α Coefficient of the Ergun equation (Equation (3.62)) Pa s m-1
αf Void fraction in particle packings
-αp Particle fraction in particle packings
-β Coefficient of the Ergun equation (Equation (3.62)) Pa s2 m-3 Pa s m-1 β Coefficient in the Koch-Hill drag force calculation
(Equation (4.6)) N m-3
β Coefficient in the Hertz contact model (Equation
(4.20))
-δ Particle overlap m
∆ Particle displacement m
∆max Maximum particle displacement m
δ∗ Maximum particle overlap during collision m
δn Normal particle overlap m
δt Tangential particle overlap m
δ0 Particle overlap during rebound m
Porosity
bulk Bulk packing porosity
i Intraparticle porosity
-ε Strain
-ε0 Strain of the free spring in the SLS model -ε1 Strain of the spring in the Kelvin or Maxwell element
-186
Chapter 13. Glossary
˙
ε2 Strain rate of the dashpot in the Kelvin or Maxwell
element s-1
η Viscosity of the dashpot in the Kelvin or Maxwell
element Pa s
ηDEM Fluid viscosity used in the DEM model Pa s
ηf Fluid viscosity Pa s
γn Normal damping constant N m-1 s-1
γt Tangential damping constant N m-1 s-1
κ Permeability of the particle packing m2
λ Compression, defines the ratio of volume of a com-pressed body to its initial volume -λ Characteristic wavelength of bulk elastic waves m
λ X-ray wavelength m
λbed Overall packed bed compression, defines the ratio of the compressed packing volume to the initial packing volume
-λr Characteristic distance used in the rolling friction
model m
µ Friction coefficient
-µd Dynamic friction coefficient
-µp Interparticle friction coefficient
-µrd Dynamic rolling friction coefficient
-µrs Static rolling friction coefficient
-µs Static friction coefficient
-µw Wall friction coefficient
-µ/ρ X-ray mass attenuation coefficient m2 kg-1
ν Particle Poisson’s ratio, defines the ratio of transverse strain to axial strain during uniaxial particle com-pression
-ν Packing Poisson’s ratio, defines the ratio of trans-verse strain to axial strain during uniaxial packing compression
-νf Kinematic fluid viscosity m2 s-1
ω Particle angular frequency s-1
Φ Packing density
-Φcrit Critical packing density
-ρ Density
-ρf Fluid density kg m-3
ρp Particle density kg m-3
ρp,dry Dry particle density kg m-3
σ Stress Pa
σ0 Stress of the free spring in the SLS model Pa
Chapter 13. Glossary
σ1 Stress of the spring in the Kelvin or Maxwell element Pa σ2 Stress of the dashpot in the Kelvin or Maxwell
ele-ment Pa
σg Gravitational stress Pa
σr Radial stress Pa
σ2z Hydrodynamic dispersion m2 s-2
τ Relaxation time s
τ Stress tensor (Equation (4.2)) m2 s-2
τp Dimensionless poroelastic relaxation time
-τR Ramp loading period s
τw Wall stress Pa
188
14. List of Figures
3.1 Elastic contact between two spheres. . . 8
3.2 Rolling friction of a sphere at the contact surface. . . 11
3.3 Viscoelastic material behavior in the relationship of stress and strain. . . 15
3.4 Schematic representation of the SLS model types. . . 16
3.5 Solvent migration in the particle induced by an external load. . . 20
3.6 Limiting conditions of viscoelastic and poroelastic relaxation. . . 21
3.7 Porosity near solid walls. . . 25
3.8 Correlation of the contact number and the packing density. . . 33
3.9 Bulk (B) and shear (G) modulus as a function of the excess packing density. 34 3.10 Visualization of force chains. . . 36
3.11 Probability distribution of contact forces in a jammed packing. . . 37
3.12 Angular distribution of contacts in a force chain network. . . 38
3.13 Modes of packing compaction. . . 39
3.14 Forces acting on an axial segment of the compressible packed bed. . . 40
4.1 Simple linear spring-dashpot model. . . 47
4.2 Normal force calculated by the Hertz and Hooke contact model. . . 49
4.3 Particle within the computational domain. . . 52
5.1 Process flow chart and micro chromatography column. . . 56
5.2 Column operating set-up. . . 57
5.3 Schematic representation of the micromanipulation method using a flat punch indenter tip. . . 60
5.4 UVM measurements of the packing behavior. . . 64
5.5 Modular structure of the simulation programs. . . 71
5.6 Exemplary CFD mesh geometry for three-dimensional simulation cases. . 72
5.7 Boundary properties of the simulation cases. . . 74
5.8 Schematic representation of the coarse grain method. . . 76
Chapter 14. List of Figures
6.1 Particle size distribution of resins SEP and TOY. . . 78 6.2 Scanning Electron Microscopy (SEM) images of resins CER, SEP and TOY. 80 6.3 SEM images of a broken Sepahrose and Toyopearl particle. . . 81 6.4 Measured force relaxation of Sepharose and Toyopearl particles at three
compression depths. . . 82 6.5 Exemplary compression force relaxation data of a Sepharose and a
Toy-opearl particle. . . 83 6.6 Quasi-static compression of chromatographic particles. . . 86 6.7 Exemplary quasi-static compression data of both chromatographic particles. 87 6.8 Normalized force relaxation responses for Sepharose and Toyopearl particles. 88 6.9 Micro Computer Tomography (Micro CT) images of a Sepharose particle
during micro manipulation. . . 90 6.10 Measured hysteresis of Sepharose and Toyopearl particles during
compres-sion load relaxation experiments. . . 92 7.1 Packing pressure drop and bed height as a function of the fluid velocity. . 96 7.2 Particle Reynolds number as a function of the fluid velocity. . . 97 7.3 Packing compression-relaxation behavior at different hydrodynamic loads. 98 7.4 CLSM images of the packing structure under hydrodynamic load. . . 99 7.5 UV-fluorescence microscopy images of the packing structure under
hydro-dynamic load. . . 100 7.6 Semi-dry packing compression-relaxation behavior at different mechanical
loads. . . 102 7.7 Wet packing compression-relaxation behavior at different mechanical loads.103 7.8 UV-fluorescence microscopy of the packing structure during mechanical
compression. . . 104 7.9 Micro CT image of a packing of CaCl2 coated TOY particles. . . 105 7.10 X-ray attenuation of different material packings as a function of the
pho-ton energy. . . 106 7.11 Micro CT image of TOY particles submerged in a contrast enhancing
agent solution. . . 107 7.12 Influence of the column diameter on the packed bed pressure drop. . . 108 7.13 Logarithmic dependency of the packing pressure drop on the column
di-ameter. . . 109
190
Chapter 14. List of Figures
7.14 Characteristic packing compression-relaxation hystereses at different col-umn diameters. . . 110 7.15 Logarithmic dependency of the packing compression on the column
diam-eter. . . 111 7.16 Simulated packing behavior as a function of the column diameter
accord-ing to the wall support model. . . 113 8.1 Comparison of nonlinear Hertz and linear Hooke particle contact models. 118 8.2 Effect of the amount of coarse graining on hydrodynamic column
perfor-mance. . . 123 8.3 Effect of the amount of coarse graining on mechnical packing compression
behavior. . . 124 8.4 Variation of the particle-particle as well as the rolling friction coefficients. 126 8.5 Comparison of measured and simulated packing pressure-flow dependency. 130 8.6 Comparision of the measured and simulated packing compression behavior
during hydrodynamic load. . . 130 8.7 Comparision of the measured and simulated packing compression behavior
during mechanical load. . . 131 9.1 Migration of Toyopearl SP-650M tracer particles in the 9.6 mm i.d.
col-umn during cyclic colcol-umn operation. . . 136 9.2 Migration of Toyopearl SP-650M tracer particles in the 12.5 mm i.d.
col-umn during cyclic colcol-umn operation. . . 136 9.3 Particle migration in the inner packing regions. . . 137 9.4 Migration of Toyopearl Butyl-650C tracer particles in the 12.5 mm i.d.
column during cyclic column operation. . . 138 9.5 Particle migration in the inner packing regions using the Butyl-650C resin.138 9.6 Migration of Toyopearl Butyl-650S tracer particles in the 9.6 mm i.d.
column during cyclic column operation. . . 139 9.7 Migration of Toyopearl Butyl-650S tracer particles in the 12.5 mm i.d.
column during cyclic column operation. . . 139 9.8 Effect of high flow velocity on the migration of small Toyopearl Butyl-650S
tracer particles in the 9.6 mm i.d. column. . . 140 9.9 Homogeneous distribution of Butyl-650S tracer particles in the packing. . 141
9.10 Simulated particle rearrangement and migration in a packing with a
bidis-perse particle size distribution. . . 142
9.11 Simulated particle migration in a packing of a polydisperse particle size distribution. . . 143
9.12 Simulated influence of particle migration on the fluid flow distribution. . 144
10.1 Simulation of the force transmission in a frictional particle packed bed. . 150
10.2 Measured variations of the axial packing density profiles of the laboratory column during different hydrodynamic loads applied after packing. . . 153
10.3 Measured long-term axial packing density profiles of columns packed ac-cording to different methods. . . 156
10.4 Measured variation of the packing density profiles from the initial profile. 157 10.5 Simulation of the axial packing density profiles during long-term operation of columns packed according to different methods. . . 159
10.6 Comparison of the simulated and measured axial packing density profiles of columns packed according to different methods. . . 160
10.7 Force probability distribution in the simulated columns packed according to different methods. . . 161
10.8 Angular distribution of contact forces in packings packed by different methods. . . 162
10.9 Simulated radial fluctuations of the axial flow velocity and porosity dis-tribution. . . 163
10.10Calculated axial hydrodynamic dispersion in columns packed by the three different packing methods during operation. . . 165
11.1 Comparison of different simulated column packing methods. . . 176
B.1 Portable process control set-up. . . 215
B.2 Microfluidic flow cell. . . 216
B.3 Sequence of a coupled CFD-DEM simulation. . . 218
C.1 5-Amino-2,4,6-triiodoisophthalic acid x-ray tomography contrast agent . 228 C.2 SLS model . . . 229