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CRYOGENIC TWOPHASE FLOW AND PHASECHANGE HEAT TRANSFER IN MICROGRAVITY By CHENGFENG TAI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 O 2008 ChengFeng Tai To my wife Xiaoling, daughter Grace and my parents. ACKNOWLEDGMENTS I would like to express my sincere gratitude to Dr. Jacob NanChu Chung for providing me the opportunity to perform this research. I can not thank him enough for being so patient and understanding for years, imparting me a lot of knowledge and giving me the best advice. I would like to thank Drs. S. Balachandar, W. E. Lear Jr., S. A. Sherif, and G. Ihas for agreeing to serve on my dissertation committee. I thank my research group members for helping with academic aspects while providing memorable company for the past years. Finally I thank my family members who have been behind me every step of the way providing their unconditional support. TABLE OF CONTENTS page ACKNOWLEDGMENT S .............. ...............4..... LI ST OF T ABLE S ................. ...............8................ LI ST OF FIGURE S .............. ...............9..... AB S TRAC T ........._._ ............ ..............._ 15... CHAPTER 1 INTRODUCTION ................. ...............17.......... ...... 1 .1 Overvi ew ............... .. .. ......... ........... ...........1 1.2 Role of Cryogenics in Space Exploration............... ...............17 1.3 All Cryogenic Systems Must Be Chilled Down ................ ...............18........... . 1.4 Background of a Chilldown Process............... .................19 1.5 Obj ectives of the Research ................ ...............19.............. 1.6 Scope and Structure .............. ...............20.... 2 PHYSICAL BACKGROUND AND LITERATURE REVIEW............... .................2 2.1 Boiling Curve and Chilldown Process............... ...............22 2.2 TwoPhase Flow Patterns and Heat Transfer Regimes .............. ...............22.... 2.3 TwoPhase Flow in Microgravity ................. ........... ...............24..... 2.3.1 Isothermal TwoPhase Flows in Microgravity .............. ...............24.... 2.3.2 TwoPhase Flow and Heat Transfer in Microgravity ................. ............. .......25 3 PROBLEM DEFINITION AND GOVERNING EQUATIONS .............. ....................2 3.1 Modeling Cryogenic Chilldown in Microgravity ................... ............ ............... ....29 3.2 Geometry, Computational Domain and Initial, Boundary Conditions ................... ..........30 3.3 Assum options .............. ...............3 1.... 3.4 Governing Equations .............. ...............32.... 3.4.1 Interfacial Conditions .............. ...............33.... 3.4.2 Nondimensionalization .............. ...............34.... 4 SOLUTION METHOD .............. ...............40.... 4. 1 Introducti on ............... ... ...... ..... ..... ....... ....... .. ............4 4.2 Fractional Step Method within the Finite Volume Framework ................. ................ ..41 4.3 Cartesian Gird Method for a Complex Geometry ............... ...............44.... 4.4 Sharp Interface Method with CutCell Approach (SIMCC)............... ................4 4.4.1 Interfacial Tracking .............. ...............47.... 4.4.2 Merging Procedure ................. ...............48.............. 4.4.3 Flux and Stress Computations in the Interfacial Region ................. ................ ..49 4.4.4 Moving Interface Algorithm ................. ...............51........... .. 4.4.4.1 Advancing the Interface .............. ...............51.... 4.4.4.2 Updating the Cells .................. .... .......... ................ ............5 4.5 Issues of Phase Change Computation in Cartesian Grid Methods ................. ................53 4.6 Heat Flux Computation at the Interface ................. ...............56........... . 4.7 Phase Change Algorithm .............. ...............58.... 4.8 Global Conservation of Mass .............. ...............62.... 5 VALIDATION OF CODE .............. ...............73.... 5 .1 Introducti on ............... ...... ........... .......... ..................... .......7 5.2 Group 1, Solver of Governing Equations and the SIMCC with the Fixed Interface........73 5.2.1 The Fictitious Interface for Evaluating the SIMCC ............... ....................7 5.2.2 Couette Flow, Fully Developed Channel Flow and Cavity Flow ..........................76 5.2.2. 1 The Couette Flow ............... .. ........... ...............77..... 5.2.2.2 The Fully Developed Channel Flow .............. ...............77.... 5.2.2.3 The Cavity Flow ............... .. .. ......... ... ..................7 5.2.3 The Fully Developed Pipe Flow with a Constant Wall Heat Flux ................... ......79 5.2.4 Flow over a Sphere with Heat Transfer............... ...............80 5.3 Group 2, Validation for Moving Interface Algorithm ................ .......... ...............81 5.3.1 Static Droplet Simulation .............. .... ...............82.. 5.3.2 Deformed Rising Bubble and Droplet............... .......... ...........8 5.4 Group 3, Validation for Accurate Mass Transfer at the Interface ................... ...............85 5.4.1 A Stationary Droplet in a Quiescent Environment with a Constant Mass Transfer Rate .................. ........... ....... ...... .............8 5.4.2 One Dimensional Phase Change Problem ................. ...............87........... .. 5.5 Summary ................. ...............89.......... ..... 6 LIQUIDGAS TWOPHASE FLOWS INT A PIPE WITHOUT PHASE CHANGE ...........1 02 6. 1 Introducti on ................. .. ....._._ .... .... .. ..........0 6.2 Isothermal LiquidGas TwoPhase Flows in a Pipe ........._._.......___ ................1 03 6.2. 1 Effect of the Reynolds Number ................. ...............104........... .. 6.2.2 Effect of the Weber Number ................. ....................... ................108 6.2.3 TwoPhase Flow Pressure Drop ................. ...............111........... ... 6.3 Grid Refinement Study .............. .... .... ......... ........ .. ... ...........11 6.4 LiquidGas TwoPhase Flows in a Pipe with Heat Transfer ................. ............... ....112 6.4. 1 Low Surface Tension Case ................. ...............113.............. 6.4.2 High Surface Tension Case ................ ...............116........... ... 6.5 Summary ................. ...............118................ 7 LIQUIDGAS TWOPHASE FLOWS INT A PIPE WITH PHASE CHANGE ...................131 7. 1 Introducti on ............... .... ... ... ....... ......... ..... ..... .........3 7.2 R508B Refrigerant with Constant Wall Temperature ................. ........................132 7.3 Constant Wall Temperature Case with Liquid Nitrogen ................ .. ....._ ..........134 7.3.1 Flow Field with the Constant Wall Temperature Case .............. ....................13 7.3.2 The Mass Transfer at Interface with Constant Wall Temperature .......................138 7.3.3 The Transient Phase Change Process with Constant Wall Temperature .............139 7.3.4 The Comparison of Nusselt Number with Phase Change Process and Constant Wall Temperature .............. .. ...... ..............14 7.4 Wall Chilldown Process by Liquid Nitrogen ................. ...............142........... .. 7.4.1 Flow Patterns during Chill down Process .............. ........... ..... ............4 7.4.2 The Wall and LiquidGas Interface Conditions during Chilldown Process.........142 7.4.3 The TimeDependent Development during Chilldown Process...........................144 7.5 Wall Chilldown Process by Liquid Hydrogen ................. ...............144............. 7.6 Summary ................. ...............147.............. 8 SUMMARY AND FUTURE WORK ................ ....___ ...............165 .... 8. 1 Sum m ary ................. ...............165...... ...... 8.2 Future Work............... ...............165. LIST OF REFERENCES ............__........... ...............167... BIOGRAPHICAL SKETCH ............ ........... ...............175.... LIST OF TABLES Table page 41 Comparison of the key issues for different numerical methods for the heat flux computati on ................. ...............65......_._. ..... 42 Comparison of exact and numerical results for the flow over sphere case............._..._... ....65 51 The error of Nusselt number by different grid resolutions in flow over a sphere case......90 52 The fluid properties of liquid and gas phases for the one dimensional phase change problem ............. ...............90..... 71 Dimensionless parameters for the nitrogen case with constant wall temperature. ..........149 72 The maximum velocity for the nitrogen case with wall chilldown and the reference cases at time=1.5 ........... ..... .._ ...............149. 73 The mass flow rate for the nitrogen case with wall chilldown and the reference cases at time= 1.5. .........._ ..... .._ ...............149.. LIST OF FIGURES Figure page 21 Typical boiling curve. ............. ...............27..... 22 Flow regimes and heat transfer regimes in a heated channel..........._ ... ......_._........27 23 Flow regimes under microgravity. .............. ...............27.... 24 Effect of the gravity on flow regimes. ............. ...............28 ............. .. 31 A simple cryogenic system. ............. ...............38..... 32 Inverted annular flow. .............. ...............38.... 33 Idealized inverted annular flow. ............. ...............38..... 34 Conjugate heat transfer. ............. ...............38..... 35 The initial shape of liquid core. ................ ........_................. ....__. ....__39 41 Nonstaggered grid system. ............. ...............65..... 42 Example of mixed structured and unstructured grid in SIMCC. ................ ................. .65 43 Example of mixed structured and unstructured grid..........._.._.. ................. ...._.._.66 44 Marker points of an oblique ellipse. ............. ...............66..... 45 Intersection points of an oblique ellipse. ........._.._.. ..........__ ....._. ...._.._.. ...66 46 Illustration of the interfacial cells and cutandabsorption procedures in the SIMCC in local situation. .............. ...............67.... 47 Cutcells of different obj ects ........._...... ...............67.._.._ .... 48 Example of cutcells of different grids. ............. ...............67..... 49 Illustration of the interfacial variables and notation. ........._._... ....___ ........._......68 410 Illustration of interfacial advancing process. ......._....._._.__ ......._.._ ..............68 411 Illustration of updating cell procedure. ....__. ...._.._.._ ......._... ...........6 412 Two probing points for second order gradient. ...._.._.._ .... .._._. ........_..........6 413 Illustration of geometry of test domain. ....__. ...._.._.._ ......._.... ..........6 414 The assumptive temperature distribution .....___.....__.___ .......____ ...........6 415 The new interface because of phase change. ........._..__......_._....._ ..............70 416 The new interface because of balance of force. .............. ...............70.... 417 The flow chart for the phase change algorithm. ............. ...............71..... 418 The global conservation of mass in gas phase .......__ ..........___ ......._._...........71 419 Interim interface construction, the initial interface ................. .......___ .........__ ..72 420 Interim interface construction, the new interface. ............. ...............72..... 421 Restructuring of markers, the original makers. ....._.. ............_ ........_._.........72 422 Restructuring of markers, the new markers ........._.__........._. ...._.._..........7 51 The continuous stress condition at marker points for the fictitious interface. ...................90 52 The algorithm for second order gradient. ............. ...............90..... 53 Eight points method for second order gradient of velocity ........._._ ....... .._. ...........91 54 Geometry of the Couette flow, cavity flow and fully developed channel with the immersed fictitious interface............... ...............9 55 Error in interfacial velocity at all the interfacial marker points (163 points) and error in mass and momentum fluxes of each cutcell (148 cutcells) in the Couette flow with radius of Eicticious interface R=0. 1....._. ........_._... ....___ ......._........92 56 Pressure contour, U velocity profie, and error in mass and momentum fluxes of each cutcell (92 cutcells) of fully developed channel flow with radius of fictitious interface R=0.05 and Re= 100. ............. ...............93..... 57 Streamline plots and U velocity profie at R=0.5 with different fictitious interface radii of cavity flow with Re=100. ........._.__...... .__. ...............94. 58 The temperature distribution of fully developed pipe flow with constant heat flux case. ........._.__...... ._ __ ...............94.... 59 The heat flux at wall of fully developed pipe flow with constant heat flux case. .............95 510 The temperature profie of fully developed pipe flow with constant heat flux case at X=0.481235 along radial direction. ............. ...............95..... 511 The temperature distributions of fully developed pipe flow with constant heat flux case with different fictitious interfaces. .............. ...............95.... 512 The temperature profile of fully developed pipe flow with constant heat flux case with a fictitious interface (R=0.3) at X=0.481235 along radial direction. ................... ......96 513 Grid and geometry of flow over a sphere with heat transfer. ............. .....................9 514 The pressure contour, the stream line and temperature contour of flow over a sphere with Re= 1, Pe= 1. ............. ...............96..... 515 Schematic of stationary droplet problem. ............. ...............97..... 516 The maximum induced interfacial velocity of stationary droplet problem.............._.._.. ....97 517 Convergent histories of the interfacial condition and the governing equations of stationary droplet problem. ............. ...............97..... 518 Pressure contour for the stationary droplet. .............. ...............98.... 519 The shape and streamline plot for a bubble with Re=10 and We=8 ................. ...............98 520 The aspect ratio plot for a bubble with Re=10.0 and We= 2.08.0............... .................9 521 The streamline and shape plot for a droplet with Re=10 and We=2. ............. .................99 522 Geometry of static droplet with a constant mass transfer rate problem. ................... .........99 523 Velocity distribution of static droplet with a constant mass transfer rate problem along radial direction at X=5. ................ ...............100.......... .... 524 Streamline plot near the interface of static droplet with a constant mass transfer rate problem ............. ...............100.... 525 Illustration of one dimensional phase change problem............... .................100 526 The location of interface and temperature distribution, contour for one dimensional phase change problem............... ...............101 61 Streamline of the isothermal case at time=6.0 with Re= 2000 and We=500. .................. 120 62 The pressure contours/di stributions at time=6.0 with Re=2000 and We=5 00. ............... 120 62 Continued ................. ...............121....._._. ..... 63 The pressure distributions along the centerline based on different Reynolds numbers at tim e=6.0. ............. ...............122.... 64 The U velocity profile at time=6.0 with Re=2000 and We=500 .............. ...................122 65 Shapes of liquid cores with different Reynolds numbers at time=6.0............._..._.. ..........122 66 The pressure contours at time=6.0 with Re=500 and We= 2.63............... ..................12 67 Pressure distribution and the curvature of interface at time=6.0 with Re=500 and W e=2.63 ................ ...............123................ 68 Streamline plot at time=6.0 with Re=500 and We=2.63. ............. ......................2 69 U velocity profile at time=6.0 with Re=500 and We=2.63. ............. ......................2 610 Shapes of the interface by different Weber numbers at time=6.0 ................. ................124 611 The development of interface with Re=500 and We=2.63 at different time steps..........125 612 The ratio of averaged local friction coefficient in the twophase zone to the fully developed singlephase gas flow friction coefficient ................. .......... ...............125 613 The computed interfaces by three different grid resolutions. ............. .....................12 614 The pressure contours, pressure and curvature distribution, streamline plot and the contour of U component contour of the case with Re=500, Pe=6500 and We=500 at tim e=5.5 .............. .. ...............126................ 615 The temperature contours of twophase plug flow and the interfacial temperature at different time steps with Re=500, Pe=6500 and We=500. ................... ...............12 616 The Nusselt number at the solid wall at different time steps with Re=500, Pe=6500 and W e=500 ................. ...............127................ 617 The comparison of interfacial temperature and Nusselt number at wall for different Reynolds number with Pe=6500 and We=500 at time=5.5 ................ ............. .......128 618 The shapes of the liquid slug at three different time steps for the case of Re=500, Pe=6500 and We=2.63 ................. ...............128._._.. ...... 619 The temperature contour of two phase plug flow and the interfacial temperature with Re=500, Pe=6500 and We=2.63at different time steps. ................... ............... 12 620 The Nusselt number at the solid wall Re=500, Pe=6500 and We=2.63 at different tim e steps. ............. ...............129.... 71 The U contours for the refrigerant R508B case at different time steps..............._._.......149 72 The local maximum U in the gas phase for the refrigerant R508B case at different tim e steps. ............. ...............150.... 73 The interfaces for the refrigerant R508B case at different time steps. ...........................150 74 The streamlines in the gas phase for the refrigerant R508B case at time=4.0 ..............151 75 The V contours for refrigerant R508B case at different time steps. ........._.... ..............151 76 The temperature contours for refrigerant R 5 08B case at different time step s................ 15 2 77 The Nusselt numbers along solid wall for the refrigerant R508B case at different tim e steps. ............. ...............152.... 78 The total mass flow rate of gas phase for the refrigerant R508B case at different tim e steps. ............. ...............153.... 79 The vector and the streamline plots for Case 2 in the nitrogen case with constant wall temperature at time= 1.5. ........._.._. ...._... ...............153.. 710 The U contours for the nitrogen case with constant wall temperature at time= 1.5. ........1 54 711 The V contours for nitrogen case with constant wall temperature at time=1.5 .............155 712 Maximum velocity for the nitrogen case with constant wall temperature of three cases. ............. ...............155.... 713 The temperature contours for nitrogen case with constant wall temperature at time= 1.5. ........._ ...... ...............156... 714 The interfaces of three cases for the nitrogen case with constant wall temperature at time= 1.5. ........._ ...... ...............156... 715 The interfaces of three different grid lengths. ....._._._ .... ... .__ ......_._........15 716 The temperature gradient of three cases along the interface at time=1.5 and the locations of markers for the nitrogen case with constant wall temperature .........._........157 717 The mass flow rate for the nitrogen case with constant wall temperature at time=1.5....157 718 The history of temperature gradient along the interface of Case 2 in the nitrogen case with constant wall temperature. ............. ...............158.... 719 The history of mass flow rate of Case 2 for nitrogen case with constant wall tem perature. ............. ...............158.... 720 The Nusselt number at wall for the nitrogen case with constant wall temperature at tim e= 1.5. .............. .. ...............159......... ..... 721 The comparison of Nusselt number by the current method and Hammouda et al.'s correlation. ............. ...............159.... 722 The U, V and temperature contours of Case 1 in the nitrogen case with wall chill down at time=1.5. .........._ ......__ ...............160.. 723 The wall temperature contours for the nitrogen case with wall chilldown at time=1.5...160 724 The temperature at solidgas interface for the nitrogen case with wall chilldown at tim e= 1.5 .............. ...............161.... 725 The temperature gradients along the interface of reference for the nitrogen case with wall chill down at time=1.5 ........._.___..... .__. ...............161. 726 The temperature history of Case 1 in the nitrogen case with wall chilldown at time= 1.5. ........._.__...... .__ ...............161.. 727 Temperature gradients along the interface of Case 1 in the nitrogen case with wall chill down at time=1.5 ........._.___..... .__. ...............162.. 728 The mass flow rates for the nitrogen case with wall chilldown and reference case. .......162 729 U contours of two cases at time=0.4 ................. ...............162............. 730 V contours of two cases at time=0.4 ............... ...............163........... .. 73 1 Temperature contours of two cases at time=0.4. ................ ...............163........... . 732 Temperature at solidgas interface for the hydrogen and nitrogen cases at time=0.4. ....164 733 Nusselt number distributions on the pipe wall for the hydrogen and nitrogen cases for both cases at time=0.4 ..........._...__........ ...............164.... Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CRYOGENIC TWOPHASE FLOW AND PHASECHANGE HEAT TRANSFER IN MICROGRAVITY By ChengFeng Tai Auguest 2008 Chair: Jacob NanChu Chung Major: Aerospace Engineering The applications of cryogenic flow and heat transfer are found in many different types of industries, whether it be the liquid fuel for propulsion or the cryogenic cooling in medical applications. It is very common to find the transportation of cryogenic flow under microgravity in space missions. For example, the liquid oxygen and hydrogen are used to power launch vehicles and helium is used for pressurizing the fuel tank. During the transportation process in pipes, because of high temperature and heat flux from the pipe wall, the cryogenic flow is always in a twophase condition. As a result, the physics of cryogenic twophase flow and heat transfer is an important topic for research. In this research, numerical simulation is employed to study fluid flow and heat transfer. The Sharp Interface Method (SIM) with a Cutcell approach (SIMCC) is adopted to handle the twophase flow and heat transfer computation. In SIMCC, the background grid is Cartesian and explicit true interfaces are immersed into the computational domain to divide the entire domain into different subdomains/ phases. In SIMCC, each phase comes with its own governing equations and the interfacial conditions act as the bridge to connect the information between the two phases. The Cutcell approach is applied to handle nonrectangular cells cut by the interfaces and boundaries in SIMCC. With the Cutcell approach, the conservative properties can be maintained better near the interface. This research will focus on developing the numerical techniques to simulate the twophase flow and phase change phenomena for one of the maj or flow patterns in film boiling, the inverted annular flow. CHAPTER 1 INTTRODUCTION 1.1 Overview The cryogenic fluids have wide application in the industry. They can be used as liquid fuel like liquid hydrogen and oxygen in the aerospace industry (Filina and Weisend 1996), as coolant like liquid nitrogen and helium in medical applications (Jha 2006). In some special cases, for example, cryogenic fluids are used in spacecrafts not only at low temperatures but also under low gravity. Liquid oxygen and hydrogen are used to power launch vehicles and helium is used for pressurizing fuel tanks (Hands 1986). For this case, the transport and storage of cryogenic fluids under microgravity is regarded as an important element of the space mission. General information about the storage of cryogenic fluids can be found in some papers (Panzarella and Kassemi 2003, Khemis et al. 2004, Plachta et al. 2006). 1.2 Role of Cryogenics in Space Exploration To explore the space to other planets such as Mars (Mueller and Durrant 1999) is one of all human being's challenges. Therefore, the effective, affordable, and reliable supply of cryogenic fluids in space mission is very important (Cancelli et al. 2005, Heydenreich 1998, Graue et al. 2000). The efficient and safe utilization of cryogenic fluids in thermal management, power and propulsion, and life support systems of a spacecraft during space missions involves the transport, handling, and storage of these fluids in reduced gravity. The uncertainties about the flow pattern and heat transfer characteristics pose a severe design concern. Therefore, the design of cryogenic fluid storage and transfer system is very important and spawns researches in several areas: for example, design of the vessel (Harvey 1974, Bednar 1986), the piping and draining system (Momenthy 1964, Epstein 1965), insulation (Riede and Wang 1960) and safety devices. Moreover, the thermofluid dynamics of twophase systems in reduced gravity encompasses a wide range of complex phenomena that are not understood sufficiently for engineering design to proceed. 1.3 All Cryogenic Systems Must Be Chilled Down When any cryogenic system is initially started, which includes rocket thrusters, turbo engines, reciprocating engines, pumps, valves, and pipelines; it must go through a transient chilldown period prior to a steady operation. Chilldown is the process of introducing the cryogenic liquid into the system, and allowing the hardware to cool down to several hundred degrees below the ambient temperature. The chilldown process is anything but routine and requires highly skilled technicians to chill down a cryogenic system in a safe and efficient manner. The reason that the highly transient chilldown process is extremely complex is because when a cryogenic liquid is introduced into a system that has reached equilibrium with the ambient, voracious evaporation occurs and a very high velocity gas mist traverses through the system. As the system cools, slugs of liquid, entrained in the gas, flow through the system in a twophase film boiling mode. As the system cools further, a liquid quenching front flows through the system and is accompanied by nucleate boiling and twophase flow. The rate of heat transfer in the nucleate boiling regime is very high and the system begins to cool down very rapidly. As the system rapidly cools down, the twophase flow passes through several flow regime transitions to singlephase liquid flow (Burke et al. 1960, Graham et al. 1961, Steward et al. 1970, Bronson et al. 1962). The inherent danger during chilldown is that twophase flows are inherently unstable and can experience extreme flow and pressure fluctuations. The hardware may be subj ect to extreme stresses due to thermal contraction and may not be able to sustain extreme pressure fluctuations from the cryogen. Efficiency of the chilldown process is also a significant concern since the cryogen used to chill down the system can no longer be used for propulsion or power generation. Therefore, chilldown must be accomplished with a minimum consumption of cryogen. As a result, it is important to fully understand the thermofluid dynamics associated with the chilldown process and develop predictive models that reliably predict the flow patterns, pressure loss, heat transfer rates, and temperature history of the system. 1.4 Background of a Chilldown Process Due to the low boiling points, boiling and twophase flows are encountered in most cryogenic operations. The complexity of the problem results from the intricate interaction of the fluid dynamics and heat transfer, especially when phasechange (boiling and condensation) is involved. Because of the large stratifieation in densities between the liquid and gas phases, the reduced gravity condition in space would strongly change the terrestrial flow patterns and accordingly affect the momentum and energy transport characteristics. Boiling and twophase flow behave quite differently when the gravity levels are varied. The proposed research will focus on addressing specific fundamental and engineering issues related to the microgravity two phase flow and boiling heat transfer of cryogenic fluids. The outcome of the research will provide fundamental understanding on the transport physics of cryogenic boiling and twophase flows in reduced gravity. 1.5 Objectives of the Research The main scientific obj ective is to seek a fundamental understanding through numerical simulations on the boiling regimes, twophase flow patterns, and heat transfer characteristics for convective boiling in pipes under reduced gravity with relevance to the transport and handling of cryogenic fluids in microgravity. The engineering obj ective is to address issues that are related to the design of systems associated with the transport and handling of cryogenic fluids, for example, pressure drop information through pipes and heat transfer coefficients. 1.6 Scope and Structure In this research, the main objective is to investigate the transportation of cryogenic fluids in microgravity with the phase change phenomena especially the flow pattern and the amount of flow rate of liquid and vapor during the transportation. The experiments about the cryogenic fluids under microgravity are not easy to be performed on the earth. In this research, numerical tools will be adopted instead of experiments. The scope of this dissertation will be: To investigate the rate of vaporization (mass loss of cryogenic liquid) based on different driving mechanism and the boundary conditions. To study the flow types based on different driving mechanism and the boundary conditions or fluid properties. To develop the necessary numerical techniques for phase change computation. The final goal is to develop a reliable numerical package which can accurately simulate the cryogenic fluid in microgravity and helps the aerospace engineers to design the most reliable cryogenic transportation system in space mission or related industries. There will be eight chapters in this dissertation. In the first two chapters, the materials about heat transfer and phasechange characteristics of cryogenic flow, impact of microgravity effect and chilldown processes are reviewed. In Chapter 3, the physical model, governing equations and interfacial conditions are listed. In Chapter 4, the numerical techniques about the solver of governing equations, moving interface algorithm, the Sharp Interface Method with Cut cell approach (SIMCC), and phase change computation will be introduced. In Chapter 5, a series of test cases are done to ensure that the current code is accurate and reliable. Chapter 6 focuses on simulation of the liquidgas twophase flow in a pipe without phase change. Chapter 7 is about the phase change and chilldown process simulation of the liquidgas twophase flow in a pipe. In the last chapter, the summarization of current work and the possible future work will be addressed. CHAPTER 2 PHYSICAL BACKGROUND AND LITERATURE REVIEW 2.1 Boiling Curve and Chilldown Process A typical boiling curve, Figure 21, shows the relationship between the heat flux (vertical axis) that the heater supplies to the boiling fluid and the heater surface temperature (horizontal axis). Based on Figure 21, a chilldown (quenching) process usually starts above point E and then goes towards point D in the film boiling regime as the wall temperature decreases. Point D is called the Leidenfrost point (Carey 1992) which signifies the minimum heater temperature required for the film boiling. For film boiling process, the wall is so hot that liquid will vaporize before reaching the heater surface and that causes the heater to be always in contact with gas. When cooling beyond the Leidenfrost point, if a constant heat flux heater were used, then the boiling would shift from film to nucleate boiling (somewhere between points A and B) directly with a substantial decrease in the wall temperature because the transition boiling is an unstable process. 2.2 TwoPhase Flow Patterns and Heat Transfer Regimes In a cryogenic chilldown process, the wall temperature is always higher than the saturation temperature of the transported liquid. Therefore, the cryogenic chilldown process in a pipe is associated with a standard twophase flow. In this process, the types of flow and heat transfer can not be determined separately as the dominant heat transfer mechanism is dictated by the flow pattern. The heat transfer also affects the flow pattern development. For the twophase flow regime, there are several possible factors that can influence it, such as the flow rate, the orientation of pipe, the fluid properties and the heat flux at the wall of pipe (Yuan et al. 2007). In Carey's work (1992), he showed the different twophase flow types in a horizontal pipe. The flow types may be bubbly flow, plug flow, stratified flow, wavy flow, slug flow or annular flow. In Dziubinski et al's work (2004), they showed the flow regimes of a twophase flow in a vertical tube and the possible flow types are bubbly flow, slug flow, churn flow, annular flow and mist flow. The main difference between the horizontal and vertical flows is the effect of gravity that causes the horizontal flow to become nonsymmetrical to the centerline of the pipe. With regard to the chilldown process in the film boiling regime, depending on the local quality and other thermohydraulic parameters, the flow regime can be either dispersed flow, annular flow, or inverted annular flow. The corresponding heat transfer regime will be dispersed flow film boiling, annular flow film boiling, or inverted annular film boiling, respectively. As the wall temperature decreases below a certain degree, the liquid phase is able to contact the wall of the pipe. The leading liquidwall contact point which is often referred to as the quenching front or sputtering region is characterized by a violent boiling process with a significant wall temperature decrease. The quenching front will propagate downstream with the flow. The heat transfer mechanism at the quenching front is the transition boiling, which is more effective than the film boiling heat transfer due to partial liquid wetting of the wall surface. This re establishment of liquidwall contact is called rewetting phenomenon and has been a research interest for several decades. Nucleate boiling heat transfer dominates after the quenching front. For a vertical pipe, the flow regime can be annular flow, slug flow or bubbly flow. While for a horizontal pipe, the flow regime is generally stratified flow. As the wall temperature decreases further, the nucleate boiling process gradually reverts back to pure convection until the wall temperature reaches the thermal equilibrium with the wall, which denotes the end of the chilldown process. Figure 22 illustrates the flow patterns and corresponding heat transfer regimes. 2.3 TwoPhase Flow in Microgravity In this research, the main purpose is to simulate the cryogenic chilldown process under reduced gravity in space to obtain the design parameters for the related equipment. The condition of low gravity has a significant impact on the twophase flow model. It is the most important factor for determining the gas/liquid interfacial dynamics. The entire flow field will now be controlled by the convection, pressure gradient and viscous effects only. There will be some contributions from the surfacetension induced forces and the presence of the solid wall. When the dynamics is changed, the flow regime will also be changed. Figure 23 is the possible flow patterns under microgravity. Figure 24 is an example to show the distinctively different flow patterns in a horizontal pipe between terrestrial and microgravity conditions. For annular flow film boiling, the effect of gravity is assessed based on the ratio of Gr /Re3 Where Gr is the Grashof number and Re is the Reynolds number. The gravity effect is measured by the natural convection contribution, which is characterized by the Gr while the forced convective film boiling is scaled by the Reynolds number. According to Gebhart et al.'s work (1988), if the velocity of gas is greater than 10 cm/s, then the Gr / Re3 is less than 0.2. Natural convection is negligible for Gr / Re3 leSs than 0.225. 2.3.1 Isothermal TwoPhase Flows in Microgravity In order to investigate the impact of gravity quantitatively, some detailed experiments under simulated microgravity conditions have been performed on an aircraft (Rezkallah and Zhao 1995, Colin et al. 1991) and on the ground in a drop tower (Mishima and Hibiki 1996) because it is too expensive to do it in space. The first report on microgravity isothermal flow pattern for a large range of liquid to gas ratios was provided by Dukler et al. (1988). In their research, they showed the flow patterns are different between the 1g and the pu g conditions. Flow patterns under microgravity can be broadly classified into three types: the bubbly, the slug and the annular flow. In the beginning, experiments could only be performed in a pipe with a very small radius. Therefore, this experimental data may not be accurate enough. In the following studies, the researchers tried to perform the experiment with larger pipes and various fluids to determine the flow patterns (Janicot 1988, Colin et al. 1991, Zhao and Rezkallah 1993). In other' s researches (Zhao and Rezkallah 1995, Rezkallah 1996), the flow patterns were classified into three regions based on the dynamics of a flow: a surface tension region (bubbly and slug flow), an inertia region (annular flow) and the transitional region (frothy slugannular flow). There are several possible factors that may influence the flow patterns under microgravity such as the pressure drop (Zhao and Rezkallah 1995), surface tension (Rezkallah 1996), flow rate, pipe diameter and viscosity (Bousman et al. 1996). In Rezkallah's work (1996), he showed the flow patterns based on different Weber numbers and there are two transition lines which divide the flow into three regions. When the Weber number of the gas phase becomes larger; the flow tends to become annular; otherwise bubbly flow and slug flow are observed. In Bousman et al.'s work (1996), they showed the flow patterns based on different superficial velocities of liquid and gas phases. When the superficial velocity of the liquid phase is larger, the flow patterns are bubbly flows and move to annular flows when the superficial velocity of gas phase is larger. 2.3.2 TwoPhase Flow and Heat Transfer in Microgravity In general, there is little heat transfer data for cryogenic flow boiling in reduced gravity. Only two reports were found. AdhamKhodaparast et al. (1995) investigated the flow film boiling during quenching of R1 13 on a hot flat surface. They used microsensors to record instantaneous heat flux and heater surface temperature during the film boiling on board a KC 135 aircraft. Antar and Collins (1997) reported flow visualizations and measurements for flow film boiling of liquid nitrogen in tubes on board KC135 aircraft. They were particularly interested in the gas/liquid flow pattern and the thermal characteristics. They identified a new gas/liquid flow pattern that is unique to low gravity: a sputtering leading core followed by a liquid filament annular flow pattern. This new flow pattern is composed of a long and connected liquid column that is flowing in the center of the pipe surrounded by a thick gas layer. The gas annulus that separates the liquid filament from the wall is much thicker than that observed in the terrestrial experiment. They attributed the filamentary flow to the lack of difference in the speed of gas and liquid phases. On the heat transfer side, they reported that the quench process is delayed in low gravity and the pipe wall cooling rate was diminished under microgravity conditions. ~Singlephase Nucleate Transition Film convection boiling boiling boiling TWall Super Heat, T~ T Figure 21. Typical boiling curve. Flow Patterns Heat transfer Region Dispersed Flow Inverted Annular Flow Single Phase Flow Film Boiling Nucleate Boiling Convective Heat _ Dansfer Figure 22. Flow regimes and heat transfer regimes in a heated channel. Annullar Flow Figure 23. Flow regimes under microgravity. Stratified Flow 1g Annular Flow *r e Pg Figure 24. Effect of the gravity on flow regimes. CHAPTER 3 PROBLEM DEFINITION AND GOVERNING EQUATIONS 3.1 Modeling Cryogenic Chilldown in Microgravity According to the literature review presented in the previous chapter, for most part of the cryogenic chilldown process in microgravity, the twophase flow is in the inverted annular flow regime with film boiling. Based on the general boiling curve shown in Figure 21, initially the state of boiling is located around point F due to a large temperature difference between the wall (room temperature ~300 K) and the saturation temperature of the cryogenic fluid (~77 K for liquid nitrogen and ~20 K for liquid hydrogen) when the cryogen first enters the pipe. The schematic of a representative cryogenic flow system is given in Figure 31. As the wall is cooled down by the cryogenic twophase flow, the boiling state will move from point F towards point E and then approach point D when the wall is cooled down further. It is noted that the route from point E to point D is followed for the current quenching process, instead of the route from point E to point C, because the wall temperature is reduced during the quenching process by the heat transfer and can not be changed independently. During the initial stage, a quenching front would form that is followed by an inverted annular flow pattern with gas phase next to the pipe wall and a liquid core in the center (Antar and Collins 1997). As the liquid vaporizes, the radius of the liquid core would decrease as it travels downstream. Figure 32 illustrates the proposed physical model of the cryogenic two phase film boiling in microgravity, a continuous inverted annular flow. There are several research publications about the inverted annular flow (Yadigaroglu 1978, Ishii and Jarlais 1987, Aritomi et al. 1990, Nelson and Unal 1992, Hammouda et al. 1997). However, the heat fluxes in their researches were set at constant values or different from this research. In this research, the temperature of the wall will vary and also the strength of heat flux will change accordingly. The physical phenomena inside the tank and other devices downstream of the pipe exit will not be included in the current simulation. Only the flow in the pipe will be considered in this research. In the absence of gravity, it is reasonable to assume that the inverted annular flow is axisymmetric. Figure 33 shows the schematic for the idealized continuous inverted annular flow that will be used as the physical model in the current numerical simulation. To complete the entire conjugate heat transfer path, the twophase flow inside the pipe must be connected to the heat source, the pipe wall. Figure 34 shows the heat transfer network of this conjugate heat transfer. The outer surface of the wall is assumed to be perfectly insulated. Therefore, on the outside of wall, the insulated boundary condition will be assigned. 3.2 Geometry, Computational Domain and Initial, Boundary Conditions Since, this is an axisymmetric computation, a rectangular domain with a width of 0.5 (pipe radius) is used to simulate the circular pipe. The length of the domain will depend on the problems. The minimum grid spacing is 0.01. The initial shape of the liquid core is a quartcircle with radius 0.38 plus a straight line 0.03 as shown in Figure 35. Initially, the pipe is filled with the stationary gas and at the entrance. A quartcircle interface is used to divide the domain into different phases. The temperatures of wall and gas phase are assigned the same initially (=1.0) and the temperature of liquid core is set the same as the temperature at inlet (=0.0). The boundary conditions for this problem can be divided into two parts: Liquid and gas phases inside the pipe: In the real application such as space mission, the mass flow rate should be most important concern in order to keep the enough thrust so that the mass flow rate in this research at the inlet is assigned as constant by an uniform velocity (=1.0). In the incompressible flow, the total mass flow rate at outlet should be equal to the mass flow rate at inlet and plus the mass flow rate generated from the liquidgas interface due to the phase change. By this idea, the total mass flow rate at outlet can be obtained. In this research, the device at the downstream is not specified so that there are not exact boundary conditions at the outlet. In numerical simulation, the extrapolation strategy is usually used for the undetermined boundary condition. In this research, the second order extrapolation is applied for the velocity boundary condition at outlet and this velocity profile will be corrected by the total mass flow rate. By this strategy, the velocity profile at the outlet should be reasonable and the conservation of mass is kept. In this research, the pressure at inlet of pipe is assigned as constant (=1.0). In Navier Stokes equations, this constant does not mean anything but a reference value and it can be any value. In the incompressible pipe flow computation, the most important thing is the pressure drop and this pressure drop should be balanced by the shear stress from wall. In this research, the exact pressure at the outlet is unknown so that the second order extrapolation is assigned to get a reasonable estimation value for pressure from the nearby locations. In the incompressible pipe flow, there are two velocity and two pressure boundary conditions at the inlet and outlet. In numerical simulation, if the exact/real boundary conations are known, the numerical boundary conations can be assigned based on this exact boundary conditions or at least one condition must be flexible. In this research, the flexible boundary condition is the pressure at the outlet. By this flexible pressure boundary condition at outlet, the correct pressure drop can be obtained to balance the shear stress from the wall. Since it is an asymmetric pipe flow computation, the symmetric condition is assigned at the centerline and nonslip condition is assigned at the wall. The temperature boundary conditions in this research are assigned constant at the inlet (=0.0) and second order extrapolation at downstream, symmetric condition at the centerline. For the temperature at wall, it depends on different cases and governed by interfacial condition. Solid wall: The boundary conditions for the left and upper sides are the insulated boundary conditions and second order extrapolation for temperature at right side. 3.3 Assumptions In this section, the assumptions in this research are made as following: The flow and heat transfer are axisymmetric in a 2D cylindrical coordinates system. The flows are incompressible for both liquid and vapor phases. In most cryogenic transport systems, the flow rate may not be very large (Mach number < 0.3) and in this research the temperature differences will be limited to moderate values such that the compressibility should be negligible. A singlecomponent inverted annular twophase flow under film boiling is taking place in a circular pipe with no body force in a gravityfree environment. Viscous dissipation is neglected due to low velocities. Constant properties in the liquid and gas phases. Fluids in both phases are Newtonian fluids. Thermal equilibrium at the liquidgas interface and thermal nonequilibrium for the gas phase. The gas phase is transparent for thermal radiation, the wall is a grey body, and thermal radiation is a surface phenomenon for the liquid phase. 3.4 Governing Equations In this research, the Sharp Interface Method (SIM) will be adopted to treat the moving interface. In the SIM, the interface is a true surface with zero volume, so the governing equations of each phase are solved separately and transport fluxes are matched between the two phases at the interface. In the cryogenic twophase flow, the mass, momentum and energy equations in the liquid and gas phases are developed separately. Based on the above assumptions, the governing equations are listed below: Liquidphase equations: plcz, +v(1 V ] (ii) = kl V2 231 Gasphase equations: V0 =0ko~~; *Solidphase equation: Pwc, = k,VT,) (33) where subscripts 'l','g' and'w' denote the liquid, gas and solid phases, respectively; I' is the velocity vector, T is the temperature, p is the density, pu is the dynamic viscosity, cp is the specific heat, k is the thermal conductivity. 3.4.1 Interfacial Conditions Force balance in the normal direction at the liquidgas interface: where it~ and 7,~ represent the viscous stress tensors in the liquid and gas phases, respectively; (7 is the surface tension coefficient; ic is the curvature of the interface; p, and p, are the pressures in the liquid and gas phase, respectively. Mass flux continuity at the liquidgas interface: mi" p,(ut untl,)n i=pg(ui unit)ni (35) where thz" is the interfacial mass flux, ii is the unit normal vector at the interface; urnt, uz and Zig are the velocities of interface, liquid and gas phases, respectively. In a twophase flow, energy must be conservative and the temperature is the same for both phases at the interface under the thermal equilibrium condition. Interfacial energy conservation condition at the liquidgas interface: Am*"= (cv; kco;()(T)"d I, 1 u,,nt)n (3 6) Interfacial temperature condition at the liquidgas interface: TI Tg = nt (3 7) where ii is the latent heat of vaporization. qlrad is the thermal radiation heat flux received by the liquid surface from the wall. Thermal radiation at the liquidgas interface: Based on the simplifying assumption stated above and also the fact that the liquid phase is entirely enclosed by the pipe wall, the thermal radiation heat flux received on the surface of the liquid phase is as follows: c)" (38) l w, Rw where E, and El are the emissivities of solid wall and liquid phase, respectively. Rwl and R, are the radii of the inner surface of pipe and liquid core, respectively and 0 = 5.6697 x 10" is the StefanBoltzmann coefficient. Continuous heat flux at solidgas interface: ar, dT n k =k + q"a~ at r = Rw (39) dr g dr adww Thermal radiation at the solidgas interface: 9 ad,w 1R (310) 1( R )+ 1 1l Rz Ew Continuous temperature at solidgas interface: T T at r = Rw, (311) Insulation on the outer surface of pipe: =O tr ~= Rw (312) where Rwo is the radius of outer surface of the pipe. 3.4.2 Nondimensionalization In this work, all the computational quantities are nondimensional. The governing equations of each phase and the interfacial conditions must be nondimensionalized first. The reference scales are characteristic length L and velocity U and the characteristic time isL/U. The characteristic length in this research is the diameter of the pipe, and characteristic velocity is the inlet velocity. The characteristic temperature scale is AT = T, Tsa,. Tw is the ambient temperature and wall initial temperature. Tsa, is the fluid saturation temperature. Based on these reference scales, the dimensionless variables are defined as I* = 2/L, t* =tU/L p' = p/p,,,d p'~= p1/p,,,, a *= e/e,,, k' = k/ k. , c,'= c /c,,, I= R/U1, p*= p/(p,,U2) T*= _,, _(Tsa Tsat where the fluid properties of the liquid phase at the inlet are used as references. After the nondimensionalization procedure, the original governing equations and the interfacial conditions can be written as follows (the asterisks for dimensionless quantities are dropped from this point on for convenience): Liquidphase equations: du 1 + V (uI) = Vp +V2U (313) dt Re dT1 +v(ljT)= 1V27 dt Pe Gasphase equations: + *ill= Up + V ( fi (314) dt plRe Cp Ps cm T pg cp; gcmiT 1 "V Bt p c p Pe1 k;k Solidphase equation: k VT) (315or( 1) where the dimensionless parameter Re is the Reynolds number and Pe is Peclet number. p,UL P; c,lUL Re= Pe= p,1 k, *Interfacial conditions: Force balance in the normal direction at the liquidgas interface: ic 1 B, a, p,p,? = + pR (27, 27t)? PI(2,zt)+ (316))2 We Re i 8 Mass flux continuity at the liquidgas interface: mi" p,(u,u nt,)n i=pg(ui unit) ni (317) Interfacial energy conservation condition at the liquidgas interface: Ja p, k,84 BT )",L (u,) (u,) (3 1 8) where AT = (T, Ts,,) and the dimensionless parameter We is the Weber number and Ja is the Jakob number. p,U L kec BT We = Ja= = cr k, A Interfacial temperature condition at the liquidgas interface: TI Tg = nt (319) Continuous heat flux at solidgas interface: aT aT 4} ,L k = k + "~ at r = Rw, /L (3 20) ar ar k, AT Continuous temperature at solidgas interface: Tw = Tg at r = Rw, /L (3 21) Insulation on the outer surface of pipe: dT = 0 at r = Rwo /L (3 22) Br In this research, the heat flux by the thermal radiation cy'a,w, ad are computed with dimension and converted to the dimensionless quantity during computation and the conversion factor isL k; AT Tnnl~ I Gas Liquid Figure 32. Inverted annular flow. idealized inverted annular flow Gas Liquid Figure 33. Idealized inverted annular flow. Other devices Figure 31. A simple cryogenic system. inverted annular flow outer surface solid wall Liquid core inner wall surface vapor liquid core surface Figure 34. Conjugate heat transfer. Strong heat flux pipe 0.41 asmen m Figure 35. The initial shape of liquid core. CHAPTER 4 SOLUTION METHOD 4.1 Introduction In earlier numerical simulations about the chilldown process, scholars have used simplified models to solve the chilldown process, such as the onedimension model (Chi 1965), the homogeneous model (Cross et al. 2002), the twofluid model (Ishii 1975, Ardron 1980, Ishii and Mishima 1984), the threefluid model (Alipchenkov et al. 2004) and the pseudosteady model (Liao et al. 2006). Also, different important correlations are developed such as the correlation for saturated boiling (Chen 1966), subcooled boiling (Gungor and Winterton 1986) and pool boiling (Steiner 1986). By these simplified modeling works, the complex governing equations and interfacial conditions can be simplified and desired results can be obtained very fast and easily. However, these modeling works need the correlations by the experiments to simplify the governing equations so they can not be applied in broad applications. Also, because of simplification, the modeling work cannot give all the necessary and realtime information about the chilldown process such as the flow patterns, and may be not very accurate for some important characteristics. Therefore, direct numerical simulation is adopted in this research. By directly solving the general governing equations, the interfacial conditions and proper initial/boundary conditions, the computation can be unsteady and all the flow characteristics can be captured. All the information can be saved and give a great help in different engineering aspects . As illustrated in Chapter 3, the complex problem of twophase flow and conjugate heat transfer with phase change does not allow any possibility of an analytical solution. In this chapter, the numerical simulation method that has been adopted for seeking the solutions is presented. The central infrastructure of this method is based on the concept of a sharp interface that truly separates the liquid phase from the gas phase. In other words, the interface is a true surface without any volume association. The key elements of the Sharp Interface Method (SIM) are listed as follows: Fixed Cartesian Grid The computational framework is built on an EulerianCartesian grid. With this underlying fixed grid system, a socalled Cutcell approach is used to treat the interfaces and boundaries that do not align with the fixed Cartesian grid. Lagrangian Moving Sharp Interface Algorithm To track the moving vaporization front, separate marker points(Nichols 1971) are used to identify the phasechange interface. These points that are connected by piecewise polynomials are employed to capture the deformation and movement of the sharp interface through the translation of these markers over the underlying Eulerian fixed grid. Fractional Step Method For each phase domain, the fractional step method is used to numerically integrate the governing equations in that phase. The overall solution is obtained by matching the mass, momentum and heat fluxes from both phases at the sharp interface. 4.2 Fractional Step Method within the Finite Volume Framework In order to best enforce the conservation laws and to treat the discontinuity at the interface, the finite volume method (Versteeg and Malalasekera 1995) is used to discretize the governing equations. By the finite volume framework, the governing equations listed in the previous chapter will be integrated over a finite control volume which is called a unit cell as shown in Figure 41. The integral forms of governing equations for an incompressible flow with constant properties are given below. Continuity equation: 17 fiS = (41) Momentum equation: CSt dV+ Sii (u 1 Sd pldS+ u dS (42) Energy equation: StdV+Ir1 T>~ (u 1 nId PeT iidS (43) where cy and cs represent the control volume and the surface of the control volume, respectively. nf is the outward normal vector from the control volume surface, fi is the velocity vector, p is the pressure and T is the temperature. In this research, a cellcentered collocated (nonstaggered) approach (Ferziger and Peric 1996) on the Cartesian grid system is adopted, the primary variables (velocity, pressure and temperature) are defined at the cell centers and the primary variables needed at the cell faces will be evaluated by interpolation from respective variables at the cell centers as shown in Figure 41. A second order accurate twostep fractional step method (Chorin 1968, Kim and Moin 1985, Zang et al. 1994) is used for advancing the solutions of the integral unsteady governing equations in time. In this approach, the solution procedure is advanced from time step "n" to "n+1" through an intermediate diffusionconvection step where the momentum equation without the pressure gradient terms is first solved and advanced in a half time step. The intermediate diffusionconvection momentum equation can be discretized as c" (44) + (Vil + WiP)> iidS where il is the intermediate velocity at the cell center and U is the velocity at center of the cell face. The cell surface velocity is used to evaluate the fluxes going in or out of a control volume. After the l'* is determined, 0', the intermediate velocity at the center of cell face, is calculated by interpolating between the respective cellcenter velocities. The first term on the right hand side is the convective term. A second order accurate AdamsBashforth scheme (Bashforth and Adams 1883) is used to discretize the convective term. The second term is the diffusive term that is discretized by the implicit CrankNicolson scheme (Crank and Nicolson 1947). This eliminates the potential viscous instability that could be quite severe in the simulation of viscous flows. Once the intermediate velocity is obtained, the pressure is obtained by the correction step: n+1 * dtliV = ~~ U" dV (45) In this pressure correction step, the final velocity u'" must satisfy the integral mass conservative equation. The integral mass conservative equation can be rewritten as the following form: I(" ' HS = 0 (46) Therefore, the integral pressure correction equation can be expressed as ( ~p"")idS 1O' ij)dS (47) CS CS Once the pressure is obtained, the intermediate velocity can be corrected and updated to obtain the final velocity by (48) The energy equation is solved by similar procedure but the predictcorrect procedure is not required because the energy equation in this research is a standard convectiondiffusion equation. The discretized energy equation is similar to Equation 44 and can be discretized as (Francois 2002) T"~ Tn 1r_ nT" 1 0, c"(49) 2Pe1 nl+~n~id For the incompressible flow with nonconstant properties, the fractional step method can work also and just needs to include the flow properties into each procedure (Ferziger and Peric 1996). 4.3 Cartesian Gird Method for a Complex Geometry In the early development of computational multiphase flow, some researchers adopted the curvilinear grid system (Ryskin and Leal 1984, Dandy and Leal 1989, Raymond and Rosant 2000, Lai et al. 2003). This approach is simple but not easy to apply. In order to describe the deformation of an interface between different phases, a very powerful grid generation is required and also, the grid has to be updated frequently to obtain the convergent solution and therefore it is very computationally intensive. In recent multiphase computational approaches, several Cartesian grid methods are broadly used such as the Sharp Interface Method (SIM) (Udaykumar et al. 2001, Ye et al. 2001), the Immersed Boundary Method (IBM) (Peskin 1977, Singh and Shyy 2007), the Front Tracking Method (Qian et al. 1998, AlRawahi and Tryggvason 2004) ,the Volume of Fluid (VOF) Method (Hirt and Nichols 1981, Pilliod and Puckett 2004), the Level Set Method (Ni et al. 2003, Tanguy and Berlemont 2005) the coupled Level Set and Volume of Fluid Method (Son 2003, Sussman 2003) and the Phase Field Method(Badalassi et al. 2003, Sun and Beckermann 2007). Based on the computational framework, Sharp Interface Method, Front Tracking Method and Immersed Boundary Method are classified under the mixed Eulerian Lagrangian category and Level Set Method, Volume of Fluid Method and Phase Field Method are in the Eulerian category (Shyy et al. 1996). In this research, the Sharp Interface Method (SIM), a branch of the mixed Eulerian Lagrangian Cartesian grid method, is adopted to handle the complex geometries. In the SIM, the Cartesian grid is designed as a background mesh, and the explicit interfaces are used to describe the shapes of the obj ects on the background grid. The interfacial dynamics associated with the moving/fixed boundaries need to be considered simultaneously. In the mixed Eulerian Lagrangian approach the interface is constructed by a sequence of marker points. With these marker points, the shape and location of the interface are determined by designated interpolation procedures, while the overall fluid flow is computed based on the fixed Cartesian grid. In the SIM, the interface is treated explicitly with zero thickness, in accordance with the continuum mechanics model. The primary variables at the interface are computed via the interfacial conditions The SIM defines the relations between the background grid and the interfaces. Because of the nonCartesian interfaces, some cells containing the interface will be cut and form the non rectangular cutcells. Special methods are needed to handle these cut cells. In this research, a Cutcell approach (Ye et al. 2001, Ye et al. 2004, Tai and Shyy 2005) is employed to treat the interface and boundary cells. In the Cutcell approach, each segment of the cutcell is merged into a neighboring cell or assigned the identity of the original Cartesian cell. Hence, even though the underlying grid is Cartesian, the cut cells are reconstructed to become the nonrectangular cells and the cutsides will form the interface. After the reconstruction, the entire grid is filled with the rectangular grid and nonrectangular grid. In this research, the shapes of the cut cells include triangle, trapezoid and pentagon. The SIM with the Cutcell approach (SIMCC) can handle the sharp discontinuity resulting from the interface formation, and can achieve higher accuracy (Ye et al. 2004, Tai and Shyy 2005). Among the EulerianLagrangian approaches, the SIMCC is the one with the best accuracy, especially for a solid boundary. Figure 42 is an example of grid system handled by SIMCC. In Figure 42, the cells far away from the object are still rectangular shapes and only the cells near the interface are modified to be nonrectangular shapes. It must be emphasized that the total number of cells will not change during the computation. 4.4 Sharp Interface Method with CutCell Approach (SIMCC) The SIMCC is adopted to solve the complex geometry in this research. For SIMCC, the underlying grid is a Cartesian grid which is cut by the explicit interfaces to form the cutcells. The entire domain will be filled with these cut and regular cells. Most cells still keep the original shapes. The SIMCC utilizing the Cartesian grid and Cutcell approach can be used to solve fluid flow and heat transfer problems involving multiphase and/or complex geometry with a high accuracy. Due to the algorithm and data management requirements the SIMCC is computationally intensive. Using SIMCC, the interface is constructed by a sequence of straight lines in the computational domain. Because the number of cells in SIMCC does not change during the computation, the matrix of coefficients of SIMCC is very similar to the curvilinear grid system. Only the coefficients of cutcells have to be modified so that the convergent speed and characteristics are very close to the curvilinear system and so it converges much faster than the unstructured grid system. For a complete set of SIMCC, there are four main procedures; the interfacial tracking, the merging procedure, the flux computations in the interfacial region and the moving interface algorithm for advancing the interface. For the interfacial tracking, the information of the interface should be input and reproduced and also the intersections between the interface and background grid must be located. These intersections will be the cutpoints and used for the next step, the merging procedure. Because of the interface, some cells are cut and can not maintain the rectangular shape anymore and therefore have to be treated specially. Using a merging technique, the fragments of cells can be merged by neighboring cells or larger fragments to form cutcells. This means the cells around the interface have to be reconstructed. For the third procedure, the cutcells around the interface may be in the form of a triangle, trapezoid or pentagon. A special interpolation scheme with higher order accuracy is required to handle the complicated cutcells to get the accurate primary variables or derivatives at the center of a cell face. The original Cartesian grid will become a mixed grid which includes most rectangular and some nonrectangular grid. Figure 43 is an example of the mixed type grid. For a fixed interface problem such as flow over a sphere, the above three techniques are enough. If the interface is not fixed but moving, the moving interface algorithm will be necessary. In the moving interface algorithm, there will be two functions to perform: To advance the interface based on the force balance in the normal direction at interface. To refresh the cell because of change of phase. By these four techniques, the SIMCC can be used to solve the moving interface problem. 4.4.1 Interfacial Tracking During this procedure, marker points are used to describe the initial position of the interface. As shown in Figure 44(A), a sequence of marker points is given initially. In this oblique ellipse case, there are 50 initial markers. These markers must be represented by a polynomial curve fitting method; also, the distance between two adj acent markers have to be adjusted based on the curve fitting method. For this research study, a quadratic curve fitting (Chapra and Canale 2002) is used so that the distance between two neighboring markers can not be too long and is set to dx/2 to maintain the shape of the obj ect. After the curve fitting, a new sequence of markers with an equal distance from each other is obtained as shown in Figure 4 4(B). The number of markers is increased to 68. The location of markers are stored as function of the arc length and represented by a sequence of quadratic functions. The intersections of the interface and the background grid are shown in Figure 45. These intersections represent the cut points of the cutcells. With this information, the normal vector of cutside can be calculated. Besides the marker points, all the primary values at interface are stored by the quadratic curve fitting also and can be represented by 5= a s2 +b~SsCc (410) The coefficients in Equation 410 can be obtained by any three points which construct a section of quadratic curve. Any quantities located at this quadratic curve can be obtained by Equation 410. 4.4.2 Merging Procedure In this study, the interface is represented by a series of piecewise quadratic curves. The governing equations will then be solved in regions separated by the interface, and the communication between these regions is facilitated by the interfacial conditions. Figure 46 illustrates the formation of interfacial cells where cells 1 to 4 are cut by an interface. According to the present Cutcell approach, the segments of an interfacial cell not containing the original cell center are merged by their neighboring cells; the segments containing the original cell center are given the same index as the original cell. For example, in Figure 46, the upper segment of cell 3 is merged into cell 5 to form a new trapezoid cell. The fraction of cell 3 with cell center becomes a new independent trapezoid cell. The main segment of cell 1 that contains the original cell center will merge the small segments of cells 4 and 2 to form a new triangular cell. The remaining segment of cell 4 containing its original cell center now becomes an independent pentagonal cell. With these cutandmerging procedures, the interfacial cells are reorganized along with their neighboring cells to form new cells with triangular, trapezoidal, and pentagonal shapes in a 2D domain. Figure 46 shows an example of the cutcells after reconstruction. Basically, when the area of a segment is less that 0.5 area of a normal cell, it will be merged. After this procedure, each newly defined cell maintains a unique index and cell center to support the needed data structure. Of course, in a 3D domain, the situation will be more complicated, and will not be discussed here. Figure 47 are two examples of cutcells of different obj ects. The obj ects here are an airfoil (NACA00 12) and a star shape. Figure 48 is another example to illustrate that the cutcells of the different grid sizes. In Figure 48(A), the grid size is 0.025 and 48(B) is 0.1. In this case, the number of cutcells of a Eine grid is larger than that in a coarse grid. It can be seen that the obj ect is represented more accurately in a Eine grid than a coarse grid. 4.4.3 Flux and Stress Computations in the Interfacial Region In Figure 49, it represents a Cartesian grid with nine cells cut by an interface. The solid squares mean the centers of cells. Because of the interface, the original ACDF cell with the cell center 1 will absorb the fragment from another cell to form a trapezoidal BCDE The original grid line AF is replaced by a section of interface BE and the original faces DF and AC are extended to become DE and BC. In the finite volume framework, the fluxes (first order derivative) at cell faces and the primitive variables at face centers must be known for the surface integration purpose. Therefore, how to obtain the highly accurate fluxes and primitive variables is very important in SIMCC. To consider a flux f across a cell face BC in Figure 49, one can construct a secondorder accurate integration procedure as follows: BCaS BAfy~~,fS AC~ BA A B C r C 4A) (411) where 4 and fe are computed at the centers of segments BA and AC respectively. Getting the value of fc is straightforward. If fe is the flow variable #, a linear interpolation can be used, yielding a secondorder accuracy. The relation can be expressed as ~CA = ~j1 + s(1 jZ) (412) x1 xA ;2= 1 c(413) If fe represents the normal gradient of flow variable #, it can be approximated by the following central difference scheme as (414) Equations 412 to 414 can not be used to estimate 4 because the neighboring cell centers are located in a region on the other side of the interface. Also, the center of segment AB is not located on the straight line connecting the two cell centers. In order to maintain a second order accuracy, a twodimensional polynomial interpolating function is adopted in the computation of for such kind of small segments. An appropriate functional form for # that is linear in X direction and quadratic in Y direction is given as S= cXY2 +c Cy + cy +c 5~ 6 + (415) the six coefficients, c, to c6, can be obtained from values at the six points (1 to 6) in Figure 4 9; thus the variable # at the center of AB can be expressed as ~An = c1x4,yj, +c Cy, + c3xyAB +c C4yB + 5 .B '6 (416) The llnoml gradient which is often needed while computing the interfaial condition, can be obtained by =czyc~~,+czy,,+c, (417) Similar approach is used to compute the flow variables or their normal gradients on the remaining segments. Once the primary variables and the derivatives at the cut side are determined, the coefficients of matrix for the cutcells can be modified based on this information and the matrix solver will be called to obtain the solution. It must be emphasized that Equation 417 is only for AB in this case. Here, the y terms are up to second order while the terms in x are linear. That means this interpolation polynomial is "quadratic" in Y direction and "linear" in X direction. For a more complicated case, a higher power polynomial may be needed but the order will not exceed two in either x or y . 4.4.4 Moving Interface Algorithm For the fixed interface problems, the techniques discussed from Section 4.4.1 to 4.4.3 constitute all the elements required to obtain the solutions. However, for most problems in multiphase flows, the interface is not fixed and therefore a moving interface algorithm is necessary. The moving interface algorithm in this research includes two functions. The first one is to advance the interface (Ryskin and Leal 1984) and the second one is to update the cells because of change of phase (Udaykumar et al. 1997). 4.4.4.1 Advancing the Interface In the unsteady multiphase computation, the interface will advance to satisfy the interfacial dynamics in each time step. In SIMCC, a "push and pull" strategy is used to determine the new location of an interface and also to satisfy the force balance at the interface. The forces acting at the interface can be resolved into the normal and tangential components. In multiphase flow computation, the order of magnitude of the normal force is much larger than that of the tangential force and therefore the displacement of interface is governed basically by the normal forces at the interface. In this "push and pull" strategy, only the normal component of force balance is considered on the interface and the new location of interface will be determined by a series of iterations. In each iteration, the residual of the force balance in the normal direction will be computed and the displacement of marker points is assumed proportional to this residual: Xn"p X"np +pH n (418) Xn" Y" +pH" 1ni~ where (X, Y) is the coordinates of marker points, nx?,n, is the components of normal vector in the X and Y direction, respectively. H is the residual of the force balance in the normal direction and p is a relaxation factor and it is an empirical value normally in the range of 0. 10.001 in this research. Figure 410 is an illustration for this interfacial advancing process. In Figure 410, a marker point A, is located at the initial interface and the residual of force balance in the normal direction n, will be computed. Based on the location of A, and the value of n,, the marker point A, will be pushed to A, In this moment, the residual of force balance in the normal direction will be checked. If the new residual H, is still large, the marker point A, will be pushed to A, Once the residual is small enough, i.e., the force balance in the normal direction achieves convergence, the interaction will be stopped and the normal component of interfacial velocity can be determined by (un,,) mt new = (Xnev. Xold a (419) (u,,), menew = (Y,, Yold)/a In this iterative procedure, the residual of the force balance in the normal direction is required to be less than 1.0E3. The normal component of the interfacial velocity will be the velocity boundary condition during the computation. This algorithm is only for the problems without phase change. Once there is phase change, Equation 419 must be modified to include the effect from phase change. 4.4.4.2 Updating the Cells After the interface is adjusted, some cells will change their phases. In Figure 411, the phase of cell center A is different from cell centers a, b, c and d initially. Once the interface advances to the new location, the phase of cell center A will change. In this case, the fluid properties of cell center A will change also such as the discontinuous primary variable of pressure. In this research, the new primary variables at the cell center A can be obtained by following steps: Select a point B on the new interface such that the line connecting A and B is normal to the interface. Point B is located at the interface and therefore, the primary variables of point B can be determined by interfacial conditions. In this normal direction, find another point C such that the distance between B and C is 1.5 ~ Find four cell centers a, b, c and d and point C is surrounded by them. To use cell centers a, b, c and d and the bilinear interpolation method to find the primary variables at point C. Use linear interpolation to obtain the primary variables at cell center A using the primary variables at points B and C. Update the fluid properties of cell center A. 4.5 Issues of Phase Change Computation in Cartesian Grid Methods In this research, the phase change phenomenon is one of the key elements. In a cryogenic twophase flow, the temperature at pipe wall or tank is much higher than the saturation temperature so that the phase change phenomenon at liquidgas interface is inevitable. The phase change phenomenon is very common in engineering applications. For the phase change computation, the most important issue is to obtain the accurate mass transfer rate generated from the interface due to the vaporization and this will rely on the accurate heat flux computation near the interface. In recent Cartesian grid methods, except the Sharp Interface Method, most of them adopt the continuum surface force model (CSF) (Brackbill et al. 1992, Shyy and Narayanan 1999) to include the surface tension in the momentum equation. By CSF, the surface tension can be modeled as a source term in the momentum equation (Sussman et al. 1994, Loirstad and Fuchs 2004, Zhang et al. 2006). In addition, the continuity and energy equations may need to be modified by adding extra source terms to include the effect of discontinuity such as mass transfer rate at interface. The fluid properties of different phase are described by a socalled indicator or Heaviside function (Dhir 2001). Therefore, only one set of governing equation is solved and there will be a smeared band of the solution across the interface. The resolution of primary variables must be poor near the interface. In Juric and Tryggvason's (1998) and Shin and Juric's (2002) work, they used the Front Tracking Method to simulate the film boiling phenomena. In their researches, the interfacial velocity urnt is obtained by the following equations: h" + (pu)i q) urnt n = (420) P1 + P, where pf and uf can be treated as an average of the velocity of liquid and gas phases near the interface. Then, the interface is advanced by following equation.  = ,+ (421) dkdt pivz Also, this interfacial velocity will be used to form the source tern in the continuity equation: Vu = f (ut) (422) This set of equations looks safe in mathematic but in fact, it is very dangerous in real physics. The trouble will come from the interpolation for uf Once phase change is very strong such as large Jakob number or large density ratio, the velocity jump at interface will be very large. With large velocity jump, uf will be inaccurate and the error may be very huge. For example, if u, =1.0 and ug =100, uf will be close to 50.5 by the interpolation and this will cause huge error. Therefore, this set of equations can be applied only when the phase change is very weak or low density ratio otherwise, the conservation of mass will be very inaccurate. In some researches about phase change by Level Set Method such as Morgan's (2005), Son and Dhir's (1998) and Son's work (2001), in order to include the volume change at liquid gas interface, they modified the continuity equation as V u = f (VH) (423) where H is a smooth step Heaviside function. By this Heaviside function, the fluid properties near the interface will be treated as continuous with 3 grid space distance. Obviously, once the density jump is small, this modified continuity equation may be able to handle the mass flux generated from the liquidgas interface. However, once density jump is very large, the conservation of mass will be inaccurate. In the general condition, the density jump between liquid and gas is on the order about 1000. In Front Tracking and Level Set Method, there are always the errors of mass because the variables and properties are treated as continuous across the interface and this does violate the real physics. In real situation, the properties of fluids are not continuous across the interface. If there is no phase change, the pressure should be discontinuous across the interface. With phase change, both pressure and velocity should not be continuous across the interface. To simulate the phase change phenomenon, the most important thing is to obtain accurate heat fluxes across the interface. These accurate heat fluxes will be transferred to be used as latent heat of vaporization for the liquid. In numerical simulations, in order to acquire the accurate heat flux at the interface, the temperature distribution must be very accurate and grid must be very fine near the interface. Also, some information about the interface must be explicit such as the normal direction unit vector and the length of the interface. Generally speaking, in order to evaluate the heat flux accurately, three conditions must be satisfied: an accurate temperature distribution near the interface, an accurate numerical method to find the temperature derivative at the interface and enough information about the interface. In the Cartesian grid method, only SIMCC can satisfy these three conditions. In SIMCC, there is no smeared band near the interface and the interface is an explicit true surface. Table 41 lists the key issues for the heat flux computation in some popular Cartesian grid methods. Based on Table 41, the SIMCC is the only method that can obtain the heat flux at interface accurately. In SIMCC, the governing equations of each phase are solved separately and there is no smeared band that distorts the actual temperature profile. Therefore, the accurate normal direction can be obtained easily and also it is easy to calculate the area of the surface of interface. This information will be very useful for evaluating the latent heat supply for the phase change. Based on these advantages of the SIMCC, therefore, SIMCC is selected to be the numerical scheme in this research. 4.6 Heat Flux Computation at the Interface In order to evaluate the heat flux accurately, the following conditions must be satisfied: Accurate temperature fields near the interface. Accurate information of the interface. Accurate numerical method to find the temperature derivative at the interface. In SIMCC, the first condition can be satisfied easily. The interface is treated as a true surface and there is no smeared band at the interface. Because the interface exists explicitly in SIMCC, the information such as the location of the interface and the unit normal vector at the interface can be obtained easily and therefore, the second condition is also satisfied. Figure 412 illustrates the numerical technique to evaluate the heat flux in this research. For each marker point, two probing points are selected. These two marker points are located in the direction of the normal vector of the interface with an equal distance. The distance from the marker point to the nearest probing point is determined by the code. The rule is that any nearest probing point can be surrounded by four cell centers in the same phase. As shown in Figure 412, probing point 1 is surrounding by cell centers 1, 2, 3 and 4 and probing point 2 is surrounding by cell centers 4, 5, 6 and 7. Once the four points are selected, the second order accurate bilinear interpolation method can be used to obtain the temperature at probing points 1 and 2. In a cryogenic flow, because the temperature of the pipe is always higher than the saturation temperature, the temperature at the interface should be equal the saturation temperature. This type of interfacial condition is also called the Dirichlet type and the value should be a fluid property. By the temperatures at the interface and at the two probing points, Taylor expansion can be used to obtain the second order accurate temperature gradient: kT, 4Tprbe1 37LIt Tprobe_2+O(x)4) dn 2Ax All the techniques in this section including those for obtaining temperatures at cell centers and the bilinear interpolation method are of second order accuracy. It can be expected that the temperature gradient at the interface can achieve at least second order accuracy also. Here, the accuracy of the current probing point method for heat flux and latent heat computation will be evaluated by a sample test case. The geometry of this test case is a rectangle with dimensions of 5 x 10 and a fictitious interface, a halfcircle with a radius R=0.5 is placed on the bottom with the center at (5, 0). The coordinate system is axisymmetric. The heat flux and latent heat at the surface of this object will be tested based on two sets of grids and assumed temperature profiles. Figure 413 is an illustration of the geometry of this test case. The two selected grids are 64 x64 and 128xl28. Two sets of assumed temperature profiles are: T = r and T = r 2 With the original point (5,0). Figure 414 is the assumed temperature distribution. In this test case, there will be several exact solutions which can be used for validation: The area of this sphere should be: A = 4xir2 = xi = 3.1415926 The temperature gradient of each temperature distribution is : dT / dr = 1 at R=0.5 The latent heat can be computed by c)"dAdt If thermal conductivity is assumed as 1.0 and time step is also 1.0, there will be the exact solution of latent heat in one time step: ii= q xAx Bt =1x xxl = zi (425) Table 42 lists the computed result and corresponding error. By Table 42, it shows that the current method for latent heat computation is reliably accurate at least for a quadratic temperature profile and the profile can be improved when the grid size becomes smaller. 4.7 Phase Change Algorithm In this research, because of the large temperature difference, the phase change phenomenon is inevitable so that the mass flow rate at the interface can not be zero. In order to truly satisfy the conservation of mass at the interface, a "discontinuous" model is adopted. In SIMCC, the normal component of interfacial velocity is determined by the interfacial displacement over At and the contribution of normal component of interfacial velocity should come from both normal force balance condition and phase change: Ax + Ax (Un ) = forcebalance phase _change (426) where Axphase~change iS the contribution comes from the phase change and the normal component of interfacial velocity can be expressed as (Un,,) = (Un,) force~balance + (1n) phasechange (427) By conservation of mass at interface and Equation 427, the normal component of the velocity near the interface of each phase can be obtained and they will be the velocity boundary condition for each phase. This is the socalled "discontinuous" model in this research. In this model, the velocities of each phase come from the conservation of mass at the interface and are discontinuous at the interface. Therefore the conservation of mass at the interface can be enforced. Based on this idea, a phase change algorithm is developed in this research that includes the following steps: Computation of heat flux at the interface locally. Computation of the corresponding mass transfer rate. Integration of the mass transfer rate by time and surface area to obtain the total latent heat and then divided by the unit latent heat to obtain the amount of vaporization. Adjust the interface based on the amount of vaporization. Heat flux computation with second order accuracy has already been addressed in the previous section; the rest of the steps will be discussed here. The local total heat flux can be expressed as aT a7T q" = kg k; q d (428) dn 1 n Then the net heat passes the interface locally in one At for one AA should be: e = q"~AAt (429) This amount of energy should be transferred to be the latent heat locally, the local mass transfer can be obtained by m trnsfer = e / ii (43 0) When the local mass transfer is counted, the interface will be adjusted by this local mass transfer. During the adjustment procedure, the reference coordinate should be fixed at the interface and the normal and tangential directions are two axis of this coordinate. Then, the new interface can be redefined as X phase _chang X, + P nx (431) Yphase chang = o + .Yy where p is the local mass transfer coefficient and can be calculated by the local mass transfer. Figure 415 shows algorithm of this step. Now, a new interface based on the phase change is defined. The next step is to call the isothermal moving interface algorithm. Because of the imbalance of force in the normal direction, the new interface will be adjusted again to achieve balance of force in the normal direction and it is the final interface of each time step. Xe phase _change Pn x (432) Ye phase _change +.I 4 The idea behind this equation is the same as the isothermal case. The difference is that in Equation 432 an initial displacement due to the phase change has already been counted before the iteration starts. Figure 416 shows the algorithm of this step. In each time step, the total displacement of interface should be the difference between the original interface and the final new interface. Therefore, the new normal component of interfacial velocity can be obtained by (u,,)x int new = (Xnew Xold )~ (433) (Un,)y it new = (Y,,, Yold)/a where (u,,)xint new and (Un,), menew represent the X and Y component of normal component of interfacial velocity for next time step. In SIMCC, if there is no phase change, the normal component of interfacial velocity will be the velocity boundary condition of each phase and this is the same as continuous model. However, because of phase change, the interfacial velocity must be modified to satisfy the conservative law. In this research, the velocity boundary at interface will be corrected by the mass conservation as (434) The two discontinuous normal components of interfacial velocities will be used to be the velocity boundary of each phase and this socalled "discontinuous" model in this research. In Equation 434, it is very clear that if there is no phase change, it will reduce to (t,,), = (un,,)1tg = 2177 mt = 217)1nt 1 = 2177)1 (43 5) Therefore, the continuous model is a special case of the discontinuous model when there is no phase change. Figure 417 is the detailed flow chart of this step. 4.8 Global Conservation of Mass The maj or difference between the current SIM and others' work is that SIM can better preserve the global conservation of mass. During numerical simulation, it is inevitable that the errors of mass will be generated by improper initial conditions or the inherent numerical errors especially in internal flows. In order to preserve the global conservation of mass, a set of conservative strategies is applied. For the gas phase, a correction of mass flow rate is applied at the outlet of the pipe. As shown in Figure 418, the inlet mass flow for the gas phase has contributions from two parts, the mass flow through the inlet and the mass flow generated at the interface. Therefore, the total mass flow rate into the gas phase should be: viz" = di"te +ril1teac = per,,e,dA2,,,,, + plrmedqn =ri #2,, (436) The mass flow which leaves the outlet is controlled by the boundary condition. If the boundary condition is unknown, usually, it can be done by higher order extrapolation or fully developed assumption once the pipe is long enough but this computed leaving mass flow rate may not equal to the mass flow rate entering the domain. Also, the errors of mass can be caused by an accumulation of numerical errors over the thousands of iterations that need to be done. Even if the mass flow rate at outlet is not accurate at beginning, an estimated value of mass flow at outlet can still be obtained: # 2", ,,,, = pto n, _ed ,,, (4 37) By the idea of conservation of mass, the entering mass flow rate should equal the mass flow rate at the outlet if the flow is incompressible and there is no other mass source. Therefore, the velocity difference can be added at outlet: Uout, = flour so + (yj",, yj2", ,,,,) /(pAo,,) (43 8) Finally, this uo,, can be used to be the velocity boundary condition and the conservation of mass in the gas phase can be preserved. The following provides the details of the conservation of mass scheme for the inflow of liquid phase. In Figure 419, the forward shaded region represents the liquid cylindrical core shape at the time step "t" that is the initial condition for the next time step. When advancing to the next time step "t+ At ", the total entering mass is known, Am which enters the pipe during the time step is equal to the velocity at the inlet multiplied by the time step size, At The interim liquid core is composed of two parts as shown in Figure 420 by a forward slashed region and a backward slashed region. The forward slashed region in Figure 420 is made identical to that in Figure 419 except that it is moved downstream while the backward slashed region is the added mass Am So the total mass of the two parts will be the mass of liquid core at the time step "t+ At ". Even though the mass is conserved but the interface shape has not been updated, so it must be recomputed based on the interfacial conditions before we complete the "t+ At step. In summary, by this strategy, the conservation of mass is strictly enforced and also, the phase interface is updated that follows the forced balance. The details on updating interface locations are based on the marker points and their movement as discussed next. During the generation of the interim interface as shown in Figure 420, the locations of the markers have to be adjusted accordingly. In the current algorithm, the total number of markers will not change for the interim interface. As shown in Figure 421, the marker points 0, 1, and 3 represent all the markers on the old interface as. During the construction of the interim interface, as stated above, because of the added mass, the markers will have to move so that point I will move to 1', 2 to 2' and 3 to 3' in Figure 422, respectively. One important rule implemented here is that Point 0 that is anchored at the inlet for the old interface will become O' for the interim interface will not move with the old liquid core, so O' remains at the inlet. As a result, the distance between O' and 1' is larger than others that calls for restructure and update the marker system with the a new set before proceeding with the determination of the final interface for time step "t+ At " Table 41. Comparison of the key issues for different numerical methods for the heat flux computation. VOF Level IBT Front SIMCC Set Tracking Explicit interface N N Y Y Y Accurate near the N N N N Y interface Fine grid near the N N N N N interface Table 42. Comparison of exact and numerical results for the flow over sphere case. Grid 64x64 Grid 128xl28 Area/ error 3.14/ 8.12E4 3.14/ 1.93E4 Latent heat /error, T = r 3.21/ 2.08E2 3.19/ 1.38E2 Latent heat /error, T = r2 3.29/ 4.64E2 3.22/ 2.57E2 Figure 41. Nonstaggered grid system. V U fc Figure 42. Example of mixed structured and unstructured grid in SIMCC. cc ev 05 04 03 D2 01 >0O D1 D2 D3 04 0 5 0 05 x Figure 43. Example of mixed structured and unstructured grid. x A x IB Figure 44. Marker points of an oblique ellipse. A) initial ,B) after curve fitting. Figure 45. Intersection points of an oblique ellipse. m5 m3 1 4 m1\ m2 interfac e 11T1If I Figure 46. Illustration of the interfacial cells and cutandabsorption procedures in the SIMCC in local situation. Figure 47. Cutcells of different objects. A) NACA0012, B) Star shape. Figure 48. Example of cutcells of different grids. A) dx=e 0.025, B)dx~ =0.1i. 67 I II~ I' I ll  t   _ I~ I ;II I l l I, t . 4m 6 Interface Figure 49. Illustration of the interfacial variables and notation. &,~~n Ay~ A Initial Interface New Interface Figure 410. Illustration of interfacial advancing process. \ du Sold interface c new interface \A Figure 41 1. Illustration of updating cell procedure. Interface Scl ui He2 Piobe 2 & Figure 412. Two probing points for second order gradient. Figure 413. Illustration of geometry of test domain. xA IB Figure 414. The assumptive temperature distribution. A) T = r B) T = r 2 69 I r .b, I int int phas e Figure 415. The new interface because of phase change. mnt phase int o / mnt new Figure 416. The new interface because of balance of force. Determine the mass transfer rate and new interface Determine final interface velo city based on imbalance of stress Check conservation of mass Modify the interfacial velocity BC of each phase Assign the fluid properties and refresh cells Fractional step method Compute the heat flux at interface Converge or not Figure 417. The flow chart for the phase change algorithm. wall inlet interface~ symmetric line Figure 418. The global conservation of mass in gas phase. outlet Figure 419. Interim interface construction, the initial interface. time step t+Atl Hquid core (shadowl+sindow2) Figure 420. Interim interface construction, the new interface. Figure 421. Restructuring of markers, the original makers. Figure 422. Restructuring of markers, the new markers. CHAPTER 5 VALIDATION OF CODE 5.1 Introduction In this chapter, several test cases with known solutions are used to validate the SIMCC code. They are divided into three groups based on the aspects of the numerical techniques. The first group is used to validate the solver of the governing equations and the SIMCC with the Eixed interface and includes the Couette flow, fully developed channel flow, cavity flow, fully developed flow in a pipe with a constant wall heat flux and flow over a sphere cases. The second group focuses on the moving interface algorithm and includes the static droplet, the rising bubble and droplet simulations. The last group focuses on the mass transfer at the interface and includes a stationary droplet in a quiescent medium with constant mass transfer rate and the one dimensional phase change problem. 5.2 Group 1, Solver of Governing Equations and the SIMCC with the Fixed Interface In this group, the interface will not be moving but Eixed in the domain and this can help to validate some numerical techniques applied in the SIMCC code, such as the solver of governing equations, the interfacial tracking, merging procedure and the flux and stress computations at the interface. 5.2.1 The Fictitious Interface for Evaluating the SIMCC In order to validate the SIMCC while taking advantage of known analytical solutions and wellaccepted benchmark cases, the entire flow Hield is divided into two regions separated by a Sictitious interface. By comparing the solutions obtained using the fictitious interface with the known solutions, the performance and accuracy of the SIMCC can be evaluated. Detailed treatment of the fictitious interface is given below. As shown in Figure 51, since region 1 and region 2 are the same fluid, the normal stress balance at the fictitious interface is: 01 = 2 (51) The shear stress balance at the fictitious interface is: zl = z2 (5 2) Continuity of the flow variables at the fictitious interface yields: uxl = ux2 (53) uyl = uy2 (5 4) P1 = P2 (55) It should be emphasized that with an interface that separates two different phases, the interfacial conditions will not be the same as the ones listed above. Furthermore, the normal and shear stress at the fictitious interface can be expressed by the stress tensors: az = H TA 1 A 2 2 (5 6) zl = n TA 1 t = n TA 2 2 (5 7) The stress tensor at point A in Figure 51, TA On the interface from either region can be written as aux, au,, au,, p +24u 4u +4u A 1 x1 141 yl3 4u +4u p +2pu Dc yx y1 A 1 aux2 x2, y2, p +2p2, 2u 2~ Dx Sy D T =(59 A 2x2 ay2 y2, #2, 2 u p + 2u p2 ay Dx SyI A The normal and tangential unit vectors at point A can be evaluated using analytical formulas: ny, _A /2(511 tx,A y,A (512) ty,A x ",A (513) For the fictitious interface case, TA 1 Should be equal to TA 2 > It is obvious that the normal and shear stresses should be a function of ilA and PA So there are two nominal equations but three unknowns. The following is a complete procedure to evaluate the interfacial conditions. First, the pressure at each marker point is determined by the bilinear and central difference interpolations. As shown in Figure 52, two imaginative points B and C, normal and equidistant from point A are defined first. Because the shape of background cells is always rectangular, there must be four cellcenters surrounding each of the points B and C. The pressures at points B and C can be determined by bilinear interpolation. Once the pressures at points B and C are obtained, central differencing can be applied to estimate the pressure at point A with secondorder accuracy. The derivatives of velocity in Equation 58 and 59 can be determined by the Taylor series expansion: f (x + x, y +Ay) = f(,y f(x,4)[,x, y)Ax + f (x, y)Ay + 1 (514) 2 [.,(x, y)Ax2 + 2 fx y~h~ yy y2 ]+O (Ax3) The procedure is similar to determining the pressure at point A. As shown in Figure 53, eight imaginative points must be chosen first. By the bilinear interpolation, the velocity on each imaginative point can be determined. The derivatives of velocity, for example, x term in Equation 58 can be expressed using Taylor series expansion as [4ux, uxG 3u, 2Ay, (515) x 2Ax2x, 2Ay,2y with a secondorder accuracy. Here, Ay, = yF G y, 1x F G, 2 H, Iy = ,, and Ax2 H I, x. To avoid the denominator becoming zero, the length of AH can not be the same du, du, du as AG By similar operations, x= flu, ) = f (uyA) and = f(uyA) at point A of each region can be obtained. By substituting all these terms back into stressbalance conditions, Equation 58 and 59, the shear and normal stresses become functions of u, and uyA .Thus with two equations and two unknowns, the interfacial velocity at point A can be obtained. 5.2.2 Couette Flow, Fully Developed Channel Flow and Cavity Flow Couette flow has a linear velocity profie and a constant pressure distribution, while the fully developed laminar channel flow has a parabolic velocity profie and a linear pressure distribution. The cavity flow is a typical benchmark case (Ghia et al. 1982). For all three cases, Eictitious interfaces are placed inside the flow domain to examine the performance of the SIMCC. As an illustration, in Figure 54(A), a circular fictitious interface with a radius of 0. 1 is immersed into a unit square with a 128 x 128 grid. This grid is used for the Couette and cavity flow calculations. In Figure 54(B), the fictitious interface with a radius of 0.05 is immersed into a 40 x 1 sized channel with a 155 x 90 grid system in a fully developed channel flow. These two cases are used to test the solver of continuity and momentum equations for the SIMCC. 5.2.2.1 The Couette Flow In this section, the errors of the primary variables at the interface and conservative properties at interface will be evaluated. The evaluations of errors here are all based on a 128 x 128 grid. The radius of the fictitious interface is 0. 1 and the center of the interface is placed at X=0.5 and Y=0.5. In this case, there are 148 cutcells generated. The exact solution for Couette flow at each cell center of the grid is U = y, V = 0 and P = Const.. The error of interface velocity at each marker point is shown in Figure 55(A). The maximum error in this case is less than 6.0E16, essentially at the roundoff level. The error in the mass flux of each cell can be expressed as e = & #dS (516) Theoretically, the summation of these errors should be zero. In this case, the total error is about 1.0E08, which is consistent with the accuracy supported by the formula. The error in the mass flux of each cutcell is shown in Figure 55(B). In a steadystate Couette flow, the unsteady, convection, pressure gradient and the diffusion terms in the momentum equations should be zero at each cell. The convective momentum flux is adopted to verify the conservation of momentum flux at the interface. The error in convective flux of each cell can be expressed as d=I #(#dS) (517) The summation of these errors should be zero. The actual total error in convective flux of all cutcells is about 1.0E7 in this case. In Figure 55(C), the error in momentum flux of each cutcell is shown. 5.2.2.2 The Fully Developed Channel Flow In Figure 56, the test results of the fully developed channel flow case are presented. As mentioned before, the channel of length 40 starts from X = 25 and ends at X =15. The grid is 155 x 90 and the Reynolds number is 100. A uniform velocity U=1.0 is assigned at the inlet and the fully developed boundary condition is assigned at outlet. A fictitious interface with a radius of 0.05 is selected and the center of the interface is fixed at X = 0.5 and Y =0.5. The total number of cutcells in this case is 92. In Figure 56(A), the pressure contour near the interface is shown. It shows that the pressure distribution is linear along the axial direction and smooth across the interface. The fully developed velocity profile at X=0.5 is shown in Figure 56(B). The maximum velocity is 1.5 which agrees with the theoretical result. Also, the parabolic velocity profile supports the accuracy of the SIMCC. The conservative properties are also examined. In Figures 56(C)(D), the errors in mass and momentum flux are shown. The summation of errors in mass and momentum fluxes are of the order of 1E07 and 1E06, respectively. 5.2.2.3 The Cavity Flow In Figures 57(A)(C), the streamlines of the liddriven cavity flow with different sizes of the fictitious interface are shown. Here, Re =100 is employed with a 128 x 128 uniform grid system. Figure 57(A) shows the case with no fictitious interface. Cases with fictitious interfaces of radii, 0. 1 and 0.2 centered at X=0.5 and Y=0.5 are shown in Figure 57(B) and Figure 57(C), respectively. In Figure 57(D), the velocity component in the X direction along X =0.5 for different fictitious interface sizes are compared with the case without a fictitious interface. These three profiles almost overlap on one another and are very smooth across the interface. The shape/location of the main vortex, size/shape of the subvortices, and velocity component in the axial direction U at X =0.5 compare favorably with benchmark results reported by Ghia et al.( 1982). 5.2.3 The Fully Developed Pipe Flow with a Constant Wall Heat Flux In this test case, a fully developed pipe flow with a constant wall heat transfer rate will be presented. The reasons for selecting this problem to be the test case are: A. there is an exact solution that can be used for validation, B. the temperature field is influenced by both convective and diffusive effects and this is enough to validate the energy equation solver. Also, a fictitious interface is again used to divide the entire domain into two regions. The thermal interfacial condition for this fictitious interface case is continuous heat flux and identical temperature at the interface.The grid system of this case is a rectangular with 1 x 0.5 and grid point is 80 x 40. Because it is a fully developed pipe flow, the velocity boundary condition is set to be a nonslip condition at the solid wall, symmetric condition at the centerline, fully developed velocity profile at inlet and outlet. The pressure should be a linear distribution along the axial direction. The heat flux at the solid wall is set at 1.0 and thermal conductivity is set as 1.0 also. The radius of tube is 0.5. The exact solution can be expressed as (Mills 1995) T 4q" 3R +r" r4 T =T (518) SkR 164 16R2 where Ts is the temperature at wall, c)" is the heat flux and k is the thermal conductivity. To avoid the error caused by the inaccurate computation of heat flux at the wall, the thermal boundary condition at the solid wall is replaced by the equivalent temperature distribution. If k is 1.0 and q" is 1.0, the temperature distribution along the solid wall can be written as T = T(0)+ 4x (519) where T(0) is the wall temperature at X=0 and is set as 0 in this case. Therefore, an equivalent wall boundary condition at the wall is: T = 4x (520) The thermal boundary condition at the centerline is set as symmetric. The thermal boundary condition at the solid wall, the inlet and outlet can be derived by the exact solution. Figure 58 shows the results of temperature distribution. To validate the result, first, the heat flux at the wall is checked to ensure that the heat flux is c)" = 1.0 Figure 59 shows the heat flux at the solid wall and it shows that there is only a small error near the inlet and outlet and the magnitude of error is about 0.0025. In order to validate the result further, the temperature profile at X=0.48125 along the radial direction is also compared and is shown in Figure 510. It shows that the result agrees with the exact solution very well. Again, a half circle fictitious interface is used for this case. There are two different radii: 0.3 and 0.4. The fluid properties are the same in each phase and the thermal interfacial conditions for this fictitious interface are: Temperature is continuous at the interface. Heat flux is continuous at the interface. Figure 511 shows the computed temperature distribution of each case. In Figure 51 1, it shows the results are the same as those without an interface. To further validate the case, the temperature profile at X=0.48125 along the radial direction for the case with a radius of 0.3 is also compared in Figure 512. Figure 512 shows the comparison and the computed results overlap with the exact solution very well. 5.2.4 Flow over a Sphere with Heat Transfer The purpose of this test case is to check the ability of SIMCC for evaluating the heat flux at the interface. In this case, two grids with different stretching, 800 x 600 and 120 x 50 are used to demonstrate the ability of SIMCC for evaluating the heat flux at the interface. The overall dimensions of the mesh are X= (60, 60) and Y=(0,60) and a half circle is placed at the center of lower boundary. Its radius is 0.5 and the center is located at (X,Y)= (0.5,0.0). Figure 513 (A) and (B) show the grid and geometry of the sphere for the case with the grid 800 x 600. In this case, the pressure boundary conditions of upper, left and right sides are zero gradient and symmetric for the lower face. The thermal boundary condition (temperature) of upper, left and right sides are 0. The velocity boundary condition at interface is nonslip and the pressure at the interface is obtained by a second order extrapolation. The temperature of the interface is set at 1.0. The velocity at far field is set as 1.0, the characteristics length is the diameter of this ball. The Reynolds number is 1.0 and the Peclet number is 1.0 also. Figure 514 are the stream line plot and pressure and temperature contours. In this case, the average Nusselt number is suggested by Whitaker(1972): Nu = 2 +(0.4 Re'/ + 0.006 Re /)Pr0 4 (521) Based on the current dimensionless parameter, the Nusselt number of the case with Re =1.0 and Pe=1.0 should be 2.406, Table 5.1 lists the calculated result. In table 5.1, dx~min means the minimum grid size of each mesh. It shows when the grid size is decreased; better results can be obtained. When the grid is denser than 800 x 600, the error will be less than 2%. In this case, it shows that a higher resolution of results is required near the interface for the heat flux evaluation. It can also be expected that based on the same gird size, SIMCC is the method with the best performance among Cartesian gird methods and this is also the reason that the SIMCC is chosen. 5.3 Group 2, Validation for Moving Interface Algorithm In the previous group of tests, the techniques for handing the stationary interface including the governing equations solver and SIMCC are validated. In this group, the interface in the validation cases will not be stationary but moving and deforming and therefore a moving interface algorithm must be imported to handle the movement of the interface. Two cases are used to validate the moving interface algorithm and they are the static droplet and the rising bubble and droplet simulations. 5.3.1 Static Droplet Simulation The first case employed here is a static droplet in a surrounding gas. The main idea here is to introduce an imbalance of pressure at the interface of a static droplet and then examine how the code reacts to the imbalance by inducing a movement at the sharp interface that results in the pressure adjustment for the interface to return to a balanced state governed by the Laplace Young equation. Therefore, for the first case, the selected problem is a stationary water droplet surrounded by its own vapor under isothermal condition in zero gravity. The density ratio is 1,605 and the viscosity ratio is 22. If there are no other force field effects and under a dynamic equilibrium, this droplet should stay stationary and the pressure difference at the interface between the liquid and vapor should be balanced by the surface tension. The geometry of this validation problem is a twodimensional domain in an axisymmetric coordinate system with length and width equal to 2.0 and 1.0, respectively. The grid arrangement is 80 x 40. A half circle mounted on the center of the lower boundary is used to represent the interface. Figure 515 is the illustration of this problem. The boundary conditions are zero gradients for velocity and pressure at right, left and upper sides and symmetric at lower side since this is an axisymmetric computation. The initial conditions for the velocity and pressure are zero everywhere that creates a pressure imbalance at the interface. In this computation, the Weber number is selected as 0.5. Since the curvature is 4.0 everywhere, the exact pressure jump between the liquid and the vapor phase should be 8.0. Figure 516 shows the maximum induced interfacial velocity during the transient adjustment period. It must be emphasized that for the incompressible flow computation, strictly speaking there should be no induced interfacial velocity for this problem because of the stationary condition and the conservation of mass. However, because of the imbalance of interfacial dynamics, an extremely small interfacial velocity is induced. This small order of magnitude velocity distribution (deformation of interface; induced interfacial velocity) should be considered a numerical disturbance and not a violation of the incompressible assumption. However, the SIMCC code can use it to adjust the pressure difference of each phase since the disturbance of velocity will cause the change of pressure in each phase. In Figure 516, the maximum induced interfacial velocity varies and achieves the steady state finally. The maximum induced interfacial velocity is about 1.0E5 in this case. Figure 517 shows the convergent histories for the governing equations and the interfacial condition. In the beginning, the interface was adjusted seriously to satisfy the governing equations and interfacial condition. In this stage both residuals are decreasing very fast. After 100 iterations, once the residue of interfacial condition (imbalance) is too small to push the interface, the interface will become stable gradually and looks like a "fixed" interface. In the meantime, the residual of governing equations must be small enough also and this means governing equations and interfacial condition reach convergence at the same time. Figure 518 is the pressure contour for this problem. It shows the pressure difference is identical to the theoretical value, 8.0 and the discontinuity at interface can be handled very well by the current method. The sharp pressure discontinuity across the interface is successfully computed. 5.3.2 Deformed Rising Bubble and Droplet For the second case, the buoyancydriven rising bubble and droplet through a quiescent liquid are selected. The test problems in the second case are classified into two groups based on the density ratio (dispersed phase to continuous phase). These groups are the low density ratio (<0.001) bubbles and higher density ratio (= 0.91) droplets used to verify the current method with a wide range of density ratios. The computational domain used in this section is a rectangle with a length of 70 and width of 38 and the grid system is 550 x 100. From X=0 to X=20 and R=0.0 to R=2.0, the grid spacing is uniform and then the grid spacing is stretched to the upper, right and left sides. Since the domain is large enough, the boundary conditions are assigned as the zero gradients for both velocity and pressure at the upper, right and left boundaries and a symmetric condition on the lower boundary (centerline). The initial condition for this case is quiescent fluids in both phases and a halfcircle with a radius of 0.5 is used to represent the bubble/ droplet and the center of the halfcircle is located at (3.0, 0.0). The time step here is 0.01. For the bubble problem (density ratio ~ 0.001), Re = 10 and We = 8.0 are assigned. In Ryskin and Leal's research (1984), the bubble is an open space without any fluid. In this computation, water is adopted to be the ambient liquid and the bubble is water vapor, so the density ratio 0.0006 and viscosity ratio is 0.045. Figure 519 is a comparison between the current computed result and the result of Ryskin and Leal. It shows that the two agree well with each other. For further verification with the same bubble of Re = 10 but different Weber numbers of We = 2.0 8.0, the results of the aspect ratio (height to width) are compared with those of Lai et al. (2004)in Figure 520. The comparison is very favorable. For the higher density ratio problem, the droplet with Re = 10 and We = 2 is adopted for comparison. The density ratio and the viscosity ratio are assigned as 0.91 and 4, respectively so the computed result can compare directly with results by Dandy and Leal (1989). Figure 521 shows that the current computed result agrees well with that of Dandy and Leal in flow structure and droplet shape. 5.4 Group 3, Validation for Accurate Mass Transfer at the Interface Tests in this group focus on validating the phase change computation at interface. The two test cases used are: a stationary droplet in a quiescent environment with a constant mass transfer rate, and one dimensional phase change problem. 5.4.1 A Stationary Droplet in a Quiescent Environment with a Constant Mass Transfer Rate For this test case, the main purpose is to evaluate the performance of current code on the velocity discontinuity at the interface between the droplet and its surrounding fluid due to mass transfer. In order to accurately evaluate the current numerical technique, a simplified case was designed where a stationary droplet is assumed to vaporize with a given constant mass transfer flux at the interface. In this way, heat transfer is not involved in the simulation and only the continuity and momentum equations are solved. For this test case, an axisymmetric cylindrical coordinate system is employed. Figure 522 is a schematic of this problem. The computational domain is a rectangle with dimensions of 10 x 5. A hemisphere with an initial radius 0.5 is placed at the center of the lower boundary. The boundary conditions of upper, right and left sides are the second order extrapolations for both velocity and pressure and symmetric at the lower boundary. In this test case, a water droplet is used and the density of water is 958.3 kg / m and 0.597 kg/m3 for the vapor. In this computation, the numerical time step is 0.1 and the constant and uniform mass flux that leaves the droplet surface, viz is assumed to be 10.0 kg/m~s It is noted that there is no internal velocity field inside the droplet because of the quiescent environment that results in an induced interface receding velocity as follows: (un ),nt =(522) The vapor velocity at the interface can be calculated by rz= Pv ((un)v (un)m) (523) Based on the assumed mass flux, the vapor velocity at the interface is 16.76 m/s. As the velocity in the vapor phase is solely due to the uniform mass from the droplet surface, the vapor velocity at any radial distance from the droplet surface can be predicted based on conservation of mass: phuA, = p2 2A2 (5 24) The velocity of vapor at different location in the vapor phase can be expressed as  2 (525) where r is the distance measured from the center of the droplet. If the known vapor velocity at the interface is used as the reference velocity, the velocity at any location in the vapor phase can be obtained. Therefore, the obtained velocities including the interface velocity, the velocities of both liquid and vapor phases at the interface and the velocities at the outer boundary of the computational domain will satisfy both the continuity equation and interfacial mass conservation condition and therefore they can be considered as the exact solution of this test case and used to validate the numerical results. Figure 523 shows the computed velocity distribution by the current code versus the exact velocity along the radial direction at X=5 after first time step. In Figure 523, the velocity is zero everywhere in the liquid phase and a huge velocity discontinuity exists at the interface. The maximum vapor velocity is located at the interface and the velocity decays by 1/r2 to the outer boundary of the domain. A comparison shows that the computed result agrees closely with the exact solution. This proves that the current code can handle the mass transfer at the interface with a velocity discontinuity with a high accuracy. Figure 524 is the stream line plot near the water droplet. In Figure 524, it shows that the streamlines radiate from the interface to the far field straight. 5.4.2 One Dimensional Phase Change Problem One dimensional phase change problem also called Stefan problem is a classic test problem and broadly used for the validation for moving boundary/interface algorithm in computational multiphase flow with phase change (Bonacina et al. 1973, Welch and Wilson 2002, Mackenzie and Robertson 2000). Using this test problem, the performance of moving boundary/interface algorithm for phase change computation especially the accuracy of mass transfer at boundary/interface can be evaluated. The entire system is shown as Figure 525. The left forward slashed block represents the wall and it will keep stationary with a constant temperature Twit The dashdot line with a distance 3(t) from the wall is the interface and separates the gas and liquid. The liquid phase and the interface are assigned the saturation temperature Ts, Once phase change happens, the liquid will be vaporized and the volume of gas will increase so that the interface will be pushed to the right side. During the vaporization, the velocity of gas is still kept as zero. Since the velocity of gas phase is zero, the original unsteadyconvectivediffusive energy equation in gas phase can be simplified as unsteadydiffusive equation: dT 82T i = a~ 0l < ; x <; 3( t) ( 5 26) where 3(t) is the location of this phase interface and a is the thermal diffusivity and defined as a = For this moving boundary/interface problem, the system of equations is closed by pCp specifying the boundary conditions: (527) ST (x = 0(t, t)= Ts,~ ,= Twazz The conservation of mass at interface: dT pgv, A = k, (5 28) g x=3(t) The theoretical solution of temperature and location of interface for this test problem can be found by Alexiades and Solomon's book (1993): 3(t) = 2j(a (529) T (x, t) = Twazz wo Tsa' er (53 0) erf(i) 2~a where erf is the error function and the parameters r can be obtained by solving the transcendental equation: re erf (r) Cp won' sa' (53 1) In this validation case, the fluid properties of each phase are assigned in Table 52, and by this set of properties, the parameter r is solved as 1.0597. Although it is a one dimensional problem, it is computed by the two dimensional code in order to apply the Cutcell approach at the interface to obtain the best resolution. For the upper and lower side, the boundary conditions are assigned symmetric to simulate one dimensional condition. The computation domain is a 1.0 x 0.02 rectangle and the grid point are 500 x 10 ,the interface is located at 3(t) = 2j(u = 0.2 initially and corresponding time is 0.089. The initial condition can be obtained by the theoretical solution. Figure 526 is the obtained results. By this set of figures, it can be seen that the present results match the theoretical results very well. In Figure 526(A) and (C), they show the accuracy of the current energy equation solver, moving interface/boundary algorithm and the mass transfer at interface. An accurate interface position can be obtained only when all the above techniques work well. In Figure 526(B), all the isothermal lines are purely vertical which means that the current Cutcell approach can achieve truly high resolution. Overall, by this test problem, the energy equation solver, the moving interface algorithm, the Cutcell approach and the accuracy of heat flux computation at interface are validated further. 5.5 Summary In this chapter, the ability of SIMCC is examined systematically. In the first group, it has been demonstrated that the current solver of governing equation is very accurate and the SIMCC only induced tiny errors at the interface. In the second group, by compared with others' results, it has been shown that the SIMCC can handle the moving interface problems well. In the last group, by the two test cases, it has been shown that SIMCC can compute the mass transfer rate from the interface correctly and adjust the interface due to the phase change very accurately. By this series of test cases, the capability of the full set of SIMCC technique to handle the current research is established. Table 51. The error of Nusselt number by different grid resolutions in flow over a sphere case. Grid Nusselt dx~ min Error Case 1 120x 50 1.14 0.05 52.3% Case 2 800x 600 2.36 0.003 1.9% Table 52. The fluid properties of liquid and gas phases for the one dimensional phase change problem. Gas 0.2 5.0 1.0 N/A Liquid 1.0 10.0 1.0 1.0 Density (p) Heat capacity ( Cp ) Thermal conductivity (k) Latent heat( (2) interface ~1 region 1 region 2 Figure 51. The continuous stress condition at marker points for the fictitious interface. / cell Figure 52. The algorithm for second order gradient. marker point > DS o~ OCI Figure 53. Eight points method for second order gradient of velocity. O Figure 54. Geometry of the Couette flow, cavity flow and fully developed channel with the immersed fictitious interface. A) Couette flow and cavity flow, B) fully developed channel flow. inteface 5E16t 4E16 1E09 1E16 50 100 150 ou50 100 Mark~ers ACutce lsB 8E08 4E08  4E08 8E08 50 100 Cutcells Figure 55. Error in interfacial velocity at all the interfacial marker points (163 points) and error in mass and momentum fluxes of each cutcell (148 cutcells) in the Couette flow with radius of fictitious artificial interface R=0.1. A) error in interfacial velocity, B) error in mass flux, C) error in momentum flux. 6; P 0+ I IS Cutcells Cutcells Figure 56. Pressure contour, U velocity profile, and error in mass and momentum fluxes of each cutcell (92 cutcells) of fully developed channel flow with radius of fictitious interface R=0.05 and Re=100. A) pressure contour near the artificial interface, B) velocity profile at X=0.5, C) error in mass flux, D) error in momentum flux. No artiflicia interface Radius=0.1 sC Radius=0.2 S0 25 0 5 0 75 Uvelocity Figure 57. Streamline plots and U velocity profile at R=0.5 with different fictitious interface radii of cavity flow with Re=100. A) streamline plot with no interface, B) with radius=0.1, C) with radius=0.2, D) velocity profile along X=0.5. U 2 0 0B 0 0 02 0 06 X Figure 58. The temperature distribution of fully developed pipe flow with constant heat flux case. Figure 59. The heat flux at wall of fully developed pipe flow with constant heat flux case. 17 R Figure 510. The temperature profile of fully developed pipe flow with constant heat flux case at X=0.481235 along radial direction. ~:~a: 1 I~ie Pl 01 (B PB X Figure 511. The temperature distributions of fully developed pipe flow with constant heat flux case with different fictitious interfaces. A) R=0.4, B) R=0.3. Radius=0.3 d J 0.1 0.2 0.3 O 4 0.5 1.9 1.7 1 6 Figure 512. The temperature profile of fully developed pipe flow with constant heat flux case with a fictitious interface (R=0.3) at X=0.481235 along radial direction. 10 ' o cr. Figure 513. Grid and geometry of flow over a sphere with heat transfer. A) the gird system, B) the interface. 1 01 O 01 1 1~ 1 X :/I  9r .1 o 2 Figure 514. The pressure contour, the stream line and temperature contour of flow over a sphere with Re=1, Pe=1. Gas 0.5 Figure 515. Schematic of stationary droplet problem. Exact solution SPresent omputation r0.02 00 '' 0 10 150 2100 Iterations Figure 516. The maximum induced interfacial velocity of stationary droplet problem. Figure 517. Convergent histories of the interfacial condition and the governing equations of stationary droplet problem. A) interfacial condition, B) governing equations. 6 * 3* X E~ Lai etal.(2004) 0 B  Present computation 07 206 04 We i I i I L.L ~ i i ~ i I Figure 518. Pressure contour for the stationary droplet. along X axis at R=0. A) entire pressure contour, B) pressure Ryskin Leal (1984)cc3 Figure 519. The shape and streamline plot for a bubble with Re=10 and We=8. Figure 520. The aspect ratio plot for a bubble with Re=10.0 and We= 2.08.0. Vapo r IZ Droplet 10 ' Figure 522. Geometry of static droplet with a constant mass transfer rate problem. Dandy & Leal (1989) Present computation Figure 521. The streamline and shape plot for a droplet with Re=10 and We=2. mPreasntCanpumblan  O Triesty 2 Z R 4 5 Figure 523. Velocity distribution of static droplet with a constant mass transfer rate problem along radial direction at X=5. Figure 524. Streamline plot near the interface of static droplet with a constant mass transfer rate problem. Gas In Liuid Ug=0iT=m Wall, T= Twan Interface, T= Tsat Figure 525. Illustration of one dimensional phase change problem. 100 Present Computation 02]01 002 003 004 Time Figure 526. The location of interface and temperature distribution, contour for one dimensional phase change problem. A) temperature distribution, B) temperature contour, C) location of the interface. CHAPTER 6 LIQUIDGAS TWOPHASE FLOWS INT A PIPE WITHOUT PHASE CHANGE 6.1 Introduction In this chapter, the main focus is the liquidgas twophase flow in a pipe without phase change. There are two studies discussed in this chapter. The first study is isothermal twophase flow, and the selected liquid is the nitrogen at 77K at 1 ATM. Since 77K is the saturation temperature of nitrogen, the nitrogen can exist as both liquid and vapor. Since the temperature is Eixed, the energy equation will not be solved and only the continuity, momentum and interfacial conditions are solved. There are two dimensionless parameters that affect the flow in this case: the Reynolds number (Re) and the Weber number ( We ). In real cryogenic flow, the Reynolds number is usually very high. Therefore only cases with higher Reynolds number are considered here. By definition, Reynolds number and Weber number can vary independently according to the surface tension. Therefore both high and low surface tension cases are possible and will be discussed. The second case is twophase flow with heat transfer but no phase change. In this case, the proposed fluids are liquid water at 273K and air at 373K at 1 ATM. Based on the assumptions, water will not be vaporized with just simple heat transfer taking place at the interface. The energy equation will be solved and another dimensionless parameter Peclet number ( Pe ) will be included into the energy equation. Similar to the isothermal case, only the cases with higher Reynolds number are considered here. Again, once the Reynolds number is decided, the Weber number may be large or small and therefore two cases of surface tension will be discussed here. The Peclet number can not be assigned independently and should be determined based on the Prandtl number (Pr). The Prandtl number is a fluid property and therefore, once Reynolds number and species of the fluid are determined, the Peclet number is determined also. 6.2 Isothermal LiquidGas TwoPhase Flows in a Pipe In this section, the computational method presented in previous chapters is employed to numerically simulate a liquid cylindrical plug translating through gaseous medium in a circular pipe. The purpose is to demonstrate the capability and accuracy of the current method for a pressuredriven internal twophase flow with a moving interface due to shape deformation and liquid phase translation. The ratios of fluid properties of two phase flows are: Pt352.4; = 15.8 The sample problems are classified into two groups. For liquidgas twophase flows, as indicated by the governing equations presented in previous chapters, the Reynolds and Weber numbers are the only dimensionless parameters needed to characterize and distinguish different fluids and flow regimes. Two maj or groups are adopted here. The first group is used to focus on the effects of different Reynolds numbers. In this group, there are four Reynolds numbers selected: 250, 500, 1000 and 2000 and the Weber number is Eixed at 500. For the first group, with the combinations of Reynolds and Weber numbers, it could represent fluids such as liquid nitrogen or water. For the first group, the obj ective is to examine the convection effects characterized by various Reynolds numbers. In the second group, the Reynolds number is Eixed as 500 and three different Weber numbers that correspond to three different strengths of surface tension are selected. These three surface tensions are equal to 1.0, 0.1, and 0.01 time the pressure difference that result in three different Weber numbers of 2.63 16, 26.3 16 and 263.16, respectively. For this group, the main challenge is to demonstrate the ability of current SIMCC code to handle the large deformations due to the surface tension effect. 6.2.1 Effect of the Reynolds Number For this group, the inflow boundary conditions for the velocities of both phases at the inlet are assigned as 1.0. While for the pressure, 1.0 is set for the vapor phase and 1.00526 ( p, = pg + curvature / We ) for the liquid phase, respectively, to include the surface tension effect between the two phases at the inlet. This strategy will be applied in all following cases to make the first point of interface is fixed at the same location in the inlet boundary during the computation. The time step in this computation is set at 0.001. First, the flow streamlines in the twophase zone and those immediately after it are plotted in Figure 61 for t = 6.0 and Re = 2000. In the twophase zone, the streamlines are almost straight and parallel as expected. The transition from the twophase zone to the singlephase pure gas zone near the front of liquid core is very smooth. The lines are slightly distorted in the vicinity of the liquid front as the gas flow is trying to fill the entire pipe interior. In this type of flow, since the velocity at wall is zero and therefore, the flow will be pushed towards the center of pipe for a fully developed parabolic profile because of the mass continuity. Therefore, the streamlines of gas phase near the front of liquid core get distorted to satisfy the continuity of mass. Figures 62(A), (B) and (C) show the pressure maps at t = 6.0 with Re = 2000. Figures 62 (B) and (C) focus on the twophase zone and single phase zone, respectively, while Figure 6 2(A) provides pressure distribution for the entire computational domain. For the single gas phase (downstream of the liquid core), Figure 62(C) shows that the pressure decreases linearly along the axial direction that is similar to a fully developed singlephase gaseous pipe flow. For the twophase flow zone as shown in Figure 62(B), the pressure driven flow patterns are displayed in the gas and liquid parts, respectively. The slight discontinuity is seen at the liquidgas interface due to surface tension, dynamic head and shear stress effects. In general, the entire domain is composed of three subdomains that all show approximately linear pressure gradients. This also shows the ability and accuracy of the current SIMCC for multiphase computations. Figure 62(D) shows the axial pressure distributions at the pipe wall, radial location R = 0.2 and the center of pipe. Basically, the three curves are very close to one another so the pressure is relatively independent of the radial location. In general, the pressure profile is composed of two linear curves (one for the twophase zone and the other for the singlephase zone) of different gradients that are j oined at the frontal area of the liquid core. The gradient in the twophase zone is much larger than that of the singlephase zone because the gas phase in this twophase zone must accelerate and create a jet profile. Also this is due to the fact that the gasphase controls the pressure drop. The sharp drop in pressure in the centerline curve is due to surface tension. For the purposes of examining the ability of the current method to accurately simulate the transition from the twophase zone to the singlephase zone, Figure 62(E) and (F) were plotted to provide three radial and three axial local enlargements of pressure distributions in the neighborhood of the liquid core frontal area. In Figure 62(E), it shows that axial pressure distributions along the centerline and at R = 0.2 are very similar to each other, except that the sharp drop discontinuities are located at different axial locations according to where the interface is positioned. As expected, the pressure distribution at the wall is continuous without any discontinuity and has a smooth transition between the two zones. The three curves all merge together after the liquid front in the pure gas zone. In Figure 62(F), three pressure distributions in the radial direction are plotted at different locations, X = 6.2, 6.4 and 6.6. For X = 6.6, basically, it is located in the pure gas phase and hence, it is virtually a straight line. At X=6.2, there is drop between R=0.350.37 and this is the location of interface. At X=6.4, the drop exists at R=0. 170. 19 and the interface is located at this section also. Inside the liquid phase, there are very slight pressure fluctuations prior to the drop and the pressure rises slightly after the drop in the gas phase. Both pressure variations are believed to be due to the changes in the gas phase immediately after the interface where the gas flow is slowing down and bending towards the center. The above can be further explained by examining Figure 62(G). Figure 62(G) is a plot of the curvature along the liquidgas interface. The plot is based on about six hundred marker points placed on the interface with uniform separation distance between adj acent points. Point 1 is at the entrance and last point is at the centerline. The curve shows that the curvature is about 5.4 at the centerline (last marker). In this computation, the effect of viscosity is very weak because of the high Reynolds number and the velocity is continuous at interface. With these conditions, the interfacial condition can be simplified as p1Pv (61) We By this simplified interfacial condition and the dimensionless parameters, it can be estimated that the order of pressure discontinuity at the interface is about r / We According to the curvature at the liquid front in Figure 62(G) and the assigned Weber number, the pressure discontinuity can be roughly estimated about 0.0108 = (5.4/500). This pressure differential is verified in Figures 62(E) and (F). In summary, the magnified pressure plots given in Figure 62 demonstrate that the current method is capable of capturing the pressure discontinuity and flow transition from a twophase condition back to a singlephase condition with fidelity and accuracy. Figure 63 shows the pressure distributions along the centerline of the pipe for We = 500 and four different Reynolds numbers: Re = 250, 500, 1000 and 2000. For each Reynolds number, the pressure distribution is composed of two straight lines of different slopes as explained before for Figure 62(D). In general, the rate of dimensionless pressure drop is inversely proportional to the Reynolds number. For the case with the highest Reynolds number of 2000, the pressure drop is only around 10% of the case with the lowest Reynolds number of 250 and this trend is the same as that in the single phase flow because the pressure difference will be used to balance the shear stress occurring at the wall. In each curve, the pressure shows a small drop at the front of liquid core and the magnitude is about 0.011 for all the cases. This is reasonable since the strength of surface tension in each case is about the same. Figure 64(A) shows the velocity profiles at different axial locations: X = 2, 4, 6 and at the exit of the pipe. Since the flow is twophase for X < 6.4, the velocity profile at any axial location comprises of two parts corresponding to the phase. In the liquid region the velocity is uniformly 1.0. In the gas phase portion, because the velocity is zero at the wall and nonslip between the two phases at the interface, the gas profile would have a jet effect as shown in Figure 64(A) and the maximum velocity is located near the liquid phase since the velocity of liquid core is about 1.0. In the twophase zone, the velocity profile relatively does not depend on the axial location. In the current method, the global conservation of mass is the key element and therefore, the global conservation of mass must be satisfied and verified. For this twophase computation, both phases are assumed incompressible. Therefore, the rate of the "volume" of fluids in and out of the pipe should be the same at any instant. This implies after the flow in the singlephase gas zone reaches the fully developed condition, the gas flow velocity profile should be identical to the conventional single phase fully developed flow in a pipe that is a parabolic profile. Since the gas flow has reached the fully developed condition well before the exit, the velocity profile named as "U at X = Exit" in Figure 64(A) and (B) should be the theoretical parabolic profile with maximum velocity of 2 at the centerline and zero at the pipe wall. The comparison between calculated velocity profile and the theoretical result is given in Figure 64(B). It is seen that the agreement is excellent. Figure 65 shows the shape of liquid cores for different Reynolds numbers at the same time, time=6.0. As expected, there is no significant difference among these cases because the strength of surface tension is about the same here as the Weber number is the same for all four cases. After 6000 time steps, the front of liquid core does show slightly different deformation due to the convection effect. 6.2.2 Effect of the Weber Number In this group of simulations, the Reynolds number is kept at 500 while the purpose is to evaluate the ability of the current method to handle larger deformations due to higher surface tension effects. The boundary conditions are very close to those used in group one except at the entrance. The radius of liquid core is 0.38 at the entrance and therefore, the curvature will be 2.63 16 = (1.0/0.3 8) in the theta direction. If the Weber number is assigned as 2.63 16, the pressure discontinuity at the interface between the two phases at the entrance should be 1.0 (Kc We ). With this procedure, once the curvature is fixed, the pressure difference can be adjusted by changing the Weber number. Based on this idea, three different Weber numbers are selected in this group: 2.6316, 26.316 and 263.16 to generate three pressure jumps, pliquld ga a, t the interface of 2.0, 1.1 and 1.01, respectively at the entrance. Figures 66(A), (B) and (C) show the pressure maps at t = 6.0 with Re = 500 and We= 2.63. Figures 66(A) and (B) focus on the twophase zone and singlephase zone, respectively, while Figure 66(C) provides pressure distribution for the entire computational domain. In general, the features in Figure 66 are similar to those in Figure 62 but both Reynolds and Weber numbers are smaller for the current plots. In this case, the pressure jump between the two phases is equal to 1.0 at the entrance. At the front of liquid core, because the surface tension is much stronger due to a larger curvature than that at the entrance, the pressure in the liquid phase at this front of the liquid core is raised to balance the larger surface tension. In Figure 66, it also shows the larger deformation due to the surface tension at the front of the liquid core. For the entrance portion, there should be no deformation because the equilibrium condition is assigned there. This shows the ability of SIMCC to handle the surface tension effect and large deformation. Figure 66(B) focuses on the pressure contour at downstream of the liquid core. It shows the downstream pressure deceases linearly that is similar with the single phase flow. Figure 67(A) provides the pressure distributions along the wall and centerline of the pipe. It shows that the pressure distribution at the wall is constructed by two linear profiles of different slopes, and this trend is very similar with the case of wall pressure distribution as shown in Figure 62(D) for different Reynolds and Weber numbers. For the liquid phase at the centerline, the pressure decreases linearly near the entrance and rises up before going through a large discontinuous drop across the interface at the front of liquid core. Figure 67(B) is the curvature along interface. Similar to Figure 62(G) in group one, Figure 67(B) shows the curvature values of the liquidgas interface. The general trend in Figure 67(B) is also similar to that in Figure 6 2(G). Near the entrance, the curvature is almost constant because only theta direction (tangential) is involved. Near the liquid core front, the curvature will increase due to the contribution from the RX plane (curvature is counted in two directions.). Figure 68 is the streamline plot for Re=500 and We=2.63 at time=6.0. Similar with the case in group one, in the twophase zone, the streamlines are relatively straight for both phases. The difference is that slightly more bending of the gas streamlines near the top of the liquid core for the current case is observed due to the liquid core deformation. Figure 69(A) shows the U velocity profile at different axial locations at time=6.0. These profiles are very similar to those in Figure 62 for group one. The velocity distribution at the exit also achieves the fully developed parabolic profile and again it shows that the current code can maintain a global conservation of mass very well. In Figure 69(B), the U velocity profile at the exit is compared with the theoretical result of a fully developed velocity profile and the two match with each other very well. Although there is a relatively large deformation, the SIMCC can still strictly maintain the conservation of mass. The effects of different surface tension strengths are shown in Figure 610 for the three different Weber numbers. It must be emphasized here that the corresponding inlet pressure boundary conditions between the two phases are also different due to differences in surface tension coefficients. In this figure, as expected it shows that the degree of liquid phase shape deformation is inversely proportional to the Weber number as the higher Weber number the lower the surface tension coefficient as a relatively flat interface corresponds to very small surface tension. It is noted that the degree of deformation is not a linear function of the Weber number. As given in Figure 610, the difference in deformation between We = 26.3 and We = 263 is much less than that between We = 2.63 and We = 26.3 even though for both cases the change in the Weber number is one order of magnitude. Figure 611 shows the liquid core deformation as a function of time as it moves downstream for the case with We = 2.63 and Re = 500. The results show that the degree of deformation increases with time as the liquid plug travels downstream. 6.2.3 TwoPhase Flow Pressure Drop As indicated in the above, the pressure gradient in the twophase flow is much higher than that in the singlephase flow. It is important to summarize the effects of different Reynolds numbers and Weber numbers on the magnitude of increase in pressure drop in the twophase zone. The ratio of the averaged local friction coefficient in the twophase zone to the fully developed singlephase gas flow friction coefficient is plotted in Figure 612 as functions of the Reynolds number and the Weber number. In Figure 612(A), it is seen that the friction coefficient ratio increases with increasing Reynolds number almost linearly. The increasing trend of the ratio with the Reynolds number is thought to be due to the increased drag between the two phases as the Reynolds number gets higher, while the trend for different Weber numbers is totally different as shown in Figure 612(B). The friction coefficient ratio decreases exponentially in the low Weber number regime and then relatively flattens out with a very small gradient. This trend is readily explained by the degree of shape deformation as shown in Figure 610: more deformation will result in higher drag between the two phases. A much larger change in shape between We = 2.63 and We = 26.3 than that between We = 26.3 and We = 263 is noted. As a result, the drop in friction coeffieient ratio is much larger between We = 2.63 and We = 26.3 than that between We = 26.3 and We = 263. 6.3 Grid Refinement Study In the numerical simulation, the results may be not accurate due to the resolution of grid especially for the coarser grid system. In this section a grid refinement study is performed to ensure the adopted resolution of grid system is fine enough in this research. The case with We=2.63 in section 6.2.2 is adopted here. There are three grid resolutions selected. The minimum grid lengths are 0.02, 0.01 and 0.005. Figure 613 shows the locations of interface by the three grids at the same time and it shows that there is not much difference between three interfaces. In order to save computational time while preserving accuracy, unless otherwise mentioned, all following simulations in this study will adopt 0.01 as the minimum grid spacing. 6.4 LiquidGas TwoPhase Flows in a Pipe with Heat Transfer In this section, for the sample calculations, the selected two phase fluids are liquid water and air at 1 ATM. The physical conditions of the twophase flow and heat transfer system are described as follows. In the beginning (t = 0) the pipe is filled with only air which is in thermal equilibrium with the wall at 373.14 K. When the transient starts (t > 0), the gate at the inlet of the pipe is opened and an annular twophase flow with water in the center core and air filling the space between the wall and the water, both phases at 273.15K, are entering the pipe at the same and uniform velocity. The wall is maintained at 373.14 K during the transient. With this maximum temperature difference between the two phases, there is no phase change and only sensible heat transfer at the liquidgas interface. The selected two phase fluids are liquid water at 273.15K and air at 373.14K. The ratios of fluid properties of two phase flows are: Pt 1055.07 # = 81.39 k = 17.99 ; = 4.17 The corresponding Prandtl number for water should be: Pr = 13.0 The time step in this computation is set as 0.001. It is noted that for the current system, there are three dimensionless parameters, Reynolds, Peclet and Weber numbers that must be specified. For the Weber number, two cases are selected, one low Weber number at 2.63 and the other high Weber number at 500. Different Weber numbers can be used to evaluate the effects of shape deformation. The Reynolds and Peclet numbers for the current study are selected as Re = 500; Pe = 6500 In the high Weber number case, extra two cases: (1) Re= 250 and Pe= 3250 and, (2) Re= 1000 and Pe= 13000 are also provided to investigate the effects of different Reynolds and Peclet numbers. 6.4.1 Low Surface Tension Case For this case, the Weber number is set at 500 such that the inertia force overwhelms the surface tension force. As a result the shape change of the liquid slug is relatively small. In Figure 614, the pressure and flow field results are presented. The bold dashed line represents the location of the liquidgas interface. Figures 614(A) and (B) show the isobaric pressure contours on the RX plane at time= 5.5. In Figure 614(A), it is seen that the pressure decreases linearly in the streamwise direction for both phases. In general, the magnitude of shear stress at the wall and the pressure jump across the liquidgas interface determine the pressure gradients in the liquid phase. Therefore the pressure of the liquid phase is larger than that of the gas phase at the same downstream location due the surface tension effect that causes a pressure jump at the interface. In the liquid phase away from its frontal area, as indicated in Figure 614(B) where the pressure contour near X=3 is focused locally, it shows that the pressure jump across the interface is very close to the theoretical value (p, Pg ~ rc / We~ 2.63/500=0.00526 as the curvature is 2.63 obtained from Figure 614(D)) since the liquid pressure is influenced and controlled by the gas phase because that the liquid phase is fully surrounded by the gas phase that carries the bulk of the inertia force. In general, there is no pressure gradient in the radial direction for both phases. At the front of liquid slug, the curvature increases to 5.5 as shown in Figure 614(D) because of the additional curvature in the RX plane and therefore, the pressure jump in the front of the liquid core can be estimated as 5.5/500=0.011 and this pressure jump can be verified in Figure 6 14(C) where the pressure distribution along the centerline is plotted. After the front of liquid core, the gas flow diffuses into a single phase flow occupying the entire pipe. For downstream locations far away from the liquid front, the pressure decays linearly as the flow adjusts into a fully developed singlephase pipe flow. Figure 614(E) is a streamline plot. In Figure 614(E), it shows that the streamlines deform in the immediate downstream of the liquid front towards to the centerline of the pipe in the singlephase region. Because of the nonslip condition and continuity requirement, the fluid near the solid wall will be pushed toward to the center of pipe and the maximum velocity takes place at the centerline when the singlephase flow is fully developed. In Figure 614(F), it shows that the velocity profile at the outlet of pipe reaches the fully developed condition. Also, in the twophase region, due to a large density difference, the gas flow can not affect the liquid substantially, therefore, the gas phase must take the form of jet flow to satisfy the continuity and it can be seen in Figure 614(F). Similarly, the shape of the liquid core will not deform too much and the interfacial velocity is close to the inlet velocity (~1.0). Based on results provided in Figure 614, it shows that the current numerical method can accurately enforce the interfacial condition, the continuity, and momentum equations for the flow field computation. Next, the heat transfer results are presented. Figure 615 is the isothermal temperature contour and the interfacial temperature distribution for the case of We=500, Re=500 and Pe=6500 at different time steps of 1.5, 3.5, and 5.5. According to Figures 615(AC), they basically show that the moving liquid slug is a heat sink with a large capacity, because the thermal capacity in the liquid phase is much larger than that in the gas phase, the temperature changes in the liquid core are much less than those in the gas phase. The heat transfer in the gas annular portion is mostly fully developed as the isotherms are parallel to one another after a very short entrance region. The isotherms then bend to become relatively parallel to the liquid slug frontal area that again shows the flow field effects and the heat sink nature of the liquid phase. Since this is an unsteady computation, the highest temperature of liquid core is located at the rightupper corner because this local region is heated by the gas phase of the highest temperature gradient. In Figure 615(D), the timedependent temperature distributions of the liquidgas interface are plotted, which basically demonstrates the heating up of the horizontal portion of liquid phase and much less heat transfer in the vertical frontal portion. The general heat transfer effectiveness from the pipe wall is measured through the dimensionless Nusselt number, Nu as defined below: h D qD Nu = (62) k, k, (T T,) where (s is the heat flux from the solid pipe wall to the gas flow and Tm is the mean flow temperature at each downstream location that includes the contribution from both phases in two phase region. The Nusselt number along the pipe wall in the downstream direction have been plotted in Figure 616 for the case of We=500, Re=500 and Pe=6500 at three instants of t = 1.5, 3.5and 5.5. As can be seen from Figure 616, the Nusselt number value is basically a reflection of the respective heat transfer mechanism that is dictated by the flow field. In the region from the entrance to around X = 0.5, the Nusselt number takes a sharp drop from a large value to reach a value of 8.9 that is the socalled entrance region. After the relatively very short entrance region, the heat transfer is fully developed and the fully developed portion is continued until reaching the frontal area of the liquid slug. After the liquid front, the Nusselt number goes through a sharp decrease again as the gas flow adjusts to the single phase condition. The adjustment portion is also very short and the Nusselt number quickly approaching the value of 3.66 which corresponds to the Nusselt number for a fully developed singlephase flow in a circular pipe with a constant wall temperature. It is worth noticing that the enhancement of heat transfer due to a liquid slug is quite extensive as that the heat transfer is increased more than 140% (Nu from 3.66 to 8.9). The gas jet flow in the twophase region raises the Nusselt number from 7.3 (pure conduction in the annulus) to 8.9. The effects of the Reynolds number on the heat transfer in the twophase region are examined in Figure 617. Figure 617(A) shows the Nusselt number distributions for Re= 250 (Pe=3250), 500 (Pe=6500) and 1000 (Pe=13000) at time=5.5. First, it shows that the entrance length is proportional to the Reynolds number even though the entrance lengths for all three cases are relatively short. For the Nusselt numbers in the fully developed region, as expected, it shows that the higher the Reynolds number, the higher the Nusselt number as the Peclet number is proportional to the Reynolds number. But the differences among the three Nusselt numbers are less than 0.03 as seen in the closeup around X = 4.7, therefore the Reynolds number effect is really small. Figure 617(B) provides the temperature profiles for the liquidgas interface for the three different Reynolds cases. The heating of the liquid slug is relatively faster for the lower Reynolds number. This is due to the fact the more heat is convected downstream to the single phase region rather than to the liquid slug for higher Reynolds numbers. 6.4.2 High Surface Tension Case For this case, the Weber number selected is 2.63 to represent the condition where the surface tension is dominant over the inertia. The Reynolds number and the Peclet number are maintained at 500 and 6500, respectively for this case. Under a large surface tension force, liquid slug would exhibit a substantial shape change near the upperright corner as shown in Figure 6 18. The result of a dominant surface tension force in a twophase flow is the roll up of the liquid slug front around the top corner. Therefore, the interface near the front of liquid core will deform to pull backward. Figure 619 displays the isothermal temperature contours and the liquidgas interfacial temperature profiles at different times of t = 1.5, 3.5, and 5.5. Basically, the general trends of the heat transfer characteristics for the low Weber number case are similar to those in the high Weber number case but there are some differences due to the shape change that will be discussed below. In Figures 619(AC), it shows that the interface near the front of liquid core deforms as the liquid plug moving downstream. Because of the substantial deformation at the front of the liquid core, the crosssectional area of gas phase between the liquid phase and solid wall decreases, that results in the local acceleration of the gas velocity. The increased gas velocity around the deformed liquid frontal area enhances not only the convective heat transfer from the pipe wall but also the heating of the liquid phase. In Figure 619(D), the higher interface temperatures at the deformed portion of the liquid slug reflect the enhanced heat transfer from the gas flow. As the liquid phase deforms more when moving further downstream, the affected interface temperatures also rise higher. The Nusselt number distribution along the pipe wall is given in Figure 620. Again, the general trends are similar to those for the high Weber number case except that there is a local peak corresponding to the enhanced heat transfer due to the deformation of the liquid slug front corner as explained above. The local heat transfer increases range from 10% at time=1.5 to 20% at time=3.5 and then to 30% at time=5.5. The general heat transfer enhancement over the pure conduction case in the annular region and the smooth transition from the twophase region to the fully developed singlephase region are all similar to those in the high Weber number case. To facilitate a close comparison between the high and low Weber number cases, Figure 6 21 is provided for this purpose. Figure 621(A) is a comparison of the interfacial temperatures for the low and high surface tension cases at the same time instant. Starting from the inlet and continuing until the deformation begins, the two curves are almost overlapped with each other because both the gas flow and the geometry are very similar. Near the deformed liquid area, there is about 50% temperature difference. At the relatively vertical front of liquid core, the temperature differences between the two cases become very small because the heat transfer is no longer affected by the shape deformation. In Figure 621(B), it is the comparison of the Nusselt number distributions. The only difference between the two cases is again due the deformation which causes a maximum 25% increase in the local Nusselt number at the peak location. 6.5 Summary In this chapter, the liquidgas twophase flows in a pipe without phase change are computed and it includes the isothermal and pure heat transfer cases. For the isothermal case, it shows that velocity fields and pressure distributions were correctly computed for both twophase and singlephase zones and a smooth transition between the two was also obtained. In the twophase zone, the liquid velocity profile is basically unidirectional and uniform while a jet profile was found in the gas phase. Almost immediately downstream of the liquid core front, the singlephase gas flow establishes the fully developed parabolic profile. The pressure is virtually independent of the radial direction and the pressure distribution in the axial direction is composed of two straight lines of different slopes for the twophase and singlephase zones, respectively. The dimensionless pressure gradient in the two phase zone is inversely proportional to the Reynolds number. The ratio of the averaged local friction coefficient in the twophase zone to the fully developed singlephase gas flow friction coefficient increases with increasing Reynolds number almost linearly, whereas this ratio decreases exponentially in the low Weber number regime and then relatively flattens out with a very small gradient for intermediate and large Weber numbers. The liquid core shape deformation is relatively insensitive to the Reynolds number, however the deformation is large for small Weber numbers and is negligible for intermediate and large Weber numbers. For the case with heat transfer, irrelevant to the Weber number, in general the moving liquid slug in the central core which induces a jet gas flow in the annulus between the pipe wall and the liquid slug serves as the main heat sink of a large capacity. The combination of the gas jet flow and the large sink capacity in the twophase flow region results in a heat transfer increase of more than 140% over that in a fully developed singlephase flow. Immediately downstream of the liquid slug, the gas flow adjusts quickly to establish as a fully developed singlephase flow and the local Nusselt number along the pipe wall reflects that process. The effect of different Reynolds numbers on the heat transfer in the twophase region is negligibly small. For the low Weber number case where the surface tension substantially overweighs the inertia force, the resultant shape change in the form of rollup and bending backwards around the upper corner of the liquid slug reduces the local gas flow area in the annulus. This flow area reduction creates a increase in the gas velocity that in turn causes a local heat transfer enhancement ranging from 10% to 30% depending on the length of time the liquid slug spends in the pipe. 08 X Figure 61. Streamline of the isothermal case at time=6.0 with Re= 2000 and We=500. P 090 Ilgg 04f 090 095 O(r~ PB1 092 041 I 09 I urrio opos osos UOPJ OgU1 0884 OBgl OBBe PB87 IUPBO I I ~I 20 Figure 62. The pressure contours/di stributions at time=6.0 with Re=2000 and We=5 00. A) entire pressure contour, B) near the entrance, C) downstream part, D) distribution at center and wall, E) at different locations along axial direction near the front of liquid core, F) at different locations along radial direction near the front of liquid core, G) curvature along interface. 120 100 2010 300 400 500 500 Markers Figure 62. Continued. * Ra=250 * Re2000l Figure 63. The pressure distributions along the centerline based on different Reynolds numbers at time=6.0. Figure 64. The U velocity profile at time=6.0 with Re=2000 and We=500. A) at different locations, B) comparison of velocity at the outlet. X $,1 S.Z E Figure 65. Shapes of liquid cores with different Reynolds numbers at time=6.0. I. 1.5 * D 10 20 5 6 5 5 4 5" * 100 200 300 400 500 500 Markrers IA Figure 66. The pressure contours at time=6.0 with Re=500 and We= 2.63. A) twophase zone, B) singlephase zone, C) entire domain. Figure 67. Pressure distribution and the curvature of interface at time=6.0 with Re=500 and We=2.63. A) pressure distribution, B) curvature distribution. 123 55 6 X 65 Figure 68. Streamline plot at time=6.0 with Re=500 and We=2.63. 15 1 25 05~ ,, .Pencomputation O 25 uU 01 02 03 04 05 A Figure 69. U velocity profile at time=6.0 with Re B) comparison of velocity at the outlet. :500 and We=2.63. A) at different locations, Figure 610. Shapes of the interface by different Weber numbers at time=6.0. O dz=0a2 A dw~o.gi * s ~ dn0. 0.7 T~I.I T~?.I T~9.1 T~II ~c T~5.1 ~Tt T~B.I ~c ii ; I,, 0 24 6 Figure 611. The development of interface with Re=500 and We=2.63 at different time steps. : :1 ~U I~M 199i4n~ RI~ Figure 612. The ratio of averaged local friction coefficient in the twophase zone to the fully developed singlephase gas flow friction coefficient. A) as a function of the Reynolds number, B) as a function of the Weber number. Figure 613. The computed interfaces by three different grid resolutions. 125 D5. 2 5 3 35 4 45 X 1 o~ ,I .. 510 15 20 0C 500 1000 / X Markem 06 2:> X=0.4 04 ~ O FullyOeeloped 15 03C  02  S4 B B U0 01 02 03 04 05 X R El F Figure 614. The pressure contours, pressure and curvature distribution, streamline plot and the contour of U component contour of the case with Re=500, Pe=6500 and We=500 at time=5.5. A) pressure contour, B) enlarged pressure contour near X=3, C) curvature distribution, D) streamline plot, E) U velocity component along radical direction. 126 = = I l i 0 * A Figure 615. The temperature contours oftwophase plug flow and the interfacial temperature at different time steps with Re=500, Pe=6500 and We=500. A) time=1.5, B) time=3.5, C) time=5.5, D) interfacial temperature. Figure 616. The Nusselt number at the solid wall at different time steps with Re=500, Pe=6500 and We=500. 127  IAl IB Figure 617. The comparison of interfacial temperature and Nusselt number at wall for different Reynolds number with Pe=6500 and We=500 at time=5.5. A) Nusselt number at wall, B) interfacial temperature. Figure 618. The shapes of the liquid slug at three different time steps for the case of Re=500, Pe=6500 and We=2.63. 01 0 1 24 6 Figure 619. The temperature contour of two phase plug flow and the interfacial temperature with Re=500, Pe=6500 and We=2.63at different time steps. A) time=1.5, B) time=3.5, C) time=5.5 and D) interfacial temperature. time=1.5 12 _ tm . time 5.5 10 ii I pure conduction Pe=0 4 single phasei 023466 Figure 620. The Nusselt number at the solid wall Re=500, Pe=6500 and We=2.63 at different time steps. 129  IAl IR Figure 621. Comparison of the interfacial temperature and the Nusselt number with different Weber numbers. A) interfacial temperature, B) Nusselt number. CHAPTER 7 LIQUIDGAS TWOPHASE FLOWS INT A PIPE WITH PHASE CHANGE 7.1 Introduction In this chapter, the text will focus on phase change computation. Based on thermodynamics, the temperature at the vaporizing liquidgas interface is assumed as the saturation temperature corresponding to the system pressure and the pipe wall temperature is set to be higher than the saturation temperature. Therefore, as the liquid phase is also at the saturation temperature without any subcooling, the heat flux reaches the liquidgas interface will be fully consumed to supply the latent heat because the temperature of liquid can not increase. As the phase change occurs, the gas mass flow rate downstream will increase due to the vaporization from the liquid surface that adds mass flows to the gas phase. Also, because of the large temperature difference assumed between the liquid phase and solid wall, the thermal radiation is considered. In the current approach, two kinds of heat transfer boundary conditions of the solid wall are considered. In the first case, the wall temperature is assumed to be constant, so that the heat flux comes from wall is relatively high and constant. This can represent the case with a very large wall heat capacity. In the more realistic situation, the wall temperature may not remain constant and it would be chilled down by the colder fluid in the pipe. In the second case, a finite wall heat capacity is assumed. It will focus on the impact of the pipe wall chilldown process on the twophase flow and heat transfer. One refrigerant R508B and two cryogenic fluids of liquid nitrogen and hydrogen are chosen as coolants. The diameter of the proposed pipe is 4 mm and inlet velocity is 10cm/s for both phases and the selected material for the wall is a Titanium alloy with p =4450 kg / m3 Cp =4200 J /(kgK), and k = 4.8 W /(mK) 7.2 R508B Refrigerant with Constant Wall Temperature In this section, the selected fluids are the liquid and gas refrigerant R508B at 208K and 248K, under 3 ATM. Its chemical formula is CHF3 / CF3CF3, 46/ 54% and its molecular weight is 95.39. Under this condition, the temperature at liquid phase is the saturation temperature for 3 ATM and will not change during the computation and the phase change should take place at the liquidgas interface. The main purpose of selecting this particular working fluid is to add a noncryogenic case but its application is quite extensive. The ratios of fluid properties for the two phases are: Pt 98.89. A! = 21.74; ki = 9.04. C~, = 1.53 The two fixed dimensionless parameters are: Pr = 3.26 Ja = 0.034 The time step in this computation is set as 0.001. In the phase change computation, there are four dimensionless parameters in the governing equations and interfacial conditions, and they are Reynolds (Re), Peclet (Pe ), Weber (We ) and Jakob ( Ja)numbers. Once the thermodynamic properties of working fluid and the temperature difference are decided, the only free dimensionless parameters are Reynolds and Weber numbers. In order to demonstrate the ability of the SIMCC code to handle the large deformation and change of volume due to the strong surface tension and phase change effects, the Reynolds and Jakob numbers are selected as 10.0 and 2.63, respectively. Figure 71 shows the contours of the Uvelocity component at time =1.0, 2.5 and 4.0. In Figure 71, the Ucomponent contours show that there are two locations with higher U velocities in each contour plot. The first location is in the gas phase near the upperright corner of liquid core. Because of the vaporization at the upper surface of liquid core, the liquid will be vaporized as gas and enters the gas phase. It can be expected that the mass flow rate of gas will be increased before entering the singlephase region so that the velocity of gas is sped up till this location and results in the maximum velocity. Figure 72 is the history of local maximum Uvelocity of the gas phase near the upperright corner of liquid core. It shows that the local maximum velocity increases as time increases but this trend is not linear. If there were no shape change for the liquid slug, then the maximum U velocity history curve would be linear as the evaporating surface area would increase linearly with time and the rate of mass evaporation is proportional to the surface area. Also the distance between the pipe wall and the liquid core would stay unchanged. In the current case, the interface shape does change substantially due to a strong surface tension effect and phase change process. In Figure 73, it shows the interface shapes at different times. At t = 4.0, the maximum distance between the pipe wall and the liquid core is 0.24. The initial (time=0) maximum distance is 0.12. This explains why the curve in Figure 72 is not linear and the slope must decrease since the cross section of area of gas phase increases. Another location with a higher velocity is at the center of the pipe. Near the wall, because of the nonslip condition and the continuity requirement, the gas is pushed to the centerline and also because of the mass flow from the twophase upstream, the velocity at the centerline will be higher than other locations. Figure 74 is the streamline plot of the gas phase at time=4.0. By the distribution of streamlines, it does show that the streamlines are pushed to the center of the pipe and this means the gas is pushed to the center of the pipe. Figure 75 shows the Vvelocity component contour at different time steps. Downstream to the front of the liquid core, the flow of gas is pushed to the center so that the maximum negative Vvelocity is located near this region. This highest negative Vvelocity component increases as time increases due to more vaporization from the liquid core. Near the centerline, the Vvelocity component will be close to 0 and that means the gas flow turns its direction because of the symmetric condition. This conclusion can also be seen in Figure 75. In Figure 75, the phase change at the interface is much stronger at the beginning due to the initial condition. As time increases, it shows that the vaporization becomes weaker. Figure 76 is the temperature contours at different time steps. To compare the gas temperatures near the front of the liquid core, it shows that the temperature gradient is much higher at time=1.0 due to the initial condition. At time=4.0, the gradient is much weaker because of the convective effect and also, the distance between the liquid core and the wall increases. It can be expected that phase change is very strong at beginning and becomes weaker as time mecreases. Figure 77 is the Nusselt number evaluated at the solidgas interface at different time steps. It shows that in general the Nusselt number decreases as time increases. By Figure 76, it is very clear that the thermal gradient decreases along the downstream direction so that the Nusselt number decreases with increased downstream location. In the single phase region, all three curves converge to the fully developed singlephase flow case very fast and the corresponding Nusselt number is 3.66. Figure 78 shows the total mass flow rate of gas phase at the pipe exit. Basically, the trend is very close to that in Figure 72. The mass transfer rate increases as time increases but the slope of curve becomes smaller as time increases. 7.3 Constant Wall Temperature Case with Liquid Nitrogen In this section, the working fluid is nitrogen with liquid and gas phases at 77K and 300K, respectively under 1ATM. Under these conditions, the temperature at liquid phase is the saturation temperature and will not change during the computation and vaporization take place at the liquidgas interface. The ratios of fluid properties of two phases are: Pt 706.91. # = 8.85 ki = 5.41. ; = 1.97 p, p, kg 'Cp, The corresponding ratio of the Prandtl number for both phases should be: 1 = 3.21 P r, The time step in this computation is set as 0.001. Three sets of dimensionless parameters are considered based on the fluid properties and geometry. They are listed in Table 71. 7.3.1 Flow Field with the Constant Wall Temperature Case Since the heat transfer and phasechange process is closely related to the twophase flow structure, a typical flow field shown in Figure 79 will be examined first that provides the velocity vector and streamline plots for Case 2 at time=1.5. Once the phase change is involved, the liquid will be vaporized and infuse into the gas stream so that the gas flow field is constantly changing with the mass flow rate and velocity increasing with increased downstream positions and the flow patterns are changed accordingly. For the liquid phase, it is a very simple unidirectional slug flow, while the gas phase is basically a continuously changing and accelerating annular jet flow due to the vaporization from the liquid surface. As seen in the streamline plot, the lines that originate from the liquidgas interface represent the vaporization mass fluxes. The streamline density gets very high near the front of the liquid slug, that results in the maximum velocity in the twophase region. After the maximum jet velocity, the gas flow diffuses into a pure singlephase region and the gas flow becomes fully developed relatively quickly. Figure 710 shows the Ucomponent velocity contours for all three cases at time=1.5. To compare the three cases, the flow patterns are very similar to one another and the maximum velocities of three cases are all located near the upperright of the liquid core which is the intersection between the twophase region and the singlephase region. These flow structures are consistent with the velocity vector plot given in Figure 79. Due to the conservation of mass, the mass flow rate of the gas phase at each downstream cross section of the pipe should be the same in the singlephase region (about X >2.0) as there is no more source of mass transfer. Therefore, the velocity decreases as the area of the cross section of the pipe increases in the singlephase region. Figure 711 is the Vcomponent velocity contours for all three cases at time=1.5. Again, they are all similar to one another also. By Figure 711, the highest vertical velocities are located on the horizontal portion of the liquidgas interface where the vaporization rate is the highest accordingly. Immediately downstream of the front of the liquid core, it shows that the V component velocities are negative and possess the maximum values because the gas must rushes downward to fill the entire pipe. Far away from the front of the liquid core, the V component decreases very fast and approaches zero as the singlephase becomes fully developed. Figure 712 gives the value of the maximum gas velocity for each of the three cases and a relatively linear relationship is found for the maximum gas velocity with the Reynolds number. Based on Table 71, the case with a lower Reynolds number is also with a lower Peclet number. By the conservation of energy, the vaporization mass flux, at liquidgas interface is proportional to Ja/Pe as given below: rz" = pg((un)g (u)nt) < u) u)nt*p( T ) (71) Pe Therefore, the case with a lower Peclet number will experience a higher evaporation rate that contributes to a higher maximum velocity. But it should also be noted that the gas temperature gradient term in the above equation would be slightly higher at a higher Reynolds number due to forced convective heat transfer in the annulus, however this effect is less significant than that due to the Peclet number. Figure 713 is the temperature profile contour of the three cases at time=1.5 For all three cases, they all belong to the category of a high Peclet condition, so the contours are very similar to one another. Under a high Peclet number, the convection is dominant as can be seen in those temperature profiles that the heat transfer in the twophase region corresponds to parallel isotherms due to a fully developed annular flow. In the singlephase region, the temperature profiles quickly establish the form of a fully developed pipe flow heat transfer. Figure 714 plots the liquidgas interface shapes for the three cases at time=1.5. As the interface shape deformation is strongly dependent on the Weber number and the three cases almost have relatively the same Weber numbers, so it can be expected only slight differences among them. This is indeed the case as shown in Figure 714. Near the entrance of the pipe, the three interfaces almost overlap. Near the front of the liquid core, intrinsically the deformation of the interfaces is more substantial for each case due to the magnitude of surface tension effect (different We numbers). Higher Weber numbers imply a lower surface tension effect and therefore, the case with the smallest Weber number (Case 3) deforms more than the other two cases. Using the liquid slug shape as a basis for the evaluation of the grid size requirement, a grid refinement test is done to ensure the results obtained will not be influenced by the grid resolution. There are three gird sizes selected and they are 0.005, 0.01 and 0.05. The parameters for Case 3 are employed for the evaluation study. Figure 715 shows the interfaces for the three grid sizes at t = 1.5. It is seen that there is not much difference among the three grid lengths and the selected grid length of 0.01 is adequate for the current numerical computations. 7.3.2 The Mass Transfer at Interface with Constant Wall Temperature In this section, it focuses on the evaporative mass transfer process at the liquidgas interface. As shown in the previous section, the mass transfer rate is represented by the difference between the velocities at the interface and accordingly the evaporation rate is dominated by the gas phase temperature gradient, the Jakob and the Peclet numbers. Previously, Figures 713 shows that temperature distributions are similar in the twophase region for all three cases. In Figure 714, there is only a small difference near the front of the liquid core where the distance between the solid wall and the liquid surface is the shortest for Case 3 that results in a highest gas phase temperature gradient locally. Figure 716 (A) is the computed temperature gradients of three cases along the interface marker points at time=1.5. Figure 716 (B) provides the locations of markers. In Figure 716(A), all three curves are similar except near the front of the liquid core, where the temperature gradient for Case 3 is larger than others because the distance between the liquid surface and the solid wall is the smallest for this case. At the entrance, because of the inlet boundary conditions, the temperature gradient is zero. At the front of the liquid core, because the distance between liquid slug surface and the solid wall becomes longer, the temperature gradient becomes smaller in all the three cases. Because the gradients of temperature of these three cases are similar along most part of the interface, the ratio of mass transfer rate at the interface for these three cases can be estimated by Equation 71. By Table 71, all three cases have the same Jakob number so that the mass transfer rate is inversely proportional to the Peclet number: thz" ac (72) Pe Since the Prandtl number is the same for all three cases, the evaporation rate should also be inversely proportional to the Reynolds number. Figure 717 is the total mass flow rates measured at the outlet of pipe for the three cases at time=1.5 and as expected the mass flow rate at the exit is inversely proportional to the Reynolds number. 7.3.3 The Transient Phase Change Process with Constant Wall Temperature In this section, the focus is on the timedependent characteristics of the phasechange process and Case 2 is used for the evaluation. Figure 718 is the temperature gradient history at the liquidgas interface along the marker points for Case 2. In Figure 718, it shows that the temperature gradients near the entrance do not change as time increases because it is controlled by the inlet boundary conditions. The peaks of temperature gradients are decreasing because the gas is chilled down by the liquid phase. The temperature gradients all drop drastically at the front of liquid core because that the distance between the solid wall and liquid surface becomes very large there. At the beginning, there is only one peak in the temperature gradient curve. Because of the deformation of the liquidgas interface due to the surface tension as shown in Figure 714, small oscillations develop near the frontal area of the liquid core. Figure 719 plots the history of mass flow rate at the exit for Case 2. It shows that the mass flow rate increases as time increases due to continuous evaporation from the liquid slug. The curve is relatively linear; therefore the rate of evaporation is increasing almost linearly during that period. Figure 720 gives the Nusselt number distributions for the three cases along the surface of solid wall at time=1.5. From Figure 713, it shows that the temperature contours of the three cases are very similar and therefore, the Nusselt numbers are also very close. They will be affected by the distance between the wall and the liquid surface. As discussed previously, the distance for Case 3 is shorter than others so that the Nusselt number for Case 3 will be larger than others. In Figure 720, it does show this trend. After the initial drop, the Nusselt number increases in the axial direction due to the increase of velocity from vaporization. Also, downstream of the front of liquid core (X>2.0), the gas phase will adjust quickly to become a fully developed single phase flow and the Nusselt number in this region reflects that process. Far away from the inlet, the effects of the liquid slug is diminished and the Nusselt number is asymptotically approaching the fully developed single phase heat transfer value. For the purpose of evaluating the heat transfer enhancement due to evaporation, the Nusselt number is plotted by a longdash line for an identical case except that there is only convection with no phasechange. The enhancement due to phase change is on the average 60%. 7.3.4 The Comparison of Nusselt Number with Phase Change Process and Constant Wall Temperature In this section, the Nusselt number in liquid nitrogen with phase change and constant wall temperature is compared. In Hammouda et al.'s paper(1997), they gave the correlation for inverted annular flow as following: Nu = 5.071/ Pr ) 0439 + 0. 0028 Pr ) 645 ReV (73) where Pr, and Re, are the Prandtl and Reynolds numbers for gas phase. In their research, the physical model is similar with the current research but there is still some difference in numerical procedures. In current research, it is a direct numerical simulation, to solve the governing equations, boundary conditions and interfacial conditions directly without any empiricism. In Hammouda et al.'s research, they used twofluid model. Their model is a semianalytic model and using empirical correlations for transport coefficients. Also, their model is one dimensional in streamwise direction and assumes the computation is steady state, fully developed condition without the entrance effect so that they can not provide dynamic information. Figure 721 is the comparison of Nusselt number by the current method and Hammouda et al.'s correlation. In Figure 721, the upper long dash line is computed by Hammouda et al.'s correlation and the short dash line is by current SIMCC code. Near the inlet, because of the entrance effect, the thermal boundary layer will be thicker so that the Nusselt number obtained by SIMCC is smaller. In Hammouda et al.'s research, their model can only work in fully developed region so there is no entrance effect in their research and the Nusselt number computed by their correlation should higher than current method. The thermal boundary thickness should become thinner as X increases because of the stronger convection effect due to vaporization in twophase zone. After the front of liquid core, the flow pattern is singlephase flow and the Nusselt number will decrease very fast and converge to the Nusselt number in singlephase zone. The correlation from Hammouda et al. can work in twophase zone but the singlephase zone. By this comparison, it helps to validate the current computation for Nusselt number. At least, the computation of Nusselt number in this research is in a reasonable order. 7.4 Wall Chilldown Process by Liquid Nitrogen In this section, the selected fluid is also the liquid and vapor nitrogen at 77k and 300K under 1 ATM. The time step in this computation is set as 0.001. The dimensionless parameters of Case 2 in the previous section are adopted in this section for the chilldown study. For the chilldown case, the wall is no longer kept at a constant temperature, whereas it is given a finite thickness and its temperature is decreasing as a result of heat loss to the cooling fluid. Therefore, computational grid system will include the wall. There will be three different thicknesses of the solid wall in this section and they are AR = Rwo RwI 0.02R (Case 1), 0.04R (Case 2), 0.08R (Case 3). 7.4.1 Flow Patterns during Chilldown Process Figure 722 includes the contours of two velocity components U and V and the temperature contour of Case 1 at time=1.5. When comparing with the Case 2 of the constant wall temperature condition in the previous section, the flow pattern is very similar with only small differences in the magnitude. Table 72 provides the maximum velocity in the twophase region for the three wall chilldown cases at time=1.5. The constant wall temperature case is also listed for comparison. Case 1 has the lowest maximum velocity because its thermal capacity is smaller than the other two which results in less amount of vaporization. Figure 723 is the wall temperature contours of three cases at time=1.5. The results show that as expected, the thinner wall is chilled down faster than the thicker one. 7.4.2 The Wall and LiquidGas Interface Conditions during Chilldown Process This section focuses on the impact from the chilldown process on the mass transfer at the interface. The generated mass flow rate at the interface is determined by the temperature gradient and the dimensionless parameters, Jakob and Peclet numbers. Figure 724 is the wall temperature distribution at the solidgas interface for the three cases at time=1.5. It shows that the wall for Case 3 is chilled down faster than others because the thickness of the wall of Case 3 is much thinner than others. Since this is an unsteady computation so that the lower temperature should be located at the entrance. Far away from the entrance, the temperature will converge to the initial temperature of 1.0 By the temperature contour given in Figure 723 and the temperature distributions at the solidgas interface provided in Figure 724, it can be expected that the temperature gradients may be smaller near the entrance. Figure 725 shows the gasside temperature gradients of the three cases along the liquidgas interface at time=1.5. In Figure 725, the temperature gradient at liquidgas interface of Case 1 is smaller than others. The temperature gradient of Case 3 is larger than others and it is very close to the reference case of the constant temperature wall condition, since the thicker the wall is, the more the energy is stored. Accordingly, the solidgas interface temperature would decrease more slowly; therefore, the temperature gradient at the liquid slug surface will be larger. In this section, the dimensionless parameters for all cases are the same so that the mass flow rate at outlet should be determined by the heat flux at the interface. Table 73 shows the mass flow rates for the three and reference cases at time=1.5 at the outlet of pipe and the results agree with the temperature gradient plotted in Figure 725 where Case 3 has the largest temperature gradient and correspondingly the most evaporation rate which gives rise to the highest mass flow rate at the exit. By Table 73, it also shows that the mass flow rate at the exit approaches to that of the case with constant wall temperature as the thickness of the wall mecreases. 7.4.3 The TimeDependent Development during Chilldown Process In this section, the timedependent system behavior during the chilldown process is presented. Figure 726 displays the temperature profiles along the inside surface at three different time steps for Case 1. The cooling of the wall is enhanced by the evaporation from the cold liquid slug and the effect due to the motion of the liquid slug is also shown by the wall surface temperature history. Figure 727 is the temperature gradients of Case 1 along the solidgas interface at different time steps. To compare with the constant wall temperature case in previous section, it shows that the temperature gradient decays faster for the chilldown case. Since the heat flux at the liquidgas interface is different for the various chilldown cases, it is clear that the mass flow rates will not be the same at the exit. Figure 728 is the history of mass flow rates for the three chilldown cases and the constant wall temperature case. At the beginning, the mass flow rates of all cases are very similar because the wall temperature/heat flux at this moment is governed by the same initial condition and the chilldown process at the wall has just started. Once the time increases, the wall is chilled down continuously so that the difference in mass flow rates will increase. For the cases with a thicker wall, the temperature of the wall decreases slower so that the mass flow rate of the case with a thicker wall is close to the case with a constant wall temperature. 7.5 Wall Chilldown Process by Liquid Hydrogen In this section, the liquid hydrogen is investigated as it has been used extensively in space applications. The selected fluids are liquid and gas hydrogen at 20.27K and 200K under 1 ATM. The ratios of fluid properties the hydrogen are: Pt 576.94 .# = 1.94 k = 0.92 ; = 0.714 Pg Ug kg Cp, For comparison purposes, the liquid and gas nitrogen case is used as a reference to measure the differences in phasechange heat transfer when hydrogen is used. The nitrogen properties are taken for 77.36K and 300K under 1 ATM for liquid and gas phases, respectively. The ratios of fluid properties for the nitrogen are: Pt 706.91. # = 8.85 ki = 5.41. ; = 1.97 p, p, kg 'Clpg The initial conditions, initial location of interface and boundary conditions are the same as those in the pervious chilldown case section. By the same geometry and boundary conditions, the four dimensionless parameters for hydrogen and nitrogen are: Hydrogen: Re = 2145; Pr = 1.08; Ja = 4.27; We = 1.71 Nitrogen : Re = 2043 ; Pr = 2.32; Ja = 0.42; We = 3.69 By these two sets of dimensionless parameters, they show that the Reynolds, Weber and Prandtl numbers are close between hydrogen and nitrogen but the Jakob numbers are quite different. The ratio of Jakob numbers for both fluids is about ten times. The thickness of the wall in this section is AR = Rwo Rwl = 0.02R. In the following, the results of hydrogen will be given side by side with corresponding ones from nitrogen for a direct comparison between these two cases. Figure 729 provides a direct comparison for the Uvelocity contours of hydrogen and nitrogen at time=0.4. In general, the two cases have very similar Uvelocity profiles. The maximum Uvelocities for the two cases are also located almost at the same location near the exit of the annular channel. But, the magnitudes are totally different and the maximum Uvelocity for hydrogen is about four times of that for nitrogen that is exclusively due to the much higher evaporation rate of hydrogen resulting from a much larger Jakob number as explained next. From previous discussion, the mass transfer rate can be estimated by mi ac (T ) (74) Pe The ratio (Ja /Pe) of hydrogen to nitrogen is 20.8 that is the main cause for the maximum Uvelocity difference. Figure 730 is the Vvelocity contours of two cases at time=0.4 and they show the same trend as that of the Uvelocity contours. The contours are similar but the magnitudes are quite different. First, the vaporization rate on the liquid slug surface is much uniform for the hydrogen case. The relatively nonuniform evaporation rate from the nitrogen liquid slug surface is due to the shape deformation whereas the hydrogen surface is very flat. Second, based on Figures 729 and 730, they show that the convection effect for the hydrogen case is much stronger so that the location of second highest Vvelocity is much close to the liquidgas interface for nitrogen case. Figure 731 is the temperature contour comparison for the two cases at time=0.4. It shows large differences between the two cases. Since the convective effect in hydrogen is much stronger due to higher evaporation rates than that in the nitrogen case. It can be expected that the temperature profile in the hydrogen case should reflect that. Due to the strong convection, the temperatures at downstream in the hydrogen case is cooler than those in the nitrogen case. Figure 732 shows the temperature distributions at the solidgas interface for both cases during chilldown process. Due to the stronger convection effects, the solid wall for the hydrogen case is cooled down much faster than that in the nitrogen case, especially near the entrance region. In the downstream region, the temperature of both cases will be close to the initial wall temperature since there is only very few heat transfer in that region. Figure 733 compares the Nusselt number distributions on the inner surface of pipe wall for the two cases. Both cases show very similar entrance region behaviors near the inlet. Similarly, because of the stronger convection effects, the thermal boundary in the hydrogen case will be thinner than that in the nitrogen case as shown in Figure 731 and this means that the Nusselt number in the hydrogen case should be larger. On the average, the heat transfer coefficient on the pipe wall is 25% higher in the hydrogen case. 7.6 Summary In summary, three working fluids with two different wall conditions were investigated in this chapter. Because of the evaporation phase change that takes place on the liquid slug surface, mass fluxes are generated from the liquidgas interface that acts as a source of mass input to the gas stream. In fluid dynamics, this generated gas does change the flow structure substantially. Because of the generated gas, the total mass flow rate of the gas phase can not be a constant at any downstream location in the twophase region. Also, there is always a very strong jet effect near the intersection between the single and two phase zones, where the maximum gas stream velocity is located. Therefore, the flow patterns with phase change are different from those without phase change. Because of the evaporation that causes a large increase in the gas stream velocity in the twophase region, a much stronger convective effect is therefore induced. This enhanced convection results in a substantial increase in the heat transfer efficiency of entire system. This enhanced convection effect can also be seen in the chilldown process where the wall temperature decreases faster in the case with a stronger convective effect. The wall chilldown case is a highly transient process as the heat transfer that supplies the latent heat for evaporation decreases with time due to decreasing heat transfer driving force, the temperature gradient between the pipe wall and the liquid slug. In the last case, a close comparison was made between liquid hydrogen and liquid nitrogen as cryogenic working fluids. In general, under similar conditions, liquid hydrogen offers a much higher evaporation rate that induces more intensive convection effects. The Nusselt number is approximately 25% higher for the liquid hydrogen case. Table 71. Dimensionless parameters for the nitrogen case with constant wall temperature. Dimensionless Parameters Re Pe Ja We Case 1 2000 4640 0.42 3.69 Case 2 1500 3480 0.42 2.76 Case 3 1000 2320 0.42 1.84 Table 72. The maximum velocity for the nitrogen case with wall chilldown and the reference cases at time=1.5. Constant Temperature Wall Chilldown Cases Case 1 Case 2 Case 3 Max Velocity 13.62 12.90 13.31 13.41 Table 73. The mass flow rate for the nitrogen case with wall chilldown and the reference cases at time=1.5. Constant Temperature Wall Chilldown Cases Case 1 Case 2 Case 3 Max Flow Rate 0.594 0.571 0.585 0.591 1205 IOQ5 H r,, ,, U ,,  381 211 Ill rro oo eat 5~5 08 SP3 151 319 07 zts taa 00 112 LL~ ii: 03 02 01 1B H 1523 012345 Figure 71. The U contours for the refrigerant R508B case at different time steps. A) time=1.0, B) time=2.5, C) time=4.0. 16 15 14 13 10 9 8 1 Time Figure 72. The local maximum U in the gas phase for the refrigerant R508B case at different time steps. 0.5 0.4 0.3 0.2 0.1 If 0 0.1 0.2 0.3 0.4 0.5 2 34 5 X Figure 73. The interfaces for the refrigerant R508B case at different time steps. Figure 74. The streamlines in the gas phase for the refrigerant R508B case at time=4.0 09 00 x1 C Figure 75. The V~~~~~~~~ cnors o ergrn 0Bcs tdffrn ieses )tm=.,B tie=.5 C) time4.0  x IC Figure 76. The temperature contours for refrigerant R508B case at different time steps. A) time=1.0, B) time=2.5, C) time=4.0. 14 12 10 4 2 Figure 77. The Nusselt numbers along solid wall for the refrigerant R508B case at different time steps. 1.1 1 0.9 S0.8 o Ll. 0.6 0.5 1 2 3 4 Time Figure 78. The total mass flow rate of gas phase for the refrigerant R508B case at different time steps. Figure 79. The vector and the streamline plots for Case 2 in the nitrogen case with constant wall temperature at time=1.5. A) vector plot and B) streamline plot. i I~ li D 3 Fixr A0 xh B otusfrtentoe aewt cntn altmeauea ie15 15Cs ,B as ,C ae3 Oi li` n ns r r~ ,6 i Is a x I I I 212 151 150 119 089 058 02~ 001 031 055 1 Iwr ,rr lur 1,9 r.m ~.,1 ~~1 uK1 uAl u.N O Figure 711. The V contours for nitrogen case with constant wall temperature at time=1.5. A) Case 1, B) Case 2, C) Case 3. 16 10 1000 1250 1500 1750 2000 Re Figure 712. Maximum velocity for the nitrogen case with constant wall temperature of three cases. ii I I I 05 x A x IB x C Figure 713. The temperature contours for nitrogen case with constant wall temperature at time=1.5. A) Case 1, B) Case 2, C) Case 3. 04  tie=.5 A) enir, ) ocl DB5 Od~ UMOrlW Ulu~~900 U~r100 ,, ,, ,, ,, o15 \\ 01 om j uvlnrroo 07 05 04 10 20 10 70 20 aR Figure 715. The interfaces of three different grid lengths. 0O 05 1 15 2 2.5 Figure 716. The temperature gradient of three cases along the interface at time=1.5 and the locations of markers for the nitrogen case with constant wall temperature. Figure 717. The mass flow rate for the nitrogen case with constant wall temperature at time=1.5. 157 4 2 O 100 200 300 400 Markers Figure 718. The history of temperature gradient along the interface of Case 2 in the nitrogen case with constant wall temperature. OB 0.55 0.5 S0.35 Tim e Figure 719. The history temperature. of mass flow rate of Case 2 for nitrogen case with constant wall single phase 18 i i 15 L e n E 12 3 Z c D 9 z 6 3 O Figure 720. The Nusselt number at wall for the nitrogen case with constant wall temperature at time= 1.5. 35 1 1 I E 20 Hammoucle et al. (1997) ~15 present 10 X result Figure 721. The comparison of Nusselt number by the current method and Hammouda et al.'s correlation. 159 Ou l l8 ~01 01llb 06 M 1 s er~ r.ir up:~, OI P :: I~ ri:i II OI P :I I~ 1)51 O11* II: I i: :...~.....~ I I b 1111.... ~) _I ~)_) _I ~ 2 j X Figure 722. The U, V and temperature contours of Case 1 in the nitrogen case with wall chilldown at time=1.5. A) U contour, B) V contour, C) temperature contour. Figure 723. The wall temperature contours for the nitrogen case with wall chilldown at time=1.5. A) Case 1, B) Case 2, C) Case 3. 160 Figure 726. The temperature history of Case time=1.5. Figure 724. The temperature at solidgas interface for the nitrogen case with wall chilldown at time=1.5 T=Conat Came l 100 200 300 400 Markers Figure 725. The temperature gradients along the interface of reference for the nitrogen case with wall chill down at time= 1.5 1 in the nitrogen case with wall chilldown at Figure 727. Temperature gradients along the interface of Case 1 in the nitrogen case with wall chilldown at time=1.5 S05 04 4 O 35 T= Const.  case 1 03C ~ case2  case 3 025 'OS O 1 5 15 Time Figure 728. The mass flow rates for the nitrogen case with wall chilldown and reference case. 0s 000511. Xr ~ XIIi ~i r A B Figure 729. U contours of two cases at time=0.4. A) hydrogen, B) nitrogen. Figure 730. V contours of two cases at time=0.4. A) hydrogen, B) nitrogen. R I I I 1.5 2 D.5 1 1.9 X Figure 731i. Temperature contours of two cases at time=0.4. A) hydrogen, B) nitrogen. 163 1.05 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 Figure 732. Temperature at solidgas interface for the hydrogen and nitrogen cases at time=0.4. 16 15 14 13 12 7 6 1 5 Figure 733. Nusselt number distributions on the pipe wall for the hydrogen and nitrogen cases for both cases at time=0.4. CHAPTER 8 SUMMARY AND FUTURE WORK 8.1 Summary In this research, a numerical package for handling phase change and chilldown for cryogenic twophase flow has been developed. The maj or contributions are the phase change computation which includes the energy equation solver, the phase change algorithm for the liquidgas interface, the conjugate heat transfer for solid and gas phase and the thermal radiation model . In Chapters 1 and 2, literature review is done and the problem is defined. In Chapter 3, the governing equations, the boundary conditions and the interfacial conditions and their dimensionless forms are listed. In Chapter 4, the main numerical techniques applied in this research are introduced and explained. All the main techniques are validated in Chapter 5 and the validation includes the solver of governing equations, the SIMCC with moving interfacial algorithm and the mass transfer computation. Chapters 6 and 7 form the core of this research and four different cases are computed and discussed. In Chapter 6, it focuses on the cases without phase change and includes the isothermal and heat transfer twophase flow in a pipe. In Chapter 7, the phase change is included with different condition: the constant wall temperature and the wall with finite thickness. 8.2 Future Work In the current research, the preliminary techniques are ready for the cryogenic twophase flow. Besides these techniques, there are still some interesting future possibilities for this research: Three dimensional interfacial tracking technique and cutcells: The current SIMCC code can only handle two dimensional axisymmetric computations. This limits the current research to axisymmetric flows. In recent researches, some scholars have introduced the three dimensional interfacial tracking technique and cutcells (Cieslak et al. 2001, Singh et al. 2005). However, currently, the researches can only handle very simple geometries and this should be a good direction for the future work. Turbulence model: The laminar flow is another limitation of the current code. The critical Reynolds number for laminar flow is aboutRe = 2000 3000 (White 1991). Once the Reynolds number becomes larger than 3000, the flow regime of pipe flow should be closer to turbulent flow and the turbulence model (Launder and Spalding 1974) is very important. Compressibility and variable fluid properties: In current code, the two phases are treated as incompressible fluids. This should be safe since the temperature gradient in the temperature contour is not very strong. 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"A Front Tracking Method for a Deformable Intravascular Bubble in a Tube with Soluble Surfactant Transport." Journal of Computational Physics, 2 14(1 ), 3 663 96. Zhao, L., and Rezkallah, K. S. (1993). "Gasliquid flow Patterns at Microgravity." hIternational Journal of2~ultipha~se Flow, 19(5), 751763. Zhao, L., and Rezkallah, K. S. (1995). "Pressure Drop in Gasliquid Flow at Microgravity Conditions." hIternational Journal ofM2dutipha~se Flow, 21(5), 837849. BIOGRAPHICAL SKETCH Chengfeng Tai was born in Changhua, Taiwan, in 1972. After receiving his Bachelor of Engineering degree in mechanical engineering from National Sun YatSen University in 1995, he received the Master of Engineering degree from in aerospace engineering from National Cheng Kung University in 1997. From 2001 he has been pursuing his Ph.D. degree in aerospace engineering at the University of Florida. His current research interests lie in computational fluid dynamics of multiphase flows and heat transfer. PAGE 1 1 CRYOGENIC TWOPHASE FLOW AND PH ASECHANGE HEAT TRANSFER IN MICROGRAVITY By CHENGFENG TAI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 ChengFeng Tai PAGE 3 3 To my wife Xiaoling, daughter Grace and my parents. PAGE 4 4 ACKNOWLEDGMENTS I would lik e to express my sincere gratitude to Dr. Jacob NanChu Chung for providing me the opportunity to perform this research. I can not thank him enough for being so patient and understanding for years, imparting me a lot of knowledge and giving me the best advice. I would like to thank Drs. S. Balachandar, W. E. Lear Jr., S. A. Sherif, and G. Ihas for agreeing to serve on my dissertation committee. I thank my research group members for helping with academic aspects while providing me morable company for the past years. Finally I thank my family members who have been behind me every step of the way providing their uncond itional support. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.........................................................................................................................9 ABSTRACT...................................................................................................................................15 CHAP TER 1 INTRODUCTION..................................................................................................................17 1.1 Overview...........................................................................................................................17 1.2 Role of Cryogenics in Space Exploration......................................................................... 17 1.3 All Cryogenic Systems Must Be Chilled Down............................................................... 18 1.4 Background of a Chilldown Process................................................................................. 19 1.5 Objectives of the Research............................................................................................... 19 1.6 Scope and Structure..........................................................................................................20 2 PHYSICAL BACKGROUND AND LITERATURE REVIEW ............................................ 22 2.1 Boiling Curve and Chilldown Process.............................................................................. 22 2.2 TwoPhase Flow Patterns a nd Heat Transfer Regim es.................................................... 22 2.3 TwoPhase Flow in Microgravity.....................................................................................24 2.3.1 Isothermal TwoPhase Flows in Microgravity ....................................................... 24 2.3.2 TwoPhase Flow and Heat Transfer in Microgravity.............................................25 3 PROBLEM DEFINITION AND GOVERNING EQUATIONS ........................................... 29 3.1 Modeling Cryogenic Chilldown in Microgravity............................................................. 29 3.2 Geometry, Computational Domain and Initial, Boundary Conditions............................. 30 3.3 Assumptions................................................................................................................ .....31 3.4 Governing Equations........................................................................................................ 32 3.4.1 Interfacial Conditions.............................................................................................33 3.4.2 Nondimensionalization......................................................................................... 34 4 SOLUTION METHOD..........................................................................................................40 4.1 Introduction............................................................................................................... ........40 4.2 Fractional Step Method within the Finite Volum e Framework........................................ 41 4.3 Cartesian Gird Method for a Complex Geometry............................................................ 44 4.4 Sharp Interface Method with CutCell Approach (SIMCC)............................................. 46 4.4.1 Interfacial Tracking................................................................................................ 47 4.4.2 Merging Procedure.................................................................................................48 PAGE 6 6 4.4.3 Flux and Stress Computations in the Interfacial Region ........................................49 4.4.4 Moving Interface Algorithm................................................................................... 51 4.4.4.1 Advancing the Interface...............................................................................51 4.4.4.2 Updating the Cells........................................................................................ 53 4.5 Issues of Phase Change Comput ation in Cartesian Grid Methods ...................................53 4.6 Heat Flux Computation at the Interface ............................................................................ 56 4.7 Phase Change Algorithm..................................................................................................58 4.8 Global Conservation of Mass...........................................................................................62 5 VALIDATION OF CODE..................................................................................................... 73 5.1 Introduction............................................................................................................... ........73 5.2 Group 1, Solver of Governing Equations and the SIMCC with the Fixed Interface ........ 73 5.2.1 The Fictitious Interface for Evaluating the SIMCC............................................... 73 5.2.2 Couette Flow, Fully Developed Channel Flow and Cavity Flow..........................76 5.2.2.1 The Couette Flow.........................................................................................77 5.2.2.2 The Fully Developed Channel Flow............................................................ 77 5.2.2.3 The Cavity Flow........................................................................................... 78 5.2.3 The Fully Developed Pipe Flow with a Constant Wall Heat Flux......................... 79 5.2.4 Flow over a Sphere with Heat Transfer.................................................................. 80 5.3 Group 2, Validation for Moving Interface Algorithm...................................................... 81 5.3.1 Static Droplet Simulation....................................................................................... 82 5.3.2 Deformed Rising Bubble and Droplet.................................................................... 83 5.4 Group 3, Validation for Accurate Mass Transf er at the Interface....................................85 5.4.1 A Stationary Droplet in a Quiescen t Environm ent with a Constant Mass Transfer Rate................................................................................................................85 5.4.2 One Dimensional Phase Change Problem.............................................................. 87 5.5 Summary...........................................................................................................................89 6 LIQUIDGAS TWOPHASE FLOWS IN A PIPE W ITHOUT PHASE CHANGE........... 102 6.1 Introduction............................................................................................................... ......102 6.2 Isothermal LiquidGas Tw oPhase Flows in a Pipe .......................................................103 6.2.1 Effect of the Reynolds Number............................................................................104 6.2.2 Effect of the Weber Number................................................................................ 108 6.2.3 TwoPhase Flow Pressure Drop........................................................................... 111 6.3 Grid Refinement Study...................................................................................................111 6.4 LiquidGas TwoPhase Flows in a Pipe with Heat Transf er.......................................... 112 6.4.1 Low Surface Tension Case...................................................................................113 6.4.2 High Surface Tension Case..................................................................................116 6.5 Summary.........................................................................................................................118 7 LIQUIDGAS TWOPHASE FLOWS IN A PIPE W ITH PHASE CHANGE................... 131 7.1 Introduction............................................................................................................... ......131 7.2 R508B Refrigerant with Constant Wall Temperature................................................... 132 7.3 Constant Wall Temperature Case with Liquid Nitrogen ................................................134 PAGE 7 7 7.3.1 Flow Field with the Constant Wall Temperature Case........................................ 135 7.3.2 The Mass Transfer at Interface w ith Constant W all Temperature....................... 138 7.3.3 The Transient Phase Change Process with Constant Wall Temperature............. 139 7.3.4 The Comparison of Nusselt Number with Phase Change Process and Constant Wall Tem perature......................................................................................................140 7.4 Wall Chilldown Process by Liquid Nitrogen.................................................................. 142 7.4.1 Flow Patterns during Chilldown Process............................................................. 142 7.4.2 The Wall and LiquidGas Interface C onditio ns during Chilldown Process......... 142 7.4.3 The TimeDependent Development during Chilldown Process........................... 144 7.5 Wall Chilldown Process by Liquid Hydrogen................................................................ 144 7.6 Summary.........................................................................................................................147 8 SUMMARY AND FUTURE WORK.................................................................................. 165 8.1 Summary.........................................................................................................................165 8.2 Future Work................................................................................................................ ....165 LIST OF REFERENCES.............................................................................................................167 BIOGRAPHICAL SKETCH.......................................................................................................175 PAGE 8 8 LIST OF TABLES Table page 41 Comparison of the key issues for diffe rent num erical methods for the heat flux computation.................................................................................................................... ....65 42 Comparison of exact and num erical results for the flow over sphere case........................ 65 51 The error of Nusselt number by different grid resolutions in flow over a sphere case. ..... 90 52 The fluid properties of liquid and gas pha ses for the one dim ensional phase change problem........................................................................................................................ ......90 71 Dimensionless parameters for the nitr ogen case w ith constant wall temperature........... 149 72 The maximum velocity for the nitrogen case with wall chilldown and the reference cases at tim e=1.5..............................................................................................................149 73 The mass flow rate for the nitrogen case with w all chilldown a nd the reference cases at time=1.5.......................................................................................................................149 PAGE 9 9 LIST OF FIGURES Figure page 21 Typical boiling curve...................................................................................................... ...27 22 Flow regimes and heat transfer regimes in a heated channel............................................. 27 23 Flow regimes under microgravity...................................................................................... 27 24 Effect of the gravity on flow regimes................................................................................ 28 31 A simple cryogenic system................................................................................................ 38 32 Inverted annular flow.........................................................................................................38 33 Idealized inverted annular flow......................................................................................... 38 34 Conjugate heat transfer.................................................................................................... ..38 35 The initial shape of liquid core.......................................................................................... 39 41 Nonstaggered grid system................................................................................................65 42 Example of mixed structured and unstructured grid in SIMCC. ....................................... 65 43 Example of mixed structured and unstructured grid.......................................................... 66 44 Marker points of an oblique ellipse................................................................................... 66 45 Intersection points of an oblique ellipse. ...........................................................................66 46 Illustration of the interf acial cells and cutand absorption procedures in the SIMCC in local situation............................................................................................................. ....67 47 Cutcells of different objects............................................................................................. .67 48 Example of cutcells of different grids.............................................................................. 67 49 Illustration of the interf acial variables and notation. .........................................................68 410 Illustration of interf acial advancing process. ..................................................................... 68 411 Illustration of upda ting cell procedure. .............................................................................. 68 412 Two probing points for second order gradient. .................................................................. 69 413 Illustration of geometry of test domain.............................................................................. 69 PAGE 10 10 414 The assumptive temperature distribution........................................................................... 69 415 The new interface because of phase change...................................................................... 70 416 The new interface because of balance of force.................................................................. 70 417 The flow chart for the phase change algorithm................................................................. 71 418 The global conservation of mass in gas phase................................................................... 71 419 Interim interface construc tion, the in itial interface............................................................ 72 420 Interim interface construction, the new interface.............................................................. 72 421 Restructuring of marker s, the original m akers................................................................. 72 422 Restructuring of markers, the new markers...................................................................... 72 51 The continuous stress cond ition at marker points for the fictitious interface. ................... 90 52 The algorithm for second order gradient........................................................................... 90 53 Eight points method for second order gradient of velocity................................................ 91 54 Geometry of the Couette flow, cavity fl ow and ful ly developed channel with the immersed fictitious interface.............................................................................................. 91 55 Error in interfacial velocity at all the interfacial m arker points (163 points) and error in mass and momentum fluxes of each cutce ll (148 cutcells) in the Couette flow with radius of ficticious interface R=0.1...........................................................................92 56 Pressure contour, U velocity profile, a nd erro r in mass and momentum fluxes of each cutcell (92 cutcells) of fully developed channel flow with radius of fictitious interface R=0.05 and Re=100............................................................................................93 57 Streamline plots and U velocity profile at R=0.5 with different fictitious interface radii of cavity flow with Re=100. ......................................................................................94 58 The temperature distribution of fully developed pipe flow with constant heat flux case. ....................................................................................................................................94 59 The heat flux at wall of fully develope d pipe flow with constant heat flux case. ............. 95 510 The temperature profile of fully developed pipe flow with constant heat flux case at X=0.481235 along radial direction. ................................................................................... 95 511 The temperature distributions of fully de veloped pipe flow with constant heat flux case with different fictitious interfaces. .............................................................................95 PAGE 11 11 512 The temperature profile of fully develope d pipe flow with constant heat flux case with a fictitious interf ace (R=0.3) at X=0.481235 al ong radial direction. ......................... 96 513 Grid and geometry of flow ove r a sphere with heat transfer. ............................................ 96 514 The pressure contour, the stream line and tem perature contour of flow over a sphere with Re=1, Pe=1................................................................................................................ 96 515 Schematic of stationary droplet problem ...........................................................................97 516 The maximum induced interfacial velo city of stationary droplet problem ........................ 97 517 Convergent histories of the interfacial condition and the governing equations of stationary droplet problem. ................................................................................................97 518 Pressure contour for the stationary droplet. ....................................................................... 98 519 The shape and streamline plot fo r a bubble with Re=10 and We=8. ................................. 98 520 The aspect ratio plot for a bubble with Re=10.0 and W e= 2.08.0.................................... 98 521 The streamline and shape plot fo r a droplet with Re=10 and We=2. ................................ 99 522 Geometry of static droplet with a constant m ass transfer rate problem............................. 99 523 Velocity distribution of st atic drop let with a constant mass transfer rate problem along radial dire ction at X=5........................................................................................... 100 524 Streamline plot near the interface of static drop let with a constant mass transfer rate problem........................................................................................................................ ....100 525 Illustration of one dimens ional phase change problem .................................................... 100 526 The location of interface and temperatur e d istribution, contour for one dimensional phase change problem...................................................................................................... 101 61 Streamline of the isothermal case at tim e=6.0 with Re= 2000 and We=500................... 120 62 The pressure contours/ distributions at tim e=6.0 with Re=2000 and We=500................ 120 62 Continued.................................................................................................................. .......121 63 The pressure distributio ns along the centerline based on different Reynolds num bers at time=6.0.......................................................................................................................122 64 The U velocity profile at time=6.0 with Re=2000 and We=500..................................... 122 65 Shapes of liquid cores with diffe rent Reynolds numbers at time=6.0. ............................ 122 PAGE 12 12 66 The pressure contours at time=6.0 with Re=500 and We= 2.63...................................... 123 67 Pressure distribution and the curvature of interface at tim e=6.0 with Re=500 and We=2.63...........................................................................................................................123 68 Streamline plot at time=6.0 with Re=500 and We=2.63................................................. 124 69 U velocity profile at time=6.0 with Re=500 and We=2.63............................................. 124 610 Shapes of the interface by different Weber numbers at tim e=6.0.................................... 124 611 The development of interface with Re=500 and We=2.63 at different time steps.......... 125 612 The ratio of averaged local friction coe f ficient in the twophase zone to the fully developed singlephase gas fl ow friction coefficient.......................................................125 613 The computed interfaces by three different grid resolutions........................................... 125 614 The pressure contours, pressure and cu rvature distribution, stream line plot and the contour of U component contour of the case with Re=500, Pe=6500 and We=500 at time=5.5....................................................................................................................... ....126 615 The temperature contours of twophase pl ug flow and the in terfacial temperature at different time steps with Re=500, Pe=6500 and We=500............................................... 127 616 The Nusselt number at th e solid wall at different ti m e steps with Re=500, Pe=6500 and We=500..................................................................................................................... 127 617 The comparison of interfacial temperature and Nusselt num ber at wall for different Reynolds number with Pe=6500 and We=500 at time=5.5............................................. 128 618 The shapes of the liquid slug at three different tim e steps for the case of Re=500, Pe=6500 and We=2.63..................................................................................................... 128 619 The temperature contour of two phase plug flow and the in terfacial temperature with Re=500, Pe=6500 and We=2.63at different time steps................................................... 129 620 The Nusselt number at the solid wa ll Re=500, Pe=6500 and We=2.63 at different tim e steps..................................................................................................................... ....129 71 The U contours for the refrigerant R508B case at differe nt tim e steps.......................... 149 72 The local maximum U in the gas phase for the refrigerant R508B case at different tim e steps..................................................................................................................... ....150 73 The interfaces for the refrigerant R5 08B case at different time steps............................ 150 74 The streamlines in the gas phase fo r the refrigerant R508B case at tim e=4.0............... 151 PAGE 13 13 75 The V contours for refrigerant R508B case at different tim e steps................................151 76 The temperature contours for refriger ant R508B case at different tim e steps................ 152 77 The Nusselt numbers along solid wall for the refrigeran t R508B case at different time steps..................................................................................................................... ....152 78 The total mass flow rate of gas phase fo r the refrigerant R50 8B case at different time steps..................................................................................................................... ....153 79 The vector and the streamline plots for Ca se 2 in the nitrogen cas e with co nstant wall temperature at time=1.5................................................................................................... 153 710 The U contours for the nitrogen case w ith constant wall tem perature at time=1.5......... 154 711 The V contours for nitrogen case with constant wall tem perature at time=1.5............... 155 712 Maximum velocity for the nitrogen case with co nstant wall te mperature of three cases.................................................................................................................................155 713 The temperature contours for nitrogen cas e with constant wall tem perature at time=1.5....................................................................................................................... ....156 714 The interfaces of three cases for the nitr og en case with constant wall temperature at time=1.5....................................................................................................................... ....156 715 The interfaces of three different grid lengths................................................................... 157 716 The temperature gradient of three cas es along the interface at tim e=1.5 and the locations of markers for the nitrogen case with constant wall temperature..................... 157 717 The mass flow rate for the nitrogen case w ith constant wall tem perature at time=1.5.... 157 718 The history of temperature gradient along the interface of Case 2 in the n itrogen case with constant wall temperature........................................................................................ 158 719 The history of mass flow rate of Ca se 2 for nitrogen case w ith constant wall temperature.................................................................................................................... ..158 720 The Nusselt number at wall for the nitroge n case with constant wall tem perature at time=1.5....................................................................................................................... ....159 721 The comparison of Nusselt number by the current method and Hammouda et al.'s correlation. .......................................................................................................................159 722 The U, V and temperature contours of Case 1 in the nitrogen case with w all chilldown at time=1.5...................................................................................................... 160 723 The wall temperature contours for the nitr ogen case with w all chilldown at time=1.5... 160 PAGE 14 14 724 The temperature at solidgas interface fo r the nitro gen case with wall chilldown at time=1.5....................................................................................................................... ....161 725 The temperature gradients along the interf ace of reference for the nitrogen case with wall chilldown at tim e=1.5.............................................................................................. 161 726 The temperature history of Case 1 in the nitrogen case with w all chilldown at time=1.5....................................................................................................................... ....161 727 Temperature gradients along the interface of Case 1 in the nitrogen case with wall chilldown at tim e=1.5...................................................................................................... 162 728 The mass flow rates for the nitrogen case with w all chilldown and reference case........ 162 729 U contours of two cases at time=0.4................................................................................ 162 730 V contours of two cases at time=0.4................................................................................ 163 731 Temperature contours of two cases at tim e=0.4.............................................................. 163 732 Temperature at solidgas interface for the hydrogen and nitrog en cases at time=0.4.....164 733 Nusselt number distributions on the pipe wall for the hydrogen and nitrogen cases for both cases at tim e=0.4...................................................................................................... 164 PAGE 15 15 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CRYOGENIC TWOPHASE FLOW AND PH ASECHANGE HEAT TRANSFER IN MICROGRAVITY By ChengFeng Tai Auguest 2008 Chair: Jacob NanChu Chung Major: Aerospace Engineering The applications of cryogenic flow and heat transfer are found in many different types of industries, whether it be the liquid fuel for propulsion or the cryogenic cooling in medical applications. It is very common to find the transportation of cryogenic flow under microgravity in space missions. For example, the liquid oxygen and hydrogen are used to power launch vehicles and helium is used for pressurizing the fuel tank. During the tr ansportation process in pipes, because of high temperatur e and heat flux from the pipe wa ll, the cryogenic flow is always in a twophase condition. As a result, the physics of cryogenic twophase flow and heat transfer is an important topic for research. In this research, numerical simulation is em ployed to study fluid flow and heat transfer. The Sharp Interface Method (SIM) with a Cutcell approach (SIMCC) is adopted to handle the twophase flow and heat transfer computation. In SIMCC, the background grid is Cartesian and explicit true interfaces ar e immersed into the computational domain to divide the entire domain into different subdomains/ phases. In SIM CC, each phase comes with its own governing equations and the interfacial condi tions act as the bridge to conn ect the information between the two phases. The Cutcell approach is applied to handle nonrectangular cells cut by the interfaces PAGE 16 16 and boundaries in SIMCC. With the Cutcell ap proach, the conservati ve properties can be maintained better near the interface. This research will focus on developing the numerical techniques to simulate the twophase flow and phase change phenomena for one of the major flow patterns in film boiling, the inverted annular flow. PAGE 17 17 CHAPTER 1 INTRODUCTION 1.1 Overview The cryogenic fluids have wide application in the industry. T hey can be used as liquid fuel like liquid hydrogen and oxygen in the aerospace industry (Filina and Weisend 1996), as coolant like liquid nitrogen and helium in medical applications (Jha 2006) In some special cases, for example, cryogenic fluids are used in spacecrafts not only at low temperatures but also under low gravity. Liquid oxygen and hydrogen ar e used to power launch vehi cles and helium is used for pressurizing fuel tanks (Hands 1986). For this case the transport and storag e of cryogenic fluids under microgravity is regarded as an impor tant element of the space mission. General information about the storage of cryogenic fluids can be found in some papers (Panzarella and Kassemi 2003, Khemis et al 2004, Plachta et al. 2006). 1.2 Role of Cryogenics in Space Exploration To explore the space to other p lanets such as Mars (Mueller and Durrant 1999) is one of all human beings challenges. Therefore, the effect ive, affordable, and reliable supply of cryogenic fluids in space mission is very important (C ancelli et al. 2005, Heydenreich 1998, Graue et al. 2000). The efficient and safe utilization of cryoge nic fluids in thermal management, power and propulsion, and life support systems of a spacecraf t during space missions involves the transport, handling, and storage of these fluids in reduced gravity. The uncertainties about the flow pattern and heat transfer characteristics pose a severe design concern. Th erefore, the design of cryogenic fluid storage and transfer system is very importa nt and spawns researches in several areas: for example, design of the vessel (Harvey 1974, Bednar 1986), the piping and draining system (Momenthy 1964, Epstein 1965), insulation (Riede and Wang 1960) and safety devices. Moreover, the thermofluid dynamics of twopha se systems in reduced gravity encompasses a PAGE 18 18 wide range of complex phenomena that are not understood sufficien tly for engineering design to proceed. 1.3 All Cryogenic Systems Must Be Chilled Down When any cryogenic sys tem is initially starte d, which includes rocket thrusters, turbo engines, reciprocating engines, pumps, valves and pipelines; it must go through a transient chilldown period prior to a steady operation. Chilldown is the process of introducing the cryogenic liquid into the system, and allowing the hardware to cool down to several hundred degrees below the ambient temperature. The ch illdown process is anyt hing but routine and requires highly skilled technicians to chill dow n a cryogenic system in a safe and efficient manner. The reason that the highly transient chilldown process is extremely complex is because when a cryogenic liquid is introduced into a sy stem that has reached equilibrium with the ambient, voracious evaporation o ccurs and a very high velocity gas mist traverses through the system. As the system cools, slugs of liquid, entr ained in the gas, flow through the system in a twophase film boiling mode. As the system cools further, a liquid quench ing front flows through the system and is accompanied by nucleate boiling a nd twophase flow. The ra te of heat transfer in the nucleate boiling regime is very high and th e system begins to cool down very rapidly. As the system rapidly cools down, the twophase flow passes through several flow regime transitions to singlephase liquid flow (Burke et al. 1960, Graham et al. 1961, Steward et al. 1970, Bronson et al. 1962). The inherent danger durin g chilldown is that twophase flows are inherently unstable and can experience extreme flow and pressure fluctuations. The hardware may be subject to extreme stresses due to therma l contraction and may not be able to sustain extreme pressure fluctuations from the cryogen. PAGE 19 19 Efficiency of the chilldown process is also a significant concern since the cryogen used to chill down the system can no longer be used for propulsion or power generation. Therefore, chilldown must be accomplished with a minimu m consumption of cryogen. As a result, it is important to fully understand the thermofluid dynamics associated with the chilldown process and develop predictive models that reliably predict the flow patterns, pressure loss, heat transfer rates, and temperature history of the system. 1.4 Background of a Chilldown Process Due to the low boiling points, boiling and tw op hase flows are encountered in most cryogenic operations. The complexity of the proble m results from the intricate interaction of the fluid dynamics and heat transfer especially when phasechange (boiling and condensation) is involved. Because of the large st ratification in densiti es between the liquid and gas phases, the reduced gravity condition in space would strongly change the terrestrial flow patterns and accordingly affect the momentum and energy transport characte ristics. Boiling and twophase flow behave quite differently when the gravity levels are varied. The proposed research will focus on addressing specific funda mental and engineering issues related to the microgravity twophase flow and boiling heat transfer of cryoge nic fluids. The outcome of the research will provide fundamental understanding on the trans port physics of cryogenic boiling and twophase flows in reduced gravity. 1.5 Objectives of the Research The m ain scientific objective is to seek a fundamental understa nding through numerical simulations on the boiling regimes, twophase flow patterns, and heat transfer characteristics for convective boiling in pipes under reduced gravity w ith relevance to the tr ansport and handling of cryogenic fluids in microgravity. The engineering object ive is to address issues that are related to PAGE 20 20 the design of systems associated with the tr ansport and handling of cryogenic fluids, for example, pressure drop information through pipes and heat transfer coefficients. 1.6 Scope and Structure In this research, the m ain objective is to i nvestigate the tr ansportation of cryogenic fluids in microgravity with the phase change phenomena especially the flow pattern and the amount of flow rate of liquid and vapor during the transportation. The experiments about the cryogenic fluids under microgravity are not easy to be perf ormed on the earth. In th is research, numerical tools will be adopted instead of experiment s. The scope of this dissertation will be: To investigate the rate of vaporization (mass loss of cryogenic liquid) based on different driving mechanism and the boundary conditions. To study the flow types based on different driving mechanism and the boundary conditions or fluid properties. To develop the necessary numerical t echniques for phase change computation. The final goal is to develop a reliable nume rical package which can accurately simulate the cryogenic fluid in microgravity and helps the aer ospace engineers to design the most reliable cryogenic transportation system in space mission or related industries. There will be eight chapters in this dissertation. In the fi rst two chapters, the materials about heat transfer and phasechange characteristics of cryogenic flow impact of microgravity effect and chilldown processes are reviewed. In Chapter 3, the physical model, governing equations and interfacial conditi ons are listed. In Chapter 4, th e numerical techniques about the solver of governing equations, moving interface algorithm, the Sharp Interface Method with Cutcell approach (SIMCC), and phase change computat ion will be introduced. In Chapter 5, a series of test cases are done to ensure that the current code is accurate and reliable. Chapter 6 focuses on simulation of the liquidgas twophase flow in a pipe without phase change. Chapter 7 is about the phase change and chilldown process s imulation of the liquidga s twophase flow in a PAGE 21 21 pipe. In the last chapter, the su mmarization of current work and the possible future work will be addressed. PAGE 22 22 CHAPTER 2 PHYSICAL BACKGROUND AND LITERATURE REVIEW 2.1 Boiling Curve and Chilldown Process A typical boiling curve, Figure 21, shows the relationship betw een the heat flux (vertical axis) that th e heater supplies to the boiling flui d and the heater surface temperature (horizontal axis). Based on Figure 21, a chilldown (quenching) process usually star ts above point E and then goes towards point D in the film boiling regi me as the wall temperature decreases. Point D is called the Leidenfrost point (Carey 1992) which signifies th e minimum heater temperature required for the film boiling. For f ilm boiling process, the wall is so hot that liquid will vaporize before reaching the heater surface and that causes the heater to be always in contact with gas. When cooling beyond the Leidenfrost point, if a c onstant heat flux heater were used, then the boiling would shift from film to nucleate boiling (somewhere between points A and B) directly with a substantial decrease in the wall temper ature because the transition boiling is an unstable process. 2.2 TwoPhase Flow Patterns and Heat Transfer Regimes In a cryogenic chilldown process, the wall tem p erature is always highe r than the saturation temperature of the transported li quid. Therefore, the cryogenic ch illdown process in a pipe is associated with a standard twophase flow. In this process, the types of flow and heat transfer can not be determined separately as the dominant heat transfer m echanism is dictated by the flow pattern. The heat transfer also affects the flow pattern develo pment. For the twophase flow regime, there are several possible factors that can influence it, such as the flow rate, the orientation of pipe, the fluid properties and the h eat flux at the wall of pi pe (Yuan et al. 2007). In Careys work (1992), he showed the different twophase flow t ypes in a horizontal pipe. The flow types may be bubbly flow, plug flow, stratif ied flow, wavy flow, slug flow or annular flow. PAGE 23 23 In Dziubinski et als work ( 2004), they showed the flow regi mes of a twophase flow in a vertical tube and the possible flow types are bubbly flow, slug flow, churn flow, annular flow and mist flow. The main difference between the ho rizontal and vertical flows is the effect of gravity that causes the horizontal flow to become nonsymmetrical to the centerline of the pipe. With regard to the chilldown process in the film boiling regime, depending on the local quality and other thermohydraulic parameters, the flow regime can be either dispersed flow, annular flow, or inverted annular flow. The corresponding heat transfer regime will be dispersed flow film boiling, annular flow film boiling, or inverted annul ar film boiling, respectively. As the wall temperature decreases below a certain degree, th e liquid phase is able to contact the wall of the pipe. The leading liquidwall contact point which is often re ferred to as the quenching front or sputtering region is char acterized by a violent boiling pr ocess with a significant wall temperature decrease. The quenching front will pr opagate downstream with the flow. The heat transfer mechanism at the quenching front is the transition boiling, which is more effective than the film boiling heat transfer due to partia l liquid wetting of the wall surface. This reestablishment of liquidwall c ontact is called rewetting pheno menon and has been a research interest for several decades. Nucleate boiling heat transfer dominates afte r the quenching front. For a vertical pipe, the flow regime can be annular flow slug flow or bubbly flow. While for a horizontal pipe, the flow regime is generally stratified flow. As the wa ll temperature decreases further, the nucleate boiling process gradually reverts back to pure co nvection until the wall temperature reaches the thermal equilibrium with the wall, which denotes the end of the chilldown process. Figure 22 illustrates the flow patterns and co rresponding heat transfer regimes. PAGE 24 24 2.3 TwoPhase Flow in Microgravity In this research, the m ain purpose is to si mulate the cryogenic chilldown process under reduced gravity in space to obtain the design parameters for the related equipment. The condition of low gravity has a significant impact on the twophase flow model. It is the most important factor for determining the gas/liquid interfacial dynamics. The entire flow field will now be controlled by the convection, pressure gradient and viscous effects only. There will be some contributions from the surfacetension induced fo rces and the presence of the solid wall. When the dynamics is changed, the flow regime will also be changed. Figure 23 is the possible flow patterns under microgravity. Figure 24 is an example to show the distinctively different flow patterns in a horizontal pipe between terrestrial and microgravity conditions. For annular flow film boiling, the effect of gravity is assessed based on the ratio of 3/Re Gr where Gr is the Grashof number and Re is the Reynolds number. The gravity effect is measured by the natural convection contri bution, which is characterized by the Gr, while the forced convective film boiling is scaled by the Reynolds numbe r. According to Gebhart et al.s work (1988), if the velocity of gas is greater than 10 cm/s, then the 3/Re Gr is less than 0.2. Natural convection is negligible for 3/Re Gr less than 0.225. 2.3.1 Isothermal TwoPhase Flows in Microgravity In order to investigate the im pact of gravity qua ntitatively, some detailed experiments under simulated microgravity conditions have be en performed on an aircraft (Rezkallah and Zhao 1995, Colin et al. 1991) and on the ground in a drop tower (Mishima and Hibiki 1996) because it is too expensive to do it in space. Th e first report on microgr avity isothermal flow pattern for a large range of liqui d to gas ratios was provided by Dukler et al. (1988). In their research, they showed the flow pattern s are different between the 1g and the g conditions. PAGE 25 25 Flow patterns under microgravity can be broadly classified into three types: the bubbly, the slug and the annular flow. In the beginning, experiments could only be pe rformed in a pipe with a very small radius. Therefore, this experimental data may not be accurate enough. In th e following studies, the researchers tried to perform the experiment with larger pipes and various fluids to determine the flow patterns (Janicot 1988, Colin et al. 1991, Zhao and Rezkallah 1993). In others researches (Zhao and Rezkallah 1995, Rezkallah 1996), the flow patterns were classified into three regions based on the dyna mics of a flow: a surface tension region (bubbly and slug flow), an inertia region (annular flow) and the transitional region (frothy slugannular flow). There are several possible factors that may in fluence the flow patterns under microgravity such as the pressure drop (Zhao and Rezkalla h 1995), surface tension (Rezkallah 1996), flow rate, pipe diameter and viscosity (Bousman et al. 1996). In Rezkallahs work (1996), he showed the flow patterns based on di fferent Weber numbers and there are two transition lines which divide the flow into three regions. When the We ber number of the gas phase becomes larger; the flow tends to become annular; otherwis e bubbly flow and slug flow are observed. In Bousman et al.s work (1996), they s howed the flow patterns based on different superficial velocities of liquid and gas phases. When the superficial velocity of the liquid phase is larger, the flow patterns are bubbly flows and move to annular flows when the s uperficial velocity of gas phase is larger. 2.3.2 TwoPhase Flow and Heat Transfer in Microgravity In general, there is little heat transfer data for cryogenic flow boili ng in reduced gravity. Only two reports were found. AdhamKhodaparast et al. (1995) investig ated the flow film boiling during quenching of R113 on a hot flat surface. They used microsensors to record PAGE 26 26 instantaneous heat flux and heater surface temperature during the film boiling on board a KC135 aircraft. Antar and Collins (1997) reported flow visualizations and measurements for flow film boiling of liquid nitrogen in tubes on board KC135 aircraft. They were particularly interested in the gas/liquid flow pattern and the thermal characteristics. They identified a new gas/liquid flow pattern that is unique to low gravity: a sputtering l eading core followed by a liquid filament annular flow patte rn. This new flow pattern is co mposed of a long and connected liquid column that is flowing in the center of the pipe surrounded by a thick gas layer. The gas annulus that separates the liquid filament from the wall is much thicker than that observed in the terrestrial experiment. They attributed the filament ary flow to the lack of difference in the speed of gas and liquid phases. On the heat transfer si de, they reported that the quench process is delayed in low gravity and the pipe wall c ooling rate was diminished under microgravity conditions. PAGE 27 27 Figure 21. Typical boiling curve. Inverted Annular Flow Single Phase Flow Dispersed Flow Convective Heat Transfer Nucleate Boiling Film Boiling Heat Transfer Region Flow Patterns Figure 22. Flow regimes and heat tr ansfer regimes in a heated channel. Bubbly Flow Annular Flow Slug Flow Figure 23. Flow regimes under microgravity. PAGE 28 28 Annular Flow g Stratified Flow 1g Figure 24. Effect of th e gravity on flow regimes. PAGE 29 29 CHAPTER 3 PROBLEM DEFINITION AND GOVERNING EQUATIONS 3.1 Modeling Cryogenic Chilldown in Microgravity According to the literature review presented in the previous chapter, for most part of the cryogenic chilldown process in microgravity, the twophase flow is in the inverted annular flow regime with film boiling. Based on the general boiling curve shown in Fi gure 21, initially the state of boiling is located around point F due to a large temperature difference between the wall (room temperature ~300 K) and the saturation temperature of the cryogenic fluid (~77 K for liquid nitrogen and ~20 K for liquid hydrogen) when the cryogen first enters the pipe. The schematic of a representative cryogenic flow system is given in Figure 31. As the wall is cooled down by the cryogenic twophase flow, the boiling st ate will move from poi nt F towards point E and then approach point D when the wall is cooled down further. It is noted that the route from point E to point D is followed for the current que nching process, instead of the route from point E to point C, because the wall temperature is re duced during the quenching process by the heat transfer and can not be changed independently. During the initial stage, a que nching front would form that is followed by an inverted annular flow pattern with gas phase next to the pipe wall and a liquid core in the center (Antar and Collins 1997). As the liquid vaporizes, the radius of the liquid core w ould decrease as it travels downstream. Figure 32 illustrates the pr oposed physical model of the cryogenic twophase film boiling in microgravity, a continuous inverted annular flow. There are several research public ations about the inverted annular flow (Yadigaroglu 1978, Ishii and Jarlais 1987, Aritomi et al. 1990, Nelson and Unal 1992, Hammouda et al. 1997). However, the heat fluxes in their researches were set at constant values or different from this research. In this research, the temperature of the wall will vary and also th e strength of heat flux will change accordingly. The physical phenomena inside the tank and other devices downstream of the pipe exit will not be included in the current simulation. Only the flow in the pipe will be considered in this PAGE 30 30 research. In the absence of gravity, it is reasonabl e to assume that the inverted annular flow is axisymmetric. Figure 33 shows the schematic for th e idealized continuous inverted annular flow that will be used as the physical mode l in the current numerical simulation. To complete the entire conjugate heat transfer path, the twophase flow inside the pipe must be connected to the heat source, the pipe wall. Figure 34 shows the heat transfer network of this conjugate heat transfer. The outer surface of the wall is assumed to be perfectly insulated. Therefore, on the outside of wall, the in sulated boundary condition will be assigned. 3.2 Geometry, Computational Domain and Initial, Boundary Conditions Since, this is an axisymmetric computation, a rectangular domain with a width of 0.5 (pipe radius) is used to simulate the circular pi pe. The length of the domain will depend on the problems. The minimum grid spacing is 0.01. The in itial shape of the liquid core is a quartcircle with radius 0.38 plus a straight line 0.03 as show n in Figure 35. Initially, the pipe is filled with the stationary gas and at the entrance. A quartcir cle interface is used to divide the domain into different phases. The temperatures of wall and gas phase are assi gned the same initially (=1.0) and the temperature of liquid core is set the same as the temperature at inlet (=0.0). The boundary conditions for this problem can be divided into two parts: Liquid and gas phases inside the pipe: In the real application such as space mission, the mass flow rate should be most im portant concern in order to keep the enough thrust so that the mass flow rate in this research at the inlet is assigned as consta nt by an uniform velocity (=1.0). In the incompressible flow, the to tal mass flow rate at outlet should be equa l to the mass flow rate at inlet and plus the mass flow rate generated from the liq uidgas interface due to the phase change. By this idea, the total mass flow rate at outlet can be obtained. In this research, the device at the downstream is not specified so that there are not exact boundary conditions at the outlet. In numerical simulation, the extrapolation strategy is us ually used for the undetermined boundary condition. In this research, the second or der extrapolation is applied for the velocity boundary condition at outlet and this ve locity profile will be corrected by the total mass flow rate. PAGE 31 31 By this strategy, the velocity profile at the outlet should be reasonable and the conservation of mass is kept. In this research, the pressure at inlet of pi pe is assigned as consta nt (=1.0). In NavierStokes equations, this constant do es not mean anything but a refe rence value and it can be any value. In the incompressible pipe flow computation, the most importa nt thing is the pressure drop and this pressure drop should be balanced by the shear stress from wall. In this research, the exact pressure at the outlet is unknown so that th e second order extrapolation is assigned to get a reasonable estimation value for pressure from the nearby locations. In the incompressible pipe flow, there ar e two velocity and two pressure boundary conditions at the inlet and outlet. In numerical simulation, if the exact/real boundary conations are known, the numerical boundary conations can be assigned based on this exact boundary conditions or at least one condition must be flexible. In th is research, the flexible boundary condition is the pressure at the outlet. By this flexible pressure boundary condition at outlet, the correct pressure drop can be obtained to balance the shear stress from the wall. Since it is an asymmetric pipe flow computation, the symmet ric condition is assigned at the centerline and nonslip condition is assigned at the wall. The temperature boundary conditions in this rese arch are assigned constant at the inlet (=0.0) and second order extrapol ation at downstream, symmetric condition at the centerline. For the temperature at wall, it depends on differe nt cases and governed by interfacial condition. Solid wall: The boundary conditions for the left and upper sides are the insulated boundary conditions and second order extrapola tion for temperature at right side. 3.3 Assumptions In this section, the assumptions in this research are made as following: The flow and heat transfer are axisymmetric in a 2D cylindrical coordinates system. The flows are incompressible for both li quid and vapor phases. In most cryogenic transport systems, the flow rate may not be very large (Mach number < 0.3) and in this PAGE 32 32 research the temperature differences will be limited to moderate values such that the compressibility should be negligible. A singlecomponent inverted a nnular twophase flow under film boiling is taking place in a circular pipe with no body force in a gravityfree environment. Viscous dissipation is neglected due to low velocities. Constant properties in the liquid and gas phases. Fluids in both phases are Newtonian fluids. Thermal equilibrium at the liquidgas inte rface and thermal nonequilibrium for the gas phase. The gas phase is transparent for thermal radi ation, the wall is a grey body, and thermal radiation is a surface phe nomenon for the liquid phase. 3.4 Governing Equations In this research, the Sharp Interface Met hod (SIM) will be adopted to treat the moving interface. In the SIM, the interface is a true surfac e with zero volume, so the governing equations of each phase are solved separa tely and transport fluxes are ma tched between the two phases at the interface. In the cryogenic twophase flow, the mass, momentum and energy equations in the liquid and gas phases are developed separately Based on the above assumptions, the governing equations are listed below: Liquidphase equations: 2 20l l ll ll l l l lpl llllu u uupu t T cuTkT t (31) Gasphase equations: PAGE 33 33 0 () ()g gg g gg ggg gpgg gpggg ggu u uupu t cT cuTkT t (32) Solidphase equation: w wpwwwT ckT t (33) where subscripts '' l,'' gand'' w denote the liquid, gas and solid phases, respectively; u is the velocity vector, Tis the temperature, is the density, is the dynamic viscosity, pc is the specific heat, k is the thermal conductivity. 3.4.1 Interfacial Conditions Force balance in the normal direction at the liquidgas interface: 22 int int()()()()lggg ll lg p puuuunnnn (34) where l and g represent the viscous stress tensors in the liquid and gas phases, respectively; is the surface tension coefficient; is the curvature of the interface; lp and g p are the pressures in the liquid and gas phase, respectively. Mass flux continuity at the liquidgas interface: int int()()ll ggmuunuun (35) where m is the interfacial mass flux, n is the unit normal v ector at the interface; intu lu and g u are the velocities of interface, liquid and ga s phases, respectively. In a twophase flow, energy must be conservative and the temperatur e is the same for both phases at the interface under the thermal equilibrium condition. Interfacial energy conservation cond ition at the liquidgas interface: int()()()llggradllmkTkTqnuun (36) Interfacial temperat ure condition at the liquidgas interface: int lgTTT (37) PAGE 34 34 where is the latent heat of vaporization. radq is the thermal radiation heat flux received by the liquid surface from the wall. Thermal radiation at the liquidgas interface: Based on the simplifying assumption stated above and also the fact that the liquid phase is entirely enclosed by the pipe wa ll, the thermal radiation heat flux received on the surface of the liquid phase is as follows: 44(1) 1 ()wsat rad wl lwwiTT q R R (38) where w and l are the emissivities of solid wa ll and liquid phase, respectively. wil R andR are the radii of the inner surface of pi pe and liquid core, respectively and 85.669710is the StefanBoltzmann coefficient. Continuous heat flux at solidgas interface: ,g w wgradwwiT T kkqatrR rr (39) Thermal radiation at the solidgas interface: 44 ,11 ()1wsat radw wi llwTT q R R (310) Continuous temperature at solidgas interface: wg wiTTatrR (311) Insulation on the outer surface of pipe: 0w woT atrR r (312) where wo R is the radius of outer surface of the pipe. 3.4.2 Nondimensionalization In this work, all the computational quan tities are nondimensional. The governing equations of each phase and the interfacial cond itions must be nondimensionalized first. The reference scales are characteristic length L and velocity U and the characteristic time is/ LU. PAGE 35 35 The characteristic length in this research is the di ameter of the pipe, and characteristic velocity is the inlet velocity. The characteristic temperature scale is s atTTT T is the ambient temperature and wall initial temperature. s atT is the fluid saturation temperature. Based on these reference scales, the dimensionl ess variables are defined as */ x xL, */ttUL, */ref */ref */ref */refkkk _/Ppprefccc */uuU, *2/()refppU, *()/() s at satTTTTT where the fluid properties of the liquid phase at the inlet are used as references. After the nondimensionalization procedure, the original governi ng equations and the interfacial conditions can be written as follows (the asterisks for dimensionless quantities are dropped from this point on for convenience): Liquidphase equations: 2 20 1 Re 1 u u uupu t T uTT tP e (313) Gasphase equations: 0 1 Re 1g gg l ll gpg lpl gpgg lpllu u uup u t c T c ck uT T tcP ek (314) Solidphase equation: PAGE 36 36 1wpl w llpwkc T T tkcPe (315) where the dimensionless parameter Re is the Reynolds number and Pe is Peclet number. Rel lUL elpl lcUL P k Interfacial conditions: Force balance in the normal direction at the liquidgas interface: 22 int int1 ()() Reg nn lgggll lg luu ppuuuu We n n (316) Mass flux continuity at the liquidgas interface: int int()()ll ggmuunuun (317) Interfacial energy conservation cond ition at the liquidgas interface: intg lllrad nn g gglT kTqL Ja uu PeknnkT (318) where () s atTTT and the dimensionless parameter We is the Weber number and Ja is the Jakob number. 2lUL We gpl lkcT Ja k Interfacial temperature conditi on at the liquidgas interface: int lgTTT (319) Continuous heat flux at solidgas interface: ,/radw wg w i lqL TT kkatrRL rrkT (320) Continuous temperature at solidgas interface: /wg wiTTatrRL (321) Insulation on the outer surface of pipe: PAGE 37 37 0/woT atrRL r (322) In this research, the heat flux by the thermal radiation radwq radq are computed with dimension and converted to the dimensionless qu antity during computation and the conversion factor is lL kT . PAGE 38 38 pipe Strong heat flux Other devices Tank Figure 31. A simple cryogenic system. Liquid Gas inverted annular flow Figure 32. Inverted annular flow. Liquid Gas idealized inverted annular flow Figure 33. Idealized inverted annular flow. inner wall surface insulated outer surface solid wall liquid core surface vapor liquid core Figure 34. Conjuga te heat transfer. PAGE 39 39 0.03 0.38 0.41 0.38X R wall gas liquid symmetric line Figure 35. The initial shape of liquid core. PAGE 40 40 CHAPTER 4 SOLUTION METHOD 4.1 Introduction In earlier numerical simulations about the chil ldown process, scholars have used simplified models to solve the chilldow n process, such as the onedimension model (Chi 1965), the homogeneous model (Cross et al. 2002), the tw ofluid model (Ishii 1975, Ardron 1980, Ishii and Mishima 1984), the threefluid model (Alipchen kov et al. 2004) and the pseudosteady model (Liao et al. 2006). Also, different important correlations are devel oped such as the correlation for saturated boiling (Chen 1966), subcooled boiling (Gungor and Winterton 1986) and pool boiling (Steiner 1986). By these simp lified modeling works, the co mplex governing equations and interfacial conditions can be simp lified and desired results can be obtained very fast and easily. However, these modeling works need the corr elations by the experiments to simplify the governing equations so they can not be applie d in broad applications. Also, because of simplification, the modeling work cannot give al l the necessary and realtime information about the chilldown process such as the flow pattern s, and may be not very accurate for some important characteristics. Therefore, direct numerical simulati on is adopted in this research. By directly solving the genera l governing equations, the inte rfacial conditions and proper initial/boundary conditions, the com putation can be unsteady and all the flow characteristics can be captured. All the information can be saved and give a great help in different engineering aspects. As illustrated in Chapter 3, the complex problem of twophase flow and conjugate heat transfer with phase change does not allow any po ssibility of an analytical solution. In this chapter, the numerical simulati on method that has been adopted for seeking the solutions is presented. The central infrastructure of this method is based on the concept of a sharp interface PAGE 41 41 that truly separates the liquid pha se from the gas phase. In other words, the interface is a true surface without any volume association. The key elements of the Sharp Interface Method (SIM) are listed as follows: Fixed Cartesian Grid The computational framework is built on an EulerianCartesian grid. With this underlying fixed grid system, a socalled Cutcell approa ch is used to treat the interfaces and boundaries that do not align with the fixed Cartesian grid. Lagrangian Moving Sharp Interface Algorithm To track the moving vaporization front, separate marker points(Nichols 1971) are used to identify the phas echange interface. These points that are connected by piecewise polynomials are employed to capture the deformation and movement of the sharp interf ace through the translat ion of these markers over the underlying Eulerian fixed grid. Fractional Step Method For each phase domain, the fractional step method is used to numerically integrate the gove rning equations in that pha se. The overall solution is obtained by matching the mass, momentum and heat fluxes from both phases at the sharp interface. 4.2 Fractional Step Method within the Finite Volume Framework In order to best enforce the conservation laws and to treat the disconti nuity at the interface, the finite volume method (Verstee g and Malalasekera 1995) is used to discretize the governing equations. By the finite volume framework, th e governing equations listed in the previous chapter will be integrated over a finite control volume which is called a unit cell as shown in Figure 41. The integral forms of governing equations for an inco mpressible flow with constant properties are given below. Continuity equation: 0csundS (41) Momentum equation: 1 Recv cs cs csu dVuundSpndSundS t (42) Energy equation: PAGE 42 42 1 ecv cs csT dVTundSTndS tP (43) where cvand cs represent the control volume and the surface of the control volume, respectively. n is the outward normal vector fr om the control volume surface, u is the velocity vector, p is the pressure and Tis the temperature. In this research, a cellcentered collocated (nonstaggered) approach (Ferziger and Peric 1996) on the Cartesian grid system is adopted, the primary variables (v elocity, pressure and temperature) are defined at the cell centers and th e primary variables needed at the cell faces will be evaluated by interpolation from respective variab les at the cell centers as shown in Figure 41. A second order accurate twostep fractiona l step method (Chorin 1968, Kim and Moin 1985, Zang et al. 1994) is used for advancing the solutions of the integral unsteady governing equations in time. In this approach, the solution procedure is advanced from time step n to n+1 through an intermediate diffusionconvection step where the momentum equation without the pressure gradient terms is first solved and advanced in a half time step. The intermediate diffusionconvection momentum eq uation can be discretized as 11 *1 3 2 1 () 2Ren nnnn cvcs n csuu dVuUnuUndS t uundS (44) where *uis the intermediate veloci ty at the cell center and U is the velocity at center of the cell face. The cell surface veloci ty is used to evaluate the fluxes going in or out of a control volume. After the *uis determined, *U the intermediate velocity at the center of cell face, is calculated by interpolating between the resp ective cellcenter velo cities. The first term on the right hand side is the convective term. A second order accurate AdamsBashforth scheme (Bashforth and Adams 1883) is used to discretize the convective term. The second term is the diffusive term that PAGE 43 43 is discretized by the implicit CrankNico lson scheme (Crank and Nicolson 1947). This eliminates the potential viscous instability that c ould be quite severe in the simulation of viscous flows. Once the intermediate velocity is obtained, the pressure is obtaine d by the correction step: 1* 1n n cv cvuu dVpdV t (45) In this pressure correction step, the final velocity 1nu must satisfy the integral mass conservative equation. The integral mass conserva tive equation can be rewr itten as the following form: 10n csUndS (46) Therefore, the integral pressure co rrection equation can be expressed as 1*()n cscs p ndSUndS (47) Once the pressure is obtained, the intermedia te velocity can be corrected and updated to obtain the final velocity by 1*1 1*1 nn cellcenter nn cellfaceuutp UUtp (48) The energy equation is solved by similar proced ure but the predictcorrect procedure is not required because the energy equation in this research is a standa rd convectiondiffusion equation. The discretized energy equation is similar to Equation 44 and can be discretized as (Francois 2002) 1 11 11 3 2 1 () 2nn nnnn cv cs nn csTT dVTUnTUndS t TTndS Pe (49) PAGE 44 44 For the incompressible flow with nonconstant properties, the fractional step method can work also and just needs to in clude the flow properties into eac h procedure (Ferziger and Peric 1996). 4.3 Cartesian Gird Method for a Complex Geometry In the early development of computational mu ltiphase flow, some researchers adopted the curvilinear grid system (Ryskin and Leal 1984, Dandy and Leal 1989, Raymond and Rosant 2000, Lai et al. 2003). This approach is simple but not easy to apply. In order to describe the deformation of an interface between different phas es, a very powerful grid generation is required and also, the grid has to be updated frequently to obtain the convergent so lution and therefore it is very computationally intensive. In recent multiphase computational approaches, several Cartesian grid methods are broadly used such as the Sharp Interface Method (SIM) (Udaykumar et al. 2001, Ye et al. 2001), the Immersed Boundary Method (IBM) (Peskin 1977, Singh and Shyy 2007), the Front Tracking Method (Qian et al. 1998, AlRawahi and Tryggvason 2004) ,the Volume of Fluid (VOF) Method (H irt and Nichols 1981, Pilliod a nd Puckett 2004), the Level Set Method (Ni et al. 2003, Tanguy and Berlemont 2005) the coupled Level Set and Volume of Fluid Method (Son 2003, Sussman 2003) and the Phase Field Method(Badalassi et al. 2003, Sun and Beckermann 2007). Based on the computational framework, Sharp Interface Method, Front Tracking Method and Immersed Boundary Method are classified under the mixed EulerianLagrangian category and Level Set Method, Volume of Fluid Method and Phase Field Method are in the Eulerian category (Shyy et al. 1996). In this research, the Sharp Interface Met hod (SIM), a branch of the mixed EulerianLagrangian Cartesian grid method, is adopted to handle the comple x geometries. In the SIM, the Cartesian grid is designed as a background mesh, a nd the explicit inte rfaces are used to describe the shapes of the objects on the background grid. The interfacial dynamics associated with the PAGE 45 45 moving/fixed boundaries need to be consider ed simultaneously. In the mixed EulerianLagrangian approach the interf ace is constructed by a sequence of marker points. With these marker points, the shape and location of the in terface are determined by designated interpolation procedures, while the overall fluid flow is computed based on the fixed Cartesian grid. In the SIM, the interface is treated explicitly with zero thickness, in accordance with the continuum mechanics model. The primary variables at th e interface are computed via the interfacial conditions The SIM defines the relations between the b ackground grid and the interfaces. Because of the nonCartesian interfaces, some cells containi ng the interface will be cut and form the nonrectangular cutcells. Special me thods are needed to handle these cut cells. In this research, a Cutcell approach (Ye et al. 2001, Ye et al. 2004, Tai and Shyy 2005) is employed to treat the interface and boundary cells. In the Cutcell approach, each segmen t of the cutcel l is merged into a neighboring cell or assigned the identity of the original Cartesian cell. Hence, even though the underlying grid is Cartesian, the cut cells are reconstructe d to become the nonrectangular cells and the cutsides will form the interface. Afte r the reconstruction, the entire grid is filled with the rectangular grid and nonre ctangular grid. In this resear ch, the shapes of the cut cells include triangle, trapezoid and pentagon. The SIM with the Cutcell approach (SIMCC) can handle the sharp discontinuity resulting from th e interface formation, and can achieve higher accuracy (Ye et al. 2004, Tai and Shyy 2005). Among the EulerianLagrangian approaches, the SIMCC is the one with the best accu racy, especially for a solid boundary. Figure 42 is an example of grid system ha ndled by SIMCC. In Figure 42, the cells far away from the object are still rectangular sh apes and only the cells near the interface are PAGE 46 46 modified to be nonrectangular shapes. It must be emphasized that the total number of cells will not change during the computation. 4.4 Sharp Interface Method with CutCell Approach (SIMCC) The SIMCC is adopted to solve the complex geometry in this research. For SIMCC, the underlying grid is a Cartesian grid which is cut by the explicit interfaces to form the cutcells. The entire domain will be filled with these cut and regular cells. Most cells still keep the original shapes. The SIMCC utilizing the Cartesian grid and Cutcell approach can be used to solve fluid flow and heat transfer problems involving multip hase and/or complex geometry with a high accuracy. Due to the algorithm and data management requirements the SIMCC is computationally intensive. Using SIMCC, the inte rface is constructed by a sequence of straight lines in the computational domain. Because th e number of cells in SIMCC does not change during the computation, the matrix of coefficients of SIMCC is very similar to the curvilinear grid system. Only the coefficients of cutcells ha ve to be modified so that the convergent speed and characteristics are very close to the curvilinear system and so it converges much faster than the unstructured grid system. For a complete set of SIMCC, there are four main procedures; the interfacial tracking, the merging procedure, the flux computations in the interfacial region and the moving interface algorithm for advancing the interface. For the interfacial tracking, the information of the interface should be input and reproduced and also the intersections between the interface and background grid must be located. These intersections will be the cutpoints and used for the next step, the merging procedure. Because of the interface, some cells are cut and can not maintain the rectangular shape anymore and therefore ha ve to be treated specially. Using a merging technique, the fragments of cells can be merged by neighboring cells or larger fragments to form cutcells. This means the cells around the interf ace have to be recons tructed. For the third PAGE 47 47 procedure, the cutcells around th e interface may be in the form of a triangle, trapezoid or pentagon. A special interpolation scheme with higher order accu racy is required to handle the complicated cutcells to get the accurate primary va riables or derivatives at the center of a cell face. The original Cartesian grid will become a mixed grid which includes most rectangular and some nonrectangular grid. Figure 43 is an example of the mixed type grid. For a fixed interface problem such as flow ove r a sphere, the above three techniques are enough. If the interface is not fixed but moving, the moving interface algorithm will be necessary. In the moving interface algorithm, th ere will be two functions to perform: To advance the interface based on the force ba lance in the normal di rection at interface. To refresh the cell because of change of phase. By these four techniques, the SIMCC can be used to solve the moving interface problem. 4.4.1 Interfacial Tracking During this procedure, marker points are used to describe the initial position of the interface. As shown in Figure 44(A), a sequence of marker points is given initially. In this oblique ellipse case, there are 50 initial mark ers. These markers must be represented by a polynomial curve fitting method; also, the distan ce between two adjacent markers have to be adjusted based on the curve fitting method. For this research study, a quadratic curve fitting (Chapra and Canale 2002) is used so that the distance between two neighboring markers can not be too long and is set to dx/2 to maintain the sh ape of the object. After the curve fitting, a new sequence of markers with an equal distance from each other is obtained as shown in Figure 44(B). The number of markers is increased to 68. The location of markers are stored as func tion of the arc length and represented by a sequence of quadratic functions. The intersections of the inte rface and the background grid are PAGE 48 48 shown in Figure 45. These intersections represent the cut points of the cutcells. With this information, the normal vector of cutside can be calculated. Besides the marker points, all the primary valu es at interface are stored by the quadratic curve fitting also and can be represented by 2asbsc (410) The coefficients in Equation 410 can be obt ained by any three points which construct a section of quadratic curve. A ny quantities located at this qua dratic curve can be obtained by Equation 410. 4.4.2 Merging Procedure In this study, the interface is represented by a series of piecewise quadratic curves. The governing equations will then be solved in regions separated by the interface, and the communication between these regi ons is facilitated by the inte rfacial conditions. Figure 46 illustrates the formation of interf acial cells where cells 1 to 4 ar e cut by an interface. According to the present Cutcell approach, the segments of an interfacial cell not containing the original cell center are merged by their neighboring cells; the segments containing the original cell center are given the same index as the original cell. For example, in Figure 46, the upper segment of cell 3 is merged into cell 5 to form a new trap ezoid cell. The fraction of cell 3 with cell center becomes a new independent trapezoid cell. The main segment of cell 1 that contains the original cell center will merge the small se gments of cells 4 and 2 to fo rm a new triangular cell. The remaining segment of cell 4 containing its orig inal cell center now becomes an independent pentagonal cell. With these cutandmerging pro cedures, the interfacial cells are reorganized along with their neighboring cells to form new cells with trian gular, trapezoidal, and pentagonal shapes in a 2D domain. Figure 46 shows an ex ample of the cutcells after reconstruction. PAGE 49 49 Basically, when the area of a segment is less th at 0.5 area of a normal cell, it will be merged. After this procedure, each newly defined cell maintains a unique index and cell center to support the needed data structure. Of course, in a 3D dom ain, the situation will be more complicated, and will not be discussed here. Figure 47 are two examples of cutcells of di fferent objects. The object s here are an airfoil (NACA0012) and a star shape. Figure 48 is anothe r example to illustrate that the cutcells of the different grid sizes. In Figure 48( A), the grid size is 0.025 and 48(B) is 0.1. In this case, the number of cutcells of a fine grid is larger than that in a coarse grid. It can be seen that the object is represented more accurately in a fine grid than a coarse grid. 4.4.3 Flux and Stress Computations in the Interfacial Region In Figure 49, it represents a Cartesian grid with nine cells cut by an interface. The solid squares mean the centers of cells. Be cause of the interface, the original A CDFcell with the cell center 1 will absorb the fragment from another cell to form a trapezoidal B CDE. The original grid line A Fis replaced by a section of interface B E and the original faces DFand A C are extended to become DE and B C. In the finite volume framework, the fluxes (fir st order derivative) at cell faces and the primitive variables at face centers must be known for the surface integration purpose. Therefore, how to obtain the highly accurate fluxes and pr imitive variables is very important in SIMCC. To consider a flux f across a cell face B C in Figure 49, one can construct a secondorder accurate integration procedure as follows: ()()BAABACCA BCBAAC f dyfdyfdyfyyfyy (411) PAGE 50 50 where BA f and AC f are computed at the centers of segments B A and AC, respectively. Getting the value of AC f is straightforward. If AC f is the flow variable a linear interpolation can be used, yielding a secondorder accuracy The relation can be expressed as 1131(1)CA (412) 1 1 13AC x x x x (413) If AC f represents the normal gradient of flow variable it can be approximated by the following central difference scheme as 13 13AC x xx (414) Equations 412 to 414 can not be used to estimateBA f because the neighboring cell centers are located in a region on the other side of the interface. Also the center of segment AB is not located on the straight li ne connecting the two cell centers. In order to maintain a secondorder accuracy, a twodimensional polynomial interpolating function is adopted in the computation of for such kind of small segments. An appropriate functional form for that is linear in X direction and quadratic in Y direction is given as 22 123456cxycycxycycxc (415) the six coefficients, 1c to 6c can be obtained from values at the six points (1 to 6) in Figure 49 ; thus the variable at the center of AB can be expressed as 22 123456ABABABBABABABABcxycycxycycxc (416) The normal gradient ABx which is often needed while computing the interfacial condition, can be obtained by PAGE 51 51 2 135ABAB ABcycyc x (417) Similar approach is used to compute the fl ow variables or their normal gradients on the remaining segments. Once the primary variables and the derivativ es at the cut side are determined, the coefficients of matrix for the cutcells can be modified based on this information and the matrix solver will be called to obtain the solution. It must be emphasized that Equation 417 is only for A B in this case. Here, the y terms are up to second order while the terms in x are linear. That means this interpolation polynomial is quadratic in Y direc tion and linear in X direction. For a more complicated case, a highe r power polynomial may be needed but the order will not exceed two in either x or y 4.4.4 Moving Interface Algorithm For the fixed interface problems, the techniqu es discussed from Section 4.4.1 to 4.4.3 constitute all the elements required to obtai n the solutions. However, for most problems in multiphase flows, the interface is not fixed and therefore a moving interface algorithm is necessary. The moving interface algorithm in this research includes two functions. The first one is to advance the interface (R yskin and Leal 1984) and the se cond one is to update the cells because of change of phase (Udaykumar et al. 1997). 4.4.4.1 Advancing the Interface In the unsteady multiphase computation, the in terface will advance to satisfy the interfacial dynamics in each time step. In SIMCC, a push a nd pull strategy is used to determine the new location of an interface and also to satisfy the fo rce balance at the interf ace. The forces acting at the interface can be resolved into the normal and tangential components. In multiphase flow computation, the order of magnitude of the nor mal force is much larger than that of the PAGE 52 52 tangential force and therefore the displacement of interface is governed basically by the normal forces at the interface. In th is push and pull strategy, onl y the normal component of force balance is considered on the interface and the new location of interface will be determined by a series of iterations. In each iteration, the residual of the force ba lance in the normal direction will be computed and the displacement of marker poi nts is assumed proportion al to this residual: 11 11 nnn newnew x nnn newnew y X Xn X Yn (418) where (,) X Yis the coordinates of marker points, x ynnis the components of normal vector in the X and Y direction, respectively. is the residual of the force balance in the normal direction and is a relaxation factor and it is an empirical value normally in the range of 0.10.001 in this research. Figure 410 is an illustration for this interf acial advancing process. In Figure 410, a marker point 0A is located at the initial interface and the residual of force balance in the normal direction 1 will be computed. Based on the location of 0A and the value of 1, the marker point 0Awill be pushed to 1A. In this moment, the residual of force balance in the normal direction will be check ed. If the new residual 2 is still large, the marker point 1A will be pushed to 2A. Once the residual is small enough, i.e., th e force balance in the normal direction achieves convergence, the interac tion will be stopped and the normal component of interfacial velocity can be determined by _int_ _int_()()/ ()()/nxnewnewold nynewnewolduXXt uYYt (419) In this iterative procedure, the residual of the force balance in the normal direction is required to be less than 1.0E3. The normal comp onent of the interfacial velocity will be the PAGE 53 53 velocity boundary condition during the computation. This algorithm is only for the problems without phase change. Once there is phase change Equation 419 must be modified to include the effect from phase change. 4.4.4.2 Updating the Cells After the interface is adjusted, some cells w ill change their phases. In Figure 411, the phase of cell center A is different from cell cente rs a, b, c and d initially. Once the interface advances to the new location, the phase of cell center A will change. In this case, the fluid properties of cell center A will ch ange also such as the discontinuous primary variable of pressure. In this research, the new primary va riables at the cell cen ter A can be obtained by following steps: Select a point B on the new interface such th at the line connecting A and B is normal to the interface. Point B is located at the inte rface and therefore, the primary variables of point B can be determined by interfacial conditions. In this normal direction, find another point C such that the distance between B and C is 1.5 x Find four cell centers a, b, c and d and poin t C is surrounded by them. To use cell centers a, b, c and d and the bilinear interpolation method to find the primary variables at point C. Use linear interpolation to obtain the primary variables at cell center A using the primary variables at points B and C. Update the fluid properties of cell center A. 4.5 Issues of Phase Change Computation in Cartesian Grid Methods In this research, the phase change phenomenon is one of the key elements. In a cryogenic twophase flow, the temperature at pipe wall or tank is much higher than the saturation temperature so that the phase change phenomenon at liquidgas interface is inevitable. The phase change phenomenon is very common in engi neering applications. For the phase change computation, the most important issue is to obtain the accurate mass transfer rate generated from PAGE 54 54 the interface due to the vaporizatio n and this will rely on the accu rate heat flux computation near the interface. In recent Cartesian grid methods, except th e Sharp Interface Method, most of them adopt the continuum surface force model (CSF) (Brackb ill et al. 1992, Shyy and Narayanan 1999) to include the surface tension in the momentum eq uation. By CSF, the surface tension can be modeled as a source term in the momentum eq uation (Sussman et al. 1994, Lrstad and Fuchs 2004, Zhang et al. 2006). In addition, the continuity and energy equations may need to be modified by adding extra source terms to include th e effect of discontinuity such as mass transfer rate at interface. The fluid properties of differe nt phase are described by a socalled indicator or Heaviside function (Dhir 2001). Th erefore, only one set of gover ning equation is solved and there will be a smeared band of the solution acr oss the interface. The resolution of primary variables must be poor near the interface. In Juric and Tryggvasons (1998) and Shin an d Jurics (2002) work, they used the Front Tracking Method to simulate the film boiling ph enomena. In their researches, the interfacial velocity intu is obtained by the following equations: intf fmun un (420) where 2lg f and f u can be treated as an average of the velocity of liquid and gas phases near the interface. Then, the interface is advanced by following equation. f fdxm nun dt (421) Also, this interfacial velocity will be used to form the source tern in the continuity equation: PAGE 55 55 int() ufu (422) This set of equations looks safe in mathematic but in fact, it is very dangerous in real physics. The trouble will come from the interpolation for f u Once phase change is very strong such as large Jakob number or large density ratio, the velocity jump at interface will be very large. With large velocity jump, f u will be inaccurate and the error may be very huge. For example, if lu =1.0 and g u =100, f u will be close to 50.5 by the in terpolation and this will cause huge error. Therefore, this set of equations can be applied only when the phase change is very weak or low density ratio otherwise, the co nservation of mass will be very inaccurate. In some researches about phase change by Level Set Method such as Morgans (2005), Son and Dhirs (1998) and Sons work (2001), in order to include the vo lume change at liquidgas interface, they modified the continuity equation as () ufH (423) where His a smooth step Heaviside function. By this Heaviside function, the fluid properties near the interface will be treated as continuous with 3 grid space distance. Obviously, once the density jump is small, this modified continuity equation may be able to handle the mass flux generated from the liquidgas interface. However, once density jump is very large, the conservation of mass will be inaccurate. In th e general condition, the density jump between liquid and gas is on the order about 1000. In Front Tracking and Level Set Method, ther e are always the errors of mass because the variables and properties are treated as continuous across the interface and this does violate the real physics. In real situation, the properties of fluids are not continuous across the interface. If there is no phase change, the pressure should be discontinuous across the interface. With phase change, both pressure and velocity should not be continuous across the interface. PAGE 56 56 To simulate the phase change phenomenon, the most important thing is to obtain accurate heat fluxes across the interface. These accurate heat fluxes will be transferred to be used as latent heat of vaporization for the liquid. In numerical simulations, in order to acquire the accurate heat flux at the interface, the temperat ure distribution must be very accurate and grid must be very fine near the interface. Also, some information about the interface must be explicit such as the normal direction unit vector and the length of the interface. Generally speaking, in order to evaluate the heat flux accuratel y, three conditions must be satisfied: an accurate temperature distribution near the interface, an accurate numeri cal method to find the temperature derivative at the interface and enough info rmation about the interface. In the Cartesian grid method, only SIMCC can satisfy these three conditions. In SIMCC, there is no smeared band near the interface and the interface is an ex plicit true surface. Table 41 lists the key issues for the heat flux computa tion in some popular Cartesian grid methods. Based on Table 41, the SIMCC is the only method that can obtai n the heat flux at interface accurately. In SIMCC, the governing equa tions of each phase are solved separately and there is no smeared band that distorts the actu al temperature profile. Th erefore, the accurate normal direction can be obtained easily and also it is easy to calculate th e area of the surface of interface. This information will be very useful fo r evaluating the latent heat supply for the phase change. Based on these advantages of the SIM CC, therefore, SIMCC is selected to be the numerical scheme in this research. 4.6 Heat Flux Computation at the Interface In order to evaluate the heat flux accurately, the following conditions must be satisfied: Accurate temperature fiel ds near the interface. Accurate information of the interface. Accurate numerical method to find the temperature derivative at the interface. PAGE 57 57 In SIMCC, the first condition can be satisfied easily. The interface is treated as a true surface and there is no smeared band at the interf ace. Because the interface exists explicitly in SIMCC, the information such as the location of the interface and the unit normal vector at the interface can be obtained easily and therefore, th e second condition is also satisfied. Figure 412 illustrates the numerical technique to eval uate the heat flux in this research. For each marker point, two probing points ar e selected. These two marker points are located in the direction of the normal vector of the interface with an equa l distance. The distance from the marker point to the nearest probing point is determined by the code. The rule is that any nearest probing point can be su rrounded by four cell centers in the same phase. As shown in Figure 412, probing point 1 is surrounding by cell centers 1, 2, 3 and 4 and probing point 2 is surrounding by cell centers 4, 5, 6 and 7. Once the four points are selected, the second order accurate bilinear interpolation method can be used to obtain the temperature at probing points 1 and 2. In a cryogenic flow, because the temper ature of the pipe is always higher than the saturation temperature, the te mperature at the interface shoul d be equal the saturation temperature. This type of interfacial condition is also called the Dirichlet type and the value should be a fluid property. By th e temperatures at the interface and at the two probing points, Taylor expansion can be used to obtain the second order accurate temperature gradient: _1int _2 2 int43 () 2probe probeTTT T Ox nx (424) All the techniques in this section including th ose for obtaining temperatures at cell centers and the bilinear interpolation method are of second order accuracy. It can be expected that the temperature gradient at the interface can ach ieve at least second order accuracy also. Here, the accuracy of the current probing po int method for heat flux and latent heat computation will be evaluated by a sample test case The geometry of this test case is a rectangle PAGE 58 58 with dimensions of 510 and a fictitious interface, a halfc ircle with a radius R=0.5 is placed on the bottom with the center at (5, 0). The coordi nate system is axisymmetric. The heat flux and latent heat at the surface of this object will be tested based on two sets of grids and assumed temperature profiles. Figure 413 is an illustration of the geometry of this test case. The two selected grids are 64 x64 and 128x128. Two sets of assumed temperature profiles are: Tr and 2Trwith the original point (5,0). Figure 414 is the assumed temperature distribution. In this test case, there will be several exact solutions which can be used for validation: The area of this sphere should be: 243.1415926 Ar The temperature gradient of each temperature distribution is :/1Tr at R=0.5 The latent heat can be computed by qdAdt If thermal conductivity is assumed as 1.0 and ti me step is also 1.0, there will be the exact solution of latent heat in one time step: 11qAt (425) Table 42 lists the computed result and corresponding error. By Table 42, it shows that the current method for latent heat computation is reliably accurate at least for a quadratic temperature pr ofile and the profile can be improved when the grid size becomes smaller. 4.7 Phase Change Algorithm In this research, because of the large temperature difference, the phase change phenomenon is inevitable so that the mass flow rate at the interface can not be zero. In order to truly satisfy the conservation of mass at the in terface, a discontinuous model is adopted. PAGE 59 59 In SIMCC, the normal component of interfacial velocity is determined by the interfacial displacement over tand the contribution of normal compone nt of interfacial velocity should come from both normal force ba lance condition and phase change: __ int() f orcebalancephasechange nxx u t (426) where se_c phahangex is the contribution comes from the phase change and the normal component of interfacial velocity can be expressed as int _()()()n nforcebalancenphasechangeuuu (427) By conservation of mass at interface and Eq uation 427, the normal component of the velocity near the interface of each phase can be obtained and they will be the velocity boundary condition for each phase. This is the socalled d iscontinuous model in this research. In this model, the velocities of each phas e come from the conservation of mass at the interface and are discontinuous at the interface. Therefore the conservation of mass at the interface can be enforced. Based on this idea, a phase change algorithm is developed in this research that includes the following steps: Computation of heat flux at the interface locally. Computation of the corresponding mass transfer rate. Integration of the mass transfer rate by time and surf ace area to obtain the total latent heat and then divided by the unit latent heat to obtain the amount of vaporization. Adjust the interface based on the amount of vaporization. Heat flux computation with second order a ccuracy has already been addressed in the previous section; the rest of the steps will be discussed here. The local total heat flux can be expressed as PAGE 60 60 g l g lr a dT T qkkq nn (428) Then the net heat passes th e interface locally in one t for one A should be: eqAt (429) This amount of energy should be transferred to be the latent heat locally, the local mass transfer can be obtained by _/transferme (430) When the local mass transfer is counted, th e interface will be adjust ed by this local mass transfer. During the adjustment procedure, th e reference coordinate should be fixed at the interface and the normal and tangential directions are two axis of this coordinate. Then, the new interface can be redefined as _phasechangox phasechangoy X Xn YYn (431) where is the local mass transfer coefficient and can be calculated by the local mass transfer. Figure 415 shows algorithm of this step. Now, a new interface based on the phase change is defined. The next step is to call the isothermal moving interface algorithm. Because of the imbalance of force in the normal direction, the new interface will be adjusted again to achieve ba lance of force in the normal direction and it is the final interface of each time step. newphasechange x newphasechange y X Xn YY n (432) The idea behind this equation is the same as th e isothermal case. The difference is that in Equation 432 an initial displacem ent due to the phase change ha s already been counted before the iteration starts. Figure 416 shows the algorithm of this step. PAGE 61 61 In each time step, the total displacement of in terface should be the difference between the original interface and the final new interface. Therefore, the new normal component of interfacial velocity can be obtained by _int_ _int_()()/ ()()/nxnewnewold nynewnewolduXXt uYYt (433) where _int_()nxnewu and _int_()nynewu represent the X and Y compone nt of normal component of interfacial velocity for next time step. In SIMCC, if there is no phase change, the normal component of interfacial velocity will be the velocity boundary condition of each pha se and this is the same as continuous model. However, because of phase change, the interfacial velocity must be modified to satisfy the conservative law. In this research, the velocity boundary at interface will be corrected by the mass conservation as int int() () () ()ln nl l gn ng gum u um u (434) The two discontinuous normal components of in terfacial velocities will be used to be the velocity boundary of each phase and this socalled discontinuous model in this research. In Equation 434, it is very clear that if there is no phase change, it will reduce to int_ intint_()()()()()ngngnnlnluuuuu (435) Therefore, the continuous model is a special case of the discontinuous model when there is no phase change. Figure 417 is the detailed flow chart of this step. PAGE 62 62 4.8 Global Conservation of Mass The major difference between the current SIM and others work is that SIM can better preserve the global conservation of mass. During numerical simulation, it is inevitable that the errors of mass will be generated by improper initial conditions or the inherent numerical errors especially in internal flows. In order to pr eserve the global conserva tion of mass, a set of conservative strategies is applied. For the gas phase, a correction of mass flow ra te is applied at the outlet of the pipe. As shown in Figure 418, the inlet mass flow for the gas phase has contributions from two parts, the mass flow through the inlet and the mass flow ge nerated at the interface. Therefore, the total mass flow rate into the gas phase should be: int intint ininleterfaceinletinlet outmmmudAudAm (436) The mass flow which leaves the outlet is c ontrolled by the boundary condition. If the boundary condition is unknown, usually, it can be done by higher order extrapolation or fully developed assumption once the pipe is long enou gh but this computed leaving mass flow rate may not equal to the mass flow rate entering the domain. Also, the errors of mass can be caused by an accumulation of numerical errors over the t housands of iterations that need to be done. Even if the mass flow rate at outlet is not accura te at beginning, an estimated value of mass flow at outlet can still be obtained: __ outguessoutbcoutmudA (437) By the idea of conservation of mass, the en tering mass flow rate should equal the mass flow rate at the outlet if the flow is incompre ssible and there is no other mass source. Therefore, the velocity difference can be added at outlet: __() / ()outoutbcoutoutguessoutuummA (438) PAGE 63 63 Finally, this outucan be used to be the velocity bounda ry condition and the conservation of mass in the gas phase can be preserved. The following provides the details of the cons ervation of mass scheme for the inflow of liquid phase. In Figure 419, the forward shaded region represents the liquid cylindrical core shape at the time step t that is the initial condition for the next time step. When advancing to the next time step t+t, the total entering mass is known,m which enters the pipe during the time step is equal to the velocity at th e inlet multiplied by the time step size, t The interim liquid core is composed of two parts as shown in Figure 420 by a forward slashed region and a backward slashed region. The forward slashed region in Figure 420 is made identical to that in Figure 419 except that it is moved downstream while the backward slashed region is the added mass m So the total mass of the two parts will be the mass of liquid core at the time step t+t. Even though the mass is conserved but the interface shape has not been updated, so it must be recomputed based on the interfacial conditions before we complete the t+t step. In summary, by this strategy, the conservation of mass is strictly enforced and also, the phase interface is updated that follows the forced balance. The details on updating inte rface locations are based on the marker points and th eir movement as discussed next. During the generation of the interim interface as shown in Figure 420, the locations of the markers have to be adjusted accordingly. In the current algorithm, the to tal number of markers will not change for the interim interface. As shown in Figure 421, the marker points 0, 1, and 3 represent all the markers on the old interface as. Du ring the construction of the interim interface, as stated above, because of the added mass, the markers will have to move so that point 1 will move to 1, 2 to 2 and 3 to 3 in Figure 422, respectively. One important rule implemented here is that Point 0 that is anchored at the inlet for the old interface will become 0 for the interim PAGE 64 64 interface will not move with the old liquid core, so 0 remains at the inlet. As a result, the distance between 0 and 1 is larger than others that calls for restructure and update the marker system with the a new set before proceeding with the determination of the final interface for time step t+t. PAGE 65 65 Table 41. Comparison of the key issues fo r different numerical methods for the heat flux computation. VOF Level Set IBT Front Tracking SIMCC Explicit interface N N Y Y Y Accurate near the interface N N N N Y Fine grid near the interface N N N N N Table 42. Comparison of exact and numerical results for the flow over sphere case. Grid 64x64 Grid 128x128 Area/ error 3.14/ 8.12E4 3.14/ 1.93E4 Latent heat /error, Tr 3.21/ 2.08E2 3.19/ 1.38E2 Latent heat /error, 2Tr 3.29/ 4.64E2 3.22/ 2.57E2 ccfc u v U V cs cv Figure 41. Nonstag gered grid system. Figure 42. Example of mixed structur ed and unstructured grid in SIMCC. PAGE 66 66 Figure 43. Example of mixed st ructured and unstructured grid. A B Figure 44. Marker points of an oblique ellipse. A) initial ,B) after curve fitting. Figure 45. Intersection point s of an oblique ellipse. PAGE 67 67 5 34 1 2 interface Figure 46. Illustration of the interfacial cells and cutandabsorption procedures in the SIMCC in local situation. A B Figure 47. Cutcells of different objects. A) NACA0012, B) Star shape. A B Figure 48. Example of cutcells of different grids. A) dx=0.025, B)dx =0.1. PAGE 68 68 A B C 1 4 2 3 5 6InterfaceD E F Figure 49. Illustration of the interfacial variables and notation. A0 A1 A2 A3 xy 123Initial Interface New Interface t 40 Figure 410. Illustration of in terfacial advancing process. n A B Cold interface new interfacea b c d Figure 411. Illustration of updating cell procedure. PAGE 69 69 Interface MarkerProbe 1 Probe 2 c1 c2 c4 c3 c5 c6c7 Figure 412. Two probing points for second order gradient. R X 5 10 1 Figure 413. Illustration of geometry of test domain. A B Figure 414. The assumptive temperature distribution. A)Tr B) 2Tr PAGE 70 70 int_oint_phase Figure 415. The new interface because of phase change. int_oint_phase int_new Figure 416. The new interface because of balance of force. PAGE 71 71 Figure 417. The flow chart for the phase change algorithm. symmetric lineinletoutlet wall interface Figure 418. The global conservation of mass in gas phase. PAGE 72 72 Figure 419. Interim interface cons truction, the initial interface. Figure 420. Interim interface construction, the new interface. Figure 421. Restructuring of markers, the original makers. Figure 422. Restructuring of markers, the new markers. PAGE 73 73 CHAPTER 5 VALIDATION OF CODE 5.1 Introduction In this chapter, several test cases with known solutions are used to validate the SIMCC code. They are divided into three groups based on the aspects of the numerical techniques. The first group is used to validate the solver of the governing eq uations and the SIMCC with the fixed interface and includes the Couette flow, fu lly developed channel flow, cavity flow, fully developed flow in a pipe with a constant wall h eat flux and flow over a sphere cases. The second group focuses on the moving interface algorithm and includes the static droplet, the rising bubble and droplet simulations. The last group focuses on the mass transfer at the interface and includes a stationary droplet in a quies cent medium with constant mass transfer rate and the one dimensional phase change problem. 5.2 Group 1, Solver of Governing Equations and the SIMCC with the Fixed Interface In this group, the interface will not be moving but fixed in the domain and this can help to validate some numerical techniques applied in th e SIMCC code, such as the solver of governing equations, the interfacial tracking, merging proced ure and the flux and stress computations at the interface. 5.2.1 The Fictitious Interface for Evaluating the SIMCC In order to validate the SIMCC while taking advantage of known analytical solutions and wellaccepted benchmark cases, the entire flow field is divided into two regions separated by a fictitious interface. By comparing the solutions obtained using the fict itious interface with the known solutions, the performance and accuracy of the SIMCC can be evaluated. Detailed treatment of the fictitious interf ace is given below. As shown in Figure 51, since region 1 and region 2 are the same fluid, the normal st ress balance at the fi ctitious interface is: PAGE 74 74 12 (51) The shear stress balance at the fictitious interface is: 12 (52) Continuity of the flow variables at the fictitious interface yields: 12 x xuu (53) 12 yyuu (54) 12 p p (55) It should be emphasized that with an interf ace that separates two different phases, the interfacial conditions will not be th e same as the ones listed above. Furthermore, the normal and shear stress at th e fictitious interface can be expressed by the stress tensors: 1_ 1_ 22 AAnTnnTn (56) 1_ 1_ 22 AAnTtnTt (57) The stress tensor at pointA in Figure 51, AT on the interface from either region can be written as 1 11 111 _1 11 1 11 1 _12 2y xx A yy x Au uu p xyx T uu u p yxy (58) 2 22 222 _2 22 2 22 2 _22 2y xx A yy x Au uu p xyx T uu u p yxy (59) The normal and tangential unit vectors at po int A can be evaluated using analytical formulas: PAGE 75 75 1/2 22 A xA AAy n xy (510) 1/2 22 A yA AAx n xy (511) ,, x AyAtn (512) ,, yAxAtn (513) For the fictitious interface case, _1 AT should be equal to _2 AT It is obvious that the normal and sh ear stresses should be a function of Au and AP So there are two nominal equations but three unknowns. The following is a complete procedure to evaluate the inte rfacial conditions. First, the pressure at each marker point is de termined by the bilinear and central difference interpolations. As shown in Figure 52, two imag inative points B and C, normal and equidistant from point A are defined first. Because the shape of background cells is al ways rectangular, there must be four cellcenters surrounding each of the points B and C. The pressures at points B and C can be determined by bilinear interpolation. On ce the pressures at points B and C are obtained, central differencing can be applied to estimat e the pressure at point A with secondorder accuracy. The derivatives of velocity in Equation 58 a nd 59 can be determined by the Taylor series expansion: 22 3,,( )( ) 1 (,)2(,)(,) 2xy xxxyyyfxxyyfxyfxyxfxyy f xyxfxyxyfxyyOx (514) The procedure is similar to determining the pr essure at point A. As shown in Figure 53, eight imaginative points must be chosen first. By the bilinear interpolation, the velocity on each imaginative point can be determined. PAGE 76 76 The derivatives of velocity, for example, x u x term in Equation 58 can be expressed using Taylor series expansion as 1 2 11 22432 432 22 22xFxGxA xHxIxA xuuuy uuuy u xy x xy (515) with a secondorder accuracy. Here, 1 FG y yy 1 FG x xx 2 H I y yy and 2 H I x xx To avoid the denominator becoming zero, the length of A H can not be the same as AG. By similar operations, ()x x Au f u y ()y yAu f u x and ()y yAu f u y at point A of each region can be obtained. By substituting all thes e terms back into stressbalance conditions, Equation 58 and 59, the shear and nor mal stresses become functions of x Au and yAu Thus with two equations and two unknowns, the interfacial velocity at point A can be obtained. 5.2.2 Couette Flow, Fully Developed Ch annel Flow and Cavity Flow Couette flow has a linear velocity profile a nd a constant pressure distribution, while the fully developed laminar channel flow has a para bolic velocity profile and a linear pressure distribution. The cavity flow is a typical benchm ark case (Ghia et al. 1982 ). For all three cases, fictitious interfaces are placed inside the flow domain to examine the performance of the SIMCC. As an illustration, in Figure 54(A), a circular fictitious interface w ith a radius of 0.1 is immersed into a unit square with a 128 128 grid. This grid is used for the Couette and cavity flow calculations. In Figure 54(B), the fictitious inte rface with a radius of 0.05 is immersed into a 40 1 sized channel with a 155 90 grid system in a fully developed channel flow. These two cases are used to test the solver of conti nuity and momentum equations for the SIMCC. PAGE 77 77 5.2.2.1 The Couette Flow In this section, the errors of the primary variables at the interface and conservative properties at interface wi ll be evaluated. The evaluations of errors here are all based on a 128 128 grid. The radius of the fictitious interface is 0.1 and the cente r of the interf ace is placed at X=0.5 and Y=0.5. In this case, there are 148 cutcells generated. The exact solution for Couette flow at each cell center of the grid is Uy 0V and .PConst The error of interface velocity at each marker point is show n in Figure 55(A). The maximum error in this case is less than 6.0E16, essent ially at the roundoff level. The error in the mass flux of each cell can be expressed as CSundS (516) Theoretically, the summation of these errors should be zero. In this case, the total error is about 1.0E08, which is consistent with the accur acy supported by the formula. The error in the mass flux of each cutcell is shown in Figure 55(B) In a steadystate Couette flow, the unsteady, convection, pressure gradient and the diffusion terms in the mome ntum equations should be zero at each cell. The convective momentum flux is a dopted to verify the conservation of momentum flux at the interface. The error in convec tive flux of each cell can be expressed as ()CSuundS (517) The summation of these errors should be zero. The actual total error in convective flux of all cutcells is about 1.0E7 in this case. In Figure 55(C), the error in momentum flux of each cutcell is shown. 5.2.2.2 The Fully Developed Channel Flow In Figure 56, the test results of the fully developed channel flow case are presented. As mentioned before, the channel of length 40 starts from X = 25 and ends at X =15. The grid is PAGE 78 78 155 90 and the Reynolds number is 100. A uniform velocity U=1.0 is assigned at the inlet and the fully developed boundary condition is assigned at outlet. A fictitious interface with a radius of 0.05 is selected and the cente r of the interface is fixed at X = 0.5 and Y =0.5. The total number of cutcells in this case is 92. In Figure 56(A), the pressure contour near the interface is shown. It shows that the pressure distribution is linear along the axial di rection and smooth across the interface. The fully developed velocity profile at X=0.5 is shown in Figure 56(B). The maximum velocity is 1.5 which agrees with the theoretical result. Als o, the parabolic velocity profile supports the accuracy of the SIMCC. The conservative propertie s are also examined. In Figures 56(C)(D), the errors in mass and momentum flux are shown. The summation of errors in mass and momentum fluxes are of the order of 1E07 and 1E06, respectively. 5.2.2.3 The Cavity Flow In Figures 57(A)(C), the streamlines of the lid driven cavity flow with different sizes of the fictitious interface are shown. He re, Re =100 is employed with a 128 128 uniform grid system. Figure 57(A) shows the case with no fictiti ous interface. Cases with fictitious interfaces of radii, 0.1 and 0.2 centered at X=0.5 and Y=0.5 ar e shown in Figure 57(B) and Figure 57(C), respectively. In Figure 57(D), the velocity component in the X direction along X =0.5 for different fictitious interface sizes are compared with the case without a fictitious interface. These three profiles almost overlap on one another and are very smooth across the interface. The shape/location of the main vortex, size/shape of the subvortices, and velocity component in the axial direction U at X =0.5 compare favorably with benchmark results reported by Ghia et al.( 1982). PAGE 79 79 5.2.3 The Fully Developed Pipe Flow with a Constant Wall Heat Flux In this test case, a fully devel oped pipe flow with a constant wall heat transfer rate will be presented. The reasons for selecting this problem to be the test case are: A. there is an exact solution that can be used for validation, B. the temperature field is influenced by both convective and diffusive effects and this is enough to valida te the energy equation so lver. Also, a fictitious interface is again used to divi de the entire domain into two regions. The thermal interfacial condition for this fictitious inte rface case is continuous heat flux a nd identical temperature at the interface.The grid system of this case is a rectangular with 1 0.5 and grid point is 80 40. Because it is a fully developed pipe flow, the velocity boundary condition is set to be a nonslip condition at the solid wall, symmetric condition at the centerline, fully developed velocity profile at inlet and outlet. The pressure should be a li near distribution along the axial direction. The heat flux at the solid wall is set at 1.0 and thermal conductivity is set as 1.0 also. The radius of tube is 0.5. The exact solution can be expressed as (Mills 1995) 224 243 16416sqRrr TT kR R (518) where s T is the temperature at wall, q is the heat flux and k is the thermal conductivity. To avoid the error caused by the inaccurate comput ation of heat flux at the wall, the thermal boundary condition at the solid wall is replaced by the equivalent temperature distribution. If k is 1.0 and q is 1.0, the temperature distribution al ong the solid wall can be written as (0)4 TTx (519) where T(0) is the wall temperatur e at X=0 and is set as 0 in this case. Therefore, an equivalent wall boundary condition at the wall is: 4 Tx (520) PAGE 80 80 The thermal boundary condition at the centerl ine is set as symmetric. The thermal boundary condition at the solid wall, the inlet an d outlet can be derived by the exact solution. Figure 58 shows the results of temperature distribution. To validate the result, first, the heat flux at th e wall is checked to ensure that the heat flux is 1.0 q Figure 59 shows the heat flux at the solid wall and it shows that there is only a small error near the inlet and outlet and the magnitude of error is about 0.0025. In order to validate the result further, the temperature profile at X=0.48125 along the radial direction is also compared and is shown in Figure 510. It shows th at the result ag rees with the exact solution very well. Again, a half circle fictitious interface is used for this case. There are two different radii: 0.3 and 0.4. The fluid properties are the same in each phase and the thermal interfacial conditions for this fictitious interface are: Temperature is continuous at the interface. Heat flux is continuous at the interface. Figure 511 shows the computed temperature di stribution of each case. In Figure 511, it shows the results are the same as those without an interface. To further validate the case, the temperature profile at X=0.48125 along the radial dir ection for the case with a radius of 0.3 is also compared in Figure 512. Figure 512 shows the comparison and the computed results overlap with the exact solution very well. 5.2.4 Flow over a Sphere with Heat Transfer The purpose of this test case is to check th e ability of SIMCC for evaluating the heat flux at the interface. In this case, two grids with different stretching, 800 600 and 120 50 are used to demonstrate the ability of SIMCC for evalua ting the heat flux at the interface. The overall dimensions of the mesh are X= (60, 60) and Y=( 0,60) and a half circle is placed at the center of PAGE 81 81 lower boundary. Its radius is 0.5 and the center is located at (X,Y)= (0.5,0.0). Figure 513 (A) and (B) show the grid and geometry of the sphere for the case with the grid 800 600. In this case, the pressure boundary conditions of upper, left and right sides are zero gradient and symmetric for the lower face. The thermal boundary condition (temperature) of upper, left and right sides are 0. The velocity boundary condition at interface is nonslip and the pressure at the interface is obtained by a sec ond order extrapolation. The temperature of the interface is set at 1.0. The velocity at far field is set as 1.0, the characteristics length is the diameter of this ball. The Reynolds number is 1.0 and the Peclet number is 1.0 also. Figure 514 are the stream line plot and pre ssure and temperature contours. In this case, the average Nusselt num ber is suggested by Whitaker(1972): 1/2 2/30.42(0.4Re0.006Re)Pr Nu (521) Based on the current dimensionless paramete r, the Nusselt number of the case with Re =1.0 and Pe=1.0 should be 2.406, Table 5.1 lis ts the calculated result. In table 5.1, dx_min means the minimum grid si ze of each mesh. It shows when the grid size is decreased; better results can be obtained. When the grid is denser than 800 x 600, the error will be less than 2%. In this case, it shows that a higher resolution of results is required near the interface for the heat flux evaluation. It can also be expected that based on the same gird size, SIMCC is the method with the best performance among Cartesian gird methods and this is also the reason that the SIMCC is chosen. 5.3 Group 2, Validation for Moving Interface Algorithm In the previous group of tests, the technique s for handing the stationary interface including the governing equations solver and SIMCC are validated. In this group, the interface in the validation cases will not be stationary but moving and deforming and therefore a moving interface algorithm must be imported to handle the movement of the in terface. Two cases are PAGE 82 82 used to validate the moving inte rface algorithm and they are the static droplet and the rising bubble and droplet simulations. 5.3.1 Static Droplet Simulation The first case employed here is a static droplet in a surrounding gas. The main idea here is to introduce an imbalance of pre ssure at the interface of a static droplet and then examine how the code reacts to the imbalance by inducing a move ment at the sharp interf ace that results in the pressure adjustment for the in terface to return to a balanced state governed by the LaplaceYoung equation. Therefore, for the first case, the selected pr oblem is a stationary water droplet surrounded by its own vapor under isothermal condition in ze ro gravity. The density ratio is 1,605 and the viscosity ratio is 22. If there are no other for ce field effects and under a dynamic equilibrium, this droplet should stay stationa ry and the pressure difference at the interface between the liquid and vapor should be balanced by the surface tension. The geometry of this validation problem is a twodimensional domain in an axisymmetric coordinate system with length and width equal to 2.0 and 1.0, respectively. The grid arrangement is 80 40. A half circle mounted on the center of the lower boundary is used to represent the interface. Figure 515 is the i llustration of this problem. The boundary conditions are zero gradients for ve locity and pressure at right, left and upper sides and symmetric at lower side since this is an axisymmetric com putation. The initial conditions for the velocity and pressure are zero ev erywhere that creates a pressure imbalance at the interface. In this co mputation, the Weber number is select ed as 0.5. Since the curvature is 4.0 everywhere, the exact pressure jump between the liquid and the va por phase should be 8.0. Figure 516 shows the maximum induced interfacia l velocity during the transient adjustment period. It must be emphasized that for the inco mpressible flow computation, strictly speaking PAGE 83 83 there should be no induced interfacial velocity for this problem because of the stationary condition and the conservation of mass. However, because of the imbalance of interfacial dynamics, an extremely small interfacial velocity is induced. This small order of magnitude velocity distribution (deformation of interface; induced interfacial velocity) should be considered a numerical disturbance and not a violation of the incompressible assumption. However, the SIMCC code can use it to adjust the pressure di fference of each phase since the disturbance of velocity will cause the change of pressure in each phase. In Figure 516, the maximum induced interfacial velocity varies and achieves the stead y state finally. The maximum induced interfacial velocity is about 1.0E5 in this case. Figure 517 shows the convergent histories for the governing equations and the interfacial condition. In the beginning, the interface was ad justed seriously to satisfy the governing equations and interfacial condition. In this stage both residuals are decreasing very fast. After 100 iterations, once the residue of interfacial condition (imbalance) is too small to push the interface, the interface will become stable gradually and looks lik e a fixed interface. In the meantime, the residual of governing equations must be small enough also and this means governing equations and interf acial condition reach converg ence at the same time. Figure 518 is the pressure contour for this problem. It shows the pressure difference is identical to the theoretical value, 8.0 and the di scontinuity at interface ca n be handled very well by the current method. The sharp pressure discon tinuity across the inte rface is successfully computed. 5.3.2 Deformed Rising Bubble and Droplet For the second case, the buoyancydriven risi ng bubble and droplet through a quiescent liquid are selected. The test problems in the s econd case are classified into two groups based on the density ratio (dispersed pha se to continuous phase). These groups are the low density ratio PAGE 84 84 (<0.001) bubbles and higher density ratio (= 0.91) droplets used to verify the current method with a wide range of density ratios. The computational domain used in this section is a rectangle with a length of 70 and width of 38 and the grid system is 550 100. From X=0 to X=20 and R=0.0 to R=2.0, the grid spacing is uniform and then the grid spacing is st retched to the upper, right and left sides. Since the domain is la rge enough, the boundary conditions are assigned as the zero gradients for both velocity and pressu re at the upper, right and left boundaries and a symmetric condition on the lower boundary (centerl ine). The initial condition for this case is quiescent fluids in both phases a nd a halfcircle with a radius of 0.5 is used to represent the bubble/ droplet and the center of the halfcircle is located at ( 3.0, 0.0). The time step here is 0.01. For the bubble problem (density ratio ~ 0.001), Re = 10 and We = 8.0 are assigned. In Ryskin and Leals research (1984), the bubble is an open spa ce without any fluid. In this computation, water is adopted to be the ambien t liquid and the bubble is water vapor, so the density ratio 0.0006 and viscosity ratio is 0.045. Figure 519 is a co mparison between the current computed result and the result of Ryskin and L eal. It shows that the two agree well with each other. For further verification with the same bubble of Re = 10 but different Weber numbers of We = 2.0 8.0, the results of the aspe ct ratio (height to width) are compared with those of Lai et al. (2004)in Figure 520. The comparison is very favorable. For the higher density ratio problem, the drop let with Re = 10 and We = 2 is adopted for comparison. The density ratio and the viscosity ra tio are assigned as 0.91 and 4, respectively so the computed result can compare directly with results by Dandy and Leal (1989). Figure 521 shows that the current computed result agrees well with that of Dandy and Leal in flow structure and droplet shape. PAGE 85 85 5.4 Group 3, Validation for Accurate Mass Transfer at the Interface Tests in this group focus on validating the phase change computation at interface. The two test cases used are: a stationary droplet in a quiescent environment with a constant mass transfer rate, and one dimensional phase change problem. 5.4.1 A Stationary Droplet in a Quiescent Environment with a Constant Mass Transfer Rate For this test case, the main purpose is to ev aluate the performance of current code on the velocity discontinuity at the interface between the droplet and its surrounding fluid due to mass transfer. In order to accuratel y evaluate the current numerical technique, a simplified case was designed where a stationary droplet is assumed to vaporize with a given constant mass transfer flux at the interface. In this way, heat transfer is not involved in the simulation and only the continuity and momentum equations are solved. For this test case, an axisymmetric cylindrical coordinate system is employed. Figure 522 is a schematic of this problem. The computational domain is a rectangle with dimensions of 10 5. A hemisphere with an initial radius 0.5 is placed at the center of the lower boundary. The boundary conditions of upper, right and left sides are the second order extrapolati ons for both velocity and pressu re and symmetric at the lower boundary. In this test case, a water droplet is used and the density of water is 958.3 3/ kgm and 0.597 3/ kgm for the vapor. In this computation, the num erical time step is 0.1 and the constant and uniform mass flux that leaves the droplet surface, m is assumed to be 10.0 2/ kgms It is noted that there is no internal velocity field inside the droplet because of the quiescent environment that results in an induced interf ace receding velocity as follows: int()n llmm u t (522) PAGE 86 86 The vapor velocity at the interface can be calculated by int()()vnvnmuu (523) Based on the assumed mass flux, the vapor veloc ity at the interface is 16.76 m/s. As the velocity in the vapor phase is so lely due to the uniform mass from the droplet surface, the vapor velocity at any radial distance from the droplet surface can be predicted based on conservation of mass: 111222uu (524) The velocity of vapor at different locati on in the vapor phase can be expressed as 2 12 2 21ur ur (525) where r is the distance measured from the center of the droplet. If the known vapor velocity at the interface is used as the refe rence velocity, the velocity at a ny location in the vapor phase can be obtained. Therefore, the obtained velocities including the interface velo city, the velocities of both liquid and vapor phases at the interface and the velocities at the outer boundary of the computational domain will satisfy both the continuity equation and interfacial mass conservation condition and therefore they can be considered as the exac t solution of this te st case and used to validate the numerical results. Figure 523 shows the computed velocity distri bution by the current code versus the exact velocity along the radial directi on at X=5 after first time step. In Figure 523, the velocity is zero everywhere in the liquid phase and a huge veloci ty discontinuity exists at the interface. The maximum vapor velocity is located at the interface and the velocity decays by 21/r to the outer boundary of the domain. A comparison shows that th e computed result agrees closely with the PAGE 87 87 exact solution. This proves that the current c ode can handle the mass transfer at the interface with a velocity discontinu ity with a high accuracy. Figure 524 is the stream line plot near the wa ter droplet. In Figure 524, it shows that the streamlines radiate from the inte rface to the far field straight. 5.4.2 One Dimensional Phase Change Problem One dimensional phase change problem also called Stefan problem is a classic test problem and broadly used for the validation for moving boundary/interface algorithm in computational multiphase flow with phase ch ange (Bonacina et al. 1973, Welch and Wilson 2002, Mackenzie and Robertson 2000). Using this test problem, the performance of moving boundary/interface algorithm for phase change co mputation especially the accuracy of mass transfer at boundary/interface can be evaluated. The entire system is shown as Figure 525. The left forward slashed block represents the wall and it will keep stationary with a constant temperature wallT The dashdot line with a distance () t from the wall is the interface and sepa rates the gas and liquid. The liquid phase and the interface are assigned the saturation temperature s atT Once phase change happens, the liquid will be vaporized and the volume of gas w ill increase so that the interface will be pushed to the right side. During the vaporization, the velocity of gas is still kept as zero. Since the velocity of gas phase is zero, th e original unsteadyconv ectivediffusive energy equation in gas phase can be simplif ied as unsteadydiffusive equation: 2 20() TT x t tx (526) PAGE 88 88 where () t is the location of this phase interface and is the thermal diffusivity and defined as k Cp For this moving boundary/interface problem, th e system of equations is closed by specifying the boundary conditions: ((),) (0,) s at wallTxttT TxtT (527) The conservation of mass at interface: () ggg x tT vk x (528) The theoretical solution of temperature and loca tion of interface for th is test problem can be found by Alexiades and Solomons book (1993): ()2 tt (529) (,) () 2wallsat wallTT x TxtT erf erf t (530) where erf is the error function and the parameters can be obtained by solving the transcendental equation: 2()wallsatTT eerfCp (531) In this validation case, the fluid properties of each phase are assi gned in Table 52, and by this set of properties, the parameter is solved as 1.0597. Although it is a one dimensional problem, it is computed by the two dimensional code in order to apply the Cutcell appr oach at the interface to obtain the best resolution. For the upper and lower side, the boundary conditions are assi gned symmetric to simulate one dimensional condition. The computation domain is a 1.00.02 rectangle and the grid point are 50010 ,the interface is located at ()20.2 tt initially and corresponding time is 0.089. The initial PAGE 89 89 condition can be obtained by the theoretical solution. Figure 526 is the obtained results. By this set of figures, it can be seen that the present re sults match the theoretical results very well. In Figure 526(A) and (C), they show the accuracy of the current energy equation solver, moving interface/boundary algorithm and the mass transfer at interface. An accurate interface position can be obtained only when all the above techni ques work well. In Figure 526(B), all the isothermal lines are purely vertical which mean s that the current Cutce ll approach can achieve truly high resolution. Overall, by this test problem, the ener gy equation solver, the moving interface algorithm, the Cutcell approach and the accuracy of h eat flux computation at interface are validated further. 5.5 Summary In this chapter, the ability of SIMCC is examined systematically. In the first group, it has been demonstrated that the current solver of governing equation is very accurate and the SIMCC only induced tiny errors at the in terface. In the second group, by compared with others results, it has been shown that the SIMCC can handle the m oving interface problems well. In the last group, by the two test cases, it has been shown that SI MCC can compute the mass transfer rate from the interface correctly and adjust the interface due to th e phase change very accurately. By this series of test cases, the capability of the full set of SIMCC technique to handle the current research is established. PAGE 90 90 Table 51. The error of Nusselt number by different grid resolutions in flow over a sphere case. Grid Nusselt _min dx Error Case 1 120x 50 1.14 0.05 52.3% Case 2 800x 600 2.36 0.003 1.9% Table 52. The fluid properties of liquid and ga s phases for the one dimensional phase change problem. Gas Liquid Density ( ) 0.2 1.0 Heat capacity (Cp) 5.0 10.0 Thermal conductivity (k) 1.0 1.0 Latent heat ( ) N/A 1.0 1212 marker pointAinterface region 1 region 2 Figure 51. The continuous stre ss condition at marker points fo r the fictitious interface. A B C cell centers Figure 52. The algorithm fo r second order gradient. PAGE 91 91 n inteface A B C D E F G H I Figure 53. Eight points method for second order gradient of velocity. A B Figure 54. Geometry of the C ouette flow, cavity flow and full y developed channel with the immersed fictitious interface. A) Couette flow and cavity flow, B) fully developed channel flow. PAGE 92 92 Markers Error 50 100 150 0 1E16 2E16 3E16 4E16 5E16 A Cutcells Errorofmassflux 50 100 2E09 1E09 0 1E09 2E09 B Cutcells Errorofmomentumflux 50 100 8E08 4E08 0 4E08 8E08 C Figure 55. Error in interfacial ve locity at all the inte rfacial marker points (163 points) and error in mass and momentum fluxes of each cutce ll (148 cutcells) in the Couette flow with radius of fictitious artificial interface R=0.1. A) error in interfacial velocity, B) error in mass flux, C) error in momentum flux. PAGE 93 93 A Y U 0 0.5 1 0 0.5 1 1.5 B Cutcells Errorofmassflux 20 40 60 80 4E08 0 4E08 C Cutcells Errorofmometumflux 20 40 60 80 4E07 2E07 0 2E07 4E07 D Figure 56. Pressure co ntour, U velocity profile, and erro r in mass and momentum fluxes of each cutcell (92 cutcells) of fully develope d channel flow with radius of fictitious interface R=0.05 and Re=100. A) pressure contour near the artificial interface, B) velocity profile at X=0.5, C) error in mass flux, D) error in momentum flux. PAGE 94 94 A B C Uvelocity Y 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Noartificialinterface Radius=0.1 Radius=0.2 D Figure 57. Streamline plots and U velocity prof ile at R=0.5 with differe nt fictitious interface radii of cavity flow with Re=100. A) str eamline plot with no interface, B) with radius=0.1, C) with radius=0.2, D) velocity profile along X=0.5. Figure 58. The temperature distri bution of fully developed pipe flow with constant heat flux case. PAGE 95 95 Figure 59. The heat flux at wall of fully deve loped pipe flow with constant heat flux case. Figure 510. The temperature profile of fully deve loped pipe flow with c onstant heat flux case at X=0.481235 along radial direction. A B Figure 511. The temperature distributions of fully developed pipe flow with constant heat flux case with different fictitious interfaces. A) R=0.4, B) R=0.3. PAGE 96 96 Figure 512. The temperature profile of fully de veloped pipe flow with constant heat flux case with a fictitious inte rface (R=0.3) at X=0.481235 along radial direction. A B Figure 513. Grid and geometry of flow over a sphe re with heat transfer. A) the gird system, B) the interface. A B C Figure 514. The pressure cont our, the stream line and temperature contour of flow over a sphere with Re=1, Pe=1. PAGE 97 97 1.0 0.5 2.0Gas Liquid Figure 515. Schematic of stationary droplet problem. Figure 516. The maximum induced interfacial velocity of stationary droplet problem. A B Figure 517. Convergent histories of the inte rfacial condition and the governing equations of stationary droplet problem. A) interf acial condition, B) governing equations. PAGE 98 98 A B Figure 518. Pressure contour fo r the stationary droplet. A) entir e pressure contour, B) pressure along X axis at R=0. Figure 519. The shape and streamline plot for a bubble with Re=10 and We=8. Figure 520. The aspect ratio plot for a bubble with Re=10.0 and We= 2.08.0. PAGE 99 99 Figure 521. The streamline and shape pl ot for a droplet with Re=10 and We=2. Vapor Droplet R 10 1 X Figure 522. Geometry of static droplet wi th a constant mass transfer rate problem. PAGE 100 100 Figure 523. Velocity distribution of static drop let with a constant mass transfer rate problem along radial dire ction at X=5. Figure 524. Streamline plot near the interface of st atic droplet with a constant mass transfer rate problem. Wall, T=TwallInterface, T=TsatGas Liquid () tx=0 T=Tsat UintUg=0T=T(x,t) n Figure 525. Illustration of one di mensional phase change problem. PAGE 101 101 A B C Figure 526. The location of in terface and temperature distribution, contour for one dimensional phase change problem. A) temperature di stribution, B) temperature contour, C) location of the interface. PAGE 102 102 CHAPTER 6 LIQUIDGAS TWOPHASE FLOWS IN A PIPE W ITHOUT PHASE CHANGE 6.1 Introduction In this chapter, the main focus is the liqui dgas twophase flow in a pipe without phase change. There are two studies discussed in this chapter. The first study is isothermal twophase flow, and the selected liquid is the nitrogen at 77K at 1 ATM. Since 77K is the saturation temperat ure of nitrogen, the nitrogen can exist as both liquid and vapor. Since the temper ature is fixed, the energy equati on will not be solved and only the continuity, momentum and interfacial condi tions are solved. There are two dimensionless parameters that affect the flow in this case: the Reynolds number (Re) and the Weber number (We). In real cryogenic flow, the Reynolds numbe r is usually very high. Therefore only cases with higher Reynolds number are considered he re. By definition, Reynolds number and Weber number can vary independently according to the surface tension. Ther efore both high and low surface tension cases are possi ble and will be discussed. The second case is twophase flow with heat transfer but no pha se change. In this case, the proposed fluids are liquid water at 273K and ai r at 373K at 1 ATM. Based on the assumptions, water will not be vaporized with just simple heat transfer taking pl ace at the interface. The energy equation will be solved and anothe r dimensionless parameter Peclet number (Pe) will be included into the energy equation. Similar to th e isothermal case, only the cases with higher Reynolds number are considered here. Again, on ce the Reynolds number is decided, the Weber number may be large or small and therefore two cases of surface tension will be discussed here. The Peclet number can not be assigned indepe ndently and should be determined based on the Prandtl number (Pr). The Prandtl number is a fluid pr operty and therefore, once Reynolds number and species of the fluid are determined, the Peclet number is determined also. PAGE 103 103 6.2 Isothermal LiquidGas TwoPhase Flows in a Pipe In this section, the computat ional method presented in previo us chapters is employed to numerically simulate a liquid cy lindrical plug translating through gaseous medium in a circular pipe. The purpose is to demonstrate the capability and accuracy of the current method for a pressuredriven internal twopha se flow with a moving interface due to shape deformation and liquid phase translation. Th e ratios of fluid properties of two phase flows are: 352.4;15.8ll gg The sample problems are classified into two groups. For liquidgas twophase flows, as indicated by the governing equatio ns presented in previous ch apters, the Reynolds and Weber numbers are the only dimensionle ss parameters needed to characterize and distinguish different fluids and flow regimes. Two major groups are a dopted here. The first group is used to focus on the effects of different Reynolds numbers. In this group, there are four Reynolds numbers selected: 250, 500, 1000 and 2000 and the Weber number is fixed at 500. For the first group, with the combinations of Reynolds and Weber numb ers, it could represent fluids such as liquid nitrogen or water. For the first group, the objective is to examine the convection effects characterized by various Reynolds numbers. In the second group, the Reynolds number is fixed as 500 and three different Weber numbers that correspond to three different strengths of surface tension are selected. These three surface tensions ar e equal to 1.0, 0.1, and 0.01 time the pressure difference that result in th ree different Weber numbers of 2.6316, 26.316 and 263.16, respectively. For this group, the main challenge is to demonstrate the ability of current SIMCC code to handle the large deformatio ns due to the surface tension effect. PAGE 104 104 6.2.1 Effect of the Reynolds Number For this group, the inflow boundary conditions for the velocities of bot h phases at the inlet are assigned as 1.0. While for the pressure 1.0 is set for the vapor phase and 1.00526 (l p = g p +curvature/We) for the liquid phase, respectively, to include the surface tension effect between the two phases at the inle t. This strategy will be applied in all following cases to make the first point of interface is fixed at the same location in the inlet boundary during the computation. The time step in th is computation is set at 0.001. First, the flow streamlines in the twophase zone and those immediat ely after it are plotted in Figure 61 for t = 6.0 and Re = 2000. In th e twophase zone, the streamlines are almost straight and parallel as expect ed. The transition from the twopha se zone to the singlephase pure gas zone near the front of liqui d core is very smooth. The lines are slightly distorted in the vicinity of the liquid front as the gas flow is trying to fill the entire pipe interior. In this type of flow, since the velocity at wall is zero and ther efore, the flow will be pushed towards the center of pipe for a fully developed parabolic profile because of the mass continuity. Therefore, the streamlines of gas phase near the front of liquid core get distorted to sa tisfy the continuity of mass. Figures 62(A), (B) and (C) show the pressure maps at t = 6. 0 with Re = 2000. Figures 62 (B) and (C) focus on the twophase zone and sing le phase zone, respectively, while Figure 62(A) provides pressure di stribution for the entire computationa l domain. For the single gas phase (downstream of the liquid core), Figure 62(C) shows that the pr essure decreases linearly along the axial direction that is similar to a fully developed singlephase gaseous pipe flow. For the twophase flow zone as shown in Figure 62(B), the pressure driven flow patterns are displayed in the gas and liquid parts, resp ectively. The slight discontinuity is seen at the liquidgas PAGE 105 105 interface due to surface tension, dynamic head and shear stress effects. In general, the entire domain is composed of three subdomains that all show approximately linear pressure gradients. This also shows the ability and accuracy of th e current SIMCC for multiphase computations. Figure 62(D) shows the axial pressure distribut ions at the pipe wall, radial location R = 0.2 and the center of pipe. Basically, the three curves are very close to one another so the pressure is relatively independent of the radial location. In ge neral, the pressure profile is composed of two linear curves (one for the twophase zone and the other for the singlephase zone) of different gradients that are joined at the frontal area of the liquid core. The gradient in the twophase zone is much larger than that of the singlephase zone because the gas phase in this twophase zone must accelerate and create a jet profile. Also th is is due to the fact that the gasphase controls the pressure dr op. The sharp drop in pressure in the centerline curve is due to surface tension. For the purposes of examining the ability of the current method to accurately simulate the transition from the twophase zone to the singlephase zone, Figure 62(E) and (F) were plotted to provide three radial and three axial local en largements of pressure distributions in the neighborhood of the liquid core frontal area. In Figure 62(E), it shows that axial pressure distributions along the centerline a nd at R = 0.2 are very similar to each other, except that the sharp drop discontinuities are located at different axial locations accordin g to where the interface is positioned. As expected, the pressure dist ribution at the wall is continuous without any discontinuity and has a smooth transition betwee n the two zones. The three curves all merge together after the liquid front in the pure gas zo ne. In Figure 62(F), thr ee pressure distributions in the radial direction are plotted at different locations, X = 6.2, 6.4 and 6.6. For X = 6.6, basically, it is located in the pure gas phase and hen ce, it is virtually a st raight line. At X=6.2, PAGE 106 106 there is drop between R=0.350.37 and this is the location of interface. At X=6.4, the drop exists at R=0.170.19 and the interface is located at this section also. Inside the liquid phase, there are very slight pressure fluctuations prior to the drop and the pressure rises slightly after the drop in the gas phase. Both pressure vari ations are believed to be due to the changes in the gas phase immediately after the interface where the gas flow is slowing down and bending towards the center. The above can be further explained by examini ng Figure 62(G). Figure 62(G) is a plot of the curvature along the liquidgas interface. The pl ot is based on about six hundred marker points placed on the interface with uniform separation distance between ad jacent points. Point 1 is at the entrance and last point is at the centerline. Th e curve shows that the curvature is about 5.4 at the centerline (last marker). In this computation, the effect of viscosity is very weak because of the high Reynolds number and the velocity is co ntinuous at interface. With these conditions, the interfacial condition can be simplified as lvpp We (61) By this simplified interfacial condition a nd the dimensionless parameters, it can be estimated that the order of pressure discontinuity at the interface is about / We According to the curvature at the liquid front in Figure 62( G) and the assigned Webe r number, the pressure discontinuity can be roughly estimated about 0.0108 = (5.4/500). This pressure differential is verified in Figures 62(E) and (F). In summary, the magnified pre ssure plots given in Figure 62 demonstrate that the current method is capable of capturing the pressure disc ontinuity and flow tran sition from a twophase condition back to a singlephase cond ition with fidelity and accuracy. PAGE 107 107 Figure 63 shows the pressure distributions along the centerline of the pipe for We= 500 and four different Reynolds numbers: Re= 250, 500, 1000 and 2000. For each Reynolds number, the pressure distributi on is composed of two straight lines of different slopes as explained before for Figure 62( D). In general, the rate of dimensionless pressure drop is inversely proportional to the Reynolds number. For the case with the highest Reynolds number of 2000, the pressure drop is only around 10% of the case with the lowest Reynolds number of 250 and this trend is the same as that in the single phase flow because the pressure difference will be used to balance the shear stress occurring at the wall. In each curve, the pressure shows a small drop at the front of liquid core and the ma gnitude is about 0.011 for all the cases. This is reasonable since the strength of surface tension in each case is about the same. Figure 64(A) shows the velocity profiles at diffe rent axial locations: X = 2, 4, 6 and at the exit of the pipe. Since the flow is twophase for X < 6.4, the veloci ty profile at any axial location comprises of two parts correspondin g to the phase. In the liquid region the velocity is uniformly 1.0. In the gas phase portion, because the veloc ity is zero at the wall and nonslip between the two phases at the interface, the ga s profile would have a jet effect as shown in Figure 64(A) and the maximum velocity is located near the liquid phase since the velocity of liquid core is about 1.0. In the twophase zone, the velocity profile relatively does not depend on the axial location. In the current method, the global conservation of mass is the key element and therefore, the global conservation of mass must be satisfied and verified. For this twophase computation, both phases are assumed incompressible. Therefore, the rate of the volume of fluids in and out of the pipe should be the same at any instant. Th is implies after the flow in the singlephase gas zone reaches the fully developed condition, the ga s flow velocity profile should be identical to the conventional single phase fully developed flow in a pipe that is a parabolic profile. Since the PAGE 108 108 gas flow has reached the fully developed condition well before the exit, the velocity profile named as U at X = Exit in Fi gure 64(A) and (B) should be th e theoretical pa rabolic profile with maximum velocity of 2 at the centerline an d zero at the pipe wall. The comparison between calculated velocity profile and the theoretical result is given in Figure 64(B) It is seen that the agreement is excellent. Figure 65 shows the shape of liquid cores fo r different Reynolds numbers at the same time, time=6.0. As expected, there is no signi ficant difference among these cases because the strength of surface tension is about the same here as the Weber number is the same for all four cases. After 6000 time steps, the front of liquid co re does show slightly different deformation due to the convection effect. 6.2.2 Effect of the Weber Number In this group of simulations, the Reynolds num ber is kept at 500 wh ile the purpose is to evaluate the ability of the current method to handle larger deformations due to higher surface tension effects. The boundary cond itions are very close to those used in group one except at the entrance. The radius of liquid core is 0.38 at th e entrance and therefore, the curvature will be 2.6316 = (1.0/0.38) in the theta direction. If the Weber number is a ssigned as 2.6316, the pressure discontinuity at the interface between the two phases at the entrance should be 1.0 (/We). With this procedure, once the curvature is fixed, the pressure difference can be adjusted by changing the Weber number. Based on this idea, three different Weber numbers are selected in this group: 2.6316, 26.316 and 263.16 to generate three pressure jumps, /liquidgas p p at the interface of 2.0, 1.1 and 1.01, respectively at the entrance. Figures 66(A), (B) and (C) s how the pressure maps at t = 6.0 with Re = 500 and We= 2.63. Figures 66(A) and (B) focus on the twophase zone and singlephase zone, respectively, PAGE 109 109 while Figure 66(C) provides pr essure distribution for the enti re computational domain. In general, the features in Figur e 66 are similar to those in Figure 62 but both Reynolds and Weber numbers are smaller for the current plots. In this case, the pressure jump between the two phases is equal to 1.0 at the entrance. At the front of liquid core, because the surface tension is much stronger due to a larger curvature than that at the entrance, the pressure in the liquid phase at this front of the liquid core is raised to bala nce the larger surface tensio n. In Figure 66, it also shows the larger deformation due to the surface tension at the fr ont of the liquid core. For the entrance portion, there should be no deformation because the equilibrium condition is assigned there. This shows the ability of SIMCC to handle the surface tension effect and large deformation. Figure 66(B) focuses on the pressure contour at downstream of the liquid core. It shows the downstream pressure deceases linearly that is similar with the single phase flow. Figure 67(A) provides the pres sure distributions along the wa ll and centerline of the pipe. It shows that the pressure distri bution at the wall is constructed by two linear profiles of different slopes, and this trend is very similar with the case of wall pressure distribution as shown in Figure 62(D) for different Reynol ds and Weber numbers. For the liquid phase at the centerline, the pressure decreases linearly near the entrance and rises up before going through a large discontinuous drop across the interface at the front of liquid core. Figure 67(B) is the curvature along interface. Similar to Figure 62(G) in group one, Figure 67(B) shows the curvature values of the liquidgas interface. The general trend in Fi gure 67(B) is also sim ilar to that in Figure 62(G). Near the entrance, the curvature is almost constant because only thet a direction (tangential) is involved. Near the liquid core front, the cu rvature will increase due to the contribution from the RX plane (curvature is counted in two directions.). PAGE 110 110 Figure 68 is the streamline plot for Re=500 and We=2.63 at time=6.0. Similar with the case in group one, in the twophase zone, the stre amlines are relatively straight for both phases. The difference is that slightly more bending of th e gas streamlines near the top of the liquid core for the current case is observed due to the liquid core deformation. Figure 69(A) shows the U velocity profile at different axial locations at time=6.0. These profiles are very similar to those in Figure 62 for group one. The velocity distribution at the exit also achieves the fully developed parabolic profil e and again it shows that the current code can maintain a global conservation of mass very well. In Figure 69(B), the U velocity profile at the exit is compared with the theore tical result of a fully develope d velocity profile and the two match with each other very well. Although there is a relatively large deformation, the SIMCC can still strictly maintain the conservation of mass. The effects of different surface tension streng ths are shown in Figure 610 for the three different Weber numbers. It must be emphasized here that the corres ponding inlet pressure boundary conditions between the two phases are al so different due to differences in surface tension coefficients. In this fi gure, as expected it shows that the degree of liquid phase shape deformation is inversely proportional to the Weber number as the higher Weber number the lower the surface tension coefficient as a relatively flat interface corresponds to very small surface tension. It is noted that the degree of deformation is not a linear function of the Weber number. As given in Figure 610, the difference in deformation between We = 26.3 and We = 263 is much less than that between We = 2.63 and We = 26.3 even though for both cases the change in the Weber number is one order of magnitude. PAGE 111 111 Figure 611 shows the liquid core deforma tion as a function of time as it moves downstream for the case with We = 2.63 and Re = 500. The results show that the degree of deformation increases with time as the liquid plug travels downstream. 6.2.3 TwoPhase Flow Pressure Drop As indicated in the above, the pressure gradient in the twophase flow is much higher than that in the singlephase flow. It is important to summarize the effects of different Reynolds numbers and Weber numbers on the magnitude of increase in pressure drop in the twophase zone. The ratio of the averaged local friction coefficient in th e twophase zone to the fully developed singlephase gas flow fr iction coefficient is plotted in Figure 612 as functions of the Reynolds number and the Weber number. In Figu re 612(A), it is seen that the friction coefficient ratio increases with increasing Reynol ds number almost linearly. The increasing trend of the ratio with the Reynolds number is thought to be due to th e increased drag between the two phases as the Reynolds number ge ts higher, while the trend fo r different Weber numbers is totally different as shown in Figure 612(B) The friction coefficient ratio decreases exponentially in the low Weber number regime and th en relatively flattens out with a very small gradient. This trend is readily explained by the degree of shape deformation as shown in Figure 610: more deformation will result in higher drag between the two phases. A much larger change in shape between We = 2.63 and We = 26.3 than that between We = 26.3 and We = 263 is noted. As a result, the drop in friction coefficient rati o is much larger between We = 2.63 and We = 26.3 than that between We = 26.3 and We = 263. 6.3 Grid Refinement Study In the numerical simulation, the results may be not accurate due to the resolution of grid especially for the coarser grid system. In this section a grid refinement study is performed to ensure the adopted resolution of grid system is fine enough in this research. The case with PAGE 112 112 We=2.63 in section 6.2.2 is adopted here. Th ere are three grid reso lutions selected. The minimum grid lengths are 0.02, 0.01 and 0.005. Figure 613 shows the locations of interface by the three grids at the same time and it shows th at there is not much difference between three interfaces. In order to save computational time while preserving accuracy, unless otherwise mentioned, all following simulations in this st udy will adopt 0.01 as the minimum grid spacing. 6.4 LiquidGas TwoPhase Flows in a Pipe with Heat Transfer In this section, for the sample calculations, the selected two phase fluids are liquid water and air at 1 ATM. The physical conditions of the twophase flow and heat transfer system are described as follows. In the beginning (t = 0) the pipe is filled with only air which is in thermal equilibrium with the wall at 373.14 K. When the transi ent starts (t > 0), the ga te at the inlet of the pipe is opened and an annular twophase flow with water in th e center core and air filling the space between the wall and the water, both phases at 273.15K, are entering the pipe at the same and uniform velocity. The wall is maintained at 373.14 K during the transient. With this maximum temperature difference between the tw o phases, there is no phase change and only sensible heat transfer at the liq uidgas interface. The selected two phase fluids are liquid water at 273.15K and air at 373.14K. The ratios of fl uid properties of two phase flows are: 1055.07l g ; 81.39l g ; 17.99l gk k ; 4.17l gCp Cp The corresponding Prandtl number for water should be: Pr13.0 The time step in this computation is set as 0.001. It is noted that for the current system, ther e are three dimensionle ss parameters, Reynolds, Peclet and Weber numbers that mu st be specified. For the Weber number, two cases are selected, one low Weber number at 2.63 and the other high Weber number at 500. Different Weber PAGE 113 113 numbers can be used to evaluate the effects of shape deformation. The Reynolds and Peclet numbers for the current study are selected as Re500 ; 6500Pe In the high Weber number case, extra two cas es: (1) Re= 250 and Pe= 3250 and, (2) Re= 1000 and Pe= 13000 are also provided to investigate the e ffects of different Reynolds and Peclet numbers. 6.4.1 Low Surface Tension Case For this case, the Weber number is set at 500 such that the inertia force overwhelms the surface tension force. As a result the shape change of the liquid slug is relatively small. In Figure 614, the pressure and flow field results are pr esented. The bold dashed line represents the location of the liquidgas interface. Figures 614(A) and (B) show the isobaric pressure contours on the RX plane at time= 5.5. In Figure 614(A), it is seen that the pressure decreases linearly in the streamwise direction for both phases. In general, the magnitude of shear stress at the wall and the pressure jump across the liquidgas interface determine the pressure gradients in the liquid phase. Therefore the pressure of the liquid phase is larger than that of the gas phase at the same downstream location due the surface tension effect that causes a pressure jump at the interface. In the liquid phase away from its frontal area, as indicated in Figure 614 (B) where the pressure contour near X=3 is focused locally, it shows that the pressure jump across the interface is very close to the theoretical value (l p g p ~/We ~ 2.63/500=0.00526 as the curvature is 2.63 obtained from Figure 614(D)) sinc e the liquid pressure is influe nced and controlled by the gas phase because that the liquid phase is fully su rrounded by the gas phase that carries the bulk of the inertia force. In general, there is no pressure gradient in the radial direction for both phases. At the front of liquid slug, the curvature increases to 5.5 as shown in Figure 614(D) because of PAGE 114 114 the additional curvature in the RX plane and therefore, the pressure jump in the front of the liquid core can be estimated as 5.5/500=0.011 and this pressure jump can be verified in Figure 614(C) where the pressure distri bution along the centerlin e is plotted. After the front of liquid core, the gas flow diffuses into a single phase flow occupying the entire pipe. For downstream locations far away from the liquid front, the pressu re decays linearly as the flow adjusts into a fully developed singlephase pipe flow. Figure 614( E) is a streamline plot In Figure 614(E), it shows that the streamlines deform in the immediate downstream of the liquid front towards to the centerline of the pipe in the singlephase region. Because of th e nonslip condition and continuity requirement, the fluid near the solid wall will be pushed toward to the center of pipe and the maximum velocity takes place at the cen terline when the singlephase flow is fully developed. In Figure 614(F), it sh ows that the velocity profile at the outlet of pipe reaches the fully developed condition. Also, in the twophase region, due to a large density difference, the gas flow can not affect the liquid substantially, th erefore, the gas phase must take the form of jet flow to satisfy the continuity and it can be seen in Figure 614(F). Similarly, the shape of the liquid core will not deform too much and the interf acial velocity is close to the inlet velocity (~1.0). Based on results provided in Figure 614, it shows that th e current numerical method can accurately enforce the interfacial condition, the continuity, and momentum equations for the flow field computation. Next, the heat transfer results are presented. Figure 615 is the isothermal temperature contour and the interfacial temperature dist ribution for the case of We=500, Re=500 and Pe=6500 at different time steps of 1.5, 3.5, and 5.5. According to Figures 615(AC), they basically show that the moving liquid slug is a heat sink with a large capacity, because the thermal cap acity in the liquid phase is much larger than PAGE 115 115 that in the gas phase, the temperature changes in the liquid core are much less than those in the gas phase. The heat transfer in the gas annular po rtion is mostly fully de veloped as the isotherms are parallel to one another after a very short en trance region. The isotherms then bend to become relatively parallel to the liquid slug frontal area that again show s the flow field effects and the heat sink nature of the liquid phase. Since th is is an unsteady computation, the highest temperature of liquid core is located at the rightupper corner because this local region is heated by the gas phase of the highest temperature gr adient. In Figure 615( D), the timedependent temperature distributions of the liquidgas interface are plotted, which basically demonstrates the heating up of the horizontal portion of liquid phase and much less heat transfer in the vertical frontal portion. The general heat transfer e ffectiveness from the pipe wall is measured through the dimensionless Nusselt number,Nu, as defined below: "()ss s ssmhDqD Nu kkTT (62) where s q is the heat flux from the solid pipe wall to the gas flow and mT is the mean flow temperature at each downstream location that in cludes the contribution from both phases in twophase region. The Nusselt number along the pipe wall in the downstream direction have been plotted in Figure 616 for the case of We=500, Re =500 and Pe=6500 at three instants of t = 1.5, 3.5and 5.5. As can be seen from Figure 616, th e Nusselt number value is basically a reflection of the respective heat transfer mech anism that is dictated by the flow field. In the region from the entrance to around X = 0.5, the Nusselt number take s a sharp drop from a large value to reach a value of 8.9 that is the socalled entrance region. After the relatively very short entrance region, the heat transfer is fully developed and the fu lly developed portion is continued until reaching the frontal area of the liquid slug. After the liquid front, the Nusselt number goes through a sharp PAGE 116 116 decrease again as the gas flow adjusts to the single phase condition. Th e adjustment portion is also very short and the Nusselt number quickly approaching the value of 3.66 which corresponds to the Nusselt number for a fully developed singlephase flow in a circular pipe with a constant wall temperature. It is worth noticing that the enha ncement of heat transfer due to a liquid slug is quite extensive as that the heat transfer is increased more than 140% (Nu from 3.66 to 8.9). The gas jet flow in the twophase region raises th e Nusselt number from 7.3 (pure conduction in the annulus) to 8.9. The effects of the Reynolds number on the he at transfer in the twophase region are examined in Figure 617. Figure 617(A) shows the Nusselt number di stributions for Re= 250 (Pe=3250), 500 (Pe=6500) and 1000 (P e=13000) at time=5.5. First, it shows that the entrance length is proportional to the Reynolds number even though the entrance lengths for all three cases are relatively short. For the Nusselt numbers in the fully developed region, as expected, it shows that the higher the Reynol ds number, the higher the Nusse lt number as the Peclet number is proportional to the Reynolds number. But the differences am ong the three Nusselt numbers are less than 0.03 as seen in the closeup around X = 4.7, therefore the Reynolds number effect is really small. Figure 617(B) provides the temperat ure profiles for the liquidgas interface for the three different Reynolds cases. The heating of the liquid slug is relativel y faster for the lower Reynolds number. This is due to the fact the mo re heat is convected downstream to the singlephase region rather than to the liquid slug for higher Reynolds numbers. 6.4.2 High Surface Tension Case For this case, the Weber number selected is 2.63 to represent the condition where the surface tension is dominant over the inertia. The Reynolds number and the Peclet number are maintained at 500 and 6500, respectively for this case. Under a large surfa ce tension force, liquid slug would exhibit a substantial shape change near the upperright corner as shown in Figure 6 PAGE 117 117 18. The result of a dominant surface tension force in a twophase flow is the roll up of the liquid slug front around the top corner. Therefore, the inte rface near the front of liquid core will deform to pull backward. Figure 619 displays the isothermal temperat ure contours and the liquidgas interfacial temperature profiles at different times of t = 1.5, 3.5, and 5.5. Basically, the general trends of the heat transfer characteristics for the low Weber number case are similar to those in the high Weber number case but there are some differences due to the shape change that will be discussed below. In Figures 619(AC), it shows that the interf ace near the front of liquid core deforms as the liquid plug moving downstream. Because of the substantial deformation at the front of the liquid core, the crosssectiona l area of gas phase between the liquid phase and solid wall decreases, that results in the local acceleration of the gas velocity. The increased gas velocity around the deformed liquid frontal area enhances not only the convect ive heat transfer from the pipe wall but also the heating of the liqui d phase. In Figure 619(D), the higher interface temperatures at the deformed portion of the liqui d slug reflect the enhanced heat transfer from the gas flow. As the liquid phase deforms more when moving further downstream, the affected interface temperatures also rise higher. The Nusselt number distribution along the pipe wall is given in Figure 620. Again, the general trends are similar to those for the high Weber number case except that there is a local peak corresponding to the enhanced heat transfer due to the defo rmation of the liquid slug front corner as explained above. The local heat transf er increases range from 10% at time=1.5 to 20% at time=3.5 and then to 30% at time=5.5. The general heat transfer enhancement over the pure PAGE 118 118 conduction case in the annular region and the smoot h transition from the twophase region to the fully developed singlephase region are all sim ilar to those in the high Weber number case. To facilitate a close comparison between th e high and low Weber number cases, Figure 621 is provided for this purpose. Figure 621(A) is a comparison of the interfacial temperatures for the low and high surface tension cases at the sa me time instant. Starting from the inlet and continuing until the deformation begins, the two cu rves are almost overlapped with each other because both the gas flow and the geometry are very similar. Near the deformed liquid area, there is about 50% temperature difference. At th e relatively vertical fr ont of liquid core, the temperature differences between th e two cases become very small b ecause the heat transfer is no longer affected by the shape deformation. In Figu re 621(B), it is the comparison of the Nusselt number distributions. The only di fference between the two cases is again due the deformation which causes a maximum 25% increase in the lo cal Nusselt number at the peak location. 6.5 Summary In this chapter, the liquidgas twophase flows in a pipe with out phase change are computed and it includes the isothermal and pure heat transfer cases. For the isothermal case, it shows that veloci ty fields and pressure distributions were correctly computed for both twophase and singlephase zones and a smooth transition between the two was also obtained. In the twophase zone, the liquid ve locity profile is basically unidirectional and uniform while a jet profile was found in the gas phase. Almost immediately downstream of the liquid core front, the singlep hase gas flow establishes the fully developed parabolic profile. The pressure is virtually independent of the ra dial direction and the pressure distribution in the axial direction is composed of two straight lin es of different slopes for the twophase and singlephase zones, respectively. The dimensionless pressure gradient in the twophase zone is inversel y proportional to the Reynolds number. The ratio of the averaged local PAGE 119 119 friction coefficient in the twophase zone to th e fully developed singlephase gas flow friction coefficient increases with in creasing Reynolds number almost linearly, whereas this ratio decreases exponentially in the low Weber number regime and then relatively flattens out with a very small gradient for intermediate and large Weber numbers. The liquid core shape deformation is relatively insensitive to the Re ynolds number, however the deformation is large for small Weber numbers and is negligible for intermediate and large Weber numbers. For the case with heat transfer, irrelevant to the Weber number, in general the moving liquid slug in the central core which induces a je t gas flow in the annulus between the pipe wall and the liquid slug serves as the main heat sink of a large capacity. The combination of the gas jet flow and the large sink capacity in the twophase flow region result s in a heat transfer increase of more than 140% over that in a fully developed singlepha se flow. Immediately downstream of the liquid slug, the gas flow adjusts quickly to establish as a fully developed singlephase flow and the local Nusselt number al ong the pipe wall reflects that process. The effect of different Reynolds numbers on the heat transfer in the twophase region is negligibly small. For the low Weber number case where th e surface tension substantially overweighs the inertia force, the result ant shape change in th e form of rollup and bending backwards around the upper corner of the liquid slug reduc es the local gas flow area in the annulus. This flow area reduction creates a increase in th e gas velocity that in turn causes a local heat transfer enhancement ranging from 10% to 30% depending on the length of time the liquid slug spends in the pipe. PAGE 120 120 Figure 61. Streamline of the isothermal case at time=6.0 with Re= 2000 and We=500. A B C D Figure 62. The pressure cont ours/distributions at time=6.0 with Re=2000 and We=500. A) entire pressure contour, B) near the entrance, C) downstream part, D) distribution at center and wall, E) at differe nt locations along ax ial direction near the front of liquid core, F) at different locations along radial direction near the front of liquid core, G) curvature along interface. PAGE 121 121 E F G Figure 62. Continued. PAGE 122 122 Figure 63. The pressure distributions along th e centerline based on different Reynolds numbers at time=6.0. A B Figure 64. The U velocity profile at time= 6.0 with Re=2000 and We=500. A) at different locations, B) comparison of velocity at the outlet. Figure 65. Shapes of liquid cores with different Reynolds num bers at time=6.0. PAGE 123 123 A B C Figure 66. The pressure cont ours at time=6.0 with Re=500 and We= 2.63. A) twophase zone, B) singlephase zone, C) entire domain. A B Figure 67. Pressure distributi on and the curvature of interfac e at time=6.0 with Re=500 and We=2.63. A) pressure distributi on, B) curvature distribution. PAGE 124 124 Figure 68. Streamline plot at time=6.0 with Re=500 and We=2.63. A B Figure 69. U velocity profile at time=6.0 with Re=500 and We=2.63. A) at different locations, B) comparison of velocity at the outlet. Figure 610. Shapes of the interface by different Weber numbers at time=6.0. PAGE 125 125 Figure 611. The development of interface with Re=500 and We=2.63 at different time steps. A B Figure 612. The ratio of averaged local friction coefficient in the twophase zone to the fully developed singlephase gas flow friction coefficient. A) as a function of the Reynolds number, B) as a function of the Weber number. Figure 613. The computed interfaces by three different grid resolutions. PAGE 126 126 A B C D E F Figure 614. The pressure contou rs, pressure and curvature distri bution, streamline plot and the contour of U component contour of the case with Re=500, Pe=6500 and We=500 at time=5.5. A) pressure contour, B) enlarged pressure contour near X=3, C) curvature distribution, D) streamline pl ot, E) U velocity component along radical direction. PAGE 127 127 A B C D Figure 615. The temperature cont ours of twophase plug flow and the interfacial temperature at different time steps with Re=500, Pe=6500 a nd We=500. A) time=1.5, B) time=3.5, C) time=5.5, D) interfacial temperature. Figure 616. The Nusselt number at the solid wa ll at different time st eps with Re=500, Pe=6500 and We=500. PAGE 128 128 A B Figure 617. The comparison of interfacial temperature and Nusse lt number at wall for different Reynolds number with Pe=6500 and We=500 at time=5.5. A) Nusselt number at wall, B) interfacial temperature. Figure 618. The shapes of the liquid slug at three different time steps for the case of Re=500, Pe=6500 and We=2.63. PAGE 129 129 A B C D Figure 619. The temperature c ontour of two phase plug flow a nd the interfacial temperature with Re=500, Pe=6500 and We=2.63at different time steps. A) time=1.5, B) time=3.5, C) time=5.5 and D) interfacial temperature. Figure 620. The Nusselt number at the solid wall Re=500, Pe=6500 and We=2.63 at different time steps. PAGE 130 130 A B Figure 621. Comparison of the interfacial temperature and the Nusselt number with different Weber numbers. A) interfacial temperature, B) Nusselt number. PAGE 131 131 CHAPTER 7 LIQUIDGAS TWOPHASE FLOWS IN A PIPE W ITH PHASE CHANGE 7.1 Introduction In this chapter, the text will focus on phase change computation. Based on thermodynamics, the temperature at the vaporiz ing liquidgas interface is assumed as the saturation temperature corresponding to the system pressure and th e pipe wall temperature is set to be higher than the saturation temperature. Therefore, as th e liquid phase is also at the saturation temperature without any subcooling, the heat flux r eaches the liquidgas interface will be fully consumed to supply the latent heat b ecause the temperature of liquid can not increase. As the phase change occurs, the gas mass flow rate downstream will increase due to the vaporization from the liquid surface that adds ma ss flows to the gas phase. Also, because of the large temperature difference assumed between the liquid phase and solid wall, the thermal radiation is considered. In the current approach, two kinds of heat transfer boundary conditions of the solid wall are considered. In the first case, the wall temperature is assumed to be constant, so that the heat flux comes from wall is relatively high and constant. This can represent the case with a very large wall heat capac ity. In the more realistic situa tion, the wall temperature may not remain constant and it would be chilled down by th e colder fluid in the pi pe. In the second case, a finite wall heat capacity is assumed. It wi ll focus on the impact of the pipe wall chilldown process on the twophase flow and heat transf er. One refrigerant R508B and two cryogenic fluids of liquid nitrogen and hydroge n are chosen as coolants. The di ameter of the proposed pipe is 4 mm and inlet velocity is 10cm/s for both phases and the selected material for the wall is a Titanium alloy with =44503/kgm, Cp=4200/()JkgK, and k= 4.8/()WmK PAGE 132 132 7.2 R508B Refrigerant with Constant Wall Temperature In this section, the selected fluids are the liquid and gas refrigerant R508B at 208K and 248K, under 3 ATM. Its chemical formula is 333/,46/54%CHFCFCF and its molecular weight is 95.39. Under this condition, th e temperature at liquid phase is the saturation temperature for 3 ATM and will not change during the computation a nd the phase change should take place at the liquidgas interface. The main purpose of selecting this particular working fluid is to add a noncryogenic case but its application is quite extensive. The ratios of fluid proper ties for the two phases are: 98.89l g ; 21.74l g ; 9.04l gk k ; 1.53l gCp Cp The two fixed dimensionless parameters are: Pr3.26 0.034Ja The time step in this computation is set as 0.001. In the phase change computation, there are four dimensionless parameters in the governi ng equations and interfacia l conditions, and they are Reynolds (Re), Peclet (Pe), Weber (We) and Jakob (Ja) numbers. Once the thermodynamic properties of working fluid and th e temperature difference are decided, the only free dimensionless parameters are Reynolds and Weber numbers. In order to demonstrate the ability of the SIMCC code to handle the large deformation and change of volume due to the strong surface tension and phase change effects, the Reynolds and Jakob numbers are selected as 10.0 and 2.63, respectively. Figure 71 shows the contours of the Uvelocity component at time =1.0, 2.5 and 4.0. In Figure 71, the Ucomponent contours show that there are two loca tions with higher U velocities in each contour plot. The first location is in the gas phase near the upperright corner PAGE 133 133 of liquid core. Because of the vaporization at th e upper surface of liquid core, the liquid will be vaporized as gas and enters the gas phase. It can be expected that the ma ss flow rate of gas will be increased before entering the singlephase region so that the veloc ity of gas is sped up till this location and results in the maximum velocity. Figure 72 is the history of lo cal maximum Uvelocity of the gas phase near the upperright corner of liquid core. It show s that the local maximum velocity increases as time increases but this trend is not linear. If ther e were no shape change for the liquid slug, then the maximum Uvelocity history curve would be linear as th e evaporating surface area would increase linearly with time and the rate of mass evaporation is pr oportional to the surface area. Also the distance between the pipe wall and the liquid core would st ay unchanged. In the cu rrent case, the interface shape does change substantially due to a strong su rface tension effect and phase change process. In Figure 73, it shows the interface shapes at di fferent times. At t = 4.0, the maximum distance between the pipe wall and the liqui d core is 0.24. The initial (tim e=0) maximum distance is 0.12. This explains why the curve in Figure 72 is no t linear and the slope must decrease since the cross section of area of gas phase increases. Another location with a higher velo city is at the center of th e pipe. Near the wall, because of the nonslip condition and the continuity requirement, the gas is pushed to the centerline and also because of the mass flow from the twophase upstream, the velocity at the centerline will be higher than other locations. Figure 74 is the streamline plot of the gas phase at time=4.0. By the distribution of streamlines, it does show that the streamlines are pushed to the center of the pipe and this means the gas is pushe d to the center of the pipe. Figure 75 shows the Vvelocity component cont our at different time steps. Downstream to the front of the liquid core, the flow of gas is pushed to the center so that the maximum negative PAGE 134 134 Vvelocity is located near this region. This highest negative Vvelocity component increases as time increases due to more vaporization from the liquid core. Near the cent erline, the Vvelocity component will be close to 0 and that means th e gas flow turns its direction because of the symmetric condition. This conclusion can also be seen in Figure 75. In Figure 75, the phase change at the interface is much stronger at the beginning due to the initial condition. As time increases, it shows that the vaporization becomes weaker. Figure 76 is the temperature contours at different time steps. To compare the gas temperatures near the front of the liquid core, it shows that the temperature gradient is much higher at time=1.0 due to the initial condition. At time=4.0, the gradient is much weaker because of the convective effect and also the distance between the liquid core and the wall increases. It can be expected that phase ch ange is very strong at beginn ing and becomes weaker as time increases. Figure 77 is the Nusselt number evaluated at the solidgas interface at different time steps. It shows that in general the Nusselt number decreases as time increases. By Figure 76, it is very clear that the thermal gradient decreases along the downstream direction so that the Nusselt number decreases with increased downstream lo cation. In the single pha se region, all three curves converge to the fully developed singlephase flow case very fast and the corresponding Nusselt number is 3.66. Figure 78 shows the total mass flow rate of ga s phase at the pipe ex it. Basically, the trend is very close to that in Figure 72. The mass tran sfer rate increases as time increases but the slope of curve becomes smaller as time increases. 7.3 Constant Wall Temperature Case with Liquid Nitrogen In this section, the worki ng fluid is nitrogen with liquid and gas phases at 77K and 300K, respectively under 1ATM. Under these conditions, the temperature at liquid phase is the PAGE 135 135 saturation temperature and will not change during the computation and vaporization take place at the liquidgas interface. The ratios of fluid properties of two phases are: 706.91l g ; 8.85l g ; 5.41l gk k ; 1.97l gCp Cp The corresponding ratio of the Prandtl number for both phases should be: Pr 3.21 Prl v The time step in this computation is set as 0.001. Three sets of dimensionless parameters are considered based on th e fluid properties and geometry. They are listed in Table 71. 7.3.1 Flow Field with the Constant Wall Temperature Case Since the heat transfer and phasechange proces s is closely related to the twophase flow structure, a typical flow field shown in Figure 79 will be examined first that provides the velocity vector and streamline plots for Case 2 at time=1.5. Once the phase change is involved, the liquid will be vaporized and infuse into the gas stream so that the gas flow field is constantly changing with the mass flow rate and velocity increasing with increased downstream positions and the flow patterns are changed accordingly. For the liquid phase, it is a very simple unidirectional slug flow, while the gas phase is basically a continuously changing and accelerating annular jet flow due to the vaporization from the liquid surface. As seen in the streamline plot, the lines that originate from the liquidgas interface represent the vaporization mass fluxes. The streamline density gets very high near the front of the liqui d slug, that results in the maximum velocity in the twophase region. PAGE 136 136 After the maximum jet velocity, the gas flow di ffuses into a pure si nglephase region and the gas flow becomes fully developed relatively quickly. Figure 710 shows the Ucomponent velocity co ntours for all three cases at time=1.5. To compare the three cases, the flow patterns are very similar to one another and the maximum velocities of three cases are all located near the upperright of the liquid core which is the intersection between the twophase region and the singlephase region. These flow structures are consistent with the velocity vector plot given in Figure 79. Due to the conservation of mass, the mass flow rate of the gas phase at each downstream cross section of the pipe should be the same in the singlephase region (about X >2.0) as ther e is no more source of mass transfer. Therefore, the velocity decreases as the ar ea of the cross section of the pi pe increases in the singlephase region. Figure 711 is the Vcomponent velocity contou rs for all three cases at time=1.5. Again, they are all similar to one another also. By Figur e 711, the highest verti cal velocities are located on the horizontal portion of the liquidgas interface where the vapor ization rate is the highest accordingly. Immediately downstream of the fr ont of the liquid core, it shows that the Vcomponent velocities are negative and possess the maximum values because the gas must rushes downward to fill the entire pipe. Far away from the front of the liquid core, the Vcomponent decreases very fast and approaches zero as the singlephase becomes fully developed. Figure 712 gives the value of the maximum gas velocity for each of the three cases and a relatively linear relationship is found for the maximum gas veloc ity with the Reynolds number. Based on Table 71, the case with a lower Re ynolds number is also with a lower Peclet number. By the conservation of energy, the vapor ization mass flux, at liquidgas interface is proportional to /JaPe as given below: PAGE 137 137 int int ''(()()) () ()gngn nng g gmuu Ja uuT Pe Ja mT Pe (71) Therefore, the case with a lower Peclet number will experience a higher evaporation rate that contributes to a higher maximum velocity. But it should also be noted that the gas temperature gradient term in the above equation would be slightly higher at a higher Reynolds number due to forced convective heat transfer in the annulus, however this effect is less significant than that due to the Peclet number. Figure 713 is the temperature pr ofile contour of the three ca ses at time=1.5 For all three cases, they all belong to the categ ory of a high Peclet condition, so the contours are very similar to one another. Under a high Pecl et number, the convection is domin ant as can be seen in those temperature profiles that the he at transfer in the twophase region corresponds to parallel isotherms due to a fully developed annular flow In the singlephase region, the temperature profiles quickly establish the form of a fu lly developed pipe flow heat transfer. Figure 714 plots the liquidgas interface shapes for the three cases at time=1.5. As the interface shape deformation is strongly depende nt on the Weber number and the three cases almost have relatively the same Weber numbers, so it can be expected only slight differences among them. This is indeed the case as shown in Figure 714. Near the entrance of the pipe, the three interfaces almost overlap. Ne ar the front of the liquid core, intrinsically the deformation of the interfaces is more substantial for each case due to the magnitude of surface tension effect (different We numbers). Higher Weber numbers impl y a lower surface tension effect and therefore, the case with the smallest Weber num ber (Case 3) deforms more than the other two cases. PAGE 138 138 Using the liquid slug shape as a basis for the ev aluation of the grid size requirement, a grid refinement test is done to ensure the results obtai ned will not be influenced by the grid resolution. There are three gird sizes selected and they are 0.005, 0.01 and 0.05. The parameters for Case 3 are employed for the evaluation study. Figure 715 s hows the interfaces for th e three grid sizes at t =1.5. It is seen that there is not much diffe rence among the three grid lengths and the selected grid length of 0.01 is adequate for the current numerical computations. 7.3.2 The Mass Transfer at Interface with Constant Wall Temperature In this section, it focuses on the evaporativ e mass transfer process at the liquidgas interface. As shown in the previous section, the mass transfer rate is represented by the difference between the velocitie s at the interface and accordingl y the evaporation rate is dominated by the gas phase temperature gradient the Jakob and the Peclet numbers. Previously, Figures 713 shows that temperature distributions are similar in the twophase region for all three cases. In Figure 714, there is only a small diffe rence near the front of the liquid core where the distance between the solid wall and the liquid surface is the shortest for Case 3 that results in a highest gas phase temperature gr adient locally. Figure 716 (A) is the computed temperature gradients of three cases along th e interface marker points at time= 1.5. Figure 716 (B) provides the locations of markers. In Figure 716(A), all three curves are simila r except near the front of the liquid core, where the temperature gradient for Case 3 is larg er than others because the distance between the liquid surface and the solid wall is the smallest for this case. At the entran ce, because of the inlet boundary conditions, the temperature gradient is zero. At the front of the liquid core, because the distance between liquid slug surface and the solid wall becomes longer, the temperature gradient becomes smaller in all the three cases. PAGE 139 139 Because the gradients of temperature of these three cases are similar along most part of the interface, the ratio of mass transf er rate at the interface for these three cases can be estimated by Equation 71. By Table 71, all three cases have the same Jakob number so that the mass transfer rate is inversely proportiona l to the Peclet number: 1m Pe (72) Since the Prandtl number is the same for all three cases, the evaporation rate should also be inversely proportional to the Reynolds number. Figur e 717 is the total mass flow rates measured at the outlet of pipe for the thre e cases at time=1.5 and as expected the mass flow rate at the exit is inversely proportional to the Reynolds number. 7.3.3 The Transient Phase Change Process with Constant Wall Temperature In this section, the focus is on the timedependent characteristics of the phasechange process and Case 2 is used for the evaluation. Fi gure 718 is the temperat ure gradient history at the liquidgas interface along the marker points for Case 2. In Figure 718, it shows that the temperature gradients near the entrance do not change as time increases because it is controlled by the inlet boundary conditions. The peaks of temperature gradients are decreasing because th e gas is chilled down by the liquid phase. The temperature gradients all drop drastically at the front of liquid core be cause that the distance between the solid wall and liquid surface becomes very large there. At the beginning, there is only one peak in the temperature gradient curve. Because of the deformation of the liquidgas interface due to the surface tension as shown in Figure 714, small oscillations develop near the frontal area of the liquid core. Figure 719 plots the histor y of mass flow rate at the exit fo r Case 2. It shows that the mass flow rate increases as time increases due to continuous evaporation from the liquid slug. The PAGE 140 140 curve is relatively linear; therefore the rate of evaporation is increasing almost linearly during that period. Figure 720 gives the Nusselt numbe r distributions for the three cases along the surface of solid wall at time=1.5. From Figure 713, it sh ows that the temperature contours of the three cases are very similar and therefore, the Nusselt numbers are also very close. They will be affected by the distance between the wall and the liquid surface. As disc ussed previously, the distance for Case 3 is shorter than others so that the Nusselt number for Case 3 will be larger than others. In Figure 720, it doe s show this trend. After the initial drop, the Nusselt number increases in the axial direction due to the in crease of velocity from vaporization. Also, downstream of the front of liquid core (X>2.0), the gas phase will adjust quickly to become a fully developed single phase flow and the Nusselt number in this region reflects that process. Far away from the inlet, the effects of the liquid slug is diminished and the Nusselt number is asymptotically approaching the fully developed si ngle phase heat transfer value. For the purpose of evaluating the heat transfer enhancement due to evaporation, the Nu sselt number is plotted by a longdash line for an identical case except that there is onl y convection with no phasechange. The enhancement due to phase change is on the average 60%. 7.3.4 The Comparison of Nusselt Number with Phase Change Process and Constant Wall Temperature In this section, the Nusselt number in liquid ni trogen with phase change and constant wall temperature is compared. In Hammouda et al .'s paper(1997), they gave the correlation for inverted annular flow as following: 0.0439 0.6455.071/Pr0.0028PrRevv vNu (73) PAGE 141 141 wherePrv andRevare the Prandtl and Reynolds numbers fo r gas phase. In their research, the physical model is similar with the current research but there is st ill some difference in numerical procedures. In current research, it is a di rect numerical simulation, to solve the governing equations, boundary conditions and interfacial conditions dir ectly without any empiricism. In Hammouda et al.'s research, they used twofluid model. Th eir model is a semianalytic model and using empirical correlations for transport coefficien ts. Also, their model is one dimensional in streamwise direction and assumes the computa tion is steady state, fully developed condition without the entrance effect so that they can not provide dynamic informa tion. Figure 721 is the comparison of Nusselt number by the current me thod and Hammouda et al.'s correlation. In Figure 721, the upper long dash lin e is computed by Hammouda et al.'s correlation and the short dash line is by current SIMCC code. Near the inle t, because of the entrance effect, the thermal boundary layer will be thicker so that the Nusse lt number obtained by SIMCC is smaller. In Hammouda et al.s research, their model can only work in fully developed region so there is no entrance effect in their resear ch and the Nusselt number computed by their correlation should higher than current method. The thermal boundary thickness should become thinner as X increases because of the stronger convection effect due to vaporiz ation in twopha se zone. After the front of liquid core, the flow pattern is singlephase flow and the Nusselt number will decrease very fast and converge to the Nusselt number in singlephase zone. The correlation from Hammouda et al. can work in twophase zone but the singlephase zone. By this comparison, it helps to validate the current computation for Nusselt number. At least, the computation of Nusselt number in this re search is in a re asonable order. PAGE 142 142 7.4 Wall Chilldown Process by Liquid Nitrogen In this section, the selected fluid is also the liquid and vapor nitrogen at 77k and 300K under 1 ATM. The time step in this computation is set as 0.001. The dimensionless parameters of Case 2 in the previous section are a dopted in this section for the chilldown study. For the chilldown case, the wall is no longer kept at a constant temperature, whereas it is given a finite thickness and its temperature is decreasing as a result of heat loss to the cooling fluid. Therefore, computational grid system will include the wall. There will be three different thicknesses of the so lid wall in this section and they are wowi R RR = 0.02R (Case 1), 0.04R (Case 2), 0.08R (Case 3). 7.4.1 Flow Patterns during Chilldown Process Figure 722 includes the contours of two velocity components U and V and the temperature contour of Case 1 at time=1.5. When comparing with the Case 2 of the constant wall temperature condition in the previous section, the flow patte rn is very similar with only sm all differences in the magnitude. Table 72 provides the maximum velocity in th e twophase region for the three wall chilldown cases at time=1.5. The constant wall temperature case is also listed for comparison. Case 1 has the lowest maximum velocity because its therma l capacity is smaller than the other two which results in less amount of vaporization. Figure 723 is the wall temperat ure contours of three cases at time=1.5. The results show that as expected, the thinner wall is ch illed down faster than the thicker one. 7.4.2 The Wall and LiquidGas Interface Conditions during Chilldown Process This section focuses on the impact from the chilldown process on the mass transfer at the interface. The generated ma ss flow rate at the inte rface is determined by the temperature gradient and the dimensionless parameters, Jakob and Peclet numbers. PAGE 143 143 Figure 724 is the wall temperatur e distribution at the solidga s interface for the three cases at time=1.5. It shows that the wall for Case 3 is chilled down faster than others because the thickness of the wall of Case 3 is much thinner th an others. Since this is an unsteady computation so that the lower temperature should be located at the entrance. Far away from the entrance, the temperature will converge to the initial temperature of 1.0 By the temperature contour given in Figure 723 and the temp erature distributions at the solidgas interface provided in Figure 724, it can be expected that the temperature gradients may be smaller near the entrance. Figure 725 shows th e gasside temperature gradients of the three cases along the liquidgas interface at time=1.5. In Figure 725, the temperature gradient at liq uidgas interface of Case 1 is smaller than others. The temperature gradient of Case 3 is larger than others and it is very close to the reference case of the constant temperature wall condition, since the thicker the wall is, the more the energy is stored. Accordingly, the solidg as interface temperature would decrease more slowly; therefore, the temperature gradient at the liquid slug surface will be larger. In this section, the dimensionless parameters for all cases are the same so that the mass flow rate at outlet should be de termined by the heat flux at th e interface. Table 73 shows the mass flow rates for the three and reference cases at time=1.5 at the outlet of pipe and the results agree with the temperature gradient plotted in Figure 725 where Case 3 has the largest temperature gradient and correspondingly the most evaporation rate which gives rise to the highest mass flow rate at the exit. By Table 73, it also shows that the mass flow rate at the exit approaches to that of the case with constant wall temperature as the thickness of the wall increases. PAGE 144 144 7.4.3 The TimeDependent Development during Chilldown Process In this section, the timedependent system behavior during the chilldown process is presented. Figure 726 displays th e temperature profiles along the in side surface at three different time steps for Case 1. The cooling of the wall is enhanced by the evaporation from the cold liquid slug and the effect due to the motion of the liquid slug is also shown by the wall surface temperature history. Figure 727 is the temperature gr adients of Case 1 along the so lidgas interface at different time steps. To compare with the co nstant wall temperature case in previous section, it shows that the temperature gradient decays faster for the chilldown case. Since the heat flux at the liquidgas interface is different for the various chilldown cases, it is clear that the mass flow rates will not be the same at the exit. Figure 728 is the history of mass flow rates for the three chilldown cases and the constant wall temperature case. At the beginning, the mass flow rates of all cases are very si milar because the wall temperature/heat flux at this moment is governed by the same initial c ondition and the chilldown process at the wall has just started. Once the time increases, the wall is chilled down continuously so that the differen ce in mass flow rates will increase. For the cases with a thicker wall, the temperature of the wall decreases slower so that the mass flow rate of the case with a thicker wall is close to the case wi th a constant wall temperature. 7.5 Wall Chilldown Process by Liquid Hydrogen In this section, the liquid hydroge n is investigated as it has be en used extensively in space applications. The selected fl uids are liquid and gas hydrogen at 20.27K and 200K under 1 ATM. The ratios of fluid properties the hydrogen are: 576.94l g ; 1.94l g ; 0.92l gk k ; 0.714l gCp Cp PAGE 145 145 For comparison purposes, the liquid and gas nitrogen case is used as a reference to measure the differences in phasechange he at transfer when hydrogen is used The nitrogen properties are taken for 77.36K and 300K under 1 ATM for liquid a nd gas phases, respectiv ely. The ratios of fluid properties for the nitrogen are: 706.91l g ; 8.85l g ; 5.41l gk k ; 1.97l gCp Cp The initial conditions, initial location of interface and boundary conditions are the same as those in the pervious chilldown case section. By the same geometry and boundary conditions, the four dimensionless parameters for hydrogen and nitrogen are: Hydrogen: Re2145; Pr1.08; 4.27Ja ; 1.71We Nitrogen : Re2043 ; Pr2.32 ; 0.42Ja ; 3.69We By these two sets of dimensionless parameters, they show that the Reynolds, Weber and Prandtl numbers are close between hydrogen an d nitrogen but the Jakob numbers are quite different. The ratio of Jakob numbers for both fluids is about ten times. The thickness of the wall in this section iswowi R RR = 0.02R. In the following, the results of hydrogen will be given side by side with corresponding ones from nitrogen for a direct co mparison between these two cases. Figure 729 provides a direct comparison for the Uvelocity contours of hydrogen and nitrogen at time=0.4. In general, the two cases have very similar Uvelocity profiles. The maximum Uvelocities for the two cases are also lo cated almost at the same location near the exit of the annular channel. But, th e magnitudes are totally different and the maximum Uvelocity for hydrogen is about four times of that for nitroge n that is exclusively due to the much higher evaporation rate of hydrogen resulting from a much larger Jakob number as explained next. From previous discussion, the mass transfer rate can be estimated by PAGE 146 146 ''()vJa mT Pe (74) The ratio (/)JaPe of hydrogen to nitrogen is 20.8 that is the main cause for the maximum Uvelocity difference. Figure 730 is the Vvelocity contours of tw o cases at time=0.4 and they show the same trend as that of the Uvelocity contours. The contours are similar but the magnitudes are quite different. First, the vaporizati on rate on the liquid slug surface is much uniform for the hydrogen case. The relatively nonuniform evaporation rate from the nitrogen liquid slug surface is due to the shape deformation whereas the hydrogen surf ace is very flat. Second, based on Figures 729 and 730, they show that the convection effect for the hydrogen case is much stronger so that the location of second highest Vvelocity is much cl ose to the liquidgas interface for nitrogen case. Figure 731 is the temperature contour comparis on for the two cases at time=0.4. It shows large differences between the two cases. Sin ce the convective effect in hydrogen is much stronger due to higher evapor ation rates than that in the nitrogen case. It ca n be expected that the temperature profile in the hydrogen case should re flect that. Due to the strong convection, the temperatures at downstream in the hydrogen case is cooler than those in the nitrogen case. Figure 732 shows the temperature distributions at the solidgas interface for both cases during chilldown process. Due to the stronger convection effects, the solid wall for the hydrogen case is cooled down much faster than that in the nitrogen case, especially near the entrance region. In the downstream region, the temperature of both cases will be close to the initial wall temperature since there is only very few heat transfer in that region. Figure 733 compares the Nusselt number distri butions on the inner su rface of pipe wall for the two cases. Both cases show very similar entrance region behaviors near the inlet. Similarly, because of the stronger convection e ffects, the thermal boundary in the hydrogen case PAGE 147 147 will be thinner than that in th e nitrogen case as shown in Figure 731 and this means that the Nusselt number in the hydrogen cas e should be larger. On the av erage, the heat transfer coefficient on the pipe wall is 25% higher in the hydrogen case. 7.6 Summary In summary, three working fluids with two di fferent wall conditions were investigated in this chapter. Because of the evaporation phase change that takes place on the liquid slug surface, mass fluxes are generated from the liquidgas interf ace that acts as a source of mass input to the gas stream. In fluid dynamics, this generated ga s does change the flow structure substantially. Because of the generated gas, the total mass flow rate of the gas phase can not be a constant at any downstream location in the twophase region. Also, there is always a very strong jet effect near the intersection between th e singleand twophase zones, where the maximum gas stream velocity is located. Therefore, the flow patterns with phase change are different from those without phase change. Because of the evaporation that causes a larg e increase in the gas stream velocity in the twophase region, a much stronger convective e ffect is therefore induced. This enhanced convection results in a substantial increase in the heat transfer efficiency of entire system. This enhanced convection effect can also be seen in the chilldown process where the wall temperature decreases faster in the case w ith a stronger convective effect. The wall chilldown case is a high ly transient process as the he at transfer that supplies the latent heat for evaporation decreases with time due to decreasing heat transfer driving force, the temperature gradient between the pipe wall and the liquid slug. In the last case, a close co mparison was made between li quid hydrogen and liquid nitrogen as cryogenic working fluids. In general, under similar conditions, liquid hydrogen offers a much PAGE 148 148 higher evaporation rate that i nduces more intensive convection effects. The Nusselt number is approximately 25% higher for the liquid hydrogen case. PAGE 149 149 Table 71. Dimensionless parameters for the nitrogen case with constant wall temperature. Dimensionless Parameters Re Pe Ja We Case 1 2000 4640 0.42 3.69 Case 2 1500 3480 0.42 2.76 Case 3 1000 2320 0.42 1.84 Table 72. The maximum velocity for the nitrogen case with wall chilldown and the reference cases at time=1.5. Constant Temperature Wall Chilldown Cases Case 1 Case 2 Case 3 Max Velocity 13.62 12.90 13.31 13.41 Table 73. The mass flow rate for the nitrogen case with wall chilldown and the reference cases at time=1.5. Constant Temperature Wall Chilldown Cases Case 1 Case 2 Case 3 Max Flow Rate 0.594 0.571 0.585 0.591 .A. .B .C Figure 71. The U contours for th e refrigerant R508B case at differe nt time steps. A) time=1.0, B) time=2.5, C) time=4.0. PAGE 150 150 Figure 72. The local maximum U in the gas phase for the refrigerant R508B case at different time steps. Figure 73. The interfaces for the refriger ant R508B case at different time steps. PAGE 151 151 Figure 74. The streamlines in the gas phase for the refrigerant R508B case at time=4.0 A B C Figure 75. The V contours for re frigerant R508B case at different time steps. A) time=1.0, B) time=2.5, C) time=4.0. PAGE 152 152 A B C Figure 76. The temperature contours for refrig erant R508B case at different time steps. A) time=1.0, B) time=2.5, C) time=4.0. Figure 77. The Nusselt numbers along solid wall for the refrigerant R508B case at different time steps. PAGE 153 153 Figure 78. The total mass flow rate of gas pha se for the refrigerant R508B case at different time steps. A .B Figure 79. The vector and the streamline plots for Case 2 in the nitrogen case with constant wall temperature at time=1.5. A) ve ctor plot and B) streamline plot. PAGE 154 154 A B C Figure 710. The U contours for the nitrogen cas e with constant wall temperature at time=1.5. A) Case 1, B) Case 2, C) Case 3. PAGE 155 155 A B C Figure 711. The V contours for nitrogen case with constant wall temperature at time=1.5. A) Case 1, B) Case 2, C) Case 3. Figure 712. Maximum velocity for the nitrogen case with constant wall temperature of three cases. PAGE 156 156 A B C Figure 713. The temperature contours for nitrogen case with constant wall temperature at time=1.5. A) Case 1, B) Case 2, C) Case 3. A B Figure 714. The interfaces of three cases for the nitrogen case with constant wall temperature at time=1.5. A) entire, B) local. PAGE 157 157 Figure 715. The interfaces of three different grid lengths. A B Figure 716. The temperature gradient of thr ee cases along the interface at time=1.5 and the locations of markers for the nitrogen case with constant wall temperature. Figure 717. The mass flow rate for the nitrogen case with constant wall temperature at time=1.5. PAGE 158 158 Figure 718. The history of temperature gradie nt along the interface of Case 2 in the nitrogen case with constant wall temperature. Figure 719. The histor y of mass flow rate of Case 2 fo r nitrogen case with constant wall temperature. PAGE 159 159 Figure 720. The Nusselt number at wall for the n itrogen case with constant wall temperature at time=1.5. Figure 721. The comparison of Nusselt number by the current method and Hammouda et al.'s correlation. PAGE 160 160 A B C Figure 722. The U, V and temperature contou rs of Case 1 in the nitrogen case with wall chilldown at time=1.5. A) U contour, B) V contour, C) temperature contour. A B C Figure 723. The wall temperat ure contours for the nitrogen case with wall chilldown at time=1.5. A) Case 1, B) Case 2, C) Case 3. PAGE 161 161 Figure 724. The temperature at solidgas interface for the nitrogen case with wall chilldown at time=1.5 Figure 725. The temperature gradients along th e interface of reference for the nitrogen case with wall chilldown at time=1.5 Figure 726. The temperature hi story of Case 1 in the nitrogen case with wall chilldown at time=1.5. PAGE 162 162 Figure 727. Temperature gradie nts along the interface of Case 1 in the nitrogen case with wall chilldown at time=1.5 Figure 728. The mass flow rates for the nitrogen case with wall chilldown and reference case. A B Figure 729. U contours of two cases at time=0.4. A) hydrogen, B) nitrogen. PAGE 163 163 A B Figure 730. V contours of two cases at time=0.4. A) hydrogen, B) nitrogen. A B Figure 731. Temperature cont ours of two cases at time=0 .4. A) hydrogen, B) nitrogen. PAGE 164 164 Figure 732. Temperature at solidgas interface for the hydrogen and nitrogen cases at time=0.4. Figure 733. Nusselt number dist ributions on the pipe wall for the hydrogen and nitrogen cases for both cases at time=0.4. PAGE 165 165 CHAPTER 8 SUMMARY AND FUTURE WORK 8.1 Summary In this research, a numerical package fo r handling phase change and chilldown for cryogenic twophase flow has been developed. The major contributions are the phase change computation which includes the energy equation solver, the phase change algorithm for the liquidgas interface, the conjugate heat transfer for solid and gas phase and the thermal radiation model. In Chapters 1 and 2, literature review is done and the problem is defined. In Chapter 3, the governing equations, the boundary conditions an d the interfacial conditions and their dimensionless forms are listed. In Chapter 4, th e main numerical techni ques applied in this research are introduced and explained. All the main techniques are validated in Chapter 5 and the validation includes the solver of governing equations, the SIMCC with moving in terfacial algorithm and the mass transfer computation. Chapters 6 and 7 form the core of this resear ch and four different cases are computed and discussed. In Chapter 6, it focuses on the cases without phase change and includes the isothermal and heat transfer twophase flow in a pipe. In Chapter 7, the phase change is included with different condition: the constant wall temperature and the wall with finite thickness. 8.2 Future Work In the current research, the preliminary tech niques are ready for the cryogenic twophase flow. Besides these techniques, there are still some interesting future possibilities for this research: PAGE 166 166 Three dimensional interfacial trac king technique and cutcells: The current SIMCC code can only handle two dimensional axisymmetric computations. This limits the current research to axisymmetric flows. In recent researches, some scholars have introduced the three dimensional interfacial tracking technique and cutcells (Cieslak et al. 200 1, Singh et al. 2005). However, currently, the researches can only handle very simple geometries and this should be a good direction for the future work. Turbulence model: The laminar flow is another limitation of the current code. The critical Reynolds number for laminar flow is aboutRe20003000 (White 1991). Once the Reynolds number becomes larger than 3000, the flow regime of pipe flow should be closer to turbulent flow and the turbulence m odel (Launder and Spalding 1974 ) is very important. Compressibility and variable fluid properties: In current code, the two phases are treated as incompressible fluids. This should be safe since the temper ature gradient in the temperature contour is not ve ry strong. However, for a more general situation, the compressibility and variable fluid properties due to the large temperat ure difference should be considered. Recently, some researchers have extended the pressurebased solver from incompressible to compressible flow (Kamakoti 2004). Adaptive grid or simple local grid refinement: In the current SIMCC code, the grid system is the Cartesian grid system. It can be st retched but still have to keep uniformity near the interface. 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"Press ure Drop in Gasliquid Flow at Microgravity Conditions." International Journal of Multiphase Flow, 21(5), 837849. PAGE 175 175 BIOGRAPHICAL SKETCH Chengfeng Tai was born in Changhua, Taiwan, in 1972. After receiving his Bachelor of Engineering degree in mechanic al engineering from National S un YatSen University in 1995, he received the Master of Engineering degr ee from in aerospace engineering from National Cheng Kung University in 1997. From 2001 he ha s been pursuing his Ph.D. degree in aerospace engineering at the University of Florida. His current research inte rests lie in computational fluid dynamics of multiphase flows and heat transfer. 