settler jces93

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Cwil.Eng. Syst, Vol. 10, pp. 207-224reprints available directly from the publisherPhotocopying permitted by license only

0 1993 Gordon and Breach Science Publishers S. A.Printed in theUnited States of America

DYNAMIC MODELLING OF SEDIMENTATIONIN THE ACTIVATED SLUDGE PROCESS

S. MARSILI-LIBELLIDepartment of Systems and Computers, University of Florence, via di S. Marta, 3-50139

Florence, Italy

(Received 19 April 1992;infinalform

21 December 1992)

Secondary sedimentation plays a fundamental role in biological wastewater treatment processes where activated

sludges arc used. In addition to water clarification and sludge compaction, the secondary settler is used as a dynamicstorage for the biomass in the system. This role is important for process control since displacing sludge from thesettler to the oxidation tank is the main way to adjust process conditions in the short time-scale. The scope of thispaper is to present a model for the time-varying behaviour of the total activated mass, taking into account thecoupling between the aeration tank and the secondary settler. Numerical simulations show that the model canreproduce any relevant feature of the real system and can be used in control strategy design for sludge management.

KEY WORDS: Sedimentation, thickening, activated sludges, wastewater treatment, mathematical modelling.

INTRODUCTION

The secondary settler plays a crucial role in biological wastewater treatment processes where

activated sludges are used, separating the sludge floes from the treated water (clarzjkation)and compacting the sludge to be returned into the aerator (thickening). But a third and most

important feature is to act as a mass storage for the activated sludge mass operating in the

system. This role is important for process control. In fact, acting on the recycle flow, the

sludge mass can be transferred from the secondav settler, where it is simply stored andinactive, back to the oxidation basin where it is active in degrading the incoming pollutant.Thus in the short time scale displacing sludge mass from the settler to the oxidation stage isone way to change process conditions.

This paper presents a dynamic model for the transfer and accumulation of sludge mass

in the secondary settler based on the theory of hindered settling. This theory, originating

some decades ago from the pioneering work ofKynch, has been widely used as a designrationale for secondary settlers (see e.g. Keinath et al., 1977 and Lauria et al., 1977), butapplications to the operational context were comparatively few. Tracy and Keinath2 pro-duced the first dynamical model using a mass balance and the Kynch sedimentation law to

derive a partial differential equation (PDE) which was then solved numerically through finite

207


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208 SMARSILI-LIBELLI

differences. Though their work neatly solved the problem from a conceptual point ofview,the resulting model was too complex to be incorporated into larger process schemes and had

the typical numerical shortcomings of PDE-based models in terms of stability and boundary

condition specifications. Stehfest3 proposed an elegant numerical method to solve theseproblems, reducing the original PDE into a single ordinary differential equation (ODE)through the method of lines. An entirely different approach was followed by Olsson and

Chapman4who used two patched black box models based on the experimental evidence

that the dynamic response of the clarifier was nonsymmetrical, i.e. the responses for flow

step increases and decreases differ. Moreover the emphasis was primarily on the clarification

aspect rather than on thickening and storage. The most recent contribution, due to Takas etuZ.~, again uses a multi-layered model as Tracy and Keinath2, but introduces the refinementof choosing the boundaries in a way consistent to the physical properties of the suspension.

Though relying heavily on the Kynch theory in deriving a settling velocity model, the

resulting model is prinarily aimed, as with Olsson and Chapman4, with clarification ratherthan thickening.The model presented in this paper is based on an ordinary differential equation and

represents an extension of a previous, more limited clarifier model included in a general

activated sludge system6. The aim of the modelling exercise is to describe the dynamics ofmass storage in the secondary settler and how this influences the sludge concentration in the

aeration tank through recycling. The model analyzes the three possible operating modes of

critical loading, underloading and overloading, assessing the implications of all three. After

briefly reviewing the Kynch theory of flocculent suspensions, a general dynamical model

of sedimentation is outlined in broad structural terms before specifying an analytical form

of the settling flux based upon the Vesilind7 equation. Later, the dependence of settlingdynamics on currently available process indicators such as the Specific Stirred Volume

Index (SSVI) is introduced with the final result of producing an operational model which

can be used to predict and control the sludge accumulation in the secondary settler and the

effect of sludge recycling in the oxidation basin. The main scope of this paper is to present

a model for the time-varying behaviour of the total activated mass in the system taking into

account the interactions between the aeration tank and the secondary settler when their

combineddynamics is considered. In this analysis the structural properties of the model and

its qualitative behaviour were considered to be pre-eminent over any precise agreement with

specific experimental data. As the model is based on physical laws and parameter valueswell established in the literature, the assessment presented later in the paper can be

considered of sufficient generality to adapt to any specific situation.

The theory of flocculent suspensions (sludges), developed primarily by Kynch and lateradvanced by Dick* and Shin and Dick, is now briefly revisited. It states that the solidflux of particles due to gravity sedimentation F, depends on the sludge density Xand itsvelocity v

F,=Xv

i

Writing a mass balance around a vertical cylinder of thickness dz between heights z andz+dz, the rate of change of the density must equal the net flow. Thus


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DYNAMIC MODELLING OF SEDIMENTATION

y = F,(z+dx) - F&z)dividing by dz yields the continuity equation

dX_dF,at az

which recalling eq. (1) can be written as

209

(2)

(3)

(4)where c=-aF$aX is defined as the upward propagation velocity of a layer of constant densityX. In this sense sedimentation can be viewed as the upward motion of increasingly thicker

layers. Conversely, the downward motion of a layer at constant concentration X can be

derived by the continuity equation

X(z+dz,t+dt)=X(z,t) (5)Expanding the left-hand-side around X(z,t) and eliminating the common term X(z,t)

yields

$dt+gdz=O (6)Equation (6) describes the dynamics at time t of a layer at height z and constant concentration

X. Solving for dz/dt and comparing with eq. (4) yieldsa x2=-$ = 5@ X = const. (7)a z

Equation (7) states that if the sedimentation velocity is a function of density alone, a layer

of given concentration X propagates with constant velocity 5 This should not be confusedwith the downward motion of a single particle (v) appearing in eq. (1).

Discontinuities occur whenever there is an abrupt change of concentration. In this case

the continuity eq. (3) no longer holds and must be replaced with a mass balance across the

discontinuity

xl(vl+u)=x,(v,+u ) (8 )

where the index 1 refers to the layer above the discontinuity and 2 to that below it.


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210 S. MARSILI-LIBELLIEquation (8) is satisfied in general for U#O. Therefore in general the discontinuity is not atrest but moves with velocity

u= F,-FZx2-x1

if the difference in concentrations is small i.e. Fr-F2 zdF, and X2-X1 gdX eq. (9) can bewritten in incremental terms to yield

.

U=_dF,_dx-5 (10)hence 5 can be regarded as the propagation velocity of an incremental discontinuity fromdensity X to X +dX. Moreover, since t=-dF,/dX then

?!&=_&Idx2 (11)The necessary and sufficient condition for the occurrence of a discontinuity is related to the

shape of the sedimentation curve as follows

2 >odgf


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DYNAMIC MODELLING OF SEDIMENTATION 211

-i.-; 6 . 0

4 . 5

3 . 0

1 . 5 1 i discontinuo& settling

0.0 I I --_-0 xi 1 0 x t 2 0

x (g I)Figure 1 Batch sedimentation curve and graphical determination of limiting (Xt) and final (X,) sludge concen-trations.

ANALYTICAL FORM OF SEDIMENTATION

So far no special hypothesis was made as to the mathematical form of the sedimentation flux

F&X), and yet some general conclusions have been drawn, especially regarding conditionsfor the existence of a discontinuity and its stability. Now an analytical expression for the

sedimentation flux Fg(X) is introduced. Among the many mathematical expressions whichwere proposed in the literature the following exponential expression will be used in the

sequel as it was shown to be in good agreement with experimental observations79-2

F, = V&e-ox (16)where V, is the limit sedimentation velocity for diluted suspensions (X-+0) and a is asedimentation parameter. This particular law was preferred to a power law as it yields better

results for low densities, although it may not lend itself to neat closed form solution as is thecase with a power law13. Substituting expression (16) into the eq. (14) yields


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7 . 5

5 . 0

2 . 5

0 . 0

3 0

2 0

1 0

0

/

/Glk f l u x/ .

/;

; x t x , ;x ( g I " )

/I

I

0 5 1 0 1 5 2 0

Figure 2 Continuous-flow sedimentation flux: a) Critical loading b) Underloading


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F = V,,Xe-ox + Xu (17)The limiting concentration Xt is obtained by vanishing the derivative of eq. (17) with respectto x

$=VOe-ox(l -aX)+u=O (18)The solution to eq. (18) is indeed a minimum since the second derivative

23 = aV0e -oX(ax-2) (19)

is always positive for X>2lo., which is the range in which a feasible solution is sought. Infact the point X=2/a represents the inflexion point of the total flux curve and is independent

of the underflow u. Equation (18) has no analytical solution but can be solved numerically

through an iteration scheme where the k-th approximation is obtained as

X =x1 I - (df/dx) (d2F/dX2)-k k - l k - l k - l= x _ u + VO(l-aX)e -aX

k - l aV0( ax-2)e -aX k - l

(20)

The feasibility of the solution depends on the underflow u. In fact a consistent (X>2/a)solution exists if and only if u is in the interval

O


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214 S. MARSILI-LIBELLI

2 0

1 5

z 1 0w -

5

0

cl ::;; ::

.a

..II .*X.i

. .

::. .

. .

. .

. .

. .

. .

. .

. .. .

: .

.* ,: ,::: /X. * , = f ( X ). * ,: I: ,

5 1 0 1 5 2 0

x ( 9 I " 1Figure 3 Convergence scheme of the iterative equation (21).

Ft = Fb = X,,zr (22)which yields X,

(23)

Tracy and Keinath14 made the assumption that the underflow concentration is not a dynamicvariable in its own right as it may be algebraically related to the underflow velocity u. In

other words it can follow immediately any change in u. However it is unrealistic to assume,

and unlikely to observe, that Xr makes abrupt changes when the recycle is changed. Thus alag can be introduced in order to model the recycle concentration as a dynamic variable

dX 6 AL=-yxr+yPdt Qr (24)


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DYNAMIC MODELLING OF SEDIMENTATION

THICKENING

215

Figure 4 Activated sludge process scheme and zone partitioning of the secondaq settler.

So far the case of a stable interface (G=O) was considered. This corresponds to a criticallyloaded clarifier where the incoming flux equals the limiting flux through the interface.

Keinath et al. demonstrated that this condition is stable in the sense that whenever therecycle is changed the system will tend to readjust the limiting flux in order to restore the

equilibrium.

The overload and underload situations are now analyzed. To do this the clarifier is

partitioned in four operational zones, as in Fig. 4. The upper zone contains clarified waterwhich flows over the weir, the build-up zone absorbs the excess flux in case of overload,

the storage zone is where the discontinuous settling occurs and sludge mass is normally

accumulated, whereas the bottom zone is where thickening from Xt to X, occurs. Now thetwo situations of underload and overload are analyzed.

Underload: such a condition occurs when the withdrawal from the underflow F, is greaterthan the limiting flux Ft sustained by the discontinuity. In this case the underflow u exceedsthe upper bound (21) and no solution to the limiting flux equation exists. This means that

dF/dX>O for all X which implies that d WdX


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DYNAMIC MODELLING OF SEDIMENTATION 217iteration path starting with X>l/a. Of course the analysis is restricted to positive values ofAF, since only in this case eq. (27) makes sense. It can be seen that the iterative scheme (28)

converges for any value of AF>O thus yielding a unique value for the sludge concentrationin the build-up zone X,,. This completes the description of the overload situation.

DYNAMIC BEHAVIOUR OF SEDIMENTATION

Having specified the general behaviour and the mathematical form of the sedimentation

process, it is now possible to incorporate this into a continuous-flow activated sludge process

including an oxidation stage and a secondary sedimentation. It should be stressed that

biological growth of the sludge mass is deliberately ignored here in order to demonstrate

how the model describes the sludge dynamics due to a mass transfer only. Of course in a

fully operational working model the sludge biodynamics consisting of growth and decayterms should be re-introduced. In addition, no wastage from the settler underflow is

considered. Hence the mass in the system is assumed to be constant. With the nomenclature

of Fig. 4 the following dynamic equations can then be written

Oxidation

Neglecting sludge kinetics and indicating the sludge concentration in the aerator as X1, amass balance yields the following dynamics

%=pXJ-p(l +r)Xi

where V is the volume of the oxidation basin, Q is the process flow rate and P-QN is thedilution rate.

Sedimentation

Assuming that the incoming mass enters the thickening zone and is immediately thickenedto Xt, a mass balance below the discontinuity yields the dynamics of the stored mass MifFi=XiQ( l+r)Ft (overloading) then

5 =XiQ( 1 + r) - AFt + AF,, (31)


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To demonstrate that model (29-30) is in agreement with the constant biomass assumption,

consider that

VXi+M=cost. (32)taking the time derivative, substituting the r.h.s. of eqs. (29-30) in place of dXi/dt and dM/dtand considering eq. (23) yields

V+Xr0r-V$(1 +r)Xi+XiQ(l+r)-AFt=*Fi-*Ft=O (33)In the case of critical loading the discontinuity is the regulating element in the loop and Ftis the mass transfer rate which the discontinuity can handle. In fact

T=O=Xiz( l +r)=Ftand

eliminating Xi yields

(35)

which coincides with eq. (23). Hence for a given flow Q the amount of mass circulating in

the system depends on the limiting flux F, and the recycle ratio r. Again the reader is warnedthat in order to demonstrate the sludge movements due to sedimentation, this model

deliberately neglects sludge growth and assumes a constant mass in the system.

SLUDGE SETTLEABILITYThe physical characteristics of the sludge influence its settling properties, which in turn

determine the overall dynamic behaviour. This dependence was already acknowledged in

section 3 where the sedimentation velocity depended on two parameters (V , and a). Thequestion arising now is whether any further relation can be established between settling

behaviour and overall sludge characteristics. In fact V, and CL are difficult to measure andthough some experimental evidence exists that they change very little in time, it would be

desirable to rely on some easily obtainable sludge index, possibly measurable on-line or at

least frequently. The most widely used sludge sedimentation parameter is the SludgeVolume Index (SVI) defined as the ratio between the volume of sludge after 30 min


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DYNAMIC MODELLING OF SEDIMENTATION 21 9

sedimentation and its dry weight. Therefore SVI is expressed in ml 8.. This index wascriticized as being density dependent and the Specific Stirred Volume Index (SSVI) was

proposed instead. To determine SSVI the sludge concentration is normalized to a prescribed

value and the slurry is not at rest in the test jar, but is subject to a normalized (1 rpm) stirring.

A detailed description of the test equipment can be found in White. The advantage of usingSSVI instead of SVI is thoroughly treated by Rachwal eta1.15 where extensive data aresupplied. These were used to perform a linear regression between SSVI and the settling

parameters V, and CL obtaining a good degree of significance, as the high value of thecorrelation p shows

V, = 9.127-0.0366 SSVI p2 = -0.9886 (37)CL = 0.277+0.0011 SSVI p2 = 0.9818 (38)

These relationships were determined dividing a total of 773 SSVI data into four groups(60+79, 80~99, lOOtl19, 120+139) and using the average V, and a values for the regres-sions (37-38). The question now arises as to what influences the settling characteristics.

Ghobrial16 states that they depend the loading conditions of the biological reactor, but givesno quantitative relationship whereas Capodaglio etaLI7 have proposed a linear relationshipbetween SSVI and F,

SSVI = 145.99 - 27.72 F, (39)This dependence was then extended to a dynamic relationship using predictive models based

on time series analysis and neural network models. Based on data both from literature and

gathered directly from medium-scale completely-mixed activated sludge plant, here an

inverse relation with the loading rate F, is used

SSVI=a,+Fcwith al and a2 numerical parameters. In the case of two medium-scale completely-mixedplants processing domestic sewage with F, values between 0.1 and 0.3 the followingnumerical values were found: a1=28.5 and al=1 1.4. This has some theoretical justificationin the fact that when the food is scarce, slow-growing filamentous bacteria, mainly respon-

sible for poor settling, take over because of their superior ability to reach for food.

Microscopic examination revealed that the presence of Sphaerotilus natans was highly

correlated with low F, spells. This was also noticed by Tsugura et al. 18, while a similarrelationship with phosphorus content was obtained by Rachwal et al.. Though Chudobaand Chudoba etaI. conclude that no general relation can be established between SSVI andF,, as this is not a primary factor influencing filamentous bacterial growth, yet for com-pletely-mixed they show that for medium-range F, values a relation such as eq. (40) canindeed be found. This can be related to soluble degradable substrate input through an

hyperbolic relation, as shown by Chudoba. The importance of SSVI in this study is tomodel the sludge blanket height, i.e. the volume taken up by a given mass of sludge. In fact


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Table 1 Process parameters

A 80 0 m Settler surface

G1200 m3 Aeration tank volume100 m3 h- Process flow

r 0.25-

recycle ratioFC 0.1 i 0.3 d- Loading factorai 28.5 ml g- SSVI parametera2 11.4 d- SSVI parameterY 0.5 h- underflow concentration time constant

this is the most important secondary settler control parameter and the one which is most

easily monitored in terms of sludge blanket height. Since the model so far considers the

accumulated sludge M as a state variable, this can be related to the blanket height through

a very simple algebraic relation is used

h=ho+&where h is the sludge blanket height and h, a reference height, for example the top of thethickening layer of Fig. 4. It should be remembered that once a SSVI values is obtained from

eq. (40) the settling parameters V, and o. are obtained through eqs. (37-38).

DYNAMIC MODEL BEHAVIOUR

In order to test the model behaviour three different situations were simulated as shown in

Fig. 6,7,8. The process parameters used in the simulation are summarized in Table 1. Sincethe aim of these simulations is to show the biomass movement in the systems, all biological

side-processes have been intentionally neglected and only mass displacements are consid-

ered. As already discussed in Sect. 4 this is an acceptable approximation in the short term,

but in the long term biological growth does represent a major contribution to sludge

dynamics and cannot be neglected. In the first example a step variation of the process flow

Q was introduced and the effect of biomass concentration and accumulated sludge is shown

in Fig. 6 with r being kept constant. An asymmetric behaviour is apparent, with a faster

response for the increasing flow. It can be seen that the step flow perturbation produces a

net build-up in accumulated mass. This could be eliminated by increasing the recycle and/or

wastage or allowing for long term sludge compression, which this model does not take into

account. Since F, is constant, the sludge blanket height is proportional to the accumulatedmass and follows the same behaviour. Due to the difficulty in accommodating the abrupt

flow increase there is a short period of overload at the leading edge ofthe flow step. Likewise,the sludge concentration in the aeration tankXi decreases due to hydraulic dilution and sodoes the return sludge X,. The simulation of Fig. 7 shows how the system responds to changes

in the recycle ratio r. At the beginning of the simulation the recycle is quickly decreased andthen slowly increased again. In the beginning a short overload occurs as the critical flux

corresponding to the new recycle is not large enough to accommodate the incoming mass.


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1 3 0

i m 1 1 0E2 9 0v )

7 0

- 7 0 . 5 r cu - L - - - - - - - - - L - ~ . _ _l . L Y 0

V 0 a * \ , , , l & I l l 'I I I I I I

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0T i me ( mi d

6

Figure 8 Influence of loading rate Fc variations on Specific Stirred Volume Index (SSVI) and Sludge blanketheight (Sbh).

Soon after the settler adjusts itself to the new condition and normal operation is resumed.

The fact that an overloaded settler can spontaneously turn into a critical one is a consequence

ofthe constant mass assumption. In fact the high input condition causing the overload cannotbe sustained for long because the recycle concentration X, decreases as a consequence ofthe reduced recycle, causing the incoming sludge concentration Xi to decrease. Eventuallyan equilibrium is reached and the system settles again at a new Ft value corresponding to thenew critical loading condition. Lastly, Fig. 8 shows the Sludge blanket height (Sbh)

dependence from the loading rate F, through the Specific Stirred Volume Index (SSVI).During the simulation F, was first increased and then decreased again. The upper part of Fig.8 shows how this reflects on the variation of SSVI through the settling parameter V, (thevariations of a were negligible). Given the rather high uncertainty and the variability of the

loading rate, a random fluctuation was superimposed to the deterministic F, variations. Itcan be seen that the effect of this disturbance is more pronounced for low values ofF,, whichcause the highest variation of the sludge blanket height.

CONCLUSION

The dynamic behaviour of the secondary settler was modelled relying on the Kynch theoryand taking into account the interaction with the oxidation tank. In this analysis the influence


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ofbiological factors such as sludge growth due to microbial activity was deliberately omitted

in order to show the role of mass transfer and limiting flux. Under this hypothesis the paper

first considers the structural properties of the model, then assesses its numerical behaviour

in describing the dynamics of the three possible loading conditions of the secondary settler

(critical, overload, underload) in a unified conceptual basis. Relying on physical evidenceand parameters drawn from the literature or obtained through direct experimentation, somesimulations were produced to show that the model was capable of describing a number of

different practical situations. Thus it can assist in the design of control strategies for the

biomass in the activated sludge system.

Acknowledgement

This research was partly supported by the Ministry of University and Scientific and

Technological Research (MURST) under contract 09/02/000056 as part of the researchproject of national interest entitled dynamic modelling of ecosystems.

References

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Kynch G. J. (1952) A theory of sedimentation Trans. Faraday Society, Vol. 48: 166176.Keinath, T. M., Ryckman M. D., Dana C. H., and Hofer D. A. (1977) Activated sludge-unified system

design and operation J. Env. Eng. Div., ASCE, Vol. 103, No. EE5: 829-849.Stehfest H. (1984) An operational dynamic model of the final clarifier Trans. MC, Vol. 6, No. 3: 160-l 64.Olsson G. and Chapman D. (1985) Modelling the dynamics of clarifier behaviour in activated sludgesystems Proc. Instrumentation and control of water and wastewater treatment and transport systems, (R. A.R. Drake ed.): 405412, Pergamon Press, Oxford.

Tak&cs I., Patry G. G., Nolasco D. (1991) A dynamic model of the clarification-thickening process Wut.Rex, Vol. 25, No. 10: 1263-1271.Marsh-Libelli S. (1989) Modelling, identification and control of the activated sludge process Adv. inBiochemicalEngineering/Biotechnology, Vol. 38: 89-148.Vesilind A. P. (1968) Discussion of Evaluation of activated sludge thickening theories, by R. I. Dick andB. B. Ewing, J. &nit. Eng., ASCE, Vol. 94: 185-191.Dick, R. I. (1970) Role of activated sludge final settling tanks J. Sun.Div., ASCE, No. SA2:423436.Shin B. S. and Dick R. I. (1980) Applicability of Kynch theory to flocculent suspensionsJ.Env. Eng. Div.,ASCE, Vol. 106, No. EE3: 505-526.White M. J. D. (1976) Design and control of secondary settlement tanks Wat. PolZut. Control, 75: 459467.Severin B. F. and Poduska R. A. (1986) Flocculant settling dynamics under constant 1oadingJ: Env. Eng.Div., ASCE, Vol. 112, No. EEl: 171-184.H&man B., Low&n M., Karlsson U., Li P. H., Molina L. (199 1) Prediction of activated sludge sedimentationbased on sludge indices Wat. Sci. Tech., Vol. 24, No. 7: 3342.

Lauria, D. T., Uunk, J. B., and Schaefer, J. K. (1977) Activated sludge process design J. Env. Eng. Div.,ASCE, Vol. 103, No. EE4: 625-645.

Tracy, K. D. and Keinath, T. M. (1973) Dynamic model for thickening of activated sludge AIChESymposium Series (Water) 70, No. 136: 291-308.Rachwal A. J., Johnstone D. W. M., Hanbury M. J., C&chard D. J. (1978) The application of settleabilitytest for the control of the activated sludge plants inBulking of the activated sludgeplants: preventative and

remedial methods (Chambers B. and Tomlinson E. J. eds.), Ellis Horwood Ltd., Chichester.

Ghobrial F. H. (1978) Importance of the clarification phase in biological process control Wat. Rex, 12:1009-1016.

Capodaglio A. G., Jones H. V., Novotny V., Feng X. (1991) Sludge bulking analysis and forecasting:

application of system identification and artificial neural computing technologies Wat. Rex, 25: 1217-1224.


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