

58

156, 1,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

3,

450.260,

1.000,

1.000,

1.000,

1.000,

.5E-3

1 ,

2,

2,

2,r

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

4,

.5E-3 .2E-4

2,

2,

2,

2,

2,

2,

2,

2,

2,r

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

.5E-3 

. 1E-8 

. 5E-3 

. 5E-7

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2,

2, 2, 1

59

60

61

4, 3, 4, 4,

74.010 414.290 450.260

72.840 293.360 397.040 450.260

72.840 293.360 414.290 450.260

450.260

105.91, 98.58, 105.91

102.42, 96.28, 104.08, 110.27

98.98, 92.78, 104.36, 110.56

95.08

62

63

156,

1.473, 1.343, 1.473

1,276, 1.176, 1.392,

1.107, 1.020, 1.310,

0.969

1.506

1.415

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

1,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

1,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

1,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2.,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,

2,f

2,

2,

2,

2,

A-24
2,

3,

2, 2, 2, 2, 2, 2,

4,

64

4,

65

81.300,

444.340

81.300,

445.790

72.770,

388.490,

89.730,

66

1.205,

0.670,

1.290,

0.579,

1.007,

0.576,

0.567,

8, 8, 9,

67

68

69

70

71

72

73

74

75

76

0

4,

4,

4,

4,

4,

4,

4,

4,

4,

4,

2,

0.1002,

1.000,

1.000,

0.1246,

-1.000,

2,

1.000,

1.000,

1.000,

132.730, 1

132.730, 1

113.410, 1

444.810

148.630, 2

1.205, 1.

1.290, 1.

0.930, 0.

0.676,

0.567, 0.

2,

0.1002,

1.000,

1.000,.

0.1246,

-1.000,

2,

1.000,

1.000,

1.000,

83.780,

83.780,

54.050,

'07.530,

205,

290,

930,

567,

234.840, 286.270, 337.300, 3

234.840, 286.270, 337.300, 3

194.690, 240.410, 286.130, 3

266.430, 325.330, 384.230, 4

1.205, 1.205, 1.205, 1.205,

1.290, 1.290, 0.830, 0.830,

0.930, .0.930, 0.930, 0.576,

0.567, 0.567, 0.567, 0.537,

88.380,

88.380,

37.310,

44.640,

2,

0.1002,

1.000,

1.000,

0.1246,

-1.000,

2,

1.000,

1.000,

1.000,

2

0.2380

1.000

1.000

0.2000

-1.000

2

1.000

1.000

1.000

A-25
LAPURW V6.0 SAMPLE INPUT DECK:

LAPURW

1

156

2

3

6 SAMPLE INPUT DECK

1 1

11

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

4

6

8

4

14

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

4

6

8

4

15

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

5

6

9

3

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

5

6

ii

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

4

5

6

ii

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

4

5

7

6

4

5

450.2600

0.55 -0.80

1 6

1

6 0.161E-03 0.111E-02 0.994E-03 0.209E-02 0.810E-03 0.203E-03

6

7

8

9

10

11

12

17

18

19

6 0.012813 0.031492 0.124571 0.328380 1.408783 3.830869

1

1 0.0

1

1 0.3632E-04

-0 .271E-02

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

A-26
1

1

1

1

1

1

1

1

0.0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

20

21

22

0.2 0.4 0.6 0.8 1.0 1.2

25.2069 20.0838 15.6386 11.8715 8.7823 6.3711 4.6379

2:

2

100 0.010

0.210

0.280

0.350

0.420

0.490

0.560

0.630

0.700

0.770

0.840

1.000

1.700

6.000

100.000

0 1 1 1

0 0 0 0

14

0.100

0.220

0.290

0.360

0.430

0.500

0.570

0.640

0.710

0.780

0.850

1.100

1.800

7.000

200.000

0.120

0.230

0.300

0.370

0.440

0.510

0.580

0. 650

0.720

0.790

0.860

1.200

1.900

8.000

0.140

0.240

0.310

0.380

0.450

0.520

0.590

0.660

0.730

0.800

0.870

1.300

2.000

9.000

0.160

0.250

0.320

0.390

0.460

0.530

0.600

0.670

0.740

0.810

0.880

1.400

3.000

10.000

0.180 0.200

0.260 0.270

0.330 0.340

0.400 0.410

0.470 0.480

0.540 0.550

0.610 0.620

0.680 0.690

0.750 0.760

0.820. 0.830

0.890 0.900

1.500 1.600

4.000 5.000

20.000 50.000

1 1 0 0

0 0~ 0 0

0 0 0 0 0 0

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1.

1 1

1 1

1 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

28

29

30

0

1.19

6 0.000 -0.500 -1.060 -1.089 -1.500 -2.000

A-27

APPENDIX B

SAMPLE LAPURX AND LAPURW OUTPUT FOR A TYPICAL BWR

B-1
OUTPUT FILE LAPURX.CSV:

LAPURX 6 SAMPLE INPUT DECK

Ch,NCH, Fuel,Power,Flow, delta-PExit, Zb

,Type, (%), (%),(bar),Quality, (m)

1, 11, 1, 39.0827, 71.7374,

2, 14, 1, 32.0004, 69.6844,

(... channels 3 to 154 deleted here)

155, 4, 3, 78.6193, 110.0030,

156, 4, 4, 83.2990, 99.9911,

0.4583,

0.4583,

0.4583,

0.4583,

0.1212,

0.0945,

0.1741,

0.2110,

1.5539,

1.8266,

1.1959,

1.0750,

OUTPUT FILE LAPURW.CSV:

LAPURW RUN: LAPURW 6 SAMPLE INPUT DECK

LAPURX RUN: LAPURX 6 SAMPLE INPUT DECK

CHANNEL STABILITY:

CH,DR,FREQ,PHASE-MARGIN,,GAIN-MARGIN

(Hz), (DEG), (Hz),, (Hz)

1, 0.00, 6.68, ******, ******,

2, 0.00, 5.79, ******, ******,

(... channels 3 to 154 deleted here)

155, 0.00, 4.00, ******, *

156, 0.00, 0.26, *****, ******,

10.41,

10.49,

5.14,

18.04,

0.32

0.34

0.41

0.39

CORE OUT-PHASE STABILITY:

REACT,DR,FREQ,PHASE-MARGIN,,GAIN-MARGIN,,AMPL,AVG

($),, (Hz), (DEG), (Hz),, (Hz), (%), (%)

0.00, 1.02, 0.53, -0.87, 0.53, 0.98, 0.53,152.84,

-0.50, 0.63, 0.50, 30.20, 0.45, 1.59, 0.54,100.00,

-1.06, 0.39, 0.48, 81.02, 0.28, 2.28, 0.54,100.00,

-1.09, 0.39, 0.48, 82.72, 0.27, 2.32, 0.54,100.00,

-1.50, 0.29, 0.47, 96.97, 0.19, 2.83, 0.55,100.00,

-2.00, 0.21, 0.47,105.34, 0.14, 3.45, 0.55,100.00,

1.47,

0.00,

0.00,

0.00,

0.00,

0.00,

OPEN AND CLOSED-LOOP TRANSFER FUNCTIONS:

FREQ,OPEN-LOOP,,CLOSED-LOOP,

(Hz), (AMP), (PHASE-DEG), (AMP), (PHASE-DEG)

0.100, 0.385E+01, 89.40, 0.591E+00,

0.120, 0.331E+01, 86.12, 0.670E+00,

55.94,

58.54,

(... frequencies 0.140 Hz to 50 Hz deleted here)

100.000, 0.788E-04,

200.000, 0.201E-04,

13.15, 0.427E+00,

6.67, 0.218E+00,

-76.77,

-83.29,

CORE IN-PHASE STABILITY:

DR, FREQ, PHASE-MARGIN,, GAIN-MARGIN,

,(Hz), (DEG), (Hz),, (Hz)

0.55, 0.44, 22.51, 0.40, 1.59, 0.49,

B-2
APPENDIX C

DETAILED DESCRIPTION OF LAPUR 6 MODIFICATIONS

C-1
CONTENTS

1. IN T R O D U C T IO N .................................................................................................................... C -4

1.1. B a ckg ro u nd .......................................................................................................................... C -4

1.1.1. Improvements Made in LAPUR 5.1 to Obtain LAPUR 5.2 ...................................... C-5

1.1.2. Thermal-Hydraulic Improvements from LAPUR 5.2 to LAPUR 6.0 ............................. C-6

2. THERMAL-HYDRAULIC IMPROVEMENTS ...................................................................... C-7

2.1. Implementation of the Variable Area and the New Friction Models'into the Code LAPUR.C-7

2.2. Modifications in LAPURX Related to the Variable Flow Area ........................................ C-8

2.2.1. Theory of the Application ......................................................................................... C-8

2.2.2. Variable Cross Flow Area and Hydraulic Diameter-LAPUR Modifications ............ C-8

2.2.3. Calculation of Cell Variables-LAPUR Modifications ................................................. C-14

2.3. Modifications in LAPURX Related to Friction Model II ...................................................... C-17

2 .3 .1. T h eo ry ........................................................................................................................ C -17

2.3.1.1. Friction M odel I ............................................................................................... C -17

2.3.1.2. Friction M odel II ............................................................................................ C -19

2.3.2. Friction Model II-LAPUR modifications .................................................................... C-20

2.3.2.1. Friction factor m odel ........................................................................................... C -20

2.3.2.2. Friction factor Model I1-LAPUR modifications .................................................. C-21

2.3.2.3. Friction m ultiplier m odel ....................................................................................... C -23

2.3.2.4. Friction multiplier Model II-LAPUR modifications .............................................. C-24

2.4. Modifications in LAPURX Related to Pressure Losses Caused By Area Changes ........... C-28

2.4.1. Gravity Losses ............................................... C-28

2.4.1.1. Gravity losses-LAPUR modifications ............................ C-29

2.4 .2. A cceleration Losses ................................................................................................... C -30

2.4.2.1. Acceleration losses-LAPUR modifications ......................... C-32

2.4 .3. Friction Losses ........................................................................................................... C -34

2.4.3.1. Friction losses-LAPU R modifications ............................ C-35

2.4.4. Irreversible Expansion Losses ................................................................................... C-38

2.4.4.1. Irreversible expansion losses-LAPUR modifications ......................................... C-39

2.4.5. Irreversible Contraction Losses ................................................................................... C-41

2.4.5.1. Irreversible contraction losses-LAPUR modifications ....................................... C-43

C-2
2.4 .6. Local Losses .............................................................................................................. C -45

2.4.6.1. Local losses-LAPUR modifications ................................................................... C-46

2.5. LAPURX 6.0 Coefficients of the Momentum Equation ........................ C-49

2.5.1. LAPUR 6.0 Nonboiling Region Coefficients .............................................................. C-50

2.5.1.1. New nonboiling region coefficients-LAPUR modifications ................................ C-53

2.5.2. LAPUR 6.0 Boiling Region Coefficients ..................................................................... C-55

2 .5.2 .1. F rictio n M odel I .................................................................................................... C -55

2.5.2.2. F riction M odel II ................................................................................................. C -62

2.5.2.3. New boiling region coefficients-LAPUR modifications ...................................... C-64

2.6. Others Modifications Made In LAPURX 6.0 ......... ................................. C-67

2.7. Modifications in LAPURW 6.0 Related To Thermal hydraulics .......................................... C-72

2.7.1. Energy Conservation-LAPUR 6.0 Nonboiling Region Coefficients .......................... C-72

2.7.1.1. LAPUR modifications ........................................................................................... C-73

2.7.2. Mass and Energy Conservation-LAPUR 6.0 Boiling Region Coefficients ................ C-73

2.7.2.1. M ass conservation ............................................................................................... C -73

2.7.2.2. Energy conservation ............................................................................................ C-74

2.7.2.3. LAPUR modifications ............................................................................................ C-79

2.7.3. Momentum Conservation-LAPUR6 Boiling Region Coefficients .............................. C-80

2.7.3.1. LAPUR 6.0 modifications..................................................................................... C-80

2.8. OTHER MODIFICATIONS MADE IN LAPURW 6.0 .......................................................... C-81

2.9. MODIFICATIONS IN SHARED FILES ................................... C-83

3. PROGRAMMING ERRORS ................................................................................................. C-87

3.1. PROGRAMMING ERRORS CORRECTED IN VERSION 5.2 .......................................... C-87

3.2. PROGRAMMING ERRORS CORRECTED IN VERSION 6.0 .......................................... C-87

R efe re nces ............................................................................................................................... C -88

N o m enclature ........................................................................................................................... C -89

C-3
CHAPTER 1. INTRODUCTION

1.1. BACKGROUND

Thermal hydraulic stability of boiling water reactors (BWRs) has been a primary concern

since the beginning of BWR technological development. The phenomena started being studied

empirically and theoretically. For many years, no BWR experienced a power oscillation event.

The evolution of the technology saw design changes and increases in power density. These

developments gave rise to stability problems. The very first power oscillation incidents occurred

in the 1980s and continued through the early 1990s. Although many reactors. suffered such

instabilities, the La Salle incident in 1988 brought the issue of BWR stability as a worldwide

safety concern to the forefront. The regulatory agency of every country required a stability

revision on their BWR plants.

General Electric, together with the BWR Owners Group (BWROG), solved the problem

temporarily by limiting the process in those regions sensitive to instabilities, creating a criterion

about how to avoid instability issues and providing the operators with better training regarding

instability.

In 1992, the incident in Washington Nuclear Plant Unit-2 (WNP-2) led the U.S. Nuclear

Regulatory Commission (NRC) to request a definitive solution to the instability problem. GE and

BWROG gave different solutions depending on the reactor; the solutions were based on the

following detection and suppression and on prevention.

In Spain, the Consejo de Seguridad Nuclear (CSN) required the implementation of these

solutions to the affected Spanish reactors. Thus, the most suitable methodology was chosen

and introduced at each plant.

From 1995 to 1998, lberdrola in cooperation with lberdrola Ingenieria Consultoria and the

Universidad Polit6cnica de Valencia (UPV) developed a method to apply the ElA methodology

using LAPUR 5, which was created by the United States' Oak Ridge National Laboratory

(ORNL) (Ref. 7), as well as a program called PAPU, created by the UPV, for generating kinetic

parameters.

The application methodology and the programs used were licensed by the CSN; lberdrola

used them to perform the stability analysis of the last three reloads at the Cofrentes Nuclear

Plant.

In 2000, ORNL and UPV collaborated on a new version, LAPUR 5.2 (Ref. 2); it was

modeled at a higher level of detail, which provided better estimations of the stability margins.

LAPUR is a code for computing the parameters that define operational stability in BWRs.

This program consists of two autonomous modules, LAPURX, which. solves the steady-state

thermal hydraulic governing equations (assuming a constant channel flow area), and LAPURW,

which computes the decay ratio in the frequency domain.

Because of the widespread safety concerns over BWR instability issues, the European

nuclear community approved the NACUSP project (2001-2004), for which lberdrola and UPV

collaborated.

C-4
The UPV and lberdrola deemed it necessary to improve LAPUR for two main reasons:

1. The new fuel designs have incorporated higher power densities and power profiles

geared to the lower part of reactor's core, as well as partial-length rods, gadolinium rods,

etc., that, although they are more efficient from a neutronic viewpoint, from a stability

viewpoint, cause the fuel to behave differently.

2. The different types of instabilities, especially the so-called out-of-phase, which has been

observed recently during the BWR operation and start-up, allow the use of tools and

applied methodologies that can more precisely estimate the stability margins, ensuring

that no instability events occur that would require a forced outage or risk facility

availability.

In pursuit of the improvements to LAPUR, it was necessary to perform certain modifications.

The thermal-hydraulic methodology will be based on LAPUR 5.2 code (Ref. 2). Its prediction

capability will be improved performing the following modifications.

1.1.1. Improvements made in LAPUR 5.1 to obtain LAPUR 5.2

The major improvements of LAPUR 5.2 with respect to LAPUR 5.1 are:

1. The maximum number of thermal-hydraulic regions was parameterized. The default

value is 200 channels. Note: LAPUR calculations with up to 200 channels were verified

against results from LAPUR 5.1 with no significant loss of precision. Calculations with

more than 200 channels are typically unnecessary due to symmetry considerations, but

are possible; however, those calculations were not verified here.

2. The maximum number of axial power shapes was parameterized so that each thermalhydraulic region could have its own axial and radial power shape. The default. value of

axial power shape regions is 200.

3. The maximum number of bundle and fuel rod types was parameterized. The default

value is 10. This allows modeling of mixed cores with up to 10 different fuel types.

4. The maximum number of frequencies was parameterized. The default value is 100

frequencies. This upgrade allowed for better definition of the transfer functions and a

more robust evaluation of the decay ratio. The old version often required two LAPURW

calculations, one to identify the frequency of oscillation using a coarse frequency mesh,

and a second calculation with a fine mesh around the oscillation frequency. With 100

frequencies, a single calculation works for most cases.

5. All relevant dimensions of internal LAPUR variables were parameterized so they could

be modified in the future. File PARAM.FI contains the definitions for all these parameters.

Note: These parameters were left at their original values, and value changes have not

been tested thoroughly; it is the user's responsibility to verify that any future changes to

these parameters do not result in significant loss of precision.

6. A well known behavior of LAPUR 5.1 was that it would report very high channel decay

ratio if the channel was very stable. The user would have to compensate for this by

manually checking the phase and gain margins. In LAPUR 5.2, this behavior was

corrected by improving the decay-ratio estimation algorithm. The problem was related to

an internal "spline fit" that LAPUR performed on the calculated transfer functions. For a

very stable. channel, the phase might cross the zero-degree level; since LAPUR

maintains the phases negative, the phase jumps from -5 deg to -355 deg. The spline fit

under those conditions cannot reproduce the discontinuity accurately and as a result,

C-5
would report a very high decay ratio. This behavior was corrected in LAPUR 5.2 to allow

the accurate calculation and reporting of low-channel-decay-ratio conditions.

7. The calculation of the average saturated boiling boundary was added at the end of the

LAPURX execution. This calculation is performed for output purposes only and does not

affect the LAPURW input files; thus, it does not affect the calculated decay ratios.

8. Two new output files are generated by LAPUR 5.2. These files have an ASCII comma

separated value (CSV) format and summarize the most relevant output information. The

file names are LAPURX.CSV and LAPURW.CSV and can be opened with most

spreadsheet programs for further manipulation of the LAPUR output data. LAPURX.CSV

contains the relative power and flow, the pressure drop, exit quality, and the saturated

boiling boundary of each channel. LAPURW.CSV contains the decay ratio and frequency

for each channel and for the core, as well as the open-loop and closed-loop transfer

functions as function of frequency.

9. The screen-input routines were removed. "Batch" is the only available mode for LAPUR

5.2. The executable files have been renamed LAPURX.EXE and LAPURW.EXE.

10. During the detailed code review, the authors found and corrected one miscellaneous

programming error.

1.1.2. Thermal-Hydraulic Improvements from LAPUR 5.2 to LAPUR 6.0

The major improvements of LAPUR 6.0 with respect to LAPUR 5.2 are:

1. The two-phase friction multipliers, even for frictional forces as for local losses, have been

improved, using updated relationships that are more realistic for the newest fuel designs.

2. The capability to model channels with variable area, in order to represent more

adequately the partial length rods of the new fuel designs.

3. Several cells in the single-phase region have been defined, and the cell variables

(densities, liquid velocities, temperatures) of the nonboiling region are calculated.

4. The local pressure losses (spacers) have been included.

5. The irreversible pressure losses due to contractions and expansions have been included.

6. During the detailed code review, the authors found and corrected three miscellaneous

programming errors.

7. Finally, the stability prediction capability of the new code was validated through data

collected from Cofrentes Nuclear Power Plant, as well as data from other plants, for

example, Ringhals and Forsmark.

C-6
CHAPTER 2. THERMAL HYDRAULIC IMPROVEMENTS

2.1. IMPLEMENTATION OF THE VARIABLE AREA AND THE NEW FRICTION MODELS

INTO THE CODE LAPUR

The current version LAPUR 5 has various model deficiencies (Ref. 5, Ref. 6); these are

listed below:

1. There is no model to evaluate local or secondary losses in the cells of the fuel elements.

Only friction losses are modeled. Therefore, it is necessary to include the local losses

through a multiplier to account for frictional losses, obtaining the equivalent pressure

drop.

2. It does not allow for the existence of a variable area in the fuel elements. Hence,

reversible losses or gaining pressures and irreversible losses have to be modeled by

means of a frictional multiplier.

3. Two-phase friction multipliers are estimated by using the Martinelli-Nelson multiplier,

which uses the Jones correction factor. This multiplier overestimates friction pressure

drop for steam qualities greater than 0.6, so it must be modified to reproduce the

available experimental data.

This section documents the implementation of new correlations for computing friction and

local losses, as well as capabilities for modeling bundles with variable cross-area in LAPUR. The

selection of basic models to implement in LAPUR is compiled in Reference 1. The friction and

local models selected are generic and do not include the proprietary information of fuel vendors.

Fuel vendors obtain from experimental data accurate friction models (single-phase friction

factors and two-phase multipliers) to use in their codes. This is a good way to obtain accurate

friction models, since they are strongly dependent on geometry and surface roughness.

However, it is more feasible for the utility to perform updates of generic models to obtain a good

agreement with data supplied by fuel vendors. With these objectives, generic friction models are

combined with free parameters to reproduce with a good degree of approximation the results

reported by vendors.

Section 2.2 presents the modifications made in LAPURX related to the variable flow area.

Section 2.3 presents the modifications made in LAPURX related to the friction model I1. Section

2.4 presents the modifications made in LAPURX related to the losses. Section 2.5 presents the

modifications made in LAPURX related to the new coefficients obtained in the momentum

equation with variable flow area. Section 2.6 presents other minor modifications made in

LAPURX related to the thermal hydraulics. Section 2.7 presents the modifications made in

LAPURW.

C-7
2.2. MODIFICATIONS IN LAPURX RELATED TO THE VARIABLE FLOW AREA

In this section, we present the following improvements:

" Theory of the application.

* Reading, storing, and preparing the new variables. These new variables are the axial

flow area and the axial hydraulic diameter of each type of channel.

" Printing new variables.

" Calculating cell variables, including the flow area and hydraulic diameter for each cell of

the nonboiling and boiling regions.

2.2.1. Theory of the Application

The current version LAPUR 5 considers, for the momentum conservation (pressure drops),

the nonboiling region as one "collapsed-cell." The new version LAPUR 6.0 computes all the

nonboiling cells variables of the momentum conservation equations.

The old LAPUR 5 does not allow for the existence of a variable cross-area in the fuel

elements. The channel has a constant axial flow area A0 (input user LAPURX data card 19). The

new version LAPUR 6.0 has the capability for modelling bundles with variable cross-area.

2.2.2. Variable Cross Flow Area and Hydraulic Diameter-LAPUR Modifications

Four subroutines have been modified:

" The subroutine INPUT: read the new variables.

" The subroutine OUTIN: print the new variables.

" The subroutine SETUP: select from the arrays the specific parameters of the channel

type IX.

" The subroutine NODE: transform the nodal parameters to the new coordinate system.

Subroutine INPUT-Read the new variables.

580 CALL INP1(NXE,NDIMV)

GO TO 6000

590 CALL INP1(NTDV,NODV)

GO TO 6000

600 CALL INP2(NTDV,NODV,DZVM,NVAMAX)

GO TO 6000

610 CALL INP2(NTDV,NODV,AVM,NVAMAX)

GO TO 6000

620 CALL INP2(NTDV,NODV,DEVM,NVAMAX)

GO TO 6000

C-8
Variable Dimension Subroutine Description Units

AVM(IBM,I) NVAMAX,NV INPUT-user Boiling cell flow area cm2

DEVM(IBM,I) NVAMAX,NV INPUT-user Hydraulic diameter cm

DZVM(JJ,JJJ) NVAMAX,NV INPUT-user Top height from bottom of the JJJ-th variable-area

axial interval of channel variable-area JJ-type NDIMV(J) NXE INPUT-user Variable-area channel type of the J-th thermal region

NODV(J) NTDV INPUT-user Number of axial intervals for the J-th type, variablearea channel

NTDV ---- INPUT-user Number of variable-area channel types

PARAMETER (NVAMAX = 10)! Max # of variable. Nrea channels

Subroutine OUTIN-Print the new variables.

58 WRITE(6, 444)

WRITE(6, 158)11, NXE , (NDIMV(J), J=1,NXE)

GO TO 200

59 WRITE(6, 159)11, NTDV , (NODV(J),J=1,NTDV)

GO TO 200

60 DO 322 JJ=1,NTDV

JI=NODV (JJ)

IF(JJ.EQ.1) WRITE (6, 160)11, (DZVM(JJ,JJJ

IF(JJ.NE.1) WRITE (6, 301) (DZVM(JJ,JJJ),J

322 CONTINUE

GO TO 200

61 DO 323 JJ=1,NTDV

JI=NODV(JJ)

IF(JJ.EQ.1) WRITE (6, 161)11, (AVM(JJ,JJJ)

IF(JJ.NE.1) WRITE (6, 301). (AVM(JJ,JJJ),JJ

323 CONTINUE

GO TO 200

62 DO 324 JJ=1,NTDV

JI=NODV (JJ)

IF(JJ.EQ.1) WRITE (6, 162)11, (DEVM(JJ,JJJ

IF(JJ.NE.1) WRITE (6, 301) (DEVM(JJ,JJJ),J

324 CONTINUE

GO TO 200

),JJJ=1,JI) JJ=1 I JI)

, JJJ=1, JI)

*J=1, JI)

*) ,JJJ=1,JI)

JJ=1 , JI)

158 FORMAT(1HO,4X, (',12, ') ',8HNXE

7112

1 / (18X, 7112)

159 FORMAT(1HO,4X,' (',12,') ', 8HNTDV

7112

1 ,/ (18X, 7112)

160 FORMAT(1H0,4X,( ',I2,') ',8HDZVM

(18X, 7F12.4) )

161 FORMAT(1H0,4X,'(',I2,') ',8HAVM

(18X,7F12.4) )

162 FORMAT(1H0,4X,'(',I2,') ',8HDEVM

(18X,7F12.4) )

- IS /10X,8HNDIMV.

I5 / 1OX,8HNODV

) = 7F12.4

= 7F12.4

= 7F12.4

/

/

/

C-9
Subroutine SETUP-Select the parameters specific to channel type IX from the arrays.

IBM=NDIMV (IX)

NV=NODV(IBM)

DO I=1,NV

DZV(I) = DZVM(IBM,I)

AV(I) = AVM(IBM,I)

DEV(I) = DEVM(IBM,I)

ENDDO

C

IF (NV.EQ.0)THEN

DZV(l) = EL

AV(1) = AO

DEV(i) = DE

ENDIF

Subroutine NODE-Transform the nodal parameters to the new coordinate system.

Cell parameters (length averaged)

C LAPUR6 1:1

C NEW LINE

DIMENSION DDV (NOVAMAX)

C LAPUR6 END

C Calculation of cell flow area and cell hydraulic diameter

C DZV(I) Height to the top of the I-th axial interval

C within the flow area is uniform.

C Input data card 60.

C DDV(I) Length of the I-th axial interval within the flow area is

uniform.

C

IF(NV.LE.1) GO TO 2510

DDV(1) = DZV(l)

DO 1260 I=2,NV

DDV(I) = DZV(I) - DZV(I-1)

1260 CONTINUE

C

L2 = 1

DO 1500 J=1,NNB

DO 1380 I=1,NV

L = I

IF(ZCOM(J+1).LE.DZV(L)) GO TO 1400

1380 CONTINUE

L = NV

1400 CONTINUE

Ll = L2

L2 = L

C-10
LL = L2 - (LI+I)

IF(LL) 1420,1440,1460

1420 DEVNBL(J) = DEV(LI)

AVNBL(J) = AV(LI)

GO TO 1500

1440 DEVNBL(J) = (DEV(LI)*(DZV(LI)-ZCOM(J))

1 +DEV(L2)*(ZCOM(J+1)-DZV(Ll)))

2 /(ZCOM(J+l)-ZCOM(J))

AVNBL(J) = (AV(L1)*(DZV(LI)-ZCOM(J))

1 +AV(L2)*(ZCOM(J+1)-DZV(LI)))

2 •/(ZCOM(J+1)-ZCOM(J))

GO TO 1500

1460 TEMPO= 0.0

TEMP1= 0.0

LL1 = Li + 1

LL2 = L2 - 1

DO 1480 I=LL1,LL2

TEMPO = TEMPO + DEV(I)*DDV(I)

1480 TEMPI = TEMPI + AV(I)*DDV(I)

DEVNBL(J) = (DEV(L1)*(DZV(LI)-ZCOM(J))+TEMP0

1 +DEV(L2)*(ZCOM(J+1)-DZV(L2-1)))

2 /(ZCOM(J+1)-ZCOM(J))

AVNBL(J) = (AV(LI)*(DZV(LI)-ZCOM(J))+TEMP1

1 +AV(L2)*(ZCOM(J+1)-DZV(L2-1)))

2 /(ZCOM(J+1)-ZCOM(J))

1500 CONTINUE

C

DO 2500 J=1,N

DO 2380 I=L2,NV

L = I

IF(ZPP(J+1).LE.DZV(L)) GO TO 2400

2380 CONTINUE

L = NV

2400 CONTINUE

L1 = L2

L2 = L

LL = L2 - (LI+I)

IF(LL) 2420,2440,2460

2420 DEVL(J) = DEV(LI)

AVL(J) = AV(LI)

GO TO 2500

2440 DEVL(J) = (DEV.(LI)*(DZV(LI)-ZPP(J))

1 +DEV(L2)*(ZPP(J+1)-DZV(L1)))/DZ

AVL(J) = (AV(LI)*(DZV(LI) -ZPP(J))

1 +AV(L2)*(ZPP(J+1)-DZV(LI)))/DZ

GO TO 2500

2460 TEMPO= 0.0

TEMP1= 0.0

LL1 = Li + 1

LL2 = L2 - 1

DO 2480 I=LL1,LL2

2480 TEMPO = TEMPO + DEV(I)*DDV(I)

C-1l
TEMPI = TEMPI + AV(I)*DDV(I)

DEVL(J) = (DEV(LI)*(DZV(LI)-ZPP(J))+TEMP0

1 +DEV(L2)*(ZPP(J+I)-DZV(L2-1)))/DZ

AVL(J) = (AV(Ll)*(DZV(Ll)-ZPP(J))+TEMP1

1 +AV(L2)*(ZPP(J+1)-DZV(L2-1)))/DZ

2500 CONTINUE

GO TO 2600

2510 CONTINUE

IF(ZNB.GT.DZV(1)) GO TO 2540

DO 2511 J=l,NNB

DEVNBL(J) = DEV(l)

AVNBL(J) = AV(l)

2511 CONTINUE

DO 2512 J=I,N

DEVL(J) = DEV(l)

AVL(J) = AV(1)

2512 CONTINUE

GO TO 2600

C

2520 CONTINUE

WRITE (6,9000) (TITLE(I),I=1,18)

WRITE (6,9100) IX

NODEGO=1 .0

RETURN

2540 CONTINUE

WRITE (6,9000) (TITLE(I),I=1,18)

WRITE (6,9300) IX,ZNB

NODEGO=1 .0

RETURN

C

2600 CONTINUE

C

Cell edge flow area

C Calculation of cell edge flow area and hydraulic diameter

IF(NV.LE.1) GO TO 5510

C

AVNBCL(1) = AV(l)

DEVNBCL(1)= DEV(l)

DO 4500 J=I,NNB

DO 4380 I=1,NV

L = I

IF(ZCOM(J+I).LE.DZV(L)) GO TO 4400

4380 CONTINUE

L = NV

4400 CONTINUE

AVNBCL(J+I) = AV(L)

DEVNBCL(J+1)= DEV(L)

4500 CONTINUE

C

C-12
AVCL(1) = AVNBCL(NNB+1)

DEVCL (1) = DEVNBCL (NNB+I)

DO 5500 J=1,N

DO 5380 I=1,NV

L = I

IF(ZPP(J+1).LE.DZV(L) GO TO 5400

5380 CONTINUE

L = NV

5400 CONTINUE

AVCL(J+1) = AV(L)

DEVCL(J+1)= DEV(L)

5500 CONTINUE

GO TO 5600

C

5510 CONTINUE

IF(ZNB.GT.DZV(1))

DO 5511 J=1,NNB+I

GO TO 5540

AVNBCL (J) = AV(1)

DEVNBCL(J) = DEV(1)

5511 CONTINUE

DO 5512 J=1,N+I

AVCL(J) = AV(1)

DEVCL(J) = DEV(1)

5512 CONTINUE

SGO TO 5600

C

5520 CONTINUE

WRITE (6,9000)

WRITE (6,9100)

NODEGO=1.0

RETURN

5540 CONTINUE

WRITE (6,9000)

WRITE (6,9300)

NODEGO=1.0

RETURN

C

5600 CONTINUE

(TITLE (I),I=1,18)

IX

(TITLE (I),I=1,18)

IX, ZNB

9300 FORMAT(IHO,10X, 42HZNB IS GREATER THAN SIGMA(DZV(I),I=1,NV).

1 /// 1OX, I10, 1PE15.5 )

C-13
Variable Dimension Subroutine Description Units

,VCL(M) N+1 NODE Boling cell edge flow area cm2

,VL(I) N NODE Boiling cell flow area - length averaged cm2

AVNBCL(J+I) NNB+1 NODE Nonboiling cell edge flow area cm2

,VNBL(l) NNB NODE Nonboiling cell flow area - length averaged cm2

DDV(l) NV NODE Length of the l-th variable-area axial interval cm

DEVCL(J+I) N+1 NODE Hydraulic diameter boiling cell edge cm

DEVL(I) N NODE Hydraulic diameter boiling cell - length averaged cm

DEVNBCL(J+ NNB+1 NODE Nonboiling cell edge hydraulic diameter cm

1 )__________

DEVNBL(I) NNB NODE Nonboiling cell hydraulic diameter - length averaged cm

DZV(l) NV SETUP Top height from bottom of the l-th variable-area axial cm

_Z()_VEU (DZVM)

7COM(J+I) NNB+N+1 OEFR & Accumulative cell edges height. ZCOM(1 )=O.O cm NODE

7NB -- BOUNDY & Position of the boiling boundary cm

7PP(J+I) N+1 fNODE Boiling cell edge height ZPP(1)=ZNB cm

2.2.3. Calculation of Cell Variables-LAPUR Modifications

The following cell variables are calculated in subroutine FRICT:

* Normalized enthalpies [cal/g] (HVNBL, HVL).

" Temperatures [°C] (TVNBL, TVL).

" Liquid dynamic viscosities [g/cm 2 s] (EMUVNBL, EMUVL).

* Densities [g/cm 3] (RHOVNBL, RHOVL).

" Liquid Velocities [cm/s] (UVNBL, UVL).

" Reynolds numbers (REVNBL, REVL).

" Friction factors (FNBL, FFL).

* Coefficient a 2 used by the Moody friction factor (A2NBL, A2L)

Subroutine FRICT-Calculate cell variables.

C LAPUR6 1:73

C NEW LINES

C Center cell variables

C

DO 400 I=I,NNB

HVNBL (I) =HO+ (HFG*XE (IX) +HF-H0) * (CCOM (I+1) +CCOM (I)) /2.

TVNBL(I)=TD(P,HVNBL(I))

EMUVNBL(I)=FPMU(P,TVNBL(I))

RHOVNBL(I)=l/VOLUME(P,TVNBL(I))

UVNBL(I)=WO/(RHOVNBL(I)*AVNBL(I))

REVNBL (I) = (RHOVNBL (I) *UVNBL(I) *DEVNBL (.I) / (EMUVNBL (I))

C-14
FNBL1(I)=FF(REVNBL(I))

A2NBL1 (I)=AF(FNBL1 (I))

C

IF (IFF .EQ. 2) THEN

FNBL2(I) = FF2 (REVNBL(I)).

A2NBL2(I) = AF2(FNBL2(I),REVNBL(I))

FNBL(I) = FNBL2 (I)

A2NBL(I) = A2NBL2 (I)

ELSE

FNBL(I) = FNBL1 (I)

A2NBL(I) = A2NBL1(I)

ENDIF

400 ENDDO

C

DO 410 I=1,NSB

HVL(I)=HF- (H(I)+H(I+1) *HFG/2.

TVL (I) =TD (P,HVL (I))

EMUVL (I) =FPMU (P, TVL (I))

RHOVL(I)=I/VOLUME(P,TVL(I))

UVL (I) =WO/ (RHOVL (I) *AVL (I))

REVL(I)=(RHOVL(I)*UVL(I)*DEVL(I)) / (EMUVL(I))

FL1(I)=FF(REVL(I))

A2LI (I)=AF (FL1 (I)) C."

IF (IFF .EQ. 2) THEN

FL2(I) = FF2 (REVL(I))

A2L2(I) = AF2(FL2(I),REVL(I))

FFL(I) = FL2(I)

A2L(I) = A2L2(I)

ELSE

FFL(I) = FL1(I)

A2L(I) = A2L1 (I)

ENDIF

410 ENDDO

C

DO 420 I=NSB+1,N

HVL (I)=HF

TVL(I)=TSAT

EMUVL (I) =FPMU (P,TSAT)

RHOVL (I) =i/VOLUME (P, TSAT)

UVL (I) =WO/ (RHOVL (I) *AVL (I))

REVL(I)=(RHOVL(I)*UVL(I)*DEVL(I)) / (EMUVL(I))

FL1 (I) =EF (REVL (I))

A2L1 (I)=AF(FL1 (I))

IF (IFF .EQ. 2) THEN

FL2.(I) = FF2 (REVL(I))

A2L2(I) = AF2(FL2(I),REVL(I))

FFL(I) = FL2(I)

A2L(I) = A2L2(I)

ELSE

FFL(I) = FL1(I)

C-15
A2L(I) = A2Ll (I)

ENDIF

420 ENDDO

Cell edges variables

C

C

C

RHOVNBCL(1)=RHOLI

DO 430 I=2,NNB

RHOVNBCL(I)=(RHOVNBL (I-I)*(ZCOM (I)-ZCOM(I-I )+

1 RHOVNBL(I)*(ZCOM(I+1)-ZCOM(I)))/(ZCOM(I+I)-

ZCOM(I-I))

430 ENDDO

RHOVNBCL(NNB+1)=RHOZNB

C LAPUR6 END

Variable Dimension Subroutine Description Units

L() N FRICT Coefficient a2 for boiling region, used by the Moody

_L(I) __N __RCTfriction factor

Coefficient a2 for nonboiling region, used by the Moody friction factor

AVL(I) N NODE Boiling cell flow area - length averaged cm2

,VNBL(I) NNB NODE Nonboiling cell flow area - length averaged cm2

NODE Normalized cumulative power generation cell edge N(From 0.0 to 1.0)

DEVL(I) N NODE Hydraulic diameter boiling cell - length averaged cm

DEVNBL(I) NNB NODE Nonboiling cell hydraulic diameter - length averaged cm

DZ NODE Boiling cell length cm

EMUVL(I) N FRICT Boiling cell liquid dynamic viscosity g/cm2 s

EMUVNBL(I) NNB FRICT Nonboiling liquid dynamic viscosity g/cm2 s

FFL(I) N FRICT Friction factor for boiling region

FNBL(I) NNB FRICT Friction factor for nonboiling region

H(+1) NSB+1 SUBOIL & Normalized subcooled cell edge enthalpy cal/g BULK H(1)=HZNB

HO ----- READY Inlet enthalpy cal/g

HF R---- READY Saturated liquid enthalpy cal/g

HFG READY R---- Evaporation enthalpy cal/g

HVL(I) N FRICT Normalized boiling cell enthalpy cal/g

HVNBL(I) NNB FRICT Normalized nonboiling cell enthalpy cal/g

N ----- INPUT-user Number of axial nodes desired in the boiling region

of the channels

NNB -... BOUNDY Number of Nonboiling cells

NSB ----- BOUNDY Number of Subcooled boiling cells

P ---- INPUT-user System pressure kg/cm2

REVL(I) N FRICT Boiling cell Reynolds

REVNBL(I) NNB FRICT Nonboiling cell Reynolds

RHOVL(I) N FRICT Boiling cell density g/cm3

RHOVNBCL(l+ NNB+1 FRICT Nonboiling cell edge density g/cm3

1)__________ ___________ __________________________________________

C-16
Variable Dimension Subroutine Description Units

RHOVNBL(I) NNB FRICT Nonboiling cell density g/cm3

TSAT ---- READY Saturated temperature °C

TVL(l) N FRICT Boiling cell temperature 0C

TVNBL(l) NNB FRICT Nonboiling cell temperature °C

UVL(l) N FRICT Boiling cell liquid velocity -no steam- (wO/rhol*a cell) cm/s

UVNBL(I) NNB FRICT Nonboiling cell liquid velocity cm/s

Wo ----- PREPAR Reference channel mass flow rate g/s

XE(IX) NXE READY & hannel exit quality SETUP _ccuuatveelldgeheiht._COM1)=00c

FCOM(J+11) NNB+N+1 COEFR & Accumulative cell edges height. ZCOM~(1)=O.0 cm

2.3. MODIFICATIONS IN LAPURX RELATED TO FRICTION MODEL II

In this section, we present the following modifications:

* Reading, storing, and preparing the new variables.

• Calculating cell variables.

2.3.1. Theory

Friction model I is the old LAPUR 5 model, and comprises the Moody Single-Phase Friction

Factor and the Martinelli-Nelson Two-Phase Friction Multiplier.

Friction model II comprises the Generic Single-Phase Friction Factor and the ChisholmBaroczy Friction Multiplier.. This friction model II will be used only if variable IFFM=2 (input-user

data on card 67) and variable IFMM=2 (input-user data on card 73).

2.3.1.1. Friction model I

The friction model I comprises the Moody Single-Phase Friction Factor and the MartinelliNelson Two-Phase Friction Multiplier. The Moody friction factor fm for one-phase flow along

smooth surfaces can be expressed by

fm = a, Re-a2 (1)

where a, and a 2 are coefficients that depend on the Reynolds number, and Re is the Reynolds

number of the flow

Re= WDHGD =G4A

pt A ILL P w

For laminar flow conditions, i.e., Re < 2300:

64

fM Re

hence, a1 = 64, a2 = 1.

(2)

(3)

C-17
For turbulent flow conditions, i.e., Re > 2300, the following transcendental equation applies:

fm-/ 2 = 2 log (Re fMl2)- 0.8 (4)

The determination of the factor fM for a given Re requires an iterative procedure.

Once fM has been determined, the coefficients of the general expression fM = a, Re-a2 can

be calculated using following expression obtained by Jones

4(log e)fM1 2 /

2 1+ 2(log e)fM112 (5)

a 1 - Rea2

where e = 2.71 the base of the natural logarithms, and log x = log10 x.

A commonly used relation for turbulent flow of water in BWRs uses a, 0.213 and a 2

=0.214.

The Martinelli-Nelson Two-Phase Friction Multiplier has been widely used in two-phase flow

pressure-drop analyses. It was originally developed for horizontal turbulent flow but has been

found reasonably accurate for vertical flows. Several modifications have been suggested to

account for flow pattern characteristics and flow rates.

A polynomial fit valid for x < 0.7 was synthesized by Jones (Ref. 7) as

4

ý2 -aj1

[n(100x+1)j

4

MN = e' (6)

where

x steam quality, and

a= pressure-dependent fitted parameters given by

8

a, = -bij Pj (7)

j=1

where

bij - fitting coefficients, giving by Jones

P1 =I

P2 = 1.42234 x10-2 p

Pj = P2 Pj-1 ; j= 3, 6; and

P =- system pressure in [Kg/cm 2

]

C-18
2.3.1.2. Friction model II

Friction model II comprises the Generic Single-Phase Friction Factor and the ChisholmBaroczy Friction Multiplier; it also allows the use of different friction models selecting specific

coefficients (Ref. 10). The equations of this model are:

64.0 (8) Re,

--- AN-[F1 +-(BN .r ON+°N (9)

FF NL Rej

FF = MAX(FF, FFLAM) (10)

where the coefficients AN, BN, CN, DN are fuel-specific and are validated by each vendor; Rr is

the relative roughness; and Re, is the liquid Reynolds number.

The two-phase friction multiplier model II uses the fit of Chisholm to the model of Baroczy

(Ref. 8). This model uses the following equations:

GMVSQ= ' F(1

BGM = GMSQ (12)

B 55.0 BGM < 9.5 (13)

[AJUSTABI. G]

B= 520.0 9.5 < BGM < 28 (14)

BGM. [AJUSTABI -G]

B15000.0 28 < BGM (15)

GMSQ . [AJUSTAB1- G]°

2 B = AJUSTAC

1 "

{1.0 + (GMSQ - 1.0). [B-AJUSTAA1 *XF° 9 • (1.0 - AJUSTAA1, XF)0 9 + AJUSTAA, .XF18J]

(16)

where Jg is the steam viscosity

ViF is the saturated liquid viscosity

PF is the saturated liquid density

C-19
Pg is the steam density

AJUSTAA1, AJUSTAB1, AJUSTAC1 are fit factors for G, XF and Global (default=1.0,

1.0,1.0)

G is the mass flux [kg/M 2 S]

XF is the flow quality

We note that the Chisholm-Baroczy model is used only to improve the pressure drop at

steady state but not to compute the perturbations around the steady-state point.

To obtain the Chisholm-Baroczy coefficients used in the momentum equations, we obtain

the relation between the Chisholm-Baroczy two-phase friction multiplier, C2B, and the MartinelliNelson two-phase friction multiplier, 2MN, times the Jones two-phase friction multiplier correction

factor, Q, at steady-state conditions denoted by the subindex 0:

IcB-MNO OMN 2MN 0 (17)

then

fr(0CB )002 QG 2_2 f _

M MN G fMfcB-MN,O MN 2- (18) OMN 2p 2p

equivalent to the LAPUR 5 model, but fCB-MN,O is calculated by LAPUR 6.0 automatically for

steady-state conditions. When computing the partial derivatives for LAPURW, the partial

derivative of this correction factor, fCB-MNo, is zero.

2.3.2. Friction Model It-LAPUR modifications

2.3.2.1. Friction factor model

Friction factor model II will be used only if IFFM=2 (input-user data on card 67). This model

needs the coefficients:

* AN, BN, CN, DN (input-user data cards 68-71)

* Rr is the relative roughness (input-user data card 72)

The new function FF2 (xfrict.for) calculates this friction factor. The partial derivate of this friction.

factor is used by LAPURW 6.0:

aFFF CN.DN 1 1 ~F22)(9

a Re1 o [1+ (BN.Rr)+ ON

T nw Reot

The new function AF2 (x_frict.for) computes the coefficient A2V for friction factor model 11.

C-20
Three subroutines are modified:

* The subroutine INPUT: read the new variables.

* The subroutine OUTIN: print the new variables.

* The subroutine SETUP: select the parameters specific to channel type IX from

the arrays.

Two new functions are added:

* The function FF2: Calculate the new friction factor for model II.

* The function AF2: Calculate the new coefficient A2V for model I1.

2.3.2.2. Friction factor model II-LAPUR modifications

Subroutine INPUT-Read the new variables.

670 CALL INP1(NTFU,IFFM)

GO TO 6000

680 CALL INP3(NTFU,ANM)

GO TO 6000

690 CALL INP3(NTFU,BNM)

GO TO 6000

700 CALL INP3(NTFU,CNM)

GO TO 6000

710 CALL INP3(NTFU,DNM)

GO TO 6000

720 CALL INP3(NTFU,RRM)

GO TO 6000

Subroutine OUTIN-Print the new variables.

67 WRITE

WRITE

GO TO

68 WRITE

GO TO

69 WRITE

GO TO

70 WRITE

GO TO

71 WRITE

GO TO

72 WRITE

GO TO

(6,

(6,

200

(6,

200

(6,

200

(6,

200

(6,

200

(6,

200

444)

167)11, NTFU , (IFFM(J),J=~1,NTFU)

168)11, NTFU ,

169)11, NTFU ,

170)11, NTFU

171)11, NTFU

172)11, NTFU

(ANM (J) , J=1, NTFU)

(BNM(J) ,J=1,NTFU)

(CNM (J) J=1, NTFU)

(DNM (J) J=1, NTFU)

(RRM(J),J=1,NTFU)

C-21
167 FORMAT(IH0,4X, ' (',I2, ')

1

168 FORMAT (H0,4X, ' (',12, ')

7F12 .4

1

169 FORMAT (H0,4X, ' (',12, ')

7F12 .4

1

170 FORMAT (H0,4X, (',12,')

7F12.4

1 fl

171 FORMAT (H0,4X,' (',I2,')

7F12 .4

1

172 FORMAT (H0,4X, ' (',12, ')

7F12 .4

',8HNTFU = 15 /10X,8HIFFM = 7112

(18X, 7112)

',8HNTFU = 15 /10X,8HANM

(18X, 7F12.4) )

',8HNTFU = I5

(18X, 7F12.4) )

',8HNTFU = i5

(18X, 7F12.4) ) ',8HNTFU = 15

(18X, 7F12.4) )

',8HNTFU = 15

/10X,8HBNM

/10X,8HCNM

/10X,8HDNM

/1OX, 8HRRM

1 ,/ (18X, 7F12.4)

Subroutine SETUP-Select the parameters specific to channel type IX from the arrays.

IFF

AN

BN

CN

DN

RR

= IFFM (IBM)

= ANM

= BNM

= CNM

(IBM)

(IBM)

(IBM)

- DNM (IBM)

= RRM (IBM)

Function FF2 (new function)-Calcu late the new friction factor for model II.

C LAPUR6 1:16

C NEW LINES

C Friction factor model II

C

C

FUNCTION FF2(RE0)

C

100

200

INCLUDE 'X common.fi'

IF(REO-2300.0) 100,100,200

FF2 = 64./RE0

GO TO 500

FF2 = AN*(I+(BN*RR)+CN/REO)**DN

CON1 =64./REO

FF2 = AMAX1(FF2,CON1)

500 CONTINUE

RETURN

END

C

C LAPUR6 END

C-22
Function AF2 (new function)-Calculate the new coefficient A2V (=AF2) for model II.

C LAPUR6 1:12

C NEW LINES

C Coefficient A2V for friction factor model II

FUNCTION AF2 (FOP,REBL)

C

INCLUDE 'X common.fi'

CON=64.0/REBL

IF(CON-FOP) 100,200,200

100 AF2 = (CN*DN)/C((+BN*RR)*REBL+CN)

GO TO 300

200 AF2 = 1.0

300 CONTINUE

RETURN

END

C LAPUR6 END

2.3.2.3. Friction multiplier model

Friction multiplier model II will be used only if IFMM=2 (input-user data on card 73). This

model needs the coefficients:

AJUSTAA1, AJUSTAB1, AJUSTAC1 (input-user data cards 74-76) (default=1.0, 1.0, 1.0)

We create a two new functions, VISC and POLY (xsubs.for), calculate the dynamic

viscosity of steam (Ref 12). In the subroutine PREPHI (xsubs.for) we calculate this friction

multiplier and its partial derivates.

Four subroutines are modified:

" The subroutine INPUT: read the new variables.

* The subroutine OUTIN: print the new variables.

" The subroutine SETUP: select the parameters specific to channel type IX from

the arrays.

* The subroutine PREPHI: calculate the new friction multiplier

Two new functions are added:

* The function VSCV: calculate dynamic viscosity of steam

• The function POLY: used by function VSCV.

C-23
2.3.2.4. Friction multiplier model II-LAPUR modifications

Subroutine INPUT-Read the new variables.

730 CALL INP1(NTFU,IFMM)

GO TO 6000

740 CALL INP3(NTFU,AJUSTAA1M)

GO TO 6000

750 CALL INP3(NTFU,AJUSTAB1M)

GO TO 6000

760 CALL INP3(NTFU,AJUSTAClM)

GO TO 6000

Subroutine OUTIN-Print the new variables.

73 WRITE (6, 444)

WRITE (6, 173)11, NTFU , (IFMM(J),J=1,NTFU)

GO TO 200

74 WRITE (6, 174)11, NTFU , (AJUSTAA1M(J),J=1,NTFU)

GO TO 200

75 WRITE (6, 175)11, NTFU , (AJUSTAB1M(J),J=1,NTFU)

GO TO 200

76 WRITE (6, 176)11, NTFU , (AJUSTAC1M(J),J=1,NTFU)

GO TO 200

173 FORMAT(IHO,4X, ' (',I2,') ',8HNTFU = 15 /10X,8HIFMM = 7112

1 ,/ (18X, 7112)

174 FORMAT(IH0,4X, ' (',I2, ') ',8HNTFU = 15 /10X,11HAJUSTAA1M=

7F12.4

1 ,/ (18X, 7F12.4)

175 FORMAT(1HO,4X, ' (',I2, ') ',8HNTFU = I5 /10X,11HAJUSTAB1M=

7F12.4

1 ,/ (18X, 7F12.4)

176 FORMAT (IHO,4X, '(',I2,') ',8HNTFU = 15 /10X, 11HAJUSTAC1M=

7F12.4

1 ,/ (18X, 7F12.4)

C-24
Subroutine SETUP-Select the parameters specific to channel type IX from the arrays.

IFM = IFMM (IBM)

AJUSTAAI= AJUSTAA1M (IBM)

AJUSTABl= AJUSTABIM (IBM)

AJUSTACl= AJUSTAClM(IBM)

Subroutine PREP H I-Calculate the new friction multiplier.

C Baroczy's two phase friction multiplier

C G - MASS FLUX (KG/M**2-SEC)

C GP - MASS FLUX (G/CM**2-SEC)

C XF - FLOW QUALITY

C VISCG - VAPOUR VISCOSITY

C VISCF - SATURATED LIQUID VISCOSITY (G/CM-SEC)

C RHOL - SATURATED LIQUID DENSITY (G /CM**3)

C RHOG - VAPOUR DENSITY (G/CM**3)

C

IF (IFM .NE. 2) GO TO 100

G = GP*10.0

RHOL = 1.0/VOLUME(P,TSAT)

RHOG = 1.0/VGP(P)

VISCG = VSCV(TSAT,RHOG)

VISCF = FPMU(P,TSAT)

C

GMSQ= (VISCG/VISCF) **0.2* (RHOL/RHOG)

BGM=SQRT (GMSQ)

C

TPFM(1) = 1.0

DTPFMDB0(1) = 0.0

DTPFMDH0(i) = 0.0

DTPFMDY0 (1) = 0.0

DO J= 2, NP1

C

IF (BGM .LE. 9.5) THEN

BB=55.0/(( AJUSTAB1*G)**0.5)

ELSEIF (BGM .GE. 28.0) THEN

BB=15000.0/(GMSQ*( AJUSTAB1*G)**0.5)

ELSE

BB=520.0/ (BGM* (AJUSTAB1*G) **0.5)

ENDIF

C

XF=X (J)

C

IF(X(J).LT.0.0) XF=0.0

C

TPFM(J) = AJUSTAC1*(1.0+(GMSQ-1.0)*

> (BB*AJUSTAA1*XF**0.9*(I-AJUSTAA1*XF)**0.9

> +AJUSTAA1*XF**1.8))

C Partial derivative

BBI =(0.9*BB*

C-25
> (1.0-2.0*AJUSTAAI*XF)/

> (XF**0.1*(l.0.-AJUSTAA1*XF)**0.1))

DTPFMDB0 (J) =AJUSTACI*AJUSTAA1* (GMSQ-1.0) *

> (BBI+1.8*XF**0.8)*DXDB0(J)

C

IF(H(J).EQ. 0.0) THEN

DTPFMDH0 (J) =0.0

ELSE

DTPFMDH0 (J)=AJUSTAC1*AJUSTAA1* (GMSQ-I.0) *

(BBl+1.8*XF**0.8)*DXDHO(J)

ENDIF

C

BB2 = -0.5*(BB*G**0.5)*(W0/A0)**((-0.5)

DTPFMDY0 (J) =AJUSTAC1* (GMSQ-1.0) *

> (AJUSTAAI*XF**0.9* (1. 0-AJUSTAA1*XF) **0.9)

> *BB2

ENDDO

C

Function VSCV (new function)-Calculate dynamic viscosity of steam.

C LAPUR6 1:53

C NEW LINES

FUNCTION VSCV (T,ROV)

C

C

C DYNAMIC VISCOSITY OF STEAM AS FUNCTION OF TEMPERATURE AND DENSITY

C DENSITY RO IN (KG/CU.M)

C TEMPERATURE T IN (C)

C DYNAMIC VISCOSITY VSCV0 IN (N S/SQ.M.)

C DYNAMIC VISCOSITY VSCV IN (G/SQ.CM. S)

C

DIMENSION A(3), F(4), G(4)

DATA A/3.53D-8, 6.765D-11, 1.021D-14/, B/.407D-7/, C/8.04D-6/,

& D/1.858D-7/, E/5.9D-10/,

& F/-.2885D-5, .2427D-7, -. 67893333333D-10, .6317037037D13/,

& G/.176D3, -. 16D1, .48D-2, -. 47407407407D-5/

C

RO=ROV*1.OE+3

V1 = B*T + C

IF(T .LE. 300.) GO TO 100

IF(T .GE. 375.) GO TO 110

VSCV0 = Vl POLY(4,F(l),T)*RO

POLY(4,G(i),T)*POLY(3,A(i),RO)*

&RO

C

GO TO 140

100 CONTINUE

VSCV0 = V1 - RO*(D - E*T)

IF(VSCVO.LT.I.D-7) VISV =1.D-7

C-26
C

C

GO TO 14

110 CONTINUE

vSCV0

140 CONTINUE

vSCv

RETURN

END

•0

= Vl + POLY(3,A(1),RO)*RO

= VSCV0*10.

C

Function POLY (new function)-Used by function VSVC.

C

C

FUNCTION POLY (N,A,X)

C

DIMENSION A(N)

C

POLY = 0.

L N

100 CONTINUE

POLY=POLY*X+A (L)

L=L-1

IF(L.GT.0) GO TO 100

RETURN

END

C

C

C

C LAPUR6 END

C-27
2.4. MODIFICATIONS IN LAPURX RELATED TO PRESSURE LOSSES CAUSED BY AREA

CHANGES

In this section, we present modifications related to the following losses:

* Gravity

" Acceleration

" Friction

* Irreversible expansion losses

* Irreversible contraction losses

* Local

2.4.1. Gravity Losses

Nonboiling region

In LAPUR 5, the nonboiling region comprises only one node. In LAPUR 6.0, the nonboiling

region has been partitioned into several cells, according to the LAPURX input data file.

For this region, the LAPUR 6.0 computes the gravity-pressure losses due to an area

change as follows:

=p(Z).-g 13hJ -Cos 0 (20)

n+2 m+1

m AP]fl =p, (m)-g.coso.[z(n + 1)z(n)] (21)

n-i

Subroutine DELP (x frict.for) computes these pressure losses [DPDHNB1(J);J=i,NNBJ.

C-28
Boiling region

For this region

azD

= [x(X(Z). (1-- 11). PF + (1-- cX(Z)).pl(Z)].g'cos0 (22)

AP] n+- [(1x(n) + oc(n + ) ( -q) PF +( ct(n) + (n +1)). P1(M)j -g cos 0 -dz (23)

Subroutine DELP (x frict.for), which has been modified, computes gravity pressure

losses at every node along the length of the channel [DPDHP1 (J);J=1,N+I] [DPDHP1(1)=O.O].

2.4.1.1. Gravity losses-LAPUR modifications

Subroutine DELP-Calculates gravity pressure losses at every node along the length of

the channel.

Nonboiling region

DPDHNB1(1) = (ZCOM(2)-ZCOM(1))*COS(TETA)*RHOVNBL(1)*0.001

DO 210 J=2,NNB

DPDHNB1(J) = DPDHNB1(J-1) +

1 (ZCOM(J+1)-ZCOM(J)) *COS(TETA)*RHOVNBL(J)*0.001

210 CONTINUE

DPDHNB2 = DPDHNB1(NNB)

DPDHNB2 = 0.0

Boiling region

DPDHP1(1) = 0.

DO 260 M=2,NP1

C LAPUR6 I:

C NEW LINES

BETM = (BETA(M)+BETA(M-1))/2.

DPDHP1(M) = DPDHP1(M-1) + COS(TETA)*DZ*0.001

1 * (BETM* (1.-ENU) *RHOL+ (I-BETM) *RHOVL(M-1))

260 CONTINUE

DPDHSB1 = DPDHP1(NSBP1)

DPDHBB1 = DPDHP1(NPI) - DPDHP1(NSBP1)

C-29
Variable Dimension Subroutine Description Units

BETA(M N+IBULK SUBOIL & Boiling cell edge void fraction (BETA(1)=0.0)

DPDHNB1(J) NNB DELP Nonboiling cell accumulative gravity pressure drop Kg/cm2

DPDHP1(M) N+1 DELP Boiling accumulative cell edges gravity pressure Kg/cm2

PHI()NIEP drop DPDHP1(1)=0.O

DZ .... NODE l Boiling cell length cm

ENU ---- READY 1-(RHOS/RHOL)

NNB ..... _BOUNDY Number of Nonboiling cells

RHOL ---- READY Saturated liquid density g/cm3

RHOVL(l) N FRICT Boiling cell density g/cm3

RHOVNBL(I) NNB FRICT Nonboiling cell density g/cm3

ETA -BLOCK _ETA_ .... DATA ngle

COM(J+I) NNB+N+1 OEFR & ccumulative cell edges height. ZCOM(1)=O.O cm

k____ _____NODE I______________I______

2.4.2. Acceleration Losses

Nonboiling region

In LAPUR 5, the nonboiling region comprises only one node. In the LAPUR 6.0 code, the

nonboiling region has been partitioned into several cells, according to the LAPURX input data

file.

For this region, the LAPUR 6.0 computes the acceleration pressure losses due to an

area change as follows:

n+2 OZ ac

n+1

AID~+ n A(rn)dýp,(n +1).A(n +1) p,(n).A(n))i

(24)

(25)

Jn-1

LAPUR 6.0 computes the cell flow areas [AVNBL(m); m=l ,NNB] and cell edge flow areas

[AVNBCL(m); m=l,NNB+I] in subroutine NODE. The values AVNBL are length-averaged.

Subroutine DELP (x._frict.for) computes these pressure losses [DPACNBl(J);J=1,NNB].

C-30
The liquid cell edge densities p1(n) are calculated in subroutine DELP, using the cell values p1(m):

First cell edge:

p,(1) =PINLET (26)

Intermediate cell edges:

p,(_ ) z(n) -2(n - 1)1 +P p(m). z(n + •-z(n))

pz(n)-z(n - 1)' + ( z(n + 1)- z(n)J (27)

2(22

_ p,(m -1)- (z(n)- z(n - 1)) + p, (m)- (z(n + 1)- z(n))

z(n -1)+ z(n + 1)

Last cell edge:

p1(NNB + 1) p,(m = NNB). (z(NNB + 1)- z(NNB))+ p,(1 _BR) DZ (28)

[z(NNB + 1)- z(NNB)] + DZ

Boiling region

For this region

'a 1 U2 2 (Z)

-azac- A(z) 0A2 A(z) (29)

1n+1 1 2 2 (B(n + 1) B(n)" (30)

SA(m)" PF *A A(n + 1) A(n))

where B is:

B- [1- 1-y2(1 _q))] (31)

and y is the slip ratio

Ug Ug (32)

U1

LAPUR 6.0 computes the cell flow areas [AVL(m); m=1,N] and cell edge flow areas

[AVCL(m); m=l,N+I] in subroutine NODE. The values AVL are length-averaged.

Subroutine DELP (xfrict.for) computes these pressure losses [DPACPI(J);J=1,N+I]

[DPACP1 (1)=0.0], which are multiplied by (0.001 /g).

C-31
One subroutine, DELP, has been modified. DELP performs the calculation of

acceleration pressure losses at every node along the length of the channel.

2.4.2.1. Acceleration losses-LAPUR modifications

Subroutine DELP-Calculate acceleration pressure losses at every node along the length

of the channel.

Nonboiling region

RHOC1= RHOLI

RHOC2= (RHOVNBL (1) * (ZCOM (2) -ZCOM (1)) +

1 RHOVNBL(2)*(ZCOM(3)-ZCOM(2)))/(ZCOM(3)-ZCOM.(i))

DPACNBI(1) = (0.001*WO**2/SG)/AVNBL(1)*

1 (1/(RHOC2*AVNBCL(2))-i/(RHOCI*AVNBCL(i))) C

DO 215 J=2,NNB-1

RHOCI= (RHOVNBL (J-1) * (ZCOM(J) -ZCOM(J-1)) +

1 RHOVNBL(J)*(ZCOM(J+1)-ZCOM(J)))/(ZCOM(J+1)-ZCOM(J- 1))

RHOC2=(RHOVNBL(J)*(ZCOM(J+1)-ZCOM(J) )+

1 RHOVNBL(J+1)*(ZCOM(J+2)-ZCOM(J+1)))/(ZCOM(J+2)-

ZCOM(O))

DPACNB1(J) = DPACNB1(J-1)+(0.001*WO**2/SG)/AVNBL(J)*

1 (1/(RHOC2*AVNBCL(J+1))-I/(RHOC1*AVNBCL(J)))

215 CONTINUE

C

J=NNB

IF (NNB==I) THEN

RHOC1=RHOLI

ELSE

RHOCI =RHOC2

END IF

RHOC2=(RHOVNBL(J)*(ZCOM(J+1)-ZCOM(J) )+

1 RHOVL(1) *DZ) /(ZCOM(J+1)-ZCOM(J)+DZ)

DPACNB1(J) = DPACNB1(J-1)+(0.001*WO**2/SG)/AVNBL(J)*

1 (1/(RHOC2*AVNBCL(J+1))-i/(RHOC1*AVNBCL(J)))

C

DPACNB2 = DPACNBI (NNB)

DPACNB2 = 0.0

Boiling region

DPACP1(1) = 0.

DO 260 M=2,NP1

C-32
CON2= 0.001*RHOL*U0**2*A0**2/SG

DPACP1 (M) = DPACP1 (M-I)+CON2/AVL(M-i)

1 * (B0 (M) /AVCL (M) -B0 (M-1) /AVCL (M-1))

260 CONTINUE

DPACSB1 = DPACP1(NSBP1)

DPACBB1 = DPACP1(NPI) - DPACP1(NSBP1)

Variable Dimension Subroutine Description Units

AO --- SETUP Channel flow area (Input data AOM(NTD)) cm2

AVCL(M) N+1 NODE Boling cell edge flow area cm2

AVL(l) N NODE Boiling cell flow area - length averaged cm2

AVNBCL(J+I) NNB+1 NODE Nonboiling cell edge flow area cm2

AVNBL(I) NNB NODE Nonboiling cell flow area - length averaged cm2

B0(M) N+1 UBOIL & B in momentum equation BULK

DPACNB1(J) NNB DELP Nonboiling cell accumulative acceleration pressure kg/cm2 'DAN_()_BEP drop

Boiling accumulative cell edges acceleration kglcm2 DPACPI_(M) N+1 DELP pressure drop DPACP1(1)=O.O

DZ ..... NODE Boiling cell length cm

NNB BOUNDY Number of Nonboiling cells

RHOL ---- READY Saturated liquid density g/cm3

RHOLI. READY Inlet liquid density g/cm3

RHOVL(I) N FRICT Boiling cell density g/cm3

RHOVNBL(I) NNB FRICT Nonboiling cell density g/cm3 BLOCK

SG --- Gravitational acceleration cm/s2

uo0 ...- PREPAR Reference channel velocity cm/s

V0 ----- PREPAR Reference channel mass flow rate g/s

COM(J+1) NNB+N+1 COEFR & ccumulative cell edges height. ZCOM(1)=O.O cm _____ _I __ NODE __________ __________

C-33
2.4.3. Friction Losses

Nonboiling region

In LAPUR 5, the nonboiling region comprises only one node. In the LAPUR 6.0 code, the

nonboiling region has been partitioned into several cells, according to the LAPURX input data

file.

For this region, LAPUR 6.0 computes the friction pressure losses due to an area change

as follows:

A P(z)=;(lu2(zl~~) AZ

f2 DH(Z) KF (33)

m+i

.in AP]= (m)"u 2 (r)" .f(m). DHKF(m). [z(n + 1)- z(n)] (34)

n-1

Subroutine DELP (xfrict.for) computes these pressure losses [DPFRNB1 (J);J=l ,NNB].

Boiling region

For this region we can use either friction model I (using the Martinelli-Nelson correlation):

AP(z) = P,(z).u (z). f() Az {d(.(z)}.KF(Z) (35) 2 D.(z) -Qzj W(5

_ pm~u(m).f~). F~m D[ {4. (n)-.(n)}+ {•.N(n + 1)-.(n +i 1)} - 2 DH(m) 2

Or friction model II (using Chisholm-Baroczy correlation):

AP(z) _ P1(Z)Y U210 -. f(Z) f42.B(z) }. KF(Z) (37) 2 DH(z)

A P(m) u (m)" f(M). _ KF(m) DZ{B(n)}+ {B(n (38) -PI 2 DH(m) [2 1(8

Subroutine DELP (x_frict.for) computes these pressure losses [DPFRP1(J);J=1,N+I]

[DPFRPI (1)=0.0].

Note that KF(Z) is the LAPUR 5 friction multiplier that was used to meet the LAPUR

pressure loss to other more-accurate codes. For LAPUR 6.0 calculations, the term {&B(z)W

replaces the old KF(Z), which must be set to 1.0.

C-34
Note that this is the formulation for steady-state LAPURX calculation. The formulation for

LAPURW is described in Sect. 2.7 and used a combination of models I and II for the derivative

terms.

Two subroutines are modified:

4 The subroutine NODE, which calculates the cell friction multiplier for the

nonboiling region

0 The subroutine DELP, which calculates friction pressure losses at every node

along the length of the channel.

2.4.3.1. Friction losses-LAPUR modifications

Subroutine NODE-Calculates the cell friction multiplier for the nonboiling region.

C Calculation of cell friction multiplier

IF(NF.LE.1) GO TO 3510

C

C DDD(I) Height to the top of the I-th axial interval within

C the friction is uniform.

C DZF(I) Length of the I-th axial interval within the friction is

uniform.

C Input data card 36

C

L2 = 1

DO 3500 J=I,NNB

DO 3380 I=I,NF

L = I

IF(ZCOM(J+1).LE.DDD(L)) GO TO 3400

3380 CONTINUE

L = NF

3400 CONTINUE

Ll = L2

L2 = L

LL = L2 - (Ll+I)

IF(LL) 3420,3440,3460

3420 EKFNBL(J) = EKFF(LI)

GO TO 3500

3440 EKFNBL(J) = (EKFF(Ll)*(DDD(Ll)-ZCOM(J))

1 +EKFF(L2)*(ZCOM(J+1)-DDD(Ll)))

2 /(ZCOM(J+l)-ZCOM(J))

GO TO 3500

3460 TEMPO= 0.0

LL1 = Ll + 1

LL2 = L2 - 1

DO 3480 I=LL1,LL2

3480 TEMPO = TEMPO + EKFF(I)*DZF(I)

EKFNBL(J) = (EKFF(Ll)*(DDD(Ll)-ZCOM(J))+TEMPO

1 +EKFF(L2)*(ZCOM(J+1)-DDD(L2-1)))

2 /(ZCOM(J+1)-ZCOM(J))

3500 CONTINUE

C-35
GO TO 3600

3510 CONTINUE

EPS=0 . 000001

IF((ZNB-DZF(1)).GT.EPS)GO TO 540

DO 3511 J=1,NNB

EKFNBL(J) = EKFF(1)

3511 CONTINUE

GO TO 3600

C

3520 CONTINUE

WRITE (6,9000) (TITLE(I),I=1,18)

WRITE (6,9100) IX

NODEGO=1 .0

RETURN

3540 CONTINUE

WRITE (6,9000). (TITLE(I),I=1,18)

WRITE (6,9200) IX,ZNB

NODEGO=1 .0

RETURN

C

3600 CONTINUE

C

Subroutine DELP-Calculates friction pressure losses at every node along the length of

the channel.

Nonboiling region

REB=REVNBL (1)

CON1= 0.001*RHOVNBL(1)*UVNBL(1)**2/(2.*SG*DEVNBL(1))

TEMP1=FF(REB)

IF(IFF .EQ. 2) TEMP1=FF2(REB)

FR1=TEMP1*CON1*EKFNBL (1)

DPFRNB1(1) = FR1*(ZCOM(2)-ZCOM(1))

C

DO 210 J=2,NNB

REB=REVNBL (J)

CON1= 0.001*RHOVNBL(J)*UVNBL(J)**2/(2.*SG*DEVNBL(J))

TEMP1=FF (REB)

IF(IFF .EQ. 2) TEMP1=FF2(REB)

FR1=TEMP1*CON1*EKFNBL (J)

DPFRNB1(J) = DPFRNB1(J-1) + FR1*(ZCOM(J+1).-ZCOM(J))

210 CONTINUE

DPFRNB2 = DPFRNB1(NNB)

DPFRNB2 = 0.0

C-36
Boiling region

DPFRPI(1) = 0.

DO 260 M=2,NPI

REB=REVL(M-1)

CON1= 0.001*RHOVL(M-I)*UVL(M-1)**2/(2.*SG*DEVL(M-1))

TEMP1=FF (REB)

IF(IFF .EQ. 2) TEMP1=FF2(REB)

FR1=TEMPI*CON1*EKFL (M-I)

TEMP2=T20 (M-I)

TEMP3=T20 (M)

IF(IFM .EQ. 2) THEN

TEMP2=TPFM (M-I)

TEMP3=TPFM (M)

DPFRP1(M) = DPFRP1(M-1) + FRI*DZ

* (TEMP3+TEMP2) /2.

ELSE

DPFRP1(M) = DPFRPI(M-1) + FR1*DZ

* (OMEGA (M) *TEMP3+OMEGA (M-I) *TEMP2)/2.

ENDIF.

260 CONTINUE

DPFRSB1 = DPFRPI(NSBPI)

DPFRBB1 = DPFRP1(NPI) - DPFRP1(NSBP1)

C-37
Variable Dimension Subroutine Description Units

DEVL(l) N NODE Hydraulic diameter boiling cell - length averaged cm

DEVNBL(I) NNB NODE Nonboiling cell hydraulic diameter - length averaged cm

DPFRNB1(J) NNB DELP Nonboiling cell accumulative friction pressure drop Kg/cm2

DPFRP1(M) N+1 DELP Boiling accumulative cell edges friction pressure Kg/cm2 DPFRP_(M) N+I ELProp DPFRP1(1)=0.0

DZ ---- NODE Boiling cell length cm

EKFL(I) N NODE Boiling cell friction multiplier

EKFNBL(J) NNB NODE Nonboiling cell friction multiplier

IFF ..... SETUP Flag for friction factor model to use

IFM ---- _SETUP Flag for friction multiplier model to use

NNB ---- BOUNDY Number of Nonboiling cells

OMEGA(M) N+I MEGAP Boiling cell edge Jones flow correction factor for

_ __ Martinelli-Nelson. OMEGA(1)=1.0

REVL(I) N FRICT Boiling cell Reynolds

REVNBL(I) NNB FRICT Nonboiling cell Reynolds

RHOVL(I) N FRICT Boiling cell density g/cm3

RHOVNBL(I) NNB FRICT Nonboiling cell density g/cm3

SG BLOCK Gravitational acceleration cm/s2 DATA

N+1 PREPHI Boiling 320(M) cell edge Martinelli-Nelson two-phase friction

ultiplier T20(1)=1.0

TPFM(J) N+1 PREPHI Boiling cell edge Baroczy-Chisholm two-phase

__PFM(__) N+I__EP _I 'riction multiplier TPFM(1)=1.0

UVL(l) N FRICT Boiling cell liquid velocity -no steam- (w0/rhol*a cell) cm/s

UVNBL(I) NNB FRICT Nonboiling cell liquid velocity cm/s

COM(J+1) NNB+N+1 OEFR & ccumulative cell edges height. ZCOM(1)=0.0 cm

_______ ______NODE _________ ___________

2.4.4. Irreversible Expansion Losses

Nonboiling region

In LAPUR 5, the nonboiling region comprises only one node. In the LAPUR 6.0 code, the

nonboiling region has been partitioned into several cells, according to the LAPURX input data

file.

For this region, the LAPUR 6.0 uses the Vennard formulation (Ref. 9) to compute the

irreversible exliansion losses due to an area change:

n+2

ln+1

&p(M) = (1-_(m))2. G2 (n)

(m) = p(m)

GY(n) -,A(n) ,A(n + 1)

G(n)=pl(n).u,(n)= Wo

A(n)

(39)

(40)

(41)

C-38
To consider that a sudden enlargement produces a pressure loss, we compute the crosssectional area variation o. The irreversible pressure loss is calculated as:

AP(m) = (1 -(m)) 2 (w(/A(n))2

2 pi(m) .- -(42)

Subroutine DELP (xfrict.for) computes these pressure losses [DPEXNB(J);J=1 ,NNB].

Note. To prevent computational noise, irreversible expansion losses are set to 0 if 1)<0.001.

Boiling region

For this region, LAPUR 6.0 uses the homogeneous two-phase multiplier formulation (Ref. 11 ):

AP(m)=(1-(m))2 2.-p, .G2(n) [ + 1P(m)1)x(m) (43) (M) •p(M) P 

-(m) A(n) = A(n + 1) (44)

e(n)= p,(n)- u,(n)- An (45) A(n)

where x(m) is'the cell edge quality.

To consider that a sudden enlargement produces an irreversible pressure loss, we compute

the cross-sectional area variation a. The irreversible pressure loss is calculated as:

AP(m) (1-c(m))2. (w°/A(n))2 1+ P (m) 1 x(m) (46)

2.p,(m) - pg()

Subroutine DELP (xfrict.for) computes these pressure losses [DPEXP(J);J= ,N].

Note: To prevent computational noise, irreversible expansion losses are set to 0 if

oao1)< 0..

One subroutine, DELP, has been modified. DELP performs the calculation of expansion

pressure losses at every node along the length of the channel.

2.4.4.1. Irreversible expansion losses-LAPUR modifications

Subroutine DELP-Calculates expansion pressure losses at every node along the length

of the channel

Nonboiling region

SIGMA=AVNBCL (1)/AVNBCL (2)

C-39
IF((1.0-SIGMA).LE.0.001) THEN

DPEXNB (1) =0.0

ELSE

TEMPO=1.0

TEMP1= (1.0-SIGMA) **2

CON1=W0/AVNBCL (1)

TEMP2=(0.001/SG)*CON1**2/(2.0*RHOVNBL(1))

DPEXNB (1) =TEMPO*TEMP1*TEMP2

ENDIF

C

DO 216 J=2,NNB

SIGMA=AVNBCL (J) /AVNBCL (J+l)

IF((1.0-SIGMA).LE.0.001) THEN

DPEXNB (J) =DPEXNB (J-1) +0.0

ELSE

TEMPO=1.0

TEMP1= (1.0-SIGMA) **2

CON1=WO/AVNBCL (J)

TEMP2=(0.001/SG)*CON1**2/ (2.0*RHOVNBL(J))

DPEXNB (J) =DPEXNB (J-1) +TEMPO*TEMP1*TEMP2

ENDIF

216 CONTINUE

DPEXNB2 = DPEXNB (NNB)

DPEXNB2 0.0

Boiling region

DPEXP(1) = 0.

SIGMA=AVCL (1)/AVCL (2)

IF((1.0-SIGMA).LE.0.001) THEN

DPEXP (2) =0.0

ELSE

TEMPO= (1.0+ (RHOVL (1)/RHOS-1) *X (1))

TEMP1= (1.0-SIGMA) **2

CON1=WO/AVCL (1)

TEMP2= (0. 001/SG) *CON1**2/ (2.0*RHOVL ())

DPEXP (2) =TEMPO*TEMP1*TEMP2

ENDIF

C

DO 266 J=2,N

SIGMA=AVCL (J) /AVCL (J+l)

IF((1.0-SIGMA).LE.0.001) THEN

DPEXP(J)=DPEXP(J-1)+0.0

ELSE

TEMPO=(1.0+(RHOVL(J)/RHOS-1)*X(J))

TEMP1= (1.0-SIGMA) **2

CON1=WO/AVCL (J)

TEMP2= (0. 001/SG) *CON1**2/ (2.0*RHOVL (J))

C-40
DPEXP(J)=DPEXP(J-1)+TEMPO*TEMP1*TEMP2

ENDIF

266 CONTINUE

.DPEXSB = DPEXP (NSB)

DPEXBB = DPEXP (N) - DPEXP (NSB)

Variable Dimension Subroutine Description Units

AVCL(M) N+I NODE Boling cell edge flow area cm2

AVNBCL(J+I) NNB+1 NODE Nonboiling cell edge flow area cm2

DPEXNB(J) NNB DELP Nonboiling dropm cell accumulative expansion pressure kg/cm2

________ drop

DPEXP(J) N DELP Boiling cell accumulative expansion pressure drop kg/cm2

N ----- INPUT-user Number of axial nodes desired in the boiling region

of the channels

NNB ----- BOUNDY Number of.Nonboiling cells

RHOS ---- READY Steam density g/cm3

RHOVL(I) N FRICT Boiling cell density g/cm3

RHOVNBL(l) NNB FRICT Nonboiling cell density g/cm3

SG ---- BLOCK Gravitational acceleration cm/s2 DATA

0 -- PREPAR Reference channel mass flow rate g/s

(M) N+1 UBOIL & Channel boiling cell edge quality. X(1)=0.0 _______ _______BULK______________________

2.4.5. Irreversible Contraction Losses

Nonboiling region

In LAPUR 5, the nonboiling region comprises only one node. In the LAPUR 6.0 code, the

nonboiling region has been partitioned into several cells, according to the LAPURX input data

file.

For this region, the LAPUR 6.0 uses the Vennard formalism (Ref. 9) to compute the

irreversible contraction losses due to an area change:

AP(m) = a(m)2 0.385 . a(1-1I G (n)

A(n)

A(n + 1)

G(n)=p,(n).u,(n)= Wo

A(n)

(47)

(48)

(49) .1

C-41
To consider that a sudden contraction produces a pressure loss, we compute the crosssectional area variation a. The irreversible pressure loss is calculated as:

AP(m) = c(m)' .0.385 (1 1) (w°/A(n))2 AP~):c(m) (50) -0385 1 (-m) 2.-p,(m)

Subroutine DELP (x frict.for) computes these pressure losses [DPCONB(J);J=1 ,NNB].

Note. To prevent computational noise, irreversible contraction losses are set to 0 if

(a- i)< 0.001.

Boiling region

For this region, LAPUR 6.0 uses the homogeneous two-phase multiplier formulation (Ref. 11):

AP(m) = Y(M)2 -.0.385. -1 - -1jJ G( [l + ___ , - )J x(m)] (51)

(T(m) - A(n) (2 A(n) (52)

G(n) = p,(n), u,(n)=- W ) (53) A(n)

where x(M) is the cell edge quality.

To consider that a sudden contraction produces a pressure loss, we compute the crosssectional area variation a. The irreversible pressure loss is calculated as:

AP(m)= o(m).2 0.385 - (w0/A(n)) 2 ,p 1(m) >1 x(m (54)

Subroutine DELP (xfrict.for) computes these pressure losses [DPCOP(J);J=1,N].

Note: To prevent computational noise, irreversible contraction losses are. set to 0 if

(a- 1)< 0.001.

One subroutine, DELP, is modified. DELP calculates contraction pressure losses at every

node along the length of the channel.

C-42
2.4.5.1. Irreversible contraction losses-LAPUR modifications

Subroutine DELP-Calculation of contraction pressure losses at every node along the

length of the channel

Nonboiling region

SIGMA=AVNBCL (1) /AVNBCL (2)

IF((SIGMA-1.0).LE.0.001) THEN

DPCONB (1) =0.0

ELSE

TEMP0=1. 0

TEMP1=SIGMA**2*0.385* (1-1/SIGMA)

CON1=W0/AVNBCL (1)

TEMP2=(0.001/SG)*CON1**2/(2.0*RHOVNBL(i))

DPCONB (1) =TEMPO*TEMPI*TEMP2

ENDIF

C

DO 218 J=2,NNB

SIGMA=AVNBCL (J) /AVNBCL (J+l)

IF((SIGMA-1.0).LE.0.001) THEN

DPCONB (J) =DPCONB (J-1) +0.0

ELSE

TEMP0=1. 0

TEMPI=SIGMA**2*0.385* (1-1/SIGMA)

CON1=W0/AVNBCL (J)

TEMP2= (0. 001/SG) *CON1**2/ (2.0*RHOVNBL (J))

DPCONB (J) =DPCONB (J-1) +TEMP0*TEMP1*TEMP2

ENDIF

218 CONTINUE

•DPCONB2 = DPCONB (NNB)

DPCONB2 = 0.0

Boiling region

DPCOP(1) = 0.

SIGMA=AVCL (1)/AVCL (2)

IF((SIGMA-1.0).LE.0.001) THEN

DPCOP (1)=0.0

ELSE

TEMPO= (1.0+ (RHOVL (1)/RHOS-1) *X(i))

TEMP1=SIGMA**2*0.385* (1-1/SIGMA)

CON1=W0/AVCL(1)

TEMP2=(0.001/SG)*CON1**2/(2.0*RHOVL(I))

DPCOP (1) =TEMPO*TEMP1*TEMP2

ENDIF

C-43
C

DO 268 J=2,N

SIGMA=AVCL (J) /AVCL (J+l)

IF((SIGMA-1.0).LE.0.001) THEN

DPCOP (J) =DPCOP (J-1) +0.0

ELSE

TEMPO= (1.0+ (RHOVL (J) /RHOS-1) *X (J))

TEMPI=SIGMA**2*0.385* (1-1/SIGMA)

CON1=W0/AVCL (J)

TEMP2=(0.001/SG) *CON1**2/(2.0*RHOVL(J))

DPCOP (J) =DPCOP (J-1) +TEMPO*TEMP1*TEMP2

ENDIF

268 CONTINUE

DPCOSB DPCOP (NSB)

DPCOBB = DPCOP (N) - DPCOP (NSB)

Variable Dimension Subroutine Description Units

AVCL(M) N+1 NODE Boling cell edge flow area cm2

AVNBCL(J+I) NNB+1 NODE Nonboiling cell edge flow area cm2

NNonboiling cell accumulative contraction pressur kg/cm2 DPCONB(J) !NNB DELP drop

DPCOP(J) N DELP Boiling cell accumulative contraction pressure drop kg/cm2

N INPUT-user Number of axial nodes desired in the boiling region

of the channels

NNB ----- BOUNDY Number of Nonboiling cells

RHOS ---- READY Steam density g/cm3

RHOVL(I) N FRICT Boiling cell density g/cm3

RHOVNBL(I) NNB FRICT Nonboiling cell density g/cm3

SG BLOCK Gravitational acceleration cm/s2 DATA I

W0 ----- PREPAR Reference channel mass flow rate g/s

(M) N+1 UBOIL & Channel boiling cell edge quality. X(1)=O.O _(M)_ N+I BULK

C-44
2.4.6. Local Losses

Nonboiling region

In LAPUR 5, the nonboiling region comprises only one node. In the LAPUR 6.0 code, the

nonboiling region has been partitioned into several cells, according to the LAPURX input data

file.

For this region, the LAPUR 6.0 computes the local losses (spacers) as:

AP(m) = KL (i). G2(n) (55)

2-p,(m)

Wo (6

G(n)=p,(n)-u,(n)- W()A(n)

Subroutine DELP (xfrict.for) computes these pressure losses [DPLLNB(J);J=1,NNB]. In

addition, we obtain a nodal "equivalent Martinelli-Nelson" local loss coefficient, which is used in

the calculation of the momentum frequency-dependent coefficients, in subroutine COEFR, which

are needed to solve the thermo-hydro-dynamic equations:

APlocal(m) (7

KL-MN(m) = AP(57t)o (i)

Boiling region

For this region, the LAPUR 6.0 computes the local losses (spacers) at steady state as:

AP(m) = KL H.[ +1 pm H ~ G (n) (8 [ g H\gm 2. p1 (m) (8

G(n)=p,(n)-u,(n) An(59) W= 

A(n)

Subroutine DELP (xfrict.for) computes these pressure losses [DPLLP(J);J=1,N]. In

addition, we obtain a nodal "equivalent Martinelli-Nelson" local loss coefficient, which is used in

the calculation of the momentum frequency-dependent coefficients, in subroutine COEFR,

needed to solve the thermo-hydro-dynamic equations:

KLMN(m)- AP°., (M) (60) APfriction Hm

C-45
Five subroutines are modified:

* Subroutine INPUT: Read the new variables.

" Subroutine OUTIN: Print the new variables.

* Subroutine SETUP: Selection of the parameters specific to channel type IX from

the arrays.

" Subroutine NODE: Transformation of the nodal parameters to the new coordinate

system.

" Subroutine DELP: Calculation of local pressure losses, at every node along the

length of the channel.

2.4.6.1. Local losses-LAPUR modifications

Subroutine INPUT-Read the new variables.

630

640

650

660

CALL INPI (NXE,NDIMLL)

GO TO 6000

CALL INPI (NTDLL,NODLL)

GO TO 6000

CALL INP2 (NTDLL, NODLL, DZLLM, NLLMAX)

GO TO 6000

CALL INP2 (NTDLL,NODLL,EKLLM,NLLMAX)

GO TO 6000

Variable Dimension Subroutin Description Unit

_ e s

Elevation measured from the channel entrance J NLL Iur of local loss JJJ. Index JJ represents the type of cm

J) L user local-loss channel

EKLLM(JJ,JJ NLLMAX,NL INPUT- Local-loss constant of local loss JJJ. Index JJ

J) L user represents the type of local-loss channel

NDIM LL(J) NXE INPUT- Local pressure-loss type of the J-th thermal ________user u e -r region g o

NLLMAX ---- param fi PARAMETER (NLLMAX = 10) ! Max # of localNLMA...paamf loss channel

NODLL(J) NTDLL INPUT- Number of axial intervals for the J-th type, localuser loss channel

NTDLL ---- INPUT- Number of local-loss channel types user I

Subroutine OUTIN-Print the new variables.

63 WRITE(6, 444)

WRITE(6, 163)11, NXE , (NDIMLL(J), J=1,NXE)

GO TO 200

64 WRITE(6, 164)11, NTDLL , (NODLL(J),J=1,NTDLL)

GO TO 200

65 DO 325 JJ=1,NTDLL

C-46
JI=NODLL (JJ)

IF(JJ.EQ.1) WRITE (6, 165)11, (DZLLM(JJ,JJJ),JJJ=1,JI)

IF(JJ.NE.1) WRITE (6, 301) (DZLLM(JJ,JJJ),JJJ=1,JI)

325 CONTINUE

GO TO 200

66 DO 326 JJ=1,NTDLL

JI=NODLL(JJ)

IF(JJ.EQ.1) WRITE (6, 166)11, (EKLLM(JJ,JJJ),JJJ=1,JI)

IF(JJ.NE.I) WRITE (6, 301) (EKLLM(JJ,JJJ),JJJ=l,JI)

326 CONTINUE

GO TO 200

163 FORMAT(1HO,4X, '(',12,') ',8HNXE = I5 /10X,8HNDIMLL

7112

1 / (18X, 7112)

164 FORMAT (IHO, 4X, '(',I2, ') ', 8HNTDV = 15 / 1OX,8HNODLL

7112'

1 ,/ (18X,. 7112)

165 FORMAT(IH0,4X, ' (',I2, ') ',8HDZLLM 7F12.4

(18X,7F12.4) )

166 FORMAT(IH0,4X,'(',I2,') ',8HEKLLM 7F12.4

(18X,7F12.4)

Subroutine SETUP-Select the parameters specific to channel type IX from the arrays.

IBM=NDIMLL (IX)

NLL=NODLL (IBM)

DO I=1,NLL

DZLL(I) = DZLLM(IBM, I)

EKLL(I) = EKLLM(IBM,I)

ENDDO

Subroutine NODE-Transform the nodal parameters to the new coordinate system

C Calculation of nodal local loss

IF(NLL.LE.0) GO TO 7510

C

11=1

C

DO 6000 I=I,NNB

EKLLNBL (I)=0.0

6000 ENDDO

C

DO 6500 J=1,NNB

DO 6380 I=Il,NLL

IF(ZCOM(J+1).LT.DZLL(I)) GO TO 6400

EKLLNBL (J) =EKLLNBL (J).+EKLL (I)

Ii=Ii+l

C-47
6380 CONTINUE

6400 CONTINUE

6500 CONTINUE

C

DO 7000 I=1,N

EKLLL (I) =0.0

7000 ENDDO

C

DO 7500 J=1,N

DO 7380 I=II,NLL

IF(ZPP(J+1).LT.DZLL(I)) GO TO 7400

EKLLL (J)=EKLLL (J) +EKLL (I)

I1=I1+1

7380 CONTINUE

7400 CONTINUE

7500 CONTINUE

C

7510 CONTINUE

Variable Dimension Subroutine Description Units

DZLL(I) NLL SETUP Top height from bottom of the I-th local loss axial interval cm (DZLLM) cm

EKLLL(J) N NODE Local-loss constant for the J-th boiling axial cell

EKLLNBL(J)NNB NODE Local-loss constant for the J-th non-boiling axial cell ___

COM(J+1) NNB+N+1 OEFR & ccumulative cell edges height. ZCOM(1)=O.O ZCOM(J+I) NNB+N+I NODE ____________________________________.0_c

Subroutine DELP-Calculates local pressure losses at every node along the length of the

channel

Nonboiling region

TEMP0=1. 0

CON1=WO/AVNBCL( 1)

TEMPI=(0.001/SG) *CON1**2/ (2.0*RHOVNBL(l))

DPLLNB (1) =TEMPO*TEMP1*EKLLNBL (1)

C

EKLFNB(I)=DPLLNB(i)/DPFRNB1 (1)

C

DO 217 J=2,NNB

TEMPO=I. 0

CON1=WO/AVNBCL (J)

TEMPI= (0. 001/SG) *CON1**2/ (2.0*RHOVNBL (J))

DPLLNB (J) =DPLLNB (J-l) +TEMPO*TEMP1*EKLLNBL (J)

C

EKLFNB (J) = (DPLLNB (J) -DPLLNB (J-1)) /

1 (DPFRNB1(J)-DPFRNB1(J-1))

217 CONTINUE

C-48
DPLLNB2 = DPLLNB (NNB)

DPLLNB2 = 0.0

Boiling region

C

C

C

TEMPO=(1.0+(RHOVL(1)/RHOS-1) *X(1))

CON1=W0/AVCL (1)

TEMP1=(0.001/SG)*CON1**2/(2.0*RHOVL(i))

DPLLP (1) =TEMP0*TEMP1*EKLLL (1)

EKLF(1)=DPLLP (1) /DPFRP1 (2)

DO 267 J=2,N

TEMPO=(1.0+ (RHOVL (J)/RHOS-1) *X (J))

CONI=WO/AVCL (J)

TEMP1=(0.001/SG)*CON1**2/ (2.0*RHOVL(J))

DPLLP (J) =DPLLP (J-1) +TEMP0*TEMP1*EKLLL (J)

EKLF (J) = (DPLLP (J) -DPLLP (J-1) /

1 (DPFRP1 (J+1) -DPFRP1 (J))

267 CONTINUE

DPLLSB = DPLLP (NSB)

DPLLBB = DPLLP (N) - DPLLP (NSB)

Variable Dimension Subroutine Description Unit

AVCL(M) N+1 NODE Boling cell edge flow area cm2

,VNBCL(J+I) NNB+1 NODE Nonboiling cell edge flow area Cm2

DPLLNB(J) NNB DELP Nonboiling cell accumulative local pressure drop kg/cm2

DPLLP(J) N DELP Boiling cell accumulative local pressure drop kg/cm2

RHOS ---- READY Steam density g/cm3

RHOVL(I) N FRICT Boiling cell density g/cm3

RHOVNBL(I) NNB FRICT Nonboiling cell density g/cm3

SG BLOCK Gravitational acceleration Cm/s2 DATA

0 ----- PREPAR Reference channel mass flow rate g/s

(M) N+1 SUBOIL & Channel boiling cell edge quality. X(1)=O.0 ________________BULK _________________ ____

2.5. LAPURX 6 COEFFICIENTS OF THE MOMENTUM EQUATION

In this section, we present the modifications made in LAPURX and related to the new

coefficients obtained for the momentum equation with variable area in the frequency domain that

will be used for the solution of LAPURW.

C-49
2.5.1. LAPUR 6.0 Nonboiling Region Coefficients

The normalized conservation momentum equation in the nonboiling region is:

A(z) uao + u0 a PF 2 A 0 " z y(2-a2 A + RNZ) Cos 0 (61)

az AO g at g azyP, A(z)) A(z)) PFAO

where H-[ is the normalized pressure

F-[ - (62)

gPF L

y is the normalized mass flow rate

W- (63)

wo

and F(z) is the single-phase friction corresponding to a fluid of density pi moving at speed up,

and for the nonboiling region is equal to:

2

F(z)=fo(z) u 0 PF (64)

2gDH(z) P,

Linearizing the conservation equation (61) around the steady-state point, and Laplace

transforming the resulting equation, and considering the crud-deposition friction multiplier KF

(Ref. 7), yields:

_LoA(Z s U 2 _ + 0 y (65) (5 f I s 5- + K 2 "j [KF (F(Z))(2 -a 2)~ A0 az g g z p0 A(z) 1310 g o (AL)ooo

Integrating equation (65) over the length of the i-th node yields:

-- AFNB = -(inZNB- 6nin)-

fu0 (Ui AO_ + A0 Liln68. {LO S9 (ZZ Aajl A•° inlYn

+u 2p__ 1 A0 I A 0A A +(1 - 1 Yn

{LOg [l=W p1,,+1 Ai+1 pj Aj)Aaj P1,1 Pin

{LO U A0 PF2 K}YIn (66)

+__K_ 11 yiof L+o g _Ain Pin

2_U PF -f i(2_a) KFj1.1 A0 1 1 A AO

L0 g PO 0 i=1 2 DHj+ Pii+i AW DHI P1 A) Aaj

+ f0,in C,in (Ain 2 (2R) in

C-50
where Aa is the length-averaged node flow area.

The irreversible pressure losses due to sudden expansions and contractions and local

losses (spacer), are not included in equation (66). These are given by:

-PF i . iK expirrk 6K Kcnirrk' 8Krk' 116 (67)

Ljg LO g i=1 = PI~aj Aaj ~Airkk k=1 k on'=1~ ,, z Fz 2 , ,~~ (7

where

Kexpirr,k = [1- cY(Zk)2 (68)

C(Zk)- A Ai (69) i+1

and Zk indicates the expansion position

Kcon,irr,k, 0.385 1 (70)

oY(Zk.) = ok. =- Ai' (71)

Ai,+1

and Zk' indicates the contraction position.

We note that the irreversible local pressure losses for the nonboiling region obtained using

equation (55) is only used at steady state, but is not used to compute the perturbations around

the steady-state point.

The irreversible local pressure loss due to the spacer in the i-th node is converted in

equivalent friction pressure losses:.

(APspaceri )0 = KLFj (APFriction) (72)

where KL-F, is the equivalent friction coefficient for local loss for the i-th node given by:

KL-F~i (APspacerj )0 KLi 2PL,' I0 (KL, )0 (73)

K-F A P~rction,, K. Gi (K f

2 PL )0

Then, including this equivalent friction pressure loss by means of the coefficient KL-F., we

obtain that the total friction coefficient in the i-th node, KF+Li, is given by:

KF+L, f, = KF;i (0 + KLF,)f, (74)

Adding the irreversible pressure losses in equation (66), we obtain

C-51
- AFINB= s IAZ. AO ) AO i]6i L°g LO 1 7 Aa , Aa,,i (75)

+ U • RF (EKACPV + EKNBPV + EKCCP + EKINP + EKLLV 6Yin

(LO g PO

where the coefficients are given by:

EKACPV = F2po [N. 1 A• 1 AO-AO j+r1 -+ 1 (76)

EKNBPV=__ Y PO f'i (2- .- a2)i AziKFL¢ KF+L 1 12'-/ -- 1 1 A A +-- 11 1 A)A (77)o

2 ' 2 - •,2 DH,,+l Pi+, Ai+, DHi P,, Ai (77)

EKCCP = AO PF KcC (78)

(Ain Pin)

( _ A° 2 2 D 2 a2n)Ln-, ,in (9

EKINP = fin Kcin 2 (79)

NNB 1_ ___i~ ___ (N 6Kr [C K

EKLLV = P ,O a, AKxpirrk YA e ZZk J+ [ak'--[ Kcon'irr'k n ' 6 Kr y (80)

In a more compact form we can write:

F NNB A

-AITNB = sIýO- Az. 1+ - L.n 6in 5 i (81) FLO [ 'Aaj) A ainjYr {CV18 y

where

2Uo PF (EKACPV + EKNBPV + EKCCP + EKINP + EKLLV) (82)

CV18 Lo g po

Thus, from equation (81) we can write

C-52
5 1 (-AiNB)

AZi + EKIP S }

[gL~o,=,. CVCV8

- NB A cv ý- KI -ArINB) (83)

CV 1 8 (-AfNB)

-{tVL S + 1}

where:

EKIPI~ (_ UJý (84)

and

t1VL CVl8 I Q AF A0 + EKIP (85) g [I oj= 'A.)

The LAPUR 6.0 new coefficients, used in subroutine COEFR, are

EKACPV = 2P [NN.p1 A,) 1 A+ A ( 1 _ 1 (86) 6, p A A P Pin

(2--,) A0 hA Ao

EKNBPV Y B fo0' (2 - a2)+ Azi K D. 1 ,. AA) +A0 (87) i=1 i+1 l~lA~ DHJi PliJ Ai. Aa

u0 FI".( A0 ±KP

VL, =cCVl8 _ z/,Az. Ao +EKIP (88) gLLo ':l Aaji) j

EKLLV A a-iKr, A Y' Z -'k [(Tk Koon~irr,' Kr 5 Yin (89)

W Pla," a ) ' k'=l k=1 

One subroutine, COEFR, is modified.

2.5.1.1. New nonboiling region coefficients-LAPUR modifications

Subroutine COEFR

TEMP1=0. 0

TEMP2=0. 0

TEMP3=0. 0

C-53
DO 570 I=1,NNB

TEMP1=TEMP1+

1 FNBL(I)*EKFNBL(I)*(EKLFNB(I)+I)*(2.0-A2NBL(I))*

2 (ZCOM(I+1)-ZCOM(I))

3 * (A0/(DEVNBCL(I+1)*RHOVNBCL(I+1)*AVNBCL(I+1))+

4 AO/(DEVNBCL(I)*RHOVNBCL(I)*AVNBCL(I)))*(A0/AVNBL(I))

TEMP2=TEMP2+

1 (AC/ (RHOVNBCL (I+1) *AVNBCL (I+1)) -

2 A0/(RHOVNBCL(I)*AVNBCL(I)))*(A0/AVNBL(I))

TEMP3=TEMP3+

1 (ZCOM(I+1) -ZCOM(I))*(A0/AVNBL(I))

570 ENDDO

EKNBPV = (RHOLI/2.0)*TEMP1

EKACPV = (2.0*RHOLI)*TEMP2

C

TEXP = 0.0

TCON = 0.0

C

DO I=1,NNB

SIGMA=AVNBCL (I) /AVNBCL (I+l)

IF((1.0-SIGMA).LE.0.001) THEN

TEXP = 0.0 + TEXP

ELSE.

TEXP=(1.0/RHOVNBL(I))*AVNBCL(I+1)/AVNBL(I)*(AO/AVNBCL(I))**2

1 *(1.0-SIGMA)**2 + TEXP

ENDIF

C

SIGMA=AVNBCL (I) /AVNBCL (I+l)

IF((SIGMA-1.0) .LE.0.001) THEN

TCON = 0.0 + TCON

ELSE

TCON=(1.0/RHOVNBL(I))*AVNBCL(I+1) /AVNBL (I)* (AO/AVNBCL (I))**2

1 *SIGMA**2*0.385*(I.-I./SIGMA) + TCON

ENDIF

C

ENDDO

C

EKLLV = RHOLI*(TEXP+TCON)

C

C18V

1. / (Q4*RHOL/RHOLI* (EKACPV+EKNBPV+EKINP+EKCPP+EKLLV))

TAULV = C18V* (TEMP3/EL+EKIP)*UO/SG

C

IF (FLOW AREA VAR.EQ.1)THEN

C18 = C18V

TAUL = TAULV

ENDIF

C-54
2.5.2. LAPUR 6.0 Boiling Region Coefficients

The LAPUR 6.0 has two friction models. Model I comprises the Martinelli-Nelson two-phase

friction multiplier. Model II comprises the Chisholm-Baroczy friction multiplier.

2.5.2.1. Friction model I

The normalized conservation momentum equation in the boiling region, using the MartinelliNelson friction model t2N and the Jones flow rate corrector Q, is:

-L DA(z) u 0 uo By2 A0

oz Ao g at g az A(z)) (90)

F(z)y(2-a2) ý2 •(A 0 + (1 cc 7)Cos 0 A(Z)

F(ZY MN ~A(z)) A0

where ]- is the normalized pressure

P

r pF- (91)

y is the normalized mass flow rate

w

y =(92)

Wo

B is:

B= 1Y (1-72 (1-n))] (93)

7 is the slip ratio

Ug (94)

ul

and F(z) is the single-phase friction corresponding to a fluid of density p, moving at speed uo, is

equal to:

2

F(z)= f0 (z) 0o PF (95)

2gDH(Z) P,

C-55
. Linearizing the conservation equation (90) around the steady-state point, and Laplace

transforming the resulting equation, and considering the crud-deposition friction multiplier KF

(Ref. 7), yields, for the Martinelli-Nelson friction model:

LAz 0 i 0n az-+ ý AAOI2B 6-

+[(KFF(z))(2 a2)4MN JAQ° +(A(z)K MN ] 0

+ rj cos 00oA) 6 + [(KF F(z))f D J acL Mo 8a (96)

SgL- A-z) a cc + [(K F F(z)) MN A z- a- 0 H

+• aL-gA(z) a H A

where H is the normalized degree of sub-cooling

H -hF -hI (97) Shfg

C-56
Integrating equation (96) over the length of the j-th node yields:

-.(5• - 5 ) =

u0 AZ. A- s + gLo Aaj

4 Uo B+ Ao Ao

gLo Aj+1 Aaj

+KF foj u° '2gLo0 + (ay MNj+AAZj J8,)

1 Ao Ao0 -

DH,j+÷ j-A.1 Aaj 6Yaj

+ 2 U20 (aBB Ao AO -AZ.Cos0

gL0o .- a4 A j+1 Aaj LO

+Kuf a4MN AZ 1 Ao Ao0 1 , +Ff0J fO , 2gLo j -0- ( Az,j+1 A a,j I •. a• )j+. AjDH

' 2gLo + DH,j+1 Aj+1 a,j

US~ 02~ Q 2 1 A0 A 0

0,] O~ 2gL0 aH )i+1 4 MN,j+l~zj DH~j~l A1+1 Aaj

u02g (CB2jJ A,0 AAO}65 +2 - A+1 j

g 2 u° Lo Aa,j A, o+ [ jl SA BLj+1 + AO A BJB

I

+KFfo~j U0 (2-a 2 )jQ1 +

2gLo ( j,

Azj

2

A° [ A°0DH 2 Aa~j Aj+-•-H,j+1 2MN'j+1 A DH O 4 MN , ..

AIDHj MJ{UYJ

gLg1o 0 A, Aa~j [ FAA1~01 a _ cc )j+ l-+ A0 -a (aB 1 a a)1 ]

+KF foj Uo02 ij 2gLo

SKFf'jgo U20Az2

~{~o2 gL 2

AZj Ao

2 Aaj

_ [AjDHIj A°0 I K.0• (a2MN N+ .A j+-•-H,j+l Ao 2e M >( C

Aa [ A0 (D •-j) 

A:1 A1 DHJ aH) 42MNJ + Aj+ DH,j+I 0 a ) + N,j+I

+ gLo u02 Aaj A° [ LjA° (aB)l +A°, (aBN) 5 Hi 1 • H 1 Ai.aH I-HjJ

where A, is the length-averaged node flow area, and:

yj_ +@j+1 2 6y~j - 2

(98)

(99)

C-57
6 _ &aj + 8oj+l

2

a' 2

(100)

(101)

The irreversible pressure losses due to sudden expansions and contractions, and local

losses (spacer), are not included in equation (98). These are given by:

U'•a F~ J A 2C +i\ K ) (Lý 5a 0 V k) =[Kexp'irrk z'Zk E K conirrk' zK'z a(t 0, aj

2gL k1 k'=1 a,j

u 2 A (1 '2N''p 2A 

+ _0 u F A( l ) I.: [Kexpirr,k 2gL0 L ai k=1

__.H

6•~Kr +.N ' [(3.72,K Kr ]/ I H¢2 5Ha, k'=1 conir a H a,j (102)

2 F + u0 j+1

2gL0 IAa,i )[ A .j expirrk Z,Zk I ( k conrrX

k'=1

6Kr ]],2 -

Z,Zk. ])I9 ýH,O,a~j Cya~j

where

Kexp~irrk = [I - a(YZk ? (103)

A (104)

and Zk indicates the expansion position

Kconirr k' = 0.385 I1, -(Zk,

Y(Zk) 0(Zk): Gk' o, :Ai,+, Ail

(105)

(106)

and zk indicates the contraction position.

We note that the irreversible local pressure losses for the boiling region obtained using

equation (58) are only calculated at steady state; the equation is not used to compute the

perturbations around the steady-state point.

The irreversible local pressure loss due to the spacer in the j-th node is converted in

equivalent friction pressure losses:

(APspacerj )0 = KLF~ (APFrildoflj )a (107)

C-58
where KL-F,j is the equivalent friction coefficient for local loss for the j-th node given by:

(APspacerj) 0 _ KLJ H. 2 2p_'j (KG 2 .. j (KF,j m )-T, K = L A F. P dcio~j 0 .. F,, ,, () N ., Q.. ji Oý K F,, i) MN~j Q.•j )0(1 )

L/2pj )0

Then, including this equivalent friction pressure loss by means of the coefficient KL-F,j, we obtain

that the total friction coefficient in the j-th node, KF+LJ, is given by:

KF+, f, = K F, ( + KL-F,)fj (109)

Adding the irreversible pressure losses in equation (98), for the Martinelli-Nelson friction

model, we obtain:

[-CV, s + CV5j + CV9j + CV13Nj ]Yaj (110)

•[CV7, C4j + CV11, + CV1 4Nj]6•a. + [CV21j + CVN21,]6Haj

-[CV6j +cv1o0]6yj -[CV8j +CVi2j]5&X,-[CV22,]6H

or in a more compact form:

-/Arl 1 (V S + CV13j + CVN13j)6yaj

+ (cv141 + CVN14j)6aaj + (CV211 + CVN21j)6Haj (111)

-CV15 6yj -CV1 6j 5aj -CV22j 5 H

where

-.Vj = UP Azj A° (112)

gLo Aaj

CV5-=4 uB j+ A (113) gL0 Aj+I Aa~j

CV9U r(2 -a 2) Q+ (M a N,j+,Azj 1 AJ Aa(1

=KF+Lj fo~1 2g L0 - 1 0 y DHj+l A1 1 +Aaj

CV7.2 g0-0 u02 (aB) aa- A° A° (115) j+I Aj+1 Aaj

C4j = -

1 Az, cos 0 (116) Lo

C-59
CV11j =KF+LjfOj 'j , 2gLo g~ Q j -- N )j+1 AZjl

1 Ao Ao

DH,j+I Aj+l Aaj

CV21w = K f+, o, ''2gLo 2__ L __~ H),j+1, Az 1 A 0 AAa+ 0 u 2gL, (B' () B 1 1 A0AAA D,j+l H A j÷1 Aj + gLo ýaH~ Aj+ A,,,

CV6 j -2 A° LB+ +B

gLoAa,j A1~1 acl Ai al

CV1 0 = KF+L~jf~j (°(2 --a2 )j fi + )

~ =, ,V 2 [A gA o ,B j +A,( B j]

•gLo Aa,,jAj+,l c J,+ AiL• ,

(117)

(118)

(119)

(120)

(121)

(122)

Azj Ao [ A° Ao 1

2 Aaj Aj+-DHj+l MN,j+ - AjDHIj MNJ

CV12 v,=K jf j . _ .Q_ Azj Ao [_ Ao (a&)N 'L'2gLo 2 Aaj AjDH,j . a + Ao (a)MN

Aj+,DH,j+i aC j+1J

CV22j KF+Lj fo 0 u Az1 Ao 2gLo 2 Aa,j

AD (-Hj•MN2 + A+ , (a-Q 1

LAJDH~jaH)J MN.J A1~1 DH~j+l a H )i+l ýPMNJ+1

(123)

+ 2 ,ro 

gL0 Aaj LA1~1 yaH) (Ijl+,(aj Aik aH__ 

U o2

CVN14 -__

j 2gLo

CVN211 - u°_

2gLo

Aýj+ "2N_" 2A [KNo2.-

A Aexp,irr,k ]K E-i kl Kcon~irrklz,rk , oc0I)H zj z~l k':l '_ 00 )o,a~j

[ Z [ K zK'r Kr. ] + Z K con ir k . Kr ]zKN.

Aa Ae xpa1rrk 6ZZk ck' ,i ka J H 0,a,j

U' 0 [aAj+ A0 •2•p K 6Kr 1+ 2 -K 8Kr 2 ZZkJ ]+ 2,Lak con,irr,k' ZZk. l H,0,aj k=1J

(124)

(125)

(126)

(127)

(128)

(129)

(130)

CV13i = CV5j + CV9j

CV14j = CV7j - C4j + CV1 li

CV1 5i = CV6j + CVI10

CV1 6j = CV8j + CV12i

C-60
The new coefficients, computed in subroutine COEFR that will be used in LAPURW, (IFMM

ODtion eaual 1. data card 73. are:

,•Vj- = U0 Azj A° gLo Aa,j

CV5j =4 uO B Ao Ao

gL-- + Aj+1 Aaj,

CV9j = KF+L,j f 2 gL0 O' a ) j j + r(2-a2 a ) 2ýNj+ lAz 1 A ° A ° 2gLo 2, R' q)j y j ) MNj, D R~j+1 A j+I A a~j

CV7j = 2 •Uo--) 2 u° cB) AL ° 0

AA

U2 2

CV1ii I= KF+LjfO~j 2g 0 ( aj ),,M

(131)

(132)

(133)

(134)

(135)

(136)

(137)

(138)

Azj 1 Ao Ao

DRHj+1 A j+A Aaj

CV21. FLjo~ O(I~ 0 1 2 0±1~j1AOA 2 uL 0 (aB = F+~fJ2gL0 aH j+ mNJ Z D H~J+ A j+1 A aj g LO aH~i j+1

Ao Ao

A j+1 Aa~j

CVJ= CV6j = 2 u°2) Ao

g Lo Aa,j

AOB

Aj+B j+1

+ -A-- Ao Bj I

=KFLfO. O r( 2 -a A E(an),IAz. [ AO N2

CV1 0 2gL0 2 j )_a 2 Aaj Aj+o D

CV8lj = KF- 2 f Oj[U0 ( B aA)j( + A2 •a

gL Aaj o Aj+ ac Aiy, +)j MNj 

A O 2 1Nj

AIDHj MNJI

(139)

CV12j KF+Ljfo2 u0 AZ1 Ao

'2gLo 2 Aaj

A0 raMNa I A0 r a2Nl 1 IAJDH~j Laa); A1+ID H,j+I ac ~ I•zJ 

V22K U AZ A o A0 a Ao (a&' 2 CV -F+LfO'j () H)jM (AD -+ ANj (MNj+1

2gLo 2 Aaj AjDHj a - AJ+1 DH, j+l

+ uL A A, ( -H j+l +A O Ij gLO Aaj [Aj 1 a aH) A1j aH

(140)

(141)

C-61
__ U"' AKr Nk + 2 ni 5KzKr ])JJ( (142) L 29A A °j Kexp,irr,k Kz kI1 k conirrk Z,Zk 1(x43) CV 2' L ,A ' A~ k=1 k'=1 I] H o,a,j

2 2 N,° n -,, 2

CVN21 - AI ai A0/N jku[Kexp'irrk Krz ]'[2 Kcikr ( (14)

2L AL~ rj [K expjirr,k 61±Zk + c k con,irr,k' Z,Zk. i'1 Hj(14

CVi 3k =CV5k +=CV9 (145)

CV143 = CV7 llj + I-C4 kCV (146)

CV15j = CV6j + CV1lO (147)

CV16i = CV8j + CV12i (148)

2.5.2.2. Friction model II

Model II comprises the Chisholm-Baroczy Friction Multiplier. This model is used when the

IFMM option is equal to 2 (input user data card 73).

We note that the Chisholm-Baroczy model is used only to improve the pressure drop at

steady state, but not to compute the perturbations around the steady-state point.

To obtain the Chisholm-Baroczy coefficients, we obtain the relation between the ChisholmBaroczy two-phase friction multiplier, C2B and the Martinelli-Nelson two-phase friction multiplier,

•MN times the Jones two-phase friction multiplier correction factor, Q:

FcBMN,j = 2 (149)

note that this factor is defined at steady state, so that the partial derivatives are zero.

Then, using this relation, the equation obtained is:

k(6n'+ - 6rj~) -AL1~

['tVj s + CV5j + CVB9j + CV13Nj]6yaj (150)

+ [CV7 -C4 + CVB1 1 + CV14Nj]6 aj + [CVB2 1 + CVN21j]SHaj

-[CV6 + CVB1 0j]6yj - [CV81 + CVB12 j ]63 - [CVB22 ]6iji

C-62
or in a more compact form:

-Aj = (tVjs + CVB13j + CVN1 3j)6Yaj

+ (CVB14j + CVN1 4 j)6-aa,j + (CV21j + CVN2 1j)6Ha,j

- CVB15i 6yj - CVB16 i 6CIj - CVB22 6 Hj

where the new Chisholm-Baroczy coefficients are:

fjU~ 2 (anQ 2~ 1 AO AO

cCVB9j =KF+L,j , , 2gLo 0 (2-a2)jjj + -a2cBM N,j+lZJ+lAzj ( aY jCBM,+ DH,j+l Aj+l Aaj

(151)

(152)

CVB1 lj = KF+L,jfoj U2 u° 0 (153) j FCB-MN,j+,a.

2g L0 (BM~ act j+1

1 Ao Ao

j DR,j+1 Aj+I Aa,j

CVB2 1 i = KF+Ljfoj U02 2(( ") ' 2gLo aH- j+l

+2 u2 (,aBI) Ao i

gL -aH )j+1 Aj+1 A

2 (

CVB O - v +L f 0 (2-a CVB1O 1 2gL0

FCM~jI2 1 Ao Ao c-MN'j+IZ jDHj+lA j+l Aa,j

a,j

2)j nj + ay ; j

(154)

(155)

Azj Ao L o FCBA MN,j+2 2 A•,jLFj+ -H•+ )MN'j+ MN~ I 

Ua2 A +JAj Hj+

CVB12j = KF+L,jf,j 2g o Az Aa

FLii2gLO 2 Aa~j

A0 FCBMN,j MNJ]

AjDH~j IN

[ A0 FBMj (FMNr ) + A0 N )I AjDH,j .Oa ) Aj+lDH,j+l - y. J+l

(156)

CVB22j = KF+Lj foj U2 j Ao

' 2gL o 2 A,,j

A AA a) I- A i D H~ aO j FCBMN,jA N,J + A Aa) j+lDH,j+l D ( OH H j+l FcBMN~j+,I)MNJ+1 I (157)

U2

+ U0

gLo

A [A O DaB J A0 (aB), Aaj AjHl +1 Aik aH I

CVB13i = CV5j + CVB9j

CVB14 = CV7j - C4j + CVB1 ll

(158)

(159)

C-63
CVB15j = CV6j + CVB1Oj (160)

CVB1 6 j = CV8j + CVB12j (161)

These new coefficients are computed in subroutine COEFR, and it will be used in LAPURW

if IFMM option is equal to 2 (input user data card 73).

2.5.2.3. New boiling region coefficients-LAPUR modifications

Three subroutines are modified:

" Subroutine COEFR.

" Subroutine PREPHI

* Subroutine OMEGAP

Subroutine COEFR

INTEGER FLOWAREAVAR

DATA FLOWAREAVAR/ 1 /

C

DIMENSION CV5 (NOBMAX) ,CV6 (NOBMAX) , CV7 (NOBMAX)

CV8 (NOBMAX)

> , CV9 (NOBMAX)

,CV1I (NOBMAX),CV11 (NOBMAX),CV12 (NOBMAX)

,CV13 (NOBMAX),CV14 (NOBMAX),CV15 (NOBMAX),CV16 (NOBMAX)

> ,CV21 (NOBMAX),CV22 (NOBMAX)

DIMENSION CVB9 (NOBMAX)

,CVB1I (NOBMAX),CVB11 (NOBMAX),CVB12 (NOBMAX)

,CVB13 (NOBMAX),CVB14 (NOBMAX),CVB15 (NOBMAX),CVB16 (NOBMAX)

> ,CVB21 (NOBMAX),CVB22 (NOBMAX)

DIMENSION CVEX (NOBMAX), CVCO (NOBMAX)

> CVN13 (NOBMAX) , CVN14 (NOBMAX), CVN21 (NOBMAX)

QA = AO/AVL(J)

QB = AO/AVCL(J)

QB1 = AO/AVCL(J+1)

QV3. = FFL(J)*UO**2/(2.*SG*DEVCL(J)) * DZ/EL

QV31= FFL(J)*UO**2/(2.*SG*DEVCL(J+1)) DZ/EL

C

CV21(J)=

QV31*T20(J+l)*DOMDHO(J+l)*EKFL(J)* (EKLF(J)+i)*QB1*QA

1 + 2.*Q4*DBDHO(J+1)*QB1*QA

CV22(J)= (QV31*(T20(J+1)*DOMDHO(J+1)*QB1)

1. - QV3*(T20(J)*DOMDHO(J)*QB))*

EKFL (J) * (EKLF (J) +i)

C-64
2 *QA/2.0 + Q4*(DBDHO(J+1)*QB1+DBDHO(J)*QB)*QA

C

IF (IFM.EQ. 2) THEN

FCBMN=T20 (J) *OMEGA(J) /TPFM(J)

FCBMN1=T20(J+1)*OMEGA(J+1)/TPFM(J+l)

CVB21 (J)=

QV31*T20 (J+1) *DOMDHO (J+1) *EKFL (J) * (EKLF (J) +1) *QB1*QA

1 *FCBMNI+ 2.*Q4*DBDHO(J+1)*QB1*QA

CVB22(J)= (QV31* (T20(J+1)*DOMDHO(J+1)*QB1*FCBMN1)

1 -QV3* (T20 (J) *DOMDHO (J) *QB*FCBMN))*

EKFL (J) * (EKLF (J) +1)

2 *QA/2.0 + Q4*(DBDHO(J+1)*QB1+DBDHO(J)*QB)*QA

ENDIF

QA = AO/AVL(J)

QB = AO/AVCL(J)

QB1= AO/AVCL(J1)

QQFV = FFL(J)*UO**2/(2.*SG)

QQAV = A2L(J)

QQFV = QQFV*EKFL(J)*(EKLF(J)+I)

QQFVO= QQFV/DEVCL(J)

QQFV1= QQFV/DEVCL (Jl)

C

QV6 = QQFV0* ( (2.-QQAV)*OMEGA(J)+DOMDYO(J) )*DZ/EL

QV60 = QQFVO* ((2. -QQAV)*OMEGA(J)+DOMDYO(J) )*DZ/EL

QV61 = QQFV1*((2.-QQAV)*OMEGA(J)+DMDYO(J) )*DZ/EL

C

CV5(J) = 4.*Q4*BO(JI)*QB1*QA

CV6(J) = 2.*Q4*(BO(JI)*QB1+BOIJ)*QB)*QA

CV7(J) = 2.*Q4*DBDBO(JI)*QB1*QA

CV8(J) = Q4*(DBDBO(Jl)*QB1+DBDBO(J)*QB)*QA

CV9 (J) =QV61*FFF1 (T20 (JI),T20 (J))*QB1*QA

CV1 (J)=QV6*FFF2 (T20 (Jl) *QB1/DEVCL (Jl), T20 (J) *QB/DEVCL (J)) *QA

CVII (J)=QQFV1*OMEGA(J) *DZ/EL*FFF1 (DT2DB0 (Jl),DT2DBO (J)) *QB1*QA

CV12 (J) =QQFV*OMEGA (J) *DZ/EL

*FFF2(DT2DBO(Jl)*QB1/DEVCL(Jl),DT2DBO(J)*QB/DEVCL(J))*QA

CV13(J) = CV5(J) + CV9(J)

CV14 (J) = CV7 (J) - C4 (J) + CV11 (J)

CV15(J) = CV6(J) + CV10(J)

CV16(J) = CV8(J) + CV12(J)

C

IF (IFM.EQ. 2) THEN

FCBMN=T20 (J) *OMEGA (J) /TPFM (J)

FCBMN1=T20 (J+1) *OMEGA(J+1) /TPFM(J+l)

C

CVB9(J) =QV61*FFF1 (T20(JI)*FCBMN1,T20(J)*FCBMN)*QB1*QA

CVB1I (J)=QV6*FFF2 (T20 (Jl) *FCBMNI*QBI/DEVCL(Jl),

C-65
T20 (J) *FCBMN*QB/DEVCL (J)) *QA

CVB11 (J)=QQFV1*OMEGA(J) *DZ/EL*FFF1 (DT2DB0 (Jl) *FCBMN1,

DT2DB0 (J) *FCBMN) *QB1*QA

CVB12 (J) =QQFV*OMEGA (J) *DZ/EL

>*FFF2 (DT2DB0 (Jl) *FCBMN1*QB1/DEVCL (Jl),

> DT2DB0 (J) *FCBMN*QB/DEVCL (J)) *QA

C

CVB13(J) = CV5(J) + CVB9(J)

CVB14(J) = CV7(J) - C4(J) + CVB11(J)

CVB15(J) = CV6(J) + CVB10(J)

CVB16(J) = CV8(J) + CVB12(J)

C

SIGMA=AVCL (J) /AVCL (J+l)

IF((1.0-SIGMA).LE.0.001) THEN

CVEX(J) = 0.0

ELSE

TEMPO= (1. 0-SIGMA) **2

TEMP1= (1. 0/SG) *U0**2/EL

TEMP2=AVCL(J+1) /AVL(J) * (A0/AVCL(J)) **2

CVEX(J) = TEMP0*TEMP1*TEMP2

ENDIF

C

SIGMA=AVCL (J) /AVCL (J+1)

IF((SIGMA-1.0).LE.0.001) THEN

CVCO(J) = 0.0

ELSE

TEMPO=SIGMA**2*0.385* (i.-i./SIGMA)

TEMP1=(1.0/SG) *U0**2/EL

TEMP2=AVCL (J+1) /AVL (J) * (A0/AVCL (J)) **2

CVCO(J) = TEMP0*TEMP1*TEMP2

ENDIF

C

CVN14 (J)= (CVEX(J)+CVCO(J))

>* (DHMDBO (J) +DHMDBO (J+1) )/2

CVN21(J)= (CVEX(J)+CVCO(J))

>* (DHMDHO (J) +DHMDH0 (J+1) )/2

CVN13(J)= (CVEX(J)+CVCO(J))*2.

> *(HM0(J)+HM0(J+1))/2

C

IF (FLOWAREAVAR.EQ.1)THEN

TAUIPV(J) = TAUIP*(AO/AVL(J))

C5 (J) = CV5 (J)

C6 (J) = CV6 (J)

C7 (J) = CV7 (J)

C8 (J) = CV8 (J)

C9 (J) = CV9 (J)

C10(J) = CV0(J)

Cll(J) = CV11(J)

C12(J) = CV12(J)

C13(J) = CV13(J)+CVN13(J)

C14(J) = CV14(J)+CVN14(J)

C15(J) = CV15(J)

C-66
CI6(J) = CVI6(J)

C21(J) = CV21(J)+CVN21(J)

C22(J) = CV22(J)

IF (IFM. EQ. 2) THEN

C9 (J) = CVB9 (J)

ClO(J) = CVBI0(J)

Cll(J) = CVBlI(J)

C12(J) = CVBI2(J)

C13(J) = CVB13(J)+CVN13(J)

C14(J) = CVBl4(J)+CVNI4(J)

C15(J) = CVBl5(J)

C16(J) = CVBI6(J)

C21(J) = CVB21(J)+CVN21(J)

C22(J) = CVB22(J)

ENDIF

ENDIF

Subroutine PREPHI-Homogeneous multiplier (QH) and partial derivatives (c )

C Homogeneous multiplier and partial derivatives

DO J= 1, NPl

HMO (J) =1+ (RHOL/RHOG-) *X (J)

ENDDO

C

DHMDB0(i) = 0.0

DHMDHO(l) = 0.0

DO J= 2, NPI

DHMDBO (J) = (RHOL/RHOG-) *DXDBO (J)

DHMDHO (J) = (RHOL/RHOG-I *DXDHO (J)

ENDDO

2.6. OTHER MODIFICATIONS MADE IN LAPURX 6.0

In this section, we present other modifications made in LAPURX 6.0.

Subroutine MAIN -LAPUR 6.0 prints a new title.

C LAPUR6 M:1

C MODIFICATION

C WRITE(*,'('' LAPUR5X RUN TITLE :''/'' ** '',18A4,1'' **')')

WRITE(*, ' ('' LAPUR6X RUN TITLE :''/'' ** '',18A4,1'' **''

C LAPUR6 END

C-67
Subroutine DRUMWl-New commons used for intermediate storage.

EQUIVALENCE (XDLI(1),DPIN1), (XFRI(i),IFFM(I))

> ,(XFR3(l),IFF),(XCTP(1),TPFM(!))

DIMENSION

DIMENSION

DIMENSION

DIMENSION

WRITE (NDISKI

WRITE (NDISK1

WRITE (NDISK1

WRITE (NDISKI

XDLl(N DLTPl)

XFRI (N FRICC)

XFR3(N FRICl)

XCTP(NCTPFM)

XDLI

XFRI

XFR3

XCTP

Subroutine FRICT-This routine is a function of the individual channel type in LAPUR 6.

C SUBROUTINE FRICT

SUBROUTINE FRICT(IX)

Subroutine DELTP-This routine is a function of the individual channel type in LAPUR 6.

C SUBROUTINE DELTP

SUBROUTINE DELTP(IX)

LAPUR 6.0 has additional pressure terms, which require an additional calculation of the

total pressure loss.

DPNB2 = DPDHNB2 + DPFRNB2 + DPACNB2 + DPEXNB2 +

DPLLNB2 +

I DPCONB2

DPSB1 = DPDHSB1 + DPFRSB1 + DPACSB1 + DPEXSB + DPLLSB

+

1 DPCOSB

DPBB1 = DPDHBBI + DPFRBB1 + DPACBB1 + DPEXBB + DPLLBB

+

1 DPCOBB

C-68
C LAPUR6 M:4

C MODIFICATION

C DPTOT

C DPFR

C DPDH

C DPAC

DPTOT1

DPFR1

DPDH1

DPACI

C LAPUR6 END

C LAPUR6 1:7

C NEW LINES

= DPIN

= DPFRIN

.DPDHIN

= DPIN

= DPFRIN

= DPDHIN

+

+f

DPNB + DPSB

* DPFRNB + DPFRSB

DPDHNB + DPDHSB

DPACNB + DPACSB

+ DPNB + DPSB

DPFRNB + DPFRSB +

DPDHNB + DPDHSB +

DPACNB + DPACSB +

+

+I

DPBB + DPR

DPFRBB + DPFRR

DPDHBB + DPDHR

DPACBB

+1

+

+ DPBB + DPR

DPFRBB + DPFRR

DPDHBB + DPDHR

DPACBB

+ DPBB1 + DPR

+ DPFRBB1 + DPFRR

+ DPDHBB1 + DPDHR

+ DPACBB1

+ DPEXBB

+ DPLLBB

+ DPCOBB

DPTOT = DPIN

DPFR = DPFRIN

DPDH = DPDHIN

DPAC -

DPEX -

DPLL =

DPCO

+

+

+

DPNB2

DPFRNB2

DPDHNB2

DPACNB2

DPEXNB2

DPLLNB2

DPCONB2

+ DPSB1

+ DPFRSB1

+ DPDHSB1

+ DPACSB1

+ DPEXSB

+ DPLLSB

+ DPCOSB

C LAPUR6 END

Subroutine INPUT-New input data reading

C READ (4,9000,END=700)

READ (4,9000,END=7700)

(TITLE (I) ,I=i,18)

(TITLE(I) ,I=1,18)

C DO 6000 I=1,60

DO 6000 I=1,80

C IF(III.LT.0.OR.III.GE.70) GO TO 800

IF(III.LT.0.OR.III.GE.80) GO TO 800

C 4 570), III

4

570,580,590,600,610,620,630,640,650,660,670,680,690,700,

5 710,720,730,740,750,760), III

C 700 CONTINUE

7700 CONTINUE

C-69
C LAPUR6 M:1

C MODIFICATION

C WRITE(*,'('' *** LAPUR5X ***''I

WRITE(*, ' (' LAPUR6X ***' '/

C LAPUR6 END

Subroutine OUTIN-New input data printing

C 4 57),II

4 57, 58 ,59 ,60 ,61 ,62 ,63 ,64 ,65 ,66 ,67 ,68 ,69

,70 ,

5 71, 72 ,73 ,74 ,75 ,76),II

C 330 FORMAT(1H1,//19H * INPUT DATA * ,60X,15H*** LAPURX

*** /!

C 1 2X, 18A4

330 FORMAT(1H1,//19H *** INPUT DATA * ,60X,.

1 22H*** LAPURX 6.0 *** // 2X, 18A4

Subroutine NODE-Prevent convergence problems during the LAPURX 6.0 initial flow iteration

C LAPUR6 M:1 I:1

C MODIFICATION AND NEW LINE

C IF(ZNB.GT.EL) GO TO 540

EPS=0. 000001

IF((ZNB-EL).GT.EPS)GO TO 540

C LAPUR6 END

C LAPUR6 M:1 I:1

C MODIFICATION AND NEW LINE

C IF(ZNB.GT.DZF(1)) GO TO 540

EPS=0. 000001

IF((ZNB-DZF(1)).GT.EPS)GO TO 540

C LAPUR6 END

Subroutine OUTST-Print new pressure drops

C LAPUR6 M:5

C MODIFICATION

C WRITE (6, 9720)

(HEAD2 (I),I=1,2) ,HEAD6,DPDHNB,DPFRNB,DPACNB,DPNB

C WRITE (6,9720)

(HEAD3 (I), I=1,2) ,HEAD7,DPDHSB,DPFRSB,DPACSB,DPSB

C WRITE (6, 9720)

(HEAD4 (I) ,I=1,2) ,HEAD6,DPDHBB,DPFRBB,DPACBB,DPBB

C-70
C WRITE (6,9710) (HEAD5(I),I=I,2),DPDHR,DPFRR,DPCONR,DPR

C WRITE (6,9730) DPDH ,DPFR ,DPAC

,DPCONT,DPTOT

WRITE (6,9720) (HEAD2(I),I=I,2),HEAD6

1

,DPDHNB2,DPFRNB2,DPACNB2,DPEXNB2,DPCONB2,DPLLNB2

2 ,DPNB2

WRITE (6,9720) (HEAD3(I),I=1,2),HEAD7

1

,DPDHSB1,DPFRSB1,DPACSB1,DPEXSB,DPCOSB, DPLLSB, DPSB1

WRITE (6,9720) (HEAD4(I),I=1,2),HEAD6

1

,DPDHBB1,DPFRBB1,DPACBB1,DPEXBB, DPCOBB, DPLLBB, DPBB1

WRITE (6,9710) (HEAD5(I),I=I,2),DPDHR,DPFRR,DPCONRDPR

WRITE (6,9730) DPDH,DPFR,DPAC,DPEX,DPCODPLL,DPCONT,DPTOT

C LAPUR6 END

C LAPUR6 M:7

C MODIFICATION

C 9700 FORMAT(lHO,///' STEADY STATE PRESSURE DROPS (KG/CM2)'

,lox,

C 1 'FLOW RATE=',lPEI3.5,' G/S CM2',

C 1///30X, 'DENSITY HEAD',4X,8HFRICTION. ,4X,12HACCELERATION

,2X,

C .2 1IHCONTRACTION ,6X, 5HTOTAL

C 9710 FORMAT(//3X,2A4,2X,13HCONFIGURATION

,2P2EI4.4,14X,IP2EI4.4)

C 9720 FORMAT(//3X,2A4,Al,lX,14HBOILING REGION

,2P3EI4.4,14X, IPEI4.4)

C 9730 FORMAT(//lX,28(IH*) ///3X,24HSUM OF ALL REGIONS

,2P5EI4.4 )

9700 FORMAT(lIH0,///' STEADY STATE PRESSURE DROPS (KG/CM2) ' ,loX,

1 'FLOW RATE=',lPE13.5,' G/S CM2',

1///30X, 'DENSITY HEAD',4X,8HFRICTION ,4X,12HACCELERATION

,2X,

2 14HEXPANSION-IRR ,2X, 5HLOCAL, 3X,15HCONTRACTIONIRR, 4X,

2 1IHCONTRACTION ,6X, 5HTOTAL

9710 FORMAT(//3X,2A4,2X,14HCONFIGURATION

,2P2EI4.4,56X, IP2EI4.4)

9720 FORMAT(//3X,2A4,Al,lX, 14HBOILING

REGION, 2P6EI4.4,14X, lPEI4.4)

9730 FORMAT(//lX,28(IH*) ///3X,24HSUM OF ALL.. REGIONS

,2P8EI4.4 )

C LAPUR6 END

RETURN

C-71
Subroutine STEADY

In LAPUR 6, subroutines FRICT and DELTP are functions of the individual channel type.

C LAPUR6 M:2

C MODIFICATION

C CALL FRICT

C CALL DELTP

CALL FRICT(IX)

CALL DELTP(IX)

C LAPUR6 END

2.7. MODIFICATIONS IN LAPURW 6.0 RELATED TO THERMAL HYDRAULICS

This section presents the modifications made in the LAPURW (frequency domain) relating

to thermal hydraulics.

2.7.1. Energy Conservation-LAPUR 6.0 Nonboiling Region Coefficients

The energy conservation equation in the nonboiling region is

Q' = p, -+ 1 A(z) ah y h,-- (162)

WO PF UO A0 at az

where Q' is the energy per unit length supply by the fuel and y is the normalized mass flow rate.

Linearizing this conservation equation around the steady-state point and Laplace

transforming the resulting equation yields:

5Q' _ P1,0 1 A(Z)_s_+ ___(6h)+Qo 5 (163) W0 PF u0 A0 a wo

The total perturbation of the heat flux from the fuel to the coolant, 6Q', can be expressed in

function of the fuel transfer functions by:

6Q'= Q0 (CBq 6q'+CCq 8T + CAq 5y) (164)

where 6q' is the normalized total power generation perturbation in the fuel, 6T, is the liquid

coolant temperature perturbation, and 8y is the normalized coolant flow rate perturbation.

We define

S- h,__ -h (165)

hz1 - ho Aho

where Ah0 is the enthalpy rise in the nonboiling region.

Integrating equation (163) over the length of the i-th node, and using equation (165) and the

total perturbation of the heat flux from the fuel to the coolant along the channel, we obtain:

C-72
6Si+1 e-edl Az 6Si +(i -e-d"Az)gi (166) dti

where

5- 1 [Q' (CBq 6q'+(CAq - 1)5y) (167)

Aho wo

The new coefficient is dij = CTAUNBV(I), used in subroutine COEFIW

di =Azis T1 Aa.i _FQ Aho CCq (168)

ZNB A C p,av

where FQj is the relative power of node i, and cp.av is the average specific heat of water in. the

NBR.

One subroutine, COEFIW, is modified

2.7.1.1. LAPUR modifications

Subroutine COEFIW

C LAPUR6 1:6

C NEW LINES

CTAUNBV(I)= (ZCOM (I+1)-

ZCOM(I)) *CSS*TAU1/ZNB* (AVNBL(I)/AO)

> -FQ (I) *DELHO/CPAVG*CCQ (I)

C

IF(FLOWAREA VAR.EQ.1)THEN

CQ1 = CTAUNBV(I)

ENDIF

C LAPUR6 END

2.7.2. Mass and Energy Conservation-LAPUR 6.0 Boiling Region Coefficients

2.7.2.1. Mass conservation

The normalized conservation mass equation in the boiling region is given by:

A 0 O _ - aa (169)

A(z) azU 0 t

Linearizing this conservation equation around the steady-state point and Laplace

transforming the resulting equation yields

A 0 0 a -y = - s 5a (170)

A(z)Oz yu(

C-73
Integrating (170) over the length of the j-th node yields:

- - A&Xi fr1~,~ (171) (8Yj =-- 6•Ya~j A- + AO s AZj 5•aaj .for j =1, NBR(1)

r A z+

where

- j + 2Ya,j 8Yj+I (172) 2

The new coefficient for the Mass Conservation, used in subroutine COEFIW, is

CDVlj = u° (173)

2.7.2.2. Energy conservation

Total fluid

The normalized conservation energy equation for the total fluid in the boiling region is given

by:

(10 - )hfg WO w0 y 1 - } (1 - 71)u 0 at A 0

where H is the normalized degree of subcooling

H- hF -h1 (175) hg

Linearizing this conservation equation around the steady-state point and Laplace

transforming the resulting equation yields:

[.l¶WhgoQ' ---{oH11] -Z[o] 6 i0iy}- HI11 (176)

o ( i-a H) - } a (1

where

1 Z _= X (177)

l-Ti

C-74
and

x1- 7 0 -,q -,[1- 7,0-')] (178)

Inserting into equation (176) the total perturbation of the heat flux from the fuel to the

coolant, equation (164), and integrating over the length of the j-th node yields:

with

005J 6a~j+ CV 61 6 X,j + CDV 7,j 8Ha~j C CEV 31j

CEV 31j= CD14j F5j +CDV 15j 56d 1±+CDV 16 j 6 Hj+CD18 6-q~

(179)

(180)

and

6yj j + 6 j+l

6y61 2

vs-, 5_ 6+6%+.

2

j 6a -6 +(H aHj+1

and the coefficients are given by:

CD 5 j ={2[cy-H(1 -ii] K1-h ] Azj Q'oj CA qj}

(181)

(182)

(183)

(184)

CDV 6 ,j= 1 2 71 (1 +H) 0 j+l ra ý~ aco XOj+

r1- u S H1,a,j

+ s aJ

1-rju0 AO ,~

Az} (185)

CDV7Tj = {-2[1 Co S Ao)aj A aj (1 - AZj

1 - uo Ao )w (186)

12 +H)+1 -- + - Azj Q[j CCqj

CD 1 4 j a -{ýaH 11 - o)]0,j+l +[(Y Hj} " - (187)

CDV ={ 1 11(1 + 1 C1 o,j+H)cc+ (1 + .- •o,j + 1T S Aaj

1-i1 uo Ao

(H'J+1 - H j ) AZJ}

2

(188)

C-75
CDV16,j = -

1 - j o0j+l 1- o1 1-juoj+ T Aaj (cLO2 1 - Az

+ r(,±H)0 (1 +l a -x)J ))(1 0o,±H , (a xJ

CD8 Q' 0,jl 0,jjA~

CD18 J ={[(1 -)wo hfg

The new coefficients for Energy Conservation, used in subroutine COEFIW, are:

(189)

(190)

COY6,j 71 2 (1 + 0,j+l-a----/ -,q (a o O~j+l

+ Q uo A 1 . Ho,a,j

1-q uO AO

Az } (191)

C OYCD['j 7 , = {~2 2 1 -1 c _o,j+ 1i- jU S , O(--l (1A a (X 0 jA, ) A ZJ

+ 12 1-q (1+H)O 1

+j t,.OH ) ojl +X (1 - 'Owo q)wo hfg hfg

(192)

CCq

71 (1(+ax

C DV151J 1 ( H o 1 )+/ --- ' k~ O c L ) 0 +

+ -•(I+H t•~ 0cc.x• °, 11 X- )o,+ 1 s Aaj (Huo• + - H+'j)Az

1-ij uO A0 2

(193)

CDV16,j = {"

1

1 -11 - [ 1 ] + sAa0

___j+ _ oj 1 Uo Ao

aO°'J+2 - (L°'J) AZj 2

(194)

+ 'n [(1 + H)Oj 1 1- 71 +

a X + H (+ 0,,ja 1tJ

Steam phase

The normalized conservation energy equation for the steam phase in the boiling region is

given by:

- )Fs wnF~0 Q 0 = ~AoOyo (1 - TI)hfg WO 0 a A -+- 0 Uo Ao a (195)

where Fs is the fraction of energy invested in steam formation, and Xs is the bubble decay

constant.

C-76
Linearizing this conservation equation around the steady-state point and Laplace

transforming the resulting equation yields:

1 a 1, F. ~ A (Z ) a a X ' H

(1-qi)wohfg aH° j Luo Ao aH ]0j

1Fs il Aaz a ]0 ~l y5+I_ u+

-w o Qz6a} L u A a -- -6 (196)

[( LU_-]j6-fo i AJz + a--(-oqjo8+- =0

Inserting into equation (196) the total perturbation of the heat flux from the fuel to the

coolant, equation (164), and integrating over the length of the j-th node yields:

CDV2 j 6y8,j + CDV3 'j 6•z,j + CDV4,j 6H--,j = CEV2 j (197)

with

CEV2 j = CDV11,j 8y5j + CDV12 j 6aj + CDV13 j 6Hj + CD171j qa,j (198)

and the coefficients are given by:

CDV2 'j Q'h0 CA.,,j AZ (2F (J + 1) + F. (j))3

2,]I I w h W I(199) C3,j~ f 

SAo+ u Uo Zj j1 1 s,j+1 + 2-o,j+1

CDV3,j fA ,Uo + 2 r yaxsj+l ,(200)

CDV4. = hfg CCqa,j AZ (2 Fs 1)• (j FAz, + -- , 2 kj+ 1

' Lk ~ )wo hf§)cp "'3 Ao uo Hj+1 (201)

rA AOz1 -q 1-i /H

CDV,, a{A1 Az- [- Oa. , 1 1.6?XsOj+i 6 -a 01 1.6,.o. 1(5•oj.+ 01j 1 + 30j)

AO UA o 2 h, C AZ' -

Q'o C1•( CAz ~ (F. (j +1) - Fs(j))o} (202)

1-i,-j w-•g h' z 6

C-77
CDV12{jSJ - ?D 1 (203) SA° uO 2.+1 +x)jl+ Ol

_ Ts, ) Q10 )o Az, . F fpoj+ F

CA3,j Z j c( j+ 1 -X i 1 i sj+l

CA° u Hj+1 H0 j 1 .wo hfg o

++1x) }X+o h cCqajAz [(Fs(j + 1)- Fs(j))0 + (T )[0X o~j+l + (1 X o,jl + l-- 1f9T.H) •w~hfg ' cp q~ "9C 6

(204)

CD17 J = { [

F s,aJ CBqaj/Azji (205)

where fp is an adjustable factor for the fraction of power invested in steam production,

The new coefficients for Energy Conservation, used in subroutine COEFIW, are:

CDV2j =-" Q'f CAqaJAZj 2Fs (j+1)+F(J)3

((206)

±AaJ T zc. .6 5 J

+Ao + - Uo- AZi o .j+1 + 2Jo7j+1

CDV3,j Ao X5 1 +2 • o, 1+}(207)

' )Wohfg 3 A0 U0 j+1 ( 

+ 1 1 w o hf9 (1 •- 'I'•) s'j+l 1 -q • H oj

Ala~J T 1 )S~~ +2C2Ji)

= [Al.j - z [oXj+l 1.6 ?,,j+l - (X 1.6 XSJI]+ (C70,+, + c-o~j) CDV1j {Ao uo 2

(209)

_____jAz Q__0 F, (.j +1) -Fs (j){

-DV1 -J w h- hJ VJ Xwo 6

CDV12'J = AAJao uI +( )[ + (210

C-78
j XsjQ0)zz f,,F s;,+l C D V13j fj A[ u ° j+ l Hj _ -( yj ) Q h fg]- " j i I

AO U o0j+l Ho j 1 wo hfg 2 2

_ax + + .x (__ 1 _ 0 N(j+(1 A 1) DE w~12r a'- C C a'-' j LF 6 ()

(211)

2.7.2.3. LAPUR modifications

One subroutine, COEFIW, is modified.

Subroutine COEFIW

C LAPUR6 1:34

C NEW LINES

C

Q3/Q2*CSS*BlI(J)

QA =

QB =

QB1=

AVL (J) /AO

AO/AVCL (J)

AO/AVCL (Jl)

+2.*XI

CDV1 (J) = -CSS*TAUA*2/(QB1+QB)

CDV2 (J) = -CB16(J) + Q8*B9(J)*QA + 2.*VO(JI)

CDV3 (J) = Q8*B5(J)*QA + 2.*DVDBO(JI)

CDV4 (J) = CB15(J) + Q8*B7(J)*QA - COPP(J)*B3(J)

+ 2.*DVDHO(Jl)

CDV6 (J) = 2.*Q6*DVDBO(JI) + Q3/Q2*CSS*BI3(J)*QA

CDV7 (J) = 2.*VO(Jl) - 2./Q2

*QA

+ COPP(J)*CCQ(I)*Q7 + 2.*Q6*DVDHO(Jl)

CDV11(J) = Q3*BO(J)*QA + VO(J1) + VO(J) -

CDV12(J) = Q8*B6(J)*QA + DVDBO(JI) + DVDBO(J)

CDV13(J) Q8*B8(J)*QA - COPP(J)*B4(J) +

+ DVDHO(J) + Q9*Q7*CCQ(I)

CDV15(J) = Q6*DVDBO(Jl) + Q5*DVDBO(J)

+ Q3*CSS*BI4 (J)/Q2*QA

CDV16(J) = VO(JI) + VO(J) - Q3*CSS*BI2(J)/Q2*QA

+ Q6*DVDHO(JI) + Q5*DVDHO(J) - 2./Q2

Q9*CAQ (I)

DVDHO (Jl)

+2.*XI

C

IF(FLOWAREAVAR.EQ.1)THEN

CD1 (J) = CDV1 (J)

CD2 (J) = CDV2 (J)

CD3 (J) = CDV3 (J)

C-79
CD4 (J) = CDV4 (J)

CD6. (J) = CDV6 (J)

CD7 (J) = CDV7 (J)

CD11(J) = CDV11(J)

CD12(J) = CDVI2(J)

CD13(J) = CDV13(J)

CDI5(J) = CDV15(J)

CD16(J) = CDV16(J)

ENDIF

C LAPUR6 END

2.7.3. Momentum Conservation-LAPUR 6.0 Boiling Region Coefficients

The momentum conservation equations used by LAPUR 6.0 are explained in Sect. 2.5.2.

The new coefficient TVj, equation (112), is used in the frequency domain (subroutine

FREQ):

c u° A° -TAUIPV(J) (212)

gL 0 Aaj

One subroutine, FREQ, is modified.

2.7.3.1. LAPUR 6.0 modifications

Subroutine FREQ

C LAPUR6 I:1

C NEW LINE

COMMON/PERTH/CDBA

C LAPUR6 END

C LAPUR6 1:4

C NEW LINES

IF(FLOW AREA VAR.EQ.1)THEN

CDP(K) = CDP(K+I) + (CSS*TAUIPV(J)+C13(J))*CDYA(K)

> + C14 (J) *CDBA (K) - C15 (J)*CDY(K) - C16 (J)*CDB(K)

ENDIF

C LAPUR6 END

C LAPUR6 1:5

C NEW LINES

IF(FLOW AREA VAR.EQ.1)THEN

CDP(J) = CDP(J+I) + (CSS*TAUIPV(J)+CI3(J))*CDYA(J)

1 + C14(J)*CDBA(J) - C15(J)*CDY(J) - C16(J)*CDB(J)

2 + C21(J)*CDHA(J) - C22(J)*CDH(J)

ENDIF

C LAPUR6 END

C-80
2.8. OTHER MODIFICATIONS TO LAPURW 6.0

This section presents other modifications to LAPURW.

Subroutine MAIN-LAPUR 6.0 Version

C. LAPURW6.1BETA

C LAPUR6 M:I

C MODIFICATION

C WRITE(*,'('' LAPUR5W RUN TITLE :''/'' ** '',18A4,'' ** ' '

WRITE(*,'('' LAPUR6W RUN TITLE :''/'' ** '',18A4,'' **

C LAPUR6 END

C LAPUR6 M:I

C MODIFICATION

C WRITE(6,'(''I'',18A4,''*** LAPUR5W*** '',24X

WRITE (6,' 1'''18A4,''* LAPUR6W ** ''24X

C LAPUR6 END

C LAPUR6 M:I

C MODIFICATION

C WRITE (*,' (/'' ** USING RESULTS FROM LAPUR5X RUN ''

WRITE (*,'(/'' ** USING RESULTS FROM LAPUR6X RUN -''

C LAPUR6 END

Subroutine COEFIW-Variable flow area

C LAPUR6 1:2

C NEW LINES

INTEGER FLOWAREA VAR

DATA FLOWAREAVAR/ 1 /

C LAPUR6 END

Subroutine DRUMRl-New commons used; Intermediate storage

EQUIVALENCE (XDLl (1),DPIN1), (XFRI (1),IFFM(1))

> ,(XFR3(l),IFF),(XCTP(1),TPFM(l))

DIMENSION XDL1(N DLTPI)

DIMENSION XFRI(N FRICC)

DIMENSION XFR3(N FRICl)

DIMENSION XCTP(NCTPFM)

READ (NDISKI) XDLI

READ (NDISKI) XFRI

READ (NDISK1) XFR3

READ (NDISK1) XCTP

C-81
Subroutine FREQ-lf FLOWAREAVAR is defined as 1, the variable flow area formalism is

used.

C LAPUR6 1:2

C NEW LINES

INTEGER FLOW AREAVAR

DATA FLOWAREAVAR/ 1 /

C LAPUR6 END

Subroutine INPUTW-LAPUR 6.0 version

C LAPUR6 M:1

C MODIFICATION

C WRITE(6,'('' ERROR.

CASES''

WRITE(6,'('' ERROR.

CASES''

C LAPUR6 END

Subroutine TRANS-LAPUR 6.0 version

LAPUR5

LAPUR6

CAN NOT

CAN NOT

HANDLE MULTICORE

HANDLE MULTICORE

C LAPUR6 M:4

C MODIFICATION

C

FREQUENCY''

C >

C

FREQUENCY''

C >

FREQUENCY''

W(IW) /6.2835

FREQUENCY''

W(IW) /6.2835

C LAPUR6 END

WRITE(*,'('' RERUN LAPUR5W

'' SPACING AROUND'', F6.2,'' Hz'')

WRITE(6,' ('' RERUN LAPUR5W

SPACING AROUND'', F6.2,'' Hz'')

WRITE(*,'('' RERUN LAPUR6W

'' SPACING AROUND'',

WRITE(6,'('' RERUN LAPUR6W

SPACING AROUND'',

WITH TIGHTER

') W(IW)/6.2835

WITH TIGHTER

') W(IW)/6.2835

WITH TIGHTER

F6.2,'' Hz'')')

WITH TIGHTER

F6.2,'' Hz'')')

C-82
2.9. MODIFICATIONS IN SHARED FILES

This section presents the modifications made in the shared files (param.fi, XW_com.fi,

W_com.fi) relative to thermal hydraulic improvements.

File Param.fi

C LAPUR6 1:4

C NEW LINES

PARAMETER (NVAMAX 10) Max # of variable area channels

(>=10)

PARAMETER (NOVAMAX = 90) Max # of axial intervals channel

90)

PARAMETER (NLLMAX = 10) Max # of local loss channel types

(>=10)

PARAMETER (NOLLMAX = 90) Max # of axial intervals 11-

channel( 90)

C LAPUR6 END

C LAPUR6 M:l

C MODIFICATION

C PARAMETER (N CEF2 = 5 + NCENMAX + NOBMAXI)

PARAMETER (NCEF2 = 5 + NCENMAX + NOBMAX1 + NOBMAX)

C LAPUR6 END

C LAPUR6 M:1

C MODIFICATION

C > + NCEMAX + 4*NCEMAX*NCENMAX)

> + NCEMAX + 4*NCEMAX*NCENMAX

C LAPUR6 END

C LAPUR6 1:2

C NEW LINES

> + 2 + 2*NCHMAX + NVAMAX + 3*NVAMAX*NOVAMAX

> + NLLMAX + 2*NLLMAX*NOLLMAX)

C LAPUR6 END

C LAPUR6 M:1

C MODIFICATION

C PARAMETER (N NOHD = 2*NOTMAXI + 4*NOBMAX1 + NOBMAX)

PARAMETER (N NOHD = 2*NOTMAXl + 4*NOBMAXl + NOBMAX

C LAPUR6 END

C LAPUR6 1:2

C NEW LINES

> + 15*NOBMAX + NOBMAXl + 16*NNBMAX

> + 2*NNBMAXl)

C LAPUR6 END

C LAPUR6 M:1

C MODIFICATION

C-83
C > + 4*NCENMAX)

+ 4*NCENMAX

C LAPUR6 END

C LAPUR6 I:l

C NEW LINE

C LAPUR6 END

+ 2 + 3*NOVAMAX + 2*NOLLMAX)

C

C

LAPUR6 1:4

NEW LINES

PARAMETER

PARAMETER

PARAMETER

PARAMETER

C LAPUR6 END

File XWcom.fi

C LAPUR6 I:l

C NEW LINE

C LAPUR6 END

(NDLTP1 = 48 + 6*NOBMAX + 16*NNBMAX)

(N FRICC = 10*NFRMAX)

(N FRICi = 10)

(NCTPFM = 8*NOBMAX1)

,TAUIPV(NOBMAX)

C LAPUR6 1:17

C NEW LINES

COMMON /DLTP1/

C LAPUR6 END

DPINI

DPNB2

,DPACNB2

,DPCONB2

,DPFRSB1

,DPLLSB

,DPDHBB1

,DPEXBB

,DPR1

,DPDH1

,DPLL

,DPCON1

,EKCP1

,DPFRP1 (NOBMAXI)

,DPDHNB1(NNBMAX)

,DPEXNB(NNBMAX)

,DPLLP(NOBMAXI)

,DPDHIN1

,DPDHNB2

,DPEXNB2

,DPSB1

,DPACSB1

,DPCOSB

,DPFRBB1

,DPLLBB

,DPDHR1

,DPFRI

,DPEX

,DPCONRI

,DPTOT1

,DPACP1 (NOBMAXi)

,DPFRNB1 (NNBMAX)

,DPEXP (NOBMAXl)

,DPCONB (NNBMAX)

,DPFRIN1

,DPFRNB2

,DPLLNB2

,DPDHSB1

,DPEXSB

,DPBB1

,DPACBB1

,DPCOBB

,DPFRR1

,DPACl

,DPCO

,DPCONT1

,DPDHP1 (NOBMAXI)

,EKCRP1 (10)

,DPACNB1 (NNBMAX)

,DPLLNB(NNBMAX)

,DPCOP(NOBMAXl)

C LAPUR6 1:6

C NEW LINES

, NTDV

,NODV (NVAMAX)

,AVM (NVAMAX, NOVAMAX)

,NTDLL

,NODLL (NLLMAX)

,NDIMV (NCHMAX)

,DZVM (NVAMAX, NOVAMAX)

,DEVM (NVAMAX, NOVAMAX)

,NDIMLL (NCHMAX)

,DZLLM (NLLMAX, NOLLMAX)

C-84
,EKLLM (NLLMAX, NOLLMAX)

C LAPUR6 END

C LAPUR6 1:16

C NEW LINES

C LAPUR6 END

C LAPUR6 1:3

C NEW LINES

, DEVL (NOBMAX)

,HVL (NOBMAX)

,RHOVL (NOBMAX)

,DEVNBL (NNBMAX)

,HVNBL (NNBMAX)

,RHOVNBL (NNBMAX)

,EKFNBL (NNBMAX)

, AVNBCL (NNBMAX1)

,DEVNBCL (NNBMAX1

,EKLLNBL (NNBMAX)

, FNBL1 (NNBMAX)

,FL1(NOBMAX)

,A2NBL1 (NNBMAX)

,A2L1 (NOBMAX)

,RHOVNBCL (NNBMAX

,EKLFNB (NNBMAX)

,NV

,DZV (NOVAMAX)

,NLL

,AVL (NOBMAX)

TVL (NOBMAX)

,UVL (NOBMAX)

, AVNBL (NNBMAX)

•EMUVL (NOBMAX)

, REVL (NOBMAX)

,TVNBL (NNBMAX) ,EMUVNBL (NNBMAX)

,UVNBL (NNBMAX) ,REVNBL (NNBMAX)

,AVCL (NOBMAXI)

),DEVCL (NOBMAXI)

,EKLLL (NOBMAX)

,FNBL2 (NNBMAX)

•FL2 (NOBMAX)

FNBL (NNBMAX)

FFL (NOBMAX)

,A2NBL2 (NNBMAX),A2NBL (NNBMAX)

,A2L2 (NOBMAX) ,A2L(NOBMAX)

,EKLF (NOBMAX)

,AV (NOVAMAX) ,DEV (NOVAMAX)

,DZLL (NOLLMAX) ,EKLL (NOLLMAX)

C LAPUR6 END

C

C

LAPUR6 1:13

NEW LINES

COMMON /FRICC/ IFFM(NFRMAX)

> , CNM (NFRMAX)

> ,IFMM(NFRMAX)

> ,AJUSTAAlM(NFRMAX

COMMON /FRICl/ IFF

,ANM(NFRMAX) ,BNM(NFRMAX)

,DNM(NFRMAX) ,RRM(NFRMAX)

,AJUSTAClM (NFRMAX)

,AJUSTABlM (NFRMAX)

,AN ,BN

> ,CN ,DN , RR

> ,IFM ,AJUSTAC1

> ,AJUSTAAl ,AJUSTAB1

COMMON /CTPFM/ TPFM(NOBMAX1)

> DTPFMDYO (NOBMAX) , DTPFMDHO (NOBMAX1)

> ,DTPFMDBO (NOBMAXI) ,DOMDH01 (NOBMAXI)

> ,HMO(NOBMAX1)

> ,DHMDBO(NOBMAX1) ,DHMDHO(NOBMAX1)

C LAPUR6 END

File W com.fi

C LAPUR6 1:5

C NEW LINES

C-85
•CTAUNBV (NOBMAX)

, CDVI (NOBMAX)

,CDV4 (NOBMAX)

,CDVII (NOBMAX)

CDV15 (NOBMAX)

,CDV2 (NOBMAX)

CDV6 (NOBMAX)

,CDVI2 (NOBMAX)

CDVI 6 (NOBMAX)

, CDV3 (NOBMAX)

, CDV7 (NOBMAX)

, CDV13 (NOBMAX)

, CDV22 (NOBMAX)

C LAPUR6 END

C LAPUR6 I:.2

C NEW LINES

C LAPUR6 END

,CTAUNBV, CDV1,CDV2,CDV3,CDV4,CDV6,CDV7

,CDV1I,CDV12,CDVI3,CDVI5,CDVI6,CDV22

C-86
CHAPTER 3. PROGRAMMING ERRORS

3.1. PROGRAMMING ERRORS CORRECTED IN VERSION 5.2

The programming errors found in LAPUR 5.1 and corrected in LAPUR 5.2 are:

Subroutine W ONECOR.for

Wrong CBD and CCD calculation

Old Logic:

CBD=CAD+NCH(IX)*( CDYX(IX) *CMX(IX)+CDQX(IX) )*Ql

CCD=CAD+NCH(IX)*( CDYX(IX)*CNX(IX)+CDTX(IX) )*Ql

Corrected Logic:

CBD=CBD+NCH(IX)*C. CDYX(IX)*CMX(IX)+CDQX(IX) )*Ql

CCD=CCD+NCH(IX)*( CDYX(IX)*CNX(IX)+CDTX(IX) )*Qi

3.2. PROGRAMMING ERRORS CORRECTED IN VERSION 6.0

Three miscellaneous errors were found in the subroutine COEFIW (Wcoef.for) of LAPUR

5.2 and were corrected in version 6.0.

Subroutine COEFIW

C LAPUR6 M:l

C MODIFICATION

C Miscellaneous error

C Q8 = (2.*FS02(J)

Q18= (2.*FS02(J)

+ FS01(J))/3.*COPP(J)

+ FS01(J))/3.*COPP(J)

C LAPUR6 END

C LAPUR6 M:2

C MODIFICATION

C Miscellaneous error

C CB15(J)

C CB16(J)

CB15(J)

CB16(J)

C LAPUR6 END

= Q8*CCQ (I) *Q7

= Q8*CAQ (I)

= QI8*CCQ(I)*Q7

= QI8*CAQ(I)

C-87
REFERENCES

1. IBERINCO Especificaciones para las tareas 1 y 2 del proyecto DROP. 22212-GN10XN-IN02.000047.00008. Feb. 2002

2. Escriv, A., J. March-Leuba, LAPUR 5.2 Verification and User's Manual. Report

NUREG/CR-6696, ORNL/TM-2000/340 (2000).

3. Linford, R. B. Analytical methods of plant transient evaluation for the General Electric Boiling

Water Reactor, NEDO-10802, Feb. 1972.

4. Muhoz-Cobo, J. L., 0. Rosell6, R. Mir6, A. Escriv6, D. Ginestar, G. Verdo, "Coupling of

density wave oscillations in parallel channels with high order modal kinetics: application to

BWR out-of-phase oscillations." Annals of Nuclear Energy, Vol. 27, pages 1345-1371.

(2000).

5. OECD NEA, State of the Art Report on Boiling Water Reactor Stability (Draft), March 1996.

6. Otaduy, P. J., J. March-Leuba. LAPUR User's Guide. NUREG/CR-5421.

7. Otaduy, P. J. Modeling of the Dynamic Behavior of Large Boiling Water Reactor Cores.

Ph.D. dissertation, University of Florida, 1979.

8. Hetsroni, G. Handbook of Multiphase Systems, pg. 2-59. Hemisphere Publishing

Corporation, 1982.

9. Lahey, R. T. Jr. and F. J. Moody. The Thermalhydraulics of a Boiling Water Nuclear Reactor

2nd Edition, pg. 277. American Nuclear Society, 1993.

10. J. P. Waggener, "Friction factors for pressure drop calculations." Nucleonics, Vol. 19,

No. 11.1961.

11. J. A. Borkowski et al., TRAC-BF1/MODI Models and Correlations. NUREG/CR-4391, 1992.

C-88
NOMENCLATURE

A

a 2

Cp

Cp.av

DH

DZ

f

fMfp

F,

G

g

H -hF -hI

hfg

h

hfg

Kc

KF

KL

L

m

N

n

NNB

Ncon

Nexp

NNB

P

Pw

Q'

Rr

Re

S

t

U

w

XF

x

w

y=-

Wo

z

ZNB

flow area

Reynolds's number constant

specific heat capacity at constant pressure

average specific heat of water in the NBR

hydraulic diameter

cell length

friction factor

Moody friction factor

.adjustable factor for the fraction of power invested in steam production

fraction of energy invested in steam formation

mass flux

gravitational acceleration

normalized degree of sub-cooling

enthapyenthalpy vaporization enthalpy

contraction coefficient for pipes (input user data card 43)

contraction coefficient at the inlet of the channels (input user data card 10)

crud-deposition friction multiplier

local-loss coefficient

channel length

cell

total boiling cells

cell edge

total nonboiling cells

total number of contractions in the channel

total number of expansions in the channel

total nonboiling cells

pressure

wet perimeter

energy per unit lenght supply by the fuel

relative roughness

Reynolds number

Laplace transform

time

velocity

mass flow rate

flow quality used in the Chisholm-Baroczy two-phase friction multiplier

steam quality

normalized mass flow rate

'height

nonboiling bulk-boiling boundary location

C-89
Greek symbols

(X void fraction

Az cell length

A increment operator

5Kr Kronecker delta

2CB Chisholm-Baroczy two-phase friction multiplier

•MN Martinelli-Nelson two-phase friction multiplier

P

g PFL

Y

P

PF

0

6q'

6y

Subscripts

0

a

F

§

i

in

NB

ZNB

Zi

bubble decay constant

normalized pressure

slip ratio

dynamic viscosity

density

Jones two-phase friction multiplier correction factor

relative density decrement in the evaporation process

angle

normalized total power generation perturbation in the fuel

liquid coolant temperature perturbation

normalized coolant flow rate perturbation

steady-state or reference conditions

averaged value

saturated conditions

steam phase

nonboiling cell

inlet conditions

boiling cell

liquid phase

nonboiling region

nonboiling bulk-boiling boundary location

nonboiling bulk-boiling boundary location

C-90
NRC FORM 335 U.S. NUCLEAR REGULATORY COMMISSION 1. REPORT NUMBER

(9-2004) (Assigned by NRC, Add Vol., Supp., Rev.,

NRCMD 3.7 and Addendum Numbers, if any.)

BIBLIOGRAPHIC DATA SHEET NUREG/CR-6958 ORNUiTM-2007/233

(See instructions on the reverse)

2. TITLE AND SUBTITLE 3. DATE REPORT PUBLISHED

LAPUR 6.0 User's Manual MONTH YEAR

10 2008

4. FIN OR GRANT NUMBER

JCN J3319

5. AUTHOR(S) 6. TYPE OF REPORT

Alberto Escriv6 Technical

Jos& Luis Mufioz Cobo

JosO Melara San Roman 7. PERIOD COVERED (Inclusive Dates)

Manuel Albendea Darriba

Jos6 March-Leuba

8. PERFORMING ORGANIZATION - NAME AND ADDRESS (If NRC, provide Division, Office or Region, U.S. Nuclear Regulatory Commission, and mailing address; if contractor,

provide name and mailing address.)

Oak Ridge National Laboratory

PO Box 2008

Oak Ridge, TN 37831

9. SPONSORING ORGANIZATION - NAME AND ADDRESS (ff NRC, type "Same as above"; if contractor, provide NRC Division, Office or Region, U.S. Nuclear Regulatory Commission,

and mailing address.)

Division of Safety Systems

Office of Nuclear Reactor Regulation

U.S. Nuclear Regulatory Commission

Washington, DC 20555-0001

10. SUPPLEMENTARY NOTES

Dr. T. L. Huang, Project Manager

11. ABSTRACT (200 words or less)

This report documents a series of programming upgrades to Version 6.0 of the LAPUR code. LAPUR, a computer program in

FORTRAN, is a mathematical description of the core of a boiling water reactor. Its two linked modules, LAPURX and LAPURW,

respectively solve the steady-state governing equations for the coolant and fuel and the dynamic equations for the coolant, fuel,

and neutron field in the frequency domain. The main upgrade in LAPUR 6 is the ability to model part-length fuel rods and fuel

spacers explicitly. General implementation descriptions are followed by a detailed description of input and output parameters of

LAPURX and LAPURW. Sample inputs are included and stability benchmarks are noted.

12. KEY WORDS/DESCRIPTORS (List words or phrases that will assist researchers in locating the report.) 13. AVAILABILITY STATEMENT

LAPUR unlimited

BWR 14. SECURITY CLASSIFICATION

BWR Stability rThis Page)

unclassified

(This Report)

unclassified

15. NUMBER OF PAGES

16. PRICE

NRC FORM 335 (9-2004) PRINTED ON RECYCLED PAPER
R P i orm.d

Federa Recycling Progrmm
UNITED STATES

NUCLEAR REGULATORY COMMISSION

WASHINGTON, DC 20555-0001

OFFICIAL BUSINESS