<%BANNER%>

Modeling Two-Phase Transport during Cryogenic Chilldown in a Pipeline


PAGE 1

MODELING TWO-PHASE TRANSPORT DURING CRYOGENIC CHILLDOWN IN A PIPELINE By JUN LIAO 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 2005

PAGE 2

Copyright 2005 by Jun Liao

PAGE 3

iii ACKNOWLEDGMENTS I would like to express my appreciation to all of the individuals who have assisted me in my educational development and in the completion of my dissertation. My greatest gratitude is extended to my supervisory committee chair, Dr. Renwei Mei. Dr. Meis excellent knowledge, boundless patience, constant encouragement, friendly demeanor, and professional expertise have been critical to both my research and education. Dr. James F. Klausner also deserves recognition for his knowledge and technical expertise. I would like to further thank Dr. Jacob N. Chung for kindly providing his experiment data of chilldown. I would like to additionally recognize my fellow graduate associates Christopher Velat, Jelliffe Jackson, Yusen Qi, and Yi Li for their friendship and technical assistance. Their diverse cultural background and character have provided an enlightening and positive environment. Special appreciation is given to Kun Yuan for his kindness providing his experiment data and insight on chilldown. I would like to further acknowledge the Hydrogen Research and Education Program for providing funding to this study. This research was also funded by NASA Glenn Research Center under contract NAG3-2750. Finally, I would like to recognize my wife Xiaohong Liao and my parents for their continual support and encouragement.

PAGE 4

iv TABLE OF CONTENTS page ACKNOWLEDGMENTS..................................................................................................iii LIST OF TABLES............................................................................................................vii LIST OF FIGURES..........................................................................................................viii NOMENCLATURE.........................................................................................................xiv ABSTRACT.....................................................................................................................xi x CHAPTER 1 INTRODUCTION.......................................................................................................1 1.1 Background............................................................................................................1 1.2 Literature Review...................................................................................................4 1.3 Scope....................................................................................................................10 2 TWO-PHASE FLOW MODELING AND FLOW BOILING HEAT TRANSFER OF CRYOGENIC FLUID................................................................................................13 2.1 Flow Regime and Heat Transfer Regime.............................................................13 2.2 Flow Models in Cryogenic Chilldown.................................................................18 2.2.1 Homogeneous Flow Model........................................................................18 2.2.2 Two-Fluid Model.......................................................................................22 2.3 Heat Transfer between Cryogenic Fluid and Solid Pipe Wall.............................26 2.3.1 Heat Transfer between Liquid and Solid wall...........................................27 2.3.1.1 Film boiling.....................................................................................27 2.3.1.2 Forced convection boiling and two-phase convective heat transfer30 2.3.2 Heat Transfer between Vapor and Solid Wall...........................................33 3 VAPOR BUBBLE GROWTH IN SATURATED BOILING...................................34 3.1 Introduction..........................................................................................................34 3.2 Formulation..........................................................................................................39 3.2.1 On the Vapor Bubble.................................................................................39 3.2.2 Microlayer..................................................................................................41 3.2.3 Solid Heater...............................................................................................42

PAGE 5

v 3.2.4 On the Bulk Liquid....................................................................................43 3.2.4.1 Velocity field...................................................................................43 3.2.4.2 Temperature field............................................................................44 3.2.4.3 Asymptotic analysis of the bulk liquid temperature field during early stages of growth...............................................................................46 3.2.5 Initial Conditions.......................................................................................50 3.2.6 Solution Procedure.....................................................................................52 3.3 Results and Discussions.......................................................................................52 3.3.1 Asymptotic Structure of Liquid Thermal Field.........................................52 3.3.2 Constant Heater Temperature Bubble Growth in the Experiment of Yaddanapudi and Kim...............................................................................56 3.3.3 Effect of Bulk Liquid Thermal Boundary Layer Thickness on Bubble Growth.............................................................................................................59 3.4 Conclusions..........................................................................................................63 4 ANALYSIS ON COMPUTATIONAL INSTABILITY IN SOLVING TWO-FLUID MODEL.....................................................................................................................64 4.1 Inviscid Two-Fluid Model...................................................................................65 4.1.1 Introduction................................................................................................65 4.1.2 Governing Equations.................................................................................67 4.1.3 Theoretical Analysis..................................................................................69 4.1.3.1 Characteristic analysis and ill-posedness........................................69 4.1.3.2 Inviscid Kelvin-Helmholtz (IKH) analysis and linear instability....72 4.1.4 Analysis on Computational Instability......................................................73 4.1.4.1 Description of numerical methods...................................................73 4.1.4.2 Code validation dam-break flow.................................................78 4.1.4.3 Von Neumann stability analysis for various convection schemes..81 4.1.4.4 Initial and boundary conditions for numerical solutions.................86 4.1.5 Results and Discussion..............................................................................87 4.1.5.1 Computational stability assessment based on von Neumann stability analysis.............................................................................................87 4.1.5.2 Scheme consistency tests.................................................................94 4.1.5.3 Computational assessment based on the growth of disturbance......95 4.1.5.4 Discussion on the growth of short wave........................................101 4.1.5.5 Wave development resulting from disturbance at inlet.................104 4.1.6 Conclusions..............................................................................................106 4.2 Viscous Two-Fluid Model.................................................................................110 4.2.1 Introduction..............................................................................................110 4.2.2 Governing Equations...............................................................................111 4.2.3 Theoretical Analysis................................................................................112 4.2.3.1 Characteristics and ill-posedness...................................................112 4.2.3.2 Viscous Kelvin-Helmholtz (VKH) analysis and linear instability113 4.2.4 Analysis on Computational Intability......................................................115 4.2.4.1 Description of numerical methods.................................................115 4.2.4.2 Von Neumann stability analysis for various convection schemes116 4.2.4.3 Initial and boundary conditions for numerical solution.................119

PAGE 6

vi 4.2.5 Results and Discussion............................................................................119 4.2.5.1 Computational stability assessment based on von Neumann stability analysis...........................................................................................119 4.2.5.2 Computational assessment based on the growth of disturbance....126 4.2.5.3 Wave development resulting from disturbance at inlet.................128 4.2.6 Conclusions..............................................................................................130 5 MODELING CRYOGENIC CHILLDOWN...........................................................133 5.1 Homogeneous Chilldown Model.......................................................................133 5.1.1 Analysis...................................................................................................134 5.1.2 Results and Discussion............................................................................136 5.2 Pseudo-Steady Chilldown Model.......................................................................140 5.2.1 Formulation..............................................................................................141 5.2.1.1 Heat conduction in solid pipe........................................................141 5.2.1.2 Liquid and vapor flow...................................................................144 5.2.1.3 Film boiling correlation.................................................................145 5.2.1.4 Forced convection boiling correlation...........................................151 5.2.1.5 Heat transfer between solid wall and environment........................152 5.2.2 Results and Discussion............................................................................155 5.2.2.1 Experiment of Chung et al.............................................................156 5.2.2.2 Comparison of pipe wall temperature...........................................157 5.2.3 Discussion and Remarks..........................................................................163 5.2.4 Conclusions..............................................................................................166 5.3 Separated Flow Chilldown Model.....................................................................167 5.3.1 Formulation..............................................................................................167 5.3.1.1 Fluid flow......................................................................................168 5.3.1.2 Heat conduction in solid pipe........................................................168 5.3.1.3 Heat and mass transfer...................................................................169 5.3.1.3 Initial and boundary conditions.....................................................172 5.3.2 Solution Procedure...................................................................................173 5.3.3 Results and Discussion............................................................................174 5.3.3.1 Comparison of solid wall temperature...........................................177 5.3.3.2 Flow field and fluid temperature...................................................181 5.3.4 Conclusions..............................................................................................186 6 CONCLUSIONS AND DISCUSSION...................................................................187 6.1 Conclusions........................................................................................................187 6.2 Suggested Future Study.....................................................................................188 LIST OF REFERENCES................................................................................................190 BIOGRAPHICAL SKETCH..........................................................................................198

PAGE 7

vii LIST OF TABLES Table page 4-1. Analytical solution for dam-break flow....................................................................80 4-2. for different discretization schemes..................................................................85 5-1. Heat and mass transfer relationship used in separated flow chilldown model........173

PAGE 8

viii LIST OF FIGURES Figure page 1-1. Schematic of filling facilities for LH2 transport system from storage tank to space shuttle external tank...................................................................................................3 1-2. The schematic of chilldown and heat transfer regime.................................................9 2-1. Schematic of two-phase flow regime in horizontal pipe...........................................14 2-2. Schematic of two-phase flow regime in vertical pipe................................................14 2-4. Typical wall temperature variation during chilldown................................................17 2-5. Schematic for homogeneous flow model...................................................................19 2-6. Schematic of the two-fluid model..............................................................................22 2-7. Schematic of heat transfer in chilldown.....................................................................27 3-1. Sketch for the growing bubble, thermal boundary layer, microlayer and the heater wall..........................................................................................................................3 9 3-2. Coordinate system for the background bulk liquid....................................................43 3-3. A typical grid distribution for the bulk liquid thermal field with 65 0 RS, 73 0 S, and 10 bR ...........................................................................................46 3-4. Comparison of the asymptotic and the numerical solutions at t =0.001, 0.01, 0.1 and 0.3 for =0 n 40 n and 71 n ......................................................................................54 3-5. Effect of parameter A on the liquid temperature profile near bubble........................55 3-6. The computed isotherms near a growing bubble in saturated liquid at t=0.01, t=0.1,t=0.3, and t=0.9............................................................................................57 3-7. Comparison of the equivalent bubble diameter eqdfor the experimental data of Yaddanapudi and Kim (2001) and that computed for heat transfer through the microlayer (1c=3.0).................................................................................................58

PAGE 9

ix 3-8. Comparison of bubble diameter, d(t), between that computed using the present model and the measured data of Yaddanapudi and Kim (2001).............................60 3-9. Comparison between heat transfer to the bubble through the vapor dome and that through the microlayer.............................................................................................60 3-10. The computed isotherms in the bulk liquid corresponding to the thermal conditions reported by Yaddanapudi and Kim (2001)..............................................................61 3-11. Effect of bulk liquid thermal boundary layer thickness on bubble growth...........62 4-1. Schematic of two-fluid model for pipe flow..............................................................68 4-2. Staggered grid arrangement in two-fluid model........................................................74 4-3. Flow chart of pressure correction scheme for two-fluid model.................................78 4-4. Schematic for dam-break flow model........................................................................79 4-5. Water depth at t=50 seconds after dam break............................................................80 4-6. Water velocity at t=50 seconds after dam break........................................................81 4-7. Grid index number in staggered grid for von Neumann stability analysis................81 4-8. Comparisons of growth rates of various numerical schemes.200 N, 5 0 la, s m ul/ 1 s m ug/ 17 and 1 0 lCFL...............................................................89 4-9. Growth rate of CDS scheme at different l gu u U f 200 N, 5 0 la, s m ul/ 1 and 1 0 lCFL....................................................................................90 4-10. Growth rate of FOU scheme at different l gu u U f 200 N, 5 0 la, s m ul/ 1 and 1 0 lCFL....................................................................................90 4-11. Growth rate of CDS scheme at different lu.200 N, s m U / 16 5 0 la, and m s x t / 1 0 .......................................................................................................92 4-12. Growth rate of FOU scheme at different lu 200 N, s m U / 16 5 0la and m s x t / 1 0 .......................................................................................................92 4-13. Growth rate of CDS scheme at different x t 200 N, s m ul/ 1, s m U / 16 and 5 0la ....................................................................................93

PAGE 10

x 4-14. Growth rate of FOU scheme at different x t .200 N, s m ul/ 1, s m U / 16 and 5 0la ....................................................................................93 4-15. Comparison of lu growth using CDS scheme on different grids. s m ul/ 1, s m ug/ 5 17 1 0lCFL and 5 0la ..............................................................95 4-16. lu using CDS scheme in the computational domain. 200 N, s m ul/ 1, s m U / 14 05 0lCFL 5 0la and s t 4 .................................................97 4-17. Amplitude of liquid velocity disturbance lu using CDS scheme. 200 N, s m ul/ 1, s m U / 14 05 0lCFL 5 0la and s t 4 ..............................97 4-18. lu using CDS scheme after 10399 steps of computation,200 N, s m ul/ 1, s m U / 5 16 1 0lCFL 5 0la and s t 2 5 .............................................98 4-19. Growth history of lu solved using CDS scheme,200 N, s m ul/ 1, s m U / 5 16 1 0lCFL 5 0la and s t 2 5 .............................................98 4-20. Growth rate of FOU scheme,200 N, s m ul/ 5 0, s m U / 16 02 0lCFL and 5 0la ..........................................................................................................100 4-21. lu using FOU scheme after 12000 steps of computation. 200 N, s m ul/ 5 0,s m U / 16 02 0lCFL and 5 0la .....................................102 4-22. Growth rate of SOU scheme. 200 N, s m ul/ 1, s m U / 16 05 0lCFL and 5 0la ..........................................................................................................103 4-23. lu using SOU scheme after 3000 steps of computation. 200 N, s m ul/ 1, s m U / 16 05 0lCFL and 5 0la ...........................................................103 4-24. Growth history of lu under different initial amplitude using FOU scheme...........104 4-25. lu propagates in the pipe with FOU at well-posed condition, quasi-steady state.107 4-26. lu propagates in the pipe with FOU scheme at ill-posed condition, quasi-steady state........................................................................................................................10 7 4-27. lu propagates in the pipe with CDS at well-posed condition, quasi-steady state.108 4-28. lu propagates in the pipe with CDS at ill-posed condition, an instance before the computation breaks down......................................................................................108

PAGE 11

xi 4-29. Comparison of growth rate between CDS and FOU schemes. 200 N, s m ul/ 1, s m ug/ 21 05 0lCFL and 5 0la .............................................................109 4-30. Schematic depiction of viscous two-fluid model...................................................111 4-31. Comparisons of growth rate of different schemes. 200 N, s m uls/ 3 0, s m ugs/ 6 ,and 1 0lCFL .................................................................................120 4-32. Comparisons of growth rate of different schemes at low k 200 N, s m uls/ 3 0, s m ugs/ 6 ,and 1 0lCFL .................................................................................121 4-33. Growth rate for CDS scheme with VKH unstable. 200 N, s m uls/ 3 0, s m ugs/ 6 ,and 1 0lCFL .................................................................................122 4-34. Growth rate for CDS scheme with VKH instability. s Pawater* 102fr, 200 N, s m uls/ 3 0, s m ugs/ 6 ,and 1 0lCFL .........................................................123 4-35. Growth rates for CDS scheme with VKH instability. s Pawater* 101fr, 200 N, s m uls/ 1 0, s m ugs/ 2 ,and 01 0lCFL .......................................................124 4-36. Growth rates for FOU scheme with VKH instability. 200 N, s m uls/ 3 0, s m ugs/ 6 and 1 0lCFL ................................................................................125 4-37. Growth rates for FOU scheme with VKH instability. s Pa ewater* 1 1f r, 200 N, s m uls/ 1 0, s m ugs/ 2 and 01 0lCFL .....................................125 4-38. Growth history of lu using CDS scheme. 200 N, s m ul/ 2, s m ug/ 0.998174 -0.0617144 98 0la and 05 0lCFL ...................127 4-39. Growth history of lu using FOU scheme. 200 N, s m ul/ 2, s m ug/ 0.998174 -0.0617144 98 0la and 05 0lCFL ...................128 4-40. Disturbance of lu propagates in the pipe with FOU and CDS schemes at VKH unstable and well-posed condition........................................................................129 4-41. Disturbance of lu propagates in the pipe with FOU and CDS schemes at both VKH unstable and well-posed condition........................................................................130 5-1. Schematic of homogeneous chilldown model.........................................................134 5-2. Schematic for evaluating film boiling wall friction.................................................135

PAGE 12

xii 5-3. Distribution of vapor quality based on the homogenous flow model......................137 5-4. Pressure distribution based on the homogenous flow model...................................138 5-5. Velocity distribution based on the homogenous flow model...................................139 5-6. Solid temperature contour based on homogenous flow model................................139 5-7. Schematic of cryogenic liquid flow inside a pipe....................................................141 5-8. Coordinate systems: laboratory frame is denoted using z; moving frame is denoted using Z ...................................................................................................................142 5-9. Schematic diagram of film boiling at stratified flow...............................................145 5-10. Numerical solution of the vapor thickness and velocity influence functions........150 5-11. Numerical solution of 0G .................................................................................151 5-12. Schematic of vacuum insulation chamber.............................................................153 5-13. Schematic of Yuan and Chung (2004)s cryogenic two-phase flow test apparatus.157 5-14. Experimental visual observation of Chung et al. (2004)s cryogenic two-phase flow experiment.............................................................................................................158 5-15. Computational grid arrangement and positions of thermocouples........................159 5-16. Comparison between measured and predicted transient wall temperatures of positions 12 and 15 at the bottom of pipe during film boiling chilldown.............160 5-17. Comparison between measured and predicted transient wall temperatures of positions 12 and 15 at the bottom of pipe during convection boiling chilldown..161 5-18. Comparison between measured and predicted transient wall temperatures of positions 12 and 15 is at the bottom of pipe during entire chilldown...................161 5-19. Comparison between measured and predicted transient wall temperatures of positions 11 and 14, which is at the bottom of pipe during entire chilldown........162 5-20. Cross section wall temperature distribution at t=0, 50, 100 and 300 seconds.......162 5-21. Computed wall temperature contour on the inner surface of inner pipe................163 5-22. Comparison between measured and predicted transient wall temperatures of positions 12 and 15 with Chen correlation (1966)................................................165 5-23. Schematic of separated flow chilldown model......................................................168

PAGE 13

xiii 5-24. Schematic of heat and mass transfer in separated flow chilldown model.............169 5-25. Flow chart of separated flow chilldown model......................................................175 5-26.Geometry of the test section and locations of thermocouples................................176 5-27. Comparison between measured and predicted transient wall temperatures of positions 12 and 15................................................................................................178 5-28. Comparison between measured and predicted transient wall temperatures of positions 11 and 14................................................................................................178 5-29. Comparison between measured and predicted transient wall temperatures of position 6 and 9 (the measured T 9 is obviously incorrect)...................................179 5-30. Comparison between measured and predicted transient wall temperatures of positions 5 and 8....................................................................................................179 5-31. Comparison between measured and predicted transient wall temperatures of position 3 and the numerical result of temperature at position 4...........................180 5-32. Comparison between measured and predicted transient wall temperatures of positions 1 and 2....................................................................................................180 5-33. Liquid nitrogen depth in the pipe during the chilldown........................................182 5-34. Vapor nitrogen velocity in the pipe during the chilldown.....................................183 5-35. Liquid nitrogen velocity in the pipe during the chilldown....................................184 5-36. Vapor nitrogen temperature in the pipe during the chilldown...............................185 5-37. Liquid nitrogen temperature in the pipe during the chilldown..............................185

PAGE 14

xiv NOMENCLATURE A dimensionless parameter for bubble growth, cross section area, surface area Ab area of vapor bubble dome exposed to bulk liquid Am area of wedge shaped interface Bo Boiling number c ratio of wedge shaped interface radius and vapor bubble radius, wave speed 1c microlayer wedge angle parameter; empirically determined CFL Courant number D diameter of pipe lD and gD liquid layer and gas layer hydraulic diameter d bubble diameter eqd equivalent bubble diameter E common amplitude factor f friction factor lof friction factor for liquid phase in homogeneous model G mass flux, amplification factor g gravity lH and gH liquid layer and gas layer hydraulic depth lh and gh liquid layer and gas layer depth

PAGE 15

xv h heat transfer coefficient FBh film boiling heat transfer coefficient poolh pool boiling heat transfer coefficient c lh, and c gh, forced convection heat transfer coefficient for liquid and gas fgh latent heat of vaporization I imaginary unit, 1f i enthalpy Ja Jacob number k thermal conductivity, wavenumber effk effective thermal conductivity L local microlayer thickness, characteristic length Nu Nusselt number m mass transfer rate between liquid and gas per unit length n normal direction p pressure 0p pressure in the liquid-vapor interface Pc Peclet number Pr Prandtl number q heat transfer rate per unit length radq radiation heat flux frcq free convection heat flux wq Heat flux from wall to fluid

PAGE 16

xvi R vapor bubble radius, pipe radius R bubble growth rate R and spherical coordinates R dimensionless radial coordinate Rb radius of wedge shaped interface 0R initial bubble radius Ra Rayleigh number Re Reynolds number r radial coordinate S suppression factor in flow nucleate boiling, perimeter RSandS stretching factor in computation T temperature satT saturated temperature wT initial solid temperature bT bulk liquid temperature t time ct characteristic time wt waiting period 0t initial time U and V averaged velocities u and v velocities u mean u velocity

PAGE 17

xvii Vb vapor bubble volume x y and z Cartesian coordinates z r and cylindrical coordinates X boundary layer coordinate Z coordinate in the direction normal to the heating surface Greek symbols thermal diffusivity, volume fraction volumetric thermal expansion coefficient tt Martinelli number T solid wall superheat superheated bulk liquid thermal boundary layer thickness, vapor film thickness dimensionless thickness of unsteady thermal boundary layer emissivity, amplitude velocity potential function for liquid flow, general variable microlayer wedge angle, azimuthal coordinate, phase angle lo friction multiplier and computational coordinates characteristic root of a matrix kinematic viscosity dimensionless temperature, azimuthal coordinate, pipe incline angle 0 initial dimensionless temperature of liquid density

PAGE 18

xviii stretched time in computation, Stefan Boltzmann constant t dimensionless time, shear stress FBt wall shear stress in film boiling regime superscripts in inner solution out outer solution quantity per unit length quantity per unit area subscripts b bubble FB film boiling eva evaporation l g and i liquid, gas, and interface i and o inner and outer pipe l liquid ml microlayer NB nucleate boiling w wall v vapor b far field condition

PAGE 19

xix 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 MODELING TWO-PHASE TRANSPORT DURING CRYOGENIC CHILLDOWN IN A PIPELINE By Jun Liao August 2005 Chair: Renwei Mei Major Department: Mechanical and Aerospace Engineering Cryogenic chilldown process is a complicated interaction process among liquid, vapor and solid pipe wall. To model the chilldown process, results from recent experimental studies on the chilldown and existing cryogenic heat transfer correlations were reviewed together with the homogeneous flow model and the two-fluid model. A new physical model on the bubble growth in nucleate boiling was developed to correctly predict the early stage bubble growth in saturated heterogeneous nucleate boiling. A pressure correction algorithm for two-fluid model was carefully implemented to solve the two-fluid model used to model the chilldown process. The connections between the numerical stability and ill-posedness of the two-fluid model and between the numerical stability and viscous Kelvin-Helmholtz instability were elucidated using von Neumann stability analysis. A new film boiling correlation and a modified nucleate boiling correlation for chilldown inside pipes were developed to provide heat transfer correlation for chilldown model. Three chilldown models were developed. The homogeneous

PAGE 20

xx chilldown model is for simulating chilldown in a vertical pipe. A pseudo-steady chilldown model was developed to simulate horizontal chilldown. The pseudo-steady chilldown model can capture the essential part of chilldown process, provides a good testing platform for validating cryogenic heat transfer correlations based on experimental measurement of wall temperature during chilldown and gives a reasonable description of the chilldown process in a frame moving with the liquid-vapor wave front. A more comprehensive separated flow chilldown model was developed to predict both the flow field and solid wall temperature field in horizontal stratified flow during chilldown. The predicted wall temperature variation matches well with the experimental measurement. It provides valuable insights into the two-phase flow dynamics, and heat and mass transfer for a given spatial region in the pipe during the chilldown.

PAGE 21

1 CHAPTER 1 INTRODUCTION One of the key issues in the efficient utilization of cryogenic fluids is the transport, handling, and storage of the cryogenic fluids. The complexity of the problems results from, in general, the intricate interaction of the fluid dynamics and the boiling heat transfer. Chilldown of the pipeline for transport cryogenic fluid is a typical example. It involves unsteady two-phase fluid dynamics and highly transitory boiling heat transfer. There is very little insight into the dynamic process of chilldown. This study will focus on the understanding and modeling of the unsteady fluid dynamics and heat transfer of the cryogenic fluids in a pipeline that is exposed to the atmospheric condition. 1.1 Background Presently there exists considerable interest among U.S. Federal agencies in driving the U.S. energy infrastructure with hydrogen as the primary energy carrier. The motivation for doing so is that hydrogen may be produced using all other energy sources, and thus using hydrogen as an energy carrier medium has the potential to provide a robust and secure energy supply that is less sensitive to world fluctuations in the supply of fossil fuels. The vision of building an energy infrastructure that uses hydrogen as an energy carrier is generally referred to as the "hydrogen economy," and is considered the most likely path toward widespread commercialization of hydrogen based technologies. Hydrogen has the distinct advantage as fuel in that it has the highest energy density of any fuel currently under consideration, 120 MJ/kg. In contrast, the energy density of gasoline, which is considered relatively high, is approximately 44 MJ/kg. When

PAGE 22

2 launching spacecraft, the energy density is a primary factor in fuel selection. When considering liquid hydrogen to propel advanced aircraft turbo engines, it is a very attractive option due to hydrogens high energy density. One drawback with using liquid hydrogen as a fuel is that its volumetric energy capacity, 8.4 MJ/liter is about one quarter that of gasoline, 33 MJ/liter. Therefore, liquid hydrogen requires more volumetric storage capacity for a fixed amount of energy. Nevertheless, liquid hydrogen is a leading contender as a fuel for both ground-based vehicles and for aircraft propulsion in the hydrogen economy. When any cryogenic system is initially started, (this includes turbo engines, reciprocating engines, pumps, valves, and pipelines), it must go through a transient chilldown period prior to 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 chilldown a cryogenic system in a safe and efficient manner. A perfect example of utilization and chilldown cryogenic system exists in NASAs Kennedy Space Center (KSC). In the preparation for a space shuttle launch, liquid hydrogen (as fuel) is filled from a storage tank to the main liquid hydrogen (LH2) external tank (ET) through a complex pipeline system (Figure 1-1). The filling procedure consists of 5 steps: Facility and orbiter chilldown. Fill transition and initial fill (fill ET to 2%). Fast fill ET (to 98%). Fill ET (to 100%). Replenish (maintain ET 100%).

PAGE 23

3 Figure 1-1. Schematic of filling facilities for LH2 transport system from storage tank to space shuttle external tank. While the engineers have a general understanding of the process in the initial fill and rapid fill stages, there has been very little insight about the process of chilldown, which is the first procedure to be initiated. There is not a single formula or computer code that can be used to estimate the elapse time during the chilldown stage if certain operating condition changes. The absence of guidelines stems from our lack of fundamental knowledge in the area of cryogenic chilldown. Many such engineering issues are present in the transport, handling, and storage of cryogenic liquid in industry applications.

PAGE 24

4 1.2 Literature Review Experimental studies: Studies on cryogenic chilldown started in the 1960s with the development of rocket launching systems. Early experimental chilldown studies started in the 1960s by Burke et al. (1960), Graham (1961), Bronson et al. (1962), Chi and Vetere (1963), Steward (1970) and other researches. Burke et al. (1960) and Graham (1961) experimentally studied the cryogenic chilldown in a horizontal pipe and in a vertical pipe, respectively. However, none of these studies provided the flow regime information in chilldown. Bronson et al. (1962) visually studied the flow regimes in a horizontal pipe during chilldown with liquid hydrogen as the coolant. The results revealed that the stratified flow is prevalent during the cryogenic chilldown. Flow regimes and heat transfer regimes in the horizontal pipe chilldown were also studied by Chi and Vetere (1963). Information on flow regimes was deduced by studying the fluid temperature and the volume fraction during chilldown. Several flow regimes were identified: single-phase vapor, mist flow, slug flow, annular flow, bubbly flow, and single-phase liquid flow. Heat transfer regimes were identified as single-phase vapor convection, film boiling, nucleate boiling, and single-phase liquid convection. Recently, Velat et al. (2004) systematically studied cryogenic chilldown with nitrogen in a horizontal pipe. Their study included: a visual recording of the chilldown process in a transparent Pyrex pipe, which is used to identify the flow regime and heat transfer regime; collecting temperature histories at different positions of the wall in chilldown; and recording the pressure drop along the pipe. Chung et al. (2004) conducted a similar study with nitrogen chilldown at relatively low mass flux and provided the data needed to assess various heat transfer coefficients in the present study.

PAGE 25

5 Modeling efforts: Burke et al. (1960) developed a crude chilldown model based on 1-D heat transfer through the pipe wall and the assumption of infinite heat transfer rate from the cryogenic fluid to the pipe wall. The effects of flow regimes on the heat transfer rate were neglected. Graham et al. (1961) correlated the heat transfer coefficient and pressure drop with the Martinelli number (Martinelli and Nelson, 1948) based on their experimental data. Chi (1965) developed a one-dimensional model for energy equations of the liquid and the wall, based on the film boiling heat transfer between the wall and the fluid. An empirical equation for predicting the chilldown time and the temperature was proposed. Steward (1970) developed a homogeneous flow model for cryogenic chilldown. The model treated the cryogenic fluid as a homogeneous mixture. The continuity, momentum and energy equations of the mixture were solved to obtain density, pressure and temperature of mixture. Various heat transfer regimes were considered: film boiling, nucleate boiling, and single-phase convection heat transfer. Careful treatment of different heat transfer regimes resulted in a significant improvement in the prediction of the chilldown time. The homogeneous mixture model was also employed by Cross et al. (2002) who obtained a correlation for the wall temperature during chilldown with an oversimplified treatment of the heat transfer between the wall and the fluid. Similar efforts have been devoted to the study of the re-wetting problem, referred to as cooling down of a hot object. Thompson (1972) analyzed the re-wetting of a hot dry rod. The two-dimensional temperature profile inside the solid rod was numerically calculated. The nucleate boiling heat transfer coefficient between the solid rod and the

PAGE 26

6 liquid was simplified to a power law relation and the heat transfer in the film boiling stage is neglected. The liquid temperature and velocity outside the rod are assumed to be constant. Sun et al. (1974), and Tien and Yao (1975) solved similar problems and obtained an analytical solution for the re-wetting. They considered different heat transfer coefficients for flow boiling and single-phase convection in order to obtain more accurate results for re-wetting problems. In those works the thermal field of the liquid is neglected and the heat transfer coefficients at the boiling and the convection heat transfer stage are over-simplified, and the results are only valid for the vertical outer surface of a rod or a tube. Chilldown in stratified flow regime, which is the prevalent in the horizontal pipeline, was first studied by Chan and Banerjee (1981 a, b, c). They developed a comprehensive separated flow model for the cool-down in a hot horizontal pipe. Both phases were modeled with one-dimensional mass and momentum conservation equations. The vapor and liquid phase mass and momentum equations were reduced to two wave equations for the liquid depth and the velocity of the liquid. The energy equation for the liquid was used to find the liquid temperature and energy equation of vapor phase was neglected. The wall temperature was computed using a 2-dimensional transient heat conduction equation and heat transfer in the radial direction was neglected. They also tried to evaluate the position of onset of re-wetting by studying the instability of film boiling. Their prediction for the wall temperature agreed well with their experimental results. Although significant progress was made in handling the momentum equations, the heat transfer correlations employed were not as advanced.

PAGE 27

7 Following Chan and Banerjees (1981 a, b, c) separated flow model, Hedayatpour et al. (1993) studied the cool-down in a vertical pipe with a modified separated flow model. The flow regime is inverted annular film boiling flow, where the liquid core is inside and the vapor film separates the cold liquid and the hot wall. This regime frequently exists in cool-down in a vertical pipe. The modified separated flow model retains the transient terms in the vapor momentum equation and the vapor phase energy equation. The procedure is the following: first, the liquid mass conservation equation is solved to obtain the liquid and vapor volume fractions. Then the vapor mass conservation equation is used to solve the vapor velocity. The vapor momentum equation is subsequently solved to obtain the vapor pressure. Finally, the liquid momentum equation is employed to find the liquid velocity. The iteration stops when the solution is converged. Although Chan and Banerjee (1981 a, b, c) and Hedayatpour et al. (1993) were successful in the simulation of chilldown with the separated flow model, their separated flow model is either incomplete or computationally inefficient. c) Issues related to two-fluid model The separated flow model is also called the two-fluid model, which consists of two sets of conservation equations for the mass, momentum and energy of liquid and gas phases. It was proposed by Wallis (1969), and further refined by Ishii (1975). Although the two-fluid model is recognized as a useful computational model to simulate the stratified multiphase flow in the pipeline, its application to the study of heat transfer in two-phase flow in the pipeline is still limited. The numerical scheme for the two-fluid model can be classified into two categories. One is the compressible two-fluid model, which can be solved by a hyperbolic

PAGE 28

8 equation solver. Examples are the commercial code OLGA (Bendikson et al., 1991), Pipeline Analysis Code (PLAC) (Black et al., 1990) and Lyczkowski et al. (1978). The other is the incompressible two-fluid model. Since the hyperbolic equation solver is not applicable to incompressible two-fluid model, several approaches for incompressible two-fluid model have emerged. One approach is to reduce the gas and liquid mass and momentum equations to two wave equations for the liquid depth and velocity, such as in Barnea and Taitel (1994b) and Chan and Banerjee (1991b). This treatment changed the properties of two-fluid model. Hedayatpour et al. (1993) approach to two-fluid model is not widely used due to lack of theoretical analysis on the convergence. Another approach is to use the pressure correction method, which was initially introduced by Issa and Woodburn (1998) and Issa and Kempf (2003) for the compressible two-fluid model. Although their pressure correction scheme is powerful for simulating the multiphase flow in the pipeline, the accuracy of the scheme is not reported. At the present, application of pressure correction scheme on the multiphase flow with heat transfer in pipeline, such as chilldown, does not exist. d) Heat transfer in chilldown A typical chilldown process involves several heat transfer regimes as shown in Figure1-2. Near the liquid front is the film boiling regime. The knowledge of the heat transfer in the film boiling regime is relatively limited, because i) film boiling has not been the central interest in industrial applications; and ii) high temperature difference causes difficulties in experimental investigations. For the film boiling on vertical surfaces, early work was reported by Bromley (1950), Dougall and Rohsenow (1963) and Laverty and Rohsenow (1967). Film boiling in a horizontal cylinder was first studied by

PAGE 29

9 Bromley (1950); and the Bromley correlation was widely used. Breen and Westwater (1962) modified Bromleys equation to account for very small tubes and large tubes. If the tube is larger than the wavelength associated with Taylor instability, the heat transfer correlation is reduced to Berensons correlation (1961) for a horizontal surface. Film boiling Cryogenic Liquid Liquid Front Wall Nucleation b oilin g Convective heat transfe r Vapor X Y Figure 1-2. The schematic of chilldown and heat transfer regime. Empirical correlations for cryogenic film boiling were proposed by Hendrick et al. (1961, 1966), Ellerbrock et al. (1962), von Glahn (1964), Giarratano and Smith (1965). These correlations relate a simple or modified Nusselt number ratio to the Martinelli parameter. Giarratano and Smith (1965) gave detailed assessment of these correlations. All these correlations are for steady state cryogenic film boiling. Their suitability for transient chilldown applications is questionable. When the pipe wall chills down further, film boiling ceases and nucleate boiling occurs. It is usually assumed that the boiling switches from film boiling to nucleate boiling right away instead of passing through a transition boiling regime. The position of the film boiling transitioning to the nucleate boiling is often called re-wetting front, because from that position the cold liquid starts touching the pipe wall. Usually the Leidenfrost temperature indicates the transition from film boiling to nucleate boiling.

PAGE 30

10 However, the Leidenfrost temperature is not steady, and varies under different flow and thermal conditions (Bell, 1967). A recent approach is to check the instability of the vapor film beneath the liquid core using Kelvin Helmholtz instability analysis (Chan and Banerjee, 1981c). Studies on forced convection boiling are extensive (Giarratano and Smith, 1965; Chen, 1966; Bennett and Chen, 1980; Stephan and Auracher, 1981; Gungor and Winterton, 1996; Zurcher et al., 2002). A general correlation for saturated boiling was introduced by Chen (1966). Gungor and Winterton (1996) modified Chens correlation and extended it to subcooled boiling. Enhancement and suppression factors for macro-convective heat transfer were introduced. Gunger and Wintertons correlation can fit experimental data better than the modified Chens correlation (Bennett and Chen, 1980) and Stephan and Auracher correlation (1981). Recently, Zurcher et al. (2002) proposed a flow pattern dependent flow boiling heat transfer correlation. This approach improves the overall accuracy of heat transfer correlation by incorporating flow pattern. Kutateladze (1952) and Steiner (1986) also provided correlations for cryogenic fluids in pool boiling and forced convection boiling. Although they are not widely used, they are expected to be more applicable for cryogenic fluids since the correlation was directly obtained from cryogenic conditions. As the wall temperature drops further, boiling is suppressed and the heat transfer is governed by two-phase convection; this is much easier to deal with. 1.3 Scope This dissertation focuses on understanding the unsteady fluid dynamics and heat transfer of cryogenic fluids in a pipeline that is exposed to the atmospheric condition and the corresponding solid heat transfer in the pipeline wall. Proper models for chilldown

PAGE 31

11 simulation are developed to predict the flow fields, thermal fields, and residence time during chilldown. In Chapter 2, visualized experimental studies on heat transfer regimes and flow regimes in cryogenic chilldown are reviewed. Based on the experimental observation, homogeneous and separated flow models for the respectively vertical pipe and horizontal pipe are discussed. The heat transfer models for the film boiling, flow boiling and forced convection heat transfer in chilldown are reviewed and qualitatively assessed. In Chapter 3, a physical model for vapor bubble growth in saturated nucleate boiling is developed that includes both heat transfer through the liquid microlayer and that from the bulk superheated liquid surrounding the bubble. Both asymptotic and numerical solutions reveal the existence of a thin unsteady thermal boundary layer adjacent to the bubble dome. In Chapter 4, a pressure correction algorithm for two-fluid model is developed and carefully implemented. Numerical stability of various convection schemes for both the inviscid and viscous two-fluid model is analyzed. The connections between ill-posedness of the two-fluid model and the numerical stability and between the viscous Kelvin-Helmholtz instability and numerical stability are elucidated. The computational accuracy of the numerical schemes is assessed. In Chapter 5, a new film boiling coefficient is developed to accurately predict film boiling heat flux for flow inside a pipe. The film boiling coefficient with the other investigated heat transfer models are applied in building chilldown models. A pseudo-steady chilldown model is developed to predict the chilldown time and the wall temperature variation in a horizontal pipe in a reference frame that moves with the liquid

PAGE 32

12 wave front. It is of low computational cost and allows for simple validation of the new film boiling heat transfer correlations. A more comprehensive separated flow chilldown model for the horizontal pipe is developed to predict the flow field of the liquid and vapor and the temperature fields of the liquid, vapor and the solid wall in a fixed region of the pipe flow. The unsteady development of the chilldown process for the vapor volume fraction, velocities of the two-phases, and the temperatures of fluids and wall are elucidated. Chapter 6 concludes the research with a summary of the overall work and discussion of the future works.

PAGE 33

13 CHAPTER 2 TWO-PHASE FLOW MODELING AND FLOW BOILING HEAT TRANSFER OF CRYOGENIC FLUID Information of heat transfer regimes and flow regimes in cryogenic chilldown obtained from the experimental study provides the foundation for modeling the heat transfer and multiphase flow in chilldown. Based on the information of flow regimes, corresponding flow models for simulating chilldown are discussed. For chilldown in the vertical pipe, homogeneous flow model is preferred due to the prevalence of homogeneous flow. For chilldown in the horizontal pipe, because the stratified flow is prevalent, two-fluid model is adopted. The heat transfer models for film boiling, flow boiling and forced convection heat transfer in chilldown are reviewed and qualitatively assessed. 2.1 Flow Regime and Heat Transfer Regime In the study of chilldown, one of the most important aspects of analysis is to determine the type of flow regime in the given region of the pipe. The flow in cryogenic chilldown is typically a two-phase flow, because liquid evaporates after a significant amount of heat is transferred from the wall to the fluid during chilldown. The two-phase flow regime is determined by many factors, such as fluid velocity, fluid density, vapor quality, gravity, and pipe size. For horizontal flow, the flow regime is visually classified as bubbly flow, plug flow, stratified flow, wavy flow, slug flow, and annular flow, as shown in Figure 2-1. For vertical flow, the flow regimes include bubbly flow, slug flow, churn flow, annular flow, as shown in Figure 2-2.

PAGE 34

14 Figure 2-1. Schematic of two-phase flow regime in horizontal pipe. Bubble Flow Slug Flow Churn Flow Annual Flow Figure 2-2. Schematic of two-phase flow regime in vertical pipe.

PAGE 35

15 The cryogenic two-phase flow is characterized by low viscosity, small density ratio of the liquid to the vapor, low latent heat of vaporization, and large wall superheat. For example, the liquid viscosity, density ratio latent heat of saturated liquid to vapor, and latent heat of the saturated water at 1 atm are 2.73E-4 Pa*s, 1610, 2256.8kJ/kg, respectively, while the corresponding data for saturated hydrogen at 1 atm are1.36E5Pa*s, 37.9, 444kJ/kg. Furthermore, film boiling, which is prevalent during chilldown, causes low wall friction. These factors combined with the complex interaction between the momentum and the thermal transportation make the two-phase flow during the chilldown to distinguish itself from ordinary two-phase flows. In the visualized horizontal chilldown experiment by Velat et al. (2004), as shown in Figure 2-3, the pressure in the liquid nitrogen Dewar drives the fluid. When the liquid nitrogen first enters the test section, a film boiling front is positioned at the inlet of test section. This film boiling front produces a significant evaporation accompanied by a high velocity vapor front traversing down the test section. If the mixture velocity is high enough due to the large pressure drop between the Dewar and the outlet of the test section, a very fine mist of liquid is entrained in the vapor flow. Immediately behind the film boiling front is a liquid layer attached to the wall. The flow regime is either the stratified flow or annular flow, depending on the flow speed, the pipe size, and the fluid properties. If the mixture velocity is high, the flow likely appears as annular flow, otherwise stratified flow or wavy flow is more common. The visual observation shows that the liquid droplets being entrained in stratified flow and wavy flow is insignificant. The nucleate boiling front follows the film boiling, indicating the end of film boiling and the cryogenic liquid starts contacting the wall. The position where the liquid

PAGE 36

16 starts contacting the wall is affected by the wall super heat, the liquid layer velocity and the thickness of the liquid layer. It is a complex hydrodynamic and heat transfer phenomenon. Usually Leidenfrost temperature indicates the transition from film boiling to nucleate boiling. If the wall temperature is lower than the Leidenfrost temperature, the vapor film cannot sustain the weight of liquid layer and becomes unstable. Therefore, the liquid starts contacting the wall, and film boiling ceases. Once the liquid contacts the wall, the nucleate boiling starts. In the nucleate boiling regime the heat transfer from the wall to the liquid is significantly larger than that in the film boiling regime, and the wall is chilled down much faster, are shown in Figure 2-4. If the nucleation sites are not completely suppressed, a region of rapid nucleate boiling is seen at the quenching front. If most of nucleate sites are suppressed by the subcooled liquid, the flow directly transforms to the forced convection heat transfer, and nucleate boiling stage is not visible. After the nucleate boiling stage, the chilldown process dramatically slows down as the convection heat transfer dominates. The wall superheat is relatively low at this stage but the heat leaking from the test section to the environment emerges. These factors lead to a lower chilldown rate. In the meantime, the liquid gradually builds up in the pipe due to less vapor generation and the friction between the liquid and the wall. The increase of the liquid layer thickness eventually leads to the transition of the flow regimes. When the liquid layer is thick enough, the stratified flow or wavy flow becomes unstable. Eventually slugs are formed and the flow transforms to the slug flow. In the final stage of chilldown, the flow is almost a single-phase liquid flow, occasionally with some small

PAGE 37

17 slugs. In this stage, the chilldown is almost completed, and the pipe wall temperature gradually reaches the liquid saturated temperature. Figure 2-3. Schematics of observed flow structures in chilldown (Velat et al., 2004). Figure 2-4. Typical wall temperature variation during chilldown. (Velat et al., 2004) Cryogenic Liquid Film Boiling Fron t Vapor Flow Cryogeni c Liqui d Film Boiling Region Vapor Flow Liquid Film Flow Liquid Film Flow Cryogeni c Liqui d Bubbly Flow Increasing Time Nucleate Boiling Front

PAGE 38

18 Chilldown in a vertical pipe is practically less important than the chilldown in horizontal pipe, due to the fact that most of cryogenic transportation pipelines are horizontal, and only a small part is vertical. The experimental study (Hedayapour et al. 1993; Laverty and Rohsenow, 1967) reveals that the flow regime is mainly bubble flow, or inverted annular flow if the vapor film of the film boiling is stable, and single-phase vapor flow and single-phase liquid flow exist at the beginning and the final stage of chilldown, respectively. 2.2 Flow Models in Cryogenic Chilldown Based on the experimental investigation, several flow regimes exist in cryogenic chilldown. At different flow regimes, the models for evaluating velocity and volume fraction of fluid are different. Two types of flow models are to be discussed in this section. First is the homogeneous flow model, which is used for modeling the chilldown in a vertical pipe, where the homogeneous flow is prevalent. Another model is the twofluid model, which is mostly used in simulating the stratified flow or wavy flow for the chilldown in a horizontal pipe. 2.2.1 Homogeneous Flow Model In the homogeneous flow model, the unsteady mass, momentum, and energy conservation equations for the mixture are simultaneously solved. The primary assumptions are: (1) single-phase fluid or two-phase mixture is homogeneous, and each phase is incompressible; (2) thermal and mechanical equilibrium exists between the liquid and the vapor flowing together; (3) flow is quasi-one-dimensional; and (4) axial diffusion of momentum and energy is negligible. Thus, the continuity equation for the mixture is

PAGE 39

19 0 ) ( ) ( z A u t A (2.1) where is the mixture density of liquid and vapor phase, u is the average fluid velocity (by the assumption of homogeneous model, both liquid and vapor velocity are u), t is time z is the vertical axial coordinate, and A is the cross section area of the pipeline. Mixture front Pipe wall Vapor bubble Liquid Figure 2-5. Schematic for homogeneous flow model. By neglecting the viscous terms, the momentum equation for the mixture becomes sin ) ( ) (fg A A z p A z p z A u u t A u f f (2.2) where p is pressure, fz P is the pressure drop due to wall friction, is the inclination angle of the pipe. For a vertical pipe, 2 The energy equation for the homogeneous model is

PAGE 40

20 S q z A i u t A iw ) ( ) ( (2.3) where i is the mixture enthalpy, wq is the heat flux from the wall to the fluid, and S is the perimeter of the pipe. If the cross section of the circular pipe is constant, the governing equations for homogenous flow are simplified to the following equations. 0 ) ( ) ( z u t (2.4) sin ) ( ) (fg z p z p z u u t u f f (2.5) A q z i u t iw 4 ) ( ) ( (2.6) The pressure drop fz P due to the wall friction is evaluated by the correlation for the homogeneous system (Hewitt, 1982). In the correlation (Hewitt, 1982), a friction multiplier 2lo is defined as ratio of two-phase frictional pressure gradient fz P to the frictional pressure gradient for a single-phase flow at the same total mass flux and with the physical properties of the liquid phase loz P i.e. 2 lo lo fdz dP dz dP (2.7) where the friction multiplier 2 lo can be calculated by

PAGE 41

21 25 0 21 1f f f g g l g g l lox xr r r (2.8) where subscribes l and g represent the liquid phase and gas phase, respectively. The single-phase pressure drop loz P is evaluated using the standard equation l lo loD G f dz dP22 (2.9) where lof is the friction factor and for turbulent flow in a pipe, it is given as 25 0079 0f l loGD fr. (2.10) in which, G is the mixture mass flux. Compared with the experimentally measured two-phase flow pressure drop, the homogenous model tends to underestimate the value of two-phase frictional pressure gradient (Klausner et al., 1990). However, it provides a reasonable lower bound of the two-phase flow pressure drop. In the film boiling regime, a layer of vapor film separates the liquid core from the pipe wall. This vapor film significantly reduces the wall friction, so that the two-phase flow pressure drop due to the friction is much lower than that in the other heat transfer regimes. To date, no correlation for the friction coefficient in the film boiling regime exists. In available chilldown studies, the vapor film is treated as a part of the mixture and Martinelli type of pressure drop correlation is used, or the wall friction is simply set to zero.

PAGE 42

22 2.2.2 Two-Fluid Model In the chilldown inside the horizontal pipe, it is assumed that flow is stratified and the liquid and the vapor flow at different velocity (Figure 2-6). Two-fluid model (Willis, 1969; Ishii, 1975) is widely used to qualitatively investigate the stratified flow inside horizontal pipeline with a relatively low computational cost compared with 2-dimensional or 3-dimensional fluid flow models. In the study of the horizontal pipe chilldown, the fluid volume fractions, velocities, enthalpies are solved with the two-fluid model. Liquid layer U Vapor layer r x Wall heat flux Pipe wall D Figure 2-6. Schematic of the two-fluid model. The basis of the two-fluid model is a set of one-dimensional conservation equations for the balance of mass, momentum and energy for each phase. The one-dimensional conservation equations are obtained by integrating the flow properties over the cross-sectional area of the flow. In this study, it is assumed that flow is incompressible as the Mach number of the gas phase is usually very low for the stratified flow. Hence, continuity equation for the liquid phase (Chan and Banarjee, 1981c) is l l l lA m u x t f (2.11)

PAGE 43

23 where is volume fraction, is density, u is the velocity, t is the time, x is the axial coordinate, and m is the mass transfer rate between the liquid phase and the gas phase per unit length; the subscript l denotes liquid. Similarly, continuity equation for the gas phase is g g g gA m u x t D, (2.12) where the subscript g denotes gas. It is noted that 1 g l (2.13) The momentum equation for the liquid phase is sin cos2 l i l i i l l l l l l i l l l l l lA u m A S A S g x H g x p u x u t t t f f f f f (2.14) whereipis the pressure at the liquid-gas interface, g is acceleration of gravity, is the angle of inclination of the pipe axis from the horizontal lane, t is the shear stress, S is the perimeter over which t acts, A is the pipe cross section area,lH is the liquid phase hydraulic depth; the subscript i denotes liquid-gas interface. The second term on the right hand side of Equation (2.14) represents the effect of gravity on the wavy surface of liquid layer. The liquid phase hydraulic depth lHis defined as l l l l l lh H (2.15) where lh is the liquid layer depth. Similarly, the momentum equation for the gas phase is

PAGE 44

24 sin cos2 g i g i i g g g g l g i g g g g g gA u m A S A S g x H g x p u x u t t t f f f f f D (2.16) wheregH is the gas phase hydraulic depth. It is defined as g g g g g gh H (2.17) where gh is the gas layer thickness. To study heat transfer, appropriate energy equations for both phases are required in the two-fluid model. Similar to the assumptions made in the homogeneous flow model, the heat conduction inside the fluid is neglected. Thus the one-dimensional energy equations for the liquid phase and the gas phase are l l l i l l l l lA q A i m i u x i t f (2.18) and g g g i g g g g gA q A i m i u x i t (2.19) where i is enthalpy, and q is the heat transfer rate to the fluid per unit length. In the two-fluid model, shear stresses lt, gt and it must be specified to close the two fluid model. There are many correlations for shear stresses for separated flow model, such as those developed by Wallis (1946), Barnea and Taitel (1976), and Andritsos and Hanratty (1987). No significant difference exists among these models except at the flow regime transition and at the high-speed flow, which will not be addressed in this study.

PAGE 45

25 Thus, widely accepted shear stress correlations by Barnea and Taitel (1994) are employed: 22l l l lU f t, (2.20) 22g g g gU f t, (2.21) 2l g l g i iU U U U f f f t, (2.22) where t is shear stress, subscripts l, g, and i represent interface between the liquid and the wall, interface between the gas and the wall, interface between the liquid and gas, respectively. Friction factors f are given by n l l lC ffRe, and m g g gC ff Re, (2.23) where lRe is defined as l l l l lD Ur Re, (2.24) where lD is the liquid hydraulic diameter l l lS A D 4 (2.25) in which lA is liquid phase cross section area, lS is the liquid phase perimeter. In Equation (2.23) gRe is defined as g g g g gD Ur Re, (2.26) where gD is vapor phase hydraulic diameter

PAGE 46

26 i v g gS S A D 4 (2.27) in which gA is vapor phase cross section area, gS is the vapor phase perimeter, and iS is the liquid-gas interface perimeter. The coefficients gC and lC are equal to 0.046 for turbulent flow and 16 for laminar flow, while n and m take the values of 0.2 for turbulent flow and 1.0 for laminar flow. The interfacial friction factor is assumed to be g if f or 014 0 if if 014 0 !gf. It is supposed that this model works in the flow boiling regime and in the forced convection heat transfer regime. However, in the film boiling stage, presence of vapor film dramatically reduces the shear stress between the liquid and the wall. In such a situation, lt should be evaluated to include the effect of vapor film layer. 2.3 Heat Transfer between Cryogenic Fluid and Solid Pipe Wall During cryogenic chilldown, the fluid in contact with the pipe wall is either the liquid or the vapor. The mechanisms of heat transfers between the liquid and the wall and between the vapor and the wall are different, as shown in Figure 2-7. Based on experimental measurements and theoretical analysis, liquid-solid heat transfer accounts for a majority of the total heat transfer. However, the liquid-solid heat transfer is much more complicated than the heat transfer between the vapor and the wall due to occurrence of film boiling and nucleate boiling. Thus, the heat transfer between the liquid and the wall is discussed first.

PAGE 47

27 2.3.1 Heat Transfer between Liquid and Solid wall The heat transfer mechanism between the liquid and the solid wall includes film boiling, nucleate boiling, and two-phase convection heat transfer. The transition from one type of heat transfer to another depends on many parameters, such as the wall temperature, the wall heat flux, and properties of the fluid. For simplicity, a fixed temperature approach is adopted to determine the transition point. That is, if the wall temperature is higher than the Leidenfrost temperature, film boiling is assumed. If the wall temperature is between the Leidenfrost temperature and a transition temperature, T2, nucleate boiling is assumed. If the wall temperature is below the transition temperature T2, two-phase convection heat transfer is assumed. The values of the Leidenfrost temperature and the transition temperature are determined by matching the model prediction with the experimental results. Film boiling Flow boiling Convective heat transfer ( li q uid ) Liquid layer Vapor layer Convective heat transfer ( va p or ) Liquid Vapor Wall heat flux Pipe wall D Thin vapor film Figure 2-7. Schematic of heat transfer in chilldown. 2.3.1.1 Film boiling Due to the high wall superheat encountered in the cryogenic chilldown, film boiling plays a major role in the heat transfer process in terms of the time span and in terms of

PAGE 48

28 the total amount of heat removed from the wall, as shown in Figure 2-4. Currently there exists no specific film boiling correlation for chilldown applications with such high wall superheat. The research starts from the conventional film boiling correlations. A cryogenic film boiling heat transfer correlations was provided by Giarratano and Smith (1965), ) ( *4 0 tt calcf Bo Nu Nu f, (2.28) where Nu is Nusselt number l FBk D h Nu (2.29) where FBh is the film boiling heat transfer coefficient and lk is the thermal conductivity of the liquid, Bo is the boiling number G h q Bofg* (2.30) where fgh is the evaporative latent heat of the fluid. In Equation (2.28), calcNu is the Nusselt number for the two-phase convection heat transfer, which can be obtained using 4 0 8 0Pr Re 023 0 calcNu, (2.31) where Re is Reynolds number of mixture and Pr is Prandtl number of vapor, tt is Martinelli number 1 0 5 0 9 01 f v l l v ttx xr r (2.32) In Giarratano and Smith (1965) correlation, the heat transfer coefficient is the averaged value for the whole cross section. Similar correlations for cryogenic film

PAGE 49

29 boiling also exist in the literature. The correlations were obtained from measurements conducted under steady state. The problem with the use of these steady state film boiling correlations is that they do not account for information of flow regimes. For example, for the same quality, the heat transfer rate for annular flow is much different from that for stratified flow. Available empirical correlations do not make such difference. Furthermore, in this study, local heat transfer coefficient is needed in order to incorporate the thermal interaction with the pipe wall. Since the two-phase flow regime information is available in the present study through the visualized experiment, it is expected that the modeling effort should take into account the knowledge of the flow regime. Suppose a liquid-gas stratified flow exists inside a horizontal pipe. Due to gravity, the upper part of pipe wall is in contact with the gas, and lower part of pipe wall is in contact with the flowing liquid. Thus, the heat transfer coefficient on upper wall is significantly different from that on the lower wall. Apparently, the local heat transfer coefficient strongly depends on the local flow condition instead an overall parameter such as the flow quality at the given location. There are several correlations for the film boiling based on the analysis of the vapor film boundary layer, such as Bromley correlation (1950) and Breen and Westerwater correlation (1962) for film boiling on the outer surface of a hot tube. Frederking and Clark (1965) and Carey (1992) correlations, for the film boiling on the surface of a sphere, are included as well. However, none of these was obtained for cryogenic fluids or for the film boiling on the inner surface of a pipe or tube.

PAGE 50

30 2.3.1.2 Forced convection boiling and two-phase convective heat transfer A pool boiling correlation for cryogens was proposed by Kutateladze (1952). The pool nucleated boiling heat transfer coefficient poolh is 5 1 626 0 906 0 5 1 5 1 750 1 282 1 10* 10 487 0 T h c p k hl v fg l p l l pool fr (2.33) where is liquid surface tension, r is viscosity, and T is wall superheat. Based on this pool boiling correlation, a convection boiling correlation was proposed (Giarratano and Smith, 1965). The heat transfer coefficient is contributed by both convection heat transfer and ebullition: pool c lh h h ,, (2.34) where c lh, is given by Dittus-Boelter equation which is used in fully developed pipe flow: l l l l c lD k h / Pr Re 023 04 0 8 0 , (2.35) where lRe is defined as l lDGr Re. (2.36) Chen (1966) introduced enhancement factor E and suppression factor S into the flow boiling correlation. The heat transfer coefficient is given pool c lSh Eh h ,. (2.37) Enhancement factor E reflects the much higher velocities and hence forced convection heat transfer in the two-phase flow compared to the single-phase, liquid only flow. The suppression factor S reflects the lower effective superheat in the forced convection as opposed to pool boiling, due to the thinner boundary condition. The value of E and S are

PAGE 51

31 presented as graphs in Chen (1966). The pool boiling heat transfer coefficient in Chen correlation is 75 0 24 0 24 0 24 0 29 0 5 0 25 0 49 0 45 0 79 000122 0 P T h g c k hv fg l l c l pool r (2.38) Chen correlation (1966) fits best for annular flow since it was developed for vertical flows. For the stratified flow regime, Chens correlation may not be applicable. At the flow boiling heat transfer, Gungor and Winterton correlation (1996) is widely used due to that it fits much more experimental data. The basic form of Gungor and Winterton correlation is similar to Chen correlation (1966), Equation (2.37). However, evaluation of E and S in Gungor and Wintertons correlation takes account for the influence of heat transfer rate by adding boiling number Bo Thus, E and S are presented as 86 0 16 1/ 1 37 1 24000 1ttBo E (2.39) and 17 1 2 6Re 10 15 1 1 1lE Sf" (2.40) The pool boiling correlation implemented is proposed by Cooper (1984) 67 0 5 0 55 0 10 12 0log 55 q M P P hr r pool f ff (2.41) The solution of heat transfer correlation in Gungor and Wintertons correlation is implicitly obtained by iteration. Although Gungor and Winterton correlation (1996) is widely used due to its good agreement with a large data set, a closer examination on this correlation shows that it is based mainly on the following parameters: Pr Re and quality x Similar to the

PAGE 52

32 development of conventional film boiling correlations, these parameters all reflect overall properties of the flow in the pipe and are not directly related to flow regimes. Thus, it cannot be used to predict the local heat transfer coefficients required in chilldown simulation. Most of existing force convection boiling heat transfer correlations do not effectively take account the influence of flow regimes and flow patterns. Recently, Zurcher et al. (2002) proposed a flow pattern dependent heat transfer correlation for the horizontal pipe. The strategy employed in Zurcher et al. (2002) is that the flow pattern is obtained using the flow pattern map at the first step. The information of flow pattern determines the part of wall contacting with the liquid or the vapor, then corresponding conventional heat transfer correlations is employed to determine the local heat transfer coefficient. The heat transfer coefficient for the whole pipe is obtained by averaging the local heat transfer coefficient along the perimeter of the pipe. Although details of the approach like flow pattern map, and correlations employed are not perfect in study of Zurcher et al. (2002), their approach to the flow boiling heat transfer is intelligible and provides insight for studying chilldown. When wall superheat drops to a certain range all the nucleation sites are suppressed. The heat transfer is dominated by two-phase forced convection. The heat transfer coefficient can then be predicated using Equation (2.35), when the flow is turbulent, or Equation (2.42), when the flow is laminar. l l c lD k h / 36 4,. (2.42)

PAGE 53

33 2.3.2 Heat Transfer between Vapor and Solid Wall The heat transfer between the vapor and wall can be estimated by treating the flow as a fully developed forced convection flow, neglecting the liquid droplets that are entrapped in the vapor. The heat transfer coefficient of vapor forced convective flow is g g g g c gD k h / Pr Re 023 04 0 8 0 , (turbulent flow) (2.43) g g c gD k h / 36 4,, (laminar flow) (2.44)

PAGE 54

34 CHAPTER 3 VAPOR BUBBLE GROWTH IN SATURATED BOILING Accurate evaluation of the nucleate boiling coefficient is a critical part of the study on the chilldown process because it provides the heat transfer rate from the wall to the cryogenic fluid. During the nucleate boiling the vapor bubble growth rate has a directly influence on the heat transfer rate. The higher the bubble growth rate, the higher the heat transfer rate. A physical model for vapor bubble growth in saturated nucleate boiling has been developed that includes both heat transfer through the liquid microlayer and that from the bulk superheated liquid surrounding the bubble. Both asymptotic and numerical solutions for the liquid temperature field surrounding a hemispherical bubble reveal the existence of a thin unsteady thermal boundary layer adjacent to the bubble dome. During the early stages of bubble growth, heat transfer to the bubble dome through the unsteady thermal boundary layer constitutes a substantial contribution to vapor bubble growth. The model is used to elucidate recent experimental observations of bubble growth and heat transfer on constant temperature microheaters reported by Yaddanapudi and Kim (2001) and confirms that the heat transfer through the bubble dome can be a significant portion of the overall energy supply for the bubble growth. 3.1 Introduction During the past forty years, the microlayer model has been widely accepted and used to explain bubble growth and the associated heat transfer in heterogeneous nucleate boiling. The microlayer concept was introduced by Moore and Mesler (1961), Labunstov (1963) and Cooper (1969). The microlayer is a thin liquid layer that resides beneath a

PAGE 55

35 growing vapor bubble. Because the layer is quite thin, the temperature gradient and the corresponding heat flux across the microlayer are high. The vapor generated by strong evaporation through the liquid microlayer substantially supports the bubble growth. Popular opinion concerning the microlayer model is that the majority of evaporation takes place at the microlayer. A number of bubble growth models using microlayer theory have been proposed based on this assumption such as van Stralen et al. (1975), Cooper (1970), and Fyodrov and Klimenko (1989). These models were partially successful in predicting the bubble growth under limited conditions but are not applicable to a wide range of conditions. Lee and Nydahl (1989) used a finite difference method to study bubble growth and heat transfer in the microlayer. However their model assumes a constant wall temperature, which is not valid for heat flux controlled boiling since the rapidly growing bubble draws a substantial amount of heat from the wall through the microlayer, which reduces the local wall temperature. Mei et al. (1995a, 1995b) considered the simultaneous energy transfer among the vapor bubble, liquid microlayer, and solid heater in modeling bubble growth. For simplicity, the bulk liquid outside the microlayer was assumed to be at the saturation temperature so that the vapor dome is at thermal equilibrium with the surrounding bulk liquid. The temperature in the heater was determined by solving the unsteady heat conduction equation. The predicted bubble growth rates agreed very well with those measured over a wide range of experimental conditions that were reported by numerous investigators. Empirical constants to account for the bubble shape and microlayer angle were introduced. Recently, Yaddanapudi and Kim (2001) experimentally studied single bubbles growing on a constant temperature heater. The heater temperature was kept constant by

PAGE 56

36 using electronic feed back loops, and the power required to maintain the temperature was measured throughout the bubble growth period. Their results show that during the bubble growth period, the heat flux from the wall through the microlayer is only about 54% of the total heat required to sustain the measured growth rate. It poses a new challenge to the microlayer theory since a substantial portion of the energy transferred to the bubble cannot be accounted for. Since a growing vapor bubble consists of a thin liquid microlayer, which is in contact with the solid heater, and a vapor dome, which is in contact with the bulk liquid, the experimental observations of Yaddanapudi and Kim (2001) leads us to postulate that the heat transfer through the bubble dome may play an important role in the bubble growth process, even for saturated boiling. Because the wall is superheated, a thermal boundary layer exists between the background saturated bulk liquid and the wall; within this thermal boundary layer the liquid temperature is superheated. During the initial stage of the bubble growth, because the bubble is very small in size, it is completely immersed within this superheated bulk liquid thermal boundary layer. As the vapor bubble grows rapidly, a new unsteady thermal boundary layer develops between the saturated vapor dome and the surrounding superheated liquid. The thickness of the new unsteady thermal boundary layer should be inversely related to the bubble growth rate; see the asymptotic analysis that follows. Hence the initial rapid growth of the bubble, which results in a thin unsteady thermal boundary layer, is accompanied by a substantial amount of heat transfer from the surrounding superheated liquid to the bubble through the vapor bubble dome. This is an entirely different heat transfer mechanism than that associated with conventional microlayer theory.

PAGE 57

37 In fact, many previous bubble growth models have attempted to include the evaporation through the bubble dome, such as Han and Griffith (1965) and van Stralen (1967). However their analyses neglected the convection term in the bulk liquid due to the bubble expansion, so the unsteady thermal boundary layer was not revealed. This leads to a much lower heat flux through the bubble dome. The existence and the analysis on the unsteady thermal boundary layer near the vapor dome were first discussed in Chen (1995) and Chen et al. (1996), when they studied the growth and collapse of vapor bubbles in subcooled boiling. For subcooled boiling, the effect of heat transfer through the dome is much more pronounced due to the larger temperature difference between the vapor and the bulk liquid. With the presence of a superheated wall, a subcooled bulk liquid, and a thin unsteady thermal boundary layer at the bubble dome, the folding of the liquid temperature contour near the bubble surface was observed in their numerical solutions. The folding phenomenon was experimentally confirmed by Mayinger (1996) using an interferometric method to measure the liquid temperature. Despite those findings, the existence of the thin unsteady thermal boundary layer near the bubble surface has not received sufficient attention. In the recent computational studies of bubble growth by Son et al. (1999) and Bai and Fujita (2000), the conservation equations of mass, momentum, and energy were solved in the Eulerian or Lagrange-Eulerian mixed grid system for the vapor-liquid two-phase flow. In their direct numerical simulations of the bubble growth process, the heat transfer from the surrounding liquid to the vapor dome is automatically included since the integration is over the entire bubble surface. They observed that there could be a substantial amount of

PAGE 58

38 heat transfer though bubble dome in comparison with that from the microlayer. However, it is not clear that if these direct numerical simulations have sufficiently resolved the thin unsteady thermal boundary layer that is attached to the rapidly growing bubble. In this study, asymptotic and numerical solutions to the unsteady thermal fields around the vapor bubble are presented. The structure of the thin, unsteady thermal boundary layer around the vapor bubble is elucidated using the asymptotic solution for a rapidly growing bubble. A new computational model for predicting heterogeneous bubble growth in saturated nucleate boiling is presented. The model accounts for energy transfer from the solid heater through the liquid microlayer and from the bulk liquid through the thin unsteady thermal boundary layer on the bubble dome. It is equally valid for subcooled boiling, although the framework for this case has already been presented by Chen (1995) and Chen et al. (1996). The temperature field in the heater is simultaneously solved with the temperature in the bulk liquid. For the microlayer, an instantaneous linear temperature profile is assumed between the vapor saturation temperature and the heater surface temperature due to negligible heat capacity in the microlayer. For the bulk liquid, the energy equation is solved in a body-fitted coordinate system that is attached to the rapidly growing bubble with pertinent grid stretching near the bubble surface to provide sufficient numerical resolution for the new unsteady thermal boundary layer. Section 3.2 presents a detailed formulation of the present model and an asymptotic analysis for the unsteady thermal boundary layer. In Section 3.3, the experimental results of Yaddanapudi and Kim (2001) are examined using the computational results based on the

PAGE 59

39 present model. A parametric investigation considering the effect of the superheated bulk liquid thermal boundary layer thickness on bubble growth is also presented. 3.2 Formulation 3.2.1 On the Vapor Bubble Consideration is given to an isolated vapor bubble growing from a solid heating surface into a large saturated liquid pool, as shown in Figure 3-1. A rigorous description of the vapor bubble growth and the heat transfer processes among three phases requires a complete account for the hydrodynamics around the rapidly growing bubble in addition to the complex thermal energy transfer. The numerical analysis by Lee and Nydahl (1989) relied on an assumed shape for the bubble, although the hydrodynamics based on the assumed bubble shape is properly accounted for. Son et al. (1999) and Bai and Fujita (2000) employed the Navier-Stokes equations and the interface capture or trace methods to determine the bubble shape. Nevertheless, the microlayer structure was still assumed based on existing models. R(t) Solid wall;heat is supplied from within or below Background bulk liquid Bulk liquid thermal boundary layer Liquid microlayer z Figure 3-1. Sketch for the growing bubble, thermal boundary layer, microlayer and the heater wall.

PAGE 60

40 In this study, the liquid microlayer between the vapor bubble and the solid heating surface is assumed to have a simple wedge shape with an angle <<1. The interferometry measurements of Koffman and Plesset (1983) demonstrate that a wedge shape microlayer is a good assumption. There exists ample experimental evidence by van Stralen (1975) and Akiyama (1969) that as a bubble grows, the dome shape may be approximated as a truncated sphere with radius ) ( t R, as shown in Figure 3-1. Using cylindrical coordinates, the local microlayer thickness is denoted by ) ( r L. The radius of the wedge-shaped interface is denoted by ) ( t Rb, which is typically not equal to ) ( t R. Let ) ( / ) ( t R t R cb, (3.1) and the vapor bubble volume ) ( t Vbcan be expressed as ) ( ) ( 3 4 ) (3c f t R t Vb (3.2) where ) (c f depends on the geometry of the truncated sphere. In the limit 1 #c, the bubble is a hemisphere and ) ( ) 3 / 2 ( ) (3t R t Vb #. In the limit 0 #c, the bubble approaches a sphere and ) ( ) 3 / 4 ( ) (3t R t Vb #. To better focus the effort of the present study on understanding the complex interaction of the thermal field around the vapor dome, additional simplification is introduced. The bubble shape is assumed to be hemispherical ( c =1) during the growth. Comparing with the direct numerical simulation technique which solves bubble shape and fluid velocity field using Navier-Stokes equation, this simplification introduces some error in the bubble shape and fluid velocity and temperature fields in this study. However, the hemispherical bubble assumption is generally valid at high Jacob number nucleate boiling (Mei, et al. 1995a) and at the early stage of low Jacob number bubble growth

PAGE 61

41 (Yaddanapudi and Kim, 2001). A more complete model that incorporates the bubble shape variation could have been used, as in Mei et al. (1995a); however, the present model allows for a great simplification in revealing and presenting the existence and the effects of a thin unsteady liquid thermal boundary layer adjacent to the bubble dome and the influence of bulk liquid thermal boundary layer on saturated nucleate boiling. The present simplified model is not quantitatively valid when the shape of the vapor bubble deviates significantly from a hemisphere. The energy balance at the liquid-vapor interface for the growing bubble depicted in Figure 3-1 is described as f f b t R R l l m r L z ml l b fg vdA n T k dA n T k dt dV h) ( ) (, (3.3) where v is the vapor density, fgh is the latent heat, lk is the liquid thermal conductivity, Tl is the temperature of the bulk liquid, Tml is the temperature of the microlayer liquid, Am is the area of wedge, Ab is the area of the vapor bubble dome exposed to bulk liquid, n is the differentiation along the outward normal at the interface, and R is the spherical coordinate in the radial direction attached to the moving bubble. Equation (3.3) simply states that the energy conducted from the liquid to the bubble is used to vaporize the surrounding liquid and thus expand the bubble. 3.2.2 Microlayer The microlayer is assumed to be a wedge centered at 0 rwith local thickness ) (r L. Because the hydrodynamics inside the microlayer are not considered, the microlayer wedge angle cannot be determined as part of the solution. In Cooper and Lloyd (1969), the angle was related to the viscous diffusion length of the liquid as

PAGE 62

42 t c t Rl b 1tan ) ( in which l is the kinematic viscosity of the liquid. A small results in ) (1t R t cb l (3.4) Cooper and Lloyd (1969) estimated 1c to be within 0.3-1.0 for their experimental conditions. A systematic investigation for saturated boiling by Mei et al. (1995b) established that the temperature profile in the liquid microlayer can be taken as linear for practical purposes. The following linear liquid temperature profile in the microlayer is thus adopted in this study f r L z t r T T t z r Tsat sat l1 , ,, (3.5) where sat s satT t z r T t r T f 0 , and Ts is the temperature of the solid heater. 3.2.3 Solid Heater The temperature of the solid heater is governed by the energy equation, which is coupled with the microlayer and bulk liquid energy equations. Solid heater temperature variation significantly influences the heat flux into the rapidly growing bubble (Mei et al. 1995a, 1995b). However, in this study, constant wall temperature is assumed so that the case of Yaddanapudi and Kim (2001) can be directly simulated. Thus, sat w sat satT T T t r T f ,, (3.6) which can be directly used in Equation (3.5) to determine the microlayer temperature profile.

PAGE 63

43 3.2.4 On the Bulk Liquid It was assumed that the vapor bubble is hemispherical in section 2.1. Furthermore, the velocity and temperature fields are assumed axisymmetric. Unless otherwise mentioned, spherical coordinates ) , ( R, as shown in Figure 3-2, are employed for the bulk liquid. r z R R(t) b' R R b R 0 1 0 1 Figure 3-2. Coordinate system for the background bulk liquid. 3.2.4.1 Velocity field Since there is no strong mean flow over the bubble, the bulk liquid flow induced by the growth of the bubble is mainly of inviscid nature. Thus the liquid velocity field may be determined by solving the Laplace equation 02 $ for the velocity potential In spherical coordinates, the velocity components are simply given by the expansion of the hemispherical bubble as

PAGE 64

44 0 0 ) ( ) (2 2 u u R R R R t R dt t dR uRD, (3.7) where dt t dR R) ( D. 3.2.4.2 Temperature field By assuming axisymmetry for the temperature fields and using the liquid velocity from Equation (3.7), the unsteady energy equation for the bulk liquid in spherical coordinates is l l l l R lT R R T R R R R T u t T sin sin 1 12 2 2. (3.8) The boundary conditions are 0 0 at Tl, (3.9) 2 at T Ts l, (3.10) ) (t R R at T Tsat l (3.11) b # bR at t T Tl) (, (3.12) where bTis the far field temperature distribution. To facilitate an accurate computation and obtain a better understanding on the physics of the problem, the following dimensionless variables are introduced, b w b l l cT T T T R R R t t, t, (3.13)

PAGE 65

45 where ct is a characteristic time chosen to be the bubble departure time, wT is the initial solid temperature at the solid-liquid interface, and bT is the bulk liquid temperature far away from the wall, which equals satT for saturated boiling. Using Equation (3.7) and Equation (3.13), Equation (3.8) can be written as t l l l l l l cR R R R R R R R R R R R R t Rsin 1 sin 1 1 12 2 2 2D D D. (3.14) In this equation, the first term on the left-hand-side (LHS) is the unsteady term, and the second term is due to convection in a coordinate system that is attached to the expanding bubble. The right-hand-side (RHS) terms are due to thermal diffusion. As shown in Chen et al. (1996) and below, the solution for l near 1 R possesses a thin boundary layer when 1%%lR RD. Therefore, to obtain the accurate heat transfer between the bubble and the bulk liquid, high resolution in the thin boundary layer is essential. Hence, the following grid stretching in the bulk liquid region is applied, &'() &'1 0 / 1 tan tan 2 1 0 / 1 tan 1 tan 1 ) 1 ( 11 1 * * f f f for S S for S S R RR R, (3.15) where RS and S are parameters that determine the grid density distribution in the physical domain and R R Rb b is the far field end of the computational domain along the radial direction.

PAGE 66

46 Typically RS~0.65 and S~0.73, and b Rranges from 5 to 25. Figure 3-3 shows a typical grid distribution used in this study. Figure 3-3. A typical grid distribution for the bulk liquid thermal field with 65 0 RS, 73 0 S, and 10 bR. 3.2.4.3 Asymptotic analysis of the bulk liquid temperature field during early stages of growth To gain a clear understanding on the interaction of the growing bubble with the background superheated bulk liquid thermal boundary layer, an asymptotic analysis for non-dimensional temperature l is presented, following the work of Chen et al. (1996). During the early stages, the bubble growth rate is high and expands rapidly so that 1%% lR R AD. (3.16)

PAGE 67

47 Thus, the solution to Equation (3.14) includes an outer approximation in which the thermal diffusion term on the RHS of Equation (3.14) is negligible and an inner approximation (boundary layer solution) in which the thermal diffusion balances the convection. Away from the bubble, the outer solution is governed by 0 12 f R R R t R Rout l out l D, (3.17) where out lis the outer solution for l in the bulk liquid. The general solution for Equation (3.17) is 3 1 31 ) ( f R t R Fout l, (3.18) as given in Chen et al. (1996). In the above F is an arbitrary function and it is determined from the initial condition of l or the temperature profile in the background bulk liquid thermal boundary layer. It is noted that the solution forout lis described by const R t R f 3 1 31 ) ( along the characteristic curve. The initial temperature profile is often written as z f0. The solution of Equation (3.17) is thus expressed as b t n f r f 3 1 3 3 0 3 01 1 cos R R R R fout l (3.19) where 0R is the initial bubble radius at ct t t !! 0. Provided the bubble growth rate is high, i.e. 1%% A, Equation (3.19) is not only an accurate outer solution for the temperature field outside a rapidly expanding bubble, but it is also a good approximation for the far field boundary condition for Equation (3.12).

PAGE 68

48 Near the bubble surface, there exists a large temperature gradient between the saturation temperature on the bubble surface and the temperature of the surrounding superheated liquid over a thin region. Therefore, the effect of heat conduction is no longer negligible in this thin region and must be properly accounted for. For a large value of A, a boundary layer coordinate X is introduced, ) ( 1A R Xf (3.20) where 1 ) (!!A is the dimensionless length scale of the unsteady thermal boundary layer. Substituting Equation (3.20) into Equation (3.14) results in sin sin 1 1 1 1 1 2 1 1 1 1 12 2 2 2 2 t t in l in l in l in l in l c in l cX A X X X A X X X X X R t R R t R D D(3.21) Neglecting higher order terms, Equation (3.21) becomes X X X R t R X X A R t Rin l c in l in l c f t t D D 3 1 3 12 2 2. (3.22) The balance between the convection term and the diffusion term on the RHS of Equation (3.22) requires 2 1 2 1f f lR R A D. (3.23) Hence Equation (3.22) becomes X X A A R t R X R t Rin l c in l in l c f t D D D 2 32 2. (3.24) The boundary conditions for the inner (boundary layer) solution are

PAGE 69

49 0 f f X at T T T Tb w b sat sat in l (3.25) b # X atin l1. (3.26) For 1!! t 2 1) (t t R + and t, R t RcD. Thus, the LHS of Equation (3.24) is small and can also be neglected. Equation (3.24) then reduces to 0 32 2 X X X in l in l for 1!! t (3.27) The solution for Equation (3.27) is sat sat in lX erf f 2 3) 1 (. (3.28) For 1!! t by matching the outer and inner solutions given by Equation (3.19) and Equation (3.28), the uniformly valid asymptotic solution of the bulk liquid temperature for the saturated boiling problem considered here is obtained, X erfc R R R R fsat l 2 3 3 1 3 3 0 3 01 1 1 cos f b t n f r f (3.29) where b w b sat satT T T T f f and erfc is the complimentary error function. Equation (3.29) is an asymptotic solution for l valid for 1!! t The asymptotic solution given by Equation (3.29) for the liquid thermal field provides an analytical framework to understand: 1) how the temperature field of background superheated bulk liquid boundary layer influences the temperature l near the vapor bubble through the function f; 2) how the bubble growth R(t) and liquid thermal diffusivity affect the liquid thermal field l through the rescaled inner variable X as

PAGE 70

50 defined in Equation (3.20) and Equation (3.23); and 3) how the folding of the temperature contours near the bubble occurs through the dependence of cos term in Equation (3.29). More importantly, from a computational standpoint, it provides: 1) an accurate measure on the thickness of the rapidly moving thermal boundary layer; and 2) a reliable guideline for estimating the adequacy of computational resolution in order to obtain an accurate assessment of heat transfer to the bubble. 3.2.5 Initial Conditions The computation must start from a very small but nonzero initial time 0t, so that ) (0tRis sufficiently small at the initial stage. To obtain enough temporal resolution for the initial rapid growth stage and to save computational effort for the later stage, the following transformation is used, 2 t. (3.30) Thus a constant time step can be used in the computation. The initial temperature profile inside the superheated bulk liquid thermal boundary layer plays an important role to the solution of l, which in turn affects the heat transfer to the bubble through the dome. There exist both experimental and theoretical studies that have considered the bulk liquid temperature profile in the vicinity of a vapor bubble. Hsu (1962) estimated the temperature profile of the superheated thermal layer adjacent to the heater surface and found the layer to be quite thin; thus the temperature gradient inside the thermal layer is almost linear. However, beyond the superheated layer the temperature is held essentially constant at the bulk temperature due to strong turbulent convection. The experimental study by Wiebe and Judd (1971) revealed similar results. It was found that the

PAGE 71

51 superheated bulk liquid thermal boundary layer thickness, decreases with increasing wall heat flux due to enhanced turbulent convection. A high wall heat flux results in increased bubble generation, and the bulk liquid is stirred more rapidly by growing and departing vapor bubbles. To estimate the superheated layer thickness, Hsu (1962) used a thermal diffusion model within the bulk liquid. Han and Griffith (1965) used a similar model and estimated the thickness to be w lt (3.31) where tw is the waiting period. The thermal diffusion model often overestimates the thermal layer thickness, as it neglects the turbulent convection, which is quite strong as reported by Hsu (1962) and Wiebe and Judd (1971). Generally, the bulk liquid temperature profile is almost linear inside the superheated thermal boundary layer, and remains essentially uniform at the bulk temperature bT beyond the superheated background thermal boundary layer. Accordingly, the initial condition for the bulk liquid thermal field used in the numerical solution is given by f z z z 0 10. (3.32) In the asymptotic solution, the discontinuity of zl in the above profile causes the solution for l to be discontinuous. For clarity, the following exponential profile is employed in representing the asymptotic solution f z exp0. (3.33)

PAGE 72

52 3.2.6 Solution Procedure An Euler backward scheme is used to solve Equation (3.14). A second order upwind scheme is used for the convection term and a central difference scheme is used for the thermal diffusion terms. After the bulk liquid temperature field is obtained, the solid heater temperature field is solved, and the bubble radius ) ( t R is updated using Equation (3.3) and Eulers explicit scheme. The information for ) ( t R is a necessary input in Equation (3.14). Although the solution for ) ( t Ris only first order accurate in time, the ) ( t Oaccuracy is not a concern here because a very small t has to be used to ensure sufficient resolution during the early stages. Typically, 410 n time steps are used. 3.3 Results and Discussions 3.3.1 Asymptotic Structure of Liquid Thermal Field To gain an analytical understanding of the liquid thermal field near the bubble and to validate the accuracy of the computational treatment for the thin unsteady thermal boundary layer, comparison between the computational and the asymptotic solutions for l near the bubble surface is first presented. As mentioned previously, the validity of the outer solution of the asymptotic analysis only requires 1 %% A, which is satisfied under most conditions due to rapid vapor bubble growth. The inner solution is valid for 1 !! t in addition to 1 %% A. The comparison is presented for bubble growth in saturated liquid with A=14000. The initial temperature profile follows Equation (3.33) and 5 0 cR in which cR is the bubble radius at ct. There are 200 and 50 grid intervals along the R and -directions,

PAGE 73

53 respectively. The grid stretching factors are 65 0 RSand 73 0 S for the computational case. Figure 3-4 compares the temperature profiles between the asymptotic and numerical solutions at 001 0 t 0.01, 0.1, and 0.3 for 0, 40n, and 71n. There are two important points to be noted. First of all, it is seen that the temperature gradient is indeed very large near the bubble surface because the unsteady thermal boundary layer is very thin. Secondly, numerical solutions agree very well with the asymptotic solutions at 001 0 t and 0.01. The excellent agreement between the numerical and analytical solutions indicates that the numerical treatment in this study is correct. At 1 0 t and 0.3, the asymptotic inner solution given by Equation (3.29) is no longer accurate, while the outer solution remains valid because 1 %% A is the only requirement. At 1 0 t and 0.3 the numerical solution matches very well with the outer solution. This again demonstrates the integrity of the present numerical solution over the entire domain due to sufficient computational resolution near the bubble surface and removal of undesirable numerical diffusion through the use of second order upwind scheme in the radial direction for Equation (3.14). The large temperature gradient near the dome causes high heat transfer from the superheated liquid to the vapor bubble through the dome. This large gradient results from the strong convection effect that is caused by the rapid bubble growth (see Equation (3.14) for the origin in the governing equation and Equation (3.19) for the explicit dependence on the bubble growth). Thus the bulk liquid in the superheated boundary layer supplies a sufficient amount of energy to the bubble.

PAGE 74

54 Figure 3-4. Comparison of the asymptotic and the numerical solutions at t =0.001, 0.01, 0.1 and 0.3 for =0n, 40n, and 71n. To capture the dynamics of the unsteady boundary layer, a sufficient number of computational grids is required inside this layer. The asymptotic analysis gives an estimate for the unsteady boundary layer thickness on the order of 025 0 14000 3 ~, which agrees with the numerical solution in Figure 3-4. At 01 0 t in Figure 3-4, the discrete numerical results are presented. There are about 23 points inside the layer of thickness025 0 This provides sufficient resolution for the R'-1 l10-310-210-11001010 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 =0o(Numerical) =40o(Numerical) =71o(Numerical) =0o(Asymptotic) =40o(Asymptotic) =71o(Asymptotic) t =0.1_ R'-1 l10-310-210-11001010 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 =0o(Numerical) =40o(Numerical) =71o(Numerical) =0o(Asymptotic) =40o(Asymptotic) =71o(Asymptotic) t =0.3_ R'-1 l10-310-210-11001010 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 =0o(Numerical) =40o(Numerical) =71o(Numerical) =0o(Asymptotic) =40o(Asymptotic) =71o(Asymptotic) t =0.001_ R'-1 l10-310-210-11001010 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 =0o(Numerical) =0o(Asymptotic) =40o(Numerical) =40o(Asymptotic) =71o(Numerical) =71o(Asymptotic) t =0.01_

PAGE 75

55 temperature profile in the unsteady thermal boundary layer. In contrast, most computational studies on the thermal field around the bubble dome reported in the open literature have insufficient grid resolution adjacent the dome, which leads to an inaccurate heat transfer assessment. Figure 3-5 shows the effect of parameter A on the asymptotic solution. When A is large, the asymptotic and numerical solutions agree very well. The discrepancy between asymptotic and numerical solutions inside the unsteady thermal boundary layer increases when A decreases. However, the outer solution remains valid for the far field even when A becomes small. R'-1 l10-310-210-1100101-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1_t =0.01 =40o10000 1000 100 10 A=14000 Asymptotic Numerical Figure 3-5. Effect of parameter A on the liquid temperature profile near bubble. The temperature contours shown in Figure 3-6 are difficult to obtain experimentally. Only recent progress in holographic thermography permits such measurements. Ellion (1954) has stated that there exists an unsteady thermal boundary

PAGE 76

56 layer contiguous to the vapor bubble during the bubble growth. Recently, Mayinger (1996) used a holography technique to capture the folding of the temperature contours during subcooled nucleate boiling. Although his study considered subcooled nucleate boiling, the pattern of the temperature distribution near the bubble dome by Mayinger (1996) is very similar to that shown in the Figure 3-6. It is expected that experimental evidence of contour folding in saturated nucleate boiling will be reported in the future. 3.3.2 Constant Heater Temperature Bubble Growth in the Experiment of Yaddanapudi and Kim In the experiment of Yaddanapudi and Kim (2001), single bubbles growing on a heater array kept at nominally constant temperature were studied. The liquid used is FC-72, and the wall superheat is maintained at 22.5 nC, so that Jacob number is 39. The bubble shape in the early stage appears to be hemispherical. To calculate the heat flux from the microlayer to the vapor bubble in the present model, the microlayer wedge angle or constant 1c in Equation (3.4) must be determined. Neither or 1c has been measured. However, the authors have reported the amount of wall heat flux from the wall to the bubble through an equivalent bubble diameter eqd assuming that the wall heat flux is the only source of heat entering the bubble. Since in the present model this heat flux is assumed to pass through the microlayer, it may be used to evaluate the constant 1c via trial and error. The superheated thermal boundary layer thickness of the bulk liquid in Equation (3.32) is also a required input. The computed growth rate ) ( t R is matched with the experimentally measured ) ( t Rin order to determine The simulation is carried out only for the early stage of bubble growth. This is because after t=6-8"10-4s the base of the bubble does not expand anymore, and the bubble shape deviates from a hemisphere.

PAGE 77

57 Furthermore, there is the possibility of the microlayer being dried out in the latter growth stages as a result of maintaining a constant wall temperature, as was observed by Chen et al. (2003). Figure 3-6. The computed isotherms near a growing bubble in saturated liquid at t=0.01, t=0.1,t=0.3, and t=0.9. Figure 3-7 shows the computed equivalent bubble diameter ) ( t deq, together with the experimentally determined equivalent ) ( t deq. In the present model, eqd is calculated using m r L z ml l b fg vdA n T k dt dV h) ( f (3.34) 0.9 0.88 0 8 5 0 8 0.77 0 7 5 0.7 0.98 0 9 6 0 9 4 0 9 2 R'-1 00.511.52 0 0.5 1 1.5 2_t =0.01 0.9 0.8 0.75 0 6 5 0.6 0 5 5 0 5 0 4 5 0 4 0 3 5 R'-1 00.511.52 0 0.5 1 1.5 2_t =0.1 0 7 0.6 0.5 0 4 0 3 0 2 5 0 2 0 9 5 0 9 0 8 5 0 8 0 7 5 0 6 5 0 1 5 R'-1 00.511.52 0 0.5 1 1.5 2_t =0.3 0 4 0.3 0 2 0 1 5 0.9 0 8 0 7 0 6 0 5 0 1 0 0 5 R'-1 00.511.52 0 0.5 1 1.5 2_t =0.9

PAGE 78

58 where 36eq bd V The heat flux includes only that from the microlayer and this allows 1c to be evaluated. For eqd, to match the measured data as shown in Figure 3-7, it requires 1c=3.0. t(s) d(t)(m)00.00020.00040.00060.0008 0 5E-05 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 0.0005deq(t)presentmodel deq(t)measurement Figure 3-7. Comparison of the equivalent bubble diameter eqdfor the experimental data of Yaddanapudi and Kim (2001) and that computed for heat transfer through the microlayer (1c=3.0). Figure 3-8 compares the computed bubble diameter ) ( 2 ) ( t R t d and those reported by Yaddanapudi and Kim (2001). In Figure 3-8, =30rm is used in addition to 1c=3.0 in matching the predicted bubble growth with measured data. The good agreement obtained can be partly attributed to the adjustment in the superheated bulk liquid thermal boundary layer thickness Because the heat transfer to the bubble (through the microlayer and through the dome) is of two different mechanisms, the good

PAGE 79

59 agreement over the range is an indication of the correct physical representation by the present model. Figure 3-9 shows the total heat entering bubble and the respective contribution from the microlayer and from the unsteady thermal boundary layer. The contribution from the unsteady thermal boundary layer accounts for about 70% of the total heat transfer. It was reported by Yaddanapudi and Kim (2001) that approximately 54% of the total heat is supplied by the microlayer over the entire growth cycle. Since, the simulation is only carried out for the early stage of bubble growth, it is difficult to compare the microlayer contribution to heat transfer reported by Yaddanapudi and Kim (2001) with that predicted by current model. At the end, the bubble expands outside the superheated boundary layer and protrudes into the saturated bulk liquid. The heat transfer from dome thus slows down. Hence, the 54% for the entire bubble growth period dose not contradict a higher percentage of contribution computed from the unsteady thermal boundary layer during the early stages. Figure 3-10 shows the computed temperature contours associated with Yaddanapudi and Kims (2001) experiment for the estimated and 1c. Folding of the temperature contours is clearly observed in the simulation for saturated boiling. 3.3.3 Effect of Bulk Liquid Thermal Boundary Layer Thickness on Bubble Growth Since the superheated bulk liquid thermal boundary layer thickness, determines how much heat is stored in the layer, it is instructive to conduct a parametric study on the effects bubble growth with varying All parameters are the same as those used in Yaddanapudi and Kims (2001) experiment except that is varied. Hence the influence of the superheated thermal boundary layer thickness on the bubble growth is elucidated.

PAGE 80

60 t(s) d(t)(m)00.00020.00040.00060.0008 0 5E-05 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 0.0005d(t)presentmodel d(t)measurement Figure 3-8. Comparison of bubble diameter, d(t), between that computed using the present model and the measured data of Yaddanapudi and Kim (2001). Here, 1c=3.0 and =30rm. t(s) heat(J)00.00020.00040.00060.0008 0 2E-06 4E-06 6E-06 8E-06 1E-05 1.2E-05 1.4E-05 1.6E-05 totalheatenteringthebubble heatfrommicorlayer heatfrombulkliquidthermalboudarylayer Figure 3-9. Comparison between heat transfer to the bubble through the vapor dome and that through the microlayer.

PAGE 81

61 Figure 3-11 shows the effect on the bubble growth rate of varying (from 1rm to 100rm). The thicker the bulk liquid thermal boundary layer, the faster the bubble grows. A large implies a larger amount of heat is stored in the background bulk liquid surrounding the bubble. It is also clear that when approaches zero, the bubble growth rate becomes unaffected by the variation of The reason is when is small, most of heat supplied for bubble growth comes from the microlayer and the contribution from the dome can be neglected. Figure 3-10. The computed isotherms in the bulk liquid corresponding to the thermal conditions reported by Yaddanapudi and Kim (2001). 0 0 0 5 0 0 5 0 1 0 3 0 5 0 7 0 9 R'-1 00.511.52 0 0.5 1 1.5 2_t=0.96ms 0 0 0 5 0 0 5 0.1 0 3 0 5 0.7 0.9 R'-1 00.511.52 0 0.5 1 1.5 2_t=0.36ms 0 0 0 5 0 0 5 0 1 0 3 0 5 0 7 0.9 R'-1 00.511.52 0 0.5 1 1.5 2_t=0.12ms 0 0 5 0 1 0 2 0 3 0 4 0 5 0 6 0. 7 0 8 0.9 R'-1 00.511.52 0 0.5 1 1.5 2_t=0.0012ms

PAGE 82

62 It is also noted that for =100rm, if the bubble eventually grows to about several millimeters, the effect of the bulk liquid thermal boundary layer is negligible on ) ( t Rfor most of the growth period except at the very early stages. Physically, this is because the bubble dome is quickly exposed to the saturated bulk liquid so that it is at thermal equilibrium with the surroundings. For small bubbles, it will be immersed inside the thermal boundary layer most of time. Hence the effect of the bulk liquid thermal boundary layer becomes significant for the bubble growth. t(s) d(m)00.00020.00040.00060.00080.001 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 =100 r m =50 r m =30 r m =10 r m =5 r m =1 r m Figure 3-11. Effect of bulk liquid thermal boundary layer thickness on bubble growth. The microlayer angle and the superheated bulk liquid thermal liquid boundary layer thickness are the required inputs to compute bubble growth in the present model. However, neither of these parameters is typically measured or reported in bubble growth experiments. It is strongly suggested that the bulk liquid thermal boundary layer

PAGE 83

63 thickness be measured and reported in future experimental studies. For a single bubble study, in the immediate neighborhood of the nucleation site should be measured. 3.4 Conclusions In this study, a physical model is presented to predict the early stage bubble growth in saturated heterogeneous nucleate boiling. The thermal interaction of the temperature fields around the growing bubble and vapor bubble together with the microlayer heat transfer is properly considered. The structure of the thin unsteady liquid thermal boundary layer is revealed by the asymptotic and numerical solutions. The existence of a thin unsteady thermal boundary layer near the rapidly growing bubble allows for a significant amount of heat flux from the bulk liquid to the vapor bubble dome, which in some cases can be larger than the heat transfer from the microlayer. The experimental observation by Yaddanapudi and Kim (2001) on the insufficiency of heat transfer to the bubble through the microlayer is elucidated. For thick superheated thermal boundary layers in the bulk liquid, the heat transfer though the vapor bubble dome can contribute substantially to the vapor bubble growth.

PAGE 84

64 CHAPTER 4 ANALYSIS ON COMPUTATIONAL INSTABILITY IN SOLVING TWO-FLUID MODEL The two-fluid model is widely used in studying gas-liquid flow inside pipelines because it can qualitatively predict the flow field with a low computational cost. However, the two-fluid model becomes ill-posed when the slip velocity between the gas and the liquid exceeds a critical value. Computationally, even before the flow becomes unstable, computations can be quite unstable to render the numerical result unreliable. In this study computational stability of various convection schemes for the two-fluid model is analyzed. A pressure correction algorithm is carefully implemented to minimize its effect on stability. Von Neumann stability analysis for the wave growth rates by using the 1st order upwind, 2nd order upwind, QUICK (quadratic upstream interpolation for convection kinematics), and the central difference schemes are conducted. For inviscid two-fluid model, the central difference scheme is more accurate and more stable than other schemes. The 2nd order upwind scheme is much more susceptible to instability for long waves than the 1st order upwind and inaccurate for short waves. The instability associated with ill-posedness of the two-fluid model is significantly different from the instability of the discretized two-fluid models. Excellent agreement is obtained between the computed and predicted wave growth rates, when various convection schemes are implemented. The pressure correction algorithm for inviscid two-fluid model is further extended to the viscous two-fluid model. For a viscous two-fluid model, the diffusive viscous

PAGE 85

65 effect is modeled as a body force resulting from the wall friction. Von Neumann stability analysis is carried out to assess the performances of different discretization schemes for the viscous two-fluid model. The central difference scheme performs best among the schemes tested. Despite its nominal 2nd order accuracy, the 2nd order upwind scheme is much more inaccurate than the 1st order upwind scheme for solving viscous two-fluid model. Numerical instability is largely the property of the discretized viscous two-fluid model but is strongly influenced by VKH instability. Excellent agreement between the computed results and the predictions from von Neumann stability analysis for different numerical scheme is obtained. Inlet disturbance growth test shows that the pressure correction scheme is capable to correctly handle the viscous two-phase flow in a pipe. 4.1 Inviscid Two-Fluid Model 4.1.1 Introduction Gas-liquid flow inside a pipeline is prevalent in the handling and transportation of fluids. A reliable flow model is essential to the prediction of the flow field inside the pipeline. To fully simulate the system, NavierStokes equations in three-dimensions are required. However, it is very expensive to simulate complex two-phase flows in a long pipe with todays computer capability. To reduce the computational cost and obtain basic and essential flow properties of industrial interest, such as gas volume fraction, liquid and gas velocity, pressure, a one-dimensional model is necessary. The two-fluid model is considered to give a realistic prediction for the gas-liquid flow inside a pipeline. The two-fluid model (Wallis, 1969; Ishii, 1975), also known as the separated flow model, consists of two sets of conservation equations for mass, momentum and energy for the gas phase and the liquid phase. Although it has success in simulating two-phase flow in a pipeline, the two-fluid model suffers from an ill-posedness problem. When the

PAGE 86

66 slip velocity between liquid and gas exceeds a critical value that depends on gravity and liquid depth, among other flow properties, the governing equations do not possess real characteristics (Gidaspow, 1974; Jones and Prosperettii, 1985; Song and Ishii, 2000). This ill-posedness condition suggests that the results of the two-fluid model under such condition do not reflect the real flow situation in the pipe. The two-fluid model only gives meaningful results when the relative velocity between the gas and liquid phase is below the critical value. However, this critical value coincides with the stability condition of inviscid Kelvin-Helmholtz instability (IKH) analysis (Issa and Kempf, 2002). Because the IKH instability results in the flow regime transition from the stratified flow to the slug flow or annular flow (Barnea and Taitel, 1994a), ill-posedness of two-fluid model has been interpreted as to trigger the flow regime transition (Brauner and Maron, 1992; Barnea and Taitel, 1994a). The computational methods for solving the two-fluid model have been investigated by many researchers. For computational simplicity, it is further assumed that both liquid and gas phases are incompressible. This is valid because most stratified flows are at relatively low speed compared with the speed of sound. To solve the incompressible two-fluid equations, one approach is to simplify the governing system to only two equations for liquid volume fraction and liquid velocity and neglect the transient terms in the gas mass and momentum equations (Chan and Banerjee, 1981; Barnea and Taitel, 1994b). A more effective method is to use a pressure correction scheme (Patanka 1980). Issa and Woodburn (1998), and Issa and Kempf (2003) applied the pressure correction scheme for the two-fluid model and simulated the stratified flow and the slug flow inside a pipe.

PAGE 87

67 When two-fluid model becomes ill-posed, the solution becomes unstable. A good discretized model should be capable of capturing the incipience of the instability point. However, numerical instability may not be the same as the instability caused by the ill-posedness. Lyczkowski et al. (1978) used von Neumann stability analysis to study a compressible two-fluid model with their numerical scheme and found that numerical instability and ill-posedness may not be identical. However, their two-fluid model lacked the gravitational term and the study focused on one specific discretization scheme and is thus incomplete. Stewart (1979), Ohkawa and Tomiyama (1995) attempted to analyze the numerical stability of an incompressible two-fluid model with a simplified model equation as an alternative. Their study showed that higher order upwind schemes yield a more unstable numerical solution than the 1st order upwind scheme. In this study, a pressure correction scheme is employed to solve the two-fluid model. It is designed to increase the computational stability when the flow is near the ill-posedness condition. The von Neumann stability analysis is carried out to study the stability of the discretized two-fluid model with different interpolation schemes for the convection term. For the wave growth rates using the 1st order upwind, 2nd order upwind, QUICK, and central difference schemes, the central difference scheme is more accurate and more stable. Excellent agreement for the wave growth rates is obtained between the analysis and the actual computation under various configurations. 4.1.2 Governing Equations The basis of the two-fluid model is a set of one-dimensional conservation equations for the balance of mass, momentum and energy for each phase. The one-dimensional conservation equations are obtained by integrating the flow properties over the cross-sectional area of the flow, as shown in Figure 4-1.

PAGE 88

68 Liquid phase Gas phase Interface Liquid velocity lu Gas velocity gu Gravity g Gas velome faction g Liquid velome faction l Pipe cross section ghlh Figure 4-1. Schematic of two-fluid model for pipe flow. Because the ill-posedness originates from the hydrodynamic instability of the two-fluid model, only continuity and momentum equations are considered in the inviscid two-fluid model. Furthermore, no mass and energy transfer occurs between two phases. Surface tension is also neglected since it only acts on small scales, while the waves determining the flow structure in pipe flows are usually of long wavelength. The gas phase is assumed to be incompressible, as the Mach number of the gas phase is usually very low for the stratified flow. Hence, the mass conservation equations for liquid phase is 0 l l lu x t (4.1) where l is liquid volume fraction,l is liquid density, lu is the liquid velocity, t is the time, x is the axial coordinate. The liquid layer momentum conservation equation is sin cos2g x H g x p u x u tl l l i l l l l l lf f f (4.2)

PAGE 89

69 whereipis the pressure at the liquid-gas interface, g is gravitational accelerator, is the angle of inclination of the pipe axis from the horizontal lane, andlH is the liquid phase hydraulic depth. It is defined as l l l l l lh H (4.3) where lh is the liquid layer depth. The second term on the right hand side of Equation (4.2) represents the effect of gravity on the wavy surface of liquid layer. The gas phase mass conservation equation is 0 g g gu x t (4.4) where g,g,gu are density volume fraction, and velocity of gas phase. It is noted that 1 g l (4.5) The momentum equation for gas phase is sin cos2g x H g x p u x u tg l g i g g g g g gf f f (4.6) wheregH is the gas phase hydraulic depth. It is defined as g g g g g gh H (4.7) where gh is the gas layer depth, 4.1.3 Theoretical Analysis 4.1.3.1 Characteristic analysis and ill-posedness It is well known that the initial and boundary conditions need to be imposed consistently for a given system of differential equations. The condition is well-posed if

PAGE 90

70 the solution depends in a continuous manner on the initial and boundary conditions. That is, a small perturbation of the boundary conditions should give rise to only a small variation of the solution at any point of the domain at finite distance from the boundaries (Hirsch, 1988). Equations (4.1, 4.2, 4.4 and 4.6) form a system of 1st order PDEs, for which the characteristic roots, of the system can be found. If s are real, the system is hyperbolic. Complex roots imply an elliptic system, which causes the two-fluid model system to become ill-posed because only initial conditions can be specified in the temporal direction. Any infinitesimal disturbance will cause the waves to grow exponentially without bound when s are complex valued. Let U be the vectorT g l lp u u) , (. Equations (4.1, 4.2, 4.4 and 4.6) can be written in vector form as ] [ ] [ ] [ C x B t A U U, (4.8) where [A], [B] and [C] are coefficient matrices, given by f f 0 0 0 0 0 0 0 1 0 0 0 1 ] [g g l lu u A (4.9a) f f g g g g g g l l l l l l g g l lu gH u u gH u u u B 2 0 cos 0 2 cos 0 0 0 0 ] [2 2, (4.9b)

PAGE 91

71 &' f f sin sin 0 0 g g Cg l. (4.9c) The characteristic roots of the system is determined by solving from the following 0 ] [ ] [ f B A. (4.10) where denotes the determinant of the matrix. Substituting Equations (4.9a) and (4.9b) into Equation (4.10) results in 0 ) 2 ( 0 cos ) ( 0 ) 2 ( cos ) ( 0 0 ) ( 0 0 f f f f f f f f f f f f f fg g g g g g g l l l l l l l g g l lu gH u u u gH u u u u (4.11) After expansion of the above determinant, the characteristic polynomial for is obtained: 0 cos 12 2 f f f f g u ul g l l l l g g g. (4.12) The roots are g g l l l g g l g l l g l g g g l l lu u g u u f f f 2sin (4.13)

PAGE 92

72 When0 g, Equation (4.13) can have real roots only if l gu u Otherwise, the two-fluid model is ill-posed (Gidaspow, 1974). If 0 / g, the well-posedness with real roots requires sin2 2 2g U u u Ul g l g g l l c l g f f (4.14) Equation (4.14) gives the critical value cU for the slip velocity U between two phases beyond which the system becomes ill-posed. The two-fluid model stability criterion from the characteristic analysis is exactly the same as that from the IKH analysis on two-fluid model by Barnea and Taitel (1994) as shown below. 4.1.3.2 Inviscid Kelvin-Helmholtz (IKH) analysis and linear instability IKH analysis (Barnea and Taitel, 1994) provides a stability condition for the linearized two-fluid model as well as useful information on the growth rate of an infinitesimal disturbance in the two-fluid model. Splitting the flow variables into the base variables and the small disturbances, such as l l ~ expressing the disturbances on the form of kx t Ilf 0 exp ~ (4.15a) kx t I ul lf 0 exp ~ (4.15b) kx t I ug gf 0 exp ~, (4.15c) kx t I ppf 0 exp ~, (4.15d) where ~ denote disturbance value, 1 f I denotes imaginary unit, is the amplitude of perturbation, 0 is the angular frequency of wave and k is the wavenumber. Substituting them into the differential governing equations (4.1, 4.2, 4.4 and 4.6), and

PAGE 93

73 linearizing the resulting equations, the following system is obtained for the disturbance amplitudes, T p g l , 0 0 cos 0 cos 0 0 0 0 f f f f f f fp g l g g l l l l l l g g l lk k u g H k k k u g H k k k u k k u 0 0 0 0. (4.16) For non-trivial solutions to exist, the following dispersion relation between the wave speed c and the angular frequency 0 must hold g g l l l g g l g l l g l g g g l l lu u g u u k c 0 f f f 2sin (4.17) It is note that the negative imaginary part of 0 determines the growth rate of disturbance. Equation (4.17) is identical to Equation (4.13), only with being replaced by c Details of the derivation for IKH stability condition can be found in Barnea and Taitel (1994). 4.1.4 Analysis on Computational Instability 4.1.4.1 Description of numerical methods In general, the governing equations (4.1, 4.2, 4.4, and 4.6) are solved iteratively. The basic procedure is to solve the continuity equation of liquid for the liquid volume fraction, and the liquid and gas phase momentum equations for the liquid and gas phase velocities. To obtain a governing equation for the pressure, Equation (4.1) and Equation (4.4) are first combined to form a total mass conservation, 0 l l g gu x u x (4.18)

PAGE 94

74 Substituting the liquid and gas momentum equations into the above leads to sin cos sin cos2 2 2 2 g x H g g x H g x u u x x p xg l g l l l g g l l g g l l (4.19) To solve the pressure equation, SIMPLE type of pressure correction scheme (Patanka, 1982; Issa and Kempf, 2002) is used in this study. A finite volume method is employed to discretize governing equation. A staggered grid (Figure 4-2) is adopted to obtain compact stencil for pressure (Peric and Ferziger, 1996). On the staggered grids, the fluid properties such as volume fractions, density and pressure are located at the center of main control volume, and the liquid and gas velocities are located at the cell face of main control volume. Figure 4-2 shows the staggered grids arrangement. Wu PuEuPp P x Velocity control volume Main control volume EpWpWEwueuwpepew Figure 4-2. Staggered grid arrangement in two-fluid model. The Euler backward scheme is employed for the transient term. The discretized liquid continuity equation becomes 00 f f w l l e l l P l P lu u t x (4.20)

PAGE 95

75 where the superscript 0 denotes the values of the last time step. The subscript P refers to the center of the main control volume, and subscripts e and w refer to the east face and west face of main control volume, respectively. The liquid velocity on the cell face is known, and the volume fraction on the cell face can be evaluated using various interpolation schemes. Among them, central difference (CDS), 1st order upwind (FOU), 2nd order upwind (SOU) and QUICK schemes are commonly used. Equation (4.4) for the gas phase is similarly discretized. The liquid momentum equation is integrated on the velocity control volume. Using similar notations, one obtains sin cos0g x g H p p u u u u u u t xP l l e l w l e w l P l w l l w l e l l e l P l l P l l f f f f f (4.21) where P e w refer to the center, east face and west face of the velocity control volume, respectively. The cell face flux is the liquid velocity, which is obtained by using central difference, and the volume fraction and liquid velocity at the cell face, which are transported variables, can be interpolated by using different schemes. It is important to note that the interpolation method used for the Equation (4.21) must be exactly the same as those for Equation (4.20). For example, if FOU is used in Equation (4.20), the cell face flux on the east face of velocity control volume in Equation (4.11) is 0 0 ,e l E l l e l P l l e l e l lu MAX u u MAX u u u f f (4.22) If CDS is used in Equation (4.20), the cell face flux on the east face in Equation (4.21) is evaluated as e l E l l P l l e l e l lu u u u u 2 (4.23)

PAGE 96

76 Using similar discretization procedure, the gas phase momentum equation is integrated: sin cos0 g x g H p p u u u u u u t xP g g e l w l e w g P g w g g w g e g g e g P g g P g g f f f f f (4.24) For convenience, the discretized mass or momentum equations are written in a general form B A A AW w E e p p (4.25) where is the variable to be solved, A is the coefficient, B is the general source term. For the pressure correction scheme, Equation (4.18) is integrated across the main control volume. The discretized equation is 0 f fw l l e l l w g g e g gu u u u (4.26) Because Equation (4.18) is obtained by combining Equation (4.1) and Equation (4.4), the discretization scheme for Equation (4.26) should be exactly the same as those for Equation (4.20) and the discretized equation of Equation (4.3). For instance, if CDS is used in Equation (4.20), it must be used in the main control volume for Equation (4.26): 0 2 2 2 2 f f W l P l w l E l P l e l W g P g w g E g P g e gu u u u (4.27) The final discretized pressure equation is obtained by substituting these two momentum equations, Equation (4.21) and Equation (4.24) into Equation (4.26). This yields b p a p a p aW w E e p p (4.28)

PAGE 97

77 e l p l l E l p l e v p v v E v p v eA A a f f 2 2, (4.29a) w l p l l W l p l w v p v v W v p v wA A a f f 2 2, (4.29b) w e pa a a f f (4.29c) 2 2 2 2e l E l P l w l W l P l e v E v P v w v W v P vu a a u a a u a a u a a b f f (4.29d) where, p represents the pressure correction value, u represents the imbalanced velocity, and pA is from the corresponding discretized liquid or gas momentum equation, Equation (4.15). The flow chart of the pressure correction scheme is shown in Figure 4-3. Similarly, the pressure correction schemes with FOU, SOU, CDS, and QUICK can be obtained. Consistently handling the discretization is critical to the reduction of numerical diffusion and dispersion. Barnea and Taitel (1994) showed that the viscosity of fluid can dramatically degrade the stability of two-fluid model through viscous Kelvin-Helmholtz stability analysis. Although the viscosity in two-fluid model appears as the body force instead of 2nd order derivative terms in the modified governing equation, it is hypothesized that the numerical diffusion and dispersion appearing as derivative in the modified governing equations produce similar impact on the stability of two-fluid model.

PAGE 98

78 Solve l andg using Equation (4.20) Solve lu andgu using Equation (4.21, 4.24) Solve p using Equation (4.28) Update lu andgu No Initial conditions End Yes t t Boundary condition If maxt t ? No Yes If lu converges? Figure 4-3. Flow chart of pressure correction scheme for two-fluid model. 4.1.4.2 Code validation dam-break flow The pressure correction scheme is first validated by computing the transient flow due to dam-break flow (Figure 4-4). The liquid flow is assumed to be over a horizontal flat surface and the flow is assumed to be one-dimensional. On the left side of the dam is a body of stationary water in the reservoir with the flat surface of height H On the right

PAGE 99

79 side of dam is a dry river bottom surface. After the dam breaks suddenly, the water in the reservoir flows to the downstream due to the gravitational force. If there are no friction between the fluid and the wall and no viscosity inside the fluid and air pressure is a constant, an analytical solution for the liquid velocity based on St Venant equation can be found (Zoppou and Roberts, 2003). The result is shown in Table 2.1. Dam Dry river plate Reservoir H x yx=0 Figure 4-4. Schematic for dam-break flow model. To solve dam-break flow, the pressure at interface, the vapor phase density and velocity are set to zero. Second order upwind scheme as the cell face interpolation scheme is implemented in the pressure correction scheme. Figure 4-5 compares water depth between the present numerical solution and the analytical solution at t=50s. Two solutions match very well except at the tail end of the liquid, where the numerical solution is smooth due to a little numerical dissipation. Figure 4-6 compares liquid velocities between the numerical and analytical solutions at t=50s. Again, these two solutions match very well except at the leading and tail ends. The discrepancy at the leading end is due to that the liquid layer is too thin and the numerical result is prone to error.

PAGE 100

80 Table 4-1. Analytical solution for dam-break flow (Zoppou and Roberts, 2003). x( position) u ( water velocity) h (water depth) gH t x f 0 u H h gH t x gH t 2 f t x gH u3 2 22 9 4 f t x gH g h gH t x 2 0 u 0 h Although only the dynamics of liquid phase is considered in the dam-break flow, it is still a solid step for validating the coupling of the pressure and liquid flow (liquid volume fraction and liquid velocity) in numerical scheme. When both the liquid and the gas phase present in the flow, the instability in two-fluid model rises due to the interaction of the liquid and the gas phase, when the slip velocity is large. Numerical instability of pressure correction scheme emerges and destroys the numerical results when the two-fluid model near ill-posedness. This numerical instability will be investigated in the next section and the code will be validated using the theoretical results of inviscid Kelvin-Helmholtz analysis (Barnea and Taitel, 1994). z(m) h(m)0500100015002000 0 1 2 3 4 5 6 7 8 9 10Numerical Analytical t=50sec t=0sec Figure 4-5. Water depth at t=50 seconds after dam break.

PAGE 101

81 z(m) velocity(m/s)0500100015002000 0 2 4 6 8 10 12 14 16 18 20Numerical Analytical t=50sec Figure 4-6. Water velocity at t=50 seconds after dam break. 4.1.4.3 Von Neumann stability analysis for various convection schemes Similar to the well-posedness of the differential equations, numerical stability is essential to solve the discretized systems. Von Neumann stability analysis is commonly used to analyze the stability of finite difference schemes (Hirsch, 1988; Shyy, 1994). ui-1 ui-0.5 ui+0.5 pi-1 pi pi+0.5 i+0.5i-1 i x Figure 4-7. Grid index number in staggered grid for von Neumann stability analysis.

PAGE 102

82 To begin with, the 1st order upwind (FOU) scheme is used as an illustrative example. For simplicity and for practical purpose, both liquid and gas velocities are assumed positive. Discretization of Equation (4.20) with FOU scheme leads to 01 12 1 2 1 f ff f fn i l n i l n i l n i l n i l n i lu u x t (4.30) Splitting the variables into base value and disturbances, the linearized equation for the disturbance l is 0 1 12 1 2 1 f f f ff f fn i l n i l l n i l n i l l n i l n i lu u u x t (4.31) where ^ denotes disturbance values. The disturbances may be expressed as Ikx n n i le E (4.32a) Ikx n l n i le E u (4.32b) Ikx n v n i ve E u (4.32c) where E is a common amplitude factor, and k is the wavenumber. Equation (4.31) is simplified to 0 1 12 1 2 11 f f f f f f I I l l I le e e u G t x, (4.33) where G is the amplification factor defined as 1fn n E E G (4.34) and is phase angle: x k (4.35)

PAGE 103

83 defined over [0, ] and x k max represents the highest resolvable wavenumber in the computational domain for the given grid. Thus ,corresponds to short wave components. The wave growth equation for the gas phase mass conservation equation is similarly obtained: 0 1 12 1 2 11 f f f f f f f I I g g I ge e e u G t z. (4.36) For the liquid momentum equation, Equation (4.21) is discretized with the FOU scheme, sin cos2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 11 1 1 1 1 g x H g p p u u u u x t u un i l l n i l n i l l n i n i l n i l n i l l n i l n i l l n i l n i l n i l n i l n i l f f f f f f f f (4.37) Linearization of Equation (4.37) leads to cos 1 1 2 1 1 12 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1n i l n i l l l l n i n i n i l n i l l l l n i l n i l n i l n i l l l n i l n i l l n i l n i l l l lH g p p u u u u u u u u u t x f f f f f f f f f f f (4.38) This equation can be rearranged as cos 1 1 1 2 1 12 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1n i l n i l l l l n i n i n i l n i l l l n i l n i l l n i l n i l l l l n i l n i l l l n i l n i l l l lH g p p u u u u u t x u u u u u t x f f f f f f f f f f f (4.39)

PAGE 104

84 The first three terms in Equation (4.39) cancel out by using the linearized liquid mass conservation equation, Equation (4.31), at grids i and i+ 1, as shown in Figure 4-4. Therefore the discretized liquid momentum equation is cos 1 1 12 1 2 1 2 1 2 1n i l n i l l l l n i n i n i l n i l l l n i l n i l lH g p p u u u u u t x f f f f f f (4.40) The gas phase momentum equation for the disturbance gu is obtained similarly: cos 1 1 12 1 2 1 2 1 2 1n i l n i l g g g n i n i n i g n i g g g n i g n i g gH g p p u u u u u t x f f f f f f (4.41) The pressure term can be canceled by combining Equations. (4.40) and (4.41), cos 1 1 12 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1n i l n i l l l g l n i g n i g g g n i l n i l l l n i g n i g g n i l n i l lH g u u u u u u u u u u t x f f f f f f f f f f f f (4.42) Substituting Equations (4.32) into Equation (4.42) leads to 0 1 1 1 1 cos1 12 1 2 1 f f f f f f ff f f f f I g g g g I l l l l I I l l g le u G t z e u G t z e e H g (4.43) Equation (4.33, 4.36, 4.43) can be written in a matrix form as 0 1 1 1 1 cos 0 1 1 0 1 11 1 1 12 1 2 1 2 1 2 1 2 1 2 1 b t n n n f f f b t n n n f f f f f f f f f f f f f f f f f f f f f f f f l g I l l I g g I I l l g l I I l I l I I g I ge u G t x e u G t x e e H g e e e u G t x e e e u G t x (4.44)

PAGE 105

85 Non-trivial solutions for T l g , exist only when the determinant of the matrix is zero. Hence, the equation for the growth rate (or amplification factor) G is 01 2 1 f fc G b G a (4.45) where a, (4.46a) f l l l g g gCFL CFL b1 1 2, (4.46b) f 2 sin 4 cos 1 12 2 2 2 l l g l l l l g g gH g x t CFL CFL c, (4.46c) and CFL are Courant numbers defined as l lu x t CFL (4.47a) g gu x t CFL (4.47b) The values of in Equation (4.46) are given in Table 4-2. Table 4-2. for different discretization schemes. Scheme 1st order upwind Ieff 1 Central difference 2 I Ie eff 2nd order upwind 2 4 32 I Ie ef f f QUICK 8 7 3 32 I I Ie e ef f f From Equation (4.45), the amplification factor can be easily found that

PAGE 106

86 ac b b a G 4 22f f (4.48) Stability requires 1 G for all 4.1.4.4 Initial and boundary conditions for numerical solutions In von Neumann stability analysis, a periodic boundary condition is implicitly assumed. In computations, such periodic boundary conditions are necessarily employed in order to provide a direct comparison. The von Neumann stability analysis is for the growth of an infinitesimal disturbance. In computations, a small initial disturbance must be properly introduced without generating additional higher harmonic noise. The best initial condition for the disturbance is that from the wave growth equation, such as given by Equation (4.44) for FOU. However, this approach makes the imposition of the initial condition too complicated, since initial conditions vary from one numerical scheme to another. A simpler but effective approach is to use the solution of inviscid Kelvin-Helmholtz analysis. Thus, if k and are specified at t=0, corresponding values for 0, l,g and p must be consistent with Equation (4.16). An initial condition that is consistent with the governing equations for the small disturbance is important for studying wave growth in the context of inviscid two-fluid model. If the initial condition is inconsistent with the original equations, unexpected higher harmonic wave components will develop. Due to possible instability, it may grow and overtake the original disturbance and make the assessment of the accuracy of the numerical scheme impossible.

PAGE 107

87 4.1.5 Results and Discussion 4.1.5.1 Computational stability assessment based on von Neumann stability analysis For well-posed inviscid two-fluid model, the small disturbance will not grow or decay so that 1 G. Comparison of stability based on the behavior of G for the FOU, SOU, CDS, and QUICK schemes will allow for an effective assessment of the accuracy (if 1 G) and instability (if1 % G) conducted for flow conditions before, near, and after the instability. It is well known that the FOU is less accurate with high numerical diffusion. High order schemes, such as SOU, CDS, and QUICK, have lower numerical diffusion (Shyy, 1994). In this study, for illustration purposes, water and air are considered and the pipe diameter is taken to be 0.078m. The computational domain is 1m long, the grid number is N= 200. The pipe inclination angle is = 0. The base values of flow variables are l = 0.5, lu=1 m/s, gu = 17 m/s. The CFL value of liquid is 0.1. Stability condition based on Equation (4.14) for the above parameters is s m U Uc/ 0768 16 Thus, the twofluid model for this condition is well-posed analytically. It serves as an ideal testing case to assess the performances of various convection schemes since the system is quite close to being ill-posed. There are two values of G given by Equation (4.48) and the larger one determines the instability. Hence, only the larger growth rate is used here. Figure 4-8 compares the growth rate G of four numerical schemes. The solid line is the theoretical IKH growth rate ( G =1). The dotted line is for the CDS scheme. It is slightly lower than one but quite close to one with a small damping at high wavenumber end. This implies the CDS is an ideal scheme to compute the two-fluid model. The

PAGE 108

88 dashed line is for the FOU scheme, which possesses excessive numerical damping at high k end. Furthermore, 1 % G at low k Thus, computations using FOU are unstable for this flow condition. The dash and dot line is for SOU scheme. Although SOU is regarded as a better scheme than FOU with less numerical diffusion, its performance for the two-fluid model is very poor. For large k the numerical diffusion of SOU is even more excessive than that of FOU. For small k the growth rate of SOU is also much larger than that of FOU. Dashed double dotted line is the growth rate of the QUICK scheme. Its numerical damping at high k is lower than that of FOU and SOU, but it is still considerably larger than that of CDS. At small k G is slightly larger than 1 indicating that QUICK is unstable as well. The reason that the growth rate of CDS is close to the analytical growth rate is probably due to a lack of 2nd order diffusion error and low dispersion error. Overall performance of FOU is better than that of SOU which suggests that the diffusion and dispersion error in the two-fluid model has much more negative impact on the stability than that in the simple convection-diffusion equation. The interpolation of QUICK is essentially linear interpolation with the upwind correction. Therefore, its numerical diffusion and stability are worse than that of CDS, but better than that of FOU and SOU. When U is smaller than the critical value cU given by the IKH stability analysis, the growth rate of all harmonic component in the computational domain are less than one. However, if cU U % the two-fluid model should be analytically ill-posed, and the growth factor for some range of k will exceed one. Figure 4-9 shows various growth rates for various value of U when the CDS is used. From numerical results, a neutral stability condition of CDS is found to be near s m UCDS c/ 0773 16, for the condition used in Figure 4-9, which is quite close to s m Uc/ 0768 16 As U further

PAGE 109

89 increases, the growth rate increases as well. The range of k for instability becomes wider. The growth rate of CDS scheme matches that of IKH only at very low wavenumber. In the high k range, numerical damping causes the growth rate to be lower than one. Figure 4-10 shows the growth rate of the FOU scheme for different values of U Unlike the CDS scheme, there is no significant change of G when U varies in the similar range. Numerical results indicate that the neutral stability for the condition shown in Figure 4-10 is s m UFOU c/ 772 14, which is much lower than the analytical value of s m Uc/ / 0768 16 The behavior of SOU and QUICK scheme is close to the FOU. The stability condition for SOU is s m USOU c/ 73 13, and for QUICK, it is s m UQUICK c/ 03 16, 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 00.511.522.533.5G FOU SOU CDS QUICK IKH Figure 4-8. Comparisons of growth rates of various numerical schemes.200 N, 5 0 la s m ul/ 1 s m ug/ 17 and 1 0 lCFL

PAGE 110

90 0.994 0.996 0.998 1 1.002 1.004 1.006 00.511.522.533.5G 12 16.0768 16.1 16.5 17 16.1,IKH 16.5,IKH 17,IKH U(m/s) IKH, U=16.0768 Figure 4-9. Growth rate of CDS scheme at different l gu u U f 200 N, 5 0 la s m ul/ 1 and 1 0 lCFL 0.8 0.85 0.9 0.95 1 1.05 00.511.522.533.5G 12 14.772 16.0768 17 18 17, IKH 18, IKH U(m/s) IKH, U=16.0768 Figure 4-10. Growth rate of FOU scheme at different l gu u U f 200 N, 5 0 la s m ul/ 1 and 1 0 lCFL

PAGE 111

91 Based on Equation (4.13) and Equation (4.17), for the given fluid properties and pipe size, only U affects IKH stability and ill-posedness. On the other hand, in Equation (4.45), the numerical stability is not only controlled by U but also by the individual liquid and gas phase velocities, grids density, and time step. Figure 4-11 shows the effect of the liquid velocity on the growth rate in CDS with s m U / 16 and m s x t / 1 0 For s m ul/ 01 0 and s m ul/ 1 0 G decreases monotonically with the phase angle. Damping appears at high k When lu increases, G at high k range rises significantly, leaving a high damping saddle at the intermediate k range. On the other hand, if U is constant, l gCFL CFL is much larger than one when lu is small and it is computationally difficult to keep both lCFL and gCFL in the moderate range, which is essential to the computational stability and accuracy. Figure 4-12 shows the effect of lu on G for the FOU scheme with s m U / 16 m s x t / 1 0 The behavior of FOU is much different from that of CDS. When lu is small, most harmonics are unstable. For a larger lu excessive numerical diffusion on the fluid flow associated with FOU scheme makes the computations stable. Figure 4-13 and Figure 4-14 show the effect of x t on G for the CDS and FOU schemes. Both show increasing numerical damping with increasing x t resulting in a decrease in G This can be explained by examining Equation (4.45c), where the last term involves the product of 2 x t and gravitational accelerator. It is well known that gravity stabilizes the stratified flow. Thus increasing x t computationally enhances the stability, if all other parameters are hold constant.

PAGE 112

92 0.993 0.994 0.995 0.996 0.997 0.998 0.999 1 00.511.522.533.5G0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1G (u=10) 0.01 0.1 1 10 Liquid velocity(m/s) Figure 4-11. Growth rate of CDS scheme at different lu .200 N, s m U / 16 5 0 la and m s x t / 1 0 0 0.2 0.4 0.6 0.8 1 1.2 00.511.522.533.5G 0.01 0.1 1 10 Liquid velocity (m/s) Figure 4-12. Growth rate of FOU scheme at different lu 200 N, s m U / 16 5 0 la and m s x t / 1 0

PAGE 113

93 0.99 0.992 0.994 0.996 0.998 1 00.511.522.533.5G0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1G( t/ x=1s/m) 0.001 0.01 0.1 1 t/ x(s/m) Figure 4-13. Growth rate of CDS scheme at different x t 200 N, s m ul/ 1 ,s m U / 16 and 5 0 la 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 00.511.522.533.5G0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1G ( t/ x=1s/m) 0.001 0.01 0.1 1 t/ x(s/m) Figure 4-14. Growth rate of FOU scheme at different x t .200 N, s m ul/ 1 ,s m U / 16 and 5 0 la

PAGE 114

94 4.1.5.2 Scheme consistency tests Consistency of a numerical scheme requires that the solution of the discretized equations tends to the exact solution of the differential equations as the grid spacing x and time step t tend to zero (Hirsch, 1988). In another word, the truncation error must approach to zero as 0 # t x for the Taylor series expansion to be valid. In the scheme consistency test, growth of an infinitesimal sinusoidal disturbance with 2 k introduced at t=0 are examined for a range of t and x The computational domain is again 1m long. Initial conditions for volume fraction, liquid and gas velocities and pressure are compatible with the results of IKH analysis, Equation (4.16). Figure 4-15 compares the growth of liquid velocity disturbance, lu using N=100, 200 and 400 at s t 5 1 The cell face interpolation scheme is CDS, s m ul/ 1 s m ug/ 5 17 0 ,1 0 lCFL and 5 0 la Because lCFL lu and gu are constant in this comparison, x t is a constant. This ensures that t goes to zero as x approaches zero. An analytical solution for wave growth by IKH analysis is also plotted in Figure 4-15 for comparison with the numerical results. With N increasing from 100 to 400, the error between the exact and numerical solutions decreases as required by consistency. Although the error with 100 N is slightly larger than that with 200 N and 400 N, the solution at 200 N is quite close to that with 400 N. This suggests that 200 N is large enough for 2 k; hence 200 N for 2 k is used unless otherwise mentioned.

PAGE 115

95 -6.00E-06 -4.00E-06 -2.00E-06 0.00E+00 2.00E-06 4.00E-06 6.00E-06 00.20.40.60.81 x(m)Disturbance (m/s)N=100 IKH N=400 N=200 t=1.5s Figure 4-15. Comparison of lu growth using CDS scheme on different grids. s m ul/ 1 s m ug/ 5 17 1 0 lCFL and 5 0 la 4.1.5.3 Computational assessment based on the growth of disturbance To validate the pressure correction scheme, comparisons between the computed wave growth rates and the analytical growth rates from the von Neumann stability analysis are presented. First we consider 2 k, N=200, s m ul/ 1 s m ug/ 15 5 0 la, 05 0 lCFL, and the computational time is t=4s. The convection scheme used is CDS. Based on IKH analysis, the disturbance should not grow. Figure 4-16 shows that at t= 4s, the disturbance of the computed liquid velocity is slightly weaker than that of the analytical solution. The phases of the analytical and numerical solutions are almost identical. This demonstrates excellent performance of CDS for the two-fluid model. Figure 4-17 shows the measured decay of the amplitude of the liquid velocity disturbance. The growth rate for each time step using CDS with 2 k is 0.999997962 based on the von Neumann stability analysis. Since it takes 16000 steps to reach t=4s, the

PAGE 116

96 ratio of the amplitude at t=4s to that t=0 is 967918 0 999997962 016000. The actual rate using CDS is 0.96807, with an error of 0.016%. Careful examination of Figure 4-17 reveals small amplitude wrinkles in the wave amplitude. The reason is that the initial condition is taken from the analytical solution of IKH analysis, which is slightly different from the solution by the CDS dispersion equation. This mismatch of the initial conditions leads to the generation of a weak high harmonic wave. Very low numerical diffusion of CDS ensures that this weak wave exists for a long time. Figure 4-18 shows wave growth for an ill-posed condition, with s m ul/ 1 s m ug/ 5 17 5 0 la, 1 0 lCFL. The relative velocity s m U / 5 16 is larger than 16.0768m/s cU and s m UCDS c/ 0773 16, so that any disturbance will grow with time analytically and computationally. The initial disturbance is introduced at t=0 with 2 k. In Figure 4-18, the computational results are presented for t=4s (after 8000 time steps) and t=5.2s (after 10399 time steps). The original long wave with 2 k is overwhelmed by a much stronger short wave at t=5.2s. In Figure 4-19, the growth history of the amplitude is presented. The initial growth stage, from t=0s to t=4s, corresponds to the growth of the initial long wave with 2 k. This is further confirmed by comparing with the analytical growth rate for 2 k. The predicted amplitude ratio based on von Neumann analysis is 22.84 from t=0 to t=4s, while the computed amplitude ratio is 22.89. After the initial growth stage, a short wave with higher growth rate takes over and becomes dominant in the numerical solution. This occurs in the stage of fast growth (s t 5 %) in Figure 4-19. For s m U / 5 16 in the present computation, the wave with the highest growth rate occurs at 0.282743max based on von Neumann analysis. If the

PAGE 117

97 1m domain is occupied by this wave, the total number of waves is ) 2 /(max N n =9, which is exactly the number of waves in Figure 4-18. -4.00E-06 -3.00E-06 -2.00E-06 -1.00E-06 0.00E+00 1.00E-06 2.00E-06 3.00E-06 4.00E-06 00.20.40.60.81 x (m)Disturbance (m/s)Analytical result Numerical result t=4s t=0s Figure 4-16. lu using CDS scheme in the computational domain. 200 N, s m ul/ 1 ,s m U / 14 05 0 lCFL 5 0 la and s t 4 3.50E-06 3.52E-06 3.54E-06 3.56E-06 3.58E-06 3.60E-06 3.62E-06 3.64E-06 00.511.522.533.544.5 t (s)Amplitude (m/s) Figure 4-17. Amplitude of liquid velocity disturbance lu using CDS scheme. 200 N, s m ul/ 1 ,s m U / 14 05 0 lCFL 5 0 la and s t 4

PAGE 118

98 -2.00E-04 -1.50E-04 -1.00E-04 -5.00E-05 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 00.10.20.30.40.50.60.70.80.91 x(m)Disturbance(m/s)t=5.2s t=4s Figure 4-18. lu using CDS scheme after 10399 steps of computation,200 N, s m ul/ 1 s m U / 5 16 1 0 lCFL 5 0 la and s t 2 5 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 01234567 t(s)Amplitude(m/s) Figure 4-19. Growth history of lu solved using CDS scheme,200 N, s m ul/ 1 s m U / 5 16 1 0 lCFL 5 0 la and s t 2 5

PAGE 119

99 Next, a comparison between the computational results using the FOU scheme and predictions from the von Neumann analysis is presented. The parameters of computation are 200 N, s m ul/ 5 0 s m U / 16 0 02 0 lCFL, and 5 0 la The flow is stable based on IKH stability analysis, but unstable based on the von Neumann stability analysis. The growth rate of FOU under the condition stated is shown in Figure 4-20. The highest growth rate occurs at max= 0.586903 with 00201 1max G. It is anticipated that this harmonic for max will grow from the round-off error and eventually dominate the computation. There should be about 19 2 /max, N n peak to peak cycles in the 1m domain. In the computation, a small amplitude sinusoidal wave with 2 kis introduced at t=0. Figure 4-21 shows the liquid velocity variation after 12000 time steps. Clearly, the short wave has overwhelmed the initial long wave. Because the short waves originate from machine level error, which has a broad spectral distribution, the amplitude and frequency of the waves are not uniform. However, the dominant wave component in Figure 4-21 is 19 n by counting number of peaks in the 1 m computational domain. This agrees very well with the result of von Neumann analysis. Furthermore, for 00201 1maxG, the amplitude can grow by a factor of 2.92x1010 in 12000 steps. Since the initial amplitude of machine level noise is of 1610fO, it is reasonable to expect the amplitude of the dominant short wave to be on the order of 610fO after 11800 time steps, which is qualitatively consistent with the results shown in Figure 4-21. Similar comparison between the predicted and computed wave growth by SOU scheme is presented next. The parameters of computation are 200 N, s m ul/ 1 s m U / 16 0 05 0 lCFL and 5 0 la The growth rate G as a function of

PAGE 120

100 is shown in Figure 4-22. The maximum of G occurs at 911062 0max with 00886 1max G The liquid velocity disturbance after 3000 computational steps is shown in Figure 4-23. The dominant wave is with 29 n, in Figure 4-22, while 29 ) 2 /(max max, N n based on von Neumann stability analysis. Similar to the case of FOU, if the initial amplitude of wave dominating SOU computation is 1610fO after 3400 steps, the wave amplitude should reach the order of 5 3000 max 1610 10f f, G. 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 1.005 00.511.522.533.5G Figure 4-20. Growth rate of FOU scheme,200 N, s m ul/ 5 0 s m U / 16 02 0 lCFL and 5 0 la It is interesting to note that the SOU scheme involves five grids points, which is not solvable by efficient Thomas algorithm (Hirsch 1988) in general. To use Thomas algorithm, five points in discretized equations must be reduced to three points and contribution from the other two points is added to the source terms. However, numerical simulation of such a deferred SOU scheme leads to a dominant wave with frequency close to that of dominant wave in FOU instead of SOU. The behavior of deferred SOU is

PAGE 121

101 unpredictable by von Neumann analysis. To obtain the solution of two-fluid model with SOU authentically, an iteration method is employed in this study to ensure that the variables at the new time step are solved simultaneously. Although the method is not efficient for the application of two-fluid model, it is employed in this study for the purpose of assessing the performance of SOU scheme. 4.1.5.4 Discussion on the growth of short wave In the last section, it is seen that the undesirable short waves emerge from the computation and destroy the original information in the computational domain because of numerical instability. This numerical instability is the character of numerical scheme and influenced by the ill-posedness of two-fluid model. Preliminary analysis shows that the unwanted short waves come from computers machine roundoff error, but the growth history of short wave is still not clear. In order to clearly demonstrate how the short wave emerges and develops during computation, another numerical experiment is conducted. For the FOU scheme used in the last section for Figure 4-20 and 4-21, a series of computations is carried out using successively decreasing initial amplitude (from s m / 104f to s m / 1012f) for the liquid velocity disturbance lu The growth of the amplitude of lu as a function of time is recorded for each initial amplitude while all other physical and computational parameters are fixed. Figure 4-24 shows the variations of the wave amplitude for all values of initial disturbance amplitude in lu In Figure 4-24, it is observed that during the initial stage, all amplitudes grow according to the G (k=2 ) in the form of nG0 in which 0 is the initial amplitude of the disturbance, G is the growth rate at k=2 based on von Neumann analysis, and n denotes

PAGE 122

102 n th time step. Since the short wave grows out of the machine round-off error independently in the form of n rGmax in which 1610 ~fOr is the amplitude of the round-off error whose exact value is uncertain, and 00201 1max G is the maximum growth rate for the FOU scheme for the present condition obtained from the von Neumann analysis. It corresponds to max= 0.586903. Clearly, smaller value of 0 requires less time (or small n ) for the round off error to take over the primary wave (k=2 ). The envelope of these computed amplitudes seem to agree well with the n rGmax denoted by the thick dash line in Figure 4-21 with an estimated value of 1610 2f" r. -2.00E-05 -1.50E-05 -1.00E-05 -5.00E-06 0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05 00.10.20.30.40.50.60.70.80.91 x(m)Disturbance (m/s) Figure 4-21. luusing FOU scheme after 12000 steps of computation. 200 N, s m ul/ 5 0 ,s m U / 16 02 0 lCFL and 5 0 la

PAGE 123

103 0.8 0.85 0.9 0.95 1 1.05 00.511.522.533.5G Figure 4-22. Growth rate of SOU scheme. 200 N, s m ul/ 1 s m U / 16 05 0 lCFL and 5 0 la -4.00E-04 -3.00E-04 -2.00E-04 -1.00E-04 0.00E+00 1.00E-04 2.00E-04 3.00E-04 4.00E-04 00.10.20.30.40.50.60.70.80.91 x(m)Disturbance(m/s) Figure 4-23. luusing SOU scheme after 3000 steps of computation. 200 N, s m ul/ 1 s m U / 16 05 0 lCFL and 5 0 la

PAGE 124

104 As computation continues, the wave amplitudes in Figure 4-24 do not become unbound. This is different from the case with the use of CDS in Figure 4-19. The difference stems from the following: 1. High numerical diffusion of FOU scheme causes decrease of U 2. When the disturbance amplitude becomes O (1), the based flow parameters are changed. The nonlinear effects in the discretized system of equations become strong so that the numerical solution may have evolved to a different stable state. The result of von Neumann analysis is no longer applicable. 1.00E-16 1.00E-15 1.00E-14 1.00E-13 1.00E-12 1.00E-11 1.00E-10 1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 00.511.522.533.54 t(s)Amplitude(m/s)growth of round-off error Figure 4-24. Growth history of luunder different initial amplitude using FOU scheme. 4.1.5.5 Wave development resulting from disturbance at inlet In the comparison of last section, periodic boundary conditions are used to match the requirement of von Neumann stability analysis. However, the applications of two-fluid model are not limited to periodic boundary conditions. In this section, wave propagation developed from an inlet disturbance is studied. The growth rate of the disturbance depends on the flow parameter and the numerical scheme. Initially small

PAGE 125

105 sinusoidal waves for l, lu and gu with 2 4 k satisfying Equation (4.49) is introduced at s t 0 At the inlet, the boundary conditions of l, lu and gu with 2 4 k satisfying Equation (4.49) is posed as function of t At the outlet, 2nd order extrapolation is employed. Figure 4-25 shows the growth of inlet disturbance by FOU scheme under well-posed condition. The computational parameters are 200 N, s m ul/ 1 s m ug/ 17 0 ,05 0 lCFL and 5 0 la The flow is well-posed and scheme is unstable for low frequency wave. Figure 4-25 clearly shows the exponential growth of a low k wave, as it propagates to the down stream. The flow under similar parameters but s m ug/ 21 is shown in Figure 4-26. The flow is ill-posed with these parameters. The major difference between the Figure 4-25 and Figure 4-26 is that the wave growth rate in Figure 4-26 is much larger than that in Figure 4-25. If the computational domain is longer enough, both computations will break down. All the behavior of waves in Figure 4-25 and Figure 4-26 agrees with the von Neumann stability analysis. Next, inlet disturbance growth with CDS scheme is studied. In Figure 4-27, the computational parameters are the same as those in Figure 4-25. It is shown that from inlet to outlet, the wave grows slowly. This reflects the accuracy of the CDS scheme. The wiggle on the wave is due to the extrapolated downstream boundary conditions. Because it is not a non-reflection boundary condition, high frequency waves are generated at downstream boundary and propagate upstream until they are bounced back by upstream boundary. Low damping rate of CDS scheme allows the high frequency waves to exist for a long time in the computational domain. Figure 4-28 shows the flow under same computational parameter as in Figure 4-26. Since the flow is ill-posed, the wave grows so

PAGE 126

106 fast that the computation breaks down while the disturbance has not reached the middle of domain. Comparison between the Figure 4-26 and Figure 4-28 shows that CDS scheme is less stable than the FOU scheme if the velocity difference is notably higher than the IKH stability criterion. This is confirmed by the comparison of growth rate of FOU and CDS (Figure 4-29) at the condition of Figure 4-26 and Figure 4-28. This feature suggests that the FOU scheme is preferred to the CDS scheme if the velocity difference between gas and liquid phase is extremely large. 4.1.6 Conclusions Numerical instability for the incompressible two-fluid model near the ill-posed condition is investigated for various cell face interpolation schemes, while the pressure correction method is used to obtain the pressure, volume fraction and velocities. The von Neumann stability analysis is carried out to obtain the growth rate of a small disturbance in the discretized system. The central difference scheme has the best stability characteristics in handling the two-fluid model, followed by the QUICK scheme. It is quite interesting to note that the excessive numerical diffusion in the 1st order upwind scheme seems to promote the numerical instability in comparison with the central difference scheme. Despite its nominal 2nd order accuracy and popularity, the 2nd order upwind scheme is much more unstable than the 1st order upwind scheme for solving twofluid model equations. Different discretization schemes for the convection term with varying degrees of numerical diffusion and dispersion cannot cause a delay the onset of instability; they often promote instability in the two-fluid model.

PAGE 127

107 -5.00E-06 -4.00E-06 -3.00E-06 -2.00E-06 -1.00E-06 0.00E+00 1.00E-06 2.00E-06 3.00E-06 4.00E-06 5.00E-06 6.00E-06 00.10.20.30.40.50.60.70.80.91 x(m)Disturbance(m/s) Flow direction Figure 4-25. lu propagates in the pipe with FOU at well-posed condition, quasi-steady state. -3.00E-03 -2.00E-03 -1.00E-03 0.00E+00 1.00E-03 2.00E-03 3.00E-03 4.00E-03 00.10.20.30.40.50.60.70.80.91 x(m)Disturbance(m/s) Flow direction Figure 4-26. lu propagates in the pipe with FOU scheme at ill-posed condition, quasisteady state.

PAGE 128

108 -1.50E-06 -1.00E-06 -5.00E-07 0.00E+00 5.00E-07 1.00E-06 1.50E-06 00.10.20.30.40.50.60.70.80.91x(m)Disturbance(m/s) Flow direction Figure 4-27. lu propagates in the pipe with CDS at well-posed condition, quasi-steady state. -1.00E-01 -5.00E-02 0.00E+00 5.00E-02 1.00E-01 1.50E-01 00.10.20.30.40.50.60.70.80.91 x(m)Disturbance(m/s) Flow direction Figure 4-28. lu propagates in the pipe with CDS at ill-posed condition, an instance before the computation breaks down.

PAGE 129

109 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 00.511.522.533.5G FOU CDS Figure 4-29. Comparison of growth rate between CDS and FOU schemes. 200 N, s m ul/ 1 s m ug/ 21 05 0 lCFL and 5 0 la The analytically predicted wave amplitude growth rate is also compared with that obtained from carefully implemented computations using various discretization schemes for the convection term. Excellent agreement between the numerical results and the predicted results is obtained for the growth of the wave amplitude and the dominant wavenumber when the computation becomes unstable. Inlet disturbance growth test shows the pressure correction scheme can correctly capture two-phase flow in the pipeline. The relation between computational instability and ill-posedness is discussed. In the presence of a small-amplitude long-wave disturbance, whose amplitude is much larger than the machine round-off error, the growth of the disturbance exactly matches the prediction of the von Neumann stability analysis when the computational stability condition is violated. In the meantime, a shorter wave emerges from the machine roundoff error, and eventually dominates the entire disturbance, which causes the computation

PAGE 130

110 to blow up. This computational instability is widely interpreted as the result of illposedness of the two-fluid model. The results of the present study suggest that the computational instability is largely the property of the discretized two-fluid model and is strongly affected by the inherent ill-posedness of the two-fluid model differential equations. Introduction of numerical diffusion and/or dispersion can significantly change the instability of the discretized system; however, such steps often yield unfavorable computational results. For solving two-fluid models, central difference is recommended since it is much more accurate and dependable than other schemes investigated. 4.2 Viscous Two-Fluid Model 4.2.1 Introduction Inviscid two-fluid model suffers from the ill-posdness problem, which coincides with the invicid Kelvin-Helmholtz instability. It is known that IKH instability in the stratified flow implies that the stratified flow is unstable and transition of flow regimes will occur. The transition can be from stratified flow to slug flow or stratified flow to annular flow, depending on other flow parameters (Taitel and Dukler, 1976; Barnea and Taitel, 1994). However, the instability of viscous two-phase flow in pipe flow, which can be described by viscous two-fluid model, comes earlier than the IKH stability in the pipe flow. Lin and Hanratty (1986, 1987) distinguished the viscous Kelvin-Helmholtz stability (VKH) from the inviscid Kelvin-Helmholtz stability. They also showed that the transition from stratified flow to slug flow is governed by the VKH stability analysis instead of the IKH stability analysis and that the VKH instability is triggered earlier than the IKH instability. In the region where the two-phase flow is VKH unstable but IKH stable, the two-fluid model is well-posed. Issa and Kempf (2003) attempted to simulate the VKH instability under the well-posed condition using the two-fluid model. They qualitatively

PAGE 131

111 captured VKH instability and the transition from stratified flow to slug flow. However, the numerical accuracy of their scheme under the VKH unstable condition is uncertain. In the last section, von Neumann stability analysis of the two-fluid model clearly shows that the flow with the IKH instability cannot be accurately captured by the numerical solution because of the indefinite growth of the disturbance under an unstable condition. In this section, the numerical instability of the viscous two-fluid model will be investigated using von Neumann stability analysis and the relation between the numerical instability and the VKH instability of viscous two-fluid model will be clarified. Furthermore, the wave growth rate obtained using the von Neumann stability analysis is used to validate the numerical scheme for the viscous two-fluid model. 4.2.2 Governing Equations In the viscous two-fluid model, as shown in Figure 4-30, the viscosity of fluid appears in the shear stresses in source terms in the fluid momentum equations. Other assumptions are the same as that of inviscid two-fluid model presented in the previous section. There is no mass transfer between the gas phase and liquid phase, and the surface tension between the two phases is neglected. Both phases are incompressible. Hence, the governing equations are as follows: Pipe cross section Liquid phase Gas phase Interface Liquid velocity lu Gas velocity gu Gravity g Gas velome faction g Liquid velome faction l gt it it lt Figure 4-30. Schematic depiction of viscous two-fluid model.

PAGE 132

112 0 l l luxt (4.49) 0 g g guxt (4.50) l i i l l l l l l i l l l l l lA S A S g x H g x p u x u ttt f f f f sin cos2, (4.51) g i i g g g g l g i g g g g g gA S A S g x H g x p u x u t t t f f f f f sin cos2. (4.52) To close the two-fluid model, the correlations for shear stress must be specified. In this study, the correlations used by Barnea and Taitel (1994) are adopted, as shown in Equations 2.20 to 2.27. 4.2.3 Theoretical Analysis 4.2.3.1 Characteristics and ill-posedness The characteristic analysis for the inviscid two-fluid model shows that the characteristic roots may be complex, which leads to ill-posedness. Similar analysis can be applied on the viscous two-fluid model. Equations 4.49 to 4.52 can be written in vector form as ] [ ] [ ] [ CxB t A U U, (4.53) where [ A ], [ B ] and [ C ] are coefficient matrices given by f f 0 0 0 0 0 0 0 1 0 0 0 1 ] [g g l lu u A (4.54a)

PAGE 133

113 f f g g g g g g l l l l l l g g l lu gH u u gH u u u B 2 0 cos 0 2 cos 0 0 0 0 ] [2 2, (4.54b) &' f f f f f g i i g g g g l i i l l l lA S A S g A S A S g C t t t t sin sin 0 0, (4.54c) The characteristic roots of the system are determined by the following: 0 ] [ ] [ f B A. (4.55) The only difference between the characteristic equation for the inviscid two-fluid model and that for viscous two-fluid model are the friction terms in vector &'C However, Equation (4.55) shows that the characteristics are not affected by vector &'C Thus, viscous terms in the viscous two-fluid model do not affect the characteristics of the two-fluid model. The criterion for ill-posedness for viscous two-fluid model remains the same as that for the inviscid two-fluid model. However, the viscous effect in &'C can affect the linear stability of the viscous two-fluid model to cause flow regime transition. 4.2.3.2 Viscous Kelvin-Helmholtz (VKH) analysis and linear instability It is known that stability of interface between the liquid phase and gas phase is attributed to the viscous Kelvin-Helmholtz instability. Barnea and Taitel (1994) showed that the velocity difference between two phases under VKH instability is less than that under IKH instability. Flow regime transition starts when the flow encounters VKH instability rather than IKH instability. VKH analysis provides not only a stability

PAGE 134

114 condition for the linearized viscous two-fluid model, but also gives growth rate of infinitesimal disturbance in the viscous two-fluid model. Governing equations (Equations 4.49-4.52) are linearized and substituted for the perturbed liquid volume fraction, liquid and gas phase velocities, and pressure given by Equation (4.15) (Barnea and Taitel, 1994). The following system is obtained for the disturbance amplitude, T p g l , ,: 0 cos cos 0 0 0 0 f f f f f f f f fp g l g g g g l g l g g l l g l l l l l l l l l l g g l lk u F i k u u F i F i g H k k u F i u F i k u F i g H k k k u k k u 0 0 0 0. (4.56) For non-trivial solutions to exist, the following dispersion equation for 0 must hold 0 22 2 f f f eki ck bi ak0 0, (4.57) where g g g g l lu u a 1, (4.58a) f g g l lu F u F b 1 1 2 1, (4.58b) f f cos 12 2g H u u cg l l l g g g l l l, (4.58c) f f l g g g l l lF u F u u F u e 1, (4.58d) where g lF F F (4.59)

PAGE 135

115 and t tsin g A S A S Fl l i i l l l lf f (4.60) t tsin g A S A S Fg g i i g g g gf f f (4.61) Therefore, dispersion relation between wave angular velocity 0 and wavenumber k is obtained as i abk ek b k c a bi ak 22 2 2f f f f 0. (4.62) The negative imaginary part of 0 determines the growth rate of disturbance. 4.2.4 Analysis on Computational Intability 4.2.4.1 Description of numerical methods Governing equations (Equations 4.49 to 4.52) are solved iteratively by the pressure correction scheme introduced in Section 4.2.3 with minor modification to include shear stress terms. The finite volume method and staggered grid were used to discretize the governing equations. We used the Euler backward scheme to discretize the transient term. Therefore, the liquid continuity equation (Equation 4.49) is integrated over the main control volume. The discretized equation is the same as Equation 4.20. Next the liquid momentum equation (Equation 4.50) is integrated over the velocity control volume. , cos0g l l l l l l e l w l e w l P l w l l w l e l l e l P l l P l lu u F g H p p u u u u u u t x f f f f (4.63)

PAGE 136

116 The cell face flux is liquid velocity, which is obtained by central difference, and the volume fraction and liquid velocity at the cell face can be interpolated using central difference, 1st order upwind, 2nd order upwind, and QUICK. Using similar discretization procedure, the gas phase momentum equation is integrated: , cos0g l l g g g g e l w l e w g P g w g g w g e g g e g P g g P g gu u F g H p p u u u u u u t x f f f f (4.64) For pressure correction scheme, the total mass constrain equation is the same as Equation (4.26). The final pressure equation is obtained by substituting two momentum equations, Equation (4.51) and Equation (4.52), into the discretized total mass constrain equation that is the same as Equation (4.26). 4.2.4.2 Von Neumann stability analysis for various convection schemes Generally, the von Neumann stability analysis for viscous two-fluid model is similar to that for inviscid two-fluid model. FOU is employed as an illustrative example. Both the liquid and gas velocities are assumed positive for simplicity and practical purpose. The wave growth equations for the liquid and gas mass conservation equations are the same as those in the inviscid two-fluid model 0 1 12 1 2 11 f f f f f f I I l l I le e e u G t x (4.65) 0 1 12 1 2 11 f f f f f f f I I g g I ge e e u G t z (4.66) For liquid momentum equation, Equation (4.51) is discretized with FOU scheme,

PAGE 137

117 , cos2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 11 1 1 1 1x u u a F H g p p u u u u x t u ul n i l n i g n i l n i l l n i l n i l l n i n i l n i l n i l l n i l n i l l n i l n i l n i l n i l n i l f f f f f f f (4.67) For the gas phase, the velocity variable is governed with the following equation , cos2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 11 1 1 1 1x u u a F H g p p u u u u x t u ug n i g n i g n i l n i l g n i l n i l g n i n i g n i g n i g g n i g n i g g n i g n i g n i g n i g n i g f f f f f f f (4.68) Combining Equations (4.67-68) to cancel the pressure term and linearizing lead to. cos 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 11 1 1x F u u F u u F H g u u u u u u u u u u t xn i l l n i g g n i l l n i l n i l l l g l n i g n i g g g n i l n i l l l n i g n i g g n i l n i l l f f f f f f f f f f f f (4.69) Substituting wave components, Equation (4.32), into Equation (4.69) leads to 0 1 1 1 1 2 cos1 12 1 2 1 2 1 2 1 f f f f f f f f ff f f f f fx u F e u G t x x u F e u G t x e e x F e e H gg I g g g g l I l l l l I I l I I l l g l (4.70) Equation (4.65, 4.66, 4.79) can be written in a matrix form as

PAGE 138

118 0 1 1 1 1 2 cos 0 1 1 0 1 11 1 1 12 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 f b t n n n f f f f b t n n n f f f f f f f f f f f f f f f f f f f f f f f f f f l g l I l l g I g g I I l I I l g l I I l I l I I g I gx u F e u G t x x u F e u G t x e e x F e e g e e e u G t x e e e u G t x (4.71) Non-trivial solutions for T l g , exist only when the determinant of the matrix is zero. Hence, the equation for the growth rate G shares the same form as the inviscid two-fluid model growth rate equation but with different coefficients: 01 2 1 f fc G b G a (4.72) where a, (4.73a) f f l l g g l l l g g gu F u F t CFL CFL b 1 1 1 1 2, (4.73b) 1 1 sin 2 sin 4 cos 1 12 2 2 2 f f l l l g g g l l l g l l l l g g gCFL u F t CFL u F t F t x t I H g x t CFL CFL c (4.73c) The values of in Equation (4.73) are given in Table 4-2. Comparing with Equation (4-46 a-c), Equations (4.73a-c) shows additional terms representing the influence of wall shear stress on the wave growth rate.

PAGE 139

119 From Equation (4.72), G can be easily found that ac b b a G 4 22f f (4.74) 4.2.4.3 Initial and boundary conditions for numerical solution Similar to the inviscid two-fluid model, periodic boundary conditions are assumed. The initial condition is given by the result of viscous Kelvin-Helmholtz stability analysis. If k and are specified at t=0, corresponding value of 0, l, g, and p must be consistent with Equation (4.56). 4.2.5 Results and Discussion 4.2.5.1 Computational stability assessment based on von Neumann stability analysis For inviscid two-fluid model, CDS has the best stability characteristics; FOU shows high numerical damping and is unstable for low k ; SOU shows excessive numerical damping and much more unstable than FOU; and the performance of QUICK is between CDS and FOU. Similar comparison will be conducted for the viscous twofluid model and the results are presented in this section. In this study, water and air are used as examples, and the pipe diameter is 0.05m. The computational domain is 1m long, the grid number is N=200, and pipe incline angle =0. Different liquid phase and vapor phase superficial velocities will be specified. It is note the base value of the liquid phase and the vapor phase velocities and volume fractions should satisfy the condition 0 F in order to maintain a steady flow. Figure 4-31 compares the growth rate G of four numerical schemes and the growth rate by VKH. The liquid superficial velocity is s m uls/ 3 0 and the gas superficial velocity is s m ugs/ 6 and 1 0 lCFL Thus, the flow is IKH stable and VKH unstable

PAGE 140

120 based on theoretical analyses. The VKH growth rate curve is flat and slightly higher than one. The growth rate of CDS is slightly lower than one but quite close to one, except at the low k where G >1. FOU scheme possesses excessive numerical damping at high k SOU shows larger numerical damping than FOU at high k Performance of QUICK scheme is between CDS and FOU. The results of Figure 4-31 generally agree with those of inviscid two-fluid model. However, shear stresses cause the flow instability to occur at lower k in viscous flow. The instability associated with the shear stresses is further illustrated in Figure 4-32, which is the enlarged low k part of Figure 4-28. 0.75 0.8 0.85 0.9 0.95 1 1.05 00.511.522.533.5GVKH CDS QUICK FOU SOU Figure 4-31. Comparisons of growth rate of different schemes. 200 N, s m uls/ 3 0 s m ugs/ 6 ,and 1 0 lCFL Figure 4-32 shows that at extreme low k the growth rates of all the schemes agree well with prediction of the VKH analysis, but when the k is slightly larger, the G profile quickly deviates from growth rate of the VKH analysis. If the flow instability are to be captured, the grid has to be extreme fine to keep the wave triggering flow instability locate at low which is x k *. It is shown in Figure 4-32 that the FOU curve is far from

PAGE 141

121 CDS, SOU and QUICK. This reflects that 1st order accuracy FOU and the other three schemes all have 2nd order accuracy. 0.998 0.9985 0.999 0.9995 1 1.0005 00.10.20.30.40.5GVKH CDS QUICK FOU SOU Figure 4-32. Comparisons of growth rate of different schemes at low k 200 N, s m uls/ 3 0 s m ugs/ 6 ,and 1 0 lCFL Next, the effect of fluid viscosity on the numerical stability is presented. Because the liquid viscosity has more influence on the stability of two-phase flow than the gas viscosity (Barnea and Taitel, 1994), the investigation focuses on the influence of the liquid viscosity. Figure 4-33 compares the growth rate of viscous and inviscid two-fluid model with CDS for air-water system. The viscosity of water is s Pawater* 10 855 03f" r. The flow is IKH stable and VKH unstable. The growth rate based on the VKH analysis is slightly higher than one. Due to the numerical damping, the major part of G of viscous two-fluid model is below one, except at low k Compared with the growth rate of inviscid two-fluid model, the effect of shear stresses on the growth rate is clearly shown. Shear stresses

PAGE 142

122 result in low G for the middle and high range of k and high G for low range of k The fact that 1 % G in the low k range leads to the instability of numerical scheme. 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999 1 1.001 00.511.522.533.5G viscous inviscid VKH IKH Figure 4-33. Growth rate for CDS scheme with VKH unstable. 200 N, s m uls/ 3 0 s m ugs/ 6 ,and 1 0 lCFL Figure 4-34 shows the effect of liquid viscosity on the growth rate. The fluid system is still air-water, but the viscosity of water is given as s Pawater* 102fr. This viscosity is much higher than the typical viscosity of water used in Figure 4-33. Thus, the growth rate based on the VKH analysis is much higher than that in Figure 4-33. The difference between the viscous and inviscid growth rate in Figure 4-31 is larger than that in Figure 4-33. Beside the value of G the k range of unstable harmonics in Figure 4-34 is larger than that in Figure 4-33. Figure 4-35 shows the amplification factor of air-water system with s Pawater* 101fr. This viscosity is one order of magnitude higher than the viscosity used in Figure 4-34. To keep the flow VKH unstable and IKH stable, the liquid and gas

PAGE 143

123 phase superficial velocities are adjusted to s m uls/ 1 0 and s m ugs/ 2 To keep both the liquid and gas Courant number moderate, 01 0 lCFL is used. It is shown in Figure 4-35 that unstable range of k is much larger than ranges in Figure 4-33 and Figure 4-34. However, the difference between the viscous growth rate and inviscid growth rate is not significantly larger than those in Figure 4-33 and Figure 4-34. Thus, the shear stresses effect is only significant at the low k range or long waves. 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999 1 1.001 00.511.522.533.5G VKH viscous Inviscid IKH Figure 4-34. Growth rate for CDS scheme with VKH instability. s Pawater* 102fr, 200 N, s m uls/ 3 0 s m ugs/ 6 ,and 1 0 lCFL Next, the effect of shear stresses on FOU scheme is presented. It is known that the numerical damping of FOU scheme is much higher than that of CDS scheme. Even in CDS scheme with low numerical damping, when the flow is VKH unstable, the numerical damping effect still makes G less than one at middle and high k due to numerical damping, as shown in Figure 4-32. Since numerical damping of FOU is much higher than that of CDS, it is anticipated that the numerical effect of FOU is much more

PAGE 144

124 substantial in viscous two-fluid model. Figure 4-36 shows that the growth rate of FOU for both the viscous and inviscid two-fluid models. The computational parameters are the same as those used in Figure 4-33. It is shown that the viscous and inviscid growth rate curves are quite close to each other and G of viscous model is slightly larger. Only at low k shear stresses effect makes significant difference between viscous and inviscid model. The growth G of viscous model exceeds one, and flow instability is thus triggered. 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999 1 1.001 00.511.522.533.5G VKH viscous Inviscid IKH Figure 4-35. Growth rates for CDS scheme with VKH instability. s Pawater* 101fr, 200 N, s m uls/ 1 0 s m ugs/ 2 ,and 01 0 lCFL Figure 4-37 shows the growth rate of FOU scheme with higher liquid viscosity. The computational parameters are the same as that used in Figure 4-35. It is clearly shown that the choice of numerical scheme has a much larger impact on the stability of the computation, and the physical viscous effect is less significant.

PAGE 145

125 0.8 0.85 0.9 0.95 1 1.05 00.511.522.533.5 G 0.85 0.9 0.95 1 1.05 00.511.522.533.5 G 0.9965 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 1.0005 00.050.10.150.20.250.3 VKH viscous Inviscid Figure 4-36. Growth rates for FOU scheme with VKH instability. 200 N, s m uls/ 3 0 s m ugs/ 6 and 1 0 lCFL 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 00.511.522.533.5 GVKH von Neumann 0.999 0.9995 1 1.0005 1.001 1.0015 1.002 1.0025 1.003 00.050.10.150.20.250.3 VKH viscous inviscid Figure 4-37. Growth rates for FOU scheme with VKH instability. s Pa ewater* 1 1 f r, 200 N, s m uls/ 1 0 s m ugs/ 2 and 01 0 lCFL

PAGE 146

126 4.2.5.2 Computational assessment based on the growth of disturbance To validate the pressure correction scheme for the viscous two-fluid model, comparisons between the computed wave growth rate and the analytical growth rate predicted using the von Neumann stability analysis are presented. In this section, the fluids used still are water and air, and pipe diameter is 0.05m. The computational domain is 1m long the grid number N=200. To maintain periodic boundary conditions for the viscous flow, the base values of the fluid volume fractions, the liquid and gas phase velocities, and the pipe incline angle should satisfy both 0 lF and 0 gF. Through solving these two force balance equations, the flow parameters are obtained. Initial conditions are compatible with the result of the VKH analysis, and the analytical solution of disturbance growth is also obtained from the VKH analysis. Figure 4-38 shows the growth history of a harmonic with 2 k by CDS. The parameters used are s m ul/ 2 s m ug/ 0.998174 -0.0617144 and 98 0 la The flow is well-posed and VKH unstable. The disturbance grows exponentially as shown in Figure 4-38. The correctness of the numerical scheme can be verified by comparing the numerical growth rate with the growth rate predicted using the von Neumann stability analysis. Based on the von Neumann stability analysis, the predicted amplitude ratio from s t 0 to s t 10 is 174.75, and the computed amplitude ratio is 176.86. The error is 1.21% in 10 seconds. Furthermore, the growth history based on the VKH analysis is also plotted in Figure 4-38. The growth rate for each time step using the VKH analysis is 1.00006491 and the growth rate for each time step using von Neumann analysis is 1.00006454. It is not surprising because the CDS has excellent numerical accuracy when it is applied to two-fluid model. In the final stage of growth, because the

PAGE 147

127 amplitude of the wave is no longer a small value, the assumption of the von Neumann stability analysis and the VKH instability analysis becomes invalid. Thus, the waves enter non-linear growth stage and the numerical growth rate no longer matches the analytical one. 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 02468101214161820 t(s)Amplitude(m/s)VK H Numerical Figure 4-38. Growth history of lu using CDS scheme. 200 N, s m ul/ 2 s m ug/ 0.998174 -0.0617144 98 0 la and 05 0 lCFL With the same computational parameters, the growth history of FOU is shown in Figure 4-39. The growth rate based on the VKH analysis for each time step is still 1.00006491 but the growth rate based on the von Neumann for each time step for FOU is only 1.00004197, which is much smaller than that using VKH analysis. The low growth rate is the result of numerical damping of FOU. Figure 4-39 presents the discrepancy in the amplitude growth between the VKH prediction and FOU scheme. The correctness of the numerical scheme can be verified using comparing the computed growth rate with predicted growth rate based on the von Neumann stability analysis. By the von Neumann

PAGE 148

128 stability analysis, the total amplitude ratio from s t 0 to s t 10 is 28.7333 and the computed total growth ratio is 28.3785. The error is 1.23% in 10s. 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 02468101214161820t(s)Amplitude(m/s)VKH Numerical Figure 4-39. Growth history of lu using FOU scheme. 200 N, s m ul/ 2 s m ug/ 0.998174 -0.0617144 98 0 la and 05 0 lCFL In this section, the computed G is compared with the predicted G using von Neumann stability analysis. The comparisons show that the pressure correction scheme is quite accurate and verifies that the FOU possesses excessive numerical damping, while the CDS has better numerical accuracy for the viscous two-fluid model. 4.2.5.3 Wave development resulting from disturbance at inlet In the previous section, periodic boundary conditions are used to match the requirement of von Neumann stability analysis. However, the applications of the twofluid model are not limited to periodic boundary conditions. In this section, an inlet boundary condition is specified and the wave propagation is predicted using the viscous two-fluid model. Similar to the initial condition in the inviscid two-fluid model, at s t 0

PAGE 149

129 small sinusoidal waves for l, lu and gu with 2 4 k satisfying Equation (4.56) are introduced. At the inlet, the boundary conditions of l, lu and gu with 2 4 k satisfying Equation (4.56) is posed as function of t At the outlet, 2nd order extrapolation is employed. Figure 4-40 shows the growth of inlet disturbance under the well-posedness and the VKH instablity. The computational parameters are 200 N, s m uls/ 3 0 s m ugs/ 6 0 and 05 0 lCFL The disturbance is expected to grow based on VKH stability analysis. The result of the CDS scheme correctly demonstrates the growth of the disturbance, while the FOU scheme damps the inlet disturbance. The FOU scheme transforms a VKH unstable flow to a steady flow, which is obviously unphysical. -6.00E-05 -4.00E-05 -2.00E-05 0.00E+00 2.00E-05 4.00E-05 6.00E-05 00.10.20.30.40.50.60.70.80.91 x(m)Disturbance(m/s) Figure 4-40. Disturbance of lu propagates in the pipe with FOU and CDS schemes at VKH unstable and well-posed condition.

PAGE 150

130 Figure 4-41 shows the growth of inlet disturbance under VKH stability condition. The computational parameters are 200 N, s m uls/ 15 0 s m ugs/ 3 0 and 05 0 lCFL under such a condition, all the wave components will decay based on the Von Neumann stability analysis. Figure 4-41 verifies the results of von Neumann stability analysis. Both CDS and FOU show that decay of waves, while CDS scheme has much less numerical damping than FOU scheme. -6.00E-05 -4.00E-05 -2.00E-05 0.00E+00 2.00E-05 4.00E-05 6.00E-05 00.10.20.30.40.50.60.70.80.91 x(m)Disturbance(m/s) Figure 4-41. Disturbance of lu propagates in the pipe with FOU and CDS schemes at both VKH unstable and well-posed condition. 4.2.6 Conclusions Numerical instability for the incompressible viscous two-fluid model near the viscous Kelvin-Helmholtz instability is investigated with various convection interpolation schemes, while the pressure correction method is used to obtain the pressure, volume fraction and velocities. The von Neumann stability analysis is carried out to obtain the growth rate of a small disturbance in the discretized system. The growth rate of all

PAGE 151

131 schemes deviates from the prediction based on the VKH instability analysis at high wavenumber range. However, the central difference scheme shows the best stability characteristics in handling the viscous two-fluid model among the investigated schemes, followed by the QUICK scheme. The 1st order upwind scheme shows excessive numerical damping in comparison with the central difference scheme. Despite its nominal 2nd order accuracy and popularity, the 2nd order upwind scheme is much more inaccurate than the 1st order upwind scheme for solving viscous two-fluid model equations. The relation between the computational instability and VKH instability near VKH instability criterion is investigated. The computational instability often appears at low wave number range, while numerical damping prevents the instability at high wave number range. The numerical instability is largely the property of the discretized viscous two-fluid model but is strongly influenced by VKH instability. To obtain an accurate numerical solution, the most accurate scheme with sufficient number of grid points is suggested. Comparisons between the predicted amplitude growth rate and the growth rate computed using the pressure correction scheme is presented. Excellent agreement between the computed results and the predictions based on the von Neumann stability analysis for central difference scheme, and 1st order upwind scheme shows the success of the pressure correction scheme in solving the viscous two-fluid model. Inlet disturbance growth test shows that the pressure correction scheme is able to correctly handle viscous two-phase flow in a pipe under different boundary conditions.

PAGE 152

132 Since the central difference scheme is the most stable and accurate scheme (Chapter 4), we used it to solve the separated-flow chilldown model (Chapter 5).

PAGE 153

133 CHAPTER 5 MODELING CRYOGENIC CHILLDOWN In this chapter, the flow and heat transfer models developed in earlier chapters are used to develop chilldown models. Three chilldown models are presented in this chapter. Homogeneous flow model focuses on the chilldown in a vertical pipe, where homogeneous flow is prevalent. A pseudo-steady chilldown model is developed to predict the chilldown time and wall temperature in a horizontal pipe at relatively low computation cost. Moreover, the pseudo-steady chilldown model servers as a testing platform for investigating and validating new film boiling heat transfer correlations. Finally, a comprehensive separated flow chilldown model for horizontal pipe is developed to predict the flow field of the liquid and the temperature fields in both the liquid and the pipe wall. 5.1 Homogeneous Chilldown Model The homogeneous chilldown model is based on the homogeneous flow model introduced in Chapter 2 and aims at modeling chilldown in the vertical section. Under such a flow condition, it is anticipated that as the liquid front propagates downward or upward, a film boiling stage exists near the liquid-gas front. After the film boiling stage, a nucleate boiling stage exists, as the wall has not been substantially cooled down. After the nucleate boiling stage, the convection heat transfer is the main heat transfer mechanism, as illustrated in Figure 5-1. Since the vapor volume fraction is not large behind the front, a homogeneous flow model is appropriate.

PAGE 154

134 Mixture front Pipe wall Vapor bubble Liquid Wall heat flux Vapor film Figure 5-1. Schematic of homogeneous chilldown model. 5.1.1 Analysis In this study, the homogeneous chilldown model, Equations 2.14 to 2-16 are solved using the SIMPLE scheme (Patankar, 1981). First, a mixture density is guessed; then, the velocity and pressure are calculated using the momentum and the continuity equations. After the velocity and pressure are obtained, the energy equation is solved for the mixture enthalpy. From the mixture enthalpy, the mixture quality and density are obtained. The updated density is reintroduced into the continuity equation to solve the new velocity and pressure. The iteration continues, until the density, velocity and enthalpy converge. Since the solid heat transfer in the homogeneous chilldown model is axisymmetrical, a two-dimensional unsteady heat conduction equation in the solid pipe is solved to obtain solid temperature. Due to insignificant heat conduction in the z direction compared with that in the radial direction, the heat conduction along the flow direction is neglected and the heat conduction in radial direction is retained.

PAGE 155

135 The heat transfer from the wall to the fluid depends on the wall superheat. If the wall temperature is higher than the Leidenfrost temperature, film boiling heat transfer exists. If the wall temperature is not high enough to support nucleate boiling, the nucleate sites are completely suppressed. Thus, the heat transfer is governed by convection. Here, the film boiling stage uses the correlation of Giarratano and Smith (1965); and the nucleate boiling uses Gungor and Winterton's (1996) correlations. Detailed discussion on the heat transfer correlation is presented in Chapter 2. The film boiling stage is a major part of chilldown heat transfer in terms of the time span, but no correlation for the friction coefficient in the film boiling regime exists. The character of the wall fraction in film boiling stage is that shear stress is small due to the vapor layer separating the liquid and the wall. However, it is an oversimplification that the wall friction is zero. In this study, a friction model based on the vapor layer thickness is proposed to qualitatively evaluate the wall friction in the film boiling regime. Vapor layer lU Liquid Figure 5-2. Schematic for evaluating film boiling wall friction.

PAGE 156

136 In the vapor film layer, the flow is assumed laminar, and the velocity profile and the temperature profile are assumed linear. Hence the wall shear stress is r r tl v v FBU u (5.1) where lU is averaged liquid velocity, is the vapor film layer thickness. It is assumed that the local heat transfer coefficient is already known from the heat transfer correlation. By the assumption of linear temperature profile in the vapor film, is calculated by v FBk h (5.2) where FBh is the local film boiling heat transfer coefficient. Substituting Equation (5.2) into Equation (5.1) yields v l FB v FBk U hr t (5.3) Therefore, the pressure drop of the homogeneous flow model in the film boiling regime can then be evaluated by D z PFB ft4 (5.4) 5.1.2 Results and Discussion The homogeneous model is applied in simulating the chilldown process of the space shuttle launch facility in NASA, where liquid hydrogen as the coolant chills the transport pipeline. The pipe is made of stainless steel and it is assumed to be adiabatic at outer surface. The inner diameter is 0.2662m and wall thickness is 1.25cm. The pipe studied is a vertical pipe with the length of 2m. The liquid hydrogen flows upward from

PAGE 157

137 the bottom of pipe to the top. Liquid hydrogen enters the pipe at velocity of 0.58m/s and quality 0 x. Initially the pipe wall temperature is at atmospheric condition. A typical set of results at s t 5 1 are shown in Figures 5-3, 5-4, and 5-5 for an instantaneous distribution of the vapor quality, pressure, and velocity after the front propagates near the end of the pipe. Z(m) quality00.511.52 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5-3. Distribution of vapor quality based on the homogenous flow model. In Figure 5-3, it is shown that the mixture front is located near m z 5 1 It is obvious that the moving speed of mixture front is higher than the speed of liquid entering the pipe. The reason is that a substantial amount of heat is transferred from the wall to the hydrogen and part of liquid hydrogen is evaporated. Thus, the density of mixture drops and the mixture velocity increases.

PAGE 158

138 Figure 5-4 shows the pressure distribution along the pipe. The pressure drop in the pure liquid region (0 x) is almost linear and is larger than that in the region mixture exists. This is due to the mixture density being lower than the liquid hydrogen and the pressure gradient mainly overcomes the gravitational force. On the other hand, if the pipe is placed horizontally, the pressure drop of the mixture should be higher than that of the pure liquid, because higher pressure gradient is to accelerate the flow when evaporation occurs. The pressure drop due to the wall friction is quite small, because the flow velocity is low in this chilldown, and the presence of the film boiling leads to lower wall friction. Figure 5-5 shows the mixture velocity distribution. The acceleration of mixture flow occurs in the middle of pipe. This is consistent with the results of the quality distribution in the pipe. Z(m) pressure(Pa)00.511.52 0 100 200 300 400 500 600 700 Figure 5-4. Pressure distribution based on the homogenous flow model.

PAGE 159

139 Z(m) velocity(m/s)00.511.52 0 0.5 1 1.5 2 2.5 3 3.5 Figure 5-5. Velocity distribution based on the homogenous flow model. Figure 5-6 shows the corresponding solid wall temperature contour at s t 5 1 The best chilling effect is at the middle of the pipe. This is because near the mixture front, the velocity of mixture is higher than that near the entrance. Thus, the heat transfer coefficient near front is larger than that near the entrance. 2 9 9 1 5 4 2 8 8 1 6 2 2 9 8 3 0 9 2 9 7 4 6 3 2 9 5 7 7 2 2 9 3 2 3 6 2 9 1 5 4 4 Z(m) radius(m)00.511.52 0.134 0.136 0.138 0.14 0.142 0.144 0.146Wall FlowDirection Vaccum CryogenicFluid Figure 5-6. Solid temperature contour based on homogenous flow model.

PAGE 160

140 5.2 Pseudo-Steady Chilldown Model Although the two-fluid model can describe the fluid dynamics aspect of the chilldown process, it suffers from computational instability for moderate values of slip velocity between two phases, which limits its application. To gain the fundamental insight into the thermal interaction between the wall and the cryogenic fluid and to be able to rapidly predict chilldown in a long pipe, an alternative pseudo-steady model is developed. In this model, a liquid wave front speed is assumed to be constant and is the same as the bulk liquid speed (Thompson, 1972). It is also assumed that steady state thermal fields for both the liquid and the solid exist in a reference frame that is moving along the wave front. The governing equation for the solid thermal field becomes a parabolic equation that can be efficiently solved. The film boiling heat transfer between the fluid and the wall is modeled with first principle. It must be emphasized that a great advantage of the pseudo-steady model is that one can assess the efficacy of the film boiling model independently from that of the nucleate boiling model since the downstream information in the nucleate boiling regime cannot affect the temperature in the film boiling regime. In other words, even if the nucleate boiling heat transfer coefficient is inadequate, the film boiling heat transfer coefficient can still be assessed in the film boiling regime by comparing with the measured temperature during the corresponding period. Once satisfactory performance is achieved for the film boiling regime, the nucleate boiling heat transfer model can be subsequently assessed. In the results section, those detailed assessments of the heat transfer coefficients are provided by comparing the computed temperature variations with the experimental measurements of Chung et al. (2004). Satisfactory results are obtained.

PAGE 161

141 5.2.1 Formulation In the pseudo-steady chilldown model, it is assumed that both the liquid and its wave front move at a constant speed U Thus, the main emphasis of the present study is on modeling the heat transfer coefficients with the stratified flow in the film boiling and forced convection boiling heat transfer regimes and the computation of the thermal field within the solid pipe. Comparisons are made with low Reynolds number data. 5.2.1.1 Heat conduction in solid pipe The thermal field inside the solid wall is governed by the three-dimensional unsteady energy equation: T r k r r T rk r r z T k z t T c 1 1 (5.5) Since the wave front speed U is assumed to be a constant, it can be expected that when the front is reasonably far from the entrance region of the pipe, the thermal field in the solid is in a steady state when it is viewed in the reference frame that moves along the wave front. Thus, the following coordinate transformation is introduced, Ut z Z (5.6) Film boiling nucleate boiling Convective heat transfer Liquid layer Liquid front z r U Vapor layer Wall heat flux Pipe wall D Thin vapor film Figure 5-7. Schematic of cryogenic liquid flow inside a pipe.

PAGE 162

142 U z r Li q iud va p or Z r Pipe wall R1 R2 Figure 5-8. Coordinate systems: laboratory frame is denoted using z; moving frame is denoted using Z Because of 2 2 2 2 Z T z T (5.7) Z T U t T (5.8) Equation (5.5) is transformed to T r k r r T rk r r Z T k Z Z T cU 1 1 (5.9) For further simplification, the following dimensionless parameters are introduced, sat w wT T T T f f d Z Z d r r 0c c c and 0k k k (5.10) where wT is the initial wall temperature, satT is the saturated temperature of the liquid, d is the thickness of the pipe wall, 0k is the characteristic thermal conductivity, and 0c is the characteristic heat capacity. Thus, Equation (5.9) is normalized as

PAGE 163

143 r k r r k r r r Z k Z Z c Pc 1 1 *, (5.11) where 0 0k Ud c Pc is the Peclet number. It is noted that Equation (5.11) is an elliptic equation. Under typical operating condition for cryogenic chilldown, Pc ~ O (102-103). The first term on the RHS of Equation (5.11) is small compared with the rest of the terms and thus can be neglected. Equation (5.11) becomes r k r r k r r r Z c Pc 1 1 *, (5.12) which is a parabolic equation. Hence, in the Z -direction, only one boundary condition is needed. In the -direction, periodic boundary conditions are used. On the inner and outer surface of the wall, proper boundary conditions for the temperature are required. For convenience, Z =0 is set at the liquid wave front. In the region of Z <0, the inner wall is exposed to the pure vapor. Although there may be some liquid droplets in the vapor that cause evaporative cooling when the droplets deposit on the wall and the cold vapor absorbs part of heat from the wall, the heat transfer due to these two mechanisms is much less than the heat transfer between the liquid and solid wall in the region of Z>0. Hence, the heat transfer for Z<0 is neglected and it is assumed that 1 at 0 Z. The computation starts from the Z =0 to b # Z, until a steady state solution in the Z -direction is reached. An implicit scheme in the Z direction is employed to solve Equation (5.12).

PAGE 164

144 5.2.1.2 Liquid and vapor flow The two-phase flow is assumed to be stratified as was observed in Chung et al. (2004). Both liquid and vapor phases are assumed to be at the saturated state. The liquid volume fraction is used to determine the part of the wall in contact with the liquid or the vapor, and is specified at every cross-section along the Z -direction based on experimental information. For the experimental conditions under consideration, visual studies (Velate et al., 2004; Chung et al., 2004) show that the liquid volume fraction increases gradually, rather than abruptly, near the liquid wave front and becomes almost constant during most of chilldown. Hence, the following liquid volume fraction variation is assumed as a function of time for the computation of the solid-fluid heat transfer coefficient, , 2 sin0 0 0 0 0t t t t t t (5.13) where 0t is characteristic chilldown time, and 0 is characteristic liquid volume fraction. Here the time when the nucleate boiling is almost suppressed and the slope of the wall temperature profile becomes flat is set as characteristic chilldown time. It is determined experimentally. The vapor phase velocity is assumed a constant. However, it was not directly measured in recent experiments (Chung et al., 2004; Velat et al., 2004). In this study, the vapor velocity is computationally determined by trial-and-error by fitting the computed and measured wall temperature variations for numerous positions.

PAGE 165

145 5.2.1.3 Film boiling correlation Due to the high wall superheat encountered in the cryogenic chilldown, film boiling plays a major role in the heat transfer process in terms of the time span and in terms of the total amount of heat removed from the wall. Currently there exists no specific film boiling correlation for chilldown applications with such high superheat. Qualitative study in Chapter 2 shows that existing film boiling correlations are not appropriate for study chilldown. Therefore, film boiling correlation for cryogenic chill-down is desired to be developed. A new correlation for cryogenic film boiling inside a tube is presented here. The schematic diagram of the film boiling inside a pipe is shown in Figure 5-9 with a cross-sectional view. The bulk liquid is near the bottom of the pipe. Beneath the liquid is a thin vapor film. Due to the buoyancy force, the vapor in the film flows upward along the azimuthal direction. Heat is transferred through the thin vapor film from the solid to the liquid. Reliable heat transfer correlation for film boiling in pipes or tubes requires knowledge of the thin vapor film thickness, which can be obtained by solving the film layer continuity, momentum, and energy equations. Liquid vapor 0 x Figure 5-9. Schematic diagram of film boiling at stratified flow.

PAGE 166

146 To simplify the analysis for vapor film heat transfer, it is assumed the liquid velocity in the azimuthal direction is zero and the vapor flow in the direction perpendicular to the cross-section is negligible. It is further assumed that the vapor film thickness is small compared with the pipe radius and the vapor flow is quasi-steady, incompressible and laminar. The laminar flow assumption can be confirmed post priori as the Reynolds number, Re based on the film velocity and film thickness is typically of 2 010 ~ 10 O. In terms of the x & y -coordinates and ( u v ) velocity components shown in Figure 5-9, the governing equations for the vapor flow are similar to boundary-layer equations: 0 y v x u (5.14) sin 12 2g y u x p y u v x u uv vf f (5.15) 2 2y T y T v x T uv (5.16) where is density, is kinematics viscosity, g is gravitation, T is vapor temperature, p is vapor pressure, and is thermal diffusivity. Subscribe v represents properties of vapor. Because the length scale in the azimuthal ( x ) direction is much larger than the length scale at the normal ( y ) direction, the v -component may be neglected. Furthermore, the convection term is assumed small and is neglected. The resulting momentum equation is simplified to sin 12 2g y u x pv vf (5.17)

PAGE 167

147 By neglecting the vapor thrust pressure and surface tension, the vapor pressure is evaluated by considering the hydraulic pressure from liquid core: f f 0 0 0 0cos cos cos cos R x gR p gR p pl l. (5.18) where 0 is the angular position where the film merges with the vapor core. The momentum equation becomes 0 sin2 2 f y u R x gv v v l (5.19) Assuming the vapor velocity profile satisfies the non-slip boundary condition 0 u at 0 y and 0 lu u at y. The vapor velocity is obtained by integrating Equation (5.19): 2* sin 2 y y R x g uv v v lf f (5.20) The mean u velocity is f R x g udy uv v v lsin 12 12 0 (5.21) Thus the u velocity is presented as function of uas f 2 26 y y u u (5.22) The energy and mass balance on the vapor film requires that ) ( u d m d y T dx h kv y fg v b t n n f fD (5.23) where k is heat conductivity, and fgh is latent heat at evaporation. If the convection terms in energy equation are neglected, the vapor energy equation is simplified as

PAGE 168

148 02 2 y T (5.24) Integrating twice and applying the temperature boundary conditions at 0 y and y yields following linear temperature profile y T T T Tsat w satf f f 1 (5.25) Introducing the temperature and velocity profile into the Equation (5.23) yields sat w v l fg v vT T gR h k R d d R f f 3) ( 12 sin (5.26) This equation has an analytical solution on the vapor thickness : 4 1 3 4 0 3 1 3 4 4 1 3sin sin 12 f f const d gR h T T k Rv l fg sat w v v. (5.27) To make the solution finite at 0 requires0 const. Thus the solution is F Ra Ja R4 16 2 (5.28) where Ja is Jacob number and Ra is Raleigh number: fg sat w v ph T T C Ja f ,, (5.29) v v v v lgD Ra f 3, (5.30) in which, pCis heat capacity, D is pipe diameter, and the F is a geometry influence factor on the vapor film thickness

PAGE 169

149 4 1 75 0 0 3 1 3 4sin sin d F. (5.31) The mean velocity u as a function of is thus sin 122 2 1 2F h gR T T ufg v v v l sat w f f (5.32) Curves for F and sin2F based on the numerical integration are shown in Figure 5-10. The vapor film thickness has a minimum at 0 and is nearly constant for 2 !. It rapidly grows after 2 %. The singularity at the top of tube when # is of no practical significance since the film will merge with the vapor core at the vaporliquid interface. The vapor velocity is controlled by sin2F which is zero at the bottom of the pipe and increases almost linearly in the lower part of the tube where the vapor film thickness does not change substantially. In the upper part of the tube, due to the increase in the vapor film thickness, the vapor velocity gradually drops back to zero at the top of the tube. Thus a maximum velocity may exist in the upper part of the tube. The local film boiling heat transfer coefficient is easily obtained from the linear temperature profile. It is 4 16389 0 Ja Ra DF k k hv v FB (5.33) The heat transfer rate per unit length from the wall to liquid is given by integrating heat flux around the wall.

PAGE 170

150 0 4 1 06 ) ( ) ( 20 G Ra Ja T T k Rd T T k qsat w v sat w v f f (5.34) where f00 1 0 d F G. (5.35) 0 1 2 3 4 5 00.511.522.533.5 radians) F2( )sin( ) F( ) Figure 5-10. Numerical solution of the vapor thickness and velocity influence functions. Figure 5-11 shows numerical solution of 0G. The average heat transfer coefficient in the tube is represented by the Nu number: 0 4 1 0 4 1 4 12034 0 6 G Ja Ra G Ja Ra T T Dk D q k D h Nusat w v v FB f f. (5.36) The Nu number is a function of liquid level angel 0. It almost linearly grows with the angle0. A further simplification is to assume that Nu is a linear function of 0.

PAGE 171

151 0 4 11763 0 Ja Ra Nu (5.37) Equation (5.37) provides a correlation to rapidly evaluate the film boiling heat transfer in a pipe or tube. If liquid volume fraction l is known, 0 can be simply calculated. Thus, Nu for the pipe is obtained. G(phi)0 0.5 1 1.5 2 2.5 3 00.511.522.533.5 phiG(phi) G(phi) Figure 5-11. Numerical solution of 0G. 5.2.1.4 Forced convection boiling correlation Several forced convection boiling correlations have been discussed in Chapter 2, including Gungor and Wintertons correlation (1996), Chens correlation (1966), and Kutateladzes correlation (1952). The quantitative comparison among these models is based on the pseudo-steady chilldown model. With pseudo-steady chilldown model, none of correlations gives a satisfactory heat transfer rate that is needed to match the experimentally measured temperature histories in Chung et al. (2004) at the forced

PAGE 172

152 convection boiling regime. Among them, Kutateladze correlation gives more reasonable results. Kutateladze correlation was proposed without considering the effect of nucleate site suppression. This obviously leads to an overestimation of the nucleate boiling heat transfer rate. Hence a modified version of Kutateladze correlations is proposed: pool c lh S h h *, (5.38) where S is suppression factor. The liquid hydraulic diameter lD in Equation (2.3) is redefined for the stratified flow: l l lS A D4. (5.39) 5.2.1.5 Heat transfer between solid wall and environment For a cryogenic flow facility, although serious insulation is applied, the heat leakage to the environment is still considerable due to the large temperature difference between the cryogenic fluid and the environment. It is necessary to evaluate the heat leakage from the inner pipe to the environment in cryogenic chilldown. A vacuum insulation chamber is usually used in cryogenic transport pipe, as shown in Figure 5-12. Radiation heat transfer exists between the inner and outer pipe. Furthermore, the space between the inner and outer pipe is not an absolute vacuum. There is residual air that causes the free convection between the inner and outer pipe driven by the temperature difference of the inner and outer pipe.

PAGE 173

153 Inner pipe Outer pipe radiation Free convection vacuum Figure 5-12. Schematic of vacuum insulation chamber. The radiation between the inner pipe and outer pipe becomes significant when the inner pipe is cooled down. The heat transfer coefficient is proportional to the difference of the fourth power of wall temperatures. Exact evaluation of the heat transfer rate between the inner pipe and the outer pipe is a difficult task. Hence, a simplified model is used to evaluate the heat transfer rate at every position of pipe. It is not quantitatively correct, but can provide reasonable estimation for the magnitude of the radiation heat transfer between pipes across the vacuum. The overall radiation heat transfer ioqbetween long concentric cylinders with constant temperature iT at inner pipe and oT at outer pipe (Incropera and DeWitt, 1990) is f f o i o o i o i i ior r T T A q 1 1 ) (4 4, (5.40)

PAGE 174

154 where the is Stefan Boltzmann constant, Ai is the inner pipe area, (ri, i) and (ro, o) are the radius and emissivity of inner pipe and outer pipe, respectively. It is assumed that the local radiation heat transfer rate per unit area on the surface of inner pipe radq is f f o i o o i o wall radr r T T q 1 1 ) (4 4 (5.41) where wallT is the local inner wall temperature, oT is the room temperature that is assumed constant in the entire outer pipe. Here the emissivity is also assumed to be constant during the entire chilldown. For the free convection heat transfer in the vacuum chamber between the inner pipe and outer pipe, Raithby and Holland correlation (1975) is used for the heat transfer rate. The average heat transfer rate per unit length of the cylinder is o i i o eff frcT T D D k q f ln 2 (5.42) where the oD and iD are outer and inner pipe diameter, T are assumed constant at inner and outer wall, effk is the effective thermal conductivity. Similar to the treatment in radiation heat transfer, the local free convection heat transfer rate per unit area on the surface of inner pipe frcq is assumed as frcq being divided by perimeter of the pipe. Thus frcq is suggested as o wall i o i eff frcT T D D D k q f ln 2 (5.43) where effk is given by Raithby and Holland (1975):

PAGE 175

155 4 1* Pr 861 0 Pr 386 0 c effRa k k, (5.44) where L o i i o cRa D D L D D Ra5 5 3 5 3 3 4ln f f (5.45) where L is the characteristic length of chamber between the inner and outer pipe defined as 2 ) (i oD D L f LRa is the Rayleigh number of the chamber 3) (L T T g Rai o Lf (5.46) where is volumetric thermal expansion coefficient. Equation (5.44) is valid when 7 210 10 * cRa. For 100 cRa, k keff,. If the rarified air density is known, the thermal conductivity k and viscosity r of the rarified air can be obtained by using Sutherlands law. The specific heat of the rarified air is assumed only a function of temperature and obtained by the average air temperature within the vacuum chamber. Since the chamber temperature is not extreme low, is obtained using ideal gas relation as T 1 5.2.2 Results and Discussion In the experiment by Chung et al. (2004), liquid nitrogen was used as the cryogen. The flow regime is revealed to be stratified flow by visual observations, as shown in Figure 5-8, and the wall temperature history in several azimuthal positions is measured

PAGE 176

156 5.2.2.1 Experiment of Chung et al. In the experiment by Chung et al. (2004), a concentric pipe test section (Figure 5-13) was used. The chamber between the inner and outer pipe is vacuum, sealed but about 20% air remained. The inner diameter (I.D.) and outer diameter (O.D.) of the inner pipe are 11.1 and 15.9 mm, and I.D. and O.D. of the outer pipe are 95.3 and 101.6mm, respectively. Numerous thermocouples were placed at different locations of the inner pipe. Some were embedded close to the inner surface of the inner pipe while others measure the outside wall temperature of the inner pipe. Experiments were carried out at the room temperature and the atmospheric pressure. Liquid nitrogen flows from a reservoir to the test section driven by gravity. As the liquid nitrogen flows through the pipe, it evaporates and chills the pipe. Some of the typical visual results are shown in Figure 5-14. The nitrogen mass flux is around 3.7E-4 kg/s and the measured average liquid nitrogen velocity is U~5 cm/s. The vapor velocity is not measured in the experiment. In this study, it is determined through trial-and-error by fitting the computed and measured temperature histories. The characteristic liquid volume fraction is 0.3 from the recorded video images. The characteristic time used in this computation is s t 1000. The Leidenfrost temperature for the nitrogen is around 180 K; hence the temperature in which the film boiling ends and nucleate boiling starts is set as 180 K. The transition temperature at which purely two-phase convection heat transfer begins is 140 K based on experimental results. The material of the inner pipe and outer pipe used in the experiment of Chung et al. (2004) are Pyrex glass with emissivity of 0.82 (based on room temperature).

PAGE 177

157 5.2.2.2 Comparison of pipe wall temperature In the computation, there are 40 grids along the radial direction and 40 grids along the azimuthal direction for the inner pipe (Figure 5-15). The results of the temperature profile at 40X40 grids and the higher grid resolution shows that 40X40 grids are sufficient. Figures 5-16 to 5-19 compare the measured and computed wall temperature as a function of time at positions 11, 12, 14 and 15, as shown in Figure 5-15. For the modified Kutateladze correlation, a proper suppression factor of 0.005 is obtained by best fit. The small suppression factor is supported by the visual observation that the majority of nucleate sites are suppressed in cryogenic chilldown (Chung et al. 2004). Likewise, the vapor velocity is 0.5m/s based on the best fit. Figure 5-13. Schematic of Yuan and Chung (2004)s cryogenic two-phase flow test apparatus. Since the governing equation for the solid thermal field is a parabolic equation in pseudo-steady chilldown model. The temperature comparison can be taken from the one regime to another following the sequence of time. Once satisfactory performance is achieved for the one regime, the subsequent regime is assessed.

PAGE 178

158 Figure 5-14. Experimental visual observation of Chung et al. (2004)s cryogenic twophase flow experiment. The comparison starts from film boiling stage at the bottom of the pipe, which is the first stage in chilldown. In Figure 5-16, the measured and predicted temperatures 12 and 15 during the film boiling chilldown are compared. Location 12 is near the inner surface of the pipe and location 15 is at the outer surface of the pipe. Thus, temperature 12 is slightly lower than temperature 15. Figure 5-16 shows both temperatures agree well with the measurements.

PAGE 179

159 Figure 5-15. Computational grid arrangement and positions of thermocouples. At the end of the film boiling chilldown, the liquid starts contacting the wall, and the wall temperature starts rapidly decreasing. Figure 5-17 shows the transition from the slow chilldown to the fast chilldown is captured correctly. During the stage of the rapidly decreasing, the computed wall temperature drops slightly faster than the measured value. The rapid decrease in the wall temperature is due to initiation of nucleate boiling, which gives significantly high heat transfer coefficient than film boiling and the forced convection heat transfer. Reasonable agreement between the computed and measured histories in this nucleate boiling regime is due to: i) the good agreement already achieved in the film boiling stage; ii) valid choice for the Leidenfrost temperature that switches the heat transfer regime correctly; and iii) appropriate modification of Kutateladze correlations. 120n T 11 T 14 T 12 T 15

PAGE 180

160 In the final stage of chilldown, as shown in Figure 5-18, the wall temperature decreases slowly, and the computed wall temperature shows the same trend as the measured one but tends to be a little lower. Figure 5-19 shows the comparison between the measured and predict temperatures at position 11 and 14 during entire chilldown. The predicted temperatures generally agree well with measured temperature, but slightly higher at the initial stage of chilldown and lower at the final stage of chilldown. Figure 5-20 shows the temperature distribution of a given cross-section at different times during chilldown. Because the upper part of pipe wall is exposed to the nitrogen vapor, the chilling effect is much reduced. The difference of chilling effect between the liquid and the vapor is also clearly shown in Figure 5-21. 100 120 140 160 180 200 220 240 260 280 300 010203040506070 t (s)T (K) T15 experimental T15 numerical T12 experimental T 12 numercal T 12 with film correlation(Giarratano and Smith 1965) Film Boiling Figure 5-16. Comparison between measured and predicted transient wall temperatures of positions 12 and 15 at the bottom of pipe during film boiling chilldown.

PAGE 181

161 50 100 150 200 250 300 657075808590 t (s)T (K) T15 experimental T15 numerical T12 experimental T 12 numercalConvection Boiling Figure 5-17. Comparison between measured and predicted transient wall temperatures of positions 12 and 15 at the bottom of pipe during convection boiling chilldown. 50 100 150 200 250 300 050100150200250300 t (s)T (K) T15 experimental T15 numerical T12 experimental T 12 numercal Figure 5-18. Comparison between measured and predicted transient wall temperatures of positions 12 and 15 is at the bottom of pipe during entire chilldown.

PAGE 182

162 50 100 150 200 250 300 050100150200250300 t (s)T (K) T 11 numerical T 11 experimental T 14 experimenatal T 14 numerical Figure 5-19. Comparison between measured and predicted transient wall temperatures of positions 11 and 14, which is at the bottom of pipe during entire chilldown. Figure 5-20. Cross section wall temperature distribution at t=0, 50, 100 and 300 seconds. 'X 'Y'-0.00500.0050.01 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 'T' 293 'X 'Y'-0.00500.005 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 'T' 262.027 258.192 254.356 250.521 246.685 242.85 239.014 235.179 231.343 227.507 223.672 219.836 216.001 212.165 208.33 'X 'Y'-0.00500.005 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 'T' 229.257 222.665 216.073 209.481 202.89 196.298 189.706 183.114 176.522 169.93 163.338 156.746 150.155 143.563 136.971 t=0s t=50s t=300s t=100s 'X 'Y'-0.00500.005 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 'T' 149.383 147.146 144.91 142.673 140.437 138.2 135.964 133.727 131.491 129.254 127.018 124.781 122.545 120.308 118.072 115.835 113.599 111.362 109.126 106.889

PAGE 183

163 -0.005 0 0.005'Y' -0.005 0 0.005' X 0 100 200 300t i m e 'T' 282.168 270.336 258.504 246.672 234.84 223.008 211.176 199.345 187.513 175.681 163.849 152.017 140.185 128.353 116.521 Figure 5-21. Computed wall temperature contour on the inner surface of inner pipe. 5.2.3 Discussion and Remarks In Figure 5-16, the wall temperature based on the film boiling correlation of Giarratano and Smith (1965) is also shown. Apparently, the correlation of Giarratano and Smith (1965) gives a very low heat transfer rate so that the wall temperature remains high. This comparison confirms our earlier argument that correlations based on the overall flow parameter, such as quality and averaged Reynolds number, are not applicable for the simulation of the unsteady chilldown. The nucleate flow boiling correlations of Gungor and Winterton (1996), Chen (1966), and Kutateladze (1952) are also compared with pseudo-steady chilldown model. Gungor and Wintertons correlation fails to give a converged heat transfer rate. Chens correlation overestimates the heat transfer rate, and causes an unrealistically large temperature drop on the wall, which results in strong oscillation of the wall temperature,

PAGE 184

164 as shown in Figure 5-22. Only Kutateladze correlation gives an acceptable heat transfer rate. However, the temperature drop near the bottom of the pipe is still faster than the measured one as shown in Figure 5-16. This may be due to the fact that most of nucleate boiling correlations were obtained from experiments of low wall superheat. However, in cryogenic chilldown, the wall superheat is much higher than that in normal nucleate boiling experiments. Another reason is that the original Kutateladze correlation does not include a suppression factor. This leads to overestimating the heat transfer coefficient. The modified correlation with the suppression factor S=0.005 gives reasonable chilldown results in Figure 5-16. This small S suggests that most of nucleate sites are suppressed. The visual study on chilldown by Chung et al. (2004) confirms that the nucleate boiling is barely seen in spite of few bulbs still existing. However, in the experimental of Velat (2004), a visible nucleate boiling stage is found and last several seconds. Furthermore, the analysis on the convection boiling heat transfer coefficients by Jackson et al. (2005) shows a substantial high heat transfer coefficient exists at the rapid chilldown stage, which cannot be achieved by the convection heat transfer, but only by the nucleate boiling. Although this modified Kutateladze nucleate boiling correlation is not a reliable correlation due to experimental specified factors, it is still useful because of qualitatively capturing the nucleate boiling heat transfer in cryogenic chilldown. Further examination of Figures 5-18 and 5-19 indicates that although we have considered the heat leak from the outer wall to the inner wall through radiation and free convection, the computed temperature is still lower than the measured temperature during the final stage of chilldown. In this final stage the heat transfer rate between the fluid and the wall is low due to the lower wall superheat. The temperature difference between the

PAGE 185

165 computed and measured values at positions 12 and 15 suggests that there may be additional heat loss, which affects the measurements but is not taken in account in the present modeling. 50 70 90 110 130 150 170 190 210 230 250 657075808590 t (s)T (K) T15 experimental T15 numerical T12 experimental T 12 numercalConvection Boiling (Chen, 1966) Figure 5-22. Comparison between measured and predicted transient wall temperatures of positions 12 and 15 with Chen correlation (1966). In this study, pseudo-steady chilldown model is developed to predict the chilldown process in a horizontal pipe in the stratified flow regime. This model can also be extended to describe the annular flow chilldown in the horizontal or the vertical pipe with minor changes on the boundary condition for the solid temperature. It can also be extended to study the chilldown in the slug flow as long as we specify the contact period between the solid and the liquid or the vapor. The disadvantage of the current pseudo-steady chilldown model is that the fluid interaction inside the pipe is largely neglected and both the vapor and liquid velocities are assumed to be constant. Compared with a more complete model that incorporates the two-fluid model, the present pseudo-steady

PAGE 186

166 chilldown model requires more experimental measurements as inputs. However, the pseudo-steady chilldown model is computationally more robust and efficient for predicting chilldown. Overall, it provides reasonable results for the solid wall temperature. While a more complete model for chilldown that incorporates the mass, momentum, and energy equations of the vapor and the liquid is being developed to reduce the dependence of the experimental inputs for the liquid velocity and trial-anderror for the vapor velocity, the present study has revealed useful insight into the key elements of the two-phase heat transfer encountered in the chilldown process which have been largely ignored. It also provides the necessary modeling foundation for incorporating the two-fluid model. 5.2.4 Conclusions A pseudo-steady chilldown computational model has been developed to understand the heat transfer mechanisms of cryogenic chilldown and predict the chilldown wall temperature history in a horizontal pipeline. The model assumes a constant speed of the moving liquid wave front, and a steady thermal field in the solid within a moving frame of reference. This allows the 3-dimensional unsteady problem to be transformed to a 2-dimensional, parabolic problem. The study shows that the current film boiling correlations for the cryogenic pipe flow are not appropriate for the chilldown due to neglecting the information of flow regime. The new proposed film boiling correlation for chilldown in the pipe shows its success in predicting the film boiling heat transfer coefficient in chilldown. The study also shows the current popularly used nucleate boiling heat transfer correlations may not work well for cryogenic chilldown. The modified Kutateladze correlation with suppression factor can accurately provide the heat transfer coefficient. With the new and modified heat transfer correlations, the pipe wall

PAGE 187

167 temperature history based on the pseudo-steady chilldown model matches well with the experimental results by Chung et al. (2004) for almost the entire chilldown process. The pseudo-steady chilldown model has captured the important features of the thermal interaction between the pipe wall and the cryogenic fluid. 5.3 Separated Flow Chilldown Model Although the two-fluid model suffers from the ill-posedness problem and VKH instability problem at large slip velocity, it is a reliable model for predicting the pipe flow with moderate slip velocity. In this section, the two-fluid model will be combined with the 3-dimensional heat conduction in the solid wall to study the chilldown in the stratified flow regime in a horizontal pipe. This model is referred to as separated flow chilldown model. The pseudo-steady chilldown model is based on the Lagrangian description. That is the observer moving along the liquid wave front. Thus, the governing equation is simplified to a parabolic equation. The wall temperature profile is function of spatial location. In contrast, the separated flow chilldown model is based on the Eulerian description, i.e., the observers location is fixed in the space. Thus, the model focuses on the temperature history in a specified spatial regime. Furthermore, the separated flow chilldown model incorporates the two-fluid model in the pipe, so it can predict the flow and thermal field of fluid in addition to the wall temperature. 5.3.1 Formulation In the separated flow chilldown model, it is assumed that the flow is stratified flow with the vapor layer on the top and the liquid layer at the bottom, as shown in Figure 523. Governing equations for the fluid flow based on two-fluid model have been given in Chapter 2, Equations (2.11, 2.12, 2.14, 2.16, 2.18, and 2.19). Unsteady three-dimensional

PAGE 188

168 heat conduction equation in cylindrical form will be used for the thermal field in the pipe wall. Appropriate initial and boundary conditions for the separated flow chilldown model will be specified. Liquid layer U Vapor layer r x Vapor film Wall heat flux Pipe wall D Figure 5-23. Schematic of separated flow chilldown model. 5.3.1.1 Fluid flow In the separated flow chilldown model, the fluid volume fractions, velocities, enthalpies are solved with the two-fluid model. Due to the significant difference between the liquid and the gas density, usually the interface velocity is close to the liquid velocity. Thus, in this study it is assumedl iu u 1. The two-fluid model can be discretized using FOU scheme, CDS scheme, or other schemes investigated in Chapter 4. To improve the numerical stability, the discretization scheme used for the energy equation should be consistent with that used for mass and momentum equations. It is also assumed that the stability characteristics of the two-fluid model are not significantly changed by the presence of heat and mass transfer terms. 5.3.1.2 Heat conduction in solid pipe The thermal field inside the solid wall is governed by the three-dimensional unsteady heat conduction equation:

PAGE 189

169 T r k r r T rk r r x T k x t T c1 1 (5.47) Equation (5.47) is discretized using Euler implicit scheme in time and CDS scheme in space. 5.3.1.3 Heat and mass transfer In separated flow chilldown model, the heat and mass transfer between the liquid and the gas (vapor core) must be specified to close the model. The schematic of heat and mass transfer in the separated flow chilldown model is shown in Figure 5-24. Liquid Gas (vapor core)0 Solid g wq, g iq, l wq, l wq, m gSlSiS R Interface Figure 5-24. Schematic of heat and mass transfer in separated flow chilldown model. In Equation (2.18), lq is the total heat transfer rate to the liquid per unit length. It consists of the heat flux from the solid wall l wq, and from the liquid-gas interface to the liquid phase in the pipe l iq, :

PAGE 190

170 i l i l l w ldS q dS q q, ,. (5.48) It must be noted that l wq, depends on the heat transfer regime between the wall and the liquid. In the boiling heat transfer stage, l wq, is part of total heat flux from the wall to the fluid, wq and the other part of wall heat flux eva wq, is to evaporate the liquid to the vapor, which is not counted in the heat flux into the liquid phase. For instance, in film boiling regime, the total heat flux wq from the wall to the fluid is given by sat w FB wT T h q f (5.49) where FBh is film boiling heat transfer coefficient and is given by Equation (5.33). The heat flux from the wall to the liquid is evaluated by the forced convection heat transfer coefficient: l sat c l l wT T h q f ,, (5.50) where c lh, is given by Equation (2.35) or Equation (2.42) depending on whether the flow is turbulent or laminar. Therefore, eva wq, in film boiling regime is the difference between the total heat flux from the wall and the heat flux into the liquid l w w eva wq q q, f (5.51) If the heat transfer is in the nucleate boiling regime, l wq, through the convection heat transfer is l w c l l wT T h q f ,, (5.52) and eva wq, through the ebullition process is l w pool eva wT T h S q f ,, (5.53) where poolh is given by Equation (2.33).

PAGE 191

171 If the heat transfer between the wall and the liquid is due to single-phase forced convective heat transfer, 0, eva wq and l w c l w l wT T h q q f ,. (5.54) The heat flux to the liquid across the interface between the liquid and vapor core is evaluated by the single-phase convection heat transfer for liquid: l i c l l iT T h q f ,. (5.55) Since only convection heat transfer exists, the evaluation of total heat flow rate into the gas phase (vapor core) is much more straightforward than that for the liquid phase. The heat flux per unit length in the pipe, denoted as gq in Equation (2.19), consists of the heat flow from the solid wall and from the liquid-gas interface: i g i g g w gdS q dS q q, ,, (5.56) where g wq, is the heat flux from the wall to the gas and g iq, is the heat flux from the interface to the gas. In the above g wq, and g iq, are evaluated using g w c g g wT T h q f ,, (5.57) and g i c g g iT T h q f , (5.58) where c gh,is the forced convection heat transfer coefficient between the solid wall and the gas, which is given by Equation (2.43) and (2.44) The total mass transfer between the liquid and the gas consists of two parts. One is by the evaporation from the liquid to the vapor on the liquid-vapor interface, whose heat

PAGE 192

172 flux is eva iq, and the other is by the ebullition on the liquid-solid interface, whose heat flux is eva wq, Thus the mass transfer rate per unit length is fg i eva i l eva w fg evah dS q dS q h q m ,. (5.59) where eva iq, is evaluated using l i g i eva iq q q, , f (5.60) The heat and mass transfer models for in the separated flow chilldown model discussed above are outlined in Table 5-1. 5.3.1.3 Initial and boundary conditions Initially the pipe is filled with the vapor and a thin liquid layer of 05 0l at the bottom to avoid computational singularity of the two-fluid model associated with setting 0l. Stratified liquid and vapor enter the pipe from the left entrance. Boundary conditions for velocity and temperature are estimated based on experimental data. The inlet volume fraction at the entrance is given by Equation (5.13). For the boundary condition at the exit of pipe, a 2nd order extrapolation is employed. For the solid wall, the initial temperature is the ambient temperature. At the both ends of x-direction, adiabatic conditions are assumed. Periodic boundary conditions are employed in azimuthal direction. Boundary conditions on the inner and outer surface of the solid wall are determined by heat transfer correlations discussed in Section 2.3. The new correlations of film boiling and flow boiling proposed in Section 5.2 are also employed.

PAGE 193

173 Table 5-1. Heat and mass transfer relationship used in separated flow chilldown model. Description Equation Remark Heat transfer rate to the liquid per unit length i l i l l w ldS q dS q q, Heat transfer rate to the gas (vapor core) per unit length i g i g g w gdS q dS q q, Heat transfer rate to evaporate liquid to vapor per unit length i eva i l eva w evadS q dS q q, l sat c l l wT T h q f , Film boiling Heat flux from wall to liquid l w c l l wT T h q f , Flow boiling, single-phase convection l w w eva wq q q, f and sat w FB wT T h q f Film boiling l w pool eva wT T h S q f Flow boiling Heat flux for evaporation between liquid and wall 0, eva wq Single-phase convection Heat flux from interface to liquid l i c l l iT T h q f , sat iT T 1 Heat flux from wall to gas (vapor core) g w c g g wT T h q f , Heat flux from interface to gas (vapor core) g i c g g iT T h q f , sat iT T 1 Heat flux for evaporation at interface l i g i eva iq q q, , f Mass transfer rate per unit length fg evah q m 5.3.2 Solution Procedure The solution procedure of the separated flow chilldown model is shown in Figure 5-25. First, the heat flux between two phases and solid wall is calculated based on the heat transfer model presented in Section 2.3 and Section 5.3. Then, the calculated heat flux is used as a boundary condition to update the solid temperature. Next, the volume fraction, fluid velocity and pressure are calculated using two-fluid model. Subsequently, the calculated flow field is combined with fluid energy equations to obtain the fluid

PAGE 194

174 temperatures. After all the flow and temperature fields are updated, the calculation goes to the next time step. The solution procedure for two-fluid model is already discussed in Chapter 4. Liquid phase and gas phase mass and momentum equations are solved iteratively until both volume fraction and velocities converge. Then the energy equations for the vapor and liquid are solved for the vapor and liquid temperature, respectively. The energy equation for the solid wall is solved by Alternating Direction Implicit (ADI) method (Hirsch, 1988). Since heat transfer coefficients are the necessary boundary conditions for the solid energy equation, the heat flux between the fluid and the wall is calculated before the solid energy equation is solved. In the boiling heat transfer stage, vapor is rapidly generated due to the large temperature difference. This leads to a large mass transfer term in the two-fluid model. It can easily cause computation to become unstable if the time step is not sufficiently small. Thus, small time step for two-fluid model is used to overcome this numerical difficulty. However, ADI method for the solid energy equation can tolerate a large time step. More importantly, the 3-dimensional nature of the solid wall energy equation implies that much more computational resources are needed for the solid wall energy equation than that for the two-fluid model. Thus, to improve the computational efficiency, the solid energy equation is only solved after several time steps for the fluid. 5.3.3 Results and Discussion With the separated flow chilldown model, not only the temperature field of the solid wall can be obtained, but also the fluid velocity and fluid temperature in the pipe. To demonstrate the feasibility of the separated flow chilldown model, the computational

PAGE 195

175 results of the separated flow chilldown model are compared with the experimental data from Chung et al. (2004). Solve heat and mass transfer from Table 5-1 Solve solid wall temperature using Equation (5.47) Solve l, lu gu, and p from two-fluid model, as shown in Figure 4-3 Solve liquid and gas temperature using Equations (2.18, 2.19) t t t endt t No Set Initial and condition (t=0s) Output End Yes Solve solid heat transfer in this time step? No Yes Figure 5-25. Flow chart of separated flow chilldown model. The experimental facility of Chung et al. (2004) is shown in Figure 5-13. The geometry of the test section is shown in Figure 5-26. The test section to be investigated is only 210mm. However, to reduce the effect of downstream boundary condition on the accuracy of two-fluid model, the length of the computational domain is set to 300mm. In

PAGE 196

176 the two-fluid model, the grid for fluid is 100. To be compatible with the grids for fluid, the grids for solid wall are 100X40X40, i.e., 100 in the x -direction, 40 in the radial direction and 40 in the azimuthal direction. 70mm 70mm 14 11 12 15 5 8 9 6 2 1 4 3 Section 1 Section 2Section 3 120 n Flow 70mm Figure 5-26.Geometry of the test section and locations of thermocouples. The vapor volume fraction at the entrance is specified according to Equation (5.13). The characteristic liquid volume fraction 0 is 0.30. The characteristic time in this computation is 1000 t The liquid nitrogen at the inlet was known to be slightly subcooled; however, the subcooled temperature is not measured. The present computation using separated flow chilldown model shows that the chilldown process is not sensitive to the initial liquid subcooled temperature. Thus a 3K subcool is assumed for the liquid nitrogen. The vapor of nitrogen at the inlet is assumed to be saturated. Following the development in the pseudo-steady chilldown model, the Leidenfrost temperature for the nitrogen is set to be around 180 K, and the temperature at which the

PAGE 197

177 nucleate flow boiling switches to single-phase convection heat transfer is 140K. In modified Kutateladzes correlation, the suppression factor S is 0.005. The visual investigation of image of chilldown suggests liquid nitrogen velocity is 0.05 m/s. The study of pseudo-steady chilldown model in Section 5.3 suggests that the vapor velocity is 0.5 m/s. These two velocities are used as the inlet boundary conditions for the liquid and gas velocities. The convection scheme for the two-fluid model is CDS. The CFL for liquid phase is 0.005. To reduce the computational cost, the solid temperature field is updated after every 5 steps for flow variables. 5.3.3.1 Comparison of solid wall temperature The comparisons between the predicted temperatures and the measured temperatures at a number of spatial positions along the flow direction are presented. First, the wall temperature histories near the entrance of the pipe are shown in Figure 5-27 and 5-28. The locations of thermocouples 11, 12, 14, and 15 are shown in Figure 5-26. Good agreement is obtained for both the bottom and upper parts of the wall. Thermocouples 5, 6, 8, and 9 are located at 70mm downstream from thermocouples 11, 12, 14, and 15. Comparison of temperature histories at positions 5, 6, 8, and 9 are shown in Figure 5-29 and 5-30. The predicted temperatures in the bottom of the pipe clearly agree well with the experimental measurements. However, in the upper part of the wall, although the trend of predicted temperature profile is close to the experimental measurements, the predicted temperature is higher than the measured one. The comparison of temperature at positions 1, 2, 3 and 4, which is near the outlet of the pipe, is shown in Figure 5-31 and 5-32. Similarly, good agreement in the bottom of the pipe is obtained, but a discrepancy in the upper part of the wall exists.

PAGE 198

178 50 100 150 200 250 300 050100150200250300 t (s)T (K) T15 experimental T15 separated flow model T12 experimental T 12 separated flow model Figure 5-27. Comparison between measured and predicted transient wall temperatures of positions 12 and 15. 50 100 150 200 250 300 050100150200250300 t (s)T (K) T 11 experimental T 11 separated flow model T 14 experimenatal T 14 separated flow model Figure 5-28. Comparison between measured and predicted transient wall temperatures of positions 11 and 14.

PAGE 199

179 50 100 150 200 250 300 050100150200250300 t (s)T (K) T 6 experimental T 6 separated flow modell T 9 experimental T 9 separated flow modell Figure 5-29. Comparison between measured and predicted transient wall temperatures of position 6 and 9 (the measured T 9 is obviously incorrect). 50 100 150 200 250 300 050100150200250300 t (s)T (K) T 5 experimental T 5 separated flow modell T 8 experimenatal T 8 separated flow modell Figure 5-30. Comparison between measured and predicted transient wall temperatures of positions 5 and 8.

PAGE 200

180 50 100 150 200 250 300 050100150200250300 t (s)T (K) T 4 separated flow model T 3 separated flow model T3 experimental Figure 5-31. Comparison between measured and predicted transient wall temperatures of position 3 and the numerical result of temperature at position 4. 50 100 150 200 250 300 050100150200250300 t (s)T (K) T 2 experimental T 2 separated flow model T 1 experimenatal T 1 separated flow model Figure 5-32. Comparison between measured and predicted transient wall temperatures of positions 1 and 2.

PAGE 201

181 Good agreement in the bottom of the pipe suggests that the treatment of the flow dynamics and heat transfer of liquid in the pipe is correct. However, mechanisms that lead to rapid chilling on the upper part of the solid wall in the downstream part of the pipe need to be investigated in separated flow chilldown model. Since the heat removal by the liquid accounts for majority of the total heat removal from the wall during chilldown, the slight discrepancy for the temperature in the upper part of the wall does not affect the applicability of the separated flow chilldown model. 5.3.3.2 Flow field and fluid temperature Comparisons of wall temperature history show that the pseudo-steady chilldown model is a reasonable model for predicting wall temperature. However, the advantage of this model lies in the capability of predicting flow field. Figure 5-33 shows the liquid nitrogen depth profile during chilldown. Since the liquid depth at entrance varies with time, from s t50 to s t100, the liquid depth rises noticeably in Figure 5-33. After s t100, the liquid depth varies much less with the time. Another significant feature is that the slope of liquid and vapor interface varies with time. At s t50, the slope of the interface is larger than the slopes at s t100 and s t150. There are two possible reasons. One is that the heat transfer in the test section is in the film boiling stage at s t50; thus, low wall friction prevents build up of liquid and thus a steeper slope exists. The other reason is the massive evaporation of film boiling causes the more loss of the liquid. It results in a thinner liquid layer. Figure 5-34 shows nitrogen vapor velocity profile in the chilldown. The vapor velocity drops near the entrance because of increasing the vapor phase volume fraction. It is clearly shown in Figure 5-34 that the vapor velocity profiles are strongly influenced by

PAGE 202

182 which heat transfer regime it is in. At s t50, s t75 and s t100, heat transfer is dominated by the boiling heat transfer. Thus, a substantial amount of the liquid is evaporated and the vapor mass flux increases significantly in the x -direction. Consequently, the vapor velocity rises because of the higher vapor mass flux. It is further noted that at s t50 the heat transfer is in the film boiling regime and at s t75 and s t100 the heat transfer is in the nucleate boiling regime. There is more evaporative mass transfer in the film boiling regime than in the nucleate boiling regime. Hence the vapor velocity at s t50 is higher than those at s t75 and s t100. At s t150, and s t300, no vapor is generated in the region of the pipe considered in the computation, so the vapor velocity is almost constant. A slight decrease in the vapor velocity near m x2 0 is observed and it is due to the increase of the vapor volume fraction near m x2 0. x(m) LiquidDepth/Diameter00.050.10.150.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 50 75 100 150 300 t(s) Figure 5-33. Liquid nitrogen depth in the pipe during the chilldown.

PAGE 203

183 x(m) Vaporvelocity(m/s)00.050.10.150.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 50 75 100 150 300 t(s) Figure 5-34. Vapor nitrogen velocity in the pipe during the chilldown. Figure 5-35 shows the liquid nitrogen velocity profile during the chilldown. At the entrance, there is a jump of the liquid velocity. This is due to the decrease of the liquid volume fraction near the entrance, as shown in Figure 5-33. Liquid accelerates along flow direction at s t50 and s t75. The reason is that the vapor velocity rapidly increases due to the evaporation so that the vapor layer drags the liquid layer through the interface shear stress. At s t100, s t150, and s t300, the liquid velocity is much lower than those at s t50 and s t75. There may be two reasons for this phenomenon. First, the liquid layer is much thicker at the final stage of chilldown than that in the early stage; second, the vapor velocity decreases with the time. Thus, the interface dragging effect is insignificant at the final stage. Nevertheless, a slight liquid velocity rise is observed at s t100, s t150, and s t300 and it is due to the decrease of liquid volume fraction along the flow direction.

PAGE 204

184 x(m) Liquidvelocity(m/s)00.050.10.150.2 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 50 75 100 150 300 t(s) Figure 5-35. Liquid nitrogen velocity in the pipe during the chilldown. Figure 5-36 shows the nitrogen vapor temperature profiles in the chilldown. Vapor temperature rises along the flow direction because the low heat capacity vapor is continuously heated by the solid wall. However, the heat transfer on the liquid-gas interface tends to reduce the vapor core temperature. These two factors lead to a flat temperature profile near the exit of the pipe. During the chilldown, because the wall temperature at a given location decreases with the time, the heat flux between the wall and the vapor also decreases with the time. In the final stage of chilldown (s t300), the vapor temperature increases slowly in x-direction than at the early stage of chilldown. The liquid nitrogen temperature profiles in chilldown are shown in Figure 5-37. Significant difference exists between the film boiling chilldown stage and other stages. In the film boiling stage the cryogenic liquid core is separated from the wall by a thin vapor film, and the film layer hinders the direct heat transfer from the wall to the liquid. Thus the heat flux entering the liquid is quite low, and the liquid temperature rises very slowly.

PAGE 205

185 In contrast, the liquid temperature rises gradually during the stages dominated by forced convection (s s s t300 150 100). Since the wall temperature continues to drop with the time, the heat flux from the wall to liquid becomes smaller and smaller. x(m) Vaportemperature(K)00.050.10.150.2 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 50 75 100 150 300 t(s) Figure 5-36. Vapor nitrogen temperature in the pipe during the chilldown. x(m) Liquidtemperature(K)00.050.10.150.2 70 71 72 73 74 75 76 77 78 79 80 81 82 50 75 100 150 300 t(s) Figure 5-37. Liquid nitrogen temperature in the pipe during the chilldown.

PAGE 206

186 5.3.4 Conclusions In this section, the separated flow chilldown model is developed that combines the heat transfer inside solid wall with the two-phase flow model for horizontal separated flow chilldown. The heat transfer models previously discussed are implemented in the separated flow chilldown model. The model can predict 3-dimensional wall temperature, as well as the essential flow properties inside the pipe, such as volume fractions, liquid and gas velocity, and pressure. The computed flow field shows that in the film boiling heat transfer stage, vapor velocity rises quickly in the pipe due to enormous fluid evaporated through boiling. In addition, liquid-vapor interface shear stress drags liquid, so liquid velocity rises as well as vapor velocity. However, in the latter stage of chilldown in a given region, liquid and vapor velocities are approaching a steady state, because boiling phenomenon no longer exists. It also shows that vapor temperature increases significantly in chilldown due to low heat capacity, and liquid temperature increases slightly. The predicted pipe wall temperature histories at different locations on the flow axis agree well with the experimental measurements on the bottom of the pipe wall but discrepancies between the prediction and measurement exist on the upper part of the wall near the outlet of the pipe. The separated-flow chilldown model is a comprehensive chilldown model with the capability of obtaining both flow properties and the wall temperature history.

PAGE 207

187 CHAPTER 6 CONCLUSIONS AND DISCUSSION 6.1 Conclusions In this dissertation unsteady flow boiling heat transfer of cryogenic fluids is studied. Proper models for chilldown simulation are developed to predict the flow fields and thermal fields. Major conclusions are 1. Flow regimes and heat transfer regimes in the cryogenic chilldown are identified by visual study. Based on the visual study and the experimental measurement, homogeneous and separated flow model for the respectively vertical pipe and horizontal pipe are presented. The heat transfer models for film boiling, flow boiling and forced convection heat transfer in chilldown are reviewed and qualitatively assessed. 2. A physical model to predict the early stage bubble growth in saturated heterogeneous nucleate boiling is presented. The structure of the thin unsteady liquid thermal boundary layer is revealed by the asymptotic and numerical solutions. The existence of a thin unsteady thermal boundary layer near the rapidly growing bubble allows for a significant amount of heat flux from the bulk liquid to the vapor bubble dome, which in some cases can be larger than the heat transfer from the microlayer. The experimental observation by Yaddanapudi and Kim (2001) on the insufficiency of heat transfer to the bubble through the microlayer is elucidated. 3. A pressure correction algorithm for two-fluid model is carefully implemented to minimize its effect on stability. Numerical instability for the incompressible two-fluid model near the ill-posed condition is investigated for various cell face interpolation schemes with the aid of von Neumann stability analysis. The stability analysis for the wave growth rates by using the 1st order upwind, 2nd order upwind, QUICK, and the central difference schemes shows that the central difference scheme is more accurate and more stable than the other schemes. The 2nd order upwind scheme is much more susceptible to instability at long waves than the 1st order upwind and inaccurate for short waves. The instability associated with ill-posedness of the two-fluid model is significantly different from the instability of the discretized two-fluid model. Excellent agreement is obtained between the computed and predicted wave growth rates. The connection between the ill-posedness of the two-fluid model and the numerical stability of the algorithm used to implement the inviscid two-fluid model is elucidated.

PAGE 208

188 4. The pressure correction algorithm is implemented to solve the viscous two-fluid model. The von Neumann stability analysis for the viscous two-fluid model by using the 1st order upwind, 2nd order upwind, QUICK, and the central difference schemes shows similar results to the inviscid two-fluid model. The central difference has the best accuracy, followed by the QUICK scheme and 1st order upwind scheme, and the 2nd order upwind scheme has the worst stability among investigated schemes. The viscous Kelvin-Helmholtz instability is significantly different from the instability of discretized viscous two-fluid model. Only the most accurate scheme with the extremely fine grid can capture the wave associated with the VKH instability. Excellent agreement between the numerical results and the predicted results is obtained for the growth of the wave amplitude. Inlet disturbance growth test shows the pressure correction scheme is capable of handling viscous two-phase flow in a pipe. 5. Current film boiling correlations for the cryogenic pipe flow are not appropriate for chilldown due to neglecting the information of flow regime. A new film boiling correlation for chilldown in the pipe is developed. It is successful in predicting film boiling heat transfer coefficient in chilldown. The study also shows the current popularly used nucleate boiling heat transfer correlations may not work well under the cryogenic condition. A modified Kutateladze correlation with suppression factor leads to a more reasonable simulation result. 6. Homogeneous chilldown model is developed to simulate the chilldown in vertical pipe where homogeneous flow is prevalent. In horizontal chilldown where separated flow dominates, pseudo-steady chilldown model is developed with the reference frame at the moving liquid wave front. This allows the 3-dimensional unsteady problem to be transformed to a 2-dimensional, parabolic problem. The pseudo-steady chilldown model can capture the essential part of chilldown and provides a good testing platform to study cryogenic heat transfer correlations for chilldown. A more comprehensive separated flow chilldown model is developed that combines the heat transfer inside solid wall with the two-phase flow model for horizontal separated flow chilldown. The computed pipe wall temperature histories at various locations match well with the experimental results by Chung et al. (2004). The separated flow chilldown model also predicts the flow field as well as the wall temperature field. 6.2 Suggested Future Study Future research efforts focus on improving the accuracy and efficiency of chilldown models. More comparisons between the computational measurements and model predictions should be performed. Another focus should be to improve cryogenic heat transfer correlations, especially the accuracy of the cryogenic film boiling and nucleate boiling. Furthermore, study on

PAGE 209

189 the transition between film boiling and nucleate boiling is necessary for cryogenic chilldown.

PAGE 210

190 LIST OF REFERENCES Akiyama, M., Tachibana, F., and Ogawa, N., 1969, Effect of pressure on bubble growth in pool boiling. Bulletin of JSME 12, 1121-1128. Bai, Q., and Fujita, Y., 2000, Numerical simulation of bubble growth in nucleate boiling effects of system parameter. Engineering Foundation Boiling 2000 Conference, Anchorage AL, May 2000, 116-135. Bailey, N. A., 1971, Film boiling on submerged vertical cylinders. Report AEEWM1051, Winfrith, U.K. Baines, R. P., El Masri, M. A., and Rohsenow, W. M, 1984, Critical heat flux in flowing liquid films. International Journal of Heat Mass Transfer 27, 1623-1629. Barnea, D., and Taitel, Y., 1994a, Interfacial and structural stability of separated flow. International Journal of Multiphase Flow 20, 387-414. Barnea, D., and Taitel, Y., 1994b, Non-linear interfacial instability of separated flow. Chemical Engineering Science 49, 2341-2349. Bell, K. J., 1969, The Leidenfrost phenomenon: a survey. Chemical Engineering Progress Symposium Series 63, 73-79. Bendiksen, K. H., Maines, D., Moe, R., and Nuland, S., 1991, The dynamic two-fluid model OLAG: theory and application. Society of Petroleum Engineering, May, 171-180. Bennett, D. L., and Chen, J. C., 1980, Forced convection for the in vertical tubes for saturated pure components and binary mixture. A.I.Ch.E. Journal 26, 454-461. Berenson, P. J., 1961, Film-boiling heat transfer from a horizontal surface. Journal of Heat Transfer 83, 351-358. Black, P. S., Daniels, L. C., Hoyle, N. C., and Jepson, W. P., 1990, Study transient multi-phase flow using the pipeline analysis code (PLAC). Journal of Energy Resources Technology 112, 25-29. Brauner, N., and Maron, D. M., 1992, Stability analysis of stratified liquid-liquid flow. International Journal of Multiphase Flow 18, 103-121.

PAGE 211

191 Breen, B. P., and Westwater, J. W., 1962, Effect if diameter of horizontal tubes on film heat transfer. Chemical Engineering Progress 58 (7), 67-72. Bromley, J. A., 1950, Heat transfer in stable film boiling. Chemical Engineering Progress 46 (5), 221-227. Bronson, J. C., Edeskuty, F.J., Fretwell, J.H., Hammel, E.F., Keller, W.E., Meier, K.L., Schuch, and A.F, Willis, W.L., 1962, Problems in cool-down of cryogenic systems. Advances in Cryogenic Engineering 7, 198-205. Burke, J. C., Byrnes, W. R., Post, A. H., and Ruccia, F. E., 1960, Pressure cooldown of cryogenic transfer lines. Advances in Cryogenic Engineering 4, 378-394. Carey, V. P., 1992, Liquid-vapor phase-change phenomena. Taylor & Francis Press, New York. Chan, A. M. C., and Banerjee, S., 1981a, Refilling and rewetting of a hot horizontal tube part I: experiment. Journal of Heat Transfer 103, 281-286. Chan, A. M. C., and Banerjee, S., 1981b, Refilling and rewetting of a hot horizontal tube part II: structure of a two-fluid model. Journal of Heat Transfer 103, 287-292. Chan, A. M. C., and Banerjee, S., 1981c. Refilling and rewetting of a hot horizontal tube part III: application of a two-fluid model to analyze rewe tting. Journal of Heat Transfer 103, 653-659. Chen, J. C., 1966, Correlation for boiling heat transfer to saturated fluids in convective flow. Industry Engineering Chemistry Process Design and Development 5, 322329. Chen, T., Klausner, J. F., and Chung, J. N., 2003, Subcooled boiling heat transfer and dryout on a constant temperature microheater. 5th International Conference on Boiling Heat Transfer, Montego Bay, Jamaica. Chen, W., 1995, Vapor bubble growth in heterogeneous boiling. Ph.D Dissertation, University of Florida. Chen, W., Mei, R., and Klausner, J., 1996, Vapor bubble growth in highly subcooled heterogeneous boiling. Convective Flow Boiling, 91-97. Chi, J. W. H., 1965, Cooldown temperatures and cooldown time during mist flow. Advances in Cryogenic Engineering 10, 330-340. Chi, J. W. H., and Vetere, A.M., 1963, Two-phase flow during transient boiling of hydrogen and determination of nonequilibrium vapor fractions. Advances in Cryogenic Engineering 9, 243-253.

PAGE 212

192 Chung, J. N., Yuan, K., and Xiong, R., 2004, Two-phase flow and heat transfer of a cryogenic fluid during pipe chilldown. 5th Int. Conf. on Multiphase Flow, Yokohama, Japan, 468. Cooper, M. G., 1969, The microlayer and bubble growth in nucleate pool boiling. International Journal of Heat and Mass Transfer 12, 895-913. Cooper, M. G., 1984, Saturation nucleate pool boiling-a simple correlation, 1st U.K. National Conference on Heat Transfer 2, 785-793. Cooper, M. G., and Lloyd, A. J. P., 1969, The microlayer in nucleate pool boiling. International Journal of Heat and Mass Transfer 12, 915-933. Cooper, M. G., and Vijuk, R. M., 1970, Bubble growth in nucleate pool boiling. 4th International Heat Transfer Conference, Paris, 5, B2.1, Elsevier, Amsterdam. Cross, M. F., Majumdar, A. K., Bennett Jr., J. C., and Malla, R. B., 2002, Model of chilldown in cryogenic transfer linear. Journal of Spacecraft and Rockets 39, 284289. Dougall, R. S., and Rohsenow, W. M., 1963, Film boiling on the inside of vertical tubes with upward flow of the fluid at low qualities, MIT report no 9079-26, MIT. Ellerbrock, H. H., Livingood, J. N. B., and Straight, D. M., 1962, Fluid-flow and heat-transfer problems in nuclear rockets. NASA SP-20. Ellion, M.E., 1954, A study of the mechanism of boiling heat transfer. Ph.D Thesis, California Institute of Technology. Ferderking, T. H. K., and Clark, J. A., 1963, Nature convection film boiling on a sphere. Advanced Cryogenic Engineering 8, 501-506. Ferziger, J. H., and Peric, M., 1996, Computational methods for fluid dynamics. Springer, Berlin. Fiori, M. P., and Bergles, A. E., 1970, Model of critical heat flux in subcooled flow boiling. Proceeds of 4th International Heat Transfer Conference 6, Versailles, France, 354-355. Fyodorov, M. V., and Klimenko, V. V., 1989, Vapour bubble growth in boiling under quasi-stationary heat transfer conditions in a heating wall. International Journal of Heat and Mass Transfer 32, 227-242. Galloway, J. E., and Mudawar, I., 1993a, CHF mechanism in flow boiling from a short heat wall I. examination of near-wall conditions with the aid of photomicrography and high-speed video imaging. International Journal of Heat Mass Transfer 36, 2511-2526.

PAGE 213

193 Galloway, J. E., and Mudawar, I., 1993b, CHF mechanism in flow boiling from a short heat wall II theoretical CHF model. International Journal of Heat Mass Transfer 36, 2527-2540. Giarratano, P. J., and Smith, R. V., 1965, Comparative study of forced convection boiling heat transfer correlations for cryogenic fluids. Advances in Cryogenic Engineering 11, 492-505. Gidaspow, D., 1974, Modeling of two phase flow. 5th International Heat Transfer Conference 7, 163. Graham, R. W., Hendricks, R. C., Hsu, Y. Y., and Friedman, R., 1961, Experimental heat transfer and pressure drop of film boiling liquid hydrogen flowing through a heated tube. Advances in Cryogenic Engineering 6, 517-524. Gungor, K. E., and Winterton, R. H. S., 1996, A general correlation for flow boiling in tubes and annuli. International Journal of Heat Mass Transfer 29(3), 351-358. Han, C., and Griffith, P., 1965, The mechanism of heat transfer in nucleate pool boiling part I. International Journal of Heat and Mass Transfer 8, 887-904. Hebel, W., Detavernier, W., and Decreton, M., 1981, A contribution to the hydrodynamics of boiling crisis in a forced flow of water. Nuclear Engineering and Design 64, 443-445. Hedayatpour, A., Antar, B. N., and Kawaji, M., 1993, Cool-down of a vertical line with liquid nitrogen. Jouranl of Thermophysics and Heat Transfer 7, 426-434. Hendricks, R. C., Graham, R. W., Hsu, Y. Y., and Friedman, R., 1961, Experimental heat transfer and pressure drop of liquid hydrogen flowing through a heated tube. NAS TN D-765. Hendricks, R. C., Graham, R. W., Hsu, Y. Y., and Friedman, R., 1966, Experimental heat transfer results for cryogenic hydrogen flowing in tubes at subcritical and supercritical pressure to 800 pounds per square inch absolute. NASA TN D-3095. Hewitt, G. F., 1982, Liquid-gas systems. Handbook of Multiphase Systems, G. Hestroni, Ed., McGraw-Hill, New York. Hino, R., and Ueda, T., 1985a, Studies on heat transfer and flow boiling part 1boiling characteristics. International Journal of Multiphase Flow 11, 269-281. Hino, R., and Ueda, T., 1985b, Studies on heat transfer and flow boiling part 2flow characteristics. International Journal of Multiphase Flow 11, 283-297. Hirsch, C., 1988, Numerical computation of internal and external flows, volume I: fundamental of numerical discretization. John Wiley& Sons, New York.

PAGE 214

194 Hsu, Y. Y., August, 1962, On the size of active nucleation cavities on a heating surface. Journal of Heat Transfer, 207-216. Incropera, F. P., and Dewitt, D. P., 2002, Fundamentals of heat and mass transfer, 5th edition, John Wiley& Sons, New York. Ishii, M., 1975, Thermo-fluid dynamic theory of two phase flow. Eyrolles, Paris. Issa, R. I., and Kempf, M. H. W., 2003, Simulation of slug flow in horizontal and nearly horizontal pipes with the two-fluid model. International Journal of Multiphase Flow 29, 69-95. Issa, R. I., and Woodburn, P. J., 1998, Numerical prediction of instabilities and slug formation in horizontal two-phase flows. 3rd International Conference on Multiphase Flow, ICMF98, Lyon, France. Jackson, J., Liao, J., Klausner, J. F., and Mei, R., 2005, Transient heat transfer during cryogenic chilldown. ASME summer heat transfer conference, San Francisco. Jones, A. V., and Prosperettii, A., 1985, On the stability of first-order differential models for two-phase flow prediction. International Journal of Multiphase Flow 11, 133148. Koffman, L. D., and Plesset, M. S., 1983, Experimental observations of microlayer in vapor bubble growth on a heated solid. Journal of Heat Transfer 105, 625-632. Kutateladze, S. S., and Leontev, A. I., 1966, Some applications of the asymptotic theory of the turbulent boundary layer. 3rd International Heat Transfer Conference 3, Chicago, Illinois, 1-6. Labunstov, D.A., 1963, Mechanism of vapor bubble growth in boiling under on the heating surface. Journal of Engineering Physics 6 (4), 33-39. Laverty, W. F., and Rohsenow, W. M., 1967, Film boiling of saturated nitrogen flowing in a vertical tube. Journal of Heat transfer 89, 90-98. Lee, C. H., and Mudawar, I., 1988, A mechanistic critical heat flux model for subcooled flow boiling based on local bulk flow conditions. International Journal of Multiphase Flow 14, 711-728. Lee, R. C., and Nydahl, J. E., 1989, Numerical calculation of bubble growth in nucleate boiling from inception through departure. Journal of Heat Transfer 111, 474-479. Lionard, J. E., Sun, K. H., and Dix, G. E., 1977, Solar and nuclear heat transfer, AIChE Symp. Serial 73 (164), 7.

PAGE 215

195 Lyczkowski, R. W., Gidaspow, D., Solbrig, C. W., and Hughes, E. D., 1978, Characteristics and stability analyses of transient one-dimensional two-phase flow equations and their finite difference approximations. Nuclear Science and Engineering 66, 378-396. Martinelli, R. C., and Nelson, D. B., 1948, Prediction of pressure drop during forced-circulation boiling of water. Transaction of ASME 70, 695-701. Mattson, R. J., Hammitt F. G., and Tong, L. S., 1973, A photographic study of the subcooled flow boiling crisis in freon-113. ASME Paper 73-HT-39. Mayinger, F., 1996, Advanced experimental methods. Convective Flow Boiling, Taylor & Francis, 15-28. Mei, R., Chen, W., and Klausner, J., 1995a, Vapor bubble growth in heterogeneous boiling I. formulation. International Journal of Heat and Mass Transfer 38, 909919. Mei, R., Chen, W., and Klausner, J., 1995b, Vapor bubble growth in heterogeneous boiling II. growth rate and thermal fields. International Journal of Heat and Mass Transfer 38, 921-934. Moore, F. D., and Mesler, R. B., 1961, The measurement of rapid surface temperature fluctuations during nucleate boiling of water. A.I.Ch.E. Journal 7, 620-624. Ohkawa, T., and Tomiyama, A., 1995, Applicability of high-order upwind difference methods to the two-fluid model. Advances in Mutiphase Flow, Elsevier Science, 227-240. Patankar, S. V., 1980, Numerical heat transfer and fluid flow. McGraw-Hill, New York. Raithby, G. D., and Hollands, K. G. T., 1975, A general method of obtaining approximate solutions to laminar and turbulent free convection problems. Advances in Heat Transfer, Academic Press, New York 11, 265-315. Shyy, W., 1994, Computational modeling for fluid flow and interfacial transport. Elsevier, Amsterdam. Son, G., Dhir, V. K., and Ramannujapu, N., 1999, Dynamics and heat transfer associated with a single bubble during nucleate boiling on a horizontal surface. Journal of Heat Transfer 121, 623-631. Song, J. H., and Ishii, M., 2000, The well-posedness of incompressible one-dimensional two-fluid model. International Journal of Heat and Mass Transfer 43, 2221-2231. Steiner, D., 1986, Heat transfer during flow boiling of cryogenic fluids in vertical and horizontal tubes. Cryogenics 26, 309-318.

PAGE 216

196 Stephan, K., and Auracher, H., 1981, Correlation for nucleate boling heat transfer in forced convection. International Journal of Heat Mass Transfer 24, 99-107. Steward, W. G., Smith and R. V., Brennan, J.A., 1970, Cooldown transients in cryogenic transfer lines. Advances in Cryogenic Engineering 15, 354-363. Stewart, B. H., 1979, Stability of two-phase flow calculation using two-fluid models. Journal of Computational Physics 33, 259-270. Sun, K. H., Dix, G. E., Tien, C. L., 1974, Cooling of a very hot vertical surface by a falling liquid film. Journal of Heat Transfer, Transaction of ASME 96, 126-131. Suryanarayana, N. V., and Merte, H., 1972, Film boiling on vertical surfaces. Journal of Heat Transfer 94, 377-384. Tennekes, H., and Lumley, J. L., 1972, A first course in turbulence. M.I.T. Press, Cambridge. Thompson, T. S., 1972, An analysis of the wet-side heat-transfer coefficient during rewetting of a hot dry patch. Nuclear Engineering and Design 22, 212-224. Tien, C. L., and Yao, L. S., 1975, Analysis of conduction-controlled rewetting of a vertical surface. Journal of Heat Transfer, Transaction of ASME, May 1975, 161165. Tong, L. S., 1968, Boundary-layer analysis of the flow boiling crisis. International Journal of Heat Mass Transfer 11, 1208-1211. Velat, C., 2004, Experiments in cryogenic two phase flow. Master thesis, University of Florida. Velat, C., Jackson, J., Klausner, J.F., and Mei, R., 2004, Cryogenic two-phase flow during chilldown. ASME HT-FED Conference, Charlotte, NC. van Stralen, S. J. D., 1966; 1967, The mechanism of nucleate boiling in pure liquid and in binary mixtures, parts I-IV, International Journal of Heat and Mass Transfer 9, 9951020, 1021-1046;10, 1469-1484, 1485-1498. van Stralen, S. J. D., Cole, R., Sluyter, W. M., and Sohal, M. S., 1975, Bubble growth rates in nucleate boiling of water at subatmospheric pressures. International Journal of Heat and Mass Transfer 18, 655-669. van Stralen, S. J. D., Sohal, M. S., Cole, R. and Sluyter, W. M., 1975, Bubble growth rates in pure and binary systems: combined effect of relaxation and evaporation microlayer. International Journal of Heat and Mass Transfer 18, 453-467. von Glahn, U. H., 1964, A correlation of film-boiling heat transfer coefficients obtained with hydrogen, nitrogen and freon 113 in forced flow. NASA TN D-2294.

PAGE 217

197 Wallis, G. B., 1969, One-dimensional two-phase flow. McGraw-Hill, New York. Weisend, J. G. II, 1998, Handbook of cryogenic engineering, Taylor & Francis, New York. Weisman, J., and Pei, B. S., 1983, Prediction of critical heat flux in flow boiling at low qualities. International Journal of Heat Mass transfer 26, 1463-1477. Westwater, J. W., 1988, Boiling heat transfer. International Communication of Heat Transfer 15, 381-400. Wiebe, J. R., and Judd, R. L., 1971, Superheat layer thickness measurements in saturated and subcooled nucleate boiling. Journal of Heat Transfer 93, 455-461. Yaddanapudi, N., and Kim, J., 2001, Single bubble heat transfer in saturated pool boiling of FC-72. Multiphase Science and Technology 12, 47-63. Zoppou, C., and Roberts, S., 2003, Explicit schemes for dam-break simulations. Journal of Hydraulic Engineering 129, 11-34. Zuber, N., Tribus, M., and Westerwater, J. W., 1961, The hydro dynamic crisis in pool boiling of saturated and subcooled liquid. International Developments in Heat Transfer, ASME, 230-236. Zurcher, O., Favrat, D., and Thome, J. R., 2002, Evaporation of refrigerants in a horizontal tube: an improved flow pattern dependent heat transfer model compared to ammonia data. International Journal of Heat and Mass Transfer 45, 303-317.

PAGE 218

198 BIOGRAPHICAL SKETCH Jun Liao was born in Hubei, China, in 1973. After receiving his Bachelor of Science degree in Turbomachinery and Refrigeration from Huazhong University of Science and Technology in 1994, he received Master of Science degree in Mechanical Engineering from Xian Jiaotong University. In pursuit of a Ph.D. degree in Aerospace Engineering, Jun Liao began his studies at the University of Florida in 2001.


Permanent Link: http://ufdc.ufl.edu/UFE0010123/00001

Material Information

Title: Modeling Two-Phase Transport during Cryogenic Chilldown in a Pipeline
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0010123:00001

Permanent Link: http://ufdc.ufl.edu/UFE0010123/00001

Material Information

Title: Modeling Two-Phase Transport during Cryogenic Chilldown in a Pipeline
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0010123:00001


This item has the following downloads:


Full Text












MODELING TWO-PHASE TRANSPORT DURING CRYOGENIC CHILLDOWN IN
A PIPELINE


















By

JUN LIAO


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


2005


































Copyright 2005

by

Jun Liao















ACKNOWLEDGMENTS

I would like to express my appreciation to all of the individuals who have assisted

me in my educational development and in the completion of my dissertation. My greatest

gratitude is extended to my supervisory committee chair, Dr. Renwei Mei. Dr. Mei's

excellent knowledge, boundless patience, constant encouragement, friendly demeanor,

and professional expertise have been critical to both my research and education. Dr.

James F. Klausner also deserves recognition for his knowledge and technical expertise. I

would like to further thank Dr. Jacob N. Chung for kindly providing his experiment data

of chilldown.

I would like to additionally recognize my fellow graduate associates Christopher

Velat, Jelliffe Jackson, Yusen Qi, and Yi Li for their friendship and technical assistance.

Their diverse cultural background and character have provided an enlightening and

positive environment. Special appreciation is given to Kun Yuan for his kindness

providing his experiment data and insight on chilldown.

I would like to further acknowledge the Hydrogen Research and Education

Program for providing funding to this study. This research was also funded by NASA

Glenn Research Center under contract NAG3-2750.

Finally, I would like to recognize my wife Xiaohong Liao and my parents for their

continual support and encouragement.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ....................................................................... .....................iii

LIST OF TABLES .............. ................ .......... ............... ............. vii

LIST OF FIGURES ............... ............... ......................... ........... viii

N O M E N C L A T U R E ........................................................................ ........................... xiv

A B S T R A C T ...............................x..................................................... .. x ix

CHAPTER

1 IN TR O D U C T IO N ............ ............................................. ........ .. ........... .. 1

1.1 B background ...................... ............... ........... ... ............................... . 1
1.2 L literature R eview ......................................................... ............ .. ............ 4
1 .3 S c o p e ...................... .. ............. .. ..................................................... 1 0

2 TWO-PHASE FLOW MODELING AND FLOW BOILING HEAT TRANSFER OF
CRY O GEN IC FLU ID.............................................................. .......................... 13

2.1 Flow Regim e and Heat Transfer Regim e.................................. .................... 13
2.2 Flow Models in Cryogenic Chilldown.................................................... 18
2.2.1 Hom ogeneous Flow M odel.................................................................... 18
2 .2 .2 T w o-F luid M odel ................................................................. ............... ... 22
2.3 Heat Transfer between Cryogenic Fluid and Solid Pipe Wall............................ 26
2.3.1 Heat Transfer between Liquid and Solid wall ....................................... 27
2.3.1.1 Film boiling ................. .............. .. .. .. ...... ..... .. ..... 27
2.3.1.2 Forced convection boiling and two-phase convective heat transfer 30
2.3.2 Heat Transfer between Vapor and Solid Wall................. .................... 33

3 VAPOR BUBBLE GROWTH IN SATURATED BOILING.............. ................ 34

3 .1 In tro d u ctio n ............ .. .............. .............................. ............... 3 4
3.2 Formulation ................ ......... ......... ........ 39
3.2.1 On the Vapor Bubble ............ .... ......... ...................... 39
3.2.2 M icrolayer............................. .............. 41
3.2.3 Solid Heater .......................................... 42









3.2.4 O n the B ulk Liquid .............. ........................................................... 43
3.2.4.1 V elocity field ... ..................................................... ... .. ....... .. 43
3.2.4.2 Tem perature field .............................. ............. ............... 44
3.2.4.3 Asymptotic analysis of the bulk liquid temperature field during early
stages of growth ...... ........................ .......... .............. 46
3.2.5 Initial Conditions .............. ..... ............. ....... ............ ................. 50
3.2.6 Solution Procedure............................................... .......................... 52
3.3 R results and D iscu ssions ................................................................ ............... ... 52
3.3.1 Asymptotic Structure of Liquid Thermal Field ......................... ............. 52
3.3.2 Constant Heater Temperature Bubble Growth in the Experiment of
Y addanapudi and K im .......................................................... .................. 56
3.3.3 Effect of Bulk Liquid Thermal Boundary Layer Thickness on Bubble
G row th .................. ............................... ....... ......... ...... 5 9
3 .4 C o n clu sio n s ............... .............................................................. 6 3

4 ANALYSIS ON COMPUTATIONAL INSTABILITY IN SOLVING TWO-FLUID
MODEL ........ ........... ........................... 64

4.1 Inviscid T w o-Fluid M odel .............. ................................................................ 65
4 .1.1 Introdu action .......................................... ...... ............ 65
4.1.2 G governing E quations ........................................... .................... ...... 67
4.1.3 Theoretical A analysis ..................................................................... 69
4.1.3.1 Characteristic analysis and ill-posedness ...................................... 69
4.1.3.2 Inviscid Kelvin-Helmholtz (IKH) analysis and linear instability.... 72
4.1.4 Analysis on Computational Instability ............................................... 73
4.1.4.1 Description of numerical methods........................... .... .............. 73
4.1.4.2 Code validation- dam-break flow ........................................ 78
4.1.4.3 Von Neumann stability analysis for various convection schemes .. 81
4.1.4.4 Initial and boundary conditions for numerical solutions............... 86
4.1.5 R results and D discussion .............................................................. ... 87
4.1.5.1 Computational stability assessment based on von Neumann stability
analysis ............................................................. 87
4.1.5.2 Schem e consistency tests........................................................... 94
4.1.5.3 Computational assessment based on the growth of disturbance...... 95
4.1.5.4 Discussion on the growth of short wave................................ .. 101
4.1.5.5 Wave development resulting from disturbance at inlet............... 104
4 .1.6 C onclu sions........................................... .................... .. .............. 106
4.2 V iscous Tw o-Fluid M odel .......................................... ........................... 110
4.2.1 Introduction..................................... 110
4.2.2 Governing Equations ................................... .. ...................... 111
4.2.3 Theoretical Analysis ...................................... .............. 112
4.2.3.1 Characteristics and ill-posedness............................... .............. 112
4.2.3.2 Viscous Kelvin-Helmholtz (VKH) analysis and linear instability 113
4.2.4 Analysis on Computational Intability ............ ............ ............. 115
4.2.4.1 Description of numerical methods.................. .. ................ 115
4.2.4.2 Von Neumann stability analysis for various convection schemes 116
4.2.4.3 Initial and boundary conditions for numerical solution............... 119









4.2.5 Results and Discussion ................... ........ .............. 119
4.2.5.1 Computational stability assessment based on von Neumann stability
a n a ly sis ............... ... ..... ............. ..... .... .. ................... 1 1 9
4.2.5.2 Computational assessment based on the growth of disturbance.... 126
4.2.5.3 Wave development resulting from disturbance at inlet............... 128
4 .2 .6 C on clu sion s........................................... .. ..................... ................. 130

5 MODELING CRYOGENIC CHILLDOWN..................................... 133

5.1 Homogeneous Chilldown Model .......................................................... 133
5.1.1 Analysis ....... ....... ............................ 134
5.1.2 R results and D discussion ...................................... ................. ... ........... 136
5.2 Pseudo-Steady Chilldown M odel... .................... .. .................. 140
5.2 .1 F orm ulation .................. ... ......................... ..... ........... .. ............ 14 1
5.2.1.1 H eat conduction in solid pipe ............. ................ ................... 141
5.2.1.2 Liquid and vapor flow ......................................... .............. 144
5.2.1.3 Film boiling correlation ...... .......... ....................... ............. 145
5.2.1.4 Forced convection boiling correlation................. .................... 151
5.2.1.5 Heat transfer between solid wall and environment........................ 152
5.2.2 Results and Discussion .................................. 155
5.2.2.1 Experim ent of Chung et al .............. ......................... ................ 156
5.2.2.2 Comparison of pipe wall temperature .......................................... 157
5.2.3 D discussion and R em arks .................................... ................................... 163
5.2 .4 C onclu sion s............................................. .. .............. .......... ....... 166
5.3 Separated Flow Chilldown M odel .............. ...... ......................................... 167
5 .3 .1 F o rm u latio n ........................................ ... ............... ....... ................ .. 16 7
5 .3 .1.1 F lu id fl ow .... .................. .. .................................. .............. 16 8
5.3.1.2 H eat conduction in solid pipe ............. ................ ................... 168
5.3.1.3 Heat and mass transfer............................. .............. 169
5.3.1.3 Initial and boundary conditions ................................... ............... 172
5.3.2 Solution Procedure............................. .............. 173
5.3.3 Results and Discussion ........................................ 174
5.3.3.1 Comparison of solid wall temperature................ .......... .... 177
5.3.3.2 Flow field and fluid tem perature .................................................. 181
5.3 .4 C onclu sion s....................................... .. ................. ...... ... .. ..... 186

6 CONCLUSIONS AND DISCUSSION ............................................................. 187

6 .1 C o n clu sio n s .............................................................................. 18 7
6.2 Suggested Future Study ....................................... .............. 188

LIST O F REFEREN CE S ................................................... ................................. 190

B IO G R A PH IC A L SK E T C H .......................................................................................... 198















LIST OF TABLES

Table page

4-1. Analytical solution for dam-break flow.......................................... .............. 80

4-2. A (O) for different discretization schem es............................................ ... ... .............. 85

5-1. Heat and mass transfer relationship used in separated flow chilldown model........ 173
















LIST OF FIGURES


Figure page

1-1. Schematic of filling facilities for LH2 transport system from storage tank to space
shuttle external tank............................................. ................ ......... 3

1-2. The schematic of chilldown and heat transfer regime. ........................................... 9

2-1. Schematic of two-phase flow regime in horizontal pipe. ...................................... 14

2-2. Schematic of two-phase flow regime in vertical pipe............................ ............ 14

2-4. Typical wall temperature variation during chilldown............................ ............ 17

2-5. Schem atic for hom ogeneous flow m odel.............................................. ... ................. 19

2-6. Schematic of the two-fluid model ................................................................. 22

2-7. Schem atic of heat transfer in chilldow n.............................................. .... .. .............. 27

3-1. Sketch for the growing bubble, thermal boundary layer, microlayer and the heater
w all ................................. ...................... ........ ......... ...... 39

3-2. Coordinate system for the background bulk liquid................ ............. .............. 43

3-3. A typical grid distribution for the bulk liquid thermal field with S,, = 0.65,
S = 0.73 and R = 10 ............ ....... .................................. ...... .. ..... 46

3-4. Comparison of the asymptotic and the numerical solutions at '=0.001, 0.01, 0.1 and
0.3 for 0 40 and 7 1 ............................................................................... 54

3-5. Effect of parameter A on the liquid temperature profile near bubble. ..................... 55

3-6. The computed isotherms near a growing bubble in saturated liquid at T=0.01,
.1, 0 .3, and 0 .9 .................... ................ ........................ .............. 57

3-7. Comparison of the equivalent bubble diameter d,, for the experimental data of
Yaddanapudi and Kim (2001) and that computed for heat transfer through the
m icrolayer (c, =3.0). ........ ...................... ................ ...................... ........ ........ .... 58









3-8. Comparison of bubble diameter, d(t), between that computed using the present
model and the measured data of Yaddanapudi and Kim (2001) ............. ............. 60

3-9. Comparison between heat transfer to the bubble through the vapor dome and that
through the m icrolayer ..................................................................................... 60

3-10. The computed isotherms in the bulk liquid corresponding to the thermal conditions
reported by Yaddanapudi and Kim (2001). ....................................................... 61

3-11. Effect of bulk liquid thermal boundary layer thickness Son bubble growth........... 62

4-1. Schematic of two-fluid model for pipe flow......... ........................................... 68

4-2. Staggered grid arrangement in two-fluid model ........................................... 74

4-3. Flow chart of pressure correction scheme for two-fluid model................................. 78

4-4. Schematic for dam-break flow model ........................... .......... ............. ................. 79

4-5. Water depth at t=50 seconds after dam break.................................. .............. 80

4-6. Water velocity at t=50 seconds after dam break .................. ............. .............. 81

4-7. Grid index number in staggered grid for von Neumann stability analysis ............ 81

4-8. Comparisons of growth rates of various numerical schemes. N = 200, a, = 0.5,
u, = Im /s, Ug = 17m /sand CFL, = 0.1 ........... ............... ...... ............ ........ 89

4-9. Growth rate of CDS scheme at different AU = Ug -u,. N = 200, a, = 0.5,
u = Im /s and C F L = 0.1 ........................... .................................. ............ ..... 90

4-10. Growth rate of FOU scheme at different AU = Ug u. N = 200, a, = 0.5,
uz = m / s and CFL1 = 0.1 ........................... .............. .................................... 90

4-11. Growth rate of CDS scheme at different u,. N = 200, AU = 16m/s, a, = 0.5, and
A t/A x = O s /m .............. ...... ............................................................... 9 2

4-12. Growth rate of FOU scheme at different u,. N = 200, AU = 16m/s, a, = 0.5 and
A t/A x = O s /m .............. ...... ............................................................... 9 2

4-13. Growth rate of CDS scheme at different At/Ax N = 200, u = Im / s,
A U = 16m /s an d a = 0 .5 .................................................................................. 93









4-14. Growth rate of FOU scheme at different At/Ax N = 200, u, = Im / s,
A U = 16m /s and a = 0.5 ............................................................................... 93

4-15. Comparison of fi growth using CDS scheme on different grids. u, = lm/s,
Ug =17.5m/s, CFL, = 0.1, and a = 0.5 ................................... .............. 95

4-16. ti using CDS scheme in the computational domain. N = 200, u, =m I/s,
AU =14m /s, CFL, = 0.05, a, = 0.5, and t =4s .............................................. 97

4-17. Amplitude of liquid velocity disturbance i, using CDS scheme. N = 200,
u = Im/s, AU = 14m/s, CFL, = 0.05, a, = 0.5, and t = 4s ............................ 97

4-18. fi using CDS scheme after 10399 steps of computation, N = 200, u = Im / s,
AU =16.5m/ s, CFL, = 0.1, a, = 0.5 and t = 5.2s. ......................................... 98

4-19. Growth history of iI solved using CDS scheme, N = 200, u, = Im/s,
AU =16.5m/ s, CFL, = 0.1, a, = 0.5 and t = 5.2s. ......................................... 98

4-20. Growth rate of FOU scheme, N = 200, u, = 0.5m/s, AU = 16m/s, CFL, = 0.02,
and a, = 0.5 ....................... .................................................. 100

4-21. ti using FOU scheme after 12000 steps of computation. N = 200,
u, = 0.5m/s,AU =16m/s, CFL, = 0.02, and a, = 0.5 ................................ 102

4-22. Growth rate of SOU scheme. N= 200, u, = Im/s, AU = 16m/s, CFL, = 0.05,
and a, = 0.5 ....................... .................................................. 103

4-23. il using SOU scheme after 3000 steps of computation. N = 200, u, =m / s,
AU = 16m / s, CFL, = 0.05, and a, = 0.5 ............ .............................. .... 103

4-24. Growth history of fi under different initial amplitude using FOU scheme........... 104

4-25. ti propagates in the pipe with FOU at well-posed condition, quasi-steady state. 107

4-26. fi propagates in the pipe with FOU scheme at ill-posed condition, quasi-steady
state ........................................................................................... 1 0 7

4-27. fi propagates in the pipe with CDS at well-posed condition, quasi-steady state. 108

4-28. ti propagates in the pipe with CDS at ill-posed condition, an instance before the
computation breaks down. ....................... ............................... 108









4-29. Comparison of growth rate between CDS and FOU schemes. N = 200, u, = Im / s,
Ug = 21m /s, CFL, = 0.05, and a = 0.5 ............................................................ 109

4-30. Schematic depiction of viscous two-fluid model........................... .... ................. 111

4-31. Comparisons of growth rate of different schemes. N = 200, u, = 0.3m I s,
Ug, = 66 m /s ,and C F L = 0. 1 ......... .................................................................... 120

4-32. Comparisons of growth rate of different schemes at low k. N = 200, u, = 0.3m / s,
U g, = 6 m /s ,and C F L = 0. 1 ......... .................................................................... 12 1

4-33. Growth rate for CDS scheme with VKH unstable. N = 200, us = 0.3m Is,
U g, = 6 m /s ,and C F L = 0. 1 ......... .................................................................... 122

4-34. Growth rate for CDS scheme with VKH instability. lwate = 10-2Pa s, N= 200,
u, = 0.3m / s, Ug = 6m /s ,and CFL, = 0.1 ....... ... ..................... ............. .. 123

4-35. Growth rates for CDS scheme with VKH instability. ,lwae = 10 -Pa s, N= 200,
u, = 0.1m/s Ug = 2m/ s,and CFL, = 0.01 ......................................... 124

4-36. Growth rates for FOU scheme with VKH instability. N = 200, u,, = 0.3m /s,
U g = 6 m /s and C F L = 0 .1 ................................................................................ 12 5

4-37. Growth rates for FOU scheme with VKH instability. ,l wae = le 1Pa s,
N =200, u, = 0.1m/s Ug =2m/s, and CFL, =0.01. ............................... 125

4-38. Growth history of u, using CDS scheme. N = 200, u, = 2m/s,
Ug = 0.998174m/s, 8 = -0.0617144, a, = 0.98, and CFL, = 0.05................... 127

4-39. Growth history of i, using FOU scheme. N = 200, u, = 2m/s,
Ug = 0.998174m/s, 8 = -0.0617144, a, = 0.98, and CFL, = 0.05 ................. 128

4-40. Disturbance of i, propagates in the pipe with FOU and CDS schemes at VKH
unstable and w ell-posed condition. ............. ................................. .............. 129

4-41. Disturbance of i, propagates in the pipe with FOU and CDS schemes at both VKH
unstable and w ell-posed condition. ............. ................................. .............. 130

5-1. Schematic of homogeneous chilldown model. ............................................... 134

5-2. Schematic for evaluating film boiling wall friction ......................... ................. 135









5-3. Distribution of vapor quality based on the homogenous flow model...................... 137

5-4. Pressure distribution based on the homogenous flow model................................. 138

5-5. Velocity distribution based on the homogenous flow model.................................. 139

5-6. Solid temperature contour based on homogenous flow model............................ 139

5-7. Schematic of cryogenic liquid flow inside a pipe............................. .............. 141

5-8. Coordinate systems: laboratory frame is denoted using z; moving frame is denoted
using Z. ............................................. 142

5-9. Schematic diagram of film boiling at stratified flow........................ ............ 145

5-10. Numerical solution of the vapor thickness and velocity influence functions........ 150

5-11. N um erical solution of G ((0 ) ................................................................. ... 151

5-12. Schematic of vacuum insulation chamber. .................................. .............. 153

5-13. Schematic of Yuan and Chung (2004)'s cryogenic two-phase flow test apparatus. 157

5-14. Experimental visual observation of Chung et al. (2004)'s cryogenic two-phase flow
experiment. ................ ............................ ............... 158

5-15. Computational grid arrangement and positions of thermocouples ...................... 159

5-16. Comparison between measured and predicted transient wall temperatures of
positions 12 and 15 at the bottom of pipe during film boiling chilldown ......... 160

5-17. Comparison between measured and predicted transient wall temperatures of
positions 12 and 15 at the bottom of pipe during convection boiling chilldown. 161

5-18. Comparison between measured and predicted transient wall temperatures of
positions 12 and 15 is at the bottom of pipe during entire chilldown. .................. 161

5-19. Comparison between measured and predicted transient wall temperatures of
positions 11 and 14, which is at the bottom of pipe during entire chilldown........ 162

5-20. Cross section wall temperature distribution at t=0, 50, 100 and 300 seconds....... 162

5-21. Computed wall temperature contour on the inner surface of inner pipe.............. 163

5-22. Comparison between measured and predicted transient wall temperatures of
positions 12 and 15 with Chen correlation (1966). .......................................... 165

5-23. Schematic of separated flow chilldown model ................................ ................. 168









5-24. Schematic of heat and mass transfer in separated flow chilldown model ........... 169

5-25. Flow chart of separated flow chilldown model................................................ 175

5-26.Geometry of the test section and locations of thermocouples .............................. 176

5-27. Comparison between measured and predicted transient wall temperatures of
positions 12 and 15. .... ........... .......................... ..................... .... .......... 178

5-28. Comparison between measured and predicted transient wall temperatures of
positions 11 and 14. ..................................................... ............. 178

5-29. Comparison between measured and predicted transient wall temperatures of
position 6 and 9 (the measured T 9 is obviously incorrect).............................. 179

5-30. Comparison between measured and predicted transient wall temperatures of
positions 5 and 8. ....................................................... .............. 179

5-31. Comparison between measured and predicted transient wall temperatures of
position 3 and the numerical result of temperature at position 4 ....................... 180

5-32. Comparison between measured and predicted transient wall temperatures of
positions 1 and 2. ....................................................... .............. 180

5-33. Liquid nitrogen depth in the pipe during the chilldown. .............. ............... 182

5-34. Vapor nitrogen velocity in the pipe during the chilldown .................................. 183

5-35. Liquid nitrogen velocity in the pipe during the chilldown ............... .............. 184

5-36. Vapor nitrogen temperature in the pipe during the chilldown............................ 185

5-37. Liquid nitrogen temperature in the pipe during the chilldown. .......................... 185


















A


Ab

Am

Bo




C,

CFL

D

D, and Dg

d

deq

E

f

flo

G

g

H, and Hg


NOMENCLATURE

dimensionless parameter for bubble growth, cross section area, surface
area

area of vapor bubble dome exposed to bulk liquid

area of wedge shaped interface

Boiling number

ratio of wedge shaped interface radius and vapor bubble radius, wave
speed

microlayer wedge angle parameter; empirically determined

Courant number

diameter of pipe

liquid layer and gas layer hydraulic diameter

bubble diameter

equivalent bubble diameter

common amplitude factor

friction factor

friction factor for liquid phase in homogeneous model

mass flux, amplification factor

gravity

liquid layer and gas layer hydraulic depth


h, and hg liquid layer and gas layer depth










h

h B

pool

h,,c and hgc


hfg


heat transfer coefficient

film boiling heat transfer coefficient

pool boiling heat transfer coefficient


forced convection heat transfer coefficient for liquid and gas


latent heat of vaporization


imaginary unit, V-

enthalpy

Jacob number

thermal conductivity, wavenumber

effective thermal conductivity


Ja

k

kif

L

Nu

1th

n

P

Po

Pc

Pr

q

qrad

q frc

q'f
qW


local microlayer thickness, characteristic length

Nusselt number

mass transfer rate between liquid and gas per unit length

normal direction

pressure

pressure in the liquid-vapor interface


Peclet number

Prandtl number

heat transfer rate per unit length

radiation heat flux

free convection heat flux


Heat flux from wall to fluid










R




R', Wand
R'

Rb

Ro

Ra

Re

r

S

S, and S,

T

Ts,

T,.


Tb


t,

to

Uand V

u and v

u


vapor bubble radius, pipe radius

bubble growth rate

spherical coordinates

dimensionless radial coordinate

radius of wedge shaped interface

initial bubble radius

Rayleigh number

Reynolds number

radial coordinate

suppression factor in flow nucleate boiling, perimeter

stretching factor in computation

temperature

saturated temperature

initial solid temperature

bulk liquid temperature


time

characteristic time

waiting period

initial time

averaged velocities

velocities

mean u velocity









Vb vapor bubble volume

x, y, and z Cartesian coordinates

z, r, and (p cylindrical coordinates

X boundary layer coordinate

Z coordinate in the direction normal to the heating surface

Greek symbols

a thermal diffusivity, volume fraction

3 volumetric thermal expansion coefficient

Xtt Martinelli number

AT solid wall superheat

6 superheated bulk liquid thermal boundary layer thickness, vapor film
thickness

6* dimensionless thickness of unsteady thermal boundary layer

c emissivity, amplitude

P velocity potential function for liquid flow, general variable

Smicrolayer wedge angle, azimuthal coordinate, phase angle

oi friction multiplier

r7 and S computational coordinates

A characteristic root of a matrix

v kinematic viscosity

0 dimensionless temperature, azimuthal coordinate, pipe incline angle

o0 initial dimensionless temperature of liquid

p density










a




'FB

superscripts

in

out






subscripts

b

FB

eva

1, g, and i

i and o

/

ml

NB

w

v

CX3


stretched time in computation, Stefan Boltzmann constant


dimensionless time, shear stress

wall shear stress in film boiling regime




inner solution

outer solution

quantity per unit length

quantity per unit area




bubble

film boiling

evaporation

liquid, gas, and interface

inner and outer pipe

liquid

microlayer

nucleate boiling

wall

vapor

far field condition


xviii















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

MODELING TWO-PHASE TRANSPORT DURING CRYOGENIC CHILLDOWN IN
A PIPELINE

By

Jun Liao

August 2005

Chair: Renwei Mei
Major Department: Mechanical and Aerospace Engineering

Cryogenic chilldown process is a complicated interaction process among liquid,

vapor and solid pipe wall. To model the chilldown process, results from recent

experimental studies on the chilldown and existing cryogenic heat transfer correlations

were reviewed together with the homogeneous flow model and the two-fluid model. A

new physical model on the bubble growth in nucleate boiling was developed to correctly

predict the early stage bubble growth in saturated heterogeneous nucleate boiling. A

pressure correction algorithm for two-fluid model was carefully implemented to solve the

two-fluid model used to model the chilldown process. The connections between the

numerical stability and ill-posedness of the two-fluid model and between the numerical

stability and viscous Kelvin-Helmholtz instability were elucidated using von Neumann

stability analysis. A new film boiling correlation and a modified nucleate boiling

correlation for chilldown inside pipes were developed to provide heat transfer correlation

for chilldown model. Three chilldown models were developed. The homogeneous









chilldown model is for simulating chilldown in a vertical pipe. A pseudo-steady

chilldown model was developed to simulate horizontal chilldown. The pseudo-steady

chilldown model can capture the essential part of chilldown process, provides a good

testing platform for validating cryogenic heat transfer correlations based on experimental

measurement of wall temperature during chilldown and gives a reasonable description of

the chilldown process in a frame moving with the liquid-vapor wave front. A more

comprehensive separated flow chilldown model was developed to predict both the flow

field and solid wall temperature field in horizontal stratified flow during chilldown. The

predicted wall temperature variation matches well with the experimental measurement. It

provides valuable insights into the two-phase flow dynamics, and heat and mass transfer

for a given spatial region in the pipe during the chilldown.














CHAPTER 1
INTRODUCTION

One of the key issues in the efficient utilization of cryogenic fluids is the transport,

handling, and storage of the cryogenic fluids. The complexity of the problems results

from, in general, the intricate interaction of the fluid dynamics and the boiling heat

transfer. Chilldown of the pipeline for transport cryogenic fluid is a typical example. It

involves unsteady two-phase fluid dynamics and highly transitory boiling heat transfer.

There is very little insight into the dynamic process of chilldown. This study will focus

on the understanding and modeling of the unsteady fluid dynamics and heat transfer of

the cryogenic fluids in a pipeline that is exposed to the atmospheric condition.

1.1 Background

Presently there exists considerable interest among U.S. Federal agencies in driving

the U.S. energy infrastructure with hydrogen as the primary energy carrier. The

motivation for doing so is that hydrogen may be produced using all other energy sources,

and thus using hydrogen as an energy carrier medium has the potential to provide a

robust and secure energy supply that is less sensitive to world fluctuations in the supply

of fossil fuels. The vision of building an energy infrastructure that uses hydrogen as an

energy carrier is generally referred to as the "hydrogen economy," and is considered the

most likely path toward widespread commercialization of hydrogen based technologies.

Hydrogen has the distinct advantage as fuel in that it has the highest energy density

of any fuel currently under consideration, 120 MJ/kg. In contrast, the energy density of

gasoline, which is considered relatively high, is approximately 44 MJ/kg. When









launching spacecraft, the energy density is a primary factor in fuel selection. When

considering liquid hydrogen to propel advanced aircraft turbo engines, it is a very

attractive option due to hydrogen's high energy density. One drawback with using liquid

hydrogen as a fuel is that it's volumetric energy capacity, 8.4 MJ/liter is about one

quarter that of gasoline, 33 MJ/liter. Therefore, liquid hydrogen requires more

volumetric storage capacity for a fixed amount of energy. Nevertheless, liquid hydrogen

is a leading contender as a fuel for both ground-based vehicles and for aircraft propulsion

in the hydrogen economy.

When any cryogenic system is initially started, (this includes turbo engines,

reciprocating engines, pumps, valves, and pipelines), it must go through a transient

chilldown period prior to 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 chilldown a cryogenic system in a safe

and efficient manner.

A perfect example of utilization and chilldown cryogenic system exists in NASA's

Kennedy Space Center (KSC). In the preparation for a space shuttle launch, liquid

hydrogen (as fuel) is filled from a storage tank to the main liquid hydrogen (LH2)

external tank (ET) through a complex pipeline system (Figure 1-1). The filling procedure

consists of 5 steps:

* Facility and orbiter chilldown.
* Fill transition and initial fill (fill ET to 2%).
* Fast fill ET (to 98%).
* Fill ET (to 100%).
* Replenish (maintain ET 100%).











- ,, ,- I ,I-F ,7r


n -I. -'









r;K;r


Calegor; III FaI Fill lo 9.8; ;


I I. .' .



4 ;7 -


F' FiL
|I,,: IL,


r ,, r

rF
F-


," r I- .F -


- .IJ


F F

, L


Figure 1-1. Schematic of filling facilities for LH2 transport system from storage tank to
space shuttle external tank.

While the engineers have a general understanding of the process in the initial fill


and rapid fill stages, there has been very little insight about the process of chilldown,


which is the first procedure to be initiated. There is not a single formula or computer


code that can be used to estimate the elapse time during the chilldown stage if certain


operating condition changes. The absence of guidelines stems from our lack of


fundamental knowledge in the area of cryogenic chilldown. Many such engineering


issues are present in the transport, handling, and storage of cryogenic liquid in industry


applications.









1.2 Literature Review

Experimental studies: Studies on cryogenic chilldown started in the 1960s with the

development of rocket launching systems. Early experimental chilldown studies started in

the 1960s by Burke et al. (1960), Graham (1961), Bronson et al. (1962), Chi and Vetere

(1963), Steward (1970) and other researches. Burke et al. (1960) and Graham (1961)

experimentally studied the cryogenic chilldown in a horizontal pipe and in a vertical pipe,

respectively. However, none of these studies provided the flow regime information in

chilldown. Bronson et al. (1962) visually studied the flow regimes in a horizontal pipe

during chilldown with liquid hydrogen as the coolant. The results revealed that the

stratified flow is prevalent during the cryogenic chilldown.

Flow regimes and heat transfer regimes in the horizontal pipe chilldown were also

studied by Chi and Vetere (1963). Information on flow regimes was deduced by studying

the fluid temperature and the volume fraction during chilldown. Several flow regimes

were identified: single-phase vapor, mist flow, slug flow, annular flow, bubbly flow, and

single-phase liquid flow. Heat transfer regimes were identified as single-phase vapor

convection, film boiling, nucleate boiling, and single-phase liquid convection.

Recently, Velat et al. (2004) systematically studied cryogenic chilldown with

nitrogen in a horizontal pipe. Their study included: a visual recording of the chilldown

process in a transparent Pyrex pipe, which is used to identify the flow regime and heat

transfer regime; collecting temperature histories at different positions of the wall in

chilldown; and recording the pressure drop along the pipe. Chung et al. (2004) conducted

a similar study with nitrogen chilldown at relatively low mass flux and provided the data

needed to assess various heat transfer coefficients in the present study.











Modeling efforts: Burke et al. (1960) developed a crude chilldown model based on

1-D heat transfer through the pipe wall and the assumption of infinite heat transfer rate from

the cryogenic fluid to the pipe wall. The effects of flow regimes on the heat transfer rate

were neglected. Graham et al. (1961) correlated the heat transfer coefficient and pressure

drop with the Martinelli number (Martinelli and Nelson, 1948) based on their

experimental data. Chi (1965) developed a one-dimensional model for energy equations

of the liquid and the wall, based on the film boiling heat transfer between the wall and the

fluid. An empirical equation for predicting the chilldown time and the temperature was

proposed.

Steward (1970) developed a homogeneous flow model for cryogenic chilldown.

The model treated the cryogenic fluid as a homogeneous mixture. The continuity,

momentum and energy equations of the mixture were solved to obtain density, pressure

and temperature of mixture. Various heat transfer regimes were considered: film boiling,

nucleate boiling, and single-phase convection heat transfer. Careful treatment of different

heat transfer regimes resulted in a significant improvement in the prediction of the

chilldown time. The homogeneous mixture model was also employed by Cross et al.

(2002) who obtained a correlation for the wall temperature during chilldown with an

oversimplified treatment of the heat transfer between the wall and the fluid.

Similar efforts have been devoted to the study of the re-wetting problem, referred to

as cooling down of a hot object. Thompson (1972) analyzed the re-wetting of a hot dry

rod. The two-dimensional temperature profile inside the solid rod was numerically

calculated. The nucleate boiling heat transfer coefficient between the solid rod and the









liquid was simplified to a power law relation and the heat transfer in the film boiling

stage is neglected. The liquid temperature and velocity outside the rod are assumed to be

constant. Sun et al. (1974), and Tien and Yao (1975) solved similar problems and

obtained an analytical solution for the re-wetting. They considered different heat transfer

coefficients for flow boiling and single-phase convection in order to obtain more accurate

results for re-wetting problems. In those works the thermal field of the liquid is neglected

and the heat transfer coefficients at the boiling and the convection heat transfer stage are

over-simplified, and the results are only valid for the vertical outer surface of a rod or a

tube.

Chilldown in stratified flow regime, which is the prevalent in the horizontal

pipeline, was first studied by Chan and Banerjee (1981 a, b, c). They developed a

comprehensive separated flow model for the cool-down in a hot horizontal pipe. Both

phases were modeled with one-dimensional mass and momentum conservation equations.

The vapor and liquid phase mass and momentum equations were reduced to two wave

equations for the liquid depth and the velocity of the liquid. The energy equation for the

liquid was used to find the liquid temperature and energy equation of vapor phase was

neglected. The wall temperature was computed using a 2-dimensional transient heat

conduction equation and heat transfer in the radial direction was neglected. They also

tried to evaluate the position of onset of re-wetting by studying the instability of film

boiling. Their prediction for the wall temperature agreed well with their experimental

results. Although significant progress was made in handling the momentum equations,

the heat transfer correlations employed were not as advanced.









Following Chan and Banerjee's (1981 a, b, c) separated flow model, Hedayatpour

et al. (1993) studied the cool-down in a vertical pipe with a modified separated flow

model. The flow regime is inverted annular film boiling flow, where the liquid core is

inside and the vapor film separates the cold liquid and the hot wall. This regime

frequently exists in cool-down in a vertical pipe. The modified separated flow model

retains the transient terms in the vapor momentum equation and the vapor phase energy

equation. The procedure is the following: first, the liquid mass conservation equation is

solved to obtain the liquid and vapor volume fractions. Then the vapor mass conservation

equation is used to solve the vapor velocity. The vapor momentum equation is

subsequently solved to obtain the vapor pressure. Finally, the liquid momentum equation

is employed to find the liquid velocity. The iteration stops when the solution is

converged. Although Chan and Banerjee (1981 a, b, c) and Hedayatpour et al. (1993)

were successful in the simulation of chilldown with the separated flow model, their

separated flow model is either incomplete or computationally inefficient.

c) Issues related to two-fluid model

The separated flow model is also called the two-fluid model, which consists of two

sets of conservation equations for the mass, momentum and energy of liquid and gas

phases. It was proposed by Wallis (1969), and further refined by Ishii (1975). Although

the two-fluid model is recognized as a useful computational model to simulate the

stratified multiphase flow in the pipeline, its application to the study of heat transfer in

two-phase flow in the pipeline is still limited.

The numerical scheme for the two-fluid model can be classified into two

categories. One is the compressible two-fluid model, which can be solved by a hyperbolic









equation solver. Examples are the commercial code OLGA (Bendikson et al., 1991),

Pipeline Analysis Code (PLAC) (Black et al., 1990) and Lyczkowski et al. (1978). The

other is the incompressible two-fluid model. Since the hyperbolic equation solver is not

applicable to incompressible two-fluid model, several approaches for incompressible

two-fluid model have emerged. One approach is to reduce the gas and liquid mass and

momentum equations to two wave equations for the liquid depth and velocity, such as in

Barnea and Taitel (1994b) and Chan and Banerjee (1991b). This treatment changed the

properties of two-fluid model. Hedayatpour et al. (1993) approach to two-fluid model is

not widely used due to lack of theoretical analysis on the convergence. Another approach

is to use the pressure correction method, which was initially introduced by Issa and

Woodburn (1998) and Issa and Kempf (2003) for the compressible two-fluid model.

Although their pressure correction scheme is powerful for simulating the multiphase flow

in the pipeline, the accuracy of the scheme is not reported. At the present, application of

pressure correction scheme on the multiphase flow with heat transfer in pipeline, such as

chilldown, does not exist.

d) Heat transfer in chilldown

A typical chilldown process involves several heat transfer regimes as shown in

Figurel-2. Near the liquid front is the film boiling regime. The knowledge of the heat

transfer in the film boiling regime is relatively limited, because i) film boiling has not

been the central interest in industrial applications; and ii) high temperature difference

causes difficulties in experimental investigations. For the film boiling on vertical

surfaces, early work was reported by Bromley (1950), Dougall and Rohsenow (1963) and

Laverty and Rohsenow (1967). Film boiling in a horizontal cylinder was first studied by









Bromley (1950); and the Bromley correlation was widely used. Breen and Westwater

(1962) modified Bromley's equation to account for very small tubes and large tubes. If

the tube is larger than the wavelength associated with Taylor instability, the heat transfer

correlation is reduced to Berenson's correlation (1961) for a horizontal surface.


Wall Liquid Front







---------------- r --- ------------ ;----r------------ -----
Cryogenic Lcuid
S/Valor




Convective Nucleation Film boiling
heat transfer boiling

Figure 1-2. The schematic of chilldown and heat transfer regime.

Empirical correlations for cryogenic film boiling were proposed by Hendrick et al.

(1961, 1966), Ellerbrock et al. (1962), von Glahn (1964), Giarratano and Smith (1965).

These correlations relate a simple or modified Nusselt number ratio to the Martinelli

parameter. Giarratano and Smith (1965) gave detailed assessment of these correlations.

All these correlations are for steady state cryogenic film boiling. Their suitability for

transient chilldown applications is questionable.

When the pipe wall chills down further, film boiling ceases and nucleate boiling

occurs. It is usually assumed that the boiling switches from film boiling to nucleate

boiling right away instead of passing through a transition boiling regime. The position of

the film boiling transitioning to the nucleate boiling is often called re-wetting front,

because from that position the cold liquid starts touching the pipe wall. Usually the

Leidenfrost temperature indicates the transition from film boiling to nucleate boiling.









However, the Leidenfrost temperature is not steady, and varies under different flow and

thermal conditions (Bell, 1967). A recent approach is to check the instability of the vapor

film beneath the liquid core using Kelvin Helmholtz instability analysis (Chan and

Banerjee, 1981c).

Studies on forced convection boiling are extensive (Giarratano and Smith, 1965;

Chen, 1966; Bennett and Chen, 1980; Stephan and Auracher, 1981; Gungor and

Winterton, 1996; Zurcher et al., 2002). A general correlation for saturated boiling was

introduced by Chen (1966). Gungor and Winterton (1996) modified Chen's correlation

and extended it to subcooled boiling. Enhancement and suppression factors for

macro-convective heat transfer were introduced. Gunger and Winterton's correlation can

fit experimental data better than the modified Chen's correlation (Bennett and Chen,

1980) and Stephan and Auracher correlation (1981). Recently, Zurcher et al. (2002)

proposed a flow pattern dependent flow boiling heat transfer correlation. This approach

improves the overall accuracy of heat transfer correlation by incorporating flow pattern.

Kutateladze (1952) and Steiner (1986) also provided correlations for cryogenic fluids in

pool boiling and forced convection boiling. Although they are not widely used, they are

expected to be more applicable for cryogenic fluids since the correlation was directly

obtained from cryogenic conditions. As the wall temperature drops further, boiling is

suppressed and the heat transfer is governed by two-phase convection; this is much easier

to deal with.

1.3 Scope

This dissertation focuses on understanding the unsteady fluid dynamics and heat

transfer of cryogenic fluids in a pipeline that is exposed to the atmospheric condition and

the corresponding solid heat transfer in the pipeline wall. Proper models for chilldown









simulation are developed to predict the flow fields, thermal fields, and residence time

during chilldown.

In Chapter 2, visualized experimental studies on heat transfer regimes and flow

regimes in cryogenic chilldown are reviewed. Based on the experimental observation,

homogeneous and separated flow models for the respectively vertical pipe and horizontal

pipe are discussed. The heat transfer models for the film boiling, flow boiling and forced

convection heat transfer in chilldown are reviewed and qualitatively assessed.

In Chapter 3, a physical model for vapor bubble growth in saturated nucleate

boiling is developed that includes both heat transfer through the liquid microlayer and

that from the bulk superheated liquid surrounding the bubble. Both asymptotic and

numerical solutions reveal the existence of a thin unsteady thermal boundary layer

adjacent to the bubble dome.

In Chapter 4, a pressure correction algorithm for two-fluid model is developed and

carefully implemented. Numerical stability of various convection schemes for both the

inviscid and viscous two-fluid model is analyzed. The connections between ill-posedness

of the two-fluid model and the numerical stability and between the viscous

Kelvin-Helmholtz instability and numerical stability are elucidated. The computational

accuracy of the numerical schemes is assessed.

In Chapter 5, a new film boiling coefficient is developed to accurately predict film

boiling heat flux for flow inside a pipe. The film boiling coefficient with the other

investigated heat transfer models are applied in building chilldown models. A

pseudo-steady chilldown model is developed to predict the chilldown time and the wall

temperature variation in a horizontal pipe in a reference frame that moves with the liquid









wave front. It is of low computational cost and allows for simple validation of the new

film boiling heat transfer correlations. A more comprehensive separated flow chilldown

model for the horizontal pipe is developed to predict the flow field of the liquid and vapor

and the temperature fields of the liquid, vapor and the solid wall in a fixed region of the

pipe flow. The unsteady development of the chilldown process for the vapor volume

fraction, velocities of the two-phases, and the temperatures of fluids and wall are

elucidated.

Chapter 6 concludes the research with a summary of the overall work and

discussion of the future works.














CHAPTER 2
TWO-PHASE FLOW MODELING AND FLOW BOILING HEAT TRANSFER OF
CRYOGENIC FLUID

Information of heat transfer regimes and flow regimes in cryogenic chilldown

obtained from the experimental study provides the foundation for modeling the heat

transfer and multiphase flow in chilldown. Based on the information of flow regimes,

corresponding flow models for simulating chilldown are discussed. For chilldown in the

vertical pipe, homogeneous flow model is preferred due to the prevalence of

homogeneous flow. For chilldown in the horizontal pipe, because the stratified flow is

prevalent, two-fluid model is adopted. The heat transfer models for film boiling, flow

boiling and forced convection heat transfer in chilldown are reviewed and qualitatively

assessed.

2.1 Flow Regime and Heat Transfer Regime

In the study of chilldown, one of the most important aspects of analysis is to

determine the type of flow regime in the given region of the pipe. The flow in cryogenic

chilldown is typically a two-phase flow, because liquid evaporates after a significant

amount of heat is transferred from the wall to the fluid during chilldown. The two-phase

flow regime is determined by many factors, such as fluid velocity, fluid density, vapor

quality, gravity, and pipe size. For horizontal flow, the flow regime is visually classified

as bubbly flow, plug flow, stratified flow, wavy flow, slug flow, and annular flow, as

shown in Figure 2-1. For vertical flow, the flow regimes include bubbly flow, slug flow,

churn flow, annular flow, as shown in Figure 2-2.























r_ -. =
..- -1
'- -



!7


Bubbly Flow



Plug Flow



Stratified Flow


WavyFlow



Slug/Internittent Flow


Annular Flow

Figure 2-1. Schematic of two-phase flow regime in horizontal pipe.


Bubble
Flow


Slug
Flow


Churn
Flow


Figure 2-2. Schematic of two-phase flow regime in vertical pipe.


Annual
Flow









The cryogenic two-phase flow is characterized by low viscosity, small density ratio

of the liquid to the vapor, low latent heat of vaporization, and large wall superheat. For

example, the liquid viscosity, density ratio latent heat of saturated liquid to vapor, and

latent heat of the saturated water at 1 atm are 2.73E-4 Pa*s, 1610, 2256.8kJ/kg,

respectively, while the corresponding data for saturated hydrogen at 1 atm arel.36E-

5Pa*s, 37.9, 444kJ/kg. Furthermore, film boiling, which is prevalent during chilldown,

causes low wall friction. These factors combined with the complex interaction between

the momentum and the thermal transportation make the two-phase flow during the

chilldown to distinguish itself from ordinary two-phase flows.

In the visualized horizontal chilldown experiment by Velat et al. (2004), as shown

in Figure 2-3, the pressure in the liquid nitrogen Dewar drives the fluid. When the liquid

nitrogen first enters the test section, a film boiling front is positioned at the inlet of test

section. This film boiling front produces a significant evaporation accompanied by a high

velocity vapor front traversing down the test section. If the mixture velocity is high

enough due to the large pressure drop between the Dewar and the outlet of the test

section, a very fine mist of liquid is entrained in the vapor flow. Immediately behind the

film boiling front is a liquid layer attached to the wall. The flow regime is either the

stratified flow or annular flow, depending on the flow speed, the pipe size, and the fluid

properties. If the mixture velocity is high, the flow likely appears as annular flow,

otherwise stratified flow or wavy flow is more common. The visual observation shows

that the liquid droplets being entrained in stratified flow and wavy flow is insignificant.

The nucleate boiling front follows the film boiling, indicating the end of film

boiling and the cryogenic liquid starts contacting the wall. The position where the liquid









starts contacting the wall is affected by the wall super heat, the liquid layer velocity and

the thickness of the liquid layer. It is a complex hydrodynamic and heat transfer

phenomenon. Usually Leidenfrost temperature indicates the transition from film boiling

to nucleate boiling. If the wall temperature is lower than the Leidenfrost temperature, the

vapor film cannot sustain the weight of liquid layer and becomes unstable. Therefore, the

liquid starts contacting the wall, and film boiling ceases.

Once the liquid contacts the wall, the nucleate boiling starts. In the nucleate boiling

regime the heat transfer from the wall to the liquid is significantly larger than that in the

film boiling regime, and the wall is chilled down much faster, are shown in Figure 2-4. If

the nucleation sites are not completely suppressed, a region of rapid nucleate boiling is

seen at the quenching front. If most of nucleate sites are suppressed by the subcooled

liquid, the flow directly transforms to the forced convection heat transfer, and nucleate

boiling stage is not visible.

After the nucleate boiling stage, the chilldown process dramatically slows down as

the convection heat transfer dominates. The wall superheat is relatively low at this stage

but the heat leaking from the test section to the environment emerges. These factors lead

to a lower chilldown rate. In the meantime, the liquid gradually builds up in the pipe due

to less vapor generation and the friction between the liquid and the wall. The increase of

the liquid layer thickness eventually leads to the transition of the flow regimes. When the

liquid layer is thick enough, the stratified flow or wavy flow becomes unstable.

Eventually slugs are formed and the flow transforms to the slug flow. In the final stage of

chilldown, the flow is almost a single-phase liquid flow, occasionally with some small







17



slugs. In this stage, the chilldown is almost completed, and the pipe wall temperature


gradually reaches the liquid saturated temperature.



Cryogenic
Liquid I Vapor Flow Increasing
Time

\Film Boiling
Front



Cryogenic Vapor Flow
Liquid

Liquid Film Flow \ Film Boiling
Region
Nucleate Boiling
Front

Cryogenic o oo o C o
Liquid Bubbly Flow Liquid Film Flow--



Figure 2-3. Schematics of observed flow structures in chilldown (Velat et al., 2004).




50


Temp 1 (T)
0 ---- Temp2(T)
-- T mp3(T)
SiTr~i. --,Eli

-50



E -100
H---



-1 50



-200
0 20 40 60 80 100 120 140 160 180 200

Time (sec)


Figure 2-4. Typical wall temperature variation during chilldown. (Velat et al., 2004)









Chilldown in a vertical pipe is practically less important than the chilldown in

horizontal pipe, due to the fact that most of cryogenic transportation pipelines are

horizontal, and only a small part is vertical. The experimental study (Hedayapour et al.

1993; Laverty and Rohsenow, 1967) reveals that the flow regime is mainly bubble flow,

or inverted annular flow if the vapor film of the film boiling is stable, and single-phase

vapor flow and single-phase liquid flow exist at the beginning and the final stage of

chilldown, respectively.

2.2 Flow Models in Cryogenic Chilldown

Based on the experimental investigation, several flow regimes exist in cryogenic

chilldown. At different flow regimes, the models for evaluating velocity and volume

fraction of fluid are different. Two types of flow models are to be discussed in this

section. First is the homogeneous flow model, which is used for modeling the chilldown

in a vertical pipe, where the homogeneous flow is prevalent. Another model is the two-

fluid model, which is mostly used in simulating the stratified flow or wavy flow for the

chilldown in a horizontal pipe.

2.2.1 Homogeneous Flow Model

In the homogeneous flow model, the unsteady mass, momentum, and energy

conservation equations for the mixture are simultaneously solved. The primary

assumptions are: (1) single-phase fluid or two-phase mixture is homogeneous, and each

phase is incompressible; (2) thermal and mechanical equilibrium exists between the

liquid and the vapor flowing together; (3) flow is quasi-one-dimensional; and (4) axial

diffusion of momentum and energy is negligible.

Thus, the continuity equation for the mixture is









S(PA) (P-A)
( + =0, (2.1)
at az

where p is the mixture density of liquid and vapor phase, i is the average fluid

velocity (by the assumption of homogeneous model, both liquid and vapor velocity are

u ), t is time z is the vertical axial coordinate, and A is the cross section area of the

pipeline.



Pipe wall


Vapor bubble

O

O 0 -- Liquid
00

00


-- Mixture front


Figure 2-5. Schematic for homogeneous flow model.

By neglecting the viscous terms, the momentum equation for the mixture becomes

S(PuA) a+ ( A) p + -p A A pgsin (2.2)
at fz dz \z )


where is pressure, is the pressure drop due to wall friction, f is the inclination
( az )f


angle of the pipe. For a vertical pipe, 3 =
2


The energy equation for the homogeneous model is









a(piA) a(puiA)
+ = q"S, (2.3)
at az

where i is the mixture enthalpy, q" is the heat flux from the wall to the fluid, and S is the

perimeter of the pipe.

If the cross section of the circular pipe is constant, the governing equations for

homogenous flow are simplified to the following equations.

a(p) a(p_)
+ = 0, (2.4)
at az

) (P ) -p ( pgsinO, (25)
at az az az

a(Ai) a(puz) 4
+ = q, (2.6)
at az IA

The pressure drop due to the wall friction is evaluated by the correlation for
The pressure dp z

the homogeneous system (Hewitt, 1982). In the correlation (Hewitt, 1982), a friction

multiplier 2o is defined as ratio of two-phase frictional pressure gradient to the
az

frictional pressure gradient for a single-phase flow at the same total mass flux and with

the physical properties of the liquid phase i.e.
w z lo

(dP
d f= 01 ,2 (2.7)
dz


where the friction multiplier 0 can be calculated by










r >-025
P0 Pg 1 -9Pg -0 25
12o 1+x l +x (2.8)
Pg t fg

where subscribes I and g represent the liquid phase and gas phase, respectively. The

single-phase pressure drop -z is evaluated using the standard equation
S)lo

=P 2 (2.9)
dz 1o Dp

where flo is the friction factor and for turbulent flow in a pipe, it is given as

025
fo = 0.079 (2.10)


in which, G is the mixture mass flux.

Compared with the experimentally measured two-phase flow pressure drop, the

homogenous model tends to underestimate the value of two-phase frictional pressure

gradient (Klausner et al., 1990). However, it provides a reasonable lower bound of the

two-phase flow pressure drop.

In the film boiling regime, a layer of vapor film separates the liquid core from the

pipe wall. This vapor film significantly reduces the wall friction, so that the two-phase

flow pressure drop due to the friction is much lower than that in the other heat transfer

regimes. To date, no correlation for the friction coefficient in the film boiling regime

exists. In available chilldown studies, the vapor film is treated as a part of the mixture and

Martinelli type of pressure drop correlation is used, or the wall friction is simply set to

zero.









2.2.2 Two-Fluid Model

In the chilldown inside the horizontal pipe, it is assumed that flow is stratified and

the liquid and the vapor flow at different velocity (Figure 2-6). Two-fluid model (Willis,

1969; Ishii, 1975) is widely used to qualitatively investigate the stratified flow inside

horizontal pipeline with a relatively low computational cost compared with

2-dimensional or 3-dimensional fluid flow models. In the study of the horizontal pipe

chilldown, the fluid volume fractions, velocities, enthalpies are solved with the two-fluid

model.


Pipe wall


S Vapor layer --
1D x
Liquid layer -


^r
Wall heat flux


Figure 2-6. Schematic of the two-fluid model.

The basis of the two-fluid model is a set of one-dimensional conservation equations

for the balance of mass, momentum and energy for each phase. The one-dimensional

conservation equations are obtained by integrating the flow properties over the

cross-sectional area of the flow.

In this study, it is assumed that flow is incompressible as the Mach number of the

gas phase is usually very low for the stratified flow. Hence, continuity equation for the

liquid phase (Chan and Banarjee, 1981c) is


a-(a, )+ (uza)=- (2.11)
at ax APz









where a is volume fraction, p is density, u is the velocity, t is the time, x is the axial

coordinate, and in' is the mass transfer rate between the liquid phase and the gas phase

per unit length; the subscript I denotes liquid.

Similarly, continuity equation for the gas phase is


a(g)+ (ugg= (2.12)
at ax Ap,

where the subscript g denotes gas. It is noted that

a, + a, = 1. (2.13)

The momentum equation for the liquid phase is

(ua,)+ a (u a, a, a=
at ax P, ax (2.14)
Ba, TS, TS mu
gcos OH, a -ag sinO- +
ax AP, AP, Ap,

where p, is the pressure at the liquid-gas interface, g is acceleration of gravity, f is the

angle of inclination of the pipe axis from the horizontal lane, 'is the shear stress, S is the

perimeter over which Tacts, A is the pipe cross section area, H, is the liquid phase

hydraulic depth; the subscript i denotes liquid-gas interface. The second term on the right

hand side of Equation (2.14) represents the effect of gravity on the wavy surface of liquid

layer. The liquid phase hydraulic depth H, is defined as


H1 =- _a (2.15)
aa, lh a'

where h, is the liquid layer depth.

Similarly, the momentum equation for the gas phase is









a (u' a ()+ I (U2 a. ag ap,
a ax P ax (2.16)
SBaa TS TS mi'U
g cos OHg 'a g sin- +-
ax Ap Ap, Ap,

whereHg is the gas phase hydraulic depth. It is defined as

ag ag
H, a (2.17)
aa lahg a,

where hg is the gas layer thickness.

To study heat transfer, appropriate energy equations for both phases are required in

the two-fluid model. Similar to the assumptions made in the homogeneous flow model,

the heat conduction inside the fluid is neglected. Thus the one-dimensional energy

equations for the liquid phase and the gas phase are

3 3 mi) 'i q,
(a + (a, + (2.18)
at ax Ap, Ap,

and

a 1fm qg
(jgag)+-(agUi)= + (2.19)
at ax Apg Apg

where i is enthalpy, and q' is the heat transfer rate to the fluid per unit length.

In the two-fluid model, shear stresses r,, rg and r, must be specified to close the

two fluid model. There are many correlations for shear stresses for separated flow model,

such as those developed by Wallis (1946), Barnea and Taitel (1976), and Andritsos and

Hanratty (1987). No significant difference exists among these models except at the flow

regime transition and at the high-speed flow, which will not be addressed in this study.









Thus, widely accepted shear stress correlations by Barnea and Taitel (1994) are

employed:


2
z, = f, (2.20)
2


Zg = fg Pg2 (2.21)


(U -U, IU g U,
,, = (2.22)
2

where ris shear stress, subscripts 1, g, and i represent interface between the liquid and the

wall, interface between the gas and the wall, interface between the liquid and gas,

respectively. Friction factors fare given by

f, = C1 Re -, and fg = C, Re,", (2.23)

where Re, is defined as


Re UID, (2.24)
fli

where D1 is the liquid hydraulic diameter

4A1
D, = (2.25)
SI

in which A, is liquid phase cross section area, S, is the liquid phase perimeter. In

Equation (2.23) Reg is defined as


PgUDg
Re = ggg (2.26)
flg


where Dg is vapor phase hydraulic diameter









4A
D = g (2.27)
S +S,

in which Ag is vapor phase cross section area, S, is the vapor phase perimeter, and S, is

the liquid-gas interface perimeter.

The coefficients C, and C, are equal to 0.046 for turbulent flow and 16 for

laminar flow, while n and m take the values of 0.2 for turbulent flow and 1.0 for laminar

flow. The interfacial friction factor is assumed to be f = fg or f = 0.014, if

fg < 0.014.

It is supposed that this model works in the flow boiling regime and in the forced

convection heat transfer regime. However, in the film boiling stage, presence of vapor

film dramatically reduces the shear stress between the liquid and the wall. In such a

situation, rT should be evaluated to include the effect of vapor film layer.

2.3 Heat Transfer between Cryogenic Fluid and Solid Pipe Wall

During cryogenic chilldown, the fluid in contact with the pipe wall is either the

liquid or the vapor. The mechanisms of heat transfers between the liquid and the wall and

between the vapor and the wall are different, as shown in Figure 2-7. Based on

experimental measurements and theoretical analysis, liquid-solid heat transfer accounts

for a majority of the total heat transfer. However, the liquid-solid heat transfer is much

more complicated than the heat transfer between the vapor and the wall due to occurrence

of film boiling and nucleate boiling. Thus, the heat transfer between the liquid and the

wall is discussed first.









2.3.1 Heat Transfer between Liquid and Solid wall

The heat transfer mechanism between the liquid and the solid wall includes film

boiling, nucleate boiling, and two-phase convection heat transfer. The transition from one

type of heat transfer to another depends on many parameters, such as the wall

temperature, the wall heat flux, and properties of the fluid. For simplicity, a fixed

temperature approach is adopted to determine the transition point. That is, if the wall

temperature is higher than the Leidenfrost temperature, film boiling is assumed. If the

wall temperature is between the Leidenfrost temperature and a transition temperature, T2,

nucleate boiling is assumed. If the wall temperature is below the transition temperature

T2, two-phase convection heat transfer is assumed. The values of the Leidenfrost

temperature and the transition temperature are determined by matching the model

prediction with the experimental results.


Pipe wall


Vapor layer -Convective heat Vapor
D transfer (vapor)
Liquid layer Liquid



Wall heat flux Thin vapor film

Convective heat Flow boiling Film boiling
transfer (liquid)


Figure 2-7. Schematic of heat transfer in chilldown.

2.3.1.1 Film boiling

Due to the high wall superheat encountered in the cryogenic chilldown, film boiling

plays a major role in the heat transfer process in terms of the time span and in terms of









the total amount of heat removed from the wall, as shown in Figure 2-4. Currently there

exists no specific film boiling correlation for chilldown applications with such high wall

superheat. The research starts from the conventional film boiling correlations.

A cryogenic film boiling heat transfer correlations was provided by Giarratano and

Smith (1965),


Nu 1Bo-04 f(tt), (2.28)


where Nu is Nusselt number

h D
Nu = (2.29)
ki

where hF is the film boiling heat transfer coefficient and k, is the thermal conductivity

of the liquid, Bo is the boiling number

Bo = q (2.30)
hfg *G

where hfg is the evaporative latent heat of the fluid. In Equation (2.28), Nucaic is the

Nusselt number for the two-phase convection heat transfer, which can be obtained using

Nucac = 0.023 ReO8*Pr 4, (2.31)

where Re is Reynolds number of mixture and Pr is Prandtl number of vapor, X, is

Martinelli number


X,,=1 x v I l (2.32)
x P,) ltJv

In Giarratano and Smith (1965) correlation, the heat transfer coefficient is the

averaged value for the whole cross section. Similar correlations for cryogenic film









boiling also exist in the literature. The correlations were obtained from measurements

conducted under steady state. The problem with the use of these steady state film boiling

correlations is that they do not account for information of flow regimes. For example, for

the same quality, the heat transfer rate for annular flow is much different from that for

stratified flow. Available empirical correlations do not make such difference.

Furthermore, in this study, local heat transfer coefficient is needed in order to

incorporate the thermal interaction with the pipe wall. Since the two-phase flow regime

information is available in the present study through the visualized experiment, it is

expected that the modeling effort should take into account the knowledge of the flow

regime. Suppose a liquid-gas stratified flow exists inside a horizontal pipe. Due to

gravity, the upper part of pipe wall is in contact with the gas, and lower part of pipe wall

is in contact with the flowing liquid. Thus, the heat transfer coefficient on upper wall is

significantly different from that on the lower wall. Apparently, the local heat transfer

coefficient strongly depends on the local flow condition instead an overall parameter such

as the flow quality at the given location.

There are several correlations for the film boiling based on the analysis of the vapor

film boundary layer, such as Bromley correlation (1950) and Breen and Westerwater

correlation (1962) for film boiling on the outer surface of a hot tube. Frederking and

Clark (1965) and Carey (1992) correlations, for the film boiling on the surface of a

sphere, are included as well. However, none of these was obtained for cryogenic fluids or

for the film boiling on the inner surface of a pipe or tube.









2.3.1.2 Forced convection boiling and two-phase convective heat transfer

A pool boiling correlation for cryogens was proposed by Kutateladze (1952). The

pool nucleated boiling heat transfer coefficient hpoo0 is

Fk 11 282 P1 750 (c 1 5 1
hP01= 0.487*10-10* (L kp) 0906 6 AT15, (2.33)
pool [hi p, 15 0906 10626


where o7is liquid surface tension, u is viscosity, and ATis wall superheat. Based on this

pool boiling correlation, a convection boiling correlation was proposed (Giarratano and

Smith, 1965). The heat transfer coefficient is contributed by both convection heat transfer

and ebullition:

h= hh,c + hpoo, (2.34)

where h,,c is given by Dittus-Boelter equation which is used in fully developed pipe

flow:

hI,~ = 0.023 Re,8 Pr,04 k,/D, (2.35)

where Re, is defined as

Re, = DG (2.36)

Chen (1966) introduced enhancement factor E and suppression factor S into the

flow boiling correlation. The heat transfer coefficient is given

h = EhlC + Shool (2.37)

Enhancement factor E reflects the much higher velocities and hence forced convection

heat transfer in the two-phase flow compared to the single-phase, liquid only flow. The

suppression factor S reflects the lower effective superheat in the forced convection as

opposed to pool boiling, due to the thinner boundary condition. The value of E and S are









presented as graphs in Chen (1966). The pool boiling heat transfer coefficient in Chen

correlation is

k079 045 049 025
hp 0.100122 : cO, P1 g AT 024AP075 (2.38)
pool 05 P 029hj024 024


Chen correlation (1966) fits best for annular flow since it was developed for vertical

flows. For the stratified flow regime, Chen's correlation may not be applicable.

At the flow boiling heat transfer, Gungor and Winterton correlation (1996) is

widely used due to that it fits much more experimental data. The basic form of Gungor

and Winterton correlation is similar to Chen correlation (1966), Equation (2.37).

However, evaluation of E and S in Gungor and Winterton's correlation takes account for

the influence of heat transfer rate by adding boiling number Bo. Thus, E and S are

presented as

E = 1+ 24000Bo 16 +1.37(1/,tX)86, (2.39)

and

1
S = 1 (2.40)
1+1.15x10-6E2 Re)17

The pool boiling correlation implemented is proposed by Cooper (1984)

hpoo1 = 55pM 12(-log p)-055M-605 q7 (2.41)

The solution of heat transfer correlation in Gungor and Winterton's correlation is

implicitly obtained by iteration.

Although Gungor and Winterton correlation (1996) is widely used due to its good

agreement with a large data set, a closer examination on this correlation shows that it is

based mainly on the following parameters: Pr, Re, and quality x. Similar to the









development of conventional film boiling correlations, these parameters all reflect overall

properties of the flow in the pipe and are not directly related to flow regimes. Thus, it

cannot be used to predict the local heat transfer coefficients required in chilldown

simulation.

Most of existing force convection boiling heat transfer correlations do not

effectively take account the influence of flow regimes and flow patterns. Recently,

Zurcher et al. (2002) proposed a flow pattern dependent heat transfer correlation for the

horizontal pipe. The strategy employed in Zurcher et al. (2002) is that the flow pattern is

obtained using the flow pattern map at the first step. The information of flow pattern

determines the part of wall contacting with the liquid or the vapor, then corresponding

conventional heat transfer correlations is employed to determine the local heat transfer

coefficient. The heat transfer coefficient for the whole pipe is obtained by averaging the

local heat transfer coefficient along the perimeter of the pipe. Although details of the

approach like flow pattern map, and correlations employed are not perfect in study of

Zurcher et al. (2002), their approach to the flow boiling heat transfer is intelligible and

provides insight for studying chilldown.

When wall superheat drops to a certain range all the nucleation sites are suppressed.

The heat transfer is dominated by two-phase forced convection. The heat transfer

coefficient can then be predicated using Equation (2.35), when the flow is turbulent, or

Equation (2.42), when the flow is laminar.

h,xc = 4.36 k /D,. (2.42)









2.3.2 Heat Transfer between Vapor and Solid Wall

The heat transfer between the vapor and wall can be estimated by treating the flow

as a fully developed forced convection flow, neglecting the liquid droplets that are

entrapped in the vapor. The heat transfer coefficient of vapor forced convective flow is

h = 0.023 *Reo8 Pr 04 k /Dg, (turbulent flow) (2.43)

hg, = 4.36 kg /Dg, laminarr flow) (2.44)














CHAPTER 3
VAPOR BUBBLE GROWTH IN SATURATED BOILING

Accurate evaluation of the nucleate boiling coefficient is a critical part of the study

on the chilldown process because it provides the heat transfer rate from the wall to the

cryogenic fluid. During the nucleate boiling the vapor bubble growth rate has a directly

influence on the heat transfer rate. The higher the bubble growth rate, the higher the heat

transfer rate. A physical model for vapor bubble growth in saturated nucleate boiling has

been developed that includes both heat transfer through the liquid microlayer and that

from the bulk superheated liquid surrounding the bubble. Both asymptotic and numerical

solutions for the liquid temperature field surrounding a hemispherical bubble reveal the

existence of a thin unsteady thermal boundary layer adjacent to the bubble dome. During

the early stages of bubble growth, heat transfer to the bubble dome through the unsteady

thermal boundary layer constitutes a substantial contribution to vapor bubble growth. The

model is used to elucidate recent experimental observations of bubble growth and heat

transfer on constant temperature microheaters reported by Yaddanapudi and Kim (2001)

and confirms that the heat transfer through the bubble dome can be a significant portion

of the overall energy supply for the bubble growth.

3.1 Introduction

During the past forty years, the microlayer model has been widely accepted and

used to explain bubble growth and the associated heat transfer in heterogeneous nucleate

boiling. The microlayer concept was introduced by Moore and Mesler (1961), Labunstov

(1963) and Cooper (1969). The microlayer is a thin liquid layer that resides beneath a









growing vapor bubble. Because the layer is quite thin, the temperature gradient and the

corresponding heat flux across the microlayer are high. The vapor generated by strong

evaporation through the liquid microlayer substantially supports the bubble growth.

Popular opinion concerning the microlayer model is that the majority of

evaporation takes place at the microlayer. A number of bubble growth models using

microlayer theory have been proposed based on this assumption such as van Stralen et al.

(1975), Cooper (1970), and Fyodrov and Klimenko (1989). These models were partially

successful in predicting the bubble growth under limited conditions but are not applicable

to a wide range of conditions. Lee and Nydahl (1989) used a finite difference method to

study bubble growth and heat transfer in the microlayer. However their model assumes a

constant wall temperature, which is not valid for heat flux controlled boiling since the

rapidly growing bubble draws a substantial amount of heat from the wall through the

microlayer, which reduces the local wall temperature. Mei et al. (1995a, 1995b)

considered the simultaneous energy transfer among the vapor bubble, liquid microlayer,

and solid heater in modeling bubble growth. For simplicity, the bulk liquid outside the

microlayer was assumed to be at the saturation temperature so that the vapor dome is at

thermal equilibrium with the surrounding bulk liquid. The temperature in the heater was

determined by solving the unsteady heat conduction equation. The predicted bubble

growth rates agreed very well with those measured over a wide range of experimental

conditions that were reported by numerous investigators. Empirical constants to account

for the bubble shape and microlayer angle were introduced.

Recently, Yaddanapudi and Kim (2001) experimentally studied single bubbles

growing on a constant temperature heater. The heater temperature was kept constant by









using electronic feed back loops, and the power required to maintain the temperature was

measured throughout the bubble growth period. Their results show that during the bubble

growth period, the heat flux from the wall through the microlayer is only about 54% of

the total heat required to sustain the measured growth rate. It poses a new challenge to

the microlayer theory since a substantial portion of the energy transferred to the bubble

cannot be accounted for.

Since a growing vapor bubble consists of a thin liquid microlayer, which is in

contact with the solid heater, and a vapor dome, which is in contact with the bulk liquid,

the experimental observations of Yaddanapudi and Kim (2001) leads us to postulate that

the heat transfer through the bubble dome may play an important role in the bubble

growth process, even for saturated boiling. Because the wall is superheated, a thermal

boundary layer exists between the background saturated bulk liquid and the wall; within

this thermal boundary layer the liquid temperature is superheated. During the initial stage

of the bubble growth, because the bubble is very small in size, it is completely immersed

within this superheated bulk liquid thermal boundary layer. As the vapor bubble grows

rapidly, a new unsteady thermal boundary layer develops between the saturated vapor

dome and the surrounding superheated liquid. The thickness of the new unsteady thermal

boundary layer should be inversely related to the bubble growth rate; see the asymptotic

analysis that follows. Hence the initial rapid growth of the bubble, which results in a thin

unsteady thermal boundary layer, is accompanied by a substantial amount of heat transfer

from the surrounding superheated liquid to the bubble through the vapor bubble dome.

This is an entirely different heat transfer mechanism than that associated with

conventional microlayer theory.









In fact, many previous bubble growth models have attempted to include the

evaporation through the bubble dome, such as Han and Griffith (1965) and van Stralen

(1967). However their analyses neglected the convection term in the bulk liquid due to

the bubble expansion, so the unsteady thermal boundary layer was not revealed. This

leads to a much lower heat flux through the bubble dome.

The existence and the analysis on the unsteady thermal boundary layer near the

vapor dome were first discussed in Chen (1995) and Chen et al. (1996), when they

studied the growth and collapse of vapor bubbles in subcooled boiling. For subcooled

boiling, the effect of heat transfer through the dome is much more pronounced due to the

larger temperature difference between the vapor and the bulk liquid. With the presence

of a superheated wall, a subcooled bulk liquid, and a thin unsteady thermal boundary

layer at the bubble dome, the folding of the liquid temperature contour near the bubble

surface was observed in their numerical solutions. The folding phenomenon was

experimentally confirmed by Mayinger (1996) using an interferometric method to

measure the liquid temperature.

Despite those findings, the existence of the thin unsteady thermal boundary layer

near the bubble surface has not received sufficient attention. In the recent computational

studies of bubble growth by Son et al. (1999) and Bai and Fujita (2000), the conservation

equations of mass, momentum, and energy were solved in the Eulerian or

Lagrange-Eulerian mixed grid system for the vapor-liquid two-phase flow. In their direct

numerical simulations of the bubble growth process, the heat transfer from the

surrounding liquid to the vapor dome is automatically included since the integration is

over the entire bubble surface. They observed that there could be a substantial amount of









heat transfer though bubble dome in comparison with that from the microlayer.

However, it is not clear that if these direct numerical simulations have sufficiently

resolved the thin unsteady thermal boundary layer that is attached to the rapidly growing

bubble.

In this study, asymptotic and numerical solutions to the unsteady thermal fields

around the vapor bubble are presented. The structure of the thin, unsteady thermal

boundary layer around the vapor bubble is elucidated using the asymptotic solution for a

rapidly growing bubble. A new computational model for predicting heterogeneous

bubble growth in saturated nucleate boiling is presented. The model accounts for energy

transfer from the solid heater through the liquid microlayer and from the bulk liquid

through the thin unsteady thermal boundary layer on the bubble dome. It is equally valid

for subcooled boiling, although the framework for this case has already been presented by

Chen (1995) and Chen et al. (1996). The temperature field in the heater is simultaneously

solved with the temperature in the bulk liquid. For the microlayer, an instantaneous linear

temperature profile is assumed between the vapor saturation temperature and the heater

surface temperature due to negligible heat capacity in the microlayer. For the bulk liquid,

the energy equation is solved in a body-fitted coordinate system that is attached to the

rapidly growing bubble with pertinent grid stretching near the bubble surface to provide

sufficient numerical resolution for the new unsteady thermal boundary layer. Section 3.2

presents a detailed formulation of the present model and an asymptotic analysis for the

unsteady thermal boundary layer. In Section 3.3, the experimental results of

Yaddanapudi and Kim (2001) are examined using the computational results based on the










present model. A parametric investigation considering the effect of the superheated bulk

liquid thermal boundary layer thickness on bubble growth is also presented.

3.2 Formulation

3.2.1 On the Vapor Bubble

Consideration is given to an isolated vapor bubble growing from a solid heating

surface into a large saturated liquid pool, as shown in Figure 3-1. A rigorous description

of the vapor bubble growth and the heat transfer processes among three phases requires a

complete account for the hydrodynamics around the rapidly growing bubble in addition

to the complex thermal energy transfer. The numerical analysis by Lee and Nydahl

(1989) relied on an assumed shape for the bubble, although the hydrodynamics based on

the assumed bubble shape is properly accounted for. Son et al. (1999) and Bai and Fujita

(2000) employed the Navier-Stokes equations and the interface capture or trace methods

to determine the bubble shape. Nevertheless, the microlayer structure was still assumed

based on existing models.


A\ Z





Bulk liquid
thermal boundary

SR(t) Background
bulk liquid




Liquid
microlayer Solid wall,heat is /
/ supplied from
within or below \



Figure 3-1. Sketch for the growing bubble, thermal boundary layer, microlayer and the
heater wall.









In this study, the liquid microlayer between the vapor bubble and the solid heating

surface is assumed to have a simple wedge shape with an angle 0<<1. The interferometry

measurements of Koffman and Plesset (1983) demonstrate that a wedge shape microlayer

is a good assumption. There exists ample experimental evidence by van Stralen (1975)

and Akiyama (1969) that as a bubble grows, the dome shape may be approximated as a

truncated sphere with radius R(t), as shown in Figure 3-1. Using cylindrical coordinates,

the local microlayer thickness is denoted by L(r). The radius of the wedge-shaped

interface is denoted by Rb (t), which is typically not equal to R(t). Let

c=Rb(t)/R(t), (3.1)

and the vapor bubble volume Vb (t) can be expressed as

4Ar
Vb(t)-= R3 (t)f(c), (3.2)
3
where f(c) depends on the geometry of the truncated sphere. In the limit c -> 1, the

bubble is a hemisphere and Vb (t) -> (2 r/3)R3 (t). In the limit c -> 0, the bubble

approaches a sphere and Vb (t) (4r / 3)R3(t).

To better focus the effort of the present study on understanding the complex

interaction of the thermal field around the vapor dome, additional simplification is

introduced. The bubble shape is assumed to be hemispherical (c=l) during the growth.

Comparing with the direct numerical simulation technique which solves bubble shape

and fluid velocity field using Navier-Stokes equation, this simplification introduces some

error in the bubble shape and fluid velocity and temperature fields in this study. However,

the hemispherical bubble assumption is generally valid at high Jacob number nucleate

boiling (Mei, et al. 1995a) and at the early stage of low Jacob number bubble growth









(Yaddanapudi and Kim, 2001). A more complete model that incorporates the bubble

shape variation could have been used, as in Mei et al. (1995a); however, the present

model allows for a great simplification in revealing and presenting the existence and the

effects of a thin unsteady liquid thermal boundary layer adjacent to the bubble dome and

the influence of bulk liquid thermal boundary layer on saturated nucleate boiling. The

present simplified model is not quantitatively valid when the shape of the vapor bubble

deviates significantly from a hemisphere.

The energy balance at the liquid-vapor interface for the growing bubble depicted in

Figure 3-1 is described as

dV aT, T,
pVhf d=- -kl Tmi dA+ f k- dAb (3.3)
Sa z=L(r) R'=R (t)

where p, is the vapor density, hfg is the latent heat, k, is the liquid thermal conductivity,

Ti is the temperature of the bulk liquid, Tm, is the temperature of the microlayer liquid, Am


is the area of wedge, Ab is the area of the vapor bubble dome exposed to bulk liquid,
an

is the differentiation along the outward normal at the interface, and R' is the spherical

coordinate in the radial direction attached to the moving bubble. Equation (3.3) simply

states that the energy conducted from the liquid to the bubble is used to vaporize the

surrounding liquid and thus expand the bubble.

3.2.2 Microlayer

The microlayer is assumed to be a wedge centered at r = 0 with local thickness

L(r). Because the hydrodynamics inside the microlayer are not considered, the

microlayer wedge angle 0 cannot be determined as part of the solution. In Cooper and

Lloyd (1969), the angle 0 was related to the viscous diffusion length of the liquid as









Rb (t)tan0 = c, vt in which v, is the kinematic viscosity of the liquid. A small 0

results in


= (3.4)
Rb (t)

Cooper and Lloyd (1969) estimated c, to be within 0.3-1.0 for their experimental

conditions.

A systematic investigation for saturated boiling by Mei et al. (1995b) established

that the temperature profile in the liquid microlayer can be taken as linear for practical

purposes. The following linear liquid temperature profile in the microlayer is thus

adopted in this study


Trrzt)=T+AT{rt -- ,l (3.5)

where ATat (r, t) = T1 (r, z = 0, t) Ts, and T, is the temperature of the solid heater.

3.2.3 Solid Heater

The temperature of the solid heater is governed by the energy equation, which is

coupled with the microlayer and bulk liquid energy equations. Solid heater temperature

variation significantly influences the heat flux into the rapidly growing bubble (Mei et al.

1995a, 1995b). However, in this study, constant wall temperature is assumed so that the

case of Yaddanapudi and Kim (2001) can be directly simulated. Thus,

AT,,t (r,t)= AT=,t = T Ts (3.6)

which can be directly used in Equation (3.5) to determine the microlayer temperature

profile.









3.2.4 On the Bulk Liquid

It was assumed that the vapor bubble is hemispherical in section 2.1. Furthermore,

the velocity and temperature fields are assumed axisymmetric. Unless otherwise

mentioned, spherical coordinates (R', y, (p), as shown in Figure 3-2, are employed for the

bulk liquid.













7/=1












Figure 3-2. Coordinate system for the background bulk liquid.

3.2.4.1 Velocity field

Since there is no strong mean flow over the bubble, the bulk liquid flow induced by

the growth of the bubble is mainly of inviscid nature. Thus the liquid velocity field may

be determined by solving the Laplace equation V2c = 0 for the velocity potential d. In

spherical coordinates, the velocity components are simply given by the expansion of the

hemispherical bubble as
=o yV











R l=1




Figure 3-2. Coordinate system for the background bulk liquid.

3.2.4.1 Velocity field

Since there is no strong mean flow over the bubble, the bulk liquid flow induced by

the growth of the bubble is mainly ofinviscid nature. Thus the liquid velocity field may

be determined by solving the Laplace equation V72 = 0 for the velocity potential 0. In

spherical coordinates, the velocity components are simply given by the expansion of the

hemispherical bubble as










uR, dR R(t) R = 0,u, u=O, (3.7)
dt R' R'


where R = dR(t)
dt

3.2.4.2 Temperature field

By assuming axisymmetry for the temperature fields and using the liquid velocity

from Equation (3.7), the unsteady energy equation for the bulk liquid in spherical

coordinates is

a T, a T (, 12 3 1 a TI
UR I a R s sin (3.8)


The boundary conditions are

a T,
T =0 at = 0, (3.9)


T, = T, at (3.10)
2

T, = Tso, at R' = R(t), (3.11)

T, = T7 (V, t) at R' --- (3.12)

where T is the far field temperature distribution.

To facilitate an accurate computation and obtain a better understanding on the

physics of the problem, the following dimensionless variables are introduced,

t- R' TI Tb
t= ,,R' ,z =T b (3.13)
t' R T -Tb









where t, is a characteristic time chosen to be the bubble departure time, T7 is the initial

solid temperature at the solid-liquid interface, and T, is the bulk liquid temperature far

away from the wall, which equals T,, for saturated boiling.

Using Equation (3.7) and Equation (3.13), Equation (3.8) can be written as

R J 1 R + a 1 k 1o
tR Jr kR2 )R' RR 2 R (3.14)R
(3.14)
a, 1 1 (sin ,
+ --= smw-
RR sinV R'2 (si

In this equation, the first term on the left-hand-side (LHS) is the unsteady term, and the

second term is due to convection in a coordinate system that is attached to the expanding

bubble. The right-hand-side (RHS) terms are due to thermal diffusion.

As shown in Chen et al. (1996) and below, the solution for 0O near R = 1 possesses

Ri
a thin boundary layer when >> 1. Therefore, to obtain the accurate heat transfer
a,

between the bubble and the bulk liquid, high resolution in the thin boundary layer is

essential. Hence, the following grid stretching in the bulk liquid region is applied,

R'=1+(R- -l){1-S, tan 1[(1- 7)tan(l/SR,)]} for 0 77 1
Sr(3.15)
V= S, tan-' tan(l/S)] for 0

where SR, and S, are parameters that determine the grid density distribution in the

R'
physical domain and R= is the far field end of the computational domain along the
R


radial direction.









Typically SR,-0.65 and S,-0.73, and R' ranges from 5 to 25. Figure 3-3 shows a

typical grid distribution used in this study.


Figure 3-3. A typical grid distribution for the bulk liquid thermal field with SR = 0.65,
S = 0.73, andR' =10.

3.2.4.3 Asymptotic analysis of the bulk liquid temperature field during early stages
of growth

To gain a clear understanding on the interaction of the growing bubble with the

background superheated bulk liquid thermal boundary layer, an asymptotic analysis for

non-dimensional temperature 0, is presented, following the work of Chen et al. (1996).

During the early stages, the bubble growth rate is high and expands rapidly so that

RR
A = -- >> 1. (3.16)
a,








Thus, the solution to Equation (3.14) includes an outer approximation in which the

thermal diffusion term on the RHS of Equation (3.14) is negligible and an inner

approximation (boundary layer solution) in which the thermal diffusion balances the

convection. Away from the bubble, the outer solution is governed by

R +oOUt = u 0, (3.17)
R at (R' ) R'

where Oo,"uis the outer solution for 0, in the bulk liquid. The general solution for

Equation (3.17) is

0"'t = F(R(t)(R'3 -1)3), (3.18)

as given in Chen et al. (1996). In the above F is an arbitrary function and it is determined

from the initial condition of 0, or the temperature profile in the background bulk liquid

thermal boundary layer. It is noted that the solution for0,outis described by

R(t)(R~3 -1)13 = const along the characteristic curve.

The initial temperature profile is often written as 00 = f : The solution of


Equation (3.17) is thus expressed as


OUt f Ro co 1+ -R' 1 1) (3.19)


where R0 is the initial bubble radius at t = to << tc. Provided the bubble growth rate is

high, i.e. A >> 1, Equation (3.19) is not only an accurate outer solution for the

temperature field outside a rapidly expanding bubble, but it is also a good approximation

for the far field boundary condition for Equation (3.12).








Near the bubble surface, there exists a large temperature gradient between the

saturation temperature on the bubble surface and the temperature of the surrounding

superheated liquid over a thin region. Therefore, the effect of heat conduction is no

longer negligible in this thin region and must be properly accounted for. For a large value

of A, a boundary layer coordinate Xis introduced,

R'-1
X (3.20)
8 (A)

where 8* (A) << 1 is the dimensionless length scale of the unsteady thermal boundary

layer. Substituting Equation (3.20) into Equation (3.14) results in

R 0a 0 R aX aOO ( + 81 -1 a ol^"-
tR tR t 3X ( (+8 x)2 8* ax
(3.21)
1 (20'n" 28' 0a" 1 1 1 ( 0a ^
+ + --- sin V/
AS*2 KX2 +8*X X A (l+8*X)2 sin/ fa/f K j

Neglecting higher order terms, Equation (3.21) becomes

R 0 _ei 1 20n R X
-R a 1 a2- I+3X 1- R aX a (3.22)
tR ar AS 2 X 2, 3tRX arT 3X

The balance between the convection term and the diffusion term on the RHS of Equation

(3.22) requires


5 = A- = (3.23)


Hence Equation (3.22) becomes

R a I a2oin+ R A lx
t = i+ 3 2 AX (3.24)
Te R cti f th i ) t X

The boundary conditions for the inner (boundary layer) solution are









Tsa-T
0"n = Osat b at X = 0, (3.25)
Tw Tb

0," = 1 at X -- (3.26)

1 R
For << 1, R(t) c t2 and -= Thus, the LHS of Equation (3.24) is small and can
tcR

also be neglected. Equation (3.24) then reduces to

S2 orn anor
/ +3X =0 for r <<1. (3.27)
aX2 IX

The solution for Equation (3.27) is

;" = (1 O,)erf(iX)+Osa,. (3.28)

For << 1, by matching the outer and inner solutions given by Equation (3.19) and

Equation (3.28), the uniformly valid asymptotic solution of the bulk liquid temperature

for the saturated boiling problem considered here is obtained,


S Rocos R 1 ( 3 -1 i -), (3.29)



T -T
where 0s,, = sa b and erfc is the complimentary error function. Equation (3.29) is an
T-T
T. Tb

asymptotic solution for 0, valid for << 1.

The asymptotic solution given by Equation (3.29) for the liquid thermal field

provides an analytical framework to understand: 1) how the temperature field of

background superheated bulk liquid boundary layer influences the temperature 0, near

the vapor bubble through the function f 2) how the bubble growth R(t) and liquid thermal

diffusivity affect the liquid thermal field 0, through the rescaled inner variable X as









defined in Equation (3.20) and Equation (3.23); and 3) how the folding of the temperature

contours near the bubble occurs through the dependence of cos y term in Equation

(3.29). More importantly, from a computational standpoint, it provides: 1) an accurate

measure on the thickness of the rapidly moving thermal boundary layer; and 2) a reliable

guideline for estimating the adequacy of computational resolution in order to obtain an

accurate assessment of heat transfer to the bubble.

3.2.5 Initial Conditions

The computation must start from a very small but nonzero initial time zT, so that

R(,r0) is sufficiently small at the initial stage. To obtain enough temporal resolution for

the initial rapid growth stage and to save computational effort for the later stage, the

following transformation is used,

T=o 2. (3.30)

Thus a constant "time step" Ao can be used in the computation.

The initial temperature profile inside the superheated bulk liquid thermal boundary

layer plays an important role to the solution of 0,, which in turn affects the heat transfer

to the bubble through the dome.

There exist both experimental and theoretical studies that have considered the bulk

liquid temperature profile in the vicinity of a vapor bubble. Hsu (1962) estimated the

temperature profile of the superheated thermal layer adjacent to the heater surface and

found the layer to be quite thin; thus the temperature gradient inside the thermal layer is

almost linear. However, beyond the superheated layer the temperature is held essentially

constant at the bulk temperature due to strong turbulent convection. The experimental

study by Wiebe and Judd (1971) revealed similar results. It was found that the









superheated bulk liquid thermal boundary layer thickness, 6, decreases with increasing

wall heat flux due to enhanced turbulent convection. A high wall heat flux results in

increased bubble generation, and the bulk liquid is stirred more rapidly by growing and

departing vapor bubbles. To estimate the superheated layer thickness, Hsu (1962) used a

thermal diffusion model within the bulk liquid. Han and Griffith (1965) used a similar

model and estimated the thickness to be

8= zat, (3.31)

where tw is the waiting period. The thermal diffusion model often overestimates the

thermal layer thickness, as it neglects the turbulent convection, which is quite strong as

reported by Hsu (1962) and Wiebe and Judd (1971).

Generally, the bulk liquid temperature profile is almost linear inside the

superheated thermal boundary layer, and remains essentially uniform at the bulk

temperature Tb beyond the superheated background thermal boundary layer.

Accordingly, the initial condition for the bulk liquid thermal field used in the numerical

solution is given by


S-- <6
,0 = z (3.32)
[o, z 8

In the asymptotic solution, the discontinuity of 0D, /Dz in the above profile causes

the solution for 0, to be discontinuous. For clarity, the following exponential profile is

employed in representing the asymptotic solution


0 = exp (3.33)
(58









3.2.6 Solution Procedure

An Euler backward scheme is used to solve Equation (3.14). A second order

upwind scheme is used for the convection term and a central difference scheme is used

for the thermal diffusion terms.

After the bulk liquid temperature field is obtained, the solid heater temperature

field is solved, and the bubble radius R(r) is updated using Equation (3.3) and Euler's

explicit scheme. The information for R(r) is a necessary input in Equation (3.14).

Although the solution for R(r) is only first order accurate in time, the O(Ar) accuracy is

not a concern here because a very small Ar has to be used to ensure sufficient resolution

during the early stages. Typically, n = 104 time steps are used.

3.3 Results and Discussions

3.3.1 Asymptotic Structure of Liquid Thermal Field

To gain an analytical understanding of the liquid thermal field near the bubble and

to validate the accuracy of the computational treatment for the thin unsteady thermal

boundary layer, comparison between the computational and the asymptotic solutions for

0, near the bubble surface is first presented. As mentioned previously, the validity of the

outer solution of the asymptotic analysis only requires A >> 1, which is satisfied under

most conditions due to rapid vapor bubble growth. The inner solution is valid for << 1

in addition to A >> 1.

The comparison is presented for bubble growth in saturated liquid with A=14000.


The initial temperature profile follows Equation (3.33) and = 0.5, in which Rc is the
R

bubble radius at t There are 200 and 50 grid intervals along the R' and y -directions,









respectively. The grid stretching factors are S = 0.65 and S, = 0.73 for the

computational case.

Figure 3-4 compares the temperature profiles between the asymptotic and

numerical solutions at z = 0.001, 0.01, 0.1, and 0.3 for V/= 0, 400, and 710. There are

two important points to be noted. First of all, it is seen that the temperature gradient is

indeed very large near the bubble surface because the unsteady thermal boundary layer is

very thin. Secondly, numerical solutions agree very well with the asymptotic solutions at

z = 0.001 and 0.01. The excellent agreement between the numerical and analytical

solutions indicates that the numerical treatment in this study is correct. At z = 0.1 and

0.3, the asymptotic inner solution given by Equation (3.29) is no longer accurate, while

the outer solution remains valid because A >> 1 is the only requirement. At z = 0.1 and

0.3 the numerical solution matches very well with the outer solution. This again

demonstrates the integrity of the present numerical solution over the entire domain due to

sufficient computational resolution near the bubble surface and removal of undesirable

numerical diffusion through the use of second order upwind scheme in the radial

direction for Equation (3.14).

The large temperature gradient near the dome causes high heat transfer from the

superheated liquid to the vapor bubble through the dome. This large gradient results from

the strong convection effect that is caused by the rapid bubble growth (see Equation

(3.14) for the origin in the governing equation and Equation (3.19) for the explicit

dependence on the bubble growth). Thus the bulk liquid in the superheated boundary

layer supplies a sufficient amount of energy to the bubble.








54







09 09 -


07 0 \\ 7 -


6 05- 6 5-


04 0
S=0001 T=0 01
0 3 0 (Numerical) =0 (Numerical)
S- V40 (Numenrcal) 0 (Asymptotic)
=71 (Numerical) 400(Numerical)
S =00 (Asymptotic) y=40 (Asymptotic)
= 40o (Asymptotic) 710 (Numerical)
v=71 (Asymptotic) -- 71 (Asymptotic)

10' 1 201 10 10? '3 101 io' 101













=01 =03
0- 00 (Numerical) \ n3 =00 (Numerical) \ \
y=400 (Numerical) y=400(Numerical) \
v710 (Numerical) v710 (Numerical)
yo00 (Asymptotic) 0FO-- =00(Asymptotic)
y=400 (Asymptotic) y=400 (Asymptotic)
710 (Asymptotic) ----- 710(Asymptotic)

Il I Il
R'-1 R'-I





Figure 3-4. Comparison of the asymptotic and the numerical solutions at r=0.001, 0.01,
0.1 and 0.3 for 100, 400, and 710


To capture the dynamics of the unsteady boundary layer, a sufficient number of


computational grids is required inside this layer. The asymptotic analysis gives an


estimate for the unsteady boundary layer thickness on the order of



6* ~ = 0.025, which agrees with the numerical solution in Figure 3-4. At
V14000


S= 0.01 in Figure 3-4, the discrete numerical results are presented. There are about 23


points inside the layer of thickness 6* = 0.025 This provides sufficient resolution for the











temperature profile in the unsteady thermal boundary layer. In contrast, most


computational studies on the thermal field around the bubble dome reported in the open


literature have insufficient grid resolution adjacent the dome, which leads to an


inaccurate heat transfer assessment.


Figure 3-5 shows the effect of parameter A on the asymptotic solution. When A is


large, the asymptotic and numerical solutions agree very well. The discrepancy between


asymptotic and numerical solutions inside the unsteady thermal boundary layer increases


when A decreases. However, the outer solution remains valid for the far field even when


A becomes small.





09 A=14000

08

10000
00


05 -

04 T=0.01
v=40'

03 I
03 Asymptotic
02 Numerical

01 -

0

103 102 10 100 10'


Figure 3-5. Effect of parameter A on the liquid temperature profile near bubble.

The temperature contours shown in Figure 3-6 are difficult to obtain


experimentally. Only recent progress in holographic thermography permits such


measurements. Ellion (1954) has stated that there exists an unsteady thermal boundary









layer contiguous to the vapor bubble during the bubble growth. Recently, Mayinger

(1996) used a holography technique to capture the folding of the temperature contours

during subcooled nucleate boiling. Although his study considered subcooled nucleate

boiling, the pattern of the temperature distribution near the bubble dome by Mayinger

(1996) is very similar to that shown in the Figure 3-6. It is expected that experimental

evidence of contour folding in saturated nucleate boiling will be reported in the future.

3.3.2 Constant Heater Temperature Bubble Growth in the Experiment of
Yaddanapudi and Kim

In the experiment of Yaddanapudi and Kim (2001), single bubbles growing on a

heater array kept at nominally constant temperature were studied. The liquid used is

FC-72, and the wall superheat is maintained at 22.5 OC, so that Jacob number is 39. The

bubble shape in the early stage appears to be hemispherical. To calculate the heat flux

from the microlayer to the vapor bubble in the present model, the microlayer wedge angle

or constant c, in Equation (3.4) must be determined. Neither 0 or c, has been

measured. However, the authors have reported the amount of wall heat flux from the wall

to the bubble through an equivalent bubble diameter deq assuming that the wall heat flux

is the only source of heat entering the bubble. Since in the present model this heat flux is

assumed to pass through the microlayer, it may be used to evaluate the constant c, via

trial and error. The superheated thermal boundary layer thickness Sof the bulk liquid in

Equation (3.32) is also a required input. The computed growth rate R(t) is matched with

the experimentally measured R(t) in order to determine The simulation is carried out

only for the early stage of bubble growth. This is because after t=6-8x10-4s the base of

the bubble does not expand anymore, and the bubble shape deviates from a hemisphere.











Furthermore, there is the possibility of the microlayer being dried out in the latter growth

stages as a result of maintaining a constant wall temperature, as was observed by Chen et

al. (2003).













S50.01 15 05 50.1





















o 05 1 15 2 05 1 15 2
R'-1 R'-1













1 1 Ri



R'-I R'-I

Figure 3-6. The computed isotherms near a growing bubble in saturated liquid at T0.01,
'0.1, 0.3, and 0.9.

Figure 3-7 shows the computed equivalent bubble diameter d,, (t), together with


the experimentally determined equivalent dq (t). In the present model, dq, is calculated


using


p h*dt =- k dA,
P uh i fg dz=L(r)


(3.34)











fz 3
where Vb = deq The heat flux includes only that from the microlayer and this allows
6


c, to be evaluated. For d to match the measured data as shown in Figure 3-7, it


requires c,=3.0.




0 0005 -

000045 - deq(t) present model
A deq(t) measurement
00004

0 00035

00003

0 00025 --

00002

000015 -

0 0001
^ -

5E-05

0 I II II I I I
00 0002 00004 00006 00008
t (s)


Figure 3-7. Comparison of the equivalent bubble diameter dq for the experimental data

of Yaddanapudi and Kim (2001) and that computed for heat transfer through
the microlayer (c, =3.0).


Figure 3-8 compares the computed bubble diameter d(t) = 2R(t) and those


reported by Yaddanapudi and Kim (2001). In Figure 3-8, =30jum is used in addition to


c, =3.0 in matching the predicted bubble growth with measured data. The good


agreement obtained can be partly attributed to the adjustment in the superheated bulk


liquid thermal boundary layer thickness Because the heat transfer to the bubble


(through the microlayer and through the dome) is of two different mechanisms, the good









agreement over the range is an indication of the correct physical representation by the

present model.

Figure 3-9 shows the total heat entering bubble and the respective contribution

from the microlayer and from the unsteady thermal boundary layer. The contribution

from the unsteady thermal boundary layer accounts for about 70% of the total heat

transfer. It was reported by Yaddanapudi and Kim (2001) that approximately 54% of the

total heat is supplied by the microlayer over the entire growth cycle. Since, the simulation

is only carried out for the early stage of bubble growth, it is difficult to compare the

microlayer contribution to heat transfer reported by Yaddanapudi and Kim (2001) with

that predicted by current model. At the end, the bubble expands outside the superheated

boundary layer and protrudes into the saturated bulk liquid. The heat transfer from dome

thus slows down. Hence, the 54% for the entire bubble growth period dose not contradict

a higher percentage of contribution computed from the unsteady thermal boundary layer

during the early stages.

Figure 3-10 shows the computed temperature contours associated with

Yaddanapudi and Kim's (2001) experiment for the estimated Sand c Folding of the

temperature contours is clearly observed in the simulation for saturated boiling.

3.3.3 Effect of Bulk Liquid Thermal Boundary Layer Thickness on Bubble Growth

Since the superheated bulk liquid thermal boundary layer thickness, 6, determines

how much heat is stored in the layer, it is instructive to conduct a parametric study on the

effects bubble growth with varying All parameters are the same as those used in

Yaddanapudi and Kim's (2001) experiment except that 6 is varied. Hence the influence

of the superheated thermal boundary layer thickness Son the bubble growth is elucidated.




















d(t) present model
* d(t) measurement


I i I I I I I I I


00002


00006


Figure 3-8. Comparison of bubble diameter, d(t), between that computed using the

present model and the measured data of Yaddanapudi and Kim (2001). Here,

c, =3.0 and 8=30pm.


1 6E-05 -


1 4E-05


1 2E-05


1E-05


8E-06


6E-06


4E-06


2E-06


total heat entering the bubble
heat from micorlayer
heat from bulk liquid thermal boudary layer


I I I


00004


00002


00006


00008


Figure 3-9. Comparison between heat transfer to the bubble through the vapor dome and

that through the microlayer.


00005


00035 -


00003


0 00025


00002


00015


00008












Figure 3-11 shows the effect on the bubble growth rate of varying S(from l1Jm to


100um). The thicker the bulk liquid thermal boundary layer, the faster the bubble grows.


A large Simplies a larger amount of heat is stored in the background bulk liquid


surrounding the bubble. It is also clear that when Japproaches zero, the bubble growth


rate becomes unaffected by the variation of The reason is when Sis small, most of heat


supplied for bubble growth comes from the microlayer and the contribution from the


dome can be neglected.
















t5 0.0012ms 5 t=0.12ms o



5 5 15 5 2
RI'-1 R'-1













t=0.36ms t=0.96ms



I 15 2 o5 1 15 2
R'-I R'-
1 5 1_5





Figure 3-10. The computed isotherms in the bulk liquid corresponding to the thermal
conditions reported by Yaddanapudi and Kim (2001).











It is also noted that for =100/m, if the bubble eventually grows to about several


millimeters, the effect of the bulk liquid thermal boundary layer is negligible on R(t) for


most of the growth period except at the very early stages. Physically, this is because the

bubble dome is quickly exposed to the saturated bulk liquid so that it is at thermal

equilibrium with the surroundings. For small bubbles, it will be immersed inside the

thermal boundary layer most of time. Hence the effect of the bulk liquid thermal

boundary layer becomes significant for the bubble growth.



0 0007

8=100 pm
00006 -


00005 =50 m


00004 8=30 pm


00003 =10 pm
6=5 pm
8=1 Pm
00002


00001



0 00002 00004 00006 00008 0001
t(s)

Figure 3-11. Effect of bulk liquid thermal boundary layer thickness Son bubble growth.

The microlayer angle 0 and the superheated bulk liquid thermal liquid boundary


layer thickness Sare the required inputs to compute bubble growth in the present model.


However, neither of these parameters is typically measured or reported in bubble growth

experiments. It is strongly suggested that the bulk liquid thermal boundary layer









thickness 6be measured and reported in future experimental studies. For a single bubble

study, Sin the immediate neighborhood of the nucleation site should be measured.

3.4 Conclusions

In this study, a physical model is presented to predict the early stage bubble growth

in saturated heterogeneous nucleate boiling. The thermal interaction of the temperature

fields around the growing bubble and vapor bubble together with the microlayer heat

transfer is properly considered. The structure of the thin unsteady liquid thermal

boundary layer is revealed by the asymptotic and numerical solutions. The existence of a

thin unsteady thermal boundary layer near the rapidly growing bubble allows for a

significant amount of heat flux from the bulk liquid to the vapor bubble dome, which in

some cases can be larger than the heat transfer from the microlayer. The experimental

observation by Yaddanapudi and Kim (2001) on the insufficiency of heat transfer to the

bubble through the microlayer is elucidated. For thick superheated thermal boundary

layers in the bulk liquid, the heat transfer though the vapor bubble dome can contribute

substantially to the vapor bubble growth.














CHAPTER 4
ANALYSIS ON COMPUTATIONAL INSTABILITY IN SOLVING TWO-FLUID
MODEL

The two-fluid model is widely used in studying gas-liquid flow inside pipelines

because it can qualitatively predict the flow field with a low computational cost.

However, the two-fluid model becomes ill-posed when the slip velocity between the gas

and the liquid exceeds a critical value. Computationally, even before the flow becomes

unstable, computations can be quite unstable to render the numerical result unreliable. In

this study computational stability of various convection schemes for the two-fluid model

is analyzed. A pressure correction algorithm is carefully implemented to minimize its

effect on stability. Von Neumann stability analysis for the wave growth rates by using the

1st order upwind, 2nd order upwind, QUICK (quadratic upstream interpolation for

convection kinematics), and the central difference schemes are conducted. For inviscid

two-fluid model, the central difference scheme is more accurate and more stable than

other schemes. The 2nd order upwind scheme is much more susceptible to instability for

long waves than the 1st order upwind and inaccurate for short waves. The instability

associated with ill-posedness of the two-fluid model is significantly different from the

instability of the discretized two-fluid models. Excellent agreement is obtained between

the computed and predicted wave growth rates, when various convection schemes are

implemented.

The pressure correction algorithm for inviscid two-fluid model is further extended

to the viscous two-fluid model. For a viscous two-fluid model, the diffusive viscous









effect is modeled as a body force resulting from the wall friction. Von Neumann stability

analysis is carried out to assess the performances of different discretization schemes for

the viscous two-fluid model. The central difference scheme performs best among the

schemes tested. Despite its nominal 2nd order accuracy, the 2nd order upwind scheme is

much more inaccurate than the 1st order upwind scheme for solving viscous two-fluid

model. Numerical instability is largely the property of the discretized viscous two-fluid

model but is strongly influenced by VKH instability. Excellent agreement between the

computed results and the predictions from von Neumann stability analysis for different

numerical scheme is obtained. Inlet disturbance growth test shows that the pressure

correction scheme is capable to correctly handle the viscous two-phase flow in a pipe.

4.1 Inviscid Two-Fluid Model

4.1.1 Introduction

Gas-liquid flow inside a pipeline is prevalent in the handling and transportation of

fluids. A reliable flow model is essential to the prediction of the flow field inside the

pipeline. To fully simulate the system, Navier-Stokes equations in three-dimensions are

required. However, it is very expensive to simulate complex two-phase flows in a long

pipe with today's computer capability. To reduce the computational cost and obtain basic

and essential flow properties of industrial interest, such as gas volume fraction, liquid and

gas velocity, pressure, a one-dimensional model is necessary. The two-fluid model is

considered to give a realistic prediction for the gas-liquid flow inside a pipeline.

The two-fluid model (Wallis, 1969; Ishii, 1975), also known as the separated flow

model, consists of two sets of conservation equations for mass, momentum and energy

for the gas phase and the liquid phase. Although it has success in simulating two-phase

flow in a pipeline, the two-fluid model suffers from an ill-posedness problem. When the









slip velocity between liquid and gas exceeds a critical value that depends on gravity and

liquid depth, among other flow properties, the governing equations do not possess real

characteristics (Gidaspow, 1974; Jones and Prosperettii, 1985; Song and Ishii, 2000).

This ill-posedness condition suggests that the results of the two-fluid model under such

condition do not reflect the real flow situation in the pipe. The two-fluid model only gives

meaningful results when the relative velocity between the gas and liquid phase is below

the critical value. However, this critical value coincides with the stability condition of

inviscid Kelvin-Helmholtz instability (IKH) analysis (Issa and Kempf, 2002). Because

the IKH instability results in the flow regime transition from the stratified flow to the slug

flow or annular flow (Barnea and Taitel, 1994a), ill-posedness of two-fluid model has

been interpreted as to trigger the flow regime transition (Brauner and Maron, 1992;

Barnea and Taitel, 1994a).

The computational methods for solving the two-fluid model have been investigated

by many researchers. For computational simplicity, it is further assumed that both liquid

and gas phases are incompressible. This is valid because most stratified flows are at

relatively low speed compared with the speed of sound. To solve the incompressible

two-fluid equations, one approach is to simplify the governing system to only two

equations for liquid volume fraction and liquid velocity and neglect the transient terms in

the gas mass and momentum equations (Chan and Banerj ee, 1981; Barnea and Taitel,

1994b). A more effective method is to use a pressure correction scheme (Patanka 1980).

Issa and Woodburn (1998), and Issa and Kempf (2003) applied the pressure correction

scheme for the two-fluid model and simulated the stratified flow and the slug flow inside

a pipe.









When two-fluid model becomes ill-posed, the solution becomes unstable. A good

discretized model should be capable of capturing the incipience of the instability point.

However, numerical instability may not be the same as the instability caused by the

ill-posedness. Lyczkowski et al. (1978) used von Neumann stability analysis to study a

compressible two-fluid model with their numerical scheme and found that numerical

instability and ill-posedness may not be identical. However, their two-fluid model lacked

the gravitational term and the study focused on one specific discretization scheme and is

thus incomplete. Stewart (1979), Ohkawa and Tomiyama (1995) attempted to analyze the

numerical stability of an incompressible two-fluid model with a simplified model

equation as an alternative. Their study showed that higher order upwind schemes yield a

more unstable numerical solution than the 1st order upwind scheme.

In this study, a pressure correction scheme is employed to solve the two-fluid

model. It is designed to increase the computational stability when the flow is near the

ill-posedness condition. The von Neumann stability analysis is carried out to study the

stability of the discretized two-fluid model with different interpolation schemes for the

convection term. For the wave growth rates using the 1st order upwind, 2nd order upwind,

QUICK, and central difference schemes, the central difference scheme is more accurate

and more stable. Excellent agreement for the wave growth rates is obtained between the

analysis and the actual computation under various configurations.

4.1.2 Governing Equations

The basis of the two-fluid model is a set of one-dimensional conservation equations

for the balance of mass, momentum and energy for each phase. The one-dimensional

conservation equations are obtained by integrating the flow properties over the

cross-sectional area of the flow, as shown in Figure 4-1.













> y Gas velome
Gas phase Gas velocity u h faction a.


Liquid velome
h Y faction a I
Liquid phase Gravity g action
Liquid velocity uz

Pipe cross section


Figure 4-1. Schematic of two-fluid model for pipe flow.

Because the ill-posedness originates from the hydrodynamic instability of the

two-fluid model, only continuity and momentum equations are considered in the inviscid

two-fluid model. Furthermore, no mass and energy transfer occurs between two phases.

Surface tension is also neglected since it only acts on small scales, while the waves

determining the flow structure in pipe flows are usually of long wavelength. The gas

phase is assumed to be incompressible, as the Mach number of the gas phase is usually

very low for the stratified flow. Hence, the mass conservation equations for liquid phase

is


(a,)+ (ua,) = O, (4.1)
at ax

where a, is liquid volume fraction, p, is liquid density, u, is the liquid velocity, t is the

time, x is the axial coordinate.

The liquid layer momentum conservation equation is


S(u,)+a ua= -- -g cos PH, -gg sin (4.2)
at ax I P A x ax









where p, is the pressure at the liquid-gas interface, g is gravitational accelerator, 3 is the

angle of inclination of the pipe axis from the horizontal lane, and H1 is the liquid phase

hydraulic depth. It is defined as

H1- a1 (4.3)
aa, lah a'

where h, is the liquid layer depth. The second term on the right hand side of Equation

(4.2) represents the effect of gravity on the wavy surface of liquid layer.

The gas phase mass conservation equation is


a (a)+a g)=0, (4.4)

where pg, ag, ug are density, volume fraction, and velocity of gas phase. It is noted that

a, + ag = 1. (4.5)

The momentum equation for gas phase is

S(Ugag)+ (ua g c H -g cosH -ggsin (4.6)
at ax Pg x ax

whereHg is the gas phase hydraulic depth. It is defined as

ag a
H =- ,- (4.7)
o g/ah aI

where h, is the gas layer depth,

4.1.3 Theoretical Analysis

4.1.3.1 Characteristic analysis and ill-posedness

It is well known that the initial and boundary conditions need to be imposed

consistently for a given system of differential equations. The condition is well-posed if









the solution depends in a continuous manner on the initial and boundary conditions. That

is, a small perturbation of the boundary conditions should give rise to only a small

variation of the solution at any point of the domain at finite distance from the boundaries

(Hirsch, 1988).

Equations (4.1, 4.2, 4.4 and 4.6) form a system of 1st order PDEs, for which the

characteristic roots, A, of the system can be found. If A's are real, the system is

hyperbolic. Complex roots imply an elliptic system, which causes the two-fluid model

system to become ill-posed because only initial conditions can be specified in the

temporal direction. Any infinitesimal disturbance will cause the waves to grow

exponentially without bound when A's are complex valued.

Let Ube the vector(a1, u,ug p)T Equations (4.1, 4.2, 4.4 and 4.6) can be written

in vector form as


[A] + [B] = [C], (4.8)
at dx

where [A], [B] and [C] are coefficient matrices, given by

1 0 0 0
-1 0 0 0
[A]= u (4.9a)




ui Oa 0 0
Ug 0 a 0

[B] = u1 + gH, cos f 2au, 0 a, (4.9b)

-u +gH cos / 0 2au, a
Pg









0

[C] = (4.9c)
-a,g sin
aOg sin 3

The characteristic roots of the system is determined by solving A from the

following

[A] [B] = 0. (4.10)

where denotes the determinant of the matrix. Substituting Equations (4.9a) and (4.9b)

into Equation (4.10) results in

S- u 0 0
-(2-g) 0 -ag 0
u, ( u, )-gH, cos f a, ( 2u ) 0 -- = 0. (4.11)

-ug ( -ug) gH cos/ 0 ag,(-2ug) g
Pg

After expansion of the above determinant, the characteristic polynomial for A is

obtained:

Pg (A-U)2 + (- 2 g) gcos =0. (4.12)
ag a, a,

The roots are

S ,+ Pgg P1 Pgsin g ( 2

A = a ag (4.13)
AP Pg
a,1 ag









When g = 0, Equation (4.13) can have real roots only if 1 = ug = u,. Otherwise, the

two-fluid model is ill-posed (Gidaspow, 1974). If g 0, the well-posedness with real

roots requires


U2= u )2 A Pg a,

Equation (4.14) gives the critical value AUc for the slip velocity AU between two

phases beyond which the system becomes ill-posed. The two-fluid model stability

criterion from the characteristic analysis is exactly the same as that from the IKH analysis

on two-fluid model by Barnea and Taitel (1994) as shown below.

4.1.3.2 Inviscid Kelvin-Helmholtz (IKH) analysis and linear instability

IKH analysis (Barnea and Taitel, 1994) provides a stability condition for the

linearized two-fluid model as well as useful information on the growth rate of an

infinitesimal disturbance in the two-fluid model.

Splitting the flow variables into the base variables and the small disturbances, such

as a, + a, expressing the disturbances on the form of

a, = exp(I(at kx)), (4.15a)

u, = E exp(I(ac kx)), (4.15b)

Ug = E exp(I(a kx)), (4.15c)

p = exp(I(a kx)), (4.15d)

where "~" denote disturbance value, I = denotes imaginary unit, cis the amplitude

of perturbation, cois the angular frequency of wave and k is the wavenumber.

Substituting them into the differential governing equations (4.1, 4.2, 4.4 and 4.6), and









linearizing the resulting equations, the following system is obtained for the disturbance

amplitudes, E_, eg, Fe ,E


w-ulk -ak 0 0
w -ugk 0 agk 0 E
H k E,
-k gcosf wc-u,k 0 =0. (4.16)
al Pi g
H k
-k H gcosp 0 co-uk ep
a, Pg

For non-trivial solutions to exist, the following dispersion relation between the wave

speed c and the angular frequency Cmust hold



( ag a1 alag
c =-- g (4.17)
k P+ Pg
a, a,

It is note that the negative imaginary part of determines the growth rate of disturbance.

Equation (4.17) is identical to Equation (4.13), only with A being replaced by c. Details

of the derivation for IKH stability condition can be found in Barnea and Taitel (1994).

4.1.4 Analysis on Computational Instability

4.1.4.1 Description of numerical methods

In general, the governing equations (4.1, 4.2, 4.4, and 4.6) are solved iteratively.

The basic procedure is to solve the continuity equation of liquid for the liquid volume

fraction, and the liquid and gas phase momentum equations for the liquid and gas phase

velocities. To obtain a governing equation for the pressure, Equation (4.1) and Equation

(4.4) are first combined to form a total mass conservation,


(ugag)+ (ua,)= 0. (4.18)
ax ax









Substituting the liquid and gas momentum equations into the above leads to

Sa, + ap a 2 (a 2 2\
u +a
9p a 1 (4.19)

+- g cos H, -a,g sin + g cosf3Hg aggsin \.
x ( ax ax

To solve the pressure equation, SIMPLE type of pressure correction scheme (Patanka,

1982; Issa and Kempf, 2002) is used in this study.

A finite volume method is employed to discretize governing equation. A staggered

grid (Figure 4-2) is adopted to obtain compact stencil for pressure (Peric and Ferziger,

1996). On the staggered grids, the fluid properties such as volume fractions, density and

pressure are located at the center of main control volume, and the liquid and gas

velocities are located at the cell face of main control volume. Figure 4-2 shows the

staggered grids arrangement.


SVelocity control
uw Ue volume





Pw Pp PE
Pw Pe
Main control w aP
volume

Figure 4-2. Staggered grid arrangement in two-fluid model.

The Euler backward scheme is employed for the transient term. The discretized

liquid continuity equation becomes


S(a,) (a )P ))+ (a, ) (a,u 2l) = 0, (4.20)
At









where the superscript 0 denotes the values of the last time step. The subscript P refers to

the center of the main control volume, and subscripts e and w refer to the east face and

west face of main control volume, respectively. The liquid velocity on the cell face is

known, and the volume fraction on the cell face can be evaluated using various

interpolation schemes. Among them, central difference (CDS), 1st order upwind (FOU),

2nd order upwind (SOU) and QUICK schemes are commonly used. Equation (4.4) for the

gas phase is similarly discretized.

The liquid momentum equation is integrated on the velocity control volume. Using

similar notations, one obtains


x ((au ,)p (a1u1 )+ (u), (a1U1), (u1 ) (a1u,) =
t ( a, (4 .2 1)
(pI, p )+ ((a, )w (ae ) )H, g cos Ax( ) g sin 8


where P, e, w refer to the center, east face and west face of the velocity control volume,

respectively. The cell face flux is the liquid velocity, which is obtained by using central

difference, and the volume fraction and liquid velocity at the cell face, which are

transported variables, can be interpolated by using different schemes. It is important to

note that the interpolation method used for the Equation (4.21) must be exactly the same

as those for Equation (4.20). For example, if FOU is used in Equation (4.20), the cell face

flux on the east face of velocity control volume in Equation (4.11) is

(a,u, )e (u, )e = (a,u, )pMAX((u, ),O)- (ao,l )EMAX(- (u, )e,0). (4.22)

If CDS is used in Equation (4.20), the cell face flux on the east face in Equation (4.21) is

evaluated as

(aeu, )P + (a1,1 )E
(a,))P + (u) ) (u, )e (4.23)
2








Using similar discretization procedure, the gas phase momentum equation is

integrated:

"- ((agug )p (agu )+ (ug ) (agug (ug (ag )
a ) (4.24)
(p + ((a)w -(a ))Hgcos/- Ax( )Pgsin .
Pg

For convenience, the discretized mass or momentum equations are written in a

general form

Ap~ + A, +E + Aw~, = B, (4.25)

where ( is the variable to be solved, A is the coefficient, B is the general source term.

For the pressure correction scheme, Equation (4.18) is integrated across the main

control volume. The discretized equation is

(aO gUg (ag g) + (aui )j (aou ), = 0. (4.26)

Because Equation (4.18) is obtained by combining Equation (4.1) and Equation

(4.4), the discretization scheme for Equation (4.26) should be exactly the same as those

for Equation (4.20) and the discretized equation of Equation (4.3). For instance, if CDS is

used in Equation (4.20), it must be used in the main control volume for Equation (4.26):

(lg )e (ag)P ( )E (g )w (g )P + (g )w
ge 2 g 2
2 2 (4.27)
(a-(u + E(1 P.
+ )( )e 2 =) \ ) 0.
2 2

The final discretized pressure equation is obtained by substituting these two

momentum equations, Equation (4.21) and Equation (4.24) into Equation (4.26). This

yields


ap p +aeE + awp = b,


(4.28)









a (a,) +(a,)E a (a)p +(a)E, a,- (4.29a)



a ( = -a a, (4.29c)
2 pPT 2 p, (A


(aj, + (a,), a (( + (a,(,,)a
W2 F





2 2 (4.29d)

+ (a l)P + (a ) (aP + (a
2 2

where, p' represents the pressure correction value, u* represents the imbalanced

velocity, and Ap is from the corresponding discretized liquid or gas momentum equation,

Equation (4.15). The flow chart of the pressure correction scheme is shown in Figure 4-3.

Similarly, the pressure correction schemes with FOU, SOU, CDS, and QUICK can

be obtained.

Consistently handling the discretization is critical to the reduction of numerical

diffusion and dispersion. Barnea and Taitel (1994) showed that the viscosity of fluid can

dramatically degrade the stability of two-fluid model through viscous Kelvin-Helmholtz

stability analysis. Although the viscosity in two-fluid model appears as the body force

instead of 2nd order derivative terms in the modified governing equation, it is

hypothesized that the numerical diffusion and dispersion appearing as derivative in the

modified governing equations produce similar impact on the stability of two-fluid model.
















































Figure 4-3. Flow chart of pressure correction scheme for two-fluid model.

4.1.4.2 Code validation- dam-break flow

The pressure correction scheme is first validated by computing the transient flow

due to dam-break flow (Figure 4-4). The liquid flow is assumed to be over a horizontal

flat surface and the flow is assumed to be one-dimensional. On the left side of the dam is

a body of stationary water in the reservoir with the flat surface of height H. On the right









side of dam is a dry river bottom surface. After the dam breaks suddenly, the water in the

reservoir flows to the downstream due to the gravitational force. If there are no friction

between the fluid and the wall and no viscosity inside the fluid and air pressure is a

constant, an analytical solution for the liquid velocity based on St Venant equation can be

found (Zoppou and Roberts, 2003). The result is shown in Table 2.1.


y

S\Dam



H



H/ Dry river plate

x=- x


Figure 4-4. Schematic for dam-break flow model.

To solve dam-break flow, the pressure at interface, the vapor phase density and

velocity are set to zero. Second order upwind scheme as the cell face interpolation

scheme is implemented in the pressure correction scheme. Figure 4-5 compares water

depth between the present numerical solution and the analytical solution at t=50s. Two

solutions match very well except at the tail end of the liquid, where the numerical

solution is smooth due to a little numerical dissipation. Figure 4-6 compares liquid

velocities between the numerical and analytical solutions at t=50s. Again, these two

solutions match very well except at the leading and tail ends. The discrepancy at the

leading end is due to that the liquid layer is too thin and the numerical result is prone to

error.











Table 4-1. Analytical solution for dam-break flow (Zoppou and Roberts, 2003).
x( position) u (water velocity) h (water depth)
x < -tgH u=0 h=H

-t


x 2t gH u=0 h=0

Although only the dynamics of liquid phase is considered in the dam-break flow, it

is still a solid step for validating the coupling of the pressure and liquid flow (liquid

volume fraction and liquid velocity) in numerical scheme. When both the liquid and the

gas phase present in the flow, the instability in two-fluid model rises due to the

interaction of the liquid and the gas phase, when the slip velocity is large. Numerical

instability of pressure correction scheme emerges and destroys the numerical results

when the two-fluid model near ill-posedness. This numerical instability will be

investigated in the next section and the code will be validated using the theoretical results

of inviscid Kelvin-Helmholtz analysis (Barnea and Taitel, 1994).




10 \ t-osec

9

8
Numerical
7 Analytical
t=50 sec
6



4

3

2

1 -

0 I
0 500 1000 1500 2000
z(m)


Figure 4-5. Water depth at t=50 seconds after dam break.