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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
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
ii
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 bench-
scale 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.
iii
Contents Page
Tables.............................................................................................................................. ix
Figures ............................................................................................................................ x
Acknowledgements........................................................................................................ xxiv
Nomenclature................................................................................................................. xxvii
1. Introduction to Flow Distribution in Packed Beds ............................................... 1
1.1 Research Motivation ..................................................................................... 2
1.1.1 Discrete Cell Model (DCM) Revisited........................................... 2
1.1.2 k-Fluid CFD Model Development and Applications ..................... 4
1.1.3 Impact of Flow Maldistribution on Packed-Bed Performance....... 4
1.2 Research Objectives ...................................................................................... 6
1.2.1DCM Revisited................................................................................ 6
1.2.2 k-Fluid CFD Model for Packed Beds............................................. 6
1.2.3 Applications of k-Fluid CFD Model .............................................. 7
1.2.4 Impact of Flow Maldistribution on Reactor Performance ............. 7
1.3 Thesis Structure............................................................................................. 7
2. Experimental Observations: Liquid Flow Distribution in Trickle Beds.............. 10
2.1 Introduction ................................................................................................... 10
2.2 Experiment-I: 2-D Liquid Flow Imaging ...................................................... 12
2.2.1 2-D Packed Bed and CCD Setup.................................................... 12
2.2.2 Imaging and Processing ................................................................. 13
iv
2.2.3 Liquid Flow Imaging...................................................................... 14
2.2.4 Experimental Results and Discussion ............................................ 14
2.2.4.1 Non-prewetted bed (dry bed) .......................................... 14
2.2.4.2 Prewetted bed (wet bed).................................................. 15
2.2.4.3 Comparison of liquid flow in non-prewetted bed
and prewetted bed........................................................................ 15
2.3 Experiment-II: Exit Flow Measurements in 3-D Bed ................................... 24
2.3.1 Experimental Objectives ................................................................ 24
2.3.2 3-D Column and Exit Flow Measurement ..................................... 25
2.3.3 Experimental Results and Discussion ............................................ 26
2.4 Conclusions ................................................................................................... 32
3. Discrete Cell Model Approach Revisited: I. Single Phase Flow Modeling .......... 33
3.1 Introduction ................................................................................................... 33
3.2 Non-Parallel Gas Flow Models ..................................................................... 34
3.2.1 Vectorized Ergun Equation Model................................................. 34
3.2.2 Equation of Motion Model............................................................. 35
3.2.3 Discrete Cell Model (DCM)........................................................... 35
3.3 Discrete Cell Model (DCM).......................................................................... 38
3.4 CFDLIB Formulation.................................................................................... 42
3.5 Modeling Results and Discussion ................................................................. 43
3.5.1 Model Packed Bed ......................................................................... 43
3.5.2 Analysis of the Energy Dissipation Equation ................................ 45
3.5.3 Comparison of DCM and CFDLIB................................................ 52
3.5.4 Comparison of DCM/CFDLIB and Exprimental Data .................. 58
3.5.5 Case Studies by DCM .................................................................... 63
v
3.6 Concluding Remarks ..................................................................................... 69
4. Discrete Cell Model Approach Revisited: II. Two Phase Flow Modeling............ 71
4.1 Introduction ................................................................................................... 71
4.1.1 Spatial Sacles in Trickle Beds........................................................ 73
4.1.2 Governing Principles for Flow Distribution................................... 73
4.2 Extended Discrete Cell Model ...................................................................... 75
4.3 Modeling Results and Discussion ................................................................. 79
4.3.1 Comparison of DCM and CFD Simulations .................................. 81
4.3.2 Effect of Liquid Distributor............................................................ 86
4.3.3 Effect of Particle Prewetting .......................................................... 90
4.4 Conclusions and Final Remarks.................................................................... 94
5. Computational Fluid Dynamics (CFD): I. Modeling Issues ................................. 96
5.1 Introduction and Background........................................................................ 96
5.1.1 CFD Applied to Multiphase Reactors ............................................ 96
5.1.2 CFD and Other Modeling Approaches to Multiphase Flow in
Packed Beds ............................................................................................ 100
5.2 Spatial and Temporal Characteristics of Flow in Packed Beds .................... 102
5.3 Structure Implementation.............................................................................. 104
5.4 k-Fluid Approach and CFDLIB Code ........................................................... 108
5.4.1 Eulerian k-Fluid Model .................................................................. 108
5.4.2 k-Fluid Model in CFDLIB ............................................................. 109
5.5 CFD Modeling Issues.................................................................................... 113
5.5.1 Significance of Terms in the Momentum Balance......................... 113
5.5.2 Closures for Multiphase Flow Equations ....................................... 115
5.5.3 Interfacial Tension Effect, Wetting Correction.............................. 120
vi
5.5.4 Effect of Mesh Size on Computated Results.................................. 123
5.5.5 Boundary Conditions ..................................................................... 129
5.6 Conclusions and Remarks ............................................................................. 130
6. Computational Fluid Dynamics (CFD): II. Numerical Results &
Compariosn with Experimental Data ......................................................................... 131
6.1 Introduction ................................................................................................... 131
6.2 Comparison of CFD Simulation and Experimental Results.......................... 133
6.2.1 Liquid Upflow in Packed Beds ...................................................... 134
6.2.2 Gas and Liquid Cocurrent Downflow in Trickle Beds .................. 142
6.3 Simulation of Feed Distribution Effects........................................................ 149
6.4 Conclusions ................................................................................................... 153
7. CFD Applications in Scale-Down and Scale-Up of Packed-Bed Reactors........... 154
7.1 Introduction ................................................................................................... 154
7.2 Model Bench-Scale Packed Beds.................................................................. 157
7.3 Modeling Results of Bench-Scale Packed Beds ........................................... 159
7.4 Statistical Nature of the Bed Structure and Flow.......................................... 171
7.4.1 Bed Structure.................................................................................. 171
7.4.2 Multiscales of Flow and Role of Various Forces. .......................... 172
7.4.3 Link of Macroscale and Cell-Scale Hydrodynamics...................... 173
7.4.4 Statistical Quantities....................................................................... 173
7.5Modeling Result and Correlation Development............................................. 174
7.5.1 Model Packed Beds........................................................................ 174
7.5.2. Capillary Force Effect ................................................................... 176
7.5.3. Porosity Distribution Effect .......................................................... 180
vii
7.5.4. Correlation Development .............................................................. 181
7.5.5. Superficial Velocities at the Inlet .................................................. 184
7.6 Conclusions and Remarks ............................................................................. 187
8. A Combined k-Fluid CFD Model and the Mixing-Cell Network Model ............. 188
8.1 Introduction ................................................................................................... 188
8.2 k-Fluid CFD Model for Flow Simulation .................................................... 192
8.3 Mixing-cell Network Model.......................................................................... 197
8.4 Concluding Remarks ..................................................................................... 203
9. Thesis Accomplishments and Future Work ........................................................... 205
9.1 Summary of Thesis Accomplishments.......................................................... 205
9.1.1 Discrete Cell Model (DCM)........................................................... 205
9.1.2 k-Fluid CFD Model........................................................................ 206
9.1.3 Mixing-Cell Network Model.......................................................... 207
9.2 Recommendations for Future Research ........................................................ 208
9.2.1 Discrete Cell Model (DCM)........................................................... 208
9.2.2 k-Fluid CFD Model........................................................................ 208
9.2.3 Mixing-Cell Network Model.......................................................... 210
Appendix Comparison between Trickle Bed and Packed Bubble Column
Reactor Performance for the Hydrogenation of Biphenyl ........................................ 211
A1 Introduction ................................................................................................... 211
A2 Reactor Models.............................................................................................. 213
A2.1 Kinetic Model................................................................................. 213
A2.2 Key Assumptions ........................................................................... 214
A2.3 Cocurrent Trickle Bed and Packed Bubble Flow Bed Model ........ 214
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A3 Results and Discussion.................................................................................. 216
A3.1 Flow Characteristics and Flow Regimes ........................................ 216
A3.2 Trickle-bed Reactor Performance .................................................. 217
A3.3 Packed Bubble Flow Reactor Performance.................................... 218
A3.4 Sensitivity of Model Parameters .................................................... 219
A4 Conclusions ................................................................................................... 220
References ...................................................................................................................... 228
Vita.................................................................................................................................. 244
ix
Tables Table Page
3-1. Dimensions of the model bed and physical properties of the fluids in the simulations
............................................................................................................................... 45
4-1. Summary of operating conditions used in flow simulations...................................... 87
5-1. Current Status of CFD Modeling in Multiphase Reactors......................................... 99
5-2. Typical Ranges of Force Rations in Two-Phase Flow in Packed Granular packing
(adapted from Melli et al., 1990)......................................................................... 114
5-3. Models for Drag Coefficients .................................................................................. 119
6-1. Statistical Description of Porosities and CFD Simulated Velocities ....................... 136
6-2. Parameters Used in the Discretization of the Radial Porosity Profile, and in the
Generation of 2D Porosity Distribution .............................................................. 149
6-3. Feed Velocities and Holdups at to Ten Sections from the Center to the Wall ........ 151
7-1. Statistical description of the porosity distribution ................................................... 176
A-1. Kinetic parameters for hydrogenation of biphenyl (Sapre and Gates, 1981).......... 221
A-2. Summary of various correlations used in this study ............................................... 221
A-3. Properties of the catalyst particles .......................................................................... 221
A-4. List of gas and liquid feed velocity......................................................................... 222
x
Figures Figure Page
2-1a. Schematic diagram of 2-D rectangular packed bed ................................................. 14
2-1b. Experimental setup using CCD video camera imaging technique........................... 14
2-2. 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 .......................................................................................... 18
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)..................... 19
2-4. Cross-sectional liquid distribution in a 3D-rectanglar non-prewetted bed of glass
beads (dp = 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 single-
point 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..................... 19
2-5. Local liquid distribution at (a) the top region and (b) the bottom region at a mass
superficial velocity of 7.04 kg/m2/s in a non-prewetted bed.................................... 20
2-6. The steady state liquid distribution in a prewetted bed at different liquid superficial
mass velocities.......................................................................................................... 20
xi
2-7. The development of finger-type liquid flow in a prewetted bed at a superficial mass
velocity of 0.74 kg/m2/s ( t: starting time, second) ................................................. 21
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)..................... 22
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. (1997) at T = 25 C, P = 1 atm.]............................................... 23
2-10. 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.]. 23
2-11. Packing image taken from the front of the 2-D rectangular packed bed ................. 24
2-12. Schematic diagram of the experimental setup for a 3D column with exit flow
measurement and periodic liquid feed controller..................................................... 27
2-13. Liquid collector with 25 individual tubes located at the bottom of the packed bed 28
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) ............................................................ 28
2-15. 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 = 0.049 m/s, uniform liquid inlet) ........................................................ 29
2-16. 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)............................... 30
2-17. 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) ........................ 31
xii
3-1. Model packed bed ('2D' rectangular as example) and velocity at each interface of cell
j. (Note that Sx,j equals to Sx+∆x ,j in the '2D' rectangular packed bed) ...................... 38
3-2. Porosity distribution of model bed (32 cells x 8 cells). ............................................. 44
3-3. Contribution of each energy dissipation rate term at each cell to the total energy
dissipation rate. V0 = 0.5 m/s (gas flow without internal obstacles); Re'= 28.51 ..45
3-4a. Contribution of each energy dissipation rate term at each cell to the total energy
dissipation rate. V0 = 0.5 m/s (gas flow with two internal obstacles at Z/dp = 30,
66); Re' =28.5 .......................................................................................................... 48
3-4b. Contribution of each energy dissipation rate term at each cell to the total energy
dissipation rate. V0 = 0.5 m/s (gas flow with two internal obstacles at Z/dp = 30,
66); Re' =28.5 (zoom-in) .......................................................................................... 48
3-5. Contribution of each energy dissipation rate term at each cell to the total energy
dissipation rate. V0 = 0.1 m/s (liquid flow without internal obstacles); Re' = 47.5 . 50
3-6. Comparison of CFD simulations and DCM predictions at a superficial velocity of 0.1
m/s at different axial positions (Z/dp) (Re' = 5.7)..................................................... 54
3-7. Comparison of CFD simulations and DCM predictions at a superficial velocity of 0.5
m/s at different axial positions (Z/dp) (Re' = 28.5)................................................... 55
3-8. Comparison of CFD simulations and DCM predictions at a superficial velocity of 0.5
m/s at different axial positions (Z/dp) (Re' = 170.1)................................................. 56
3-9a. Comparison of superficial velocity (Vj) between CFD and DCM predictions for gas
flow in the Reynolds number (Re') range of 5 to 171. ............................................. 57
3-9b. Comparison of relative interstitial velocity (Uj/U0) between CFD and DCM
predictions for gas flow in the Reynolds number (Re') range of 5 to 171. (U0 =
V0/εB) ....................................................................................................................... 58
xiii
3-10a. Comparison of predicted interstitial velocity component in the Z direction (Uz) by
two methods in liquid up-flow system: liquid superficial velocity V0 = 0.1 m/s (Re'
= 47.5). ..................................................................................................................... 59
3-10b. Comparison of predicted interstitial velocity component in X direction (Ux) by
two methods in liquid up-flow system: inlet liquid superficial velocity V0 = 0.1 m/s
.................................................................................................................................. 60
3-11a. Influence of gas feed superficial velocity on DCM predicted cell interstitial
velocity profiles........................................................................................................ 61
3-11b. Effect of particle Reynolds number (Rep) on the calculated relative cell superficial
velocity profile inside a bed using DCM ................................................................. 61
3-12. Comparison of experimental data of Stephenson and Stewart (1986) and CFDLIB
simulated results for relative velocity in a packed bed with D/dv = 10.7 and dv =
0.7035 cm (cylindrical particles). Physical properties of liquid: Liquid -B for
condition at a Rep of 5, ρ = 1.125 g/cm3; µ = 0.474 g cm/s. Liquid -C for condition
at a Rep of 80, ρ = 1.027 g/cm3; µ = 0.114 g cm/s ................................................... 62
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) (U0 =
120.5 cm/s); (velocity vector plotting). .................................................................... 65
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) (U0=120.5 cm/s)
(point source inlet from left side, inlet point superficial velocity is of 4.0 m/s)
(velocity vector plotting).......................................................................................... 66
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
xiv
plates are placed in this bed, one is located at Z/dp of 66 (at the left side), another
is at Z/dp 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). ................................. 67
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, ψρG
G gPZ
=
1 ∆∆
is plotted). Two obstacle
plates are placed in this bed, one is located at Z/dp of 66 (at the left side), another is
at Z/dp 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) ..................................... 68
4-1. Tne coordinate system and velocity conventions for α phase in the cell .................. 76
4-2a. Local porosity distribution in model bed; Random internal porosity (0.36 ~ 0.44).
Darker color corresponds to higher porosity............................................................ 80
4-2b. Average porosity profiles in X and Z directions in model bed................................ 80
4-3a, 3b, 3c and 3d. Comparison of the predicted gas interstitial velocities (relative) at the
specific axial level by DCM and CFD. Ul = 0.00148 m/s (UF); Ug = 0.05 m/s (UF);
Completely prewetted packed bed. (The relative interstitial velocity is defined as
the local interstitial velocity (Vi) divided by the overall interstitial velocity (V0).
The value of V0 in this case is equal to 0.1205 m/s). ............................................... 83
4-3e. Comparison of the predicted gas interstitial velocities (relative) for all the cells by
DCM and CFD. Inlet superficial velocities (uniform): Ul = 0.00148; Ug = 0.05m/s;
Completely prewetted packed bed ........................................................................... 84
4-4. Comparison of predicted liquid holdup at specific levels by DCM and CFD. Single
point source liquid inlet: Ul = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05m/s;
Non-prewetted packing ............................................................................................ 85
xv
4-5a. Liquid holdup distribution with single liquid point source inlet (located at No. 5
cell from left) by DCM. Ul = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05m/s;
Non-prewetted packing ............................................................................................ 87
4-5b. Liquid holdup distribution with two liquid points source inlet (located at No. 3 cell
and No. 6 cell from left) by DCM. Ul = 0.00148 m/s (Ul (PS2)=0.00592 m/s); Ug =
0.05m/s; Non-prewetted packing ............................................................................. 88
4-5c. Liquid holdup distribution in whole domain of the non-prewetted packed bed with
uniform liquid distributor by DCM. Ul = 0.00148 m/s; Ug = 0.05m/s..................... 89
4-5d. Comparison of liquid flow maldistribution calculated by DCM along the bed for
different liquid distributors. Ul = 0.00148 m/s; Ug = 0.05m/s. ................................ 90
4-6a. Liquid holdup distribution in the whole domain of the completely prewetted packed
bed (f = 1). Ul = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05 m/s; Point liquid
distributor (PS1)........................................................................................................ 92
4-6b. Liquid holdup distribution at specific levels (Z/dp) in the completely prewetted
packed bed (f = 1), Ul = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05 m/s; Point
liquid distributor (PS1); Overall liquid holdup = 0.0758.......................................... 93
4-6c. Liquid holdup distribution at specific levels (Z/dp) in the completely non-prewetted
packed bed (f = 0), Ul = 0.00148 m/s (Ul (PS1) = 0.01184 m/s); Ug = 0.05 m/s; Point
liquid distributor (PS1); Overall liquid holdup = 0.0716.......................................... 93
5-1. Generated pseudo-Gaussian distribution of porosity under three constraints: (1) ε0 =
0.36; (2) Longitudinally averaged radial porosity profile (white filled circles)
reported by Stephenson and Stewart (1986). (Dr = 7.6 cm, dp = 0.703 cm, Section
size = 0.05R = 0.19 cm). (a)-contour plot; (b)-radial profiles; (c)-histogram
(standard deviation of porosity, σB = 12% ε0). ...................................................... 108
xvi
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. ...................................................................................................... 112
5-3. Comparison of Xkl 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). ............................. 117
5-4. 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. ................................................................................................. 118
5-5. Comparison of the calculated capillary pressure values from two different
expressions, Eq 5-17a and 5-17b for air-water system (dp = 0.003m; θs =0.63). .. 120
5-6. Simulated liquid upflow velocity component, Vx contour (a) and profiles (b) using
mesh1 (10 × 20) and mesh2 (20× 40) at a superficial velocity of 10 cm/s ............ 125
5-7. Simulated liquid upflow velocity component, Vz contour (a) and profiles (b) using
mesh1 (10 × 20) and mesh2 (20× 40) at a superficial velocity of 10 cm/s ............ 126
5-8. Initial solid volume fraction distribution at 10 ×15 section-discretization (section size
= 1.0 cm) for gas-liquid cocurrent downflow simulation (zoom: x = 0 ~ 4; z = 4 ~
8)............................................................................................................................. 127
xvii
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)..................................................................... 128
5-10. Effect of the mesh sizes (a, b, c) on the cell-scale gas holdup values ................... 129
6-1a. 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 6.1. ................................................................ 136
6-1b. 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 6.1. ...................................... 137
6-2a. Comparison of longitudinally averaged radial velocity profiles at different Reynolds
numbers and experimental data of Stephenson & Stewart (1986). ........................ 138
6-2b. 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). ...................................................................... 139
6-3a. 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) ............. 140
6-3b. Frequency distribution of axial interstitial velocity (Re = 280): RN1-CFD
simulation based on random porosity 1 (ε: std/µ = 0.0916/0.3534; Vx: std/µ
=1.879/0.2034; Vz: std/µ = 3.864/7.0915); RN2-CFD simulation based on random
porosity 2 (ε: std/µ = 0.0916/0.3534; Vx: std/µ =1.879/0.2034; Vz: std/µ =
3.864/7.0915). Exp. –Experimental data reported by Stephenson and Stewart (1986)
................................................................................................................................ 141
xviii
6-4a. Discretization of the radial porosity profile into sectional porosity values (dp =
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%..................................................................... 143
6-4b. Solid volume-fraction distribution generated based on the data in Table 2 in a pilot
scale packed bed..................................................................................................... 145
65. 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..................................................................................................................... 146
6-6. 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.......... 147
6-7. 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...................................................................................................................... 148
6-8a. Comparison of liquid holdup distribution under nonuniform (left) and uniform
(right) feed conditions at Ul0 = 0.295 cm/s, Ug0 = 22 cm/s .................................... 151
xix
6-8b. Comparison of gas holdup contour and gas interstial velocity vector plot under
nonuniform (left) and uniform (right) feed conditions at Ul0 = 0.295 cm/s, Ug0 = 22
cm/s ........................................................................................................................ 152
7-1. 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). ......................................................... 159
7-2. 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 Ug0 = 6
cm/s and Ul0 = 0.3 m/s at steady state operation. ................................................... 162
7-3a.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 (Ul0
= 0.05 cm/s; Ug0 = 6.0 cm/s). ................................................................................. 163
7-3b.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 (Ul0 = 1.0
cm/s; Ug0 = 12.0 cm/s). .......................................................................................... 163
7-4a. Relative interstitial velocity profiles of the gas and liquid phase, porosity profile in
the radial direction at low flow rates (Ul0 = 0.05 cm/s; Ug0 = 6.0 cm/s). ............... 164
7-4b. Relative interstitial velocity profiles of the gas and liquid phase, porosity profiles in
the radial direction at low flow rates (Ul0 = 1.0 cm/s; Ug0 = 12.0 cm/s). ............... 164
7-5. Histogram of the relative interstitial velocities of the gas and liquid phase at low flow
rates (1: Ul0 = 0.05 cm/s; Ug0 = 6.0 cm/s) and high flow rates (2: Ul0 = 1.0 cm/s; Ug0
= 12.0 cm/s)............................................................................................................ 165
xx
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 Ug0 =6 cm/s
and Ul0 =0.3 cm/s. .................................................................................................. 167
7-7. (a) Solid volume-fraction (THE1 = 1.0 - 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)... .................................................................................................... 169
7-8. Comparison of cross sectional liquid holdup profiles at different axial locations
under steady (filled squares) and unsteady state operation ((a), Z= 1.8 cm; (b), Z=
18.9 cm; (c), Z= 26.1 cm); (c), Z= 28.8 cm) (!-15s; !-25s; "-40s; "-55s).. .. 170
7-9. Trickle bed and model bed with 500 cells ............................................................... 175
7-10. 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 Ul = 0.3 cm/s, Ug = 6.0 cm/s ......................... 178
7-11. 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 Ul = 0.3
cm/s, Ug = 6.0 cm/s (G-gas, L-liquid) ................................................................... 179
7-12. Contours of solid volume fraction (=1.0-porosity) distribution of model beds (II,
III, IV) for CFDLIB ............................................................................................... 181
7-13. Contours of CFDLIB simulated liquid volume fraction (holdup) distribution in
model beds (II, III, IV) ........................................................................................... 182
7-14. 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
xxi
7-15. Standard deviation (S.D.) of the liquid holdup distribution vs. standard deviation
of the bed porosity at two wetting limits at Ul = 0.3 cm/s, and Ug = 6.0 cm/s....... 184
7-16. Liquid holdup (filled squares) (Holub et al., 1992) and particle external wetting
efficiency from correlation (empty circle) (Al-Dahhan & Dudukovic, 1995)....... 185
7-17. Liquid holdup values from CFDLIB simulations (mean +/- S.D.) and Holub
correlation (1992): εB = 0.399; dp = 3 mm; Ug = 0.03 m/s (hL: liquid holdup) ..... 186
8-1a. Two-dimensional interconnected cell network ...................................................... 191
8-1b. Fluid superficial velocities and concentrations of species i at interface of the cell j,
where C3 i,j = C4 i,j due to the well-mixing ............................................................... 191
8-2a. Porosity distribution at spatial resolution of 1 cm (dp = 3 mm, L = 50 cm, D = 10
cm).......................................................................................................................... 193
8-2b. Histogram of porosity distribution (Gaussian distribution) used in the k-fluid CFD
model...................................................................................................................... 194
8-3a. Simulated liquid superficial velocity component (Ux, m/s): Ul0 = 0.003 m/s; Ug0 =
0.06 m/s .................................................................................................................. 195
8-3b. Simulated liquid superficial velocity component (Uz, m/s): Ul0 = 0.003 m/s; Ug0 =
0.06 m/s .................................................................................................................. 196
8-4. Computed cell scale mass transfer coefficient (kls, cm/s): Ul0 = 0.003 m/s; Ug0 = 0.06
m/s .......................................................................................................................... 197
8-5. Cells with different inflow and outflow configurations........................................... 200
8-5. Concentration contour of species B in the liquid phase (m = 0.0; n = 1.0; r = 1.0),
CBl, 0 = 5.4 kmol/m3, Ul0 = 0.003 m/s; Ug0 = 0.06 m/s............................................ 201
xxii
8-7. Longitudinally averaged concentration profile of species B and liquid velocity
component (Uz) profiles in the X direction. (CBl - filled circle; Uz – blank square;
CBl, 0 = 5.4 kmol/m3; Ul0 = 0.003 m/s; Ug0 = 0.06 m/s).......................................... 202
8-8. Calculated concentration profiles of species B at k = 1.E-04 m3/kg.s, n = 1.0, m =
0.0 by (i) plug flow; (ii) ADM (De = 2.53E-04 m2/s calculated from Sater and
Levenspiel, 1966, Pe = 5.92); (iii) Mixing-cell network model; (iv) ADM (adjusted
De = 1.5E-04 m2/s, Pe = 10). CBl, 0 = 5.4 kmol/m3; Ul0 = 0.003 m/s; Ug0 = 0.06 m/s
................................................................................................................................ 203
A-1. Contacting pattern in the trickle flow regime and bubble flow regime .................. 222
A-2. Effect of pressure on the flow regime in downflow packed bed using the flow chart
of Larachi et al. (1993)........................................................................................... 223
A-3. Species concentration profiles along the reactor. D = 11.3 m; Ul = 0.0075 m/s; Ug =
0.06 m/s; P = 70 atm; T = 603K; CaL : hydrogen; CbL : biphenyl; CcL : product .... 223
A-4. Effect of operating pressure on the exit biphenyl conversion and global
hydogenation rate (Re). H= 4 m, D = 11.3 m ........................................................ 224
A-5. Effect of liquid superficial velocity on the exit biphenyl conversion and global
hydrogenation rate (Rg). H = 4 m; D = 11.3 m...................................................... 224
A-6. Effect of liquid superficial velocity on the exit biphenyl conversion in the upflow
packed bed.............................................................................................................. 225
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; Ul =
0.0075 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 = 0.0144 s-1, 0.1× (ka)ls = 0.129 s-1; for the down-
flow mode, 0.1×(ka)gl = 0.00344 s-1, 0.1× (ka)ls = 0.0338 s-1. (For the directly
xxiii
calculated values of mass transfer coefficients, the corresponding exit biphenyl
conversion for up-flow is 72.35%, and for down-flow, 71.56%) .......................... 226
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 ............................................................................................. 227
xxiv
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 project brimmed with
challenges, which made my first-year life in the United States full of excitant. Many
enjoyable hours were spent with my co-worker, M. R. Khadilkar, from whom I have
learned so much in professional attitude, which made the project 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.
xxv
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. Z. Xu, J. Xue 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 enjoy the three-month internship at DuPont
Company sponsored by Conoco Inc. in 1999. 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 1989. Thank you for
xxvi
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
xxvii
Nomenclature In Chapter 2 dsp the radius of the liquid channel (i.e., filament)
G gas superficial mass velocity, kg/m2/s
H length of paced bed, m
L liquid superficial mass velocity, kg/m2/s
P pressure, atm
R radius of packed column, m
r radial distance from the center, m
T temperature, °C
Vav cross-section averaged liquid volumetric flow rate, m3/s
V measured liquid volumetric flow rate, m3/s
In Chapters 3 & 4 a constant in Leverett’s function (= 0.48 for air-water system)
aj, working variable (E1(1-εj)2µα/(ραε3jdp
2))
b constant in Leverett’s function (= 0.036 for air-water system)
bj working variable (E2(1-εj)/(ε3jdp))
D width of model bed, m (= 0.072 m), 8 cells
dp particle diameter, m (= 0.003 m)
dv equivalent diameter of particle (m)
E1, E2 Ergun constants
Ev,j mechanical energy dissipation in the cell jth, J/s (based on Vj)
Ev,j,α energy dissipation rate of phase α in the jth cell, J/s (based on Vj,α)
xxviii
Ev, bed total mechanical energy dissipation rate in the bed, J/s
f particle wetting factor
f1,, f1,j resistance factor, 150(1-εj)2µ/(ρ εj3
φ2dp2 )
f2, f2,j resistance factor, 1.75(1-εj) ρ/(εj3 φdp)
Gaα, j Cell Galileo number of the α phase, gdP3εj
3/(µα2(1-εj)3)
gi, i=x,z gravitational acceleration in the i direction, (gx = 0; gz = 9.8 m/s2)
H height of model bed, m (= 0.288 m), 32 cells
N total number of the cells (8 × 32 = 264)
Nc number of cells in each row
Pc pressure at the center of the cell, N/m3
P0 pressure, dyn/cm2
Pz pressure in the z direction, N/m2
kp0 pressure ( pp k −0 non-equilibrium pressure)
∆P/∆Z pressure drop per unit cell length, N/m3
r number of cells in each row (X direction)
R Radius of packed beds, m
Re’ Reynolds number, V0dpρ/6/(µ (1-εB))
Rep particle Reynolds number, V0dPρ/µ
Reα, j cell Reynolds number of the α phase, VαdP/(µα(1-εj))
si cell face area at a given coordinate direction i ,m2
Sw, j liquid saturation in cell j (εL,j /εj)
Ti energy dissipation rate due to the inertial term, J/s
Tk energy dissipation rate due to the kinetic term, J/s
Tv energy dissipation rate due to the viscous term, J/s
Uj local interstitial velocity, m/s (=Vj / εj)
u0 material velocity, (cm/s)
ku material k interstitial velocity, cm/s (ρkuk ≡ < αkρ0u0 >)
ku' fluctuating part of material k interstitial velocity, cm/s
xxix
U0 input interstitial velocity, m/s (=V0 / εB)
V velocity vector
Vj, α superficial velocity of phase α in the jth cell, m/s
(Vol)j volume of cell j , m3 (=∆Z×∆X× ∆Y for rectangular bed)
Vc, j volume of the cell j, m3 (=S z,j×∆Z)
Vj superficial velocity in the jth cell, m/s
V0 input superficial velocity, m/s
∆Z,∆X,∆Y size of the cell, m (in this work, 3dp= 0.009 m)
Greek Letters
kα material indicator (=1 if material k is present; =0 otherwise)
kα! material derivative
εB bed porosity (= 0.415)
εj porosity in the jth cell
εj, α holdup of phase α in the jth cell
σ liquid surface tension, N/m (0.072 for water)
γi, 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:1.8×10-5 Pa s; liquid:1.0×10-3 Pa s)
µα viscosity of phase α in the bed, Pa s (µL=1.0e-3 Pa s; µG=1.8e-5 Pa s)
θ k material k volume fraction (θ k = < α k > )
τ0 deviatiric stress
ρ density of fluid, kg/m3 ( gas: 1.2 kg/m3; liquid: 1000 kg/m3)
ρk density of material k, g/cm3 (≡ < αkρ0 >)
ρα density of phase α in the bed, kg/m3 (ρL=1000 kg/m3; ρG=1.2 kg/m3)
ψG local gas flow dimensionless pressure drop, ψρG
G gPZ
=
1 ∆∆
xxx
ψα,j dimensionless pressure-drop for phase α, (= ∆ ∆P Lg
α
αρ/
+1)
Subscripts
X x coordinate for the rectangular cell or bed
Z axial coordinate along the length of bed
< > ensemble average (note: cross-sectionally average in Eq.5-15)
In Chapters 5 & 6 Agl, Ags, Aks parameters defined in Table 5-3
Bgl, Bgs, Bks parameters defined in Table 5-3
Bo Bond number
Ca Capillary number
C1,j, C2,j inflow concentration of species i in j-cell
C3,j, outflow concentration of species i in j-cell
dp particle diameter, m
de equivalent diameter of particle
dmin minimum equivalent diameter of the area between three spheres in contact
( pdd5.0
min 5.03
−=
π)
Dr diameter of packed bed, m
E1, E2 Ergun constants (E1 = 180; E2 = 1.8)
f particle wetting factor
F pressure factor
FD(k-l) Drag between phases k and l
g gravity, 9.81m/s2
H height of model bed, m
J J-function
kr relative permeability parameter in Eq (5-17)
ls size of section, m
xxxi
n time step number
p, P pressure, N/m3
Pc capillary pressure, N/m3
PG pressure in gas phase, N/m3
PL pressure in gas phase, N/m3
P0 pressure, dyn/cm2
kp0 pressure (=k
k pθ
α >< 0 )
r radial position in cylindrical coordinate, m
R Radius of packed beds, m
Rep Reynolds number, V0dpρ /µ
Sk saturation of phase k
S.D. standard deviation
t time, s
THE1, 2, 3 solid, liquid and gas volume fraction
u0 material velocity, (cm/s)
ku material k interstitial velocity, cm/s (ρkuk ≡ < αkρ0u0 >)
ku' fluctuating part of material k interstitial velocity, cm/s
U0 input superficial velocity, m/s (=V0 × εB)
U1,j, U2,jinflow velocity of j-cell
U3,j, outflow velocity of j-cell
Ug gas superficial velocity, m/s
Ug0 gas feed superficial velocity, m/s
Ul liquid superficial velocity, m/s
Ul0 liquid feed superficial velocity, m/s
V0 input interstitial velocity, m/s (=U0 / εB)
Vg gas interstitial velocity, m/s
Vg0 gas feed interstitial velocity (= Ul0 / εB), m/s
Vl liquid interstitial velocity, m/s
xxxii
Vl0 liquid feed interstitial velocity (= Ug0 / εB), m/s
Vx interstitial velocity component in horizontal or radial direction, cm/s
Vz interstitial velocity component in axial direction, cm/s
V velocity vector
Z axial position, cm or m
Xkl momentum exchange coefficient between phases k and l
Greek Letters
kα material indicator (=1 if material k is present; =0 otherwise)
kα! material derivative
εB mean porosity of packed bed
εg gas holdup
εl liquid holdup
εl0 static liquid holdup
φ particle shape factor, (φ = 1 for Spherical particle)
µ viscosity of fluid, Pa s
kθ material k volume fraction (θ k = < α k > ), k = G, L, S
0Lθ static liquid holdup
τ0 deviatiric stress
ρ density of fluid, kg/m3
ρk density of material k, g/cm3 (≡ < αkρ0 >)
ρ0 material density, kg/m3
ρg density of gas phase, kg/m3
ρl density of liquid phase, kg/m3
σs surface tension
σB standard deviation of porosity distribution
< > ensemble average
∇⋅ divergence
xxxiii
ϕ angular coordinate
In Chapters 7 dp particle diameter, m
E1, E2 Ergun constants (E1 = 180; E2 = 1.8)
f fractional wetting value
f (xj) probability density function
FD Drag force
g gravity, cm/s2
P0 pressure, dyn/cm2
S.D. standard deviation
t time, s
V2 interstitial liquid velocity components, cm/s
V3 interstitial gas velocity components, cm/s
Vr superficial relative velocity based on gas flow, as defined in Eq (7d), cm/s
Vx, Vz interstitial velocity components, cm/s
u0 material velocity, (cm/s)
|ukl| slip interstitial velocity between phase k and phase l, cm/s
ku material k interstitial velocity vector, cm/s
ku' fluctuating part of k interstitial velocity vector, cm/s
U0 input superficial velocity, cm/s
Ul0 liquid superficial velocity, cm/s
Ug0 gas superficial velocity, cm/s
x horizontal position in x-z coordinate
Xkl momentum exchange coefficient between phase k & l
xj variable of system
z axial position in x-z coordinate
xxxiv
Greek Letters
α1, α2, α3 parameters
kα material indicator (=1 if k is present; =0 otherwise)
kα! material derivative
εB bed porosity
ε section porosity
ε(r) longitudinally averaged radial porosity profile
ε(z) cross-section averaged porosity profile
θ k material k volume fraction (θ k = < α k > )
τ0 deviatiric stress
ρ density of fluid, kg/m3 ( gas: 1.2; liquid: 1000)
ρk density of material k, g/cm3 (≡ < αkρ0 >)
µ mean value
µα viscosity of phase α
σs surface tension
σB standard deviation of porosity distribution
σl standard deviation of liquid holdup
γ1 Skewness of statistical data
γ2 Kurtosis of statistical data
< > ensemble averaged
In Chapter 8 a0 a basis for cell cross-section area, m2
ak k interface area of cell, m2
aL gas-liquid mass transfer area per unit cell volume, m2/m3
CAg,0 concentration of A in the feed gas, kmol/m3
CAg,k concentration of A in the gas phase enter the cell, kmol/m3
CAg,out concentration of A in the gas phase leaving the cell, kmol/m3
xxxv
CAl,k concentration of A in the liquid phase enter the cell, kmol/m3
CAl,out concentration of A in the liquid phase leaving the cell, kmol/m3
CAs concentration of A at particle surface, kmol/m3
CBl,0 concentration of B in the feed gas, kmol/m3
CBl,k concentration of B in the liquid phase enter the cell, kmol/m3
CBl,out concentration of B in the liquid phase leaving the cell, kmol/m3
CBs concentration of B at particle surface, kmol/m3
De effective intraparticle diffusivity of the species, m2/s
HA Henry's law solubility coefficient of A, Ag/Al
kr reaction-rate constant [m3/kg.s][m3/kmol] m+n-1(i.e., 1.e-4)
k* dimensionless rate constant
kl liquid-film mass transfer coefficient, m/s
kg gas-film mass transfer coefficient, m/s
ks solid-film mass transfer coefficient, m/s
KL overall gas-liquid mass transfer coefficient, m/s, defined as
( ) ( ) ( )ALgAALlALL akHakaK
111 +=
Sp external surface area of a catalyst particle, m2
t time, s
Ug superficial gas velocity, m/s
Ul superficial liquid velocity, m/s
Vc volume of cell, m3
Vp volume of a catalyst particle, m3
xAl dimensionless concentration of A in liquid phase
xBl dimensionless concentration of B in liquid phase
xBs dimensionless concentration of B at particle surface
yAg dimensionless concentration of A in gas phase
yAs dimensionless concentration of A at particle surface
xxxvi
Greek Letters
αA, αB dimensionless gas-liquid mass transfer coefficients defined by Eq (6)
βA parameter defined as Eq (6)
ε void fraction in the jth cell
η catalyst effectiveness factor, defined as ηφ φ φ
= −
1 13
13tanh
ηCE catalyst particle wetting efficiency
γ stoichiometric coefficient of A in the reaction
γA, γB parameters defined as Eq (6)
ρ density of the catalyst particle, kg/m3 (=ρp = 2500 kg/m3)
φ Thiele modulus, defined as ( )φ ρ=
∫−
VS
kA B D r c dcP
PP j I
mj In
e c
Cj I
, , ,
.,
20
0 5
Ω Reaction rate per unit volume of catalyst particle, kmol/m3.s
In Appendix Ci,L, concentration of species i in liquid phase
ci,L, dimensionless concentration of species i in liquid
DEL,i diffusivity of species i in liquid phase
G gas mass superficial velocity, kg/m2/s
Ug gas superficial velocity, m/s
uSL, Ul liquid superficial velocity, m/s
ki 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/m2/s
T temperature, K
P operating pressure, atm
Pe Péclet number
Qg gas volumetric flow rate, m3/s
xxxvii
Ql liquid volumetric flow rate, m3/s
ri reaction rate based on the species I
Rv global hydrogenation rate, mol/s/m3
X conversion of biphenyl, %
Greek Letters
αG,L dimensionless parameter
αL,S dimensionless parameter
β1,,β1, dimensionless parameter
ξ dimensionless position
Subscripts
A hydrogen
B biphenyl
C cyclohexylbenzene
D H2S
e equilibrium
i input
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 subjects 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 major 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
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 structure-
flow relation is the main concern when one uses this relationship in design practice.
The major 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 1.1.1 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
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 justification 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
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.
1.1.2 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
Swaaij, 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.
1.1.3 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),
5
and other models based on liquid flow maldistribution (Funk et al., 1990), the stagnant
liquid zones (Rajashekharam 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
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 Objectives The overall objectives of this study are outlined below. Details of the
implementation of each are discussed in Chapter 2 through Chapter 9.
1.2.1 DCM Revisited The objectives 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.
1.2.2 k-Fluid CFD Model for Packed Beds The objectives 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.
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.
1.2.3 Applications of k-Fluid CFD Model The objectives 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.
1.2.4 Impact of Flow Distribution on Reactor Performance The objective
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 mixing-
cell 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
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
objectives 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
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.
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 subject 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 just treats the gas, liquid and even the solid as
continuous but penetrating phases. In fact, such a model has been developed not only for
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
justifying 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 computer-
tomography (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 zooming-
in 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 pilot-
scale 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
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
2.2.1 2-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
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.
2.2.2 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 adjustable 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.
14
2D Rectangular Bed I2D Rectangular Bed I
72 mm
Liquid
Gas
Liquid out
Gas vent
Pres
sure
dro
p I
Pres
sure
dro
p I
Pres
sure
dro
p II
Pres
sure
dro
p II
LiquidLiquidPoint SourcePoint Source
Pack
ing
with
3 m
m sp
here
sPa
ckin
g w
ith 3
mm
sphe
res
280
- 300
mm
280
- 300
mm
Gas - LiquidGas - LiquidSeparatorSeparator
Figure 1. 2D packed-bed I
LiquidLiquidUniform InletUniform Inlet
Liquid Gas
2.2.3 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/m2/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.
2.2.4 Experimental Results and Discussion 2.2.4.1 Non-prewetted bed (dry bed)
The experiments were run in non-prewetted beds at the liquid mass velocity range
from 0.74 to 11.12 kg/m2/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
2D Bed with two phase flow inlet and outlet
CCD Camera
Monito Compute
Background
Figure 2-1a Schematic diagram of 2-D rectangular packed bed.
Figure 2-1b Experimental setup using CCD video camera imaging technique.
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), dsp, increases in the increase of liquid superficial mass velocity, L.
The relationship between L and dsp, however, does not follow the ‘square’ rule (i.e., L ∝
dsp2). 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 jet, 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 jet 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 jet by using the
modified Ergun equation. Once this disc type of initial liquid distribution is formed, the
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 2-2.
2.2.4.2 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.
2.2.4.3 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 trickle-
bed 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
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.
18
L = 0.74 kg/m2/s L = 1.48 kg/m2/s L = 3.52 kg/m2/s L = 7.04 kg/m2/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
19
L = 0.74 kg/m2/s L = 3.52 kg/m2/s L = 7.04 kg/m2/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 = 0H = 0 H = 0H = 0
H = 12H = 12 H = 12H = 12
L = 1 kg/m2.s; G = 0.05 kg/m2.s
Figure 2-4. Cross-sectional liquid distribution in a 3D-rectanglar non-prewetted bed of
glass beads (dp = 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 single-
point 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.
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/m2/s in a non-prewetted bed.
L= 1.48 3.52 7.04 (kg/m2/s)
Figure 2-6. The steady state liquid distribution in a prewetted bed at different liquid superficial mass velocities.
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/m2/s (t: starting time, second).
22
(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).
0
50
100
150
200
250
0 3 6 9 12 15 18 21 24X (dp)
Inte
nsity
of i
mag
e
non-prewetted bedprewetted bed
Axial position: 6 cm down from topdp = 0.3 cm
a. Non-prewetted bed b. prewetted bed
23
0.00.51.01.52.02.53.03.54.0
0.0 2.0 4.0 6.0 8.0Time (hr)
Ra
x 10
8 (gm
ol/c
m3 .s
)
prewetted bed
nonprewettedb d
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].
1.0
1.5
2.0
2.5
3.0
3.5
0.0 2.0 4.0 6.0 8.0
L (kg/m2.s)
Ra
x 10
8 (gm
ol/s
.cm
3 ) prewetted bed
nonprewetted bed
Figure 2-10. 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].
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
2.3.1 Experiment Objectives 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 objectives 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
25
(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
2.3.2 3D Column Setup and Exit Flow Measurement The schematic diagram of the 3D column setup is shown in Figure 2-12. 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/m2/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 just 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/m2/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 adjusted 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.
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/Vav,
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 2-12.
2.3.3 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 2-14. 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
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 2-17.
Figure 2-12 Schematic diagram of the experimental setup for a 3D column with exit flow
measurement and periodic liquid feed controller.
air Tape
1
2
3
4
5
6 Manomet
Timer
Collector and Measuring
Tank
A B
Pum
Packing: 3 mm Glass
28
Figure 2-13 Liquid collector with 25 individual tubes located at the bottom of the packed
bed.
0.000.501.001.502.002.503.003.504.00
1 3 5 7 9 11 13 15 17 19 21 23 25
Tube #
V / V
av
L=0.5 kg/m2.sL=1.0 kg/m2.sL=5.0 kg/m2.sL=10 kg/m2.s
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).
1
2
4
3 5
6 7
89
10
11
12
13 14
15 16
17 18 19
20
21
22 23
24
25
29
0.00
1.00
2.00
3.00
4.00
1 3 5 7 9 11 13 15 17 19 21 23 25
Tube #
V / V
av
L=3.0 kg/m2.sL=5.0 kg/m2.sL=7.0 kg/m2.s
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 = 0.049 m/s, uniform liquid inlet).
30
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 0.2 0.4 0.6 0.8 1(r/R)2
V / V
av
40 (ON)/20(OFF)
30/30
20/40
Figure 2-16. 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.
31
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 0.2 0.4 0.6 0.8 1(r/R)2
V / V
av
40(ON)/20(OFF)
30/30
20/40
Figure 2-17. Effect of time split in On/Off periodic 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).
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
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 major 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 jets 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.
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; Ziolkowska and Ziolkowski, 1993; Cheng and Yuan, 1997; Bey and Eigenberger,
1997; Subagyo et al., 1998). It was noted, however, that in industrial packed beds, some
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 3.2.1 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).
( )VffVP 21 +=∇− (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).
( )[ ] 0VfflnVV 21 =+∇×−×∇− (3-2)
0=⋅∇ V (3-3)
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 two-
and three-dimensional flow in packed beds.
3.2.2 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 adjusting factor, Ziolkowska and Ziolkowski
(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).
3.2.3 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
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).
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 justification 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 objectives 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.
38
3.3 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 (Vz and Vx) 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
X-Z coordinates for either two dimensional rectangular (2D) or three dimensional axi-
symmetric 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 (Si) and the cell volumes (Vc,j).
Figure 3-1. Model packed bed ('2D' rectangular as example) and velocity at each
interface of cell j. (Note that Sx,j equals to Sx+∆x ,j in the '2D' rectangular packed bed).
X
Z
VZ, j
VZ+∆Z, j
VX, j VX+∆X, j
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 jth cell, the rate of energy dissipation in X-Z coordinates can be
expressed by (Eq. 3-4).
−
Φ+
+
⋅
∑
=inj
iiii s
sVs
sVPs
sV
sV
,
4
1
2
21 ρρ
0,
4
1,
2
=
−
Φ+
+
⋅
∑
= jVjoutj
iiiEsVsVPsVVεε
ρεεε
ρ (3-4)
where the superficial velocity (Vj) and the corresponding energy dissipation rate for the
cell (EV,j) are used. Rearrangement of Equation (3-4), by substituting the expression for
the area for each cell interface, yields Equation (3-5).
−+−=
∆+∆+∆+∆+
jSVSVSVSVjVE ZZZZZZ
jj
XXXXXX
j
33
2
33
211
2, εε
ρ
( ) ( ) ZZjZZZZZjZZXXjXXXXXjXX SVPSVPSVPSVP ∆+∆+∆+∆+∆+∆+ −+−+ ,,,,
( ) ( ) ZZjZZZZZjZZXXjXXXXXjXX SVSVSVSV ∆+∆+∆+∆+∆+∆+ Φ−Φ+Φ−Φ+ ,,,,ˆˆˆˆ ρρρρ (3-5)
The difference in potential energy terms ( ∆EP , 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.
( ) ( ) ZZjZZZZZjZZXXjXXXXXjXXP SVSVSVSVE ∆+∆+∆+∆+∆+∆+ Φ−Φ+Φ−Φ=∆ ,,,,ˆˆˆˆ ρρρρ ≅ 0 (3-6)
The pressure terms at each cell interface (e.g. PZ and PZ+∆Z), 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 Z direction, as an
example, the desired relationships can be written as follows.
2Z
ZP
czPZPZZ
∆
∆∆−+=
∆+
(3-7)
40
2Z
ZP
czPZZPZZ
∆
∆∆−−=∆+
∆+
(3-8)
Rearranging Equation (3-5), by substituting Equations (3-6), (3-7) and (3-8), gives:
−+−=
∆+∆+∆+∆+ jSVSVjSVSVE ZZZZZZ
jXXXXXX
jjV
332
332,
112 εερ
( ) ( )ZZjZZZjZCZXXjXXXjXCX SVSVPSVSVP ∆+∆+∆+∆+ −+−+ ,,,,
∆∆−+
∆∆−+
∆∆−+
∆∆−+ ∆+
∆+∆+
∆+jZZ
ZZjZ
ZjXX
XXjX
X
j VZPV
ZPV
XPV
XPVol
,,,,2 (3-9)
For each cell, we can write Equation (3-10) based on the mass balance as follows
( ) ( ) 0,,,, =−+− ∆+∆+∆+∆+ ZZjZZZjZXXjXXXjX SVSVSVSV (3-10)
Since the magnitudes of PCZ and PCX have to be the same at the central point of the cell j,
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 (X and Z) will be used to simplify the equations. For example, for
the Z direction, we have
j,Zj,Zj,2j,Zj,1Z
VVfVfZP1 +=
∆∆
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.
−ερ+−
ερ=
∆+∆+∆+∆+ jSVSVjSVSV21
j,VE ZZ3
ZZZ3Z2
jXX
3XXX
3X2
j
( ) jCjXXXXjjXXjjXjXjjXj VVVfVfVVfVf ,,2
,22
,,1,2
,,22
,,1 ∆+∆+∆+ ++++
41
( ) j,Cj,ZZj,ZZj,j,ZZj,j,Zj,Zj,j,Zj, VVVfVfVVfVf ∆+∆+∆+ ++++ 22
21
22
21 (3-12)
In Equation (3-12) f 1,j and f 2,j are Ergun coefficients (Ergun, 1952) defined as follows
( )( ) 32
2
1
1150
jP
jj, d
fεφε−µ
= (3-13)
( )( ) 32
1751
jP
jj, d
.f
εφε−ρ
= (3-14)
In this study, we use the 'universal values' (E1=150, E2=1.75) to calculate f1,f and
f2,f as done by most other investigators (Vortmeyer and Schuster, 1983; Stanek, 1994;
Bey and Eigenberger, 1997, etc.). Although E1 and E2 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) subject 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
1jj,Vbed,V EMinEMin (3-15)
( ) ( ) 0=−+− ∆+∆+∆+∆+ jZZZZZZjXXXXXX SVSVSVSV (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 Vx and Vz for each cell in the bed.
42
3.4 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:
>=<∇+ kkkkk ut
αρρ∂
∂ρ!. (3-17)
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:
=⋅∇+ kkkkk uu
tu
ρ∂
∂ρ (rate of change in k th phase momentum)
><+ ku αρ !00 (net mass exchange source of k)
><∇⋅− kkk uu ''0ρα (multiphase Reynolds stress)
pk ∇−θ (accln. by the equilibration pressure)
><∇⋅+ 0τα k (accln. due to average material stress)
)( 0 pp kk −∇− θ ( accln. by nonequilibrium pressure)
gkρ+ (accln. by body force)
>∇⋅−−<+ kIpp ατ ])[( 00 (momentum exchange terms) (3-18)
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
3.5.1 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'
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: 0.360 ~ 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 (X direction) and allow free-slip in the parallel direction (Z
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).
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 Z/dp of 66 (at the left side),
another is at Z/dp 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 = 0.072 m; dp = 0.003 m ρG = 1.2 kg/m2; ρL = 1000 kg/m2;
H = 0.288 m µG=1.8 x 10-5 Pa.s; µL=1.0 x 10-3 Pa.s;
3.5.2 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 (Ti), viscous loss (Tv), and kinetic energy loss
46
(Tk). 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 j are given below.
jSVSVjSVSVT ZZZZZZj
XXXXXXj
j,i
∆+∆+∆+∆+ −
ερ+−
ερ= 33
233
2 22 (3-19)
( ) j,Cj,ZZZZj,j,Zj,Zj,j,XXXXj,j,xj,Xj,j,k VVVfVVfVVfVVfT ∆+∆+∆+∆+ +++= 22
22
22
22 (3-20)
( ) j,Cj,ZZj,j,Zj,j,XXj,j,Xj,j,V VVfVfVfVfT 21
21
21
21 ∆+∆+ +++= (3-21)
-1.00E-040.00E+001.00E-042.00E-043.00E-044.00E-045.00E-046.00E-047.00E-048.00E-049.00E-04
0 48 96 144 192 240 288cell number
Ener
gy d
issi
patio
n ra
te, J
/s
TiTkTv
Figure 3-3. Contribution of each energy dissipation rate term at each cell to the total
energy dissipation rate. V0 = 0.5 m/s (gas flow without internal obstacles); Re’ = 28.5;
sJTn
i /1085.6 5256
1
−
=×−=∑ ; sJT
nk /1037.1 1
256
1
−
=×=∑ ; sJT
nv /1087.6 2
256
1
−
=×=∑ ;
( ) sJTTTn
vki /10056.2 1256
1
−
=×=++∑ (the cell number is counted from the top left of the
bed in the X direction)
47
As derived earlier, the pressure drop term is substituted by Tk,j and Tv,j 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 V0 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 (Rep) used in some
studies by multiplying it with a factor of ( )6 1−εB (~3.51 in this study). It was found that
when no internal obstacles are present and the flow is nearly parallel, the inertial term
(Ti) is negligible compared to the other two terms (Tk and Tv). The viscous term (Tv) is
about one third of the total energy dissipation rate, and the kinetic term (Tk) 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 (Ti) is still negligible compared to the other two
terms (Tk and Tv) except in the very proximity of the obstacles as shown in Figure 3-4b.
The values of Tk and Tv are scattered, but of the same order. Higher values of Tk 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 (Tk) is seen to be dominant in the energy dissipation rate per unit cell at
liquid superficial velocity, U0 of 0.1 m/s (Re' = 47.5). The inertial term (Ti) is not
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 justifies the
substitution of the pressure drop by the Ergun equation terms (Tk and Tv) 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).
49
-1.00E-03
0.00E+00
1.00E-03
2.00E-03
3.00E-03
4.00E-03
5.00E-03
6.00E-03
7.00E-03
0 48 96 144 192 240 288
Cell number
Ener
gy d
issi
patio
n ra
te, J
/sTiTkTv
Figure 3-4a. Contribution of each energy dissipation rate term at each cell to the total energy dissipation rate. V0 = 0.5 m/s (gas flow with two internal obstacles at Z/dp = 30,
66); Re' =28.5. sJTn
i /1022.1 4256
1
−
=×−=∑ ; sJT
nk /1015.2 1
256
1
−
=×=∑ ;
sJTn
v /10905.0 1256
1
−
=×=∑ ; ( ) sJTTT
nvki /1006.3 1
256
1
−
=×=++∑ (The dashed line region will
be re-illustrated in Figure 4b).
-1.00E-040.00E+001.00E-042.00E-043.00E-044.00E-045.00E-046.00E-047.00E-048.00E-049.00E-04
0 48 96 144 192 240 288Cell number
Ener
gy d
issi
patio
n ra
te, J
/s
TiTkTv
Figure 3-4b. Contribution of each energy dissipation rate term at each cell to the total
energy dissipation rate. V0 = 0.5 m/s (gas flow with two internal obstacles at Z/dp = 30,
66); Re’ =28.5; Ebed = ( ) sJTTTn
vki /1006.3 1256
1
−
=×=++∑ .
50
-1.00E-03
0.00E+00
1.00E-03
2.00E-03
3.00E-03
4.00E-03
5.00E-03
6.00E-03
0 48 96 144 192 240 288
Cell number
Ener
gy d
issi
patio
n ra
te, J
/sTiTkTv
Figure 3-5. Contribution of each energy dissipation rate term at each cell to the total
energy dissipation rate. V0 = 0.1 m/s (liquid flow without internal obstacles); Re' = 47.5;
sJTn
i /1079.5 4256
1
−
=×−=∑ ; sJT
nk /101.9 1
256
1
−
=×=∑ ; sJT
nv /1053.1 1
256
1
−
=×=∑ ;
( ) sJTTTn
vki /10624.10 1256
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 (Rep) of 600 (Vl = 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 (Rep), which may vary with particle diameter. For instance, the
critical Reynolds number range of 150 to 300 was reported by Jolls and Hanratty (1969)
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
Nijemeisland 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 X and Z, Pcx and Pcz, 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
(Pcx and Pcz) based on the two dimensional velocity solution (Vz and Vx) and verified that
they do have the same values at the center of each cell as required, although the pressure
drop in the X and Z 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,
V0 = 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 = 256packing zone + 8supporting 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.
52
3.5.3 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 170.9 respectively. Three sets of results at different
Reynolds number are shown in Figures 3-6, 3-7 and 3-8 at different axial positions (Z/dp
= 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:
53
(Relative) local superficial velocity 0V
VUU j
jj
jj =><
=εε
(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' = 170.9 at V0 = 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.
54
Z/dp =4.5 (from top)
0.10
0.30
0.50
0.70
0.90
1.10
1.30
0 3 6 9 12 15 18 21 24
X/dp
rela
tive
velo
city
0.100.150.200.250.300.350.400.450.50
CFDDCMporosity
Z/dp =19.5
0.100.300.500.700.901.101.30
0 3 6 9 12 15 18 21 24X/dp
Rel
ativ
e ve
loci
ty
0.10
0.20
0.30
0.40
0.50
z/dp=34.5
0.100.300.500.700.901.101.30
0 3 6 9 12 15 18 21 24X/dp
rela
tive
velo
city
0.100.150.200.250.300.350.400.450.50
Z/dp=49.5
0.1
0.3
0.5
0.7
0.9
1.1
1.3
0 3 6 9 12 15 18 21 24
X/dp
Rel
ativ
e ve
loci
ty
0.100.150.200.250.300.350.400.450.50
Z/dp=64.5
0.10
0.30
0.50
0.70
0.90
1.10
1.30
0 3 6 9 12 15 18 21 24
X/dp
Rel
ativ
e ve
loci
ty
0.100.150.200.250.300.350.400.450.50
Z/dp=79.5
0.100.300.500.700.901.101.30
0 3 6 9 12 15 18 21 24
X/dp
Rel
ativ
e ve
loci
ty
0.10
0.20
0.30
0.40
0.50
Figure 3-6. Comparison of CFD simulations and DCM predictions at a gas superficial velocity of 0.1 m/s at different axial
positions (Z/dp) (Re' = 5.7). Left axis is relative cell superficial velocity; Right axis is cell porosity. 54
55
Z/dp =4.5 (from top)
0.10
0.30
0.50
0.70
0.90
1.10
1.30
0 3 6 9 12 15 18 21 24
X/dp
rela
tive
velo
city
0.100.150.200.250.300.350.400.450.50
DCMCFDporosity
Z/dp =19.5
0.100.300.500.700.901.101.30
0 3 6 9 12 15 18 21 24X/dp
Rel
ativ
e ve
loci
ty
0.10
0.20
0.30
0.40
0.50
z/dp=34.5
0.100.300.500.700.901.101.30
0 3 6 9 12 15 18 21 24X/dp
rela
tive
velo
city
0.100.150.200.250.300.350.400.450.50
Z/dp=49.5
0.1
0.3
0.5
0.7
0.9
1.1
1.3
0 3 6 9 12 15 18 21 24
X/dp
Rel
ativ
e ve
loci
ty
0.100.150.200.250.300.350.400.450.50
Z/dp=64.5
0.10
0.30
0.50
0.70
0.90
1.10
1.30
0 3 6 9 12 15 18 21 24
X/dp
Rel
ativ
e ve
loci
ty
0.100.150.200.250.300.350.400.450.50
Z/dp=79.5
0.100.300.500.700.901.101.30
0 3 6 9 12 15 18 21 24
X/dp
Rel
ativ
e ve
loci
ty
0.10
0.20
0.30
0.40
0.50
Figure 3-7. Comparison of CFD simulations and DCM predictions at a gas superficial velocity of 0.5 m/s at different axial
positions (Z/dp) (Re' = 28.5).
55
56
Z/dp =4.5 (from top)
0.100.300.500.700.901.101.30
0 3 6 9 12 15 18 21 24
X/dp
rela
tive
velo
city
0.10
0.20
0.30
0.40
0.50
DCMCFDporosity
Z/dp =19.5
0.100.300.500.700.901.101.30
0 3 6 9 12 15 18 21 24X/dp
Rel
ativ
e ve
loci
ty
0.10
0.20
0.30
0.40
0.50
z/dp=34.5
0.100.300.500.700.901.101.30
0 3 6 9 12 15 18 21 24X/dp
rela
tive
velo
city
0.10
0.20
0.30
0.40
0.50
Z/dp=49.5
0.1
0.3
0.5
0.7
0.9
1.1
1.3
0 3 6 9 12 15 18 21 24X/dp
Rel
ativ
e ve
loci
ty
0.100.150.200.250.300.350.400.450.50
Z/dp=79.5
0.100.300.500.700.901.101.30
0 3 6 9 12 15 18 21 24X/dp
Rel
ativ
e ve
loci
ty
0.10
0.20
0.30
0.40
0.50
Z/dp=64.5
0.100.300.50
0.700.901.101.30
0 3 6 9 12 15 18 21 24
X/dp
Rel
ativ
e ve
loci
ty
0.100.150.200.250.300.350.400.450.50
Figure 3-8. Comparison of CFD simulations and DCM predictions at a gas superficial velocity of 3.0 m/s at different axial
positions (Z/dp) (Re' = 171).
56
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 Vz (velocity component in the Z
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 Vz than CFD as shown in Figure 10a. Correspondingly, a lower prediction of Vx
(velocity component in X 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.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
V from CFD (m/s)
V fro
m D
CM
(m
/s)
-10%
+10%
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.
58
0.5
0.7
0.9
1.1
1.3
1.5
0.5 0.7 0.9 1.1 1.3 1.5U/U0 from CFD
U/U
0 fro
m D
CM
-10%
+10%
(b)
Figure 3-9b. Comparison of the relative interstitial velocity (Uj/U0) between CFD and
DCM predictions for gas flow in the Reynolds number (Re') range of 5 to 171. (U0 =
V0/εB ).
3.5.4 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
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).
20.0
22.5
25.0
27.5
30.0
20.0 22.5 25.0 27.5 30.0Uz(DCM), cm/s
Uz(
CFD
), cm
/s
+10%
-10%
Figure 3-10a. Comparison of predicted interstitial velocity component in the Z direction
(Uz) by two methods in liquid up-flow system: liquid superficial velocity V0 = 0.1 m/s
(Re' = 47.5).
60
-4.0
-2.0
0.0
2.0
4.0
-4.0 -2.0 0.0 2.0 4.0
Ux (DCM), cm/s
Ux
(CFD
), cm
/s
Figure 3-10b. Comparison of predicted interstitial velocity component in X direction (Ux)
by two methods in liquid up-flow system: inlet liquid superficial velocity V0 = 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, Rep) 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 Rep 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 Rep higher than 500.
61
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 3 6 9 12 15 18 21 24
X/dp
uj,
m/s
0.15
0.25
0.35
0.45
0.55
0.65
poro
sity
0.1m/s (Rep=20) 0.5m/s (Rep=100)porosity
Figure 3-11a. Influence of gas feed superficial velocity on DCM predicted cell interstitial
velocity profiles.
0.7
0.8
0.9
1.0
1.1
1.2
0 3 6 9 12 15 18 21 24X/dp
Vj/V
0
0.15
0.25
0.35
0.45
0.55
0.65
poro
sity
0.1m/s (Rep=20) 0.5m/s (Rep=100)3.0m/s (Rep=600) porosity
Figure 3-11b. Effect of particle Reynolds number (Rep) on the calculated relative cell
superficial velocity profile inside a bed using DCM.
62
0.3
0.6
0.9
1.2
1.5
1.8
0 4 8 12 16 20cell # (interval #)
Rel
ativ
e ce
ll su
perf
icia
l ve
loci
ty
0.200.250.300.350.400.450.500.550.600.650.70
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5r (cm)
poro
sity
Vz/V0 (Exp.) Vz/V0 ( Rep = 5) (Cal.)Vz/V0 (Rep = 80) (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/dv = 10.7 and dv =
0.7035 cm (cylindrical particles). Physical properties of liquid: Liquid -B for condition at
a Rep of 5, ρ = 1.125 g/cm3; µ = 0.474 g cm/s. Liquid -C for condition at a Rep of 80, ρ =
1.027 g/cm3; µ = 0.114 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/dp ratio of 10.7. Velocity was measured inside a bed of
cylindrical particles (dv = 0.7035 cm) with liquid flows of very different physical
properties ( ρ µL L; ). To simulate the experimental bed, a 2D axi-symmetric bed in
cylindrical coordinates (r-Z) is chosen in CFDLIB simulation. The radial spatial
discretization (Nc = 20) is the same as that used as the viewing zone for collecting each
63
experimental data point (i.e a space interval of ∆ = 0 05. R ). 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 (Vz/V0) and the measured data at Rep 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 3.5.3. 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 two-
dimensional velocity field by DCM/CFDLIB with two-dimensional experimental data of
flow distribution at the cell scale.
3.5.5 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
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 (ψρG
G gPZ
=
1 ∆∆
) in the case of the side gas feed. Higher values of ψG are evident
in the entry and obstacle regions. In contrast, lower values of ψG are apparent in the
corner regions.
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) (U0=120.5
cm/s); (velocity vector plotting).
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) (U0=120.5
cm/s) (point source inlet from left side, inlet point superficial velocity is of 4.0 m/s)
(velocity vector plotting).
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 Z/dp of 66 (at the left side), another is at Z/dp 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).
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, ψρG
G gPZ
=
1 ∆∆
is plotted). Two obstacle
plates are placed in this bed, one is located at Z/dp of 66 (at the left side), another is at
Z/dp 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).
69
3.6 Concluding Remarks 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
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.
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 major
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; Zimmerman 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
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 objective 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.
73
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, time-
consuming computations required to determine liquid flow on the particle-scale
(Zimmerman 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.
4.1.2 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
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 (Zimmerman
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.
75
4.2 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
jth 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 (εj, 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+
+
⋅
∑
=inj
iiii s
sVs
sVPs
sV
sV
,
4
1
2
21
ααα
ρρ
0,,
4
1,
2
=
−
Φ+
+
⋅
∑
= αααεε
ρεεε
ρjVj
outj
iiiEsVsVPsVV (4-1)
76
VZ, j, αααα
VX, j, αααα VX+∆∆∆∆X, j, ααααCell j
∆∆∆∆X=3dP
VZ+∆∆∆∆Z, j, αααα
∆∆∆∆Z=3dPX
Z
Figure 4-1. The coordinate system and velocity conventions for the α phase in the jth
cell
For a 2D rectangular cell (j) as depicted in Figure 4-1, the mechanical energy
dissipation rate of phase α, EV, j , α , can be written in the discretized form as below
−+−=
∆+∆+∆+∆+ α
α
αα
α
αα ε
ρε
ρ,
3332,
,
332,
,,1
2 jZZZZZZj
jXXXXXXj
jV SVSVSVSVE
( ) ( )[ ]( ) jjXXjXXjjjXjXjjj
VolVVbaVVba 2,,,,,
2,,,,,
3
,
1αααααα
αε ∆+∆++++
+
( ) ( )[ ]( ) jjZZjZZjjjZjZjjj
VolVVbaVVba 2,,,,,
2,,,,,
3
,
1αααααα
αε ∆+∆++++
+
( ) ( )[ ]( ) − + + ++ +g V V g V V VolX X X X j X Z Z Z Z j Z j∆ ∆, ,cos cos
α αγ γ2 2 2 (4-2)
A discretized form of the macroscopic mass balance equation can be similarly written as
( ) ( ) 0,, =−+− ∆+∆+∆+∆+ αα ρρρρ jZZZZZZjXXXXXX SVSVSVSV (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.
77
This is essentially a nonlinear, multivariable minimization problem as given in Equation
(4-4) subject to the mass balance constraint (Equation 4-3) for each phase in cell j, 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 adjacent to the wall are zero in the
direction normal to the wall).
Minimize: ∑∑= =
=2
1 1,,,
αα
N
jjVbedV EE (4-4)
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 (εj,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).
∆
∆=
∆
∆L
PL
P jLjG ,, (4-5a)
Then pressure drops are expressed in terms of dimensionless pressure drop
functions (ψG for gas, ψL for liquid).
( )1,, −Ψ=
∆∆
jGLjG g
LP
ρ (4-5b)
( )1,, −Ψ=
∆∆
jLLjL g
LP
ρ (4-5c)
Substitution of Equations (4-5b) and (4-5c) into equation (4-5a) yields equation (4-6)
( )11 ,, −Ψ+=Ψ jGL
GjL ρ
ρ (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).
+
−
=ΨjG
jG
jG
jG
jLj
jjG Ga
EGa
E
,
2,2
,
,1
3
,,
ReReεε
ε (4-6a)
78
+
=Ψ
jL
jL
jL
jL
jL
jjL Ga
EGa
E
,
2,2
,
,1
3
,,
ReReεε
(4-6b)
Thus, substitution of ψG, j and ψL, j (from Eq 4-6a and 4-6b) into Equation (4-6)
yields a nonlinear equation in terms of phase holdups (εj,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
( )fPPP jCjLjG −=− 1,,, (4-7)
where the capillary pressure, Pc,j, 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).
( ) ( )jWPj
jjC sJE
dP ,
5.01,
1σ
εε−
= (4-8a)
( )
−+=
jW
jWjW s
ssJ
,
,,
1ln036.048.0 (4-8b)
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, Pc,j)
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).
79
( ) ( ) ( )LsJ
fEdL
PL
P jW
Pj
jjLjG
∆∆
−−
+∆
∆=
∆∆ ,5.0
1,, 1
1σ
εε
(4-9)
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.
[ ] ( ) ( ) ( )( )LsJ
ssbfE
djW
jWjWPjL
jjG
L
GjL ∆
∆
−
−−
+−Ψ+=Ψ ,
,,
5.01,, 1
11
11 σερ
ερρ (4-10)
when ψG, j and ψL, j are obtained from Equations 4-6a and 4-6b as before.
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 0.415 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 0.406 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 (nC) and 32 cells in length
(nR) and is 1 cell thick (nT). Each cell has a size of 3 dp × 3 dp × 3 dp (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.
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.
0.390.400.410.420.430.440.45
0 8 16 24 32 40 48 56 64 72 80 88 96
x/dp or z/dp
poro
sity
axial (Z direction)radial (X direction)
Figure 4-2b. Average porosity profiles in X and Z directions in model bed.
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
∑=
−=
N
i
ii
VV
AAmf
1
2
00
1 (4-11)
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.
4.3.1 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 justification 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 non-
linear 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
82
Equation of continuity:
>αρ<=ρ∇+∂
∂ρkkkk
k u.t
! (4-12a)
(Accumulation) (Convection) (Mass source)
Equation of momentum:
gp'u'u.uu.tu
kkkk0kkkkkk ρ+∇θ−>ρα<−∇=ρ∇+
∂∂ρ
(Accumulation) (Convection) (Reynolds stress) (Mean pressure) (Body force)
>αρ<+>α∇τ−−<+ k00k00 u].I)pp[( !!
(Exchange term) (Mass source)
)pp(. 0k0k −θ∇−>τα<∇+ (4-12b)
(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
( ) ( )lkkllklkD uuXF −=− θθ (4-13)
where the Xkl 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 (Z/dp)
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%.
0
0.3
0.6
0.9
1.2
1.5
0 3 6 9 12 15 18 21 24X/dp
Rel
ativ
e ite
rstit
ial
velo
city
CFDDCM
Z/dp = 15
0
0.3
0.6
0.9
1.2
1.5
0 3 6 9 12 15 18 21 24X/dp
Rel
ativ
e in
ster
stiti
al v
eloc
ity
CFDDCM
Z/dp = 30
(4-3a) (4-3b)
0
0.3
0.6
0.9
1.2
1.5
0 3 6 9 12 15 18 21 24X/dp
Rel
ativ
e in
ters
titia
l vel
ocity
CFDDCM
Z/dp = 45
0
0.3
0.6
0.9
1.2
1.5
0 3 6 9 12 15 18 21 24X/dp
Rel
ativ
e su
perfi
cial
vel
ocity
CFDDCM
Z/dp = 75
(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. Ul = 0.00148 m/s (UF); Ug = 0.05
m/s (UF); Completely prewetted packed bed. (The relative interstitial velocity is defined
as the local interstitial velocity (Vi) divided by the overall interstitial velocity (V0). The
value of V0 in this case is equal to 0.1205 m/s).
84
0.5
0.7
0.9
1.1
1.3
1.5
0.5 0.7 0.9 1.1 1.3 1.5Relative velocity by CFD simulation
Rel
ativ
e ve
loci
ty b
y D
CM
mod
elin +13%
-13%
Figure 4-3e. Comparison of the predicted gas interstitial velocities (relative) for all the
cells by DCM and CFD. Inlet superficial velocities (uniform): Ul = 0.00148; Ug =
0.05m/s; Completely prewetted packed bed.
85
0.00
0.05
0.10
0.15
0.20
0.25
0 3 6 9 12 15 18 21 24
X/dp
Liqu
id h
oldu
p"Z/dp=74""Z/dp=62""Z/dp=50""Z/dp=74""Z/dp=62""Z/dp=50"
Solid line: CFDLIBDash line: DCM
Figure 4-4. Comparison of predicted liquid holdup at specific levels by DCM and CFD.
Single point source liquid inlet: U L = 0.00148 m/s (Ul (PS1)= 0.01184 m/s);
Ug = 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.
86
4.3.2 Effect of Liquid Distributor
Three types of liquid inlet distributors: single point source (PS1), two points
source (PS2) 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 PS1, and two for PS2). 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.
87
Table 4.1 Summary of operating conditions used in flow simulations
Ul = 0.00148 m/s Ug = 0.050 m/s (superficial velocity)
Completely prewetted bed Completely nonprewetted bed
Gas inlet Liquid inlet Gas inlet Liquid inlet
UF PS1, PS2; UF UF PS1
UF: uniform
PS1: single point source located at top layer* at No. 5 cell
PS2: 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 = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05m/s;
Non-prewetted packing.
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 = 0.00148 m/s (Ul (PS2)=0.00592 m/s);
Ug = 0.05m/s; Non-prewetted packing.
89
Figure 4-5c. Liquid holdup distribution in whole domain of the non-prewetted packed
bed with uniform liquid distributor by DCM. Ul = 0.00148 m/s; Ug = 0.05m/s.
90
0.0
0.51.0
1.5
2.02.5
3.0
0 9 18 27 36 45 54 63 72 81 90
z/dp (from top)
mf
PS1(non-prewetted)PS2(non-prewetted)PS1(prewetted)UF(non-prewetted)
Figure 4-5d. Comparison of liquid flow maldistribution calculated by DCM along the
bed for different liquid distributors. U L = 0.00148 m/s; Ug = 0.05m/s.
4.3.3 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 (PS1) (actually one
cell inlet). For further comparison, Figures 4-6b and 4-7c show the predicted liquid
holdup at the specific levels (Z/dp = 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
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.
92
Figure 4-6a. Liquid holdup distribution in the whole domain of the completely prewetted
packed bed (f = 1). U L = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05 m/s; Point
liquid distributor (PS1).
93
0.0500.070
0.0900.110
0.1300.150
0.1700.190
0.2100.230
0.0 6.0 12.0 18.0 24.0
X/dp
Liqu
id h
oldu
p
Z/dp=94.5Z/dp=73.5Z/dp=61.5Z/dp=49.5
Figure 4-6b. Liquid holdup distribution at specific levels (Z/dp) in the completely
prewetted packed bed (f = 1), U L = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05 m/s;
Point liquid distributor (PS1); Overall liquid holdup = 0.0758.
0.0500.070
0.0900.1100.130
0.1500.1700.190
0.2100.230
0.0 6.0 12.0 18.0 24.0
X/dp
Liqu
id h
oldu
p
Z/dp=94.5Z/dp=73.5Z/dp=61.5Z/dp=49.5
Figure 4-6c. Liquid holdup distribution at specific levels (Z/dp) in the completely non-
prewetted packed bed (f = 0), U L = 0.00148 m/s (Ul (PS1) = 0.01184 m/s); Ug = 0.05 m/s;
Point liquid distributor (PS1); Overall liquid holdup = 0.0716.
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
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 just the flow distribution but also reactor performance. At this point, we re-
emphasize 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.
96
Chapter 5
Computational Fluid Dynamics
(CFD): I. Modeling Issues 5.1 Introduction and Background
5.1.1 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
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 Swaaij, 1998). In particular, one of the promising methods is the so-
called 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 reactive-
flow 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 Swaaij (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 (Delnoij 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 Swaaij, 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
98
closures for averaged equation models used to predict large scale flows in industrial
reactors.
99
Tabl
e 5.
1 C
urre
nt st
atus
of C
FD m
odel
ing
in m
ultip
hase
reac
tors
R
eact
ors
Cur
rent
Fea
ture
s and
Fut
ure
Cha
lleng
es
Prog
ress
Sa
mpl
e C
FD W
ork
Bub
ble
Col
umns
Tw
o-flu
id E
uler
ian
mod
el
Mix
ed E
uler
ian-
Lagr
ange
mod
els
Vol
ume
of F
luid
(VO
F) m
odel
for s
ingl
e ga
s bub
ble
risin
g M
ostly
lim
ited
to b
ubbl
y flo
w
Futu
re c
halle
nge:
chu
rn-tu
rbul
ent f
low
mod
elin
g.
Rea
sona
ble
!!!
Soko
lichi
n &
Ei
genb
erge
r, 19
94
Del
noij
et a
l., 1
997
Pan
et a
l., 2
000
G-L
Stir
red
Tank
s Tw
o-flu
id m
odel
(Sna
psho
t app
roac
h, M
RF
mes
h, S
lidin
g m
esh)
Fu
ture
ch
alle
nges
: A
ccur
ate
mod
elin
g of
th
e im
pelle
r; av
aila
bilit
y of
loca
l flo
w d
ynam
ic i
nfor
mat
ion
and
the
rang
e of
di
sper
sed
phas
e ho
ldup
from
exp
erim
ents
; tur
bule
nce
mod
elin
g
Littl
e !
R
anad
e &
van
den
Akk
er,
1994
R
anad
e &
D
eshp
ande
, 19
99
G-S
Flu
idiz
ed B
eds
e.g.
bub
blin
g, so
lid
riser
s
Two-
fluid
mod
el w
ith si
mpl
e so
lid rh
eolo
gy
Two-
fluid
mod
el w
ith k
inet
ic th
eory
D
iscr
ete
parti
cle
appr
oach
Fu
ture
cha
lleng
es: R
efin
ed m
odel
for p
artic
le-p
artic
le, p
artic
le-
wal
l int
erac
tions
; cou
ple
with
reac
tion;
Pre
dict
ion
of fl
ow re
gim
e tra
nsiti
on.
Goo
d !!!!
Si
ncla
ir &
Jack
son,
198
9 D
ind
& G
idas
pow
, 199
0 N
ieuw
land
et a
l., 1
996
G-S
or L
-S P
acke
d B
eds
Two-
fluid
mod
el w
ith 3
D m
esh
for i
nter
stiti
al d
omai
n Tw
o-flu
id m
odel
s with
rand
om p
oros
ity d
istri
butio
n St
ruct
ural
pac
king
with
hea
t tra
nsfe
r Fu
ture
ch
alle
nges
: G
eom
etric
al
com
plex
ity;
avai
labi
lity
of
expe
rimen
tal d
ata
for v
alid
atio
n.
Littl
e !
Lo
gten
berg
&
D
ixon
, 19
98
Cha
pter
3 in
this
thes
is
G-L
-S P
acke
d B
eds
e.g.
tric
kle
beds
Tw
o-flu
id m
odel
with
rand
om p
oros
ity d
istri
butio
n Fu
ture
ch
alle
nges
: G
eom
etric
al
com
plex
ity;
parti
al
wet
ting
conc
ern;
flo
w h
isto
ry d
epen
denc
e; a
vaila
bilit
y of
exp
erim
enta
l da
ta fo
r val
idat
ion
Ver
y Li
ttle
Cha
pter
4 in
this
thes
is
To b
e di
scus
sed
in t
his
Cha
pter
Sl
urry
reac
tors
e.
g. G
-L-S
stirr
ed
tank
s, sl
urry
bub
ble
colu
mn
Stan
dard
k-ε
turb
ulen
ce m
odel
(FLO
W3D
)-Stir
red
tank
Tw
o-flu
id m
odel
Fu
ture
cha
lleng
es: A
vaila
bilit
y of
exp
erim
enta
l dat
a fo
r va
lidat
ion;
diff
icul
t to
capt
ure
the
mic
rosc
opic
phe
nom
ena
(e.g
., pa
rticl
e ac
cum
ulat
ion
near
the
G-L
inte
rfac
e).
Ver
y lit
tle
Ham
ill e
t al.,
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5
G-L
-S fl
uidi
zed
beds
No
99
100
5.1.2 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
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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 one-
dimensional 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 two-
dimensional 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 fluid-
fluid 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-
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mesh within the pore space (Logtenberg and Dixon, 1998). These simulations were
limited to the tube with very low column to particle diameter ratio (Dr/dp = 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-
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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 volume-
fraction 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 0.0016 (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
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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 0.02-0.3 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
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 (Wijngaarden
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, σB, 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, lv and the variance of section porosity distribution, σB
(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
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.
107
(a)
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.0 1.0 2.0 3.0 4.0r (cm)
sect
iona
l por
osity
(b)
108
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60
5
10
15
20
25
30
35
40
45
50
(c)
Figure 5-1. Generated pseudo-Gaussian distribution of porosity under three constraints:
(1) ε0 = 0.36; (2) Longitudinally averaged radial porosity profile (white filled circles)
reported by Stephenson and Stewart (1986). (Dr = 7.6 cm, dp = 0.703 cm, Section size =
0.05R = 0.19 cm). (a)-contour plot; (b)-radial profiles; (c)-histogram (standard deviation
of porosity, σB = 12% ε0).
5.4 k-Fluid Approach and CFDLIB Code
5.4.1 Eulerian k-Fluid Model In the Eulerian k-fluid approach, the different phases or materials are treated
mathematically as interpenetrating continua (Ishii, 1975). The derivation of the
conservation equations for mass, momentum and enthalpy is performed by using either
the volume averaging or ensemble averaging technique to describe the time-dependent
motion of fluids and track volume fraction distribution of each phase. In the ensemble
averaging technique, the probability of occurrence of any one phase in multiple
realizations of the flow is given by the instantaneous volume fraction of that phase at that
point. Sum total of all volume fractions at a point is identically unity (Anderson and
Jackson, 1967).
109
The microscopic flow structure such as local detailed flow structure is achievable
only by direct numerical simulation (DNS), which is limited to relatively low Reynolds
and Schmidt numbers (Kuipers and van Swaaij, 1998). Since DNS is not possible for
most industrial gas-liquid flows, several authors (Sokolichin and Eigenberger, 1994;
Ranade, 1995; Pan et al., 2000) have used the Eulerian k-fluid model for simulation of
the dynamic gas-liquid flow by modeling subgrid/local phenomena and simulating the
large-scale phenomena. It is well known that the successful applications of such
simulations in multiphase flow are mainly dependent on the appropriate closure laws for
the inter-phase transport of mass, momentum and energy (Delnoij et al., 1997; Kuipers
and van Swaaij, 1998). For modeling multiphase flow in packed beds, additional effort is
needed, as discussed above, to properly implement the porosity structure of packed beds
into the model equations.
In this work, the packaged computational fluid dynamics code, CFDLIB,
developed by Los Alamos National Laboratory (Kashiwa et al., 1994), has been adopted
as a transient multiphase flow simulation tool using a k-fluid model for the simulation of
multiphase phase flows in packed beds. The key aspects of the k-fluid model and the
main features of CFDLIB code are described below.
5.4.2 k-Fluid Model in CFDLIB CFDLIB is a library of hydrocodes that share a common numerical solution
algorithm, and a common data format (i.e., ‘block-structured’). There is a hierarchy to the
codes in CFDLIB, which depends on the complexity of the systems dealt with (from
multispecies, multiphase and compressible system to single species, single phase and
incompressible system). The time dependent mass, momentum and energy conservation
equations were derived using ensemble-averaging technique, and were cast in the integral
form, and the solution is based on the finite-volume method (FVM) (Kashiwa et al.,
1994).
In the finite-volume approach, the physical domain is subdivided into small
volumes, and the dependent variables are evaluated either at the center of the volumes
(cells) or at the corners (vertices) of the volumes. In the CFDLIB codes, the physical
110
domain is divided into the main computational subunits: blocks, sections and cells. A
block is a logical rectangular portion of the meshes, having left, right, and bottom and top
boundaries. In each mesh direction within the block there are several sections. In each
section, the material data is constant and is specified in the input file before initializing
the calculation. Therefore, the material data distribution information can be introduced
into the computational domain through each section. Moreover, the code allows unequal
section size within a single block. All these features allow inputting a detailed porosity
distribution data as an initial condition in the input file. The code can discretize each
section into cells; each cell has four vertices and four faces for 2D simulation treated
here. Each interior face is common to two cells, and each interior vertex is common to
four cells. Figure 5-2 illustrates how to assign the logical block, sections and cells from
the physical block in the 2D CFDLIB code, respectively. It is noted that in the two-
dimensional CFD computation in CFDLIB code, the 2-D cells still represent three-
dimensional volumes. For example, in 2-D Cartesian coordinate, the cells have a nominal
depth of ∆z = 1 length unit; in 2-D cylindrical coordinate, the cells have a normal depth
of ∆ ϕ =1 radian.
For the system with more complicated geometry, several blocks are needed,
which can be linked together in logical space by boundaries that are completely
transparent to the flow (Johnson et al., 1997). This finite-volume method used in
CFDLIB has an obvious advantage over a finite-difference method if the physical domain
is highly irregular and complicated, since arbitrary volumes can be utilized to subdivide
the physical domain. Since the integral equations are solved directly in the physical
domain, no coordinate transformation is required. Also the mass, momentum, and energy
are automatically conserved, since the integral forms of the governing equations are
solved (Tannehill et al., 1997).
The ensemble-averaged conservation equations that serve as the basis for k-fluid
model in CFDLIB are:
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Mass equation (continuity):
kkk u
tρ
ρ⋅∇+
∂∂
>=< kαρ !0 (5-1)
The terms on the left hand side of Equation (5-1) 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. kα represents the net rate at which
material k is being created. In present study this term is equal to zero since no phase
exchange, reaction or mass transfer is considered at this stage.
Momentum equation:
=⋅∇+∂
∂kkk
kk uutu
ρρ
><+ ok uαρ !0 (Mass exchange source)
><∇⋅− ''0 kkk uuρα (Reynolds stress)
>∇<+ 0pkα (Pressure term)
>⋅∇<+ 0τα k (Stress term)
gkρ+ (Body force) (5-2)
After decomposing the pressure and stress terms in terms of pressure acceleration
and material stress divergence, one can get the following momentum equation:
=⋅∇+∂
∂kkk
kk uutu
ρρ
><+ ok uαρ !0 (Mass exchange source)
><∇⋅− ''0 kkk uuρα (Reynolds stress)
pk ∇−θ (Pressure term)
( )pp kk −∇− 0θ (Mean pressure)
( )[ ] kIpp ατ ∇⋅+−−<− 00 (Momentum exchange)
><∇⋅+ 0τα k (Average stress)
gkρ+ (Body force) (5-3)
112
To close Equation (5-3), the closure models for computing the Reynolds stress and the
momentum exchange terms are needed. Such closure problems can be resolved either by
phenomenological models (e.g., Ergun equation), or by the formula from the
microstructure flow element (e.g., DNS), or from an original transport equation (e.g.,
Lattice Boltzmann simulation). In the flow modeling of packed beds discussed in this
chapter, the phenomenological closure formulas are used.
(a) (b) (c)
Figure 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.
Cell Block
Section
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5.5 CFD Modeling Issues To adopt the CFDLIB code to the present flow problem in packed beds, some
subroutines related to the closures and phase pressure calculations need to be added and
modified to carefully take into account the essential physics of the system.
5.5.1 Significance of Terms in the Momentum Balance To numerically describe the flow pattern at different scales, one has to formulate
the governing flow equations with different dependent basic force terms: inertial force,
viscous force, capillary force, gravitational force and turbulence related force, etc. For
example, one has to take Reynolds stress term into consideration in the fine-mesh CFD
modeling with high gas flow rate (Logtenberg and Dixon, 1998). In fact, the existence of
microscale turbulence in porous media has been detected by several experiments by
point-wise probes (Jolls and Hanratty, 1966; Latifi et al., 1989). For the macroscopic
flow modeling in packed beds, however, the contribution of the Reynolds stress term to
the fluid momentum equation is not important as discussed in Chapter 3 because when
averaging a number of local (random) signals within a representative elementary volume
(e.g., a cubic cell contain a cluster of particles), the microscopic turbulence is smoothed
out. More discussions of micro- and macro-scale turbulence modeling in porous media
are available elsewhere (Lage, 1998).
The size and the shape of packing elements determine the void structure of the
granular assembly, which further affects the contribution of each basic-force on flow
distribution. Table 5.2 lists the typical ranges of these force ratios in gas and liquid flow
through the packed granules. Obviously, oil displacements are capillary-dominated
creeping flows. In packed beds used as separation columns with large packing elements
(e.g., 10 ~ 30 mm Pall rings and Rasching rings), both the gravity and inertial forces are
important, whereas the liquid distribution patterns are not much sensitive to the wetability
of the packing surface (see Bemer and Zuiderweg, 1978) due to negligible capillary
force. However, in trickle beds, the particle sizes are typically in the range of 0.5 to 3
mm, all the forces contribute to the flow distribution, and the influence of particle
114
external wetting on liquid distribution is significant (Lutran et al., 1991; Ravindra, et al.,
1997a; also see Chapter 2). This implies that even for describing the same macroscale
flow pattern in the overall packed bed, the contribution of each basic-force may be of
different magnitude depending on the different characteristic radii of the flow passages.
Table 5.2 Typical ranges of force rations in two-phase flow in packed granular packing
(adapted from Melli et al., 1990)
Packings Particle dp, m
Re inertial/viscous
Ca viscous/capillarity
1/Bo capillarity/gravity
Porous media: oil displacement; chromatography
10-7 - 10-4 10-9 - 10-2 10-7 - 10-3 102 - 109
Trickle-beds 10-3 - 10-2 10-2 - 103 10-1 – 10 10-1 - 10 Separation columns
10-2 – 10-1 10 - 105 10 -102 10-3 - 10-1
Reynolds number: kkpP Ud µρ≡Re ; Capillary number: SLLUCa σµ≡ ;
Bond number: ( )[ ] SGLp gdBo σρρ −≡ 2
The inertial effect in modeling of flow in porous media has been the topic of
debate for many years (Stanek, 1994; Lage, 1998). The numerical modeling of single
flow phase in packed beds has shown that the contribution of the inertial term to the total
(mechanical) energy dissipation rate is negligible compared to the viscous term and to the
kinetic term except in the regions of structural obstacles which change the flow direction
sharply (see Chapter 3). This agrees with the experimental findings of gas flow through
the packed beds with obstacles (Choudhary et al., 1976). In trickle beds with two-phase
cocurrent flow, the increase in gas and liquid flow rates can generate high inertial forces
exerted in the bulk fluids, which further contributes to the growth of interfacial waves
and to the bed-scale destabilization of the trickle flow regime. On the other hand, the high
inertial force causes the gradients in liquid saturation, and further result in the capillary
force at gas and liquid interface, which contributes to the attenuation of interfacial waves
and to the stabilization of the trickle flow regime. The simulation of 1-D bed using a two-
115
fluid model has shown that the inertial forces of fluids play an important role in the
mechanism of flow transition from trickling to pulsing flow (Grosser et al., 1988; Attou
and Ferschneider, 2000).
In summary, the Reynolds stress term is not important in determining the
macroscale flow pattern in packed beds with a particle size of 10-4 to 10-2 m, however,
other basic forces (i.e., inertial, viscous, gravity and capillary forces) are normally of a
similar order of magnitude so that they have to be taken into account in the flow
equations in a proper way.
5.5.2 Closures for Multiphase Flow Equations The volume averaging technique for equations of motion leads to the well known
closure issue for some of the terms associated with fluctuating variables and source terms
in which some of the forces acting on a representative permeable volume need to be
modeled. For single-phase flow through porous media, several studies used the effective
viscosity idea of Brinkman, and lumped the forces acting on the fluid phase of the
permeable medium into an effective viscous force (Bey and Eigenberger, 1997). In this
work, we compute the drag forces due to fluid-particle and fluid-fluid interactions based
on the phenomenological models developed in bench-scale hydrodynamics experiments.
Moreover, the magnitude of the drag force is expressed as a product of a user defined
exchange coefficient, Xkl, phase volume fractions, θl, θk and relative interstitial velocity
of the two phases k and l as below
( ) ( )lkkllklkD uuXF −=− θθ (5-4)
Clearly, the essential part of Eq (5-4) is to determine the exchange coefficient values, Xkl.
Thus far, there are several models capable of providing Xkl values: namely, the
relative permeability model (Saez and Carbonell, 1985), the single slit model (Holub, et
al., 1992), the two-fluid interaction model (Attou et al., 1999) as tabulated in Table 5.3.
Note that the interaction force between the gas and the liquid phase was neglected in both
single slit and relative permeability models indicating zero shear stress at the gas-liquid
interface. This may be true only for the case where gas and liquid flows are both low, and
116
the flow system is located ‘deep’ in the trickling flow regime. Otherwise, one does need
to take into account this force in the hydrodynamics computations because experimental
studies of Larachi et al (1991) and Al-Dahhan and Dudukovic (1994) have shown that
gas flow can exert considerable influence on the hydrodynamics of the trickle-bed
reactor, especially at high operating pressure and/or high gas velocity. Hence, Attou et al
(1999) included the gas-liquid interaction force (see Equation 5-9) in the 1-D two-fluid
model based on an ‘annular flow’ model in which the gas and liquid phases are separated
by a smooth and stable interface. Good predictions of liquid holdup and pressure-drop
were claimed in their paper. In our earlier CFD simulations of two phase flow (see
Chapter 4), either no interaction was assumed or a drag formula of a single-sphere in
fluid was used as an approximation for the momentum gas-liquid exchange coefficient,
Xgl. In this work, the gas-liquid interface drag formula for Xgl developed by Attou et al
(1999) has been chosen due to more appropriate physical basis.
The expressions for other two-phase momentum exchange coefficients, Xkl, are
written in similar format in Table 5.3. The comparisons of these expressions for given
sectional porosity and particle size are shown in Figure 5-3. For a given gas superficial
velocity of 6.0 cm/s, an increase in liquid superficial velocity causes Xgs to increase and
Xls to decrease. Moreover, the relative permeability model gives relatively higher values
of Xgs and Xls than either the single slit model and the two-fluid interaction model.
Hence, the relative permeability model gives lower predictions of liquid holdup than the
single slit model at the same flow conditions as shown in Figure 5-4. The two-fluid
interaction model yields the same values of Xls as the slit model, and provides
intermediate values of Xgs, between the slit model and the relative permeability model.
The effect of liquid velocity on the Xgl value is not significant. Due to the lack of detailed
flow velocity and volume-fraction distribution data together with known bed structure at
certain section scale, we are not able to establish the best drag expression for Xgl at the
present time. However, we can appreciate how these drag expressions affect the
predictions of the liquid holdup and pressure gradient at bed scale, which will be
presented in Chapter 6. The full validation of the best drag expression, in principle, will
be possible by using MRI technique to obtain the needed experimental data.
117
0
5
10
15
20
0 1 2 3 4 5 6 7 8
Ul, superficial velocity, kg/m2/s
X gs,
X gl
x1.0
4
0.0
2.0
4.0
6.0
X ls
x1.0
6
Xgs(A) Xgs (H) Xgs (SC)
Xgl (A) Xls (H) Xls (SC)
Xls (A)
Figure 5-3. Comparison of Xkl values from different models [Ug = 6 cm/s]: A- Two-fluid
interaction model (Attou et al., 1999); H- Single slit model (Holub et al., 1992); SC-
Relative permeability model (Saez and Carbonell, 1985).
The overall particle external wetting efficiency can be calculated by the
correlation of Al-Dahhan and Dudukovic (1995) based on the superficial velocities of gas
and liquid as well as particle parameters etc. As shown in Figure 5-4, at the bed scale, the
particle external partial wetting does exist at a low liquid superficial mass velocity (L
<5.0 kg/m2/s). Therefore, the drag formulations derived from the double-slit model in
which two interconnected slits: wet and dry slit, are assumed (see Iliuta et al., 2000),
could provide a reasonable alternative, because it is possible then to account for particle
wetting in the multiphase drag calculations. However, the computation may become
cumbersome due to too many equations involved in the double-slit model. Furthermore,
118
how much improvement can be gained is still uncertain, because based on the
comparisons of these drag models in predictions of global hydrodynamics quantities
Larachi et al (1999) and Iliuta et al (2000) concluded that all of these models fit the
experimental data to about the same degree of accuracy.
0.0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Ul, kg/m2/s
Liqu
id h
oldu
p
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Wet
ting
effic
ienc
y
hp: S & C modelhp: Holub modelwt: S & C modelwt: Holub model
Figure 5-4. 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.
119
Table 5.3 Models for drag coefficients Single Slit Model (Holub et al, 1992) Fluid-Particle: (Xls and Xgs)
( )( ) kkkSkkkSkS u
UBUAXε
ρµ α −+=
112 (5-5)
( )23
2
11
pkkS d
EAϑ
ε−= ; ( )pk
kS dEB 32
1ϑ
ε−=
Relative Permeability Model (Saez and Carbonell, 1985) Fluid-Particle: (Xls and Xgs)
( )( ) kkkSkkkSkS u
UBUAXε
ρµ α −+=
112 (5-6)
( )28.4
8.121180pg
gS dA
ϑεε−= ; ( )
pggS d
B 8.4
8.118.1ϑ
εε−=
( ) 43.2
0
0
23
21180
−−−=
ll
l
pllS d
Aθθθε
ϑε ; ( ) 43.2
0
0
3
18.1
−−−=
ll
l
pllS d
Bθθθε
ϑε
Two-Fluid Interaction Model (Attou et al., 1999) Fluid-Particle: (Xls and Xgs)
( )( ) llllslllslS u
UBUAXε
ρµ−
+=1
12 (5-7)
( )23
21180pl
lS dA
ϑε−= ; ( )
pllS d
B 3
18.1ϑ
ε−=
( )( ) g
gggsgggsg
gS uUBUAX
ερµ
εθ
−+=
112 (5-8)
( ) 32
23
2
111
180
−−−
=gpg
ggS d
Aθε
ϑθ
; ( ) 31
3 111
8.1
−−−
=gpg
ggS d
Bθε
ϑθ
Fluid-Fluid: (Xgl)
( )( ) lg
rggsrgglg
gl uuUBUAX
−−+=
ερµ
εθ
112 (5-9)
gsgl AA = ; gsgl BB = ; lggr uuU −=θ
120
200
240
280
320
360
0 0.1 0.2 0.3liquid holdup
Cap
illary
pre
ssur
e, P
a
J functionAttuou
Figure 5-5. Comparison of the calculated capillary pressure values from two different
expressions, Eq 5-17a and 5-17b for air-water system (dp = 0.003m; θs =0.63)
5.5.3 Interfacial Tension Effect, Wetting Correction Direct and indirect liquid flow visualizations have shown that the effect of pre-
wetting of the packing on liquid distribution is significant (Lutran et al., 1991; Ravindra,
et al., 1997; see also Chapter 2). The analysis of basic forces outlined in Section 5.1 also
confirms this result. In general, liquid holdup and particle wetting efficiency are reduced,
and liquid rivulets are favorably formed when introducing trickle flow into the dry
packing. Moreover, the liquid distribution in packed beds is a function of flow history
(Lutran et al., 1991; Ravindra, et al., 1997a). It means that the wetting state of the particle
surface affects the upcoming flow distribution significantly. Experimental studies have
established that the gas and liquid interfacial tension forces and the packing wettability
are not only responsible for the liquid flow maldistribution (see Lutran et al., 1991), but
are also responsible for the hysteresis observed in the pressure-drops and liquid holdup
measured during cocurrent and countercurrent flow in packed beds (see Levec et al.,
121
1988). To numerically capture these flow phenomena, one must consider the interfacial
tension effect and packing wettability on the pressure calculations.
When two immiscible fluids (e.g., gas and liquid) are in contact with each other,
interfacial tension causes the fluids to have different pressures. This discontinuity in
pressure between fluids is know as the capillary pressure, Pc. Specifically, we define the
capillary pressure between gas (G) and liquid (L) as:
CLG ppp =− (5-10)
At a pore scale the capillary pressure can be expressed by
+=
21
112dd
p SC σ (5-11)
The characteristic lengths d1 and d2 are further described in terms of particle diameter,
porosity and the minimum equivalent diameter of the area between three particles in
contact as well as pressure factor, F, as given in Eq 5-12b (see Attou and Ferschneider,
2000).
+
−
=L
G
pG
SSC F
ddp
ρρ
θθ
σmin
3111
12 (5-12a)
L
G
L
GFρρ
ρρ
1.881+=
for ρG/ρL< 0.025 (5-12b)
The capillary pressure also could be expressed through the permeability concept together
with the correlation of the experimental data in various porous media. Grosser et al.,
(1988) proposed the following expression for calculating the capillary pressure
( )5.01
,
−
=k
Jp SSLC
θθθσ (5-13)
where σ is the surface tension, k is the permeability of the porous media, which is related
to the Ergun constant (E1) and the equivalent particle diameter (de) for viscous flow in
packed beds; J is a dimensionless function obtained from the experimental data of
various sand samples with air and water (Leverett, 1941) as given below
122
( ) eS
SS
dE
k θθθ−
=
−
11 5.0
15.0
(5-14)
( )
−−+=
L
LSSLJ
θθθ
θθ 1ln036.048.0, (5-15)
Therefore, the capillary pressure is a function of liquid holdup, and the gradient of the
capillary pressure depends on the gradient of liquid holdup in the packed bed. From the
experimental observation of liquid distribution in prewetted and non-prewetted beds (see
Chapter 2), complete pre-wetting of the particle surface can greatly reduce the gradient of
liquid holdup in the packed bed, which considerably reduces the capillary pressure effect
on liquid distribution. For the modeling of macroscale flow, Equation (5-10) is further
modified by incorporating the particle wetting factor, f, or external particle wetting
efficiency (a fraction of external particle area wetted by liquid) as given below
( ) CGL pfpp −−= 1 (5-16)
By substituting Equation (5-12), or Equations (5-13), (5-14) and (5-15), into Equation (5-
16), one gets Equations (5-17a) and (5-17b), respectively.
( )
+
−
−−=L
G
pG
SSGL F
ddfpp
ρρ
θθ
σmin
3111
112 (5-17a)
( ) ( )
−−+
−−−=
L
LS
eS
SSGL d
Efpp
θθθ
θθ
σ1
ln036.048.01
15.0
1 (5-17b)
It means that the macroscale capillary effect is negligible when the particles are
completely externally wetted (f = 1.0) whereas this effect could be significant when the
particle surfaces are completely dry (f = 0), as one can see from the experiment results
with the non-prewetted beds in Chapter 2.
In the CFD flow simulation the value of particle wetting factor at each cell scale
can be evaluated based on the cell scale flow velocities and local pressure-gradient using
the correlation of Al-Dahhan and Dudukovic (1995) for the external wetting efficiency of
particles in trickle beds.
123
9/1
3/1 1
1Re104.1
+∆
−=
L
LL
GagZP
f ρε
(5-18)
The simulations reveal that in cases of partial particle wetting (f < 1.0), the contribution
of macroscale capillary pressure on liquid flow distribution is significant. It is also
expected that the hysteresis observed in the pressure-drops and global liquid holdup at
bed scale is due to the hysteresis in liquid flow distribution. This in turn is caused by the
capillary pressure hysteresis and different particle wetting status in liquid imbibition and
drainage experiments.
The difference of the calculated capillary pressure values by two expressions,
Equations (5-17a) and (5-17b) does exist even for the same system of air-water as shown
in Figure 5-5. It is noted that Equation (5-17b) was originally derived based on the
experimental data for air-water flow through consolidated porous media (such as sands,
Leverett, 1941). This expression was suggested for flow through packed beds (Grosser et
al., 1988). The derivation of Equation 5-17a was based on the local linear momentum
balance law, applied to the gas-liquid interface, in which the effect of gas density was
incorporated through F(ρG/(ρL), and was claimed to be suitable for elevated pressure
system (see Attou and Ferschneider, 2000). In fact, there have been no direct experiments
designed for validation of these two expressions for trickle beds. For the present time, the
J-function expression (i.e. Eq 5-17b) is used for the air-water system, and the expression
of Attou and Ferschneider (2000) can be used for other systems, particularly at elevated
operating pressures.
5. 5.4 Effect of Mesh Size on Computed Results In general, the grid is a discrete representation of the continuous field phenomena
that one wants to model. The accuracy and numerical stability of the simulation depends
on the choice of the grid (Tannehill et al., 1997). In CFDLIB code the finite volume
method is used to discretize the conservation equations. At this stage, we utilized two
different 2-D coordinate systems for the grid cells: Cartesian and Cylindrical coordinates.
124
In the 2-D cylindrical coordinates, we assume that there is no dependence on the θ-
direction. Although the flow in a cylindrical column does distribute in three-dimensions,
the radial and vertical distributions of the flow play a more important role than the θ-
direction distribution in determining the reactor performance (Stanek, 1994). In addition
to discretization in space, an explicit temporal discretization scheme is utilized in the
code, so the solution proceeds with respect to a sequence of discrete time, t n, where n is
the cycle number (n = 0, 1, 2, …). The time step ∆t n = t n+1 - t n varies from cycle to
cycle.
As discussed in Section 3, there is certain relationship between the section size
and the standard deviation of sectional porosities. After the section size is chosen, the cell
size needs to be specified by further space discretization. Once a converged solution has
been obtained it is essential to assess the invariance of the computated results with
respect to the section discretization. Particularly for the flow in packed beds, various
flow scales and structural scales exist which make the selection of the grid size important
to generating meaningful computational results. To test the dependence of the solution on
the grid size, two simulation runs, one with a fine grid, and another with coarse grid size
were performed. Figures 5-6 and 5-7 show the comparison of the steady state liquid
upflow interstitial velocity components (Vx, Vz) in the forms of contour and transversal
profiles. Apparently, the flow patterns do no vary with changing the cell size from 1.0 cm
to 0.5 cm. However, the relatively detailed flow characters are obtained in the fine-cell
simulation. Figure 5-8 shows the initial solid volume fraction at section size of 1 cm for
two-phase cocurrent down-flow simulation at the gas and liquid superficial velocities of
0.001 m/s and 0.05 m/s, respectively. If one zooms in a specific square area, say x = 0 ~
4, and z = 4 ~ 8, one can see the sectional flow patterns are similar but the cell scale flow
texture becomes more detailed when the cell size is reduced from 1.0 cm to 0.25 cm as
shown in Figure 5-9. The mesh size independent gas holdup at section scale are shown in
Figure 5-10.
125
XZ
0 2 4 6 8 100
2
4
6
8
10
12
14
16
18
20
CFDLIB97. 2
T=8.994E+01
N=60001
X
Z
0 2 4 6 8 100
2
4
6
8
10
12
14
16
18
20
U243.532.521.510.50
-0.5-1-1.5-2-2.5-3
CFDLIB97.2
T=8.994E+01
N=60001
CFDLIB97.2
T=4.498E+01
N=60001
V=10cm/sLiquidupflow Ux
(a)
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
0 1 2 3 4 5 6 7 8 9 10
X
Vx (I
nter
stiti
al v
eloc
ity),
cm/s
z=10 cm (mesh1) z =11 cm (mesh1) z1=10 cm (mesh2)z2=10.5 cm (mesh2) z3=11 cm (mesh2)
(b)
Figure 5-6. Simulated liquid upflow velocity component, Vx contour (a) and profiles (b)
using mesh1 (10 × 20) and mesh2 (20× 40) at a superficial velocity of 10 cm/s.
VX
126
X
Z
0 2 4 6 8 100
2
4
6
8
10
12
14
16
18
20
CFDLIB97. 2
T=8.994E+01
N=60001
X
Z
0 2 4 6 8 100
2
4
6
8
10
12
14
16
18
20V2
-2-3.04167-4.08333-5.125-6.16667-7.20833-8.25-9.29167-10.3333-11.375-12.4167-13.4583-14.5-15.5417-16.5833-17.625-18.6667-19.7083-20.75-21.7917-22.8333-23.875-24.9167-25.9583-27
CFDLIB97.2
T=8.994E+01
N=60001
CFDLIB97.2
T=4.498E+01
N=60001
V=10cm/sLiquidupflow
Uz
(a)
-30.0-29.0-28.0-27.0-26.0-25.0-24.0-23.0-22.0-21.0-20.0
0 1 2 3 4 5 6 7 8 9 10
X
Vz (I
nter
stiti
al v
eloc
ity),
cm/s
z=10 cm (mesh1) z =11 cm (mesh1) z1=10 cm (mesh2)z2=10.5 cm (mesh2) z3=11 cm (mesh2)
(b)
Figure 5-7. Simulated liquid upflow velocity component, Vz contour (a) and profiles (b)
using mesh1 (10 × 20) and mesh2 (20× 40) at a superficial velocity of 10 cm/s.
VZ
127
THE10.650.6442860.6385710.6328570.6271430.6214290.6157140.610.6042860.5985710.5928570.5871430.5814290.5757140.57
CFDLIB97.2
T=9.000E+01
N=91354
Figure 5-8. Initial solid volume fraction distribution at 10 ×15 section-discretization
(section size = 1.0 cm) for gas-liquid cocurrent downflow simulation
(zoom: x = 0 ~ 4; z = 4 ~ 8).
Solid volume-fraction
128
X0 1 2 3 4
CFDLIB97.2
T=9. 000E+01
N=91354
CFDLIB97.2
T=6. 000E+01
N=226091
CFDLIB97.2
T=5. 000E+01
N=80586
CFDLIB97.2
T=9. 000E+01
N=91354
X0 1 2 3 4
CFDLIB97. 2
T=9.000E+01
N=91354
CFDLIB97. 2
T=6.000E+01
N=226091
CFDLIB97. 2
T=5.000E+01
N=80586
CFDLIB97. 2
T=9.000E+01
N=91354
CFDLIB97. 2
T=5.000E+01
N=80586 (a) (b)
X0 1 2 3 4
THE30.30.2857140.2714290.2571430.2428570.2285710.2142860.20.1857140.1714290.1571430.1428570.1285710.1142860.1
CFDLIB97. 2
T=9.000E+01
N=91354
CFDLIB97. 2
T=6.000E+01
N=226091
CFDLIB97. 2
T=5.000E+01
N=80586
CFDLIB97. 2
T=9.000E+01
N=91354
CFDLIB97. 2
T=5.000E+01
N=80586
CFDLIB97. 2
T=6.000E+01
N=226091 (c) Figure 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).
Gas volume-fraction
129
0.210
0.220
0.230
0.240
0.250
0.260
0 1 2 3 4 5
gas
hold
up
mesh-(a)mesh-(b)mesh-(c)
Figure 5-10. Effect of the mesh sizes (a, b, c) on the cell-scale gas holdup values
5.5.5 Boundary Conditions
It is well known that the quality of flow distribution at the top boundary can have
a profound influence on the bed dynamics (Christensen et al., 1986). Szady and
Sundaresan (1991) experimentally examined the effects of the top boundary and the
bottom boundary on the hydrodynamics in a pilot-scale trickle bed. They found that both
the top and bottom boundaries affect the flow characteristics in the trickling regime of
flow, such as overall pressure-gradient and liquid saturation. They also affect the onset of
pulsing. In the flow simulations of interest in this work, special care has been taken in
setting all the boundary conditions. The stationary boundary conditions are used in most
flow simulations with steady state feed conditions in packed beds. This includes specified
inflow velocities of fluids, zero velocity-gradient for outflow, reflective boundary
condition (or symmetry) at the center, and no-slip condition for the wall(s). However, for
the case of periodic liquid feed or gas feed case (so called ‘on-off’ flow modulation), the
130
inflow velocity is specified to vary with time. For a 2-D cylindrical coordinate (a
cylindrical bed), a reflective-wall boundary is used for the left side of the logical block
(i.e., the center-line of the column); for a 2-D Cartesian coordinate (a rectangular bed), a
no-slip wall condition is used for the right side of the block. All the boundaries are
treated as Eulerian boundaries since these boundaries are stationary in space even in the
periodic operation. In Chapter 6, we present the simulated flow distributions at both
steady state and unsteady state flow feed conditions and show how the top boundaries
affect the multiphase flow distributions in the entire packed bed.
5.6 Conclusions and Remarks
The Eulerian k-fluid model has been adopted to model the macroscale multiphase
flow in packed beds in which the geometric complexity of bed structure is resolved
through statistical implementation of sectional porosities, and the complicated multiphase
interactions are evaluated using the Ergun type of expressions which were developed
based on bench-scale hydrodynamics experiments. The effect of particle wetting on flow
distribution is incorporated in the phase pressure computations. The drag formulas for
fluid-particle and fluid-fluid interactions are examined and discussed. The drag exchange
coefficients for the solid particle and fluid, Xks, is obtained based on the models of Holub
et al (1992, 1993) or Attou et al (1999) with E1 of 180 and E2 of 1.8. The drag exchange
coefficient for gas and liquid is calculated by the model of Attou et al (1999). The J-
function is used to calculate the capillary pressure term. Due to the relationship between
the section size and the variance of the sectional porosity, the selection of the section-size
has to follow a certain relation, which is expected to be available by analyzing the full 3-
D porosity distribution data of MRI (Sederman et al., 1997) and of computer tomography,
CT of high spatial resolution. The dependence of the simulated sectional scale flow
pattern on the cell size has been examined for both liquid-solid and gas-liquid-solid and it
was demonstrated that a grid independent macroscopic flow structure can be obtained by
the k-fluid model although the more detailed flow field may be obtained with a finer grid
(i.e., small cell size).
131
Chapter 6
Computational Fluid Dynamics
(CFD): II. Numerical Results &
Comparison with Experimental Data 6.1 Introduction
In recent years, computational fluid dynamics (CFD) has become an important
tool in studies of multiphase flows. It is expected that CFD modeling will become more
pervasive in the design of multiphase reactors as researchers in both academic and
industrial communities are intensifying their efforts in this area. So far the CFD approach
has been used to simulate single phase flow within relatively simple geometries and to
compare the results to those obtained from experiments (Kuipers and van Swaaij, 1998;
Lahey and Drew, 1999; Pan et al., 2000). More complex systems such as multiphase flow
in packed beds has not been studied in detail by the CFD approach due to the complex
geometry of the tortuous pore space and the complicated fluid-fluid and fluid-particles
interactions. A new strategy for flow modeling in packed beds is devised by
implementing the statistical description of the bed structure into the CFD model and by
using the drag forces that have been developed and discussed in Chapter 5. A
multidimensional k-fluid Eulerian model has been adopted and executed in the
framework of the CFDLIB package from Los Alamos National Laboratory. The details of
132
this code library are available elsewhere (Kashiwa et al., 1994; Johnson, 1996; Johnson et
al., 1997).
Since the void space and the flow distribution in random packed beds are
intrinsically statistical in nature, a statistical approach to porosity distribution description
in the bed certainly has advantage over the conventional deterministic mean porosity
assignment everywhere in the bed and/or the use of longitudinal-averaged radial porosity
profile (Stanek, 1994; Bey and Eigenberger, 1997; Yin et al., 2000). Up to date, there
have been few comparisons between the CFD results and the measured flow data in the
bed with randomly packed particles, with different liquids in the upflow mode
(Stephenson and Stewart, 1986). Those comparisons, however, were limited to the use of
longitudinal-averaged radial velocity profiles at different particle Reynolds numbers (Rep
= 5, 80), and no statistical quantities of the two-dimensional velocity field were reported
in Chapter 3. In this Chapter, we expand the above comparison study to include the
statistical quantity comparison such as the probabilities of sectional liquid velocity
distribution.
The entrance (or feed) distribution of fluid(s) is controlled by the distributor
design and the top layer of the packing. The effects of feed distribution on the
macroscopic flow structure were found significant in experiments (Christensen et al.,
1986; Szady and Sundaresan, 1991) and numerical simulations (Anderson and Sapre,
1991). The use of inert large particles as the top layer is rather common in commercial
packed columns. In fact, Moller et al (1996) found that compact ceramic cylindrical
tablets, TK-10, on the top of the packed beds of 1/16 inch cylindrical extrudates have a
positive effect on the liquid distribution (e.g., the top layer compensates for the poor inlet
distribution). However, a negative effect on liquid distribution (e.g., enhancing rivulet
flow) was reported by Szady and Sundaresan (1991) in the packed bed of 3 mm glass
spheres topped with a 10 cm layer of 6 mm Rashig rings. The discrepancy in the effect of
the top layer might be due to the different particle structures used in the two studies for
the top layer, but, definitely, the down-stream flow distribution is sensitive to the upper
boundary of flow, particularly in the trickling flow regime (Szady and Sundaresan, 1991).
Any uneven feed distribution due to the distributor or the top layer can cause a change in
133
the downstream flow pattern. In this Chapter, we intend to explore such flow phenomena
numerically by computing the flow pattern based on the k-fluid CFD model.
This Chapter has been organized in following manner. First, we present a
comparison of CFD predictions and experimental data for liquid upflow in packed beds.
Then we report some comparisons of CFD computations and the measured liquid holdup
and pressure-drop in a pilot-scale trickle beds with gas and liquid cocurrent downflow.
We report only global quantities due to the lack of data on spatial distribution of these
quantities. The second part presents some simulation results regarding the effects of feed
distribution, at steady state conditions, on downstream flow distribution. A summary and
conclusions follow at the end of the Chapter.
6.2 Comparison of CFD and Experimental Results Since the CFD approach to flow modeling presented in Chapter 5 has to be based
on known porosity distribution at a certain scale, a full-comparison of CFD predictions
and experimental data is possible only if the data for distribution of porosity, flow
velocity, and phase volume-fraction are available on the same scale. The lack of such sets
of experimental data in packed beds has made the validation of the current CFD model
impossible. While magnetic resonance imaging (MRI) has recently shown some promise
in providing volume- and velocity-distribution data in packed beds (Sederman et al.,
1997), and it has been illustrated well for the validation of Lattice Boltzmann simulation
for single phase flow (Martz et al., 1999), for multiphase flow of interest in this work,
there is no suitable data in the open literature. What we found in the literature so far is
few experimental results, which could be used for partial validation of our CFD
simulation results. For example, Stephenson and Stewart (1986) presented the
longitudinally averaged radial profile of porosity in a packed bed of spheres and the
corresponding liquid velocity profiles obtained from cylindrical packed beds using a
marker tracking method (i.e., optical measurement). Data were given for several particle
Reynolds numbers within the range from 5 to 280 in the beds with Dr/dp = 10.7 and L/dp
= 20.6. The statistical information on the axial interstitial velocity distribution was also
134
reported in the paper. In at least a partial validation of the numerical results of CFD
simulations, we reported the comparison of CFD predicted velocity profiles and the
measured profiles of Stephenson and Stewart (1986) at particle Reynolds numbers of 5
and 80 for the liquid upflow case in Chapter 3. For a multiphase flow system such as gas-
liquid flow in a packed bed, there is no suitable data for a similar comparison. What we
can do is to look at the CFD predictions of the global hydrodynamic quantities, such as
liquid holdup and pressure-gradient, and compare these quantities quantitatively with
experimental data. For the predicted distributed quantities, a qualitative comparison is the
only choice as the data for the distribution of porosity, multiphase velocities and phase
volume-fraction are available only based on computational results obtained at different
conditions.
6.2.1 Liquid Upflow in Packed Beds Since there was no 2-D porosity distribution data reported in Stephenson and
Stewart (1986), the comparison of CFD predictions for liquid upflow in packed beds was
limited to the longitudinally-averaged radial axial velocity profiles, in which only one-
dimensional (i.e., radial) variation of porosity was considered in the 2-D flow simulation.
Although velocity profiles that agree well with experimental data were achieved in
Chapter 3, the simulated velocity results based on the radial porosity profile could not
capture the reported statistical information with axial interstitial velocity reported by
Stephenson and Stewart (1986). Clearly, there is a need to use a 2D variation in porosity
in the CFD flow simulation in order to get comparable statistical results for velocity, but
unfortunately, these is no such data available in the original paper.
Continuing the discussion initiated in Chapter 5, a packed bed can be treated as a
network of interconnected sections with certain section size (see Figure 5-2 in Chapter 5).
The sectional porosities are normally distributed in a pseudo-Gaussian manner except
when the sectional size is very small, where a pseudo- binomial distribution might be
expected. In this study, based on the mean porosity and reported longitudinal-averaged
radial porosity profile with two different standard deviations, we generated two sets of
pseudo-Gaussian porosity distributions, namely, RN1 and RN2. The sectional porosities
135
together with the longitudinal-averaged porosity values from Stephenson and Stewart
(1986) are plotted in 2-D cylindrical coordinates (r, z) as shown in Figures 6-1a and 6-1b.
The heterogeneity of the RN2 bed is clearly higher than that of RN1 bed, as one can see
from the standard deviations of the two porosity distributions in Table 6.1. Figure 6-2a
shows the comparison of the predicted and the measured longitudinally-averaged radial
axial velocity profiles for RN1 section porosity assignment as well as with the CFD result
from 1-D porosity variation assignment in Chapter 3 at different Reynolds numbers.
Similarly, Figure 6-2b gives the comparison for the RN2 sectional porosity assignment
case. A similar conclusion regarding the predictions of the radial axial liquid velocity
profile can be drawn from these two figures as in Chapter 3. This is expected since the
longitudinally averaged porosity profile determines the longitudinally averaged velocity
profile. However, significant differences do exist in the predicted statistical information
of sectional liquid velocities based on two different porosity distributions RN1 and RN2
in the beds. As one can see from Figures 6-3a and 6-3b, the histogram of predicted liquid
axial velocity in the RN2 bed is much closer to the experimental data than that in the
RN1 bed. This implies that the sectional porosities in the experimental beds are much
more spread than a narrow Gaussian distribution and are much closer to the RN2 bed
shown in Figure 6-1b. This is most likely caused by the fact that cylindrical particles
were used in the experiments of Stephenson and Steward (1986), and larger spread in the
porosity distribution is expected with cylindrical particle than with spherical particles
(Bey and Eigenberger, 1997). Thus, if we know the mean porosity and the statistics of the
porosity distribution, we can predict the statistics of the velocity distribution. In fact, our
CFD simulations reveal that if we fix the mean porosity and the variance of the porosity
distribution, we get the same variance of the flow velocity if we keep the same operating
conditions.
136
Table 6.1. Statistical description of porosities and CFD simulated velocities Two random packed beds RN1 RN2 Porosity mean = 0.3527
S.D. = 0.0420 mean = 0.3534 S.D. = 0.0916
Axial interstitial velocity, Vz, cm/s S.D. / mean (Re = 5) S.D. / mean (Re = 280)
2.0012/ 6.6740 6.9313/ 29.8377
3.8640 / 7.0915 12.0708 / 31.3548
Radial interstitial velocity, Vx, cm/s S.D. / mean (Re = 5) S.D. / mean (Re = 280)
0.4379/ 0.1029 1.2752/ 0.2645
1.8790/ (-0.2034) 7.7352/ (-1.5758)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0r, cm
sect
iona
l por
osity
Figure 6-1a. Generated sectional porosities (RN1) plotted in the radial direction (blank
diamonds) and the longitudinally averaged radial porosity profile of Stephenson &
Stewart (1986) (blank circles). Statistics of the RN1 distribution are given in Table 6.1.
137
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0r, cm
sect
iona
l por
osity
Figure 6-1b. Generated sectional porosities (RN2) (blank diamonds) plotted in the radial
direction and the longitudinally averaged radial porosity profile of Stephenson & Stewart
(1986) (blank circles). Statistics of the RN2 distribution are given in Table 6.1.
138
0.3
0.6
0.9
1.2
1.5
1.8
0 0.76 1.52 2.28 3.04 3.8r (cm)
Rel
ativ
e su
perf
icia
l vel
ocity
0.20
0.30
0.40
0.50
0.60
0.700.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
r/dp
poro
sity
Uz/U0 (Exp.) Uz/U0 (Re =280) RN1Uz/U0 (Re=5) RN1 Uz/U0 (Re=5) PAporosity (Exp.)
Figure 6-2a. Comparison of longitudinally averaged radial velocity profiles at different
Reynolds numbers and experimental data of Stephenson & Stewart (1986).
Statistics of the RN1 bed are available in Table 6.1; PA bed: sectional porosities are only
varying in the radial direction.
139
0.3
0.6
0.9
1.2
1.5
1.8
0 0.76 1.52 2.28 3.04 3.8r (cm)
Rel
ativ
e su
perf
icia
l vel
ocity
0.20
0.30
0.40
0.50
0.60
0.700.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
r/dp
poro
sity
Uz/U0 (Exp.) Uz/U0 ( Re =280) RN2Uz/U0 (Re=5) PA Uz/U0 ( Re=5)RN2porosity (Exp.)
Figure 6-2b. 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.
140
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
-5.0 0.0 5.0 10.0 15.0 20.0 25.0
Axial Interstitial Velocity Vz, cm/s
n/nt
RN1RN2Exp. (Re = 5)
Figure 6-3a. 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).
141
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
-10.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0Axial Interstitial Velocity Vz, cm/s
n/nt
RN1RN2Exp. (Re = 280)
Figure 6-3b. Frequency distribution of axial interstitial velocity (Re = 280):
RN1-CFD simulation based on random porosity 1 (ε: S.D./µ = 0.0916/0.3534; Vx: S.D./µ
= 1.879/0.2034; Vz: S.D./µ = 3.864/7.0915).
RN2-CFD simulation based on random porosity 2 (ε: S.D./µ = 0.0916/0.3534; Vx: S.D./µ
= 1.879/0.2034; Vz: S.D./µ = 3.864/7.0915).
Exp. –Experimental data reported by Stephenson and Stewart (1986).
142
We conclude, based on the above simulations, for single phase flow, that the k-
fluid model can predict not only the longitudinally averaged radial profiles of axial
velocity but also provide the statistical information on fluid velocity distribution provided
that the following information on bed structure are all available.
(i) Mean porosity
(ii) Longitudinally averaged radial porosity
(iii) Sectional porosity distribution type and variance
The mean porosity and the longitudinally averaged radial porosity profile are
obtainable by experiments and are also predictable by various empirical correlations in
the literature (Benenati and Brosilow, 1962; Muller, 1991; Bey and Eigenberger, 1997).
The statistical properties of the porosity distribution are functions of particle size, shape,
column diameter as well as the packing method, which can, in principle, be developed
through 3D sphere-packing computer simulations (see Tory et al., 1973) and extensive
MRI or CT measurements of packed-bed structures (see Baldwin et al., 1996; Sederman
et al., 1997).
6.2.2 Gas and Liquid Cocurrent Downflow in Trickle Beds One should note that the above comparisons are limited to the system which
consists of a fixed solid phase and a saturated flowing fluid, typical examples of which
are (i) gas flowing through fixed beds; (ii) liquid upflow through packed beds (e.g.,
Stephenson and Stewart, 1986). For the packed beds with gas and liquid two-phase flows
(e.g. gas-liquid cocurrent downflow in trickle beds), the competition of gas and liquid for
the fixed cavities between solid particles makes the liquid distribution much more
complicated than the saturated single phase flow distribution. In this Section, we intend
to partially validate the CFD two-phase flow predictions by comparing them with the
experimental data of overall liquid saturation and overall pressure gradient. The case of
two-phase flow in a pilot-scale trickle bed with relatively high gas superficial velocity
(e.g. 0.22 m/s) is chosen to assess the model capability in scaling up trickle bed reactors.
The simulations were based on a cylindrical column having an internal diameter (I.D.) of
0.152 m. The packing part is 1.50 m high and consists of 3 mm spherical particles. The
143
measured voidage of ~0.37 was used as the mean porosity. The working fluids were air
for gas phase and water for the liquid phase. The experimental data was obtained from
the paper by Szady and Sundaresan (1991), in which the overall liquid saturation and
pressure gradient data were reported in packed beds with similar dimensions in both
trickling and pulsing flow regimes.
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.8 1.6 2.4 3.2 4 4.8 5.6 6.4 7.2
(R-r), cm
Poro
sity
sectional valueMueller correlation
Figure 6-4a. Discretization of the radial porosity profile into sectional porosity values
(dp = 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 S.D./mean = 20%, 15%, 10%, 10%,
10%, 10%, 10%, 10%, 10%, 10%.
Before computing the two-phase flow using the k-fluid CFD model, one needs to
generate a multi-dimensional porosity distribution at a certain sectional size as discussed
in Chapter 5. For an axisymmetric cylindrical column, a two-dimensional porosity
distribution, ε(r, z), is needed for flow simulation. Based on the measured mean-porosity
and the longitudinally averaged radial porosity profile curve calculated by Mueller’s
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.3 0.6 0.9 1.2
144
model (Mueller, 1991), one can discretize along the radius, r, into several annular
sections, and calculate the sectional porosities by integrating the radial porosity profile
curve as shown in Figure 6-4a. It is known that the oscillation of the porosity profile is
pronounced in the wall zone which is about 3~4 particle diameters distance from the
wall. The variance of porosities in the wall region is expected to be higher than that in the
core region. Table 6-2 lists the parameters used in the discretization of the annular
sections, and Figure 6-4b displays the solid volume-fraction distribution of the generated
packed bed, in which the porosities of 1500 (=10 × 150) sections are represented by a 2D
pseudo-Gaussian distribution with the standard deviations given in Table 6-2. There are
two annular sections in the radial direction (r) in the wall region with relatively small
section size. The section size in longitudinal direction (z) is 0.01 m. At the top boundary
of the bed, it was assumed that a uniform gas and liquid feed distribution is attained as
claimed in the experiments (Szady and Sundaresan, 1991). Figures 6-5a and 6-5b show
the simulated liquid and gas volume-fraction distribution at gas superficial velocity of
0.22 m/s and liquid superficial velocity of 0.0045 m/s. Relatively high liquid and gas
holdups appear in the wall region where the porosities are high due to the interference of
the wall.
In Figures 6-6 and 6-7, we compare the CFD predictions of the overall liquid
saturations and pressure gradients with the experimental data of Szady and Sundaresan
(1991) at different liquid superficial velocities. We also plotted the calculated values
from the single-slit model (Holub et al., 1992) and the relative permeability model (Saez
and Carbonell, 1985). As discussed in Chapter 5, there is a need to predetermine the two
Ergun values (i.e. E1 and E2) experimentally for using Holub’s model to calculate the
overall liquid holdup and pressure-drop. Similarly, the static liquid holdup, εL0, has to be
determined in order to calculate εL and (∆Ρ⁄∆z) in Saez and Carbonell’s model. The
values obtained by Holub’s model shown in Figures 6-6 and 6-7 are based on the
measured values, E1 = 215 and E2 = 1.4. Two sets of data from Saez and Carbonell model
are based on both measured static liquid holdup (εL0 = 0.022) and the correlation-
estimated value (εL0 = 0.05), respectively. The mean values for liquid saturation and
pressure gradient from the CFD model are plotted. Note that there was no need for using
145
the measured Ergun constants and the measured static liquid holdup in the CFD two-fluid
simulations in which the momentum exchange coefficients, Xgs and Xls, are calculated
from Holub model with constant E1 (= 180) and E2 (= 1.8), and Xgl is from Attou et al
(1999). The detail discussion of Xkl calculations was given in Chapter 5.
0
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100
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Figure 6-4b. Solid volume-fraction distribution generated based on the data in Table 2 in
a pilot scale packed bed.
146
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20
40
60
80
100
120
140
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CFDLIB97.2
T=9.500E+01
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(a) (b)
Figure 6-5. 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.
147
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1
Liquid superficial velocity, cm/s
Liqu
id S
atur
atio
n
CFD modelHolub modelExperimentSaez & Carbonell model (exp)Saez & Carbonell model (cal)
Figure 6-6. 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.
148
0
4
8
12
0 0.2 0.4 0.6 0.8 1
Liquid superficial velocity, cm/s
Pres
sure
Gra
dien
t (kP
a/m
)
decreasing Vlincreasing VlCFD modelHolub modelSaez & Carbonell model (exp)Saez & Carbonell model (cal)
Figure 6-7. 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.
149
Both bed-scale models (Holub model, Saez and Carbonell model with calculated
εL0) give unsatisfactory predictions of the pressure gradient. The k-fluid CFD model and
Saez and Carbonell’s model with measured εL0 provide more reasonable predictions for
the pressure gradient and better prediction for liquid saturation as one can see from
Figures 6-6 and 6-7. The comparison of the k-fluid CFD model predictions with
additional experimental data for overall liquid holdup and pressure gradient can be
performed in a similar way to fully assess how good the k-fluid CFD model is. The onset
of the natural pulsing was observed experimentally at the liquid superficial velocity of 0.8
cm/s at the given gas superficial velocity of 22 cm/s. It seems reasonable that the k-fluid
CFD model should produce agreeable predictions of the overall hydrodynamics quantities
only in the trickling regime (< 0.6 cm/s) for which proper closures were well provided.
As we discussed in Chapter 5, the interactions between the fluid and particles, fluid and
fluid become very complicated at flow transition regime and in pulsing flow regime
which remain a challenge for researchers (Szady and Sundaresan, 1991).
Table 6.2. Parameters used in the discretization of the radial porosity profile, and in the
generation of 2D porosity distribution Two Regions Wall Core Section number and size in r
2, 0.6 cm 8, 0.8 cm
Section number and size in z
150, 1 cm 150, 1 cm
Radial position (from center to wall), cm
7.6, 7.0 6.4, 5.6, 4.8, 4.0, 3.2, 2.4, 1.6, 0.8
Longitudinal-averaged sectional porosity
0.411, 0.363 0.363, 0.365,0.362,0.362,0.363,0.364,0.362,0.366
Ratio of std/mean 0.2, 0.15 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1
6.3 Simulating Feed Distribution Effects There have been many discussions about the role of feed distribution on the flow
distribution inside packed beds in the literature, particularly for gas liquid cocurrent
downflow in the trickling flow regime where well designed gas and liquid distributors are
important for achieving good flow distribution. Beyond this general concern, however,
150
there have been some discrepancies reported on how the feeding of the liquid and gas
affects the downstream flow pattern, particularly in the quantitative sense. In fact, there
are many parameters, which contribute to the feed distribution effect. In most absorption
columns packed with relative large elements (~ 2 - 10 cm), the inertial force and gravity
play an important role in causing significant wall flow, but the particle wetting seems not
to be a significant factor for large-size packing (Stanek, 1994). However, in most trickle
beds with relative small porous particles (~0.5 – 5.0 mm), the capillary force and particle
partial external wetting become important in determining the flow distribution so that
significant feed distribution effects on flow and wetting were found.
As a preliminary study of this topic, we provide a set of numerical results that
describe the simulated flow distribution at steady state liquid and gas feed. Three types of
liquid inlet distributors: single point source, two-point source and uniform distributor
have been tested in numerical simulations using the discrete cell model (DCM) approach
based on the minimization of total energy dissipation rate, as presented in Chapter 4. The
effect of liquid feed distribution was observed to be significant in the upper half of the
bed, and less pronounced at depths exceeding 50 particle diameters (15 cm) for total bed
length of 96 particle diameters. Since the steady-state simulations in Chapter 4 were
limited to a bench scale 2D rectangular packed bed, it is desirable to see if those effects
are retained in a cylindrical pilot-scale packed bed. Hence, we test the effect of a
nonuniform steady state liquid feed distribution in the pilot-scale trickle bed used above
(see Figures 6-4a and 6-4b) on downstream two phase flow distribution using the CFD k-
fluid simulation. The averaged feed superficial velocity for the top ten sections is 0.295
cm/s for liquid and 22 cm/s for gas. Table 6-3 lists the sectional velocities and volume-
fractions of the top layer ten sections for flow simulation with a nonuniform feed
distribution. Such uneven liquid feed distribution might result from the improper design
of the liquid distributor or due to the improper use of the packing top-layer.
151
Table 6.3. Feed velocities and holdups at top ten sections from the center to the wall
Section 1 2 3 4 5 6 7 8 9 10
Vl
εl
0.885
0.363
0
0
0
0
0.738
0.373
0.738
0.391
0
0
0
0
0
0
0.295
0.150
0.295
0.100
Vg
εg
0
0
33.0
0.389
33.0
0.362
0
0
0
0
36.67
0.360
36.67
0.370
36.67
0.364
22.0
0.218
22.0
0.391
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N=86718 Figure 6-8a. Comparison of liquid holdup distribution under nonuniform (left) and
uniform (right) feed conditions at Ul0 = 0.295 cm/s, Ug0 = 22 cm/s.
152
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N=96643 Figure 6-8b. Comparison of gas holdup contour and gas interstial velocity vector plot
under nonuniform (left) and uniform (right) feed conditions at Ul0 = 0.295 cm/s, Ug0 = 22
cm/s.
The gas and liquid flow maldistribution are detected at downstream locations for
a nonuniform gas and liquid feed distribution, listed in Table 6-3, and show that the
overall liquid holdup decreases by ~ 11% in case of a nonuniform distribution (from
0.1084 for uniform inlet to 0.0968 for nonuniform inlet). Moreover, the maldistribution is
more significant in the upper 25 cm portion of the packed bed, although the effect does
propagate throughout the whole packed bed. Figure 6-8a exhibits the comparison of the
liquid holdup distribution at nonuniform feed condition (left plot) and uniform feed
153
condition (right plot). Figure 6-8b displays the gas holdup contour plot and gas interstitial
velocity plot at nonuniform and uniform feed condition. At high gas superficial velocity,
more gas flow is encountered in the wall region due to the higher porosity as shown in
Figure 6-8b (right plot). The nonuniform feed of gas and liquid make the gas
maldistribution worse as one can see from Figure 6-8b (left plot).
6.4 Conclusions The comparison of the k-fluid CFD simulation and the experimental results has
been performed for both liquid upflow and gas-liquid cocurrent downflow in packed
beds. The effects of feed flow distributions have been simulated for a packed bed at
steady state flow conditions. The following conclusions are reached:
(1) The k-fluid CFD model can capture the longitudinally averaged radial axial liquid
velocity profile and the statistical features of the 2-D sectional velocity
distribution provided that the following information on bed structure are all
available: (i) mean porosity, (ii) longitudinally-averaged radial porosity and (iii)
sectional porosity distribution type and its variance.
(2) For two phase flow system reported in Szady and Sundaresan (1991), the
predictions of the k-fluid CFD model on overall liquid saturation and pressure
gradient are comparable with experimental data and with phenomenological
hydrodynamics models developed for reactors of modest scale.
(3) The k-fluid CFD model can simulate the feed distribution effects on flow
distribution at downstream locations.