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ANL-83-75 ANL-83-75
USER'S MANUAL FOR THE SOD1UM-WATER
REACTION ANALYSIS COMPUTER CODE SWAAM.II
by
Y. W Shin, C. K. Youngdahl, H. C. Lin,
B. J. Hsieh, and C. A. Kot
4404 n Pe '0.4ot"
APPLIED TECHNOLOGY
Any further distribution by any holder of this document or of the datatherein to third parties representing foreign interests, foreign governments.foreign companies and foreign subsidiaries or foreign divisions of U. S.companies should be coordinated with the Deputy Assistant Secretary forBreeder Reactor Programs, U. S Department of Energy.
ARGONNE NATIONAL LABORATORY, ARGONNE, ILLINOIS
Operated by THE UNIVERSITY OF CHICAGOfor the U. S. DEPARTMENT OF ENERGYunder Contract W.31-109-Eng•38
DISCLAIMER
This report was prepared as an account of work sponsored by anagency of the United States Government. Neither the UnitedStates Government nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied, or assumesany legal liability or responsibility for the accuracy, com-pleteness, or usefulness of any information, apparatus, product,or process disclosed, or represents that its use would not infringeprivately owned rights. Reference herein to any specific com-mercial product, process, or service by trade name, trademark,manufacturer, or otherwise, does not necessarily constitute orimply its endorsement, recommendation, or favoring by theUnited States Government or any agency thereof. The views andopinions of authors expressed herein do not necessarily state orreflect those of the United States Government or any agencythereof.
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Distribution Categories:LMFBR--Structural Materials andDesign Engineering: AppliedTechnology (UC-79Th)
LMFBR--Safety: AppliedTechnology (UC-79Tp)
ANL-83-75
ARGONNE NATIONAL LABORATORY9700 South Cass Avenue
Argonne, Illinois 60439
USER'S MANUAL FOR THE SODIUM-WATERREACTION ANALYSIS COMPUTER CODE SWAAM -II
by
Y. W. Shin, C. K. Youngdahl, H. C. Lin,B. J. Hsieh, and C. A. Kot
Components Technology Division
August 1983
APPLIED TECHNOLOGY
Any further distribution by any holder of this document or of the data thereinto third parties representing foreign interests, foreign governments, foreigncompanies and foreign subsidiaries or foreign divisions of U. S. companiesshould be coordinated with the Deputy Assistant Secretary for Breeder ReactorPrograms, U. S. Department of Energy.
Page
1
1
ABSTRACT
I. INTRODUCTION
TABLE OF CONTENTS
II. GENERAL DESCRIPTION OF CODE STRUCTURE AND USER OPTIONS 3
A. Physical Systems and Phenomena to be Modeled 3
B. Main Modules of SWAAM-II and Their Interactions 4
C. Main Options for Use of SWAM-II 8
III. THEORETICAL FOUNDATION OF SWAMI-II 10
A. Water-Side System Module 10
1. Field Equation Solution by the Two-stepLax-Wendroff Scheme 11
2. Junction and Boundary Condition Solution by theIntegral Method of Characteristics 12
B. Reaction Zone Analysis Module 14
1. Set of Governing Equations 152. Closure of the Governing Equations Set 233. Computation of the Equation Set 24
C. Sodium-Side Modeling 29
1. One-Dimensional Sodium System Dynamics Module 292. Relief System Filling Module 403. Two-Dimensional Sodium Flow Module 44
D. Structural Dynamics and Fluid-Structure Interaction 47
1. Elastoplastic Rupture Disk Dynamics Module 472. Fluid-Structure Interaction Scheme at Rupture Disk 513. Coupling Models for Double-Disk Assemblies 534. Shell Dynamics 56
E. Fluid Property Calculations 57
1. Water 572. Nitrogen Gas 613. Liquid Sodium 61
IV. INPUT DESCRIPTION 63
A. Input Data 63
iii
TABLE OF CONTENTS (Coned)
lags
1. Input Data for2. Input Data for3. Input Data for
Calculation) 4. Input Data for
Run A (Sodium-Side) 63Run B (Water-Side) 77Run C (Two-Dimensional Sodium
81
Run D (Shell Deformation) 83
B. Notes on Input Data and System Modeling
86
1. General 86
2. RUNA (SODSID) 87
3. RUNB (WATSID) 92
V. BRIEF SUMMARY OF SWAAM-II VALIDATION 95
A. Validation Using LLTR Data 96
B. Validation Using SWAT-3 Data 100
VI. ARRAY SIZES AND ALTERATIONS TO PROGRAM STORAGE 106
VII. CONCLUDING REMARKS 112
ACKNOWLEDGMENTS 113
REFERENCES 114
v
LIST OF FIGURES
Figure
1 Major Modules of SWAAM-II 4
2 Grid System for Interior Procedure 11
3 Break Boundary 13
4 Solution Methodology for Break Boundary 13
5 Water-Side System Junction and Boundary Conditions 14
6 Stoichiometric Coefficients Diagram for the General Reaction
Equation 18
7 Reaction-Bubble/Sodium Response Coupling Model 20
8 Typical Bubble-Size Histories 23
9 Time Step Management and Module Interaction Schemes 29
10 Finite-Difference Grid 32
11 Finite-Difference Grid for Boundary Node 35
12 End-Node Characteristics 41
13 Characteristic Cone and Mesh Net. Bicharacteristics 1P, 2P,3P, and 4P are the Integration Paths 46
14 Finite-Difference Grid at Rupture Disk Boundary for EqualTime Steps
51
15 Numerical Treatment of Rupture Disk Boundary for UnequalTime Steps
53
16 Region Boundaries for Approximate Computation of Water
Properties 59
17 Schematic Diagram of LLTR Series-II Test Facility 96
18 CRBRP Prototype Rupture Disk Assembly 97
19 SWAAM-I Model for LLTR Series-II Tests 98
20 Validation Results for LLTR Series-II Test Arl 99
21 Validation Results for LLTR Series-II Test A72 101
22 SWAT-3 Facility Schematic Flow Diagram 102
23 SWAT-3 Water-Injection System 103
LIST OF FIGURES (Cont'd)
Figure
24 Rupture-Sleeve Design of SWAT-3 Injection System 103
25 SWAAMHI Model for SWAT-3 Run-6 TEST 104
26 Comparison of Early Pressure History of SWAT-3 Run-6 at P1113with SWAAM-I Prediction (A = 2.6, B = 0.65, W = 20.60 105
27 Combined Effects of A and the Early Leak Rate on Pressure106History of SWAT-3 Run-6 at P1113
TABLES
Table I'Agft
1 Phase Changes of Reaction Products 20
2 Expressions Used to Represent Internal Energy and SpecificVolume at Region Boundaries 60
3 Array Size Limitations for Sodium-Side Computation (RUNA = T).. 107
4 Array Size Limitations for Water-Side Computation (RUNE = T)... 109
5 Array Size Limitations for Two-Dimensional Sodium-Side(RUNC = I) and Shell Dynamics (RUND = T) 110
6 Sharing of Labeled COMMON Among Sodium-Side Subroutines(RUNA)
111
7 Sharing of Labeled COMMON Among Water-Side Subroutines(RUNE) 111
vi
I
USER'S MANUAL FOR THE SODIUM-WATERREACTION ANALYSIS COMPUTER CODE SWAAM-II
by
Y. W. Shin, C. K. Youngdahl, H. C. LinB. J. Hsieh, and C. A. Kot
ABSTRACT
The computer program SWAAM-II performs analysis of the
transient flow, the coupled bubble dynamics, and the fluid-
structure interaction for the early wave-propagation effects
resulting from a large scale sodium-water reaction in an LaBR
steam generator system and the intermediate heat transport
system. The first production version, SWAAM-/ (issued in 1979),
contains code capabilities suitable for analysis of the CRBR
system and the Atomics International straight-tube steam
generator design. SWAM-II is a more recent version that
includes new code capabilities developed for post-CUR
applications, including the National Large Scale Prototype
Breeder and the helical-coil-tube steam generator design.
SWAMI-II also includes all improvements and error corrections
made since the first issuance of the SWAM-I code. This user's
manual contains the governing equations on which the various
constituent models are based, the input data description needed
to run the program, and the status of the code validation to
date. The report also discusses additional needs for
development of new code capabilities in anticipation of future
design requirements.
I. INTRODUCTION
Tube failure in an LIKFBR steam generator can result in a water/steam
leak flow contacting the liquid sodium, producing an exothermic chemical
reaction with sudden generation of a large amount of hydrogen gas. The
pressure pulses thus produced can exert large forces on the structural
members. The design of the steam generator system and the intermediate heat
transport system (IHTS) therefore must consider the effects of potential
sodium-water reactions to ensure structure/ integrity and, further, to
provide means for mitigating the pressure effects. A large class of events
covering the entire spectrum of possible scenarios must be considered. The
most important consideration in the definition of a sodium-water reaction
2
event, however, is the one that defines the amount of water leak, commonly
referred to as the design-basis leak (DBL). DBLs associated with the
hypothetical sudden break of tubes, called double-ended-guillotine (DEG)
breaks, are generally considered the most severe event - the one that
generates the highest possible pressure loadings. It is this type of large
leak event for which the SWAAM-II code is designed. The SWAAM-II code is
based on rigorous modeling of the leak flow blowdown, the fluid hammer
effects in the liquid sodium, the interactive dynamics of the sodium-water
reaction and the hydrogen bubble growth, and the fluid-structure interaction
in the IHTS piping and the steam generator shell. The emphasis in SWAAM-II
is placed on the initial wave propagation effects of the flow transients and
the associated fluid-structure interaction, where the time domain of
applicability generally is less than one second for typical system scales.
As part of the National Steam Generator Development Program at the
U. S. Department of Energy, Argonne Components Technology Division initiated
development of the series of SWAAM (Sodium Water Advanced Analysis Method)
codes in the early 1970s. Emphasis was first placed on analysis of the
short-term wave propagation phase of the sodium-water reaction event.
Various independent modules were developed and were then integrated into the
first production version SWAAMHI Code 11], issued in 1979. Shortly
thereafter, SWAAM-I was installed at various steam generator vendor
organizations where it has remained operational. SWAANHI capabilities were
oriented toward the Clinch River Breeder Reactor and the straight-tube steam
generator (Atomics International's design).
After the issuance of SWAAM-I, development of additional codecapabilities continued. Many new capabilities were needed for designanalysis of large reactor designs in the National LSPB Program. Vendor
experience with SWAAM-I had revealed the need for certain code improvements
and development of various user-convenient features. Application of SWAAM-I
to the helical-coil-tube steam generator system also necessitated additional
code features. Validation of SWAAM-I treatment with respect to the cover-
gas space in the helical-coil-tube steam generator design was performed
using SWAT-3 data. Other code features were validated using the Large LeakTest Rig (LLTR) Series-II data. It was felt desirable to update thedocumentation of the SWAAM-I code with respect to these new developments.
This report describes the second production version of the code,
denoted by SWAAM-II. Section II describes the general code structure andthe various options for use of the code. The input requirements for use ofSWAAM-II are given in Section II/. Section IV highlights the theoretical
bases of the various constituent code modules and the coupling between the
3
modules. A summary of the extent of SWAAM-II code validation is presented
in Section V. The changes in array sizes in COMMON and DIMENSION statements
needed to treat larger systems or reduce storage requirements are described
in Section VI. Finally, the conclusions of the report are presented in
Section VII.
II. GENERAL DESCRIPTION OF CODE STRUCTURE AND USER OPTIONS
A. Physical Systems and Phenomena to be Modeled
SWAAMHII is intended to analyze the pressure pulse propagation in ISM
piping systems resulting from instantaneous failure of a steam generator
tube. The systems and components involved in the transient event include
the faulted steam generator, the intermediate heat transport system (IHTS),
rupture disks mounted on the IHTS piping or steam generator, a sodium-water
reaction products (SWIP) relief system connected to the rupture disks, and
the steam system piping that feeds the broken tube.
The physical phenomena modeled include
Thermochemical dynamics of the sodium-water reaction, including phasechanges of the reaction products,
Propagation of rarefaction waves through the steam system caused by thesudden depressurization at the break, including associated phase changes ofthe water and the dynamic coupling with the reaction products bubblepressure,
Pressure-pulse propagation in the sodium in the faulted steam generator andIHTS resulting from the bubble expansion, including the effects ofcavitation and inelastic deformation of the piping,
Dynamic deformation and failure of the rupture disks, including largegeometry changes, inelastic strains, and coupling to the sodium dynamics,
Filling of the relief system piping, with coupling of the wave propagationin the filling system to that in the IHTS, and
Dynamic deformation of the steam generator shell caused by the expandingreaction products bubble and pressure pulses in the sodium.
Gross motions of the piping caused by the transients in the water and sodium
systems along with the associated feedback effects are not modeled in SWAAM-
The steam generator is assumed to be long relative to its diameter so
that the initially spherical reaction products bubble becomes piston-shaped
as it grows. Angular variations of the pressure and velocity fields in the
steam generator are ignored, so the analysis is two-dimensional at most.
SODIUMSYSTEM
DYNAMICS
RUPTUREDISK
DYNAMICS
STEAM GENERATORSHELL
DYNAMICS
RELIEFSYSTEMFILLING
4
The rupture disks are assumed to be the spherical cap type, and either
single membrane or double membrane disk assemblies may be modeled.
Computational models for fluid transient interactions with many types
of junctions and boundaries are included for both the sodium and water
systems. Modeling of components with complex flow passages, such as the
intermediate heat exchanger (IHX), is left to the judgment of the user, but
SWAAM-II contains a variety of input options intended to facilitate the
construction of complex models from one-dimensional computational elements.
B. Main Modules of SWAAM-II and Their Interactions
SWAAMHII consists of eight major modules (Fig. 1) that interact to
analyze the effects of a large leak event in an LMFBR steam generator
system. By "module" we mean a set of subroutines that can be grouped
together conveniently to perform one specific aspect of the total
analysis. In addition to operating together, most of the modules can be
conveniently run separately or in subgroups to enable the user to
concentrate on parts of the analysis or system.
STEAMTABLE
WATERSYSTEM
DYNAMICS
TWOSODIUM
DYNAMICS
SODIUM/ WATERREACTION AND
BUBBLE DYNAMICS
Fig. 1. Major Modules of SWAMI-II
5
The water system dynamics module computes the transient two-phase flow
of water in the steam generator tubes and steam system piping resulting from
a sudden double-ended-guillotine (DEG) break in a tube. The initial
condition of the water in the system can be subcooled water, a steam/water
mixture, or superheated steam, and it need not be uniform. The rapid
depressurization of the system produces space and time dependent blowdown of
water/steam into the reaction zone. A model of the water system is
constructed by the user from one-dimensional tubes and a variety of junction
types. A finite-difference technique based on the two-step Lax-Wendroff
scheme is used for the calculations at interior points, and a method of
characteristics technique is used to compute the solution at junctions. The
needed thermodynamic properties and their derivatives are obtained from the
steam tables module. The interaction of the water system module and the
reaction zone module is coupled because the water flow rate determines the
rate of energy release and production of gaseous products, while the
pressure in the resultant bubble influences the water blowdown.
The steam tables module is based on a formulation by Keenan et al. [21
that is used to compute thermodynamic properties at various points on the
saturation line. Cubic splines are then used to approximate property values
at intermediate points on the solution line, and computations in the two-
phase region are performed in terms of values for liquid and vapor.
Thermodynamic properties in the subcooled liquid or superheated vapor
regions are determined by using a transfinite interpolation technique. This
combination of methods gives accurate results for a small computational
effort. Various combinations of properties can be used as independent and
dependent variables as needed by the water system dynamics module. SWAM-II
also contains nitrogen gas subroutines for analysis of simulation tests
where nitorogen injection is used in place of water injection.
The sodium-water reaction and bubble dynamics module computes the
thermochemical reaction of the water leaving the broken tube with the sodium
in a steam generator and the mechanical interaction of the resultant
reaction products bubble with the sodium and water systems. The reaction
calculation takes into account the various possible combinations of reaction
products, phase changes of these reaction products, and consumption of
sodium at the bubble interface. The bubble temperature is computed from the
energy balance, rather than being an input parameter. The reaction bubble
has a spherical shape initially, but is converted to a cylindrical shape as
it grows in the steam generator shell. The ordinary differential equations
governing the chemical reaction and bubble growth are coupled to the rate of
water Injection as computed in the water dynamics module and to the dynamics
of the sodium system as determined by its inertia and compressibility. The
6
bubble module also makes use of the steam tables module because the energy
contained in the unreacted water or steam remaining in the bubble is
accounted for in the bubble energy balance.
The sodium dynamics module computes the pressure transient in the
sodium in the steam generator and IHTS resulting from the sudden growth of
the reaction products bubble. It uses the one-dimensional method of
characteristics applied to a fluid hammer formulation. Cavitation produced
by rarefaction waves in the system is treated using a column separation
technique. The effect of elastic-plastic deformation of pipe cross-sections
on local wave speed is accounted for. Pipe friction, gravitational effects,
and convective velocity terms are included in the formulation. Because of
the variety of junction types and internal options available, the user can
model piping system configurations of any desired complexity and the
internal flow passages of system components. Sodium properties, piping
material properties, and pipe friction factors are provided by subroutines
in the module. The sodium dynamics module is coupled to the bubble dynamics
module, the rupture disk dynamics module, and the two-dimensional sodium
dynamics module.
The rupture disk dynamics module computes the dynamic elastic-plastic
deformation of single or double membrane spherical-cap rupture disks in
response to pressure transients in the sodium system. A rupture disk
membrane is assumed to fail when the buckled membrane touches the knife edge
behind it. Several options are available for computing the interrelated
failures of double membrane disks. A corotational finite element method is
used to compute the dynamic response. The pressure and velocity of the
sodium at the disk are coupled to the disk forces and motion, and cavitation
at the interface is computed if the membrane pulls away from the fluid.
After the rupture disk breaks, the sodium begins to flow through the opening
and the relief system filling module is activated. The rupture disk
dynamics module has the option of using an instantaneous disk model, where
the disk fails at a prescribed pressure or time.
After a rupture disk failure, the relief system filling module computes
the filling of the relief system pipes and the transient pressure waves in
the filled part of the system. The same numerical method is used as in the
one-dimensional sodium system dynamics module with a special treatment given
to the end of the moving fluid column. The pressure transient calculation
in the two modules is thereby completely coupled. Multiple relief systemscan be modeled; however, they may not intersect because the current version
of SWAAM-II does not allow for sodium entering a pipe from both ends.
7
The two-dimensional sodium dynamics module provides detailed treatment
of the region near the reaction sone in a faulted steam generator. A method
of bicharacteristics applied to • fluid hammer formulation is used to obtain
the solution for pressure transient propagation in a two-dimensional grid.
Spatial details of the pressure field around the bubble and at the steam
generator shell are computed in a finite-length region by the two-
dimensional sodium dynamics module for a given bubble pressure and volume.
Coupling of the bubble to the one-dimensional sodium system dynamics
calculation through an intermediary two-dimensional region was considered
and examined, but the spatial resolution required during the early part of
the transient, when the bubble is still small, necessitated a subgrid
structure ouch smaller than the normal grid. Therefore, this coupling
scheme was abandoned and a direct coupling of the bubble to the one-
dimensional sodium dynamics was adopted. However, coupling of the one-
dimensional and two-dimensional domains at their interfaces at the ends of
the two-dimensional region is an option.
The steam generator shell dynamics module computes the dynamic elastic-
plastic deformation of the shell or flow shroud produced by the time-
dependent pressure field in the faulted steam generator. A method-of-
characteristics technique is used to solve an endochronic theory of
viscoplasticity formulation of the shell response. Either one-way or two-
way coupling is available between the two-dimensional sodium dynamics module
and the shell deformation module.
As indicated in Fig. 1, two way coupling is available between all the
dynamics modules in SWAAM-II. The modules for water system dynamics,
sodium-water reaction and bubble dynamics, sodium system dynamics, rupture
disk dynamics, two-dimensional sodium dynamics, and shell dynamics all have
their intrinsic time steps. However, only one of these is input by the
user, and the others are set internally by the code to accommodate
compatible solutions at the interfaces between modules. For example, the
time step for the sodium system module may be typically thirty-five times
larger than the time step for the rupture disk module coupled to it. The
calculation for the fluid at the interface is then divided into thirty-five
substeps to permit a coupled interaction with the disk dynamics.
As discussed below, the modules may be run in a variety of combinations
to enable the user to concentrate on various aspects of the large leak
effects problem. The input requirements are arranged so that it is not
necessary to provide input data for modules that are not being used. Node
spacings, consistent time steps, wave speeds, friction factors, and fluid
and structural material properties are all computed internally.
8
C. Main Options for Use of SWAAWII
SWAAM-II is grouped for operational purposes into four main options,
denoted as RUNA, RUNB, RUNC, and RUND, which may be run independently or in
various combinations. The modules contained in these options are as
follows:
RUNA (SODSID) includes the sodium-water reaction and bubble
dynamics module, the one-dimensional sodium system dynamics
module, the rupture-disk-dynamics module, the relief system
filling module, and the steam-tables module.
RUNE (WATSID) includes the water system dynamics and steam tables
modules.
RUNC (NA2D) is the two-dimensional sodium dynamics module.
RUND (SHELL) is the steam-generator shell dynamics module.
The most general run is activated by RUNA = RUNE = RUNC = RUND = T,
which uses all the SWAAM-II modules. Another important combination is given
by RUNA = RUNE = T and RUNC = RUND = F; this omits the two-dimensional
treatment in the faulted steam generator and its shell deformation, but
includes the sodium-water reaction and the dynamics of the water and sodium
systems.
If RUNA = T and RUNE = F, a sodium-system transient is computed without
a water-side calculation. A prescribed water injection rate history can be
input to the sodium-water reaction calculation in RUNA in place of the RUNB
computation, if desired. Alternatively, the sodium-water reaction can be
omitted by not specifying any bubble junction in the sodium system;
prescribed pressure histories then can be input at various points in the
sodium system to provide pulse sources for the transient calculation.
Each of the modules in RUNA can be run independently or with minimal
use of some of the other modules. The sodium system dynamics module can be
operated as a standard fluid hammer code by omitting any bubble junctions.
Not specifying any relief system piping eliminates the use of the relief
system filling module, and not specifying any dynamic rupture disk junctions
eliminates the use of the rupture disk dynamics module. The option is
included to use water properties rather than sodium properties if a
waterhammer calculation is desired, e.g., to model a water loop simulation
of a sodium system. The rupture disk dynamics module can be exercised if a
9
sodium system of at least one pipe is specified. Exercising the relief
system filling module requires a rupture disk and at least a one-pipe sodium
system.
The water system dynamics can be analyzed separately by choosing RUNB •
T and RUNA RUNC • RUND F. This case corresponds to a system blowdown
problem with a prescribed constant bock pressure.
RUNC T may be opted without RUM or RUNB to compute the two-
dimensional detals of the pressure distribution around the bubble for an
input bubble history. When this option ie chosen, RUND may be included to
compute the associated shell response also.
Finally, RUND • T may be run singly to obtain the shell response to a
prescribed pressure loading history typical of a sodium/water reaction
event.
The sodium/water reaction and bubble dynamics module is treated as a
junction condition of the sodium system dynamics module and cannot be used
singly. However, only a minimal sodium system need be included. A minimal
water system also can be included or replaced by a prescribed water
injection rate history.
SWAANHI uses three additional logical parameters to provide
supplemental options:
REACT: If REACT - T, the leak flow is water/steam that activates
the sodium-water reaction in the bubble-dynamics routine. If REACT
F, the injected fluid is nitrogen gas to model a gas-driven event
simulation.
CALINK: Link parameter between RUNA (SODSID) and RUNC (NA2D). If
CALINK T, SODSID and NA2D are coupled through boundary conditions
at both ends of the two-dimensional region. If CALINK • F, NA2D uses
nonreflective boundary conditions at the ends.
CDLINK: Link parameter between RUNC (NA2D) and RUND (SHELL). CDLINK
T provides a two-way coupling between NA2D and SHELL. Shell motion
is fed back to the sodium transient. CDLINK F gives a one-way
coupling. Sodium pressure is supplied from NA2D to SHELL to compute
the shell response, with no feedback of shell motion to NA2D.
10
Use of the seven logical parameters as described above provides SWAAMH
I/ users with the capability to analyze not only SWR problems, but special
effects requiring only certain parts of SWAAM-II. Additional flexibi litY is
provided through the use of input parameters that are not logical
parameters; these are discussed in Section IV.
III. THEORETICAL FOUNDATION OF SWAM-II
A. Water-Side System Module
The water-side system module calculates the transient flow of
water/steam taking place in the tube side of the steam generator following a
tube break. The piping system generally consists of many pipe sections
connected to each other or to certain other boundary conditions. The
piping-network flow is modeled by a one-dimensional two-phase flow for the
individual pipes, and the flows between pipes and between a pipe and the
ambient or other system component are coupled through junction and boundary
conditions, respectively.
The homogeneous equilibrium model (HEM) of two-phase flow is
considered:
a 2TiOn) + p) pg cose - T
and3
+ -a
in(PE + P)] = Pgu cose .3t ax
Here, 0 is the pipe inclination angle, r the wall friction, and Q the wall
heat transfer; the total energy E is defined by
1 2E = i +
2
where i is the internal energy. Note that the energy equation does not
include the terms involving axial heat conduction. In a rapid transient
flow, the axial heat conduction term generally is small and hence is
neglected. The HEM equations are identical to those of a single-phase
flow. In SWAM-TI code, options are available for both the water/steam two-
phase flow and the single-phase flow of nitrogen gas. The reason for the
(4)
-01
11
single-phase nitrogen gas option is that some of the Large Leak Test Rig
(LLTV) tests employed nitrogen gas to simulate the leak flow and the SWAMI-II code was used to analyze the test results.
The numerical technique used
to solve the network flow in
SWAAM-II is a hybrid technique [3]
that combines the two-step Lax-
Wendroff scheme [4] for the field
equation solution and the integral
method of characteristics [5] for
the junction and boundary
conditions. The numerical grid
system with a constant spacing (Ax
const) is shown in Fig. 2.
Fig. 2. Grid System for Interior
Procedure
1. Field Equation Solution by the Two-Step Lax-Wendroff Scheme
The REM equations can be written in the following conservative form:
aU aftTt- s
where U, F, and S are three-element vectors given by
(5)
'• [PU)
PE
• ; F
PU
+ p j;
u( PE + p)
S •
0
pg cos() r
ogu cosi) + Q
(6)
Application of the two-step Lax Wendroff scheme to the above conservative
form of equations leads to the following explicit difference equations:
1 1 1U
1 0
2—( U
A + U
C ) +
2— 0(7C - FA ) + 4— 0Ax(SA + Sc) ' ( 7)
1U2 —2
U + U8) + 2— 0(F - F ) + 4 00x(SA + SB)• (S)AA11
and
Up UA
+0(F1 - F2 ) +
2— 0Ax(S
1 + S
2)
' (S)
(10)
1 2
where 0 = At/Ax. The subscripts A, B, C, P, 1, and 2 refer to the points
shown in Fig. 2. U 1 and U2 are the half time step (1/2 At) values used to
calculate the new time step values U p . The limitation on the time step is
the usual CFL criterion for the explicit difference scheme:
The sonic speed c and the fluid velocity u vary among all grid points. The
time step At therefore is chosen such that the CFL criterion is satisfied
for all grid points in the system at any step of calculation. The above
describes the procedure to calculate the interior grid points of all pipe
segments.
2. Junction and Boundary Condition Solution by the Integral Method of
Characteristics
The HEM equations are written here in characteristic form as follows:
!L= 1
•t.(uT + Q) along T-characteristic dx = udtd Or
ItE 4. duPc
t. L12 , ,
, ut + Q) + c( pg cos 6 - T)dt dt P s ai) P
along R-characteristic dx = (u + c)dt,and
_ du _ 1 rIEdt dt 31.)P (UT + Q)
c( pg cos 8 - r)
along S-characteristic dx = (u -c)dt.
Here, s is the entropy, T the absolute temperature, and (ap/ai) athermodynamic quantity of the water/steam obtained from the steam tables.
The integral method of characteristics uses a time step much smaller than
the general time step used for the interior field solution procedure, i.e.,
At/n, where n = 5 to 50, depending on the circumstances. The reason for the
small time step is that the gradients of the flow properties can be veryhigh near the junction or the boundary and the usual approximation of thecompatibility equations for the full time step is too crude in many cases oftwo-phase flow. In the case of a subcooled water blowdown, for example,
sudden evaporation or flashing occurs at the pipe exit where the flow
remains two-phase, while a short distance upstream the flow is still in
subcooled state. Many conventional difference schemes for these problems
failed or were only partially successful due to the large errors introduced
(12)
(13)
•nnnnn••••
CO) PHYSICAL CONFIGURATION
Ibl WAVE DIAGRAM
Fig. 3. Break Boundary
-+ S1 + S.1.)3 p
13
because of the large impedance (pc)
gradients [6,71. The sub-time-step
integral method can best be illustrated
by an example for the break boundary
shown in Fig. 3. Points A and B lie on
the current time line where the solution
is known, and the task is to find the
boundary condition (the pressure,
velocity, and entropy) at the advanced
point P. The 1'-characteristic, Eq. II,
is just advanced to obtain s and the
average entropy i is formed by
(14)
which next is used to advance the I-characteristic, Eq. 12 (S-characteristic
equation does not apply in this case). The scheme used to advance the it-
characteristic equation is described schematically in Fig. 4. The R-
characteristic equation is integrated in the decreasing pressure until the
path meets either the sonic line or the back-pressure line. The intercept
P, as shown in Pig. 4, is the solution state for the boundary point at the
new time step. The starting
point R of Fig. 4 corresponds
SONIC LINE to point R of Fig. 3 on the
initial time line. The
integration paths are
represented in Fig. 4
schematically as straight
lines, but they are usually
curved severely. This is the
reason vhy a small time step
is needed in the integration
of the R-characteristic
equation. Note that once the
.1 entropy is determined by
Eq. 14, the R-characteristic
R-CHARACTERISTIC
Fig. 4. Solution Methodology for
Break Boundary.
equation involves two vari-
ables only - pressure and
velocity.
14
The same integral procedure is employed in other junction and boundary
condition options available in SWAAM-II. Currently, twelve different
junction and boundary condition options are available, as shown in Fig . 5.
The details of these formulations and the accuracies attainable from these
formulations can be seen from Refs. 3, 5, and 8. The dummy junction enables
the user to keep each of the pipe sections within a reasonable length so
that the number of nodes in each pipe section can be reasonably small. This
helps reduce the requirement for computer memory space. The inline rupture
disk acts as a closed end until the pressure reaches a failure pressure for
the disk, from which time the junction acts as a dummy junction, area
change, or the orifice-in-pipe junction, depending on the user input
specification.
—• BREAK END (OUTFLOW)
—I, BREAK END W / ORIFICE
RESERVOIR
RESERVOIR W/ ORIFICE
4-4 DUMMY JUNCTION
ORIFICE IN PIPE
AREA CHANGE
AREA CHANGE W /ORIFICE
NONREFLECTING END
CLOSED END
TEE JUNCTION
INLINE RUPTURE DISK
Fig. 5. Water-Side System Junction and Boundary Conditions
B. Reaction Zone Analysis Module
The theoretical basis for the SWAAM-II reaction zone analysis (RZA)
module is an improved version of the Ttegonning [9] model. Tregonning
considers the reaction bubble dynamics and the incompressible flow response
surrounding the bubble. The incompressible flow response is replaced by the
compressible flow response of the sodium system in SWAAMHII. Another
important modification made to the Tregonning model is that the reaction
rate equation is simplified. Tregonning's attempt to relate reaction rate
to hydrodynamic mixing length did not appear to offer much promise,
1 5
especially in view of the little knowledge we have to date concerning the
reaction kinetics of sodium and water. It was the philosophy of SWAM-II
development that a simple reaction model based on simple geometric
considerations be correlated with the large scale test data, and that the
constants thus obtained be used in modeling the reaction rates.
The basic assumptions made in formulating the SWAAM-II RZA module are:
The reaction bubble is in both thermal and mechanical equilibrium.
The reaction bubble consists of reaction products and unreacted
water but not pure sodium.
The bubble energy is born by all bubble constituents, including
the phase transition of the reaction products.
The reaction bubble is assumed to be spherical initially but to
convert later to a cylindrical (pancake) shape.
All gas phases present in the bubble are ideal gases.
The equations describing the dynamics of the reaction bubble under the above
assumptions are a set of ordinary differential equations with time as the
independent variable. The set of equations describing the RZA module are
discussed below.
1. Set of Governing Equations
a. Energy Equation
d r-i m' 4' m'C(T - T -A (AR + a_h + F1) dm'
dt ref N dt
- d;+ h — - q f - qs - pV .
dt
The left side of Eq. 15 represents the increase in total bubble energy,
while the right side represents heat input, heat loss, and expansion work.
The variables involved are defined as follows:
n unreacted water mass in the bubble,
internal energy of unreacted water in the bubble,
m' n reacted water mass,
n the total heat capacity of the reaction products,
C (a' + as)Cav,
(15)
16
a' = mass of hydrogen gas generated per unit mass of water reacted,
a s= mass of condensed phase reaction products per unit mass
of water reacted,
Cav = average specific heat of the reaction products,
= reference temperature to measure all energy quantities,TrefAH = heat of reaction,
aN= mass of sodium per unit mass of water reacted,
hN= enthalpy of sodium,
= enthalpy of injected water,
qf = heat loss at the bubble/sodium interface,
qs = heat loss to solid inclusions such as the tube bundle, and
p,T,V = bubble pressure, temperature, and volume.
The expanded form, written below, is more convenient for purposes of
computation:
-du 4. nrodT[All+ ah + - c(T - T
Nn dm'maT dt = ref dt
- (u - qf - qs - pV •
The energy equations presented here do not contain terms representing the
phase change of the reaction products, Na 20 and NaOH. These terms are added
to the equations later when the computational forms of the equations are
discussed.
b. Equation of State
T a )V = m'(RNa'— + -J2 + my,
P Ps
where
RH = gas constant for hydrogen gas,
p s = average density of condensed phase reaction products, and
v = specific volume of unreacted water in the bubble.
The differential form of the state equation used in the computation is
' dm -dvpV = (Re T +ps
r4. 4.
dt '
m o p AI - col dP
-H dt -H p dt
(16)
(17)
(18)
17
c. Reaction Rate Equation
tiLA fdm'
dt
▪
V
The left side of Eq. 19 represents the rate at which the water mass
undergoes the sodium—water reaction. The reaction rate is expressed as
being proportional to the amount of unreacted water in the bubble (i)
available for reaction and the flame surface area (or the bubble/sodium
interface area), and as being inversely proportional to the bubble volume.
A is the proportionality constant, treated in SWAAMr-II as an input constant.
d. Water Mass Equation
—dm' , du dmdt dt dt ' (20)
where dm/dt is the inter injection rate calculated by the water —side system
module, and a is the total injected water mass. A coupling is maintained
between the RZA module and the water system module. The bubble pressure is
the back pressure for the break flow boundary condition of the water system
module, while the injected water determines the source term for the RZA
module.
e. Heat Loss to Structures
Acis
•
. h5 (17I )V(T — Ts ) , (21)
where
hs • heat transfer coefficient,
As n heat transfer area, and
Tstemperature of the solid structure contained in the bubble.
The heat loss from the bubble to the immersed solid structures is treated
simply, as shown above. The temperature of the structure Ts is assumed
constant.
f. Heat Loss at the Plane Surface
qf• hf
Af(T — TN)
(22)
where
h f 0 heat transfer coefficient,
Af 0 bubble/sodium interface area or flame area, and
TN sodium temperature.
(19)
18
g. Reaction Chemistry Equation
A Na + H2 = B H
2 + C
INa
20 + C
2NaOH + C
3NaH + AH . (23)
The most general form of the reaction chemistry equation (above) is
considered in SWAAM"II. The stoichiometric coefficients A, B, C. C2, and
C3 are not all independent, but must satisfy the following conditions for
conservation of individual element masses:
A = 2B+ 2 C •3
C 1 = 2B + C3 - 1
C2 = -28 C3
Values of B and C3 are chosen first, and then the rest of the coefficients
are determined by Eq(s). 24. NaH has a low disassociation temperature, and
therefore C3 = O. Hence, the hydrogen conversion ratio B is the only
required input parameter. Figure 6 is used to determine A, C I , and C2
values for given values of B and C3 (C3 = 0 in SWAAM-II).
Fig. 6. Stoichiometric Coefficients Diagram for the General ReactionEquation
( 24)
19
The heat of reaction AR is found from the heats of formation for the
reactants and reaction products involved in Eq. 23 as follows:
1 [ r. Ako+ °AN n Li
oMi
20
n'auf(Na20) + C
2Ah
f(Na0H) JAhf(NaH) - nf(H20) 1 '
( 25)
where C I , C2 , C3 are the stoichiometric
reaction equation 23, Mu A is the molecular.2.are the respective heats of formation at the
25.0(Ah°f(Na) n Ah°f(H
2)
n 0) . The JANAY
to find the following heats of formation:
coefficients appearing in the
weight of water and the All's
standard condition of 1 atm and
Thermochemical Tables are used
Ah° 9f(Na
20)
n 9.90 kcal/sole
-1Ahf(Na0H)
01.90 kcal/mole
(26)
f(NaH) n 13.49 kcal/mole
Ah
n 68.32 kcal/sole .Ahf(H
20)
Hence, the heat of reaction AR per unit mass of water reacted can be
expressed by
1till--(18.016 99.9C 1
+ 101.9C2 + 13.49C
3 - 68.32) kcal/kg(H
20) . (27)
The phase transition of the condensed phase reaction products Na 20 and NaOH
also is considered in the heat balance of the bubble. These compounds
either absorb or liberate heat as they undergo change of phase and, due to
the latent heats, the temperature of the bubble remains constant during the
phase change. This will be discussed more in detail later when the
computational aspects are described. Table 1 shows the phase transition
temperatures and the latent heats involved in the possible phase changes.
h. Bubble Dynamics/Sodium-Response Coupling Equations
The mass and momentum interaction at the reaction bubble/sodium
interface provide the needed coupling between the RZA module and the sodium-
side system module.
The interface mass relationship is
aim
Af n +
PN
m f (28)
This equation relates the
bubble dynamics (quantities
such as p, a, da/dt, andd2 a/dt 2) with the surrounding
sodium pressure pR, which is
the pressure at the shell
radius (R) location, as shownin Fig. 7. pR also is thepressure in the sodium-side
BUBBLE system at the boundary that
interfaces the reaction zone.Fig. 7. Reaction-Bubble/Sodium
Response Coupling Model
20
where
a - bubble radius (x = a after the shape conversion),
UN = velocity, normal to Af , of sodium surrounding the bubble, and
PN = sodium density.
Table 1. Phase Changes of Reaction Products
Phase ChangeTemperature
(°C)Latent Heat(kcal/kg)
Na20(s) Na20( 1132.06 183.81
Na0H(s) t Na0H(t) 319.11 37.96
Na0H(L) Na0H(g) 1389.56 946.94
The first term on the right side represents the contribution to the bubble
growth due to mass transfer of the surrounding sodium. The momentum
equation for the sodium immediately surrounding the spherical bubble is
based on the potential flow (incompressible flow) solution. The simple
solution for an expanding bubble in an infinite fluid medium given by Lamb
[101 (also discussed in Zaker and Salmon [11]) was extended to include the
effects of a finite region [1]. The finite-region momentum equation is
2p _ IIVLA 11_ z11 (2114101a)21 (29)R ) 12 R 2 Lit)
(30)
2 1
Equations 28 and 29 are further coupled to the response of the sodium-
side system as follows. Because the bubble junction can be a multiple pipe
connection (up to three pipes, as can be seen later in this section) the
response of the sodium adjacent to the reaction bubble must be expressed in
terms of equivalent velocity. The equivalent velocity u n is defined by
where the subscript i refers to the pipes connected to the reaction zone.
The C- characteristic for the sodium at the reaction zone/sodium interface
is written for the individual pipe i:
PR r P -Ur= ui + Gdt + GSA]
where the and ;Li. are the pressure and velocity of the neighboring point
In pipe i, G is the friction term, and GSA is the gravity term. The
equivalent velocity un is now expressed in terms of the sodium response in
each of the connecting pipes by combining Eqs. 30 and 31 as follows:
N AiP y 7--r- / lAiR kl0e;
(32a)n
Ai
where the Riemann constant i is defined by
_ iZin -u + Gdt + GSA . (32b)
Equation 32a provides the relationship between the pressure and the velocity
at the reaction zone/sodium interface. This is the additional condition
that needs to be considered in the solution of the reaction bubble
dynamics. The quantity UN of Eq. 28 is related to un by the following:
(31)
22
R22 un (for spherical bubble)
2a
UN = u
n(for pancake bubble) .
(33)
i. Bubble-Shape Conversion
The choice of criteria for conversion of the bubble from its original
spherical shape to its later pancake shape was a difficult subject. The
TRANSWRAP-II code [12] attempted, in an early stage of its development, a
shape conversion scheme involving an arbitrary choice of the time of
conversion. This conversion scheme introduced an error in the
redistribution of flow properties, accompanied by nonphysical
discontinuities in their values. Hence the scheme was not very
successful. The rigorous formulation for the reaction bubble dynamics
described above permits calculation of all physically meaningful variables
in a way that preserves continuity during shape conversion. The criteria
adopted here are simply conservation of the volume and the flame area of the
bubble.
If "a" denotes the bubble radius before the conversion and "x" the
bubble size after the conversion, the criteria for conversion of the bubble
shape require that the following conditions be met:
a
•
=.1A-41i-and
* 1*x = — a
3
where F is a geometry factor; F = 1/2 for the hemispherical bubble andF = 1 for all other cases (N>2). The superscript (*) in these equations
refers to the time of conversion. The conversion criterion, Eq. 34,
determines the maximum size of the spherical bubble. For the hemispherical
bubble (i.e., leak at tubesheet) or the two pipe connection (N = 2),a = R/1/7- . Equation 35 indicates that there is a discontinuity in the
bubble size parameter at the time of shape conversion. Because bubble size
does not enter the governing equations explicitly, the discontinuous
behavior of the bubble size parameter does not appear in any other
variables. Figure 8 shows typical bubble growth and shape conversion for
two different cases of water injection rates and their comparisons with a
(34)
(35)
23
24
20
piston model. These are the results
obtained earlier in the module
development stage when the
sensitivity of the RZA module was
. 16
aId 12
CASE I CASE 2 studied.
It is desirable to keep the maximum
spherical bubble radius a* less than
p.
O the shell radius R (a* 0 R is needede
•
to avoid the possible mathematical
singularity in Eq. 29). This limits
the total number of pipes connecting
NSTON to the bubble junction to a maximum
of three. The following defines theio 2° effective shell radius for all
TIME ms possible cases of pipe connections:
Fig. 8. Typical Bubble-Size Histories
AfN * 1, 2, or 3 . (36)
2. Closure of the Governing Equations Set
In the above, the set of equations is discussed that is needed to solve
the reaction bubble dynamics and the coupling of the reaction bubble
dynamics with the sodium system adjacent to the bubble. Here, the closure
problem, that is, whether there are a sufficient number of equations to
solve for the unknown quantities, is discussed. A close examination of the
equations above reveals that there are nine main equations and nine
variables. Hence the closure requirement is satisfied. The nine equations
are Eqs. 16, 18-22, 28, 29, and 32; the nine variables are u, p, m', i, a,
PR, on, ; a , and sq f . Not discussed above is the state equation for water (or
the steam tables) which relates all water/steam properties to the two chosen
variables u and p.
The form of the nine equations, however, is not convenient for
numerical computation. The derivatives of the water state variables other
than the two chosen variables u and p, for example, need to be expressed in
terms of u and p. Moreover, a number of substitutions can be made to reduce
the number of variables in the system.
24
3. Computation of the Equation Set
In the above, the equations are presented in a form to emphasize the
physical aspects of the reaction bubble dynamics and its coupling with the
sodium system, and these equations are not necessarily in a form convenient
for numerical calculations. In the following, substitutions are made to
reduce the number of equations (and the number of variables), and the
equations are rewritten in the form used in the coding of the SWAAM-II RZA
module. In addition, terms resulting from the phase change of the reaction
products Na20 and NaOH, described in Table 1, are included in the energy
equation. This slightly modifies the equation set with additional equations
and variables.
The enthalpy and pressure are chosen as the two variables to describe
the water state, and all other state variables are expressed in terms of h
and p as follows:
du _ dh dv vdpdt dt PaT dt '
dT = (DT) dh
▪
(DT ) lipdt hp dt , ap)h dt
anddv _ (3v ) dh
▪
(3v) dEdt dt 3p)tt dt
The final energy and state equations, including the phase change terms and
involving only the two chosen variables, h and p, for water properties, areas follows:
Dv DT dh av , aT aul- p(irOp ] + c(-z)p } + 1-13;[p(-Wh + v] + m C(TI-3.1111 dt
-= [AH + 1011,1 +C(T T "
)]dm'ref dt (h Pv h) dt
(37)
(38)
(39)
dmIL
dm2X- A
Hlt dt AH2/ dt
dm(411 An 1 __A
2X -21g) dt
and
Ada- h (--E ) V(T -T_) - h
fAf(T T
N) - pA —sit
f dts (40)
P PR r 3 2a +)4I( cla) 2
PN
L 2 R 2‘1‘dtd
2a
dt2
a( -
(42a)
and
25
((e'RHa' 112g R2)(gdp P171( Mpli (111 1(21iRHai
Nffl1.11 _ ..:117.A 1 LIE2g
R 2 11 ‘a p ) h p v-lap/hi dt
as)dm'
dm+ -A III
dt- (Re'T + - pvTIT - (R-T --P-- (41)
ps v f dt ')
where
mit mass of Na20 in liquid phase,
m21 • mass of NaOH in liquid phase,
u2g
n mass of NaOH in gas phase,
Alia • latent heat of phase change of Na 20 from solid to liquid,
61121 latent heat of phase change of NaOH from solid to liquid,
AH2glatent heat of phase change of NaOH from liquid to gas,
C - total heat capacity of the reaction products, a'C' + a l C I +a 2 C2 ; the specific heats C t , C2 are functions of phase masses,i.e., C I C l ( m it , m18 ), C2 C221 , m2g , m28 ), and
asn total volume of all condensed phase reaction products, i.e., at
s the standard condition where m2g n O.
The bubble dynamics and sodium response equations, 28, 29, and 32, arecombined to obtain the following equations:
For the spherical bubble,
1 P n
rI(A da _ N dua , N
R N Ai
27 R
)
2 F f dt p
N dt L +ZIAiU ;
iw
Tp7iT-1.1
(42b)
for the pancake bubble,
dx dm' 4. 1 N A
i
dt - Af
oN
dt • [ 11 (K)1=1A
NX Z
iAi
.1=1
(43)
26
1=1
The phase change of the reaction products Na 20 and NaOH introduces
three new variables to the system: m i x, m21, and m2g . Therefore, there
must be three additional equations besides those already discussed above.
When a phase change takes place, the bubble temperature is at one of the
three phase change temperatures, Tml , Tm2 , or Tg2 , shown in Table 1, and the
temperature does not change until the phase change is completed. Therefore,
at a phase change,
dTdt '
which, using Eq. 38, is equivalent to
d (21.-)1,h 3p dt (3T) dt •
ah )1)
(44a)
(44b)
The following additional relations are needed for the seven differenttemperature regimes:
(1) T < Tm2 (no phase change)
dmlt dm din
dm2g _ 0
dt dt dt - •
(2) T = Tm2 (solid-to-liquid phase change of Na0a)
dmIt dm2g_ 0 .
dt dt (46)
(3) T T < Tml (no phase change)
and
dmIt dm2g
dt dt - 0
(45)
27
dm2t dm'dt 4
•
2 dt •
(4) T Tim (solid-to-liquid transition of Na20)
de2 t dm'dt 44
•
2 dt
dm2ift . 0dt
(5) Tml < T < Tg2 (no phase change)
dmIt dm'dt
•
dt
des 2 I dm'dt a
•
2 dt •
(6) T Tg2 (liquid-to-gas phase change of NaOH)
dalit dm'dt
•
dt
dm2g . dm' dm2tdt 42 dt dt •
(7) T2g < T (no phase change)
dmlt dm'
dt
•
dt
- o (51)
dm2gdm.'
dt 42 dt •
and
and
and
and
(47)
(48)
(49)
dm2tdt
28
Note that there are only two relationships in each of the three phasetransition regimes above. The third needed relationship is Eq. 44.
The eight equations actually solved in the RZA module are the five
equations 19, 20, 40, 41, 42 (or 43) plus three equations from 44 and 45-51
depending on the temperature regime. The eight variables involved in these
equations, for which the equations are to be solved, are m', h, p, a (or
x), m, mu, and m2g . Therefore the closure of the equation system is
satisfied. The bubble radius a in Eqs. 40 and 41 is replaced by x after the
bubble shape converts to pancake.
With the exception of Eq. 42, the system of equations to be solved is
composed of first-order ordinary differential equations. The second-order
equation (Eq. 42) is expressed in terms of WO first-order differential
equations. The resulting nine equations are then solved by the first-order
ordinary differential equation solver GEARDV. The nine variables are
defined in the subroutines DIFFUN and BUBDYN as follows:
Y(1) = m
Y(2) = m'
Y(3) = h
Y(4) = p
Y(5) = a (or x)
(52)Y(6) = i (or i)
Y(7) = mit
Y(8) =
Y(9) = m2g.
The system of equations describing the bubble dynamics and the coupling
to the sodium system response represents an initial value problem. The
system possesses a singular behavior in the limit as the bubble size
approaches zero, i.e., a+0. The initial conditions for the bubble condition
are rather arbitrary, and the solution for the first few steps for the
assumed set of initial conditions usually exhibits a nonphysical erraticbehavior. To avoid the unusual erratic solutions and thus to provide a
smooth starting of the initial-value problem, an internal routine is written
that solves the simplified version of the nine equation system under a
number of simplifying assumptions. The details of this internal procedure
are not discussed here because the initial singularity is integrable and the
exact initial conditions do not have any physical significance. Theinternal routines serve well in providing a smooth start of the overalltransient.
METHOD OF CHARACTERISTICSEXPLICIT
(WATER SIDE)
29
The RZA module interacts with the water side module and the sodium side
module, and different time steps are used in each module. The management of
the three time steps and the matching of the time levels between the modules
at some points of the computation is an important task. Figure 9 shows the
time step management scheme adopted in SWAAM-II. The GEARDV uses a time
step (6t) that is such smaller than the sodium-side time step At.
Therefore, one sodium-side step is subdivided into many GUM steps. For a
close interaction between the RZA and the sodium-side module, however, the
Riemann constants are obtained for each of the GEARDV steps, as shown in
Fig. 9. The injection rate dm/dt used in the RZA module is the latest value
obtained from the waterside module. The water-side time step is either
nearly the same or slightly greater than the sodium-side step. Hence, the
water side solution is first advanced for the back pressure (bubble
pressure) available currently at the sodium side step. Then the injection
rate is used for all GEARDV steps until a new value is available for the
injection rate. This scheme of time step management has proved satisfactory
in all SWAAM-II applications to date.
Fig. 9. Time Step Management and Module Interaction Schemes
C. Sodium-Side Modeling
1. One-Dimensional Sodium System Dynamics Module
This module is based on the PTA-2 code [13-15] developed earlier at ANL
to analyse pulse propagation in reactor piping systems. PTA-2 combines the
capabilities of previous codes in the Pressure Transient Analysis series;
30
these are PTAC [161, which uses a column-separation model to treat the
effect of cavitation on pulse propagation, and PTA-1 [17-18], which uses a
fluid-structure interaction model that includes the effect of pipe
plasticity on pressure transients. All codes in the series are based on a
fluid-hammer formulation using the one-dimensional method of characteristics
applied to a fixed time and space grid. Pipe friction, nonlinear velocity
terms, fluid compressibility, and wave-speed dependence on pipe deformation
are included in the formulation. The codes are capable of treating complex
piping networks and include a variety of junction types. Pipe network
connections, node spacings, fluid properties, mechanical properties of
typical piping materials, flow areas, friction factors, and wave speeds are
computed internally.
A detailed treatment of either cavitation or structure-fluid
interaction in a large piping network would require a computational effort
that would be incompatible with the use of a pressure transient code as a
design tool. Consequently, relatively simple computational models for
cavitation and pipe plasticity effects on pulse propagation were developed
that are consistent with a one-dimensional treatment of the system. The
intent was not to model the complex nonequilibrium thermodynamic processes
involved in cavitation or the dynamic structural response of the piping to
transient loads, but to incorporate features of these phenomena that have
the strongest influence on pulse propagation in the fluid. Despite the
simplicity of both models, the agreement between available experimental data
and code computations [14, 15, 183 is very good and well within the
experimental accuracy limit.
In modeling the effect of pipe plasticity on pulse propagation, we
neglect all waves traveling through the pipe wall and assume the pipe to be
sliced into a series of unconnected rings. Consequently, bending moments,
axial forces, and pipe inertia are neglected, the pipe response is quasi-
static, and deformations are not required to be continuous functions of
axial position. As a result of these assumptions, the only influence of
pipe deformation on transient propagation in the fluid is through its effect
on local wave speed. Wave speed is no longer just a function of fluid
properties, but now also depends on pipe properties, pipe-deformation
history, circumferential stress, and direction of loading. Consequently, it
can vary with time and position along the pipe, and provision is made in the
computational scheme to accommodate this variation.
Detailed descriptions of the pipe plasticity model, the various
junction-type models, and the general code structure for the PTA series are
given in Ref. 13. The governing equations and numerical procedure for the
(58)
31
fluid-hammer formulation, including pipe plasticity effects, are summarized
below.
The characteristic equations of one-dimensional fluid hammer theory are
+ G(u) + g sina n 0dt pc dt
which holds along the positive characteristic C+ given by
dx n (u + c)dt (54)
and
du _ 1 AR+ G(u) + g sina • 0
dt pc dt
which holds along the negative characteristic C- given by
dx (u c)dt .
Where appearing, u and p are fluid velocity and pressure, t is time, x is
the axial coordinate along the pipe, p is fluid density, a is the angle of
the pipe with the horizontal (positive upward), c is wave speed, g is the
acceleration of gravity, and G(u) is the pipe friction term defined by
G(u) ft+LD (57)
where f is the Darcy-Weisbach friction factor and D is the pipe inner
diameter.
The wave speed for an elastically deforming pipe is constant; for a
plastically deforming pipe, it is allowed to vary with position and time and
is given by
+ 1 dap(53)
(55)
(56)
where K is the bulk modulus of the fluid, H is the pipe-wall thickness, and
a and are circumferential stress and strain in the pipe wall. The stress
is in equilibrium with the local fluid pressure and is computed from
paag2H •
(59)
c+
to 1- At
32
For typical piping material, the slope of the stress-strain curve
depends on the current stress, previous stress history, and the sign of the
current stress variation, i.e., whether plastic loading or elastic unloading
is occurring. The history effect is accounted for by keeping track of the
highest stress previously attained at each node point in the piping system.
If the solution for pressure and
fluid velocity is known at a time
to , the solution at a later time t o+ At can be found through the
relations between du/dt and dp/dt
that hold along the characteristict,RA B 5
curves. Expressing Eqs. 53 and 55
in finite-difference form for CX C-characteristics intersecting atPa
point P gives (Fig. 10)
Fig. 10. Finite-Difference Grid
pp PeeP = YA 'and
(60)-PP - pcB
uP = Y
B '
where
+rYA
pA + pcA [ uA
- (GA + g sina)At]
YB = pB - pc;[uB - (GB + g since)At]
c+A is the average wave speed along the C+ characteristic between points A
and P, and ci is the average wave speed along the C - characteristic betweenpoints B and P. The solution for pp and up at an uncavitated interior nodeP is thus given by
YAcB + Y c
+B A
c+ + c
A B
YA - y
B U =p
+ p(cI + c'.;)
(6 I )
PP -
(62)
33
If the region of the pipe being computed is deforming elastically, the wave
speeds are constant along the characteristics, and an explicit solution is
obtained for the intersections A and B of the characteristics with the grid
and for the solution up and pp. If the pipe is deforming plastically, the
locations of the intersections, the wave speeds at A. B, and P. and the
solutions up and pp are found by an iterative procedure.
When the pressure anywhere in the systems falls below the equilibrium
vapor pressure, a vapor cavity forms and the transients on both sides of the
cavity are essentially decoupled. Although the cavity probably will not
form instantaneously at the equilibrium condition, the assumption that it
does may be a good approximation for inception of cavity formation. To be
consistent with the one-dimensional flow model, the cavity is represented as
an idealized fluid-column separation with two free surfaces. Once the flow
separation takes place and as long as the columns remain separated, the
pressure is set equal to the vapor pressure for the prevailing
temperature. Again, the cavitation model, as such, does not consider the
nonequilibrium phenomena associated with the complex cavitation process, but
emphasizes the strongest effect on transient-pressure propagation. The
present model 114, 16, 19, 20) differs from other models of column
separation in that it allows occurrence of cavitation anywhere in the
system. This permits a detailed description of inception, growth, and
collapse. Growth of the cavitation region is represented by a region of
successive cavitated nodes, and in this manner the reduction in propagation
speed in the cavitated region is achieved as a natural outcome of the
computational procedure to represent the correct timing of cavity
collapse. The transient cavitation mode/ described here provides
conservative estimates of the generated pressure pulses.
Following the model assumption, a cavity forms at a node when the
computed pressure falls below the vapor pressure, and the cavity collapses
when the size of the cavity shrinks to zero. The cavitated node becomes a
dual-velocity node, and the cavity size is determined by the relative
position of the two interfaces. For convenience, the cavity is assumed to
be fixed at the node where it originated.
During the computational procedure described following Eq. 10,
cavitation condition may be detected at point P of Fig. 10, for example.
Then the pressure is set equal to the vapor pressure, and the interface
velocities up and wp are computed according to a fixed-pressure boundary
condition:
34
PP = Pcav
YA - p
cavu -
PoicA
and
-YPcav
w -P
B
The cavity size or column separation Sp is calculated from
Sp = SQ + ÷[(wp + wQ ) - (up + uQ nAt . (64)
The cavity is assumed to have collapsed when Sp = 0. The computation then
reverts to Eq. 10, and the interface velocities become identical (up = wp).
The Courant-Friedrichs-Lewy (CFL) criterion t21] for convergence and
stability of the finite-difference scheme used here requires that the time
step At and axial grid spacing Ax for a pipe satisfy
Ax (c + lul)At . (65)
Because the time step is the same for the entire system and the wave
speed varies from pipe to pipe if the pipes deform, Ax must be selected for
each pipe so as to satisfy the above inequality. We take
Ax < Ax Ax2 '1
where Ax1 is chosen to be large enough that the CFL criterion will not be
violated for reasonable velocity increases and Ax2 is chosen to be small
enough that pulses are not excessively smeared by interpolation
inaccuracies. Input pipe lengths are altered automatically if the
inequalities are not initially satisfied. If violation of the CFL criterion
is imminent during program execution, the time step At is reduced.
Typical finite-difference grids for boundary nodes are shown in
Fig. 11, where Fig. ha indicates a last-node pipe end and Fig. lib a first-
node pipe end. SWAANHII has the logical structure to treat any arrangement
of pipe ends; i.e., pipes can be connected at either their first- or last-
node end to any junction, and multibranch junctions can connect any
combination of pipe ends. For the sake of brevity, equations will be
presented for only one arrangement for each type of junction.
(63)
(66)
at Fig. II. Finite-Difference Gridfor Boundary Node
(b)ial
35
For a pipe with its last-node end connected to a closed-end junction,
up wr 0
PP YA 'and
Sp • 0 ,
provided YA pcsv. If the junction cavitates, then
PP m Pcav
P 0
Y -A Peav
-P
Pc+A
and
SP
SQ
- ( uP + u
Q )6t .
2
For a pipe with its last-node end connected to a constant-pressure
boundary at pressure pc,
PP . PC
YA - p
C U W
P P+ 'Pc
A
and
sp 0
(67)
(68)
(69)
36
It is assumed that the prescribed pressure is greater than Pcav' so
cavitation cannot occur at this type of junction. Several pipes may be
connected to each constant pressure boundary.
The pulse-source junction is similar to a constant-pressure boundary,
except that the prescribed pressure at the junction is time dependent. The
source pulse is input as a table of pressure-time pairs, and linear
interpolation is used within the table. Several pipes may be connected at
each pulse source, and several pulse sources may be specified.
SWAAM-II has an elaborate finite-element treatment of deformable
rupture disks, described in Section HID. A simple instantaneous rupture
disk boundary also is available, using either a prescribed failure time or a
prescribed failure pressure. This instantaneous rupture disk junction is
treated as a rigid closed-end boundary (Eqs. 67 and 68) until the prescribed
failure condition is attained. Afterward it is treated either as a constant
pressure boundary at a prescribed back pressure or as one of the available
two-pipe boundaries if a relief system is attached to it.
The pressure and volume in the gas space in a surge tank junction are
related by a pV Y = constant law. The volume change in a time step is
computed from the end-node velocities at time t in the pipes connected to
the tank. Then the new gas pressure is used to compute new velocities at
time t At, using Eq. 69. Several pipes may be connected to each surge
tank. An option also is available to have an instantaneous-type rupture
disk mounted on the gas space; in this case, only one pipe may be connected
to the junction.
The far end junction is a boundary that transmits pressure waves out of
the system without reflecting them. This is accomplished by putting the
fluid velocity and pressure at the far-end junction equal to their values at
the adjacent node of the pipe; e.g., if the last-node end of a pipe isconnected to the junction, then (see Fig. 2)
PM' up um, wp wm. (70)
The impedance discontinuity junction connects two pipes with the same
flow area but differing wall thicknesses or material properties. For
example, let the last-node end of one pipe, denoted by subscript m, be
connected to the first-node end of another pipe, denoted by subscribt n, ata junction. Then
37
Y c
_ + Y c
Am Bn Bn Am PPm PPn +
c + cAm Bn
upm wps wpn . wpn A: Bn
P(CAM CB-11)
Y - y
(71)
and
Spm Spn . 0 ,
provided ppn exceeds peav . Otherwise, cavitation occurs at the junction and
PPm PPn Pcav '
YAm pcav
'Pmpc Am
Pcav
- YBn
wPn -PCBn
w u 0Pn Pn
IfSpa -s - likupm + uQm + uQinditit ,
andIf
S S +Pn Qn 2
--uw + w )At . Pn Qn
For systems with any short pipes and a few long pipes, input of the
desired time step may result in violation of the limit on maximum number of
computational nodes per pipe. Rather than raise this limit and increase
core storage requirements, it may be expedient to break the long pipes into
two or more pipes by inserting dummy junctions. A dummy junction also might
be useful for identifying the location of a pressure transducer or reserving
a location for inserting a tee that connects to a subsystem whose effect on
the main system will be determined later. The computations for up and pp at
the dummy junction are identical to those at the impedance discontinuity and
are performed by the same subroutine.
SWAAM-I/ has three treatments of area-change junctions: no pressure
drop, standard energy loss, and prescribed energy loss. Assume, for
example, that the area-change junction (expansion or contraction) connects
(72)
38
the last-node end of pipe m to the first-node end of pipe n, and that the
flow is from pipe m into pipe n. Then, provided no cavitation occurs, the
energy balance at the junction gives
20 2-u + p = -e u2 + p — K2 Pm Pm 2 Pn Pn 2 ranuft
where K the energy loss coefficient for flow in the assumed
direction. Flow continuity requires
Am u
pm = AnuPn '
where Am and An are the pipe flow areas.
Using Eq. 74, the energy balance can be rewritten as
2PPm 2 P6 uPm = PPn
where
6 = I - R2 - K
mnand (76)
R = A/Am n
.
The characteristic equations 60 combined with Eqs. 74 and 75 give fourequations for the unknowns ppm, upm, ppn, and upn , assuming no cavitationoccurs at the junction. Their solution is
uPm w
•
Pm+ 8
2yd
upn = w
•
pn = R
•
upm
p YAm
P•
eAmuPm
(77)
and PPn =
•
Bn P• e B- nuPn '
Spm = SPn = 0 .
In the above,
-a = c + Rc
BnAm
(73)
(74)
(75)
R c Y + c+
Y-
Bn Am Am BnPPm a PPn
.
39
and
(78)
Y = Y )P‘ Am Bn)•
For the area change junction with no pressure drop, 6 = 0 and Eq(s). 77
simplify to
wuPm
. Pm 28 '
upn = wpn = Rupm
(79)
and
Spm -s - o0 .If the pressure calculated from Eq(s). 79 drops below the cavitation
pressure, it is set equal tocav and the velocities on either side of thePJunction are computed from Eq(s). 63.
The area change junction with standard steady-flow energy losses has
Km given by 1221
Kmin = (1 - 102 if R < 1
and
(80)
Kmn = 0.45 R(R1) if R > 1
The prescribed energy loss junctions uses Kin and Km supplied by input
data; i.e., the user has to prescribe energy loss coefficients for both
directions of fluid flow through the junction.
Several treatments of cavitation at area change junctions with pressure
drops were investigated. The most obvious treatment was to maintain the
pressure on the low pressure side of the junction atcavn and use the energy.- equation (75) and characteristic equations (60) to determine the other
unknowns. This gives plausible results if the net fluid flows are low.
However, when the relief system is opened and velocities are high, Eq. 75
predicts a large pressure difference across the junction. If the high
pressure side of the junction is the downstream side (6 > 0), this results
in high accelerations on the downstream fluid column and gross movement of
the cavity downstream from the computational node. In effect, fluid with
b y velocity is evaporated at one edge of the cavity and fluid with high
velocity is condensed at the other. Because this violates physical laws and
40
the assumptions of the analysis, a different cavitation treatment is used
for the pressure drop junctions. Overall flow continuity through the
junction is assumed (Eq. 74), but the energy equation is ignored. The
pressure on the low pressure side is set to D and the characteristic
equations and Eq. 74 are used to compute the pressure on the high pressure
side, upm , and upn . This procedure is continued until the use of the energy
equation would again predict a pressure above pcav on the low pressure side
of the junction.
The generalized tee junction connects an arbitrary number of pipes.*
The pressure is the same for all pipe ends at the tee and is given by
pp = [ AiYAm/cAm + AnY13n/c;11 ]/[ / Alm /cIm + All /c-Bn ] (81)
The summations over m refer to connected last-node ends and those over n
refer to connected first-node ends. The fluid velocities at the pipe ends
are then found from Eqs. 60. If Eq. 81 predicts a value of pp less than
cavt column separation is assumed to occur and Eqs. 63 are used.P
The simple pump model used here treats the pump as a tee junction, with
the pump head added to the pressure at the pump end of the pipe representing
the outlet of the pump. More complex models of a pump can be constructed
from pipe elements and various junctions, if desired.
The temperature-dependent properties of sodium used in SWAAM-II were
taken from Golden and Tokar [23], and the Nuclear Systems Materials Handbook
[24] was used for the high-temperature piping materials.
2. Relief System Filling Module
The calculation of sodium filling into the relief system and the wave
propagation in the moving fluid column is initiated by the failure of arupture disk in the IHTS system. The transient pressure and velocity
distributions in filled pipes in the relief system are computed by the
sodium system dynamics module described in Sec. IV.C.1, using the one-
dimensional method of characteristics applied to a fluid hammer
formulation. A partially filled pipe is computed in the same way up through
the next-to-last grid point before the end of the sodium column; the special
calculations at the last "wet" grid point and at the end of the column are
*In the current version of SWAMI-II, six pipes may be joined at the tee;however, this number can be increased easily (see Sec. VI).
40. X
CASE
•
CASE 2
I.
S
CASE 3
4 1
decribed below. After the transient pressure and velocity distributions at
the end of a time step At have been determined, the end of the sodium column
is moved a distance uIAt' where ul is the velocity of the liquid/gas
interface. This may involve moving the interface past a grid point or
filling a pipe and moving the interface into the next pipe in the system.
P
The special transient calculationtoa Al
at the end of the sodium column has
three cases (see Fig. 12), which
depend on the location of the
interface I-I' within the grid.4 Sodium occupies the region to the
left of I-I', and computations at
grid points to the left of PQ are
performed as described in Sec.
III.C.1. The pressures and
velocities at points R, Q, and I,
and the interface location x i are
assumed to be known; in particular,
pi is the back pressure in the
relief system, which is assumed to
be constant, and uTis the4interface velocity. The pressure
pp and velocity up at the grid
point P and the new interface
velocity u .r, can then be computed
4 .10 by applying Eqs. 53-56 along the
characteristic lines shown in Fig.
12 and interpolating explicitly or
implicitly in the time-space
grid. The resulting equations are4
given below. Details of the
derivation are given elsewhere
[25).
Fig. 12. End-Node Characteristics
Using the notation defined in Sec. TV.C.1, the solution at point P at
time to + At can be expressed in the form
and
and
42
_ +cBYA + c
AFB
O
P P - +CB + c
A0
and
YA - PP (82)
up -+ f
P cA
where FB and 41 depend on the location of the interface.
Define a dimensionless interface velocity 3 and interface location y at time
to by
(83)
-
_X 1 xc?
Y _cBAt
and
FB
= Ti L(1 + OpI - PQ
- pcB
- u
Q + pc
B(G
I + g sina)At] .
Case 2: (1 - 02 )12 <Y 4 1- a
+= 1
B Y P I - PQ + PcB-(u
I - u
Q )]
+ I f 20pI + (1 + Op
Q + pc-
BP2u
I O+ ( I + u
Q113I -
+ pc;(G, + g sina)At .
Case 3: / - $ < y
+= 1
Case I: 0 < y 4 (1 - 02)/2
1 - 32 - Y (0 -
I(84)
(85)
and
43
1 - 0FB VP/ - Pcs( u i u0)]
+ PQ - PcelQ + pc s (GI + g siea)ht .
If y 4 1 + 0, the interface velocity at to + At is given by
1 1 I 1fUr ° 1-71 1 - (1 + 0 - Y)pp + ypcl i + ( 1 + 0 - y)up
pcs
PI+ Yu(
) - 1(0 + g sina)At - .
Pc B
If Y 1 + 0, lir, is found from
1 4' u 08(PQ
- + u() - u/ ) + u/ - (GE + g sina)At .PC
B
The new location of the interface at the end of the time step is
• x i + ur At .
If x I is beyond the end of the pipe, the excess fluid is put into the next
pipe in the relief system. If the junction is a tee with M unfilled
branches connected to the filled pipe, the interface velocity in the
unfilled branches is given by
ui - %An / At (90)
where the subscript n refers to the filled pipe.
The junction types available for the relief system are the
multibranched tee, the closed end, the far end, the impedance discontinuity,
the dummy junction, the area change with or without pressure drop, and the
prescribed energy loss junction. The treatments are the same as for the
main sodium system, except that there is allowance for the possibility of
partial filling between nodes.
(86)
(87)
(88)
(89)
44
3. Two-Dimensional Sodium Flow Module
The flow in the vicinity of the reaction bubble during the early stage
is expected to be multidimensional. The fluid forces acting on internal
structural members near the bubble were considered best calculated by a
multidimensional method. It was decided that two-dimensional axisymmetric
and Cartesian geometry modeling be implemented in SWAAM-I and the method of
characteristics be used in the computation. This module of SWAAM-II is
Identical to that of SWAAM-I. A discussion of the two-dimensional sodium
flow module is included for completeness.
The governing equations solved are the conservation equations of mass,
momentum and energy with the waterhammer approximation (ap/ao = c2 =
const.). For the assumed two-dimensional flow, the following three
equations (two momentum and the continuity) are sufficient to describe the
waterhammer phenomenon [26]:
au au au 1 22+ 177c + —p ax Fx = ° (91)
av. a y av 1 iEat + u-57, + + —p ay + Fy ° (92)
and
+ + + 2P- + 1,17Pc -) 0 , (93)at ax ay ax + ay y
where c is the sonic speed and v a geometry parameter that takes the value 0
for Cartesian and 1 for cylindrical geometry. Thus the equations apply to
both Cartesian and cylindrical geometries. F x and Fy are the friction termsIn the respective flow directions. The three-equation set, Eqs. 91-93,
contain three variables - u, v, and p. The density and sound speed are
assumed to be constants, and the friction terms are functions of the flow.
Equations 91-93 are solved by the method of characteristics. Unlike
the one-dimensional case for the sodium system module discussed in the
preceding sections, the two-dimensional method of characteristics is a weak
formulation [26,27]. That is, instead of the unique characteristic lines in
the 1-D formulation, an infinite number of rays on the characteristic cone
are the wave-propagation paths, and any of the rays may be chosen as theintegration path. Hence, in the characteristic form of the two-dimensional
waterhammer equations, there can be infinitely many compatibility
equations. A good choice of the needed equations, however, determines theaccuracy of the numerical procedure.
45
By a linear combination of the governing equations and appropriate
geometric identification of wave-propagation paths, the compatibility
equations along the bicharacteristics (rays on the characteristic cone along
which the disturbances propagate) can be written as follows 1261:
1 SIR - in°cos° du
- - -.4! + c[sin2 O LL u+ 'v-) + cos 2 6(21 +3x y y
4)—oc dt dt ' dt ' a y,au 3v)1- sine cosik-- + ---) j n F cos') + F sine
x .3x ay Y
Here, the drivative d/dt is taken along a bicharacteristic whose orientation
is defined by the angle 6, as shown in Fig. 13; d/dt is defined by
d a- c (coge eine ?at .=,
dt eY •
The bicharacteristic is already linearized because of the small fluid
velocities as compared to the sonic speed, i.e., u, v(<c. The
chatacteristic cone shown in
Fig. 13 is for the linearized
bicharacteristics.
The task that remains is the choice of specific bicharacteristics.
Reference 28 shows that the choice of bicharacteristice corresponding to 6 n
0, w/2, w, and 3w/2 gives rise to a numerical scheme allowing the greatest
time step size, allowing calculation of the most accurate results. The four
bicharacteristics thus chosen yield the following numerical scheme for any
interior grid point:
u n,1 lu1 + u + —(1 p - p
2 ) - 6t[(F )
1 + (F )
2 n •
2 2 pc 1 x x '
,• —2- tv3 + V4 +-(p3 p4 ) - 6t[(Fy ) 3 + ( Fy)411
and1
P IP1 +1,2 + Pc(11 1- u2 ) Pc6t[(Fx ) 1 - (Flt)211
- pc2 AtOM + v ,ay Y
(94)
(95)
(96)
(97)
(98)
where the subscripts 1, 2, 3, and 4 refer to the base points of the
bicharacteristics, as shown in Fig. 13. The velocities u and v are
46
Fig. 13. Characteristic Cone and Mesh Net.Bicharacteristics 1P, 2P, 3P, and4P are the Integration Paths
calculated first in the entire field, and p is calculated next. The partial
derivative av/ay on the right side of Eq. 98 is evaluated numerically
using the new values of v. The term v/y also is based on the new value of
v. The time step restriction on the above numerical scheme is
4r AT
ul ' c + vi-[Minimum (
c + AT I)] •5
This indicates that the numerical procedure discussed above allows a time
step that is approximately 80% of the Courant time step.
At boundaries where not all bicharacteristics are available, the
boundary conditions substitute for the unavailable bicharacteristics. At an
x-boundary, for example, either the bicharacteristic 1 (6 = w) or the
bicharacteristic 2 (6 = 0) falls outside the computing domain. Hence, the
boundary condition u = 0, or any other value if a moving structure, is used
in place of the unavailable compatibility equation.
At
In SWAANHI, a length of the faulted steam generator vessel containing
the leak site is modeled by the two-dimensional module. The pressures at
(99)
4 7
the nodes of the two-dimensional region contained in the reaction bubble are
equated with the bubble pressures. Then the velocities u, v at the source
points are calculated by the two-dimensional scheme discussed here, using
Eq.. 96 and 97. This scheme for modeling of the source pressures is a
simple method that does not require the tracking of the bubble/sodium
interface.
The two-dimensional module is coupled to the bubble module and to the
one-dimensional sodium system module. As described in the earlier section
about the reaction zone analysis module, the bubble dynamics module is
coupled directly to the one-dimensional sodium system module, not through
the two-dimensional module. This scheme of bubble dynamics coupling to the
sodium system response is considered more accurate. In SWAAM-I, therefore,
the two-dimensional module calculation is performed in parallel with the
one-dimensional sodium system module calculation. The user thus has the
option of skipping the two-dimensional calculation without affecting the
one-dimensional sodium system calculation.
D. Structure Dynamics and Fluid-Structure Interaction
1. Elastoplastic Rupture Disk Dynamics Module
The rupture disk is a portion of a thin spherical cap with its edges
fixed and the convex side subjected to hydrodynamic pressure loading. Under
this type of loading, the disk is potentially unstable and can snap through
to form a new configuration. Eventual relief of the fluid energy on the
convex side of the disk is attained when the disk displacement is large
enough to pass through the cutting-knife structure. The exact process of
tearing and opening the disk is a complicated phenomenon and is difficult to
treat precisely. Therefore, the disk is assumed to open fully when its
displacement reaches a specified value. Rigorously sought here is the
feedback to the fluid dynamics (from the fluid-structure interaction), and
the fluid loading of the disk that leads to opening of the disk.
The loading and response of the disk are assumed to be axisymmetric. A
corotational finite-element method is used to compute the dynamic
response. Equations of motion are obtained by applying the virtual work
principle to individual elements used to obtain the final governing
equations. To simplify the presentation of the basic solution method, the
fluid force is assumed to be known. In the actual computation, however, an
iterative procedure is used that solves the disk equations and fluid
equations simultaneously (see Sec. III.D.2).
A A
f"ext
od" r [ u + p 6v]dS ,
Px (105)
48
In the corotational coordinate system of each element, the x axis is
the line connecting the end points of the element, the y axis is normal to
the x axis, and the origin is at the first end of the element. The shape
functions are expressed in terms of the corotational coordinates. Then the
transverse and axial displacements for an element are assumed to be cubic
and linear functions of x, respectively:
v(x) = ao + a
lx + a
2 x2 + a
3x3 (1 00)
and
u(x) = bo + b
lx
(101)
where the coefficients ai and bi can be uniquely determined from the nodal
displacements and rotations. The axial and circumferential strains are
obtained, using classical shell theory:
and
ex(x,y) 3u(x) y a2 v( ;) 132/ 4 h/2
ax
ce(x,y) u(rX)2
(102)
(103)
where r is the distance from point (x,y) to the axis of symmetry and h is
the thickness of an element.
The equivalent internal nodal force fi nt can be written in the
corotational coordinate system in terms of the stresses in the x and 0directions 0 and ue, asx
fint
dd = f [axSe
x(d) + a
0 8e
6 (d i )1dV
(104)
where di is the nodal displacement including rotation and V is the volume ofintthe element. Transformation of the nodal force f i to that of the global"intsystem f
iis done easily since the rigid-body rotation of the element
can be computed from the global nodal displacement of the element.
extThe equivalent external nodal force fican be computed by
where d i is the global nodal displacement, p x and py,, are the surfacetractions in the global x and y directions, u and v are the total
displacements in the global x and y directions, and S is the area on which
the traction is applied. By the principle of virtual work, equilibrium of
forces at a node can be written in terms of global coordinates as
49
'ext ^int Aft- f
imidi • (106)
where mi is the lumped element mass associated with d i at node i and thesuperscript dot means time derivative.
The equation of motion is obtained by summing the contributions from
all elements with a node connected to the node i. It can be written
^est mint " "F F Mi Di1
where
-exA t r extF L L
ij f
lint . L iintL
Mi
/ Lijm '
Lij
d1
.
and
(107)
008
Here, Lij is the connective matrix and the summation is carried over all
elements. More details of the corotational finite-element method and the
constitutive equations are available in Refs. 29 and 30.
The equation of motion (Eq. 107) is integrated using a central-
difference technique:
1 2 " ,D1 (t + 6t) Di (t) + (At)D(t) + --at) D1(t) .
2
alb ab
% a ext, %MD
1(t + At) is F kt + At, - F kt + At)
1.fD (t + At) D(t) + -NtliD %ti + D
t(t + At))
2
(109)
-
where At is the step size. The displacement at advanced time, t + At, is
determined by information at current time, t. Once the displacement"int
Di(t + At)is obtained, strain, stress, F Mit
9 and F' be determinedi
by the method of corotational coordinates as outlined above.
The conventional constitutive equations derived for small-displacement
be extended large-strainproblems can easily to large-displacement, problems
because rigid-body motions are eliminated in the corotational coordinate
system. The rate of deformation V in the corotational coordinate system is
chosen as the measure of strain and is related to the strain by
and
V = c 1(1 —cx
)x x
(110)
V8 = 8/(1 + ).
0
Therefore, the stress-strain relation for a linearly elastic material is
given by,•••n
x1 v
[VV
(111)• ) 1- v
2a v
8
50
Equations 109 describe a stepwise integration procedure that requires
no iteration. The Courant criterion for numerical stability applies to this
procedure and serves as a guide for estimating the maximum step size.
Because of the nonlinear nature, such as the large displacement and non-
linear material properties, the step size should be less than the Courant
step.
where E is the elastic modulus and v is Poisson's ratio. The condition of
plane stress in the direction transverse to the element is used in deriving
this equation. If the material has undergone plastic deformation, the
effect of plasticity also must be included.
In this analysis, we use elastoplasticity theory with a linear,
isotropic hardening law. The von Mises yield criteria and the Prantl-Reuss
flow rule are adopted. The stress-strain relation, including plastic flow,is given by
I ;
V - VP Ix x
X I - v2
[1 1
v 1 f. V - VP9
(112)
where the detailed procedure for obtaining Arfc and lig is described in Refs.29 and 30. In application, we first compute Vx and Ve from the strain anddisplacement. Then, a trial state of stresses, a 8 , is computed by theelastic law given by Eq. 111 and substituted into the yield function
f o2 + (1 2 - a a .x 9 x 0 (113)
(a) (b)
51
These stresses are correct if f 4 K 9 because the stress state is inside the
yield surface. If f > K 9 the stress state is outside the yield surface andmodifications of oft and a; must be found such that the condition f K is
satisfied. The rates of deformation corresponding to these modifications
are subtracted from Vx and Ve, respectively, and the resulting rates of
deformation are substituted into Eq. 112 to obtain the correct stress state.
The fluid system connected to the rupture disk is a one-dimensional
model, and the rupture disk is a two-dimensional description. Hence, an
adjustment is needed in transferring information during the coupling. The
pressure is considered to apply uniformly over the convex side of the disk;
the disk motion is averaged over the base area of the disk for feedback to
the fluid side. Average velocity is defined as the rate of change of the
volume generated by the disk during its motion, divided by its base area.
Details of the treatment are available in Ref. 31 and its validation in Ref.
32.
2. Fluid-Structure Interaction Scheme at Rupture Disk
Consider the interaction of pressure transients with a rupture disk
located at either the last node (Fig. 14a) or the first node (Fig. 14b) of a
pipe. In either case, only one characteristic intersects the boundary and
therefore only one relationship is available to solve for the two unknowns
(pressure and velocity). The second equation must pertain to the rupture-
disk response itself. Because the fluid and the rupture disk interact
strongly, their responses are coupled [33, 34].
Fig. 14. Finite-Difference Grid at Rupture DiskBoundary for Equal Time Steps
+pp + pcA
uP = Y
A(114)
52
For a rupture disk located at the last pipe node, forward differencing
yields the finite-difference expression
(see Eqs. 60 and 61). To be compatible with the one-dimensional fluid-
transient analysis, the pressure pp acting at the rupture-disk surface is
assumed to be uniform. Then the response of the rupture disk may be
expressed as
wp = R(pp) . (115)
The dual velocity convention is used at the interface node, i.e., up is the
fluid velocity and wp is an average disk velocity. The function R(pp) is
determined numerically using the finite-element procedure described in Sec.
IV.D.1.
If the junction does not cavitate, the fluid is able to follow the
structural displacement, and the two velocities at the interface are
equal. In this case, the fluid and rupture disk response are coupled
completely and the solution must be obtained by an iterative procedure.
Then Eqs. 114 and 115 can be rewritten as
+pP = Y
A - pc
A uPand (116)
u = wP = R(p ) .
P P
The iteration procedure used to solve Eq(s).116 is a simple resubstitution
scheme in which the value of the velocity from the preceding iteration is
used to compute a new value of pressure, which is then used in the finite-
element computation to calculate a new value of velocity, etc.
The impedance mismatch at the boundary may lead to inertial overshoots
of the rupture disk that the fluid cannot adjust to before reaching its
vapor pressure. In this case, the junction becomes cavitated and the
column-separation technique outlined earlier must be employed. Because the
junction pressure is now known (Pp = Pcav' Egg* 114 and 115 are solved--) directly to obtain the velocities up and wp, respectively, without
Iteration. The size of the cavity formed between the fluid column and the
structural boundary is computed using Eq. 64. When this cavity vanishes,
the calculation reverts to the coupled and iterative solution procedure.
The discussion above has assumed that the time steps for the solution
of the fluid and structural problems are the same. However, for reasons of
53
numerical stability, the time step
for the rupture disk calculation may
be two orders of magnitude smaller
than that required for the fluid
AI
//// / 4
//
//
/ / //
_LSI
calculation. The necessary
modification of the solution
procedure is illustrated in Fig.15. The approach is to use therupture disk time step 6t only at
the fluid-structure boundary and
compute the remainder of the fluid/ / /
/ #1 1 / __I_ el system using the appropriate fluid
time step 6t. Because all variablesalong the time line t to are known
1•14.A NA, A,"*1 *1
Fig. 15. Numerical Treatment of
Rupture Disk Boundary
for Unequal Time Steps
at the node points, the values of
the variables at the intersection
points of the characteristics with
that line (i.e., at points Ai.,
A2 AK AN) are obtained by using linear interpolation. The
solution for each rupture-disk time step 6t then proceeds in the same manner
as that outlined above for equal time steps. For each fluid step, the totalnumber of rupture disk time steps is given by
• At
(117)
Because the rupture disk may undergo rapid oscillations, the fluid at the
boundary may experience cavitation and cavity recollapse several timesduring a single fluid time step. This phenomenon has caused no difficulties
in applying the fluid-structure interaction model.
For a rupture disk located at the first node of a pipe, the solution
procedure is analogous; however, the relationships along the negative
characteristic must be used (see Fig. 14b).
3. Coupling Models for Double-Disk Assemblies
A typical rupture disk assembly has two disks in series with a gas
space between them. The dynamic response of the second disk in the assembly
is strongly affected by the details of the tearing of the first disk and the
obstructed flow of the sodium through the constricted and irregular
opening. Experimental evidence indicates that there may be wrapping of the
first disk around the knife edge accompanied by a delayed and incomplete
tearing of this disk. Computations that assume an instantaneous and
AX
54
complete opening of the first disk when it strikes the knife edge are likely
then to underestimate the time required for the moving fluid column to fail
the second disk in the assembly. SWAAM-II has several options available for
modeling the interaction between the first and second disk failures.
For the standard computation, the first disk is assumed to open fully
and instantaneously when it strikes the knife edge. The gas between the
disks is compressed according to the law pVY = constant. The pressure and
volume changes of the gas are coupled to a finite-element computation of the
dynamics of the second disk, using a procedure similar to that described in
Sec. III.D./ and III.D.2.
SWAAMII provides the option of inputting data to model the dynamic
fracture process of the first disk as a moving variable orifice. Let R(t)
be the ratio of the time-dependent opened area of the first disk to its
total area:
R(t) = Ri + (Rf - Ri) T A , 0< T < 1
R(t) = R f T > 1 , (118)
with t - ti
tf - t
where t i is the time when the first disk hits its knife edge, t f is the timewhen it has opened as far as it is going to, R i is the initial area ratio towhich the disk opens instantaneously at time t i , Rf is the final area ratioattained at time t f, and $ A determines the shape of the time-dependent areachange. The time t i is computed from the finite-element disk dynamics asdescribed previously, and R i , Rf , O A, and t f ti are input parameters. Let
U(t) be the time-dependent mean velocity of the unfractured portion of thefirst disk and Ui be the computed value of U when the disk hits the knifeedge. We take
and
U(t) = U1 [ 1 + (68 - 4)r + (3-60 v)r 2] , 0 < r 1
(119)U(t) = 0 , r > 1
where Ov is an input parameter that determines the shape of the time-
dependent disk velocity. The form of the coefficients in Eq. 119 is such
that the average value of U for the time interval t f - t i is BvUi.
1K n - I)
2 ,
o RCc
(121)
55
Steady-flow orifice relations are used to represent the effect of the
partially open disk on pulse transmission. The pressure drop is given by
Kop
2P1 m P2 2 uF
where pl is the pressure upstream of the first disk, p 2 is the pressure inthe gas space between the disks, and up is the flow velocity through the
partially open first disk. The loss coefficient Ko is given by
(120)
where Cc is the contraction coefficient. Miller 135) gives curves for both
Ko and Cc ; a reasonable fit to the average curve for C c is
Cc = 1 - 0.365 1 - R . (122)
Equations 118-122, the characteristic equations (60), and the gas
compression law are combined to predict the dynamic loading on the second
disk (see Ref. 36 for details). Then the finite- element treatment
described in Sec. III.D.1 is used to compute the motion of the second disk,
that is assumed to fail when it strikes its knife edge.
Another option available in SWAANHII is to use the standard calculation
for the first disk response and then to use an empirical pressure-time
relation in place of a calculation to describe the response of the second
disk. A table of pressure/time data is input and is used after the first
disk hits its knife edge. The last data point corresponds to the failure
time of the second disk. The data table should be based on experimental
evidence or an independent, more detailed computation of the disk assembly
behavior.
Finally, a simple technique that increases the time between failures of
the two disks but maintains the essentials of the standard disk dynamics
calculation is to artificially increase the volume of the gas space between
the disks. Because the delay time between failures is governed mainly by
the filling of the gas space by the moving sodium column, increasing the
available volume has the desired effect of increasing the delay time without
affecting the method of computing the disk behavior.
56
4. Shell Dynamics
The dynamics of the shell structure surrounding the two-dimensional
sodium transient region are developed based on the endochronic theory of
viscoplasticity proposed by Valanis [37]. The difference between
conventional plasticity and endochronic theory is that the latter does not
require a yield function as used in the flow theory. The flow theory is
based on the existence of an initial yield surface coupled with an assumed
hardening rule used to obtain subsequent yield surfaces. Calculations
relating to yield-surface behavior and logical checks on the position of the
yield surface are costly in computer time and storage requirements.
Endochronic plasticity, through its formulation in terms of a timelike
measure, which itself is a material property, dispenses with the need for a
yield surface and so offers potential saving of computer time.
In this section, the endochronic theory of plasticity is applied to the
axially symmetric motion of a finite-length, circular cylindrical shell
subjected to an arbitrary pressure pulse applied on its inner surface.
Consider a thin shell with mean radius R and thickness H. Let u and w
denote the average displacement in the axial x and radial r directions,
respectively, at time t of a cross section at distance x from a reference
section. Then the equations of motion in the x and r directions can bewritten as
aax
axr = au+ n
3x ' at(123)
3axr p(x,t) a6 _ aw
ax H R P Bt '
and
where p is the density, p
normal stress components in
the shear stress. The
coordinates are
is the hydrodynamic pressure, 4z:i x and 00 are thethe x and 6 directions, respectively, and a xr isstrain-displacement relations in cylindrical
1 awE - -xr 2 ar
and
(124)
57
The constitutive equations for the problems described here can be obtained
as follows:
3o aoaux
+ - d n d —at 1 3x 2 R
Dx
and
3ox
ao8 3u
ax atd3 iTt - d4
Daxr 3w w
Dt - 0 3x a5 R '
(125)
where u o is the shear modulus and (1 1 , d 2 , d3 , d4 , and d 5 are material
parameters involving both the material constants and the deformation
history.
These equations are then solved by the method of characteristics.
Equations along the characteristics are obtained as follows:
1/2xr
dox t pc du
dx + F dt, along AI n tc t[ °1 dt-
2)11)
dx ch1/2
doxr tp cs dw - F
2 dx - F
3 dt, along — n tc (126)dt
and
do e n do - —a— de - F de along dx 0 .x 1 + v x 4 e'
The coefficients F 1 0 F2 , F3, and F4 are functions of the material parameters
involved, E0 is the elastic modulus, and v is Poisson's ratio.
The characteristic equations together with the appropriate boundary
conditions provide solutions for the problem described here. For more
details of their derivation and discussion, see Ref. 38.
E. Fluid Property Calculations
1. Water
The basis for the water property computation is a formulation according
to Keenan [2] that relates the Helmholtz function to temperature and
58
specific volume. This single formulation for both liquid and vapor to
100Mloa and 1570K provides good agreement with available data on
thermodynamic equilibrium properties [39]. A subprogram "REG2" contains
this formulation: temperature and specific volume must be input, and the
Helmholtz function, together with its first and second derivatives, is given
as output. All thermodynamic equilibrium properties, such as enthalpy and
entropy, are linear functions of these variables. Transport properties,
such as viscosity and thermal conductivity, are not derivable from the
Helmholtz function and must be computed separately.
When the independent variable pair (defining the water property state)
is other than temperature and specific volume, an iteration process is used
to compute the values of temperature and specific volume corresponding to
the available independent variables. In all such cases, Newton's method is
used, with derivatives being calculated from Keenan's formulation. Thus all
water property computations will produce consistent values (within a
specific convergence criterion), irrespective of the choice of independent
variables. In addition to providing internal consistency, this scheme has
the advantages of ease in accommodating additional choices of independent
variables and requiring very little computer-memory storage. The
disadvantage of the scheme is that the complexity of Keenan's formulation
can result in excessive computer time for water property computation. As
described below, aproximate techniques are employed to compute in the two-
phase region and, as an option, to reduce computation time in single-phase
regions.
Water property values according to Keenan's formulation are taken to be
"exact" and are used as a basis for approximation schemes. As mentioned
above, the use of Keenan's formulation can involve considerable expenditure
of computer time so provision is made for less precise computation of water
properties. Approximation can be achieved easily with good precision along
the saturation line, where properties are continuous and are functions of a
single independent variable rather than two independent variables, as isgenerally the case.
Water properties along the saturation line are represented with the use
of cubic splines [40]. The saturation line is divided into segments within
which the property is approximated as a cubic polynomial of the independent
variable. The polynomial coefficients are calculated so that both the
function and its first derivative are continuous from segment to segment
along the saturation line. Then computation in the two-phase region is done
in terms of values of liquid and vapor on the saturation line. When aproblem is run, values along the saturation line are first calculated from
Fig. 16. Region Boundaries for
Approximate Computation of
Water Properties
59
Reenan's formulation, and cubic spline representations along the saturation
line are made for subsequent use. Also cubic spline coefficients thus
produced may be stored for use in future problems.
For a single phase (liquid or vapor), approximation is done in selected
subregions. Before a problem is run, subregions are chosen where most
property computation is expected to be done during execution of the
problem. Approximate property values computed during execution within these
subregions may be used when lower precision is acceptable, or may be taken
as initial values of the solution set for the interpolation scheme mentioned
above, using Keenan's formulation directly.
The approximation method chosen
for the single-phase region is
a surface-mapping technique
known as "transfinite
interpolation" Pell. Briefly,blending functions are used
that define a surface within a
four-sided boundary in terms of
values On the region
boundaries. For example, it is
desired to calculate pressure
when given values of internal
energy and specific volume inthe superheat region shown in
Fig. 16. Dummy variables Z1
and Z2 are introduced that
form a unit square in the Z
plane. A transformation is
sought between the variables
(e, v) and (Z 1 , Z2 ). The
region boundaries are expressed
in terms of the dummy
parameters (Z1 , Z2 ), as listed
in Table 2. The transforma-
tions for (Z 1 , Z2) in terns of
v) as found from the
expressions in Table 2 are
60
and
z i = (v - vA)/v c vit)
(127)
Z2 = (e - eAD
)/(eBC - e
AD).
Table 2. Expressions Used to Represent Internal Energy and
Specific Volume at Region Boundaries
Parameter
Segmenta
Z2 = 0 e(Z1, 0) = eAD(Z1)
v(Z1 , 0) = vA + Z1 (vr, - VA) AD
Z2 = 1 e(Zi, 1) = e(Z1)
v(Z i , 1) = vB + Zi ( vc vB) BC
Z1 = 0 e(0, Z2) = eA + Z2(eB - eA) v(0, Z2) = VA AB
Z1 = 1 e(1, Z2) =eD + Z2 (ec eD) v(1, Z2 ) = vc CD
aSee Fig. 16.
The blending function for pressure in terms of Z I and Z2 is of the form
P(Z I ,Z2) = (1 - Z 1 )P(0,Z2) + Z 1 P(1,Z2) + (1 - Z2 )P(Z 1 ,0) + Z2P(Z1,/)
- [(1 - Z 1 )(1 - Z2 )P(0, 0) + (1 - Z I )Z2P(0,1) (128)
+ z1 (1 - Z2)P(1
' 0) + Z
1Z2P(1 ' 1)]
where Z1 and Z2 are found from the above expressions in terms of internalenergy and specific volume. Functions P(0, Z2 ), P(1, Z2 ), P(Z i , 0), andP(Zi , 1) are pressures corresponding to values of internal energy and
specific volume (hence, also dummy parameters Z1 and Z as indicated inTable 2) on region boundaries AB, CD, AD, and BC, respectively, of Fig. 16.
Thus, any equilibrium property of superheated vapor may be readily
written in the form given above when four additional parametized curves are
provided for each new property. Variables along the region boundaries are
expressed by cubic splines. Property values used to derive the cubic-spline
coefficients are generated using Keenan's formulation. Pressure may thus be
calculated from extremely simple algebraic relations and computational time
reduced at the expense of precision.
61
2. Nitrogen Gas
The equation of state selected for nitrogen gas is the Nobel-Abel form,
p(v - • RT , (129)
where p is the absolute pressure, v is the specific volume, b is the co-
volume, R is the gas constant, and T is the absolute temperature. This
equation is a modified form of the perfect gas law, which uees a co-volume
parameter to improve its applicability at pressures of a few thousand pounds
per square inch. Specific heats and the ratio of specific heats are treated
as constants. Thus the following relationships apply for other state
variables:
. T P• C xn— - R &—P
To
Po
(130)u • Cv(T To)
i(_
kPvc1 v _ b
and
where s is the entropy, u is the internal energy, Cp is the specific heat at
constant pressure, Cv is the specific heat at constant volume, k is the
ratio of specific heats, and c is the velocity of sound. The constants used
are
b • 0.00607 ft3/1b,
R - 0.07092 Btu/lb °R,
k 1.4,
C kt/(k-1), and
CvR/(k-1),
and the reference state is
po - 14.696 psis,
To491.7 °R,
uo • 0, and
so 0.
3. Liquid Sodium
(131)
(132)
Temperature-dependent properties of liquid sodium are computed from
correlations recommended by Golden and Tokar [231. The specific weight y of
62
sodium in lb/ft 3 , calculated from Eq. 2.1 of Ref. 23, is
y = 59.566 - 7.9504 x 10-3T - 0.2872 x 10
-6T2
+ 0.06035 x 10-9 11 , 2080 F T 25000F
the corresponding density p in lb-sec2/ft4 is
P = Y/g
(133)
(134)
where g = 32.2 ft/sec 2 is the acceleration of gravity. The dynamic
viscosity g in lb-sec/ft 2 , using Eq. 5.19a of Ref. 23, is calculated from
g = (exp(2.303[1.0203 + 397.171(T + 460)
(135)
- 0.4925 log io(T + 460))))/(3600 x 32.2)
where T is in degrees Fahrenheit. Golden and Tokar recommend a linear
dependence of sound speed on temperature. Based on tabulated values in
their Appendix E, co (in ft/ sec) is calculated from
co = 8285 - 2187(T - 210)12290 . (136)
63
IV. INPUT DESCRIPTION
A. Input Data
FORTRAN
Card No. Name
Format Description
1
A4, 7L4 Choice of main options.
LABEL ABCD is the required label.
RUNA RUNA T: Sodium-side computation
(SODSID) is opted.
RUNA F: SODS/D is not opted.
RUNS RUNS - T: Water-side computation
(WATSID) is opted.
RUNS F: WATSID is not opted.
RUNC RUNC T: Two-dimensional sodium-side
flow transient (NA2D) is opted.
RUNC F: NA2D is not opted.
RUND RUND T: Shell dynamics (SHELL) is
opted.
RUND F: SHELL is not opted.
CALINK CALINK • T: Boundary link between
SODSID and NA2D is opted.
CALINK F: No link is opted.
CDLINK CDLINK T: Boundary link between NA2D
and SHELL is opted.
CDLINK F: No link is opted.
REACT REACT .• T: Reactive water-injection
analysis.
REACT F: Nonreactive nitrogen-
injection analysis.
1. Input Data for Run A (Sodium-Side)
Al A4, 714 Sodium system parameters.
LABEL £001 is the required label.
NPIPM Number of pipes in main system.
Number
having
Number
having
Number
Number
of thin-shell
single membran
of thin-shell
double membran
of pumps.
of prescribed
rupture disks
e.
rupture disks
e.
pressure pulse
A3
LABEL
PINIT
QINIT
TEMP
PCAV
A4, 6F8.0
64
FORTRANCard No. Name Format
NJUNM
NPIPR
NJUNR
NRLFC
KFLUID
KFRIC
Description
Number of junctions in main system.
Number of pipes in relief system.
Number of junctions in relief system.
Number of relief systems.
If KFLUID = 1, the fluid is sodium. If
KFLUID = 2, the fluid is water.
If KFRIC = 0, pipe friction is
neglected. If KFRIC = 1, pipe friction
is included.
Sodium system parameters.
A002 is the required label.
Number of surge tanks with gas space.
Number of constant-pressure boundaries.
Number of rupture disks (instantaneously
opening).
sources.
Number of prescribed energy-loss
junctions.
A2 A4, 814
LABEL
NOSRG
NOCPB
NOINRD
NOSGRD
NODBRD
NOPUMP
NOPULS
NOENLS
Sodium system parameters.
A003 is the required label.
Initial system pressure, psig.
Initial system flow rate, ft3/sec.
System temperature, °F.
Cavitation pressure, psig.
6 5
FORTRAN
Card No. Name Format Description
PRELF Relief system back pressure, pug.
PGAGZ Gauge pressure zero on absolute scale,
psia.
A4 A4, 2F8.0 Problem parameters.
LABEL A004 Is the required label.
DT Time step, seconds.
TFIN Time at which calculation terminates,
seconds.
A5 A4, 1514 Output parameters. XPRTS and KPPRTS are
input as pairs, 140NPRTS.
LABEL A005 is the required label.
NPRTS Number of output specification ranges,
14NPRTS47.
KPRTS(E) Starting cycle number for printout of
results at frequency KFPRTS(R).
KFPRTS(K) Frequency of printout for cycles
starting at KPRTS(K). If KFPRTS(K)..1,
results for every cycle between RPRTS(K)
and KPRTS(K+I) are printed. If
KFPRTS(0. 5, results for every fifth
cycle are printed, etc.
A6 Set Main system pipe data, 141ANPIPM.
Ma A4,614, 678.0
LABEL AO6A is the required label.
LPIPE(L) Pipe identification number.
JI(L) First-node junction number of pipe.
JN(L) Last-node junction number of pipe.
MAT(L) Material number for pipe.
66
FORTRAN
Card No. Name Format Description
MAT = 1: Type 304 stainless steel at
elevated temperature.
MAT = 2: Type 316 stainless steel at
elevated temperature.
MAT = 3: Bilinear stress-strain
relation for carbon steel.
MAT = 4: Type 304 stainless steel at
room temperature.
MAT = 5: Bilinear stress-strain
relation for 2.25Cr-lMo at 600°F.
MAT = 6: Rigid pipe wall.
INCOND(L) If INCOND=0, initial conditions for pipe
are set to system conditions input on
Card A3. If INCOND= 1, initial
conditions for pipe are input on Card
A6 b.
IPRIN(L) Detail of printout for pipe. If
IPRIN= 1, results are printed for every
node. If IPRIN=3, results are printed
for every third node, etc. If IPRIN=0,
no results are printed for this pipe.
If INPRIN>99, results for only the end
nodes are printed.
D(L) Inner diameter of pipe, inches.
11(L) Wall thickness of pipe, inches.
PLNGTH(L) Pipe length, feet.
RRF(L) Relative roughness of pipe wall.
ALFA(L) Pipe angle with horizontal (positive
upward), degrees.
A(L) Flow area of pipe, in 2 . If A=ITD2 /4, set
A=0 and it will be computed from D.
67
FORTRAN
Card No. Name Format Description
A6b A4, 4P8.0 Omit this card for pipe LPIPE(L) if
/NC0ND(L)0.
LABEL A06b is the required label.
POI(L) Initial pressure in pipe at first-node
end, psig.
PON(L) Initial pressure in pipe at last-node
end, pug.
U01(L) Initial velocity in pipe at first-node
end, ft/sec.
UON(L) Initial velocity in pipe at last-node
end, ft/sec.
Cards A6a and A6b are input in pairs for
each pipe for which A6b is needed.
Al A4,514,6F8.0 Relief system pipe data, 14L4NPIPR.
LABEL A007 is the required label.
LPIPE( Pipe identification number.
First-node junction number.
JN(L) Last-node junction number.
MAT(L) Material number.
IPRIN(L) Detail of printout for pipe.
D(L) Inner diameter, inches.
Wall thickness, inches.
PLNGTH(L) Pipe length, feet.
L) Relative roughness of pipe wall.
ALFA(L) Pipe angle with horizontal (positive
upward), degrees.
A(L) Flow area of pipe, in2'
A8 A4, 1614 Junction type description Cards for main
system, 14ANJUNM. Eight junctions per
card.
LABEL A008 is the required label.
68
FORTRAN
Card No. Name Format Description
JUN(J)
JTYPE(J)
Junction identification number.
Junction type:
JTYPE = 1: Area change, no pressure
drop.
JTYPE = 3: Tee junction (three to six
branches).
JTYPE = 4: Pump junction.
JTYPE = 5: Variable-pressure surge tank
with gas space.
JTYPE = 6: Acoustic-impedance
discontinuity (no area change).
JTYPE = 6: Dummy junction.
JTYPE = 7: Closed end.
JTYPE = 8: Constant-pressure boundary.
JTYPE = 9: Far end (nonreflecting).
JTYPE = 10: Instantaneous rupture disk
with prescribed failure pressure or
failure time.
JTYPE = /1: Single membrane thin-shell
rupture disk.
JTYPE = 12: Double membrane thin-shell
rupture disk.
JTYPE = 15: Prescribed pressure pulse
source.
JTYPE = 17: Bubble junction for sodium-
water reaction.
JTYPE = 18: Bubble junction for gas-
injection source (nonreacting).
JTYPE = 22: Area change, standard
pressure drop.
JTYPE = 23: Prescribed energy loss.
69
FORTRAN
Card No. Name Format Description
A9 A4,I614 Junction type description cards for
relief system, I(J4NUUNR. Eight
Junctions per card. Omit if NJUNR=0.
LABEL A009 is the required label.
JUN(J) Junction identification number.
JTYPE(J) Junction type:
JTYPE = I: Sudden expansion or
contraction.
JTYPE = 3: Tee junction (three to six
branches).
JTYPE = 6: Acoustic-impedance
discontinuity (no area change).
JTYPE = 6: Dummy junction.
JTYPE = 7: Closed end.
JTYPE = 8: Constant-pressure boundary.
JTYPE = 9: Far end (nonreflecting).
JTYPE = 22: Area change, standard
pressure drop.
JTYPE = 23: Prescribed energy loss.
A10 A4, 214 Relief system connection points,
14R(NRLFC. Omit if NRLPC = 0.
LABEL A010 is the required label.
JRFI(K) Junction number in main system where
relief system is attached (must be at
last-node end of pipe).
JRF2(10 Corresponding junction in relief system
(must be at first-node end of pipe).
All
A4,14,6F8.0 Data for surge tanks with gas space,
140N0SRG. Omit if NOSRG = O.
LABEL
A011 is the required label.
70
FORTRAN
Card No. Name Format Description
JSRG(K) Junction number to which the surge tank
is connected.
GSRG(K) Gas compression exponent.
PSRG(K) Initial gas pressure, psig.
VSRG(K) Initial gas volume, ft3.
ASRG(K) Cross-sectional area of surge tank,
ft2 . If ASRG = 0.0 is input, ASRG will
be set to 1.0 ft2.
PBRG(K) Burst pressure of rupture disk on gas
space, psig. If no rupture disk is to
be modeled, input PBRG = 0.0.
PRLG(K) Back pressure behind rupture disk, psig.
Al2 A4,I4,F8.0 Data for constant-pressure boundaries,
1<K5NOCPB. Omit if NOCPB = O. Up to
six pipes may be connected at boundary.
LABEL A012 is the required label.
JCONP(K) Junction number at constant-pressure
boundary.
PCONP(K) Pressure at constant-pressure boundary,
psig.
A13 A4,I4,3F8.0 Data for instantaneous rupture disks,
l<K<NOINRD. Omit if NOINRD = O.
LABEL A013 is the required label.
JRD(K) Junction number at which the rupture
disk is connected.
PRDB(K) Failure pressure of rupture disk, psig.
PRDG(K) Back pressure behind rupture disk, psig.
TRDB(K) Specified time of rupture-disk failure,
seconds. If TRDB(K) = 0.0, pressure
criterion for failure is used.
71
FORTRAN
Card No. Name Format Description
£14 Set
Data for single-membrane rupture disks,
lc ONOSGRD, input on Card Sets
14*-14d. Omit if NOSGRD n O.
£14. A4,14,1,8.0,214
LABEL Al4A is the required label.
JTS11D(K) Junction number at rupture disk.
PRACK(K) Back pressure behind rupture disk, psig.
KINITW If KINIT n 1, initial stress
distribution is computed from system
pressure.
If KINIT n 0, initial stresses in disk
are not computed.
NTPRD(K) NTPRD 0 corresponds to single membrane
model.
NTPRD>0 is number of data points for
second membrane prescribed pressure-time
response after first membrane failure.
A14b A4,I4,7F8.0
LABEL Al4B is the required label.
NUNEL(E) Number of elements representing the
disk. Eight elements are recommended
for a normal calculation. Eight to 50
elements are permissible.
RAD(K) Radius of curvature of disk, inches.
DIA(K) Diameter of base area of disk, inches.
TH(K) Thickness of disk, inches.
YN(K) Elastic modulus of disk material, psi.
PSSNR(K) Poisson's ratio of disk material.
GAWK) Density of disk material, lbs/ft3.
CKNIFE(K) Clearance between cutting-knife edge and
disk center, inches.
72
FORTRAN
Card No. Name Format _Description
A14c A4,3F8.0
LABEL A14C is the required label.
SICK) Yield pressure of disk material, psi.
If SY(K) = 0, elastic response is
assumed internally.
EP(K) Plastic modulus in bilinear stress-
strain relationship, psi.
SU(K) Ultimate strength of disk material, psi.
A14d A4,8F8.0 UcKINNTPRD(K). Omit if NTPRD(K) = O.
Four data pairs per card.
LABEL Al4D is the required label.
TRDPR(K,KK) Time data point for prescribed second
membrane response, seconds.
PRDPR(K,KK) Pressure data point for prescribed
second membrane response, psig.
A15 Set Data for double-membrane rupture disk
assemblies, l<K<NODBRD, input on Card
Set 15a-15F. Omit if NODBRD = O.
A15a A4,I4,4F8.0,
LABEL 214 A15A is the required label.
JRDUB(K) Junction number at which the double-
membrane disk assembly is connected.
PBACK1(K) Initial pressure in the gas space
between membranes, psig.
PBACK2(K) Back pressure behind the second membrane
(same as the relief-system pressure),
psig.
VRDUB(K) Gas-space volume between the membranes,
cubic feet.
73
FORTRAN
Card No. Name Format Descrietion
GRDUB(K)
Polytropic exponent representing the
compression process of the gas in the
gas space.
KINIT1(K)
If KINITI n 1, initial stress
distribution is computed from system
pressure.
If KINIT1 n 0, initial stresses are not
computed.
KOPEN(K)
If KOPEN n 0, first membrane opens
completely and instantaneously upon
hitting knife edge.
If KOPEN n 1, data are input on Card
A15f to model the failed first membrane
as a variable orifice.
Al 5b
A4,I4,7F8.0 Data for the first membrane.
LABEL Al5B is the required label.
NUMELl(K)
RAD1(K)
DIA1(K)
TH1(K)
Description is the same as for the
TM! (K)
single—membrane disk, Card A14b.
PSSNR1(K)
GAM1(K)
CKNF1(K)
Al5c A4,3F8.0 Data for the first membrane.
LABEL Al5C is the required label.
STICK) Description is the same as for the
EP1(1) single—membrane disk, Card A14c.
SU1(K)
74
FORTRAN
Card No. Name Format Description
And
A4,I4,7F8.0 Data for the second membrane.
LABEL A15D is the required label.
NUMEL2(K)
RAD2(10
DIA2(10
TH2(10
Description is the same as for the
YM2(10
single-membrane disk, Card A14b.
PSSNR2(K)
GAM2(K)
CKNF2(K)
A15e A4,3F8.0 Data for the second membrane.
LABEL Al5E is the required label.
SY2(K) Description is the same as for the
EP2(10 single-membrane disk, Card A14c.
SU2(K)
A15f A4,5F8.0 Omit if KOPEN(K) = O.
LABEL A15F is the required label.
TOPEN(K) Time duration for first membrane to open
from AINL to AFIN, seconds.
AINL(10 Ratio of initial open area to total area
of failed first membrane.
AFIN(10 Ratio of final open area to total area
of failed first membrane.
BTAR(K) Area variation parameter.
BTVE(K) Membrane velocity-decay parameter.
A16 A4,214,F8.0 Data for pumps, 14 1NNOPUMP. Omit if
NOPUMP = O.
LABEL A016 is the require label.
JPUMP(K) Pump junction number.
LPUMP(K) Pump discharge-pipe number.
75
FORTRAN
Card No. Name FormatDescription,
HEAD(K) Pump head (constant), psi.
Al7 Set
Data for K pulse sources, 14K4NOPULS,
input on Card Set Alla, A17b. Omit if
NOPULS n O.
Ails A4,214
LABEL AllA is the required label
JPULS(K) Junction number at pressure pulse
source.
NPULD(R) Number of data points for pulse source.
Al7b
A4,8F8.0 14KX5NPULD(K). Four data pairs per
card.
LABEL A17B is the required label.
TSRC( K dat) Time data point for pulse source,
seconds.
PSRC(K,KK) Pressure data point for pulse source,
psi g.
A18 A4,314,2F8.0 Data for energy-loss junctions
14K4NOENLS. Omit if NOENLS • O.
LABEL A018 is the required label.
JENLS(K) Junction number at energy loss.
LENLSL(K) Left-hand pipe number at junction.
LENLSR(K) Right-hand pipe number at junction.
CKL(K) Energy-loss coefficient for flow from
left to right.
CKR(K) Energy-loss coefficient for flow from
right to left.
76
Card No. Name Format Description
A19 A4,5F8.0 Reaction bubble dynamics data. Omit if
there are no injection-source junctions
(JTYPE = 17 or JTYPE = 18).
LABEL A019 is the required label.
LAMBDA Sodium-water reaction-rate coefficient,
ft/s.
HS Heat-loss parameter to inert
surroundings, Btu/ft3/°F/s.
HF Heat-loss parameter to liquid sodium at
flame front, Btu/ft3/°F/s
BR Moles of hydrogen generated per mole of
water reacted.
C3R Moles of NaH generated per mole of water
reacted.
A20 Set
Water injection rate table input on Card
Set A20a, A20b.
Omit if RUNB = T or if Card A19 is
omitted.
A20a A4, 14, F8.0
LABEL A20A is the required label.
NINJIN Number of data points.
PBINL Initial bubble pressure, psig.
A20b A4, 6F8.0 14K4NINJIN. Two data triplets per card.
LABEL A20B is the required label.
TIMEIN(K) Time value for water injection rate
table, seconds.
MDOTIN(K) Water injection rate at time TIMEIN,
lb/sec.
SSHIN(K) Stagnation enthalpy of injected water,
ft2/sec2.
77
FORTRAN
Card No. Format Description
A2I
A4,314 Bubble connections. Omit if RUNC 0 F.
LABEL A021 is the required label.
JBUBL Junction number at reaction bubble.
LBUBLI Left-side pipe number at bubble.
LBUBLN Right-side pipe number at bubble.
2. Input Data for Run B (Water-Side)
BI
A4,6I4 Water system parameters.
LABEL B001 is the required label.
NTUBE Number of tubes in the water-side
system.
NJCN Total number of junctions.
NORFJ Number of junctions having orifices.
NRSV Number of reservoirs.
NRDSK Number of rupture disks.
KSNIC If KSNIC • / is specified, supersonic
outflow is not allowed. It is replaced
by sonic conditions for outflow
junction. If KSNIC 0, supersonic
outflow is permitted.
52 A4,5118.0 Water system parameters.
LABEL 9002 is the required label.
DX Node spacing in all tubes, feet.
PRESO Initial system pressure, psig.
TEMPO Initial system teperature, 'F.
QUALO Initial system quality. If 04QUAL041,
PRESO and QUALO are used to determine
the initial state. If QUALM, then
PRESO and TEMPO are used.
FLOWO Initial system mass-flow rate, lb/sec.
78
FORTRAN
Card No. Name Format _Description
B3 A4,3F8.0 Omit if RUNA = T.
LABEL 3003 is the required label.
TFIN Time at which calculation terminates,
seconds.
PBREAK Constant back pressure at break, used
for stand alone computation, psig.
PGAGZ Gauge pressure zero on absolute scale,
psia.
B4 A4, 1514 Output parameters. ICOUT and IFOUT are
input as pairs, 1(K<NTOUT.
LABEL B004 is the required label.
NTOUT Number of output specification ranges,
l(NTOUT47.
ICOUT(K) Starting cycle number for printout of
results at frequency IFOUT(10.
IFOUT(K) Frequency of printout for cycles
starting at ICOUT(K). If ICOUT(K) = 1,
results for every cycle between ICOUT(K)
and ICOUT(K + 1) are printed. If
IFOUT(K) = 5, results for every fifth
cycle are printed, etc.
B5 Set Tube data, 14L<NTUBE, input on Card Set
B5a, B5b.
B5a A4,614,4F8.0
LABEL BO5A is the required label.
LTUBE(L) Tube number.
LSYS(L) System number to which tube number LTUBE
belongs. For main system, LSYS = 1 must
be input. For subsequent systems, LSYS
must be sequential, i.e., 2, 3, 4.
79
FORTRAN
Card No. Name Format Description
JCN1(L) Junction number at first node of tube.
JCNN(L) Junction number at last node of tube.
NI(L) Number of initial-condition data points
for tube. If Ni n 0, initial conditions
input on Card 62 are used for tube.
NDOUT(L) Node increments for output for tube.
For instance, if NDOUT n 5, results for
every fifth node are printed.
TUBLEN(L) Tube length, feet.
DIAM(L) Tube diameter inches.
RRUP(L) Relative roughness of tube wall.
ALFA(L) Tube angle with horizontal (positive
upwards), degrees.
B5b
A4,5F8.0 Initial-conditon data, K Cards,
lc ONI(L). Omit if NI(L)-0.
LABEL 8058 is the required label.
PIN(K) Initial pressure at X n DIST, psig.
TIN(R) Initial temperature at X n DIST, °F.
QIN(K) Initial quality at X n DIST.
FLOWIN(K) Initial flow rate at X n 01ST, lb/sec.
DIST(K) Distance from first-node end of tube,
feet.
B6 A4,16I4 Junction type description Cards,
14AMJCN. Eight junctions per Card.
LABEL 8006 is the required label.
JCW(J) Junction number.
JCNTYP(J) Junction type for junction number
JCN(J):
JCNTYP n 1: Break end (fully open).
JCNTYP n 2: Break end with orifice.
8 0
FORTRAN
Card No. Name Format Description
JCNTYP = 3: Reservoir.
JCNTYP = 4: Reservoir with orifice.
JCNTYP = 5: Dummy junction.
JCNTYP = 6: Orifice.
JCNTYP = 7: Sudden area change.
JCNTYP = 8: Area change with orifice.
JCNTYP = 9: Nonreflecting end.
JCNTYP = 10: Closed end.
JCNTYP = 11: Tee junction (3 branches)
B7 A4,I4,2F8.0 Data for orifices, 141NNORFJ. Omit if
NORFJ = O.
LABEL B007 is the required label.
JORF(K) Junction number where orifice is
located. It should be identified as
type 2, 4, 6, or 8 on Card B6.
ORFDIA(K) Orifice diameter, inches.
ORFTC(K) Orifice time constant to represent time-
dependent orifice area opening.
B8 A4,I4,6F8.0 Data for reservoirs, 1 4 K5NRSV. Omit if
NRSV = O.
LABEL B008 is the required label.
JCWRSV(K) Junction number at reservoir.
RVP(K) Reservoir pressure, psig.
RVT(K) Reservoir temperature, °F.
QRSV(K) Reservoir quality.
VRSV(K) Volume of reservoir, cubic feet. If
VRSV(K) = 0, the reservoir is treated
internally as a constant pressure and
temperature boundary.
8 I
FORTRAN
Card No. Name Format Description
FRSV(K)
GRSV(K)
Fraction of volume VRSV(K) occupied by
noncondensible gas. If FRSV(K) n 0,
only one component.
Ratio of specific heats or polytropic
constant of the noncondensible gas.
39 44,2I4,F8.0 Data for rupture disks, 14K4NRDSK. Omit
if NRDSK = O.
LABEL 3009 is the required label.
JCNRD Junction number.
LRDSYS System number that becomes active after
failure of the rupture disk.PRDSK Rupture pressure, psig.
3. Input Data for Run C (Two-Dimensional Sodium Calculation)
Cl
A4,614,F8.0 Problem parameters.
LABEL C001 is the required label.
MX Number of grid points in x direction
(axial).
NY Number of grid points in y direction
(radial).
IPRT Frequency of result printout. If IPRT =
1, every step; if 5, every fifth step,
etc.
NPRT Number of initial steps to be skipped
before printer output starts. If NPRT =
100, first 100 step results are not
printed, etc.
NXOUT Increment in x nodes for printer
output. If NXOUT = 1, every node result
in x; if 5 every fifth node result, etc.
NYOUT Increment in y nodes for printer output.
82
FORTRAN
Card No. Name
Format
Description
YL
Length in y dimension of the
computational domain, feet.
C2 A4, 614 Baffle description card. Two baffles of
either center-open or periphery-open
type are modeled. Spacer plates with
uniform flow holes are not to be
considered here. 14K<2. Data for both
baffles are input on the same card.
LABEL C002 is the required label.
IBAF(K) I location (x-grid number) of baffles.
K = 1 for first baffle and K = 2 for
second baffle. If IBAF(K) = 0, no
baffle.
JBAF(K) Starting J location (y-node number of
solid area of baffle K.
JBAF2(K) Ending J location (y-node number) of
solid area of baffle K. The area not
covered by this range of J locations
will be the open area for flow passage.
C3 A4,6F8.0 Omit if RUNA = T.
LABEL C003 is the required label.
XB Axial location of bubble center, feet.
XL Length in x of the computational region,
feet.
RHO Fluid density, lb/ft3.
Fluid sonic speed, ft/s.
PO Initial pressure (uniform), psig.
UO Initial x velocity (uniform), ft/s.
Initial y velocity is set uniformly to
zero internally.
8 3
FORTRAN
Card No. Name Format Description
C4 A4,I4,F8.0 Omit if RUNA n T.
LABEL C004 is the required label.
NPTSIN Number of entry points for prescribed
bubble-condition input.
TFIN Finish time of computation, seconds.
C5 A4,6F8.0 Table for bubble pressure and volume
histories, 14K4NPTSIN. Omit if RUNA
T. TWo data triplets per card.
LABEL C005 is the required label.
TIMEIN(K) Time entries for corresponding bubble-
pressure and -volume inputs, seconds.
PBUBIN(K) Bubble pressure, psig.
VBUBIN(K) Bubble volume, cubic feet.
4. Input Data for Run D (Shell Deformation)
D1 A4,6I4,F8.0
LABEL 0001 is the required label.
MAT Integer parameter for choice of shell
material properties. If MAT n 1,
properties of 2-1/4Cr-1M0 steel at 316°C
(600°F) are provided internally. If MAT
n 2, input must be made by the user.
(See Card D3.)
II Number of nodes in axial coordinate.
IX Frequency of printout of results in
axial coordinate.
IT Frequency of printout in time step. If
IT n 1, every step; if 5, every fifth
step, etc.
84
FORTRAN
Card No. Name Format Description
IB/
Axial boundary condition at I = 1:
I81 = 1: Stress free.
I81 = 2: Nonreflecting.
IB1 = 3: Hinge supported.
I81 = 4: Plane of symmetry.
IBII
Axial boundary condition at I = II:
IBII = 1: Stress free.
IBII = 2: Nonreflecting.
IBII = 3: Hinge supported.
IBII = 4: Plane of symmetry.
TH
Shell thickness, inches.
D2 A4,7F8.0 Input for stand-alone computation. Omit
if RUNC = T.
LABEL D002 is the required label.
XL Length of shell, feet.
Mean radius of shell, inches.
TF Finish time of computation, seconds.
PMAX Pressure parameter for input loading-
shape history, psig.
TCONST Time constant for input loading-shape
history, seconds.
RI Constant parameter for x distribution,
ft-2 .
R2 Constant parameter for time
distribution, s-1.
Loading-shape history functional:
p(x,t) = PMAX ( t ) exp(-R1 x2)TCONSTfor t < TCONST,
p(x,t) = PMAX exp(-R2 exp(-R1 x2)
for t > TCONST.
85
FORTRAN
Card No. Name Format Description
D3 A4,5F8.0 Material properties. Omit if MAT ,• 1 on
Card Dl.
LABEL 0003 is the required label.
EO Young's modulus, psi.
EN Tangent modulus of material at large
strain, psi.
SO Intercept of the asymptotic straight
line of stress-strain curve with stress
axis (slightly less than the yield
strength), psi.
NU Poisson's ratio.
RHO Density of shell material, slugs/in3.
8 6
B. Notes on Input Data and System Modeling
1. General
For convenience and to minimize confusion in the input descriptio n , the
one-dimensional flow channels in the sodium system are referred to as
"pipes" while those in the water system are referred to as "tubes".
The identification numbers for pipes, tubes, and junctions can be
assigned arbitrarily, and consecutive numbers need not be used. The water
and sodium systems are numbered independently, so there can be a junction
#10 in the water system and a different junction #I0 in the sodium system.
However, pipe and junction numbers cannot be duplicated between the sodium
system and relief system.
The positive direction for velocity in pipes and tubes is from the
first-node end toward the last-node end.
A pipe or tube angle is positive if the pipe or tube slopes upward from
its first-node end toward its last-node end.
All input and output pressures are in psig, except for PGAGZ on Cards
A3 and B3, which is in psia. PGAGZ locates the origin of the gauge scale on
the absolute scale. The system dynamics calculation depends only on
pressure differences and can be based on gauge pressures. However, the
water/steam properties and the perfect gas law computations used for gas
spaces both need absolute pressures; PGAGZ is added to the system pressure
to get absolute pressure when needed.
The array dimensions that determine the size of the system that can be
analyzed are summarized in Sec. VI. The alteration of these dimensions to
handle more complex systems or reduce core storage is discussed there also.
Each input card has an identification label in the first four
columns. The program checks this label against the label it is expecting
and stops the calculation if they do not match. This labeling is included
to prevent the misordering of input cards or the omission of needed data.
Generally, the input requirements for SWAAM-II are set up so that it is
not necessary to input data that are not needed for a particular problem.
For instance, if there is no rupture disk in a system, no rupture disk data
cards are needed. Consequently, simple system models can be analyzed using
very little input data.
87
2. RUNA (SODSID)
The program expects all relief system pipes to fill from their first-
node end toward their last-node end, so the node end designations must be
made accordingly on Cards Al. For the same reason, multiple relief systems
must be parallel, i.e., nonintersecting, so that the direction of filling of
each relief system pipe can be defined uniquely. The main system pipe
connected to a rupture disk that forma the interface with a relief system
must have its last-node end connected to the rupture disk. All other end
node designations in the main sodium system are arbitrary.
Each interface with a relief system has two junction numbers--one on
the main sodium system side and one on the relief system side; this
connectivity is identified on Card A10. The main system junction connected
at a rupture disk has a rupture disk junction type specified on Card M.
The first junction in the relief system connected to the same disk has a
different junction number and has the junction type (specified on Card A9)
which the junction assumes after the disk fails.
The water system can be replaced by a prescribed water injection leak
rate history by letting RUNBA. F and including Card A20. To use prescribed
pressure pulses rather than a sodium-water reaction as the source of the
pressure transient, omit the bubble junction (JTYPE .. 17) and use pressure
pulse source junctions (JTYPE15). Up to three different pulses can be
input and up to six pipes can be connected at each pulse source junction.
Card Al. The array sizes* in SWAAM-II require that 1 < NPIPM + NPIPR
< 65, 1 < NJUNM + NJUNR < 66, and 0 < NRLFC < 10.
KFLUID 1 is the standard computation. KFLUID 2 may be used to
model water loop simulations of piping transients.
URIC 1 is the standard computation. URIC 0 may be used to omit
the pipe friction effect without setting all the wall roughnesses to zero.
Card A2. The input quantities on this card indicate the number of each
type of junction that requires special input data. These input data are
then given on Cards All through A18. Storage allotments are such that
NOSRG, NOCPB, NOINRD, and NOPUMP cannot exceed 10; NOSGRD, NOD8RD, and
NOPULS cannot exceed 3; and NOENLS cannot exceed 25.
*See Section VI for information on altering these storage allotments.
88
Double membrane rupture disks where the response of the second disk is
prescribed rather than being calculated are included in NOSGRD rather than
NOD BRD.
Card A3. Initial conditions are set to PINIT and QINIT for each pipe
in the main (HITS) system for which initial conditions are not input
individually on Card A6b. The initial pressure is set to PRELF for each
relief system pipe.
The initial flow, QINIT, has the same sign convention as velocity, so
the designation of first and last node ends must be made so as to assure
flow continuity if QINIT > O. Care should be used in specifying a nonzero
value for QINIT if the pipes are not connected in series. For branching
systems, a nonzero QINIT probably will give an initial unbalance of flows at
tees and result in large initial pressure spikes at these junctions.
Sodium properties are computed for temperature TEMP. The structural
properties of temperature-dependant piping materials MAT=1 and MAT=2 are
computed at TEMP also.
PCAV is the pressure at which sodium cavitation is caused by a
decompression wave; usually it is equal to -PGAGZ. To suppress the
cavitation treatment, set PCAV equal to a large negative number.
PGAGZ is the location of zero gauge pressure on the absolute scale. If
zero gauge pressure is atmospheric pressure, then PGAGZ = 14.7 psis. PGAGZ
is needed for computing the compression of gas spaces, where absolute
pressures are required.
Card A4. The time step DT is used in conjunction with the
computational stability criterion to determine the node spacing DX in each
pipe and, thereby, the number of nodes NNODE in each pipe. Pipe lengths are
adjusted to become an integral multiple of DX. If NNODE exceeds IMAX = 100
in any pipe, DT is increased by factors of two until NNODE < 100 for every
pipe. If this is undesirable, the user can either split up the longer pipes
in the system, using dummy junctions, or increase IMAX, as described in Sec.
VI.
Card A6a. If IPRIN(L) = 3, for example, results are printed for the
first, fourth, seventh, tenth, etc., axial nodes. Results for the last node
also are printed.
R9
The diameter D and flow area A of each pipe can be input individually
to permit modeling of components where A 0 vD2 /4. The diameter is used in
computing the friction faster for pipe losses, the area is used in computing
flow continuity at junctions, and the ratio of diameter D to thickness H Is
used to compute the wave speed in the fluid in the pipe. Consequently, it
may be possible to contrive values of D, H, and A to model an irregularly
shaped channel. If A is left blank, it will be computed from A n 1D2/4.
ALFA is used in computing the gravity head for nonhorizontal pipes.
Card A6b. This card is included only for those pipes for which INCOND
n 1. Linear interpolation is used to set the initial pressure and fluid
velocity at interior nodes of the pipe.
Card A7. This card gives the pipe data for each relief system pipe and
contains the same information specified for the main sodium system piping on
Card 6a except for INCOND(L). The initial pressure in each relief system
pipe is set to PRELF, input on Card A3.
First and last node designations in a relief system must be such that
each pipe will fill from the first-node end toward the last-node end.
Card AB. Each closed end (JTYPEn7) and far end (JTYPEn9) junction must
have only one pipe connected to it. The area change junctions (JTYPE n
1,22,23), acoustic impedance discontinuity (JTYPE n6), and dummy junction
(JTYFEn6) must have two pipes connected to them. The bubble junctions for
the sodium water reaction (ITYPEn 17) and inert gas injection (JTYPEn 18) can
have up to three pipes connected to them. Each tee (JTYPEm3), pump
(JTYPEn4), surge tank* (JTYPEn5), constant pressure boundary (JTYPE n8), and
prescribed pulse source (ITYFEn 15) can have up to six pipes connected to it.
Each rupture disk (JTYPEn 10,11,12) can have only one main sodium system,
pipe connected to it, which must be the last-node end if there is a relief
system attached. The connection to the relief system is specified on Card
£10.
Both pipes connected at an acoustic-independance discontinuity or dummy
junction (.JTYFEn6) must have the same flow area. In the case of the
*A surge tank that has a rupture disk on its gas space may have only onepipe connected to it (see "Card All" discussion).
90
acoustic impedance discontinuity, however, they need not have the same
thickness or be made of the same material. The dummy junction is useful for
breaking up long pipes or for identifying a point where instrumentat ion is
located in making comparisons with experimental results.
Card A9. The relief system junction connected to the rupture disk must
be a first-node end. Its junction type is the type the junction assumes
after the rupture disk fails--usually JTYPE=1, 6, 22, or 23. If it is
JTYPE=23, remember to count this prescribed energy loss junction in with the
total NOENLS specified on Card A3.
Card All. Up to six pipes may be connected to a surge tank when no
relief system is attached. If a relief system is attached via a rupture
disk on the gas space, only one pipe connection is allowed; moreover, the
last-node end has to be connected to the surge tank junction.
GSRG is the exponent in the PVY gas compression process.
The quantities ASRG, PBRG, and PRLG are needed only if there is a
rupture disk on the gas space. After disk rupture at pressure PBRG, the
junction becomes a constant pressure boundary at pressure PRLG. The height
of the gas space at disk rupture, computed as the ratio of final volume to
cross-sectional area, is printed in the output.
Card A13. If TRDB > 0 is specified, the disk fails automatically at
that time, regardless of the applied pressure. If TRDB = 0 is specified,
the disk acts as a rigid closed end until the pressure reaches PRDB. After
failure, the junction becomes a constant pressure boundary at pressure PRDG.
Card A14. JTYPE=11 can represent either a single-membrane spherical
cap rupture disk if NTPRD=0 or a double-membrane spherical cap rupture disk
with a prescribed second membrane response if NTPRD > 0.
If no relief system is attached, the rupture disk is replaced after
failure by a constant pressure boundary at pressure PBACK. If there is an
attached relief system, the rupture disk junction is replaced by the
associated relief system junction from Card A10.
The disk is assumed to fail when the membrane hits the cutting knife
edge, i.e., when the disk center is displaced the distance CKNIFE.
NTPRD > 0 gives the number of data points in a prescribed pressure
history for second membrane behavior beginning after the first membrane
91
fails. TRDPR is measured from the time of failure of the first membrane,
and the second membrane is assumed to have failed at the end of the time
data points. Linear interpolation for pressure is used for times between
the tabulated points.
Card 15. The gas between the membranes is compressed according to the
law PVY n constant, where the initial pressure and volume are PBACKI and
VRDUB, respectively, and Y • GRDUB.
If no relief system is attached, the rupture disk junction is rep/aced
after both membranes fail by a constant pressure boundary at pressure
PBACK2. If there is an attached relief system, the rupture disk junction is
replaced by the associated relief system Junction from Card AIO.
If KOPEN 0, the first membrane is assumed to open fully when it
strikes the knife edge. If KOPEN n 1, data are input on Card A151 to model
the dynamic failure process of the first membrane as a moving variable
orifice, as described in Sec. 111.0.3. Referring to the input description
for Card Al5f and Eqs. 118 and 119,
TOP EN
AINL
AFIN
BTAR
and
EWE
By. (136)
The delay time between membrane failures can be increased by increasing
VRDUB on Card 15a.
Card A16. The pomp junction is computed as a tee, with the pump headHEAD added to the computed pressure at the entrance to the pump discharge
pipe identified by LPUMP.
Card A17. Linear interpolation is used to determine pressure at times
between the tabulated points. The pulse pressure at source K is set to
PSRC(K,1) for times less than TSRC(K,1) and to PSRC(K, NPULD(K)) for times
greater than TSRC(K, NPULD(K)).
Card A18. The left-hand, right-hand designation is arbitrary and is
used to tie the prescribed loss coefficients with the computed direction of
flow.
92
If an energy loss junction is the first junction in a relief system,
one of the associated pipes LENLSL or LENLSR is the relief system pipe
connected to the junction and the other is the main sodium system pipe
connected to the rupture disk junction.
Card A19. The currently recommended value of LAMBDA is 5.0; the
recommended values of BR and C3R are 0.65 and 0.0, respectively.
3. RUNB (WATSID)
General initial water conditions are input on Card Bl, and initial
conditions for particular pipes are input on Cards B5b. In both cases,
pressure, temperature, and quality are input. If the quality is between
zero and one, inclusive, the input temperature is ignored, the input
pressure and quality determine the state of the water, and the initial
temperature is set to the saturation temperature for the input pressure. If
the input quality is greater than one, it is ignored, and the input pressure
and temperature are used to determine the initial state of the water. The
saturation temperature for the input pressure is computed. If the input
temperature is less than the saturation value, the appropriate subcooled
liquid state is determined and the initial quality is set to zero
automatically. If the input temperature is greater than the saturation
value, the appropriate superheated vapor state is computed and the initial
quality is set to one automatically.
Care must be taken if the initial conditions vary along a pipe. There
is no problem if all points in the pipe are in the same phase region. For
example, if a different superheated vapor state is prescribed at each end of
a pipe, the program will ignore the input qualities (>1) and use linear
interpolation to determine the temperature and pressure at intermediate
nodes. However, if one end of a pipe initially is prescribed to be in the
two-phase mixture region and the other in the superheated vapor region, the
program will interpolate between the input values of pressure, temperature,
and quality to obtain values of these quantities at intermediate nodes. For
all points with interpolated qualities greater than one, it will assign
superheated vapor states at the interpolated temperature. At the first
point where the interpolated quality drops below one, it will abruptly drop
the temperature to the saturation value. This is not likely to be the
intended initial state profile along the pipe. The best way to handle this
situation is to input initial conditions at the intermediate point where the
phase regions change. In the previous example, the program would then
interpolate the quality between the two-phase end and the intermediate input
9 3
point and interpolate the pressure and temperature to give superheated vapor
states along the rest of the pipe.
RUNB can be used without RUNA to model the blowdown of a water system
by replacing the sodium system and the sodium-water reaction with a constant
pressure at the break junction, input on Card 83. On Card 1, set REACTnT to
obtain water properties.
Experimental configurations using rupture disk failures to initiate an
event can be simulated by dividing the water system into multiple systems
separated by rupture disks, using LSYS on Card 85a and disk data on Card 89.
Card I. Array storage* is such that 1 < NTUBE < 25, 1 < NJCN < 26,
0 < NORFJ < 26, 0 < NRSV < 2, and 0 < NRDSK < 8. The quantities NORFJ,
NRSV, and NRDSK indicate the number of each type of junction that requires
special input data; these data are then given on Cards 87, 88, and 89,
respectively.
Card 82. Initial conditions are computed from PRESO, TEMPO, QUALO, and
FLOW for each pipe in the water system for which initial values are not
input individually on a Card B5b. Determination of the initial thermo-
dynamic state of the water from the input pressure, temperature, and quality
is included in the general notes on RUNB.
The positive direction for initial mass flow is the same as that for
velocity, i.e., from the first-node end toward the last-node end.
Consequently, if FLOWO > 0, the node end designations on Card 85a must be
made to assure continuity of flow and avoid head-on collisions. The
designation of node ends is immaterial if FLOWO n O.
A nonzero value for FLOWO will cause problems if the water system pipes
are not connected in series because of the imbalance of flows at tees.
Card 83. This card is needed if RUNA is not used. It supplies TFIN
and PGAGZ, which otherwise would be input on Cards A4 and A3, and a constant
back pressure PBREAK at the tube break, which otherwise would be a time-
dependent quantity computed by the sodium-water reaction module.
Card 85s. If the water system is divided into subsystems by rupture
disks for experiment simulation, LSYS is used to indicate the subsystem to
*See Sec. VI for information on altering these storage allocations.
94
which each tube belongs. If there are no subsystems, every tube should have
LSYS = 1.
The tube numbers and node end designations are arbitrary.
'MEILEN/DX must not exceed 70 because of the array allocations for tube
nodes. If this causes a problem, the longer tubes can be broken up, using
dummy junctions; DX (on Card 82) can be increased; or the array dimensions
can be increased, as described in Sec. V/.
Card B5b. If NI(L) = 1, all nodes of the tube have the same initial
conditions, as determined from PIN(1), TiN(1), QIN(1), and FLOWIN(1);
DIST(1) can be any point along the tube in this case. As discussed before,
the pressure and quality determine the state if 0 < QIN < 1, and the
pressure and temperature determine the state if QIN > 1.
If NI(L) > 1, all nodes before the first prescribed point of a tube are
assigned the same initial conditions as the first prescribed point;
similarly, all nodes after the last prescribed point are assigned the same
initial conditions as the last prescribed point. Linear interpolation of
the input quantities is used to determine initial conditions at nodes
between prescribed points.
Card 86. Junction types 1, 2, 5, 6, 7, and 8 must have two tubes
connected to them; junction types 3, 4, 9, and 10 must have only one tube
connected to them; and junction type 11 is a three tube junction.
Only one break junction (JCNTYP - I or 2) may be used in the water
system.
Card 87. Each junction identified as type 2, 4, 6, or 8 on Card B6
must have a Card 87.
If an orifice does not vary with time, put ORFTC = 0.0. If ORFTC >
0.0, the time-dependent orifice area A is given by
t4
A(t) = 7D2
-- , t < to to
10)2- t > to
where D - ORFDIA and to -- ORFTC.
(137)
Card 88. The reservoir junction can be used to model either a
constant-state boundary or a tank occupied partially by water and partially
95
by a noncondensible gas. As before, the initial state of the water is
determined from the input pressure and temperature if the quality is greater
than one, and from the pressure and quality if the quality is between zero
and one, inclusive.
V. BRIEF SUMMARY OF SWAAM"II VALIDATION
In addition to the validation of the individual modules performed
during the early stages when the modules were developed, the integrated
SWAAA"I code has been validated against large scale test data from both the
LLTR Series II and the SWAT-3 experiments. The results of the validation
apply also to the SWAAM-II code, and they are summarized briefly here. The
LLTR tests employed a water-injection system that was simple to analyze;
hence the LLTR data served well as the basis of the intended code
validation. The LLTR A-1 test 142, 431 was a nitrogen-injection test; it
was a useful data base for isolating the bubble dynamics and its interaction
with the surrounding sodium from the generally complex situation where the
sodium-water reaction also takes place. The good agreement shown in the
comparison of the code prediction with the A-1 tests gave confidence in the
SWAAM"I bubble dynamics modeling capability and the interactive sodiumsystem response (441. Then validation 136, 45) continued with the reactive
A-2 test [461.
The SWAT-3 test facility employs a piping loop and IHTS components muchcloser in scale and complexity to a real plant, and the SWAT-3 data allowcomprehensive validation of the sodium system response. In addition, the
SWAT-3 data are for a helical-coil steam generator with a cover gas, while
the LLTR data are for a straight-tube steam generator with no gas space. So
SWAT-3 data serve as the basis for validation of the code capabilities for
the helical-coil steam generator type with cover gas space. Unfortunately,
the SWAT-3 test employs a complex water injection system that does not
permit a simple leak-rate calculation. As a result, the early leak rate
(when the source pressure peak occurs in the steam generator) is difficult
to calculate and only limited validation of the source term determination is
possible. SWAT-3 Run-3, -5, and -6 data (47-49) were analyzed 150, 511 by
SWAMI-I, and the results of the Run-6 data analysis [51) are summarized
briefly here. The discussions that follow are based on the summary paper
presented at the Second Joint U.S./Japan LMFBR Steam Generator Seminar in
June 1981 152).
+.5-Nsr.,
INATEIN-RTEAAILINE
LARGE LEAK iNJECTOR DEVICE MUD'eisToN BELLOWS
1N
207A
LLTv
IV, PURGE
1.30AL.201
Fig. 17. Schematic Diagram of LLTR Series-II Test Facility
RPT
RO.3
MN
L 303
I. 204
96
A. Validation Using LLTR Data
LLTR Series-II Tests A-1 and A-2 were used to validate the SWAMI-I
code. Test A-la employs a rupture disk system that has only one membrane;
Test A-lb used two membranes. The presence of the second membrane in
general had little effect on the system pressure transients and the source
pressure history. Other than the difference in the rupture disk system, the
two tests are both nitrogen injection tests and essentially identical. Test
A-2 is a one double-ended guillotine (DEC) test for the subcooled water,
evaporator start-up condition of the CRBR design. The LLTR test
configuration consists of essentially the reaction vessel LLTV, the water
injection system, and the relief system, as shown in Fig. 17. The LLTV is
prototypic to the CRBR steam generator in vessel diameter. The piping
diameter also is nearly prototypic to CRBR IHTS piping. However, in other
respects, LLTR is not prototypic to CRBR. The double-membrane rupture disk
system used in LLTR is shown in Fig. 18.
—dv-411.4.4.111.1111 MEALwe WW1
201/417 11fF
-114 VP A/41404
CINTER ELICTROOEFLUSH WITH IA
LAW 01141.
< SE A1Ri :leAS
LED. WIPU
IF HEMMED
97
Fig. 18. CRBRP Prototype Rupture Disc Assembly
The SWAAM-I models for the water injection system and the sodium side
systems are shown in Fig. 19. In both the A-1 and A-2 tests, the SWAAM-I
code was first run to obtain pretest predictions before the test results
became available )44, 45). Later a posttest analysis of the A-2 test was
performed with a modified bubble-modeling parameter (the reaction rate
coefficient A) to obtain better agreement between the calculated and
experimental source pressure histories. Test A71 results are compared with
the SWAAAHI pretest prediction results in Fig. 20. Only the calculated
nitrogen injection rate is available; it was not measured, as typically is
the case with large leak tests. The agreement for the source pressure
history and the pressure history just upstream of the rupture disk (junction
23) are excellent, as shown in Fig. 20. The buckling of the rupture disk in
the SWAMI-I prediction occurs somewhat earlier than the test results
19
24 P521
12
Fig. 19. SWAAM-I Model for LLTR Series-II Tests
(b) Sodium-Side System
YID
0 4010163-3T— ITS!
• AMC saga itioTI
*X •
AG
MOD
° ° 4.,.° 00? * °
--- •44,1MAP64140.40. — —
m m sio1116.1=1.4W
(c) Pressure History at
Juntion 23
99
(a) Leak Rate Result
(b) Source Pressure History i
Pig. 20. Validation Results for LLTR Series-II Test A-I
1 0 0
indicated. Handbook values were used for the description of the rupture
disk properties, and the small discrepancy is attributed to possible
inaccuracies in the rupture disk input data.
The reactive Test A-2 results and the pretest and posttest SWAMI-I
calculations are shown in Fig. 21. The typically good agreement observed
for Test A71 also is shown here for the Test A-2 calculations. The source
pressure history agrees well throughout the 100 ms transient. The complex
pressure history at the rupture disk location also shows generally good
agreement. But the failure of the second rupture disk membrane in the
experiment is significantly later than SWAAM-I predicts. This discrepancy
was studied in a great detail, but it remains to be explained (see Ref. 36).
The cavitation phenomenon occurring near the closed end of the LLTR
upper piping (location P524) is well predicted. The pressures calculated in
the pretest analysis are results for A = 30 ft/s, and are in general
slightly higher than the pressures obtained in the posttest analysis, where
A = 5 ft/s was used. The value of A = 30 ft/s used in the pretest analysis
was chosen such that the calculated bubble temperature was - 2200°F maximum,
which is approximately the maximum temperature measured in earlier
experiments. The choice of A .-- 5 ft/s in the posttest analysis was made by
matching the peak source pressure. This choice yielded a SWAMI-I calculated
maximum bubble temperature of - 1700°F, which agrees closely with the bubble
temperature actually measured in Test A-2.
B. Validation Using SWAT-3 Data
The SWAT-3 Test Facility employs elaborate piping and IHTS components,
as shown in Fig. 22. The primary purpose of the test facility was to
demonstrate the safety and the design adequacy of the Japanese Monju reactor
steam generators with respect to large-leak sodium-water reactions. As
mentioned briefly above, the SWAT-3 water injection system employs initially
empty piping and a rupture sleeve as shown in Figs. 23 and 24. The water
pipe splits at the top of the steam generator to divide the flow into the
coil region and the downcomer region. Initially, the entire piping
downstream of Valve V-501 (see Fig. 23) is empty (nearly in vacuum). The
rupture sleeve shown in Fig. 24 is installed at the injection point in the
helical coil bundle region, as shown in Fig. 23. As the valve is opened,
the piping becomes pressurized by the influx of high pressure water/steam
from the water heater. When the pressure at the rupture sleeve exceeds its
set pressure, the rupture sleeve breaks and the leak starts.
00 Ml 30 ;0 00 403 70 El
TI ILT.Ji 1.1.1 —SEC
0
VONRWMP
- —
Mite SUM
IN
nimmmu-sw
(c) Pressure History at Location 23 (d) Pressure History at Location 24
' (a) Leak Rate Result
(b) Source Pressure History
Fig. 21. Validation Results for LLTR Series-II Test A-2
Fig. 22. SWAT-3 Facility Schematic Flow Diagram
000
rI
,
.:J.
Ilk _ 4 AM] ler 1 Adak eii
or-iiralliMMINCIIMMIMAIIMIIIIIIII
COM t I
. 'mum • III II III III Mr= OW'I ilk
*NOas . r____41
CROSS -SECTION A - A
103
Fig. 23. SWAT-3 Water-Injection System
Fig. 24. Rupture-Sleeve Design of SWAT-3 Injection System
PK500114
P6500215
P6500316
80502
1?PK5004
18P65005
19
DOVINCOMER
104
The condition in the injection piping at the time of leak initiation
can be quite dynamic, and this can influence the early leak rate
significantly. Moreover, calculating the water-filling transient to obtain
the initial conditions needed for calculating the leak rate was very
difficult [51]. So the measured quasisteady leak flow rate (20.68 lb/s for
Run-6 Test) was used, and the effects of the uncertainty in the early leak
rate were studied parametrically.
The following discussion includes only the Run-6 data analysis
performed at ANL [51]. The Run-3 and Run-5 data were analyzed by Babcock
and Wilcox [50] and the results of their effort are very similar to those
obtained at ANL.
The SWAMI-I models for the water side and sodium side systems are shown
in Fig. 25. Due to the difficulties associated with obtaining the initial
conditions for the water side system blowdown calculations as described
above, the exact solution obtained from the water side system model is not
used. Instead, the measured quasisteady leak rate (20.68 lb/s) is used for
the sodium side transient calculation. Transducer Location P1113 is
1120TANK
V501 V502 P50010 0 0 0 0 0 0
BRANCHING RD501 (
;
POINT 9
0 12COIL 0) 0 0I I 1 0
13 213-,...-313
(a) Water Injection System (b) Sodium Side System
Fig. 25. SWAMI-I Model for SWAT-3 Run-6 Test
closest to the leak location (1.53 ftbelow the injection point). The
pressure history calculated by SWAAM"
I for the same location is comparedwith the test data in Fig. 26. Thepeak presssure and the pressure risetime are in good agreement, but the
SWAMI
\ ,, rA_:\
-
calculated pressure after the peak isgenerally higher than the test
30111 3 %IA data. This is partly due to the
inadvertent presence of non-IS
TINILW ILI]
S" condensible gas in the pipeline
Oa I
I
105
Fig. 26. Comparison of Early
Pressure History of
SWAT"3 Run-6 at PI113
with SWAM-I Prediction
(A 2.6, B - 0.65,
W 20.68)
between the reaction vessel and the
superheater, causing a different
system response to the reaction
bubble dynamics and growth (see Refs.
49 and 51 for more details). The
result shown in Fig. 26, which shows
good agreement in the source pressure
peak and the rise time, was for the
modeling parameters A 2.6 ft/s and
B 0.65.
The choice of A • 2.6 ft/s differs from the value A 5.0 ft/s that was
used in the posttest analysis of LLTR Series-II Test A-2. So it was of
interest to examine how the uncertainty in the early leak rate can affect
the parameter A while maintaining the same peak source pressure.
Specifically, the interest was to establish the possibility of a A value of
5.0 ft/s being appropriate for SWAT-3 as well as LLTR. The leak tube in the
SWAT-3 injection system is likely to be filled initially with a less dense
fluid at a lower pressure than predicted for the quasisteady equilibrium.
Hence the initial leak rate was reduced by 502 and increased with time
linearly to the quasisteady value of 20.68 lb/s at t 20 ms, but A 5.0
ft/s was used. The result is compared with the reference case (A 2.6
ft/s, 1 • 20.68 lb/s) in Fig. 27; the results are almost identical. Thisindicates that there is a strong possibility that A 5.0 ft/s may be the
appropriate constant for both the LLTR system and the SWAT-3 system. It
should be noted, however, that this point has yet to be confirmed by a more
rigorous study of the SWAT-3 water injection system. In the meantime, the
choice of A 5.0 ft/s appears to be a reasonable choice for design
applications for all types of steam generators.
A = 5.0 AND /NITIALLY LOWERED LEAK RATE
/ ' 0
IJ ' f) /-'\ n, .,,f ' \ i' 1‘ Pn f
1;\ 1I \l,) t100
106
10
TI ME.M I LLI—SEC
Fig. 27. Combined Effects
of A and Early
Leak Rate on
Pressure History
of SWAT-3 Run-6
at P1113
VI. ARRAY SIZES AND ALTERATIONS TO PROGRAM STORAGE
The array dimensions that determine the complexity of the systems that
can be analyzed by SWAAMHII are given in Table 3 for RUNA, Table 4 for RUNE,
and Table 5 for RUNC and RUND. These dimensions were chosen to permit
reasonably detailed analysis of an LMFBR system without an excessive amount
of computer core storage. In anticipation of the need to alter some array
dimensions, all the arrays containing the dimension of a given system
parameter were grouped together in labeled COMMON blocks, which are listed
in Tables 3-5. The current maximum value of each of these dimensions is
specified in a DATA statement in the MAIN program and also is listed in
Tables 3-5. The subroutines that contain these COMMON blocks are listed in
Table 6 and 7 for RUNA and RUNS, respectively.
For example, if you want to increase the allowable number of constant
pressure boundaries in the sodium system from 10 to 14, you would need to
change MXCONP in DATA in MAIN from 10 to 14 and change every 10 in COMMON
/S0D7/ to 14 in the subroutines listed under SOD7 in Table 6.
Table 3. Array Size Limitations for Sodium-Side Computation (RUM • T)
DescriptionFORTRAN
NameInput MinimumCard Allowed
Maximum
Allowed
Name ofMax. ValueIn DATA
COMMONName
Total pipes in mainand relief system
NPIPM +NPIPR Al 1 65 KXPIPE
SOD1,SOD2
Total junctions in NJUMN +main and relief systems NJUNR Al 1 66 MXJUNC SOD3
Axial nodes in a pipe NNODE Computed 2 100 ?NODE SOD1
Pipes connected at amultibranched junction(JTYPE = 3,4,5,8,15) 1 6 MXBRCH SOD3
Pipes connected atbubble junction(JTYPE • 17,18) 1 3a
Pipes connected atJTYPE = 7,9,10,11,12,13 1 1 a
Pipes connected atJTYPE = 1,6,22,23 2 28
Output specificationranges NPRTS AS 1 78
Relief systems NRLFC Al 0 10 MXRELF SODIl
Surge tanks NOSRG A2 0 10 MXSURG SOD8
Constant pressureboundaries NOCPB Al 0 10 MXCONP SOD7
Table 3. Array Size Limitations for Sodium-Side Computation (RUNA = T) (Conted)
FORTRANName
InputCard
MinimumAllowed
MaximumAllowed
Name ofMax. Valuein DATA
COMMONName
NOINRD A2 0 10 MXRDPI SOD5
NOSGRD A2 0 3a
NTPRD A14a 0 30 MXRDPR SOD9
NODBRD A2 0 3a
NOPUMP A2 0 10 MXPUMP SOD6
NOPULS A2 0 3 MXSRCE SOD4
NPULD A17a 1 25 MXSRCD SOD4
NOENLS A2 0 25 MXENLS SOD10
NINJIN A20a 1 200 MXINJN SOD12
Description
Instantaneous rupturedisks
Single-membranerupture disks
Data points for input ofsecond disk response afterfirst disk failure
Double-membranerupture disks
Pumps
Pulse sources
Data points per pulsesource
Energy loss junctions
Data points for prescribedwater injection
aNot conveniently altered.
Table 4. Array Size Limitations for Water-Side Computation (RUNE - T)
FORTRAN
Name
Input Minimum
Card Allowed
MaximumAllowed
Name ofMax. Value
in DATA
COMMONName
NTUBE 111 1 25 MXTUBE NATI,
WAT2
NJCN Ill 1 26 MXJUNS WAT3
NODES Computed 2 71 MXNODS WAT1
NI 355 0 20 MXINTL WAT4
NORFJ BI 0 26 MXJUNS WAT3
NRSV 131 0 2a
NRDSK 111 0 8a
NRDSYS Computed,NRDSK + 1
1 9a
86 0 1 a
2 28
3 38
NUT 34 1 7a
Description
Tubes
Junctions
Nodes per tube
Data points for pipeinitial conditions
Junctions havingorifices
Reservoirs
Rupture disks
Subsystems connectedby rupture disks
Break junctions
Tubes connected tobreak junction
Tubes connected to tee
Output specification
ranges
allot conveniently altered.
110
Table 5. Array Size Limitations for Two-Dimensional Sodium-Side(RUNC = T) and Shell Dynamics (RUND = T)
DescriptionFORTRANName
InputCard
MinimumAllowed
MaximumAllowed
COMMONName
Nodes in axialdirection NX Cl 2 61a BL5, BLSHL
Nodes in radialdirection NY Cl 2 21a BL5, BL21
Baffles C2 0 2a BL21
Data points forprescribed bubbleconditions NPTSIN C4 1 10 PBTBL
Nodes in axialdirection II D1 2 61a SHELL3,
SHELLS
aNot conveniently altered.
Table 6. Sharing of Labeled COMMON Among Sodium-SideSubroutines (RUNA)
Labeled COMMON Subroutines
SOD1, SOD2, SOD3
MAIN, SODIN, SODINL, SODSTP, SODOUT, SODSID,INTRP,FAREND, AREACH, AREAC2, ENGLOS, TEE, PUMPIRPED, CLOSED, CONSTP, SURGE, RDINST, RDSNGL,RIMUHL, MOVINT, INTJUN, INTNOD, PRESSO, BUBBLE
SOD4 MAIN, SODIN, SODINL, SODSTP, SODOUT, SODSID,PTIME
SODS, S009, SOD11 MAIN, SODIN, SODINL, SODOUT, SODSID
SOD6, SOD7 MAIN, SODIN, SOD/NL, SODSID
SODEI MAIN, SODIN, SODINL, SODSTP, SODOUT, SODSID,SURGE
SOD10 MAIN, SODIN, SODINL, ENGLOS, INTJUN
S0012 MAIN, SODIN, BUBBLE
Table 7. Sharing of Labeled COMMON Among Water-SideSubroutines (RUNB)
Labeled COMMON Subroutines
WAT1
MAIN, WATIN, WATSTP, WATOUT, WATSID, OUTFLO,RSVEND, /NFL°, TUBEND, TUBERD, ORIFIC, SIZCHG,
TUBTEE
WAT2
MAIN, WATIN, RSVINL, WATSTP, WATOUT, WATSID,BREAK, OUTFLO, RSVEND, INFLO, TUBEND, TUBERD,ORIFIC, SIZCHG, TUBTEE, LOSS
WAT3
MAIN, WATIN, RSVINL, WATSTP, WATSID, BREAK,RSVEND, TUBEND, TUBERD, ORIFIC, SIZCHG, TUBTEE
WAT4
MAIN, WATIN
112
VII. CONCLUDING REMARKS
This report describes the theoretical basis, numerical modeling
techniques, and user requirements for the most recent version of the large
leak sodium-water reaction analysis computer code SWAAM-II. The report also
describes the extent of code validation with respect to predicting the
response of the steam generator and intermediate heat transport system to a
large leak sodium-water reaction. Both the U.S. large scale LLTR tests and
the Japanese SWAT-3 tests were used in validating the code. The results
show that SWAAM-II is capable of predicting the effects of large leak
sodium-water reaction events, for both straight tube and helical coil tube
steam generator designs, with accuracy sufficient for most design
applications. The geometric complexities and physical sizes of the LLTR and
SWAT-3 test facilities are reasonably close to the real plant systems.
Hence it is our position that SWAAM-II is adequately validated with respect
to complexities in system geometry and size of the steam generator vessel
and piping. In some other respects, however, several needed improvements
have been identified.
The double-membrane rupture disk system that is used in the CRBR design
employs a low-pressure gas space between the two disk membranes. The
relative change in the gas volume during the early disk dynamics of the
first membrane prior to its opening is small enough to be ignored. Once the
membrane opens and sodium fills the gas space, the gas pressure rises and
the dynamics of the second membrane follows. The pressure differential
across the first disk opening (treated as an orifice) was computed from
steady-state flow data. It was found in the analysis of the LLTR Series-II
A-2 test that SWAAM-I was able to predict the disk dynamics and the fluid-
structure interaction well up to the first disk opening. However, the
filling of the gas space and hence the opening of the second disk was
predicted to be much earlier than the test data showed. The flow in the
empty relief piping subsequent to the second disk opening is treated as an
idealized fluid column with a distinct sodium interface, and steady flow
friction factors and junction losses are used. This treatment predicts
sodium velocities in the relief line that are much higher than measured in
the tests. These two areas need modeling improvements.
Another area that needs improvement is in the water side module of SWAAM-
II. SWAAM-II is not able to calculate the high-strength shock flow problem
of the SWAT-3 water injection system during the initial filling flow in the
empty piping. Accurate definition of the flow distribution in the injection
piping at the time of the rupture sleeve opening is necessary to calculate
the early leak rate, which in turn allows accurate calculation of the source
I 13
pressure and early system transient. The SWAT-3 tests provide valuable data
for helical-coil tube steam generator design; the improvement in the water
side module is needed to make full use of the SWAT-3 test data.
Design of LKFBR steam generators, the /NTS, and the relief systems with
respect to sodium-water reactions must consider the entire spectrum of
possible leak scenarios. The current SWAAM-II code is designed primarily
for double-ended-guillotine (DEG) leaks, and validation was limited to thistype of leak. Another, perhaps more probable, scenario involves a small
leak that persists for a significant length of time and finally leads tolarge leaks. This leak scenario calls for the code to calculate the
pressure transient in the generally bubbly flow of sodium and hydrogen
gas. Unlike the linear behavior in pure liquid sodium, the mixture of
sodium and hydrogen gives rise to nonlinear behavior where compression waves
tend to become sharper while rarefaction waves tend to flatten. In the
simple fluid-hammer case of pure sodium, the wave shape is preserved unless
a geometric disturbance is encountered.
Another major addition to SWAAM-II to be developed in the near future
is the consideration of fluid-structure interaction between the pressure
transient and the gross motion of the piping. The piping motion induced by
the flow pressures and flow turning can significantly alter the pressures in
the pipe. Piping restraints designed without consideration of the
fluid/structure interaction can be unnecessarily very costly. A simple
engineering approach is being pursued where an adequate means to extract the
main effects of the piping motion on the pressure transients is sought to
improve the method of pressure transient computation. Then the fluid forces
thus calculated can be input to existing structural computer codes.
ACKNOWLEDGEMENTS
We thank Mt. Carl E. Ockert of the U.S. Department of Energy for his
support, encouragement, and many helpful discussions given during the course
of this work. We thank Mr. Robert S. Zeno, Director of Components
Technology Division, for his continued special interest in this work. We
wish to acknowledge the contributions of Dr. Gregory Berry, Dr. James Daley,
MA. Beverly Sha, and Dr. R. A. Valentin of ANL, and Mr. Thomas Eichler and
Mt. Arne Wiedermann of ATResearch, who were members of the team that
developed SWAAM-I and who provided advice and consultation in the
development of SWAAM"-II. Thanks are also due to MA. Shari Zussman for her
editing and Mrs. Emma L. Berrill for typing the report.
1 1 4
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1 1 5
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82-4 (Feb. 1982).
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118
Internal:
E. S. BeckjordC. E. TillR. AveryR. A. LewisR. S. ZenoR. A. ValentinG. S. RosenbergP. R. HuebotterC. A. KotC. K. Youngdahl (25)Y. W. Shin (25)M. W. Wambsganss
Distribution for ANL-83-75
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External:
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