nureg/cr-2913 sand82- 1935 r4 printed january 1983 · nureg/cr-2913 sand82-1935 r4 two-phase jet...
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NUREG/CR-2913SAND82- 1935R4Printed January 1983
©
Two-Phase Jet. Loads
G. G. Weigand, S. L. Thompson, D. Tomasko
Prepared bySandia National LaboratoriesAlbuquerque, New Mexico 87185 and-Livermore, California 94550
for the United States Department of Energyunder Contract DE-AC04 76DP00789
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REGULATORY COMMISSION
NOTICEThis report was prepared as an account of work sponsored by anagency of the United States Government. Neither the UnitedStates Government nor any agency thereof, or any of their employ-ees, makes any warranty, expressed or implied, or assumes anylegal liability or responsibility for any third party's use, or theresults of such use, of any information, apparatus product orprocess disclosed in this report, or represents that its use by suchthird party would not infringe privately owned rights.
Available fromGPO Sales ProgramDivision of Technical Information and Document ControlU.S. Nuclear Regulatory CommissionWashington, D.C. 20555andNational Technical Information ServiceSpringfield, Virginia 22161
NUREG/CR-2913SAND82-1935
R4
TWO-PHASE JET LOADS
G. G. WeigandS. L. Thompson
D. Tomasko
Date Published: January 1983
Sandia National LaboratoriesAlbuquerque, New Mexico 87185
operated bySandia Corporation
for theU. S. Department of Energy
Prepared forMechanical/Structural Engineering Branch
Division of Engineering TechnologyOffice of Nuclear Regulatory Research
U. S. Nuclear Regulatory CommissionWashington, D. C. 20555
Under Memorandum of Understanding DOE 40-550-75NRC FIN No. A-1216
ABSTRACT
A model has been developed for predicting two-phase, waterjet loadings on axisymmetric targets. The model ranges inapplication from 60 to 170 bars pressure and 70 0 C subcooledliquid to 0.75 (or greater) quality -- completely covering therange of interest in pressurized water or boiling waterreactors. The model was developed using advanced two-dimensional computational techniques to solve the governingequations of mass, momentum, and energy. The model displays ina series of tables and charts the target load and pressuredistributions as a function of vessel (or break) conditions;this enables fast yet accurate "look up" for answers. For manysituations of practical interest, the model can predict sub-cooled and saturated loadings in excess of the simple controlvolume upper bounds of 2 PoAe for nonflashing liquid and1. 2 6 PoAe for steam. Also, the results indicate that thearea of the loading on a flat target is often larger thanassumed by simplier models. Finally, approximate models aregiven for estimating two-phase jet flow, expansion character-istics, shock strengths, and stagnation pressures. Theseapproximate models could be used for estimating pressures ontargets not specifically addressed in this study.
iii/iv
CONTENTS
Page
1.0 Introduction ............................................ 1
2.0 Two-Phase Jet-Load Model: Conclusions and Application.. 3
2.1 Model Description .................................. 3
2.2 Model Application .................................. 12
3.0 Characteristics of Two-Phase Jets ....................... 19
3.1 Generalized Two-Phase Jet Behavior .................... 19
3.2 Idealized Configuration ............................ 21
3.3 Governing Equations ................................ 26
3.4 Breakflow Conditions ............................... 29
3.5 Standing Shock at the Target ....................... 33
3.6 Pipe Exit Core ..................................... 40
3.7 Far Field Relaxation to Ambient Pressure .............. 42
3.8 External Choke ..................................... 45
4.0 Two-Phase Jet Computer Load Model ....................... 47
4.1 Computer Model ..................................... 47
4.2 Some Consequences of the Exit Core .................... 50
4.3 Modifications of CSQ for the Two-Phase JetImpingement Study .................................. 50
4.4 Model Calculations ................................. 54
4.5 Data Base Calculations ............................. 69
5.0 Thermodynamic and Critical Flow Properties ................. 75
V
CONTENTS (Continued)
6.0 Approximate Models ......................................
6.1 Centerline Free-Jet Expansion Model ................
6.2 Centerline Target Pressure Model ...................
6.3 Target Radial Pressure Model .......................
7.0 Engineering Model .......................................
7.1 Centerline Stagnation Pressure Model ...............
7.2 Radial Stagnation Pressure Model ...................
7.3 Experimental Verification ..........................
References ...................................................
Appendix A Two-Phase Jet Load Model .........................
Appendix B Centerline Target Pressure Distributions .........
Appendix C Two-Phase Jet Exit Core Lengths ..................
Appendix D Water Properties and HEM Critical Flow Properties
Page
87
87
95
103
107
107
114
118
136
139
266
273
276
vi
NOMENCLATURE
A Area
AE Break area
Cs Sound speed (sonic velocity)
D Break diameter
E Internal energy
Fr Force on target,
G Mass flux
GE Break mass flux
H Enthalphy
Ho Stagnation enthalpy
L Length from break exit to target
Lc Length of exit fluid core
M Mass flow rate
P Pressure
PA Ambient pressure
Po Stagnation pressure
PT Target pressure
R Target Radius
RA Target Radius where PT = PA
r Radial coordinate direction
S Entropy
so Stagnation entropy
T Temperature
To Stagnation temperature
vii
Tsat Saturation temperature
ATo Degrees of subcooling ATo = Tsat - T
t Time
tc Time constant for exit fluid core
V Velocity
X Steam:water quality
X0 Stagnation steam:water quality
z Axial coordinate direction
a Parameter in load model
Parameter in load model
p Density
viii
1.0 INTRODUCTION
A nuclear power plant must be designed to ensure that theconsequences of a pipe break (large or small) will be mitigated toprevent damage of any safety equipment or systems. A majorconcern in the event of a pipe break is the loading uponsurrounding structures, equipment, etc., caused by the two-phasejet expanding from the break. These high energy jets have thepotential to cause wide spread destruction.
In an attempt to prevent failures due to jet loading on any ofthe safety equipment or control systems, a complex system of jetdeflectors, snubbers, and pipe restraints has been installed inplants. The design basis for much of these structural supportscan be traced to Moody's one-dimensional jet load model 1 .Various interpretations of the actual zone of influence of the jetfor this model have been made by others 2 , 3 , but the one-dimensional force model from Reference 1 is nearly alwaysapplied. The error that could result from the one-dimensionalapproximation or the arbitrariness in deciding the zone ofinfluence of the jet can result in costly overdesign, poorutilization of limited space inside of containment, or underdesignof restraint/barrier systems.
The use of simple one-dimensional modeling is inappropriatefor two-phase jet load calculations; the jet is a complicatedmultidimensional flow. The high pressure and high temperaturefluid that exits the break expands with supersonic velocitiesdownstream of the break. Upon encountering a target (or obstacle)a shock wave forms in the flow field, and it is the thermodynamicproperties downstream of this shock that determine the pressurefield and load on the target. A multidimensional analysis, whichis capable of treating strong shocks, is required to evaluate thethermodynamic properties downstream of these shocks.
Sandia National Laboratories, with the support of the U. S.Nuclear Regulatory Commission, has completed a study using modernmultidimensional computational methods to evaluate the two-phasejet load on target geometries. The governing equations of mass,momentum, and energy were solved with a high resolution Eulerianmethod for all calculations. The calculations form a computa-tional data base for evaluating jet and target pressures foraxisymmetric target geometries. This data base covers the rangeof pressures, temperatures, and distances to the target present inboth pressurized water (PWR) and boiling water (BWR) reactordesigns. A two-phase jet load model, which provides both pressureand load distributions, was developed using the computational database. In addition to the load model, approximate models weredeveloped for estimating the jet expansion, shock, and target load.
1
At the beginning of this modeling effort, it was generallybelieved that current analysis methods 3 were overly conserva-tive. Considerable design and manufacturing cost reductionswould result from a more accurate model, and overall safetycould be improved by a reduction in the complexity of thesystem. This was not the case. For many situations .ofpractical interest, the new model can predict subcooled andsaturated loading in excess of the simple control volume upperbounds currently employed 3 . The results also indicate thatthe area of the loading on a flat target is often larger thanassumed by the simplier model.
Persons only interested in applying the two-phase jet loadmodel should go directly to Chapter 2; the remaining chapterscontain modeling discussions and derivations and model verifica-tion. Chapter 2 contains a discussion of the load model and thefigures and charts that display the model; furthermore, twotutorial examples are provided.
Chapter 3 provides a discussion of the general character-istics of two-phase jets; their general behavior, geometry,governing equations, boundary conditions, and expansioncharacteristics are given. In Chapter 4 the computer model isdeveloped and the computational data base is given; severalexamples, taken from the data base, are graphically illus-trated. Chapter 5 provides charts for evaluating the thermo-dynamic properties of water and critical flow conditions.Chapter 6 contains the derivations and discussions of theapproximate models: free jet expansion, standing shock, andtarget load. Model comparisons with both the computational database and experimental data are given. Finally, Chapter 7provides and discusses the details of the final load model; inaddition, model versus experimental data comparisons areincluded. Four appendices provide the data needed for routineapplication of the model.
2
2.0 TWO-PHASE JET LOAD MODEL: CONCLUSIONS AND APPLICATION
This chapter and Appendices A, B, C, and D provide the jetload model. The model, at the request of the Nuclear RegulatoryCommission, is given in the form of charts that provide theradial pressure and load distributions as functions of the breakflow stagnation conditions and distance to the target. Chapter 7describes the theoretical methods utilized in developing thismodel. This chapter will first discuss the load model and thenconclude with an example calculation and tutorial discussion ofthe model. In this study, the development of a two-phase jetload model, we only considered initial conditions that result intwo-phase flows (steam:water mixtures) at pressures above ambientconditions. Cold water jets were not considered, although partsof the model describe some of the behavior exhibited by highlysubcooled, nonflashing jets.
2.1 Model Description
This section gives a brief summary of the load model.Appendix A contains a complete set of charts showing targetpressure and load distributions for break flow stagnationconditions of
60 to 170 bars
0 to 70 degrees of subcooling or}0 to .75 quality * I
and L/D's ranging between 0.50 to about 10. Appendix B containsthe centerline target pressure distributions for the aboveconditions. Appendix C contains figures describing the jetexit-core length (Lc/D), and Appendix D contains figuresdescribing the jet inlet-region thermodynamics and flow andassociated water properties charts.
Figure 2.1 shows the geometry for this model; the geometry isaxisymmetric. A list of the nomenclature used in Figure 2.1 andthe figures describing the load model is given below as aconvenience to the reader.
Ae break exit areaD break exit diameter
Fr force on targetGe exit mass flux
L length from break exit to targetLc length of break exit core, see Sections 3.6 and 4.2
*A larger range of qualities was considered in this study; a largerrange is not reported because the normalized target pressuredistribution is a weak function of quality for Xo > .75.
3
I-.a-
IR --
TARGET
Figure 2.1 Two-Phase jet load model geometry.
4
PO stagnation pressure at break
PT target pressureATO subcooling of stagnation temperature at break
Xo stagnation quality at breakR radius measured on the target
Figures 2.2 through 2.7 illustrate the features of the model.
Figures 2.2 and 2.5 give the target pressure distribution inbars as a function of target radius for stagnation conditions ofPo = 130 bars with ATo = 350C and Xo = 0.333 respectively.The target L/D associated with each curve is listed in the upperright hand corner of the figure; the lowest L/D value correspondsto the uppermost cu-re, e.g., in Figure 2.2 the curve with thehighest centerline pressure is for L/D = 1.50 and the curve withthe lowest centerline pressure is for L/D = 10.0.
Figures 2.3 and 2.6 give the target load distribution as afunction of target radius for the same conditions noted above; infact, Figures 2.2 and 2.5 are the integral of the pressuredistributions in Figures 2.3 and 2.6. Fr is the total normalforce acting on a circular disk target of radius R. Theseintegrations are valid for target radii up to the point where thepressure on the target equals the ambient pressure. Far-fieldphenomena, the consequences of neglecting them, and the detailsof the load integration are given in Section 7.2. Figures 2.4and 2.7 are composite contours of the target pressure. Thesefigures display the extent of the exit core along with a letterindicator for pressure on a Cartesian grid of length to target(L/D) versus target radius (RADIUS/D): A = 1 bar, B = 2.5 bars,C = 5 bars, D = 10 bars, E = 15 bars, F = 20 bars, G = 25 bars, H= 30 bars, etc. Any target within the exit core (the shadedarea) will be loaded by the vessel stagnation pressure, Po.These curves are only informational and useful for quick scopingcalculations; serious calculations should always be performedusing Figures 2.2, 2.3, 2.5, and 2.6.
Use of the two-phase jet load model (figures in Appendix A)is not limited to targets with large radii. Except for a smallsubsonic region of flow near the center of the target, flow onthe target is supersonic, thus edge effects are negligible. Inother words, for nearly all problems of interest the solution isindependent of target radius; one simply evaluates targetloadings using. the appropriate target R/D.
In this model there are three items that must be noted.First, when subcooling existed (ATo > 0 0C) and thetarget was located close to the break (L/D small) the loadspredicted exceeded the simple control-volume limit ofFr/PoAe = 2. This occurs because of the wide jet expansionfrom the break and because of far-field, supersonic phenomena.In subcooled cases the expansion is large enough to produce
5
U)
I-a-
0:
C)
I-
130. --
120.
110.
100.
90.0 0
80.0
70.0 0
60.0 0
50.0
40.0
30.0
20.0
10.0
0.000.0
Figure 2.2
1.0 2.0 3.0 4.0
RADIUS / 0
5.0
Target pressure distributions for stagnationconditions of Po = 130 bars and ATo = 35 0 C.The target L/D associated with each curve islisted in the upper-right-corner key; thelowest L/D value corresponds to the uppermostcurve.
6
0
N,
LLL
4.00 -
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1 .75
1 .50
1 .25
1.00
.750
.500
.250
0.00 "0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / 0
Figure 2.3 Target load distributions for stagnationconditions of Po = 130 bars and ATo = 351C.The target L/D associated with each curve islisted in the upper-right-corner key; thelowest L/D value corresponds to the uppermostcurve.
7
a
7 - (BARSA= I .0B= 2.5C= 5.0D= 10 .0E= 15. 0F= 20. 0G= 25.0H 30. 01= 35. 0J= 40. 0K= 45. 0L 50. 0M= 55.0N= 60. 00= 65.0P= 70. 00= 75.0R= 80. 0
5+
4+
0
N 3--
IJ
2+ F I B
S F I C
I MS P £ B
C S
CS
-J
1-
Po0 130 ATo= 35 Lc /D= 1. 14
0 U-M 1lllllllV , i I - - | I | | | |
M 1 2 3RADIUS
4D0
I I T5 6
Figure 2.4 Composite target pressure contours as afunction of target L/D and target RADIUS/Dfor stagnation conditions of Po = 130 barsand ATo = 35 0 C. Smooth lines connecting likealphabetic letters form an approximate pressurecontour corresponding, in bars, to the pressureversus alphabetic letter key. This contourplot is approximate and is only informational;it is useful for scoping calculations.
8
130. I ,
120. PO=130 X0 = .33 Lc /D= .50
1 10. L /DI. 00
100. :1.25
1.75
m 90.0 12.00• /2. 00
80.0 2 3.503.50
S70.0 4.00
M5.00D60.0 -7. 50
UJ 50 0al.
40.0LAJ
CO 30.0
20.0
10.0
0.000.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
Figure 2.5 Target pressure distributions for stagnationconditions of Po = 130 bars and Xo = 0.33.The target L/D associated with each curve islisted in the upper-right-corner key; thelowest L/D value corresponds to the uppermostcurve.
9
4.00 T
3.75 PO 130 Xo= .33 LC /D= .50
3.50, L/D3.25 1.00
1 .25
3.00 1.501.75
2.75 2.002.50
2.50 3.00
2.25 3.504.00
2.00 5.00o 7.50Q- 1.75
< 1.50 .
1.25
•. 1.00 -
.750
.500
. 250 3
0.000. . 1.0 2.0 •3.0 4.0 5.0 6.0
RADIUS D 0
Figure 2.6 Target load distributions for stagnationconditions of Po = 130 bars and Xo = 0.33.The target L/D associated with each curveis listed in the upper-right-corner key; thelowest L/D value corresponds to the uppermostcurve.
10
74-
6-
BARS1.02.55.0
10.015.020.025. 030 . 035. 040. 045. 050. 055. 060. 065.070.075.080.0
5t
4± 6
a
2-
-I V
£ 0
-. V 1 0
MSV 1 0
' ~J2~.I 1 0
C a
C a
C 0
A
C 9
IPO 130
''0X 0 = .33 L /D=
I I. 50
U "ll• I I I I I I I I I'-'
6II
~1 2RADI
3us
4/D
5 6
Figure 2.7 Composite target pressure contours as afunction of target L/D and target RADIUS/Dfor stagnation conditions of PO = 130 barsand XO = 0.33. Smooth lines connecting likealphabetic letters form an approximatepressure contour corresponding, in bars, tothe pressure versus alphabetic letter key.This contour plot is approximate and is onlyinformational; it is useful for scoping cal-culations.
11
velocities with components opposite to the break flow. Secondly,because the target area increases as the square of the radius, atlarge radii a small error in pressure can result in a significantload change; consequently, some judgment must be used inevaluating the loads on large targets with large L/D's. Finally,because the computational data base used to develop this modelwas sparse for low pressure, highly subcooled flows, some careshould also be exercised when making load predictions for highlysubcooled flows (Lc/D > 3.0, where Lc is the jet corelength given in Figure 3.14.)
2.2 Model Application
This section provides both an example of applying the abovemodel and a tutorial. There are two principal methods ofapplication. For both, the stagnation conditions at the breakposition are the important parameters. When only the peak loadsare of interest, it is reasonable to assume that the stagnationconditions are those in the pipe or vessel prior to the pipebreak. Since these conditions are the model's independentvariables, the load and pressure distribution can be readdirectly from the graphs in Appendix A.
When there is interest in the time-dependent behavior or pipefriction effects, then the procedure becomes more difficult. Thereality that faces the model user in this case is that pipe breakinformation will be supplied by system codes such as TRAC 4 ,RELAP 5 , and RETRAN 6 . These codes provide blow-down historiesof the break flow and some thermodynamic properties upstream ofthe break. The bridge linking these system code histories andthe two-phase jet model has been performed and is detailed inChapter 5. Briefly, we developed a set of thermodynamic chartsplus a set of HEM critical flow charts that relate the staticthermodynamic properties just upstream of the break and the breakflow to the stagnation properties. It is not necessary that thesystem code use the HEM model. The two following examplesillustrate the use of the figures in Appendices A, B, C, and D byboth of the above methods.
Consider the case where the two-phase jet load was ofinterest at two points in a blow-down calculation. The systemcode supplied the following break flow and thermodynamic statejust upstream of the break:
1. Ge = 6.15 Kg/cm2 s 2. Ge = 1.71 Kg/cm2 s
P = 120.0 bars P = 50.9 bars
T = 598.5 OK T = 538.1 OK
p = 0.653 g/cm3 P = 0.0663 g/cm3
12
For the first point the entropy is read from Appendix !.Figure D.l, S = 35 x 106 ergs/g OK. Then using the break flow(Ge) and the entropy the stagnation temperature is located inFigure D.4, To = 600.4 OK. Finally, the stagnation pressure islocated in Figure D.6, Po = 150 bars. The saturation temperatureat 150 bars is Tsat = 615.4 OK, thus the stagnation conditionsat the break for the first point are
PO = 150 bars, ATo = 15 0 C
The target pressure and load distributions for this stagnationcondition were found in Appendix A and are given in Figures 2.8 and2.9. Consider an axisymmetric target with R/D = 3 and L/D = 1.From Figure 2.8 the target pressures at R/D's of 0, 1, 2 and 3 are66.5, 25.0, 6.0, and 2.0 bars, respectively; from Figure 2.9 thetotal cumulative loads (Fr/PoAe) at R/D's of 0, 1, 2, and 3are 0., 1.08, 1.9, and 2.16, respectively.
For the second point the entropy is read from Figure D.3,S = 40.5 x 106 ergs/g OK. Then using the break flow and theentropy the stagnation temperature is located in Figure D.5,To = 568 OK and lays in the two-phase region. Finally, thestagnation quality is located in Figure D.7, Xo = .333. Thus thestagnation conditions at the break for the second point are
PO = 80 bars, ATo = 0 °C, Xo = 0.333
The target pressure and load distributions for this stagnationcondition were found in Appendix A and are given in Figures 2.10 and2.11. Consider the same axisymmetric target with R/D = 3 and L/D =1. From Figure 2.10 the target pressures at P./D's of 0, 1, 2 and 3are 30.75, 9.5, 2.25, and 1.0 bars, respectively; from Figure 2.11the total cumulative loads (Fr/PoAe) at R/D's of 0, 1, 2 and 3are 0, .81, 1.3, and 1.35, respectively.
These two examples illustrate the use of the tables andcharts in Appendix A and Appendix D. Both examples convenientlyfell on existing curves. When this is not the case linearinterpolation is recommended.
13
150.
140.
130.
- 120.
< 110.
100.
" 90.0
" 80.0
D 70.0LO
• 60.0
a- 50.0
w 40.0LD
< 30.0
20.0
10.0
0.000.0 1.0 2.0 3.0 4.0 5.0
RAD I US / D
Figure 2.8 Target pressure distributions for stagnationconditions of P0 = 150 bars and To = 15 0 C.The target L/D associated with each curve islisted in the upper-right-corner key; thelowest L/D value corresponds to the uppermostcurve.
14
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
-. 2.000
0_ 1.75
< 1.50
1.25
IL 1.00
.750
.500
.2500.00 -A
0,.0 1.0 2.0 - 3.0 4.0 5.0 6.0
RADIUS / D
Figure 2.9 Target load distributions for stagnationconditions of Po= 150 bars and ATo = 15 0 C.The target L/D associated with each curveis listed in the upper-right-corner key; thelowest L/D value corresponds to the uppermostcurve.
15
U)
a:
0-.
L&iC)
80.
75.
70.
65.
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
Figure 2.10 Target pressure distributions for stagnationconditions of Po = 80 bars and X0 = 0.33.The target L/D associated with each curveis listed in the upper-right-corner key; thelowest L/D value corresponds to the uppermostcurve.
16
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
0-
4)
2.00
1.75
1 .50
1.25
1.00
.750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / 0
Figure 2.11 Target load distributions for stagnationconditions of P = 80 bars and X = 0.33.The target L/D associated with e2ch curveis listed in the upper-right-corner key;the lowest L/D value corresponds to theuppermost curve.
17/18
3.0 CHARACTERISTICS OF TWO-PHASE JETS
3.1 Generalized Two-Phase Jet Behavior
In this study we found that the flow field for two-phasejets impinging on axisymmetric targets is extremelycomplicated. However, regions of this complicated flow fieldcan be explained using simple concepts, e.g., isentropic flow,normal shock theory, HEM choked flow, and others. Figure 3.1illustrates the expected pressure behavior along the centerlineof the jet. Figure 3.1a is a spatial illustration of thepressure field in a two-phase, impinging jet. The heavy solidlines in Figure 3.1a describe solid boundaries that depict thevessel, break nozzle, and target. The small plot on the rightin Figure 3.1a describes the pressure-expansion in the vesseland piping and in the jet outside the vessel. The conditions atthe dashed vertical line are the conditions at the break (pipeexit). The section of the graph to the left of the verticaldashed line corresponds to conditions outside of the pipe exit.Moving to the left corresponds to an increase in the target topipe spacing. The points labeled 2,3, and 4 in 3.1a correspondto the pressures for one pipe spacing given the conditions atpoints 0 and 1. Figure 3.1b shows an actual pressure-expansioncalculation for Po = 100 bars, ATo = 0 oC.
In the jet flow field there are three natural divisions ofthe field; each will be described in detail in later sections.There is a nozzle (or break) region where the flow chokes. Inthis region there is a core at choked flow thermodynamicproperties that projects a distance downstream of the nozzledepending upon the degree of subcooling. Downstream of thisregion there is the free jet region. Here the jet expandsalmost as a free, isentropic expansion. The flow is supersonicthroughout this entire region. The free jet region terminatesat a stationary shock wave near the target. This shock wavearises because of the need for the target to propagate pressurewaves upstream and, thus, produce a pressure gradient that willdirect the fluid around the target. Downstream of the shock isthe target region where the local flow field imposes thepressure loading on the target.
The pressure Po is the vessel stagnation pressure, andP1 is the choke pressure in the break nozzle. The pressureP2 denotes a family of isentropic expansion pressuresdescribing the centerline jet expansion upstream of the shockwave, and P 3 describes a family of pressures downstream of theshock wave that correspond to a normal-shock Hugoniotcalculation from the jet-expansion state P 2 . Finally, P 4describes the family of stagnation pressures that correspond toeach of the P3 normal-shock Hugoniot calculations. Notice
19
TARGET
SHOCK
70
1 ( CHOK~Ea.. 4
II
'I
IIII
P1 I - expons ion
PC
VESSEL
I
-.-* "EXPANSION"125.
r~100.
75.0m 50.0
S25.0
0.00 L-
10-2 10- 1 10*0 10* 1FREE JET PRESSURE, P2 (bars)
10.
Figure 3.1 Illustration of the expected pressurebehavior along the centerline of a two-phase jet impinging on targets.
20
that the shock wave causes a substantial stagnation pressureloss for highly expanded jets. This stagnation pressure loss isdue to the non-isentropic processes in the shock and should notbe confused with the very small viscous and entrainment losses.Figure 3.1b shows a set of actual jet pressure families for Po =
100 bars and saturated liquid conditions.
3.2 Idealized Configuration
The idealized configuration used in this study is shown inFigure 3.2. A large vessel is uniformly filled with water withknown thermodynamic properties. An exit pipe of diameter D isthe only flow path out of the vessel. A target is located adistance L from the end of the pipe; the target has no curvatureand has a radius that is large enough so that the fluid exitingits edge has expanded to containment pressure.
This piping configuration is equivalent to an offsetguillotine break. The second pipe exit will be ignored for thisstudy. It is assumed that there is no interaction between thetwo jets. Of course this could not be the case very early in thebreak history when the pipe sections are separating. However,this phase of the problem is beyond the current scope of thiseffort. Performing calculations early in the break history wouldrequire a number of specific configuration properties: piperigidity, restraint positions, and other items necessary todetermine pipe whip.
This idealized configuration assumes that the exit pipeinitially has fluid with the same thermodynamic state as thevessel, i.e., there is no initial motion. At time zero the pipeexit opens instantaneously to the outside or containmentenvironment. The exit flow history that would be expected isillustrated in Figure 3.3 The initial steps in the flow are theresult of the wave propagation from the pipe exit to the vesseland return. The length of time involved is approximately 2 Lp/Cswhere Lp is the exit pipe length and Cs is the sound speed inthe fluid. Multiple steps of decreasing amplitude would beexpected.
After these initial steps, the maximum exit flow willdevelop. Further decreases in the flow would result as thevessel depressurizes. If the vessel is sufficiently large, thenthere will be a relatively long period of time where the rate ofchange in the flow is small. It is this period of the blowdownhistory that will be of main interest in this study. Points inthis blowdown correspond to different initial vessel conditions.
21
TARGET
1I
I
-PLARGE VESSEL
PO (IT. or X.)
I
Figure 3.2 Idealized two-phase jet loads geometry
0
LI)
LO
TIME
Figure 3.3 Typical break flow history Cinstantaneousdouble-ended guillotine break)
22
The flow conditions in the jet outside of the pipeadjust to the exit flow conditions much faster than the rateof change of conditions in the vessel. This is a consequenceof the extremely high velocities that are developed as thejet expands downstream of the exit. For this reason, thetwo-phase jet was assumed to be quasi-steady, i.e., the timescale of the jet was assumed to be much smaller than the timescale of the vessel blowdown. The jet properties and loadscan now be calculated while holding the vessel propertiesconstant.
Using the quasi-steady approach makes the problemtractable. An unsteady model, which would be needed tocalculate the initial transient, would need to consider notonly the break and the target but also the flow in the vesseland associated piping. The time required on modern computersto carry out a large number of these calculations is simplyprohibitive. It should be made clear that the modeling codesused in this study have the capability of evaluating theentire time-dependent blowdown. These type of calculations,however, require several hours of CDC 7600 time to advancethe solution through the period of interest. Thequasi-steady approach, on the other hand, allows the blowdownto be calculated independent of the jet problem; thetime-dependent behavior of the vessel (reactor system) isanalyzed over small time intervals using steady statecalculations.
At this point an example that demonstrates the validityof using the quasi-steady assumption will be given. Figure3.4 shows a typical blowdown history 7 ; this history isactual data from a Marviken 8 free jet test. Figure 3.5 isan axial velocity distribution, computed with the model code,for the conditions of Figure 3.4 at about 5 sec -- a pointwhere the pressure slope in the blowdown history was steep.The nozzle exit velocity was about 6000 cm/sec (196 ft/sec),and the downstream velocity was about 35,000 cm/sec (1148ft/sec), thus an average axial velocity would be about 20,000cm/sec (656 ft/sec). A particle leaving the nozzle wouldtherefore travel about 200 meters (656 ft) in one second, orstated in another way, the time needed for a particle leavinga 1 ft diameter nozzle to traverse an L/D of about 6 is ninethousandths of a second (0.009 sec). Clearly the time scalein the flow outside of the nozzle is much different than thetime scale of the blowdown given in. Figure 3.4.
In this report, only two-dimensional axisymmetricgeometries (targets) are considered. The extension of thesemodels to other types of targets is beyond the scope of thisinvestigation. However, the development of the HEM critical
23
Co
a. Ball valve cloSu"e
a
-10 0 10 20 30 40 50 60 10 60 90 100TIME IS)
Figure 3.4 Typical blowdown vessel pressure history -- Marviken 1vessel pressure history from Reference 7.
40.00
+ 37.50
35.00
32.50
30.00
27.50
25.00L,)
u2 22.50
20.00
17.50U0-- 15.00LLJ
.. 12.50
x 10.00
7.500
5.000
2.500
0.0000.00 20.0 40.0 60.0 80.0 100. 120.
AXIAL POSITION (CM)
Figure 3.5 Axial velocity distribution computed with.the model code for the vessel conditionsof Marviken 1 (Figure 3.41 at 5s.
25
flow model, the development of a method for considering thetotal pressure behind standoff shocks, and the development ofmethods for calculating the free jet properties are tech-nologies that can be applied immediately to perform upperlimit calculations for any target geometry. These featuresdepend only on the thermodynamic properties of water.
3.3 Governing Equations
In the early part of the two-phase jet loads program,considerable effort was expended in finding a suitable modelfor describing the region between the pipe break and thetarget. This effort was twofold: a) use existing computercodes on available data and test problems, and b) use simpleengineering equations and correlations to estimate jeteffect's. One-dimensional models were developed from firstprinciples for the two-phase jet 9 ,1 0 . These models provedincapable of predicting the behavior of the jet because ofthe strong multidimensional behavior of the jet near thenozzle' exit. It was only after considerable amounts offinite difference code calculations had been completed thatprogress was made in this area.
An expensive and lengthy code assessment, using a fewtwo-phase jet problems was conducted; the results are givenin References 9, 11, and 12. BEACON/MOD2, 1 3 BEACON/MOD3, 1 4
SOLA, 1 5 RELAP, 5 CSQ, 1 6 and TRAC 4 were all evaluated fortheir capability to treat various aspects of this problem. Thefinal conclusion reached in the code evaluation was that noneof the above codes, with the exception of CSQ, was capable ofaccurately treating the desired two-phase jet configurationwithout considerable code modifications; CSQ would only requireminor modification. Because CSQ calculations also had provento be the most reliable over the widest range of parameters andinitial conditions, it was selected for use in this study.Nodalization studies also indicated that the meshing requiredto treat this problem was far in excess of the capabilities ofany of the codes except CSQ.
The CSQ code is a finite difference code which solves ahomogeneous, equilibrium set of equations for a specified flowfield; the code has multi-material capabilities and uses anaccurate equation of state for water 1 . Our choice of anequilibrium system of equations is discussed near the end ofthis section.
The governing equations are the conservation of mass,momentum and energy. These relations written in cylindricalcoordinates for axisymmetric configurations are:
bM + .0 + ON_+ t = 0, (3.1)bt Or 6z
26
where the vectors are:
PVr2
PVr+PM r
PVrVz
pVr (E+P/ p)
PVz
PVzVr=r 2
pVz+P
pVz (E+P/p)
p
PVrU r pVz
pE
and
0
0
In these relations p is the density, V is the velocity, P isthe pressure, E is the specific internal energy, t is the time,r is the radius from the center line of symmetry, and z is theaxial coordinate. These equations when combined with thethermodynamic equation of state form a closed system ofequations. For the present purpose, the equation of statesimply-relates the values of P, E and p. Two independentthermodynamic variables determine the complete statedescription.
27
The above relations may be simplified under certainconditions. When heat transport with the fluid is notimportant, one of the above relations can be eliminated infavor of the statement that the entropy (S) at a fluid fixedlocation is constant. For uniform initial conditions, thisimplies that the space fixed entropy value is also constantat positions where equation (3.1) is applicable, thus
S = so . (3.2)
It should be noted that these relations do not apply at allpositions in a jet field. Modifications are necessary forthe description of nonisentropic behavior such as shock waves.
For the steady state situation in which all of the timederivatives in (3.1) vanish, there is another interestingsolution. In vector form the energy equation becomes
v•V(H + • V'V = 0V " VV) (3.3)
where H is the enthalpy and V is the velocity. It thenfollows that
H + i V'V = HO = constant . (3.4)
Additionally, when the steady state form of equation (3.1) isused the absolute length dependency may be scaled from theproblem. In other words the form of the governing equationsremains unchanged when distance scales are modified. Forexample, if the scaled position coordinates
r =_Lr and z' = z•- (3.5)
are introduced into the steady state form of (3.1) then theform of the resulting equations are the same as before thetransformation. The scaling distance n vanishes from the
28
expressions. This scaling property can be employed in theengineering model to eliminate one of the independentparameters (jet size). The obvious choice in this problemfor n is n1 = D (the exit pipe diameter).
Most applications of these equations require numericalsolution. In general, the equation of state relation isquite complex and not suited for linearized treatment. Themain problem area is the phase change regions where verylarge discontinuities in some thermodynamic relations areencountered. The numerical methods used in the detailcalculations for this study were developed in other studiesat Sandia National Laboratories. 1 6
As earlier noted, this above system of equations is anequilibrium system. Questions have always been raised aboutthe validity of such a model when it is applied to situationswhere nonequilibrium conditions may exist. Our previoustwo-phase jet experience shows that the equilibrium rela-tions, when properly applied in multidimensional geometries,are capable of accurately describing the two-phase jetwithout empirical correlation. It should be noted here thatmany times in the past what was thought to be nonequilibriumeffects were actually multidimensional effects. 1 8 Finally,the equilibrium behavior should be completely investigatedbefore expensive nonequilibrium calculations are attempted.
There are some additional properties of the aboverelations that can be developed and are of interest to thepresent problem. Choke flow is considered in the nextsection and shock wave relations are considered in Section3.5.
3.4 Breakflow Conditions
The present computational models developed for analyzingtwo-phase jets require the flow conditions at the pipebreak. The types of breaks addressed in this report resultin what is commonly referred to as choked or critical flow.The calculation of two-phase choked flow for the purpose ofestimating breakflow rates is a subject that has hadconsiderable attention in the past 1 ,20,21. As a result,numerous models have been published and debated. Becausethere is a scarcity of both existing and planned two-phase,choked flow experiments 2 2 , there is no universal consensusabout which of the choked flow models is best. In thisstudy, the homogeneous equilibrium flow model (HEM) is used.This model was selected because it is thermodynamicallyconsistent with the jet model, and moreover, earlier criticalflow studies 1 0 showed that the HEM model provides thefluid's thermodynamic state at the break and is applicable toa wide range of initial conditions.
29
The properties of choked flows are a thermodynamicconsequence; consequently, the geometry of the situation neednot be considered until the final step. This last step is,however, very important when multidimensional choking occursand the local geometry can have significant impact upon thebreak-flow rate. Reference 10 reported a good example of theeffect geometry has. Several guard pipe calculations wereperformed where the two-dimensional structure at the breaklocation was considered. The result was a homogeneousequilibrium flow rate that was about forty percent less thanthe flow predicted using the HEM critical flow model, appliedas a one-dimensional model, i.e., a discharge coefficientMs =: 0.60.
Discharge coefficients were not used in this study atany time to correct flow rates for geometry effects. Thispractice has no thermodynamic justification since the fluid'sthermodynamic state at the break would also need to becorrected. This is rarely done and cannot be done in athermodynamically consistent manner.
Consider a flow in a pipe or a nozzle. The mass fluxvector is defined by
G = pV , (3.6)
where p is the density and V is the velocity. The totalmass flow in the pipe can be written as
M fG dA (3.7)
where dA is the normal surface unit vector. The surface inquestion can be any surface which covers the pipe exit.
For homogeneous-equilibrium, inviscid flow
1H ( V- and S=S (38)H° 2 o '1 (38
where H is the enthalpy and S is the entropy. Note that theflow is isentropic. The mass flux may now be expressed as afunction of the density and the, enthalpy by combiningequations (3.6) and (3.8), thus
30
G a p r2 (Ho-H) • (3.9)
For the HEM model, the mass flux depends.only on thethermodynamic state of the fluid; consequently, the maximummass flux (critical flow) must also only depend upon thefluid's local thermodynamic state.
Choking is defined as the state' in the fluid wherechanges in the downstream properties do not alter the massflow rate or any properties upstream of the break. The fluidat this state (or position) is supersonic, i.e., themagnitude of the fluid velocity vector is equal to or exceedsthe local sound speed. Now if the upstream fluid is subsonic(the situation that exists in power reactor piping), thenchoking will occur at the position of the transition tosupersonic flow:
[..< i switches to[VV_> I • (3.10)Cs Cs--
An absolute equality was specifically avoided in the aboveequation. Moreover, when the flow involves a phase change anequality does not exist and many of the acceptabledefinitions for choked flow breakdown because of thediscontinuous behavior in the property derivatives at thesaturation line. Consider the generally acceptabledefinition of choked flow
_r -o0 (3.11)oP/s
This definition seems to express mathematically the abovedefinition for choked flow; however, equation (3.1i) may notbe valid at a phase boundary. This is easily seen from the,following derivation.
Consider the square of equation (3.9) and differentiatethis relation with respect to density while holding theentropy constant. The result is
31
G s =2 (Ho-H) P-p (3.12)
The enthalpy can be expressed as a function of internalenergy, E, pressure, P, and density, p
H-E+PP P8p) P-) (3.13)
Thus equation (3.12) becomes
a ) (H0 H 8P (3.14)
Subs:titution of equation (3.6) and (3.9) into equation (3.14)and following some rearrangement gives
V 8P
BG) Wf 0 - j (3.15 )V P pS /
where Cs is the local sound speed. This above expression,the one generally applied to locate a choke point, willobviously fail the test in equation (3.11) when the soundspeed behaves discontinuously.
A computer code was developed which determines the chokelocation defined by equation (3.10) for the various singleand -;nixed phases of water. Additionally, the code willgene:cate the fluid properties all along the isentrope inquestion. As long as no shocks are encountered in the flowfield downstream of the choke point, this isentrope remainsvalid and represents the allowable thermodynamic states forthe expanding fluid. This result is not insignificant,because if the jet is assumed to be isentropic, thethermodynamic behavior of the jet becomes known. What willnot be known is the spatial dependence of these thermodynamicstates. Two examples are given.
32
Figure 3.6 shows the flux versus density behavior whichwould result along an isentrope that crosses a phaseboundary. This curve was produced using
G - p= 2 (Ho-H) , S a constant. (3.16)
The initial stagnation conditions (Po, TO) were 100 barsand saturated liquid. The density and mass flux marked bythe broken line is the choke point. Figure 3.7 shows theV/Cs ratio as a function of density for the same isentrope(S = 3.3598 x 107 ergs/g°C). In this region of the waterphase diagram, where the sound speed is continuous, note thatthe choke occurs at V/Cs = 1. Equation (3.11) would haveworked satisfactorily for this case.
Now consider the case where the water is initiallysubcooled. Figure 3.8 shows the mass flux versus densitybehavior for Po = 100 bars and ATo = 35 0 C. For thiscase, equation (3.15) becomes negative at the saturatedliquid line without going through zero due to thediscontinuous behavior of the sound speed there, thus chokingoccurs at this point. Figure 3.9 shows the discontinuousbehavior in V/Cs that occurred with choking. Note thevalue of V/Cs (V/Cs = 2.6) at the choke point and alsonote the behavior of V/Cs downstream (P < Pcrt) of thecritical flow point. Equation (3.11) would not have workedfor this case.
3.5 Standing Shock at the Target
Simple modeling and CSQ calculations have shown theexistence of a shock wave that stands a small distance fromthe target. The existence of such a shock wave was notunexpected; their existence and strength is well documentedin single phase flows 2 3 . These types of shock waves existbecause the fluid is required to deaccelerate rapidly in avery short distance. A shock wave may be characterized as aninstantaneous, irreversible compression of the fluid. Theenergy for performing this rapid compression comes from thefluid's upstream kinetic energy; consequently, since theprocess is irreversible, the kinetic energy leaving the shockis less than would have existed for an isentropiccompression. In other words the entropy changes (increases)across an adiabatic shock and the stagnation pressuredecreases.
33
C)
3.15
3.,0
2.5
2.0
1.5
1.0
I~oLI0-3 10-2 l0o-
p (g/cm3)
I 040
Figure 3.6 HEM mass flux as a function of density forstagnation conditions of P0 = 100 bars andsaturated liquid, equation (3.16). Chokingoccurs at the vertical broken line.
Cl)C~)N.
4.
3.
2.
1 .
0.10 .4 -2 -I
p0 10g 3p (g/cm3
1 0
Figure 3.7 HEM velocity to sound speed ratio for stagnationconditions of P0 = 100 bars and saturated liquid.
34
U)
tUi
0Y
7.
6.
5.
4.
3.
2.
1.
0.-4 -3 -2 -I
10 102 10P (g/cm3
] 0 ÷
Figure 3.8 HEM mass flux as a function of density forstagnation conditions of P = I00 bars andAT° = 35 0c.
3.0
2.0
1.0
C-PT = 35 bro0A!T° 3 35°C
0 . . .| I I I . . .I i i i . . I , i i , • .
0.010-4 -3 lo-
2
p (g/cm3 )o- I 01 0
Figure 3.9 HEM velocity to sound speed ratio for stagnationconditions of Po = 100 bars and AT° = 35 0 C.
35
If this is the case, then the homogeneous equilibriumflows in Section 3.5 become the input for the shockcalculation. For the present discussion, consider the shockto be a normal shock. This assumption is quite reasonablesince (a) we are endeavoring at this point to consider whathappens to peak centerline stagnation pressure, and (b)independent of target shape, the shock is normal in characterat some point (flow lines perpendicular to the shock).
Consider then the two-phase flow of a homogeneous,equilibrium, and isentropic fluid shown in Figure 3.10. Thefollowing equations express conservation of mass, momentumand energy for a stationary normal shock:
pUVU = PDVD , (3.17)
2 2Pu + PUVU PD + PDVD (3.18)
and
ED - EU =2 \PD+PU - -DJ (3.19)
where p, V, P, E are the density, velocity, static pressureand internal energy, respectively, and the subscripts U and Dindicate upstream and downstream of the normal shock. Theseabove equations are simply the Rankine-Hugoniot relationsexpressed for a stationary adiabatic shock. A code wasdeveloped to solve the above normal shock equations given theHEM input from Section 3.5
Consider the target and pressure nomenclature given inFigure 3.11. Figure 3.12a shows a sample calculationperformed where the stagnation conditions were Po = 100bars and saturated liquid. Figure 3.12b shows a samplecalculation performed where the stagnation conditions werePO = 100 bars and a subcooling of 350C. These curvesshow the pressures that would exist due to an isentropicallyexpanding jet at the general point just upstream of the shock(P 2 ), the point just downstream of the shock (P 3 ), andthe stagnation point on the target (P 4 ). The three
36
STREAM TUBE BOUNDARY
PU' P U VU
m
-~ I
-~ I
I -
D D D
H
SHOCK WAVE
Figure 3.10 Illustration showing the control surfacefor a stationary normal shock.
37
ITARGET
SHOCK
7 01 (CHOKEt
IL
P1- exponsion
Figure 3.11 Pressure definitions and shock wave definitionfor the centerline shock model.
38
325.
U)
a4
100o. PO 100 BARS
75.0 T Tsat 0
50.0
25.0
0.00 II0"' 10' 10 le20Io 10 *
ISENTROPIC EXPANSION PRESSURE, P 2 (bars)
(a)
U)
U)
U)
125.
100.
75.0
50.0
25.0
0.00 L_10.2 0-110* 0 10* 10* 2
ISENTROPIC EXPANSION PRESSURE, P2 (bars)
(b1
Figure 3.12 Calculation of the shock pressure P andstagnation pressure behind the shocl P 4for the isentropic expansion pressureP and for vessel stagnation conditions
a) P = 100 bars, ATo = 0°C and b)P = 10? bars, AT° = 35°C.
39
corresponding pressures are all plotted at the same abscissaposition. Note the rather significant (and rapid) fall off instagnation pressure on the target as the jet expands. For bothcurves when P 2 is about one to five bars static pressure, themagnitude of the shock is such that the stagnation pressurebehind the shock is only slightly higher than the approachstatic pressure, P 2 .
Figure 3.13 shows a typical centerline pressuredistribution for a two-phase free jet calculated using the CSQcode. This calculation models a Marviken 8 free jet test; thedata points are actual Marviken data. Note the rapid decreasein static pressure that occurs axially in the free jet. Thisbehavior confirms the pressure behavior given in Figure 3.12 andis the result of thermodynamic phenomena and not viscous dissi-pation phenomena. Also, this behavior in a free jet coupledwith the above shock model indicates that for many flows atseveral diameters away from the break exit, the stagnationpressure on a target would not be significantly different fromthe static pressure of the free jet.
The shock structure has implications beyond this two-phasejet loads study. Experimentalists need to consider theinfluence of such a shock on their pressure measuringinstruments.
3.6 Pipe Exit Core
The fluid exiting the pipe break will always be in asupersonic flow state for the conditions of interest in thisstudy. This will create a conical exit core that will remain atcritical flow conditions beyond the nozzle. This is a result ofmultidimensional behavior and has nothing to do with non-equilibrium effects.
The length of this exit core, Lc, depends on the time ittakes for a pressure wave to travel from the outer edge of theexit nozzle to the center of the flow. This time is
tc = D/2Cs , (3.20)
where Cs is the sound speed at the exit. If V1 is the exitflow velocity, then the core length is
Lc = V1 tc . (3.21)
Combining (3.20) and (3.21), we find that
Lc/D = Vl/2Cs . (3.22)
The point on the centerline Lc from the pipe exit is the firstpoint downstream of the pipe break where pressure information
40
1.0
HEM choke pressure
Pstatic 0.5 0Ptotal
0 I I I I
0 ID 2D
Axial Distance
Figure 3.13 Typical axial pressure distribution in atwo-phase jet calculated using CSQ 1 6 .Also shown are Marviken 8 free jet test data.
41
originating outside of the pipe would be communicated (via wavestraveling at Cs) to the fluid that exited at the pipe center.For distances along the centerline less than Lc, the thermo-dynamic state of the fluid is the same as that inside the pipe.This conical core region with length Lc is shown in Figure3.14. The structure of the core region has several dominatingfeatures that will be discussed in Section 4.2.
There are several things that should be noted about(3.22). First, since the critical flow parameters arethermodynamic in nature, it follows that Lc/D is also purely athermodynamic property. Secondly, Lc/D will always be greaterthan or equal to 1/2 because the flow is sonic or supersonic atthe choke surface. In Chapter 4, charts will be provided thatshow Lc/D = 1/2 for saturated stagnation conditions and Lc/D >1/2 for subcooled conditions. For highly subcooled conditions,the Lc/D can be large.
3.7 Far Field Relaxation to Ambient Pressure
In the flow field far away from the nozzle (large radii),the pressure must relieve to the ambient pressure, PA. Theflow is supersonic and this relaxation to PA occurs after theflow has expanded to pressures below PA. This over-expandedregion of the flow is terminated by a standing shock followed bya subsonic relaxation to PA. Figure 3.15 illustrates theexpected pressure loading on a flat target. In Figure 3.15 thescales have been purposely distorted to emphasize theoverexpansion and shock. This far field shock wave is generallylocated at large radial positions for the two-phase problems ofinterest here. The exact position is, however, highly dependenton the exit flow conditions, target location, and ambientpressure.
The pressure involved in the overshoot and shock isrelatively small (< 1 bar). However, because of the largeradii and large associated areas, the total integrated load canbe appreciably affected by the far field shock phenomena.
No attempt was made in this study to accurately model thefar field shock and flow behavior. It was present in all of thecalculations but adequate numerical resolution was not usedbecause of calculational expense considerations.
The load distributions were only evaluated up to the pointwhere the pressure on the target reached ambient conditions.Further details about the load calculations performed in thisreport and their relationship to the far field shock phenomenaare given in Section 7.2.
42
7ARGE-7
D =2R
L "I
T C
L R
PI
PC To' •0
VESSEL
Figure 3.14 Illustration of conical core regionwith equations for its length, L .
43
E-1
PA\
RADIUS/D
Figure 3.15 Illustration showing the expected pressureloading on a flat target for a two-phase,flashing jet.
44
3.8 External Choke
Up to this point, choking was assumed to occur in thesystem piping. This may not always be the case. As the targetis moved closer and closer to the pipe exit, a point is reachedwhere the choke point will move from inside the pipe to anexternal surface formed between the edge of the pipe and thetarget. The external choke phenomena is highly two-dimensionaland it occurs almost instantaneously as the target moves towardthe pipe exit. The flow can no longer maintain the chokesurface in the piping. The shock that occurs near the targetmoves up into the vessel and an external choke surface forms.
Under these circumstances the idealized model introduced inSection 3.3 loses its validity. The external choking problemwas not extensively investigated in this study; solutions ofthis type were beyond the present scope. However, this type ofcalculation can be performed without modification to thecomputer model. The total pressure at the centerline of thetarget would be Po; therefore one interim approach would be toassume a uniform pressure distribution of magnitude Po on thetarget for some appropriate radius.
45/4 6
4.0 TWO PHASE JET COMPUTER LOAD MODEL
Once it was established that CSQ 1 6 would be modified (seeSection 3.5) for the two-phase jet problem a matrix of calculationswas developed to establish a computational data base. This database covers the following range of initial conditions
pressures of 60 to 170 bars
{subcooling of 0 to 70 °C1qualities of 0 to 0.75
L/D's of 0.50 to 15.0
This chapter contains the description of the computer model andpresents typical data-base calculations and their consequences.
4.1 Computer Model
The advanced, two-dimensional computer code CSQ was used todevelop the two-phase jet load data base. The calculations wereconducted for the geometry shown in Figure 4.1.,
Briefly, the model can be described as a steady constantproperty vessel blowing down through a nozzle of diameter D. Thejet develops outside of the nozzle and strikes a target at adistance L from the pipe exit. The break flow from the vessel isevaluated using the HEM critical flow model; consequently, the breakflow only depends upon the vessel stagnation properties. (Thedischarge coefficient at the nozzle exit is assumed to be unity.)The wall loading then becomes a function of four independentvariables:
PT= Pt(R/D, Po, To, L/D) (4.1)
or
PT= Pt(R/D, Po, Xo, L/D) , (4.2)
where PT is the pressure on the target, R is the radius (measuredon the target), Po is the stagnation pressure, To is thestagnation temperature, and Xo is the stagnation quality.
Figure 4.2 defines three regions within the geometry of Figure4.1. Region I, inside the pipe, consists of an inlet boundary zonewhere the flow conditions are driven using the HEM critical flowvelocity and thermodynamic properties (PI, TI, etc.).Throughout this entire region, the flow is uniform and nozzlegeometry effects are neglected.
47
MODEL
GEOMETRY
01-~
R -+TARGET
Figure 4.1 Model Geometry for Two-Phase Jet LoadCalculations
48
i TARGET
LT I
IIID = 2R
RV1
C S
II
P1
PR
P 0 . T O , 100
VESSEL
-J
Figure 4.2 Illustration of vessel and target showing thethree computational regions: I, HEM boundary;II, Exit Core; III, Jet Environment
49
Region II, the pipe exit core, consists of a conical core offluid which remains at HEM critical flow conditions beyond thenozzle (or break). This core of fluid is traveling at supersonicvelocity and was discussed in Section 3.6.
Region III, the jet environment, is the main solution region.In this region, the inviscid equations of motion are solved; theresult is not only the wall pressure distribution, but thevelocities, temperatures, pressures, etc., for each point in RegionIII. As earlier noted in Section 3.4, there is a distinct shockstructure present near the wall in Region III. Our calculationsdefine both the strength and shape of this shock.
4.2 Some Consequences of the Exit Core
There are three distinct (and obvious) possibilities that canoccur with the exit core region, Region II: Lc greater than, equalto, or less than L. Consider first the case where Lc > L.Here, theoretically, the exit core touches the wall. In practice,there is a weak shock and the fluid recovers, for all practicalpurposes, the vessel's total pressure Po. Using the HEM model,once given an L/D ratio and the system's stagnation properties, itis possible to determine if the peak pressure on the target will bethe vessel pressure, i.e., if Lc > L. The charts in Figures 4.3and 4.4 give Lc/D as a function of stagnation pressure andtemperature; degree of subcooling is also shown. Consider the casewhere L/D = 2.0 and two operating points were Po = 100 barsATo = 70 0 K and Po = 100 bars, To = 560 K. For the firstoperating point Lc/D is about 2.7, thus the peak target pressurewould be 100 bars. For the second operating point, Lc/D is about1.0 and the flow field would have a strong shock, thus the peaktarget pressure would be less than 100 bars. Since L/D = 2, anyvessel stagnation properties that generate an Lc/D > 2 wouldrecover the stagnation pressure Po at the target centerline.
The case where Lc = L should theoretically recover the vesselpressure Po. This case has not been thoroughly investigated inthis study because the gradients in this region are very large andwould require a very fine grid. However, calculations on eitherside of this point show clearly that as Lc approaches L, the peakpressure approaches Po.
In the cases where Lc < L, a strong standing shock exists inthe flow field near the wall and, depending upon the amount of jetexpansion (see Figure 4.2), there is a loss of total pressure.
4.3 Modifications of CSQ for the Two-Phase Jet Impingement Study
The CSQ 1 6 code is a multi-purpose code that was developed atSandia National Laboratories to handle a wide spectrum of materialresponse problems. The next few lines, taken from the introductionin Reference 16, briefly describe the code's versatility.
50
5.0
4.0 '
35P
3.0 SO
0 2.5
_ 2 .0
1 . 5 ( S I P
1.0
. 50 -/- o0C•- - • 1
0.0
0.00 40.0 80.0 120. 160. 200.
P (bar s)0
Figure 4.3 Two-phase, jet-exit core (Lc/D) as a functionof stagnation pressure and stagnation tempera-ture or subcooling.
51
15.
14./
13. / o
12.11. K €,
iO
10.
9.0
8.0
\ 7.0 MO
-J 6.049
5.0 SO "
4.0 SJO
3.02.0S6
0 .0 -0 i 0
0.00 40.0 80.0 120. 160. 200.
P (baors)0
Figure 4.4 Two-phase, jet-exit core (Lc/D) as a function ofstagnation pressure and stagnation temperature orsubcooling.
52
CSQ is a FORTRAN program for computation of two-dimensionalmaterial response. An Eulerian finite difference method is employedin either rectangular or cylindrical coordinates. Models areincluded to treat an extremely wide range of problems, from soliddynamics to totally stripped plasmas. Portable editing and graphicsprograms are available to aid in analyzing the computed behavior.
Because the models in CSQ are general and apply to a wide rangeof problems with different time scales, it is possible to increasethe computational speed for a given problem by selectively modifyingthe code to best suit the problem of current interest -- two-phasejets.
Modifications of CSQ were performed as part of this study;additionally, a version of CSQ was used which incorporated asecond-order-accurate numerical convection algorithm for performinginter-cell transport of mass, momentum, and energy. 2 4 Themodifications performed and a slight change in input parametersresulted in over a factor of 20 in calculation speed for thetwo-phase jet problem. Typical calculations with 10 to 25 cellsacross the break exit require about 20 minutes of CDC 7600 CPU timebefore the code reaches convergence in the jet field; thisillustrates the importance of the factor of 20 increase in speed.The two main areas responsible for most of the speed increase are:
(1) improved numerical methods for the equation of statedescription for water in two-phase states;
(2) changes in input parameters for the viscosity computation;
Several other modifications to the standard CSQ code were alsoincluded for user convenience for these calculations, but the abovewere the only modifications related to computational speed andaccuracy.
For the jet impingement calculations, CSQ was modified to usethe thermodynamic description (equation of state, EOS) of water fromReference 17. The analytic form used is a complex function ofdensity and temperature. Mixed-phase states were evaluated usingthe Gibbs phase rules with an iteration on the single phasesurface. The end result was that an average of about 10single-phase EOS evaluations were required for each mixed-phasestate. Since for two-phase jets, a large fraction of the totalcomputational mesh is mixed phase, the code was using a very largefraction of the total execution time in EOS evaluations.
This problem was solved by carefully fitting the thermodynamicproperties along the mixed-phase boundaries. A very dense array ofpoints and standard interpolation methods was used. All requiredEOS data were included so that no evaluations of the single phase
53
functions were required for any mixed-phase state. The net resultwas an increase in overall computational speed by a factor of about2.5.
The second change concerned the use of viscosity in CSQ. CSQwas developed in projects where strong shock waves propagatedthrough much of the two-dimensional grid. To smooth the results inthe vicinity of the strong shock waves, the viscosity coefficientswere given values considerable higher than the real physicalviscosity. This is the standard procedure for this type ofcalculation (strong shocks) and results in very sharp wave frontsbeing spread over a number of computation cells.
The existence of a standing shock wave positioned very close tothe target was discussed in Section 3.5. The large viscositycoefficients, which are the default input in CSQ, caused the needfor a large number of cells near the target to obtain a mesh sizeconverged solution. Since the computational time of the code is alinear function of the number of cells and the time step is Courantlimited, the fine resolution was expensive. (See also Reference 24for further discussion and examples of the effect these viscositycoefficients have on run time and nodalizations.)
The extent of the viscosity "difficulty" was discovered during aseries of tests to determine mesh sizes for the final impingementmatrix. A separate study was conducted to determine the appropriatecorrection. The study found that minor changes to the code inputwould eliminate the problem; no other code changes were necessary.A factor of approximately eight in overall computational speed wasrealized for this change. Nodalizations as coarse as AZ/DAr/D - 1/20 are now being used (20 cells across the pipediameter). The larger viscosity coefficients required less thanhalf of that cell dimension.
4.4 Model Calculations
Table 4.1 shows the variable ranges selected for study; note,however, that initial calculation experience has shown that some ofthe flow combinations are outside of the model's capabilities, i.e.,double choking, etc.
Figure 4.5 shows a typical coarse grid which is used to initiatethe calculations; finer zoning is later used to obtain fullyconverged solutions. Although for the present modeling purposesonly the target pressures are of interest, the code actuallyprovides the flow solution between the break and the target. Twoexamples will be provided. Axial position 0.0 is the break-exitplane.
Example one has a stagnation pressure and temperature of 170bars and 555.52 K (70 °C subcooled). The target L/D is 2.0 andLc/D is 1.98; thus we expect to recover nearly the initial
54
Table 4.1
Two-Phase Jet Model Calculational Matrix Base
ý4 j
_d _P
Wed
a)04H
subcooling, LTo( 0 C)
70, 50, 35, 15, 0
qualities 0, 0.01, 0.05, 0.1, 0.2,0.333, 0.75, 0.99
Q)
pressure,Po 60, 80, 100, 130, 150, 170a(bars)
0"I target position 0.5, 0.75, 1.0, 1.25, 1.50,
L/D 1.75, 2.0, 2.5, 3.0, 4.0, 8.00
55
SPACE MESH
4.0
3.0 t
2.0
0-44IJ
0a. 1.0
...............
............
............
. . . . . . . .. . . .
...........
!!jam0.0
-1.0
0. 1.0 2.0 3.0 4.0 5.0
Radius/D
Figure 4.5 Typical coarse computational grid used inCSQ model calculations.
56
stagnation pressure at the target centerline. Figure 4.6 shows theresulting pressure contours and Figure 4.7 shows the densitycontours. The ordinate and abscissa in these two figures are thenondimensional axial and radial coordinates. The target is locatedat L/D = 2.0. Figure 4.8 shows the velocity, sound speed,temperature, and pressure on the target as a function of radius. Ofparticular interest is the pressure. The distribution of pressurewith radius is typical of the distributions observed when Lc/D > L/D-note how rapidly the pressure falls off in the radial direction.Figure 4.9 shows the density, temperature, and pressure along thejet's centerline; note that large pressure gradients exist near thetarget and that the centerline target pressure is indeed about 170bars.
Example two has a stagnation pressure and temperature of 150bars and 615.39 K (0oC subcooling). The target L/D is 2.0 andLc/D is 0.5. In this example, the strong shock structure willappear. Figure 4.10 shows the pressure contours. and Figure 4.11shows the density contours. These contours show a very rapidsupersonic expansion of the jet just outside of the break. This isthe same behavior observed for free jets. The expansion of the jetis very rapid; within about a pipe diameter beyond the exit core,the static pressure has dropped to less than 5 bars. Figure 4.12shows the velocity, sound speed, temperature, and pressure on thetarget as a function of the target radius. The bell shaped curvefor the pressure, observed in numerous calculations, was expected.Figure 4.13 shows the density, temperature, and pressure along thejet's centerline. In the pressure curve the shock, which occursnear the wall, is clearly visible. Figure 4.14 displays thevelocity field. The length of each arrow is proportional to thevelocity magnitude. Notice the well-defined shock where thevelocity field changes direction discontinuously. Figure 4.15 showsthe target pressure distribution (including the stagnationcenterline value) on a scale which includes the vessel stagnationpressure, P0 = 150 bars. The shock structure, which stands offfrom the target, has a significant impact upon the local magnitudeof the load. Finally, Figure 4.16 is a three-dimensionalillustration of the pressure field; the jet expansion followed by acompression is clearly shown. The last curve in the foreground inFigure 4.16 is the wall pressure distribution.
The possibility of using some dimensionless correlation torepresent the computer results was investigated. To date this hasnot been successful. This is mainly attributable to the change inbehavior of the solution in the region of the shock and the changein the types of wall pressure distributions seen when the value ofLc/D becomes greater than or slightly smaller than L/D. The fluidupstream of the shock is a two-phase mixture while the downstreamfluid is pure liquid. The fluid near the center of the target isalso in a pure liquid state. The fluid flashes as it expandsradially along the target. This results in a pronounced change ofthe shape of the target pressure distribution as shown in Figure 4.8.
57
4.0
E = 100 barsF = 90G = 80H = 70I = 60J = 50K = 40
L = 35 barsM = 30N = 250 = 20P = 15Q = 10R= 5S = 2.5
3.04
0-.4
0a.,
-4
2.0
1.0
0
-1.0
0 1.0 2.0 3.0 4.0 5.0
Radius/D
Figure 4.6 Pressure contours for subcooled stagnationproperties (P = 170 bars, AT = 70 C). Notethat the exit core extended to the wall,thus stagnation conditions were present atthe center of the target.
58
4.o
A = .800 g/Cmr3
B = .700C = .600D = .500E = .400F = .300
G = .200 g/cm3
H = .100I = .050J = .010K = .005L = .002
3.0 +
0
004
2.0
1.0
0
-1.0
a 1.0 2.0 .3.0 4.0 5.0
Radius/D
Figure 4.7 Density contours for subcooled stagnationproperties (Po = 170 bars, ATo = 70 C). Notethat the exit core extended to the wall, thusstagnation conditions were present at thecenter of the target.
59
. . . . . . . . . . . . . .
50,00043 40,00%.,4 W0 30,000-
• 20,000'
10,000
00 100,000
0-. 75,000
(UIN 50,000
25,000,
0~500•
400
4 J
0 300W004 200
Ei~01
150
• i00W g04
50
0
0 1.0 2.0 3.0 4.0 5.0
Radius/D
Figure 4.8 Velocity magnitude, sound speed, temperatureand pressure distributions along the radiusof the target for subcooled vessel stagnationconditions (P0 = 170 bars, nT = 70 C) where,the exit core extended to the wall.
60
b,
0.75
0.50
0.25
0.
0
a)
.4J
Iý4
0)
1.4
W
500.
4Q0.
300.
200.
100.
150.
100.
50.
0.
0. 0.5 1.0 1.5 2.0
Axial Position/D
Figure 4.9 Density, temperature, and pressure distributionsalong the centerline of the two-phase jet forsubcooled vessel stagnation conditions (Po = 170bars, AT = 70 0 C). Note that the exit coreextended to the wall and that the stagnationconditions at the target (L/D = 2.0) are thevessel conditions.
61
4.0
EFGHIJ
KL
100 bars90807060504035
MN0pQRS
30 bars25201510
52.5
3.0 t
04.)
0--4
x
2.0
1.0
0
-1.0a
0 1.0 2.0 3.0 4.0 5.0
Radius/D
Figure 4.10 Pressure contours for saturated liquidstag-nation conditions (Po = 150 barsATo 0°C). Lc/D = 0.50
62
4.0
E = .400 g/cm 3
F = .300
I = .050 g/cm3
J = .010
3.0 G = .200
H = .100
K = .005
L = .002
09-q4)
0
a-4
2.0
1.0
a
0 1.0 2.0 3.0 4.0 5.0
Radius/D
Figure 4.11 Density contours for saturated liquidstagnation conditions (Po = 150 bars,ATo = 00 C). Lc/D = 0.50
63
75,000
>I-• 3 u 50,000U wI
S 25,000
0
30,000
'04 •" 20,000
= u 10,0000-
0500
400
M 300
0-- 200
E 100
~15 0
(L)•• 10
5
004
1.0 2.0 3.0 4.0 5.0
Radius/D
Figure 4.12 Velocity magnitude, sound speed, temperatureand pressure distributions along the radiusof the target for saturated liquid stagnationconditions (Po = 150 bars, ATo = 00 C).Lc/D = 0.5
64
0.
0.20
O. . l
0.10
0.0
500.0
400.
S300.41
S200.
100.
0.
100.U
75.
0 50.
25.
0.
0 0.5 1.0 1.5 2.0
Axial Position/D
Figure .4.13 Density, temperature, and pressure distri-butions along the centerline of the two-phasejet for saturated liquid stagnation conditions(Po = 150 bars, ATo = 0°C). Lc/D = 0.5
65
VELOCITY
2.0
1.5
1.00O
A 4 * 'P 0 V V v v - P - - - epaa .
....44... ... ... .....Pp a a a ii ./1/4/4 4 4 t//t4t 0 0 W V
" @""t44,tt d V, , ,, ,' -. a •- -.zz•, -e -e - -i -"-
.... 7.z zz 7. Z. X" , 1ZZ ZZZ
: -n -- ..- ..-
* fff/////2///22/I -I,.I.I..
// //" - " ." ... .- - .-F F / / / / '/// " A'4' ' -5--s--'' -- .'...-- - . - --- • --
A A A A FA F A f / X1A A A4 A A A A 9/ Of Af 0,0----- --.- a- a aAAI A A A A 4 1 ;f X~ Or ' 'a
AA A A A A A 49/Id~a a ~ - a -
A A A A I A # 9 dA~ ~ '''--
A A A A A A A 4 A A
A A A A A A A A A &
0.5
0.
0. 0.5 1.0
Radius
1.5 2.0 2.5
Figure 4.14 Velocity field for a two-phase, impinging jetat stagnation conditions of Po = 150 bars andsaturated liquid. L/D = 2.0 and the axis arethe axial and radial coordinates measured incentimeters. Velocity magnitude is proportionedto the length of each vector.
66
180.0
160.0
140.0
120.0
100.0
80. 00
60.00
I I I I I I I I I
'C
w
w
0.-w
CEI.-.
40.00
20.00
I I I I I0 0.000
Do ,*.00 1.0 2.0 3.0 4.0 5.0
Radius/D
Figure 4.15 Target pressure distribution as a function ofthe radial position for L/D = 2.0 and for vesselstagnation conditions of Po = 150 bars andsaturated liquid (AT = 01.
67
Figure 4.16 Three-dimensional pressure field contourfor L/D = 2.0 and for vessel stagnationconditions of Po = 150 bars and saturatedliquid.
68
4.5 Data Base Calculations
In this section a small number of illustrative data basecalculations will be given; this section will be kept brief becausethe calculations shown are a small subset of the figures in AppendixA. These figures illustrate the range of calculations performed.
Figure 4.17 shows the effect of target position with respect tothe pipe break; the figure shows the target pressure distributionsfor several L/D ratios. The vessel stagnation conditions werePo = 150 bars and 350 C subcooling. The curve for L/D = 1 is anexample where L/D was less than Lc/D (Lc/D = 1.07); note thatthe wall stagnation pressure was Po. The change in the shape ofthe pressure profile for L/D = 1 at about R/D = 0.50 is due to aphase change in the fluid at that point. Figure4.18 shows thestatic pressure behavior along the jet's centerline for the sameflow and geometric conditions of Figure 4.17. This figure shows anoverlay of five centerline pressure distributions as a function ofnondimensionalized axial position for five different targetlocations (L/D = 1.00, 1.25, 1.50, 2.00, and 4.00). Note that eachcurve ends with the stagnation pressure on the target. The jetbehaves characteristically as a free jet up to the shock, whichoccurs near the target. Figure 4.19 illustrates three-dimensionallyan overlay of both the wall and centerline pressure distributionsgiven by Figures 4.17 and 4.18. The rapid fall in centerlinestagnation pressure on the wall is observed in all L/D studies. Forthis particular group of calculations, P 4 /Po = .17 at L/D = 2.0and P 4 /Po = .05 at L/D = 4.0 where P 4 is the centerline stagnation(wall) pressure. This trend is easily explained by the shock theoryintroduced earlier in Section 3.5.
Figure 4.20 shows the effect the upstream stagnation pressure,Po, can have on the target's pressure distribution; the figureshows the wall pressure distribution as a function of the radialposition. The vessel pressure Po was varied between 60 and 170bars, L/D was 2.0 and there was a constant 35 0 C subcooling.
Figures 4.21 and 4.22 show the effect of subcooling for twodifferent vessel pressures. Figure 4.21 shows the target pressuredistributions at L/D = 2.0 for Po = 170 bars and several differentstagnation temperatures; Figure 4.22 shows the wall pressuredistributions at L/D = 2.0 for Po = 130 bars. The centerlinepressure distributions that correspond to the target pressuredistributions given in Figure 4.21 are given in Figure 4.23. Noticethe effect the stagnation conditions have on the break (pipe exit)pressure; again these distributions are characteristic of free jetbehavior except for the points near the shock at the target.
69
IO0.
160.
140.
120.
W 100.
W80.0
90.0C.
40.0
20.0
0.000.0 1.0 2.0 3.0
RADIUS/D
4.0 5.0
Figure 4.17
200.
180.
ISO.-160.
tn
C 140.
. 120.InIn
100.0.
80.0-J
e 60.0
w 40.0
20.0
Wall pressure distribution for several L/D ratiosand for vessel stagnation conditions P = )50 barsand 350 subcooled liquid. o
0.00L .. .•0.00 .500 1.00 1.50 2.00 2.50 3.00 3.50 4.00
AXIAL POSITION / 0
Figure 4.18 Overlay of five centerline pressure distributionsas a function of the non-dimensional axial positionshowing the effect of target position (L/D).
70
200
160
120,
100
.80.
60
'10
20
Po = 150 Bars
T-T = -35 0 C
5
Figure 4.19 3-D illustration of five wall pressure distributionsas a function of target radius each at a different L/D.
too.
160.
140.
120.U1
100.
o.oA.
--j 60.0
40.0
20.0
0.000.0 1.0 2.0 3.0
RADIUS/D
4.0 5.0
Figure 4.20 Stagnation pressure effect onpressure distributions at L/D
target (wall)= 2.
71
200. '. . ..
ISO. PO 70 ARS L/D 2.0
160.
140.40 T T-SAT "700 C .c/D1.98
120. 1-T SATs 50 C , LC/D 1.39wC" -T - TSAT -350 C . LC/DO ",.02
T-TSAT &C 1. -00.50. 90.0 0.333 L/D-0.50-J
dI 60.0
40.0
20.0
0. 000.00 .500 1.00 1.50 2.00 2.50 3.00 3.50 4.00
RADIUS/D
Figure 4.21 Effect of subcooling on target (wall pressureat Po = 170 bars and L/D 2.
200.. . . . .
oo. Po -130 BARS I/D - 2.0
160.
14.-T - T - -7o0C , L/D - 2.29,--T - TSAT " -350 C LC/D = 1.14!- 120.
--T - TSA. " O. C , LC/D- 0.50
I 0
•, 60.0
40.0
20.0
0.00 . . . . . . . .0.00 .500 1.00 1.50 2.00 2.50 3.00 3.50 4.00
RADIUS/D
Figure 4.22 Effect of subcooling on target (*wall) pressureat Po = 130 bars and L/D = 2.
72
200.
180.
wU)
w0r
wz
wI--zwL)
160.
140.
120.
100.
80.0
60.0
40.0
20.0
0.00 , , I •0.00 .500 1.00 1.50 2.00 2.50 3.00 3.50 4.00
AXIAL POSITION / D
Figure 4.23 Effect of vessel subcooling on centerlinepressure at L/C = 2.0 and for a vesselstagnation pressure Po = 170 bars.
73/74
5.0 THERMODYNAMIC AND CRITICAL FLOW PROPERTIES
In this chapter we show figures that provide a descriptionof the thermodynamic behavior of water. These charts are veryaccurate, and the relations used to develop them are used in boththe CSQ model described in Chapter 4 and the approximate modelthat will be developed in Chapter 6. The defining relations forcritical flow were developed in Chapter 3.
The HEM critical flow model is defined by the governingequations (3.8), (3.9) and (3.10). This model provides aconsistent (isentropic) thermodynamic connection of theproperties and flow upstream of and at the break. The criticalflow charts relate the mass flux and the break thermodynamicproperties to the upstream stagnation conditions.
These charts are given in Figures 5.1 through 5.9. Figures5.1, 5.2, and 5.3 provide the properties of subcooled andsaturated water for the pressure and temperature ranges ofinterest in this report. Figure 5.1 gives the temperature as afunction of pressure (I bar = .1 MPa) and entropy (1 erg/g°K10-7 kJ/kg OK) for subcooled water;. Figure 5.2 gives thetemperature as a function of the density (1 g/cm3 = 103
kg/m 3 ) and entropy for saturated water. Figure 5.2 alsoincludes a bounding 200 bar isobar. Figure 5.3 gives thetemperature as a function of density and entropy but at lowdensities (p = 0.001 to 0.1 g/cm3 ). Figures 5.4 through 5.9relate various thermodynamic properties to the HEM mass flux.Figures 5.4 and 5.5 show the HEM mass flux (I kg/cm2 s = 10-4
kg/m 2 s) as a function of the entropy and stagnationtemperature; Figure 5.4 emphasizes the region with subcooledstagnation conditions while Figure 5.5 emphasizes the region withsaturated stagnation conditions. Figure 5.6 shows the HEM massflux as a function of the stagnation pressure and the stagnationtemperature; these curves are used where the stagnationconditions are saturated. Figure 5.7 and 5.8 show the HEM massflux as a function of the stagnation quality and stagnationtemperature; these curves are used where the stagnationconditions are saturated. Figure 5.9 shows the HEM mass flux asa function of the stagnation density and the stagnationtemperature, and Figure 5.10 shows the critical flow density as afunction of the stagnation density and stagnation temperature.These charts provide a means for finding the break flow and breakproperties when given the stagnation or upstream properties, orfor finding the stagnation properties when given the break flowand break or upstream properties. For example, the waterproperties figures (5.1, 5.2, and 5.3) can be used to determinethe entropy just upstream of the break. Then using this entropyand the break flow rate the stagnation temperature is located in
75
WATER700 .
650.
600.
550.
500.
450.I--
400.
350.
300.
250.0.00 40 . 0 80.0 120. 160. 200.
P (bor s )240.
Figure 5.1 Thermodynamic properties of water.Temperature as a function of pressureand entropy for a range of pressure andentropy that. emphasizes subcooledconditions.
76
WATER700
650.
600.
550.
500.
450.
400.
350.
300.
250.
I--
0. .2 .4 .6 .8
,o ( g/cm 3
1.
Figure 5.2 Thermodynamic properties of water.Temperature as a function of density andentropy for a range of density and entropythat emphasizes saturated conditions.
77
WATER700 .
650.
600.
550.
500.
450.
400.
350.
300.
250
1 -3 -2 - 1I0 I0 / 0M3,o (g9/cm)
Figure 5.3 Thermodynamic properties of water.Temperature as a function of densityand entropy for 10w densities.
78
20.
18.
16.
14.
04 12.E
O0)
.a)
0
10.
8.0
6.0
4.0
2.0
0.00.00 5.00 10.0 15.0 20.0 25.0 30.0 35.0 40.0
S( 10* ergs/9 K)
Figure 5.4 HEM mass flux as a function of entropy andstagnation temperature for a range of entropythat emphasizes subcooled stagnation conditions.
79
(J~(~4
E0N0)
0)
5.0
4.5
4.0 -
3.5 -
3.0 -
2.5 -
2.0 -
1.5
1.0
.50
0.0 -
0.0 10. 20. 30. 40. 50. 60. 70.
S( 10÷ er 9s/9 K )
Figure 5.5 HEM mass flux as a function of entropy andstagnation temperature for a range of entropythat emphasizes saturated stagnation conditions.
80
20.
18.
16.
14.
12.E
10.
•8.0CD
6.0
4.0
2.0
0.0000.00 40.0 80.0 120.
P (bors)0
160. 200.
Figure 5.6 HEM mass flux as a function of stagnationpressure and stagnation temperature.
81
-@rnq~aduie; uoT4PuB6qS PUPIA.TTpnbh UOT4VU6PS j0 uo~flDun; P sp xnlT SsPW WHHL~ * 9aaBTa
0X
"t9,Vp a,
or,
(D
0 z
z,
0"0 V
1 .
.9
.8
.7
.6
.5
C4
EU
0)
5- .4
CD
.3
.2
.I
0.0. .2 .4 .6 .8
X0
HEM mass flux as a function of the stagnationquality and stagnation temperature.
1 .
Pigure 5.8
83
*aznjvadtuI8 uoqI ub~s pup AqrsuapuOT4Puvs 9q4~ 40 uoTq.3uf v sp XflT; ssvUI WSH 6*S 9im5Tj
WU /6 ) cIY"I9 ,D,
0110
or.
0 8
0 (0
Z)
o 3
O0.
.
.9
.8
,7
.-. .6
0 5
o• .4
Q .3
.11
• 2~-- , 400 •
035_0 1 I I1 1 -I I- LI I I
0. .2 .4 .6 .8 1.3
p3 ( g/cm )
Figure 5.10 HEM critical flow density as a function ofstagnation density and stagnation temperature.
85
Figures 5.4 and 5.5. Finally, using the stagnation temperatureand the break flow, Figures 5.6 or 5.7 are used to find thestagnation pressure or stagnation quality (whichever isapplicable). Although this procedure is given here in graphicform, it is not difficult to computerize the entire procedure.
86
6.0 APPROXIMATE MODELS
The data base established using CSQ was sparse because ofthe wide range of initial conditions investigated with theresources available. Since the data base was sparse it becamenecessary to interpolate (curve fit) within the computationaldata base to arrive at complete sets of loading curves for a widerange of desired initial conditions. Simple interpolativestrategies proved to be unsuccessful; the computational data basewas too sparse and the two-phase jet impingement solutions weretoo complicated. Consequently, several approximate models, whichcontained much of the physics required to describe the basicphenomena occurring in the two-phase jet, had to be developed;these models were then used to perform the interpolation withinthe computational data base. This chapter describes jet andshock models for calculating both axial and target pressuredistributions. These approximate models when combined arecapable of predicting, with less accuracy than the modelintroduced in Chapter 2, two-phase jet loads, and are useful forfast scoping calculations.
6.1 Centerline Free-Jet Expansion Model
This model approximates the free jet expansion along thecenterline of a two-phase jet.
The geometry for this model is given in Figure 6.1. Adetailed description of this geometry was given in Chapter 3.Briefly, we are assuming the blowdown is quasi-steady. Thevessel pressure Po is the upstream stagnation pressure; P1 isthe break pressure; P 2 is the centerline pressure in the jetupstream of any shock phenomena; P3 is the pressure in the jetdownstream of the shock; and finally, P 4 is the stagnationpressure at the target. Homogeneous-equilibrium (HEM) chokedflow conditions are assumed for the break.
Consider the problem of determining the centerline pressurein an expanding jet for a general point 2, P2- Since thispoint is upstream of the standing shock wave, it is essentiallyunaffected by the target's presence. (Once the full state ofconditions are known at 2, then the centerline properties at 3and 4 are completely determined provided the shock standoffdistance is small.)
A large number of both free jet and target calculations wererun as a part of this two-phase jet study. Results from thesecalculations and detailed discussions were given in Chapter 4.From observing the basic behavior of the thermodynamic propertycontours and the centerline pressure behavior we found that thefollowing relation approximates the free jet expansions for flowswith saturated stagnation conditions:
87
TARGET
SHOCK
T\U
J \I
PI
Poe
V
To 9 0
ESSEL
-JLi
Figure 6.1 Geometry and definitions for the freejet expansion model and the centerlineshock model.
88
D2 )2 , z 1(PV) 2 - (PV)l ( , z D (6.1)
where p is the density of the two-phase mixture, V is the axialvelocity, z is the axial distance to point 2, and the subscripts 2and 1 refer to conditions at point 2 and the pipe exit, respectively(see Figure 6.1).
Arguments can be made for an approximate derivation of equation(6.1) if two assumptions are made about the two-phase jetexpansion. Neither is absolutely true, but the errors appear tocancel. First assume that the jet has a well defined boundarydescribed by a 450 expansion angle, then assume that the axialcomponent of the mass flux is not a function of radius. These twoassumptions are illustrated in Figure 6.2. Mass conservation thenyields
(pVA) 2 = (pVA) 1 , (6.2)
where A is the cross-sectional area of the jet. Since
A 72 (6.3)
and
A2 = z2 , (6.4)
equation (6.2) will reduce to equation (6.1). Relation (6.4) usesthe 450 expansion angle. Again, note that this relation is onlyapproximately true along the axis of symmetry.
When subcooled stagnation conditions exist, the model inequation (6.1) does not account for the core length Lc. Forsubcooled stagnation conditions, the core length is greater thanD/2. (The HEM core length Lc was discussed and a method forcalculating Lc was given in Section 3.6.) Equation (6.1) can bemodified to account for this behavior. This new approximateexpression applicable in the range D/2 < Lc < z is given by
(pV) 2 = (pV) 1 (D/2z) 2 , (6.5)
1 - (Lc/Z)P E1 - (D/2LI) 2 ]
89
Ia
D - X
t
Z..L-a- Z
2
450
VLC~
Figure 6.2 Illustration showing the geometry anddefinitions used for developing thealgebraic free jet expansion model.
90
where a is a constant that must be evaluated using comparisonswith data or calculations. A value of 8 = 14 describes most ofthe calculations of interest. In both the simple model, equation(6.1), and the modified model, equation (6.5), HEM choked flowproperties exist for 0 < z < Lc. For saturated stagnationconditions, Lc = D/2.
The procedure for using the above relations to determine thefree field jet parameters is straightforward. The exit conditionsare calculated using the HEM critical flow model described inChapter 3 or the charts in Chapter 5. The other relations whichapply to the free field fluid are
1 2Ho = Hz + Vz (6.6)
and
so = Sz (6.7)
where H is the enthalpy and S is the entropy; the model assumes anisentropic expansion. The three relations [(6.1) or (6.5)], (6.6)and (6.7) can be solved with the thermodynamic equation of state forwater to complete the description of all thermodynamic variables andthe velocity along the axis of symmetry. The numerical method issimple but was performed on a computer because of the nonlinear,complicated equation of state for water. (This same computerprogram was used to develop some of the thermodynamic charts shownin Chapter 5.)
Figure 6.3 shows the results from a sample calculation using(6.1) for a vessel stagnation pressure of Po = 150 bars and 350 Cof subcooling (ATo = 350 C). These initial conditions wereselected to emphasize that equation (6.1) should be carefullyapplied when the stagnation conditions are not saturated. The curvelabeled P2 is the free jet expansion pressure calculated usingequation (6.1); curves P3 and P4 will be discussed later in thisreport. Figure 6.4 shows the results obtained using equation (6.5)for the same stagnation conditions. The correction for the effectof the HEM boundary core is evident upon inspection. Finally,Figure 6.5 shows the comparison of the curves P 2 and P 4 fromFigure 6.4 with the 2-D axisymmetric results from Section 4.5. Theapproximate model, P2 from equation (6.5), compares quite wellwith the 2-D calculations for the entire length of the expansion.
91
200.
180. PO 150 BARST1-0Ts = -35°C
160. Lc/D = 1.07
140.
120. P4
P3100.
.P2
CL 80.0 P1
60.0
40.0 -
20.0
0.000.00 .500 1.00 1.50 2.00 2.50 3.00 3.50 4.00
L/D
Figure 6.3 Pressure variation along the axis ofsymmetry of a two-phase jet for the freejet, equation (6.1), and shock models.P is the vessel pressure; P1 is the exitHEM pressure, P 2 is the jet pressure,equation (6.1); P 3 is the Hugoniot pressure;and P 4 is the target pressure,
92
200.
0
CL
180.
160.
140.
120.
100.
80.0
60.0
40.0
20.0
0.00 L0.00 .500 1.00 1.50 2.00 2.50 3.00 3.50 4.00
L/D
Figure 6.4 Pressure variation along the axis ofsymmetry of a two-phase jet for the freejet, equation (.5), and shock models.PO is the vessel pressure; P 1 is the exitHEM pressure, P 2 is the jet pressure,equation (6.5); P 3 is the Hugoniot pressureand P 4 is the target pressure.
93
200.
18 0 . '0 0 "'oT -TSAT = -350C
160. Po
1 .... 2-D COMPUTER RESULTS
120. - SIMPLE MODELa 0: 100. P1
C 80... :
60.0 -
40.0 "
20.0 "' • P2
0.00 .,. ...
0.00 .500 1.00 1.50 2.00 2."50 3.00 3.50 4.00
L/D
Figure 6.5 comparison of centerline results fromSection 4.5 Figure 4.18 with the free jet,equation (6.5), and the shock models.Curves for P and P 4 are identical tothe curves ShOWn in Figure 6.4.
94
The above jet expansion model, using equation (6.1), has beencompared to experimental data from the small-scale Ontario-Hydrotests 2 5 and from the near full-scale Marviken tests 8 . Thesedata comparisons are presented without the benefit of labeled axesbecause the data at this writing was proprietary. Figures 6.6 to6.13 show model and data comparisons for Ontario-Hydro tests forthree nozzle diameters and three different initial stagnationconditions. For these comparisons the ambient pressures wereunavailable to us; consequently, the model pressures at large z/Dmay represent expansion beyond the experimental ambient value.Figures 6.9 to 6.13 show the model and data comparisons for theMarviken tests; the experimental data was taken from pressurehistories recorded in tests 1-3. Thus, the model was applied in aquasi-steady fashion. The conditions in the vessel were assumed tobe the stagnation conditions at the pipe exit; the justification formaking these assumptions is given in Chapter 3. For these Marvikendata comparisons several different times in the pressure historieswere selected to cover the range of slightly subcooled to fullysaturated stagnation conditions.
The above data comparisons produce reasonably good agreementwith data over a wide range of operating conditions; the model alsocompares favorably with the 2-D analytical predictions from Section4.4.
6.2 Centerline Target Pressure Model
As the expanding jet approaches the target (or any stationaryobject) a standing shock wave forms near the target. Along thecenterline this shock is normal to the target. The shock wave formsbecause the two-phase jet cannot sense the target's presence. Thesupersonic velocities in the jet prevent pressure signals from thetarget from propagating upstream. At a shock, kinetic energy isirreversibly exchanged for thermal energy; thus, there is a decreasein stagnation pressure across the shock.
At the target center we assume that the shock is normal andcoincident with the target surface. Normal shock equations give thepressures P 3 and P 4 behind the shock. For a stationary normalshock in a two-phase, homogeneous, and equilibrium fluid thegoverning equations of mass, momentum, and energy are given as
(6.8)P2 V2 = P3 V3 ,
2 2P2 + p2 V2 = P 3 + p3V 3 , (6.9)
95
w
U,Sd
Sda
a.
0.0010.00
ONTARIOoHTDR0 30
o.00 1-0.00
Z LOCATION ONTARIO-HYDRD 33 Z LOCATION
Figure 6.6 Free jet model comparisonwith Ontario-Hydro data.
Figure 6.7 Free jet model comparisonwith Ontario-Hydro data.
IL 4n
acW
%0
0.000 AY000
ONTARIO-HYDRO 26
0.000.00
"ARVIKEN IZ LOCATION Z LOCATION
Fiqure 6.8 Free jet expansion comparisonwith Ontario-Hydro data.
Figure 6.9 Free jet expansion comparisonwith Marviken data.
2.089E#91 SECONDS
(ftI',a"a.
a"
In
Inw
0.00 L-'0.00
0.000.00
MARVIKEN IMAl VIKEN I Z LOCATION
Free jet model comparisonwith Marviken data.
Z LOCATION
Figure 6.10 Figure 6.11 Free jet model comparisonwith Marviken data.
hiwIL In
0.00 L-A0.00
NARVIKEN 2
0.00 L0.00
Z LOCATION NARVIKEN 3 Z LOCATION
Figure 6.12 Free jet model comparisonwith Marviken data.
Figure 6.13 Free jet model comparisonwith Marviken data.
and
E3 - E2 (P3 + P2 ) -'(6.10)(P2 P3
where p is the density, P is the pressure, V is the velocity, E isthe internal energy and the subscripts 2 and 3 refer to points 2 and3 in Figure 6.1.
State 2 in equations (6.8) to (6.10) is the expansion stategiven by equations [(6.1) or (6.5)], (6.6) and (6.7). The system ofequations (6.8) to (6.10), then can be solved for state 3 (P 3 ) onthe material Hugoniot. Once we know state 3, the total pressuredownstream of the shock, P 4 , is simply the isentropic stagnationpressure.
Equations (6.8) to (6.10) have been solved for the free jetexample given in the last section (Po = 150 bars, ATo = 35 0 C) ;these results are given in Figures 6.3, 6.4, and 6.5. Figures6.3 and 6.4 show P 3 plus the stagnation (target) pressure P 4calculated using each of the simple jet model equations. Figure6.5 shows a comparison of the target pressure P 4 calculatedusing the simple model and the target pressures from the 2-Dcalculations of Section 4.5. (Note that the 2-D curve is anoverlay of five entire centerline distributions; P 4 is thefinal point on each of the broken curves).
A comparison between the stagnation pressure predicted withthe combined free jet expansion and shock model and Marvikenexperimental data 8 is given in Figures 6.14 to 6.16. The dataare shown without the benefit of labeled axes for the reasonalready given; however, a blow down typical of Marviken free jettests was given earlier in Figure 3.4. This blow down is illus-trative of Marviken stagnation pressures versus time at thebreak. Again, the engineering models show good agreement. TheMarviken data given in Figures 6.14 to 6.16 are the stagnationpressures behind the shock that formed in front of the pressurerecording instrument; therefore the stagnation pressure in thefree jet was close or equal to the vessel stagnation pressurePo. These data comparisons are further proof that viscouseffects in the jet are small and that shock waves in the jetfield are responsible for the stagnation pressure losses.Ultimately in an unconfined free jet there will be an over-expansion and subsequent shock wave formed allowing the jet toadjust to the ambient pressure conditions. (This particularphenomena was discussed in more detail in Section 3.7).Neglecting these shock waves leads to a misinterpretation of thestagnation conditions in two-phase jets.
100
H0H
hi
I0-
I..S
a.IL
0.-Fz
0.000.0 I-0.00
HMnVIKEN 3 Z LOCATION NARVIKEN 3 Z LOCATION
Figure 6.14 Stagnation Pressure ModelComparison with Marviken Data
Figure 6.15 Stagnation Pressure ModelComparison with Marviken Data
40
e.0 L-0.00S
NARVIKEN 3 Z LOCATION
Figure 6.16 Stagnation Pressure ModelComparison with Marviken Data
102
6.3 Target Radial Pressure Model
A review of the computational data base showed that theradial pressure loadings on a target were not nondimensionallysimilar. Subcooling and target spacing (L) had a stronginfluence on the shape of the pressure distribution.Consequently, one universal pressure profile will not describethe radial pressure behavior. In this section a model, whichcombines the free-jet expansion and shock wave features of theprevious two sections, will be given for the radial pressuredistribution on axisymmetric targets.
As indicated in Section 6.2 a shock forms in front of thetarget. At the center of the target the shock behaves as anormal shock but at target radii greater than zero the shockbecomes oblique to the flow. Oblique shocks are governed by thesame equations as normal shocks when using the component ofvelocity normal to the oblique shock.
Figure 6.17 illustrates the geometry for the radial model.Although CSQ calculations showed that there was a slightcurvature to the bow shock (see Figure 4.14), the radial pressuremodel assumes that the target and shock are coincident. Theproperties upstream of the shock (P 2 , T2 , V2 , etc.) areevaluated with the centerline expansion model, equation (6.1),using the expansion length of L' where
L' = L2 + R2(6.11)
The properties downstream of the shock are evaluated with thenormal shock model, equations (6.8), (6.9), and (6.10) using thecomponent of velocity normal to the oblique shock. Finally, thepressure on the target is found by isentropically stagnating thevelocity normal to the wall behind the shock. The above processis repeated for the various radii until the pressure on thetarget reaches the ambient pressure; ambient pressure is thenassumed beyond this point.
Figure 6.18 shows a comparison between the simple radialpressure model and radial pressure profiles from thecomputational data base (CSQ calculations); the case shown hereis for stagnation conditions Po = 150, ATo = 350C and forL/D's of 1.25, 1.5, 2.0 and 4.0. The agreement shown in Figure6.18 is typical of a wide range of stagnation conditions. Thecomparison is best at large L/D's where the target is away fromthe strong multi-dimensional effects near the break exit.Additionally, at small L/D's the standoff distance of the bowshock becomes significant with respect to the target spacing Land the shock wave exhibits a significant amount of curvature.
103
DLTARGET
SHOCK
<,
Figure 6.17 Geometry for the radial load model(approximate model).
104
I.
.9 - engineering model
.8 0 CSQ data base
.7
a) .5
).4-Ii
L4
0.0 1.0 2.0 3.0 4.0 5.0
RADIUS/D
Figure 6.18 Radial pressure model (approximate)comparison with the computational database (CSQ calculations).
105/106
7.0 ENGINEERING MODEL
The two-phase jet problem was too difficult to treat withapproximate models, but at the same time it is impractical to runa two-dimensional, Eulerian finite difference calculation eachtime load calculations are made.
In Chapter 6 several approximate models were introduced.These models compared well with both data and CSQ calculations.This chapter describes the development of procedures forinterpolating within the sparse computational data base using theapproximate models. The result of this modeling was adescription of the centerline pressure distribution and radialtarget pressure distribution for a wide range of vesselconditions and target spacings. These results have been providedin graphical form (see Appendixes A and B) and in dimensionalvariables when possible, thus the result forms a quick referenceset of loading and pressure charts for a range of parametersapplicable to most PWR's or BWR's.
7.1 Centerline Stagnation Pressure Model
The centerline stagnation pressure model was obtained byfitting the analytical data (CSQ data base) with the approximatemodel, equation (6.5), centerline function
(PV) 2 = (PV) 11 (D/2z) (7.1)
S /Z) [1- (D/2L) ]
The parameters a and a were determined using a nonlinearleast squares surface fitting technique 26 where
a = a (Po, ATo, L/D, Lc/D)(7.2)
lim =2 , lim a = 2
L/D-- AT-*O
and
8= (Po, ATo, L/D, Lc/D) . (7.3)
The resulting centerline pressure model is given in Figures 7.1through 7.6 (see also Appendix B). Figures 7.1 through 7.6 show the
107
= ~fl RAR~1.1 a 1 1 1 1I • I I I ... I
1.0 -A &TO " 70*I &TO z 50
o ATO = 35
.90 - ATO = 5
x X = 0.333
.80 - x+ 0 0.750
.70
0 .60
' .50N
a..40
.30 -
.20
.10
0.00.0 2.0 4.0 6.0 8.0 10.
L/D
Figure 7.1 Centerline target pressure as a function of thedistance to the target (L/D) and the stagnationproperties. Comparison of the jet load modelwith computational data base.
108
P._ = 80 PARSP = 80 BARS.
A TO = 701.0 + &TO = 50
0 ATO = 35V A•To 1.5
.90 A0 &TO = 0
x X = 0 .333
.80 + x 0 .750
.70
0 .600_
N .50N0.
.40
.30
.220
o10
0.00.0 2.0 4.0 6.0 8.0 10.
L/D
Figure 7.2 Centerline target pressure as a function of thedistance to the target (L/D) and the stagnationproperties. Comparison of the jet load modelwith computational data base.
109
P : inn PARSI .I - I a I - IIII I I I
1.0 A &T 0 70+ ATo = 500 TO = 35.9 ATO = 15
90 TO = 0x X 0 .333
0
. 80 * + = 0.750
.70
o .60
\ý .50N
.40
. 30
.20
.10
0.00.0 2.0 4.0 6.0 8.0 1
L/D
Figure 7.3 Centerline target pressure as a function of thedistance to the target (L/D) and the stagnationproperties. Comparison of the jet load modelwith computational data base.
0.
110
TTT
TaPOUI PPOT 4a 9q4 JO UOSTapduioo -saTqadold
UOTqubu~s Gq;4 Pule (GI/r) -495aie; GT4 04 ~ulesirpaqq; go uo-r;ounj P sv aingsaad qa6.jpq 9UTTaaquDD tIL Gan6b.I
1010 80 "90 *'0 z0"00 0.0
lot.*
OI,"N
"0
09- °
OL"a
08"
06"
0"It
I., ISdve Oct =d
P_= 15fl PARS1~~ = 15 ... A.RS-I . I 1 1 I
1 T AT0 : 701 AT = 50
0 & T 35
.9 ATO : 15&"T : 0
x 0o 0.333.80+ X :0.750.80
.70
o .60 "
'- .50N
a-.40
.30
.20
.10
0.00.0 2.0 4.0 6.0 8.0 10.
L/D
Figure 7.5 Centerline target pressure as a function of thedistance to the target (L/D) and the stagnationproperties. Comparison of the jet load modelwith computational data base.
112
"0G
EIT
•aspq ew4p TPUOT4pqnduoo qq.Tm
TPOUI pPoT 49C[ eqi Jo UOSTapduioD "saTqa;doadUoTqpUbpqs eqq PUP (cl/r) ;4fxI2; ati oq goup-sip@T4 JO UOTqounj P sp oanssotd qebaxE; auTixouGD 9"L @InBTa
0/-
1 0 "8 0 "9 0 0 " , 0 "0
OZ°
0 f
0r.
09,
N
"u09" 0
OL
0 ~08"0£/0
0 017
SC °017. 0_S o _
I1 .SIV8 OLI = d
centerline-target, stagnation pressure as a function of distancefrom the break (L/D) for stagnation pressures between 60 and 170bars and both saturated and subcooled stagnation temperatures(ATo = 70 °C to Xo = .75). The data symbols in the figuresindicate centerline target pressures taken from the CSQ data baseand used to obtain values for a and a in equations (7.2) and(7.3). Note the very rapid fall off in target pressure when L/D isslightly larger than Lc/D. This behavior can easily account forwide disparity in experimental results that were run under"identical" conditions. Additionally, these results re-emphasizethe results of Section 4.2 where it was shown that powerfullydestructive pressures occur on targets existing at L/D's less thanLc/D.
7.2 Radial Stagnation Pressure Model
The analytic radial load model introduced in Section 6.3 coupledwith a curve fitting procedure forms the radial stagnation pressuremodel. This model provides a family of profiles that are a functionof the break stagnation conditions and L/D. This approximate modelwas shown to agree well with data and the computational data basefor saturated stagnation conditions (agreement best at large L/D's,L/D > 4). However, for subcooled stagnation conditions that havesmall L/D's the comparison between the approximate models and theCSQ data base was not good. Curve fitting was performed to increasethe comparison between the approximate model and the CSQ data base.A brief description of the radial model follows; details of theapproximate models were given in Chapter 5.
The centerline model in Section 6.2 uses z, the coordinatelength normal to length along the target, to evaluate thethermodynamic conditions just upstream of the shock. Then thenormal shock equations and isentropic compression relations are usedto evaluate the stagnation conditions downstream of the shock. Theradial stagnation pressure model uses equation (6.1) and assumes anisentropic expansion, thus
S = SO = constant (7.4)
and
(PV) 2 ( V) D 2 * ' z* > 1: D ,(7.5)2 11z
where z* is the absolute value of the vectorial sum of the twocoordinate lengths (z,r) describing points on the target, thus
z* = v/ z2 +r r2.
Figure 7.7 shows the geometry used in the radial jet expansionmodel. The oblique shock surface found in all of the baseline
114
D Z=L
TARGET
SHOCK
qcý*
Figure 7.7 Geometry for the radial two-phasejet load model.
115
CSQ calculations was modeled as a planer surface normal to thebreak centerline at a distance L from the pipe break, i.e., theshock surface and target surface are coincident. The conditionsdownstream of the shock are evaluated using
P2 V2 sinO = P3 V3 sinO
P2 + P2 V2 sin2 o = P 3 + P3 V2 sin2 o (7.6)2 3 "
where = tan-I (z/r) ; z = L, 0 < r < R
The stagnation conditions on the target are evaluated using theconditions downstream of the shock and by imposing an isentropiccompression where the velocity normal to the target (V 3 sinO)is brought to rest. The resulting pressure distributions werenormalized using the centerline pressure evaluated with theengineering model; the final pressure distributions were thencalculated from these normalized curves using the centerlinepressure model in Section 7.2.
Target loads are then calculated by integrating thestagnation pressure distribution
R (7.7)Fr = 27 f (PT - PA)rdr ; 0 < R < RA
0
where PT Target PressurePA Ambient PressureRA Radius where PT = PA
The ambient pressure, PA, in this study was held fixed at 1.0bar. Therefore, the radial pressure curves were only integrated outto a R = RA where PT = 1.0 bar. For all radii greater than RAthe pressure was assumed to be constant and equal to PA. This isonly an approximation of the actual behavior of the target pressurefor r > RA; the actual behavior was discussed earlier in Section3.7 and is illustrated in Figure 7.8a. The pressure on the targetdrops below PA because of an over expansion of the jet flow; thisover expansion of the flow is followed by a shock wave to providefor merging of the pressure in the jet with the ambient condition.The area on the target where (PT - PA) is negative acts toreduce the target load as shown in Figure 7.8b. In this region ofthe target a small negative value of (PT - PA) can have asignificant effect on the load calculation because of the largeareas associated with radii greater than RA. This effect wasneglected in this model and all model load calculations were haltedwhen the maximum load, illustrated in Figure 7.8b, was reached. The
116
RADIUS/D
(a)
RADIUS/D
(b)
Figure 7.8 Illustration showing the two-phase jet
behavior at large radii. Note the shock
wave that forms because the jet expandsbeyond the ambient pressure.
117
1.0 bar pressure cut off occurred mostly in the range 4 < R/D < 10,depending upon the initial conditions, but the shock occurs atlarge R -- P/D > 10 to 20. The calculations were cut off atthe 1.0 bar limit because to accurately compute the solutions outto and beyond the shock would require a large grid, thus greatlyincreasing the costs to produce one point in the data base. Alsothe shock location does not easily scale and would ultimatelylead to another parameter in the load model. Finally, becausethe flow is supersonic at the point where PT = 1.0 bar acorrection for ambient pressures other than 1.0 bar would be tosimply add to the load, equation (7.5), the value (1.0 - PA) RR2 .This correction is only valid when (1.0 - PA) is small.
Comparison of this radial model to the CSQ baseline calculationsshowed an excellent agreement of both radial profiles and radialloading for saturated flow at the break; however, in the case ofsubcooled flows when L/D was small (< 4) the agreement between themodel and baseline calculations was poor. To correct the model forsubcooled flow at small values of L/D a parametric fittingprocedure, similar to the one used in Section 7.2, was used toimprove the comparison between the baseline CSQ data base and theload model. Basically, the procedure was to minimize the leastsquared error of the target load between the CSQ data base and theload model by sliding the model's pressure distribution along theradial axis an amount AR where
lim AR = 0,
R--* 0
and for R>>0, AR---o-constant.
The final model was presented in Chapter 2 and detailed charts fora wide range of PWR and BWR applications are provided in Appendix A.
One final note concerns the sensitivity of the load curves atlarge radii. Because the area is large at large radii, a smallchange or error in the pressure will result in a significant changein the cumulative load (-fApp 2 ). Therefore some care isnecessary when applying the load model to large targets, especiallyfor targets with large L/D's.
7.3 Experimental Verification
Experimental verification of the final two-phase jet load modelis not straightforward because of the small amount of experimentaldata that exists with stagnation pressures in the 60-170 bar range.Also, most data in the 60-170 bar range was recorded in small scalefacilities. A certain amount of the experimental data base, such asMarviken data 8 or Ontario Hydro data, 2 5 was utilized inverifying the methods used to obtain the approximate model, but
118
cannot be used to verify the final load model because the pressuresare too low. The data taken at the higher pressures and used hereto verify the load model are reported in References 27, 28, and 29.Reference 27 reports Kraftwerk Union (KWU) data for steady impingingjets operating at saturated conditions for stagnation pressuresvarying between 30 to 100 bars; a schematic illustrating the testarrangement is shown in Figure 7.9. References 28 and 29 reportblow-down tests performed by Battelle-Frankfurt for virtuallyidentical initial conditions of 140 bars and 30 0 C subcooled(- 3000C) ; a schematic illustrating the test arrangement isshown in Figure 7.10. Additional discussion of both of these setsof data can be found in Reference 9.
Figures 7.11 to 7.14 show the KWU data comparisons. Thestagnation conditions for these data were nominally 100 bars andsaturated liquid; however, in each case the actual data showedvessel stagnation conditions between 92 and 96 bars. Because thelisted experimental pressure uncertainty was about 10% and becausethere is some total pressure loss between the vessel and the nozzle,the exact stagnation pressure is not precisely known; however, itdoes range between 80 to 100 bars. Therefore, in Figures 7.11 to7.14 target pressure distribution curves for stagnation conditionsof 80 to 100 bars and saturated liquid are plotted. Figures 7.11 to7.14 show the data comparisons for L/D's of 1, 2, 3 and 5,respectively; the comparison is very good. These data were notconnected, in any way, with the model development -- only the CSQdata base was used. Additionally, the good comparison between themodel and experiment confirms the nondimensional behavior of the jetfor different L's and D's indicated by the equations in Section 3.3.
The data comparisons for the Battelle-Frankfurt data were moredifficult to perform than the KWU data comparisons; this was becausethe blow-down network was complicated (consisting of two inter-connected vessels), and the measurements reported in References 28and 29 did not provide the necessary upstream total conditions. Thefacility diagram in Figure 7.10 showed two vessels -- a pressurevessel and a surge flask (small vessel). It is the stagnationconditions in the small vessel that govern the load on the target.There is, however, no reporting of the conditions in the smallvessel in References 28 and 29. Additionally, a clear definition ofthe target L/D is not possible.
Figure 7.15 shows the nozzle geometry. CSQ calculations(Chapter 4) have shown that highly subcooled jet flows expand atlarge half angles, most with a half angle greater than 80 to 90degrees. However, the geometry in Figure 7.15 shows that the jet isconfined by a nozzle half angle of 30 degrees for 40 mm (0.4 D). Asa result the nozzle has an effective L/D that lays between L/D = 2.0and L/D = 2.4 depending upon the vessel conditions; the higher the
119
C)
-Rapid -closing valve NW 67,pneumatic-operaled, closingtime 0.5 sec.
Figure 7.9 Kraftwerk Union blow down and two-phasejet load test facility (see Reference 27).
Auxiliary CirculationSystem
Impact PlateD - 600 mm, A - 240 mm7 Pressure Taps3 Force Transducers
Ru ptu re Site'~D =100mm
v-'--ýMeasurlng RingN" imj pt sTElectric
HeaterH
H -Exit PipeD - 150 mm, L- 1.5 m
\Exit PipeD =200mm, L =15m
Surge Flask0.87m3
5.3m3 Pressure Vessel
Figure 7.10 Illustration showing the Battelle-Frankfurt RS-50test system.
10 0 . I I I I I I I 1 ,
0KWU DATA COMPARISON , RPT. KWU-R 512/NRC-477P 0 = 80-100 BARS. SATURATED LIQUID. L/D = I
50MM AND 65MM NOZZLES
80.0 0
P 0 = 100 BARS. AT 0 = 0. (MODEL)
70.0 PO = 80 BARS. AT 0 = 0. (MODEL)0 KWU DATA. 50MM NOZZLE
M0 KWU DATA. 65MM NOZZLE
w 60.0
m 50.0
CL 40. 0 000
UJ cLD
30.0 0
20.0
10.0
0 .00 0l
0.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
Figure 7.11 Comparison between KWU data and the two-phasejet load model for L/D = 1.
122
L)
LUJ
CL
Ld0
100.
90.0
80.0
70.0
60.0
50.0
40.0
30.0
20 . 0
i I I
KWU DATA COMPARISON . RPT. KWU-R
P 0 = 80-100 BARS. SATURATED LIOUI
50MM AND 65MM NOZZLES
512/NRC-477D. L/D = 2
P0 = 100 BARS.
PO = 80 BARS.
0 KWU DATA. 50MM
0 KWU DATA. 65MM
&T 0 = 0.At 0 = 0.
NOZZLE
NOZZLE
( MODEL )( MODEL )
10.01
0.00 Of i
0.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
Figure 7.12 Comparison between KWU data and the two-phase jetload model for L/D = 2.
123
100.
90.0
80.0
70.0
60.0
50.0
KWU DATA
P0 = so-I
COMPARISON . RPT. KWU-R 512/NRC-477
00 BARS. SATURATED LIQUID. L/D = 3
PO = 100 BARS.PO = 80 BARS.
0 KWU DATA. 65MM
AT 0 0.AT 0 = 0.
NOZZLE
(MODEL(MODEL
)
LLIJ
(I)
ri-CD
Lo
LLJ
0:
n.-
40. 0
30. 0
20.0
10.0
0.00 I I -~
0.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
Figure 7.13 Comparison between KWU data and the two-phase jetload model for L/D = 3.
124
100.
90.0
80.0
I I I f I
KWU DATA
P0 = 0o-ICOMPARISON . RPT. KWU-R 512/NRC-477
00 BARS. SATURATED LIOUID. L/D = 5
0-
w
W-
W
0~
PO = 100
P0O = 80
0 KWU DATA.
BARS.
BARS,
50MM
&T 0 = 0.&T0 = 0.
NOZZLE
(MODEL)(MODEL)
70.0
60 .0
50.0
40.0
30.0
20.0
10.0
0.00I 1~ P0
0.0 1.0 2.0 3.0
/ D
4.0 5.0
RADIUS
Figure 7.14 Comparison between KWTJ dataload model for L/D = 5.
and the two-phase jet
125
Drag Disc _T-
Gamma-Ray Densitometer
F e 5-143in mm-noz
Figure 7.15 Battelle-Frankfurt RS-50 nozzle configuration.
126
degree of subcooling the closer this effective L/D will come to2.0. The effect of this nozzle design coupled. with the targetlocation leads to a situation where the target pressures (or load)are very sensitive to the degree of subcooling. For exampleconsider conditions not unlike those in the Battelle blow downs:Po = 80 bars, ATo = 350C to 500 C. When this range of.thermodynamic conditions is examined using Figure 4.3 or Figure 7.2,we find that (L/D - Lc/D) is small, and the slope of thecenterline pressure curve (Figure 7.2) is steep and extremelysensitive to subcooling. This effect is readily seen in theBattelle data. In the Battelle Cll experiment (Reference 28) andC12 experiment (Reference 29) the thermodynamic conditions upstreamof the break are, within experimental accuracy, identical.However, downstream of the break on the target there aredifferences in target pressure, measured at the same time andlocation in the blow down, of 30 bars or more (see Figures 7.17 and7.18). These differences are caused by the sensitivity of theexperiment to subcooling and subsequent differences in effectiveL/D. The extreme sensitivity displayed by Figures 7.1 to 7.6 of thetarget conditions for subcooled flows at small L/D to small changesin thermodynamic conditions can explain widely varying targetpressures or loads for "identical" vessel conditions. In factFigures 7.1 to 7.6 explain experimental results that have in thepast been incorrectly interpreted as dynamic effects.
As indicated above the experimental reports Cll and C12 did notcontain the small vessel conditions; the data' comparisons were,therefore, performed by two methods. First, experimental pressuresand temperatures measured near the small vessel were assumed to berepresentative of the small vessel conditions, and secondly, thesmall vessel conditions were taken from a CSQ model of the Battelleexperiment.
The data comparisons to be shown here are the centerlinepressure on a target as a function of the blow-down time, t. Theload model theory used in making these comparisons is given inSection 7.1. Although some radial profile information exists forthe Battelle experiments, these comparisons were not performed. Thecenterline pressures are the most difficult to predict and willgovern any radial comparisons.
In case one the pressure in the pipe connecting the large andsmall vessels (PS 194) and the temperature at the break exit (TS270)were assumed to be representative of the small vesselconditions(jet stagnation conditions); these two curves are given in Figure7.16. The error introduced by this assumption is unknown. Thepressure trace, PS194, was subject to considerable fluctuation whilethe temperature, TS270, in the exit was relatively constant.Consequently, the data comparisons were performed by plotting thetarget pressure resulting from the upper-and lower bound on the PS
127
1.9
P
IIaal
,1.3
.5 -f.
.6-
ZElt (g) 1 CXP-IzczvorrsuT s .000 1
qygNSUCMSHR. :C 12HCSS-STELLE 17?ePO1Sj4r27
(a)
.UP &S3.,
4
C aA.-
*1 . ................ . ....... ...... ....
f t I
- i --- - -- f- - -.J0 .'t.i ! .. 'l 1.50 2 .99
21!? (9) 0 EXP aVtRSUCI4SMN. C It zmIorrsET 1 .600 aftCSS-STELLE qS,,*TS2?@V52 101,a2TSOSeSF
(b)
Figure 7.16 Pressure and temperature measured forexperiment RS-50-C12 and used to evaluatethe upper and lower bound stagnationconditions for Figures 6.17 and 6.18.(a) PS194, (b) TS270.
128
194 pressure trace; moreover, the results were plotted for a targetL/D of 2.0 (effective L/D equal to 2.0). Figures 7.17 and 7.18 showthese comparisons for the ClI and C12 experiments; these curves showthe load model predictions plotted on the ClI and C12 targetpre-ssure histories taken from References 28 and 29.. Thesecomparisons illustrate the strong influence of subcooling(pressure). This comparison is actually quite good when oneconsiders the wide variation in the target data for C1I and C12experiments, which have "identical" initial conditions. The shapeof both curves is predicted, and for the most part the ClI amplitudeis predicted. In Figures 7.17 and 7.18 the pressure uncertaintyresulted in widely differing predictions ( 70 bars) for the initialportion of the blowdown. This result shows the sensitivity ofpressure loading results to the initial thermodynamic conditions forsubcooled states. Furthermore, these results show that thediscrepencies between tests C1l and C12 are most likely due to smallthermodynamic differences in their initial conditions and not anydynamic effects.
The second comparison technique was to use the small vesselconditions predicted with a CSQ model of the Battelle-Frankfurt blowdown. 1 0 These calculations were performed early in the two-phasejet program, and the results showed excellent agreement withpressure and temperature data in the large tank and system pipingand the break flows showed good agreement. The centerline targetpressures predicted with the load model using the small vesselconditions from the CSQ calculation are given in Figures 7.19 and7.20 for CIlI and C12, respectively. Again, the model predictionsare plotted on the ClI and C12 target pressure histories taken fromReferences 28 and 29. The comparison is given by L/D's of 2.0 and2.4, thus bounding the range of effective L/D. Again, in view ofthe wide variation that exists between the C1l and C12 target data,the comparison is quite good; the shape of both curves is predictedand for the most part the amplitude in C1I is predicted.
Finally, Figures 7.21 and 7.22 show the load model pressureprediction for the saturated portion of the blow down (t > .3 to .4 s).The agreement is excellent. Recall that the effective L/D will benearest L/D = 2.4 in the case of saturated, small-vessel conditions.
129
p ATTELLE-FRANKFURT C1I TARGET PRESSURE
LU
c-nLU
LO
*--
z
LU
110.
100.
90.0 0
80.0 0
70.0 0
60.0 0
50.0
40.0
30.0 0
20.0 0
10.0
0.000.0 .40 .80 1.2 1.6 2.0
TIME ( 10 S)
Figure 7.17 Two-phase jet load model and Battelle-FrankfurtRS-50-Cll data comparison. Stagnation conditionswere approximated using RS-50 measurements;upper and lower bound approximations were made.
130
8JNTTELLE-FRANKFURT C12 TARGET PRESSURE
LOl
M
Ln
LO
0:
LOCK
z6-4
zLUU)
110.
100.
90.0 0
80.0
70.0 0
60.0
50.0
40.0 0
30.0
20.0
10.0
0.00 /
0.0 .40 .80 1.2 1.6 2.0TIME ( 101 S)
Figure 7.18 Two-phase jet load model and Battelle-FrankfurtRS-50-C12 data comparison. Stagnation conditionswere approximated using RS-50 measurements;upper and lower bound approximations were made.
131
BATTELLE-FRANKFURT C11 TARGET PRESSURE
(n
U)
Cr,
0~IY-
z-LJ
w
80.
75.
70.
65.
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.0 .40 .80 1.2 1.6
TIME (10-1S)2.0
Figure 7.19 Two-Phase jet load model and Battelle-FrankfurtRS-50-Cll data comparison. Stagnationconditions were provided from a CSQ model ofthe Cll experiment.
132
8JATTELLE-FRANKFURT C12 TARGET PRESSURE
cn
0:
U,
6-4
zWJa:
80.
75.
70.
65.
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.0 -
0.0 .40 .80 1.2 1.6 2. 0
TIME ( 10 S)
Figure 7.20 Two-phase jet load model and Battelle-FrankfurtRS-50-C12 data comparison. Stagnation conditionswere provided from a CSQ model of the C12experiment.
133
BATTELLE-FRANKFURT C11 TARGET PRESSURE
(n
crn
LU
IO
LUJ
CY
LJ
z-J
LU
9-.
LU.
.U
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.0 L.
0.0 .40 .80 1.2 1.6 2.0
TIME (S)
Figure 7.21 Two-phase jet load model and Battelle-FrankfurtRS-50-Cll data comparison. Stagnationconditions were saturated and were providedby measurements at the exit nozzle.
134
BATTELLE-FRANKFURT C12 TARGET PRESSURE
(In
Ln
NC
0-4
LJ
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.0 L-0.0 .40 .80 1.2 1.6 2.0
TIME (S)
Figure 7.22 Two-phase jet load model and Battelle-FrankfurtRS-50-C12 data comparison. Stagnationconditions were saturated and were providedby measurements at the nozzle exit.
135
REFERENCES
1. Moody, F. J., "Fluid Reaction and Impingement Loads," SpecialtyConference on Structural Design of Nuclear Plant Facilities,Vol.. , Chicago, Illinois, December, 1973.
2. American National Standard: Design Basis for Protection ofLight Water Nuclear Power Plants Against Effects of PostulatedPipe Rupture, ANSI/ANS-58.2-1980, American Nuclear Society,LaGrange Park, IL, 1980; or see
Watts Bar Nuclear Plant Final Safety Analysis Report, Section3.6, Tennessee Valley Authority; or see
Grand Gulf Nuclear Station Final Safety Analysis Report,Section 3.6, Middle South Utilities Systems.
3. U. S. Nuclear Regulatory Commission, Standard Review Plan forthe Review of Safety Analysis Reports for Nuclear Power Plants,LWR Edition, NUREG-0800, Section 3.6.2, Office of NuclearReactor Regulation, July, 1981.
4. TRAC-PlA: An Advanced Best Estimate Computer Program for PWRLOCA Analysis, LA-7777-MS (NUREG/CR-0665), Safety CodeDevelopment Group, Energy Division, Los Alamos ScientificLaboratory, Los Alamos, NM, May 1979.
5. RELAP4/MOD6, A Computer Program for Transient Thermal/HydraulicAnalysis of Nuclear Reactors and Related Systems, User'sManual, CDAP TR-003, Idaho National Engineering Laboratory,January 1979.
6. RETRAN - A Program for One-Dimensional TransientThermal-Hydraulic Analysis of Complex Fluid Flow Systems, EPRICCM-5, Energy Incorporated, Idaho Falls, Idaho, 1978.
7. Centi, R., Faravelli, M., Gaspari, G. P., and Lo Nigro, F.,"Fog Test Facility Performances, Programs, and Results,"Workshop on Jet Impingement and Pipe Whip, Genoa, Italy,June 29-July 1, 1981, Ansaldo Impianti, SIS-81-R-06, July 1981.
8. The Marviken Project, "The Marviken Full-Scale Jet ImpingementTests, Fourth Series," Report Series MXD-200s, Joint ReactorSafety Experiments in the Marviken Power Station, Sweden,1980-1981.
9. Thompson, S. L., Thermal/Hydraulic Analysis Research ProgramQuarterly Report, April-June 1980, SAND80-1091, Sandia NationalLaboratories, Albuquerque, NM, August 1980.
136
10. Thompson, S. L., Thermal/Hydraulic Analysis Research ProgramQuarterly Report, July-September 1980, SAND80-1091, SandiaNational Laboratories, Albuquerque, NM, November 1980.
11. Tomasko, D., Two-Phase Jet Loads Fiscal Year 1979 AnnualReport, SAND 80-0267, Sandia National Laboratories,Albuquerque, NM, March 1980.
12. Thompson, S. L., Thermal/Hydraulic Analysis Research ProgramQuarterly Report, January-March 1980, SAND80-1091, Sandia
- National Laboratories, Albuquerque, NM, May 1980.
13. Broadus, C. R., James, S. W., Lee, W. H., Lime, J. F., andPate, R. A., BEACON/MOD2: A CDC 7600 Computer Program forAnalyzing the Flow of Mixed Air, Steam, and Water in aContainment System, CDAP-TR-002, Idaho National EngineeringLaboratory, December 1977.
14. Broadus, C. R., et. al., BEACON/MOD3: A Computer Program forThermal/Hydraulic Analysis of Nuclear Reactor Containments -Users Manual, NUREG/CR-1148, EGG-2008, EG&G Idaho Inc., IdahoFalls, Idaho, April 1980.
15. Hirt, C. W., Romero, N. C., Torrey, and Travis, J. R., SOLA-DF:A Solution Algorithm For Nonequilibrium Two-Phase Flow,LA-7725-MS, NUREG/CR-0690, Los Alamos Scientific Laboratory,Los Alamos, NM.
16. Thompson, S. L., CSQ-II - An Eulerian Finite Difference Programfor Two-Dimensional Material Response, Part I MaterialsSection, SAND77-1339, Sandia National Laboratories,Albuquerque, NM, January 1979.
17. Keenan, J. H., Keys, F. G., Hill, P. G., and Moore, J. G.,Steam Tables, Wiley-Interscience Publications, New York, 1978.
18. Travis, J. R., Hirt, C. W., Rivard, W. C., "MultidimensionalEffects in Critical Two-Phase Flow," Nuclear Science andEngineering, Vol. 68, pp. 338-348, 1978.
19. Moody, F. J., "Maximum Flow Rate of a Single Component,Two-Phase Mixture," Trans. of ASME, J. of Heat Transfer, Vol.87, pp 134-142, 1965.
20. Moody, F.J., "Maximum Two-Phase Vessel Blowdown From Pipes,"Trans. of ASME, J. of Heat Transfer, Vol. 88, pp. 285-295, 1966.
21. Henry, R. E. and Fauske, H. K., "The Two-Phase Critical Flow ofOne-Component Mixtures in Nozzles, Orfices, and Short Tubes,"Trans. of ASME, J. of Heat Transfer, Vol. 93, 1971.
137
22. Hall, D. G., An Inventory of the Two-Phase Critical FlowExperimental Data Base, EGG-CAAP-5140, Idaho NationalEngineering Laboratory, EG&G Idaho, Inc., Idaho Falls, ID, 1980.
23. Liepmann, W. H., and Roshko, K., Elements of Gasdynamics, JohnWiley and Sons, Inc., New York, 1957.
24. McGlaun, J. M., Improvements in CSQII: An Improved NumericalConvection Algorithm, SAND82-0051, Sandia NationalLaboratories, Albuquerque, NM, January 1982.
25. Shin, K. S., "Ontario-Hydro Jet Tests - Results of Free JetTests," Technical Memorandum, CWTM-067-FD, Westinghouse Canada,Inc., Atomic Power Division, July 1980.
26. More, J. J., "The Levenberg-Marquardt Algorithm, Implementationand Theory," Numerical Analysis, G. A. Watson (ed), LectureNotes in Mathematics 630, Springer-Verlag, 1977.
27. Eichler, R., Kastner, W., Lochner, H., and Riedle, K., SafetyProject in the Area of Light Water Reactors, Studies onCritical Two-Phase Flow, NRC-477 (translation, 1978) KWU-R-512,Kraftwerk Union, FRG, 1975.
28. Vorhaben RS 50, Untersuchung der Vorgange in einen mehrfachunterteilten Containment beim Bruch einer Kuhlmittelleitungwassergekuhlter Reaktoren, Technischer Bericht BF RS50-32-CII-I, Battelle-Frankfurt, FRG, 1976.
29. Vorhaben RS 50, Untersuchung der Vorgange in einen mehrfachunterteilten Containment beim Bruch einer Kuhlmittelleitungwassergekuhlter Reaktoren, Technischer Bericht BF RS50-32-C12-1, Battelle-Frankfurt, FRG, 1976.
138
APPENDIX A
TWO PHASE JET LOAD MODEL,
This appendix contains the target pressure distribution, loaddistribution, and target pressure contours for two-phase jetsimpinging on axisymmetric targets. A complete description of thesecharts is provided in Chapter 2. Chapter 2 also providesinstruction concerning their application.
139
w
w
a-i
Ix
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.0 1.0 2.0 3.0 4.0 5.0
RADIUS
F I GURE A. I TARGET PRESSURE DISTRIBUTIONS
140
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
. 2.000
a- 1.750)1.50
1.25
. 1.00
.750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.2 TARGET LOAD DISTRIBUTIONS
141
74-
6-
5-
4
(BARSAn I .08= 2.5Cu 5.0D 10 .0E= 15.0F= 20. 06= 25.0Ha 30. 01= 35.0JJ= 40. 0K = 45.0L= 50.0M- 55.0N 60. 00= 65.0P= 70. 0D= 75.0R= 80 .0
a
20
o0c a
A
- a
a I c a
- Fl S c 9
Po= 60 Xo= .75 L c/= 50-I I1II11H,..
1 2R
3 4AOIUS / D
5 6
FIGURE A.3 COMPOSITE TARGET PRESSURE CONTOURS
142
60.
55
m
a-
U)Lo
a-.
Lo
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
F I GURE A. 4 TARGET PRESSURE DISTRIBUTIONS
143
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
-. 2.000
n 1.75
< 1.50
LL
1.25
1.00
.750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.5 TARGET LOAD DISTRIBUTIONS
144
7+
6-
5-
4-
(BARS)A= 1 .0B= 2.5C= 5.0D= 10.0E= 15.0F= 20.0G= 25.0H= 30 . 01= 35.0J= 40.0K= 45.0L= 50.0M= 55.0N= 60.0D= 65.0P= 70.0Q= 75.0R= 80. 0
It
a
003
c a2+
- S
U S
-'La
C U
c 9
a
0
eau IN@ f c a a
P 060 Xo= .33 Lc /D=I I I l I I
. 50vE II ' -.-- - ----6---S 2 3RADIUS
4ID
5 6
FIGURE A.6 COMPOSITE TARGET PRESSURE CONTOURS
145
-LO
I-
(.n(n,LUJ
LUJ0D
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.0 1 .0 2.0 3.0 4.0
RADIUS / D
TARGET PRESSURE DISTRIBUTIONS
5.0
FIGURE A.7
146
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
.. 2.000
o. 1.75
< 1.50
1.25
L 1.00
.750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.8 TARGET LOAD DISTRIBUTIONS
147
7
6
5
4
C
. 3
-J
2-
1-
(BARS)A= 1.0B= 2.5C= 5.0D= 10.0E= 15.0F= 20. 0G= 25.0H= 30. 01= 35.0J= 40. 0K = 45. 0L= 50. 0M = 55.0N= 60. 00= 65.0P= 70. 00= 75.0R= 80. 0
I 1%
0
a c S
Po= 60 ATo=I I I I
0 L /D= .50I I I I I I
m llilillN m • • m m • I | I |Ifl~ll11 h T I I I I
1 2 3RAO I US
I 1
4/0D
I 6 I5 6
FIGURE A.9 COMPOSITE TARGET PRESSURE CONTOURS
148
U,
m0.
rn
m
LLJ
I-
I--
I--
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00 0 1.0 2.0 3.0 4.0
RADIUS / D
5.0
F I GURE A. 10 TARGET PRESSURE DISTRIBUTIONS
149
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
.750
.500
.250
0.00
0a-a,
1..
0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A. 11 TARGET LOAD DISTRIBUTIONS
150
7-
6-
(BARS)A= 1 .08= 2.5C= 5. 0D 10 .0E= 15. 0F - 20. 06= 25.0N- 30. 01= 35. 0J= 40. 0K = 45.0L- 50.0M= 55. 0N= 60. 00= 65.0P= 70.00= 75.0R= 80.0
5-
4- 0
0
0
- I S
£ S
'IS
c S
c S
c 9
I
P0 = 60 ATo = 15 L /0= .88iallllllll---- I I I
2I 3RAOIUS
4/D
5 6
FIGURE A. 12 COMPOSITE TARGET PRESSURE CONTOURS
151
V)
LuJU,
LuJ
C)
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A. 13 TARGET PRESSURE DISTRIBUTIONS
152
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1 .750
a.
0 1.50
1.25
L 1.00LL
.750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A. 14 TARGET LOAD DISTRIBUTIONS
153
A
7-
6-
5-
4
BARSA= 1.0B= 2.5C= 5.0D= 10 .0E= 15. 0F= 20. 0G= 25. 0H= 30. 01= 35.0J= 40. 0K= 45.0L= 50. 0M= 55. 0N= 60 .00= 65.0P= 70. 00= 75.0R= 80 0
- a A
£
00
0 A
2
0*Po0 60 A To= 35 Lc /D= 1.68
1illll11l
2 3 4 5 6RA 0I US / D
FIGURE A. 15 COMPOSITE TARGET PRESSURE CONTOURS
154
60.
55.
C,,
Lii
Lii
Lii
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.
I i I II
Po= 60 ATo= 50 Lc/D= 2.41
L/D3.003.504.005.007.50
10.00
IIiIIII I I
0 1.0 2.0 3.0.
RADIUS / D
4.0 5.0
FIGURE A.16 TARGET PRESSURE DISTRIBUTIONS
155
4 .00 1 1 1 1 ,
3.75 P = 60 ATo= 50 Lc/D=
3.50
3.25
3.00
2.75
2.50
2.25
_ 2.000
a- 1.75
1.50
1.25
i. 1.00LL
.750
.500
.250
0-.000 ,0 1.0 2.0 3.0 4.0
RADIUS / D
FIGURE A.17 TARGET LOAD DISTRIBUTIONS
5.0 6.0
156
7-
6-
5-
4-
( BARS)A- 1.0
B9 2.5Ca 5.00D 10. 0E- 15. 0F.r 20. 06a 25. 0H- 30. 01= 35.0J 40. 0K=: 45.0L= 50.0M= 55.0N= 60.00 = 65.0P= 70.00 = 0= 75.0R= 80.0
Po= 60 ATo= 50 L /D= 2.41
D
2
10
0* lll 1 2 3RAOIUS
4/D
5 6
FIGURE A.18 COMPOSITE TARGET PRESSURE CONTOURS
157
0V)
mU)
ci,C)
0
I--
n-
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.
Po= 60 ATo= 70 L /D= 3.72
L/D4.005.007.50
10.00
11II I I i I
0I I I i i i I I
1 .0 2.0 3.0 4.0
RADIUS / D
TARGET PRESSURE DISTRIBUTIONS
5.0
FIGURE A.19
158
0a-
a)
n
1C
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1 .50
1.25
1.00
.750
.500
.250
0.000
Po= 60 ATo3
7 L
70 Lc /D= 3.72
L/D4.005.007.50
10.00
I I I I I I1 I
.~: T ... ...0 1.0 2.0 3.0
RADIUS
4.0 5.0 6.0
/ D
F I GURE A. 20 TARGET LOAD DISTRIBUTIONS
159
7
6
5
(BARSA 1.0B= 2.5C= 5.0D= 10.0E= 15.0F: 20.0G= 25.0H= 30.01= 35.0J= 40.0K= 45.0L= 50.0M= 55.0N= 60.00= 65.0P= 70.0Q= 75.0R= 80.0
8
4+ c a A
0
N 3
2
1
P 0 = 60 ATo= 70 Lc /D= 3.72
01 2 3 4
RADIUS / D5 6
FIGURE A.21 COMPOSITE TARGET PRESSURE CONTOURS
160
LA
ci-
w
LALAw0~
Lii
LD
I-
80.
75.
70.
65.
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.0 1.0 2.0 3.0 4.0 5.0
RADIUS / 0
FIGURE A. 22 TARGET PRESSURE DISTRIBUTIONS
161
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.7500)
< 1.50
1.25
L. 1.00
.750
.500 [
.250
0.00 -0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.23 TARGET LOAD DISTRIBUTIONS
162
7-
64-
(BARSA= 1 .08= 2.5C= 5.00= 10.0E= 15.0F= 20.0G= 25.0H= 30.01= 35.0J= 40.0K = 45.0L 50. 0M= 55.0N= 60. 0O= 65.0P= 70. 00= 75. 0R= 80. 0
5-
4-
0
25
0
S
B
aC S
C S
C S
C S A
F £ S
S. SF6 S
Poh~k - 80 XO 75 L /0DC
.50m
I I I I I I
III 2 3RAOIUS
4/D
5 6
FIGURE A.24 COMPOSITE TARGET PRESSURE CONTOURS
163
80.
LnV)LLJ
0-
L~JC)
75.
70.
65.
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.0 1.0 2.0 3.0 4.0 5.0
RADIUS / 0
F I GURE A. 25 TARGET PRESSURE DISTRIBUTIONS
164
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1 .75
1 .50
1.25
1.00
.750
.500
.250
0.00
0CL
a)
LL
0-.0 1 .0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
F I GURE A. 26 TARGET LOAD DISTRIBUTIONS
165
7-
6±
(BARSA= 1 .08= 2.5C= 5. 0D= 10 .0E= 15.0F; 20. 0G= 25. 0H= 30. 01= 35. 0J= 40. 0K= 45.0L= 50. 0M= 55.0N= 60 .00= 65.0P= 70. 00= 75. 0R= 80 .0
5-
4
0
, 3
S
S2-
0*
€ S
C S
C U
C S.5
9 a a95.
A
C 0
P 0 80 Xo=.33 L /0=
i I I I.50
Illl Ill irB=
R2RAflI
3 4 5 6US / 0
FIGURE A. 27 COMPOSITE TARGET PRESSURE CONTOURS
166
0-
c-
nn
0--
0.
(I)U')
n-
w.
80.
75.
70.
65.
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.0 1.0 2.0 3.0 4.0
RADIUS / 0
TARGET PRESSURE DISTRIBUTIONS
5.0
FIGURE A.28
. 167
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1 .75
1 .50
1 .25
1.00
.750
.500
0a.
L.U-L
0
250
.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A. 29 TARGET LOAD DISTRIBUTIONS
168
7+
6-
5-
(BARS)A= 1 .08= 2.5C= 5.0D= 10.0E= 15.0F= 20.0G= 25.0(= 30 •H= 35.0
J= 40.0K = 45.0L =50 .0M =55.0N 6o.r0= 65.0P= 70. 00= 75.0R= 80. 0
4- *I
90
3-
2-
1-
BI
9 a
cc, .U 0
a a jus a *- aS
Ilk P0 = 80 ATo =* • ' l i
0 Lc/D= 504rn11I111, I
1 2 3RAO I US
4/D
5 6
FIGURE A.30 COMPOSITE TARGET.PRESSURE CONTOURS
169
80.
75.
70.
Lo~
m
CC
LU'
w
LuC.Dw:
65.
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00. 0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A.31 TARGET PRESSURE DISTRIBUTIONS
170
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
-. 2.000
CL 1.75
< 1.50
1.25
•. 1.00
.750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
F I GURE A. 32 TARGET LOAD DISTRIBUTIONS
171
7-
6-
5-
4
(BARSA 1 .0B8 2.5C= 5. 0D= 10 .0E= 15.0F= 20. 0G= 25.0H 30.01= 35. 0J 40. 0K 45.0L 50. 0M 55.0N= 60 .00= 65. 0P = 70. 00= 75.0R= 80. 0
0
a A
C 1%
- S
£ S
-9 6 S
691 S
-J iuee i 0
S
S
S
A
c
c
1- a
LkP 0 80 tATO 15 LCI0=I I I
.77
U •IIIlIIIII•i II I
12R
2R 3AOI US
4/ D
I I I
5 6
FIGURE A.33 COMPOSITE TARGET.PRESSURE CONTOURS
-172
m
a-
u;
C,)V)
0-
LJC)
80.
75.
70.
65.
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.0-0.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A.34 TARGET PRESSURE DISTRIBUTIONS
173
0CL
LL
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
.750
.500
.250
0.00 '0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.35 TARGET LOAD DISTRIBUTIONS
174
7-
6-
5-
4
(BARS)A= 1 .08= 2.5C= 5.0D= 10. 0E= 15. 0F= 20 .0G= 25. 0H= 30. 01= 35. 0J= 40. 0K= 45. 0L= 50. 0M= 55.0N= 60. 00= 65.0P= 70. 0Q= 75. 0R= 80 0
0
a a
- C
",3-
- a
2 - a C S
C S
a
F a 0
Po= 80 ATo= 35 Lc/D= 1 .. 44
0 Jll|l|lll•
is iU 2 3RADIUS
4/D
5
FIGURE A.36 COMPOSITE TARGET PRESSURE CONTOURS
175
80.
I-
Lii
w
0-.
75.
70.
65.
,60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A.37 TARGET PRESSURE DISTRIBUTIONS
176
0
0-
LL
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1 .50
1.25
1.00
.750
.500
.250
0.O0 L
0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.38 TARGET LOAD DISTRIBUTIONS
177
7
6
5
4
A=z8 =
0=
D=E.4F=
H=
'.1
K=
(BARSS1.02.55.0
10.015.020 .025.030.035.040.045.050 .055.060.065.070.075.080.0
0 a
90
_J
C S
0 9
2
1
0
P0= 80 ATo= 50 Lc /D= 2.04
2 3RAOIUS
4/D
5 6
FIGURE A.39 COMPOSITE TARGET PRESSURE CONTOURS
178
U)
0-
U)U)u-i0-
ILLC)
80.
75.
70.
65.
60.
55.
50.
45.
40.
35.
30.
25.
20.
15.
10.
5.0
0.00
II I I I I I I
P0 = 80 ATo= 70 Lc/D= 3.08
L/D3.504.005.007.50
10.00
_ _ _ _ _ _ il I
.0 1.0 2.0 3.0 4.0
RADIUS / D
TARGET PRESSURE DISTRIBUTIONS
5.0
FIGURE A.40
179
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
.. 2.000
CL 1.75< 1.50
1.25
.. 1.00
.750
.500
.250
0.000.
Po= 80 ATo= 70 Lc /D= 3.08
L/D3.504.005.007.50
10.00
0 Il. 0 2.0 3.0 4.0
RADIUS / D
TARGET LOAD DISTRIBUTIONS
5.0 6.0
FIGURE A.41
180
7
6
5
4
0
N 3
_J
A=
0=
6=
E=
F=
G=
(BARS1.02.55.0
10.015.020. 025. 030. 035.040. 045.050 .055.06o 065. 070. 075.0BO. 0
0
0
2
0 80 ATo= 70 L /0= 3.08
I 2 3 4 5 6
RAOIUS I D
FIGURE A.42 COMPOSITE TARGET PRESSURE CONTOURS
181
100.
90.0
80.0LA
m0 70.0
CL 60.0
LO
LO
50.0
40.0
30.0
20.0
10.0
0.00F .0
FIGURE A.43
1.0 2.0 3.0 4.0
RADIUS / D
TARGET PRESSURE DISTRIBUTIONS
5.0
182
4 .0 0 1 1 1 1 1 ,
3.75 Po=I00 Xo .75 Lc/D=
3.50
3.25
3.00
2.75
2.50
2.25
.. 2.000
1.75
,c 1.50
1.25
i. 1.00IL.
* 750
.500
.250
0 . 0 0 -0.0 1.0 2.0 3.0 4.0
RADIUS D 0
FIGURE A.44 TARGET LOAD DISTRIBUTIONS
5.0 6.0
183
74-
0
N,
6-
5-
4
3-
2-
0-
a
a
0
(BARSA= 1 .08= 2.5C= 5.0D= 10 .0E= 15. 0F= 20. 06= 25.04= 30. 01= 35. 0J= 40. 0K= 45. 0L= 50.0M= 55.0N 60.00= 65.0P= 70.00= 75.0R= 80.0
£ 0
Is
- S
S
- I S
S IL S
C S
c a
A
£
IhP 0 =10 0 Xo= .75 L /D=I i • i i
.. 50U 1111111k
Imi 2RAE 0I
I
3US
4I/D
5 6
FIGURE A.45 COMPOSITE TARGET PRESSURE CONTOURS
184
100.
90.0
80.0
a 70.0
"60.0
wýlxM,(n
wIx
50.0
40.0
30.0
20.0
10.0
0.000
F I GURE
.0 1.0 2.0 3.0 4.0
RADIUS / 0
5.0
A. 46 TARGET PRESSURE DISTRIBUTIONS
185
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1 .50
1.25
1.00
.750
.500
.250
0.00
L
0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A. 47 TARGET LOAD DISTRIBUTIONS
186
7.
6-
(BARSA= 1 .08= 2.5C= 5.00= 10.0E= 15.0F= 20.0G= 25.0H 30.01= 35..0J= 40.0K= 45.0L= 50.0M'= 55.0N= 60.00= 65.0P= 70. 0Q= 75.0R= 80.0
4-
a A
0
a
I
2+ a
- £ S
B II S
- IMBI £ S
cc
c
c
S
S
S
S
S
A
0~
Po= 1 0 0 Xo= .33 Lc/D= * 50FiIIIIIIIhI I I I I I I I I I II I I I
I I I j I
1 2 3RA0 I US
4/D
5 6
FIGURE A.48 COMPOSITE TARGET PRESSURE CONTOURS
187
100.
90.0
80.0c-fl
m 70.0
60..013.
Cr)
Lo
50.0
40.0
30.0
20.0
10.0
0.0
FIGURE A.49
1.0 2.0 3. 0 4.40 50
RADIUS / D
TARGET PRESSURE DISTRIBUTIONS
188
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.750
0.
< 1.50
1.
1 .25
1.00
.750
.500
. 250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.50 TARGET LOAD DISTRIBUTIONS
189
74-
6-
5-
(BARSA= . 08= 2.5C- 5.0D= 10. 0E= 15. 0F= 20. 0G-= 25.0H-- 30. 01= 35.0J= 40. 0K = 45. 0L= 50. 0M= 55. 0N= 60. 00= 65. 0P= 70. 0Q= 75. 0R= 80. 0
8
44
- 9
3c
a
c 9
€ S
2-• I SI
A
api a t
1 +- 1 * a c 0
P 0 10
ATo0 0 L /D=I I I I
50i I
0 * 4 41111111
2Jl~U 3 24 5 6RAOIUS / D
FIGURE A.51 COMPOSITE TARGET PRESSURE CONTOURS
190
100.
90.0
80.0Lo
M 70.0
- 60. 0a-
U)Lo
Loi
50.0
40.0
30.0
20.0
10.0 "
0.
FIGURE
0 1.0 2.0 3.0 4.0
RADIUS / D
5.0
A.52 TARGET PRESSURE DISTRIBUTIONS
191
A-
N
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1 .75
' 1.50
1.25
L 1.00
.750
.500
.250
0.000.0
FIGURE A.53
1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / 0
TARGET LOAD DISTRIBUTIONS
192
7-I-
6-
5-
4-
(BARSAn I .0B= 2.5C= 5.0D= 10.0E= 15.0F= 20.0G- 25. 0H. 30.01= 35.0J- 40.0K- 45.0L 50.0Ma 55. 0N- 60.0O= 65.0P= 70.00- 75.0R= 80.0
9
9 A
C S
S
- U S
WSU5
S
S
c
9c
S
S
S
A"
0- - adUSP I 0 C U
P0 =100 ATo= 15 L /D=I I
.70I AIIIIIII. I II I I
1 2 3RAOIUS
4/D
5 6
FIGURE A.54 COMPOSITE TARGET PRESSURE CONTOURS
193
100.
90.0
80.0
70.0
LO
ý-60. 0CL •
LýJ
LO.
CLJ
ui
50.0
40.0
30.0
20.0
10.0
0.000 0 1.0 2.0 3.0 4.0 5.0
RADIUS
FIGURE A.55 TARGET PRESSURE DISTRIBUTIONS
194
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1 .750
n~
< 1.50
N.
L.L
1.25
1.00.750
.500
02 5 0
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.56 TARGET LOAD DISTRIBUTIONS
195
7-
6-
5-
4-
(BARSA= 1.0B= 2.5Cz 5.0D= 10.0E= 15.0F= 20.06= 25.0H= 30 .01= 35.0J= 40.0K= 45.0L= 50.0M = 55.0N= 60 .00= 65.0Pz 70.0Q= 75.0R= BO .0
A
1%
0
- I I a
2- a 1.0 £
FI 41'
IS0 1 0 0
T= 31A
A To= 35 Lc /0= 1. 29IIIItIIIII I I I I I I I
I 2 3 4RAOIUS / D
5 6
FIGURE A.57 COMPOSITE TARGET PRESSURE CONTOURS
196
100.
90.0
80.0Cr)Ln-
m 70.0
"-60.0CL.
Ln(n
a-
50.0
40.0
30.0
20.0
10.0 k
0.00-0.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
F I GURE A. 58 TARGET PRESSURE DISTRIBUTIONS
197
0
aI
LL
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1 .75
1.50
1.25
1.00
.750
.500
.250
0. 00 L0.0 1.0 2.0 3.0 "4.0 5.0 6.0
RADIUS / D
FIGURE A.59 TARGET LOAD DISTRIBUTIONS
198
a
7-
6-
5-
4-
0\ 3
(BARSA= 1 .0B= 2.5C= 5.0D= 10.0E= 15.0F= 20.0G= 25.0H= 30.01= 35.0J= 40.0K= 45.0L= 50.0M= 55.0N= 60.00= 65.0P= 70.0Q= 75.0R= 80.0
A
. c S
0
a 0
2-
0L c/D= 1.80P 0 10o ATo 50
I I I
a 1i , 2 3I I I I I I4 5 6
RAOIUS / D
FIGURE A.60 COMPOSITE TARGET PRESSURE CONTOURS
199
(F,a::
9-0~
U0
C.
C
100 .
90.0
80 .0
70.0
- 60.0
"50.0
S40.0I.A.1
LoJ30.,
20.
10.
5.0
0.0000 1.02.0
3.0mRADIUS / D
F I GURE A. 61 T ARGET PRESSURE, DISTRIBUTION.S
200
0
0-
LL
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
I .50
1.25
1.00
.750
.500
.250
0.000. 0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
6.0
FIGURE A.62 TARGET LOAD DISTRIBUTIONS
201
7-
6-
5-
4-
3-"3-
U
(BARS)An 1.08= 2.5C= 5.00= 10 .0E= 15.0F= 20. 0G= 25.0H= 30. 01= 35.0J= 40. 0K = 45.0L= 50.0M= 55. 0N= 60. 0O= 65.0P= 70.00= 75.0R= 80. 0
0 A
£
a 0
2-
0 KPo= 100 tTO= 70. Lc/D= 2.68=1 iii
2 '3R A) I US
4/D
5 6
FIGURE A.63 COMP.OSITE TARGET PRESSURE.CONTOURS
202
130.
120.
110.
100.
90.0
U)
4:
t- 80 . 0
LoLoLii
LiiLo4l:
70.0
60. 0
50.0
40.0
30.0
20.0
10.0
0.00 -0.0
FIGURE A.64
1.0 2.0 .3.0 4.0
RADIUS / 0
TARGET PRESSURE DISTRIBUTIONS
5.0
203
4.00
3.75
3.50 i
3.25
3.00
2.75
2.50
2.25
2.00
1 .75
' I I I I f I I
Po=130 Xo= .75 Lc /D= .50
L/D
1.001 .251.50I.752.002.503.003.504.005.007.500
0-
< 1.50
N, 1.25
. 1.00LA-
750
.500
.250
n on i0.0o i1.0 2.0 3.0 4.0 5.'0 6.0
RADIUS / 0
FIGURE A.65 TARGET LOAD DISTRIBUTIONS
204
E7
6-
5-
4-
3' 3-
(BARSA= 1.0B= 2.5C= 5.0D= 10.0E= 15.0F= 20.0G= 25.0H4 30.01= 35.0J= 40.0K= 45.0L= 50.0M= 55. 0N= 60.0O= 65.0P= 70.0Q= 75.0R= 80.0
- e
9
A
2
7
I I
I a
9 £ 0
MIP 0 A
- iAIt .. I N c 0
P 0 13 0 Xo= .75 .L1/D=0 I. I
~.50.
i I I .4 5 6
RAOIUS / D
FIGURE A.66 COMPOSITE TARGET PRESSURE. CONTOURS
205
0-
LJ
ir~ul)V)
9-
LiiCD
130.
120.
110.
100.
90 .0
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.000.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A.67 TARGET PRESSURE DISTRIBUTIONS
206
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
_ 2.000
o, 1.75,)1.50
1.25
• 1.00LAL
750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.68 TARGET LOAD DISTRIBUTIONS
207
a
7-
6-
5-
4
(BARS)A= 1 .0B= 2.5C= 5.0D= 10 .0E= 15.0F= 20. 0G= 25.0H= 30 .0I= 35. 0J= 40.0K= 45.0L= 50 .0M= 55. 0N= 60. 00= 65.0P= 70. 00= 75.0R= 80. 0C•
-J
9
0 0
.+ O
S S
U
C S
C U
It
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a. a S C S
.-
P 0 13 0
a
Xo= .33 Lcl/ = .50
U •I IIII II¢ImL
2 3 4 5 6
RA)IUS I D
FIGURE A.69 COMPOSITE TARGET PRESSURE-CONTOURS
208
130.
120.
110.
I 00.
90. 0
t--80.0n-
c-n
a-
LuLD
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.000.0 1 .0 2.0 3.0 4.0 5.0
RADIUS / D
F I GURE A.70 TARGET PRESSURE DISTRIBUTIONS
209
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
.750
.500
.250
0.00
0a-
0
N
LLL
0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / 0
FIGURE A.71 TARGET LOAD DISTRIBUTIONS
210
TTI
sdfloiNOO 38lsS38d 13gdVi 311ISOdWOO2L. *V 3m n 9 1
0/9 j7I I I I I
s n i u vz, N K mqlINNTNN
| | ! n | | | | | m .•lJnnlnnI I I I I
09 =0/ .=0
0 I .0dN9*0
-TV
V6
6
U
2 3a j A9 t '
2 a I d
a
S
S
I
r-
0 08 =8
0Or.9 = 00 09 =N
0OGC I10 OC =H
0 oz =9
o0o0i a00~ GGr Z0 1 V)
at,
a
-9
/L
L0
U)
uLJ
wa_
ui-
130.
120.
110.
100.
90.0
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.000 0 1 .0 2.0 3.0 4.0
RADIUS / D5.0
FIGURE A.73 TARGET PRESSURE DISTRIBUTIONS
212
4 .0 0 1 1 1 , I I 1 ,
3.75 P 0 =130 ATo= 15 Lc/D= .63
3.50 L/D
3.25 1.001.25
3.00 1.501.75
2.75 2.00
2.50 2.503.00
2.25 3.504.00
2.00 5.00o 7.50
a- 1.75 - 10.00
I1.50 "
1.25
.. 1.00
.750 -
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS D 0
FIGURE A.74 TARGET LOAD DISTRIBUTIONS
213
7-
6+
(BARS)Ar 1 .08= 2.5C= 5.0D= 10.0E= 15.0FI 20.06= 25.0H- 30.01 35.0Jx 40.0K1 45.0L" 50.0Mu 5.5.0N- 60 .0O 65.0P- 70.0ON 75.0Ra 80. 0
5.
4+ 8
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9 5 S
- S- 9 S S
j i.gu g S
- K~MJ3N*P £ S
IC
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S
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1-~
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.63
0 II
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FIGURE A.75 COMPOSITE TARGET PRESSURE CONTOURS
214
130.
120.
110.
0:
CALo
%
100.
90. 0.
z-80.0
70.0
60.0
50.0
40.0
30.0
20.0 0
10.0
0.00 -0.0
FIGURE
1.0 2.0 3.0 4.0
RADIUS I D
TARGET. PRES-SURE DISTRIBUTIONS
5.0
A. 76
215
0a_
L.
4.00
3.75
3.50
3.25
3.00
2.75.
2.50
2.25
2.00
1.75
1 .50
1.25
1.00
.750
.500
.250
0. 000 .0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
6.0
FIGURE A. 77 TARGET LOAD DISTRIBUTIONS
216
£
7+
6±
5-.
4"
(BARSA= 1 .09= 2.5C= 5.0D= 10.0E= 15.0F= 20.06= 25.0H= 30.012 35.0J= 40.0K= 45.0L2 50.0M= 55.0N= 60.00= 65.0P= 70 . 0Q= 75.0R= 80.0
a
00
It
2-
0'
V~ I Sc
a 9 I S cS
-a I1 $ w a S
ATo= 35 LC/0= 1.1.4Po0 = 30U 111111111
'1 i 2 3I i i
4 5 6RAOIUS / D
FIGURE A.78 COMPOSITE TARGET PRESSURE CONTOURS
217
130.
120.
110.
II I I I I
P 0 = 13 0 AT0-= 50 L I 1D=1 .57
U)
U,)
0-
wi
Lo
100.
90.0
80.0
70.0
60. 0
50.0
40.0
30.0
20.0
10.0
0.00
L/D2.002.503.003.504.005.007.50
10.00
0.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A.79 TARGETPRESSURE DISTRIBUTIONS
218
4. 00
3.75 Po=130 ATo= 50 Lc/D= 1.57
3.50 L/D
3.25 2.00
.50
3.00 3.003.50
2.75 4. 00
2.50 5. 00" 7.50
2.25 10 00
. 2.00
0- 1.75
,G 1.50
1.25
1 1.00
.750
.500 -
.250 -
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.80 TARGET LOAD DISTRIBUTIONS
219
S
74
6-
5-
4-
(BARSA 1 .08= 2.5C= 5.0D= 10.0E 12.5.0F= 20.0G= 25.0H= 30'.01= 35.0J= 40.0K= 45.0L= 50.0M= 55.0N= 60. 00= 65.0P= 70.0G= 75.0R= 80.0
6
I
t0
0 9
a 9
2+ 0 o 9 0
1-
P0 = 130I I
ATo= 50 LclD.=I I i i II
I .57
U foul lii,!: -
R)2 3RADIUS
4/D
5 6
FIGURE A.81 COMPOSITE TARGET PRESSURE CONTOURS
220
U)
LkJ
(f)U)u-i0-
Ia--
LI-
1 30.
120.
110.
100.
90 . 0
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.000 0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
F I GURE A.82 TARGET PRESSURE DISTRIBUTIONS
221
4 . 0 0 , , , I I I I I ,
3.75 P 0 =130 ATo= 70 Lc/D= 2.29
3.50 L/D
3.25 2.003.00
3.00 3.504.00
2.75 5.007.50
2.50 10.00
2.25
2.00
o. 1.75
1.50
1.25
,. 1.00
.750
.500 -
.250 -
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.83 TARGET LOAD DISTRIBUTIONS
222
7-
5-
4
3-
U
0
(BARSA= 1 .0B= 2.5C= 5.0D= 10.0E= 15. 0F= 20. 0G= 25. 0H= 30. 01 35. 0J= 40. 0K= 45. 0L= 50. 0M = 55.0N= 60. 00= 65. 0P= 70. 00= 75. 0R= 80. 0
9C
I D
2
0~
a IwI I6 a 0
ATo0 70 Lc /D= 2.29I i i i I i i
P0 =130II I
Rf2 3R An Ius
4/D
I 1 I I5 6
FIGURE A.84 COMPOSITE TARGET PRESSURE CONTOURS
223
Li
0::c-nLLIay-
LiJC)
150.
140.
130.
120.
110.
100.
90.0
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.000.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A.85 TARGET PRESSURE DISTRIBUTIONS
224
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1 .75
1.50
1.25
1.00
.750
.500
.250
0.00
00~
G)
I.
0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.86 TARGET LOAD DISTRIBUTIONS
225
74
6-
5-
4
(BARSA= 1 .0B8 2.5C= 5..D= 10 .0E= 15.0F= 20. 0G= 25.0H= 30. 01= 35.0J= 40. 0K= 45.0L= 50 .0M= 55.0N= 60. 00= 65.0P= 70.0Q= 75.0R= 80.0
9
C
9 S
S-2 S
2-
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P 6 5
- S P £ 5
INSP £ S
-NIaJINSP £ S
C
c
c
€
S
S
S
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A
A
k~r P~O= 1150 X 0=.75 L /D=
I I I I
.50i I
Mni[liiru i i i i i i i | i | |•lllll1[li I I I I I
1 2 3RAOIUS
I I
4ID
I I I I
5 6
FIGURE A.87 COMPOSITE TARGET PRESSURE CONTOURS
226
150.
140.
130.
120.
110.
100.
C.,
;_ 90. 0
w0-
CD
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.000.0 1.0 2.0 3.0 4.0
RADIUS / D
5.0
F I GURE A. 88 TARGET PRESSURE DISTRIBUTIONS
227
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
.. 2.00
o1 1.75
1.50
1.25
L. 1 .00
.750
.500
.250
0.000.0
FIGURE A.89
1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / 0
TARGET LOAD DISTRIBUTIONS
228
7+
6±-
BARSA= 1 .08= 2.5C= 5. 0D= 10 .0E= 15. 0F= 20. 0G 25.0H 30. 01= 35.0J= 40. 0K= 45.0L 50 .0M= 55.0N= 60 .00= 65.0P= 70 .0Q= 75.0R= 80. 0
5- It
4+c a
0EC
3-
0 9
2-
0
- I S
I I S
- SI I S
J £51 I S
ULAJIMS# I S
C
c
c
c
c
S
S
S
A
A
A
m
Pthh=£ P 150 X0= .33. Lc/DO .50I
I I I I I I
I 2 3. 4 5 6RAOIUS / D
FIGURE A.90 COMPOSITE TARGET PRESSURE CONTOURS
229
150.
140.
130.
120.
110.
100.
U.)
'- 90 . 0
Li
ix
CL
Lo
80.0
70.0
60.0
50.0
40.0
30. 0
20.0
10.0
0.000.0 1.0 2.0 3.0 4.0
RADIUS / D
5.0
FIGURE A.91 TARGET PRESSURE DISTRIBUTIONS
230
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1 .50
1.25
1.00
.750
.500
.250
0.00
LL
0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.92 TARGET LOAD DISTRIBUTIONS
231
7
6
5
4
A a
F:
H:
I:K:
N:0:Ps
(BARS)1.02.55.0
10 .015.02O 025.030.035.040 .045.050 055.06O 065.070.075.0so . 0
a
a
-J 0"
2 's1- ~. ..4... ~ * * £ a
4 £ S
1 SAUdI £ B
c
c
C
C
0
OS
S
a
0 .Lc/0= .50
C
P0 = 150 AT 0 =
01 2 3
RAI0 1 US4
ID5 6
FIGURE A.93 COMPOSITE TARGET PRESSURE CONTOURS
232
0-
ar-
0C'CL
LLJ
150.
140.
130.
120.
110.
100.
90.0
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.000.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
F I GURE A. 94 TARGET PRESSURE DISTRIBUTIONS
233
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.000
CL. 1.75
< 1.50
1.25
. 1.00Lu
.750
.500
.250-0.. 00
0.00.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
F I GURE A. 95 TARGET LOAD DISTRIBUTIONS
234
7-
6+
(BARS)A= 1 .0B= 2.5Cm 5. 0D= 10.0E= 15.0F= 20. 0G= 25.0H= 30. 0I= 35.0J= 40 .0K= 45.0L= 50 .0M= 55.0N= 60.00= 65.0P= 70.0Q= 75. 0R= 80. 0
5-
4
a
3+ I
U-J
2- * £ S
9 6 3
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d IWSU £ S
-. b3&RJiNSP £ S
inauaw., I S
C
C
€
C
S
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I A
a
k. PO=150 ATO= 15 Lc/D=I II
. 590 IllIlll[i I I I I I I I I I I I-71 I 1
2 3RA I US
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FIGURE A.96 COMPOSITE TARGET PRESSURE CONTOURS
235
150.
140.
130.
120.
110.
100.
Lo
m
C- 90 00 •
LZ
CL
Locr
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0 -0.00
0.0 1 .0 2.0 3.0 4.0
RADIUS / D
TARGET PRESSURE DISTRIBUTIONS
5.0
.FIGURE: A.97
236
4.00
3.75 Po=150 ATo= 35 Lc/D=
3.50
3.25
3.00
2.75
2.50
2.25
2.0001_ 1.75
< 1.50
1.25
1.00U_
.750
.500
.250
0.000.0 1.0 2..0 3.0 4.0
RADIUS D 0
FIGURE A.98 TARGET LOAD DISTRIBUTIONS
5.0 6.0
237
a
7
6
5
4
A=
0=
6=
D=E=
(BARS1.02.55.0
10.015.020. 025.030 .035.040.045.050.055.060. 065.070.075.080.0_
-i 9
L=M=N=
• • 0=
P=
R=
3 A* A
*
35 L /D= 1.07
2 . : I a a
40M IN p b
c
C
0
Po=150 ATo=
01 2 3
RAOIUS4
/D5 6
FIGURE A.99 COMPOSITE TARGET PRESSURE CONTOURS
238
150.
140.
130.
120.
110.
I I I I I I I I .
P0 = 150 AT0 = 50 L /D= 1.47
LA
mTr
L/D2.002.503.003.504.005.007.50
10.00
100. -
ý-90. 0a- •
LI)
W~9-
uw
80.0
70.0
60 .0
50.0
40.0
30.0
20.0
10.0
0.000.0 1.0 2.0 3.0 4.0 5.0
RADIUS / 0
FIGURE A. 100 TARGET PRESSURE DISTRIBUTIONS
239
0
CL
LL
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
.750
.500
.250
0.00 .L
0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A. 101 TARGET LOAD DISTRIBUTIONS
240
9
74-
6+
(BARSAs 1.08= 2.5C 5.0D,, 10 .0Ex 15. 0F'- 20. 0G- 25. 0H- 30. 01" 35. 0
it J 40. 0K x 45. 0La 50. 0M a 55.0Na 60. 0On 65.0Ps 70. 00- 75 0R- 80.0
5-
4-
C
0
I
- a 0
2+ 0 a 0, 1 S 0
P0 =150 ATo= 50 Lc/D= 1.47
0 i i P i i i - -
RADI 3RAO)I US
4 5 6/D
FIGURE A.102 COMPOSITE TARGET PRESSURE CONTOURS
241
150.
140.
130.
120.
110.
100.
LA
ý- 90. 00-
wLu
Lou-i
I.-
LiiLo
80.0
70.0
60 .0
50.0
40.0
30.0
20.0
10.0
0.000 0 1 .0 2.0 3.0 4.0 5.0
RADIUS / 0
FIGURE A. 103 TARGET PRESSURE DISTRIBUTIONS
242
4 .0 0 1 1 1 1 1 1 1 ,
3.75 Po =150 ATo = 70 Lc/D= 2. 12
3.50 L/D
3.25 2.002.50
3.00 3.003.50
2.75 4.00
2.50 5.00• 7.50
2.25 10.00
- 2.000
CL 1.75S1 .50
1.25
. 1.00S.750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.104 TARGET LOAD DISTRIBUTIONS
243
7-
6-
5-
4-
BARS)A= 1 .08- 2.5C= 5.00= 10.0E= 15.0F= 20.0G" 25.0.Nu 30.01= 35.0J= 40.0K 45.0Lz 50.0M= 55.0N= 60.00= 65.0P= 70.0Q= 75..0R= 80.0
a
a
9 C S A
0
._J
0 c a
- N a f 9 0 C S . A
2 *-I M fANS a it 0 eC S
P0 =150 ATo= 70 Lc/0=I ~ I I I I I
2. 12I I•lllllll[•-"III
I3I '1 2 3RAD I US
4/D
5 6
FIGURE A.105 COMPOSITE TARGET PRESSURE CONTOURS
244
0-
LUJ
cr-f
wC-
Ir-
180.
160..
140.
120.
100.
80.0
60.0
40.0
20.0
0.000.0 1 .0 2.0 3.0 4.0
RADIUS / D
TARGET PRESSURE DISTRIBUTIONS
5.0
FIGURE A.106
245
4.00
0a-
a)
N.
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50 -
1.25
1.00
.750
.500
. 250
0.00 L0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A. 107 TARGET LOAD DISTRIBUTIONS
246
7
6
5
4
3
AzBa
EmF=
Lm
Nm0=Pa
(BARS)1.02.55.0
10.015.020.025.030 .035.040.045. 050 .055. 06o .065.070. 075.080 .0
9
0
-J 0
2 -- i c
c
c
€
a
S
S
U
.75 L /0= .50
9 9
P0 = 170 Xo=0
I 2 3RAD I US
4D0
5 6
FIGURE A. 108 COMPOSITE TARGET PRESSURE CONTOURS
247
V)
0-
Sý
LIJ
0:
(PILO
180.
160.
140.
120.
100.
80.0
60.0
40.0
20.0
0.000.0 1.0 2.0 3.0 4.0 5.0
RADIUS / 0
TARGET PRESSURE DISTRIBUTIONSFIGURE A.109
248
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
. 2.000
a- 1.75
< 1.50
1.25
. 1.00
750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 i.,5.0 6.0
RADIUS / D
FIGURE A. 110 TARGET LOAD DISTRIBUTIONS -
249
A
7-
6-
5-
4
(BARSA 1 .0B= 2.5Cc 5.0Da 10.0Ez 15.0F= 20.0G= 25.0H= 30.01u 35.0Js 40.0K= 45.0Lm 50.0Mc 55.0N= 60. 0O= 65.0P= 70.0O= 75.0R= 80.0
0
A
S.
2- -F a 3 C 99 C
- U S .6 £ aa JIMOP £ S
t a
C I A
1 + 111,4 IN ad F a S C 9
P~h~ P0 17 0 Xo= .33 L /D=0 C
.50-=iiiiii1iN
1 2 3 4 5 6RAflIUS / 0
FIGURE A. 111 COMPO.SITE TARGET PRESSURE CONTOURS
250
180.
160.
Ln
a-
Ln
Lu
cI-
Lo
9r-
140.
120.
100.
80.0
60.0
40.0
20.0
0.000 0 1 .0 2.0 3.0 4.0
RADIUS / D
5.0
F I GURE A. 112 TARGET PRESSURE DISTRIBUTIONS
251
4.00
3.75
3.50
3.25
3.00
2.7"5
2.50
2.25
2.0,00
0_ 1.75
< .50
1.25
•. 1.00
.750
.500
.250
0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A. 113 TARGET LOAD DISTRIBUTIONS
252
a
7-
6-
(BARSA= 1 .0B= 2.5C= 5.0o= 10.0E= 15.0F= 20.0G= 25.0H= 30.01= 35.0J= 40 .0K = 45.0L 50.0M= 55.0N= 60.00= 65.0Pc 70 . 0Q= 75.0R= 80.0
5+
0
4.
3-
2-
1*
9
C a
a
- S C £
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c
c 9
A
£
A
A
1 P0 = 170 AT0= 0 Lc
.50* 'I 7
0 I I I 1 I - I -" I - -
1 2 3 4 5RAIIUS D 0
"T I
6
F I GURE A. 114 COMPOSITE TARGET PRESSURE CONTOURS
253
Lu
LIZ
LUJ
(.-
180.
16 0
140.
120.
100.
80.0
60.0
40.0
20.0
0.000 0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A. 115 TARGET- PRESSURE DISTRIBUTIONS
254
4.00
0
a-0)
N.
3.75
3.50
3.25
3.00
2.75 -
2.50
2.25
2.00 .
1.75
1.50
1.25
1.00
.750
.500
. 250
0.00 LA0.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
6.0
F I GURE A. 116 TARGET LOAD DISTRIBUTIONS
255
- I
7-
6
(BARS)A- 1 .0Be 2.5Ca 5.0Do 10.0Ea 15.0Fa 20.0G6 25. 0Ho 30. 0
a 35. 0J" 40. 0Ka 45. 0La 50. 0mu 55.0No 60. 0D0 65.0Ps 70 . 0Qu 75.0R" 80.0
5+c
4+
00
c I
2-
- S
-F I S
* ~ I S
-4 U S P I ~S
leg.., I I
-PSb*~6JIM* I I S
IC
c
IS
SI
£
a
C
k P = 170i 0
ATo= 15I 4 I
Lc D= .57I I I .0- I I I 1 I I I
1 2 3 4RAr'IUS / 0
I ; I I
5 6
FIGURE A. 117 COMPOSITE TARGET PRESSURE CONTOURS
256
180.
160.
U,
Lii
LA
Lii0:
140.
120.
100.
80.0 0
60.0 -
40.0
20.0
0.,000.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A. 118 TARGET PRESSURE OISTR.IBUIONS
257
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
.. 2.000
0. 1.75
< 1.50
1.25
S.LL
1.00
.750
.500 -
.250
0.00 -0.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A.119 TARGET LOAD DISTRIBUTIONS
258
7+
6-
(BARSA= 1 .08= 2.5C= 5.0D= 10 0E= 15. 0F= 20. 0G= 25. 0H= 30. 01= 35. 0
I J= 40. 0K = 45. 0L= 50. 0M= 55.0N= 60. 00 = 65. 0P= 70. 00 Q= 75. 0R= 80. 0
5t
4
0
\ 3
9
a
0
2-it a I w
S
0
C
C
I
A
J
Po0 = 70 ATo= 35 Lc/D= 1.02
0 I I I I1 al ill i - 2 3RAO I US
4/D
5 6
FIGURE A. 120 COMPOSITE TARGET PRESSURE CONTOURS
259
V')
L:J
I-
Lo
180.
160.
140.
120.
100.
80.0
60.0
40.0
20.0
0.000.0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A. 121 TARGET .PRESSURE DISTRIBUTIONS
260
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
-. 2.000
0. 1.75
< 1.50
1 .25
L 1.00
750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / D
FIGURE A. 122 TARGET LOAD DISTRIBUTIONS
261
a
7-
6-
5-
4
(BARS)A= 1 .08= 2.5C= 5.0D= 10 .0E= 15. 0F= 20. 0G= 25. 0H 30. 01- 35. 00= 40. 0K= 45. 0L= 50. 0M= 55. 0N= 60. 0
A 0= 65. 0P= 70. 0
s 0= 75. 0R= 80. 0
5
a
M
0
2-
0
" 1 9 £ 0
ATo= 50 Lc/D= 1.39II I I I I I
P = 170I IIIIliIIl I I I I I I
2 3RAO I US
I I I I I I4 5 6
/ D
FIGURE A.123 COMPOSITE TARGET PRESSURE CONTOURS
262
Li
(n
LLJ0~
CC
180.
160.
140.
120.
100.
80.0
60.0
40.0
20.0
0.000 .0 1.0 2.0 3.0 4.0 5.0
RADIUS / D
FIGURE A. 124 TARGET PRESSURE DISTRIBUTIONS
263
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.750
1. 50
N
1~Lh~
1.25
1.00
.750
.500
.250
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0
RADIUS / 0
FIGURE A.125' TARGET"LOAD DISTRIBLTITONS
264
7-
6-
5-
4
3-
2
1
0'
A
(BARSA= 1 .08= 2.5C= 5.0D= 10.0E= 15.0F= 20. 0G= 25. 0H 30. 01= 35.0J= 40. 0K 45. 0L= 50. 0M= 55. 0N= 60. 00= 65.0P= 70. 00= 75. 0R= 80.0
I
c a
- S CC-
a
S a%.M~k nx a" 0 S B
kPo= 170 AT0o= 70 Lc/D=I. I I I I
1 .98
1II 2RAO)
I
3I US
4/D
5 6
FIGURE A. 126 COMPOSITE TARGET PRESSURE CONTOURS
265
APPENDIX B
CENTERLINE TARGET PRESSURE DISTRIBUTIONS
This appendix contains a series of figures showing thecenterline target pressures as functions of the distance to thetarget and stagnation properties. A complete description of thesefigures is provided in Chapter 6.
266
P- = 60 BARS1 .I11
1.0
.90
.80
.70
.60
.50
. 40
0
N
NL
*30
.20
.10
1... 0 .0
0.0
Figure B.1
2,.0 , 4.0 6,.0 8..0 :' 10.
L/D
Centerline target pressure as a function ofthe distance to the target (L/D).:and thestagnation properties.
267
P, = 80 BARS1.1
1.0
.90
.80
.70
0
Na-
.60
.50
.40
.30
.20
.10
0.0 --
0.0 2.0 4.0 6.0 8.0 10.
L/D
Figure B.2 Centerline target pressure as a function of thedistance to the target (L/D) and the stagnationproperties.
268
P^ = 100 BARS1.1
1.0
.90
.80
.70
0 .60,.
.50N
a. r.40
. 30
.20
.10
0.0 -
0.0 2.0 4.0 6.0 8.0 10.
L/D
Figure B. 3 Centerline target pressure as a function of thedistance to the target (L/D) and the stagnationproperties.
269
P. = 130 BARSJý. 1
1.0
.90
.80
.70
00L
Nn
.60
.50
* 40
* 30
.20 -
.10
0.0 2.0 4.0 6.0 8.0 10.
L/D
Figure B. 4 Centerline target pressure as a function of thedistance to the target (L/D) and the stagnationproperties.
270
P- = 150 BARS1 . 1
1.0
.90
.80
.70
0
N
a_
.60 "
.50 "
.40 "
.30 -
* 20 -
* 10
0.00 .0 2.0 4.0 6.0 8.0 10.
L/D
Figure. B. 5 Centerline target pressure as a function'of thedistance to the target (L/D) and the stagnationproperties.
271
P- = 170 BARS1. I
1.0
.90
.80
.70
0
IN
N0_
.60
.50
.40
. 30
.20 -
.10
0.0 -
0 . 0. 2.0 4.0 6.0 8.0 10.
L/D
Figure B. 6 Centerline target pressure as A function of thedistance to the target (L/D) and the stagnationproperties.-
272
APPENDIX C
TWO-PHASE JET-EXIT CORE LENGTHS
This appendix contains the jet exit core length (Lc/D) forvarious vessel conditions. A complete description of these curvesis given in Sections 3.6 and 4.2.
273
5.0
4.5
4.0
3.5
3.0
2.5
2.0
0IN,
U_J
1.5
1.0
.50
0.0 00.00 40.0 80.0 120.
P (bar s)0
160. 200.
Figure C.1 Two-phase, jet-exit core (L /D) as a function ofstagnation pressure and stagnation temperatureor subcooling.
274
15.
0C-,
14.
13.
12.
11.
10.
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0 -a-0.00
Figure C.2
40.0 80.0 120.
P ( bars)0
160. 200.
Two-phase, jet-exit core (Lc/D) as a function ofstagnation pressure and stagnation temperature orsubcooling.
275
APPENDIX D
WATER PROPERTIES AND HEM CRITICAL FLOW PROPERTIES
This appendix contains figures showing the thermodynamicproperties of water and the HEM critical flow for water as afunction of thermodynamic conditions. A description of the curvesis given in Chapters 3 and 5.
276
700 .
650.
600.
550.
WATER
/ ~0.0:
37.S
30.0
-27.5
500 a
450.
400.
350.
300.
22.5
20.0
25.0
S( 1O*ergs /9 K~
17.5
15.0-
12.5
10.0
7.0S---5.0
-2.5
250 I I I I I Ia
0.00 40 . 0 80.0 120. 160. 200. 240 .
P (bor s )
Figure D. 1 Thermodynamic properties of water.Temperature as a function of pressure andentropy for a range of pressure and entropythat emphasizes subcooled conditions.
277
WATER700 I
200 bar isobar
650.
600. go 33/ 35.0
550.
-- 27.5
500.
22.5-
450. ~20.0I-17.5
400.S5.01
, , .- 12 .5 -
350. S( 10*6ergs/9 K 10.0-
7.5
300. 5.02.!
250 . I , , I ,0. .2 .4 .6 .8
p(g/cm )
Figure D.2 Thermodynamic properties of water.Temperature as a function of density andentropy for a range of density and entropythat emphasizes saturated conditions.
1.
278
WATER700.
650.
600.
550.
500.
450.
400.
350.
300.
250
103 10-2 10-
3
,0 ( /cm
Figure D. 3 Thermodynamic properties of water.Temperature as a function of density andentropy for low densities.
279
20.
18.
16.
14.
12¢• 12.
CD
10.
8.0
6.0
4.0
2.0
0.0 -
0.00 5.00 10.0 15.0 20.0 25.0 30.0 35.0 40.0
S( 10 ergs/9 K)
Figure D.4 HEM mass flux as a function of entropy andstagnation temperature for a range of entropywhich emphasizes subcooled stagnation conditions.
280
5.0
09
E
C)
0)
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
.50
0.0 -0.0 10. 20. 30. 40. 50. 60.
* 6 KS( 10 ergs/9 K)
70.
Figure D.5 HEM mass flux as a function of entropy andstagnation temperature for a range of entropywhich emphasizes saturated stagnation conditions.
281
20.
18.
16.
14.
12•_ 12.
E0'N0)
CCD
10.
6.0 "
4.0
2.0
0.0 -
0.00 40 .0 80.0 120.P ( bars)
0
160. 200.
Figure D.6 HEM mass flux as a function of stagnationpressure and stagnation temperature.
282
-anqpraduia qI uoqubqs UPuX4TTpflb UOTPU5P4S 30 uoT~oufl P sp xflT3 SsPtu WRH LQC aib
0x
"t9,P, a"0
0 0
0~I
02
CD
0 (0
0 0
r.07
I .
.9
.8
.7
(I)
N
E0
c~)
CD
.5
.4
.3
.2
.1
0.
0.
Figure D. 8
.2 .4 .6 .8
x0
HEM mass flux as a function of the stagnationquality and stagnation temperature.
284
l.
*@an;qpu~cduiq uoL42ubpqs pup AqTsuapUoTqpubpqs @ilq go uoqioun; P SP xflTj SSPUI WRH 6-C aan6
Tj
w 0/ 6 0
.!9.D, a"0
0.
m-
0 (0
0
0"a,
1.
.9
.8
.7
.6r
E 5
,4
.2.2400
• • T(Oik• = 300
0. .2 .4 .6 .8 1.
S(g/cm 3 )
Figure D.10 HEM critical flow density as a function ofstagnation density and stagnation temperature.
286
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287
NRC FORM 335 1. REPORT NUMCER (AAssigned by DDC)U.S. NUCLEAR REGULATORY COMMISSION SN 213BIBLIOGRAPHIC DATA SHEET NUREG/CR-2913
4. TITLE AND SUBTITLE (Add Volume No., if appropriarv) 2. (Leave blank)
TWO-PHASE JET LOADS3. RECIPIENT'S ACCESSION NO.
7. AUTHOR(S) G. G. Weigand, S. L. Thompson, 5. DATE REPORT COMPLETED
D. Tomasko MONTH YEAR
-__January 19839. PERFORMING ORGANIZATION NAME AND MAILING ADDRESS (Include Zip Code) DATE REPORT ISSUED
Organization 9444 MONTH YEAR
Sandia National Laboratories January 1983P. 0. BOx 5800 6. (Leave olank
Albuquerque, New Mexico 871855. (Leave blank)
12. SPONSORING ORGANIZATION NAME AND MAILING ADDRESS (Include Zip Code)
U. S. Nuclear Regulatory Commission 10. PROJECT/TASK/WORK UNIT NO.
Office of Nuclear Regulatory ResearchMechanical/Structural Engineering Branch 11. FINNO.
Division of Engineering Technology A-1205Washingt-on. DC 20555 I__
13. TYPE OF REPORT PERIOD COVERED (Inclusive dars)
Technical15. SUPPLEMENTARY NOTES 14. (Leave olank)
16. ABSTRACT (200 words or less)
A model has been developed for predicting two-phase, waterjet loadings on axisymmetric targets. The model ranges inapplication from 60 to 170 bars pressure and 700C subcooledliquid to 0.75 (or greater) quality -- completely covering therange of interest in pressurized water or boiling waterreactors. The model was developed using advanced two-dimensional computational techniques to solve the governingequations of mass, momentum, and energy. The model displays ina series of tables and charts the target load and pressuredistributions as a function of vessel (or break) conditions;this enables fast yet accurate "look up" for answers. For manysituations of practical interest, the model can predict sub-cooled and saturated loadings in excess of the simple controlvolume upper bounds of 2PoAe for nonflashing liquid and1.26PoAe for steam. Also, the results indicate that thearea of the loading on a flat target is often larger thanassumed by simplier models. Finally, approximate models aregiven for estimating two-phase jet flow, expansion character-istics, shock strengths, and stagnation pressures. Theseapproximate models could be used for estimating pressures ontargets not specifically addressed in this study.
17. KEY WORDS AND DOCUMENT ANALYSIS 17a. DESCRIPTORS
17b. IDENTIFIERS/OPEN-ENDED TERMS
18. AVAILABILITY STATEMENT 19. SECURITY CLtSS (This report) 21. NO. OF PAGESUnci 492
20. SECURIT.ý.JSs (Th,s pa•e) 22. PRICES
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