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One-dimensional modeling of secondary clarifiers using a concentration and feed velocity-dependent dispersion coefficient

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One-dimensional modeling of secondary clarifiers using a concentration and feed velocity-dependent dispersion coefficient
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Watts, Randall W
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x, 72 leaves : ill. ; 29 cm.

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Activated sludge ( jstor )
Blankets ( jstor )
Data lines ( jstor )
Dispersions ( jstor )
Dynamic modeling ( jstor )
Flow velocity ( jstor )
Modeling ( jstor )
Parametric models ( jstor )
Solids ( jstor )
Velocity ( jstor )
Dissertations, Academic -- Environmental Engineering Sciences -- UF
Environmental Engineering Sciences thesis, Ph. D
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non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1996.
Bibliography:
Includes bibliographical references (leaves 68-71).
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Typescript.
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Vita.
Statement of Responsibility:
by Randall W. Watts.

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ONE-DIMENSIONAL MODELING OF SECONDARY CLARIFIERS USING A
CONCENTRATION AND FEED VELOCITY-DEPENDENT DISPERSION
COEFFICIENT











By

RANDALL W. WATTS


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1996














ACKNOWLEDGMENTS


Funding for this study was provided by Gainesville Regional Utilities and the U.S.

Geological Survey through USGS matching grant #C91-2237. Additional funding was

provided by the Engineering Research Center for Particle Science and Technology at the

University of Florida, The National Science Foundation (NSF) grant #EEC-94-02989, and

the Industrial Partners of the ERC.

The operational and instrumentation personnel (B. Braun, S. Byce, C. Caldwell, P.

Davis, R. Dare, B. Gandy, J. Jones, B. Rossie, B. Snyder) and management (J. Cheatham,

B. McVay, J. Regan) of the Kanapaha Water Reclamation Facility are thanked for their

cooperation and assistance with the experimental program.

I would like to express my gratitude to the members of my committee: Dr. Paul

Chadik, Dr. Oscar Crisalle, and Dr. Kirk Hatfield. Their instruction, assistance, and

friendship are greatly appreciated. I would also like to thank Dr. Bill Wise for completing

my committee at short notice during Dr. Hatfield's absence. I would especially like to

thank my committee chair, Dr. Ben Koopman, and cochair, Dr. Spyros Svoronos.

Working with them has been a great experience. They have set a fine example of

synergistic collaboration. Their guidance, instruction, and friendship are greatly

appreciated.















TABLE OF CONTENTS


page


ACKNOWLEDGMENTS .......

LIST OF TABLES ..........


LIST OF FIGURES ................................


ABSTRACT ......................


CHAPTERS


1 INTRODUCTION ................... ... .. 1

2 ONE-DIMENSIONAL MODELING OF SECONDARY
CLARIFIERS USING A CONCENTRATION AND FEED
VELOCITY-DEPENDENT DISPERSION COEFFICIENT .. .. 4

Introduction 4
Initial Model Development ........... .......... 9
Application of Clarifier Model to the Pflanz Data ... .. ... 19
Experimental Measurements on a Full-Scale Clarifier .. 22
Application of Clarifier Model to KWRF Data and Further
Model Development ............. .... 26
Conclusions ................... ........ 40

3 CALIBRATION OF A ONE-DIMENSIONAL CLARIFIER
MODEL USING SLUDGE BLANKET HEIGHTS .. .... 42

Introduction ....................... ... 42
Description of Clarifier Model ..... ...... .... ... 43
Materials and Methods ............... ... ... 50
Results and Discussion .......... ... .. .. .... 53
Conclusions ................... ........ .. 64













4 CONCLUSIONS .......... ................ 66

REFERENCES ... ...................... ........ 68

BIOGRAPHICAL SKETCH .......................... 72















LIST OF TABLES


Table page

2-1 Performance of models with concentration-dependent dispersion
functions in comparison to the Takacs et al. (1991) model, as applied to
Pflanz data ........ ........... ..... 20

2-2 Estimated parameters for 50-layer model with concentration-
dependent dispersion functions when fitted to Pflanz data .. .. ...... 21

2-3 Operational variables for KWRF loading tests which achieved
steady blanket levels ..... ................ 30

2-4 Results for models with Dma constant across all cases and for the
Dmx model with Dmax fitted individually for each case ... ... 34

2-5 Parameters resulting from FVDDma and FVDDmax-Cit-P fit across
nine KWRF cases for which steady blanket levels were obtained ...... .. 37

2-6 Results for FVDDax model fit across the nine KWRF cases for
which steady blanket levels were obtained ... ..... .... 38

2-7 Comparison of model predictions to clarifier loading test results ...... .. 39

3-1 Comparison of predicted effluent concentrations, RAS
concentrations, and sludge blanket heights to measured values for
test period C ...... ........... ................. 56

3-2 Comparison of predicted effluent concentrations, RAS
concentrations, and sludge blanket heights to measured values for
test periods A and B .. ........... ......... 59

3-3a Comparison of model predictions of success and failure to test
results for test period A ................. ......... .. 61








3-3b Comparison of model predictions of success and failure to test
results for test period B ............ ..62

3-3c Comparison of model predictions of success and failure to test
results for test period C .... .. ........... 63














LIST OF FIGURES


Figure ge

2-1 Example of clarifier concentration profile obtained using total
limiting flux constraint .................. ............... 6

2-2 Comparison of 10- and 20-layer versions of the Takacs et al. (1991)
model using parameters estimated for the 10-layer version of the model
applied to case 1 of the Pflanz data .......... 10

2-3 Comparison of 10- and 50-layer versions of the Takacs et al. (1991)
model, with optimal parameters estimated for each version applied to case 1
ofthe Pflanz data .. .................. .. 10

2-4 Comparison of concentration profiles obtained by 20-layer versions
of the Takacs et al. (1991) model (eq. (3)) and the model with dispersion
(eq. (8)) to experimental data (case 1 of the Pflanz data). Parameters
reported for the 10-layer model of Takacs et al. (1991) were employed to
generate model fits.. ................. 14

2-5 Clarifier geometry used for initial model development. Clarifier is
divided into n layers .................. .. 17

2-6 Pooled data from batch settling tests after discarding outliers.
Model line represents fit of Vesilind equation with Vo = 182.9 m/d and
b = 0.3055 m3/kg ......... ............. .. ..25

2-7 Schematic diagram of full-scale secondary clarifier at the Kanapaha
Water Reclamation Facility, showing shroud, bottom conical section and
model discretization ........... ............... .. 27

2-8 Results of typical clarifier loading tests: (a) test in which a steady
blanket level was achieved (case 4), (b) test in which blanket continued
rising throughout experimental period (case 13). .... ..... 31










2-9 Comparison of fits achieved using Dmax, FVDDmax, and FVDDmx-
Ccrit-P models. Data are from nine loading tests on a full-scale clarifier
at the Kanapaha Water Reclamation Facility in which a steady sludge
blanket level was achieved and concentration profiles were measured.
(Dashed line = fit ofD,,ax model, dotted line = fit ofFVDD,,x model,
solid line = fit of FVDDmax-Cct-P model) ......... .. 33

2-10 Variation of Dx with feed velocity. Data points were estimated
on a case-by-case basis using the Dmax model. The line represents the fit
of the proposed feed velocity-dependent expression for Dm to the
computed Dmax values. ................. .. .. 35

3-1 Schematic diagram of clarifier at KWRF .. .......... .. .. 46

3-2 Comparison of clarifier model predicted concentration profiles to
measured profiles. Heavy vertical lines represent measured blanket
heights; light vertical lines represent model predicted blanket heights. .. 55

3-3 Comparison of predicted blanket heights to measured values from
test period C. .. ....... .................... ..... 57

3-4 Comparison of predicted blanket heights to measured values from
test periods A and B. ..... .. .. .. ........... ..... 58















Abstract of Dissertation Presented to the Graduate School of the University of Florida in
Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

ONE-DIMENSIONAL MODELING OF SECONDARY CLARIFIERS USING A
CONCENTRATION AND FEED VELOCITY-DEPENDENT DISPERSION
COEFFICIENT

By

RANDALL W. WATTS

May, 1996

Chair: Ben Koopman
Cochair: Spyros A. Svoronos
Major Department: Environmental Engineering Sciences

A one-dimensional model of activated sludge secondary clarifiers with a dispersion

term dependent on concentration and feed velocity was developed. The model provides

predictions of effluent and underflow suspended solids concentrations and sludge blanket

height. Data collected from a full-scale clarifier at the Kanapaha Water Reclamation

Facility in Gainesville, FL, were used to evaluate the model. Better matches to observed

concentration profiles were achieved with the current model than with a gravity-flux-

constraining model. In addition, the model, when calibrated using concentration profile

data from experiments in which the sludge blanket reached steady levels, successfully

predicted the outcomes of the five experiments during the test period which exhibited

continuously rising blankets. These failures to reach steady blanket levels were not

predicted by either limiting total solids flux theory or the gravity-flux-constraining model.








Since concentration profile data are not readily available under normal plant

operating conditions, the ability of the model to be calibrated using sludge blanket height

data was investigated. The clarifier model was coupled with an algorithm for predicting

sludge blanket height. The model was successfully calibrated using blanket heights,

effluent suspended solids and return activated sludge concentrations, and the measured

Vesilind settling equation parameters Vo and b. Model validity was confirmed by

comparing predictions of the calibrated model against separate sets of data. Out of forty

clarifier loading tests for which the system could conclusively be determined as overloaded

(a clarifier failure) or operating acceptably (a successful state), the blanket height-

calibrated model correctly predicted the outcome of thirty-seven. Two experimental

successes were predicted as clarifier failures, whereas one experimental failure was

predicted as a success. In contrast, a gravity-flux-constraining model and limiting total

solids flux theory predicted success for all cases, and thus failed to predict the eight cases

of clarifier overloading.














CHAPTER 1
INTRODUCTION


The process of separating solids from wastewater effluent in the secondary clarifier is

critical to the optimal operation of activated sludge systems. The secondary clarifier

performs two functions in this capacity. It thickens the sludge to a high concentration for

recycle back to the bioreactors, and it clarifies the wastewater effluent reducing effluent

suspended solids and effluent biochemical oxygen demand (BOD) due to effluent solids.

Because clarification is an integral part of the activated sludge system, it is useful to have a

reliable clarifier model that can be incorporated with existing biological process models for

application in design, optimization, and control of activated sludge systems.

The objective of the current research was to develop and validate a clarifier model

capable of predicting sludge blanket heights and effluent and underflow suspended solids

concentrations that could be easily combined with an activated sludge biological process

model. Full-scale clarifier loading tests were conducted at the Kanapaha Water

Reclamation Facility (KWRF) in Gainesville, FL, to collect data for model development,

calibration and validation. The settling characteristics of the KWRF sludge were

determined by conducting batch sludge settling tests in parallel with the loading tests. The

experiments were conducted during three intervals over a six month period.








The developed model employs a dispersive flux term in which the dispersion

coefficient is a function of both solids concentration and influent velocity. The

development of this dispersion function and calibration and testing of the model are the

topics of Chapter 2. Initial model development was based on the model of Hamilton et al.

(1992) which has a dispersive flux term with a constant dispersion coefficient and the

model ofTakics et al. (1991) which applies a constraint on the gravity flux. A simple

implementation of the model with a dispersion coefficient dependent only upon

concentration was calibrated separately to three sets of steady-state concentration profile

data reported in the literature (Pflanz 1969), and the resulting model fits were compared

to those of the Takacs et al. (1991) model. The model was also modified to reflect

structural characteristics of the KWRF clarifiers and calibrated to nine sets of steady-state

concentration profile data from KWRF clarifier loading tests conducted during one of the

experimental periods. In analysis of the model fits, it was found that model performance

could be substantially improved by incorporating a dependence on influent velocity in the

dispersion coefficient function. The calibrated model with the concentration and feed

velocity-dependent dispersion term was used to simulate the six remaining KWRF clarifier

loading tests from the same experimental period (one which reached steady state and five

which did not reach steady state due to clarifier overloading). The model's ability to

predict clarifier failure due to solids overloading was compared to that of both the gravity-

flux-constraining model and limiting total solids flux theory.

The requirement for concentration profile data to calibrate a clarifier model is

problematic since these data are not generally available under normal plant operating








conditions. Therefore the use of sludge blanket heights and effluent and underflow

suspended solids concentrations to calibrate the model was investigated. This is the topic

of Chapter 3. The algorithm for calculation of sludge blanket height from Hamilton et al.

(1992) was modified to yield better agreement with blanket heights observed in the field,

and the model incorporating the modified algorithm was calibrated using sludge blanket

height data from nine KWRF steady-state loading tests conducted during one of the

experimental periods and the measured sludge settling equation parameters. The resulting

model parameters and the experimentally measured sludge settling equation parameters

were applied in simulations of steady-state loading tests from the other two experimental

periods to validate the model. Additional simulations of KWRF loading tests which did

not reach steady state from all three experimental periods were performed to compare the

model's ability to predict clarifier failure due to solids overloading to that of both limiting

total solids flux theory and the gravity-flux constraining model.














CHAPTER 2
ONE-DIMENSIONAL MODELING OF SECONDARY CLARIFIERS USING A
CONCENTRATION AND FEED VELOCITY-DEPENDENT DISPERSION
COEFFICIENT


Introduction


The limiting total solids flux concept is used for the design of sludge thickeners and

the thickening region of activated sludge secondary clarifiers, where the total solids flux is

the sum of the solids flux due to gravity settling and the solids flux due to bulk downward

movement of liquid. This concept originated with Coe and Clevenger (1916), who

suggested that if a layer in a suspension has a lower total solids-handling capacity than the

overlying layer, it will be unable to discharge solids as rapidly as they are received and will

therefore grow in thickness. If a given layer has a higher total solids-handling capacity

than the layer above, its thickness will decrease or remain infinitesimal. The layer with the

lowest total solids-handling capacity therefore limits the throughput of the thickener. If

the thickener is overloaded, this layer (which contains the limiting solids concentration)

will ultimately reach the liquid surface. Yoshioka et al. (1957) and Hasset (1964)

developed graphical procedures for computing the value of the limiting total solids flux

that are accepted for secondary clarifier design (Vesilind, 1968; WPCF, 1985; Metcalfand

Eddy, 1991). A feature common to these approaches is the postulate that the settling

velocity of sludge particles in the hindered settling regime is a function only of the local








suspended solids concentration, as proposed by Kynch (1952) in his modeling of batch

settling. He used the method of characteristics to solve a partial differential equation

(PDE) model.

A family of one-dimensional, dynamic clarifier models was developed based on limiting

total solids flux theory (Bryant, 1972; Tracy and Keinath, 1973; Lessard and Beck, 1993).

These models adjust the thickness of the layer with the limiting solids concentration so as

to satisfy the limiting total solids flux constraint. As a result, they give steady-state

concentration profiles having four distinct values in the clarifier (Fig. 2-1). Above the feed

layer, the solids concentration is very low. The feed layer has an intermediate

concentration that is less than the feed concentration. Below the feed layer, the sludge

blanket has a concentration equal to the limiting concentration, whereas the concentration

of the bottom layer will be higher, as set by mass balance. Petty (1975) solved, using the

method of characteristics, a partial differential equation model for the clarifier and raised

questions as to whether the limiting flux is appropriate in all cases.

A second family of models is based on a modification of the limiting flux constraint

(Stenstrom, 1976; Hill, 1985; Vitasovic, 1986, 1989; Takics et al., 1991). Rather than

constraining all layers above the bottom layer to concentrations less than or equal to the

limiting concentration, they constrain only the gravity flux term. This entails setting the

value of the downward gravity flux from a given layer in the thickening zone to the

minimum of the gravity flux calculated for that layer and the gravity flux calculated for the

layer below. This approach avoids the necessity of calculating the limiting flux and limiting

concentration and gives a more realistic concentration profile in the thickening zone.
























5



4-



6 3


4 2



1 -




0 4000 8000 12000

Concentration, mg/L



Fig. 2-1. Example of clarifier concentration profile obtained using total limiting flux
constraint.








An alternative approach for obtaining a realistic concentration profile is to add a

dispersive flux term in the mass balance for each layer (Anderson and Edwards, 1981; Lev

et al., 1986; Hamilton et al., 1992). Adding a dispersion term converts the model equation

from a hyperbolic to a parabolic PDE which eliminates problems with multiple solutions

encountered using the hyperbolic equation. Anderson and Edwards (1981) included a

dispersion term in their model for peripheral feed clarifiers. It is noteworthy that the

dispersion coefficient was not constant, varying with position. Lev et al. (1986) extended

the analysis of Petty (1975) to include the clarification zone and noted, as had Petty

(1975), that the limiting flux constraint has a limited range of validity and that its

imposition could lead to erroneous results. They included a dispersion term in a dynamic

clarifier model and reported that it yielded correct dynamic behavior. Hamilton et al.

(1992) modeled a pre-denitrification process, employing a constant dispersion coefficient

in the secondary clarifier component, and proposed a method for calculating sludge

blanket height.

Alternative approaches emphasize the interaction between solid and liquid phases at

high solids concentrations. Hartel and Popel (1992) postulated that settling velocity is

affected both by underlying layers in the thickening zone and by overlying layers in the

compression zone, as well as the local suspended solids concentration. They employed a

correction function that reduces the settling velocity applied in the thickening zone based

on location in the clarifier relative to the feed layer and the position of the compression

concentration. They defined the compression concentration position as the point of

transition between hindered settling and compression and gave a procedure for calculating

it. In their model, the gravity flux is the product of the correction function, the calculated








settling velocity based on concentration, and the layer concentration. George and Keinath

(1978) added a liquid phase mass balance describing the change in the upward velocity of

displaced fluid with depth, and included in their model a settling velocity equation that

depended on the local concentration gradient as well as the local concentration. It was

still necessary to impose a limiting flux constraint due to the model's inability to predict

rising blankets under overloaded conditions (Hill, 1985; George, 1976). Others have also

questioned the Kynch proposition that settling velocity depends solely on local solids

concentration (Tiller, 1981; Fitch, 1983; Font, 1988). In another approach, thickening is

viewed in terms of transport of mass and momentum in a non-rigid saturated porous

medium (Kos, 1977; Kos and Adrian, 1974; Landman et al., 1988; Leonhard, 1993; Tiller

and Hsyung, 1993).

The work of Takacs et al. (1991), which employs the gravity flux constraint, is notable

in that it presents an excellent match to the full-scale data set of Pflanz (1969). In the

present study, it is shown that this approach can be reinterpreted as modeling with a

concentration-dependent dispersion coefficient. The two interpretations give differing

results if the number of layers into which the clarifier is divided is changed. It is shown

that for finer discretizations, the dispersion interpretation leads to better fits with

experimental data. The expression for the concentration-dependent dispersion coefficient

is then simplified without decreasing model performance. The final model modification

was to incorporate a dependence of the dispersion coefficient on clarifier feed velocity.

The model was applied to data from experiments we conducted on a full-scale clarifier at

the Kanapaha Water Reclamation Facility in Gainesville, FL.








Initial Model Development


Takacs et al. (1991) modified the Vesilind (1968) equation for settling velocity to

account for the fact that the settling velocity decreases as concentration approaches zero.

They used this equation in a model employing the gravity flux constraint. To test their

model, they used three cases of experimental data presented by Pflanz (1969). Takics et

al. (1991) reduced the two-dimensional data sets to one-dimensional forms with respect to

depth. As each of the reduced data sets involved ten depths, they modeled the clarifier as

consisting often layers. Their model gave good predictions of effluent suspended solids

as well as excellent matches with the solids concentration in the thickening zone. It was

found, however, that when the number of layers in their model is increased to twenty

without changing model parameters, the model performance deteriorates considerably

(Fig. 2-2). Furthermore, at finer discretizations (e.g., 50 layers) model performance is

worse than with a discretization of 10 layers, even with parameters optimally fitted for that

level of discretization (Fig. 2-3). Ideally, model performance should improve with an

increase in the degree of discretization. In the following, the model of Takics et al.

(1991) is analyzed and an alternative approach is developed that achieves this objective.

A mass balance on the thickening zone, using the Takacs et al. (1991) expression for

the gravity settling velocity without a flux constraint, gives:


6z dCi /dt = (Q,/A) Ci1 (Qu /A) Ci + V,i.i Ci.i V., Ci (1)


V,,i = min{ Vo [exp(-b(Ci-Cmi)) exp(-bp(Ci-Cmin))] Vma }








1E5


1E4'


1E3
0

lE2,
0

1El


1EO


0.5 1
0.5 1


Depth, m


Fig. 2-2. Comparison of 10- and 20-layer versions of the Takics et al. (1991) model using
parameters estimated for the 10-layer version of the model applied to case 1 of the
Pflanz data.


1E5-


1E4-


1E31


1E2-


1El t


1EO


Depth, m

Fig. 2-3. Comparison of 10-and 50-layer versions of the Takacs et al. (1991) model, with
optimal parameters estimated for each version applied to case 1 of the Pflanz data.


Takacs 10 layer


Takacs 20 layer


Takacs- 10


Takacs 50 layer fit


_








where Ci is the ith (from the top) layer suspended solids concentration, Cmin is the

nonsettleable suspended solids concentration, Vo and b are the Vesilind (1968) settling

parameters, bp is a settling parameter characteristic of low suspended solids

concentrations, Vnx is the highest settling velocity achieved by sludge flocs, V,i is the

gravity settling velocity from layer i, Q. is the underflow flow rate, A is the clarifier cross-

sectional area, and 5z is the layer thickness. Incorporating the constraint on gravity flux

(i.e., setting the value of the downward gravity flux from a given layer to the minimum of

the gravity flux calculated for that layer and the gravity flux calculated for the layer

immediately below) modifies eq. (1) to:


6z dCj /dt = (Q. /A) Ci- (Q. /A) Ci + min[V,i-1 Ci1, V.,, Ci] min[V,,i Ci, V,il Ci+l] (3)


An examination of the Takacs model as applied to the Pflanz data shows that the

constraint on the gravity flux becomes active at a certain layer in the thickening zone and

remains active throughout all lower (i.e., with higher i) layers. In that region, the material

balance for each layer becomes:


6z dCi /dt = (Q, /A) Ci,. (Q. /A) Ci + V,,iC Vi+lCii+ (4)


An alternate approach is to add a dispersion term to eq. (1), in which case the mass

balance for layer i is:


8z dCi /dt = (Q. /A) Ci.- (Q. /A) Ci + Vi-.Ci-1 Vs,iCi D.i-1, (Ci-Ci.i)/5z

+ Di,i.+ (Ci+,-Ci)/6z (5)








where D;.i,i is the dispersion coefficient for the dispersive flux from layer i to layer i-1 and

Di,j1 the coefficient for the flux from layer i+1 to layer i. Eq. (5) becomes identical to eq.

(4) if one uses the following concentration-dependent expression for the dispersion

coefficient:


Di,i+l(Ci,Ci+l) = 6z {V.,iCi Vs,i+lCi+l }/(Ci+l Ci) (6)


The agreement with the gravity flux constraint is complete if one imposes the physical

constraint


Di,i+l(Ci,Ci+l) > 0 (7)


This is because the expression for the dispersion coefficient becomes negative exactly

when the gravity flux constraint becomes inactive. With the above constraint applied to

the dispersion coefficient, eq. (5) becomes equivalent to eq. (3) throughout the clarifier.

Eq. (6) implies that the dispersion coefficient disappears as the layer thickness 5z -> 0.

This is not physically correct. In terms of the original limiting gravity flux formulation, as

6z -> 0, eq. (3) takes the form of eq.(1) throughout the clarifier, i.e., the constraint on the

gravity flux disappears everywhere.

To correct this problem, eqs. (6) and (7) are modified to


Di,i+1(Ci,Ci+l) = max[a(V,,iCi Vs,i+lCi+l)]/(Ci+, Ci) 0] (8)


i.e., the new parameter a replaces 6z. In this case as 6z 0, the dispersion coefficient D

converges to the finite quantity








d(V, C)
D(C)=max{-a ,0} (9)
dC


If the number of layers into which the clarifier is subdivided changes, so too will the

layer thickness 6z. In the model with dispersion (eq. (8)), a will remain constant,

whereas the constrained gravity flux approach (eq. (3)) is equivalent to modifying a

according to the change in 6z. Thus the two approaches will provide different results if

the level of discretization is changed. Figure 2-4 presents concentration profiles obtained

by each approach for the Pflanz data (first case) with a discretization of 20 layers using the

parameters that Takacs et al. (1991) reported for the 10-layer discretization. It is clear

that eq. (8) provides a better fit. Cases 2 and 3 of the Pflanz data provide similar results

(data not shown). It is concluded from this result that the Takacs et al. model would

benefit by removing its dependence on the level ofdiscretization. This uncoupling is

accomplished by recasting the equations resulting from the gravity flux constraint as a

dispersive flux term, as presented in eq. (8).

It would be advantageous to simplify the expression for the concentration-dependent

dispersion coefficient (eq. (8)). Since Cmin, exp(-bp (Ci-Cmni), and the Vmax constraint are

only significant in layers of low concentration where the first argument of the max operator

in eq. (8) is negative and therefore not used, these terms can be neglected, yielding


Di,i+l(Ci,Ci+,) = max{ Vo [exp(-b Ci) Ci exp(-b Cil) Ci+I]/(Ci+, Ci), 0} (10)



IfCij+ is close to Ci, exp(-b C;i1) C,+1 and exp(-b Ci) Ci can be approximated by their first

order Taylor series expansions about the geometric mean of the concentrations















1E5

Equiv. D 20 layer [eq. (8)]
1E4


1E3
0
Takacs 20 layer [eq. (3)]
( IE2
0 *
8 *

IE1


1EO-i
0 0.5 1 1.5 2 2.5
Depth, m


Fig. 2-4. Comparison of concentration profiles obtained by 20-layer versions of the
Takacs et al. (1991) model (eq. (3)) and the model with dispersion (eq. (8)) to
experimental data (case 1 of the Pflanz data). Parameters reported for the 10-layer
model of Takacs et al. (1991) were employed to generate model fits.








(Ci,i+1= C C+1 ). In this case, eq. (10) becomes

Di,i+l(Ci,CiQl) = max{a Vo (b Ci,+1-1) exp(-b Ci,i+,), 0} (11)


The first argument of the max operator in the above expression becomes negative for

Ci,i+1 < 1/b and attains a maximum when Ci,+1 = 2/b. It makes physical sense that at high

concentrations (>2/b) the dispersion coefficient decreases with increasing concentration,

as eq. (11) implies. It does not, however, make physical sense that the dispersion

coefficient decreases as the concentration decreases at low concentrations (< 2/b), as eq.

(11) also implies. Therefore Di,i+1(Ci,Ci+1) is set equal to its maximum (= a Vo exp(-2))

for concentrations Ci,+1 < 2/b. The expression for the dispersion coefficient now

becomes:


Sa Vo (bCu+, -1) exp(-b C1 +,) for Ci,+i > 2/b
Di'+(CiC+1)= Vo texp(-2) for C,1 <2/b (12)



The above expression involves parameters from the settling equation (Vo, b) in addition to

the introduced parameter a. The dispersion parameters can be decoupled from the settling

parameters by introducing P = b, Ccit = 2/b, and Dmax = a Vo exp(-2), in which case eq.

(12) is replaced by:


SDii+Dmax [1+ P(Cl+(i, Cjt)]exp[-P3(C,+1j -Ccr)] for C,.,+1 > Ct (13)
Dmax for Ci1i+1 < Cent








Fitting p and Ciet instead of computing them from b gives more degrees of freedom and

therefore potentially better fits, but at the expense of having extra estimated parameters.

The above expression (eq. (13)) for the dispersion coefficient D is the simplest

function that has the following features:

* Sets D equal to a constant D,, for low concentrations (less than Cent)

* Decreases D exponentially with increasing C for high concentrations. The physical

justification for this is that viscosity increases with increasing suspended solids

concentration.

* Provides for a smooth transition between the constant D region and the exponential

decay region. The contribution of the factor 1+P(C-Cnt) is to eliminate a

discontinuity corerr) in the slope at C = Coit.


The complete model equations as applied to a cylindrical clarifier

(Fig. 2-5) are now presented, with Qf denoting the flow rate into the clarifier, Qe the

effluent flow rate, Qu the underflow (return activated sludge) flow rate, A the cross-

sectional area, Uq the overflow velocity (Q,/A), and Ub the underflow velocity (Qu/A).

The clarifier is subdivided into layers of thickness 6z, with numbering from top to bottom.

In this geometry, the effluent concentration (C.) will be the concentration of the first

layer, whereas the return activated sludge (RAS) concentration (Cu) will be the

concentration of the bottom layer.


For the top layer (i = 1):


6z dCi/dt = Uq C2 UqCI Vs,1CI + DI,2 (C2 Ci)/ 5z


(14)

















i s Qe
I----Q
Ce e






feed layer







Q Q
Cu u


Fig. 2-5. Clarifier geometry used for initial model development. Clarifier is divided into n
layers.


A

N
















.- .(i =.n) __.


Qf I--
Cf








For the ith layer in the clarification section:


6z dCi/dt = Uq CiI+ UqCi + Vs,i-lCi.- V,,iCi Di-l,i (Ci-Ci-1)/5z + Dii+1 (Ci+i-Ci)/Sz


For the layer receiving clarifier feed:


6z dCi/dt = (Qf/A) Cf Uq Ci UbCi + V,,i-Ci-1 V,,iCi Di-,,i (Ci-Ci-l)/6z


+ Di,i+1 (Ci+l-Ci)/6z


(15)


(16)


For the ith layer in the thickening zone:


5z dCi/dt = Ub Ci-i Ub Ci + V,,i-,Ci.- V,iCi Di-,i (Ci-Ci-1)/8z + Di,i+ (Ci,+-Ci)/6z (17)


And finally for the bottom layer:


5z dCi/dt = Ub Ci-1 UbCi + Vs,i-1Ci-1 Di-l,i (Ci-Ci-l)/8z


(18)


The above equations are essentially the result of discretizing the parabolic partial

differential equation

aC ac aG; a ac
-CUC OG 8 _C
=U +- [D(C) (19)
at az az z (19)


where U = Uq in the clarification section, U = Ub in the thickening zone, and G, is the
gravity settling flux.








Application of Clarifier Model to the Pflanz Data


Table 2-1 presents the fit of the previously presented model to the three sets of Pflanz

data as modified by Takics et al. (1991). Parameters were estimated using the Levenberg-

Marquardt algorithm (Marquardt, 1963; Press et al., 1989; Cuthbert, 1987). The

objective function for parameter estimation was the sum of the squares of the relative

errors (SSRE) between observed and model concentrations. As was done by Takacs et al.

(1991), a separate set of parameter values was estimated for each case.

The clarifier was discretized into 50 layers and the feed layer was set in a position

consistent with that chosen by Takacs et al. (1991) in their 10-layer discretization. The

table shows the fit with the dispersion expression of eq. (13) for three cases:

* One dispersion parameter estimated (Dx). The other two parameters are calculated

from settling parameters as implied by eq. (12), i.e., P = b and Cnt = 2/b. We refer to

this as the Dma model.

* Two dispersion parameters estimated (Dma and Cent) with P = b. We refer to this as

the Dmax-Crit model.

* Three dispersion parameters estimated (Dmx, Cnt and 0). We refer to this as the Dmx-

Corit-P model.

As can be seen from Table 2-1, the Dmx model did quite well (SSREs 0.087, 0.067

and 0.077 for the three respective cases of the Pflanz data) in relation to the 10-layer

Takacs et al. (1991) model (SSREs 0.275, 0.254 and 0.157). Recall that the Takacs et al.

(1991) model performed worse for a 50-layer discretization (Fig. 2-3). Some

improvement is obtained by the Dmax-Ct model (SSREs 0.085, 0.060 and 0.056), while








Table 2-1. Performance of models with concentration-dependent dispersion functions in comparison to the Takacs et al. (1991) model,
as applied to Pflanz data
Takacs et al. (1991) Model with concentration-dependent dispersion term
10 layers 50 layers
Dmax-Ccrit-B model Dmax-Ccrit model Dmax model
Depth Mean cone. Model Rel. error Prediction Rel. error Prediction Rel. error Prediction Rel. error
Case (m) (mg/L) (mg/L) (%) (mg/L) (%) (mg/L) (%) (g) (%)
0.11 9.0 9.1 1.1 8.4 -7.2 8.4 -7.1 8.4 -7.0
1 0.34 10.7 11.2 4.7 11.8 10.5 11.8 10.6 11.9 10.8
0.57 13.6 14.1 3.7 15.0 10.0 15.0 10.1 15.0 10.2
0.79 23.8 19.5 -18.1 19.8 -16.8 19.8 -16.7 19.8 -16.7
1.02 35.0 33.8 -3.4 30.1 -13.9 30.2 -13.8 30.1 -13.9
1.25 66.6 96.6 45.0 73.3 10.1 73.4 10.2 73.2 9.9
1.48 787 707 -10.2 770 -2.2 770 -2.2 772 -1.9
1.70 5281 4619 -12.5 5292 0.2 5289 0.2 5301 0.4
1.93 10022 9124 -9.0 9793 -2.3 9787 -2.3 9553 -4.7
2.16 12487 12353 -1.1 12354 -1.1 12354 -1.1 12354 -1.1
SSRE* = 0.275 SSRE = 0.085 SSRE = 0.085 SSRE = 0.087
0.11 15.6 15.7 0.6 13.9 -11.1 13.9 -11.1 13.7 -12.3
2 0.34 14.8 18.9 27.7 17.0 15.0 17.0 15.0 17.1 15.8
0.57 21.8 23.6 8.3 21.1 -3.1 21.1 -3.2 21.4 -2.0
0.79 29.9 32.9 10.0 29.6 -1.1 29.6 -1.1 29.7 -0.7
1.02 58.8 59.2 0.7 55.8 -5.2 55.8 -5.1 54.6 -7.2
1.25 274 187 -31.7 273 -0.5 273 -0.4 277 1.2
1.48 933 826 -11.5 970 4.0 971 4.1 957 2.6
1.70 5264 6130 16.5 5122 -2.7 5117 -2.8 5190 -1.4
1.93 10482 10700 2.1 10469 -0.1 10443 -0.4 10128 -3.4
2.16 12100 13767 13.8 13779 13.9 13779 13.9 13779 13.9
SSRE = 0.254 SSRE = 0.060 SSRE = 0.060 SSRE = 0.067
0.11 30.7 30.8 0.3 30.5 -0.5 30.5 -0.5 30.9 0.8
3 0.34 41.4 42.9 3.6 43.8 5.7 43.8 5.8 43.6 5.3
0.57 59.4 58.5 -1.5 57.0 -4.0 57.0 -4.0 56.5 -4.8
0.79 88.6 87.4 -1.4 80.5 -9.2 80.5 -9.2 79.7 -10.0
1.02 136 164 20.7 143 5.8 143 5.9 143 5.5
1.25 568 481 -15.3 576 1.5 576 1.5 580 2.2
1.48 1274 1378 8.2 1305 2.5 1306 2.5 1376 8.0
1.70 6999 8309 18.7 6674 -4.6 6675 -4.6 6065 -13.3
1.93 10614 11901 12.1 11196 5.5 11195 5.5 10221 -3.7
2.16 12893 15238 18.2 15239 18.2 15239 18.2 15238 18.2
SSRE = 0.157 SSRE = 0.056 SSRE = 0.056 SSRE = 0.077
*Sum of squares of relative errors













Table 2-2. Estimated parameters for 50-layer model with concentration-dependent dispersion functions when fitted
to Pflanz data
D.m-Ccit-0 model Dmax-Crit model D x, model
Parameter Case 1 Case 2 Case 3 Case 1 Case 2 Case 3 Case 1 Case 2 Case 3
Vo (m/d) 413 662 229 413 664 229 465 1000 227
V,,x (m/d) 141 235 141 141 235 141 141 184 141
bp (m3/kg) 2.08 4.19 1.90 2.08 4.19 1.90 1.96 2.18 2.07
b (m3/kg) 0.444 0.305 0.276 0.444 0.305 0.276 0.524 0.514 0.296
Cm (g/m3) 0.79 9.63 6.35 0.78 9.64 6.34 0.80 8.67 8.94
Dmx (m2/d) 3.49 18.8 6.14 3.50 18.9 6.14 3.47 12.3 6.54
Cit (g/m3) 9,481 11,019 10,562 9,495 11,710 10,562 *
p (m3/kg) 0.433 0.144 0.276 *
Feed layer 32 30 29 32 30 29 32 30 29
a 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
* calculated from parameter b








the Dmax-Ccrit-P model offers no further improvement. The estimated parameters are

given in Table 2-2. A 10-layer discretizaton of the model with dispersion (eq. (13)) was

also investigated and gave SSREs of 0.258, 0.090, 0.187 when the D x model was used

and 0.250, 0.089, 0.185 when the Dax-Crit model was employed (data not shown). As

expected, these SSREs were higher than obtained with the 50-layer discretization.



Experimental Measurements on a Full-Scale Clarifier


Field tests were conducted at the Kanapaha Water Reclamation Facility (KWRF) to

collect data for testing the clarifier model. The KWRF is a 38,000 m3/d (10 Mgal/d) plant

utilizing the Ludzack-Ettinger process for nitrification, denitrification, and carbon

oxidation (Ludzack and Ettinger, 1962). Clarifier loading tests were conducted using one

of the plant's four secondary clarifiers. Batch sludge settling tests were carried out in

parallel with the loading tests to provide data for determining settling equation parameters.


Clarifier Test Procedure

The secondary clarifiers were 28.96 m (95 ft) in diameter with a 3.66 m (12 ft)

sidewall depth and a 4.6 m (15 ft) depth at the center. The influent entered at a central

feedwell that was bounded by an annular baffle (shroud) extending from above the liquid

surface to 2.44 m (8 ft) below the liquid surface. Effluent left the system over peripheral

and radial weirs. The clarifier feedwell occupied approximately 28 percent of the cross-

sectional area. RAS was removed continuously via a rotating multiple-pipe suction system

with four draw-offs approximately 0.23 m (9 inches) above the bottom and spaced at

distances of 1.5, 5.2, 8.6, and 12.2 m from the center of the clarifier. Waste activated








sludge (WAS) was removed by periodic pumping from a central sump. During the loading

tests, the influent flow rate to the clarifier was controlled by flooding the distribution box

that fed the four clarifiers and then adjusting the flow rate manually using the in-line valves

between the distribution box and the four clarifiers. Flow adjustments were made with

reference to a hand-held meter that displayed the effluent flow rate of the test clarifier.

The underflow rate was controlled by adjusting the speed of the RAS pump with reference

to a flow meter on the RAS line.

A loading test was initiated by setting the influent and underflow flow rates to selected

values and typically lasted 8-10 hours. The sludge blanket height was measured at 15

minute intervals using a 5 cm ID transparent plastic tube (Sludge Judge). In loading tests

where a steady-state was achieved as judged from the blanket height, data collection was

continued for at least two hours into the steady-state period. In tests where a

continuously rising blanket was observed, data collection was ended when the sludge

blanket approached the effluent weirs. Mixed liquor flowing into the clarifier, clarifier

effluent, RAS and WAS were sampled hourly: mixed liquor starting at the beginning of

the test and the remaining streams beginning at the time that the sludge blanket reached a

stable height or when it became apparent that the sludge blanket would not stop rising.

Samples were stored on ice until analysis for suspended solids, which was performed on

the same day as the loading test. Concentration profiles were determined in selected tests

by collecting samples at 0.61 m intervals through the clarifier depth. The sampling

apparatus allowed collection of samples at three depths simultaneously. Concentration

profiles were measured at one point inside the shroud and another point halfway between

the shroud and the peripheral wall of the clarifier.








Batch Settling Tests

A water-bath enclosed six-column settling apparatus was constructed for the batch

sludge settling tests based on the design of Wahlberg and Keinath (1986). Tests were

carried out at sludge concentrations ranging from 2000 to 14000 g/m3. The

concentrations were achieved by mixing RAS, mixed liquor, and clarifier effluent in

selected ratios. Samples of the secondary effluent, mixed liquor, and RAS collected for

the settling test were taken for later suspended solids analysis. Sludge in the columns was

mixed for 5 minutes immediately prior to the settling test using compressed air introduced

through air stones. After mixing, the interface height was recorded every two minutes

until the compression phase was reached. Total test duration ranged from 30 minutes to

2.5 hours depending on sludge concentration.

The settling velocity (V,) at each initial suspended solids concentration (C) was

determined from the slope of data points lying along the initial linear portion of the

interface height versus time curve. The expression (V, = Vo e"b ) (Vesilind 1968) was

used in finding the Vesilind parameters Vo and b by least squares linear regression on the

logarithms of the settling velocities and the corresponding sludge concentrations, as

recommended by Daigger (1995). (A settling test refers to the six settling trials carried

out simultaneously in the multi-column apparatus.) A total of seven settling tests were

carried out over the three week experimental period. Values of Vo and b were found for

each test and averaged. One settling test gave values of Vo and b that were more than two

standard deviations from the mean of all seven Vo and b values. Data from this test were

rejected, then velocity versus concentration data from the six accepted tests were pooled

(Fig. 2-6) and a linear regression was performed to obtain a single set of Vesilind




















103
Vo = 183 m/d
b = 0.306 m3/kg
r2 = 0.947


102



0
0 N
> 101 7






100 III I I
0 4000 8000 12000

Concentration, mg/L


Fig. 2-6. Pooled data from batch settling tests after discarding outliers. Model line
represents fit ofVesilind equation with Vo = 182.9 m/d and b = 0.3055 m3/kg.








parameters for the test period (Vo = 182.9m/d and b = 0.3055 m3/kg).


Additional Measurements

The 60-minute nonsettleable suspended solids concentration (Method 2540F, APHA

et al., 1992) was measured in duplicate. A sample of mixed liquor was settled for 60

minutes in a plastic one L graduated cylinder, then a supernatant volume of 150 mL was

withdrawn for subsequent TSS analysis. The mean of the 60-minute nonsettleable

suspended solids concentrations was 3.2 g/m3. This was higher than the TSS of a

significant number ofclarifier effluent grab samples. The mean of these samples (2.4

g/m3) was therefore used as the nonsettleable suspended solids concentration for

simulations. Samples for total suspended solids analysis were filtered through glass fibre

filters having an average pore size of 1.2 plm (Whatman GF/C). Filtered residues were

dried to constant weight.


Application of Clarifier Model to KWRF Data and Further Model Development


The clarifier model presented in the previous section assumes that the clarifier can be

regarded as a cylinder with underflow removed uniformly from the bottom. In the KWRF

clarifier, underflow is removed by a hydraulic suction system which has intake pipes

spaced along the sloping floor of the bottom, conical section. Because the pipe intakes are

at different depths, the concentrations of sludge withdrawn at the different depths will be

substantially different, and therefore RAS removal cannot be modeled as if it were

withdrawn from a single layer. The clarifier model was therefore modified by including a

conical section at the bottom of the main cylindrical section (Fig. 2-7) The conical section



















Q, --
Cf


_-::7: .....-.--.-- _----- 7__ ~ -

--__4 (il= 1)


Shroud

As
*.. ........ .... .... .... "








--- (i=n) -----


-N-. Qe
Ce






A


Q =Qu


Qr2


Qn+2 =Qn+- Qrn+2


Qn+p-2=n+p-3 -Qrn+p-2
A
n+p-2 =



i = n+p


Fig. 2-7. Schematic diagram of full-scale secondary clarifier at the Kanapaha Water
Reclamation Facility, showing shroud, bottom conical section and model
discretization.








is divided into p layers, whereas the cylindrical section is divided into n layers. In each of

the layers of the conical section, sludge is withdrawn at a rate equal to the change in

cross-sectional area from the top to the bottom of the layer multiplied by the underflow

velocity:


Qr,i = (Ai-Ai,1)Ub (20)

This results in a constant underflow velocity throughout the conical section. The total

RAS flow rate (Q.) is the summation of the withdrawals from the layers of the conical

section:

n+p
Q = I1Q (21)
i=n+l r.I

The RAS concentration was calculated as the flow-weighted average of the concentrations

of layers in the conical section.

The model was also modified to account for the presence of a shroud in the upper

section of the clarifier. The cross-sectional area of the clarifier available for overflow in

the upper region of the clarifier (i.e., from 0 to 2.44 m below the water surface) is 72% of

the cross-section below the shroud (Fig. 2-7).

Based on observations of density current flows in prototype scale clarifiers (Andersen,

1945), the feed in the clarifier model was input to the layer above the first layer having a

concentration greater than the feed concentration. This was accomplished by the

following recursive procedure. A position for the feed layer was initially assumed. The

concentration profile was then calculated and used to update feed layer position. The

latter two steps were repeated until convergence was achieved. In rare cases the above








procedure did not converge to a single layer and instead began to oscillate between two

adjacent layers. In those cases, the higher of the two layers was chosen as the feed layer.

Data from nine clarifier loading tests at the KWRF in which a steady-state blanket was

attained and the concentration profile was measured were used to calibrate and evaluate

the model. These tests involved a range ofunderflow and overflow rates (Table 2-3).

During the same experimental period, five other tests failed to reach a steady state because

of overloading (i.e., continually rising blanket) and in one other test a steady state was

reached but the concentration profile was not measured. An example of a loading test in

which a steady blanket level was achieved is shown in Figure 2-8a, whereas Figure 2-8b

shows an example of a test in which the blanket continued rising throughout the

experimental period.

Model parameters were determined as follows: Vo, b, and Ci, were obtained by

analysis of batch settling data and clarifier effluent samples, as described previously. The

parameters bp, V,,a, and Dmax were obtained by least-squares nonlinear regression on

concentration profiles as explained below. The remaining parameters (Ceot and P) were

either estimated by nonlinear regression on concentration profiles or calculated from the

experimentally determined b (P = b and Cit = 2/b). The effluent (overflow) and RAS

concentrations used in fitting were the average values over the period when the system

was determined to be at steady-state, whereas the remaining points in the concentration

profile were from a single set of measurements taken near the end of each loading test.

To determine model parameters for the Pflanz data (Takics et al., 1991), the sum of

the squares of relative errors in concentration was used as the performance measure. This

was selected because Takics et al. (1991) reported the quality of their fits in terms of

















Table 2-3. Operational variables for KWRF loading tests which achieved steady blanket levels


Effluent
Case flow rate
m3/d
1 11,396
2 14,942
3 18,923
4 18,748
5 18,815
6 18,840
7 22,400
8 18,776
9 24,418


RAS
flow rate
m3/d
9,447
9,460
5,662
9,428
9,448
11,320
9,450
13,172
9,451


Waste
flow rate
m3/d
91
91
91
91
91
91
91
91
91


Influent
flow rate
m3/d
20,934
24,492
24,676
28,266
28,354
30,251
31,940
32,039
33,960


Feed
cone.
mg/L
4,053
3,972
3,801
4,130
3,664
3,994
3,560
3,787
3,444


RAS
conc.
mg/L
8,877
9,890
14,592
11,534
10,890
9,984
10,738
8,752
11,690


Sludge blanket
height
m
1.09
1.37
1.88
1.77
1.68
1.78
2.00
1.71
3.22












Case 4

effluent flowrate

A


250


10 ,o 0 000


.iO .ram nnpnlp


0 100 200 300 40(
Time, min




Case 13 *

blanket height


S* effluent flowrate

A


RAS flowrate
Iva m als mE a m I a m a m

I I I I S


50


100


150


200


Time, min


Fig. 2-8. Results of typical clarifier loading tests: (a) test in which a steady blanket level
was achieved (case 4), (b) test in which blanket continued rising throughout
experimental period (case 13).


blanket height
***** *000*

RAS flowrate
mum um ma m1


4



3


2



1



0





4


3
.4r





1


0








relative errors in concentration. An alternative performance measure is the sum of the

squares of the errors in the logarithms of concentrations (SSELC). This measure provides

for better fits in the thickening (high concentration) section at the expense of somewhat

worse fits in the clarifying section (low concentration). As the relative errors in

concentration measurements are higher at low concentrations, this is a desired tradeoff.

Therefore parameter fitting with the KWRF data was based on the SSELC.

The dashed lines in Figure 2-9 show the Dmax model fit to the KWRF data. These

curves were generated by a single set of model parameters for all nine cases. The fits in

some of the cases (cases 1, 2, 4, 6) approximated quite well smooth curves that could be

drawn to represent the measured data. In other cases, particularly at high loadings (cases

7, 8 and 9), the fits are poor. The total SSELC for the Dx model (Table 2-4) was 29.2.

The Dmx-Cerit model and the Dmax-Cerit-3 model gave somewhat better fits overall

(SSELCs of 24.9 and 21.8, respectively), but still performed poorly in cases of high

loading (Table 2-4). If, however, Dmx is allowed to vary from case to case, satisfactory

fits can be obtained for all cases, even with the D,mx model (Table 2-4, last column). This

leads to the investigatation of a possible dependence of Dax on clarifier loading. There is

physical justification for correlating dispersion to velocity. For example, Taylor in his

classic paper (Taylor, 1953) reports a quadratic dependence. Figure 2-10 shows the

individually fitted Dmax versus feed velocity Vf (= Qf/A). It is observed that the equation

SDi + y(Vf-Vf, )2 ifVf Vf, l
Dmx= < (22)
[ D1 ifVf< Vf,i










1 I 1E4 E4

IE3 IE3 1E3

I I IE2 1E2

El.... ---------- ------------ ---- E ----

1EO 1EO IEO------0----


1E5 I 1E5 IES




I-E
Case 4 Case 5 Case 6









1E 5 ]----------------------- --------- ------------ -------------------
IE4 I --E4 1E4




IE 5 E
Case 7 I Case 8 Case 9





E4Depth from water surface mE4









were measured. (Dashed line = fit of a/ model, dotted line = fit of FVDDmax model, solid line = fit ofFVDDmax-Ccnt-b model)
IE3 / 1E3 1IE3

IE2 IE21 1E2
IEI IEI- IEI i


IE EO IE 1EO
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
Depth from water surface, m

Fig. 2-9. Comparison of fits achieved using Dmax, FVDDmax, and FVDDmax-Crit-b models. Data are from nine loading tests on a full-
scale clarifier at the Kanapaha Water Reclamation Facility in which a steady sludge blanket level was achieved and concentration profiles
were measured. (Dashed line = fit of Dmax model, dotted line = fit ofFVDDmax model, solid line = fit ofFVDDmax-Crit-b model)


"I I ase
Casel


Case 3















Table 2-4. Results for models with Dx constant across all cases and for the Dma model with
Dma fitted individually for each case
Sum of Squares of Errors in Logarithms of Concentrations
Case D,, model Dn-Cit model Dmax-Ccrt-i model D. model with
individually fitted DmI
1 0.882 1.618 2.314 1.084
2 0.323 0.803 1.610 0.148
3 4.347 1.783 0.812 0.126
4 0.245 0.331 1.212 0.073
5 1.254 0.799 0.440 0.327
6 0.757 0.547 0.525 0.148
7 4.381 3.577 1.921 0.385
8 1.450 0.560 0.184 0.110
9 15.533 14.918 12.754 1.274
Total 29.172 24.935 21.772 3.675














20
D, = 3.95 m2/d
g = 0.0676 m
V,, = 38.6 m/d

15




Q 10




5

*


0 I I I I
25 30 35 40 45 50 55
Vf, m/d
Fig. 2-10. Variation of Dmx with feed velocity. Data points were estimated on a case-by-case basis using
the Dmax model. The line represents the fit of the proposed feed velocity-dependent expression for Dmx
to the computed Dmax values.








fits the relationship between Dmx and Vf quite well. The model sum of errors squared is

only 6% of the total variation in the fitted D.x values.

Two versions of our clarifier model which incorporated the feed-velocity dependence

were further investigated: one in which 0 and Cit are calculated from settling parameters

and one in which 0 and Cit are found by model fitting. These are referred to as the feed-

velocity-dependent Dx (FVDDmax) and FVDDmx-Cft-0 models, respectively. Fits of

the FVDDmax and FVDDmax-Crit-0 models to the KWRF data were carried out using

SSELC as the objective function (see Table 2-5 for parameter values). These fits are

shown by the dotted (FVDDmax) and solid (FVDDax-Crit-P) lines in Figure 2-9. The fits

with these models are greatly improved in most of the cases over those with constant Dmx.

In most cases, the FVDDmax model gave results almost indistinguishable from those

obtained with the FVDDmx-Ccit-P model. Table 2-6 compares profiles from the FVDDmx

model to measured data and gives values for measured and predicted RAS concentrations.

RAS concentrations were less than concentrations at the bottom of the clarifier (i.e., layer

50) because RAS was drawn off at depths throughout the bottom conical section. The

total SSELC for this fit was 5.05. The SSELC for the fit of the FVDDma-Cit- model

was 4.34 (data not shown). For comparative purposes, a fit was also carried out on the

KWRF steady-state data using the Takics et al. (1991) model, with the feed layer

determined according to the recursive procedure outlined earlier, a conical section at the

bottom, and 50 layers. This model gave a SSELC of 45.8.

The predictive capability of the FVDDmax and FVDD,,-Ccit- models was evaluated

by comparing their output to the results of the extra six cases from the experimental

period, none of which was used in model development. Both the FVDDmax and















Table 2-5. Parameters resulting from FVDD!, and FVDDmax-Crt-P fit across nine
KWRF cases for which steady blanket levels were obtained


Parameters


Model


FVDDmx ,FVDDm-Crit-b
Vmx 173.5 171.9 m/d
bp 28.8 27.2 m3/kg
D, 2.95 3.477 m2/d
g 0.0446 0.0507 d
Vf,1 34.7 37.0 m/d
Ceit 11654 g/m3
b 0.00287 m3/kg
* calculated from parameter b










Table 2-6. Results for FVDDmax model fit across the nine KWRF cases for which steady blanket levels were obtained
Logarithm of Concentration (concentration in mg/L)
Case Effl. layer 7 layer 14 layer 20 layer 30 layer 37 layer 42 layer 44 layer 49 RAS SSELC
1 Data 0.77 0.58 0.72 0.82 1.58 1.24 3.72 3.74 4.02 3.94
Model 0.44 0.50 0.54 0.56 0.65 1.32 3.71 4.01 4.21 3.94
Error -0.33 -0.07 -0.18 -0.25 -0.92 0.08 -0.00 0.26 0.19 0.00 1.18
2 Data 0.77 0.63 0.74 0.94 0.82 2.14 3.99 3.96 4.15 3.99
Model 0.49 0.57 0.62 0.65 0.76 1.77 3.87 4.05 4.22 4.00
Error -0.28 -0.05 -0.12 -0.28 -0.05 -0.36 -0.11 0.08 0.06 0.01 0.34
3 Data 0.74 0.63 0.63 0.61 1.27 3.96 4.10 4.12 4.30 4.16
Model 0.55 0.68 0.76 0.83 1.42 3.93 4.17 4.21 4.29 4.21
Error -0.19 0.05 0.12 0.21 0.15 -0.02 0.06 0.09 -0.00 0.04 0.14
4 Data 0.71 0.79 0.96 0.83 1.24 3.56 3.99 4.00 4.24 4.06
Model 0.57 0.69 0.78 0.87 1.49 3.66 4.03 4.09 4.21 4.08
Error -0.14 -0.09 -0.18 0.03 0.24 0.10 0.03 0.09 -0.03 0.02 0.14
5 Data 0.64 0.58 0.65 1.34 1.22 3.66 3.97 3.96 4.15 4.03
Model 0.57 0.68 0.75 0.82 1.16 3.18 3.95 4.05 4.19 4.03
Error -0.07 0.10 0.10 -0.51 -0.05 -0.48 -0.01 0.09 0.03 -0.00 0.53
6 Data 0.78 0.61 0.70 0.95 1.11 3.45 3.83 3.91 4.02 3.99
Model 0.60 0.72 0.82 0.93 1.76 3.61 3.96 4.03 4.15 4.02
Error -0.18 0.10 0.11 -0.01 0.64 0.16 0.13 0.11 0.13 0.02 0.55
7 Data 0.73 0.58 0.63 1.09 3.41 3.69 3.99 3.94 4.22 4.03
Model 0.70 0.87 1.06 1.41 3.36 3.88 4.03 4.08 4.17 4.07
Error -0.02 0.29 0.42 0.32 -0.04 0.18 0.04 0.13 -0.05 0.04 0.43
8 Data 0.83 0.74 0.82 0.88 1.45 3.49 3.74 3.82 4.11 3.94
Model 0.62 0.74 0.86 1.00 1.93 3.55 3.89 3.96 4.10 3.96
Error -0.21 -0.00 0.04 0.11 0.47 0.05 0.15 0.14 -0.00 0.01 0.33
9 Data 0.69 0.89 3.54 3.56 3.69 3.81 4.04 3.97 4.16 4.06
Model 1.01 1.47 2.60 3.40 3.77 3.95 4.05 4.09 4.16 4.08
Error 0.32 0.57 -0.93 -0.16 0.08 0.14 0.01 0.11 0.00 0.02 1.37












Table 2-7. Comparison of model predictions to clarifier loading test results

Test conditions Test resultt Was clarifier failure predicted
by the model?
Case Uq Ub MLSS Total solids FVDDmx FVDDax- Takics et Limiting total
flux Ci-P al. (1991) solids flux
m/d m/d mg/L kg/(m2 d)
1 17.31 14.48 4053 129 Success No No No No
2 22.69 14.50 3972 148 Success No No No No
3 28.74 8.74 3801 142 Success No No No No
4 28.47 14.46 4130 177 Success No No No No
5 28.57 14.49 3664 158 Success No No No No
6 28.61 17.33 3994 183 Success No No No No
7 34.02 14.49 3560 173 Success No No No No
8 28.51 20.14 3787 184 Success No No No No
9 37.08 14.49 3444 178 Success No No No No
10 28.18 14.49 3885 166 Success No No No No
11 37.21 14.47 4044 209 Failure Yes Yes No No
12 39.90 14.49 3444 187 Failure Yes Yes No No
13 37.25 20.24 3987 229 Failure Yes Yes No No
14 37.26 26.01 3983 252 Failure Yes Yes No No
15 36.78 27.45 3618 232 Failure Yes Yes No No
t Measured Vo and b were 182.9 m/d and 0.3055 m3/kg, respectively
Success indicates that a steady-state sludge blanket level was observed; Failure indicates that the sludge
blanket continued rising throughout the test








FVDDma-Ccit-P models correctly predicted failure for the five cases in which a continually

rising blanket was observed experimentally and also predicted success for the "tenth"

steady-state case for which a concentration profile was not available. Table 2-7 compares

the predictions of the FVDDma and FVDDax-Ccit-0 models for all 15 cases to the

experimental observations and also to predictions of the Takics et al. (1991) and total

limiting solids flux (e.g., Coe and Clevenger, 1916) models. Notably, the latter two

models predicted success in all of the cases where failure was observed. White (1976) and

Ekama and Marais (1986) observed that total limiting flux theory can overpredict clarifier

thickening capacity by as much as 20%. Reduction of the calculated limiting flux by 20%

would classify cases 11 and 12 as overloaded, case 13 as borderline, and cases 14 and 15

as still underloaded. In the latter two trials, the underflow velocity (Ub) exceeded the

critical underflow velocity, i.e. the plot of total solids flux vs. concentration had no

minimum, only an inflection point. In these cases, the limiting total solids flux was

computed according to the procedure used by White (1976) for such situations.



Conclusions


As a result of the modeling and experimental work carried out in this study, the

following conclusions can be drawn:

* Although clarifier models incorporating a constraint on gravity flux can provide

excellent simulation of experimental concentration profiles, the flux constraint

effectively disappears as the level of discretization is increased








* The gravity flux constraint can be recast as a concentration-dependent dispersion term,

which improves the ability of the model to fit experimental data as the level of

discretization increases

* A further advantage of the dispersion model is that the effect of feed velocity on

clarifier thickening performance can be accounted for

* Total limiting solids flux theory as well as models incorporating a gravity flux

constraint can fail to predict overloading of clarifiers.

* To be most valuable, the clarifier model should be integrated with a model of the

activated sludge process under investigation, using for example the IAWPRC

Activated Sludge Model No. 1 (Henze et al., 1986). In this way, the integrated model

can be employed to simulate the impacts of varying flow rates and feed compositions

on both biochemical and sludge thickening performance.















CHAPTER 3
CALIBRATION OF A ONE-DIMENSIONAL CLARIFIER MODEL USING SLUDGE
BLANKET HEIGHTS



Introduction

There has been considerable work on developing easy-to-use models for secondary

clarifiers. One group of models is based on application of limiting solids flux theory

(Tracy and Keinath, 1973; Bryant, 1972; Lessard and Beck, 1993). Another approach to

clarifier modeling is to impose a constraint not on the total solids flux but only on the

gravity settling flux term (Stenstrom, 1976; Vitasovic, 1989; Takics et al., 1991). This

approach can yield a realistic solids concentration profile with respect to depth. An

alternative approach to the above models is the introduction of a dispersion term in the

clarifier model equations (Anderson and Edwards, 1981; Lev et al., 1986). This can also

give a realistic solids concentration profile that can be used as the basis for predicting

sludge blanket height (Hamilton et al., 1992). Recently, the Hamilton et al. (1992)

dispersion model was modified by incorporating dependence of the dispersion coefficient

on both feed velocity and local solids concentration (see Chapter 2).

Work reported in this paper improves the Hamilton et al. (1992) algorithm for

predicting sludge blanket heights. It subsequently shows that the clarifier model can be

calibrated without measurements of the suspended solids profile along the depth of the

clarifier. The calibration uses as measure of fit a weighted combination of the error








squared in blanket height, the error squared in effluent suspended solids concentration,

and the error squared in the return activated sludge (RAS) concentration. The model was

calibrated using a set of data collected at the Kanapaha Water Reclamation Facility

(KWRF) in Gainesville, Florida, then validated against two other KWRF data sets

collected during different time periods.



Description of Clarifier Model

Dispersion Coefficient Expression

As is the case with Hamilton et al. (1992), the model incorporates a dispersion

coefficient D. In this case, however, D is dependent on the suspended solids

concentration. The dispersion coefficient expression has the following characteristics:

a) Decreasing value with increasing concentration, at high solids concentrations. This

could perhaps be attributed to the concomitant increase in viscosity. As the dispersion

coefficient should always be positive, an exponential decrease towards zero is reasonable.

b) A constant value at low solids concentrations.

c) A smooth transition from the constant dispersion region to the exponential decay region

at a certain concentration, which will heretofore be referred to as Cit.

The dependence of D on the solids concentration C is

D= [1 + P(C-Cent)]exp[-/f(C-C,,,)] for C > C,(
D(C) DforC ,. for C < C ,

dD
Note that in (1) the top expression yields D(Crit) = Dm, and dD(Cct) = 0, i.e., we have
continuity and smoothness at C=Ct.
continuity and smoothness at C=C,rit








In the model, the clarifier is divided into layers numbered from top to bottom, and

the dispersion term carries material from a layer of concentration Ci to a layer of

concentration Ci1. In equation (1) the concentration C that affects the dispersion

coefficient is taken to be the geometric mean of C and CQ.-, Ci-.1. It should be remarked

that the above concentration-dependent dispersion coefficient can be linked to the Takacs

et al. (1991) model as shown in Chapter 2.

The dispersion coefficient was also correlated to velocity. There is physical

justification for this, as Taylor (1953) found a quadratic dependence of the dispersion

coefficient on velocity. It is shown in Chapter 2 that fits at different loadings were

markedly improved by allowing Dmax to vary as a function of the feed velocity Vf. The

following functional dependence:



F Di + y (Vf- Vf,)2 if VVf V
D.x = (2)
[ Di ifVf


gave good results. The full dispersion coefficient expression is given by equation (1) with

Dmax calculated using equation (2).



Settling Velocity Equation

The equation proposed by TakBcs et al. (1991) was used to model gravity settling

velocity as a function of concentration. This equation subtracts an exponential term from








the commonly used Vesilind (1968) equation resulting in decreasing settling velocity with

decreasing concentration in the low concentration region:



Vs,i = min{Vo [exp(-b(Ci-Cmn)) exp(-bp(Ci-Cmi))] Vx } (3)


In this equation, V,,i is the gravity settling velocity from layer i, Ci is the layer suspended

solids concentration, Ci,n is the nonsettleable suspended solids concentration, Vo and b are

the Vesilind (1968) settling parameters, bp is a settling parameter characteristic of low

suspended solids concentrations, and Vmx is the highest settling velocity achieved by

sludge flocs.



Model ofKWRF Clarifier

A schematic of the KWRF test clarifier is presented in Figure 3-1. The model

accounts for the presence of an annular baffle (shroud) in the upper cylindrical section of

the clarifier (from 0 to 2.44 m below the water surface). The cross-sectional area

available for overflow (Ak) in this region is 72% of the cross-section area (A) in the lower

cylindrical section (between the shroud and the cone) of the clarifier.

The bottom of the clarifier consists of a conical section from which RAS is removed

by a hydraulic suction system. Because the pipe intakes are spaced at different depths

along the sloping bottom, the concentrations of sludge withdrawn at the different depths

will be substantially different. Therefore, sludge removal should not be modeled as if it



















Qf --
Cf


Shroud


~czzZZIIlzzz


(i = n)


Ce Qe


Q =Qu


Qn+2 Q n+l Qrn+2


Q n+p-2 Qn+p-3 "Q r n+p-2


i = n+p


Fig. 3-1. Schematic diagram ofclarifier at KWRF.


~"~""~"""~"~~


QMMMM


An~p-2 1
An~p-


Qrn+p


Cn+p








were withdrawn from a single layer. As Figure 3-1 shows, the conical section is divided

into p layers and the cylindrical section into n layers. In the model, sludge is removed

from each layer of the conical section at a rate equal to the change in cross-sectional area

from the top to the bottom of the layer multiplied by the underflow velocity:


Qr,i = (Ai-Ai+l)Ub (4)


Here Qr,i is the rate of flow removed from layer i, A, is the area at the top of a layer in the

conical section, and A+i1 is the area at the bottom of the layer. The underflow velocity Ub

in the lower sections of the clarifier is the ratio of the sum of the RAS and waste activated

sludge flows, Qu, to the area of the lower cylindrical section, A. Note that

n+p
Qu= E Q ri and that allocation of the withdrawal flows according to eq. (4) maintains a
i=n+l

constant underflow velocity throughout the conical section. The RAS concentration will

be the flow-weighted average of the concentrations of layers in the conical section.

Let the thickness of the layers be 6z and let the layers be numbered from top to

bottom. Mass balances on suspended solids give the equations below.



For the top layer (i = 1):

5z dC1/dt = Uq C2 UqCI VisC + D1,2 (C2 C)/ 6z (5)



For the ith layer in the region above the feed layer:

8z dCi/dt = Uq Cil UqCi + Vs,i-.Ci- Vs,iCi- Di.,i (Ci-Ci.l)/6z + Di+i (Ci+1-Ci)/8z (6)








For the layer receiving clarifier feed:

6z dCi/dt = (Uq+Ub) Cf- Uq Ci UbCi + V.,i.-Ci- V,,iCi- Di-1,i (Ci-Ci-l)/6z

+ Di,i+l (Ci+- Ci)/6z (7)


For the ith layer in the region below the feed layer:

6z dCi/dt = Ub Ci1 Ub Ci + Vs,i-.Ci-l Vs,iCi Di-,i (Ci-Ci-.)/6z + Di,i+l (Ci+1-Ci)/6z (8)


And finally for the bottom layer:

6z dCi/dt = Ub C-1 UbCi + V,i-,Ci-. Di-l,i (Ci-Ci-l)/5z (9)



Here Uq is the overflow velocity (= QeA, in the upper cylindrical region or QJA in the

lower cylindrical section with Qe being the effluent flowrate), Ub the underflow velocity

(= QJAc in the upper cylindrical section or Q/A below the shroud), and Di-,i = D( C,.-, ).

To calculate the location of the feed layer, an initial position for the feed layer was

assumed and the concentration profile calculated. The location of the feed layer was then

chosen as the layer above the first layer having a concentration greater than the feed

concentration. This is consistent with observations of density current flows in prototype

scale clarifiers (Andersen, 1945). Subsequently, the concentration profile was

recomputed, and the feed layer reassigned. The latter steps were repeated until

convergence was achieved. In rare instances when the above procedure did not converge

to a single layer and instead began to oscillate between two adjacent layers, the higher of

the two layers was chosen as the feed layer.








Sludge Blanket Algorithm

The sludge blanket algorithm was modified from Hamilton et al. (1992), who

calculated the blanket height as the height corresponding to the maximum rate of change

in the slope Pi of the solids concentration versus depth profile. The modified algorithm

uses in place of Pi a relative concentration slope:

Ri =(C- C,)/5z (10)
(C, + Ci-_)/2

which is the concentration slope divided by the average concentration between the

adjacent layers. This gives higher predicted blanket heights than those calculated with the

Hamilton et al. (1992) algorithm.

The height hi (measured from the bottom of the conical section) of the interface

between model layers having the maximum relative concentration slope R, is located, then

a quadratic interpolating polynomial is used to find a smooth curve passing through this

point and two adjacent points (Ri.i, hi.1; Ril, hi+,). When the maximum Ri is at the

interface between the bottom layer and the layer above it, R;,+ (at hi,l = 0) cannot be

calculated by eq. (10). It is then assigned a value of zero to ensure that a positive blanket

height is obtained. The sludge blanket height (SBH) is calculated as the location of the

maximum of the interpolated polynomial. This results in

(m21Ri1 + m22R + m23Ri,+)
SBH- (11)
2(m,,Ri_ + m,Ri + m,3Ri+l)

where mij is the ijh element of the matrix









h hi_ 12
1 hi -1
M = h h 1 (12)
ih + hi+ 1



Materials and Methods

Clarifier Loading Tests

Clarifier loading tests were conducted at the 38,000 m/d (10 Mgal/d) Kanapaha

Water Reclamation Facility (KWRF) using one of the plant's four secondary clarifiers.

The clarifier was 28.96 m (95 ft) in diameter with a 3.66 m (12 ft) sidewall depth and a 4.6

m (15 ft) depth at the center.

During a loading test, influent and RAS flow rates were controlled at selected values.

Sludge blanket height was measured every 15 min using a 5 cm ID transparent plastic tube

(Sludge Judge, NASCO Inc., Ft. Atkinson, Wisconsin). Data collection was continued for

two hours after reaching a steady blanket level. Tests where a continuously rising blanket

was observed were ended when the sludge blanket approached the effluent weirs. Influent

mixed liquor was sampled hourly throughout each test, whereas clarifier effluent, RAS,

and waste activated sludge (WAS) were sampled hourly once a steady blanket level was

reached or when the sludge blanket approached the effluent weirs. Samples were stored

on ice until analysis for suspended solids later in the day.

To determine concentration profiles, samples were collected at 0.61 m intervals

through the clarifier depth. The sampling location was halfway between the shroud and

the peripheral wall of the clarifier. An additional location was inside the shroud.








Batch Settling Test Procedure

Batch sludge settling tests were conducted in a water-bath enclosed six-column

settling apparatus (Wahlberg and Keinath, 1986). RAS, mixed liquor, and clarifier effluent

were mixed in selected ratios to obtain suspended solids concentrations ranging from 2000

to 14000 mg/L. Columns were mixed for 5 min prior to each test using compressed air.

Interface height in each column was recorded every two minutes until the compression

phase was reached.

Settling velocity (V,) at each initial suspended solids concentration (C) was found by

least squares linear regression on the initial linear portion of the interface height versus

time curve. Settling parameters Vo and b in the expression (V, = Vo e"b ) (Vesilind, 1968)

were found by least squares linear regression on the logarithms of the settling velocities

and the corresponding sludge concentrations (Daigger, 1995). Four settling tests were

carried out during experimental period A, six settling tests during period B, and seven

during period C. (A settling test refers to the six settling trials carried out simultaneously

in the multi-column apparatus.) Settling test data were screened by comparing the Vo and

b from each individual test to the preliminary mean Vo and b for all tests in the respective

test period. Tests with Vo or b more than 2.0 standard deviations away from the

preliminary mean parameter values from that period were rejected. Accepted data from

the respective test periods (only one test was rejected) were pooled and parameters of the

Vesilind expression were found by linear regression. Parameters were Vo = 149.9 m/d and

b = 3.921 x 104 L/mg for period A, Vo = 152.5 m/d and b = 3.213 x 104 L/mg for period

B, and Vo = 182.9 m/d and b = 3.055 x 104 L/mg for period C.










Additional Measurements

The mean of grab samples from the clarifier effluent (collected for use in the batch

settling tests) was used as the nonsettleable suspended solids concentrations, Cm, for

simulations. These values were 3.9 mg/L for period A and 2.4 mg/L for both periods B

and C. Samples for total suspended solids analysis were filtered through glass fibre filters

having an average pore size of 1.2 itm (Whatman GF/C). Filtered residues were dried to

constant weight.



Parameter Estimation

Model parameters (Vmax, bp, D1, y, Vf,i) were estimated using the Levenberg-

Marquardt algorithm (Marquardt, 1963; Press et al., 1989) with scaling according to

Cuthbert (1987). The objective function weighted three components: the sum of the

squares of the relative errors in sludge blanket height, the sum of the squares of the

relative errors in RAS concentration, and the sum of the squares of the relative errors in

effluent concentration. The three components were weighted 80%: 10%: 10%,

respectively, since the blanket height provides the greatest amount of information about

the shape of the concentration profile.








Results and Discussion

Clarifier Loading Tests

A total of 43 clarifier loading tests were carried out during three experimental

periods (A, B, C) over a time span of 6 months. Out of 13 tests in period A, nine reached

steady-state conditions as judged from steady blanket levels. Eleven of the 15 tests in

period B and 10 of the 15 tests in period C also reached steady-state conditions. Mass

balance closure errors under steady-state conditions ranged from 0.9% to 18.6%, with a

median error of 9.0%. Overall mean influent suspended solids concentrations ranged

between 3200 and 4170 mg/L, whereas effluent suspended solids concentrations under

steady-state conditions ranged from 1.6 mg/L to 8.7 mg/L. Effluent suspended solids

were not correlated with total solids loading (P<0.05). Steady-state RAS concentrations

ranged between 6330 and 14600 mg/L.



Model Calibration

The clarifier model was calibrated using steady-state data (sludge blanket height,

effluent suspended solids and RAS concentrations) from test period C. The estimated

model parameters were Vnx = 172.0 m/d, bp = 2.7587 x 10-2 L/mg, Di = 4.835 m2/d,

y = 2.500 x 10-2 d, and Vf, = 32.88 m/d. Model profiles exhibited a zone of rapidly

changing suspended solids concentration between regions of slowly changing

concentration near the top and bottom of the clarifier, and were generally consistent with

measured data (Fig. 3-2). As Table 3-1 shows, the model tended to overestimate RAS

concentration slightly (median error = 4.7%). The error in RAS concentration predictions








is essentially set by the mass balance closures of the respective experiments. The model

tended to underestimate effluent suspended solids concentration by as much as 3.3 mg/L,

and the correlation of the model estimates to the measurements was poor. This is not

surprising since, in the tests, effluent suspended solids were not correlated with clarifier

loading. (This holds since experiments were never carried out to loss of blanket.)

Sludge blanket heights are represented in Figure 3-2 by vertical lines, light solid for

model-calculated heights and heavy solid for measured heights. Model sludge blanket

heights fall in the region of rapidly changing concentration, as indicated by the model

profile. Model blanket heights differed from measurements by no more than 0.21 m, and

were within 0.08 m of measurements in most tests. A linear regression of model-

calculated sludge blanket heights to the measurements gave a slope of 1.04 (Fig. 3-3).



Model Validation

The model as calibrated using test period C data, along with Vo, b, and Cn measured

in each period, was run to predict sludge blanket heights of tests in periods A and B. The

relationship between predicted and measured heights, for the tests in which a steady

blanket was observed, is shown in Figure 4. In period A, for all nine tests the model

blanket heights within 0.43 m (Table 3-2) with median absolute error of 0.16 m. For ten

of the eleven tests of period B, the model predicted blanket height within 0.36 m of the

measurement. The one exception was the test with the lowest underflow velocity (case B-

7), in which the model overpredicted the height by 2.1 m. The median absolute

























II

0. d


















*
1.1.. 1 .11 .i i~ i .... >J>'i.0


F 0


t:




a*
1


0


a













0 4
o C
g 0^








o S-
gs a












0 0
S



a 0 9




Co











CO C
- 1
0 -y


'/LU 'UoIWJlua3uOD


4 qrs


0

0i

YS


0
^ .




i
I*


~34 L~L~














Table 3-1. Comparison of predicted effluent concentrations, RAS concentrations, and sludge blanket
heights to measured values for test period C
Measurements Model predictions and relative errors*

effluent conc. RAS conc. blanket height
effluent RAS SBH
Case cone. conc. prediction rel. error prediction rel. error prediction rel. error
(mg/L) (mg/L) (m) (mg/L) (%) (mg/L) (%) (m) (%)
C-1 6.0 8877 1.09 2.7 -55.2 8892 0.2 1.07 -1.7
C-2 5.9 9890 1.37 3.0 -49.9 10181 2.9 1.18 -14.1
C-3 5.6 14592 1.88 3.5 -38.3 16292 11.7 1.93 3.0
C-4 5.2 11534 1.77 3.5 -32.8 12257 6.3 1.70 -4.4
C-5 4.4 10890 1.68 3.4 -21.8 10883 -0.1 1.47 -12.6
C-6 6.1 9984 1.78 3.6 -40.3 10581 6.0 1.70 -4.3
C-7 5.4 10738 2.00 4.5 -17.2 11907 10.9 2.03 1.3
C-8 6.9 8752 1.71 3.8 -44.9 9142 4.5 1.70 -0.4
C-9 4.9 11690 3.22 7.0 42.4 12239 4.7 3.19 -0.8
*Model was calibrated using data from test period C




















regrets. slope = 1.04



4-
4 r 2 = 0.990







2



'-

predictions regression --- 1:1 line
0 I I
0 1 2 3 4
Measured blanket height (m)


Fig. 3-3. Comparison of predicted blanket heights to measured values from test period C.
(Model was calibrated using blanket height data in addition to RAS and effluent
suspended solids concentrations from test period C.)
























4 regres. slope = 1.11
r = 0.653










0
/




2-*
*.












predictions regression -1:1 ine
0 I
S1 2 3 4
Measured blanket height (m)


Fig. 3-4. Comparison of predicted blanket heights to measured values from test periods
A and B. (Model parameters from calibration to test period C data were used in the
simulations.)
simulations.)











Table 3-2. Comparison of predicted effluent concentrations, RAS concentrations,
and sludge blanket heights to measured values for test periods A and B
Measurements Predictions *

effluent RAS blanket effluent RAS blanket
Case cone. conc. height conc. conc. height
(mg/L) (mg/L) (m) (mg/L) (mg/L) (m)
A-1 7.6 11358 1.86 4.5 11589 1.95
A-2 7.2 6691 1.35 4.0 7347 1.19
A-3 5.4 8313 1.41 4.3 8872 1.39
A-4 6.9 8313 1.97 4.4 9138 1.54
A-5 7.2 7152 1.56 4.3 7859 1.29
A-6 8.7 6332 1.59 4.3 7404 1.29
A-7 6.9 9790 2.29 5.1 10219 2.18
A-8 5.9 9075 2.01 4.9 9692 1.86
A-9 3.0 8806 2.16 4.9 9525 1.79
B-1 3.6 8383 1.11 3.2 7830 1.15
B-2 3.9 9155 1.52 3.8 10376 1.49
B-3 3.6 9679 1.98 4.8 11186 2.00
B-4 3.2 9681 2.36 4.8 11165 2.00
B-5 1.9 10170 2.23 4.8 10975 1.99
B-6 2.7 9271 1.21 3.2 10152 1.25
B-7 3.6 12788 2.22 40.8 15340 4.34
B-8 1.9 7774 2.08 5.4 9216 2.02
B-9 1.6 7324 1.89 5.0 7721 1.77
B-10 2.5 9040 2.23 5.4 9053 2.01
B-11 4.9 7470 1.20 3.2 8588 1.19
*Model parameters, with the exception of Vo, b, and Cmin, were those found from
test period C fit








error of the eleven tests was 0.06 m. The linear regression slope for all the predictions in

periods A and B was 1.11 (Fig. 3-4).

Table 3-2 also shows the predictions ofRAS and effluent suspended solids

concentrations for the tests in periods A and B. The model tended to over predict RAS

concentrations in a manner consistent with the mass balance closures of the respective

experiments. The range of predicted effluent suspended solids concentrations for periods

A and B, excluding case B-7, was 3.2 5.4 mg/L with a median (including B-7) of 4.65.

In comparison, the range of the measured suspended solids concentrations was 1.6 8.7

mg/L with median 3.75 mg/L. Other than having values of the correct order of

magnitude, the model predictions do not correlate well with the measured values.

Tables 3-3a, 3-3b, and 3-3c give the results of the clarifier tests and classify each test

according to whether the applied loading was acceptable or not under the test operating

conditions. Since field tests could not be carried out to the point of actual failure, the end

result was judged on the basis of sludge blanket behavior. Experimentally observed sludge

blankets that were rising at a rapid rate (> 0.1 m/h) near test termination were considered

to be the result of clarifier overloading, whereas steady or falling blanket levels near test

termination were considered to result from acceptable loading levels. Tests ending in

slowly rising blankets were considered inconclusive.

Also reported in Table 3-3 are the results of simulations with the present model

under the loading and operating conditions of each test run. Vo, b, and Cm were the

measured values for each of the three test periods, whereas all other model parameters

were those found in test period C. Model results were classified according to the
















Table 3-3a. Comparison of model predictions of success and failure to test results for test period A
Test Period A Field test results Model runs with test period C Takacs model runs Solids flux analysis
parameters
Solids flux Acceptable Steady or Blanket Acceptable Effluent Steady Acceptable Effluent Steady Acceptable Fraction of
Case Uq Ub a MLSS applied loading? b final SBH rise rate loading? cone. SBH loading? cone. SBH loading? limiting
flux
(m/d) (m/d) (mg/L) (kg/m2 d) (m) (m/hr) (mg/L) (m) (mg/L) (m) (%)
A-1 17.4 8.7 3893 102 Y 1.86 Y 4.5 1.95 Y 4.65 1.24 Y 86.7
A-2 11.5 14.2 4071 105 Y 1.34 Y 4.0 1.19 Y 4.04 0.95 Y 63.2
A-3 17.0 11.5 3611 104 Y 1.40 Y 4.4 1.39 Y 4.51 1.05 Y 71.7
A-4 17.3 14.4 4166 132 Y 1.98 Y 4.4 1.54 Y 4.56 1.06 Y 78.9
A-5 17.0 14.3 3622 114 Y 1.55 Y 4.3 1.29 Y 4.49 0.96 Y 67.8
A-6 17.0 17.2 3739 128 Y 1.58 Y 4.3 1.29 Y 4.49 0.96 Y 68.0
A-7 22.7 14.3 3971 148 Y 2.29 Y 5.1 2.18 Y 5.35 1.33 Y 88.1
A-8 22.6 14.3 3789 140 Y 2.01 Y 4.9 1.86 Y 5.22 1.24 Y 83.6
A-9 22.6 14.4 3731 138 Y 2.16 Y 4.9 1.79 Y 5.19 1.18 Y 82.3
A-10 17.2 17.3 4078 141 Y (1.86)c -0.059 Y 4.4 1.42 Y 4.53 1.05 Y 74.6
A- 1 23.5 14.3 4119 156 Y (2.50) -0.047 N 174 4.45 Y 5.63 1.60 Y 93.3
A-12 22.9 14.4 4043 151 I (2.59) 0.042 Y 5.2 2.44 Y 5.44 1.42 Y 90.1
A-13 28.6 14.3 3731 161 N (3.26) 0.133 N 498 4.45 Y 6.46 1.97 Y 95.9

a Velocities calculated using full clarifier cross-sectional area (A = 658.5 m2)
by = yes (steady or falling blanket), N = no (blanket rising at > 0.1 m/h), I = inconclusive (blanket rising at < 0.1 m/hr)
C Blanket height at test termination
















Table 3-3b. Comparison of model predictions of success and failure to test results for test period B
Test Period B Field test results Model runs with test period C Takacs model runs Solids flux analysis
parameters
Solids flux Acceptable Steady or Blanket Acceptable Effluent Steady Acceptable Effluent Steady Acceptable Fraction of
Case Uq Ub MLSS applied loading? b final SBH rise rate loading? cone. SBH loading? cone. SBH loading? limiting
flux
(m/d) (m/d) (mg/L) (kg/m2d) (m) (m/hr) (mg/L) (m) (mg/L) (m) (%)
B-1 17.3 14.4 3573 114 Y 1.1 Y 3.2 1.15 Y 3.4 0.96 Y 55.1
B-2 22.6 14.3 4049 150 Y 1.5 Y 3.8 1.49 Y 4.1 1.15 Y 72.9
B-3 28.5 14.4 3777 162 Y 2.0 Y 4.8 2.00 Y 5.0 1.24 Y 78.7
B-4 28.7 14.4 3757 162 Y 2.4 Y 4.8 2.00 Y 5.0 1.24 Y 78.6
B-5 28.8 14.4 3680 159 Y 2.2 Y 4.8 1.99 Y 5.0 1.24 Y 77.2
B-6 17.5 8.6 3388 89 Y 1.2 Y 3.2 1.25 Y 3.4 0.86 Y 61.9
B-7 28.4 8.6 3645 135 Y 2.2 N 40.8 4.34 Y 5.4 1.40 Y 94.2
B-8 28.3 20.1 3841 187 Y 2.1 Y 5.4 2.02 Y 4.9 1.17 Y 73.4
B-9 28.6 20.1 3205 156 Y 1.9 Y 5.0 1.77 Y 4.8 1.09 Y 61.5
B-10 28.7 20.1 3744 183 Y 2.2 Y 5.4 2.01 Y 4.9 1.20 Y 72.1
B-11 16.8 14.3 3978 124 Y 1.2 Y 3.2 1.19 Y 3.4 0.96 Y 56.9
B-12 34.5 20.1 3744 205 N (3.2) 0.337 N 289 4.45 Y 6.0 1.42 Y 80.6
B-13 34.2 20.1 3698 201 N (3.2) 0.178 N 195 4.45 Y 5.9 1.34 Y 79.2
B-14 31.4 20.1 3775 195 I (3.1) 0.095 Y 8.1 3.02 Y 5.4 1.25 Y 76.7
B-15 31.3 24.1 3842 213 I (3.2) 0.057 Y 12.9 3.48 Y 5.4 1.24 Y 81.1
* Velocities calculated using full clarifier cross-sectional area (A = 658.5 m2)
b Y= yes (steady or falling blanket) N = no (blanket rising at >= 0. I m/h), I = inconclusive (blanket rising at < 0.1 m/hr)
c Blanket height at test termination












Table 3-3c. Comparison of model predictions of success and failure to test results for test period C
Test Period C Field test results Model runs with test period C Takacs model runs Solids flux analysis
parameters
Solids flux Acceptable Steady or Blanket Acceptable Effluent Steady Acceptable Effluent Steady Acceptable Fraction of
Case Uq Ub a MLSS applied loading? b final SBH rise rate loading? cone. SBH loading? cone. SBH loading? limiting
flux
(m/d) (m/d) (mg/L) (kg/m2-d) (m) (m/hr) (mg/L) (m) (mg/L) (m) (%)
C-1 17.3 14.3 4053 129 Y 1.1 Y 2.9 1.09 Y 3.1 0.96 Y 56.2
C-2 22.7 14.4 3972 148 Y 1.4 Y 3.3 1.25 Y 3.6 1.06 Y 64.4
C-3 28.7 8.6 3801 142 Y 1.9 Y 4.0 2.05 Y 4.6 1.21 Y 90.2
C-4 28.5 14.3 4130 177 Y 1.8 Y 4.0 1.79 Y 4.3 1.24 Y 77.5
C-5 28.6 14.3 3664 158 Y 1.7 Y 4.0 1.54 Y 4.3 1.24 Y 68.8
C-6 28.6 17.2 3994 183 Y 1.8 Y 4.2 1.73 Y 4.3 1.24 Y 70.7
C-7 34.0 14.3 3560 173 Y 2.0 Y 5.1 2.03 Y 5.0 0.93 Y 75.3
C-8 28.5 20.0 3787 184 Y 1.7 Y 4.3 1.69 Y 4.3 1.15 Y 64.3
C-9 37.1 14.4 3444 178 Y 3.2 Y 7.2 2.98 Y 5.5 0.94 Y 77.5
C-10 28.2 14.4 3885 166 Y 1.8 Y 3.9 1.62 Y 4.3 0.95 Y 72.3
C-11 37.2 14.3 4044 209 N (3.7) 0.421 N 400 4.45 Y 5.8 1.76 Y 91.3
C-12 39.9 14.4 3444 187 N (3.1) 0.379 N 77.8 4.44 Y 6.0 1.02 Y 81.7
C-13 37.2 20.1 3987 229 N (3.7) 0.353 N 181 4.44 Y 5.6 1.51 Y 79.7
C-14 37.3 25.9 3983 252 N (3.4) 0.444 N 211 4.44 Y 5.5 0.95 Y 79.0
C-15 36.8 27.3 3618 232 N (3.7) 0.223 Y 18.3 3.60 Y 5.3 1.34 Y 73.1

a Velocities calculated based on full clarifier cross-sectional area (A = 658.53 m2)
b Y= yes (steady or falling blanket), N = no (blanket rising at >= 0.1 m/h), I = inconclusive (blanket rising at < 0.1 m/hr)
SBlanket height at test termination








predicted effluent suspended solids. Effluent concentrations of less than 20 mg/L were

considered to represent acceptable loading levels. "Overloaded" cases were characterized

by effluent concentrations in excess of 70 mg/L. Predictions of the Takacs et al. (1991)

model, which was considered to be the best available by Grijspeedt et al. (1995), as well as

limiting solids flux theory (Coe and Clevenger 1916; Yoshioka et al. 1957) are given in the

table for comparison. The sludge blanket algorithm was incorporated in a program

implementing the Takacs et al. (1991) model, and the model was calibrated to KWRF test

period C data using procedures identical to those described above. Simulations of the

other test periods were performed with the test period C fitted parameters and Vo, b, and

C.,m for the given period. The total limiting solids flux was calculated based on the

measured Vo and b values for the appropriate test periods.

Out of 40 conclusive loading tests, the present model correctly predicted the

outcome of 37. Two experimental successes were predicted as failures (cases A-11 and

B-7), whereas one experimental failure was predicted as a success (case C-15), but with a

relatively high blanket height and effluent solids concentration. In contrast, the TakAcs et

al. (1991) model predicted success for all cases (which is also consistent with limiting

solids flux theory), and thus incorrectly predicted the eight experimental failures.



Conclusions

The model utilized in the present work introduces an algorithm for locating the top

of the sludge blanket based on the point of greatest relative concentration slope. This

algorithm is computationally efficient and reliable in matching experimentally measured

blanket heights. Incorporation of the blanket algorithm in the model enables calibration





65


using measured blanket heights instead of concentration profiles throughout the clarifier

depth. The model validation carried out in this research was based on extensive full scale

plant data sets. Based on these data, the model developed in this work appears to be more

reliable than limiting solids flux theory or the model ofTakics et al. (1991) in predicting

clarifier failure due to solids overloading.














CHAPTER 4
CONCLUSIONS


As a result of the modeling and experimental work carried out in this study, the

following conclusions can be drawn:

* Clarifier models incorporating a constraint on gravity flux have been shown to provide

good fits to experimental concentration profiles, but the flux constraint is dependent

upon the level of model discretization. The flux constraint therefore effectively

disappears as the level of discretization is increased.

* The gravity flux constraint can be recast as a concentration-dependent dispersion term

which improves the model's ability to fit experimental data as the level of model

discretization is increased.

* The ability of the model to fit concentration profiles collected in steady-state full-scale

clarifier loading tests over a range of solids loading was substantially improved by

inclusion of dependence on influent velocity in the dispersion function.

* Incorporation of the algorithm for the determination of sludge blanket height in the

model enables calibration using measured blanket heights instead of concentration

profiles.

* Model validation carried out in this research was based on extensive full-scale plant

data sets. The model was calibrated using data from nine steady-state clarifier loading

tests from one experimental period, and was validated against data from twenty

66








steady-state clarifier loading tests from two different experimental periods. The model

was further tested in simulations of fourteen additional loading tests from all three

experimental periods.

* Model predictions of sludge blanket heights and underflow suspended solids

concentrations were generally good. Effluent suspended solids concentration

predictions were of the correct order of magnitude but did not correlate with measured

concentrations.

* The model developed as a part of this research outperformed both total limiting solids

flux theory and the gravity-flux-constraining model in the prediction of clarifier failure

due to solids overloading.















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BIOGRAPHICAL SKETCH


Randall W. Watts graduated from Fort Pierce Central High School in Ft. Pierce,

Florida, in 1978. After serving six years in the U.S. Navy, he received an A.A. degree

with highest honors from Indian River Community College in Ft. Pierce, Florida, in May

1986. He entered the University of Florida in August 1986 and received a Bachelor of

Science degree with high honors from the Department of Chemical Engineering in August

1989. He began graduate studies at the University of Florida in the Department of

Environmental Engineering Sciences. His major area of study was wastewater treatment,

and he received a Master of Engineering degree in environmental engineering in December

1992. After completion of his doctoral degree at the University of Florida in

environmental engineering, he plans to work in industry.














I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.


Ben Koopman, Chai
Professor of Enviro mental Engineering
Sciences


I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.


Spros Svoronos, Cochairman
Professor of Chemical Engineering


I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.


Paul Chadik
Assistant Professor of Environmental
Engineering Sciences


I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.


Oscar Crisalle
Assistant Professor of Chemical
Engineering














I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.


Bill Wise
Associate Professor of Environmental
Engineering Sciences


This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.


May 1996
M Winfred M. Phillips
Dean, College of Engineering




Karen A. Holbrook
Dean, Graduate School




















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