COMPUTATIONAL MODELING OF FALLING LIQUID FILM FREE SURFACE EVAPORATION

A Thesis Presented to The Academic Faculty

by

Emmanuel O. Doro

In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Mechanical Engineering

Georgia Institute of Technology August 2012

Copyright c Emmanuel O. Doro 2012 COMPUTATIONAL MODELING OF FALLING LIQUID FILM FREE SURFACE EVAPORATION

Approved by:

Dr. Cyrus K. Aidun, Advisor Dr. Jeff Hsieh Mechanical Engineering Chemical & Biomolecular Engineering Georgia Institute of Technology Georgia Institute of Technology

Dr. Mostafa Ghiaasiaan Dr. Preet Singh Mechanical Engineering Material Science & Engineering Georgia Institute of Technology Georgia Institute of Technology

Dr. Yogendra Joshi Dr. Chris Verrill Mechanical Engineering Principal Scientist Georgia Institute of Technology International Paper

Date Approved: 14 June 2012 To Funmi & Nnenne

iii ACKNOWLEDGEMENTS

I would like to express my gratitude to the Institute of Paper Science and Technology

(IPST) at Georgia Institute of Technology for funding this research through the PSE

Graduate Research Fellowship. I would like to thank my advisor Dr. Cyrus K. Aidun for his guidance and input on my research. I would also like to thank Dr. Mathias

Gourdon at Chalmers University of Technology - Sweden for his collaboration on experimental validation.

iv TABLE OF CONTENTS

DEDICATION ...... iii

ACKNOWLEDGEMENTS ...... iv

LIST OF TABLES ...... ix

LIST OF FIGURES ...... x

NOMENCLATURE ...... xiv

SUMMARY ...... xix

1 INTRODUCTION ...... 1 1.1 Research Objectives ...... 1 1.2 Specific Aims ...... 2 1.3 Motivation ...... 3 1.4 Falling Liquid Films ...... 4 1.4.1 Literature Review ...... 4 1.4.2 Falling Film Parameters ...... 7 1.5 Thesis Outline ...... 10

2 METHODOLOGY ...... 11 2.1 Mathematical Model ...... 11 2.1.1 Navier-Stokes Equations ...... 12 2.1.2 Interface Evolution (VOF) ...... 13 2.1.3 Modified VOF Equation ...... 14 2.1.4 Modified Conservation Laws ...... 15 2.1.5 Boundary and Initial Conditions ...... 16

3 NUMERICAL METHOD ...... 18 3.1 Finite Volume Method ...... 18 3.1.1 Domain Discretization ...... 18 3.1.2 Discretization of Conservation Law ...... 19

v 3.1.2.1 Transient Term ...... 20 3.1.2.2 Gradient Term ...... 20 3.1.2.3 Convection Term ...... 21 3.1.2.4 Diffusion Term ...... 21 3.1.2.5 Source Term ...... 21 3.1.3 Discretization of Navier-Stokes Equations ...... 22 3.1.4 Discretization of Phase Equation ...... 24 3.2 PISO Algorithm ...... 24 3.3 Numerical Solution Sequence ...... 25

4 DYNAMICS OF BACKFLOW IN FALLING LIQUID FILMS . 27 4.1 Background ...... 27 4.2 Mathematical Model ...... 28 4.3 Numerical Validation ...... 29 4.3.1 Correlations for 2-D Falling Films ...... 29 4.3.2 2-D Falling Film Experiments ...... 30 4.4 Analysis of Simulation Results ...... 32 4.4.1 Interfacial Waves and Backflow Evolution ...... 32 4.4.2 Dynamics of Backflow ...... 39 4.4.2.1 Open Vortex ...... 39 4.4.2.2 Solitary-Capillary Waves ...... 41 4.4.2.3 Solitary-Capillary-Capillary Waves ...... 44 4.4.3 Buoyancy and Viscous Dissipation ...... 48

5 WAVY-LAMINAR FALLING LIQUID FILM EVAPORATION 49 5.1 Background ...... 49 5.2 Mathematical Model ...... 51 5.2.1 Governing Equations ...... 52 5.2.2 Evaporation Source Terms ...... 52 5.2.3 Boundary and Initial Conditions ...... 53

vi 5.3 Numerical Validation ...... 53 5.3.1 Stefan Problem ...... 54 5.3.2 Evaporation Heat Transfer Coefficient ...... 55 5.4 Analysis of Simulation Results ...... 59 5.4.1 Free Surface Waves Evolution ...... 59 5.4.2 Wave Properties Compared to 2-D Correlations ...... 61 5.4.3 Streamwise Thermal Regions ...... 61 5.4.3.1 Growing Wall Temperature ...... 62 5.4.3.2 Near-Uniform Wall Temperature ...... 63 5.4.3.3 Fluctuating Wall Temperature ...... 67 5.4.4 Evaporation Rates along Waves ...... 69

6 FALLING FILM EVAPORATION OF BLACK LIQUOR .... 71 6.1 Background ...... 71 6.2 Mathematical Model ...... 73 6.2.1 Governing Equations ...... 74 6.2.2 Evaporation Source Terms ...... 75 6.2.3 Phenomenological Crystallization Model ...... 76 6.2.4 Crystallization Parameters ...... 78 6.2.5 Black Liquor Transport Properties ...... 79 6.2.6 Boundary and Initial Conditions ...... 80 6.3 Numerical Validation ...... 81 6.3.1 Evaporation Heat Transfer Coefficient ...... 81 6.3.2 Exit Dry solids Mass Fraction ...... 83 6.4 Analysis of Simulation Results ...... 84 6.4.1 Black Liquor Free Surface Waves ...... 84 6.4.1.1 Effect of Dry Solids ...... 85 6.4.1.2 Secondary Instability and Wave-Breaking ...... 88 6.4.1.3 Evolution of 2-D Perturbation ...... 91

vii 6.4.2 Wave Induced Transport in Evaporating Film ...... 93 6.4.2.1 Influence on Temperature and Species ...... 93 6.4.2.2 Implications for Crystallization and Scaling . . . . . 96 6.4.2.3 Accumulative Effect of Waveforms ...... 99

7 CONCLUSIONS AND FUTURE RECOMMENDATIONS ... 102 7.1 Conclusions ...... 102 7.1.1 Backflow Dynamics ...... 102 7.1.2 Thermal Analysis of Falling Film Evaporation ...... 103 7.1.3 Falling Film Evaporation of Black Liquor ...... 103 7.2 Future Recommendations ...... 104 7.2.1 Non-Newtonian Black Liquor Rheology ...... 104 7.2.2 Crystallization Model ...... 104 7.2.3 Lagrangian Dynamics of Crystals ...... 105

REFERENCES ...... 106

viii LIST OF TABLES

4.1 Wave properties of simulated water falling film ...... 30 5.1 Transport and heating conditions for falling film evaporation simulations 57 6.1 Mean values of typical components of black liquor dry solids . . . . . 78

6.2 Physical properties of Na2CO3 relevant to species transport model . . 79 6.3 Flow and heating conditions for black liquor falling film evaporation simulations ...... 81

ix LIST OF FIGURES

2.1 Schematic diagram of a wavy falling film: u, v are x, y velocity com- ponents, h is wave height and g is gravitational acceleration...... 12 3.1 Section of cylindrical computational flow domain discretized using struc- tured finite volume cells...... 18 3.2 Parameters in finite volume discretization ...... 19

4.1 Simulated non-dimensional maximum wave height Nhp and non-dimensional wave speed Nuw for water falling films ...... 31 4.2 Comparison of numerical and experimental data by Dietze et al. (2009) for time traces of film thickness and streamwise velocity ...... 32 4.3 Time traces of film thickness and streamwise velocity from falling film simulation of DMSO solution ...... 33 4.4 Surface wave profiles for simulated water falling film at Re = 69 and different inlet disturbance frequencies...... 34 4.5 Film free surface wave profiles for (a): Solitary-capillary waves and (b): sinusoidal waves both at Re = 69...... 34

4.6 Normalized film thickness δ/δN , liquid film streamwise velocity u and ∂p normalized liquid film streamwise pressure gradient px(= ∂x )/γ across a solitary wavefront with Re = 69 and f = 30Hz ...... 35

4.7 Normalized film thickness δ/δN , liquid film streamwise velocity u and ∂p normalized liquid film streamwise pressure gradient px(= ∂x )/γ for the wave front in figure 4.6 after traveling further downstream...... 36 4.8 streamlines at the solitary wavefront corresponding to flow conditions in figure 4.7 (a) and (b) respectively ...... 38

4.9 Normalized film thickness δ/δN , liquid film streamwise velocity u and ∂p normalized liquid film streamwise pressure gradient px(= ∂x )/γ across the wavefront with Re = 69 and f = 70 Hz...... 38

∂p 4.10 Normalized liquid film streamwise pressure gradient px(= ∂x )/γ across the wavefront for Re = 69, f = 30 Hz and 70 Hz sampled at 20 µm and 0.0001 m from the wall...... 40

+ 4.11 Normalized maximum positive streamwise pressure gradient (px) /γ, ∂u and normalized wall shear stress τw(= µl ∂y |y=0)/γδN measured at the wavefront of a solitary wave (a) and sinusoidal wave (b) ...... 40

x 4.12 Streamlines and velocity ve