WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLOGY DEPARTMENT OF CHEMICAL ENGINEERING

Transcription

1 WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLOGY DEPARTMENT OF CHEMICAL ENGINEERING FLOW DISTRIBUTION AND ITS IMPACT ON PERFORMANCE OF PACKED-BED REACTORS by Yi Jiang Prepared under the direction of Prof. M. H. Al-Dahhan & Prof. M. P. Dudukovic A dissertation presented to the Sever Institute of Washington University in partial fulfillment of the requirements for the degree of DOCTOR OF SCIENCE December, 2000 Saint Louis, Missouri, USA

2 WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLOGY DEPARTMENT OF CHEMICAL ENGINEERING FLOW DISTRIBUTION AND ITS IMPACT ON PERFORMANCE OF PACKED-BED REACTORS by Yi Jiang Prepared under the direction of Prof. M. H. Al-Dahhan & Prof. M. P. Dudukovic December, 2000 Saint Louis, Missouri, USA Packed-bed reactors are used in numerous industrial applications. Extensive efforts in academic and industrial research have been made to improve the understanding of the hydrodynamics and modeling the effect of particle-scale and bed-scale phenomena on reactor performance. The studies conducted so far have been limited to phenomenological approaches focusing on the global and mean quantities and utilizing only the mean properties in model development. Many of these models are homogeneous and pseudo-homogenous in nature, whereas few models consider the heterogeneity of the bed structure. This research shows that the performance of packed-bed reactors could be better modeled by properly accounting for the heterogeneity of the bed structure and of the flow. The first part of this study is focused on modeling of flow distribution in packed beds. Two modeling approaches are developed for simulating single-phase and

3 multiphase flow distribution in packed beds. One is the discrete cell model (DCM) approach, which essentially rests on the minimization of the total mechanical energy dissipation rate proposed by Holub (1990). Another approach is the k-fluid computational fluid dynamics (CFD) approach, which solves the ensemble-averaged Naiver-Stokes equations for stationary solid phase and flowing phase(s) with appropriate closures. The predictions of flow distribution by these two different approaches (DCM and CFD) are comparable for single- and two-phase flow systems. An agreement between the predicted flow velocities and the experimental data in the literature is also achieved. The k-fluid CFD modeling approach is superior in computational efficiency particularly for a packed bed of large size. It is capable of simulating the system with different inlet flow distributions (e.g., uniform and nonuniform; steady state and unsteady state). A statistical implementation of bed porosity distribution into the model has been developed and adopted in both DCM and CFD flow simulations. This shows significant promise in our ability to predict the flow structure in the bed. The second part of this study is devoted to utilizing the flow distribution information in industrial practice, and to quantifying the impact of flow distribution on reactor performance. From an engineering point of view, the developed flow distribution models have proven useful in the following applications: Quantification of the relationship between bed structure, flow distribution and operating conditions. Obtaining the multiphase flow structure in a bench-scale packed bed by performing flow simulation, interpretation of the irregular or scattered benchscale experimental data, and exploration of scale-down issues. Development of a combinational modeling scheme for flow and reaction in packed-bed reactors based on the mixing-cell network concept, in which the simulated flow results can be used as input data for the cell network model. Such a modeling strategy is definitely useful for the diagnostic analysis of the operating units because of its capability of providing the mapping information on the bulk flows and the species concentration for a given kinetics. ii

4 Contents Page Tables... ix Figures... x Acknowledgements... xxiv Nomenclature... xxvii 1. Introduction to Flow Distribution in Packed Beds Research Motivation Discrete Cell Model (DCM) Revisited k-fluid CFD Model Development and Applications Impact of Flow Maldistribution on Packed-Bed Performance Research Obectives DCM Revisited k-fluid CFD Model for Packed Beds Applications of k-fluid CFD Model Impact of Flow Maldistribution on Reactor Performance Thesis Structure Experimental Observations: Liquid Flow Distribution in Trickle Beds Introduction Experiment-I: 2-D Liquid Flow Imaging D Packed Bed and CCD Setup Imaging and Processing iii

5 2.2.3 Liquid Flow Imaging Experimental Results and Discussion Non-prewetted bed (dry bed) Prewetted bed (wet bed) Comparison of liquid flow in non-prewetted bed and prewetted bed Experiment-II: Exit Flow Measurements in 3-D Bed Experimental Obectives D Column and Exit Flow Measurement Experimental Results and Discussion Conclusions Discrete Cell Model Approach Revisited: I. Single Phase Flow Modeling Introduction Non-Parallel Gas Flow Models Vectorized Ergun Equation Model Equation of Motion Model Discrete Cell Model (DCM) Discrete Cell Model (DCM) CFDLIB Formulation Modeling Results and Discussion Model Packed Bed Analysis of the Energy Dissipation Equation Comparison of DCM and CFDLIB Comparison of DCM/CFDLIB and Exprimental Data Case Studies by DCM iv

6 3.6 Concluding Remarks Discrete Cell Model Approach Revisited: II. Two Phase Flow Modeling Introduction Spatial Sacles in Trickle Beds Governing Principles for Flow Distribution Extended Discrete Cell Model Modeling Results and Discussion Comparison of DCM and CFD Simulations Effect of Liquid Distributor Effect of Particle Prewetting Conclusions and Final Remarks Computational Fluid Dynamics (CFD): I. Modeling Issues Introduction and Background CFD Applied to Multiphase Reactors CFD and Other Modeling Approaches to Multiphase Flow in Packed Beds Spatial and Temporal Characteristics of Flow in Packed Beds Structure Implementation k-fluid Approach and CFDLIB Code Eulerian k-fluid Model k-fluid Model in CFDLIB CFD Modeling Issues Significance of Terms in the Momentum Balance Closures for Multiphase Flow Equations Interfacial Tension Effect, Wetting Correction v

7 5.5.4 Effect of Mesh Size on Computated Results Boundary Conditions Conclusions and Remarks Computational Fluid Dynamics (CFD): II. Numerical Results & Compariosn with Experimental Data Introduction Comparison of CFD Simulation and Experimental Results Liquid Upflow in Packed Beds Gas and Liquid Cocurrent Downflow in Trickle Beds Simulation of Feed Distribution Effects Conclusions CFD Applications in Scale-Down and Scale-Up of Packed-Bed Reactors Introduction Model Bench-Scale Packed Beds Modeling Results of Bench-Scale Packed Beds Statistical Nature of the Bed Structure and Flow Bed Structure Multiscales of Flow and Role of Various Forces Link of Macroscale and Cell-Scale Hydrodynamics Statistical Quantities Modeling Result and Correlation Development Model Packed Beds Capillary Force Effect Porosity Distribution Effect vi

8 Correlation Development Superficial Velocities at the Inlet Conclusions and Remarks A Combined k-fluid CFD Model and the Mixing-Cell Network Model Introduction k-fluid CFD Model for Flow Simulation Mixing-cell Network Model Concluding Remarks Thesis Accomplishments and Future Work Summary of Thesis Accomplishments Discrete Cell Model (DCM) k-fluid CFD Model Mixing-Cell Network Model Recommendations for Future Research Discrete Cell Model (DCM) k-fluid CFD Model Mixing-Cell Network Model Appendix Comparison between Trickle Bed and Packed Bubble Column Reactor Performance for the Hydrogenation of Biphenyl A1 Introduction A2 Reactor Models A2.1 Kinetic Model A2.2 Key Assumptions A2.3 Cocurrent Trickle Bed and Packed Bubble Flow Bed Model vii

9 A3 Results and Discussion A3.1 Flow Characteristics and Flow Regimes A3.2 Trickle-bed Reactor Performance A3.3 Packed Bubble Flow Reactor Performance A3.4 Sensitivity of Model Parameters A4 Conclusions References Vita viii

10 Tables Table Page 3-1. Dimensions of the model bed and physical properties of the fluids in the simulations Summary of operating conditions used in flow simulations Current Status of CFD Modeling in Multiphase Reactors Typical Ranges of Force Rations in Two-Phase Flow in Packed Granular packing (adapted from Melli et al., 1990) Models for Drag Coefficients Statistical Description of Porosities and CFD Simulated Velocities Parameters Used in the Discretization of the Radial Porosity Profile, and in the Generation of 2D Porosity Distribution Feed Velocities and Holdups at to Ten Sections from the Center to the Wall Statistical description of the porosity distribution A-1. Kinetic parameters for hydrogenation of biphenyl (Sapre and Gates, 1981) A-2. Summary of various correlations used in this study A-3. Properties of the catalyst particles A-4. List of gas and liquid feed velocity ix

11 Figures Figure Page 2-1a. Schematic diagram of 2-D rectangular packed bed b. Experimental setup using CCD video camera imaging technique Effect of liquid superficial mass velocity on liquid rivulet flow at single-point inlet in a non-prewetted packed bed: the radius of the liquid rivulet increases with the liquid superficial mass velocity Effect of liquid irrigation rates on the local radial spreading of the liquid rivulet at a point source inlet in the non-prewetted bed (ROI Size: 3 cm 2 cm) Cross-sectional liquid distribution in a 3D-rectanglar non-prewetted bed of glass beads (d p = 1.6 mm), [From Ravindra et al (1997)]. Upper left- at the top layer with single-point liquid inlet; lower left- at the layer 12 cm far from the top with singlepoint liquid inlet; upper right- at the top layer with single-line liquid inlet; lower right- at the layer 12 cm far from the top with single-line liquid inlet Local liquid distribution at (a) the top region and (b) the bottom region at a mass superficial velocity of 7.04 kg/m 2 /s in a non-prewetted bed The steady state liquid distribution in a prewetted bed at different liquid superficial mass velocities x

12 2-7. The development of finger-type liquid flow in a prewetted bed at a superficial mass velocity of 0.74 kg/m 2 /s ( t: starting time, second) Liquid flow distribution in (a) non-prewetted bed and (b) prewetted bed with singlepoint liquid inlet without gas flow. Part (c) shows the image intensity profiles at specific vertical position (z = 6 cm from the top) in cases (a) and (b) Transient behavior of reaction rates in non-prewetted and prewetted beds for oxidation of SO2 with the active carbon particles as catalyst. [Data are extracted from Ravindra et al. (1997) at T = 25 C, P = 1 atm.] Dependence of the global reaction rate on liquid velocity in non-prewetted and prewetted beds (uniform liquid inlet) for oxidation of SO2 with active carbon catalyst. [Data are extracted from Ravindra et al (1997) at T = 25 C; P = 1 atm.] Packing image taken from the front of the 2-D rectangular packed bed Schematic diagram of the experimental setup for a 3D column with exit flow measurement and periodic liquid feed controller Liquid collector with 25 individual tubes located at the bottom of the packed bed Liquid flow measurements in the non-prewetted bed: dimensionless liquid flow velocity data from 25 individual tubes at different liquid superficial mass velocities (H = 6 ft, G = 0.0 m/s, uniform liquid inlet) Individual points measurements in the prewetted bed: dimensionless liquid flow velocity data from 25 individual tubes at different liquid superficial mass velocities (H = 6 ft, G = m/s, uniform liquid inlet) Effect of time split in On/Off periodic operation mode on liquid radial profiles with uniform liquid inlet (H=2.0 ft, G=0.049 m/s, L = 1.5 kg/m2/s) Effect of time split in On/Off periodic operation mode on liquid radial profiles with point source liquid inlet (H=2.0 ft, G=0.049 m/s, L = 1.0 kg/m2/s) xi

13 3-1. Model packed bed ('2D' rectangular as example) and velocity at each interface of cell. (Note that S x, equals to S x+ x, in the '2D' rectangular packed bed) Porosity distribution of model bed (32 cells x 8 cells) Contribution of each energy dissipation rate term at each cell to the total energy dissipation rate. V 0 = 0.5 m/s (gas flow without internal obstacles); Re'= a. Contribution of each energy dissipation rate term at each cell to the total energy dissipation rate. V 0 = 0.5 m/s (gas flow with two internal obstacles at /d p = 30, 66); Re' = b. Contribution of each energy dissipation rate term at each cell to the total energy dissipation rate. V 0 = 0.5 m/s (gas flow with two internal obstacles at /d p = 30, 66); Re' =28.5 (zoom-in) Contribution of each energy dissipation rate term at each cell to the total energy dissipation rate. V 0 = 0.1 m/s (liquid flow without internal obstacles); Re' = Comparison of CFD simulations and DCM predictions at a superficial velocity of 0.1 m/s at different axial positions (/d p ) (Re' = 5.7) Comparison of CFD simulations and DCM predictions at a superficial velocity of 0.5 m/s at different axial positions (/d p ) (Re' = 28.5) Comparison of CFD simulations and DCM predictions at a superficial velocity of 0.5 m/s at different axial positions (/d p ) (Re' = 170.1) a. Comparison of superficial velocity (V ) between CFD and DCM predictions for gas flow in the Reynolds number (Re') range of 5 to b. Comparison of relative interstitial velocity (U /U 0 ) between CFD and DCM predictions for gas flow in the Reynolds number (Re') range of 5 to 171. (U 0 = V 0 /ε B ) xii

14 3-10a. Comparison of predicted interstitial velocity component in the direction (U z ) by two methods in liquid up-flow system: liquid superficial velocity V 0 = 0.1 m/s (Re' = 47.5) b. Comparison of predicted interstitial velocity component in direction (U x ) by two methods in liquid up-flow system: inlet liquid superficial velocity V 0 = 0.1 m/s a. Influence of gas feed superficial velocity on DCM predicted cell interstitial velocity profiles b. Effect of particle Reynolds number (Re p ) on the calculated relative cell superficial velocity profile inside a bed using DCM Comparison of experimental data of Stephenson and Stewart (1986) and CFDLIB simulated results for relative velocity in a packed bed with D/d v = 10.7 and d v = cm (cylindrical particles). Physical properties of liquid: Liquid -B for condition at a Re p of 5, ρ = g/cm 3 ; µ = g cm/s. Liquid -C for condition at a Re p of 80, ρ = g/cm 3 ; µ = g cm/s a. Interstitial velocity field in a packed bed with two internal obstacles and gas uniform feed from the top at a superficial velocity of 0.5m/s (Re' = 28.5) (U 0 = cm/s); (velocity vector plotting) b. Interstitial velocity field in a packed bed with side gas feed (top-left) and internal obstacles. Inlet gas mean superficial velocity: 0.5m/s (Re' = 28.5) (U 0 =120.5 cm/s) (point source inlet from left side, inlet point superficial velocity is of 4.0 m/s) (velocity vector plotting) a. Pressure field in a packed bed with two internal obstacles and gas uniform feed from the top at a superficial velocity of 0.5m/s (Re' = 28.5) (The relative values of pressure with respect to the inlet operating pressure are plotted). Two obstacle xiii

15 plates are placed in this bed, one is located at /d p of 66 (at the left side), another is at /d p of 30 (at the right side). The width of the obstacle plate (i.e. the length that it protrudes into the bed) is half of the width of bed (4 cells) b. Dimensionless pressure drop in a packed bed with two internal obstacles and a gas point feed from top-left side at an equivalent feed superficial velocity of 0.5m/s (Re' 1 P = 28.5) (Dimensionless pressure drop, ψg = is plotted). Two obstacle ρ plates are placed in this bed, one is located at /d p of 66 (at the left side), another is at /d p of 30 (at the right side). The width of the obstacle plate (i.e. the length that it protrudes into the bed) is half of the width of bed (4 cells) G g 4-1. Tne coordinate system and velocity conventions for α phase in the cell a. Local porosity distribution in model bed; Random internal porosity (0.36 ~ 0.44). Darker color corresponds to higher porosity b. Average porosity profiles in and directions in model bed a, 3b, 3c and 3d. Comparison of the predicted gas interstitial velocities (relative) at the specific axial level by DCM and CFD. U l = m/s (UF); U g = 0.05 m/s (UF); Completely prewetted packed bed. (The relative interstitial velocity is defined as the local interstitial velocity (V i ) divided by the overall interstitial velocity (V 0 ). The value of V 0 in this case is equal to m/s) e. Comparison of the predicted gas interstitial velocities (relative) for all the cells by DCM and CFD. Inlet superficial velocities (uniform): U l = ; U g = 0.05m/s; Completely prewetted packed bed Comparison of predicted liquid holdup at specific levels by DCM and CFD. Single point source liquid inlet: U l = m/s (U l (PS 1 )= m/s); U g = 0.05m/s; Non-prewetted packing xiv

16 4-5a. Liquid holdup distribution with single liquid point source inlet (located at No. 5 cell from left) by DCM. U l = m/s (U l (PS 1 )= m/s); U g = 0.05m/s; Non-prewetted packing b. Liquid holdup distribution with two liquid points source inlet (located at No. 3 cell and No. 6 cell from left) by DCM. U l = m/s (U l (PS 2 )= m/s); U g = 0.05m/s; Non-prewetted packing c. Liquid holdup distribution in whole domain of the non-prewetted packed bed with uniform liquid distributor by DCM. U l = m/s; U g = 0.05m/s d. Comparison of liquid flow maldistribution calculated by DCM along the bed for different liquid distributors. U l = m/s; U g = 0.05m/s a. Liquid holdup distribution in the whole domain of the completely prewetted packed bed (f = 1). U l = m/s (U l (PS 1 )= m/s); U g = 0.05 m/s; Point liquid distributor (PS 1 ) b. Liquid holdup distribution at specific levels (/d p ) in the completely prewetted packed bed (f = 1), U l = m/s (U l (PS 1 )= m/s); U g = 0.05 m/s; Point liquid distributor (PS 1 ); Overall liquid holdup = c. Liquid holdup distribution at specific levels (/d p ) in the completely non-prewetted packed bed (f = 0), U l = m/s (U l (PS 1 ) = m/s); U g = 0.05 m/s; Point liquid distributor (PS 1 ); Overall liquid holdup = Generated pseudo-gaussian distribution of porosity under three constraints: (1) ε 0 = 0.36; (2) Longitudinally averaged radial porosity profile (white filled circles) reported by Stephenson and Stewart (1986). (D r = 7.6 cm, d p = cm, Section size = 0.05R = 0.19 cm). (a)-contour plot; (b)-radial profiles; (c)-histogram (standard deviation of porosity, σ B = 12% ε 0 ) xv

17 5-2. Block, sections and cells in CFDLIB for packed bed modeling: (a) physical block, (b) logical block consists of a number of sections, (c) a section consists of a cell or a number of cells Comparison of kl values from different models [Ug = 6 cm/s]: A- Two-fluid interaction model (Attou et al., 1999a); H- Single slit model (Holub et al., 1992); SC- Relative permeability model (Saez and Carbonell, 1985) Effect of liquid superficial mass velocity on liquid holdup (hp) and particle external wetting efficiency (wt) at a gas superficial velocity of 6 cm/s. Holub model (Single slit model, see Holub et al., 1992); S & C model (Relative permeability model, see Saez and Carbonell, 1985). Particle external wetting efficiency values (wt) were calculated by the correlation of Al-Dahhan and Dukovic (1995). wt-s & C model means the pressure-drop value used in calculating wt value was from S & C model; wt-holub model means the pressure-drop value used in calculating the wt value was from Holub model Comparison of the calculated capillary pressure values from two different expressions, Eq 5-17a and 5-17b for air-water system (d p = 0.003m; θ s =0.63) Simulated liquid upflow velocity component, V x contour (a) and profiles (b) using mesh1 (10 20) and mesh2 (20 40) at a superficial velocity of 10 cm/s Simulated liquid upflow velocity component, V z contour (a) and profiles (b) using mesh1 (10 20) and mesh2 (20 40) at a superficial velocity of 10 cm/s Initial solid volume fraction distribution at section-discretization (section size = 1.0 cm) for gas-liquid cocurrent downflow simulation (zoom: x = 0 ~ 4; z = 4 ~ 8) xvi

18 5-9. Gas phase holdup contours and gas interstitial velocity vectors in the 4 4 cm zone marked in Figure 5-9 (a) cell size =1.0 cm; (b) cell size = 0.5 cm; (c) cell size = 0.25 cm (zoom: x = 0 ~ 4; z = 4 ~ 8) Effect of the mesh sizes (a, b, c) on the cell-scale gas holdup values a. Generated sectional porosities (RN1) plotted in the radial direction and longitudinal averaged radial porosity profile of Stephenson & Stewart (1986). Statistics of the RN1 distribution are given in Table b. Generated sectional porosities (RN2) plotted in the radial direction and longitudinally averaged radial porosity profile of Stephenson & Stewart (1986). Statistics of the RN1 distribution are given in Table a. Comparison of longitudinally averaged radial velocity profiles at different Reynolds numbers and experimental data of Stephenson & Stewart (1986) b. Comparison of longitudinally averaged radial velocity profiles at different Reynolds numbers and experimental data of Stephenson & Stewart (1986). Statistics of the RN2 bed are available in Table 6.1; PA bed: sectional porosities are only varying in the radial direction) a. Frequency distribution of axial interstitial velocity (Re = 5): RN1-CFD simulation based on random porosity set 1; RN2-CFD simulation based on random porosity set 2; Exp. Experimental data reported by Stephenson and Stewart (1986) b. Frequency distribution of axial interstitial velocity (Re = 280): RN1-CFD simulation based on random porosity 1 (ε: std/µ = /0.3534; V x : std/µ =1.879/0.2034; V z : std/µ = 3.864/7.0915); RN2-CFD simulation based on random porosity 2 (ε: std/µ = /0.3534; V x : std/µ =1.879/0.2034; V z : std/µ = 3.864/7.0915). Exp. Experimental data reported by Stephenson and Stewart (1986) xvii

19 6-4a. Discretization of the radial porosity profile into sectional porosity values (d p = 3mm): From the wall to the center: sectional mean = 0.411, 0.363, 0.363, 0.365, 0.362, 0.362, 0.363,.364, 0.362, 0.366; sectional std/mean = 20%, 15%, 10%, 10%, 10%, 10%, 10%, 10%, 10%, 10% b. Solid volume-fraction distribution generated based on the data in Table 2 in a pilot scale packed bed Simulated phase volume-fraction distribution at liquid superficial velocity of 0.45 cm/s and gas superficial velocity of 22 cm/s in a pilot-scale packed bed. (a) liquid; (b) gas Comparison of CFD k-fluid model and other phenomenological models prediction of liquid saturation with the experimental data of Szady and Sundaresan (1991) (gas superficial velocity is 22 cm/s). The f values used in CFD modeling are evaluated by the particle external wetting efficiency correlation by Al-Dahhan and Dudukovic (1995). exp- use measured static liquid holdup (0.022) in Saez & Carbonell model; cal- use the correlation-estimated value (0.05) in Saez & Carbonell model Comparison of CFD k-fluid model and phenomenological models prediction of pressure gradient with the experimental data of Szady and Sundaresan (1991) (gas superficial velocity is 22 cm/s) The f values used in CFD modeling are evaluated by the particle external wetting efficiency correlation by Al-Dahhan and Dudukovic (1995). exp- use measured static liquid holdup (0.022) in Saez & Carbonell model; cal- use the correlation-estimated static liquid holdup (0.05) in Saez & Carbonell model a. Comparison of liquid holdup distribution under nonuniform (left) and uniform (right) feed conditions at U l0 = cm/s, U g0 = 22 cm/s xviii

20 6-8b. Comparison of gas holdup contour and gas interstial velocity vector plot under nonuniform (left) and uniform (right) feed conditions at U l0 = cm/s, U g0 = 22 cm/s Bench-scale cylindrical packed-bed and its porosity description (a) computer generated 2D axisymmetric solid volume fraction distribution; (b) radial porosity profile, ε (r); (c) axial porosity profile, ε (z) Contours of packed bed structure and corresponding hydrodynamic parameters: (a) solid holdup-the1; (b) liquid holdup-the2; (c) gas holdup-the3; (d) axial liquid interstitial velocity-v2; (e) axial gas interstitial velocity -V3; (f) pressure at U g0 = 6 cm/s and U l0 = 0.3 m/s at steady state operation a.Relative interstitial velocity profiles of the gas and liquid phase, profiles for porosity, gas and liquid volume fraction in the radial direction at low flow rates (U l0 = 0.05 cm/s; U g0 = 6.0 cm/s) b.Relative interstitial velocity profiles of the gas and liquid phase, profile for porosity, gas and liquid volume fraction in the radial direction at low flow rates (U l0 = 1.0 cm/s; U g0 = 12.0 cm/s) a. Relative interstitial velocity profiles of the gas and liquid phase, porosity profile in the radial direction at low flow rates (U l0 = 0.05 cm/s; U g0 = 6.0 cm/s) b. Relative interstitial velocity profiles of the gas and liquid phase, porosity profiles in the radial direction at low flow rates (U l0 = 1.0 cm/s; U g0 = 12.0 cm/s) Histogram of the relative interstitial velocities of the gas and liquid phase at low flow rates (1: U l0 = 0.05 cm/s; U g0 = 6.0 cm/s) and high flow rates (2: U l0 = 1.0 cm/s; U g0 = 12.0 cm/s) xix

21 7-6. Liquid holdup distribution in a periodic liquid inflow model (15s-on and 45s-off) (left) and steady sate model (right) in 1-inch cylindrical packed bed at U g0 =6 cm/s and U l0 =0.3 cm/s (a) Solid volume-fraction (THE1 = Bed Porosity) distribution in the model 2D rectangular bed; (b) liquid holdup (THE2) contour at steady state liquid feed; snapshot of liquid holdup (THE2) contours at (c) t =15s; (d) t = 25s; (e) t = 40s; (f) t = 55s from start of the liquid ON cycle (left) in comparison with steady state holdup contours (right) Comparison of cross sectional liquid holdup profiles at different axial locations under steady (filled squares) and unsteady state operation ((a), = 1.8 cm; (b), = 18.9 cm; (c), = 26.1 cm); (c), = 28.8 cm) (!-15s;!-25s; "-40s; "-55s) Trickle bed and model bed with 500 cells Transverse averaged profiles of porosity (hard line), liquid holdup (square) and liquid saturation (least line) vs. longitudinal position (z) at different wetting states (a) f = 0.0; (b) f = 0.5; (c) f = 1.0 at U l = 0.3 cm/s, U g = 6.0 cm/s Distribution of gas and liquid interstitial velocity components in non-prewetted bed (f = 0) (up-2 rows plots) and in prewetted bed (f = 1) (low-2 rows plots) at U l = 0.3 cm/s, U g = 6.0 cm/s (G-gas, L-liquid) Contours of solid volume fraction (=1.0-porosity) distribution of model beds (II, III, IV) for CFDLIB Contours of CFDLIB simulated liquid volume fraction (holdup) distribution in model beds (II, III, IV) Standard deviation (S.D.) of the liquid holdup distribution from CFD simulations and from Eq (7-9) calculations vs. bed wetting factor (f) in model Bed-II, III, IV183 xx

22 7-15. Standard deviation (S.D.) of the liquid holdup distribution vs. standard deviation of the bed porosity at two wetting limits at U l = 0.3 cm/s, and U g = 6.0 cm/s Liquid holdup (filled squares) (Holub et al., 1992) and particle external wetting efficiency from correlation (empty circle) (Al-Dahhan & Dudukovic, 1995) Liquid holdup values from CFDLIB simulations (mean +/- S.D.) and Holub correlation (1992): ε B = 0.399; d p = 3 mm; U g = 0.03 m/s (hl: liquid holdup) a. Two-dimensional interconnected cell network b. Fluid superficial velocities and concentrations of species i at interface of the cell, where C 3 i, = C 4 i, due to the well-mixing a. Porosity distribution at spatial resolution of 1 cm (d p = 3 mm, L = 50 cm, D = 10 cm) b. Histogram of porosity distribution (Gaussian distribution) used in the k-fluid CFD model a. Simulated liquid superficial velocity component (U x, m/s): U l0 = m/s; U g0 = 0.06 m/s b. Simulated liquid superficial velocity component (U z, m/s): U l0 = m/s; U g0 = 0.06 m/s Computed cell scale mass transfer coefficient (k ls, cm/s): U l0 = m/s; U g0 = 0.06 m/s Cells with different inflow and outflow configurations Concentration contour of species B in the liquid phase (m = 0.0; n = 1.0; r = 1.0), C Bl, 0 = 5.4 kmol/m 3, U l0 = m/s; U g0 = 0.06 m/s xxi

23 8-7. Longitudinally averaged concentration profile of species B and liquid velocity component (U z ) profiles in the direction. (C Bl - filled circle; U z blank square; C Bl, 0 = 5.4 kmol/m 3 ; U l0 = m/s; U g0 = 0.06 m/s) Calculated concentration profiles of species B at k = 1.E-04 m 3 /kg.s, n = 1.0, m = 0.0 by (i) plug flow; (ii) ADM (De = 2.53E-04 m 2 /s calculated from Sater and Levenspiel, 1966, Pe = 5.92); (iii) Mixing-cell network model; (iv) ADM (adusted De = 1.5E-04 m 2 /s, Pe = 10). C Bl, 0 = 5.4 kmol/m 3 ; U l0 = m/s; U g0 = 0.06 m/s A-1. Contacting pattern in the trickle flow regime and bubble flow regime A-2. Effect of pressure on the flow regime in downflow packed bed using the flow chart of Larachi et al. (1993) A-3. Species concentration profiles along the reactor. D = 11.3 m; Ul = m/s; Ug = 0.06 m/s; P = 70 atm; T = 603K; C al : hydrogen; C bl : biphenyl; C cl : product A-4. Effect of operating pressure on the exit biphenyl conversion and global hydogenation rate (Re). H= 4 m, D = 11.3 m A-5. Effect of liquid superficial velocity on the exit biphenyl conversion and global hydrogenation rate (Rg). H = 4 m; D = 11.3 m A-6. Effect of liquid superficial velocity on the exit biphenyl conversion in the upflow packed bed A-7. Comparison of biphenyl conversion profile and hydrogen concentration profile in liquid phase in the up-flow and down-flow packed beds. H = 4 m; D = 4 m; U l = m/s; Ug = 0.12 m/s; P = 70 atm; T = 648K; mass transfer coefficients used: for the up-flow mode, 0.1 (ka) gl = s -1, 0.1 (ka) ls = s -1 ; for the downflow mode, 0.1 (ka) gl = s -1, 0.1 (ka) ls = s -1. (For the directly xxii

24 calculated values of mass transfer coefficients, the corresponding exit biphenyl conversion for up-flow is 72.35%, and for down-flow, 71.56%) A-8. Sensitivities of the model with respect to gas-liquid, liquid-solid mass transfer coefficients in trickle-bed reactor at 605 K, 70 atm. with flow conditions: Ug = 0.06 m/s; H=32 m; D=4 m xxiii

25 Acknowledgments This thesis is a result of over three-years of research carried out in the framework of packed-bed studies at the Chemical Reaction Engineering Laboratory (CREL), Washington University in St. Louis. The visions, ideas, models and results contained in this thesis could not have been achieved without the supports of many people inside and outside CREL. I wish to express my deep gratitude to two advisors Prof. M. H. Al-Dahhan and Prof. M. P. Dudukovic for their guidance and freedom while working towards this thesis. Their encouragements of independent thoughts made the final thesis possible. Special thanks to my co-advisor, Prof. M. P. Dudukovic for his constant patience and support, and for challenging me to improve my writing and critical reviewing skills, which made my publications possible. It has been my pleasure to work at CREL, the place has provided me great opportunities to experience the developments of various multiphase flow reactors and to learn extensively from great colleagues inside and outside CREL. My first assignment at CREL was truly memorable. The assignment was sponsored by Monsanto Company to investigate the feasibility of using high-pressure trickle-bed reactors to substantially improve the productivity of complex reaction networks. The proect brimmed with challenges, which made my first-year life in the United States full of excitant. Many enoyable hours were spent with my co-worker, M. R. Khadilkar, from whom I have learned so much in professional attitude, which made the proect successful and later on was helpful in several aspects of my thesis work. Thanks to Dr. R. Kahney, Dr. Sh. Chou, and Dr. G. Ahmed at Monsanto Company for our invaluable discussions, and for their helpful suggestions. xxiv

26 My special thanks is to Dr. R. A. Holub who initialized the flow distribution study at CREL as a part of his doctoral research, and provided me the original program of the discrete cell model (DCM), which allows me to revisit and to extend this approach. I greatly appreciate his contribution to the development of the original DCM, and his kindness in agreeing to sit on my thesis committee. I would like to thank Prof. P. A. Ramachandran for our invaluable discussions, for his helpful suggestions and for acting as my thesis committee member. Thanks also to Prof. R. A. Gardner from the Department of Mechanical Engineering for being on my thesis committee, for taking interest in my work and providing useful comments and suggestions. I wish to thank Dr. P. L. Mills at DuPont Company for his invaluable advice regarding many interesting issues in trickle-bed research. Particularly important for the progress of this work were the numerous discussions I had with my past and present colleagues at CREL. My sincere gratitude goes to Dr. M. R. Khadilkar, Dr. S. B. Kumar, Dr. J Chen and Dr. S. Roy, whose help and encouragement were invaluable throughout my stay at Washington University. I sincerely acknowledge the help and assistance offered by all the past and present members of CREL including Dr. A. Kemoun, Dr. K. Balakrishnan, G. Bhatia, J. A. Castro, P. Chen, Dr. S. Degaleesan, Dr. P. Gupta, J. Mettes, Dr. J. Lee, K. Ng, B-Ch. Ong, Dr. Y. Pan, N. Rados, A. Ramohan, Dr. Y. Wu, Dr.. u, J. ue and many others. I also wish to thank the entire Chemical Engineering Department, particularly the secretaries for their help with numerous formalities. I am also glad that I was able to enoy the three-month internship at DuPont Company sponsored by Conoco Inc. in Special acknowledgements go to Dr. Tiby Leib of DuPont Company and Dr. Harold Wright of Conoco Inc. for this great opportunity and for their professional guidance, which allowed me to gain invaluable experiences in multiphase reactors other than packed-bed reactors. I wish to express my deep gratitude to Prof. G. Gao in China, who has given me constant support in the professional development since I worked as an assistant professor at Jiangsu Institute of Petrochemical Technology (JIPT) in Thank you for xxv

27 introducing me to CREL, and thank you for numerous help you offered during the last ten years. Last, but not least, I would like to thank my wife, Feixia, who supported my decision to embark in graduate studies and fulfill my career goals, despite the significant changes it involved in our lives. She has also endured many long hours waiting for me to come home from the lab, and has provided stability to our family by taking charge of our home and our daughter s education. I thank my daughter, Sheri, for her understanding and encouragement to finish this thesis. My deep thanks to my parents, brothers and sisters in China for their prayerful supports both in my decision to go on to graduate studies and to start my new career at Conoco Inc. Yi Jiang Washington University, St. Louis December, 2000 xxvi

28 Nomenclature In Chapter 2 d sp G H L P R r the radius of the liquid channel (i.e., filament) gas superficial mass velocity, kg/m2/s length of paced bed, m liquid superficial mass velocity, kg/m2/s pressure, atm radius of packed column, m radial distance from the center, m T temperature, C V av V cross-section averaged liquid volumetric flow rate, m3/s measured liquid volumetric flow rate, m3/s In Chapters 3 & 4 a constant in Leverett s function (= 0.48 for air-water system) a, working variable (E 1 (1-ε ) 2 µ α /(ρ α ε 3 d 2 p )) b constant in Leverett s function (= for air-water system) b working variable (E 2 (1-ε )/(ε 3 d p )) D width of model bed, m (= m), 8 cells d p particle diameter, m (= m) d v equivalent diameter of particle (m) E 1, E 2 Ergun constants E v, mechanical energy dissipation in the cell th, J/s (based on V ) E v,,α energy dissipation rate of phase α in the th cell, J/s (based on V,α ) xxvii

29 E v, bed total mechanical energy dissipation rate in the bed, J/s f particle wetting factor f 1,, f 1, resistance factor, 150(1-ε ) 2 µ/(ρ ε 3 φ 2 d p 2 ) f 2, f 2, resistance factor, 1.75(1-ε ) ρ/(ε 3 φd p ) Ga α, Cell Galileo number of the α phase, gd P 3 ε 3 /(µ α 2 (1-ε ) 3 ) g i, i=x,z gravitational acceleration in the i direction, (g x = 0; g z = 9.8 m/s 2 ) H height of model bed, m (= m), 32 cells N total number of the cells (8 32 = 264) N c number of cells in each row P c pressure at the center of the cell, N/m 3 P 0 pressure, dyn/cm 2 P z pressure in the z direction, N/m 2 k p 0 pressure ( p k p 0 non-equilibrium pressure) P/ pressure drop per unit cell length, N/m 3 r R number of cells in each row ( direction) Radius of packed beds, m Re Reynolds number, V 0 d p ρ/6/(µ (1-ε B )) Re p particle Reynolds number, V 0 d P ρ/µ Re α, cell Reynolds number of the α phase, V α d P /(µ α (1-ε )) s i cell face area at a given coordinate direction i,m 2 S w, liquid saturation in cell (ε L, /ε ) T i T k T v energy dissipation rate due to the inertial term, J/s energy dissipation rate due to the kinetic term, J/s energy dissipation rate due to the viscous term, J/s U local interstitial velocity, m/s (=V / ε ) u 0 material velocity, (cm/s) u k material k interstitial velocity, cm/s (ρ k u k < α k ρ 0 u 0 >) u' k fluctuating part of material k interstitial velocity, cm/s xxviii

30 U 0 input interstitial velocity, m/s (=V 0 / ε B ) V velocity vector V, α superficial velocity of phase α in the th cell, m/s (Vol) volume of cell, m 3 (= Y for rectangular bed) V c, V V 0 volume of the cell, m 3 (=S z, ) superficial velocity in the th cell, m/s input superficial velocity, m/s,, Y size of the cell, m (in this work, 3d p = m) Greek Letters α k material indicator (=1 if material k is present; =0 otherwise) α! k material derivative ε B bed porosity (= 0.415) ε ε, α σ porosity in the th cell holdup of phase α in the th cell liquid surface tension, N/m (0.072 for water) γ i, i=x,z the angle of each axis with horizontal plane "Φ the gravitational potential, φ particle shape factor, (φ = 1 for Spherical particle) µ viscosity of fluid, Pa s ( gas: Pa s; liquid: Pa s) µ α viscosity of phase α in the bed, Pa s (µ L =1.0e-3 Pa s; µ G =1.8e-5 Pa s) θ k material k volume fraction (θ k = < α k > ) τ 0 deviatiric stress ρ density of fluid, kg/m 3 ( gas: 1.2 kg/m 3 ; liquid: 1000 kg/m 3 ) ρ k density of material k, g/cm 3 ( < α k ρ 0 >) ρ α density of phase α in the bed, kg/m 3 (ρ L =1000 kg/m 3 ; ρ G =1.2 kg/m 3 ) ψ G local gas flow dimensionless pressure drop, ψ xxix G 1 P = ρ G g

31 ψ α, dimensionless pressure-drop for phase α, (= Pα / ρ α L ) + 1 g Subscripts x coordinate for the rectangular cell or bed axial coordinate along the length of bed < > ensemble average (note: cross-sectionally average in Eq.5-15) In Chapters 5 & 6 A gl, A gs, A ks parameters defined in Table 5-3 B gl, B gs, B ks parameters defined in Table 5-3 Bo Bond number Ca Capillary number C 1,, C 2, inflow concentration of species i in -cell C 3,, outflow concentration of species i in -cell d p d e d min D r particle diameter, m equivalent diameter of particle minimum equivalent diameter of the area between three spheres in contact = ) π ( d min 0. 5 d p diameter of packed bed, m E 1, E 2 Ergun constants (E 1 = 180; E 2 = 1.8) f particle wetting factor F pressure factor F D(k-l) Drag between phases k and l g gravity, 9.81m/s 2 H height of model bed, m J J-function k r relative permeability parameter in Eq (5-17) l s size of section, m xxx

32 n time step number p, P pressure, N/m 3 P c capillary pressure, N/m 3 P G pressure in gas phase, N/m 3 P L pressure in gas phase, N/m 3 P 0 pressure, dyn/cm 2 k p 0 pressure (= r R α > ) θ < p k 0 k radial position in cylindrical coordinate, m Radius of packed beds, m Re p Reynolds number, V 0 d p ρ /µ S k S.D. saturation of phase k standard deviation t time, s THE1, 2, 3 solid, liquid and gas volume fraction u 0 material velocity, (cm/s) u k material k interstitial velocity, cm/s (ρ k u k < α k ρ 0 u 0 >) u' k fluctuating part of material k interstitial velocity, cm/s U 0 input superficial velocity, m/s (=V 0 ε B ) U 1,, U 2, inflow velocity of -cell U 3,, outflow velocity of -cell U g U g0 U l U l0 gas superficial velocity, m/s gas feed superficial velocity, m/s liquid superficial velocity, m/s liquid feed superficial velocity, m/s V 0 input interstitial velocity, m/s (=U 0 / ε B ) V g V g0 V l gas interstitial velocity, m/s gas feed interstitial velocity (= U l0 / ε B ), m/s liquid interstitial velocity, m/s xxxi

33 V l0 V x V z V kl liquid feed interstitial velocity (= U g0 / ε B ), m/s interstitial velocity component in horizontal or radial direction, cm/s interstitial velocity component in axial direction, cm/s velocity vector axial position, cm or m momentum exchange coefficient between phases k and l Greek Letters α k material indicator (=1 if material k is present; =0 otherwise) α! k material derivative ε B ε g ε l 0 ε l φ mean porosity of packed bed gas holdup liquid holdup static liquid holdup particle shape factor, (φ = 1 for Spherical particle) µ viscosity of fluid, Pa s θ k material k volume fraction (θ k = < α k > ), k = G, L, S 0 θ L static liquid holdup τ 0 deviatiric stress ρ density of fluid, kg/m 3 ρ k density of material k, g/cm 3 ( < α k ρ 0 >) ρ 0 material density, kg/m 3 ρ g density of gas phase, kg/m 3 ρ l density of liquid phase, kg/m 3 σ s σ B surface tension standard deviation of porosity distribution < > ensemble average divergence xxxii

34 ϕ angular coordinate In Chapters 7 d p particle diameter, m E 1, E 2 Ergun constants (E 1 = 180; E 2 = 1.8) f fractional wetting value f (x ) probability density function F D Drag force g gravity, cm/s 2 P 0 pressure, dyn/cm 2 S.D. standard deviation t time, s V 2 V 3 Vr interstitial liquid velocity components, cm/s interstitial gas velocity components, cm/s superficial relative velocity based on gas flow, as defined in Eq (7d), cm/s V x, V z interstitial velocity components, cm/s u 0 material velocity, (cm/s) u kl slip interstitial velocity between phase k and phase l, cm/s u k u' k U 0 U l0 U g0 x kl x z material k interstitial velocity vector, cm/s fluctuating part of k interstitial velocity vector, cm/s input superficial velocity, cm/s liquid superficial velocity, cm/s gas superficial velocity, cm/s horizontal position in x-z coordinate momentum exchange coefficient between phase k & l variable of system axial position in x-z coordinate xxxiii

35 Greek Letters α 1, α 2, α 3 parameters α k material indicator (=1 if k is present; =0 otherwise) α! k material derivative ε B ε ε(r) ε(z) bed porosity section porosity longitudinally averaged radial porosity profile cross-section averaged porosity profile θ k material k volume fraction (θ k = < α k > ) τ 0 deviatiric stress ρ density of fluid, kg/m 3 ( gas: 1.2; liquid: 1000) ρ k density of material k, g/cm 3 ( < α k ρ 0 >) µ mean value µ α viscosity of phase α σ s σ B σ l γ 1 γ 2 surface tension standard deviation of porosity distribution standard deviation of liquid holdup Skewness of statistical data Kurtosis of statistical data < > ensemble averaged In Chapter 8 a 0 a basis for cell cross-section area, m 2 a k k interface area of cell, m 2 a L gas-liquid mass transfer area per unit cell volume, m 2 /m 3 C Ag,0 concentration of A in the feed gas, kmol/m3 C Ag,k concentration of A in the gas phase enter the cell, kmol/m 3 C Ag,out concentration of A in the gas phase leaving the cell, kmol/m 3 xxxiv

36 C Al,k concentration of A in the liquid phase enter the cell, kmol/m 3 C Al,out concentration of A in the liquid phase leaving the cell, kmol/m 3 C As concentration of A at particle surface, kmol/m 3 C Bl,0 concentration of B in the feed gas, kmol/m3 C Bl,k concentration of B in the liquid phase enter the cell, kmol/m 3 C Bl,out concentration of B in the liquid phase leaving the cell, kmol/m 3 C Bs concentration of B at particle surface, kmol/m 3 D e effective intraparticle diffusivity of the species, m 2 /s H A k r Henry's law solubility coefficient of A, A g /A l reaction-rate constant [m 3 /kg.s][m 3 /kmol] m+n-1 (i.e., 1.e-4) k* dimensionless rate constant k l k g k s K L liquid-film mass transfer coefficient, m/s gas-film mass transfer coefficient, m/s solid-film mass transfer coefficient, m/s overall gas-liquid mass transfer coefficient, m/s, defined as 1 = 1 ( K La L ) A ( kl a L ) A H A ( k g al ) A S p external surface area of a catalyst particle, m 2 t time, s U g U l superficial gas velocity, m/s superficial liquid velocity, m/s V c volume of cell, m 3 V p volume of a catalyst particle, m 3 x Al x Bl x Bs y Ag y As + dimensionless concentration of A in liquid phase dimensionless concentration of B in liquid phase dimensionless concentration of B at particle surface dimensionless concentration of A in gas phase dimensionless concentration of A at particle surface 1 xxxv

37 Greek Letters α A, α B dimensionless gas-liquid mass transfer coefficients defined by Eq (6) β A parameter defined as Eq (6) ε void fraction in the th cell η catalyst effectiveness factor, defined as η = φ tanh3 φ 3 φ η CE γ catalyst particle wetting efficiency stoichiometric coefficient of A in the reaction γ A, γ B parameters defined as Eq (6) ρ density of the catalyst particle, kg/m3 (=ρ p = 2500 kg/m 3 ) C I, V P m n φ Thiele modulus, defined as φ = ρpka, IBI, 2 Dec, r() c dc SP Ω Reaction rate per unit volume of catalyst particle, kmol/m3.s In Appendix C i,l, c i,l, D EL,i concentration of species i in liquid phase dimensionless concentration of species i in liquid diffusivity of species i in liquid phase G gas mass superficial velocity, kg/m 2 /s U g gas superficial velocity, m/s u SL, U l liquid superficial velocity, m/s k i rate constant (ka) gl gas-liquid mass transfer coefficient. 1/s (ka) ls liquid-solid mass transfer coefficient. 1/s L liquid superficial velocity, kg/m 2 /s T temperature, K P operating pressure, atm Pe Péclet number Q g gas volumetric flow rate, m 3 /s xxxvi

38 Q l liquid volumetric flow rate, m 3 /s r i reaction rate based on the species I Rv global hydrogenation rate, mol/s/m 3 conversion of biphenyl, % Greek Letters α G,L α L,S dimensionless parameter dimensionless parameter β 1,,β 1, dimensionless parameter ξ dimensionless position Subscripts A hydrogen B biphenyl C cyclohexylbenzene D H 2 S e equilibrium i input xxxvii

39 1 Chapter 1 Introduction to Flow in Packed Beds Fluid flowing through packed grain-like material constitutes a large part of our natural environment as well as a substantial fraction of man-made processes. The heterogeneity of the packed media and its impact on global transport properties have been important subects of study in various science and engineering disciplines including hydrology, oil recover, chemical engineering, composite material processing, biology, and medicine (Bideau and Hansen, 1993; Stanek, 1994; Keller, 1996; Helmig, 1997; Ingham and Pop, 1998; many others). Although there is a broad spectrum of length scales involved in such a multidisciplinary research field, a certain similarity does exist in different disciplines in both geometrical aspects and the flow transport phenomena. Thus, the research result derived from one discipline makes a certain contribution in other related disciplines. The work described in this thesis has been carried out in the framework of chemical reaction engineering, particularly for catalytic packed-bed reactors, in which the uniformity of the flow field is important in assessing reactor performance. The understanding and prediction of the flow structure (i.e. pattern) are crucial for improving the yield of chemical reaction. Moreover, the significance of this research is due to the maor applications of packed beds in petroleum, petrochemical and biochemical processes in terms of number, capacity and annual value of products (Sie and Krishna, 1998). To systematically study the bed structure and flow phenomena in packed beds, it is necessary to clarify several important issues. For example, due to various interaction forces that exist in the system and contribute to the flow-pattern formation in packed

40 2 beds, it is essential to assess the relative importance of the interaction forces before making any approximation in the flow equations. Because of the statistical nature in both voidage structure and flow distribution in packed beds, it is reasonable to come up with a statistical methodology for describing the spatial distributions of voidage space and fluid flow. How to effectively represent the real 3-D structure in 2-D coordinates is one of the basic issues to be resolved before preaching the virtues of 2-D flow simulation. In the scale-up and scale-down of packed-bed reactors, the scale-dependency of the structureflow relation is the main concern when one uses this relationship in design practice. The maor goal of this study is to conduct systematic theoretical modeling of fluid flow in packed beds, and to assess the impact of flow distribution on the reactor performance. In addition, some experimental work has been performed to validate the model development. The first part focuses on developing the flow distribution model using either engineering approach or a computational fluid dynamics method. The second part focuses on the applications of the developed flow distribution models in the scale-up and scale-down of packed beds as well as in modeling of reactor performance. 1.1 Research Motivation Discrete Cell Model (DCM) Revisited Modeling the complex fluid dynamics in packed bed reactors is important in design and scale-up. Previous studies have developed various models to predict the single phase or two phase flow distribution in packed beds. Although they have provided the insights into fluid flow distribution at a certain level, most of them still could not capture a number of experimental observations. Furthermore, some of available models are too complicated for applications, others are too simple to reflect the actual flow textures inside packed beds. It is desirable to develop a model that can capture most of the important experimental observations and predict the results within engineering accuracy. Based on the assumption that the flow is governed by the minimum rate of total energy dissipation in the bed, Holub proposed a discrete cell model (DCM) for single phase and two phase flow distribution (Holub, 1990). In DCM, the packed bed is treated

41 3 as a number of interconnected cells, and for each cell, one can write the mechanical energy dissipation rate for all phases. The model structure is so clear that it provides potential for further upgrading of the model. For example, it has been noticed that the original DCM results were based on a small size two-dimensional packed bed due to the limitation of computational power. Moreover, the model could not distinguish the liquid flow distribution in prewetted and non-prewetted beds because of the lack of consideration of the capillary force in the model equations. Our current research overcomes those obstacles and extends the utility of DCM. The main assumption of DCM is that the flow is governed by the minimum rate of total energy dissipation in the bed. The theoretical ustification for this assumption has been provided only for linear systems, in which the fluxes and driving forces have a linear relationship, and rests on the principle of minimization of entropy production rate (Jaynes, 1980). For non-linear systems, examples can be constructed for which the 'principle of energy minimization' does not hold and, hence, that demonstrates that it is not a general 'principle' at all (Jaynes, 1980). Nevertheless, this energy minimization approach was reported to be valid for some classes of nonlinear systems such as particle flow in circulating fluidized beds (Ishii et al., 1989; Li et al., 1988, 1990). Hence, for any specific nonlinear system one needs to conduct a detailed verification study before considering 'energy minimization' as the governing principle for flow distribution (Hyre and Glicksman, 1997). Regarding the flow distribution in packed beds, it is necessary to revisit DCM by examining how well can this 'principle' be used to describe the flow. This can be done by comparing the results of the DCM either to accepted solutions of the ensemble-averaged momentum and mass conservation equations or to reliable experimental data. Unfortunately, there is very few experimental data for single phase velocity profiles inside packed beds available in the literature due to the limitations on the non-intrusive velocity measuring techniques (McGreavy et al., 1986; Stephenson and Steward, 1986; Peurrung et al., 1995). Fortunately, recent advances in understanding of multiphase flow and development of robust computation codes make the extension of this work as well as the verification of DCM predictions feasible. Such comparison study should generate a better appreciation of what the concept of minimization of the total

42 4 energy dissipation rate can and cannot do. The intent of this part of our study is not to replace the fluid dynamic simulations by the minimization of total energy dissipation rate, but to examine whether an alternative of engineering accuracy to a CFD model exists and can be used k-fluid CFD Model Development and Applications The superiority of the k-fluid CFD model in computational efficiency, particularly for large-scale packed beds, has motivated us to undertake such model development. Although there have been many studies in utilizing the CFD approach to simulate the flow pattern in fluidized beds and bubble column reactors (Kuipers and van Swaai, 1998), there is no detailed study of CFD in the multiphase packed beds because of the difficulty in incorporating the complex geometry (e.g. tortuous interstices) into the flow equations, and the difficulty in accounting for the fluid-fluid (gas-liquid) interactions in presence of complex fluid-particle (e.g., partial wetting) contacting. The intent of this part of study is to find an efficient way to solve the above problems. To be successful in scaling up multiphase packed beds, it is important to quantify the flow structures in bench-, pilot- and commercial-scale reactors by either flow measurements or reliable numerical flow simulations. Once we ensure that the k-fluid CFD model can predict the macroscopic flow structure with engineering accuracy, then we can conduct extensive numerical flow modeling in the beds of different sizes. Those flow simulation can help us to understand how the flow distribution varies with the reactor size; how the relationship of bed structure and flow pattern varies with the operating conditions Impact of Flow Distribution on Packed-Bed Performance Many phenomenological reactor models for multiphase packed beds have been developed and utilized for several decades by assuming simple flow patterns without solving the momentum balances (El-Hisnawi et al., 1982). To account for the non-ideal flow patterns in reactor modeling, efforts made in the literatures include a two-region cell model (Sims et al., 1994), a cross-flow model (Tsamatsoulis and Papaynnakos, 1995),

43 5 and other models based on liquid flow maldistribution (Funk et al., 1990), the stagnant liquid zones (Raashekharam et al., 1998), and the one-dimensional variations of gas and liquid velocities along the reactor (Khadilkar, 1998) etc. The ways used to incorporate the multiphase flow pattern, however, do not make these models suitable as a diagnostic tool for operating units, which are normally operated under conditions not amenable to the model assumptions. In principle, the performance of multiphase reactors can be predicted by solving the conservation equations for mass, momentum and (thermal) energy in combination with the constitutive equations for species transport, chemical reaction and phase transition. However, because of the incomplete understanding of the physics, plus the nature of the equations- highly coupled and nonlinear, it is difficult to obtain the complete solutions unless one has reliable physical models, advanced numerical algorithms and sufficient computational power. Although the full probability density function (PDF) method has some promises in solving the single-phase reactive flow (Fox, 1996), for most multiphase reactive flows, the challenge exists in both numerical technique and physical understanding. The use of direct numerical simulation (DNS) on single particle and single void scale in micro-flow modeling requires complete characterization of solids boundaries and voids configuration, which is obviously undoable for a massive packed bed. To focus on the macroscale flow distribution, a statistical method for implementing the porosity distribution has potential for success in multiphase flow modeling using ensemble-averaged equations of motion (i.e., k-fluid computational fluid dynamics, CFD, model) because both porosity and flow structures are statistical in nature. The intent of this part of our study is to utilize the simulated flow distribution results by the k-fluid CFD or DCM to assess the impact of flow pattern on the reactor performance for a given kinetics. For the systems in which the flow patterns are not substantially affected by reaction, the sequential modeling of flow and reaction(s) is a good alternative for quick evaluation of the reactor performance based on flow considerations. Such sequential modeling scheme of flow and reaction allows one to deal with the packed beds with a complex flow pattern and complicated chemical kinetics. The modeling results provide

44 6 the map of both flow and species concentration distribution in packed beds, which are particularly valuable for the diagnostic analysis of the operating commercial reactors. 1.2 Research Obectives The overall obectives of this study are outlined below. Details of the implementation of each are discussed in Chapter 2 through Chapter DCM Revisited The obectives of this study are: To analyze the contribution of each energy dissipation term in DCM equations, and to compare its predictions of single phase flow with the k-fluid CFD simulation results and available experimental data, and to fully assess the applicability of this engineering approach for single phase flow modeling in packed beds. To extend the DCM for predictions of gas and liquid two phase flow distributions in trickle beds. The developed model has the ability to take into account the state of particle external wetting and the distributor effects. It is desire to compare the predictions of the extended DCM with the k-fluid CFD simulation and other independent modeling results, and to reach same conclusion regarding the application of this model k-fluid CFD Model for Packed Beds The obectives of this study are: To analyze the importance of each basic force in ensemble-averaged conservation equations of mass and momentum, such as inertial force term, Reynolds stress term, gravity term and capillary force term. To develop a statistical method to implement the porosity distribution in the k- fluid model simulation.

45 7 To establish the way to compute the momentum exchange coefficients used in the k-fluid CFD model. To compare the predictions of the k-fluid CFD model with the experimental data available in the literature Applications of k-fluid CFD Model The obectives of this study are: To perform a series of numerical flow simulations using the k-fluid CFD model to quantify the relationship between the bed structure, flow distribution and particle external wetting at different operating conditions. To perform the flow simulation in bench-scale packed beds in order to provide a basis for interpretation of irregular experimental data Impact of Flow Distribution on Reactor Performance The obective of this study is to develop a methodology for modeling flow and reaction in multiphase packed-bed reactors. The model can provide the mapping information on both flow and species concentration in the entire reactor domain. Based on the concept of the mixingcell network, a combinational modeling scheme is to be developed in which the k-fluid CFD model or the DCM model can provide the detail flow distribution information, then the mixing-cell network model can provide the distribution information of species concentration for a given kinetics. 1.3 Thesis Structure To be consistent with the thesis format requirement, and also for convenience of the reader, the thesis has been organized in the following manner: each Chapter is written as a full manuscript which consists of (i) introduction of the topics, (ii) results and discussion, and (iii) conclusions. In the course of flow distribution study, there have been many opportunities to work on other aspects of packed-bed reactors, which are relevant to the flow pattern directly or indirectly. One of such typical research accomplishments

46 8 is documented as Appendix. The comparison study of the commercial scale trickle-bed reactor performance under upflow and downflow conditions is performed, in which the effect of large-scale flow pattern on the reactor performance is discussed. Although the Chapters are independent as topics, they are structured in a certain logical sequence. The main body of this thesis consists of Chapter 2 to Chapter 8, in which the experimental and modeling results are presented and discussed in detail. Chapter 1 is a general introduction in which the general motivation and the overall obectives of this study are given. In Chapter 9, the thesis accomplishments are summarized; the issues that deserve future research efforts are highlighted. To obtain a better physical understanding of particle external wetting and its impact on the formation of liquid structures in packed beds, in Chapter 2, we present some experimental observations of liquid distribution in pseudo two-dimensional and real three-dimensional packed beds, which provide a physical background for developing the flow distribution model. In Chapters 3 and 4, an engineering approach, discrete cell mode (DCM) (Holub, 1990) has been revisited and extended. The main assumption used in DCM, that the flow distribution is governed by the minimum total energy dissipation rate in packed beds, has been examined in detail. In Chapter 3 we focus on the single-phase flow system, whereas in Chapter 4 we deal with the two-phase flow system such as that in trickle-bed reactors. The flow distribution results obtained by the DCM approach have been compared with the solutions of ensemble-averaged Naiver-Stokes equations (i.e., k-fluid CFD model), and with the experimental results. A reasonable agreement is achieved for engineering applications. In Chapters 5 and 6, we focus on the development of the k-fluid model in the framework of computational fluid dynamics (CFD) for the prediction of macroscopic flow pattern in packed beds. A statistical method is developed for implementing the complex bed structure in the k-fluid CFD model. Several important issues in using the k- fluid model for packed beds are discussed in Chapter 5. The numerical results of the k- fluid CFD model at steady state and unsteady state feed conditions are presented in

47 9 Chapter 6. The comparison of the model predictions with experimental data is also provided. In Chapter 7, two case studies demonstrate the applications of the k-fluid CFD simulations in scale-down and scale-up of multiphase flow packed beds. The first case study presents the multiphase flow simulation in bench-scale packed beds for the first time. The simulation results provide valuable insights on the distributions of velocity, pressure, and phase holdup, which are useful in interpreting scattered experimental data. In the second case study, the quantitative relationships among bed structure, operating condition, particle external wetting, and the resultant flow distribution are developed in a statistical manner through a series of the k-fluid CFD simulations. This work revealed that the contribution of capillary forces to liquid maldistribution is significant in the case of partial particle wetting; however, the effect of porosity non-uniformity in packed beds can be reduced if the particles are prewetted well. In Chapter 8, we focus on the impact of flow distribution on packed-bed performance. A combinational modeling strategy of flow and reaction in packed-bed reactors has been developed based on the concept of the mixing-cell network. Such a methodology provides an efficient way to utilize the flow information obtained by DCM or CFD simulation in the prediction of reactor performance. The spatial mapping information on the bulk flow and the species concentration are valuable in the diagnostic analysis of the operating commercial units.

48 10 Chapter 2 Experimental Observations: Liquid Flow Distribution in Trickle Beds 2.1 Introduction Trickle-bed reactors with cocurrent gas and liquid downflow have found various applications in petroleum, petrochemical and biochemical industries. Gas and liquid distribution play an important role in determining the reactor performance. To develop an advanced model for the design of new units and for the diagnostic analysis of operating units, the bed structure and flow distribution need to be incorporated into the reactor model. In fact, several researchers have shown that the prediction of packed bed performance can be improved if the nonuniformities of the bed structure are properly accounted for (Lerou and Froment, 1977; Delmas and Froment, 1988; Daszkowski and Eigenberger, 1992). Because of the complex structure of the interstitial space between particles plus the complicated interactions between particles and fluids, reliable flow distribution modeling in trickle beds has been the challenging subect for several decades. In the literature, certain approximations have been made in solving the flow equations, particularly for flow in a commercial scale packed bed. For example, the k-fluid model, based on the volume-averaged or ensemble-averaged Navier-Stokes equations, has shown promise in dealing with the flow in packed beds, because it can avoid solving for the tortuous particle boundaries, and ust treats the gas, liquid and even the solid as continuous but penetrating phases. In fact, such a model has been developed not only for

49 11 one-dimensional (1-D) trickle beds to predict the global hydrodynamics and flow regime transition (Attou et al., 1999; Attou and Ferschneider, 2000), but also for simulating the flow in 2-D beds (Anderson and Sapre, 1988). It has been realized that the progress in using the k-fluid model in packed beds relies on better closures for momentum exchange coefficients and efficient ways for implementing the porosity distribution information into the model. To establish reliable formulae for computing the various momentum exchange coefficients and for describing the porosity distribution, well-designed fine-scale experimental studies using advanced techniques are essential. The non-invasive monitoring of macroscopic flow pattern provides the physical mirror for validation of large-scale flow modeling. Moreover, in the beginning of model development, experiments even using conventional techniques can still offer useful evidence for ustifying the importance of each term in the model equations. The motivation for the experimental study presented in this Chapter is to obtain some experimental evidence of the effects of particle wetting and inflow-operating mode on the liquid distribution in packed beds. The indirect flow visualization techniques such as radioactive computertomography (CT), magnetic resonance imaging (MRI) and electric capacity tomography (ECT) have shown the capabilities of obtaining the spatial distribution of multiphase flows at certain resolution (Lutran et al., 1991; Kantzas, 1994; Toye et al., 1997; Chaouki et al., 1997; Sederman et al., 1997; Reinecke et al., 1998). Nevertheless, the direct flow visualization, such as digital imaging technique, at some cases, are valuable for studying the parameter dependence and monitoring the course of flow development. By zoomingin and out the region of interest (ROI), one can obtain the information at different scale. In this Chapter, we present some experimental observations of liquid distribution in a bench-scale pseudo two-dimensional (2-D) rectangular packed-bed and in a pilotscale three-dimensional (3-D) cylindrical packed-column. The flow modeling results based on the same dimensions of these two packed beds are given in Chapters 4 and 6. A Charge-Couple-Device (CCD) video camera was used to visualize the liquid texture in the 2-D transparent packed bed at both bed scale and particle scale by simply

50 12 zooming-in and out the ROI. To track the development of the finger-type liquid texture after introducing the liquid into the bed, both the bed scale and particle scale images were recorded during the most of liquid flow development. The particle prewetting effects were confirmed in both 2-D and 3-D packed beds. The following issues have been targeted: In the 2-D bench-scale rectangular packed-bed, we focus on Particle prewetting effect at particle and bed scales on liquid flow distribution Liquid texture development at trickling flow condition Causes of liquid filament formation In the 3-D pilot-scale cylindrical packed-bed, we focus on Particle prewetting effect at particle and bed scales on liquid flow distribution Liquid distributor effect on the bed scale liquid distribution Unsteady state liquid feed on the bed scale liquid distribution 2.2 Experiment I: 2-D Liquid Flow Imaging D Packed Bed and CCD Setup A 2-D rectangular packed-bed was made of Plexiglas with a height of 30 cm, a width of 7.2 cm and a thickness of 1.25 cm as shown in Figure 2-1a. The schematic diagram of the experimental setup for visualizing the liquid flow using a computer-based CCD image technique is depicted as Figure 2-1b. The packing consists of glass beads of 3mm in diameter. The packing height of the bed is about 27 cm. This setup can be operated with gas and liquid co-current flows and with different liquid distributors (e.g., single-point liquid inlet and multi-point liquid inlets). The gas feed is uniform during all experimental runs. Working fluids are air and colored water (in black) at room temperature (~25 C). Pressure drops with and without including the collector plate, are

51 13 measured by manometers. The video imaging was taken during the experiment running from the front side and the rear side of the bed, and then was processed after that Imaging and Processing In general, the computer-based CCD video imaging system consists of a computer with a plug-in image acquisition board (e.g., DT-3851 from Data Translation Co.; IMAQ PCI-1408 from National Instruments Co.) and a CCD video camera. At times, different lighting apparatus may be necessary to condition the image for easier image processing or to illuminate the scene under low-level light conditions. Technically, one may think of lighting as analogous to signal conditioning. If the scene is properly lighted (conditioned) then the image is easier to process. The image acquisition board uses a high-speed analog to digital converter to digitize the incoming video signal. With the emergence of multimedia, image acquisition hardware has become less expensive and more powerful. Application software, which may be a graphical or text-based language, controls the image board as well as processes and displays the incoming video. The system applied in this study includes a Sony CCD TRP-279 video camera connected to the DT-3851 image acquisition hardware, an adustable lighting background, and a high-speed Pentium-II computer installed with Global Lab image processing software. This setup allows showing the acquisition of full live liquid flow video during single experiment run.

52 14 Liquid Uniform Inlet Liquid Gas 2D Rectangular Bed I Liquid Gas Liquid Point Source Background Monito Compute Pressure drop I Pressure drop II Packing with 3 mm spheres mm CCD Camera Gas - Liquid Separator Gas vent Liquid out 2D Bed with two phase flow inlet and outlet 72 mm Figure 1. 2D packed-bed I Figure 2-1a Schematic diagram of 2-D rectangular packed bed. Figure 2-1b Experimental setup using CCD video camera imaging technique Liquid Flow Imaging A set of experiments was conducted at different liquid superficial mass velocities, in the range of 0.5 to 11.0 kg/m 2 /s. To confirm the particle prewetting effect on liquid texture, the experiments were performed in nonprewetted bed and prewetted bed at the same flow conditions. The particle scale flow images were obtained by zooming in the region of interest (ROI). This allows us to see how the particle scale liquid flow pattern varies with time and with the superficial liquid feed velocity Experimental Results and Discussion Non-prewetted bed (dry bed) The experiments were run in non-prewetted beds at the liquid mass velocity range from 0.74 to kg/m 2 /s. Figure 2-2 shows the effect of liquid superficial velocity on liquid channel ling (rivulet) flow as single-point liquid inlet was used. Clearly, the liquid

53 15 from the distributor follows the previously established flow path without making any new rivulet while the superficial liquid velocity increases. The radius of the liquid channel or rivulet (i.e., filament), d sp, increases in the increase of liquid superficial mass velocity, L. The relationship between L and d sp, however, does not follow the square rule (i.e., L d 2 sp ). When liquid superficial velocity increases from 0.74 to 3.52 kg/m2/s as shown in Figure 2-2, the liquid saturation of the channel gradually increases. When the liquid superficial velocity is doubled to 7.04 kg/m2/s, an apparent spreading of the liquid filament takes place since the center of the filament has already saturated the voidage space. If one looks at the specific region by zooming on ROI in Figure 2-2, as one can see from Figure 2-3, the liquid easily occupies the interstitial voidage without radially spreading in the non-prewetted bed. A similar liquid flow pattern was reported in Ravindra et al. (1997a) for non-prewetted beds as given in Figure 2-4, except for one difference. In Figure 2-4 the radius of the liquid rivulet increases along the liquid path through the packing from the top to the bottom. However, in our experiments, the radius of the liquid rivulet decreases along the liquid path downwards in the vertical direction. Such a difference in the rivulet paths could be caused by the different particle sizes [1.6 mm in Ravindra et al (1997a); 3 mm in this work], or perhaps by the different surface tension of the liquid due to the different color additives used [organic material in Ravindra et al. (1997a); the black inorganic color material in this work], and by the different diameters of the liquid inlet tubes. By setting the ROI at the top layer and the bottom layer, it has been observed that the relatively large radius rivulet at the top of the bed is mainly caused by the liquid inlet et, as shown in Figure 2-5(a). At high mean irrigation rates the packing immediately below the point inlet is not only completely filled by the liquid, but, part of the liquid actually can not penetrate into the void available below the mouth of the nozzle and spreads radially over the top surface of the bed. The narrow et thus effectively transforms into a disc whose radius depends on the mean irrigation rate and the geometry of the packing. A similar experimental observation has also been reported and analyzed by Stanek (1977) in which an attempt has been made to calculate the radius of the disc distributor obtained from the central et by using the modified Ergun equation. Once this disc type of initial liquid distribution is formed, the

54 16 liquid channel keeps flowing through the bed, which is not exactly straight except at very high liquid irrigation rate. The liquid saturation at bottom of the bed (see Figure 2-5b) is obviously higher than that at the top of the bed (see Figure 2-5a) due to the surface tension effect. The radial liquid spread decreases along the flow path from the top to the bottom and the liquid droplet or channel becomes more filament type. Different trends in liquid rivulet path observed in Ravindra et al (1997a) as shown in Figure 2-4 could be due to the small particle size (1.6 mm). The effect of liquid superficial mass velocity on the radius of the liquid rivulet is also clear: the higher the liquid irrigation rate is, the larger the rivulet radius is as seen in Figure Prewetted bed (wet bed) Before these experiments were started, the bed was prewetted by flooding it with clear (non-colored) water. The bed was then allowed to drain until no liquid dropped out. Figure 2-6 shows the steady state liquid distribution in the 2-D prewetted bed at different liquid superficial mass velocities with single liquid inlet. Liquid textures become complicated, and more liquid spreading and more particle wetting is observed. Figure 2-7 shows how the liquid texture is developed after starting the liquid irrigation. Apparently, while the liquid follows the established paths, the new liquid paths are also formed, and eventually, a tree-type steady state flow textures are generated after a certain time period Comparison of liquid flow in non-prewetted and prewetted beds The significant difference in liquid flow texture is clearly shown in Figure 2-8. One can further examine the intensity profiles of the two images at a specific axial position, 6 cm away from the top (see Figures 2-9 and 2-10). Clearly, more liquid spreading occurs in the prewetted bed whereas liquid rivulet flow is the dominant flow pattern. The impact of such a difference in liquid textures on the performance of tricklebed reactor has been experimentally demonstrated by Ravindra et al (1997b) through the oxidation of sulfur dioxide with the active carbon particles as catalyst. It was found that the reaction took a long time to reach the steady state in the non-prewetted bed, and the

55 17 global reaction rate was lower than that in the prewetted bed at the same liquid superficial mass velocity. So far, the presented experimental work is limited to the bench-scale 2-D rectangular bed at steady state flow condition. It is believed that the liquid flow distribution in a 3-D cylindrical column is different from those obtained in a 2-D rectangular bed. As one can see from an image of particle packing in the 2-D bed (see Figure 2-11), by counting the particles, it was found that there is no significant porosity difference between the central region and the wall regions because most of the particle confinement arises due to the front and rear walls of the bed. In other words, such 2-D packed bed can be a representation of the packing zone with relatively uniform porosity at a scale of two or three particle diameters. Hence, the liquid flow distribution observed in such a 2D bed presents the flow situation inside large scale packed beds. To examine the similar parameter effects and unsteady state operation in a 3-D column, we conducted flow experiments using exit flow measurement in a pilot scale cylindrical column packed with the same size of glass beads.

56 18 L = 0.74 kg/m 2 /s L = 1.48 kg/m 2 /s L = 3.52 kg/m 2 /s L = 7.04 kg/m 2 /s (ROI Size: 10cm 6 cm) Figures 2-2 Effect of liquid superficial mass velocity on liquid rivulet flow from single-point inlet in a non-prewetted packed bed: the radius of the liquid rivulet increases with the liquid superficial mass velocity. 18

57 19 L = 0.74 kg/m 2 /s L = 3.52 kg/m 2 /s L = 7.04 kg/m 2 /s (a) (b) (c) Figure 2-3. Effect of liquid irrigation rates on the local radial spreading of the liquid rivulet at a point source inlet in the non-prewetted bed (ROI Size: 3 cm 2 cm). H = 0 H = 0 H = 12 H = 12 L = 1 kg/m 2.s; G = 0.05 kg/m 2.s Figure 2-4. Cross-sectional liquid distribution in a 3D-rectanglar non-prewetted bed of glass beads (d p = 1.6 mm). [From Ravindra et al (1997a)]. Upper left- at the top layer with single-point liquid inlet; lower left- at the layer 12 cm far from the top with singlepoint liquid inlet; upper right- at the top layer with single-line liquid inlet; lower right- at the layer 12 cm far from the top with single-line liquid inlet.

58 20 a. Top region b. Bottom region Figure 2-5. Local liquid distribution at (a) The top region and (b) the bottom region at a mass superficial velocity of 7.04 kg/m 2 /s in a non-prewetted bed. L= (kg/m 2 /s) Figure 2-6. The steady state liquid distribution in a prewetted bed at different liquid superficial mass velocities.

59 21 t = 5 s t = 11 s t = 25 s Figures 2-7. The development of finger-type liquid flow in a prewetted bed at a superficial mass velocity of 0.74 kg/m 2 /s (t: starting time, second).

60 22 a. Non-prewetted bed b. prewetted bed Intensity of image non-prewetted bed prewetted bed Axial position: 6 cm down from top dp = 0.3 cm (dp) (c) Figure 2-8. Liquid flow distribution in (a) non-prewetted bed and (b) prewetted bed with single-point liquid inlet without gas flow. Part (c) shows the image intensity profiles at specific vertical position (z = 6 cm from the top) in cases (a) and (b).

61 23 Ra x 10 8 (gmol/cm 3.s) prewetted bed nonprewetted b d Time (hr) Figure 2-9. Transient behavior of reaction rates in non-prewetted and prewetted beds for oxidation of SO2 with the active carbon particles as catalyst. [Data are extracted from Ravindra et al. (1997b) at T = 25 C; P = 1 atm]. Ra x 10 8 (gmol/s.cm 3 ) 3.5 prewetted bed 3.0 nonprewetted bed L (kg/m 2.s) Figure Dependence of the global reaction rate on liquid velocity in non-prewetted and prewetted beds (uniform liquid inlet) for oxidation of SO2 with the active carbon particles as catalyst. [Data are extracted from Ravindra et al. (1997b) at T 25 C; P = 1 atm].

62 24 wall core Figure 2-11 Packing image taken from the front of the 2-D rectangular packed bed. 2.3 Experiment II: Exit Flow Measurement in 3D Bed Experiment Obectives The particle scale and bed scale liquid textures have been visualized and presented in Section 2.2 in which the particle prewetting effect on liquid distribution has been clearly demonstrated. These liquid flow observations, however, have been limited to two dimensional bench scale packed beds under steady state liquid feed condition and with single point liquid inlet because of the technique limitation (e.g. using colored liquid). Obviously, we need to confirm those liquid distribution phenomena such as particle wetting effect in a pilot scale cylindrical column. Moreover, we would like to gain some information on liquid distribution under unsteady state liquid feed such as periodic liquid feeding. Thus, the obectives for the experiments based on exit flow measurement are as follows: (i) Verify the particle prewetting effect on liquid distribution in a pilot scale cylindrical column (ii) Explore the possibility of using exit flow measurement to detect the difference in liquid distribution at steady state liquid feed and at periodic liquid feed

63 (iii) In periodic liquid feed, examine the effect of liquid cycle split ratio (i.e., the ratio of liquid ON time and liquid OFF time) on the radial liquid velocity profiles D Column Setup and Exit Flow Measurement The schematic diagram of the 3D column setup is shown in Figure The pilot scale column was made of Plexiglas with an inner diameter of 5 5/8 inch (14.3 cm) and a height of 6ft (2 m). The same size of glass beads as used in the 2D bed (i.e., 3 mm) was employed as packing. The total height of the packing was varied in the range of 2 ft to 6 ft. The fluid media used were air and water at room temperature (~25 C). Two types of liquid distributors were used: uniform and a point source. There are 182 holes with a mean diameter of 0.6 cm on the uniform distributor (about 33 % of the area of the distributor is open). Gas enters the reactor through the separate tubes located high than the level of the liquid inlet. In addition, attention has been paid to the proper design of gas distributors to avoid maldistribution problem. A uniform liquid inlet distribution was obtained when the liquid superficial velocity is beyond 1.0 kg/m 2 /s in the presence of gas flow. The performance of the liquid distributor was checked using the same procedure reported in Kouri and Sohlo (1987, 1996): by locating the distributor ust above the liquid collector and measuring the liquid distribution of the distributor by the liquid collecting annulii. At a low liquid inlet irrigation rate such as 0.5 kg/m 2 /s, it was found that the uniform liquid initial distribution could not be obtained due to the wettability of Plexglass materials. Higher liquid irrigation rate and/or gas flow can eliminate the above problem to get a uniform initial liquid inlet. For investigation of periodic operation, a solenoid valve is fixed in close proximity to the reactor inlet and is connected to the flexible timer that regulates the on/off operation of the valve within an adusted cycle. The timer handles a wide range of on/off cycles, which can range from 0.1 to 2000 seconds. For the off-time period we used a bypass to let the water go through the pump back to the feed tank. In the present study, three types of time split of the ON/OFF setting are tested such as ON/OFF (defined as SR) = 20 sec/ 40 sec.= 0.5; 30 sec./30sec.= 1.0; 40 sec./20 sec. = 2.0.

64 26 At the bottom of the column is a collector connected to 25 flexible tubes. The collector has five rings to separate the water in the radial direction and also walls to separate the liquid in these rings to get the liquid distribution in the azimuthal direction (see Figure 2-13), therefore, the liquid distribution at the bottom of the column was measured at each set of conditions by collecting the liquid flow in 25 different sections of the cross-sectional area. Most of the radial liquid distribution data is based on the six annular sections. As recommended by Kouri et al (1996), the liquid radial distribution data reported in this Chapter are expressed as dimensionless liquid velocity, V/V av, against the square of the dimensionless radius, (r/r) 2, as the square of the dimensionless radius is proportional to the area of the sampling section under consideration. The axial pressure drop data along the column were measured for a couple of experiments using a water manometer as shown in Figure Experimental Results and Discussion Since there were significant differences in liquid textures in the prewetted and non-prewetted 2D packed beds with single point liquid inlet were observed. We performed similar experiments in the 3D column with non-prewetted particles and prewetted particles, and then determined the liquid distribution from 25 individual tubes located at the bottom of the cylindrical column. As shown in Figures 2-14 and 2-15, the same conclusion about the particle prewetting effect in 3D packed column can be drawn as established for 2D column. The liquid paths in non-prewetted bed are relatively stable, even when increasing the liquid superficial mass velocity from 0.5 kg/m2/s to 10 kg/m2/s, as shown in Figure More uniform liquid distribution is found in the prewetted bed, which seems independent of liquid flow rate (see Figure 2-15). Figures 2-16 and 2-17 show the measured radial liquid velocity profiles at different cycle split ratios (SR = 0.5, 1.0 and 2.0) with a uniform liquid inlet (Fig. 2-16) and with a point liquid inlet (Fig 2-17). In the uniform liquid inlet case, the small cycle split ratio (SR = 20-on/40-off = 0.5) yields more uniform radial liquid flow distribution in the time-averaged sense. For a given time-averaged liquid superficial velocity, the

65 27 smaller SR value means higher superficial velocity during the liquid ON period. More liquid is driven by the higher momentum of liquid flow to the radial direction, and causes the maximum velocity position to move in the direction of the wall (see Figure 2-16). For the point source liquid inlet, there is no conclusive effect of SR value on liquid radial spreading as shown in Figure A B Timer Tape Manomet 6 5 air Packing: 3mmGlass Tank Collector and Measuring Pum Figure 2-12 Schematic diagram of the experimental setup for a 3D column with exit flow measurement and periodic liquid feed controller.

66 Figure 2-13 Liquid collector with 25 individual tubes located at the bottom of the packed bed. V / V av L=0.5 kg/m2.s L=1.0 kg/m2.s L=5.0 kg/m2.s L=10 kg/m2.s Tube # Figure 2-14 Liquid flow measurements in the non-prewetted bed: dimensionless liquid flow velocity data from 25 individual tubes at different liquid superficial mass velocities (H = 6 ft, G = 0.0 m/s, uniform liquid inlet).

67 29 V / Vav L=3.0 kg/m2.s L=5.0 kg/m2.s L=7.0 kg/m2.s Tube # Figure 2-15 Liquid flow measurements in the prewetted bed: dimensionless liquid flow velocity data from 25 individual tubes at different liquid superficial mass velocities (H = 6 ft, G = m/s, uniform liquid inlet).

68 30 V / V av (ON)/20(OFF) 30/30 20/ (r/r) 2 Figure Effect of time split in On/Off periodic mode on liquid flow radial profiles with uniform liquid inlet (H=2.0 ft, G=0.049 m/s, L = 1.5 kg/m2/s) with uniform liquid inlet.

69 31 V / Vav (ON)/20(OFF) 30/30 20/ (r/r) 2 Figure Effect of time split in On/Off periodic mode on liquid radial profiles with point source liquid inlet (H = 2.0 ft, G = m/s, L = 1.0 kg/m2/s). The comparison of the radial liquid flow exit profiles at steady state and periodic operations, reveals no significant difference of the model of operation. However, such exit flow measurement cannot give the flow distribution information inside the column and can only provide the time-averaged radial flow profile at exit. Moreover, many factors may contribute to such measurement, in particular, the arrangement of collecting tubes, because unequal flow resistances in individual tube can results in redistribution of

70 32 liquid flow, and change the liquid flow profiles completely. Hence, the experimental data obtained by exit flow measurement shown in this Section are inconclusive except for the effect of particle prewetting. 2.4 Conclusions The first maor goal of this study is to model multiphase flow distribution in packed bed reactors. The liquid flow observations presented in this Chapter indeed provide helpful information for the formulation of flow equations, which will be utilized in Chapters 3, 4 and 5. This information are summarized as follows: (i) The flow distribution experiments by direct liquid flow visualization in 2-D bed and by indirect exit-flow measurement in 3-D column have demonstrated the significant particle external wetting effect on the formation of liquid texture. The particle scale liquid textures indicate the apparent influence of capillary pressure on the liquid spreading. Proper implementation of the capillary force in the flow model equations is important. (ii) The liquid flow textures in the prewetted bed and non-prewetted bed are apparently different. Filament flow is dominant in the non-prewetted bed whereas the film liquid texture exists in the well-prewetted packed bed. (iii)in general, inflow distributors play an important role in flow distribution. The proper design of the liquid ets is essential in determining the liquid distribution at the top layer of the packing. (iv) Exit flow measurement is not a recommended way to compare the liquid flow distribution at steady-state liquid feed and in periodic flow mode. For the liquid ON/OFF mode with uniform liquid inlet, the cycle split ration (SR) does have an impact on the time-averaged radial liquid flow profiles. There is no effect of the SR values on the time-averaged radial liquid flow profiles when the point source liquid inlet is used. The numerical flow simulation definitely can contribute to the understanding of dynamic flow pattern at quantitative way.

71 33 Chapter 3 Discrete Cell Model Approach Revisited: I. Single Phase Flow Modeling 3.1 Introduction Gas flow through packed beds is commonly encountered in industrial applications involving mass or/and heat transfer both with and without chemical reaction. Complete understanding of the gas flow distribution in packed beds is of considerable practical importance due to its significant effect on transport and reaction rates. It was shown that reaction and radial heat transfer can only be modeled correctly if the radial nonuniformities of the bed structure are properly accounted for (Lerou and Froment, 1977; Delmas and Froment, 1988; Daszkowski and Eigenberger, 1992). Therefore, over the years, a number of studies investigated the radial variation of the axial gas velocity in packed beds. This included axial velocity measurement at various radial positions, measurement of radial porosity profiles (Morales et al., 1951; Schwartz and Smith, 1953; Benenati and Brosilow, 1962; Lerou and Froment, 1977; McGreavy et al., 1986; Stephenson and Stewart, 1986; Volkov et al., 1986; Peurrung et al., 1995; Bey and Eigenberger, 1997), and modeling of the radial variation of axial velocity (Schwartz and Smith, 1953; Stanek and Szekely, 1972; Cohen and Metzner, 1981; Johnson and Kapner, 1990; iolkowska and iolkowski, 1993; Cheng and Yuan, 1997; Bey and Eigenberger, 1997; Subagyo et al., 1998). It was noted, however, that in industrial packed beds, some

72 34 nonuniformities either due to the presence of internal structures (Bernier and Vortmeyer, 1987a, 1987b), or due to irregular gas inlet design (Szekely and Poveromo, 1975) could cause the flow not to be one dimensional and the gas velocity to vary in both radial and axial direction. Such two dimensional flow is called non-parallel flow in the literature (Stanek, 1994). Hence, for industrial applications of packed beds, it is certainly important to be able to effectively model the non-parallel gas flow. In general, three types of mathematical models have been developed for the treatment of non-parallel gas flow in packed beds. They are briefly summarized below. It should be noted that our goal here is simulation and prediction of single phase flow on a bed scale, i.e. the capture of the gas velocity profile on a scale of a couple of particles, not on the scale of the individual tortuous passages in the bed. We are not attempting to model the flow on a particle scale but to find the means for effectively computing the bed scale flow distribution provided the voidage distribution is known. 3.2 Non-Parallel Gas Flow Models Vectorized Ergun Equation Model This model is based on the assumption that a packed bed can be treated as a continuum. Therefore, it is assumed that the Ergun equation can be used in the differential, vector form as shown by Equation (3-1). ( f V) P = V f1 + 2 (3-1) The intent is to utilize the empirical Ergun equation, which is shown to hold well for overall pressure drop in macroscopic beds with unidirectional flow, for an infinitesimal length of the bed and apply it in the direction of flow. For an incompressible fluid, applying the curl operator ( ) to Equation (3-1) yields Equation (3-2), which is a vector equation containing the velocity vector V as the only dependent variable. The components of the velocity vector also have to satisfy the continuity Equation (3-3). [ ln( f + f V) ] 0 V V 1 2 = (3-2) V = 0 (3-3)

73 35 The solution for the velocity components can be obtained by solving Equations (3-2) and (3-3). A number of investigators (Stanek and Szekely, 1972, 1973, 1974; Szekely and Poverromo, 1975; Beminger and Vortmeyer, 1987a) utilized this method to model twoand three-dimensional flow in packed beds Equations of Motion Model In principle, the mass conservation (continuity equation) and momentum balance (Navier-Stokes equations) can be solved for the flowing phase provided the solid boundaries are precisely specified. Such direct numerical simulation (DNS), however, is beyond reach at present for large industrial scale packed beds (Joseph, 1998). By employing the effective viscosity as an adusting factor, iolkowska and iolkowski (1993) and Bey and Eigenberger (1997) tried to develop a mathematical model for the interstitial velocity distribution based on the Navier-Stokes equations, but porosity was only considered as a function of radial position in such models. To take into account the complex fluid-particle interactions and the multi-dimensional variation of bed voidage in packed beds, a k-fluid (interpenetrating fluid) model provides a viable alternative (Johnson et al., 1997). By ensemble averaging, the continuity and momentum equations for the flowing phase are formulated in a multi-dimensional form and the interphase interaction is described via an appropriate drag correlation. The resulting equations can be solved via packaged computational fluid dynamics codes such as CFDLIB (Kashiwa et al., 1994) Discrete Cell Model (DCM) This two-dimensional model is based on the concept that the bed may be represented by a number of interconnected discrete cells (Holub, 1990), with the bed porosity allowed to vary in two directions from cell to cell. The fluid flow is assumed to be governed by the minimum rate of total energy dissipation in the packed bed (i.e. flow follows the path of the least resistance). Ergun equation is assumed to be applicable to

74 36 each cell. Therefore, the solution for velocity at each cell interface can be achieved by solving the non-linear multi-variable minimization problem. Although the vectorized Ergun equation model (Stanek and Szekely, 1972) has provided a good description for non-parallel gas flow (1D axial flow), it is still difficult to capture the nonuniformity of flow at the cell scale (few particles). It is also cumbersome to model the flow in beds with an internal random porosity profile because of the difficulties in assigning discrete porosity values to points in a continuum. Another difficulty of this model is the inability to set a no-slip boundary conditions at the walls. The validity of the vectorized form of the Ergun equation was demonstrated only by comparison of the predicted exit velocity profile with experimental measurements. This kind of comparison is only reasonable for the parallel flow system that exhibits no effect of the packing support plate on the flow. Because of the above considerations, the discrete cell model was formulated as an alternative that may offer advantages in solving these problems. For example, the cell model is capable of capturing the non-parallel flow on a cell scale (few particles) due to the character of the cell model. The appropriate voidage can be assigned easily for each cell and the no-slip wall condition can be simulated by the extra cell method (the detail discussion will be given later). It is assumed that the Ergun equation is applicable at the cell scale. This assumption is reasonable because the original Ergun equation was derived from the experimental measurements in small laboratory-scale packed beds (Ergun, 1952). The cell size has to be small compared to the bed scale (i. e., bed diameter), to obtain the desired resolution of the bed properties and flow distribution, but large compared to the particle scale (i.e., particle diameter) in order to apply the Ergun equation (1952) to each discrete cell. The appropriate cell dimensions that satisfy these criteria were discussed by Vortmeyer and Winter (1984), who concluded that homogeneous models of packed bed heat transfer failed in beds with a tube to particle diameter ratio less than three. While this conclusion was not reached for the exact situation considered here, a minimum linear dimension of about three particle diameters for each cell can be considered appropriate (Holub, 1990).

75 37 The second assumption of DCM is that the flow is governed by the minimum rate of total energy dissipation in the bed. The theoretical ustification for this assumption has been provided only for linear systems, in which the fluxes and driving forces have a linear relationship, and rests on the principle of minimization of entropy production rate (Jaynes, 1980). For non-linear systems, examples can be constructed for which the 'principle of energy minimization' does not hold and, hence, that demonstrates that it is not a general 'principle' at all (Jaynes, 1980). Nevertheless, this energy minimization approach was reported to be valid for some classes of nonlinear systems such as particle flow in circulating fluidized beds (Ishii et al., 1989; Li et al., 1988, 1990). Hence, for any specific nonlinear system one needs to conduct a detailed verification study before considering 'energy minimization' as the governing principle for flow distribution (Hyre and Glicksman, 1997). Regarding single phase flow distribution in packed beds, it is necessary to revisit DCM by examining how well can this 'principle' be used to describe the flow. This can be done by comparing the results of the DCM to either accepted solutions of the ensemble-averaged momentum and mass conservation equations or to reliable experimental data. Unfortunately, there is very few experimental data for the velocity profiles inside packed beds available in the literature due to the limitations on the non-intrusive velocity measuring techniques (McGreavy et al., 1986; Stephenson and Steward, 1986; Peurrung et al., 1995). Thus, the obectives of this study are (i) to perform a series of numerical comparison studies of DCM predictions and CFD two-fluid model simulations, (ii) to compare the numerical results of DCM/CFD with the limited experimental data available in the literature, and finally, (iii) to reach a conclusion regarding the applicability of the minimization of energy dissipation concept in modeling single phase flow distribution in packed beds. Another motivation for this study is the fact that the concept of minimization of the rate of energy dissipation was never tested against the solution of the full set of equations of motion for a non-parallel flow system. Now, we provide such a test for flow distribution in packed beds. The results should generate a better appreciation of what the concept of minimization of the total energy dissipation rate can and cannot do.

76 Discrete Cell Model (DCM) The discrete cell model based on the minimization of energy dissipation rate presented and discussed here is adapted from the concept originally proposed by Holub (1990). Although a 2D-model bed is considered here, its extension to 3D axi-symmetric cases is readily accomplished. The 2D rectangular model bed shown in Figure 3-1 is divided into a number of cells, each of which is assumed to have uniform porosity within itself and have two fluid velocity components (V z and V x ) at each cell-interface. The porosity can vary from cell to cell. The rate of energy dissipation for each cell can then be derived from the macroscopic mechanical energy balance and results in Equation (3-4) in - coordinates for either two dimensional rectangular (2D) or three dimensional axisymmetric cylindrical (3D) situation. The differences in Equation (3-4) for 2D rectangular and 3D axial symmetric cylindrical bed are the expressions for the interface areas (S i ) and the cell volumes (V c, ). V, V, V +, V +, Figure 3-1. Model packed bed ('2D' rectangular as example) and velocity at each interface of cell. (Note that S x, equals to S x+ x, in the '2D' rectangular packed bed).

77 39 The detail derivation of DCM Equations was given in Holub (1990), and rests on the macroscopic mechanical energy balance. Here we give the main steps of these derivations. For the th cell, the rate of energy dissipation in - coordinates can be expressed by (Eq. 3-4). + Φ + = in i i i i s s V s s V P s s V s V, ρ ρ 0, 4 1, 2 = + Φ + = V out i i i E s V s V P s V V ε ε ρ ε ε ε ρ (3-4) where the superficial velocity (V ) and the corresponding energy dissipation rate for the cell (E V, ) are used. Rearrangement of Equation (3-4), by substituting the expression for the area for each cell interface, yields Equation (3-5). + = S V S V S V S V V E , ε ε ρ ( ) ( ) { } S V P S P V S V P S V P ,,,, ( ) ( ) { } S V S V S V S V Φ Φ + Φ Φ +,,,, ˆ ˆ ˆ ˆ ρ ρ ρ ρ (3-5) The difference in potential energy terms ( E P, Eq. 3-6) (shown as the last two terms in Eq.3-5) can be considered negligible ( E P 0) for gas flow at normal or low pressure since the gravitational force on the gas is very small. ( ) ( ) { } P S V S V S V S V E Φ Φ + Φ Φ =,,,, ˆ ˆ ˆ ˆ ρ ρ ρ ρ 0 (3-6) The pressure terms at each cell interface (e.g. P and P + ), however, can be considered to be equal to the pressure at the cell center plus the pressure gradient between the center and the interface multiplied by the appropriate distance. For the direction, as an example, the desired relationships can be written as follows. 2 P cz P P + = + (3-7)

78 40 2 P cz P P = + + (3-8) Rearranging Equation (3-5), by substituting Equations (3-6), (3-7) and (3-8), gives: + = S V S V S V S V E V , ε ε ρ ( ) ( ) C C S V S V P S V S V P ,,,, V P V P V P V P Vol,,,, 2 (3-9) For each cell, we can write Equation (3-10) based on the mass balance as follows ( ) ( ) 0,,,, = S V S V S V S V (3-10) Since the magnitudes of P C and P C have to be the same at the central point of the cell, the substitution of the mass balance Equation (3-10) into Equation (3-9) eliminates the central pressure term. To completely eliminate the pressure terms from Equation (3-9), the body force terms, represented by the pressure gradient, can be replaced by an appropriate drag force model which relates pressure drop to the local superficial velocity. In this work, a specially abbreviated form of the Ergun equation (Ergun, 1952) for each coordinate direction ( and ) will be used to simplify the equations. For example, for the direction, we have,, 2,, 1, V V f V f P 1 + = ρ (3-11) where the pressure loss per unit cell is caused by simultaneous viscous and kinetic energy losses. The resulting expression for calculating the energy dissipation rate per unit cell can be obtained, as shown by Equation (3-12), and the total energy dissipation rate for the entire bed is then obtained by the summation of Equation (3-12) over all the cells. ε ρ + ε ρ = S V S V S V S V 2 1 V, E ( ) C V V V f V f V V f V f,, 2 2, 2, 1,, 2, 2, 2,,

79 ( f V + f V V + f V f V V ) } 1,, 2,,, 1, +, + 2, +, +, VC, (3-12) 41 In Equation (3-12) f 1, and f 2, are Ergun coefficients (Ergun, 1952) defined as follows f 1, f 2, ( 1 ε ) 2 3 ( φd ) ε 2 150µ = (3-13) P ( 1 ε ) 1. 75ρ = (3-14) 3 ( φd ) ε P In this study, we use the 'universal values' (E 1 =150, E 2 =1.75) to calculate f 1,f and f 2,f as done by most other investigators (Vortmeyer and Schuster, 1983; Stanek, 1994; Bey and Eigenberger, 1997, etc.). Although E 1 and E 2 values can vary from macroscopic bed to bed due different structures of the packing in the bed (MacDonald, et al., 1979), this effect can be accounted for by the assignment of a non-uniform porosity distribution instead of using the average porosity value for the bed. The complete model for determining the gas flow distribution in the bed requires the minimization of the rate of total energy dissipated with the cell velocities as variables. It is a nonlinear, multivariable minimization problem (Eq.3-15) subect to mass balance constraint for each cell (Eq.3-16, based on constant fluid density assumption), and constraints for bed boundaries. The setting of cell boundary conditions reflect the internal structural nonuniformities and operating conditions. In other words, this model can predict the gas flow distributions in packed beds with various operating conditions (i.e. side gas feed) and with different internal structural nonuniformities. N [ E V,bed ] = Min[ E V, ] Min (3-15) = 1 ( S V S ) + ( V S V S ) = 0 V (3-16) The subroutine DN0ONF from the International Mathematical Statistics Library (IMSL) was used to solve this constrained nonlinear minimization problem and obtain the fluid velocity components V x and V z for each cell in the bed.

80 CFDLIB Formulation CFDLIB, a library of multiphase flow codes developed by Los Alamos National Laboratory (Kashiwa et al., 1994), has been used to obtain the results for comparison with the DCM predictions. The solution algorithm is a cell-centered finite-volume method applied to the time-dependent conservation equations (Kashiwa et al., 1994). The governing equations that serve as the basis for the CFDLIB codes are: Equation of continuity: ρ k t. ρ u =< ρ α! > (3-17) + k k k k The terms on the left hand side of Equation (3-17) constitute the rate of change in mass of phase k at a given point, and the term on the right hand side is the source term due to conversion of mass from one phase to the other. In present study this term is equal to zero since no phase change, reaction or mass transfer is considered in this cold flow modeling. Equation of momentum: ρ k u t k + ρ u k u k k = (rate of change in k th phase momentum) < ρ 0 u 0α! > (net mass exchange source of k) + k < α u ' u' > (multiphase Reynolds stress) k ρ 0 k k θ k p (accln. by the equilibration pressure) < α k τ > (accln. due to average material stress) + 0 θ ( 0 p) ( accln. by nonequilibrium pressure) ρ k k p k + g (accln. by body force) < [( p0 p) I τ 0 ] α > (momentum exchange terms) (3-18) + k

81 43 This set of equations is exact with no approximations other than the ensemble averaging used in the two fluid model approaches (Ishii, 1975). The special case of one fixed phase (the catalyst bed) has been incorporated in the code for single phase flow simulation (Kumar, 1995). In Equations (3-17) and (3-18), the mass source term is considered as zero due to absence of reaction or interphase transport. The important term is the interphase momentum exchange term, which is modeled by the choice of the appropriate drag closure. Contribution of Reynolds stress can be ignored for most cases for flow through packed beds. The detail discussions of this term will be given later. One of the advantages of CFDLIB is that there are options for specifying user defined drag forms based on each combination of the phases under consideration. In this study, the same drag force formulation as used in the Ergun equation is employed for both CFDLIB (Exchange term in Eq.3-18) and DCM simulations. This is a realistic drag correlation at the cell scale as mentioned earlier, and it has been used by many other investigators point-wise in packed beds (Vortmeyer and Schuster, 1983; Stanek, 1994; Song et al., 1998; etc.). CFDLIB code also allows the choice of velocity and pressure boundary conditions for inflow, outflow and free slip or no slip at the wall boundaries. To keep the consistency with the discrete cell approach used in DCM, the spatial discretization of the model bed is the same in both methods as the cell scale (few particles). Regarding the dependency of the flow simulation result on the grid size, one will see in Chapter 5, that the macro-scale velocities simulated by the k-fluid CFD model are grid independent. The comprehensive discussions of CFD modeling are given in Chapter 5 in which the detail implementations of bed structure and interaction forces are presented. 3.5 Modeling Results and Discussion Model Packed Bed The model bed used for this numerical study is a two dimensional packed bed with a predetermined pseudo-random porosity distribution as shown in Figure 3-2. The average porosity of this bed is 0.415, and was obtained experimentally in an identical '2D'

82 44 rectangular bed with spherical particles of 3 mm diameter (see Chapter 2). The porosity profiles in the internal region of the bed were generated by a computer program under certain constraints (Range: ~ 0.440; mean: 0.406), which is fairly close to that obtained by dumping spheres into beds (Tory et al., 1973). A relatively higher porosity of 0.44 was assigned to the wall and the support plate regions based on the typical porosity profiles reported in the literature (Benenati and Brosilow, 1962; Haughey and Beveridge, 1969). The dimensions of the model bed and of the cells as well as physical properties of the fluid (gas) are given in Table 3-1. The bed walls are considered to be impermeable in the normal direction ( direction) and allow free-slip in the parallel direction ( direction). In order to implement the no-slip boundary conditions in DCM, the ghost cell method can be used in which an extra column of cells outside the bed can be set and assigned an extremely low porosity (i.e. less than 0.01). Thus, the effect of bed wall and no slip boundary condition on gas flow could in principle be considered in this way. It should be noted that the use of DCM is not limited to spherical particles. It can be applied to any shape of particles by taking into account the particle shape factor, φ, in Equations (3-13) and (3-14).

83 45 Figure 3-2. Porosity distribution of model bed (32 cells x 8 cells): Total average porosity: 0.415; internal region: 0.36~0.44 (random distribution); wall region: 0.44; Two limits (0.36 and 0.44) correspond to the dense packing and loose packing porosity. When two obstacle plates are placed in this system, one is located at /d p of 66 (at the left side), another is at /d p of 30 (at the right side) as marked in the above figure. The width of the obstacle plate (i.e. the length that it protrudes into the bed) is half of the width of bed (4 cells). Table 3-1 Dimensions of the model bed and physical properties of the fluids in the simulations. Dimension Properties D = m; d p = m ρ G = 1.2 kg/m2; ρ L = 1000 kg/m2; H = m µ G =1.8 x 10-5 Pa.s; µ L =1.0 x 10-3 Pa.s; Analysis of the Energy Dissipation Equation As shown in Equation (3-12), there are three terms contributing to the total energy dissipation rate per unit cell: inertial loss (T i ), viscous loss (T v ), and kinetic energy loss

84 (T k ). The contribution of the gravitational potential term has been ignored for gas flow due to the low density of the fluid (this term is accounted for when liquid flow is considered, see Eq 3-5). The expressions for these three terms in cell are given below. T T T i, k, V, ρ = 2ε = 2 V 3 S V 3 ρ 3 + S + + V S V 2 2ε 3 + S ( f, V, Vx, + f 2, V + V +, + f 2, V, V, + f 2, V + V +, ) VC, ( f V + f V + f V f V ) V } 46 (3-19) 2 (3-20) = (3-21) 1,, 1, +, 1,, + 1, +, C, Energy dissipation rate, J/s 9.00E E E E E E E E E E E Ti Tk Tv cell number Figure 3-3. Contribution of each energy dissipation rate term at each cell to the total energy dissipation rate. V 0 = 0.5 m/s (gas flow without internal obstacles); Re = 28.5; Ti = J / s ; Tk = J / s ; Tv = J / s ; n= n= ( Ti + Tk + T v ) = J / s (the cell number is counted from the top left of the n= 1 bed in the direction) 256 n= 1

85 As derived earlier, the pressure drop term is substituted by T k, and T v, to eliminate the pressure term (see Equations 3-9 and 3-11). This is rigorously true only when inertial terms are zero and no source terms due to interphase transport are present in the continuity equation. Hence, we still consider the inertial terms in Equation (3-12) so as to account for flow with abruptly changing direction. The significance of this term is examined for a low density gas flow (where it is expected to be negligible), a high density liquid flow, and gas flow with internal obstacles (where it can approach in magnitudes the other terms). For a non-parallel gas flow test case (Reynolds number, Re of 28.5), Figure 3-3 shows the contribution of each energy dissipation rate term to the total energy dissipation rate. One should note that the Reynolds number (Re') in this paper is defined on the basis of the input superficial velocity V 0 and the inverse of the specific surface of particles as the length scale (see Notation) which is the same as that in Stanek (1994). It can be converted to the particle Reynolds number (Re p ) used in some studies by multiplying it with a factor of 61 ( ε B )(~3.51 in this study). It was found that when no internal obstacles are present and the flow is nearly parallel, the inertial term (T i ) is negligible compared to the other two terms (T k and T v ). The viscous term (T v ) is about one third of the total energy dissipation rate, and the kinetic term (T k ) is two thirds of the total energy dissipation rate. However, when two obstacle plates are placed in the above packed bed to create significantly nonparallel flow (see Figure 3-2), their effect on the total energy dissipation rate per unit cell is significant as shown in Figure 3-4a. The total energy dissipation rate is almost 50% higher compared to the one without the internal obstacles. The inertial term (T i ) is still negligible compared to the other two terms (T k and T v ) except in the very proximity of the obstacles as shown in Figure 3-4b. The values of T k and T v are scattered, but of the same order. Higher values of T k are observed at the obstacle regions as shown in Figure 3-4b. It is clear that internal obstacles make the gas flow more non-uniform. The possibility of a dominant inertial term was examined for a case of high density fluid by simulating a saturated liquid flow case. Here, the kinetic term (T k ) is seen to be dominant in the energy dissipation rate per unit cell at liquid superficial velocity, U 0 of 0.1 m/s (Re' = 47.5). The inertial term (T i ) is not 47

86 48 significant even in this case as shown in Figure 3-5. It can be concluded that the inertial term is not important except in the obstacle region which is in agreement with the simulations reported in the literature (Choudhary et al., 1976). This also ustifies the substitution of the pressure drop by the Ergun equation terms (T k and T v ) and elimination of the pressure term from the equation completely. In general, however, it is still advisable to include the inertial term in the formulation of the total energy dissipation per unit cell to account for those highly non-uniform flow situations in which the inertial terms could be important in affecting the nature of flow (Choudhary et al., 1976).

87 49 Energy dissipation rate, J/s 7.00E E E E E E E E E Cell number Ti Tk Tv Figure 3-4a. Contribution of each energy dissipation rate term at each cell to the total energy dissipation rate. V 0 = 0.5 m/s (gas flow with two internal obstacles at /d p = 30, ); Re' =28.5. Ti = J / s ; Tk = J / s ; 256 n= Tv = J / s ; ( Ti + Tk + Tv ) = J / s (The dashed line region will n= n= 1 be re-illustrated in Figure 4b). 256 n= 1 Energy dissipation rate, J/s 9.00E E E E E E E E E E E Cell number Ti Tk Tv Figure 3-4b. Contribution of each energy dissipation rate term at each cell to the total energy dissipation rate. V 0 = 0.5 m/s (gas flow with two internal obstacles at /d p = 30, 256 i k v /. n= ); Re =28.5; E bed = ( T + T + T ) = J s

88 E-03 Energy dissipation rate, J/s 5.00E E E E E E E Ti Tk Tv Cell number Figure 3-5. Contribution of each energy dissipation rate term at each cell to the total energy dissipation rate. V 0 = 0.1 m/s (liquid flow without internal obstacles); Re' = 47.5; Ti = J / s ; Tk = J / s ; Tv = J / s ; n= n= ( Ti + Tk + Tv ) = J / s. n= 1 Regarding the contribution of the Reynolds stress term to the cell-scale velocity distribution in packed beds, we performed CFDLIB simulations of liquid up-flow at a high particle Reynolds number (Re p ) of 600 (V l = 0.2 m/s) with and without turning on a simple turbulence model based on the mixing-length concept (using particle diameter as a sample of mixing-length). The relative differences in simulated liquid velocity profiles in the two cases are negligible (less than 0.1 %). This implies that the contribution of the Reynolds stress term to the cell-scale (i.e. 0.9 cm = 3 particles) flow distribution in packed beds is negligible. However, such term may become important if one attempts to model the local particle scale (less than one particle diameter) flow field. As a matter of fact, the transition between laminar and turbulent flow regime occurs at a certain particle Reynolds number range (Re p ), which may vary with particle diameter. For instance, the critical Reynolds number range of 150 to 300 was reported by Jolls and Hanratty (1969) 256 n= 1

89 51 for particles of 1.27cm in diameter while Latifi et al. (1989) reported the range of 110~370 for 0.5 cm diameter glass beads. This data was locally measured by using micro-electrodes with a diameter of 25 µm (Latifi et al., 1989) and reflects the flow behavior in the interstitial space in packed beds. The recent fine-mesh CFD simulation by Niemeisland et al. (1998) did find the stronger turbulent eddies in the gaps in between the spheres at higher Reynolds number flow conditions. In the development of the DCM, we made use of the fact that pressure for the orthogonal directions and, P cx and P cz, has the same magnitude at the center of the cell. We could then eliminate the central pressure term from the expression for the energy dissipation rate per unit cell (Eq 3-9) by using the mass balance for each cell. In order to ensure that this formulation is consistent, we have back calculated the central pressure (P cx and P cz ) based on the two dimensional velocity solution (V z and V x ) and verified that they do have the same values at the center of each cell as required, although the pressure drop in the and directions may have different values. Due to the nonlinearity of the equations (cubic in velocity), another important consideration is the uniqueness of the velocity obtained by the solution of the minimization problem solved in DCM. To examine this, different starting guess values varying over two orders of magnitude were used for a test case (input superficial velocity, V 0 = 0.1 m/s). For this case, starting guessed values anywhere between 1.0 m/s to +1.0 m/s converged to a unique solution for velocity based on the minimum total energy dissipation rate. With regard to the computational efficiency of DCM, for the cases considered in this study (total cell number: 264 = 256 packing zone + 8 supporting plate ; the corresponding number of variables in the optimization is 569), the computation time is comparable with that required to execute the CFDLIB code with identical discretization. It is noted, however, that simulation of a case with a larger number of cells would require a more effective non-linear multi-variable optimization algorithm to get better computational efficiency.

90 Comparison of DCM and CFDLIB The verification of DCM predictions can be obtained by comparing them with the fluid dynamic model solutions (CFDLIB) under identical physical and operating conditions. For the simplified case of 'parallel flow' Stanek (1994) argued that the solutions for velocity obtained by the two methods, Differential Vectorial Ergun Equation Model (based on momentum equation) and minimum rate of energy dissipation method are identical in both limiting ranges of the Reynolds number (fast flow, i.e. Re' 150, and slow flows, i.e. Re' 1.5). This conclusion was reached by comparing the analytical solutions of the two methods. In the transition region (1.5 < Re' < 150), however, the minimum rate of energy dissipation method yielded smaller velocities (Stanek and Szekely et al., 1974; Stanek, 1994) than the vectorized Ergun Equation. As mentioned earlier, the rate of energy dissipation term due to inertia was ignored in the differential vectorized Ergun equation model (Kitaev et al., 1975). For the case of two dimensional ''non-parallel flow'', which is of interest in this study, the conclusions regarding the applicability of the minimum rate of energy dissipation concept in providing a comparable solution for the gas velocity at cell scale need to be reconsidered. However, analytical solutions of the fluid dynamic equations for "non-parallel flow" are unavailable; therefore, the numerical results from computational fluid dynamic solution (CFDLIB) are used for verification of the DCM simulation results. In order to compare them effectively, the same operating conditions and the same structure of the bed are used in the simulations. To cover a wide range of Reynolds numbers, three sets of superficial gas velocity of 0.1, 0.5, 3.0 m/s are chosen. The corresponding Reynolds numbers (Re') are 5.7, 28.5, and respectively. Three sets of results at different Reynolds number are shown in Figures 3-6, 3-7 and 3-8 at different axial positions (/d p = 4.5, 19.5, 34.5, 49.5, 64.5, 79.5 from the top of bed). Following the work of Stephenson and Stewart (1986), and Cheng and Yuan (1997), we use the (relative) local superficial velocity (dimensionless superficial velocity) defined as:

91 53 (Relative) local superficial velocity = U < U ε ε > V = V 0 (3-22) i.e, the local interstitial velocity times the local porosity (for single phase flow) divided by the cross-sectionally averaged superficial velocity as given in Equation (3-22). It is found that the simulated local (i.e. cell scale) dimensionless gas superficial velocity profiles by both DCM and CFD at each given axial position track the porosity profile very well. Higher local porosity corresponds to higher local velocity. The difference in prediction between DCM and CFD simulation was found to be less than 10 % over the whole range of Reynolds numbers (Re = 5 ~ 171) as shown in Figures 9a and 9b. It is also shown that the dimensionless local superficial velocities vary in the range of 0.8 to 1.2 for the given system with a porosity variation of a cell scale of 0.9cm. All this indicates that the velocity profiles from DCM and CFDLIB compare well at three different Reynolds numbers. Reasonable comparisons of the two modeling approaches are achieved even at high Re' number (Re' = at V 0 = 3.0 m/s). This implies that DCM based on the minimum rate of energy dissipated can provide us with gas velocity predictions comparable to those obtained by CFD, which rests on ensemble-averaged mass and momentum conservation equations.

92 54 /dp =4.5 (from top) /dp =19.5 z/dp=34.5 relative velocity CFD DCM porosity /dp Relative velocity /dp relative velocity /dp /dp=49.5 /dp=64.5 /dp=79.5 Relative velocity /dp Relative velocity /dp Relative velocity /dp Figure 3-6. Comparison of CFD simulations and DCM predictions at a gas superficial velocity of 0.1 m/s at different axial positions (/d p ) (Re' = 5.7). Left axis is relative cell superficial velocity; Right axis is cell porosity. 54

93 55 /dp =4.5 (from top) /dp =19.5 z/dp=34.5 relative velocity DCM CFD porosity /dp Relative velocity /dp relative velocity /dp /dp=49.5 /dp=64.5 /dp=79.5 Relative velocity /dp Relative velocity /dp Relative velocity /dp Figure 3-7. Comparison of CFD simulations and DCM predictions at a gas superficial velocity of 0.5 m/s at different axial positions (/d p ) (Re' = 28.5). 55

94 56 /dp =4.5 (from top) /dp =19.5 z/dp=34.5 relative velocity DCM CFD 0.30 porosity /dp Relative velocity /dp relative velocity /dp /dp=49.5 /dp=79.5 /dp=64.5 Relative velocity /dp Relative velocity /dp Relative velocity /dp Figure 3-8. Comparison of CFD simulations and DCM predictions at a gas superficial velocity of 3.0 m/s at different axial positions (/d p ) (Re' = 171). 56

95 57 To examine the effect of fluid density and gravity, the calculations by both methods were repeated for liquid flow through a liquid-saturated bed. In practice, this would be the case of liquid up-flow through a packed bed. It should be noted that the gravity term has now to be accounted for (see Eq. 3-5) because Equation 3-6 is not satisfied for liquid flow. The difference in prediction of V z (velocity component in the direction) between DCM and CFD simulation was found to be less than 10 % for the liquid superficial velocity of 0.1 m/s (Re' = 47.5). DCM yields a 1~2% lower prediction of V z than CFD as shown in Figure 10a. Correspondingly, a lower prediction of V x (velocity component in direction) was found in CFD as show in Figure 3-10b. This implies that in a liquid-solid system the prediction by DCM is a little bit more sensitive to the bed structure such as porosity distribution than CFD. From a practical point of view, this feature of DCM prediction for liquid flow does not diminish its appeal as the method of providing a reasonable solution. This also implies that DCM is applicable for modeling of the high pressure gas-solid systems in which the density of the gas is high. 4.0 V from DCM (m/s) % -10% V from CFD (m/s) Figure 3-9a. Comparison of superficial velocity between CFD and DCM predictions for gas flow in the Reynolds number (Re') range of 5 to 171.

96 U/U0 from DCM % -10% U/U0 from CFD (b) Figure 3-9b. Comparison of the relative interstitial velocity (U /U 0 ) between CFD and DCM predictions for gas flow in the Reynolds number (Re') range of 5 to 171. (U 0 = V 0 /ε B ) Comparison of DCM/CFDLIB and Experiment Data As discussed earlier, most experimental studies in the literature reported the velocity profiles at the bed exit (Morales et al., 1951; Szekely and Poveromo, 1975; Bey and Eigenberger, 1997 etc.). They provide the data only for validating the model prediction for the velocity profile downstream of the bed (see Bey and Eigenberger, 1997; Subagyo et al., 1998). For non-parallel flow system of interest in this study, the exit velocity profile cannot represent the flow behavior inside the bed (Lerou and Froment, 1977; McGreavy et al., 1986). Hence, experimental data inside packed beds is needed to perform the proper comparison of DCM/CFDLIB predictions and experimental results. The liquid velocity profile inside the bed of Stephenson and Stewart (1986) is useful for such a comparison because both porosity and velocity data were reported in

97 59 their paper. However, the data is still inadequate for a very rigorous comparison of the numerical simulation and experiments since only one set of data was reported, and this was an ensemble-averaged result based on a large number of 'cell' measurements. Nevertheless, for lack of better data, this information has been used by others for model validation (Cheng and Yuan, 1997; Subagyo et al., 1998). The single phase flow distribution data of McGreavy et al (1986) inside the packed bed is only good for a qualitative test of numerical simulations because the corresponding porosity data was not reported (see Figure 7 in McGreavy et al., 1986) % Uz(CFD), cm/s % Uz(DCM), cm/s Figure 3-10a. Comparison of predicted interstitial velocity component in the direction (U z ) by two methods in liquid up-flow system: liquid superficial velocity V 0 = 0.1 m/s (Re' = 47.5).

98 Ux (CFD), cm/s Ux (DCM), cm/s Figure 3-10b. Comparison of predicted interstitial velocity component in direction (U x ) by two methods in liquid up-flow system: inlet liquid superficial velocity V 0 = 0.1 m/s. Figures 3-11a and 3-11b display the DCM results indicating the effect of fluid superficial inlet velocity (or particle Reynolds number, Re p ) on the velocity profile inside the bed, which are qualitatively comparable with the experimental data of McGreavy et al (1986) (see Figures 7 and 8 in that paper). The high velocity zones match the high voidage regions, as would be expected, and as the flow rate increases the magnitude of these peaks become more pronounced. Figure 3-12b is also comparable with the recent independent modeling result of Subagyo et al (1998) (See Figure 9 in their paper). The same conclusions are evident as reported by Subagyo et al (1998) that for Re p less than 500, the velocity profile is dependent on the particle Reynolds number. On the other hand, the effect of the Reynolds number on the velocity profile is no longer significant for Re p higher than 500.

99 61 u, m/s /d p porosity 0.1m/s (Rep=20) porosity 0.5m/s (Rep=100) Figure 3-11a. Influence of gas feed superficial velocity on DCM predicted cell interstitial velocity profiles. V/V /dp 0.1m/s (Rep=20) 0.5m/s (Rep=100) 3.0m/s (Rep=600) porosity porosity Figure 3-11b. Effect of particle Reynolds number (Re p ) on the calculated relative cell superficial velocity profile inside a bed using DCM.

100 62 Relative cell superficial velocity r (cm) cell # (interval #) porosity Vz/V0 (Exp.) Vz/V0 (Rep = 80) (Cal.) Vz/V0 ( Rep = 5) (Cal.) porosity (Exp.) Figure 3-12 Comparison of experimental data of Stephenson and Stewart (1986) and CFDLIB simulated results for relative velocity in a packed bed with D/d v = 10.7 and d v = cm (cylindrical particles). Physical properties of liquid: Liquid -B for condition at a Re p of 5, ρ = g/cm 3 ; µ = g cm/s. Liquid -C for condition at a Re p of 80, ρ = g/cm 3 ; µ = g cm/s. The quantitative comparison of our CFD numerical simulations (CFDLIB) has been carried out with the data of Stephenson and Stewart (1986) in which the velocity and voidage data were obtained by using optical measurements for Reynolds numbers of 5 and 80 in beds with D/d p ratio of Velocity was measured inside a bed of cylindrical particles (d v = cm) with liquid flows of very different physical properties ( ρ ; µ ). To simulate the experimental bed, a 2D axi-symmetric bed in L L cylindrical coordinates (r-) is chosen in CFDLIB simulation. The radial spatial discretization (N c = 20) is the same as that used as the viewing zone for collecting each

101 63 experimental data point (i.e a space interval of =0. 05R ). Regarding the dependency of flow simulation result on the grid size, one will see in Chapter 5, that the simulated macro-scale velocities by k-fluid CFD mode are grid independent. In addition, no-slip wall boundary and liquid gravity effect are accounted for in the simulations. The experimentally reported radial porosity profile is used in CFDLIB simulation. Figure 3-12 shows the comparison of CFDLIB simulated relative superficial velocity profile (V z /V 0 ) and the measured data at Re p numbers of 5 and 80. Good agreement is achieved. This implies that even for a cell size less than a particle diameter, the CFDLIB code can still provide a reasonable prediction of the velocity profile. The same agreement between DCM and the experimental data of Stephenson and Stewart (1986) is expected since DCM and CFDLIB have always provided results with 10% of each other as discussed in Section One should note that the above comparison of CFDLIB and the experimental data still rests on the one-dimensional porosity variation in the radial direction. Because of the lack of the two-dimensional measured porosity distribution and velocity distribution data reported in the paper by Stephenson and Stewart (1986), it is impossible to conduct the full comparison of simulated twodimensional velocity field by DCM/CFDLIB with two-dimensional experimental data of flow distribution at the cell scale Case Studies by DCM Since the validity and accuracy of DCM are established in the previous sections, DCM can be used in engineering applications as demonstrated in the case studies considered here. Because of the discrete nature of DCM, boundary conditions can be easily set. The local variation of porosity can also be accommodated readily. It is possible to use DCM to model two- or three-dimensional non-parallel flow fields. Two cases are considered to demonstrate these claims: (i) A bed with pseudo-random porosity distribution and internal obstacles was considered; (ii) Two types of gas flow inlets (top and side gas inlets) are examined using the DCM method. Velocity vector plots and pressure and dimensionless pressure drop contour plots are shown in Figures 3-13a and 3-13b, 3-14a and 3-14b, respectively. Figures 3-13a and 3-13b illustrate the dependency

102 64 of the gas velocity field on the internal structure nonuniformities inside the beds (i.e. two internal obstacle plates) and the effect of irregular gas feed (i.e. side gas input) as well as of the pseudo-random porosity distribution. No vortices appear in the vicinity of the obstacle plates or at least they are not larger than the cell size. The predicted results for velocity are almost symmetric with respect to the obstacle plate, which is in good agreement with entrance region, but also in downflow regions as shown in Figure 3-13b at given operating conditions, although no effect could be detected at the exit. Therefore, it is difficult to draw the proper conclusion about the effect of side gas feeding on the flow field based only on the exit velocity measurements (Szekely and Poveromo, 1975). It is expected, however, that the effect of side feed will depend on the magnitude of the side feed gas velocity. The full pressure field in the packed bed with two internal obstacles is shown in Figure 3-14a. Higher local pressure drop occurs at the regions around internal obstacles. This is not surprising due to the higher velocity and higher flow resistance in these regions. Figure 3-14b shows the dimensionless local pressure drop (ψ G 1 P = ) in the case of the side gas feed. Higher values of ψ ρ G are evident G g in the entry and obstacle regions. In contrast, lower values of ψ G are apparent in the corner regions.

103 65 Figure 3-13a. Interstitial velocity field in a packed bed with two internal obstacles and gas uniform feed from the top at a superficial velocity of 0.5m/s (Re' = 28.5) (U 0 =120.5 cm/s); (velocity vector plotting).

104 66 Figure 3-13b. Interstitial velocity field in a packed bed with side gas feed (top-left) and internal obstacles. Inlet gas mean superficial velocity: 0.5m/s (Re' = 28.5) (U 0 =120.5 cm/s) (point source inlet from left side, inlet point superficial velocity is of 4.0 m/s) (velocity vector plotting).

105 67 Figure 3-14a. Pressure field in a packed bed with two internal obstacles and gas uniform feed from the top at a superficial velocity of 0.5m/s (Re' = 28.5) (The relative values of pressure with respect to the inlet operating pressure are plotted). Two obstacle plates are placed in this bed, one is located at /d p of 66 (at the left side), another is at /d p of 30 (at the right side). The width of the obstacle plate (i.e. the length that it protrudes into the bed) is half of the width of bed (4 cells).

106 68 Figure 3-14b. Dimensionless pressure drop in a packed bed with two internal obstacles and a gas point feed from top-left side at an equivalent feed superficial velocity of 0.5m/s (Re' = 28.5) (Dimensionless pressure drop, ψ G 1 P = is plotted). Two obstacle ρ plates are placed in this bed, one is located at /d p of 66 (at the left side), another is at /d p of 30 (at the right side). The width of the obstacle plate (i.e. the length that it protrudes into the bed) is half of the width of bed (4 cells). G g

107 3.6 Concluding Remarks 69 A discrete cell approach for modeling single phase flow in packed beds was analyzed by considering the contributions of the individual terms in the equation for the rate of energy dissipation. The inertial term in the energy dissipation rate per unit cell is negligible compared to the kinetic term and viscous term except in the regions of structural obstacles. Even in the presence of obstacles the overall inertial term in the total energy dissipation rate is still not important. Reynolds stress term can be ignored due to negligible contribution of this term to cell-scale (i.e. particle diameter scale) fluid velocity distribution. DCM can also be applied for liquid up-flow prediction and gas flow at high operating pressure. A numerical comparison study based on DCM and CFDLIB approaches for the non-parallel flow field has been carried out to verify the DCM approach which rests on the assumption that flow is governed by the minimum rate of total energy dissipation in packed beds. A reasonable agreement between predictions of these two methods is achieved over a wide range of Reynolds numbers for gas flow. It was found that the local superficial velocities track the local bed porosity well. Lower flow resistance produces higher local superficial velocity. The cell superficial velocity with respect to the cross-sectionally averaged superficial velocity varies in the range of 0.8 to 1.2 for the case considered in this study. It is not our intent to advocate the use of DCM instead of CFD. The purpose of this study was to indicate that the model so frequently used by engineers, which is based on minimization of the total rate of energy dissipation, indeed works for flows in packed beds in the sense that it provides acceptable solutions of engineering accuracy. The method is relatively simple to use and contains only those physical terms that are deemed important in a particular situation. Using the Ergun equation to describe the pressure drop velocity relation at the cell level is apparently successful enough in describing flows in packed beds for a wide range of Reynolds numbers, fluid densities and velocities. That is the main message of this paper. It should also be clear that our intent was not to obtain the refined more precise flow field in the presence of internal obstacles in the packed beds, which could be done by local mesh refinement in direct numerical simulation

108 70 (DNS), but rather to describe the gross flow pattern, which is of interest for quickly evaluating packed-bed reactor performance, at the same level of discretization via DCM and CFD. Only such comparisons are reported. An agreeable comparison of numerical simulations (DCM and CFDLIB) and experimental velocity data inside a bed is achieved both qualitatively and quantitatively. Additional experimental efforts in obtaining the experimental data for multi-dimensional porosity and fluid velocity distributions are needed to further verify these numerical models and enhance our understanding of flow distribution within beds with complex internal structural nonuniformities.

109 71 Chapter 4 Discrete Cell Model Approach Revisited: II. Two Phase Flow Modeling 4.1 Introduction Trickle bed reactors with gas-liquid cocurrent downflow have been widely used in hydrogenation, hydrodesulfurization and other hydrotreating processes. One of the maor challenges in the design and operation of this type of reactor is the prevention of liquid flow maldistribution which causes portions of the bed to be incompletely wetted by the flowing liquid. This results in an underutilized catalyst bed and, hence, reduces reactor performance and productivity, particularly for liquid limited reactions at low liquid mass velocities. Consequently, conventional reactor models that assume a uniform wetting efficiency throughout the reactor are found to over-predict the reaction rate (Funk et al, 1990). The solution to this problem requires a quantitative understanding of flow maldistribution at different scales in trickle beds. A number of models of the liquid distribution have been developed in the past two decades based on different concepts or governing principles (Herskowitz et al., 1979; Crine et al., 1980; Stanek et al, 1981; immerman and Ng, 1986; Ahtchi-Ali and Pedersen, 1986; Fox, 1987; Melli and Scriven, 1991; Marchot et al., 1992). Although these efforts have provided insights into liquid flow distribution at a certain level, these models cannot simulate a number of experimental observations. For example, they cannot account for prewetting of the bed, which is known to have a marked influence on liquid

110 72 flow distribution (Lutran et al., 1991; Ravindra et al., 1997a). Thus, there is a need to incorporate such effects in an engineering model that can reflect these experimental observations in the model predictions. The obective of this study is to develop a phenomenological, user friendly, model for prediction of liquid and gas flow distribution in trickle-bed reactors. The developed model should be able to capture the experimental observations, and have acceptable engineering accuracy. Since the trickle bed is treated as a number of inter-connected cells, the flow distribution model developed in this study is called 'discrete cell model' (DCM). The gas and liquid distribution is assumed to be governed by the minimization of total energy dissipation rate (Holub, 1990). The interactions between phases can be incorporated into the model in terms of the capillary pressure and the particle surface wetting factor, etc. The motivation for this study was provided by the fact that minimization of the total energy dissipation rate has been used frequently in engineering models, yet the results from such an approach were always accepted with a degree of scepticism as not being based on fundamentals. In this study, our goal is also to compare the results obtained from the application of the total rate of energy dissipation "principle" (DCM) to those that arise from solution of more fundamental momentum and mass balance formulations (i.e. Computational Fluid Dynamics, CFD) and, of course, to experimental evidence (i.e. photo images by Ravindra et al., 1997a). The intent is not to replace computational fluid dynamics (CFD) simulations by the minimization of total energy dissipation rate, but to examine whether an alternative of acceptable engineering accuracy exists to CFD in flow modeling in trickle beds and can be used more convenient. Before presenting the strategy involved in the DCM development, and discussing the superiority of DCM to other models, it is necessary to summarize the previous liquid distribution models in the literature with reference to their governing principles and spatial scales considered.

111 4.1.1 Spatial Scales in Trickle Beds Since different spatial scales exist in a packed-bed (i.e., bed-scale, cell-scale, particle-scale), there is no question that a flow distribution model based on different spatial scales will require computation at different levels. At one extreme, timeconsuming computations required to determine liquid flow on the particle-scale (immerman and Ng., 1986), limit the application of this model to a small size bed, although this model can reflect partial catalyst wetting on the particle scale. On the other hand, a bed-scale model, which divides the bed into several regions (i.e., central region and wall region etc.), is too simplistic to capture the important features of the flow field. Therefore, Holub (1990) assumed that a packed bed could be represented as a number of interconnected cells. Each cell consists of a few particles, and each cell has uniform structure and physical properties. The cell size has to be small compared to the bed scale (i. e., bed diameter), to obtain the desired resolution of bed properties and phase distribution, but large compared to the particle scale (i.e., particle diameter), to apply the existing phenomenological hydrodynamic models developed in lab-scale packed-beds (i.e. two phase Ergun equations, Holub et al., 1992, 1993). The appropriate cell dimensions to satisfy these criteria were discussed by Vortmeyer (1984) who concluded that homogeneous models of packed bed heat transfer failed in beds with a tube to particle diameter ratio less than three. While this conclusion was not reached for the exact situation considered here, a minimum linear dimension of three particle diameters for each cell can be considered appropriate (Holub, 1990). Figure 4-1 represents a typical two dimensional cell (which consists of nine particles) with the velocity convention and coordinate system used in DCM formulations. The nature of such a discrete cell model allows us to obtain flow distribution information (at few particles scale) with reasonable computational efforts Governing Principles for Flow Distribution Because of the complexity of two phase flow distribution in trickle-bed reactors, a number of models have been developed, in which different governing principles for flow distribution have been assumed. The diffusion model assumed that the irrigation flux

112 74 satisfies the diffusion equation, which is then solved for a variety of inlet flow distributions (Stanek et al., 1981; Stanek, 1994). This model was not capable of predicting phase separation phenomena that occur in trickle beds. The percolation approach has been used by many researchers to model the flow in trickle beds (Crine et al. 1979, 1980; Larson et al. 1981; Melli and Scriven, 1991; Marchot et al., 1992). The model assumed that the flow distribution in the bed is due to a stochastic process. The liquid was distributed on the network by randomly choosing the bonds in the structure that have flowing liquid. While the model had the merit of representing the liquid structure as discontinuous in the bed, the predictive ability is questionable for the small size of particles since a direct relationship does not exist between the network and bed structure. In a computer generated packed bed of equally sized spheres, the sphere-pack model predicts the liquid distribution based on developed wetting criteria (immerman and Ng, 1986). The model was able to predict liquid coring, but gas flow was not included in the model. The model was also limited to the case of initially dry spherical particle surface. The effect of particle prewetting could not be accounted for. In the discrete interconnected cells model (DCM) (Holub, 1990) addressed in this study, it was assumed that the flow can be determined by the minimum rate of total energy dissipation in the packed-bed (i.e. flow follows the path of the least resistance). The known porosity variation in the bed could readily be incorporated into DCM by inputting cell porosity values. The type of liquid and gas distributors (i.e. point source; uniform source; irregular source for two fluids) is accounted for by setting two phase velocity values at inlet cell boundaries. To consider the effect of particle wetting state on the liquid distribution (i.e. prewetted bed and nonprewetted bed), which has been observed in experimental studies (Lutran et al., 1991; Ravindra et al., 1997a), the contribution of capillary pressure has been incorporated into the original DCM, and reported as an extended DCM in this study. The superiority of the extended DCM to other flow distribution models can be attributed to its ability to consider (i) the effect of bed structure nonuniformity (two dimensional porosity variations; internals in packed bed); (ii) the effects of gas and liquid distributors; (iii) the effect of particle prewetting.

113 Extended Discrete Cell Model The discrete cell model (DCM) for packed beds was originally proposed and formulated by Holub (1990). The full analysis and detailed implementation of individual aspects of DCM for single phase modeling (gas flow or liquid upflow) have been presented in Chapter 3. In this Chapter, DCM is applied to two phase flow modeling in trickle beds. Since the detail formulation is available in Chapter 3 and Holub (1990), only the key model equations and the parts pertinent to two phase flow are presented here. The equation for the macroscopic mechanical energy balance for phase α in the th unit cell can be expressed in continuum form by Equation (4-1), and is based on the following key assumptions: (i). Each unit cell of the bed has a uniform porosity (ε, which can vary from cell to cell), and constant phase holdup as well as constant phase properties; (ii). Steady-state flow distribution is considered in the entire bed and fluids are incompressible; (iii). No phase change occurs at the gas-liquid interface. The contribution of chemical reaction to the flow distribution is not accounted for in this model. + Φ + = in i i i i s s V s s V P s s V s V, α α α ρ ρ 0,, 4 1, 2 = + Φ + = α α α ε ε ρ ε ε ε ρ V out i i i E s V s V P s V V (4-1)

114 76 V,, α V,, α V +,, α Cell =3d P V +,, α =3d P Figure 4-1. The coordinate system and velocity conventions for the α phase in the th cell For a 2D rectangular cell () as depicted in Figure 4-1, the mechanical energy dissipation rate of phase α, E V,, α, can be written in the discretized form as below + = α α α α α α α ε ρ ε ρ, ,, 3 3 2,,, 1 2 V S V S V S V S V E ( ) ( ) [ ]( ) Vol V V b a V V b a 2,,,,, 2,,,,, 3, 1 α α α α α α α ε ( ) ( ) [ ]( ) Vol V V b a V V b a 2,,,,, 2,,,,, 3, 1 α α α α α α α ε ( ) ( ) [ ]( ) } g V V g V V Vol,, cos cos α α γ γ (4-2) A discretized form of the macroscopic mass balance equation can be similarly written as ( ) ( ) 0,, = α α ρ ρ ρ ρ S V S V S V S V (4-3) A detailed derivation of each term in Equation (4-2) is available in Holub (1990). To simulate the flow distribution, the two phase velocities at each cell interface are obtained by minimization of the total energy dissipation rate over the entire bed domain.

115 77 This is essentially a nonlinear, multivariable minimization problem as given in Equation (4-4) subect to the mass balance constraint (Equation 4-3) for each phase in cell, and additional constraints to reflect gas and liquid velocities at the bed inlet and at other boundaries of the bed (i.e., phase velocities in the cell adacent to the wall are zero in the direction normal to the wall). Minimize: 2 N E V, bed = EV,, α (4-4) α = 1 = 1 The subroutine DN0ONF from the International Mathematical Statistics Library (IMSL) was used to solve this nonlinear multivariable minimization problem. To get phase velocities from the above equations, we have also to solve for cell phase holdup (ε,α ) corresponding to a set of assumed phase velocities. This can be done by equating pressure drops in the gas and liquid phase (in absence of capillary pressure). P, L PL = L G, (4-5a) Then pressure drops are expressed in terms of dimensionless pressure drop functions (ψ G for gas, ψ L for liquid). P G, L P L, L ( Ψ 1) = ρ g (4-5b) L G, ( Ψ 1) = ρ g (4-5c) L L, Substitution of Equations (4-5b) and (4-5c) into equation (4-5a) yields equation (4-6) Ψ ρ ( Ψ 1) G L, = 1 + G, ρl (4-6) To relate these pressure drop functions to cell flow velocities, two phase flow Ergun equations (Holub et al., 1992, 1993) as given in equations (4-6a) and (4-6b) are used. This is equivalent to utilizing the concept of relative permeability discussed by Saez and Carbonell (1985). Ψ G, ε = ε ε L, 3 E1 ReG GaG,, E2 Re + Ga 2 G, G, (4-6a)

116 Ψ L, ε = ε L, 3 E1 Re Ga L, L, + E 2 Re Ga 2 L, L, 78 (4-6b) Thus, substitution of ψ G, and ψ L, (from Eq 4-6a and 4-6b) into Equation (4-6) yields a nonlinear equation in terms of phase holdups (ε,α ) and cell phase velocities which can be readily solved if flow velocities are known. The essential part of extended DCM is the treatment of the drag which takes into account capillary pressure. The pressure difference in the gas and liquid phase were correlated with the capillary pressure (Grosser et al., 1988) and a particle wetting factor, f, as ( f ) P, P, = P, 1 (4-7) G L C where the capillary pressure, P c,, in packed bed can be written as Equation (4-8) in terms of the well-known Leverett s J-function (Leverett, 1941) as suggested by Grosser et al. (1988). ( 1 ε ) ( s ) 0.5 P C, E1 σ J W, ε d P J = (4-8a) 1 s 0 (4-8b) ( ) W, s = + W, ln sw, When complete external particle wetting occurs (f = 1) the pressure difference between the gas and liquid phase disappears. This is the case treated in the original DCM (Holub, 1990). The pressure difference reaches a maximum (equal to the capillary pressure, P c, ) when the wetting factor f is equal to zero (completely non-prewetted case). Depending on the cell-scale, liquid velocity and cell porosity the f value is somewhere in the range of zero to one, which can be exactly calculated by the correlation for the particle external wetting efficiency (Al-Dahhan and Dudukovic, 1995). Thus, phase holdup is solved for by equating the difference of pressure drops in gas and liquid phases to the capillary pressure times the factor (1-f) as given in Equation (4-9).

117 ( 1 ε ) ( s ) PG, PL, J 0.5 W, = + E1 σ ( 1 f ) L L ε d L P (4-9) 79 Similarly to Equation (4-6), we can get Equation (4-10) in terms of dimensionless pressure drop functions, and from it we can solve for liquid holdup. Ψ L, [ Ψ 1] ( 1 ε ) ρg 0.5 = 1+ G, + E1 σ ρ ρ ε d L L P ( 1 f ) s W, b ( 1 s ) W, ( s ) J W L when ψ G, and ψ L, are obtained from Equations 4-6a and 4-6b as before., (4-10) 4.3 Modeling Results and Discussion The modeling results are presented for a '2D' rectangular bed, 7.2 cm (width) 28.8 cm (height) 0.9 cm (thickness), referred henceforth as model bed, which has an average bed porosity of corresponding to the value measured experimentally. To examine the cell porosity effect on flow distribution, the internal porosity profiles were specially designed by using a pseudo- random porosity distribution generated by a computer program with given constraints (i.e. porosity is kept in the range of 0.36 to 0.44 with an average of for the inner bed region away from the walls while higher porosity (0.44) is assigned to the wall region: 3 particles next to the wall) (see Figure 4-2a). Two transverse locations with low average porosity were deliberately designed as plotted in Figure 4-3b. This bed is divided into 8 cells in width (n C ) and 32 cells in length (n R ) and is 1 cell thick (n T ). Each cell has a size of 3 d p 3 d p 3 d p (0.9 cm 0.9 cm 0.9 cm) as depicted in Figure 4-1. The bed walls are considered to be impermeable boundaries. The liquid inlet distribution was assigned as: uniform, single point source and as two points source to simulate different liquid distributors. The inlet distribution for the gas phase was assigned as uniform in all the case studies.

118 80 Figure 4-2a. Local porosity distribution in model bed; Random internal porosity (0.36 ~ 0.44). Higher porosity of 0.44 at the walls. Darker color corresponds to higher porosity. porosity axial ( direction) radial ( direction) x/dp or z/dp Figure 4-2b. Average porosity profiles in and directions in model bed.

119 81 To quantify the liquid flow maldistribution, it is necessary to compare the deviation from a uniform velocity profile in term of the liquid maldistribution factor, mf, defined as mf = N i= 1 2 A i V i 1 (4-11) A0 V0 When the liquid flow distribution is uniform over the bed cross-section, mf is equal to zero and mf increases as the distribution becomes less uniform. The effect of different parameters (i.e. a state of particle prewetting, liquid distributor type, particle size) on flow distribution can be quantitatively described by the value of mf. The axial mf profile reflects the effect of bed depth on the flow distribution Comparison of DCM and CFD Simulations The main assumption of DCM is that the flow is governed by the minimum rate of total energy dissipation in the bed. The complete theoretical ustification for this assumption has been provided for linear relationships between fluxes and driving forces and rests on the principle of entropy maximization (Jaynes, 1980). In Chapter 3, for nonlinear systems, particularly non-parallel gas flow or liquid up-flow in the packed beds (where the local phase holdup is equal to the local porosity), agreeable numerical comparisons of DCM and CFD (using CFDLIB code as described below) have been achieved (Jiang et al., 1998). The difference between DCM and CFDLIB simulations was found to be always within 10 % over a wide range of Reynolds numbers. Nevertheless, it is desirable to compare the predictions of these two methods for the gas-liquid two phase flow system which is of interest in trickle flow. For this purpose, the Computational Fluid Dynamics code, CFDLIB developed by Los Alamos National Laboratory (Kashiwa et al., 1994), has been used to obtain the results for comparison with the DCM. The governing equations that serve as the basis for the CFDLIB codes and drag closures used in the simulation are given below

120 Equation of continuity: ρ t k +. ρ u (Accumulation) (Convection) k k = < ρ α! > (4-12a) k (Mass source) k 82 Equation of momentum: ρ k u t k +. ρ k u k u k =. < α ρ k 0 u' k u' k > θ p (Accumulation) (Convection) (Reynolds stress) (Mean pressure) (Body force) + < [( p0 p)i τ0 ]. α! k > + < ρ 0u 0α! k > (Exchange term) (Mass source) k + < α τ > θ (p p) (4-12b). k 0 k 0 (Average stress) (Non-equilibrium pressure) This set of equations is exact with no approximations other than the ensemble averaging used in the two-fluid model approach. One of advantages of CFDLIB is that it treats the packed bed case specifically and has options for user defined drag force formulation. Boundary conditions for inflow, outflow, and free/no slip at the reactor walls can be directly specified (Kumar, 1994). In this study, a user defined drag formulation is incorporated in simulating the drag between the stationary solid phase and each of the flowing phases in terms of phase fractions and relative velocity given for any combination of phases k and l as given below F D ( k l ) = k θ l kl ( u k u l ) θ (4-13) where the kl is modeled by the modified Ergun equation (Holub et al., 1992, 1993) with universal Ergun constants in this study. The drag between flowing phases has been ignored. This drag form is the same as that used in DCM simulation. To keep the consistency with the discrete cell approach used in DCM, free-slip boundary conditions are used for the reactor walls in CFD simulation. The spatial discretization of the model bed is also the same in both methods. For a given set of operating conditions, Figures 4-4a ~ 4-4d display the predicted relative gas flow interstitial velocity profiles by CFD and DCM at different heights (/d p ) + ρ k g

121 83 in the bed. Comparison of the complete data set at all heights is plotted in Figure 4e. Quantitatively, the agreement between the two model predictions for gas flow is good, and the differences in prediction of gas flow in all the cells by CFD and DCM are less than 13%. Relative iterstitial velocity CFD /dp = 15 DCM /dp Relative insterstitial velocity CFD /dp = 30 DCM /dp (4-3a) (4-3b) Relative interstitial velocity CFD /dp = 45 DCM Relative superficial velocity CFD /dp = 75 DCM /dp /dp (4-3c) (4-3d) Figures 4-3a, 3b, 3c and 3d. Comparison of the predicted gas interstitial velocities (relative) at the specific axial level by DCM and CFD. U l = m/s (UF); U g = 0.05 m/s (UF); Completely prewetted packed bed. (The relative interstitial velocity is defined as the local interstitial velocity (V i ) divided by the overall interstitial velocity (V 0 ). The value of V 0 in this case is equal to m/s).

122 Relative velocity by DCM modelin % -13% Relative velocity by CFD simulation Figure 4-3e. Comparison of the predicted gas interstitial velocities (relative) for all the cells by DCM and CFD. Inlet superficial velocities (uniform): U l = ; U g = 0.05m/s; Completely prewetted packed bed.

123 Liquid holdup "/dp=74" "/dp=62" "/dp=50" "/dp=74" "/dp=62" "/dp=50" 0.05 Solid line: CFDLIB Dash line: DCM /dp Figure 4-4. Comparison of predicted liquid holdup at specific levels by DCM and CFD. Single point source liquid inlet: U L = m/s (U l (PS 1 )= m/s); U g = 0.05m/s; Non-prewetted packing. The comparison of predicted liquid holdup at the specific levels of bed is shown in Figure 4-4 for the case of a single point source liquid inlet (PS). From the engineering point of view, the comparison of predicted liquid holdup by both methods is reasonable particularly in the central core. The difference in the prediction of liquid holdup by two methods (CFD and DCM) at locations far from the central flow indicates that the DCM seems to be more sensitive to local porosity values than CFD, and also predicts more liquid spreading. These numerical results can be qualitatively compared to experimental observations presented in Figure 2-8, which illustrates that rivulet flow is affected by variations in the local porosity which causes it not to flow straight down through the bed.

124 4.3.2 Effect of Liquid Distributor 86 Three types of liquid inlet distributors: single point source (PS 1 ), two points source (PS 2 ) and uniform distributor (UF) have been used to demonstrate the effect of liquid distributors on the liquid distribution in a non-prewetted packed bed. The boundary (inlet) values of the liquid superficial velocities at the top cell layer of the bed were assigned to keep the same volumetric liquid feed rate for all types of liquid distributors studied. With liquid point source inlets, as shown in Figure4-5a for single inlet and Figure 4-5b for two inlets, it was found that the number of liquid channels (rivulets) formed in the non-prewetted packed bed corresponded to the number of liquid point sources (e.g one for PS 1, and two for PS 2 ). This observation is qualitatively reflected in the result shown in Figure 2-8. With the uniform liquid inlet, as shown in Figure 4-6c, however, uniform liquid distribution occurs only in the entrance region, then channel (rivulet) flow forms in the downstream region due to the nonuniform porosity and capillary pressure effect. Under the chosen set of operating conditions (in Table 4.1), an onset of formation of liquid channels (i.e., phase segregation) is seen at a depth of 2 cm and formation of distinct rivulets occurred at a bed depth of 8 cm. These rivulets meandered, merged, and split as experimentally observed by Ravindra et al. (1997). It can be concluded that liquid rivulet flow is typical of non-prewetted beds. These DCM simulation results are qualitatively comparable with direct flow visualizations (Figure 4-2a and Ravindra s et al photo observations, 1997). A comparison of the calculated liquid maldistribution factor (mf) along the bed for different distributors is presented in Figure 4-5d. The effect of liquid distributor on liquid flow maldistribution is significant in the upper half of the bed (in non-prewetted beds) and is less pronounced at depths exceeding 50 particle diameters (15 cm) for total bed length of 96 particle diameters.

125 Table 4.1 Summary of operating conditions used in flow simulations 87 U l = m/s U g = m/s (superficial velocity) Completely prewetted bed Completely nonprewetted bed Gas inlet Liquid inlet Gas inlet Liquid inlet UF PS 1, PS 2 ; UF UF PS 1 UF: uniform PS 1 : single point source located at top layer* at No. 5 cell PS 2 : two points source located at top layer at No. 3 and No. 6 cell *There are 8 cells on the top layer from No.1 to No.8 Figure 4-5a. Liquid holdup distribution with single liquid point source inlet (located at No. 5 cell from left) by DCM. U L = m/s (U l (PS 1 )= m/s); U g = 0.05m/s; Non-prewetted packing.

126 88 Figure 4-5b. Liquid holdup distribution with two liquid points source inlet (located at No. 3 cell and No. 6 cell from left) by DCM. U L = m/s (U l (PS 2 )= m/s); U g = 0.05m/s; Non-prewetted packing.

127 89 Figure 4-5c. Liquid holdup distribution in whole domain of the non-prewetted packed bed with uniform liquid distributor by DCM. U l = m/s; U g = 0.05m/s.

128 90 mf PS1(non-prewetted) PS2(non-prewetted) PS1(prewetted) UF(non-prewetted) z/dp (from top) Figure 4-5d. Comparison of liquid flow maldistribution calculated by DCM along the bed for different liquid distributors. U L = m/s; U g = 0.05m/s Effect of Particle Prewetting Experimental observations have corroborated the fact that the effect of particle prewetting on liquid distribution is significant and causes more liquid spreading in 3D rectangular beds compared to non-prewetted bed (Lutran et al., 1991; Ravindra et al., 1997a). The CCD video images in Figure 2-8 also confirm the same finding in a pseudo- 2D bed. It is known that lower capillary pressure (by lower liquid surface tension, lower contact angle (θ) at the three-phase contact line) causes more particle wetting in the bed, and accordingly, causes an increase in overall liquid holdup (Levec, et al., 1986). In order to predict these experimental observations, we have incorporated a particle surface etting factor (f) into DCM as described earlier. Figure 4-6a shows the liquid holdup distribution in the entire completely prewetted bed with a single point liquid inlet (PS 1 ) (actually one cell inlet). For further comparison, Figures 4-6b and 4-7c show the predicted liquid holdup at the specific levels (/d p = 94.5; 73.5; 61.5; 49.5 from bottom) in the completely prewetted bed (f = 1) and in non-prewetted bed (f = 0), respectively. More liquid

129 91 spreading is evident in the prewetted bed whereas the effect of capillary pressure on liquid holdup distribution is apparent in the non-prewetted bed, where the pressure difference between the gas and liquid phase exists and prevents liquid from spreading. This is the reason for liquid channel (rivulet) flow formation in the non-prewetted bed. If the whole bed is pre-wetted with liquid, thin liquid films will be formed around the particle surfaces, in addition to the pores of the particles being filled by liquid, even when the liquid is drained off. These liquid films nullify the effect of capillary pressure and help spreading of the incoming liquid. Therefore, as expected, more liquid spreading in a prewetted bed is observed as shown by DCM simulation in Figure 4-6a. The overall liquid holdup in the prewetted bed is 6% higher than in the non-prewetted bed at the same operating conditions (as seen in Figures 4-6b and 4-6c). The increase in predicted overall liquid holdup by DCM is in agreement with Levec's et al experimental finding (1986). It is also of interest to consider Figure 4-6d, which shows that the liquid flow distribution with one point source liquid inlet in the prewetted bed is better in most of the bed (except the inlet region) than that obtained by two points source liquid inlet in the non-prewetted bed. This also corroborates the evidence of better reactor performance in prewetted beds. The only detrimental consequence of prewetting is liquid wall flow which occurs in the case of the completely prewetted bed with uniform liquid inlet (Figure 4-5c), since liquid spreads more easily until it reaches the wall and then continues along it resulting in the observed wall flow. If we consider one cell (three particles in this case) next to the wall as the wall zone, the magnitude of the wall flow is about 20-30% higher than the central region flow. These results for wall flow are only slightly different if ghost cells are created next to the wall to set a zero slip velocity at the wall. This has also been tested through the CFD simulations with slip boundary and with no-slip boundary conditions.

130 92 Figure 4-6a. Liquid holdup distribution in the whole domain of the completely prewetted packed bed (f = 1). U L = m/s (U l (PS 1 )= m/s); U g = 0.05 m/s; Point liquid distributor (PS 1 ).

131 93 Liquid holdup /dp /dp=94.5 /dp=73.5 /dp=61.5 /dp=49.5 Figure 4-6b. Liquid holdup distribution at specific levels (/d p ) in the completely prewetted packed bed (f = 1), U L = m/s (U l (PS 1 )= m/s); U g = 0.05 m/s; Point liquid distributor (PS 1 ); Overall liquid holdup = Liquid holdup /dp /dp=94.5 /dp=73.5 /dp=61.5 /dp=49.5 Figure 4-6c. Liquid holdup distribution at specific levels (/d p ) in the completely nonprewetted packed bed (f = 0), U L = m/s (U l (PS 1 ) = m/s); U g = 0.05 m/s; Point liquid distributor (PS 1 ); Overall liquid holdup =

132 94 Although two extremes of external particle wetting were considered here, in reality this parameter takes the value between two limiting cases (f = 0 and f = 1.0) depending on the particle surface and fluid properties of system. It is expected that the value of surface wetting factor, f, is associated with three phase interfacial-phenomena, such as liquid-solid contact angle, liquid surface tension, particle internal porosity etc. Local liquid vaporization may also cause local particle wetting non-uniformity, and further affect the value of f. It is also believed that the differences between non-porous and porous particles are reflected in different values of f, and thus cause different liquid distribution as observed experimentally by Ravindra et al., (1997a). The surface wetting factor (f) used here can be evaluated through the correlation of the particle external wetting efficiency which has been widely used in the literature (Al-Dahhan and Dudukovic, 1995). The flow simulations based on two limiting values of f (zero and one) essentially cover the range of possible liquid distribution at given operating conditions (see Figures 4-5a and 4-6a), which is valuable in examining possible trickle-bed scale-up and design. 4.4 Conclusions and Final Remarks An extended discrete interconnected cell model (DCM) was developed for simulation of two phase flow in trickle-bed reactors. Due to the nature of DCM, structural nonuniformities and different liquid inlet distributors can be readily incorporated into the model. Particle wetting characteristics are accounted for in the model by introducing the particle wetting factor (f) which allow us to distinguish between the flow patterns in prewetted and non-prewetted beds. The model predicted results are quantitatively comparable with those obtained from computational fluid dynamic codes (CFDLIB). Simulated liquid holdup distribution data qualitatively agree with the flow visualization experiments, which has not been achieved by other available models. Two bounds (corresponding to the completely prewetted and completely non-prewetted catalyst) of the liquid flow distribution at given operating conditions can be provided by the DCM model. The effect of liquid distributor on liquid flow distribution is significant in the

133 95 upper half of the bed. In regard to the computational efficiency of DCM, which is essentially formulated as a non-linear multi-variable optimization problem, more effective optimization algorithms are desirable for industrial scale problems with a large number of cells (as compared with only 256 cells used for our model bed). The advantage of DCM will become more apparent when we utilize it to compute not only ust the flow distribution but also reactor performance. At this point, we reemphasize that DCM is not suggested as a replacement for CFD. However, it is shown here that when one is interested only in the coarse structure of the flow pattern, DCM can provide answers comparable to those obtained by CFD.

134 96 Chapter 5 Computational Fluid Dynamics (CFD): I. Modeling Issues 5.1 Introduction and Background CFD Applied to Multiphase Reactors The performance of multiphase reactors, in principle, can be predicted by solving the conservation equations for mass, momentum and (thermal) energy in combination with the constitutive equations for species transport, chemical reaction and phase transition. However, because of the incomplete understanding of the physics, plus the nature of the equations- highly coupled and nonlinear, it is difficult to obtain the complete solutions unless one has reliable physical models, advanced numerical algorithms and sufficient computational power. Hence, in the past several decades, Residence Time Distribution (RTD) together with the macromixing and micromixing models have been the primary tool in reactor modeling used to characterize the nonideal flow pattern and mixing in the reactor without solving the complete flow velocity field (Levenspiel, 1972). The disadvantage of such an approach is that it cannot be adopted well to serve as a diagnostic tool for operating units, which normally need to be operated under conditions not amenable to the above simplified analysis. To achieve this goal, one has to solve the complete multi-dimensional flow equations coupled with chemical species transport, reaction kinetics, and kinetics of

135 97 phase change. Fortunately, computational fluid dynamics (CFD) has made great progress during the last few years, and has been applied to chemical processes (Trambouze, 1993; Kuipers and van Swaai, 1998). In particular, one of the promising methods is the socalled full Probability Density Function (PDF) model for single-phase reactive-flow systems (Pope, 1994; Fox, 1996). For most multiphase reactive-flow systems, however, the challenge still exists in both numerical technique and physical understanding. Reasonable progress has been made for multiphase cold-flow systems and few reactiveflow systems via CFD modeling. The features and the problems encountered in the current CFD modeling of multiphase reactors have been summarized in Table 5.1, which clearly indicates that more effort is needed in applications of CFD in gas-liquid stirred tanks, gas-liquid-solid packed-beds (e.g., trickle-beds), gas-liquid-solid fluidized beds and slurry reactors. For a detailed discussion of these topics, one is encouraged to consult the recent comprehensive review by Kuipers and van Swaai (1998). Based on the growing applications of CFD in multiphase flow systems, it is expected that the role of CFD in the future design of multiphase reactors will increase substantially and become common engineering practice. So far, a consensus emerges with regard to the following issues: It is unrealistic to hope for a universal CFD code that applies to all multiphase flow problems (Johnson, 1996). Even for one type of multiphase reactor such as bubble column, a hierarchy of models is more likely to have successful impact (Delnoi et al., 1997). Experimental validation of CFD results for several benchmark multiphase flows is essential to the widespread acceptance of CFD in multiphase reaction engineering (Kuipers and van Swaai, 1998; Dudukovic et al., 1999). Properly formulated two-fluid model is able to capture most of large-structure characteristics of multiphase flow (Lahey and Drew, 1999; Pan et al., 2000) The solutions from direct numerical simulation (DNS) (Joseph, 1998) and Lattice Boltzmann simulation (LB) (Sankaranarayanan et al., 1999; Manz et al., 1999) can provide an improved understanding of flow microstructure, and are a tool for obtaining

136 98 closures for averaged equation models used to predict large scale flows in industrial reactors.

137 99 99 Table 5.1 Current status of CFD modeling in multiphase reactors Reactors Current Features and Future Challenges Progress Sample CFD Work Bubble Columns Two-fluid Eulerian model Mixed Eulerian-Lagrange models Volume of Fluid (VOF) model for single gas bubble rising Mostly limited to bubbly flow Future challenge: churn-turbulent flow modeling. G-L Stirred Tanks Two-fluid model (Snapshot approach, MRF mesh, Sliding mesh) G-S Fluidized Beds e.g. bubbling, solid risers G-S or L-S Packed Beds G-L-S Packed Beds e.g. trickle beds Slurry reactors e.g. G-L-S stirred tanks, slurry bubble column Future challenges: Accurate modeling of the impeller; availability of local flow dynamic information and the range of dispersed phase holdup from experiments; turbulence modeling Two-fluid model with simple solid rheology Two-fluid model with kinetic theory Discrete particle approach Future challenges: Refined model for particle-particle, particlewall interactions; couple with reaction; Prediction of flow regime transition. Two-fluid model with 3D mesh for interstitial domain Two-fluid models with random porosity distribution Structural packing with heat transfer Future challenges: Geometrical complexity; availability of experimental data for validation. Two-fluid model with random porosity distribution Future challenges: Geometrical complexity; partial wetting concern; flow history dependence; availability of experimental data for validation Standard k-ε turbulence model (FLOW3D)-Stirred tank Two-fluid model Future challenges: Availability of experimental data for validation; difficult to capture the microscopic phenomena (e.g., particle accumulation near the G-L interface). G-L-S fluidized beds No Reasonab le!!! Little! Good!!!! Little! Very Little Very little Sokolichin & Eigenberger, 1994 Delnoi et al., 1997 Pan et al., 2000 Ranade & van den Akker, 1994 Ranade & Deshpande, 1999 Sinclair & Jackson, 1989 Dind & Gidaspow, 1990 Nieuwland et al., 1996 Logtenberg & Dixon, 1998 Chapter 3 in this thesis Chapter 4 in this thesis To be discussed in this Chapter Hamill et al., 1995

138 CFD and Other Modeling Approaches to Multiphase Flow in Packed Beds Packed-beds have been extensively used in petroleum, petrochemical and biochemical applications (Dudukovic et al., 1999). The stationary packing in the columns can be either active catalyst for chemical reaction systems or an absorbent in a separation column. Depending upon the application there are multiple configurations available for packed beds with gas and liquid flows: cocurrently downward (i.e., trickle-bed), cocurrently upward (i.e., packed bubble column) and counter-currently flows (e.g., catalytic distillation column). The criteria for choosing the proper flow direction have been established, and the evaluation of the effect of flow direction on reactor performance has also been performed (Wu et al., 1996; Khadilkar et al., 1996). Since most of these models rely on assumed ideal flow patterns and are one dimensional, the accurate prediction of multiphase flow pattern (i.e. spatial and temporal distributions) in the packed beds is still an unresolved issue, which is an obstacle to advanced reactor model development. Multiphase flow modeling in packed beds is a challenging task because of the difficulty in incorporating the complex geometry (e.g. tortuous interstices) into the flow equations, and the difficulty in accounting for the fluid-fluid (gas-liquid) interactions in presence of complex fluid-particle (e.g., partial wetting) contacting. Moreover, until recently, the lack of noninvasive experimental techniques suitable for validating the numerical results was also a detrimental factor in numerical model development due to lack of reasonable validation. The earliest flow models of packed beds focused on the bed-scale flow pattern without considering the detailed heterogeneities of the bed structure. The 'diffusion' model (Stanek et al., 1974) and porous media model (Anderson and Sapre, 1991) are examples of such an approach. To account for the statistical nature of the bed structure, a 'percolation based' model was adopted to predict the flow pattern in packed beds (Crine et al., 1979). These models provided certain predictions of the overall quantities that were found comparable with the experiments; however, they could not yield much insight into the flow distribution in the beds. A discrete cell model (DCM) approach evolved from the

139 101 assumption made by Holub (1990) that flow distribution is governed by the minimum total energy dissipation rate. In the recently updated DCM, as presented in Chapter 4, a statistical assignment of cell porosity values and the incorporation of the interfacial tension force related to the particle wetting and inflow distributors has been accomplished for two-phase flow in trickle beds. The quantitative predictions of liquid upflow in packed beds by the DCM approach compare well with the available experimental data and other independent numerical methods as presented in Chapter 3. However, the numerical scheme of multivariable non-linear minimization used in DCM often leads to low computational efficiency when dealing with a large packed bed with small cell dimensions. Direct numerical simulation (DNS) on single particle and single void scale requires complete characterization of solids boundaries and voids configuration, which is difficult to obtain for a massive packed bed. Statistic implementation of porosity distribution for a large size packed bed is proper for modeling of the macroscopic flow field. For example, to consider the interactions between fluid and particles a global flow model in packed beds, a k-fluid model, resulting from the volume averaging of the continuity and momentum equations, has been developed and solved for a onedimensional representation of the bed at steady state, and at isothermal non-reaction conditions (Attou et al., 1999). It provided reasonable predictions for global hydrodynamic quantities such as liquid holdup and pressure-drop. A similar k-fluid model, based on the relative permeability concept, was used to compute the twodimensional flow without considering porosity variation and without solving for the solid phase. The simulated liquid flow pattern qualitatively agreed with experimental observation (Anderson and Sapre, 1991). It seems that the Eulerian-Eulerian two-fluid model is a rational choice for flow simulation in packed beds if good closures for fluidfluid and fluid-particle interactions can be found. Moreover, the geometrical complexity of packed beds can in a certain sense be avoided in a two-fluid model, since there is no need to deal with the exact boundaries of particles, and since one treats the solid phase as a penetrated continuum. A study has been made to resolve the flow field at fine scale and CFD simulations were conducted of heat-transfer in a tubular fixed-bed using a 3-D fine-

140 102 mesh within the pore space (Logtenberg and Dixon, 1998). These simulations were limited to the tube with very low column to particle diameter ratio (D r /d p = 2~3) and with large particle size (e.g., 5cm). Obviously, it is impossible to adopt such an approach for a massive commercial packed bed, or even for a bench-scale trickle bed packed with small particles (e.g., 0.5~3 mm). Hence, one has to discover an efficient way to implement the bed structure into the flow model. It is most desirable to retain all the statistical characteristics of the pore space but without introducing the real pore structure, since the exact 3-D interstitial pore-structure varies with repacking the bed, even with the same particles and using the same packing method, although the mean porosity may retain the same value. In this work, we introduce a statistical description of the bed structure into a k- fluid model framework. In order to properly consider the effect of the solid phase on gas and liquid flows, the k-fluid model is applied to the gas, liquid and solid phase simultaneously while turning off the momentum equation for the solid phase, so that the initial volume fraction distribution of the solid phase is retained. The work accomplished is presented in two subsequent Chapters. In this Chapter (Chapter 5) the various issues related to the k-fluid model implementation in packed-beds were discussed, and the current state of the art for closures is presented. The multi-scale and statistical nature of flow is illustrated and the choice of the grid size and boundary conditions is discussed. Chapter 6 presents selected numerical simulation results based on the model presented in Chapter 5, and discusses the comparison of the numerical results with available experimental data and recommends the future research-focus and a methodology for utilizing the modeling results in packed-bed analysis and design. 5.2 Spatial and Temporal Characteristics of Flow in Packed Beds There are many structure and flow parameters responsible for the flow distribution in packed beds such as porosity distribution in the bed and the inlet flow velocity distribution (see Chapter 4). For example, in a system saturated with single-

141 103 phase flow (e.g., gas flow, liquid upflow case), the spatial variation of porosity is the essential parameter in determining the spatial distributions of fluid velocity and volumefraction variations. In a system with gas and liquid two-phase flow, the additional parameters affecting the liquid distribution are the state of particle external wetting, the interaction between phases, distributor design, etc. It is believed that there exists a quantitative relation between flow, bed structure and operating conditions of the system. Moreover, since the flow distribution/maldistribution can be observed at different spatial scales (Melli et al., 1990; Wang et al., 1998), it has been suggested that different scales be used to describe the corresponding flow phenomena. This so-called multiscale nature of the flow in packed beds results from the multiscale heterogeneities of bed structure. In packed beds, two complementary spaces coexist: the grain space and the porous space (i.e. cavity). The pore size, defined by the radius of the largest sphere, which can be put inside that cavity, depends on particle size, shape and packing method. For porosities of 0.36 to 0.4 obtained from monosize spheres, the pore size is in the range of 0.38R to 0.44R (R: radius of particle). As reported in several studies on packed beds, the mean porosity is reproducible for a given packing method with the standard deviation of only (Cumberland and Crawford, 1987). The longitudinally averaged radial porosity profile follows a certain oscillatory pattern due to the confines of walls, which can be predicted in terms of particle size, shape and column to particle diameter ratio (Benenati and Brosilow, 1962; Mueller, 1991; Bey and Eigenberger, 1997). Although typical bed structural information such as the above is available, it is not sufficient to predict the complete spatial distribution of flow in packed beds. Additional information on porosity distribution in 3-D or at least 2-D, wall effect, entrance and exit effect on flow are needed before going on to flow simulation. For both steady state and dynamic flow simulation in packed beds, the temporal behavior of flow has to be considered. In a two-phase flow trickle-bed in which gas is the continuous phase whereas the liquid is trickling down through the packing (i.e. trickle bed) at low superficial velocity, so-called trickle-flow regime in the literature, the bed-scale liquid flow pattern is rather stable whereas the local scale interstitial flow within the pore space still fluctuates in a chaotic motion. Once the gas and liquid

142 104 superficial velocities increase to a certain level, and the flow reaches so-called high interaction pulsing regime, even the macroscale liquid pattern becomes unstable: liquid rich-zone and gas-rich zone move alternately through the bed with a certain frequency. Such a macroscale flow fluctuation pattern can be also generated through the periodic input of the flows, which has been shown to enhance the reactor performance (Khadilkar et al., 1999). The experimental exploration of these spatial and temporal flow variations in packed beds are definitely important, but it is difficult for a single technique to capture both spatial and temporal behavior of flow simultaneously with high resolutions (Reinecke et al., 1998). For example, the magnetic resonance imaging (MRI) can provide a good spatial resolution of mm, but it is not suitable to measure dynamics of the flow such as these encountered in pulsing flow regime due to the temporal resolution problem. The electric capacitance tomography (ECT) gives a temporal resolution of a millisecond, but with relatively poor spatial resolution at this stage (Reinecke et al., 1998). On the numerical flow modeling side, a similar trend exists. We do not expect to use a single model to obtain information on a variety of spatial and temporal scales of flow, but we are to obtain one level of flow information through one particular model. In these two Chapters, we focus on the macroscale flow pattern at steady state operating conditions. We do explore the dynamic flow behavior of large-scale structures under periodic operating condition by including flow modulation to examine the possible improvement of the liquid distribution, but we do not intend to model the flow dynamics in the natural pulsing flow regime, which involves complex flow dynamic mechanism (Tsochatzidis and Karabelas, 1995). 5.3 Structure Implementation The implementation of porosity distribution in flow simulation increases the level of difficulty in packed beds as compared to other multiphase reactors. So far, this issue has been tackled in a deterministic and simplified manner to a large extent. For example, either uniform porosity or the radial porosity variation is considered in the model of the

143 105 bed (Bey and Eigenberger, 1997; Yin et al., 2000). In some cases, a multi-zone porosity assignment was used (Stanek, 1994). Since the 3-D interstitial pore space varies with repacking the bed, the porosity distribution possesses a statistical nature (Wingaarden and Westerterp, 1992) and the use of statistical description of the porosity structure in the flow model has considerable potential for success (Crine et al., 1992). In this Section we discuss how to partition the 3-D pore space into sections and what will be the type of the section porosity distribution, since in the flow simulation of a volume-averaged k-fluid model, one needs to assign the initial solid phase volume fraction to each section. Depending on the section size chosen for the partition, the section porosity values follow a certain probability density function (p.d.f). That means that the p.d.f. is section size dependent. For example, the measured section porosity data from a cylindrical column packed with 3-mm monosize spheres has exhibited a Gaussian distribution at a section size of 3 mm (Chen et al., 2000). However, a nearly binomial type of section porosity distribution was found by MRI measurement at a section size of 180 µm (Sederman, 2000). In principle, a quantitative relationship of the section size and the variance of section porosity distribution, σ B, can be developed through extensive MRI measurements of packed beds. Obviously, this relationship varies with particle shape and packing method. Thus, for a certain size of a section, a set of pseudo random section porosities can be generated based on the following constraints Mean porosity (measurable) Longitudinally averaged radial porosity profiles (correlation available) Correlation of section size, l v and the variance of section porosity distribution, σ B (obtainable by MRI, Sederman, 2000). Figure 5-1a shows a sample contour plot of 2-D section porosity distribution in r-z coordinates, which was generated under the constraints of a mean porosity of 0.35, a longitudinal averaged radial porosity profile (see Fig.5-1b) measured by Stephenson and Stewart (1986) and a pseudo Gaussian distribution with a variance of 12% mean porosity (see Fig.5-1c). The tail shown at high porosity range in Figure 5-1c indicates the effect of

144 106 walls on porosity. Such porosity generation process under constraints has certain analogue to particle repacking process in practice. It is noted that based on the mean porosity and the radial profile of section porosity, for a given section size, many possible probability density functions exist. The third constraint is definitely needed for generating a realistic porosity distribution. Moreover, although the example used provided an illustration for a 2-D distribution case, the approach is applicable for the 3-D porosity distribution case.

145 107 (a) sectional porosity r (cm) (b)