modelling of clogging in laboratory column experiments
TRANSCRIPT
Modelling of Clogging in Laboratory Column Experiments Conducted with Synthetic Land fdl Leachate
by Andrew J. Cooke
Faculty of Engineering Science Civil Engineering
Submitted in partial fuifilment of the requirements for the degree of
Master of Engineering Science
Faculty of Graduate Studies The University of Western Ontario
London, Ontario, Canada September 1997
OAndrew J. Cooke 1997
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ABSTRACT
A numericd model has been developed for predicting the rate of clogging in
column experiments conducted using synthetic landfi11 leachate as part of a broader study
of the clogging processes occurrïng in landfill leachate collection systems. Consideration
is given to biological growth and biochemically driven minerai precipitation. The model
represents the porous media flow system as a collection of elements in which each element
acts as a separate, fixed film reactor. By applying biological process and geotechnical
engineering concepts to the system, the model predicts the substrate utilization, growt h
and decay of biofilm, and accumulation of inert biomass and calcium carbonate on the
porous media at any tirne, or position, dong the length of the column. The model is
applied to two series of experiments and is show to be successfiil in predicting substrate
removal, biofilm thickness and porosity.
Keywords: Numerical modelling, Clogging in drainage systems, Synthetic leachate,
Porous mediq Biofilm, Mineral precipitation, Landfill.
ACKNOWLEDGEMENTS
The author wishes to express his appreciation and gratitude to his supervisor, Dr.
R Kerry Rowe, who provided patient and continual guidance, advice and encouragement
throughout the duration of this research.
The author would also like to acknowledge Dr. Bruce E. Rittmann for his insight,
guidance, and encouragement.
Sincere appreciation is expressed to 1. Fleming, M. Armstrong, L. Hrapovic, Dr.
R. Cullimore, S. Millward, and J. vanGulck for their assistance and for perforrning the
laboratory column tests with synthetic leachate that are modelled in this thesis. In
addition, R. Brachman provided much advice and support, and the faculty, staff and
graduate students at the Geotechnical Research Centre were always available for
assistance when needed; their contributions are much appreciated.
This study was supported by the Natural Sciences and Engineering Research
Council of Canada under Collaborative Research Grant CPG 0 163097.
Finally, the author wishes to thank his family and fkiends for their love and support
throughout the course of this study.
TABLE OF CONTENTS
CERTIFICATE OF EXAMINATION . . . . . . . . . . . . . . . . . . .
ACKNOWLEDGEMENTS . . a . .
TABLEOFCONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . .
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . xix . . . . . . .
CaAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 LANDFILLING 1 1.2 DRAINAGE SYSTEM CLOGGINC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 COLUMN TESTS 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 O ~ C T M 5 1.5 THESIS OUTLINE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
CHAPTER 2 MODELLiNG TBE BIOLOGICAL CLOGGING IN COLUMN EXPERlMENTS FED SYNTHETTC LEACaATE . . . . . . . . . . 10 2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 B A ~ E R I U M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Dimensions and Classification . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Growth 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 FORMATION OF A BIOLOGICAL GROWTH ENVIRONMENT . . . . . . . . 15 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Chernical and Biochemical Oxygen Demand . . . . . . . . . . . . . 16 2.3.3 Landfil1 Leachate and the L.C.S. Environment . . . . . . . . . . 16
2.3.3.1 Leachate Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.3 -2 The L.C. S . Environment . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.4 Synthetic Leachate and the Column Test Environment . . . . 18 2.4 THE CLOGGING PROCESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.2 Biological Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4.7.5 Numencal Solution of Rhombic Model Shape for Volume and Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.7.6 Corrective Model Solutions for Porosity and Specific . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface 53
3.4.8 Derivation of the Special Cap Mode1 Solution . . . . . . . . . . . 55 3.4.8.1 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.8.2 Specific Surface 56 3.4.9 Derivation of Final Corrected Equations . . . . . . . . . . . . . . . . 57
3.4.9.1 Base Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4.9.2 Corrected Equations for Cubic Packing . . . . . . . . . . . . . 58
. . . . . . 3.4.9.3 Corrected Equations for Orthorhombic Packing 58 3 -4.9.4 Corrected Equations for Tetragonai-Sphenoidal Packing
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 . . . . . . 3.4.9.5 Corrected Equations for Rhombohedral Packing 61
. . . . . . . . . . . . . . . . . . . . 3.4.10 Note Regarding Taylor et al . (1990) 62 3.5 FILM TH~CKNESS AT CLOGG~NG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5.1 Pore Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 S ~ Y 64
CHAPTER 4 . . . . . . . . . . . . . . . . . . . . . . . . . . CLOGGING MODEL: FORMULATION 81
4.1 ~NTRODUC~ION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 REPWSENTATION OF COLUMN FLOW . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 . . . . . . . . . . . . . . . . . . . 4.2.2 Representation of the Porous Media 82
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Boundary Conditions 83
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Advection Algorithm 83 4.2.4.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
. . . . . . . . . . . . 4.2.4.2 Caiculation of Fluid Travel Time Delays 84 . . . . 4.2.4.3 Management o f Calculated Effluent Concentrations 85
4.4 BIOLOGICAL PROCESSES WITHXN ELEMENT . . . . . . . . . . . . . . . . . . . . 88 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4.2 Process Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.3 Biofilm Idealization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.4 Calculation of Substrate Flux . . . . . . . . . . . . . . . . . . . . . . . . . 90
. . . . . 4.4.5 Calculation of Biofilm and Mineral Growth and Loss 94 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5.1 Introduction 94
. . . . . . . . . . . . 4.4.5.2 Non-Steady Growth and Loss of Biofilm 95 4.4.5.3 Minerai Precipitation and Inert Biomass Accumulation . 96
. . . . . . . . 4.4.6 Calculation of Porosity and Speeific Surface Area 99 4.4.7 Convergence to Revised Effluent Substrate Concentration
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.4.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.4.7.2 Application of a Second Substrate . . . . . . . . . . . . . . . . 101 4.4.7.3 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IO1
vii
LIST OF TABLES
Table
2.1
3.1
3.2
3 -3
3 -4
Description Page
Conditions afEecting rnethanogenesis in landfills (Farquhar. 1989) . . 18
Characteristics of Packing of Uniform Spheres . . . . . . . . . . . . . . . . . 38
Surnrnary of Critical Film Thicknesses in Terms of 2Lt / d, . . . . . . . 45
Corrective Models and Corresponding Film Thickness (2Lt / d, ) . . . 48
Surnrnary of Film Thicknesses at Pore Discontinuity in Tems of .. 2L,/dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculated variables for advection algorithm example . . . . . . . . . . . . 87
Porosity and Specific Surface Equation Summary . . . . . . . . . . . . . . 121
Summary of Convergence Routine Details . . . . . . . . . . . . . . . . . . . . 122
Experiment Flow Rates and Muent Substrate Concentration . . . . . 130
. . . . . . . . Given Mode1 Parameters (Rittmann and McCarty. 1980) 131
Assumed Mode1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Parameter Sensitivity Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Directly Measured Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . 147
Indirectly Measured Input Paramet ers . . . . . . . . . . . . . . . . . . . . . . . 150
Assumed Mode1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Variable Mode1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Composition of Synthetic Keele Valley Leachate (Millward. 1997) . . 163
Surnrnary of reported Monod kinetic constants for anaerobic degradation of propionic and acetic acids by various mixed cultures ' 164
Summary of Critical Film Thicknesses in Terms of 2Lt I d, . . . . . . . . 235
LIST OF FIGURES
Figure Description Page
1.1 Typical two liner, two leachate collection system Iandfill. . . . . . . . . . 8
1.2 Granular leachate collection system drainage layer and, (Inset) unsaturated and saturated zones of the drainage path with dominant flow paths shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Common idealization of biofilm structure (Characklis and Marshall, 1990). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 (a) Formation of iron sulphide on sulphate reducing bacteria, (b) Formation of carbonate on methane bacteria (Brune et al., 199 1). 3 5
3.1 The four stable, regular packing arrangements, illustrated as faces of the unit cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Variation of porosity with film thickness using equations by Taylor et al. (1990). Symbols indicate mode1 breakdown due to the occurrence of volume overlaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Variation of normalized specific surface area with film thickness using equations by Taylor et al. (1990). Symbols indicate model breakdown due to the occurrence of volume overlap. . . . . . . . . . . . . 68
3.4 Verification of length of radius of interface circle at any contact point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Verfication of film thickness at onset of spherical cap overlap for cubic packing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3 -6 Verification of film thickness at complete occlusion of cubic packing. . . . . . . . . . Double hatched area indicates overlap of spherical caps. 71
3 -7 Verification of film thickness at occurrence of first overlap for orthorhombic packing. Hatched area indicates spherical cap at two contacts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.8 Overlap shapes used in corrective models. The shapes depicted, named according to the face in which they occur, are (a) Square, @) Rhombic, and (c) Special Rhombic Spherical Cap Overlap. . . . . . . . 73
3.9 Diagram of shapes for volume and surface area integration. . . . . . . . 74
3.10 Curve fit to normalized volume of overlap as a fùnction of d'Et for the square corrective model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
xii
Curve fit to normdized surface area of overlap as a function of a/R . . . . . . . . . . . . . . . . . . . . . . . . . . . for the square correction model.
Curve fit to normalized volume of overlap as a function of aR for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the rhombic corrective model.
Curve fit to normalized surface area of overlap as a function of aR for the rhombic corrective model. . . . . . . . . . . . . . . . . . . . . . . . . .
Corrected variation of porosity with film thickness. Symbols indicate the beginning of corrective models, pore discontinuity and curve tennination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Corrected variation of normalized specific surface with film thickness. Symbols indicate the beginning of corrective models, pore
. . . . . . . . . . . . . . . . . . . . . . . . discontinuity, and cuve termination.
(a) Column test apparatus, (b) flow field column test modelling, (c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . one element (or "segment").
Method of retrievd of effluent concentrations for advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . algonthm.
Flow diagram of processes performed for each element of the flow field during each tirnestep (optionai secondary substrate processes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . omitted).
(a) Idealized biofilm and thicknesses, (b) substrate concentration . . . . . . . . . profiles (modified fiom Rittmann and McCarty, 1980a).
. . . . . . . . . . . . . . . . . Flow diagrarn of primary processes of model.
Effect of timestep length dt on predicted effluent Acetate . . . . . . . . . . . . . concentration for test BC 1 using mean pararneters.
Effect of timestep length dt on predicted effluent Acetate . . . . . . . . . . . . . concentration for test BC3 using mean parameters.
Effect of number of segments on predicted effluent Acetate . . . . . . . . . . . . . concentration for test BC 1 using mean pararneters.
Measured Acetate concentration profile dong the length of the column at steady-state and initiai predictions using mean pararneters for test BCl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured Acetate concentration profile along the length of the colurnn at steady-state and initial predictions using mean pararneters fortestBC3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S..
Xlll
Measured Acetate concentration profile dong the length of the colurnn at steady-state and fined predictions using Ks = 4.48 mg/L
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . and b = 0.254 d" for test BCI.
Measured Acetate concentration profile dong the length of the column at steady-state and fitted predictions using Ks = 4.48 mglL
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . and b = 0.254 d" for test BC3.
Measured biofilm thickness profile dong the length of the column at steady-state and fitted predictions using Ks = 4.48 mgR. and b = 0.254 d" for test BC1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured biofilm thickness profile along the length of the column at steady-state and fined predictions using Ks = 4.48 mg/L and b = 0.254 d" for test BC3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured Acetate concentration profile along the length of the column at steady-state and effect of difision layer thickness LI on fitted predictions using Ks = 4.48 mgL and b = 0.254 d" at 28 days
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for test BC3.
Measured Acetate concentration profile along the length of the column at steady-state and effea of diffusion layer thickness L, on fitted predictions using Ks = 4.48 mgL and b = 0.254 d" at 28 days for test BC3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured Acetate concentration profile along the length of the column at steady-state and effect of diffusion coefficient D, on fitted predictions using Ks = 4.48 m g L and b = 0.254 d-' at 28 days for testBC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured Acetate concentration profile dong the length of the column at steady-state and effect of detachment method on fitted predictions using Ks = 4.48 mg/L and b = 0.254 d-' at 28 days for testBC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured calcium carbonate removed versus COD removed and calculation of calcium carbonate yield coefficient Y, by linear regression for colurnns fed synthetic leachate. . . . . . . . . . . . . . . . . .
Reported measured half-velocity coefficient, Ks, versus system temperature for bacterial growth in propionate and acetate (various sources, see Table 5.10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reported measured maximum specific substrate utilization rate, q, versus system temperature for bacterid growth in propionate and
. . . . . . . . . . . . . . . . . . . . acetate (various sources, see Table 5.10).
xiv
Reported measured yield coefficient, Y, versus systern temperature for bacterid growth in propionate and acetate (various sources, see
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 5.10).
Reported measured decay coefficient, b, versus system temperature for bacterial growth in propionate and acetate (various sources, see
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table S. 10).
Measured and initiaily predicted nomalized effluent concentration using q = 4 mgCOD1mgVS-d for propionic acid and q = 4 mgCOD/mgVS-d for acetic acid for colurnn tests fed synthetic
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . leachate.
Measured and initially predicted porosities using q = 4 mgCOD/mgVS-d for propionic acid and q = 4 mgCOD/mgVS-d for acetic acid after 220 and 270 days of operation of columns fed synthetic leachate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured normalized etnuent concentration and model prediction using q = 3 mgCOD/mgVS-d for propionic acid and q = 4 mgCOD/mgVS-d for acetic acid for column tests fed synthetic
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . leachate.
Measured porosities and model predictions using q = 3 mgCOD1mgVS-d for propionic acid and q = 4 mgCOD1mgVS-d for acetic acid after 220 and 270 days of operation of colums fed synthetic leachate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured normalized effluent concentration and the effect of timestep length dt on mode1 predictions using q = 3 mgCOD/rngVS-d for propionic acid and q = 4 mgCOD/mgVS-d for
. . . . . . . . . . . . . . acetic acid for column tests fed synthetic leachate.
Measured porosities and the effect of timestep length dt on model predictions using q = 3 mgCOD1mgVS-d for propionic acid and q =
4 mgCOD/mgVS-d for acetic acid after 220 and 270 days of . . . . . . . . . . . . . . . . . . operation of columns fed synthetic leachate.
Measured normalized effluent concentration and the effect of the number of segments on model predictions using q = 3 mgCODImgVS-d for propionic acid and q = 4 mgCOD1mgVS-d for
. . . . . . . . . . . . . . acetic acid for column tests fed synthetic leachate.
Measured porosities and the effect of the number of segments on model predictions using q = 3 mgCODIrngVS-d for propionic acid and q = 4 mgCOD/mgVS-d for acetic acid after 220 and 270 days of operation of columns fed synthetic leachate. . . . . . . . . . . . . . . . . . .
Measured normalized effluent concentration and mode1 predictions using q = 3 mgCODImgVS-d for propionic acid and q = 3.9 mgCOD1mgVS-d for acetic acid for column tests fed synthetic
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . leachate.
Measured porosities and model predictions using q = 3 mgCOD/mgVS-d for propionic acid and q = 3.9 rngCOD1mgVS-d for acetic acid after 220 and 270 days of operation of columns fed
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . synthetic leachate.
(a) Measured porosities and model predictions using q = 3 mgCOD1mgVS-d for propionic acid and q = 3.9 mgCODlmgVS-d for acetic acid d e r 130 days of operation and @) porosity profiles at
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 day intervals.
Variance of predicted propionate degrader film thickness at specified . . . . . . . . . . . . . . . . . . . . . . . . . . . . column heights (Ht) over tirne.
Variance of predicted acetate degrader film thickness at specified . . . . . . . . . . . . . . . . . . . . . . . . . . . . column heights (Ht) over tirne.
Variance of predicted total inactive film thickness at specified . . . . . . . . . . . . . . . . . . . . . . . . . . . . column heights (Ht) over time.
Variance of predicted effluent COD concentrations of propionic . . . . . . . . . . . . . . . . . acid, acetic acid, and the total COD over time.
Predicted (a) propionate and @) acetate COD concentration profiles . . . . . . . . . . . . . . . . . . . . . . . . along the column at 50 day intervals.
Predicted (a) propionate and (b) acetate degrader film thickness . . . . . . . . . . . . . . . . . . profiles along the column at 50 day intervals.
Predicted propionate and acetate degrader film thickness profiles . . . . . . . . . . . . . along the column at (a) 220 days and (b) 270 days.
Variance of predicted detachment coefficient b' at specified column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . heights (Ht) over time.
Predicted propionic, acetic and total COD profiles along the column . . . . . . . . . . . . . . . . . . . . . . . . . . . at (a) 220 days and @) 270 days.
Predicted inactive and total film thickness profiles dong the column . . . . . . . . . . . . . . . . . . . . . . . . . . . at (a) 220 days and (b) 270 days.
Variance of predicted total and inactive film thickness at the influent (Seg. 1) and effluent (Seg. 13) ends of the
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . column over time.
5.41
5 -42
5 -43
5 -44
Al. 1
Al .2
A1 -3
A1.4
Variance of predicted total film thickness at specified column heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . wt)overtime
Variance of predicted total active film thickness at specified column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . heights (Ht) over time.
Variance of predicted detachment rates for propionate degraders at specified colurnn heights (Ht) over time. . . . . . . . . . . . . . . . . . . . . .
Variance of predicted detachment rates for acetate degraders at specified column heights (Ht) over time. . . . . . . . . . . . . . . . . . . . . .
Venfication of length of radius of interface circle at any contact point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Verification of film thickness at onset of sphencal cap overlap for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic packing.
Verification of film thickness at complete occlusion of cubic packing. Double hatched area indicates overlap of spherical caps. . . . . . . . . .
Verification of film thickness at occurrence of first overlap for orthorhombic packirig. Hatched area indicates sphencal cap at two contacts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Verification of film thickness at occurrence of second overlap for orthorhombic packing. Double hatched area indicates overlap of
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sphencal caps.
Verification of film thickness at pore space occlusion for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . orthorhombic packing.
Verification of film thickness at onset of special sphencal cap overlap for tetragonal-sphenoidai packing.
Verification of film thickness at onset of complex overlaps for tetragonal-spheroidal packing. Special spherical cap shown in centre ofsection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Venfication of film thickness at pore occlusion for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tetragonal-sphenoidal packing.
Verification of film thickness at onset of complex overlaps for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rhombohedral packing.
Verification of film thickness at onset of pore occlusion for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rhombohedral packing.
LIST OF APPENDICES
Appendix Title Page
Al DETERMINATION OF CRITICAL FILM THICKNESSES OF THE POROSITY AND SPECIFIC SURFACE MODELS . , . . . 2 19
NOMENCLATURE
Symbol Definition Dimension
Radial distance to overlap
Specific surface of porous media
Initiai, clean media specific surface
Specific surface calculated using Taylor et al. (1990)
Biornass decay coefficient
Total biornass loss coefficient
Biomass detachment coefficient
Diameter of porous media particle, Stone or sphere
Molecular diffision coefficient of the substrate in the Liquid
Molecular diffusion coefficient of substrate 1 in the liquid
Molecular difision coefficient of substrate 2 in the liquid
Dimensionless molecular difision coefficient of the substrate in the iiquid
Molecular difision coefficient of the substrate within the biofilm
Molecular diffusion coefficient of substrate 1 within the biofilm
Molecular diffision coefficient of substrate 2 within the biofilm
Dirnensionless moIecular difiùsion coefficient of the substrate within the biofilm
Fraction of biomass degradable due to decay
Flux of substrate into the biofilm
Flux of the substrate 2 (propionic acid) into the biofilm
Dimensionless flux of substrate into the biofilm
Segment number
Detachent coefficient
Endogenous decay coefficient
Maintenance coefficient
Half - maximum rate substrate concentration
Half-maximum rate substrate concentration for substrate 1
Haif-maximum rate substrate concentration for substrate 2
Thickness of active biofilm
Dimensionless thickness of the active biofilrn
Revised active biofilm thickness
Active biofilm thickness caused by substrate 1
Active biofilm thickness caused by substrate 2
Revised active biofilm thickness caused by substrate 2
Critical film thickness
Thickness of inactive film layer
Revised inactive film thickness
Total initial active biofilrn thickness
Inactive biofilm thickness caused by substrate 1
Inactive biofilm thickness caused by substrate 2
Revised inactive film thickness caused by substrate 2
Thickness of diffusion layer
Dimensionless thickness of diffusion layer
Total film t hickness (L, = Lfa + L,J
Number of contact points
Porosity
Porosity calculated using Taylor et al. (1 990)
Number of time steps elapsed as fluid travels fiom beginning of column to the end of segment k
Number of segments
Portion of Ieachate which is substrate 2
S b
S b *
S c
S c
s,, sC.2
Matrix colurnn number for advection routine
Maximum specific rate of substrate utilkation
Maximum specific rate of substrate utilization of substrate I
Maximum specific rate of substrate utilization of substrate 2
Substrate flow
Rate of endogenous decay of biomass
Biomass growth rate
Biomass loss rate
Rate of loss of substrate for ce11 maintenance
Net biofilm growth rate
Substrate utilization rate
Rate of bacterial growth
Radius of the surface of a sphere coated uniformiy with film
Reynolds number
Growth limiting substrate concentration
Substrate concentration of the bulk Buid
Dimensionless substrate concentration in the bulk liquid
Shmidt number
Effluent substrate concentration for a segment
Revised effluent substrate concentration
Effluent concentration of secondary substrate (pro pionic acid)
Substrate concentration within the active biofilm
Total influent substrate concentration
Infiltration substrate I concentration
Infiltration substrate 2 concentration
Refractory influent substrate concentration
Substrate concentration at the active biofilm surface
DirnensionIess substrate concentration at the active biofilm surface
Substrate concentration at the base of the active biofiIm
Influent substrate concentration
Average time required for nuid to move from one end of a segment to the other
System temperature
Real number of time steps elapsed as fluid travels from beginning of colurnn to the end of segment k
Temperature of the reference parameter
Volume
Biomass concentration
Density of active biofilm
Density of inactive film layer
Maximum yield coefficient
Yield coefficient for calcium carbonate precipitation
Packing factor
Effectiveness factor
Interior angle of the overlap shape
Temperature factor for a parameter
Absolute viscosity
Time step length
Specific growth rate
Maximum specific growth rate
Standard biofilm depth dimension
Convergence tolerance factor
CHAPTER 1
INTRODUCTION
The most cornrnon method of disposing of municipal and industrial wastes is by
landfilling. Of the approximately 10 million tonnes of municipal solid waste generated in
Ontario, 90% is disposed of in about 1,400 active landfills in Ontario (MOEE, 1992).
Water which percolates through the landNl leaches compounds fkom the decomposing
waste creating a contarninated fluid called leachate. The leachate which escapes fiorn
landfiils may have a negative impact on local groundwater and surface waters. Where
natural attenuation is not sufficient to assure protection of the environment, a barrier
system is conaructed pnor to landfilling to diminish the escape of leachate.
Municipal solid waste landfills typicaliy include a bamer systern consisting of a
leachate collection system or systems and either a natural or engineered liner (or liners)
(see Figure 1.1). A liner rnay be made of compacted clay, a geomernbrane, a geosynthetic
clay liner, or a combination of a geomembrane and either compacted clay or geosynthetic
clay liner. Due to the low hydraulic conductivity of the liner, leachate will build up over
the liner. To prevent this build up, a Ieachate collection system (L.C.S.) is constructed on
the liner. The L.C.S. typicdly consists of a senes of perforated pipes ( co~ec ted to
manholes to allow cleaning) in a granular layer (sand, grave1 or crushed stone). The
L.C.S. is expected to collect most (or ail) of the leachate generated and to controi the
height of the leachate mound on the base of the landfill, thereby minimizing the potential
for contamination of ground water (by migration through the liner) or surface water (by
leachate seeps). The L.C.S. drains the leachate to one or more locations where it is
rernoved for treatment.
Failure of leachate collection systems, including pipes and surrounding drainage
matenal, may be caused by clogging (due to extreme sedimentation, chemical
precipitation, biochernical precipitation or biological growth) or pipe faiiure (eg. due to
separation, breakage or deterioration) (McBean et al, 1995). Clogging has been found to
be a very cornrnon cause of failure of the L.C.S. (Brune et al., 199 1). A L.C.S. is said to
have failed due to clogging when there is a physicai buildup of material in the collection
pipes, drainage layer or filter layer that causes the leachate head to exceed the design
value. Clogging due to sedimentation, and pipe failure, can be controlled by appropriate
design and, in the case of pipes, maintenance. This thesis, therefore, will focus on
clogging due to the presence of microorganisms causing biological growth and
biochernical precipitation. As indicated by Ramke (1989), the mechanisms of biological
clogging include slime growth, filamentous growth, biomass formation and femc
incrustations. Also attributed to biological processes were the bacterial production of
sulphide and carbonate precipitations.
Once a clogged L.C.S. can no longer control the height of the leachate mound
acting on the base of the landfill, the mound will increase and this will generally result in
increased advective contaminant transport through the underlying banier. Clogging cm
also increase the incidence of leachate seeps ( lateral leakage of leachate to surface
waters), and in extreme cases, the stability of the landfill may be compromised. Since the
L.C.S. also removes contaminants fiom the base of the landfill, clogging may reduce its
ability to divert contarninants fiom the underlying hydrogeological system. The reliability
of the leachate collection system is of particular importance since excavation and
replacement are generally not feasible.
The time at which the collection system clogs and the magnitude of the leachate
mound at subsequent times may be critical in terms of assessing the potential impact of the
landfill on groundwater. If the L.C.S. operates without failure for long enough, the
concentration of contarninants in the landfill leachate will reduce and will have negligible
impact if released to the environment (see Rowe, 199 1, 1995 and Rowe et al., 199%).
However, if the collection system fails prematurely, the concentration of contarninants in
the leachate that is now transported through the liner system ma7 cause unacceptable
impact.
Contaminant transport models designed for landfills may incorporate the modelling
of the effect of failure of the leachate collection system on contaminant migration to
adjacent aquifers (Rowe and Booker, 1995, 1997a,b). The time at which this failure may
occur is an important input to these models, and current estimates are based on relatively
cmde caiculations (Rowe et al., 1994).
Exhumation of a portion of the L.C.S. at the Keele Valley Landfill (Rowe et al.,
1997a) has indicated that there are two zones in the collection system. A Iower saturated
zone where the clogging was greatest, and an overlying unsaturated zone where some @ut
less) clogging occurs (see Figure 1.2). It may be hypothesized that the lower portion of
the collection system will decrease in hydrauiic conductivity with tirne, due to clogging,
and that once this clogs the leachate will then be diverted higher up in the system creating
a new saturated zone in the previously unsaturated portion of the leachate collection
system. In order to understand and model the clogging of the leachate collection system,
it is first necessary to understand and model the clogging in the saturated portion of the
system. Furthemore, since the saturated system in real landfills contain many variables,
including variable grainsize of the granular material and variable leachate quality it would
be desirable to first test the model of any such system against tests where the grainsize and
leachate characteristics were controlled before testing it against real systems ( where
greater vanability and uncertainty is to be expected).
Several investigators have performed experiments simulating the passage of landfill
leachate through the saturated portion of the granular drainage layer using laboratory
colurnns packed with porous media fed actual or synthetic leachate (ie. Brune et al., 1991;
Paksy et al., 1995). It is also noted that numerous scientists in the field of wastewater
treatment have performed column expenments which, while not simulating the L.C.S., are
otherwise similar in many ways and provide penpheral contributions to the study and will
be referenced, where appropriate, through the thesis.
To allow increased focus on the biological causes of clogging, a synthetic (man-
made) leachate may be substituted for actual leachate, thus providing relatively uniform
source substrate concentration. The substrate concentration is the concentration of the
nutrient which limits the rate of bacterial growth. It's considered that this type of
controlled column experiment provides the best way of collecting data for testing the
modelling of biological processes occurring in the L.C.S.
1.4 OBJECTIVE
The objective of this thesis is to develop, irnplement and test a numencal model to
predid the clogging of porous media in column experiments conducted using synthetic
landfill leachate as the first step towards predicting the rate of clogging in the drainage
blanket in a landfill leachate collection system. Specifically, the objectives (and original
contribution) of this thesis are to :
derive corrective formulas to allow use of a published geometric mode1 for
caiculation of porosity and specific surface area of film-coated porous media for
situations involving clogging of granular systems;
develop a numerical mode1 of clogging that incorporates:
a a novel relationship between bacterial activity and mineral precipitation so
as to allow simulation of the contribution of each to clogging;
the representation of bacterial utilization of both a primary and secondary
substrate;
a variable mass transfer resistance, advection, and temperature dependencies
of parameters;
develop a method for expenmentally denving a yield coefficient for mineral
precipitation;
examine how well the model simulates the behaviour observed in well controlted
column tests performed by Rowe et ai. (199%);
illustrate the shift in system dominance over time fiom one of biofilm growth to
one of rnineral precipitation for a specific case.
THESIS OUTLIIW
Chapter One has defined the problem. In Chapter Two readers are given
background information regarding the growth of bactena, the formation of biofiirns, the
relationship between bacteria and mineral accumulation, and the proposed structure of the
ideaiized film used by the model. Since published geometric models for the porosity and
speci£ic surface of a porous media fail when there is a significant biofilm, Chapter Three
describes the development of a geometric model that allows the calculation of porosity
and specific surface area of granular media coated with thick film. The ciogging model is
described in Chapter Four, including the method by which the system is defined and
substrate transported, the biofilrn and rnineral film growth and loss routines, the iterative
procedures required to process each element, and the management of the system of
elements as the program marches forward in time. Chapter Five applies the mode1 to two
sets of published column test data: in one case a short duration experiment using a single
substrate and no inactive film growih; in the other case a long duration experiment in
which a synthetic leachate is represented by two substrates and inactive film accumulation
is simulated. The final chapter, Chapter Six, presents the conclusions of this work and
recornmendations for future extension of this work towards the ultimate goal of prediction
of leachate collection system clogging.
Figure 1 .1 : Typical h o liner, Mo leachate collection system landfill.
Primary Leachate
Cover 1
Leac Yion
Waste
Natural Soil
J . . . . . . ' Aquifer . ,
CHAPTER 2
MODELLING THE BIOLOGICAL CLOGGING IN COLUMN EXPERIMENTS FED SYNTHETIC LEACHATE
Landfill leachate collection systems have been found to clog by biological
processes in landfills (Bass, 1986; Brune et al, 1991; McBean et al, 1993; Rowe et al,
1995a,b ) and in expenments simulating landfill conditions (Brune et al. 199 1 ; Paksy et al,
1995).
A survey conducted by Arthur D. Little Inc. (Bass, 1986) found that of 23 U. S.
waste disposal facilities investigated, biological growth was a problem at four sites and
biochemical precipitation was a problem at one site. One site, for example, contained a
30m long biological growth mass which packed a leachate collection pipe. Bass concludes
that although few cases of biological or biochemical clogging of the drainage @ers of a
L.C.S. were found, the conditions beneath landfill facilities are such that these processes
cannot be mled out as possible clogging mechanisms.
A survey by Brune et al. (199 1) of 23 Geman sanitary landfills with Ieachate
collection systems (between 1987 and 1989) found that more than half of the cases
investigated had drainage pipes and drainage layers which contained consolidated,
insoluble deposits. Impairment of the drainage system ranged corn moderate deposits on
the pipe bottom to incrustation of the whole drainage layer. The formation of these
incrustations were aîtributed to anaerobic bacteria. This conclusion was based on
experimental and in-field observations which showed that the anaerobic rnicroorganisms
(present in landfill leachate at high concentrations) easily colonize drainage system
surfaces and this is accompanied by the precipitation of inorganic deposits around the
bacteria. The aggregate of bacteria and inorganic deposit can accumulate to the degree
that clogging occurs. The deposits were found to mainly contain calcium and iron
combined with sulphur and carbonate. A significant arnount of organic material was found
in al1 deposits.
As indicated by Rowe et al. (199Sa), significant clogging has occurred in the
leachate collection system at the Keele Valley Landfll in Canada. The clog matenal
typically consisted of 21 to 27% calcium, 38 to 54% carbonate, 1 to 10% iron and 2 to
7% silica.
McBean et al. (1993) found that al1 the essential features required for rnicrobial
growth were present in leachate collection systems. They report the results of an
excavation of a landfill in Peterborough, Ontario, where biomass accumulation was found
to be the cause of leachate mounding and subsequent leachate seeps.
Paksy et al. (1995) performed a laboratory experiment in which synthetic leachate
was passed through a column modelling a landfill drainage system. It was concluded that
under simulated landfill drainage system conditions bacterial populations cm rapidly
develop on granular materials and due to this growth a gradua1 reduction in drainable pore
volume with time will occur. It was found that the pore reduction leveled off, Ieading to
the conclusion that microbial growth may not continue indefinitely for the test condition
they examined.
Coiumn tests performed by Brune et ai. (1 99 l), using real landfill leachate and
drainage rnaterial (gravel), showed that a highly loaded leachate caused extensive clogging
corresponding to that observed in landfills. The chernical composition and microscopie
structure of the clog material formed was similar to that formed in landfill sites. Over the
approximately 10 to 15 month period of the tests, the permeability of coarse drainage
material (grain size 16 - 32 mm) showed no significant change, while finer drainage
material (grain size 8 - 16 mm) showed a significant reduction. Use of fine and well
graded material (2 - 4 mm and 1 - 32 mm respectively) resulted in considerable loss of
permeability.
These case histories demonstrate that biological clogging of the L.C.S. occurs and
that anaerobic bacterial activity is suspected to be the dominant process. Colurnn tests
have provided the opportunity to observe the processes under controlled conditions.
This chapter discusses clogging in colunin tests fed synthetic leachate by first
describing the bacterium involved, the processes of biological clogging, bacterial growth
models, and the groundwork for the proposed model. To aid in the understanding of the
processes of biological clogging, the structure and metabolic activity of bacterium are first
discussed. The means by which an environment favourable to biological growth cm be
formed is then explored. Thirdly, bacterial adhesion and the clogging process are
discussed. Fourthy, bacterial growth models are addressed and kinetic parameters are
defined, and finally the basis for the distribution of the bacteria by the mode1 is discussed.
2.2 BACTERIUM
2.2.1 Dimensions and Classification
According to Brune et al. (1 99 l), anaerobic bacteria are the primary
microorganism involved in the biological clogging of landfill leachate collection systems.
Bacteria form various shapes and sizes, are unicellular, and most are bound by a rigid ceIl
wall (Charackiis and Marshall, 1990). Many have diameters in the 0.3 to 3 pm range and
Iengths in the 1 to 10 pm range (cocci, bacilli, vibnos) or up to 50 pm long (spinlla) or
greater than 100 pm long (filamentous forms) (Metcalf and Eddy, 199 1). The bacteria ce11
may secrete extracellular polymers which can fom an attached, highiy hydrated capsule or
create a fiee, viscous, soluble slime (Characklis and Marshall, 1990). The composition of
the bacterial ce11 has been found to be 80% water and 20% dry matenals, with 90% of the
dry materials being organic (Metcalf and Eddy, 1 99 1 ).
The dominating bactena in wastewater are usually classified as facultative
anaerobic chemo heterotrophs and obligate anaerobic chernoheterotrop hs, while
chemoautotrophs are also present (McBean et al, 1995). According to Metcalf and Eddy
(1991), a 'chemoautotroph' obtains energy fiom the oxidation of reduced inorgonic
compounds such as ammonia, nitrite and sulphide and carbon from carbon dioxide while a
'chemoheterotroph' usually denves energy and carbon from the oxidation of organzc
compounds. OMigateiy anaerobic bacteria can oniy exist in an environment devoid of
oxygen whilefuczîltative anaerobic bacteria cm grow in the presence or absence of
oxygen.
2.2.2 Growth
The following description of biological growth is based on Characklis and Marshall
(1990), unless othenvise noted, and is provided as background for the reader.
Bioenergetics is the transformation of energy through cellular metabolism. The
process of metabolisrn must be maintained in order to conserve the structure and viability
of the bacteria. The metabolism of bacteria relies upon reactions of oxidation-reduction.
In the process, reduced energy donors (electron donors), are oxidized allowing some
available energy to be stored by the bacterium in a usefùl form. The bacteria of interest
have metabolisms which may utilize organic compounds or reduced inorganic compounds
as energy sources.
Bacterial growth may be defined as the increase in the number of cells or mass of
the entire collection of cells from the initial state. Reproduction is largely by binary
fission, but some bactena reproduce by budding or fragmentation of filamentous growths.
If al1 growth requirements are met a rapid increase in growth will occur.
Bacteria have nutntional and environmental requirements which must be met in
order for growth to occur. Nutntionally, the bacteria must have sources of energy,
carbon, nutrients (ie. nitrogen, phosphorus) and may require organic nutrients for ce11
synthesis. With respect to the bacterial environrnent, the proper range of pH and
temperature among other factors must be met. Specific bacteria groups have fairly narrow
temperature ranges and diEering optimal temperatures. Bacteria have an optimum pH
range fYom 6.5 and 7.5 (Metcalf and Eddy, L 99 1).
2.3 FORMATION OF A BIOLOGICAL GROWTH ENVIRONMENT
2.3.1 Introduction
McBean et al. (1993) found that al1 the essential features required for rnicrobial
growth were present in leachate collection systems including microorganisms: organics as
food; nutrients such as nitrogen and phosphorus; an acidic (in the early stages) to slightly
basic (in the late stages) environment; and temperatures and rnoisture contents conducive
to growth. The establishment of an environment that is conducive to biological growth
and biochemicai precipitation begins with the decomposition of solid waste, coupled with
the percolation of infiltrating water creating a contaminated water called leachate. This
leachate percolates through and is collected by the leachate collection system for removal
from the landfill. The following sections define two rnethods of measuring the organic
content of wastewater, discuss the characteristics of Ieachate and the environmental
conditions created in the L.C.S., and discusses synthetic leachate and the column test
environment.
2.3.2 Chemical and Biochemical Oxygen Demand
Chemical oxygen demand (COD) and biochernical oxygen demand (BOD) are both
parameters which measure the organic content of wastewaters. According to Metcalf and
Eddy (1991). COD is the measurement of the oxygen equivalent of the organic matter that
can be oxidized using a strong oxidizing agent in an acidic medium. BOD is the
measurement of the dissolved oxygen utilized by microorganisms in the biochemical
oxidation of organic matter. The ratio of B0D:COD is often used to show the
degradability of the organic matter in wastewater. Since the COD oxidizing process is
more efficient than the BOD process, which oxidizes only the readily biodegradable
organics, a high ratio of BOD to COD would indicate a highly degradable leachate. ORen
the concentration of organic constituents is considered the limiting factor in microbial
growth (ie. the substrate), and therefore the values of COD and BOD may be critical.
2.3.3 Landfill Leachate and the L.C.S. Environment
2.3 -3.1 Leachate Quality
Due to the heterogeneity of the waste, different phases of decomposition will likely
occur simultaneously in the landfill at different locations until closure, and after closure the
hydrolysis, acetogenic and methanogenic phases will continue simultaneously for several
years until hydrolysis and acid anaerobic phase decomposition is gradually diminished.
Leachate, therefore, may travel through or be produced by zones or layers in the waste of
varying decomposirg phases and so will not simply have the characteristics of one phase,
but be a mixture of the intermediate and end products of different phases and include
constituents of non-biological origin, and rnay include large concentrations of
microorganisms (McBean et al., 1995).
Crawford (1985) stated that the leachate of a young landfill ( les than 2 to 4 years
after closure) is likely to contain high concentrations of organic acids, arnmonia and total
dissolved solids (TDS). The organic acids will rnostly be fatty acids of low molecular
weight, for example acetic acid, and some fùlvic or other more complex organic acids.
The young landfiil leachate's high COD/TOC value reflects the unoxidized state of organic
carbon in the leachate. Inorganic components such as dissolved saits may be present in
concentrations greater than 10,000 mg/l, and high concentrations of sodium, calcium,
chloride, sulphate and iron rnay be present. The pH will be in the low 6 to 7 range
(McBean et ai., 1995). The leachate contains large microbial populations with the
majonty being bacterial, but also including other pathogens such as vimses, protozoa and
helminths (Lu et al., 1985).
2.3.3.2 The L.C.S. Environment
From the previous sections it can be seen that the leachate which flows through the
drainage system will contain the elements conducive to biological growt h. Leachate,
especiaily young landfill leachate has been found on exiting the L.C. S. to contain the
bacteria responsible for decomposition and the organic substrates and nutnents required to
sustain them. Landfill conditions, though, may not be optimum for bacterial growth, as
shown in Table 2.1, compiled by Farquhar (1989) for representative southem Ontario
landfills.
Table 2.1 : Conditions affecting methanogenesis in landfills (Farquhar, 1989)
Parameter - - - . - -
Optimal Representative ranges for Value southern Ontario landfills
Temperature (OC) 35 10-20 at depth > 2 m
PH 7.2 5.5 - 6.5 (young leachate)
Moisture content (% wet weight) Saturation 20 - 30 at placement
Oxidation reduction potential (mV) < - 330 > - 330 at placement
Nutrients Sufficient Phosphorous deficient
Toxicants None Ammonia and certain metais
It can be seen that landfill leachate, especially of southem Ontario landfills, does
not produce an environment which optimizes bacterial growth. It must be noted, though
that the majonty of these comparisons are for methanogenic bactena and that many other
varieties of bacteria exist in the landfill system, and each of them have differing optimum
values of environmental factors. It should also be noted that the comparison in Table 2.1
is an averaging of measured values, and therefore a specific landfill may produce more
favourable conditions. Likewise, more favourable conditions may occur at a particular
point in a landfill (eg. see Barone et al., 1997). To conclude it can be said that while the
conditions in leachate collection systems may not be optimal for bacterial growth, the
requirements exist, and growth is possible.
2.3.4 Synthetic Leachate and the Column Test Environment
Control of the composition of the synthetic leachate and other column test factors
d o w s a column test environment to simulate L.C.S. conditions which are average, or
optimal for bacterial growth. The pH of the synthetic leachate, or temperature of the
system, for example, may be controlled such that microorganism growth is optirnized.
The porous media used in a colurnn test may be selected to relate closely to the L.C.S. by
using actual drainage layer material (such as Brune et al, 199 1) , or may be selected to
relate closely to a mode1 by using, for example, glas spheres (such as Millward, 1997,
Rowe et al., 199%). Synthetic leachate usually does not contain a significant number of
microorganisms, thus the porous media is often inoculated with actual landfill leachate
prior to testing. This provides a source of microorganisms available to consume substrate
frorn the synthetic leachate and multiply throughout the system.
2.4 THE CLOGGING PROCESS
2.4.1 Introduction
The previous section showed that the conditions required for rnicrobial growth
were present in landfill leachate and synthetic landfill Ieachate. It was shown that while
typical leachate does not provide optimum bacterial growth conditions, the potential for
biological growth and biological precipitation does exist, and that synthetic leachate may
be substituted and controlled to optimize this potential. This section discusses the two
biological clogging processes to be modelled: biological growth in the fom of biofilms,
and biochernically driven precipitation.
2.4.2 Biologieal Growth
2.4.2.1 Introduction
The primary cause of biological clogging is believed ( Bass, 1986; Brune et al.,
199 1; Paksy et al., 1995 ) to be the growth of bactena and the subsequent production of
bioslime (extracellular polymers, or exopolymers) in the porous media. The majority of
bactena and bioslirne grow in the form of biofilms (Flemming, 1993), and therefore the
formation, structure and effects of biofilms in porous media will be discussed in detail.
2.4.2.2 Bacterial Attachent and Biofilm Formation
The following is a list of steps fiom bacterial transport to biofilm development
(derived from Cullirnore, 1993):
1) Bacteria are usually negatively charged and are attracted to positive surfaces.
Other causes of attachment may be of greater significance since positively charged
surfaces do not dominate the system.
2) The bactena ce11 becomes anchored to the surface by extending polymers (long
chained molecules) which make the primary attachent.
3) Cells reproduce and colonize the surface by "jumping" or "tumbling" or may
simply clump together. They spread rapidly on fnendly surfaces.
4) A biofilm is formed when the structures in which the various microorganism
species CO-exist are bound together with a water-retaining polyrneric matrix.
5 ) The biofilm extracts substrates and bioaccumulates fiom the passing water. The
bioaccumulates are not utilized, but accumulate within the biofilm. Some comrnon
accumulates are non-degradable organics and metallic ions (ie. iron, manganese,
copper, aluminum, and zinc).
6) The biofilm, initially a randomized mixture of rnicroorganisms, becomes more
organized over time. Due to competition and other factors some rnicroorganism
strains are elirninated, some become dormant, and some stratification occurs.
Besides being sirnply attracted to surfaces, it has been thought that it is
advantageous for starved bactena to colonize on surfaces and it has been found that
starved bacteria are more adhesive than well-fed bacteria and some can utilize substrates
at the surface (Characklis and Marshall, 1990).
2.4.2.3 The Biofilm
Characklis and Marshall (1990) stated that "A biofilm consists of cells
irnrnobilized at a substratum and fkequently ernbedded in an organic polymer matnx of
microbial origin ... it is a surface accumulation, which is not necessarily uniform in time or
space.. . [and]. .may be composed of a significant fraction of inorganic or abiotic substances
held together by the biotic matrùc."
Biofilm Structure
The latest studies of biofilm structure have led to the belief (Costerton et ai, 1995)
that with adhesion, structural molecules of bactena change allowing the synthesis of
exopolysaccharides (bioslime). This synthesis, combined with ceIl division, leads to the
development of microcolonies encloseci in dense slime and attached to the colonized
surface.
Studies at the Centre for Biofilm Engineering in Montana, U.S.A. @ebeer et al.
1993; Debeer et al., 1994; Stoodley et ai., 1994) found that confocal scanning laser
rnicroscopy allowed detailed examination of live biofilms under flow conditions. These
works revealed the structure of biofilms to be heterogeneous and consisting of round ce11
clusters separated by channels. It was shown that flow existed within the voids (channels)
while the interior of the ce11 clusters remained stagnant. Flow velocities within the biofilm
were found to be related to physical parameters such as the local structure and biofilm-to-
voids ratio. This fiow was found to travel around and over ceIl clusters, and even against
the direction of bulk flow. It is thought by Debeer et al. (1993) that due to convection
within the biofilm the mass transfer rate fiom the bulk liquid may be significantly higher
than previously believed.
Based on these observations Lewandowski et al. (1995) concluded that a new
conceptual mode1 of biofilm where microorganisms are organized in clusters separated by
interstitiai voids must be accepted. Liquid flow in the charnels was found to extend to
the substratum. Three microscopie methods were used by Stewart et a1 (1995) to provide
a p i a r e of the stmcture which was consistent between methods and previous studies
above, and which noted that patches of bare substratum could be seen even though the
biofilm was several hundred microns thick in places. It was found (Costerton et al, 1995)
that early clusters are cone-shaped while later clusters may develop mushroom shapes,
with water channels existing very near to the substratum. It was also noted that the
channels are wide enough to allow the passage of 0.3 pm latex beads.
The biofilm system, idealized by Charackiis and Marshall (1990), is made up of
five compartments (see Figure 2.1): the gas, the bulk liquid, the surface film, the base film,
and the substratum. The gas cornpartment, wbich may not be present in some biofilm
systems, provides for aeration and/or removal of gaseous products of the biofilm. The
bulk liquid provides substrates to the biofilm. The surface film is the transition zone
between the bulk liquid and the base film and may extend to the substratum, or may not
exist at dl . In the surface film advective transport dominates. The base film consists of a
rather structured accumulation of microbiai cells where rnolecular diffusion transport
dominates. The film portion of the biofilm, composed of the base film and surface film, is
composed of two phases including a continuous liquid phase containing some dissolved
and suspended particulates and a solid phase composed of rnicroorganisms, extracellular
rnaterial or inorganic particles. The substratum is generally an impermeable, nonporous
material, in our case the stones of the L.C.S. or g las beads.
It is important to note that while this biofilm representation recognizes the
existence of a liquid phase within the biofilm little was known of the possible significance
of this Iiquid phase until very recently. The mode1 adopted for this research, based on this
representation, assumes a continuous solid phase bacterial growth on the substratum and
diffusion to be the only manner of mass transport within the biofilrn. Since the biofilm
may grow rapidly, it may expand into interstitial spaces forming a semi-permeable bamer
to flow. The biofilrn is deemed semi-permeable because of the existence of
interconnections between the biofilms. Due to hydraulic forces penodic sloughing of
biomass occurs and this matenal may colonize surfaces downstream.
2.4.3 Biochemical Precipitation
2.4.3. l Introduction
It is accepted ( Bass, 1986; Paksy, et al., 1995) that the main cause of biological
clogging is formation of biomass, but in the long term, due to questionable permeability of
the biofilm and slime, it may be the build-up of biochemically produced precipitates that
cause zones of near impermeability. Biochemical precipitates are solid deposits formed
with the aid of the metabolic activity of bacteria. The following provides explanation of
biochemical precipitation formation in landfill systems.
2.4.3 -2 Precipitation Formation
Bass (1986) concluded that the principal biochernical precipitates are Fe(OH), and
FeS although manganese compounds may be also involved. The processes involved
depends primarily on the availability of dissolved ffee ions and on the presence of iron-
reducing bacteria. It was found that biochernical precipitate is generdly mixed with the
bioslime. Usually the precipitates formed by biological processes are different in fom and
structure than chernical precipitates, and Bass concludes that they are more effective in
clogging.
Brune et al. (1 99 1) postulate that the precipitation of incnistation material is
caused by:
(a) The formation of iron sulphide on sulphate reducing bacteria (Figure 2.2(a)). Fe3' is
reduced to ~ e " by iron reducing bactena while sulphate is reduced to sulphide by sulphate
reducing bacteria. Due to this bioreduction the region immediately around the bacteria
becomes more alkaline and leads to precipitation of sulphur as an insoluble metal sulphide
(Le. iron sulphide).
(b) The formation of carbonate on methane bacteria (Figure 2.2(b)). Calcium and other
metals are first mobilized in the waste due to the acidity of the young leachate. Methane
and sulphate reducing bacteria, consurning hydrogen ions, cause a local increase in pH and
the disturbance results in hydrogen carbonate combining with the mobilized metals to
produce precipitates of calcium carbonate and other metal carbonates on the bacteria
surfaces.
Rittmann et al. (1996), utilizing data fî-om mesocosm studies using actual landfill
leachate and drainage materials, aiso concluded that COD removal by rnicroorganisms
allowed or accelerated the precipitation of calcium carbonate (CaCO,). Because most of
the COD removed was acetic acid, a moderate strength acid is converted to CO, with
about 50% dissolving as a weaker acid. It was concluded that the resulting increase in pH
and total carbonate allows or accelerates CaCO, precipitation.
2.5 BIOFILM GROWTH MODELLING
2.5.1 Bacterial Growth Models
The mode1 developed in this thesis was based upon the assumption that a unifonn
biofilm is formed on the media within which utilization of substrate and subsequent growth
and decay occur. According to Baveye and Valocchi (1989), two other types of
mathematical models have been adopted for predicting bactenal growth and biologically
reacting solute transport in saturated porous media. One other class of models assumes
that the bacteria grow in small discrete colonies attached to the porous media and these
colonies either grow in size, or increase in number while another class of models makes no
assumption of rnicroscopic configuration and distribution in the porespaces. Growth and
decay of the biofilm are govemed by the kinetics of the substrate / biofilrn reactions,
discussed next .
2.5.2 Kinetics of Biofilm Growth
The following is based on Characklis and Marshall, 1990 (except where noted).
2.5.2.1 Introduction
Expressions of the rates of the fundamental biological processes including growth,
maintenance. decay and death are derived through the combination of rate expressions and
stoichiometnc ratios (ie. yields). Since the rates of these fundamental processes are
difficult to measure, observable rates such as specific substrate removal rate, specific
oxygen removal rate and specific biomass production rate must be used to arrive at an
expression. The form of a rate equation may be based on empincal curve fit or theoretical
analysis of a mechanistic model. Environmental factors are reflected in the rate
coefficient. The values of rate coefficients must be determined experimentally and
typically involves the study of the infiuence of concentration on the rate assuming al1 else
constant.
The most common expressions describe unstmctured kinetic models, with the
biotic component characterized by mass and/or cell numbers only, with little attention
given to state or species diversity. Balanced growth (ie. steady-state in an open system )
and a single species are often assumed.
2.5.2.2 Growth Rate Equations
The required conditions for rnicrobial growth include a viable inoculum, an energy
source, nutrients for synthesis and a suitable physicochernical environment.
Two exarnples of growth rate equations are:
Exponential Growth
The rate of microbial growth is proportional to the biomass present:
rdy = CL^
where p = specific growth rate (t-')
& = biomass concentration (M LJ )
r, = rate of bacterial growth (M L-3 t-' )
This expression is usefùl as long as environmental conditions remain constant and the
constitution of the biomass remains constant (balanced growth). The specific growth rate,
p, is afFected by many environmental variables especially the substrate, but also the
temperature, pH, ionic strength and concentration of inhibiting substances.
Saturation Rate Eauation
The most widely used expression for describing the rate of microbial growth as a
function of nutnent concentration is aaributed to Monod. The equation, describing
specific growth rate lirnited by substrate concentration, descnbes a rectangular hyperbola:
where Ks = rate (saturation) coefficient, aiso called the half velocity constant, the
substrate concentration at one-half the maximum growth rate (M L")
S = growth limiting substrate concentration (M L")
p, = the maximum specific growth rate (t")
The Monod equation (Monod, 1949) can be considered to provide a transition between
limiting cases for very low concentrations and very high concentrations:
High concentration, P = CL,, for S )) Ks
Low concentration,
At low concentrations the rate is first order with respect to substrate
concentration. The organisms still have significant reaction potential, therefore an
increase in substrate supply will cause an increase in growth rate. At high concentrations,
the rate is independent of concentration (zero order) because the cell can no longer
assimilate the substrate.
If Equation 2.2 is substituted into Equation 2.1 for the value of p, the growth rate
can be expressed as:
The quantity of new cells produced has been found to depend upon the
concentration of substrate, as implied by the yield coefficient Y, defined as the ratio of the
mass of cells formed to the mass of substrate consumed. The rate of the utilization of
substrate for ce1 growth is expressed as:
where r,u = substrate utilization rate (M L-3 V' )
Y = maximum yield coefficient (Mx M i 1 )
Often the term p, 1 Y in Equation 2.4 is replaced by the term ij which is defined as the
maximum rate of substrate utilization per unit mass of microorganisms:
If the term ij is substituted into Equation 2.4 the following expression is created:
2.5 -2.3 Maintenance Rate Equations
Microorganisms require energy to maintain existing structures and for process
such as movement, therefore maintenance rate reflects a diversion of substrate away from
synthesis or growth, therefore maintenance decreases the overall yield of the cells from
substrate. The maintenance rate tenn is part of the substrate depletion term
where = maintenance coefficient (M M" t-')
Often the overall decrease in ceU mass due to maintenance, predation (destruction
by a second population) and death is combined and called endogenous decay. The
endogenous decay term is formulated as
rd = -kdXf rZe8]
where k, = endogenous decay coefficient (t-') (Metcalf and Eddy, 199 1).
2.6 IDEALIZATION OF BIOFILM STRUCTURE BY PROPOSED MODEL
2.6.1 Uniformity of Film Thickness
It was decided that the model's objective of predicting the degree of clogging of
the porous media would be best met by prediction of the porosity of the media. In order
to allow geometric calculation of porosity, the media is represented using simple,
repeating arrangements of spheres. The representation is discussed in Chapter 3. To
incorporate the change in porosity due to the accumulation of biomass, the biomass is
assumed to grow on the spheres at a uniform thickness throughout the subregion analysed.
The assumption of uniform (and tiius continuous) biofilm growth has been
assumed by several authors including Taylor et al. (1990) and Taylor and JaEe (1990a,b,c)
and Cunningham et al. (1991). Other authors, such as Vandevivere and Bayveye
(1992a,b), McBean et al. (1993). Stewart et al. (1995) and Murga et al. (1995) assume the
growth is discontinuous and non-uniform. By analysis of the works of several of these
authors, Rittmann (1 993) concluded that the biofilms were continuous for highly loaded
surfaces, while discontinuous under situations of low loading. Since usuaiiy much of the
porous media of the collection system is subjected to a high substrate loading, it is
expected that the spheres would be covered in a continuous film. For the purposes of this
thesis, the thickness of the film is taken to be relatively uniform to examine how well
processes can be modelled based on this assumption.
2.6.2 Film Layering
The mode1 must simulate the accumulation of both biomass as biofilm and
biochemically driven precipitate. The proposed mode1 assumes that two distinct films are
formed. One film, called the active biofilm, is a uniform layer of active biomass in which
substrate travels by difision, substrate is utilized, and biomass growth and loss occurs. A
second film, called the inactive film is a layer consisting of non-degradable inert biomass
and precipitate (calcium carbonate). It is assumed that the inactive film adheres to the
porous media surface, and the active film adheres to the inactive film. It is assumed that
each film has a distinct thickness and spatially unifom density. The inactive film,
consisting predominately of calcium carbonate, will have a considerably greater density
than the active layer.
Several researchen (tawrence et al., 199 1 ; Masuda et al., 199 1; Zhang and
Bishop, 1994; Ohashi and Harada, 1994 ) have found that the density of biofilm changes
with depth. Zhang and Bishop (1994) found that the density of bottom layers were 5 to
10 times greater than those in the top layers, and the ratio of living cells to total biomass
decreased significantly from the top layers to the bottom layers. These findings
substantiate the proposed model's separation of the two layers which places more dense
matter in a layer below the active layer. The distribution of living cells also provide some
substantiation for the incorporation of the inert biomass into the lower, inactive film layer.
2.6.3 Film Permeability
Several authors ( Taylor and J&e, 1990qb,c ; Taylor et al., 1990 ; Cunningham et
al., 199 1; Vandevivere and Baveye, 199îa,b ; Baveye et ai., 1992 ; McBean et al., 1993 ;
Wanner et al., 1995) have stressed the critical role that bacteria, in the form of biofilms,
play in reducing the permeability of porous media. Disagreements exist between authors
regarding the manner in which permeability is affected by biofilm growth.
Taylor et al. (1990) and Cunningham et al. (199 1) mode1 permeability reduction by
the decrease in pore diameter caused by biofilm accumulation (neglecting slime
production) on the pore walls. Overall biofilm thickness, therefore, is the critical factor.
Vandevivere and Baveye (1992a,b) and McBean et al. (1993) concluded that biofilm with
slime (exopolymers) accumulating at pore restrictions were the critical factor. Still others
believe permeability reduction is a function of the fiction factor attributed to the
accumulation of biofilm and slime at all points within the pore (Rittmann, 1993).
The proposed model, in an effort to allow flexibility with regards to this issue,
would allow the active biofilm layer to be either permeable or impermeable, that is, the
thickness of the active layer could be chosen to not be used in caiculating changes in
porosity. The inactive film is assumed to be impermeable to flow, and therefore the film
thickness of this layer is always included in the calculation of porosity.
2.7 MODELLING BIOLOGICAL CLOGGING IN THE L.C.S.
This chapter has demonstrated though published case histories (Bass, 1986; Brune
et al, 199 1; McBean et al, 1993; Rowe et al, 1995a,b ) that biological clogging in the
leachate collection systems of landfills occurs, that the dominant process is suspected to be
the growth of anaerobic bacteria, and that column tests have been utilized to study the
phenornenon involved. Under favourable conditions, rapid growth of bacteria can occur.
The favourable conditions for microbial growth in the L.C.S. are established by the
percolation of leachate, containing the products of the decomposition of the waste above.
As indicated by McBean et al. (1 993). the L X . S. environment provides: microorganisms,
organics for consumption, nutrients, and pK temperature and moisture levels conducive
to growth. In a column test fed synthetic leachate the conditions for bacterial growth may
be met or optimized by controlling leachate composition and other system factors.
The majority of bacteria and bioslime grow in the form of biofilms (Flemming,
1993) which fix themselves to the granular media of the L.C.S. and may form a semi-
permeable barrier to leachate flow. Attributed to the growth of bacteria is the build-up of
biochernical precipitates, solid deposits fonned with the aid of the metabolic activity of
bacteria. The growth of such deposits has been discussed by Brune et al. (1991) and
Rittmann et al. (1 996). An idealized biofiim structure was developed for the proposed
modei which incorporates an inactive layer consisting of non-degradable inert biomass and
the biochemical precipitate attached to the porous media, and an active layer of biomass
adhering to the surface of the inactive film. The formulas for calculating the porosity and
specific surface area of the porous media foiiowing film build-up are discussed in Chapter
3.
Figure 2.1 : Cornmon idealkation of biofilm structure (Characklis and Marshall, 1 990).
Acidic Environment - - caf Acidic Environment
Figure 2.2(a,b): (a) Formation of iron sulphide on sulphate reducing bacteria, (b) Formation of carbonate on methane bacteria (Brune et al.. 1991).
CHAPTER 3
POROSITY AND SPECIFIC SURFACE MODELS
A mode1 was required that could calculate the porosity and specific surface of a
volume of attachent media with various biofilm thicknesses. The model rnust ailow
varied initial porosities or packing of the clean attachent media. The model should have
the best possible accuracy within the eniire range of film thicknesses, including the film
thicknesses as the porosity and specific surface near zero.
3.2 ASSUMPTIONS
It was assumed that the attachment media could be idealized as perfect spheres of
equal diarneter with a repeating, consistent packing arrangement. The film is assumed to
be a single, impermeable layer of unifonn thickness in the derivation of the models.
3.3.1 Packing Arrangements
The mode1 assumes the structure of the porous media can be simulated by spheres
of equal diameter. Any manner of arranging solid spheres in which each sphere is held in
place by gravity and supported by tangent contact with neighbouring spheres is considered
itspacking. II will be assumed that the porous media can be represented by a regrrlm or
geometrically sysiematic packing arrangement, that is, the Iayers, or sets of rows of
spheres repeats its arrangement in both directions within that plane.
Assurning that the structure of the porous media is represented by stable, regular
packing arrangements, it has been found that only four such arrangements exist for spheres
of equal diameter ( Graton and Fraser, 193 5 ). The four arrangements are cubic,
orthorhombic, tetragonal-sphenoidal and rhombohedral. These arrangements are named
&er the shape of the 3-dimensionai, 6 sided box formed by joining the centres of eight
spheres. This box is called the unit cell and is the smallest portion of a stack of layers of
spheres which completely represents the packing and distribution of voids throughout the
entire arrangement. The unit ce11 for each arrangement may be descnbed completely by
illustrating the three sides or faces of the ce11 as shown in Figure 3.1. The shapes of the
faces are called square, with a facial angle of 90°, rhombic, with a facial angle of 60" and
the third face shape is neither square nor rhombic and will be called a specid rhombic
face, with a facial angle of 75 O 3 1 '.
3.3.2. Packing Arrangement Characteristics
For each packing arrangement the spheres have a unique number of contact points
with neighbouring spheres, m, and the unit ce11 has a unique volume, see Table 3.1. By
summation of the fractions of the spheres in the corners of the unit ce11 it cm be proven
geometncally that each unit ce11 contains exactly the volume of one solid sphere (0.52 d:
where $ is the diameter of the sphere) with the remainder of the volume of the unit ce11
being pore space. The volume of the unit void space can thus be found by subtracting the
volume of one sphere fiom the volume of the unit cell. The porosity of each packing
arrangement can be determined by dividing the volume of the unit void space by the
volume of the unit cell. The porosities are listing in Table 3.1. The packing factor, am , a
dimensioniess value characterizhg the packing arrangement and a required parameter for
further calculations of porosity and specific sudace, is calculated by dividing the volume of
the unit ce11 by d,) (Taylor, IWO).
Table 3.1 : Characteristics of Packing of Uniform Spheres
Packing
Cubic
Orthorhombic
Tetragonal- Sp henoidal
d, = diarneter of a sphere Graton and Fraser (1935), Cadle (1965), Taylor et al. (1 990).
Rhombo hedral
The mode1 assumes that the initial, clean media porosity for the simulation can be
measured or estimated. The initiai specific surface, k0, the surface area provided by the
porous media per unit volume (L" ) when clean, may be calculated from the initial
porosity and particle diarneter, assuming the porous media is represented by ideal spheres.
The specific surface is critical as a measurement of the surface area available for
attachment by biofilm. The initial specific surface for al1 elements of the flow field is
caiculated using (Taylor et al., 1990):
Nwnber of Contact points, m
6
8
10
12
Volume of Unit Ce11
4' n/ 2 43 0.75 d i
1 / J Z ~ '
Porosity, n (%)
- - - --
47.64
39.54
30.19
Packing Factor. am
1
d 7 / 2
O. 75
25.95 1 / a
where d, is the particle diameter.
The fùnctions derived in this chapter calculate the porosity and specific surface of
the film-coated media for only the four possible regular packing arrangements. Each of
these two tùnctions requires the number of contact points, m. and packing factor, am, to
describe the packing arrangement. Lf the initial estimated porosity does not correspond to
one of the four regular packing arrangements, the porosity and specific surface may be
estimated by interpolating between values obtained for the packing arrangements that
bracket the estimated initial porosity.
3.4 DERIVATION OF POROSITY AND SPECIFIC SURFACE MODELS
3.4.1 Introduction
Of particular interest in this thesis, is the change in porosity and specific surface
assuming that a film of thickness L, grows on the surface of a sphere of diameter dp.
These properties of the media will be calculated geometrically based on an arbitrary film
thickness, that is, with no consideration of the growth requirements of the bacteria. Pore
discontinuity causing the isolation of pore space, for example, will have no effect on film
thickness in the isolated pore. It is also important to recognize that as the film grows on
each sphere, allowance must be made for the fact that, near the contact points there will be
'overlapping' of the film on adjacent spheres and one must avoid double counting in
assessing the actual porosity and surface area. The following sections will discuss the
existing models for porosity and specific surface (Taylor et al. 1990) and the authors
modifications to these models.
3.4.2 Taylor et aL(1990) Mode1
Taylor's equations for caiculating porosity and specific surface are derived by
calculating the volume of the unit cell, the volume of solids in the unit cell, and the surface
area of the solids. The volume of the unit ce11 is a function of the packing factor and
sphere diameter,
as listed in Table 3.1. With the volume of the unit ce11 known, the calculation of porosity
requires calculation of the total volume of solids (sphere + film) in the unit cell, while the
calculation of the specific surface requires calculation of the surface area of the sphere and
film in the unit cell. Since the packing arrangements are geometrically systematic, the
volume and surface area of the 8 quarter hernispheres of spheres making up the unit ce11
are equal to an entire single sphere of equal size.
The total volume of solids in the unit ce11 is based on a calculation by Deb (1969)
which calculates the volume of a solid sphere and its film coating taking contact points
between spheres into account :
The first term of this equation is the total volume of a solid sphere and film coating
assurning uniform coverage and the second term is the sum of the volumes of each
Merical cap of film where accumulation cannot occur due to contact with neighbouring
spheres and which is therefore subtracted fiom the total. A sirnilar equation is used to
calculate the sufiace area in each unit cell.
The following equations of Taylor et al. (1990) were used to calculate the variance
of porosity, n, and specific surface, A, , with film thickness:
The equations calculate porosity and specific surface as a function of the film thickness in
terms of non-dimensional 2 L j 4 , and account for the packing arrangement using the
packing factor q , and number of contact points, m. The equation results are plotted in
Figures 3.2 and 3.3 . In subsequent discussions the thickness of the film will be expressed
in normalized terms of 2Lt /d, for mathematical simplification.
3.4.3 Error in the Taylor et aL(1990) Model
Figure 3 -2 shows a plot of the effect of the film thickness on porosity for various
packing arrangements based on Equation 3.4. It can be seen that porosity decreases with
increasing film thickness to the equation's local minima, and after this thickness is reached
the equations calculate porosities to be increasing with increasing film thickness. This
does not make physical sense. From physical reasoning, it follows that there must be
some film thickness at which the pore space is completely filled, and the porosity is equal
to zero. Since the porosities at each of the minima are greater than zero it may be
concluded that some error is inherent in the assumptions or denvation of the equations by
Taylor et al. (1990).
Figure 3.3 shows a plot of the eEect of film thickness on specific surface for
various packing arrangements. The specific surface decreases with increasing film
thickness until the specific surfaces become zero. Although the specific surface plots do
not indicate obvious error, some error is suspected due to the sirnilarities in the derivation
of the equations, as discussed Iater. Figure 3.3 illustrates that the specific surface becomes
zero, indicative of completely filled pores, while the porosity expression discounts
completely filled pores.
3.4.4 Source of Error in Taylor's Equations
A spherical cap is illustrated as the hatched area in Figure 3.4 for any contact
point. The sphencal cap has a circular base centred at and tangent to the contact point,
and a height equal to the film thickness, Lt. The line DB is the diameter of the circular
base of the cap. A spherical cap is a three dimensional volume of film which is subtracted
from the total volume of sphere plus film and the multiplier m represents the number of
contact points for a specific packing factor. The specific surface expression utilizes a
sirnilar equation in which the total surface area of a sphere covered by film is calculated
and the surface area of the sphencal cap (excluding the base) at each contact point is
subtracted.
In the Plan view in Figure 3.5 the hatched area indicates two of the six sphencal
caps subtracted from each sphere's total volume for cubic packing. Figure 3.6 shows the
same spheres after further accumuiation. The double hatched area in this figure reveals an
overlapping of spherical caps. If Equation 3.5 is used when spherical caps have
overlapped the overlapped volume is subtracted from the total solid volume for each cap,
therefore subtracting the same space twice when it should be subtracted oniy once. The
simple calculation of Deb (1969), Equation 3 -3, therefore becomes inaccurate when the
film thickness increases to such an extent that the spherical caps overlap.
Figure 3.7 depicts the film thickness at the onset of overlap. On inspection it may
be mistakenly assumed that since this overlap coincides with pore discontinuity of the face
it is insignifiant, but, as section SS ' shows, the complete filling of ail of the void space
occurs only after greater accumulation has occurred. The inaccuracy of Equation 3.5 for
large thicknesses causes an underestimation of the total volume of the film covered sphere
in aii subsequent calculations. Similarly, an underestimation of surface area occurs. In
order to correct for the overlapping of volumes, the volume and surface area of the
overlapped space is to be calculated geometrically. The limitations of the equations by
Deb (1969) and Taylor et al. (1990) have not previously been noted in the literature other
than the erroneous argument by Taylor et al. (1990) that the solution breaks down due to
the isolation of the remaining pore space. Since the mode1 is geometnc (and does not
consider hydraulic conductivity) the question of whether pores are isolated (while an
important practical issue ) has no bearing on the breakdown of the solution.
3.4.5 Determination of Critical Film Thickriesses
3.4.5.1 Introduction
Critical film thicknesses are thicknesses at which the geometrical basis of the
porosity or specific surface mode1 rnust change or complete filling of the pore space,
cdled pore occlusion, has occurred. The critical film thicknesses occur when (a) spherical
caps first overlap, (b) spherical caps overlap at a different region at a greater thickness, (c)
overlaps overlap, and (d) pore space is completely füled. As film thickness increases on an
initidy clean sphere, overlap will occur earliest in the face with the most contact points,
since as the number of contact points in the face increases, the closer the spherical caps are
to each other and the thinner the film thickness required for volumes to overlap. The face
with the most contact points is the cntical face. The critical face for the cubic packing
arrangement is the square face while the criticai face for the other three packing
arrangements is the rhombic face (see Figure 3.1 and imagine the packing arrangement of
each face repeating around a single sphere to sum the contact points for each face).
3.4.5.2 Critical Film Thickness Summary and Interpretation
As already illustrated for cubic packing in Figures 3.4 - 3.7, the critical film
thicknesses can be geometrically derived for al1 four packing arrangements (see Appendix
Al for the detailed geometric proof for the four arrangements ). A summary of these
cntical film thicknesses is given in Table 3.2:
Table 3.2 - Summary of Critical Film Thicknesses in Terms of ZL, / d,
Packing Arrangement
Name
Cubic
Orthorhombic
At thicknesses greater than 2L& = 0.4 142 for cubic and ZLjd,, = 0.1547 for the other
packing arrangements the fist overlap occurs. For larger values of 2L&, the calculation
of the volume and surface area of a sphere covered with film is underestimated by
Equation 3.3 and the similar equation for surface area. Using the methods of Taylor et al.
(1 990) the porosity is overestimated and the specific surface underestimated. This point
of theoretical breakdown and thus beginning of inaccurate results is shown as the "first
occurrence of overlap" in Figures 3.1 and 3.2 for each packing arrangement. It may be
expected that by correcting for the overestimated porosity it will be possible to bring the
curves to zero porosity.
Number of Contact Points
(1)
6
Tetragonal- Sphenoidal
Rhombo hedral
The geometnc identification of the film thickness required to reach complete pore
occlusion can now be used to ver@ that there is also an error in the specific surface
equation by Taylor et al. (1 990) since the thicknesses at which the equation calculates a
specific surface of zero does not correspond to the geometricaily derived film thickness at
pore occlusion.
8
First Cap Overlaps
(2)
0.4132
10
12
O. 1547
Second Cap Overlaps
(3 1
-
0.1547
0.1547
0.4 142
Complex Cherlaps (4
0.732
0.5275 0.5275
0.2247
0.2247
CompIete Pore Occlusion
(5 1
0.732
0.2639
0.2247
0.3228
0.4 142
3.4.6 Corrective Models Required
3.4.6.1 Introduction
In order to obtain the porosity and specific surface for a given film thickness it is
necessary to provide a correction to the solution of Taylor et al. (1990). The corrective
models and the corresponding film thicknesses at which they are to be used are
summarized in Table 3.3, and discussed in the following subsections.
3 A6.2 Square Model
The first cap overlap of the cubic arrangement and the second cap overlap of the
orthorhombic arrangement result in overlapped space with sirnilar shapes. The cross-
section of the centre of this shape is depicted in Figure 3.6 (Plan). The model used to
calculate the volume and specific surface of this shape will be called the Square model
since it occurs in the square face of the packing arrangement. Focussing only on the shape
of one overlap on one film coated sphere, it cm be seen in Figure 3.6 that the shape is a
section of a sphere created by perpendicular cuts tangent to the sphere. A three
dimensional depiction of this shape is given in Figure 3.8(a).
3 -4.6.3 Rhombic Model
The first cap overlap of the orthorhombic, tetragonal-sphenoidal, and
rhombohedral packing arrangements result in overlapped space which is similar in shape.
The cross-section of the centre of the shape is shown in Figure A I S (Plan). The
corrective mode1 for calculating the volume and specific surface area of this shape will be
called the Rhombic model because it occurs in the rhombic face of the sphere
arrangement. It c m be seen in Figure A1 -5 and in the three dimensional drawing in
Figure 3.8(b) that this shape is sirnilar to the Square model shape, except that the cutting
planes intersect with an inside angle of 120°, but still lie tangent to the contact points.
3.4.6.4 Special Cap Model
The tetragonal-sphenoidd packing arrangement has, in addition to the rhombic
overlap corresponding to its rhornbic face, a special cap overlap corresponding to its
special rhombic face. The cross-section of the shape of this overlapped volume is shown
in Figure AL8 (Section) and is sïmilar to the regular spherical caps which occur at each
contact point. The simple model required to account for the volume and surface area of
this overlapped space is called the Special Cap model. A three-dimensional depiction of
this shape is given in Figure 3.8(c). The use of the Special Cap model differs frorn the
Square and Rhombic models because, in contrast to these models, the volume and surface
area of the shape is to be subtracted fiom the total, not added. This is because the overlap
creating a special cap is an overlap of actual film while the other overlap models are for
the overlap of contact point volumes which is already accounted for by the Deb (1 969)
equation. A special cap is essentially a contact point overlap like those which occur
between al1 spheres, but which occurs at a specific film thickness in the special rhombic
face of the tetragonal-sphenoidal racking arrangement.
3.4.6.5 Geometric Complexity
Geometric complexity of the cubic and orthorhombic packing arrangements either
did not occur or coincided with pore occlusion, thus there was no need for a corrective
mode1 beyond that given in colurnn (4) of Table 3 -3. However, for the tetragonal-
sphenoidal and rhombohedral packing arrangements, for film thicknesses between those
given in colurnn (5) and (7) of Table 3.3 it was not possible to get an exact geometric
solution due to the geometric complexity of the overlaps for this range of film thickness.
Thus for the tetragonal-sphenoidai and rhombohedral packing arrangements it is assurned
that upon correction of the porosity and specific surface values using the above three
models, a iinear relationship may be employed for calculation within the region of
complexity to the known point of pore occlusion.
Table 3.3 - Corrective Models and Corresponding Film Thickness ( 2L, 1 d,)
Packing Arrangement
Name
Cubic
Orthorhombic
Tetragonal- Sphenoidal
3.4.7 Derivation of the Square and Rhombic Model Solutions
3.4.7.1 Introduction
In order to correct the porosity and specific surface equations, the volume and
surface area of the overlapped shape as a function of film thickness, sphere size, and
packing arrangement must be derived. The overlap shape is depicted at the top of Figure
3.9. Note that in this figure the sphere of radius R is the shape of the surface of the film
First Cap Overlaps
(1)
0.4142
O. 1547
0.1547
0.1547
Second cap
Overlaps
(3
- 0.4 152
0.2247
0.2247
Corrective Mode1
(2)
Square
Rhombic
Rhombic
Rhornbic
Corrective Mode1
(4
NA
Square
Special Cap
NA
Corrective Mode1
(6)
N A
NA
L inear
L inear
Geometric- all~
CompIex Overlaps
(5 )
0.732
0.5275
0.2649
0.2247
Complete Pore
Occlusion
(7)
0.732
0.5275
0.3228
0.4 142
coating an ideal sphere (not shown) centred at the origin. Using symmetry about the y = O
plane, half of the shape is defined as the portion of the sphere of radius R formed by the
intersection of the y = O plane with a plane intersecting at y = 0, z = a, and interior angle
8. Due to symmetry about the y=O plane the interior angle 0 is half of the actual angle,
and the volume calculated is half the actual volume. Control of the intenor angle is
required because for the square model the angle is 90" while for the rhombic model the
angle is 120". This also would allow for calculation of other shapes if required.
3.4.7.2 Limits of Integration
The dotted lines in Figure 3.9 (Top) show the projection of the y > O side of the
shape on the z = O plane. This projection is an ellipse on the plane, and thus, in order to
calculate the limits of integration, the equation of the ellipse must be derived. The
diagram in the middle of Figure 3 -9 illustrates the cross-section at the x = O plane, the
bounds for z and the distance to the centre of the projected ellipse from the ongin. By
equating the equations of z, and knowing the location of the centre of the ellipse, the
equation of the ellipse on the z = O plane was solved. This allowed the calculation of the
length of the major and rninor axes of the ellipse, shown at the bottom of Figure 3.9, and
required for the limits of integration.
The integral of the volume was found to be
where the lirnits are
b = <jsin29 (R' - a ' s in28) - a s i n û c o s û i'
and the integral of the surface area is
where lirnits b, and b, are the sarne as for the volume, a is the distance from the centre of
the sphere to the intersection of the cut with the y = O plane and R is the radius of the
sphere. The integrals could not be solved analytically and were evaluated numerically as
described below.
3.4.7.3 Numerical Analysis of Integrals
The integrals of Equations 3.6 and 3.8 were numerically evaluated using the
program MathCad. For a particular solution model, since 8 is constant, the volume or
surface area of the overlap shape depends only on radiai lengths a and R. It was
hypothesized (and subsequently verified) that if the volume is normaiized with respect to
R~, one function would describe the normalized volume as it varies with a/R for al1
applicable values of a and R. A sirnilar finding was applied to the surface area normalized
with respect to R ~ .
In order to calculate the applicable range of a/R , it must be converted to terms of
2L,/dp. The applicable range of 2Ljdp for the square mode1 in exact terms is:
while for the rhombic model the range is
Geometrically it is known that R and a for the integration in terms of tL/d, are
and
a = - d p if square model, J2
Therefore the ratio d R in terms of 2L/dp is
and the applicable range of the square model is
while the applicable range of the rhombic model is
3.4.7.4 Numerical Solution of Square Mode1 Shape for Volume and Surface Area
To solve the square model shape the angle 0 was set to d 4 (45"). To check for
the independence of the normalized volume or surface area of the length R, calculations
were performed for R values of 0.00 1, 0.01, 0.1, 1, 2, 5, and 10. A set of values of the
ratio a/R were selected from the applicable range. The length a was then calculated for
each value of R. The results of the numerical analysis evaluating the normalized volume of
the overlap shape used to correct the square model are s h o w in Figure 3.10. tt was
found that the normalized volume was independent of the value of R. To obtain a ciosed
fom expression, a fourth order polynomial was fit to the results, and the coefficients are
given in Figure 3.10. The curve fit was impiemented using the cornputer program
SigrnaPlot which utilizes the Marquardt - Levenberg algorithm. The results of the
numerical analysis for the surface area of the shape are shown in Figure 3.1 1. Sirnilar to
the volume andysis, it was found that the normalized area was independent of the value of
R. A fourth order polynomial was fit to the results, and the coefficients are given in
Figure 3.1 1.
3 -4.7.5 Numerical Solution of Rhombic Model Shape for Volume and Surface Area
For the rhombic model shape, the angle was set to d 3 (60"). The values of R
were the same as those used in the square model shape analysis. A set of a/R values
within the applicable range for the rhombic model shape were selected. The normalized
volume results of the numerical integration are shown in Figure 3.12. The coefficients of
the fourth order polynomiaf fit to the data is given in Figure 3.12. The surface area
results are plotted the coefficients of the fourth order polynomial that provide a fit to the
results are given in Figure 3.13. Both the normalized volume and area were found to be
independent of the choice of R.
3 A7.6 Corrective Model Solutions for Porosity and Specific Surface
The volume and surface area of the overlap shape are descnbed by the following
equations:
where Vol is volume, S is surface area, and b[#] represents the calculated coefficients.
The equations for correcting the volume and specific surface equations for the two models
were denved by substituting Equation 3.13 for a/R and the coefficients for each model
shape and by division by the ce11 volume, a&,'. The square mode1 correction equation for
each overlap for porosity is
and the square model correction equation for each overlap for specific surface is
The rhombic mode1 correction equation for each overlap for porosity is
and the rhombic model correction equation for specific surface is
It should be noted that V, and VR are not volumes, but dirnensionless terms (one overlap
volume correction divided by unit ce11 volume). S, and SR are corrections to the specific
surface with dimension CL-'].
3.4.8 Derivation of the Special Cap Model Solution
3.4.8.1 Porosity
The Special Cap Modei required a calculation of the porosity contribution
(correction value) for the special cap. The volume, V, of a spherical cap is calculated
using
where R is the radius of the sphere plus film thickness, and h is the height of the cap. For
this analysis,
and
where on the right side of Equation 3.24, the term in the brackets is the radius of the
sphere plus film, and the subtracted tenn is the length to the midpoint between the two
spheres in this direction, see Figure A1.8 (Section). Substitution of Equations 3 -23 and
3.24 into Equation 3.22, and division by the volume of the unit cell, a,%', results in the
correction factor to be applied to the porosity equations:
where V,, is not a volume, but a dimensionless term of solid (ie sphere and film) volume
divided by total unit ce11 volume.
3 A.8.2 Specific Surface
The Special Cap Mode1 required a simple calculation of the specific surface
contribution of the special cap. The basis is the surface area, S, of a spherical cap,
where R and h are previously defined in the porosity derivation, and where the surface
area excludes the surface area of the circular base of the cap. Substitution of Equations
3.23 and 3.24 and division of the surface area by the unit volume, a,,,d,', results in the
specific surface contribution of this shape given by this equation:
3.4.9 Derivation of Final Corrected Equations
3.4.9.1 Base Equations
For al1 four packing arrangements the underlying porosity equation (Taylor, 1990)
i s
and underlying specific surface equation is
where the calculated maximum film thickness under which the equations may be applied, is
and where n,, is the porosity and 4,, the specific surface for this range of film
thicknesses using Taylor's equations, a, is the packing factor, m is the number of contact
points, L, is the total thickness of film presumed to be impermeable and d, is the average
Stone diameter. The packing arrangement corresponding to each m value is listed in
column 1 of Table 3.
3.4.9.2 Corrected Equations for Cubic Packing
For the cubic packing arrangement (m = 6) with film thicknesses greater than the
specified lirnit stated in Equation 3 -30 the porosity and specific surface equations and
applicable range are
where V, and S, are calculated using Equations 3.18 and 3.1 9.
The corrected variation of porosity with film thickness using Equation 3.28 and
3.3 1 is illustrated in Figure 3.14. The corrected variation of specific surface with film
thickness using equations 3.29 and 3.3 1 is illustrated in Figure 3.1 5. A symbol is used to
indicate the point at which a critical thickness is reached, and the equation is changed. It
can be observed that the equations converge to porosities and specific surfaces of zero at
the same film t hickness t hat was geometrically calculated.
3 -4.9.3 Corrected Equations for Orthorhombic Packing
The porosity and specific surface equations and applicable film thickness range for
the orthorhombic packing arrangement (m = 8) for film thicknesses greater than the
specified lirnit stated in Equation 3.30 are
- As - As, Tay + 6 % J
-
and
where n is the corrected porosity and A, is the corrected specific surface area for the film
thickness range given, and V, ,V, , SR and S, are calculated using Equations 3.20, 3.18,
3 -2 1 and 3.19 respectively.
A plot of the corrected porosity results using Equations 3.28, 3 .32 and 3.33 are
shown in Figure 3.14. The corrected specific surface results are shown in Figure 3.15.
Two symbols dong the curve indicate the critical points at which the equations employed
were changed from Equation 3.28 or 3.29 to 3.32 and fiom Equation 3.32 to 3.33.
Similar to the Cubic curves, the corrected equations reach porosities and specific surfaces
of zero at the same film thickness as was calculated geometncally.
3 -4.9.4 Corrected Equations for Tetragonai-Sphenoidal Packing
For Tetragonai-Sphenoidai packing the equations describing porosity and specific
surface for film thicknesses greater than the limits given for m=10 in Equation 3.30 are
and
where n is the corrected porosity and A, is the corrected specific surface area for the given
film thickness range, and V, ,SR , V,, , and S,, are caiculated using Equations 3.20, 3.21.
3.25 and 3.27.
A plot of the corrected porosity results using Equations 3.28, 3 -34 and 3.35 are
show in Figure 3.14. The corrected specific surface results, using equations 3.29, 3.34
and 3 -3 5 are shown in Figure 3.15. The first two symbois along the curve indicate the
criticai points at which the equations employed were changed from Equation 3.28 or 3.29
to 3.34 and fiom Equation 3.34 to 3.3 5 . On the curves for both porosity and specific
surface an open triangle filled with a dot indicates the limit of special cap corrective
model. It can be seen that this limit occurs at low values of porosity and specific surface,
and are close to, but not quite at, the film thickness at which pore occlusion occurs.
A linear relationship estimating the remaining values of porosity and specific
surface to pore occlusion may be applied since this portion of the curve is not very
significant. The majority of any clogging calculation will be for lesser film thicknesses.
Using the calculated porosity and specific surface at the limit of Equation 3.3 5, and the
film thickness at which pore occlusion occurs, the equations estimating the remaining
portion of the curves are:
The linear completion of the curve is show in Figures 3.14 and 3.15.
3.4.9.5 Corrected Equations for Rhombohedral Packing
For rhornbohedral packing the equations calculating porosity and specific surface
with film thicknesses greater then the limits stated in Equation 3.30 are
where n is the corrected porosity, and A, is the corrected specific surfaces for the given
film thickness range, and V, , and SR are calculated using Equations 3.20 and 3.2 1 .
The corrected porosity results of Equations 3.28 and 3.37 are plotted in Figure
3.14, and the corrected specific surface values of Equations 3 -29 and 3.37 are plotted in
Figure 3.15. An open triangle indicates the beginning of the corrected curve and a dotted
open triangle indicates its limit. The limit occurs at Io w values of porosity and specific
surface, and close to, but not at, the point of pore occlusion. Using the same reasoning as
with the tetragonal-sphenoidd curves, a linear estimation was applied. Using the
calculated porosity and specific surface at the limit of Equation 3.37, and the film
thickness at which pore occlusion occurs, the equations estimating the remaining portion
of the curves are given by:
The linear completion of this curve is s h o w in Figures 3.14 and 3.15.
3.4.10 Note Regarding Taylor et al. (1990)
In the work by Taylor et al. (1990) fiom which the porosity and specific surface
expressions are taken the break down of the theory is amibuted to the filling of the narrow
passageways between the spheres and the isolation of enclosed pore space, and not to the
geometric complexity caused by the overlapping of film volumes at the contact points. In
the case of the cubic and rhombohedral packing arrangements these film thicknesses
coincide, that is, the film thickness at which pores become isolated equals the film
thickness at which film becomes overlapped. With the other two packing arrangements,
pore discontinuity occurs after the first occurrence of film overlap at the contact points. It
is true, therefore, that for two of the four packing arrangements the mode1 by Taylor et al.
breaks down at the film thickness that the pore spaces become isolated, but it does not
break down because of this action, and in the other packing arrangements it breaks down
at smaller thicknesses than pore discontinuity. Taylor et al. determined the film
thicknesses of pore discontinuity to be the sarne as the averages listed in Table 3.2 except
the orthorhombic arrangement was given a value to 0.24. It is suggested that the authors
erroneously averaged one square layer and two rhombic layers instead of two square
layers and one rhombic layer in calculating this Nm thickness.
3.5 FILM THICKNESS AT CLOGGING
3.5.1 Pore Discontinuity
Pore discontinuity is the obstruction to flow within the openings between the
granular media. The first occurrence of pore discontinuity will occur where the pore
space is at its minimum, which occurs in the faces of each amangement. The film
thickness at which the void space in the plane of each face becomes filled is equal to the
critical film thickness at which the overlap of film occurs in the face as calculated
geometricaily in Appendix Al . The film thicknesses are summarized in Table 3.4, column
1, for each face of each packing arrangement.
Table 3.4 - Summary of Film Thicknesses at Pore Discontinuity in Terms of ZL, I d,
Name 1 Pore Discontinuity 1
Cubic
Orthorhombic
Since flow in the model is in only one direction in each element, and since the faces
0.5 142
I
in each packing arrangement have no directional representation in the model, it can not be
0.4 142
Rhombo hedral 1 O. 1547
determined whether having one or even two faces filled obstructs flow. Since each face
0.4 142
0.1547 1 0.1547 1 O. 1547
0.4 142
Note : the values 0.4 14 and 0.155 are from fi - 1 and 2 lJf respectively.
0.4 142 O. 4 142
O. 1547 0.3277
separately may not be used to determine total pore discontinuity, it may be assumed that
the film thickness at which the average void space of the faces is filled rnay represent the
occurrence of pore discontinuity for a particular packing arrangement. The average film
thicknesses are listed in Table 3.4, column 2.
Mathernatically, using the average film thickness to represent pore discontinuity
may result in two faces of the unit ce11 being open, as in the case of the orthorhombic
packing arrangement. It must be emphasized, therefore, that the different packing
arrangements are used simply to represent the porosity and specific surface of the granular
media. For al1 packing arrangements the average film thickness at which pore
discontinuity occurs will be the limiting, or highest film thickness reached before the layer
is deemed a clogged or discontinuous layer and normal flow through is discontinued. The
point of pore discontinuity is illustrated in Figures 3.14 and 3.15 using small filled circles.
The cntical film thickness used in the model is found by interpolating between the cntical
film thicknesses of the two packing arrangements bounding the estimated initiai porosity.
The porosity and specific surface area of a porous media afler the accumulation of
a known thickness of film may be calculated using equations based on a model in which
the media is represented by regularly packed spheres of equal diameter. The porosity and
specific surface model is based on the work of Taylor et aL(1990). Film thicknesses were
geometrically detemiined above which Taylor's equations calculated erroneous volumes
and surface areas, and thus erroneous porosity and specific surface values. For sorne
packing arrangements a significant region of error could be found between the maximum
applicable film thickness and the geornetrically calculated thickness at which the porosity
and specific surface becomes zero. The errors were found to be caused by the double
counting of overlapping volumes and surface areas. Equations were derived to be used
within the region oferror for each of the four packing arrangements by deriving equations
calculating the erroneous volumes and sunace areas as a function of the packing
arrangement, sphere size and film thickness.
Cubic I Orthorhombic Three Square Faces Two Square Faces One Rhomblc Face
Tetragonal-Sphenoidal One Speclal Rhombic Face Two Rhombic Faces
Rhombohedral Three Rhombic Faces
1
:igure 3.1 : The four stable, regular pocking arrangements, illustateci as foces of the unit cell.
Packing First Volume Overlap Arrangement Occurence - Cubic O Square - - Orthorhombic Cl Rhombic -- Tetragonal-Sphenoidal A Rhombic
1 Rhombohedral v Rhombic 1
Film Thickness, 2LJdd,
Figure 3.2 : Variation of porosity with film thickness using equations by Taylor et al. (1990). Symbols indicate modal breakdown due to the occurrence of volume overlaps.
Packing First Volume Overlap Arrangement Occurence
Cubic Orthorhombic Tetragonal-Sphenoidal Rhombohedral
Square Rhombic Rhombic
v Rhombic 1
Film Thickness, 24d,
Figure 3.3 : Variation of nonnalized specific surface area with film thickness using equations by Taylor et al. (1990). Symbols indicate model breakdown due to the occurrence of volume overlap.
Figure 3.4 : Verification of length of radius of interface circle at any contact point.
SECTION
- - - - -
Figure 3.5 : Verification of film fhickness at onset of spherical cap overkip for cubic packing.
CTI
Figure 3.6 : Verification of film thickness at complete occlusion of cubic packing. Double hatched area indicotes overiop of spherical caps.
Figure 3.7 : Verifkation of film thickness at occurrence of first overlap for orthorhom bic packing. Hatched area indicates spherical cap at two contacts.
Figure 3.9 : Diagram of shapes for volume and surface area integrdion.
Square Model(8 = rr/4 )
1 I 1 Numerical solution with R=0.001,0.01,0.1,1,2, 5, 10
- Cuwe fit to R = 1 -
-- Calculated Coefficients
I bO = -0.491 757 bl = 5.002605 b2 = -1 1.7991 81 b3 = 10.554990 b4 = -3.266650
Figure 3.10 : Cuwe fit to normalized volume of overiap as a function of a/R for the square corrective model.
Numerical solution wi R = 0.001, 0.01, O. 1, Curve fit to R = 1
Square Model ( e = 1
Calculated Coefficients
I bO = 50.31 5497 b l = -220.075399 b2 = 370.982948 b3 = -282.654845 b4 = 81.431 397
Figure 3.1 1 : Curve fit to normalized surface area of overiap as a function of alR for the square correction model.
R = 0.001, 0.01, 0.1, 1, Curve fit to R = 1
I 1 l Rhombic Model (e = r1/3 )
I Calculated Coefficients
I
- bO = 0.01 51603 b l = 4.098545 b2 = -1 1.8408021 b3 = 11.3171685 b4 = -3.590046
Figure 3.12 : Cuwe fit to normal for the rhombic corrective mod
zed volume of overiap as a function of a 1 R $1.
Numerical solution with R = 0.001, 0.01,0.1, 1, 2, 5, I O Curve fit to R = 1
I I I
Rhombic Model ( e = n13 ) I
Calculated coefficients
I
Figure 3.1 3 : Cunre fit to normalize a/R for the rhombic corrective moc
d surface area of overlap as a function of Iel.
Packing
\ Corrective Model Curve
Arrangement Rhombic Square S. Cap Linear End
Pore Discontinuity
- - Orthorhombic O El -- Tetragonal -
Sphenoidal A ....... Rhombohedral v
Film Thickness, 24d,
Figure 3.14 : Corrected variation of porosity with film thickness. Symbols indicate the beginning of corrective models, pore discontinuity and curve termination.
Packing Corrective Model Curve Arrangement Rhombic Square S. Cap Linear End
Cubic O a - - Orthorhombic O m -- Tetragonal -
Sphenoidal A A ....... Rhornbohedral v
Pore Discontinuity
Film Thickness, 24d,
Figure 3.15 : Corrected variation of normalized specific surface with film thickness. Symbols indicate the beginning of corrective models, pore discontinuity, and curve termination.
CHAPTER 4
CLOGGING MODEL: FORMULATION
The objective of this chapter is to develop and implement model for predicting the
rate of clogging in laboratory column experiments. The model combines concepts
associated with transient anaerobic, fixed film biologicai processes (such as those used in
wastewater treatment) with concepts of geotechnical engineering involving fluid flow
through saturated porous media. The model uses a time marching algorithm to mode1 the
evolution of the infiuent and efnuent organic concentration, biofilm thickness, inert biofilm
plus mineral film thickness and porosity at any position or time.
4.2 REPRESENTATION OF COLUMN FLOW
4.2.1 Introduction
The model assumes a column test is an experiment in which landfill leachate (or a
synthetic leachate) is passed through a cylinder packed with granular media, entenng the
column fiom a single port at one end and exiting from a single port at the other (see
Figure 4.1 a). The column is discretized into a nurnber of elements (also called segments
hereafter). The manner in which the system is subdivided into elements is theflowfield
controlling fluid flow in the system. The column test flow field is rnodelled as 'one
dimensional' advective flow, that is, fiow is fiom the infiuent end of the column through a
single line of segments along the length of the colurnn as illustrated schematically in Figure
Each segment, or "element" is assumed to act as a separate, fixed film reactor.
Acting as a reactor, each element reduces the concentration of substrate in the fluid as the
chemicai energy is converted partially into bacteria ceIl mass. This growth of biomass
foms an active film layer on the porous media. The bactenal activity aids in the formation
of a second film, an inactive film which consists of inert biomass and solid precipitate. The
modelling of these processes are detailed in Section 4.4. The active film grows on the
outer-most surface, whether that be the media or inactive film, and the inactive film is
assumed to fom directly on the porous media. The idealization of the biofilrn is described
in fiil1 in Section 4.4.3.
The influent substrate concentration to a segment (except the first segment) is
equal to the emuent substrate concentration of the previous segment in the series, as
s h o w in Figure 4.1 c. The transport of substrate fiom one segment to the next is purely
by advection. Saturated flow conditions are assumed for al1 segments. Flow through this
system is controlled by the segment with the greatest clogging; once a critical film
thickness is reached in the segment the flow stops (this corresponds to what is actually
observed in these experiments, Armstrong (pers. corn.)).
4.2.2 Representation of the Porous Media
The porous media of the flow system is represented using a volume of ideal
spheres of equal diameter. The diameter of the spheres is assumed to be equal to the
average diameter of the media while the packing arrangement of the spheres is related to
the initial porosity of the media. This idealization of the porous media allows geometric
calculation of the porosity and specific surface as a function of the film thickness on the
porous media. Details are given in Chapter 3.
4.2.3 Boundary Conditions
The boundary conditions involve a specified initial influent flow rate and
concentration. The effluent flow rate must (for reasons of continuity) be the same as the
idluent however the effluent concentration is calculated by the mode1 (and not
prescribed). The initial condition is zero substrate and a specified thickness of biofilm in
each segment (as discussed in Section 4.4.5.2). The leachate is assumed to provide the
substrate for biofilm growth, but not bacteria.
4.2.4 Advection Algorithm
4.2.4.1 Introduction
Flow (and hence substrate) is transferred from one segment to another over time
using a time stepping process described below. As noted above, the influent concentration
of substrate to the first segment is a boundary condition (which could vary with time). For
each subsequent segment, the effluent concentration of the neighbouring upstream
segment is the concentration of the influent to this segment. To simulate the movernent of
the treated flow through the column segments consideration is given to the time required
for the flow to move fiom the influent end of the column to the effluent end of each
segment.
4.2.4.2 Calculation of Ruid Travel Tirne Delays
The average tirne required for fluid to move from one end of a segment k to the
other end is given by
where V is the volume of the segment, n is the clean media porosity of the element, and Q
is the flow. If tirne steps are of length At, the total number of t h e steps elapsed as fluid
travels @y advection) fiom the beginning of the colurnn to the end of a segment k is T,
(the time delay in fiactions of timesteps) where
where p is the number of segments. and where T, = O.
It is assumed that the biomass growth and treatment of leachate in the segment
does not occur until the siug of leachate that entered the segment reaches the end of the
segment and hence these values are rounded up to the nearest whole number to arrive at
the number of tirnesteps required to pass through the column to the effluent end of
segment k, called Nk . Rounding the total time delay to the effluent end of each segment,
instead of rounding the time delay within each segment minimires the accumulation of
round-off error along the length of the column. Nk is rounded up using
As an example, Table 4.1 gives the calculated values of T, and N, for a 65 cm long column
divided into 13 segments, and n = 0.382, V = 10 1.53 cm3, At = 0.2 d, Q = 1 120 cm3/d .
This is the column to be modelled in Section 5.3.
4.2.4.3 Management of Calculated Effluent Concentrations
Figure 4.2 illustrates how the advection algorithm manages the effluent
concentrations over time. in this figure, Sc indicates a calculated effluent concentration
for a segment, and the arrows indicate the timestep from which the influent is retrieved.
The figure shows the effluent history for the column up to the completion of timestep 5
(1 .O days). In the figure it can be seen that on timestep 1 the first 5 segments calculate an
effluent concentration, as indicated by each having an S. value, and this calculation is
based the effluent concentration received from each upstream segment as s h o w by the
arrows. The remaining segments have fluid travel time delays of greater than 1 timestep,
therefore the first slug of leachate does not reach these segments during timestep 1, and
thus retum zero effluent values. On timestep 2 the first 5 segments treat a new slug of
leachate, while segments 6 to 1 1 treat the slug from the previous timestep since during this
timestep the slug travels past the end of segment 11. The diagonal arrow in the row
between segments 5 and 6 illustrates that between timestep 1 and 2 the slug of leachate
h m segment 5 passes through segment 6 and then segments 7 through 1 1.
The program keeps record of the required effluent concentrations of the previous
timestep for use in the current timestep. The bracketed matrix illustrates the values which
must be stored in order to calculate the effluents for timestep 5 for this example. The
columns of stored effluent concentrations in this matrix are numbered fiom 0, and in this
case the algorithm calculates a required storage of 2 columns of data, numbered O and 1.
The column on the far right of this matrk (in this case colurnn number 1 of the matrix)
contains the calculated effluent values of the current timestep, timestep 5, while al1 other
columns in the matrix contain the recorded effluents of previous timesteps (in this case
only column O of the matrix, timestep 4, is required).
In order to retrieve the correct recorded effluent concentration fiom the matrix, its
location must be calculated. Since each effluent concentration in the matrix is defined by
row (equal to segment number) and colurnn, and the segment number for retrieval of an
influent to a segment is always the previous segment number, only the column number
must be calculated. The delay values N, are used to calculate the column number location
in the matrix of recorded concentrations corresponding to each segment. This column
number, P, is calculated using:
The P, values for the column test example are given in Table 4.1. By inspection of Figure
4.2 and Table 4.1 it can be seen that in the current timestep (tirnestep S ) , the values of P,
indicate the matnx colurnn number fiom which each segment retrieves an infiuent value,
either O or 1, where a O indicates the slug concentration was calculated during the
previous timestep, and a 1 indicates the slug concentration was calculated dunng the
current timestep. For example, segment 5 receives the effluent of segment 4 from column
P, = 1 (which is the previously calculated effluent fkorn this timestep), while segment 6
receives the effluent of segment 5 from column P, = O (which was calculated in the
previous timestep).
Table 4.1 : Calculated variables for advection aigorithm example.
1 1 Segment Number 1
This advection algorithm allows the location of each segment's influent to be
precalculated so that repetitive computations cm be ornitted. Once a timestep is
complete, the record shifis its columns by one timestep to the right, thus making the just
completed timestep the previous timestep. This algorithm allows for future expansion, for
example, the influent concentration may be made variable and retrieved from the
concentration record, or the fiow may be distributed along the length of the series of
segments like a drainage layer. If the total length of the flow field is short andfor the flow
is high and the time step length is sufficiently long, there may be no delay applied.
4.4 BIOLOGICAL PROCESSES WITHIN ELEMENT
4.4.1 Introduction
The application of synthetic landfill leachate to each element of the flow field
encourages biological processes within the element. The model assumes that the supply of
organic compounds in the synthetic leachate provides a substrate for biological growth
and the formation of biofilm on the granular media. The substrate is considered to be the
single required nutrient which is not available in excess, and therefore limits the rate of
biofilm growth. This section deals with the processes occumng within a single element
dunng a single time step only. The routines required to impiement each element's
processes within the structure of the system's flow field and with respect to time, are
covered in Section 4.5.
The rate of biofilm and mineral film growth in a layer will depend on the bio-
kinetic parameters, influent COD concentrations, amount of flow, fluid mass density, the
physical dimensions of the flow field, the average diameter of the granular media, the
choice of packing arrangement, the system temperature and film properties such as
densities and mineral accumulation yield rate.
This section outlines the idealization of the films, and then, following the model
procedure, discusses the calculation of substrate flux, biofilm and mineral growth and loss,
and the new porosity and specific surface area. The iterative routine incorporating the
previous 3 topics to denve the revised effluent concentration is discussed next, and is
followed by a second iterative routine utilizing repeated calculations of the revised effluent
concentration to converge on a total loss coefficient. Lastly, the section discusses the
convergence critena and routines used.
4.4.2 Process Description
A non-steady state biofilm has a thickness that changes with time. Growth of ce11
matter is proportional to the flux of substrate into the biofilm, while losses occur due to
bacterial detachment and respiration decay. Non-steady biofilm growth and loss due to a
single substrate can be modelled using an algorithm modified fiom Rittmann and McCarty
(198 1) and Rittmann and Brunner (1984). The non-steady state growth and loss
algorithm is combined with equations to model the change in porosity and specific surface
are% mass balance equations, and detachment equations. Due to non-linearity, two
coupled iterative procedures are required to derive each layer's effluent COD
concentration, film thickness, porosity, and specific surface area at each time increment.
A simplified flow diagram of the procedures incorporated to model the biological
processes in one element during one time step is illustrated in Figure 4.3.
4.4.3 Biofilm Idealization
For model simplification, the biofilm is idealized as shown in Figure 4.4(a). The
active portion of biofilm is assumed to have a unifonn thickness of L, (L where L is units
of length), and a uniform density of X, (Mx L~~ where Mx is mass of bactena). A liquid
diffision layer of thickness L, (L) represents a theoretical unifonn layer of stagnant liquid
adjacent to the biofilm which is void of microorganisms and offers substrate mass
transport resistance between the bulk liquid and the biofilm surface. The bulk liquid is
assumed to be perfectly mked. The inactive film lies between the active film and the
attachment media and is assumed to have a uniform thickness of L, (L), and a unifonn
density of X, (M,L-~ where MT is total mass of solids).
Figure 4.4(b) depicts the substrate concentration profile through the biofilm.
Syrnbols Sb, S,, S, and S, in Figure 4.4@) represent the substrate concentrations in the
bulk liquid, at the outer surface of the biofilm, within the biofilm, and at the inactive film,
respectively (ail have units of Ms L - ~ where Ms is the mass of substrate). It is assumed
that the substrate concentration within the biofilm, S , only varies in the z direction, the
direction normal to the surface of the biofilm. It is also assumed that the substrate neither
difises into, or is utilized by, the inactive film and attachment media.
The biofilm growth and loss mode1 will include five processes occumng
simultaneously. The five processes are: mass transport of the substrate across the
difision layer, difision of the substrate in the biofilm, the utilization of the substrate for
ce11 growth, the growth of active cells due to this utilization, and the loss of biomass due
to detachment and decay of ce11 matter.
4.4.4 Calculation of Substrate Flux
The third step indicated on the flow diagram, Figure 4.3, is the calculation of the
flux of substrate into the biofilm. By considering substrate utilization kinetics similar to
Monod kinetics and molecular difision, Atkinson and Davies (1974) developed
approximate solutions for flux, J, into a biofilm, as a function of biofilm constants and
thickness, L, and surface substrate concentration S, :
where J is the substrate flux into the biofilm (Ms L" T'), (î is the maximum specific rate of
substrate utilization (Ms MyL TI), Ks is the half-maximum rate substrate concentration
(Ms LJ), and q is an effectiveness factor relating the ratio of actual flux to the flux if the
film were entirely penetrated at concentration S,. In dimensionless ternis (Rittmann and
McCarty, 198 1),
where the new dimensionless variables are defined as folIows:
and where z is the standard bioflm depth dimension in which
where D is the molecular diffusion coefficient of the substrate in the liquid (L2 T-l) and D,
is the molecular diffusion coefficient of the substrate within the biofilm (L* TI).
The effectiveness factor, q, was found (Atkinson and Davies, 1974) to be related
to S, and L, and the relationship, in dimensionless terms (Rittrnam and McCarty, 198 1)
was
where
Due to rnass transfer resistance, the substrate concentration at the biofilm surface,
S,, is not equal to the bulk substrate concentration, Sb. While the bulk substrate
concentration is assumed to be a known value at each iteration containing this routine, the
surface concentration is unknown. Mass transfer resistance is expressed in the mode1 by
employing an effective diffusion layer where it is assumed mass transport resistance
between the bulk Liquid and the biofilm surface occurs in its entirety. Assurning a one
dimensional diffision layer, Fick's first law gives
which may be converted to dirnensionless terms and combined with Equation 4.6a to yield
where the new dimensionless terrns are defined
A quadratic expression may be denved fiom Equation 4.94 and the quadratic formula
solution according to Rittmann and McCarty (198 1) is
An iterative procedure can be used with Equations 4.10 and 4.7a to calculate an
acceptable effectiveness factor, q, and dirnensionless expression of the surface substrate
concentration, s,' . Equations 4.6a and 4.6b rnay then be used to calculate the flux into
the biofilm. The steps taken were:
1. Convert inputs D, L, Sb and L, into dimensionless f o m using Equations 4.6c,
4.6d, 4.9b and 4 . 9 ~ .
2. Initialize the effectiveness factor q by setting the first estimate using
3. Calculate initial guess of S,' using Equation 4.10.
4. Calculate q' using Equations 4.7a and 4.7b.
5 . Check for convergence of q with q' to the given tolerance.
6. If convergence has not occurred, set q = q' and repeat steps 3 to 6.
7. Upon convergence, calculate J' using Equation 4.6a and J using Equation 4.6b.
A routine has thus been outlined which determines the substrate flux into the biofilm given
the kinetic constants, a known bulk substrate concentration and a known biofilm
thickness.
4.4.5 Calculation of Biofilm and Mineral Growth and Loss
4.4.5.1 Introduction
The calculation of change in thickness of the active biofilm and change in thickness
of the mineral and inen film, which added together f o m the inactive film, for the current
time step is step four as indicated in the flow diagram in Figure 4.3.
4.4.5.2 Non-Steady Growth and Loss of Biofilm
The expression of non-steady growth and loss of biofilm developed by Rittmam
and Brunner (1984) differs fi-om other non-steady models due to the fact that it
distinguishes between decay Losses and shearing losses.
The biomass growth rate, r, (M, L3 T-'), was expressed by
where Y is the tme yield coefficient (M, M i 1 ) , and the biomass loss rate, r,, (Mx L" T-')
was expressed by
where b' is the first-order biofilm loss coefficient (TI). The growth and loss of biofilm
routine may be derived by first combining Equations 4.12 and 4.13 so that for an
increment of time,
where r,, is the net biofilm growth rate. This equation may be simplified to become
(fittmann and McCarty, l98Oa, Rittmann and Brunner, 1984)
where t is time (T) and which may be integrated with respect to time, and if the tirne
incrernent, A t is small, J and b' do not change significantly, and Equation 4.15 becomes
Equation 4.16 can be used to calculate changes in biofilrn thickness for finite time steps.
By irnplementing Equation 4.16, the calculation of the new active film thickness in the
model is found using:
where the flux, I, used in the equation is calculated as outlined in Section 4.4.4 and the
biofilm thickness, L, , is the thickness calculated in the previous time step for this element.
The solution for L, is marched fonvard in time by incrernents of At. A very small initial
estimation of L, is required for the first time step since the calculation of substrate flux, J,
in Equation 4.17 requires that a non-zero thickness of biofilm exists. The initial film
thickness suggested by Rittmam and Bmmer (1984) and used throughout this thesis is
0.05 pm, which according to Rittmann and Brumer (1984) represents about 5% surface
coverage for a monolayer of bacteria.
4.4.5.3 Mineral Precipitation and Inert Biomass Accumulation
The composition of the inorganic portion of clog material has been found (Brune
et al. (1991) , Rowe et al. (1995a)) to be predorninately calcium carbonate, CaCO,.
Rittmann et al. (1996), reported that there is a relationship between the rnicrobial
oxidation of COD to inorganic carbon to the mass of calcium carbonate precipitated out
of landfill leachate. Laboratory experiments perforrned using landfill leachate collection
Stone infiltrated with landfill leachate fiom the KVL indicated that the removal of COD
(mostly acetic acid), and its partial substitution with H2C03 (bicarbonate), resulted in
major increases in pH and total carbonate, which together caused a large increase in CO,"
concentration, allowing or accelerating CaCO, precipitation. From data obtained from
Rowe et al. (1995a) a linear relationship between COD removal aiid CaCO, precipitation
may be derived and a yield coefficient, Y,, may be calculated (Rowe et al, 1995a; Fleming,
pers. comm.).
Calculation of Y, requires reguiar measurement of influent and effluent COD and
CaCO, concentrations. The concentration of CaCO, is calculated by measurement of total
hardness and expressing it as an equivalent mass of calcium carbonate. The COD removal
is calculated by subtracting the mass of effluent COD from the mass of infiuent COD, and
similady, the CaCO, precipitated is calculated by subtracting the CaCO, in the effluent
from the CaCO, in the influent. The COD removed and CaCO, precipitated Vary as the
colurnn operates. The value of Y, for the column test is detemined by plotting the COD
removed versus the CaCO, precipitated. The dope of a straight line passing through the
plotted experimental data points and the origin is equal to Y,. If the plotted data is not
linear, but becomes asymptotic to some maximum CaCO, concentration, the system may
be calcium deficient, and the yield coefficient may oniy be applicable to the COD removal
that fdls within the linear portion of the plotted data.
Utilizing the calculated yield coefficient for the specific system and substrate the
rate of mineral precipitation of CaCO, is given by
where J is the flux of COD into the active biofilm.
The accumulation of inert biomass rnay be found by assuming a fraction of the
active biornass, f,, is degradable due to decay. The remaining fraction, (1 - fJ, is not
degradable. The decay coefficient b represents the fraction of the biomass which dies off,
and therefore the portion (1 - fJb dies off, but does not degrade, and is thus assumed to
become inert. The rate of accumutation of inert biomass is
where b is the decay coefficient of the active biofilm.
The new thickness of the inactive film is found by the addition of the new mineral
and inert film accumulations as calculated in Equations 4.18 and 4.19 and implemented in
the mode1 using:
where X, is the total density of the biofilm.
4.4.6 Calculation of Porosity and Specific Surîace Area
The calculation of new d u e s of porosity and specific surface area are the fifth
step in the flow diagram in Figure 4.3. The new porosity n, and specific surface A, d e r
the accumulation of biofilm, is calculated using equations based on the sphere model
representing the porous media as previously described in detail in Chapter 3.
The porosity and specific surface model is based on the work of Taylor, 1990. In
Chapter 3 film thicknesses were geometncally determined above which Taylor's equations
calculated erroneous volumes and surface areas, and thus erroneous porosity and specific
surface values. For some packing arrangements a significant region of error could be
found between the maximum applicable film thickness and the geometncally calculated
thickness at which the porosity and specific surface becomes zero. By calculating the
erroneous volumes and surface areas, equations were derived to be used within the region
of error for each of the four packing arrangements. The derivation of the limits,
modification equations, and final corrected equations are discussed in detail in Chapter 3
and Appendix Ai.
The porosity and specific surface can be calculated assuming the active biofilm is
permeable or impermeable. If the active biofilm is considered permeable, the porosity
depends upon only the inactive (inert plus minerai) film thickness, while if the active
biofilm is considered impermeable, the porosity depends upon the inactive film thickness
plus the active biofilm thickness. The film thickness L, is the total thickness of the film on
the sp heres assuming t hese considerations
For each of the four packing arrangements Table 4.2 (page 12 1) sumrnarizes the
corrected equations used to calculate the porosity and specific surface and the film
thickness bounds in which they must be used. Interpolation between these functions is
used to derive porosities and specific surface areas for packing arrangements which do not
correspond to one of the four known fùnctions.
4.4.7 Convergence to Revised Effluent Substrate Concentration
4.4.7.1 Introduction
By making an initial guess of Sc and considenng the non-steady growth and loss of
biofilm equation, the change in porosity and specific surface, and mass balance, the revised
effluent substrate concentration S, can be calculated using an iterative procedure. This
process is illustrated in Figure 4.3 by steps 2 through 7 which begins with an initial
estirnate of S, and ends when the estimated value of the current iteration converges with
the calculated value. Convergence of the effluent concentrations indicates the balancing of
the flux of biomass and mineral ont0 the porous media, the surface area available, total
biomass losses, and substrate removal from the system. This section discusses this
iterative routine.
4.4.7.2 Application of a Second Substrate
Leachate may include a number of potential substrates, however the dominant
substrates will usually be acetic and propionic acid. The model, therefore, was designed
to not only model a single substrate, but also a mixture of these two acids. The utilization
of each acid is considered separately, and thus the convergence routine outlined in this
section is repeated to caiculate the effluent concentration corresponding to the second
substrate. The schematic in Figure 4.3 omits this repetition.
Each substrate has its own set of kinetic parameters, but shares a single decay
coefficient, b, which, in the absence of contrary information, is used as the first
approximation of the decay coefficient for the biofilm as a whole. The detachment part of
the total loss coefficient is assumed to be proportional to the relative arnounts of each
substrate. Each substrate contributes to separate active and inactive film thicknesses, but
a single value of porosity and specific surface is calculated fiom the sum of the thicknesses
of the films. Besides input parameters, the only difference between the processing of two
substrates occurs in the mass balance used to calculate the effluent concentration of each
substrate, discussed next.
4.4.7.3 Mass Balance
In order to consider changes in substrate flux and specific surface area which occur
due to film growth, the effluent concentration is recalculated using a mass balance. The
mass balance for calculating the effluent concentration of a single substrate, or for the
propionic acid in a mixture of acetic and propionic acid is
JVA, Swm = s, - -
Q
where S, is the iduent concentration of the specific substrate, J is the flux of the specific
substrate, 4 is the specific surface area, V is the volume of the element and Q is the flow
t hrough the element .
For the acetic acid in a mixture of acetic and propionic acid a different mass
balance equation is required to incorporate the degradation of propionic acid to acetic
acid. Since a product of the utilization of propionic acid is acetic acid, this acetic acid is
available for consumption, and thus must be incorporated into the acetic acid mass
balance. The addition of acetic acid to the mass balance is performed through the use of
the calculated substrate flux from propionic acid calculated in the previous step (step 2 in
the procedure in Section 4.4.8.4). The value 0.5714 is the ratio of the mass ( in COD) of
acetic acid created per mass of propionic acid degraded (Fùttmann, pers. comm.). The new
effluent substrate concentration for acetic acid, S,,, is calculated using :
J V A , JVA, Se+, = S, + 0.5714-0- - -
Q P
On the nght side of this equation the subtracted term is the concentration of the substrate
removed fiom the flow by substrate flux into the acetate degrader biofilm and the middle
term is the additional substrate concentration acquired ffom the degradation of propionate
into acetic acid @y the propionate degraders).
4.4.7.4 Approximation of the Buik Substrate Concentration
The calculation of the substrate flux and thus rate of growth or loss of biofilm
described in the previous sections requires the bulk substrate concentration as input. The
bulk substrate concentration is the substrate concentration applied to the biofilm uniformly
throughout a segment. It is assumed that the bulk substrate concentration is equal to the
log-mean of the infiuent and effluent substrate concentration of the element. This
calculation is shown in Equation 4.23 in step 2 in Section 4.4.7.5. According to Wang et
al. (1986), the log-mean concentration represents the equivalent average reactor
concentration when a first-order reaction takes place in a plug-flow reactor. As shown by
the rapid removal of substrate over a relatively short distance from the inlet, the colurnn
tests modelled in this thesis undergo zero order reactions only in the segments nearest the
inlet, and first order substrate removal dominates the remaining length. The log-mean
concentration has also been used to model colurnn tests by Rittmann et al. (1986) and
Wirtel et al. (1992). The application of the log-mean will require further study, but is used
in the model as a first approximation of the bulk substrate concentration.
The substrate concentration applied to al1 regions of the element is equal to this
bulk substrate concentration. The assumption is valid so long as the difference between
the influent and effluent concentrations remain relatively small. This assurnption may be
met by making elements small, and increasing the number of elements, thus reducing the
removal by each element.
i o a
4.4.7.5 Convergence on Revised Effluent Substrate Concentration
The concentration of the effluent of an element can only be established by adopting
an iterative process. The iterative steps are:
1. Initialize effluent substrate concentration variables Se- , Se,, for convergence,
and set the initial estimate of the effluent concentration using:
Se = So Time step = 1
se = se.,- 1 Time step > 1
where S,,, is the effluent concentration of this element from the previous time
step.
2. Calculate the bulk substrate concentration
3. Invoke substrate flux routine (Section 4.4.4) to calculate J fiom kinetic constants
and L, and Sb.
4. Calculate the new active and inactive film thicknesses using Equations 4.17 and
4.20.
5 . Invoke the porosity and specific surface mode1 desctibed in Chapter 3 and
calculate the new values of porosity and specific surface fiom the new active and
inactive film thicknesses. See Table 4.2 for a sumrnary of the modified porosity
and specific surface equations.
6 . Mass balance. Calculate the efnuent concentration S, using Equation 4.2 1 if the
current substrate is a single substrate or the propionic acid portion of a mixture.
Calculate using Equation 4.22 if the current substrate is acetic acid as part of a
mixture.
7. Check for convergence of Se within given tolerance. Convergence is discussed in
detail in Section 4.4.9.
8. If Se has not converged, and the number of iterations is less than 10, set S, = S,,,
and repeat steps 2 to 8.
9. If Se has not converged, then recalculate Se using a binary chop algorithm and
repeat steps 2 to 8.
The resetting of S, to S,,, in step 8 allows quick convergence which occurs when
the initial estimate of S, is close to the final value of Sc. Slow converging solutions which
occur when Se is not close to the final value of S. converge faster using a binary chop
aigonthm (Step 9) implemented if 10 or more iterations are required. This is discussed in
detail in Section 4.4.9. Given the biofilm properties, influent substrate concentration S,,
total Ioss coefficient, b', and values of La, L,, n, A,, and Se from the previous time step for
a given segment, new values of L, ,L,, J, n, 4, and S, can then be calculated (specific to
the substrate if two are modelled).
4.4.8 Convergence on the Biofilm Total Loss CoefFicient
4.4.8.1 Introduction
It is necessary to establish the biofilm total loss coefficient for each segment, at
each tirne step (see Figure 4.3, noting that the optional second substrate is ornitted). The
initiai estimate of the total loss coefficient, b', and diffusion layer thickness, L, (if
considered variable), are calculated based on the properties of the element in the previous
time step. The effluent concentration (for each substrate, if required) can then be
calculated using these parameters and the steps outlined in Section 4.4.7. The model
recalculates b' and L, based on the new properties of the element for this time step, and
the process is repeated until b' converges. The calculation of the parameters L, and b',
and convergence routine follow.
4.4.8.2 Calculation of the Diffision Layer Thickness L,
Introduction
The diffusion layer is a stagnant layer of Iiquid which is assumed to be void of
microorganisms and through which substrate is transported by molecular diffusion. The
model allows the difision layer thickness to be recalculated each iteration, or a fixed
value rnay be used.
The Difision Layer Thickness
The difision layer thickness may be estimated fiom principles of mass and
momentum transfer according to Skelland (1 974), by utilizing the equation
where j, is the Colbum j factor for mass and momentum transfer, S, is the Shenvood
number, & is the Reynold's number, S, is the Shmidt number, and v is the empty bed
(superficial) velocity. A correlation developed by Wilson and Geankoplis (Skelland.
1974) for caiculating the j factor for mass transfer between liquids and deep beds of
packed spheres is
for 0.00016 i q s 55, 165 s Sc 70,600 ,0.35 3 n i 0.75. From equations 4.24 and
4.25 an equation for the thickness of the boundary layer can be derived :
This equation has been applied by Liu et al. (1991) and Suidan (1986) to calculate L, in
modelling column tests using light aggregate and packed spheres, respectively, as biofilm
attachent media.
Variability of L,, with Clogging
The calculation of the Reynold's nurnber or Shenvood number does not account
for the effects of film buiidup on a bed of packed spheres therefore Equation 4.26 is
suspected to become Iess accurate as the element nears clogging.
Umealistic Values of L,
At low fluid velocities and large particle sizes such as those used in colurnn tests
modelling leachate collection systems, Equation 4.26 ofien results in unrealistically high
values of thickness L,,, such as thicknesses larger than the critical film thickness (which
would mean that at this thickness the stagnant difision layers could cause pore
discontinuity). The unrealistically high thicknesses may be due to the fact that the mass
and momentum transfer relationships have been derived from laboratory expenments using
small particles and very high fluid velocities relative to the values required for columns
representing leachate drainage systems. While coupling large particle size with small fluid
velocities such as those of this research results in Reynold's numbers which fa11 into the
allowable range for the relationships, it may do so due to the particle size, and when these
values are substituted into Equation 4.26 the result is a high value of L, . Due to the
possibility of unrealistic values of 4, computer implementation of the mode1 interface
calculates an estimate of L, using Equation 4.26 based on the known problrtm parameters
but then allows the user to choose to use either this calculated value or use a specified
fixed value.
4.4.8.3 Calculation of the Total Loss Coefficient b'
Introduction
The calculation of the decay coefficient b', or total loss rate coefficient, is
performed assuming it includes losses due to ce11 decay or maintenance, b, and detachrnent
by any measurable means, b,:
The value of b can be obtained from the literature, while the loss due to detachment, b, is
often a calculated value based on the ceIl removal process believed to be responsible in
this system under analysis. A number of expressions have been developed to calculate b,
for specific applications, but few may be applied to a wide range of conditions. For this
reason two methods of calculating b, have been considered.
Shear-Stress Model
The theory behind the first method of caiculating b, is that shear-stress is the
primary mechanism of biofilm detachrnent. A relationship between shear stresses and
biofilm detachrnent was derived by Rittmam (1 982) based on data collected by Trulear
and Characklis fiom experiments measuring the rate of biofilrn loss within an amular
reactor rotated at different speeds. The expression denved for detachrnent due to shearing
for L, > 0.003 cm was
where a is the shear stress, and for L, s 0.003 cm
b, = 8 . 4 2 ~ 1 0 -' O O - ' ~
where the shear stress o is
in which u is the superficial velocity (cdday). The shear stress calculation is applicable
for
Growth Rate Mode1
A second theory proposed that detachment rate is directly related to biofilm
growth rate in terms of its substrate utilization rate, yield, and biofilm thickness (Peyton
and Characklis, 1993). Monopopulation and rnixed-population biofilm data fiom the
study was combined with other experimental observations with the same cultures of
bacteria in order to correlate the expression. The relationship denved for the detachment
rate, &, was
where k, is the detachment rate coefficient, A,, is the biofilm surface area, Si is the influent
substrate concentration and S, is the bulk substrate concentration. From this expression
the detachment coefficient b, may be found to be
where k, should be found experimentally. For the model the data compiled by Peyton and
Characklis (1993) was used to calculate detachment rate coefficients of 454 cm-' and 63
cm-' for single and mixed population bacteria cultures respectively. If this method is
selected, the model assumes that the bacteria culture is a mixed population and thus k, is
set to 63 cm".
4.4.8.4 Steps to the Satise Biofilm Total Loss Coefficient
The procedures required to obtain the biofilm total loss coefficient (with o v e ~ e w
given in Figure 4.3) are as follows:
1. Calculate the total biofilm loss coefficient, b', and diffusion layer thickness, L,,
based on previous values of n, As and L, and flow characteristics.
2. Solve the effluent concentration for substrate 1 (Section 4.4.7). Use the kinetic
pararneters for substrate 1. Calculates J, L, ,,,. L,,, n, 4 and Sc.
3 . Solve the effluent concentration for substrate 2 (optional) (Section 4.4.7). Use the
kinetic pararneters for substrate 2. Calculates I, , LBLn- LS.Lnm, n, 4 and S,,.
The flux of substrate 1 (propionic acid) is required to include the degradation of
propionic acid to acetic acid in the mass balance.
4. Recalculate b ', and b.
Check for convergence of b' and b', (see Section 4.4.9 regarding convergence).
If b 'and bt2 do not meet convergence criterion, set b ' = bI2 and repeat steps 2 to 6.
Check for clogging of the element by checking that the total impermeable film
thickness is less than L,,, and A, is greater than zero.
Segment calculation is complete, go to next segment in line.
Convergence
4.4.9.1 Introduction
In the preceding sections, discussion of convergence has been greatly simplified to
allow attention on other processes. This section discusses aspects of the convergence
methods not described in detail. The parameters solved by convergence are:
the total loss coefficient of a layer, b', (Section 4.4.8),
effluent concentration of a layer, Se, (Section 4.4.7),
I effectiveness factor of the substrate flux, q, (Section 4.4.4)
Each convergence routine follows the same pattern. Maximum and minimum limits are
placed on the dependent pararneter, and an initial estimate of this parameter is made.
Using this estimated parameter as input to various equations, the parameter is
recalculated. If the estimated and calculated versions of the pararneter meet specific
convergence critena, the solution has been found and the convergence routine ends. If the
criteria were not met, adjustments are made to the limits such as increasing the minimum,
or decreasing the maximum (as listed in column 6 of Table 4.3, page 122) such that upon
further iteration, the lirnits converge. A new estimation of the parameter is made either
dependent on the limits or the previously calculated parameter. The routine repeats. The
details of each convergence routine are given in Table 4.3.
4.4.9.2 Maximum Number of Iterations
The procedures have been simplified by assuming that the routines eventually
converge on a solution. In actuality, under certain conditions convergence criterion may
never be met or the solution may converge at a very slow rate. In order to allow
automatic or manual adjustments to be made to the convergence conditions a Iimit must
be placed on the maximum number of iterations so that the particular convergence routine
cm be halted. If the maximum number of iterations is reached, the mode1 terminates.
4.4.9.3 Convergence Criterion
The convergence criterion for each routine is basically the same. The parameter is
assumed to have converged when the relative change in the parameter is less than some
given tolerance. The condition for the convergence of the effectiveness factor, q, for
example, is
where q, is the previous estimate of the effectiveness factor, and e, is the convergence
tolerance for the effectiveness factor.
4.4.9.4 Rapid versus Stable Convergence
Using the calculated parameter fiom one iteration as the estimate to the next
iteration is the fastest method of convergence, but this method may cause various
instabilities. The more stable, but slower converging method of revising the initial
estimate of the parameter is to use the calculated average of the limits of the parameter,
where the limits converge (as descnbed in Section 4.4.9.1). The convergence routines for
the total loss coefficient and effluent concentration are designed to change from a rapid to
a stable method if specific conditions are met which indicate instability.
For convergence of the total loss coefficient, an instability occurs when oscillating,
or recursive results are calculated. For example, a cyclical sequence can develop where
input (a) gives output (b) which then gives output (a) and the process repeats without
convergence. The mode1 has been programrned to detect non-converging cyclical
sequences and change from using the rapid convergence aigorithm to the less problernatic
algorithm involving averaging of the parameter limits, see Table 4.3, Column 7.
When using the effluent concentration calculated fi-om one iteration as the estimate
of actual effluent concentration, convergence to the final value may, under certain
conditions, be extremely slow. Under these conditions the rate of change of the estimated
effluent concentration is so small that the maximum number of iterations is reached before
the convergence criteria is met. The routine is prograrnrned to allow only 10 iterations
using the so called 'rapid convergence algorithrn' and if convergence is not reached after
10 iterations the convergence routine changes to the more stable averaging method.
These convergence problems generally only occur when the input parameters do
not well represent the physical system.
4.5 COMPLETE MODEL PROCEDURE
4.5.1 Procedure
The complete procedure of the mode1 accepts the necessary input, initializes
variables and arrays of variables according to the values entered, and repeats the
caiculations of each segment dong the flow field for each time step analysed.
The algorithm adopted is illustrated schematically in Figure 4.5 and involves:
Entry of input parameters.
Variable Precalculations: temperature dependent variables, fluid absolute viscosity,
coefficients of substrate difision in the biofilm variables dependent on the
packing arrangement, element dimensions, influent flow and substrate
concentration and biofilm growth constants. Details are given in Section 4.5.2.
Initiaikation of Arrays: Set a11 elements (segments) to have the same initial
porosity, specific surface and initial biofilm thickness, and precalculate the arrays
for simulating advection. Details are given in Section 4.5.3.
Time step = 1.
Segment number: k = 1.
Perform calculations for segment k. The influent concentration is govemed by the
advective routine.
Repeat step 6 for each segment (k = k + 1).
8. For advective flow algorithm. shift columns of values in the record of effluent
concentrations (see Section 4.2.4.3).
9. Repeat steps 6 through 8 for each time step.
4.5.2 Variable Precalculations
4.5.2.1 Temperature Dependency
Calculate temperature dependent pararneters fiom reference parameters and
temperatures if required. Temperature dependent pararneters calculated are K,,, K,,, 4,.
q2, Do,ir Da2 , and b. Temperature dependency is calculated using the expression (Metcalf
and Eddy, 199 1):
where T is the system temperature in OC, TT is the temperature the reference parameter
was measured at, in OC, PT is the parameter at temperature T T . P, is the parameter at
the reference temperature T r a c and h is the temperature factor for the parameter.
4.5.2.2 Absolute Viscosity
The absolute viscosity of the fluid can be specified or calculated assuming the
viscosity of the fluid to be that of water. A formula for calculating absolute viscosity
variance with temperature was derived using regression and published data (Metcalf and
Eddy, 199 1). Using this denvation, viscosity variance due to temperature is accounted for
b y :
where T is temperature in O C , and y is the absolute viscosity (mg cm-' d-'). The absolute
viscosity is required in the caiculation of the diffision layer thickness and detachment
coefficient if the shear-stress mode1 is applied.
4.5.2.3 Substrate Difision in the Biofilm
The molecular diffision coefficient for the substrates in the active biofilm, D,,, and
D,, , may be obtained from published values or calculated fiom the diffision coefficient of
the substrates in the free liquid, Do,, and Do,Z. When D, is calculated, D, is assumed to be
80% of the coefficient of difision of the substrate in the fiee solution.
4.5.2.4 Packing Arrangement Dependencies
The initial specific surface area, k0, is calculated from the particle diarneter and
initial porosity using Equation 3.1. The number of contact points, m, and packing factor,
a,, corresponding to the packing arrangements bounding the initial porosity are set. The
number of contact points and packing factor corresponding to the initial porosity for the
four regular packing arrangements are given in Table 3.1. The critical film thickness, Lt&,
is calculated by interpolation of the critical film thicknesses of the bounding packing
arrangements. The critical film thickness is the film thickness at which the element
analysed is considered clogged. Derivation of this parameter is discussed in Section 3.5.
The critical film thicknesses are listed for each packing arrangement in Table 3.4, colurnn
4.5 -2.5 Conversion to Mode1 Units and Miscellaneous Calculations
The segment length (cm) is calculated from the column length (cm) and the
nurnber of segments. The cross-sectionai area of flow (cm2) is calculated fkom the column
diameter. The half-velocity constants of the two substrates, K,, and K,, , entered in (mg
L-') are converted to (mg cm"). The standard biofilm depth dimensions for each substrate
(cm) are calculated using Equation 4.6E
4.5 -2.6 Substrate Flow and Concentration
The infiltration flow (cm3 d") is converted from units of (L d-'). The infiltration
substrate concentration (mg cm3) for substrates 1 and 2, SwI and S,, respectively, are
calculated using:
and
where P is the portion (%) of the leachate which is substrate 2, S , is the total substrate
concentration of the infiuent leachate (mg L-') , and SR,, is the concentration of refractory
substrate in the leachate (mg L-'). If any substrate concentration is calculated to be a
negative value, it is assumed to be zero.
4.5.3 Initialbation of Arrays
4.5.3.1 Element Properties
The mode1 requires that the granular media of each element begins with a relatively
thin active biofilm thickness for each substrate, L,, and L,, so that initial utilization of
substrate and subsequent biomass accumulation c m occur (see Section 4.452). This
initialization is performed using
and
where L,, is the total initial active biofilm thickness (cm). The initial film thickness for
each substrate is calculated to have the same proportions as the substrates cornprising the
total concentration of substrate. If only one substrate is considered, P is zero and
therefore L,,, for each element is zero. The initial inactive film thicknesses, L,, and LP2 of
each element are set to zero. The initial specific surface area of the elements are set to the
value calculated using Equation 3.1.
4.5.3 -2 Advection Arrays
The precalculated arrays required for storage of effluent concentrations and
retrieval of influent concentrations to simulate advection through the segments are
outlined in Section 4.2.4.
4.6 CHAPTERSUMMARY
A mode1 was developed for the prediction of the change in porosity in colurnn test
experiments. The porous media system was represented by a senes of segments in which
each segment was assumed to act as an anaerobic fixed-film reactor. An advection
algorithm was developed to retrieve influent concentration values fiom previous timesteps,
thus simulating time delay. Biofilm and mineral growth equations are derived, and are
implernented within an iterative routine for convergence on an effluent concentration.
This routine is implemented into a second iterative routine for convergence upon the total
loss coefficient. This routine is repeated for each segment and each timestep, with the
effluent of each segment being the influent to the next downstream segment (after some
time delay).
Table 4.2 : Porosity and Specific Surface equation summary
b
Packing Arrangement -
(ml
Cubic (6 )
Film Thickness (2L/d,) 1 Porosity Equation -
Lower Bound Bound
Specific Surface Equation
Table 4.3 : Summary of Convergence Routine Details
Convergence Routine
Total loss coefficient
Efïïuen! concentration
Initial Settings and Estimate of
Parameter
b' = calculated
Sc,, = calculated
q = calculated
Critical Parameter
1 s' Iteration
Condition
Convergence Cri terion Adjusimcnl of Liinits Revision of Initial Gucss
Jb', - b'I / b' < q,, I b' = bIl 1 ;bl, c b') and (b' < b1,,,3 then b',, = b'
if IbfmU- btm,( I bl,< 0.01(tbP) if
if oscillating, b' = 0.5 (bl,+ b',,)
Start Element Anaiysis
1 1
Initial Estimate of Total Los Coefficient
- - -
l&i Estimate of ~ff&nt Concentration 1
Calculate Substrate Flux
l Calculate Porosdy ond Specific Surface
I . Calculate Revised Effluent Concentration
from Substrate Mass Balance I
\ NO
Concentration ?
1 Calculate Revised Total Loss Coefficient 1
NO
Los Coefficient ?
Go to Next Element
Figure 4.3 : Flow diagram of processes performed for each element of the flow field during each timestep (optional secondary substrate processes ornitteci).
Input o f Variables s 1
Precalculations, Conversion of Variables to Model Units
lnitialization of Arrays
Segment Anaiysis
/
YES Additional Segments ?
NO
YES Additional Timesteps ?
NO
stop
Figure 4.5 : Flow diagram of primary processes of model.
CHAPTER 5
APPLICATION OF MODEL
5.1 INTRODUCTION
This chapter outlines two applications of the model. The first is to a fully
documented, short term colurnn reactor expenment reported by Rittmann and McCarty
(1980). In this test, a single substrate which does not contribute to mineral precipitation
was used, and kinetic coefficients were denved experimentally. Predicted and measured
acetate concentrations and biofilrn thicknesses are presented. Since a single substrate
which does not form mineral precipitate is examined, it is possible to test, and if necessary
isolate any problems with the core biokinetic components of the complete model, before
examining the complex case of biofilrn growth and minera1 precipitation examined in the
second application discussed below.
The second application of the model is to a saturated column test conducted using
synthetic leachate (Millward, 1997; J. vançiulck, pers. comrn.). The synthetic leachate is
represented by two substrates, and inciudes significant quantities of dissolved rninerals
which have the potential for mineral precipitation. This application of the model provides
comparison of predicted and measured effluent concentrations over time and porosity
profiles dong the column measured at specific times.
5.2 APPLICATION TO RITTMANN AND MCCARTY (1980) DATA
5.2.1 Introduction
The rnodel was first applied to the results of colurnn experiments reported by
Eüttmann and McCarty (1980). This work was chosen because the experiments are well
documented for modelling purposes, and test only the core components of the clogging
model. The experiments were ideal for modelling purposes since
a. they were intended to veriQ a sirnilar biofilm growth mode1 and therefore the work
includes most of the required modelling parameters and significant results,
b. many of these parameters were determined by R i t t m a ~ and McCarty using
independent experirnents,
c. since glass beads were used, the media was well represented by the model,
d. the choice of influent allows representation by a single substrate (sodium acetate),
e. there is negligible inactive film growth over the shon duration of the test.
5.2.2 Experiments
The column reactor tests of R i t t m a ~ and McCarty (1980) were performed using a
glass column which was 12 cm in length, and 2.5 cm in diameter. The column was
designed with sampling ports at 1, 2,4,6, 8, and 10 cm fiorn the influent port. Glass
beads, 0.3 cm in diameter, were packed into the column at an initial porosity of 34%. The
colurnn tests were operated in upflow mode. Before each expenment, the reactor was
inoculated uniformly with about 10 mg C of e~chment culture, bacteria cultivated from
primary sewage. The rate-limiting substrate thereafter was carbon labeled sodium acetate.
The biofilm cultures, named BC1 and BC3 by Rittmann and McCarty, were developed at
the flow rates and acetate concentrations listed in Table 5.1.
Table 5.1 : Experiment Flow Rates and Influent Substrate Concentration
- -
Parameter BC 1 BC3 Units
Q 1.58 6.676 Iitre 1 day
S, 7.2 3 -6 mg 1 litre
This continuous feeding was maintained until steady-state occurred, as indicated
by stable substrate removal with time. According to Rittmam and McCarfy (1 980), and
Rittmann (pers. co rn . ) deve!opment of steady-state required about 3 to 4 weeks of
reactor operation. Once at steady-state, each test was terminated and sarnples were then
taken by syringe and by removal of beads f?om various locations along the column. The
bacteria was sheared off of the beads and the resulting water was assayed for organic
carbon. Organic carbon was determined using a hybnd Oceanography International -
Dohrrnann instrument. Bacterial carbon was measured as non-soluble organic carbon. In
this marner the profiles through the column of measured acetate concentration and biofilrn
thickness at steady-state were developed.
5.2.3 Mode1 Parameters
Table 5.2 lists the parameten denved by Rittmann and McCarty (1980) to be used
to mode1 the column tests. These parameters will not be challenged. These parameters
include the dimensions of the reactor and the experimentally rneasured bio-kinetic
properties. Due to the multiple evaluations performed, the kinetic constants include
standard deviations. The parameters for calculating the temperature dependence of kinetic
constants and diffusion coefficients were not required in this application of the model.
Table 5.2 : Given Mode1 Parameters (Rittmann and McCarty, 1980)
pp - -- -
Parameter Value Units -- - - -- - -
Length of column 12 cm
Diameter of colurnn 2.5 cm
Particle diameter 0.3 cm
Initial porosity 0.34
Ks 3.9 * 0.58 mg / litre
4 20 4.4 mg / mg C-day
b 0.204 * 0.049 day -' Y 0.07 1 0.007 m g C l m g
% 2.5 mg C / cm3
Do 1 .O9 cm' / day
Table 5.3 lists parameters required by the cornputer model which were not
published, but could be given an assumed value. It was assumed that the reactor
temperature is close to room temperature, at approxirnately 20" C (Rittmann, pers.
comm.). Selection of the system temperature has little significance since the kinetic
constants were experimentally denved at the system temperature. This eliminates the need
for recalculation of parameters using temperature dependency factors, and the oniy other
temperature dependent variable is the viscosity of the fluid. The refractory concentration
is assumed to be zero since there is no significant non-degradable substrate. The calcium
carbonate yield coefficient was taken to be zero since there was insufficient calcium to
allow CaCO, precipitation. Since no inactive film buildup is expected, the inactive film
density was taken to be a large, arbitrary value so that, coupled with a calcium carbonate
yield rate of zero, the accumulated thickness of inactive film would be sufficiently close to
zero.
Table 5.3 : Assumed Mode1 Parameters
Parameter Value Units
Temperature 20 OC
Refiactory concentration O mg / litre
y, O mg 1 mg
10000 mg / cm3
Table 5.4 Iists the input parameters which were varied. Parameters such as the
number and length of timesteps, and number of segments were initially estimated, and then
the optimum value was established as discussed later. The sensitivity to the choice of
parameters such as permeability, diffusion layer thickness, intemal diffusion coefficient and
detachment method was examined. According to current theory it was assumed that in its
initial stages the active biofilm may be considered permeable. The diffusion layer
thickness, L, , is calculated by the model using a slightly different equation than used by
the model proposed by Rittmam and McCarty (1980) as discussed shortly.
Table 5.4 : Parameter Sensitivity Study
Parameter Predorninately Unit s Assumed Value
- - - - --
Number of timesteps 1000
Length of timestep 0.04 day
Pemeability of biofilm Penneable
Number of segments 24
b Equation 4.26 cm
D , 0.87 cm2 I day
Detachment mode1 Shear
5.2.4 Defining the Required Number of Timesteps and Segments
5.2.4.1 Tirnesteps
The length of the timestep used in the model is a numencal parameter which can
infiuence the calculated results. Lf a smdl enough tirnestep is used then the active film
thickness of the previous timestep provides a reasonable basis for calculating the film
growth of the current timestep. The smaller the timestep is, the smaller the change in film
thickness between timesteps, and the greater the accuracy until the solution converges to
values that cease to be dependent on the timestep size. Using timesteps that are too long
will result in an overestimation of effluent concentration. The optimum timestep length is
the largest value below which further decreases no longer produce a significant change in
results, since this provides the required accuracy while rninirnizing the number of timesteps
required and thus shortenhg run-time.
The mode1 was run for experirnents BC 1 and BC3 using the parameters listed in
Tables 5.1 - 5.4 above for various timestep lengths and for a test duration of 40 days. For
pararneters where Rittmann and McCarty provide a mean and standard deviations, the
mean values were used. Figures 5.1 and 5.2 show the effect of the timestep length on the
plot of effluent concentration versus time for BC 1 and BC3 respectively for the timestep
lengths of 1 .O, 0.2.0.04, and 0.008 days. For both experiments, a reduction in timestep
length fiom 1 .O to 0.2 days caused a significant shift in the calculated effluent acetate
concentration versus time plot to the left by approxirnately 2.5 days. Further reduction in
timestep length fiom 0.2 to 0.04 days resulted in plots that were shified to the lefi by a
significantly smaller amount (0.5 days), and a further decrease in timestep length fiom
0.04 to 0.008 days had little effect. Thus, a timestep length of 0.04 days (approx. I hour)
was found to be adequate for subsequent modelling.
While Rittmam and McCarty (1980) assumed that steady-state had been reached
after about three weeks of operation and had terminated the tests between 3 to 4 weeks,
Figure 5.1 shows that for this set of input pararneters, the mode1 predicts a gradua1
increase in effluent concentration afier the initial drop in effluent concentration, and tme
steady-state conditions are not reached within the 40 day period of analysis. Steady-state
was assumed to have been reached when there was no significant change in substrate
removal (Rittmann, pers. cornrn.). If these predictions are accurate, it shows that
experiment termination may have been prernature, but also that the rate of change of
effluent concentration after 3 or 4 weeks may have been small enough that changes in
effluent concentration were insignificant.
5.2.4.2 Number of Segments
The properties of each segment (or element) of the column are assumed to be the
same throughout the segment. The bulk substrate concentration providing nutnent to the
biofilm in each segment is taken as the log mean of the influent and effluent substrate
concentrations as suggested by Wang et al. (1986) and as descnbed in Section 4.4.7.4.
This approximation is valid so long as the dserence between the influent and effluent
concentrations (ie. the substrate concentration removed), remains relatively small. In
order to improve model accuracy by reducing the substrate concentration removed by
each segment, the nurnber of segments must be increased and length of segment reduced.
This ailows approximation of the bulk substrate concentration.
The optimum segments length is selected in a manner sirnilar to the optimum
timestep length by decreasing the segment length until there is an insignificant change in
results for any additional reduction. It is important that in assessing the convergence on
the required segment length (and hence number of segments) that the model be mn for the
entire duration of the test being modelled. This is because at early times in the expenment
there is Little difference in the growth of biofilm between segments, and therefore little
removal. As the elapsed time increases, there is an increase in the difference between the
thickness of the biofilm in each segment and hence greater removal by each segment, thus
requiring smaller segments. The number of segments may be selected such that values are
caiculated at specific locations dong the colurnn, but the selected number of segments
must be greater than the optimum number of segments as defined above.
The mode1 was run for test BC 1 using the parameters listed in the tables above for
various other numbers of segments for a test duration of 40 days using rnean parameters.
Figure 5.3 illustrates the effect of the number of segments on the effluent concentration
versus tirne. The figure shows the results for 1,2,4,6, 12 and 24 segments and it c m be
seen that there is no significant difference between the results obtained using 12 segments
and 24 segments and as few as 6 segments would have provided acceptable results. The
results for 1, 2 and 4 segments were accurate at early times, but become inaccurate after
about 10 days.
5.2.5 Results and Discussion
5 -2.5.1 Predicted Concentration Profiles Using Mean Parameter Values
Figure 5.4 shows the measured and predicted substrate concentration through the
column after three weeks for test BC 1. The leveling off of the rneasured acetate
concentration indicates that substrate removal predominately occurs in the first 4 cm of
the reactor. The remainder of the column has a substrate concentration so low that there
is no significant biofilm activity, and thus, since no further substrate is removed, the
concentration remains constant. The predicted acetate concentrations along the column
obtained using the mean parameter values are shown for 21 and 28 elapsed days. Two
times were exarnined because there is some uncertainty regarding the elapsed time at
which the measurements were made ( "about 3 weeks of operation" as reported, and
Rittmam (pers. comrn.) has indicated that it was 3 - 4 weeks). The elapsed operation time
before termination and sarnpling also rnay have differed between colurnns BC L and BC3.
There is no significant difference between the predicted profiles at 2 1 and 28 days for
BC 1. The predicted concentration profile obtained using the mean parameters
overestirnates the acetate concentration in the first-in-line segments of the first 1.5 cm,
then underestimates the acetate concentration of the remaining portion of the reactor. The
predicted results become relatively constant with length along the reactor.
Figure 5.5 shows the rneasured substrate concentration at steady-state and the
predicted concentrations through the column after 2 1 and 28 days for test BC3. The
measured substrate concentrations from this expenment do not become constant within
the 12 cm of the reactor. This is due to the greater flow rate which decreased the
detention tirne in each segment, thus decreasing the rernoval by the early-in-line segments,
and allowing substrate to be transported to segments further from the source. The
predicted concentration profiles for 2 1 and 28 days are very close together, and correctly
predict that the test does not reach a constant minimum concentration. Using the mean
parameters the mode1 overestimates the first 2 cm, and underestimates the remaining 10
cm.
5 -2.5.2 Final Fitted Predicted Concentration Profiles
By slightly adjusting the kinetic constants within the range defined by the standard
deviation of each parameter, the predicted profile of test BC3 was fit to the measured data
using K, = 4.48 mgAitre and b = 0.254 dayeL . These kinetic constants were then used to
mode1 test BC 1. Figure 5.6 shows the concentration profiles for 2 1 and 28 days using the
revised input parameters. It was observed that the predictions for 21 and 28 days remains
close, and the first 1.5 cm is still overestimated, but the revised concentration profile
provides a better fit to the measured data. The mode1 correctly predicts the occurrence of
a constant minimum concentration.
Figure 5.7 shows the predicted concentration profiles using the kinetic constants
denved for test BC3. The predicted results for 2 1 and 28 days separate by a maximum of
about 0.3 mg/litre. Through the first 2 cm of the reactor, the measured data is
overestimated at both 2 1 and 28 days, but the remaining measured concentrations lie
between the 2 1 and 28 day profiles.
5.2.5.3 Predicted Biofilm Thickness Profiles
The measured biofilm thicknesses at steady-state and predicted biofilm thicknesses
at 21 and 28 days for test BCI are shown in Figure 5.8. By cornparison of the measured
values in Figures 5.8 and 5.6 it can be seen that where significant removal of substrate
occurs, in the first 2 cm, there is very thick biofilrn. This biofilm thickness rapidly
decreases over the remaining 10 cm. The predicted biofilm thicknesses, calculated using
the revised kinetic constants, also show this rapid decline. There is little difference
between the predicted biofilm thicknesses after 21 and 28 days. The biofilm thicknesses
are overestimated over the first I cm, and are underestimated over the range 2 to 12 cm.
Figure 5.9 shows the measured steady-state biofilm thicknesses and predicted
biofilm thicknesses after 21 and 28 days for test BC3. By inspection of the measured
substrate concentrations for test BC3 in Figure 5.7, it can be seen that the gradua1 increase
in substrate removal which occurred throughout the reactor is matched with a measured
gradud decline in biofilm thickness indicating utilization of substrate along the entire
length of the column. The predicted biofilrn thicknesses provides a good fit to the
measured values. There is a great difference between the predicted thicknesses at 2 1 and
28 days in the first 2 cm, which is followed by very similar predicted results from 2 to 12
cm length. This indicates the rîpid growth occurring in the early-in-line segments. The
measured data fa11 between the predicted profües at 21 and 28 days in the first 2 cm and
remain very close from 2 to 12 cm. This suggests that the measurements may have been
taken between 21 and 28 days.
5 -2.5 -4 Parameter Significance
Figure 5.10 shows the sensitivity of the model to the permeability of the active
biofilm for test BC3. The figure shows the effect of assuming the active biofilm is either
impermeable or permeable (that is, whether the active film should, or should not, be used
in the calculation of porosity). The predicted results for 28 days show a slight increase in
removal throughout the reactor, but indicate that for this model run, the permeability of
the active biofilm is insignificant.
Figure S. 1 1 illustrates the effect of using different rnethods of calculating the
difision layer thickness. If not set to a fixed thickness, the model applies the difision
layer thickness calculation (Suidan, 1986 and Lui et al., 1991) discussed in Section
4.4.8.2. The model discussed by Rittmann and McCarty, 198 1, uses a similar equation to
calculate a fixed layer thickness. This equation gives a fixed difision layer thickness of
0.0 1 58 cm, as compared to the model denved initial thickness of 0.0 129. The use of the
fixed (siightly greater) difision layer thickness results in an increase in the predicted
substrate concentrations (a decrease in removal), which is expected since an increase in
the diffusion layer thickness causes a decrease in the substrate concentration available for
utilization by the biofilm.
In order to calculate the difision coefficient of the substrate within the biofilm,
Rittmann and McCarty, as well as many other modellers, suggest the use of a DJD0 ratio
of 0.8, meaning that the difision of substrate into the biofilrn, D,, is 80% that of the
diffusion of the substrate in fiee water, Do- The 0.8 ratio was determined by Williamson
and McCarty (1 976) for nitri@ng aerobic biofilms. Recent work (Yu and Pinder, 1994,
Pavlostathis and Giraldo-Gomez, 1991) has suggested much Iower D& ratios for the
difision of acetate in biofilms. A Df/Do ratio of 0.3 1 reported by Yu and Pinder was used
to obtain the results shown in Figure 5.12 for D, = 0.338 cm2/d and a ratio of 0.8 to obtain
the results shown for D, = 0.872 cm2/d. For this case it cm be seen that the choice of
diffusion coefficient has negligible effect over the range examined.
The sensitivity of the model to the method of calculation of the detachment
coefficient, b, was studied. The previous predictions were performed using the shear-
stress method, discussed in Section 4.4.8.3. Prediction using the growth rate method of
calculating b, ( also in Section 4.4.8.3) were also obtained and the results are compared in
Figure 5.13 and it can be seen that the growth rate method results in higher predicted
substrate concentrations (a decrease in removal). For this test, the choice of detachment
model did not significantly affect the results.
5.2.5.5 Hydrodynamic Dispersion
According to Golla and Overcarnp (1 WO), the small disagreement between the
measured and predicted results shown in Figures 5.6, 5.7, 5.8 and 5.9 may be due in part
to the neglect of dispersion by the model. Golla and Overcamp (1990), also working with
test BC1 data, concluded that in comparing a plug flow rnodel and a model which included
longitudinal dispersion, the model which included longitudinal dispersion showed good
agreement while the plug flow model overpredicted the concentration and biofilm
thickness at the inlet of the reactor and underpredicted these variables downstream.
As s h o w in Figures 5.6 and 5.8 , this was the case for the plug flow mode1
developed in this thesis for test BC 1, but, as indicated in Figures 5 .7 and 5.9, not the case
for test BC3. According to dispersive theory, the effects of mechanical dispersion should
be more pronounced in test BC3 because of the substantially higher flow, leading to the
conclusion that dispersion may not be the cause of the overpredicted concentration and
biofilm thickness at the iniet of the reactor and underpredicted values downstream.
Instead, this phenomenon is likely caused by omission of some other modelling factor or
(less likely) expenmental error, such as neglecting biological activity O C C U ~ ~ ~ in the
influent tubing between the point of influent concentration measurement and the first
measurement location within the reactor. Such biological activity would result in an
overestimated influent concentration to the reactor, and cause the phenomenon described
above. It is unknown whether regular cleaning of the tubing was performed, or what
effect regular cleaning would have on substrate removal within the tubing. Modelling the
growth of bacteria within the tubing of the apparatus may correct this situation, and
improve the accuracy of the results.
5.2.6 Conclusions for Rittmann and McCarty (1980) Experimenb
The model predictions of the concentrations and biofilm thicknesses were found to
provide a reasonably good fit to the experimental data for the two different flow rates and
initial concentrations. The effect of the choice of parameters where there was some
uncertainty was generally found to be insignificant. A better comparison may have been
possible had a more precise elapsed time to test termination been published. Neglect of
biofilm growth in the influent tubing to the reactors by the model may cause some
disagreement between measured and predicted results. In summary, the model was found
to provide a reasonable simulation of the acetate removal and biofilm growth for a short
duration expenment using a single substrate for which inactive film build-up may be
neglected.
5.3 APPLICATION TO SYNTHETIC LEACHATE COLUMN EXPERIMENTS
5.3.1 Introduction
Column experiments conducted using glass beads and synthetic leachate provide
the next level of complexity against which the proposed model can be tested. Like the
Rittmann and McCarty experiments (discussed in the previous section), the synthetic
leachate column tests (Millward, 1997, vanGulck, pers. cornrn.) simplify modelling
because the media (glass beads) are well represented by the sphere rnodel. However in
this case the leachate is more complex involving two primary substrates (propionic and
acetic acid) and significant concentrations of C a and so that mineral precipitation
may be expected.
As compared to actual raw leachate, synthetic leachate provides an idluent with a
relatively constant value of COD, pH, Eh, and VFA ratio and substantiaily less particulate
matter (which may provide additional clogging due to particles attaching to the biofilm).
In the absence of experimentally derived kinetic constants for the leachates at the test
temperatures, the applicability of the mode1 will be studied by estimating the kinetic
constants fiorn reported values indicated in literature.
5.3.2 Experiments
The leachate reactor (column) tests were performed using columns constructed
from 5.08 cm intemal diameter, 76.0 cm long PVC pipe. Influent and effluent valves were
attached to the columns at 2.0 and 66.5 cm from the base respectively. The columns were
designed to allow gas escape and piezometric readings at various elevations. The colurnns
were packed from the influent valve to effluent valve with 0.6 cm diameter glass beads.
The experiment was nin in triplicate using 3 identical columns.
The composition of the prepared synthetic leachate was based on samples taken
from Keele Valley Landfill between June and August 1993. The composition of the
mixture is described in Table 5.9 (page 163) fiom Millward (1997). Synthetic leachate
provides the substrate for bacteria, but, unlike actud leachate, does not contain a
significant population of bacteria. Thus it was necessq to seed the column with a
bacterial population at the oiitset of the experiment. This inoculation process was
performed by initially operating the columns using a mixture of KVL and synthetic
leachate, and then gradually increasing the proportion of synthetic leachate until the
mixture was 1 00% synthetic after approximately 12 days.
The column tests were inoculated and operated in an upflow direction with a target
continuous flow rate of 1 .O L/d ; controlled by a peristalic pump. The average fluid and
column temperature (infiuent and effluent) was 22 * 1 O C . Influent and effluent samples
were regularly taken for water quality measurements which included COD, calcium
hardness, biological activity reaction test ( B A R F ) , temperature, pH and Eh. Drained
porosity was measured at various times. At the tirne of modelling? these column
experiments had not been terminated and therefore autopsies had not been performed.
5.3.3 Modelling the System
The synthetic leachate contains acetic, propionic and butyric acids as organic
substrates in a known ratio of 75: 1 (intended concentrations of 7000 mg& 5000 mg/L
and 1000 mg/L respectively). Since the portions of acetic and propionic acids were
similar, and significantly greater than the concentration of butyric acid, it was assumed that
no one substrate would dominate in the system. For this reason, the system was modelled
using the two substrate option, with acetic acid as one substrate, and the mixture of
propionic acid and butyric acid to act as the second substrate (simulated as propionic
acid). This mode1 simulates the breakdown of propionic acid to acetic acid.
5.3.4 Measured Input Parameters
5.3.4.1 Introduction
The basic, fixed mode1 pararneters are Iisted in Table 5.5. Details regarding the
measurement of these pararneters are given below.
5.3 -4.2 Fluid Temperature, Reactor Length, Diameter, Particle Size, Initial Porosity
The average temperature of the fluid was measured fiom the influent and effluent
temperatures taken prior to each water quality measurement. The reactor length is the
length dong which attachent of biofilm occurs. This length was taken as the length from
the Muent port to effluent port (approximately 65 cm). The reactor diameter was the
average measured inside diameter of the colurnn, and the particle sire was the measured
average diameter of the clean glass beads. The initiai clean bead porosity was measured
before testing using multiple measurements of the volume of water drained (the "drained"
porosity).
5.3.4.3 Flow Rate and Influent Concentration
For modelling purposes the flow rate was taken to be the average of the flows
obtained by regularly measuring the accumulated effluent volume and dividing by the
elapsed time. The influent concentration was taken as the average measured COD
concentration.
5.3.4.4 Calcium Carbonate Yield Coefficient, Y,
As discussed in Section 4.4.5.3, a calcium carbonate yield coefficient may be
calculated from the COD rernoval and CaCO, precipitated for a particular set of
conditions. The infiuent and effluent COD concentrations and infiuent and efluent
calcium hardness were measured regularly. This data was used to calculate the COD
removed and CaCO, precipitated for each sampling time. The COD removed was plotted
against the CaCO, precipitated as shown in Figure 5.14. The slope of the straight line
passing through the experimental data and the ongin was to be the yield coefficient, Y,. A
single yield coefficient was derived fiom the data fiom the three sirnilar experiments.
5.3.4.5 Percent Acetic Acid
The average concentrations of acetic, propionic and butyric acids in the synthetic
leachate (approx. 7000 mg acetic 1 L ; 5000 mg propionic 1 L ; 1000 mg butyric / L )
were converted to concentration in terms of COD per litre (approx. acetic: 7399
mgCODL; propionic: 7565 mgCoDa; butyic: 18 18 mgCODL). The total calculated
COD concentration fi-orn these sources was approximately 16,782 mgCOD/L, which
compares well with the measured average infiuent concentration of 17,070 mgCODL.
The portion of the COD fiom the acetic acid source was caiculated to be approximately
44%.
Table 5.5 : Directly Measured Input Parameters
-
Pararneter Value Units
Temperature 23i1 OC
Reactor length 65 cm
Colurnn diameter 5.08 cm
Bead diameter 0.6 cm
Porosity 0.382 - Flow (avg.) 1.12 L / day
[niluent concentration (avg.) 17070 mgCOD / L
Calcium carbonate yield rate, Y, 0.2 1 mgCaCo, / mgCOD
Percent acetic 44 YO
5.3.5 Indirectly Measured Input Parameters
5 -3.5.1 Refiactory Concentration
The refractory concentration, or non-degradable influent concentration, was
measured from batch tests to be no greater than approximately 500 mgCOD/L for actual
landfill leachate. To account for the significantly lower expected mass of non-degradable
organics in synthetic leachate, the value was estimated as 200 mgCODL
5.3 -5.2 Density of the Active and Inactive films, qct, and Xci
The active density is required to calculate the thickness of the active film, as shown
in Equation 4.17 (Section 4.4.5.2) and the inactive density is required to calculate the
thickness of the inactive film, as shown in Equation 4.20 (Section 4.4.5.3). The active
density is the mass of volatile solids (volatile solids are presumed to be ceIl matter) per
cubic centimetre of active film. The inactive density is the mass of non-volatile solids per
cubic centimetre of inactive film. The data required to calculate the film densities along
the length of each column can be measured during the autopsy performed immediately
after the terrnination of each test.
Since at the tirne of this anaiysis the synthetic leachate columns had not been
terminated and autopsied, the autopsy results of a sirnilar column experirnent run using
actual landfiii leachate was used (Armstrong, pers. comm.). The "sirnilar" column
experiment was performed in duplicate at approximately the same temperature, flow rate
and organic loading. The autopsy process determines, at regularly spaced heights along
the column, the mass of water, non-volatile solids and volatile solids per bead using
modified Hach DiU2000 Spectrophotometer Handbook procedures (Hach, 1993). Also
determined, by specific gravity tests, are the bulk density of the film, the dry density of the
film material (volatile and non-volatile), and the dry density of the non-volatile solids (eg.
ash). From this data, film thichesses and densities may be estimated. To the time of
writing, the autopsy procedure could not isolate the properties of the active film fiom the
inactive film. Thus, for the purposes of density measurement, the structure of the film was
idealized as two films: an active film consisting of ail of the volatile solids and water, and
an inactive film consisting of the non-volatile solids.
Idealizing the films in this marner aliows the active biofilrn thickness and density to
be calculated using published methods, such as those of Rittmann et al. (1 986) as
described in the following. Using this method the biofiim is assumed to be approximately
99% water by weight, the active film thickness Lt, cm be calculated using
where W, is the mass of evaporated water per bead, p is the density of water at 2 1 OC, and
A is the surface area of one bead. The active film density, was calculated using
where B, is the mass of volatile solids per bead.
The active film density used for modelling purposes was taken as the average
active f lm density in the two KVL leachate columns autopsied, where the active film
density in a column was calculated as the average of three samples taken From the top (or
outlet) half of the column, from 30cm to 65 cm from column base. This value was used
because it was hypothesized that the top portion of the column contained samples which
better represented the active biofilm density. Near the inlet the decreased water content
(indicating a decrease in active film) and increased volatile content means it is less likely
that the volatile solids are only contained in the active film, and therefore less accurately fit
the idealized active film. The lower active film densities measured at the top of the
column also better represent the active densities during the duration of the test. The
measured active density input parameter is given in Table 5.6.
Since it was assumed that al1 the non-volatile solids were located in the inactive
film, the inactive film density was measured by performing a specific gravity test on the
oven-dried ash remaining after the volatization of the film material. The measured inactive
density input parameter is given in Table 5.6.
Table 5.6 : Indirectly Measured input Parameten
Parameter Value Units
Refractory concentration, S, 200 mgCOD 1 L
Active film density, Xf, 70 mgVS 1 cm3
Inactive film density, xi 2700 mgTS 1 cm3
5.3.6 Assumed Mode1 Parameters
5.3.6.1 Introduction
The assumed model parameters are listed in Table 5.7. These parameters were
determined f?om research of published data and application to the particular system
modelled. The parameters are discussed in greater detail next.
5.3.6.2 Permeability
Due to the high active biofilm density measured, and the high precipitate
accumulation (especially at later times), the active biomass is considered impermeable.
5.3 -6.3 Kinetic Constants Ks, Y, and b
The kinetic constants Ks, 4, Y, and b were not directly measured. A summary of
reported values for kinetic constants corresponding to anaerobic degradation of acetic and
propionic acid by a variety bacteria is given in Table 5.10 (page 164). These values are
also plotted in Figures 5.15, 5.16, S. 17 and 5.18. Cornparison of kinetic values is made
difficult due to the variability in the environmentai and operational conditions (eg. pH,
organic loading), the mode of operation (eg. batch vs. continuous) and the lack of
advanced instrumentation in older studies (Pavlostathis and Giraldo-Gornez, 199 1). In the
absence of experimentally derived kinetic constants specific to the system modelled,
average kinetic values will be utilized, but only to show the applicability of the model
within reasonable ranges of kinetic constants.
The substrate type and temperature will be used to select kinetic constant values
for input to the model. Lawrence and McCarty (1969) found that fiom conversion of
volatile fatty acids the maximum specific substrate utilization rate, q, and half-velocity
coefficient, Ks, varied with temperature while the yield coefficient, Y, and decay
coefficient, b. were relatively unaffected. Lin et al. (1 987) found that for a VFA mixture,
there was a slight decrease of Y with temperature. Figure 5.15 illustrates the decrease in
the half-velocity coefficient with increasing temperature. The increase in 4 with increasing
temperature reported by Lawrence and McCarty ( 1969) is not clear in Figure 5.16. The
yield rate and decay coefficient show little evidence of variance with temperature in
Figures 5.17 and 5.18.
Since the half-velocity coefficient Ks has an established temperature variance,
separate acetic and propionic coefficients were estirnated for 22 O fluid temperatures from
the reported measured values. These values are listed in Table 5.7. The maximum
specific substrate utilization rate 4 was varied to fit the predicted data to the measured
data, providing 4 remained within estimated, reasonable temperature dependent limits,
assumed to be 1.5 to 8 mgCOD/mgVSS-d at 22'. The yield and decay coefficients, Y and
b, were fixed values calculated using the averages of the population of reported values for
each coefficient (see Table 5.7).
5 -3 -6.4 Coefficients of Difision
The coefficients of diasion for the substrates in fiee solution and within the
biofilm were obtained fi-om literature. The intemal difision coefficient, D, and diffusion
ratio DF,, , published by Yu and Pinder (1993) for acetate and propionate were used to
establish the free difision coefficient for each substrate. The interna1 difision
coefficients arid fiee solution dif'fiision coefficients for the substrates modelled are Iisted in
Table 5.7. In the absence of temperature dependence factors for diffusion coefficients, the
values, measured at 35 OC, were applied without modification. Note that these values
correspond to DdD, ratios of 4 1 % and 3 1%, which , as discussed previously in Section
5.2.5.4, are significantly lower than the ratio of 80% suggested by Williamson and
McCarty (1 976).
Table 5.7 : Assumed Model Parameters
Pararnet er Value Units
Penneability of biofilm
Ks (Propionic)
Y (Propionic)
K, (Acetic)
Y (Acetic)
b (Substrate mixture)
Do (Propionic)
D, (Propionic)
Do (Acetic)
Df ( Acetic)
Detachment model
Impermeable
2800
0.042
1700
0.03 8
0.066
1.27
0.52
1.5
0.47
Growth
- mgCOD / L
mgVS I mgCOD
mgCOD / L
mgVS / mgCOD
d- ' cm2 / d
cm2 / d
cm2 / d
cm2 / d
-
5.3.7 Variable Model Parameters
5.3.7.1 Introduction
A number of numerical parameters wiil be varied to optimize the accuracy of the
model fit while the value of 4 will also be varied over a reasonable range t o examine its
effect on model fit. Table 5.8 lists the initial estimate of these parameters and the value
used in the final analysis.
5.3.7.2 Timesteps and Segments
The number of timesteps, length of timesteps, and number of segments were
established using the same approach as described in Section 5.2.4.1 for the Rittmann and
McCarty data. The optimization will be discussed in Section 5.3 -6.
5.3 -7.3 Maximum Specific Substrate Utilization Rate, 4
The initial estimates of the maximum specific substrate utilization rates were based
on temperature dependence, as discussed in Section 5.3 -6.3.
Table 5.8 : Variable Mode1 Parameters
Parameter Initially Optimum or Units Estimated Best Fit
Value Value
Number of timesteps 300 1500
Length of timestep 1 0.2 day
Number of segments 13 13
4 (Acetic) 4 3.9 mpCOD 1 rngVS-d
5.3.8 Predicted Results Based on Initial Parameter Estimates
The predicted and measured variation in the normalized effluent COD
concentration (ie effluent COD 1 influent COD) with time were compared as shown in
Figure 5.19. It can be seen in Figure 5.19 that the measured effluent concentrations of the
3 column tests were nearly identical. The results show an initial penod of approximately
100 days of negligible removal (a "lag" phase), followed by an approximately 60 day drop
in effluent concentration and then a relatively steady period of between 0.45 to 0.65
normalized COD. In addition, the predicted and measured drained porosity along the
length of the column afler 220 and 270 days of operation were compared as shown in
Figure 5.20. Figure 5.20 (a) and (b) show that the porosity at the column inlet (near the
bottom of the column) decreased more rapidly than at the outlet (near the top).
In non-linear modelling of the form being conducted here the numerical parameters
(timestep and segment length) can oniy be optimized if the physical parameter altered
provide a reasonable approximation to the measured normalized COD concentrations.
The fit shown in Figure 5.19 is reasonable for the first 120 days, but gives a poor
prediction between 120 and 270 days. To aid in understanding why this situation occurred
the maximum specific substrate utilization rate, 4, was varied. Based on experience
modelling aceticlpropionic acid mixtures, it was known that the initial lag period is
strongly dependent on the acetic acid kinetics, while the steady state portion was strongly
dependent on the propionic acid kinetics. Since the initial lag penod was well predicted by
the model, but the COD removal of the measured steady-state portion was overestimated
by the model (see Figure 5.19), it was expected that the maximum specific substrate
utilization rate of the acetic acid was reasonable but that the value for propionic acid was
too high. To examine this the value for acetic acid was held constant while that for
propionic acid was decreased fiorn 4 to 3 mgVS/mgCOD-d. The revised predictions are
shown in Figure 5.2 1 and Figure 5.22. Since the predicted effluent concentration now fit
the measured values reasonably well, the number of timesteps and segments could be
optimized.
5.3.9 Defining the Required Number of Timesteps and Segments
5.3 -9.1 Timesteps
As discussed previously in Section 5.2.4.1, the optimum timestep length must be
found such that funher decreases in timestep length gave an insignificant change in the
results. The input parameters used to predict the curves plotted in Figures 5.21 and 5.22
were used with timestep Iengths of 1 .O, 0.2 and 0.04 days to produce the results shown in
Figures 5.23 and 5.24. The reduction in the timestep from 1 -0 to 0.2 days reduced the
predicted lag time before the steep concentration drop, but has no effect on the predicted
effluent concentration during the lag time or steady state periods. Reducing the timestep
From 0.2 to 0.04 days gave an insignificant change in predicted effluent concentration.
Similarly, the predicted porosity decreased due to a decrease in timestep from 1.0 to 0.2
days, but did not change significantly when the timestep is reduced from 0.2 to 0.04 days.
The predicted decrease in porosity is greatest at the inlet end of the column at the earlier
time of 220 days. It was concluded that 0.2 day tirnesteps provide reasonable predictions
of effluent concentration and porosity.
5.3.9.2 Number of Segments
The previous model runs were perfonned assurning the column could be modelled
using 13 segments, each 5 cm in length. As discussed in Section 5.2.4.2, model accuracy
is improved by minimiung the volume of the segments. In order test the sensitivity of the
predicted results to the number of segments an analysis was performed using 26 segments
each 2.5 cm in length with a timestep of 0.2 days. The predicted effluent concentration
and porosities at 220 and 270 days are show in Figures 5.25 and 5.26 and it can be seen
that there was no significant difference in effluent concentration or porosity. It was
concluded that a timestep of 0.2 days and a segment length of 5.0 cm (13 segments) were
reasonable for the purpose of predicting the behaviour of the columns.
5.3.10 Results and Discussion
5.3.10.1 Prediction of Measured Data
Inspection of the measured and predicted porosities s h o w in Figure 5.26 indicates
that using the current input parameters, porosities at the inlet are underestimated at 220
days, but otherwise the predicted values provide a reasonable fit to the data at both 220
and 270 days. To improve the predicted porosities, the maximum specific substrate
utilization rate of the acetic acid portion of the mixture was decreased fiom 4 to 3 -9
mgVS/mgCOD-d. Figures 5.27 and 5.28 show the predicted and rneasured effluent
concentrations as they Vary with time and porosities at 220 and 270 days using the new
utilization rate. There was an insignificant effect on effluent concentration and a small
increase in porosity. The predicted effluent concentrations (Figure 5.27) fit very well to
the measured data. The predicted porosities (in Figure 5.28) fit reasonably well to both
the measured porosities at both 220 days and 270 days.
At earlier times, in particular 130 days, the porosity is not as well predicted (Figure
5.29a). The average measured drainable porosity ranges between approximately 0.35 at
the efnuent port to 0.30 at the influent port while the mode1 predicts a porosity of about
0.36 which is relatively constant along the length of the column.
5.3.10.2 Predicted Chronological Ecology of the Synthetic Leachate Column System
O - 50 davs : Acetate and propionate degrader lae phase
In the period up to 50 days there was negligible growth of the biofilm due to
propionate and acetate degraders as s h o w in Figures 5.30 and 5.3 1 and inactive film as
show in Figure 5.32. This period of negligible growth is known as a "lag" phase. This
resulted in negligible decreases in propionic and acetic COD through the column (see
Figures 5.33 and 5.34).
50 - 100 davs : Propimate lu acetate degrader ~ o w t h b a
In this period the lag phase of the propionate degraders continued and there was
nedigible inactive film accumulation as show in Figures 5.30 and 5.32. Acetate
degraders began to grow as shown in Figure 5.3 1, giving a relatively constant biofilm
thickness throughout the length of the column (see Figure 5.35b). This uniform growth of
acetate degraders resulted in a linear decline in acetate COD concentration along the
column, and a slight decrease in effluent acetate COD concentration as shown in Figures
5.33 and 5.34b.
100 - 150 davs : Prooionate d e d e r Iw acetate degrader erowth
The lag phase of the propionate degraders continued and little inactive film
accumulated (see Figures 5.30 and 5.32). The growth rate of acetate degraders increased
substantiaily along the column (see Figure 5.3 I), but remauied relatively unifonn, with
only slightly more growth at the inlet end of the column (see Figure 5.35b). A large
decline in acetate COD along the column could be seen in Figure 5.34b, nearly constant
removal occurring along the length of the column. The result was a steep decline in
effluent acetate COD, and a deciine in etnuent total COD, as shown in Figure 5.33.
150 - 300 d a ~ s - Propionate degrader growth. acetate demader growth and decline
(a) 150 - 225 days - Decline of acetate degraders near outlet, rapid growth of
acetate degraders at inlet
Over the period of 150 to 225 days a decline in acetate degraders occurred
between 20 and 65 cm fiom the rniddle of the co1umn to the outlet end, as shown
in Figure 5.3 1, and Figure 5.35b by cornparison of 150 and 200 day profiles. The
decline was likely due to acetate deficiency caused by the great accumulation of
acetate degraders near the inlet dunng this period, as shown in Figures 5.3 1 and
5.35b. The acetate deficiency near the outlet is indicated by the relatively constant
acetate COD concentration (zero removal) at 200 days above 30 cm in Figure
5.34b.
@) 175 - 300 days - Propionate degrader growth phase near outlet
At approximately 175 days the growth phase of the propionate degraders
began as shown in Figure 5 -30. It can be seen in Figures 5.35a and 5.36 that the
growth occurred mostly in the latter 20 - 65 cm of the column. The restriction of
propionate degraders fiom near the inlet was likely due to Ioss rate control, as the
high rate of growth of acetate degraders near the inlet (discussed in part (a),
above) caused great detachment of both acetate and propionate degraders, as
illustrated in Figure 5.37. The propionate growth phase results in a decrease in
propionate COD dong the column, as shown in Figure 5.34% and in effluent
propionate COD in Figure 5.33. The degradation of the propionic acid resulted in
production of acetic acid (see Figure 5.38) where the total COD concentration in
the effluent remained relatively constant in a range of about 9000 to 9500
mgCOD/L (see Figure 5.33).
(c) 250 - 300 days - Second phase of acetate degrader growth at outlet
A second penod of growth of acetate degraders began near the outlet as
shown in Figure 5.3 1 and Figure 5.35b after about 250 days. This new growth is
caused by the acetate production by the late growth of propionate degraders near
the outlet while most of the acetate From the influent has been utilized before
reaching this part of the column (see Figures 5.34b and 5.38).
(d) Film accumulation
Inactive film growth occurred at a nearly linear rate (see Figure 5.32) and
with most accumulating near the inlet (see Figures 5.32, 5.39 and 5.40). Inactive
film growth is dependent on the flux of substrate into the active film, not on the
thickness of the active film, therefore the linear rate of the inactive film growth
(see Figure 5.40) indicates an increasing flux, even afler the acetate degraders
have plateaued (in segment 1 in Figure 5.3 1, for example).
The variation in total film and active film thickness with time are s h o w in
Figures 5.4 1 and 5.42 respectively. Near the influent port there is rapid active film
growth from 125 to 200 days and slower growth d e r that. In contrast the total
film thickness increased (from 125 days) almost linearly with time until clogging
occurred. The rate of total film growth is much slower elsewhere indicating that
there is significant biofilm growth occumng near the influent pon.
5.3.10.3 Biomass Detachment
In developing the model it was assumed that suspended biomass in the influent
could be neglected, and that the re-attachent of detached biomass could be neglected.
The first assumption was tested by measuring the biomass content of the influent, which
resulted in negligible biomass since the influent was synthetic. The second assumption is
difficult to test, since while detached biomass in the effluent may be measured, the arnount
which re-attached within the column is unknown. Figures 5.43 and 5.44 show the
predicted rate of ce11 detachrnent in mass of volatile solids per day for the separate
degraders in the system (note the scale change). Since detachrnent and film thickness are
dependent on the substrate flux, the predicted trends of these two properties are nearly
identical, as indicated by cornparison of Figures 5.43 and 5 -44 to Figures 5.30 and 5.3 1.
Upon review of these figures it was concluded that fbrther research may be required on
this subject as it seemed that detached biomass becomes sigruficant at times of clogging,
and may re-attach downstream.
5.3.11 Conclusions for the Synthetic Leachate Experiments
The model provided quite good predictions of effluent COD and column porosity
for the synthetic leachate experiments examined. For this analysis, the film densities
measured from similar autopsied column expenments, difision coefficients from
Literature, and kinetic coefficients taking into account substrate type and system
temperature, provided adequate input parameters. From cornparison of Figures 5.19 to
5.22 it was observed that the mode1 may be sensitive to some input parameters (in this
case <î). The parameters adopted al1 lie within the typical range based on published values.
Dunng penods of rapid growth, it may be difficult to predict porosity as shown in Figure
5.29(a) and 5.29(b).
The model provides a logical history of the ecologj of the system incorporating
bacterial growth and minerai accumulation, and the interaction between two bacterial
populations. A more detailed synthetic leachate column test study in which the separate
COD concentrations of the two substrates were measured, effluent biomass was rneasured
and autopsies were performed would allow many of the predicted phenornenon to be
compared to that observed in the laboratory.
Table 5.9 Composition of Synthetic Keele Valley Leachate (MilIward,1997)
COMPONENT CONCENTRATION (mg/L)
Acetic (Ethanoic) Acid 7000
Propionic (Pmpanoic) Acid 5000
Butyric (Butanoic) Acid 1000
NaNO, 50
NaHCO, 3012
CaCI 2882
M&12 x 6H20 3114
MgSO4 156
NtI,HC03 243 9
co(NH32 695
Na$ x 9H,O (Eh adjusiment & reducing medium preparation)
Trace metal solution (TSM) 1 ml per 1L of synthetic leachate
Trace Metal Solution
FeSO, 2000
H3BQ 50
ZNSO, x 7H,O 50
CuSO, x 5H,O JO
MnSO, x 7H,O 500
96% concentrated H,SO, (An&) 1 ml
Table 5.10 Summary of reported Monod kinetic constants for anaerobic degradation of propionic and acetic acids by various mixed cultures '
Reference Process ' T Ks Y 9 b " C mgCOD/L mgVSS/ mgCOD/ d-'
mgCOD me;VSS-d Propionic A cid O'Rotuke, J.T.. 1968
Lawrence and McCarty, 1969 Lawrence and McCarty, 1969 Chang et al., 1983 t Gujer and Zehnder, 1983 t Heyes, RH., and Hall,RJ., 1983
Heyes, RH., and Hal1,RJ.. 1983
Whitmore et al., 1985 t Siegrist et al., 1993 t
Acetic A cid
ORourke, J.T.. 1968
Lawrence and McCarty, 1969 Lawrence and McCarty, 1969 Lawrence and McCarty, 1969 Kugelrnan and Chin, 197 1 t Cappenberg, T. E., 1975
van den Berg, 1977
van den Berg, 1977
van den Berg, 1977
Massey and Pohland, 1978 Smith, M.R and Mah, RA., 1978 Zehnder, A.J.B. et al., 1980 Wandrey,C. and Aivasidis,A., 1983 Chang et al., 1983 t Noike,T., et al., 1985 Noike,T., et ai., 1985
Noike,T,, et ai.. 1985
Siegrist et al., 1993 pt al 1994h t W S f f 1813
' Compiled from summaries by Pavlostathis and Giraldo-Gomez (199 1) and Vavilin and Lokshina (1996).
t referenoes listed by Vavilin and Lokshina (1996). al1 others listed by Pavlostathis and Giraldo-Gomez (1 99 1).
C = chemostat data, B = batch data, N = , C/S = simulation mode1 used with chemostat data. average value.
* 4 calcuiated fiom =
Predicted, dt = 1 .O d .,....... Predicted, dt = 0.2 d --. Predicted, dt = 0.04 d -..- Predicted, dt = 0.008 d
20
Elapsed Time (days) Figure 5.2 : Effect of timestep length dt on predicted effluent Acetate concentration for test BC3 using mean parameters.
1 Segment 2 Segments
-- 4 Segments - - 6 Segments - 12 Segments - - 24 Segments
20
Elapsed Tirne (days)
Figure 5.3 : Effect of number of segments on predicted effluent Acetate concentration for test BC1 using mean parameters.
Measured - Predicted at 21 days .... . Predicted at 28 days
4 6 8
Length Along Column (cm)
Figure 5.5 : Measured Acetate concentration profile along the length of the column at steady-state and initial predictions using mean parameters for test BC3.
I 1
Measured - Fitted prediction at 21 days - ..... Fitted prediction at 28 days
L
I
I
4 6 8
Length Along Column (cm)
Figure 5.6 : Measured Acetate concentration profile along the length of the column at steady-state and fitted predictions using Ks = 4.48 mglL and b = 0.254 d-' for test BCI.
Measured - F itted prediction at 21 days ..... Fitted prediction at 28 days
Length Along Column (cm) Figure 5.7 : Measured Acetate concentration profile along the length of the column at steady-state and fitted predictions using Ks = 4.48 mglL and b = 0.254 d-' for test BC3.
Length Along Column (cm) Figure 5.9 : Measured biofilm thickness profile along the length of the colurnn at steady-state and fitted predictions using Ks = 4.48 mglL and b = 0.254 d" for test BC3.
Measured - Biofilm permeable . . . . . Biofi lm impermeable
I
Length Along Column (cm) Figure 5.10 : Measured Acetate concentration profile along the length of the column at steady-state and effect of biofilm permeabitity on fitted predictions using K, = 4.48 mglL and b = 0.254 d" at 28 days for test BC3.
Measured - LI variable (L, = 0.0129 cm initially)
. . L, fixed (LI = 0.0158 cm)
4 6 8
Length Along Column (cm) Figure 5.11 : Measured Acetate concentration profile along the length of the column at steady-state and effect of diffusion layer thickness LI on fitted predictions using K, = 4.48 mglL and b = 0.254 d" at 28 days for test BC3.
" Il Il -
f e o
Measured - Shear detachment ..... Growth detachment
Length Along Column (cm) Figure 5.13 : Measured Acetate concentration profile along the length of the column at steady-state and effect of detachment method on fitted predictions using Ks = 4.48 mglL and b = 0.254 d-' at 28 days for test BC3.
26 28 30
Temperature (OC) Figure 5.15 : Reported measured half-velocity coefficient, KSI versus system temperature for bacterial growth in propionate and acetate (various sources, see Table 5.10).
P Propionate A Acetate
26 28 30
Temperature (OC) Figure 5.17 : Reported measured yield coefficient, Y, versus system temperature for bacterial growth in propionate and acetate (various sources, see Table 5.10).
! 1 Column SOI L 0 Column S02 r Column S03 - Predicted
150
Time (days) Figure 5.19 : Measured and initially predicted normalized effluent concentration using q = 4 mgCODlmgVS-d for propionic acid and q = 4 rngCOD/mgVS-d for acetic acid for column tests fed synthetic leachate.
I
Column S01 - 0 Column S02 v Column S03 - Predicted
150
Time (days) Figure 5.21 : Measured norrnalized effluent concentration and model prediction using q = 3 mgCODImgVS-d for propionic acid and q = 4 mgCOD1mgVS-d for acetic acid for column tests fed synthetic leachate.
Column SOI 0 Colurnn S02 r Column S03 - Predicted, 13 Segments
-- Predicted, 26 Segments
O 50 100 150 200 250
Time (days) Figure 5.25 : Measured normalized effluent concentration and the effect of the number of segments on mode using q = 3 mgCODlmgVS-d for propionic acid and q = 4 mgCODlmgVS-d for acetic acid for column tests fed leachate.
I predict ions synthetic
H Column SOI 0 Column S02 t v Column S03 1 1 - Predicted (Fitted) 1
Elapsed tim = 220 days "Pr
0. O O. 1 O. 2 0.3 0.4
Drained Porosity
Column SOI v O O Column S02 O r a r Column S03 1 - Predicted (Fitted) 1 1
O. 0 O. 1 O. 2 O. 3 0.4
Drained Porosity Figure 5.28 : Measured porosities and model predictions using q = 3 mgCODlmgVS-d for propionic acid and q = 3.9 mgCODlmgVS-d for acetic acid after 220 and 270 days of operation of columns fed synthetic leachate.
Propionate ~eiraders
- Ht = 2.5 cm (Seg. 1) - Ht = 12.5 cm (Seg. 3) -- Ht = 22.5 cm (Seg. 5) -- Ht = 32.5 cm (Seg. 7) . . . . Ht = 42.5 cm (Seg. 9) - - Ht = 52.5 cm (Seg. 11) - .- Ht = 62.5 cm (Seg. 13)
150
Time (days)
Figure 5.30 : Variance of predicted propionate degrader film thickness at specified column heights (Ht) over tirne.
Acetate ~ e ~ r a j e r s 1
Ht = 2.5 cm (Seg. 1) - Ht = 12.5 cm (Seg. 3) -- Ht = 22.5 cm (Seg. 5) -- Ht = 32.5 cm (Seg. 7) . . Ht = 42.5 cm (Seg. 9) - - Ht = 52.5 cm (Seg. 11) - . - Ht = 62.5 cm (Seg. 13)
150
Time (days)
Figure 5.31 : Variance of predicted acetate degrader film thickness at specified column heights (Ht) over time.
150
Time (days)
Detachment Cc efficient, b'
Figure 5.37 : Variance of predicted detachment coefficient b' at specified column heights (Ht) over time.
-
Ht = 2.5 cm (Seg. 1) - - Ht = 12.5 cm (Seg. 3)
-- Ht = 22.5 cm (Seg. 5) -- Ht = 32.5 cm (Seg. 7) . . Hl = 42.5 cm (Seg. 9)
- - Ht = 52.5 cm (Seg. 11)
I - - Ht = 62.5 cm (Seg. 13) I I /
Propionic COD . . . . * Acetic COD Total COD
O 4000 8000 12000 16000
COD Concentration (mgCOD/L)
O 4000 8000 12000 16000
COD Concentration (mgCODIL)
Figure 5.38 : Predicted propionic, acetic and total COD profiles along the column at (a) 220 days and (b) 270 days.
I - Ht = 2.5 cm (Seg. 1) Total - -- Ht = 2.5 cm (Seg. 1) Inactive -.. Ht = 62.5 cm (Seg. 13) Total
- . . . . . . . Ht = 62.5 cm (Seg. 13) Inactive
150
Time (days) Figure 5.40 : Variance of predicted total and inactive film thickness at the influent (Seg. 1 ) and effluent (Seg. 1 3) ends of the column over time.
Total Film -
Ht = 2.5 cm (Seg. 1) - Ht = 12.5 cm (Seg. 3)
- -- Ht = 22.5 cm (Seg. 5) -- Ht = 32.5 cm (Seg. 7)
. , Ht = 42.5 cm (Seg. 9) - - - Ht = 52.5 cm (Seg. il) - . - Ht = 62.5 cm (Seg. 13)
*
---- - .- . 7
150
Time (days) Figure 5.41 : Variance of predicted total film thickness at specified column heights (Ht) over time.
Total Active Filh
Ht = 2.5 cm (Seg. 1) - Ht = 12.5 cm (Seg. 3) -- Ht = 22.5 cm (Seg. 5) -- Ht 32.5 cm (Seg. 7)
, . . Ht = 42.5 cm (Seg. 9) - . - Ht = 52.5 cm (Seg. Il) - . - Ht = 62.5 cm (Seg. 13)
150
Time (days)
Figure 5.42 : Variance of predicted total active film thickness at specified column heights (Ht) over time.
Acetate Degraders
-- Ht = 2.5 cm (Seg. 1) - Ht = 12.5 cm (Seg. 3) -- Ht = 22.5 cm (Seg. 5) - -- t i t = 32.5 cm (Seg. 7) . . . Ht = 42.5 cm (Seg. 9) - - - Ht = 52.5 cm (Seg. 11) - . . - Ht = 62.5 cm (Seg. 13)
/'
/ / -----
150
Time (days)
Figure 5.44 : Variance of predicted detachment rates for acetate degraders at specified colurnn heights (Ht) over time.
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1 INTRODUCTION
This chapter provides a sumrnary and the conclusions of the work presented in this
thesis, and provides recornmendations for further model development.
6.2 SUMMARY AND CONCLUSIONS
The key processes involved in the clogging in othenvise well maintained Iandfill
leachate collection systems may be simulated under controlled laboratory conditions using
synthetic leachate and columns packed with porous media. A model was designed to be
applicable to the saturated laboratory experiments as a first step towards modelling the
clogging in a complete leachate collection systems. The model focused on the prediction
of clogging due to biological growth and biologically driven mineral precipitation since
with current design and maintenance practices most other causes of clogging can be
controlled in the L.C. S. or laboratory experiment .
As indicated in Chapter 2, current research has s h o w that under favourable
conditions rapid growth of bactena in the form of biofilms, and mineral deposits formed in
association with bacterial activity occur on the granular media of the L.C. S., and that this
phenornenon may be simulated using laboratory colurnn expenments. An idealized film
structure consisting of an outer active layer of biomass and an inner inactive layer of inert
biomass and biochernical precipitate was developed.
In Chapter 3 a geometric model developed by Taylor et al. (1990) was modified to
ailow accurate calculation of porosity and specific surface area for granular media coated
with a known thickness of film. The mode1 assumes that the granular media cm be
represented by a regularly packed ideal spheres of equal diameter. Under these conditions
only four stable packing arrangements exist, each with a unique clean sphere porosity.
The fûnctions used for calculation of porosity and specific surface area are interpolated
f?om the four known functions given the initiai clean stone porosity and current film
thickness.
The model is described in Chapter 4. The fiow system is discretized into
subregions where each subregion, or element, is assumed to act as a separate, k e d film
reactor. Within each element organic substrate is converted by bacteria into additional ce11
mass, with the remaining substrate being the infiuent to the next element. The column test
consists of a single series of saturated elements with a single influent source at one end,
and a single efnuent port at the other. The system initially contains zero substrate and a
specified uniform layer of inoculum biofiim. An innuent flow rate and substrate
concentration is applied to the first segment of the system. An advective transport
algonthrn using a time stepping process is applied to mode1 the movement of substrate
dong the length of the colurnn.
B i o f h growth and loss due to a single substrate was modelled using an algorithm
modified fiom Rittmann and McCarty (1981) and Rittmann and Brumer (1984). An
additional substrate was modelied by repeating these algorithms for a second substrate
with its own growth parameters and concentration, and allowing each substrate to
contnbute to a separate active and inactive film thickness. The growth and loss algorithm
is combined with equations to model the change in porosity and specific surface area. mass
balance equations, and detachment equations. Due to non-linearity, two coupled iterative
procedures were required to derive the effluent COD concentration, film thickness,
porosity, and specific surface area at for an element at a tirnestep. These calculations were
repeated for each element, fiom the influent to efnuent end of the column, and each series
of element calculations was repeated each timestep.
In Chapter 5 two applications of the model are addressed. In the first case the
model is applied to the published data of Rittmann and McCarty (1 980). In this
expenment a single substrate (acetate) was used, and kinetic parameters were derived in
independent experiments. Using the same kinetic parameters for the two sets of data for
experiments at dEerent flow rates and influent concentrations provided a reasonable fit to
both measured acetate concentrations and biofilrn thicknesses dong the length of the
column. Due to the chernistry of the infiuent and the short duration of the experirnent, it
was not necessary for the model to simulate the effects of inactive film growth, clogging,
or an additiond substrate.
In the second case the model was applied to a colurnn fed synthetic landfill
leachate performed by Millward (1997), Armstrong (pers. comm.) and J. vanGulck (pers.
comm.). The experiments were nin to a high degree of clogging, were expected to
contain considerable mineral precipitation, and the kinetic coefficients had to be obtained
from published reports. Two substrates (acetic and propionic acid) were used to represent
the synthetic Ieachate. Using estimated kinetic parameters the model predicted effluent
COD concentrations and porosity profiles dong the length of the columns which fit
reasonably well to experimental data. The model was found to be sensitive to the kinetic
parameters used to fit the data, the maximum specific substrate utilization rate 4.
Additional research is required to derive kinetic parameters for landfill leachate, synthetic
leachate, propionic acid and acetic acid under the temperatures and loadings expected
within the L.C. S.. The model provides a Iogical history of the ecology of the system
incorporating bacterial growth and mineral accumulation, and the interaction between two
bacterial populations.
6.3 RECOMMENDATIONS
6.3.1 Introduction
The clogging of leachate colIection systems is a serious concern and any
improvements which can be made to increase the accuracy of predictions should be
pursued. This thesis described a model for simulation of saturated colurnn experiments
conducted using synthetic Ieachate with negligible suspended soiid content and was
intended as the first step towards developing a complete model for leachate collection
systems. Further work, therefore, should be focused on both predicting the clogging in
laboratory column expenrnents conducted with real leachate (which contains bacterial
loading and suspended solids), and on extending the model towards the clogging of
granular drainage layers. The recommendat ions, t herefore, are broken into groups:
Drainage Layer Model Changes, for extending the model to represent leachate collection
systems, General Model Changes, for better prediction of clogging of any porous system,
and Additional Research suggests aspects of the current topic requiring improvement.
6.3.2 Drainage Layer Model Changes
The mode1 could be extended to simulate the clogging of the leachate collection
system by :
(a) modelling one drainage path from the top of the dope to the pipe using a senes
of segments, with each segment containhg multiple elements,
@) distributing the flow so that the upper unsaturated element in each segment
receives flow, and after treatment delivers it to horizontal flow elements, with this
flow from the segment entering the horizontal flow element of the downstrearn
segment where it is mixed with the downward flow from this segment, and so on,
(c) aliowing the upper, unsaturated portion of the drainage layer to be represented
by elements within which biofilm growth and loss is given special consideration
and flow is downward,
(d) allowing the number of saturated elements dong the length of the siope to be
chosen so that the height of saturation corresponds to the arnount of flow through
the zone. The use of variable heights of saturation will require further research.
(e) upon clogging of a saturated element in this scenario, flow could be stopped
within the element, and diverted over it by adding saturated elements within the
unsaturated zone
6.3.3 General Model Changes
6.3 -3.1 Spatial Variation of Particular Parameters
The model could be altered to allow elements to be given specific initial porosity
and particle (or stone) diameter, as opposed to a single porosity and particle diameter for
ail elements. This alteration could be applied to either a column or drainage layer flow
field. This would ailow the modelling of clogging of systems of varying stone sizes, such
as clear stone underlain by sand, a design used at KVL. Such a change would also require
changes to the present advection algo rithm.
Specific to the drainage layer flow field, the downward influent flow rate and
substrate concentration could Vary from segment to segment. This would allow modeliing
of a drainage layer which is, for example, covered with an unequal height of waste, thus
producing unequal flow rates and substrate concentrations dong the length of the slope.
6.3.3.2 Temporal Variation of Particular Parameters
Typicaily, measured influent flow rates, substrate concentrations and temperatures
from experiments or field studies do not remain constant. For these situations the model
could be altered to allow entry of data files containing the variance of the parameter with
time, or the choice of a time dependent fiinction which describes the variance. Penods of
low or high flow or concentration cm have a great impact on the system, and this impact
is not represented when a fixed average value is used. Allowing the influent flow rates or
substrate concentrations to change with tirne will require changes to the present advection
algorithm.
6.3 -3.3 Bactend Transport
The model could simulate bacterial transport and attachment within the system.
Presently bacterial transport is neglected, but should be modelled if significant biomass is
measured in the influent to the system, or significant biomass is detaching fiom the porous
media.
6.3.3.4 Advection Algorithm
The current advection algorithm could be changed so that the substrate delay is not
precalculated, but instead recalculated based on changes in the porosity of the elements.
Many of the other model changes listed would require changes to the advection algorithm.
6.3.3.5 Representation of Additional Systems
The model could be expanded to represent systems with different geometry, flow
directions or attachrnent media. For example, by sirnpiifjmg the model the clogging of a
leachate collection pipe transporting leachate could be simulated, and with additional
modification the clogging of the pipe perforations could be modelled. The addition of
other attachrnent media such as geotextiles in drainage layers or metal screens in column
tests could be represented by solving the geometry of the film accumulation.
6.3.4 Further Research
6.3.4.1 Colurnn Expenments
The verification of the modelling of acetic and propionic acid could be improved
using a column test similar to that of Millward and vanGuluck, but in which only acetic
and propionic acids are used (ie. no butyric acid), the COD of each acid should be
measured instead of only the total COD, and concentrations could be measured along the
length of the column over tirne. The autopsy of such a test should involve differentiation
of the bacterial populations to verify the predicted spatial distribution of the propionate
degraders and acetate degraders if possible.
6.3.4.2 Kinetic Constants
Considerable research is required in the area of the growth kinetics of landfill
leachate and landfill leachate constituents such as acetic and propionic acids. To date the
study of growth kinetics has been for use in wastewater treatment at high temperatures,
and therefore little is known of growth kinetics in the temperature range of leachate
collection systems (approx. O to 20 O C ) . The model may be sensitive to growth kinetics,
even within reasonable ranges as provided fiom literature. An expenmental method
should be developed for determination of these input parameters, and the dependence of
the kinetic constants to temperature and substrate loading should be determined.
6.3.4.3 Field Studies
Verification of the model with regards to leachate collection system clogging in
actual field conditions would require many parameters which are currently difficult to
ohtain for a specific case (such as infiuent concentrations and flow rates). An exhumation
of a clogged L.C.S. rnay yield critical information. Modelling a more controlled system
representing a L.C.S., such the mesocosm studies of Rowe et al. (1995a) is also
suggested.
6.3 -4.4 Idealization of Films
The idealization of the film requires further study. In particular, the density and
distribution of cells and mineral deposits on the porous media over time grown frorn amal
and synthetic leachate should be studied. The relationship between these hdings and the
utilization of substrate and detachment of film matter rnay provide an improved simulation
of the processes at work.
6.3 -4.5 Additional Parameter Uncertainties
The following parameters require fbrther research: (a) the permeability of the
biofilm, (b) the moisture content and dependencies in an unsaturated zone if a drainage
layer model is developed, and (c) the effective difision layer thickness for large media
under low flow conditions such as those of a L.C.S..
6.4 OVERALL CONCLUSION
A model for prediction of biologically induced clogging in column expenments fed
synthetic leachate has been developed as a first step toward developing a model for
predicting clogging in landfill leachate collection systerns. Full verification of the model
will require further research however the cornparison of experimental and predicted
behaviour presented in this thesis provides encouragement that the portion of the mode1
that deals with saturated flow as presented herein represents a substantial step fonvard.
DETERMINATION OF CRITICAL FILM THICKWESSES OF THE POROSITY AND SPECIFIC SURFACE MODELS
A1.1 Introduction
Al . 1 . 1 Critical Film Thicknesses
Critical film thicknesses are thicknesses at which filling of the porespace has
occurred, or the geometrical basis of the porosity or specific surfhce mode1 must change.
The cntical film thicknesses occur when (a) spherical caps first overlap, (b) spherical caps
overlap at a diEerent region at a greater thickness, called a spherical overlap, (c) overlaps
overlap, and (d) at the thickness at which pore occlusion occurs. As film thickness
increases on an initially clean sphere, overlap will occur earliest in the face (or plane) of
the packing arrangement with the most contact points, since as the number of contact
points in the face increases, the closer the spherical caps are to each other and the thimer
the biofilm thickness required for volumes to overlap. The face with the most contact
points is the critical face. The critical face for the cubic packing arrangement is the square
face while the critical face for the other three packing arrangements is the rhombic face
(see Figure 3.1 and imagine the packing arrangement of each face repeating around a
single sphere to sum the contact points for each face).
Al . 1.2 The Contact Interface Radius, C R
To simplify the calculation of the critical film thicknesses, the radius of the contact
interface should be predetermined. The contact interface is the division between a
sphere's film and a neighbouring sphere's f lm coinciding with the base of the spherical
cap, in the shape of a circle centred at the contact point between the spheres. Figure
Al. 1 shows a cross section of the spherical cap and the contact interface, DB. The C R
varies only with sphere size and film thickness, therefore the equation is consistent for al1
packing arrangements. The CIR is determined using :
- C o n t a c t interface R a d i u s = C I R = AB
AB = J - Z F
C I R = d(2 + - ( + j 2
CIR = JL;'-+ d F L c
where points 4 B, and C are points on Figure A l . I .
AL2 Cubic Packing Arrangement
A1.2.1 First Overlap, Cubic Packing Arrangement
The film thickness at which the first overlap of spherical caps occurs in the cubic
packing arrangement may be determined geometrically, Figure A1.2 depicts the spheres,
film, and film interface at the onset of spherical cap overlap. Using this figure it can be
seen that for a particular sphere, the onset of cap overlap occurs when the sphencal caps
meet at the intersection of the lines tangent to two neighbouring contact points. The
critical film thickness, in tems of 2Ld4 , can be determined as the thickness at which the
contact interface radius reaches the radius of the sphere, thus:
C I R = Sphere radi us
L L c = fl - 1 P o s i t i v e r o o t : - d,
The normalized film thickness at which spherical caps overlap, and errors in the original
equations begin for the cubic packing arrangement, is 0.4142. The number of overlaps
requinng correction is 12.
A 1.2.2 Pore Occlusion, Cubic Packing
The film thickness at which the pore space becomes completely filled with film
may also be determined geornetrically. The onset of this state, also known as pore
occlz~sion, is s h o w in Figure A1.3. Section SS' in Figure A1.3 depicts the contact
interfaces of the eight spheres at this film thickness as diagonally hatched circles. On
inspection of Section SS ' it can be determined that pore occlusion occurs when the
contact interfaces meet at the centre of the square created by joining the contact points of
the eight spheres. The critical film thickness can be found by equating the contact
interface radius to the distance to the centre of square formed by the four contact points:
CIR = Distance t o centre of
contact point square
L l i t =JS - 1 P o s i t i v e m o t :
Thus at a normalized film thickness of 0.732, pore occlusion occurs and hence the
porosity and specific surface go to zero. No other critical thicknesses occur between the
In overlap and pore occlusion, although it can be proven that the overlapped volumes
overlap with one another (tip of horizontal overlap with tip of vertical overlap) at exactly
the sarne thickness as pore occlusion occurs.
AL3 First Overlap, Non-cubic Packing Arrangements
Each of the three remaining packing arrangements, orthorhombic, tetragonal-
sphenoidal, and rhombohedral, have a rhombic face. In the following discussion and
figures for each of these packing arrangements the rhombic face is considered to be the
horizontal face, show in the Plan view of each figure. As discussed in section Al . 1.1 it is
in this face that the first overlap of spherical caps occurs for these packing arrangements.
Figure A1 -4 (Plan) shows the rhombic face of the orthorhombic packing
arrangement at the onset of spherical cap overlap and Figure A1.5 plan) shows the same
spheres after overlap has occurred. The doubled hatched area in Figure A 1.7 (Plan)
indicates the cross-section of the overlapped volume. By comparison of Figure A1 -4 with
Figure A1.2 of the square face of the cubic arrangement, and with the understanding that
the spheres are of equal diarneter and are accurately illustrateci, it can be observed that the
film thickness at which overlap occurs in the rhombic face is significantly less than that of
the square face. By comparing the Plan views of Figures A 1 -3 and A 1.5 it may also be
observed that the cross-section of the overlapped space (the doubled hatched area) is a
diferent shape due to the difference in the geometry of the face. This difference is
discussed in Chapter 3.
Using Figure A1.4 (Plan) it c m be determined that for a particular sphere in a
packing arrangement containing a rhombic face the onset of the first sphencal cap overlap
occurs when the caps meet at the intersection of the lines tangent to two neighbouring
contact points. This intersection occurs at the centre of the equilateral triangle fonned by
joining the centres of the three spheres in the face. Since the spherical caps overlap when
the contact interface radii meet at this centre, the critical film thickness is determined by
equating :
CIR =
2 L t - Posi t i ve r o o t : - -
d,
Distance from contact p o i n t
to t r i a n g l e centre
Therefore the nomalized film thickness at which the first overlap of sphencal caps occurs
for the non-cubic packing arrangements is 0.1547.
A1.4 Orthorhombic Parking Arrangement
A1 -4.1 Second Overlap, Orthorhombic Packing
Figure A1 -4 illustrates the rhombic face (Plan) and one square face (Section) of the
orthorhombic packing arrangement at the onset of overlap in the rhombic face. The six
spherical caps around the circumference of the sphere in the rhombic face will overlap (2
are shown in the Plan view), while the two remaining sphericai caps at the top and boaom
are not at the onset of overlap. One of the caps is depicted in the Section. Figure A 1.5
(Section) illustrates the same sections at the onset of overlap in the square face. This
event is considered the second overlap for this packing arrangement since it occurs
between spherical caps of different contact points than the first overlap. Since the
geometry of this face is exactly the same as that of the 3 square faces of the cubic
arrangement, it cm be proven that this cntical film thickness occurs at the same film
thickness as proven in Section Al .2.l. The nomalized cntical film thickness, therefore,
at which the second overlap occurs in the orthorhombic packing arrangement is 0.4142.
12 overlaps must be replaced.
A 1.4.2 Pore Occlusion, Orthorhombic Packing
It was detemined that the onset of pore occlusion is best depicted by a section
perpendicular to the rhombic face, tangent to the spheres of the rhombic face at the
contact points of the spheres as illustrated in Section SS ' of Figure A 1.6. Grap hically,
pore occlusion in this section occurs when the film thickness of the two spheres on the left
intersect with the contact interfaces of the spheres on the nght. Assurning this intersection
occurs at d42 above the centre of the spheres in the Plan, the critical thickness is that at
which the sum of the lengths from points A to B and B to C equal the total distance from
points A to C. From the Plan view, the length AC can be found to be 6 / 2 d, , the length
AB is the contact interface radius, and the length BC can be calculated from the known
sides of the right triangle BCD, where BD equals CR. The film thickness is found by
equating :
d2
After manipulation, L-' + d p L c - = O 3
Pos i t ive root: - -
Thus the normalized film thickness at which pore occlusion of the orthorhombic packing
arrangement occurs is 0.5275.
A1.5 TetragonaCSphenoidal Packing Arrangement
A 1.5.1 First Overlaps, Tetragonal-Sphenoidal Packing
The first film overlaps in the tetragonal-sphenoidal packing arrangement occur at
the sarne critical film thickness as the first film overlaps of the orthorhombic packing
arrangement since both arrangements contain a rhombic face as its critical face. The first
overlaps occur at a normalized film thickness of 0.1547. The tetragonal-sphenoidd
arrangement includes 2 rhombic faces, each contributing 6 overlaps for a total of 12
overlaps.
A 1.5.2 Second Overlap, Tetragonai-Sphenoidal Packing
Figure A1.7 illustrates the tetragonai-sphenoidai packing arrangement at the film
thickness at which a second overlap occurs. The second overlap is the result of the
growth of film on the non-contacting spheres between the closest corners of the special
rhombic face, depicted in the top left and bottom right spheres in the section shown in
Figure A1 -7. The section in Figure A1 -8 shows the shape of the overlap after fiirther
growth as a cap-shaped hatched area. The overlap is a volume similar in shape to the
spherical caps at the contact points, but originating at a tangent plane which is not in
contact with the sphere, but at the midway point between the spheres. It is therefore
called a special sphericai cap. Using the section shown in Figure AIS, it can be seen that
this second overlap begins when the film thickness passes this rnidway distance. The
critical film thickness at which this second film overlap occurs may be determined,
therefore, by equating the sphere and film radius with the rnidway point of the short
diagonal, see Figure A 1.8 (Section). :
R a d i u s of sphere + f i l m = Midway p o i n t of d i a g o n a l
The normalked thickness at which the second overlap of the tetragonal-sphenoidal
packing arrangement occurs is 0.2247. Four of these special overlaps occur for each unit
cell.
A 1.5 -3 Complex Overlaps, Tetragonal-Sphenoidal Packing
As the radius of the special spherical cap increases with film growth it eventually
rneets and overlaps with the regular spherical caps. The onset of this event is depicted in
the section in Figure A1 -8. The volumes of the complex shapes created by the overlap of
the special and regular sphetical caps could not be determined by simple methods. The
film thickness at which they occur is required so that the remaining porosities and specific
surfaces of larger film thicknesses can be simplified. In Figure A1.8 (Section) the length
AB represents the radius of the sphere plus film thickness, the length BC represents the
radius of the base of the special sphencal cap (sirnilar to C R the contact interface radius
of the regular spherical caps) and lengths AC and CD are geometrically known lengths for
this packing arrangement. The radius of the specid spherical cap can be determined using:
Radius of special
spherical cap, =
The complex overlaps occur when the radius of the sphere plus film thickness of the
bottom left sphere intersects with the radius of the special spherical cap, BC, on Iine AC.
The critical thickness is the film thickness at which the sum of the radius of the special
spherical cap and the radius of the sphere plus film equals the known length AC. The
verification is as follows:
( R a d i u s of sphere + f i l m ) +
Radius of s p e c i a l cap = Half l e n g t h of d i a g o n a l
2L t After m a n i p u l a t i o n , -
dP
Thus at a normalized film thickness of 0.2649 the tetragonal-sphenoidal packing
arrangement becomes too cornplex to mode1 exactly. Frorn this thickness to pore
occlusion the porosity and specific surface will be assurned to vary linearly with film
thickness.
A 1.5.4 Pore Occlusion, Tetragonal-Sphenoidal Packing
In order to find the film thickness at which pore occlusion occurs a cross-section
of the packing arrangement clearly showing the largest pore space is given in Section BB
of Figure A1.9. Note that Section BB is of Section AA in Figure A1 -9, and Section AA
is indicated in the Plan in Figure A1 .a. The shaded circle in the centre of Section BB is
the base of the speciai spherical caps of spheres on opposite sides of the cross-section (ie.
Imagine a sphere out of plane, sitting in the 'hopper' created by the 4 spheres shown).
Due to the symrnetry of the arrangement, the spheres responsible for the special spherical
caps out of plane are the sarne distance apart as those non-contacting pairs on the left and
right in the section shown. Due to this fact, the filling in of the pore space dong the
horizontal centre line, AB fiom the centre, A to radius B. is equal to that ftom the centre
of the spherical caps on either side, filling from C to B. The onset of pore occlusion can
be determined by solving for the film thickness at which the sum of the radius of the centre
sphencal cap base, AB, and the radius of the side spherical cap, BC, where AB equals BC,
equals half the diameter of the sphere.
2 x S p e c i a l sphere cap r a d i u s = Sphere r a d i u s
P o s i t i v e r o o t :
Pore occlusion of the tetragonal-sphenoidal packing arrangement occurs at a nonnalized
film thickness of 0.3229.
A1.6 Rhombohedral Packing Arrangement
A 1.6.1 First Overlap, Rhombohedral Packing
The cntical face of the rhombohedral packing arrangement is the rhombic face, and
therefore, similar to the orthorhombic and tetragonal-sphenoidai arrangements the cntical
thickness at which the first film overlap occurs is 0.1547. Since the rhombohedral
packing arrangement has 3 rhombic faces, al1 contact interfaces have 4 overlaps each, each
beginning at the critical thickness. A total of 24 overlap volumes must be accounted for.
A 1 -6.2 Complex Overlaps, Ethornbohedral Packing
As film thickness increases in the rhombohedral packing arrangement a critical
thickness is met at which multiple overlaps of various shapes, including overlaps of
overlaps, occurs. The film thickness at which these complex overlaps occur is the critical
thickness at which exact porosity and specific surface calculation will terminate, and a
linear relationship to pore occlusion will begin. By inspection of Figure Al . 10, Section
SS ', the occurrence of a complex overlap is illustrated as the sphencal cap on the bottom
left sphere (hatch filled) intersects with the vertical plane through the centre of the sphere
at point A. At this thickness the remaining 2 spherical caps at the top of this sphere (out
of section) also converge at the 'peak', point A. An increase in film thickness will cause
overlapping and create complex volumes to be corrected for. The overlap of the contact
interface circle at the top of the section with the film of the bottom left sphere is a cross-
section of the intersection of the outsf-section sphencal cap overlaps. The radius of the
sphencal cap at which the intersection occurs cm be geometrically denved as illustrated in
the section in Figure Al. 10. The cntical film thickness may be found by equating:
Jz Spherical cap r a d i u s , C I R = ? d p
P o s i t i v e r o o t : F
The normalized critical film thickness at which complex overlaps occur begin is 0.2247.
A 1 -6.3 Pore Occlusion, Rhombohedral Packing
In order to calculate the critical film thickness at which pore occlusion occurred
the thickness of the film at points along the line joining the contact points in section SS
were studied. It was assurned that pore occlusion occurred at the film thickness at which
the radius of the spherical cap intersected with the film thickness between the non-
contacting spheres in the section. Since it was assumed that the critical thickness rnay
cause an overlap between the non-contacting spheres, the length fiom the centre along the
line joining the contact points was caiculated as the radius of a special spherical cap
between the non-contacting spheres:
R a d i u s of s p e c i a l
spheri cal cap, == ,/-
It was assumed that pore occlusion, therefore, happened when dong the line joining the
contact points D and E, the addition of the radius of the regular sphencal cap, EB to the
radius of the special sphericd cap, EB, equalled the known length of the line joining the
contact point to the midpoint. Thus:
CIR + Radius of special cap = Midway point of diagonal
2 4 Positive Root: - = fi - 1 dP
The nomalized cnticai film thickness at which pore occlusion occurs, therefore, is 0.4142.
By inspection of Figure Al. 1 1 and mathematical verification, it is noted that the critical
film thickness results in a film intersection coinciding with the intersection of the lines
joining the contact points and the spheres, and therefore there is no overlap of the film
between the non-contacting spheres, and the special cap radius is zero at the critical film
thickness.
A1.7 Critical Film Thickness Summary and Interpretation
Table Al. 1 surnmarizes the critical fiim thicknesses cdculated in this appendix. It
can be seen fkom Table Al. 1 that at thicknesses greater than 2L/d, = 0.4142 for cubic and
2 L j 4 = 0.1547 for the other packing arrangements, the first overlap occurs and the
calculation of the volume and surface area of a sphere covered with film becomes
underestirnated using the base volume and surface area calculation methods of Taylor et
al. (1990). This causes the porosity to be overestimated and the specific surface
underestimated at these thicknesses. These points of first theoretical breakdown and thus
beginning of inaccurate results are indicated by a syrnbol for each packing arrangement for
each curve in Figures 3.1 and 3.2. Each additional cntical film thickness corresponds to a
change in the equation required to calculate the volume or surface area until the cntical
film thickness at which pore occlusion occurs is reached. These equations are derived in
Chapter 3.
Table AL1 - Summary of Critical Film Thicknesses in Terms of TL, / d,
Packing Arrangement
Name
Cubic
Orthorhombic
Tetragonai- Sphenoidal
Rhombohedral
Nurnber of Contact Points
(1)
6
8
10
12
First Cap Overlaps
(2)
0.4 142
O. 1547
O. 1547
O. 1547
Second Cap Overlaps
(3 )
0.4 142
0.2247
0.2247
Complex Overlaps
(4)
0.732
0,5275
0.2649
0.2247
Complete Pore Occlusion
( 5 )
0.732
0.5275
0.3228
0.4 142
Figure A l .1 : Verification of length of radius of interface circle at any contact point.
SECTION
-- ~~~~~~
Figure A1.2 : Verifkation of film thickness at onset of spherical cap overlap for cubic packing.
Figure A1.3 : Verification of film thickness at cornplete occlusion of cubic packing. Double hatched a r a indicates overlap of spherical caps.
orthorhombic packhg. Hatched area indicates spherical cap at h o contacts.
orthorhombic packing. Double hatched area indicates overlap of spherical caps,
Figure A l .6 : Verifkation of film thickness at pore space occlusion for orfhorhombic packing .
overlap for tefragonal-sphenoida packing.
Figure A l .8 : Verification of film thickness at onset of complex overkps for tetragonal-spheroidal packing. Special spherical cap shown in centre of section.
sphenoidal packing.
SECTION S S'
Figure A l . 10 : Verification of film thickness at onset of complex overlaps for rhombohedral packing.
Figure A l . 1 1 : Verification of film fhickness at onset of pore occlusion for rhombohedral packing.
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