settler to the oxidation tank is the main way to adjust process conditions in the short time-scale. The scope of this paper is to present a model for the time-varying ...
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DYNAMIC MODELLING OF SEDIMENTATION IN THE ACTIVATED SLUDGE PROCESS S . MARSILI-LIBELLI Department 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 dynamic storage for the biomass in the system. This role is important for process control since displacing sludge from the settler to the oxidation tank is the main way to adjust process conditions in the short time-scale. The scope of this paper is to present a model for the time-varying behaviour of the total activated mass, taking into account the coupling between the aeration tank and the secondary settler. Numerical simulations show that the model can reproduce 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
f
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 and inactive, 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 is one 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 of Kynch’, has been widely used as a design rationale for secondary settlers (see e.g. Keinath et al., 1977 and Lauria et al., 1977), but applications to the operational context were comparatively few. Tracy and Keinath2 produced 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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differences. Though their work neatly solved the problem from a conceptual point
of view,
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 these problems, 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 Chapman 4 who 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 et uZ.~, again uses a multi-layered model as Tracy and Keinath2, but introduces the refinement of 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 rather than 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 of mass 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 settling dynamics 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 combined dynamics 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 values well 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 later advanced by Dick* and Shin and Dick’, is now briefly revisited. It states that the solid flux of particles due to gravity sedimentation F, depends on the sludge density X and its velocity v F,=Xv Writing a mass balance around a vertical cylinder of thickness dz between heights z and z+dz, the rate of change of the density must equal the net flow. Thus
ic
”
DYNAMIC MODELLING OF SEDIMENTATION
209
y= F,(z+dx) - F&z)
(2)
dividing by dz yields the continuity equation dX_dF, at az
(3)
which recalling eq. (1) can be written as
(4) where c=-aF$aX is defined as the upward propagation velocity of a layer of constant density X. 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) yields
ax
2 = - $ =5@
X = const.
(7)
az
.
*
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 confused with 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)
d
I
I
where the index ‘ 1’ refers to the layer above the discontinuity and ‘2’ to that below it.
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Equation (8) is satisfied in general for U#O. Therefore in general the discontinuity is not at rest but moves with velocity u=
F,-FZ x2-x1
if the difference in concentrations is “small” i.e. Fr-F2 z dF, and X2-X1 g dX eq. (9) can be written in incremental terms to yield
U=_dF,_
dx-5
(10)
hence 5 can be regarded as the propagation velocity of an incremental discontinuity from density X to X + dX. Moreover, since t=-dF,/dX then
?!&=_&I dx2
, 6 .
(11)
The necessary and sufficient condition for the occurrence of a discontinuity is related to the shape of the sedimentation curve as follows
2 >odgfO for all X which implies that d WdXl/a. Of course the analysis is restricted to positive values of AF, 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 concentration in 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 decay terms 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, a mass 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 the dilution rate. Sedimentation Assuming that the incoming mass enters the thickening zone and is immediately thickened to Xt, a mass balance below the discontinuity yields the dynamics of the stored mass M if Fi=XiQ( l+r)Ft (overloading) then
I
5 = XiQ( 1 + r) - AFt + AF,,
(31)
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S. MARSILI-LIBELLI
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/dt and 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 Ft is the mass transfer rate which the discontinuity can handle. In fact T=O=Xiz(l +r)=Ft and (35) eliminating Xi yields
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 warned that 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 SETTLEABILITY The 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). The question 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 and though 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 Sludge Volume Index (SVI) defined as the ratio between the volume of sludge after 30 min
DYNAMIC MODELLING OF SEDIMENTATION
219
sedimentation and its dry weight. Therefore SVI is expressed in ml 8.‘. This index was criticized 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 using SSVI instead of SVI is thoroughly treated by Rachwal et a1.15 where extensive data are supplied. 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 the correlation p shows
1
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 regressions (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 gives no quantitative relationship whereas Capodaglio et aLI7 have proposed a linear relationship between 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,+F c with al and a2 numerical parameters. In the case of two medium-scale completely-mixed plants processing domestic sewage with F, values between 0.1 and 0.3 the following numerical values were found: a1=28.5 and al=1 1.4. This has some theoretical justification in the fact that when the food is scarce, slow-growing filamentous bacteria, mainly responsible 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 similar relationship with phosphorus content was obtained by Rachwal et al.“. Though Chudoba” and Chudoba et aI.” conclude that no general relation can be established between SSVI and F,, as this is not a primary factor influencing filamentous bacterial growth, yet for completely-mixed they show that for medium-range F, values a relation such as eq. (40) can indeed 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 to model the sludge blanket height, i.e. the volume taken up by a given mass of sludge. In fact
S. MARSILI-LIBELLI
220
Table 1 Process parameters
A G r FC
ai a2
Y
800
m’
1200 m3 m3 h-’ 100 0.25 0.1 i 0.3 d-’ ml g-’ 28.5 11.4 d-’ 0.5 h-’
Settler surface Aeration tank volume Process flow recycle ratio Loading factor SSVI parameter SSVI parameter underflow concentration time constant c
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 the thickening 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. Since the 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 considered. 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 accumulated mass 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 tank Xi decreases due to hydraulic dilution and so does 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 and then 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.
.
DYNAMIC MODELLING OF SEDIMENTATION
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