uv .. ~-.-...gon m 55‘5 I‘M... -__.. . i --.._ ..._..-.‘-_.‘_-. -“- “Ann-U 5’? fir s- fz. :‘ r‘flfi')‘ $0.1 fl)": {4.7. 4-4: 13:1! .7 sting,“ i c yang": “-29 iv 6. Lm'mow‘sa This is to certify that the thesis entitled The Effect of Elbow Restraint on Pressure Transients presented by Robert Stephen 0twell has been accepted towards fulfillment of the requirements for Doctoral degree in Civil Engineering Major professor Date 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution MSU LIBRARIES -— \v RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. THE EFFECT OF ELBOH RESTRAINT OH PRESSURE TRANSIENTS By Robert Stephen Otwell A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil and Sanitary Engineering 1984 ABSTRACT THE EFFECT OF ELBOH RESTRAINT ON PRESSURE TRANSIENTS By Robert Stephen Otwell Transient pressure in piped liquid is a function of structural restraint at an elbow. Uhen supported rigidly, the elbow causes no appreciable alteration of the pressure transient generated by rapid valve closure. However, if the support is relaxed, significant pressure alteration is observed, with peak pressures being greater than the traditional Joukowsky pressure rise. Elbow motion, driven by the axial stresses in the pipe and the fluid pressure, causes the alteration. A numerical model is developed and verified with experimental data. The one-dimensional equations of continuity and momentum for the liquid and pipe wall are solved by the method of characteristics. At an elbow, coupling is introduced by continuity relationships. The translation of attached piping at an elbow is represented by an added stiffness term, and solved simultaneously with the characteristic equations. Comparison is shown between experimental data and predicted results. The equations are normalized and dimensionless parameters are identified that describe the liquid-pipe interaction. ACKNOWLEDGMENTS The author wishes to express his gratitude to the members of his doctoral committee for their assistance; Reinier Boumeester, Merle Potter, and Sidney Stuckenbruck. In particular, the author wishes to thank David Higgert, chairman of the committee, under whose advice and supervision the work was carried out, and committee member Frank Hatfield, for his encouragement and guidance throughout the work. The author is indebted to the Department of Civil and Sanitary Engineering for teaching and research assistantships, and to the National Science Foundation for their financial support. The author also wishes to thank his sister Carol for help in preparation of the manuscript, and his wife Laura for her interest and encouragement during the project. ii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES NOMENCLATURE CHAPTER 1 INTRODUCTION 1.1 Problem Introduction 1.2 Background 1.2.1 Strain-related Coupling 1.2.2 Pressure Resultants 1.2.2.1 Periodic Forces 1.2.2.2 Non-periodic Forces 1.3 Scope CHAPTER 2 EXPERIMENT 2.1 Experimental Setup 2.1.1 Motivation 2.1.2 Pipe and Liquid Parameters 2.1.3 Design Considerations 2.1.3.1 Piping 2.1.3.2 Liquid Setup .4 5 Acquisition 2.1 P p 2.1. D t 2. 2. 2. 2.2 Experimental Results 2.2.1 Introduction 2.2.2 Case A: Stiff S stem 2.2.3 Case 8: Axial S iffness iii .1 Pressure Transducers .2 Accelerometers .3 Analog to Digital Conversion o-u-u-t m - 010101 90 vi viii 00’ Oh NH 0-5 12 12 12 14 16 16 16 17 22 22 23 23 24 24 26 29 2 4 Case C: Bending Stiffness 2.5 Case D: Combined Effects .2.6 Case E: Valve Unrestrained 2 7 Discussion 2.8 Uncertainty NNNNN CHAPTER 3 THEORY 3.1 Introduction 3.2 Four-Equation Model CHAPTER 4 NUMERICAL STUDY 4.1 Method of Characteristics 4.1.1 Navespeeds 4.1.2 Compatibility Equations 4.1.3 Initial Conditions 4.1.4 Boundary Conditions 4.1.4.1 Reservoir 4.1.4.2 Valve 4.1.4.3 Stiff Support 4. L 4. 4 Elbow 4.2 Numerical Analysis 4.2.1 Comparison to Experimental Results 4.2.1.1 Case A: Stiff System 4. 2. L 2 Case 8: Axial Stiffness 4. 2. L 3 Case C: Bending Stiffness 4.2.1.4 Case D: Combined 4.2.2 Parametric Study 4.2.2.1 Poisson Ratio 4. 2. 2. 2 Dimensionless Parameters 4. 4. 4. 3 Stiffness Coefficients CHAPTER 5 SUMMARY AND CONCLUSIONS APPENDIX LISTINGS OF COMPUTER PROGRAMS A.1 CONTRL A.2 SAMPL A.3 LIQPIP LIST OF REFERENCES iv 30 33 36 36 37 38 38 38 43 43 45 46 49 49 so so so 52 55 56 57 57 57 62 62 62 65 68 71 74 74 78 so 90 LIST OF TABLES No. Title Page 1. Stiffness Coefficients 56 2. Pipe Material Constants 66 2 O \IO’U’Owat-e 17. 18. LIST OF FIGURES Title Schematic of Experimental Setup Valve (Top View) Pressure Transducers and Valve Stiff Support Elbow and Accelerometers Structural Restraint for Four Experimental Setups Pressure Response for Case A, Showing Full Pressure Rise Pressure Response for Case A, Triggered by Pressure ransducer . Pressure and Elbow Velocity Responses for Case 8 10. 11. 12. 13. 14. 15. 16. Pressure and Elbow Velocity Responses for Case C Pressure and Elbow Velocity Responses for Case D Pressure Response for Case E Pipe Element Characteristic Representation Elbow Schematic Comparison of Experimental and Predicted Pressures for Case A Comparison of Experimental and Predicted Pressures and Elbow Velocities for Case B Comparison of Experimental and Predicted Pressures and Elbow Velocities for Case C vi P_ag_e; 18 20 20 21 21 28 28 31 32 34 35 42 51 51 58 59 ’12-. 19. 20. 21. 22. 23. 24. LIST OF FIGURES (cont'd) Title Comparison of Experimental and Predicted Pressures and Elbow Velocities for Case D Comparison of Predicted Pressures at P1 for Case B with Variable Poisson Ratio Force ratio versus rle The Inverse of the Dimensionless Parameter J versus rle Assumed Deflected Shapes Comparison of Pressure and Elbow Velocities for Case C with Variable Stiffness vii Page 61 64 64 67 67 7O A1,A2 nan +rt-‘h n I i- ‘U PXXQIMMG 01.02 deft/3375 NOMENCLATURE Navespeed in infinite medium, m/s Cross-sectional area, m2 Coefficient matrices Liquid wavespeed in pipe, m/s Pipe axial wavespeed, mls Characteristic lines Pipe wall thickness, m Young's modulus of elasticity, Pa El(1-v2), Pa Dimensionless parameter (equation 23) Dimensionless parameter (equation 15) Bulk modulus of liquid, Pa Stiffness coefficient, N/m Pipe length, m Pressure, Pa Dimensionless parameters (equations 22, 23) Pipe radius, m Elbow radius, m Constant relating initial flow and pressure Time, 3 Transformation matrix Pipe axial displacement, m viii NOMENCLATURE (cont'd) Pipe axial velocity, mls V Liquid velocity, mls Vr Relative velocity, mls w Pipe radial displacement, m H Dimensionless parameter (equations 14 and 16) x Distance along pipe axis, m 2 Matrix of dependent variables 8 Strain A Characteristic roots v Poisson ratio 6 Density, kglm3 0 Stress, Pa T Dimensionless valve opening Subscripts f Liquid t Pipe x x-direction y y-direction 6 Circumferential direction ix Chapter 1 INTRODUCTION 1.1 Problem Introduction Piping systems used for transfer of pressurized liquids operate under time varying conditions imposed by pump and valve operation. Unsteady pressures and flows result, which are known as liquid transients, waterhammer, or surges. Traditionally, to analyze the unsteady behavior of the liquid, the equations of motion and continuity of the liquid are solved without regard to the motion of the piping. The transients propagate at the acoustic velocity, or wavespeed, of the liquid in the pipe. The diameter, wall thickness, and elastic modulus of the piping are used in computing the wavespeed, after which the liquid is assumed to be flowing through a straight, rigid pipe. Recently there has been concern that the transient behavior of liquid in a piping system that is neither rigid nor straight may differ from that predicted by a traditional rigid pipe analysis. It is reasoned that the dynamic forces exerted by the liquid at fittings (elbows, tees, valves, and 1 reducers) where flow direction or area changes can set the pipe in motion and the feedback between the liquid and pipe can cause alteration of the liquid behavior. Some investigators have suggested that this alteration is either negligible or that a rigid pipe analysis would provide a conservative estimate of the transient pressure because of the transfer of energy out of the liquid and into the structure. However, experiments have shown that in some systems, the response of the piping amplifies transient pressure in the liquid. Motion of fittings is caused by dynamic forces in the liquid and pipe wall. The amplitude and velocity of motion are functions of the restraint provided by the attached piping and supports. It is apparent that the analysis of liquid in a piping system must include information on the piping structure itself. A coupled liquid-pipe analysis must consider structural parameters. 1.2 Background In the late 1800's Joukowsky [1] determined that the wavespeed of liquid in a pipe, and hence the speed at which liquid discontinuities propagate, was related to the relative circumferential stiffness of the pipe. This wavespeed is an "apparent" wavespeed and is less than the true wave propagation speed in an infinite liquid. Joukowsky assumed that pressure is uniform across any cross section and that the radial dilation of the pipe is equal to the static dilation that would be caused by the pressure. That is, he neglected the radial inertia of the liquid, mass of the pipe wall, and axial and bending stresses in the wall. Since Joukowsky's time, much work has been done on liquid transients in piping systems. Horks by Hylie and Streeter [2], and Chaudhry [3] outline analytical techniques to solve many types of problems with various boundary conditions. The remaining literature review will concentrate on liquid-pipe interaction. Contained liquid interacts with piping in several ways: internal pressure causes circumferential strain of the pipe wall; pressure resultants act at locations where flow direction or area change; high steady flow rates induce lateral pipe motion; and, the transverse acoustic modes in the liquid may interact with shell modes of the pipe wall. This thesis will only consider two mechanisms of dynamic interaction between the contained liquid and the piping: 1) strain-related effects which occur axially along pipe reaches; and 2) pressure resultant effects, where coupling occurs only at fittings. 1.2.1 Strain-related coupling Strain-related or Poisson coupling results from the transformation of circumferential strain caused by internal pressure to axial strain due to Poisson's ratio: 6 = v e (1) Uhere ex and e are axial and circumferential strain and v 6 is Poisson's ratio. Skalak [4] extended Joukowsky's method to include Poisson effects. The extension consisted of treating the pipe wall as an elastic membrane to include the axial stresses and axial inertia of the pipe. For sudden valve closure, an axial tension wave was found to propagate in the pipe wall at a wavespeed near that of the pipe material, hence a 'precursor' wave travels ahead of the main pressure wave in the liquid. The axial tension is a Poisson effect in response to pipe dilation caused by the pressure transient. An increase in liquid pressure due to the tension wave was identified, but this increase was small because it was caused by the contraction in pipe diameter due to the tension wave: it is a second order Poisson effect. Thorley [5] completed a study very similar to Skalak's, including experimental validation of the theory. Uilliams [6] also conducted a similar study and found that the longitudinal and flexural motion caused damping that was greater than the viscous damping in the liquid. The researchers discussed did not include the radial inertia of the liquid or pipe wall. Comments were made on damping due to radial motion as the wave traveled through the pipe and the damping was found to be small. Halker and Phillips [7] proposed a new theory that included the radial inertia of the pipe wall because they were interested in transients of very short duration. They formulated a one-dimensional, axisymmetric system of six equations that included the radial and axial equations of motion of the pipe wall, two constitutive equations for the pipe wall, and the equation of motion and continuity for the liquid. They reported that the method 'retains much of the rigor of the axisymmetric, two-dimensional approach" of Lin and Morgan [8]. They found their method is ideal for transients where the generation of the pressure pulse occurs in several microseconds, and that the classical waterhammer equations are adequate for longer pulse lengths. The researchers discussed identified two important waves in a straight length of pipe, one in the pipe wall and one in the liquid, and they identified interaction between the liquid and pipe. None of these studies considered the possibility that a fitting, such as an elbow, might move in response to the precursor wave and alter the transient response of the liquid. 1.2.2 Pressure Resultants At fittings where the pipe area or direction changes, the pressure resultant acts as a localized force on the pipe. The discussion of past research concerning pressure resultants will first consider periodic liquid forces, then non-periodic forces. 1.2.2.1 Periodic Forces Blade, Lewis, and Goodykoontz [9] in 1962 were among the first to report on the alteration of the liquid behavior due to the motion of an elbow. They studied harmonic loading on a single elbow and reported the elbow provided coupling between the pipe and liquid but caused no appreciable reflection or attenuation. Hood [10] also studied harmonic loading of a pipe structure analytically by representing the structure as a single degree of freedom spring-mass. He found that the natural frequencies of the liquid were shifted, especially when the frequency of the loading was near one of the natural frequencies of the supporting structure. Davidson and Smith [11], interested in the generation of noise due to harmonic loadings from pumps, developed an eight-equation system to solve for the eight degrees of freedom in the plane of a single elbow. The eight degrees of freedom consisted of an axial and radial pipe displacement, an axial liquid displacement, and a rotation, at each end of the elbow. An experimental setup verified the theory. In a later paper, Davidson and Samsury [12] studied a more complex system with three elbows connected by short straight lengths. In both studies they identified significant coupling between the pressure waves in the fluid and the pipe. Hatfield, Uiggert, and Otwell [13] developed a general solution procedure to study fluid-pipe interaction with harmonic loadings. The eigen solution of the supporting pipe structure is obtained from an existing finite element structural program. The modal responses are then coupled to the liquid analyses by a method known as component synthesis. This method was validated experimentally in a follow-up paper [14]. Phillips [15] studied the reflection and transmission of harmonic liquid loads at elbows by coupling the liquid and pipe equations. He found reflection and transmission coefficients of about 15! and 85!, respectively, for sharp and gentle bends for a frequency range of loo-10,000 hz. At higher frequencies no alteration was found. The model accounted for bending, shear and axial forces in the pipe, axial displacements, and liquid pressures. No comment was made as to the reason for the alteration around the elbows. 1.2.2.2 Non-periodic Forces This dissertation will concentrate on the alteration of the liquid behavior due to non-periodic forces. The following is a review of the literature for that class of problem. Hood [16] investigated the effect of pipe motion on the pressure generated by rapid valve closure. He studied analytically and experimentally the effect of structural motion on the liquid behavior as a function of structural parameters and valve closure rate. His experiment involved a straight pipe with a pressure tank upstream and a simply-supported beam connected to a slip ring at the other end, with a branch and valve immediately upstream of the beam. The valve was slammed and the pressure resultant forces set the beam in motion. It was concluded that the beam response could significantly alter pressures. Hood and Chao [17] set up an experimental apparatus that included an elbow between two 6 m lengths of copper pipe, with a quick closing valve downstream and a constant pressure reservoir upstream. Results were obtained with the elbow restrained by a supporting structure and with the elbow unrestrained. For the restrained case the pressure rise resembled the traditional Joukowsky prediction. For the unrestrained (case, there was alteration of the pressure response in the form of a oscillation about the pressure observed for the restrained case. This alteration was thought to be caused by motion of the elbow driven first by a stress wave in the pipe and then by the pressure wave. An analytical method was devised that used the measured structural velocities from the experiment as input into flow conservation relationships at the elbow in the liquid model. Favorable comparisons were shown between this analysis and the experiment. Ellis [18] developed an analytical procedure to couple liquid equations with the equations for axial motion of piping. Coupling took place at fittings such as valves, elbows and branches, and mass and stiffness were lumped at the fittings. The Poisson effect was not included in the analysis. Schwirian and Karabin [19], Giesecke [20], Otwell [21], and Higgert and Hatfield [22] all developed general analytical techniques to couple liquid and pipe equations in order to study the dynamics of general piping systems. In all these studies, coupling was imposed only at fittings, and the effect of support and piping stiffness was shown to be significant. In contrast, Swaffield [23] found that the influence of an elbow on liquid behavior was solely dependent on its dimensions: radius of curvature to pipe diameter ratio and included angle. He experimentally determined reflection and transmission coefficients of 201 and 80$ respectively. Pipeline restraint was found to have no effect. 10 1.3 Scope It is apparent that although there is much evidence suggesting that the behavior of liquid in piping can be influenced by the piping system that contains it, the actual mechanisms governing the alteration are not fully understood. Two types of coupling have been identified but their significance and their relationship to each other have not been quantified. The studies related to Poisson effects dealt only with straight pipes. At pipe fittings, pressure resultant forces have been shown to be important in vibrating piping; typical methods of solution include coupling only at fittings, ignoring the Poisson effect. Little experimental validation has been attempted. The main objective of this study was to design and build an experiment that isolated the important parameters of liquid-pipe interaction. Structural restraint of an elbow was the independent variable. Very stiff supports were used for the control case of an immobile elbow. An elbow restrained only by the axial stiffnesses of the connecting pipes was studied so that the Poisson effect could be observed. Elbows restrained by the flexural stiffness of short pipe lengths were studied to observe a more flexible system. 11 A numerical model was developed that incorporates structural parameters necessary to represent the coupling mechanisms. The model simplifies the six-equation Halker and Phillips [7] development by ignoring the radial inertia term. The resulting equations are easily implemented in a numerical solution procedure. At an elbow, coupling is introduced by continuity relationships and the translation of attached piping can be represented by an added stiffness term. Comparisons are shown between predicted responses and experimental data for different restraint conditions. Chapter 2 presents the design of the experiment and results. Chapter 3 contains the theoretical development, and Chapter 4 the numerical formulation, comparison with experimental results and an investigation into the important parameters involved in liquid-pipe interaction. Chapter 5 provides a summary discussion and conclusions. Chapter 2 EXPERIMENT 2.1 Experimental Setup 2.1.1 Motivation The purpose of the experiment was to enable observation of the alteration of a generated pressure transient for varied structural restraints at an elbow. From this the important pipe parameters influencing the alteration could be determined. One extreme was to fix the elbow rigidly and determine if geometric aspects of the elbow caused significant alteration of the liquid behavior. Once that was established, the alteration of the liquid behavior as a function of the structural restraint could be determined. Before discussing the copper pipe experiment and results, a previous experiment using polyethylene pipe will be discussed. Many of the design considerations incorporated in the copper pipe experiment were a result of knowledge gained from the original experiment. The original experiment was designed to isolate a single elbow. Polyethylene pipe was used because of its high flexibility. 12 13 The flexibility was desired for two reasons: 1) the elbow restraint depended on pipe properties only to a small degree; and 2) the liquid wavespeed would be low and the vibrations could be observed before reflections from boundaries returned to the elbow. The pipe was wrapped on a circular (radius 2.5 m) frame. At the middle of the length, the pipe came off the frame and was straight for 3 m to a 90 degree elbow. Then another 3 m straight pipe came back to the frame. Problems resulted from the mounting system; the frame itself moved as the transient traveled through the pipe. This was not anticipated as the 2.5 m radius was chosen to minimize the pressure resultant forces. Also there was a gradual rise in the pressure response at the valve, along with vibrations of the valve itself, which made observing the structural effects attributable to the elbow very difficult. The objective was to mount supports with different stiffnesses at the elbow to create varied structural restraint. The first case considered was one without any support at the elbow. The elbow and attached piping were suspended by wire supports. It was expected that the elbow would move in the direction of the resultant pressure when the pressure transient from slamming the valve traveled upstream to the elbow. However, the initial direction of movement was parallel to the downstream pipe leg and in the 14 direction of shortening the leg. This was determined to be from the precursor tension wave which pulled the elbow before the pressure wave arrived. This observation, although puzzling at first, turned out to be the major motivation for the bulk of the remaining dissertation, including the design of the copper pipe experiment and a more thorough search of the literature on the precursor wave leading to an analytical method incorporating both_ the precursor and pressure resultant effects. 2.1.2 Pipe and Liquid Parameters This section discusses the different parameters considered in the design of the pipe experiment. Since the experiment was designed to vary the restraint of the pipe, and not the type of pipe or contained liquid, many of the parameters were constants throughout the experiments. The pipe parameters are: Pt-density gx-axial stress E -Young's modulus u -axial velocity v ~Poisson's ratio w -radial displacement r -inside radius e -wall thickness R -elbow radius The experiment consisted of a piping layout using 1 inch copper pipe with standard fittings. Because of that, pt, E, v, r, e, and R were all fixed and could not be varied. Axial velocity and axial stress were the important variables 15 to study. The axial velocity and axial stress are dependent on the restraint imposed on an elbow. The axial velocity was measured at the elbow with accelerometers. The radial displacement of the pipe is a function of the internal pressure and was not restrained. The liquid parameters are: Pf-density p -pressure K -bulk modulus V -axial velocity Cf-wavespeed The mean density and bulk modulus remained constant. Hater was the contained liquid. The liquid wavespeed was determined by measuring the time delay between two pressure transducers at a known distance apart. The liquid wavespeed was found to be 1270 mls. The dynamic pressure was the important dependent liquid variable. The pressure was monitored by pressure transducers at several locations. The steady-state liquid velocity was controlled by varying the mean pressure drop between two pressure tanks. During the experiments, environmental temperature and humidity were monitored. Their variations were found to be too small to affect significantly the observed pressures and accelerations. 16 2.1.3 Design Considerations Certain variables must be controlled during the experiments. To do this, the design incorporates the following constraints: 2.1.3.1 Piping The valve and piping outside the reach to be investigated must be rigidly supported in all directions. The reach to be investigated needs to be supported, ideally by supports that contribute negligible stiffness, inertia and damping for motion in the plane of the elbow. The lengths of pipe upstream of the elbow must be long enough to permit observation of the elbow's vibration before the reflection returns from the upstream end. Two elbows are incorporated so that a combination of restraints is possible, including one that allows translation of a section of pipe between the elbows. 2.1.3.2 Liquid The liquid flow is controlled at the upstream and downstream ends of the pipe. The upstream end is a constant pressure tank and the downstream end is a valve connected to another pressure tank. 'The valve closes rapidly to cause an abrupt pressure rise so that the Poisson effects and pressure resultant effects on the elbow can be differentiated. The 17 valve closes in a time period such that the sum of the valve closure time plus the tension wave travel time to the elbow is less than the pressure wave travel time from the valve to the elbow. This will ensure that the full Poisson effect will take place before the pressure wave begins arriving. The generated pressure transient is a function of the steady-state flow velocity V0, so there must be control of the steady-state pressure drop across the system to control the velocity. 2.1.4 Pipe Setup Figure 1 is a schematic of the pipe setup. There is a total of 47.9 meters of 1 inch (nominal) diameter copper pipe between the upstream pressure tank and the valve. The pipe material constants (obtained from the manufacturer) are: pt=8940 kglm’, E=117 GPa, 9:0.34, r=13 mm and e=1.27 mm. The system has a total of six elbows, with R=20.6 mm. The elbow to be studied is elbow 1. Rigid connections anchored to concrete walls or floor are used at elbows 3, 4 and 5, and as needed for elbows 1 and 2, depending on the restraint desired. The two main test sections of copper pipe (L1 and L2) are suspended by wire hangers to allow movement in the plane of elbow 1 and to keep the pipe from sagging and hence changing the axial stiffness characteristics. Hhere necessary, the pipes are hung within wooden enclosures for protection. 18 gauom peacoe.coaxu Lo u_aaeonom .u usam.m s X N 8: m>_a> ss v\‘ i. .1 ma Ha .Amw . . s zos_m x: zopm . as .e zos_s . as m 36s,“ sm.as _as6h sk.~ as N soaps o“.0s es ms so.“ ms mm.m as mo.m ms ms so.s sms as m. ems :6 so a~.~s as s sps as m spa Ase aspects saws 19 The valve is a critical part of the setup as it has to close very quickly. The design of the valve is similar to one used by Hood [17]. As shown in Figure 2, the valve consists of an inlet chamber with an orifice connected to another chamber in the main valve block with an outlet 90 degrees from the inlet. The orifice has a hard rubber washer glued on the upstream side. The two pieces of the valve body are joined by four bolts and sealed with an O-ring. Running through the orifice and out the end of the block is a brass rod that enlarges to a diamond shape in front of the orifice and a ball outside the block. Hhere the rod goes through the block there is a Teflon sleeve. Operation of the valve is manual: the rod is pulled through the valve, and as the diamond-shaped seat approaches the orifice, differential pressure develops and slams the seat against the rubber washer. The valve has a closure time of approximately 4 ms. The valve body itself is bolted to the floor with two bolts and threaded expansion anchors (Figure 3). The valve closure, though hand actuated, has been found to be repeatable. The rigid supports (Figure 4) are designed to resist axial pipe motion. The supports are made from aluminum blocks. In each block a hole was drilled to a diameter equal to the outside diameter of the pipe and then cut at the centerline of the hole. The cut removed 1 mm of material. The pipe is held in place by bolting the upper and lower pieces of the block together. The supports are then bolted to the floor. 20 O-ring Teflon Rubber Washer \\\ \§\\ Sleeve :: -—€I>' J . O %M\ Figure 2. Valve (Top View) Figure 3. Pressure Transducers and Valve 21 Figure 4. Stiff Support Figure 5. Elbow and Accelerometers 22 An air purge is located just upstream of elbow 4, the highest elevation on the piping system. This consists of a short, small diameter copper tube with a valve soldered to the copper pipe. Air can be bled off when the system is under pressure. 2.1.5 Data Acquisition To obtain information on the behavior of the experimental setup, dependent variables must be recorded as a function of time. This is accomplished by measuring pressure and acceleration with transducers. The transducer converts the pressure or acceleration to voltage which is amplified and transferred to a recording device. Data were obtained on pressure at two locations, and on acceleration in orthogonal directions at one location. The information then could either be viewed on a storage oscilloscope or recorded on a digital computer. 2.1.5.1 Pressure Transducers PCB Piezotronics Model 111A26 quartz pressure transducers were used to measure the dynamic pressure at two locations (P1 and P2) as shown on Figure 1. These transducers were chosen because of their high frequency response and because they are acceleration compensated. Between the transducers and recording device was a PCB Model 480006 battery power unit. The transducers were mounted by tapping a brass block 23 per PCB specifications and then soldering it to the pipe. The hole was drilled so the tip of the transducer would mount flush with the inside of the pipe. One Sensotec Model '8' strain gage type pressure transducer was mounted at the valve for preliminary setup and testing. This transducer measured the static and dynamic pressure. It was also used as a trigger for the recording devices. The PCB (left) and Sensotec (right) transducers are shown in Figure 3. 2.1.5.2 Accelerometers PCB Model 302A quartz accelerometers were used to measure the movement of elbow 1. The accelerometers were mounted on a brass block that was soldered to the inside of the elbow (Figure 5). Between the transducer and recording device was a PCB Model 480A08 integrating power unit. It could be switched to output either the acceleration or velocity by electronic integration. 2.1.5.3 Analog to Digital Conversion Data were recorded and stored by a Digital Equipment Corporation (DEC) POP-11I02 mini-computer. Transducer voltages were run through an AID converter and stored as digital quantities. The hardware includes a DEC Model ADVll-C AID converter and a DEC Model KHVll-C real-time clock. Software was developed to control the AID converter and clock. A FORTRAN control program CONTRL (listing in 24 Appendix) drives an assembly language sampling subroutine SAMPL (listing in Appendix). Input data into CONTRL consist of the sampling rate, number of samples, and the channels to sample. The sampling begins with activation of a Schmitt trigger by the Sensotec transducer. SAMPL is an interrupt driven, clocked sampling subroutine that can take samples at a maximum rate of 10 kHz on one channel. The samples are stored in an array, then transferred back to CONTRL to be converted to pressure and velocity units and stored on a floppy disk to be printed or plotted. A Tektronix Model 5103N oscilloscope was used for visual monitoring of the transducers. 2.2 Experimental Results 2.2.1 Introduction The following sections contain the results obtained from four different restraint conditions. The results consist of the pressure response at P1 (the valve) and at P2, and the axial velocities of elbow 1 for various structural restraints. Figure 6 shows the configurations for the four cases. From elbow 3 upstream to the pressure tank the system is the same for all cases. 25 W1 X3 lZl'i Egg] Valve :3 Stiff Support Figure 6. Structural Restraint for Four Experimental L1 Case A L2 P2 L ’ L3 1‘9 L1 u Case B L2 11” l e TV L1 0.36 mF—H Case C L2 1 0.36:rn:|;-_>i'>2 ’ L3 L1 Case D L2 0.36 m P2 ' L3 Setups 26 The first setup, Case A, has all the elbows supported rigidly in all directions. For the second setup, Case 8, the stiff supports were removed at elbow 1 so that it is restrained by the axial stiffnesses of legs L1 and L2. The third setup, Case C, has two stiff supports placed 0.36 m from elbows 1 and 2. Axial translation of leg L2 is restrained by the bending stiffness of the two short lengths. This arrangement is less stiff than Case 8 because pipes are much stiffer axially than in flexure. The fourth setup, Case D, represents a combination of Case 8 and Case C. The final setup discussed, Case E, is the stiff system (Case A) with the support at the valve removed to observe liquid-pipe interaction caused by a fitting other than an elbow. 2.2.2 Case A: Stiff System Figure 7 shows the pressure response at locations P1 and P2 for the stiff system. The interval of the pressure rise at P1 (hence the valve closure time) is approximately 4 ms. On subsequent plots, the data acquisition process will be triggered by the Sensotec transducer at the valve. Therefore, the initial 2 ms of pressure rise will not show at P1. A common assumption of one-dimensional transient analysis procedures is that there is no alteration of the liquid transient behavior as it travels around a bend. The 27 experiment was designed so that this assumption could be verified on a system with a truly fixed elbow, which was the reason for placing the two stiff supports at elbow 1, the main test elbow. Figure 8 shows the responses triggered by the transducer which can be used for comparison with the experiments to be described later. The sampling frequencies for Figures 7 and 8 are 1 kHz and 2 kHz, respectively. The higher sampling frequency shows a pressure spike due to flutter as the valve seats against the rubber washer. The pressure spike does not affect the subsequent pipe response. The remaining transducer data for Case 8 through Case F shown on Figures 9-12 were also sampled at 2 kHz. The pressure response at P1 (the valve) on figure 8, after the pressure spike, is essentially flat, with only small amplitude, high frequency oscillations. If a reflection of the pressure pulse were to be generated at the elbow, it would arrive at the valve at approximately 20 ms. This is not apparent: the oscillations in this region are a few percent of the initial pressure rise. In the 40-60 ms range larger oscillations occur. These are from the upstream elbows that could not be supported as rigidly. If there is to be no alteration of the transient, then not only must there be no reflection from the elbow, but no losses at the elbow. Therefore, the pressure at P2 should be the same magnitude as at P1. 28 2000 -— E .x 1000 - L” o: :3 -— U) (I) 1.1.1 a: Q' 0 l l I I 0 30 60 90 120 TIME (ms) Figure 7. Pressure Response for Case A, Showing Full Pressure Rise 2000 -1 .— A J 4 I) \ 3 1000 -— i ‘k :1 P1 3P2 \ .s .‘ ~, a: _ | ' D i ‘ 5; ' 1 Lu ' \ o: ' v1}; 0' 0 — “_.a/*” i ~\~' I l l l 0 20 40 60 80 - TIME (ms) Figure 8. Pressure Res onse for Case A, Triggered by Pressure ransducer 29 This is shown to be the case by the dashed line of Figure 8. The average pressures over the pressure surge for P1 and P2 are within 0.3% of each other, showing that the transient is propagated through the elbow with virtually no alteration. If this system were analyzed with a traditional liquid transient model, the predictions would be very similar. The pressure rise predicted by a traditional model is equal to the product of prfVO, which is equal to 1500 kPa in Figure 8. 2.2.3 Case 8: Axial Stiffness As shown on Figure 6, the stiff supports on both sides of elbow 1 are now removed and the elbow is restrained in its plane by the axial stiffness of leg L1 in the x-direction and the axial stiffness of leg L2 in the y-direction. An examination of Figure 9 shows that the pressure response at P1 is significantly different from that of Case A. Elbow motion is initially in the negative x-direction and is driven by the precursor stress wave; this action results in an increase in the liquid pressure. The pressure increase propagates to the valve and is recorded at 13 ms. The elbow motion at 10 ms is the result of the primary pressure pulse driving the elbow in the positive x and y-directions, resulting in a decrease in pressure, as shown on the P1 curve at 20 ms. The elbow then continues to vibrate at the natural frequencies of the fundamental axial modes of reaches L1 and L2 of the piping, and the pressure response 30 is a combined effect of the resultant velocities. The maximum pressure, occuring at 44 ms, is 22% above that which would occur in a piping system with no motion (ofoVO). The minimum pressure, occuring at 23 ms, is 34% below ofoVe. The maximum velocity of the elbow is 0.27 mIs and the maximum displacement is about 0.5 mm. 2.2.4 Case C: Bending Stiffness For the third case discussed, as shown on Figure 6, the two elbows and the connecting reach L2 are allowed to translate in the y-direction by the flexibility of the attached piping. Because of the relatively large axial stiffness of these short attached lengths, elbow motion in the x-direction is negligible as shown on Figure 10. The effect of the precursor pipe stress on elbow 1 is minimal for this case because of the support. In the y-direction, the frequency of vibration is much lower than that of Case 8. The elbow moves first in the positive y-direction when the pressure pulse arrives at the elbow (10 ms). Then as the pulse moves towards elbow 2, there is an elastic return of elbow 1 combined with the pressure pulse arriving at elbow 2 to drive leg L2 in the negative y-direction. The elbow's motion can again be shown to alter the pressure responses. At P1, after the initial pressure rise, there is a pressure decrease beginning at 20 ms caused by the initial movement of elbow 1 in the positive y-direction. 31 2000 fl 4 .’ . A. o «v. E i .7 \“ JV \w \ “\ 'i.’ \_‘ 3"; 1 000 — P1 " \\ .12 ~« ,jPz ( Lu 1 \ O! — I ‘ :3 i ‘ m i ‘ 3 I, k\ : 0 _ ----“"" Kahls‘. ' T l 1 TIME (ms) 0 . 5 — Tn‘ . E _. u, >- E \\ i“ I \ I S 0 o 0 — -~ ‘| I. ‘1‘ 'l’ \ . \‘ ’1‘ ' ‘|. h “ ' ’A'I. 11>; ', I II WV! \J \‘ 3 7‘ \N "V 3 'l u 0 . | 3 u.y ‘1 I Lu _ J - O . 5 - ° 2 ° 4 0 6 0 8 0 TIME (ms) Figure 9. Pressure and Elbow Velocity Responses for Case 8 32 2000 -H A‘ R\ \\\._,A'\\ _ r- ,‘t‘ ’ L | 1‘5 \, $\’J’ A g . \L 3 1000 — R 1‘ :1 P1 {p2 ,\ Lu 1' \ g — I ‘ I \ m , \ E? I Mn 0. 0 _ -..._/’ awn \' l I l T 0 20 40 60 80 TIME (ms) 0.5 —1 T; u \ E _ >.. .—— {3 S 0.0 — I.“ > ‘3 c> cn _] - LLJ '0-5 l l l l O 20 4O 60 80 TIME (ms) Figure 10. Pressure and Elbow Velocity Responses for Case C 33 The pressure rise that follows is caused by the combination of two effects: 1) the initial translation of leg L2 in the positive y-direction will cause a pressure increase propagating from elbow 2 back to the valve; and 2) the translation of leg L2 in the negative y-direction will cause a pressure increase propagating from elbow 1 to the valve. The combined effects create a steep pressure increase at P1 at 30 ms. The maximum pressure at P1 is 25% above PfoVO and the minimum is 351 below. 2.2.5 Case 0: Combined Effects As shown on Figure 6, the support near elbow 1 is now removed so that it is restrained by the axial stiffness of leg L1 in the x-direction and the flexural stiffness of one 0.36 m length in the y-direction. For the elbow movement shown in Figure 11, the x-direction velocity is similar to that of Case 8, and the y-direction velocity is similar to that of Case C. The resulting pressure shows the high frequency component of Case 8 superimposed on the low frequency component of Case C. The pressure oscillation is the greatest of all three cases. The maximum pressure at 46 ms is 331 greater than prfVo, and the minimum at 27 ms is 441 below. 34 2000 - T; / fig - H J, 1". Jh‘ ‘“ " H J bu / 1“ LLJ ' J» ‘ a: l \ :3 1000 — I \l 5; P1 If ‘ u: f k s. _. 1P2 ‘ .. : \ J, ”a": 0 -— _“__,r 7‘ \ T l T l o 20 40 50 30 TIME (ms) 0.5 .1 7; /fl"\ “7 g It ' \\ v — I O IN \ l U : .1 ‘\ X I, ‘\ a ‘. ” E 0 o o 'l-IIIII ” " "I ‘ > ‘n g; \ 1 co \ I' .1 \\ ’ 1.1.: —l I up P” \J’ -0.5 . l I l T 0 20 4O 60 80 TIME (ms) Figure 11. Pressure and Elbow Velocity Responses for Case 0 PRESSURE (kPa) 35 2000— ] l l I o 20 40 60 80 TIME (ms) Figure 12. Pressure Response for Case E 36 2.2.6 Case E: Valve Unrestrained Figure 12 shows the pressure response due to relaxing the restraint at the valve. The system is set up as in Case A except that the bolts holding down the valve were loosened. The resulting stiffness of the valve in the x-direction is due to the axial stiffness of leg L1 and the restraint provided by attached piping between the valve and pressure tank. This arrangement shows liquid-pipe interaction at a fitting other than an elbow. At P1, the pressure rises initially to a value less than ofoVO, due to the valve moving in the negative x-direction. The valve then moves forward and the pressure rises then falls with the oscillations quickly dying out because of damping from the valve and attached piping. ‘ 2.2.7 Discussion Two important observations can be made from the experimental results: 1. If an elbow is fully restrained, there is no observable alteration of a pressure transient travelling through the elbow. 2. If an elbow is not fully restrained, there can be significant alteration of the pressure transient. The alteration is related to the direction and amplitude 37 of the motion of the elbow, and is, therefore, dependent on the mechanical characteristics of the piping and pipe support structure. 2.2.8 Uncertainty The pressure transducers and accelerometers were calibrated by the manufacturer, PCB Piezotronics. The manufacturer estimates that the pressure transducers are accurate to within 131 full scale, resulting in an uncertainty within 145 kPa. The accelerometers are accurate to within 131 full scale, resulting in an uncertainty within 1.015 mls for velocity. These estimates cover the estimated errors between the measurement source and the recording device. Another possible source of error is the conversion of the analog voltages to digital quantities by the AID converter. The uncertainty of that process is 30.03%. The clock is accurate to within 10.011. The copper pipe was manufactured by American Brass Company. They provided tolerances for the inside diameter and wall thickness of the pipe. The inside diameter is manufactured' to within 10.4% or 10.1 mm and the wall thickness is within :3! or 10.04 mm. The measurements of the pipe lengths are accurate to within 11!. Chapter 3 THEORY 3.1 Introduction Internal pressure strains piping circumferentially and axially due to the Poisson effect and due to pressure resultants at fittings such as elbows. The development of a coupled liquid-pipe analysis procedure must include these interactions between the piping and the liquid. A four~equation model is presented that solves for the dependent variables: liquid pressure p, liquid velocity V, axial pipe stress 0x, and axial pipe velocity 0. 3.2 Four-Eguation Model Halker and Phillips [7] developed a six-equation model that consists of the one-dimensional continuity and momentum equations for the liquid, and the axial and radial momentum equation and two constitutive equations for the pipe wall. The method was developed to study situations where the generation of the transient could be as fast as a few 38 39 microseconds. For transients more typical of waterhammer waves generated by valve slam, Halker and Phillips suggested that the inertial term in the radial momentum equation could be neglected and that the classical waterhammer theory was adequate for transient propagation in straight pipes. The following four-equation model is obtained by neglecting the radial inertia term in Halker and Phillips' model. The main advantage of the simplification is that the time step in the numerical solution can be increased considerably over that of the six-equation model because accurately representing the radial dilation of the pipe wall requires an extremely small time step. The underlying assumptions are one-dimensional flow with uniform p and V over the cross-section, and negligible fluid friction. The pipe is assumed to be linearly elastic, isotropic, prismatic, round and thin-walled. Figure 13 shows the pipe element used for the six-equation model description. The six equations are listed and then simplified to a four-equation system. Two equations represent the continuity and momentum relations for the liquid: --+-—+-=o (2) p - + -— = 0 (3) 40 The axial and circumferential stress-strain relationships for the pipe wall are: * Bu w o,=£<—+v-> (4) 3x r * w 3u 06: E ( — + v -— ) (5) r 3x where E* = EI(1- 02) and 09 is the circumferential stress. The equations of motion for the pipe in the axial and radial directions are: Box at -- - ot - = 0 (6) 3x 3!: 3w 7 p re - = rp - 0 e ( ) t at e Neglecting the radial momentum term on the left side of equation 7 for waterhammer waves, the circumferential stress can be evaluated for in terms of the pressure: 09 = - P (8) Combining equations 8 and 5 to eliminate circumferential stress gives: Bu v- ) (9) 3X .0 II '9!“ I"! A '31! + 41 The time derivatives are taken of equations 4 and 9, w is then solved for in equation 9 and substituted into equation 2 and 4, resulting in the two following equations: 1 2r 3p 36 3V (“+“*)—’2V-+-=0 (10) K eE at 3x 3X 30x ad rv 3p -—--E---—=0 (11) at 3x eat Equations 3, 6, 10, and 11 are linear, first-order, hyperbolic, partial differential equations describing the behavior of four variables, p, V, 0 and 0, which are X" functions of distance and time. These expressions are an improvement over the classical waterhammer theory since they include dynamic coupling between the liquid and the pipe wall. The coupling exists through the Poisson ratio as seen in equations 10 and 11. The solution of these equations is presented in Chapter 4. 42 Pi e Dilated p I—H—i e Pipe Elongatei Figure 13. Pipe Element Chapter 4 NUMERICAL STUDY 4.1 Method of Characteristics This section describes the solution of the four-equation model presented in the preceding chapter. Equations 3, 6, 10, and 11 are transformed from partial differential equations into ordinary differential equations by the method of characteristics [24]. Characteristic roots are found, and then compatibility relations can be found that are valid along characteristic lines. For the numerical study, the equations are presented in dimensionless form. The parameters are non-dimensionalized as follows: X taf V P x*: -- ’ t* = --— ’ v* = —- ’ p* = , r r V0 p a V . f f 0 (12) * 0 2e ,* u 2e 0' = ' u = where o is the axial stress, af =IK70f and the asterisk superscript represents non-dimensional values. For the remaining development the superscript is dropped for 43 clarity. Equations 3, 6, 44 10 and 11 in dimensionless form are: 3p 3V _+—:0 ax at 30 BO - - H - = 0 8x 3t 3p 80 3V J -— — B —— + —- = 0 at 3x 3x as 3p 1:.) -— — 2v - - - - = 0 at 3t H 3x af 112 where H = -- . at =(E/Dt) a t 2rK 2 1/2 J =(1+-—<1-V)) eE W‘D B = f H eot In matrix form, equations 13-16 are: 0 1 O O 0 0 -H O J 0 O 0 -2v 0 0 1 [ A1 I { 2 }t p 1 0 0 0' p V O 0 0 1 V . + . u 0 1 -B O u 0_ 0 O -1/H 0 o t x + [ AZ 1 { Z }x = { 0 } (13) (14) (15) (15) (17) (13) (19) = { 0 } (208) (20b) A1 and A2 are coefficient matrices and Z is a column vector containing the dimensionless dependent variables. The 45 subscripts t and x represent partial differentiation. The characteristic roots ( A) can be obtained by equating the determinate to zero, that is: [ A2 - A A1 ] = 0 (21) 4.1.1 Havespeeds The characteristic roots are the wavespeeds in the liquid-pipe system. The solution of equation 21 yields four roots. The first two roots are the dimensionless liquid wavespeeds, Cf: 01 (22) wavespeeds, Ct: C H 02 (23) A =+ =4»— 3.4 -t ‘0.) ZrK 1I2 where H = ( 1 + -- ) (24) eE V2 2rpf 1’2 01 = ( 1 + -——-———— ) (25) 9(1'qIDt 02 2r of 1/2 02 =( 1 ' ) (26) e qil'qipt 2 2 q=HIH (27) 46 For many piping systems, 01 and 02 are close to unity and the wavespeeds can be simplified: 1 C = t ’ (23) f J H Ct = t "' (29) N J For thin-walled copper pipe filled with water equation 28 differs from equation 22 by less than 2%, and equation 29 differs from equation 23 by less than 1%. Note that equation 28 is the same wavespeed as obtained with traditional analyses considering only the liquid equations and assuming that the pipe is anchored throughout against axial movement; see Hylie and Streeter [2]. 4.1.2 Compatibility Equations In order to eliminate one of the differential operators in equation 20 a linear transformation can be made with a matrix [T] which possesses a non-vanishing determinate: [T] [41} {th + [TI [42} {Z}x = 0 (30) A useful form of the transformation matrix is: [T] [42} = i A] [Ti [Al] (31) where [ A] is a diagonal matrix composed of the characteristic roots. Let [T] [A1] = [AS] and equation 30 becomes: 47 [AS] {Z}t + [1.] [AS] {Z}x = 0 (32) which can be written as: d [AS] - {Z} = 0 (33) dt dx valid along the characteristic directions [ dt ] = {A } Solving equation 31 for the transformation matrix [T] and substituting into equation 33 results in the following four dimensionless compatibility equations: dp dV cfw 2 d6 1 2 do - 1 Cf - t -- (l-Ql ) - - - (1'01 ) - = 0 (34) dt dt 2v dt 2v dt dx valid along -— = 1 Cf, and; dt dp dV ctw 2 66 1 2 do - 1 Ct " t -- (1- 4 02 ) - - - (1- q 02 l - = 0 (35) dt dt 2v dt 20 dt dx valid along - = 1 Ct dt Equations 34 and 35 can be simplified by using the assumptions inherent in equations 28 and 29: dp dV -— 1 Cf - = 0 (36) dt dt dp dV H du 1 do -— 1 ct - : -—— (1- q) - - —- (1- q) -— = o (37) dt dt ZVJ dt 2v dt Figure 14 shows the characteristic representation in the x-t plane. Equations 34 and 35 are integrated along their respective characteristic lines: the two lines with positive 48 slope are designated as C+ characteristics, and the two with negative slopes C- characteristics. The following dimensionless finite difference equations result: pP- pB+ cf (VP'VB) + cfwof (LP- 03) - Gf (Op- 03) = 0 (38) Pp' PD‘ Cf (Vp'VD) ' CfMGf (HP' IID) ‘ Gf (OP' 00) = 0 (39) pP- pA+ ct (VP-VA) + ctwot (ép- BA) - 6t (op- 0A) = o (40) Pp“ pc- c. (Vp’Vc) - ctuet (5,- 5c) - 6: (Up- °cl = o (41) where Gf = (1-of)/(2\>) , Gt = (1- q 022)/(2V) (42) Solution of equations 38-41 consists of dividing the pipeline into elements. The points at the end of the elements are either intermediate locations where information is desired or boundaries that require additional information to solve equations 38—41. For intermediate sections, unknown values at location P are found by simultaneous solution of equations 38-41. For typical piping systems, Ct/Cf is not an integer so timeline interpolations must be made at locations where the characteristic lines fall between time steps. The time step and pipe lengths are chosen so that values for the pipe equations (equations 40 and 41) are not interpolated. Interpolated values are needed for equations 38 and 39, shown as points 8 and D on Figure 14. At boundaries, only two characteristic lines are available, so two additional relationships must be known to solve for the four unknowns. 49 4.1.3 Initial Conditions To begin the solution procedure, initial values of the dependent variables must be established along the pipe. The initial steady-state velocity V0 is input as data. The initial pressure p0 is found by calculating the steady-state pressure drop through the valve. Since liquid friction has been neglected, p0 remains constant throughout the pipe. The initial axial velocity H0 is set equal to zero, assuming that the pipe is initially at rest. The initial axial stress 00 is caused by the resultant steady-state pressure at the elbows. 4.1.4 Boundary Conditions The solution of the four-equation system has been described for the intermediate sections of the pipeline. At locations where there are additional constraints imposed on the liquid and/or pipe, boundary conditions must be included in the solution. If the constraint is at the upstream or downstream ends of the piping system, two known conditions must be added to the C- or C+ characteristic equations. If the constraint is not at an end, it is treated as an intermediate section with length equal to zero. Different values of the dependent variables for each side of the section are possible. Following is a description of the reservoir, valve, stiff support, and elbow boundaries. 50 4.1.4.1 Reservoir The upstream reservoir adds two known constants to be solved with the C- equations: p=p0, and u=0 Equations 39 and 41 can then be solved simultaneously for V and 0. 4.1.4.2 Valve The valve imposes the non-linear relationship between V and P: 1/2 V = T S (P) (43) where I is a dimensionless number representing the valve opening: 1:1 for the initial setting, and 1:0 for the valve closed. S is a constant relating the initial flow and pressure. An additional constraint is again u=0 and equations 38 and 40 are solved simultaneously with equation 43. 4.1.4.3 Stiff Support The stiff support imposes the following conditions: p1=p2, V1=V2, 01:0, and u2=0; where 1 and 2 are the upstream and downstream sides of the support. Equations 38-41 are then solved simultaneously for the unknowns. TIME 51 At DISTANCE Figure 14. Characteristic Representation Figure 15. Elbow Schematic 52 4.1.4.4 Elbow As presented in Chapter 2, the motion of an elbow alters the behavior of the liquid contained within the pipe. As the elbow moves, it can act to increase or decrease the liquid pressure depending on the direction of motion. The relationship between the liquid and elbow motion is derived from the conservation of mass for a translating control volume. The amount of motion is determined by the equation of motion of the elbow. The conservation of mass can be written for the control volume shown in Figure 15: d + - S of d V'+ f oer°fi dA = 0 (44) dt cv cs where V, is the relative velocity between the liquid and control volume. The elbow is assumed to be rigid and the liquid within it incompressible. Therefore, the time rate of change term can be neglected, resulting in the following relationship: For a constant diameter pipe: (v1 - 6x) = (v2 - 6,) (46) 53 V1 and V2 are the upstream and downstream liquid velocities and ux and I, are the elbow velocities in the x and y directions, respectively. Elbow motion is assumed to be planar. The amplitude of elbow displacement is determined by the equations of motion for the elbow. The four-equation model is solved for the liquid and pipe variables along a straight length of pipe. The axial stiffness of the pipe is included in the equations. To represent the restraint caused by the stiffness of the piping attached perpendicular to the straight length, a simple model is used that lumps stiffness at the elbow, as shown on Figure 15. The equation of motion can be written: p Af - 0 At = K u (47) where subscripts on cross-sectional area A are f for liquid and t for pipe wall. The lateral liquid momentum force acting on the elbow is neglected in this presentation because of low steady-state flow rates. For situations where the flow rates are a significant percentage of the liquid wavespeed, the lateral momentum force should be included. The coefficient K is the discrete stiffness used to represent the translation of the attached piping. It is assumed that the attached length of pipe bends in single 54 curvature (assuming support conditions of one end fixed and the other end free to translate). Any damping attributed to the attached piping is neglected. The mass of the attached piping is neglected for the following reasons: 1) for a short length of pipe, its lateral restraint due to inertia is small compared to its lateral restraint due to stiffness; and 2) for a long length of pipe, the mass can be neglected because during the short duration of observation in this study, the flexural disturbance travels only a short distance; therefore, only a small portion of the pipe length is displaced. For longer duration events, or for intermediate lengths of pipe, the inertia should be included in the analysis. To solve the elbow boundary, equation 46, equation 47 (imposed in the two orthogonal directions), and equations 38-41 are solved simultaneously with the addition of the following condition: p1=p2. Equation 47 is integrated by the trapezoidal rule (see Craig [25]) before combining with the other equations. Note that setting u=0 at the reservoir, valve, and stiff support implies that the restraint of the pipe at those points is infinitely stiff. It is possible to represent instead the actual structural restraint imposed at that point mathematically, and then solve the resulting system of equations. The representation of the supports mathematically is beyond the scope of this thesis. 55 The numerical model was developed to determine if the pressures and elbow velocities recorded in the experiment could be predicted with a relatively simple model. Representing the attached piping by a lumped stiffness was possible because the lengths of pipe were extreme, either very long or very short. For a more general piping system, a finite element model of the piping and pipe support structure could be used (see Hatfield, et al. [13]) to provide a more comprehensive model. In the structural analysis, care would need to be taken to avoid including axial modes of the pipe that are already included in the four-equation model. 31; Numerical Analysis The procedures set forth in section 4.1 have been implemented in a computer program called LIQPIP (listing in Appendix). Input data consist of the liquid and pipe properties, valve closure time and loss coefficient, initial flow velocity, and the stiffness coefficients used to represent the attached piping. The configuration of the piping system is input as a series of elements. The three possible element types are a straight length of pipe, an elbow, and a stiff support. The program is able to handle any number of elbows and stiff supports. 56 4.2.1 Comparison to Experimental Results The four experimental cases presented in Chapter 4 were simulated with LIQPIP to verify the mathematical model. The steady-state velocity used as input into LIQPIP was calculated from the initial pressure rise from the experimental data. The stiffness coefficients used in the predictions are shown in Table 1. Stiffness coefficients for the translation perpendicular to the long lengths (L1 and L2) are approximated by zero. For the 0.36 m lengths, the coefficients were calculated by the stiffness method (see section 4.2.2.3 for further discussion). The axial stiffness of all the pipe lengths is included in the four-equation model. Table 1 Stiffness Coefficients Case Direction Elbow K LNIpL A x 1 Infinite y 1 Infinite B x 1 0 y 1 O C x l O Y 1 94,800 2 2 O V 2 94,800 D O "< N‘< X NNHH 0 88,700 57 4.2.1.1 Case A: Stiff System Figure 16 shows the predicted pressures at the valve, P1, versus the experimental data. The discrepancies are from the valve flutter at 4 ms and oscillations in the 40-70 ms range as discussed in section 2.2.2. The response predicted by LIQPIP is not flat as would be predicted by a traditional two-equation model, the variation being caused by the second-order Poisson effect. 4.2.1.2 Case 8: Axial Stiffness Figure 17 shows the comparison of pressure, P1, and comparison of elbow 1 velocities for Case B. The phase and the amplitude of the velocity oscillations are predicted accurately. The pressure oscillations resulting from the elbow's movement are predicted slightly higher than the experimental data. 4.2.1.3 Case C: Bending Stiffness Figure 18 shows a general agreement for the predicted pressure response and for the y-direction elbow velocities. The predicted y-direction elbow velocity is slightly out of phase with the experimental data, suggesting that the estimated stiffness K was too small. 1.1.1.)! I i 58 2000 - 7; 22 1000 — ,, Predicted ; a _J ---Experiment l O —- L l T l l 0 20 40 60 80 TIME (ms) Figure 16. Comparison of Experimental and Predicted Pressures for Case A 2000 1000 P1 (kPa) 0.25 0.0 ux (m/s) -O.25 0.25 0.0 uy (m/s) -0.25 59 : r“ 1| 1 ' A 1 :i ' ‘3 \J l l ,4 Predicted ---Experiment l i l I l L l l l l 20 40 60 80 TIME (ms) 1’ I - " I l V: I F U l i . I I 4 '” I ' I . ’1 , , “ II ‘p’h’b” i 1’ V, i I I (I l l l 7 20 40 60 80 TIME (ms) Figure 17. Comparison of Experimental and Predicted Pressures and Elbow Velocities for Case 8 60 2000 - I 1 I 0 21000 — O. x a: — Predicted -'--Experiment 0 — l- T l l T O 20 40 60 80 TIME (ms) 0.5'- A. , — I] l‘ pvn‘1~\ E “I ‘ ”Mfyl‘l 1’ J \\ E I \ f I v 0. 0 — ) I *\ >5 \ I ‘1 '3 Vi I \ L f’ 1 h , "‘ — F“ I, bwhld\ —O.5 l l l W 0 20 40 60 80 TIME (ms) Figure 18. Comparison of Experimental and Predicted Pressures and Elbow Velocities for Case C 61 2000 - i; I/T'Ax I; i ”ii [Jars I, \ I’V‘I _ i ’ i e 1 '7’ f; , f ’ ’va a: I ‘ v 1000 "' l \ _. ii T O. l —i l -———-Predicted I - - -Experiment ', 0 - \‘k 0.25 '3 \ E V 0.0 X 0:3 -0.25 0.50 ’3 \ E 0 0 >4 0: - '0 ° ° I I I I I 0 20 4O 60 80 TIME (ms) Figure 19. Comparison of Experimental and Predicted Pressures and Elbow Velocities for Case D 62 4.2.1.4 Case 0: Combined Figure 19 shows small discrepancies in amplitude for the pressure comparison. The x-direction elbow velocity is predicted accurately. The y-direction elbow velocity again is slightly out of phase. 4.2.2 Parametric Study This section examines some of the important parameters involved in liquid-pipe interaction. The reason for presenting the wavespeeds and compatibility equations in dimensionless form is to reduce the number of parameters needed to describe the behavior of the liquid-pipe system. This section will discuss Poisson ratio effects, the dimensionless parameters that resulted from the equation development, and the stiffness coefficients. 4.2.2.1 Poisson Ratio The first parameter discussed is Poisson's ratio. As shown in Table 2, the ratio varies from 0.3 for steel to 0.5 for PVC. From the compatibility equations, its importance as a coupling parameter is clearly seen. If Poisson's ratio is set equal to zero, inferring no transformation between axial and circumferential strain, equation 34 becomes uncoupled from equation 35. Equation 34 becomes identically equation 36, and multiplying equation 35 by v to avoid division by 63 zero, equation 35 becomes: d0 do :(1-9)--(1-4)—=0 (48) dt dt Equations 36 and 48 are uncoupled equations and are similar to those used by Ellis [19]. Figure 20 shows a comparison of predicted pressures at P1 for Case 8, with and without Poisson coupling (Poisson's ratio is set equal to zero} for the dashed line). The first pressure rise at 14 ms disappears as would be expected because it is caused by the precursor. Hith the equations uncoupled, no precursor exists. The magnitude of the resulting oscillations is also reduced. To gain an understanding of the magnitude of the precursor stress wave, it was desirable to compare the pressure resultant forces and stress forces at an elbow. This is accomplished by comparing the two dynamic forces that drive an elbow as shown in equation 47. A simple system was analyzed with a length of pipe between an elbow and valve. The valve was closed instantaneously and the stress wave (the precursor wave generated by the Poisson effect) and pressure wave were computed. Figure 21 shows the ratio of stress forces in the pipe wall (Ft) to pressure forces in the liquid (Ff) versus rle. 64 2000 ‘- [x ‘11, I I I, ‘x A — -~\ ‘ - / ‘\ g; 1000 —- i :3 I m I m l o: I 0- “ v = 0.34 L ¢ --- \) = 0.0 l l o s I I I l . [I l l l l 0 20 40 60 80 TIME (ms) Figure 20. Comparison of Predicted Pressures at P1 for Case 8 with Variable Poisson Ratio q Steel 0'4 ‘ Copper \ —— Aluminum .1 \\ .1 — -- — PVC t —- \ \ — —— — Polyethylene LL.“ \\ ~ .0 0 2 __I \N.‘ \-- S e — M \._ g \___-_ E 8 _ “ht—"l — .~— E 0.0 1 l j 7 0 5 10 15 20 r/e Figure 21. Force Ratio versus rIe 65 The metal pipes show a stress force equal to 15-201 of the pressure force. In the plastic pipes, the stress force can be as large as 40$ of the pressure force for low rle. The steel, copper, and aluminum pipes show very little dependence on rIe. It should be pointed out that viscoelastic behavior may occur for the polyethylene pipe, a property that has not been included in this study. 4.2.2.2 Dimensionless Parameters Table 2 lists constants for five different common pipe materials and the dimensionless parameters H, 01, and 02. The parameter H, the ratio of liquid and pipe material wavespeeds in an infinite medium, has been calculated using water at 200 C as the contained liquid. The parameter 01, shown for three rle ratios, affects mainly the two plastic pipes; it is very close to 1.0 for the metal pipes. The parameter Q2 is very near 1.0 for all cases shown. 66 Table 2 Pipe Material Constants Material Steel Copper Alpminpm PVC Polyethylene ot(kgIm) 7900 8940 2700 1300 940 v 0.30 0.34 0.33 0.50 0.46 E (GPa) 211 117 70 2.5 0.86 at (mls) 5160 3620 5090 1390 960 H 0.29 0.41 0.29 1.07 1.55 rle 01 5 0.995 0.990 0.986 0.864 0.881 10 0.992 0.982 0.978 0.865 0.884 20 0.986 0.973 0.968 0.865 0.886 02 5 1.000 1.000 1.001 1.015 1.010 10 1.001 1.002 1.001 1.008 1.006 20 1.001 1.003 1.001 1.004 1.003 Figure 22 is a plot of the inverse of the dimensionless parameter 3 versus rIe for the different pipe materials. This term represents the effect of the circumferential wall the liquid 22, value of 1IJ=1, stiffness on wavespeed. From equation assuming Ql=1, for a the dimensionless non-dimensionalized liquid), 1/J=1, 1/J wavespeed Cf=1. Since the wavespeed is infinite 22, as (the wavespeed in an for by af Cf=af. As shown in Figure rle decreases, approaches 1 (hence the pipe wall is getting stiffer and its the the For aluminum, rIe=20. the wavespeed decreases). 301 at effect on wavespeed is reduced by about For plastic pipes, the wavespeed is reduced by as much as 90%. 67 1.0 - —.I -——-—— Steel ~‘—_‘—“°"--—-—_ \\ Copper 1 . —-- Aluminum 3- 0.5 "'I ‘ — "—PVC \ - —— - Polyethylene \\~ ‘u.-~‘~. —-i \ \ “.\-- \~~~ \-_~_ 0'0 I j I I 0 5 10 15 20 r/e Figure 22. The Inverse of the Dimensionless Parameter J versus rle ‘ N=3.58 \‘ 0 36 m 1k «x \— —————— —\ l— —————— TI \ N=3 \ \ N=12 \ l l ) Vb. tn VR s“ Figure 23. Assumed Deflected Shapes 68 4.2.2.3 Stiffness Coefficients The representation of the attached piping at an elbow by a stiffness coefficient was used as a first approximation of the dynamic effects of that portion of the piping system that is not included in the four-equation model. One stiffness coefficient can be input for each elbow axis. "In calculating the stiffness coefficient for the short lengths of pipe in Case C and D, the pipe was assumed to be anchored at one end to a stiff support with no rotation possible at that point. The flexural stiffness of the pipe is then calculated as follows: K = N E1“? (49) where E is Young's modulus, I is the moment of inertia of the pipe cross section, and L is the length of the pipe. The constant N ranges from 3 when assuming the elbow end of the pipe to rotate freely (treating the elbow as a hinge), to 12 when assuming no rotation of the pipe at the elbow. The actual rotational fixity is a function of the elbow's geometry and material properties, and the length of attached straight pipes. For Case C and D, the elbows are attached to a long length of pipe (L2=7.6 m). Because of this, the elbows were assumed to be rigid, with any rotation coming from the flexibility of the attached pipes. Case C and D were analyzed by the stiffness method (see Hhite, Gergely, 69 and Sexsmith [26]). N is equal to 3.58 for Case C and 3.35 for Case D. Figure 23 shows the stuctural configuration and deflected shape used in the calculation of the coefficient for Case C (N=3.58) and for the extreme values of N equal to 3 and 12. Figure 24 shows the effect of N on the predicted pressures and elbow velocities for Case C. 2000 - II \ l‘\ , . - I \ - / _‘ l \ \\\ I ’r; lI \\ / \\ . E v! \ / ‘ ,u __ \ if, 1000 \ 53 P1 \ E1 __ M} '——-—N = 3.00 i) N = 3.58 l. 0 ___, — — — — N = 12.0 I T l 7 l 0 20 40 60 80 TIME (ms) 0.5 - - u .\ g 7 "I y /'\ ' “ \ V I l E \‘ \ ll \ ./l 1 ' 23 ‘ I \ I 1 \ __, 0.0 -- 1‘ \ l I \ L” I l I \ ‘ g \\ ’ (I I \ \ g \ I \ / I _ \/ j \/ \\ ~ \ a! '0.5 ‘l I I —| O 20 40 60 80 TIME (ms) Figure 24. Comparison of Pressures and Elbow Velocities for Case C with Variable Stiffness Chapter 5 SUMMARY AND CONCLUSIONS The objective of this thesis was to determine the effect of elbows on a generated pressure transient. It has been demonstrated that the elbow motion is the most important factor in altering the dynamic pressure. The elbow motion is driven by the axial stresses in the pipe and by the liquid pressure. For an elbow that is fully restrained, no significant alteration of the pressure transient occurs. Elbows that are not fully restrained can cause significant alteration of the pressure transient. An analytical technique was developed that couples liquid and pipe equations to model the interaction. For a straight pipe length, the one-dimensional equations of continuity and momentum for the liquid and pipe wall are solved by the method of characteristics. At an elbow, continuity relationships and an added stiffness component are solved simultaneously with the characteristic equations. Four experimental configurations were used for verifying the model. The amount of elbow restraint was varied in each setup. In Case A, all the elbows were rigidly supported. 71 72 The resulting pressure response was similar to the traditional Joukowsky pressure rise. The predicted pressure response matched the experimental data. In Case 8, the supports were removed from the main test elbow. The elbow was then restrained by the axial stiffness of the two connecting pipes. The model predicted the elbow velocities and pressure responses accurately. Elbow motion in Case 8 was initially due to a precursor wave. The precursor wave is an axial tension wave that is generated by strain-related coupling. The precursor wave travels at approximately the wavespeed in the pipe material. Case C was designed to investigate the translation of a pipe length between two elbows. Stiff supports were located 0.36 m away from each elbow. The pipe between the two elbows could then translate as a function of the structural restraint of the short lengths. The predicted results modeled the general behavior of the elbow velocities and pressure response. The fourth case studied, Case 0, combined the axial and translational modes of cases 8 and C. In all the cases where the elbow was not restrained, significant alteration of the pressure response occurred. Pressures 33$ higher than the traditional Joukowsky pressure rise were recorded. These results clearly show a need for an increased awareness of the potential for liquid-pipe interaction. The upper limit on the amplification of dynamic pressure by the 73 elastic response of piping is unknown. In the design of piping systems that are subject to transient pressures and are not supported rigidly, engineers need to consider possible interaction. Rigidity is difficult to achieve, and usually undesirable, in actual pipe hardware and even small flexibility leads to interaction. Future experimental investigations are needed on more complicated piping configurations with conventional supports. Great care should be taken in documenting the flexibility of the support structure. Damping, which was negligible in the experimental part of this work, could be significant in some systems. The numerical model predicts the experimental results accurately but needs improvement to be useful for the analysis of a general piping system. Possible improvements could consist of a more complete and general modal representation of the piping structure, material damping coefficients, and damping at discrete supports. The formulation should also be capable of handling a greater variety of hydraulic boundary conditions. Continuing research into liquid-pipe interaction will lead to improved design of piping by reducing the uncertainties in existing design-analysis procedures. APPENDIX 74 A.1 CONTRL Listing CeeeeeeeeeeummeeeeeeCONTRL_FORemeemmeeeemeeeeemmeanneeeeeeeee CONTROL PROGRAM FOR MULTICHANNEL A/D SAMPLING NRITTEN BY BOB OTNELL. JUNE 1982 MODULES: SAMPL(N(1).NSMPL.NTICK.NRATE.ICHAN.NCHAN.IERR) SAMPLES DATA ON CLOCK INTERUPT AFTER INITIAL TRIGGER N(1)- SAMPLE BUFFER NSMPL - NUMBER OF SAMPLES TO TAKE NTICK - NUMBER OF TICKS BETWEEN SAMPLES NRATE . CLOCK RATE (0-7) OcSTOP 1-1 MHZ 2&100 KHZ 3‘10 KHZ 481 KHZ 5-100 H2 bBST1 7-LINE FREG(60 HZ) ICHAN - AID CHANNEL TO SAMPLE (0-15) NCHAN - NUMBER OF CHANNELS TO SAMPLE IERR & NUMBER OF SAMPLING ERRORS LINKING INSTRUCTIONS FOR MODULES: LINK CONTRLoSAHPL OOOOOIBOOOOOOOOOOOOOOOOOOC'IOO LDCICALGI FNAHEilS) DIMENSION VilOOO):ACC(250).P(3.250) INTEGER N(1000) URITE(7:100) 100 FORMAT(' BEGIN EXECUTION OF CONTRL:’./) DD 194 J-1:250 194 ACC(J15O. DO 195 131:3 DO 195 J-1:250 195 P(1.J)=O. DO 196 I-1.1ooo 196 ViI)-O. C c eaeemmemeeeeeeesAMPLINO SECTIONeemeaeeeeeeeeae C . HRITE(7.105) 105 FORMATt’OENTER NUMBER OF SAMPLES/CHANNEL:’.$) READ(5.110) NSMPL 110 FORMAT(15) HRITE(7:120) 120 FORMATt'OENTER CLOCK RATE (O-7)'.$) READ(5:110) NRATE C DELT’S ARE IN NS IF(NRATE.EG.1) DELT-.OO1 IF(NRATE.EG.2) DELT8.01 IF(NRATE.EG.3) DELT=.1 IFCNRATE.EO.4) DELT-1.0 IF(NRATE.EG.5) DELT‘10.0 "RITE(70130) 130 FORMAT('OENTER NUMBER OF CLOCK TICKS/SAHPLE:’.S) READ(5.1101 NTICK ' 140 145 200 149 150 160 l") 000 N F‘ F.) \l O O P.) H i a ugnoonnom (“D [J (I! (n OOOF’) O“! '3 75 DELTSDELTfiNTICK HRITE(7.140) FORMAT(’OENTER FIRST CHANNEL TO SAMPLE:’;$) READ(5;IIO) ICHAN HRITE(7:145) FORMAT(’OENTER NUMBER OF CHANNELS TO SAMPLEz’TS) READ(5;IIO) NCHAN NSMPT-NSMPL*NCHAN IPLOTBO ZERO OUT DATA BUFFER; DO 200 I-1,1000 N(I)=0 URITE(7.149) NRITE<7.150)NRUN.NSMPL.NTICK.NRATE FORMAT(///,' ififiiO§§§§§9...}.99*iiifiiifiiiifiifififfififi§§’,//) FORMAT(1X.’CALLING SAMPLING SUBROUTINE. NRUN I’.I5.///. 1’ NSMPL =’.IS.//.’ NTICK-’oIS://.’ NRATEI’oI5./) NRITE(7.160) NCHANpICHAN FORMAT(? SAMPLING ’.12.' CHANNELS STARTING ON CHANNEL ’.IZ./) START TO SAMPLE INTO BUFFER I: CALL SAMPL(N(1).NSMPL.NTICK.NRATE:ICHAN.NCHAN.IERR) NRITE(7;I70)IERR FORMAT(’0**********SAMPLING FINISHED**********'.//p IIS.’ - A/D ERRORS ENCOUNTERED’) FORMAT(AI) DO 221 I-IpNSMPT V(I)-(N(I)-2048)/400. fii§§§QQ§§§G§O§DATA CONVERSIONiOfi§§§9§§*§*§§* PUT VOLTAGES INTO ACCELERATION(VELOCITY) AND PRESSURE ARRAYS HRITE(7.300) FORMAT(’ DID YOU TAKE ACCELERATION DATA(Y/N)?’.$) READ(5,220)IACC IF(IACC.NEBIHY) GO TO 320 IPLOT=J HRITE(7.305) FORMAT(’ HAS IT INTEGRATED(Y/N)?’.C) READ(5.220)IVEL VOLTAGE TO ACCELERATION(OR VELOCITY) PCB QUARTZ ACCELEROMETERS TRANSDUCERS 5712 AND 5713 ACCELERATION IF(IVEL.NE.IHY)FTSEC2=3ZI4. VELOCITY(FT/SEC) IF(IVEL.EG.1HY)FTSEC2=1.202 VELOCITY(M/SEC) IF(IVEL.EG.IHY)FTSEC2‘.3664 K30 DU 310 J-IINSHPTINCHAN K3K+l ACCELERATION(OR VELOCITY) BUFFER ACC(K)=V(J)*FTSEC2 NPRI‘P GO TO 325 NPRI=I KK'O DD 330 I'NPRI’NCHAN Kan 76 KK=KK+J VOLTAGE TO PSI PCB QUARTZ PRESSURE TRANSDUCERS TRANSDUCER 3808 IF(KK.EG.I)PSI-9é.1 VOLTAGE TO KPa IF(KK.EG.1)PSI-662.é TRANSDUCER 3809 IF(KK.EG.2)PSI=93.3 IF(KK.EG.2)PSI=643.2 TRANSDUCER 3810 IF(KK.EG.3)PSI=9B.4 IF(KK.EG.3)PSI=678.4 DO 330 J'I.NSMPT.NCHAN KIK+I 30 P(KK.K)-V(J)*PSI ()0 0000 (70 PRINT SAMPLE DATA nonw HRITE(7.265) 265 FORMAT(’ODO YOU HANT TO SEE VALUES(Y/N)?’.$) READ(5.220)IPRR IF(IPRR.NE.1HY) GO TO 229 IF(IACC.NE.IHY) GO TO 420 HRITE(7.AIO)(ACC(I).III.NSMPL) 410 FORMAT(SP9.2) 420 KK=O DO 430 J8NPR1.NCHAN KK=KK+J 430 NRITE(7.410)(P(KK.I).I=1.NSMPL) C C C§§O§O§§§§§§VELOCITY PLOTS§§§§§§§§§§§O§Q§ C 229 IF(IVEL NE.IHY) GO TO 894 WRITE(7.SIO) BIO FORMAT(’ODO YOU WANT A VELOCITY PLOT(Y/N)?’.$) READ(5.220)IVL IF.NSAMPL,NTICK.NRATE.ICHAN.NCHAN.ERROR) IBUF-SAMPLE ARRAY NSMPL-NUMBER OF SAMPLES NTICK-NUMBER OF CLOCK TICKS/SAMPLE NRATEsCLOCK TICK RATE ICHAN-FIRST CHANNEL NUMBER NCHAN-NUMBER OF CHANNELS TO BE SAMPLED ERROR-NUMBER OF ERRORS uHILE SAMPLING .- \- ‘I \' V. ‘1 .0 ‘1 ‘- .GLOBL SAMPL ‘ L] O 0 O O O O .HORD O ADVECI‘4OO ADVEC2=402 ERVEC1-404 ERVEC2=406 ADSR=177000 ADSR1=177001 ADBR=177002 CLKSR=170420 CLKBR=I70422 TTPDB=I77566 CLR ERROR aINITIALLIZING THE AID ERROR COUNT TO 0 CLR DFLG sINITIALLIZING THE DONE FLAG MOV 2(R5).ADDR IBEGINNING ADDRESS OF SAMPLE OUTPUT BUFFER MOV QAtRS).RO BNUHBER OF SAMPLES MOV G6(R5).RI sT-fl OF CLOCK TICKS NEG R1 5 MOV R1.¢¢CLKBR sPUT -T INTO CLOCK BUFFER MOV 910(R5).TEMPCK sCLOCK RATE ASL TEMPCK :SET UP CLOCK RATE ASL TEMPCK s BITS 3-5 ASL TEMPCK 3 BIC #177707.TEMPCK sZERO OTHER BITS BIS 020002.TEMPCK sCLOCK STATUS: sREPEATED INTERVAL sSTART HHEN SCHMIDT TRIGGER 2 FIRES MOV G12(R5).TEMPAD iGET FIRST CHANNEL NUMBER BIC 0177bOOnTEMPAD BZERO OTHER BITS SNAB TEMPAD BSHAP BYTES BIS GOAOIAOoTEMPAD sSET UP A/D STATUS: JENABLE REAL TIME CLOCK aTNTFRRUPT UHFN AID IS DONE AGAIN. ISRI: ISRZ: SERVEI: SEPVZZ: SERV29: ERR. STOP: MOV GISR1.G*ADVEC1 MOV GSAOTQGADVECE MOV GERRTQGERVECI MOV *3‘OIQ*ERVEC2 MOV QI4(R5).NCHAN MOV NCHANTCOCHAN MOV TEMPADTQGADSR MOV ROTCOUNT MOV TEHPCKD..CLKSR UAIT TST DFLG BEG AGAIN RTS PC MOV GOO7TGGTTPDB MOV GISRzoGGADVECI nov QRADBRTIADDR ADD Q2.ADDR DEC COCHAN BEG SERV29 INCB euADSRI BIS 01.e«ADSR TSTB csAoSR BMI SERv21 JMP senvzz DEC COUNT, BEG STOP nov TEMPAD.Q#ADSR nov NCHAN.COCHAN RTI INC ERROR BIC GIOOZOOTADSR BIC NZOGTQRCLKSR RTI CLR GQCLKSR MOV ERROP.Q16(R5) MOV GITDFLG RTI .END SAMPL '79 iINTERRUPT FOR AN A/D CONVERSION ERROR aSET UP BEEP ISR VECTOR TPRIORITY 7 ISET UP AID ERROR ISR VECTOR sPRIORITY 7 iGET NUMBER OF CHANNELS TO SAMPLE aSET UP CHANNEL COUNTER sLOADING A/D STATUS REGISTER sMAXIMUM NUMBER OF SAMPLES iLOADING CLOCK STATUS REGISTER THAITING FOR AN INTERRUPT sARE HE FINISHED ? sBACK FOR MORE HAITING IRETURN TO THE MAIN PROGRAM 5 TBEEP WHEN SAMPLING BEGINS ISET UP A/D DONE ISR VECTOR iA/D DONE SERVICE ROUTINE IMOVE A/D SAMPLE TO THE BUFFER iPOINT TO THE NEXT BUFFER ADDRESS JALL CHANNELS SAMPLED iNOTINCREMENT CHANNEL iSTART NEXT SAMPLE iSAMPLE DONE? sYES, GO GET IT 5N0 HAIT SOME MORE iDECREMENT SAMPLE COUNT 5ENOUGH SAMPLES TAKEN ? 3ND. SET UP A/D AGAIN TRESET CHANNEL COUNTER TRETURN FOR MORE A/D SAMPLES ON CLKOV l sA/D ERROR SERVICE ROUTINE iCOUNTING THE NUMBER OF AID ERRORS sCLEAR ERROR CONDITION ;CLEAR THE OVERFLON FLAG I sSTOP THE CLOCK iPASSING THE NUMBER OF ERRORS TO FORTRAN sSIGNAL THAT ALL SAMPLES ARE TAKEN sCLEANING UP REMAINING INTERRUPT 80 A.3 LIQPIP Listing *iid'fitlrfiOfii‘IR-IKGVIGL I OP I P --l_ I GU I D-P I PE / SDOFGMIuIm“§§§§§§*§§§§§Q§§§§§§ WRITTEN BY BOB OTHELL. SPRING 1983 LAST UPDATE NOV. 1983 LIQPIPS IS A COUPLED FLUID/PIPE ANALYSIS PROGRAM 1. A FOUR EQUATION MODEL CONSISTING OF THE ONE DIMENSIONAL EQUATIONS OF CONTINUITY AND MOMENTUM FOR THE FLUID AND PIPE HALL ARE SOLVED BY THE METHOD OF CHARACTERISTICS(MOC) 2. STRUCTURAL RESTRAINT CAUSED BY PIPE SUPPORTS AND ATTACHED PIPING IS REPRESENTED AS THO SDOF SPRING/MASS/DASHPOT SYSTEMS AT EACH ELBOH THE ELBOH BOUNDARY CONSISTS OF SOLVING 7-EQUATIONS(4 MOC.I CONTINUITY. AND 2 SDOF EQUATIONS OF MOTION) SIMULTANEOUSLY. THE EQUATIONS OF MOTION ARE INTEGRATED BY THE AVERAGE ACCELERATION METHOD AS OUTLINED IN ”STRUCTURAL DYNAMICS" BY CRAIG (PG 148) 3. ANOTHER INTERMEDIATE BOUNDARY CONSISTS OF A STIFF SUPPORT HHERE UDOT IS SET EQUAL TO ZERO 4. THE UPSTREAM BOUNDARY IS A CONSTANT HEAD RESERVOIR 5. THE DOHNSTREAM BOUNDARY IS A VALVE THE PROGRAM IS SUBDIVEDED INTO THE FOLLOWING SECTIONS 1. MAIN PROGRAM -CONTROL OF TIME AND SUBROUTINES 24 READP -INPUT PIPE AND FLUID DATA 3. READS -INPUT SDOF DATA 4. PIPE -SOLUTION OF 4-EQUATION MODEL 5. PRINT -PRINTING SUBROUTINE b. PLOT -PLOTTING SUBROUTINE THE MAIN VARIABLES ARE: INDEPENDENT X -DISTANCE ALONG PIPE AXIS T -TIME DEPENDENT P(X.T) -FLUID PRESSURE V(X.T) -FLUID AXIAL VELOCITY SGX(X.T)-PIPE AXIAL STRESS UDT(X.T)-PIPE AXIAL VELOCITY CONSTANTS CF -FLUID HAVESPEED CT -PIPE HAVESPEED DT -TIMF STEP TC -VALVE CLOSURE TIME V0 -INITIAL FLOH VELOCITY PRES -INITIAL PRESSURE R -INSIDE RADIUS OF PIPE ET -HALL THICKNESS K -FLUID BULK MODULUS E -MODULUS OF ELASTICITY KNU -POISSON’S RATIO RHOF -FLUID DENSITY RHOT -PIPE DENSITY SK -SDOF STIFFNESS SM -SDOF MASS SC -SDOF DAMPING OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO COMMON/PASS/JJ.T.SK(4).SM(A).SC(4) p 0 000000100 000 [U 0 c. C 99 81 COMMON/P/P(15.50).V(15:50).SGX(15.50):UDT(15.50). IXL(15),NTYP(15) COMMON/CONS1/PIIXLMINDPRESDSDTHAXDTCIDTIKOUNTIKPLOTONSTDY! INDTNELMTNELBWTIPIoIPZ.1P3,IUI:IU2:J1.J2:JS:J4:J5:PJOUK COMMON/PLOT/TPLT(200).UPLT(2.200).PPLT(3»200).KK LOGICAL*1 FNAME(15) CALL GTLIN(FNAME.’ ENTER PRINT FILE NAME ’) OPEN(UNIT-1.NAME-FNAME.TYPE-’NEW’) WRITE(I.IO)FNAME FORMAT(4X.’FILENAME IS ’oAIS) CALL READP IF(NELBW.NE.O)CALL READS STEADY—STATE SETUP -TIME JJ -COUNTER FOR MOC JJC-COUNTER FOR KOUNT AND KPLOT -COUNTER FOR PLOTTING T-O. JU‘I JJC=I KK=I CALL PIPE CALL PRINT NRITE(7.1) FORMAT<4X.’DO YOU WISH TO CONTINUE WITH TRANSIENT 150LUTIDN(Y/N)?II‘) READ(5.2)IRUN FORMAT(AI) IF(IRUN.NE.IHY) GO TO 99 TRANSIENT SOLUTION T=T+DT IF(T.GT.TMAX) 00 TO 99 uu-uav: JJC-JJC+1 CALL PIPE IF(JJC/KOUNT*KOUNT.EG.JJC)CALL PRINT IF(KPLOT.EG.O) co TO so IFiJJC/KPLOT*KPLOT.NE.JJC) 00 TO 30 KK=KK+J TPLT(KK)=T*IOOO. SUBTRACT PRES SO DYNAMIC PRESSURE IS PLOTTED PPLT(1.KK)‘(P(IP1.JJ)-PRES)*S PPLT(2.KK)=(P(IP2.JJ)-PRES)*S PPLT(3.KK)8(P(IP3.JJ)-PRES)&S MULT UDOTI BY -1 TO MODEL ACCELEROMETERS UPLT(1.KK)8-UDT(IUIoJJ) UPLT(2.KK)8UDT(IU2.JJ) GO TO 30 CONTINUE CLOSE(UNIT=I) IF(KPLOT.NE.O)CALL PLOT CALL EXIT END C*‘l’§§§§§§§fi§O§*§*§**§**§if.*§*.O§§*§§§§i§§‘INI'MNININI’{*SNIG’OG-II’OGGOQQGQGOINIfiOfiifffi‘tfii SUBROUTINE READP COMMON/P/P(I$o50).V(15o50)oSGX(15.50).UDT(I$.SO). 1XL(15):NTYP(IS) COMMON/CONSTIPI.XLMIN.PRES.S.TMAX.TC:DT.KOUNT.KPLOT.NSTDY. IND.NELM.NELBW.IPIoIPZTIPS.IUI»IU2:J1:J2:J3oJ4oJ5:PJOUK WRITE(7.I) HRITE'I1.I) IO M “(J v 82 FORMAT(/.4X.'***TIME DATA***‘./.4X.’INPUT TMAX(MS).TC(MS). INSTDY’) READ(5.*)TMAX.TC:NSTDY WRITE(1.IO)TMAX.TC:NSTDY FORMAT(6X.FS.2.F10.2:IS) WRITE(7.2) WRITE(1a16) FORMAT(/.4X.’***PIPE DATA***';/:4X:’INPUT NUMBER OF IELEMENTS ’.$) FORMAT(/:4X.’***PIPE DATA***’) READ(5.§)NELM WRITE(7;3) WRITE(I.I7) FORMATi/T4XT’INPUT ELEMENT TYPE: LENGTH(M)’) FORMAT(21X.’TYPE LENGTH’) XLTOTBO. XLMIN=IOOO. DO 5 IBIoNELM WRITE(7.A)I FORMAT(4X.’ELEMENT'TISTEXTG) READC5.§)NTYP(I).XL(I) XLTOT‘XLII)+XLTOT IF(XL(I).LT.XLMIN.AND.XL(IJ.NE.O)XLMIN=XL(I) CONTINUE DO 20 I-ITNELM WRITE(1.12)I.NTYP(I).XL(I) FORMAT(4X.'ELEMENT'TISTSX.15.F10.3) CONTINUE NRCH=O NELBW=O NSTIFF=O DO 6 IBITNELM IF(NTYP(I).EQ.I)NRCHINRCH+1 IF(NTYP(I).EQ.2)NELBW=NELBW+I IF(NTYP(I).EQ.3)NSTIFF.NSTIFF+1 WRITE(7T7)NRCH.NELBW.NSTIFF.XLTOT WRITE(I.7)NRCH.NELBW.NSTIFF.XLTOT FORMAT(/.4X.’THERE ARE’IIBI’ PIPE REACHES’OISOI ELBOWS. I AND’oISo’ STIFF SUPPORTS’I/94X!’ THE TOTAL PIPE LENGTH 1 IS’.F7.S./) WRITE(7;S) FORMAT(4X.’INPUT JOUKOVSKY PRESSURE RISE(KPA) ’.$) READ(5,*)PJOUK RETURN END C f§§§§§*******.********§.‘...********.****************.**§****‘M'M'N'i‘f‘i*‘l’ M SUBROUTINE READS COMMON/PASS/JJ:T.SK(4).SM(4).SC(4) COMMON/CONST/PI.XLMIN.PRES:S.TMAX.TC:DT.KOUNT.KPLOT.NSTDY. INDTNELM.NELBW.IPI.IPZaIPS.IUI.IU2.JI.JE.J3.JA-J5:PJOUK ND=2*NELBW WRITE(7.I)ND FORMAT(4X.’***SDOF DATA***’.//.4X.’INPUT FOR’.I3.’ SDOFS’) WRITE(7:2) WRITE(I.6) FORMAT(5X.'INPUT STIFFNESS. MASS. AND I DAMPING FACTOR’OI) FORMAT(15X.’STIFFNESS MASS DAMPING’./) DO 4 I‘ITND WRITE(7.3)I FORMATtAXo’DOF ’.IS.2X,$) READ(5.*)SK(I).SM(I).SC(I) WRITE(I.5)I:SK(I).SM(I).SC(I) FORMAT(4X.’DOF ’IISDEXD3F10.3) IF(SK(I).EQ.O)SK(I)¢I.E-9 TFIQMIT) FG OTRMlTTxI F-9 83 IF(SC(I).EG.O)SC(I)‘I.E-9 CONTINUE RETURN END C iii-*QQI‘O..."IT’S".'9‘}...§*.§”O§§§“§*§§“§§§Q§§I’*§ ifiifli’fi’iifl'l’l’I'MO'Ilfiffii SUBROUTINE PIPE COMMON/PASS/JJ.T.SK(4).SM(4).SC(4) COMMON/P/P(15.90).V(15o50):SGX(15.50):UDT(15.50). IXL(15).NTYP(15) COMMON/CONST/PI.XLMIN.PRES.S.TMAX:TC:DT:KOUNT.KPLOT.NSTDY; INDONELHINELBuIIpxolpzlIP31IUllIUZDJlIJBIJGIJ4IJ$IPJDUK COMMON/PLOT/TPLT(200).UPLT(2.200).PPLT(3.200).KK DIMENSION JALGT(15).JALGF(15).THETA(15):SKST(4).SKM(4).E1(2) I.E2(2).E3(2).E4(2).EB(2).DE(2).UO(4).UIO(4).UZO(4) REAL KnKNU DATA STATEMENT IS USED FOR PIPE AND FLUID DATA FLUID--WATER¢25C. PIPE-I“ TYPE L COPPER DATA RIETIKIEDKNUORHOFIRHOTI.0130.0012702.2E901.17E11 1. . 0034. 99B. 2: S900. / IFIT.GT.O)GO TO 30 NSFT-zs NSFT2=2fiNSFT JJMX=0 ESTR-E/(I.-KNU*KNU) TAU=1. CDAO=.OOOI PI=3.1416 RM=R+ET/2 ZI-RHOT/(2.*RHOF) 22-E/(2.*K)+RM/ET 23-2.*RHOT*RM/ET CF-SGRT({-22-21+SQRT((ZS-21)O§2+ZSOKNU**2/RHOF))/ 1(23/E*(KNU**2-I.)-RHOT/K)) CT-SGRT((-22-21-SGRT((22-21)9§2+23*KNUi*2/RHOF))/ I(23/E*(KNU**2-1.)-RHOT/K)) TMAx-TMAX/IOOO. Tc-TCIIooo. s-1./10009 DT-XLMIN/CT DTESDTPDT NPTs-NELM+1 NELMItNELM-I TSTDY-NSTDYfinT TC-TC+TSTDY XMULTPCT/CF DO 9 I-I.NELM JALGT(I)-XL(I)/XLMIN ALGF-XMULTPJALGT(I) JALGF(I)-ALGF IF(JALGF(I>.GT.JJMX)JJMXIJALGF(I) THETA(I)-ALGF-JALGF(I) uRITEt7.I7)XL(I).xLMIN.THETA(I).JALcT CONTINUE AF-PI*R*R AP-PI*((R+ET)**2-R*R) GAI(RHOFGCF*CF*(2.ORM/ESTR+ET/K)-ET)/(KNUPR> GCI(RHOF*CT§CT*(2.ORM/ESTR+ET/K)-ET)/(KNU§R) GCA-GC/GA BI-RHOFCCF CI-RHOTéCFocA B2-RHOFOCT C2-RHOTQCTRGC RC-(Rfi—CDTIIRI-C‘l) 84 GCAI‘I.-GC/GA GMch-GA B2I-B2/BI 25-(B2-GC9BI/GA)/(AF§GCAI) C CONSTANTS FOR ELBOW BOUNDARY IF(NELBW.EQ.O) GO TO 12 DI-I.+GA§AF/AP D2-I.+GC*AF/AP DO 10 I-IDND SKST(I)-SK(I)+2.*SC(I)/DT+4.*SM(I)/DT2 IO SKM(I)'I.-SK(I)/SKST(I)-SM(I)*4./(DT2*SKST(I)) DO II IIITNELBW II-2*I-I I2-2*I EI(I)IBI-GA*SC(II)/(AP*SKM(II))-C1 E2(I)IB2-GCRSC(II)/(AP*SKM(II))-C2 E3(I)IGAOSC(I2)/(AP*SKM(12))+C1-B1 E4(I)-GC*SC(I2)/(AP*SKM(I2))9C2-B2 EBiI)-B2*E1(I)-BI*E2(I) II DE(I)'D2*EI(I)-DI*E2(I) “RITE(7I15)R0ET0KOEORHOFDRHOTOKNU 15 FORHAT(/o 4X; 'R: ET: K» E3 ’0 2FB. 5: 25:0. 3. I: 14X. 'RHOF. RHOT: KNU: 'o2F7. 1. F6. 3) WRITE(I.15)R:ET.K.E.RHOF.RHOT.KNU *****STEADY-STATE SETUP***** hvfi()fit5 '2 PJOUKfiPJOUK/S VO'PJOUKI(RHOF*CF) PRES‘RHOFfi(AFGVO/CDAO)§*2/2. WRITE<7aI3)VO:CF:CT.PRES*S.PJOUK*S 13 FORMAT(/.4X.'VO- '.Fb.3o’M/S. CFoCT- ’.2F7.Ia'M/S’./. IAX.’PRES:PJOUK= '.F6.3.F7.1.'KPA’) WRITE(I.13)VO.CF.CT.PRES*S.PJOUK*S CVPIVOiVOiAFfiAF/(2.*PRES) SGXOIPRES*AF/AP FORCIO-PRES*AF-SGXO*AP FORC2OB-FORC10 INITIALIZE ARRAYS (Tatfi DO 20 I‘laNPTS V‘Io 1)=VO P(III)IPRES SCX(I»I)-SCXO 2O UDT(I.I)'O. DO 25 I8ITND UO(I)=O. UID(I)‘O. 25 U20(I)=O. TPLT(I)'O. UPLT(I:I)3O. UPLT(2:I)=O. PPLT(IoI)-O. PPLT(2:I).O. PPLT(3:I)=O. RETURN C C*****TRANSIENT SOLUTION***** 30 CONTINUE IF(NELBW.EQ.O) GO TO 41 C C ELBOW BOUNDARY C C J TR COUNTER FOR FlRDW 0000 85 J51 IS COUNTER FOR FIRST SDOF JS2 IS COUNTER FOR SECOND SDOF II IS COUNTER FOR UPSTREAM POINT I2 IS COUNTER FOR DOWNSTREAM POINT JSI:-1 J80 DO 40 I=2TNELM1 IF(NTYP(I).NE.2) GO TO 40 JIJ+I JSI-JSI+2 JS2=JSI+I IIBI I2=I+I JTI‘Jd-JALGT(I-1) IF(JTI.LT.I)JTI=1 dT2=JJ-JALGT(I+1) IF(JT2.LT.I)JT2=I JFI‘JJ-JALGF(I-I) IF(JFI.LT.I)JFI=I JF2'JJ-JALGF(I+1) IF(JF2.LT.I)JF2=1 JFII‘JFI-I IF(JFII.LT.I)JFII=I JF21=JF2~I IF(JF21.LT.I)JF2I=I THETIcTHETAtl-I) THET2=THETA(I+1) T11:1.-THET1 T12=1.-THET2 PR=TII*P(I-I.JFI)+THETI*P(I-I:JFII) VR-TII*V(I-IoJFI)+THET1*V(I-IIJFII) SGXR-TII*SGX(I-IoJFI)+THETI*SGX(I—I:JFII) UDTRITII*UDT(I-I.JFI)+THETI*UDT(I-1.JFII) PSBTI2*P(I2+I.JF2)+THET2*P(I2+I.JF2I) VS¢T12*V(I2+I:JF2)+THET2*V(I2+I:JF21) SGXSITI2*SGX(I2+I.JF2)+THET2*SGX(I2+I:JF2I) UDTSBTI2*UDT(I2+I:JF2)+THET2*UDT(I2+I:JF2I) CPI-PR+BI*VR-CI*UDTR+GA*SGXR CP2-P(I-I.JTI)+B2*V(I-IpJTI)-C2*UDT(I-IaJTI)+GC*SGX(I-I.JTI) CMI‘PS-B1*VS*CI*UDTS+GA*SGXS CM2=PII2+1.JT2)-B2*V(I2+I:JT2)+C2*UDT(12+I.JT2)+GC*SGX(IZ+I.JT2) FORCI-P(II;JJ-I)*AF-SGX(II.JJ-I)*AP-FORCIO FORC2=-P(I2oJJ-I)*AF+SGX(12.JJ-I)*AP-FORC20 CN1=(4.*SM(JSI)/DT+2.*SC(JSI))*UIO(JSI)+2.*SM(JSI)*U20(JSI) CN2=(4.*SM(JS2)/DT+2.*SC(JSZ))*UIO(JS2)+2.*SM(JS2)*U20(JS2) CTIU-U2O(JSI)+4.*(-UO(JSI)-UIO(JSI)*DT)/DT2 CT2I-U2O(JS2)+4.*(-UO(JS2)-UIO(JS2)*DT)/DT2 CFI'UO(JSI)+(CNI-FORCI)ISKST(JSI) CF2-UO(JS2)+(CN2-FORC2)ISKST(JS2) SKCI-SK(JSI)PCFI+SM(JSI)*(CTI+4.*CFI/DT2)+FORCIO SKC2-SK(JS2)iCF2+SM(JS2)*(CT2+4.*CF2/DT2)+FORC2O FI-CPI+GA*SKCI/(APOSKM(JSI)) F2-CP2+GC*SKCI/(APGSKM(JSI)) EF-E2(J)*FI-F2*EI(J) , Fafi-CM1+GA*SKC2/(APiSKM(J52)) F4--CM2+GC*SKC2/(AP*SKM(JSZ)) P(II:JJ)8(E4(J)*(BI*EF+EB(J)*F3)-(B2*EF+EB(J)*F4)*E3(J))/ I((EB(J)*D2+B2*DE(J))*E3(J)-E4(J)*(EB(J)GDI*BI*DE(J))) P(IZ.JJ)-P(II.JJ) UDT(I2:JJ)--(P(IIoJJ)*(DI+BI*DE(J)/EB(J))+EF*BI/EB(J)+F3)/E3(J) V‘IzlJJ)-UDT(12IJJ)-(P(110JJ)*DE(J)+EF)/EB(J) UDT(II.JJ)-(-P(II.JJ)*DI-V(12.JJ)§BI+UDT(I2oJJ)GBI+FI)/EI(J) V(II.JJ)8V(I2.JJ)+UDT-PRES JT2=JJ-JALGT(1) IFIJT2.LT.I)JT2=I JF2RJJ-JALGF(1) IF(JF2.LT.I)JF2‘1 JF213JF2-I IF(JF2I.LT.I)JF2III THET2=THETA(I) TI2*1.-THET2 PS-TI2*P(2.JF2)+THET2§P(2oJF2I) VSBTI2*V(2oJF2)+THET2§V(2:JF2I) SGXS-TI2RSGX(2IJF2)+THET2*SGX(2.JF21) UDTSBTI2*UDT(2.JF2)+THET2OUDT(2.JF2I) CMI‘PS-B19V5+CI*UDTS+GA*SGXS CM2-P(2.JT2)-B2*V(2.JT2)+C2*UDT(2:JT2)+GC*SGX(2,JTZ) SGX(I.JJ)-(P(I.JJ)*(I.-BZI)-CM2+B2I*CMI)/(B2I*GA-GC) V(I.JJ)=(P(1:JJ)-CM1+GA*SGX(I.JJ))/B1 C STIFF SUPPORT DO 50 I-2TNELM IF(NTYP(I).EQ.3)GO TO 45 IF(NTYP(I).EQ.2)GO TO 50 IF(NTYP(I-I).EQ.1)GO TO 47 GO TO 50 45 II‘I I2=I+1 JTI‘Jd-JALGT(I-1) IF(JTI.LT.I)JTI=I JT2=JJ-JALGT(I+I) IF(JT2.LT.I)JT2=I JFIBJJ-JALGF(I'1) IF(JF1.LT.I)JFI=I JF2‘JJ-JALGF(I+I> IF(JF2.LT.I)JF2=I JFIISJFI-I IF(JF11.LT.I)JFII-I JF21=JF2-I IF(JF21.LT.I)JF2I=1 THETIBTHETA(I*I) THET2=THETA(I+I) TIIBI.-THETI TI2-I.-THET2 PR-T11*P(I-1.JF1)+THETI*P-B2*V(I+IoJT2)+C2*UDT(I+IoJT2)*GCOSGX(I+I.JT2) SGX(II.JJ)=((CP2+CM2)/2.-(CPI+CMI)/2.)IGM P(II.JJ)8(CP2+CM2)/2.-GC*SGX(II:JJ) UDT(II.JJ)-(P(II.JJ)*(B2I-1.)+CP2-B2I§CPI+SGX(IIoJJ)* 1(B2IGGA-GC))/(B2I§CI-C2) V(II.JJ)-(CPI-P(II.JJ)+CI*UDT(II:JJ)-GA*SGX(II:JJ))/BI CONTINUE VALVE IF(T.GT.TSTDY) GO TO 60 CVfiCVP GO TO 63 IF(T-TC) 61:62.62 TAU=(I.-SQRT(T/TC))**2 CV-TAU*TAU*CVP GO TO 63 TAU’O. CV-O. UDT(NPTS.JJ)‘O. .ITI IJJ-JAL CT ( NFLM': C C C 69 7O 88 IF(JTI.LT.I)JTI=I JFI’JJ-JALGF(NELM) IF(JFI.LT.I)JF1=1 JFII‘JFI-I IF(JF11.LT.I)JFII=I THETI-THETA(NELM) TIIBI.-THETI PR-TII*P(NELM.JF1)*THETI*P(NELMTJFII) VR-TII*V(NELM.JFI)+THETI*V(NELM:JFII) SGXR-TIIOSGX(NELMoJFI)+THETI*SGX(NELM;JFII) UDTR-TII*UDT(NELM.JFI)+THET1*UDT(NELM;JFII) CPI-PR+BI*VR-CI*UDTR+GA*SGXR CP2-P(NELMoJTI)+B2*V(NELM.JT1)-C2*UDT(NELM:JT1)+ IGC*SGX(NELM.JT1) 243(CP2-GCA*CPI)/GCAI V(NPTS.JJ)-(-CV§25+SQRT((CV*25)**2+2.*CV*24))/AF P(NPTS.JJ)-24-ZS*V(NPTS.JJ)*AF SGX(NPTS.JJ)-(CPI-P(NPTS.JJ)-BI*V(NPTS.JJ))/GA RESET PARAMETERS IF‘JJ.LT.NSFT2)RETURN IF(JJ/NSFT.NSFT.EQ.JJ) GO TO 69 RETURN DO 70 I-1INPTS DO 70 J‘IoNSFT P(I;J)-P(IpJ+NSFT) V(IaJ)BV(I:J+NSFT) SGX(IIJ)-SGX(IIJ+NSFT) UDT(I:J)'UDTII:J+NSFT) JJ‘JJ-NSFT RETURN END C***§§§**Q**.Q.§§***O§§O§Q§§G§§§§Ofiififffifififififfiif§§§§§*§§§*ifiiffifi 100 M 110 130 10 SUBROUTINE PRINT CONNON/PASS/JJ.ToSK‘R):SN‘4)nSC(4) COHMON/P/P(ID:50).V(15»50):S°X(ID»50):UDT(15:50)I IXL(15)INTYP(15) COMMON/CONST/PI:XLMINaPRESoSaTMAXTTCoDT:KOUNToKPLOTaNSTDYa 1NDINELHINELBuDIPIJIP20IP30IUlD1U20J1DJ20J3DJ43JsJPJOUK IF(T.CT.O)CO TO 10 URITE(7:I) FORMAT(4XT’***PRINT AND PLOT DATA***’:/.4XT’INPUT KOUNT 1(PRINT DATA INCREMENT)’:$) READ(5:*)KOUNT URITE(I:IOO)KOUNT FORNAT(/I4XO’KOUNT= ’015) WRITEI712) URITE(1:2) FORMATIAX:'INPUT 5 POINTS FOR PRINTOUT INFORMATION') READ(5.*)JI:J2:J3:J4:J5 HRITE(IoIIO)JI:J2:J3:J4:JD FOR"AT(4XIDID) HRITE(7:3) FORNAT(4Xo’INPUT KPLOTIPLOT DATA INCRENENT)'o/4Xo'IF NO PLOTS I ARE “ANTED: KPLOT‘O’02XI‘) READ(5.*)KPLOT IF‘KPLOT.EG.O)OO TO IO WRITE(7:4) WRITE(I:4) FORMAT(4XT’INPUT 3 POINTS FOR PRESSURE PLOTS: AND 2 IPOINTS FOR ELBOW VELOCITY PLOTS’oI) READ‘50*)IP101P2IIPSDIUIDIUQ "RITE(1I130)IPIIIP2IIPBOIUIIIUZ FORMAT(4X:5I5) URTTFT7.?OTT/Q 89 WRITE(I:20)T/S 20 FORMAT(/4Xa'TIHE":FIO.3:’HS') URITE(7:30)P(JI:JJ)*S:P(J2:JJ)*S:P(J3:JJ)*S:P(J4:JJ)*S: 1P(JD: \J‘J).SD V‘JI- OJ): V(J2v \JJ) , V(J3o \JJ): V‘J4o JU)» V‘JSo JJ) o ISSX(J5»JJ)*S:UDT(JI:JJ):UDT(J2:JJ)IUDT(J30JJ):UDT(J40JJ)I IUDT‘JSTJJ) URITE(I:30)P(JI:JJ).SoP‘JzuJJ)’S:P(J3:JJ)*S»P(J40JJ)*S: IP‘JD:JJ)§S:V(JI»JJ):V(J2»JJ)»V(J3:JJ):V(J4:JJ)IV(J5:JJ): ISGX(JIoJJ)’S:SCX(J2:JJ).SoSOX‘JG:JJ)’S:S°X(J4:JJ)§So ISGX(J5oJJ)*SIUDT(JI:JJ):UDT(J2:JJ):UDT(J3:JJ)»UDT(J4:JJ): IUDT(J5:JJ) 30 FORMAT(/o(5E12.3)) RETURN END c*§§**§§**I‘§*§*§**ON".§§*§*§.QO**§“*.***“§..m.§§.“*§§.§.******§f".*‘I SUBROUTINE PLOT COMMON/PLOT/TPLT(200).UPLTI2.200):PPLT(3-200).KK LOOICAL‘I FNAHE‘IS) C C ELBOW VELOCITY PLOTS C 190 WRITE(7.200) 200 FORMAT(’ DO YOU WANT ELBOW VELOCITY PLOTS(Y/N)?’.$) READ(5.210)IU 210 FORMAT(A1) IF(IU.NE.IHY) GO TO 290 D0 270 181:2 CALL GTLIN(FNAME,’ ENTER UDOT FILENAME’) OPEN(UNIT'I,NAME-ENAMETTYPE3’NEW') WRITE(I:220) 220 FORMAT(’ 5 UDOT FILE’) DO 250 J-IrKK WRITE(I:240) TPLTtJ)aUPLT(IoJ) 240 FORMAT(’RD’.2GIS.7) 250 CONTINUE . WRITE(I:260) 260 FORMAT(’ED’) CLOSE(UNIT=I) 270 CONTINUE C C PRESSURE PLOTS C 290 WRITE(7.300) . 300 FORMAT(’ DO YOU WANT PRESSURE PLOTS(Y/N)?’.S) READ(5.2IO)IPR IF(IPR.NE.IHY) GO TO 999 DO 380 181.3 CALL GTLIN(FNAME:’ ENTER PRESS FILENAME’) OPEN(UNIT-I.NAME-FNAME.TYPE-’NEW’) WRITE(I.320) 320 FORMAT(’ ; P FILE’) DO 360 J'lIKK WRITE(I.350) TPLT(J):PPLT(I:J) 350 FORMAT('RD’.2GIS.7) 360 CONTINUE WRITE(1.370) 370 FORMAT('ED’) CLOSE(UNIT~I) 380 CONTINUE 999 CONTINUE RETURN END LIST OF REFERENCES 10. LIST OF REFERENCES Joukowsky, N. E., translated by 0. Simin as "Hater Hammer,“ Proceedin 3 American Hater Horks Association, Vol. 74, 1904, pp. 341-424. Hylie, E. 8., Streeter, V. L., Fluid Transients. McGraw-Hill, 1978. Chaudhry, M. H., Applied Hydraulic Transients, Van Nostrand Reinhold Co., 1976. Skalak, R., 'An Extension of the Theory of Hater Hggmfré' Trans. ASML, Vol. 78, No. 1, 1956, pp. Thorley, A. R. 0., ‘Pressure Transients in Hydraulic Pipelines," ASME Journal of1 Basic Engineering. Vol. 91, Sept. 1969, pp. 453- 461. Hilliams, D. J., I'Haterhammer in Non-Rigid Pipes: Precursor Haves and Mechanical Damping,“ Journal of Mechanical En ineerin Science Institute 0L ggngilcal Engineers. Vol. I9, No. 6, I977, pp. Halker, J. 8., Phillips, J. H. “Pulse Propagation in Fluid Filled Tubes,“ Journal of Applied Mech. Trans. ASME March 1977, pp. 31- 3S. Lin, T. C., and Morgan, G. H., “Have Propagation Through Fluid Contained in a Cylindrical, Elastic Shell, ' Journal of Acoustical Society of America, Vol. 28(6), 1536, pp. 1153- 1173. Blade, R. J., Lewis, H., and Goodykoontz, J. H., "Study of a Sinusoidally Perturbed Flow in a Line Includin a 90 deg16 Elbow with Flexible Supports,“ NASA Technica Note0211962. Hood, D. J. “A study of the Response of Coupled Liquid Flow- Structure Systems Subjected to Periodic Disturbances,“ Jour. of Basic Engr.. Trans. ASME, Vol. 90, Dec. 1968, pp. 532-540. 90 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 91 Davidson, L. C. , and Smith, J. E. ,“Liquid- -Structure Coupling in Curved Pipes,I The Shock and Vibration Bulletin, No.40, Part 4, Dec. 1969, pp. 197- 207. Davidson, L. C., and Samsury, D. R. ,‘Liquid-Structure Coupling in Curved Pipes-II," The Shock and Vibration Bulletin, No. 42, Part 1, Jan. 1972, pp. 123-125. Hatfield, F. J., Higgert, D. C., and Otwell, R. S., “Fluid-Structure Interaction in Piping by Component Synthesis,“ ASME Journal of Fluids Engineering. Vol. 104, No. 4, Sept. 1982, pp. 318- 325. Hatfield, F. J., Davidson, L. C., and Higgert, D. C.,"Acoustic Analysis of Liquid-Filled Piping by Component Synthesis. Experimental Validation and Examination of Assumptions,‘I Fluid Transients and Fluid- Structure Interaction, ASHE PVF- Vol. 64, ‘1982, PP 153-115 Phillips, J. H.,'Reflection and Transmission of Fluid Transients at Elbows," TAM Report No. 425, University of Illinois at Urbana-Champaign, May 1978. Hood, D. J. “Influence of Line Motion on Haterhammer Pressures,“ Journal of Hydraulics Division Proc. ASCE, Vol. 96, May 1969, pp. 941- 959. Hood, D. J., and Chao, S. P., “Effect of Pipeline Junctions on Haterhammer Surges," ransportation Engineering Journal Proc. ASCE, Vol. 97, Aug. 1971, pp. 441- 456. Ellis, J., 'A Study of Pipe- Liquid Interaction Following Pump Trip and Check- Valve Closure in a Piping Station,“ Proc. Third Intl. Conf. on Pressure Surges, Vol. 1, BHRA Fluid Engr., Canterbury, England, March 1980, pp. 203-220. Schwirian, R. E., and Karabin, M. E., I'Use of Spar Elements to Simulate Fluid-Solid Interaction in the Finite Element Analysis of Piping System Dynamics," S m osium on Fluid Transients and Structural Interactions in Pip ng Systems. ASHE, June 1961 pp. - 1. Giesecke, H. D., 'Calculations of Pipin Response to Fluid Transients Including Effects of F uid/Structure Interaction,“ Sixth:lnterngtional Conference on Structural Mechanics in Reactor Technology. North-Holland Publishing Co., August 1981. 21. 22. 23. 24. 25. 26. 92 Otwell, R. 8., "The Effect of Elbow Translations on Pressure Transient Analysis of Piping Systems," Fluid Transients and Fluid-Structure Interaction. ASHE PVP-Vol. 64, 1982, pp. 127-136. W9?ert . C., and Hatfield J. 'Time Domain na ysis of Fluid- Structure Interaction in . Hulti- -Degree- -of Freedom Piping System, ' Proc. 4th Intl. Conf. on Pressure Surges. BHRA, Sept. 1983. Swaffield, J. A. “The Influence of Bends 0n Fluid Transients Propagated in Incompressible Pipe Flow,“ Proc. Inst. Mechnical Engrs., Vol. 183, Pt. 1,No. 29, 1968-69, pp. 603-614. Forsythe, G. E. and Uasou, H. R. Finite Difference Methods for Partial Differential Eguati ons, John Riley and Sons, Inc. 1960. C3319, R. R., Structural Dynamics, John Wiley & Sons, 1 1. Hhite, R. N., Gergely, P., Sexsmith, R. 6., Structural Engineering, John Hiley & Sons, 1976 TV millilii‘fllillislliil'liill flliillillil H illilalliiliies 31293 03103 7215