mathematical model of anaerobic processes applied to the anaerobic sequencing batch reactor

8/18/2019 Mathematical Model of Anaerobic Processes Applied to the Anaerobic Sequencing Batch Reactor

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MATHEMATICAL MODEL OF ANAEROBIC PROCESSESAPPLIED TO THE ANAEROBIC SEQUENCING BATCH REACTOR

by

Yale Yunsheng Zheng

A thesis submitted in conform ity w ith the requirements

for the degree o f Doctor o f Philosophy

Graduate Department o f C iv il Engineering

University o f Toronto

©Copyright by Yale Yunsheng Zheng 2003

roduced with permission of the copyright owner. Further reproduction prohibited without permission.



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Mathematical Model o f Anaerobic Processes Applied To the Anaerobic

Sequencing Batch Reactor

Doctor of Philosophy, 2003

Yale Yunsheng ZhengGraduate Department o f Civil Engineering

University of Toronto

Abstract

A mathematical model o f anaerobic processes described in this thesis contains two

integrated sub-models: a biolog ical model describing the anaerobic degradation o f complex

substrates and a gravitational settling model for solids-liquid separation in wastewater treatment

processes.

Major developments in this biolog ical model are the incorporation o f new hydrogen

product regulation functions o f glucose degradation and new hydrogen inh ibition functions o f

propionate and butyrate degradation. The new hydrogen product regulation functions are derived

from the rate equation o f ordered single-displacement enzymatic reactions having two substrates.

The new hydrogen inh ibition functions are developed from the thermodynamic basis o f

propionate and butyrate degradations. The model was applied to simulate different

configurations o f anaerobic processes operated under different conditions and provided good

agreement w ith literature data. One o f the contributions o f this model is that it provides good

predictions o f the microbial populations o f different metabolic groups, which are not w ell

predicted by other mod els in the literature.

A gravitational settling model was developed for predicting the solids concentration

profile in the zone-settling and compression regimes o f a gravity thickener. In this model, the

effective solids pressure is a function o f solids concentration and the rate o f change o f solids

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concentration. The model was applied to differen t situations o f gravity settling processes and

provided good agreement with literature data. The advances o f this model are its predictive

ab ility fo r vertical solids pro files for both zone-settling and compression regimes, and its use o f

model parameters that are independent o f operating conditions.

The integrated model was applied to evaluate anaerobic sequencing batch reactors

(AnSBRs). The simulation results indicate that, in general, the influent strength and the reactor

mixed liquid volatile suspended solids (MLVSS) concentration have a positive effect on the

maximum organic loading rate o f AnSBRs, while the fill/cy c le time ratio has a negative effect on

the maximum organic loading rate. The optimum fill volume/total volume ratio depends on

influent strength, MLVSS, and fill/cycle time ratio.

ii i

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Acknowledgements

I would like to thank my supervisor Prof. David M. Bagley, who provided excellent

supervision, constructive criticisms, and unreserved patience with my progress. I would also like

to thank my other committee members Prof. Barry J. Adams, Prof. Elizabeth A. Edwards, and

Prof. Brent Sleep, whose guidance, assistance, and review are invaluable.

I would also like to thank John Shizas and Jerry Lalman, who were always w illin g to help

w ith this project.

Financial assistance was provided by the Natural Sciences and Engineering Research

Council o f Canada (NSERC) in the form o f post-graduate scholarships, the Ontario M inis try o f

Economic Development, Trade, and Tourism, the Ontario M inis try o f Energy, Science, and

Technology, the Centre for Research in Earth and Space Technology, an Ontario Centre o f

Excellence, and the Unive rsity o f Toronto.

Lastly, I wou ld like to thank my fam ily (my w ife, Arikun Zhao; my son, Jeffrey Zheng;

and my mother-in-law, R uijin Wu) fo r bearing with me.

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Table of Contents

Abstract ii

Acknowledgements iv

Table o f contents v

L ist o f tables ix

List o f figures x i

List o f symbols xv

L ist o f abbreviations xx i

1 Introduction 1

1.1 Anaerobic wastewater treatment 1

1.2 Mathematical modeling o f anaerobic processes 2

1.3 Objectives 5

1.4 Publications 6

2 Literature review 7

2.1 Mathematical models for anaerobic processes 7

2.2 Experimental investigations on anaerobic treatment 11

2.2.1 Performance o f differen t reactor configurations 11

2.2.2 Granulation 12

2.2.3 Effect o f low temperature 14

2.3 Gravity settling 16

3 Bio logica l model development 18

3.1 M otivation for new developments 18

3.2 Hydrogen inhibition for propionate and butyrate degradation 21

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3.3 Hydrogen partial pressure product regulation 30

3.3.1 Regulation functions for glucose degradation 30

3.3.1.1 Lactic acid production 3 0

3.3.1.2 Butyric acid production and acetic acid production 34

3.3.2 Regulation functions for lactic acid degradation 36

3.4 Computation 39

3.4.1 Implem entation for CSTRs and AnSBRs 39

3.4.2 Numerical method 41

4 Model ve rifica tion 45

4.1 Model verifica tion w ith data from Bagley and Brodkorb (1999) 45

4.1.1 Description o f the experimental data 45

4.1.2 Model calibration 47

4.1.3 Model valida tion 54

4.2 Model verification w ith data from K im (2000) 58

4.2.1 Description o f the experimental data 58

4.2.2 Model simulation 59

4.3 Model verifica tion w ith data from Denac (1988) 63

5 Rate lim iting step o f anaerobic treatment 67

5.1 Rate lim itin g step o f anaerobic processes 68

5.2 M icrob ial population distribution 78

5.3 Discussion 83

5.4 Conclusions 84

6 S imulation o f the startup o f anaerobic reactors 85

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6.1 Introduction 85

6.2 Simulation o f the startup o f UASB reactors 86

6.3 Simulation o f the startup o f AnSBRs 90

6.4 Discussion 92

7 Modeling o f the gravitationa l settling process 95

7.1 Development o f the gravitationa l settling model 96

7.1.1 Governing equation for gravitational settling 96

7.1.2 Effective solids pressure, Ps 97

7.1.3 Em pirical functions for K/ and K 2 101

7.2 Batch settling 103

7.2.1 Governing equation for batch settling process 103

7.2.2 Num erical formulation 104

7.2.3 Simulation o f interface height versus time 106

7.2.4 Simulation o f dynamic concentration profiles 112

7.2.5 Sensitivity o f compression parameters 116

7.3 Steady state secondary clarifie r 117

7.3.1 Governing equation for steady state secondary clarifie r 117

7.3.2 Steady state solution 118

7.3.3 Validation o f steady state solution 121

7.4 Ve rtical solids profile in UASB reactors 127

8 Evaluating the AnSBR w ith simulation 129

8.1 Organic loading rate 129

8.2 Design and operational parameters o f AnSBRs 130

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List of Tables

Table 3.1: Hydrogen partial pressure product regulation constants 3 7

Table 3.2: Kine tic constants used in the model 43

Table 3.3: Symbols for soluble and particulate components (Bagley and Brodkorb, 1999) 44

Table 4.1: Operating conditions (Bagley and Brodkorb, 1999) 46

Table 4.2: Predicted maximum substrate utilization rate(g CO D/L/d) 54

Table 4.3: Operating conditions (K im, 2000) 58

Table 4.4: Input biomass composition for each run 63

Table 4.5: Steady state operational and performance parameters 65

Table 4.6: Predicted microb ial population distributions 65

Table 5.1: Lim itation on ind ividua l substrate degradation 73

Table 5.2: Sensitivity analysis o f model parameters on estimated maximum SOLR 76

Table 5.3: Comparison o f anaerobic sludge activity (g substrate/g VSS/d) 80

Table 5.4: Comparison o f reactor performances 80

Table 5.5: Predicted m icrob ial population distributions 81

Table 6.1: Predicted seed sludge com position 86

Table 6.2: OLR and HR T (from Tay and Yan, 1996) 87

Table 6.3: Startup time simulation conditions and results 89

Table 6.4: Simulation conditions 90

Table 6.5: Startup time requirement 91

Table 7.1: Model parameters for various suspensions 107

Table 7.2: Operating conditions and model parameters for the experiment by George and Keinath

(1978) 122

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Table 7.3: Operating conditions and model parameters for the experiment by Pflanz (1969) 124

Table 8.1: Box-Behnken response surface design 132

Table 8.2: Factor values for flocculen t sludge 133

Table 8.3: Factor values for granulated sludge 134

Table 8.4: Simulation results o f maximum OLR for flocculent sludge 137

Table 8.5: Parameter estimates and effect test fo r flocculent sludge 138

Table 8.6: Simulation results o f maximum OLR for granulated sludge 145

Table 8.7: Parameter estimates and effect test fo r granulated sludge 146

Table 8.8: S imulation conditions 150

Table 8.9: M icrob ial population distributions 151

Table A l: Stoichiometric coefficients (v,y) for soluble components (Bagley and Brodkorb, 1999)

176

Table A2: S toichiometric coefficients ( ia ) for particulate components (Bagley and Brodkorb,

1999) 177

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Figure 4.14: Volatile fatty acid concentrations vs. time for Run 4. 62

Figure 4.15: Vola tile fatty acid concentrations vs. time for Run 5. 62

Figure 4.16: Comparison between the simulated and experimental results for the accumulation o f

organic acids (Denac et al., 1988; Costello et al., 1991b). 66

Figure 5.1: Lim itation on sludge loading rate imposed by ind ividua l constituents. 74

Figure 5.2: Hydrogen partial pressure versus SOLR. 77

Figure 6.1: Comparison o f specific methanogenic activ ity during reactor startup (Data from Tay

and Yan, 1996). 88

Figure 6.2: Predicted effluent COD during startup. 89

Figure 6.3: SM A versus time for the startup o f an AnSBR. 92

Figure 7.1: Force balance over incremental volume o f suspension [Adapted from Fitch (1979)].

97

Figure 7.2: Response o f solid-water m atrix o f thickness i f under effective solids pressure, Ps.

99

Figure 7.3: Zone settling velocity for aluminum hydroxide floes [Calculated from Bhargava and

Rajagopal (1990) results]. 110

Figure 7.4: Interface height versus time for aluminum hydroxide floes. I l l

Figure 7.5: Zone settling velocity for bentonite [Calculated from Bhargava and Rajagopal (1990)

results]. I l l

Figure 7.6: Interface height versus time for bentonite. 112

Figure 7.7: Zone settling velocity for desanded fraction from a gold ore pulp (Scott, 1968).

114

Figure 7.8: Interface height versus time fo r desanded fraction from agold ore pulp. 115

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Figure 7.9: Height-concentration profiles for desanded fraction from a gold ore pulp at different

settling times. 115

Figure 7 . 1 0 : Sensitivity analysis on and ri 2 based on the simulation in Figure 7 . 8 . 1 1 6

Figure 7 . 1 1 : Solids concentrationprofiles for a calcium carbonate suspension. 123

Figure 7 . 1 2 : Solids concentration profile for an activated sludge suspension. 1 2 4

Figure 7 . 1 3 : Solids concentration profile for a high ly loaded activated sludge suspension. 1 2 6

Figure 7 . 1 4 : Solids concentration profile for a UASB reactor. 1 28

Figure 8 . 1 : Contour plot o f maximum O L R for flocculent sludge (S jn low; t y /y low). 1 3 9

Figure 8 . 2 : Contour plot o f maximum O L R for flocculent sludge (S jn low; t j j t r medium)

1 4 0

Figure 8 . 3 : Contour plot o f maximum O L R for flocculent sludge (S jn low; t j j t r high) 1 4 0

Figure 8 . 4 : Contour plot o f maximum O L R for flocculent sludge (S jn medium; t y Jtr low)

141

Figure 8.5 : Contour plot o f maximum O L R for flocculent sludge (S jn medium; fy /f r medium)

141

Figure 8.6: Contour plot o f maximum O L R for flocculent sludge (S jn medium; t j / t r high)

1 4 2

Figure 8 . 7 : Contour plot o f maximum O L R for flocculent sludge ( S i „ high; t j j t r low) 1 4 2

Figure 8.8: Contour plot o f maximum O L R for flocculent sludge (S m high; t j j t r medium)

14 3

Figure 8 . 9 : Contour plot o f maximum O L R for flocculent sludge (S in high; t j l t r high) 14 3

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148

149

149

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Figure 8.10: Individua l VFAs versus time ( t f f tr - 0.5/22)

Figure 8.11: Individua l VFAs versus time ( t f j t r =12.5/10)

Figure 8.12: Individua l VFAs versus time ( /y j t r = 20.5/2)



List of Symbols

A = cross-sectional area o f the reactor (m2)

A, B = substrate concentrations (mol/L )

A, B = integration constants

Bv = volumetric organic loading rate (g CO D/L/d)

Bx= specific organic loading rate (g COD/g COD VSS/d)

b - constant

bj = decay constant o f microorganism group X, (d_1)

b, = overall decay constant (dH)

C = solids concentration (kg/m3)

Co = in itia l solids concentration (kg/m3)

Ch C2= constants

Cc = c ritica l concentration (kg/m3)

Ce= effluent solids concentration (kg/m3)

Cu = underflow solids concentration (kg/m3)

A = degradation rate (g COD/L/d)

E = enzyme

ES = enzyme substrate complex

f = fraction o f total COD that is consumed as the constituent o f interest

g = acceleration due to gravity (m/s2)

H = in itial thickness o f a thin solids m atrix (m)

j = vertical spatial step

K = equilibrium constant

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K\ = reciprocal o f the hydraulic conductivity o f a solids matrix ( ----— )m

K 2 = ability o f a solid m atrix to squeeze out liqu id under pressure ( Pa - h )

Kj = inh ibition constant

KiM 2 xp, Kimxb = hydrogen inhibition parameters (atm)

Km= ha lf velocity constant (m ol/L)

Krfh, Km.- l, K rla = product regulation constants (atm)

Kg . = half-velocity constants (mol/L)

k\ = coefficient used in the exponential interface settling velocity model (m/h)

k l, k3 = forward rate constants

leg Ii 2 = coefficien t used in the effective solids pressure model ( —g - )

m h

k2, k4 = reverse rate constants

k-RFB,K fl, kpaA, = product regulation constants

L = height o f supernatant-suspension interface (m)

Lo = in itial height o f suspension (m)

Lj = volumetric substrate loading rate (g COD/L/d)

L, = total organic loading rate (g COD/L/d)

M = modulus o f elasticity (Pa)

NAD = concentration o f NAD+ (mol/L)

NA DH = concentration o f NA DH (mo l/L)

n = time step

n 1 = coefficient used in the exponential interface settling velocity model (m /kg)

-3« 2 = coefficient used in the effective solids pressure model (m /kg)

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P = product

P = dynamic fluid pressure (Pa)

Ps= effective solids pressure (Pa)

P,,_ = hydrogen partia l pressure (atm)

Qin = influent flow rate (L/h)

Q ou t = effluent flow rate (L/h)

rNAD= ratio o f NA DH to NA D+

rs = sum o f the biolog ica l reaction rates that produce or consume S

S = constituent concentration in the reactor

S = substrate

SA= acetic acid (g COD/m3)

SAi = sludge ac tivity (g COD/g COD VSS/d)

SB= butyric acid (g COD/m3)

Sc = readily degradable carbohydrate (g COD/m3)

SF= readily fermentable monomer; e.g., glucose (g COD/m3)

SH= dissolved hydrogen (g COD/m3)

S, = inert organic compounds (nonbiodegradable) (g COD/m3)

Sin = influent concentration (g COD/m3)

S,„ = the influent COD concentration (g COD/m3)

SL= lactic acid (g COD/m3)

SM= dissolved methane (g COD/m3)

SM A = sludge methanogenic ac tivity (L CHVgVSS/d)

SP= propionic acid (g COD/m3)

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Ss = slow ly degradable complex organics (g COD/m3)

t = time (h)

tc= total cycle time (h)

td= decant time (h)

tr = react time (h)

ts= settle time (h)

U = bu lk (fluid and solids) flow velocity (underflow ve locity) (m/h)

Ue= bulk (fluid and solids) up-flow velocity (m/h)

U = specific substrate utilization rate (g COD/g COD VSS/d)

Uj = specific substrate utilization rate (g COD/g COD VSS/d)

u = solids velocity (m/h)

Uf = flu id ve locity (m/h)

V - solids settling veloc ity relative to the tank wall (m/h)

V= reaction rate (g COD/L/d)

Vf = fi ll volume (L)

Vm= maximum reaction rate (g COD /L/d)

Vmj = maximum substrate utilization rate (g COD/L/d)

VR= reaction volume (L)

v = solids settling velocity relative to bulk flow, positive by convention (m/h)

vo = interface settling ve locity in batch settling tests (m/h)

vs= interface settle velocity (m/h)

V, = total reactor volume (L)

XA= aceticlastic methanogenic organisms (g COD/m3)

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X B= butyric acid acetogenic organisms (g COD/m3)

X F= heterotroph ic-hydrolytic organisms (g COD/m3)

X H= hydrogenotrophic methanogenic organisms (g COD/m3)

X, = inert component o f lysed biomass (g COD/m3)

X L= lactic acid acidogenic organisms (g COD/m3)

X P= propion ic acid acetogenic organisms (g COD/m3)

X s = biodegradable component o f lysed biomass (g COD/m3)

X, = total solids in the system (g COD/m3)

YF = yield o f heterotrophic-hydrolytic organisms (g COD VSS/g COD)

7, = yield (g COD VSS/g COD)

Y, = overall yield (g COD VSS/g COD)

Yrhod = theoretical product yield

z = vertical coordinate, positive up (m)

T = mass-action ratio

Sj = substrate removal efficiency

s, = overall removal efficiency

r/i.sx = settling efficiency

ij, = inh ibition factor

} j u l2 = hydrogen inh ibition function factor

rjiiAc = regulation factor for acetate production from glucose degradation

t I rm = regulation factor fo r butyrate production from glucose degradation

rim;. = regulation factor fo r propionate production from glucose degradation

Tjin.-A= regulation function for acetate production from glucose degradation

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7/,,,,/j = regulation func tion for butyrate production from glucose degradation

7jRhL = regulation function for lactate production from glucose degradation

'Hrla = regulation function for acetate production from lactate degradation

rjliLP = regulation fun ction fo r propionate production from lactate degradation

0 - hydrau lic retention time (d)

6X = solids retention time (d)

•7

Pf = density o f fluid (kg/m )

p, = growth rate o f microorganism group X t (g COD/L/d)

• • 3ps= density o f solids (kg/m )

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List of Abbreviations

ABR = anaerobic baffled reactor

AD M1 = Anaerobic D igestion Model No. 1

AF = anaerobic fi lte r

AFF = anaerobic fixed film

AnSBR = anaerobic sequencing batch reactor

COD = chemical oxygen demand

EGSB = expanded granular sludge bed

EMP = Embden-Meyerhoff-Pamas pathway

FIRT = hydraulic retention time

IAW Q = International Association on Water Quality

IW A = International Water Association

MLVSS = mixed liqu id vo latile suspended solids

N AD+ = nicotinamide adenine dinucleotide

OLR = organic loading rate

SEM = scanning electron microscopy

SMA = sludge methanogenic activity

SOLR = specific organic loading rate

SRT = solids retention time

SVI = sludge volume index

UASB = up -flow anaerobic sludge blanket

VF A = volatile fa tty acid

VSS = vo latile suspended solids

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Chapter 1 Introduction

1.1 Anaerobic wastewater treatment

Anaerobic digestion is one o f the oldest bio log ica l wastewater treatment

processes, having first been used more than a century ago (Pavlostathis and Giraldo-

Gomez, 1991). Anaerobic digestion has become the most common method o f sludge

stabilization. Because of the growing emphasis on energy conservation and recovery as

well as other environmental concerns related to land disposal o f wastewater sludges,

anaerobic digestion is expected to continue to play a major role in municipal sludge

processing and treatment o f other complex concentrated wastes (i.e., high concentration

o f biodegradable organics) such as agricultural wastes (e.g., plan t residues, animal waste)

and food-processing wastewaters (Pavlostathis and Giraldo-Gomez, 1991).

Over the past 30 years the popularity o f anaerobic wastewater treatment has

increased as public u tilities and industries have utilized its considerable benefits (Azbar et

al., 2001). Primary advantages o f anaerobic treatment include: 1) reduction o f waste

biomass disposal costs; 2) reduction o f installation space requirements; 3) conservation o f

energy, ensuring ecological and economical benefits; 4) minimization o f operational

attention requirement; 5) e lim ination o f off-gas air pollution; 6) biodegradation o f aerobic

non-biodegradables; 6) provision o f seasonal treatment; 7) reduction o f chlorinated

organic toxicity levels (Speece, 1996). Recognition o f the advantages o f anaerobic

processes, over alternative aerobic processes, has led to the development o f new

anaerobic process configurations. Common configurations o f anaerobic reactors include

1




2

the upflow anaerobic sludge blanket (UASB) reactor, the anaerobic baffled reactor

(ABR), and the anaerobic filte r (AF).

The anaerobic sequencing batch reactor (AnSBR) developed by Dague et al.

(1992) is one of the promising configurations fo r the anaerobic treatment o f high strength

organic wastes. The main advantages associated with AnSBRs are operational fle x ib ility ,

better effluent qua lity control, the absence of secondary settlers, no liqu id and solids

recycling, potential to select a specific microbial population, and plug flo w kinetics (Za iat

et al., 2001). The AnSBR can maintain a high concentration o f slow-growing

methanogenic bacteria in the system, through which process stability is improved (Dague

et al., 1992). Recently AnSBRs have been extensively studied (Chang et al., 1994; Dague

and Pidaparti, 1992; Ng, 1988; Suthaker et al., 1991); however, industrial application o f

this process has not been established, since several fundamental features and

technological aspects remain to be investigated (Zaiat et al., 2001). After a thorough

review o f AnSBR technology, Zaiat et al. (2001) proposed further studies on the

optimization o f operating and design parameters to make the AnSBRs feasible for

industrial applications. One goal o f this study is to determine optimal operating and

design parameters for the AnSBR, and mathematical modeling is the approach employed

to achieve this goal.

1.2 Mathematical modeling of anaerobic processes

A mathematical model is a convenient tool fo r understanding the process,

defining its solution, and optimizing the design and operation o f wastewater treatment

processes (Masse and Droste, 2000; Fernandez et al., 1993; Henze et al., 1987 and 1995).

The mathematical models allow extrapolation o f the design space to conditions beyond

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4

et al. (1999) focused on population dynamics in anaerobic reactors, it excluded the

hydrogenotrophic methanogens, the controlling organisms for anaerobic processes.

Hydrogen was assumed to be instantaneously converted to methane when produced.

The existing models may not be able to provide a reasonable prediction o f

population dynamics. For example, the predicted percentage o f propion ic acid-consuming

bacteria in the total biomass was close to zero after long-term simulation using the

Bagley and Brodkorb (1999) model. Costello et al. (1991b) did not present the

concentrations o f each bacteria group under the in itia l steady-state condition. However,

the authors reported that the concentration o f some groups o f bacteria approached zero

when the model iterated towards the steady-state conditions (initial conditions for

overload test simulation). For their simulations, an arb itrarily small influen t concentration

o f 1 mg/1 was set for each group o f bacteria.

In addition to lim its to the ir predictive capabilities, the existing models may also

experience numerical challenges. Bagley and Brodkorb (1999) used a Cash-Karp fifth -

order Runge-Kutta algorithm w ith adaptive step sizing to solve the system o f nonlinear

differential equations. A Runge-Kutta algorithm was the common algorithm used by

other investigators as w ell (Masse and Droste, 2000; Costello et al., 1991b; Kie ly et al.,

1997). The Runge-Kutta algorithm is not stable for s tif f sets o f equations. However,

because the anaerobic systems typ ica lly have s im ilar rates o f consumption and

production for key intermediates, the nonlinear differential equations used to model them

may be s ti ff (Batstone et al., 2002). An alternative algorithm to Runge-Kutta may be

required.




5

Solids-liquid separation is an integral component o f biolog ical wastewater

treatment processes. This solids-liquid separation is traditionally achieved by gravity

sedimentation (Ekama et al., 1997). Most anaerobic reactor configurations do not make

use o f the conventional c larifier fo r solids-liquid separation. The application o f anaerobic

models could be lim ited w ithout a sub-model for so lids-liquid separation.

1.3 Objectives

The objectives o f this thesis are to:

1. Extend the Bagley and Brodkorb (1999) AnSBR model to a general model for the

anaerobic degradation process to make this model more comparable and

compatible w ith other models.

2. Develop and incorporate into the model a hydrogen partial pressure product

regulation function and hydrogen partial pressure inhibition function that better

accounts fo r the thermodynamic constraints o f anaerobic systems.

3. Implement an improved numerical routine (better than the Cash-Karp fifth-order

Runge-Kutta algorithm) to solve the coupled first-order ordinary differential

equations fo r the model.

4. Develop a gravitationa l settling model applicable for both zone settling and

compression, which is needed for the so lid-liquid separation process.

5. V erify the model using extensive sources o f data in the literature: 1) from

laboratory AnSBRs, 2) from long-term operation o f continuous high-rate

anaerobic reactors, 3) from the dynamic response o f a continuous reactor

subjected to step changes in loading, 4) from the startup o f UASB reactors.




6

6. Examine the optimal design and operational parameters o f AnSBRs through

model simulation.

1.4 Publications

Several sections o f this thesis have been published in refereed journals. Sections

7.1, 7.2, and 7.3 o f Chapter 7 have been published in the Journal of Environmental

Engineering, ASCE (Zheng and Bagley, 1998; Zheng and Bagley, 1999).




Chapter 2 Literature Review

2.1 Mathematical models for anaerobic processes

Mosey (1983) developed the firs t complex model to explain the patterns o f

volatile acid production in the anaerobic digestion process. In this model, he defined and

incorporated the regulation and inh ibition o f the hydrolytic bacteria, and the prop ionic

and butyric acid acetogenic bacteria. Based on the fact that the uptake rates and product

distribution o f some species o f bacteria were regulated by hydrogen gas (Iannotti et al.,

1973; Kaspar and Wuhrmann, 1978), Mosey (1983) proposed that the various mixtures o f

acetic, propionic, and butyric acids in an anaerobic digester under stress are the response

o f the acid-forming bacteria to changes in the redox potential o f their growth medium

brought about by changes in the trace concentrations o f hydrogen in the digester gas. He

further proposed that the obligate hydrogen-utilising methane bacteria are the controlling

organisms for the redox potential o f the anaerobic digestion under normal circumstances.

The level o f hydrogen in the reactor is linked to the redox reactions o f the pyridine

nucleotides (characterized by the nicotinamide adenine dinucleotide redox couple NAD+-

NADH). Hydrogen in the gas phase o f the reactor determined the ratio o f oxidized to

reduced NAD within the bacteria, which in turn regulated and inhibited the metabolic

reactions within the bacteria that were coupled to the NAD+-NADH redox reaction.

Assuming that the half-reaction,

NAD+ + H + + 2e~ « • NADH (2.1)

and the h a lf reaction,

2 H + + 2 e ~ ^ H 2 (2.2)

7




9

For example, Costello et al. (1991a) developed a dynamic mathematical model o f the

high-rate anaerobic treatment process by incorporating advances made in the modeling

and study of the general anaerobic degradation process. A major change was the

mod ification o f the anaerobic ecosystem to include lactic acid bacteria. The proposed

metabolic pathway o f the glucose bacteria was modified to produce lactic acid rather than

propionic acid, while the lactic acid bacteria produced either acetic or propionic acid

according to the level o f hydrogen in the biogas. This was supported by the experimental

evidence o f Pipyn and Verstraete (1981) and Eng et al. (1986) who showed that lactic

acid accumulates after a sudden increase in the loading o f a readily degradable substrate

to an anaerobic reactor. Bagley and Brodkorb (1999) applied the International

Association on Water Qua lity (IAW Q) (Henze et al., 1987 and 1995) approach, which

ex plicitly allows for inclusion o f multiple populations o f microorganisms performing

different metabolic activities, to systematically develop a model describing

microbiological processes occurring in an AnSBR. The model was validated using data

obtained from operation o f a bench-scale batch reactor treating glucose as the substrate.

Formation and consumption o f intermediate products, including lactate and VFAs, are

predicted, as is system pH. Masse and Droste (2000) developed a comprehensive model

o f anaerobic digestion o f swine manure slurry in a sequencing batch reactor. This model

has sim ilarities to models developed by Mosey (1983) and Costello et al. (1991a), but the

hydrogen effect on metabolism is based on the dissolved hydrogen concentration instead

o f gaseous concentrations. Batstone et al. (2000a) extended the Costello et al. (1991a)

model by incorporating hydrolys is o f particulates and long chain fatty acid (3-oxidation.




10

Recently, the Anaerobic Digestion Model No. 1 (AD M1) was proposed by the

International Water Association (IW A) task group for mathematical modeling o f

anaerobic digestion processes (Batstone et al., 2002). This model includes the formation

and consumption o f carbohydrates, proteins, and fats, while most other models consider

carbohydrates only. Lactate is not considered as an intermediate in this model. This

model incorporates the free ammonia inh ibition o f aceticlastic methanogens in addition to

hydrogen and pH inhib ition .

Some models in the literature do not include the regulation and inhibition effect o f

hydrogen, such as the models developed by Skiadas et al. (2000) and Merkel et al.

(1999). In these models, the product regulation factors from glucose degradation are

constants and are independent o f hydrogen partial pressure. Unfortunately, ADM 1

(Batstone et al., 2002) is one o f these models. Skiadas et al. (2000) developed a dynamic

model for the anaerobic digestion o f glucose in the periodic anaerobic baffled reactor. In

this model the acidogenic bacteria consume glucose and produce an unknown

intermediate (fin al and/or intrace llular) product, as w ell as lactic acid. They observed a

significant COD balance deficit considering the substrate (glucose) and known products

(VFAs, lactate, and biogas) during the in itia l stage o f a batch experiment. From these

observations they argued that accumulation o f some undetermined intermediate products,

as well as accumulation of intracellular intermediate products of bacterial metabolism, is

taking place during the in itia l stage o f the batch experiments. Different groups o f

acidogenic bacteria convert lactate and an intermediate product to a m ixture o f acetate

and propionate. This model does not include the hydrogen-utilizing bacteria and does not

consider the production o f methane from hydrogen by hydrogen-utilizing bacteria. No




11

inhibition factors were considered. Merkel et al. (1999) developed a mathematical model

for the anaerobic digestion process with particular attention to population dynamics. In

the model the microbial populations were grouped into substrate-specific organisms.

However, hydrogenotrophs were not considered as a distinct group in this model. The

model simulation results were validated w ith experimental data obtained by a

combination o f in situ hybridization techniques and epifluorescence microscopy.

There are also many other sim plified models fo r anaerobic processes proposed in

the literature. For example, the model developed by Kiely et al. (1997) for anaerobic

digestion considers the production o f methane as occurring in two stages; that o f

hydrolysis/acidogenesis producing acetate and that o f aceticlastic methanogenesis

producing methane. The model considers the inhibition caused by ammonia in the growth

kinetics o f methanogenic bacteria.

2.2 Experimental investigations on anaerobic treatment

2.2.1 Performance of different reactor configurations

Anaerobic processes for wastewater treatment have been extensively studied.

Many experimental investigations focused on the comparisons o f the performance o f

different reactor configurations. Tay and Zhang (2000a) studied the s tab ility o f three

typical high-rate anaerobic treatment systems (anaerobic fluidized bed reactor, AF, and

UASB) subjected to various disturbances (organic loading rate shock, hydraulic loading

rate shock, bicarbonate supplement absence shock, underload shock, and toxic shock).

The three reactors showed different resistances to different shocks. The anaerobic

fluidized bed reactor showed good resistance to all the shocks, while the UASB exhibited




12

good performance in almost all the shocks except the toxic event. The AF seemed to

perform the worst in all the shock tests except the toxic shock. A fuzzy stability index for

the evaluation o f stability o f high-rate anaerobic systems was proposed by Tay and Zhang

(2000b). Perez et al. (1998) compared the performance o f two high rate anaerobic

treatment systems (anaerobic filte r and fluidized bed). They concluded that the anaerobic

fluid ized bed system was more effective than the anaerobic filte r system.

Yeh et al. (1997) investigated the performance o f an anaerobic rotating bio logical

contactor under differe nt flo w rates and influen t organic strengths. They showed that the

removal efficiencies increased as the hydraulic retention time (HRT) increased or the

influen t COD decreased. Show and Tay (1999) examined the influence o f support media

on biomass growth and retention on anaerobic filters. Their results indicated that media

surface texture and porosity have a sign ificant impact on anaerobic filte r performance.

Support media o f open-pored surfaces and high porosity were recommended.

2.2.2 Granulation

Granulation is an important feature for some anaerobic treatment processes. The

granule enhances solids-liquid separation and helps to retain the slow-growing anaerobic

microorganisms in the reactors. Jhung and Choi (1995) evaluated the characteristics o f

the waste in the development o f sludge granulation and the operation o f two differen t

reactors, UASB and a downflow anaerobic fixed film (AFF) reactor. The ir results

indicated that sludge granulation was influenced more by the characteristics o f the waste

used than by the reactor type itself. Microscopic observations, including scanning

electron microscopy (SEM), revealed that microbial compositions in the two reactors




13

were essentially identical. A complex carbohydrate waste, having a higher CO D/volatile

acids ratio, produced filamentous microbes and these appeared to promote granulation.

Thaveesri et al. (1995) examined the form ation o f anaerobic granular sludge on

wastewater from sugar-beet processing in UASB reactors by addition o f high-energy

substrates and varying the reactor liquid surface tension. Granular sludge growth only

occurred when there were sufficient high-energy substrates. A low reactor liquid surface

tension increased granular yield. O ’Flaherty et al. (1997) investigated the influence o f

feed composition and liquid upflow velocity on the microbiological and physico

chemical properties o f sludges developed in anaerobic up flow hybrid reactors. Their

results showed that high-energy substrates are not a prerequisite for granulation, although

the presence o f sugars has been found to promote granulation. Elevated up flow velocities

promoted granulation (O’Flaherty et al., 1997). Grootaerd et al. (1997) reported that

adding carrot pu lp waste product as a granular growth supplement was not successful.

Tay et al. (2000) proposed a theory for the molecular mechanism o f sludge

granulation. They suggested that the bacterial surface dehydration caused by proton

translocating activity initiates sludge granulation. The overall granulation process

included four stages: dehydration o f bacterial surfaces, embryonic granule form ation,

granule maturation, and postmaturation. Teo et al. (2000) investigated the effects o f some

factors on granular strength. Both calcium ion and surfactants were found to strengthen

the granular structure. In the physiolog ical pH range o f 5.5-11.0, the lower the pH value,

the stronger the granule. Both the metabolic inhibitors, iodoacetic acid and sodium

fluoride, and the proton translocator, carbonyl cyanide m-chlorophenyl-hydrazone,




15

acetogenic activity at mesophilic temperature (35°C), even after incubation for many

months at psychrophilic temperatures. Hardly any selection or enrichment for

psychrophilic bacteria had taken place, but the mesophilic bacteria initially present were

metabolizing at lower rates. Mesophilic inocula incubated at lower temperatures in batch

bioassays produced more propionate than acetate due to a greater reduction in the rate o f

the syntrophs compared to the methanogens at low temperatures.

Fernandez et al. (2000) showed that a stably performing anaerobic bioreactor

exhibited tremendous variation in the composition o f its microb ial community. W u et al.

(2001) showed that at 15°C, a cellulose-fermenting methanogenic microbial culture was

dominated by Methanosaeta and at 30°C by Methanosarcina. They found that the 15 and

30°C culture lines behaved differently, indicating that the com position o f the acetoclastic

archaeal community indeed affects the functional performance at two different

temperatures. Their study showed that functionally similar but structurally different

methanogenic archaeal communities can have a decisive effect on the reaction o f a

methanogenic system to temperature shifts. However, the composition o f the bacterial

community can also have strong effects. They speculated that the low temperatures

selected against propionate-utilizing microorganisms.

Kettunen and Rintala (1998) studied the treatment o f municipal land fill leachate

at 13-23°C using a pilot-scale UASB reactor. Their results indicated that a UASB reactor

was feasible for the on-site treatment o f municipal landfill leachate at temperatures as

low as 13°C. Mesophilic digested sewage sludge adapted well to low temperature.




16

2.3 Gravity settling

The International Association on Water Quality (IAW Q) presented a scientific

and technical report on secondary settling tanks (Ekama et al., 1997). This report

summarized all new, important, and significant developments in the secondary settling

tank area, including theory, modeling, and design and operation. It was recommended

that, for better secondary settling tank performance, intentiona l optimization o f the

clarification, thickening and sludge storage functions o f secondary settling tanks is

required in addition to the design o f the secondary settling tank external dimensions, such

as area, depth, and recycle flow.

Namoli and Mehrotra (1997) proposed a model to simulate the solids profile in

the sludge blanket o f the UASB reactor. According to their de finition , the UASB reactor

consists o f a sludge bed in the bottom, a sludge blanket in the m iddle, and an interna l

settler supplemented w ith a gas-solid separator on the top. This model considered the

solid diffus ion induced by gas bubbles. The model predictions were validated w ith data

from literature. They showed that the model facilitates the optimization o f reactor

dimensions and the desludging schedule.

Chancelier et al. (1997) presented a theoretical analysis on the characterization o f

the steady states o f the secondary settler and its relations w ith the lim iting flu x theory.

Their results allowed insights into the lim iting flux theory, especially when the settler is

overloaded.

Ka rl and Wells (1999) developed a numerical model o f gravitational

sedimentation and thicken ing from the governing two-phase flow equations fo r the liqu id

and solid phases. Constitutive relationships describing the physical properties o f the




17

sludge were required to solve the model numerically. These constitutive properties were

determined experimentally and by model calibration. The model was calibrated and

verified using vertical solids profile data from batch settling tests.

Vanderhasselt and Vanrolleghem (2000) compared two means o f obtaining the

parameters for the zone-settling velocity models: ( 1 ) the traditional approach using zone

settling velocity data obtained from a dilution experiment and ( 2 ) a direct parameter

estimation method relying on a single batch settling curve. The latter was achieved by

fitting a one-dimensional 50-layer settling model, which incorporated a zone-settling

model, to single batch settling curves. When the dynamics o f sludge blanket descent were

fast, the second method failed to derive zone-settling velocity model parameters that are

consistent w ith the first method. I t was concluded that the second approach is not ready

for practice.

Chatellier and Audic (2000) proposed a model for wastewater treatment plant

clarifier simulation. This model was based on the layer model proposed by Takacs et al.

(1991). However, the settling velocity equation was modified based on the hypothesis

that the solids flux through all layers o f the cla rifier is constant even for non-steady-state

conditions.




19

pressure on the degradation o f propionate. When the hydrogen pa rtial pressure is higher

than the thermodynamic lim it, their model s till predicts very high propionate degradation

(> 80% o f uninh ibited degradation rate). For the models o f Bagley and Brodkorb (1999)

and AD M1 (Batstone et al., 2002), the thermodynamic lim it was under consideration

when the inhibition constant was chosen. However, their functions have a flat shape in

the normal range o f hydrogen partial pressure for the anaerobic process. These functions

predict that the degradation rates are not much different when the hydrogen partial

pressure is higher or lower than the thermodynamic lim it. Therefore these functions

cannot represent the deteriorating effect o f hydrogen partial pressure on the degradation

o f propionate. The hydrogen inhib ition function used in this study, which is derived in the

following section, is shown in Figure 3.1 for comparison.

The glucose degradation products distribute among propionate, butyrate, and

acetate. Generally, it is believed that the hydrogen partial pressure affects these

distributions. Figure 3.2 plots propionate yield versus hydrogen partial pressure predicted

by the hydrogen regulation functions used by the models in the literature. Propionate

yield is the amount o f propionate produced from 1 un it o f glucose COD degraded. I t can

be seen from Figure 3.2 that for the models o f Costello et al. (1991a), Masse and Droste

(2000), and Bagley and Brodkorb (1999), little propionate is produced when the

hydrogen partial pressure is in the normal range o f anaerobic treatment. Therefore, there

w ill be very little growth o f propionate acetogens. As a result, these models w ill predict

that very few propionate acetogens w ill be present in the anaerobic biomass. In addition,

according to Batstone et al. (2002), the regulation function developed by Costello et al.

(1991a) could not be used consistently with a variety o f experimental data sets. AD M1




20

(Batstone et al., 2002) uses a constant value for the propionate yield. The propionate

yield predicted by the regulation function used in this study, which is derived in the

following section, is shown in Figure 3.2 for comparison.

0.0E+00 2.0E-05 4.0E-05 6 .0E-05 8.0E-05 1.0E-04

Hydrogen Partial Pressure (atm)

Figure 3.1: Comparison o f hydrogen inhibition function used in different models.

A , Costello et al. (1991a) and Masse and Droste (2000); B, current model; C, Bagley and

Brodkorb (1999); D, Batstone et al. (2002).

thermodynamic limit

1




21

0.4

2

CD

g 0.2 oQ .

2Q_

0.0E+00 2.0E-05 4 .0E-05 6.0E-05 8.0E-05 1.0E-04Hydrogen Partial Pressure (atm)

Figure 3.2: Comparison o f propionate yields predicted by d ifferent models. A ,

Costello et al. (1991a) and Masse and Droste (2000); B, current model; C, Bagley and

Brodkorb (1999); D, Batstone et al. (2002).

3.2 Hydrogen inhibition for propionate and butyrate degradation

Enzyme kinetics w ill be used to approximate the situation for propionate and

butyrate degradation. The reaction scheme for a single substrate enzyme reaction can be

assumed to be (Snoeyink and Jenkins, 1980)

k \ k i

S + E o E S & P + E (3.3)k2 k4

where S is the substrate; E is the enzyme; ES is the enzyme substrate complex; P is the

product; k l and k3 are forw ard rate constants; and k2 and k4 are reverse rate constants.

The rate o f reaction should be related to the concentration o f ES complex because,

for the reaction to take place, ES must be formed. Therefore ES must be determined in

terms o f the total amount o f E-containing species and substrate concentration.




23

v = _ m 14

v =

- - = ( « ( * 2 + “ W ) W ] + M [ P ] + k 2 + B

■— = (MB[s]~ k2k4[p] ) ~ ^ B ------------

dt v k\[s] + k4[p] + k2 + k3

V = - ^ J k 3 [ S ] - ™ * [ P ] dt I k\

[4[,s]+ M [ p ] + * 2 ± ML J k \ l J k\

(3.12)

(3.13)

(3.14)

dt H ± « + [S] + M jp j

k\ 1 J k l L 1

(3.15)

Defining the maximum reaction rate Vmas (Snoeyink and Jenkins, 1980)

v. = »[£], (3.16)

and lumping kinetic constants (Snoeyink and Jenkins, 1980)

k l + k3K,„ =

k\ (3.17)

Equation 3.15 can then be rewritten as

V =dS _ " V 1 k3 k\ i

dt K m+ [S] + ̂ [ P ] k\

(3.18)

According to Fersht (1985), k2 » k3 most o f the time, and for the reaction to

proceed, [S'] > — — [p], therefore it can be concluded that [s] » — [/*] (because — isk3 k\ k\ k3

large). Therefore Equation 3.18 can be sim plified to

V = - ds

dt (3.19)




24

k1 k4r Because k l » k3 , --------- |PJ may not be negligible. This is indeed the case for

propionate and butyrate degradation (Hoh and Cord-Ruwisch, 1996).

Equation 3.19 can be rewritten as

(3.20)

where

, k l k4 [p ]Vl.Hl = 1 _ T T T T T rd

k3 kl [S'](3.21)

The four rate constants in equation 3.21 combine to give the equilibr ium constant K (Hoh

and Cord-Ruwisch, 1996)

r. kl k3K = -------- (3.22)

k l k4

According to Hoh and Cord-Ruwisch (1996)

IP] U = r (3.23,

where T is the mass-action ratio (actual ratio o f [products] over [substrates]) used in the

calculation o f the overall change o f Gibbs free energy (A G ') (A G ' = RT ln(T IK)) .

Therefore, equation 3.21 can be rew ritten as

tjl M2= i - r / K (3.24)

As an example, the energy reaction equation fo r propionate degradation is

CH 3CH 2COOH + 2 H 20 - > CH3COOH + C 0 2 + 3H2 (3.25)

Then

r { c h , c o o h }{ c o 2}{ h 2?

{ c h 3c h 2 c o o h )(3.26)




25

and the equilibrium constant K is expressed as

(3.27)

Equations 3.24 and 3.26 are equations describing the product inh ibition effect o f

propionate degradation. Equation 3.26 indicates that all degradation products, including

acetic acid, carbon dioxide, and hydrogen, can inh ib it the degradation o f propionate.

However the variation o f hydrogen concentration has the most significant effect on T due

to the third power in the expression. Therefore the effects o f acetate and propionate are

minimal, compared to the effect o f hydrogen. Th is was theoretically examined by Hoh

and Cord-Ruwisch (1996) and experimentally examined by Mosche and Jordening

(1999). In order to s im plify the expression for the production inh ibition function,

Equation 3.24 is rewritten as (assuming that the hydrogen gas and liquid phases are in

equilibrium):

where K/jqxp is an inhibition parameter. The following expression can be derived for

K/,h2 xp from Equations 3.24, 3.26, and 3.28:

where K\\ is the Henry’s Law constant.

Equation 3.30 shows that the inhibition parameter, K^mxp, is a function o f the

equilibrium constant as well as the actual activities o f other constituents. I f needed, this

\ 1 ,H 2 XP J i f P H2 ̂ 7f[ | |2XP (3.28)

t 1i ,h 2 xp i f P H2 > -K),H2XP (3.29)

I ,H2XP

\ c h ,c o o h }{ c o 2\ (3.30)




26

parameter could be calculated each time there is a change. However, for many situations,

the inhibition parameter can be approximately constant, and can be calculated by

assuming typ ical concentrations o f the constituents. The inhibition parameter, Kj H2XP’

is equivalent to the thermodynamic lim it o f propionate degradation. When the hydrogen

partial pressure is at the thermodynamic lim it, there should be no propionate degradation

(i.e., rjj H2XP = 0 ). Therefore, it can be seen from Equation 3.28 that the inhibition

parameter equals the thermodynamic lim it o f propionate degradation. For example, the

thermodynamic lim it o f propionate degradation occurs at Pm o f 4.0x10- 5 atm at 22°C

(Bagley and Brodkorb, 1999), thus

•£l,H2XP = 4.0x10“ 5 atm (3.31)

Similarly, the energy reaction equation fo r butyrate degradation is written as

CH 3CH2CH2COOH + 2 H 2 0 ^ 2 CH3COOH + 2H2 (3.32)

and the corresponding hydrogen partial pressure inhibition function is written as

Vl,H2XB ~ 1r P ^1HI

\̂ -I,H2XB J i f Pm ^ î,h 2xb (3.33)

Vi..H2XB ~ 0 if Pm > Î,H2XB (3.34)

where Kimxs is the inhibition parameter for butyrate degradation

The thermodynamic lim it o f butyrate degradation occurs at Pm o f 3.0x10” 4 atm at

22°C (Bagley and Brodkorb, 1999), thus

Î ,H2XB = 3. Ox 10- 4 atm (3.35)




27

There is a discontinuity in the expressions o f rj{ H2 as given by Equations 3.28,

3.29, 3.33 and 3.34. This w ill cause a numerical problem fo r the computer simulation,

and the expressions can be smoothed as follows.

The actual hydrogen inhibition functions used for propionate in the model are

rll,H2XP = 1-

r P ^r H 2

K I,H2XP i f Pm ^ O ^X i^x p (3.36)

VlMlXP - X X T i f ^H2 > 0.97Kl ,H2XP (3.37)“H2 ~ L 2

where C; and C 2 are constants to be determined. The coefficien t 0.97 is arbitrary.

When Pm equals 0.97 K ff /2Xp, the r h H2XP calculated from Equations 3.36

and 3.37 are the same, therefore

- 1.

v

^■9^K I,H2XP

KI,H2XP = 0.087327 (3.38)

0-97K I tH 2X P- C 2

and the derivatives o f P ]j j2Xp with respect to Pm from both equations are the same,

therefore

C, r 0.97K j u2XP A

v K I,H2XP

2.8227(3.39)

K I,H2XP {o.97K1H 2X P- C 2)2 K I,H2XP

Solving Equations 3.38 and 3.39 for C/ and C 2 gives

Q =0 .00270167^ H2XP (3.40)

C2 = 0.9390626KIH 2X P (3.41)

Using the same procedure, the actual hydrogen inhibition functions used for

butyrate in the model are




28

V l , H 2 X B

1 ,H2XB y

/ \

i f Pm ^ 0.97A'im2xb (3-42)

0.001800418if I , H 2 X B

V l , H 2 X B ~ P h i - 0 . 9 3 9 5 3 6 0 S 2 K i h 2 x b i f Ph2 > 0.97^i;h2xb (3.43)

Equations 3.36, 3.37, 3.42 and 3.43 resolve the discontinuity problem for the

inhibition functions. Therefore, the inh ibition functions are numerically satisfactory.

Figure 3.3 plots the hydrogen inhibition function for propionate degradation and

Figure 3.4 plots the hydrogen inhibition function for butyrate degradation. When

compared to the functions used in other models (see Figure 3.2 for propionate), these new

functions are closer to the actual step function desired.

1

oo03

o 0.5

0

0 0.00005


0.0001

Figure 3.3: Hydrogen inhibition function for propionate degradation.




29

v.

3o

05LL.

§ 0.5■4~>

ZZ c

0 0.0003 0.0006


Figure 3.4: Hydrogen inhibition function for butyrate degradation.




30

3.3 Hydrogen partial pressure product regulation

Enzyme kinetics can serve as a model for the kinetics to be developed for product

regulation. The general rate equation for ordered enzymatic reactions having two

substrates, where A is the leading substrate, is (Lehninger, 1975):

where V is the reaction rate; V„, is the maximum reaction rate; A and B are substrate

3.3.1 Regulation functions for glucose degradation

3.3.1.1 Lactic acid production

The acidogenic organisms ferment glucose into acetic, butyric, or lactic acid

through the Embden-Meyerhoff-Parnas pathway (EMP) with pyruvic acid as intermediate

(Figure 3.5). The sum of the product regulation functions for glucose equals 1 to maintain

mass conservation, as indicated in the follow ing equation:

where tjppi is the regulation function fo r lactate production; rjRpB is the regulation

function for butyrate production; and t ]R pa is the regulation function fo r acetate

production.

For the formation o f acetyl-CoA from pyruvic acid shown in Figure 3.5 (Reaction

1), nicotinamide adenine dinucleotide (NAD+) is utilized as a coenzyme to accept

electrons from pyruvic acid and the leading substrate A can be defined as NA D+ w ith B

as pyruv ic acid. For the formation o f lactic acid from pyruvic acid (Reaction 2), NADH

V = (3.44),S m m _|_ m ^ ^

A B A B

concentrations; and K$ , , and are half-velocity constants.

rhm. + Brfb + V ri - a ~ 1(3.45)




31

acts to deliver electrons to pyruvic acid. The leading substrate A can be defined as NADH

with B as pyruvic acid. Because the conversion from glucose to pyruvic acid is a fast

process (Mosey, 1983), B may be assumed to be in excess (B » K “ ). Then Equation

3.44 reduces to:

V =

+ i

(3.46)

Glucose

'NAD

, NA DH

2 [Pyruvic ac id]'

(2)

lNAD

B rf l

2 [Lactic acid]

NA D+ NAD H

2[Acetyl-CoA]~

'NADH 'NADH

( 4 )

^NAD

V rf r B rf a

Butyric acid

( 3 )

2 [Acetic acid]

Figure 3.5: The degradation pathway o f glucose (Mod ified from Costello et al.,

1991a). The numbers are referred to in the text.

Then, per Figure 3.5, the rate o f formation o f acetyl-CoA from pyruvic acid is

written as (subscript 1 for reaction 1; A = NA D+):

V„,V,

ml T̂NAi

ml

NAD

(3.47)

+ 1




32

and the rate o f formation o f lactic acid from pyruvic acid is w ritten as (subscript 2 for

reaction 2; A - NADH):

V2 =m2

K NADH

m2

(3.48)

+ 1N A D H

where N AD is the concentration o f N AD + (mol/L); and NA DH is the concentration o f

NADH (mol/L).

D ividing Equation 3.47 by Equation 3.48 gives:

r r NADH

ml +1

V\ v mi N A D H Vm2 K ™ +1

(3.49)

N A D

K NADH

m2 ■ + 1

v = K i N A D H v

V K NAD 2 m2 mX | |

N A D

(3.50)

Defining V - Vx + V2 , where V is the total removal rate o f pyruvic acid, then

V =

K NADH \

m2

1+K „ i N A D H

+ 1

V,m2 i f NAD

^ - + 1N A D

V-, (3.51)

V r fl

K NADH 'N

m2 + 1

1+ v m\ N A D H

v K nadm2 ml ^

N A D

(3.52)

According to Mosey (1983) (Equation 2.3)

N A D H

N A D= 1500P,

H 2 (3.53)

Defining N A D T = N A D + N A D H , then




33

NAD =1

1+ 1500P,NADT

H 2

NADH = — 5 Q Q /" 2 NADT

1+ 1500P,HI

Substituting Equations 3.54 and 3.55 into 3.52 gives

P rf l =

1+ 1500P, A-'H2-K™DH +NADT

1 + Vm 1S00P„

F„ 2 (l + 1500P„2)K,%D +NADT

For a normal anaerobic system operated w ith PH2 in the range o f 1.0x10 6

atm(Speece, 1996):

So:

1+ 1500/)^ «1

P rf l =

K™DH +NA DT

1+K , 1500PH2

V,ml + NADT

P rf l + -1 V.ml K

NADH

m2 x , Vmi NADT

v Vm2 NADT + K T 1500PH2 Vm2 NADT + K N J D,

1 + Vm] NADT

Pm,Vm2 NAD T + K Z

NAD

1 V.ml K NADH

ml

1500 F„ , 2 N AD T + K T H2 +

1 + ml NADT

Vm2 NADT + K mX NAD

Therefore


(3.54)

(3.55)

(3.56)

l.OxlO - 4

(3.57)

(3.58)

(3.59)

(3.60)



34

k P „ _ K RF I,r H2 n

t1r f l ~ ~ p ~ T k ~ ( }H2 "r i '-RFL

where

° , , V„, NADT {3'62)

V„2 NADT + K ™

i t / ts NADH

1 m\ _______ ml ______

1500 V 2 NADT + K%?D

K «"■=- : V NADT 1 + — ----------------------

Vm2 NADT + K ™

Equation 3.61 is the product regulation function for lactic acid production from

glucose degradation.

3.3.1.2 Butyric acid production and acetic acid production

According to Figure 3.5, the rate o f formation o f acetic acid from acetyl-CoA can

be written as (subscript 3 for reaction 3):

r3 = r„3 (3.64)

because there is no NAD+ /NADH involved and still assuming that the substrate, acetyl-

CoA in this case, is in excess.

The rate o f formation o f butyric acid from acetyl-CoA is written as (subscript 4

for reaction 4; A = NADH ):

V a = k n a d h (3.65)

^ m 4NADH

Therefore,

V3 _ vm3

+ 1

y NADH m4 j

NADH v J

(3.66)




35

Vi =

V,m3

V,m4

K NADH mA

NADH + 1 (3.67)

Defining V = V3 +V4 , where V is the total removal rate o f acetyl-CoA, then

V = 1+K m3

r V N AD H A

v.mA

LmA

NADH + 1 VA (3.68)

Vt v 1+ "'3

V m A

K m4

NADH + 1

J J

(3.69)

Substituting Equations 3.54 and 3.55 into 3.69 gives

Av

1+ 1500P,

l + Vm3 1500PH2 H2 K™DH + NADT

V,mA NADT (3.70)

Considering 1+ 1500Pf/2 « 1 per Equation 3.57, Equation 3.70 is sim plified as

l + v m3H 2

— = -------------- — ----------------- (3.71)V -t rr is NA DH y J

1 'm3 mA

Therefore

where

PH 2 + ’1500 Vm4 NADT

1 + Vm3

PmA

V k P y 4 _ RFB H I

V ~ P H2 +K * fb

]r —

" 'RFB ~

i + F-V.mA

(3.73)




36

NADH * Vm, Kmt

_ 1500 Vm4 NADT RFB ~

1 + K ,(3.74)

V.mA

Equation 3.72 is the regulation function for butyric acid production from acetyl-CoA.

Considering the production o f acetyl-CoA from glucose, the regulation func tion fo r

butyric acid production from glucose degradation can be written as

V rfb 1 -k P 11 m A h 2

PH2 + Km j

k P RFB H I

PH2+ k rfb

(3.75)

and the product regulation function for acetic acid production from glucose degradation is

written as

V r i - a

k p j _ R̂FL1hi

P H 2 + K r f l J \

k p\ _ 1U‘B H 2

P + K 1 H 2 T a^ r f b y

(3.76)

3.3.2 Regulation functions for lactic acid degradation

Lactic acid acidogenic organisms ferment lactic acid to either propionic acid or

acetic acid (Equations 3.77 and 3.78).

CH 3CHOHCOOH + H20 ->. CH3COOH + C 0 2 + 2 H2 (3.77)

CH 3CHOHCOOH + H 2 -> CH3CH2COOH + H20 (3.78)

Similar to the derivation for glucose degradation, the production regulation

functions for lactic acid degradation can be written as

’ H r i .p

k P RIJ3 H 2

P + K 1 H 2 T R IJ ’

k P _ 1 _ RLP1 H 2

H r l a ~ 1PH2 + K RLp

(3.79)

(3.80)




37

where t j r l p is the regulation function for propionate production from lactate degradation;

rjRIA is the regulation function for acetate production from lactate degradation; and kRfA

and K ria are constants.

Figure 3.6 illustrates the hydrogen partial pressure product regulation from

glucose fermentation and Figure 3.7 illustrates the hydrogen partial pressure product

regulation from lactic acid degradation. The regulation constants are summarized in

Table 3.1. The constants were developed by comparison to Bagley and Brodkorb (1999)

data.

Table 3.1: Hydrogen partial pressure product regulation constants

Parameters Description Value3

k RFL Regulation o f glucose fermentation 1 . 0

K rfl (atm) Regulation o f glucose fermentation 5.0x10- 5

Jr RFB Regulation o f glucose fermentation 0.7

K rfh (atm) Regulation o f glucose fermentation l.OxlO -4

If RLP Regulation o f lactic acid fermentation 0.45

K rlp (atm) Regulation o f lactic acid fermentation l.OxlO - 5

“ Developed from Bagley and Brodkorb (1999)




38

0.8lactate

c o

| 0.4O)CDOd

0.2

acetate

butyrate

0 0.0001 0.0002 0.0003 0.0004


Figure 3.6: Hydrogen regulation o f product distribution in the degradation o f glucose to

acetate, lactate, and butyrate.

0.8

%to

Ll_

acetate0.6

co-4—'

_ro=5CDCDCd

0.4propionate

0.2

0 0.0001 0.0002 0.0003 0.0004


Figure 3.7: Hydrogen regulation o f product distribu tion in the degradation o f lactate to

acetate and propionate.




39

3.4 Computation

3.4.1 Implementation for CSTRs and AnSBRs

The overall mass balance for a completely mixed reactor is

(3.81)

where VR is the reaction volume; Qm is the influent flo w rate; Q out is the effluent flow

rate; Sm is the influent concentration; S is the constituent concentration in the reactor; and

rs is the sum o f the biolog ical reaction rates that produce or consume S. W ith the

exceptions identified in this chapter, the form for rs is taken from Bagley and Brodkorb

(1999). Equation 3.81 is valid for both soluble components and particulate components.

For continuous reactors (such as CSTRs and UASBs), VR, QIN, and Q out can

remain constant, and Qm equals Qouf, Equation 3.81 is simplified to

The AnSBR has five distinct stages: fill, react, settle, decant, and waste. VR, Q m ,

and Q ou t vary w ithin an AnSBR cycle. Equation 3.81 can be sim plified fo r the differen t

stages. For the fill stage,

(3.82)

Qour ~ 0 (3.83)

and

(3.84)

Therefore, Equation 3.81 is simplified to

(3.85)




40

For the react stage, Q m equals 0, Q ou t equals 0, and VR remains constant,

therefore, equation 3.81 is sim plified to

For the settle stage, the m ixing is stopped and liquid -so lids separation occurs.

Assuming that m icrobial reactions continue only in the settled volume and no reaction

w ill occur in the supernatant, an effective V r can be described by the follow ing equation:

where vs is the interface settle velocity; and A is the cross-sectional area o f the reactor.

For the reactor as a whole, Q ou t equals 0 in the settle stage. For the settled

reaction volume, the settle stage is equivalent to having an outflow rate o f vsA. If the

settling efficiency is represented by t ]FSX, the effluent particulate component

concentrations (the particulate components remaining in the supernatant) can be written

as

Therefore, for particulate components X, substituting Equations 3.87 and 3.88 into

Equation 3.81 (considering Qm =0and Qom = vsA ) gives

~ d t = ~ v ^ ’1"s- 'x + r ' <3,89)

while for soluble components, Equation 3.81 is simplified as (settle efficiency equals 0)

(3.87)

(i-nfAx (3.88)

dS — = r\ (3.90)




41

For the decant stage, assuming that m icrob ial reactions occur only in the settled

volume, for the reaction volume, Q j N equals 0, Q ou t equals 0, and VR remains constant.

Equation 3.81 is simplified to

" T = r s ( 3 -9 1 )at

For the waste stage

Q,n = 0 (3.92)

and

=S o u r (3-93)

Therefore, Equation 3.81 is sim plified to

, —* ..— = rs (3-94)at

3.4.2 Numerical method

The model has 18 processes and 19 components (Bagley and Brodkorb, 1999),

giving rise to a set o f 19 coupled first-order ordinary d iffere ntial equations. As this is a

s tif f set o f equations, it is solved num erically using the sem i-imp licit extrapolation

method (Press et al., 1992). The stoichiometry matrices are presented in Appendix A . The

computer program was coded in ANSI C and executed on a desktop PC. The source code

is presented in Appendix B.

The general form o f a set o f N coupled first-order ordinary d ifferen tial equations

is (Press et al., 1992):

= i = (3.95)at

where the functions f on the right-hand side are known.




42

The computer code for the semi-implicit extrapolation method is available in

Press et al. (1992) except for two user-supplied routines. The first user-supplied routine

computes the right-hand side derivatives o f Equation 3.95 (or dy^ty/dt). This routine is

the routine derivs in Appendix B. This user-supplied routine is straightforward and the

Runge-Kutta method requires this user-supplied routine only. The second user-supplied

routine computes the Jacobian ma trix; the Jacobian matrix is the matrix o f the partial

derivatives o f the right-hand side o f Equation 3.95 w ith respect to y (d f /By ). This routine

is the routine jacobn in Appendix B.

The current model is bu ilt on the model o f Bagley and Brodkorb (1999). The

computer code was revised so that the model also works for continuous reactors. The

kinetic constants used in the model are listed in Table 3.2. The kinetic constants reported

in the literature vary significantly (Pavlostathis and Giraldo-Gomez, 1991). Except for

the kinetic constants in Table 3.2 and the hydrogen inhibition and regulation parameters

in Table 3.1, the model parameters are the same as Bagley and Brodkorb (1999). Table

3.3 is a lis t o f symbols for soluble and particulate components, which w ill be referred to

in rest o f this thesis. A ll model parameters are presented in the sample input files in

Appendix C. The computer code checks the COD balance at each computational step.

Therefore, the overall COD balance closure is ensured for model simulations.

Dissolved hydrogen concentration is more correctly used in the thermodynamic

relations governing metabolism (Masse and Droste, 2000). The hydrogen partial pressure

used in the current model is calculated assuming equilibrium from the dissolved

hydrogen concentration divided by Henry’s Law constant. Therefore, the hydrogen




43

partial pressure in this model represents the dissolved hydrogen concentration reported in

the un it o f atm and should not imply gas-liquid equilibrium.

Table 3.2: Kinetic constants used in the model

Reaction3 k

(gCOD/gXCOD/d)

K s

(gCOD/m3)

Y

(gXCOD/gCOD)

b

(d -1)

Sp to S a, Sb, or Sl 49.4b 2 2 .5b 0.07 0 .0 2 d

Sl to S a or Sp 34.6 d 36.5d 0.064 0 .0 2 d

S a to Sm 6 .1 ° 165.0° 0.058c o © L O o

Sp to S a 5.3C 60.0° 0.059° 0 .0 1 °

Sb to S a 5.3C 13.0° 0.067° 0.027°

Sh to Sm 24.7a 0 .0 1 2 d 0 .2 2 d 0.088b

aRefer to Table 3.3

b Pavlostathis and Giraldo-Gomez, 1991

c Lawrence and McCarty, 1969

d Costello et al., 1991b




44

Table 3.3: Symbols for soluble and particulate components (Bagley and Brodkorb, 1999)

Symbol Description

Sc Readily degradable carbohydrate

Ss Slowly degradable complex organic

Si Inert organic compounds (nonbiodegradable)

Sf Readily fermentable monomer; e.g., glucose

Sl Lactic acid

Sb Butyric acid

Sp Propionic acid

S a Acetic acid

Sm Dissolved methane

Sh Dissolved hydrogen

x F Heterotrophic-hydrolytic organisms

X L Lactic acid acidogenic organisms

X B Butyric acid acetogenic organisms

x P Propionic acid acetogenic organisms

X A Aceticlastic methanogenic organisms

X h Hydrogenotropic methanogenic organisms

X s Biodegradable component o f lysed biomass

X i Inert component o f lysed biomass

S, soluble component; X, particulate component; all units in g COD/m




Chapter 4 Model Verification

The current model is based on the model o f Bagley and Brodkorb (1999), which

has been validated (Bagley and Brodkorb, 1999). The new developments for the

hydrogen inhibition and regulation functions described in Chapter 3 are intended to

improve the model predictions o f m icrobial population. This chapter verifies the new

model w ith a wide range o f experimental data from the literature. Long-term model

simulation, including steady-state simulation, is the best way to verify the model

predictions o f microbial growth. The current model simulations o f the experimental data

o f Bagley and Brodkorb (1999) show that the current model maintains the predictive

ability on VFAs, COD, pH, and gas production.

4.1 Model verification with data from Bagley and Brodkorb (1999)

4.1.1 Description of the experimental data

The laboratory AnSBR was fabricated from clear Plexiglass® and Tygon® tubing.

The reactor had a total liq uid volume o f 13 L w ith an internal diameter o f 15 cm. The

reactor was operated with a 12.0 L total liquid volume, 6.0 liter fill volume, and 6.0 liter

settled volume. The reactor was seeded w ith anaerobic sludge from the anaerobic digester

o f the Toronto (Ontario, Canada) Main Wastewater Treatment Plant. The average volatile

suspended solids (VSS) concentration in the reactor was 7000 mg COD VSS/L. The

synthetic wastewater feed consisted o f glucose as the sole COD source, sodium

bicarbonate as alka linity , and necessary nutrients and trace metals. M ixin g was conducted

w ith liquid re-circulation and the reactor was maintained at 22°C (Bagley and Brodkorb,

1999).

45




46

Two different experiments were conducted in the laboratory. The operating

conditions are listed in Table 4.1. The following variables in the reactor were measured

every hour throughout the experimental run: lactic acid, acetic acid, propionic acid,

butyric acid, pH, COD, methane gas production, and total gas production.

Table 4.1: Operating conditions (Bagley and Brodkorb, 1999)

Parameter Case 1 Case 2

Influent concentration (mg CO D/L) 2 0 0 0 4000

Organic loading rate (g COD/L/d) 1 . 0 2 . 0

F ill time (hours) 0.4 0.4

React time (hours) 2 2 2 2

Settle time (hours) 1 . 0 1 . 0

Decant time (hours) 0 . 6 0 . 6

Hydraulic retention time (hours) 48 48

Because in itia l concentrations o f ind ividu al microbial groups are not readily

measured directly, they are normally quantified by fitting the model predictions to

experimental measurements on VFAs, COD, and gas production (Bagley and Brodkorb,

1999; Masse and Droste, 2000; Skiadas et al., 2000). Case 1 served as the base cond ition

for estimating the in itia l biomass concentrations, the unmeasured “ loca l” model input

parameters. The same initial biomass concentrations were then used to simulate Case 2.

Overall COD balance closure was reported for these experimental data (Brodkorb, 1998).




47

4.1.2 Model calibration

Initial biomass concentrations were quantified by fitting the model prediction to

the experimental data on glucose, lactate, butyrate, propionate, and acetate. The Xs was

fixed at 50 mg COD/L follow ing Bagley and Brodkorb (1999) and Xi made up the rest o f

the measured VSS o f 7000 mg COD/L. A grid search method was used (because o f its

sim plic ity) to find the best fi t between the model predictions and experimental data. The

search parameter range was from 0 to 200 mg COD/L and the coarseness o f the grid was

5 mg COD/L. The objective function for the optimization was the minimum residual sum

o f squares between the model prediction and experimental measurements on glucose,

lactate, butyrate, propionate, and acetate. Initial biomass concentrations were (in mg

COD/L): XF, 115; X L, 85; X B, 5; XP, 60; X A, 100; X H, 175; Xs, 50; X,, 6410. The

hydrogen regulation parameters were also determined in this process. The total active

biomass consists o f 7.7% o f VSS. The total active biomass predicted by Bagley and

Brodkorb (1999) w ith the same set o f experimental data was 47.9% o f VSS. Shizas

(2000) reported a value o f 17.4% active biomass for a sim ilar sludge through

experimental measurement. Because anaerobic digester sludge contains a high content o f

non-viable biomass, the total active biomass predicted by the current model is more

reasonable than that by Bagley and Brodkorb (1999).

Figure 4.1 presents the comparison between the predicted and measured COD,

glucose, and total VFA concentrations for Case 1. The model predictions match the

experimental data very well w ith r values ranging from 0.78 to 0.92. Due to the short f il l

time, glucose concentration increased rapid ly w ith a peak value o f about 900 mg COD/L.

The glucose was completely degraded w ithin approximately 4 hours. The total VFA




49

500

* Acetate

a Butyrate

Propionate

Lactate400

_i

Qg 300

D)E

200&5

I 100

0 84 12 16 20 24

Time (hours)

Figure 4.2: Vo latile fatty acid concentrations vs. time fo r Case 1. Lines - model

predictions; symbols - measurements o f Bagley and Brodkorb (1999).

Similar predictions were made by Bagley and Brodkorb (1999). Though the r 2

values from the current model are lower than those (range from 0.87 to 0.99) o f Bagley

and Brodkorb (1999), the current model maintains the base performance o f the model in

predicting glucose and VFAs.

The comparison between the simulated and measured pH is presented in Figure

4.3. The model prediction matches the measured data very well. The good f it is prim arily

contributed to the good prediction o f VFA concentrations in Figure 4.2, as calculation o f

pH from known acid concentrations is a relatively well understood physical-chemical

process. The initial pH increase is due to the higher alkalinity and lower dissolved carbon

dioxide in the feed. The pH decreased as VFAs accumulated in the reactor and pH

increased again as VFAs were consumed.




50

7.5

xCL

6.5

0 4 8 12 16 20 24Time (hours)

Figure 4.3: pH versus time fo r Case 1. Line - model prediction; symbols - measurements

o f Bagley and Brodkorb (1999).

2.0

_ i

<DE 1.0.3O>COroO 0.5

0.0

0 84 12 16 20 24

Time (hours)

Figure 4.4: Cum ulative gas production versus time fo r Case 1. Lines - model

predictions; symbols - measurements of Bagley and Brodkorb (1999).




51

Figure 4.4 presents the comparison between the simulated and measured

cumulative gas production for methane and carbon dioxide. The model prediction

matches the measured data well.

Hydrogen partial pressure is an important variable for the model; it regulates the

product distribution o f glucose and lactic acid degradation and inh ibits the degradation o f

propionic and butyric acids. Though hydrogen was not measured in the original work, the

model-predicted hydrogen partial pressure is presented in Figure 4.5 for better

understanding o f the model behavior. The hydrogen spikes up quickly as glucose is added

and keeps accumulating until the depletion o f glucose. As a consequence o f the high

hydrogen partial pressure during the degradation o f glucose, lactic acid was the prim ary

product o f glucose degradation (see Figure 4.2). The hydrogen partial pressure dropped

when glucose consumption was complete and remained at the lower level until lactic acid

consumption was complete. The hydrogen partial pressure reached a level o f about

2.2x10- 5 atm after the lactic acid degradation was complete. Although degradation o f

propionic acid produces hydrogen, the hydrogen production rate from propionic acid

degradation was very low because o f the low degradation rate o f propion ic acid

(compared to the rate o f lactic acid and glucose degradation; see Figures 4.1 and 4.2).

Both the current model and the model o f Bagley and Brodkorb (1999) can

simulate the experimental data o f glucose, lactate, and VFAs very w ell w ith adjustable

model input o f in itia l biomass concentrations. Both models predict the pH and gas

production rate well. The two models are similar in predicting the measured constituents.

This is as expected as the measured constituents were the objective o f model

optimization. The major difference between these two models is the model predictions on




52

components that were not measured, such as initial biomass and hydrogen partial

pressure.

2.0E-04

1.5E-04 -

■p1.0E-04 -

? 5.0E-05 -

0.0E+00

0 4 8 12 16 20 24

Time (hours)

Figure 4.5: Predicted hydrogen partial pressure vs. time for Case 1.

The predicted hydrogen partial pressure by Bagley and Brodkorb (1999) has the

same pattern as shown in Figure 4.5, but the current model predicts a lower hydrogen

partial pressure than their model did. For example, the hydrogen partial pressure is about

2.2x10” 5 atm for the current model when the time is greater than 9 hours (see Figure 4.5),

compared to 6 x l0 - 5 atm for their model. As propionate degradation was observed during

that period (see Figure 4.2), the hydrogen partial pressure should be lower than the

thermodynamic lim it o f propionate degradation. Therefore the current model gives a

more reasonable prediction o f hydrogen partial pressure. The initia l spike o f hydrogen

partial pressure is about 10 times less than what was predicted by Bagley and Brodkorb

(1999). This is the consequence o f the changes incorporated in the current model. No data




53

are available to justify which prediction is better. Both models predict that the hydrogen

partial pressure rises during the glucose-consumption phase. However, because the

hydrogen partial pressure was 1 0 times higher in the original model, when the hydrogen

partial pressure is plotted, it looks flatter during the glucose-consumption phase.

For short-term simulation, the effect o f initia l biomass concentration and the

specific substrate utilization rate can be combined as one factor and represented by the

maximum substrate utiliza tion rate.

VmA = kiX i (4-1)

where Vm/ is the maximum substrate utiliza tion rate (g COD /L/d). Maximum substrate

utiliza tion rates, instead o f concentrations o f individu al metabolic groups, were predicted

in the models o f Masse and Droste (2000) and Skiadas et al. (2000). A disadvantage o f

this is that the models can neither be used for long-term simulation nor predict the

microbial growth. Because the specific substrate utilization rates used in the current

model are different than those used by Bagley and Brodkorb (1999), comparison o f

maximum substrate utilization rates (instead o f concentrations o f ind ividual metabolic

groups) between the current model and the model by Bagley and Brodkorb (1999) shows

the difference between these two models. Table 4.2 compares the predicted maximum

substrate utilization rates between these two models. The maximum substrate utilization

rates for glucose and acetate are close for both models. The maximum substrate

utilization rate for hydrogen is higher for the current model, and this explains the lower

hydrogen partial pressure simulated in the current model. The maximum substrate

utilization rates for butyrate, propionate, and lactate are lower for the current model. The

actual substrate utiliza tion rates for glucose, lactate, propionate, and butyrate predicted by




54

the two models were close, as both models were fitted to the same set of data. This

indicates that the original model has lower hydrogen inhibition factors for propionate,

butyrate, and lactate under the conditions examined (the pH inhibition functions are the

same for both models). Both models agree that the low aceticlastic methanogen

concentration in the reactor is the reason for poor acetate degradation in the reactor.

However, the current model predicts that low concentration o f propionate acetogens is

the reason for poor propionate degradation, while the original model predicts that there

are sufficient propionate acetogens in the reactor, but the high hydrogen partial pressure

inhibits the degradation o f propionate. The same argument for propionate applies to

butyrate.

Table 4.2: Predicted maximum substrate utilization rate (g COD/L/d)

Model Glucose Lactate Butyrate Propionate Acetate Hydrogen

Current 5.68 2.94 0.03 0.32 0.61 4.32

Original 5.50 12.50 0.15 3.00 0.38 0.46

4.1.3 Model validation

The in itia l biomass concentrations calibrated in Case 1 were then used to simulate

Case 2 following Bagley and Brodkorb (1999). No modifications to the model parameters

were made except for the influent COD. Figure 4.6 presents the simulated and measured

COD, VFA, and glucose concentrations. Figure 4.7 presents the simulated and measured

results for ind ividua l VFAs. The simulated results generally agree w ith experimental

data, although model predictions for case 2 were sligh tly poorer than for case 1 .




56

1200

♦ Acetate Propionate

LactateButyrate _i

O 800 oO)E

c/) 400JQ3CO

12 16 20 240 4 8

Time (hours)

Figure 4.7: Vola tile fatty acid concentrations vs. time fo r Case 2. Lines - model


0.0003

E03

0.0002 -

c0O)2

* f I

-o 0.0001 -

o'u 2CL

8 12 16 20 240 4

Time (hours)

Figure 4.8: Predicted hydrogen partial pressure vs. time for Case 2.




57

7.5

xQ .

6.5

0 4 8 12 16 20 24

Time (hours)

Figure 4.9: pH versus time for Case 2. Line - model prediction; symbols - measurements

o f Bagley and Brodkorb (1999).

6.00

d 4.00 -<DE

O>

g 2.00 -

CH.

0.00

0 4 8 12 16 20 24

Time (hours)

Figure 4.10: Cumulative gas production versus time for Case 2. Lines - model





58

4.2 Model verification with data from Kim (2000)

4.2.1 Description of the experimental data

The laboratory AnSBR used by Kim (2000) was the same used by Brodkorb

(1998). The reactor was operated with a 12.0 L total liquid volume, 6.0 liter fill volume,

and 6.0 lite r settled volume. The reactor was seeded with granulated anaerobic sludge

(Champlain Industries, Cornwall, ON). Average volatile suspended solids (VSS)

concentration in the reactor was 10,050 mg COD VSS/L. The synthetic wastewater feed

consisted o f lactose as sole COD source, sodium bicarbonate as alkalinity, and necessary

nutrients and trace metals. M ixing was conducted w ith gas re-circu lation and the reactor

was maintained at 22°C (K im , 2000).

Table 4.3: Operating conditions (K im , 2000)

Parameter Run 1 Run 2 Run 3 Run 4 Run 5

Organic loading rate (g COD/l/d) 2 . 1 2 . 1 2 . 1 2 . 1 2 . 1

F ill time (hours) 9 5 3 1 0 6

React time (hours) 1 5 7 1 2 16

Settle time (hours) 1.5 1.5 1.5 1.5 1.5

Decant time (hours) 0.5 0.5 0.5 0.5 0.5

Total cycle time (hours) 1 2 1 2 1 2 24 24

Hydraulic retention tim e (hours) 24 24 24 48 48

Number o f cycles 9 9 9 3 2

Five separate experimental runs were conducted to evaluate the performance o f

the AnSBR w ith respect to varying operational parameters. The operating conditions are

listed in Table 4.3. These experiments were conducted in order from Run 1 to Run 5. The




59

rector was starved for 1 to 2 days between runs, to allow maximum conversion o f

degradation products to biogas prior to the next run. For Runs 1, 2, and 3, samples were

taken during every other cycle. The measured variables included pH, lactose, glucose,

VFAs, and soluble COD. The hydrolysis substrate utilization rate for lactose measured by

K im (2000) was 109.8 gCOD/gXCOD/d.

4.2.2 Model simulation

Each run was multi-cycle, and a shift in microorganism populations was expected

between runs due to microbial growth and decay. Therefore biomass concentrations

predicted from the previous run were used as initial biomass concentrations for the

current run.

Initial biomass concentrations were predicted by fitting the model prediction to

the experimental data on propionate and acetate only. Data from Run 1 were used for th is

prediction. The measured lactose, glucose, lactate, and butyrate concentrations were

generally very low even during the fill stage. Therefore they were not used in the model

prediction, as a small difference between the predicted and measured values resulted in a

poor r value. Initia l biomass concentrations that provided an optimum f it to the measured

propionate and acetate data were (in mg C OD/L): X F, 115; XL, 85; X B, 5; X P, 60; X A,

100; X h, 175; Xs, 50; Xi, 6410. The most sensitive biomass concentrations for this

optimization are X h and X A.

Figure 4.11 presents the comparison between the predicted and measured

propionate and acetate for Run 1. The model predictions match the experimental data

very w ell with r 2 values o f 0.86 for propionate and 0.78 for acetate. The concentrations o f




60

lactose, glucose, lactate, and butyrate predicted by the model are very low ( < 1 0 mg

CO D/L), which is in agreement w ith experimental data o f K im (2000).

The predicted biomass concentrations from Run 1 were used as input for Run 2

simulations. The input biomass concentrations o f the follow ing runs were from the output

o f its preceding run. Figures 4.12, 4.13, 4.14, and 4.15 present the comparison between

the predicted and measured propionate and acetate for Runs 2 through 5. Overall the

model predictions match the experimental measurements very well. Considering the

length o f the simulation and the w ide range o f operational conditions, the model

predictions are satisfactory. The fit between the model prediction and measurement data

for Runs 2 to 5 can be improved i f the in itia l biomass concentrations are predicted by

optimization. Masse and Droste (2000) identified different initial biomass concentrations

for different tests.

400« Acetate

° Propionate

300 -

u>E 200 -

-§ 100

0 24 48 12072 96

Time (hours)

Figure 4.11: Vo latile fatty acid concentrations vs. time fo r Run 1. Lines - model

predictions; symbols - measurements o f K im (2000).




61

400

Acetate

Propionate

300QOO03

E 200

00oo

0303

CO

■§ 100< Z>

0 24 48 72 96120

Time (hours)

Figure 4.12: Vola tile fatty acid concentrations vs. time for Run 2. Lines - model

predictions; symbols - measurements o f Kim (2000).

600

Acetate

Propionate

oo _ J

§ 400 OO)E,

co 2 0 0JOdcn

0 24 48 72 96 120

Time (hours)

Figure 4.13: Vola tile fatty acid concentrations vs. time fo r Run 3. Lines - model

predictions; symbols - measurements o f K im (2000).




62

Q

OOo>E,

<t>

to

JD

CO

1000

Acetate

Propionate800

600

400

200

0

24 48

Time (hours)

72

Figure 4.14: Vola tile fatty acid concentrations vs. time for Run 4. Lines - model


1000

Acetate

Propionate800

o

8 600 o>E

& 400cc

(/) _Q

w 200

0 24 48

Time (hours)

Figure 4.15: Vo latile fatty acid concentrations vs. time for Run 5. Lines - model





63

Several factors m ight contribute to the discrepancy between the model predictions

and measurement. First, the starvation period between runs was not simulated because the

exact lengths o f each starvation period are not available. Second, sign ificant amounts o f

lactose (influe nt substrate) were detected for Run 2 only (K im , 2000), ind icating an

anomaly in performance during Run 2.

Table 4.4 presents the input biomass composition for each run. The biomass

composition for Run 1 was quantified fittin g the model prediction to the experimental

data. The input biomass o f other runs was the output o f its preceding run. These results

indicate that the shift in microorganism populations is significant for the condition

simulated.

Table 4.4: Input biomass composition fo r each run

Run No. Xs

(%)

X,

(%)

X F

(%)

x P

(%)

X L

(%)

X A

(%)

X H

(%)

1 0.7 29.2 32.0 0.07 1 2 . 1 2 . 6 3.4 19.9

2 6.3 25.0 32.1 0.14 11.3 2.4 5.6 17.1

3 8.9 2 1 . 8 32.2 0 . 2 0 1 1 . 6 2.4 7.3 15.5

4 1 0 . 2 18.3 32.4 0.32 13.2 2 . 6 8 . 6 14.3

5 10.9 16.0 32.7 0.41 13.8 2.7 9.6 13.9

4.3 Model verification with data from Denac et al. (1988)

In the previous sections, the in itia l concentrations o f ind ividual m icrobial

populations were obtained by fitting the model simulation results to experimental

measurements. Though techniques for partitioning the total biomass have been proposed




64

in the literature (Maillacheruvu and Parkin, 1996; Merkel et al., 1999), direct

measurement o f the microbial population d istribution is very d ifficu lt. Fortunately, i f the

reactor was operated under steady state, the concentrations for different microbial

populations can be predicted by steady-state model simulation.

The reactor was a laboratory-scale fluidized bed reactor in the study o f Denac et

al. (1988). The pH was maintained at a value o f 7.0 by the automatic addition o f sodium

hydroxide. The reactor was operating in itia lly under steady-state conditions. Steady-state

operational and performance parameters are summarized in Table 4.5. Some o f the

parameters listed in Table 4.5 are simulated results from the model. The model predicted

VSS microbial population distributions are listed in Table 4.6.

For steady-state simulation, the initial biomass concentrations were set and the

model was run long enough to reach steady state. The solids wasting rate was adjusted to

reach the desired solids retention time. Though the initial biomass concentrations have no

effect on steady-state performance, convergence improves i f they are set at higher values.

Figure 4.16 shows the sim ulation results o f step changes in the influen t substrate

concentration from 2240 mg COD/L to 7839 mg COD/L. This corresponds to an organic

loading rate (OLR) o f 31.4 g CO D/L/d. Figure 4.16 compares the simulation result and

Denac et al. (1988)’s experimental data (from Costello et al., 1991b). The model gives a

reasonable prediction o f acetic and butyric acid, but poor prediction o f prop ionic acid.

Costello et al. (1991b) have simulated Denac et al.’s (1988) experiment. Compared to the

study o f Costello et al. (1991b), where only trends o f VFA accumulation were predicted,

the current model improves the predictions o f acetate and butyrate significantly.




65

Table 4.5: Steady state operational and performance parameters

Parameter Value

Liquid volume1 (L) 3.6

Gas volume1 (L ) 0.425

Temperature1 (°C) 35.0

Liquid residence time1 (day) 0.25

Reactor pH 1 7.0 (constant)

Influent glucose1 (mg COD/L) 2240

Organic loading rate (g/L/d) 8.96

Solids retention time1 (day) 1 1

Volatile suspended solids2 (g COD/L) 16.4

Effluent acetate2 (mg COD/L) 73.2

Effluent propionate2 (mg COD/L) 57.6

Effluent butyrate2 (mg COD/L) 5.7

Experimental conditions (Data o f Denac et al., 1988; From Costello et al., 1991b).

2 Model prediction.

Table 4.6: Predicted steady-state microbial population distributions

Xs X, X F x P X L X B X A X H

(%) (%) (%) (%) (%) (%) (%) (%)

14.6 3.4 34.3 3.1 10.9 2.3 13.3 18.1

The discrepancy between the simulated and experimental concentration o f

propionate might be attributed to the initial steady-state condition simulated, which was




66

not verified because data are not available. The model fi t can be improved i f the in itia l

biomass concentrations are obtained by optimization.

1500

♦ acetate ° propionate x butyrate

O)E 1000co»

2+Jcd)ocoo

500

240 48 72

Time (hours)

Figure 4.16: Comparison between the simulated and experimental results for the

accumulation o f organic acids (Denac et al., 1988; Costello et al., 1991b).

A common disadvantage among the models in the literature is that the models are

applied to dynamic situations without being validated for steady-state conditions. For

example, Costello et al. (1991b) and Batstone et al. (2000b) did not report any model

prediction for steady-state conditions. A reasonable sequence is that the model is

validated with steady-state conditions first, and then it is applied to dynamic situations,

such as shock loading and a single AnSBR cycle. The fo llow ing chapter is an effort to

validate the model w ith steady-state reactor performance data.




Chapter 5 Rate Limiting Step of Anaerobic Treatment

Anaerobic treatment involves multiple series and paralle l reactions converting

complex substrates to end products. These reactions proceed at different rates; some

reactions are fast, while others are slow. Identifying the rate lim iting step is important for

the design organic loading rate o f anaerobic reactors. For example, i f the rate lim iting

step is oxygen gas transfer fo r aerobic treatment, the design organic loading rate w ill

depend on the oxygen gas transfer rate. The rate lim iting step is also important fo r the

modeling o f anaerobic processes. With the rate lim iting step identified , the model can be

simplified without sacrificing its predictive ability. More attention should be paid to

modeling the rate lim iting step, thus providing better model performance.

For anaerobic digestion, when suspended solids are the main source o f COD,

hydrolysis is often the rate-lim iting step (Gossett and Besler, 1982). The suspended solids

must be hydrolyzed into simpler molecules prior to being degraded by the acetogenic

microorganisms. For high rate anaerobic treatment, when the substrate is prim arily

composed o f soluble components, many researchers proposed that aceticlastic

methanogenesis is the rate lim itin g step, due to the fact that aceticlastic methanogens are

slow growing bacteria (Andrew s and Graef, 1971; Henze and Harremoes, 1983).

However, propionate buildup had often been observed as the firs t sign o f reactor overload

(Bjonsson et al., 1997). This implies that aceticlastic methanogenesis may not be the rate

lim iting step for high rate anaerobic treatment.

Anaerobic sludge is a m ixed culture o f many metabolic groups o f bacteria. The

degradation rates are a positive function o f the amount o f active biomass o f each

ind ividua l group present in the sludge. Therefore, the rate lim iting step is strongly related

67




69

where b, is the decay constant o f microorganism group X, (d-1).

Equation 5.2 is applicable for the overall anaerobic degradation. Therefore the

solid retention time can also be written as

(53)

where X t is the total solids in the system (g COD/L); et is the overall removal efficiency;

Lt is the total organic loading rate (g COD/L/d); Yt is the overall yield (g COD VSS/g

COD); and bt is the overall decay constant (d_1).

Equation 5.3 is an equivalent o f the general expression o f solids retention time for

a CSTR at steady-state, which is w ritten as (Pavlostathis and Giraldo-Gomez, 1991):

6 x = ----- (5.4)UYt-bt

where U is the specific substrate utiliza tion rate and is written as

U = - ^ — (5.5)K s + S

A t steady state, substrate loading rate equals substrate u tilization rate, hence

£tLt - UXt (5.6)

Therefore, Equation 5.3 can be converted to Equation 5.4. Equation 5.3 emphasizes the

aspect that the availability o f substrate lim its the growth rate o f microorganisms. This is

important fo r the analysis o f anaerobic processes, as some metabolic groups, such as

lactate acetogens, have high substrate utilization rates but the substrate is limited.

From a mass balance on solids,

X t ^ X i (5.7)i




70

and

btX t = '£ b iX i (5.8)i

However,

Lt ^ (5.9)i

and

(5.10)i

The loading rates may be counted more than once because products are not being

subtracted. In other words, on a loading rate basis, in all likelihood:

H L i > L t (5.11);

For the completely mixed anaerobic system, each metabolic group in the reactor

has the same solids retention time and is the same as the overall solids retention time.

Therefore, the follow ing relationship holds

^ ^ -------- (5.12)S i L ^ - b i X i £t L t Yt —btX t

Equation 5.12 is rewritten as

^ f L - b i = ^ f L - b t (5.13) A i A t

The decay rates o f each metabolic group are very close. In consideration o f Equations 5.7

and 5.8, it is expected that bt is o f the same magnitude as b,. To s im plify the analysis, it is

assumed that bt = bt (the influence o f this assumption is minimized fo r the estimation o f

maximum SOLR, as bt is negligible when L t / X t is large), and that all substrates are




71

completely consumed, which means £[=1 and st = \ . Therefore equation 5.13 is

simplified to

X. = ' ' X, (5.14)l j ,

Equation 5.14 indicates that the fraction o f an individu al metabolic group is

proportional to the product o f substrate and growth yield for an anaerobic system,

because in anaerobic systems different groups o f bacteria consume different substrates.

This is not the case when d ifferen t groups o f bacteria compete for the same substrate. In

that case the fraction o f an individual group o f bacteria is proportional to the maximum

specific growth rate // (Henze and Harremoes, 1983). An example o f the latter is the

denitrification process.

Now define

L i = f i L t (5.15)

where f is the fraction o f total COD that is consumed as the constituent o f interest. For

example, f i is the fraction o f total COD that is consumed as lactate

( f i = Yrhod (1 - Yp )7/m-i X where Yp is the yield o f heterotrophic-hydrolytic organisms (g

COD VSS/g COD) and Yphod is the theoretical product yield (see Bagley and Brodkorb,

1999). For lactate production from glucose, Yrhotj = 1. X f ) * 1 because fo r fermentative

organisms, f p = 1. The biomass mass balance closes, however, because Yp accounts for

the fact that incom ing COD is transformed into both biomass and products.

A t steady state, the overall degradation rate Dt must be greater than or equal to the

overall loading rate Lt. Otherwise substrate w ill accumulate. The same is true for every

intermediate, therefore, at steady state




72

D; > L; (5.16)

where D, is the degradation rate (g COD/L/d).

The general form o f D, can be expressed as

Di = ViUfXt (5.17)

where iji is an inhibition factor; and U, is the specific substrate utilization rate as

expressed in Equation 5.5 (g COD/g COD VS S/d).

Combining Equations 5.16 and 5.17 gives

where Bx is the SOLR (g COD/ g COD VSS/d).

Equation 5.20 describes the lim itation imposed on the SOLR by an ind ividu al

intermediate degradation product in the multi-step anaerobic process. For the successful

operation o f the anaerobic process, the condition described by Equation 5.20 must be

maintained for every constituent.

The loading lim itatio n expression for any substrate can be readily determined

using Equation 5.20. Table 5.1 (column 2) presents the SOLR lim itation expression for

each major substrate in the anaerobic system. The pH inhibition term is not included as

ViU iXi > Lt (5.18)

And substituting Equation 5.14 into Equation 5.18 gives

(5.19)

Equation 5.19 is further s implified to

(5.20)




73

neutral pH could be maintained. However, the hydrogen partial pressure inhibition term

r/i cannot be neglected.

The hydrogen inhibition function for propionate degradation was derived in

Chapter 3 and is written as

V l x i x p -1 A P ^ *H2

K I , H 2 X P

(5.21)

ConsideringK p + S P

<1, the SOLR (Bx) lim itation equation for propionate can be

written as

/ \ 3 ~

< ^ k P 1 -

P 1 H I

Y, y K / H 2XP ,(5.22)

Other equations are simplified appropriately and are presented in the third column

o f Table 5.1.

Table 5.1: Lim itation on individua l substrate degradation

Substrate Lim itation Equation Sim plified Lim itation Equation

HydrogenBx <

Y H k H S H

Yt K h + S hB x <

Y H k H S H

Yt K h + S h

Propionic acid Yp kpS p

Bx-Tl Y ^ Pn,-H2xe 1 -( P 1H2

K, H2XP

Acetic acidBx <

Y a kASA

Yt K a + S a

Butyric acid Yb kBSB x v v P l , H 2 X B

r t k b + ^ bB.r 4 * <

\ 2

H 2

V I ,H2XB y




74

3

COif)>

0

DB

0 0.00005


0.0001

Figure 5.1: Lim itation on specific organic loading rate imposed by ind ividua l

constituents. Letters are described in the text.

Figure 5.1 plots the SOLR (Bx) versus hydrogen partial pressure according to the

sim plified lim itation equations in Table 5.1. Except Yt, all parameters in those equations

were reported earlier in Chapter 3. Y, is 0.18 gXCOD/gCOD (Pavlostathis and Giraldo-

Gomez, 1991). Line H is the upper lim it o f SOLR for hydrogen degradation. A t very low

substrate concentration the Monod equation becomes first order, therefore the upper lim it

o f SOLR for hydrogen consumption is a positive linear function o f the hydrogen partial

pressure; in other words the hydrogen degradation rate increases with the increase o f

hydrogen partial pressure in the reactor. Line P is the upper lim it o f SOLR fo r propionate

degradation. The upper lim it o f SOLR fo r propionate decreases w ith the increase o f

hydrogen partial pressure and is close to zero when the hydrogen partial pressure reaches

the thermodynamic lim it for propionate degradation. Line B is the upper lim it o f SOLR




76

determined fo r a ±10% change in each parameter. The results o f the analysis are

presented in Table 5.2.

Table 5.2: Sensitivity analysis o f model parameters on estimated maximum SOLR1

Parameter kH K h Y h Y t Y P kp K i,H2XP

Change 10% 1.04 0.89 1.04 0.89 0.98 0.98 1.04

Change -10% 0 . 8 8 1.05 0.87 1.08 0.94 0.94 0 . 8 8

The hydrogenotrophic methanogens have a very high maximum specific substrate

utilization rate compared to the hydrogen production rate in an anaerobic system.

Normally the hydrogen level can be kept very low in an anaerobic system and the

degradation o f hydrogen is kept in the first-order region o f the Monod equation. For an

anaerobic reactor operated under steady state, the hydrogen production rate equals the

hydrogen consumption rate, therefore the following relationship holds

y H k H s H

Bx = (5.23)Yt K h + S h

Figure 5.2 plots hydrogen partial pressure versus SOLR. Figure 5.2 illustrates the

operation o f an anaerobic system under steady state. The ac tivity o f hydrogen consuming

methanogens determines the hydrogen level in the reactor. When SOLR is low, the

reactor w ill be operated at a low hydrogen level. W ith the increase o f SOLR, the

hydrogen level increases along the line in Figure 5.2. In other words, the anaerobic

system responds to an increase in SOLR by increasing the hydrogen level. The hydrogen

degradation rate depends strongly on the hydrogen concentration in the reactor, w ith a

higher hydrogen concentration leading to a higher hydrogen degradation rate. When the

SOLR is lower than or equal to the critical point C shown in Figure 5.1, all other




77

microbial groups are able to completely consume their substrates. When the SOLR is

higher than the critical value, the elevated level o f hydrogen partial pressure inhib its the

degradation o f propionic acid. The propionate acetogens in the system w ill not be able to

completely consume the propionate produced in the system. Many researchers have

shown that build-up o f prop ionic acid is the first sign o f reactor overload (Bjonsson et al.,

1997). Therefore, propionate degradation is the rate-lim iting process for anaerobic

treatment.

0.0001

_ 0.00005 -

O)

-O

2 310

SOLR (g COD/g COD VSS/d)

Figure 5.2: Hydrogen partial pressure versus SOLR.

Figure 5.1 also predicts the responses o f an anaerobic system under differen t

levels o f overloads. When the reactor is subject to low-strength overload (the SOLR is

greater than 0.9 mg COD/ mg COD VSS/d (point C) but less than 1.9 mg COD/mg COD

VSS/d (point D)), propionic acid w ill be the primary COD in the effluent, w hile acetic

acid and butyric acid in the effluent w ill be low. When the reactor is subject to severe

overload (the SOLR is close to or greater than 1.9 mg COD/ mg COD VSS/d), the

concentrations o f all VFAs w ill be high in the effluent.




78

The hydrogen inhibition function derived in this study (Equation 5.21) was used

in the analysis above. I f the hydrogen inhib ition function used by ADM 1 (Batstone et al.,

2002) is applied, the conclusion about the rate-lim iting step w ill be the same. However,

the estimated maximum SOLR w ill be 0.13 g COD/g COD VSS/d, assuming that other

parameters remain the same. This value is much lower than the reported value o f Fang

and Chui (1993). This also confirms that the hydrogen inhibition function used by ADM 1

is inappropriate.

5.2 Microbial population distribution

The glucose degradation products are distributed among acetate, propionate, and

butyrate. Some models (Batstone et al., 2002; Skiadas et al., 2000; Merkel et al., 1999)

use constant coefficients to model the product distribution. Other models (Mosey, 1983;

Costello et al., 1991a; Bagley and Brodkorb, 1999; Masse and Droste, 2000) suggest that

the product distribution coefficients vary w ith hydrogen partial pressure in the reactor. In

both cases the coefficient values have to be determined. Equations 5.14 and 5.15 relate

the fraction o f a metabolic group dire ctly w ith its correspondent substrate loading rate.

Therefore the production and consumption rate o f an intermediate product, such as

propionate and butyrate, can be estimated from the microb ial population distributions o f

the sludge. This provides a method to verify the product distribution coefficients

(constant or a function o f hydrogen partial pressure) used in the model. It is very d ifficu lt

to directly measure the individual metabolic groups in the mixed culture. However, the

fraction o f a metabolic group can be estimated from a sludge activ ity test, which is easy

to perform. The sludge activity can be related to the specific substrate utilization rate k as

SA, = M i (5.24) A t




80

Table 5.3: Comparison o f anaerobic sludge activity (g substrate/g VSS/d)

Hydrogen Propionate Acetate Glucose

Measurement1 0.40 0.13 0.91 1.99

Prediction 0.49 0.15 1.39 2 0 . 0

Shen and Guiot (1996).

Table 5.4 compares other reactor performance data between the model prediction

and measurement (Shen and Guiot, 1996). In general the model prediction agrees well

w ith the measurements. The predicted biomass washout rate is lower than the measured

value, indicating that the yield and decay values for the model which were determined

from other data (Table 3.2) may not be completely appropriate for these experiment

conditions. The amount o f alka linity added was not reported. The input a lka linity was

adjusted to maintain the simulated reactor pH the same as the reactor pH value reported.

Table 5.4: Comparison o f reactor performances

Item Measurement1 Prediction

pH 6.78 6.79

COD removal rate (%) 95.4 97.7

Biomass washout rate (g VSS/L/d) 0.52 0.23

CH4 in the biogas (%) 43.6 52.8

CO2 in the biogas (%) 49.1 44.5

Gas flow rate (L/d ) 8.49 10.28

CH4 yield (% CODin) 64.1 8 6 . 6

Shen and Guiot (1996).




81

Table 5.5: Predicted microbial population distributions

SRT

(d)

SOLR

(g COD/g COD VS S/d)

Xs

(%)

x ,

(%)

X F

(%)

x P

(%)

x L

(%)

X B

(%)

x A

(%)

X H

(%)

100 0.10 16.6 25.5 29.8 2.9 6.6 1.3 9.3 8.0

50 0.13 18.4 15.2 33.1 3.1 7.6 1.6 10.9 10.1

10 0.35 15.7 4.1 34.3 3.1 10.2 2.2 13.3 17.1

8 0.42 14.6 3.4 34.3 3.1 10.9 2.3 13.3 18.1

Table 5.5 shows the simulation results o f microb ial population d istribu tion under

different SOLRs and SRTs. The reactor was assumed to be a continuous reactor, and

glucose was the sole COD source for these simulations. The SOLR affects the

distribution o f m icrobial populations. The reactor with a lower SOLR w ill have lower

biomass production and wasting rates and relatively high biomass concentration.

Therefore, lower SOLRs mean a longer solids retention time at steady state. It is known

that for anaerobic digesters w ith long solids retention times, a considerable portion o f the

digester VSS content is non-viable biomass (Skiadas et al., 2000). This is in agreement

with the prediction in Table 5.5.

The microbial population distributions in Table 5.5 are also in agreement with the

kinetic data reported by Aguilar et al. (1995). They measured the kinetic data for the

degradation o f VFAs by glucose pre-grown sludge, which was from a continuous

laboratory digester operating at 18 days o f hydraulic retention time and fed fo r three

years w ith synthetic medium containing glucose (10 g/L). They reported that the

maximum substrate utilization rates (Vm = kX) by 50 mL o f sludge were 46.8 m g/L/h




(49.92 mg C OD/L/h) for acetic acid and 11.84 mg/L/h (17.92 mg CO D/L/h) for propionic

acid (the VSS was not reported). It can be deduced from Equation 5.24 that

This ratio is in agreement w ith the predictions shown in Table 5.5.

Sludge methanogenic activ ity (SM A) reflects the potential o f anaerobic sludges to

convert soluble substrate into methane and carbon dioxide. The substrate used for the

SM A test is usually acetate or a mixture o f VFAs (James et al., 1990). The SMA o f the

sludge can be calculated as follows i f the substrate is acetic acid:

where SMA is the sludge methanogenic activity (L CHVgVSS/d).

James et al. (1990) measured the SMAs o f the sludge from 5 UASB reactors, 3

treating medium-strength wastewaters and 2 treating low-strength wastewaters. They

reported that the SMAs were in a range o f 0.20-0.40 L CHVg VSS/day. Because the

simulations cannot be applied to those reactors. However, the SMAs calculated for the

sludges in Table 5.5 range from 0.28 to 0.40 L CHTgVSS/d respectively, which is in

agreement with the measurement o f James et al. (1990).

kAX A = 49.92 = . g(5.25)

kpXp 17.92

and

(5.26)

(5.27)

detailed operational conditions o f the reactors (James et al., 1990) are not available, d irect




83

5.3 Discussion

As was pointed out in Chapter 1, some models (Costello et al., 1991a; Bagley and

Brodkorb, 1999) predict very low concentrations o f propionate acetogens (XP) in the

sludge. Equation 5.14 explains this problem. Those models used a product regulation

function for glucose degradation that produces very little propionate under low hydrogen

partial pressures, which is the normal operational condition o f an anaerobic reactor.

Using the regulation equations and parameter values provided in Costello et al. (1991a)

and Costello et al. (1991b), it can be estimated that the fraction o f total COD that is

consumed as propionate (fp) is 0.00045 (neglecting growth) when the hydrogen partial

pressure is l.OxlO - 5 Pa, and is 0.032 when the hydrogen partial pressure is l.O x ltT 4 Pa. If

fp =0.00045, then X p / X t =0.00015 by Equations 5.14 and 5.15 using the yield

coefficients o f the current model.

The ADM1 (Batstone et al., 2002) uses fixed regulation parameters (independent

o f hydrogen partial pressure). The recommended stoichiometric yie ld for propionate

(fpro.su ) is 0.27. The f P w ill be sligh tly lower than 0.27 when growth is considered. I t can

be estimated that X p / X t is around 0.09 for the ADM1 model. This value is close to the

value predicted in Table 5.5.

The ha lf velocity constant o f propionate degradation is low er than that o f acetate

degradation. Under normal operating conditions (when the reactors are not overloaded),

the effluent contains more acetate than propionate. This might mislead researchers to the

conclusion that aceticlastic methanogenesis is the rate lim iting step.




84

5.4 Conclusions

The following conclusions are made from these simulations:

1. The rate lim itin g step for high rate anaerobic treatment is propionate degradation.

2. The maximum SOLR is estimated to be near 0.97 gCOD/ gCOD VSS/d, but it is a

function o f the specific k inetic constants, yields, and hydrogen inh ibition function

used. The maximum organic loading rate w ill depend on the amount o f active

sludge a reactor can retain.

3. The sludge microb ial population distribution is a reflection o f the amount of

intermediate product produced and consumed. Therefore it is useful for choosing

appropriate glucose degradation product distribution coefficients.

4. The model provides a good prediction o f m icrobial population distributions.




Chapter 6 Simulations of the Startup of Anaerobic Reactors

6.1 Introduction

The startup process is very important fo r the operation o f anaerobic reactors and

has been investigated by many researchers (Morvai et al., 1992; Ghangrekar et al.,

1998;Tay and Yan, 1996). Most o f these studies were laboratory examinations, and these

studies enhance the understanding o f the startup process.

The seed sludge for startup o f anaerobic reactors is normally from an anaerobic

sludge digester because o f its a va ilab ility. The anaerobic sludge digester biomass

contains a considerable portion o f non-viable biomass (Skiadas et al., 2000), and the main

reasons for this are as follows: First, there is a large amount o f inert biosolids in the feed

to an anaerobic sludge digester. The inert biosolids m ix w ith the active biomass in the

digester and decrease the percentage o f active biomass in the sludge. Second, anaerobic

sludge digesters are operated under long solids retention times and low specific organic

loading rates.

The startup o f anaerobic reactors is a process to achieve its design organic loading

rate. From Chapter 5, it is known that the maximum organic loading rate is related to the

amount o f active biomass in the reactor. Therefore, the startup o f an anaerobic reactor is a

process o f accumulation o f active biomass. This can be done by im prov ing the quality o f

the sludge (percent o f active biomass in the sludge) and increasing the quantity o f sludge

(amount o f sludge retained in the reactor). The quantity o f sludge a reactor can retain is

strongly related to the settling characteristics o f the sludge. The seed sludge from an

anaerobic sludge digester is flocculent sludge, which has very poor settleability. More

85




86

sludge can be retained in the reactor only when the sludge in the reactor is granulated,

since granulated sludge has very good settling characteristics.

The SMA represents the percentage o f aceticlastic methanogens in the sludge.

Aceticlastic methanogenesis is the last step o f the series o f reactions happening in

anaerobic degradation. The growth o f aceticlastic methanogens indicates the growth o f

other groups o f bacteria as well. Therefore, the SM A could be a good indicator o f the

total active biomass in the sludge. On the other hand, sludge activity on glucose may not

be a good indicator for the total active biomass, because latter reactions may not proceed.

The improvement o f sludge quality during the startup process is simulated in this

chapter. This s imulation confirms the predictive ability for m icrobial growth o f the

current model. The granulation process is not simulated but some postulates are presented

in the discussion section.

6.2 Simulation of the startup of UASB reactors

The seed sludge composition used in the following simulations is shown in Table

6.1. This composition was predicted in Chapter 4 using the experimental data by

Brodkorb (1998).

Table 6.1: Predicted seed sludge composition

Xs (%) X, (%) X F(%) X P(%) X L (%) X B (%) X A (%) X H (%)

0.71 91.57 1.64 0 . 8 6 1 . 2 1 0.07 1.43 2.5

Tay and Yan (1996) examined microbial granulation in UASB reactors. The

synthetic wastewater they examined contained glucose, peptone, and meat extract as

carbon sources. The seed sludge was from an anaerobic sludge digester. The startup

operation was guided by maintaining a high microb ial load index o f about 0.8. The




87

microbial load index is defined by the ratio o f the specific organic loading rate applied to

the specific methanogenic activity measured. The reactor was operated for 6 months. The

influent CODs were 1000, 2000, 5000, and 10000 mg/L, but only the run with 1000 mg/L

influent COD was simulated in the following. The initial sludge concentration was 6.06 g

YSS/L. The sludge concentration was 6.5 g VSS/L after 30 days, 9.0 g VSS/L on day 90,

and 18 g VSS/L at the end o f the operation. The pH was 6.85. Table 6.2 lists the loading

history fo r the 6 months operation.

Table 6.2: OLR and HRT (from Tay and Yan, 1996)

Day 0-4 5-17 18-32 33-74 75-127 128-180

HRT (hour) 28 18 8 4 2 . 8 2

OLR (g COD/L/d) 0 . 8 6 1.3 3 6 8 . 6 1 2

A model s imulation was conducted for this experimental run. Figure 6.1 compares

the simulated and measured SMA during the startup o f the UASB. The model predicts

the improvement o f SM A during startup time very well. The discrepancy between the

model simulation and measurement might be attributed to the seed sludge composition

used in the simulation. Tay and Yan (1996) reported the SMA o f the ir seed sludge was

0.02 L CH4/g VSS/d. The SMA o f the seed sludge in Table 6.1 is 0.04 L CH4/g VSS/d.

Tay and Yan (1996) also examined 2000, 5000, and 10000 mg/L influent COD

with the same startup strategy. The 10000 mg/L run was interrupted because o f sludge

washout caused by an overnight fast-running pump. Both the simulation and

experimental results for 2000 and 5000 mg/L influent COD are almost the same as for

1000 mg/L influent COD (Figure 6.1), though the operating conditions were different;

therefore, they are not shown.




88

0.8

5 n f i -■

o

— Simulation

° Experiment

0

0 30 60 90 120 150 180

Time (days)

Figure 6.1: Comparison o f specific methanogenic activ ity during reactor startup (Data

from Tay and Yan, 1996).

Ghangrekar et al. (1998) investigated the effect o f loading rate on the startup time

o f a UASB reactor. The synthetic wastewater contained sucrose as the carbon source. The

inoculum was from an anaerobic sludge digester. The reactors were operated at a constant

OLR and SOLR during operation for about 90 days. The pH in the reactor was

maintained at 6 .8 . The OLR and SOLR values were in the range 1.5 to 9.0 g COD/L/d

and 0.1 to 0.6 g COD/g VSS/d, respectively. The time o f startup was the time required fo r

each reactor to achieve steady-state COD removal e fficiency (Ghangrekar et al., 1998).

Table 6.3 summarizes the simulation conditions and results. The measured startup

times by Ghangrekar et al. (1998) are also listed in Table 6.3 for comparison. They are in

good agreement w ith simulation. Figure 6.2 plots the simulated effluent COD versus time

during startup operation.




89

Table 6.3: Startup time simulation conditions and results

Simulation No. SI S2 S3 S4 S5

SOLR (g COD/gVSS/d) 0 . 1 0 . 2 0.26 0.3 0 . 6

OLR (g COD/l/d) 1.5 3.0 4.0 4.5 9.0

Influent COD (mg/L) 1 0 0 0 1 0 0 0 2 0 0 0 3000 3000

Hydraulic retention time (hour) 16 8 1 2 16 8

Initial sludge concentration (gVSS/L) 15 15 15 15 15

Simulated startup time1 (day) 15 2 0 23 25 55

Experimental startup time2 (day) 16, 2 0 2 1 , 2 1 19,21,22 30,50 45,45

Model prediction

M ultip le measurements by Ghangrekar et al. (1998)

2000

_iQO

OCT>E

'g 10004-»

5-t-*c0ocoO

0 10 20 30 40 50 60

Time (days)

Figure 6.2: Predicted effluen t COD during startup. From bottom to top: S I, S2, S3, S4,

S5.




90

6.3 Simulation of the startup of AnSBRs.

A startup simulation was conducted fo r an AnSBR. The simulation conditions are

presented in Table 6.4. The seed sludge was assumed to be the anaerobic digester sludge

as shown in Table 6.1 and glucose was assumed to be the carbon source. The startup

strategy was as follows. The initial OLR was 0.5 g/L/d. When the effluent COD

concentration was below 150 mg COD/L, the OLR was increased to 1.0 g/L/d. From then

on, the OLR was increased by an increment o f 1.0 g/L/d by increasing the influent COD

when the effluent COD was below 150 mg/L. The target VSS concentration was 11,000

mg COD/L.

Table 6.4: Simulation conditions

Parameter Value

Cycle length (hours) 24

F ill time (hours) 12.5

React time (hours) 1 0

Settle time (hours) 1

Decant time (hours) 0.5

HRT (hours) 48

F ill volume (L) 6

Settle volume (L) 6

Initial VSS (mg COD/L) 7,000

Target VSS (mg COD /L) 1 1 , 0 0 0

Target organic loading rate (g COD/L/d) 5.0




91

Table 6.5 presents the s imulation results o f startup time requirement for the

AnSBR. It can be seen from Table 6.5 that it takes a longer time to reach the target

effluent criteria for the initial startup period. The time requirement decreases with the

follo w ing stepwise increase o f organic loading rate. For example, it took 13 days to

upload the reactor from 1 g COD/L/d to 2 g COD/L/d, w hile it took only 5 days to upload

from 4 g COD/L/d to 5 g COD/L/d.

Table 6.5: Startup time requirement

OLR (g COD/L/d) 0.5 1 . 0 2 . 0 3.0 4.0 5.0

Loading duration (days) 3 14 13 9 6 5

Figure 6.3 illustrates the improvement o f SMA over time during the startup

process. Comparing Figures 6.1 and 6.3, it can be seen that the simulation results for

SM A versus time are very close for the UASB and AnSBR even though d ifferent startup

strategies have been used. An implica tion o f this is that AnSBRs m ight have a startup

time requirement very close to UASBs. Another imp lication is that the startup o f an

anaerobic reactor can be accomplished by monitoring the effluent COD instead o f

monitoring the SMA. Measuring effluent COD is easier than measuring SMA, so the

startup process can be simplified.




92

0.8

§ 06 co >.O)

f 0.4 O

_i

<0.2

co

100 150500

Time (days)

Figure 6.3: SMA versus time fo r the startup o f an AnSBR.

6.4 Discussion

The SM A o f digester sludge is about one twentieth o f the SM A o f high rate

anaerobic reactor sludge (Tay and Yan, 1996). The estimated maximum SOLR for a high

rate anaerobic reactor is 0.97 g COD/ g COD VSS/d (Chapter 5). Therefore the maximum

SOLR applied to the in itia l period o f anaerobic reactor startup should be about 0.05 g

COD/g COD VSS/d, otherwise the growth o f propionate acetogens w ill be inhibited. The

optimal OLR applied to the in itia l period o f anaerobic reactor startup depends on the VSS

concentration in the reactor. As reactor startup is a matter o f accumulation o f active

biomass, the OLR applied during the startup time should be high for maximum

microorganism growth. On the other hand, overload should be avoided so that the growth

o f microorganisms is not inh ibited. This is in agreement with the startup strategy

proposed by Tay and Yan (1996).




93

The accumulation o f active biomass through increasing the percentage o f active

biomass in the sludge is simulated above. The tota l amount o f sludge in the reactor can be

increased significantly when the sludge is granulated. The following discussion shows

that these two processes may be related; it is postulated in the following that granulation

is a consequence o f the increase o f the percentage o f active biomass in the sludge.

Granulation is a process in which a flocculent biomass, which is a conglomerate

w ith a loose structure, forms w ell-defined pellets or granules, which are packed densely

(W irtz and Dague, 1996). MacLeod et al. (1990) reported that granules were three-

layered structures. The internal core o f the granules was composed mainly o f aceticlastic

methanogens. The middle layer consisted o f hydrogen producing acetogens and hydrogen

consuming methanogens. The exterior layer o f the granules contained acidogens and

hydrogen consuming methanogens.

The complete mechanism o f granulation is not completely understood, though

many theories fo r granulation have been proposed in the literature (Tay et al., 2000). I t is

reported that the reactor liquid surface tension, substrate composition, and hydraulic

condition (up-flow velocity o f UASB) a ffect the granulation process (Grootaerd et al.,

1997; Thaveesri et al., 1995; O’Flaherty et al., 1997). Many investigations in the

literature focus on enhancement of the granulation process. I t is reported that polymer

addition enhances the granulation process (Ong et al., 2002; W irtz and Dague, 1996).

Sludge granulation has been widely observed in high rate anaerobic reactors with

different configurations. However, sludge granulation does not happen in anaerobic

sludge digesters. The major difference between high-rate anaerobic reactor sludge and

anaerobic digester sludge is that the anaerobic digester sludge contains a high content o f




94

non-viable biomass. It is reasonable to postulate that anaerobic digester sludge has the

driving force to be granulated. What prevents anaerobic digester sludge from granulation

is its high content of non-viable biomass. The high content of non-viable biomass of

digester sludge is due to the inert biomass in the feed. When the sludge is fed w ith

soluble substrate and has sufficient time for growth, granulation is a natural process for

anaerobic sludge. This postulate is in agreement w ith the experimental results o f Tay and

Yan (1996). Their experimental results indicate that granulation initiated after about 30

days o f operation when the SM A is increased (Figure 6.1).

The model does not differentiate between granulated sludge and flocculent

sludge. For granulated sludge, mass transfer might have significant effect on the substrate

utilization rates (Pavlostathis and Giraldo-Gomez, 1991). The model is applicable to

granulated sludge w ith the assumption that the mass transfer is not the rate-lim iting step.




Chapter 7 Modeling of the Gravitational Settling Process

The gravitational settling model presented in this chapter is stimulated from the

need for a batch settling model for the settle stage o f AnSBRs. This model is valuable fo r

design and operation o f secondary settlers, the most popular application o f gravitational

settling in the activated sludge treatment process. This model may also be applicable for

predicting the solids profile in UASBs, which is an important aspect o f UASB reactor

design (Narnoli and Mehrotra, 1997).

Secondary settlers are the most sensitive parts o f the activated sludge treatment

process (Chancelier et al., 1997). The design o f a secondary settler includes the

specification o f a surface area and depth (Ekama et al., 1997). The design criterion for

settler surface area and its underlying principle, i.e., the solids flux theory, have been

extensively studied in the literature (Ekama et al., 1997). In contrast, studies on

secondary settler depth requirement are lim ited. Some design rules, such as the design

rules recommended by the Institute for Water Pollution Control (IWPC (SA Branch),

1973) and the Great Lakes and Upper Mississippi R iver Board o f State Sanitary

Engineers (GLUMRB, 1968), did not even include a design criterion fo r secondary settler

depth (Ekama et al., 1997).

The Abwassertechniche Vereinigung (ATV, 1976) procedure presented the most

detailed design criterion fo r the secondary settler depth (Ekama et al., 1997). In this

procedure the depth o f the secondary settler was viewed as fou r horizontal layers o f

different depth: (1) the clear water zone, (2) the separation zone, (3) the sludge storage

(zone settling) zone, and (4) the thickening zone. In this procedure, the thickening zone

depth is calculated empirically; it is calculated from the volume fraction occupied by the

95




96

sludge after 30 min o f settling in the 1-litre measuring cy linder i f the in itia l test sludge

concentration is the same as the reactor sludge concentration. For a better design o f

secondary settler depth, knowledge o f the solids pro file in the zone-settling and

thickening zones is required.

The model presented in this chapter assumes ideal one-dimensional settling and

can pred ict the solids pro file in the zone-settling and thickening zones o f secondary

settlers under that condition. The work presented in this chapter can be used to improve

the design o f secondary settler depth. A more direct application o f this model is to the

settle stage o f AnSBRs. The material in Sections 7.1, 7.2, and 7.3 has been presented in

Zheng and Bagley (1998) and Zheng and Bagley (1999), and is reproduced by permission

o f the publisher, ASCE.

7.1 Development of the gravitational settling model

7.1.1 Governing equation for gravitational settling

The one-dimensional force balance over an incremental volum e o f suspension

w ith a unit area and thickness dz (Figure 7.1), with w all effect neglected, leads to (Fitch,

1979):

g ( y T dP dP̂ Du

f ^ - p ^ + Tz + ^ = c - D t+ p <

f c ^1 - —

. a Dt f (7.1)

where g is the gravitational acceleration, ps is the density o f solid particles, pr is the

density o f liqu id, C is the solids concentration, P is the dynamic fluid pressure, Ps is the

effective solids pressure, and Du/Dt and D u f / D t are the local accelaration o f the solids

and liquid, respectively. The z coordinate is defined as positive up.




97

The local acceleration terms are significant only when there is sudden change in

velocity, which is not expected in the compression zone of a thickener. Therefore, the

local acceleration terms are removed, and Equation 7.1 is simplified to (F itch, 1979)

(7.2) p s oz oz

dP PsH dz

Figure 7.1: Force balance over incremental volume o f suspension [Adapted from Fitch

(1979)]

The dynamic fluid pressure gradient can be related to the solids settling velocity

through Darcy's law fo r flo w through porous media (Cho et al., 1993; Fitch, 1993):

dP = - = (7.3)dz

where K\ is the reciprocal o f hydrau lic conductivity and v is the settling ve locity o f the

solid particle relative to bulk fluid flow.

7.1.2 Effective solids pressure, Ps

In a solid matrix, stress is transmitted mechanically by particle contact. The

equation describing the stress-strain relationship fo r a solid matrix is:




98

(7.4)

where M is the modulus o f elasticity, H is the in itia l height o f the solid matrix and Ps is

the effective solids pressure.

Stress transmittance also occurs in a solid-water matrix through particle contact.

Additionally , stress can be transmitted hydrodynamically between particles approaching

each other (Dixon, 1978). The effective solids pressure for the solid-water matrix, then,

arises from a combination o f both mechanically and hydrodynamically transmitted stress

(Fitch, 1979). Under the normal conditions o f gravity settling, however, the

hydrodynamic stress contribution would be expected to exceed the mechanical stress

contribution because the volume fraction o f solids is relative ly small compared to the

total matrix volume.

This hydrodynamic stress is a function o f the approaching rate o f particles.

Furthermore, in contrast to a pure solids matrix, water in the suspension matrix is

squeezed out during compression. As the water is removed, it exerts a friction force on

the solids that opposes consolidation (Dixon, 1978). Both the hydrodynamic stress and

the water frictio n force are functions o f the rate o f compression, dH/dt. Therefore a

constitutive equation for Ps is postulated to be:

where K 2 is a constant for a given suspension.

For a un it area o f a thin suspension matrix undergoing compression (Figure 7.2), a

mass balance on the solids gives:

HC = constant (7.6)




99

where C is the solids concentration in the suspension matrix.

Differentiating Equation 7.6 w ith respect to time gives:

d(HC) d f iC i n dC Q

dt dt dt

which allows Equation 7.5 to be rewritten in terms o f the solids concentration:

dC 1P. = K ,

dt C

(7.7)

(7.8)

H C+dC

- d H

J 2

^ ~ - d H

J 2

Figure 7.2: Response o f solid-water m atrix o f thickness //un der e ffective solids pressure,

Ps(C represents concentration)

Several models for Ps have been proposed in the literature (Bustos and Concha,

1988; Kos, 1977; Landman et al., 1988; Fitch , 1993). A common feature o f these models

used in the modeling o f thickeners is that Ps is a function o f concentration alone.

Equation 7.8 suggests that Ps is a function o f both the loca l concentration, and the rate o f

change in concentration, in accordance w ith D ixon's hypothesis (1978). A n advantage o f

Equat ion 7.8 is that i t predic ts Ps= 0 in t he zone -se t tl ing reg ime , whe re dC/dt = 0 , even

at high concentrations. Other models for Pspredict e ither dPJdz = 0 or Ps& 0 in the zone

settling regime, the latter being counterintuitive.




100

Substituting Equations 7.3 and 7.8 into Equation 7.2 provides an expression in

terms o f solids concentration and solids settling velocity:

/d

— i p , - P , ) c - K , v + V f C ) = 0 (7-9> A

I f U is defined as the bu lk flow velocity (relative to the tank wall), and V is the solids

particle velocity relative to the tank wall, then v can be expressed as

v = V - U (7.10)

Note that v, V, and U are positive down.

To sim plify Equation 7.9, recognize that

j c _ s c + acdzdt dt dz dt

Additiona lly, the solids settling veloc ity can be written as

V = ~ — (7.12)dt

which upon substitution into Equation 7.11 gives

^ = s c _ v ? c (713)

dt dt dz

For one-dimensional settling across a unit area, the continuity equation for the

solids is

S ^ = 8j T C ) = 3 V c + v 3C (?14)

dt dz dz dz

Equation 7.13 then becomes

^ = (7.15)dt dz

Substituting Equations 7.10 and 7.15 into Equation 7.9 gives




101

( 8 —

— ( p , -p , ) c - K , ( V - U ) + J =0 (7.16)Ps d z

which can be rearranged to give:

+ (F _ ^ ) + - p f )c = 0 (7.17)2 & 2 dz dz lV ' Ps s 1^ ’

Equation 7.17 is the general dynamic equation governing gravity compression, as

long as the loca l acceleration term is negligible.

7.1.3 Empirical functions for Ki and K 2

The constant K\ from Equation 7.3 is the reciprocal o f hydrau lic conduc tivity o f a

solids-water matrix. I t is a function o f porosity or concentration alone and is independent

o f flow velocity. The value o f K\ for a given matrix can be determined from batch

settling tests.

The initial interface settling velocity during a batch settling test represents the

zone-settling velocity for the suspension at a given concentration. During zone-settling,

concentration gradients are absent (Cho et al., 1993; Fitch, 1993) so the momentum

equation (Equation 7.2) can be rewritten as

-^-(p , - p , ) c + ^ = o P -11*)Ps d z

where

(7.19)d z

and vo is the interface (zone) settling velocity.

The interface (zone) settling velocity can be described by the exponential model

(Vesilind, 1968)




102

v0 = kx exp(-w,C) (7.20)

where k\ and n\ are experimentally determined constants.

Substituting Equations 7.19 and 7.20 into Equation 7.18 and solving fo r K\ gives

^ gU-P/JCexpfaC) A , — ( / . z l )

PsK

K 2 reflects the ab ility o f a solid matrix to squeeze out liqu id under pressure.

Similar to K\, K 2 is a function o f porosity or concentration alone, and is independent o f

the compression rate. The relationship o f K 2 to concentration has not been experimentally

determined. However, both dynamic hydraulic force and effective solids pressure result

from the fluid friction acting on the solids. Therefore, it is reasonable to postulate that K 2

depends on C sim ilarly to how K\ does, based on the sim ilarity between the equations

describing dynamic hydraulic force and effective solids pressure. The following

constitutive equation for K 2 is proposed:

g f c . - p ^ e x p („2C) (?22)

Psk2

where 122 and are coefficients to be determined, and their values depend on the

characteristics o f the indiv idua l solids-liquid mixture. W hile both m and depend on the

compression characteristics o f the suspension matrix, 122 reflects the relative ease o f

compression o f the same sludge at different concentrations, while k 2 reflects the relative

ease of compression o f different sludges at the same concentration. Values for ri 2 and &2

are determined by fitting the model to solids profiles. These two parameters should be

independent o f operating conditions o f secondary clarifiers.

The expression for K 2 is not exactly the same as that for K\. One o f the Cs in the

K\ expression is changed to C2. This makes intuitive sense as K 2 is expected to have a




103

greater concentration dependence than K\. Furthermore, C exp(ii2C) can be

mathematically estimated as Cexp(«2C) with limited error within certain C ranges by

adjusting «2.

7.2 Batch settling

7.2.1 Governing equation for batch settling process

For batch settling,there is no bulk flow, and (7.17) simplifies to

d2V dK2 dV g ( V- n^ ^ ^ ^ +— (A - Pf F = 0 -23)

oz oz oz ps

This can be restated in terms o f solids concentration and solids particle velocity

by substituting (7.21) and (7.22) into (7.23).

C / d2V — exp{n2C ) - ^ + k2 oz

2 + n2C t dC exp(«2C) —

SF_exp(2]C )f + ]= 0 (?24)

dz k] k2 dz

Note that (7.23) is completely general for batch settling. I f different expressions for K\

and Ki are used, (7.24) w ill be different.

During batch settling, the continuity equation and (7.24) apply in the domain (/2)

between the bottom o f the settling basin and the supematant-suspension interface:

n = M os:SIJ <7-25>

where L is the height o f the supematant-suspension interface measured at any time from

the column bottom.

The initial conditions are given by

Cl = Cn (7.26)l0<z</.o,/=0 0 v '

V\ = 0 (7.27)l o < z < / . , 2 =o v '




104

where Lo is the in itia l height o f a suspension and Co is the initial solids concentration. The

in itia l solids concentration isassumed to be un iform ly distributed between 0 and Lo.

Equation 7.27 was not usedbecause the local acceleration terms wereneglected.

A t the bottom o f the sedimentation basin, solids accumulate so one boundary

condition occuring at z = 0 is:

V\ n = 0 (7.28)lz=0,;>0 v '

A t the supematant-suspension interface, the effective solids pressure is zero:

dV K2— = 0 (7.29)

dz

providing another boundary condition at the interface height, L:

= 0 (7.30)dV_

dz z = L , l > 0

Equation 7.24 and the continuity equation are solved numerically using a

sequential finite difference approach to provide the concentration pro file in a batch settler

w ith respect to time and height.

7.2.2 Numerical formulation

The momentum balance equation (7.24) must be solved simultaneously w ith the

one-dimensional continuity equation (7.31) using a finite-difference scheme.

^ = + (7.31)dt dz dz

For the momentum balance equation, a central difference scheme is applied to the spatial-

derivative terms.

d2v = r; +l- 2 v ; +v:_

dz2 Az(7.32)




105

dc _ c;+1- c;dz Az

d V = ^ L = K L

dz 2Az

and the concentration at Zj is approximated by

(7.33)

(7.34)

C =c n + c nW +1 + W (7.35)

where « denotes the time step; j denotes the spatial step; and Cj is the average

concentration in the region between lines zy and zr /.

Applying the finite -differen ce equations (7.32) through (7.35) to (7.24) gives the

following approximation

-expC” +C" S+i v ^ - 2 v ; + v;_t

Azl

+

C” + C" 2 + n J -

-exp n- C" +C

s~m s~m

S +1 ~ S

Az

F " - F "yJ+1 %-i

2Az(7.36)

exp

v — F f +1 = 0kx

W ith a known concentration (C) fie ld, boundary conditions and initia l conditions,

(7.36) can be solved for the settling velocity (V) field. Then the settling velocity field is

used as shown below to solve (7.31) and predict the concentration fie ld at time n+1.

For (7.31), a forward difference scheme is applied to the time-derivative term:

dc C f - C j

dt At (7.37)

A central difference scheme is used for the spatial-derivative o f concentration




and a backward difference scheme is used for the spatial-derivative o f the ve locity.

d v _ v ; - v ;_,

dz Az

where the velocity is approximated by

(7.39)

v" +V nV = - L ^ (7.40)

2

App lying the difference equations (7.37) through (7.40) to (7.31) gives the

following approximation

✓-*/?+1 s i n j zn , r r n s-iti rr n j y n T/ n

S ~ S _ vJ+Vj - 1 C7'+1 S -1 , vi yj - 1 (7.41) At 2 2Az j Az

which is solved fo r the concentration fields at time step n+1.

The height o f the supernatant-suspension interface is updated according to the

following equation

L"+' = Ln- AtV? (7.42)

where Z" +1 and IT are the heights o f the supernatant-suspension interface at time steps

n+1, and n, respectively, and V" is the settling velocity at the top o f the

supernatant-suspension interface at time step n.

7.2.3 Simulation of interface height versus time

The experimental measurements o f Bhargava and Rajagopal (1990) were used to

show that the model can simulate, w ith one set o f model parameters, the height variation

o f the supernatant-suspension interface w ith time. These experimental measurements




107

included batch settling results for various suspended materials w ith different in itia l

suspended solids concentrations.

The zone settling velocities (vo) for individual solids concentrations were

determined from the linear section o f the interface height versus time plots. The

parameters for the Vesilind equation (k\ and n\) were then determined from zone settling

velocity data. Figure 7.3 shows a plot o f the zone settling velocity (vo) versus solids

concentration (Q for one suspension (aluminum hydroxide floes) examined by Bhargava

and Rajagopal (1990). The solids concentrations ranged from 1.643 g/L to 4.381 g/L. The

k\ and n\ were determined by nonlinear regression.

Table 7.1: Model parameters for various suspensions

Type o f suspended material k\

(m/h)

n\

(m3/kg)

k2

(kg/m 4/h)

n2

(m3/kg)

Alum inum hydroxide floes 2.57a 0.98a 3.6xl05b 0.5 b

Bentonite 1.61a 0.049a 1 .8 x l 0 7b 0 . 0 1 b

Desanded gold ore pulp 0.43c 0.0072° 1 .8 x l 0 5d 0.06 d

aDetermined from zone se ttling data presented by Bhargava and Rajagopal (1990)

b Determined as described in the text using data presented by Bhargava and Rajagopal

(1990)

c Determined from those zone settling data w ith in itia l solids concentrations greater than

45.8 kg/m3 presented by Scott (1968)

d Determined as described in the text using data presented by Scott (1968)

k2 and n2 were quantified by fitting the model prediction to the experimental data

shown in Figure 7.4. Due to the complexity o f Equation 7.24, regression techniques were




108

not used to estimate kj and nj. Instead, ki and rii were determined by means o f

optimization. The objective function for the optimization was the minimum residual sum

o f squares between the model prediction and experimental measurements. As there were

only two factors (&> and ni) in this optimization, the response-surface methodology

(specifically, grid search method) was used. A series o f simulations was conducted and k-i

and ni were systematically varied to provide the minimum residual sum o f squares. The

k 2 and values for the aluminum hydroxide floes were 3 .6xl0 5 kg/m4/h and 0.5 m3/kg,

respectively (Table 7.1), and are constant for a given suspension regardless o f initia l

solids concentration.

The model simulation results for aluminum hydroxide floe compare well to the

experimental measurements (Figure 7.4, r 2 = 0.99 overall). The experimental

measurements show both zone settling in itia lly and compression as time proceeds. The

model predicts the interface heights for suspensions in either settling regime and

seamlessly moves from one regime to another.

This model worked very well for other suspensions tested by Bhargava and

Rajagopal (1990). For example, Figure 7.5 shows the zone settling velocity versus solids

concentration for a bentonite suspension, while Figure 7.6 shows the comparison between

the model prediction and experimental measurement o f interface height versus time for

bentonite, w ith six d ifferent in itia l suspended solids concentrations. The model

parameters used are listed in Table 7.1. As Figures 7.4 and 7.6 indicate, the model

provides very good predictions o f batch settling processes for various inorganic

suspensions under differen t in itia l solids concentrations.




109

The in itia l linear section o f a batch settling test corresponds to zone settling. The

compression terms (the terms with K2 in Equation 7.23) used in this model disappear in

the zone settling regime where dVjdz- 0 . Therefore, the prediction accuracy o f this

model in the initial linear section in Figures 7.4 and 7.6 depends entirely on the suitability

o f the equation used for the zone settling velocity. The discrepancy between the model

prediction and experimental measurement in the initial linear section in Figure 7.4,

especially at lower in itia l solids concentrations, is due to the relatively poor f it o f the

Vesilind equation that was used to describe zone settling velocity (Figure 7.3). Because

the model describes compression settling well, it appears to work better for higher initial

solids concentrations that have quicker zone settling sections.

As indicated previously, several models have been suggested to describe the zone

settling velocity o f a suspension. While the Vesilind equation is used in this study, an

alternative zone settling expression (e.g. Islam and Karamisheva, 1998; Cho et al., 1993)

could have been used. Equation (7.21) would be re-derived for the zone settling

expression, leading to a different expression for (7.24). The compression terms remain

the same as does the solution procedure. A user o f this model should select the zone

settling expression that best fits the particular data o f interest.

The four parameters (k\, n\, k2 and n2) were determined in two steps above. A ll

fou r parameters may be determined in one step by conducting a series o f simulations and

systematically varying the four model parameters to provide the minimum residual sum

o f squares. When only one batch settling curve is available, this method could be used,

although it is not recommended for the fo llow ing reasons. First, batch settling tests are

commonly used for determ ining the zone settling velocities and so collection o f the




110

required data is straightforward. Second, k\ and n\ have different physical meanings than

£ 2 and «2 - Determining all four parameters with one data set may introduce correlation

between them that is not physically legitimate. Finally, this approach requires a priori

selection o f a zone settling expression. The suitability o f the selected expression would

not be independently tested.

E

&o0

50)

d)(/)<D

1

v0 = 2.57exp(-0.98C)

R2 = 0.96

0.01

52 3 40 1

Solids concentration (kg/m3)

Figure 7.3: Zone settling velocity for aluminum hydroxide floes [Calculated from

Bhargava and Rajagopal (1990) results].




I l l

0.4

Initial solids concentration (kg/m )

□ 4.381 o 3.490 a 2.747

x 2.640 x 2.312 o1 .6430.3E

D)

i? 0.2a)o XV

\ o x

&c

3002401801 2 0600

Time (min)

Figure 7.4: Interface height versus time fo r aluminum hydroxide floes. Symbols -

Experimental measurement (Bhargava and Rajagopal, 1990), Solid lines - Model

prediction.

E,

oo

5JD*3CDC/5

cuco

N

v0 = 1.61exp(-0.049C)

R2 =0.99

0.01806040200


Figure 7.5: Zone settling velocity for bentonite [Calculated from Bhargava and Rajagopal

(1990) results].




112

0.4

jET 0.3

£g><D

© 0.2

I1

0.1

0

0 60 120 180 240 300 360

Time (min)

Figure 7.6: Interface height versus time fo r bentonite. Symbols - Experimental

measurement (Bhargava and Rajagopal, 1990), Solid lines - Model prediction.

7.2.4 Simulation of dynamic concentration profiles

The batch settling model can also predict the vertical solids concentration profile

during the batch settling process. Laboratory measurements from Scott (1968) were used

to examine this capability. The suspension examined was the desanded fraction from a

m illed gold ore pulp, consisting mainly o f hydromuscovite w ith a fair amount o f very

fine quartz and small quantities of chlorite and pyrophyllite. The slurry was flocculated

by means o f lime. The batch settling runs fo r the suspensions categorized as intermediate

test pulp by Scott (1968) were simulated for this work. The initial concentration was 70

kg/m3, and the initial height was 1.43 m. As indicated by Scott (1968), an induction

period o f 24 min was required for the floccu lation process before the pulp developed a




113

constant settling rate. Therefore the starting point for model comparison was at the end of

the flocculation process, 24 minutes after the start o f the batch settling test.

Figure 7.7 shows the plot o f zone settling ve loc ity versus solids concentration for

this suspension from Scott (1968). For this suspension, when the solids concentrations

were greater than 45.8 kg/m , the zone settling velocity was described by the Vesilind

equation. The k\ and n\ values estimated from data with solids concentrations greater than

45.8 kg/m in Figure 7.7 are 0.43 m/h, and 0.0072 m /kg, respectively. Because the init ia l

concentration was 70 kg/m 3 in the batch settling test o f interest, the zone settling

veloc ities fo r solids concentrations less than 45.8 kg/m were not required.

The interface height versus time results for this test are shown in Figure 7.8. A

series o f simulations were conducted and kj and « 2 were systematically varied to provide

the minimum residual sum o f squares between the model prediction and experimental

measurements o f the interface height versus time data in Figure 7.8. The values

determined were:

kj = 1 .8 x 1 0 5 kg/m4/h, nj= 0.06 m3/kg

The model provides an extremely close fi t to the interface height versus time data with an

r 2 equal to 1 .0 0 .

Using the parameters quantified from interface settling data, the model simulated

the vertical solids concentration profile over time (Figure 7.9). The model predictions

agree reasonably well w ith the results measured by Scott (1968), w ith an r value o f 0.87.

An improved r 2 could have been achieved by using the measured results in Figure 7.9

directly to find the ki and « 2 values. However, in practice, concentration versus height

profiles at different times are rarely measured. The results presented in Figure 7.9




114

indicate that the model provides reasonable predictions o f the solids concentration

profiles using parameter values estimated from interface settling information which is

much easier to obtain.

The effective solids pressure expression, Equation 7.5, was proposed for

compression processes where the volume fraction o f solids are low enough that the

hydrodynamically transmitted stress exceeds the mechanically transmitted stress. When

the solids concentration is very high, the mechanically transmitted stress (by particle

contact) becomes the main mechanism o f stress transmittance, and the model’ s agreement

w ith data would be expected to decrease. This may be the case in Figure 7.9 as solids

concentrations increase to 150 kg/m .

.c

E

&oo$O)

c(1)cnCDc

Data used

v0 r 0.43exp(-0.0072C)i

! R2 = 0.98

0.01

250100 150 2000 50


Figure 7.7: Zone settling velocity for desanded fraction from a gold ore pulp (Scott,

1968). Experimental data w ith in itia l solids concentration greater than 45.8 kg/m3 are

used in determining the Vesilind equation parameters.




115

— Model fit

x MeasurementE

CDO

00.5

c

150010005000

Time (min)

Figure 7.8: Interface height versus time for desanded fraction from a gold ore pulp.

Symbols - Experimental measurement (Scott, 1968), So lid lines - Model prediction.

<— Initial heightSettle Time

16 min

□ 16 min

o 56 min

a

116 min x 151 min

56 min

116 min

E£ 151 min05

'(Dx Initial

0.5 -

200100 1500 50


Figure 7.9: Height-concentration profiles for desanded fraction from a gold ore pulp at

different settling times. Symbols - Experimental measurement (Scott, 1968), Solid lines

- Model prediction.




116

7.2.5 Sensitivity of compression parameters

The parameter K2 has a clear physical meaning: it represents the ease w ith which

water is removed from a solids-liquid matrix under pressure. As K 2 increases, water is

less easily removed from the matrix and the rate at which the solids concentration

increases under a given pressure declines as indicated in Equation 7.8. The relationship o f

K 2 to £ 2 and « 2 is indicated in Equation 7.21. As increases or £ 2 decreases, K 2 increases

and water is less easily removed from the so lid-liqu id m atrix.

1.5

g 1

£5?<33

o 0.5

‘t:£ c

0

0 200 400 600 800 1000

Time (min)

Figure 7.10: Sensitivity analysis on k2 and ri 2 based on the simulation in Figure 7.8. For

central line, k2 = 1 .8 x l 0 5 kg/m 4/h, n2 = 0.06 m3/kg.

The practical implications o f changes in k 2 and n2 are examined in Figure 7.10.

The predicted interface height versus time curves for the suspension examined in Figure

7.8 are plotted with values o f k 2 and n2 that have been varied by 25% from the values

presented in Table 7.1. The values for k\ and n\ were held constant at the values shown in

Table 7.1. The interface height predictions were more sensitive to n2, as expected from

Equation 7.21 (n2 occurs w ithin the exponential term). The slope o f the zone settling

n2 increases by 25%

k2 decreases by 25%

k2 increases by 25%

n2 decreases by 25%




117

portion o f the interface height curve was not affected because the compression terms in

the model disappear in the zone settling region.

As predicted from a consideration o f Equations 7.8 and 7.21, decreasing ki or

increasing ni reduced the ease with which water is removed from the suspension. A

suspension w ith relatively lower kj or higher yij values would undergo compression at a

lower solids concentration and achieve a decreased ultimate compressed solids

concentration. Therefore, the bentonite suspension, w ith the largest kj and lowest m w ill

compress the most readily and to the highest concentration during gravity settling while

the aluminum hydroxide floe compresses poorly (Table 7.1). In practice, suspensions

with smaller ki and larger nj values would require deeper thickeners to achieve target

underflow concentrations.

7.3 Steady state secondary clarifler

7.3.1 Govering equation of steady state secondary clarifier

In the case o f steady state gravity settling in a secondary cla rifier , the follo w ing

relationship holds for the zone below the feed point:

where Cu is the underflow solids concentration, and U is equal to bulk underflow rate

divided by the clarifier tank area.

A t steady state, both Cu and U are constant, and Equation 7.17 can be rewritten as:

(7.43)C

K 2CUU d 2C | K2CuU fdC }2 dK2 CUU dC

C 2 dz2 C3 i dz J dz C2 dz dz C 2 dz (7.44)

Substituting Equations 7.21 and 7.22 into 7.44 and simplifying gives




1 , r ,d2c n2 r J d C ' 2 — exp(772CJ— + — exp(n2C)iv 2 CiZ #C2 dz

+exp{n]C )Cu- C C

c„ c n

Equation 7.45 is solved in the following section.

7.3.2 Steady state solution

Equation 7.45 may be rewritten as:

-exp(n2 C)rdC^2

v dz j + — exp(«2C)

d 2C C exp(n,C) Cu-C

dz2 C U C„

and then rearranged to give:

' d c VKdz J

d C + — —= k2 exp(~n2C)

dz ,C UU c u J

Let

then

s = d -£ dz

d 2C d

dz dz

f dC}

\dz j

dS dS dC _ dS _ 1 d(s2)

dz dC dz dC 2 dC

Substituting Equations 7.48 and 7.49 into Equation 7.47 gives

n,S2 +1 d(s2)

2 dC - k2 exp(- n 2C)

f C exp(« ,C)Cb-C n

yCnU c 11 J

Equation 7.50 is rewritten as

4 ? 2) _ dC

2k2 exp(~n2C)r C exp{nxC )C u- C ^yCuU c

- l n 2S 2 u J

Note that for an equation w ith the form o f

dy

dx = f( x ) + by

0 (7.45)

(7.46)

(7.47)

(7.48)

(7.49)

(7.50)

(7.51)

(7.52)




119

where/(x) is a function o f x and b is a constant, the solution is

y = B exp(6 x) + exp(fex) Jexp(-for)f{x)dx (7.53)

where B is an integration constant.

Equation 7.51 has the same form as Equation 7.52. I f S2 is viewed as y, then

b = - 2 n. (7.54)

and

/ ( c ) = 2k2 exp(—n2C)C exp(«,C) Cu-C

yCuU c,(7.55)

u J

Therefore the solution fo r Equation 7.51 is written as

S2 = ,4exp(-2«2C) +

exp(-2« 2C ) j| exp(2n2C')J^2/ exp (-n 2C)j

where A is an integration constant.

Equation 7.56 can be reduced to

C exp(nlC)Cu-C

Kc j i kx q T ,\dC

(7.56)

S1 = A exp(-2n2C) + 2k2 exp(- 2n2C) J

and it is further reduced to

exp(w2C)C exp(nlC)Cu-C

KC*U c.u

dC (7.57)

Olr S2 - Aexp(-2n2C) + ^ ^ exp(- 2n2 C) J[exp( « 2 C )C }fC

+ ■

2 k2

K 2 k2

KCU

exp(-2n 2C)J{exp[(«, + n2 )C]}<7C

exp(- 2n2 c ) j{ c ex p[(n, + n2 )c]}dC

(7.58)

Performing the integration term by term gives




120

S2= A exp(-2n2C) + exp(- 2n2C)exp(n2C )^ Y ~ ~CUU n2

lk 2 exp(-2« 2C)exp[(ft1 +« 2 )cU — -— r (7.59)v A j J l I *>2 vy J v A | J |_ \ \ 2 / J / \

k} (nl +n2)

+ ~ ~ exp (- 2«2C)exp[(«, + n2)c ]{n\ ^ \ 1

kfu \n,+n2)

Equation 7.59 can be reduced to

S2= A exp(-2n2C) + | ) exp(_ 2ki — exp[(«, - n2 )c ]kx{n ,+n2)

+ 2 . [ ( y n 1 )C - 1] exp[^ _ ^ )c ]

£,C>, +«2j

and it is fu rther reduced to

S2 = Aexp(-2n2C) + 2k"1'f f 2̂ 2 ^ e xp (-n2C)CuUn2

+ ̂ ^ X c - c J z i ] exp[(wi_ yf2)c]

+n2)

The physically meaningful solution is given by

dC

dz

\

2k-,(n-,C —i) / \ A exp(-2n2C) H —----- -— exp(- n2C)CuUni

+

(7.60)

(7.61)

(7.62)

*iC H(wi + « 2 ) 2

where ̂ 4 is an integration constant. An exp licit analytical solution o f Equation 7.62 is not

available. However, with all the parameters known, Equation 7.62 can be integrated

numerically to give the C versus z relationship.

The integration constant A has to be determined from boundary conditions. In the

case that there is a distinctive zone-settling region above the compression zone, the

follow ing method can be used to evaluate the integration constant A. The zone-settling

concentration is the critical concentration Cc for the clarifier. W ith zone settling, there is




121

no concentration gradient and no effective solids pressure so Px= 0 and dCjdz = 0 .

Equation 7.62 can then be rearranged for A, w ith Cc replacing C.

A - 2kj [(«, + n , X ^ - C.) +1] exp[(H| + jc j (?63)

I ii v ^ l + « 2 ) u ^ 2

For the c ritical concentration Cc, the fo llow ing relationship holds:

(klQxp(-n ,Cc) + U)Cc = C uU (7.64)

7.3.3 Validation of the steady state solution

Two sets o f experimental results from the literature were used to validate the

steady-state solution (Equation 7.62). The first results are laboratory data for the

thickening o f a calcium carbonate suspension (George and Keinath, 1978). Two

conditions are examined (Table 7.2). The underflow rate for both conditions was

provided by the authors. The underflow concentration, Cu, was provided for condition 4.

A value o f Cu for condition 3 was estimated from the solids flu x and underflow rate,

assuming, as stated by the authors, that no appreciable quantity o f solids was transmitted

above the feed point. Values for k\ and n\ (Table 7.2) were estimated from the batch

settling data provided by least-squares, non-linear regression o f Equation 7.20 using the

Levenberg-Marquardt algorithm (Press et al., 1992) written in the C language on a

personal computer. These values were then used for both conditions.

Because a distinctive zone-settling regime was formed above the compression

regime, Equation 7.63 could be used to evaluate the integration constant A. Estimates o f

the critica l concentration, Cc, from Equation 7.64 agree well w ith the actual Cc, which is

the constant concentration in the zone settling regime (Figure 7.11). The parameters ki

and ni, like k\ and n\, should be constant for the same suspension matrix, regardless o f




122

system operating conditions. Therefore, values for £ 2 and « 2 (Table 7.2) were estimated

from the results o f condition 3 by least-squares, non-linear regression o f Equation 7.62

(numerically integrated using the Romberg algorithm [Press et al., 1992)]) and then

applied to condition 4.

Table 7.2: Operating conditions and model parameters for the experiment by George and

Keinath (1978)

Condition Ua

(m/h)

c ua

(kg/m3)

k ,b

(m/h)

m b

(m3/kg)

k2b

(kg/m4 /h)

n2b

(m3/kg)

Ccc

(kg/m3)

3 0.61 35.73.0 0.084 2.09xl05 0.15

14.9

4 0.33 50.9 35.0

aFrom or ca culated from George and Keinath (1978)

bBest fit parameters from non-linear regression

“Calculated using Equation 7.64

There is good agreement between the measured data and the model results (Figure

7.11), w ith r 2 values o f 0.83 fo r condition 3 (which was used to determine & 2 and ni) and

0.81 fo r condition 4. The concentration profiles indicate the presence o f both zone-

settling and compression regimes. The model simulates both regimes and different

operating conditions w ith one set of parameters.




124

Table 7.3: Operating conditions and model parameters for the experiment by Pflanz

(1969)

Condition Ua

(m/h)

Cua

(kg/m3)

k ,b

(m/h)

n ib

(m3/kg)

k2c

(kg/m4 /h)

n2c

(m3/kg)

Cca

(kg/m3)

Low load 0.38 12.9 2 . 6

Medium load 0.38 1 2 . 8 1 1 . 0 0.43 3.62xl04 0.64 2.5

High load 0.38 14.7 2.9

aFrom Pflanz (1969)

bEstimated from SVI (Pflanz, 1969) using equations from Hartel and Popel (1992)

cBest fi t parameters from non-linear regression

dCalculated using Equation 7.64

E

o-Q

E

O)<DX

1

0.8feed point

0.6

o low load

x medium load

— model fit

0.4

0.2

0

151050

Solids concentration (kg/m )

Figure 7.12: Solids concentration profile for an activated sludge suspension. Points are

from Takacs et al. (1991). Line represents the best-fit model prediction. Operating

conditions are shown in Table 7.3.




125

The settling characteristics o f the sludge were not provided in the o riginal paper,

so non-linear regression o f Equation 7.20 could not beused todetermine k\ and n\.

However, the sludge volume index (S VI) o f thesludge wasreported as 80 mL-g_1.

Empirical equations exist in the literature for relating k\ and n\ to SVI. For example,

Hartel and Popel (1992) provide the following expressions (Equations 7.65 and 7.66)

based on extensive data. These equations are used for illustration.

k, =17.4 exp(-0.0113SVI) + 3.931 (7.65)

nx =-0 .9834 exp(-0.0058LSF/) + 1.043 (7.66)

The concentration profile below the feed point suggests only compressive settling.

There was no distinctive zone-settling regime observable in the data. Therefore, the

integration constant A cannot be evaluated using Equation 7.63 but must instead be

estimated along with £ 2 and by non-linear regression o f Equation 7.62. The low load

and medium load conditions were essentially identical with respect to concentration

profile, so the combined data from those two conditions were used to estimate A, k̂ and

ri 2 - The experimental data and the best-fit curve for the combined conditions are shown in

Figure 7.12. The r 2 value o f 0.98 is extremely high, as would be expected when fitting

three parameters to a small quantity o f measurements. The parameter estimates are

included in Table 7.3.




126

£ 0.8

1 0.6-DE2 0.4

M—

£

■S 0.2 x

feed point

high load

model fit

10 1550


Figure 7.13: Solids concentration profile for a highly loaded activated sludge suspension.

Points are from Takacs et al. (1991). Line is the model prediction using the parameters

determined from Figure 7.12.

Because the high loading condition was performed with the same sludge and in

the same cla rifier, only one set o f parameters should be used. The model pred iction

versus the experimental measurements for the highly loaded condition is shown in Figure

7.13. Although the r 2 is sign ificantly low er than before (0.55), the fi t o f the model is s till

reasonable, especially considering the paucity o f data in the compressive regime used for

determining the parameter values. Other models for the same experimental data (Watts et

al., 1996; Takacs et al., 1991) developed separate sets o f parameter values fo r each

loading condition, in contrast to the model developed here which uses only one set o f

parameter values for a ll three operating conditions.

Ideal one-dimensional settling was assumed for thickeners in this study. The flo w

patterns in real thickeners may not achieve this assumption, in w hich case m ulti

dimensional models o f the flu id dynamics may be required. By inco rporating the




127

effective solids stress, the work developed here improves the performance o f ideal one

dimensional models. This work should also be applicable for two-dimensional model, but

its va lidity requires further examination.

7.4 Vertical solids profile in UASB reactors

For a steady state UASB, the follow ing relationship is va lid

C U V = (7.67)

C

where Ceis the effluent solids concentration and Ue is the up-flow velocity. Substituting

Equations 7.67, 7.21, and 7.22 into Equation 7.17 gives an equation similar to Equation

7.45 with Ce and Ue replacing Cu and U, respectively. The difference is that since Ue is

upward, it is negative, wh ile U is positive.

1 / n \d 2C n2 f c / cY exp{nxC )C e- C C — exp («2C )— y + -^ e x p (n2C) — + — V ------ Y 7 7 T T = 0 (7'68>

dz Vdz ) K] (Le C'eUe

Following the same procedure presented in Section 7.3.2, the solution for Equation 7.68

is

dC_

dz

^exp(-2n 2C) + 2k2(n2C 0 exp(_ n̂ CeU nl

e L 2 (7.69)

+ M ^ U z 7 H l exp[(„ , _ „ 2 )c ]

*iC e(«, +n2f

The measurement o f a UASB solids pro file by Yan and Tay (1997) is used here to

illustrate the application o f the current model for the prediction o f UASB solids pro files.

For their measurement in Day 30, the up-flow velocity was 0.08 m/h and the effluent

solids concentration was 0.1 kg/m3. Since only the volatile suspended solids were

reported; the suspended solids were calculated from VSS (assuming that SS/VSS equals

1.4). No settling data were provided in Yan and Tay (1997) for quantification o f model




128

parameters k i, n i, k2, and n2. For the purpose o f illustra tion, the parameters in Table 7.3

for aerobic sludge are used instead. The sludge had not granulated yet on Day 30.

The integration constant A and the solids concentration on the bottom o f the

reactor (boundary condition) were obtained by fitting the model prediction to the data.

Figure 7.14 shows the best-fit curve and the experimental data. The r value is 0.84. Yan

and Tay (1997) also provided the solids profiles on Day 90 and Day 180. These two sets

o f data are not simulated here because changes o f settling characteristics are expected due

to sludge granulation. A differen t set o f model parameters (k j, n i, k2, and n2) must be

used to model those two sets o f data. Further va lidations o f the current model are needed

for the simulation o f UASB solids profiles after granulation.

D)a>■CL-o-*—>oCOa>

0.4

0.3

0.2

measurement

model fit0.1

0

30 4010 200

Solids concentration (kg/m )

Figure 7.14: Solids concentration profile for a UASB reactor. Points are from Yan and

Tay (1997); line is the model prediction.




Chapter 8 Evaluating the AnSBR with Simulation

This chapter investigates the e ffect o f design and operational parameters on the

performance o f AnSBRs. AnSBRs have been operated w ith a wide range o f parameter

values. For example, the total cycle time examined has varied from 2 hours (Welper et

al., 1997) to 8 weeks (Masse and Droste, 2000). Conflicting operational strategies have

also been used in these studies. For example, Schmit and Dague (1993) kept the fill stage

as short as possible, while Bagley and Brodkorb (1999) recommended a slow fill

strategy. It is the purpose o f this chapter to see how many factors interact w ith each other

and to identify the optimum design and operational parameters for AnSBRs under

different situations.

8.1 Organic loading rate

The organic load ing rate is a good indicator o f the performance o f an anaerobic

reactor. The objective o f these simulations is to maximize the organic loading rate o f

AnSBRs. For AnSBRs, the organic load ing rate is defined as

ByJ - ' W (8.1)‘ c

where Bv is volum etric organic loading rate (g COD/L/d); Sm is the influent COD

concentration (g COD/L); V/is the fill volume (L); V, is total reactor volume (L); and

tc is total cycle time (day). tc is the sum o f the fi ll time (//), react time (tr), settle time (4

),

and decant time (4 ).

The hydrau lic retention time o f an AnSBR is defined as

0 = — (8.2)

vf ! vt

129




130

where 0 is the hydraulic retention time (day). The organic loading rate varies w ith many

design and operational parameters o f AnSBRs.

8.2 Design and Operational parameters of AnSBRs

Many factors affect the maximum loading o f AnSBRs. The operational factors

include flow recycle, m ixing, f il l rate, pH, temperature, fil l volume, etc. The design

factors include the total volume o f the reactor, the configuration o f the reactor, m ixing

configuration, etc.

The factors selected for this analysis are MLVSS, f il l volume, fill/rea ct time ratio,

and influen t COD. Reactor MLVSS is one o f the most important factors for the design o f

AnSBRs. The analysis presented in the Chapter 5 shows that the maximum load ing an

anaerobic reactor can achieve is a positive function o f the amount o f VSS the reactor can

retain. However, for AnSBRs, higher MLVSS means a longer settle time requirement,

which may decrease the organic loading rate. F ill volume also has an impact on the

loading treatable by AnSBR. It can be seen from Equation 8.1 that a larger fill volume

increases the theoretical OLR o f AnSBRs. On the other hand, a larger fi ll volume

requires a longer settle time and react time, which may decrease the loading o f AnSBRs.

Bagley and Brodkorb (1999) showed that fill/react time ratio is an important factor for

the operation o f AnSBRs. Though in fluent COD is a constraint factor rather than a design

and operational factor, it is included to see how it interacts w ith other factors in terms o f

maximum organic loading rate.

8.3 Response surface designs

Factorial design is a very good technique fo r examining the response o f a system

to changes in factors. Factorial design reduces the number o f experiments to obtain the




131

desired information. Response surface designs are bu ilt for fittin g a curved surface to a

continuous factor so that the factor values for the minimum or maximum response can be

sought. One o f the commonly used response surface designs is the Box-Behnken design.

The Box-Behnken design is a quadratic design, and it requires three levels for each

factor. The Box-Behnken design for four factors requires 27 runs, wh ile a fu ll three-level

factorial design for four factors requires 81 runs. Table 8.1 presents the Box-Behnken

design table prepared using JMP® software running on a Windows-compatible personal

computer. In Table 8.1, + or 1 means high level, 0 means medium leve l, and - or -1

means low level. Runs 25, 26, and 27 have the same factor pattern (0000). For an

experimental system, triplicates provide additional information about the experimental

variation in the system. However, the same inpu t pattern w ill have same output for

current computer simulations. Therefore pattern (0000) w ill be run only once. A

quadratic regression model o f N factors can be described by the fo llow ing equation:

y = b + f lblX , + £ l f ibllX ,X , (8.3)(=1 / = ! j = i

Two sets o f simulations were carried out based on the sludge characteristics. It

was assumed that the sludge is flocculent fo r the firs t set o f simulations, and granulated

for the second set. The major difference between the granulated sludge and flocculent

sludge is that the granulated sludge has very good settling characteristics. For example,

the interface settling ve locity w ith an initia l sludge concentration o f 10 g CO D/L is 0.18

m/h for flocculent sludge and 2.56 m/h for granulated sludge, based on the settling

parameters used in the model. Therefore, the reactor can retain much higher MLVSS

w ithin a given settle time for granulated sludge.




133

Table 8.2 lists the three level values o f each factor fo r the firs t set o f simulations

(floccu lent sludge). The high level o f f il l volume ( V f ) is ha lf o f the react volume V r,

which has been used by Bagley and Brodkorb (1999). The low level o f V f is 1/6 V r,

which has been used by Dague and Pidaparti (1992). The middle level is the average o f

the high level and low level. The high level o f MLVSS is 12 g COD/L. This value was

chosen in consideration o f the settle time requirement. Based on the settling parameters

used in the model, the reactor requires 14.7 hours to settle when the fill volume is 0.5Vr

and the MLVSS is 12 g COD/L. H igher MLVSS w ill require a very long settle time. The

low level o f MLVSS is 6 g COD /L, and the middle level is 9 g COD/L. The high level o f

fill/react ratio is 1:1 and the low level is 1:5. The middle level is the average o f high and

low.

Table 8.2: Factor values for flocculent sludge

Level MLVSS (g COD/L) Vf /Vt ! fr Sin (mg COD/L)

Low 6 1 / 6 1:5 2 , 0 0 0

Medium 9 1/3 1 : 2 6 , 0 0 0

High 1 2 1 / 2 1 :1 1 0 , 0 0 0

Table 8.3 lists the three leve l values o f each factor fo r granulated sludge. The high

level of V f is ha lf o f the react volume V r, the low level of V f is 1/6 V r, and the middle

level is the average o f the high level and low level. The middle level o f MLVSS is 40 g

COD/L. The MLVSS was 42 g COD/L (30 g VSS/L) in the laboratory experiment of

Angenent and Dague (1995), where the highest organic loading rate was achieved in the

literature. The high level o f MLVSS is 60 g COD/L, and the low level is 20 g COD /L.




134

The high level o f fill/react ratio is 3 :1 and the low level is 1 :3 . The middle level is the

average o f high and low ( 1 : 1 ) . The values o f S jn were chosen in consideration o f loading

rate and hydraulic retention time.

Table 8.3: Factor values for granulated sludge

Level MLVSS (g COD/L)Vf / Vt tf l tr

Sin (mg COD/L)

Low 2 0 1 /6 1 :3 1 0 , 0 0 0

Medium 4 0 1 /3 1:1 2 0 ,0 0 0

High 6 0 1 /2 3 :1 3 0 ,0 0 0

8.4 Simulation conditions

For all the simulations conducted, the reactor was assumed to be a 12 L anaerobic

sequencing batch reactor receiving glucose as the sole carbon source. A ll nutrients were

assumed to be present in excess. It was assumed that an appropriate amount o f a lka lin ity

is present in the influent to maintain neutral pH (the pH is not fixed). The reactor has a

height o f 0.68 m (up to the reactor liqu id level). I t was assumed that the decant rate is 10

L/h, and therefore the decant time is a function o f fil l volume. For example, i f the decant

volume (same as fill volume) is 2 L, then the decant time (td) is 0.2 hour. The settle time

is a function o f MLVSS and fi ll volume:

tf = f{MLVSS,Vf ) (8.4)

For the flocculent sludge, the settle time is calculated using the batch settling model

developed in Chapter 7. The settle parameters used are as follows: ki=10.5 m/h, ni=0.43

L/g COD, k2=7200 kg/m4/h, and n2=0.01 L/g COD. These values are chosen based on the

experimental results o f Higgins (2001), assuming that anaerobic biosolids w ill settle

roduced with permission o f the copyright owner. Further reproduction prohibited without permission.



135

sim ilarly to flocculated aerobic bioso lids. For granulated sludge, the interface settle

velocity is estimated from the settling velocity data o f Sung and Dague (1995) w ith k i=

8.94 m/h and ni=0.0766 L/g COD. As the compression data were not available for

granulated sludge, a constant interface settling rate was assumed throughout the settle

stage for a given MLVSS. It was assumed that the sludge in the reactor has a uniform

vertical concentration at the beginning o f the settle stage.

A ll simulations meet the same effluen t criteria o f 50 mg COD/L o f VFAs. For

each simulation run, the fill time and react time were adjusted while maintaining the

fill/react ratio to meet the effluent criteria. The total cycle time was allowed to vary

accordingly. Each simulation was run to steady state, which was defined as an effluent

COD variance w ithin 0.1 mg CO D/L after 20 consecutive cycles. The maximum organic

loading rate for the factors examined was then calculated using Equation 8.1 w ith the

appropriate total cycle time. The program maintains a preset target MLVSS automatically

by adjusting the solid wasting rate. The SRT was calculated based on the MLVSS and

wasting rate. The statistical analysis was conducted using JMP® software.

8.5 Simulation results

8.5.1 Flocculent sludge

The simulation results for maximum organic loading rate are presented in Table

8.4. For the flocculent sludge, the highest organic loading rate is 5.9 g/L/d (run 12), when

the MLVSS is high (12,000 mg COD/L), fill volume is medium (4 L), fill/react ratio is

medium (1:2), and influent COD is high (10,000 mg/L). The lowest organic loading rate

is 3.0 g/L/d (run 4), when the MLVSS is high (12,000 mg COD/L), fill volume is high ( 6

L), fill/react ratio is medium (1:2), and influent COD is medium (6,000 mg/L). The total




136

cycle time ranges from 2.2 hours (run 21) to 26.5 hours (run 24). The hydraulic retention

time ranges from 11.4 hours (run 5) to 65.4 hours (run 10). SRT ranges from 17.9 days

(run 17) to 54.9 days (run 4). The specific organic loading rate ranges from 0.25 (run 4)

to 0.66 (run 17) g COD/g COD/d.

Table 8.5 presents parameter estimates for the standard least squares response

surface model from JMP®. The analysis o f variance fo r this response surface model gives

an F ratio o f 11.99 and Prob >F o f 0.000058. The F ratio evaluates the effectiveness o f

the model. I f the probability associated with the F ratio is small, then the model is

considered a better statistical fit for the data than the response mean alone. Prob >F is the

observed significance probab ility o f obtaining a greater F ratio by chance alone i f the

specified model fits no better than the overall response mean. Observed significance

probabilities o f 0.05 or less are often considered evidence o f a regression effect. The

probability value o f 0.000058 indicates that the four-variable response surface model

w ith two factor interactions provides a very good f it o f simulation data.

The effects o f each term are also presented in Table 8.5 as sum o f squares. The

influent COD has a significan t effect on the maximum loading rate o f AnSBRs, wh ich is

reflected by a large sum o f squares. This observation can be explained using Equation

8.1. Lower influent COD requires a short total cycle time to reach a higher organic

loading rate, i f the fi ll volume remains the same. The fraction o f time used for settle and

decant increases with the decrease o f total cycle time. As a result, the efficiencies o f

AnSBRs decrease. This result indicates that AnSBRs are more suitable fo r high strength

wastewater in terms o f reactor loading rate. The reactor MLVSS also has a significant

effect on the maximum loading rate o f AnSBRs.




137

Table 8.4: Simulation results of maximum OLR for flocculent sludge

Run MLVSS

(g/L)

Vf /Vt tf ! tr

(g/L)

tc

(hours)

HRT

(hours)

SRT

(days)

OLR

(g/L/d)

SOLR

(g/g/d)

1 6 1 / 6 1 / 2 6 6.9 41.4 19.8 3.5 0.58

2 6 1 / 2 1 / 2 6 19.1 38.2 2 2 . 6 3.8 0.63

3 1 2 1 / 6 1 / 2 6 4.6 27.6 25.8 5.2 0.43

4 1 2 1 / 2 1 / 2 6 23.7 47.4 54.9 3.0 0.25

5 9 1/3 1/5 2 3.8 11.4 26.4 4.2 0.47

6 9 1/3 1/5 1 0 16 48 2 1 . 1 5.0 0.56

7 9 1/3 1 /1 2 4.5 13.5 35.2 3.6 0.40

8 9 1/3 1 / 1 1 0 17.9 53.7 26.0 4.5 0.50

9 6 1/3 1 / 2 2 4.9 14.7 24.5 3.3 0.54

1 0 6 1/3 1 / 2 1 0 2 1 . 8 65.4 19.8 3.7 0.61

1 1 1 2 1/3 1 / 2 2 5.2 15.6 49.4 3.1 0.26

1 2 1 2 1/3 1 / 2 1 0 13.5 40.5 24.6 5.9 0.49

13 9 1 / 6 1/5 6 4.6 27.6 18.4 5.2 0.58

14 9 1 / 6 1 /1 6 5.8 34.8 26.9 4.1 0.46

15 9 1 / 2 1/5 6 15.1 30.2 25.2 4.8 0.53

16 9 1 / 2 1 /1 6 17.5 35 32.8 4.1 0.46

17 6 1/3 1/5 6 1 2 . 1 36.3 17.9 4.0 0 . 6 6

18 6 1/3 1 /1 6 15.5 46.5 25.8 3.1 0.52

19 1 2 1/3 1/5 6 9.0 27.0 26.5 5.3 0.44

2 0 1 2 1/3 1 /1 6 1 0 . 6 31.8 35.8 4.5 0.38

2 1 9 1/3 1 / 2 2 2 . 2 13.2 29.1 3.6 0.40

2 2 9 1 / 6 1 / 2 1 0 7.7 46.2 19.4 5.2 0.58

23 9 1 / 2 1 / 2 2 6.7 13.4 37.0 3.6 0.40

24 9 1 / 2 1 / 2 1 0 26.5 53.0 27.3 4.5 0.50

25 9 1/3 1 / 2 6 9.4 28.2 2 1 . 8 5.1 0.57




138

Table 8.5: Parameter estimates and effect test for flocculent sludge

Term Estimate Sum o f squares % o f Sum o f squares

Intercept 5.110 Not applicable Not applicable

MLVSS 0.489 2.871 15.7

Vf/V, -0.258 0.796 4.3

t f / t r -0.383 1.756 9.6

Sin 0.621 4.625 25.3

MLVSS * MLVSS -0.707 2.663 14.5

Vf /Vt * MLVSS -0.618 1.525 8.3

Vf /vt * Vf jv t -0.419 0.937 5.1

t f tr * MLVSS 0.018 0 . 0 0 1 0 . 0

t f / t r *Vf /Vt 0.105 0.044 0 . 2

l (r * tf l fr -0.204 0 . 2 2 2 1 . 2

Sin * MLVSS 0.613 1.501 8 . 2

Sin * Vf / Vt -0.150 0.090 0.5

Sin * t f j t r 0.030 0.004 0 . 0

Sin * Sin -0.489 1.276 7.0

Figures 8.1, 8.2, and 8.3 show the contour plots for the maximum loading rate

when the influent COD is low (2,000 mg COD/L). It can be seen that for low strength

influent, the optimum design MLVSS is around medium (9,000 mg COD/L), and the




139

optimum fi ll volume w ill be sligh tly lower than the medium value (4 L ). For the fill/rea ct

ratio, the maximum OLR decreases from low fill/react ratio to medium fill/react ratio but

then increases as fill/react ratio increases further. This suggests a non-linear relationship

between the maximum OLR and the fill/react ratio. Table 8.5 indicates that there is a

strong interaction between fill volume and MLVSS. Reactors with different MLVSS

levels have different optimum f il l volumes. For example, i f the reactor MLVSS is high, a

lower fi ll volume is better, while i f the reactor MLVSS is low , a higher fi ll volume is

better.

0.5 -

£4—

>

-0.5 -

1.5 -1 -0.5 0 0.5 1 1.5

MLVSS

Figure 8 .1: Contour plot o f maximum OLR fo r flocculent sludge (S jn low; t f j t r low)




140

0.5

-0.5 -■

1.5 -1 -0.5 0 0.5 1 1.5

MLVSS

Figure 8.2: Contour plot o f maximum OLR for floccu lent sludge (Si„ low ; t f j t r

medium)

0.5

>

-0.5

1.5 -1 -0.5 0 0.5 1 1.5

MLVSS

Figure 8.3: Contour plo t o f maximum OLR fo r flocculen t sludge (Sjn low; t f j t r high)




0.5 -

£>

-0.5 -

-1.5

1.5 -1 -0.5 0 0.5 1 1.5

MLVSS

Figure 8 .4 : Contour plot o f maximum O L R for flocculent sludge (S jn medium; tj- jt r

low)

0.5 -■

>

-0.5 -

-1.5

- 1 . 5 -1 - 0 . 5 0 0 . 5 1 1 .5

MLVSS

Figure 8 .5 : Contour plot of maximum O L R for flocculent sludge (S jn medium; t f Jtr

medium)




142

0.5 -

£5

-0.5 -

1.5 -1 -0.5 0 0.5 1 1.5

MLVSS

Figure 8 .6 : Contour plot o f maximum OLR fo r flocculen t sludge (Sjn medium; tf j tr

high)

0.5

£>

-0.5

1.5 -1 -0.5 0 0.5 1 1.5

MLVSS

Figure 8.7: Contour plot o f maximum OLR for flocculent sludge (Sjn high; t y j t r low)




143

0.5

£>

-0.5

1.5 -1 -0.5 0 0.5 1 1.5

MLVSS

Figure 8 .8 : Contour plot o f maximum OLR for flocculent sludge (Sin high; /y / tr

medium)

0.5 -

-0.5 -

1.5 -1 -0.5 0 0.5 1 1.5MLVSS

Figure 8.9: Contour p lot o f maximum OLR fo r flocculent sludge ( S j „ high; t f j t r high)




144

Figures 8.4, 8.5, and 8 . 6 show the contour plots for the maximum loading rate

when the influent COD is medium (6,000 mg COD/L). It can be seen that for medium

strength influent, the optimum design MLVSS is slightly less than the high value (12,000

mg CO D/L), and the optimum fi ll volume w ill be close to the lower value (2 L). For the

fill/react ratio, the lower, the better.

Figures 8.7, 8 .8 , and 8.9 show the contour plots for the maximum loading rate

when the influent COD is high (10,000 mg COD/L). For high strength influent, high

MLVSS, low fill volume, and low fill/react ratio are the optimum conditions.

8.5.2 Granulated sludge

The simulation results for the maximum organic loading rate are presented in

Table 8 .6 . For the granulated sludge, the highest organic loading is 30.2 g/L/d (run 19),

when the MLVSS is high (60,000 mg COD/L), fill volume is medium (4 L), fill/react

ratio is low (1:3), and influent COD is medium (40,000 mg/L). The lowest organic

loading rate is 9.7 g/L/d (run 18), when the MLVSS is low (20,000 mg COD/L), fill

volume is medium (4 L), fill/react ratio is high (3:1), and influent COD is medium

(40,000 mg/L). The cycle time ranges from 4.1 hours (run 21) to 44.3 hours (run 2). The

hydraulic retention time ranges from 19.8 hours (run 5) to 130.8 hours (run 10). SRT

ranges from 16.3 days (run 17) to 33.8 days (run 20). The specific organic loading rate

ranges from 0.39 (run 11) to 0.70 (run 17) g COD/g COD/d.




145

Table 8.6: Simulation results of maximum OLR for granulated sludge

RunMLVSS

(g/L)vf ! vt t f / t r

s»,

(g/L) (hours)

HRT

(hours)

SRT

(days)

OLR

(g/L/d)

SOLR

(g/g/d)

1 2 0 1 / 6 1 /1 40 14.7 8 8 . 2 21.3 10.9 0.54

2 2 0 1 / 2 1 /1 40 44.3 8 8 . 6 25.5 1 0 . 8 0.54

3 60 1 / 6 1 /1 40 6 . 1 36.6 27.0 26.2 0.44

4 60 1 / 2 1 /1 40 17.7 35.4 32.4 27.1 0.45

5 40 1/3 1/3 2 0 6 . 6 19.8 19.6 24.2 0.61

6 40 1/3 1/3 60 17.8 53.4 17.1 27.0 0.67

7 40 1/3 3/1 2 0 8.7 26.1 30.2 18.4 0.46

8 40 1/3 3/1 60 25 75 27.9 19.2 0.48

9 2 0 1/3 1 /1 2 0 14.6 43.8 23.7 1 1 . 0 0.55

1 0 2 0 1/3 1 / 1 60 43.6 130.8 2 2 . 0 1 1 . 0 0.55

1 1 60 1/3 1 / 1 2 0 6 . 8 20.4 33.3 23.5 0.39

1 2 60 1/3 1 /1 60 16 48 25.5 30.0 0.50

13 40 1 / 6 1/3 40 6 . 2 37.2 16.6 25.8 0.65

14 40 1 / 6 3/1 40 8.5 51 26.4 18.8 0.47

15 40 1 / 2 1/3 40 18.4 36.8 20.4 26.1 0.6516 40 1 / 2 3/1 40 26.3 52.6 32.2 18.3 0.46

17 2 0 1/3 1/3 40 2 2 . 8 68.4 16.3 14.0 0.70

18 2 0 1/3 3/1 40 33 99 27.0 9.7 0.48

19 60 1/3 1/3 40 1 0 . 6 31.8 23.7 30.2 0.50

2 0 60 1/3 3/1 40 13.1 39.3 33.8 24.4 0.41

2 1 40 1 / 6 1 / 1 2 0 4.1 24.6 24.4 19.5 0.49

2 2

401 / 6 1 /1

60 10.764.2 20.5 22.4 0.56

23 40 1 / 2 1 / 1 2 0 11.9 23.8 29.2 2 0 . 2 0.50

24 40 1 / 2 1 / 1 60 34.1 6 8 . 2 26.8 2 1 . 1 0.53

25 40 1/3 1 / 1 40 15.6 46.8 25.3 20.5 0.51




146

Table 8.7 presents parameter estimates for the standard least squares response

surface model. Compared to the flocculent sludge, reactors with granulated sludge have a

much higher maximum OLR. The intercept for the regression model is 20.5 for

granulated sludge, versus 5.1 for flocculent sludge. The simulation results are in

agreement w ith the analysis in Chapter 5; the amount o f active biomass in the reactor

determines the maximum OLR o f the reactor, while the amount o f sludge a reactor can

retain depends on the settling characteristics o f the sludge.

Table 8.7: Parameter estimates and effect test for granulated sludge

Term Estimate Sum o f squares % o f Sum o f squares

Intercept 20.510 Not applicable Not applicable

MLVSS 7.839 734.43 79.80

vf ! vt -0.008 0 . 0 0 1 0 . 0 0

tf ! tr -3.212 123.84 13.46

Sin 1.159 16.12 1.75

MLVSS * MLVSS -2.052 22.50 2.44

Vf /Vt * MLVSS 0.233 0 . 2 2 0 . 0 2

V f /V t * V f / V t 0.238 0.30 0.03

t f / t r * MLVSS -0.355 0.50 0.05

t f / t r *Vf /Vt -0.213 0.18 0 . 0 2

t f / t r * t f j t r 1.348 9.70 1.05

S^ * MLVSS 1.601 10.30 1 . 1 2

S i n * Vf /Vt -0.493 0.97 0 . 1 1

Sin * t f l t r -0.480 0.92 0 . 1 0

Sin * Sin 0.271 0.39 0.04




147

8.6 Discussion

The simulations presented above show that influent COD and MLVSS are two

major factors that affect the organic loading o f AnSBRs, which is expected. The

interesting finding is that the influent COD affects the optimal MLVSS and optimum fill

volume in terms o f maximum organic loading rate. From the analysis presented in

Chapter 5, it is known that the loading rate o f anaerobic systems is a positive function o f

reactor MLVSS concentration. Therefore the maximum organic loading rate a reactor can

achieve depends on the maximum amount o f MLVSS the reactor can retain. U nlike

continuous anaerobic reactors, AnSBRs have a distinct settle cycle; i f the settle time is

long enough, AnSBRs can retain higher VSS concentrations comparable to other reactor

configurations. However, longer settle times decrease the loading o f AnSBRs. There is a

tradeoff between the settle time and VSS concentration in the reactor and the f il l volume.

Bagley and Brodkorb (1999) predicted that a long fill cycle is beneficial for the

AnSBR system. The influent COD examined by Bagley and Brodkorb (1999) was 2000

mg/L. The simulation results are in agreement with the prediction o f Bagley and

Brodkorb (1999) w ith respect to low influent COD conditions. However, the simulations

show the maximum organic loading rate increases w ith the decrease o f fill/react ra tio for

medium and high strength influen t COD.

To further examine the effect o f fill/react ratio on the operation o f AnSBRs,

additional simulations were carried out. Figures 8.10, 8.11, and 8.12 present the

simulation results o f indiv idua l VFAs versus time under different fill/rea ct ratios while

other conditions are the same. The common conditions for these simulations are

summarized in Table 8 .8 . The effluent CODs fo r these simulations are w ith in 70 to 80




148

mg COD/L. Figure 8.10 shows that a short f il l time results in the accumulation o f

propionic and acetic acids right after the fill stage ends. For a medium fill time the

propionic acid accumulates while the acetic acid concentration remains low (Figure

8.11). For a long fill time both propionic acid and acetic acid concentrations are flat and

low throughout the AnSBR cycle (Figure 8.12). The operation o f AnSBRs under a long

f i ll time and short react time is close to a continuous reactor.

2500

lactate2000 -

Q

g 1500 JD)E

1000 - -

ropionate

£ 2

"go

X!1 3

CO

acetate

500

butyrate

2412 16 200 84

Time (hours)

Figure 8.10: Individua l VFAs versus time ( t j / tr = 0.5/22)




149

800

propionate^ 600

oOO

E 400

acetate

butyrate

242012 1680 4

Time (hours)

Figure 8.11: Individu al VFAs versus time ( t f j t r = 12.5/10)

100

acetateQ

OOO)E

propionate

40

CO-Q

w 20lactate

butyrate

2416 201280 4

Time (hours)

Figure 8.12: Individua l VFA s versus time ( t f j t r = 20.5/2)




150

Table 8.8: Simulation conditions

Item Parameter Value

1 Cycle length (hours) 24

2 Settle time (hours) 1

3 Decant time (hours) 0.5

4 HRT (hours) 48

5 F ill volume (L) 6

6 Settle volume (L) 6

7 Settled VSS (mg CO D/L) 2 0 , 0 0 0

8 VSS (mg COD/L) 1 0 , 0 0 0

9 Influent COD (mg COD/L) 8000

1 0 Organic loading rate (g COD/L/d) 4

1 1 Specific sludge loading rate (g COD/g COD VS S/d) 0.4

1 2 Solids retention time (days) 21.4

The predicted microbial population distributions are presented in Table 8.9. The

microbial population d istribution o f a continuous reactor operated under the same loading

rate is also presented for comparison. Table 8.9 shows that a short fill time results in

higher percentages o f Xp and X l. This can be expected from the analysis presented in

Chapter 5, as a short fi l l time results in a spike o f hydrogen partial pressure. The high

hydrogen partial pressure regulates the product distribution o f glucose consumption and

produces more lactate and propionate. As a consequence, the reactor w ill accumulate

more lactate and propionate bacteria after long-term successful reactor operation. Though




151

short f il l t ime results in a higher production o f propion ic acid from glucose degradation,

the AnSBRs are able to degrade the propion ic acid by increasing the percentage o f

propionic acid consuming microorganisms.

However, when the reactor is in its initial startup stage or when the reactor is

overloaded, the propionic acid produced is not consumed completely, and the growth o f

propionic acid consuming microorganisms is limited.

Table 8.9: M icrob ial popu lation distributions

Fill/react Xs

(%)

x ,

(%)

X F

(%)

x P

(%)

X L

(%)

X B

(%)

X A

(%)

X H

(%)

0.5/22 12.4 2.5 25.0 10.3 2 1 . 1 1.4 11.9 15.4

12.5/10 14.8 3.1 30.6 5.1 12.5 2.5 13.9 17.5

20.5/2 14.4 3.3 32.2 4.2 1 1 . 0 2.4 14.4 18.1

CSTR 14.7 3.4 32.9 3.8 10.4 2.3 14.4 18.1

Although the simulations show low fill/react time ratios being preferred, high

fill/rea ct ratios may nevertheless be beneficial fo r the operation o f AnSBRs in several

aspects. A low fill/react time ratio increases the peak VFA concentration appearing at the

end o f the fi ll stage. This increases the alkalinity requirement in the influe nt to neutralize

the pH. Also, the high VFA concentration within the AnSBR cycle makes it susceptible

to overload. The pH drops sign ifican tly (below 6.5) at the end of the fi ll stage, and the

reactor pH buffer capacity drops significantly. Therefore, for a reactor in its startup stage

a high fill/react ratio is beneficial. A reactor operated with a high fill/react ratio w ill also

be more resilient to an inh ibitor entering the system. Another potential problem w ith the




Chapter 9 Summary and Conclusions

9.1 Summary

A major disadvantage o f existing anaerobic models (Costello et al., 1991a; Bagley

and Brodkorb, 1999) is their poor prediction o f microbial population. This is due to the

use o f inappropriate hydrogen partial pressure regulation and inh ibition functions, as has

clearly been shown in this thesis. Anaerobic Digestion Model No. 1 (AD M 1) (Batstone et

al., 2 0 0 2 ) also reported that the hydrogen partial pressure product regulation functions

described by Mosey (1983) and further developed by Costello et al. (1991a) could not be

used consistently with a variety o f experimental data sets. No hydrogen regulation

function is used in the AD M 1.

New hydrogen partial pressure regulation functions are derived from the rate

equation o f ordered single-displacement enzymatic reactions having two substrates

(Lehninger, 1975) and the equation describing the relationship between the oxidation

state o f the NADH/NAD+ couple and hydrogen partial pressure (Mosey, 1983). New

hydrogen partial pressure inhibition functions are developed from a thermodynamic basis

for propionate and butyrate degradations. Other models (Batstone et al., 2002; Costello et

al., 1991a; Bagley and Brodkorb, 1999; Masse and Droste, 2000) used a non-competitive

inh ibition function for hydrogen partial pressure inhib ition o f propionate and butyrate

degradation. The new inhibition functions presented in this thesis are supported by recent

development o f the inhibition mechanism o f propionate degradation by hydrogen partial

pressure (Hoh and Cord-Ruwisch, 1996).

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The sem i-implicit extrapolation method (Press et al., 1992) was implemented in

this study to solve the coupled first-order ordinary differential equations for the model.

Although the Runge-Kutta algorithm was used in other models (Masse and Droste, 2000;

Costello et al., 1991b; Kiely et al., 1997), the formulated equation set from the model is a

s tiff set o f equations (Batstone et al., 2002), for wh ich the Runge-Kutta algorithm is

unstable and introduces integra tion error.

The model is validated by a variety o f experimental data from the literature. I t is

shown that the model gives good predictions o f COD, VFAs, pH, and CH4 and CO2

production rate for AnSBRs. The model gives a better prediction than the Costello et al.

(1991b) model o f the response o f a CSTR subjected to step changes in the substrate

loading.

Mass balance analysis was carried out on each metabolic group o f bacteria and its

correspondent substrate for anaerobic reactors operated under steady state. This analysis

revealed that propionate consuming acetogenesis, instead o f aceticlastic methanogenesis,

is the rate lim iting step for anaerobic treatment. This finding is in agreement w ith

experimental observations reported in the literature (Bjonsson et al., 1997) that

propionate buildup is the firs t sign o f reactor overload. The maximum specific loading

rate was estimated from this analysis, which could be a design criterion for anaerobic

reactors. Many anaerobic organisms, such as glucose acidogens and lactate acetogens, are

capable o f producing several products. However, the distributions among these products

are not well quantified, i.e. the product regulation functions or coefficients are not well

quantified. This analysis reveals the correlation between production rate o f intermediate

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155

(.L,•) and its correspondent sludge activity (SA,). Therefore the product regulation

coefficients or functions could be determined or verified by the sludge activity data.

The sludge compositions predicted from steady state simulations agree w ell w ith

a variety o f experimental data (sludge activities, sludge methanogenic activ ity) in the

literature. This indicates that the model predicts the microb ial population very w ell; other

models are poor in predicting the m icrobial population. The s imulation o f the startup o f

an anaerobic reactor further illustrates the predictive ability on microbial population by

the current model. The simulated sludge methanogenic activ ity versus time agrees well

w ith experimental data from the literature for the startup o f an anaerobic reactor.

The solids-liquid separation process is important for anaerobic reactors, though

many configurations o f anaerobic reactors, such as AnSBR and UASB, do not have

separate solids- liquid separation devices. The volum etric organic loading rate o f

anaerobic processes is lim ited by the quantity o f active biomass that a reactor can retain,

as the maximum specific organic loading rate remains constant. The solids-liquid

separation during the settle stage o f AnSBRs is simulated using a dynam ic model

developed for the gravitational settling process (Zheng and Bagley, 1998). This settling

model was validated by literature data on vertical solids concentration profiles for both

laboratory and field-scale clarifiers operated under steady state, and was validated by

literature data on the batch settling process (Zheng and Bagley 1999). This settling model

is applied to predict the vertical solids profile o f UASB reactors.

The Box-Behnken response surface design is applied to evaluate the performances

o f AnSBRs under different design and operational parameter values. Four factors were

selected, including MLVSS, fill volume, fill/react time ratio, and influent COD. Two sets

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156

o f simulations were carried out based on the sludge characteristics: floccu lent sludge and

granulated sludge. The highest OLR achieved in these simulations is 5.9 g/L/d for

flocculent sludge and 30.2 g/L/d for granulated sludge. The simulation results show that

the influent strength constrains the maximum organic loading rate that AnSBRs can

reach. Therefore it is an important factor to be considered in the design o f AnSBRs. The

reactor MLVSS is a very important factor for the maximum loading o f AnSBRs. The

optimum MLVSS concentration depends on influent strength and sludge settling

characteristics. The optimum fill volume depends on other factors such as influent COD

and reactor MLVSS. Short fill/react ratios have a positive effect on maximum organic

loading rate.

The current model is based on the model o f Bagley and Brodkorb (1999). M ajor

developments o f the current model compared to the model o f Bagley and Brodkorb

(1999) include:

• Current model works for continuous reactors as well as for AnSBRs

• Current model implements new hydrogen partial pressure product regulation

functions.

• Current model implements new hydrogen partial pressure inh ibition functions.

• Current model implements the sem i-imp licit extrapolation method to solve the

ordinary differential equations.

The new developments improve the performance o f the model, for example, through

better prediction o f m icrobial population and improved numerical s tab ility, and are

validated by extensive experimental data from the literature.

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157

9.2 Conclusions

The following conclusions are made from this study:

1. Hydrogen regulation functions incorporated in the model represent the product

distribution o f glucose degradation we ll.

2. The hydrogen inh ibition functions incorporated in this model represent the inhib ition

mechanisms o f propionate and butyrate degradation.

3. Propionate degradation is the rate lim iting step for anaerobic treatment.

4. The microbial population distributions are a reflection o f the amount o f intermediates

produced and consumed fo r anaerobic processes operated under steady state.

5. The influen t strength constrains the maximum organic loading rate that AnSBRs can

reach. The reactor MLVSS is a very important factor for the maximum loading o f

AnSBRs; in general, the reactor M LVSS has a positive effect on maximum OLR. The

optimum fi ll volume o f AnSBRs depends on other factors such as influen t COD and

reactor MLVSS.

6 . Though the simulations show that a short fill/react ratio has a positive effect on

maximum OLR under the conditions simulated, short fill/rea ct ratios may not always

be best because o f potential negative effects on the stab ility o f AnSBRs, increased

operational complexity, and possible effects on sludge granule quality.

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Chapter 10 Engineering Significance and Suggestions for

Future Research

10.1 Engineering significance

The materials presented in this thesis are valuable for engineers and researchers

who engage in anaerobic process research, development, modeling, design, operation,

and optimization. These materials are also valuable for engineers and researchers who

engage in gravity settling process (e.g. secondary clarifier) research, modeling, design,

and operation.

10.1.1 Operational significance

The model developed in this study can be applied to full-scale anaerobic reactors

to predict the response o f the reactor under varying input conditions, such as influe nt

COD and HRT. The model can be applied to evaluate the potential maximum OLR o f

anaerobic reactors. The results from this study can be used for the startup o f anaerobic

reactors. This study showed the startup process can be achieved by monitoring the

effluent COD instead o f specific methanogenic activity.

The results from this study can be used to optimize the operation o f existing

anaerobic reactors. For example, this study showed that high specific organic load ing rate

has a positive effect on the volumetric organic loading rate, but has a negative effect on

the stability o f the reactor. The specific organic loading rate o f existing anaerobic reactors

can be examined for optimum specific organic loading rate applied. This can be achieved

by adjusting the applied vo lumetric organic loading rate or by adjusting the VSS (SRT) in

the reactor. Also, this study shows that a short fill/react ratio for AnSBRs is good for

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maximum volumetric organic loading rate, but increases the influent alkalinity

requirements. The fill/react ratio could be examined for its optimum value.

10.1.2 Design significance

The results from this study can be used for the design o f anaerobic reactors. For

example, this study derives the maximum specific organic loading rate a reactor can

achieve, which implies that specific organic loading rate can be a design criterion for

anaerobic reactors. The knowledge from this study can be used for the development o f

new configurations o f anaerobic reactors. For example, this study shows that the

volum etric organic loading rate can be increased i f the hydrogen partial pressure in the

reactor can be decreased through engineering methods. An example o f these methods is

adding more hydrogen utilizing methanogens to the reactor. Another example is

designing a three-stage anaerobic reactor. The first stage is for acidogenesis and produces

VFAs and hydrogen. The hydrogen is fed to the second stage to produce

hydrogenotrophic methanogens. The VFAs from the first stage and hydrogenotrophic

methanogens from the second stage are fed to the th ird stage for methanogenesis.

The gravitational settling model developed in this thesis can be used to verify the

empirical crite ria used for the design o f secondary settler depth in engineering practice.

New design criteria for the secondary settler depth can be derived from this study. Direct

applications o f the gravitational settling model include 1 ) determining the upflow

velocity o f UASBs, 2) determining the settle time requirement for AnSBRs.

10.2 Suggestions for future research

In addition to carbohydrates, proteins and lipid s are common components o f

wastewater and the hydrolysis products o f composite particulate material. The current




160

model considers the degradation o f carbohydrates only. Further development o f this

model should include proteins and lipids. It is recommended that advanced optimization

techniques be employed for model calibration and that a better objective function (e.g.

the percent error o f estimate used in Masse and Droste, 2000) be chosen for the

optimization.

The current model requires a large number o f parameters, inc luding kine tic and

stoichiom etric parameters. The values o f these parameters reported in the literature vary

significantly between studies. I t is recommended to determine those parameters that w ill

impact significa ntly upon the performance o f the current model using model simulation.

It is postulated in this study that AnSBRs have similar startup time requirements

to UASB reactors. It is recommended to verify this postulate experimentally.

It is postulated in th is study that the content o f active biomass in the sludge is an

important factor for sludge granulation. It was observed that granular biomass has a

higher specific activ ity than flocculent biomass (W irtz and Dague, 1996; Lettinga et al.,

1980). It is postulated in this study that sludge granulation is a consequence (not the

cause) o f improved sludge activ ity; this postulate needs further examination.

The gravity settling model developed in this study was applied to steady state

secondary clarifiers. It is recommended to apply this model to predict the dynamic

response o f secondary clarifiers under varying loading conditions.

In the empirical design for compression depth o f secondary clarifiers (ATV,

1976), the compression depth is independent o f other design and operational parameters

o f the secondary clarifiers. The experimental data o f George and Keinath (1978) clearly

indicate that the compression depth depends on the operational parameters, such as




161

underflow solids concentration (Figure 7.13). It is recommended that new design criteria

for compression depth be developed using the current gravity se ttling model.




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Wat. Res., 34(15), 3725-3736.

8 6 . Snoeyink, V. L. and Jenkins, D. (1980). Water chemistry. W iley, New York.

87. Speece, R. E. (1996). Anaerobic biotechnology fo r industrial wastewater. Archae

Press, Nashville, Tenn.

8 8 . Stumm, W., and Morgan, J. J. (1981). Aquatic Chemistry. 2nd Ed., W iley & Sons,

New York.

89. Sung, S. and Dague, R. R. (1995). “ Laboratory studies on the anaerobic

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90. Suthaker, S., Polprasert, C., and Droste, R. L. (1991). “ Sequencing batch

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Technol., 23, 1249-1257.

91.Takacs I., Patry G. G. and Nolasco D. (1991). “A dynamic model o f the

clarification-thickening process.” Wat. Res., 25(10), 1263-1271.

92. Tay, J. H. and Yan, Y . G (1996). “ Influence o f substrate concentration on

microbial selection and granulation during startup o f upflow anaerobic sludge

blanket reactors.” Water Envir. Res, 68(7), 1140-1150.




173

93. Tay, J. H. and Zhang, X. (2000a). “ Stab ility o f high-rate anaerobic systems. I:

Performance under shocks.” J. Envir. Engrg., ASCE, 126(8), 713-725.

94. Tay, J. H. and Zhang, X. (2000b). “ Stab ility o f high-rate anaerobic systems. II:

Fuzzy stability index.” J. Envir. Engrg., ASCE, 126(8), 726-731.

95. Tay, J. H., Xu, H. L., and Teo, K. C. (2000). “Molecular mechanism o f

granulation. I: H+ Translocation-dehydration theory.” J. Envir. Engrg., ASCE.

126(5), 403-410.

96. Teo, K. C., Xu , H. L ., and Tay, J. H. (2000). “ Molecular mechanism o f

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411-418.

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Appendix A Stoichiometry

Tables A l and A2 are stoichiometry metrices for the model (see Table 3.3 for S

and X and see Bagley and Brodkorb, 1999 for further definitions).

175




Table Al: Stoichiometric coefficients( ia ) for soluble components (Bagley and Brodkorb, 1999)

Process 1,SC 2,Sf 3,Ss 4,Si 5,S a 6 ,SP 7,Sl 8 ,Sb 9,S M X C / 3

o' 1 1 ,Sc0 2

1 Hydrolysis: Xs to soluble fxsc fxSF fxss fxsi2 Hydrolysis: Sc to Sf - 1 fsCF fscs fsci3 Hydrolysis : Ss to Sf fsSF - 1 fssi4 Growth: Sf to S a - 1

Y XFA

Y 1 A,FA

Y XFA

Y 1 H , FA

Y 1 XFA

Y 1 COl,FA

Y 1 XFA

5 Growth: Sf to Sl —1

Y XFL

Y,„fl

Y 1 XFL

6 Growth: Sf to Sb —1

Y 1 XFB

Y 1 B,FB

Y 1 XFB

Y 1 H,FB

Y 1 XFB

Y 1 C02,FB

Y 1 XFB

7 G rowth: Sp to S a Y 1 A,PA

Y 1 XFA

- 1

Y 1 XPA

Y H J’A

Y 1 XPA

Y 1 COl,PA

Y 1 XPA

8 Growth: Sl to S a Y 1 A,LA

Y 1 XIA

- 1

Y 1 XIA

Y 1 H,LA

Y 1 XIA

Y 1 COl,LA

Y 1 XIA

9 Growth: Sl to Sp Y 1 p ,lp

Y 1 XLP

- 1

Y 1 XLP

- y 1 HJ.P

Y 1 XLP

1 0 Growth: Sb to S a Y 1 A,BA

Y 1 XBA

- 1

Y 1 XBA

Y H ,BA

Y 1 XBA

1 1 Growth: S a to Sm - 1

Y 1 XAM

Y M,AM

Y 1 XAM

Y 1 COl,AM

Y 1 XAM

1 2 Growth: Sh to Sm Y 1 M , H M

Y 1 XHM

- 1

Y 1 XHM

- y C02,HM

Y 1 XHM

13 Cell lysis fBSC fBSF fBSS fBSI

176



Table A2: Stoichiometric coefficients (v,y) for particulate components (Bagley and Brodkorb, 1999)

Process 1,XS 2 , Xi 3 ,X f 4, X P 5, X L 6 , X B 7 ,X a 8 ,X h

1 Hydrolysis: Xs to soluble - 1

2 Hydrolysis: Sc to Sf

3 Hydrolysis: Ss to Sf

4 Growth: Sf to S a 1

5 Growth: Sf to Sl 1

6 Growth: Sf to Sb 1

7 Growth: Sp to S a 1

8 Growth: Sl to S a 1

9 Growth: Sl to Sp 1

1 0 Growth: Sb to SA 1

1 1 Growth: S a to Sm 1

1 2 Growth: Sh to Sm 1

13 Cell lysis: X f to Xs, Xi, Sc, Sf, Ss, and Si fBXS fBXI - 1

14 Cell lysis: Xp to Xs, X i, Sc, Sf, Ss, and Si flBXS fBXI - 1

15 Cell lysis: X l to Xs, Xi, Sc, Sf, Ss, and Si fBXS fBXI - 1

16 Cell lysis: XB to Xs, Xi, Sc, Sf, Ss, and Si fBXS fBXI - 1

17 Cell lysis: X a to Xs, Xi, Sc, Sf, Ss, and Si fBXS fBXI - 1

18 Cell lysis: X h to Xs, Xi, Sc, Sf, Ss, and Si fBXS fBXI - 1

177



Appendix B Source Code

The source code is divided in to the fo llow ing list o f files:

Header file : Asbr.h

File 1: phdw.c

File 2: inp.c

File 3: ode.c

File 4: stiff.c

File 5: sim.c

File 6 : der.c

File 7: jac.c

File 8 : settle.c

178




Header file: asbr.h

# i n c l u d e < s t d d e f . h >

# i n c l u d e < s t d l i b . h >

# i n c l u d e < s t d i o . h ># i n c l u d e < m a t h . h ># i n c lu d e < s t r i n g . h >

# d e f i n e T IN Yo de 1 . 0 e - 1 0 / / s e t s y s c a l i n ODEINT, i f y=0.0 & dy dx =0 .0# d e f in e T IN Yrh o 1 .0 e -1 0 / / o r i g i n a l l y = l e - 1 0 , a v o i d exc

ess c a l c , f o r t i n y ##def ine FREE_ARG char *#def ine YVAR 22

i n t e g r a t i o n c a l c .#def ine ZVAR 24

t o y k ee p & p r i n t e d# d e f in e P 21r a t i o n )# d e f i n e MAXKEEP 16005

t i m e s t e p a r r a y s

# d e f i n e MAXSTP 10000#def ine CONTIN 0.97

i d e f i n e KMAXX 7# d e f in e IMAXX (KMAXX+1)#de f ine SAFE1 0 .25

#def ine SAFE2 0.7#def ine REDMAX 1.0e-5#def ine REDMIN 0.7# d e f i n e T I N Y 1 . 0 e - 3 0#def ine SCALMX 0.1

/ / # c o mp o ne n ts i t o b e us e d i n

/ / # c o mp o ne n ts i t o b e s t o r e d

/ / # p r o c e s s e s j ( us ed i n i n t e g

/ / a l l o c a t e s memory f o r

# d e f i n e F MA X( a, b) ( m a x a r g l = ( a ) , m a x a r g 2 = ( b ) , ( m a x a r g l ) >(maxarg2) ? \

(maxarg l ) : (maxarg2) )/ / # d e f in e FMAX(a,b ) ( (a ) > (b) ? (a) : ( b ) ) / / n o t

used

# d e f i n e F M IN (a ,b ) ( m i n a r g l = ( a ) , m i n a r g 2 = ( b ) , ( m i n a r g l ) <( m i n a r g 2 ) ? \

( m i n a r g l ) : ( m i n a r g 2 ) )

# d e f i n e S IG N(a ,b ) ( (b ) >= 0 .0 ? f a b s (a ) - f a b s ( a ) )

179




#define SQR(a) ((sqarg=(a))==0.0?0.0:sqarg*sqarg)

# d e f i n e t r u e 1# d e f i n e f a l s e 0

# d e f i n e NO 5000# d e f i n e TOL L l e - 6

t yp e d e f s t r u c t

{

d o u b l e c o n e ;d o u b l e d i s s ;

} A c i d ;

t yp e d e f s t r u c t

{

d o u b l e c o n e ;d o u b l e k x ;d o u b l e k s ;d o u b l e b x ;

} M i c r o ;

/ * d e r * /v o i d d e r i v s ( d o u b l e x , d o ub le y [ ] , d ou b l e d y d x [ ] , d o ub le * *

nu, \ d o u b le k i n [ ] , d o u b le s t o [ ] , d o u b l e y i n f [ ] , \d o ub le q f ,d o u b l e q d ,d o u b le n f s x , d o u b l e n f r , d o u b l e

v r t 1 , \d o u b l e g P H 2 , d o u b l e g P C 0 2 , d o u b l e g P C H 4 , d o u b l e m i x )

r

v o i d R ATE(doub le * r ,d o u b l e y [ ] , d o u b l e * * n u ,d o u b l e k i n [ ] , d

o u b l e s t o [ ] , \d o u b l e g P H 2 , d o u b le gP C 0 2 , d o u b le gP C H 4 ,d o u bl e v r t l

, d o ub le n f r , d o u b l e m i x ) ;

v o i d RHO(double * rh o , d o u b l e y [ ] , d o u b l e k i n [ ] , d o ub le s t o [ ], \

d o u b l e C H , d o u b l e P H 2 , d o u b l e g P H 2 , d o u b l e g P C 0 2 , d o ub l e gP CH 4,dou ble v r t l , d o u b l e m i x ) ;

d o u b l e P H ( d ou b l e C N a n e t, d o u b l e K w , d o u b l e C T C 0 3, \

180




double KaH2C03, Acid *ac,int num) ;

/ * j a c * /v o i d ja c o b n ( d o u b l e x , d o ub le y [ ] , d o ub le d f d x [ ] , d o u b l e * *

d f d y , \

d o ub le * * n u ,d o u b le k i n [ ] , d ou b le s t o [ ] , \d o ub le q f , d o u b l e q d , d ou b le n f s x , d o u b l e n f r , d o u b l e

v r t l , d o u b l e m i x ) ;

v o i d D RATEDY(double * * d f d y , d o u b l e y [ ] , d o u b l e * * n u , \d ou b le k i n [ ] , d ou b le s t o [ ] , d ou bl e v r t l , d o u b l e n f r ,

d o ub le m i x ) ;

v o i d DRHODY(double * * d r h o d y , d o u b l e y [ ] , d o u b l e k i n [ ] , d o u b l

e s t o [ ] , \d o ub le C H ,d ou ble P H2 ,d ou bl e v r t l , d o u b l e m i x ) ;

/ * s im * /v o i d s im p r (d o u b l e y [ ] , d o u b l e d y d x [ ] , d ou b le d f d x [ ] , d ou b l e

* * d f d y , i n t n, d o u b l e x s , \d ou bl e h t o t , i n t n s te p ,d o u b l e y o u t [ ] , \d ou ble * * n u ,d o u b le k i n [ ] , d ou b l e s t o [ ] , d o u b l e y i n

f [ ] , \d o ub le q f ,d o u b l e q d ,d o u b le n f s x , d o u b l e n f r , d o u b l e

v r t 1 , \d o u b l e g P H 2 , d o u b l e g P C 0 2 , d o u b l e g P C H 4 , d o u b l e m i x ,

\v o i d ( * d e r i v s ) ( d o u b l e , d o u b l e [ ] , d o u b l e [ ] , d o u b le * *

,\d o u b l e [ ] , d o u b l e [ ] , d o u b l e [ ] , \d o u b l e , d o u b l e , d o u b l e , d o u b l e , d o u b l e , \d o u b l e , d o u b l e , d o u b l e , d o u b l e ) ) ;

v o i d l u b ks b ( d o u b le * * a , i n t n, i n t * i n d x , d o ub le b [ ] ) ;

v o i d l ud c mp (d ou b le * * a , i n t n, i n t * i n d x , d o u b l e * d ) ;

/ * s t i f f * /v o i d s t i f b s ( d o u b l e y [ ] , d o u b l e d y d x [ ] , i n t n v ,d o u b le * x x , d ou b le h t r y , d o u b l e e p s , \

d ou b le y s c a l [ ] , d ou bl e * h d i d ,d o u b l e * h n e x t , \d ou b l e * * n u ,d o u b le k i n [ ] , d ou ble s t o [ ] , d ou b l e y i n

f [ ] , \

181




double qf,double qd,double nfsx,double nfr,doublevrtl,\

d o u b l e g P H 2 , d o u b l e g P C 0 2 , d o u b l e g P C H 4 , d o u b l e m i x ,

\v o i d ( * d e r i v s ) ( d o u b l e , d o u b l e [ ] , d o u b l e [ ] , d o u b l e * *

, \d o u b l e [ ] , d o u b l e [ ] , d o u b l e [ ] , \d o u b l e , d o u b l e , d o u b l e , d o u b l e , d o u b l e , \d o u b l e , d o u b l e , d o u b l e , d o u b l e ) ) ;

v o i d p z e x t r ( i n t i e s t , d o u b l e x e s t , d o u b l e y e s t [ ] , d o u b le y z [

] , d o u b le d y [ ] , i n t n v ) ;

/ * o d e * /v o i d o d e in t (d o u b l e y s t a r t [ ] , d ou b le x l , d o u b l e x 2 , \

d o ub le e p s , d o u b le h i , d o u b l e h m i n , \d o ub le * * n u , d o u b le * k i n , d o u b l e * s t o , d o u b l e * y i n f ,

\d o ub le q f , d o u b l e q d ,d o u b le n f s x , d o u b l e n f r , d o u b l e

v r t 1 , \d o u b l e g P H 2 , d o u b l e g P C 0 2 , d o u b l e g P C H 4 , d o u b l e m i x ,



v o id n r e r r o r ( c h a r e r r o r _ t e x t [ ] ) ;

/ * i n i t * /

v o i d i n i t ( d o u b l e * *n u ,d o u b le * y i n f , d o u b l e * y i n i , d o u b l e *s

t o , \d o ub le * k i n , d o u b l e * p r o p , d o u b le * g p p ) ;

v o i d r e a d _ i n i _ f i l e ( c h a r * f i l e , d o u b l e * a c o n c , i n t n a c , d o u b l

e * g p p , i n t n g p ) ;

v o id r e a d _ f i l e (c h a r * f i l e , d o u b l e * a r , i n t n i t e m ) ;

v o i d ps e y (d o u b le * s t o ) ;

v o i d F in d _ nu ( do u b le * * n u , d o u b l e * s t o ) ;

v o id t i m f l ( d o u b l e * p r o p ) ;

182




void pywc(double *yt,double *gp,double *kin);

void pyre(double *yt,double *gp,double *kin);

/ * s e t t l e * /i n t s e t t ( d o u b l e * h g t , d o u b l e s d t , d o u b l e d z , d o ub l e * v , d o u b l

e * c o n c ) ;d o u b l e c o n p h ;

183




File 1: phdw.c

# i n c l u d e " a n s b r . h "/ * co2 1 . 9 2 , h 2 , 4 . 5 0 ch4 1 . 4 9 * /

v o i d u a s b( d ou b le * q f , d o u b l e * q d , d o u b le * n f s x , d o u b l e * n f r ,

\d o ub le * m i x , i n t * s t a g e , d o u b l e t c , d o u b l e *p r o p , d o u

b l e * q s ) ;v o i d s b r a f ( d o u b l e * t c , d o u b l e * v r t l , d o u b l e *g PH 2 , do ub le * gPC02, dou b le

*gPCH4 , \d o u bl e q f , d o u b l e q d , d o u b l e d t , d o u b l e * z) ;

v o i d e x t r a ( d o u b l e * z , d o u b l e * y , d o u b l e q f , d o u b l e q d ,d o u b l e

n f s x , d o u b l e

n f r , \d o u b l e v r t l , d o u b l e g P H 2 , d o ub l e g P C 0 2 , d ou b l e gPCH4

, d o u b l em i x ,d o u b le s t a g e , \

d o ub le d t , d o u b l e v g a s , \d ou b le * k i n , d o u b l e * y i n f , d o u b l e * g p p ) ;

v o id p r i n t t o f i l e ( i n t j , d o u b l e d t , d o u b l e * z ,d o u b l e * y , F I L E

* f o ) ;i n t h v t ( d o u b l e h o , d o u b le t h , d o u b l e d h , d o ub l e c o , d o u b le d t

, d o u b l et t , d o u b l e * h t ) ;d o ub le * s e t f q d ( i n t n um ,d ou ble d t , d o u b l e * h t ) ;

v o i d m a i n ( v o i d )

{

i n t i , j , s t a g e , n c y c l e = 0 /d o ub le y i n f[ Y V A R + 1 ] , y i n i [ YVAR+1] , y [ Y V A R + l ] ;

/ * i n f l u e n t , i n i t i a l a nd m ix Y v a r . * /d ou b l e s t o [ 5 2 ] , k i n [ 8 3 ] , p r o p [3 9 ] , * * n u , g p p [ 6 ] ;d ou ble gP H2 ,gP C02 ,g PC H4 ,q f, q d , n f s x , n f r , m i x , v r t l ;

d o u b l e g h , g o , g c ;d o ub le z [ ZVAR+1] , t c , d t ;c ha r * o u t _ f i l e = " a s b r . o u t " ; / * u t p u t f i l e , f i

n a lc o n c e n t r a t io n s . * /

d o u b le h = 0 . 1 , h m i n = 0 .0 ; / * t i m e s

t e p h i n h o u r s

* /

d ou b le e p s= 1 .0 e -4 ; / * e r r o r s e n s i t i v

i t y * /d ou bl e h t [ 8 2 ] , t e m , * q s ;

F I L E * f o ;

184




f o = f o p e n ( o u t _ f i l e , " w " ) ;f p r i n t f ( f o , " j , T ime, S ta g e ,

S f ,Ss ,

S I ,

Na,Salic, "

X f ,

ot ,X t o t , "

0 2 ,

S i , "

Sb,Sm, Sh, "

Xp,X I , Xb ,

PCH4,

PH2,

03,

f f ,

g e , \ n " ) ;

CH2C03,CVFA,

PC02,"

TSS, "

CODbal,

C O D t o t , "

Sa,

Sn,

Xs,

Xa,

rCH4,

Qg,pH,

logPH2,

SVFA,

/ * d y na m ic memory a l l o c a t i o n

Sc,

Sp,

Sco3,

X i ,

Xh,

rH2,

V r , "CNa,

S e f f ,

SCOD,

7

St

rC

CHC

Xe

Sta

e ) ) ;

n u = ( d o u b l e * * ) m a l l o c ( (YVAR+1) * s i z e o f ( d o u b l e * ) ) ;

f o r ( i= 0 ; i< =Y VA R; i+ +)n u [ i ] = (d ou b l e * ) m a l l o c ( ( P + l ) * s i z e o f ( d o u b l

i n i t ( n u , y i n f , y i n i , s t o , k i n , p r o p , g p p ) ;t e m =0 ;/ * fo r ( i= 1 5 ; i < = Y V A R ; i+ + ) t e m = t e m + y i n i [ i ] ;

i = h v t ( 0 . 7 2 , 0 . 3 6 , 0 . 0 1 , t e m / 1 0 0 0 / k i n [ 5 1 ] * p r o p [ 3 ] / p r o p [ 1 ] , p r op [10] , 8 , h t ) ;i f ( i = = 0 ) e x i t (1) ;p r o p [ 1 8 ] = i * p r o p [ 1 0 ] ;

q s = s e t f q d ( i , p r o p [ 1 0 ] , h t ) ; * /

185




q s = (d o u b le * ) m a l lo c ( 1 0 0 * s i z e o f ( d o u b l e )) ;f o r ( i = l ; i < 1 0 0 ; i + + ) q s [ i ] = p r o p [ 2 ] / p r o p [ 1 8 ];/ * t h e a bo ve t w o l i n e s comment o u t t h e s e t t l e mode

1*/gPH2=gpp [1 ] ;

g P C 0 2 =g p p [ 3 ] ;gPCH4=gpp [2 ] ;c o n p h = p r o p [ 3 5 ] ;i f ( p r o p [ 3 6 ]< 0 .5 )

{v r t l = p r o p [ 3 ] ;

}e l se i f ( p r o p [ 3 6 ]< 1 .5 )

{

v r t l = p r o p [ 2 ] + pro p [ 3 ] ;p r o p [ 2 5 ] = p ro p [ 2 5 ] * p r o p [ 3 ] / v r t l ;

}d t = p r o p [ 1 0 ] ;t c = 0 ;f o r ( i = l ; i < = 1 4 ; i + + )

{y [ i ] = y i n i [ i ] ;

}f o r ( i = 1 5 ; i< =Y V AR ; i+ + )

{

i f ( p r o p [ 3 6 ] < 0 . 5 )

{y [ i ] = y i n i [ i ] ;

}e l s e i f ( p r o p [ 3 6 ]< 1 .5 )

{y [ i ] = y i n i [ i ] * p r o p [ 3 ] /vrtl;

}

}

p r i n t f ( " I n i t s u c c e s s \ n " ) ;f o r ( j = l ; j < = 1 0 0 0 0 ; j+ + ) / * j i s

a t i m e s t e ph e r e . * /

{

uasb (&q f , &qd, Senfsx, & n f r , &m ix , & s ta g e , t c , p r

o p , q s ) ;o d e i n t ( y , 0 , d t , eps , h, h m i n , n u , k i n , s t o , y

i n f , \

q f , q d , n f s x , n f r , v r t l , g P H 2 , gPC02, gPCH4, m ix , d e r i v s ) ;e x t r a ( z , y , q f , q d , n f s x , n f r , v r t l , g P H 2 , g P C 0 2 ,

186




\

gPCH4, m i x ,s t a g e , d t , p r o p [ 5 ] , k i n , y i n f , g p p ) ;s b r a f ( & t c , & v r t 1 , & g h , & g o ,& g c , q f , q d , d t , z ) ;p r i n t t o f i l e ( j , d t , z , y , f o ) ;

i f ( t c - ( p r o p [ 1 2 ] + p r o p [ 1 4 ] + p ro p [ 1 8 ] + p r o p [21] ) > - 0 . 0001)

k o . o o o i :

{

i f ( z [ 2 ] > p r o p [2 5 ])

{f o r ( i = 1 5 ; i < =Y V A R ;i + + )

{i f ( p r o p [ 3 6 ] - 0 . 0 00

{

y [ i ] = y [ i ] * ( v r t l - p r o p [ 4 ] ) / v r t 1 ;

1 . O O O K O . 0 0 0 1 '

y [ i ] = y [ i ] * ( 1 - p r o p [ 4 ] / p r o p [ 3 ] ) ;

}e ls e i f ( p r o p [ 3 6 ] -

}}t c = 0 ;n c y c l e = n c y c l e + l ;p r i n t f ( " % d c y c l e e n d \ n " , n c y c l e ) ;

i f (n c y c l e > = p r o p [ 9 ] )

{p r i n t f ( "COD= % l f \ n " , z [ 1 ] )

r

b r e a k ;

}

i f ( z [ 1 ] < =p r o p [1 6 ] ) b r e a k ;

}/ * i f ( t c - ( p r o p [ 1 2 ] + p r o p [1 4 ] + p ro p [ 1 8 ] + p r o p [21

] ) > - 0 . 0 0 0 1 )

{

i f ( p r o p [ 2 3 ] < 0 . 0 0 0 1 ){

i f ( z [ 2 ] > p ro p [ 2 5 ])p r o p [ 2 3 ] = p r o p [ 4 ] / p r o p [ 2 4 ] ;

}

1 8 7




i f (t c - ( p r o p [ 12 ] + p r o p [ 14 ] + p r o p [ 18 ] 4 - p r o p [21 ] 4 - p r o p [23 ] ) >-0 . 00 0 1 )

{

c y c l e ) ;

n " , z [ 1] )

r o p [ 1 3 ]

t c = 0 ;

n c y c l e = n c y c l e 4 - l ;

p r i n t f ( " % d c y c l e e n d \ n " , n

i f ( n c y c l e > = p r o p [ 9 ] )

{p r i n t f ( " C O D = % l f \

b r e a k ;

}i f ( z [ 1 ] < = p r o p [ 1 6 ] ) b r e a k ;p r o p [ 1 2 ] = ( p r o p [ 1 ] - v r t l ) / p

p r o p [ 2 3 ] = 0 ;

}} * /

}f c l o s e ( f o ) ;f o r ( i= 0 ; i< = Y V A R ; i+ + ) f r e e ( n u [ i ] ) ;f r e e ( n u ) ;f o r ( i= 15 ; i<=YVAR; i - f- f)

{i f ( p r o p [ 3 6 ] < 0 . 5 )

{y [ i ] = y [ i ] ;

}e l s e i f ( p r o p [ 3 6 ] < 1 .5 )

{y [ i ] = y [ i ] / p r o p [ 3 ] * v r t l ;

}}f o = f o p e n ( " a s b r _ i n i . r s t " , " w " ) ;

f o r ( i = l ; i < = Y V A R ; i + + )f p r i n t f ( f o , " % l f \ n " , y [ i ] ) ;

f p r i n t f ( f o , "% 8 . 7 1 f \ n % 8 . 7 1 f \ n % 8 . 7 1 f \ n % 8 . 7 1 f \ n % 8 . 7 1 f \ n " , z [7

] , z [ 6 ] , z [ 8 ] , gp

p [ 4 ] , gpp [ 5 ] ) ;f c l o s e ( f o ) ;p r i n t f ( "P ro g r a m e n d \ n " ) ;

} / * -----------------------------End of M a i n - - - - - - - -- - - - - - - -- - - - - - - -- - - - - - -- - - - - - - -- - - - - - - -- - - - - - -- - - - - - - -- - - - - - - -- - - - - - - -- - - - - - -- - - - - - - -- - - - - - - -- - - - * /

1 8 8




v o i d u as b( d ou b le * q f , d o u b l e * q d , do u b l e * n f s x , d o u b l e * n f r ,

\d ou bl e * m i x , i n t * s t a g e ,d o u b l e t c , d o u b l e * p r o p ,d o u

b l e * q s )

{d o u b l e s v = 0 . 0 0 0 1 ;i n t t i ;

/ * f i l l s t a g e * /i f ( p r o p [ 1 2 ] - t c > s v ) / * s a f e t y v a lu e t o a v o id r

ea l numberp r o b l e m s * /

{* q f = p r o p [ 1 3 ] ;* q d = 0 ;* n f s x = p r o p [ 3 3 ] ;* n f r = p r o p [ 28 ] ;* m i x = p r o p [ 3 4 ] ;

* s t a g e = l ;

}/ * --------- REACT st a g e * /

e l se i f ( p r o p [ 1 2 ] + p ro p [ 1 4 ] - tc > s v )

{

* q f = 0 ;* q d = 0 ;* n f s x = p r o p [ 3 3 ] ;* n f r = p r o p [ 2 9 ] ;* m i x = p r o p [ 3 4 ] ;

* s t a g e = 2 ;

}

/ * --------- SETTLE s t a g e ( n o t e : r e a c t i o n o c c u r s i n s e t t l e d ph

a se o n l y ) * /e l se i f ( p r o p [ 1 2 ] + p r o p [ 1 4 ] + p r o p [ 1 8 ] - t c> s v )

{* q f = 0 ;t i = c e i l ( ( t c - ( p r o p [ 1 2 ]+ p r o p [ 1 4 ] ) ) / p r o p [ 10]

+ 0 . 5 ) ;* q d = q s [ t i ] ; / * p r o p [ 2 2 ] ; * /* n f s x = p r o p [ 3 3 ] ;

* n f r = p r o p [ 3 0 ] ;

* m i x = p r o p [ 3 4 ] ;* s t a g e = 3 ;

}/ * ----------- DRAW s t a g e ( n o t e : r e a c t i o n o c c u r s i n s e t t l e d p h as

e o n l y ) * /e l s e i f ( p r o p [ 1 2 ] + p r o p [ 1 4 ] + p ro p [ 1 8 ] + p r o p [ 2 1 ] - t c > s

189




V)

{

* q f = 0 ;* q d = 0 ;* n f s x = p r o p [ 3 3 ] ;

* n f r = p r o p [ 3 1 ] ;* m i x = p r o p [ 3 4 ] ;* s t a g e = 4 ;

}

/ * ----------- WASTE s ta ge (no te : r e a c t i o n o cc u rs i n s e t t l e d pha

se o n l y ) * /e l s e / * & & ( X t o t > = X w as t e ) ) t r i g g e r i n m ai n * /

{* q f = 0 ;* q d = p r o p [ 2 4 ] ;* n f s x = 0 ;* n f r = p r o p [ 3 2 ] ;* m i x = p r o p [ 3 4 ] ;* s t a g e = 5 ;

}i f ( p r o p [ 3 6 ] - s v< sv ) r e t u r n ;

* q f = p r o p [ 2 ] / p r o p [ 6 ] ;* q d = * q f ;* n f s x = p r o p [ 3 3 ] ;* n f r = p r o p [ 2 9 ] ;

* m i x = p r o p [ 3 4 ] ;/ * * s t a g e = 2 ; * /i f ( p r o p [ 3 6 ] - l - s v < s v ) r e t u r n ;

r e t u r n ;} / * ---------------------------------------End o f D e r i v s F u n c t i o n * /v o i d s b r a f ( d o u b l e * t c , d o u b l e * v r t l , d o u b l e * gP H2 ,d ou bl e *g

PC02, doub le*gPCH4, \

d o u b l e q f , d o u b l e q d , d o u b l e d t , d o u b l e * z )

{* t c = * t c + d t ;* v r t l = * v r t l + ( q f - q d ) * d t ;

* g P H 2 = z [ 7 ] ;* g P C 0 2 = z [ 8 ] ;* g PC H 4 = z [ 6 ] ;

}

v o i d e x t r a ( d o u b l e * z , d o u b l e * y , d o u b l e q f , d o u b l e q d , do u b l e

n f s x , d o u b l en f r , \

190




*tb/ta;/ * z [ 8 ] = t b - z [ 6 ] - z [ 7 ] ; * /

}

e l s e

{

tb=gPCH4+gPH2+gPC02;

z [6] = ( z [ 3 ] * d t * k i n [ 7 4 ] * k i n [ 7 5 ] + g P C H 4 * ( v g a s - t a ) ) / v g a s * ( g p p [5 ] - g p p [ 4 ] ) / g p p

[5 ] ;

z [ 7 ] = ( z [ 4 ] * d t * k i n [ 7 4 ] * k i n [ 7 5 ] + g P H 2 * ( v g a s - ta ) ) / v g a s * ( g p p [5] - g p p [ 4 ] ) / g p p [

5] ;

z [8] = ( z [ 5 ] * d t * k i n [7 4 ] * k i n [7 5 ] +gPC02 * ( v g a s - t a ) ) / v g a s * ( g p p [5 ] - g p p [ 4 ] ) / g p p[ 5 ] ;

/ * z [ 8 ] = t b - z [ 6 ] - z [ 7 ] ; * /

}t b = g p p [ 5 ] - g p p [ 4 ] ;

z [ 9 ] = k i n [ 7 4 ] * k i n [ 7 5 ] * ( z [ 3 ] + z [ 4 ] + z [ 5 ] ) / t b ;z [ 1 0 ] = v r t l ;

z [ 1 1 ] = - l o g l 0 ( t c h / 1 0 0 0 . 0 ) ;

t a = 0 ;f o r ( i = 0 ; i < = 3 ; i+ + )

{t b = a c [ i ] . d i s s / ( t c h + a c [ i ] . d i s s ) * a c [ i ] . co

n c ;

t a = t a + t b ;

}

z [12] = y [ 1 3 ] / k i n [ 6 4 ] - t a ;z [13] = y [ 1 2 ] / k i n [ 6 3 ] * k i n [ 7 0 ] / ( k i n [7 0 ] + t c h ) ;z [ 1 4 ] = y [ 1 2 ] / k i n [ 6 3 ] * t c h / ( k i n [ 7 0 ]+ t c h ) ;

t a = 0 ;

f o r ( i = 0 ; i < = 3 ; i + + ) t a = t a + a c [ i ] . c o n e ;z [ 1 5 ] = t a ;t a =0 ;f o r ( i = 1 7 ; i < = 2 2 ; i + + ) t a = t a + y [ i ] ;

z [ 1 6 ] = t a / k i n [ 5 2 ] + y [ 1 5 ] / k i n [53] + y [ 1 6 ] / k i n [ 5 4 ] ;i f ( y [ 1 0 ] > 0 )z [ 1 7 ] = l o g l 0 ( y [ 1 0 ] / k i n [ 6 0 ] / k i n [ 7 3 ] ) ;

e l s e

{z [ 1 7] = 0;

192




}

z [ 1 8 ]= 0 ;z [ 1 9 ]= 0 ;

i f ( s ta ge == 4)

{

ta =0 ;f o r ( i = l ; i < = 8 ; i+ + ) t a = t a + y [ i ] ;z [ 1 8 ]= t a ;

t a = 0 ;f o r ( i = 1 5 ; i < = 2 2 ; i + + ) t a = t a + y [ i ] * ( 1 - n f s x ) ;z [ 1 9 ] = t a ;

}

/ ^ i m pl em e nt l a t e r * /z [ 2 0 ] = 0 ;

z [21]=0;

t a = 0 ;f o r ( i = 5 ; i < = 8 ; i + + ) t a = t a + y [ i ] ;z [ 2 2 ] = t a ;

z [ 2 3 ] = z [ l ] + y [ 1 1 ] / 6 4 . 0 * 1 4 . 0 ;z [ 2 4 ] = s ta g e;

}

v o id p r i n t t o f i l e ( i n t j , d o u b l e d t , d o u b le * z ,d o u b l e * y , F I L E

* f o )

{ i n t i ;f p r i n t f ( f o , " % 5 d , " , j ) ;f p r i n t f ( f o , "% 6 . 2 1 f , " , j * d t ) ;f p r i n t f ( f o , " % 5 . I l f , " , z [24] ) ;f o r ( i = l ; i < = YVAR; i++ ) f p r i n t f ( f o , " % 1 2 . 7 1 f , " , y [ i ] ) ;

f o r ( i = l ; i <= Z V AR ; i+ + ) f p r i n t f ( f o , "% 1 2 . 7 1 f , " , z [ i ] ) ;f p r i n t f ( f o , " \ n " ) ;r e t u r n ;

}

i n t h v t ( d o u b l e h o , d o u b l e t h , d o u b l e d h , do u b l e c o , d o ub l e d t

, d o ub l et t , d o u b l e * h t )

{

i n t i , n u m , l a y ;d o u b l e * c o n , * v ;d o u b le h ;

193




n u m = c e i l ( t t / d t ) ;l a y = c e i l ( h o / d h ) ;

c on =( do ub le * ) c a l l o c ( ( l a y + 2 ) , s i z e o f ( d o u b l ev =( do ub le * ) c a l l o c ( ( l a y + 1 ) , s i z e o f ( d o u b l e ) )f o r ( i = l ; i < = l a y + l ; i + + ) c o n [ i ] = c o ;

h t [ 0 ] = ho;h=ho;f o r ( i = l ; i < = n u m ; i + + )

{i f ( s e t t ( & h , d t , d h , v , c o n ) ==0) r e t u r nh t [ i ] = h ;

i f ( h < t h ) b r ea k ;

}f r e e ( c o n ) ;f r e e ( v ) ;r e t u r n i ;

}

d ou b l e * s e t f q d ( i n t n um ,d ou bl e d t , d o u b l e * h t )

{i n t i ;d o u b l e t e r n , * q s ;t em=1 2 / 0 . 7 2 / d t ;

q s =( do ub le * ) c a l l o c ( n u m + 1 , s i z e o f ( d o u b l e ) ) ;f o r ( i = l ; i < n u m ; i + + )

{qs [ i ] = (h t [ i —1 ] - h t [ i ] ) * t em;

}q s [n u m ]= ( h t [ n u m - 1 ] - 0 . 3 6 ) * te m;r e t u r n q s ;

}

194




File 2: inp.c

# i n c l u d e " a n s b r . h "

v o i d i n i t ( d o u b l e * * n u , d o u b l e * y i n f , d o u b l e * y i n i , d o u b l e *st o , \

d o u b l e * k i n , d o u b l e * p r o p , d o u b l e * g p p ){

F I L E * f o ;c h a r * i n i _ f i l e = " a s b r _ i n i . i n " ; / * i n p u t f

i l e names * /c h a r * i n f _ f i l e = " a s b r _ i n f . i n " ;c h ar * s t o _ f i l e = " a s b r _ s t o . i n " ;c h ar * k i n _ f i l e = " a s b r _ k i n . i n " ;c ha r * p r c _ f i l e = " a s b r _ p r c . i n " ;

i n t i ;r e a d _ f i l e ( p r c _ f i l e , p r o p , 3 8 ) ;r e a d _ f i l e ( k i n _ f i l e , k i n , 8 2 ) ;

r e a d _ i n i _ f i l e ( i n f _ f i l e , y i n f , Y VAR ,g p p , 5 ) ;p y w c ( y i n f , g p p , k i n ) ;/ * i f ( y i n f [ 1 4] < 0. 1 ) p y w c ( y i n f , g p p , k i n ) ;f o = f o p e n ( " i n f . r s t " , " w " ) ;f o r ( i = l ; i < =Y VA R ; i ++ )

f p r i n t f ( f o , " % l f \ n " , y i n f [ i ] ) ;

f o r ( i = l ; i < = 5 ; i + + )f p r i n t f ( f o , " % 6 . 5 f \ n " , g p p [ i ] ) ;

f c l o s e ( f o ) ; * /i f ( p r o p [ 3 8 ] < 0 . 5 )

{r e a d _ i n i _ f i l e ( i n i _ f i l e , y i n i , Y V A R , g p p , 5 ) ;p y r e ( y i n i , g p p , k i n ) ;

}e l s e

{

i n i _ f i l e = " a s b r _ i n i . r s t " ;

r e a d _ i n i _ f i l e ( i n i _ f i l e , y i n i , Y V A R , g p p , 5 ) ;

}r e a d _ f i l e ( s t o _ f i l e , s t o , 5 1 ) ; / * r e a d s t o , k i n , p r c *

/p s e y ( s t o ) ;

F i n d _ n u ( n u , s t o ) ; / * c a l l f i n d nu * /

t i m f l ( p r o p ) ; / * p r oc e ss s ta ge t im e s * /r e t u r n ;

/ * - Read I n i t i a l R e ac to r and i n f l u e n t C o n c e n t r a t i o n s f r omI N I , I N f i n p u t

195




f i l e * /v o id r e a d _ i n i _ f i l e (c h a r * f i l e , d o u b l e * a c o n c , i n t n a c ,d o u b le * g p p , i n t

ngp){

i n t i = l , j = l ;c h a r t e m p [ 8 0 ] ;F I L E * f p ;f p = f o p e n ( f i l e , " r " ) ;i f ( fp == NULL )

{p r i n t f ( "Can n o t open %s. \ n " , f i l e ) ;

e x i t ( 1 ) ;

}w h i le ( ! f e o f ( f p ) )

{f g e t s ( t e m p , 8 0 , f p ) ;i f ( t e m p [ 0 ] ! = ' / ' &&temp[ 1 ] ! = ' / ' & &temp[ 0 ] ! = '

\0 ' ){

i f ( i < =n ac )

s s c a n f ( t e m p , " % l f " , & a c o n c [ i+ + ] ) ;e l se s s c a n f ( t e m p , " % l f " , &g

p p [ j + + ] ) ;}

i f ( j > n g p ) b re ak ;

}f c l o s e ( f p ) ;

}/ * -------------------- Read C on s t an ts f ro m i n p u t f i l e : s t o , k i n , p r c

*/

v o i d r e a d _ f i l e ( c h a r * f i l e , d o u b l e * a r , i n t n it em )

{i n t i = l ;c h a r t e m p [ 8 0 ] ;F I L E * f p ;f p = f o p e n ( f i l e , " r " ) ;i f ( fp == NULL )

{p r i n t f ( "Canno t open % s . \ n " , f i l e ) ;

e x i t ( 1 ) ;}w h i l e ( ! f e o f ( f p ) )

{f g e t s ( t e m p , 8 0 , f p ) ;i f ( t e m p [ 0] ! = ' / ' & & te mp [1] ! = ' / 1& & te m p[ 0] ! = '

196




\ o -

}

s s c a n f ( t e m p , " % l f " , & a r [ i + + ] )i f ( i > n i t e m ) b re ak ;

}f c l o s e ( f p ) ;

C a l c . P r o d u c t / S u b s t r a t e E ne rg y Y i e l d s/*----------

on GrowthY i e l d s

s ince Yp ,sp = YThOD* fe , where fe = 1 - f s and f s

COD u n i t s )

*/

v o i d p s e y ( d o u b l e * s t o )

{s t o [26] = s t o [ 2 6 ] * ( 1 . 0 - s t o [ 17 ]) ;

ens S f => Sa

* /s t o [27]s t o [28]

s t o [ 2 7 ] * ( 1 . 0 - s t o [ 1 7 ]s t o [ 2 8 ] * ( 1 . 0 - s t o [ 1 7 ]

7

s t o [29] = s t o [ 2 9 ] * ( 1 . 0 - s t o [18]

Sf => SI

s t o [ 3 0 ] = s t o [ 3 0 ] * ( 1 . 0 - s t o [18]s t o [31] = s t o [ 3 1 ] * ( 1 . 0 - s t o [18]

s t o [ 3 2 ] = s t o [ 3 2 ] * ( 1 . 0 - s t o [19]

Sf => Sb7

s t o [33] = s t o [ 3 3 ] * ( 1 . 0 - s t o [19]s t o [ 3 4 ] = s t o [ 3 4 ] * ( 1 . 0 - s t o [19]

s t o [ 3 5 ] = s t o [ 3 5 ] * ( 1 . 0 - s t o [20]

Sp =>

s t o [3 6 ] = s t o [ 3 6 ] * ( 1 . 0 - s t o [20]s t o [37] = s t o [ 3 7 ] * ( 1 . 0 - s t o [20]

s t o [38] = s t o [ 3 8 ] * ( 1 . 0 - s t o [21]

t a t e S I =>Sa * /s t o [3 9 ] = s t o [ 3 9 ] * ( 1 . 0 - s t o [21]s t o [4 0 ] = s t o [ 4 0 ] * ( 1 . 0 - s t o [21]

i o n a t eSa * /

s t o [ 4 1 ] = s t o [ 4 1 ] * ( 1 . 0 - s t o [ 22 ])

197

based

Yxsp (

A c i d o g

* Prop

* Lac

* n o t




e : S1 +Sh => Sp * /

s t o [4 2 ] = s t o [ 4 2 ] * ( 1 . 0 - s t o [ 2 2] )

s t o [43] = s t o [ 4 3 ] * ( 1 . 0 - s t o [ 2 2 ] )

s t o [4 4 ] = s t o [ 4 4 ] * ( 1 . 0 - s t o [ 2 3] )r a t e Sb => Sa

/s t o [4 5 ] = s t o [ 4 5 ] * ( 1 . 0 - s t o [ 2 3] )s t o [ 4 6 ] = s t o [ 4 6 ] * ( 1 . 0 - s t o [ 2 3 ] )

/ * B u t y

s t o [4 7 ] = s t o [ 4 7 ] * ( 1 . 0 - s t o [ 2 4] )ano gen s Sa =>

Sm 7

s t o [ 4 8 ] = s t o [ 4 8 ] * ( 1 . 0 - s t o [ 24 ])

/ * M e t h

s t o [49] = s t o [ 4 9 ] * ( 1 . 0 - s t o [ 2 5 ] ) ;ano gens Sh =>Sm/ * /

s t o [ 5 0 ] = s t o [ 5 0 ] * ( 1 . 0 - s t o [ 2 5 ] ) ;s t o [5 0 ] = s t o [ 5 0 ] + s t o [ 5 1 ] * s t o [ 25 ] ;

h+Sco2 =>X h , ( s y n t h e s i s ) * /

/ * M e t h

/ * n o t e : S

/*C02 consumed as C-sour c e * /

r e t u r n ;

}

/ * ---------------------- FIND_NU: C a l c u l a t e s nu ( s t o i c h i o m e t r i c c o e f f i c i e n t s ) -------

*/

v o i d F i n d _ n u ( d o u b l e * * n u , d o u b l e * s t o ) / * nu c ha ng ed t o GLOBAL VARIABLE

* /{

i n t i f j ; / * i i s c omponent , j i s p ro c es s */

f o r ( j = l ; j < = P ; j + + ) / * n o te : n u [ i ] [ j ] * /f o r ( i = l ; i < =Y V A R ; i + + ) n u [ i ] [ j ] = 0 .0 ;

/ ^ i n i t i a l i z e d */ n u [ 1 ] [ 1 ] = s t o [ 7 ] ;

c S u b s t r a t e

* /n u [ l ] [2 ] = - 1 . 0 ;f o r ( i = 1 3 ; i < = 1 8 ; i ++ )

/ * S c , C o m p l e x O r g a n i

1 9 8




n u [ l ] [ i ] = s t o [ 3 ] ; / * I mm ed i at e C e l l L y s is P ro d u c t s

* /nu

nunununu

l a t e I n e r t s

* /}n u [ 2 ] [ 1 ]

[ 2] [ i ] ^

[ 3 ] [ i ][ 4 ] [ i ][ 1 5 ] [ i ][ 1 6 ] [ i ]

s t o [ 8 ] ;

s t o [ 4 ] ;

s t o [ 5 ] ;s t o [ 6 ] ;

= s t o [ 1 ] ;= s t o [ 2 ] ;

a b l e S u b s t r a t e

* /nu 2] 2] = s t o [ 12 ] ;nu 2] 3] = s t o [ 1 5 ] ;

nu 2] 4] = - 1 . 0 / s t o [ 1 7 ] ;PH2 * /

nu 2] 5] = - 1 . 0 / s t o [ 1 8 ] ;nu 2] 6] = - 1 . 0 / s t o [ 1 9 ] ;

nu 3] 1] = s t o [ 9 ] ;

l e S o l u b l eS u b s t r a t e /

nu 3] 2] = s t o [ 1 3 ] ;nu 3] 3] = - 1 . 0 ;nu 4] 1] = s t o [ 1 0 ] ;

n e r t s * /

nu 4] 2] = s t o [ 1 4 ] ;nu 4] 3] = s t o [ 1 6 ] ;

nu 5] 4] = s t o [2 6 ] / s t o [17]

* /nu 5] 7] = s t o [ 3 5 ] / s t o [20]nu 5] 8] = s t o [ 3 8 ] / s t o [21]

nu 5] 10] = s t o [ 4 4 ] / s t o [23]nu 5] 11] = - 1 . 0 / s t o [ 24 ] ;nu 6] 7] = - 1 . 0 / s t o [ 20 ] ;

* /nu 6] 9] = s t o [ 4 1 ] / s t o [22]

nu 7] 5] = s t o [ 2 9 ] / s t o [18]

* / nu 7] 8] = - 1 . 0 / s t o [ 21] ;

r t i o n e d by PH2

* /nu 7] 9] = - 1 . 0 / s t o [ 2 2 ] ;nu 8] 6] = s t o [ 3 2 ] / s t o [19]

/ * X i , P a r t i c u

/ * S f , R e a d i l y Fe rm en t

/ * p r o p o r t i o n e d by

/ * Ss, S l o w ly D eg rad a b

/ * S i , S o l u b l e I

199




* /

* /

n u [ 8 ] [ 10] = - 1 . 0 / s t o [ 23 ] ;n u [ 9 ] [1 1 ] = s t o [ 4 7 ] / s t o [ 24 ] ; / * Sm, Metha ne

n u [ 9 ] [ 1 2 ] = s t o [ 4 9 ] / s t o [ 2 5 ] ;

n u [ 9 ] [ 2 0 ] = - 1 . 0 ; / *g as t r a n s f e r * /

PH2

by PH2

nu [10] [4]

nu [ 10] [5]

* /n u [ 1 0 ] [ 6 ]nu [10] [7 ]nu [ 10] [8]

* /n u [ 1 0 ] [ 9 ]

= s t o [ 2 7 ] / s t o [17]

= s t o [ 3 0 ] / s t o [ 18]

= s t o [ 3 3 ] / s t o [19]= s t o [ 3 6 ] / s t o [20]= s t o [ 3 9 ] / s t o [21]

/ * Sh, Hyd rogen *

/ * r h o p r o p o r t i o n b y

/ * r h o p r o p o r t i o n

/ * S l + Sh - > Sp * /

* /

/ * Sn, S o l u b l e N

= - s t o [ 4 2 ] / s t o [22]n u [ 1 0 ] [10 ] = s t o [ 4 5 ] / s t o [ 2 3 ] ;n u [ 1 0 ] [ 1 2 ] = - 1 . 0 / s t o [ 2 5 ] ; / * S co 2+ Sh -> Sm

n u [ 1 0] [ 2 1 ] = —1 . 0 ;n u [ 1 1 ] [ 1 ] = s t o [ l l ] ;

i t r o g e n COD

* /f o r ( i = 4 ; i < = 1 2 ; i + + ) n u [ l l ] [ i ]

/ *p r o p o r t i o n e d b y PH2 * /

n u [ 1 2 ] [4 ] = s t o [ 2 8 ] / s t o [17]

= - s t o [ 1 1 ] ;

x i d e * /n u [ 1 2 ] [ 5 ]

PH2 * / nu [ 12] [6]nu [ 12] [7]nu [ 12] [8]

PH2 * /n u [ 1 2 ] [ 9 ]

= s t o [ 3 1 ] / s t o [18]

= s t o [ 3 4 ] / s t o [19]= s t o [ 3 7 ] / s t o [20]= s t o [ 4 0 ] / s t o [21]

/ * Sco2 , Ca rbon D io



= s t o [ 4 3 ] / s t o [22 ] ;n u [ 1 2 ] [ 1 0 ] = s t o [ 4 6 ] / s t o [ 2 3 ] ;

n u [ 1 2 ] [11] = s t o [ 4 8 ] / s t o [ 2 4 ] ;n u [ 1 2 ] [12] = - s t o [ 5 0 ] / s t o [25]

// * g a s t r a n s f e r * /

n u [ 1 2 ] [ 1 9 ] = - 1 . 0 ;

/ * n u [ 1 3 ] [ j ] = 0 .0 ;r e n c e

N a + n e t , g C a C 0 3 / m 3 )

n u [ 1 4 ] [ j ] = 0 . 0;y ( v a r i a b l e ,

g C a C 0 3 / m 3 ) * /

/ * Sco2+Sh-> Sm *

SNa, A l k a l i n i t y ( r e f e

S a l k , A l k a l i n i t

2 00




/ * P a r t i c u la t e C ompo ne nt s ( gC0D/ m3) * /n u [ 1 5 ] [1] = ■- 1 . 0 ;

* /n u [ 1 7 ] [4] = 1 . 0 ;a * /

n u [ 1 7 ] [5] = 1 . 0 ;n u [ 1 7 ] [6] = 1 . 0 ;

n u [ 1 7 ] [13] = - 1 . 0 ; 0 0 \ —1 3 C

[7] = 1 . 0 ;

n u [ 1 8 ] [14] = - 1 . 0 ;n u [ 1 9 ] [8] = 1 . 0 ;

n u [ 1 9 ] [9] = 1 . 0 ;

n u [ 1 9 ] [15] = - 1 . 0 ;n u [ 2 0 ] [10] = 1 . 0 ;

n u [ 2 0 ] [16] = - 1 . 0 ;n u [ 2 1 ] [11] = 1 . 0 ;

n u [ 2 1 ] [17] = - 1 . 0 ;n u [22] [12] = 1 . 0 ;

* /n u [ 2 2 ] [18] = - 1 . 0 ;r e t u r n ;

- End

1 . 0 ; / * r h o p r o p o r t i o n e d b y PH2

* /

* /

* /

* /

* /

s * /

nogens

} / * ---------------------------------------End o f F i nd _ nu f u n c t i o n * /

/ * p r o ce s s s t a ge t i m e s * /v o i d t i m f l ( d o u b l e * pr op )

{i f ( p r o p [ 12 ]= =0 .0 ) p r o p [1 2 ] = p r o p [ 2 ] / p r o p [ 1 3 ] ; / *

f i l l t im e i ft f = 0 . 0 * /

e ls e p r o p [ 1 3 ] = p r o p [ 2 ] / p r o p [ 1 2 ] ; / * o r f i l l r a t e if t f i ss p e c i f i e d * /

i f ( p r o p [ 14 ]= =0 .0 ) p r o p [ 1 4 ] = p r o p [ 1 5 ] ; / * s e t m a x. re ac t t i me i f

t r = 0 ( S r s t o p ) * /i f ( p r o p [ 18 ]= =0 .0 ) p r o p [ 1 8 ] = p ro p [2]/ ( p r o p [ 2 0 ] * p r o p

[ 1 9 ] ) ; / * s e t t l et im e i f t s = 0 * /

e l s e p r o p [ 1 9 ] = p r o p [ 2 ] / ( p r o p [ 2 0 ] * p r o p [ 1 8 ] ) ; / * o r se t t l i n g

201




v e l o c i t y i f t s i s s p e c i f i e d * /

i f ( p r o p [ 2 1 ] = =0 .0 ) p r o p [ 2 1 ] = p r o p [ 2 ] / p r o p [ 2 2 ] ; / * ca l e . d r aw t i m ei f t d = 0 * /

e l s e p r o p [ 2 2 ] = p r o p [ 2 ] / p r o p [ 2 1 ] ; / * o r draw r a t e i

f t d i ss p e c i f i e d * // * i f ( t w == 0 . 0) t w = v w/ qw ;

c a l c , w a s t et i m e i f t w = 0. 0

e l s e qw = v w / t w ; o r w a s t e r a t e i f t w i ss p e c i f i e d * // * i f ( p r o p [ 23 ]= =0 .0 ) p ro p [ 2 3 ] = p r o p [ 4 ] / p r o p [ 2 4 ] ;

e l s e p r op [ 2 4 ] = p r o p [ 4 ] / p r o p [ 2 3 ] ; * /p r o p [ 4 ] = c e i l ( p r o p [ 4 ] / ( p r o p [ 1 2 ] * p r o p [ 1 0 ] ) ) * ( p r o p [1

2 ] * p r o p [ 10 ]) ;

p r o p [ 2 3 ] = f l o o r ( p r o p [ 2 3 ] / p r o p [ 1 0 ] + 0 . 5 ) * p r o p [ 1 0 ] ;p r o p [ 2 4 ] = p r o p [ 4 ] / p r o p [ 2 3 ] ;p r o p [ 2 3 ] = 0 ;p r o p [ 6 ] = p r o p [ 1 2 ] + p r o p [ 1 4 ] + p r o p [ 18] + p r o p [ 2 1 ] ;

/ * i f ( r t i m e == 0 .0 ) r t im e = t t o t * n c y c l e ; / / d e t e rm i ne ru nt im e i fr t i m e = 0 . 0e ls e n c yc le = r t i m e / t t o t ; / / o r # c y c l e s i f r t i me i s s p e c i f i e d

* /p r o p [8] = p r o p [ 6 ] * p r o p [ 9 ] ;r e t u r n ;

}/ * --------------------- C a l c u l a t e I n f l u e n t A l k a l i n i t y & pH b as ed onI n p u t s a n dVFAs — * /

v o i d p y wc ( do u bl e * y t , d o u b l e * g p , do u b l e * k i n )

{d ou bl e t c h , t a , t b ;i n t i ;

A c i d a c [ 4 ] ;

y t [ 9 ] = g p [ 2 ] * k i n [ 5 9 ] * k i n [ 7 2 ] ;y t [ 1 0 ] = g p [ 1 ] * k i n [ 6 0 ] * k i n [ 7 3 ] ;

/^assume CTC03 = CNanet* // * y t [ 1 2 ] = y t [ 1 3 ] / k i n [ 6 4 ] * k i n [ 6 3 ] ; * /

/ * t h i s i s more r e a l i s t i c * /

202




y t [ 1 2 ] = gp [ 3 ] * k i n [ 6 3 ] * k i n [ 7 1 ] * ( l e - 4 + k i n [ 7 0 ] ) / l e - 4 ;a c [ 0 ] . c o n c = y t [ 5 ] / k i n [ 5 5 ] ;a c [ 0 ] . d i s s = k i n [ 6 6 ] ;a c [ 1 ] . c o n c = y t [ 6 ] / k i n [ 56 ] ;a c [ l ] . d i s s = k i n [ 67 ] ;

a c [ 2 ] . c o n c = y t [ 7 ] / k i n [ 5 7 ] ;a c [ 2 ] . d i s s = k i n [ 6 8 ] ;a c [ 3 ] . c o n c = y t [ 8 ] / k i n [ 5 8 ] ;a c [ 3 ] . d i s s = k i n [ 6 9 ] ;

tch=PH (y t [13 ] / k i n [64 ] , k in [65 ] , y t [ 12 ] / k i n [ 63 ] , k i n [7 0 ] , a c , 4) ;/* p r i n t f ( "WPh=%e\n" , - l o g l O (t c h / 1 0 0 0 . 0 ) ) ; * /

t a = 0 ;

f o r ( i =0 ; i < = 3 ; i + + )


n c;t a = t a + t b ;

}y t [1 4 ] = (y t [ 1 3 ] / k i n [ 6 4 ] - t a ) * k i n [ 6 4 ] ;

}

v o i d p y r e ( d o u b l e * y t , d o u b l e * g p , d o u b le * k i n )

{d ou bl e t c h , t a , t b ;i n t i ;

A c i d a c [ 4 ] ;

y t [ 9 ] = g p [ 2 ] * k i n [ 5 9 ] * k i n [ 7 2 ] ;y t [ 1 0 ]= g p [ 1 ] * k i n [ 6 0 ] * k i n [ 7 3 ] ;

/^assume CTC03 = CNanet* /

/ * y t [ 1 2 ] = y [ 1 3 ] / k i n [ 6 4 ] * k i n [ 6 3 ] ; * /y t [ 1 2 ] = gp [ 3 ] * k i n [ 6 3 ] * k i n [ 7 1 ] * ( l e - 4 + k i n [ 7 0 ] ) / l e - 4 ;a c [ 0 ] . c o n c = y t [ 5 ] / k i n [ 5 5 ] ;a c [ 0 ] . d i s s = k i n [ 6 6 ] ;a c [ 1 ] . c o n c = y t [ 6 ] / k i n [ 56 ] ;a c [ 1 ] . d i s s = k i n [ 6 7 ] ;a c [ 2 ] . c o n c = y t [ 7 ] / k i n [ 5 7 ] ;

a c [ 2 ] . d i s s = k i n [ 6 8 ] ;a c [ 3 ] . c o n c = y t [ 8 ] / k i n [ 5 8 ] ;a c [ 3 ] . d i s s = k i n [ 6 9 ] ;

t c h = P H ( y t [ 1 3 ] / k i n [ 6 4 ] , k i n [ 6 5 ] , y t [ 1 2 ] / k i n [ 6 3 ] , k i n [7 0 ] , a c , 4 ) ;

203




/ * p r i n t f ( " I n i t P h = % f \ n " , - l o g l O ( tc h /1 0 0 0 . 0 ) ) ; * /t a = 0 ;f o r ( i = 0 ; i < = 3 ; i+ + )


n c ;t a = t a + t b ;

}y t [ 1 4 ] = ( y t [ 1 3 ] / k i n [ 6 4 ] - t a ) * k i n [ 6 4 ] ;

}

204




File 3: ode.c

# i n c l u d e " a n s b r . h "v o id o d e in t (d o u b l e y s t a r t [ ] , d o u b l e x l , d o u b l e x 2 , \

d o u b l e e p s , d o u b l e h i , d o u b l e h m i n , \

d ou bl e * * n u ,d o u b l e * k i n , d o u b l e * s t o , d o u b l e * y i n f ,\

d o ub le q f , d o u b l e q d , d o u b l e n f s x , d o u b l e n f r , d o u b l ev r t 1 , \

d o u b le g PH 2 , d o u b le g PC 0 2 , d o u b le g PC H 4 , d o u b le m ix ,

\

A

v o id ( * d e r i v s ) ( d o u b l e , d o u b l e [ ] , d o u b l e [ ] , d o u b l e 1

d o u b l e [ ] , d o u b l e [ ] , d o u b l e [ ] , \d o u b l e , d o u b l e , d o u b l e , d o u b l e , d o u b l e , \d o u b l e , d o u b l e , d o u b l e , d o u b l e ) )

i n t n s tp , i ;d o u b l e x , h n e x t , h d i d , h ;d o u b l e * y s c a l , * y , * d y d x ;

y s c a l = ( d o u b l e * ) m a l l o c ( ( Y V A R + 1 ) * s i z e o f ( d o u b l e ) ) ;

y = ( d o u b l e * ) m a l l o c ( (YVAR+1) * s i z e o f ( d o u b l e ) ) ;d y d x = ( d ou b l e * ) m a l l o c ( (YVAR+1) * s i z e o f ( d o u b l e ) ) ;x = x l ;h = S IG N ( h i , x 2 - x l ) ;

f o r ( i = l ; i < = Y V A R ; i + + )y [ i ] = y s t a r t [ i ] ; / * s e t s o d e in t

y [ i ] asi m p o r t ed y s t a r t [ i ] * /

f o r ( n s t p = l ; n s t p < = M A X S T P ; n s t p + + )

{( * d e r i v s ) ( x , y , d y d x , n u , k i n , s t o , y i n f , \

q f , q d , n f s x , n f r , v r t 1 , gPH2, gPC02, gPCH4, m i x ) ;

f o r ( i = l ; i <= YV AR ;i ++ )

y s c a l [ i ] = f a b s ( y [ i ] ) + f a b s ( d y d x [ i ] * h ) + T I N Y o d e ; / * T I N Y o d e = l e -3 0 , s e e * . h * /

i f ( ( x+ h - x2 ) * ( x + h - x l ) >0 . 0) h =x 2 - x ;s t i f b s (y , d yd x , YV AR ,& x , h , ep s , y s c a l , & h d i d , &

h n e x t , \

205




n u , k i n , s t o , y i n f , q f , q d , n f s x , n f r , v r t l , gPH2, gPC02, gPCH4, m ix ,d e r i v s ) ;

/ /( * r k q s ) ( y , d y d x , & x , h , e p s , y s c a l , & h d i d , &h n e x t , n u , p r o p , k i n , \

/ / s t o , p C H , v r t 2 , y i n f , t c , v r t 1 , t , d e r i v s ) ; / * ca l l s r kq sf u n c . * /

i f ( ( x - x 2 ) * ( x 2 - x l ) >=0 .0 ) / * i f x2 n

o t r e a c h e d ,c o n t i n u e l o o p i n g * /

{f o r ( i = l ; i < = Y V A R ; i + + )

y s t a r t [ i ] = y [ i ] ;

/ * r e t u r n s n e wy s t a r t [ i ] f r o m c a lc . y [ i ] * /

f r e e ( d y d x ) ; / * (as

ma in y [ i ] )

* /f r e e ( y ) ;f r e e ( y s c a l ) ;r e t u r n ; / * f u

n c t i o n m u s te nd t h i s wa y! * /

}i f ( f a b s ( h n e x t ) <= hmi n)

n r e r r o r ( " S te p s i z e t o o s m a l l i n o

d e i n t " ) ;h = h n e x t ;

}n r e r r o r ( " T o o many st ep s i n r o u t i n e o d e i n t " ) ;

} / * -------------------------------- End o f O d e i n t f u n c t i o n * /

v o id n r e r r o r ( c h a r e r r o r _ t e x t [ ] )

{f p r i n t f ( s t d e r r , " \ n N u m e r ic a l Rec i pe s r u n - t i m e e r r o

r . . . \ n " ) ;f p r i n t f ( s t d e r r , " % s \ n " , e r r o r _ t e x t ) ;f p r i n t f (s t d e r r n o w e x i t i n g t o s y s t e m . . . \ n " ) ;

e x i t ( 1 ) ;

} / * -------------------------------------End o f n r e r r o r f u n c t i o n * /

206




File 4: stiff.c

# i n c l u d e " a n s b r . h "s t a t i c d ou bl e * * d , * x ;

s t a t i c d o u bl e s q ar g ;d o u b l e m i n a r g l , m i n a r g 2 , m a x a r g l , m a x a r g 2 ;

v o id s t i f b s ( d o u b l e y [ ] , d o u b l e d y d x [ ] , i n t n v , d o u b l e *x x , d ou b l eh t r y , d o u b l e e p s , \

d ou bl e y s c a l [ ] , d ou bl e * h d i d , d o u b le * h n e x t , \d ou bl e * * n u ,d o ub l e k i n [ ] , d ou bl e s t o [ ] , d ou b l e y i n

f [ ] , \d o ub l e q f , d o u b l e q d , do u b le n f s x , d o u b l e n f r , d o u b l e

v r t l , \d o u b le g PH 2 , d o u b le g PC 0 2 , d o u b le g PC H 4 , d o u b le m ix ,


, \d o u b l e [ ] , d o u b l e [ ] , d o u b l e [ ] , \

d o u b l e , d o u b l e , d o u b l e , d o u b l e , d o u b l e , \d o u b l e , d o u b l e , d o u b l e , d o u b l e ) )

{/ / v o i d j a c ob n ( do u b le x , d o u b l e y [ ] , d o u b l e d f d x [ ] , d o ub le * * d f d y , i n tn) ;v o id s i m p r (d o u b le [ ] , d o u b l e ( ] , d o u b l e [ ] , d o u b l e * * , i n t , d ou b l

e, \ d o u b l e , i n t , d o u b l e [ ] , \d o ub l e * * , d o u b l e [ ] , d o u b l e [ ] , d o u b l e [ ] , \d o u b l e , d o u b l e , d o u b l e , d o u b l e , d o u b l e , \d o u b l e , d o u b l e , d o u b l e , d o u b l e , \v o i d ( * d e r i v s ) ( d o u b l e , d o u b l e [ ] , d o u b l e [ ] , d o u b l e * *


v o i d p z e x t r ( i n t i e s t , d o u b l e x e s t , d o u b l e y e s t [ ] , d ou b l ey z [ ] , d o u b l e d y [ ] , i n t nv) ;

i n t i , i q , k , k k , k m ;s t a t i c i n t f i r s t = l , k ma x , k o p t , n v o l d = - l ;s t a t i c d ou bl e e p s o l d = - l . 0 , x new;

207




e s t j

b l e )

b l e ) )

1 ]

d ou b le e p s l , e r r m a x , f a c t , h , r e d , s c a l e , w o r k , w r k m i n , x

d o ub le * d f d x , * * d f d y , * e r r , * y e r r , * y s a v , * y s e q ;s t a t i c d o u b l e a [ I M A X X + l ] ;s t a t i c d o u b l e a l f [ K M A X X+ 1 ] [KMAXX+1] ;

s t a t i c i n t n s e q [ I MAXX+ 1] = {0 , 2 , 6, 1 0, 14 , 2 2, 34 , 5 0, 70

i n t r e d u c t , e x i t f l a g = 0 ;d = ( d o u b l e * * ) m a l l o c ( ( n v + 1 ) * s i z e o f ( d o u b l e * ) ) ;

f o r ( i = l ; i < = n v ; i + + )d [ i ] = ( d o u b l e * ) m a l l o c ( ( KMAXX+1) * s i z e o f ( d o u

d f d x = ( d o u b l e * ) m a l l o c ( ( n v + 1 ) * s i z e o f ( d o u b l e ) ) ;

d f d y = ( d o u b l e * * )m a l l o c ( ( n v +1 ) * s i z e o f ( d o u b l e * ) ) ;f o r ( i = l ; i < = n v ; i + + )

d f d y [ i ] = ( d ou b le * ) m a l l o c ( ( n v+ 1 ) * s i z e o f ( d o u

e r r = ( d o u b l e * ) m a l l o c ( (KMAXX+1) * s i z e o f ( d o u b l e ) ) ;

x = ( d o u b l e * ) m a l l o c ( ( K M A X X + 1 )* s i ze o f (d o u b l e ) );

y e r r = ( d o u b le * ) m a l l o c ( ( n v + 1 ) * s i z e o f ( d o u b l e ) ) ;y s a v = (d o u b l e * )m a l l o c ( ( nv+ 1) * s i z e o f ( d o u b l e ) ) ;y s e q =( d ou b le * )m a l l o c ( ( n v + 1 ) * s i z e o f ( d o u b l e ) ) ;

i f ( e p s ! =e ps o ld | | n v != n v o ld )

{

* h n e x t = x n e w = - l . 0e2 9;e p s l = S A F E l * e p s ;a [ 1 ] = n s e q [ 1 ] + 1 ;f o r ( k = l ; k<=KMAXX;k++) a [ k + 1 ] = a [ k ] + n s e q [ k+

f o r ( i q = 2 ; i q< =K MA XX ;i q+ +)

{f o r ( k = l ; k < i q; k ++ )

a l f [ k ] [ iq ] = p o w ( e p s l , ( ( a [ k +1 ] - a [ i q + 1 ] ) / ( ( a [ i q + 1 ] - a [ 1 ]+ 1 . 0)* (2 * k+1) ) ) ) ;

}

e p s o l d = e p s ;n v o l d = n v ;a [ 1 ] + =nv ;f o r ( k = l ; k<=KMAXX;k++) a [ k + 1 ] = a [ k ] + n s e q [ k +

208




l ] ;

f o r ( k o p t = 2 ; k op t< KM AX X; ko pt ++ )i f ( a [ k o p t + 1 ] > a [ k o p t ] * a l f [ k o p t - 1 ] [

k o p t ] ) b r e a k ;

kma x= ko p t ;}h = h t r y ;f o r ( i = l ; i < = n v ; i + + ) y s a v [ i ] = y [ i ] ;

j a c o b n ( * x x , y , d f d x , d f d y , n u , k i n , s t o , q f , q d , n f s x , n f r ,v r t 1 , m i x ) ;/ / j a c o b n ( * x x , y , d f d x , d f d y , n v ) ;

i f ( * x x ! = x n e w | I h ! = ( * h n e x t ))

{f i r s t = l ;kop t=kmax;

}r e d u c t = 0 ;

f o r ( ; ; )

{f o r ( k = l ; k <=kmax; k++)

{x n e w = ( * x x ) + h ;i f ( x n e w = = ( * x x ) )

{

p r i n t f ( " x n e w = %f ,h=%e \ n

" , x n e w , h ) ;

e x i t ( 1 ) ; n r e r r o r ( " s te p s i z e u n d e r f

l o w i ns t i f b s " ) ;

}

s i m p r ( y s a v , d y d x , d f d x , d f d y , n v , * x x , h , n s e q [ k ] , yseq , \

n u , k i n , s t o , y i n f , q f , q d , n f s x , n f r , v r t 1 , g P H 2 , g P C 0 2 , g P C H 4 , m i x ,d e r i v s ) ;

x e s t =S Q R ( h / ns e q [ k ] ) ;

p z e x t r ( k , x e s t , y s e q , y , y e r r , n v ) ;i f ( k !=1)

{e r rmax=TINY;f o r ( i = l ; i <= nv ; i + + )

e r r m a x = F M A X ( e r r m a x , f a b s ( y e r r [ i ] / y s c a l [ i ] ) ) ;

209




1.0/(2 * km+1))

e r r m a x / = e p s ;k m = k - l ;

e r r [ k m ] = p o w ( e r r m a x / S A F E l ,

}

i f ( k ! = l && ( k > = k o p t - l | | f i r s t )){

i f ( e r r m a x < l . 0 )

{e x i t f l a g = l ;b r e a k ;

}i f ( k = = k m a x | | k = = k o p t + l )

{r ed =S AF E2 / e r r [ km]

b r e a k ;

}e l s e

i f ( k = = k o p t & & a l f [ k o p t -1 ] [ k o p t ]< e r r [ k m ]

{

}e l s e

i f ( k = = km a x & & a l f [ k m ] [ k m a x - 1 ] < e r r [k m ] )

{

r e d = l . 0 / e r r [ k m ];b r e a k ;

r e d = a l f [ k m ] [ km ax -1 ] * S A F E 2 /e r r [ km ] ;

[km] )

1 ] / e r r [ km]

b r e a k ;

}e ls e i f ( a l f [ k m ] [ k o p t ] < e r r

{

r e d = a l f [ k m ] [ k o p t -

b r e a k ;

}}

i f ( e x i t f l a g ) b r e a k ;red= FMIN( red,R EDMIN)red=FMAX(red,REDMAX)h * = r e d ;r e d u c t = l ;

210




}

*xx=xnew;* h d i d = h ;f i r s t = 0 ;w r k m i n = l . 0 e 3 5 ;f o r ( k k = l ; k k< =k m; kk+ +)

{f a c t = F M A X ( e r r [ k k ] , SCALMX);w o r k = f a c t * a [ k k + 1 ] ;i f ( wo rk < wr km i n)

{s c a l e = f a c t ;w rkm in = w o rk ;k o p t = k k + l ;

}}* h n e x t = h / s c a l e ;

i f ( ko pt >= k& & k o p t ! = km ax && ! r e d u c t )

{f a c t = F M A X ( s c a l e / a l f [ k o p t - 1 ] [ k o p t ] , SCALMX)

i f ( a [ k o p t + 1 ] * f a c t < = w r k m i n )

{* h n e x t = h / f a c t ;k o p t + + ;

}}f r e e ( y s e q ) ;f r e e ( y s a v ) ;

f r e e ( y e r r ) ;f r e e ( x ) ;f r e e ( e r r ) ;f r e e ( d f d x ) ;

f o r ( i = l ; i < = n v ; i + + )

{f r e e ( d f d y [ i ] ) ;f r e e ( d [ i ] ) ;

}f r e e ( d f d y ) ;f r e e ( d ) ;

v o i d p z e x t r ( i n t i e s t , d o u b l e x e s t , d o u b l e y e s t [ ] , d ou bl e yz [] , d ou bl ed y [ ] , i n t nv)

{i n t k l , j ;

211




d o u b l e q , f 2 , f l , d e l t a , * c ;c = ( d o u b l e * ) m a l l o c ( ( n v + 1 ) * s i z e o f ( d o u b l e ) ) ;

x [ i e s t ] = x e s t ;f o r ( j = l ; j < = n v ; j + + ) d y [ j ] = y z [ j ] = y e s t [ j ] ;i f ( i e s t = = l )

f o r ( j = l ; j < = n v ; j + + ) d [ j ] [ 1 ] = y e s t [ j ] ;e l s e

{f o r ( j = l ; j < = n v ; j + + ) c [ j ] = y e s t [ j ] ;f o r ( k l = l ; k l < i e s t ; k l + + )

{

d e l t a = 1 . 0 / ( x [ i e s t - k l ] - x e s t )f l = x e s t * d e l t a ;f 2 = x [ i e s t - k l ] * d e l t a ;

f o r ( j = l ; j < = n v ; j + + )

{

q = d [ j ] [ k l ] ;d [ j ] [ k l ] = d y [ j ] ;d e l t a = c [ j ] - q ;

d y [ j ] = f l * d e l t a ;c [ j ] = f 2 * d e l t a ;y z [ j ] + =d y [ j ] ;

i f ( y z [ j ] < = 0 ) y z [ j ] = T I N Y ;

}}f o r ( j = 1 ; j < = n v ; j ++) d [ j ] [ i e s t ] = d y [ j ]

}f r e e ( c ) ;

}

212




File 5: sim.c

# i n c l u d e " a n s b r . h "

v o i d s i m p r ( d o u bl e y [ ] , d ou bl e d y d x [ ] , d ou bl e d f d x [ ] , d ou b l e* * d f d y , i n t n ,d o u b l e x s , \

d ou bl e h t o t , i n t n s te p ,d o u b l e y o u t [ ] , \d ou b l e * * n u ,d o ub l e k i n [ ] , d ou bl e s t o [ ] , d ou bl e y i n

f [ ] , \d ou b l e q f , d o u b l e q d ,d o u b l e n f s x , d o u b l e n f r , d o u b l e

v r t 1, \d o u b l e g P H 2 , d o u b l e g P C 0 2 , d o u b l e g P C H 4 , d o u b l e m i x ,

\v o i d ( * d e r i v s ) ( d o u b l e , d o u b l e [ ] , d o u b l e [ ] , d o ub l e * *

,\ d o u b l e [ ] , d o u b l e [ ] , d o u b l e [ ] , \d o u b l e , d o u b l e , d o u b l e , d o u b l e , d o u b l e , \d o u b l e , d o u b l e , d o u b l e , d o u b l e ) )

{v o i d l u b k s b (d o u b l e * * a , i n t n, i n t * i n d x , d ou b l e

b [ ] ) ;v o i d l ud cm p (d o ub le * * a , i n t n, i n t * i n d x , d o u b l e * d

) ;i n t i , j , n n , * in d x ;d ou bl e d , h , x , * * a , * d e l , * y t e m p ;i n d x = ( i n t * ) m a l l o c ( ( n + 1 ) * s i z e o f ( i n t ) ) ;

a = ( d o u b l e * * ) m a l l o c ( ( n + 1) * s i z e o f ( d o u b l e * ) ) ;f o r ( i = l ; i < = n ; i + + )a [ i ] = ( d o u b l e * ) m a l l o c ( ( n + l ) * s i z e o f ( d o u b l e )

) ;d e l = ( d o u b l e * ) m a l l o c ( ( n + 1 ) * s i z e o f ( d o u b l e ) ) ;y t e m p = ( d o u b le * ) m a l l o c ( ( n + 1 ) * s i z e o f ( d o u b l e ) ) ;h = h t o t / n s t e p ;

f o r ( i = l ; i < = n ; i + + )

{f o r ( j = l ; j < =n ; j + + ) a [ i ] [ j ] = - h * d f d y [ i ] [ j ] ;a [ i ] [ i ] = a [ i ] [ i ] + 1 . 0 ;

}

l u d c m p ( a , n , i n d x , & d ) ;

f o r ( i = l ; i < = n ; i + + )y o u t [ i ] = h * ( d y d x [ i ] + h * d f d x [ i ] ) ;

213




l u b k s b ( a , n , i n d x , y o u t ) ;f o r ( i = l ; i < = n ; i + + )

y t e m p [ i ] = y [ i ] + ( d e l [ i ] = y o u t [ i ] ) ;x= xs+ h ;( * d e r iv s ) ( x , y t e m p , y o u t , n u , k i n , s t o , y i n f , \

q f , q d , n f s x , n f r , v r t 1 , gPH2, gPC02, gPCH4, m i x ) ;/ / ( * d e r i v s ) ( x , y t e m p , y o u t ) ;

f o r ( n n = 2 ; n n < = n s t e p ; n n + + )

{

f o r ( i = l ; i < = n ; i + + )y o u t [ i ] = h * y o u t [ i ] - d e l [ i ] ;

l u b k s b ( a , n , i n d x , y o u t ) ;f o r ( i = l ; i < = n ; i + + )

y t e mp [ i ] += ( d e l [ i ] + =2 . 0 * y o u t [ i ] ) ;x+=h;

( * d e r i v s ) ( x , y t e m p , y o u t , n u , k i n , s t o , y i n f , \q f , q d , n f s x , n f r , v r t 1 , g P H 2 , g P C 0 2 , g P C H 4 , m i x ) ;

/ / ( * d e r i v s ) ( x , y t e m p , y o u t ) ;

}f o r ( i = l ; i < = n ; i + + )

y o u t [ i ] = h * y o u t [ i ] - d e l [ i ] ;l u b k s b ( a , n , i n d x , y o u t ) ;f o r ( i = l ; i < = n ; i + + )

{y o u t [ i ] + = y t e m p [ i ] ;i f ( y o u t [ i ] < = 0 ) y o u t [ i ] = T I N Y ;

}f r e e ( y t e m p ) ;

f r e e ( d e l ) ;f o r ( i = l ; i < = n ; i + + )f r e e ( a [ i ] ) ;

f r e e ( a ) ;f r e e ( i n d x ) ;

v o i d lu dc mp (d ou b l e * * a , i n t n , i n t * i n d x , d o u b l e *d)

{i n t i , i m a x , j , k;d o u b l e b i g , d u m , s u m , t e m p ;

d o u b l e * v v ;

v v = ( d o u b l e * ) m a l l o c ( ( n + 1 ) * s i z e o f ( d o u b l e ) ) ;* d = l . 0 ;f o r ( i = l ; i < = n ; i + + )

{b i g = 0 . 0 ;f o r ( j = l ; j < = n ; j ++)

214




{

i f ( ( t e m p = f a b s ( a [ i ] [ j ] ) ) > b i g ) b ig =

temp;

}

i f ( b i g = = 0 . 0 ) n r e r r o r ( " S in g u l a r m a t r i x i n

r o u t i n el u d c m p " ) ;

v v [ i ] = 1 . 0 / b i g ;

}

f o r ( j = l ; j < = n ; j + + )

{

f o r ( i = l ; i < j ; i + + )

{

s u m = a [ i ] [ j ] ;f o r ( k = l ; k < i ; k + + ) s u m - = a [ i ] [ k ] * a [ k

] [ j ] ; a [ i ] [ j ] =sum;

}

b i g = 0 . 0 ;f o r ( i = j ; i < = n ; i + + )

{

s um = a[ i ] [ j ] ;f o r ( k = l ; k < j ; k++)

s u m- =a [ i ] [ k ] * a [ k ] [ j ] ;a [ i ] [ j ] =sum;i f ( ( d u m = v v [ i ] * f a b s ( s u m ) ) >=b ig )

{

b ig=dum;i m a x = i ;

}}i f ( j ! = imax)

{f o r ( k = l ; k < = n ; k + + )

{d u m = a [ i m a x ] [ k ] ;

a [ i m a x ] [ k ] = a [ j ] [ k ] ;a [ j ] [ k ]=dum;

}

* d = - ( * d ) ;v v [ im ax ] = v v [ j ] ;

}

i n d x [ j ] = imax ;i f ( a [ j ] [ j ] — 0 .0 ) a [ j ] [ j ] =T INY;i f ( j ! = n )

215




}

v o i d

{

b [ j ]

d u m = l . 0 / ( a [ j ] [ j ] ) ;f o r ( i = j + 1 ; i< = n ; i + + ) a [ i ] [ j ] * = d u m

}

}

f r e e ( v v ) ;

l ub k sb ( do u b l e * * a , i n t n , i n t * i n d x , d o u b l e b [ ] )

i n t i , i i = 0 , i p , j ;doub le sum;f o r ( i = l ; i < = n ; i + + )

{i p = i n d x [ i ] ;

s u m = b [ i p ] ;b [ i p ] = b [ i ] ;

i f ( i i ) f o r ( j = i i ; j < = i —1 ; j+ + ) s u m - = a [ i ] [ j ]

e l s e i f ( su m ) i i = i ;b [ i ] = s u m ;

}f o r ( i = n ; i > = l ; i - - )

{sum=b[ i ] ;f o r ( j = i + l ; j < = n ; j ++) s u m -= a [ i ] [ j ] * b [ j ] ;b [ i ] = s u m / a [ i ] [ i ] ;

}

216




a c [ 0 ] . c o n c= y [ 5 ] / k i n [ 55 ] ;a c [ 0 ] . d i s s = k i n [ 6 6 ] ;a c [ 1 ] . c o n c = y [ 6 ] / k i n [ 5 6 ] ;

a c [ 1 ] . d i s s = k i n [ 6 7 ] ;a c [ 2 ] . c o n c= y [ 7 ] / k i n [ 57 ] ;

a c [ 2 ] . d i s s = k i n [ 6 8 ] ;a c [ 3 ] . c o n c= y [ 8 ] / k i n [ 58 ] ;a c [ 3 ] . d i s s = k i n [ 6 9 ] ;c h= PH (y [ 1 3 ] / k i n [ 64 ] , k i n [ 6 5 ] , y [ 1 2 ] / k i n [ 6 3 ] , k i n [70]

f 3.C / 4 ) /hh = y [ 1 0 ] / k i n [ 6 0 ] / k i n [ 7 3 ] ;

R H O ( r ho , y , k i n , s t o , c h , h h , g P H 2 , g PC02, gPCH4, v r t 1,m ix ) ;

f o r ( i = l ; i <= YV AR ;i ++ )

{ r [ i ] =0 .0 ;r a t e = 0 . 0 ;f o r ( j = l ; j < = P ; j + + ){ ra te

/ * n f r i sr e a c t i o n e f f i c i e n c y * /

r [ i ]

}}f r e e ( r h o ) ;

f r e e ( a c ) ;r e t u r n ;

} / * ---------------------------------------End o f F i n d _ r f u n c t i o n * /v o i d RHO( doubl e * r h o , d o u b l e y [ ] , d o u b l e k i n [ ] , d o ub l e s t o [ ]

,\d o u b le C H , d o u b le PH 2 , d o u b le g PH 2 , d o u b le g PC 0 2 , d o u

b l eg PC H4 ,d ou bl e v r t l , d o u b l e m ix )

{

i n t j ;

d o u b l en r H 2 X f 1 , n r H 2 X f a , n r H 2 X f b , n r H 2 X l a , n r H 2 X l p , n i H 2 X f , n i H 2 X l , n i H2Xp, \

n i H 2 X b , n i p H X f , n i p H X a , n i p H X h , t x , t a , t b ;

n r H 2 X f 1 = k i n [ 7 8 ] * PH2/ ( k i n [ 4 8 ] +PH2) ;

n r H2X fa = ( k i n [ 4 8 ] + ( 1 - k i n [ 7 8 ] ) * PH2) / ( k i n [ 4 8 ] +PH2)

218

= n u [ i ] [ j ] * r h o [ j ] * n f r ;

= r [ i ] + r a t e ;




* ( k i n [49] + ( 1 - k i n [ 7 9 ] ) * PH 2) / ( k i n [ 4 9 ] +PH2) ;n rH 2 Xf b = ( k i n [ 4 8 ] + ( 1 - k i n [ 7 8 ] ) * PH2) / ( k i n [ 4 8 ] +PH2)

*

k i n [7 9 ] * P H 2 / ( k i n [ 4 9 ] +PH2) ;

n rH 2X l a = ( k i n [ 5 0 ] + ( 1 - k i n [ 8 0 ] ) *PH2) / ( k i n [ 5 0 ] +PH2)f

n rH 2X l p = k i n [ 8 0 ] * P H 2 / ( k i n [ 5 0 ] +PH2) ;

n iH 2 X f = k i n [ 8 1 ] * k i n [ 4 4 ] / ( k i n [ 4 4 ] +PH2) ;

n iH 2 X l = k i n [ 8 2 ] * k i n [4 6 ] / ( k i n [ 4 6 ] +PH2 ) ;

t x = P H 2 / k i n [ 4 5 ] ;

i f ( t x C C O N T I N )

{n iH2 Xp = l - p o w ( t x , 3 ) ;

}e l s e

{

t a = ( l -CONTIN*CONTIN*CONTIN) * ( l -CONTIN*CONTIN*CONTIN) /CONT

I N / C ON TI N / 3 ;t b =4 * C ON TI N / 3 - l / 3 / C ON TI N / C ON TI N ;n i H 2 X p = t a / ( t x - t b ) ;

}t x = P H 2 / k i n [ 4 7 ] ;i f ( tx CC ON TIN )

{n i H2Xb = l - p o w ( t x , 2 ) ;

}e l s e

{t a = ( l - C O N T I N * C O N T I N ) * ( l - CO NT I N* CO NT IN ) / CO

N T I N / 2 ;t b = 3 * C O N T I N / 2 - l / C ON TI N/ 2;n i H 2 X b = t a / ( t x - t b ) ;

}

n ip H Xf = ( k i n [ 3 5 ] / ( k i n [35 ] + C H) ) * ( C H / ( k i n [ 37 ] +C H ) )

* ( ( k i n [35 ] + k i n [ 3 6 ] ) / k i n [ 3 5 ] ) *( ( k i n [37] +

k i n [ 3 6 ] ) / k i n [ 3 6 ] ) ;n ip HXa = ( k i n [ 3 8 ] / ( k i n [38] + CH) ) * ( C H / ( k i n [4 0] +

2 1 9




CH) )* ( ( k i n [38] +

k i n [39] ) / k i n [ 3 8 ] ) * ( ( k i n [ 40] + k i n [ 3 9 ] ) / k i n [ 39 ] ) ;n ipHXh = ( k i n [ 4 1 ] / ( k i n [41] + C H) ) * ( C H / ( k i n [ 43 ] +

CH) )

* ( ( k i n [41] +k i n [ 4 2 ] ) / k i n [ 4 1 ] ) * ( ( k in [ 4 3 ] + k i n [ 4 2 ] ) / k i n [ 42 ] ) ;

r h o [ 1] = k i n [1] * ( y [ 1 5 ] / y [ 1 7 ] ) / ( k i n [ 2 ] +y [ 1 5 ] / y [17] ) *

y [ 1 7 ] ;r h o [ 2 ] = k i n [3] * y [ 1 ] / ( k i n [ 4 ] + y [ 1 ] ) * y [ 1 7 ] ;

r h o [ 3 ] = k i n [ 5 ] * y [ 3 ] / ( k i n [ 6 ] + y [ 3 ] ) * y [ 1 7 ] ;

r h o [ 4 ] = n r H 2 X f a * k i n [ 7 ] * s t o [ 1 7 ] * y [ 2 ] / ( k i n [ 8 ] + y [ 2 ] ) *

y [ 17 ] ;r h o [ 4 ] = r h o [ 4 ] * n i p H X f * n i H 2 X f ; / * x f ph2 i n h i b i t i

o n a d d e d * /r h o [ 5 ] = n r H 2 X f l * k i n [ 1 0 ] * s t o [ 1 8 ] * y [ 2 ] / ( k i n [ 1 1 ] + y [ 2]

) * y [ 1 7 ] ;r h o [ 5 ] = r h o [ 5 ] * n i p H X f * n i H 2 X f ; / * x f ph2 i n h i b i t i

o n a d d e d * /r h o [ 6 ] = n r H 2 X fb * k i n [ 1 3 ] * s t o [ 1 9 ] * y [ 2 ] / ( k i n [ 1 4 ] + y [2]

) * y [ 1 7 ] ;r h o [ 6 ] = r h o [ 6 ] * n ip H X f * n iH 2 X f ; / * x f ph2 i n h i b i t i o

n a d d e d * /r h o [ 7 ] = k i n [ 1 6 ] * s t o [ 2 0 ] * y [ 6 ] / ( k i n [ 1 7 ] + y [ 6 ] ) * y [ 18] ;

r h o [ 7 ] = r h o [ 7 ] * n i H 2 X p * n i p H X f ;r h o [ 8 ] = n r H 2 X l a * k in [ 1 9 ] * s t o [ 2 1 ] * y [ 7 ] / ( k i n [ 2 0 ]+ y [ 7]

) * y [ 1 9 ] ;r h o [ 8 ] = r h o [ 8 ] * n i H 2 X l * n i p H X f ;r h o [ 9 ] = n r H 2 X l p * k in [ 2 2 ] * s t o [ 2 2 ] * y [ 7 ] / ( k i n [ 2 3 ] + y [7]

) * y [ 19] ;r h o [ 9 ] = r h o [ 9 ] * n i H 2 X l * n i p H X f ;r h o [ 1 0 ]= k i n [ 2 5 ] * s t o [ 2 3 ] * y [ 8 ] / ( k i n [2 6 ] + y [ 8 ] ) * y [20]

t r h o [ 1 0 ] = r h o [ 1 0 ] * n i H 2 X b * n i p H X f ;r h o [ 1 1 ] = k i n [ 2 8 ] * s t o [ 2 4 ] * y [ 5 ] / ( k i n [ 2 9 ] + y [ 5 ] ) * y [ 2 1 ]

f

r h o [ 1 1 ] = r h o [ 1 1 ] * n i p H X a ;

r h o [ 1 2 ] = k i n [ 3 1 ] * s t o [ 2 5 ] * y [ 1 0 ] / ( k i n [ 3 2 ] + y [ 1 0 ] ) * y [ 2

2 ];

r h o [ 1 2 ] = r h o [ 1 2 ] * n i p H X h ;r h o [ 1 3 ] = ( n r H 2 X f a * k i n [9 ] + n r H 2 X f l * k i n [12 ] +n r H 2 X f b * k i n [ 1 5 ] ) * y [17] ;

r h o [ 1 4 ] = k i n [ 1 8 ] * y [ 1 8 ] ;r h o [ 1 5 ] = ( n r H 2 X l a * k i n [21] + n r H 2 X l p * k i n [ 2 4 ] ) * y [19 ]

220




r h o [ 1 6 ] = k i n [ 2 7 ] * y [ 20 ] ;

r h o [ 1 7 ] = k i n [ 3 0 ] * y [ 2 1 ] ;r h o [ 1 8 ] = k i n [ 3 3 ]* y [ 2 2 ] ;

/ * n e w y a l e * /

r ho [ 1 9 ] = m i x * k i n [ 3 4 ] * ( CH /( CH +k in [7 0 ] ) * y [ 1 2 ] / k i n[63] -g P C 0 2 * k i n [ 7 1 ] ) * k i n [ 6 3 ] ;

r h o [ 2 0 ] = m i x * k i n [ 7 7 ] * ( y [ 9 ] / k i n [59] - g PC H4 *k i n[72 ] ) * k i n [ 5 9 ] ;

r h o [ 2 1 ] = m i x * k i n [7 6 ] * ( y [ 1 0 ] / k i n [ 6 0 ] - g P H 2 * k in [7 3 ] ) *

k i n [60] ;f o r ( j = l ; j < = P ; j + + )

r h o [ j ] = r h o [ j ] / 2 4 .0 ;r e t u r n ;

} / * ---------------------------- End o f F i n d _ rh o f u n c t i o n * /

/ * e n d o f z h l s t d f . c f i l e * /

d o u b l e P H ( d o u b l e C N a n e t , d o u b l e K w , d o u b l e C TC 03 , \d o u b l e KaH2C03, A c i d * a c , i n t num)

{i n t i , m , k ;d o u b l e e r r m a x l = 0 . 0 , e r r m a x 2 = 0 . 0 , e p sg = l e - 3 , T C H ,

T C a l k ;

d o u b l e c y , t e m p a ;d o u b l e C a l k o l d , C H o l d , t e m p b , t e m p c ;i f ( c o n p h > 3 & & c o n p h < 1 0 )

{

r e t u r n p ow ( 1 0 . 0 , 3 - c o n p h ) ;

}T C H = l e - 4 ;

C a l k o l d = C N a n e t ;f o r ( i = l ; i < = 1 0 ; i + + ) / * l o o p w i l l s to p a f t e r max.

10 i t e r a t i o n so r b e f o r e * /

{t e m p a =0 ;f o r ( k =0 ; k <= n u m - l ; k + + )

{ cy = a c [ k ] . d i s s / (T C H + a c [ k ] . d i s s ) *

a c [ k ] . c one ;/ * d i s s o c . c o n c . ( mo l/ m3 ) o f VFAs * /

t empa=tempa+cy ;

}

221




T C a l k = C N a n e t - t e m p a ;

e r r m a x 2 = 0 . 0 ;

C H ol d = TCH; / * s t a r t g uess pH =7.0

* /

f o r ( m= l; m<= 10 ;m+ +) / * l o o p w i l l s t o p a f t er m a x . 1 0

i t e r a t i o n s o r b e fo r e * /

{

t e m p a =0 ;f o r ( k =0; k <= n u m - 1 ; k ++ )

{

c y = T C H / ( T C H + a c [ k ] . d i s s )

* a c [ k ] . c one;/ * d i s s o c . c o n c . ( mo l /m 3) o f VFAs * /

t em pa =t em pa +a c[ k ] . d i s s * c y

r

}

/ * s o lv e q u a d r a t i c e q u a t i o n f o r C

H * /tempb = TCalk ;

tempc = -(Kw + KaH2C03*TCH/(KaH2C

03+TCH)*CTC03

+ t e m p a ) ;TC H = - t e m p b / 2 . 0 + s q r t ( p o w ( t e m p b

,2 .0 ) -

4 . 0 * t e m p c ) / 2 . 0 ;e r r m a x 2 = f a b s ( T C H / C H o l d - 1 . 0 ) ;

i f ( ( e r rm a x 2 /e p s g ) <= 1 .0) b re a k ;

/ * e p s g = l e - 3n o r m a l l y s a m e a s r q k s * /

CHold = TCH;} / * e n d pH c a l c , l o o p * /e r rm a x l = f a b s ( T C a l k / C a l k o l d - 1 . 0 ) ;i f ( ( e r r m a x l / e p s g ) < = 1 .0) b re a k ; / *e p s g = l e

-3 norm.paramete r same as rqks * /

C a l k o l d = T C a l k ;} / * e n d A i k . c a l c , l o o p * /

r e t u r n T C H ;}

222




File 7: jac.c (user-supplied routine jacobn)

# i n c l u d e " a n s b r . h "v o i d ja c o b n ( d o u b l e x , d o u b l e y [ ] , d o ub l e d f d x [ ] , d o ub l e * *

d f d y , \d ou b l e * * n u ,d o u b l e k i n [ ] , d ou bl e s t o [ ] , \

d o ub l e q f , d o u b l e q d ,d o u b l e n f s x , d o u b l e n f r , d o u b l ev r t 1 , d ou bl e

mix)

{

i n t i ;D RA TE DY (d fd y, y, nu , k i n , s t o , v r t 1 , n f r , m i x ) ;

f o r ( i = l ; i < = 1 4 ; i + + )

{d f d x [ i ] = 0 ;d f d y [ i ] [ i ] = d f d y [ i ] [ i ] - q f / ( v r t l t ( q

f - q d ) * ( x - 0 ) ) ;

}

f o r ( i = 1 5 ; i < = Y V A R ; i + + )

{d f d x [ i ] = 0 ;d f d y [ i ] [ i ] =

d f d y [ i ] [ i ] + ( q d * n f s x - q f ) / ( v r t l + ( q f - q d ) * ( x - 0 ) ) ;

}r e t u r n ;

v o i d DRATEDY( doubl e * * d f d y , d o u b l e y [ ] , d o u b l e * * n u , \d ou b le k i n [ ] , d ou b le s t o [ ] , d ou b l e v r t l , d o u b l e n f r ,

d o u b l e m i x )

{i n t i , j , k ;d o u b l e * * d r o d y ;d o u b l e c h , h h ;

A c i d * a c ;a c = ( A c i d * ) m a l l o c ( 4 * s i z e o f ( A c i d ) ) ;

a c [ 0 ] . c on c= y [ 5 ] / k i n [ 5 5 ] ;

a c [ 0 ] . d i s s = k i n [ 6 6 ] ;a c [ 1 ] . c on c= y [ 6 ] / k i n [ 5 6 ] ;a c [ l ] . d i s s = k i n [ 6 7 ] ;a c [ 2 ] . c on c= y [ 7 ] / k i n [ 5 7 ] ;a c [ 2 ] . d i s s = k i n [ 6 8 ] ;a c [ 3 ] . c on c= y [ 8 ] / k i n [ 58 ] ;

223




ac[3].diss=kin[69];ch=PH(y [13]/kin[64],kin[65],y[12]/kin[63],kin[70]

,ac,4) ;hh = y [ 1 0 ] / k i n [ 6 0 ] / k i n [ 7 3 ] ;

d r o d y = ( d o u b l e * * ) m a l l o c ( ( P + l ) * s i z e o f ( d o u b l e * ) ) ;f o r ( i = l ; i < = P ; i + + )

d r o d y [ i ] = ( d o u b l e * ) c a l l o c ( (YVAR+1) , s i z e o f (d o u b l e ) ) ;

DRHODY( dr od y, y, k i n , s t o , c h, h h , v r t l , m i x ) ;f o r ( i = l ; i < = Y V A R ; i + + )

{

f o r ( j = l ; j < = Y V A R ; j + + )

{

d f d y [ i ] [ j ] = 0 ;f o r ( k = l ; k < = P ; k + + )

{

d f d y [ i ] [ j ] = d f d y [ i ] [ j ] + n u [ i ] [ k ] * d r o d y [ k ] [ j ] * n f r ;

}}

}f r e e ( a c ) ;f o r ( i = l ; i < = P ; i + + ) f r e e ( d r o d y [ i ] ) ;f r e e ( d r o d y ) ;r e t u r n ;

}

v o i d DRHODY(double * * d r h o d y , d o u b l e y [ ] , d o u b l e k i n [ ] , d o u b l

e s t o [ ] , \d o u b le C H ,d o ub le P H 2 ,d ou b le v r t l , d o u b l e m ix )

{

i n t i = l , j = l ;d o u b l e

n r H 2 X f 1 , n r H 2 X f a , n r H 2 X f b , n r H 2 X l a , n r H 2 X l p , n i H 2 X f , n i H 2 X l , n i H

2Xp, \n i H 2 X b , n i p H X f , n i p H X a , n i p H X h ;

d o u b l ed r f l , d r f a , d r f b , d r l a , d r i p , d i f , d i l , d i p , d i b , t e m p , t e m p i , t e m p 2

, t x , t a , t b ;

n r H 2 X f l = k i n [ 7 8 ] * P H 2 / ( k i n [ 4 8 ] + PH2) ;

d r f l = k i n [ 7 8 ] * k i n [ 4 8 ] / ( k i n [4 8 ] +PH2) / ( k i n [4 8 ] +PH2) / k i n [ 6 0 ] /

k i n [ 7 3 ] ;

224




nrH2Xfa = (kin[48]+(1-kin[78])*PH2)/(kin[48]+PH2)*

( k i n [4 9 ] + ( l - k i n [ 7 9 ] ) * P H 2 ) / ( k i n [ 4 9 ] +PH2) ;t e m p = ( k i n [ 4 8 ] + PH2)* ( k i n [ 4 8 ] +PH2) * ( k i n [ 4 9 ] +PH2) * (k

i n [4 9 ] + PH 2) ;t e m p l = ( 1 - k i n [7 8 ] ) * ( k i n [ 4 9 ] + ( l - k i n [ 7 9 ] ) *PH2) +

( 1 - k i n [7 9 ] ) * ( k i n [ 4 8 ] + ( 1 - k i n [ 7 8 ] ) * P H2 );t e m p l = t e m p l * t e m p ;

t e m p 2 = ( k i n [ 4 9 ] + k i n [ 4 8 ] + 2 * P H 2 ) *( k i n [ 4 8 ] + ( l - k i n [ 7 8 ] ) * P H 2 ) * ( k i n [ 4 9 ] + ( 1 - k i n [ 7 9 ] ) *P H 2 ) ;

d r f a = ( t em p l - te m p 2) / t e m p / k i n [ 6 0 ] / k i n [73] ;

n rH 2X fb = ( k i n [ 4 8 ] + ( 1 - k i n [ 7 8 ] ) * PH2) / ( k i n [ 4 8 ] +PH2)*

k i n [7 9 ] * P H 2 / ( k i n [ 4 9 ] +PH2) ;

t e m p l = ( 1 - k i n [ 7 8 ] ) * k i n [7 9 ] * P H2 + k in [7 9 ] * ( k i n [ 4 8 ] + ( 1 - k i n [78]) * P H 2 ) ;

t e m p l = t e m p l * t e m p ;t e m p 2 = ( k i n [ 4 9 ] + k i n [ 4 8 ] + 2 * P H 2 ) *

( k i n [ 4 8 ] + ( 1 - k i n [ 7 8 ] ) * P H 2 ) * k i n [7 9 ] * PH2;d r f b = (t em p l- te m p2 ) / t e m p / k i n [ 6 0 ] / k i n [ 7 3 ] ;

n r H2 X la = ( k i n [ 5 0 ] + ( 1 - k i n [ 8 0 ] ) *PH2) / ( k i n [ 5 0 ] +PH2)

d r l a = ( ( 1 - k i n [ 8 0 ] ) * ( k i n [ 5 0 ] + P H 2 ) - ( k i n [ 5 0 ] + ( 1 - k i n [ 8 0 ] ) * P H 2 )

) / ( k i n [ 5 0 ] +PH2) / ( k i n [ 5 0 ] + P H 2 ) / k i n [ 6 0 ] / k i n [7 3 ] ;

n rH 2X lp = k i n [ 8 0 ] * P H 2 / ( k i n [ 5 0 ] + PH2 );

d r l p = k i n [ 8 0 ] * k i n [ 5 0 ] / ( k i n [ 5 0 ] +PH2) / ( k i n [ 5 0 ] +PH2) / k i n [ 6 0 ] /

k i n [ 7 3 ] ;

n i H 2 X f = k i n [ 8 1 ] * k i n [ 4 4 ] / ( k i n [ 4 4 ] +PH2) ;

d i f = - k i n [ 8 1 ] * k i n [ 4 4 ] / ( k i n [ 4 4 ] +PH2) / ( k i n [ 4 4 ] +PH2) / k i n [ 6 0 ] /k i n [ 7 3 ] ;

n iH 2X l = k i n [ 8 2 ] * k i n [ 4 6 ] / ( k i n [ 4 6 ]+PH2 ) ;

d i l = - k i n [ 8 2 ] * k i n [ 4 6 ] / ( k i n [ 4 6 ] +PH2) / ( k i n [ 4 6 ]+PH2) / k i n [ 6 0 ] /

k i n [ 7 3 ] ;

225




t x = P H 2 / k i n [ 4 5 ] ;i f ( tx CCONT IN)

{

n iH2Xp = l - p o w ( t x , 3 ) ;d i p = - 3 * p o w ( t x , 2 ) / k i n [ 4 5 ] / k i n [ 6 0 ] / k i n [7 3 ] ;

}e l s e

{

t a = ( l -CONTIN*CONTIN*CONTIN)* ( l -CONTIN*CONTIN*CONTIN) /CONT

IN/CONTIN/3 ;tb=4 *C ON TI N/ 3- 1 / 3 / CONTIN/CONTIN;n i H 2 X p = t a / ( t x - t b ) ;d i p = - t a / ( t x - t b ) / ( t x - t b ) / k i n [ 4 5 ] / k i n [ 6 0 ] / k

i n [ 7 3 ] ;

}

t x = P H 2 / k i n [ 4 7 ] ;i f ( tx CCONT IN)

{

n i H2Xb = l - p o w ( t x , 2 ) ;

d i b = - 2 * t x / k i n [ 4 7 ] / k i n [ 6 0 ] / k i n [ 7 3 ] ;

}

e l s e

{

t a = ( l -C O NT I N *C O NT I N ) * ( l -CO N TI N *C ON TI N ) / CO

N T I N / 2 ;t b = 3 * C O N T I N / 2 - l / C O N T I N / 2 ;

d i b = - t a / ( t x - t b ) / ( t x - t b ) / k i n [4 5 ] / k i n [ 6 0 ] / ki n [ 73 ] ;

}

n ip HX f = ( k i n [ 3 5 ] / ( k i n [35] + CH) ) * ( CH /( k i n [37 ] +

CH) )* ( ( k i n [35] + k i n [ 3 6 ] ) / k i n [ 3 5 ] ) *

( ( k i n [ 37] +k i n [ 3 6 ] ) / k i n [ 3 6 ] ) ;

n i pHXa = ( k i n [ 3 8 ] / ( k i n [ 38] + C H) ) * ( C H /( k i n [40 ] +

CH) )* ( ( k i n [38] +

k i n [ 3 9 ] ) / k i n [ 3 8 ] ) * ( ( k i n [40] + k i n [ 3 9 ] ) / k i n [ 3 9 ] ) ;n i pHXh = ( k i n [ 4 1 ] / ( k i n [ 41] + C H) ) * ( C H / ( k i n [ 43] +

CH) )* ( ( k i n [41] +

k i n [42] ) / k i n [ 4 1 ] ) * ( ( k i n [ 43] + k i n [ 4 2 ] ) / k i n [ 4 2 ] ) ;

226




/ / d r h o d y [ 1 ] = k i n [ 1 ] * ( y [ 1 5 ] / y [ 1 7 ] ) / ( k i n [ 2 ] + y [ 1 5 ] / y [17

] ) * y [ 1 7 ] ;d r h o d y [ 1 ] [ 1 5 ] = ( k i n [ 1 ] * k i n [ 2 ] ) / p o w (k i n [ 2 ] + y [ 1 5 ] / y [

1 7 ] f 2 ) ;

d r h o d y [1] [ 1 7 ] = k i n [ l ] * y [ 1 5 ] * y [ 1 5 ] / p o w ( y [ 1 7 ] * k i n [2]

+ y [ 1 5 ] , 2 ) ;/ / d r h o d y [ 2 ] = k i n [ 3 ] * y [ 1 ] / ( k i n [ 4 ] + y [ 1 ] ) * y [ 1 7 ] ;

d r h o d y [ 2 ] [ l ] = k i n [ 3 ] * y [ 1 7 ] * k i n [ 4 ] / ( k i n [ 4 ] + y [ 1 ] ) / ( ki n [ 4 ] + y [ 1 ] ) ;

d r h o d y [ 2 ] [ 1 7 ] = k i n [ 3 ] * y [ l ] / ( k i n [ 4 ] + y [ l ] ) ;/ / d r h o d y [ 3 ] = k i n [ 5 ] * y [ 3 ] / ( k i n [ 6 ] + y [ 3 ] ) * y [ 1 7 ] ;

d r h o d y [ 3 ] [ 3 ] = k i n [ 5 ] * y [ 1 7 ] * k i n [ 6 ] / ( k i n [ 6 ] + y [ 3 ] ) / ( ki n [ 6] +y [ 3] ) ;

d r h o d y [ 3 ] [ 1 7 ] = k i n [ 5 ] * y [ 3 ] / ( k i n [ 6 ] + y [ 3 ] ) /

/ /d r h o d y [ 4 ] = n r H 2 X f a * k i n [ 7 ] * s t o [ 1 7 ] * y [ 2 ] / ( k i n [ 8 ] + y [ 2 ] ) * y [17]

* n i p H X f * n i H 2 X f

d r h o d y [ 4 ] [ 2 ] = n r H 2 X f a * k i n [ 7 ] * s t o [ 1 7 ] * y [ 1 7 ] * k i n [ 8 ] / p o w ( k i n [8 ] + y [ 2 ] , 2 ) * n i pH X f * n i H 2 X f ;

d r h o d y [ 4 ] [ 1 0 ] = k i n [ 7 ] * s t o [ 1 7 ] * y [ 2 ] / ( k i n [ 8 ] + y [ 2 ] ) * y [ 1 7 ] * n i pH X f * ( d r f a * n i H 2X f + n r H 2 X f a * d i f ) ;

d r h o d y [ 4 ] [ 1 7 ] = n r H 2 X f a * k i n [ 7 ] * s t o [ 1 7 ] * y [ 2 ] / ( k i n [ 8 ] + y [ 2 ] ) * n

i p H X f * n i H 2 X f * ni H 2 X f ;

/ /d r h o d y [ 5 ] = n r H 2 X f l * n i p H X f * k i n [1 0 ] * s t o [ 1 8 ] * y [ 2 ] / ( k i n [ 1 1 ]+ y [2 ] ) * y [ 1 7 ] *n i H2

X f ;

d r h o d y [ 5 ] [ 2 ] = n r H 2 X f l * n i p H X f * k i n [ 1 0 ] * s t o [ 1 8 ] * y [ 1 7 ] * k i n [ 1 1 ]/ p o w (k i n [ 1 1 ] +y[ 2 ] , 2 ) * n i H 2 X f ;

d r h o d y [ 5] [ 1 0 ] = n i p H X f * k i n [ 1 0 ] * s t o [ 1 8 ] * y [ 2 ] / ( k i n [11] + y [ 2] ) *

y [ 1 7 ] * ( d r f l * n iH 2 X f + n r H 2 X f l * d i l ) ;

d r h o d y [ 5 ] [ 1 7 ] = n r H 2 X f l * n i p H X f * k i n [ 1 0 ] * s t o [ 1 8 ] * y [ 2 ] / (k i n [11

] + y [ 2 ] ) * n i H 2 X f

227




//

d r h o d y [ 6 ] = n r H 2 X f b * n i p H X f * k i n [ 1 3 ] * s t o [ 1 9 ] * y [ 2 ] / ( k i n [ 1 4 ]+ y [2 ] ) * y [ 1 7 ] *n i H 2X f ;

d r h o d y [ 6 ] [ 2 ] = n r H 2 X f b * n i p H X f * k i n [ 1 3 ] * s t o [ 1 9 ] * y [ 1 7 ] * k i n [14]/ p o w (k i n [ 1 4 ] +y

[ 2 ] , 2 ) * n i H 2 X f ;

d r h o d y [ 6 ] [ 1 0 ] = n i p H X f * k i n [ 1 3 ] * s t o [ 1 9 ] * y [ 2 ] / ( k i n [ 1 4 ] + y [ 2 ] ) *y [ 1 7 ] * ( d r f b * n i

H 2 X f + n r H 2 X f b * d i l ) ;

d r h o d y [ 6 ] [ 1 7 ] = n r H 2 X f b * n ip H X f * k in [ 1 3 ] * s t o [ 1 9 ] * y [ 2 ] / ( k i n [14

] + y [ 2 ] ) * n i H 2 X f

r

//

d r h o d y [ 7 ] = k i n [ 1 6 ] * s t o [ 2 0 ] * y [ 6 ] / ( k i n [ 1 7 ] + y [ 6 ] ) * y [ 1 8 ] * n i H 2 X

p * n i p H X f ;

d r h o d y [ 7 ] [ 6 ] = k i n [ 1 6 ] * s t o [ 2 0 ] * y [ 1 8 ] * k i n [ 1 7 ] / p o w (k i n [ 1 7 ] + y [6 ] , 2 ) * n i H 2 X p * ni p H X f ;

d r h o d y [7 ] [ 1 0 ] = k i n [ 1 6 ] * s t o [ 2 0 ] * y [ 6 ] / ( k i n [ 1 7 ] + y [ 6 ] ) * y [18 ] *ni p H X f * d i p ;

d r h o d y [ 7 ] [ 1 8 ] = k i n [ 1 6 ] * s t o [ 2 0 ] * y [ 6 ] / ( k i n [ 1 7 ] + y [ 6 ] ) * n i H 2 X p *

n i p H X f ;/ /d r h o d y [ 8 ] = n r H 2 X l a * n i H 2 X l * n i p H X f * k i n [ 1 9 ] * s t o [ 2 1 ] * y [ 7 ] / ( k i n[ 2 0 ] + y [ 7 ] ) * y [ l

9] ;d r h o d y [ 8 ] [ 7 ] = n r H 2 X l a * n i H 2 X l * n i p H X f * k i n [ 1 9 ] * s t o [ 2 1

] * y [ 1 9 ] * \k i n [ 2 0 ] / p o w ( k i n [ 2 0 ] + y [ 7 ] ,

2 ) ;

d r h o d y [ 8 ] [ 1 0 ] = n i p H X f * k i n [ 1 9 ] * s t o [ 2 1 ] * y [ 7 ] / ( k i n [ 2 0 ] + y [ 7 ] ) *

y [ 1 9 ] * \

( n r H 2 X l a * d i l + n i H 2X l * d r l a ) ;

d r h o d y [8 ] [ 1 9 ] = n rH 2 X l a * n i H 2 X l * n i p H X f * k in [ 1 9 ] * s t o [ 2 1 ] * y [ 7 ] /( k i n [ 2 0 ] + y [ 7 ] )

228




/ /d r h o d y [ 9 ] = n r H 2 X l p * n i H 2 X l * n ip H X f * k in [ 2 2 ] * s t o [22] * y [ 7 ] / ( k in[ 2 3 ] + y[ 7 ] ) * y [ l9] ;

d r h o d y [ 9 ] [ 7 ] = n r H 2 X l p * n i H 2 X l * n ip H X f * k in [ 2 2 ] * s t o [2 2

] * y [ 1 9 ] * \k i n [ 2 3 ] / p o w ( k i n [ 2 3 ] + y [ 7 ] ,

2 ) ;

d r h o d y [ 9 ] [ 1 0 ] = n i p H X f * k i n [ 2 2 ] * s t o [ 2 2 ] * y [ 7 ] / ( k i n [ 2 3 ] + y [ 7 ] ) *y [ 1 9 ] * \

( n r H 2 X l p * d i l + n i H 2X l * d r l p ) ;

d r h o d y [ 9 ] [ 1 9 ] = n r H 2 X l p * n i H 2 X l * n i p H X f * k i n [ 2 2 ] * s t o [ 2 2 ] * y [ 7 ] /( k i n [ 2 3 ] + y [ 7] )

r

/ /

d r h o d y [ 1 0 ] = n i H 2 X b * n i p H X f * k i n [ 2 5 ] * s t o [ 2 3 ] * y [ 8 ] / ( k i n [ 2 6 ] + y [8] ) * y [ 2 0 ] ;

d r h o d y [ 1 0 ] [ 8 ] = n i H 2 X b * n i p H X f * k i n [ 2 5 ] * s t o [ 2 3 ] * k i n [ 2 6 ] * y [20]/ p o w (k i n [ 2 6 ] +y[ 8 ] , 2 ) ;

d r h o d y [ 1 0 ] [ 1 0 ] =

n i p H X f * k i n [ 2 5 ] * s t o [ 2 3 ] * y [ 8 ] / ( k i n [2 6 ] + y [ 8 ] ) * y [ 2 0 ] * d ib ;

d r h o d y [ 1 0 ] [ 2 0 ] = n iH 2 X b * n i p H X f * k i n [ 2 5 ] * s t o [ 2 3 ] * y [ 8 ] / ( k i n [2 6

] + y [ 8 ] ) ;/ / d r h o d y [1 1 ] = n ip H X a * k i n [2 8 ] * s t o [ 2 4 ] * y [ 5 ] / ( k i n [ 2 9 ] +y[ 5 ] ) * y [ 2 1 ] ;

d r h o d y [ 1 1 ] [ 5 ] = n i p H X a * k i n [ 2 8 ] * s t o [ 2 4 ] * k i n [ 2 9 ] * y [ 2 1 ] / p o w ( k in [ 2 9 ] + y [ 5 ] , 2) ;

d r h o d y [ 1 1 ] [ 2 1 ] = n ip H X a * k i n [2 8 ] * s t o [ 2 4 ] * y [ 5 ] / ( k i n [29 ] + y [ 5 ] ) ;

/ / d r h o d y [1 2 ] = n ip H X h * k i n [3 1 ] * s t o [ 2 5 ] * y [ 1 0 ] / ( k i n [ 3 2] +

y [ 1 0 ] ) * y [ 2 2 ] ;

d r h o d y [ 1 2 ] [ 1 0 ] = n ip H X h * k in [ 3 1 ] * s t o [ 2 5 ] * k i n [ 3 2 ] * y [ 2 2 ] / p o w (k

i n [ 3 2 ] + y [ 1 0 ] ,2) ;

d r h o d y [ 1 2 ] [ 2 2 ] = n i p H X h * k i n [ 3 1 ] * s t o [ 2 5 ] * y [ 1 0 ] / ( k i n [3 2 ] + y [ 1 0 ] ) ;/ / d r h o d y [ 1 3 ] = ( n r H 2 X f a * k i n [9] + n r H 2 X f l * k i n [12 ] +n r H 2 X f b * k i n [ 1 5 ] ) * y [ 1 7 ] ;

229




d r h o d y [ 1 3 ] [ 1 0 ] = ( d r f a * k i n [ 9 ] + d r f l * k i n [ 1 2 ] + d r f b * k i n

[ 1 5 ] ) * y [ 1 7 ] ;d r h o d y [ 1 3 ] [ 1 7 ] = n r H 2 X f a * k i n [9] + n r H 2 X f l * k i n [12 ] +

n r H 2 X f b * k i n [ 1 5 ] ;

/ / d r h o d y [ 1 4d r h o d y [ 1 4

/ / d r h o d y [ 1 519] f

d r h o d y [ 1 5d r h o d y [ 1 5

/ / d r h o d y [ 1 6

d r h o d y [ 1 6

/ / d r h o d y [ 1 7d r h o d y [ 1 7

/ / d r h o d y [ 1 8d r h o d y [ 1 8

d r h o d y [ 1 9* / k i n [63] * /

d r h o d y [ 2 0d r h o d y [ 2 1f o r ( j = 1;

] = k i n [ 3 3 ] * y [ 2 2 ] ;

* /*/

{f o r ( i = l ; i <= YV AR ;i ++ )

{d r h o d y [ j ] [ i ] = d r h o d y [ j ] [ i ] / 2 4 .0 ;

}

}

r e t u r n ;

230




File 8: settle.c (batch settling model)

# i n c l u d e < s t d i o . h ># i n c l u d e < s t d l i b . h >

# i n c l u de < s t r i n g . h ># i n c l u d e < m a t h . h ># d e f i n e SWAP(a,b) { t e m p = ( a ) ; ( a ) = ( b ) ; ( b ) = t e m p ; }

#def ine NR_END 1#def ine FREE_ARG char*# d e f i n e NMSP 360 / * n u m b e r o f s t e p s i n d t * /

/ * now u i n m /s C in kg /m3 * /

c o n s t d o u b l e g = 9 . 8 1 ;c o n s t d o u b l e k l = 1 . 6 2 3 e - 3 ; / * m /s e c * /c o n s t d o u b l e n l = 0 . 5 0 8 ;c o n s t d o u b l e k2 = 2 0 0 0 . 0 ; / * 3 . 6 4e 4 P a . h

* /c o n s t d o u b l e n 2 = 0 . 6 ;c o n s t d o u b l e p h o s = 1 0 5 0 . 0 ; / * k g/ m

3 * /c o n s t d o u b l e p h o l = 1 0 0 0 . 0 ;char *emsg ;

i n t c r m at (d o u b l e * a , d ou b l e * b , d o u b le * c ,d o u b l e * r i t e r , u n s

i g n e d i n tn u m l a y e r , d o u b l e d z , \

d o ub l e d t , d o u b l e * v , d o u b l e * co nc )

r i n t c f i e l d ( u n s i g n e d i n t n u m la y e r , d ou b l e * v , d o u b l e * v n , d ou

b l e* c o n c , d o u b l e d t , d o u b l e d z ) ;i n t c o m pe q u( u ns ig n ed i n t n u m l a y e r , d o u b le d z , d o u b l e d t , d o u

b l e* c o nc , do u b le * v ) ;i n t t r i d a g ( d o u b l e * a , d o u b l e * b , d o u b le * c ,d o u b l e * r , d o u b l e

* u , \u n si gn ed i n t n ) ;

i n t s e t t ( d o u b l e * h g t , d o u b l e s d t , d o u b l e d z , do u b l e * v , d o u b l

e *conc){

i n t i , l a y , n um step ;d o u b l e d t = 1 . 0 ;n u m s t e p = f l o o r ( s d t * 3 6 0 0 / d t ) ;

f o r ( i = l ; i <= n u m s t e p ; i+ + )

231




{l a y = f l o o r ( * h g t / d z ) ;i f ( c o m p e q u ( l a y , d z , d t , c o n e , v ) ==0) r e t u r n 0;* h g t = * h g t - v [ l a y ] * d t ;

}

r e t u r n 1 ;}

/ * c r e a te s o l u t i o n m a t r i x a [ ] , b [ ] , c [ ] * /

i n t c r m at (d o u b le * a , d o u b le *b , d o u b le * c , d o u b l e * r i t e r , u n s

i g n e d i n tn u m l a y e r , d o u b l e d z , \d o ub l e d t , d o u b l e * v , d o u b l e * co nc )

{u n si gn e d i n t i ;d o u b l e c t , t p l , t p 2 ;c t = ( c o n e [ 1 ] + c o n c [ 2 ] ) / 2 . 0 ;

i f ( p h o s - c t = = 0 )

{s t r c p y (e m s g , "E r r o r 1 i n c r m a t " ) ;

r e t u r n 0 ;

}t p l = p h o l / g / ( p h o s - p h o l ) / c t * ( p h o s / ( p h o s - c t ) ) * ( p h o s /

( p h o s - c t ) ) ;t p 2 = ( p h o s + p h o l * ( p h o s + c t ) / ( p h o s - c t ) ) / ( p h o s - p h o l ) ;b [ 1] = 1 / ( d t * g ) + 2. 0 * c t / k 2 ,lre xp ( n 2 * c t ) / d z / d z + e x p ( n l * c

t ) / k l ;c [ 1 ] = - c t / k 2 * e x p ( n 2 * c t ) / d z / d z \

- ( ( 2 . 0 + n 2 * c t ) / k 2 * e x p ( n 2 * c t ) * ( co n e [ 2 ] - c o n e [ 1 ] ) / d z + t p 2 * v [ 1]

/ g ) / 2 . 0 / d z ;

r i t e r [ 1 ] = v [ 1 ] / ( d t * g ) + 1 .0 + t p l * v [ 1 ] * v [1] * ( c o ne [ 2 ] - c

o n e [ 1 ] ) / d z ;

f o r ( i = 2 ; i c n u m l a y e r ; i + + )

{c t = ( c o n e [ i ] + c o n c [ i + 1 ] ) / 2 . 0 ;i f ( p h o s - c t = = 0 )

{s t r c p y (e m s g , "E r r o r 1 i n c r m a t " ) ;

r e t u r n 0 ;}

t p l = p h o l / g / ( p h o s - p h o l ) / c t * ( p h o s / ( p h o s - c t ) ) * ( p h o s / (p h o s - c t

) ) ;t p 2 = ( p h o s + p h o l * ( p h o s + c t ) / ( p h o s - c t ) ) / ( phos

232




- p h o l ) ;

a [ i ] = - c t / k 2 * e x p ( n 2 * c t ) / d z / d z

\

+ ( ( 2 . 0 + n 2 * c t ) / k 2 * e x p ( n 2 * c t ) * ( c o n e [ i + 1 ] - c o n e [ i ] ) / dz \

+ t p 2 * v [ i ] / g ) / 2 . 0 / d z ;

b [ i ] = 1 . 0 / ( d t * g ) + 2 . 0 * c t / k 2 * e x p ( n 2 * c t ) / d z / d z + e x p ( n l * c t ) / k l ;c [ i ] = - c t / k 2 * e x p ( n 2 * c t ) / d z / d z

\

- ( ( 2 . 0 + n 2 * c t ) / k 2 * e x p ( n 2 * c t ) * ( c o n e [ i + 1 ] - c o n e [ i ] ) / d z + t p 2 * v [i ] / g ) / 2 . 0 / d z ;

r i t e r [ i ] = v [ i ] / ( d t * g ) + l . 0 + t p l * v [ i ] * v [ i ] * ( c o n e [ i + 1 ] - c o n e [ i ]

) /d z ;

}i = n u m l a y e r ;c t = c o n c [ i ] ;a [ i ] = - 2 . 0 * c t / k 2 * e x p ( n 2 * c t ) / d z / d z ;

b [ i ] = 1 . 0 / g / d t + e x p ( n l * c t ) / k l + 2 . 0 * c t / k 2 * e x p ( n 2 * c t ) /d z / d z ;

r i t e r [ i ] = v [ i ] / ( d t * g ) +1 . 0 ;r e t u r n 1 ;

}

i n t c f i e l d ( u n s i g n e d i n t n u m la y e r , d ou b l e * v , d o u b l e * vn , d ou

b l e* c o n c , d o u b l e d t , d o u b l e d z )

{u n si gn ed i n t i ;d o u b l e * c n e w ;cn ew =( do ub le * ) m a l l o c ( ( n u m l a y e r + 1 ) * s i z e o f( d o u b l e )

) ;c n e w [ 1 ] = c on c [ 1 ] + ( v [ 1 ] + v n [ 1 ] ) * ( co ne [ 1 ] + c o n c[ 2 ] ) / 4 .

0 / d z * d t ;

f o r ( i = 2 ; K n u m l a y e r ; i ++ )

{

c n e w [ i ] = c o n c [ i ] + ( v [ i ] + v n [ i ] ) * ( c o n e [ i ] + c o n c [ i + 1 ] ) / 4 . 0 / d z * d

t \

- ( v [ i —1 ] + v n [ i - 1 ] ) * ( co ne [ i —1 ] + c o n c [ i ] ) / 4 . 0 / d z * d t ;

}

233




i = n u m l a y e r ;c ne w[ i ] = c o n c [ i ] + (v [ i ] + v n [ i ] ) * ( c o ne [ i ] + c o n c [ i + 1 ] ) /

4 . 0 / d z * d t \

- ( v [ i —1 ] + v n [ i —1 ] ) * ( co n e [ i —1 ] + c o n c [ i ] ) / 4 . 0 / d z * d t ;

f o r ( i = l ; i < = n u m l a y e r ; i + + )

{

i f ( c n e w [ i ] < 0 )

{s t r c p y (e m s g , " E r r o r 1 i n c f i e l d " ) ;r e t u r n 0 ;

}

c o n e [ i ] = cn e w [ i ] ;

}

f r e e ( c n e w ) ;r e t u r n 1 ;

}

i n t c o m pe q u( un si g ne d i n t n u m l a y e r , d o u bl e d z , d o u b l e d t , d o u

b l e* c o n c , d o u b l e * v )

{

u ns ig n ed i n t i ;d o u b l e * a , * b , * c , * r i t e r , * u ;

r i t e r = ( d o u b l e * ) m a l l o c ( ( n um la ye r+ 1) * s i z e o f ( d o u b l

e) ) ;a = ( do u b l e * ) m a l l o c ( ( nu ml ay er +1 ) * s i z e o f ( d o u b l e ) ) ;b = (d o ub le *) m a l l o c ( ( n u m l a y e r + 1 ) * s i z e o f ( d o u b l e ) ) ;

c = (d o ub l e * ) m a l l o c ( ( n u m l a y e r + 1 ) * s i z e o f ( d o u b l e ) ) ;u =( do u b l e * ) m a l l o c ( ( n u m l a y e r + 1 ) * s i z e o f ( d o u b l e ) ) ;i f ( c r m a t ( a , b , c , r i t e r , n u m l a y e r , d z , d t , v , c o n e ) ==0) r

e t u r n 0 ;i f ( t r i d a g ( a , b , c , r i t e r , u , n u m l a y e r ) ==0) r e t u r n 0;i f ( c f i e l d ( n u m l a y e r , v , u , c o n e , d t , d z )==0) r e t u r n 0;f o r ( i = l ; i < = n u m l a y e r ; i + + )

{i f ( u [ i ] < 0 )

{s t r c p y ( e m s g , " E r r o r 1 i n c omp eq u" )

r r e t u r n 0 ;

}v [ i ] = u [ i ] ;

}

f r e e ( a ) ;

234




f r e e ( b ) ;f r e e ( c ) ;f r e e ( u ) ;f r e e ( r i t e r ) ;r e t u r n 1 ;

}

i n t t r i d a g ( d o u b l e * a , d o u b l e * b ,d o u b le * c , d o u b le * r , d o u b l e

* u , \u n s i gn e d i n t n)

{u ns ig n ed i n t j ;d o u b l e b e t , * g a m ;gam = ( d o u b l e * )m a l l o c ( ( n+ 1) * s i z e o f ( d o u b l e ) ) ;i f ( b [ 1 ]= = 0 . 0 )

{ s t r c p y ( e m s g , " E r r o r 1 i n t r i d a g " ) ;

r e t u r n 0 ;

}b e t = b [ 1 ] ;u [ 1 ] = r [ 1 ] / b e t ;

f o r ( j = 2 ; j < = n ; j + + )

{gam[ j ] = c [ j - 1 ] / b e t ;b e t= b [ j ] - a [ j ] * ga m [ j ] ;i f ( b e t = = 0 . 0 )

{

s t r c p y ( e m s g , " E r r o r 2 i n t r i d a g " ) ;r e t u r n 0 ;

}

u ( j ] = ( r [ j ] - a [ j ] * u [ j - l ] ) / b e t ;

}f o r ( j = ( n - 1 ) ; j > = 1 ; j — )

u [ j ] - = ga m [ j + 1 ] * u [ j + 1 ] ;f r e e ( g a m ) ;

r e t u r n 1 ;

235




Appendix C Sample Input Files

Sample input files are listed as follows:

File 1: asbr inf.in

File 2: asbr_ini.in

File 3: asbr__prc.in

File 4: asbr_kin.in

File 5: asbr sto.in

236




File 1: asbr inf.in (influent condition)

0 . 0 Sc y i n f [ 1 ]1000 .0 Sf y i n f [ 2]0 . 0 Ss y i n f [ 3]0 . 0 S i y i n f [4]

0 .0 Sa y i n f [5]0 . 0 Sp y i n f [6]0 .0 SI y i n f [7]0 .0 Sb y i n f [ 8]0 . 0 Sm y i n f [ 9]

0 .0 Sh y i n f [ 10]30. 0 Sn y i n f [ 11]0 .0 Sco3 y i n f [ 12]3400 .0 SNaNET y i n f [ 13]0 .0 S a l k y i n f [14]0 .0 Xs y i n f [15]

0 .0 X i y i n f [16]0 .0 X f y i n f [17]0 .0 Xp y i n f [ 18]0 .0 X I y i n f [ 19]0 .0 Xb y i n f [ 20]

0 .0 Xa y i n f [21]0 .0 Xh y i n f [22]

l e - 1 0 PH2 y i n f [ 23]0 .55 PCH4 y i n f [ 24]0 .3 PC02 y i n f [25]0 .027 PH20 y i n f [ 26]

1 .0 PATM y i n f [ 27]

237




File 2: asbr ini.in (initial reactor condition)

0 .0 Sc y [1]0 .0 Sf y [2]0 .0 Ss y [3]0 .0 S i y [4]0 .0 Sa y [5]0 .0 Sp y [ 6 ]0 .0 SI y [7 ]

0 .0 Sb y [ 8 ]0 .0 Sm y [ 9 ]0 .0 Sh y [10]0 .0 Sn y [ i i ]0 .0 Sco3 y [12]3000 .0 SNaNET y [13]0 .0 S a l k y [14]1 00 .0 Xs y [15]12800 .0 X i y [16]

2 3 0 . 0 X f y [17]120. 0 Xp y [18]170. 0 X I y [19]10. 0 Xb y [20]2 0 0 . 0 Xa y [21]350 .0 Xh y [22]0 .00000001 PH2 y [23]

0 .52 PCH4 y [24]

0 .44 PC02 y [25]

0 .027 PH20 y [26]1 .0 PATM y [27]

238




File 3: asbr_prc.in (process values)

12 v t o t p r o p [ l ]6 v f p r o p [ 2]

6 v s p r o p [ 3]0 . 5 v w p r o p [ 4 ]2 . 0 v g a s p r o p [ 5]

5 0. 0 t t o t p r o p [ 6 ]1 . 0 qg p r o p [ 7]0 .0 r t i m e p r o p [ 8]40 n c y c l e p r o p [ 9]0 .1 d t p r o p [ 10]1 .0 d t k e e p p r o p [ 11]1 2. 5 t f p r o p [ 1 2 ]1 .0 q f p r o p [ 1 3 ]20 t r p r o p [ 1 4 ]4 8 . 0 t r m a x p r o p [ 15 ]150 S r s t o p p r o p [ 16 ]

2 1 . 0 S rc omp p r o p [ 1 7]1 t s p r o p [ 1 8 ]6 . 0 vO p r o p [ 1 9]1 . 76 7 1 a r e a p r o p [ 2 0 ]0 .5 t d p r o p [ 2 1 ]2 . 0 q d p r o p [ 2 2 ]

0 . 1 t w p r o p [ 2 3 ]1 . 0 q w p r o p [ 2 4 ]

500000 Xwas te p r op [2 5 ]

7 500 00 X t a r g e t p r o p [ 2 6 ]1 .0 n f r p r o p [ 2 7 ]

1 .0 n f r f p r o p [ 2 8 ]1 .0 n f r r p r o p [ 2 9 ]0 .1 n f r s p r o p [ 3 0 ]0 .1 n f r d p r o p [ 31 ]0 .1 n f r w p r o p [ 3 2 ]0 . 9 n f s x p r o p [ 33]10 m i x p r o p [ 3 4 ]0 pH p r o p [ 3 5 ] 4 - 1 1 / 0 ; f i x / v a r y0 s b r p r o p [ 3 6] 0 / l ; A SB R / U A S B0 s e t t l e p r o p [ 37] 0 / 1 ; f i x / m o d e l

0 n o t c o n t i n u e p r o p [ 38 ] 0 / l ; . i n / . r s t

239




File 4: asbr kin.in (kinetic constants)

0 . 1 0 khxs k i n [1 .0 Kxs k i n [

2 . 0 khsc k i n [100. 0 Ksc k i n [1 . 5 khss k i n [1 000 .0 Kss k i n [

49 .4 k x f a k i n [24. 6 K s f a k i n [0 .02 b x f a k i n [49.4 mu xf 1 k i n [24. 6 K s f 1 k i n [0 . 0 2 b x f 1 k i n [49 .4 mux fb k i n [24. 6 K s f b k i n [

0 .02 b x f b k i n [5 .3 muxpa k i n [60. 0 Kspa k i n [

0 .01 bxpa k i n [34. 6 mux l a k i n [3 6 . 5 K s l a k i n [0 . 0 2 b x l a k i n [34. 6 m u x l p k i n [

3 6 . 5 K s l p k i n [0 .02 b x l p k i n [

5 .3 muxba k i n [13. 0 Ksba k i n [0 .027 bxba k i n [6 .1 muxam k i n [

1 5 0 . 0 Ksam k i n [0 .0 3 6 bxam k i n [24 .7 muxhm k i n [0 .0 12 Kshm k i n [0 .088 bxhm k i n [

1 .0 KlaC02 k i n [1 . 0 e - 2 K i l p H X f k i n [1 . 0 e - 4 K i n p H X f k i n [1 . 0 e - 6 K i h p H X f k i n [

3 . Oe- 4 K i l p H X a k i n [1 . 0 e - 4 KinpHXa k i n [

3 . Oe- 5 KihpHXa k i n [1 . 0 e - 3 K i l p H X h k i n [1 . Oe- 4 KinpHXh k i n [1 . Oe- 5 KihpHXh k i n [

240

1 ]2]

3]4]5]6 ]

7]8 ]9]1011

12

1314

151617

181920

2122

232425262728

293031

323334

353637

383940414243




1000 K i H 2X f k i n [44]4 . Oe-5 KiH2Xp k i n [45]3 . Oe-4 K i H2 X l k i n [4 6]3 . Oe-4 KiH2Xb k i n [47]

5. Oe-5 K r H 2 X f l k i n [48]

I — 1 O( D 1 Kr H 2X f2 k i n [49]1. Oe-5 KrH2Xl k i n [50]1 .416 i v s s k i n [51]1. I l l i t s s b m k i n [52]

1. I l l i t s s x s k i n [53]0. 900 i t s s x i k i n [54]64 .0 i m o l a k i n [55]

112 .0 i m o l p k i n [56]

96.0 i m o l l k i n [57]160. 0 i r r tolb k i n [58]

64 . 0 imo lm k i n [59]

16 .0 i m o l h 2 k i n [60]

61. 0 i m o l h c o 3 k i n [61]

62.0 i m o l h 2 c o 3 k i n [62]

44 .0 i m o l c o 2 k i n [63]

50. 0 i m o l a l k k i n [64]0 . 6 8 1 e - l l Kw k i n [65]1 . 7 53e-2 Kaa k i n [66]1 . 338e-2 Kap k i n [67]

1 . 3 5 7 e - l K a l k in [ 68]

1 . 542e-2 Kab k i n [69]4 . 1 60e-4 KaH2C03 k i n [70]

36 .7647 KhC02 k i n [71]

1 .42 21 KhCH4 k i n [ 72 ]0 .8035 KhH2 k i n [73]

8 . 2 0 6 e - 5 Rgas k i n [ 74 ]

295 Temp k i n [75]23 .5 KlaH2 k in [7 6]

7 .8 KlaCH4 k i n [77]

1 .0 k r H 2 X f 1 k i n [78]

0 .7 k r H 2 X f 2 k i n [79]

0 .45 k r H 2 X l k i n [80]1 .0 k i H 2 X f k i n [81]

1 .0 k i H 2 X l k i n [82]

241




File 5: asbr sto.in (stoichiometric)

0. 63 f b x s s t o [0 .07 f b x i s t o [0 . 10 f b s c s t o [

0 . 01 f b s f s t o [0 . 10 f b s s s t o [

0 . 09 f b s i s t o [0 . 40 f x s c s t o [

0 . 10 f x s f s t o [

0 .50 f x ss s t o [0 . 0 0 f x s i s t o [0 .0875 f x s n s t o [1 .00 fscf s t o [0 .00 fscs s t o [

0 . 00 f s c i s t o [

1 .00 f s s f s t o [0 . 00 f s s i s t o [

0 .07 Y x f a s t o [

0 . 0 6 Y x f l s t o [0 . 0 6 Y x f b s t o [

0 . 0 5 9 Yxpa s t o [0 .064 Y x l a s t o [

0 .064 Y x l p s t o [0 .067 Yxba s t o [0 .058 Yxam s t o [

0 .22 Yxhm s t o [

0 .667 Ya fa s t o [

0 . 333 Y h f a s t o [0 .4 58 Y c o 2 f a s t o [

1 .000 Y l f l s t o [0 . 000 Y h f 1 s t o [0 . 00 0 Y c o 2 f 1 s t o [0 .833 Yb fb s t o [0 .167 Y h f b s t o [

0. 458 Y c o 2 f b s t o [0 .571 Yapa s t o [

0 . 4 2 9 Yhpa s t o [

0. 393 Yco2pa s t o [0. 667 Y a l a s t o [0 . 33 3 Y h l a s t o [0 .458 Yco21a s t o [1. 167 Y p l p s t o [

0 . 16 7 Y h l p s t o [

0 .000 Yco21p s t o [

242

1 ]2 ]

3]

4]5]

6 ]

7]8 ]9]10

11

12

1314

151617

1819

20

21

22

2324

25

2627

28

29303132333435

3637

383940414243




0 . 800 Yaba s t o [ 44 ]

0 .200 Yhba s t o [ 45 ]

0 .000 Yco2ba s t o [46]

1 .000 Ymam s t o [47]

0 6875 Yco2am s t o [48]

mathematical model of anaerobic processes applied to the anaerobic sequencing batch reactor

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