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ONEDIMENSIONAL MODELING OF SECONDARY CLARIFIERS USING A CONCENTRATION AND FEED VELOCITYDEPENDENT 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 #C912237. 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 #EEC9402989, 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 ONEDIMENSIONAL MODELING OF SECONDARY CLARIFIERS USING A CONCENTRATION AND FEED VELOCITYDEPENDENT DISPERSION COEFFICIENT .. .. 4 Introduction . . . 4 Initial Model Development ........... .......... 9 Application of Clarifier Model to the Pflanz Data ... .. ... 19 Experimental Measurements on a FullScale Clarifier .. 22 Application of Clarifier Model to KWRF Data and Further Model Development ............. .... 26 Conclusions ................... ........ 40 3 CALIBRATION OF A ONEDIMENSIONAL 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 21 Performance of models with concentrationdependent dispersion functions in comparison to the Takacs et al. (1991) model, as applied to Pflanz data ........ ........... ..... 20 22 Estimated parameters for 50layer model with concentration dependent dispersion functions when fitted to Pflanz data .. .. ...... 21 23 Operational variables for KWRF loading tests which achieved steady blanket levels ..... ................ 30 24 Results for models with Dma constant across all cases and for the Dmx model with Dmax fitted individually for each case ... ... 34 25 Parameters resulting from FVDDma and FVDDmaxCitP fit across nine KWRF cases for which steady blanket levels were obtained ...... .. 37 26 Results for FVDDax model fit across the nine KWRF cases for which steady blanket levels were obtained ... ..... .... 38 27 Comparison of model predictions to clarifier loading test results ...... .. 39 31 Comparison of predicted effluent concentrations, RAS concentrations, and sludge blanket heights to measured values for test period C ...... ........... ................. 56 32 Comparison of predicted effluent concentrations, RAS concentrations, and sludge blanket heights to measured values for test periods A and B .. ........... ......... 59 33a Comparison of model predictions of success and failure to test results for test period A ................. ......... .. 61 33b Comparison of model predictions of success and failure to test results for test period B ............ ..62 33c Comparison of model predictions of success and failure to test results for test period C .... .. ........... 63 LIST OF FIGURES Figure ge 21 Example of clarifier concentration profile obtained using total limiting flux constraint .................. ............... 6 22 Comparison of 10 and 20layer versions of the Takacs et al. (1991) model using parameters estimated for the 10layer version of the model applied to case 1 of the Pflanz data .......... . 10 23 Comparison of 10 and 50layer versions of the Takacs et al. (1991) model, with optimal parameters estimated for each version applied to case 1 ofthe Pflanz data .. .................. .. 10 24 Comparison of concentration profiles obtained by 20layer 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 10layer model of Takacs et al. (1991) were employed to generate model fits.. ................. 14 25 Clarifier geometry used for initial model development. Clarifier is divided into n layers .................. .. 17 26 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 27 Schematic diagram of fullscale secondary clarifier at the Kanapaha Water Reclamation Facility, showing shroud, bottom conical section and model discretization ........... ............... .. 27 28 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 29 Comparison of fits achieved using Dmax, FVDDmax, and FVDDmx CcritP models. Data are from nine loading tests on a fullscale 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 FVDDmaxCctP model) ......... .. 33 210 Variation of Dx with feed velocity. Data points were estimated on a casebycase basis using the Dmax model. The line represents the fit of the proposed feed velocitydependent expression for Dm to the computed Dmax values. ................. .. .. 35 31 Schematic diagram of clarifier at KWRF .. .......... .. .. 46 32 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 33 Comparison of predicted blanket heights to measured values from test period C. .. ....... .................... ..... 57 34 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 ONEDIMENSIONAL MODELING OF SECONDARY CLARIFIERS USING A CONCENTRATION AND FEED VELOCITYDEPENDENT DISPERSION COEFFICIENT By RANDALL W. WATTS May, 1996 Chair: Ben Koopman Cochair: Spyros A. Svoronos Major Department: Environmental Engineering Sciences A onedimensional 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 fullscale 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 gravityflux 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 gravityfluxconstraining 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 thirtyseven. Two experimental successes were predicted as clarifier failures, whereas one experimental failure was predicted as a success. In contrast, a gravityfluxconstraining 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. Fullscale 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 steadystate 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 steadystate 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 velocitydependent 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 fluxconstraining 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 steadystate 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 steadystate 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 gravityflux constraining model. CHAPTER 2 ONEDIMENSIONAL MODELING OF SECONDARY CLARIFIERS USING A CONCENTRATION AND FEED VELOCITYDEPENDENT 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 solidshandling 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 solidshandling capacity than the layer above, its thickness will decrease or remain infinitesimal. The layer with the lowest total solidshandling 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 onedimensional, 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 steadystate concentration profiles having four distinct values in the clarifier (Fig. 21). 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. 21. 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 predenitrification 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 nonrigid 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 fullscale data set of Pflanz (1969). In the present study, it is shown that this approach can be reinterpreted as modeling with a concentrationdependent 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 concentrationdependent 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 fullscale 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 twodimensional data sets to onedimensional 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. 22). 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. 23). 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(CiCmi)) exp(bp(CiCmin))] Vma } 1E5 1E4' 1E3 0 lE2, 0 1El 1EO 0.5 1 0.5 1 Depth, m Fig. 22. Comparison of 10 and 20layer versions of the Takics et al. (1991) model using parameters estimated for the 10layer version of the model applied to case 1 of the Pflanz data. 1E5 1E4 1E31 1E2 1El t 1EO Depth, m Fig. 23. Comparison of 10and 50layer 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,i1 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.Ci1 Vs,iCi D.i1, (CiCi.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 i1 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 concentrationdependent 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 24 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 10layer 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 concentrationdependent dispersion coefficient (eq. (8)). Since Cmin, exp(bp (CiCmni), 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 1EOi 0 0.5 1 1.5 2 2.5 Depth, m Fig. 24. Comparison of concentration profiles obtained by 20layer 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 10layer 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,+11) 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(CCnt) is to eliminate a discontinuity corerr) in the slope at C = Coit. The complete model equations as applied to a cylindrical clarifier (Fig. 25) 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 IQ Ce e feed layer Q Q Cu u Fig. 25. 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,ilCi. V,,iCi Dil,i (CiCi1)/5z + Dii+1 (Ci+iCi)/Sz For the layer receiving clarifier feed: 6z dCi/dt = (Qf/A) Cf Uq Ci UbCi + V,,iCi1 V,,iCi Di,,i (CiCil)/6z + Di,i+1 (Ci+lCi)/6z (15) (16) For the ith layer in the thickening zone: 5z dCi/dt = Ub Cii Ub Ci + V,,i,Ci. V,iCi Di,i (CiCi1)/8z + Di,i+ (Ci,+Ci)/6z (17) And finally for the bottom layer: 5z dCi/dt = Ub Ci1 UbCi + Vs,i1Ci1 Dil,i (CiCil)/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 21 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 10layer 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 DmaxCrit model. * Three dispersion parameters estimated (Dmx, Cnt and 0). We refer to this as the Dmx CoritP model. As can be seen from Table 21, 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 10layer 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 50layer discretization (Fig. 23). Some improvement is obtained by the DmaxCt model (SSREs 0.085, 0.060 and 0.056), while Table 21. Performance of models with concentrationdependent dispersion functions in comparison to the Takacs et al. (1991) model, as applied to Pflanz data Takacs et al. (1991) Model with concentrationdependent dispersion term 10 layers 50 layers DmaxCcritB model DmaxCcrit 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 22. Estimated parameters for 50layer model with concentrationdependent dispersion functions when fitted to Pflanz data D.mCcit0 model DmaxCrit 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 DmaxCcritP model offers no further improvement. The estimated parameters are given in Table 22. A 10layer 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 DaxCrit model was employed (data not shown). As expected, these SSREs were higher than obtained with the 50layer discretization. Experimental Measurements on a FullScale 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 LudzackEttinger 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 multiplepipe suction system with four drawoffs 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 inline valves between the distribution box and the four clarifiers. Flow adjustments were made with reference to a handheld 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 810 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 steadystate was achieved as judged from the blanket height, data collection was continued for at least two hours into the steadystate 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 waterbath enclosed sixcolumn 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 multicolumn 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. 26) 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. 26. 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 60minute 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 60minute 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. 27) 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+p2=n+p3 Qrn+p2 A n+p2 = i = n+p Fig. 27. Schematic diagram of fullscale 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 crosssectional area from the top to the bottom of the layer multiplied by the underflow velocity: Qr,i = (AiAi,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 flowweighted 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 crosssectional 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 crosssection below the shroud (Fig. 27). 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 steadystate 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 23). 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 28a, whereas Figure 28b 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 leastsquares 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 steadystate, 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 23. 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. 28. 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 29 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 24) was 29.2. The DmxCerit model and the DmaxCerit3 model gave somewhat better fits overall (SSELCs of 24.9 and 21.8, respectively), but still performed poorly in cases of high loading (Table 24). 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 24, 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 210 shows the individually fitted Dmax versus feed velocity Vf (= Qf/A). It is observed that the equation SDi + y(VfVf, )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 IEO0 1E5 I 1E5 IES IE 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 ofFVDDmaxCcntb 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. 29. Comparison of fits achieved using Dmax, FVDDmax, and FVDDmaxCritb 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 ofFVDDmaxCritb model) "I I ase Casel Case 3 Table 24. 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 DnCit model DmaxCcrti 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. 210. Variation of Dmx with feed velocity. Data points were estimated on a casebycase basis using the Dmax model. The line represents the fit of the proposed feed velocitydependent 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 feedvelocity 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 velocitydependent Dx (FVDDmax) and FVDDmxCft0 models, respectively. Fits of the FVDDmax and FVDDmaxCrit0 models to the KWRF data were carried out using SSELC as the objective function (see Table 25 for parameter values). These fits are shown by the dotted (FVDDmax) and solid (FVDDaxCritP) lines in Figure 29. 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 FVDDmxCcitP model. Table 26 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 FVDDmaCit model was 4.34 (data not shown). For comparative purposes, a fit was also carried out on the KWRF steadystate 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 25. Parameters resulting from FVDD!, and FVDDmaxCrtP fit across nine KWRF cases for which steady blanket levels were obtained Parameters Model FVDDmx ,FVDDmCritb 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 26. 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 27. 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 CiP 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 steadystate sludge blanket level was observed; Failure indicates that the sludge blanket continued rising throughout the test FVDDmaCcitP models correctly predicted failure for the five cases in which a continually rising blanket was observed experimentally and also predicted success for the "tenth" steadystate case for which a concentration profile was not available. Table 27 compares the predictions of the FVDDma and FVDDaxCcit0 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 concentrationdependent 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 ONEDIMENSIONAL CLARIFIER MODEL USING SLUDGE BLANKET HEIGHTS Introduction There has been considerable work on developing easytouse 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(CCent)]exp[/f(CC,,,)] for C > C,( D(C) DforC 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 concentrationdependent 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(CiCmn)) exp(bp(CiCmi))] 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 31. 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 crosssectional area available for overflow (Ak) in this region is 72% of the crosssection 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+p2 Qn+p3 "Q r n+p2 i = n+p Fig. 31. Schematic diagram ofclarifier at KWRF. ~"~""~"""~"~~ QMMMM An~p2 1 An~p Qrn+p Cn+p were withdrawn from a single layer. As Figure 31 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 crosssectional area from the top to the bottom of the layer multiplied by the underflow velocity: Qr,i = (AiAi+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 flowweighted 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 (CiCi.l)/6z + Di+i (Ci+1Ci)/8z (6) For the layer receiving clarifier feed: 6z dCi/dt = (Uq+Ub) Cf Uq Ci UbCi + V.,i.Ci V,,iCi Di1,i (CiCil)/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.Cil Vs,iCi Di,i (CiCi.)/6z + Di,i+l (Ci+1Ci)/6z (8) And finally for the bottom layer: 6z dCi/dt = Ub C1 UbCi + V,i,Ci. Dil,i (CiCil)/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 waterbath enclosed sixcolumn 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 multicolumn 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 steadystate 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 steadystate conditions. Mass balance closure errors under steadystate 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 steadystate 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). Steadystate RAS concentrations ranged between 6330 and 14600 mg/L. Model Calibration The clarifier model was calibrated using steadystate 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 102 L/mg, Di = 4.835 m2/d, y = 2.500 x 102 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. 32). As Table 31 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 32 by vertical lines, light solid for modelcalculated 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. 33). 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 32) 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 31. 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) (%) C1 6.0 8877 1.09 2.7 55.2 8892 0.2 1.07 1.7 C2 5.9 9890 1.37 3.0 49.9 10181 2.9 1.18 14.1 C3 5.6 14592 1.88 3.5 38.3 16292 11.7 1.93 3.0 C4 5.2 11534 1.77 3.5 32.8 12257 6.3 1.70 4.4 C5 4.4 10890 1.68 3.4 21.8 10883 0.1 1.47 12.6 C6 6.1 9984 1.78 3.6 40.3 10581 6.0 1.70 4.3 C7 5.4 10738 2.00 4.5 17.2 11907 10.9 2.03 1.3 C8 6.9 8752 1.71 3.8 44.9 9142 4.5 1.70 0.4 C9 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. 33. 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. 34. 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 32. 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) A1 7.6 11358 1.86 4.5 11589 1.95 A2 7.2 6691 1.35 4.0 7347 1.19 A3 5.4 8313 1.41 4.3 8872 1.39 A4 6.9 8313 1.97 4.4 9138 1.54 A5 7.2 7152 1.56 4.3 7859 1.29 A6 8.7 6332 1.59 4.3 7404 1.29 A7 6.9 9790 2.29 5.1 10219 2.18 A8 5.9 9075 2.01 4.9 9692 1.86 A9 3.0 8806 2.16 4.9 9525 1.79 B1 3.6 8383 1.11 3.2 7830 1.15 B2 3.9 9155 1.52 3.8 10376 1.49 B3 3.6 9679 1.98 4.8 11186 2.00 B4 3.2 9681 2.36 4.8 11165 2.00 B5 1.9 10170 2.23 4.8 10975 1.99 B6 2.7 9271 1.21 3.2 10152 1.25 B7 3.6 12788 2.22 40.8 15340 4.34 B8 1.9 7774 2.08 5.4 9216 2.02 B9 1.6 7324 1.89 5.0 7721 1.77 B10 2.5 9040 2.23 5.4 9053 2.01 B11 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. 34). Table 32 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 B7, was 3.2 5.4 mg/L with a median (including B7) 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 33a, 33b, and 33c 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 33 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 33a. 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) (%) A1 17.4 8.7 3893 102 Y 1.86 Y 4.5 1.95 Y 4.65 1.24 Y 86.7 A2 11.5 14.2 4071 105 Y 1.34 Y 4.0 1.19 Y 4.04 0.95 Y 63.2 A3 17.0 11.5 3611 104 Y 1.40 Y 4.4 1.39 Y 4.51 1.05 Y 71.7 A4 17.3 14.4 4166 132 Y 1.98 Y 4.4 1.54 Y 4.56 1.06 Y 78.9 A5 17.0 14.3 3622 114 Y 1.55 Y 4.3 1.29 Y 4.49 0.96 Y 67.8 A6 17.0 17.2 3739 128 Y 1.58 Y 4.3 1.29 Y 4.49 0.96 Y 68.0 A7 22.7 14.3 3971 148 Y 2.29 Y 5.1 2.18 Y 5.35 1.33 Y 88.1 A8 22.6 14.3 3789 140 Y 2.01 Y 4.9 1.86 Y 5.22 1.24 Y 83.6 A9 22.6 14.4 3731 138 Y 2.16 Y 4.9 1.79 Y 5.19 1.18 Y 82.3 A10 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 A12 22.9 14.4 4043 151 I (2.59) 0.042 Y 5.2 2.44 Y 5.44 1.42 Y 90.1 A13 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 crosssectional 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 33b. 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) (%) B1 17.3 14.4 3573 114 Y 1.1 Y 3.2 1.15 Y 3.4 0.96 Y 55.1 B2 22.6 14.3 4049 150 Y 1.5 Y 3.8 1.49 Y 4.1 1.15 Y 72.9 B3 28.5 14.4 3777 162 Y 2.0 Y 4.8 2.00 Y 5.0 1.24 Y 78.7 B4 28.7 14.4 3757 162 Y 2.4 Y 4.8 2.00 Y 5.0 1.24 Y 78.6 B5 28.8 14.4 3680 159 Y 2.2 Y 4.8 1.99 Y 5.0 1.24 Y 77.2 B6 17.5 8.6 3388 89 Y 1.2 Y 3.2 1.25 Y 3.4 0.86 Y 61.9 B7 28.4 8.6 3645 135 Y 2.2 N 40.8 4.34 Y 5.4 1.40 Y 94.2 B8 28.3 20.1 3841 187 Y 2.1 Y 5.4 2.02 Y 4.9 1.17 Y 73.4 B9 28.6 20.1 3205 156 Y 1.9 Y 5.0 1.77 Y 4.8 1.09 Y 61.5 B10 28.7 20.1 3744 183 Y 2.2 Y 5.4 2.01 Y 4.9 1.20 Y 72.1 B11 16.8 14.3 3978 124 Y 1.2 Y 3.2 1.19 Y 3.4 0.96 Y 56.9 B12 34.5 20.1 3744 205 N (3.2) 0.337 N 289 4.45 Y 6.0 1.42 Y 80.6 B13 34.2 20.1 3698 201 N (3.2) 0.178 N 195 4.45 Y 5.9 1.34 Y 79.2 B14 31.4 20.1 3775 195 I (3.1) 0.095 Y 8.1 3.02 Y 5.4 1.25 Y 76.7 B15 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 crosssectional 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 33c. 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/m2d) (m) (m/hr) (mg/L) (m) (mg/L) (m) (%) C1 17.3 14.3 4053 129 Y 1.1 Y 2.9 1.09 Y 3.1 0.96 Y 56.2 C2 22.7 14.4 3972 148 Y 1.4 Y 3.3 1.25 Y 3.6 1.06 Y 64.4 C3 28.7 8.6 3801 142 Y 1.9 Y 4.0 2.05 Y 4.6 1.21 Y 90.2 C4 28.5 14.3 4130 177 Y 1.8 Y 4.0 1.79 Y 4.3 1.24 Y 77.5 C5 28.6 14.3 3664 158 Y 1.7 Y 4.0 1.54 Y 4.3 1.24 Y 68.8 C6 28.6 17.2 3994 183 Y 1.8 Y 4.2 1.73 Y 4.3 1.24 Y 70.7 C7 34.0 14.3 3560 173 Y 2.0 Y 5.1 2.03 Y 5.0 0.93 Y 75.3 C8 28.5 20.0 3787 184 Y 1.7 Y 4.3 1.69 Y 4.3 1.15 Y 64.3 C9 37.1 14.4 3444 178 Y 3.2 Y 7.2 2.98 Y 5.5 0.94 Y 77.5 C10 28.2 14.4 3885 166 Y 1.8 Y 3.9 1.62 Y 4.3 0.95 Y 72.3 C11 37.2 14.3 4044 209 N (3.7) 0.421 N 400 4.45 Y 5.8 1.76 Y 91.3 C12 39.9 14.4 3444 187 N (3.1) 0.379 N 77.8 4.44 Y 6.0 1.02 Y 81.7 C13 37.2 20.1 3987 229 N (3.7) 0.353 N 181 4.44 Y 5.6 1.51 Y 79.7 C14 37.3 25.9 3983 252 N (3.4) 0.444 N 211 4.44 Y 5.5 0.95 Y 79.0 C15 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 crosssectional 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 A11 and B7), whereas one experimental failure was predicted as a success (case C15), 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 concentrationdependent 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 steadystate fullscale 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 fullscale plant data sets. The model was calibrated using data from nine steadystate clarifier loading tests from one experimental period, and was validated against data from twenty 66 steadystate 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 gravityfluxconstraining model in the prediction of clarifier failure due to solids overloading. REFERENCES Andersen, N.E. (1945) Design of final settling tanks for activated sludge. Sewage Works J. 17, 5065. Anderson, H.M. and Edwards, R.V. 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(1968) Design of prototype thickeners from batch settling tests. Wat. Sew. Works 115, 7, 302307. Vitasovic, Z.Z. (1986) An Integrated Control Strategy for the Activated Sludge Process. Ph.D. dissertation, Rice University, Houston, TX. Vitasovic, Z.Z. (1989) Continuous settler operation: a dynamic model. In Patry, G.G. and Chapman, D. (Eds.), Dynamic Modeling and Expert Systems in Wastewater Engineering. Lewis Publishers, Chelsea, MI, pp. 5981. Wahlberg, E.J. and Keinath, T.M. (1986) Development of settling flux curves using SVI. Paper presented at 59th Ann. Conf., Wat. Pollut. Control Fed., Los Angeles, CA. 71 White, M.J.D. (1976) Design and control of secondary settlement tanks. Wat. Pollut. Control 75, 459467. WPCF (1985) Clarifier Design. Wat. Pollut. Control Fed., Washington, D.C., MOP FD8. Yoshioka, N., Hotta, Y., Tanaka, S., Naito, S, and Tsugami, S. (1957) Continuous thickening of homogenous flocculated slurries. Chem. Eng. (Tokyo) 21: 6674. 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 LD 1780 199k UNIVERSITY OF FLORIDA II I1262II II I08555 0597 3 1262 08555 0597 
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