Zusammenfassung / Abstract ...................................... v
Nomenclature ................................................. xvii
List of Figures ............................................. xxvii
List of Tables ............................................. xxxiii
1 Introduction ................................................. 1
1.1 Relevance of the study .................................. 1
1.2 Objectives of the thesis ................................ 4
1.3 Outline ................................................. 5
2 Basic theory and concepts .................................... 7
2.1 Historic perspective on the modelling of
biodegradation .......................................... 7
2.2 Pore-scale modelling .................................... 8
2.3 Mathematical description of reactive transport .......... 9
2.3.1 Transport of reactive species ................... 11
2.3.1.1 Advection .............................. 11
2.3.1.2 Dispersion ............................. 13
2.3.2 Reaction kinetics ............................... 13
2.4 Bioavailability ........................................ 16
2.5 Upscaling .............................................. 18
2.6 Numerical solution of the system ....................... 22
2.6.1 Discretization of the time derivative ........... 22
2.6.2 Grid generation ................................. 23
2.6.3 Finite Element Method ........................... 24
2.7 The software toolbox UG ................................ 30
3 Upscaling of the advection-diffusion-reaction equation
with Michealis-Menten kinetics .............................. 31
3.1 Introduction ........................................... 32
3.2 Conceptual Model ....................................... 36
3.2.1 Mathematical description ........................ 36
3.2.2 Geometrical description ......................... 40
3.2.3 Scenarios considered for calculations ........... 41
3.2.4 Upscaling of the pore-scale processes ........... 42
3.2.5 Numerical scheme ................................ 44
3.3 Upscaling and Analytical Methods ....................... 46
3.3.1 First-order kinetics ............................ 46
3.3.1.1 Uniform velocity field ................. 47
3.3.1.2 Parabolic velocity field ............... 48
3.3.2 Michaelis-Menten kinetics ....................... 50
3.3.2.1 Uniform velocity field ................. 50
3.3.2.2 Parabolic velocity field ............... 52
3.3.3 Synopsis of analytical methods .................. 54
3.4 Results and Discussion ................................. 55
3.4.1 First-order kinetics with uniform velocity
field ........................................... 55
3.4.2 First-order kinetics with parabolic velocity
field ........................................... 58
3.4.3 Michaelis-Menten kinetics with uniform
velocity field .................................. 60
3.4.4 Michaelis-Menten kinetics with parabolic
velocity field .................................. 63
3.4.5 Equivalent Michaelis-Menten parameters .......... 65
3.4.6 Synopsis ........................................ 70
3.5 Summary and Conclusion ................................. 72
4 A Linear Exchange Model for the description of
bioavailability at the pore scale ........................... 75
4.1 Introduction ........................................... 77
4.2 Theory ................................................. 80
4.2.1 Biodegradation at the pore scale ................ 80
4.2.2 Pore-scale geometry ............................. 81
4.2.3 Mass transfer described by a linear exchange
term ............................................ 81
4.2.4 Dimensionless description ....................... 84
4.2.5 General approach ................................ 87
4.3 Results and Discussion ................................. 89
4.3.1 General behavior of the mass-flux coefficient
jtr ............................................. 90
4.3.2 Using a constant estimate for jtr ............... 93
4.3.3 Comparison with macroscopic Michaelis-Menten
kinetics ........................................ 95
4.3.4 Relevance of pore-scale bioavailability
restrictions .................................... 96
4.3.5 Comparison to experimental data ................. 98
4.3.6 Implications for practical applications ......... 99
5 Influence of biofilm growth and pore-width variations on
effective degradation rates ................................ 101
5.1 Introduction .......................................... 102
5.2 Theory ................................................ 104
5.2.1 Scale of Interest .............................. 104
5.2.2 Governing Equations ............................ 104
5.2.3 Channel Domain ................................. 105
5.2.4 Sinusoid domain ................................ 106
5.2.5 Scaling analysis ............................... 107
5.2.6 Scenarios considered for the calculation ....... 109
5.3 Analytical Methods .................................... 111
5.3.1 Scenario I ..................................... 112
5.3.2 Scenario II .................................... 113
5.3.3 Scenario III ................................... 114
5.4 Results and Discussion ................................ 118
5.4.1 Scenario I ..................................... 118
5.4.2 Scenario II .................................... 119
5.4.3 Scenario III ................................... 120
5.4.4 Scenario IV .................................... 122
5.4.5 Contributions of the new reactive-transport
model .......................................... 123
5.5 Summary and Conclusion ................................ 125
6 Synthesis .................................................. 127
6.1 Summary and conclusions ............................... 127
6.2 Outlook ............................................... 130
Bibliography ............................................... 133
A Appendix of Chapter 3 ...................................... 151
A.l Development of the upscaled solution .................. 151
A.1.1 Separation of the variables .................... 151
А.1.2 Analysis of the case of first-order kinetics
with a uniform velocity field .................. 156
A.1.3 Analysis of the case of first-order kinetics
with a parabolic velocity field ................ 158
A.1.4 Analysis of the case of Michaelis-Menten
kinetics ....................................... 160
В Appendix of Chapter 4 ...................................... 163
B.l Analytical estimate for jtr ........................... 163
B.2 Effective bioavailability ............................. 166
B.3 Experimental setup .................................... 167
Danksagung .................................................... 171
Selbstständigkeitserklärung ................................... 173
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