.“.I.“n) |1"-"‘ ‘.‘g'.\‘fi ‘H." \' " “ ' .' . .. ~ ‘4II«Mudfi'mflmHuxfiiw‘fl‘I'e‘. :u x. - PIPELINE TRANSIENTS IN HIGHLY SATURATED FLUIDS WITH GASEOUS AND VAPOROUS CAVITATION Thesis for the Degree of M. S. MICHIGAN STATE UNIVERSITY MARK J. SUNDQUIST 1977 L [B I: A R Y Michigan State University ABSTRACT PIPELINE TRANSIENTS IN HIGHLY SATURATED FLUIDS WITH GASEOUS AND VAPOROUS CAVITATION BY Mark J. Sundquist Recently, hydraulic transients in pipes which include multi-phase flows and gas release have received considerable attention. A lack of understanding of the gas release phenomena in turbulent pipe flow has resulted in difficulty to correctly model gas release during transient conditions. In addition, due to the variable wave speed associated with these mixtures, questions arise as to the selection of a numerical technique to accurately model the governing differential equations. In this study a mathematical model is developed to simulate hy- draulic transients in two-component, gas-liquid mixtures including dis- tributed gas evolution. The numerical technique is the method of char- acteristics using specified distance and time intervals. Due to the variable wave speed, an interpolation scheme is developed which reduces the severity of interpolation errors as compared to the conventional interpolation scheme. To obtain a quantitative understanding of the nature of gas release under turbulent conditions, the experimental phase of this study includes the measurement of the amount of gas released from agitated water satur- ated with carbon dioxide gas in a closed container. This experiment provides a basis from which a gas release model is developed for use in Mark J. Sundquist the numerical simulation of transient pipe flow. Pressure transients are recorded from a 400 ft. copper pipeline in.which water saturated with carbon dioxide is the transported fluid. Initial steady-state pressures are above the gas saturation pressure of the water resulting in essentially one-phase flow. A quick valve closure initiates the transient resulting in column separation and, owing to gas release, a developing two-component flow situation. Numerical simulations are conducted for the laboratory conditions. Good agreement of peak pressures and timing of the primary pressure peaks are obtained; however, higher order pressure fluctuations travel- ing along the pipe are not simulated, possibly due to inaccuracies in predicting the evolution of gas along the pipe. PIPELINE TRANSIENTS IN HIGHLY SATURATED FLUIDS WITH GASEOUS AND VAPOROUS CAVITATION BY 3“ Mark JéYSundquist A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil and Sanitary Engineering 1977 ACKNOWLEDGMENTS The writer wishes to express sincere appreciation to his major professor, Dr. David C. Wiggert, for his invaluable guidance, encour- agement and support and to Dr. David A. McIntosh for his suggestions and interest. The writer also wishes to thank Dr. E. B. wylie of the University of Michigan for his helpfulness and interest in this study. Appreciation is also expressed to the University of Michigan for their generosity in extending to this study the use of the Van Slyke appar- atus, dead weight tester and copper coil. This study was supported in part by the National Science Founda- tion, Grant No. 74-06653, and by the Department of Civil Engineering of Michigan State University. The writer expresses his appreciation of this financial support. The writer also wishes to acknowledge the numerous friendships acquired over the past years for making the stay at this university a most memorable and enjoyable experience. ii TABLE OF CONTENTS LIST OF TABLES. . . . . . . . . . . . . . . . . . . . . . ‘ LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . NOMENCLATURE. . . . . . . . . . . . . . . . . . . . . . . Chapter I. II. III. IV. INTRODUCTION 0 O O O O O O O O O O O C O C I O O O 1.1 Description of the Problem. . . . . . . . . . 1 O 2 Literature ReView O O C O O O O O O O O O O O 1.3 Scope of Investigation. . . . . . . . . . . . EQUATION DEVELOPMENT . . . . . . . . . . . . . . . 2.1 The Conservation of Mass of the Liquid Phase. 2.2 The Conservation of Mass of the Gas Phase . . 2.3 Mixture Momentum. . . . . . . . . . . . . . °. NUMERICAL ANALYSIS . . . . . . . . . . . . . . . . 3 1 Determination of the Characteristic Equations 3 2 Finite Difference Relations . . . . . . . . . 3.3 Integration of the Pathline Characteristic. . 3 4 Interpolation Scheme. . . . . . . . . . . . . 3 5 Modeling Aspects. . . . . . . . . . . . . . . LABORATORY APPARATUS AND INSTRUMENTATION . . . . . 4.1 Gas Release Experiment. . . . . . . . . . . . 4.1.1 Description of Apparatus and Instru- mentation. . . . . . . . . . . . . . . 4.1.2 Experimental Procedure . . . . . . . . 4.2 Hydraulic Transient Experiment. . . . . . . . 4.2.1 Description of Apparatus and Instru- mentation. . . . . . . . . . . . . . . 4.2.2 Experimental Procedure . . . . . . . . COMPARISON AND DISCUSSION OF EXPERIMENTAL AND THEORETICAL RESst 0 O O O C C O O O C O O I O O 0 iii Page vi (Dbl-i 10 11 14 16 20 21 ‘26 33 35 49 53 S3 54 56 60 60 69 73 Chapter VI. LIST OF APPENDI A. B. 5.1 Gas Release Experiment. . . . . . . . 5.1.1 Conditions Investigated. . . . . . 5.1.2 Discussion of Results and Modeling Implications . . . . . . . 5.2 Pipeline Transients . . . . . . . 5.2.1 Conditions Investigated. . 5.2.2 Comparison of Results. . . SUMMARY AND CONCLUSIONS. . . . . . . . REF ERENCB O O O 0 O C O O O O O I O 0 CES I O O O C O O O O O O O O O O O O O DETERMINATION OF CHARACTERISTIC EQUATIONS. COMPUTER PROGRAM AND SAMPLE OUTPUT . iv Page 73 74 78 86 86 87 102 105 107 107 112 LIST OF TABLES Table 1 Important Variables used in Numerical Experiment Comparing Increased At/Ax Ratio with Conventional Ratio. . . . . . . . . 2 Summary of Gas Release Experiments . . . . . . 3 Summary of Transient Pipeline Experiments. . . 4 Gas Release Variables used in Numerical Model. Page 40 77 88 e9 Figure 10 11 12 LIST OF FIGURES Control Volume for Conservation of Mass Of Liquid 0 O O O O O O O O O O O O O O O O O 0 Control Volume for Conservation of Mass Of Gas 0 O O O O O O O O O O O O O O C O O O O 0 Control Volume for Mixture Momentum Equation 0 O O O O O O O O O . O C O O O O O O 0 wave Speed versus Void Fraction for Bubbly Two-Phase Flow in a One Inch Copper Pipe. . . . Definition Sketch Showing Intersection of C+, C' and Pathline Characteristic Curves in the X't Plane 0 o o o o o o o o o o o o o o o o o 0 Characteristic Lines on the x-t Grid with Specified Distance and Time Intervals . . . . . Definition Sketch Illustrating Interpolations taken within One Distance Increment . . . . . . Definition Sketch Showing Possible Interpola- tions within One Distance Increment Associated with Variable wave Speed. . . . . . . . . . . . Definition Sketch Illustrating Interpolations taken Beyond One Distance Increment . . . . . . Definition Sketch Showing Possible Interpola- tions Associated with Increased At/Ax ratio for a Variable Wave Speed Situation . . . . . . Comparison of Pressure Response at the Valve for n=1 and n=2 with the Pipe Divided into 20 Computing Sections . . . . . . . . . . . . . Comparison of Pressure Response at the Valve for n=1 and n=2 with the Pipe Divided into 20 Computing Sections. . . . . . . . . . . . . . . vi Page 12 15 17 27 28 30 37 37 39 39 42 43 Figure 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Comparison of Pressure Response at the Valve for n=1 and n=2 with the Pipe Divided into 40 Computing Sections . . . . . . . . . . . . . . Comparison of Pressure Response at the Valve for Two-Phase Flow with the Lax-Wendroff‘ Two-Step Scheme and the Method of Characteristics With n=1 and n=2 C O O O O O O O O O O O O O C O 0 Comparison of Pressure Response at the Midpoint of the Pipe for Two-Phase Flow with the Lax- Wendroff Two-Step Scheme and the Method of Char- acteristics with n=1 and n=2. . . . . . . . . . . Interpolations taken at Boundary with Increased At/AX Ratio 0 o o o o o o o o o o o o o o o o o 0 Definition Sketch of Vapor Void Occurring at a Computing Node. . . . . . . . . . . . . . . . . . Detailed Schematic of Gas Release Experiment. . . Close-up View of an Agitation Grid. . . . . . . . View of Gas Release Experiment under Test Con- dit ions 0 O O O O O O I O O O O O O 0 O I C O O O Close-up View of Lucite Test Cylinder During Actual Testing. 0 O O O O O O O O O O O O 0 O O View of Van Slyke Dissolved Gas Measuring Apparatus . . . . . . . . . . . . . . . . . . . . Van Slyke Apparatus Schematic . . . . . . . . . . Comparison of Van Slyke Measurements versus Henry ' S Law 0 O O O O I O O O O O O O O O O O O 0 Typical Wire Gage Calibration Curve . . . . . . . Typical Data Trace Showing Water Elevation Rise in Measuring Tube during a Gas Release Experiment 0 O O O O O I O O O O O O O O O I O 0 0 Detailed Schematic of Experimental Transient Pipeline. . O O O C O C Q C O I O O C I O O 0 . O View of Copper Coil and Lucite Pipes. . . . . . . Close-up View of Elbow Meter and Solenoid- Actuated Valve in the Upstream Location . . . . . vii Page 44 46 47 50 50 55 57 57 58 58 59 61 61 62 64 65 65 Figure 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 Elbow Meter Calibration Curve . . . . Flow Resistance Curve for Copper and Lucite Pipes. . . . . . . . . . . Close-up View of Miniature Strain Gage Transducer Mounted in the Lucite Pipe 0 O O O O O O O O O O O O O O O 0 Photograph of Electronic Data Gather- ing System which includes Amplifier, Storage Oscilloscope, Mini-Computer, D/A Converter and X—Y Plotter . . . . Schematic of Electronic Data Gather— ing system 0 O O O O O O O O O O O O 0 Actual Data Trace of Transient Pressures at Valve as Recorded on the x-Y PlOtter o o o o o o o o o o o o 0 Results of Gas Release Experiment for AP = 10 p81. 0 O O O O O O O O O O 0 Results of Gas Release Experiment for Ap = 20 psi. . . . . . . . . . . . . 'Results of Gas Release Experiment for Ap = 30 psi. . . . . . . . . . . . . Results of Gas Release Experiment for AP = 40 p81. 0 O O O O O O I O O O 0 Gas Production Rate Curve used in Numerical Model . . . . . . . . . . . Comparison of Measured Transient Pressures with Theoretical Results for Experimental Rm 1 O O O O O O O O O O O O O O O 0 Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run 4 . . . . . . . . . . . . . . . . Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run 5 O O O O O O O O O O O O O O O 0 Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run 8 I O I O I O O O O O O I Q 0 O 0 viii Page 66 67 68 68 70 71 80 81 82 83 85 90 91 92 93 Figure 45 46 47 48 Page Comparison of Measured Transient Pressures with Theoretical Results for Experimental Rm 9 O O O 0 O O O O C O O O O O O O O C O O O O O O O O 94 Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run 15 I O O O O O O O O O Q C O O I O C O O C C O O O O O 95 Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run 16 O O O O O O O O O O O O O O O O I O O O I O O O O O 96 Comparison of Measured Transient Pressures with Theoretical Results for Experimental 97 Run 14. . . . . . . . . . . . . . . . . . . . . . . . . . ix NOMENCLATURE pipe cross-sectional area L2 wave speed in pipe LT-1 total height of oscillation L grid point on x-t plane characteristic curves grid points on x-t plane pipe constraint factor gas concentration in water ML.-3 gas concentration at bubble surface ML.-3 gas concentration far away from the bubble ML”3 pipe diameter L diffusion coefficient I LZT-1 pipe elastic modulus ML-J'Tm2 pipe wall thickness L Force in the x-direction in the momentum equation MLT-2 Darcy-weisbach friction factor ”hydraulic gradeline L bulk modulus of elasticity of the gas ML-':l"l‘-2 bulk modulus of elasticity of the liquid MLDIT'.2 universal gas constant grid point on x-t plane length of pipe segment L 119 grid point on x-t plane moles of gas in bubble number of distance increments used in interpolation scheme polytropic gas constant grid point on x-t plane internal pipe pressure reference pressure fluid pressure vapor pressure of fluid difference between gas saturation pressure of the fluid and the instantaneous pressure bubble radius location on x-t grid where the C+ characteristic intersects a distance line location on x-t grid where the C- characteristic intersects a distance line fluid temperature reference temperature time specified time increment of x-t grid mixture velocity or velocity of right side of vapor void velocity of left side of vapor void relative velocity of fluid to a gas bubble volume volume of gas bubble velocity vector grid point on x—t plane distance along pipe xi Ax specified distance increment of x-t grid L location on the x-t grid where a characteristic curve intersects the boundary grid point on x-t plane elevation above datum ' L void fraction Henry's Law solubility gas constant gas production rate M'l‘ud'L-3 unit weight of liquid TIL-QT-2 unit weight of mixture ML-ZT-2 interpolation location between fixed nodes interpolation location used in conventional interpolation scheme angle of pipe inclination characteristic roots density of gas ML-3 density of liquid . ML-3 surface tension MT-2 pipe wall shear stress ML-lT-2 flow parameter or dummy variable of integration frequency T-1 xii CHAPTER I INTRODUCTION 1.1 Description of the Problem The transient analysis of a piping system is of importance when designing for peak pressures and flows. Through the operation of system components such as valves and pumps, pressure wave propagation will take place in the system. This may result in the local pressure at some point reaching vapor pressure of the liquid initiating vapor cavity growth. This phenomenon is commonly referred to as column separation. The im- pending collapse of the cavity may result in large pressures which may damage the pipe or other system components. The presence of even a small amount of entrained gas can signifi- cantly change the transient response. If any entrained gas is present or if the fluid has the capacity to release gas during the transient, an analysis which neglects this effect will predict transient pressures which are more severe than will actually occur. In a pipe, two types of cavitation can be distingushed. Accord- ing to Henry's Law, if at any time the pressure falls below the satura- tion pressure of the liquid, the dissolved gas will evolve out of solu- tion in the form of free gas. Conversely, if the pressure is greater than the existing gas saturation pressure, any free gas will dissolve into solution until equilibrium is reached. However, the evolution rate is faster than the solution rate. Therefore, during a transient in which the pressure-time history oscillates about the gas saturation pressure of the fluid, a condition of rectified diffusion will exist resulting in a net gain of free gas with each cycle. The presence of this free gas results in a two-phase flow situation. “The celerity of this mixture is reduced due to the added compressibility of the gas and can be smaller than either the gas or liquid celerities. This reduced wave speed will lead to attentuated pressure peaks and phase changes. This process of evolution of free gas is called gaseous cavitation. In addition to gaseous cavitation, if the fluid pressure falls to its vapor pressure, vaporous cavitation will occur. This is an explo- sive process which results from the vaporous growth of sub-macroscopic nuclei containing liquid vapor and free gas. This phenomenon accounts for the flashing behind a negative pressure wave and in the most severe situation, for column separation. As outlined by Van Wijngaarden [22] the rate of growth for a sin- gle bubble neglecting viscous forces and assuming the growth is slow is estimated to be b kT £13 4 pf-pv + (§)(0/R) dt where Vb is the bubble volume, t is time, k is the universal gas con- stant, T is the temperature, pf is the fluid pressure, pV is the vapor pressure of the fluid, 0 is the coefficient of surface tension, R is the bubble radius and N is the moles of gas in the bubble. The quantity dN/dt is the molar flux of gas into the bubble and represents the diffusive process of gaseous cavitation. As described by Van Wijngaarden, this quantity is dependent upon the diffusion co- efficient, the bubble radius, the transport velocity around the bubble and the gas concentration gradient in the liquid. If the denominator in the right-hand side of the above equation does not vanish, then the growth of the bubble is governed by the gas- eous diffusion term. These growth rates will have finite values and can be termed gaseous cavitation. If, however, the denominator is zero, which implies that -3 3 Rue Pf = Pv then the growth rate would become infinite. This pressure then marks the threshold for vaporous cavitation. For an accurate description of vaporous cavitation, higher order terms should be included in the growth equation. Then a finite growth rate of the bubble at the threshold vaporous cavitation pressure would result. However, these growth rates would still be very large as com? pared to the gaseous cavitation growth rates. Parkin and Kermeen [15] have found that the time required for vaporous cavitation is of the order of a few microseconds as compared with hundreds of milliseconds to achieve any significant gaseous cavi- tation. However, vaporous cavitation does enhance gaseous cavitation by creating large bubbles in which the mass transfer of gas can occur. Since portions of a pipe may experience vapor pressure conditions for extended periods, such as severe column separation cases, significant gas release can occur. The importance of considering gaseous cavitation and its effects during a pressure transient depends upon the liquid being transported, the type of gas, and the extent to which it is dissolved into the liquid. In some cases, such as short lines transporting water which has a small dissolved air content, unless steady state pressures are below the gas saturation pressure, it is probable that very little gas is released and essentially one-phase flow exists. In this case,ai waterhammer analysis neglecting gas release should accurately predict the system response. In long lines where the transient pressures may exist below the gas saturation pressure for extended times, or in highly soluble solutions such as water and carbon dioxide, significant gas release may occur and should be considered to correctly model the physical system. The effects of released gas on hydraulic transients has lacked rational interpretations. In part, this is caused by a lack of trans- ient data which shows significant gas release. It is hOped that this study gives insight into the effects of released gas through the pre- sentation of comprehensive laboratory data. In addition, the numerical aspects, which are concerned with accurately modeling the phenomena in- volved, are hoped to provide a reliable means of predicting gas release transients. 1.2 Literature Review It is well accepted that the presence of submacroscopic nuclei is required to initiate vaporous and gaseous cavitation[:6,15,17] . These nuclei can be small gas-filled pockets in the crevices of the container or gas pockets on particles within the free stream of the flow. Epstein and Plesset [4] developed solutions for the rate of growth of a bubble by diffusion neglecting translational motion of the bubble. Parkin and Kermeen [15] improved on their approach by deriving equations to handle bubble growth by convective diffusion. They established that the growth time is several orders of magnitude greater when considering the effects of water transport around the bubble than without transport. They also noted that these growth times are very much greater than comparable vaporous growth. Parkin and Kermeen also collected data on the growth of bubbles attached to a body submerged in a free stream. It was suggested that the diffusive growth of these small gas bubbles then served as nuclei for vaporous growth as they were swept downstream. They were able to predict the data within an order of magnitude. Also included in their experimental investigation was the measurement of the minimum pressure achieved by water just before vaporous cavitation inception. They found that ordinary water in the tunnel did experience a tensile stress. The tension required for inception increased as the free-stream velocity increased. Once cavitation occurred, the tension was relieved suggest- ing the possibility that tension in the water is required, if for only an instant, to initiate vaporous cavitation. Van Wijngaarden [22] tried to improve the analytic approach to con- vective diffusion by including the effects of surface tension and vapor preSsure. In his development he showed that tensile stresses can be reached before vaporous cavitation due to the surface tension of the nuclei. He obtained a slightly better agreementcfifthe data obtained by Parkin and Kermeen. Schweitzer and Szebehely [17] conducted experiments in which they measured gas release in a closed container with an agitated air-liquid interface. Using an indirect method in which the pressure of the air volume of the container was measured, they found that the gas released exponentially with time for a step decrease in pressure. Included in their formulations is the presence of an experimentally determined time constant which characterizes the rate of release. This time constant is a function of the air-liquid volume ratio in the container. In studying the transient pressures on the downstream side of a closing valve in a kerosene pipeline, Swaffield [20] included the in- fluence of air release. He assumed that the volume of fluid to re- lease air was just that which passed through the valve since closure began. He also assumed no resolution and a simple adiabatic compres- sion and expansion of the air involved. His experimental data showed transient pressures that were always above vapor pressure indicating that vaporous cavitation did not occur and that the column separation void behind the valve contained mostly air caused by gas release. In his model he used a complicated boundary condition at the valve to handle the effects of the released gas at that point. The model in- corporated the method of characteristics using a constant wave speed, and Henry's Law was used to calculate the volume of air released. He obtained better agreement with experimental data when including air release than with just a vapor-filled cavity. To improve on Swaffield's model, Driels [3] used the results of Schweitzer and Szebehely in predicting transient pressures in a kero- sene-filled pipeline. He utilized the same assumptions and boundary conditions as Swaffield but included Schweitzer's experimental time constant to determine the time-dependent nature of gas release. He was successful in predicting the first pressure rise but had less suc- cess in predicting the magnitude and timing of the secondary peaks. ‘Kranenburg [8,9] developed a model in which be accounted for gas release into existing nuclei. He describes two flow regimes, one be- ing column separation which is characterized by free surface flow, the other as cavitating flow which is of a bubbly nature due to the presence of dispersed gas and vapor bubbles. In his analytical develop- ment he considers the gas release which occurs into an individual nucleus; then assuming the number of nuclei present, their size, and relative velocity to the free stream, calculates the gas release occurring through- out the pipe during transient conditions. He utilizes a Lax—wendroff scheme with a variable wavespeed due to its ability to spread a develop- ing shock over a number of mesh points. He attained good agreement with experimental data from a long pipeline in which water with some dissolved air content was used as the working fluid. A pump failure and restart were used to initiate the transients. In studying the experimental re- sults of Swaffield [20] and Baltzer [2], he concluded that gas release has no great influence in cases where only cavitating flow occurs, where- as the influence is considerable in cases where column separation come bined with cavitating flow exist. Martin, Padmanabhan and Wiggert [14] investigated hydraulic trans- ients in two—phase, bubbly, air-water mixtures. Their laboratory experi— ments were concerned with initial void fractions that were large as compared to a developing, two-phase flow due to gas release. Included in their equation development is the provision for a gas release rate. In their study both the Lax-Wendroff scheme and the method of character- istics were used and compared. Due to the convergence of like charac- teristics during the formation of shocks, the Lax-wendroff scheme was favored. However, the two schemes produced identical results in cases where no shock developed. Good agreement was obtained with two-phase experimental data where there was no gas production. In a more recent study Tullis, Streeter and WylieIZZI] developed two models to simulate gas release during pressure transients. To- account for the variable wave speed the method of characteristics using a characteristics grid was employed with occasional interpolations to straighten the grid. A constant gas release rate was assumed whenever the pressure dropped below the gas saturation pressure. In the second model, the air was concentrated at the computing sections while keeping the full liquid wave speed between the computing nodes, thereby allow- ing the method of specified time intervals to be used. The two models produced nearly identical transients in cases where the same amount of air was released. It was shown that by concentrating the air at com- puting points, the effective overall system wave speed can be reduced thereby correctly simulating the variable wave speed phenomenon. Thus, they concluded that the air does not have to be homogenously dispersed to accurately model gas release transients. They also concluded that in cases where little air is released there may be an apparent thermo- dynamic loss associated with the vaporization process. To account for this effect a linear dissipation was added to the model. With such dissipation, satisfactory agreement with experimental data in which only small amounts of air were released was attained. 1.3 Scope of Investigation As cited in the literature review, little is known about the pro- cess and time rates of gas release occurring in a closed conduit. This can be seen by the large number of arbitrary values used in past models describing the rate of gas release. To attain a better under- standing of this problem, a bench-top experiment was conducted in which a closed vessel completely filled with gas-saturated water was subjected to a step decrease in pressure under agitated conditions. Measurement of the amount of gas released provided insight for approximating a rate of gas release function for turbulent flow conditons in a closed circuit. ‘A laboratory investigation was conducted to observe the effects of released gas on hydraulic transients. Tests were carried out in a 400 ft. long copper pipe of one inch diameter with a quick-closing solenoid- actuated valve. The transported fluid was water which was saturated with carbon dioxide. Since CO2 is much more soluble in water than air, (approximately 47 times more soluble at 20°C according to Henry's Law) significant gas release during the transient-occurred. Systematic changes of pertinent variables, primarily the dissolved gas content and initial pressure provided important information regarding the effects of gas release on the system response. The analytical aspects are concerned with the development of the governing partial differential equations which describe the transient flow regime, including the distributed evolution of gas and the subse- quent two-phase flow. The method of characteristics is then used to numerically solve the governing equations. Attempts are then made to numerically predict the transient flows observed in the experiments. CHAPTER II EQUATION DEVELOPMENT To describe the analytical nature of two-phase transients, the conservation of the mass of the liquid and gas phases along with the conservation of momentum of the mixture will be developed using the con- trol volume approach. Unlike the usual waterhammer analysis where one is concerned with only the fluid, the presence of the gas entrained in the fluid requires the added consideration of its behavior in the con- trol volume. In addition, the production or resolution of any free gas in the control volume must also be considered. These equations will then form a basis from.which a numerical solution can proceed to obtain values for the dependent variables velocity, pressure and void fraction as functions of the independent variables distance and time. The following assumptions are made with respect to the flow re- gime: 1. The flow is one dimensional. 2. The fluid mixture is of a homogeneous two-phase, bubbly nature. 3. The difference in pressure across a bubble surface can be ignored. 4. The momentum of the gas phase relative to the liquid phase is small and therefore neglected. Hence, the liquid and bubble velocities in the development are the same. 5. The average cross-sectional values of void fraction, mixture velocity and phase densities can be used, that is; 10 where A is the cross-sectional area and ¢ represents the fluid property. HenCeforth, the symbol <> will be omitted and average properties will be assumed unless otherwise stated. 2.1 ‘The Conservation of Mass of the Liquid Phase The conservation of mass for the liquid component requires that the time rate of increase of mass within a control volume is just equal to the net rate of mass inflow to the control volume. Figure 1 shows the mass flow of liquid into a control volume of length fix. The flow is normal to the control surface at surfaces 1 and 2. Applying the law of conservation of mass to the liquid in the con- trol volume, as outlined by Yadigaroglu and Lahey [24], results in the differential form 8 5]:- [ogfl- 00A] +—— Bax-Eogfl a)Au] - -FA . (1) where F is the gas production rate per unit volume of mixture, pg is the liquid density, A is the cross-sectional area, u is the liquid velocity and a is the void fraction, defined as the ratio of the volume of the gas to the control volume. Expanding Equation (1) and grouping derivatives results in '3u+ +3A 1 3(1’-0 ) 3(1 '43) ax+ A“ +312) + (1 - a) [‘1 3x + at ] + pz “fl 3x 3t p£(l - a) (2) 12 xe memmcoo Low msapo> _oeucou nu x qg.lm + <3Aa-pvaq ._ weaned 13 Identifying.the total derivative with respect to time as d .. _a_ e. .3; - “‘3x + 8t (3) Equation (2) becomes . . do §2+l§é+ l d(l-a)+_]_-_ 2' = —I' (4) 3x A dt (1 — a) dt p2, dt 92(1 - a) Considering the bulk modulus of elasticity of the liquid as d dp/dt K= —P-= 1 pi do, pl dpz/dt (5) where dp/dt is the total derivative of the pressure with respect to time, then from Equation (5) it is seen that ifl=ie2 ' (6, pg dt K1 dt The second term of Equation (4) describes the radial deformation of the pipe. By considering the radial stresses and strains and using Poisson's ratio it can be shown [19] that l; éé;= 22. AB A dt Ee dt (7) where D is the pipe diameter, c is the pipe constraint factor that char- acterizes the anchorage of the pipe, E is Young's modulus of the pipe and e is the thickness of the pipe wall. Substitutions of Equations (6) and (7) into Equation (4) yields 12.9 .1. 92 fliml slum—'1‘ 8 [Ee+K ]dt+3x (1-a) dt 02(1-01) () 14 Equation (8)-describes, in differential form, the conservation of mass of the liquid, including the Compressibility of the liquid and pipe wall, of a two-phase gas-liquid mixture in which gas generation can occur. 2.2 The ConservatiOn of MaSS of the Gas Phase Consider the control volume of length 6x of Figure 2. Represented is the mass flow rate of gas into the control volume. Applying the law of conservation of mass to the gas in the control volume, as outlined by Yadigaroglu and Lahey [24], results in 3 3 3t- (pg a A) + 33-:— (pg a Au) = FA (9) Expanding Equation (9) and grouping derivatives results in Using Equation (3) to recognize the total derivatives with respect to time yields d- 39. IE iii. _1____pfi=._1'_ 3x+A dt+a dt‘l-pg dt pga (11) Substituting Equation (6) as it applies to the gas phase and Equation (7) into Equation (11) yields [a (12) ale. ”,9 _1_ .122 22 9.2 I [Kg+Ee]dt+3x+a Equation (12) describes, in differential form, the conservation of mass of the gas, including the compressibility of the gas and pipe wall, for 15 .mm c we mme *0 cowpm>mecou Lou... mE:_.O> FOLHCOU m xo m Foeucou xe :3 xIM + E .m mesmwm 18 Equation (15) by 5x and expanding yields 3_ 2A- 3.2 22. 3x (pA) + Ym A sin 0 To HD + p 3x pm A at + uApIn 3x 3 3 I + u 7&- (Apm) + u 5; (u Aom) (16) By the combination of the continuity Equations (1) and (9), it can be shown that the last two terms on the right-hand side of Equation (16) sum to zero. In addition, since we are dealing with very low void frac- tion situations in which u << 1, then pg u << p2(l - a) and the mixture density and mixture unit weight can be approximated by ‘D I 1%(1-w (N) .< I! - 02 (1 - a)g (18) The wall shear stress is generally taken as the steady-state shear stress for the same velocity; To = 91(1 - a) f u lul /8 (19) where f is the Darcy-Weisbach friction factor based on one—component flow [13]. The absolute value of the velocity is taken such that the shear stress opposes the direction of theaflow. Substituting Equations (17), (18) and (19) into Equation (16) and rearranging yields :12___L___3_e_f_-_u_l_1 = :0 dt + p£(1 _ a) ax+ 2D g sin 0 0 ( ) 19 Equation (20) is the differential form of the conservation of momentum of the gas and liquid mixture.‘ Together with the continuity equations for the gas and liquid, a set of three equations have been described which form a basis from which a numerical solution can proceed. The equations are recognized to be of the hyperbolic type, hence they can be transformed into total derivatives and integrated along character- istic lines. In this manner the equations can be solved simultaneously to obtain values for the dependent variables of velocity, pressure and void fraction. CHAPTER III NUMERICAL ANALYSIS In the last chapter the governing differential equations for two- phase transient pipe flow including gas release were presented. The present task is to develop a numerical technique to obtain accurate values from the simultaneous solution of the governing equations while maintaining numerical errors within acceptable limits. Except for the presence of nuclei, the initial flow in a piping system may be essentially single phase. As the transient pressuresdrop below the gas-saturation pressure, gas evolution takes place resulting in a two-phase flow regime. To numerically handle this situation, the grid of characteristics could be used to find explicit solutions, with- out interpolation errors, of pressure, velocity and void fraction. However, as the void fraction grows, the changing wave speed may badly distort the grid making the simultaneous intersection of the three characteristic equations almost impossible. Since we are dealing with a developing two-phase flow, the void fractions are probably small enough such that shock conditions do not occur. The convergence of like Characteristics would signify shock development. If this does occur the Hugoniot relations must be incorporated into the model to handle this internal discontinuity thereby making the programming difficult. The Lax—Wendroff two-step scheme has the capability to spread a developing shock over a number of mesh points by the introduction of 20 21 a numerical viscosity. This scheme is an explicit finite—difference technique of second-order accuracy which requires that the constitutive equations be written in so-called conservation form. Kranenburg [9] found that the amount of numerical damping introduced by this scheme to be acceptable in cases where the nonlinearity does not exceed certain bounds. However, in cases where the pressure dropped to vapor pressure or during shock conditions, additional numerical viscosity was needed to suppress oscillations and instabilities caused by the nonlinearity of the problem. This numerical viscosity was introduced by the addition of a smoothing operation to those nodes in which the solution oscillations were above a prescribed amount. In light of the undesirable aspects of the two above schemes, the method of characteristics with specified distance and time increments is the numerical technique chosen for this study. Since the computing nodes are located at fixed distance locations, this technique has a particular advantage in the integration of the pathline characteristic along a time line. Ames [1] states that this method will pass weak shocks in a pseudo manner. However, interpolation errors resulting in numerical dissipation of the wavefront are inherent with this method if the cerer- ity is changing in the pipe. To minimize these errors, an interpolation technique, similar to that proposed by Vardy [23], was developed which attempts to reduce them to an acceptable level. Evidence will be pre- sented which demonstrates that interpolations taken outside of a dis- tance increment may reduce the numerical damping associated with inter- polations. 3.1 Determination of the Characteristics Equations The governing equations developed in Chapter 2 form a system of 22 three quasi-linear, hyperbolic, partial differential equations. Due to their hyperbolic nature they can be transformed into ordinary differen- tial equations along characteristic lines. To find these characteristic equations, the basic equations will first be put into a more suitable form. With the substitutions __L 122 b2“K+Ee (21) 2 __1- PE. bg - Kg + Ec (22) and eliminating dp/dt from Equations (8) and (12) results in 519'... .l.__.]'_)[ 1 + 1]-122=I‘[_1_2__+_1___] dt b bg b£(l-a) b8 a 8x p2(l-a)b£ pg b a 2 g -l l 1 —— + —— . (23) [ b£(1-a) b8 u] Eliminating %% from Equations (8) and (12) yields - 512-512 ___1___ .1. s- ___1 ___1 (b2 b8) dt dt I: (l-o) + a I' 01(1-01) + pg“ (24) The hydraulic gradeline H is defined as H = P/pgg + z (25) where z is the elevation above a reference datum. Solving Equation (25) for p and differentiating with respect to x and t gives 31)., 25-3.3.2. 3x pig ( 3x 8x ) (26) 23 .32, éfl__3_z_ g 2.11 at 923(2):; at) 013 at (27) Equations (26) and (27) were attained assuming Bo 9. an at 301 z) 3x<d¢ = f (¢O)E¢1 - ¢OJ (52) ¢. Referring to Figure 5, the two compatibility relations, Equations (48) and (50), can be integrated along their characteristics to give the first-order approximations ) (uP - ur) + (HP - Hr) + Era}, - tr) = o (53) r (film 27 l I I 1* .1 Pressure = 4200 psf ‘ 1 I 7000 2000 1000 '3? d} H .U 700 G) (D O. U) m > $2 300 100 50 Figure 4. i. 1 1 I l fi—J 0 0.2 0.4 0.6 0.8 1.0 Void Fraction Nave Speed versus Void Fraction for Bubbly Two-Phase Flow in a One Inch Copper Pipe. 28 Pathline Figure 5. Definition Sketch Showing Intersection of C+. C- and Pathline Characteristic Curves in the x-t Plane. 29 4%) (UP -US) + (H? —HS) +fis(tp - t3) = o (54) where S E=(E§-F-aB/E) A (55) i=(E§-§+aB/E) (56) To improve the numerical accuracy a second-order finite difference approximation can be used. The approximat101, known as the trapazoid rule, is given by ¢1 I f (0)00 = ¢O huh: Ef<¢o) + f<¢l)] E01 - 00] (5?) Integrating Equations (48) and (50) along their characteristics us- ing Equation (57) results in the second-order finite difference equations: 1 a a 1 " " - E [( E) + (‘3) ](uP - ur) + (HP - Hr) +§[Gr + GP] At - 0 r P (58) l a a 1 " " '5 [(E) +(%) ](“P'“s)+(Hp'Hs)+§[HS+H‘P]At=O s P (59) Consider a grid of fixed distance and time increments superimposed on the x-t plane, an; in Figure 6. The objective is to determine values of the unknowns at a new time step. Since the wave speed is much greater than the velocities, approximations to the characteristic Equations (47), (49) and (51) will be made as dx/dt=0, dx/dt=a, dx/dt=-a, respectively. If at times t all values along the pipe are known, then since the character- istics pass through node P, the simultaneous solution of the three 30 .m~m>ewueH we?» use moempmwo uaeeaumam gas: were a-x as“ so mazes aasmerasuaeagu .m masses “Qua. pa + p 31 difference equations will lead to the solution of values at P. Using the first-order Equations (53) and (54) and eliminating Hp yields “p'[< r+( )US-Hir OI In: (”In V + A calm V I__J l g..- E A (film V : _HS - (Er - fis)At]: (60) Eliminating up from Equations (53) and (54) gives ‘6 E -1 -— — - - (—) - (-) ] g __ _g _ g "r [a r a s [us(a)8 ur(a)r Hs(a)8 +Hr( e:- ) - ErAt( g- ) - EBA“ % ) ] (61) r r 8 To obtain second-order accuracy for the velocity at node P, Hp is eliminated from Equations (58) and (S9) to yield [< 1,]‘1 [« r 01.: ‘+[< Second-order accuracy for the head at P can be obtained from either )+2( (film (film (film (film (film a —— — — .— L0 ) ] “s + 2(Hr-HS)-At ( Gr + GP - HS - H?) (62) s P OI In Equation (58) or (59). Solving for Hp in Equation (59) results in 1 HP = Hs +-§' [( ) + ( ) ] (uP - us) - %- EH; + if] At (63) s P rfllm (film 32 The finite-difference equations must now be applied to the various boundary conditions that are encountered. At the ends of the pipe, one of the compatibility relations cannot be used. Therefore, the type of boundary condition supplies the additional information as its influence on the flow. ' At the upstream boundary, if the head is known the velocity can be determined from Equations (54) and (59) to give E a uP=uS+( )(HP-Hs+fi30t) (64) 8 )+( 8 0P=us+2 [( (film (film -1 _ _ ) ] [Hp-HS +% (118 + HP)A t] (65) P If the velocity is known, Equations (54) and (59) can be solved for the head to obtain "12”.“ > (“p-“5"fis“ (66’ (film ) + (g) ](uP-us) -% [is +F—LP]At (67) C s P CHIN At the downstream boundary, if the head is known the velocity can be determined from Equations (53) and (58) to be uP=ur+(g) [Hr-HP-ErAt] (68) r -l a 1 - - ) + ( E ) ] [-H.P + Hr - 2(Gr + GP)At] (69) uP-ur'i-Z [( r P CHIN If the velocity is known then Equations (53) and (58) can be solved for 33 the head to obtain _ .3 - _'— HP - H - ( E_) (uP ur) Gr At (70) (film HP-Hr-% [( )+(%) ](uP-ur 4%[Er+EP]Ac (71) 3.3 Integration of the Pathline Characteristic Attention is now focused upon the integration of the pathline char- acteristic Equation (46), which relates the dependence of the void frac- tion to the head. Making the assumptions K £>>Kg and p£(l-a)>>pg a, the coefficient terms E and F can be approximated as K rig-L, (72) O 8 8 P K 3 E—-—£— - u sin 6 (73) pgpgg Substituting Equations (72) and (73) into Equation (46) yields K K I' _fii.§2. §§.= _B__._ 74 u dt+ngdt pgo "99.381116 () Differentiating Equation (25) with respect to time; Ella—1 SIP... dt pgg dt u sin 6 (75) and substituting into Equation (74) gives K K P J§2+§B-._g__ (76) d dt dt — a pg 34 Furthermore, using the polytropic gas relation the compressibility of a gas can be shown to be K = n'p (77) where n' is the polytropic gas constant. In addition, the ideal gas law can be written as pg = p/kT (78) where k is the universal gas constant and T is the absolute temperature. Using Equations (77) and (78), Equation (76) becomes gg_+'lw.dp = kT F dt (79) a 11 p pa For a polytropic process it can be shown [18] that T= T (80) where po and T0 are a known pressure and temperature at some time. Substituting Equation (80) into (79) yields do 1 .92 -1/n' k To -07).? p =L__a (——————1_1/n, ) 1‘ dt (81) p0 1/n' Multiplying by ap results in k T 1/n' .g 1/n'-1 _ o p do + n' p dp — ( ;—I:IZET ) F dt (82) 0 Equation (82) is equivalent to k To )=(——1—_I7'IF)Pdt .(33) p0 d(mpl/n' Equation (83) is valid only along dx/dt = u. As previously mentioned, since this slope is very large with respect to the absolute magnitude 35 + .. of the C and C characteristics, it is assumed as an approximation, that the equation is also valid along a time line. Integrating Equa- tion (83) from point C to point P of Figure 6 yields . P up ppl/n - ac pclln' = (k To po(1/n'-1)) [C F dt (84) Thelast term on the right-hand side of Equation (84) can be approximated as P th=lAt(I' +I') (85) C 2 C P Substituting Equation (85) into (84) and rearranging yields l/n' l-l/n' PC k To pc At oP=(--) [aC+PC ('13—) 7(rc+rp) (86) Equation (86) can be used to obtain values for the void fraction at a new time. Together with the first-order Equations (60) and (61) and second-order Equations (62) and (63), the three unknown variables can be solved at any interior point provided that the variables are known at the previous time. 3.4 Interpolation Scheme If the wave speed is constant, the method of specified distance and time increments can produce an accurate solution without interpo- lation damping [19]. With reference to Figure 7, if a constant wave speed is present throughout the pipe, the grid could be arranged such that points r and s coincide with points L and M. Since the unknowns have been solved at points L and M at the previous time step, the characteristics transmit exact information from points L and M to 36 point P. However, if points r and 3 do not coincide with L and M, in- formation from r and 3 must be transmitted to node P. An approximation to the unknowns at r and s can be found by interpolating between points L, C, and M. This information is then used to obtain unknown values at P. This interpolation leads to numerical damping and dissipation of the wave front [19]. In a pipe where the wave speed is changing, it is impossible to avoid interpolations when using a fixed grid. As the celerity changes, the location of r and s will change. In the conventional interpolations scheme with changing wave speed, the grid is arranged such that points r and s are within the points L, C and C, M. In this manner the Courant condition is always satisfied, since a A t Ax. 5 1 (87) The linear interpolation equations as outlined by Lister [12] for the above case are 0r = 06 - c'(¢c - 0L) (88) ¢ = 0C5 c'(¢c - 0M) (89) s where ¢ is the variable being interpolated such as velocity, void frac- tion, or pressure head and where aAit c' = Ax '; 0 s c’ s 1 (90) The term C' indicates the position of r and s. As c' approaches zero the interpolation points approach C and as c' approaches one, r and s approach L and M. A value of c' = 1 represents no interpolations since the characteristics then coincide with points L and M. 37 Figure 7. Definition Sketch Illustrating Interpolations ‘taken within One Distance Increment. low a I high a’ g T ./ “At |\ _L x/L -O h I x/L -l.0 Figure 8. Definition Sketch Showinq Possible Interpolations within One Distance Increment Associated with Variable Wave Speed. 38 With this scheme the maximum wave speed that is to be attained con- trols the grid size. This usually requires a very small grid size and fixed At/Ax ratio. Figure 8 depicts this situation. It characterizes a.pipe which at one end exhibits a low wavespeed while the wavespeed at the other end is near its maximum value. This condition may exist due to gas release in liquids at regions of low pressure. It is apparent that severe interpolations will occur in the low wave speed regions. In these regions a larger time step is desirable but is prohibited due to the limiting condition of Equation (87). However, as suggested by Vardy [23], it is not necessary to stay within one distance increment. It is possible to increase the time step such that points r and 8 lie outside the nodes adjacent to node P. Referring to Figure 9, interpolations for points r and 3 may now be taken between L and C', and between C" and M. We can now define n as the number of distance increments to point L or M: n = IFIX (aAt/Ax) + l (91) where IFIX (aAt/Ax) is the integer equivalent of aAt/Ax. Again defining C to be the relative position between the nodes L and C' or C" and M: c=aE-n+1 (92) The interpolation equations are -8- II ¢Cv ' C(¢Cv - ¢L) (93) (P = ¢Cn - C(‘PCH - ¢M) (94) Notice that when n = 1, equations (92), (93) and (94) reduce to the 39 At L / cl c C" \ . 12—4 a S AX *- aAt Figure 9. Definition Sketch Illustrating Interpolations taken Beyond One Distance Increment. ‘low a x/L =0 high a __T__ At x/L 81.0. Figure 10. Definition Sketch Showing Possible Interpolations Associated with Increased at/Ax ratio for a Variable Wave Speed Situation. 4O conventional interpolation Equations (88), (89) and (90). Application of an increased time step to a system which experiences regions of low and high wave speed is shown in Figure 10. The grid is now closer to the preferred At/Ax ratio in the low wave speed region. In the high wave speed region the characteristics span over several distance increments; however, interpolations can still be made between the nodes closest to r and s. In light of the above discussion, perhaps by increasing the time step and allowing interpolations to take place outside of the first distance increment, reduced interpolation damping may occur. To investi— gate this, the following numerical experiments are provided. A pressure transient is simulated in a 400 foot horizontal fric- tionless line by the instantaneous closure of a valve. A constant head reservoir is situated at the other boundary. The initial velocity is 2.0 fps and the reservoir elevation is 500 feet of head. In the analysis a constant wave speed is used. The fixed grid is adjuSted such that the conditions described in Table l are simulated. Table 1. Important Variables Used in Numerical Experiment Comparing In- creased At/Ax Ratio with Conventional Ratio Number of Computing Figure sections n a time (sec) 11 10 l O.25,0.50,0.75,1.0 21.7 11 10 2 " 8.5 12 20 1 " 83.6 12 20 2 " 27.1 13 4O 1 " 325.4 13 4O 2 " 99.3 41 In summary, interpolation damping was directly compared for inter- polations taken within the first distance increment (n=1) and interpo- lations taken within the second distance increment (n=2). The simulation was run for three different number of sections of pipe with four C values. Figure 11 represents the total length of pipe divided into 10 distance increments; Figure 12 represent 20 sections; and Figure 13 is the 40 section case. The dimensionless head at the valve is plotted against the dimensionless time. The exact solution in each figure is the n=1 and c = 1.0 situation, since this represents no interpolation. The amount of damping that occurs for each case can be visually deter- mined by comparison with the ; = 1.0 case. In observing the n = 1 cases only, it is evident that the amount of interpolation damping decreases as C approaches one and as the number of sections of the pipe increases. The damping is very severe in the ten section simulations and is also undesirable in the 40 section runs. In viewing the n = 2 cases, an improvement is seen. In every in- stance, the n = 2 interpolations produced much less numerical damping as compared to the n = 1 case. As can be witnessed in Figure 13, the damping associated with 40 sections is acceptable. Also, it seems that the degree of interpolation plays a small role on the damping as the graphs for the different C values nearly coincide.l The results are clear. By increasing the time step and interpo- lating outside of the nodes adjacent to the node being solved, a re- duction of the numerical damping associated with interpolations results when compared to the conventional grid. This also has an added benefit of reduced computing cost associated with the larger time step as can be seen from the computing times tabulated in Table 1. 41:60 I-BO 0100 Figure ll. 42 Comparison of Pressure Response at the Valve for n=l and n=2 with the Pipe Divided into l0 Computing Sections. 43 rl=| |./ha:-=2()’ C \ V) m (‘5 A1" I 1 M M n82 L/ax'ZO C I 1.80 I I T I I I I I 0.00 0.50 1.00 1.60 2.00 2.60 3.00 3.50 #3200 Figure 12. Comparison of Pressure Response at the Valve for n=l and n=2 with the Pipe Divided into 20 Computing Sections. 44 n-l Llax=40 5' . V) /\ m g; m i (l M n=2 Llax=40 g '— 1:00 1 1 20 M 333 PRESSURE than 0.40 I I I I ' ‘ . .oo 0.00 0150 1100 1160' 2.00 2.60 3.00 a so 4 Figure 13. Comparison of Pressure Response at the Valve for n=l and n=2 with the Pipe Divided into 40 Computing Sections. 45 A numerical experiment was also conducted for a variable wave speed situation. In this experiment, a pressure transient was simulated in a 400 foot horizontal lucite pipe of one-inch diameter by an instantaneous valve closure. The flow regime was a bubbly, two-phase, air-water mix- ture with a void fraction of 0.0354 as referenced to atmospheric pres- sure. Adibatic expansion and compression of the air was assumed. The initial velocity was 5.0 fps and the upstream reservoir pressure was 55 psia. A friction factor of .022 was used. Based on the maximum wave speed attained during the transient, the time step was adjusted such that the n = 1 and n é 2 interpolations schemes were simulated. The wave speed varied in the simulation from about 440 fps to 1050 fps. Therefore, interpolations ranged from g = 0.42 to 1.0 for the n = 1 case and for the increased time—step situation ranged from C = 0.84, n I l to C = 1.0, n = 2. In addition, the Lax-Wendroff two-step scheme was used to model the identical transient [13]. In each case, 40 computing sections were used. Figures 14 and 15 are the plotted results of the pressure at the valve and midpoint of the pipe for the three runs. In comparing the results for the pressure at the valve, the n = 1 and the Lax-Wendroff results show a nearly identical pressure trace while the n = 2 case exhibits a slightly sharper response with less smoothing. In viewing the midpoint pressures, the Lax-Wendroff scheme pro- duced slightly more smoothing than the n=1 case, with n=2 displaying the least amount of smoothing. From this experiment it is evident that the n g 2 case produced less numerical damping of the pressure computations than either n = 1 or the Lax-Wendroff scheme. If one were to perform a formal error analysis similar to that used by Liggett and Cunge [11], less damping of higher frequency components would be seen with n é 2. It seems LAX-WENDROFF 200k Po MOC n-l 200k Po MOC n52 ZOO-ILPO o 2 4 é a {o TIME IN SECONDS Figure l4. Comparison of Pressure Response at the Valve for Two-Phase Flow with the Lax-Wendroff Two-Step Scheme and the Method of Characteristics with n=l and n52. 47 LAX-WENDROFF 200 It Po MOC n=l 2000: Pa J T MOC n52 200k Po J 5 a 4 a a no TIME IN SECONDS Figure 15. Comparison of Pressure Response at the Midpoint of the Pipe for Two-Phase Flow with the Lax-Wendroff Two-Step Scheme and the Method of Characteristics with n=l and n52. 48 reasonable to conclude that in a changing wave speed situation the nu? merical damping can be reduced by taking At/Ax ratios which are larger than that dictated by Equation (87), thereby producing a more accurate solution. Since Martin et a1 [14] and Kranenburg [9] found the numeri- cal damping associated with the Lax-Wendroff scheme to be acceptable when comparing computed results to experimental data, it would seem that the method of characteristics as applied to the fixed grid with increased At/Ax ratios would be a competitive scheme for dealing with changing wave speed situations. In addition, this scheme could be modi- fiedtxiaccomodate a changing time step which would minimize the amount of interpolation error that occurs over that time step. Vardy [23] also showed the advantages of an increased At/Ax ratio in a numerical experiment in which an air duct is connected to two reservoirs. The pressure was reduced in one reservoir and the duct midpoint pressure-time history was observed. Using the convential grid he found an accurate solution with an acceptable amount of interpola- tion error was simulated at a Ax = 0.4 meters. He then increased Ax to 1.6 meters and by using a At/Ax ratio greater than that permitted by Equation (87) obtained a result that was indistinguishable from the Ax = 0.4 meter case, thereby greatly reducing the computing cost. When interpolating with increased At/Ax ratios, attention must be paid to points which are near the boundary. Figure 16 shows the char- acteristic as it intercepts the boundary. Values at point Y are needed to solve for values at point P. Interpolating between points 2 and X can be done to accomplish this. The interpolation equation for this case would be _ _ n"Ax _ ¢Y ‘ Iz At a (¢z ¢x ) (95) 49 where n" is the number of distance increments to the boundary. A more accurate approach could be taken as follows. Interpolations between points C" and X could be made to attain values for 3. Values for Y could then be solved by writing the compatibility relations be- tween 3 and Y. 3.5 Modeling Aspects A specific numerical technique has been described which will simu- late two—phase hydraulic transients including gas release by using the method of characteristics. An algorithm has been devised and by the use of a high speed digital computer, values for the velocity, pressure and void fraction along the pipe can be calculated. The purpose of this section is to point out some important aspects of the numerical model. To minimize numerical errors, the equations of second-order accuracy will be employed. This entails solving the first-order equations to find initial values of head, velocity and void fraction. These values are then used in the second-order equations to obtain more refined answers. Also, to reduce interpolation errors, an increased time step will be used as proposed in the previous section. It is believed that by taking these measures the numerical damping is reduced to an acceptable level. The gas release aspect will be handled in the following manner. Whenever the pressure at a node is below the gas saturation pressure, some mass of gas is allowed to evolve out of solution in a distributed manner throughout that computing section. This is accomplished through the gas generation term P, which is present in the compatibility rela- tions. F is defined as the time rate of change of mass of gas per unit volume of mixture. It is a function which recognizes the time dependent 50 Z Y C" X S e n" AX 'l Figure 16. Interpolations taken at Boundary with Increased At/AX Ratio. \ / c' L/ UL¢—-— +_U Figure 17. Definition Sketch of Vapor Void Occurring at a Computing Node. 51 nature of gas release. However, the rate of gas release is not well documented in the literature. In Chapter V the results of a labora- tory experiment will be presented in which the rates of gas release in supersaturated solutions are determined. These results are used to approximate the time dependent nature of gas release occurring in the pipe through the selection of a suitable gas release function. In modeling column separation phenomenon, which occurs when the pressure is calculated to be below the vapor pressure, the pressure at that node is set to vapor pressure. By bringing the C- and C+ char- acteristics up to this point, the only unknown in their respective come patibility equations is the velocity. Therefore, a velocity can be calculated for each side of the node as represented in Figure 17. With these two velocities, a vapor void can be accounted for by a simple continuity balance of the flow leaving the node. Using the first-order finite difference Equations (53) and (54) the velocities on each side of the cavity are E. a )r Gr At + ur (96) uL g (:§')r [yr - HP] - ( _ E - E - u-(a)S [up HS]+(a)HSAt+uS (97) 8 Using the second-order finite difference Equations (58) and (59) the velocities on each side of the cavity are -l "r?- [<§> +<§)] [-H?+Hr-% (5H5?) M] M. <98) C r C P u=2[(9—) +(3)]-l[ 1(E +-) A + (99) ES E P “p‘Hs+‘2' s “p t “S 52 Since velocities are known from the previous time step, a second-order approximation to the cavity volume can be made: 1 (Cavity Volume)t+At (Cavity Volume)t + 2 AAt [:(u-uL)t+At +'(u-uL)t:] (100) Using Equation (100) the size of the vapor void can be monitored. The pressure is kept at vapor pressure until the cavity closes. When a cavity exists at a node, the question arises as to what velocity should be transmitted along the characteristics to the next time step. In this study, the averaged value of u and uL is sent along the C- and C+ characteristics [16]. An alternate approach would be to transfer uL along the C“ characterisitic and u along the C+ characteris- tic. However, the former method seems to admit the presence of some momentum transfer across the void and does not produce the instabilities that are sometimes associated with transmitting two velocities from the same node. CHAPTER IV LABORATORY APPARATUS AND INSTRUMENTATION The observation of gas release occurring under controlled conditions and the gathering of transient pipe data which exhibits gas release are essential to this study. The following is a detailed description of the experimental apparatus and procedures used to obtain the required data. 4.1 Gas Release Experiment As previously cited, the time rates of gas release occurring in a supersaturated liquid has not been extensively researched. Schweitzer and Szebehely [l7] conducted detailed experiments to determine the amount of air evolved from water in a closed container with a constant air- liquid volume ratio. Their measurement technique was to monitor the in- creasing air pressure above the free surface and relate that information to the amount of evolved air. They found that the air release rate de- cayed exponentially with time for a step decrease in pressure and was dependent upon an experimentally determined half-life of evolution. They also found that this half-life was dependent upon the air-liquid .volume ratio in the container. In a developing two-phase pipe situation, ' there is essentially no initial air volume except for the presence of nuclei. Indeed, as gas is released, the air-liquid volume ratio would change with time. Therefore, there is some question as to the direct use of Schweitzer and Szebehely's findings in the present study. Alter- nately, the equations describing the mass transfer of gas into a single 53 54 bubble as developed by Parkin and Kermeen [15] or Van Wijngaarden [22] could be used. However, several additional unknown constants, such as the number of nuclei present and their radii, would be introduced. To obtain a more realistic picture of the manner in which gas is released in a filled, closed container under turbulent conditions, a bench-top experiment was conducted. From this, gas release rates were obtained that gave an order of magnitude approximation to the gas genera- tion term appearing in the compatibility relations. 4.1.1 Description of Apparatus and Instrumentation A schematic of the experimental set-up is shown in Figure 18. The test section is composed of a lucite cylinder of 7.5 inch diameter and height of 14 inches. Extending vertically into the cylinder is an MTS- actuated rod to which has been attached a series of three, 1/4 inch square mesh grids spaced at 4-inch intervals from one another. The grid system, in which one grid is shown in Figure 19, can be oscillated sinusoidally over a range of frequencies and amplitudes to produce a de- sired fluid agitation, or grid turbulence. Attached to the test section by means of 0.25 inch copper tubing is a reservoir in which water can be saturated with 002 or air at elevated pressures. Also attached to the test cylinder is a series of three vertical lucite tubes of 1.0, 1.5 and 2.0 inch I.D. The test section can be directly connected to any one of these tubes by a quick-opening valVe. With the test cylinder filled with water saturated with gas at some elevated pressure and isolated from the reservoir, quickly opening a valve to one of the lucite tubes would cause a step decrease in pres- sure to atmospheric in the test section initiating gas release. As bubbles start to form they will displace the fluid volume which will be 55 r- r— FIEASIIIING TUBE RESISTANCE WIRE T’r—ms wmauuc munch [Ill/1 :smmou omo meme / 35.56:— X I105": F—Lucne cvuuoen 'o ////e///// / / , - some rooms 1"- QUICK mm vuve c. VALVE l I /////////////////////////////g Figure l8. Detailed Schematic of Gas Release Experiment 56 reflected in the water surface rise within the measuring tubes. By monitoring this water surface rise, a gas release-time history can be obtained. Figure 20 shows an overview of the entire set-up while under test conditions. The reservoir can be seen in the lower right. Figure 21 is a close-up of the test cylinder under test conditions. The measurement devise to record the water surface level in the lucite measuring tube consisted of a 0.001 inch diameter tungsen resis- tance wire looped through the interior of the tube. This wire then served as one leg of a Wheatstone bridge. As water ascended in the tube, the resistance associated with this leg of the bridge changed, thereby creating a voltage potential across the bridge which could then be re- corded. The bridge was activated by a Sanborn carrier-preamplifier, which also amplified the response and recorded the signal on a strip— chart recorder. A Van Slyke apparatus, shown in Figure 22 and schematized in Figure 23 was used to monitor the dissolved gas content of the water. The apparatus was modified such that dissolved gas contents up to 200% by volume could be measured. To check the accuracy of this device, the results of several tests in which CO2 was bubbled through the water until equilibrium conditions existed were checked against Henry's equilibrium law. Figure 24 shows this comparison and exhibits the close agreement between the measured and theoretical value. 4.1.2 Experimental Procedure The actual gas release experiments could only be conducted after the accuracy and stability of the wire gage was assured. Since the resistance characteristics of the gage varied from day to day, the gage was physi- ' cally calibrated before and after each series of tests to determine if 57 Figure l9. Close-up View of an Agitation Grid. V 9. N R E ‘x '5 "6‘ g r C1 i “'5 ’2 .‘A 9;: A if 4' A? .*l . r s .‘ r1! ’1 If, a Q '1. 4 ' l L I ‘ I - Figure 20. View of Gas Release Experiment under Test Conditions. 58 .mzumemae< mcweamemz mew uo>pommwo oxzpm ee> mo som> .NN acacia .uewpmoh _mapo< mavens cmucwpzu amok op_o=m mo 3ow> azuomopu .L....w....»..‘ 1.14.... v ....... .‘v. S..\ K .13....“ ., by. ._N messed 59 I ¢{1%400 mm ’ TO RESERVOIR /MANOMETER t H9 T RESERVOIR THERMOMETER (‘6) GLASS WATER [JACKET iCHflfll. "TYGON' \Tuemo a) VJ \K'TYGON" TUBINGV Figure 23. Van Slyke Apparatus Schematic. 60 any variance occurred. Small changes did occur; however, compensation for these changes were considered in the plotting of the data. A typi- cal calibration curve is shown in Figure 25. As can be seen, the cali- bration is nonlinear. A typical run consisted of the following procedure. The water in the reservoir was saturated with CO2 by bubbling the gas through the water for a 40—minute to one-hour period at an elevated pressure accomb plished by the adjustment of a needle valve. A sample was then intro- duced directly into the Van Slyke apparatus and the gas content tested to insure that the water was saturated at that pressure. The valve between the test cylinder and reservoir was then opened and water was introduced into the test-section. During this time, both the test cy- linder and reservoir pressures were above the gas-saturation pressure, preventing any release of gas during the transfer. The gas in the test cylinder was bled out through a needle valve in the top of the cylinder during the filling. Once filled, it was then isolated from the reservoir. The grid system was then oscillated at a given frequency and amplitude. Next, the quick acting valve connecting the test cylinder to a measuring tube was opened, resulting in a step decrease in pres- sure to atmospheric. The water was then in a supersaturated condition and gas evolution began which displaced the fluid volume resulting in the rise of the water surface in the measuring tube. During this time the wire gage was monitoring the water surface elevation. Figure 26 displays an actual experimental record. 4.2 Hydraulic Transient Experiment 4.2.1 Description ongpparatus and Instrumentation A pipeline has been assembled which consists of 400 feet of coiled, 61 200 - g ,__ 150 L 2 LIJ i.— Z O U ., 100 9 S o HENRYS LAW g 5' o - MEASURED VALUES m g} 50 - C: 0 I 1 I 4 0 5 10 15 20 PRESSURE [PSIG] Figure 24. Comparison of Van Slyke Measurements versus Henrys Law. 30 - N ' O I —.l O ELEVATION [INCHES] 0 2 4 5 8 10 CHART DIVISIONS Figure 25. Typical Wire Gage Calibration Curve. 62 .peoeweoaxm ommopmm mac n merino one» acmezmmmz cw ommm cowue>mFu some: memzoem month memo Fmowaxp .mm mesmwm delfiww Ith.II|W. II. ICIW ,. ..H..... .. . .71.. . no. ... . . .."l ..fl. . . . .. . —. . ...ls. . II rIIIIv- ‘1 nt.l 7' .L ".‘i.;.‘ l -0 I I I o--.g-—.. I I I .1 I I I o I I —-¢ I I I —-—< . .. I o - - I - I- -..--- A o———-r b I . n I I «r-..1_'--.- . .... w. _r . . -. ., - m. .. 7“ .... H. , . . 3.2. . ....._.;:..:y.. -H ..H .... g 471* ... ....H... ...M .7. .. 33 .1: .... . , H . .3... ... r if . II to IIL. .1 1 1| b .v . y . u I _ _ . _ .. . j . _ . _ . fl 1 _ 0 _ u _ E. . . . .u — . . _. _ . _ . _ . _. . . _ . 1 _ A _ — _ _ .. . . . . . 7:..- ._ _, _ . __i. . _. . . . I .. . A ~ . . . ..I. . .. . w, * m. “I .1! .. . . x. .. . a): 2.4%.. r .1. W 1-. _1 .w JIJIIHIIHILIIIIIIIII . 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T . u .U n ., . _ .1 .._ m ...l. U n. H . u m .u.-..:.. A . . x 7.... w _. .... ... T .. . A 3.. L. 1.; . . I - - I-I -1144.) 1.1.3.0- .. . ..-H II... I ”.1.-- IWTI 4 I . -INI .--II - .. I“- - .iI.-1..-I...- -1..- IIIII.--- I I- ... . m 3 it . i... .... a a . . is.-. .. . L . M w... .. . . .1 u o I --l II-_..I.--N.I.-»II NI : TI .T.I ..Ifl. I4. I. -I J) ...»- W. TIMIIIHTI 4-1.-- . . .I I..- II--- I -. I .. -... m i u .g ...n. w . r. a “ham ...m u ...... : .. H. T g” 23:... m“. ..... . m.” ..H ..m N H. w w u: 61. I .-II-..J.F.. ”I- -I fl-..~.-17.31. . ... . fl. - ..:.; .... ......I. I .. : . .mi . w. . o-......-I. l-«.-I.I a... --Il......-A-In -. p .. . ... ... .. .. . . . ... a . . .. .... .... ... . .. . . ....u.0. ”in «n i. A 7.. x2... .LW...... :17: 3...... m ....”q-..uw...7.:. .. Hindi.» .... - ”-.....n .4171. IqianIIi. a i: .. H.... . ..m. In... ..... a . . MILAJITWIW .. -7 . . .. . IIIJIIAIIHIJIm WT. ...»... 1.. . a. ...,” . 1.51%...- 21.1%...” . .n... . 3 $21.17; an ..... . ... . u ....H 22”.-.“ . * x ...-H: ”-1 L31 T.1 I v .. .. bur F v. t ......h ... I.>v.... .IH .ICI: H“ .IMIW. $.II.9.'.IPIU-.. .v I ...M I . I u n ...?...n p. I ..... LIPNNHIII ... . f- ... m 3.. l. ...w m in? ... 5. ms“. .5 1......w....m.fim. - ... ...- new. w P. .. 1 . 1L.» .. - f P1 FM -. 3t .. Us knittt» T». 4?. $3.3? 3. :m a . 3” H? .4. ”J W 1 MI. . r 3*; v “Twitmni I. 7 ..I. .. 63 one-inch I.D. copper pipe. At each end of the coil has been connected 12.2 foot sections of one—inch I.D. lucite pipe for visualization pur- poses. Reservoirs located at the upstream and downstream ends of the loop, having the capacity to be pressurized, complete the system. Figure 27 is a schematic of the apparatus and Figure 28 shows the coil and lu- cite sections. The upstream reservoir has been constructed such that the water can be saturated with C02 or air. The Van Slyke apparatus was used to determine dissolved gas contents. A quick-acting solenoid- activated valve which can be located in either an upstream or downstream location was used to initiate the transients. Measurements of the clos- ing action of the valve showed a closure time of 20 to 30 milliseconds. Velocity measurements were made by the use of an elbow meter attach- ed to an inverted air-water manometer. The elbow meter was calibrated volumetrically. Figure 29 shows the elbow meter leading to the manometer and the valve in the upstream position. The calibration curve for the meter in the upstream position is shown in Figure 30.‘ To determine the frictional characteristics of the pipeline, piez- ometer taps were installed at the ends of the copper coil and lucite sections. Steady state losses associated with a range Of flows were found using manometers. Figure 31 shows the measured friction factors versus Reynolds number for the copper and lucite pipes. Five transducer mountings were placed within the system to accept the placement of flush-mounted, miniature Sensotec strain-gage pressure transducers. The transducers were activated by a four channel Sensotec U-4 amplifier. The tranducers were statically calibrated within the system. They were also temperature compensated by the application of a thin layer of silicone rubber over the diaphragm. Figure 32 shows 64 .mcmpmawa pcwmmcmsk Popcmsmgmaxm we umpmsmgum nwamumo Sum.— .5 3a .5 Eoo.~£ EOEUmm mtuag fl” W< w“. < :53. an //I|\\\ Eoc.mu mamUDOmZ._<> th n... <— Euvm. mu 0 Ea. 5...; ZO:.Um_m mCUDa ZOZUmm mmmmOU fiN' q .NN ogsmwu /////////// // ou I I 1 6 ® / a ~:O>mmmmm OmNEDmmmmm / mmOZZ>U m<0 ....- b ‘_- 5- 3:‘ “1., . i 3 E 1k”-- { , ...“. --s...‘.‘. FigUre 28. View of Copper Coi] and Lucite Pipes. Figure 29. Ciose-up View of Elbow Meter and Solenoid-Actuated Vaive in the Upstream Location. 66 We o6 .m>g=u cowumgamFmo swam: zonpm ”veg az~ag=u mucmpmwmmm zap; .Pm mgamwm «mg: 2.525”— o8.S 82 2: d d 4‘ d H q 1 J u d d + q 1 1 q d d1 d d 1 J u —.°o° ... f o .. 8.9 O. '3‘ 'x O u l N . I o. . T . I. ma 9 j 0 o I I O In . 1 7: ...—m: 5.3:.— ..u 0 mm: 5&8 co 7 . mec P D D P p b P b rb r p r D P p h b r. D p b I 801W! NOILDIEH Figure 32. 68 :.r0"'. .:."." |-.L - . :0... . v0".o- ..‘.. o .-.b¢ ’ oo“'. a.". Q .,o". y ..:.. i. .... .4 0‘ .0 av: ... 3...:30 0 .“' .4 .0 . o . .. -n Close-up View of Miniature Strain Gage Transducer Mounted in the Lucite Pipe. 9. “-Q ‘ ', '1“ - b ‘,‘ ‘--‘ \L‘“- ‘Nra L.‘. | ! . . «\‘I \§ ‘ ‘2;7\“v“ "x .‘ . , 3:?! 5“ 9. “ . ES” ,. I \ "“‘ ‘ IR": K 3 %§‘ ‘ ' 1:: rs. “ 1 m “ . _ i‘g‘-"“"_“\-.— 75:? fifi‘igfig‘ - ‘ "tins-5“}: ‘ ~.~ 'cx'fi' ‘1 N tuv. 1‘»? \ _ 'W’V .\_‘—‘$,\\ " ’ \ ‘xfiku 5* Figure 33. Photograph of E1ectronic Data Gathering System which includes Amplifier, Storage Osciiloscope, Mini-Computer, D/A Converter and X-Y Piotter. 69 a transducer mounted in the lucite pipe. A Computer Automation mini-computer was programmed to sample the amplifier output voltages every 10 milliseconds and store these values on a floppy disc. Upon command, these voltage values were then sent to a Hewlett-Packard X-Y recorder to obtain a permanent, visual record of the experimental run. The transducer signals were also sent to a Tek— tronix storage oscilloscope for quick visualization purposes. Figure 34 is a schematic of the electronic set-up and Figure 33 shows the amr plifier, oscilloscope, mini-computer and X-Y recorder. 4.2.2 Experygental Procedure Carbon dioxide gas was bubbled through the upstream reservoir with the reservoir pressure adjusted by the throttling of a needle valve. water samples were directly introduced into the Van Slyke apparatus to determine the dissolved gas content. When the desired degree of satura- tion.was obtained, the reservoir was kept at a constant pressure by a pressure regulator connected to the building air system. Records of the room and water temperature along with the atmospheric pressure were also maintained. Transducers were installed at the valve and the midpoint locations of the pipe. The calibration of the transducers was checked periodically throughout the testing and were found to remain stable with very little drift. Steady-state flOW'WaS then established in the system. The desired velocity was attained through the adjustment of a throttle valve. Steady- state pressures throughout the pipe were always above the gas saturation pressure of the water; therefore, the initial flow was essentially one- phase. Considerable care was taken to insure that all gas bubbles were 70 .emumam newcwcamu memo owcoauumpm mo uwpmemgom .em mczmmm _u _ mmemm>zou «meaazou -Eamoomm I _ r .3 , -HzE _ . _ mmHmHmmz< uaoumomgHumo umm>Hmo mmmozomzzou a\< 71 2.u-'fl L , um. All, ‘ q‘o'lf| ‘0'. .cmuuopa >ux we» so kucoumm mm m>~m> um mmcammmca acmwmcmgp mo mung» memo Pmauo< .mm mesmec I 9 .--... 4-4-44. 0'- “f. “...” IIO‘D— 72 bled out of the system before an experimental run was conducted. The mini-computer was then activated to start sampling moments be- fore the valve switch was thrown. The mini-computer next sampled the voltage output from the transducers for 8.05 seconds. The stored data could then be transferred to the X-Y plotter. This data-gathering system and procedure produced accurate results and provided large plots with good resolution of the transducer output. Figure 35 shows an actual data trace from the X-Y recorder. In the figure, the first 4.025 seconds of the transient are plotted below the second half of the trace. CHAPTER V COMPARISON AND DISCUSSION OF EXPERIMENTAL AND THEORETICAL RESULTS The culmination of the present study on column separation and gas release in highly saturated fluids takes place in this chapter. The first section deals with the results of the gas release experiment and the influence of these results on the modeling of the gas release phe- nomenon. Next, the observed experimental results obtained from the hy- draulic transient study along with the associated computer-simulated, theoretical comparisons are presented. A discussion of these comparisons and the effects of gas release is included. 5.1 Gas Release Experiment As discussed earlier, the purpose of these experiments were to ob- tain, in a preliminary way, an order of magnitude approximation to the gas production rate to be used in the pipe transient modeling. These experiments did provide insight into the gas production rates and the time-dependent nature of gas release occurring in a closed container with grid turbulence present. However, one should be cautious in directly applying these results to an actual pipeline situation due to differing turbulence conditions and possible scaling differences. In this light, the gas release rates used in the model are recognized not as a precise description of the gas release phenomena in turbulent pipe flow, but rather as a more refined approximation than what is presently available in the literature. 73 74 5.1.1 Conditions Investigated To gain an understanding of some of the important variables which may influence gas release it is of benefit to review some present theory. In an attempt to improve on an initial study by Parkin and Kermeen [15], Van Wijngaarden [22] investigated convective diffusion of gas into small bubbles attached to a submerged body. As reviewed in Chapter One, assume .ing slow growth and neglecting viscous forces, Van Wijngaarden describes the time rate of volume change for a single bubble to be b kT dN 1. dt pf-pv + 3(CI/R) (101) The term dN/dt is the molar flux of gas into the bubble. The con- centration gradient of the gas in the fluid provides the driving poten- tial for diffusion. This gradient is caused by the difference in concen- tration far away from the bubble c; and the equilibrium concentration c'o at the bubble surface. The mass flux is related to the component 8c'/3n of this gradient normal to the bubble surface by SE.; , 3c' dt f D 3n ds (102) where D' is the diffusion coefficient and ds is a surface element. Attention is now given to the calculation of dN/dt. The concentra- tion distribution in the field can be described by the diffusion equa- tion. For the bubble size and growth rates studied by Van Wijngaarden, the diffusion process was approximately steady; therefore, the governing 75 equation would be (103) where v is the velocity vector in the region around the bubble. For the solution of Equation (103) with the boundary conditions c' = c'0° at in- finity and d' a c; at the bubble surface, the velocity distribution around the bubble must be known. This presents a very complicated prob- lem in which some approximations must be made. One particular model proposed to represent this mass transfer into the bubble is described by Levich [10]. In summary, Levich argues that there is no velocity bound- ary layer on the bubble in the usual sense. Therefore, assuming invis- cid flow, the velocity distribution around the bubble can be calculated. Also, he observed that the diffusion takes place in a narrow region around the bubble as compared to the bubble radius; therebysimplifying Equation (103). Utilizing this observation and using the inviscid flow assumption to calculate a velocity distribution from the oncoming ve- locity u'; Equation (102) can be integrated to yield' $15 = Me; - <=,'.)<2wR3u'I>')L5 (104) Using Henry's Law, the concentration difference 9; - 9g can be re- lated to the difference in the saturation pressure of the fluid and the instantaneous pressure. Letting Ap be equal to this pressure difference, Equation (104) becomes 33:- = 48Ap(2nR3u'p')1i (105) 76 where B is the Henry's Law proportionality constant. Equation (105) was also used by Kranenburg [9] in modeling gas release during hydraulic transients. Equation (105) describes the mass influx into a single bubble. It states that the bubble growth rate is directly proportional to Ap, R3/2, (U')15 and (D')%. Applying Equation (105) locally to every bubble within a specific volume of fluid could be used to estimate F; the total gas production rate per unit volume of mixture. Therefore, theI‘term.would also be expected to be proportional to the above variables. In this study, no attempts were made to correlate the observed gas release data with Equation (105) . In the laboratory, the parameters that were varied were Ap and the turbulence level. The turbulence level was varied by the adjustment of the frequency of grid oscillation, thereby changing the convective trans— port around the bubble which was related to changing the slip velocity u'. A series of tests were then conducted as described by Table 2 with w being the frequency of oscillation of the grid system. water was saturated with CO2 gas at pressures of 10 psig, 20 psig, 30 psig and 40 psig. In the test cylinder, the pressure was then dropped to atmospheric conditions. At each Ap value, five experimental runs were made at dif- ferent frequencies of grid system oscillation. In each case, the full amplitude was maintained at a constant 0.5 inches; thus, turbulence levels were solely dependent on the frequency. The data was then hand-smoothed such that the numerical 77 Table 2. Summary of Gas Release Experiment. Full Water Atmos. Run Ap w Amplitude Temp. . Pressure No. (psi) (Hz) (in.) (°c> . (in. Hg) 1 10 0 0.5 18.5 29.25 2 10 1 0.5 18.5 29.25 3 10 3 0.5 18.5 29.25 4 . 10 5 0.5 18.5 29.25 5 10 7 0.5 .18.5 29.25 6 20 0 0.5 18.0 29.25 7 20 1 0.5 18.0 29.25 8 20 3 0.5 18.0 29.25 9 20 ‘5 0.5 19.0 29.48 10 20 7 0.5 19.0 29.48 11 30 0 0.5 19.0 29.48 12 30 1 0.5 19.0 29.48 13 30 3 0.5 19.0 29.48 14 30 5 0.5 19.0 29.48 15 30 7 0.5 19.0 29.48 16 40 0 0.5 20.0 29.32 17 40 1 0.5 20.0 29.32 18 4O 3 0.5 20.0 29.32 19 40 5 0.5 20.0 29.32 20 40 7 0.5 20.0 29.32 78 differentiation of the data to determine actual P values would also produce a somewhat smooth curve. This numerical differentiation was done by computer using a simple weighted differencing technique between data points. Figure 36 shows the experimentally measured void fraction and the associated time rates for Ap = 10 psi. Figures 37, 38, and 39 are the results for Ap - 20, 30, and 40 psi, respectively. 5.1.2 .Discusgigngof Results and Modelingglmplications Before discussing the analysis of the gas-release data, possible experimental and measurement errors that may have occurred are examined. These errors were primarily due to the design of the particular apparatus and measurement technique. Some possible sources of error are: 1. Upon the quickropening of the valve to depressurize the test cylinder, an initial surge occurred in the measuring tube due to the compressibility of a small amount of entrapped air remaining at the top of the test cylinder which could not be bled out. ‘This initial surge complicated the obtaining of accurate measurements within the first two seconds. 2. The nonlinearity of the resistance wire gage exhibited a more sensitive calibration at higher water surface levels than lower levels, thus further complicating early-time measurements. 3. At the larger Ap values, the pressure in the test cy- linder may not have remained constant at atmospheric. A rise in pressure may have occurred due to the re- sistance associated with the small 2" copper line connecting the test cylinder to the measuring tube. 79 4. When running the Ap = 40 psi tests, there was con- siderable gas release occurring in the water enter- ing the measuring tube causing slug flow to evolve therein. Some observations of the actual testing are important to note. Upon the initial depressurization of the test cylinder, small bubbles could be seen to form on the walls of the cylinder and on the grid system. These bubbles would then grow, break loose and rise to the top of the cylinder. After a short time, extensive bubble growth could be observed throughout the cylinder. The pretest condition of the test cylinder also had a large bearing upon the experiment. It was observed during the preliminary testing that much more gas release occurred during a test when the test cylinder and agitation grid system were completely dry before the test was conducted, as opposed to the prior wetting of these components. This .seems to indicate that there were more nucleation sites available for gas diffusion when the components were originally dry. In the experiments described in Table 2, these components were thoroughly wetted before testing. Due to constraints on the profitable use of time, a check on the repeatability of these tests was not Conducted. However, preliminary testing with water saturated with air showed that repeatable results were not usually obtained. In examining Figures 36 through 39 certain consistencies are notic- able. For each Ap value examined, more gas release occurred at increas- ing frequencies of agitation. It is apparent that the translational fluid motion around the bubbles increased with higher frequencies, thus enhancing the evolution of gas. Also, another expected result was for a given frequency, the amount of gas released increased with increasing 0.08 0.12 VOID FRACTION * 10" 0.04 c9,00 0.30 0.10 0.20 L .00 80 AP=lOpol “:7 (0:5 (0:3 //7 “=1 ‘_ ,,»/f’ w=o r’ v ”I 1 up 4.00 8.00 12.00 16.00 20.00 TIME IN SECONDS 493109" GAS PRODUCTION RATE [slugs/sec-fta] * 10's cp.00 i100 8100 12.00 _ 1h.OO 26.00 TIME IN SECONDS Figure 36. Results of Gas Release Experiment for Ap=l0 psi. 0.04 'VOIO FRACTION 0.02 . 01.12 c,OJ.OO 0.08 0.04 81 AP: 2090i .00 4100 8100 12.00 13.00 zb.oo TIME IN ssconos 493209“ A GAS PRODUCTION RATE [slugs/sec-fta] * Io" cp.00 .00 4.00 3100 IE.OO, I%.oo 25.00 TIME IN seconos Figure 37. Results of Gas Release Experiment for Ap=20 psi. VOID FRACTION GAS PRODUCTION KAI! [slugs/sec-ft3] * 10" 82 Q 9 oi " avt7' 00., A9330 pol O (085 (4): N c? (n=l,, a” .. ,1, (0:0 O c.’ cb.OO 2100 4100 6100 8100 Ib.OO TIME IN SECONDS A P = 30 pol N ‘3 cb.OO 2100 4300 6300 - 8100 15.00 Figure 38. TIME IN SECONDS Results of Gas Release Experiment for Ap=30 psi. VOID FRACTION cas- nonuc'non urn [slugs/sec-ft3] * lo'4 83 O ‘3 o‘ AP=4OPH «n=7 3; 0.1 was «1:3 (n=l ea ,,/ 0. (41:0 0‘ O /"/ c3 , cb.oo 2.00 4100 sion aioo Ib.oo TIME IN SECONDS AP=4ODM O ‘? od 0 7 . O. E! O" O O. . cb.oo i100 3100 6100 8100 Ib.OO TIME IN SECONDS Figure 39. Results of Gas Release Experiment for Ap=40 psi. 84 Ap values. A third important trend can be noted in observing the actual P values. The gas production rate initially increased with time with the Ap - 30 psi and 40 psi cases reaching a constant limiting value. In reference to Equation (105), this initial increase would be an expec- ted result. Since the gas production rate is proportional to the diameter of the bubbles, a smaller rate would be expected at the begin? ning of the experiment when only very small diameter bubbles exist. The reason for the limiting rates in the Ap = 30 psi and 40 psi runs is uncertain. It may have been due to a pressure rise in the test cylinder due to the extensive gas release associated with these runs or perhaps a steady state situation may have been achieved since the bubble size was restricted in a sense that the bubbles could only become so large before they would ascend and collect at the top of the cylinder. In light of the above findings, it was decided to numerically model gas release with the curve described by Figure 40, in which.F was normalized by Ap. Normalizing the actual data curves did produce an order of magnitude range of values that were used to describe this curve. In the figure, GAMl is the normalized ordinate at time equal zero and GAM2 is the normalized ordinate at time equal to TS. These three vari- ables were entered into the computer program to determine the slope of the gas production rate, the limiting value and the time at which it ‘was reached. The procedure used in the a1gorithm.was as follows. If the pressure at a node was calculated to be below the gas saturation pressure of the fluid, a r/Ap value was calculated according to the total amount of time a supersaturated condition existed at that node since the beginning of the transient. Multiplying by the instantaneous Ap resulted in a P for that node and time. If the pressure was calculated to exist r/Ap I GAMZ‘ GAMl 85 TS Figure 40. TOTAL TIME OF SUPERSATURATION [sec] Gas Production Rate Curve used in Numerical Model. 86 above the gas saturation pressure, no resolution of the gas was assumed and F was set to zero. Upon returning to a supersaturated condition, ordinate values were again calculated using the total amount of time that node had experienced supersaturation, which implies starting from the point on the time axis where ordinate values were last calculated. When the time TS was reached, the limiting value of GAM2 was used. How- ever, in the simulations conducted in this study, in which a TS value of four seconds was used, values from the Sloped portion of the curve were always used since a supersaturated condition never existed for that length of time. It is often difficult to obtain repeatable results during hydraulic transient testing. Several factors may cause this including differences in the number and sizes of available nuclei between experimental runs. In this study, the amount of gas release occurring during the transient was also affected by these variables which could not be controlled. The actual GAMl and GAMZ values used in the modeling were in the range of 1.0E-8 to 1.5E-7. These were the approximate normalized values for the w 8 l and w = 3 data runs. Values were selected from this range with regard to the apparent gas release characteristics of a particular ex- perimental run. 5.2 Pipeline Transients '5.2.l' Conditions Investigated The capability to place the solenoid-actuated valve at either an upstream or downstream location combined with the ability to control important variables created a combination of possible laboratory condi- tions that could have been investigated. To systematically compare laboratory results, the three variables that were manipulated were the 87 dissolved gas content, reservoir pressure and valve position. Transient data was gathered from both upstream and downstream valve closures. With the valve located in the downstream position, the steady- state velocity was maintained at 2.41 fps for all data runs. The up- stream reservoir was varied between three pressure levels, approximately 10 psig, 25 psig and 40 psig. The dissolved gas content of the water ranged from approximately 2% by volume, which was water whichluulnot been saturated with CO and contained mostly dissolved air, to 175% 2 which was almost totally dissolved CO gas. 2 When upstream valve closures were conducted, a steady-state velocity of 2.50 fps was used. The downstream reservoir pressure was varied be- tween approximately atmospheric pressure to 10 psig and 20 psig with the dissolved gas content ranging from approximately 2% to 145% by volume. Table 3 describes the experimental runs conducted. 5.2.2 'Comparison of Results Figures 41 through 44 show the comparisons cfifthe experimental data and the theoretical results for Experimental Runs 1, 4, 5 and 8, which were downstream valve closures. Comparisons for the upstream valve clo- sures are exhibited in Figures 45 through 48; which correspond to Experi- mental Runs 9, 15, 16 and 14. Although numerical comparisons were con- ducted for all the experimental runs, these were judged as representative examples of the predictions. Shown in the figures is the pressure at the valve and midpoint oftimapipe. Table 4 shows the GAMl, GAMZ and TS values used in the numerical predictions. A polytopic constant of one was used for all predictions which implies isothermal compression and expansion of the gas. Before discussing the predictions, examining only the experimental 88 oo.mN ma mm.0N mea om.N : mHIN ma oo.mN ma H¢.OH mqa om.N : mHIN NH oo.wN ma HN.NN cm om.N : mHIN NH cm.wN ma oN.HH om om.N : mHIN ea oe.wN ma mH.m on om.N : mNIN ma OO.mN MH mmoHN m¢ OmoN : NHIN «H oo.mN NH oo.oa ma om.N : NHIN ma eo.mN ma Nw.m mm om.N : NHIN NH ®0.0N MH HmoHN N Om.N : NHIN HH oo.mN ma mN.HH N om.N : NHIN OH oo.mN ma mo.o N om.N amouumop NHIN m ma.mN ca Hm.N¢ med H¢.N : HNIN w wH.aN ma qo.eN mNH H¢.N : HHIN N ma.aN ma 0¢.N¢ mm Nq.N : HHIN o wH.mN ea ma.mN mm Hq.N : HHIN m ma.aN 0H mm.NN mm H¢.N : HHIN a ma.mN ca He.Nq N H¢.N = NNIN m wH.mN ma HapoN N H¢.N : HNIN N wH.mN 0H mN.HH. N N¢.N ewouumosoa NHIN H Awm .dwv AUOV Awwmav Aucoouoov AmamV aoaumooq NNmH pooauz snowmoum ououmuooaoe unannoum ucouaoo huaooao> o>Hu> mama com owuosamoaud umumz uao>uowom mow .muaoawuomxm onwaomam udoamamua mo humaabm .m ~23 89 Table 4. Gas Release Variables Used in Numerical Model. Experimental GAMl 4 GAM2 TS Run (Slugs/sec-ft ) (Slugs/sec-ft ) (sec) 1 0.0 0.0 - 4 2.0E-8 5.0E-8 4.0 5 5.0E-8 1.2E-7 4.0 8 7.0E-8 1.5E-7 4.0 9 0.0 0.0 - 15 1.0E-8 6.0E-8 4.0 16 4 3.5E-8 1.0E-7 4.0 14 7.0E-8 1.5E-7 4.0 90 r‘* ”'I I Run I EXPERIMENT q no ... _ — — THEOREflCAl ..o J F . "0 l | I I um: I l II 4 |. m I I q ' I F' . no I I I I I .0 . 40 I I I I I l I 7 an I J_J l l (a). x . 424.4 n. rmnsun IN pfll % E ____:_.__———a ——-—-— “—— essssfis /~ ‘ h M‘ (b) x-zna n ' I I L o 19 2o ’ TIME IN SECONDS Figure 41. Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run 1. 91 T T I [UN 4 EXPERIMENT _——-—--— THEORETICAL (a) x: 424.4 N. l mssuu III In). (b) xszuatt l.0 2.0 3.0 TIME iN SECONDS Figure 42. Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run 4. 92 EXPERIMENT ‘— . THEORETICAL reissue: III pl- l (b) x.2I2.2 n. to 2.0 3,0 TIME IN seconos Figure 43. Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run 5. 93 m T T 1 L- RUN I thIERINENI ___— THEORETICAL .- | q I . I I I q I 1 \ (a) x . 424.4 II. PRESSURE IN poi. ,’\ - I ‘ ’- .II \ ' I. ‘ i v l( b) n 2I2.2 II. O . In . 29 so TIME IN SECONDS ’ Figure 44. Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run 8. 94 .m cam Pavemewtmaxu to» mupammm quwpmtomcw gum: mmtzmmmta easemente outammmz mo cemwnuaeou .mv weaned 82003 2_ Us; _ILuI RT. . ( 5:4...4 .., .7 II. we £96 5. A; i "59 NI 330553 (I J ON 8 on . . . o8 . c i can . assauzpllllll .. 3... 35.5.8. . o .22. . . p . r , p - ————— —-_—— ‘. \.—. 95 T T . I fl RUN Is EXPERIMENT 140 4 — —— TNEORETICAL - 100 80 T 60 ' 4O - 201,1 140 120 100 PRESSURE IN pd. ' (b) "213.2 N. I—lr ‘ 1 o 1.6 Eli 5:0 4.3 TIME IN SECONDS Figure 46. Comparison of Measured Transient Pressures with Theortical Results for Experimental Run l5. 96 I I _- EXPERIMENT I ——-—— THEORETICAL ‘ 160 140 120 100 D PRESSURE IN psIa p.- C! O \ ‘s s‘ a . ‘- "T A J 1 1 1 j ‘ (b) All213.2 it. O 1:6 2:0 To 4% ~ TIME IN SECONDS Figure 47. Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run l6. 97 EXPERIMENT .2 a. z :4 (a) x I 0.0 ii. 5 g, 340*- -I I 120 I‘ .. I I00 I, A ‘ I | I“! So I - ‘ I \\ I ‘ [A 60 I I l I 1 ’1 l l I i I I . 40 I l \ I I I \ 20 .. \ 4.. 1 I“ ’ \I ‘ T \’ ‘I O P .. (b) “213.2 II. I I . . o I.O 2.0 . 3.0 TIME IN SECONDS Figure 48. Comparison of Measured Transient Pressures with Theoretical Results for Experimental Run 14. 98 data does provide insight into the effects of gas release on the trans- ient. Figures 41 and 45 show experimental runs in which the water was not saturated with 002.. These runs where characterized by sharp pres- sure changes and vapor pressure conditions being reached at the valve several times. In these runs, it is doubtful if any significant gas release occurred. In examining the runs in which the water was saturated to some degree with CO2 gas, the effects of released gas become apparent. Some of the characteristics of these runs are a much smoother pressure trace and during the initial column separation at the valve, a slightly rising pressure until the return of high pressure conditions. Also common to these runs, apparently due to significant gas release during the first lowbpressure cycle, is the reaching of vapor pressure at the valve only during the first low-pressure cycle. Visual observations of the transient flow in the lucite pipe re- vealed the presence of tiny cavitation bubbles that immediately formed throughout the pipe during the initial column separation period. These bubbles would disappear upon the return of high pressure and would again become visible during periods of low pressure. In the low gas content runs (2% by volume), when the transient had been completely damped, a few very small bubbles were seen to remain in the pipe suggesting the release of gas during the transient. In the higher gas content runs, Ithe size and number of these bubbles were considerably larger. Motion pictures were then taken of the transient flow in the lucite pipe. From these pictures it was seen that in the highergas content runs the bub- bles did not completely collapse upon the return of the first high- pressure conditions, whereas, within the resolution of the camera, the bubbles were seen to completely disappear in the low-gas content runs when high-pressure conditions existed. 99 In examining the predicted results for Runs 1 and 9, good agreement was obtained for the pressure at both the valve and midpoint. These runs were associated with low-gas content water. In the modeling of these runs, no gas release was allowed, and the compatibility equations reduced to the usual one-phase waterhammer equations. This suggests that good results can be obtained from the simpler one-phase model in cases were insignificant gas release occurs. The prediction of the downstream valve closures, Runs 4, 5, and 8 also showed good agreement in the peak pressures and phasing at both the valve and midpoint locations. However, the theoretical results did not reproduce the higher frequency pressure oscillations that were present between peak pressures which were observable at the midpoint of the pipe. In examining the upstream.va1ve closures with gas release, varied success was achieved. Runs 15 and 16 showed a fair agreement while Run 14 exhibits larger predicted pressures than actually occurred. Again, the higher order pressure fluctuations at the midpoint were not predicted. It is apparent that closer agreement was obtained with the valve located downstream. Since the valve required a finite time to close, vapor pressure may have occurred behind the valve before it completely closed complicating the transient. Indeed, in the numerical model, vapor pressure was calculated at the valve before complete closure occurred. Since this caused difficulties in the model, all upstream valve closures were modeled as instantaneous. This inconsistency be- tween the modeling and reality may have been a factor complicating the attaining of better comparisons of the upstream valve closures. 100 As previously mentioned, the experimental results showed the pass- age of small pressure fluctuations along the pipe during periods of re- duced pressure which the theoretical results failed to reproduce. The reason for this is not clear. It is possible that the oscillations were smoothed out due to numerical damping primarily caused by interpolations. However, to suppress these oscillations a considerable amount of damping would be needed. Since much of the detail and phasing of the primary pressure peaks were predicted, it seems reasonable to conclude that the numerical damping that did occur was not significantly large enough to smooth these pressure fluctuations. A.more plausible explanation would seem to lie in the modeling of the gas release phenomena. The numerical model simulating gas release was based upon the results of the gas release experiment. The actual gas release occurring in the pipeline may have taken a different form due to several factors. In the gas release experiment a grid turbulence was generated throughout the duration of the test. 'Obviously, turbulence conditions experienced in the transient pipeline would be different and could be described as a decaying turbulence with the turbulence level dependent on the oscillating velocity field; this would imply a different type of gas release. Also, the gas release characteristics of water when vapor conditions are reached are not known. Perhaps the opening of vapor voids increases the gas release occurring at that point due to the increased gas-liquid interface. In addition, the numerical model did not account for the evolution of gas into the vapor pockets. In the model it was assumed that the gas was released in a distributed manner throughout the pipe with vapor cavi- ties forming at computing nodes. Perhaps the inclusion of gas evolution 101 into the vapor voids would more closely model the actual situation, thus resulting in better_agreement. CHAPTER VI SUMMARY AND CONCLUSIONS The primary objective of this study was to provide a better under- standing of the effects of gas release on hydraulic transients and to develop a mathematical model to quantitatively predict transient pipe flow which experiences gas release. Along with modeling the distributed release of gas throughout the pipe, the phenomenon of column separation was also taken into account. The theoretical findings were then compared with experimentally obtained pressure data. The results and conclusions of this study are: (a) The differential equations which characterize two-phase (b) hydraulic transients including gas release are recognized to be a set of three, nonlinear, hyperbolic equations which are amenable to solution by the method of character- istics. A sophisticated mathematical model can be developed using the method of characteristics with specified distance and time intervals. Evidence is presented which shows that reduced interpolation errors occur by the application of At/Ax ratios larger than those conventionally used. Due to the reduced numerical damping, this method appears to be competitive with other finite-difference schemes such as the Lax-Wendroff two-step scheme in dealing with vari- able wavespeed situations. Another advantage of this 102 (e) (d) (e) 103 method is the capability to integrate the pathline characteristic along a time line. The gas release experiment provides insight into the evaluation of gas in a closed container. By control- ling Ap and the grid turbulence, certain gas release trends are observed. It is recognized that the evo- lution of gas under turbulent pipe conditions is a complicated process and that the results of this ex- periment are not a precise description of gas release occurring during transient pipeflow; however, it does provide a basis from which a gas release model is developed along with supplying an order of magnitude range for the gas release variables that are used in the model. Experimental data obtained from the transient pipeline apparatus provides insight into the effects of gas release when.water is saturated to some degree with carbon dioxide gas. Systematically changing the gas content and reservoir pressure reveals that in cases of higher dissolved gas content pressure peaks are slightly attenuated, a smoother response is obtained and the timing of the peaks are delayed. The comparisons of the experimental data and theoreti- cal results obtained from the mathematical model show good agreement in the predictions of the timing and magnitude of the maximum pressure peaks when using a . gas release model based upon results of the gas release 104 experiment. However, higher frequency pressure os- cillations traveling along the pipe during periods of reduced pressure are not simulated. This discrep- ancy is attributed to inconsistancies between the actual gas release that occurred in the pipe and that described by the model. In addition, it was found that in cases where insignificant gas release occurs, the standard one-phase waterhammer approach which accounts for the formation of vapor voids at computing nodes utilizing a simple continuity balance adequately predicts transient pressures. It is recommended that future studies of the effect of released gas on hydraulic transients be concerned with acquiring a better understand- ing of the nature of gas release during turbulent pipe flow conditions. Further gas release testing should probably be done on a prototype scale and also be concerned with the number and size of available nuclei. In addition, incorporating the thermodynamic aspects of the vaporization process would be desirable since heat conduction, condensation and other thermodynamic considerations may be significant factors in describing and predicting the occurance of column separation and cavitation bubbles. It is also recommended that future laboratory studies utilize a longer pipe. This would increase the wave travel time resulting in longer dura- tions of supersaturated conditions, thus permitting the evolution of other more insoluble gas, such as air, to be studied. LIST OF REFERENCES 10. 11. 12. LIST OF REFERENCES Ames, W. F., Nonlinear Partial Differential Equations in Engineering, Academic Press, New York, 1965. Baltzer, R. A., "A Study of Column Separation Accompanying Transient Flow of Liquids in Pipes," Ph.D. Dissertation, University of " Michigan, February 1967. Driels, M. R., "An Investigation of Pressure Transient in a System Containing a Liquid Capable of Air Absorption," ASME Paper No. Epstein, P. S., and Plesset, M. 8., "On the Stability of Gas Bubbles in Liquid-Gas Solutions," Journal of Chemical Physics, Vol. 18, No. 11, Nov. 1950, pp. 1505-1509. Forsythe, G. E. and Wasow, W. R., Finite-Difference Methods for Partial Differential Equations, John Wiley and Sons, Inc., New York, 1960. Hall, J. W., "Nuclei and Cavitation," Journal of Basic Engineering, ASME, December 1970, pp. 681-688. Kalkwijk, J. P., and Kranenburg, C., "Cavitation in Horizontal Pipe- lines due to Water Hammer," Journal of the Hydraulics Division, ASCE, 21, HYlO, October 1971. Kranenburg, C., "The Effect of Free Gas on Cavitation in Pipelines Induced by water Hammer," Proceedings First International Con- ference on Pressure Surges, BHRA Fluid Engineering, Cranfield, Bedford,Eng1and, 1972. Kranenburg, C., "Gas Release during Transient Caviation in Pipes," Journal of the Hydraulics Division, ASCE, 100, HYlO, October 1974. Levich, V. C., Physicochemical Hydrodynamics, Prentice-Hall, New Jersey, 1962. Liggett, J. A., and Cunge, J. A., "Numerical Methods of Solutions of the Unsteady Flow Equations," in Mahmood, K., and Yevjevich, V., eds., Unsteady_Flow in Qpen Channels, Vol. 1, Water Resources Publications, Fort Collins, Colorado, 1975, pp. 89-182. Lister, M., "The Numerical Solution of Hyperbolic Partial Differ- ential Equations by the Method of Characteristics," in Ralston, A., and Wilf, H. S., eds., Mathematical Methods for Digital Computers, John Wiley and Sons, Inc., New York, 1960, pp. 165- 179. 105 l3. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 106 Martin, C. S., and Padmanabhan, M., "The Effects of Free Gas on Pressure Transients," Final Report, Research Grant GKr38570, National Science Foundation, June, 1976. Martin, C. S., Padmanabhan, R., and Wiggert, D. C., " Pressure wave Propagation in Two-Phase Bubbly Air-Water Mixtures," Pro- ceedings Second International Conference on Pressure Surges, BHRA Fluid Engineering, Cranfield,-Bedfoni,Eng1and, 1976. Parkin, B. R., and Kermeen, R. W., "The Role of Convective Air Diffusion and Liquid Tensile Stresses During Cavitation Incep- tion," Proceedings IAHR Symposium on Cavitation and Hydraulic Machinery, Dendai, Japan, 1962. Safwat, H. H., and de Kluyer, J. P., "Digital Computations for Water Hammer-Column Separation," Proceedings First International Con- ference on Pressure Surges, BHRA Fluid Engineering, Cranfield, Bedford, England, 1972. Schweitzer, P. H., and Szebehely, V. G., "Gas Evolution in Liquids and Cavitation," Journal of Applied Physics, Vol. 21, December 1950, pp. 1218-1224. Streeter, V. L., Fluid Mechanics, 5th ed., McGraw-Hill Book Company, New York, 1971. Streeter, V. L., and wylie, E. B., Hydraulic Transients, McGrawbflill Book Company, New York, 1967. Swaffield, J. A., "A Study of the Influence of Air Release on Column Separation in an Aviation Kerosene Pipeline," The Institution of Mechanical Engineers, Proceedings 1972, Vol. 186 56/72. pp. 693-703. Tullis, J. P., Streeter, V. L., and wylie, E. B., "waterhammer Analysis with Air Release," Proceedings Second International Con- ference on Pressure Surges, BHRA Fluid Engineering, Cranfield, Bedford, England, 1976. Van Wijngaarden, L., "0n the Growth of Small Cavitation Bubbles by Convective Diffusion," International Journal of Heat and Mass Transfer, Vol. 10, No. 2, February 1967, pp. 127-134. Vardy, A. E., "On the Use of the Method of Characteristics for the Solution of Unsteady Flows in Networks," Proceedings Second In- ternational Conference on Pressure Surges, BHRA Fluid Engineer- ing, Cranfield, Bedford, England, 1976. ’ Yadigaroglu, G., and Lahey, R. T. Jr., "0n the Various Forms of the Conservation Equations in Two-Phase Flow," International Journal of Multiphase Flow, Vol. 2, 1976, pp. 477-494. APPENDICES APPENDIX A DETERMINATION OF CHARACTERISTIC EQUATIONS APPENDIX A DETERMINATION OF CHARACTERISTIC EQUATIONS [5] The system of Equations (42) are of the form [B] {w}t + [A] {w}x + {d} = o (A.l) where {w} is a column vector of unknowns with the subscripts x and t referring to the partial derivatives with respect to distance and time and where [B], [A] and {d} may be functions of x, t, and w. Equation (A.1) is subjected to a linear transformation with a nonvanishing deter- minant so that a new system of similar form is obtained which is equiva- lent to the original one. The coefficients of the transformation may depend on x, t, and w. If [T] is such a transformation, multiplying Equation (A.1) by [T] results in [T] [BJMt + [T] [A] {w}x + [T] {a} = o (A-Z) A convenient form of the transformation is one in which [T] [A] = [c] [T] [B] (A.3) * where [C] is a diagonal matrix. If we set ET] [B] = [A j and [T] {d} . * = {d } then Equation (A.2) becomes * . . * . * [A J {w}t + [C] [A ] {w}x + {a } = o (A.4) Equation (A.4) indicates that every equation in the system now contains 107 108 differentiation in one direction only, that is, dx/dt = [C]. By the transformation we have eliminated one of the differential operators and reduced the system of equations (A.l) t0'a simpler system of ordinary differential equations. To calculate [C], the condition of equality of indiviual rows in Equation (A.3) represents a system of homogeneous linear algebraic equations with the unknowns [T]. For a non-trivial solution to exist, it is necessary that det (EAJ - [c] [3]) = 0 (A.5) Letting [C] e AEI], where [I] is the identity matrix, allows Equation (A.5) to be written as det ([A] - AEI] [B]) = O (A.6) The A values are termed the characteristic roots or the eigenvalues. If A are all real and distinct roots, the system of equations (A.1) is .hyperbolic. Once [C] is determined by the solution of Equation (A.6), then [T] can be found by substitution of [G] into Equation (A.3). Using the transformation matrix, the system of ordinary differential equations can be found by the substitution of [T] into Equation (A.4). The following is the application of this procedure to the system of Equations (42). Step A: Determination of eigenvalues [C]. Applying Equation (A.6) to the system of Equations (42) yields det ([A] -, AU] [3]) = o u -A E = o (A.7) 109 Expanding Equation (A.7) results in (u - on. - n2 -KEE]= o in which the eigenvalues are found to be >’ II Step B: Determination of transformation matrix ET]. From Equation (A.3) it is known that E)L5 7’ _ _f' A F _- ‘11 1:12 :13 F_u A 0:] u + (A ‘21 ‘22 ‘23 ‘31 ‘32 ‘33 _J b (A.8) (A.9) Expanding Equation (A.10) yields 8 f ( C E) (tll O +Et ‘12 ‘13 ‘22 ‘23 ‘32 ‘31;l 13) “A ‘ l -A t l-(A C E) (A.10) 21 k k ‘32 (A ESE) (t31 + Et33) -A t31+(A C E) 110 Equation (A.ll) can be rearranged into the following form: .——— -—- l 1 Luca)” o mama)". Vtu‘ _ -- 4: ' A -(AC E) 0 . 1:12 _"__.k:1 . 0 0_ _-;_(A_C_E), .1 _ _l ’ t13 L = O (A.12) I . I I t 22‘ I .I Q. .0 ............ ‘23 ‘31 .x GE)" 0 .32 0 E (AEE) t J L Bil The components of the transformation can now be determined. 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