UC Irvine UC Irvine Electronic Theses and Dissertations Title Liquid-Gas Heat and Mass Transfer at Supercritical Pressures Permalink https://escholarship.org/uc/item/6w0178qd Author Poblador Ibanez, Jordi Publication Date 2018 License https://creativecommons.org/licenses/by-nc-nd/4.0/ 4.0 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, IRVINE Liquid-Gas Heat and Mass Transfer at Supercritical Pressures THESIS submitted in partial satisfaction of the requirements for the degree of MASTER OF SCIENCE in Mechanical and Aerospace Engineering by Jordi Poblador Ibanez Thesis Committee: Professor William A. Sirignano, Chair Professor Derek Dunn-Rankin Professor Said E. Elghobashi 2018 c 2018 Jordi Poblador Ibanez TABLE OF CONTENTS Page LIST OF FIGURES iv LIST OF TABLES vi NOMENCLATURE vii ACKNOWLEDGMENTS ix ABSTRACT OF THE THESISx 1 Introduction1 1.1 Literature review.................................4 1.1.1 Subcritical liquid jet injection......................5 1.1.2 Transition from subcritical to supercritical state............9 1.1.3 Supercritical liquid jet injection..................... 13 1.2 Objectives..................................... 20 2 Problem Statement and Governing Equations 22 2.1 Definition of the problem............................. 22 2.2 Governing equations............................... 23 2.3 Thermodynamic relations............................ 26 2.3.1 The equation of state........................... 27 2.3.2 An equation for enthalpy......................... 35 2.3.3 An equation for fugacity......................... 38 2.3.4 The enthalpy of vaporization....................... 39 2.3.5 An equation for entropy......................... 40 2.3.6 Diffusion mass flux............................ 41 2.3.7 Viscosity and thermal conductivity................... 45 2.3.8 Surface tension.............................. 45 2.4 Matching conditions............................... 47 2.5 Methodology comparison............................. 50 3 Numerical Method 52 3.1 Discretization of the equations.......................... 52 3.2 Algorithm..................................... 56 ii 3.3 Solution method for the matching conditions.................. 59 3.3.1 Computation of gradients at the interface................ 64 3.4 Boundary conditions and initial conditions................... 65 3.5 Grid independence study............................. 66 4 Results 71 4.1 Phase equilibrium at high pressures....................... 71 4.2 The diffusion process: oxygen/n-decane mixture................ 73 4.2.1 Pressure effects.............................. 73 4.2.2 Temporal evolution............................ 79 4.2.3 Analysis of the equation terms...................... 81 4.2.4 Phase change analysis.......................... 84 4.3 Instability analysis: Kelvin-Helmholtz...................... 91 5 Conclusions 98 5.1 Discussion..................................... 98 5.2 Further research.................................. 100 Bibliography 101 Appendix A. Physical Properties 107 Appendix B. SRK-EoS Derivatives 108 Appendix C. Development of the Energy Equation 111 Appendix D. Fluid Properties Models 113 Appendix E. Validation of the Models 126 Appendix F. Nitrogen/n-Octane Mixture Results 136 iii LIST OF FIGURES Page 1.1 Disintegration modes as a function of the liquid Reynolds and Weber numbers (source [1]).....................................6 1.2 Re = 1,600 and We = 230,000 and gas-to-liquid density ratio 0.5 at different times (source [2])..................................8 1.3 Effect of increasing Re number (from 320 on the left to 1,600 on the right) for a fixed We number (230,000) (source [2]).....................9 1.4 Liquid nitrogen injection into gaseous nitrogen at 4 MPa (top), 3 MPa (center) and 2 MPa (bottom) (source [3])......................... 10 1.5 View of the liquid iso-surface for a planar liquid sheet varying We with all other parameters fixed (source [4])........................ 12 1.6 Image sequence at a fixed position of liquid nitrogen injected into helium at 5.5 MPa (source [3])................................ 12 1.7 Results for nitrogen injected into carbon dioxide showing isocontours of dif- ferent properties (source [5])............................ 16 1.8 Interfacial density profiles and thicknesses for the low-temperature (left) and high-temperature (right) interface states of the n-dodecane/nitrogen mixture (source [6])..................................... 18 2.1 Sketch of the interface problem.......................... 23 2.2 Mass and energy balances across the interface.................. 49 3.1 Control volume discretization of the 1-D domain................ 53 3.2 Sketch of the proposed time integration..................... 58 3.3 Flow diagram of the solution algorithm..................... 58 3.4 Approach to compute gradients at the interface................. 65 3.5 Grid independence study: Variable profiles for test cases C5, C8 and C9... 69 4.1 Phase equilibrium results for different pairs of species.............. 72 4.2 Oxygen mass fraction and fluid density on each side of the interface for the oxygen/n-decane mixture............................. 75 4.3 Fluid velocity and temperature distribution for the oxygen/n-decane mixture. 76 4.4 Deviations from the ideal case for the oxygen/n-decane mixutre........ 78 4.5 Temperature and liquid velocity at the interface for the oxygen/n-decane mix- ture......................................... 78 4.6 Variation of interface properties with pressure for the oxygen/n-decane mixture. 79 iv 4.7 Temporal evolution of the temperature distribution and the fluid density for the oxygen/n-decane mixture at p = 50 bar................... 80 4.8 Temporal evolution of the temperature distribution and the fluid density for the oxygen/n-decane mixture at p = 150 bar.................. 81 4.9 Total and local derivatives of the species continuity equation and the energy equation for the oxygen/n-decane mixture.................... 82 4.10 Analysis of the terms of the energy equation for the oxygen/n-decane mixture. 83 4.11 First Law results for the oxygen/n-decane mixture at different pressures... 87 4.12 Mesh refinement effects on thermodynamic laws for the oxygen/n-decane mix- ture......................................... 88 4.13 Temporal evolution of enthalpy for the oxygen/n-decane mixture at different pressures...................................... 89 4.14 Temporal evolution of entropy for the oxygen/n-decane mixture at different pressures...................................... 90 4.15 Temporal evolution of integrated entropy for the oxygen/n-decane mixture at different pressures................................. 90 4.16 Sketch of the Kelvin-Helmholtz instability (source [7])............. 91 4.17 Kelvin-Helmholtz instability for oxygen/n-decane mixture at p = 10 bar and injection velocity 10 m/s (Case A)........................ 94 4.18 Kelvin-Helmholtz instability for oxygen/n-decane mixture at p = 10 bar and injection velocity 100 m/s (Case B)........................ 94 4.19 Kelvin-Helmholtz instability for oxygen/n-decane mixture at p = 150 bar and injection velocity 10 m/s (Case C)........................ 95 4.20 Kelvin-Helmholtz instability for oxygen/n-decane mixture at p = 150 bar and injection velocity 100 m/s (Case D)........................ 95 v LIST OF TABLES Page 1.1 Independent groupings appearing in the liquid breakup phenomena......6 1.2 Groupings eliminating velocity dependence in the liquid breakup phenomena.6 2.1 Binary interaction coefficients for the SRK-EoS obtained with Soave et al. model........................................ 35 2.2 Methodology comparison between different works................ 50 3.1 Test cases for the grid independence study (a).................. 66 3.2 Grid independence study: Interface temperature and absolute error compared with C9....................................... 67 3.3 Grid independence study: Interface gas density and absolute error compared with C9....................................... 67 3.4 Grid independence study: Interface liquid density and absolute error com- pared with C9................................... 68 3.5 Test cases for the grid independence study (b).................. 70 3.6 Grid independence study: Domain length influence............... 70 4.1 Analyzed cases for the Kelvin-Helmholtz instability............... 93 4.2 Fluid properties at the interface.......................... 93 4.3 Results of Kelvin-Helmholtz instability for the oxygen/n-decane mixture... 96 vi NOMENCLATURE [Pi] Parachor of species i αi;Si Functions in the SRK equation of state Γ Thermodynamic factor λ Thermal conductivity, W/(m K) µ Viscosity, kg/(ms) !i Acentric factor of species i Φi Fugacity coefficient of species i ρ Density, kg/m3 σm Surface tension of the mixture, N/m A; B Constants in the SRK equation of state ai; bi Attractive and repulsive parameter of species i c Growth rate of Kelvin-Helmholtz instability, 1/s D Fickian diffusion coefficients matrix, m2/s −1 di Driving force of species i due to diffusion of species, m 2 Dij Binary diffusion coefficient between species i and j, m /s fi Fugacity of species i H Enthalpy of the mixture, J h; h¯ Specific enthalpy and molar enthalpy of the mixture, J/kg or J/mole ¯ hi; hi Specific enthalpy and molar enthalpy for species i, J/kg or J/mole 2 Ji Diffusion mass flux of species i, kg/(m s) kij Binary interaction coefficient between species i and j vii MW Molecular weight, kg/mole N Number of species p Pressure, Pa Ru Universal gas constant,