THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINIERING VALVE STROKING TO CONTROL TRANSIENT FLOWS IN LIQUID PIPING SYSTEMS Thomas P. Propson 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 September, 1970 IP-834

To my wire and parents

ACKNOWMLEDGMNTS The author would liie to express his gratitude to Professor Victor L. Streeter, Chairman of the committee, for his suggestion of this area of research and his guidance and inspiration throughout the author's graduate studies; to Professor E. Benjamin Wylie for his deep interest and the many valuable suggestions provided in numerous discussions of this topic; to Professor Ernest F. Brater for his generous counsel and unfailing encouragement throughout the years; and to Professors Walter R. Debler and Robert B. Keller for their interest and cooperation throughout the course of study. The author was exceedingly fortunate to have at his disposal an excellent laboratory system that enabled him to obtain convincing experimental verification of some of the new concepts presented in this investigation. While several individuals assisted with the development and construction of the experimental system, especial thanKs are due Mr. Joel L. Caves, a former graduate student at the University. Mr. Caves was primarily responsible for improving the original design and for effecting numerous modifications in the instrumentation and operation of the system, and the author acknowledges his indebtedness to him. Without the use of the excellent facilities and services available at the Computing Center, this study would not have been possible. The author is particularly grateful to The University of Michigan for the use of these services and for the generous donation of computing time. During part of his graduate studies the author was the recipient of a Ford Foundation Engineering Faculty Development Loan and wishes to express his gratitude for that support. The author is also indebted to the iii

National Science Foundation for financial support during the latter stages of this investigation and for the occasional provision of computing time. For assistance in the preparation of the manuscript, the author is indebted to Mrs. Jeanette Dennis of the South Dakota School of Mines and Technology. Appreciation is also due the staff of the Industry Program of the College of Engineering, The University of Michigan, for final preparation and publication of the manuscript. iv

TABLE OF CONTENTS Page ACKNOVI EDGEISTS............... *. *. iii LIST OF FIGURES................,......... viii LIST OF TABLES...........*.............. xii LIST OF PLATES..................... xii NOMENCIAE........................ xii I INTRODUCTION........................ 1 1.1 Literature Review................... 1 1.2 Scope of Investigation................ 12 It BASIC CONCEPTS AND EQUATIONS OF THE TRANSIENT PHENOMENON. 14 2.1 Fundarmental Equations.............. 14 2.2 The Method of Characteristics... 15 2.2.1 Domain of Dependence...... 16 2.2.2 Characteristic Initial-Value Problem...... 18 2.2.3 Boundary-Data Problems.. a.,I 20 2.3 Simplified Equations and Finite-Difference Solutions. 25 2.4 Interior-Point and Boundary-Condition Equations.... 29 2.4.1 Points Interior from a Space-Liee Arc 30 2.4.2 Left-End Boundary Conditions.. 32 2.4.3 Right-End Boundary Conditions.34 2.4.4 Points Interior from a Time-LiKe Arc...... 36 2.5 Complex Systems.................... 38 2.5.1 Series-Junction Boundary Condition, 39 2.5.2 Branching-Junction Boundary Condition..... 2 2.6 initial and Steady-State Conditions.......... 44 2.6.1 Initial Conditons............... 45 2.6.2 Final Steady-State Conditions......... 48 2.7 Chapter Surmmry...........*.... 51 v

TABLE OF CONTENTS (CONT'D) Page III VALVE-STROKING CONTROL OF SIMNPLE PIPING SYSTEMS.... 52 3.1 System with an Upstream Constant-Head Reservoir.... 52 3.1.1 Specification of the Variable Upstream Velocity........... ~..... ~ 54 3.1.2 Study of the Frictionless System....... 57 3.1.3 Study with Friction Included 62 3.2 System with a Downstream Constant-Velocity Condition.. 68 3.2.1 Specification of the Variable Downstream Head.. 70 3.2.2 Study of the Frictionless System. 71 3.2.3 Study with Friction Included... *.. *... 76 3.3 System with an Upstream Constant-Relationship Condition.... *.................. 78 3.3.1 Specification of the Upstream Variables.... 80 3.3.2 Preferred Solution................... I.. 82 IV VALVE-STROKITNG CONTROL OF COMIhPLEX PIPING SYSTEMS...... 89 4.1 Series System.................. 89 4.1.1 Evaluation of the Existing Technique...... 91 4.1.2 Preferred Solution. D........... 96 4.2 Branching System............... 101 4.2.1 Evaluation of the Existing Technique..... 101 4.2.2 Alternate Solution.............. 106 4.3 Passive Systems...............111 V LABORATORY EXPERIiTENTAL SYSTEM... *............... 118 5.1 General Description of System...... * 118 5,2 System Calibration........ l,....... 128 5.3 Experimental Procedures e........Islo 134 VI COMPARISON AND DISCUSSION OF THEORETICAL AND EXPERIM4ENT1L.1 Simple ystemt.................... 137 6.2 Series System.................... 17 6.3 Btranch ing Sy steims....... ** *. 147 vi

TABLE OF CONTENTS (CONT'D) Page VII SUiARY AEI CCONCLUSIONS................... 154 APPE'TDIX A COISVERGENCE OF ITERATION SOLUTIONS............. 160 B SOLUTII P1OCEDURES............... 166 C'17- tIENT DJURATION CALCUATIONS........ 176 EFERENCES................. *....v 180 vii

LIST OF FIGURES Figure Page 1 Domains in the x-t plane in which the solution of the initial-value problem can be established....... 17 2 Domains in the x-t plane in which the solution of the characteristic initial-value problem can be established......................... 19 3 Domain of dependence on (a) a space-like and a time-like arc, and (b) a space-like and two timelike arcs................. 21 4 Domain of dependence on (a) a space-like and a timelike arc, and (b) the limiting case of a space-like and a time-like arc..................... 24 5 Intersection of the C+ and C characteristics in the x-t plane..................,...... 27 6 Notation for finite-difference equations.......... 27 7 The double-staggered grid of characteristics in the x-t plane. Notation for (a) a space-like interior point, (b) a left-end boundary point, (c) a right-end boundary point, (d) and (e) time-like interior points........ 31 8 Series-Junction boundary condition: (a) definition sketch, (b) solution grids and notation........... 40 9 Branching-Junction boundary condition: (a) definition sketch, (b) solution grids and notation..43 10 Regions where solution is dependent upon initial conditions. Notation for (a) a steady-state interior point, and (b) a steady-state boundary point........ 46 11 Region of final steady-state solution............ 49 12 Simple system with constant-head reservoir: (a) definition sketch, (b) x-t plane and notation.... 53 13 Region of established uniform surge condition........ 56 14 Simple frictionless system with constant-head reservoir. x-t planes for (a) tf > 4L/a, and (b) tf < 4L/a...... 59 15 Valve-stroking control of a simple system with constant-head reservoir...................66 viii

LIST OF FIGURES (CONT'D) Figure 16 Simple system with constant-velocity condition: (a) definition sketch, (b) x-t plane and notation... 69 17 Region of established uniform transient condition..... 72 18 Simple frictionless system with constant-velocity condition. x-t planes for (a) tf > 4L/a, and (b) tf < 4L/a........................ 74 19 Valve-stroking control of a simple system with constant-velocity condition...... 77 20 Simple system with constant-relationship condition: (a) definition sketch, (b) x-t plane and notation.... 79 21 Simple system with constant-relationship condition. x-t plane for preferred solution..............83 22 Valve-stroking control of a simple system with constantrelationship condition................... 86 23 Definition sketch of series system with upstream constant-head reservoir.................. 90 24 x-t plane and notation for series system with upstream constant-head reservoir............ 92 25 x-t plane and notation for series system with upstream constant-head reservoir -- preferred solution...... 97 26 Valve-stroking control of a series system with upstream constant-head reservoir............... 99 27 Definition sketch of branching system with upstream constant-head reservoir................... 102 28 x-t plane and notation for branching system with upstream constant-head reservoir..... 103 29 Valve-stroking control of a branching system with upstream constant-head reservoir -- existing procedure... 109 30 Valve-stroking control of a branching system with upstream constant-head reservoir -- alternate procedure... 110 31 x-t plane and notation for branching system with upstream constant-head reservoir -- passive system..... 113 ix

LIST OF FIGURES (CONT'D) Figure 32 Definition sketch of parallel system with upstream constant-head reservoir................ 116 33 Schematic of laboratory system.............. 120 34 Schematic of control valve and servo system......... 124 35 Flow-resistance relationship for the 2000-ft. copper pipe............... 129 36 Flow-resistance relationship for the 400-ft. copper pipe.......................130 37 Relationship between valve stem position and valve coefficient........................131 38 Relationship between valve stem position and card height....................... 132 39 Relationship between wavespeed and head for the 2000-ft. copper pipe.............. 135 40 System variables versus time. Simple system. Case I -- closure in 3L/a................. 139 41 System variables versus time. Simple system. Computed results of a linear tau-time closure to compare with Case I................... 140 42 System variables versus time. Simple system. Case II -- opening in 3L/a................ 142 43 System variables versus time. Simple system. Computed results of a linear tau-time opening to compare with Case II.............. 143 44 Head at valve versus time. Series system -- Case I.... 145 45 Head at valve versus time. Series system -- Case II.... 146 46 System variables versus time. Branching system -- Case I.......................150 47 System variables versus time. Branching system. Computed results of a linear tau-time closure to compare with CaseI.................. 151

LIST OF FIGURES (CONT'D) 48 System variables versus time. Branching system -- Case II........................... 152 49 System variables versus time. Branching system. Computed results of a linear tau-time closure to compare with Case II.................. 153 1-A Graphical representation of the method of successive approximations.1.....................161 1-B Solution procedure for one datum on each boundary..... 167 2-B Solution procedure for one datum on each boundary -- alternate style....................... 169 3-B Solution procedure for both data on one boundary -- active case........................170 4-B Solution procedure for both data on one boundary -- passive case.............. *... 172 5-B Solution procedure for mixed boundary-data problem -- active case.1........... 73 6-B Solution procedure for mixed boundary-data problem -- passive case........................ 175 1-C Graphical representation of the numerical evaluation of the time-duration integral............... 177 xi

LIST OF TABLES Table I Resume of analytical computer results........... 67 II Comparison of analytical computer results.......... 85 III Comparison of analytical computer results -- series system................100 IV Resume of orifice calibration data.............133 V Resume of pertinent data -- simple system........138 VI Resume of pertinent data -- series system........147 VII Resume of pertinent data -- branching system........149 LIST OF PLATES Plate I Experimental system...................119 II Pressure regulator and relief valve.............122 III Pressure transducer, transducer mounting block, and piezometer ring......................125 IV Speed controller, motor, plastic cam, motion transmitter, and valve positioner...................126 V Control valve, diaphragm actuator, micrometer, and displacement transducer..................127 xii

NOMENCIAITURE Symbol Meaning Units A Cross-sectional area of pipe ft.2 A,A' Denotes point position in the x-t plane A Subscript denoting grid-point location a Celerity of transient pressure pulse ft./sec. B A ratio; B = a/g sec. B,B' Denotes point position in the x-t plane B Subscript denoting grid-point location C A constant term C Subscript denoting grid-point location C+,C- Chasracteristic curves in the x-t plane C1,C2,C3 Collection of Known terms D Diameter of pipe ft. D Subbscript denoting grid-point location zE b 3Subscript denoting grid-point location F Variable function denoting head loss per reach a3s a iunction of velocity; F=-h(V) f(t. F aSubscript denoting grid-point location F( ) Functional relationship of the indicated a, gu3me nnt f Darcy-Weisbach friction factor Subscript denoting final conditions f'( ) Functional relationship cf the indicated argument G Subscript denoting grid-point location G( ) Functional relationship of the indicated argument xiii

NOIECLATURE (CONT' D) Symbol Meaning Units G'( ) Ordinary derivative of function with respect to the indicated argument g Acceleration of gravity ft./sec.2 H Piezometric head as a function of distance and time; H = H(x,t) ft. H Subscript denoting grid-point location H( ) Head as a function of the indicated argument ft. H'( ) Ordinary derivative of head with respect to the indicated argument H0 Initial steady-state value of head at valve ft. o Hf Final steady-state value of head at valve ft. H Extreme value of head desired in system ft. H' Extrerme value of head obtained in system ft. m H' Head at downstream side of orifice ft. r HGL Hlydraulic Grade Line h,(V) Total head Aoss as a function of the velocity; h,(V) = fLV/D2g ft. I, I' Integer counter i Subscripted integer counter j Subscript denoting conditions at junction K A constant term 1/2 Ko Orifice coefficient ft. /sec. K Valve coefficient ft. sec. Ke 1Minor-loss coefficient at pipe entrance L Total length of pipe ft. m Subscript denoting extreme conditions xiv

NOIMENCIATURE (CONT' D) Symbol Meaning Units N Number of reaches per pipe N Subscripted integer counter n Subscripted integer counter 0,0' Denotes point position in the x-t plane o Subscript denoting initial conditions P Pressure; P = P(x,t) lb./ft. PP,' Denotes point position in the x-t plane P Subscript denoting grid-point location Q-1 Volumretric flow rate; = Q(x,t) ft.3/sec. Q(t)'Volurmetric flow rate at a point as a function of time only ft. sec. it A region of the unique solution of a botundary-data problem;Subsc.-iptX denoting conditions at reservoir St;ereay State t Independent variable time sec. t Subscript denoting partial differentiation tc Time of complete valve closure sec., zti DDuration of transient condition in system sec. V Uci'ean cross-sectional velocity as a function oP distance and time; V = V(x,t) L't./sec. V( ) Velocity as a functicn of tihe indicated a rgnuent f t./sec. V'( ) Ordinaryr derivative of velocity with respect to the indicated argument v Subscript denoting conditions at valve x Independent variable distance t. XV

NOMENCIATURE (CONT' D) Symbol Meaning Units x Subscript denoting partial differentiation Z Elevation of pipe centerline above datum ft. a Angle between pipe centerline and the horizontal Dimensionless time ratio Y Specific weight of fluid lb./ft. 3 AIl Change in piezometric head ft. AlH ia/ximum head rise at valve ft. m Ah(V) Head loss per reach of pipe as a function of the velocity ft. At Vertical grid height sec. At' Time increment less than a complete At sec. AV Change in fluid velocity ft./sec. ax Horizontal grid waidth ft. o Curve in the x-t plane T Dimensionless variable valve relationship 0 Curve in the x-t plane Curve in the x-t plane 1,2,3 Sub script denoting pipe or valve number III, III Region in the x-t plane xvi

I. fINTRODUCTION The elimination or effective control of objectionable transients in the operation of fluid systems is becoming an increasingly important consideration in the design of hydroelectric generating stations, water supply systems, cyclic components of otherwise continuous flow processes, and hydraulic systems developed for industrial and commercial applications. With the ever-increasing utilization of the modern high-speed digital computer for rapid analysis and feedbaci operations, eventual on-line computer control of large and complex hydraulic systems is rapidly becoming a distinct possibility. Historically, transient control has proceeded on the basis of analysis rather than design; within the past decade, however, procedures have been developed which permit a design, or synthesis, approach. These procedures not only permit a more formal and systematic treatment but also result in more elegant and sophisticated control techniques than were previously attainable. This study attempts to identify underlying principles that govern any design approach, to formalize the existing valve-stroKing theory, and to extend this theory to include a greater number of boundary conditions in simple and complex piping systems. o.1 Literature Review The control of undesirable transient pressures and velocities In fluid systems has been the subject of intensive study by all investigators of the waterhammer phenomenon. This phenomenon has been analyzed by arithmetic methods since the turn of the century, by graphical methods developed about forty years ago, and starting about 1960, by numerical -1

-2methods particularly suited for solution on the high-speed digital computer. With a few notable exceptions, prior to the early 1960's control of objectionable transients was achieved by utilizing the available analysis procedures. In general, changes of the appropriate positions of the valves and other boundary control devices are assumed as functions of time; the system is analyzed for those assumed controller motions; and if the hypothe(8) sized motions prove to be inadequate, other, presumably slower boundary motions are assumed. The analysis process is continually repeated until a satisfactory solution is obtained. Even for relatively simple systems, reliance upon such tedious and restrictive procedures ranges from being cumbersome and inconvenient at best to totally inadequate at worst. Because of the severe limitations imposed by the analysisoriented procedures, investigators have attempted to develop a synthesis, or design, approach to the determination of proper control-valve operation in transient pipe flow. It would appear that Streeter (26,27,283031) has been the greatest single contributor to the development of suitable synthesis procedures, and he has called this area of study "Valve StroKing". In its most general form, valve stroking may be defined as the synthesis of external boundary conditions to create a desired transient. Hence the term also refers to control by means other than a valve -- for example, the specified variation in elevation of the free surface in a reservoir. Implicit in this definition is that the desired transient is subject to one or more elements of control: i.e., the extreme values of pressure are limited to physically acceptable ones, the period of the transient may be arbitrarily determined, residual transients are eliminated or reduced in magnitude, etc.

-3The earliest reference in the available literature to the fundamental distinction between the analysis and synthesis procedures appears (14) to be in an article written by Knapp. Considering the design of automatic shutoff valves installed in distribution systems to terminate the flow of water in case of a sudden rupture of the pipeline, he was the first to enunciate this distinction: "It is the usual practice to assume the time and closure law of the valve and then to checK the maximum surge caused by the valve closure. It is the purpose of this paper to determine the characteristics and the hydraulic design of self-acting shutoff valves in such a rmnner that the waterhammer,..., remains within permissible limits." Hiis solution was to provide a large quicK-closing main valve whose control mechanism could be of extremely simplified design. It;:ould be closed in no more than 2L/a seconds following the rupture, where L is the distance from the valve to the upstream constant-head reservoir and a is the speed of the pressulre pulse. To control the rise in pressure that would occur wnen the negatively-directed low pressure wave reflects bjacK to the valve as a positively-directed normal pressure wave accompanied by a significantly higher velocity of the fluid, he proposed the installation of a smaller bypass valve with a more complex control mechanism. Directing his attention to the design of the control motion for this bypass valve and using the graphical method developed by Allievi, he'las able to limit the resulting pressure rise to a predetermined nrhaximum value. 2le only possible criticism of his procedure is that upon complete closure of the bypass valve an undesirable residual transient

still exists in the pipeline. However, it would be more than twenty-five years before it was recognized that this residual transient could be eliminated, and considering the state of the art at that time (1937), Knapp's contribution was both remarkable and significant. In his classical study first published in France in 1949, (2) Bergeron considered the comparatively more familiar problem of the complete closure of a valve at one end of a pipeline which terminates in a constant-head reservoir at the other end. His solution was similar to Knapp's in that he was able to limit the resulting pressure rise to a predetermined maximum value. Again, after the liquid column in the pipe has been brought to rest, a significant pressure gradient still exists in the fluid which results in the creation of an undesirable residual transient. Nevertheless, he demonstrated that the resulting transient pressures are less severe than would be created if either of two other techniques are used to determine the valve motion for the same time of closure: valve closure such that (1) the fluid velocity at the valve is reduced linearly from its initial steady-state value to zero, and (2) the effective area of the valve is reduced linearly from the initial open position to zero. With regard to this latter closure, perhaps it should be mentioned that linear valve-closure studies are abundant in the literature (Ref. 2,18,19); by their essential character, however, such studies are analysis oriented and generally only provide useful standards for comparison with the results of comparable synthesis investigations. Kerr(10l11,2) also studied the problem of valve closure in a simple system (a single pipeline with a valve downstream, a constant-head

-5reservoir upstream). He concluded that the valve closure motion ideally should be such that the fluid velocity at the valve reduces linearly from its initial steady-state value to zero. In one of his studies he compared the maximum pressure rise created by a closure according to his suggested criterion with that created by a closure in which the stem position of the valve is changed linearly in the same time. He reported that for a flow reduction from 5 feet per second to zero in a time of closure equal to lOL/a the maximum pressure rise created in the former case was 60 feet and in the latter case was 210 feet. Such disparate results are understandable when one recognizes that, for the majority of valves usually installed in water distribution systems, during the first 50% of the stem motion the reduction in flow is negligible and that for even 75% of the stem motion reduction in the flow generally is on the order of only 10%. Kerr suggested that one practical solution wvould. be the design of valves utilizing a two-speed closure: a rapid initial linear stem motion to some intermediate partially closed position followed by a much slower linear stem motion to full closure. Kerr also studied the effect of valve closure from a partially opened position using the same valve motion and rate of closure for the valve that are determined by the linear velocity-reduction criterion. He noted that under some circumstances the maximum pressure rise resulting from the partial closure is worse than that created during the corresponding full closure condition. Other investigators who also recognized that valve closure in a simple pipe can be achieved without exceeding a predetermined maximum pressure rise were Streeter and Ruus, with the

-6latter's goal being the development of a theory for the optimum closure of hydraulic turbine gates. Their methods are identical to the procedures developed earlier by Knapp and Bergeron: in one round-trip wavetravel time the head at the valve is increased linearly from its initial steady-state value to the prescribed maximum value; that maximum is then maintained until the required fluid deceleration occurs and the valve is closed. As in the earlier studies cited above, upon complete closure of the valve a residual transient still exists in the pipeline. ZQruba(35) referred to a "perfect" relationship for closing a valve and credited it to the worK of KrivcenKa. Although this reference is unavailable for review, it is evident that KrivEenKa's closure method is identical to those of Knapp, Bergeron, Streeter, and Ruus. (21) Ruus appears to be the first investigator who attempted to modify his procedure to accommodate the effect of fluid friction. He was able to arrive at a solution by utilizing the existing techniques (Ref. 2,18) for incorporating frictional effects into the graphical method of analysis. The first study which succeeded in eliminating the objectionable (26) residual transient described above was that of Streeter. Again considering a simple frictionless pipe and aided by the visual insight afforded by utilization of the dependent variable graph (Allievi's graphical method), he presented a procedure by means of which the downstream valve motion can be determined which would create a controlled transient between any desired initial and final steady-uniform flow conditions. The essential features of his procedure are as follows: 1. During the first one round-trip wave-travel time

-7in the pipe, the flow is organized such that the velocity is uniform and a predetermined straight hydraulic grade line is established, sloped positively or negatively depending upon whether the flow is being decreased or increased. 2. During the second phase, the uniform grade line is maintained and the flow is decelerated or accelerated uniformly. 3. During the final one round-trip wave-travel time, the hydraulic grade line and velocity are brought to their final steady-unif-orr values. As a consequence of this procedure, the transient condition created is a highly controlled one. The extreme values of pressure can be predetermined and will not be exceeded, all pressures and velocities are linear functions of time and distance, the velocities are not permitted to reverse, anda the transtent ceases w;;hen the valve motion ceases. Included in that worK were techniques by which friction corrections could be applied to the preceding solution and also a study which.. applied the basic procedure to several reaches of pipe of constant diameter but with a varying wall thickness. In that study Streeter explicitly restricted the time of valve closure to a duration of 4L/a or greater. For situations involving complete closure of the valve, the time of closure tc can be related to the naximura prescribed head rise at the valve AH by the following relationm ship: VL t = --- + _t c a g~l c -a

-8where V0 represents the initial steady-uniform velocity in the pipe and g is the acceleration of gravity. Considering the earlier procedure which prescribes the maximum head rise at the valve but which also results in the creation of residual pressure fluctuations, the following relationship between valve closure time and maximum head rise can be obtained: L VL 2L t L gAH(t > -). c a gAgH c - a Comparing the two methods for the same time of valve closure, the ratio of the maximum pressure rise created by closure according to the older procedure to the maximum pressure rise sustained during closure according to Streeter's suggested technique is given by,a-2 a-1 (a > 4), where the common time of closure is t = aL/a. Thus the older procedure has the advantage of creating a lesser maximum rise in pressure, whereas the newer procedure succeeds in eliminating the objectionable residual fluctuations in pressure. When compared to a closure predicated on the linear velocityreduction criterion studied by Bergeron and suggested by Kerr, Streeter's recommended closure procedure is clearly superior. Not only does it eliminate the residual transient, but also the ratio of the maximum pressure rise created by closure according to the former procedure to the maximum pressure rise sustained during closure according to Streeter's procedure may be shown to be 2 - - (s > 4), which has a value equal to or in excess of 1.0.

-9(27) Streeter subsequently broadened his procedure when he presented a study demonstrating that in one round-trip wave-travel time any Known unsteady condition in a frictionless pipe can be organized into a condition of uniform velocity and predetermined straight hydraulic grade line. Once this condition is established, the procedure is identical to that described above. This concept was applied to the control of the transient condition created following the failure of power to a centrifugal pump. VThile the problem of transient control in centrifugal pump systems has long intrigued numerous investigators of' the waterhammer phenomenon, Streeter's study appears to be the only existing one to suggest a suitable synthesis procedure. Other studies (e._g, a recent investigation by Kinno(3)) are generally restricted to the development of guidelines for valve operation that have been obtained using the existing analysis techniques. (28) Streeter further extended these early valve-stroking concepts to include the control of decreasing or increasing flow in frictionless series pipe systems. In the latest and most significant study reported to date, (30) Streeter substantially broadened the scope of the valve-stroking principle by developing a.theory which provides for an exact solution when frictional effects are included. This method superseded the earlier technique(2628) which was only an approximation and which yielded poor results when the effect of fluid friction was significant. The procedTrhe developed was first applied to the problem of changing from some initial to some final steady-uniform flow condition in a simple pipe. The essential features of the suggested procedure are identical to that

-10of the earlier study(26'28) for the frictionless case: in one roundtrip wave-travel time the flow is organized such that the velocity is uniform and a straight hydraulic grade line is established; these conditions are maintained during a central phase of the transient; during the final one round-trip wave-travel time the velocity and grade line are brought to their final steady-uniform values. Again the transient condition created is highly controlled: extreme values of pressure are identified and not exceeded, pressures and velocities are nearly linear functions of time and distance, the velocities are not permitted to reverse, and the transient ceases when the valve motion ceases. As in his earlier study, Streeter res ricted the time of valve closure to a duration of 4L/a or greater. The method was then readily extended to complex systems by apportioning the flow changes among the branches as a linear relation of the initial to the final steady-Luiform values in each pipe. By combining these two orocedures, values of head and velocity may be determined at each external boundary in the system. SThyus, motion of the valves or other adjustable boundary conditions can be determined at those control points to create the desired transient. This general procedure was formulated by considering the manner in which the method-of-characteristics solution of the unsteady flow equations proceeds on the independent variable graph (the x-t plane). Essential to this procedure was the realization that if Kn(Uown values of velocity and head exist for a specified period of time along one boundary, the characteristics equations can be employed to determine the history of the transient throughout the x-t plane.

-11Streeter also presented a suggested valve-stroxing procedure for a single pipeline where one boundary condition is a Known constant relationship between velocity and head, and he concluded the study with a description of experimental verification of the valve-stroKing principle applied to a simple system. Streeter and Wylie(31) incorporate much of the preceding worK in their discussion of the topic; in addition, they provide explicit procedures f'or the application of the valve-stroKing concept to series, branching, parallel, and complex piping systems. Zaruba(34,35) appears to be the only other recent investigator who has attempted to develop suitable synthesis procedures, and his worK parallels some of Streeter's studies. He suggested a procedure for determining the proper control-valve closure motion to create a predetermined maximum pressure rise in frictionless series pipe systems, and that procedure is identical to Streeter's.(28) In his study Zaruba restricted the time of valve closure to a duration of' N t > 4 ZL-, c i=lal where N represents the number of pipes connected in series. (Although Streeter didn't explicitly restrict the valve closure time in his treatment of the same problem, that restriction is at least implicit in his procedure also.) Zaruba's most recent effort was the development of a closure procedure for a simple pipeline where frictional effects are important; again, the procedure that he suggested is identical to Streeter's.(33)

-121.2 Scope of Investigation This study of valve stroking has been divided into two major phases. The first phase includes: (1) an examination of the fundamental equations that form the foundation of all modern computer-oriented transient analysis and synthesis investigations, and (2) the development of additional synthesis procedures to supplement the existing valve-stroKing theory for simple and complex piping systems. The second phase concerns a description of the laboratory experimental system and the subsequent laboratory studies which verify some of the new concepts presented in this thesis. The method-of-characteristics solution of the transient flow equations is examined in Chapter II. A fundamental property of the characteristic equations -- that their application is not restricted to advancing the resulting numerical solutions forward in time only -- is presented, and this property is demonstrated to be of essential importance in the formulation of the valve-stroKing theory. The existing valve-stroKing procedures for simple piping systems are evaluated in Chapter III. It is demonstrated that, theoretically at least, the minimum time for control valve motion in such systems is 2L/a. The consequence of this significant development relative to the control of a system consisting of a single pipe with an upstream constant-head reservoir is presented. A suitable valve-stroking procedure for a pipe in which a constant velocity forms one boundary condition is developed. The chapter concludes with an examination of a third type of simple system, a single pipe in which one boundary condition is a Known constant relationship between head and velocity. A valve-stroKing procedure is

suggested that permits a predetermined extreme value of pressure to be developed and maintained at the valve during a central phase of the transient. In Chapter IV attention is directed to the development of valvestroking procedures for elementary complex systems. The existing method for a series system is evaluated; it is demonstrated that this procedure rarely allows the extreme value of pressure encountered in the system to be identified a priori. An alternative technique is suggested that overcomes this disadvantage. Branching systems are next considered and the same disadvantage is found to exist in those systems. Again, an appropriate alternative procedure is suggested. The chapter concludes with the introduction of the concept of lassive valve stroing and the application of this concept to the development of a suitable valve-stroKing procedure for a representative branching system. Chapter V concerns a description of the laboratory experimental system, and the discussion of the results of the various laboratory studies are included in Chapter VI. Experimental verification of the applicable valve-stroKing theory has been obtained and is presented for a simple system (a closure and an opening, both in 3L/a), a series system (two closure studies), and a passive branching system (two closure studies). In Chapter VII the principal conclusions derived from this investigation are summarized.

II. BASIC CONCEPTS AND EQUATIONS OF E E TRANSIENT PHENOMENON The basic partial differential equations describing the unsteady flow of liquids in elastic closed conduits are presented in this chapter, and their solution by the method of characteristics is indicated. Fundamental uniqueness properties of the characteristic solutions are examined relative to the concept of domain of dependence and to selected characteristic initial-value and boundary-data problems in the solution plane. The basic equations are then reduced to a simplified form and a second-order finite-difference solution of the simplified equations is proposed. Computational schemes suitable for execution on a high-speed digital computer are presented for both transient analysis and synthesis procedures. The manner in which the existing initial conditions influence the transient solution is discussed, and the singular fashion by which the final steady-state solution may be established without residual transient fluctuations is indicated. 2.! Fundamental! Equations The fundamental partial differential equations governing the unsteady flow of liquids in rigid closed prismatic circular conduits have been developed from considerations of mass conservation and the equation of motion and are, respectively 2 V + VH + VH + Vsina O (1) VV + + fVVV+ 0 (2) x + Vt + gHx 2D The independent variables t and x represent time and distance along the pipe, respectively, and when used as subscripts denote partial differentiation. The dependent variables V and H are the velocity and the

-15elevation of the hydraulic grade line above some fixed datum. The pressure, P. and H are related by the expression P = y(H-Z), Z being the elevation of the centerline of the pipe above the datum and Y the specific weight of the fluid. The variable f represents the Darcy-Weisbach friction factor, D the pipe diameter, and a the angle between the horizontal and the centerline when measured positively downward. In these equations the pipe is assumed to be full with the minimum pressure always greater than vapor pressure. The velocity is assumed to be one-dimensional with a uniform distri-bution across the cross-section, and pressure is considered similarly with a value equal to that existing at the centerline. The liquid and pipe walls are assumed to be perfectly elastic, and the friction is evaluated assuming that the instantaneous value of wall shear stress is equal to the shear stress sustained during the corresponding condition of steady flow. In the friction term the absolute value of V is introduced to maintain the direction of the shear force opposite to tthat of the ve-locity. 2.02 The etr'eLhod of' CharacteristicsL Equations (1) and (2) are two simultaneous quasi-linear partial differential equations of the first order. They are of the hyperbolic tyope and are amenable to solution oby the methlod of characteristics (Ref. 1,6,16,22). These solutions are d + + d + i Vsina = 0 (3) dt a dt 2D a C+ dx I V + (3a) dt

dY_ dH + ~_ I V Vsinc - - 0 (4) dt a dt 2D a dx = V-a (4a) dt Thus, the original two simultaneous partial differential equations have been transformed into four ordinary differential equations. The significance of these equations may be explained as follows: along a curve in the x-t plane defined by Equation (3a) -- the C+ characteristic curve -- Equation (3) is valid, and along a curve defined by Equation (4a) -- the C- characteristic curve -- Equation (4) is valid. According to the derivation, every solution of the original system of Equations (1) and (2) satisfies this characteristic systemn of Equations (3) to (4a). It may be shown that the converse is also true.(6) Before proceeding further with a discussion of the techniques that can be developed to construct numerical approximations to the solution of these equations, attention is now directed to a summary of some fundamental uniqueness properties of the characteristic solutions. For a thorough discussion of these properties and proofs of the corresponding uniqueness theorems, the reader is referred to the worm of' Courant and Friedrichs. (6) 2.2.1 Domain of Dependence With reference to Figure 1, let a curve 9 be given in the x-t plane and along 9 continuous values of V(x,t) and H(x,t) are arbitrarily prescribed. It is assumed that the curve 9 with its prescribed values of V and H has nowhere a characteristic direction. The C+ characteristic which intersects the currve 9 at A and the C- characteristic which intersects 9 at B intersect at point P. Then it may be shown that the values of the solutions V, IH at the point P depend only upon the initial values

i a I~~~~/ I~~~~~~~P I ue?~ O8n i h -tpaei i~ ~ ~~wihteslto fteiiil 9au rbe a eetbihd

-18of V and H on the section of 9 intercepted by the two characteristics through P. This interval on the curve 9 intercepted by the two characteristics is called the domain of dependence of the point P. From this notion, the uniqueness of the solution of Equations (1) and (2) in the region ABP bounded by the two characteristics through the point P and the domain of dependence AB cut out by them from the initial curve 9 may be demonstrated. In the preceding discussion no side of the initial curve 0 is distinguished. Therefore, the interval LAB on the curve 0 is also the domain of dependence for the point P', and the solution of Equations (1) and (2) in the region ABP' bounded by the two characteristics C+ and Cand the interval AB also exists and is likewise unique. Therefore, a unique solution of the initial-value problem exists on both sides of the curve 0. Intrinsic to the validity of this concept is the recognition that in formulating Equations (3) to (4a) from Equations (1) and (2) no explicit or implicit constraints relative to time exist. Thus, solutions may be advanced forniard or backward in time, as the case may be, and the only consideration relative to the existence of unique solutions is the orientation of initial-value curves in the solution plane and the nature of the initial data existing on those curves. 2.2.2 Characteristic Initial-Value Problem In the preceding section, the initial-value problem of a noncharacteristic curve was considered. In this section the characteristic initial-value problem is examined. With reference to Figure 2, suppose 9 is a C+ characteristic. Then Equation (3) demonstrates that along 0 the values V(x,t) and H(x,t) cannot both be prescribed arbitrarily since Equation (3) is an ordinary differential equation in V and H along 0 and

-19X A CI- / C'* P K Figure 2. Domains in the x-t plane in which the solution of the characteristic initial-value problem can be established.

-20establishes a relation between the dependent variables. Consequently, only one function may be prescribed along 9 and, at a single point, the value of the other function. In many applications the initial-value problem is posed not for a non-characteristic initial curve but for initial data along two intersecting characteristic arcs. This characteristic initial-value problem is formulated for the characteristic differential equations as follows. Let one assume that compatible values of V(x,t) and H(x,t) exist along two characteristic segments, a C+ characteristic 9 and a C- characteristic I, as in Figure 2. The C+ characteristic which intersects the curve X at A and the C- characteristic which intersects the curve 0 at B intersect at point P. Then it may be shown that the values of the solutions V,H at the point P depend only upon the intial values along AOB intercepted by the two characteristics through P. Again, this interval AOB is called the domain of dependence of P and the uniqueness of the solution of Equations (1) and (2) in the region AOBP can be demonstrated. As before, no side of the intersection of 9 and 0 is distinguished. Therefore, if' compatible values of V(x,t) and H(x,t) exist along 9 and $, then the interval B'OA' is the domain of dependence of the point P' and the solution of Equations (1) and (2) in the region B'OA'P' also exists and is liKewise unique. 2.2.3 Boundary-Data Problems In this section uniqueness properties relative to problems in which the initial data are prescribed on two or more non-characteristic arcs are examined. Referring to Figure 3(a), a and 0 are two noncharacteristic arcs meeting at point O and enclosing an angular region R of the x-t plane. Assigning an arbitrary direction to each of the two families of characteristics (in this example both directions are assigned

-21C PCRR I8 \' I~~B o O (b) Figure 3. Domain of dependence on (a) a space-like and a time-like arc, and (b) a space-like and two time-like arcs.

-22as positive, where positive and negative are associated with increasing and decreasing values of time, respectively), it is seen that two characteristics issuing from points on G enter region R while only one characteristic issuing from points on 0 enters R. Then it may be stated that two data on @ -- both V(x,t) and H(x,t) -- and one datum on 0 -- either V(x,t) or H(x,t) -- determine the solution in R. If the data are prescribed in this manner, then it may be shown that the solution of Equations (1) and (2) in the region R is unique. Again, the designation "domain of dependence of the point P" for the segment AOB is justified. Before further characterizing these problems which possess a unique solution, it becomes convenient to introduce the concepts of spacelite and time-liKe directions. A direction is called space-liKe if both characteristic directions with dt > 0 lie on the same side of it; a direction is called time-liKe if it separates the characteristic directions with dt > 0. In this last example, for instance, @ is space-liKe and: is time-liKe. Essential to the formulation of these boundary-data problems is the following observation: the number of initial data prescribable on an arc depends upon the number of characteristics which, when drawn from a point on the arc in the arbitrarily assigned directions of the characteristic families, enter the region of interest. The fashion in which these concepts may be extended to three boundaries is demonstrated in Figure 3(b). Arbitrarily assigning positive directions to each of the two families of characteristics, then for a unique solution to exist in the region fR two data must be prescribed on the space-lite arc G and one datum each on the time-liKe arcs ( and P. Again the segrnent AOO'B is designated as the domain of dependence of the point P. It is pertinent to note here that when the arcs X and * are

-23parallel to the t-axis and the arc 9 is parallel to the x-axis, then the region R assumes the familiar configuration encountered in the conventional transient analysis solutions. A central concept in the consideration of such boundary-data problems is that the assigned directions of the two families of characteristics are arbitrary. If, as in Figure 4(a), the positive direction is assigned to the C+ characteristic family and the negative direction to the C- characteristic family, then it is observed that two characteristics issuing fror points on the time-liKe arc 0 enter R while only one characteristic issuing from points on the space-liKe arc 9 enters R. Again, it may be shown that if two data are prescribed on 0 and one datum on 9 the solution of Equations (1) and (2) in the region R exists and is unique. As before, the segment AOB is designated as the domain of dependence of the point P. In Figure 4(b) a limiting case of the last problem is considered. As before, two data are prescribed on the time-liKe arc X and one datum (or two compatible data) on the space-liKe arc 9 which now is a characteristic curve. Again, the solution of Equations (1) and (2) exists in R and is unique, and the segment AOB is the domain of dependence of the point P. A particularly significant consequence of this latter type of bouldary-data problem is illustrated in Figure 4(b) and is that along any other time-liKe arc in R (for example, the arc 4 ) no datum may be prescribed. Arc P lies within the region of the unique solution of the given boundary-data problem, and hence the solution along P is unique and dependent upon the data prescribed on, and 9. When 0 and 4 are parallel to the t-axis and 0 coincides with a characteristic curve, this last boundary-data problem will be shown to be of fundamental importance in the formulation of the valve-stroking theory.

-240 A x S-[ / 8 / Figure 4. Domain of dependence on (a) a space-like and a time-like arc, and (b) the limiting case of a spacelike and a time-like arc.

-252.3 Simplified Equations and Finite-Difference Solutions In Equations (1) and (2) the nonlinear convective terms and the V sin a term can be shown(31) to have a negligible effect upon the consequent solutions and are often deleted. (At least one Justification is that these terms do not include time derivatives and yet are not commonly incorporated into the corresponding steady-state solutions.) The simplified continuity equation and equation of motion are then obtained and are, respectively 2 " —V + H =0 (5) g x t V+H 1 (6) V + gH + fv M 0 (6) t ~ x 2D The corresponding characteristic ordinary differential equations obtained from the solution of Equations (5) and (6) by the method of characteristics then become dV + dH + fV 0 (7) dt a dt 2D Cd dx = a (7a) dt dV "' dH +.fVV... (8) dt a dt 2D C>. dx __ -a(8a) -a Again, the significance of these equations parallels that of the earl]ier set;: along a straight line of slope dx/dt = a in the x-t plane Equation (7) is valid, and along a straight line of slope dx/dt = -a Equation (8) is valid. (In this development and throughout the remainder of this investigation, unless otherwise indicated, the wave speed a will be considered to be constant for a given pipe.)

-26Numerical approximations to the solution of Equations (7) to (8a) may be constructed by utilizing either the linear first-order or the trapezoidal second-order finite-difference approximations.(16) Although the first-order approximations are widely used and have been demonstrated to (31) be generally adequate, for reasons to be explained later some motivation does exist for the employment of the second-order approximations. The formula for this approximation is X1 f (x)dx'2 [f(xo) + f(x1)] (xl - xO). xO With reference to Figure 5, it is assumed that all conditions are Known at points A and B. The C+ characteristic through A and the Ccharacteristic through B intersect at point P where conditions are unKnown and are to be determined. Equations (7) to (8a) are now multiplied by dt and integrated utilizing the second-order finite-difference approximation. The results are HP-HA l(AD - ( ) V VA + A + + P (t - t) = (9) C+ Xp- xA = a(tp- t) (9a) Hp -HPB i vi fp p (t- tB) (10) Vp - VB B (2D 2D B0 xP- xB =-a(tp- tB) (loa) where the subscripts are used to define the location of the Known or unkznown quantities, and the constant B = a/g. S-ince the friction factor f may ordinarily be determined as a function of velocity, f = f(V), then these four equations are sufficient to determine the values of the four

-27C X _4 8 Figure 5. Intersection of the C+ and C characteristics in the x-t plane. D F E tH... X Figure 6. Notation for finite-difference equations.

-28unknowns Vp, Hp, xp, and tp. There are several ways (1622'31) of employing the set of Equations (9) to (10a) to obtain an orderly approximate numerical solution to the original set of simplified partial differential equations. One possibility is to use a double-staggered grid of characteristics. Such a grid is particularly simple in this case and, in fact, is identical with a grid of specified-time intervals. Since Equations (9a) and (lOa) are independent of V and H, the grid can be immediately determined and is independent of the repeated solutions of Equations (9) and (10). If a pipe of length L is considered to be divided into N equal reaches, then p - XA = CB - W = Ax = L/N. From Equations (9a) and (lOa), tp - tA = tp - tB = At = Ax/a = L/(aN). Therefore, for a particular pipe and an arbitrarily selected value of N, the grid assumes a rectangular shape with a constant mesh spacing defined by Ax and At above. As a convenience in reducing the friction term in the finitedifference equations and in order to better appreciate its significance, one such term is now examined, for example 2DAVA (t t ) Since tp - tA = L/(aN), this term now becomes AVA VA L 1 fAL A 2D aN B DN 2g Thus, this term reduces to =Ahf(VA) where Ahf(VA) is the corresponding steady-siate head loss sustained over a section of pipe of length L/N when the velocity of' the fluid flowing is VA. The variable F may be defined to represent this function which, for a specific pipe, can be determined as a irnonm function of velocity only: F = Ahf(V),

-29As has already been observed and demonstrated with regard to the characteristic ordinary differential equations, no constraints relative to time exist. Therefore, in addition to Equations (9) and (10) a set of companion equations that are the result of finite-difference integrations along the negatively-directed characteristic lines also exist. These equations are virtually identical to the respective equations associated with the positively-directed characteristics, differing only in the signs of the friction terms. With reference to Figure 6 and incorporating the notation for the friction term as proposed above, these four final finitedifference equations are HP-HA Fp+FA c +/, (11a) VP - VA + B 2B =O C P-HB F +F V _. - HPBHB + FP+FB= 0 C (lb) P B B 2B Hp-HE Fp+FE Vp - VE + B 2B C (lc) H-HD Fp+FD.d) V -v - P D P D (lld) VP VD B 2B Tnese equations and the corresponding grid arrangement for which they are valid will be used throughout the remainder of this investigation to numerically model and evaluate the transient phenomenon. 2.41 Interior-Point and Boundary-Condition Equations In Section 2.2 uniqueness properties of the characteristic solutions were discussed, and selected characteristic initial-value and boundary-data problems were examined. In this section attention is directed to the fashion by which Equations (lla) to (ild) may be utilized to obtain the desired numerical solutions to such problems. Discussion of the particular computational Lrcedures involved in these

-30solutions will be deferred until the need for a specific procedure is encountered in subsequent phases of this investigation. 2.4.1 Points Interior from a Space-LiKe Arc In Figure 7 the double-staggered grid of characteristics is established in the solution plane. Arbitrarily assigning the positive direction to both characteristic families, it is assumed that the required values of V(X,t) and H(x,t) are IKnown or prescribed along the space-like line t = to (or at least at the N+1 intersections of the vertical grid lines with the line t = to). Then Equations (lla) and (11b) may be combined to solve for the desired values of V and H at all interior (nonboundary) grid points situated on the line t - to + At, as in Figure 7(a). Subtracting Equation (llb) frorm (lla) and rearranging, the resulting expression for the desired value of Hp is HA+HB (VA-VB) FA-FB (12a) Hp 2 + B 24 This value of Ip may then be substituted into Equation (lla); after rearrangement, the desired expression for Vp is Hp-HA Fp+FA VP = VA B A 2B (2b) Since Fp is a function of Vp, this latter equation must be solved by iteration. By basing the first value of Fp upon the value of V existing at point A (ioe., initially evaluating Vp utilizing the first-order finitedifference approxinmation), and successively improving this value using the previously calculated value of Vp, the solution is readily convergent. (See Appendixe A.)

-31Id) l- H(e) 0 P f =to -. 0 2 3 N-3 N- 2 N-I N Section No. x=O x=L Figure 7. The double-staggered grid of characteristics in the x-t plane. Notation for (a) a spacelike interior point, (b) a left-end boundary point, (c) a right-end boundary point, (d) and (e) time-like interior points.

-32e.4,e Left-End boundary Concaltons If the positive direction is arbitrarily assigned to both characteristic families, then it has been observed that one datum along each of the two time-lixe boundary arcs is essential to the existence of a unique solution in the region of interest. This datum may be either a Known value of velocity or head, or it may be some Known relationship between these two variables. However the case may be, this auxiliary condition must be determined exterir tIo the Dipe and must always reflect the physical behavior of the particular terminal boundary device under consideration. With reference to Figure 7(b), for example, let one consider that a constant-head reservoir is located at the upstream (left) boundary of the pipe. The unsteady form of the energy equation(29) may be applied to a control volume of fluid between the reservoir surface and the pipe entrance. By assuming that the time rates of change of mass and internal energy existing within this control volume are negligible, the resulting expression obtained for flow from the reservoir into the pipe is r 2 V 2 H H + -+k — r P 2g e2g in which H is the elevation of the reservoir surface above the arbitrarily r selected datum end K is the minor-loss coefficient for the entrance. (fTnis e coefficient is evaluated assuming that its value is the same as would exist during the corresponding condition of steady flow; this assumption is completely analogous to that concerning the unsteady value of the wall shear stress.) The above relationship may be further simplified, since quite frequently the Kinetic-energy and minor-loss terms are inconsequential and therefore not incorporated into the corresponding steady-state solutions.

-33Then this simplified boundary condition for the time-liKe line x = O is Hp = Hr (13a) and has been experimentally verified in numerous studies.(5'15'30 931) This value may then be substituted into the only available finite-difference equation, Equation (11b); the consequent expression for Vp is HP-HB Fp+FB (13b) Vp = VB + B B - 2B This latter equation must be solved by iteration; the same technique that was outlined for the solution of Equation (12b) is equally applicable here. For flow from the pipe into the reservoir, Equations (13a) and (13b) are still valid; this is true even if' the Kinetic-energy and minor-loss terms are significant, since for this situation K =-1. e If the elevation of the reservoir, instead of being constant, varies as a Known function of time, the only modification required is to incorporate this variation into Equation (13a): i.e. Hp = Hr(t). Another example of a typical upstream boundary device would be that of a fixed orifice located just inside the entrance of the pipe from a constant-head reservoir. As with the previous example, a quasi-steady relationship may be developed to model its behavior. Again assuming that the time rates of change of mass and internal energy existing within the adjacent fluid are negligible and that the orifice discharge coefficient is the same as would exist for the corresponding steady-state condition,(5 31) the result is Vp = KKo Al for flow from the reservoir into the pipe or Vp =-Ko,Ip-TiF"

-34for flow from the pipe into the reservoir; the parameter K, assumed to be constant in this investigation, must incorporate the areas of the pipe and orifice, the discharge coefficient of the orifice, and other constant terms. Solving these relationships simultaneously with Equation (llb), the resulting expressions for V are P -C1 +.1 + 4(.2 - K..F/2) (i2a) P 2 or C1 -C - 4(C, - 2F/2) P 2b) in which Ci = Ko B and C2 Ko (BVB + Hr-HB-FB/2): i.e., the C's represent a collection of constant terms that are dependent only upon Known values existing at the previous time step. As before, these equations must be solved by iteration. Equation (14a) is valid for flow from the reservoir into the pipe and is:e~d,henever the eor~ession CK2 -. F/2 has a positive value; Equation (14b) is valid for flow from the pipe into the reservoir and is used whenever C -K 2F /2 is negative. This determination must be rmade during each successive trial of the iteration procedure. The ultimate value of V may then be substituted into Equation (llb) P to obtain the desired value of H: Fp +FB(14 Hp =HB + B(Vp- VB) + 2 ('c) 2.4.3 Right-End Boundary Conditions Again assigning the positive direction to both characteristic families and referring to Figure 7(c), consider that a control valve is located at the downstream (right) boundary of the pipe. The valve may be analyzed as an orifice,(15,18,29,30,31) and as with the previous example, a

-35quast-steady relationship may be developed to model its behavior. Again assuming that the time rates of change of mass and internal energy existing in the adjacent fluid are negligible, that the valve discharge coefficient is the same as would exist for the corresponding steady-state condition, and that the Kinetic-energy of the fluid within the pipe is small, the result is VP = Kv G P and is valid if the valve discharges into the atmosphere, if the centerline elevation of the valve is selected as the arbitrary datum, and if the value of H is always greater than zero (gage), thus eliminating the possibility of air bacKflow through the valve. The parameter K is assumed to be a v Known function of time (or at least Known at every intersection of the horizontal grid lines with the time-liKe line x = L), and it must incorporate the areas of the pipe and valve, the discharge coefficient of the valve, and other constant terms. This relationship may be solved simultaneously with the only available finite-difference equation, Equation (lla), with the result that -C1 +C 4(c2 - K Fp/2) (15a) V 2 in which C1 K2B and C2 = K2(BVA + HA - F/2); as with the previous examples, an iteration solution is required. Substitution of the resulting value of Vp into Equation (11a) yields the desired value of HD: P AP F +F Hp = HA B(Vp - V) P (5b) If, instead of a control valve, a fixed orifice is located at the downstream boundary of a pipe discharging into the atmosphere, the preceding analysis must only be modified to the extent that K, must be

-36replaced by the fixed-orifice parameter Ko Another possible downstream boundary condition would exist if the velocity is some Known function of time (for example, at a dead end in which the pressure is always greater than vapor pressure, V(L,t) = o); then Vp = V(L,t). (16a) From Equation (11a) the desired expression for Hp is Fp+FA (16b) Hp =HA - B(Vp - VA) 6b) In a similar fashion, if the downstream head is a Known function of t (an obvious example is provided by the presence of a downstream reservoir, in which case H(L,t) = Hr), Hp = H(L,t) (17a) and H=-H A F_+F_ (17b) vP VA B 2B Once again, an iteration procedure is utilized in obtaining the desired solution for Vp. 2.4.4 Points Interior from a Time-LiKe Arc In Section 2.2 it was observed that no constraints relative to the characteristic directions exist. With reference to Figure 7(d) for example, if the positive direction is arbitrarily assigned to the Ca family of characteristics and the negative direction to the C family, then it was noted that both V(x,t) and H(x,t) must be Known or prescribed along the time-liKe line x = O in order for a unique solution to exist in the region

-37interior from the left-end boundary. Assuming that these values are Known along that line (or at least at every intersection of the horizontal grid lines with the boundary), then Equations (lla) and (l1d) may be combined to solve for the desired values of V and H at all grid points located along the line x = Ax. Adding Equations (lla) and (lld) and rearranging, the resulting expression for the desired value of Vp is VA+VD HA-HD FA-FD (18a) vP -2 2B -- 4B This value of V may then be substituted into Equation (Ila); after rearrangement, the desired expression for Hp is FP+FA (18b) Hp = HA - B(Vp- VA) - 2 (8b Scrutiny of these equations reveals at least one motivation for the utilization of the second-order finite-difference approximations: unliKe the corresponding solutions involving space-liKe points, neither solution requires an iteration procedure. This is in direct contrast to the situation encountered awhen the first-order approximations are utilized, as was the case in the previous valve-stroKing investigations. (30,31) Similarly, if values of both V(x,t) and H(x,t) are Known or prescribed at all grid points along the line x = L, then the desired values of V and H at all grid points located on the line x = L - Ax may be determined, as in Figure 7(e). Adding Equations (llb) and (lic) and rearranging, VB+VE HB-E FB-FE (19a) Vp 2 2B BB Again, after substitution of this value of Vp into Equation (11b),

-38F +FB HP HB +B(Vp - VB) + 2 These latter two pairs of equations will be of fundamental importance in the formulation of suitable transient synthesis procedures. Since in this system of equations the computation of the desired values of the dependent variables originates at one time-liKe boundary and procedes across the grid to the other boundary, the necessity for the development of auxiliary boundary-condition equations analogous to those developed in Sections 2.4.2 and 2.4.3 is eliminated. In the discussion of relevant boundary-data problems presented in Section 2.2.3, the general case of this latter type of problem was considered. There it was observed that, in addition to the two data prescribed along a time-liKe arc, the Knowledge of one datum along a space-liKe arc is usually essential to the existance of a complete solution. However, as will be demonstrated subsequently in this investigation, the space-liKe bounding arcs encountered will be characteristic lines along which two compatible data will always exist. Therefore, the necessity to develop supplementary equations essential to the determination of the desired solution is likewise eliminated. 2.5 Complex Systems Although the preceding discussion has been restricted to the development of transient-calculation techniques suitable for a single pipe, the transient response of more complex piping systems can be readily analyzed as well. The transient calculations for each component pipe of a system involve the solution of interior-point equations and boundary-condition equations. As has been demonstrated, at every computational time step the

-39determination of both dependent variables at all interior points is independent of conditions existing at each boundary at that particular instant. Furthermore, if one end of the pipe terminates in an external boundary, i.e., one in which none of the other piping components of the system are connected, then conditions at that boundary may be determined using either the appropriate set of equations developed in Sections 2.4.2 or 24.43, or similar equations developed to model the behavior of the specific external boundary device under consideration. What remains, therefore, is merely to consider the analysis of conditions existing at internal boundaries - - interior junction points at which two or more of the system component pipes are connected. In the following sections two typical internal boundary situations are analyzed, and the necessary boundary-condition equations are developed. 2.5.1 Series-Junction Boundary Condition With reference to Figure 8, let one first consider the analysis of conditions existing at the junction of two pipes connected in series. (Although in the figure the pipes are depicted as having dissimilar diameters, the following analysis is equally applicable to any combination of discontinuity of diameters, head-loss properties, or wave speeds.) Assigning the positive direction to both characteristic families and assuming that all conditions are Known along the line t=to, two equations, Equation (Ila) for pipe 1 and Equation (llb) for pipe 2, are available to relate the four unknowns Vpl1 Vp2, Ipl, and HP2. The two additional necessary relationships may be developed by application of the unsteady continuity and energy equations to a control volume of fluid adjacent to the junction. Again assuming that the time rates of change of mass and

Pipe P Pipe 2 "2 Xi (a) Junction to lto + at I t- to ~'A'-2 N.-f N. N-2 N,-l NO O 2 Section Nos (b) Figure 8. Series-junction boundary condition: (a) definition sketch, (b) solution grids and notation.

internal energy existing within this small volume are negligible and that the Kinetic-energy and minor-loss terms are inconsequential, the resulting equations are AlVp =A2Vp P1 2P and HP HP2 Hp where Al and A2 are the respective areas of pipe 1 and pipe 2. Combining these four relationships to first eliminate the unKnown velocity terms, the result is C3-C4-C1F /2+C2FP /2 1 2 (20a) Hp = C +C 1 2 where C A/B1, C2 A2 /B, C3 = A V C(HA 1 FA/2), and C4 = 2 VB2 - C(H + FB /2). This value of Hp may then be substituted into the respective finite-difference equations; the results are C3-C1(H+Fp /2) (2Ob) V = P1 Al and C4+C2(H-P Fp/2) (20c) P2 A2 Since the Fp's are functions of the respective Vp's, the above set of three equations must be solved by iteration. By initially evaluating F p at VA 1 1 and Fp at VB (again, basing the initial evaluations upon the first-order 2 2 finite-difference approximations) and successively improving these values using the previously calculated values of Vp and Vp, the solution of Equations (20a) to (20c) is readily convergent. (See Appendix A.)

One final observation is relevant at this point; this boundarycondition calculation now imposes a restriction upon the heretofore arbitrary grid spacing, since At1 must equal At2. Therefore, it follows that L1 L2 alN1 a2N2 In order for N1 and N2 to be suitable integer numbers, minor adjustment of the pipe lengths and/or wave speeds may be required. In reality, however, this presents no real problem, since in most applications wave-speed values generally are not precisely Known.(30) 2.5.2 Branching-Junction Boundary Condition Now consider the analysis of conditions existing at the branching junction of Figure 9. Again assigning the positive direction to both characteristic families and assuming that all conditions are Known along the line t = to, three equations, Equation (lla) for pipe 1 and Equation (l1b) for both pipe 2 and pipe 3, are available to relate the six unknowns VP, VP, Vp, Hp, Hp, and Hp. The three additional necessary relation-1 2 3 1'2 3 ships may be obtained from the simplified continuity and energy equations, as before, and are A1Vp =A2VP + A3Vp and HP - Hp = HP P Combining these six relationships to first eliminate the unKnown velocity terms, the result is C4-C 5-C6-CF /2+C2FP2/2+C3FP /2 ________________________________ _ 2(2la) P C1+C+C+C 1a2 3

X Pp t Pipe 2 (at) Junction A. PI Pie 3 Pipe 9 B b t o A-2 Section Nos. Pipe 2 (b) Figure 9. Branching-junction boundary condition: (a) definition sketch, (b) solution grids and notation.

-44where C1 =A1/1 C 2 A2/B2, C3 = A3/B3, C4 A1 VA + C1(HA FA /2), I 1 5 A2 VB - C2(HB2 + FB /2), and C6 = A3 VB - C3 (HB3 + FB /2). 2 2 2 3 3 3 Substitution of this value of Hp into the respective finite-difference equations yields: V C4-C1 (HP+Fp/2) (21b) V A C5 +C2(Hp-Fp2/2) (21c) VP = A - 2 2 C6+C3(Hp-Fp3/2) (21d) 3 V = P3 A3 Again, this set of fourn equations must be solved by iteration; the same technique that was outlined for the solution of Equations (20a) to (20c) is equally applicable here. Boundary-condition equations for other internal boundary configurations can be readily developed in a comparable manner. 2.6 Initial and Steady-State Conditions In concluding this examination of the basic concepts and equations of transient flow, attention inevitably turns toward a consideration of two somewhat related topics: initial conditions and steady-state conditions. Tne fi-lrst topic is of interest because of the fashion by which the initial conditions influence and restrict the subsequent transient analysis or synthesis solutions; the second is important because of the dependence of one of the criteria of valve stroking -- elimination of residual transient

-45fluctuations -- upon a thorough understanding of the steady-state phenomenon. 2.6.1 Initial Conditions With reference to Figure 10, consider that the values of both dependent variables are initially Known along the space-liKe line t = to ( or at least at the N + 1 intersections of the vertical grid lines with the line t = to). Then from the discussion in Section 2.2.1, the line t = to is the domain of dependence of the region bounded by the positive characteristic line extending upward and to the right from the point t = to, x = 0 and by the negative characteristic line extending upward and to the left from the point t = to, x = L. Therefore, a unique solution exists within this region and is independent of subsequent control-device operation at either boundary. In addition to the initial existence of both variables along the line t = to, it is possible that one of the boundary relationships may be invarient. Again referring to Figure 10, suppose that along the left-end boundary one of the two dependent variables (or a relationship between the two) has a constant value. Then from the discussion in Section 2.2.3, a unique solution exists within the triangular region bounded by the line t = to, the left-end boundary, and the negative characteristic line which extends upward and to the left from the point t = to, x = L and intersects the line x = 0 at t = tN. As before, the solution is unaffected by subsequent control-device operation at the rightend boundary. Although not intrinsic to the preceding discussion, the consideration of initial steady-state conditions is also of interest. With reference to Figure 10(a), let one assume that steady-state conditions exist along the line t = to. Then VA = VB = VC = Vo, where Vo is the initial steady-uniform velocity in the pipe; HA = HC - Fo and HB = HC + Fo, where Fo is the function

-46N*2 tN+. tN,'i- I. I i I -I-:N-2 II'2 Section Nos x=O x=L Figure 10. Regions where solution is dependent upon initial conditions. Notation for (a) a steady-state interior point, and (b) a steady-state boundary point.

F evaluated at Vo. (These latter relationships follow directly from the elimination of the time-derivatives from Equations [5] and [6] and from the basic definition of F.) Incorporation of the above relationships into Equation (12a) and subsequent elimination of canceling terms yields Hp = HC' After substitution of this result into Equation (12b), FP Fo Vp+ Fp P 2B Vo 2B Since the function F is a strictly monotone-increasing function(3) of the velocity V, the only possible conclusion to be derived from this latter equation is that VP = VO Therefore, the point P has been demonstrated to be an initial steady-state point. Since the selection of points A, B, and C was completely arbitrary, then every point within the domain of dependence of the line t = to is an initial steady-state point. If, in addition to the initial existence of steady-state conditions along the line t = to, one of the boundary relationships is invarient, then the entire region within the domain of dependence of that boundary and the line t = to is a region of initial steady-state conditions. For example, assume that a constant-head reservoir is located at the upstream boundary of the pipe. With reference to Figure 10(b), VB = Vo, HB = Hr Fo, and Hp = H. After substitution of these identities into Equation (13b), F F Vp + 2B = V0 + 2B

-48As before, the only possible conclusion is that VP = V0 Therefore, from these interior-point and boundary-point evaluations, it is evident that the region of initial steady-state conditions is bounded by the line t = to, the left-end boundary, and the negative characteristic line which extends upward and to the left from the point t = to, x = L and intersects the left end boundary at t = tN. Coincidentally, this last example also demonstrates a second interpretation of the characteristic lines: lines along which disturbances in the flow are physically propagated (Ref. 1,6). 2.6.2 Final Steady-State Conditions As indicated previously, a most significant criteria of valve stroKing is the establishment of the final steady-state conditions in a piping system without the existence of residual transient fluctuations. Initially, of course, the transient flow condition originates when either one or both boundary conditions compatible with sorne initial steady-state conditi Lon are al'tered. Only upon cessation of these boundary modifications (that is, only after these boundary conditions have attained their final steady-state character) will the transient phenomenon subsequently begin to asymptotically (both mathematically and physically) approach the final steady-state solution due to the combined physical processes of reflection of the transient pressure waves and the action of viscous effects. The singular fashion by which the final steady-state solution may be established without this asymptotic process is indicated in Figure 11. Assuming that one has the freedom to arbitrarily select the charactekristic directions (in Chapter III the validity of this assumption is verified), the

tN#2 N - P ___ N- 2 A t 2 I 4E i I dX I o0o x=L Figure 11. Region of final steady-state solution.

-50positive direction is assigned to the Ca family and the negative direction to the C- family. Recalling that both variables must then be Known or prescribed along the left-end boundary, the velocity is arbitrarily prescribed to be Vf, the desired final steady-uniform velocity in the pipe, and the head as IIf, the final steady-state value compatible with Vf and either the boundary condition existent at that boundary or other Known flow conditions. Then V VG V Vf and HA = HG = HD = Hf. Incorporation of these relationships into Equation (18a) and subsequent elimination of canceling terms yields Vp= Vf. After substitution of this result into Equation (18b), Hp= Hf - Ff J where Ff is the function F evaluated at Vf. Thus, the point P has been demonstrated to be a final steady-state point. Since the selection of points A, D, and G was completely arbitrary, then final steady-state conditions must exist at every grid point located along the line x = Ax on or above the positive characteristic line which extends upward and to the right from the point t = to, x = 0 and intersects the right-end boundary at t = tN. This latter argument being true for the line x = Zhx, then the same reasoning must hold for the line x = 2Ax, and so on. Therefore, every grid point within the domain of dependence of that section of the left-end boundary for which t > t must be a point of final steady-state conditions, ultimately including every point located on the right-end boundary beginning at t = tN.

-512.7 Chapter Summary The transient synthesis procedures developed in subsequent phases of this study will draw abundantly upon the concepts presented in this chapter. As will be demonstrated, an unusual amount of flexibility exists relative to the arbitrary selection of the directions of the two characteristic families. It is possible, for example, to specify both directions as positive during an early stage of the controlled transient and then to arbitrarily assign the negative direction to one of the families during a later stage. In this investigation a valve-stroKing synthesis will refer to the determination of the unknown dependent variables at the two boundaries (and, therefore, the boundary control adjustments required to produce the desired transient condition) either when both variables are specified at one boundary or one variable is specified at each boundary. A characteristics analysis will refer to the determination of the complete history of the transient condition when the unsteady boundary conditions determined by the valve-stroring synthesis are imposed upon the system, and as such it will demonstrate the validity of the valve-stroking computations. Two broad classifications of valve stroking will be considered in this study. Active valve stro0ing refers to the synthesis and subsequent operation of a system in which the period of the entire transient may'be arbitrarily selected. In a passive valve stroxing synthesis and operation, a similar degree of control over the period of the transient is not possible. The digital computer programs essential to this investigation were executed on the IBM 7090/1410 system at the University of Michigan Computing Center.

III. VALVE-STROKING CONTROL OF SIMPLE PIPING SYSTEMS Having presented the fundamental properties and suitable techniques for numerical evaluation of the cliaracteristic solutions in Chapter II, consideration is now given to the application of these concepts and equations to the control of the transient phenomena in simple pipipig systems. 3.1 System with an Upstream Constant-Head Reservoir Attention is first directed to the most frequently encountered of the simple systeims: a single pipe originating at an upstream constanthead reservoir and terminating at a downstream control valve. With reference to Figure 12, cnsider that it is desired to modify flow conditions in the system from some initial steady-uniform condition (SSo) to some final steady-uniform condition (SSf) such that at time t = tf the final steady-uniform condition is established without the presence of subsequent resicdual transient fluctuations. From the discussions presented in Sections 2.2.3, 2.6.1, and 2.6.2, and with reference to Figure 12(b), the following observations are relevant: (1) Because of the presence of the invarient upstreamboundary relationship, H( 0,t) = H, the region of the initial steadyuniform condition is as indicated. (2) If the positive direction is now arbitrarily assigned to the Cu characteristic family and the negative direction to the C- characteristic family, then both data must be prescribed alcong the t-ime-liKe upstream boundary for a unique solution to exist in the region of interest (recognizing that the requisite compatible data do exist along the space-liKe cnharacteristic line bounding the initial steady-state region). (3) If the velocity Vr(t) is prescribed -52

'uoiqs;ou pus aueid -}-x (q) tqo~axs Uolqwu Lip (e):aTOsA~Sal peaq-cuwe.suoo q-Tm! uwasXs a-lCdus'Z1I ajnsa (q) 7 0 /7 (.VA:a4ndwo euozt /uisuovJ 9'7- _ ON_______ ____(/ I - 0H117 — ~!~i

-54along the upstream boundary such that at time t >tf-L/a, Vr(t) = Vf, then the final steady-uniform condition will be established as desired. (4) Lastly, no datum may be prescribed along the downstream boundary since it lies within the region of the unique solution of the given boundary-data problem. Therefore, the solutions Vv(t), Hv(t), and thus Kv(t) of the valve are solely dependent upon the initial conditions and the data prescribed along the upstream boundary. Having established the essential element of this problem -- the necessity of specifying the upstream velocity as some function of time -- the next logical step would be to speculate about how best to specify this transient boundary velocity in order to produce the m.ost orderly and controlled transition between the initial and final steadyuniform conditions. 3.1.1 Specification of the Variable Upstream Velocity Recognizing that along the upstream boundary the hydraulic grade line has a constant value ('thus HIt(O,t) = 0), one may investigate the consequence of requiring that Ht(x,t) = 0 throughout the flow. With reference to Equation (5), if Ht = O, then Vx = 0. Consequently H = 1(x) + C and V = V(+) +K are the most general relationships that IH and V may satisfy (C and K are constant terms). Then Hx = IP(x) and Vt = V'(t). Subsequent substitution into the simplified equation of motion, Equation (6), yields V'(t) + fv-IV = -gH'(x) 2D Since for a given pipe and fluid f = f(V), the left-hand side of the above equation contains only time-dependent terms while the righthand side contains only distance-dependent terms. Thus, each side must

-55be equal to a constant and the above expression is now identified as the incompressible surge equation where H'(x) is the slope of the hydraulic grade line. If the extreme permissible value of head desired at the valve is designated as HIm (maximum for closure situations, minimum for situations involving valve openings), this constant slope of the hydraulic grade line becomes H'(x) = (Hm-Hr)/L. Thus dV (H m-r) fV_|V (22) dt -g L 2D is suggested as the technique for specifying Vr(t) along the left-end boundary. One may next investigate the consequences implied by utilization of Equation(22). With reference to Figure 13, assume that the velocity at every grid point along the left-end boundary has been specified according to the second-orderl finite-difference approximation to Equation (22): (Hm-Hr) F +F.i+ (22a) V V.- v i+l i NB 2B (This approxitration follows frora the definition of the function F and from B = a/g, L/Ax = N; it is completely analogous to the comparable finite-diffLerence approximations oL' the characteristic equations and is solved utilizing the same iteration technique outlined for solution of the equations of Chapter II.) Thus H H HD = H and A G D Fr (Hm-Hr) FA+FG G A NB 2B (Hm-Hr) FG+FD v V - ) D G NB 2B

-56tN- 0 2 -A C -- __ _/ _ G,, -N_2 r I Ix'to 0 1 2 3 N-3 N-i N Section Nos. x=O x= L Figure 13. Region of established uniform surge condition.

-57Incorporation of these relationships into Equation (18a) and subsequent elimination of canceling terms yields VP = After substitution of this result into Equation (18b), (Hm-Hr ) Hp= H + r N Therefore, conditions at the point P have been demonstrated to satisfyr the incompressible surge conditions: Vx, = 0, and iH = a constant. Since the selection of points A, D, and G was completely arbitrary, then surge conditions must exist at every grid point located along the line x = ax on or above the positive characteristic line wShich extends upward and to the right from the point t = to, x = 0 and intersects the right-end boundary at t = t. This latter argument being true for the line x = Ax, then the same reasoningC must hold for the line x = 2Ax, and so on. Therefore, every grrid point withi n the doLmain of dependence of that section of the left'-end boundary over which the velocity has been arbitrarily prescribed using Equation (22a) must be a point of incompressible surge conditions. This ultimately includes every point along the right-end boundary lying within the aforemenltioned domain of dependence, beginning at t = tN, and the head at the boundary must then have the prescribed extreme value H 3.12 Study of the Frictionless System For a system in which viscous effects are so insignificant that they may be safely neglected, the concepts investigated above may

-58be extended and equations describing the entire history of the transient condition throughout the system may be developed. Although such a study would appear to have limited value, ultimately it will be demonstrated that the basic history of the transient condition is only slightly modifled by the presence of significant viscous effects; therefore, a thorough analysis of the functionless system provides valuable insight into the nature of the transient condition developed in the system in which viscous effects are important. For a given situation (all physical properties and initial and final steady-uniform conditions specified), the time of the complete transient tf is a function of the arbitrarily selected value of H.. For the frictionless system, Equation (22) may be integrated. The initial condition is V = Vo when t = L/a, the final condition is V = Vf when t = tf - L/a, and the result is L (Vo-Vf) (23) t= 2L/a + -_ r - ) I g (H-H In view of the fact that t > 2L/a, the quantity(' -Hr)must have the same sign as(Vo-Vf). The basic pattern of the transient condition imposed upon the system is considerably different for values of t > 4L/a than it is for values of t., between 2L/a and 4L/a. Referring to Figure 14(a), one may first consider the pattern for to> 4L/a. Since conditions in Zone II already have been demonstrated to satisfy the incompressible surge equations, for the frictionless system these equations, after integration and evaluation with the appropriate initial conditions, are

I: C\1 ~~~~~~~~~~~~~~~~~C)`~-.X 4.) mo x I 0 -~~~~ a) d 8 S OIt O ox ~o., xco "~~~~~~ aI, ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~r, A t3 11u3C) Oa-r.r-Ia) C~~~~~~~~~~C u\ ~ ~ ~ ~ ~ ~ ~ ~ I 0Sk \d -d ~d ed IC\ b ~.)~

-60(Hm-Hr) V(t) = Vol - g L (t - L/a), (24a) (H -H) H(x) = H + x (24b) r L Next by solving Equations (18a) and (18b) for a point (x,t) in. Zone I simultaneously with Equation (24a) evaluated at time tr and the condition that t = tr - x/a, (Hm-H) x-L (24c) V(x,t) = V- m L (t +- (2c) o 2 L a (H-H) xL H(x,t) = H (t + a ( (24d) r 2 L a Equations describing the history of the transient condition throughout Zone III may be developed in a similar fashion. These equations are V(xt) = V+ (H-H) + -L (24e) ( 2 L ( f a ) (Hm-Hr) x-L) (24f H(x,t) = Hr + 2 (tf - t +a(24 Evaluation of these equations at the right-end boundary enables one to explicitly determine the time-dependent value of the valve coefficient required to produce the desired transition between the initial and final steady-uniform conditions: (Hm-Hr) V _ m. t K (t) = 2 L 0 < t < 2L/a, (25a) (H -H ) m r K ( = V g -g L (t-L/a) = 2L/a < t < t - 2L/a, (25b) m (H -Hr) V2....L (tf -t) K (t) = - if t -2L/a < t < t. (25c) r2d a L r (t, -t)

These valve equations are the dimensional equivalents to Streeter's dimensionless valve equations. (2631) The limiting case for which the preceding analysis is valid is for tf = 4L/a. This case is of especial interest; the apex of the surge zone (now a triangle) just extends to the right-end boundary at t = 2L/a. The value of Hm necessary to produce this situation is (Vo-Vf ) (26) H =H +B m r 2 One may now consider the transient pattern that develops for values of tf between 2L/a and 4L/a. As may be readily seen from Figure 14(b), an analysis of the conditions developed in Zones I, II, and III performed in an identical fashion as before would yield the same results. Thus, Equations (24a) to (24f) are still valid. Then solving Equations (18a) and (18b) for a point (x,t) in Zone IV, the results are Vo+Vf (24g) 2 (V-V f) H = H + B f (24h) The Zone I and Zone III valve coefficient equations previously developed are still applicable, except that for this situation Equation (25a) is only valid during O< t<tf-2L/a, and Equation (25c) is only valid during 2L/a <t <tf. The constant valve coefficient that obtains during the central phase of the transient condition is then V +V Vo+Vf K = 2 t - 2L/a < t < 2L/a. (25d) V (V-V ) In concluding this examination of the frictionless system, one last observation is relevant. One recognizes that Equations (24h) and (26) are identical. Thus, Equation (24h) represents the maximum

-62(or minimum) value of head that can possibly be developed throughout the system when the proposed valve-stroKing control procedures are utilized. Even if an Hm were selected in excess of the value prescribed by Equation (26), thereby resulting in a value of tf less than 4L/a, the extreme value of head developed throughout the system would still be in accordance with Equation (24h). This immediately implies that if the system can physically accomodate that extreme value, then the minimum possible time of valve-stroking control in a simple system is 2L/a, not the 4L/a restriction of the previous investigations cited in the literature review. Finally, it is also pertinent to note that most discussions (Ref. 18,19,31) of the waterhammer phenomenon conclude that for "rapid" valve closures in a frictionless system (i.e., tf<2L/a) the maximum value of head rise developed at the valve is given by AH = AV. g This is the so-called "JouKows.1y surge". Yet by using a closure procedure predicated upon the concepts of this section, it is theoretically possible to close a valve in exactly 2L/a seconds, while at the same time restricting the subsequent head rise in the system to only one-half of that value. 3.1.3 Study with Friction Included For a system in which viscous effects are not negligible, explicit equations describing the entire history of the transient condition can not be developed and one must resort to numerical solutions utilizing the appropriate equations of Chapter II. For the valve-stroKing analysis, for example, the left-end boundary velocity at each grid point is specified according to Equation (22a), and Equations (18a) and (18b)

-63are used to advance the resulting solutions across the double-staggered grid of characteristics to the right-end boundary; the value of the valve coefficient of each right-end boundary grid point can then be calculated. For the characteristics analysis used to confirm the valve-stroKing analysis, interior-point conditions are evaluated with Equations (12a) and (12b), the left-end boundary conditions with Equations (13a) and (13b), and the right-end boundary conditions with Equations (15a) and (15b). Specific details of these solution procedures are described in Appendix B. A considerable amount of flexibility exists relative to Equation (22a). Either a value of Hm can be arbitrarily prescribed and the correspondinc value of tf numerically evaluated, or t4 can be prescribed and Hm then evaluated (see Appendix C). In either case the central surge zone will develop within the domain of dependence of the left-'nand boundary as discussed in Section 3.1.1. For values of tf between 2L/a and 4L/a this zone will be of triangular shape, as in the comparable frictionless system. Designating as H' the value of head created at the apex of this zone and developed m at time t,/2, then from Equation (24b) a (Hm-Hr ) H' H + (t( - 2L/a) (27) m r 2 L f since x = (t,-2L/a). In situations involving valve closures (i.e., V> V),'the above expression for H' represents the maximum value of head developed in the system. The maximum value of head encountered at tie valve occurs at t = 2L/a and is marginally less than the above value. (Recall that in the frictionless case the head developed at the apex of the surge zone and the constant head obtained in Zone IV were identical; in

this case, however, wall shear stresses accentuate the fluid deceleration, thus permitting somewhat lower values of head to exist in Zone IV.) In similar situations involving valve openings, the minimum value of head does occur at the valve at t = 2L/a and is marginally less than H' Yet another possibility should be mentioned relative to the arbitrary selection of tf. One can always specify tf and then specify that the upstream velocity vary linearly between V and Vf during the time L/a < t < tf - L/a. Such a strategy results in a very simple and straightforward computational procedure; for values of tf between 2L/a and 4L/a, valve-stroking control following this procedure may be preferred because of the simplicity of the procedure and because the extreme value of head developed in the system is essentially the same in either case. For values of tf significantly greater than 4L/a, specification of the upstream velocity according to Equation (22a) is probably preferred because, for the same value of tf, the extreme value of head developed in the system is less than is obtained utilizing the linear specification. As previously indicated, the minimum time for valve-stroking control of this simple system is tf = 2L/a. Although of doubtful practical interest because of the instantaneous valve motions required, it does provide valuable insight into the one restriction inherent in the valvestroking theory. In closure situations the maximum head (H' ) develops m at a point just inside the left-end boundary at t = L/a. (In this case the entire transient zone is a Zone IV triangle.) Conditions at this point may be evaluated with Equations (24g) and (24h), since the distance over which the friction terms of the finite-difference equations have to be integrated is negligibly small. To create this one-half reduction in velocity at the reservoir at t = L/a, a greater velocity reduction has to

-65occur at the valve because of the line-packing (9'31) phenomenon that exists in any real fluid system. In extremely long pipelines these viscous effects could be sufficiently significant to require that negative velocities develop at the valve during a central period of the transient condition. These negative velocities would be incompatible with the physical operation of a simple control valve and could only be produced by other control devices (i.e., an accumulator) operated in conjunction with the valve. A computer program which included both the valve-stroking analysis and the confirming characteristics analysts was written to illustrate the effect of system coneration according to the principles suggested in the preceding sections. Results of several representative valve-closure studies are illustrated in Figure 15. For convenience, the valve relationships illustrated are the dimensionless tau relationships (Ref. 18,31), obtained by dividing the instantaneous value of the valve coefficient by the initial steady-state value. The pertinent systemi parameters are: H = 100 feet, V 5 feet per second, Vf = O L = 4000 feet, O a = 3200 feet per second, f =.025, and D = 1 foot; thus L/a = 1.25 seconds, Ho = 61.2 feet, and H. 100 feet. The significant values of head prescribed or developed in the system are summarized in Table 1. These results illustrate remarKably well the fundamental differences in the transient condition developed for various values of tf; especially, well illustrated is the distinction between system control in closure times less than 4L/a and control in times greater than 4L/a. In summarizing the results of this section, the varlve-stroKing procedure proposed creates an extremelyr controlled transient condition

'a tOAas1a p aq-4uWSUo q%.TA muasA's alTduTs V Jo oTuo iuWO UXMoas-aTlA'* axnlTi spuooaS u/ 9w/i 0 8 9 O I zI 0 ii l iD/79_j /r\' \ ___..~~' ---. —40 0/ ___..........oo. lb -99

-67TABLE I RESUMIE OF ANALYTICAL COMJPUTER RESULTS tf Hm H Max. H at Valve second s feet feet feet 2.50 -- 348.4 342.7 3.75 584.2 342.1 339.8 5.00 336.0 336.0 336.0 7.50 212.3 -- 212.3 throughout the system: all velocities and pressures are nearly linear functions of time and distance, an arbitrarily selected extreme value of pressure is maintained at the valve during a central phase of the transient condition, and the final steady-uniform condition is established without the development of residual transient fluctuations. Lastly, it is pertinent to note that although the preceding treatment has been predicated on the assumption that steady-uniform conditions originally exist in the system, the theory developed here is not necessarily restricted to such situations. With reference to the discussion of Section 2.6.1 and Figure 12(b), if the values of both dependent variables are initially unown along the line t = O, then a unique solution does exist within the region labeled "SSO" (regardless of whether or not it is a steady-state region). Hence, the velocity Vr(L/a) has an established value and Equation (22) or (22a) may then be utilized to prescribe V (t) along the left-end boundary. Again the surge r condition is established throughout the domain of dependence of the left-end boundary, the extreme value of head is maintained at the valve during the central phase of the transient condition, and the final steady-uniform condition can be established as desired.

-683.2 S stem with a Downstream Constant-Velocity Condition As another example of a system for which a valve-stroKing procedure may be developed, consider the system illustrated in Figure 16. Assume that it is desired to modify the upstream reservoir elevation from an initial elevation Hr to a final elevation Hr while maintaining o rf a constant velocity Vo at the downstream valve. As in the first study it will be assumed that steady-uniform conditions originally exist in the system (although, as in the former case, that restriction is not essential to the development of the theory) and that at time t = tf the final steady-uniLorn condition is to be established without the presence of subsequent residual transient fluctuations. It will be assumed that the elevation of the reservoir can be precisely controlled to satisfy any arbitrary function of time. From the discussions referred to in Section 3.1 and with reference to Figure 16(b), the following observations parallel those of Section 3.1: (1) Because of the presence of the invarient downstreamboundary relationship, V(L,t) = Vo, the region of the initial steadyuniform condition is as indicated. (2) If the negative direction is now arbitrarily assigned to the C+ characteristic family and the positive direction to the C- characteristic family, then both data must be prescribed along the time-liKe downstream boundary. (3) If the head Hv(t) is prescribed along the downstream boundary such that at time t >tf - L/a, Hv(t) = Hf, then the final steady-uniform condition will be established as desired. (4) No datum may be prescribed along the upstream boundary since the solutions Vr(t) and Hr(t) are solely dependent upon the initial conditions and the data prescribed along the downstream boundary. Having again established the essential element of this problem — the need to specify the downstream head as some function of time -- one

-69NrItal HGL ~ g- Fina/ HGL H (o) f-L/a Specify'. Transient Zone V Compute: ---- (t) Hr (t).| Vr (tt L/a 0 0 L (b) Figure 16. Simple system with constant-velocity condition: (a) definition sketch, (b) x-t plane and notation.

-70again can speculate about how best to do this in order to produce an orderly and controlled transition between the two steady-uniform conditions. 3.2.1 Specification of the Variable Downstream Head Recognizing that along the downstream boundary the velocity has a constant value (thus Vt(L,t) = 0), one may investigate the consequence of requiring that Vt(x,t) = 0 throughout the flow. If V. = O, then V = V(x) + C is the most general relationship that V can satisfy (C is a constant). Then V = V'(x) and from Equation (6), gH'(x) = - rlvl 2D It is apparent that H'(x) is also a function of x only, since f = f(V). This equation may then be integrated with respect to x and the result is some function involving only x, F(x). Therefore the most general relationship that H may satisfy is H = F(x) + G(t) + K (K is a constant). Thus Ht = G'(t) can be a function of t only. The continuity equation, Equation (5), then has the form 2 a V' (x) = -G'(t). g Again, for a function involving only x to be equal to a function involving only t, the two functions must be equal to a constant. Since G'(t) must thus be constant and represents the rate at which the hydraulic grade line changes elevation, dH H f-H~ dH f o 2L/a (28) is suggested as the technique for specifying H_(t) along the right-end boundary.

-71Again, one may now investigate the consequences implied by utilization of Equation (28). With reference to Figure 17, VB = V V V and (Hf-Ho ) HH = HA + (tf-2L/a) At, (Hf-H ) H = H + (t f'2La ) t E H At. Incorporation of these relationships into Equation (19a) and subsequent elimination of canceling terms yields 1 (H -H0) P Vo +B (t-2L/ At After substitution of this result into Equation (19b), Fp+Fo Hp =HH + 2 Therefore, conditions alt the point P have been demonstrated to satisfy the restrictions of the preceding analysis: Vt = O, Vx = a constant, and Ht = a constant. Since the selection of points B, E, and H was completely a-bitrary, these results permit a conclusion analogous to that of Section 3.1.1: every grid point within the domain of dependence of that section of the right-end boundary over which the head has been arbitrarily prescribed using Equation (28) must be a point at which the specified uniform transient condition is established. 3.2.2 Study of the Frictionless System As in the first system considered in this chapter, equations describing the entire history of the transient condition throughout a

-72tN-+2 I X - -., —-_ 4 - tN tl ---..' --'i...1 —''90 O / 2 3 N-3 N-2 N- N Section Nos. x= 0 x-L Figure 17. Region orf established uniform transient condition.

-73frictionless system may be developed. Again, they provide insight into the transient condition developed in the system in which viscous effects are important. As before, the basic pattern of the transient condition imposed upon the system is considerably different for vilues of tf > 4L/a than it is for values of tf between 2L/a and 4L/a. Referring to Figure 18(a), one may first consider the pattern for t f > 4L/a. Since conditions in Zone II already have been demonstrated to satisfy the specially restricted forms of Equations (5) and (6), for the frictionless system these equations, after integration and evaluation with the appropriate initial conditions, are 1 (Hf-H0) (L-x) V(x) = Vo + (t2La -x) (29a) (Hf-Ho) H(t) = (tf-2L/a) (t - L/a) (29b) Next by solving Equations(19a) and (19b) for a point (x,t) in Zone I simultaneously with Equation (29b) evaluated at time tv and the condition that t = t - (L-x)/a, (H -H 0) V(xt) = V +2 (t - x) (29c) o 2B (t f-2L/a) - a 1 (Hf-Ho) H(x,t) = H + - (t f x) a (29d) Equations describing the history of the transient condition throughout Zone III may be developed in a similar fashion. These equations are V(xt) = V + 1 ( L/) (tf x

'~e/S > 4 (q) Pm'uei B < a (E) JIoj snTmId 4-x uno~T puoo 4!TOOTIA -4tm4SUOo G4TAu L4suas sa8UOS TUOT.J aTduWIS *R GIn5l3T (q) (0) x x 7 0 7 O 0% o 7/7 0/7 I/72i'x / A O/7 7-_1 97 OA iT~~~ 4;7- / WI

-751 XH) H(x,t) = H --' (t t - a) (29f f 2 (tf-2L/a) f a Evaluation of these equations at the left-end boundary enables one to determine the time-dependent value of H required to produce the r desired transition between the initial and final steady-uniform conditions: 1 (Hf-Ho) H (t) = H + - 2 t o < t < 2L/a, (30a) (Hf-Ho) H(t)= H-2L/a (t - L/a) 2L/a < t < t - 2L/a, (30b) i (Hf-Ho) Hr(t) = Hf 1 (t-2L/a) (tf - t) tf - 2L/a < t < tf (30c) One may now consider the transient pattern that develops for values of tf between 2L/a and 4L/a. As may be readily seen from Figure 18(b), an analysis of the conditions developed in Zones I,II, and III performed in an identical fashion as before would yield the same results. Thus, equations (29a) to (29f) are still valid for their respective zones. Solving Equations (19a) and (19b) for a point (x,t) in Zone IV, the results are (Hf-Ho) V= V + f. (29g) 0 2B H +H H = f (29h) 2 The Zone I and Zone III equations previously developed for H r(t) are still valid, except that for this situation Equation (30a) is only valid during 0 < t < t - 2L/a, and Equation (30c) is only valid during 2L/a < t < tf. The constant value of H that obtains during the central phase of the transient condition is then given by Equation (29h):

H +H H= f t -2L/a < t < 2L/a. (29h) Lastly, the time-dependent right-end boundary valve coefficient may be determined by evaluating Equation (29b) at x = L: V K (t) = L/a < t < t - L/a (31) V. -H) + (t-2L/a) (t - L/a) Since the boundary velocity is invarient and the method of specifying the transient elevation of the hydraulic grade line at the valve is independent of viscous effects, Equation (31) is perfectly general and is not restricted only to the frictionless system. 3.2.3 Study with Friction Included For a system in which viscous effects are not negligible, one must again resort to numerically-oriented solutions. For the valvestroking analysis, the right-end boundary head at each grid point is specified according to Equation (28), the unsteady valve relationship is computed with Equation (31), and Equations (19a) and (19b) are used to advance the solutions across the grid to the left-end boundary. For the characteristics analysis, interior-point conditions are evaluated with Equations (12a) and (12b), the left-end boundary conditions with Equations (13a) and (13b), and the right-end boundary conditions with Equations (15a) and (15b). A computer program which included both the valve-stroking analysis and the confirming characteristics analysis was written to illustrate the effect of system operation according to these principles. Results of several representative studies are illustrated in Figure 19. Again, the dimensionless valve relationships are presented. The pertinent

200 /75 ~ 150 es / i /25 lr,.0 X i -.7 -.6_ 0 2 4 6 8 /0 T/me in seconds Fi gure 19. Valve-stroking control of a simple system with constant-velocity condition.

-78system parameters are: VO = 3.5 feet per second, H = 100 feet, H = 200 o f feet, L = 3200 feet, a = 4000 feet per second, f =.035, and D = 0.5 feet; thus L/a = 0.8 seconds, Ho = 57.4 feet, and Hf = 157.4 feet. The results again confirm the theory; in every case the downstream velocity is maintained at a constant 3.5 feet per second throughout the history of the transient condition and the final steady-uniform condition is established as desired. Of particular interest is the leading case of tf = 2L/a. This is analogous to the one round-trip valve-stroking control of the previous study. Again, the instantaneous control motions required by the theory would appear to exclude most practical applications. In summary, the proposed valve-stroking procedure creates a highly controlled transient condition throughout the system: all velocities and pressures are nearly linear functions of time and distance, the transient elevations of the hydraulic grade line are everywhere bounded between the initial and final steady-state grade lines, and the final steadyuniform condition is eatablished without the development of residual transient fluctuations. 3.3 System with an Upstream Constant-Relationship Condition As a third example of a simple system for which a valve-stroking procedure may be developed, consider the system illustrated in Figure 20. Instead of the simple upstream boundary relationship provided by the presence of the constant-head reservoir, the upstream boundary condition is now the constant-relationship condition of the fixed orifice: v =K ff — H'. r o r r (Before proceeding further, it should be emphasized that any other boundary condition involving a fixed relationship between the dependent variables could be considered, and furthermore the relationship need not

-79Hr~jrFinal HGL r HI H;Initial -GL H (0) f ff -L/a Tronsient t Ze (t)Compute, Specify. H1(f) HI'(t) K(t SSvo V0 O L x (b) Figure 20. Simple system with constant-relationship condition: (a) definition sketch, (b) x-t plane and notation.

-80be in equation form; a tabular relationship would be entirely acceptable. The H-V relationship of a centrifugal pump provides an obvious example.) Again consider that it is desired to modify flow conditions in the system from some initial steady-uniform condition (SSo) to some final steady-uniform condition (SSf) such that at time t = tf the final steadyuniform condition is established without the presence of subsequent residual transient fluctuations. (Again, the assumption of an initial steadyuniform condition is adopted for convenience; it is not intrinsic to the theory.) With reference to Figure 20(b), it is evident that the analysis of the similar boundary-data problem presented in Section 3.1 is equally applicable here; it will not be repeated. Again, the crucial element in the development of a suitable control theory would appear to be that of specifying either of the upstream dependent variables as some arbitrary function of time; the other variable would then be completely specified by the prescribed fixed-orifice relationship. 3.3.1 Specification of the Upstream Variables An examination of the governing partial differential equations conducted in a similar fashicanto those of the previous studies fails to provide any meaningful criteria for upstream variable specification for this system. it is impossible, for example, to establish and maintain a uniform hydraulic grade line throughout the system during a central phase of the transient because of the variable head condition that prevails on the downstream side of the orifice. However, one can arbitrarily specify either of these upstream variables as some prescribed function of time. Following this direction, let us specify that the upstream velocity vary linearly between the initial steady-uniform value (Vf):

-81(v f-vo) Vr(t) = V + (f2L/a) (t - L/a) L/a < t < t < tf- L/ao (32) (Recall that in the frictionless constant-head reservoir system the upstream boundary velocity is a linear function of time. ) As with the previous studies, equations describing the entire history of the transient condition throughout a frictionless system may then be developed. While again providing insight into the condition developed in the system in which viscous effects are important, these equations are rather lengthy and will not be presented here. Relative to the extreme value of head developed throughout the system, these equations demonstrate that for tf > 4L/a the extreme value occurs at the valve at t = tf - 2L/a; for values of tf between 2L/a and 4L/a a zone analogous to Zone IV of Figure 14(b) is developed throughout which the extreme condition is constantly maintained. For a system in which viscous effects are not negligible, one must again resort to numerically-oriented solutions. For the valvestroking analysis the left-end boundary velocity is specified according to Equation (32), the corresponding head is computed from the prescribed fixed-orifice relationship, and Equations (18a) and (18b) are used to advance the solutions across the grid to the right-end boundary; the value of the valve coefficient at each right-end boundary grid point can then be calculated. The confirming characteristics analysis is identical to those previously discussed, except that the left-end boundary conditions are evaluated using Equations (14a) to (14c). Scrutiny of typical results generated by a program written to illustrate this suggested valve-stroking procedure reveals that whenever tf > hL/a the extreme value of head encountered throughout the system does develop at the valve at t = t - 2L/a. For situations involving

-82rapid valve closures in times less than 4L/a, the maximum value of head developed in the system occurs at the apex of Zone IV at time tf/2. As in the earlier study, the maximum value encountered at the valve is marginally less than the system maximum. Pertinent system parameters and results of a typical study are presented in Section 3.3.2. Scrutiny of these results also reveals a significant deficiency in this procedure: unlike the procedure developed in Section 3.1, specification of the upstream boundary velocity according to Equation (32) does not permit an arbitrarily selected extreme value of pressure to be developed at the valve. If a predetermined extreme value is not to be exceeded, then the appropriate value of t and, the consequent valve motion required to produce the desired transient condition can only be determined by numerical experimentation. The alternate procedure suggested in the next section overcomes this obvious disadvantage. 3.3.2 Preferred Solution Recall that in the constant-head reservoir system of Section 3.1 the suggested valve-stroking procedure resulted in the following head-time variation at the valve: (1) a nearly linear transition from H to H o m during the first 2L/a seconds; (2) the prescribed value of H maintained m at the valve during the central phase of the controlled transient condition; (3) a nearly linear transition from H to Hf during the last 2L/a m f seconds. This results suggests the following alternate strategy to the procedure outlined in Section 3.3.1. From the discussion presented in Sections 2.2.3, 2.6.1, and 2.6.2, and with reference to Figure 21, if the positive direction is arbitrarily assigned to each characteristic family, one datum must be prescribed on each of the time-like boundaries. The fixed-orifice

~uoInflos pa iajaid woJ atreTd.-x'uoT4Tpuoo dtqsuoT.Iva-qtXW~SUOO qITA mwa8sXs aTdurFS *IT a.nSTlj x 7 0 (y0/- 7 /usuJ /7A;anduow 0/7 i 74? flSS, alloz

-_84relationship provides the upstream datum. One may arbitrarily specify that the downstream head change linearly during the first 2L/a seconds to an arbitrarily selected extreme valve H and that it be maintained at m that value during the central phase of the transient condition: (H -Ho) H (t) = H + 2L t 0 < t < 2L/a, (33a) H (t) = H 2L/a < t < tf- 2L/a. (33b) A unique solution then exists in the region enclosed by the time-like boundaries, the space-like line t = 0, and the negative characteristic line extending from the right-end boundary at t = tf - 2L/a and intersecting the left-end boundary at t = tf - L/a. This constant value of Hm is prescribed along the right-end boundary until such time that the corresponding value of velocity obtained L/a seconds later at the left-end boundary equals the final steady-uniform velocity (Vf). At that point one may now arbitrarily assign the positive direction to the C+ characteristic family and the negative direction to the C- characteristic family and specify that along the upstream boundary Vr(t) = Vf. Thus the final steady-uniform condition will be established as desired, and a unique solution will exist in the traingular region bounded by the negative characteristic line (the dashed line illustrated in Figure 21) and by the positive characteristic line bounding the final steady-state region. The necessary numerical solutions again utilize the appropriate equations of Chapter II. The downstream head is prescribed according to Equations (33a) or (33b); the downstream velocity is determined from Equation (I7b). Interior point conditions are then evaluated diagonally along a C- characteristic utilizing Equations (12a) and (12b), and the left-end boundary conditions are evaluated using Equations (14a) to (lhc).

-85This solution procedure is terminated when the left-end velocity equals Vf. The velocity along the boundary is then specified to be Vf and Equations (18a) and (18b) are used to advance the solutions in the upper triangular region of the transient zone across the grid to the valve boundary. Specific details of the solution procedure are described in Appendix B. The confirming characteristics analysis is identical to that described in the preceding section. A computer program which included both the valve-stroking analysis and the confirming characteristics analysis was written to illustrate the effect of system operation according to these concepts. Results of two typical studies are illustrated in Figure 22; results of several studies are summunarized in Table II and compared with the results of the computer studies described in Section 3.3.1. The pertinent system parameters are: Hr = 165 feet, VO = 4 feet per second, Vf = O, L = 3220 feet, a = 3220 feet per second, f =.025, K =.8, and D = 0.5 feet; thus L/a = 1.00 second, H' = 140 feet, H' = 165 feet, H = 100 feet, and H = 165 r r o f 0 rf feet. TABLE II CO'PASRISON OF ANALYTICAL COMPUTER RESULTS Upstream Velocity Specification Do.mstrearn Head Specification iax. l- at MaXo H in Max. H Max. H tf Valve System t. Specified Obtained seconds feet feet seconds feet feet 3.00 344.2 346.2 3.96 344.2 340.2 4.00340.i2 340.2 3.99 340.2 340.0;.00 T258.7 258.7 5.58 258.7 258.7 8 *~~_. oo 22 8.8 6.97 228.8 228.8 10.00 213.4 23.4 3.7 213.4 213.4

-86. 400 e = 3.96 300 =8/.07 zoo Zr e in /00.75 =3r -96.50 8.. 0 2 4 6 8 /0 Time in Sec cds Figure 22. Valve-stroking control of a simple system with constant-relationship condition.

-87These results illustrate in a most remarkable way the differences in the transient conditions obtained utilizing the two procedures. This is particularly true for the slower valve closures since the head developed at the valve by the procedure outlined in Section 3*3.1 continually rises during the central phase to a maximum value 2L/a seconds before the cessation of the valve motion, whereas the maximum value developed by the preferred procedure proposed in this section is maintained at the valve during the entire central phase of the transient condition. Thus it is not unexpected that, for the same mximu value obtained, the time of valve operation required by the preferred procedure can be considerably less than that required by the forrier one. The distinction between system control in closure times less than 4L/a and control in times greater than 4L/a is again illustrated rather well. Note that in the preferred solution technique, a direct analogy exists with the control procedures developed for the constant-head reservoir system. In either case an extreme value of downstream head is selected that may not be attained if, for the specific system conditions that obtain, the consequent value of t is less than 4L/a. Since the maximuml value f head developed in the system is essentially the same for control times between 2L/a and tL/a, valve-stroKing control following the upstream-velocity specification procedure is probably preferred (because of the simplicity of the computational procedure) for systems that can physically accomodate such rapid closures. In summary, the proposed valve-strocing procedure creates a highly controlled transient condition throughout the system: unliike the previous stud ies (30,31) of constant-re lationship boundary- cond ition systems, a predetermined extreme value of pressure can be developed and

-88maintained at the valve during a central phase of the transient condition; again, the final steady-uniform condition is established without the presence of subsequent residual transient fluctuations.

IV. VALVE-STROKING CONTROL OF COMPLEX PIPING SYSTEMS The valve-stroKing concepts and procedures essential to the deterministic control of the transient phenomenon in the simple piping systems of the last chapter may be readily extended to develop suitable control techniques for the elementary complex systems as well. 4.1 Series System Attention is first directed to the most frequently encountered of the elementary complex systems: a system of two pipes connected in series with the upstream pipe originating at a constant-head reservoir and the downstream pipe terminating at a control valve. With reference to Figure 23, again consider that it is desired to modify flow conditions in the system from some initial steady-state condition to some final steadystate condition such that at time t = t.. the final steady-state condition is established without the presence of subsequent residual transient fluctuations. (Throughout this chapter, as before, the assumption of an initial steady-state condition is adopted for convenience only.) Recall now that in the development of the control techniques for the simple systems of Chapter III, the essence of the problem -- the really crucial element. — was the necessity to specify one of the boundary dependent variables as an arbitrary function of time. It is instructive to note that a generalized criterion exists relative to either transient analysis or synt'lhesis investigations of all complex systems: if there are m vunnown boundary dependent variables and n available equations, then m-n of the variables must be speciAfied to obtain the desired solution. The only restriction is that such specification may not violate one of the governing equations: i.e., specification of all of the -89

gum-O........ 17a HGL WNW/b HGL Hr H. H. H~ ---— Pioe-IL- -f-Ho Figure 23. Definition sketch of series system with upstream constant-head reservoir.

-91velocities at an interior junction would violate the Junction continuity equation. In the series system of Figure 23, for example, there are a total of six unknown variables (the velocity at the reservoir, two velocities and the head at the junction, and the velocity and head at the valve) and only five available equations (two characteristic equations for each pipe and the series-junction continuity equation). Therefore, specification of one of the unlmno'wn variables completely specifies the transient condition imposed upon the system. I.1.1 Evaluation of the Existing Technique From an extension of the discussions presented in Sections 2.2.3, 2.6.1, and 2.6.2, and with reference to Figure 24, the following observations are relevant: (1) Because of the presence of the invarient reservoir-boundary relationship, Hl(O,t) = Hr, the regions of the initial steady-uniform conditions are as indicated. (2) If the positive direction is now arbitrarily assigned to the C+ characteristic family and the negative direction to the C- characteristic family, then both data must be prescribed along the time-lit-.e reservoir boundary for a unique solution to exist in the region of interest of pipe 1; since both data are thereby established at the upstream boundary of pipe 2 (the existence of the conmrn boundary head and the series-junction continuity equation), a unique solution likewise exists in pipe 2. (3) If the velocity Vr(t) is prescribed along the reservoir boundar-y such that at tine t > t-L/a -L2/a2, Vr(t) = V], then the desired final steady-state condition will be established tr-oughnout the system as illustrated. (4) Again, no datum may be prescribed along the dowmstream boundary since it lies within the region of the unique solution of the given boundary-data problem. The

f 1 ss. Yf f4 -L2/a ZI Compute: V Pf VV (f) 44r -L,/o, -L,/o, rTransient Trnsient Hv f 2LI'L /O/ 162' Surge Compute: Known: Surge f -2L,/oI,-2yei2 Zone Zone P Speify., N V. Vr t N H (f) A7P L~ 2L/l 2L0 0 0~~~~~0 L-1 / t L7igre24 Zone p a Zone Sst /O 23 0 LI 0 L2 X/ X2 Fi gure 24. x-t plane and notation for series system with upstream constant-head reservoir.

-93solutions Vv(t), HV(t), and Kv(t) are solely dependent upcn the initial conditions and the data prescribed along the upstream reservoir boundary, Streeter and Wylie(31) suggest that the reservoir-boundary velocity be specified according to Equation (22), except that the extreme permissible value of head desired of the value (Hm) is replaced by the extreme permissible value of head desired at the Junction (H ). Since mj the discussion presented in Section 3.1.1 is equally applicable here, the existence in pipe 1 of a central zone of incompressible surge conditions is assured. While the strategy is reasonable and certainly acceptable, experience indicates that during rost transient conditions the extreme value or pressure sustained in the system obtains at the valve, not at the Junction, and as before it would be desirable to identify and limit that value a priori. Recognizing that at the junction the head developed during the central phase of the transient condition is the constant value H -- thus conditions during that time are identical to those provided by the presence of a constant-head reservoir -- one may speculate about whether a central surge zone is propagated into pipe 2. If this be the case, then the incompressible surge equations for each pipe can be related to permit the selection of Hm. in terms of Hm and the dilemma J is favorably resolved. If a surge zone is propagated into pipe 2, then conditions in each pipe must satisfy Equation (22). Each equation may be multiplied by the respective area of the pipe and equated, since Q1 = Q2: Ad- / L~r, r h = -g L1 -H H + h (V H +hf(V,) H + hmj2 The friction term taies the indicated form since L fVIV = h () g 2D fh(V)

-94. After rearrangement, A1 A2 A2 A1 hL1 (V1) L h4 (V) H - H (H -H). (34) 1i1 2 2 2 j 1 3 Since the right-hand side of the above equation must be a constant, then 1hf A ( 1) constant. But this constant must be zero, since hf (0) = h, (O) = 0. Therefore 1'2 A1 A2 L f A L fA (35a) 1 ~22 f2 is the necessary relationship between the viscous resistance properties of each pipe essential to the establishment of a zone of incompressible surge conditions in pipe 2. If the Darcy-Weisbach friction factors can be considered to have a constant value for each pipe, then the above criterion becomes fl f2 (35b) D D One irnediately recognizes that either of the above criteria rarely,would be satisfied. If they are, however, then the desired relationship between H and I may be obtained by rearrangement of the mj right-hand side of Equation (34): A1 A Al 2 L1 r L2 M (36) 1 2 in A1 A L1 L2 One does recognize that a frictionless system satisfies the criterion of Equation (35b) and thereby does permit extreme system pressures ito be limited to arbitrarily established values. Relative to the frictionless system, equations describing the entire history of the

-95transient condition throughout the system may be developed analogous to those of the simple system of Section 3.1.2. These equations are so lengthy (because of the several parameters involved) and numerous (because of the compound possibilities of values of tf and relative values of L/a in each pipe) that they have diminished utility for providing insight into the controlled transient condition developed in the viscous system; they will not be presented here. For a system in which viscous effects are not negligible, one must again depend upon numerically-oriented solutions. For the valvestroking analysis the reservoir-boundary velocity is specified according to Equation (22) (modified only to the extent that H replaces H ), and m. Equations (G1a) and (18b) are used to advance the solutions across the grid of pipe 1 to the junction boundary. The common junction-boundary head and continudity equation establish the junction boundary solutions for pipe 2, and Equations (18a) and (18b) again permit the calculation scheme to advance to the valve boundary; the value of the valve coefficient at each grid point can then be determined. The confirming characteristics analysis is virtually identical to that of the simple systems, except that Equations (20a), (20b), and (20c) must now be utilized to evaluated all conditions at the junction. A computer program was again written to illustrate the effect of system operation according to the principals suggested in this section. Pertinent system parameters and results of a typical study are presented in Section 4.1.2. In yet another study the friction factors here selected according to Equation (35b); the validity of that criterion was confirmed and a central surge zone did develop in pipe 2 as illustrated in Figure 24.

-964.1..2 Preferred Solution Having demonstrated the inherent deficiency of the existing technique for control of the series system, attention is now directed toward the following alternate strategy: recalling the preferred solution technique developed in Section 3.3.2, and with reference to Figure 25, the positive direction is arbitrarily assigned to each characteristic family and the downstrean head is specified to change linearly from H to Hm during the first 2Ll/al+2L2/a2 seconds and maintained at that value during the central phase of the transient condition: (H -H )t (t) Ho + 2 0 < t < 2Ll/al + 2L2/a2 (37a) v o 2L /a + 21'2 - 11 22' m 11 L t2 2t H(t) Hm2/a2 < t < tf 2L /a + 222 - 2L/a- 2L2/a2 * (37b) A unique solution again exists in the lorer regions of the transient zones cclose6 by, the tirze-line boundaries, the space-lire line t = 0, and the rnegative (dashed) characteristic lines extending from the valve to the reservoir. Again, the constant value of H is prescribed along the valve m boundary until such time that the correspondirng value of velocity obtained L1 /a1+L2/a2 seconds late- at the reservoir boundary equals the final steady-wuiform velocity (V1 ). At that point one may again assign the positive direction to the C+ characteristic family and the negative direction to the C- family, specify that along the upstream boundary V ('t) = V1l, established the final steady-state condition as desired, and obtain unique solutions in the upper triangular region of the transient zone of pipe 1 and the upper trapezoidal region of pipe 2.

5$ Comnpute: L' Compute. f L2 1r2 vv V (f.) Known: v (t) LA |L-L /a Transient / ( Zone Zone 2L/2 2 Compute: Compute. Specify. 2(1)-" l.() H. () V. (t) "v (t),H. (t) rlonlV et)n c0 mpute Compute:Specifyt.. 12 ~X Figure 25. x-t plane and notation for series system with upstream constant-head reservoir — preferred solution.

-98The necessary numerical solutions are patterned after those of Section 3.3.2. The head at the downstream valve is prescribed according to Equations (37a) or (37b); the downstream velocity is determined from Equation (17b). Interior point conditions are then evaluated diagonally along a C- characteristic as before, and the series-junction boundary cond itions are evaluated using the appropriate equations. Again the cornmputations proceed diagonally along the C- characteristic of the upstream pipe, and the reservoir-boundary conditions are evaluated using Equations (13a) and (13b). This solution procedure is terminated when the reservoir-boundary velocitr equals V,. The velocity along the boundary is then specified to be Vr, and the remainder of the solution is obtained in the upper regions of the transient zones in a fashion identical to trat of the preceding series-system vralve-stroKing analysis. Specific details of the solution procedures are described in Appendix B. The confirming characteristics ana0lysis is 4identical to that described in the preceding section. A computer program which included both the valve-stroking analysis and the confirming characteristics analysis was written to illustrate the e-Lfect of system operation according to these concepts. Results of two ltypical studies are illustrated in Figure 26; results of several studies are summarized in Table III and compared with the results of the computer studies described in Section 4.1.1. The pertinent system parameters are: H 125 feet, V0 2.56 feet per second, V = 4.00 1 02 feet per second, V = V = 0, L1 = 3500 feet, L2 4800 feet, Al = 3500 fl f2 - feet per second, a2 = 4000 feet per second, fl =.022, f2 =.020, D = 1.25 feet, and D2, = 1.00 feet: thus L1/a1 + L2/a2 = 2.2 seconds, H. = 118.7 feet, H = 94.9 feet, and H = H = 125 feet. O f

ITOAoaSaa pdt-,a-u4tsuoo uroaa~sdn q,~T waqsXs saTJas W JO TOaguoO 2UTXO.IS-AT *A'9g aenTIl spuo-3S u! aW.L Oz 9/ i/ 8 t 0 0 I | \= K......_ _..... -1 \ 000 N! 1 \.,,0 7 _ __ I I__ I lo -66

-100TABLE III COMPARISON OF ANALYTICAL COMPUTER RESULTS -- SERIES SYSTEM Junction Head Specification Downstream Head Specification Max. H at Max. H at Meax. H Max. H tf Junction Valve tf Specified Obtained seconds feet feet seconds feet feet 8.80 189.2 315.2 8.76 315.2 313.3 10.00 172.7 273.2 9.99 273.2 273.2 12.00 159.6 232.9 11.96 232.9 232.9 14.00 152.0 209.4 13.78 209.4 209.4 16.00 147.0 193.9 15.72 193.9 193.9 These results again confirm the theory and illustrate the differences in the transient conditions obtained utilizing the two procedures. The somewhat more rapid closure times permitted by the valve-head specification technique (for the same ultimate maximum value of head developed in the system) are obtained for the same reasons as they were in the system of Section 3.3. Especially noteworthy is the obvious extension of the rapid control concepts of the simple systems to the series system as well. As illustrated by the first case presented in Table III, an extreme value of head was selected that was not attained since the consequent value of tf was less than 4L1/al + 4L2/a2. In surmary, the proposed valve-stroKing procedure creates a controlled transient condition throughout the system and permits the final steadyr-state condition to be established without residual transient fluctuations. UnliKe the previous procedure, the solution technique proposed in this section permits a predetermined extreme value of pressure to be developed and maintained at the valve during a central

-101phase of the transient condition. Furthermore, the procedure does not depend upon the presence of a constant-head upstream reservoir; it is perfectly general and could be applied to any series system with a Known constant-relationship upstream boundary condition. Lastly, it may be applied to any number of pipes in series. In such cases the minimum value of ta, is Z 2L/a, and all other values of valve operation time discussed in this section must be modified accordingly. 4.2 Branchins System Another example of an elementary complex system for which a valve-strosing procedure may be developed is provided by the branching system illustrated in Figure 27. Again consider that a controlled transient condition is to be develooed between Known initial and final steady-state conditions in the sys tem such that residual transient fluctuations are eliminated. An analysis of the system indicates that a total of nine unKnown boundar- dependent variables exist, wihereas only seven equations are available. Thus, specification of two of the unknown variables will be required to completely specify the transient condition imposed upon the system. 4.2.1 Evaluation of the Existing Technique With reference to Figure 2,3, the existing solution technique (Ref. 30,31) for this system is first examined. Because of the invarient reservoir-boundary relationship, IIH (0,t) = Hr, and the subsequent procedure for specification of velocities at the junction, the regions of the initial steady-uniform conditions are as illustrated. Arbitrarily assigning the positive direction to the C+

Final HCfGL az~~~~~~~~~~~~~~ Yf~~~~~~~~~moH~~~~~~~0 3f~~~~~N Initi,71 HGL Hr Hr H. ~ Iflo pip ee FigrUe 27. Defifitic. sKetch of branching system vith uPsrear ~s,,,Oonstant-head reservm~r

tf If 2 2Sf Compute.: VV VrnV V2 (t) Ir~ L7/9, -L~a~, Trasient TrIansient Ht oyie 2 tf -2L, /a,- L2/a12 Transiet t Surge Compute: Known: i 1 23 Zone I()4Surge tfL//-2L4 /t Specify VHj (t) Zone? Zone 4I 2L, /a+2L/aVr(I)- H) (t) Specify. * L lo/o V O.)j L,/oL2/2 2 J1Known. LI /a/ 21a22 Zone 1Zone 3V ss ss ss~~~ss L2/a2' v.30 0 L 0 L2 0 L0 Figure 28. x-t plane and notation for branching system with upstream constant-head reservoir.

characteristic family and the negative direction to the C- family, the reservoir-boundary velocity is specified according to Equation (22) exactly as it was for the series system: an extreme permissible value of head desired at the junction is specified. Thus a unique solution exists in pipe 1, and by specifying that at time t >t - a - L2/a2, Vr(t) = V1, the final steady-uniform condition in the pipe is established as desired. Directing attention now to the branching junction, three of the four essential boundary data are now established; the fourth must therefore be specified (the second of the two permissible boundary-data specifications). Tne flow distribution at the junction is determined by arbitrarily specifying that the change in flow in pipe 2 is to be proportional to the change in floew in pipe 1: Q (t) - Qj (t) - Q1 t2 20 Q.1 0 1 (38) Q2 Q2 Q1 Q1 ~f Q20 f 0 (Application of the continuity equation at the junctton demonstrates that the change in flow in pipe 3 is similarly proportional to the change in flovw in pipe 1.) Since both data are thereby established at the upstream boundaries of each pipe, unique solutions exist, and the solutions of the dependent variables and unsteady valve coefficients may be determined at each downstream valve boundary. Again the final steady-uniform conditions will be established as desired, since at t > t - L /a2 the head and velocity at the upstream boundary of each pipe is the final steadyuniform value. As with the existing technique for the series system, the same objection to this strategy may exist because the extreme values of pressure sustained in the system usually develop at the valves, not the

-105junction; again it would be desirable to identify and limit those values a priori. Since conditions in pipe 1 of this system are identical to those encountered in the series system, one may speculate about the possible propagation of surge zones into the downstream pipes as before. If a surge zone is propagated into pipe 2, for example, conditions in each pipe must again satisfy Equation (22). Equation (38) may be reduced to the form Q2(t) = CQl(t) + K, where -Q2 Q2 C= K= = -CQ0 f 10 0 0 Then from Equation (22), A1 A2 C -(H Hr hf (V1)) L2 (H H -H h (V2)) C1 jH H+hf1 2 2 2 since dQ/dt = C *~ dQ/dt; H is the extreme value of head desired at valve 2. After rearrangement, A1 A2 A2 A1 C h(V) (V2) -- L(v (H - H ) - C (H - H). (39) 1 1 2 2 2 2 j 1 j Since the right-hand side of the above equation must be a constant, then A l A2 f-CQ +K( C ihf1 _h 2) hf2 CA2K constant. This constant may be evaluated, since the first term above is zero when Q= O. Therefore L, Q,(Q1 A

is the necessary relationship between the viscous resistance properties of each pipe essential to the establishment of a zone of incompressible surge conditions in pipe 2. In terms of the Darcy-1Weisbach friction factors, the above criterion becomes f f D13 D23 One again recognizes that the above criteria would probably never be satisf-ied b. flow conditilons prevailing in a real systea.. A system in which viscous effects are so insignificant that they rmay be safel-y neglected would satisfy'the criteria, of course, and the desired relationship between Hm and H can be obtained by rearrangement o-' the right-hand side of Equation (39): A A m. A A j C-+ 2 1 2 Since f1 cond itions in pipe 3 are sirilar to those in pipe 2, a surge zone woul.d also bc propagated into pipe 3; a sim-ilar relaticnnhip involving H and II liKe;tise would exist. The lesser of the two computed m. r~l~ 3 O values of H could then be used to specify the rescrvoir-boundary Il'. ve loc i t 4.2.2 Alternate Solution Let oc- assume that -.-sysen parameters are such that it would be desirable to l imit the extreme value of head developed at one of the valves to a predetermined value. lAgain,ith reference to Figure 28, assurae that the extreme value permitted at valve 2 is to be HI. Then if the velocity V: (t) at the junction is specified by means of the

-107incompressible surge equation over the interval L2/a2 < t < t -L2/a2 and not br Equation (38), then the discussion presented in Section 3.1.1 assures the propagation of a central zone of incompressible surge conditions into pipe 2 as illustrated (assuming, of course, that the surge zone is of sufficient duration to propagate to the valve boundary). Thus dV _v 2, zg; cm2 -Ha 2V 2 2-V2 i dt L2 m2 ( 2D2 is suggested as an alternate procedure for specifying the flow into pipe 2 at the junction. Each of the surge equations can then be integrated to determine the duration each is utilized to specify the appropriate boundary velocity, and the results may be equated since the time of specification at the reservoir is 2L /a1 seconds less than the time of specification at the junction: V1 Vd 2L1 L f dV1 -L f dV al ~ Vr ( H H h 1) ) g ( (H - -H + hf (V2 )) (43) V m r f 1 V m m f 2 10 1 20 2 2 Since every parameter of Equation (43) is Known except H, Hm. may be determined. (The solution is a numerical one -- see Appendix C.) A computer program was written to illustrate the effect of system operation according to either of the suggested techniques. For the valve stroaing analysi1s the reservoir-boundary velocity is specified according to Equation (22) (again by replacing Hm with Ha ), and EquaJ tions (18a) and (1Sb) are used to advance the solutions across the grid of pipe 1 to the junction boundary. Either Equation (38) or (42) is utilized to specify the junction-boundary velocity in pipe 2; the junction-boundary velocity in pipe 3 is then determinable from the

-108continuity equation. Equations (18a) and (18b) again advance the solution schemes to each valve boundary permitting the value of both valve coefficients to be determined at the respective boundary grid points. The confirming characteristics analysis is identical to that of the series system, with the exception that Equations (21a), (2lb), (21c), and (21d) must now be utilized to evaluate all conditions at the junction. Results of a study comparing the two techniques are illustrated in Figures 29 and 30. The pertinent system parameters are: H = 100 r feet, V = 3.52 feet per second, V = 5.00 feet per second, V = Vf = _ 2 3 3 2.00 f.32 eet per second, V =.320 feet, fl 2 L2 = 3200 feet, L3 = 1800 feet, al = 3600 feet per second, a2 = 4000 feet per second, a3 = 3000 feet per second, fl =.018, f2.020, f3 =.025, Di = 1.25 feet, D2 = 1.00 feet and D3 = 0.50 feet; thus L1/al+ L2/a2 = 1.8 seconds, Hj = 90.0 feet, H2 = 65.2 feet, H3 = 84.4 feet, ~0 0 0 HI = =I2 99.9 feet, and H 94.3 feet. -3f The study was arranged to compare the results of the two techniques for the same value of tf). This value was determined by establishing H = 175.0 feet; H1 was then m2 mj determined to be 136.7 feet and tf was subsequently found to be 12.52 seconds. These results again confirm the theory and illustrate the differences in the transient conditions obtained utilizing the two Drocedures. Control according to the existing technique results in a maximum head developed at valve 2 of 187.6 feet; control according to the alternate technique suggested here permits the maximum value to be limited to the preselected 175.0 feet. One should note, however, the considerable difference in conditions imposed upon the valve boundary

~aanpaoooxd 2u-TsTxa —.tTOAJsoa. pvaq- utsuoso umra;rsdn q.lT wuaqcsas 2uTt3usjq e Jo To4uo0o 2SuTXQOs-aATA'6Z aal'Td spuo9o u/! tW.1L OZ 9/ 1/ 8 t O0 09' /|9A/DA 001 I: I i ID 00 4? 0 / O IO/ A~ eAIDA -60m

4-10200Valve 2 150. -.. F/ -.. Valve 3 I c 4. 50'I 2.00 I I Vo/ve 2.50 Vl / 0 4 8 /2 /6 20 Time in Seconds Figure 30. Valve-stroking control of a branching system with upstream constant-head reservoir — alternate procedure.

-111of pipe 3. (The constant head and velocity developed there by the existing technique during a central phase of the transient condition is only a circumstance of the data. Since V3 V = 2.00 feet per second,.3 3f specification of the junction flows according to Equation (38) results in V. (t) being 2.00 feet per second throughout the entire transient condiJ3 tion. Since the head at the junction is the prescribed constant Hmj during the central phase, these two constant variables combine to propagate a zone of steady-uniform conditions into pipe 3.) In summary, either valve-stroking procedure creates a controlled transient condition throughout the system and again permits the final steady-state condition to be established without the presence of residual transient fluctuations. UnliKe the existing technique, the alternate technique proposed in this section does permit a predetermined extreme value of pressure to be developed and maintained at one valve during a central phase of the transient condition. The technique does, however, impose a less orderly condition of operation upon valve 3 and thercbr demonstrates what would appear to be a fundamental axiom of transient control in complex systems that must always be appreciated: the more sophisticated and elegant the desired control techniques are, the more sophisticated the control devices will have to be. L P.3 ssive Systems As an instructive example of the concept of passive valve strolinG, consider the following: assume that it is desired to develop a controlled eransint condition between Known initial and final steadystate conditions in the branching system of Figure 27 subject to the additional operational constraint that a constant outflow be maintained

-112through valve 3 during the entire history of the imposed transient condition. (Recall that in the example reported in the last section both the initial and final steady-uniforn velocities through valve 3 were 2.0 feet per second; however, that value was not mintained throughout the entire transient.) This imposed constraint now reduces the number of unKnown boundary dependent variables to only eight, while the seven original equations are still available. Here, specification of only one of the unlnow-n variables will be required to completely specify the controlled transient operation of the system. WJiith reference to Figure 31, the initial steady-uniform regions of each pipe are as indicated due to the presence of the invarient rese-voir-boundary relationship, HE(O,t) = HIr,and the invarient valveboundary rel].ationship, V3(L3,t) = V3. Arbitrarily assigning the positive directionr to the C+ characteristic farmily and the negative direction to the C- family, the reservoir-boundar- xrelocity could then be specified according to Equaticn (22) following th-e ex- sting procedLure for complex systems: an ext-reme permissible valule of head desired at the junctiocn is spec:i;fied. Again, a unique solution exists in pipe 1, and denoting as tr the tim.e at which Vr(t) becomes thereafter equal to V1, the final steady-uni-forn zone in pipe I is as illustrated in the figure. Attention is no:r directed to pipe 3. One bowldary datum has now been establishled at eachn bLoundar.r -- the junction-boundary head, I{ (t), available from the solution in pipe 1 and the steady valve~ J boundary velocity, V3(t) = V3o. Thus a unique solution exists in pipe 3. After t = t + L1/al, the junction-boundary head becomes the final steady-state value (H. ). Note, however, that unliKe the situations considered in the previous examples or pipe 1 of this example, in which

SS/f If tr + LI/0/ Transient tr Trransient oTrnsient Known., Compite: Known., Knopn t H N 3-L/o Surge K(t) Compute: Compute: 2 Vt) 3H Specify: Zone 3 Ht r (1) Zone 13(1/ 2L,/7 +LS/o2 1 Compute: Zone L /laLk16~ Zone VWt) II 22 1; ss H.(t ss'0 I 2.o Lp'4+L3/73 o L, O LP O2 2o 5 0 V3. 0 4L0 L2 0L3 X/ X2 X5 Figure 31. x-t plane and notation for branching system with upstrean: -onstant-head reservoir — passive system.

a final steady-state zone is forced by the creation of two steady-state boundary data at one boundary, no final steady-state zone is immediately created in pipe 2. As discussed in Section 2.6.2, only after t tr + L1/a1 will the transient condition subsequently begin to asymptotically approach the final steady-unifrom conditions in pipe 3. UnliKe the previous situations considered, the duration of the transient condition imposed upon pipe 3 (and thereby pipe 2) can not be arbitrarily specified and can only be determined by the appropriate numerical evaluations advanced upward on the double-staggered grid of characteristics (see Appendix B). The calculations may be halted thereafter, and the valve coefficient-time relat;ionship may then be determined from the solutions V3(t) and H3(t) computed at the right-end boundary. Wlith reference to pipe 2, both data are thereby established at its upstream boundary (the existance of the common boundary head and the branching-junction continuity equation), and thus a unique solution exists in the pipe; the valve coefficient-time relationship may then be computed from the Known solutions V2(t) and II2(t). Therefore, the entire transient condition for this system has been prescribed and valve motions determined that would create the desired condition. Of the system, pipe 1 and its transient condition would be considered as active elements; the period of the transient zone may be arbitrarily selected and the nature of the transient itself can be predetermined. Pipe 3, its prescribed transient condition, and the valve would be considered as passive elements; the period and nature of the controlled transient are not readily predictable. In addition, the valve, instead of creating a transient condition, merely reacts to that condition in order to maintain the required constant outflow (hence the

term "passive"). Pipe 2, its transient condition, and valve would have to be considered as pssive elements also; the period of the transient is dependent upon the transient condition in pipe 3, its nature is not readil-y predictable, and the valve, even though it creates the initial disturbance, must necessarilyr respond to the passive character of the transient created in pipe 3. It is pertinent to note that although the passive condition of this example was a consequence CfL an arbitrarily imposed operational constraint, any constant-condition boundary constraint at the downstream end of pipe 3 would have created the same condition; th1e poesence of a constant-head reservoir, a fixed orifice, a centr fuga' pump, or a dead end are all cormmonplace exampleas. Also note that the passive condition of the example is independent of the specification of the one free unionownrro bounoary vanriable. It would hsave been possible, efor exam!ple, to select pine 3 as the active wipe by specifying the transicnt head at the valve according to Equation (28) of Section 3.2.1. hrilat strategy wouldc alter the sequence o' 2alc-lation' s of course, but wtould not change the essential, passive character. of the system. An e2:xample 6O a systen tha+ is inherently passive is provided byr the last of the elemrzentary coAmple sysstems, the parallel system of Figur'e 32. Analysis of this system indicates that a total of eleven unktnovwn boundary dependent variables exist, whereas only ten equations are available (two characteristic equations for each pipe plus the two branching-junction continuity equations). The existing technique(3) for valve-strorting control of the system is to specify the reservoirboundarS vrelocity according to Bquation (22) by again selecting the

I~ ~._._.~~~~~~R na/ HGL I/nitialH W H.' HfH F 32. — Pipe s/ os i u - r FPipe 3 2 Figure 32. Definition sketch of parallel system with upstream constant-head reservoir.

-117extreme value of head to be developed at the upstream Junction. The transient condition in pipe 1 is then completely specified and the pipe is an active element of the system. With reference to pipes 2 and 3, the number of uninown boundary variables and available equations now both equal f'ive; therefore unique solutions exist in these pipes, and as with the preceding example, the duration of the transient condition imposed upon the remainder of the system -- the passive elements -- can only be determined by subsequent numerical evaluation. Relative to the raore extensive complex systems, the following generalized observation can be made: all multiple series systems are active systems; multiple branching systems may be active if a constantrelationship boondary condition exists at only one of the external boundaries of th.e system, and all other multiple branching systems must necessarily be passive; all multiple loop systems must be passive systems. Experimental confirmation of the concept of passive valvestro~ting control of a branching system is provided in Chapter VI, and a discussion of proposed alternate valve-stroKing techniques for the elementary lpassive systems is included in the recommendations for further stud-y discussed in Chapter VII.

V. IABORATORY EXPERItENTAL SYSTEM In order to obtain experimental confirmation of the previously published valve-stroKing theory,(30'31) an experimental system had been constructed in the G. G. Brown Fluids Engineering Laboratory of the University of Michigan. This system was used to obtain additional verification of some of the valve-stroKing concepts presented in this investigation. 5.1 General Description of System The laboratory system (see Plate I) was comprised of three principal elements: the reservoir and piping systems, the control valve and servo systems, and the various instrumentation devices essential to system calibration and transient operation. A schematic of the flow system is illustrated in Figure 33. One of the building open-channel_ floor drains was utilized as the sump; its dimensions were approximately 14 inches wide, 18 inches deep, and 36 feet long. lWater was supplied to the reservoir from the sump by means of a Worthington Type 1.I4 0 turbine pump. The bypass valve on the downstream side of the pump permitted the water level in the reservoir to be maintained at an essentially constant level over the intended range of system operating conditions. The high-pressure reservoir was constructed from a 2-foot diameter, 5-foot high welded galvanized steel tanK. This reservoir was about half filled with water, and compressed air occupied the space above the water surface. The air pressure in the tanK was regulated utilizing a Fisher Type 98LD low pressure differential relief valve; the pressure on the upper side of the valve diLaphragm was regulated using a Bellofram -118

-119W s Zw9ZW ~5ff~~~ii E'Ia) i 9l i i T ~4.,tn.),I ~ ~~~~ ~ I>

QPiezometer ring Relief valve * Pressure transducer Air supepjy. "-Reservo/r U$&fer ge/ass 400' P,~~~~~~~~~~~e ~Needle valve 9~1~ Ht — 400' Pipe —3~v ~Need// $ T~~ 200 F' Pipe 2000' P3lem.o Contfrol valve 8 Servo system Supp ly line - To sonp oss alve From wsmp Figure 33. Schematic of laborator-j system.

-121Type 10A pressure regulator (see Plate II). Of the piping system, the upstream pipe consisted of 2000 feet (nominal) of.95-inch ID copper tubing; the wall thicKness was 1/16 of an inch. An identical pipe formed one of the downstream branching pipes, and the other downstream pipe consisted of 400 feet (nominal) of 1.00 inch ID copper tubing. The wall thickness of this pipe was 7/64 of an inch. The 2000-foot pipes were each constructed from 100-foot segments carefully joined together by soldered sleeve connections and loosely coiled in a double helix (a 1000-foot inner coil, followed by a 1000 foot outer coil) on a 6-foot diameter frame constructed of'"Unistrut" steel members. The 400-foot pipe was similarly constructed from 100-foot segments and coiled in a helix on a 5-foot diameter wooden frame. Earlier (15,25) studies' had demonstrated that this configuration would produce no significant distortion of the transient pressure waves. At the downstream end of the 400-foot pipe, a 3/4-inch Crane needle valve was installed at the same elevation as the control valve (the arbitrary datum) to provide the function of a fixed orifice. The system was very versatile: by closing a valve in the 400foot pipe at the branching junction, a simple system could be obtained. (The actual length of this system, including all connector pipes, was determined to be 4030 feet. That value was used in the simple-system analyses reported in Chapter VI. The nominal lengths were used, however, in the series and branching system analyses, because of the heretofore mentioned D-At computational constraint.) Finally, by connecting the 400-foot pipe between the reservoir and the two 2000-foot pipes, a series system was obtained.

-122E; - t::2:0: i::::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-:i~~~c~:i:::......... Plate II. Pressure regulator and relief valve.

-123A schematic of the control valve and low-pressure pneumatic servo system is illustrated in Figure 34. A MinariK Model TF-63SJ6 speed controller was used to regulate the speed of a 1/50-horsepower Bodine motor. This motor in turn rotated the 5.5-inch diameter cam platform on which was mounted a plastic card (see Plate IV). The variable vertical displacement of this card was converted by a Fisher Type 3561 motion transmitter into an equivalent air-pressure signal. Based upon this signal, a Fisher Series 3560 valve positioner would then appropriately position the Fisher Series 6S7A diaphragm control valve (1/4-inch microflute type). (See Plates IV and V.) Four sets of 2-inch diameter piezometer rings were attached to the piping system. The first ring was attached to the downstream 2000foot pipe about 15 inches upstream from the control valve (see Plate III); the second was located at the branching junction; the third was placed just downstream from the reservoir; and the fourth was positioned on the 400-foot pipe about three feet upstream from the needle valve. 100-inch reservoir-type differential manometers were then connected to these rings and used in the various system and instrumentation calibrations. The manometer fluid was mercury; by connecting the other leg of each manometer to the building constant-head tanK (at a measured elevation of 42.2 feet above the datum), somewhat higher pressures were capable of being recorded than would have been possible otherwise. Tso Drjnisco APT25-3C strain-gage type pressure transducers were used to monitor the transient pressures at the control valve and junction. The valve transducer was located about 12 inches upstream from the control valve and the junction transducer on an auxiliary 30-inch pipe suspended f'rom the junction. (The length of this pipe was so negligible compared to

Air supply Motion transmitter Cam Diaphragm actuator Speed 7 L| Motor controller Micrometer Valve poslv1one _ Dsplacsemet transducer Figure 34. Schematic of control valve and servo system.

-125A- 411,::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~itoaiain: is Plate III. Pressure transducer,, transducer mounting block, and piezometer ring.

-126Plate IV. Speed controller, motor, plastic cam, motion transmitter, and valve positioner.

-127I....... I Plate V. Control valve, diaphragm actuator, micrometer, and displacement transducer.

-128. the rest of the system that no subsequent buffering of the imposed transient condition resulted.) Each transducer was mounted nearly flush with the inside wall of the pipe (see Plate III). Sanborn Model 311 transducer amplifier-indicator units were used to supply the input voltage needed to activate the Wheatstone bridge circuit of each transducer and to amplify the response signal. The output of the amplifier was then recorded on photosensitive paper utilizing a Sanborn Model 4500 directwriting optical oscillograph. Fin.ally, a micrometer and Sanborn Model 24DCDT-050 displacement transducer (LVDT) were mounted to measure and record, respectively, the vertical stem motion of the control valve (see Plate V). An H-Lab Model 6204A power supply was used to supply the input voltage to the transducer, and its signal was also recorded utilizing the Sanborn optical oscillograph.,.2 Ssterm Calibration Calibr-ation data essential to establish the frictional-resistance characteristics of the piping system, the discharge coefficients of the cozntirol valve and orifice, and the relationship between plastic card hcight, control-valve stem position, and valve coefficient were obtained'.or the full range of anticipated system operation. The differential manometers weore used to measure the heads, weigh-tant observations were used to determine the flow velocities, and a calibrated metal cam template and the micrometer wHere used to est;ablish the essential valve and servo relationships. The flow-resistance relationships for the 2000-foot pipes and thle 400-foot pipe are illustrated in Figures 35 and 36, respectively; the card height-stem position-valve coefficient relationships are itllustrated in Figures 37 and 38. Thel orifice calibration data for the two cases presented in Chapter VI are summarized in Table IV. Since no apparent trend

-12970 LIt!i f 60 50 40 0 0.5 1.0 /5 2.0 2.5 Velocity in Feet per Second Figure 35. Flow-resistance relationship for the 2000-ft. copper pipe.

~adTd Jaddo3o'qJ-00 aq Joj d TsuoTcaW aa3ts4sa1-SJ-AOTd'9~ a9n 9, puooeS Jsd /s I u!/ X//oA/0 7 0~ g'l 07 ~'0 0 i i/ i ii i ii iio o rvr r X~~~~~~~~~~~~~~~~~Z I / to~,.i..... 1 1...... ~

/2 IX05 I _w X He 0.8 q) - - _ _ _ _ _ __, 0.6 1 0.4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. 04_ __ll 0.025.050.075. /00.125.150 Valve Coefficient Figure 37. Relationship between valve stem position and valve coefficient.

-132/.2 0,. 0.9. -.. Card Height in Centimeters Figure 38. Relationship between valve stem position and card height. position and card height.

-133exists in either set of data, the average values were used in the respective theoretical studies of the next chapter. Earlier studies had established the wavespeed in the 400-foot pipe as being between 4250 and 4350 feet per second; values in this range TABLE IV RESUME OF ORIFICE CALIBRATION DATA Case I Case II Velocity Head K~ Velocity Head Ko ft./sec. Aft. ft. /2/sec. ft./sec f. ft. 1/2/sec 1.834 32.14.2024.348 136.8.0298 1.822 80.99.2024 338 136.4.0290 1.812 80.83.2015.343 136.6.0294 1.792 80.52.1997.341 136.4.0292 1.786 79.63.2001.337 136.1.0289 i.775 77.84.2011.341 135.0.0294 1.750 75.33.20 6.337 133.2.0292 1.713 72.34.2014.330 i30.8.0289 1.686 69.o0.2029.331 127.0.0294 1.618 75.17.2004.326 122.6.0294 1.588 61.13.2031.315 116.7.0292 1.562 58.98.2033.314 113.4.0295 1.510 56.73.2001.305 110.4.0290 i.484 )51. 58.2009.302 106.3.0293 i. 457 52.44.20+12.299 102.9.0295 i.4483 51.34.2020.297 100.5.0296 Arve. =.2015 Ave. =.0293

were used accordingly. Previous evidence had indicated that the wavespeed in the 2000-foot pipes was somewhat pressure-dependent, probably due to the presence of small amounts of free gas in the water and to the fact that the relatively thin-walled pipe was slightly out-of-round at lower pressures. By timing the advance of a small disturbance from the valve transducer to the junction transducer, at various static pressure levels, this relationship was established (as illustrated in Figure 39), and was used as a guide in the selection of wavespeed values for the theoretical studies reported in Chapter VI. 5.3 Experimental Procedures The calibration data of the last section was entered into the pertinent computer programs of Chapter VI in tabular form; essential information was then computed from this tabular data as necessary utilizing the technique of parabolic interpolation. Program output included computer (Calcomp) plots of the theoretical head versus time relationships at the valve and junction, the control-valve stem position versus time relationship, and a drawing of the plastic card profile. This profile was traced on a sheet of plastic and cut with a scissors to the required shape. A short lead-in section was provided to allow the turntable to accelerate to its uniform rotational speed before the control action was to be initiated; a similar run-out section was also provided. After an elaborate and systematic procedure to bleed the entire system and manometers of entrapped air was completed, the signals from the two pressure transducers were then calibrated utilizing the respective manometers, and conveniently scaled using the gain and zero

2500 __ _ _ _ __ _ __ 5 I I /500 - —._ _ - _ _ / 2500 25 5 75 /00 /25 /S0 z75 Static Head in Feet of Wafer Figure 39. Relationship between wavespeed and head for the 2000-ft. copper pipe. Figure 39. Relationship between wavespeed and head for the 2000O-ft. copper pipe.

-136adjustments of the recorder; the signal from the displacement transducer was calibrated using the micrometer and similarly scaled. The plastic card was then mounted on the turntable, and the speed controller was adjusted to establish the proper time of valve operation. After all instrumentation had been calibrated and checKed, the setting of the pressure relief valve was modified, if necessary, to establish the desired value of reservoir head. (In every study of this investigation, a value of 140 feet - corresponding to 93.3 inches manometer reading -- was used.) The speed controller would then be activated, causing the valve to initiate the transient condition. After the cessation of the transient, the recorded data would then be inspected. Of particular interest was the control-valve stem position versus time relationship. Since the card height versus stem position relationship was not altogether stable, and was also unable to account for the dynamic lag in the servo equipment, subsequent modification of the plastic card profile usually was necessary. The above procedure would then be repeated until the cam profile had been modified such that the desired stem position versus time relationship was truly obtained. The head versus time relationships thus obtained are the onesreported in the several studies of Chapter VI. After the desired valve closure relationship was obtained, several more runs were usually made, and in every case the results were highly repeatable.

VI. COMPARISON AND DISCUSSION OF THEORETICAL AND EXPERIMENTAL RESULTS This study of valve stroKing culminates in a comparison between the theoretical results derived from the appropriate mathematical models and the experimental laboratory results obtained in the system described in Chapter V. Experimental verification of the pertinent valve-stroKing theory has been obtained and is presented for a simple system, a series system, and a passive branching system. 6.1 Simple System Since the recognition that the minimum time for the valvestroKing control of a simple system is 2L/a, not the previously reported 4L/a, is one of the new concepts presented in this investigation, experimental verification was deemed to be a desirable addition to the previously reported confirming laboratory studies. Because of the presence of the upstream pressure transducer at the approximate midpoint of the 030-froot pipe (the junction of the basic system), it twas decided to investigate a closure situation (Case I) and an opening (Case II), with tf for each being exactly 3L/a. (Recall from the discussion of Section 3.1.3 that in the closure case the maximum value of head created in the system should then be developed at the midpoint.) The essential elements of' the computer analysis are identical to those discussed in Section 3.1.3. Initial and final steady-uniform velocities were arbitrarily selected for each study, and the value of Hm necessary for tf to be equal to 3L/a was determined by utilizing the procedure described in Appendix C. Pertinent system parameters for each case are summarized in Table V. -137

TABLE V RESU1E OF PERTINENT DATA -- SIMPLE SYSTEM Case I Case II t (sec.) 4.44 4.50 a (ft./sec. ) 2725 2685 Vo " 1.10 0.10 V 0. 10 1.00 H (ft.) 140.00 140.00 r Hi... 123.45 139.51 jo H 139.51 126.30 H " 106.90 139.02 H " 139.02 112.60 H 212.68 53.00 H' 176.34 96.50 m Extreme H in System (ft.) 176.34 86.60 A comparison between the theoretically predicted and experimentally obtained results for Case I are presented in Figure 40. Again ior convenience, the dimensionless tau-time relationship of the valve is illustrated. Note that during the central L/a seconds the value was required to open slightly, a requirement readily attainable on the laboratory control system. Considering the assumptions implicit in the development of the basic theory (especially the assumption of a onedimensional velocity uniformly distributed across the cross-section, and the assumptions relative to the instantaneous values of wall shear stress and valve coefficient), the probability of slight experimental error in the various system calibrations, and that somewhat inexact values of wavespeed were utilized in the theoretical analysis, the

/95 /80 /65 150 l O0 1/35,75 ) Head at va/ye, computed Head at valve, experimental /20 ----- IHead at midpoint, computed -.50 - \0- Head d midpoint, experimento/al Tau of valve 105 " - \.25 T= 4.44 sec. 900 2 4 6 8 /0 /2 /4 /6 /8 Time in Seconds Figure 40. System variables versus time. Simple system. Case I — closure in 3L/a.

195 /60 120 2. 48/2/6 /65 A / /50 Time in Seconds ~/2/ Figure 41. System variables versus time. Simple system. Computed results of a linear tau-time closure to compare with Case I.valve,, ----— Head at midpoint /05 _.__. I.__. 9 0......._ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ 0 2 4 6 8 /0 /2 /4 /6 /6 Time in Seconds Figure ~4l. System variables versus time. Simple system. Computed results of a linear tau-time closure to compare with Case I.

-141results are considered to be excellent. Because of the response characteristics of the transducers and the opportunity for slight error in the transducer calibration and scaling operations, the experimental data is estimated to be accurate within 0.50 feet. Computer results of a comparable linear valve closure in the same time are illustrated in Figure 41; Figures 40 and 41 provide a most remarkable comparison of the results of the two closure strategies. Comparison between the theoretical and experimental results for Case II are presented in Figure 42. Again, good agreement is obtained for much of the duration of C the transient condition; only at the end do the experimentally obtained pressures at the valve and midpoint overshoot the final steady-state values. Since the same computer program wqas utilized in the theoretical analyses for both Case I and Case II, and the experimental studies were conducted concurrently, the discrepancy between the predicted and observed results can not be attributed to program or system calibration ecrors. There is, however, co insiderable evidence to suggest that the observed discrepancy is due to the effect of gas liberation duing the time that system pressures are substantially below the initial static level. The net effect of such liberation is a lower value of wavespeed;(31) a lesser average wavespeed persisting during the latter phase of the transient would produce the observed conditions. Contractor(5) observed and reported the same phenomenon accompanying a sudden reduction in pressure conditions in an earlier study. Because of the limitations on tire available for the laboratory investigations, the probable difficulty in readily developing an improved mathematical model of the phenomenon, and the fact that such an effort would have been decidedly tangential to the stated objectives of this

/50 140 130 - =4.50sec. /20 o 00 H he.75 Head at valve, cc*apwted Head of valve, experimental /00' Head at midpoint, computed.5 - - ~~Head at midpoint,. experimental Tau of valve 60 _ ____.2__ 80 iiI 0 1I' 0 2 4 6 8 /O 12 /4 16 /8 Time in Seconds Figure 42. System variables versus time. Simple system. Case II — opening in 3L/a.

/50 /40 /30 /20 I/O t%4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~L. /00 Head at midpoint 803 0 2 4 6 8 /0 /2 /4 1 Time in Seconds Figure 43. System variables versus time. Simple system. Computed results of a linear tau-time opening to compare with Case II.

-144investigation this aspect was not pursued further. One should recognize, however, that the minimum system pressures were held within established limits and that, in the case of valve openings, residual transient fluctuations are usually of negligible magnitude and duration. 6.2 Series System* Experimental verification of the existing technique for valvestroKing control of a series system has also been obtained. The essential elements of the computer analysis are identical to those discussed in Section 4.1.1. Initial and final steady-uniform velocities in the upstream (400-foct) pipe were arbitrarily selected for each study, and the prescribed values of H were similarly established; velocities in m. J the downstream (4000-foot) pipe and all steady-state values of piezo-.ctric head were then obtained from the appropriate steady-state equations. ThOne ratios of length to wavespeed for each pipe were such that N1 was selected to be 1 and N2 to be 16. Other pertinent system parameters for each case are summarized in Table VI. Comparisons between the theoretically predicted and experimentally obtained results for these two cases are presented in Figures 44 and 45, respectively. Again, the results are considered to be excellent. The somethat better agreement obtained in Case II motivated the use of the Case II wavespeed values in the theoretical analyses reported in the next section. * The theoretical and experimental studies of this section are the wor~ cof r. Joel L. Caves, and the author is indebted to him for these re suits.

/60_ /50 /40 6 -828 sec. /30 Q~120 //O I:,, Compiled Experimenta/ /00 901 0 2 3 4 5 6 7 8 9 /0 Time in Seconds Figure 44. Head at valve versus time. Series system — Case I.

/B60 _a —-rI i lo /60 / =5.0 8sec. 0I J /0 _ _ ___ -_ I 0 / 2 3 4 5 6 7 Time In Seconds Figure 45. Head at valve versus time. Series system — Case II.

-147TABIE VI RESUME OF PERTINENT DATA -- SERIES SYSTE14 Case I Case II t, (sec.) 8.28 5.08 al (ft./sec.) 4344 4262 a2 2709 2664 2" Vo 1 1.10 1.10 1 V " 0.10 0.10 V "t 1.22 1.22 02 V r2 ".11.11 H (ft.) 140.0 140.0 r H " 95.3 95.3 H 138.7 138.7 H 141.5 145.5 6.3 Branching SLstem Since the introduction of the passive valve-stroting concept is one of the original contributions of this investigation, experimental verification was also desired. The essential elerments of the computer analysis are identical to those di-scussed in Section 4.3, except that the fixed-orifice boundarycondition equations (Equations (15a) and (15b), modified) must be utilized at the right-end boundary of pipe 3 (the 400-foot pipe). Initial and final steady-unilform velocities in thle upstream (2000-foot) pipe were arbitrarily selected for each study, and the prescribed values of Hm were similarly established. Both velocities and all steady-state values of piezorzetric head were then obtained for pipe 3 by solving the appropriate steady-state equations utilizing the method of interval halving.

(For references and a brief discussion of this method, see Appendix C). All steady-state conditions in pipe 2 were then directly determilnable* Nl1and N2 were selected to be 8, N3 to be 1. Pertinent system parameters for each case are summarized in Table VII. Comparisons between the theoretically predicted and experimentally obtained results for these two cases are presented in Figures 46 and 48, respectively. Computer results of the comparable linear valve closures are illustrated in Figures 47 and 49, respectively, and again provide interesting comparisons of the results of the two closure strategies. In each case, the slight discrepancy between the theoretical and experimental initial conditions results from the frictional-resistance calibrations. Since the time lapse between system calibration and the subsequent experimental studies was frequently several days, changes in ambient laboratory temperature within that period of time undoubtedly contributed to changes in the resistance properties of the system. Nevertheless, the results are considered to constitute excellent verification of the passive valve-stroKing concept. Considering the significant viscous effects encountered in this system, particularly in the high-throughflow situation of Case I (eighty-odd feet of loss sustained in pipe 1 alone), these results parenthetically demonstrate the intrinsic accuracy of the characteristic equations in the analysis of the transient phenomenon in viscous liquid piping systems.

-149TABLE VII RESUIE OF PERTINENT DATA -- BRANCHING SYSTEM Case I Case II tr (sec.) 5.72 7.13 al (ft./sec.) 2664 2664 a2 ) 2664 2664 a3 " 4262 4262 V 2.60 1.70 1 V " 2.10.50 -1 V) ".96 1.37,Vi.~13.12 -2 v 1.48.30 03 vo 7 1.78.34;- 3 H ( ft. ) 140.00 140.00 r H. " 5905 103.80 u. Hi. 85.60 135 98 H2 46.31 78.80 ZHi Xt {384.95 135.35 I3It 1 53.64 105.35 H 77.93 135.46 32f H It 92.00 148.00 In

//O /00 re IIIIII 90 l0 - ",ea't~c/n eprmn 80 z00 70.75 o2 TieHead af valve, computed..... 4Head at valve, experime ntal Head of junctian, ccnlouted.50 -O -~ ~'" Head at/wuction, experimental Tau of valve 50 5 =5. 72 sec. -0 - --- -) - - 40 0 0 2 4 6 8 /0 12 14 /6 18 Time in Seconds Figure 46. System variables versus time. Branching system — Case I.

I/O /00 90'~.80 L7O ___________ Head at valve - - ~~Head at junct ion 60 50 40 0 2 4 6 8 /0 /2 /4 /6 /8 rime in Seconds Figure 47. System variables versus time. Branching system. Computed results of a limear tau-time closure to compare with Case I.

w80 /65 /59 /35 -mmm m-, _______ _______ _______ - _ ____ _ 1.00,/20.75 r;) Head at valve, computed Head avle, experirne'ta/ /05 Head at /unctic%, ccmputed50 - - ~~Head at junction,, experimental -Tu of valve 7,' 713 se c. 75 _ _ _ _ 0 0 2 4 6 8 /0 /2 /4 /6 18 Time'in Seconds Figure 48. System variables versus time. Branching system -- Case II.

/65 /50 /20 /05 A l /Head at valve Head at ji7ction 90 75 0 2 4 6 8 /0 12 /4 /6 /8 Time in Seconds Figure 49. System variables versus time. Branching system. Computed results of a linear tau-time closure to compare with Case II.

VII. SUMMARY AND CONCLUSIONS The primary objective of this study was to interpret and evaluate the existing valve-stroking theory in terms of the basic principles governing the transient phenomenon in liquid piping systems and to modify and extend the theory to include a greater number of operational constraints in the desired control of the simple and elementary complex piping systems. The fundamental property of the characteristic equations relative to the arbitrary selection of the characteristic directions was shown to be of essential importance in the formulation of the pertinent theory. Three unique classes of problems —one datum at each boundary, two data at one boundary, and the mixed boundary-data problem —were each demonstrated to be instrumental in the development of procedures appropriate to the desired control action and system configuration. Valve-stroking control of the three types of simple systems was considered in detail; the recognition that the duration of the transient condition imposed upon such systems may be arbitrarily selected has been presented, and the minimum time for control valve motion in such systems was demonstrated to be 2L/a, not the 4L/a previously reported. The consequence of this development relative to the control of a system consisting of a single pipe with an upstream constant-head reservoir has been discussed. A valve-stroking procedure for a pipe in which a constant velocity forms one boundary condition has been developed that permits complete control of the transient pressures developed in the system. The third type of simple system, a single pipe in which one boundary condition is a known relationship between head and velocity, has been examined and a valve-stroking procedure has been developed that permits a predetermined extreme value of

-155pressure to be developed and maintained at the valve during a central phase of the transient condition. The deficiency of the existing control procedure for the active complex systems has been demonstrated, and alternate techniques have been proposed that do allow the extreme system pressures to be established a priori. Finally, the concept of a passive system has been introduced and its implications in the control of the more extensive complex systems has been discussed. Specific conclusions which can be drawn from this study, and which hopefully contribute to the development of systematic control procedures in transient pipe flow, are summarized. 1. A considerable amount of flexibility exists relative to the determination of valve-stroking procedures for the simple systems. Either the duration of the transient or the extreme system pressure may be arbitrarily selected. For systems which can physically accomodate the rapid closures (tf _< 4L/a), control following the duration-specification procedures is preferred because of the relative simplicity of the computational procedures; for systems in which slower closures must be utilized, the extreme-head procedure is preferred because of the direct nature of the desired solution. 2. Direct procedures can be established to limit extreme system transient pressures to acceptable values in the active complex systems. 3. Valve-stroking control of the passive complex systems is less direct; the duration and nature of the transient condition imposed upon the system can only be determined by numerical evaluation.

-1564. Because of the persistent changes in velocity that are developed in a system controlled by valve-stroking principles, the second-order finite-difference approximations to the characteristic ordinary differential equations are preferred, as they more accurately model the effects of viscous resistance to the flow. This is, to the author's knowledge, the first study in which the second-order approximations have been so extensively utilized. Because of the unusual circumstance that prevails relative to these solutions — iteration solutions (which are shown to be readily convergent) being required whenever the characteristic directions are both selected as positive, direct solutions otherwise —the program execution time of the second-order system is not significantly increased over the first-order system. (For the programs written during the course of this study, the maximum increase in program execution time was less than 25 percent.) 5. The experimental verification of some of the more advanced valve-stroking concepts presented in this study, together with the previously reported laboratory studies, demonstrates the efficacy of the theory and the intrinsic accuracy of the characteristic equations in the evaluation of the transient phenomenon in viscous liquid piping systems. Although no direct consideration of the philosophies or procedures involved in the eventual on-line feed-forward computer control of large and complex distribution systems was attempted in this study, it should be recognized that the ultimate development and application of on-line control techniques will be dependent upon the procedures discussed herein and suitable extension of these procedures yet to be

-157investigated. In this regard, the extreme downstream-head specification technique developed in this study appears to be particularly significant; further investigation into the application of this technique to the control of the following elementary complex systems is recommended: 1. The active branching system discussed in Section 4.2. The head at each downstream valve could be prescribed to change linearly from the initial value to the extreme value during the time the initial disturbance travels to the reservoir and is reflected bacK to the valve. The head at each valve would then renain at its respective preselected value during the central phase of the transient condition. When the velocity at the reservoir becomes the final steadystate value, that final value would then be specified along the reservoir boundary. The flow distribution at the junction in the upper region of the transient zone would then ave to be specified; it appears liitely that a technique sirilar to the existi1ng procedure could be utilized. The remainder of the imposed transient condition would then be evaluated as before. 2. The passive branching system discussed in Section 4.3. The head at the valve would be prescribed and the transient condition in the lower region of the transient zone evaluated in a fashion similar to that for the active case. After the reservoir velocity becomes the final steady-state value, the analrsis of system conditions in the upper region are identical to that discussed in Section 4.3. 3. Tlhe parallel system. Again the head at the valve would be prescribed and conditions in the lower region of the

-158transient zone could be evaluated. After the reservoir velocity attains its final steady-state value, the final value is again specified at the reservoir and conditions in pipe 1 (the active element) completely determined. Conditions may then be determined in pipes 2 and 3. The solution advances in a spiral fashion around the two pipes if the value of L/a in each pipe is not identical; otherwise the equations must be solved simultaneously. Conditions in pipe 4 may then be determined. As with the preferred solution developed for the series system, one should recognize that the above procedures would be perfectly general and could be applied to any system with a Known constant-relationship upstream boundary condition. A recent development that has immediate valve-stroKing impli(32) cations is Strceter's study in which he was able to avoid the restriction on the grid spacing of complex systems heretofore imposed by the Ax-At constraint. His solution is to arbitrarily select a value of At which then establishes the grid spacing for each system element, Axi = aiAt. The last, short reach of each pipe (and the total length of short pipes) are then modeled using either the lumped equations or the implicit equations, which are then incorporated into the appropriate boundary-condition equation of the respective pipe. It appears liKely that similar techniques could be developed for the valve-stroKing analyses of complex systems; such an investigation is also recommended for further study. Lastly, two other investigations deserve especial mention since each could be relevant to future extensions of the valve-stroKing theory for liquid piping systems. Wylie(33) has developed techniques

-159for the control of the transient phenomenon in open-channel systems; the computational schemes he developed could be applicable to the analysis and control of piping systems in which the wavespeed is timedependent. Compelling evidence exists that this is the case in systems constructed of such plastic pipes as Polyvinyl Chloride (PVC). With the ever-expanding utilization of these newer materials in large distribution systems, the analysis and control of undesirable transients in their operation will become an essential design consideration. Stoner(23) has studied the application of valve-stroxing theory to the control of gasdistribution systems; he has developed computational procedures for complex multiple loop systems which could be applicable to the liquid systems. In either case, however, one must recognize that the physical aspects of those two systems are decidedly dissimilar to those of the liquid system. The relatively minor inertial considerations of the former differ considerably from the pre-eminent inertial considerations of the latter, and the effective control of inertial effects must necessarily be the essential priority in tie development of suitable control strategies for the liquid systems.

APPENDIX A CONVERGENCE OF ITERATION SOLUTIONS The interior-point and external boundary-condition equations developed in Chapter II which can not be solved directly may be readily solved by an (17) iterative procedure based upon the method of successive approximations.(7) As an example, consider the solution for the velocity at a spaceliKe interior point: Hp-HA F+FA (b) v = V PA _P A- (l2b) P A B 2B This equation is ol the form V = G(V) and its desired solution is unique since G(V) is a strictly monotone-decreasing function of V. (Recall that the function F is a strictly monotone-increasing function of V, and that it is the only unknown term on the right-hand side of the equation.) The solution procedure is depicted geometrically in Figure 1-A, with the arrows indicating the pattern of the iterations. The desired solution Vp lies at the intersection of the 45~0 line with the curve described by G(V). A trial value of Vp is selected, indicated by Vp, and the next value is computed by utilizing this first estimate: V =G(V ) 2 1 The next approximation is then vp = G(Vp ), 3 2 and so on, until the nth approximation (or the nth iterate, as it is often called) is Vp = G(Vp ). n n-l

-161G(V).'/ VP/VV / I I I I l\ /, I I i I I Fi e.-A. rap i representation of the Figumethode -Af successive raphial representoximatons. the method of successive approximations.

-162Delaying for a moment the question of rapidity of convergence, attention is now directed to the more basic question of convergence itself: i.e., do the Vp converge to the desired solution Vp as n increases? As n (17) would appear to be evident from Figure 1-A and can be proven analytically, the sufficient condition on G(V) for this desired convergence is 0 < IG'(V) < 1. (A-l) One can investigate Equation (12b) relative to this criterion, since G'(V) F(V) 1 d (Li 2B 2B dV DN2g/ 1 12fLV LV2'(v)_ 2B \DNLg DN2g F 2 f'(V)) 2B V f Since the derivative of f with respect to V is generally a negative number (the only exception being for flow in the critical zone, and then the value of the function F is not large), this term can be safely neglected in an assessment of the anticipated maximum value of IG'(V)J. Thus |G'(v) | V(A-2) BV Now consider the following evaluation: for the 2000-ft pipes of the laboratory experimental system, the maximum head loss that can be reasonably extrapolated from the calibration data is 92.5 ft at a velocity of 2.8 ft/sec. For a = 2664 ft/sec, N=8, and g = 32.2 ft/sec2,

-163which is safely within the criterion established by Equation (A-l). The only external boundary-condition equations which can not be solved directly and in which the friction term does not appear in the same form as that of Equation (12b) are those involving an orifice or a valve. In these, the solution of a quadratic in V is encountered. For example, consider the solution for the velocity of flow through a valve located at the downstream boundary of the pipe: + ~2 + 4(c - K 2F /2) r 1+ C + (c2 V (15a) P 2 Again, convergence can be demonstrated a priori since K 2F'(V) |G'(V) -v 2C12 + 4(C2 - K 2Fp/2) Therefore K F' (V) -2C2 since, in order to ensure positive velocities through the valve, C2-Kv Fp/2 can not be a negative number. Finally, since C1 = KV2B, IG'(V)l S F'(V) G'(V)I 2 < B which has already been demonstrated to adequately satisfy the convergence criterion. In the event that G'(V) might not satisfy the convergence criterion established by Equation (A-l), several alternatives are available: (1) reduction of the time increment At, thus creating an equivalent reduction in Ac and F, and, therefore, G'(V); (2) treating f as a constant, thus permitting the direct solution of a quadratic in V; or if neither of the above two alternatives

are acceptable, (3)the simultaneous solution of two nonlinear equations in V and f by some suitable numerical procedure. With regard to rapidity of convergence, that the method of successive approximations does not converge as rapidly as other possible solution techniques is readily acknowledged. Generally speaKing, the modified method of successive approximations, Newton's method, the Newton-Raphson method, or the method of regula falsi will oftentimes converge more rapidly than the former. Nevertheless, the method of successive approximations is preferred because of its relative simplicity and ease in programming, particularly when the original estimate is reasonably accurate in any case. (Recall that the original evaluations are always based upon the equivalent first-order finitedifference approximations.) In the various computer programs essential to this investigation, a maximum of only eight iterates were necessary to produce values of head and velocity accurate to two and three decimal places, respectively. UnliKe the interior-point and external boundary-condition equations, convergence of the internal boundary-condition equations can not be demonstrated a priori. For example, consider the solution for the conditions existing at the junction of two pipes connected in series: C3-C 4- C1F/2+C2FP /2 C3'C 4 1Pi 2 (20a) Hp =...... HP Cl+C2 3-c1 (Hp+Fp /2) VPA 1 (20b) 1 1 4C+C2( p-Fp2/2) 2 2 (20c)

-165Equations (O20a) to (20c) are three simultaneous nonlinear algebraic equations in three unxnovms. While several solution techniques (7,17) have been shown to be convergent for systems of linear equations, one generally must resort to numerical experimentation in the case of the nonlinear equations. In this investigation, the solution procedure adopted is based upon the Gauss-Seidel (single step) iteration within successive approximations. Again, relative simplicity and ease in programming motivated its selection, and the procedure is as follows: (1) Fp and F are initially evaluated at 1 2 VA and VB 2, respectively; (2) Hp is computed from Equation (20a); (3) Vp is computed from Equation (20b) using the initial value of Fp and the previously computed value of Hp; Vp is computed from Equation (20c) in a similar fashion; (4) F. and F are then re-evaluated at the previously computed values of V and Vp, respectively; (5) the process is then continually repeated until the 2 desired convergence obtains. As with the interior-point and external boundarycondition equations, a maximum of only eight iterations were necessary to produce values of head and velocity accurate to two and three decimal places, respectively.

APPENDIX B SOLUTION PROCEDURES Orderly and systematic procedures essential to the valvestroKing and characteristics analyses may be developed and catagorized according to the particular boundary-data problem and system considered. With reference to Figure 1-B, assume that initial conditions are Known along each grid point on the space-liKe boundary and one datum on each grid point of the time-liKe boundaries. Then conditions at point 1 may be evaluated by utilizing the appropriate left-end boundary-condition equation, interior points 2, 3,..., N-l, N by utilizing Equations (12a) and (12b), and point N+1 by utilizing the appropriate right-end boundarycondition equation. The cycle is then repeated as indicated; if a total of I such cycles exist, then the total number of grid points at which conditions are evaluated is I(N+1). This procedure is intrinsic to all characteristics analyses, of course, and occasionally is employed in the valve-stroKing analysis of a passive system when complete boundary data are available prior to the start of the computational scheme (for example, as they are for pipe 3 of the laboratory branching system). Note that even though the initial conditions may be steady-uniform, the simple and direct nature of the procedure eliminates the need to establish the steady-state zone prior to the beginning of the computations. It will develop during the computational procedure as discussed in Section 2.6.1. In the valve-stroKing analyses of parallel and multiple loop systems, necessary boundary data is frequently generated sequentially, one boundary at a time, as the computation scheme advances around the loop from one pipe, to the next, etc. Again, if one datum is eventually -166

-16TA!~~d x i t IT N+2 Nf3 _ l 2N+1 2N+2 / _ _ N-1 N N*I O 2 N3 N N-2 N-/ N Section Nob. Figure 1-B. Solution procedure for one datum on each boundary.

.),68established at the next grid point on each boundary, the computational procedure for such a pipe is illustrated in Figure 2-B. The same sequence and equations are used as before, but the computations advance diagonally upward along the appropriate characteristic lines. Here it becomes expedient to establish the initial zone a priori. If the pipe is subdivided into N equal reaches, the number of initial calculations is given by the sum of the arithmetic progression N+1 + N + N-l +... + 3 + 2 + 1. The sum of this progression is(4) (N+1)(N+2)/2. Again the total number of computational points in the transient zone is I(N+1) unless the pipe is contiguous to a system element of the mixed-boundary data type (see below); in such cases the total number of points is (I+l)(N+l). The fundamental procedure essential to the valve-stroting analyses of rm:st of the elementary active systems of Chapters III and IV is illustrated in Figure 3-B. The necessary data are prescribed along the appropriate boundary, and again it becomes expedient to establish the initial and final steady-state zones a priori. Conditions at the time-liKe interior points are then evaluated using the appropriate equations -- Equations (18a) and (l8b),or Equations (19a) and (19b) -- along each successive vertical grid line as indicated, ultimately extending the computations to the other boundary. If I represents the number of grid points along the boundary at which the transient boundary data are specified, then the total number of interior points at which conditions must be evaluated is given by the sum of the progression I+2 + I+4 +... + I+2(N-1) + I+2N.

- NAi3 3 Section No. Figure 2-B. Solution procedure for one datum on each boundary — alternate style.

-170/ I/+6..2..+5 _ 0 / 2 N N-/ N Section No. Figure 3-B. Solution procedure for both data on one boundary — active case,

-171The sum of this progression is N(I+N+l). If both data are prescribed along one of the boundaries, as in Figure 4-B, but the pipe is a passive element of a system (for example, pipe 2 of the laboratory branching system), then no final steady-state zone may be arbitrarily established. The solutions along the boundary are terminated after a given period of time, and the transient zone is established on and below the dashed line of the figure as indicated. Again, if I represents the number of grid points along the boundary at which the transient boundary data are specified, then the transient zone consists of a total of ILlx time-liKe interior points. Note that in this case computations may either be advanced along successive vertical grid lines as indicated., or they may be advanced diagonally downward along characteristic lines. (For example, the first diagonal would consist of the points 1, I+1, etc.) In the valve strolking analyses of parallel and multiple loop systems, this latter technique must be used if the necessary boundary data are generated -equcntially, as is frequently the case. The solution procedure essential to a valve-stroKing analysis involving the downstream-head specification technique of Sections 3.3.2 and 4.1.2 is illustrated in Figure 5-B. Again the initial zone is first established. Conditions at point 1 may be evaluated by utilizing the appropriate right-end boundary-condition equation, interior points 2, 3,..., N-l, J again by utilizing Equations (12a) and (12b), and point N+l by utilizing the appropriate left-end boundary-condition equation. After a period of time this scheme is terminated, and both boundary data are then specified along the left-end boundary; the remainder of the solution procedure is thereafter identical to that of Figure 3-B. (The dashed characteristic line delineates the boundary between the two zones.) If

ase83 aT ssed --.JCpunoq auo uo v'sp qtoq Joj ainpaooid uoTanioS *'[- aJnrT.'oN UO!,/39$ N I/-N -N, / 0 7AA I- \/-I; Z-j — * —- /+.,,. lw

*~ s'o DA!~ — a alaqo.td ~p-ixTrnTmoq paxim woo ainpaooid uo!TnLoS'g-4 aannl,'ON Uo!/4aS N I-N -N & I 0 I-N't l t L! b+'I / Of~

I represents the number of left-end boundary grid points of the transient zone below the dashed line, and I' the number of boundary grid points above the line, then the transient zone consists of a total of I'xJ1 + (N+1)(I+1+l) points at which conditions must be evaluated. One should recognize that conditions must be evaluated twice on the first diagonal characteristic above the dashed line; the dashed line would rarely coincide with the grid points irlmediately above it, and its position can only be established aftcr the last c-y-cle of computations along the diagonal hnas advanced to the furthest upstream boundary. Thne ex:tension of this latter solution proceure to a passive element o-f a system is illuistrated in Figure 6-B; it would be intrinsic to the studies proposed in Chapter VII. The procedure in the lower region of tie transient zone is identical to that of Figure 5-E and in the upper r:~egion to that of FiGuzre 4-B. As with the previous discussion, coriputaticns in the upper zone rmy advance either along successive vertical rid lines or dowtnw.arcd along diagonal characteristic lines. Tine transient zone contains a total of I'TN + (I+l)(N+l) points at which conditions must be evalmated.

-175I!.2 ZILI1 0 1..1... 1 _,N ~ N O / 2 N-2 N-/ N Section No, Figure 6-B. Solution procedure for mixed bounrldary-data problem — passive case.

APPENDIX C TRANSIENT DURATION CALCULATIONS Whlen the transient condition imposed upon the system is specif ied by selection of the extreme permissible value of head desired at the valve, the duration of the transient (tf) of the simple system of Section 3.1 may be determined by integration of Equation (22). Thus tf = 2L/a- dV (CV I-(HmHr) + (cl o IL m r 2D Since tf > 2L/a, close examination of Equation (22) reveals the following restriction on H: for V >>Vf, Hm>Hf; for Vo< Vf, Hm<HfH The above integral may be evaluated numerically in the following manner: With reference to Figure 1-C (G(V) has the indicated shape since the friction term is a strictly-monotone increasing function of V), recall the second-order (trapezoidal) finite-difference approximation to Equation (22): (Hm-Hr) Fi +Fi+ (22a) V V i+l i NB - 2B Beginning at Vo, Equation (22a) is solved (by iteration) to establish V1 such that the approximate area under the curve of G(V) is equal to At, the constant time increment of the solution grid. V2, V3,..., Vi.l1 Vi, Vi+l are established in the same way. (This procedure, of course, thereby establishes the upstream boundary velocity at each point of the solution grid.) The scheme is terminated when the last velocity calculated is less than Vf for closure situations, or exceeds Vf for situations involrving valve openings. The time At' then may be approximated by -176

ID 3 L3 VVI _ _// v _ Vi'01 f Iq Vk-2 2 0 Figure 1-C. Graphical representation of the numerical evaluation of the time-duration integral.

At' = - At (C-2) Vi -Vi+1 If I represents the number of grid points along the boundary at which the transient velocity data are thereby specified, then the duration of the transient condition is tf = 2L/a + IAt + At'. (C-3) Even though the upstream velocity at each grid point is established using a different technique, Equation (C-3) is equally applicable to the downstream-head specification procedure of Section 3.3.2. Of course, the equation must be appropriately modified to include the wave-travel times of the downstream pipes in the series and branching systems of Chapter IV. The manner by which tf may be arbitrarily selected and the appropriate value of Hm subsequently determined is now considered. From Figure 1-C it is evident that a change in the value of Hm would merely raise or lower the curve of G(V) relative to its position illustrated. Therefore, the area under the curve between V0 and Vf -- and hence the value of tf -- must be a strictly-monotone function of Hm. Thus, the value of H that would yield the prescribed value of tf can be readily determined by the method of interval halving (i7) applied to the solution procedure outlined for the determination of tf. This method (also called the bisection method (2931) or Bolzano's method(7)) permits an exceedingly simple and direct computer-oriented solution when the desired root may be isolated in a Bnown interval, as is the case here. (Recall the restriction on Hm cited above. The other bound on the solution interval can be established without difficulty.)

-179A direct extension of this procedure is utilized to determine the uninown head to be prescribed at the Junction (Hm ) in Equation (43) of the branching system analysis. For closure situations, for example, the left-hand side of the equation is a strictly-monotone decreasing function of H while the right-hand side is a strictly-monotone increas-'a. ing function. For any specific Slow situation encountered, the limits on the solution interval again can be readily established.

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