sensitivity analysis of a one-dimensional convection-diffusion model for secondary settling tanks

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SENSITIVITY ANALYSIS OF A ONE-DIMENSIONAL CONVECTION-DIFFUSIONMODEL FOR SECONDARY SETTLING TANKSLiesbeth Verdickt a , Ilse Smets a & Jan Van Impe aa Department of Chemical Engineering , BioTeC, BioprocessTechnology and Control, Katholieke Universiteit Leuven , Leuven,BelgiumPublished online: 25 Jan 2007.

To cite this article: Liesbeth Verdickt , Ilse Smets & Jan Van Impe (2005) SENSITIVITY ANALYSIS OFA ONE-DIMENSIONAL CONVECTION-DIFFUSION MODEL FOR SECONDARY SETTLING TANKS, ChemicalEngineering Communications, 192:12, 1567-1585, DOI: 10.1080/009864490896179

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Sensitivity Analysis of a One-DimensionalConvection-Diffusion Model for Secondary

Settling Tanks

LIESBETH VERDICKT, ILSE SMETS, ANDJAN VAN IMPE

Department of Chemical Engineering, BioTeC, BioprocessTechnology and Control, Katholieke Universiteit Leuven,Leuven, Belgium

One-dimensional models describing the flow and settling process in secondary settlingtanks have great potential with respect to process control. Most of these models arebased on a conservation equation that is discretized by dividing the settler into a fixednumber of horizontal layers. In this article, the focus is on convection-diffusion mod-els, which—in contrast to other one-dimensional models—ensure mesh-independentconcentration profiles. The sensitivity of a prototype convection-diffusion model withrespect to its parameters and the loading and operational conditions is studiedby means of steady-state simulations. Based on the results, the importance of eachof the model parameters is assessed. For each combination of parameter values, load-ing conditions, and operational variables considered in the study, the minimumnumber of layers required to obtain a practically mesh-independent concentrationprofile is determined on the basis of a newly developed objective criterion. Thisanalysis leads to the identification of those factors having a large influence on thenumerical behavior of the model.

Keywords Biological wastewater treatment; Secondary settling tank; Modeling;One-dimensional model; Sensitivity analysis

Introduction

The secondary settling tank plays a crucial role in activated sludge wastewater treat-ment systems, combining the tasks of (1) compacting the sludge to be returned to thebiodegradation tanks (thickening), (2) separating sludge flocs from the treatedeffluent (clarification), and (3) sludge storage during peak flows. Failure at the levelof any of these functionalities can lead to serious process deterioration. Therefore,models that can adequately describe the flow and settling process occurring in thesecondary settler are important tools for design, control, and simulation of the acti-vated sludge system. A wide range of models have been developed during the pastdecades, progressing from low-complexity conceptual models over static and dynamicone-dimensional approaches to complex two- and three-dimensional hydrodynamicmodels.

Address correspondence to Jan Van Impe, Department of Chemical Engineering,Bioprocess Technology and Control, Katholieke Universiteit Leuven, W. de Croylaan 46,B-3001 Leuven, Belgium. E-mail: [email protected]

Chem. Eng. Comm., 192:1567–1585, 2005Copyright # Taylor & Francis Inc.ISSN: 0098-6445 print/1563-5201 onlineDOI: 10.1080/009864490896179

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Dynamic one-dimensional models are very convenient for process control and forsimulations in combination with activated sludge models since they approximate thesludge balance over the settling tank well without requiring too much computercapacity. Most of them are based on the solids flux theory introduced by Kynch(1952), which assumes that the settling process can be described entirely by a conser-vation law for the sludge concentration. To make the theory operational in computerprograms, the settler is split up into a fixed number of layers n for which the conser-vation equation is discretized. In most cases, a hyperbolic conservation equation isapplied, neglecting diffusion mixing due to turbulent fluid motion in the tank. Theresulting models, however, exhibit a serious drawback: they are unable to describeconcentration gradients in steady-state (Queinnec and Dochain, 2001). Indeed,numerical solution of the continuity equation in a mathematically rigorous way showsthat the steady-state solution is piecewise constant (Burger et al., 2004; Chancelieret al., 1994; Diehl, 1996), which is not in accordance with experimental observations.

To overcome this shortcoming, different modifications to the model have beenproposed. The most popular one is the introduction of a flux restriction at the dis-crete level (see, for example, Dupont and Dahl, 1995; Otterpohl and Freund, 1992;Tak�aacs et al., 1991). This strategy, however, results in an inconsistency with respectto the number of grid layers applied in the discretization scheme, caused by the factthat the effect of the flux restriction disappears when the number of layers tends toinfinity (Diehl and Jeppsson, 1998). A mathematically more sound way to obtainrealistic concentration profiles is extending the hyperbolic conservation equationwith a diffusion term (see, for example, Hamilton et al., 1992; Lee et al., 1999; Wattset al., 1996). The resulting convection-diffusion models are known to be consistentwith respect to the discretization accuracy, i.e., the concentration profiles becomemore accurate as the number of layers increases, until they become practically meshindependent. Krebs (1995) suggested that, when using a one-dimensional sedimen-tation model, a sensitivity study of the model outputs with respect to the numberof layers should be performed. Hamilton et al. (1992) conducted such a sensitivityanalysis for their convection-diffusion model and concluded that 24 layers are suf-ficient for their model to give reliable results. Lee et al. (1999) demonstrated thattheir convection-diffusion model gives good results when at least 20 layers are used.However, both studies are based on a limited set of loading and operational con-ditions, and, therefore, caution should be applied when using these values under dif-ferent conditions than those for which they were derived.

In this article, a profound sensitivity analysis of the steady-state behavior of aprototype convection-diffusion model with respect to the loading conditions, theoperational variables, and the model parameters is performed. The outcome is usedto assess the importance of each of the model parameters. Second, the number oflayers nmin required to obtain practically mesh-independent concentration profilesis determined for each of the simulations performed. Based on the results, factorshaving a large influence on the required number of layers are identified.

Convection-Diffusion Model

Conservation Equation

Themodel under study is a prototype convection-diffusionmodel, in which the follow-ing conservation equation describes the solids transport through the secondary settler:

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@X

@tþ @J

@z�D � @

2X

@z2¼ sðtÞ � dðz� zf Þ ð1Þ

with sðtÞ ¼ Xf �Qf

A

and z is the vertical coordinate, directed downwards,X is the solids concentration, J isthe convection flux, s(t) is a source termmodeling the feed inlet, d is the delta function,zf is the vertical coordinate of the feed inlet,Qf (t) is the flow rate of the feed,Xf (t) is thesolids concentration of the feed,A is the cross-sectional area of the settler, andD is the(constant) diffusion coefficient.

Solids Flux

The total convection flux J consists of a bulk flux Jb and a settling flux Js. The solidsflux due to bulk flow is

Jb ¼ q � X ð2Þ

where q is the bulk flow velocity. When the horizontal cross section A is assumed tobe constant over the entire depth, q is dependent only on whether the observed crosssection is in the underflow region (under the inlet level) or in the overflow region(above the inlet level), i.e.,

q ¼�qov ¼ �Qe=A in the overflow region

qun ¼ Qun=A in the underflow region

�ð3Þ

whereQe andQun are the effluent and underflow flow rates, respectively (see Figure 1).Qe and Qun can be obtained from the following equations (under the assumption thatwastage of sludge is negligible):

Qun ¼R

1þ R�Qf and Qe ¼

1

1þ R�Qf ð4Þ

with R the recycle ratio, the main operational variable of the secondary settling tank.The settling (gravitational) flux is

Js ¼ vs � X ð5Þ

where vs is the settling velocity of the sludge particles. The solids flux theory assumesthat the sludge settling velocity is dependent only on the local solids concentration X.To relate the settling velocity to the solids concentration, the double-exponentialfunction proposed by Tak�aacs et al. (1991) is applied:

vsðXÞ ¼ max½0;minðv00; v0 � expð�rh � ðX � XminÞÞ � v0 � expð�rp � ðX � XminÞÞÞ� ð6Þ

with Xmin ¼ fns � Xf

The function contains five model parameters: v0 and v00 are theoretical and practical

maximum settling velocities, fns is the fraction of non-settleable solids in the feed,

Sensitivity Analysis of a 1D SST Model 1569

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and rh and rp are parameters associated with the settling behavior in the hinderedsettling zone and at low solids concentrations, respectively.

The double-exponential settling velocity equation is an extension of the classicalVesilind equation:

vsðX Þ ¼ v0;V � expð�KV � XÞ ð7Þ

in which v0,V and KV are calibration parameters. Tak�aacs et al. (1991) extended thisequation because it overestimates the settling velocity at low solids concentrations.The settling velocity according to Equations (6) and (7) is plotted as a function ofthe solids concentration in Figure 2.

Discretization

Equation (1) is discretized by dividing the settler into a fixed number of layers n ofequal height h, within which the concentration of solids is assumed to be constant.The updates of the concentrations in the layers are made in accordance with a massbalance at the layer interfaces, resulting in the following set of n ordinary differentialequations:

. Top layer (layer 1)

h � dX1

dt¼ qov � X2 � qov � X1 � Js;1 þD � X2 � X1

hð8Þ

. i-th layer in overflow zone (2� i�m� 1)

h � dXi

dt¼ qov � Xiþ1 � qov � Xi þ Js;i�1 � Js;i þD � Xiþ1 � Xi

h�D � Xi � Xi�1

hð9Þ

Figure 1. Flow scheme of a secondary settling tank according to the one-dimensionalapproach.


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. Feed layer (layer m)

h � dXm

dt¼ Qf

A� Xf � ðqov þ qunÞ � Xm þ Js;m�1 � Js;m

þD � Xmþ1 � Xm

h�D � Xm � Xm�1

hð10Þ

. i-th layer in the underflow zone (mþ 1� i� n� 1)

h � dXi

dt¼ qun � Xi�1 � qun � Xi þ Js;i�1 � Js;i þD � Xiþ1 � Xi

h�D � Xi � Xi�1

hð11Þ

. Bottom layer (layer n)

h � dXn

dt¼ qun � Xn�1 � qun � Xn þ Js;n�1 �D � Xn � Xn�1

hð12Þ

with Js,i ¼ vs(Xi) �Xi for i ¼ 1,. . ., n� 1. The effluent concentration Xe is assumedto be equal to X1, the concentration in the top layer, and, analogously, the under-flow concentration Xun is assumed to be equal to Xn, the concentration in the bot-tom layer. These boundary conditions have been chosen because they are appliedin all but a few of the publications available on one-dimensional modeling of sec-ondary settling tanks.

Simulation Analysis

Model System

Prototype Settling TankAs a prototype secondary settling tank, the settler configuration proposed by theCOST 682 benchmark is used. The COST benchmark (Pons et al., 2001; http://www.ensic.inpl-nancy.fr/COSTWWTP/) is a fully defined simulation protocol for

Figure 2. Settling velocity vs as a function of the solids concentration according to the Vesilindand the Tak�aacs settling velocity equations.


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an activated sludge system including plant layout, simulation models, and modelparameters, originally meant for the evaluation of control strategies. The geometri-cal characteristics of the secondary settling tank are listed in Table I.

Parameter ValuesThe nominal values for the parameter values are also included in Table I. Realisticvalues for the five parameters of the settling velocity function are taken from theCOST benchmark. The nominal value for the diffusion coefficient is 13m2=d (takenfrom (Hamilton et al., 1992)).

Loading and Operational ConditionsA logical choice of loading and operational conditions would be to use the loadingand operational conditions applied in the COST benchmark. However, the bench-mark focuses on control aspects of the biological system and has therefore beenchosen as a very robust system with respect to the settling process. The range of load-ing conditions applied in this study is wider than the one proposed in the COSTbenchmark, with feed concentrations Xf ranging from 1000 to 5500 g=m3 and feedflow rates Qf from 3000 to 71000m3=d. As a result, situations in which the secondarysettling tank becomes overloaded are included in this study. The nominal valueapplied for the recycle ratio R is 1, and wastage of sludge is neglected.

Simulations

Steady-State SolutionSteady-state concentration profiles are obtained by performing a dynamic simula-tion with constant loading and operational conditions until the concentration profileno longer changes as a function of time. The initial state is a constant zero concen-tration profile. The number of layers applied in the discretization scheme is takensufficiently large (n equal to 2000) to exclude any significant influence of the number

Table I. Numerical values adopted in the simulation study

Settler configuration

Value Unit

A Cross-sectional area 1500 m2

H Total tank depth 4 mzf Vertical coordinate of the inlet 1.8 m

Nominal values of the model parameters

v0 Theoretical maximum settling velocity 474 m=dv00 Practical maximum settling velocity 250 m=drh Hindered zone settling parameter 0.000576 m3=grp Flocculent zone settling parameter 0.00286 m3=gfns Non-settleable fraction 0.00228D Diffusion coefficient 13 m2=d


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of layers on the obtained concentration profiles. As an alternative, one could try tosolve the set of steady-state (algebraic) equations for the unknown concentrations.

Determination of nmin

For each simulation performed, the number of layers nmin required to obtain a prac-tically mesh-independent concentration profile is determined by means of an objec-tive criterion. A schematic presentation of the assessment procedure is provided inFigure 3. The criterion is based on a comparison of the concentration profilesobtained for n equal to subsequent ten-folds (ntest equal to 10, 20, 30, . . .) with thereference profile obtained for n equal to 2000. The difference between the concen-tration profile for ntest and the reference concentration profile is quantified by asum of squared errors, which is calculated after a rescaling of the data. A thresholdvalue is then applied to the sum of squared errors divided by ntest-1, and nmin is

Figure 3. Schematic presentation of the criteria applied for the determination of nmin, the num-ber of layers required to obtain mesh-independent concentration profiles.


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defined as the lowest value of ntest for which this sum is smaller than the thresholdvalue.

Results

Loading Conditions

In this section, the evolution of the steady-state concentration profile as a functionof the loading conditions Xf and Qf is examined for the nominal parameter valuesand the nominal value of the recycle ratio. For each combination of Xf and Qf,the minimum number of layers required to obtain mesh-independent concentrationprofiles is determined. The results are used to assess the influence of the loading con-ditions on the required number of layers.

Influence on the Steady-State Concentration ProfileThe steady-state concentration profiles obtained as a function of the loading con-ditions are categorized into four classes based on the sludge blanket height (SBH),which is defined as the vertical position from which the solids concentration Xexceeds 2000 g=m3. The four classes are shown in Figure 4. The aim of this classi-fication is to facilitate visualization of the evolution of the steady-state concentrationprofile as a function of the loading conditions in a two-dimensional graph, asillustrated in Figure 5. When the loading conditions are very low, a so-called Type1 concentration profile is obtained: sludge accumulates only at the bottom of thetank. Somewhat higher loading conditions give rise to a profile of the second type,with sludge accumulation in the underflow zone. A Type 3 profile is obtained as soon

Figure 4. Classification of the steady-state concentration profiles based on the sludge blanketheight.


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as the sludge blanket reaches the overflow zone. Finally, when the loading conditionsare very high, a profile of the fourth type is obtained: the settling tank is overloadedand sludge leaves the tank with the effluent.

The only goal of this classification is to facilitate visualization of the evolution ofthe concentration profile as a function of the loading conditions, and the limitingvalues of the sludge blanket height applied to distinguish between the differentclasses are chosen somewhat arbitrarily. Sludge accumulation in the settling tankis considered to be small when the whole sludge blanket is within the lowest 0.5mof the tank, i.e., when the sludge blanket height is larger than 3.5m. The secondlimiting value, 1.8m, is the feed level. The third limiting value, 0.25m, is chosenbecause simulations revealed that at this value significant amounts of solids startto leave the settling tank with the effluent.

Influence on the Numerical BehaviorFor each combination of loading conditions, the minimum number of layers nmin

required to obtain a mesh-independent concentration profile is determined. Theloading conditions requiring the highest values for nmin are indicated in Figure 5using white markers, and the corresponding values of nmin are provided in the legend.It can be seen that the operational status of the secondary settling tank has a stronginfluence on nmin, the highest number of layers being required when the settler is onthe verge of becoming overloaded, i.e., at the transition from a Type 3 (dark grey onthe graph) to a Type 4 concentration profile (black on the graph). If the model is toprovide mesh-independent concentration profiles for the entire range of loading con-ditions considered, for a certain set of parameter values p0 and a certain recycle ratioR0, the highest value obtained for nmin as a function of the loading conditions has tobe retained for use in the discretization step:

nminðp0 ;R0Þ ¼ max½nminðXf ;Qf Þ�p0 ;R0ð13Þ

Figure 5. Classification of the steady-state concentration profiles as a function of the loadingconditions for the nominal parameter values. The loading conditions requiring the highestvalues for n are indicated using white markers, and the corresponding values of nmin areprovided in the legend.


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This means that for the nominal parameter values, the number of layers applied inthe discretization scheme should be at least 70.

In the following sections, the sensitivity of the steady-state solution of the modelwith respect to its parameters and the main operational variable, the recycle ratio R,is addressed by means of figures similar to Figure 5. In addition, Table II providesthe sensitivities of the effluent and underflow concentrations with respect to themodel parameters and the recycle ratio. In this table, a distinction is made betweenTypes 1, 2, and 3 and Type 4 concentration profiles as the sensitivities of the effluentand underflow concentrations depend strongly on the fact whether the sludgeconcentration in the effluent is significant (Type 4 profiles) or not (Types 1, 2, and3 profiles).

Diffusion Coefficient

Influence on the Steady-State Concentration ProfileFigure 6 illustrates the evolution of the steady-state concentration profile as a func-tion of the loading conditions for four values of the diffusion coefficient D. Compar-ing the graphs reveals that the transition from a Type 1 to a Type 4 concentrationprofile, i.e., from a highly underloaded to an overloaded state, becomes more abruptand occurs at higher loading conditions if the value of D is decreased. The effect ofthe diffusion coefficient on the steady-state concentration profile is illustratedfurther in Figure 7, in which the evolution of the steady-state profile as a functionof D is presented for Xf equal to 5000 g=m3 and Qf equal to 37000m3=d. Figure 7clearly illustrates that the sludge blanket height declines if the value of the diffusioncoefficient is decreased. In the ultimate case where D equals zero, a piecewise con-stant concentration profile of Type 1 is obtained since the parabolic conservationequation reduces to a hyperbolic equation, which cannot describe concentrationgradients in steady-state.

As can be concluded from Table II, the diffusion coefficient strongly influencesthe effluent concentration Xe, no matter what the operational status of the settlingtank. The underflow concentration Xun, however, is influenced significantly onlyby the value of the diffusion coefficient for Type 4 profiles. This insensitivity of

Table II. Sensitivity of the effluent concentrationXe and the underflow concentrationXun for the model parameters and the recycle ratio

Xe (¼X1) Xun (¼Xn)

Type 4 Types 1, 2, 3 Type 4 Types 1, 2, 3

D þþ þþ þ 0v0 þþ þþ þ 0rh þþ þ þþ 0rp 0 þþ 0 0fns 0 þþ 0v00 0 0 0 0R þþ þ þ þþ

þþ ¼ large influence, þ ¼ moderate influence, 0 ¼ negligible influence.


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Figure 6. Type of steady-state concentration profile as a function of the loading conditionsfor four values of D. In each graph, the loading conditions requiring the highest values for nmin

are indicated using white markers, and the corresponding values of nmin are provided inthe legend.

Figure 7. Evolution of the steady-state concentration profile as a function of the diffusion coef-ficient D for Xf ¼ 5000 g=m3 and Qf ¼ 37000m3=d. The values applied for D are indicated.


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the Types 1, 2, and 3 profiles with respect to the diffusion coefficient can beexplained by examining the following steady-state mass balance over the settlingtank:

Qf � Xf ¼ Qe � Xe þQun � Xun

¼ 1

1þ R�Qf � Xe þ

R

1þ R�Qf � Xun ð14Þ

For Types 1, 2, and 3 concentration profiles, the first term on the right-hand side canbe neglected because Xe is very small compared to Xun, leading to the followingexpression for the underflow concentration:

Xun ¼1þ R

R� Xf ð15Þ

It can be concluded that the underflow concentration for Types 1, 2, and 3 concen-tration profiles is influenced significantly only by the feed concentration Xf and theoperational variable R.

Influence on the Numerical BehaviorIn Figure 6, the highest values obtained for nmin as a function of the loading con-ditions are indicated. The previous observation that those loading conditions forwhich the settler is on the verge of becoming overloaded require the highest numberof layers clearly remains valid if the value of the diffusion coefficient is changed. Foreach D value considered, the minimum number of layers required to obtain mesh-independent concentration profiles for the entire range of loading conditions isdetermined according to Equation (13). The resulting (D, nmin)-couples are plottedin Figure 8. Clearly, the lower the diffusion coefficient is, the higher the numberof layers has to be to ensure mesh-independent concentration profiles. An excellentdescription of the observed relationship between nmin and D is given by the followingexponential function:

nmin ¼ 75þ expð�1:409 �Dþ 6:12Þ ð16Þ

Note that for D ¼ 8m2=d, one combination of loading conditions gives rise to anextremely high value for nmin: nmin of 150 for Xf ¼ 5000 g=m3 and Qf ¼ 40400m3=d.

Figure 8. nmin as a function of the diffusion coefficient for the range of loading conditions con-sidered in this study.


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This value was not considered in the determination of nmin as a function of D sinceapplying a finer grid to the (Xf, Qf) region around it reveals that values higher than80 occur only in a very narrow zone at the transition from a Type 3 to a Type 4concentration profile (see Figure 9).

Settling Velocity Parameters

In this section, the sensitivity of the steady-state model behavior with respect to theparameters of the Tak�aacs settling velocity function (Equation (6)) is examined. Foreach of these parameters, simulations have been performed with 0.75, 1, and 1.25times its nominal value.

Influence on the Steady-State Concentration ProfileVesilind Related Parameters. The parameters v0 and rh of the Tak�aacs

settling velocity function correspond to the Vesilind parameters v0,V and KV (seeEquation (7)). Figure 10 shows that both of these parameters have a large influenceon the settling velocity for the entire range of solids concentrations occurring in thesettling tank. As a result, the influence of v0 and rh on the evolution of the steady-state concentration profile as a function of the loading conditions (see Figure 11)is considerable: changing the value of v0 and especially of rh clearly induces a largeshift in the loading conditions for which the settler becomes overloaded. Similarly,Table II reveals the high sensitivity of the steady-state behavior for v0 and rh: bothparameters influence the effluent concentration for all types of concentration profiles

Figure 9. Application of a finer grid to (a part of) the transition region from Type 2 to Type 4concentration profiles for D ¼ 8m2=d, in which the observed outlier for nmin (Xf ¼ 5000 g=m3,Qf ¼ 40400m3=d) is situated. The loading conditions requiring the highest values for n areindicated with white markers and the corresponding values are provided in the legend (valueshigher than 80 are indicated directly in the graph).


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and the underflow concentration for Type 4 profiles. To explain the insensitivity ofthe underflow concentration for Types 1, 2, and 3 concentration profiles, the sameline of reasoning can be followed as above for the insensitivity with respect to thediffusion coefficient (see Equation (15)).

Parameters Added By Tak�aacs. The parameters rp, fns, and v00 were introduced

into the Vesilind equation by Tak�aacs et al. (1991) to obtain a realistic estimationof the settling velocity for low solids concentrations. Figure 12 shows their influence

Figure 11. Sensitivity of the evolution of the steady-state concentration profile as a function ofthe loading conditions with respect to v0 (top) and rh (bottom). In each graph, the loading con-ditions requiring the highest values for nmin are indicated.

Figure 10. Sensitivity of the settling velocity with respect to v0 (left) and rh (right).


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on the evolution of the settling velocity as a function of the solids concentration. It isclear from the graphs that they influence the settling velocity significantly only at lowsolids concentrations. As a result, their impact on the evolution of the steady-state

Figure 12. Sensitivity of the settling velocity with respect to rp (top), v00 (middle), and fns

(bottom). The graph for fns zooms in on the low concentration range because the three curvesare indistinguishable on a plot of the entire concentration range.


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Figure 13. Sensitivity of the evolution of the steady-state concentration profile as a function ofthe loading conditions with respect to the recycle ratio. In each graph, the loading conditionsrequiring the highest values for nmin are indicated.


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profile as a function of the loading conditions is negligible (results not shown).Nevertheless, the importance of rp and fns becomes clear by inspection of Table II:rp and fns have a strong effect on the effluent concentration Xe as long as Xe remainslow (i.e., for Types 1, 2, and 3 concentration profiles) because of their influence onthe settling velocity for very low solids concentrations. For v0

0, on the other hand, nosignificant influence on the effluent or underflow concentration could be detected.This is caused by the fact that v0

0 influences the settling velocity for solids concentra-tions only near the top of the double-exponential function, a concentration rangethat is not strongly represented in the secondary settling tank (the concentrationprofiles observed are usually very steep as a function of z in this concentration range).

Influence on the Numerical BehaviorOnce again, the highest number of layers is required when the settler is on the verge ofbecoming overloaded. The settling velocity parameters have no significant influenceon the number of layers required to describe the entire range of loading conditions(see Figure 11 for v0 and rh).

Recycle Ratio

Influence on the Steady-State Concentration ProfileThe recycle ratio is the most important operational variable of secondary settlingtanks and as such is the most interesting candidate for use as a control variable forthe settling process. Therefore, the sensitivity of the steady-state behavior of the modelwith respect to the recycle ratio R is assessed. In the decoupled case considered in thisstudy, the recycle ratio solely quantifies the ratio of the underflow to the overflow flowrate (see Equation (4)), whereas in a full plant, changing the recycle ratio would alsoinfluence the feed flow rate and feed concentration to the settling tank.

The evolution of the steady-state profile as a function of the loading conditionsfor 0.75, 1, and 1.25 times the nominal value of R is presented in Figure 13. Increasingthe value of the recycle ratio induces a large shift upwards of the loading conditionsfor which the settler becomes overloaded. This was to be expected, since for a largerR, a larger part of the inlet flow is directed towards the bottom and a smaller parttowards the top of the settler (see Equation (4)). The high sensitivity of thesteady-state behavior with respect to the recycle ratio is illustrated further by Table II:R influences the effluent and underflow concentrations for all types of concen-tration profiles. In contrast to the model parameters, R influences the underflowconcentration for Types 1, 2, and 3 concentration profiles, which was to be expectedfrom Equation (15).

Influence on the Numerical BehaviorOnce more, it is clear that the highest number of layers is required when the settler ison the edge of becoming overloaded (see Figure 13). The recycle ratio has no signifi-cant effect on the number of layers required to obtain mesh-independent concen-tration profiles for the entire range of loading conditions considered. One outlier(n ¼ 100 for R ¼ 1.25) is not considered because zooming in on the region aroundit reveals that it lies in a very narrow zone in which high nmin values are obtained(results not shown).


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Conclusions

In this article, the sensitivity of the steady-state behavior of a prototype convection-diffusion model for secondary settling tanks with respect to its model parameters andthe loading and operational conditions has been assessed. The steady-state concen-tration profiles are shown to be highly sensitive with respect to the diffusioncoefficient, so calibration of this parameter is of utmost importance when usingthe model. This explains why calibration of the diffusion coefficient has received a lotof attention in the literature (see, for example, Hamilton et al., 1992; Lee et al., 1999;Watts et al., 1996). The sensitivity of the model with respect to the parameters of theTak�aacs settling velocity function has been examined. A distinction has been madebetween the parameters corresponding to the Vesilind parameters and the extraparameters introduced by Tak�aacs et al. (1991) when they extended the Vesilind equa-tion. The parameters corresponding to the Vesilind parameters, v0 and rh, stronglyinfluence the mass balance over the settling tank. Changing the value of one of theseparameters induces a large shift in the loading conditions for which the settlerbecomes overloaded. The parameters induced by Tak�aacs et al., on the other hand,hardly influence the evolution of the operational status of the settling tank as a func-tion of the loading conditions. Nevertheless, two of them, rp and fns, have a pro-nounced effect on the effluent concentration Xe as long as Xe remains low. Theinfluence of the remaining settling parameter v0

0 on the model output is insignificant.These results indicate that the Vesilind equation will suffice to describe the settlingvelocity of the sludge when the aim of the model is to predict when the tank becomesoverloaded and=or to predict the underflow concentration. The latter might be thegoal when the model is used in combination with an activated sludge model. TheTak�aacs settling velocity is the better option if prediction of the effluent concentrationis required as well, although it might be suggested to omit v0

0 from the equation.After addressing the sensitivity of the model output with respect to the model para-meters, its sensitivity with respect to the main operational variable, the recycle ratio,has been considered. The recycle ratio is shown to have a strong influence on themass balance over the settler and on the effluent and underflow concentration, mak-ing it a prime candidate for use as a control variable.

Finally, the numerical behavior of the model has been studied. The convection-diffusion models are known to give concentration profiles that are practically mesh-independent if the number of layers applied in the discretization step is equal to orgreater than a certain number nmin. The required number of layers nmin has beendetermined for each combination of loading conditions, parameter values, and oper-ational variables using an objective criterion. The value of nmin is shown to be highlysensitive to the operational status of the settler, the highest number of layers beingrequired when the settler is on the verge of becoming overloaded. In addition, it isillustrated that nmin decreases exponentially as a function of the diffusion coefficient.Therefore, caution has to be taken when using values for nmin proposed in literaturebased on simulations for a certain secondary settling tank with a certain parameterset, for a limited range of loading conditions (see, for example, Hamilton et al., 1992;Lee et al., 1999). When implementing a one-dimensional sedimentation model, it isadvisable to determine nmin for the settling tank under study and the relevant rangeof loading conditions.

The study performed provides a clear view on the influence of each of the modelparameters separately on the steady-state model output. An interesting prospect for


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further research is to examine the effect of changing combinations of parametersrather than changing them one after another. Due to the highly nonlinear characterof the model, and especially of the settling velocity function, correlations between theparameters may arise. As a result, the effect of changing combinations of parametersmight be considerably different from the superposition of the individual effects.

Acknowledgments

Liesbeth Verdickt is a research assistant and Ilse Smets is a postdoctoral fellow, bothwith the Fund for Scientific Research-Flanders (FWO). Work was supported in partby Projects OT=99=24, OT=03=30, and IDO=00=008 of the Research Council of theKatholieke Universiteit Leuven and the Belgian Program on Inter-university Polesof Attraction, initiated by the Belgian Federal Science Policy Office. The scientificresponsibility is assumed by its authors.

References

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Chancelier, J. P., Cohen de Lara, M., and Pacard, F. (1994). Analysis of a conservation PDEwith discontinuous flux: A model of settler, SIAM J. Appl. Math., 54(4), 954–995.

Diehl, S. (1996). A conservation law with point source and discontinuous flux function mod-elling continuous sedimentation, SIAM J. Appl. Math., 56(2), 388–419.

Diehl, S. and Jeppsson, U. (1998). A model of the settler coupled to the biological reactor,Water Res., 32(2), 331–342.

Dupont, R. and Dahl, C. (1995). A one-dimensional model for a secondary settling tankincluding density current and short-circuiting, Water Sci. Technol., 31(2), 215–224.

Hamilton, J., Jain, R., Antoniou, P., Svoronos, S. A., Koopman, B., and Lyberatos, G.(1992). Modeling and pilot-scale experimental verification for predenitrification process,J. Environ. Eng., 118(1), 38–55.

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dynamics of complex biological processes, AIchE J., 45(10), 2245–2268.Otterpohl, R. and Freund, M. (1992). Dynamic models for clarifiers of activated sludge plants

with dry and wet weather flows, Water Sci. Technol., 26(5–6), 1391–1400.Pons, M., Copp, J., Jeppsson, U., and Spanjers, H. (2001). Benchmarking the evaluation of

control strategies in wastewater treatment: The four-year story of the cost benchmark,in Computer Applications in Biotechnology, eds. D. Dochain and M. Perrier, CAB8,231–237, International Federation of Automatic Control, Kidlingten, U.K.

Queinnec, I. and Dochain, D. (2001). Modelling and simulation of the steady-state of second-ary settlers in wastewater treatment plants, Water Sci. Technol., 43(7), 39–46.

Tak�aacs, I., Patry,G.G., andNolasco,D. (1991).Adynamicmodel of the clarification-thickeningprocess, Water Res., 25(10), 1263–1271.

Watts, R. W., Svoronos, S. A., and Koopman, B. (1996). One-dimensional modeling ofsecondary clarifiers using a concentration and feed velocity-dependent dispersioncoefficient, Water Res., 30(9), 2112–2124.


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sensitivity analysis of a one-dimensional convection-diffusion model for secondary settling tanks

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