THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING A STUDY OF COLUMN SEPARATION ACCOMPANYING TRANSIENT FLOW OF LIQUIDS IN PIPES Robert A. Baltzer A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Civil Engineering 1967 February, 1967 IP-768

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ACK2TOWLE1GMETNT S The writer is indebted to Professor Victor L. Streeter, graduate committee co-chairman and thesis advisor, for his suggestion of this topic and for his inspiration, incisive guidance, and patience; to Professor Ernest F. Brater, committee co-chairman, for his timely counsel, encouragement, and unfailing support; and to Professors Lawrence C. Maugh, William P. Graebel, and James W. Daily, committee members, for their interest and cooperation throughout the course of the study. Aippreciation is also due Professors Arthur G. Hansen and Russell F. Dodge, who have since left The University, for their interest during the early phases of the study. The writer is appreciative of the sound advice and craftsmanship rendered by Messrs. George L. Geisendorfer and Waldemar G. Buss, who assisted with the construction of the experimental apparatus. A word of gratitude is also extended to many unnamed individuals, including fellow graduate students, who have contributed to this study in various ways. Without the use of the excellent facilities and the services available at The University of Michigan Computing Center, this study would not have been possible. The writer is particularly grateful to The University for the use of these facilities and for the valuable experience gained thereby. For assistance with the collection and evaluation of the experimental data and for aid with the preparation of the manuscript, the writer is indebted to his wife, Lisa. Appreciation is also due ii

the staff of +,the riuduslry Progrsn. of the College of Engineering for finai preparation and publication of the manuscript. The study was supported in part by a fellowship tendered by Thile University of Michigan ftIom National Science Foundation Grant No. h4O d The writer expresses his appreciation of this financial support. iii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.......................... ii LIST OF TABLES............... vi LIST OF FIGURES................................... a.. vii SYMBOLS..................................... d. xi CHAPTER I INTRODUCTION............................................. 1 Description of Problem................................ 1 Review of Literature.....5...................00.. 3 Scope of Investigation................................ 7 II THEORETICAL ANALYSIS.11 III ANALYTIC INTERPRETATION AND EVALUATION PROCEDURES...... 20 Equations of Transient Motion -- Pipe Flowing ~~~~~~~~~Full ~..e *** *eee*22 Equation, of Continuity...........................e 23 Equation of Motion........................... 41 Characteristics Equations........................... 44 Finite Difference Solution............O,. 46 Boundary Conditions.......... e *oe....o......... 54 Equations of Transient Motion -- Horizontal Pipe Flowing Partially Full................ e 57 Equation of Continuity...................... o....o 58 Equation of Motion.... * a *. *. *. 0o 0 a *. a 61 Characteristics Equations....................... 64 Finite Difference Solution...... o................. 66 Boundary Conditions........................... 71 IV COMPUTER SIMULATION....... 0 0 e o 0........... 75 Flow System............e... d......................... 75 Boundary Conditions................................... 76 Sequence of Operations............................... 83 Digital Computer Program,...... 84 iv

TABLE OF CONTENTS (CONT'D) Page CHAPTER V EXPERIMENTAL APPARATUS AND LABORATORY INVESTIGATION...... 91 General Description of Pipe System.................... 91 Instrumentation............... 0 97 System Calibration.................... 101 Experimental Procedures,......................... 106 VI COMPARISON AND DISCUSSION OF EXPERIMENTAL AND THEORETICAL RESULTS................ 111 Conditions Investigated...........................111 Comparison of Results.........................e 114 Pressure Rises,............................ 114 Column-Separation Voids............. 122 Significance of Findings.............................. 135 VII CONCLUSIONS.......... 0..........o.0................. 145 SELECTED REFERENCES...............e 148 APPENDIX I MAIN COMPUTER PROGRAM.................................... 153 II SUBROUTINE PROGRAMS USED WITH MAIN COMPUTER PROGRAM...... 168 III EXPERIMENTAL DATA; RUNS NUMBERS 25 AND 29.,,......... 182 IV COMPUTER SIMULATED RESULTS............................ 194

LIST OF TABLES Table Page I Copper Pipe Properties.................................. 95 II Plastic Pipe Properties.................................. 96 III Summary of Experimental Runs....................... 113 Tables in Appendix III I Experimental Run Number 25 Pressure-Rise Data at Gate Valve..................................... 183 II Experimental Run Number 25 Column-Separation Void Data....... 186 III Experimental Run Number 34 Pressure-Rise Data at Gate Valve.......................................... 188 IV Experimental Run Number 34 Column-Separation Void Data...................................... 192

LIST OF FIGURES Figure Page 1 Photograph of a Typical Vapor Cavity Accompanying Column Separation.................................... 12 2 Schematic Time-Sequence Representation of Column Separation and Pressure Pattern in a Pipe System Undergoing Transient Flow............................... 14 3 Definition Sketch Depicting Transient Conditions in Pipe Segment Flowing Full.............................24 4 Definition Sketch Depicting Stressed Element Located Within Pipe Wall................................. 28 Curves Defining Relationship Between Pipe Constraint Coefficient, c, and the Ratio of Outside to Inside Pipe Radii, b/R, for Various Conditions of Constraint and Different Values of Poisson's Ratio.......... 38 6 Definition Sketch Showing Intersection of C+a and C- Characteristics Curves in the x-t Plane........47 7 Definition Sketch Showing a Space-Time Grid Superimposed on the x-t Plane................................ 49 8 Definition Sketch Showing Relationship Between Space-Time Grids and Characteristics Curves at a Pipe Junction....................................... 56 9 Definition Sketch Illustrating Element of FreeSurface Flow in a Pipe................................... 60 10 Definition Sketch Illustrating Domain of Dependence Governed by Characteristics Passing Through Grid Points L and R.......... 0........... 8.... 72 11 Schematic Representation of Laboratory Flow System Used in Experimental Investigation...................... 77 12 Definition Sketch Showing a Parabolic Cylinder Intersecting a Circular Cylinder.............81 13 An Abridged Flow Diagram Setting Forth the Sequence of Operations Used to Simulate Transient Pipe Flow and the Accompanying Column Separation...................85 14 Detailed Schematic of Laboratory Flow System............. 93 vii

TABLE OF FIGURES (Cont'd) Figure Page 15 View of Constant-Head Weir Box and Loosely Coiled Copper Pipe on Wooden Frame........................ 94 16 View of Coiled Copper Pipe, Plexiglas Pipe, SolenoidOperated Gate Valve, and Assorted Instrumentation......... 98 17 Top View of Plexiglas Pipe and Solenoid-Operated Gate Valve................................................ 98 18 Close-Up View of Principal Test Section Showing Depth Gages and Pressure Transducer Mounted in Plexiglas Pipe.~................................................... 100 19 Enlarged View of Sample Depth Gage....................... 100 20 Diagram of Circuit Designed for Use with a Miniature Wave Gage...................................... 102 21 Close-Up View of Dual-Channel Oscillograph Units, Camera-Equipped Oscilloscope, and Bridge Amplifier Unit. o.....................................103 22 Flow-Resistance Relationships for the Copper and Plexiglas Pipes..........................................104 23 Transient Pressure-s, Photographically Recorded from the Oscilloscope Cathode-Ray Tube During Laboratory Run Number 19, are Typical of the Experimental Pressures Observed at the Gate Valve..................... 108 24 The Time-Dependent, Oscillograph Data Trace Shown at Top is Typcial of the Free-Surface Flow Depths Observed at Gage Number 1 During Periods of Column Separation in the Pipe................................... 109 25 Comparison of the Computer-Simulated, Transient Pressures with the Experimentally-Determined Pressures Observed at the Gate Valve During Laboratory Runs Numbers 25 and 34........................115 26 Detailed Comparison of the Computer-Simulated Initial Pressure Rise with the Corresponding Experimentally-Determined Initial Pressure Rises for Laboratory Runs Numbers 25 and 34................117 V l l

LIST OF FIGURES (Cont'd) Figure Page 27 Detailed Comparison of the Computer-Simulated Second Pressure Rise with the Corresponding Experimentally-Determined Second Pressure Rises for Laboratory Runs Numbers 25 and 34............... 118 28 Transient Pressures Observed at Gate Valve During Laboratory Run Number 25........................... 119 29 Transient Pressures Observed at Gate Valve During Laboratory Run Number 34........................... 119 30 An Isometric Representation of the Computer-Simulated Transient Pressures at the Gate Valve and Concurrent Free-Surface Profiles at Selected Distances from the Gate Valve as Determined from Theoretical Considerations..........o................ 123 31 An Isometric Representation of the ExperimentallyDetermined Transient Pressures at the Gate Valve and Concurrent Free-Surface Profiles at Selected Distances from the Gate Valve as Observed During Laboratory Run Number 25............................................. 124 32 An Isometric Representation of the ExperimentallyDetermined Transient Pressures at the Gage Valve and Concurrent Free-Surface Profiles at Selected Distances from the Gate Valve Due to Column Separation During Laboratory Run Number 34..o.....oo........................ 125 33 Comparison of the Computer-Simulated and ExperimentallyDetermined, Time-Dependent, Free-Surface Profiles at Selected Gages During Periods of Column Separation in the Pipe....................................... 126 34 Computer-Simulated Sequence of Water-Surface Profiles for the Initial Period of Column Separation in the Pipe..............................................00. 127 35 Computer-Simulated Sequence of Water-Surface Profiles for the Second Period of Column Separation in the Pipe...................................................... 128 36 Advancing Column-Separation Void Photographed During Run Number 19........................................ 130 ix

LIST OF FIGURES (Cont'd) Figure Page 37 Advancing Column-Separation Void..........................130 38 Column-Separation Void in a State of Suspended Motion Just Prior to Retreat and Collapse Against Valve.........131 39 Diagram Relating Static Pressure to Bubble Size..........139

SYMBOLS Symbols Units Descriptive Identification 2 A ft. Cross sectional area, pipe flowing full A ft.2 Cross segmental area, pipe flowing partially full a ft./sec. Celerity of transient pressure pulse, pipe flowing full a Subscript denoting ambient condition b ft. Outside pipe radius C+,C- Characteristicscurves C'C,C3 cT Constants of integration c,c, C2, C3 Coefficients of pipe support and restraint c Subscript denoting circumferential direction D ft. Inside pipe diameter E lb./ft.2 Modulus of Elasticity of pipe wall e,exp Napierian base f Darcy-Weisbach frictional resistance term f Function notation g ft./sec.2 Acceleration of gravity g Subscript denoting gaseous condition H ft. Piezometric head (P/pg + z) H Coordinate axis Hf ft. Head loss due to resistance to flow i Counting integer or indexing subscript J ft.lb. Parameter denoting air mass in bubble K lb./ft.2 Bulk modulus of elasticity of liquid k Counting integer or indexing subscript; usually refers to pipe segment xi

SYMBOLS (Cont'd) Symbols Units Descriptive Identification L Left grid point on x,t plane; frequently used as subscript L Length of an undefined pipe segment M Middle grid point on x,t plane; frequently used as subscript nm sec. Linear multiplier coefficient + A/gT; pipe flowing partially full m Subscript denoting x-position -1 n sec. Linear multiplier coefficient + g/a; pipe flowing full P Grid point and intersection point of characteristicscurves on x,t plane; frequently used as subscript fto Wetted perimeter; pipe flowing partially full p lb./ft. Internal pipe pressure p lb./ft. Internal bubble pressure p Subscript, denotes x-length of parabolic void q lb./ft.2 External pressure on pipe R Right grid point on x,t plane; frequently used as subscript R fto Inside pipe radius R ft.lbo/lb./Deg. Gas constant for air RH ft. Hydraulic radius; pipe flowing partially full r ft. Radial distance r ft. Cavitation nucleus radius r Subscript denoting radial direction T ft. Free surface width; pipe flowing partially full xii

SYMBOLS (Cont'd) Symbols Units Descriptive Identification T Coordinate axis t sec. Time t Subscript denoting travel time u ft./sec. Mean velocity; pipe flowing partially full -V ft 3 Volume v ft./sec. Mean velocity; pipe flowing full v Subscript denoting vapor condition X Coordinate axis x ft. Distance measured in longitudinal space direction Z Coordinate axis z ft. Depth of flow; pipe flowing partially full Intersection point of C+ and t-grid on x,t plane; frequently used as a subscript r ft./sec.2 Equation of motion; pipe flowing full r ft./sec. Equation of continuity; pipe flowing full r ft./sec. Equation of continuity; pipe flowing partially full r ft./sec. Equation of motion; pipe flowing partially full E Intersection point of C- and t-grid on x, t plane; frequently used as a subscript ft./ft. Strain o Radial angle subtended by pipe wall element o A decimal parameter o Gradient angle of pipe (xx = - sine) A ft./sec. Linear sum of F3 and F4 xiii

SYMBOLS (Cont'd) Poisson's ratio ft. Radial increment of displacement p slugs/ft.3 Liquid density lb./ft 2 Stress lb./ft. Surface tension T lb./ft.2 Boundary shear stress at pipe wall Coefficient correcting K for changing pressure cP tL/r X Coefficient for change in pipe size Coefficient for change in pipe size crr 2 ft./sec. Linear sum of Fl and F2 E~ ~Reynolds number xiv

CHAPTER I INTRODUCTION Engineering treatment of the phenomenon of the liquid column separation frequently accompanying transient pipe flow has lacked rational interpretation in terms of the governing fluid dynamics. This study of column-separation attempts to provide better insight into the mechanics and dynamics of the phenomenon while establishing a more rational physical basis for its interpretation and analysis. Description of Problem Consider the liquid pressure at a point in a pipeline which is flowing full. Should this point pressure decrease, regardless of cause, to the vapor pressure of the flowing liquid, a vapor cavity will form. This vapor cavity is sometimes referred to as a vapor column. The phenomenon of vapor cavity formation is commonly referred to as a "column separation, meaning separation or interruption of the liquid column. It often occurs following rapid closure of a valve in a pipeline which is flowing full. Nevertheless, column separation may also occur under other circumstances, for instance, following propagation of a transient wave of low pressure into an elevated portion of a pipeline flowing full. Such a low-pressure wave, for example, could be caused by a sudden pump failure in a liquid transmission or distribution system. The severe damage which the high pressures associated with collapse of the vapor column can inflict upon the pipe system causes column separation to be a problem of significant engineering concern. -1

Certain features associated with column separation have been the cause of speculation in the past. The physical laws, for example, governing the shape and movement of the vapor column have not been adequately describedo Moreover, the nature of the collapse of such a vapor column is even less well defined. Definition of these physical laws together with a broader understanding of vapor column collapse would be of appreciable aid in providing enlightened engineering design and troublefree operation of both large and small, closed-conduit, hydraulic systemso The problem of column separation is clearly an integral part of the much broader problem of transient liquid flow in both open and closed conduits; that is, flow in which the velocities and pressures vary with location and with time. In pipes the phenomenon of transient flow of liquids is commonly referred to by the imprecise, lay term, "water hammer." However, column separation has a unique and distinctive feature which sets it apart from the ordinary water-hammer type phenomenon. In the portion of pipe momentarily occupied by the vapor cavity, any flow of liquid occurring beneath the cavity appears to be free-surface flow, in other words, flow in which gravity would be the principal governing factor. Thus, transient flow conditions occur throughout a system composed of both closed-conduit flow (pipe flowing full) and open-channel flow (pipe flowing partially full). However, in that phase of the problem involving transient, open-channel flow, the pressure variations which occur with time and location are manifested by fluctuations of free surface head (depth of liquid) above a datum. The motivation for the present study was the belief that pertinent information regarding the various undetermined aspects of column

-3separation could be qualitatively defined and quantitatively determined through the development of an experimentally-verified, mathematical model simulating transient, liquid, pipe flow with column separation. Review of Literature Study of transient flow in both open and closed conduits is not a new area for research in fluid dynamics in any sense of the word. To be sure, water hammer occurring in pipe flow was a topic of interest to many of the noted hydraulicians of the early and middle 19th century. However, rational physical interpretation and true understanding of this closed conduit phenomenon did not evolve until 1898 when Joukowsky(29) published his treatise on the subject. His work, an outgrowth of the experimental studies and mathematical analysis which he conducted while supervising design of the then new Moscow waterworks, provided many of the basic concepts and relationships essential to comprehension and further study of closed conduit, transient flow. Some four years later Allievi(l) expanded and greatly extended Joukowsky's analytic work. However, Allievi did not publish his rather extensive mathematical and graphical treatment of water hammer until 1913. Nevertheless, this work became the truly definitive study from which much of the modern day interpretative analysis of closed conduit, transient flow can be traced. Meanwhile, during the decades after the mid 19th century, the fraternity of French hydraulicians, among them Saint-Vennet, Dupuit, Bresse, and Boussinesq, were concerning themselves with the analysis of open-channel hydraulics. In 1871 Saint-Vennet(45) presented a paper to the French Academy of Sciences which contained a form of the equation of motion for unsteady flow in open channels. Six years later in 1877 the Academy

-4(7) the third secpublished Boussinesq's outstanding essay on hydraulics, the third section of which presents an analysis of wave stability and wave propagation celerity in open channels. Taken together, these contributions provided the foundation necessary for the present day study of transient flow in open conduits. Although the rudimentary dynamics of transient flow in both open channels and closed conduits had been more or less correctly appraised and partially formulated by the early years of the 20th century, techniques for the analysis of the phenomena still remained to be developed and applied. Drawing extensively upon Allieve's graphical methods, Bergeron(6) and others expanded and applied these techniques in order to approximate solutions for various problems involving electrical, mechanical, as well as fluid transients. It is entirely anachronistic, however, to discover that in 1860, the German mathematician Riemann(41'489'6) had already developed and used the then unnamed method of characteristics to achieve an explicit solution of a second order partial differential equation depicting soundwave propagation in steady, two-dimensional air flowo An interlude of nearly 30 years passed before Massau discovered Riemann's work and transformed his solution into a graphical technique in 1889. Later, in 1900, after extending the graphical technique to apply to first order partial differential equations as well, Massau(36) published a fully-documented study of his application of the method of characteristics. It is of particular interest to note that his presentation included examples specifically treating transient, open-channel flow. Yet, despite Massau's important contribution, the potential value of the method of characteristics as a powerful tool for the explicit solution of problems concerning both open and closed conduit transient flows, remained largely unrecognized.

In fact, the basic nhnysical and mathematical similarity of the two flow situationst —he transient, pressure-wave phenomenon in pipes on the one hanad and the transient, gravity-wave phenomenon in open channels on the other —was only vaguely appreciated. (58) Meanwhile, Thomas, i.n a particularly noteworthy paper, attacked the open-channel, flood-routing problem as truly a problem in transient flow. Starting with the fundamental partial differential equations for unsteady flow, he outlined a tentative procedure for their numerical solution. Stoker, treating the same problem, outlined an implicit procedure(56) based upon an iterative solution of the partial difference equations. Dronkers(l5) and Schonfeld,(47) studying theclosely related problem of tide-induced, transient flow in estuaries, outlined a solution technique employing power series as well as one based upon characteristics. Chow(10) presented a graphical solution based upon the method of characteristics as first developed by Lin. However, all of the above-mentioned solution techniques for handling problems in both open and closed conduit transient flow suffer from one common and very serious drawback; namely, an overwhelming amount of very tedious, exacting, and extremely time-consuming computation. This formidable drawback has been all but eliminated within the past decade by the introduction and rapid development of the high speed digital computer and sophisticated computer programming languages. It is now economically practicable as well as technically feasible to solve a great variety of transient flow problems by digital computer techniques as evidenced by the recent technical literature.(4' 5)'2130) 53 554)55)

During the first half of the 20th century Allieve's analysis of water hammer was refined and the graphical techniques which he and Bergeron had devised were tediously applied to a variety of practical, closed-conduit, transient-flow problems. (2,28,37,39,42,52) It was the treatment of certain of these practical problems which began to focus attention upon the specific phenomenon of liquid column separation and the associated free-surface flow taking place beneath the resulting vapor cavity.(3l'43) A search of the literature, however, evidences little actual investigation of the mechanics of column separation in transient flow systems. Recent publications by Li(3 33) and by Walsh(34) are notable exceptions. In 1962 Li presented a theoretical treatment of column separation in a pipe of rectangular cross section with frictionless flow. Although several different separation conditions were considered, cavity behavior was treated as an isolated phenomenon rather than a phenomenon integrally dependent upon the transient flow conditions prevailing throughout the entire conduit system. No experimental verification of the analysis was presented in the paper. The chief result of this study appeared to be that the form of the separation void has little bearing on the magnitude of the resulting (8) pressure decay. Heath also studied column separation accompanying the rapid closure of a valve located at the end of a long pipe. However, flow in the pipe was assumed to be frictionless and an equivalent friction loss was inserted in the graphical analysis. In a subsequent paper Li and Walsh reported their investigation of the maximum pressure resulting from the collapse of the vapor cavity. Escande(l8) treats column separation occurring in a penstock subject to instantaneous valve closure by using graphical means. However, neither

-7the growth nor the collapse of the vapor cavity, nor the dynamics of the flow beneath the cavity is adequately treated in the investigations mentioned above. What is perhaps more significant is that the cavity resulting from the liquid column separation was not interpreted as a manifestation of free-surface, open-channel, transient flow occurring adjunctly and simultaneously with closed conduit transient flow in the same pipe system. In recent years considerable interest has developed concerning the dynamic conditions and liquid properties prevailing at the inception of cavitation. As a result of their investigation of the roles of air diffusion and liquid tensile stresses upon incipient cavitation, Parkin and Kermeen(38) have shown that air diffusion in flowing water is responsible for the growth of microscopic bubbles on the solid boundaries in a region where the pressure is slightly greater than vapor pressure. Moreover, the microscopic air bubbles serve as nuclei for the explosive growth of vapor cavities as the pressure decreases, according to Ripken and Killen.(44) Other apparent properties of incipient cavitation are presented by Stepanoff and Kawaguchi(50) and by Hooper.(27) Scope of Investigation This study of column separation may be divided into two major phases. The first phase treats the theoretical analysis and analytical interpretation of the problem of transient liquid flow with column separation; the second phase concerns the accompanying experimental investigationso A theoretical interpretation of the problem of column separation is presented in Chapter II. The fundamental, quasilinear, partial differential equations representing one-dimensional, transient, liquid motion

-8in pipes as well as in open channels are derived and reviewed in Chapter IIL. The friction losses encountered in both types of flow are included in the respective sets of equations. Techniques for evaluating these sets of equations, based upon numerical interpretations of the method of characteristics for first-order, hyperbolic, partial differential equations, are also developed and presented in Chapter III. The two sets of partial differential equations together with the respective evaluation techniques provide the basic building blocks used to formulate the mathematical model simulating transient liquid flow with column separationo Chapter IV is devoted to formulation of this model in terms of an operational program for high-speed digital computer. The functions and operating sequence of the model are presented in a flow chart. The digital computer program together with all major subroutines and supporting programs are presented in Appendices I and II, respectivelyo The flow system employed in the second or experimental phase of the study was specifically designed to facilitate the laboratory measurement of column separation. A rapid-closing, solenoid-operated, gate valve was employed in order to produce column separation and to fix its initial occurrence within the system to a face of the gate in the solenoid-operated valve. Although column separation could certainly have been induced at the downstream face of the valve gate, the incipient formation of the vapor cavity would tend to take place while the gate was still in the process of closingo Moreover, flow separation on the downstream face of the gate could seriously disturb the liquid flow pattern. Clearly, these conditions could lead to distortion of the ultimate shape of the vapor cavity. In order to avoid this possibility, a flow system was selected

-9consisting of a long pipe leading from a constant-level, low-head reservoir to an elevated position. The solenoid gate-valve was situated at the end of an elevated, horizontal section of pipe. A short return pipe connected the valve to a sump. Flow was maintained by syphon action such that column separation would occur without cavity distortion at the upstream face of the gate of the solenoid-operated valve following abrupt valve closure. The separation phenomenon was achieved once the initial wave of high pressure produced at the moment of valve closure had propagated from the valve to reservoir, and the counterpart, low-pressure wave had propagated back to the valve. Water was used as the liquid throughout the laboratory investigation. Column separation was confined to the elevated section of horizontally-mounted pipe. A detailed discussion of the experimental apparatus, of the instrumentation used in measuring column separation, and of the measurements themselves is presented in Chapter V. In order to acquire specific information about column separation from the mathematical model, it must be activated and numerically evaluated by the digital computer for the particular set of bounding parameters delineating the experimental flow system being investigated in the laboratory. Chapter VI presents a comparison of the analytic and experimental results obtained for various laboratory conditionso A discussion of the findings and some concluding remarks are also contained in Chapter VI. Chapter VII briefly summarizes the principal conclusions derived from the study. All apparatus used in the experimental phase of the study was assembled and all experimental work conducted in the G. G. Brown Fluids Engineering Laboratory at The University of Michigan. The digital computer

-10program and associated subroutines were written in MAD, the Michigan (3) Algorithm Decoder language, and executed on an IBM 7090/1410 digital computing system at The University of Michigan Computing Center.

CHAPTER II THEORETICAL ANALYSIS Contrary to what one might deduce from literal interpretation of the term "column separation," complete physical interruption of liquid flow in a horizontal pipe does not take place for the magnitudes of transient pressures and velocities ordinarily encountered in horizontal closed conduit fluid systems. In fact, once nascent development of a vapor cavity has occurred at a point in the pipe, the cavity expands and propagates in the direction of flow as an elongated "bubble." A typical example of a vapor cavity resulting from column separation in water flow in a horizontal pipe is shown in Figure 1. While the vapor cavity occupies the upper volume segment within the pipe, liquid continues to flow with a free surface in the lower volume segment. With identification and recognition of the typical physical circumstances commonly encountered with column separation, two key questions immediately come to mind. These questions are the following: (a) Once a vapor cavity has formed and the liquid has pulled away from the uppermost, inner surface of the pipe, is not the flow which is occurring beneath the cavity simply free-surface, gravity flow? (b) If time is measured from the moment of column separation, must not the volume of the vapor cavity at any subsequent instant be equivalent to the accumulated volume of liquid discharged through the pipe just ahead of the cavity? Deductive reasoning and rational analytic interpretation of the column separation phenomenon seem to indicate that the answer to both of these questions is unquestionably affirmative. -11

Figure -1. Photograph-of-a-Ty-ical-Vapor-Cavity-Accompanying-Column-Separation. Figure 1. Photograph of a Typical Vapor Cavity Accompanying Column Separaticn

Although a surface tension mechanism must appreciably influence the actual separation of the liquid from the inner surface of the pipe wall, the effects of this mechanism can only be very.local and temporal at best. Once separation has occurred and a free liquid surface exists, the remaining forces —namely, gravity and fluid friction forces —must be the principal forces acting upon the flow. Of course, the flowing liquid continues to have momentum. Therefore, one may hypothesize that the flow occurring beneath the vapor cavity is simply unsteady, open-channel flow in a circular conduit. More specifically, the free-surface, gravity flow produced by column separation is hypothesized to be analogous to a negative surge wave in an open channel of circular section. Meanwhile, pipe flow ahead of the vapor cavity is assumed to continue as full-pipe flow. One may further hypothesize that transient conditions continue to govern the flow in this portion of the pipe system. However, the magnitude of the pressure fluctuations must rapidly diminish as a result of the change in the boundary conditions introduced by the vapor cavity. In fact, liquid motion in the portion of the pipe which continues to flow full must approximate surge flow. By way of illustration consider a reservoir and a quick-closing gate valve connected by a long, horizontal pipe. Let the reservoir be exposed to atmospheric pressure as shown in Figure 2. The mean velocity of flow in the pipe system is v while the pressure is p = y(ho - hf) in which y is the specific weight of the liquid, ho is the head of liquid i the reservoir, and hf the head loss created by fluid friction in the pipe. If the gate valve at point C is abruptly closed at time to, transient flow conditions are created in the pipe. A positive (high pressure) wave

Reservoir Pressure Pressure Diagram TIME Atmospheric Pressure Vapor Pressure_ < to Water Surface A 0 Reservoir PiPe V (XI to) / RPAP VP- V to WS A v B C RP AP VP wS AI _;v B v=O C/ RP AP VP-' 12 WS A O _ B v=O C/ RPAP WSA~ —'VP t3 WS A. B v=O C RPAP WS A V____ - _ v B D C/ RP - AP.. VP-. 5.....- - w S 2 -'' —A- 0=' ___-__ v-D Cavity C _?.. - -- c/ RP --- AP " VP..... WS- iT- BC Figure 2. Schematic Time-Sequence Representation of Column Separation and Pressure Pattern in a Pipe System Undergoing Transient Flow.

-15caused by the rapid deceleration of the liquid propagates from point C toward the reservoir at point A as indicated by the wave front at B At time tl flow from the reservoir continues with undiminished velocity toward B. However, between B and C the velocity is reduced to nearly zero; the pressure, which greatly exceeds p, compresses the liquid and tends to expand and, in some instances, elongate the pipe segment. When the positive pressure wave reaches A at time t2, the pressure is abruptly reduced to the sustaining pressure of the reservoir. The compressed liquid in the pipe begins to expand and starts flowing toward A o The pressure between A and B returns to p as the pressure-drop, delimited by the wave front B propagates toward C at time t3. However, when the pressure drop reaches C,.the entire column of liquid, AC, is moving toward A with velocity v. Because the gate at point C remains closed, the water column abruptly decelerates and the pressure decreases below p. A negative (low pressure) wave tends to develop, but because the pressure becomes less than the vapor pressure of theliquid, a vapor cavity is assumed to occur at the uppermost point on the valve gate at C and column separation takes place at time t4. The cavity expands until at time t5 the transient velocity in the segment of pipe flowing full, that is, segment AD, becomes zero. Meanwhile, unsteady, free surface flow has been occurring beneath the cavity in the pipe segment DC. However, the differential pressure between the reservoir and the vapor cavity causes the liquid column, AD, to flow toward C as indicated at time t6, thereby initiating the imminent collapse of the vapor cavity. At time t7 the vapor cavity collapses and conditions in the system are again equivalent to those which occurred at t0 o The

-16phenomenon will continue to repeat itself, although diminished in magnitude by the energy losses caused by fluid friction, until the pressure fails to drop below vapor pressure and column separation fails to occur. The above description of the sequence of physical events —though admittedly somewhat oversimplified and idealized —provides a basic familiarity with the transient-flow, column-separation phenomena as encountered in the experimental system. A more penetrating and detailed analysis of these phenomena leads one to hypothesize circumstances which could bring about the genesis, growth, and collapse of the vapor cavity anticipated at time t4 in the above system. Because the causative drop in pressure first takes place at the gate valve, it is reasonable to assume that cavitation will originate at the point of lowest pressure on the upstream face of the valve gate. The actual point of origin is probably at the topmost intersection of the vertical valve gate with the upstream port in the valve itself. As the pressure rapidly drops below atmospheric pressure and approaches the vapor pressure of the water, one or more undissolved microscopic air cavities —air nuclei —attached to the uppermost edge of the gate or to particulate matter in the flow begin to expand. Molecules of free gaseous air coming out of solution in the immediate vicinity of the point of lowest pressure enter the bubble and help to increase its sizeo Mutual coalescence takes place between adjacent bubbles and the-larger resultant bubbles continue to expand until one is of sufficient magnitude that the difference between its internal pressure and the decreasing external pressure is enough to offset the surface-tension pressure which is inversely proportional to its size. Once this threshold size is reached,

-17the bubble will expand explosively as a water-vapor-filled cavity. Calculations show that the threshold bubble diameter must exceed approximately 8 x 10-4 inches (20 microns) in order for vapor cavity expansion to occur at pressures normally encountered in practical situations. It is quite probable that for a fleeting moment the water undergoes tensile stress (as demonstrated(38) in recent water tunnel experiments) until the nascent bubble can sustain explosive cavitation after which the pressure reverts to the vapor pressure of the liquid water. The complete hypothesized sequence of events, starting with the rapid drop in pressure and concluding with the onset of the explosive growth of the vapor cavity, occurs, of course, in a very brief time span —probably a few milliseconds in duration at most. During the same time interval in which cavitation is initiated, the horizontal column of water (denoted by AC, Figure 2) starts to undergo deceleration of its flow upstream, toward the reservoiro This deceleration, of which the pressure drop responsible for inception of cavition is but a manifestation, first occurs in the water adjacent to the gate valve. Although the considerable momentum of the water moving as a column.tends to cause it to separate from the vertical face of the gate, the change in momentum caused by the deceleration of the flow is ordinarily insufficient to completely overcome gravitational effects. As a result the tendency for column separation at the gate is satisfied by the explosive growth of the nascent air bubble into a water-vapor cavity occupying the topmost segment of the pipe. Gravity causes some of the decelerating flow to remain in the bottom segment of the pipe and in contact

-18with the gate. The upstream flow in the pipe continues to decelerate as the front of the pressure wave, B, delimiting the drop to vapor pressure propagates toward the reservoir. Meanwhile the vapor cavity, advancing (D) behind this wave front and seemingly tearing the water from the topmost inner surface of the pipe wall, expands to occupy a volume equivalent to the volume vacated by the receding full-pipe flow. Surface tension may have some nominal effect upon the shape of the advancing edge of the cavityo Ultimately, the forces created by the frictional resistance to flow and by the differential pressure between the reservoir and the vapor cavity bring water column AD to a momentary rest. At the same time the free-surface, transient flow taking place beneath the cavity (DC) gradually diminishes to a minimum. The vapor cavity will achieve its maximum expansion at the moment of rest. But because the differential pressure force continues to act, the water in column AD now reverses direction and begins to flow downstream toward the gate valveo Conditions are now right for collapse of the vapor cavity. As column AD accelerates in the downstream direction, one hypothesizes that the interface between the vapor cavity and the water column becomes a surging, free-surface front. Although surface tension is believed to be a factor of nominal overall effect, it is assumed to aid reattachment of this free surface to the pipe wall. Moreover, the total volume of the cavity decreases at the same rate at which full-pipe flow is occurring at D. At the instant of total collapse of the water-vapor cavity, the abrupt deceleration of the water column by the closed gate valve once again produces high pressures at time t7. The minute volume-of gaseous air which initiated

-19cavity formation is readily forced back into solution, and, except for the loss of energy expended to overcome flow friction, conditions in the system at time t7 are analogous to those at time to. Now that the theoretical concepts of transient pipe flow with column separation have been hypothesized and explicitly set forth, the analytic conditions believed to describe and govern these phenomena can be ascribed. This is done in the following chapter.

CHAPTER III ANALYTIC INTERPRETATION AND EVALUATION PROCESS Derivations of the basic differential equations describing transient liquid motion, first in a pipe flowing full and then in a horizontal pipe flowing partially full, are given in this chapter. Flow is simulated mathematically according to conventional one-dimensional analysis in which the longitudinal space-dimension and time are the independent variables governing analysis. Several underlying assumptions are needed to provide a foundation and a starting point from which to undertake theoretical analysis of column separation in transient pipe flow. These assumptions concern the physical properties of the liquid, the physical properties of the pipe material, and the kinematics of the flow: (a) The fluid used in the flow system is elastic and of homogeneous density when in its liquid state. (b) The minute fluctuations of the liquid vapor pressure, resulting from the latent heat of vaporization and occurring at the instant of vapor-cavity formation or collapse, are insignificant by comparison with the difference between atmospheric pressure and vapor pressure33 Therefore, these pressure fluctuations may be, and in fact, are disregarded. (c) The pipe is constructed of a sectionally homogeneous, isotropic, elastic solid in which the stresses never exceed the yield point of the material. -20

-21(d) The velocity and the pressure in the pipe when flowing full are considered to be uniformly distributed over any cross section transverse to the pipe, with the result that the flow may be treated as one-dimensional, closed-conduit flow. (e) The velocity in the pipe when flowing partially full is considered to be uniformly distributed over any transverse cross segment of flow. Moreover, vertical accelerations of the water surface are considered to be insignificant, implying that hydrostatic pressure prevails, and thereby permitting the flow to be treated as one-dimensional, open-channel flow. (f) The frictional resistance encountered with transient flow, whether it be in a closed conduit or in an open channel, is assumed to be proportional to some power of the mean velocity in a cross section (or segment) transverse to the flow. The extent to which frictional resistance may be dependent upon the transient character of the flow is believed nominal(14'46) and is disregarded. (g) Surface tension forces in the regions where the free water surface intersects the solid, pipe-wall boundary are assumed to be of insignificant overall effect and are disregarded. This assumption is somewhat less valid for pipes of small diameter. In deriving each of the sets of partial differential equations according to the assumptions set forth above, elements of water are considered which are bounded by two fixed planes normal to the-longitudinal

-22axis of flow. Discharge and velocity are assumed to be positive when water is flowing in the positive x direction. A pipe having finite wall thickness is considered in the derivation of the equations describing transient liquid motion in the pipe flowing full. The laws of conservation of mass and conservation of momentum govern the derivation of the equation of continuity and-the equation of dynamic equilibrium, respectively, for both regimes of flowo Once the sets of partial differential equations describing transient liquid motion have been derived, they are recognized to be of the hyperbolic type and are then transformed into sets of total differential equations which are valid along their respective "characteristics" curves. These sets —one set for the pipe flowing full and another for the pipe flowing partially full —are then expressed as sets of finite difference equations readily amenable to solution by high-speed, digital computer. The actual finite difference technique used in developing the computer solution is an adaptation of the specific time interval procedure advanced by Hartree and others. (l2l19)220)2125'26'35) Initial values obtained from steady flow conditions, together with expressions for the various boundary conditions encountered at the valve, at junctions between pipes, and at the reservoir make an explicit solution possible. Equations of Transient Motion —Pipe Flowing Full Transient motion in a pipe flowing full is a function of two independent variables. These variables are the space and time coordinates, x and t, respectively. Let x be the axial location measured along the pipe from some arbitrarily fixed reference point. Two quasi-linear partial differential equations are written to represent the transient

-23velocities and pressures in the pipeo Once derived these relationships are transformed into a set of four total differential equations which are the so-called "characteristic" equations. The characteristic equations are subsequently written as finite difference equations after which numerical methods are introduced and a solution technique developed to permit their evaluation by high-speed, electronic digital computer. The appropriate boundary conditions which govern the solution are examined and set forth. Equation of Continuity The equation of continuity states that the net mass inflow per unit time into a pipe segment is equal to the time: rate of mass increase taking place within the segment. This statement is premised upon the concept of the conservation of mass. A typical control segment of a pipe that is flowing full is shown in Figure 30 If v is the average velocity and p the density of the liquid flowing into the segment (Figure 3) then the mass influx per unit time is pAvy, where A is the cross-sectional area at (. At cross section ( the efflux of mass from the segment is given by pAv + (pAv) dx 6x The net mass inflow into the segment is pAv - [pAv + a(pAv) dx] ax or (pAv) dx. (1) ax The total mass in the segment is pAdx o Therefore, the time rate of mass increase occurring within the segment is

-24L Bdx A+ dx H ax \ x0 P \ /z + [p(H-z)+fpAx (H-Z} d! Datum Figure 3. Definition Sketch Showing Transient Conditions in Pipe Segment Flowing Full.

-25a(pAdx) (2) (2) Equating the expressions given by Equation (1) and (2) according to the definition above, one has _ (pAv) x = (pAdx) () dx = t Normally, x is independent of t and, therefore, is zero. However, if the pipe is supported in such a manner that the pipe segment is subject to axial strain, ~x, then t dx. In this expression ~ represents the strain; the subscript x, the direction. Consequently, expansion of Equation (3), subsequently divided by the mass of the fluid element, pAdx, results in the expression A, Px At Pt dflx x+ Ax p + t+- + dx O (4) x + A p A p dt or by rearranging (vpx+pt) + (VAx+At) + vx + x 0 (5) P x A dt Here the subscripts x and t denote partial differentiation with the respective variable. However, since dp =p dx + op dt 3x dt at or simply dp= vx + Pt, dt and, similarly, because dA't= vA + At

-26Equation (5) may be written 1 dp+ 1 dA + Vx + drlx (6) (6) p dt A dt dt The essential task, now, is to represent the liquid density and the pipe cross-sectional area changes with respect to time as functions of the velocity of the flow and the pressure. First, by recalling that the bulk modulus of elasticity for liquids denoted by K is defined as K - d-A (7) d-/- dp/p' where p is the pressure and e is the volume, one can rewrite Equation (7) as dp _ Edp(8) P K Subsequent differentiation with respect to time gives 1 dp _1 (9) p dt K dt K dt in order to treat the change in the cross-sectional area of the pipe, consider the circumferential strain to be given by Fc o Therefore, the increase in the cross-sectional area with time is given by dA d'c (2tR)R d 2R2 = 2A ddt dt (R)R dt dt or 1 dA 2 dc (10) A dt dt Thus, Equation (6) becomes + 2 c + vx + x O (11) K dt dt dt

-27The magnitudes of the various terms in the continuity equation clearly depend upon the radial or the circumferential strain. But the circumferential strain depends upon the physical characteristics of the pipe itself, as well as upon the restraint to deformation imposed by the particular manner in which it is supported. In order to evaluate the terms drc/dt and drx/dt, a general expression must be developed in which the physical characteristics of the pipe and its support conditions are treated as parameters. The relationship between stress and strain is commonly given by Stress/E = Strain, where E is the modulus of elasticity of the pipe wall material. Moreover, stresses in any one direction create strains in the two other normal directions according to Poisson's ratio, p. Thus, the following relationships can be written to express the radial, circumferential, and axial pipe strains in terms of their appropriate stresses: r 1 [-r(<c+ax)] (12) =1 [x ((13) Qc E C_4( r ax)] E [1rx4(cr+:c)] (14) The term, a, denotes stress; the subscripts r, c, and x denote the principal directions. These relationships provide the connecting link which will subsequently allow Equation (11) to be rewritten in terms of velocities and pressures.(9,40,59) Consider the stressed element located within the pipe wall shown in Figure 4. The element, which is denoted by abcdefgh, has the length, the curved-width, and the thickness dimensions, dx, rdO, and dr,

-28do-r'r+ dv. dr Ire dr~~~~'~d bC~~~~~~~~~' Figure 4. Diefinition Sketch Depictin Stressed Element Located Within Pipe Wall. Transverse Cross-Section is Shown on the Left and Three Dimensional Enlargement is Shown on the Right.

-29respectively. In Figure 4, dO is the radial angle subtended by the element, r is the radial distance to the element measured from the axis of the pipe, R is the inside pipe radius, and b is the outside pipe radiuso When considering the open-ended element shown in Figure 4, one must be mindful of two basic conditions which are assumed valid: (a) The axial strain created by the radial stress is constant. (b) Displacement perpendicular to the axis is radial and dependent only upon the radius. The unstrained radial distance to the particle is r. Let its strained radial distance be given by r + ~ where ~ is the total increment of displacement in the radial direction. Thus, the radial strain is dr while the circumferential strain is r The axial strain is a constant, rx = constant The summation of forces acting upon the element abcdefgh, in which the outward direction of the radial bisector in Figure 4 is taken to be positive, is dO dO dar arrde + acdr -7 + acdr 2 (ar + d dr)(r+dr)de = O or

-30rdar d 2 ardO + cdrde - a rd - drd drdO (dr) d = 0 r c r r dr dr or, after combining like terms, eliminating higher order terms, and dividing by drdO dc ac- r - r = 0 c ~Jr dr Reduced to simplest terms d(rcr) c dr (15) Equations (12-14) can be rewritten as follows: CO - [ (ac+x) = rE = E d_ (16) drr(16) a - [ r(r+x) = ncE = E _ (17) ax - (rar+rc)= xE = constant. (18) However, by rewriting Equation (17) as r c -[ (ar+Cx)j = Et and then differentiating with respect to r, {c-4(Cr+ax)} + d {ac-4(cr+x)} = E dr, (19) dr dr one can eliminate E d- from Equation (16) and (19). This is now done. Let a = arr in Equation (16) and (19): dr X J r dr t d21 d?+_ d _ dc4x d-r dr -xd 2 r dr drxr X - dr - d-xrr dr -x

-31d2V d_ dV drx r + = ir- o (20) dr2 dr r dr Differentiation of Equation (18) with respect to r produces dax d(crr+c=c) - t= 0 dr dr or, written in terms of V, dac i 1 d*( t+ d2V- - (21) dr r dr r2 dr2 Therefore, using Equation (21) one may write dax 2 dd V d2 4r _- k -- + r dr dr r dr2 which when substituted into Equation (20) produces the homogeneous linear differential equation d2V 1 dV V id2 + 1 d _ ) (1-2)r = 0 (22) dr2 r dr r-.By letting V = rcp for which V' = rcp' + cp and V" = rcp" + 2cp', Equation (22) may be solved in the following manner: rep" + 2cp' + cp' + - _ P- 0 r r (T, + p 0 o r The integration factor for this equation is exp [ r dr] = exp(3 ) eln r r5 thus r3Cp" + 3r2cp' = o (23) for which the exact differential is r3Cp'. In order for Equation (23) to be valid r3cp' must equal a constant. Thus,

-32P Cl,3 and C1 C2 qO fop'dr = 2 + C3 = + C3 But because p = r, it is possible to write r C2 -+ rC3 and since = r r r C2 Jr = C3 + 2' (24) r and from Equation (15) d(rar) C2 a'c= d C3 - (25) dr r Once the constants C3 and C2 are evaluated, the circumferential and radial stress terms will become known. Let ar = -p when the internal pipe pressure is p and the inner radius is R. By the same token let oa = -q when the pressure acting upon the pipe externally is q and the outer pipe radius is b Then one can write C2 C2 R2_b2 P q 2 -2 (R2b2) b R RC2 or (p-q)R2b2 C2 R. (26) Consequently, evaluation of C3 is readily accomplished using Equation (24).

- 33_p C3 2 2-qR2 (R2-b2)R2 or c3 = R -pb2 +pb 2-qb = qb2-2 R2_b2 R2-b2 Thus, the radial and circumferential stress relationships maybe written 1 [.2 2 R2b2 (~] (28) r b2R2 [PR- qb2 + 2 (q-p) (28) 1 pR2 b R2b2 (29) C b2-R2 R- qb- 2 -p Note, also, that the combined stress term, Cr + a 2 (pR2-qb2) bc _R2 is independent of both the circumferential and the axial positions; thus, o'x is truly a constant at any cross section as originally postulated. Clearly, there are several methods by which a section of pipe may be supported. But it should be just as apparent that the tolerable, pressure-induced strains are governed in large measure by the support conditions. Using Equation (12) and (14), one can write the relationship given in Equation (11) in terms of the appropriate stress conditions~ d t E1d-~ -- /d - + - V + + 0 (30) K dt + +dt M dt dt x E [dt dt dt/ Let the external pipe pressure, q, be considered zero. Equation (30) may then be evaluated for the three commonly encountered conditions of pipe support described below: Condition I. Assume a section of pipe anchored at one end, but otherwise free to deform both radially and axially with respect

-34to this fixed location. In order to determine the deformation which occurs within the pipe let r = R. Then, from Equations (29) and (28), respectively, it is apparent that pR2 b2+ ac =p221 +R] (31) c b2-R2 R2 while rbR2 b2' The axial stress, which is determined by dividing the end pressure force by the cross sectional pipe area, is given by 2 2 x it (b 2-R2) (b2-R2) Therefore, Equation (30) can be written _ d dp| R + b dI R b 11 ___ — + 1 b dK E dt b2-R2 dt b2_R2 R2 or 1 +'R 3 + 2b 6 262 )] dt )K E}]+ =0 x or simply dEt[K R + 1 = 0 (34) 2R b where c = (1.5 - 3 + (1+4)) (35) b+R R Note that differentiation of R and b with respect to time t results in insignificant terms which are not included above.

-35Condition IIo Assume a section of pipe anchored throughout its length such that axial strains cannot occur, but otherwise free to undergo radial deformation. In order to determine the internal deformation within the pipe let r = R. Then, Equations (29) and (28) give pR = b2R2 (36) and pR2 F b2 (37) Because no strain is permitted in the axial direction, the axial stress must be R2 ax = 2ip b2 R2 (38) according to Poisson's ratio. In this instance Equation (30) becomes d tL dt + bB J- -I-2 -+ 2Li1 + Vx 0 K dt Edt b22 R2 dt bE-R R or pl- + 2 + x 22 = 0 dt[K E b2-R B2 R2 R2 or simply 4 + 1 i c21 + Vx = 0 (59) dtLK E b-R 2R - - 2 b2 where c2 -R (1 - 2i b2 (+ (40)

-36 - Condition III. Assume a section of pipe with numerous expansion joints or their equivalent, such that radial deformation may occur without the occurrence of axial stresses or strains. Let r = R in order that the inner deformation may be found. Then, Equations(29) and (28) give bR2 2 2 (41) b -Ra and - b 1 - (42) b -R R From the conditions stated ax = 0 (43) and Equation (30) becomes 1 dp + 2 gdp ( R 2 1 + b 2 R2 b 2 K dt E dt 2 2 dt 2_2 2 b -R R b-R R or F+2 2 ~ 2 -(1+~ J+ 2lj] v = rl + 2 2R t1 +b b + 11 x dtLK E bR R RI or simply +J1 R 3 + V (45 dtLK Eb-R X( where c3 2R (1- L + b2 (46) 3 b+R R2

-37Thus, to summarize, the continuity equation may be written +E CR -o x 0(47) dtLK E (b-R) + for which the parameter, c, in the three particular mounting conditions investigated above is given by the following: 22 ( 1 b+R z1.5 - 3[ + (+)) c2 R= - - - +b2 (1+ (4o) 2RC3 [ 2 (.)- * (46) b+R R - Halliwell,(24) following a different procedure, derived the same results at approximately the same time. For the limiting condition when b -- R and the pipe wall becomes thin, the three parameters reduce to simply the following: Cl = 2[ - (48) C2 = 2[l - i (49) 3 = 2. (50) The variation of the coefficient, c, is shown as a function of the ratio of the outside to inside pipe radii for various commonly encountered values of Poisson's ratio in Figure 5. Each of the three conditions of support is considered. By referring back to Figure 3 it is apparent that the pressure acting at cross section O is given in terms of the static head p = pg(H-z)

-38Poissons Ratio pL =.25 2 Condition II. /-Conditions I and m Poi son's Ratiol / =.30 Condition I 2 Condition m Condition I Pois on's Ratio -. 35 Condition 2I _'Condition II Condition IMI Condition I I 2 3 4 5 6 C Figure 5. Curves Defining Relationship Between Pipe Constraint Coefficient, c, and the Ratio of Outside to Inside Pipe Radii, b/R, for Various Conditions of Constraint and Different Values of Poisson's Ratio.

- 39Since dp = -p dx + ap = ap v + ap (51) dt x dt xt t x at it is possible to determine and' as follows: x= pg(Hx-Z x) + g(H-z)px (52) = pg(Ht-O) + g(H-z)p (53 However, ax = - sinG; Equation (52) thus becomes = pg(Hx++sin) + g(H-z)p (54) Integration of Equation (8) results in,p =Kln p + c' Subsequent partial differentiation of p with respect to x and t gives Px = K PX P and Pt = K pt Thus, one may write Px = px= Pg (Hx+sine) + g (H-z)p. P K K and 1 - pt = (Ht) + ( (H-z)Pt KPt p - K or, after gathering like terms Px = P2g(Hx+sine)(K-pg(H-z)) (55)

-40and Pt P g t(K-pg(H-z) (56) Equations(55) and (56) may be substituted into Equations(53) and (54) and after gathering like terms; at pg Ht 1 + pg (K-pg(H-z)) and ~a pg(Hx+sino) [ + g (K-pg(H,)) 6X (K-pg(H-z))] Thus, using the above expressions for 2 and a, one may rewrite Equation (51) as = pg [(Hx+sine)v + Ht + pg (K-pg(H-z)) (7) dt (K-pg(H-z)) Consequently, the continuity equation, Equation (47), may be written pg (Hx+sine)v + Ht] + R (9) [ i + v 0 + C 1 +r Kpg +=0 o:r simply Ht + v(Hx+sine) a x - (58) g where a (5) + E b-RI) and = [1+p1g(Hz)] (6o0)

-41The term a is an expression for the celerity of the pressure pulse through the pipe system when flowing full; the term 0 compensates for the change in the bulk modulus of elasticity with increasing pressure. For water the term 0 remains approximately unity for moderate pressures. For the range of pressure heads encountered in this study the magnitude of 0 will be assumed to be unity. Equation of Motion According to Newton's second law of motion, the net flux of momentum from the control segment and the time rate of change of momentum within the segment must be instantaneously equivalent to the resultant of the external forces acting on the segment. Consider again the control segment of a circular pipe flowing full shown in Figure 3. Recall that the area at cross section O is A and the area at cross section v is A + A dx. The gradient of the pipe is downward (negative gradient) at an angle 0. Flow occurs in the positive x-direction. The mass of the element is given by p(A + -1 HA dx) dx The acceleration in the positive x-direction is given by the total derivative of the velocity with respect to time dv av dx + av av av +-= v-+ dt ax at at ax at The product of the mass and the acceleration is pAv dx + pA dx+- pv-aA a- (dx2 + p - aA av (dx) PAV ~ ~2 x~ 2r+ A x+ t

-42or, after eliminating higher order terms, simply pAvvxdx + pAvtdx (61) The pressure force acting upon the cross section at Q is pgA(H-z) o (62) At cross section ( the pressure force is -[pgA(H-z) + (pgA(H-z))xd (63 ) An additional pressure force acting upon the expanding cross section of the pipe between ( and Q is [pg(H-z) + (pg(H-z))xdx Axdx (64) Fluid resistance to motion is interpreted as a boundary shear force and denoted by -T2tRdx, (65) where T is the fluid shear stress at the pipe wall. The body force due to gravity is pg sin A + (A t dx) dx (66) The summation of Equations(62) through (66) is the resultant of the forces acting upon the control segment in the x-direction: pgA(H-z) - pgA(H-z) - pgA(H)xdx - pgAsinedx - pg(H-z)Axdx + pg(H-z)Axdx + pg(Hx+sine)Ax(dx)2 - T2rtRdx + pgsinAdx + pg 2 Ex (dx)

-43.or upon the elimination of higher order terms - pgAHxdx - 2TrRdx. (67) The equation of motion may now be written by equating Equations(61) and (67), - pgAHxdx - 2TcRdx = pAvvxdx + pAvtdx (68) Now then, for dynamic equilibrium of the flow in the pipe, the stress-created shear force at the pipe wall must equal the so-called frictional head loss per unit length, L, that is T2nTR = R2 = g R2 AkL AL where Hf is the friction loss in terms of head loss. The Darcy-Weisbach expression for head loss per unit length resulting from friction is AHf = f jyj iAL 2R 2g Consequently, the shear stress is T = pf 8 Therefore, substituting Equation (69) into Equation (68) and dividing by pgAdx results in the relationship Hx + f2gD (v VX + Vt) = 0 70) The absolute value sign is introduced into the frictional term so that it has the proper sign depending on the direction of flow.

Characteristics Equations The partial differential expressions, Equations(58) and (70), derived in the previous section and representing transient liquid motion in a pipe flowing full, may now be rewritten as rl = vt + V Vx + gHx + fIL =0 (71) r2 = - avx + Ht+ vHx + vsine =0. (72) g These two equations form a set of simultaneous, quasi-linear, hyperbolic, partial differential equations of the first order in two independent and two dependent variables. Again the subscripts x and t indicate partial differentiation with respect to these independent variables. The expressions Fl and F2 may be combined linearly such that = l + nr2 (73) or 2 Q = vt + (v + n- ) vx + nHt + (g+nv)Hx + f vvj + nvsin' = 0 (74) Now, if v = v(x,t) and H = H(x,t). are solutions of Equations(71) and (72), then dv = dx + and dH = -H dx +- H ax at ax at or dv dx dH dx (75) dt vt + Vx dt and dt t + Hx d From inspection of Equation 74 it is readily apparent that it may be rewritten

d+ dH + vllv + nvsine = o (76) dt dt 2D provided dx v + na _ g+nv () Tt V + 9 n (77) dt g n or in other words provided dv n2 dt Vt + (v + g ) vx and dt= Ht + ( + v) Hx From Equation (77) one finds n2a2 = g2 or n = + g (78) - a By substituting Equation (78) into Equations(76) and (77), one has the pair of relations f2dt = dv + -- dH + 2 vlvldt + g vsinedt = (79) a 2D a C+ dt 1 (80) dx= v+a and dt = dv - g dH + f vlvjdt - a vsinedt = 0 (81) a 2D a -dt v1 (82)

-46To summarize what has been accomplished, Equations(79) and (81) are two separate total differential equations for which solutions for v and H may be determined along the families of characteristics curves occurring in the x-t plane. These curves are defined by the total differential expressions given by Equations(80) and (82), respectively. In other words, the characteristics curves represent the paths along which infinitesimally small velocity and pressurehead differences propagate. Equations (79) and (80) are applicable along a positively directed characteristicscurve, while Equations(81) and (82) are applicable along the negatively-directed characteristics. Finite Difference Solution Equations (79-82) may now be solved by means of a finite difference technique. If the step size of the increments used in the solution is small, one can solve by a first order approximation for which X1 f f(x)dx e f(xo)(xl-xo) xo otherwise, a second order approximation technique requiring iteration should be considered. A first order technique is used throughout this development. Consider the fact that for a set of two hyperbolic differential equations the characteristics curves at a point have real and distinct directions; therefore, the curves must intersect at the point. Let Pi be a point on the x-t plane as shown in Figure 6. Two characteristicscurves from the families of positively and negatively directed characteristics curves are shown passing through Pi

-47t CaE Figure 6. Definition Sketch Showing Intersection of C+ and C- Characteristics Curves in the x-t Plane.

The objective of this solution technique is to provide a means for obtaining values of velocity and head (pressure) at selected locations throughout the x-t plane. Figure 7 shows a portion of a rectangular grid in the x-t planeo The two pairs of characteristicsequations, Equations (79-82), representing transient motion in a pipe flowing full may be written in the following manner: dt - dx v+a C+ dv + g dH + f vvJ + vsinl dt = a 2D a dt - dx = 0 v-a C dv - dH + 2D vIvI - vsine dt = 0 When written in incremental form these equations become (tp-tcl) - (va) (xp-x) = 0 (83) (vp-vc) + (Hp —y) - + v|v + vsinO) (tp-t) = 0 (84) (tp-te) - (1) (xp-xe) = 0 (85) (vp-vC) - (Hp-HE) ag +2 vlvI g vsinG (tp-te) 0 (86) where the subscripts P. a, and E refer to locations on the x-t plane in Figure 7. The grid in this figure has specified incremental distance and time intervals, nAx and At. The points Po and Pn are boundary points. Note that time tL = ta = tM = tE = tR; also, note that distance Xp XM

-49-.ti+, X- - t I 1 I 1+1 i, Po Characteristics Curves are Shown Passing Through Grid Intersection Points, P, P, and t i-i ti R 4 txi-2 \ 1 Xi'; "'n-i X

-50Clearly, the exact location of Pi which is at a grid point is known with respect to x and t. However, the locations of a and E although known with respect to t, are unknown with respect to x and need to be so determined. The values of v and H at the points a and C need to be determined in order to determine vp and Hp. It is assumed that v and H are known at specific point (L,M,R, etc.) along the x-axis at the preceding times ti_1, ti_ 2, etc., either by having been initially given or previously calculated. Linear interpolation is used to determine v and H at a and E, thus vX-vL VM-VL VM-VL (87) x-xL xM-xL A= x and VR-VE VR-VM VR-VM (88) XR-XE XR-XM AX In order to determine va, vE, IH, and He one must note, first of all, that the two characteristics equations, namely, Equations (83) and (85), can be written xM - <ay = (v+a)aAt (89) and xE - xM = -(v-a)EAt (90) These equations describe C+ and C_, respectively, at G and E: Equations (87) and (88) may be respectively rewritten as

-51( aL) A- x = Xa - XL (91) VM-VLand vR-V' ) x = R- XE (92) Inspection of Figure 7 clearly reveals x - xL = ax - (xM-X ) or, following substitution of Equation (89), x - L = Ax - (v+a)oAt into which substitution of Equation (91) yields (VMIVL Ax x= AX (v+a)OAt VM- VL ~~ = vL + (VM-L)[ - (v+a) or by approximation V = VL + (VmVL)[l - (v+a)M A] (93) Figure 7 also reveals that XR - XE = Ax - (XE-xM) or, by substituting Equation (90), XR - xE = Ax + (v-a)E At, into which substitution of Equation (92) yields VE = yE + (vM-vEl +x (94)

The linear interpolation expressions used to determine HE and H. are x -HL _HHL _ M- HL XCe-xL XMXL AX and HRHE ~ HZR-H I XR-X XR-XM AX x The same general procedure used to determine v. and v. is followed to determine Ix and He, with the result that a = HL + (HM-HL)[1 - (V+a)M (95) and H H v-a) At (96) HE = HR + (HM-HR)[1 + (v-a)M ] (96) In summary, Equations (93)-(96), which define vi, vE, HI, and HE may be simply written Va = VM - (VM-VL)(V+a)M A (97) At ~ = HM - (HM-HL)(v+a)M A (98) At vE = vM + (VAVR)(v-a)M t (99) At HE = HM + (HM-HR)(V-a)M (100) Note that (v+a)a " (v+a)M and (v-a)E e (v-a)M are assumed to be entirely valid approximations provided the grid of points is closely spaced. Using the four equations for Vaz, iH, vE, and He, one can rewrite Equations (84) and (86) in preparation for determining vp and Hp:

-53(VpV ) + (Hp- H9(a + ( Vc iJ)At + v vasinOAt = 0 (101) ~(vp-v) (Hp-H9(gll + ( vve}L t - () v sineAt = 0 (102) To solve for vp and Hp, Equations(101) and (102) are first added, 2vp- (vHE I a-u(. + (H)M(V I +VI I vE I )Lt + (v) iM(-v )sinent 0 VP =2 (v,+v,) +'(H-He) f ( U)Iva I +v v )tl ( v- ()vIv,)sineGt (103) and then subtracted, VU + v+epaM (+ 2HH(aM + M(vlval-vE v. I| )At + ( (va+v,)sinoit = O E: a 1 (xH (f 1a (va vcl-vv )At -i P 2(V E-V )( + (11y+HE ) I v E )t - 2(vx+vE )sin it (104) Thus, using Equations(103) and (104) it is now possible to determine the velocity and pressure head, vp and Hp, at each interior point on the x-t grid. The evaluation process advances with time throughout the bounded x-region. It should be noted that the size of the increments Ax and At cannot be selected independently of each other. On the contrary, their size is integrally dependent. Since the finite difference solution scheme which has been outlined above converges to the exact characteristics solution as Ax approaches zero, one can conclude the following: (a) the values of v and H at any point P within the region of existence of the solution are determined solely by the initial values prescribed along

-54the segment of the x-axis subtended by the two characteristics passing through P, and (b) the two characteristics through P are themselves determined solely by the initial values on the segment. The segment of the x-axis determined by a and E is, thus, the domain of dependence of the point P as shown in Figure 6. Therefore, when LAx is arbitrarily selected, At is automatically limited to an interval no larger in magnitude than that given by the relationship At v ix In other words, the point P must lie within the region bounded by the intersecting characteristics curves through L and R and the subtended segment of the x-axis. Boundary Conditions The finite difference solution technique used with the characteristicsequations must now be extended to include the various boundary conditions which are to be encountered. Consider the situation at the right edge of Figure 7 in which only the positively directed characteristicscurve C+ is shown. One condition at point Pn is assumed to be given. This given condition may be either vp or Hp. In order to calculate the unknown condition, it is necessary to first calculate vC and H. E The given boundary condition can then be used to determine the remaining unknown condition. Equations (97) and (98) are used to determine E, and vi. If vp is the known boundary condition at point Pn, Equation (101) can be rearranged to give Hp, n

= - (vpvj(a ) - (vcv +( {) vcsinO]( )At. (105) On the other hand, if Hp is given, this same equation may be rearranged to provide vp at point n, = a -) -v v~~ ( vHfsin JAt. (106) v = va a (Hp-HaRM ~t2D val a [}M }(106)a At the left end of the x-t grid shown in Figure 7, the C_ characteristicsgovern the unknown boundary value. Again one begins by calculating v. and HE from Equations(99) and (100). When vp is the known boundary value at point P1, Equation (102) can be rearranged to provide Hp at P1 Hp = HE: + (vP-~v )() +(vIvEi - vEsin ](a) At (107) and when vp is the given boundary value at P1, Equation (102) is similarly rearranged to give vp = ve + (Hp-H)(a)M -e th (bar vaesinj at (108) Now then, Equations(105-108) provide the boundary values appropriate for use in conjunction with the general Equations (103) and (104). Attention must be given to the boundary conditions occurring at the junction between two pipes of different sizes and different materials. A typical portion of the x-t grid at such a junction is shown in Figure 8. In addition to the four boundary equations, Equations (105-108), already developed above, two other relationships prevail. These relationships are (1) the continuity-of-mass relationship, vIDI = vIiD2I, and (2) the energy-equilibrium relationship, HI=HII. Minor losses at the

-56Pipe I Pipe II t3'' I \I's,,,\ \, \ \ _'I,' / / II, /V 0 It I Ln-3 I,n-2 I n- Xn Figure 8. Definition Sketch Showing Relationship Between Space-Time Grids and Characteristics Curves at a Pipe Junction.

-57junction are ignored. Using the boundary equations one can derive an expression in vPI and HpI by adding Equations(105) and (106), vp + HpIV)Ma + HaM VGV M At - a sin M1A V~Ia IMI a;aMI A2D I - vsin gMI (109) Similarly, Equations (107) and (108) may be subtracted and combined to give Hp(,,) | MII At + vsinv G At = X (110) Now, substituting vPI = vpII DII and Hpi = HPII into Equation (109), 7 YP= (I Di)2 + HpII a( M and rewriting Equation (110), X = VPII - HPII( }Mi i one can eliminate vpII DI DI HPIIa[( a)MI(DL) a MI, ] Di -) X or 2 full, that is to say, flowing-with a free surface, is comparable in many ways to the previously discussed transient liquid motion in a pipe flowing

-58full. Analytic treatment of the free surface flow as presented below will more clearly identify the similarities as well as the principal differences between the two types of transient motion. Transient flow in a horizontal pipe having a free liquid surface is a function of two independent variables. These variables are the space and time variables, x and t, respectively; and are exactly comparable to the independent variables used previously with a pipe that is flowing full. Relative position in space is denoted by x as measured axially along the pipe. The gate valve is arbitrarily designated the point of zero reference. For pipe flow with a free surface the two dependent variables are the mean velocity of the free surface flow, u, and the depth of flow, z. In order to represent the variable velocities and depths, two quasi-linear, partial differential equations are developed and then transformed into four total differential equations —the characteristics equations of transient, open-channel flowo After being rewritten as finite difference equations in a manner comparable to the one used with the equations for the pipe flowing full, numerical methods are again introduced to enable evaluation by high-speed, digital computer. The governing boundary conditions are examined and interpreted analytically. The section concludes with a brief comparison of the similar analytic properties of transient, free-surface, pipe flow and transient liquid motion in a pipe flowing full. Equation of Continuity The net inflow of mass into a segment of free surface flow must equal the rate of mass increase taking place within the segment per unit

time. This statement sets forth the conservation of mass concept upon which the equation of continuity is based. Figure 9 shows an element of free-surface flow in a pipe. The element is bounded on the left and right by cross sections G and Q, respectively, For u equal to the mean velocity and A equal to the cross segmental area at cross section (, the mass influx through this segment is pAu. At cross section Q, flow leaves the element; therefore, the efflux of mass is given by -[pAu + (pAu)xdx]. The net increase of mass in the control element per unit of time is F(pAu) DAu - Au + dx] or -p(Au)xdx (112) The rate of mass increase occurring in the control segment per unit time is (A + (A + dx))pdxt or A[(A+ At + At + A)dx] or finally pAtdx (113) after higher order terms are eliminated.

-60~ 0' //// iI ///Iv/ //// z/ u \ ~~~~~~~~~~~~~~~I I /s E X~~~~~~~~~~~~~~~~~~I I I Surface < A /~~~~4 dx u z/;~~~ o s~-~~.;~s s rn Figure 9. Definition Sketch Illustrating Element of FreeSurface Flow in a Pipe. Transverse Cross-Segment is Shown On the Left and Longitudinal Flow Profile is Shown On the Right.

-61The continuity equation may now be set forth by equating expressions (112) and (113) above: - p(Au)xdX p(At)dx or puAx + pAux + pAt = 0. (114) Note that the small pressure changes associated with the changes in depth accompanying open-channel flow have no significant effect upon liquid density. Therefore, p is independent of both x and t The cross segmental area is a function of the flow depth, in other words, A = A(z), and thus, dA = Tdz where T is the surface width 2 zD-z and D is the inside pipe diameter. Moreover, A 6A 6z T 6z Ax = d - ax - =Tzx, (115) and by the same token At Tzt (116) Equation (114) may now be written as uzX + T U + zt = (117) which is the form of the continuity equation used in subsequent parts of the analysis. Equation of Motion The net flux of momentum from the control element and the time rate of change of momentum within the element must be instantaneously equivalent to the resultant of the external forces acting on the element.

-62The flux of momentum through cross section G in Figure 9 is actually an influx of momentum into the control element, namely -p(Au). At section Q the efflux of momentum is +p(Au2) + p dx. Therefore, the net flux of momentum from the control element is + p (A2) dx (118) ax The time rate of increase of momentum within the control element is [p{A + (A + dx t + (u + dx 7 dx t or simply P(Au)t dx (119) after all higher order terms have been eliminated. The net hydrostatic pressure force, as is clearly illustrated in Figure 9, is -pg(A + a dx)(yz dx) Once again higher order terms are eliminated to give -pgAzxdx. (120) The flow resistance force may be attributed to the shear stresses developed at the pipe wall. It is given by -TPwdx, (121) where Pw is the wetted perimeter and T the boundary shear stress.

By equating the terms depicting a change in momentum to the force terms, one obtains the expression + p(Au2)dx + p(Au)tdx - PgAzdx - TP dx or 2pAuuxdx + pu Axdx + pAutdx + puAtdx + pgAzxdx + TPwdx = 0 Division by pAdx reduces this expression to simply 2 Ax At T(122) uu + u A+ut +u - + gzx + - = 0 (122).uuxA PRH The variable RH represents the hydraulic radius. Because the shear stress can be defined as T = pf 8L (69) where f is again the pipe friction-factor noted previously, the flow resistance term may be written TPWdx = f pululPwdx 8 after which, division by pAdx leaves 1T -fuJ u (123) pRH 8RH Equation 122 can now be rearranged to include this expression, 2T T + + o (124) gzx +u z ~uLzt + 2uux + Ut+ fuul =( ) W + u 7 z ~ ~ ~ ~ 8R. A o P T

-64Write the continuity equation (Equation (117)) with each term multiplied A by u2 T Zx + u T zt + uux = 0 (125) A A Subtraction of Equation (125) from Equation (124) reduces the momentum equation to the form gZx + uu +ut+ fJ 0 (126) 8R which is used in the subsequent parts of this analysis. Characteristics Equations The set of simultaneous, quasi-linear, hyperbolic, partial differential equations which describes transient liquid movement in a pipe flowing partially full, is rewritten below F =uzx + zt +A ux = 0 (127) r4 = gZx + uux ++ ut + ful = 0 (128) Linear combination of the expressions F3 and F4 produces A = r3 + mr4 (129) or A = mut + A + mu) U+ t + (u+mg)zx + mfulu (130) The dependent variables u and z are functions of the space and time variables x and t by assumption. As a result, if u and z are, in fact, solutions of Equations (127) and (128), then

du = dx + dt ax at or du au dx + au dt ax dt at' and dz = dx + dt or dz az dx z dt -x dt at Therefore, Equation (130) can be written du dz f A = m dt m umu| = 0 (131) provided A + mu dx _ T m(u+mg) 2) dt m m(13 Equation (132) is valid if du ut + + u ux(133) dt mT and d = Zt + (u+mg) Zx (1.34) dt If Equation (132) is now solved for m, one finds that m= + (135)

-66 - By substituting Equation (135) into Equations(131) and (132), two pairs of linear relations are determined, namely, |0 = du: + dz + 8 ululdt = Adt (136) C dt +1 dt x= 7 (137) dx u +~ and:0 = Idu + dz R ululdt = Adt (138) C_ (dt = 1 (139) which are total differential equations. These four equations are the characteristics equations governing the transient velocities and depths in a horizontal circular conduit flowing partially full. In retrospect, Equation (136) simply states that there is a constant, velocity-depth relationship for a point whose motion in the x t-plane is characterized by Equation (137). Likewise, a similar relationship exists between Equation (138) and (139). In other words, Equations (137) and (139) define the families of solution curves along which Equations(136) and (138) are valid. Finite Difference Solution The technique devised for finite difference solution of the characteristics equations depicting free-surface flow in a pipe is analogous to the technique described previously for solution of the characteristics equations representing liquid motion in a pipe flowing full. The size of

the stepping increment used in the solution is made small enough to insure that first order approximation is adequate. The fact that any point on the x,t-plane is an intersection for a specific positive curve and a specific negative curve belonging to the respective families of characteristics curves associated with partial differential equations of the hyperbolic type is the basis for the solution technique. Figure 6 shows the intersection of a positive characteristics curve with a negative characteristics curve at Pi. Given the necessary initial and boundary conditions, it is the purpose of the solution technique to permit evaluation of mean velocity, u, and depth, z, at point Pi ~ In more general terms it must permit evaluation of u and z at any network of selected grid points on the x,t-plane, such as the rectangular grid illustrated in Figure 7. Characteristics Equations, Equations (136)-(139), may be rewritten as follows: dx dt - = 0 C( -+ (140) A f dz + jdu/ + T{u~udt= 0 dx dt - d 0 C u T (141) Adz -W' du -1T 8f uJuldt = 0. gT gT 81H If written in finite difference form these expressions become (tp-ta) - (xp-x9)/(u + X-) = 0 (142) T

-68(zp-zc) + (up-ua) + (f g)4# uu)'(tp-ta) = o (143) (tp-tc) - (Xp-Xc)/(u -)E = 0 (144) (Zpz ) X A (UpU) - (XA)( 8 uH UuuI) C(tP-te) = 0 (145) where the subscripts P, a, and E refer to locations on the x,t-plane in Figure 7. Values of u and z are assumed known, i.e., given initially or computed previously at all the grid points along the x-axis at prior to time ti_l o Although the points on the grid are separated by specified space and time intervals, the locations of a and e while known with respect to time are unknown with respect to x o These locations must be determined, If the solution is to advance with respect to time so that up and zp can be determined at time ti, the appropriate values of u and z at a and c must be found firsto This may be done using linear interpolation: uac-uL UM-UL uM-uL (UL146) xry-XL XM-XL Ax UR-Us uR-UM UR-UM R - - R _ R M(147) XR -Xs XR-XM 1x Now, by recognizing that xp = xM, and tM = ta = tc, Equations (142) and (144) can be written XM - x = (u + (tp-t) = (u + ) At (148) and X - xM = -l-( l)(tp-t~) = -(u _- At, (149)

which represent the increment slopes of C+ and C- at Qd and c respectively. Equations (146) and (147) may be rewritten: (u -uL) zx = x - XL = Ax - (xM-xa) (150) UR-UM} Ax = xR - x. = Alx - (x -xM) (151) Substitution of Equation (148) into Equation (150) and Equation (149) into Equation (151) gives Equations (152) and (153), respectively: "a - uL = (uM)uL ( u+ TJ j (152) uR- uE (uR-uM)[l+ -(U + ] (153) By using linear interpolation to determine za and z, and then by reasoning in an analogous manner, the equations of depth complementing Equations(152) and (153) are found: Z - zL = (ZM-ZL)[l l- u + A (154) ZR - Z = (zR-zM)[l + (u- (155) Hence, the desired values of u and z at Qd and E become known: = UM + (ULUM) + (156) Za = ZM + (ZL-ZM)(U A x (157) uE = uM + -u)u (-(158) zc = ZM + (zH -zR) gu -1 x (159) ZM ~~WR' ~TJM A~X

-70Note that it is assumed that (u + ) ( + )M and (u (u - M which is quite reasonable for a closely spaced rectangular grido Mean velocity and depth at point Pi can now be computed. Equations (143) and (145) are rewritten (ZpZ + (u- ) + T uiu I At = 0 (160) (Zp-Zc)- ug(u-U) - )8 UU) At=, (161) gT - TJM 8R then added to find zp, 2zp = (Zz + (u-u)( )M - ( )Mcul oIuEIue( I) t Z =- (Z_~+ZE ) + ( fu-ue ) )M ( H)MI ) Mt (162) and subtracted to find up 2up = u + uE + (Za-zc) A ) 8RH (ucuI +uEu ) At up -= 2 (U+UE) + 2 (Z ZE +u- I +uE ) IMt (163) Clearly, it is now possible to determine the velocity and depth of the free surface flow at any interior point on the x-axis not only at time ti but at all times after ti simply by repetitively using Equations (156)(159) together with Equations (162) and (163). As previously pointed out while developing the finite difference solution for the pipe flowing full, the size of the increments ax and At are interdependent. The same is true of the Ax and At increments

-71of free-surface pipe flow. With reference to Figure 6, the segment of the x-axis bounded on the left at a and on the right by c is the domain of dependence of the point Pi. Thus, whenever nAx is arbitrarily selected, At is limited to an increment of magnitude no larger than At < Another way of expressing this is to specify that any point P at which up and zp are to be computed must lie within the domain of dependence bounded by the characteristics curves, C+ and C-, passing through L and R, respectively. Figure 10 graphically illustrates the significance of this requirement. Boundary Conditions At the right edge of Figure 7 where x = xn only one characteristicscurve, C+, is shown. This curve passes through the boundary point Pn. One value, either up or zp is assumed to be given at this point. By first calculating ug and z. from Equations(156) and (157), the given boundary value can then be used to determine the unknown value. Suppose that up is the known value. Then Equation (160) can be rewritten to give zp, = - (up-ua)( - uli At. (164) If Zp is the known boundary condition, then Equation (160) can be rewritten uP = Ua - (Zp-Za( ) (g-uMui) At. UN (165) A~ ~uu ^ ~

-72t /C I+ ~ t+9 lmatx. C+ Ax Figure 10. Definition Sketch Illustrating Domain of Dependence Governed by Characteristics Passing Through Grid Points L and R. Point P Must Lie Within Region Bounded by Line Segment LR and Characteristics Curves C+ and C-.

-73At the left edge of Figure 7 where only one characteristics curve, C-, passes through PO the analogous boundary condition equations can be found using Equations(158) and (159) with Equation (161), first for up known and then for zp known: Zp = ZE + (Up-ue) + ( ului t)M (166) up = uE + (zEp-ZE) M (8R-H (i ul t. (167) Before concluding this chapter, it is certainly worth noting the very pronounced similarity between the equations of transient motion for the pipe flowing full and the equations of transient motion for a horizontal pipe with free surface flow. The latter is analogous to unsteady, open-channel flow. By comparing Equation (72) with Equation (127) and Equation (71) with Equation (128), one becomes clearly aware of this similarity. The similarity is further evidenced by comparing the respective sets of characteristics equations, namely, Equations (79)-(82) and Equations (136)-(139). Note that in the closed pipe the celerity of the internal pressure wave is 1 K E b-R On the other hand in a pipe undergoing flow with a free surface, the celerity of the infinitesimal surface wave —a pressure wave in the external sense —is given by the expression. In each instance the celerity is affected by the particular properites of the liquid. It is also affected by the particular liquid-solid and/or liquid-gas interface conditions.

-74When the pipe is flowing full the celerity is a function of the pipe constraint and elasticity as well as the liquid properties, whereas, when the pipe is flowing partially full the celerity is a function of the shapegeometry of the flow and of the liquid properties~ Nevertheless, both expressions are dimensionally similar. When one recognizes that transient flow conditions in a pipe flowing full on the one hand and in a pipe flowing with a free surface on the other hand are basically very similar hydrodynamic phenomena, then the similarity of the respective analytic treatments and mathematical results is not surprising. However, recognition of the hydrodynamic similarity has been delayed perhaps, by the traditional division of hydrodynamics into "pipe-flow" problems and "open-channel" problems.

CHAPTER IV COMPUTER SIMULATION Column separation precipitated by transient flow in a pipe was interpreted theoretically in Chapter II. Its two principal regimes of liquid motion were the subjects of analysis which included the development of numerical processes for their evaluation in Chapter III. In this chapter the theoretical concepts of transient-flow, column separation phenomenon, and the mathematical treatment of its two regimes of liquid motion are used to formulate a working mathematical model for simulating the phenomenon by digital computer. The specific boundary criteria applicable to the model are discussed in detail. The sequence of operations followed in the model is described and presented by means of a flow chart. The structure of the computer program is discussed briefly. The chapter concludes with several general remarks about the operational characteristics of the simulation model. Flow System The flow system envisioned for computer simulation of the transient flow, column-separation phenomenon is physically similar to the example system considered in Chapter II (See Figure 2). A constanthead reservoir and a quick-closing gate valve are assumed to be connected by a long length of pipe. The reservoir is open to atmospheric pressure. The gate valve is capable of instantaneous closure. Its elevation, however, is variable with respect to that of the reservoir. The long length of pipe connecting the reservoir to the valve is composed of two (or more) pipe segments, each of which may be of unique diameter, -75

-76wall-thickness, length, composition, and may have different frictionloss properties as well as support fixity. The pipe segment connected to the gate valve is the one in which column separation is anticipated. This segment remains horizontal at all times; the other segments are free to assume any inclined position. Because axial distance along the pipe system is measured from the valve, this segment is denoted as segment No. 1. Figure 11 shows a schematic drawing of the flow system. Initially, the gate valve is assumed to be fully open, thereby permitting steady flow throughout the system. At some time prior to the reference or starting time, to, the velocities and pressure heads on the x,t-plane at the particular points, Xk,i, are determined from steady-state conditions. These quantities are the initial conditions used to begin simulation of the transient-flow, column-separation phenomenon in the computer model. The notation, k, is an integer subscript denoting the particular pipe segment. Boundary Conditions The purpose of this discussion is to identify and set forth the controlling boundary conditions which delimit a unique, well-defined solution of the mathematical model. Without an explicitly stated set of boundary conditions, accurate computer simulation of the transientflow, column-separation phenomenon observed in a prototype system would be impossibleo Prior to closure of the gate valve two boundary conditions govern steady-state flow in the system: pressure head at the entrance to the pipe and the velocity of flow through the gate valve. Since the constant head reservoir is unable to sustain a pressure other than its

-77Segment No.2 Segment No.l 1 W. S. To Sump Figure 11. Schematic Representation of Laboratory Flow System Used in Experimental Investigation.

-78own static pressure head, Hk is known and constant throughout time. The velocity through the gate valve is equivalent to simply the discharge divided by the area of the cross section at the valve; therefore, vl 0 is known, also. Hk,n and vl 0 are "given" boundary conditions which are used with Equations(105) and (108) in the numerical evaluation process to determine vkn and Hi 0 Transient-flow conditions are initiated by closure of the gate valve. This occurs at time to. Therefore, vl10 is set to zero at time to and remains zero until column separation takes place. However, the pressure head, Hk,n, at the reservoir remains unchanged and continues to be the second boundary condition needed to define this phase of the flow. When the pressure head (H1,0) at the gate valve falls to the vapor pressure of the water, column separation takes place. At the instant of column separation, new boundary conditions must be defined for the pipe flowing full and another set of boundary conditions introduced to describe conditions governing the free-surface flow in the pipe. For the pipe flowing full the pressure head at the reservoir, Hkn, remains unchanged. However, the boundary condition which formerly governed the condition of flow at the gate valve is now located at the moving interface separating the full flow regime from the free-surface flow regime within the pipe. This boundary condition is H1,0 set equal to the water-vapor pressure head where the subscript zero now refers temporarily to the position of the interface. Thus, Hk,n and Hi,0 are used with Equations(108) and (106) to determine vkn and vl 0 at the boundaries of the full-flow system.

- 79One of the conditions governing the free-surface flow occurring in pipe segment No. 1 is the boundary condition at the gate valve. Because the gate valve is closed there can be no flow (with a free-surface or otherwise) through the valve; thus, the boundary condition is simply that the velocity u0 equal zero. However, a second boundary condition, which suitably pertains to the condition of flow at the interface between the two regimes of liquid motion is not so easily reconciled. A condition is needed prescribing either the velocity of flow or the depth of water at the leading edge of the free surface interface. But because the forefront of the cavity is itself moving with time, this task is formidable. Repeated laboratory observations of column separation revealed that the leading edge of the cavity appeared to separate from the topmost, inner surface of the pipe wall in a manner which remained basically unchanged regardless of the prior velocity of flow. In fact, the foremost edge of the cavity appeared to maintain a profile of more-orless constant parabolic shape (See Figure 1). Although the mechanism of liquid separation from the pipe wall is not investigated in this study, local surface tension is thought to be the principal factor governing its behavior. The mechanism of liquid separation from the pipe wall, while fundamentally a three-dimensional flow process, can be adequately treated as a one-dimensional flow process provided the pipe diameter is not too small. Thus, if the foremost tip of the advancing vapor cavity is assumed to actually have a parabolic profile, a means is at hand by which the boundary condition, Zm, can be prescribed.

-80In the simulation model the shape of the advancing tip of the vapor cavity is assumed to be analytically represented by the similarly shaped void subtended by a parabolic cylinder which intersects at right angles with a circular cylinder, i.e., the pipe, as shown in Figure 12. The equation of the parabola on the x, z-plane is (z - 2R(1-0))2 = 4xp(-OR), wherein the focal distance, cb, is OR; the latus rectum, ad, is 4OR; and the vertex, located on the Z axis,,is offset from the X axis by the distance 2R(1-0). The horizontal length of the subtended void as measured from the vertex of the parabolic cylinder is the variable, Xp, while 0 is a preselected decimal parameter. The depth, z, at any point on the surface of intersection depicting the free-surface profile at the forefront of the cavity is given by z = 2R(1-0) + Rx. (168) The hypothesis advanced in Chapter II asserts that the volume rate-of-growth of the vapor cavity is equivalent to the full-flow discharge that occurs in the pipe immediately ahead of the cavity. Thus, since vl 0 can be found by computation and since the pipe diameter is known, the equivalent volume rate-of-growth of the cavity can be determinedo The subtended parabolic void representing the forefront of the cavity takes form at the gate valve, that is, at x0. It expands in volume and length with time. As soon as its volume is sufficient to insure an overall length greater than Ax, thus causing it to extend

<f Parabolic Cylinder Trace OR OR _ 2R(I-8) Circular Pipe Cylinder Figure 12. Definition Sketch Showing a Parabolic Cylinder Intersecting a Circular Cylinder. The Shape of the Common Volume Resulting from this Intersection is Considered Representative of the Shape of the Void Typically Encountered at the Leading Edge of the Vapor Cavity.

-82beyond x1, the value of z0 and z1 can be determined from Equation (168). As previously stated uO equals zero. The velocity u1 is arbitrarily set equal to vl0. which although an approximation must be very nearly correct. Thus, the initial velocity and depth of the embryonic, free-surface flow at xo and x1 are now known at some time ti on the x,t-planeo These values serve as "tgiven conditions" with which the characteristics evaluation process can be initiated to compute the changing values of velocity and depth at subsequent intervals in time. Meanwhile, of course, the actual vapor cavity is continuing to lengthen and increase in total volume. At x0 the boundary condition u0 = 0 remains unchanged. Although the characteristics evaluation process can be used to simulate the free-surface depths and flow velocities, the boundary depth, Zm, farthest from the gate valve (in other words, the depth at the grid point nearest the cavity tip) must be determined. This boundary depth is computed recursively with time by means of the parabolic-void approximation. However, it is worth noting that this approximation affects only the tip-most portion of the simulated vapor cavity, chiefly that portion of the cavity extending beyond the farthest, free-surface grid point on the x axis. The parabolic void approximation is used to approximate the depth during both the growth and collapse phase of the cavity simulation process. Thus, with the exception of some slight distortion which perhaps may be introduced by the boundary condition imposed at the forefront of the vapor cavity, a well-defined technique for mathematically simulating the column separation accompanying transient flow in pipes is available.

-83Sequence of Operations The model employed to mathematically simulate the phenomenon of transient flow in a pipe and the accompanying phenomenon of liquid column separation consists of a multitude of individual operations performed in a specifically prescribed, logical sequence. The operations may be grouped into general categories which are related to particular functions. These functional categories are listed sequentially: (a) Introduction, predefinition, and initialization of specific parameters, constants, and control factors used in the simulation modelo (b) Establishment of the theoretical flow system including initialization of the pre-transient regimen for the pipe system (pipe flowing full)o (c) Delineation of the boundary conditions and operating controls determining the transient flow phenomenon in the pipe system. (d) Successive calculation of the transient velocities and depths throughout the system with respect to time while the pipe is flowing full. The method of characteristics is used to carry out these calculations. Special means are employed to apply the method when the increment Ax equals the overall length of a pipe segment. (e) Determination of the occurrence of column separation by means of test conditions. Certain control parameters are set up in order to integrate an additional sequence of operations to be followed during periods of separationo

(f) Predefinition of functions and computation of initial values and boundary conditions governing free-surface flow. (g) Computation of transient free-surface velocities and depths associated with the growth and collapse of the vapor cavityo The computations are carried out by the method of characteristics for free-surface flow concurrently with the calculation for the pipe flowing full. (h) Reinitialization of both major and minor control parameters and boundary conditions following collapse of the vapor cavity. An abbreviated schematic flow diagram depicting the sequence of operations followed in the simulation process is presented in Figure 13. Much of the mathematical detail has been eliminated from the diagram in order to emphasize the logic of the simulation process. Digital Computer Program The digital computer program used to transmit the statement of the model simulating the transient-flow, column-separation phenomenon to the computing machine follows the skeletal structure& outlined in the flow diagram. The actual computer program is composed of a main program and several subprograms, which when integrated make up the whole program. The main program performs the functions outlined in the flow diagram. Consequently, it is structured similarly. Eleven subprograms or subroutines were written to perform various operations required by the main program. These subroutines are called upon as neededo In addition, six internal subroutines are defined and embedded in the main program.

DFFINE INTEGER,; \ /\ < READ CONSTAMTS READ DIMENSIONAL READ PIPE r READ START \ DRFSTATEMENT LABEL, DIMENSIOMS /MATRICES IN AND PHYSICAL DATA DESCRIBING FRICTION,SURGE,AND PROGRAM AND'FORMAT OF MATRIX PROGRAM DATA DESCRIBING ANALYTIC FLOW CHARACTERISTICS CONTROL VARIABLEIAES COMMON FLOW SYSTEM DATA DATA PART I COMPUTE STATEA, t STATEB COMPUTE PRINT PRINT PRINT S COMPUTE CP(1) + CONSTANTS DIMENSIONAL P FRICTION, PROGRAM MOFSEP, -- JNI AND PHYSICAL DATA AND CONOL J, MARK, ~c~)~~AM COIRO CJAR CP(2) DATA DESCRIB ING DATA VEEE DESCRIBIG ANTIC CS DATA cU FLOW FLOW SYSTEM STAMPTEA STATEB t DE-DELTT(2) PART I SEC. A "I t~~~~~~~~~~_ t IS DEL9T(1) t I __NTFOMA N- -O J —1 J=1 FCR PRESSURE ISE~~~~~~ -IS~~ ~ -~DDELT'(2? HEAD, N>35? |>2? I>2? VELOCITY, N=N+l J=J+1 J=J+I ELEVATION, f f (1) f AND INCLINATION |PRINT LIST COMPUTE ZW 1 COMPUTE OF REYNOLDS | COMPUTE REYNO A(J),DEIX(J)U AOG(J), I NUMBERS AND Ivw(J), GOA(J),ARD rc FRICTION HLOS(J),AND MU(J) I COEFFICIENTS DLI()v COMPUTE ALPHA(N) I I I EL(N) NN(1)+MN t COMPUTE r ~~~~~~~~~~~~COMPUTE MARS ALPHA(N)AI~y(B AIMD IS N<O? I I I Y /I S \ I I I COMPUTE I I b EL(N) COMPUTE IS ~~~~~~ ~ ~ ~~~COMPUTE ALP HA (N) _|N=N-1 I. I r~ MARSKOC3 I~ALPHA(N) II IAND COMPUTE I * r I t kf~~~~~~~ -DELX(2) AND AND t EL(N) ALPA(N) IARS< t o/|EL(N) COMISUTE f HABS 1 IS | | ) hN (1)? N~~~~~~~~~ARS<LOC2? I=+ t COMPUTE N, PRINT I; OPT EIP (1, I) H (1.- 1) I H(, ),V(1' I), AND LN=E I T \ J I V11,I),ADD THETA(1,I) THEM (1, I) Figure 13. An Abridged Flow Diagram Setting Forth the Sequence of Operations Used to Simulate Transient Pipe Flow and the Accompanying Column Separation.

PART I - SEC. PART I - SC. C I. I > U N ( 2 )? |AOGG0 HP(2,NA(2)) | |2JN? \?| f ft COWU`TE COMPUTE Z t~,~ ~ ~~ 4 1 1 1( f ~~~~~~~~VP(l I), PRINT I, COMPUTE N., TOFOFF, H(2, I), Es(2, I),. AND SBITOF V(2,I),AND H(2,I,. TA(2,I) V(2,I),AmD THEIA(2,I) ITTIME OF GATE VALVE CLOSURE VP(V,O), PSI COMPUTE L b w [ c~ I @ ~ t C MN(J)<B? ABD M>MOF~~~~~~~~~~~~~~~~~~~~~~~~~I~OSEP ~IPII VAL, HAL, COPUEHP(lO)-ELP(lO0) O PST _PARVEP, I - SEC. PSI is VRE9 COMPUTE is t ~~~~~~~HP(J10) r?|t | MPUEC H t | Cf A~~~~~~~~~ CAP IS (J(JOIC, MNJ-) fS J>1? ~~CHI, HVAPCR+ELP(J,O) -V J1M'I~~HPJO A+ FJ-1)EP )' PSI, CS ~~PA~Figure 3. (Cont' d) I ~ ~~~~ COMPUTE VL, t =OMPumI t COMPUTE HL, Vml INVEP, EEPP'VP (J,I) IS VR,HR EEP fI+ f f COPT H 1 0 EP f t ~~~~~HP (J,I) < HVAPOR? is ca~~~~~~~pv~~~~~~~ r HP(J,~~~~~~~~t CMPT (,1)< MrCR IS J>1? CH~CI, +L(J, I) AND HP (j, I)~~~~~~~VPJII)H Figure 13. (Cont'd)

COMPUTE PSI t' COMPUTE ~ ~ ~ OMUT HP(j, I l) IPR:NT colI' I! PART I -SEC. E COMPUTE VEEE REGARDING COMPUTE T RI~o 9 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ IS J<2~~~~~~~~~~~~~~~~~EOCTE COMPUTE H, t D IN..I I t I HHLD, JE 2, MN1(2)D <1 t SYSTE I ~ co~ ~ If /> -,- | < i f t r lL I-O u ~~~~~PRINT' TIMEMFS AND M<J J CO < T| f m<mOF+ IS: ~~~~OF FLOW*I <(ICPFTO)> I>MN(J)? 9 ~~REVERSAL AT INCp COM PUTE I 14 J H(J' I) COMPTY [S~~~~~~~ I ~~~~MIN S~~~~~~~~~~~~~~~~~~~~I MfJ Qm/ OUTINE UTE JJ Q VELOC f VELOCITIESI (v -cl COPUT T SUBOUIN PRTRUL AND PRESSE YUj I E AND PRESSN-PRESRE HEADS H /EADS \ / \, 1E~~~~~~~~~~~~~~~TROUGHOUT THROUGHOUTY SYSSYS_.. X HHOLD, Jl 0 RIT =1 PIN OHHOLD D f f (1,O)>10? HEADS HEADS t f f 03~~~~~~~~~ \ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t FigureU~ 13.I~ ( I~ Cont'OE d )GOUT l lr~, I) I,'I' 0MPUTPT]SE SYTE Figure 13. (Cont'd)

PAT _ sI COMPUTE PRINT TIM INTERIMO) LE~~g~FA FUNCTIONS: t. (1.),U'(2) C1PT,KKK ITOFSEP AND (CYCLE TOPAPVMISW=CCP, QACrSQA~OGT, CIPI,KI Ki r H' —P,(o, 0) Ks, s PoRs, T —-Ip c(Oi I ool - \~~r (1 0) H(1,0) CIFOFCOLUMNJ is BUlBVOLIIXlBC pSS RGEQ 3-1,0),j SEPARAICION MIIC, LAMBDA,?,HC DELTC, DELXC,.......COUNT,YY,JHK Z(K)=D(1) PRINT COMMENI COMPUTE MANTD DELXC,KK OFNTID Is KK.-6 1~~~~~~~~~~~~ I (K9 IPP6 P CO l RINTT VALUES COMPUTE SPIEK, f |_y t COMPUTE MAAT COME MUC CF VAPOR I PRK,RK, C PUTZP) K( t DELXCKK ~~~~PRESSURE, FVA,FVB, IEOA N ImaIEMENT S UZZ >I,?J f ~~~~~~~~~~~~~~~~~~SI2Z,ANDK=+ -~ S KKa3 A KD K61 CC|FI~ I f ~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~COMPUTE UP(K ) (~ PAT II - SEC. B -, I CO 8 ICI1 E, t r11raa tS~ f t WI OF DEPTH AND f f Co m AN KKO 1 ND K<m K Ip COMPUTE Jr VELOCITY IN OPT M oam AD? KO (Mff~l) JuQ,1VI VAPOR CAVITIY Z(I MN' U(K) Z(K) II~(KVOID,.STAT3EE Fi e 1 (COU t t K —O 1?R*R!~,~,, 1COMPUTE UL, t f ZLI UM, Z K I1 f COMPUTE URZR CO~~~~~~~~~~~~~~~~CMPUT?YY COMPUTE QPUTVU0lpl gc:[oD IS aOI~P<OP IS Yr>Ot OUL:FLO. UAL, ZAL,UEP, COMPUTE QPml COMPUTE ZEP,UALLY, UIP, EP, U(g),Z(E, UEPPY, CFFQ, ['~: vZ2,T:(K+a), l zP(K*l) Figure 13. (Cont'd)

COMPUTE COMPUTE STATEC, Z(MK),KK,PMK W 9 ~~~~~~~)it f t I IK=K+1 t 7717 0 LO'? SATE='f C| COMPUTE f (K)<Z(K-1 COMPUTE PART II - SEC. C COMPUTE COMPUTE IS f COMPUTE NX, OMPUTE PRN COMPUTE 1)1 OUTFL0 INFLOW, (I+JHK) JHK,YY, XX MOCOL, COMPUTA- MOFSEP,JJ KKPMK iTo)~so' ( —'-~ _MOFCOLO O —-' —1NO CYCLE AT VP(-,o),VOID, TOCOL WHICH VAPOR EEE, MARK ICAVITY COLLAPont'SE PRINT PRINT PRTINT PROGRAM PROGRAM PROGRAM PROGRAM CONTROL 8 t CONTTOL $1 4x ERROR''1 CONTROL PETYINPORMAT I OP PR STATEMENT STATEMENT STATEMENT STATEMENTS I1CUTOFF"'MAXUOM' REGARDING XE REGARDING Figure 15. (Cont'd)

-90 - Consider the fact that in the particular prototype flow system being simulated, the elapsed time, starting with the moment transient flow is initiated, continuing through the ensuing formation of the first vapor cavity, and concluding with its collapse, is approximately two or three seconds. If the digital computer could simulate this portion of the phenomenon in less than actual time, it would be performing in socalled "fast time." If it required exactly the same number of seconds it would be performing the simulation in "real time." If the computer required longer than the actual time to complete the simulation run, it would be performing in "slow time." Because the ratio of the actual time to that required to simulate the separation phenomenon was approximately 1 to 240, the digital computer simulated column separation in slow time. The computer program (the main program and special subroutines) is written in the Michigan Algorithm Decoder language commonly known as MAD. This language is an enriched Algol base compiler language which has been designed for implementation on IBM 7090 series computing systems having 32K words of main memory. Certain of the operational characteristics of the program experienced for the particular system configuration investigated are tabulated below: (a) Required locations in main memory Main program —8,139 words Special subroutines —1,639 words Complete program —17,390 words (b) Total comilation time of complete program —127 seconds Complete listings of the main program and of the subroutines are given in Appendices I and II, respectively.

CHAPTER V EXPERIMENTAL APPARATUS AND LABORATORY INVESTIGATION Experimental investigation of the column separation phenomenon comprises the second major phase of this study. It serves to complement the theoretical analysis which culminated in the technique for computer simulation of column separation discussed in Chapter IV. Experimental investigation was carried out in a specially constructed pipe system located in the G. G. Brown Fluids Engineering Laboratory on the North Campus at The University of Michigan. This chapter describes the pipe system and the special instrumentation and apparatus used in the laboratory investigation. It also treats the determination of the resistance-to-flow characteristics of the pipe system as well as the calibration of the instrumentation used to measure transient pressure and depth-of-flow. The chapter concludes with a summary of the procedures followed in conducting the experimental measurements. General Description of Pipe System The laboratory flow system was comprised of four principal elements: a constant-head weir box; approximately 439 feet of heavy-duty, one-inch inside diameter copper pipe; approximately 8.5 feet of one-inch inside diameter plexiglas (cast acrylic resin) pipe; and a solenoidoperated, quick-closing gate value. The weir box was connected to the gate valve by means of the copper pipe and the plexiglas pipe. The plexiglas pipe served as the test section in which the anticipated column separation could be observed. It was therefore connected directly to the -91

-92gate valve. Water was supplied to the weir box from the laboratory supply and returned to a sump by means of a copper pipe and a heavy duty hose, Siphon action was used to maintain flow through the system. A detailed sketch of the flow system is presented in Figure 14The constant-head, weir box was designed to minimize the fluctuations in head caused by changes in the discharge in the pipe system. Two sharp-crested, horizontal, overflow weirs were provided such that if a maximum flow velocity of 3 ft./sec. in the system were suddenly reduced to zero, the maximum increase in head in the box would be less than.011 fto A manifold was used to introduce water into the weir box with minimum turbulence, The copper pipe was composed of six segments carefully joined together to form a continuous conduit. A short segment only 1.25 feet in length and integrally attached to the constant-head weir box, serves as the inlet, The next four segments, each approximately 100 feet in length, were joined by soldered sleeve connections. To conserve space these segments were loosely coiled in a helix 5 feet in diameter and supported by a finger-like wooden frame mounted vertically upon a mobile hexagonal platform. The pipe was not constrained by the frame, therefore the coils could freely adjust to imposed strains. Figure 15 shows the constant-head weir box and the coiled pipe on the supporting frame. The final segment of copper pipe extended approximately 37.5 feet from the top of the coil to its junction with the plexiglas pipe. The properties of the various segments of copper pipe used in the system are listed in the table below:

LL(f) 8.77 LL(2) 439.19 LLoc I 2 4.021.00, 39.00 118.00 LOC 2= 444.00 LEL 22.0-18 HEL= 44-120. 34.719, 27.980 LOCI LL(I) MEL, = 26.328030 WSEL = 22.548 S014oi HEL Pr~~~Pexiglas pip 0 330 Pirmee PreSSUre LOc2 RigTransducer Piezometr Ring LL(2) Cole LOC3Copper Pipe J WSEL W. S. Resenteo Piezornefer LEL neservoir Ring Throttle Valve Figure 14. eaedc Detile Scematic ofLboa Laortry Fjo-w Systera

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..........~~~~~~~~~ I:::::...........'........ v_:::::::i-:: it:i-iiiiiii iiiii 0 ii s By,iii: E ii: iiii-iii:~~~~~~~~~~~~~~~~ ~ iaL-i~~~~~~~~~~~~~~~~~~~i~~~~~~ii~~~~iT'ii~~~~~~~~i~~~~:~~~~~~~:L~~~~~f:~ ~ ~ ~~;.~i,,~~....:.. Figure 15. View of Constant-Head Weir Box and Loosely Coiled Copper Pipe on Wooden Frame.

-95TABLE I COPPER PIPE PROPERTIES Inside Wall Modulus of Segment Length Diameter Thickness Density Elasticity ft. ft. ft. b./ft.3 lb. ft.2 x 1010 Inlet 1.25.0824.00532 546.2448 1 100.00.0836.00529 558.2448 2 100.16.0831.00535.2448 3 100.19.0832.00529 558.2448 4 100.13.0816.00540 558.2448 Connector 37.46.0828.00528 558.2448 Total or Average Values 439-.19 0829.00532 558.2448 Because the properties of the copper pipe segments were similar with respect to each other, they were treated as one continuous pipe having average characteristics. The plexiglas pipe was constructed of two short lengths of pipe cemented together to form a continuous tube 8.44 feet in length. Plexiglas flanges were machined and fitted to the ends of the pipe, thereby permitting connection to a similar bronze flange attached to the copper pipe on the one hand, and to another bronze flange attached to the gate valve on the other hand. O-rings inserted between the flanges in machined chamfers provided leak-proof connections. A steel frame was used to hold the pipe firmly in position. Although the pipe could still accommodate radial strains, the frame curtailed axial strains. The cast plastic pipe was assumed to be isotropic. Its properties are listed in the table following:

-96TABLE II PLASTIC PIPE PROPERTIES Inside Wall Modulus of Length Diameter Thickness Density Elasticity 8 ft. ft. ft. lb./ft.3 lb./ft.2 x 10 8.77.0828.0211 73.63.5112* *Methyl Methacrylate (acrylic resin) at 26~C. The steel frame used to secure the plexiglas pipe also served as a mounting for the quick-closing, solenoid-actuated gate valve. Because this frame was designed to permit rotational adjustments about a horizontal axis, the value was positioned so that the axis of rotation passed through the center of the gate opening. However, all investigations undertaken in this study were conducted with the plastic pipe in the horizontal position. The solenoid-actuated valve was normally in the open position. When actuated by the solenoid the sliding gate was pushed downward until the circular passage in the gate was no longer aligned with the inlet and outlet ports. The time-of-closure was found to vary from less than 20 milliseconds to nearly 60 milliseconds, depending upon the phase position of the AC voltage at the moment the solenoid coil was energized. Clearly, the instantaneous valve closure assumed in the theoretical phase of the study was not achieved in the laboratory. However, because the round-trip travel time of the main pressure wave was in fact equivalent to instantaneous valve closure.

-97Because the flow system was operated under syphon conditions, the drain hole at the bottom of the valve chamber was submerged in a reservoir of water. Figures 16 and 17 show the plexiglas pipe, the solenoid-operated gate valve, and the steel supporting frame from two different vantage points. Figure 18 provides a close-up view of the principal test section in the plexiglas pipe, the solenoid-actuated gate valve, and the supporting frame and mounting bracket. A one-inch inside diameter copper pipe was used to return the flow from the valve to a lower elevation. A heavy-duty, one-inch inside diameter rubber hose connected to this pipe returned the flow to the laboratory sump, thereby completing the flow system. Instrumentation Three sets of piezometer rings were attached to the copper pipe system. The first ring was located just beyond the entrance to the pipe system at the weir box; the second was placed immediately ahead of the junction between the copper and the plastic pipes; and the third ring was positioned just beyond the gate valve. A removable, fourth plexiglas piezometer ring (shown hanging from the dual manometer stand in Figure 16) was available for insertion at the junction between the plexiglas pipe and the gate valve, When connected to the appropriate manometers these piezometer rings were used to determine the friction loss characteristics of the flow system. A Dynisco unbonded strain-gage type transducer was used to determine the transient pressures at the gate valve. This transducer, which was mounted nearly flush with the inside wall of the pipe and

-98Figure 16. View of Coiled Copper Pipe, Plexiglas Pipe, Solenoid-Operated Gate Valve, and Assorted Instrumentation. Plexiglas Pipe and Gate Valve are Mounted on Steel Frame Clamped to Structural Column. Figure 17. Top View of Plexiglas Pipe and Solenoid Operated Gate Valve.

-99normal to the longitudinal axis, was temperature compensated throughout its full 0.300 psia pressure sensing range. Its range of frequency response was 0-12,000 cps, The transducer is partially visible on the back side of the pipe in the center of Figure 18. An Ellis BAM-1 bridge amplifier was used not only to supply the input voltage needed to activate the four arms of the Wheatstone bridge circuit in the transducer, but also to amplify the response signal. The output of the amplifier was displayed on a Tektronics Model 565 dual beam oscilloscope equipped with a polaroid camera. In order to determine the magnitude and shape of the vapor cavities a series of miniature wave gages were very tediously installed throughout the length of the plexiglas pipe, Each gage, consisting of.002-inch diameter nickel-chrome resistance wire threaded in a "W"-like pattern through the walls of the pipe and cemented in place with epoxy resin, could serve as a single leg in a variable conductance Wheatstone bridge circuit. The gages were vertically oriented in a plane normal to the axis of flowo The first gage, located as close to the gate valve as possible, was.342 foot from the near face of the gate while the second gage was.500 foot from the face. The next 17 gages were spaced at 2-inch intervals, the following 16 gages at 3-inch intervals, and the remaining gages at 6-inch intervals, A total of 37 gages were installed in the plexiglas pipe. Figure 19 shows a sample miniature wave gage typip cal of the gages installed in the plexiglas pipes Several of the actual gages are visible in Figures 16-18. Two Sanborn Twin-Viso oscillographs equipped with A.C. amplifiers (2400 cps) were used to activate the gages, amplify the responses, and record the resulting signals. Four identical bridge circuits were provided

-100Figure 18. Close-Up View of Principal Test Section Depth Gages and Pressure Transducer Mounted in Plexiglas Pipe. I,~~~~~~~~~~~~~~~ |~~~~~~~~~~~~~ii~.~~ iz?igur 1_ ls-pVe fPicplTs eto et

in order that any four gages could be monitered simultaneously. A schematic diagram of the bridge circuit is shown in Figure 20, The two Sanborn units were interconnected and their circuits slightly modified in other ways to eliminate beat frequencies and minimize crosssignal feedback. Figure 21 provides a closeup view of the two oscillograph units, the camera-equipped oscilloscope, and the bridge amplifier unit used in the study, System Calibration The unique frictional resistance characteristics of the laboratory flow system were investigated for the full range of anticipated rates of discharge. The head losses in both the plexiglas pipe (pipe segment 1) and the copper pipe (pipe segment 2), were determined for 55 different rates of discharge~ Two differential manometers'were used to measure the head losses, while weigh-tank observations were used to determine the flow velocities, These data, together with the appropriate kinematic viscosity data, were used to obtain the steady-state, frictional flow-resistance term f, and the associated Reynolds number, Figure 22 is a graphical representation of the typical flow resistance relationships obtained by plotting the respective data versus the associated Reynolds numbers, While the relationships shown in Figure 22 admittedly represent steady state flow resistance characteristics, the flow resistance relationships for transient flow conditions are assumed to be equivalent. Before undertaking the investigation of the column separation accompanying transient flow in the experimental laboratory apparatus, both the pressure-sensing instrumentation and the wave or depth-sensing

-102Signal Source Recording 2400 cps.,5v. Galvonometer I ~~~~~~~~GI SANBORN TI IUNIT I -- I l 4472 mfid. EXTERNAL I 1471 BRIDGE UNIT I I Wave Gage Figure 20. Diagram of Circuit Designed for Use with a Miniture Wave Gage.

Figure 21. Close-Up View of Dual-Channel Oscillograph Units, Camera Equipped Oscilloscope, and Bridge Amplifier Unit.

1.01.5 Pipe Segment 2 - Copper Pipe Pipe Segment I - Plexiglas Pipe.2.05.02.01 100 200 500 1000 2000 5000 10)000 20p00 50,000 R 4= 4 V U Figure 22. Flow-Resistance Relationships for the Copper and Plexiglas Pipes.

-105instrumentation had to be thoroughly checked, adjusted, and fully calibrated, In fact, these operations had to be repeated frequently in order to insure that all equipment was functioning properly and that the observed readings were indeed meaningful and accurate~ The pressure transducer was carefully checked for accuracy of full-scale reading using a dead-weight gage tester, Then the proper scaling was established on the Ellis bridge amp:lifier and on the oscilliscope to give the proper beam deflections on the dual beam cathode ray tube, The dead weight gage tester was used to determine the desired scaling~ Maintenance and calibration of the depth sensing gages proved to be a tedious, complex, and often difficult task. Although these gages were all of similar design, each one had unique calibration characteristics which had to be taken into account. Calibration of these gages consisted of establishing (or reestablishing) the zero flow or dry pipe adjustment of the appropriate oscillograph channels, Next the full-flow reading was used to adjust the various oscillograph channels for proper scaling with a particular gage. Another reading had to be taken to compensate for the pressure condition in the pipe. This reading was subsequently used as an adjustment applied to the recorded values during appropriate periods in timeo The time required to adjust, calibrate, and stabilize the operation of the wave gages ranged from one or two hours to as much as a day or longer. Fortunately, once proper operation was achieved only minor adjustments were required between successive column separation experiments in a sequence of investigations0

-106Experimental Procedures Actual column-separation experiments could be undertaken only after all electronic equipment had operated sufficiently long to insure its stability and after all instrument calibration had been successfully accomplished. The first step in an experiment was to prime the pipe system and thereby establish syphon flow. The priming operation was done by temporarily attaching a hose to the inlet of the system in the weir box. Water drawn under pressure from the laboratory constant head tank was used for priming and for flushing out entrapped air, All piezometers were bled to eliminate air which may have become trapped in the piezometer rings. Flow in the system was regulated by means of a throttle valve in the sump return pipe, The steady-state discharge was determined by using a stop watch and a scale to time and weigh, respectively, several sampled quantities of water passing through the system, Water temperature readings and mercury barometer readings were recorded at frequent intervals throughout an experimental period, To initiate transient flow and the subsequent column separation, the recording oscillographs were started and the shutter of the polaroid camera attached to the oscilloscope was opened immediately prior to the instant the solenoid valve was switched shut, When the valve closed, the self-triggering, beam-sweep circuits in the dualsbeam oscilloscope caused the trace of the pressure wave sensed by the transducer at the gate valve to be recorded on film. The upper beam was used to record the individual pressure peaks superimposed, one upon the other, using an expanded time scale. The lower beam recorded the entire sequence of pressure peaks separated by periods of column separation during a -second sweep interval, The trace of transient pressures observed at the gate valve and

-107shown in Figure 23 for laboratory run No. 19 is typical of the data obtained from the pressure transducer. In this figure a major division in the vertical direction is equivalent to 55 psia, A major division in the horizontal direction on the upper scale is equivalent to.05 second; on the lower scale,.5 second. Meanwhile, the oscillographs recorded the depth of flow at selected points within the travel range of the cavitation void. Figure 24 shows a typical set of data obtained simultaneously on one of the twin-channel, strip-chart oscillographs during laboratory run No,- 33, The upper trace represents the depth of flow observed at miniature wavegage Noo 1; the lower trace, the transient pressures at the gate valve. The trace of the transient pressure was frequently recorded on one of the four oscillograph channels to assure the simultaneity of time base for all recording media, One-second tick marks are visible along the lower edge of the chart in Figure 23; the major chart divisions are 5mm., the minor divisions are 1 mm. so that the chart speed was 100 mm. per second. Simultaneous strip-chart traces representing the depth of flow at two other wave gages were recorded on the second, twin-channel oscillographo On certain laboratory runs photographs of the cavitation void were taken. Once column separation had ceased and the transient pressure wave had diminished due to friction, the solenoid valve was opened and the recorded data was removed from the recording instruments and marked for subsequent identification. The system could then be reinitialized

Figure 23. Transient Piessures, Photographically Recorded from the Oscilloscope Cathod-Ray Tube During Laboratory Run Number 19, are Typical of the Experimental Pressures Observed at the Gate Valve.

in the Pipe. The Data Trace Shown at Bottom Represents the Simultaneously ObLaboratory Run Number 33- Depth-of-Flow Data Traces, Not Shown But Similar to Oscillograph. 1, It........S -4 -1.......... Oscillograph.

and made ready for the next experimental run. It was found essential to recheck all instrument calibrations before starting each new laboratory experimental run, The experimental results are presented in the next chapter and compared with the theoretical results obtained from computer simulation for the same physical conditions.

CHAPTER VI COMPARISON AND DISCUSSION OF EXPERITMENTAL AND THEORETICAL RESULTS The study of column separation accompanying the transient flow of liquids in pipes culminates in this chapter with comparisons of the laboratory-observed, experimental results and the computer-simulated, theoretical results derived from the mathematical model. The ability to throttle the syphon-flow velocity and to vary the elevation of the solenoid-operated valve throughout a range of approximately 26 feet, as well as intrinsic fluctuations in the atmospheric pressure and in the water temperature offered a limitless combination of experimental conditions for laboratory investigation. However, practical constraints on the time which could be profitably expended analyzing laboratory data and economic sanctions on the use of computer time, limited this study to the investigation of only a few selected conditions. These conditions are briefly presented in the first section. A detailed comparison of the experimental and theoretical results obtained for one particular set of investigated conditions is presented in the subsequent section. Analysis and discussion of the similarities and dissimilarities between the theoretical and experimental pressure rises are followed by a similar appraisal of the theoretical and experimental column-separation voids. The significance of these findings is discussed in the final section, Conditions Investigated Experimental data were gathered for three distinct sets of laboratory conditions. These conditions differed, one from another, because of differences in the elevation of the solenoid-operated gate valve, differences in the initial steady-state rate of flow, or because of both, -111

-112However, minor changes in atmospheric pressure and in water temperature, along with slight fluctuations in the initial rate of flow resulted in subset conditions within two of the principal sets of conditionso Table ITITT is a summary of the 34 data runs considered in the experimental phase of the study. The grouping of the various data runs according to laboratory conditions is tabulated below: (a) Group I: Runs No. 1-10 Valve Elevation (HEL) = 27.980 feet Location (Loc 1) = 39.00 feet Initial Steady-state Discharge (Q) -0.0107 cubic feet per second (b) Group II: Runs Noo 11-25, 31-34 Valve Elevation (HEL) -= 34719 feet Location (Loc 1) = 18.00 feet Initial Steady-State Discharge (Q) -0.0107 cubic feet per second (c) Group III~ Runs No. 26-30 Valve Elevation (HEL) = 37.719 feet Location (Loc 1) = 18.00 feet Initial Steady-State Discharge (Q) =-0.0078 cubic feet per second These same parameters, together with other parameters describing the nonvariable experimental conditions, were used as input data to the computer program. Computer simulations were then carried out for the three laboratory conditionso

TABLE III SUMMARY OF EXPERIMENT~ RUNS Runt Date Configuration Water Atmos. Rho Nu Q Oscilloscope Oscillograph Data PhotoI Max. Rise Time I t No. 1964 J HEL LOCI Temp. Pressure p Data — Gage Gage Gage Extent 2nd 13rd! ft ft i Deg. F ft Slugs/ft3 ft2/sec 10-5 ft3/sec feet see see __ [ 1 3-3 27.98039.00 32.683 1.936 1.033 -.01077 Yes 1 2 3 No - 1.81 3.12 2 3-3 27.980139.00 72o 32.683 1.936 1.033 Jl -'01077 Yes 1 2 3 No 1.06 1.793.08 3 3-3 27.98039.00 72o 32.682 1.936 1.033 i 01077 Yes 1 2 3 No 1.03 1.763.04 3-3 72~ 27.980z39.00! 72o 32.681 1.936 1.033 I-.01077 Yes 1 2 3 No 1.01 1.76 3.03 5 3-3 27.98039.00! 72o 32.680 1.936 1.033'-.01077 Yes 1 2 3 No.96 1.72 2.88 3-3 27.98039.00! 72o 32.680 1.936 Yes 1 2 1.033 1-.01077 3 No.92 1.68 2.83 3-4 27.98039-00 t o: 75 32.528 1.935.995 -.01058 Yes 1 2 3 No.99 1.76 2.99 8 3-4 27.98039.00! 75~ 32.528 1.935 -995 -.01058 Yes 1 2 3 No - 1.69 2.97 3-4 27 98039 O0 I 75~ 32.528 1 935 995 - 01058 Yes 1 2 3 No 96 1 69 2 83 ~ ~ I * ~ ~ ~ ~ ~ 10 3-4 27.98039.00 ~i 75~ 32.528 1.935.995 -.01058 Yes 1 2 3 No.90 1.68 12.78 11 3-7 34.719 18.00 I 75~ 33.116 1.935.995 -.01073 Yes 1 4 ~ Yes 1.16 1 9813 34, 12 3-7 34.719 18.00 75~ 33.116 1.935.995 -.01073 Yes 1 4 3 No 1.14 2.00!3.41 13 3-7 34.71918.00 75~ 33.126 1.935.995 -.01073 Yes 1 4 3 No 1.11 1.95i3.22 14 3-7 34.71918.00 75~ 33.134 1.935.995 -.01073 Yes 1 4 3 Yes 1.09 1.91i3.20' 15 3-7 34.71918.00 75~ 33.152 1.935.995 -.01073 Yes 1 4 3 No 1.08 1.91i3.18 16 3-7 34.719 18.00 I 75~ 33.170 1.935.995 -.01073 Yes 1 3 4 No 1.18 1.96 ~.27 17 3-7 34.71918.00 75~ 33.188 1.935.995 -.01073 Yes 1 3 4 Yes 1.22 1.92 18 3-7 34.71918.00 75~ 33.2O6 1.935 -995 -.01073 Yes 1 2 4 No 1.23 1.91 3.17 19 3-7 34.71918.00 75~ 33.224 1.935.995 -.01073 Yes 1 2 4 Yes 1.26 2.07 3.45 20 3-7 34.71918.00 75~ 33.239 1.935 -995 -.01073 Yes 1 2 4 No 1.13 1.903.12 21 3-7 34.719 18.00 33.258 1.935.995 -.01073 Yes 1 2 4 No - 1.87 3.06 22 3-10 I 75~ 34.71918.00 72o 32.421 1.936 1.033 -.01054 No 1 2 4 No 1.13 2.043.37 23 3-10 34.71918.00 72o 32.421 1.936 1.033 -.01054 Yes 1 2 4 No 1.21 2.053.40 24 3-10 34.719 18.00 72~ 32.421 1.936 1.033 -.01054 Yes 1 2 4 No 1.25 2.05 3.40i 25 3-10 34.71918.00 720 32.421 1.936 1.033 -.01054 Yes 1 2 4 No 1.19 2.113.32 26 3-10 34.719 18.00 1.033 - 1.64 2.75 7~ 32.421 1.936.00777 Yes 1 2 4 No t 27 3-10 34.71918.00 72~ 32.421 1.936 1.033 -.00777 Yes 1 2 3 Yes - 1.63 2.75 28 3-10 34.71918.00 720 32.421 1.936 1.033 -.00777 Yes 1 2 3 No.82 1.63 2.73 29 3-10 34.719 18.00 72o 32.421 1.936 1.033 -.00777 Yes 1 2 4 No.88 1.66 2.79 30 3-10 34 719!18 O0 I 72~ 32.421 1.936 1.033 -.00777 Yes 1 2 4 No.78 1.62 2.72 31 3-10 34.719 f18. 00 72o 32.421 1.936 1.033 -.01054 Yes 1 5 3 No 1.28 2.05 3.43 32 3-10 34 719!18 O0 I 72o 32.421 1.936 1.033 -.01054 Yes 1 5 4 No 1.24 2.03 3.42 33 3-10 34.719i18.00 I 72o 32.421 1.936 1.033 -.01054 Yes 1 5 4 Yes 1.22 2.05 3.42 34 3-10 34.719 i18.00 72~ 32.421 1.936 1.033 -.01054 Yes 1 5 4 Yes 1.18 1.882.9?

-114Comparison of Results In the theoretical analysis presented in Chapter II, the assumption was made that the pipe flow ahead of the void created by column separation would continue to be full-pipe flow. That this assumption is, in fact, not valid became apparent rather early in the experimental phase of the study. While the free-surface, vapor-filled, column-separation void formed as anticipated, small gaseous cavities —bubbles —were observed to occur ahead of the void and throughout the remaining length of the plexiglas pipe. Moreover, with continued propagation of the low-pressure wave toward the constant head reservoir, similar bubbles were heard to occur throughout most of the length of the copper pipe. With the iminent collapse of the column-separation void and with the subsequent creation of a new high-pressure wave propagating toward the reservoir, the cavitation bubbles were observed to diminish, first, and then, to disappear completely (or almost completely), respectively. The occurrence of vapor-cavity bubbles ahead of the void has a significant bearing on all aspects of the transient-flow, column-separation phenomenao This significance will become fully evident in the comparisons and detailed discussion which follow. Two experimental runs, Numbers 25 and 34, were selected from the laboratory data in Group II for detailed comparison with the corresponding computer-simulated results. These runs were considered to be entirely representative of the other experimental runs comprising Group II. Pressure Rises Figure 25 presents a time-comparison of the transient, absolute pressures occurring at the gate valve as derived from the mathematical

Theoretical Result 300 200 - -: 0.2.4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 Experimental Result j 300 No.25 200 1T I SECONDS ir U) 0.2.4.6.8 ID 12 14 1.6 1.8 2.0 2.2 2.4 2.6 2,8 3,0 3.2 3.4 3.6 Experimental Result 200 - Atmospheric Pressure 0::::j::::~::::I:* d~i:::,:,:::j:~::jg............. O.2.4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 22 2.4 2.6 28 3.0 3.2 3.4 3.6 TIME IN SECONDSFigure 25. Comparison of the Computer-Simulated, Transient Pressures with the Experimentally Determined Pressures Observed at the Gate Valve During Laboratory Runs Numbers 25 and 34.

-116model and from laboratory measurements (experimental runs Numbers 25 and 34). Time is measured from the instant the gate valve was closed. For each of the three examples shown, column separation occurs at the valve during the time intervals between the successive pressure rises. Figures 26 and 27 provide detailed overlay comparisons of the computer-simulated and experimentally-determined initial pressure rises and second pressure rises, respectively. Figures 28 and 29 are reproductions of the respective oscilloscope photographs on which the absolute pressure-head data for runs Numbers 25 and 34 were recorded. The numerical data used to plot the experimental results of Figures 25, 26, and 27 are tabulated in Appendix III; many of the data used in plotting the theoretical results are contained in Appendix IV. Comparative study of Figures 25-27 discloses several factors of note. Principal among these factors are the following~ (a) In broad terms the overall appearances of the three pressure diagrams shown in Figure 25 are similar; abrupt pressure rises separate intervals of subatmospheric pressure during which time column separation occurs. (b) Despite some minor pressure fluctuations and local pressure differences evident in the experimental results, the three initial pressure rises closely resemble each other in time of occurrence, as well as in shape, duration, and magnitude of the pressure rise. (c) The experimental second pressure rises, while manifesting the same minor pressure fluctuations and local pressure differences noted in the initial experimental rises, compare favorably in shape and duration with the corresponding

-117Theoretical 0 -. Experimental, 300- No. 25 i~ J;, \1 Experimental, No.34 30; t! * < t Atmospheric iT i 0 LL 200 I Li LI 0.10.20.24 TIME IN SECONDS Figure 26. Detailed Comparison of the Computer-Simulated Initial Pressure Rise with the Corresponding ExvperimentallyDetermined Initial Pressure Rises for Laboratory Runs Numbers 25 and 34.

-118300 Theoretical Experimental, No. 25 Experimental, No. 34 - - Atmospheric Pressure --- - w I W Iw CL Theoretical 3.10 3.20 3.30 Experimental, No.25 210 2.20 2.30 Experimental, No.34 1.90 2.00 2.10 TIME IN SECONDS Figure 27. Detailed Comparison of the Computer-Simulated Second Pressure Rise with the Corresponding ExperimentallyDetermined Second Pressure Rises for Laboratory Runs Numbers 25 and 34.

-119-— Rl -IrnIi1m ~ l ll.......a ~~~~~~~~~~~~~~~~~~~~~~~~liIIlkC I I l I I /11 Figure 28. Transient Pressures Observed at Gate Valve During Laboratory Run Number 25. -i -— m- - l mhhhhhhi..i..1. II' -I J' Figure 29. Transient Pressures Observed at Gate Valve During Laboratory Run Number i4.

-120theoretical pressure rise. However, the magnitudes of the experimental pressure rises are appreciably less than is the magnitude of the theoretically-determined rise. (d) The recurrence intervals between successive, experimental pressure rises are of appreciably less duration (35-40 percent less) than are the corresponding intervals determined theoretically. (The third theoretical pressure rise, although not shown in Figure 25, was determined to occur beginning at approximately 5.09 seconds after valve closure.) (e) Appreciable differences occur between the experimental data of laboratory runs Numbers 25 and 34 with regard to the magnitudes of the pressure rises and the duration of the recurrence intervals. Yet, both runs were made ostensively under the same laboratory conditions. (f) Minor pressure fluctuations appear superimposed on all of the experimental pressure rises. These fluctuations have an apparent duration ranging from 0.0068 to 0.0072 seconds. (g) Although the duration of both the experimentally and theoretically determined initial pressure rises is very close to 0.2 seconds, a slight tendency toward increased duration appears to develop in the second and subsequent experimental rises. This tendency appears to accompany the deterioration in the abruptness with which the pressure rises collapse. Among the factors listed above two very significant findings are apparent. One is the considerably shorter duration of the experimental, pressure-rise recurrence interval as compared with the duration of the

-121theoretically-predicted interval. The second is the more rapid rate at which the magnitudes of the second and all successive pressure rises diminish, again as compared with the theoretical findings. Without even comparing the experimentally and theoretically-determined column-separation voids, it is apparent that the duration of these voids must also be considerably shorter than anticipated from theory. It is probable that the extent of their travel will prove to be similarly reduced. Taken together the two complementary findings are strong evidence suggesting that during periods of column separation the rate of energy dissipation which occurs in the flow ahead of the void is much higher than anticipated. The obvious question is, "why?" There are several conditions which lead one to believe that a higher energy-dissipation rate could be a direct outgrowth of the "bubble" or "plug" flows taking place in the pipe ahead of the column-separation void, However, a thorough discussion of these circumstances is deferred to a subsequent section. The laboratory flow system was designed to minimize sources of potential disturbance to pressure wave propagation. The intent, of course, was to assure a clean, uncluttered wave form. It was for this reason that syphon flow was preferred over flow induced by a turbine pump located at the outlet of the system. Nevertheless, not all disturbance-producing elements could be eliminated, Using an expression for the round-trip time-of-travel of a pressure wave in fully confined flow, namely, the expression tt = 2 Lt/al, one finds that the distance, Lt, from the transducer to a disturbance having a propagation and return interval, tt of 0.0066 to 0.0072 seconds ranges from 8.1 to 8o9 feet. The term al represents the celerity of propagation of a pressure disturbance in the

plexiglas pipe. This places the source of the disturbance at or just beyond the junction between the copper and plexiglas pipes. The superimposed, minor, pressure fluctuations appearing on the experimental pressure-rise data are believed to be caused by pressure reflections from the small annular chamber of the piezometer ring attached to the copper pipe and situated 0.3 foot from the pipe junction (8.7 feet from the pressure transducer). Column-Separation Voids Figure 30 presents a modified isometric diagram showing the absolute pressures at the gate valve, as well as the column-separation void profiles at five successive distances measured from the valve, all with respect to time, The data on this diagram were derived entirely from the mathematical model by digital computer simulation. Figures 31 and 32 are the companion diagrams presenting the corresponding experimentally-derived data from laboratory runs Number 25 and Number 34, respectively. In each of the diagrams the depth of flow, z, in the pipe is measured on the vertical plane passing through the axis of pipe and normal to the bottom inner pipe surface, Figure 33 presents a more detailed comparison of the superimposed depth profiles for both the experi mentally-determined and theoretically-determined voids with respect to time, In Figures 30-33, the column-separation voids are compared on the basis of profiles in the z - t plane at a particular location, x The voids can also be studied as phenomena occurring in the x z plane at a selected time t o Figures 34 and 35 present time-sequence

z.oea.060 -040 THEORETICAL RESULT -0823. -00.0833 ~'i,.06 0 t!,, ~ -~~ ~~~040... oC'083 0 06 0 -~~~~04 0.o20 ~'0 0,0 H I 25~~~~~ 00 0o'83 i'.060~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.040~~-~:.~;:I:Ili~lil:~~:()~~::j~i~; ~ ~ --- iil::i;~~.:if:::i ~ilfii~i~ilililiic a r.0(~~0',, 10 250~~~~~~~~~~. 200~~~~~~~~~~~~~~~~~~. C~~~~~~~~~~~~~~~~~~~~~~~~~~ 50~~~~~~~~~~~~~~~~~~~~~~~~~~~. ~~~~~~~~~~M 4. 4~~:::i~:jj~ /0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 1?0 Fig'~~~~~~~~~~~~e J0. Gate Valve as Determined fro Theoretical Considerations G/:;I:::::;if al~~~~~~~~:i::s Figre 0.An somtrc Rprsenatin f te CmpterSiulaed rasiet Pesure a Gate Valve and'Concurrent Free-Surface Profiles at Selected Distances from t]~~~~~~~~~~~~~~~~~~:~:~';:::~::::::.. ~~.;~:~.:.:::;rt: Gate Valve as Determined from Theoretical Considerations.

EXPERIMENTAL RESULT NO. 25.0.08,3.02040. 0 C3V, / VGL.02d3 Q ~006008.o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o~~~~~~~~~~~~~~~ o0d..083O 350, I ~. =250 ~, 200 I / 150 La 1 009 0 L,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ co Iso' f,'0,~ ~~~/0 -~.....'~' w. /~~~~~~~~~~~~~.0 /.Z~~~~/'9-_C0*,0S ~ ~ ~ ~ 6 B J~ Figure 31. An I sometri c Repre sentation O~f the Experimentally- Determined Transient Presssue at the Gate Valve and Concurrent Free-Surface Profiles at Selected Distances fo the Gate Valve as Observed During Laboratory Run Number 25.

.oe3-~/ r~~ 060..040 EXPERIMENTAL RESULT i.o0o20 o NO. 34 -0'00 7: _7 -020 O;: ~~~~~~~~~~~~~~~~~~,; /50,~t0s, ~,.o~ -00 10 / 18 o-~2 - e' ~~~~~~~~~~~~~~~~~ZA 31 3oo',,,,~~~~~~~~~~~~~~~~~~~~~~i- -i ~,~/ —c o-V;OS /'~~: 4oo6 2,,o Fgr32 AnIoer RersnainoteEprmnal-eeieTrnin e/ te e e C r F S c o a e d t th Gate Vv', o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;.;.~50,'~ Ii,.~1 P 350~~. Pi, Fiue3.A smti ersetto fteEprmnal-Dtrie rnin rsue

LUI Distances,X, of Numbered Gages Measured in Feet..oe03<.03X=I.000 5 X=.000 >.070 - X=.833 X.833 2.060 x =.667 0.060 -.667 Orr' tX — 6 ~C ~X67 LL.040 X;.500.030 X =.342 (I) wL.020 X -.500 Theoretical Result o.010 X=.342 z. < 0- Z o 03 0.2 4.6.8 1.0 12 14 6 18 2.02 24 26 2.8 3.0 3.2 3.4 3.6 3.8 4.0 42 4.4 4.6 4.8 5.0 a 0.083w L t-.070 W.060 I X=.342 -J.050 LLJ O3.040 X=.$33 H.030 X=.500 Experimental Result Q.020 0X=.342 I No.25 -.0 10 r Ur I 0 I I W~~~~~~~~~~~~~~~~~~~ U_ 0.2 4.6.8 1.0 12 14 1.6 s2.0 22 2.4 26 28 30 32 3A 3.6 3.8 4.0 42 4.64.8. z.083 NV -.070' I 34 \\ I~~~~~~~~~~~.060 J 342,.050., 010 X 1..000 5.0 0..._.0830 i' O.070 ~".0 40 Ui 0.2 4.6.8 1.0 1.2 1.4 1.6 1.8 2) 2-2 2.4 2.6 2.8 3.0 3,2 3A U &B3 4.0 4.2 4.4 ~4.. TIME IN SECONDS Figure 33. Comparison of the Computer-Simulated and Experimentally-Determined, Time-Dependent, Free-Surface Profiles at Selected Gages During Periods of Column Separation in the Pipe.

t=.204.295 406 499 611 703.815.907 1000 1.112 1.204 1.316 1.408, -.02 0.2 4.6.8 1.0 1.2 14 1.6 1.8 2.0 2.2 2.4 2.6 DISTANCE,x,IN FEET FROM VALVE GATE Valve gate face Pipe wall t=3.116 3.097 3.004 2.893 2.800 2.707 2596 2.503 2.411 2.299 2.206.08.02.06.114 1.81? DEPTHzT,.095 1.910 1.705 IN FEET.04 - 1.613 1.501.03 - 2.003.02 - 1.91 2.299 2.707 901 1.613-1L705 2.114 2.503 0.2.4.6.8 LO 1.2 1.4 1.6 1.8 2.0 22 2.4 2.6 DISTANCE,x,IN FEET FROM VALVE GATE Figure 34. Computer-Simulated Sequence of Water-Surface Profiles for the Initial Period of Column Separation in the Pipe. Individual Profiles are Identified According to the Time of Their Occurrence, t, in Seconds After Closure of the Gate Valve.

t= 3.334 3425 3.518 3.629 3.723 3.815 3.927 4.038.08.07 96 DEPTH, z.05 IN FEET.94.03.02.01 - THEORETICAL RESULT 0.2 4.6.8 1.0 12 1A 16 1.8 2.0 2.2 2.4 2.6 DISTANCE,x,IN FEET FROM VALVE GAT E Valve gate face Pipe wall t=5.021 4.929 4.836 4.725 4.632.08 907.06 DEPTH, z,.05 4.520 IN FEET.4 -4.428 4.224.03 4.316.01 4.224-4.316 4131 0 0.2.4.6.8 1.0 1.2 1.4IA 1.6 1.8 2.0 2.2 2.4 2.6 DISTANCE, x, IN FEET FROM -VALVE GATE Figure 35. Computer-Simulated Sequence of Water-Surface Profiles for the Second Period of Column Separation in the Pipe. Individual Profiles are Identified According to the Time of Their Occurrence, t, in Seconds After Closure of the Gate Valve.

-129families of profiles derived from the mathematical model by computer simulation. Figure 34 presents the time-sequence of the water-surface profiles for the initial period of column separation; Figure 35, the watersurface profiles for the second period of column separation. Note that in both figures dissimilar horizontal and vertical scales were used. In each figure the growth sequence of the theoretical void is depicted in the top diagram while its collapse sequence is traced in the bottom diagram. Unfortunately, there were insufficient experimental data with which to plot the companion experimental curves. However, Figures 36, 37, and 38, as well as Figure 1, are typical photographs of the column-separation voids observed in the laboratory. Each of the partial profiles shown. in these figures may be thought of as one profile out of a family of profiles similar to the theoretically-determined profiles of Figure 34. Because the photographs shown in Figures 36, 37, and 38 were taken at close range using a camera with a very shallow depth of field and because the inverted tape measure was positioned behind the plexiglas pipe, the apparent distances of points in the void from the gate valve are somewhat distorted and have only approximate significance. Most of the data used in Figures 30-35 are tabulated in Appendixes III and IV. The principal factors noted from comparing the theoreticallyderived and experimentally-determined column-separation voids are cited below: (a) The similarities which exist between the corresponding computer-simulated and experimentally-determined void profiles shown in Figures 30-34 are limited largely to similarities in the gross sense of the term. The shapes of the initial

-130Figure 36. Advancing Column-Separation Void Photographed During Run Number 19. Inverted Scale Gives the Approximate Distance to the Inner Face of the Gate Valve in Inches. Figure 37. Advancing Columtn-Separation Void. Notice the Cavitation Bubbles Ahead of the Void and the Ripple Behind the Void.

"* —;:::'':iiiiiii:iii~iii j:::::::::i~~:l:::-::l:::::::-::-:::::: ~:::::.:::::::-::::::":-:i:irf-::: i,i:ii~iiiiiai;i'i;.;:i:::i:::::i:i:i::::::\::i:::::.:. ii- iii:: i i:-::~~iiii- iii:ii::-:::::-ii:::..:::'iiii ii': —:'::'''':':'-'-''-ii::.:::::.: —::::::::::1:::::'-::::: i i:':::-:-:: —:::;~i'r'i':":-'i,;i~~iiiiii:iiii-~:iiiii:iiii:iii i.ii'-iiiji lit-ii-i-i-i:::-: -i i:.:: -iii::-i:: —:_: —liliii'~iiiii-: __:-:::::_:i:::::_::i::::(-:::_:i:_::::::_::,i::i:i::i:i-::i-iii —i:i-i::::::::: - -::-:ii': --- -:: -:::-._:::::;;::_ —::1::::_.::-:::-:-:;;:: iii.:.,i:i::iiii-i:i-iii i:::::::.::::::::i:_:::i: ~:;:::-::::: iii~iiiiiiliiiiiiiiiiiiiijiiijiiiiiiiiii i-iiiiiiiiiiii~iiiiiiii iiiiiiiii'iiiiiLi:~ijiiiii'-'-:: —':'i''i'il' iiiiiiiiiiiiii:iiiiiii:ii;iiiiiii iiiliii.li:iiiliiieiii':' iiiliiiiii'ii:::':::-:-::::::::'::ii;:::i'iiiiiiiiii;ii`iiiiiiiiii'::::i'-:i'i:i'::~'-'''''":''''-'''-':'::'''::::-: -'' —-:iiiiiii-::i i-i-i-i:i-i-.:.. -,_:i:-iili:::- -::: -- -—::i:.ii-: — -:-:::::::::.::::-:-:..:..:::. ii-i- -—':-::;': ii:. -.: I-I:: -i-_-:::-:i::i;rii::::::::ii-:-:-iiiiiiii:::::: -::::::::iiiiii::-::-::::: i:-::-_:_:::i:ii::-iiii-i:i-i:i-i:iiiii:i.i i: iii...:::.: iiiiii-i —:,:~:,i::~ii iiiiiiiFi:iii:i'i:i-j-i:i:-j-i-j:iiiiii:::i:i:-:::_:_-::_-:-:::::i i-i:-i::-:: -:i:::-:i'::i'i:~i:i~::ii: iiiiiiliiiii~ii-iiiiii:-iii —-i-i-i:,:i —-iiiii:i iii ii -ii:- -iiiiiii''''''-': —iii:i:iii:ii -:: —ii:iii: —i:i i-i i:_:i-i —::::::::: iiiiiiii:ii:llib::iI:i:i::a:::::: i:i-i::_lcii::ii:::iiiiiii:lii:/iliii:j::::::::::::::::::'::::::::~::i:::i:::::-:_:::::::i:::i:i-iiiiiiii::::iii::: i:i-i:-:c:::ii::ii:-::iiii~i-.::i iiiiiii-ii-i-iciiiiii:-ii-ii::iiiriii:ii:i:::::::i:i:i:i:i::::::ii:i::~::iijliiiii::-:::::::::::::::.:.:.:::_ iii:iiiuii;isiiii:iiiiiiii-~:iii:::., -iiiiiiiiii:iiiiir:i i:i-i:i:i-i-ii;i-iii:iii:i:iii.i-i-ixii iiiiiiiiiiiiii:iiii i:i'':i: —iiiiiiii:i —:i:iii:iliii:i:.ii:':::':'::::::::-: iiiiiii:ii:'i-'''::: -::- -:- -,-i-::-_-:-:::-:::.: -i: i:i:iii:i:-::i:::::i:::-:-::::: -::::::_:_:::_::::::;:___::::::::_:::::::_::_::::::::;:::::~::~:' 1 tJ::::::~:::i:-::::::::::::::'iioii:i~iii~iiii Figure 38. Column-Separation Void in Xtate of Suspended Motion Just Prior to Retreat and Collapse Against Valve.

-132experimentally-determined void profiles, especially the profiles recorded at wave gage Number 1, have an identifiable resemblance to the corresponding profiles determined from theory despite topical differences and differences in duration. (b) The shortened duration of the experimentally-determined voids is coincident with the reduced experimental recurrence interval noted in the previous section. As anticipated the extent of travel of the experimental columnseparation void is much reduced -- approximately 50 percent reduced during the initial column-separation cycle -- from that which is predicted by the mathematical model. The experimental maximum extent of travel for each of the initial column-separation voids comprising Group II data is given in Table III. (c) The depth of free-surface flow observed in all laboratory runs, but specifically as shown for runs Numbers 25 and 34, is much greater than that derived from theory. This factor, together with the reduced extent of travel, dictates that the transient volumes of the experimentallydetermined voids are very much smaller than the predicted volumes. (d) The differences between the experimental void profiles recorded at the various wave gages during laboratory runs are even more pronounced than the differences noted for the comparisons of the experimental pressure-rise data.

-133For example, the two experimental z - t profiles recorded at gage Number 1 have a fair resemblance to each other. Both voids form and collapse at approximately the same times respectively. Yet the associated profiles at gage Number 4 are certainly less comparable. (e) The leading front of the experimentally-determined void is less bulbous than the front computed for the simulated void. Moreover, the front of the laboratory-produced void is usually followed by one or more surface ripples not predicted by the mathematical model (see Figures 1, 36-38). (f) Small ripples appear on the free-surface of the experimental flow throughout the void. These ripples are not manifest on the theoretically-determined free surface profile. (g) On the basis of comparisons with the theoretical conditions, the second experimental column-separation void has, proportionally speaking, an even smaller extent of travel than does the first experimental void, The single most significant finding evident from the list of factors above is that the experimental void is of less physical magnitude and of less duration than is predicted by the simulation model, The shorter extent-of-travel, the greater depth of free-surface flow which conversely results in a void of shallower depth, the shorter recurrence interval, as well as the proportionally diminished extent-of-travel during the second void cycle, all lend additional support to the belief that during column separation, the rate of energy dissipation occurring in the flow ahead of the void is much higher than anticipated,

The small differences in the appearance of the leading edge of the theoretically and experimentally-determined void profiles are not unexpected. The boundary condition used with the simulation model to represent separation of flow from the pipe wall at the leading edge of the void is premised on a one-dimensional concept of the flow process. Moreover, the effects of surface tension are not considered. Thus, differences -- even greater differences than those observed -- were anticipated. Moreover, as the separation front of an advancing void moved in the pipe, a thin film of water was observed to flow around and laterally down the inner surface of the pipe and rejoin the free-surface just in back of the void front. This flow, entering the free-surface at either side of the pipe, is believed to be the cause of the rather pronounced ripple trailing the separation front of an advancing columnseparation void (see Figures 36 and 37). The factor(s) causing the other small ripples appearing on the experimental void profiles is unknown. One can speculate, however, that these ripples may have been produced by some slight pressure fluctuations occurring ahead of the separation front or by some slight anomaly in the pipe system. Comparisons of the simulated results and the experimental results were not carried out in as great detail for the laboratory conditions of Groups II and III. However, the same general results were observed, The experimental results obtained in the laboratory had three common characteristics:

-135(a) Although the first pressure rise compared favorably in magnitude, form, and duration with the simulated pressure rise, the second pressure rise was of lesser magnitude than its simulated counterpart. (b) The duration of the first experimental void was considerably shorter than the duration determined from the mathematical model. (c) The extent-of-travel of the first experimental void was considerably shorter than predicted by simulation. Significance of Findings The significance of the theoretical and experimental findings cannot be interpreted properly without considering the apparent effects that cavitation and the resulting bubble flow have upon the column separation phenomena. Thus, it is of value at this point to describe in some detail the flow conditions observed in the laboratory during separation and to briefly review the role of gas nuclei and gaseous diffusion in incipient cavitation. The formation of bubble cavities ahead of the main void was observed regularly during periods of column separation for all of the laboratory conditions investigated. The bubble cavities were observed to develop generally near the pipe wall in the topmost quadrant of the plexiglas pipe and in the portion of the flow-wake immediately behind the individual wires of the wave gages. Bubble cavities were never observed to form in the flow in the lower half of the pipe. Formation occurred very rapidly; so rapidly, in fact, that it was difficult to perceive that it occurred progressively along the pipe with time. The

-136ultimate size of these bubble cavities varied from nearly spherical bubbles having an approximate diameter of less than.01 foot to larger, elongated bubbles exceeding 0.3 foot in length (see Figures 1, 36, 37, 38). No variation in bubble distribution throughout the length of the plexiglas pipe could be visually detected during any given occurrence, However, the number and ultimate size of the bubble cavities did vary appreciably from one experimental run to another, even under ostensibly unchanged flow conditions, Sometimes many larger bubbles developed; at other times the bubble cavities would be predominantly smaller and more numerous It is noteworthy that the smaller bubble cavities appeared to move more or less with the flow whereas the larger ones moved rather sluggishly and tended to "roll" along if they moved at all. Yet, with collapse of the first column-separation void the bubble cavities collapsed and disappeared or at least diminished to a subvisual level, The bubble cavities reappeared during the second and subsequent column separation voids, However, after the collapse of the third or fourth column separation void, the largest bubble cavities did not always disappear completely, Occasionally, a few very small bubbles could be observed remaining in the flow, It has been fairly well established that under ordinary engineering conditions the inception of cavitation in water results from the growth of submacroscopic nuclei containing liquid vapor and undissolved (free) air.(l316,38,44) These nuclei may be present in the water, or they may be attached to the boundary surfaces or to particulate matter

-137in water. For a small spherical bubble in static equilibrium in water, one may write the- equation Pg + Pv Pa = 2C -r (169) where Pa is the local ambient pressure, Pv is the liquid vapor pressure, pg is the gas partial pressure, a is the surface tension, and r is the bubble radius. The pressure, pg, of a known weight of a perfect gas at constant temperature is wRT g where w is the weight of the gas, T is its absolute temperature, V is its volume, and R is the gas constant (53o35 ft. lb./lb./Deg.Rankin). Thus, this expression can be written 3wRT J g 4h1r3 r3 where J = 3wRT/4x is a parameter indicative of the weight of the gas in a bubble at a particular temperature. Equation (169) becomes J 2a Pa- Pv = (170) r3 r The minimum pressure at which this equilibrium relationship is valid is found by differentiating with respect to r: d(Pa - Pv) _ 3J + 2u =0 dr r2

-138Therefore, the critical pressure, r*, is r* )1/2 Consequently, the minimum pressure, (Pa - Pv)min = p*' is P* = (Pa - Pv)min (3J/2o)3/2 (3J/20)l/2 or simply - 4c P* = (Pa - Pv)min = 3r* (171) for which the critical pressure is below the vapor pressure. If the ambient pressure is decreased even slightly, the bubble (water-vapor-air nucleus) becomes unstable and grows without bound. At greater than critical pressures the bubble is stable and assumes an equilibrium radius according to Equation (170). Figure 39 shows the static relationship between the pressure head and the bubble radius for various masses of gas as given by the parameter J. From this figure it is clear that in order for small diameter bubble nuclei to expand, the pressure must be less than the vapor pressure. Moreover, Strasberg(51) shows that bubbles having a diameter of less than 20 microns (6.6 x 10-5 foot) require negative absolute pressure for cavitation inception. These simple bubble-nuclei equilibrium arguments are only generally relevant to the observed cavitation phenomenon, since the dynamic effects of an expanding, moving bubble are ignored. Moreover, the arguments assume constant temperature and constant air mass. Nevertheless, these argumients do provide a basis from which to appraise the laboratory

-1393.0 2.5 2.0 — 0.5 Ii I.0 F5.0 U- ~~~~~~~~~~10.0 -O( IMOJ 7 Q-O (ft.Ib.) I- ~~~~~~~~~~~~~~(720r) Id~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 50. 0 Iii~~~~~~~~~~~~~~~~ UL z Li- 7500 0 Id I W~~~~~~~~~ U) C/ / [Id En,. t o,!//// 0 - CRITICAL RADIUS i,i 1.01.1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 BUBBLE RADIUS~r,IN FEETx I4 Figure 59. Diagram Relating Static Pressure to Bubble Size.

The water used in the experimental phase of the study was obtained from the main recirculated supply available in the laboratory. It was found to be nearly saturated with air under atmospheric conditions. The fact that the water must have been in a supersaturated state at the point in the laboratory flow system at which column separation occurred might lead one to presume that the cavitation may have been gaseous cavitation. However, Parkin and Kermeen(38) determined that even with convective diffusion aiding gaseous cavity growth, the time required for such growth is measured in hundreds of milliseconds, whereas, the time required for true vaporous cavitation is of the order of a few microseconds. Moreover, as noted by Strasberg(51) true vaporous cavitation is virtually independent of the dissolved air content of the water, except insofar as the dissolved air may influence the size and growth of subcritical nuclei. Vaporous cavitation with its explosive bubble growth requires the pressure to fall below the liquid vapor pressure, if only for an instant. Gaseous cavitation, on the other hand, can occur at a pressure above the vapor pressure provided this pressure is maintained long enough to sustain the comparatively slow diffusion process. Thus, three factors indicate that the main column separation void, as well as the cavitation occurring ahead of it are, in fact, of vaporous origin: (a) the almost instantaneous explosive character of bubble-cavity formation, (b) the insufficient time for gaseous cavitation, and (c) the complete (or nearly complete) disappearance of both the separation-void and bubble cavities with the sudden reoccurrence of high pressures. Recent studies by Ripken and Killen(44) offer a possible explanation for the appearance of the vaporous bubble cavities near the top, inner surface of the pipe and in the wakes behind the wires of the

wave gages, Their research revealed that vorticity and boundary-layer turbulence promote the diffusive growth of the entrained air-water-vapor nuclei, thereby promoting vaporous cavitation in these regions at the onset of subcritical pressures. From the laboratory observations and information pertaining to cavitation phenomena presented above, it appears that the mechanism of column separation and the accompanying bubble cavitation occurring ahead of the separation void may be described as follows: The initial high pressures in the pipe are relieved by flow of water toward the reservoir. When the motion of flow extends to the interface between the water column and the closed gate valve, tensile stresses tend to form in the water adjacent to the gate and the local pressure abruptly decreases below ambient pipe pressure. Minute quantities of dissolved air leave solution in free, molecular form and coalesce with the already present submacroscopic air-water-vapor nuclei. Pressures at the gate valve continue to decrease and to propagate from the valve toward the reservoir as a low-pressure wave. The rate of propagation is of the order of 2,5 to 405 x 103 feet per seconds Concurrently, a submacroscopic nucleus in the vicinity of the topmost region of the water-column, gate-valve interface is subjected to less than the critical equilibrium condition and expands explosively, The result is formation of a columnseparation void, The void continues to expand and propagate from the gate valve toward the reservoir, but with a celerity of only a few feet per second, Other nuclei situated between the propagating, low-pressure, wave and the much more slowly advancing separation void encounter less than critical conditions and expand explosively, also. These, then, are the bubble cavities observed to occur ahead of the void,

-142Contrary to the assumption of Chapter II that flow ahead of the separation void would be full-pipe flow, the bubble cavities produce a highly transient, metastable, two-phase flow characterized by intricate flow geometry and by ever-changing boundary conditions. The outlook for a rational interpretation of the energy dissipation rate under such conditions is dismal at best. However, the rate of energy dissipation that occurs under somewhat similar flow conditions is known to be greater than for comparable single-phase, steady-state, full-pipe flow(57) This is so, first of all, because of the complex character of the two-phase flow pattern and, second, because of the energy losses associated with cavitation itself,(33) Nevertheless, the treatment of energy dissipation in two-phase flow, either by rational or empirical means, is beyond the scope of this study. In order to test the notion that a higher rate of energy dissipation would shorten the duration of the theoretically-determined column separation intervals and would thereby bring the computer-simulated results into closer agreement with the experimental findings, the frictionloss coefficient in the simulation model was arbitrarily adjusted, The computer program was temporarily modified so that whenever the absolute pressure head at any point in the pipe fell below 12 feet of water during periods of column separation, the friction coefficient at that point was increased by a factor of 3, The simulated results obtained with this arbitrary modification did indeed verify the notion; the duration of the first column separation interval was decreased, the distance of travel of the separation void was reduced, and the magnitude of the second pressure rise was brought more nearly into agreement with the laboratory finding. The simulation was not carried beyond the second pressure rise.

From a qualitative point of view, the better agreement between the theoretical and experimental results tends to support the arguments for increased energy dissipation. Quantitatively speaking, however, the arbitrary increase in the friction coefficient is of no significance because the column separation phenomenon is dependent upon the following related factors: (a) The nucleation characteristics of the water -- specifically the size spectrum and quantity of submacroscopic nuclei. (b) The vorticity and turbulence-generating characteristics of the particular pipe. (c) The diameter of the pipe and the velocity of the flow in the pipe. (d) The magnitude and occurrence characteristics of the negative pressure initiating column separation0 Although iteration methods could be used to determine the friction coefficients which would bring the theoretical findings into very close agreement with each of the experimental results, such coefficients would be highly empirical and would have very little transfer value to other flow conditions, Thus, the significance of this discussion may be summarized as follows: The theoretical representation of the column separation phenomena depicted by the computer model is a limiting condition, During actual column separation under typical transient flow conditions, the separation. void(s) that form will be of shorter duration and lesser extent, and the pressure rises after the initial rise will be of lesser magnitude than predicted by the mathematical model, The actual column

-144separation phenomena will tend to approach the theoretically predicted conditions derived from the model only when a deficiency of air, watervapor nuclei in the water inhibits the full development of two-phase flow ahead of the void.

CHAPTER VII CONCLUSIONS The primary objective of the study was to provide better insight into the fluid dynamics of liquid column separation. The approach used was to develop a rational, generally applicable, mathematical model with which to numerically simulate transient flow, including liquid column separation, and thereby duplicate the essence of the phenomena without actually attaining reality. The findings derived from simulation were then systematically compared with the corresponding findings obtained from concurrent experimental investigations conducted in the laboratory. The conclusions drawn from the study are: (a) The two sets of partial differential equations, describing the transient pressure waves in a pipe flowing full on the one hand, and transient, free-surface, gravity-type waves in the partially-full pipe on the other hand, are both of the nonlinear, hyperbolic type and directly amenable to solution by the method of characteristicso The appropriate effects of fluid friction upon each type of transient flow are readily included in the solution, (b) By means of the appropriate boundary conditions and by considering other controlling physical parameters, a sensitive articulate, and highly sophisticated mathematical model representing the transient-flow, column-separation phenomena can be created, -145

(c) Experimental laboratory studies of transient flow with column separation reveal that vaporous cavitation occurs throughout much of the length of the pipe under conditions of supposedly full-pipe flow ahead of the void. As a direct result of this cavitation, flow is temporarily transformed into highly transient, metastable, two-phase flow. (d) Comparison of computer-simulated and experimentallydetermined transient flows accompanied by column separation reveals that the mathematical model does indeed reproduce the characteristics of the phenomena, e.g. the time sequence of pressure rises at the closed valve separated by column separation voids. The magnitude, duration, and form of the first simulated pressure rise compared very favorable with its experimentally-derived equivalent. However, after the first experimental pressure rise the subsequent rises are observed to diminish in magnitude more rapidly with time than their simulated counterparts. Moreover, the duration and extent-of-travel of the successive experimental voids are also considerably less than predicted from the mathematical model. (e) An overall similarity in shape is apparent between the experimental voids and their model-derived counterparts. However, free-surface ripples and other small perturbations are evident on the free surface of the experimental voids that do not appear on the rather idealistically smooth free surface of the simulated voids.

-147(f) The higher rate of energy dissipation associated with the two-phase flow produced by cavitation ahead of the void is responsible for the shortened duration and extent of the experimental voids and for the diminished magnitude of the experimental pressure rises subsequent to the first rise, Whereas the model assumes viscous energy dissipation governed by the velocity of the full-pipe flow, the twophase, cavitation-producing flow is accompanied by thermodynamic as well as more vigorous viscous energy dissipation. (g) Computer-simulated transient flows determined from the mathematical model represent a limit or maximum condition which will be approached only when cavitation ahead of the void is nominal. Hence, the model has direct engineering application in the determination of design criteria. Future study of transient-flow accompanied by column-separation should be directed toward incorporating the thermodynamic aspects of liquid column separation into the mathematical model, Vaporization, heat conduction, and condensation are factors of greater significance as a result of the occurrence of cavitation ahead of the separation void. Further experimental investigation is desirable using larger diameters of pipe and higher velocities, It is recommended that for any future experimental study stroboscopic photography be used to determine the form of the separation voids rather than the wave gages used in this study. Such a procedure would remove the possible influence of the wave gages on the formation of cavitation, provide more data on the void, and eliminate troublesome instrumentation requiring repeated calibration,

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-15137. O'Brien, Mo P., and Hickox, G. H., Applied Fluid Mechanics, New York: McGraw-Hill Book Co., 1937, p. 242-255. 38. Parkin, B. R., and Kermeen, Ro W., The Roles of Convective Air Diffusion and Liquid Tensile Stresses During Cavitation Inception, Proceedings, IAHR - Symposium on Cavitation and Hydraulic Machinery, Sendai, Japan, 1963, p. 17-35. 39. Parmakian, John, Waterhammer Analysis, New York: Prentice-Hall, Inc., 1955, 145 p~ 40, Prescott, John, Applied Elasticity. London: Longmans, Green and Co., 1924, p. 326-332. 41. Reddick, H. W., and Miller, F. H., Advanced Mathematics for Engineers. New York: John Wiley and Sons, Inc., 1957, p. 110, 114, 139-143. 42, Rich, G. R., Hydraulic Transients. 2nd Edo, New York: Dover Publications, Inc., 1963, p. 1-73, 166-187o 43o Richard, R. T., "Water-Column Separation in Pump Discharge Lines", Transactions, American Society of Mechanical Engineers, New York, Vol. 78, 1956, p. 1297-1306, 44. Ripken, J. F., and Killen, J. M., Gas Bubbles; Their Occurrence, Measurement, and Influence in Cavitation Testing, Proceedings, IAHR - Symposium on Cavitation and Hydraulic Machinery, Sendai, Japan, 1963, p. 37-57. 45. Saint Venant, JO C, B, de, Theorie du Mouvement non-permanent des eaux, Institute de France, Academie des Sciences Comptes Rendus, Vol. 73, Paris, (July 1871), p. 147-237. 46, Schonfeld, J, C., "Resistance and Inertia of Flow of Liquids in a Tube or Open Canalt", Applied Science Research, Section A, Vol.. 1, No. 13, Netherlands, (1948), p. 169-197. 47. Schonfeld, Jo C,, Propagation of Tides and Similar Waves,'s-Gravenhage, Netherlands, Staatsdrukkerji Vitgevenijbedijf, 1951, p. 31-69, 48. Solkolnikoff, I. S., and Redheffer, R, M., Mathematics of Physics and Modern Engineering. New York: McGraw-Hill Co., Inc., 1958, p. 509-521, 49, Sommerfield, Arnold, Partial Differential Equations in Physics. New York: Academic Press, 1949, p. 52-55, 50, Stepanoff, Ao Jo, and Kawaguchi, K., Cavitation Properties of Liquids, Proceedings, IAHR - Symposium on Cavitation and Hydraulic Machinery, Sendai, Japan, 1963, p. 71-85,

51o Strasberg, M., Undissolved Air Cavities as Cavitation Nuclei. First International Symposium on Cavitation in Hydrodynamics, Report No. 6, National Physics Laboratory, Teddington, England, 1955, 19 po 52, Streeter, Vo L., Fluid Mechanics. 3rdo Edo, New York: McGraw-Hill Book Co,, Inc,, 1962, pe, 211-222 53~ Streeter, V. L., and Lai, Chintu, Waterhammer Analysis Including Fluid Friction, Proceedings, American Society of Civil Engineers, Ann Arbor, Michigan, Vol. 88, No. HY3, 1962, p. 79-112. 54~ Streeter, V. L., Keitzer, W. F., and Bohr, D. F., Pulsatile Pressure and Flow Through Distensible Vessels, Circulation Research, American Heart Association, Vol. XIII, No, 1, 1963, p. 3-20~ 55, Stoker, JO J,, Numerical Solution of Flood Prediction and River Regulation Problems, New York University, Institute for Mathematics and Mechanics, NYU-200, 1953, 36 p. 56. Stoker, J. J., Water Waves. New York: Interscience Publishers, Inc., 1957, p. 291-314, 451-509. 570 Tek, M. R., Two-Phase Flow: Handbook of Fluid Dynamics. Section 17, edited by Vo L. Streeter, New York: McGraw-Hill Book Co., Inc., 1961, p 1-39. 58. Thomas, Ho A., The Hydraulics of Flood Movements in Rivers, Pittsburgh) Carnegie Institute of Technology, 1934, p. 9-45~ 59, Timoshenko, S., Strength of Materials. Part II. New York: Van Nostrand Co,, Inc., 1930, p. 528-533. 60, Tollmein, W,, Theory of Characteristics (translated from German), National Aeronautics and Space Administration, 1942, 28 p.

APPENDIX I MAIN COMPUTER PROGRAM A complete syntactical listing of the MAD language computer program for simulation of transient flow accompanied by column separation in the pipe system shown in Figure 14 is presented below. The program, which is designated MAIN, is divided into several subsections, each of which is identified and briefly described by comments regarding its primary function. These subdivisions also correspond to the subdivisions of the flow diagram of Figure 13. All internal subroutines are included in the listings. External subroutines, however, are listed syntactically in Appendix II. -153

$ COM1PILE FAD,PRII"'T C)RJECT,EXECIJrEPU NCH []BJECTI/O DUMP MAIN 001 070785 04/06/64 10 27 54.8 PM $ FIILL DlJUMP MAD (12 MAR ~1964 VERSIIJN) DRIJGRAM LITING......... THIS PROGRAM IS A THEORETICAL ANALYSIS (iF THE PHENOMENA OF LIQUII. COL'UM. SEPARATIION IN A HOtRIZONTAL PIPE AS ENCOUNTERED UNDER CONDITIONS OF TRANI'SIENJT FLOW. THE PROGRAM HAS BEEN WRITTEN IN''MAD", THE MICHIGA4N.LGLRITHM DECODER LANGUAGE ANU EXECUTED O(N THE IiHM 7090 AT THE UNIVERSITY OF MICHIGAN C3MPUTING CENTER. INTEGER I,J,K,L,M,N,MNGAPMAXLIM,CUTOFF,SHIJTnF,PRTLIMPRTLFRQ *001 0,PRTCYC, INCVPTMUFSEPtMOFCOL,JJ, JJQtKKt JHK,INCtMK FACTORtC, *001 I SPRK,IPRK,LPRK,FVAiVB,PMK,MARIK,COUJT *001 STATEMENT.LAHEL STATEA,STATCH-i,STA4TEC,STATED,ATEE *002 FORMAT VARIA3LE FVA,FVi *003 DI'M'INIUON V(Z1lO,ADIIl),H(210,ADIM).,VP(210,ADIM),HP(210,ADIM), *004 C EL (21)),THET(21O,ADIM),ALPtI4A(200),l2),B(2),AOG(2 ), *004 1 LL(2),MN(2),ELTT(Z) DELXI),MU(2),A(2),ZETA(2),CP), *004 2 GOA(2), IVEL(2),HLf)S(2),U(l),Z (100),UP( 100),ELP(2OADIM) *004 PRO;R'AM COMMO;) R(2),XItD12),PAREANU,GAPF(102,r)IM),ZP(100), *005 ( SPRK, IPRKtLPRK,FVA,FVBISTATDNYORK,PLNNSYSUZ (11) *005 VECTOR VAI. UES DIM = 2,1,51 *006 VECTOI' VALUES ADIM = 2,1,105 *007 UNITEr) CONTINUE'00 READ FORMAT CARD,RHtI,KA,VMAX,GAP,HMAR,NU,Q *009 REAl' FOIIRMAT PLANHEI-,MF-,LELWSEL,LOCI,LUC2,LOC3 o010 RCAr) F'OUMAT PIPEIEIl),ZETAI ),D(1),B( l ),R(IlLL)LI(),MN4() *011 READ rlORMAT PIPE2,E(2),ZETA(2),D(2)B(2) tR(2),LL(2),MN(2) *012 READ Af ORMAT CONTRL,MAXL[M,CUTOFF, INCVPTSHUTOF,PRTLIMPRTFR *013 O,PRTCYC *013 READ FORMAT QUANT,INC,HVAPn,,XI,FACTOR,IOTAZOV *014 READ FORMAT FRICL,F(I,O)....F(1,35) *015 READ FORMAT FRIC2,F(20)...F(2,35) *016 READ FORMAT CSURG,SIJZ(O).. SUZ(10) *017 CPI ) = 1.-ZETA( )*( (11 )-RI) )/R 1)) C18 CP(2) = 1.-ZETA(2)*( (B(2)-R(2) )/R(2) )-ZETA(2)*ZETA(2)*(B(2)/R *019 u (2)) *019 PRINT COMMENT $1 *020 0 GIVEN I NFORMATION $ *020 PRINT COMMENT $-GIVEN PHYSICAL DATA AND CONSTANTS $ *021 PRINT FORMAT GI VEN, HO,KA,VMAX, HBAR,Q *022 PRINT COMMENT $ODIMENSIONAL DATA DESCRIBING ANALYTIC MODEL $ *023 PRINT FORMAT SCHEMEtHEL,MEL,LEL, WSEL,LOC1,L0C2,LOC3 *024 PRINT COMMENT $OPIPE CHARACERISTICS AND RELATED DATA S *025 PRINT FORMAT TUBE1,L( ),ZETAIl),D(1),B(1),R(1)LL(1),MN(1) *026 PRINT FORMAT TUBE2),(2),ZETA(2),D2),B(2)R(2R(2 2),MN(2) *027 PRINT COMMENT $OPROrGRAM OPERATION AND CONTROL DATA $.*028 PRINT FORMAT I)RDERS,MAXLIM,CUTOFF,INCVPT,SHUTOF,PRTLIM,PRTFRQ *029 0 PRTCYC 029 PRINT COMMENT SOFRICTION VALUES FUR CONSECUTIVE REYNOLDS NUMB *030 0 ERS FROM 100 TO 35100 IN INCREMENTS OF 1000. $ *030 *. * * * *

PART I - PART ONE or THIS P!RUGRAM IS PRIMARILY CONCERNED WITH THL TRANlSIENT CONDITIO;S I.N THE FLOW SYSTEM (OR PORTION THEREOF) WHICH IS ALWAYS FLOWING FULL. SEVERAL STATEMENTS OF INTRODUCTION AND INITIALIZATION FOLLOW. WHE'EVER M;4( t).E.2 -031 S TA TEA=M ICHGN *032 S TATEh=MONTAN *033 OTHERWISE *034 STATEA=MISSIP *035 S TA'B=MA I NE *036 END OF CONDITIONAL *037 MOFSEP = MALIM *038 JJ = tAXLIM *039 MARK=F *040 VEEE=O. *041 THROUGH STATESFUR 4=0, IN.G.35 *C42 REYO=10 0. + ( N*1000.) *043 STATES PRINT FORMAT RESFAC,RE'YNOtNFI,N),N,F2,NN) *044 THROUGH ALASKA, FOR J = t1,1J.G.2 *045 ZW=) ( J ) *046 A(J) = S(ORT.(1./(RHO*( (1./KA)+(D(J)*CPIJ))/(E(J )J*(IJ)-RIJ) *047 0 )) ))) 047 OELX(J) = LL(J)/MqN(J) *048 IVEL(J) = Q/(3.14L59*R(J).P.2) *049 HLOS(J) = FFC.(Zw,IVEL(J),J)*DELX(J)* *050 0 IVEI. (J).P.2/I0(J)-64.32) *050 ALASKA D'LTT(J) = DELX(J)/(.ASS.VMAX+.ABS.A(J)) 051 H WHENEVER OELTT 1).GE.OELTT(2) *052 DELr = )CITT 2) *053 UTHERW I SE *054 OELT = OELTT(1) *055 END OF CONDI IONAL *056 THROUGH ALABAM, FOR J = 1.1,J.G.2 *057 AOG(J) = A(J)/32.16 *058 GUAIJ) = 3R.16/A(J) *059 ALABAM MU(J) = DELT/DELX(J) *060 PART [ - SEC.A. THIS SECTION OF PROGRAM SETS UP THE THEORETICAL FLOW SYSTEM BEING INVESTIGATED AND DELINEATES THE APPROPRIATE POINT ELEVATIONS AND SLOPES AS WELL AS'THELVELOCITIES AND HYDRAULIC-GRADIENT HEADS UNDER STEADY STATE CONDITIONS. PRINT COMMENT $1 *061 O COMPUTED RESULTS $ -061 PRINT COMMENT $- COMPUTED VALUES OF ABSOLUTE PRESSURE HEAD, V *062 0 ELOCITY, ELEVATION, AND ANGLL OF PIPE INCLINATION AT TIME T = *062 1 0. $ *062 THROUGH ARIZON, FOR N = MN(1)+MN(2),-I,N.L.O *063 MARS = (N-MN(1) )*DELX(2)+LL( 1) *064 WHENEVER MARS.GE.LOC3 *065 ALPHA(N) = 0. *066

[I (`'I) = L L L *067 O0 tI FiN EV!0R r OAR S.GL.LC 3-I) LLX 2IA). OMARS.L.LOC3 *06b ALPHAIN) = A'~CII.((MEL-LFL)/(LGC3-LOC2)) *069 FL.N) = IEL+SP;.(A(ALPIIA( )) (LOC3-MARS) *070 CR IrfE!,iFVCR MARS.GE.10C,2 *07 1 ALPHA(A) = ALP1A4N+1) *072 EL(A) = EL(N+1)+SIN.(ALPHA(N))*DELX(2) *U73 (IR WHENFVFR MARS.GE.1)C2-UiELX(2).AND.NARS.L.L(JC2 *074 ALPHAV!) = AR(CS I.((HEL-M-.EL)/(LUC2 -LOC1l)) *U75 L(N) = NIsL+SINi.(Al-PHA(N))*((1OC2-MARiS) *076 OR VHL~IFVER MARSS.GE.LCI21 *077 ALPHA(M) = ALPHA(N+1) *07b'L(N) = EL(!N+1)+SU.-t4.(A'LHA(N))*OELX(2) *0 79 (ITHERIISE *080 ALPHA() = 0. *081 EL(N) = HEL *082 ARIZON END UF CUONITIUNAL *083 HAPS = iSfE +HfHAR *084 THROUGH CALIF, FOR I = O,1,I.G.MN(1) *085 N = I *086 ELP(1,1=11E(N) *087 H( 1, I)=HiABS-HLOS(2)*+MN(2)-HLOS(1)*( (MN(1)-N) *081 V(1,I) = IVEL(1) *089 THETA(l,I) = ALPtiA(N) *090 CALIF PRINT FORMAT MEMOIH(1,I),I,1V(1I)191 ELP(1,I),,THETA(iI) *091 THROUGH NCAROI FOR I = 0,19I1G.MN(2) *092 N = l+MN(1) *093 ELP(2,I)=EL(N) *094 H(291)=HA8S-HLOS(2)*(fN(2)-I) *095 V(2,I) = IV'L)?2) *096 H THETA(2,I) = ALPHAI-N) *097 \J NCANO PRINT FORMAT RANDUM, I,H(2,1)tlV(2,I), IELP(2,1)91,THETA(2, *098 0 I) *098 PRINT COMMENT $1 VALUES OF KEY VAP.IAB3LES USED THROUGHOUT THE *099 O COMPUTATIONS IN PART I. * *099 PRINT kESULUS A(l))A(2),ELX( 1),DELX(2) OELTTI)1)DELT(2), *100 u nELT, MU(I)tMU(2),CP(1),CP(2),A(UG(l),AOG(2),GOA(1),GGA(2) *100 VP(1,0) = V'(lO) *101 HPI2,NN(2))=H(?9,MN(2)) *102 PART I - SEC.1. THIS SECTION OF THE PROGRAM CONTROLS THE HOUNDARY CONDITIONS USED TO INITIATE TRANSIENT FLOW IN THE THEORETICAL SYSTEM AND TO CREATE THE VACUOUS CAVITY PHFNO"ENA. IT IS PRIMARILY INTENDED TO CONTROL THE SEQUENCE OF OPERATIONS WITH RESPECT TO TIME THROUGHOUT THE DURATION OF THE TRANSIENT PHENOMENON. TEXAS THROUGH ARKANSFDR M=l+MARK,1,M.G.MAXLIM *103 WHENEVER M,G.CUTUFF *104 TRANSFER TO WVIRG *105 OR WHENEVER M.E.SHUTOF *106 VP(L,0)=O. *107 TOFOFF = M*DELT *108 PRINT FORHAT OBDE,TOFDFFM *109 SHUTOF=O *110 END OF CONDITIONAL *111

PARi I SCC.C. THIS SECTIIJN OF THE PROGRAM COMPUTES THE VELOCITIES ANFD THE HEADS THiROUGHOUT THL X-T PLANE BY THL METHOr OF CHARACTERISTICS FOR THE SPECIAL CASE (IF THE FULL LENGTH OF A PIPE BEING EQUIVALENT TO AN INCREMENTAL LENGTH, THAT IS, EITHER THE LENGTH OF INCREMENT MN(1) = LL(1) OR THE LENGTH OF INICREMENT MN(2) = LL(2). HAWAII THROUGH COl ORAFOR 1=1.1, J.G.2 *112 Z = (J) 113 hHENEVER MN{J).L.2 *114 VL=V(J,O) *115 HL=V-H(J,O) *116 VR=V(J, 1) *117 HR=( J, I ) *118 VAL=VR+(VL-VR)*(VR+A J)I )U(J) *119 HAL=HR+(HL-HR )* ( VR+A J)MU) *120 VEP=VL+(VL-VR)*(VL-A(J))*MUIJ) *121 HEP=HL+ (HL-HR)* ( VL-A(J) )*MU(J) *122 WHENEVER J.L.2.AN). M. GE. MOFSEP *123 VP ( J,O)=VEP+ (HP ( J,O)-HEP)*GOA(J)-FFC. (ZW,VL,J)* *124 O { ( VL*.ABS.VL ) / ( 2.*D)))*DELT+VEP*SIN.(THETA *124 1 (J,O) )*(GDA( J )*I)ELr *124 PSI=VAL+HAL*(GOA(J))-FFC.(ZW,VR,J)* *125 O (( VR*.ABS.VR)/ (2.*D(.I)))*DELT-VAL*SIr. ( THCTA(J, 1)) *125 1 *(GOA(I))*DELT *125 OJR WHENEVER J.L.2 *126 HP(J,O)=HEP+(VP(J,0)-VEP)*(AOG(J))+FFC.(ZW,VL,J)* *127 O ((VL*.ABS.VL)/(2.*D(J)))*DLLT*AOG(J)- *127 1 VEP*SIN.(THETA(J,O))*DELT *127 WHENEVER (HP(I,O)-ELP(1,O)).L.HVAPOR,TRANSF-ER TO FL *128 O O' I0 *12d PSI=VAL+HAL*{GOA(J))-FFC.(ZW,VR,J)* *129 O (( VRABS.VR)/(2.*D( J)))*ELT-VAL*S I N.(( THETA(J,)) *129 1 *(G()A(J))*DCLT *129 OR WHENEVER J.G.1 *130 CHI=VEP-HEP*(GOA(J))-FFC.(ZW,VL,J)* *131 C ((VL*.ABS.VL)/(2.*D(J)))*DELT+VEP*SIN.(THETA(J,0) *131 1 *(GOA(J))*DELT *131 HP(J,O)=(1./((GOA(J))*((D(J-1)/D(J)).P.2.)+GOA(J))) *132 O *~PSI*((U(J-1)/D(J) ).P.2.)-CHI) *132 WHENEVER HP(J,O).L.HVAP)R+ELP(J,O).AND.M.GE.MOFSEP, *133 O HP(J,O)=HVAPOR+ELP(J,O) *133 VP(J,O)=CHI+HP(JO)*(GOA(J)) *134 HP(J-1,MN(J-1))=HP(JU) *135 VP(J-1,NJ- -1) )=VP(J,0 )*((D(J)/D(J-1) ).P.2.) *136 PSI=O. *137 CHI=O. *138 VP(J,1)=VAL-(HP(J,1)-HAL)*(GOA{J)) *139 0 -FFC.(ZW,VR,J)*((VR*.ABS.VR)/ *139 1 (2.*D(J)))*DELT-VAL*SIN.(THETA(J,1))*DELT*G(A(J) *139 END OF CONDITIONAL *140 TRANSFER TO COLORA *141 END OF CONDITIONAL *142 * * * * * *+

PA:"1V I - SEC.D. THIS SECTION OF THE PROGRAM COMPUTES THC VELOCITIES AND THE H-iEADS THitUUGHOUT THE X-T PLANjE YV THIE METHOD OF CHARACTLRISTICS FOR THL GENERAL CASL9 THAT IS, FOR THE LENGTH LL(l) EQUJIVALENT To) AN INTEGRAL IMULTIPLE, MN(11, OF THE INCREMENTAL LFNGvTH DELXI1), AND THiE LE:4ITH LL(2) EQUIVALENT TO AN INTEGTRAL MULTIPLE, MN(?), OF THE INCREMLNTAL LENGTH 0ELX(2). THROUGH COL))RAFOR I=O,1,I.G.Mj(J)-2 *143 VL=VIJ, ) *144 HL=HIJ.I) *145 VM=V(IJ,1+1) *146 HM=H(JI+I) *147 VR=V J,1+2) *14b HR-=1-II J, 1+2) *149 OETLRfA4hATION OF PRESSURE AT IOULNARY CONOITION X = 0, OR AT JU.JCTION BETWEEN PIPES. WHENEVER I.L.1 *15u VEP=VL+(VL-VM)*(VL-A(J))*MU(J) *151 HEP=HL+(HL-HM)*(VL-A(J))*MUIJ) *152 WHENEVER J.L.2.AND.M.GE.MO FSEP.153 VP(JtI)=VEP+(HP(JI)-HEP)*GOA(J)-FFC. (ZwVLJ) *154 0 *((VL*.AiS.VL)/(2.*D)IJ)))*DLLT+VLP*SIN.( *154 1 THETA(JIl))*GUA(J)*DELT.154 OR WHENEVER J.L.2 *155 HP(JI)=HEP+(VP(JI)-VEP)*(AOG(J)) *156 FO +FFC.(LWPVLJ)*((VL*.AbS.VL)/ *156 1 (2.*D(J)))*DELT*AOG(J)-VEP*SIN.(THETA(J,)I*DELT *1156 WHENEVER (IIP(1,0)-EIPII,0)).LHVAPORTRANSFER TO 15 1 o FLORID *157 OR WHENCVLR J.G.1 *158 CHI=VEP-HEP*GO)A(J)-FF;.(ZWVLtJ) *159 o *((VL*.ABS.VL)/(2.*D(J)l)*DELT+VEP* *159 SIN.(THETA(Jo,))*GOA(J)*DELT *159 HP(J,I)=(l./I(GOA(J-1))*((D(J-1)/D(J)).P.2.)+ *160 O GOA(J)))*(PSI*( (O(J-)/)(J)).P.2.)-CHI) *160 WHENEVER HP(JI).L.HVAPOR+ELP(JI).ANO...GE.MOFS *161 O EPHP(JI)=HVAPUR+ELP(CJI) *161 VP(JI)=CHIeHP(JI)*(GOA(J)) *162 HP(J-1qMN(J-1))=HP(J,1) *163 VP(J-1,MN(J-1))=VP(J,I)*((D(J)/D(J-1)).P.2.) *164 PSI=O. *165 CHI=0. *166 END OF CONDITIONAL *167 END OF CONDITIONAL *168 COMPUTATION OF PRESSURES AND VELOCITIES AT INTERIOR POINTS. VAL=VM+(VL-VM)*(VM+A(J) )*1U(J) *169 HAL=HM+(HL-HM)*(VM+A(J))*MU(J) *170 VEP=VM+(VM-VR')*(VM-A(J))*MU(J) *171 HEP=HM+(HM-HR)*(VM-A(J))*MU(J) *172 VALLY = VAI..ABS.VAL *173 VEPPY = VEP*.ABS.VEP *174 FFCC = FFC.(LWvVMJ) *175

V(IJ, I+1)=O.5*(VAL+VEP)+0.5*(HAL-HCP)*GOA(J) *176 u -FFCC*((VALLY+VEPPY)/(4.*D(J)))*DELT-o.5*SIN. -176 1 (THETA(J,I+1))*(VAL-vEP)*GOA(J)*O)ELT *176 HP(J, I+1)=0.5*(VAL-VcP)*AOG(J) +O.5* (HAL +fEIP) *177 0 -FFCCAOG(J)M ( (VALLY-VEfPY)/(4.*I)(J)) *)ELT-0.5*S IN. 177 I (THIET(J,I++1) )*(VAL+V\/P)*OELT *177 WHENEVER HP(J,I+L1 ).1.VAPR+ELP(J, +1 ) AN.M. Gt.MQFSLP *178 0,HP ( J,I+1 ) =HVAPO+LLP J,I ) *178 DErERMINA4Ar TON [IF PRESS(IRE AT BOUN[DARY CONDIT ION X = LL(1) + LL(2), OR AT JUJ:CTI10, BETviEEN PIPES. WHENEVER I.GE.MN(J)-2 *179 VAL=VR+IVM-VR)*(VR+A(J))*MU(J) *180 HAL=HR+IHM-HR)*(VR+A(J))*IU(J) *181 WHLNEVER J.L.2 *182 PSI=VAL+HAL*(GOIA(J ) )-FFC. (ZW,VR,J *183 C *(VR*.ABS.VR)/(2.*lJ) ) )*DELi'-VAL *183 1 SIN.(THETA(J,I+2) )*GA(J)*DELT *183 OR WHENE/LR J.G.1'184 VP(JI+2)=VAL-(HP(2,MN(J) )-HAL) *GOA(J) 185 O -FFC. (LWVR,J)* (VR*.ABS.VR)/ 185 i (2{.*D(J)) )*ELT-VAL*SIN.(TiiETA(J, 1+2))*GIOA(J)* *185 2 I)ELT *185 END OF CUNDITIONAL *186 ElJD OF CONDITIONAL *187 COLORA CONTINUE * 188 HHOLD = Ht 1,0) 189 HOLDD = V(2,MI4(2)) *190 MOVC FIORWARD DELTA T IN THE T-DIRECTION ON X-T PLANE. THROUGH SCAROFOR J=1,1,J.G.2 *191 THROUGH SCAROFOR I=O,l,I.G.MN(J) *192 V(J, I )=VP(J,I) *193 SCARO Hi(J,I)=HP(J,I) *194 PART I - SEC.E. THIS SECTION OF THE PROGRAM GOVERNS THE PRINTING OF COMPUTED RESULTS AND PROVIDES CERTAIN SWITCHING OF OPERATIONS TO PART II. WHENEVER V(2,MN(2))/HOLDD.L.O.,PRINT FORMAT BASSUN,M *195 0 aDELT+DELT/2. *195 WHENEVER M.E.MOFSEP *196 VEEE=VEEE+V(1,O) *197 PlINT COMMENT $0 THE VELOCITIES AND ABSOLUTE HEADS THR *198 0 OUGHTOUT THE REMAINDER OF THE FULL-FLOWING PIPE SYSTEM ARE AS *198 1 LISTED BELOW. $ *198 TRANSFFR TO STATEB *199 OR WHENEVER M.GE.MOFSEP.AND.M.L.JJ *200 VEEE=VEEE+V( 1,O) *201 TRANSFER TO ARKANS *202 O(R WHENEVER M.GE.MOFSEP.AND.M.E.JJ *203 JJ=M+FACTOR *204 VEEE=VEEE+V( 1,O) *205 TRANSFER TO GEORGI *206

IOWA CUNTIiJUE *?07 VEEE=O. *208 WIlENEVER M.LofIFSLP+(INCVPT*FACTOR),TRANSFER TO RHODLI *209 WHENEVER M.NE.JJQTRANSFER TO ARKANS *210 RHO!)EI T=DELT*M -21t PRINT FORMAT CELLLJ,T,M *212 PRINT COMPENT $0 VAPUR CAVITY $ *213 EXECUTE PRTLAW.(PMK) *214 NJERSY PRINT FORMAT HARP,(K=SPRKIPRK,K.G.LPRKK*DELXC)t, 215 0 (K=SPRK,IPRKK.G.LPRK,KI,(K=SPRK,IPAKK.G.LPRKU(K)), *215 I (K=SPRKIPRK,K.G.LPRKZ(K)) *215 TRANSFER TO STATED *216 PENNSY EXECUTE PRTJL.I(PMK) *217 TRANSFER TO NJERSY *218 NYORK JJO=M+(PRTCYC*FACTOR) *219 PRINT COMMENT SO FULL-FLOWING PIPE SYSTEM $ 220 TRANSFER TO STATEB *221 MONTAN PRINT FliRMAT FLUTE,V(1,0),Vl,1l),V(l1,2)tV12,03)V(220) *222 V(2,40).V(2,6O)V(280), V (2,8 100))(H(1,0)-ELP(1,0)), *222 1 (H(1, 1)-ELP(,1)), (H( 12)-EL(,2 H(,G-ELP( 1,23 (H(2,03-ELP(2,0)), *222 2 (H(2,20)-CLP2?,20)),(tH(2,40)-ELP(2,40)),(H12,60)-ELP(2 *222 3,60)),III(2,80)-ELPIZ,BO)leIH(2,L0O)-ELP(2.100) )*222 WHENEVER M.L.MARK,TRANSFER TO TEXAS *223 TRANSFER TO ARKANS *224 MAINE PRINT FORMAT [JRGAN,Vll,0),V(ll1)tV(2,O),V(2t10), *225 O YV(2,20),V(2,30),V(2,40),V(2,50),(H(1,O)-ELP(1,0)), *225 1 (H(1,l)290(1,1)),(H12,0)-ELP(2,0)),1H12,10)-ELP(2,1O) *225 2 ),(H(2,20)-ELP(2,20)),(H(2,30)-ELP(2,30)),(H(2,40)-ELP *225 3 (2,40)),(H(2,50)-ELP(2,50)) *225 WHENEVER M.E.MARK,TRAI4SFER TO 1EXAS *226 0C TRANSFER TO ARKANS.227 O E;OD OF CONDITIONAL *228 WHENEVER (H(1,O)-ELP(1,03).L.(HVAPOR+1.).OR.(HHOLD-HI1,O *229 O )).G. 10.TRANSFER TO DELAWA.229 WHENEVER M.L.PRTLIP,TR4NSFcR TO DELAWA *230 WHENEVER M/PRTFRQ.tE.M/(PRTFRQ*I.),TRANSFER TO ARKANS *231 DELAWA T=DELT*M *232 PRINT FORMAT TUBA. TM *233 TRANSFER TO STATEA *234 MICHGN PRINT FORMAT FLUTE,V(1,O),V(1I1.,V(1,2), )v2,OI.V(2,20) *235 0,V(2,40),V(2,60,V(,80),V(2,100), (HI,O )-ELP 1,G)). *235 1 (H(1.1)-ELIP(1,1)),IH(1,2)-ELP(1,2)),(H(2,O)-ELP(2.0)), *235 2 (H(2,20)-ELP(220)),(H(2.40)-ELP(2,40)),(H(2,60)-ELP(2 *235 j,60)) (H(2,80)-ELP(2,80)),(H(2.100)-ELP(2,100)) *235 TRANSFER TO ARKANS *236 MISSIP PRINT FORMAT ORGANV(1,0),V(111),V(2O),V1(2,10),V(2,20), *237 O V(2,30),V(2,40),V(2,50),(H(1,0)-ELP(1,O)),(H(1,1)-ELP *237 I (1,1)),(11(2,0)-ELP(2,0)),(H(2.10)-ELP(2,10)),(H(2,20)- *237 2 ELP(2,20)),(H(2,30)-ELP(2,3O)),(H(2,40)-ELP(2,40)),(H(2,.237 3 50)-ELP(250G)) *237 ARKANS CONTINUE *238 TRANSFER TO CONNEC *239 PART II - PART TWO OF THIS PROGRAM IS CONCERNED WITH THE PARTIALLY FULL PIPE FLOW (OPEN-CHAINEL FLOW) ENCOUNTERED DURING THE PRESENCE OF A VACUOUS CAVITY. MORE SPECIFICALLY, IT IS CONCERNED wITH THE FORMATION AND COLLAPSE OF A VAPOR CAVITY. SEVERAL INTRODUCTORY AND DEFINITIVE STATEMENTS

FOLLOV. FLORII morFSEP=m *240 T UF SE P= MOE SEP *EL T *241 PRINT FORMAT QRUM,TOFSEPMOFSEPTOFSLP-TOFOFF *242 V(10) =V(1,1) *243 H()1,O)=HVtiPf)R4ELP1,0) *244 HP 1,O)=HVAPOR+ELP(19,) *245 J=l *246 THRJOUGH NDAK.)JTFOR K=O,1,K.G.INC *247 NCAKOT 7(K) 1) *248 UJ(o)=;. *249 U) )=V( I,O)*ZOV *250 U(2)=V(1,O)*ZOV *251 PAREA=3.14159*R(1).P.2. *252 JP(O)=0. *253 MIT=.015 *254 LAMt~DA=SORT. PAREA*32.16/MINIT) *255 PrLLTC=FACTOR*DELT *256 r)ELXC= r)ELrTC*(.A3S.V Ax+.AS.LAMbOAA) *257 OCITFL)(=DELTC*V) I O)*ZOV*PARLA *258 STATEC-KANSAS *259 STATEL t.KLAH)J *260 C UNiT = *261 YY=O. *262 JHK=O *263 DEFINITIONS OF INTERNAL FUNCTIONS INTERNAL FUNCTION TOPWTH. (Z) =2.*SQRT. (Z*DI 1 )-Z.P.2.) *264 INTERNAL FUNCTION So AGOTr.IL)=SQRT.(PFPA.I, J)*32.16/TOPWTH.(Z *265 0 )) *265 INTERNAL FUNCTION SOAOGT.IL)=SQRT.CPFPA.(ZJ)/(TOPWTH.(Z)*32. *266 O 16)) *266 INTERNAL FUNCTIUN HOHVOL.IX)=PARVOIL.IA ARK.)XJ),J) *267 INIERN. 1;kAC. FONICTION OUbEQH b.(X)=BUBVOL.(X)-OUTFLU *268 INTERNAL FUNCTION SURGEO.(NA)=SIMPWV.IOELXCMKINXJYY)- *269 O INFLOW *269 PART II - SEC.A. THIS SECTION OF THE PROGRAM CONSIDERS THE FORMATION OF THE VAPOR CAVITY. THE INITIAL GROwTH OF THE CAVITY IS CONSIDERED TO OCCUR AS A FRICTIONLESS, NEGATIVE SURGE WAVE IN A PARTIALLY FULL-FLOWING, CIRCULAR, OPEN CHANNEL. SEPARATION OF THE LIQUID AT THE TOP OF THE PIPEWALL IS APPROXIMATED BY A PARAHOLIC SURFACE. X=IISECT.).OOl*DELXC,10*DELXCBUBEQtO.,IOTAVIRGNA) *270 KK=X/I)LLXC *271 wHENLVLR KK.GE.9 *272 TRANSFER TO NMEXCO *273 OR WHENEVER KK.GE.6.AND.KK.L.9 *274 OELXC=3.*DELXC *275 KK=X/DELXC *276 PRINT COMMENT SOTHE MAGNITUDE OF DELXC IS TRIPLE THE MINIMU *277

0 M VALUE. $. *277 (OR WHENEVER KK.GE.3.ANP.KK.L.6 *278!)LXC=2.*r)ELXC *279 KK=X/DELXC *280 PRIM!l COMMENF SOTHE MAGNIIUDE OF DELXC IS DOUBLE THE MINIMU *281 0 M VALUE. $ *281 OTHERWISE *282 PRINT COMMENT $0 THE MAGNITUDE OF DELXC REMAINS THE MINIMUM *283 O VALUE. S *283 END OF CONDITIIONAL *284 MUCH=DELTC/DELXC *285 PRINT COMMENT $0 GIVEN DATA AND COMPUTED VALUES PERTINENT TO *286 0 COMPUTATION OF FLOW FOLLOWIJG COLUMN SEPARATION. S *286 PRINT FORMAT AMOUNT,HVAPOR,DELXC,DELTCtMARK,MUCH,FACTOR,XI, *287 O INCIOTA,PAREA *281 SP;<K=) *288 IPRK=I *289 LPRK=KK *290 FVA=KK *291 FVB=KK+1 *292 SSUZZ.=USURG. (DJ)-2.*XI*R(J) J) *293 KENTUC ZP(O)=PARK.(X,J)+R( ) *294 THROUGH VPMONT, FOR K=1,1,K.G.KK *295 ZP(K)=PARK.(X-(K*DELXC),J)+Rt1[ *296 UP(K)=V(,O)*ZOV+USURG. (Z(K).J)-SSUZZ *297 VRMONT 4HENLVER UP(K).L.O.,UP(K)=O. *298 THROUGH SDAKOT, FOR K=O,1,K.G.KK *299 U(K)=UP{(K) *300 SCAKOF Z(K)=ZP(K) *301 TRANSFER TO STATEE *302 H OKLAHO JJ=M+(FACTOR-1) *303 JJQ=JJ *304 I' VO I=OUTFLO *305 PRINT COMMENT $- THE INITIAL VELOCITIES AND DEPTHS IN THE VAP *306' 0 OR CAVITY ARE LISTED BELOW. $ *306 PRINT FORMAT HARP, (K=SPRKIPRKtK.G.LPRKK*DELXCI,(K=SPRK, IPRK *307 0,K.G.LPRKK),(K=SPRK,IPRKK.G.LPRKUIK))}(K=SPRKIPRKtK.G.LPR *307 1 KZ(K)) *307 MK=O *308 TRANSFER TO HAWAII *309 PART II - SEC.R. THIS SECTION OF THE PROGRAM TREATS THE GROWTH AND COLLAPSE OF THE VAPOR CAVITY. AGAIN THE METHOD OF CHARACTERISRICS IS UTILIZED, THIS TIME AS APPLIED TO OPEN-CHANNEL FLOW, IN ORDER TO DETERMINE THE VELOCITIES AND DEPTHS WITHIN THE CAVITY..NON-LINEAR QUANTITIES INCLUDING FLOW RESISTANCE DUE TO FRICTION ARE INCLUDED. GEORGI MK=MK+KK *310 WHENEVER MK.E.O.AND.KK.E.O *311 J=1 *312 QOUT=DELTC*(VEEE*ZOV/FACTOR)*PAREA *313 OUTFLO=VOID+QOUT *314 X=BISECT..001*DELXC,3.*DEXC3.DELXCB UBEQ. IOTAVIRGNA *315 KK-X/DELXC *316 VOID=OUTFLO *317

PMK=MK+KK -.*318 COUN' =COUNT+ 1 *319 WHENEVER COUNT.GE.3,TRANSFER TO WASH *320 STATEE=RHODCI *321 TRANSFER TO KENTUC *322 OR WHLNEVER MK.LE.O.AND.KK.L.O *323 TRANSFER TO TENNSE *324 END OF CONDITIONAL *325 J=l *326 U(I:K+1)=U(MK) *327 Z(MK+1 )=Z(MK) *328 THROUGH INDIAN,FOR K=O,1,K.G.MK-1 *329 UL=U(O') *330 ZL=Z(K) *331 UM=U (K+1) *332 ZM=L(K+1) *333 UR=U(K+2) *334 ZR=Z(K+2) *335 WHENEVER K.L.1 *336 QLZL=SQAGOT. (ZL) *337 kJEP=UL+ (UL-UM) * (UL-QZZL) *M(Ctl *338 ZLP=ZL+(ZL-ZM) * (UL-ZL )*MUCCH *339 ZP(K)=ZEP+(UP(K)-UEP)*SQAOGT. (ZL)+FFC. (ZL,UL,J) *340 O *UEP*.ABS.UEP*SQAOGT.(ZL)*DELTC *340 TRANSFER TO NHAMP *341 OTHERWISE *342 NHAMP QZZM=SQAGOT.(ZM) *343 UAL=UM+(UL-UM)*(UM+QZZM)*MUCH *344 ZAL=ZM+ (ZL-ZM) * (UM+QZZM)*MUCH *345 JUEP=UM+ UM-UR) * (UM-QZZM) *MUCH *346 H ZEP=ZM+(ZM-ZR ) * UM-QZZM ) *MUCH *347 UALLY=UAL*.ABS.UAL *348 UEPDY=LUEP*.ABS.UEP *349 CFFQ=FFC.(ZMUMJ) *350 VZZM=SQAOGT. ( ZM) *351 UP(K+1)=O. 5*UAL+UEP)+O. 5* (ZAL-ZEP) /VZZM-O.5*CFF(UALLY *352 O +UEPPY)*DELTC *352 ZP(K+1)=0.5*(ZAL+ZEP)+O.5*(UAL-UEP)*VZZM-O.5*CFFQVVZZM* *353 O (UALLY-(JEPPY)*DELTC *353 INDIAN END OF CONDIfIONAL *354 THROUGH IDAHOFOR K=O,1,K.G.MK *355 U(K)=UP(K) *356 IDAHO Z(K)=ZP(K) *357 QOUT=DELTC*(VEEE*ZOV/FACTOR) *PAREA *358 VOID=VOID+QOUT *359 WHENEVER QOUT.L.O.,TRANSFER TO STATEC *360 WHENEVER YY.G.0.,TRANSFER TO OREGON *361 KANSAS OUTFLO=VOID-SIMPQ.(J,DELXCMK) *362 TRANSFER TO OHIO *363 OREGON OUTFLO=VOID-(SIMPQ.(J,DELXC,MK-1)) *364 MK=MK-1 *365 VAPOR CAVITY GROWTH IS DETERMINED BY THE FOLLOWING STATEMENTS. OHIO WHENEVER OUTFLO.L.O.,TRANSFER TO UTAH *366 WHENEVER STATEC.E.NEBRSK *367 STATEC=KANSAS *368 JHK=O *369 YY=O. *3'70

EN!) OF CO'DJ[TIOIAL *371 X=t[IS1Et. (.q0*UEL^C,6. 1 *)ELAC, HUBEB., IiTA, V[RGNA *372 WHLNEVER X.LE.OELXC *373 7(MK)=PARK.IX,J)+,!I) *374 KK=O *375 PMK=MK *376 TRANtSFER TOr IO()W *377 ENU OF CO;IDI I OAL 3A78 KK=X/I)ELXC *379 THROUGH II.LNJI, f-fiR K=MK+1,1 K.G.M KK *380 (K)=PARK. (X-(I)ELXKC*(K-MK),J )+R() *381 WHE.iEVER Z(K).L.Z(K-1).ANn.K.,JE. (fiK+KK),Z(K)=Z(K-1) *382 U(K)=V( I,0)*ZOV+USURG.(Z(K),J)-SSIJUZZL *383 ILLiNOI WHENEVER U(K).L..,U(K)=C. *384 PMK=MK+KK *385 TRANSFER TO IOWA *386 VAPOR CAVITY SHRIA4KAC IES 1DETERMI4ED BY TIlE SUBSEQUENT S ATEMENTS. NEBRSK WHENLV/CR VOID.L.O,TRANSFER TO TLNASE *387 OUTFLI=V[)I-( S IMPQ. (J, DELXC,MK )-SIMPWV. (t)ELXC,Mk,YYJ,0. )) ) *38 UTAH INFLOW=-OUTFLO *389 STATEC=.NEBRSK *390 WHLNLVECR MK.L.(l+JH!(),TRANSFER TO IO(WA *391 NX=BISECT. (O.,5. 1*DELXC,SURGEQ., IITA,WISCIJN) *392 JHK= ( NK+YY )/DELXC * 39 3 YY= ( NX+YY)-( JHK*DEL(C ) * 394 KK=-JHK * 395 PMK=MK+KK *396h TRANSFER TO IOWA *397 PART II - SEC.C. [HIS SECTION OF THE PROGRAM RECORDS THE COLLAPSE OF THE VAPOR CAVITY AND) CARRIES [)UT THE REINITIALIZATIC.N AhND SWITCHING N4ECESSARY FOR CONTINUED COMPUTATION OF THE FULL-FLOWING, TRANSIENT PRESSURES AND VELOCITIES THROUGHOUT THE ENTIRE SYSTEM. TENNSE CONTINUE *398 MOFCOL=M *399 TOFCUL=DELT*MOFCOL *400 PRINT FORMAT P ICCLO, TOFC(OL, MOFCOL, TOFCOL-TOFSE P, TOFCOL-TOFOFF *401 MOFSEP=MAXL IM+2 *402 JJ=MAXL I + 2 *403 VP( 1,0 ) =O. *404 VOID=O. *405 VEEE=O. *406 MARK=M *407 TRANSFER TO STATEB *408 WVIRG PRINT COMMENT sOCOMPUTATION HAS BEEN HALTED BY CUTOFF. $ *409 TRANSFER TO WYOMNG *410 VIRGNA PRINT COMMENT O$ SUBROUTINE'BISECT' IS UNABLE TO FIND ROOT.$ *411 TRANSFER TO WYOMNG *412 WISCON PRINT COMMENT $0 SUBROUTINE'BISECT' COULD NOT FIND ROOT NX.$ *413 TRANSFER TO WYOMNG *414 CONNEC PRINT COMMENT SOCOMPUTATION HAS BEEN HALTED BY EXCEEDING MAXL *415

O IM. $.415 TRANSFER TO RY.IMNE, *416 NMEXCO PRIAT C(GM'IA11 S' -OTHE'AG4I Ii nF KK EXCEEDS N~IIE TIMES OELXC *41 1 () HICH IS CONSIL)LRED TO [RE EXCESSIVE. COMlPUTATION IS BEING il.417 I ALT0D FOK-O THIS RELVS!P4 5.417 TRANSTLE TO 4YOiMN-37 *416 WASH PRHIN COMMEAT IOTHF VAPOR CAVITY IS OF INSUFFICIENT MAGNJTUJE *4419 C TO iARRENT IU",THLR INVLSTIGATIO1-. COMPUTATION IS BEINJG HAL *4'-j I EU [1Ff (HISC RlrA AF).. *419 3FYOMNC cn JTI'F1 *42u VECE,' VALUES CARD =$F4.2,S5,ElO.4,S5,F-4.2,S6,15,S6,F6.3, *421 U S6 CA 7 Sl,F6.4 *s *421 VECTOR VALUt-c PLAN =$F7.3,SZ,61F7.3qS3).5 *422 VECTOR( VALUES PIPLI =$EIO.4,S4,F4.3,SI,3(F6.5,S4),F7.3,53,13 *423 O *s *423 VECTOR VALUES PIPE?2 =$L1.4,54,F4.3,Sl,3(F6.5,S4),F7.3,S3, 13 *424 0 *$ 0424 VECTOR VALUES JUANT = $13,S7,F6.3,S4,F4.3S4,[2,S4vF6.5,S4,F4 *425 O.2 * *425 VECTOR VALULS CONTRL = $2(114,S6),(II,59),(12tS3),2(14,6), *426 0 ( I1,S9) ) $ *420 VECTOR VALUES GIVENA =$1i1 tS2,6HRH(J F5.39,57,SHKA = E10.4S6, *427 o 7 HV MMAX = F4.2,56,7HfSliAP = F6.39, 8,4HI) = F6.4 *s *427 VECTOR VALUtS SCHE' —ME- =511H),S2,6f1f-,L = F7.3,P55tHMEL = F7.39 *428 O SfFGHLFL = F7.3,S4,fHiWSEL = Fl.3t"4,7HL0C1 =!-73,tS4,7HLflc/'= *428 I F7.3,S5,7HiLnci = Fl.3 *S.426 VECTOR VALUES TOOEl =1H,S2,IHE(1) = E1C.4,54,1O HZETA(1) = F *423 O 4.3,S4,7h1(l) = F.5,S4,7HB( l) = F6.5,S5,7HRMIL = F6.5,S58HL *429 I LMI) = F7.3,S4,HHMN(l) = 13 *$ *429 VECTOR VALUPS TOBE2 =Slf,52,t47HE(2) = E10.4,S4,IOHZETA(M) = F *430 F) 4.3,S4,7HD(2) = F6.',S4, 7HB(2) = r6.5,S5,7HR(2) = F6.5vS5,86L *430 I L(2) = F7.3,S4,IFMNI?2) = 13 *S *430 VECTOR VALUES FRIC1=$12(F5.4,S1) *5 *431 VECTOR VALUES FRIC2=S12(F5.4,S1) * *432 VECTOR VALUES CSUORG=$F2.1,10iSlF6.5).5 *433 VECTOR VALULS RESFAC=$IH,t23,l5HREYNULDS NO. = V6.GSL554HF(.434 O 1,12,46) = F5.4,S15,4HF(2,12,4H) = F5.4*S.434 VECTOR VAL UE-S MEMO =% Si1,4HH(lI33v4H) = F7.3,S15,4HV(l,13,4H *435 0 ) 1 F6.3,S15,6HJELP(1,I3,4H) = F7.3,51 —' 3,8HTHETA(1,I3,4H) = F6. *435 1 3.5 *435 VECTOR VALUES RANDOM =$ Sll4HH(21I3,4H) = F7.3,S15,4HV(2,13, *436 O 4H) = F6.3,515,6HELFL(2,13,4H) = F7.3,Sl3,BHTHETA(2,13,4H) = F *436 1 6.3 *$ *436 VECTOR VALULS TUBA = SLHOSL,12HAT TIME T = F8.5,16H SECONDS, *437 O AFTER 14,896 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AN *437 1 0) PRESSURES AT KEY POINTS THROUGHOUT THE /1l S1I,35HFLOW SYST *437 2 EM ARF AS TABULATED BELOW. *437 VECTOR VALUES FLIJTL =$1HO,54,4HLH')C.,57,5H11,O),58,5H(1,1),58 *436 O,5H( 1,2),S8,5H(2,0),S8,6 H (21,?0),S76H12,40)57,6H(2l,6UtSl6H *438 1 (2,80),S7,7H(2-,LCo) /IH,S5,lHVt6,9(S2,F8.4,S3) /1H,S5,IH *438 2 Hi,56,9F52,FFU.4,S3) *$ *438 VECTOR VALUES ORGAN = $IHO,S4,4HLOC.,56,5H11,O),10,5H1i(1,1),S *439 o 1O,5H)2,O),S1O,66H((2,10),9,6H((2,2G),59,p6H(2,3D)59,6H2,40),S *439 1 9,6H(2,50) /1H,S5IlHVS5,7(52,F8.4,S5),52,F8.4,52 /IH,S5,IH *439 2 litS5,7(S2,F8.4iS5),S2,F8.4,S3 *5 *439 VECTOR VALUIES OBOE = $lH-9Sl,83HTHE GATE VALVE AT LOCATION J *440 O = 1,I = 0 HAS BEEN INSTANTA.NEOJSLY CLOSED AT TIME T = F8.5, *440 1 34H, SECONDS, OR AT OPERATION NUMBER 12,1H. *$ *440 VECTOR VALUES ORDERS=$IH,S2,31HMAXIMUM ALLOWABLE ITERATIONS *441 O = 14,S3,29tHiCOMPUTATIDN CUTOFF LIMIT = I4S3926HINITIAL CAV *441

1 IlY PRIINTOUT = I1,S3,22f1VALVE SHUTOFF POINT = 12 /1I1,S2,31Hi *441 2 NITIAL PRINTOUT LIMII PnINT = 14,53,29HPRINTOUT FREQUENCY CON *411 3 TRUL = I4,S3,26HCAVITY PRINTOUT CYCLE = I1 *$ *441 VECTOR VALULS DfRUM=11H-,S1,GAHCULUMN SEPARATIUN HAS OCCURRED *44? O AT THF GATE VALVE AT TIME T = F8.5,30H SECONDS (OPERATION CYC *442 1 LE NU. 14v13H). THIS IS F8.5/1H tSI,40HSECONDS AFTER CLUSUR *442 2 E OF THE GATE VALVE. *s *442 VECTOR VALUES AMOUNT = S1H,S2,9HHVAPOR = F7.4qSI5,8HOELXC = *443 U F7.4,SL5,8HOELTC = F8.5,S10,RHMARK = I4,S12,7HMUCH = F6.4/'443 1 1H,S2,lOHFACTOR = IZvS22,5HXI = F7.4,S1796HIINC = I3tS *443 2 15,7HIUTA = F5.4,SI1,8HPAREA = F7.5 *$ *443 VECTOR'JALOLS PICCLO = I.H-,SL,54HTHE VAPOR CAVITY HAS COMPLE *444 O TELY COLLAPSED AT TIME T = F8.5,30H SECOiJDS (MPERATIOU' CYCLE'444 1 M = 14,13H). THIS IS F8.5,9H SECONDS /1H iSL,38HAFFCR COLUM *444 2 N SEPARATION OCCURRED, AND F8.5,32H SLCONDS AFTLR VALVE WAS C *444 3 LOSED. *$ *444 VECTOR VALUES CELLO = SIHHSIIZHAT TIME T = F8.5,16H SECONDS *445 O, AFTER 14,87H COMPLETE CYCLES (OF OPERATION, THE VELOCITIES A'445 1 ND DEPTHS IN THE VAPOR CAVITY AS CELL AS /IH,S1,89HVL'LCITIC *445 2 5 AND PRESSU1RES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE *445 3 AS TAN3ULATED BELOW. *$ *445 VECTOR VALUES HARP = $LHOS4,SHOISTY.,1,F6.4O#FVA'(52,F6.4)/1 *446 O H,54,4HLOC.,9'FV8' (5,12,SI)/1H,5,lHUS204FV13' (2,F6.3)II *446 1 55,11HZS?-,'FVB'(S3,F5.3) *$ *446 VECTOR VALUES HASSOrI = $lH-,SI,87HTHE D[RECTION OF FLUW AT TH *447 O E RESERVOIR END OF THE PIPE SYSTEM HAS REVERSED AT TIME T = F'447 1 8.5,9H SECONDS.'$ *447 TRANSfER TO UNITED *448 END OF PROGRAM *449 ON

MAC PROGRAMTYPE 25 MAR 1964 (ALL NUMBERS ARE OCT t~L) NO. OF LOCATIONS 17735 TRA Vt-CTOR SIZE 00926 TRA VECTOR STARTS 00'952 ENTRY PT. 05657 ERASABLE STARTS 77777 VARIABLE STORAGE (A:ARP, AYC:COMKONE:ERASABLED[GIT=MODE ) AOIM 00454 A 1 FVB 10332 Ct LL 03145 A C PIPE2 03451 A 1 UEPPY 04235 0 ALABAM 00406 4 GAP 10011 C1 LOCI 03146 0 PLAN 03455 A I UEP 04236 0 ALASKA 00406 4 GEOR, G[ 00414 4 LOC2 03147 0 pMK 03456 1 UL 04231 0 ALPHA 00765 A 0 GIVEN 02204 A I 1003 03150 0 PRTCYC 03457 I UM 04240 0 AMOUNT 01026 A I GOA 02207 A b LPRK 10330 CI PRTFKO 03460 I UNITED 00443 4 AOG 01031 A 0 HABS ('32210 0 MAINE 00421 4 PRTLIM 03461 1 UP 04405 A O AR IZON 00406 4 HAL 02211 0 MARK 03151 I PSI 03462 0 UR 04400 0 ARKANS 00407 4 HARP 02237 A I MARS 03152 0 ~[IUT 03465 O U 04553 A D A 01034 A 0 HAWa, I I 00415 4 MAXLIM 03153 I Q 03464 0 UTAH 00444 4 BASSON 01060 A 1 HBAR 02240 0 NIL 03154 0 L)UANT 03474 A l VALLY 04554 0 B 01063 A 0 HEL 02241 0 MEMO 03176 A I QZZL 03415 0 VAL 04555 0 CALIF 00406 4 HEP 02242 0 MICHGN 00422 4 QZZM 03476 0 VEEE 04556 0 CARD 01075 A 1 HHOLD 02243 0 MINT 03177 0 RANgUM 03520 A I VEPPY 04551 0 CELLO 01146 A I HL(1S 02246 A O HISSIP 00423 4 RESFAC 03536 A I V~P 04560,D CFFQ 01147 0 HL 02247 0 MK 03200 I REYNU 03537 0 VIRGNA 00445 4 CHI 01'150 0 - HM 02250 0 MN 03203 A 1 RhCDEI 00434 4 VL 04561 0 COLORA 00410 4 HOLDD 02251 0 MOFCOL 03204 I RitO 03540 0 VMAX 04562 0 CONNEC 00411 4 HP 02574 A 0 MOFSEP 03205 I R 10002 AC9 VM 04563 0 CONTRL 01160 A 1 HR 02575 0 MONrAN 00424 4 SCARU 00406 4 VOIU 0456~ 0 COUNT 01161 I H 03120 A 0 ~4 03206 I SCHEME 03565 A I VP 05107 A 0 CP 01164 A 0 HVAPOR 03121 0 MUCH 0'3207 0 SDAKOT 00406 4 ~RMONT 00406 4 CSURG 01170 A 1 IDAltO 00406 4 MU 03212,~ 0 SIiUTOF 03566 I VR 05113 0 CUTOFF 01171! ILLN{!I 00406 4 NCAP, O 00406 4 SPRK 10326 C[ V 05433 A 0 OELAWA 00412 4 INC 03122 I Nt)AKOT 00406 4 SSUZZ 03567 0 VZZM 05434 0 DELTC 01172 0 INCVPT 03123 1 NEBRSK 00425 4 SEATEA 00435 4 WASH 00446 4 OELT 01173 0 INDIAN 00406 4 NHAMP 00426 4 STATI:B 00436 4 WISCON.00447 4 DELTT 01176 A 0 INFLOW 0312,. 0 NJERSY 00427 4 STAi'EC 00437 4 WSEL 05435 0 -—,1 DELXC 01177 0 IOTA 03125 0 NMEXCO 00430 4 STATED 10333 C4 WVIRG 00450 4 I OELX 01202 A 0 IOWA 00416 4 N 03213 1 STA[[E 00440 4 WYOMNG 00451 4 DIM 01205 A I IPRK 10327 C1 NU 10010 CO STATLS 00406 4 XI 10003 C'O DRUM 01245 A 1 I 03126 I NX 03214 0 SUZ 10.351 ACO X 05436 0 D 10006 ACO IVEL 03131 A 0 NYORK 10334 C4 TENNSE 00441 4 YY 05437 0 ELP 01570 A 0 JHK 03132 1 OBOE 03245 A I TEXAS 00442 4 ZAL 05440 0 EL 02113 A 0 JJO 03133 I OHIO 00431 4 TIIErA 04112 A 0 ZEP 05441 0 E 02116 A 0 JJ 03134 I OKLAHO 00432 4 TOFCOL 04113 0 ZETA 05444 A 0 FACTOR 02117 1 d 03135 l ORDERS 03322 A I TOFOFF 04114 0 ZL 05445 0 FFCC 02120 0 KANSAS 00417 4 OREGON 00433 4 TOFSEP 04115 0 ZM 05446 0 FLORID 00413 4 KA 03136 0 ORGAN 03363 fi 1 T 04116 0 ZOV 05447 0 FLUTE 02157 A 1 KENTUC 00420 4 OUTFLO 03364 0 TtlBA 04156 A 1 ZP 10325 ACO FRICt 02162 A I KK 03137 I PAREA 10007 CO TUBE1 04204 A I ZR 05450 0 FRIC2 02165 A i K 03140 I PENNSY 10335 C4 TUBE2 04232 A 1 Z 05615 A 0 F 10160 ACO LAMBDA 03141 0 PICCLO 03433 fi 1 UALLY 0423'5 0 ZW 05616 0 FVA 10331 CI LEL 03142 0 PIPE1 03442 A 1 UAL 04234 0 FUNCTION DICTIONARY.03310 00352 0.01300 00353 0.0130I 00354 0.03311 00355 0 ARCSIN 00356 0 BISECT 00357 0 FFC 00360 0.PCOMT 00361 0.PRINT 00362 0.PRSLT 00363 0.READ 00364 0 PARK 00365 0 PARVOL 00366 0 PFPA 00367 0 PRTLAW 00370 0 PRTRUL 00371 0 SIMPQ 00372 0 SIMPWV 00373 0 SIN 00374 0 SQRT 00375 0 SYSTEM 00376 0 USURG 00377 0 BUBEQB 00400 0 BUBVOL 00401 0 SQAGOT 00402 0 SQAOGT 00403 0 SURGEQ 00404 0 TOPWTH 00405 0 ABSOLUTE CONSTANTS 05617 +203506314631 05620 +203606314631 05621 +202614631463 05622 +000000000003 05623 +202600000000 05624 +000000000006 05625 +000000000011 05626 +167406111564 05627 +172753412172 05630 +204500000000

APPENDIX II SUBROUTINE PROGRAMS USED WITH MAIN COMPUTER PROGRAM A syntactical listing for each of the 11 MAD language external subroutines used in the computer program called MAIN (See Appendix I) is presented below. The subroutines are listed according to name and function: Name Function 1l PARK Given a distance measured from the leading edge of the separation void, the subroutine returns the depth of flow based on the parabolic free surface (See Figure 12) described in Chapter IV. 2. PARLN Given a depth on the free surface flow, the subroutine returns the distance to the leading edge of the separation void based on the parabolic surface (See Figure 12) described in Chapter IV. 30 PFPA This subroutine determines the cross-sector area of a pipe flowing partially full. The computation is based on flow depth. 4~ BISCT This subroutine is used to determine the exact length of a separation void that extends at least n, but not (n+l) increments of length. The exact distance of the leading edge of the void from the n-th increment is computed using equilibrium between void volume and total outflow. 50 FFC Given the depth of flow, the velocity of flow, and the pipe segment identification, this subroutine returns the friction coefficient based upon Reynold's number. If the depth of -168

-169Name Function of flow is equal to the diameter of the pipe, the coefficient is that for pipe flow; if less than the diameter of the pipe, then the coefficient is for free surface flowo 6. SIMPQ Given the number of length increments, AXC, from the valve to the grid point nearest the separation void, the function returns the total volume of the void. 7. PARVOL Given the depth of flow at a point less than an increment of length from the leading edge of the void, this subroutine returns the volume of the subincremental void determined by the parabolic surface. 8. PINTR Given the number of increments, AXC, from the valve to the separation void, the subincremental distance to the leading edge of the void, and the size of the increments, the subroutine computes and returns the parabolically interpolated depth of flow from known depths. 9. SMPVL Given the negative distance, NX, along with other controlling parameters, this function gives the incremental volume removed from the void during collapse. 10. PRINT This function, with PRTLAW and PRTRUL providing double entry points, controls the print format to accommodate the data produced. 11. USURG This function is used to approximate the velocity at the first grid point behind the separation void. The velocity is prorated according to depth-of-flow using the velocity ahead of the void and the free-surface propagation velocityo

$ CCMPILE MAC,PRINT CP'JECT,XLCU1EI t.iPUN. (UJ[CT P: / 12 1. MAO (12 FAR 1964 VERSICN) PkGGRA~ LISTING EXTERNAL FUNCTION (CL) *~Cl INTEGER J,L PRCGRAM COPMCN RI2IXILI2),P~kEA, NLCAI'F(lC2,LI~),P C) o SPRK,IPRKLPRKFVAVFvE1 TATEL,,NYCRKPFNNSY,SuL(1) VECTOR VALUES DIM = 2,1,61 C 4 ENTRY TC PARK. J=L X=C ALP-A = (1.-2.*XI) K=ALPHA*R(J)-SQkT.(4.*XI*R(JI*X) FUNCTION RETURN K ENC OF FINCTI c H -1

$ CCMPILE ACIPI N r (N BJECT, XXECUTE!,DUP,pIJCH C1JCT P L.I 7 / AC (12 MAR 1964 VERSIDN ) PXC: ~A/,' LISTI......... EXTERNAL FUNCTIfON (In,L) *CL I N r EG L,J P R (, G R A,! Clti('', (), I, iv) ( 2 ), P. t.,~L,,F (A,,Z,A: F.,i ), ZA i) ( 1, s I 0 SPRK, IRKLP RKtFVAFVVP, STAUTE! A, N Y.,Kt PF',Y\yl, SUZ i Z ) VECTCR VALUES DIP = 2,1,51 ENRY (3 PARL, N. J=L Kt3 ALP-. = ( 1.-2.*X I ) X=(K-ALPHA* R(J) ). P.2. / (4. X I*r(J ) ) FUNCTION RE[I..RN X r END OF FUNCFI 1; ICN H

$ CCMPILF PAC,PRINT O8JECT,EXCCUTE,0DUN.P,UNCH Ci3JePFPA. C01 C7,121' 8 ~4/ CI4 12 2&.b 4C MAC (12 MAR 1i64 VERSICN) PRUGRAM LISTING EXTERNAL FUNCTICN (PL) + INTEGER JL CC 0 PROGRAM COMMON R(2),XID(2),PAREA,NL,GAP,F( 102,LIM),LP(1CC), O SPRK,IPRK,LPRK,FVA,FVBSTATEO,tJYCRKPE nNSYSLL(11) VECTOR VALUES DIP = 2,9151 *uC4 ENTRY TO PFI'A. *o J=L * C C6 R*P *C37 ALPHA=2.*R(J)*R(J)*ARCSIN.ISURT.(B/(2.*R(J)) ) )+(B-R(J) )*5(2 o T.(2.*R(J)*R-U*B) *ocH FUNCTION RETURN ALPHA *c09 END OF FUNCTION *0 C1 H R)

S COMPILE MAC,PRINT OBJECT, PUNCH CBJECT, EXECUTE, CUMP tISCT.01 07!:121F 04/04/64' 12 31.1 AM MAD (12 MAR 1964 VERSION) PROGRAM LISTING......... EXTERNAL FUNCTIOiN (L,R,F.,LIi,K) *0C1 STATEMENT LABEL K *602 ENTRY TO BISECT. *C03 XL=L *0C4 XR=R *C05 IOTA = LIM *cc6 FL = F.(XL) *C07 THROUGH DRLM, FOR H=XR-XL,-H/2.,H.L.(XR-XL)/32. *008 THROUGH CRUM, FOR X=XL,H,(X+H).G.XR *C09 DRUM WHENEVER FL*F.(X+h).L.C, TRANSFER TC GCNG *(.1Ifl TRANSFER TO K *011 GONG THROUGH TUBA,FOR H=H/2.,-H/2.,FL*F.(X+I:).G.0. *012 TUBA hHENEVER H.L.IOTA, FUNCTION RETURN X *013 X=X+H *C14 TRANSFER TO GONG *015 END OF FUNC1 IN *01 L-.

$ CCMPILE MAE, PR.INT OBJECT, EXECUTE, 01?'NC-D3JFCT 1FG. 001C' 07 211 04/0',t64 ~ I ~i MAO (12 MAR 1964 VERSICOJI PROG1CA!M LISTINN-. EXTCRNAL FUNCTIGN IP,V,S) OI CPR' A M C!OM C:N R (2,X I t)D( 2),!13A RI, N 1, G:A P, F I 0 2~i, I L' I.,) 0 SPRKI!'NK,LPRK,FVA,FVDtSTATEF,1VCY(RK/,PE'\:~SYSUL(1l) VECTOR VALUES DIM = 2,1,51.*0 INrEGER FlI,K,S,!.;AP C(4 ENTRY TO FFC.*.C F.l=P.6 K=S C H=GAP VdFENEVER HZ.GE.C(K) TRANSFER TO MIADRID *L 10 OR WHENEVFR I-L.GE.R(K) ~i CR WHENEVER i-L.L.R(K) WEJP=2.*R(K)*ARCCOS.((R(K)-IiL)/R(K)) END OF CCNDITITONAL *. A=PFPA. (H-il ) *06 HYCRAC=A/WE TP *17 REY=HYORAC*4.*.AFBS.V/NU.i TRANSFER Ta LAPAI,1 MACRID REY=U(K)*.ABS.V/NU LAPAZ WHENEVER REY.L.64. *021 W =1. *E22 DR WHENEVER RCY.L. 100. *tu23 W = 64./REY *024 OR WHENEVER REY.L.3510C. *C25 X = REY-ICC. * C-26f I = X/H *C027 ETA = (X-H*I)/H- * -),2 B WHENEVER I.E.O 02 1 =I * 030C ETA = ETA - 1. *i031 ENC OF CONDITIONAL *032 W=F(KtlI)+(ETA/2. )*(F(K,1+1)-F(K, 1-1) ).(ETA/2. )*E1A*(F(K,I -0 333 O 1I).'F(K, I-1)-2.*F(K, 1)) *C33 OR WHENEVER REY.GE.35100..AND.K.G.1 *034 W =.C247 *035 OTHERW ISE *036 W =.0324 *037 END OF CONDITIONAL *03F. WHENEVER Hi.GE.C(K) *0Q39 FUNCTION RETURN W *040 OTHERW ISE *041 hW=W/(8.*HYORAD) *C042 FUNCTInN RETURN WW *0.43 END OF CONDITIONAL *044 END OF FUNCTION *13 045

$ CCOMPILE MAC,PRINT OBJECT,EXECUTE,DUMP,PUNCH' CBJECT SIMPQ.C0 07(21F 4'/6', 6'. 12 43.3 AM MAC (12 MAR 1964 VERSION) PROGRAM LISTING EXTERNAL FUNCTION (L,M,N') INTEGER N, IK,LK,K,NN,GAP,J,L STATEMENT LABEL SATURN,APOLLO,AG.ENA PROGRAM COMMCN R(2),XI,D(2),PAREA,NU,GAP,F(102,{iI]),LP(1OC), *CC4 0 SPRK,IPRK,LPRK,FVA,FVB,STATED),NYORK,PENNSY,SUL(11) *C04 VECTCR VALUES DIM = 2,1,51 CO ENTRY TO SIMPQ. *CG, J=L. C 37 N=NN * CO' MMu=M *009 AEVEN=O. *01(, ACCC=C. C011 OAA=PAREA-PFPA.(ZP(O),J) -*012 NAA=PAREA-PFPA.(ZP(.N),J) *01 WFENEVER N/2.E. (N-1)/2 *14 SATURN=APOLLO C15 K=1 *$Ij6 MM=MM*O.5 *017 OTHERWISE *0 H SATURN=AGENA *019 K=2 *02C END OF CONDITIONAL *021 THROUGH THOR,FOR LK=KK,LK.G.N-1 *022 AAAA=PAREA-PFPA.(ZP(LK),J) *023 THOR AEVEN=AEVEN+AAAA.i 2 4 THROUGH ZEUS,FOR IK=1,K,IK.G.N *025 TRANSFER TO SATURN *,2(, APOLLO ZA=(ZP(IK-1)+ZP(IK))*O.5 *027 TRANSFER TO PLUTO *028 AGENA ZA=ZPIIK) *029 PLUTO BBEH=PAREA-PFPA.(LA,J) *030 ZEUS AOOD=AOCD+RERB *031 V=(IM/3.)*IUAA+NAA+4.*AODD+2.*AEVEN) *032 FUNCTION RETLRN V *033 END OF FUNCTION *034

$ COMPILE MAC, PRINT OBJECT, EXECUTE, DUMP, PUNCH CIJECT PRVOL.OL C702 1 04/04/64 9 12 4).5 AM MAD (12 MAR 1964 VERSION) PROGRAM LISTING. EXTERNAL FUNCTION (AL) *001 PROGRAM COMM-ON R(2),XI,0(2),PAREANUGAPF(102, DW),LP(GOO), *002 0 SPRK-,IPRKLPRKFVAFVB,STATEONYORKPENNSYSUZ(11) *002 VECTOR VALUES DIM = 2,1,51 *003 DIMENSION KK(10)tXX(10),AA(10) *C04 INTEGER JL,,GAP *0+5 ENTRY TO PAKVOL. *006 K=A *007 J=L *008 X=PARLEN.(KJ) *C09 DELX=X/10. *010 AODD= O.; 1 AEVEN=O. *012 THlROUGH YUKONFOR I=0,1,.G.1O *013 XX(I)=CELX*l *014 KK(I)=PARK.(XX(I),J) *615 AA(I)=PAREA-PFPA.(KK(I)+R(J),J) *016 WHENEVER L/2.E.(I-1)/2.AND.I.NE.O C17 AODC=ACDD+AA(I) *Qjp OR WHENEVER I.NE.O.AND.I.NE.1O *019 AEVEN=AEVEN+AA(I) *C20 YUKON END OF CONDITIONAL *021 V=(DELX/3.)*(AA(O)+4.*AOOD+2.*AEVEN+AA(10)) *022 FUNCTION RETURN V *023 EN FFUNCTION *023 END OF FUNCTIUN *024 ON

$ COMPILE MAC,PRINT OeJECT,EXECUTE,DUMP,PUNCH OBJECT PINTR.01 0702IF 04/04/64 3 12 55.U AM MAD (12 MAR 1964 VERSION) PRUGRAM LISTING. EXTERNAL FUNCTION (H,I,L) *0G1 INTEGER MK,I *CC2 PROGRAM COMMCN R(2),XID(2),PAREA,NU,GAPF(102,CIM),ZP(1CO), *003 0 SPRK,IPRK,LPRK,FVA,FVB,STATED,NYCRK,PENNSY,SLJZ(11) *C03 VECTOR VALUES DIM = 2,1,51 *004 ENTRY TO PINTRP. *CO5 X=L *CO6 MK=I *007 CELXC=h *008 TH=(X/DELXC) *009 WHENEVER MK.E.O *0Cl MK=1 *01l TH=TH-1. *012 END OF CONDITIONAL *013 Z=ZP(MK)+(TH/2.)*(ZP(MK+I)-ZP(MK-1))+TH*TH*.5*(ZP( K+I)+ZP(M *014 O K-1)-2.*ZP(MK)) *014 FUNCTION RETURN Z *015 END OF FUNCTION *016!

S CCMPILE MACPRINT ORJECTEXECUTEDUM~,PUNCH OdJECT SMPVL.01 0 7;1i I 8 04/C'i/64 9 12 59.9 AM MAD (12 MAR 1964 VERSION) PROGRAM LISTING. EXTERNAL FUNCTION (HI,M,K,YE) *c01 INTEGER JKMKIIMKHK *002 STATEMENT LABEL SUNEARTHURANUS *003 PROGRAM COMMON R(2),XID(2),PAREANUGAPF(102,OIM),ZP(100), *004 0 SPRKIPRKLPRKFVAFVBSTATED,NYORKPENNSYSUL(11)'.04 VECTOR VALUES DIM = 2,1,51 *005 ENTRY TO SIMPWV. *00f) OELXC=H'CC7 MK=I I:0 NX=M *009 J=K'010 YY=YE *011 WHENEVER NX.LE.O.,FUNCTION RETURN 0. *012 IVK=((NX+YY)/DELXC)+1.'013 XX=IMK*OELXC-(NX+YY) *014 ZZZ=PINTRP.(DELXCMK-IIKIKXX) 15 ZZ=PINTRP.(DELXCtMK-1,OELXC-YY) *016 ACCO=O. *017 AEVEN=O. *018 XYX=XX *019 THROUGH JUPTERF-OR 4K=1,19,HK.G.9 *020 XYX=XYX+(NX/10.)'021 H WHENEVER XYX.G.OELXC'022 IMK=IMK-1'023 XYX=XYX-DELXC *024 ENO OF CONDITIONAL'025 Z=PINTRP.(DELXCMK-IMKXYx) *026 WHENEVER (HK+.1)/2.E.HK/2'027 SUN=EARTH'028 OTHERWISE *029 SUN=URANUS'030 END OF CONDITIONAL'031 TRANSFER TO SUN'032 EARTH AAAA=PAREA-PFPA.(ZtJ) *033 AEVEN=AEVEN.AAAA'034 TRANSFER rO JUPTER *035 URANUS BBBB=PAREA-PFPA.(Z,J) *036 AOCO=AOCO+BeeB *037 JUPTER CONTINUE'038 OAA=PAREA-PFPA.(ZZZJ) *039 NAA=PAREA-PFPA.(LZJ)' *040 Va(NX/30.)*(OAA+4.*AOOO+2.*AEVEN4NAA)- *041 FUNCTION RETURN V'042 END OF FUNCTION'043

$ COMPILE MACPRINT OCJECT,EXECUTE,DUMPPUNCH OBJECT PRINT.CL 07021E (C4I4/64 9 13 6.6 AM MAO (12 MAR 1964 VERSION) PROGRAM LISTING......... EXTERNAL FUNCTICN (M) *CCI INTEGER M,MK,SPRK, IPRK,LPRK,FVA,FVB *002 STATEMENT LABEL STATED,PENNSY,NYCRK GC3 PROGRAM COMMON R(2),XI,0(2),PAREA,N,GAP,F ( 102,Ir,),,; ( 100 ),C 0 SPRK,IPRK,LPRK,FVA,FVB,STATCE,NYCRK,PENNSY,SU(11) *004 VECTOR VALUES DIM = 2,1,51 *005 FORMAT VARIABLE FVA,FVB *,C6 ENTRY TO PRTLAW. * 07 MK=M.COg WHENEVER MK.LE.14 *CC') SPRK = C *r 1, IPRK = 1 *G11 LPRK = MK *012 FVA = MK C 13 FVe = MK+1 14 STATED = NYORK *015 OR WHENEVER VK.G.14.AN.VIK.LE.29 *016 SPRK = 0 *G17 IPRK = 1 *G 1 LPRK = 14 *(19 FVA = 14 *020 FVB = 15 *021 STATED = PENNSY *022 - OR WHENEVER MK.G.29.ANC.VK.LE.44 *0)23 2 SPRK = 0 *024 IPRK = 2 *C02i LPRK = 28 *026 FVA = 14 *02 7 FVt = 15 *028 STATED = PENNSY *029 OR WHENEVER MK.G.44.ANCD.K.LE.59 *030 SPRK = 0 *031 IPRK = 3 *032 LPRK = 42 *C33 FVA = 14 *034 FVB = 15 *035 STATED = PENNSY *036 OR WHENEVER MK.G.59.ANC.MK.LE.74 *037 SPRK = 0 *038 IPRK = 4 *039 LPRK = 56 *640 FVA = 14 *041 FVB = 15 *042 STATED = PENNSY *043 OR WHENEVER MK.G.74.ANO.MK.LE.89 *044 SPRK = 0 *045 IPRK = 5 *046 LPRK = 70 *047 FVA = 14 *048 FVB = 15 *049 STATED = PENNSY *050 END OF CONDITIONAL *051 TRANSFER TO SCRIBE *052

ENTRY TO PRTRUL. *053 PK - M *u54 WHENEVER MK.G.14.AND.MK.LE.29 *C55 SPRK = 15 -*056 IPRK = 1 *057 LPRK = MK *05S FVA = MK-15 *059 FVB = MK-14 *('6 STATED = NYORK, 61 OR WHENEVER tK.G.29.ANO.MK.LE.44 *. 6 SPRK = 30 6-63 IPRK = 1 *604 LPRK = MK *065 FVA = MK-30 *, 6FVB = MK-29 r.067 STATED = NYORK *06P OR WHENEVER MK.G.44.ANC.MK.LE.59 *069 SPRK = 45 *07C IPRK = 1 *071 LPRK = MK *072 FVA = MK-45 *07t FVB M PK-44 *074 STATED = NYORK *075 OR WHENEVFR MK.G.59.ANC.MK.LE.74 *076 SPRK = 60 *077 IPRK = 1 *C'78 LPRK = MK *079 FVA = PK-60 *G80 FVB = MK-59 *08l 0: STATED = NYORK *082 0 OR WHENEVER MK.G.74.AND.MK.LE.89 *C83 SPRK = 75 *084 IPRK = I *085 LPRK = MK *"86 FVA = MK-75 *087 FVB = HK-74 *088 STATED = NYORK *089 END OF CONDITIONAL *090 SCR IB:E CCNTINUE *091 FUNCTION RETURN *092 END OF FUNCTION *093

$ COMPILE MAC, PRINT OBJECT, EXECUTE, CUMP, PUNCH ObJECT 0Us(;.UI 7021M I 4/04/64 9 13 15.7 AM MAO (12 MAR 1964 VERSIUN) PROGRAM LISTING EXTERNAL FUNCTION (QL) INTEGER IJL *( 0 2 PROGRAM COMMON Ri(2)XI,D(O2),PANRE-ANUGAPtF(I IM),ZPi-PL ), *(CC3 0 SPRKIPRK,LPRKFVAFVBTSTATENYC',K KPENNSYSUi(11) VECTOR VALUES DIM = 2,1,51 oIt ENTRY T) USURG. i J=L * C06 Z=Q *Co 7 CEL=C(J)/10 * 10 0 B MEXICO ThROUGH MEXICOFCR I=O,1,OEL(I+1).G.L *009 TH=7-OEL* I * I C WHENEVER I.L.0 *011 1=1 *012 Th=TH-1. *C13 END OF CONOITICN L * [14 U=SUZ(I)+(TH/2.)*(SULII+1)-SUL(I-1)) +TF*JH*5.*(sULI 1+11+ *015 0 SUZ(I-1)-2.*SUZII)) *015 COEFF=2.*SQRT.(64.32*RIJ)) I016 FUNCTION RETURN U*COEFF *011 END OF FUNCTION *018 H co H

APPENDIX III EXPERIMENTAL DATA; RUNS NUMBERS 25 and 34 The experimental data obtained in the laboratory fromBRuns Numbers 25 and 34 (Group II conditions) are presented in tabular form in this appendix. Tables I and II give the pressure-rise data at the gate valve and column-separation void data, respectively, for Run Number 25. Tables III and IV provide the corresponding data for Run Number 34. Time is referenced to the moment of valve closure. Head in feet of water is measured with respect to absolute pressure conditions. -182

TABLE I EXPERIMENTAL RUN NJUMBER 25 PRESSURE-RISE DATA AT GATE VALVE Time Head Time Head Seconds Feet of Water Seconds Feet of Water.0012 17.3.1362 314.5.0073 135.8.1382 319.9.0110 180.7.1421 308.7.0134 235.4.1444 308.4.0147 248.9.1478 295.8.0166 259.8.1526 291.4.0187 255.7.1559 281.2.0194 269.7.1588 282.2.0212 278.8.1650 272.7.0247 267.3 11685 275.4.0276 280.2.1713 268.6.0294 280.2.1738 269.3.0324 275.4.1803 260.6.0359 291.1.1828 263.2.0379 295.5.1869 255.7.0410 286.3.1912 258.5.0447 297.2 o1974 248.9.0494 283.2 o1991 242.8.0507 290.4.2007 237.7.0532 291.1.2185 41.8.o06o 283.6.2201 31.2.o096 294.8.2235 13.8 0615 293.1.2276 11.3.0625 285.3 --.0643 280.2 2.0524 11.3.0667 283.2 2.0568 76.4.o0694 285.6 2.0581 67.2.0721 282.9 2. 0599 83.5.0746 288.0 2.0631 32.9 ~0769 287.3 2.0663 11.3.0795 299.2 2.0718 16.8.0828 295.1 2.1155 16.1.0846 303.3 2.1188 28.2.0868 306.7 2.1218 87.3.0938 305.0 2.1233 111.1.0997 310.4 2.1250 126.7.1032 313.-5 2.1259 120.2.1079 324.3 2.1280 43.1.110o6 320.3 2.1302 83o9.1149 327.0 2.1306 146.4.1200 318.6 2.1324 130.1.1221 324.7 2.1331 155.2.1266 313.5 2.1365 119.9 ~1307 319.6 2.1387 152.8

TABLE I (Cont'd) Time Head Time Head Seconds Feet of Water Seconds Feet of Water 2.1400 153.2 2.3031 151.5 2.4112 148.8 2.3080 153.6 2.1443 166.4 2.3103 152.8 2.1478 149.4 2.3118 160.9 2.1509 171.5 2.3147 165. 2.1530 171.5 2.3178 165. 2.1558 161.7 2.3213 154.5 2.1593 174.2 2.3374 47.9 2.1625 168.8 2.3427 34.3 2.1660 174.9 2.3488 15.5 2.1703 169.1 2.3565 11.3 2.1744 177.6 - 2.1777 165.7 3.3303 11.3 2.1799 169.1 3.3344 81.1 2.1821 165.7 3.3358 68.9 2.1843 167.4 3.3376 77.8 2.1865 164.0 3.3384 65.5 2.1887 174.6 3.3416 50.9 2.1909 168.6 3.3434 84.9 2.1933 174.2 3.3466 90.3 2.1958 187.8 3.3500 80.5 2.1990 182.7 3.3528 76.8 2.2024 195.3 3.3591 111.4 2.2056 188.5 3.3619 110.7 2.2088 198.0 3.3656 126.3 2.2122 188.8 3.3704 115.8 2.2159 199.4 3.3735 123.3 2.2196 198.0 3.3766 120.6 2.2230 206.5 3.3804 125.3 2.2277 201.4 3.3851 119.8 2.2312 204.7 3.3895 123.3 2.2365 202.8 3.3988 118.4 2.2391 208.2 3.4026 126.3 2.2431 202.4 3.4068 126.3 2.2449 207.5 3.4103 130.7 2.2494 206.5 3.4148 133.1 2.2537 209.2 3.4179 138.5 2.2579 208.5 3.4210 136.4 2.2609 199.4 3.4242 141.5 2.2641 201.4 3.4285 140.9 2.2681 194.9 3.4309 143.7 2.2718 196.1 3.4347 143.7 2.2750 189.9 3.4385 150.3 2.2836 191.2 3.4413 148.3 2.2879 185.8 3.4450 149.3 2.2924 1 74.9 3.4492 140.9

-185TABLE I (Cont'd) Time Head Time Head Seconds Feet of Water Seconds Feet of Water 3.4556 140.2 4.4582 109.0 3.4685 140.2 4.4651 109.7 3.4719 142.6 4.4697 112.4 3.4762 136.4 4.4790 112.1 3.4834 130.7 4.4883 104.6 3.4890 130.3 4.5000 86.3 3.4982 130.0 4.5082 108.0 3.5062 132.4 4.5184 100.9 3.5100 131.3 4.5296 85.9 3.5179 123.2 4.5365 81.5 3.5222 125.3 4.5415 83.5 3.5281 124.9 4.5481 75.1 3.5344 123.2 4.5593 49.6 3-5379 116.4 4.5678 41.8 3.5485 70.3 4.5785 35.7 3.5566 47.2 4.5932 29.5 3.5628 35.0 4.6214 21.8 3.5770 30.2 4.6644 14.4 3.6048 11.3 4.6663 11.3 4.3460 11.3 4.3506 62.8 4.3519 67.6 4.3535 76.8 4.3566 79.1 4.3596 83.2 4.3623 84.2 4.3653 66.3 4.3675 89.3 4.3732 93.1 4.3776 92.0 4.3826 96.1 4.3863 94.8 4.3898 99-5 4.3954 86.1 4.3987 98.2 4.4009 95.1 4.4043 99.2 4.4076 96.1 4.4112 101.5 4.4117 104.9 4.4216 103.6 4.4251 107.7 4.4335 102.9 4.4419 106.3 4.4460 110.7

-186TABLE II EXPERIMENTAL RUN NUMBER 25 COLUIMN-SEPARATION VOID DATA Time Depth of Flow -- Feet Time Depth of Flow -- Feet Seconds Gage Seconds Gage o No. 1 No. 2 No. 4 No. 1 No. 2 No. 0.00.083.083.083 1.10.052.066.083 0.20.083.083.083.12.052.065.083 0.30.083.083.083.14.051.o64.083.32.081.083.083.16.051.064.081.34.082.083.083.18.050.064.076.36.081.083.083 1.20.050.064.o69.38.080.083.083.22.o49.062.062 0.40.079.083.083.24.049.060.o04.42.079.083.083.26.048.060.051.44.078.080.083.28.047.o60.055.46.076.078.083 1.30.047.061.o65.48.075.074.083.32.048.063.0o64 o.50.073.071.083.34.048.o65.060.52.071.068.083.36.o49.068.057.54.o69.067.083.38.050.067.058.56.067.070.083 1.40.051.065.062.58.064.075.083.42.052.064.o64 0.60.062.o069.083.44.053.063.063.62.059.067.083.46.053.064.062.64.057.063.083.48.053.o65.060.66.058.055.083 1.50.052.065.056.68.o60.052.083.52.051.067.0o4 0.70.058.050.083.54.050.068.053.72.057.051.083.6.0o49.o69.o04.74.055.052.083.58.048.068.0o54.76.052.0o6.083 1.60.048.068.055.78.050.056.083.62.048.067.055 0.80.o050.053.083.64.048.064.0o6.82.050.0o6.083.66.048.062.058.84.o49.o06.083.68.048.062.059.86.o49.059.083 1.70.o49.063.o60.88.050.058.083.72.050.062.061 0.90go.050.061.083.74.o49.062.064.92.052.062.083.76.050.063.073.94.052.069.083.78.051.065.083.96.053.070.083 1.80.051.064.083.98.053.071.083.82.o050.064.083 1.00.052.071.083.84.o050.o69.083.02.053.071.083.86.050.075.083.04.0o4.068.083.88.057.083.083.o06.053.067.083 1.90go.079.083.083.08.052.066.083.92.081.083.083

-1 87 - TABLE II (Cont'd) T~ime Depth of Flow Feet Depth of Flow -- Feet' Gage Gage Second.s Gage Seconds Gage' No. 1 No. 2 No. 4 e No. 1.2 NO..94.083.083.083 3.50.083.083.083.96.083.083 83.083 5.00.083.083.083.98.o83.o83.o83 2.00.083.083.083 2.10.083.083.083 2.20.083.083.083 2.30.083.083.083 2.40.083.083 i.083 2.50.083.083.083 2.6 083.083.083.62.083.083.083.64.083.083.083.66.082.083.083.68.080.083.083 2.70.078.083.083.72.076.083.083.74.074.083.083.76.072.083.083.78.070.083 1.083 2.80.066.083.083.82.064.083.083.84.063.083.083.86.064.083.083.88.064.083.083 2.90.067.083.083.92.o069.083.083.94.070.083.083.96.070.083.083.98.o69.083.083 3.00.068.083.083.02.068.083.083.04.068.083.083.06.068.083.083.o8.070.o83.083 3.10.073.083.083.12.076.083.083.14.079.083.083.16.080.083.083.18.079.083.083 3.20.078.083.083.22.079.083.083.24.081.083.083.26.083.083.083.28.083.083.083 3.30.083.083.083 3.40.083.083.083

-i88TABLE III EXPERIMENTAL RUN NUMBER 34 PRESSURE-RISE DATA AT GATE VALVE Time Head Time Head Seconds Feet of Water Seconds Feet of Water.0028 i7.6.1260 308.2.0074 183.8.1314 311.1.0105 195.6.1350 303.1.0121 181.7.1385 302.2.0140 180.3.1404 293.3.0156 273.5.1429 291.9.0186 256.3.1449 274.9.0211 273.2.1474 278.3.0237 249.8.1494 275.0.0273 280.9.1519 268.2.0308 261.7.1542 266.4.0337 283.0.1554 260.8.0364 280.9.1600 259.0.0392 281.5.1633 249.5.0412 277-7.1659 250.4.0436 280.6.1705 243.0.0462 291.9.1739 245.4.0490go 275.9.1774 241.5.0523 291.9.1808 239.2.0555 278.9.1833 238.6.0592 290.1.1846 238.3.0635 276.5.1879 240.7.o658 278.6.1900 240.7.o692 278.6.1910 234.1.0712 275.6.1929 231.2.0733 280.9.1941 225.0.0763 277.4.1962 209.9.0782 284.5.2046 136.4.0821 285.7.2128 50.9.0839 294.0.2147 36.6.0871 292.5.2171 24.5.0899 300.5.2206 11.3.0942 296.9 -.0987 312.3 1.8818 11.3.1016 305.2 1.8850 87.6.1038 314.1 1.8873 47.0.1071 311.1 1.8908 47.6.1109 312.6 1.8961 120.2.1154 321.2 1.8995 122.2.1187 312.6 1.9005 133.5.1210 317.0 1.9031 133.5

-189TABLE III (Cont'd) Time Head Time Head Seconds Feet of Water Seconds Feet of Water 1.9046 142.1 2.0921 83-7 1.9066 145.0 2.0969 60.9 1.9092 140.3 2.1046 39.6 1.9108 148.3 2.1149 24.5 1.9130 155.4 2.1303 11.3 1.9145 159.2 -- _ 1.9168 156.3 2.9938 11.3 1.9256 163.1 2.9977 79-3 1.9290 167.8 3.0024 80.2 1.9315 166.6 3.0048 90.5 1.9336 169.9 3.0101 90.5 1.9354 167.5 3.0160 97.4 1.9380 166.0 3.0227 93.8 1.9484 159.5 3.0256 97-9 1.9569 161.9 3.0333 92.9 1.9572 166.0 3.0409 92.0 1.9642 172.0 3.0499 95.0 1.9687 169.6 3.0596 98.2 1.9735 167.-5 3.0687 106.8 1.9780 175.2 3.0711 107.7 1.9800 172.9 3 -0759 112.7 1.9858 183.8 3 0792 114.2 1.9879 182.6 3.0816 118.7 1.9944 190.9 3.0910 120.1 2.0021 188.2 3.0965 123.7 2.0043 190.0 3.1096 124.6 2.0159 192-7 3 1192 121.0 2.0232 189.1 3.1286 115.7 2.0264 181.4 3.1363 111.0 2.0321 72.0 3.1391 113.3 2.0341 72.6 3.1477 113.0 2.0400 167.0 3.1545 113.0 2.0446 164.3 3.1609 118.7 2.0497 162.2 3.1718 121.9 2.0539 160.0 3.1756 121.6 2.0585 154.8 3.1872 100oo.6 2.0611 148.3 3.2052 55-9 2.0640 141.5 3.2173 38.7 2o0676 134.7 3.2256 32.5 2.0719 133-5 3.2468 27.2 2.0752 133 5 3.2602 11.3 2.0775 133. -9... 2.0835 131.1 3.8050 11.3 2.0868 118.7 3.8056 16.5

-190TABLE III (Cont'd) Time Head Time Head Seconds Feet of Water Seconds Feet of Water 3.8274 11.3 4.0872 93.2 3.8582 11.3 4.0912 89.4 3.8659 20.9 4.0958 80.2 3.8723 45.5 4.0986 73.7 3.8813 29.8 4.0136 65.4 3.8864 50.6 4.1124 57-7 3.8942 44.9 4.1229 51.5 3.9009 63.6 4.1299 42.3 3.9069 59.4 4.1389 41.4 3.9095 66.9 4.1423 41.4 3.9208 73.1 4.1507 34.6 3.9232 71.3 4.1714 29.8 3.9262 67.4 4.1799 30.7 3.9282 68.0 4.1900 19.5 3.9303 71.6 4.1984 11.3 3.9344 76.o0 - 3.9353 74.3 4.6409 11.3 3.9376 71.9 4.6426 16.9 3.9399 73.4 4.6567 24.5 3.9426 76.0 4.6759 14.8 3.9454 75.1 4.6785 27.0 3.9491 74.8 4.6939 35.8 3.9517 78.4 4.7003 47.3 3.9550 76.3 4.7144 44.0 3.9591 79-.0 4.7208 50.9 3.9593 84.0 4.7516 57-1 3.9659 81.7 4.7580 64.5 3.9711 79-1 4.7772 66.6 3.9775 81.1 4.7875 70.4 3.9813 84.6 4.7958 73.1 3.9870 81.7 4.7982 70.7 3.9912 87.0 4.8018 71.9 3.9951 84.9 4.8057 72.5 4.0000 88.2 4.8105 73.7 4.0053 86.1 4.8151 71.9 4.0120 88.5 4.8208 74.8 4.0174 66.7 4.8239 74.3 4.0217 90.8 4.8303 75.5 4.0279 90.2 4.8342 78.1 4.0298 93 5 4.8469 82.0 4.0446 90.5 4.8531 80.2 4.0467 88.2 4.8627 80.2 4.0o40 87.6 4.8799 79.3 4.0657 92.6 4.8913 74.8 4.0708 93.5 4.9009 71.9 4.0809 95.6 4.9113 72.5

-191TABLE III (Cont'd) Time Head. Time Head Seconds Feet of Water Seconds Feet of Water 4.9156 70.1 4.9210 62.4 4.9318 55.3 4.,9432 50.0 4.9515 44.6 4.9613 44.6 4.9675 41.1 4.9762 40.5 4.9878 36.7 5.0000 19.8 5.0118 27.8 5.0285 25.4 5.0439 18.0 5.0618 13.0 5.0740 11.3

-192TABLE IV EXPERIMENTAL RUN NUMBER 34 COLUMN-SEPARATION VOID DATA Time Depth of Flow -- Feet Time Depth of Flow -- Feet Seconds NgSeconds1o_ Gg 4 ___ No1 N 4 No. No. No. 4 No. 5 0.00.083.083.083.98.o48.079.083 0.20.083.083.083 1.oo0.048.073.081.22.083.083.083.02.047.065.079.24.083.083.083.04.047.0o4.079.26.083.083.083.06.046.051.078.28.080.083.083.08.045.050.078 0.30.078.083.083 1.10.045.o06.078.32.076.083.083.12.o04.067.077.34.073.083.083.14.044.063.077.36.071.083.083.16.043.057.073.38.073.083.083.18.043.053.068 o.40.076.083.083 1.20.043.053.063.42.077.083.083.22.042.052.06.44.073.083.083.24.043.052.0o8.46.070.083.083.26.043.053.052.48.068.083.083.28.043.055.052 0.50.064.083.083 1.30.043.0o8.059.52.059.083.083.32.043.064.062.54.0o4.083.083.34.044.063.06.56.051.083.083.36.044.062.056.58.o49.083.083.38.043.057.054 o.6o.o49.083.083 1.40.044.08.051.62.051.083.083.42.044.04.051.64.051.083.083.44.044.052.052.66.050.083.083.46.043.050.052.68.o49.083.083.48.043.046.053 0.70.048.083.083 1.50.043.o49.o04.72.o46.083.083.52.043.o49.055.74.045.083.083.54.043.0o4.o56.76.044.083.083.56.043.054.056.78.044.083.083.58.043.052 xo68 0.80.043.083.083 1.60.043.o04.083.82.o43.083.083.62.043.055.083.84.042.083.083.64.044.058.083.86.043.083.083.66.044.072.083.88.043.083.083.68.044.083.083 0.90go.044.081.083 1.70.044.083.083.92.045.081.083.72.044.083.083.94.047.081.083.74.044.083.083.96.048.080.083.76.044.083.083

-193TABLE IV (Cont'd) Time Depth of Flow - Feet Time Depth of Flow -- Feet Gage Gage Seconds Seco Nndns Seconds o. 1 No. 5 No. 1_ No. 4 No. 5 1.78.044.083.083.06.079.083.083 1.80.o45.083.083.08 ~ 079.083.083.82.052.083.083 3.10.082.083.083.84.071.083.083.12.083.083.083.86.081.083.083.14.083.083.083.88.083.083.083.16.083.083.083 1.90.083.083.083.18.083.083.083 2.00.083.083.083 3.20.083.083.083 2.10.083.083.083 3.30.083.083.083 2.20.083.083.083 3.40.083.083.083 2.30.083.083.083 3.50.083.083.083 2.40.083.083.083 5.00.083.083.083.42.081.083.083.44.079.083.083.46.077.083.083.48.075.083.083 2.50.074.083.083.52.072.083.083.54.070.083.083.56.068.083.083.58.063.083.083 2.60.057.083.083.62.058.083.083.64.o60.083.083.66.060.083.083.68.061.083.083 2.70.063.083.083.72.063.083.083.74.063.083.083.76.063.083.083.78.o63.083.083 2.80.063.083.083.82.063.083.083.84.064.083.083.86.064.083.083.88.o65.o83.083 2.90.065.083.083.92.064.083.083.94.o64.083.083.96.063.083.083.98.063.083.083 3.00.063.083.083.02.o65.083.083.04.o69.o83.o83

APPEND IX IV COMPUTER SIMULATED RESULTS The results obtained from digital computer simulation of column separation accompanying transient pipe flow for Group II laboratory conditions are presented in this appendix. Photo reduced copies of the line-printer output list the data in sequence with respect to time, However, only the first 2,000 cycles (3.704 seconds of the simulated phenomena) are presented. Certain events which occur during the course of the simulation are of particular interest. These events are identified according to time and to cycle number: Time in Seconds Cycle No. Event 0 Computation starts while steady-state conditions prevail. ~00557 3 Gate valve is instantaneously closed,.10299 55 High pressure wave reaches reservoir; direction of flow at reservoir reverses..03711-.1855'7 At valve, pressure continues to increase slightly as pipe is "packed"..19299 104 Direction of flow at valve reverses and pressure begins to decrease. ~20413 110 Negative absolute pressure occurs at gate valve and initial separation void forms. Free-surface flow throughout void is toward reservoir. -194

-195Time in Seconds Cycle No. Event 1.16723 629 Flow within void reverses direction at point 0.384 foot from gate valve. Reversal propagates in both directions from this point with subsequent cycles. 1.48270 799 Column separation void reaches its maximum extent, 2.30 + feet measured from gate valve, and hovers at this approximate distance. 1.62837 869 Flow direction at reservoir reverses. Flow starts toward valve. 2.11363 1139 Void starts to retreat from its maximum extent. 3013426 1689 Initial void collapses. Second high pressure rise occurs, 3.23540 1743 High pressure wave reaches reservoir; direction of flow at reservoir reverses, 3.15468-3 32911 Pressure continues to increase at valve as pipe is "packed". 3~33468 1797 Negative absolute pressure occurs at the gate valve and the second separation voids forms.

GIVEN INF(JRMATILON GIVEN PHYSICAL DATA AND CONSTANTS RHO = 1.930 KA =.4J66E 08 VMAX = 3.50 HBAR = 33.224 0 = DIMENSICNAL CATA LOF-CRI0ING ANALYTIC MODEL HEL = 34.71I4 MEL = 26.328 LEL 22.078 WSEL = 22.514 LUCI = 18. L', L& 44.jU LUC3 444.u;i0 PIPE CHARACERISTICS AND RELATED DATA E(l) =.5112E 08 ZETA(I) =.280 0(1) =.06278 0(1) =.06250 Rh ) = *u4413 LII) =.17 MfI41) =I E(2) =.24480 10 ZE4AI2) =.300 0(2) =.08292 8(2) =.04678 R(M) =.0414t, LL(2) = 43.. 1i( M'L I So PROGRAM OPERATION AND CONTROL DATA MAXIMUM ALLOWASLE ITERATIOAS = 4003 COMPUTATIUN CUTOFF LIMIT = 1600 INITIAL CAMIVY PrIJITijTuT 4 vAL/t,I1UTJFF POINU =3 INITIAL PRINTOUT LIMIT POINT = 4 PRINTO)UT FREQUENCY CONTROL = 10 CAVITY PRIN4TtluT CYCLL FRICTION VALUES FOR CONSECUTIVE REYN4OLDS NUMbERS FROM 100 TO 35100 IN INCREM-ENTS OF IGOC. REYNOLDS'JO. = 100 Ff1. 0) =.7000 FI', u).40CU REYNOLDS ND. = 1100 F(1t 1) =.1400 Fl?, 1) u.65 REYNOLDS JO. = 2100 F(L, 2) =.0925 Ff2, 5 6 ) REYNOLDS NO. = 3100 FIt. 3) =.0735 F12, 3) z.0445 RZEYNOLDS NO. = 4100 Ff1. 4) =.0630 Fl4, 4) =.0385 REYNOLDS NO.,= 5100 Ff1, 5) =.0570 Ff2, 5) = U350 REYNOLDS NO. = 6100 f(1, 6) =.0525 Ff2, 6) =.0328 RFYNOLDS NO. = 7100 Ff1, 7) =.0490 F(Z, 7) =.0318 REYII)OLDS a7. = 8100 Ff1, 8) =.0470 Fl?, 8) =.312 RE'YNOLDS NO. = 9100 Ff1, 9) =.0454 F(2, 9) =.0311 REYNOLDS'O. = 10100 F(1.10) =.0438 F(291U) =.0309 REYNOLDS ND. = 11100 F(1,11) =.0422 F(2t11) =.0306 REYAOLDS NO. = 12100 FIl12) =.0412 F(2,1?) =.0302 RLYJ-LIS NO40. = 13100 F(1.13) =.0402 F(2,13) =.0299 REYNOLDS 0JO. = 14100 Ff1it4) =.0393 F(2v14) =.0296 LEYNOLOS NO. = 15100 F(1915) =.0386 F(2.15) =.0292 REYAOLDS NO. = 16100 F(lI,6) =.0380 Ff2,16) =.u289 REYNOLDS NO. = 17100 F(1.17) =.0374 F(2.17) =.0287 RtEYJOLDS NO. = 18100 Ff1.18) =.0368 F(2.18) =.0284 RE Y )LS NO. = 19100 Ff1.19) =.0362 F(2919) =.0280 REY.J'3LDS iD0. = 20100 Ff1.20) =.0357 Ff2.20) =.0278 RiLYNOLDS NO. = 21100 F(lv21) =.0352 F(2,21) =.0275 REY'JOLOS'JO. = 22100 Ff1.22) =.0348 F(2.22) = U0272 REYNOL'D4nS ND. = 23100 Ff1.23) =.0345 Ff2.23) =.0270 REYNOLDS NO. = 24100 Ff1.24) =.0342 F(2.24) =.0268 RLE YOL S NNO. = 25100 F(1.25) =.0339 F(2.25) =.0266 REYNOLDS NO. = 26100 Ff1.26) =.0337 Ff2.26) =.0263 REYNOLDS ND. = 27100 F(1.27) =.0334 F(2.27) =.0261 REYNOLDS NO. = 28100 F(1.28) =.0332 F(2.28) =.0259 REYNOLDS NO. = 29100 F(1.29) =.0330 F(2.29) =.0257 REYNOLDS 140. = 30100 Ff1.30) =.0329 F(2.30) =.0255 REYNOLDS NO. ='31100 F(l131) =.0328 F(2.31) =.0253 REYNOLDS NO. = 32100 F(1.32) =.0327 F(2.32) =.0251 REYNOLDS NO. = 33100 Ff1.33) =.0326 F(2.33) =.0249 REYNOLDS NO. = 34100 Ff1.34) =.0325 F(2.34) =.0248 REYNOLDS NO. = 35100 Ff1.35) =.0324 F(2.35) =.0247

COMPUTEO RFSULTS H(E, 0) = 46.246 V(l, O) =-1.988 ELP(X, 0) = )e.?[c) TtlETA(E, 0) =.000 h(l, 1) = 46.491 V([, [) =-E.)78 ELP(I, E) = 34.?19 THETA(E, 1) =.000 HI2, O) = 46.491 V(2, O) =-l.981 LLP(2, O) = )4.7[9 rHLTA(2, G)=.003 H(2, [) = 46.67! V(2, L) =-1.981 ELP{2, E)= 34.719 THETA(Z, E) =.000 H[2, 2) = 46.803 V(2, 2) =-t.9~1 ELP(2, 2) = 3l,,.02~ I HETA(Z, Z)=.3l,,9 H(2, 3) = 47.049 V(2, 3) =-[.981 ELP(2, 3) = 29.1)~ THETA(Z, 3) =.329 H(2, 4) = 41.236 V(2, 4) =-1.981 ELP(2, ~+) = 26,.)t~8 THLTA(2, 4) =.32~J H(Z, 5) = 47.422 V(2, 5) =-1.98! ELP(2, 5)= Zb. Z f6 THETA(l,,, 5) =.011 H(2, 6) = 47.F)'38 VIZ, 6) =-E.~)8[ ELP{2, 6) = 2~.L42 THETA(Z, 6) -.011 H(2, 7) = 47.794 V(2, 7) =-1.981 ELP(2, 7) = 26.049 THFTA(2, 1) —'.011 H(2, ~) = 47.960 V(2, 8) =-1.981 ELP(2, 8) = 25.95b THeTA(Z, b):.OIE h(2, <~) = 48.166 V(2, 9) =-1.981 ELP(2, 9) = 25.d62 THETA(Z, 9)=.Oil H(2, 10) = 48.352 V(2, 10) =-1.981 ELP(2, lOT = Y5.169 THETAI2, 10):.011 Hi2, 11): 4a.5)9 V(l,,, 1[) =-1.98[ ELP(2, 11) = 25.616 THETA(Z, 11) --.011 H(2, 12) = 48.7?5 V(2, [2) =-l.98l ELP(2, [2) = 25.~82 IHETA(2, 12) =.Oil H(2, 13): 48.'~11 V(2, 13) =-E.981 ELP(2, 13) = 25.~,89 THETAll,', 15) =.011 H(2, ta) = 49.007 V(2, 14):-t.981 ELP(2, t4) = 2S.5~6 THEI'A(Z, 14) -.011 K(2, 15) = /,9.283 V(2, /5) =-E.981 [LP(2, t5): 25.502 THETA(2, 15)'".OE1 H(2, 16) = 49.~+69 V(2, E(5) =-E.981 ~LP(2-, 16) = 25.20~) THETA(2, lb) =.Oil H(Z, 17) = 49.655 V(2, 17) =-1.981 ELP(2, 17) = 25.[16 [HETA(2, 17) = oOll H(2, 18) = 49.8/+l V(2, 18) =-1.981 ELP(2, 18) = 25.~j22 THETA(Z, 18) =.011 H(2, 19) = 50.028 V(2, 19) =-E.981 ELP(2, 19) = 24.929 [HETA(Z, 19) =.011 h(2, 20) = 50.214 V(2, ZO) =-1.981 t:LP(2, 20) = 24..~3G THETA(Z, 20) = o01[ I H(2, 21) = 50.400 V(2, i,'1) =-l.9aE ELP(2, 21) = 24.74,_' THETA(2, 21) =.01! }, H(2, 22) = 50.586 V(Z, 22):-t.981 ELP(2, 22) = 24.649 THETa(2, 22) =.011 y~) H(2, 23) = 50.712 V(2, 23) =-1.961 ELP(2, 23) = 24.'~56 THETa(2, 23)=.011 — ~ H(2, 24) = 50.e~58 V(2, 24) =-1.981 ELP(2, 24) = 24.462 THETA(2, 24):.01E I HI2, 25! = 51.t44 V(2, 25) =-1.981 ELP(2, 25} = i24.~6~ THETA(2, 25) =.OEE Hi2, 26) = 51.3~1 Vl2, 26) =-1.981 ELP(2, 26) = 24.216 THETA(2, 26) =.01[ H(2, 27): 5~.517 V(2, 27) =-l.981 ELP(2, 21): 24.182 THETA(2, 21) =.011 H(2, 28) = 51.703 V(2, 28) =-1.981 ELP(2, 28) = 24.089 THeTA(Z, 2~) =.DIE 1~(2, 29): 51.889 V(2, 29) =-1.981 ELP(2, 29) = 23.996 THETA(2, 29) =.Oil H(2, 30) ='32.075 V(2, 30) =-1.981 ELP(2, 30): 23.902 THETA(2, 3D) =.011 H[2, 31) = 52.261 V(2, 51) =-1.981 ELP(2, 31) = 23.809 THETA(Z, 31) = o011 H(Z, 32) = 52.447 V(2, 32) =-1.981 [LP(2, 32): 23.116 THETA(2, 32):.OIE H(2, 31~) = 52.634 V(2, 33) =-1.98l ELP(2, 33) = 23.623 [HETA(2, 33) =.0[1 H(2, 34): 52.820 V(2, 34) =-1.981 ELP(2, 34) = 23.529 THETA(Z, 34) =.01! H(2, 35) = 53.006 V(2, 35) =-1.98l ELP(2, 35) = 23.436 THETA(Z, 35) =.011 H(2, 36) = 53.192 V(2, 36) =-1.981 [LP{2, 36) = 23.343 THETA(2, 36) =.011 H(2, 37) = 53.378 V(2, 37) =-1.981 ELP(2, 37) = 23.249 THETA(2, 37) =.011 H(2, 38): 53.564 V(2, 38):-1.981 ELP(2, 38) = 23.156 THETA(2, 38):.01! H(2, 39) = 53.750 V(2, 39) =-1.981 ELP(2, 39) - 23.063 THLTA(2t 39) =.011 H(2, 40) = 53.9.37 V(2, ~+0) =-1.981 ELP(2, 40) = 22.969 THETA(Z, 40) =.011 H(2, 41) = 54.123 V(2, 41) =-l.981 ELP(Z, 41) = 22.816 THETA(2, 41) =.011 H(2, 42) = 54.309 V(2, 42) =-1.981 ELP(2, 42) = 22.183 1HETA(2, 42) =.011 HI2, 43) = 54.495 V(2, /,3) — 1.981 ELP(Zt 43): 22.089 TH[;TA(2, 43) = o011 H(2, 44): 54.681 V(2, 44) =-1.981 ELP(Zt 44) = 22.59G THETA(2, 44) = o011 H(2, 45) = 54.867 V(2, 451:-1.981 ELP(Zt 45) = 22.503 THETA(2, 45) =.Oil H(2, 46) = 55.053 V(2, 46) =-1.981 ELPIZ, 46) = 22.409 THETA(2, 46) =.Oil H(2, 47) = 55.240 V(2, 47) =-1.981 ELP(2, 47) = 22.316 THETA(2, 47) =.011 H(2, 48) = 55.426 V(2, 48):-1.981 ELP(2, 48) - 22.223 THETA(2, 48) =.011 H(2) 49) = 55.612 V(2, 49):-1.981 ELP(Zt 49) = 22.129 THETA(2, 49) =.011 H(2t 50): 55.798 V(2, 50):-1.981 ELP(2, 50): 22.078 THETA(2, 50) =.000

VALUE4 IF KEY VAkIAIJLLS uSEU THKiUGCH0UF THE UMUTATI11NS IN PART I. All) = 44).8407, A(2) = 47299.95945, DELX(1) = 8.7l1.O, DELX(2) 8 VELTI(L) 3.57A14C0-3, UELTT(2) = 1.8!569LF-G3, UELT a 1.655691-n3, MUl) = -1Li5712 MU =2) 2. 1 1 2.4E-04 CP(L) =.8dIL93, CPI() a A8 Cul, AG () =.1 Ag0Gl(i) 1=47.'0190, GGA(1) = 1312 GOA(2) - 799289 AT lIME I =.00186 sECONDS, AFTER I COMPLETE CYCLES OF OPERAII1l1, THE VLLOCITIES AND PRESSIJRLS AT KC-Y POIVrs T1HN'U-h1UJT T~ FLOw SYSTA' ARE AS TABULATEU EELfl6. LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (L,50) V -1.98b1 -l.v881 -1.4814 -1.9814 -1.9814 -1.9814 -1.9614 t. I H 17.6737 17.92 16 17.9216 22.0031 24.bt"80 27.6124,0.4L68 31.1790 AT TIME T =.00371 SECONDS, AFIER 2 COMPLEIF CYCLES OF OPERAI IO4, THE V&ELLOCITIES AND PRESSURLS A KEY POLNIS ThRP:U,HjUU THFLOw SYSTEM ARE AS TA8ULATEU HLLOW. LOC. (1910) (191) (2,0) (29101 2( I23O (2930 )~4~ V -1.9881 -1.9H881 -.9814 -1.981: -1.9814 -1.9614 -1.9814 -1.9814 H L7.b73h 17.9211 17.9217 22.0038 24.H082 21.61e5 0.416 9 33.179 - THE GATE VALVE AT LOCATION J = lv1 -= HA S BEEN INSTANTANErOUSLY CLOsio Al TMIF T =.0C5579 SECONUS, OR AT OPErATI04 NHMhER 3. AT lIME T.0C557 SEC0NDS, AFUER 3 CUMPLETE CYCLES OF OPEaATIflvv THE VCLOCITIES AND PRESSURES AT KEY POINTS THrs;U4sisUI ii FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (10) (til) (2,0) (2,10) (2 LL) (2930) (2,40) (2,5)H V.0000 -1.9881 -1.9814 -1.9814 -1.9814 -1.981.4 1.9U4 -1.4814 H 169.1229 11.9218 17.9218 22.0039 24.dC083 27.6127 30.4170 33.1740 AT TIME T =.01856 SECONDS, AFTER 10) COMPLEIE CYCLES OF OPERATION, THE VELOCITIES AN) PRESSU RES AT KEY POINIS THRL'Ui-I3JUT I FLOW SYSTEM ARE AS TABULATE) BELOW. LOC. (1,0) (1,1) (2,0) (Q,10) (29,C) (2,30) (2I40) ( C,5.) V.0000 -.2578 -.2569 -1.9u14 -1.9814 -1.9614 -1.9814 -1.9814 H 277.4224 271.9850 271.985U 22.0C53 24.8091 27.6135 30.4178 3-5.179E AT TIME T =.03711 SECONDS, AFTER 20 COMPLETE CYCLES OF OPERATIO.N,. THE VELOCITIES AND PRESSURES AT KEY POINTS THR3U,;HDUT I FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (I1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.0000 -.0219 -.0218 -.2637 -1.9814 -1.9814 -1.9814 -1.9814 H 307.8768 307.4195 307.4795 275.0641 24.8108 27.6146 30.4178 3Js.1190 AT TIME T =.05567 SECONDS, AFTER 30 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THkuIirHiOu'T tilL FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2l20) (2930) 12940) (2,50) V.0000 -.0040 -.0039 -.0282 -.2705 -1.9814 -1.9814 -i.9814 H 311.0562 311.0418 311.0418 310.6210 276.8696 27.6152 30.4179 33.1790 AT TIME T =.07423 SECONDS, AFTER 40 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AN) PRESSURES AT KEY POINTS THROUSHOUT Irf FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2,2C) (2930) (2940) (2950) V.0000 -.0026 -.0026 -.0103 -.0346 -.2773 -1.9814 -1.9814

ii 3,12b1 ~ 312.1759 3L2.1159 3.1. 1264u 28478b73 3&.'184 Al I IME 1.09278 SECCI)DS, AfIE- J COMPLET1 CYELES0FIPERArIi, I-L V:.LO I It'S 4hJ. * ~ ~ Kit.Y POtNJT' T-rik H. 0 I FLOW )YsfLM ARE AS TAULLTEOC CELO]W. LOC. 1.32( (,1) (2,0) ( loT ( (2,3C) (2,',C I C iV.o00 - 25 -. C -.Js -.6 lbu -.40) -4 U II.3 1 3. 1 08o 3 1 3. 1248 3 1 3. 124 8 315.3 I1.8 1 4.3493 2.', 7 THE 0I1KRCTIO~N OF FL-fi ArT ltiH rFSi1'_:RVt)R i:,U TOF PI!" 3 RIPL Y,IEIM HAS RL iU Al 1i.-RiE 1 =I AT TIME T =.1113 4 CONDs, AFTER LU COMPLEIC CYCLE.: OF 1PIERATIu-4, FHE VLLOCIT[ES A'iL PRFSSI.JPS_'j AFT KEY P0ov1L:) Til *uoi I FLOW sYaTEM ARE AS T~oULATEU 3FLOW. LOC. (Iu) (I,1) (2,0) (,13) (2,20) (2,310) (2P4C) V.OCJ000 u2 5 -.2. -.0383 u2 -.23 1 -.0473 3.C18 H 314. L418.314. C 78 3 14. 0 5 I310.2t).[845 31 7. 9111.3 1.2143 33.17) Af TIME T.1299J SECONUD, AFtER 70 COMPtLErt CYCLE. OF OPE RAT 1O., IhE VI-LOCITI ES A"C PRE SSUXLSA rAF KtY PONS rH-e U,.H;JT I4 - FLOW SYSTEM ARE AS IAOULAFEU EELOIw. LOG. (1,0~) (1,1) (22,0) (2,10) (2 ) ( 2, 302 ) (2,4) (,) V.0000 QC-.02 - 0o24 -.uO88 -.1 151 -.021b 1.65 79 18767 H 314.9733 3 14.98 o 314.9886 317.1995 3 18.1314 319.0495 Y1.4492 33.17)0 AT TIME T =.14846 SECONDS, AFTER 80 COMPLETE CYCLES OF OPERAT(U'4, IhE VILOCITIES Ac) PRESSURoS Al KEY POINTS THRiUAOJ Tm FLOW SY.TEM ARE AS TABULATED BELOW. LOC. (iG) (1,1) (2,0) (2,10) (2,2C) (2,30) (2,40) V.0000 -_. b0 -.00/4 -.uJ86 -.0151 1.6602 1.8964 L.91? H 315.9036 315.9188 315. 918 316.1297 319.0628 72.4635 3o.9225 33.1790 AT TIME T =.16701 SELONDS, AFTER 9C COMPLETE CYCLES OF (JPERATIOiL, THE VELOCITIES AND PRESSURES AT KEY POINTS TH:1U.HOUT TV fLOw SYSTEM ARE AS TABULATED BELOW. LOC. (1to) (1,1). (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.0000 -.0024 -.0024 -.OCBo 1.6611 1.8969 1.9147 l.916~ H 316.8337 316.5485 316.845.319.0588 73.2625 37.8110 34.3139.33.179 Al HIME r =.1b557 SELONDS, AFTER 100 COMPLETE CYCLES OF OPERATION, THE VLELOCITIES AND PRESSURES Al KEY PJIN[.S THRlUHOUT T9 FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (111) e2,0) (2,10) (2,20) (2930)' (2,40) (2,50) V.0000 -.0024 -. 024 1.660 1.8973 1.9151 1.9164 1.9Ibs H 317.7625 317.7769 317.7769 74.0599 38.6820 35.1923'34.1157 33.1790 AT TIME I.19485 SECONDS, AFTER 1.05 COMPLETE CYCLES OF OPERATION, FIlE VELOCITIES AND PRESSURES AT KEY POINTS THiU8()UT Hi FLOW SYSTEM ARE AS TABULA[ED BELOW. LOG. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (i,50) V.0000 1.2830 1.2786 1.8467 1.9115 1.91o4 1.9168 1.9168 H 259.2695 220.2949 220.2949 47.0015 36.6331 35.0356 34.1033 33. AT TIME T =.194 70 SECONDS, AFTER 106 COMPLETE CYCLES OF OPERATION, IHE VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THO FLOW SYSTEM ARE ASTAdULATED BELOW. LOC. (1,0) (1,1l) (2,0) (291U) (2,20) (2,30) (2,40) (2,50) V.0000 1.4479 1.4430 1.8627 1.9126 1.9164 1.9167 1.9168 H 188.4125 177.1988 177.1988 44.6576 36.4565 35.0225 34.1024 33.1790

AT lIME T =.19856 sECONDS, AFTER 107 COMPLETE CYCLES OF UPERATIGN, THiE VELOGI llES AND PRESSURCS AT KEY POINIS Thi'' UW'OUI l:i FLOW SYSTEM ARE AS rAdULATED BELOW. LOC. (1.,) (1il) (2,C) (2,10) { 2120) (2,30) (240) [ 5) V.0000 1.4354 1.4305 1.8751 1.9137 I.91b66 l.91b8. H 125.4320 126.6930 126.693U 42.85;1; 36.5190 35.0116:34. 1,15. _. 17'1. AT TIME T =.20041 SECO.'NDS, AFTER 13f COMPLEIC CYCLES OF (OPERATI0:4, IHE V~LOCITIES ANO PRESSURtS. T KEY QoINTS T' U'iIU[,' FLOW SYSTEM ARE AS TABULATEOD t6ELOW. LOC. (1,0) (1,1) (2,0) (2,10) (Z,20) (2,30) (2,40) C) V.U000 1.3430 1.3335 1.8846 1.9144 1.9166 1.9168 1. P18 H 69.4124 76.2915 76.2915 41.4577 3o.2138 35.0039 34.1008.7179. AT TIME T =.20227 SECOlNLS, AFFER 109 COMPLETE CYCLES UF OPERAl ION, THE VELOCITIES AND PRESSURtS AT KFY POINTS TH'-:' LJH);.UT i. FLOW SYSTEM ARE AS TABULAIEL BELOW. LOC. (1,0) (1,1) (2,0) (2,10) 2 I,2) (2,3G) (2,40) {2,66) V.0000 1.2163 1.2122 1.892C 1.9150 1.9168 1.9169 1.9169) H 19.9497 29.2550 29.2550 40.3820 36.1313 34.9972 34. ILO 3.1796 CCOLUMN SEPARAfION HAS OCCURRED AT THE GATE VALVE AT lIME f =.ZU41i SECONDs (OPERATION CYCLE NO. 1l.). THIS 1s.1956 SECONDS AFTER CLOSURE OF THE GATE VALVE. THE MAGNITUDE OF DELXC REMAINS THE MINIMUM VALUE. GIVEN DATA AND COMPUTED VALUES PERTINENT TO COMPUTAItION OF FLliOW FOLLOWING CULUMN SEPARATION. I HVAPOR = 11.3000 DELXC = 0640 DELTC =.00928 MARK = J MUCH -.145' ) FACTOR = 5 XI =.0250 INC = 90 IOTA =.0P;1 PAREA =.00538 0 0 THE INITIAL VELOCITIES AND DEPTHS IN THE VAPOR CAVITY ARE LISTED BELtI. DIST..0000.0640 LOC. 0 1 U.000 1.053 ~.061.070 THE VELOCITIES AND ABSOLUTEL HEADS THROUGHTOUT THE REMAINDER OF THE FUiLL-FLOWING PIPE SYSTEM ARE AS LISTED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V 1.0935 1.2728 1.2685 1.8976 1.9154 1.9168 1.9169 1.9169 H 11.3000 15.5826 15.5826 39.5523 36.0688 34.9927 34.0998 33.1790 AT TIME Tr=.21155 SECONDS, AFTER 114 COMPLETE CYCLES OF OPERATION, THIE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINIS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY 0IST..0000.0640 LOC. 0 1 U.000.812 Z.054.066 FULL-FLOWING PIPE SYSTEM LOC. (1,0) ( 111) (2,0} (2,10) (2,20) (2,30) (2,40) 12,50) V 1.3537 1.4227 1.4179 2.0872 1.9165 1.9170 1.9170 1.917. H 11.3000 11.3000 11.3000 63.7753 35.9304 34.9820 34.0986 33.1790

Ar TIME i =.22083 SEC(INDSt AFTER 11' COMPLEIT CYCLES OF OPERAIIO N, Te1 VELOCITIES AND DEPTHS IN1 THE VAPOR.AVITY 3A WELL'~ VELOCITIES AN1D PkESSU,{ES AT KEY POI(NTS THROUGHOUT T1t: FLOW SYSTEM AL AS TABULATED BELOW. VAPOR CAVITY DIlT..COOG.C640.1280 LOC. 0 1 2 Li.oui.635 1.296 Z.051.uJ8.Go7 FULL-FLUWING PIPE SYSTLN LOC. (1,o) (11) (1, (2,0) ( ) (2,20) (2,30) (2,40) (2,50) V 1.459+ 1.4696 1.4647 1.733C 1.9171 1.9173 1.9173 1.91 3 H 11.300j 11300 i 11.3 00) 11. )3000 35.8734 34.970h8 34.0979 3. 179 Al' TIME T =.23011 SECCINOS, AFTE`i 124 COMPLETE CYCLE' OF OPERAIIOU,: THE VLOCITIES AND DEPTHS 1N THE VAPOR CAVITY:s WELL,s VELOCIrtIE AAID PKESUJRCS AT KEY P0IFi' THROUGHOUT THE FLO)W SYSTEM ARE AS TABULATED BELUW. VAPOR CAVITY DIST..00O0.0040.i280.1920 LOC. 0 1 2 3 U.000.543 1.120 1.501 L.048.0U2.05.076 FULL-FLOWING PIPE SYSIEM IR LOC. (1,6) (1,1) (2,0) (2,10) (2,(;) (2,30) (2,40) (2,50) V 1.5011 1.5325 1.5273 1.7275 2.!926 1.9174 1.9174 1.9174 H H 1 t. 3000 1 1. 30 11. 3000 11. 300 61.6408 34.9756 34. 0981 33. 1790 Af TIME r =.23938 SECCONOS, AFTER 129 COMPLETE CYCLES OF UPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY 4,S WELL AS VELOCITIES AND PRESSUR(ES AT KEY POINTS TIIROUGHOUT THE FLOW SYSTEM AKt AS TABULATED BELOW. VAPOR CAVITY UIST..0000.0640.1280. 1920 LOCG 0 1 2 3 U.000.479.988 1.328 Z. 46.048.053.366 FULL-FLOWING PIPE SYSTEM LOG. (1,() { 161) (2, 0 ((2,10) (2,20 ) (2,30) (2,40) (2,50) V 1.5418 1.5404 1.5352 1.7120 1.7491 1.9176 1.9176 I.)176 H 11.3000 11.3245 11.3245 11.3000 11.3000 34.9742 _ 34.0977 33.179i AT TIME T =.25794 SECONDS, AFTER 139 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPLR CAVITY AS WELL,A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULAIED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560 LOG. 0 1 2 3 4 U.000.392.798 1.082 1.304 Z.042.043.046.058.069 FULL-FLOWING PIPE SYSTEM

LOG. (I10) 1 (2,0) (2,10) (2,9') (2,30) ( 2 1 4;5' V 1.5854 1.839 1.5?16 1.691 1. 7449 1.7555 1.9174 1.'i 17 H 11.3000 11..,67 11.3867 11.300Q 1 1.300 11.3000 34.0973 3.17 9 AT TIME T = 216(50 SECOJNDS, AFTER 149 CCMPLETC CYCLES OF OPERATION, I'HE VELOCITIES AND DEPTHS IN IHE VAPOR CAVITY \s iFLL ~S VELOCITIES ANO PRESSURES AT KIY PUI;'TS THROUGHOUT ThE FLOW SYSTEM ArL AS TAMULATED BELOW. VAPOR CAVITY DIST..0000.0640.1260.1920.2560.323io LOC. 0 1 2 3 4 U.000.333.665.897 1.142 1.453 L.039.039.042.051.060.072 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (L,5C) V 1.6156 1.6139 1.6085 1.6891 1.7400 1.7516 1.7618.91z2 H 11.3000 11.3809 11.3809 11.300iU 11.30u 11.30u0 11.3000i j.5179 AT TIME T.29505 SECONDS, AFTER 159 COMPLETE CYCLES OF OPERATIUN, THE VELOCITIES AND DEPTHS IN THE VAPOR EAVITY iS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARL AS TABULATED BELON. VAPUR CAVITY DIST..0000.0640 1280.0920.2560.3200.3840 LOC. 0 1 2 3 4 5 6 U.000.288.564.761.979 1.227 1.475 Z.036.037.039.046.054 o0b2.072 FULL-FLOWING PIPE SYSTEM 0 LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2940) (2,50) V 1.6376 1.6361 1.6306 1.6855 1.7347 1.7478 1.7579 (.8710 H 11.3000 11.3676 11.3b16 11.3000 11.3000 11.3000 11.3(00 33.1790 AT TIME T.31361 SECONDS, AFTER 169 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479 LOC. 0 1 2 3 4 5 6 7 U.000.252.485.657.851 1.067 1.280 1.491 Z.034.035.037.043.049.0s5.063.074 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2930) (2,40) (2,50) V 1.6540 1.6527 1.6472 1.6839 1.7293 1.7440 1.7369 1.6356 H 11.3000 11.3574 11.3514 11.3000 11.3000 11.3000 13.8824 33.1790 AT TIME T =.33217 SECONDS, AFTER 179 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES Al' KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119 LOC. 0 1 2 3 4 5 6 7 8 U.000.221.420.573.747.942 1.131 1.319 1.504

Z.032.033.035.040.045.051.057.0u3.074 FULL-FLOWING PIPE SYSTrEM LOC. (1,0) (1,i) (2,0) (2,10) (2Z)2) (2,30) (2),!') V 1.6662 1.6652 1.6595 1.6845 1.24C 1.7276 1.6144 1. 6? H 11.3000 11.3472 11.3412 11i.3000 11.3000 12.99 j4,.4613 J3. t AT TIME T =.35073 SECONUDS, AFIER 189 COMPLETE CYCLES OF OPERATION, ltiE VELOCITIES ANLi DEi'liiS IN THF VAPOR CAVITY: WLLL'S VELOCITIES ANU PRESSURES Al KEY POINTS I'HROUGHlOUT THE FLOW SYSTEM ARL AS TABULATED 6ELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4419.5119.5759 LOC. 0 i 2 3 4 5 6 7 8 9 U.000.194.366.50.662.839 1.012 1.187 1.335 1.615 Z.031.032.034.038.042.047.052.,J8.065.01l FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,) (2,0) (, IC) (2,2) (2,3C) (2,40) t(,tO) V 1.6753 1.u745 1.66[88 1.6874 1.1145 l.6053 1.5940 1.5934 H 11.3000 11.3368 11.3368 11.3000C 11.563 29.4638 32.2588 J i.171: AT TIME T =.36928 SECONDS, AFTER 199 COMPLETE CYCLES OF OPERATIUN, IHE VELOCITIES AND DEPTHS IN lliE VAPOR CAVITY AS hELL.~ VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM AltL AS IABULATED BELOW. VAPOR CAVITY DIST.. 000oo.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399 (O LOC. 0.l 2 3 4 5 6 7 8 9 10 0 U.000.171.321.446.591.752.12 1.u74 1.212 1.440 1.682 Z.030.031.033.036.040.044.049.C53.060.C64.C78 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,2n) (2,30) (2t40) (2,50) V 1.6822 1.6815 1.6758 1.6923 1.5893 1.5816 1.5845 1.585.s H 11.3000 11.3295 11.3295 11.4256 27.'t391 30.7421 32.2148 J3.1792 AT TIME T =.38784 SECONDS, AFTER 209 COMPLETE CYCLES GF RiPERATIGN, THE VLLOCITIFS ANl) DEPIHS IN THE VAPOR CAVITY As WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOw SYSTEM ARE AS TABULAITD BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560 5200. 3840.4479.5119.5759.639' LOC. 0 1 2 3 4 5 6 7 8 9 10 U.)000.150.282.396.531.679.826.977 1.108 1.309 1.4'6 7.029.030.032.035.038.042.046.u50.055.0)8.066 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2?50) V 1.6874 1.6868 1.6811 1.5880 1.5691 1.5687 1.5730 1.5756 H 11.3000 11.3215 1.3215 27.5549 29.2013 30.1870 31.7095 3.3.179:3 AT TIME T =.40640 SECONDS, AFTER 219 COMPLETE CYCLES OF OPERATION, TlHE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY 4S WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY

LI ST..0000.0640.1280.1920.2560.320J.3840.44 79.5119.5759.6399. D39 LOC. 0 1 2 3 4 5 7 8 9 1G 11 U.000.131.248.354.479.615.753.ou2 1. 18 1.203 1.316 1. 358 Z.028.029. Ojl.034.037.040.043.047.u52.0,4.061..67 FULL-FLOWING PIPE SYST[M LOC. (lC) (11l) (2,0) (2,10) (2,20) (2,30) (2,40) (Lb5c) V 1.4891 1.4970 1.4919 1.5666 1.3075 1.5607 1.5599 1. 56' H 1 1.3000 12.4i81 12.4187 29. 174 GC.2893 JU. 1734 1. 1983 3. 1 71: AT TIME I =.42495 SECONDS, AFTER 229 COMPLE-T CYCLELS OF OPERATlON, THE VLOCITIIES A:40 DEPTHS IN IHF VAP.R CAVITY:5 WLLI,, VELOCITIES AND PRESSURES AT KEY POI'JTS THROUGHOUl TrIE FLOW SYaT[I; AKL As TABULATED BELOW. VAPOR CAVITY 0DIST..0000.0640.i28G.1920.2560. 32 i).3840.4479.5119.!759.6399.1 39.7679 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 U.000.114.219.317.434.560.688.818.939 1.112 1.212 1.Z72 1.297 L.028.028.030.033.C35.0,,8. 41.u45.;,49.051.057..664.073 FULL-FLOWING PIPE SYSTFi LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,5C) V 1.4592 1.4587 1.4537 1.4 -19 1.5582 1.5588 1.5486 1.5444 H 11.3000 11.4113 11.4113 16.4861 3G.j760 1.29:)1 J1.6835 33. 179 AT TIME T =.44351 SECONDS, AFTER 219 COMPLETE CYCLES OF OPERATION, THE VELOilIIES AND DEPTHS iN THE VAPiR CAVITY',S WLL F o VELOCITIES AND PRESSURES AT KEY POINIS THROUGHOUT THE FLOw lYSTtM Ait A: FABULAITD BELOw. i VAPOR CAVITY O OIST..0000.0640.1280. 1920.2560.3ZOO.3840.4479.5119. 759.6399.7039.7679 LOC. 0 i 2 3 4 5 b 7 8 9 10 11 12 U.000.100.194.285.394.512.631.152.869 1.031 1.122 1.118 1.213 Z.027.028.029.032.034.031.04C.v43.046.043.053 -. 60.065 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (Z,20) (2,30) (2,40) (2,50) V 1.4495 1.4494 1.4446 1.4459 1.4637 1.5462 1.5434 1.5365 H 11.3000 11.iC42 11.3042 13.73u4 17.5533 31.5763 33.3C69 33.1790 AT TIME r =.46207 SECONDS, AFTER 249 COMPLETE CYCLES OF OPERAflUO, THIE VELOCITIES AND DEPIHS lN THE VAPOR CAVITY AS WELL As VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM At- AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4 479.5119.5759.6399.7039.76r9.8319 LOC. 0 1 2 3 4 7 6 7 8 9 10 11 12 li U.000.087.171.258.359.469.581.094.807.958 1.041 1.093 1.143 1.220 Z.027.027.029.031.033.035.038.041.044.046.051.057.063.ob FULL-FLOWING PIPE SYSIEM LOC. (1,0) (111) (2O0) (2t10) (2,20) (2t30) (2,40) ( V,50) V. 1.4423 1.4422 1.4373 1.4365 1.4345 1.4489 1.5342 1.5423 H 11.3000 11.3037 11.3037 13.6583 15.2969 19.6201 33.1056 33.1790 AT TIME T.48062 SECONDS, AFTER 259 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY.S wELL AS

VELOCITIES AND PKESSURtES Al' KEY P'II'4TS TtHROUGHOUT THE FLOv bYSTEM A4K. AS TABULATED BELOv,. VAPOR CAVITY DIST..0000.0' 40.128J0.1920.2560.*3Zu)O.3143.4419.5119.5759.639.. T'39.7619.8ts19. 9:9 LOC. 0 1 2 3 4 6 / 8 9 1 It 12 i 1 U *.0 * 7. 152.233.328.430. 35.c/,L..752.892.8 1. " 15 1.3011 1.1,I 1. 322 Z.026.327.028.030.032.054. 37.,39. 42.044.C48.-54.060b.. 7i FULL-FLOWING PIPE SYSTJ-r LOC. (1,C) (1,1) (2,0) (2,) (2,2) (2,30) (/Z,4) (,) V 1.4342 1.4337 1.4'8 8 1.42o1 1.4218 1.4230 1.4483 1.:,31 H 1 1.3 St.;o 1 1.340 i 11.3401 1 4l. 4, 1. 1289 16. 922 19. 9 1. 79, AT TIME T =.49911.jEC!INDO, AFTEP, 269 COMPLETE CYCLL I OF 1OPER;ATIT-li [liE VLLOCITILS ANDl DEPTH-S IN IH- VAPOR CVITY 5 itL. AS VELOCITIES ANID PRLSSURES AT KEY POliiTS THIROUGHOUT TilE FL,) SYSTEN AKR.- AS TAbU)LA[T-D tELl)h. VAPOR CAVIY 1 DIST..u000. 04J.12820. 19 ZO.2560..2;0D.384C.4. 19.51 1.5759.6399.;.7619. 33 1.8J59 LOC. 0 1 2 3 4, 6 1 P 9 1 12 l., 14 U.000.O6.134.211.390.396.495.:'b.IC2.833.962.:)44 1.092 1. O.'J i L Z.'26.02 7.028.02.031.:J3.03..J8.'040.C42.046._ 5.07..l..: 4 DIST..9599 LOC. 15 U 1.422 Z.075 FULL-FLOWING PIPE SYSTt I LOC. (1,O) I1,L) (2,0) (2, (2,YOI I2,3C) (2,40) (Y,23, V 1.4224 1.4207 1.4159 1.4142 1., 147 1.4213 1.4212 1.3551 H 11.3000 11.4422 11.4422 14.682o 15.7351 15.72iJ 11.0824,#3.179U AT TIME T =.51774 SEiLCONOJ, AFfER 219 COMPLETE CYCLES OF UJPLRATIuAIJ ItE VELO.ITIES ANJ DEPTHS IN TniE VAPOR CAVITY'IS WcLL A, VELOCITIES AND PRESSURES AT KEY PO'INTS fltNOUGHOUT THE FL)OW SYSTF' AK,' A3 TAHJULATEU SELOn. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.384'0.4,t 19. 5119.5759.6399.1u39.1619.813 49.89 ) LOC. 0 t 2 3 4 5 6 1 d 9 12 11 12 1J 14 U.000 8..120.19.275.365.458.554.b57.778.841.879.937 1.;'8 1.130 Z.026.C2o.027.02).031.032.034.u37. 39.041.045.550.055.058.j6i DIST..9599 LOC. 15 U 1.266 L.065 FULL-FLOWING PIPE c YSTrM LUG. (11t) L1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V 1.4066 1.4053 1.400o 1.4046 1.4138 1.4129 1. 387 1.3115 H l 1~3003, 11.4560 11.4560 14.3098 14.6786 15.9303 29.2934 33. 790 AT TIME T =.53629 sECUINDsb AFTER 289 COMPLETE CYCLES OF OPERATION, THE VELOCITIES ANU DEPTHS IN THE VAPLIR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINftS THROUGHOUT THE FLOW SYSTEMt ARE AS TABULATED 8ELOW.

VAPOR CAVITY UIST..0000.0640.1280.1920.2560 *32u0.3840.4419.5119.5759.6399.70339.7679.83L3.899 LOG. 0 1 2 3 4 5 6 7 8 9 11 12 14 U.00 0.51.107.114.253.337.42,5.A7.616.721.784.826.878 9) 1*b.O2.1- 026.027.028.030.032.033.C36.038.039.043 j.4b.053.Q-6 01ST..9599 1.0239 LOG. 15 16 U 1.173 1.190 L.062.068 FULL-FLOWING PIPE SYSTCM LOG. (1,0) (1.1) (2,0) (2.10) (2,2)) (2.3C) (2,40) (45'u) V 1.3968 1.3961 1.3914 L.4002 1.4029 1.3216 1.3037 1.332' 11.3U0U 11.3402 11.3402 12.7375 14.5118 28.1920 )2.0049 33.179S AT TIME T.55485 SECONDS, AFTER 299 COMPLEFE CYCLEs OF OPERITIuN, THE VLLOCITIES AND DEPTW. IN THE VAPOR CAVIfY'S iLL LS VELOCITIES AND PRESSURES AT KEY POINTS IHROUGHOUF ThE FLOW SYSTEM AjiL AS TABULAitO POLOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4419.5119.5759.6399.1039.'1619.II.89 LOC. 0 1 2 3 4 5 6 7 i 9 10 iI 12 - 1' U.000.045.095.158.233.312.395.483.8.680.731.165.823 2 1 1.025.026.027.028.029.C31.033 35.36.038.042 L47.051 DIST..9599 1.0239 1.0879 LOG. 15 16 17 U 1.99 1.129 1.233 0 Z.059.064.073 FULL-FLOWING PIPE SYSTEM LOG. (1,0) (1,1) (2,03 (2,10) (2,2C) 2,30) (2,40) (,5 i) V 1.3959 1. 354 1.3907 1.393L 1.3085 1.2942 1.2954 1 29c, H 11.3000 11.3219 11.3219 13.1104 26.1961 30.5246 32.1145 3i.1740 AT TIME T.57341 SECoNDS, AFTER 309 COMPLETE CYCLES OF OPERAI ION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM AR_ AS TABULATED 8ELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5159.6399.7)39.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 e 9 1G 11 12 Li 14 U.000.039.086.144.214.289.368.452.543.636.682.714.772.8j9.951 L.025.026.026.027.029.030 32 U34.035.03r.041.045.049.052.054 01ST..9599 1.0239 1.0879 1.1519 LOG. 15 16 17 18 U 1.035 1.010 1.145 1.394 L.057.062.064.076 FULL-FLOWING PIPE SYSTEM LOG. (1,0) (1,1) (2,0) (2,10) (2,20) (2930) (2,40) (2,50) V 1.3948 1.3939 1.3892 1.30b6 1.2849 1.2825 1.2865 1.2885 H 11.3000 11.3595 11.3595 25.8330 29.0624 30.1181 31.7431 33.1790

AT TIME I =.59197 SECONDS, AFTER 319 COMPLETE CYCLES OF OPERATION, ThE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY IS.ELI. EL VELOCITIES AND PRES3URES AT KEY POINTS TIiRO)UGFHOUT THE FLOW SYSTEM ARL AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.89:,9 LOC. 0 1 2 3 4, 6 7 8 9 10 11 12 lj 14 U.000.035.077.132.198.269.343.424.511.595.636.667.725. ~8I.89' Z.u25.025.026.027.C028.G30.031.G33.034.036.040.044.047.u)., K2 DIST..9599 1.0239 1.0879 1.1519 LOC. 15 16 17 18 U.977 1.014 1.090 1.215 Z.055.059.Co2.067 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,2C) (2,30) (2,40) (?,50) V 1.2292 1.Z372 1.2330 1.2817 1.2806 1.2773 1.2756 1.2771 H 11.3000 13.6268 13.6268 28.4951 29. 1469 30.2834 31.2286 33.179g AT TIME T =.61052 SECONDS, AFTER 329 COMPLETE CYCLES OF OPERATION, IHE VLLOCITIES AND DEPTHS IN THE VAPOR CAVITY 4S WELL AS VELOCITIES AND PRESSURES AT KEY POIN;TS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..OCO0.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 1 14 U.000.031.O70.121.182.250.321.~98.480.556.593.624.682.7t.50 I Z.025.025.026.027.028.029.031.032.034.035.039.0 43.046.C468.05 DIST..9599 1.0239 1.0879 1.1519 1.2159 LOC. 15 16 17 18 19 U.922.961 1.036 1.127 1.179 L.053.057.060.063.075 FULL-FLOWING PIPE SYSTEM LOC. ( 10l) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V 1.1796 1.1799 1.1759 1.2074 1.2741 1.2738 1.2680 1.2628 H 11.3000 11.4860 11.4860 18.8599 29.7135 30.8549 31.7614 33. 1790 AT TIME T =.62908 SECONDS, AFTER 339 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT ThE FLOW SYSTEM ARK AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000.028.064.111.168.233.300.j74.452.520.552.583.642.722.804 Z.025.025.026.026.027.029.030.031.,333.035.038.042.044.047.049 DIST..9599 1.0239 1.0879 1.1519 1.2159 LOC. 15 16 17 18 19 U.871.911.985 1.Jb7 1.069.1.052.056.058.061.067 FULL-FLOWING PIPE SYSTEM LOC. (1,0) ( 11) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50)

V I.1704 1.1702 1.1663 1.1688 1.2010 1.2649 1.2b11 H 11.3000 11.3108 11.315 14.0419 20.0131.31.18b3 s2.8405 8.1790 Al TIME i =.6476 9 SECGNzON, AFTER 349 COMPLETE CYCLES OF GPERATION, THE VLLOCITIES AND DEPTHS IN THE VAPOR C4VITY:S'CLL LS VELOCITIES AND PRESSUlJES AT KCY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED 6CLOw. VAPOR CAVITY DIST..0000.0640.i280.1920.2560.3200.3840.4419.5119.5759.6399.1039.7679 *83I') *89: LOG. 0 1 2 3 4 5 6 1 8 9 10 11 12 13 19 U.000.025.058.102.156.217.282.352.425.485.514.546.60b.6.3.76o L.025.025.025.026.027.028.029.o3L.032.034.037.041.043.45. 01ST..9599 1.0239 1.U879 1.1519 1.2159 1.2799 LOC. 15 16 17 18 19 zt U.823.865.937.999 1.029 1.002 L.050.054.056.059.064.074 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2930) (2,40) V 1.1646 1.1644 1.1605 1.1600 1.160(; 1.1886 1.2559 1.2594 H 11.3000 11.3082 11.3082 13.7555 15.5715 22.0399 32.6b405 3..1790 AT TIME I =.66619 SECONUS, AFTER 359 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY fS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM AKE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.89(9 LOG. 0 1 2 3 4 5 6 7 8 9 10 11 12 I 14 U.000.023.053.094.144.202.265.331.399.4152.479.511.5 1.'.71 Z.025.025.025.026.027.028.029.30.032.034.037.640.042 4.0't 01ST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 LOC. 15 lb 17 18 19 20 U.778.821.890.946.966.950 1.049.052.055.058.062.066 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2t2C) (2,30) (2,40) (2,50) V 1.1581 1.1578 1.1539 1.1518 1.1479 1.1514 1.1873 1.25/5 H 11.3000 11.3260 11.3260 14.1139 15.7920 17.0812 22.4666 33.1790 AT TIME T =.68475 SECONDS, AFTER 369 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEN ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8399 LOC. 0 1 2 3 4 5 6 1 8 9 10 11 12 Ii 14 U.000.022.049.087.134.189.249.312.374.421.446.479.538.60 68 1.025.025.025.026.026.027.028 U30.031.033.036.039.041.u93.045 D1ST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 LOC. 15 16 17 18 19 20 21 U.736.780.846.896.913.911 1.060 1.048.051.053.056.061.065.068

FULL-FLOWING PIPE SYSTFM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (Z,30) (2,4u) (2,5() V 1.1483 1.1477 1.1438 1.1418 1.1433 1.1466 1.1488 1.15Is H 11.3000 11.3983 11.3983 14.6361 15.6215 16.2337 17.7232 AT TIME T =.70331 SECONDS, AFTER 37) COMPLETE CYCLES OF OPERAFION, THE VELOCITIES AND DEPTHS iN THE VAPJR CAVITY AS WELl. AS VELOCITIES AND PRESSURES AT KEY POINTS IHROUGHOUT THE FLOW SYSTEM AKE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 1 1 U.000.020.046.081.125.176.234.Z94.3O5.392.415..49.507.5l1.045 L.025.025.025.025.026.027.028.029.031.033.035.U38.040.L..044g DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4J78 LOC. 15 16 17 18 19 20 21 22 U.696.741.803.848.863.885.987 1.134 Z.047.050.052.055.059.063.065.077 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (,Z2C) I2,30) (2,40) (Y,5C) V 1.1348 1.1343 1.1305 1.1354 1.1406 1.1407 1.0156 L1.~45 H 11.3000 11.4250 11.4Z50 14.1822 15.0792 16.2673 26.9541 33.1791) AT TIME T =.72186 SECONDS, AF1ER 389 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY S5 W~LL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM AKE AS TABULATED BELO. I VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.89-9 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 1s 14 U.000.019.043.075.116.165.220.276.328.364.386.422.4 19.556.608 Z.024.025.025.025.026.027.028.u29.030.032.035.037.039.1. DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4078 LOC. 15 16 17 18 19 20 21 22 U.658.704.763.802.817.849.930 1.U09 Z.046.048.051.054.058.061.063 06b7 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2920) (2,30) (2940) (2,50) V 1.1290 1.1285 1.1247 1.1294 1.1328 1.0699 1.0379 1.0355 H 11.3000 11.3318 11.3318 13.1504 14.8322 25.7526 31.7118 33.1790 AT TIME T =.74042 SECONDS, AFTER 399 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS iN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000.018.040.071.109.155.207.260.306.337.359.396.452.516.575 Z.024.024.025.025.026.026.027.028.030.032.034.037.039.04C.042 DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4718 LOC. 15 16 17 18 19 20 21 22 23

U.623.668.723.759.774.812.886.951.966 2.045.047.050.053.057.060.061.064.074 FULL-FLOWING PIPE SYSTFM LOC. (1,0) (1,1) (2,0) (2,103 (2ZO) (2930) (2,40) ( 50) V 1.1287 1.1282 1.1244 1.1253 1.0591 1.03C5 1.0299 I.0313 H 11.3000 11.3259 11.3259 13.4314 23.1803 30.2245 32.0139 33.l79& AT TIME T =.7589'8 SECOND S, AFTER 409 COMPLE1E CYCLES OF OPERATIONJ, THE VELOCITIES AND DEPTHS IN TIIE VAPOR CAVITY is LLL As VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPUR CAVITY D1ST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7.0139.7679.8319.890) LOC. 0 1 2 3 4 5 6 7 8 9 LO 11 12 1I I' U.000.017.038.066.102.145.194.244.285.312.334.371.427.488.044 2.024.024.025.025.025.026.027.328.029.031.034.036.038.039 DIST..9599 1.0239 1.u879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4718 LOC. 15 16 17 18 19 20 21 22 23 U.589.634.686.717.734.717.846.897.912 2.044.046.048.052.055.058.060.ub3.065 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (111) (290) (2,10) (2,20) (2,30) (2,40) (2,50) V 1.1258 1.1253 1.1215 1.0564 1.0233 1.0192 1.0229 1.24 H 11.3000 11.3302 11.3302 22.9968 28.7725 30.0390 31.7349 33.1790 AT TIME T =.77753 SECONDS, AFTER 419 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPIiHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES ANO PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. o VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679 e8319 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000.016.036.063.096.136.182.L28.265.288.310.349.403.461 2.024.024.024.025.025.026.021 028 2Z9.031.033.035.037 C.39.041 DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799' 1.3438 1.4078 1.4718 1.5358 LOC. 15 16 17 18 19 20 21 22 23 24 U.558.602.649.677.697.743.808.854.874.900 2.043.045.048.051.054.057.058.061.064.068 FULL-FLOWING PIPE SYSTEM LOC.' (10) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V 1.0110 1.0118 1.0084 1.0200 1.0166 1.0158 1.0137 1.0155 H 11.3000 14.2594 14.2594 27.8842 29.2493 30.2844 31.2491 33.1790 AT TIME T =.79609 SECONDS, AFTER 429 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY -AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8909 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 P4 U.000.016.035.059.090.128.171.213.245.266.288.327.380.416.485 2.024.024.024.025.025.026.026.027.029.031.033.035.036.038.040

01ST..9599 1.0239 1.j879 1.1519 1.2159 1.2799 1.3438 1.4U-78 1.4718 1.5358 LOC. 1 5 1 0 17 la 19 23 21. 2l2 23 24 u.528., 71.614.640.662.710.771.812.835.862 Z.042.044.047.U50.053.055.057.060.363.U65 FULL-FLOWING PIP~E SYSIEMh LOC. (1,0) (1,1) (2,0) 12,10) (2,20) 2,3r) (2,40) (;I,50) v.9232.9239. 92.38.9688 I.C125 1.0111 1.C084 I1."I0031 H 11.3003" 11.5783 11.5783 21.8117 29.3944 30.4585 31.7'108 3i.1790 AT TIME T.81465j SE(20N00, AFTER 439 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVIIY AS WFLLLS VELOCITIES AND PRESSURES AT KEY POI-NTS THROUGHOUT THE FLOW SYSTEM AKE AS FABULATED BELOW. VAPOR CAVITY DIST..0000.3640.1280.1920.2560.3200.3840.4419.5119.5759.6399.70.39.7679.8319.399 LOC. 0 1 2 3 4 5 6 7 8 9 13c 11 12 I. 14~ U.000.015.033.056.085.121.160.191.226.246.268.307.359.411 45 ~.024.024.024.024.025.025.026.327.029.031.033.34.036 I2~ *3 01ST..9599 1.0239 1.U879 1.1519 1.2159 1.2799 1.3438 1.4018 1.4718. 1.5358 1.5998 LOG. 15 16 17 18 19 20 21 22 23 24 25 U.499.541.580.604.629.678.735.173.198.823.857 ~..041.044.046.049.052.054.056 G059.062.065.069 FULL-FLOWING PIPE SYSTEM LOC. (190) (1#1) (2,0) 42,10) (2,20) (2930) (2,40) (?,50) V.9126.9125.9094.9136b.9635 1.0052 1.0306 1.0014DH H- 11.3000 11.3163 11.3163 14.4106 23.0477 30.8719 32.4266 31.1790 AT TIME T.83321 SECONDS, AFTER 449 COMPLETE CYCLES OF OPERATIONp THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELLIA VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE As TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.51I19.5759.6399.7039.7679.8319.85 LOC. 0 1 2 3 4f 5 6 7 8i 9 10 11 12.13 1 U.000.015 o.032.054.081.11.3.149.183.208.226.250.21188.338.3 ~3 L.024.024.024.024.025.025.026.027.028.030.032. 034.035.0il. DIST..9599 1.0239 1.3879 1.1519 1.2159 1.2799 1.3438 1.43~78 1.4118 1.5358 1.5998 LOC. 15 1b 17 18 19 20 21 22 23 24 25 U.472.511.547.570.598.647.700.7 5.761.785.809 Z.041.043.045.048.051.053.055.u58.061.064.066 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1*1) (2,0) (2,10) (2,20) (2930) (2,40) (k,S0) V.9079.9078.9047.9043.9067.9532.9982.9980 1H 11.3000 11.3128 11.3128 13.8469 15.9368 25.0393 32. 3248 33.1790 AT TIME T=.85176 SECONDS, AFTER 459 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN iHE VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY 01ST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.85 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 P

U.000.u14.031.051.077.107.139.168 190.208.232.271.318 8 2.024.024.024.024.025.025.026.L27.028.C30. 32.C33.0.0.3 D1ST..9599 1.0239 1.0879 1.519 1.2159 1.2799 1.3438 1.4u78 1.4718 1.5358 1.5998 1.Uo38 LOG. 15 16 Li 18 19 20 21 i2 23 24 25 5 O U.446.483.516.538.569.617.667.luG.125.751.781.902 L.040.042.045.047.050.052.054.6u7.060.062.064.o75 FULL-FLOWING PIPE SYSTEM LOL. (1,) (1,1) (2,0) (2,10) (2920) (2,3C) (214C) (/,5() V.9026.9024.8994.89(8.8942.9000.9509 H 11.3000 11.3304 11.3304 14.1162 15.8521 17.4250 25.8606 AT TIME T =.87032 SEC0NDj, AFFER 469 COMPLETE CYCLE.. OF OPERATIO,4 IiTE VELOCITICS AND DEPTOHS IN THE VAPUR C4VITY *\ WELL As VELOCITIES AND PRESSURES Al KEY POINTS THROUGIHOJUT THE FLOW SYSTEM ARL AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4419.5119.5759.6399.7j39.7679.83*9.8'b9 LOC. 0 1 2 3 4 5 6 7 3 9 to 11 12 I; Ic U.000.014.030.049.073.100 129.154.13 190.216.254.299 3,34.3tiv z.024.024.024.024.024.025.026 to27.028.030.031 (.t33.034 38 DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4u78 1.4718 1.5358 1.5998 1.0638 IDE. 15 16 17 18 19 20 21 22 23 24 25 2b U.420.456.486.508.541.588.634.665.b90.718.745.802 L.040.042.044.047.049.051.053.656.059.061.064.068 FULL-FLOWING PIPE SYSTEM H LOG. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (~,56) V.8947.8944.8914.8893.8912.8920.8971.4042 H 11.3000 11.3866 11.3866 14.6112 15.6034 16.6907 18.3b61 33.1790 AT I IME I.b8888 SECONDS, AFTER 479 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY Aa WELL 4S VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATOD KELOW. VAPOR CAVITY 0IST..0000.0640.1280.1920.2560.3200.3840.4419.5119.5759.6399.7039.7679.8319.8909 LOG. 0 1 2 3 4 5 6 7 8 9 1U 11 12 1. 14 U.000.014.029.047.069.094.119.4O.157.174.200.238.282.324.3 2.024.024.024.024.024.025.025.026.028.030.031.033.034.0.35.037 DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4018 1.4718 1.5358 1.5998 1.6638 1.1278 LOG. 15 16 17 18 19 20 21 Z2 23 24 25 26 27 U.396.429.457.480.514.560.603.b32.657.685.715.159.882 1.039.041.043.046.048.050.052.055.58.060.063.065.077 FULL-FLOWING PIPE SYSTEM LOG. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.8830.8828.8798.8849.8871.8883.8456.1999 H 11.3000 11.4223 11.4223 14.1505 15.4495 16.5468 24.0387 34.179C AT TIME I.90743 SECOND4, AFTER 489 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A~ VELOCITIES AND PRESSURES AT KEY POINTS TiiROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY

UIST. 0000. 640 6 1280.1920.2560.3200.3840 *4474.5119.5759 6. II;' 39 f6 79.r 9 LOC. 0 1 2 3 4 5 b 7 9 Iu i1 12I? U.000. 14.028.045.C. 65.087.109.ALz.l4t.158.165.223.264.4.3 L.024.024.024.024.C24.23 025 *i26.6 8.29.Cs -2.033. 01ST..9o99 1.0239.L.879 1.151 1.2159 1.27'9 1. 34 38.4l.77 1.4(18 1.3!)8 1.598 I.: i.1218 LOC. 15 lb 17 1d 19 2 0 21 J2 2 24 25 26 21 U.373.404.429.45i.488.533. 572.6C,.c~5.653..:.720.785 1.038.041.043.045.047.049. Ct1.54.057.059.062.;4 FULL-FLUWING PIPE SYSTEM LOC. (1) (1,1) (2,C) (2,10) (2.2C) (2,3t) 12,4') V.8797.6793.8(63.8777.820C.8409.7)13 7'2 HI 11.3000 1i.3236 11.3236 13.5408 1.L 9 6 22.1689 1.3638 3 i 7 AT TIME 1 =.9259) SECONDS, AFTER 499 COMPLETE CYCLES OF OPERATION, THE VtzLOCITICS AFt) DEPIH-lj IN IHE VAP!IR CAVITY r.S AZLLI VELUCITIES AND PRESSURES AU KEY POI!4TS THROUGHOUT THE FLOhi SYSTEM ARE A. TABULATE.- BELOW. VAPOR CAVITY UIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.639.7039.7679. 8 LOC. 0 1 2 3 4 5 6 1 8 9 lb i 12 1. _- i U.000.013.028.044.062.061.100.114.127.144.171.208.48.28.: 1 L.024.324.024.024.04.025.025.u26.028.029.031.2s)2.033.035 U1ST..9599 1.0239 1.u879 1.1519 1.2159 1.2799 1.3438 1.4L78 1.4718 1.5358 1.5998 1.6638 1.1278 1.?9.6 LOC. 15 16 17 16 19 20 21 e2 23 24 25 26 27 4, U.350.379.403.428.463.506.543.b9.594.623.655.689.74C.8718 z.038.040.042.045.047.049.051.053. 56.058.061.063.065 I FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2930) (24)5) V.8786.8780.8150.8749.8318.7853.7b26.1820 H 11.300i 11.3375 11.3315 13.5141 20.8342 29.8760 01.9428 3.j179% AT TIME i =.94455 SECONDO, AFTER 509 COMPLETL CYCLES UF OPERATION, THE VELOCITIES AND DEPThS IN THE VAPOR CAVITY!S w0LL Aj VELOCITIES AND PRESSURES AU KEY POINTS THROUGHOUT ThE FLOW SYSTEM ARE AS TABULAIED BELOn. VAPOR CAVITY UIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.83 1 4 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 U 14 U.000.03.027.042.058.075.090.102.113.130.158.194.232.27.20 ~.024.024.024.024.024.024.025.026.028.029.30.31.033.0j4.u DIST..9599 1.0239 1.U879.1.1519 1.2159 1.2799 1.3438 1. 40 78 1.4118 1.5358 1.5998 1.6b38 1.7278 1.7918 LOC. 15 16 17 18 19 20 21 ~2 23 24 25 26 27 213 U.328.355.378.403.439.480.514.539.564.593.626.661.698.7o6 1.038.040.042.044.046.048.050.052.055.057.060.062.064.Lub FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2920) (2,30) (2,40) (?,50) V.6756.8753.8723.8293.7785. 775.7768 7787 H 11.3000 11.3239 11.3239 19.8778 28.2585 30.0019 31.7341 33.1793 AT TIME T =.96310 SECONDS, AFTER 519 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A~ VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW.

VAPOR CAVITY DIST..0000.0640.1280.1920.25b0.3200.3640.4i 79.5119.5759.639') 7; 3 9.7679.13'. 89 LOG. 0 2 3 4 5 6 7 8 9 10C 11 12 U1.000.C13.32.04)..055.069.08 1..0.IuO.117.145.180.217 L.024.024.024.024.024.0~4.025 U626.J27.029.030.)31.032 DIST..'1599 1.0239 19.879 1.151 1.2159 1.2799 1.3438 1.4Q78 1.4718 1. 538 1.5998 1.6638 1.7278 1. I) I 8 I i LOG. 15 16 17 18 19 20 21 ~2 23 24 25 26 21 2 2 U.307.331.354.380.416.455.486.511.536.565.598.633.669. 71u 1.037.039.041.043.045.041.049.U52.054.056.059).061.063 785 FULL-FLOWING PIPE SYSTLM LOC. (1,0) (1,1) (2,0) (2,10) (209 2) (2930) (Zt40) (~, 5) V.8070.8031.BJ04.7762.7711.7700.7b90.719 H 11.3000 13.7547 13.7541 27.3066 29.u369 30.1183;1.2620 3).1790 AT TIME 1 =.96166 SECONDS, AFTER 529 COMPLEIE CYCLES OIF OPERATION1 THE VELOCITIES AND DEPiHS IA THE VAPOR CAVITY A. ),ELL A VELUCITIES AND PRESSUkES AT KLY POINTS IHROUGHOUT THE FLDW SYSTEMl. ARL 4A3 TABULATED BELOW.. VAPOR CAVITY DIST..0000.0640.i280.1920.2560.3200.3840.4479.5119.5759.6399 I7 39.7679.8319.89s9 LO. 0 1 2 3 4 5 6 1 8 9 10 11 12 1. i4 U.000.013 025.038.051.063.072 U(9.068.105.133.167.201.21i.266 1.024.024.024.024.024.024.025.06.027.029.030.031.032.l4.035 01ST..9599 1.0239 1.u879 1.1519 1.2159 1.2799 1.3438 1.4C76 1.4718 1.5358 1.5998 l.o638 1.1278 1.7918 1.8558 LOC. 15 16 17 18 19 20 21 22 23 24 25 26 2? 28 49 U.286.309.331.358.394.430.459.483.508.538.571.606.641.674.1180 L.037.39.041.043.045.047.049.051.053.056.058.060.062. 72 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (191) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.6839.6861.6838.7424.7671.7666.7641.7603 H 11.3G00 11.8018 11.8018 24.1979 29.1653 30.3164 31.6060 33.179c AT TIME T = 1.00022 SELONDSt AFTER 539 COMPLETE CYCLES OF UPERATIO.,4 THE VOLOCITIES AND DEPTHS IN FHE VAPOR CAVITY AS wELL AS VELOCITIES AND PRESSURES Ar KEY POINTS THROUGHOUT THE FLOW SYSTCM AiL AS TA8ULAT D ELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LO. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000.012.025.037.048.057.064.669.077.094.121.154.187.216.242 L.023.023.024.024.024.024.025.%26.(27.028.029.031.032.33 DIST..9599 1.0239 1.u879 1.1519 1.2159 1.2799 1.3438 1.4JI8 1.4718 1.5358 1.5998 1.6638 1.7278 1.7918 1.8558 LO. 15 16 17 18 19 20 21 22 23 24 25 26 27 2E 29 U.266.287.310.337.372.406.433.457.482.512.545.579.612.635.663 1.037.038.040.042.044.046.048.u50.(053.055.057.059.061.064.68 FULL-FLOWING PIPE SYSTCM LOC. (1,0) (1,1) (290) (2,10) (2,20) (2930) (2,40) (2,50) V.6704.6706.6683.6755.1380.7618.7577.7574 H 11.3000 11.3493 11.3493 14.9698 25.4916 30.6508 32.2528 J3.1790

AT TIME T = 1.01877 SECONDS, AFTER 549 COMPLETE CYCLES OF OPERATION, THE VELOCITIES A:AD DEPIHS IN THt VAPOR CAVITY f.S wELL AS VELOCITIES AND PKESSUlES AT KEY POINTS TIHiROUGHOUT THE FLOW SYSTEM AKt AS TABULATED BELOh. VAPOR CAVITY OIST..0000.0640.1280.1920.256.3200.3840.4419.5119.5159.6399.7039.7679.8319. 8 5 ) LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 1t 14 U.000.012.024.035.044.051.055.059.066.083.110.141.112.Z;u.225 L.023.023.023.024.024.024.025.626. 27.028.029.03(:.032.53.s35 DIST..9599 1.0239 1.7 159 1.1519 1.2159 1.2799 1.3438 1.4 78 1.4718 1.5358 1.5998 1.6638 1.7278 1. 7918 1.858 LOC. 15 16 17 18 19 20 21 2Z z25 4 25 26 27 2 ~9 U.247.267.289.317.350.382.408.>1.457.487.520.553.583.6 1.621 Z.036.038.040.042.044.045.048.050.L52.054.0,.0u5.060.0o3.Co5 FULL-FLOWING PIPE SYSrTM LOC. (1,0) (1,1) (2,C) (2,10) (2,20) (2,30) (2,40) It,50) V.6664.6663.6640.664C.6699.7292.7551.1553 H 11.3000 11.3205 11.3205 13.9370 16.4854 27.4400 32.2629 3~.1793 AT TIME T = 1.03733 SECONDst AFTER 559 COMPLETE CYCLES OF OPERAIION, THE VELOCIIIES AND DEPTHS IN THE VAPUR CAVITY.1 WELL AS VELOCITIES AND PRESSUR'ES AT KEY POUITS THROUGHOUT THE FLOW SYSTEM AKL AS TAHULLATED BELOW. VAPOR CAVITY D1IST..OO0.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6b68 i.791o LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2u. 7 U.00O.023.040.047.057.099.1.58.07.247.297.359.407.463. L 7.57 Z.023.023.024.025.027.029.031.u34.038.041.045.049.053.Gi.b62 L.J DIST. 1.9198 LOC. 30 U.605 Z.071 FULL-FLOWING PIPE SYSTEM LOC. (It,O) (1,1) 12,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.6620.6619.6597.6584.6554.6634.7270.1528 H 11.3000 11.3340 11.3340 14.1129 15.8997 18.1262 28.4228 31.1790 AT TIME T = 1.05589 SECONDS, AFTER 569 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPtHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TA8ULATLD BELO". VAPOR CAVITY DIST..0000.1280.2560. 840.5119.6399.7619.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.66 8 1.7918 LOC. 0 2 4 6. 8 10 12 14 16 18 20 22 24' 26 28 U.000.021.036.039.048.089.144.190.229.278.337.383.440.5i']2.541 Z.023.023.024.025.07.029.031.034.038.041.045.049.053.051.6 DIST. 1.9198 LOC. 30 U.560 Z.068 FULL-FLOWING PIPE SYSTEM LOC. (1,0) t1,1) (2,0) (2,10) (2,20) (2,30.) (2,40) (",50)

v.5 54. 65 32.6 510C.6520.5 3.3. 6t1I I I 3'co~ 1 1. 3 PR5 1 4. 5726 1 5. 7 521 1 6 89)9 1 4. I1b2 1 71? Al' lIME I = I * 0 44i, A'TCI< 57)j COMPLEriw. CYCLES OF f0PERAT1614, THE VELOCITIFS AiM) OLPIOS IN (HF VAP,0-~ C V II Y S.o: vLL I VELOCITIES ASW) PkE SUL'r,_~ 4f KE1Y'(3I~TS TH(~OUGHOUT THE FLojw SYSTEM ARt AS TABULATED BLELOw. VAPOR CAVITY 1S T.0ou.12.1.256 J.3840'.211 9.6399.16179.r59 I.C239 1. 15 19 1.2 7:,9 L.4 L78 1.53 5 8 I66A b LOC. 0 2 4 6 6 1G 1 2 14 16 1 8 26 2 2 24 U. (10 0.0 20. -32. )37.0.19.01)r. 130.114 t.aII.260. 3 i.3b1.4 17.1 z. 02 3.03. J -;4. 02o,.02 1.029.0(31I.034.0,3 7.04 1. 04',.Q48.0-52. 0IST. 1.9193 LOC. 30 O.538 1.065 FULL-FLOWING PIPE sysrcM LOC. lp)(1)(2,0) (2,10) (2,22) (2,30)(,)(L52 V.6453.645 i.64.31 t.b468. (4 8.6499.6253 J5: H 11.3000 1 1.4 289 1 1.4 269 14.29bZ 15. 5721 1 6.74 33 2 1.69 73 31 79 AT TIME T 1.09300 SEE0DNDS, AFTER 5)3) COMPLETE CYCLES OF OPEkAIIO0,q THE VELOCITILS AND DEPTHS IN THiE VAPOR CAVITY n ELE L VELOCITIES AND PRESSURIES Al KEY POP:,TS TI1ROUC;AHOUT TI-I FLOw SYSTEM AKL' As TABIULATED B-LOvw. VAPOR CAVITY D1ST. 00U0.1280.2560.3840.5119.b39-).7t879.6-i59 1.0239 1.1519 1.2799 1.40(78 1.5358 1.66_)8 17911 5 LOC. 0 2 4 6 8 10 12 14 lo6 18 20 22 24 20 H U.000.619.02 8.024.032.6b9.117.i -j7.194.242.294..339.395.45 -O Z.023.023.02~~~~~~~~')4.022).021 C; 2.031.u34.%137.040.044 vu48.051.0) 20 UIST. 1.9198 1.9838 LOC. 30 31 U).513.585 L.065.069 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (292C) (2,30) (2,40) (2 5C;) V.6430.6423.6402Z.6410.6448.6212.5590 52 H 11.3000 11.3465 11.3465 1.3.7081 15.2893 2Co. 355 1 30.8202 3i.L?9o AT TIME T 1.11156 SECGNO-~, AFTER 599 COMPLETE CYCLES OF OPERATIUN, THE VELOCITIES AND DEPTHS IN4 THE VAPOR CAVITY.S WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARL AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.66ib 1.79i~ LOG. 2) 2 4 6 8 10 12 14 16 18 20Q 22 24 ~. 2 U.000.017.024.018.024.060.104,.141.177.2 25..274.318.374. 4 6.6 3 Z.023.023.024.025.027.028.031.;633.037.040.043.047.051. C' DIST. 1.9198 1.9838 LOG. 30 31 U.490.535 Z.064.066

FULL-FLOwING PIPE SYSITEo LOC. (1,) (1,1) (2, (,10) 2,0 2,2) (2.30 (2,40) 4I,) V.6408 ~.404+.6383.6382.6133.5541.5479. u473 H 1 1.C3000 11.3257 11. 2 57 1 J. 6 195 18.7i 7 2 2. 3393 31.8684 t.1791, AT TIME T = 1.13012 SECLAJNU., AFTER 0)9 C).OMPLEIF CYCLLE OF OPERATI('9, THE VLLOCITIES AfND DEPIHs 1I4 THE VAPJR CA4I[Y S W.LL iA VELOCITIES AND PRESSURES A KLY PCOINTS THROUGHOUF THE FLOw SYSTEM AKc AS FADULATED BFLO,. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.d059 1.0239 1.1519 1.2799 1.4,,)78 1.5358 1.6o6;C 1.191J LOG. 0 2 4 6 8 10 12 i4 l6 18 2( Z22 24 2..;5 U.000.916.019.011.017.051.091.126.lo t.208.254./98.353.4 2.4 1 z.023.023.024.025.027.028.030.033.036.040.443.041.050.G. 5 DIST. 1.9198 1.9838 2.A478 LOG. 30 31 32 U.471.510.638 L.063.065.0 16 FULL-FLOWING PIPE SYSTEM LUC. (1,0) (1,1) (2,0) (2,10) (12,I) (,30) (2,4C.) (,t) V.6384. 381. 630O.6107.5477.5402.5428.5438 H 11.3000 11.3204 11.3204 11. 3568 27.6433 29.9815 31.7420. 179T,5 AT [TIME I = 1.14867 SECONDS, AFTER 619 COMPLETE CYCLES OF OPERAI&i'4, THE VLLOCILTIES AiiD UEPTH3 IN THE VAPOR CAVITY,'S WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THIROUGHOUT THE FLOW SYSTEM ARt AS [ABULATED 6ELOWh. I ro VAPOR CAVIT [Y -.1 DIST..0000.1280.2560.3840.5119.6399.7679.8b'59 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7916 LOC. 0 2 4 6 8 LO 12 14 16 18 26 22 24 26 20 U.000.014.014.005.011.042.078.L1O1.145.192.235.279.332.317.AC2 Z.023.023.024.02s.027.028.030.033.036.039.043.046.050.Ct 4.65w 01ST. 1.9198 1.9838 2.U478 LOG. 30 31 32 U.451.477.543 Z.063.004.069 FULL-FLOWING PIPE SYSTEM LOC. (1,u) (1,1) (2,0) (2,10) (2,20) (2,3C) (2,40) (i,.50I V. 60G7.5963.5943.5457.5 376. 5365.5361.5379 H 11.3000 12.9274 12.9274 26.5941 28.8506 30.0483 S1.3356 J:3.1790 AT TIME 1 = 1.16723 SECONUS, AFTER 629 COMPLETE CYCLES OF OPERATION, THE VFLOCITIES AND DEPTHS IN THE VAPOR CAVITY AS wELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THiE FLOW SYSTEM AKRt AS TARULATELD BELOW. VAPOR CAVI TY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 U.00o0.011.010 -.001.005.033.066.095.130.175.217.260.312.353.379 Z.023.023.024.025.027.028.030.033.036.039.042.046.049.C53.0,8 DIST. 1.9198 1.9838 2.0478 LOC. 30 31 32

U.429.453.499 Z.C62.Co4.067 FULL-FLOWING PIPE SYSTEM LOC. (I1,U) (1,I1) (2,tG) (210) (2,20) (2,3C) (2,40) (, ) V.46C5.4642.4u26.5213.5345.bS 3..bJ.'44 H 11.3000 12. 3300 12.33j6 25.6961 Z8.9998 3C.2044 1. 52'8 3.i1' Al TIME r = 1.1857) SECONDS, AFTER 639 COMPLETE CYCLE. OF OPERATI0Cm, THE VELOLITIES AND DEPThS IfiN tE VAPI:R CAVI rY', i.FL, o VELOCITIES AND PRESSURES AT KEY POINrTS TtiROUGHiOUT THE FLOW SYSTEM ARL A1 TABULAIEI) BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7619.b959 1.0239 1.1519 1.2799 i.4 -7t8 1.53h l.oct 1.?Jl7 LOC. 0 2 4 b 8 10 12 14 16 18 2U 22. 4? U.000.C09.005 -.0U7 -.O01.OZ5..54.u;81.115.159.199.242.292.J':.)o Z.023.023.024.025.026.028.030.U33.036.939.042.L40.049., 3'.: DIST. 1.9198 1.9838 2.0418 LOC. 30 31 32 U.408.431.467 Z.061.063.066 FULL-FLUWING PIPE SYSTEM LOC. I1,0) (1,1) (2,0) (2,101 (2,29) (2,30( (2,4c) (,:.5 I V.4394.4396.43Z82.4516.5173.5290.5259.52'2 H 11.3000 11.3992 11.3992 16.0394 27.0570 30.4780 32.0881 33. 1796 AT TIME T = 1.20434 SECONDS, AFTER 649 CO)MPLEIE CYCLES OF OPERATIUb, THE VELOCITIES AND DEPTH5 iN THE VAPUR CAVITY -, ELL W[L VELOCITIES AND PRESSURES AT KEY POINTS THKOUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. - VAPOR CAVITY DIST..0000.1280.2560.3840.5119.b399.7679.8959 1.G239 1.1519 1.2799 1.4078 1.5358 1.6628 1.7'14 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2 eN U.000.007.000 -.013 -.007.017.042.u67.1C1.143.182.224.272.3,.6.531 Z.023.023.024.025.026.028.030.b33.036.039.042.045.049.S33. - 7 DIST. 1.9198 1.9838 2.0478 LOC. 30 31 32 U.386.409.441 Z.061.062.065 FULL-FLOWING PIPE SYSTEM LOC. (1,0} (1,11 (2,0) (2,10) (2,20) (2,30) (2,4}) (2,50) V.4352.4352..4337.4343.4469.5097.5233.523 t H 11.3000 11.3296 11.3296 14.0531 17.5409 28.9463 S2.1696 3i.1790 AT TIME T = 1.22290 SECOND, AFTER 659 COMPLETE CYCLES OF OPERATION, THE VLOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638d 1.? 1L LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2, 23 U.000.004 -.005 -.018 -.013.008.031.054.087.127.165.107.252.234.316 Z.023.023.024.025.026.028.030.033.035.038.042.045.048.C C2. (6

DIST. 1.9198 1.9838 2.0418 2.1118 LOC. JO i 32 33 U.365.388.397.393 L.060.6 2.065.070 FULL-FLOWING PIPE SYSTLM LOC. (I1,C) (1l) (2,C) (2,10) (2,25) (2,,,) (2,40) ({,5,) V.4315.4314.4J,$u.429.).4268.4467 50 13.52 J H 11.300U 11!.400. 11.34CC 14.1Oo8 151.')b1 19.25:2 50.C 86 3J3.179: AT TIME T = 1.24146 SECONDS, AFTtER 66') COMPLEIE CYCLE5 OF OPERA AIICO, IHE V,-LOCITIES AN\D DEPTH) IN THiE VAPOR CAVITY Is WILL AS VELOCITIES AND PRESSUr)ES AT KEY POINTS [HROUGHOUT TIlE FLOw SYSTEM AKL AS TABULAIED BELOn. VAPOR CAVI Y DIST..OuCO.1280.2560.3840.5119.6399.1679.89,9 1.0C39 1.1519 1.27~9 1.4;74 1.5358 1.bo6o s L.791 LUC. 0 Z 4 6 8 1iC 12 14 Ito id 20 2 24 2L Z, U.000.00G1 -.009 -.023 -.018.Quo.OLC., 41. 73.112.1'9.191.233.2:.2)1 C.023.023.C024.02.026.028.030.oJz.C35.038.C41.644.048. -.C 56 UIST. 1.9198 1.9838 2.0478 2.1118 LOC. 30 31 32 53 U.345.3,2.356.358 L.059.C6.C065.068 FULL-FLOWING PIPE SYSILt LOC. (l,) (1,1) (2,0) (20)(2,10),2) (2,30) (Z,40) (Z,5;) V.4261.4261.4246.4221,.4228.424:.4389.49 1 j 1 H 11.3000 11.3779 11.3/f9 14.5192 15.8181 17.1111 20.3308 3.3 179- Ar TIME r = 1.26001 SECO:iNa, AFTER o79 COMPLETE CYCLE- OF tPERAIIKjA, ItHE VELOCITIES AND DEPTHIS IN IHF VAPO!R CAVITY.:.S iLL AS VELOCITIES AND PRESSUl:ES AT KEY POINITS THROUGHO(JU THE FLlOW SYSTEI: Ar(L AS [ABJLATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.2119.b399.7679.U:-59 1.023 519.1519 1.2799 1.478 1.5358 1.66 1.7918 LO;. 0 2 4 6 8 10 12 14 1 t 18 2 22 24 24. i J U.C(o -.001 -.014 -.021 -.024 -.003.)09.v29.60.697.133.1 75.214.IZ'1.278 Z.023.023.624.025.026.028.030.,32. 035 8.041.044.048.,d.05 DIST. 1.9198 1.9838 2.0478 2.1118 LOC. 30 31 32 33 U.323.333.327.338 Z.059.061.064.0hb6 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,2C) (2,3O) (2,40) (1,50) V.4172.4172.4158.4185.4202.4210.4086.3569 H 11.30ou 11.433: 11.4333 14.3670 15.b751 16. 895 C;0.2529 33.1790 AT TIME T = 1.27857 SELNUSD, AFTER 089 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTIHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PKESUct<E AT KEY POINTS THOllUGHOUI THE FLOW SYSTEM AiK AS TABULATLE iELOW. VAPOR CAVITY DIST..000U.120O.2')60.3840).5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4U78 1.5358 1.663 8 1.7918 LOG. 0 2 4 6 8 10 12 i4 16 18 20 22 24 2. 28

U. I 0 -.004 -.019 -. 032 -.029 -.016 -.002.u17. 0i 7.08Z.1i8.158.195 2. L.023.J23.024.025.026.02 8.030.32.C35.038.041 *.:44 *047 L j * CIST. 1.9198 1.983j8 2.o'u78 2.1118 LOC. 30 31 32 33 U.299.335.306.324 L.059.061.064.065 FULL-FLOWING PIPE SYSIEM LOC. (19o) (1,1) (2,0) (2910) (2I2C) (2930) (2t4o) ce.5c V.4138.4t34.4120.4134.4167.4C44.3391 326 1 H 11.3006 11.3260 11.3260 13.86b3 15.442S 1t.808s 2A). 91 J.117 9 AT TIME i = 1.29713 SECIONLJS, AFTER 699 COMPLEIL CYCLES OF OPERATION, THE VELOCITIES All) DEPTHS IN TIC VAP R CAVItY is WELL A, VELOCITILS AND PRESSURES Ar KEY POINTS 111ROUGHIJUT THE FLOW SYSTEm ARC AS FABULATID BFLOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119).6399.7679.8959 10239 1.1519 1.2799 1.4L78 1.5358 1.t.68, 1.79~3 LOC. 0 2 4 6 8 10 [2 14 16 18 20 22 24 % U.000 -.007 -.023 -.036 -.034 -.C24 -.013. i'5.134.667.lu.142.116.2./4 1.023.i323.U24.UZ5.0~6.028.u30.,2 35 u3s.041.L 44.047 I.i34 01ST. 1.9198 1.9838 /2.6478 2.1118 2.1757 LOC. 30 31 32 33 34 U.275.280.290.30.362 1.058.061.063.065.072 FULL-FLOWING PIPE SYSTEM P0 LOC. (1,I) (11L) 12,0) (2,10) (2,20) (2,30) (2,40) (? 5u) V.4120.4117.4 13.4103.3977.3349.3220.3214 H 11.3000 11.3203 11.3203 13.6808 Ib.9 951 28.2810 31.7649 33.1790 AT TIME T = 1.31569 SECON.S, AFTER 709 COMPLETF CYCLES OF OPEKATION, THE VELOCITIES AND DEPTHS IN THE VAPJR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS 1IROUGHOUT THE FLOW SYSTEM AKL AS TABULATED BELOW. VAPOR CAVITY 01ST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 ~8 U.000 -.011 -.028 -.040 -.039 -.032 -.023 -.006.022.053.088.126.158 Iiis.223 1.023.023.024.025.026.028.030.u 3.035.031.040.043.047.0 L 01ST. 1.9198 1.9838 2.0478 2.1118 2.1757 LOC. 30 31 32 33 34 U.252.257.268.274.312 7.058.060.063.065.068 FULL-FLOWING PIPE SYSTEM LOC. 11,0) (1,1) (2,0) (2,10) (2920) (2930) (2,40) (2,50) V.4100.4098.4084.3947.3287.3154.3172.317), H 11.3000 11.3211 11.3211 15.7171 26.5036 29.9410 31.7518 33.1790 AT TIME T = 1.33424 SECONDS, AFTER 719 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AlS WELL A., VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOw. VAPOR CAVITY

U1ST..0000.1280.2560.3840.5119.6399.7679. 89 5 9 1.;239 1.1519 1.2799 1.4078 1.5358 1.6638 I.? LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2 6. U.000 -.014 -.032 -.044 -.044 -.040 -.033 -.17.(C9.040.13.110.140.Loi.2J4 Z.023.023.024.025.026.028.030 j3z2.035.037.04O.243.047 Uu4 UIST. 1.9198 1.98 38 2.o478 2.1118 2.1757 LOC. 30 31 32 33 34 O.229.235.246.258.290 7.058.060.062.064.065 FULL-FLOWING PIPE SYSTEM LOG. (1,0) (14) (2,0) (2,10) (2,20) (2,30) (2940) (Z,50) V.3900.3868.3855.3270.3125.3109.3113.313t H 11.3000 12.2530 12.2530 25.399o 28.6512 29.9(21 31.3982 J. 1790 AT TIME T = 1.35280 SECONDS, AFTER 729 COMPLETE CYCLES OF OPERATIONr, iHE VE:LOCITIES ANU DEPHS IN THE VAPOR GAVITY AS WELL As VELOCITIES AND PRESSURES AT KEY POINTS 1HROUGHOUT THE FLOW SYSTEM AKL AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3640.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4378 1.5358 1.66j8 1.7918 LOG. 0 2 4 6 8 10 12 14 16 18 20 22 24 2-. /8 U.000 -.017 -.036 -.048 -.050 -.047 -.043 -.u28 -.003.026.059.094.122 I. L.023.024.024.025.026.028.030.j32.034.031.040 043.046 C;I.53 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 LOG. 30 31 32 33 34 35 U.207.214.226.246.252.279 Z.058.060.062.064.066.070 I' FULL-FLOWING PIPE SYSTEM LOG. (1,0) (1,1) (2,0) (2910) (2,20) (2,30) (2,40) (2,50) V.2547.2580.2571.3034.3092.3084.3067.3048 H 11.3000 12.9740 12.9740 26.4569 28.8703 30.1084 31.4416 33.1790 AT TIME T = 1.37136 SECONDS, AFTER 739 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPIHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSUR{ES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARL AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOG. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 U.000 -.020 -.040 -.052 -.055 -.055 -.052 -.038 -.014.013.045.078.105.133 166 7.023.024.024.025.026.028.030.032.034.037.040.043.046.050 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 LOG. 30 31 32 33.34 35 U.185.194.208.224.229.216 z.057.060.061.063.066.075 FULL-FLOWING PIPE SYSTEM LOG. (1,0) (1,1) (2,0) (2,10) (2920) (2,30) (2,40) (2,50) V.2165.2169.2161.2395.2993.3051.3018.3005 H 11.3000 11.'.523 11.4523 17.7517 27.9177 30.3390 31.9299 33.1790 AT TIME T = 1.38991 SECONDS, AFTER 749 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW.

VAPOR CAVIrY UIST..00CO.128L.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.665b 1.791u LO(. 0 2 4 6 8 10 12 14 16 18 20 22 24 2t U.000 -.023 -.044 -.056 -.060 -.063 -.061 -.048 -.026.001.032.063.088.1t7.147 Z.023.024.024.025.026.028.030.032.034.037.04G.043.046.-','.,3 DIST. 1.9198 1.9838 2.0418 2.1118 2.1757 2.2397 LOC. 30 31 32 33 34 35 U.165.175.189.200.189.187 z.05.059.C61.063.066.072 FULL-FLOwING PIPE SYSIEM LOC. (1,6) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (/,50) V.2116.2116.2109.2123.2355.2928.2988.2989 H 11.3000 11.3398 11.3398 14.2229 19.2499 29.741! 32.1175 3.179C0 AT TIME T = 1.40847 SECONUS, AFTER 759 COMPLETE CYCLES OF OPERATION, TIlE VELOCITIES AND DEP1HS IN IHE VAPOR CAVITY iS WELL AS VELOCITIES AND PRESSURES Al KEY POINTS THROUGHOUT THE FLOW SYSTEM AKR AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.66.6 1.7918 LOG. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 2o U.000 -.026 -.048 -.060 -.065 -.070 -.070 -.058 -.037 -.012.018.047.071.l1:.128 Z.023.024.025.025.026.028.030.032.C34.037.039.042.046.u,9.03 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 LOC. 30 31 32 33 34 35 U.145.156.169.171.161.168 Z.057.059.061.063.066.069 I FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.- 2085.2085.2078.2070.2060.2295.2899.2971 H 11.3000 11.3470 11.3470 14.1206 16.C774 21.0602 31.0366 3.1 790 AT TIME T = 1.42703 SECONDS, AFTER 769 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPUR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOw SYSTEM ARE AS TABULATED BELOw. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 U.000 -.029 -.051 -.064 -.071 -.077 -.079 -.068 -.048 -.024.005.032.055.084.109 L.023.024.025.025.026.028.030.032.034.037.039.042.046.049.053 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 LOC. 30 31 32 33 34 35 U.126.137.146.144.141.155 Z.057.059.061.063.066.067 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (I1,1) (20) (2,10) (2,20) (2,30) (2,40) (2,50) V.2041.2040.2034.2015.2010.2033.2281.2809 H 11.3000 11.3776 11.3776 14.4783 15.-325 17.4039 22.1983 J3.1799

AT TIME T 1.44558 SECONDS, AFTER 7lri COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY s wELL AS VELOCITIES AND PRESSURES Al KEY POINTS THROUGHOUT THE FLOW SYSTEM AkL AS TABULATED BELOw. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7o79.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7I8 LOC. U 2 4 6 8 10 12 14 16 18 20 22 24 2 28 U.oo000 -.031 -.055 -.061 -.076 -.085 -.087 -.077 -.059 -.036 -.008.017.040.0ab.090 Z.024.024.025.025.026.028.030.032.034.037.039.042.045.049.}sZ DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 LOC. 30 31 32 33 34 35 U.107.117.122.120.126.146 Z.056.058.061.063.065.066 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.1965.1965.1958.1974.1988.1996.1946.159) H 11.0o0o 11.4337 11.4337 14.4673 15.8046 17.0710 19.6157 33.1790 AT TIME T = 1.46414 SEC[)NOD, AFTER 789 COMPLETE CYCLES OF OPERATION, tHE VELOLIIIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4G78 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2 2 U.000 -.034 -.058 -.071 -.082 -.092 -.095 -.087 -.070 -.047 -.021.002.025.051.i71 Z.024.024.025.025.027.028.030.u32.034.036.039.042.045.049.052 DlST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 LOC. 30 31 32 33 34 35 U.088.096.098.100.115.138 Z.056.058.061.063.065.065 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.1918.1916.1909.1931.1959.1901.1309.1083 H 11.3000 11.3168 11.3168 14.0380 15.6G60 18.0134 28.0898 33.1790 AT TIME T = 1.48270 SECONDs, AFTER 799 COMPLETE CYCLES OF OPERATIUN, IHE VELOCITIES AND DEPIHS IN THE VAPOR CAVITY AS WELL AS VELO ITIES ANt PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATFD BELOW. V/APOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 U.000 -.036 -.061 -.075 -.087 -.099 -.103 -.696 -.080 -.059 -.035 -.013.010.O35.033 i.024.024.025.026.027.028.030.032.034.036.039.042.045.048.052 0IST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U.068.074.077.083.106.131.190 Z.056.058.060.063.064.064.078 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50)

V 19(04.90.3.ILf..18I-5.1844.-12 73.13.1 / H 10 0 0 1l. i220 11.3220 lI.7328 16.2'447 26.62I1 31.33 3.1 AT TIME 1 = 1.50112s SECUN'IDS, AFJER 809 COMPLLT- CYCLCF. OF CPPCR.riIoN., THE VrL(CLTIES ANDI DLPI(I. IN THE VAPOR CAVITY ft.s CtL. 3s VELOCITIES ANJ PRESSU2Lz AT KEY POINTS IHOUUHOuIT TiF FLOW SYSIEM ARE AS TAbULAIED 3E1O.i VAPOR CAVITY DIST..0000.1280.2560.3840.)119.6399.7uH9 d.d9 1.039 1. 151 1.2?)'.4L7E 1.5358 1. oc, 1.7ih LOG. 0 4 0 I 10 12 14 I b is 2j 22 2 s U.000 -.038 -.064 -.0)79 -.093 -.106 I.II -.1u 5 -. 90 -.070 -. -.027 -.00.5 ic. Z.024.024.025 C26.027.0~8a.030 v32 34.036. C39.,42.045.. DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOG. 30 31 32 33 34 35 36 U.0,49.053.C57.070.092.C88.122 L.056 0$.0G60.062.064.066.6 71 FULL-FLOWING PIPE SYS[Es LUC. (1,0) (2,1) 12,0) (2,10) I2,) (2,3(1 (2,4o) V.1889 I Z15.oI.I8Cos.iz0.,09 79.0984 H 11.3000 11.3265 1 1.320 14.b038 24.(56I 29687'c. 51.7099 33.179 AT TIME T 1.5i1981 SECONO3j, AFTER 819) COMPLETt CYCLE, (OF OPERATI&'1, 115 V1LOCIT (ES All DFPtH, IN tHE VAP-R CA%1TY >-S WELL L VELOCITIES AND PRESSURES AT KEY POINTS ThR0~UGtiOUT THE FLi; SYSTPE' A.{ AS TA61ULATEU 8CLlUw. VAPOR CAVITY UIST..0000.1280.2560.3840.5119.0399.7679,.d9 1.039 1.1519 1.279) 1.4178 1.5358 1.66-8 1.79l16 LOC. 0 2 4 6 8 10 (2 14 16 18 2; 2 24 2 s () U.000 -.040 -.067 -.083 -.u9G -.112 -. 118 -.113 - 10'0 -.81 -.Lb1 -,.41 -.02. 1( -k 7L,3.024.024.025 26.027.028.030 3.034.036.u 39.045.04.2 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOG. 30 31 32 33 34 35 36 U.029.033.040.055.070.066.093 7.056.058.060.062.064.066.068 FULL-FLUWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2910) (2,20) (2,30) (2,40) (2,50) V.1783.1763.1757.1194.0,94_.919.0932.094? H 11.3000 11.6291 11.8291 23.6314 28.4522 29.9069 31.4685 33. 1790 AT TIME T = 1.5383 7 SECGND0N3 AFTER 829 COMPLETE CYCLES OF PLPERATION, THE VELOCITIES AND DEPtHS IN THlE VAPOR GAVITY o.s WELL As VELOGITIES AND PRESSURES AT KEY POINTS THROUGHOUT THiE FLUW SYSTEM ARlc AS TABULATOD LLO0W. VAPOR CAVITY U1ST..0000.1280.2560.3840.5119.6399.7679.6959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7916 LOG. 0 2 4 6 8 10 12 14 16 18 20 22 24 2 k, 2 U.000 -.042 -.070 -.087 -.104 -.119 -.126 -.122 -.110 -.092 -.073 -.055 -.034 I -.0 I.024.024.025.026.027.028.030.032.034.036.039.042.045.0*8 cUs2 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOG. 30 31 32 33 34 35 36 J1.009.014.024.038.048.053.078 2.056.058.060.062.063.065.067

FULL-FLOWING PIPE SYSTEM LUC. (1,0) (1,1) (2,0) (,U) ) (2,20) ) (,50) V.0634.;64 t.~ (:6' 5. J89s.:,90,.~ 0.8C 2 H 1 1.300. 4 13.3542 1 3'+2 2. fu3 28. 1927 33. C14.. i 790 AT TIME I = 1.55b69 SECCND;s,, AFTE;R 839 COMPLICT CYCLF-j OF UPEiATIiUM,. I li V:LOCITIS AhD DEPI h IN IH. VWP,,K CVl I Y:4S wFLL,j VELOCITIES AN)D PRESSlJU(ES AT KLY POINTS 1Hr-OUGIifU!T THE FLOW SYSTAiT, AKr A, IAt8ULAIErD BELOn. VAPOR CAVITY DIST..0000.12;O.2560.3284,.511 ).039C.7079..1519 1.2799 i.: 7>3 1.5'3j8 1.co6 I 1. i18 LOC. 0 2 4 6 H 1t 1Z 14 16 i8 20 22 24, U.000 -.043 -.072 -.091 -.1V9 -.126 -.133 -.1 -.120 -.123 -. 0 6 -.:69 -.049 -. i -. 19 Z.024.024.025.0U26.o2 7.2 3.C30.32.034.036.0, 9.C42.0/t5..,:) DIST. 1.9198 1.9838 2.0478 2.1757 2.23-)7 2.3017 LOC. 3C 31 32 33 34 3) 36 U -.010 -.0,3.007.020.029.040.0o4 L.C56.058.0bO.061.o63.065.067 FULL-FLOWING PIPE:YSICh LOC. ( ) (12,C2, (2 (,1) (2,C) (2,10) (2 2 ) (.230) (2,43) (2,53) V -. 000. 00u4.0:4. 356.,4'.0867.0839) H 11.30C0 11.5429 11.5429 19.81it 28.o19.30.,454 31.79)2 533.179 AT TIME T = 1.S7548 S5C0N)St AFTER 849 CCMPLFT- CYCLES Of- OPERAIIU/, Ili( VLUCiTIES ANDf) DEPTHS iN THE VAPOR CAVITY,S Wt-LL AS VELOCITIES AND PRESSURES AT KCY POINTS THtROUGHOUT TliE FLOWl SYStIE ARe AS TAI3ULA1LD iBELOW. VAPOR CAVITY DIST..0300.1280.2560.3840.:119.6399.7679.d,59 1 2Z39. 1119 1.2799 1.4078 I.b358 1.6638 1. 1918 LOC. 0 2 4 6 8 1l 12 14 b1 18 20 22 24 2, 26 U.000 -.045 -.C75 -.095 -.115 -.132 -.141 -.t 9 -.129 -.114 -.098 -.82 -.063 -.:d -. L.024.025.025.026.C7.028. Ct30.u32.C:{4.036.039.j42.045.3Otd *.052 CIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.C28 -.020 -.009.002.012.022.C45 Z.056.058.059.061.U63.065.068 FULL-FLOWING PIPE SYSTtM LOC. (11,) (1,t) (2,0) (2,10) (21,0) ( 2,30) (2v,4) (2 50) V -.0071 -.u07O -.0070 -.0041.032u.0786.002.a8C3 H 11.3000 11.3497 11.3497 14.427C 21.2862 30.1146 32.1040 3.1190 AT TIME T = 1.59404 SEC(NDaS, AFTER 859 COMPLETE CYCLES OF OPERAlION, TIlE V&LOCITIES AND DEPIHS IN THE VAP3R CAVITY AS WELL As VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEN ARL AS TAbULATL-D BELOw. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.6959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 Z2 24 26 28 U.000 -.046 -.077 -.099 -.120 -.138 -.148 -.147 -.139 -.125 -.111 -.095 -.078 -.064 -.u55 Z.025.025. 02 5.026.027.028.030.j32.034.036.039.042.045.G48.j52 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36

U -.045 -.037 -.026 -.015 -.005.001.025 z.056.057.C59.061.063.065.068 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2940) (15,5) V -.0101 -.0101 -.0100 -.0106 -_.098.0256.0751. C70s h 11.3000 11.3478 11.3478 14.0936 16.L931 23.1648 31.5450 3,,179( AT TIME T = 1.61260 SECONDS0 AFTER 869 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN IHE VAPOR CAVITY'-S WELL As VELOCITIES AND PRESSURES Ar KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY D1ST..0000.1280.2560.3840.5119.6399.7679.6959 1.0239 1.1519 1.2799 1.4.078 1.5358 1. 6636 1.7918 LOC. 0 2 4 o 8 10 12 14 16.18 20 22 24 20 2R U.000 -.047 -.080 -.104 -.126 -.145 -.155 -.155 -.148 -.136 -.123 -.108 -.093 -L -. 13 1.025.025.025.026.021.029.030.u32.034.036.039.042.045 8 u~ 2 UIST. 1.9198 1.9838 2.U478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.062 -.053 -.042 -.032 -.024 -.019.004 7.055.057.0.59.061.063.065.068 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2910) (2,20) (2930) (2,40) (2,50) V -.0140 -.o141 -.0140 -.0157 -.0176 -.0132.0243.06 9 H 11.3000 11.3723 11.3723 14.3902 15.9682 17.6371 24.3009 33.1790 - THE DIRECTION OF FLOW Al' THE RESERVOIR END OF THE PIPE SYSTEM HAS REVERSEo AT TIME T = 1.62837 SLCONDS. o AT TIME T = 1.63115 SECONDS, AFTER 879 COMPLETE CYCLES OF OPERATIUN, IHE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 - 12 14 16 18 20 22 24 26 28 U.000 -.048 -.082 -.108 -.132 -.151 -.162 -.163 -.157 -.147 -.135 -.121 -.107 -.097 -.090 L.025.025.025.026.027.029.030.032.034.036.039.042.045.046.o52 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34' 35 36 U -.078 -.069 -.059 -.050 -.043 -.039 -.015 L.055.057.059.061.063.065.068 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (290) (2,10) (2,20) (2,30) (2,40) (2,50) V -.0208 -.0208 -.0208 -.0204 -.0191 -.0183 -.0183 -.0299 H 11.3000 11.4253 11.4253 14.5249 15.8343 17.1009 19.3257 33.1790 AT TIME T = 1.64971 SECONDS, AFTER 889 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW* VAPOR CAVITY DIST.OCOO.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918

LOC. 0 2 4 6 8 10 12 14 L6 l8 20 22 24 21 ch U.000 -.049 -.085 -.112 -.137 -.157 -.169 -. 11 -.166 -.158 -.147 -.134 -.122 -.114 -. Z.025.025.025.026.027.029.030.u32.034.036.039.042.045.(C48.b2 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.094 -.085 -.076 -.068 -.062 -.058 -.033 L.055.057.059.061.063.065.067 FULL-FLOWING PIPE SYSTEM LUC. (1,0) (1,1) (2,0) (2,10) (2,2C) (2,30) (2,40) (2,50) V -.0272 -.0272 -.0271 -.0242 -.Q21t -.0242 -.u0724 -.1043 H 11.3000 11.3258 11.3258 14.1481 15.6581 17.5199 26.0000 33.1790 AT TIME I = 1.66827 SECONDS, AFTER 899 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEK, ARE As TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6658 1.7918 LOC. 0 2 4 6 8 10 12 i4 16 18 20 22 24 26 28 U.000 -.050 -.087 -.116 -.143 -.163 -.176 -.179 -.176 -.168 -.159 -.147 -.137 -.1 3i -.12 Z.025.025.026 G.26.027.029.030.o32.034.036.039.042.045.C48.052 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.110 -.101 -.093 -.086 -.081 -.014 -.050 z.055.057.059.061.063.065.067 FULL-FLOWiNG PIPE SYSTEM R) LOC. (1,0) (1.1) (2,0) (2,10) (2,20) (2,3G) (2,40) (2,50) V -.0284 -.0284 -.0283 -.0284 -.U293 -.0757 -.1121 -. 1 15 H 11.3000 11.3249 11.3249 13.7378 15.8326 24.5375 31.4009 33.1790 AT TIME T - 1.68682 SECONDS, AFTER 909 COMPLETE CYCLES OF OPERATIUNt THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 U.000 -.0OO -.090 -.121 -.148 -.170 -.183 -.187 -.185 -.179 -.171 -.161 -.152 -.146 -.139 Z.025.025.026.026.028.029.030.032.034.036.039.042.045.048.0 2 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.125 -.117 -.110 -.104 -.098 -.089 -.064 Z.055.057.059.061.063.064.066 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (230) (2,40) (2,50) V -.0298 -.0298 -.0297 -.0334 -.U0822 -.1171 -.1183 -.1179 H 11.3000 11.3268 11.3268 14.2852 22.5982 29.7017 31.7650 33.1790 AT TIME T = 1.70538 SECONDS, AFTER 919 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW.

VAPOR CAVITY UIST..0000.1280.26.3840.5119.6399.7679.8959 1.0239 1. 1519 1.2799 1.4078 1.5358 1.66)8 L. 1)16 LOC. Q 2 4 6 8 10 12 14 16t 18 20 22 24 a Q. U.300 -.051 -.092 1.125 -. 154 -.176b -. 189 -195 -. 194 -.190 -.183 -. 174 L. 17 -.162 -.15 L.025.025.026.026.028.029.030.o32.0,34.036.039.042.045.: *0 01ST. 1.9198 1.9838 2.W478 2.1118 2.1757 2.2391 2.3037 LOC. 30'31.32 33 34 35 36 U -. 141 -.133 -. 127 -.121 -.114 -.101 -.077 2.055.051.059.061.062.064.066 FULL-FLOWING PIPE SYSTEM LUG. (1,0) (1,1) (2,0) (2,10) (2,20) (2930) (2,4C) (~0 V -.0356 -.0367 -.0366 -.0834 -.1211 -.1249 -.1229 -. 1216 H 11.3000 11.5838 11.5838 21.4456 28.1431 29.6333'31.5236 3i.179b AT TIME T 1.72394 SECONDS, AFTER 929 COMPLETE CYCLES OF UPIERAT1ON9 THE VELOCITIES AND DEPTHSi 1!4 THE VAPOR LAVITY ~S WELLLA VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.12o0.2560.3840.5119.6399.7679.8959 1.u239 1.1519 1.2799 1.4078 1.5358 1.6638 1.791 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 Z..) 26 U.000 -.051 -.094 -.129 -.159 -.182 -.196 -.Z03 -.204 -.201 -.195 -.187 -.182 -.Lid -.17 1.025'02.6.2.01.028.09.3 02.34.036.039.L42.045.'b.5 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2391 2.3031 LOC. 30 31 32 33 34 35 36 U -.157 -.150 -.144 -.137 -.128 -.112 -.088 Z.055.057..059.061.062.064.065 FULL-FLOWING PIPE SYSTEM V -.1225 -.1234 -.1230 -.1243 -.1261 -.1269 -.1282 -1279 H 11.3000 13.3604 13.3604 26.7106 28.6886 29.9662 31.2889 31.1790 AT TIME T 1.74249 SECOND.), AFTER 939 COMPLETE CYCLES OF OPERATION, THE VfLOLITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE As TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.89t59 1.6239 1.1,519 1.2799 1.4078 1.5358 1.66)8 1.791 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 08 U.000 -.052 -.097 -.134 -.164 -.188 -.203 -.211 -.2e13 -.2 11 -.207 -.201 -.197 -.194 -. 18 L.026.026.026.027.028.029.031.032.034.036.0-39.042.045.0 48.02 DIST. 1.9198 1.9838 2.0478 2.1118 2.1751 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.173 -.166 -.160 -.152 -.141 -.127 -.102 Z.055.057.059.060.062.064.067 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (21,0) (2,10) (2,20) (2930) (2,40) v5. V -.2133 -.2114 -. 2107 -.1658 -.1301 -.1295 -.1319 -. 134f6 H 11.3000 11.7681 11.7661 21.8961 28.5358 30.1442 31.6630 33.1790

Al TIME T = 1.76105 SECONDS, AFTER 949 COMPLETE CYCLES O[F OPERATIU,, THE VELOCITIES AND DEPIHS IN rTHE VAPOR CAVITY b,ELL As VELOCITIES AND PRESSURES A1 KEY POINTS THROUGHOUI THE FLo]w SYSTEfMi A,<r AS IABDILATED BELOw. VAPOR CAVITY DIST..0000.1280.2560.J840.5119.6399.7679.d959 1.0239 1.1519 1.2199 1.418 1.5358 1.66 d 1. 7ii LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2, U.000 -.O2z -.099 -.138 -.170 -.194 -.210 -.219 -.223 -.222 -.21 -14 -.212 -.2 9 -.2L Z.026.026.026.027.028.029.031.3jZ.034.037.0).42,.045.. Ji DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2391 2.3031 LOC. 30 31 32.33 34 35 36 U -.188 -.182 -.176 -.167 -.155 -.149 -.123 Z.055.057.059.060.062.064.068 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,2C) (2,30) (2,4L) ('S50) V -.2231 -.2230 -.2223 -.2166 -.1691 -.1351 -.1361 -. l3 H 11.3000 11.J692 11.3692 14.8627 23.3541 30.2334 32.0771 32.179 AT TIME T = 1.1 7961 SECflNUS, AFTER 95'9 COMPLETO CYCLES OF OPERATION, THE VFLOCITIES ANU DEPTHS IN THE VAP3R CAVITY A.S wELL AS VELOCITIES AND PRESSURES AT KEY POINTS ttiHROUGI1OfUT THE FLOw SYSTEm ARE AS TAfiULATED BELOw. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.bi'59) 1.0239 1.1519 1.2799 1.407T L.5358 1.66`8 1.791b LOC. 0 2 4 6 & 1t0 12 14 1t 18 20 22 24 20t 28 U.000 -.053 -.101 -.142 -.175 -.200 -.217 -.Z27 -.232 -.231 -..228 -.227 -.2.4 -.21 Z.026.026.026.027.028.029.031.u32.& 34.037.039.342.045.0.d.t52 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.204 -.198 -.190 -.181 -. 173 -. 1 17 -.150 Z.055.057.059.060.062.064.010 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,t30) (2,40) (?50) V -.2262 -.2262 -.2254 -.2257 -.2211 -.1757 -.1392 -.1373 H 11.3000 11.3494 11.3494 14.0925 16.5508 25.2894 31.7929 33.17%j AT TIME T = 1.79816 SECONDS, AFTER 969 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOw. VAPOR CAVITY DIST..0000.1280.2560.3843.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 20 28 U.000 -.054 -.104 -.146 -.181 -.206 -.224 -.235 -.242 -.244 -.243 -.242 -.242 -.2;39 -.230 Z.026.026.026.027.028.029.031.032.034.037.039.042.045.048.052 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.219 -.213 -.205' -.197 -.195 -.208 -.178 Z.055.057.058.060.062.065.070 FULL-FLOWING PIPE SYSTEM LOC. (110) (1,1) (2,0) (2,10) 12,20) {2,30) (2,40) (2,50)

V -.2298 -2299 7. 2291 -.2305 -.2323 -.2258 -1769-.12 H 11. 3000 I1. 3689 11.3689 14. 3138 16.0177 18.1034 26.4348 3~179CL AT TIME r 1. 81672 SECONDS, AFTER 979 COMPLETE CYCLES OF OPERATIUN, THE VtLOCITIES AND DEPIIHS IN THE VAPOR CAVITY AS5 WELL A VELUCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE As TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.6959 1.0239 1. 1519 1.2799 1.4U78 1.5358 I. 6638. 74i s LOC. 0 2 4 6 8 10 12 14 18 i 20 22 24 2 O.000 -.054 -.106 -. 151 -.186 -.212 -.231 -.243 -.251 -.254 -.2t5s -.255 -.256 - Z3 - 241 1.026.026.027.027.028.030.031 0 33.035.037.039.o42.045.c 9. DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.234 -.227 -.220 -.215 -.221 -.237 -.207 Z.055.0C)7.058.060.063.065.070 FULL-FLOWING PIPE SYSTEM LOG. (1,0) (1,1) (2,0) (2,10) (2,120) (2,30) (2,40) (2,95c) V -.2358 -.2358 -.2351 -.2357 -.2346 -.2336 -.2290 -.2166 H 11.3000 11.4168 11.4168 14.5704 15.8665 17.1563 19.5244 33.1790 AT TIME T i.83528 SECONDS, AFTER 989 COMPLETE CYCLES OF 11PERAT IU-N, THiE VELOCITIES AINO DEPTHS IN THF VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS IHROUGHOUT THE FLOW SYSTEM AKL AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.51 19.6399.7679.8959 1.0239 1.1519 1.2799 1.4U78 1.5358 1.66.38 1.79180 LOC. 0 2 4 6 8 10 12 14 16 1b 20 22 24 2 6 28 iJ U.000 -.055 -108 -.155 -.191 -.218 -.238 -.252 -.260 -.265 -.2b7 -.269 -.271 -.2cI -.259 Z.026.026..027.027 C.28.030.031.U33.035.037.039.042.045.049.052 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2391 2.3037 LOC. 30 31 32 33 34 35 36 U -.248 -.242 -.236 -.237 -.248 -.264 -.233 Z.055.057.059.061.063.066.070 FULL-FLOWING PIPE SYSTE'M LOC. (1,0) (1,1) (2,0) (2,10) (2920) (2930) (2940) (2,50) V -.2433 -.2431 -.2423 -.2391 -.2370 -.2378 -.2731 -.3155 H 11.3000 11.3702 11.3102 14.2479 15.71.02 17.2875 23.9181 33.1790 AT TIME T= 1.85384 SECONDS, AF-TER 999 COMPLETE CYCLES OF OPERATION,4 THE VELOCITIES AND DEPIHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7916 LOC. 0 2 4 6 8 10 12 14 16 18 20 22- 24 26 28~ U.000 -.056 -.111 -.159 -.1l97 -.225 -.245 -.260 -.270 -.276 -.279 -.283 -.285 -.281 -.273 1.027.027.027.028.029.030.031.0-33.035.037.04.0.042.046.0-49.5 01ST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.263 -.257 -.255 -.261 -.274 -.287 -.255 z.055.057.059.061.064.066.070

FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,2C) (2,30) (2,40) (2,50) V -.2446 -.2446 -.2438 -.2435 -.2423 -.2762 -.3240 -.3294 H 11.3000 11.-j279 11.3219 13.7911 15.6695 22.44Z4 30.9519 33.179. AT TIME i = 1.87239 SECONDS, AFTER 1009 COMPLETE CYCLES OF OPERATION, [iE VELOCITIES AND DEPJ-IS 1 THE VAPOR CAVITY WS WELL As VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARt AS TABULATED dELOw. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4278 1.5358 1.6638 1.7918 LOC. 0 2 4 6 a 10 12 14 16 18 20 22 24 2e 25 U.000 -.057 -.113 -.163 -.2C2 -.231 -.252 -.268 -.279 -.286 -.291 -.296 -.298 -.295 -.28? L.027.027.027.028.029.030.031.4.33.035.037.0,0.043.046 L492 DIST. 1.9198 1.98308" 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.277 -.274 -.276 -.285 -.299 -.306 -.275 7.055 -.057.059.061.064.066.069 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V -.2459 -.2459 -.2450 -.2469 -.2825 -.3282 -.3326 -.332H 11.3000 11.3279 11.3279 14.0277 20.4983 29.3040 31.7436 33o1790 AT TIME T = 1.89095 SECONDS, AFTER 1019 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS I THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS TH(ROUGHOUT THE FLOW SYSTEM ARE AS TABULATED RELOW. ro VAPOR CAVITY UIST..0000.1280.2560.3840.5119.6399.7679.6959 1.0239 1.1519 1.27I9 1.4078 1.5358 1.66i8 1.7918 LOC. 0 2 4 6 8 10 12 14 6 1.8 20 22 24 26 28 U.000 -.058 -.115 -.167 -.207 -.237 -.259 -.276 -.289 -.297 -.304 -.310 -.312 -.39 -.301 7.027.027.027.028.029.030.031.ij33.035.037.040.043.046.049.052 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.293 -.293 -.298 -.310 -.320 -.320 -.291 L.055.057.059.062.064.066.069 FULL-FLOWING PIPE SYSTEM LOC. (1rO) (111) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V -.2494 -.24993 -.2491 -.2838 -.3326 -.3388 -.3367 -.3357 H 11.3000 11.4574 11.4574 19.2867 27.6346 29.7982 31.5748 33.1790 AT TIME T = 1.90951 SECONDS, AFTER 1029 COMPLETE CYCLES OF OPERAJIUN, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4.6 8 10 12 14 16 18 20 22 24 26 28 U.000 -.059 -.118 -.171 -.213 -.243 -.267 -.285 -.298 -.307 -.316 -.323 -.325 -.322 -.315 1.027.027.027.028.029.030.032.033.035.037.040.043.046.049.052 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36

U -.310 -.313 -.321 -.332 -.338 -.332 -.304 2.056.03)8.060.0b2.065.bb.0o9 FULL-FLOWING PIPE SYSTOM LOC. (1,0) (1,1) (2,0) (2,10) (2,2C') (2,36) (2,4u) V -.3093 -.3110 -. HO -.334o -.402 -.3411 -.3419 -.341. H 11.3000 13.L638 13eC636 26.3155 28. 54 -9.9 G6 -1.2765 33.1792 AT TIME f = 1.92806 SECd'4D5, AFIER 1039 COMPLETE CYCLEa UF [i'PLKAT ('., l THEI VLOCITIES A'I) DEPTI. I1I TH1E VAPtSR CAVITY A3S oWELL. ~ VELOCITIES AND PRESSURES AT KEY PUIN5 HI ROU-GtOUT T-F FL'1i aYSTE6 AK: A3 rABULAIl1):LLOh. VAPOR CAVilY D1ST..0000.iZ80.2560.3840.5119 u.399.7b79 b-.)89 l.u239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7418 LOC. 0 2 4 b 8 10 12 14 16 18 Zu 22 24 2 U 0U0o -.060 -.120 -.175 -.218 -.250 -.i74 -./r3 _j C7 -318 -. 3 -.336 -.339 -3 -.2 L.027.027.028.028.(29.030.032 3 3.5.038.-4.u43.046.0,i9 U1ST. 1.9198 1.9838 2.u478 2.1118 2.157 2.2397 Z.3031 LOC. 30 31 32 33 34 35 36 U -.328 -.334 -.344 -.352 -.353 -.348 -.319 L.056.058.Oo0.063.065.066.071 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2, 2) (2,30) (2,40) (?,5E) V -.4192 -.4167 -.4152 -.3668 -.3431 -.3433 -.3454 -.3482 H 11.3000 12.1458 12.1458 23.6475 28.5b87 3.0617, 31.5522 3~.1790 AT TIME T 1.94662 SECONOz, AFTER 1049 COMPLEIE CYCLES OF UPERATION, THO VELOCITIES AND DEPIH'Zo IN IHE VAPOR CAVITY AS WELL A. VELOCITIES AND PRESSURES AT KEY POINtS THIROUGHIOUT THE FLOW SYSTEM AkL AS TAHULATED BELOw. no VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.89:9 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOG. 0 2 4 6 8 10 12 14 16 18 20 22 24 2o 2N U.000 -.061 -.123 -.179 -.223 -.256 -.281 -.iGI -.316 -.329 -.341 -.349 -.352 -.34t9 -.344 L.027.02?.028.028.OZ9.030.032.033.035.038.040.043.046.049.052 DIST. 1.9198 1.9838 2.Co478 2.1118 2.1757 2.2397 2.3031 LOC. 30 31 32 33 34. 35 36 U -.347 -.355 -.365 -'.370 -.370 -.377 -.344 L.056.058.061.063.065.067.074 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2tZ0) (2930) (2,40) (2,50) V -.4358 -.4357.-.4342 -.4236 -.3698 -.3474 -.3495 -.3497 H 11.3000 11.3997 11.3997 15.7186 25.1366 30.2342 32.0049 a3.1790 AT TIME T = 1.96518 SECONDS, AFTER 1059 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WCLL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARL AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 141519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 U.000 -.062 -.125 -.183 -.229 -.262 -.289 -.309 -.326 -.341 -.353 -.362 -.364 -.362 -.359 z.028.028.028.029.029.031.032.034.036.038.041.643.046.049.053

DIST. 1.9198 1.9838 2.u478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.367 -.376 -.384 -.388 -.393 -.415 -.377 Z.C56.059.061.063.065.068.075 FULL-FLOWING PIPE SYSIEM LOC. (1,0C) (1,1) (2,0) (2,10) (2,2C) (2,3C) (2,40) (,5C) V -.4388 -.4388 -.4373 -.4371 -.4277 -.3759 -.3517 _. 3 01) H 11.3000 11.3578 11.3578 14.1744 17.3844 27.0906 31.9033 S I.1796 AT TIME T = 1.98373 SECONDS AFTLR 1069 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGhOUl THE FL(OW sYSTEM AKL AS TABULATED BELOW. VAPOR CAVITY DIST..COO0.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.',078 1.5358 1.6638 1.7'I[ LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 U.000 -.063 -.128 -.187 -.234 -.269 -.296 -.31 7 -.336 -.352 -.366 -.375 -.377 -.3 5 -.35 Z.028.028.028.029.030.031.032.634.036.038.041.044.047.C56.GO3 DIST. 1.9198 1.9838 2.0418 2.1118 2.1757 2.2391 2.3037 LOC. 30 31 32 33 34 35 36 U -.388 -.397 -.403 -.409 -.424 -.450 -.410 Z.057.059.061.063.066.069.017 FULL-FLOWING PIPE SYSTEM LOC. (10) (1,01) (2,0) (2,10) (2,20) (Z,30) (2,40) (2,50) V -.4417 -.4418 -.4403 -.4414 -.4432 -.4319 -.3171 -.3537 H 11.3000 11.3729 11.3729 14.29b61 16.1354 19.0744 28.3190 33.1790 AT TIME T = 2.00229 SECONDS, AFTER 1079 COMPLETE CYCLES OF OPERATIU, THE VELOCITIES AND DEPTHS IN THE VAPR CAVITY 45 WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS ]ABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.6959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7)18 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 U.000 -.0u5 -.130 -.191 -.239 -.275 -.303 -.326 -.345 -.363 -.378 -.387 -.390 -.389 -.393 Z.028.028.028.029.030.031.032.034.036.038.041.044.047.050.053 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.408 -.416 -.424 -.436 -.456 -.480 -.439 z.057.059.062.064.067.070.073 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2.40) (2,50) V -.4465 -.4466 -.4451 -.4463 -.4456 -.4444 -.4337 -.4033 H 11.3000 11.4138 11.4138 14.6107 15.9865 17.3721 20.4129 33.1790 AT TIME T = 2.02085 SECONDS, AFTER 1089 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

U.000 -.066 -133 -195 -.244 -.282 -.310 -~.334 -.355 -.375 -.390 -.399 -.402 -.4 3.1 Z.028.028.029.029.030.031.032.034.636.039.041.1044.04 7 6.C u 01ST. 1.9198 1.9838 2.U478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.428 -.437 -.448 -.465 -.485 -.491 -.456 1'.058.060.062.065.068.070.0-72 FULL-FLOWING PIPE SYSfEM LOC. (1,0) (1,1) (2,0) (2910) (2,20) (2,30) (2940) ( V -.4538 -. Ji53 7 -.4522 -.4493 -.4475 -.4415 -.4704 -.5134 H 11.3000 11.4108 11.4108 14.3797 15.6478 17.3228 22.2716 3J. 179t] AT TIME T= 2.03940 SECONDS, AFTER 1099 COMPLETE CYCLES~ OF OPERAT1ONa, THE VFLOCITIES ANOJ DEPTH-S IN [HE VAPOR CAVITY;45 wELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUt ThE FL~k SYSTEM Akt AS TABULAIED BELOW. VAPOR CAVITY DIST..0000..1280.2560.3840.5119.6399.7679.8959 1.0239 1. 1519 1.2799 1.4378 1.5358 1.6638 1.7)18 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2e~ 2 U.000 -.068 -.136 -.199 -.250 -.288 -.317 -.,343 -.366 -.386 -.402 -.412 -.415 -.4LB -.2 1.028.028.029.029.030.031.033 (034.036.039.041.044 -.047 C.0.c' 01ST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.448 -.459 -.474 -.493 -.504 -.499 -.468 1.058.060.063.066.068.070.012 FULL-FLOWIlNG PIPE SYSTEM ) LOC. (1,0) 11)(2,0) (2,10) (2920) (2930) (2,40) 12,50) V -.4556 -.4556 -.4-541 -.4533 -.4511 -.4735 -.5Z70 -.5374 H 11.3000 11.3328 11.3328 13.42fl 15.7162 20.7366 30.1073 33.1790, AT TIME T 2.05796 SECONDS, AFTER 1109 COMPLETE CYCLES OF OPERATION, THiE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM AAE~ AS TABULAFED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.191 LOC. 0 2 4 6 8 10 12 14 16 1a 20 22 24 26 28 U.000 -.069 -.138 -.203 -.255 -.294 -.325 -.351 -.376 -.398 -.414 -.424 -.428 -.434 -.44 Z.029.029.029.030.036.031.033.035.031.039.042.044.047.050.5 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.470 -.484 -.501 -.515 -.517 -.505 -.477 1.059.061.064.066.068.070.071 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1).(2,0) (2,10) (2*20) (2,30) (2,40) (2,50) V -.4565 -.4565 -.4549 -.4559 -.4792 -.5305 -.5403 -5407 H 11.3000 11.3359 11.3359 13.9471 18.8064 28.4793 31.6749 33.1790 AT.TIME T = 2.07652 SECONDS, AFTER 1119 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY

DIST..0000.1280.2560.3840.5119.6399.7619.8959 1.0239 1.1519 1.2799 1.4C78 1.5358 1.6bJ, 1.716. LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2 2 8 U.000 -.071 -.141 -.207 -.260 -.301 -.332 -. 60 -.386 -.4L9 -.426 -.436 -.441 -.450 -.468 z.029.029.029.030.030.032.033.35.037.039.042.045.048.O 1. 155 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 30 U -.493. -.509 -.525 -.532 -.527 -.511 -.485 Z.059.062.064.066.068.069.071 FULL-FLOWING PIPE SYSTEM LOC. (1,G) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V -.4585 -.4587 -.4572 -.48C0 -.5351 -.5460 -.5441 -.5433 H 11.3000 11.4010 11.4010 17.4818 Z6.b883 29.7345 31.5942 jA.1790) AT TIME T = 2.09508 SECONDS, AFTEi 1129 COMPLETC CYCLES OF ()PERATIOU, THE VELOCITIES AND DEPTHS LN TIE VAPOR CAVITY AS wELL Ab VELOCITIES AND PRESSURES AT KEY POINTS IlIRDUGHOUT THE FLOW SYSTEM AKt AS TABULATtD BELOW. VAPOR CAVITY DIST.,.0000.1280.2560.3840.5119.6399.7u79.6959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2. 4 6 8 10 12 14 ib 18 20 22 24 26 28 U.000 -.072 -.144 -.211 -.266 -.307 -.340 -.370 -.397 -.421 -.438 -.448 -.455 -.461 -.488 Z.029.029.029.030.031.032.033'.035.C37.039.042.045.048.001.'55 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.517 -.533 -.544 -.545 -.533 -.516 -.492 Z.060.062.065.067...'.068.069.070 0 FULL-FLOWING PIPE SYSTEM LOC. (1,0) 1,1) (2,0) (2,0) (22,0) (2,320) (230,40) (2,50) V -.4968 — -.4993 -.4976 -.5363 -.5475 -.5487 -.5489 -.5476 H 11.3000 12.6260 12.6260 25.3979 28.3979 29.8040 31.2817 33.1790 AT TIME T = 2.11363 SECONDS, AFTER 1139 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW..VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.1918 LOC. 0 2 4 6 8. 10 12 14 16 18 20 22 24 2Z 28 U.000 -.074 -.146 -.214 -.271 -.313 -.347 -.379 -.408 -.432 -.449 -.460 -.469 -.485 -.809 Z.029.029.03030.030.031.032.033.035.037.040.042.045.048.052.056 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 2.3037 LOC. 30 31 32 33 34 35 36 U -.541 -.554 -.560 -.554 -.539 -.521 -.499 Z.060.063.065.067.068.069.070 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V -.6116 -.6092 -.6071 -.5642 -.5499 -.5504 -.5522 -.5545 H 11.3000 12.5776 12.5776 24.8170 28.5141. 29.9445 31.4307 33.1790 AT TIME Tr 2.13219 SECONDS, AFTER 1149 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW.

VAPOR CAVITY DIST..0000.1280.2560.3840.5119.o399.7679.di59 1.0239 1.1519 1.27t9 L.4078 1.5358 1.6635'.791, LOC. 0 4 6 b 1C 12 14 16 18 20 22 24 2: U.000 -.075 -.149 -.218 -.276'-.319 -.355 -.388 -.418 -.443 -.461 -.473 -.485 -.: 4 -. Z.029.030.030.030.031.032.034.u35.038.040.043.2-45.048.-.,-, DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 LOC. 30 31.32 33 34 35 U -.563 -.572 -.572 -.561 -.544 -.525 Z.061.063.oC5.066.068.069 FULL-FLOWING PIPE SYST EM LOC. (1,0) (1,1) (2,0) (Zl10) (2,20) (2,3() (2,40) (t,50) V -.6411 -.o408 -. 6 386 -.620s -.5671 -. 553J -. 5559 -. 556 H 11.3000 11.4443 11.4443 16.9970 2g.3724 3C.14u2 31.8823 J3.1 fu AT TIME T = 2.15075 SECONOS, AFTER 1159 COMPLETF CYCLEs OF OPERATIUN, THE VELOCITIES AND DEPTHS IN IHE VAPLOR CAVITY IS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGtHOUT THE FLOW SYSTEM ARL AS TABULATcD dELO1s. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7619.8959 1.0239 1.1519 1.2799 1.4)17 1.5358 1.66bJ6 1.7~ 1; LOC. O 2 4 6 8 10 12 14 16 18 2Z 22 24?2 z2; U.000 -.077 -.152 -.222 -.281 -.326 -.364 -.398 -.429 -.455 -.413 -.486 -.500 -.5; -.'5 Z.030.030.030.031.031.032.034.Uj6.:8.040.043.046.049.;,.: 37 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 LOC. 30. 31 32 33 34 35 U -.583 -.587 -.581 -.567 -.549 -.530 Z.061.063.065.066.067.068 a'~ FULL-FLOWING PIPE SYSTEM LOC. (t,O) (1, ) (2,0) (2,10) (2,20) (2,C0) (2,40) (2,50) V -.6444 -.6444 -.6422 -.6414 -.6238 -.5725 -.5579 -. 574 H 11.3000 11.3694 11.36'94 14.2936 18.6471 28.3178 31.9302 3i,. 1790 AT TIME T = 2.16930 SECONDS, AFTER 1169 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR C4VITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARt AS TABULATED bELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.66j8 1.7918 LOC. 0 2 4 6 8 10 12 14 i6 18 20 22 24 2o 2;i U.000 -.079 -.155 -.226 -.286 -.332 -.372 -.408 -.440 -.466 -.484 -.499 -.517 -.5b,4 -.577 Z.030.030.030.031.032.033.034.036.038.041.043.046.049.30:3.os7 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 2.2397 LOC. 30 31 32 33 34 35 U -.599 -.598 -.587 -.571 -.553 -.534 Z.062.064.065.066.067.068 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,2C) (2,30) (2,40) (:,50) *V -.6467 -.6467 -.6445 -.6454 -.6467. -.6282 -.5740 -. 5590 H 11.3000 11.3794 11.3794 14.2968 16.2542 20.4626 29.6657 33.1790

AT TIME I = 2.167846 SEC)ND5, AFTER 1179 COMPLETF CYCLEs OF OPERATIOLt, THE VgLOCITIES AND DEPIHS IN THE VAPOR CAVITY iS whLL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED 6ELOh. VAPOR CAVITY VIST..0000.1280.2560.3840.5119.6399.7679.~959 1.~Oz 9 1.1519 1.27I9 i.~478 1.5358 1.~6ob 1.791t LOC. 0 2 4 6 8 1C 12 14 to 18 20 z2 24 ~, 2 U.000 -.080 -.158 -.230 -.291 -.339 -.3 1 s -.48 -51 -.477 -.496 -.h513 -.55 -.05 -.598 Z.030.030.031.031.032. 33.034.03o ~. 8.;41.043. 4o.050. u4.068 U1ST. 1.9198 1.9838 2.~478 2.1118 2.1757 LOC. 30 31 32 33 34 U -.612 -.606 -.593 -.576 -.557 Z.C062.064.065.066.067 FULL-FLOWING PIPE SYSI EM LOC. (1,0) (1t1) (2,0) (2,10) (2,2C) (2,30) (2,40) (1,5C) V -.6505 -.6506 -.6484 -.6499 -.6498 -.646U -.6291 -.5904 H 11.3000 11.4136 11.4136 14.6165 16.1107 17.6190 21.7788 33.17)9S AT TIME T = 2.20642 SECONDS AFTER 1189 COMPLETE CYCLEs OF OPERATION, THE Vt-LOC1TIES AM) DEPTHS IN THE VAPOR CMITY As iLL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARL As TABULATED BELOW. VAPOR CAVITY 1DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6656 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2~ 26 U.000 -.082 -.161 -.234 -.296 -.346 -.389 -.'28 -.462 -.489 -.509 -.528 -.553 -.58 -.618 ~.030.031.031.031.032.033.035.036.G39.041.044.o47.050.C4..59 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 LOC. 30 31 32 33 34 U -.623 -.613 -.597 -.579 -.560 Z.062.064.065.066.067 FULL-FLOWING PIPE SYSTEM LOC. (1,0) t(1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V -.6571 -.6570 -.6548 -.6527 -.6512 -.6508 -.6642 -.6989 H 11.3000 11.4375 11.4375 14.5052 15.9816 17.4266 21.1833 33.1790 AT TIME T = 2.22497 SECONDS AFTER 1199 COMPLETE CYCLES OF OPERATIONt THE VFLOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATLED BELOW. VAPOR CAVITY DIST..0000.1280.Z560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 U.000 -.084 -.164 -.238 -.301 -.353 -.398 -.439 -.473 -.500 -.521 -.543 -.573 -.6f8 -.63 z.031.031.031.032.032.033.035.037.039.041.044.041.051.05!.059 DIST. 1.9198 1.9838 2.0478 2.1118 2.1757 LOC. 30 31 32 33 34 U -.631 -.618 -.601 -.583 -.564 Z.062.063.064.065.066 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50)

V -.6598 -.6598 -.6575 -.6561, -.6537 -.6673 -7203 -. 37.) H 11.3000 11. 3434t 11.3434 14.0820 15.8217 19.5380 28. 8432 3~.;.79j) AT TIME IT = 2. 24 353 SECONDS, AFTER 1209 COMPLETE CYCLES OF OPERATION, THE VELOC-ITIES AND DEPTHS IN TH-E VAPOR CAVITY;'.s WELL ~ VELOCITIES AND PRESSURES Ar KEY POIN4TS THROUGHOUT THE FLOW SYSTEM AAE AS TABULATED BFLOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6038 1.791 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 f' z U.000 -.085 -.167 -.242 -.306 -.360 -.4U8 -.4t49 -.484 -.511 -.534 -.560 -.593 -.6/i9 -.64' 1.031.031.031.032.032.034.035.037.039.042.044.047.051.035., 01ST. 1.9198 1.9838 2.0478 2.1118 LOC. 30 31 32 33 U -.637 -.622 -.605 -.586 Z.062.063.064.065 FULL-FLOWING PIPE SYSTEM LOC. (It)o1,) (2,10) (2t10) (2,201 (2,30) (2,40) (.',50) V -.6602 -.6602 -.6580 -.6585 -.6722 -.7230 -.7408 -.7417 H 11.3000 11.3476 11.3476 13.9383 17.6319 27.2133 31.5558 i$3.1790 AT TIME T = 2.26209 SECONDS, AFTER 1219 COMPLETE- CYCLES OF OPERATIUNt THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM AkE AS TABULATED BELOW. VAPOR CAVITY D1ST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918r LOC. 0 2 4 6 8 1 0 1 2 14 16 1 8 20 22 24 2 O 20 U.000 -.087 -.170 -.246 -.311 -.368 -.417 -.459 -.495 -.523 -.548 -.577 -.613 6-t6'b -.6b~ Z.031.031.032.032 C033.034.035.037.040.042.045.048.052.056.06. DIST. 1.9198 1.9838 2.0478 2.1118 LOC. 30 31 32 33 U -.642 -.626 -.608 -.589 1.062.063.064.065 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2920) (2,30) (2,40) (2,50) V'-.6612 -.6614 -.6592 -.6740 -.7277 -.7455 -.7444 -.743f H 11.3000 11.3800 11.3800 16.1671 25.3051 29.6309 31.5895 33.1790 AT TIME T =2.28064 SECONDS,, AFTER 1229 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY As WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY D1ST..0000.1280.2560.3840.5119.6399.7619.8959 1.0239 1. 1519 1.2799 1.4078 1.5358 1.6638 1.191 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2 b 28 U.000 -.088 -.173 —.250 -.317 -.376 -.426 -.470 -.506 -.535 -.563 -.595 -.634 -66 6 -.67 2.032.032.032.032.033.034.036.038.040.042.045.048.052.Ot)7.06 01ST. 1.9198 1.9838 2.0478 2.1118 LOC. 30 31 32 33 U -:.646 -.629 -.611 -.592 2.062.063.064.065

FULL-FLOWING PIPE SYSTEM LOC. (1,G) (1,1) (2,0) (2,10) (2,20) (2130) (2,40) (I",5) V -.6842 -.6864 -.6840 -.7282 -.7472 -.7490 -.7484 -.7471 H 11.3000 12.2208 12.2208 23.9988 28.1455 29.6797 31.2974 3..1790 AT TIME r = 2.29920 SECONDS, AFTER 1239 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPJR CAVITY AS wELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARL A3 TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1. 918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 f8 U.000 -.090 -.176 -.254 -.322 -.383 -.436 -.481 -.517 -.548 -.518 -.615 -.655 -.b1 -.b77 Z.032.032.032.033.033.034.036.038.040.043.046.049.053.C07.7 6~ DIST. 1.9198 1.9838 2.0478 LOC. 30 31 32 U -.649 -.632 -.614 2.062.063.064 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V -.7906 -.7891 -.7865 -.757u -.7495 -.7501 -.7517 -.7531 H 11.3000 12.9027 12.9027 25.462b 28.3708 29.8115 31.3115 33.1790 AT TIME T = 2.31776 SECONDS, AFTER 1249 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARK AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560,.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC..0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 U.000 -.092 -.178 -.258 -.328 -.392 -.446 -.491 -.528 -.561 -.594 -.634 -.674 -.693 -.682 Z.032.032.033.033.034.035.036.038.041.043.046.050.054.057.060 DIST. 1.9198 1.9838 2.0478 LOC. 30 31 32 U -.652 -.634 -.616 Z.062.063.064 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,11) (2,0Y (2,10) (2,20) (2,30) (2,40) (2,50) V -.8380 -.8372 -.8344 -.8076 -.7599 -.7522 -.7548 -.756A H 11.3000 11.5367 11.5367 18.5778 27.1348 30.0018 31.7335 33.1790 AT TIME T = 2.33632 SECONDS, AFTER 1259 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 L4 16 18 20 22 24 26. 28 U.000 -.093 -.181 -.262 -.334 -.400 -.456 -.502 -.540 -.574 -.612 -.655 -.691 -.703 -.687 Z.033.033.033.033.034.035.037.039.041.043.047.050.054.058.060 DIST. 1.9198 1.9838 LOC. 30 31

U -.655 -.637 7.062.063 FULL-FLOWING PIPE SYSTEM LOC. (II,) (1,1) (2,0) (2,10) (2,2 0) (2,30) (2,40) V -.8419 -.8418 -.8390.-.8371 -.8101 -.7645 -.7567 -.756' H 11.3000 11.3862 11.3862 14.5126 20.Z363 29.0617 31.91.03 3Z).179 AT TIME T = 2.35487 SECONDS, AFTER 12b9 COMPLETE CYCLES OF OiPERATIoJ, 11-iE VELOCITICS AND DEPTHS IN THE VAPUR CAVITY AS iELL.~ VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARL AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7o79.8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.66.8 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2~ ~E U.000 -.095 -.184 -.266 -.341 -.408 -.466 -.213 -.552 -.589 -.630 -.674 -.707 -.111 -.69 L.033.033.033.034.034.035.037.039.041.044.041.051.055 C.58 DIST. 1.9198 1.9838 LOC. 30 31 U -.657 -.639 7.062.062 FULL-FLOWING PIPE SYSTEM LOC..(1,0) (1,1) (290) (2,10) (2,20) (2930) (2,40) (U,50) V -.8436 -.8436 -.8407 -.8414 -.8414 -.8144 -.7661 -.7572 H 11.3000 11.3881 11.3881 14.3262 16.4645 22.1720 30.5557 33.1790 AT TIME T = 2.37343 SECONDS, AFTER 1279 COMPLETE CYCLES OF OPERATION, itE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. Q VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679.0959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7918 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2c, 28 U.000 -.096 -.187 -.271 -.347 -.417 -.476 -.524 -.564 -.604 -.649 -.693 -.720 -.717 -.693 7.033.033.034.034.034.036.037.039.042.044.048.051.055.C58.060 DIST. 1.9198 LOC. 30 U -.659 7.061 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V -.8464 -.8465 -.8436 -.8451 -.8457 -.8429 -.8147 -.7757 H 11.3000 11.4159 11.4159 14.6162 16.2652 17.9853 23.5111 33.1790 AT TIME T = 2.39199 SECONDS, AFTER 1289 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL As VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.1280.2560.3840.5119.6399.7679..8959 1.0239 1.1519 1.2799 1.4078 1.5358 1.6638 1.7916 LOC. 0 2 4 6 8 10 12 14 16 18 20 22 24 2u 28 U.000 -.097 -.190 -.275 -.354 -.426 -.486 -.535 -.578 -.621 -.668 -.711 -.731 -.722 - 7.034.034.034.034.035.036.038.040.042.045.048.052.055.058.06w

DIST. 1.9198 LOC. 30 U -.660 Z.061 FULL-FLOWING PIPE SYSIEM LOC. (1,G) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,5() V -.8519 -.8519 -.8490 -.8479 -.8465 -.8459 -.8522 -.8718 H 11.3000 11.4521 11.4521 14.b320 16.1370 17.6091 20.6720 33.1790 AT TIME T = 2.41054 SECLINDU. AFTER 1299 COMPLETE CYCLES OF OPEKATION, THE VtLOCITIES AND DEPTHS IN THE VAPOR CAVITY aS wELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY D01ST..0000.0640.1280.1920.2560.3200.3840.4419.5119.5759.6399.7039.7679.8319.8959 LUC. 0 1 2 3 4 5 6 7 8 9 1t 11 12 1 14 U.000 -.050 -.099 -.147 -.193 -.237 -.280 -.321 -. 361 -.399 -.435 -.467 -.496 -.543 -.5,7 Z.034.034.034.034.034.034.034.o35.C35.036.036.037.038.09.i4L DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4118 1.5358 1.5998 1.6638 1.7278 1.7916 1.8558 LOC. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 U -.569 -.591 -.614 -.638 -.663 -.688 -.710 -.727 -.737 -.740 -.736 -.726 -.712 -.o6)b -.679 2.041.042.0C44.045.047.049.051.053.054.056.057.058.059.Cbu. 06 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (I11) (2,0) (2910) (2,20) (2,30) (2,40) (2,50) V -.8555 -.8554 -.8525 -.8504 -.8481 -.8558 -.9028 -.9284 H 11.3000 11.3669 11.3669 14.2522 15.9768 18.8183 27.2958 33.179t AT TIME T = 2.42910 SECONDS, AFTER 1309 COMPLETE CYCLES OF OPERATIOl, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED 6ELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000 -.051 -.100 -.149 -.196 -.241 -.284 -.i27 -.368 -.407 -.444 -.477 -.507 -.534 -.559 Z.034.034.034.034.035.035.035.035.036.036.037.038.038.039.040 DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4718 1.5358 1.5998 1.6638 1.7278 1.7918 1.8558 LOC. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 U -.582 -.606 -.631 -.656 -.682 -.706 -.727 -.741 -.748 -.747 -.740 -.728 -.714 -.698 -.681 Z.042.043.044.046.048.050.051.053.055.056.057.058.059.059.060 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,401 (2,50) V -.8554 -.8554 -.8525 -.8527 -.8597 -.9048 - -.9318 -.9337 H 11.3000 11.3609 11.3609 13.9909 16.9296 25.6358 31.3362 33.1790 AT TIME T 2.44766 SECONDS, AFTER 1319 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

U.000 -.051 -.102 -.151 -.199 -.245 -.289 -.333 -.376 -.416 -.487 -.518 -.5e5 -.57I Z.035.035.035.035.035.035.035.036.036.036.037.038.039.l4c.0 DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4718 1.5358 1.5998 1.6638 1.7278 1.798b t.8k58 LOC. 15 16 11 18 19 20 21 22 23 24 25 26 27 23 29 U -.596 -.622 -.648 -.675 -.701 -.724 -.742 -.153 -.756 -.752 -.743 -.730 -.715 -.699 -.682 L.042.043.045.047.048.050.052.053.055.056.057.058.059.059 C FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2920) (2,30) (2940) (2,50) V -.8556 -o8551 -.8528 -.8617 -.9091 -.9354 -.9356 -.9352 H 11.3000 11.3765 11.3765 15.3105 23.6230 29.4119 31.5529 33.1793 AT TIME T = 2.46621 SECONDS, AFTER 1329 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED 8ELOW. VAPOR CAVITY DIST..0000.0640.128 0.1920.2560.3200.3840.4479.5119.5759.6399.7039.7619.8319.8959 LOC. 0 1 2 3 4 5 6 1 8 9 10 LI 12 13 14 U.000 -.052 -.103 -.153 -.202 -.249 -.295 -.339 -.383 -.424 -.463 -.497 -.529 -.551 -.584 Z.035.035.035.035.035.035.036.036.036.037.038.038.039.04u. DIST,.9599 1.0239 1.u879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4118 1.5358 1.5998 1.6638 1.7278 1.7918 LOC. 15 16 17 18 19 20 21 22 23 24 25 26 27 2z U -.611 -.638 -.666 -.694 -.719 -.741 -.756 -.763 -.763 -.756 -.746 -.732 -.716 -.730 Z.043.044.046.047.049.051.052.,;54.055.056.057.058.058.059 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (I,) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) IX V -.8685 -.8700 -.8671 -.9091 -.9372 -.9399 -.9388 -.9375 H 11.3000 11o9074 11.9074 22.2577 27.7677 29.5328 31.3038 33.1790 AT TIME T 2.48477 SECONDS, AFTER 1339 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119 o5759.6399.7039.1679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000 -.052 -.104 -.155 -.205 -.253 -.300 -.346 -.391 -.433 -.472 -.508 -.540 -.570 -_.98 1.036.036.036.036.036.036.036.036.037.037.038.039.040.041.042 01ST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4718 1.5358 1. 59,98 1.6638 1.7278 1.7918 LOC. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 U -.627 -.655 -.684 -.712 -.737 -.756 -.767 -.771 -.768 -.759 -.747 -.733 -.717 -.701 1.043.045.046.048.050.051.053.054.055.056.057.058.058.059 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50).V -.9573 -.9571 -.9539 -.9423 -.9398 -.9405 -.9418 -.9423 H 11.3000 13.0125 13'.0125 25.6151 28.1583 29.6531 31.2018 33.1790 AT TIME T = 2.50333 SECONDS, AFTER 1349 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY

DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.89 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000 -.053 -.106 -.158 -.208 -.258 -.306 -.353 -.399 -.442 -.482 -.518 -.552 -.583 -.613 Z.036.036.036.036.036.036.036.037.031.038.038.039.040.041.042 DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4718 1.5358 1.5998 1.6638 1.7278 LCO. 15 16 17 18 19 20 21 22 23 24 25 26 27 U -.b43 -.673 -.703 -.730 -.753 -.768 -.776 -.777 -.771 -.761 -.749 -.734 -.718 1.044.045.047.049.050.052.053.054.055.056.057.058.058 FULL-FLOWING PIPE SYSTEM.LC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (U,50) V -1.0232 -1.0219 -1.0185 -.9843 -.9457 -.9417 -.9441 -.9460 H 11.3000 11.7135 11.7135 20.2098 27.5041 29.8263 31.5689 33. 119 AT TIME T = 2.52188 SECONDS, AFTER 1359 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000 -.054 -.107 -.160 -.212 -.263 -.312 -.360 -.407 -.451 -.492 -.530 -.564 -.57 629 1.036.036.036.036.036.037.037.337.038.038.039.040.041.042.043 DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4718 1.5358 1.5998 i.6638 1.7278 LOC. 15 16 17 18 19 20 21 22 23 24 25 26 27 U -.660 -.692 -.721 -.747 -.767 -.779 -.783 -.781 -.174 -.163 -.749 134 -.718 l.044.046.048.049.051.052.053.054.055.056.057.057.058 FULL-FLOWING PIPE SYSTEM LOC. (110) (1,1) (2,0) (2,10) (2,20) (2.3-.) (2,40) (2950) V -1.0291 -1.0290 -1.0256 -1.0215 -.9860 -.9492 -.9459 -.9458 H 11.3000 11.4079 11.4079 14.9158 21.9053 29.4225 31.8437 33.1790 AT TIME T =2.54044 SECONDS, AFTER 1369 COMPLETE CYCLES OF OPERAIION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 6 7 8 9 10 11 12 13 14 U.000 -.054 -.109 -.162 -.215 -.267 -.318 -.308 -.416 -.461 -.502 -.541 -.577 -.611 -.645 Z.037.037.037.037.037.037.037 U038.038.039.039.040.041.042.043 DIST. 9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4718 1.5358 1.5998 1.6638 LOC. 15 16 17 18 19 20 21 22 23 24 25 26 U -.678 -.710 -.739 -.762 -.779 -.788 -.789 -.785 -.776 -.764 -.750 -.734 1.045-.047.048.050.051.052.054.054.055.056.057.057 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) - (2,10) (220) (2,30) (2,40) (2,50) V -1.0302 -1.0302 -1.0268 -1.0272 -1.0248 -.9900 -.9509 -.9458 H 11.3000 11.3991 11.3991 14.3929 16.8689 23.9492 31.0785 33.1790 AT TIME T = 2.55900 SECONDS. AFTER 1379 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS [N THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW.

VAPOR CAVITY IST1..0000.o640.1280.1920.2560.34i 0.3840.4479.s119.5759.6399.7639.7679 8.[.89, LOC. 0 1 2 3 4 5 - 8 9 10 L I 12 U.000 -.655 -., 169 -.219 -.272 -.324 -.i75 -.424 -.4(0 -.513 -.553 -.591 -.6L7 Z.037.o37.037.037.037.037.08.u-8.0;39.039.040.J41.042.... DIST..9399 1.(239 1 8 79 1.1519 1.2159 1.2799 1.3438 1. t:8 1.4118 1.5358 1.5998 LOG. 15 16 17 lb L9 2 i2 2? 23 24 25 U -.696 -.726 -.755 -.116 -.789 -. (94 -.793 -.(87 -.117 -.(64 -.750 7.046.647.049.053.052.053.054.o!>5.056.051 FULL-FLOWLING PIPE SYSTEM LUIC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,46)( V -1.0321 -1.0321 -1.0286 -1.030o -1.0311 -1.0262 -.9b97 515 H.11.~00 11.4Z13 11.4i13 14.0286 16.4476 18.5609 25.3549. 7 179u AT TIME T = 2.51756 SECONU,) AFTER 1389 COMPLETE CYCLEs OF OPERATIONt, IHE VILOCITIES ANI) DEPTHS IN THE VAPDR CAVI1Y 4S wELL 40 VELOCITIES AND PRESSUkES Ar KEY POINTS THROUGHOUT THE FLOW SYSTC!i Atr- AS TABULATED BCLOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.63'99 1039.1679. ii9.8 LOC. 0 1 2 3 4 9 6 I 8 9 10 11 12 13 14 U.000 -.096 -.112 -.168 -.223 -.218 -.331 -.383 -.433 -.46i -.524 -.566 -.605 3 -.6 1.038.033A.038. 038.038.038.038.oi9.039.040. 04.041.042..3 Ct5 DIST..9599 1.0239 1.u879 1.1519 1.2159 1.2799 1.3438 1.4078 1.4718 1.5358 1.5998 LOG. 15 16 17 18 19 20 21 22 23 24 25 I -.715 -.745 -.770 -.788 -.797 -.800 -.79b -.78ts -.777 -.704 -.149 Z.046.048..049.051.052.053.054.055.c55.656.051 FULL-FLOWING PIPE SYSTEM LOG. (1,0) (1,1) (2,0) (2,10) (2,20) (2,3c) (Z,40) (2,50) V -1.0363 -1.0363 -1.0328 -1.032- t-1.314 -1.0301 -1.0310 -1. V333 H 11.3000 11.4606 11.4606 14.1526 16.32C5 17.8663 zO.7364 33.1790 AT TIME T = 2.59611 SECONDS, AFTER 1399 COMPLETE CYCLES h F OPERATION, THE VLLOCITIES ANU DEPTHS IN IHE VAPOR CAVITY AS WELL As VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM A.iL AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.92-0.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOG. 0 1 2 3 4 5 6 7 8 9 10 11 [2 13 14 U.000 -.057 -.113 -.170 -.227 -.283 -.338 -.391 -.442 -.491 -.536 -.580 -.621 -.660 -.698 7.038.038.038.038.038.038.039.039.040.040.041.042.043 "t.4 45 DIST..9599 1.0239 1.0879 1.1519 [.2159 1.2799 1.3438 1.4078 1.4718 1.5358 LOG. 15 16 17 18 19 20 21 22 23 24 U -.732 -.761 -.783 -.797 -.804 -.803 -.798 -.789 -.777 -.764 1.047.048.050.051.052.053.054.055.055.056 FULL-FLOWING PIPE SYSTEM.LOC. (1,0) (1,1) (2,0) (2,10) (2t20) (2930) (2,40) (2,50) V -1.0402 -1.0401 -1.0366 -1.0342 -1.0321 -1.0361 -1.0739 -1.15 H 11.3000 11.4010 11.4010 14.4315 16.1723 18.4930 25.7190 33.1790

AT TIME r = 2.61467 SECONDO, AFTLR 1409 COMPLECT CYCLES OF =IVERA[TIU, tlHE VLLOCITIES AO DEPIhS IN THE VAPW1 CAVITY S WELI. As VELOCITIES AND PRESSurES AT KEY POINTS TH.ROUGHOUI THE FLOW SYSTEM ARL AS TABULATLD BELOW. VAPOR CAV 1Y JIST..0000.G543.1280.192. 2560.2u0C.3340.4,79.5119.5759.6399.7039.1679.83ic.895) LOC. U0 2 3 4 5 6 7 6 9 t u 11 12 Li i, U.000 -.0D7 -.115 -.173 -.232 -.289 -.345 -.400 -.452 -.5C2 -.549 -.594 -.637 -.618 -.r1 z.039.33).0.39.0 O9.03 9.039.09. i440.041.;41. 42.043..5.'t JIST..9599 1.02j9 1.0879 1.1519 1.2159 1.2799 1.3438 1.41,78 1.4718 1.5358 LOC. 15 16 17 18 19 20 2i 22 23 24 U -.749 -. 775 -.794 -.805 -.808 -.806 -.199 -.789 -.777 -.763 Z.048. 49.050.051.05;.053.054.u55.055.056 FULL-FLOWING PIPE SYSTLM LOC. (1,O) (1I1) (2,0) (2,10) (2,20) (2,30) (2,40) (t,50) V -1.039' -1.0397 -1.0362 -1.G362 -1.C389 -1.(752 -1. 1104 -1.114'/ H 11.3000 11.3745 11.3745 14.1007 16.6(21 23. 9')89 u J.9378 3. 794 AT TIME T = 2.63323 5EC(INO.,, AFTER 1419 COMPLETE CYCLES OF OPERATIlON, IHE VELOCITIES ANU UFPjTHS IN THE VAPOR CAVIIY;: wLl As VELOCITIES ANO PRESSURES Al KEY POINTS THROUGHOUT THE FLOw SYSTEM AKL AS TABULATLED HELOw. VAPOR CAVITY 1IST..0000.0640.1280.1920.2560.3200.3840.4419.5119.5759.6399.1039.7619.63i9.89',t LOC. 0 1 2 3 4 5 6 7 8 9 10 I 12 1, 14 U.000 -.058 -.117 -.177 -.236 -.295 -.352 -.408 -.462 -.513 -.562 -.b09 -.654 -.6)_ -.13i Z.039.039.039.039.039.0 39.040.u40.041.041.042.u43.044. 5.047 DIST..9599 1.0239 1.u879 1.15t9 1.2159 1.2799 1.3438 1.4u 8 1.4718 LOC. 15 16.17 18 19 20 21 22 23 U -.764 -.788 -.803 -.811 -.812 -.808 -.800 -. 189 -.1777 Z.048.049.051.052.053.053.054.054.L65 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (Z,50) V -1.0393 -1.0394 -1.0359 -1.0409 -1.0790 -1.11l9 -1.1155 -1.1153 H 11.3000 11.3828 11.3828 14.8199 21.9025 29.0082 31.4830 3. 1790 AT TIME T = 2.65178 SECONDS, AFTER 1429 COMPLETE CYCLES OF OPERAIION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABIULATED BELOw. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.639).7 39. 7679.8319.8959 LOC. 0 1 2 3. 4 5 6 7 8 9 IC 11 12 1j 1 4 U.000 -.059 -.119 -.180 -.241 -.301 -.360 -.417 -.472 -.525 -.576 -.625 -.671 -.71i -.149 Z.040.040.040.040.040.040.040.041.041.042.043.044.045.0'b.j41 DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 1.4678 1.4718 LOC. 15 16 17 18 19 20 21 22 23 U -.778 -.799 -.811 -.815 -.814 -.809 -.800 -.789 -.776 Z.049.050.051.052.053.053.054.054.055 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50)

v -I.C453 - 1. 44 -L.04 i3 -1.ul8b -1.114c -1.1192 -1.1177 -1.1171 H I1. 3CJ0 1 1. 69/ l.I.b9 ~.4353 27.1362 29.3761 ~ 1.2J15.3.7AT TIME T = 2.67034 j0(WfJA0,, AFTOR 14t39 COMPLETI CYCLE. OF OF C0RAICI 4 Itil V:-LOCITICS 4:4 0)EPItin IN THE V4PC0- Lj'-VITY.' > ELL ItS VELOCITIES AND PRESSU<tES AT KEY POIATS THiROUGH1OUTl THE FL,_J $YSlTUM ARE AS TAUlJLAT_ FkLBvLOW. VAPUR CAVITY U1ST..T OO.0640.1280.1.920 2.SC 2 3jy,4o 1.- 9 1.5 759.6,39v.u39.7679 3.$O1) LOC. 0 1 2 3 4 0 7 9 9 lo 11 12 1' 1 U 000 -.jo -.12 2 -.184 -.246.3C( I4 -.2.3 -. 38 -591 -41 -.668 -l-.o L 4.040.C40 04C G40.040 41.641.41.J42.042.(043.J44.045.I UIST..9599 1.0239 1 0 879 1. 159 1.2159 1. 749 I' 1to 1.3It 3'LOC. 15 16 I 18 1 9 2 k; 21 zz O -.790 -.807 -.8A-6 -.81 -16 -.u -.799 -.188 7.049.050.0 51.02.053.303.C54, FULL-FLOWING PIPE SYSTEM LOC. (II() (1,1) (20I) Io) (21) (2,33) K (2,4C) ( ) V -1.1135 -1.1143 -1.1-1.1161 -1.1186 -1.1194 -1.1203 H 11.3000 12.9582 12.9i. 82 25.3032 27.Qe36 29.4o64.31.1C96 3. 11 AT TIME T = 2.68890 SECONOZ, AFTER 1449 GiMPLL It CYCLES, OF tJPEv, TIU.~, THL VLLOC II[LS ANJD DEPTHJ IN THE VAPOR CAVITY 5S WELL As VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT ThE Fi-io SYS1L-7:1 A.A( A) TAFULAILU BELOWt. VAPOR CAVITY DIST..0000.0640.1280.1920.2566.3200.3d40.4479.519.5759.639).7039.7679.831.9.89 LOC. 0 1 2 3 4 7 9 9 10 11 12 II I U.000 -.062 -.124 -.188 -.251 -.314 -37o _.436 -.495 -.552 -.607 -.658 -.705 -14c -.777 7.041.041.041.041.641.041.041.642.042.043.044.045.046'7 j47 DIST..9599 1.0239 1.j879 1.1519 1.2159 1.2799 1.3438 1.4C78 LOC. [5 lb 17 18 19 20 21 z2 U -.800 -.814 -.820 -.821 -.816 -.809 -.798 -.786 7.050.051.051.052.053.053.654.654 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) 12,0) (2,10) (2,10) (2,30) (2I40) (2,50) V -1.1935 -1.1918 -1.1818 -1.1503 -1.1213 -1.1197 -1.1218 -1.1239 H 11.3000 11.9708 11.9708 21.6408 21.5661 29.6221 31.3970 33.1790 AT TIME T = 2.70745 SECONDs, AFTER 1459 COMPLETE CYCLEs OF OPERATIliN, ImE VELOCITIES ANU DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES Al KEY POINTS THROUGHOUT THE FLOW SYSTEM ARt AS YAHULAT6D 8LOw. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.15119.5759.6399.7039.7679.8319.8959 LOG. 0 1 2 3 4 5 6 7 a 9 10 11 12 13 14 U.000 -.063 -.127 -.192 -.257 -.321 -.385 -.447 -.!07 -.566 -.623 -.675 -.722 -.750 -.789 ~.041.041.041.042.042.042.042.43.43.044.045.046.047.8.049 DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 1.3438 LOL. 15 16 17 18. 19 20 21 U -.808 -.819 -.823 -.822 -.816 -.808 -.797 L.050.051.052.052.053.053.054

FULL-FLOWING PIPE SYSTFM LOG. (1,C) I) (2,0) (2,10) (2,2zi) (Z,3C0) (2,40) {(,5) V -1.2040 -1.2 39 -1.1998 -1.1922 -1.1512 -.1239 -1.1233 -I3.123 H 11.3000 11.4340 11.4340 15.573) 2:.4037 29.5156 51.731'.i 1T' AT TIME T = 2.12601 SECOND3, AFTER 1469 COMPLETE CYCLES OF ()PERAfIIUJ, rEil VtLOCITIES AND DEPPItiS IN Hlt- VAPtRR CA\VIY'bs LL v A VELOCITIES AND PR'SSUJRCS AT KEY POINTS THROUGHOUT THE FLOW SYSTEP. ARE AS TABULAIED RBLOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5159.7639.76(19.3 lV. e8959 LOC. 0 1 2 3 4 5 6 7 8 9 lu 11 12 I Is, U.000 -.365 -.130 -.196 -.262 -.328 -.394 -.458 -.. 1 -.581 -. 39 -.692 -.737 -.7 i -. 718 Z.042.042.042.042.042.042.043.u43. 44.045.C45.C46.048.C'. DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2199 1.3438 LOC. 15 16 1 1 19 2.0 21 U -.815 -.823 -.825 -.822 -.816 -.807 -. 196 Z.050.051.052.052.053.053.054 FULL-FLOWING PIPE SYSTEM LOC. (1,O) (1,1) (2,0) (2,11G) (/,) (12,30) (2,40) (2,5C) V -1.2046 -1.2046 -1.2005 -1.200. -1.1943 -1.1541 -1.1255 -1.1221 H 11.3000 11.4117 11.4117 14.4963 11.j449 25.5356 il.3413 5 1.79.: AT TIME T = 2.74457 SECONDS) AFTER 1479 COMPLEFE CYCLES OF OPERATIU.4, THE VELOCIIIES AND DEPTHS IN THEF VAPOR CAVITY 5S IWEL AL VELOCITIES AND PRESSURES AT KEY POINTS TR1KOUGHOUT THE FLOw SYSTEMP ARE AS TAIULAILD BiLOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3843.4'79.5119.5159.6399.7039.1679.8319. 3'9 LOC. 0 1 2 3 4 5 6 7 8 9 1l 11 12 1. 14 U.000 -.066 -.133 -.201 -.268 -.336 -.403 -.470 -.534 -.597 -.655 -.707 -.75C -. 7.1 -.830 Z.043.043.043.043.043.043.043.044., 45-.045.046.'47.048. (o9.uO DIST..9599 1.0239 1.0879 1.1519 1.2159 1.2799 LOC. 15 16 17 18 19 20 U -.820 -.826 -.826 -.822 -.815 -.805 Z.051.051.052.052.053.053 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (230} (20) (2,40) (L,50) V -1.2055 -1.2055 -1.2014 -1.2026 -1.2039 -1.1956 -1.1539 -1.1275 H 11. 3000 11.4288 11.4288 14.6608 16.6484 19.4132 21.0511 33.1790 AT TIME T = 2.76312 SECONDS, AFTER 1489 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPIHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOw. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOG. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000 -.068 -.136 -.206 -.275 -.344 -.414 -.482 -.549 -.613 -,671 -.722 -.762 -.792 -.812 Z.043.043.043.043.044.044.044.045.045.046.047.048.049.050.056 DIST..9599 1.0239 1.0879 1.1519 1.2159 LOC. 15 16 17 18 19

U -.bki -.h27 -.SLO -.821 -.813 L.1.cs.C2.0,2.Cs3 FULL-FLOWING PIPE S YSLI LO. (1,0) (1,1) (2,0) (I,1o) (2, ) (2,30) (2,40) (,5C) v -1 Ui33 -1.2083 -I.2&'2 -1..._.7 -Il.;39 -132029 -1.1173 -1.1847 H 11. 3000; 1 1. q 6' 7 1 1. 4657 L4.851~ 7C b1. 96 18.1865 21.3343 3.3.L 79C AT FIME T = 2.b616b SECiONDs, AFIER 1499 COiMPLEI[ CYCLCS, F!,bV:PE-0IWLj., iE,1i VtiOCITIES AND DEPTHS IN THE VAPOR C4VITv AS w!ELL 4 VELOCITIES AND PRESSURES AT KEY POINTS Ti;ROUGHOUI THF FLi;v SYS[Eo A.-'- A: lAHI AtLAftD BELOW. VAPOR CAVIIY U1ST..0000.0640.1280.1920.2560.3~Lj.394J.44 7~.i19.5759.6399.1039.7679.819.) LOC. 0 1 2 3 4 6 7 9 to 11 12 Ii 14 U.000 -.070 -.140 -.211 -.282 -.3~3 -.415 -.IJ -.264 -.628 -.686 -.735 -.773 -.80 - 0 ~.044.044.j44.044.044.045 4A5.u40.6.047.048.049.049 DIST..9599 1.0239 1.879 1.1519 1.2159 LOC. 15 16 17 18 19 U -.825 -.828 -.825 -.820 -.811 1.051.052 052.052.053 FULL-FLOWING PIPE SYSTEM 10C. (I1l) (1,1) (20) (L,1J) (2,u) (2930) (2,4C) (2,SC) V -1.2121 -1.2119 -1. 209 -1.205o -1.Z37 -1.2056 -1.2335 -1.266i H 11.3000 11.4361 11.4361 14.6131 16.3961 18.4501 24.3569 33.1792 AT TIME T = 2.80024 SECONDS, AFTER 1509 COMPLETE CYCLEs OF OPFERATIJi, THE VCLOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL As VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTlE AaI AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.32O.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 b 7 8 9 10 11 12 U.000 -.072 -.144 -.217 -.289 -.363 -.,437 -.509 -.519 -.643 -.700 -.746 -.781 -.8-5 -.dL Z.045.045.045.045.045.045.046.046.047.048.048.049.050 DIST..9599 1.0239 1.0879 1.1519 LOC. 15 16 17 18 U -.826 -.827 -.824 -.818 1.051.052.052.052 FULL-FLOWING PIPE SYSTEM LOC. (1,0) ( 1,) (2,0) (2,10) (2,20) (2930) (2,40) (2,50) V -1.2115 -1.2115 -1.2074 -1.2069 -1.2072 -1.2341 -1.2745 -1.281) H 11.3000 11.3880 11.3880 14.2556 16.5336 22.5453 30.2951 33.1791 AT TIME T = 2.81879 SECONDS, AFTER 1519 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8909 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000 -.074 -.148 -.223 -.297 -.313 -.449 -.523 -.594 -.657 -.712 -.755 -.787 -.809 -.821 L.045.045.046.046.046.046.047.047.048.049.049.050.050 0) 1

DIST..9599 1.0239 1.0879 1.1519 LUG. 15 16 17 18 U -.827 -.827 -.823 -.816 L.051.032.052.052 FULL-FLOWING PIPE SYSTEM LUC. (1,0) (1,1) (2,0) (2,10) (2,2() (2,30) (2,40) (2,50) V -1.2104 -1.2104 -1.2663 -1.2090 -1.2371 -1.275b -1.2823 -1.2824 H 11.3000 11.3939 11.3939 14.5853 20.3851 28.3168 31.3833 3j.1790 AT TIME T = Z.83735 SECONDS, AFTER 1529 COMPLETE CYCLES OF OPEKATIOui, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY.5 WLLL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM AL- AS TABULAILD BELOW. VAPOR CAVITY DI ST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7 () 39.7679.831.9.895 LOG. 0 1 2 3 4 5 6 7 8 9 10 11 12 I 4 I U.000 -.076 -.152 -.229 -.306 -.384 -.462 -.:37 -.638 -.670 -.722 63 -.792 -.811 -.622 7.046.046.046.047.047.047.C48 L48.049.349.050.050.051 DIST..9599 1.0239 1.0879 LOC. 15 16 17 U -.826 -.825 -.820 L.051.052.052 FULL-FLOWING PIPE SYSTEM LOG. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V -1.2129 -1.2135 -1.2094 -1.Z364 -1.774 -1.2852 -1.2637 -i.282 R) H 11.3000 11.5704 11.5704 18.7790 26.3465 29.1987 31.2601 3?.179Q AT TIME T = 2.85591 SECONDS, AFTER 153'9 COMPLETE CYCLES OF OPERATION, IHE VELOCITIES AND DEPiHS IN THE VAPOR CAVITY AS> WO-LL VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4419.5119.s h9.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U.000 -.078 -.157 -.236 -.315 -.395 -.475 -.551 -.621 -.682 -.731 -.769 -.795 -.813 -.822 1.047.047.047.047.048.048.049.049 050.050.050 u51.051 0.5 DIST..9599 1.0239 LOC. 15 16 U -.825 -.823 7.051.052 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,ZO) (2,30) (2,40) (2,50) V -1.2604 -1.2618 -1.2576 -1.2775 -1.2842 -1.2852 -1.2856 -1.285C0 H 11.3000 12.6650 12.6650 24.5675 27.5651 29.2671 31.0381 33.1790 AT TIME T = 2.87447 SECONDS, AFTER 1549 COMPLETE CYCLES OF OPERATION, THE VELOCITIES-AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 B 9 10 11 12 13 14

U.000 -.081 -.162 -.243 -.325 -.407 -.488 -.564 -.633 -.691 -.738 -.773 -.797 -.813 -.81 I.048.048.048.U48.049.049.~,5O.050,.050.051.051.051.051. 51.I 51 DIST..9599 1.0239 LOC. 15 10 U -.823 -.8i0 z.051.0152 FULL-FLOWING PIPE SYSTCM LUC. (1,0) (1,1) (2,0) (2,10) (2,2C) (Z,30) (2,40) (250) V -1.3466 -1.3448 -1.3403 -1.3051 -1.2853 -1.2847 -1.2865 -1.28it H 11.3G00 12.2553 12.2553 22.7221 27.485b Z9.4040 31.2278 331790, AT TIME T = 2.89302 SELON0-~, AFTER 1559 COMPLETE CYCLES OF OPERATION, THE VrELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A VELOCITIES AND0 PRESSURES Al KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TAB(JLATED BELLOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5159.6399.7039.7679.8319.8195 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 I4 U.000 -083 -.167 -.251 -.336 -.419 -.501 -.516 -.643 -.699 -.743 -.775 -.798 -.F8l2 -.89 1.049.049.049.049.050.050.051.u5I.051.051.051.u51.051.clI.I.5 DIST..9599 LOG. 15 U -.820 Z.051 FULL-FLOWING PIPE SYSTEM LOG. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (,0 V -1.3652 -1.3650 -1.3604 -1.3478 -1.3054 -1.2865 -1.2874 -1.2879 H 11.3000 11.4732 11.4732 16.4923 24.5791 29.4463 31.5843 33.1790 AT TIME T 2.91158 SECONDS, AFTER 1569 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AN1I DEPIHS IN [HE VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUF THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY 01ST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.1039.7679.8319.895 LOG. 0 1 2 3 4 5 6 7 8 9 10 II 12 1s 14 U.000 -.086 -.173 -.260 -.346 -.432 -.513 -..387 -.651 -.704 -.746 -.776 -.797 -.810 -.t~ Z.050.050.050.050.051.051.052.052.052.052.052.052.052 I11.5 FULL-FLOWING PIPE SYSTEM LOC.' (1,0) (1,1) (2,0) (2,P10) (2,20) (2,3C) (2,40) (250) V -1.3655 -1.3655 -1.3609 -1.3605 -1.3487 -1.3080 -1.2880 -1.2865 H 11.3000 11.4255 11.4255 14.6376 18.4959 26.7764 31.4396 33.1790 AT TIME T = 2.93014 SECONDS, AFTER 1579 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119 -.5759.6399.1039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1 U.000 -.090 -.179 -.269 -.357 -.443 -.524 -.596 -.658 -.708 -.747 -.176 -.796 -.8 —8 -.81 z.054.051.051.052.052.052.053.053.053.052.052.052.052.02 01

FULL-FLOWING PIPE'SYSTEM LOC. (1,C) (1,1) (2,0) (2,10) (2,20) (2,30) (2,4C) (2,50) V -1.3655 -1.3h55 -1.3609 -1.3618 -1.3629 -1.3499 -1.3070 -1.2880 H 11.3000 11.4372 11.4372 14.1094 16.8631 20.5323 28.4325 33.179U AT TIME T = 2.94869 SEION.D, AFTER 1539 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND OEPTHS IN THE VAPOR CAVITY S WELI. AS VELOCITIES AND PRESSURES AT KEY POINTS THROUnGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOw. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4419.5119.5759.6399.7039.7679.8319 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 12 1i U.COO -.393 -.186 -.Z78 -.368 -.454 -.533 -.603 -.662 -.710 -.747 -.775 -.793 -. 84 Z.052.052.052.053.053.053.054.053.053.053.052.052.052.052 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (Z,50) V -1.3670 -1. 671 -1.3b24 -1.3633 -1.3629 -1.3617 -1.3496 -1.3212 H 11.3000 11.4694 11.4694 14.9386 16.1486 18.5498 22.3570 3J.1790 AT TIME T = 2.96725 SECONDS AFTER 159) COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY A WELL AS VELOCITIES AND PRESSURES AT KEY POINTS IHKOUGHIUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.38J40.~4419.5119.5759.6399.7039.7679 LOC. 0 1 2 3 4 5 6 7 8 9 1u 11 12 U.000 -.097 -.193 -.287 -.378 -.464 -.541 -.b608 -.665 -.711 -.746 -.772 -.790 Z.054.054.054.054.054.054.054.054.054.053.053.052.052 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V -1.3702 -1.3701 -1.3655 -1.3636 -1.3620 -1.3627 -1.3816 -1.4107 H 11.3000 11.4652 11.4652 14.7863 16.6261 18.5165 23.3525 33.1790 AT lIME T = 2.98581 SECONDS, AFTER 1609 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039 LOC. 0 1 2 3 4 5 6 7 8 9 10 11 U.000 -.100 -.200 -.296 -.388 -.472 -.547 -.612 -.666 -.710 -.744 -.769 Z.055.055.055.055.056.056.055.055.054.054.053.052 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,101 (2,20) (2,30) (2,40) (2,50) V -1.3698 -1.3698 -1.3652 -1.3642 -1.3633 -1.3818 -1.4234 -1.435t H 11.3000 11.4037 11.4037 14.4329 16.6148 21.4132 29.4021 33.1790 AT TIME T = 3.00436 SECONDS, AFTER 1619 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039

LOC. 0 1 2 3 4 5 6 1 9 10 11 U.000 -.104 -.206 -.304 -.396 -.478 -.551 -.613 -.665 -.707 -.740 -.163 L.056.056.057 U057.057.057.056.u56.055.054.0~3 * FULL-FLOWING PIPE SYSIEM LOC. (1,0) (1,1) (2,0) (2,10) (2,21)) (2,30) (2,4C) e(lC) V -1.3680 -1.3680 -1.3634 -L.3649 -1.3839 -1.4237 -1.4355 -[.436 H 11.3000.11.4067 11.4067 14.510s 19.2061 27.3993 31.2485 i,.17 7 AT TIME T = 3.02292 SECONDS, AFIER 1629 COMPLETE CYCLES OF OPERATION, TIE VELOCITIES AND DEPTHS IN THE VAPljR CAVITY \s NELL L. VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED HCLOW. VAPOR CAVITY U1ST..0000.0640.1280.1920.2560.32O.3840.4479.5119.5759.6399 LOC. 0 1 2 3 4 5 6 1 6 9 10 U.000 -.107 -.212 -.311 -.402 -.482 -.552 -.c,12 -.662 -.703 -.734 A.058.058.058.058.058.058.057.56 C55.054.053 FULL-FLOWING PIPE SYSTEM LOC. (110) (1,1) (2,0) (2910) (2920) (2,30) (2,40) (2,50) V -1.3682 -1.3685 -1.3639 -1.3830 -1.4250 -1.4374 -1.4363 -1.4356 H 11.3000 11.5021 11.5021 17.4414 25.2543 29.0100 31.2125 33.1794 AT TIME T = 3.04148 SECONDS, AFTER 1639 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPIHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES Ar KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759 (' LOC. 0 1 2 3 4 5 6 7 8 9 U.000 -.111 -.217 -.317 -.406 -.484 -.551 -.609 -.658 -.696 7.060.060.060.059.059.059.058.057.056.055 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,11) (2,0) (2,10) (2920) (2930) (2,40) (2,50) V -1.3990 -1.4005 -1.3958 -1.4237 -1.4362 -1.4375 -1.4374 -1.4365 H 11.3000 12.3746 12.3746 23.4820 27.2115 29.0619 30.9842 33.1790 AT TIME T = 3.06003 SECONDS, AFTER 1649 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL As VELUCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119 LOC. 0 1 2 3 4 5 6 7 8 U.000 -.113 -.221 -.320 -.407 -.483 -.549 -.604 -.650 7.061.061.061.061.060.060.059.057.056 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2910) (2,20) (2,30) (2,40) (2,50) V -1.4827 -1.4814 -1.4764 -1.4487 -1.4362 -1.4363 -1.4378 -1.4392 H 11.3000 12.4905 12.4905 23.4025 27.2831 29.1846 31.0707 33.179s AT TIME T = 3.07859 SECONDS, AFTER 1659 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT K'EY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW.

VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479 LOC. 0 1 2 3 4. 5 6 7 U.000 -. 114 -.223 -.321 -.406 -.480 -.543 -.596 Z.063.063.063.062.062.060.059.U58 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (220,30) (2930) (2,40) (2,50) V -1.5125 -1.5120 -1.5069 -1.4885 -1.4487 -1.4365 -1.4381 -1.43'JO H 11.3000 11.5465 11.5465 17.6046 25.3887 29.2911 31.4192 33.179Y AT TIME T = 3.09715 SECONDS, AFTER 1669 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN IHE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUr THE FLOw SYSTEp ARC AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200 LOC. 0 1 2 3 4 5 U.000 -.114 -.222 -.318 -.402 -.474 Z.065.065.065.064.063.061 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (11,1) (2,0) (2,10) (2,30) (2,40) (2,50) V -1.5129 -1.5129 -1.5078 -1. 5066 -1.4885 -1.4504 -1.4371 -1.4369 H 11.3000 11.4427 11.4427 14.8464 19.6534 27.6347 31.4404 33.1790 AT TIME T = 3.11571 SECONDS, AFTER 1679 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPIHS IN THE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARKt AS TABULATED BELOw.. VAPOR CAVITY DI1ST..0000.0640.1280.1920 LOC. 0 1 2 3 U.000 -.112 -.218 -.313 Z.068.067.067.065 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) 12,10) (2,20) (2,30) (2,40) (2,50) V -1.5121 -1.5121 -1.5070 -1.5077 -1.5081 -1.4895 -1.4492 -1.4364 H 11.3000 11.4483 11.4483 14.7761 17.1268 21.8428 29.4501 33.1790 - THE VAPOR CAVITY HAS COMPLETELY COLLAPSED AT TIME T = 3.13426 SECONDS (OPERATION CYCLE M = 1689). THIS IS 2.93014 SECONDSAFTER COLUMN SEPARATION OCCURRED, AND 3.12870 SECONDS AFTER VALVE WAS CLOSED. LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (230) (2,3040) (2,50) V -1.5125 -1.5125 -1.5074 -1.5085 -1.5086 -1.5066 -1.4879 -1.4613 H 11.3000 11.4752 11.4752 15.0048 16.9719 18.9788 23.6648 33.1790 AT TIME T = 3.13612 SECONDS, AFTER 1690 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THRiOUHOUT TH FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2120)!2,30) (2,40) (2,50) V.0000 -1.5127 -1.5076 -1.5085 -1.5084 -1.5068 -1.4916 -1.4673 H 126.5298 11.4788 11.4788 15.0196 16.9606 18.9115 23.1907 33.1790

Ar TIME T = 3.15468 SECONDS, AFTER 1100 COMPLCTC CYCLES OF OPERATIGN, THE VELOCITIES AND PRESSURES AT &EY POINTS ThJRjU;HijJUT I FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,c) (1,1) (2,0) (2,101 (2,10) (2,30) (2,40) (9,5C) V.0000 -.0919 -.0)16 -1.5083 -1.5069 -1.5072 -1.5217 -1.5451 H 222.3961 220.4687 Zi0.4b8f 14,.9212 16.8446 L8.8396 23.1672 33. 174C AT TIME I = 3.11323 SECIINDS, AFTER 1710 COMPLLTE CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THk('UItOUI inH0 FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (290) (2,10) (2,20) (2,30) (2,40) (2,50) V.0000 -.0077 -.0C77 -.0939 -1.507U -1.5217 -1.5610 -(.5750 H 233.3888 233.2271 233.2211 222.9844 16.8012 21.0872 28.8351 3,.1790 AT TIME T = 3.19179 SECONDS, AFTER 1726 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THRN)UJHOUT 1W FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2,10) (2,30) (2,40) (2,50) V.0000 -.0011 -.0011 -.0103 -.1127 -1.5605 -1.5753 -1.5761 H 234.4422 234.4238 234.4238 235.8169 226.6305 26.7563 31.1035 33o1790 AT TIME T = 3.21035 SECONDS, AFTER 1730 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THR1UGHOUT 1T FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (i1,) (2)0) (2,10) (2920) (2,30) (2940) (2,50) V.0000 -.0011 -.0017 -.0196 -.C676 -.1704 -1.5756 -1.5750) H 234.8549 234.8720 234.8720 239.3619 245.1795 235.9953 31.1333 33.1790 AT TIME T = 3.22890 SECONDS, AFTER 1740 COMPLETE CYCLES OF OPERATION, [HE VELOCITIES AND PRESSURES AT KEY POINTS THROUSHOUT 1HZ FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2930) (2,40) (2,50) V.0000 -.0150 -.0150 -.0589 -.0770 -.0868 -.1145 -1.5751 H 237.6604 237.8206 237.8206 245.4985 248.1561 248.9576 237.4691 33.1740 -- THE DIRECTION OF FLOW AT THE RESERVOIR END OF THE PIPE SYSTEM HAS REVERSED Al TIME T = 3.23540 SECONDS. AT TIME T = 3.24746 SECONDS, AFTER 1750 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THkRUGHOUT tt FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2930) (2,40) (2,50) V.0000 -.0288 -.0287 -.0724 -.0781 -.0808 -.0904 1.2190 H 247.7477 247.1045 247.7045 248.4821 249.2789 250.2749 250.4436 33.1790 AT TIME T = 3.26602 SECONDS, AFTER 1760 COMPLETE CYCLE. OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THROUSHOUT 1W FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.0000 -.0104 -.0104 -.0479 -.0762 -.0816 1.3095 1.3883 H 255.5118 255.3711 255.3711 252.7516 249.9986 250.7663 46.5190 33.1790 AT TIME T = 3.28457 SECONDS, AFTER 1770 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT 1Hz FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2940) (2,50) V.0000 -.0021 -.0021 -.0143 -.0516 1.3108 1.3936 1.3998 H 257.3105 257.2807 257.2807 258.1508 254.2275 46.7309 34.0492 33.1790 AT TIME T = 3.30313 SECONDS, AFTER 1780 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THRCUIHOUT 1HZ

FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.0000 -.0012 -.0012 -.0057 1.3692 1.4201 1.4610 1.3989 H 257.8703 257.8616 257.8616 260.0297 55.3635 38.003Z 33.4494 33.1790 AT TIME T 3.32169 SECONDS, AfTER 1790 COMPLETE CYCLES OF OPERATION, THE VILOCITICS AMNU PRESSURS AT KEY POINTS THK-RiU.HOUT rFLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2,2C) (2,30) (2,4C) (2,50) V.0000 -.0016 -.0016 1.3787 1.4622 1.4589 1.4252 1.4022 H 258.4284 258.4250 258.4250 56.8706 44. 148 42.0606 37.1541 3i.179( AT TIME T = 3.32540 SECGNDS, AFTER 1792 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINIS ThrI)UOHOUT IIFLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1)'2,0) (2,10) {2, 20) {230) { 2,40) (t5 0) V.0000.9842.9809 1.4153 1.4652 1.4623 1.4330 l.4067 H 211.3071 183.4390 183.4390 51.4954 43.9692 42.5557 38.3408 53.1790 AT TIME T = 3.32725 SECONDS, AFTER 1793 COMPLE1E CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THR`ULHOUT H; FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2tO) (2,10) (2,2C) (230C) (2940) (2,50) V.0000 1.1050 1.1013 1.4276 1.4661 1.4635 1.4368 1.4098 H 158.0020 149.7719 149.7719 49.6803 43.8552 42.7476 38.9261 33.1790 AT TIME r = 3.32911 SECONDS, AFTER 1194 COMPLETE CYCLES OF OPERATIONI THE VELOCITIES AND PRESSURES AT KEY POINIS THROIUSrIUUT TiFLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1{C) (11i) (2,0)!2,10) (2,20) (2,30) (2,40) (2,50) -l V.0000 1.0954 1.0917 1.4372 1.4669 1.4646 1.4406 1.4138 H 110.1061 111.0738 111.0738 48.2775 43.7719 42.9110 39.4920 33.1790 AT TIME T = 3.3309: SECONDS, AFTER 1795 COMPLETE CYCLES OF OPERATION, THE VLLOCITIES AND PRESSURES AT KEY POINTS THROUHOUT TH FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V.0000 1.0256 1.0222 1.4445 1.4674 1.4653 1.4442 1.418t H 67.3567 72.5709 72.5709 47.1905 43.7079 43.0444 40.0207 33.1790 AT TIME T = 3.33282 SECONDS, AFTER 1796 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THt FLOW SYSTEM ARE AS TABULATED BELOW. LOC. (1,0) {1,1) (2,0) (2,10) (2, 20) (230) (2,40) (2,50) V.0000.9293.9262 1.4502 1.4678 1.4660 1.4477 1.4238 H 29.5638 36.6482 36.6482 46.3516 43.6630 43.157S 40.5059 33.1790 -pCOLUMN SEPARATION HAS OCCURRED AT THE GATE VALVE AT TIME T = 3.33468 SECONDS (OPERATION CYCLE NO. 1797). THIS IS 3.32911 SECONDS AFTER CLOSURE OF THE GATE VALVE. THE MAGNITUDE OF DELXC REMAINS THE MINIMUM VALUE. GIVEN DATA AND COMPUTED VALUES PERTINENT TO COMPUTATION OF FLOW FOLLOWING COLUMN SEPARATION. HVAPOR = 11.3000 DELXC =.0640 DELTC =.00928 MARK. = 1689 MUCH =.1450 FACTOR = 5 XI =.0250 INC = 90 IOTA =.0001 PAREA =.00538 THE INITIAL VELOCITIES AND DEPTHS IN THE VAPOR CAVITY ARE LISTED BELOW.

UIST..0000.0640 LOC. 0 1 U.000 Ih)7 L.063.0(14 THE VELOCITIES ANGU AbSOLUTE _HCAbs iHROLHITUUT lti;: R -MAI'4iJER OF THE FULL-FLO;NING PIPC 4YsIFM ARE As LISTiZ EDEIJL(W. LOC. I,) (11) (2,0) (2, 1) (zLo) 12,3c) (2,4L V.7 5 6 o.9it53.9421 1.4546 1.468& 1.4t64 1.4 0 H 11.300U 22.e504 22.25)4 45.7016 43.6286 43.2463 (,0.319 3:. _('1 AT FIME T 3 i.342L0 SECUNDs, AFTEk 1801l COMPLETE CYCLES (IF OPERATIUN, I HE VEI' OCITIES AND DEPTHS IN TH1L VAPiOiR CAVITY sb rL" 5 VELUCITIES AND PRESSURES AT KEY -'Of IS TtiROUGHOUT THE FLnw SYSTEts AiE AS TABUiLATED rELOR. VAPOR CAVITY 01ST..0000.06,40 LOG. 0 1 U.000.589 1.058.C68 FULL-FLOWING PIPE SYSIEM LOG. (1,u) (1,0) (2,0) (U710) (2,2c) (2,3c) (,4). V.9856 1.0475 1.0440 1.6061 1.4683 I.467L 1.4626 1.45' H 11.3000 [I.3000 11.3)00 65. 1 7o7 43.5751 43.422 41.9 2 C9 11 AT TIME T = 3.35138 SFCUN[)s, AFTEk 1896 COMPLErE CYCLES OF OPERA! IUir THE VELOCITIES AND DLPl)H4 iN IHL VAPOIR CAVITY * wELL L VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT ThiE FLOW SYSTEM ARL AS TABULAIO BCLLOvi. VAPOR CAVITY D1ST..0000.0640.1280 LOC. 0 1 2 U 0CO0.468.9u6 7.056 C062.071 FULL-FLOWING PIPE SYSTEM LOG. (110) 11,1) 12,0) 12,10) (2I20) 12,30) (2,40) (2,50) V 1.G693 1.797 1.0761 1.2425 1.4680 1.4675 1.4777 1.4963 H 11.3000 11.3000 11.3000 11.3000 43.5981 4 3. 55s19 41.2864 33.1 79s AT TIME T = 3.36066 SECONDS, AFTER 1811 COMPLEJE CYCLES OF OPERATIUN, THE VLELOCLTIES AND DEPlm!S IN THE VAPOR CAVITY Az WELL AS VELOCITIES AND PRESSURES AT KEY POINIS THROUGHOUT THE FLOW SYSTEM ARE AS FAHULAIED BELOW. VAPOR CAVITY 01ST..0000.0640.1280 LOC. 0 1 2 U.000.400.791 L.053.058.066 FULL-FLOWING PIPE SYSTEM LOC. (1,0) 11,1) (2,0) (2,10) (2,20) (2,3.0) (2,40) (U,15) V 1.1003 1.1181 1.1143 1.2398 1.6081 1.4701 1.4969 1. 526"' i H 11.3000 11.3000 11.3000 11.3000 64.3438 43.1690 38.9629 33.1190 AT TIME T = 3.36994 SECONDS, AFTER 1816 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPIHS IN TH1 VAPOR CAVITY As WELL As

VELOCITIES AND PKESSURES AT KEY PUINTS IHI-OUGHTUT THE FLOIW SYSIEM ARt AS TABULAIED bEL0W. VAPOR CAVITY DIST..0000.0640.1280.1920 LOC. 0 1 2 j U.cGO.349.694.965 L.051.055.Cb2.072 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (11l) (2,0) (2,11) (2,0) (2,30) (2,4.;) (,50) V 1.1277 1.1310 1.1272 1.2327 1.2479 1.4797 1.5158 1.J 12 H 11.300 11. 300C 11.3000 L1.300C 11.4358 41.7385 36.3', 0. 3. 7 AT TIME T = 3.38849 SECONDS, AFTiR 1826 COMPLETF CYCLES OF OPERATIGN, TlE VLLOCITICS AND DEPTHS IN THE VAP'(J CAVITY.s WFLL o VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT TtIE FLOW SYSTEM ARE AS TABULATED BLLOo,. VAPOR CAVITY DIST..CO000.0640.1280.1920.256C LOC. 0 1. 2 3 4 U.000.276.555.799 1.000 Z.048.050.056.063.074 FULL-FLOWING PIPE SYSTEM LOC. (1,G) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,5C) V 1.1623 1.1638 1.1599 1.2277 1.2503 1.3299 1.5329 1.5352 1 H 11.3000 11.3000 11.3000 11.3000 11.JOOC 11.3C00 53.7061 3. 1790 \Jn AT TIME T =3.40705 SECUNDS AFTLR 1836 COMPLETE CYCLES OF OPEKATI-U.N THE VELOCITIES AND DEPTHS IN [HE VAPOR CAVITY AS WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARL AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560 LOC. 0 1 2 3 4 U.000.227.458.670.849 Z.046.C47.052.058.065 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2t30) (2,40) (2,50) V 1.1873 1.1881 1.1840 1.2268 1.2542 1.3283: 1.3836 1,5346 H 11.3000 11.3000 11.3000 11.3000 11.3000 11.3000 11.3000 33.1790 AT TIME T = 3.42561 SECGNDS, AFTER 1846 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY'S WELL AS VELOCITIES AND. PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200 LOC'. 0 1 2 3 4 5 U.000.191.385.570.741.951 Z.044.045.049.054.060.065 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (210) (2,10) (2,20) (2,30) (2,40) (2,50)

L. 2C60 1.2063 1.2022 1.2289 1.2,;79 1. 32 12 1.3b.44' H 11.31G 1 1.3000 11. 300 11.30;3 113.0O 11.3)00 11.300.s.l. 79v AT TIME I = 3.44416 SE"()NOs, AFUEK 1850 COMPLEFE CYCLES OF OPERATIu J, THE VFLUCIrIES AND DEPTiiS I' ThE iAPiOR CVLTY Y wFLI VELOCITIES AND I'PESSURES AT KEY POINT [tHROUGHOUT THE FLOW SYSTEM AeKL AS TABULATED B1LOW. VAPOR CAVI fY DIST..0000.C640.1280.1920.2560.3200.3840 LOC. 0 t1 2 3 4 b U.000.163.328.491.654.836.977 z.043.044.046.051.056.060.008 FULL-FLOWING PIPE SYSTEM LO. (11) (1) ) ( ) () (2,30) (10) (2) (23) (,40) (2,5C) V 1. 220 1.2202 1.2161 1.2351 1.2612 1.3265 1. 33 8 i.Z5 H 11.3000 11.300 11.3600 11.300C 11.300C 11. 30 17.1039 33.17 95 AT TIME 1 = 3.46272 SECONDS, AFTER 1866 COMPLETE CYCLES OF CPERATIOL, THE VLLOLITIES ANt DEPIHS IN IHE VAPOR LAVITY'S W-LL tS VELOCITIES AND PRESSU(JES AT KEY POI7NTS THRUUGHOUT TliHE FLOW SYSTEM AKc AS TABULATED BELOW. VAPOR CAVITY )IST..0000.0640.1280.19zc.2560.3200.3840.44 79 LOC. 0 1 2 3 4 5 O 1 U.OCO.140.282.427.579.741.870 1.u',6 Z.041.042.045.048.053.057.063.u08 FULL-FLOWING PIPE SYSIM LOC. (1,C) (1,1 ) (2,0) (Z110) (I,2u) (2,301 (2,4C) (2,50) V 1.308 1.2310 1.2268 1.243 1.2641 1. 3091 1.2306 1.229 H 11.3000 11.3000 11.30000 11.5296 11.3U06 12.7346 29.4578 3.17190 AT TIME r = 3.48128 SEC4DUS, AFTER 1876 COMPLETE CYCLES OF OPERATIiON IHE VLLOCITICS AND DEPTiS IN THE VAPOR CAVITY AS WELL A= VELOCITIES AND PRESSURES AT KLY POINTS THIIROUGH'IUT lTHE FLIW S'YSTEMN AKL AS TABULATED BELUr. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119 LOC. 0 1 2 3 4 5 6 7 8 U.CO0.121.245.375.515.61L.789.95t 1.077 Z.040.C41.043.046.050.054.059.u63.u73 FULL-FLOWING PIPE SYSIEM LOC. (1,C) ( 11) (2,0) (2,1O) (2,20) (2,30) (2,40) (2,50) V 1.2395 1.2398 1. 2 56 1.2511 1.2669 1. 1998 1.2007 1.2062 H 11.3000 11. 3000 11. 3o00 11.6535 11.3000 24.7443 28.8404 3.lr190 AT TIME T = 3.49983 SECONDS, AFTER 1886 COMPLtTE CYCLEs OF OPERATIUN, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY.-S WELL A VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUI THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4419.5119 LOC. 0 1 2 3 4 5 6 7 8 U.000.106.215.332.460.593.718.8i5.971 Z.039.040.042.045.048.051.056.s59.065

FULL-FLOWINGG PIPEt SYSI.Ml LOC. ( 1, ) ( 1) (2,0) (2,10 ) (2,20 (,) (,4 ) (, V 1.2'2, 7 1.2474 1. 2432 1.2581 1.182o 1.115 1. 15 1. 172, H 11. 30.00 1. 000 11. 3000 11. 7124 22.C072 4. 925 9'8.5106.. 1 7. AT FIME r = 3.51839'EiUNDU, AFTER 185'6 COMPLETF CYCLES OF O(PERATIO.i, TiHE VELOCITIFS AND DEPTI-1i IN rTHL VAPOR CAVITY'IS CLL:VELOCITIES AND PRESSURJES Al KEY POINTS THROUGHtOUT THEI- FLOW SYS1EM ARE AS TAHULATCI BELO,. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.)759 LOC. 0 1 2 3 4 5 6 8 9 U.000.092.190.296.413.534.655.795.892 1.021 Z.039.039.041.043.046.049.53.026.061.065 FULL-FLOWING PIPE SYSITEM LOC. (110) (1,1) (2,0) (2,t10) (2,0) (2,30) (2,40) (, 5J) V 1.2540 1.2545 1.25C2 1.1896 1.1734 1.l158 1.1475.1l45$ H 11.300 11.300() 11.300i 22.8171 24.3702 25.7683 29.296{) 0j, L T9I AT TIME T = 3.5369t' SEC(iND., AFTER 1906 COMPLETE CYCLE. O)F OPERAIIGN, THE VtLOCITIES AND DEPI-IS IN THE VAPOR CAVITY' WELL;: VELOCITIES AND PRESSURES AT KEY POINTS HKROUGHOUF THE FLO)w SYSTEM ARc AS TABULAIED 6ELOw. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4419.5119.5759.6399 LOC. 0 1 2 3 4 5 6 7 e 9 1 U.000.0:I1.169.265.372.484.6OC.71/.822.941 1.100 Z.038.039.040.042.045.047.051. u54.058.062.012'J, FULL-FLOWING PIPE SYSFEM LOC. (1,1) (1,1) (2,0) (2t10) (2,20) (2,3C) (2,40) (2,50) V 1.1254 1.1317 1.1278 1.1732 1.1657 1.1454 1.1285 1.1221 H 11[.;000 12.2281 12.2 U1 24.1137 26.5588 28.7198 30.4703 3. 1790 AT TIME T = 3.555.C SECOINUS, AFTER 1910 COMPLETJ' CYCLLS OF OPERATION, IHE VtLOCITIES AND OEPTIl-j IN THtE VAPOR CAVITY.AS WELL A, VELOCITIES AND PRESSURES AT KEY POINTS TttROUGHOIUT TrtE FLOhw SYSTEM ARL AS TABULATED BELOW. VAPOR CAVITY UIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5159.6399 LOC. 0 1 2 3 4 5 6 1 8 9 10 U.000.072.151.238.337.441.550.o64.759.867.977 Z.038.038.039.041.043.046.049.052.056.059.065 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,10) (2,20) (2,30) (2,40) (2,50) V 1.1010 1.1012 1.0975 1.1043 1.1454 1.1356 1.1207 1.1118 H 11.3000 11.4200 11.4200 17.2708 28.4421 31.2384 32.6376 33.17 0 AT TIME T = 3.57406 SECONDS, AFTER 1926 COMPLETE CYCLES OF OPERATION\i THE VELOCITIES AND DEPITHS IN THE VAPOR CAVITY As WELL AS VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4479.5119.5759.6399.7039

LOC. C. 1 2 3 4 5 6.1 8 9 1 ~1 UI.000.,b4.13 5.216.3C6;432.5U5.610,.7C11.8 L2.08t7. 9 3 I, *3 7..,3 7.)338.040.042 *04 4.04 7.0DO.054.057.;~02. b 7 FULL-FLOWING 1'1P-E SYSV'.M LOC. (1,cI) ( 11) (12)(21) (2, L)2 12) (23.)(,4) V I.08,006 1. o04 1.0 767 L. 1 01o 1.(.746 1. 120,71113.16 H L1I. 3 o 0- 1'1. 609 8 11L.60-)8 17.0467 21I.96)98 3Z.35zl 3 3. 9754.331 70: AT TIME T 3.592oZ SiCOONOS, AFTL, 191c0 COMPLErt- CYCLES OF OPEK{ATION,41 THE VitLOCITIES ANI) DEPIIHS IN THE VAPOR C'.VITY v LLL VELLUCITIL-S AND) VI't_1CSURUS AT KEY POJIN'TS 1HROUGHOUT TviE FLOW SYSTEM ARC_ AS IAIU)LATOI)8COD VAPOR~ CAVI[Y D1ST..0C00 J.064 0.12bo.1920.2560.32 C0.3840.4479.5119. Ii7,19.6 39)4 7139.7679 L0C. 0 1 2 3 4 5 6 7 8 9 1 1 1 12 U.000.058.122 19o.279.369.465.t.61.649.742.51.,.t b9.690 1.031.0 37.038.03)4.041.043.046.048 L152.055. 659.364.014 FULL-FLOWING PIPE SYSTEM LOC. I1,0) (1,1) (2,0) (2,10) (2,I2 (2,93 )1 (294%;) (:-,5 V 1.V%523 1.0494 1.0-,4 58 1. 4 72 1..',45 8, 1. 05o2 1. I Ibo 1.126 H 1 1.3(00u 11.ti259 I1I. 8239 17.58Z6 20.;1693 Z4.7340 32.9.348 j, AT ([ME T= 3. u111 3:SEL(JP!iS, AFTER' 1946 LOMIPLEIF CYCLES (iF OPERATILY-l, THE VELOCIIIES AND) DEI'Ih~ 14 THE VAP~jR CAVITY -,lS wkdLL VCLLICITIES AN0 PRO:!SSOCES AT KtY POIN1S fi-,ROUGoHO0)T THEO FLOW SYSTEM AKE AS> TAL'uLATi-O HCLOw. VAPOR CAVITY DIST..0000.3640o.1260.1920.2560.32'oO.3840.4479.5119.olV.399.7139.7679Q LOG. C 1 2 4 5 6 1 8 9 1 0 IL 12 L).0C). C 2.111.1).255.33).428. 1M.601L.686.149.786.824 L.036.. 7.7.03).04.04Z.044.4 7 30.L53.0.07.o~3.066 FULL-FLOWiNG PIPE SYSlON, LOC. (1,o) (11)(23 (2,1) 2,20; (2,30) (2,4U) (,5 0, V 1.L219 I.1o2 1 3 1.0178 1.021o 1.0~31;) 1.L441 1.0655 I.113. H 1 1.31,C, 1I. 5)16 11.5916.17.0528 20.350b 21.6104 24.0208 3i179. AT TIME T =3.629)73 330 AFTER 1')06 COMPLE'TE CYCLES OF OPERATIi;4, TH:E VELOCITIES A,,I DEPTHS IN4 THE VAPOR CAVITY A~S ELL1 \ VELOCITIES A~u PKESSUK6ES Ai' KOY PO1.45 IHROUGHiOUT THE FLOW SYSTC8~ A.{6 AS ['AbULAILI) I3CLOW. VAPOR CAVITY UIST..5000.6 40.1.260.1920.2560.3200.3840.4'v19.5119.5759.6399.7; 339.7679.63. 9 LOC. 3 1 24 0 6 1 8 9 Jo II 12 O.0(.4 7.101.163.234.312.395.478.5!57.634.689. 130.781.95 Z.036.AO.03 7.039.039.041.043.64 6.348.051.0s6.,6I.064.>. FULL-FLOWiNG PIPE SYSTEM LOG. (1,0) (1ll) (2,0) (2,.10) (2,20) (2,30) (2,40) SO,5) V.9986.s~988.9954 1.001it 1.0201 1.0384 1.0444 1.03,55 H 11.3000 11.5046 11.5046 15.6446 17.6866 19.6543 21.9354 31) AT TIME T = 3.64829 SECONDs,- AFTER 1966 COMPLETE CYCLES OF OPERATION, THE VELOCITIES AND DEPTHS IN THE VAPOR CAVITY A WELL A VELOCITICS AND PkESSURES AT KEY POINTS rHROUGHOIJI THE FLOW SYSTEM ARE AS TABULATED BELOWN.

VAPOR CAV ITY LIST..0OO0.0640.1280.1920.2560.3,,00.3840.4, 79.s119.5159.63V9.7.39.7679. 6i i.,J LOC. 0 1 2 3 4 5 o 7 8' ) I i; 12 U.COO.043.C92.150.216.z88.365.442.516.586.o3.O?7.74q.. t,.J Z.CI0.C30.336.63!..039.040.042. z44.4 t 7.00.C t..r,.2.i,5. FULL-FLOWING PIPE SYSIEN LOC. (1,0) (1,1) (2,0),1) (2,Zi) (2,3 C) (,') 2,'i ) V.9885.9876.9843.993') 1.t(92 1.0205.9180.97.: H 11.3060 11. 3625 11.3625 13.4260 14.ml?) t.7 1.02 3 2 13. 315 31. 179 AT TIME = 3.66685 SEC (J ND.S, AFTER 1976 COMPLETE CYCLES OF O.JPERATIOiU, lHL V[LOCITIES AND DEPTH,') IN THE VAPOr< CAVITY:s wtLL t VELOCITIES AND) PRESSURES AT KEY POINT1S THROUGHOUF THE FL(OIW SYSTEM ALf A: IABULAI':D BELOW. VAPOR CAVITY LIST..C'OCO..o40.1280.1920.25 60.. 200.i840.4479.5119.5159.6399.-3.1.7679.8.,.: 9'J LOL. C 1 2 3 4 5 6 7 8 9 1 i l 12 1. U.000.0'..085.138.199.266.338.409.478.54C.585 3, 30.697.7,9.7 3 z.035.3o.036.037.038.040.C41.43.- 46.049.03.u51.8o0.(,3. oS FULL-FLOWING PIPE SYS-TE, LOC. (1,0) (1 1) 2,0) (12,lO) (2,20) (2,3 0) (2,45) (l5,5) V.9882. 94 81.8'47.9933.9945.9498. 946'9.9519 H 11.3000 11 3C.306 11.3066 12.2039 13.7745 24.1265 29.2893 33.17'90 AT TIME T = 3.68540 SECONs,. AFIER 1986 COMPLETE CYCLE:. OF OPERATIO"-, THE VFLOCITI~S AND DEPTH, IN TIHE VAPOR CAVITY.'S wELI. AS 0 VELOCITIES AND PRESSURES AT KEY POIN4TS THROUGHOUT THE FLOw SYSTEM AkE AS TABULAILUD 3ELOw. VAPOR CAV I TY DIST1..0000.0640.1280.1920.2560.i200.3840.44 9.5119.5759.6599.7o3%9.7619.8319.8959: LOC. 0 1 2 3 4 5 6 1 83 9 16, 11 12 I 14 U.000.037.079.128.185.247.313.319.442.498.539.587.655.7'9.873 Z.035.035.036.036.038.039).G41.I 4 3.'045.048.C t2. C 55.058. C,1. 64 DIST..9599 LOC. 15 U.991 Z.075 FULL-FLOWING PIPE SYSTEM LOC. (1,0) (1,1) (2,0) (2,0 ) (2,2) (2,30) (2,40) (4,5(:) V.9917.9920..9886.9927.9542.9212.9232.9235 H 11.3000 11.3000 11.3000 13.0802 21.3382 25.0180 23.5210 33.1793 AT TIME T = 3.7396 SECONDS AFTER 1996 COMPLETE CYCLES OF OPERAIIUN, TlE VCLOLITIES AND DEPliS IN THE VAPOR CAVITY AS WELL VELOCITIES AND PRESSURES AT KEY POINTS THROUGHOUT THE FLOW SYSTEM ARE AS TABULATED BELOW. VAPOR CAVITY DIST..0000.0640.1280.1920.2560.3200.3840.4419.5119.5159.6399.7039.7679.8319.8959 LOC. 0 1 2 3 4 5 6 7 8 9 16 11 12 13 14 U.000.034.073.118.171.229.290.:351.408.458.497.546.614.885.154 Z.035.035.035.036.037.038.040.042.044.047.051.054.057.059.062

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