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A simple, conceptual mathematical model for the activated sludge process and its variants
Wat. Res. Vol. 23, No. 12, pp. 1535-1543, 1989 Printed in Great Britain. All rights reserved
0043-1354/89 $3.00+0.00 Copyright © 1989 Pergamon Press plc
A SIMPLE, CONCEPTUAL MATHEMATICAL MODEL FOR THE ACTIVATED SLUDGE PROCESS A N D ITS VARIANTS NANDAN PADUKONE1 a n d GRAHAM F. ANDREWS2 JDepartment of Chemical Engineering, North Carolina State University, Raleigh, NC 27695-7905 and 2Biotechnology Unit, Idaho National Engineering Laboratory, P.O. Box 1625, Idaho Falls, ID 83415, U.S.A. (First received February 1988; accepted in revised form June 1989)
Abstract--A simple structured kinetic model is applied to the activated sludge process. The objective is less to predict exact process performance than to illustrate some of the possibilities and difficulties in producing a comprehensive model for all the process variants. The rate equations are chosen so as to reduce to the Monod equation during balanced growth. Because these rate equations are linear, the cell growth and substrate uptake in a stirred tank can be defined exactly in terms of the average composition of the biomass. It is shown that this is not valid for other forms of rate equations. The stored substrate to protoplasm ratio in the fiocs is found to decrease with increasing mean cell residence time. If extracellular biopolymers are included in the stored substrate this corresponds qualitatively to observations of poor flocculation in extended aeration. The model is also applied to the contact stabilization process and is found to be in agreement with the essential process variables. Key
words--activated
sludge, mathematical modelling, contact stabilization, structured kinetic
models
NOMENCLATURE k~ = k2 = ka = km= K= n= q=
Subscripts
substrate uptake rate constant (l/rag h) growth rate constant (h - t ) cell decay rate constant (h -~ ) maintenance coefficient (h- ~) Monod half saturation constant (mg/l) number concentration of flocs in influent (1- ~) specific rate of substrate uptake (rag substrate/mg protoplasm h) S = liquid-phase substrate concentration (mg/l) V = reactor volume (1) XA = protoplasm concentration (mg/l) M A = mass of active biomass in a fioc Xs = stored substrate concentration M s = mass of stored substrate in a floc y = ratio of stored substrate to active biomass in a floc= M s / M h YAv= Xs/XA = average composition of floes in reactor yb = biomass composition during balanced growth (approximate) y ~ = b i o m a s s composition during balanced growth (exact) Y = cell yield (mg protoplasm/mg substrate) = specific growth rate (h-~) 0 = hydraulic residence time (h) 0x = mean cell residence time (h). Dimensionless variables E = ( S " + I)/ymY k'm = km Y/k2ym S" = k I YS /k2y m = S / K t' = k2Ymt X'Ao = kiXAo/k2ym = X A o / Y K O" = k2ymO Y' = Y /Ym "
c = contact tank influcnt value m = maximum value s = stabilization tank. 0 =
INTRODUCTION A m a t h e m a t i c a l model is a useful tool for the optim u m design a n d control of any process, mainly because the effects o f adjusting the o p e r a t i n g variables can be studied far m o r e quickly o n a c o m p u t e r t h a n by doing experiments. This is particularly true of a process like activated sludge biological wastewater t r e a t m e n t which m a y take days to reach a new steady-state after one of the process variables is changed. T h e need for a n accurate model of this process has long been recognized, a n d m a n y have been p r o p o s e d a n d tested ( G o o d m a n a n d Englande, 1974; G a u d y a n d K i n c a n n o n , 1977; Lawrence a n d M c C a r t y , 1980; C h e n a n d H a s h i m o t o , 1980). Even the simplest models, based o n M o n o d kinetics for cell growth ( G r a d y a n d Lim, 1980) offer a useful guide to sizing aeration tanks a n d illustrate the i m p o r t a n c e o f m e a n cell residence time as a control parameter. M o r e a d v a n c e d versions like t h a t produced by the I A W P R C task g r o u p (Henze et al., 1987) m a y ultimately allow a rational choice to be m a d e between process variants (contact stabilization, extended aeration etc.) for a particular wastewater.
1535
1536
NANDAN PADUKONEand GRAHAM F. ANDREWS
A truly comprehensive model would be extremely complex. Some of the difficulties that have been incorporated into one or more models are as follows: (1) As with all wastewater treatment processes the substrate is a complex and variable mixture of chemicals. Some of these may not be biodegradable and those that are may be degraded simultaneously or sequentially (Vavilin and Vasiliev, 1985). This is normally dealt with by the "pseudo-single solute" assumption by which the substrate concentration is defined in terms of a single measurement such as BOD or TOC. Of particular importance for activated sludge is the distinction between soluble organic material, which is biodegraded normally, and colloidal material, which adsorbs rapidly on the biomass and is subsequently slowly degraded (Fujie et al., 1988). (2) Both the concentration and composition of wastewaters change with time. Thus, although quasi-steady state models can be quite successful, only a transient model can give a truly accurate description of the process. {3) The biological flocs are also complex and variable mixtures of different species of heterotrophic bacteria (including filamentous organisms implicated in sludge bulking), autotrophic nitrifying bacteria, protozoa, extra-cellular polysaccharide material produced by the bacteria to bind them together, adsorbed colloidal substrate and inert material produced by decay of the cells or by adsorption of non-biodegradable wastewater constituents (both particulate and soluble; Tsezos and Bell, 1988). An unstructured kinetic model that considers these flocs just as "biomass" can never give an accurate description of their behaviour, so many types of structured models have been proposed. Some attempt to describe the interaction between the various species of bacteria with the many types of substrate in the wastewater (Vavilin and Vasiliev, 1985; Benefield and Mole, 1984). Mikesell (1981, 1984) focussed his attention on the production of biopolymers in the floc. Work on the contact stabilization process has stressed the importance of substrate stored in the flocs in the contact tank and then oxidized in the stabilization tank, although the complexities inherent in the use of these structured models have not always been appreciated [see the discussion by Orhon (1977) of the model proposed by Benefield and Randall (1976)]. The IAWPRC model (Henze et al., 1987) divides the flocs into active heterotrophic biomass, active autotrophic biomass, slowly degradable adsorbed colloidal matter and inert matter.
(4) The secondary settler is an integral part of the activated sludge process and must be included in a comprehensive process model (Busby and Andrews, 1975; Sheintuch, 1987). This requires knowledge of how the settling velocity of the floes depends on their size and composition and how these depend on process variables such as loading and sludge residence time. This is an area in which only generalizations are presently available; the presence of filamentous bacteria is associated with bulking sludge, contact stabilization tends to produce good settling behaviour while extended aeration may generate small floes that are hard to settle. Until new experimental techniques (Da Hong and Gaczarcyzk, 1988) measure settling velocity as a function of floc composition and structured kinetic models can predict floc size and composition as a function of proce3s variables (including shear in the aeration tank) most models will continue to treat the settler as a "separation point" with no biomass in the overflow. They will provide little information about the common problem of settler overloading. The complexities listed above mean that a workable comprehensive model will need to be assembled from the work of individual researchers, each working on one facet of the problem. The original objective of the work reported here was to develop the simplest possible kinetic model for the contact stabilization process that could incorporate both the storage/metabolism hypothesis for why this process works (Ulrich and Smith, 1951: Jones, 1970) and the growth/decay model of Orhon and Jenkins (1973). The subsequent development of the model was found to illustrate some more general problems and possibilities in the use of structured models, and to suggest a way in which a floc's settling is related to its composition. MODEL DEVELOPMENT
A complete model consists of four things; a description of its "structure", that is the components into which the substrate and the biomass are to be divided; a set of rate equations describing how rapidly one component is coverted into another by the various physical and biological processes; a description of the stoichiometry showing how much of a component is consumed or generated by a process: and finally a set of conservation equations describing the mass balance for each component in the particular type of reactor chosen. Structure and stoichiometry
The simplest structured kinetic model for activated sludge describes the substrate by a single parameter (BOD) and divides the biomass into an active cell (protoplasm) fraction and a stored substrate fraction
Model for activated sludge processes (Andrews and Tien, 1977; Busby and Andrews, 1975). The "reactions" can be described by:
Substrate S
upt~k~ Stored substrateq
XS
1537
Applying intuition to the process of adsorption of colloids onto the floc suggests that the adsorption
Growth
Active cell mass (protoplasm)
t~/ Y
XA
Maintenance
~ Respiration
+ Respiration products
(1) products.
k~
Two categories of "stored substrate" are usually identified, substrate which is adsorbed on the floc and intracellular storage products such as PHB and glycogen. However some part of the extracellular biopolymers that hold the floc together should also probably be included. The polymers are mainly polysaccharides and it seems reasonable to suppose that, if faced with starvation, the bacteria would re-hydrolyse them to make substrate. The distinction between the three categories of stored substrate is then mainly one of time scale. If flocs are aerated in the absence of fresh substrate for a few hours, as in a stabilization tank, the adsorbed material and intracellular storage products are consumed but there is little change in floc volume. If aeration is continued for tens of hours, as in aerobic digestion, the polymers are degraded and there is a marked reduction in volume. Rate equations
The rates in equation (1) (q,#/Y, km) are the specific rates (i.e. per unit active biomass) with which substrate is being taken up, used for growth and used for maintenance, respectively. Rate equations describe how these processes are affected by the substrate concentration, the condition of the floc etc. In choosing a set of rate equations there are three sources of information. The first is our understanding and intuition about how the process behaves. Second, and much less used, is our accumulated knowledge about how bacteria behave during "balanced growth". Balanced growth can be defined as the situation where the conditions to which the floc is exposed change so slowly that the composition of the biomass always remains perfectly acclimated to them. In this model the composition of a floc is described by the variable y = Ms/MA (i.e. the ratio of stored substrate to active biomass in a floc) and, mathematically, balanced growth corresponds to dy/dt ~ O. If we require that during balanced growth # is described by the Monod equation, then this imposes severe constraints on the form of rate equations that can be adopted. The third source of information is experimental data. Unfortunately this ultimate test may not be as conclusive as we would like; models based on two different sets of rate equations can sometimes describe a single set of data.
rate per unit of active biomass should be proportional to the surface area not already occupied by adsorbed material and to the colloid concentration in the liquid. The resulting rate equation has the form: q = k~S(1 - Y/Ym).
(2)
This form also seems reasonable for soluble substrate and is therefore adopted for the total substrate uptake. Ym is the maximum amount of stored substrate, as adsorbed material, intracellular storage products and bio-polymers, that can be associated with a unit of active biomass. Since the model [equation (1)] sees the bacteria growing on the stored substrate, it is reasonable to make the specific growth rate proportional to the amount of stored substrate available to a unit of protoplasm: # =k2y.
(3)
Balanced growth
Balanced growth occurs when the external conditions to which a cell is exposed (S in the model) change so slowly that its composition (y in the model) remains perfectly acclimated to them. For wastewater treatment we should also specify that a floc has been exposed to a set of conditions for long enough so that its microbial composition (the ratios of the different types of micro-organisms it contains) has reached a steady-state that is optimal for the particular wastewater. There are several ways to find what a particular set of rate equations predicts about balanced growth. The most convenient is to follow what happens when a fioc particle containing a mass MA0 of active biomass and Ms0 of stored substrate enters a completely mixed tank where it is exposed to a constant substrate concentration S. Note that these values are not concentrations but the mass per floc, and that if the floc breaks up in the tank then they include the total active biomass and stored substrate in all the resulting fragments. These masses will change with the time, t, that the floc spends in the reactor.
1538
NANDANPADUKONEand GRAHAMF. ANDREWS
The mass conservation equations for this floc are:
shows that they are identical, with equivalent parameters
active biomass: kd =km Y dMA
dt
(4a)
= taMA
tam = k2.)'m+km Y K = k2Ym/k ~Y.
stored substrate: (4b)
dt --MA q - - Y boundary condition: MA=MA0
Ms=Ms0
at t =0.
These can be reduced to a single equation for the floc composition y = M s / M A. Substituting from rate equations (2) and (3) and converting to dimensionless variables gives this equation as: dv'
"-dt'
=
Eye,
-
Ey'
S'-km
y(,---
-
y,2
E=
S ' "q- 1
S'+I
(5)
YYm
Physically, we know that there will be lag period during which the floc will become acclimated to the (constant) conditions. After this lag period y ' will remain constant at a "balanced growth" value y ~, which can be found by setting dy'/dt = 0 in equation (5) and solving the resulting quadratic equation [the same result can be obtained by solving equation (5) and letting t ' ~ ~ , but this is unnecessarily complicated] (6) Since YYm is of order 0.2 the inequality ( S ' q'- 1) 2 -
YYm<< -
( s ' - k;.)
(7)
will certainly be true both for large and small (S '-~ kin) substrate concentrations. It follows that y~<
tab=k2yb=k2Ym -
(8)
Comparing this with the usual Monod model with a cell-decay term: tab
tamS K+S
kd
(9)
(10)
This equivalence is very useful both as a demonstration that the assumed form of the rate equations is feasible (it does not prove they are correct; other forms may also reduce to the Monod equation) and as a way of finding values for the model parameters from known values of the Monod parameters. Typical values for the Monod parameters for the floc in a system treating domestic sewage have been taken as K = 35 mg carbon/l, tam = 0.1 h -1. kd = 0.0025 h -j, Y = 0.30. Assuming a value of Ym= 0.4 (Andrews and Tien, 1977) allows us to calculate from equations (9) that k~ =0.0t l/(mgh), k 2 = 0 . 2 6 h -~, k m= 0.008 h ~. These values have been used in subsequent analyses. The fact that the maintenance term in the structured model becomes a biomass decay term in the unstructured model is not unreasonable. In the structured model "biomass" consists of both protoplasm and stored substrate, so the consumption of stored substrate for maintenance does represent a net decay of biomass. Mass consert~ation equations in a stirred tank
The reactors in an activated sludge system are usually modelled as completely mixed tanks or a series of such tanks. However, a rigorous analysis of the problem of combining structured kinetic models with the wide residence time distribution of stirred tanks shows that the apparent simplicity of such models can be misleading. The difficulty lies in the floc composition, given in this model by the parameter y. Taking a sample of biomass from the reactor or the effluent (they would be the same in a completely mixed reactor) would show some flocs had been in the reactor for a very short time, so that their composition had hardly changed from that of the incoming flocs (Y0)- Other flocs will have been in the reactor for a very long time and their composition will approach the quasi-steady state value (Yb) given by equation (8). Most of the flocs will have compositions distributed between these two extremes. With such a wide variation of floc composition [unless Y0 =Yb as it is in a complete-mix activated sludge (CMAS) system if metabolic activity in the settler is ignored] how can we calculate the substrate uptake rate? Consider the simplest case, a completely mixed aeration tank whose influent contains a concentration n flocs/l all of which are identical in size and composition. It will be necessary to assume that if a floc breaks up due to shear or the wash-off of new cells from the surface, then all fragments from a given floc leave the reactor at the same time. Since the residence
1539
Model for activated sludge processes time distribution function for complete mixing is
mathematical fact that the average value of a function
e-'/°/O (Levenspiel, 1974), the number of floes that
f ( x ) in some range of x equals the function evaluated at the average value of x, only if f ( x ) is a linear
have spent a time between t and (t + dt) in the reactor is Vn e -"° dt/O (all fragments resulting from a single floc being counted as one). The total concentrations of active biomass and stored substrate in the reactor are related to the mass in each floc particle by: XA= ~
MA e-'/°dt
X s = ~n f ; Mse-'/°dt.
(11)
The specific growth rate and substrate uptake rate (/~ and q) for a particular floe particle are functions of the floe composition (y) and therefore vary from floc to floc depending on the time that the floe particle has spent in the reactor (t). The total growth rate and substrate uptake rate must be found by integrating over all possible values of t. The mass balance equations are therefore: substrate: So - S = n
f;
M Aq e-'/° dt
(12a)
X k - XA0 = n ; ; MA# e -t/° dt
(12b)
active biomass:
function. For all other types of rate equation, setting the average values of q and # equal to the values at the average biomass composition can only be approximately correct. This caution applies to, among others, the IAWPRC model in which the hydrolysis rate of adsorbed organics is a non-linear, Monod-type function of y (called Xs/Xau by Henze et al., 1987). Using the "average composition" approximation with these more complex rate equations may be the only way to make the problem mathematically tractable [solving equations (4) for the IAWPRC model would be difficult], but it is not exact and would introduce an unknown amount of error. Note that in an ideal plug flow tank all the floes have identical residence times so no distribution of floe compositions will develop and modelling becomes, at least from this viewpoint, simpler. Unfortunately a real tank not only has a mixing condition somewhere between the extremes of plug flow and complete mixing, but also may have a distribution of floc sizes and compositions in the influent. A rigorous model for this situation would be impossibly complex. Equations (12) and (13) give the solution in dimensionless variables as" XA -
stored substrate:
Xs -- Xso = n
f0 (
M A q -- ~
f0
MA#e-'/°dt =k2n
k2n
f;
f;
MAye-t/°dt
M s e -'/° dt = Ok2Xs = Ok2ymXA
(13a)
n f f M Aq e-'/°dt =klSn fo~ M a ( 1 - y ) e-t/°dt = klSn f f M a e - ~ / ° - Ms - - e -I, o dt Ym =k'SnO[XA---~m]=OXAkIS(1--YA~V']ym/" (13b) This proves that it is valid to evaluate q and/~ using the rate equations for a floc [equations (2) and (3)] and the average value of floc composition YAV XS/XA" However, it must be emphasized that this is only true because the rate equations adopted here are linear in both XA and X s. It results from the =
1 =
Y~v =
Equations (12) are true for any form of the rate equations. The rigorous procedure for solving them would be to solve equations (4) for y and X A as functions of time and to use the resulting expressions to evaluate the integrals in (12). Fortunately the specific form of the rate equations used here [equations (2) and (3)] allows a considerable simplification of this procedure.
n
-
XA0
So
--
S
-
-
(14a)
-
1 -- O'y'AV
y~ + EO'y~, 1 + O'(E
1 = O'X'Ao
(14b)
+y~)
1 -Y~v ,
l -- OyAv
.
(14C)
As expected, equation (14b) gives Y~v = Y~ for a short residence time (0 -~ 0), and it reduces to Y~v = Y(, for very long residence times (0 ~ ~ ) if E >>y~. (Recall that the balanced-growth situation discussed previously reduced to y ~ = y~, if E>>y~,; the condition is slightly different for Y~v because it is affected by the residence time distribution of the floes.) In the aeration tank of a conventional CMAS system y~ = Y~v (assuming no metabolic activity in the settler) and (14b) reduces to the exact balanced-growth equation for y ' [a quadratic equation with the solution given by equation (6)]. Equations (14) contain five dimensionless parameters, the hydraulic residence time, 0, the amount and condition of the incoming cells, X~0 and y~, and E and y~, which are related to the ultimate "storage capacity" of the biomass and its "storage capacity" under balanced growth conditions. Obtaining numerical solutions is complicated by the fact that E and y~, both depend on S'. The best procedure depends on what values are specified. In a design situation the required S ' value is known so E and y~, can be calculated knowing Ym and the Monod parameter values. Then if the condition of the incoming biomass
1540
NANDAN PADUKONE and GRAHAM F. ANDREWS
is specified, )'~,v can be calculated from 14(b) for a given residence time, and 14(c) gives the relationship between the inlet substrate concentration $6 and the amount of biomass X~0 required to achieve the given S' value. If the objective is to calculate the S' value for given values of 0', S~, X~,0, y~, Ymand the Monod parameters, then y l v can be eliminated between 14(b) and (c) leaving a complex quadratic equation for S'. FLOC COMPOSITION AND SLUDGE SETTLING
The only type of activated sludge system in which all growth is balanced is the conventional, CMAS system operating at steady state. A floc may be recycled around the system but it encounters the same S value throughout the aeration tank and, if metabolic activity in the settler is ignored, all around the recycle loop. It therefore remains perfectly acclimated in a way that is not possible in a contact stabilization unit, or in a plug-flow system in which conditions vary along the aeration tank and flocs acclimated to the aeration tank effluent conditions are recycled to the inlet. If there are no cells in the settler overflow, the growth rate of active biomass in the C M A S system (VpXA) must be balanced by the wastage rate. Using equation (3) for p gives the mean cell residence time as:
1
0~= ~ .
(15)
When combined with the previous discussion about including extra-cellular biopolymer in the stored substrate this equation may clarify the poor settling behaviour and appearance of pin-point floc observed at very long mean cell residence times (Bisogni and Lawrence, 1971). As 0x --, ~ , y ~ 0, there are no polysaccharides to hold the flocs together and therefore no settling. This could never actually happen, even with no deliberate wasting of biomass, because it would be a self-limiting process. If flocculation stopped, more cells would appear in the settler overflow thus reducing the effective value of 0X, increasing y [equation (11)] and allowing flocculation to resume. Real extended aeration systems would operate at this limit. This highly simplified picture of the effect of sludge age on floc composition and thus on sludge settling is complicated by the presence of the three different categories of stored substrate discussed previously. These can be distinguished qualitatively in terms of time scales. Adsorbed colloids are short-term storage, quickly adsorbed and quickly degraded. Intracellular compounds such as PHB represent medium term storage. They require a greater investment of time and energy for the cell to manufacture, and would not be used until no more liquid-phase or adsorbed substrate was available. Extracellular polymers are long-term storage produced late in the growth cycle perhaps only after the cell contains significant
amounts of intracellular storage products. This would explain the apparent correlation between bioflocculation and PHB content reported by Crabtree et al. (1966) and the results of Parsons and Dugan (1971). The extracellular polymers are relatively resistant to biodegradation by non-acclimated cells but "old" cultures become acclimated to them resulting in floc dispersion (Pavoni et al., 1972). This suggests a systematic variation of stored substrate composition with 0~. At small 0x it is virtually all adsorbed and intracellular material because the cells have no time to produce extracellular polymers. This accounts for the poor settling observed with very small 0x in conventional systems and for why it can be improved by adding a sludge stabilization tank to provide the necessary time for converting one form of stored substrate to another. At larger 0x there is less total stored substrate [equation (15)] but a higher proportion of it is biopolymer so the flocs can be more cohesive and settle better. At the limit of extended aeration virtually all the stored substrate may be biopolymer, but enzymes are being induced to depolymerize it, and there is not enough of it to ensure good settling (Bisogni and Lawrence, 1971). This very qualitative discussion shows a general agreement between what the model predicts about the composition of the flocs, and what is known about their settling behaviour. The agreement could obviously be improved if interchanges between different categories of stored substrate were incorporated into the model. However, quantitative predictions about settling would require modelling the floc size, which depends not only on its composition and cohesiveness but also on the intensity of shear in the reactor. CONTACT STABILIZATION
Unlike the conventional CMAS and extended aeration systems, contact stabilization (and to a lesser extent plug flow and step-feed systems) is specifically designed to take advantage of unbalanced growth. Substrate stored in the contact tank is degraded in the stabilization tank resulting in large changes in floc composition. Unstructured kinetic models can give little useful information but the ability to predict, at least semi-quantitatively, the performance of a contact stabilization system provides a useful first test for a structured model. A contact stabilization system consists of two tanks, usually modelled as completely mixed reactors. Mass balances can be written either over one tank and the entire system, or over each tank separately. The latter approach is preferred because the equations [14(a), (b) and (c)] are the same for each tank. Only the parameter values need to be changed. The contact tank
Equations (14) are plotted in Figs 1 and 2 as S/So and y vs dimensionless space time, 0', with the condition of the incoming cells (stored substrate
Model for activated sludge processes concentration, y~), as a parameter. The inlet concentrations S~ = 33.7 and X~0 = 160 are typical of those for a system treating domestic sewage (Metcalf & Eddy, 1977; Committee on Water Pollution Management, 1980). It should be noted that these are not concentrations based on the raw wastewater or the recycle stream but on the concentrations of substrate and cells when these two streams are mixed. Consider first a conventional system. This has a space time, t = 5 h (this is the actual space time = tank volume/total inflow rate including recycle flow) which, with the parameter values stated previously, corresponds to 0' = 0.5. Figure 1 shows that the condition of the incoming biomass (value of y0) has virtually no effect on the substrate removal. It follows that aerating the recycle stream to decrease Y0 under these conditions would be unproductive. However, if the space time, t, is reduced to 0.5 h or less, as it is in the contact stabilization process, the above conclusion will no longer be valid. Suppose 0' is set at 0.05. If the recycle sludge is not aerated then we require that the sludge entering the tank has the same composition as that which leaves it (y~ = y ' ) again assuming no activity in the settler. Figure 2 shows that at 0' = 0.5 this happens at y ' = 0.4. Now, going back to Fig. 1, it can be seen that the corresponding substrate removal is approx. 82%. By aerating the recycle stream in a stabilization tank most of the stored substrate will be either converted to biopolymer (see previous section) or removed from the sludge, altogether making Y0 smaller although not zero since this would imply deflocculation. As Y0 decreases the substrate removal in the contact tank increases towards 90% (Fig. 1) so there is now a definite advantage to aerating the recycled flocs. It is clear from Fig. 1 that this advantage increases rapidly as the space time of the contact tank decreases.
1541
1.0
S~=33.7 x~o= 16o
0.9 .~_ 0.8 P
0.7 0.6 0.5 0.4
"8
o 0.5
'~ 0.2 u_ 0.1
~
~o.4 i
,
,
,
,
,
,
,,
0.2
Oo
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0._0 0.45 0.50 Space time (8')
Fig. 2. Biomass composition leaving the contact tank.
The stabilization tank The major difference between the stabilization tank and the contact tank is that SG is lower and -¥~,0 is higher in the stabilization tank. Figures 3 and 4 show the predictions of the model for X~ = 375 which is typical for the stabilization tank. With this high cell/substrate ratio it is not surprising that Fig. 3 shows rapid removal of substrate from the liquid phase. (Note that, as in Fig. 1, the model predicts that S drops not to zero but to kin.) Figure 4, which is more important for design, shows how large a stabilization tank needs to be to remove specific amounts of stored substrate from the sludge (the same graph, with an extended abscissa, could in principle be used for the design of aerobic digestion systems). The system
"-
lo [
s0= 33T
0.9 ~
X~O=160
0.7'
1.0
o,m -i: 2 0.5 ~ 0 . 4 0.4 ~ 0 . 2 °.°
I \\l~-~--
0
~ono~ . . . . .
0.05 0.10 0.15 0.20 0.z5 0.30 0.35 0.40 0.45 0.50
Space time (8') Fig. 1. Substrate c o n c e n t r a t i o n leaving the contact tank.
To illustrate how Figs 1--4 describe the performance of the whole system, assume a contact tank size of 0 ' = 0.05 (corresponding to 0 = 0.05 h) and assume that y~ = 0.05 is the smallest desirable value consistent with proper flocculation of the incoming flocs. Figure 1 predicts 89% removal which fixes the inlet substrate concentration to the stabilization tank as S~ = 3.6 which is the value for which Figs 3 and 4 were drawn. The biomass composition leaving the contact tank, y ' = 0.15 (Fig. 2), becomes the inlet value for the stabilization tank. Interpolating on Fig. 4 shows that Y0 = 0.15 and Y~v = 0.05 requires a space time of approx. 0' = 0.2 or 0 = 2 h. The value of 5 h typical for the stabilization tank seems too conservative in this case. An interesting feature of this analysis is that it does not contain the usual parameters of recycle ratio, R, and mean cell residence time, 0x. These quantities can be calculated from mass balances at the wastage
1542
NANDANPADUKONEand GRAHAMF. ANDREWS 1.0
point, and the point where the recycle stream mixes with the incoming wastewater (concentration S~,) g0A1 + R ) = S~, + RSs XAoAI + R) = RXAs
XAc (I +YAvc)
0x =
(16c)
XA0cX~0s XA~ XA~
1
0.6
~o51 04
However 0~ does not appear as a central feature of the model in the way that it does for analyses of conventional CMAS systems. In fact, application of the 0x concept to the contact stabilization process represents a convenient extrapolation from experience with the CMAS system (Alexander et al., 1980) which has little theoretical justification since the underlying mechanisms (balanced vs unbalanced growth) are different. Greater basic insight into the operation of the contact stabilization system may show that its design and control should in fact be based on some other parameter. CONCLUSIONS Structured kinetic models can give some quantitative insight into the effect of the operating conditions on the composition of the biomass and thus on its settling behaviour. The mathematics can be simplified by lumping adsorbed colloidal material, intracellular storage products and extracellular polysaccharides into a single "stored substrate" term but this obscures some important, time-dependent differences between these categories. When choosing the rate equations for a structured kinetic model, the form to which they reduce
10
os ~'~'~'-
0.2
I.L
10
01
0.4 0.0
I 005
I 0.10
I I 015 020 Space
I 025 time
I I I I 0.30 0.35 0.40 0 4 5
102 0.50
(8')
Fig. 4. Biomass composition leaving the stabilization tank.
under balanced growth conditions, where floc composition is a function only of its local environment, must be considered. This is particularly important for modelling the conventional CMAS process in which the floes are inherently in balanced growth. The contact stabilization process involves unbalanced growth, and the combination of structured kinetic models and floc residence time distributions found in stirred tanks must be treated cautiously. The usual approach of basing the overall substrate uptake and growth rates on the average floc compositions is inexact for non-linear rate equations. A kinetic model has been presented that reduces to the Monod equation under balanced growth conditions, and for which the "average composition" approach is exact. It is believed to be the simplest model capable of giving a realistic description of the contact stabilization process. REFERENCES
s~=3.6
0.9
X;,o=37~
~1~ o.e
0.7 ~ 0.6 0u 0.5
~y~ = 1.O /0.8 0.4 ~ ! !O.6 .ID 0.3
0.4 02
0.2
0.0
0"1~ I ~ 1 0
XA0=575
0.7 c o0
XA0c ( 1 + YAVs)
S0=3.6
0.8
(16b)
0c+0~
b.
0,9
(16a)
0,05 010
=
t
i
=
~
,
~
I
0.15 0 . 2 0 0 . 2 5 0 . 3 0 0.35 0 , 4 0 0.45 0 . 5 0 Spoce
time
(8')
Fig. 3. Substrate composition leaving the stabilization tank.
Alexander W. V., Ekama G. A. and Marais G. R. (1980) The activated sludge process, Part 2. Application of the general kinetic model to the contact stabilization process. Wat. Res. 14, 1737 1747. Andrews G. F. and Tien C. (1977) New approach to bacterial kinetics in wastewater. J. envir. Engng Div. Am. Soc. cir. Engrs 103, 1057-1074. Benefield L. D. and Randall C. W. (1976) Design procedure for a contact stabilization activated sludge process. J. Wat. Pollut. Control Fed. 48, 147-152. Benefield L. and Mole F. (1984) A model for the activated sludge process. Biotechnol. Bioengng 26, 352-361. Bisogni J. J. and Lawrence A. W. (1971) Relationships between biological solids retention time and settling characteristics of activated sludge. War. Res. 5, 753-763. Busby J. B. and Andrews J. F. (1975) Dynamic modelling and control strategies for the activated sludge process. J. Wat. Pollut. Control Fed. 47, 1055-1080. Chen Y. R. and Hashimoto A. G. (1980) Substrate utilization kinetic model for biological treatment processes. Biotechnol. Bioengng 22, 2081-2095.
Model for activated sludge processes Committee on Water Pollution Management of the Environmental Engineering Division (1980) Engineering design variables for activated sludge processes. J. envir. Engng Div., Am. Soc. cir. Engrs 106, 473-503. Crabtree K., Boyle W., McCoy E. and Rohlich G. A. (1966) A mechanism of floc formation by Zooglea ramigera. J. Wat. Pollut. Control Fed. 38, 1968-1980. Da Hong J. and Ganczarczyk J. L. (1988) Stroboscopic determination of settling velocity, size and porosity of activated sludge flocs. War. Res. 21, 257-262. Fujie K., Tsubone T., Moriya H. and Kubota H. (1988) A simplified kinetic model to simulate soluble organic substances removal in an activated sludge aeration tank. War. Res. 22, 29-36. Gaudy A. F. Jr and Kincannon D. F. (1977) Comparing design models for activated sludge. Wat. Swge Wks 123, 66-70. Goodman B. L. and Englande A. J. (1974) A unified model of the activated sludge process. J. War. Pollut. Control Fed. 46, 312-332. Grady C. and Lira H. (1980) Biological Wastewater Treatment: Theory and Applications. Dekker, New York. Henze M., Grady C. P. L., Gujer W., Marais G. V. R. and Matsuo T. (1987) A general model for single sludge wastewater treatment systems. War. Res. 21, 505-515. Jones P. H. (1970) A mathematical model for the contact stabilization modification of the activated sludge process. Adv. Wat. Pollut. Res. 5, Paper II-5. Lawrence A. W. and McCarty P. L. (1980) Unified basis for biological treatment and design. J. sanit. Engng Div., Am. Soc. civ. Engrs 96, 757-778.
WR 23/12--F
1543
Levenspiel O. (1974) Chemical Reaction Engineering, p. 260. Wiley, New York. Metcalfe L. & Eddy H. P. (1977) Wastewater Engineering. Metcalf & Eddy. Mikesell R. (1981) Aerobic bacterial metabolic processes. J. envir. Engng Div., Am. Soc. cir. Engrs 107, 1261-1275. Mikesell R. (1984) New activated sludge theory: steady state. J. envir. Engng 110) 141-149. Orhon D. M. (1977) Discussion on design procedure for contact stabilization activated sludge process. J. Wat. Pollut. Control Fed. 49, 865-869. Orhon D. M. and Jenkins D. (1973) The mechanism and design of the contact stabilization activated sludge system. Adv. War. Pollut. Res. 7, 353-366. Parsons A. B. and Dugan P. R. (1971) Production of extracellular polysaccharide matrix by Zooglea ramigera. Appl. Microbiol. 21, 657-661. Pavoni J. L., Tenney M. W. and Echelberger W. F. (1972) Bacterial exocellular polymers and biological floculation. J. War. Pollut. Control Fed. 44, 414-431. Sheintuch M. (1987) Steady-state modelling of reactorsettler interaction. War Res. 21, 1463-1472. Tsezos M. and Bell J. P. (1988) Significance of biosorption for the hazardous organics removal efficiency of a biological reactor. Wat. Res. 22, 391-394. Ulrich A. H. and Smith M. W. (1951) The biosorption process of sewage and waste treatment. Swge lndust. Wastes 23, 1248-1253. Vavilin A. and Vasiliev L. (1985) A multispecies model of biological treatment with sequential pollutant oxidation. Biotechnol. Bioengng 27, 491-497.
0043-1354/89 $3.00+0.00 Copyright © 1989 Pergamon Press plc
A SIMPLE, CONCEPTUAL MATHEMATICAL MODEL FOR THE ACTIVATED SLUDGE PROCESS A N D ITS VARIANTS NANDAN PADUKONE1 a n d GRAHAM F. ANDREWS2 JDepartment of Chemical Engineering, North Carolina State University, Raleigh, NC 27695-7905 and 2Biotechnology Unit, Idaho National Engineering Laboratory, P.O. Box 1625, Idaho Falls, ID 83415, U.S.A. (First received February 1988; accepted in revised form June 1989)
Abstract--A simple structured kinetic model is applied to the activated sludge process. The objective is less to predict exact process performance than to illustrate some of the possibilities and difficulties in producing a comprehensive model for all the process variants. The rate equations are chosen so as to reduce to the Monod equation during balanced growth. Because these rate equations are linear, the cell growth and substrate uptake in a stirred tank can be defined exactly in terms of the average composition of the biomass. It is shown that this is not valid for other forms of rate equations. The stored substrate to protoplasm ratio in the fiocs is found to decrease with increasing mean cell residence time. If extracellular biopolymers are included in the stored substrate this corresponds qualitatively to observations of poor flocculation in extended aeration. The model is also applied to the contact stabilization process and is found to be in agreement with the essential process variables. Key
words--activated
sludge, mathematical modelling, contact stabilization, structured kinetic
models
NOMENCLATURE k~ = k2 = ka = km= K= n= q=
Subscripts
substrate uptake rate constant (l/rag h) growth rate constant (h - t ) cell decay rate constant (h -~ ) maintenance coefficient (h- ~) Monod half saturation constant (mg/l) number concentration of flocs in influent (1- ~) specific rate of substrate uptake (rag substrate/mg protoplasm h) S = liquid-phase substrate concentration (mg/l) V = reactor volume (1) XA = protoplasm concentration (mg/l) M A = mass of active biomass in a fioc Xs = stored substrate concentration M s = mass of stored substrate in a floc y = ratio of stored substrate to active biomass in a floc= M s / M h YAv= Xs/XA = average composition of floes in reactor yb = biomass composition during balanced growth (approximate) y ~ = b i o m a s s composition during balanced growth (exact) Y = cell yield (mg protoplasm/mg substrate) = specific growth rate (h-~) 0 = hydraulic residence time (h) 0x = mean cell residence time (h). Dimensionless variables E = ( S " + I)/ymY k'm = km Y/k2ym S" = k I YS /k2y m = S / K t' = k2Ymt X'Ao = kiXAo/k2ym = X A o / Y K O" = k2ymO Y' = Y /Ym "
c = contact tank influcnt value m = maximum value s = stabilization tank. 0 =
INTRODUCTION A m a t h e m a t i c a l model is a useful tool for the optim u m design a n d control of any process, mainly because the effects o f adjusting the o p e r a t i n g variables can be studied far m o r e quickly o n a c o m p u t e r t h a n by doing experiments. This is particularly true of a process like activated sludge biological wastewater t r e a t m e n t which m a y take days to reach a new steady-state after one of the process variables is changed. T h e need for a n accurate model of this process has long been recognized, a n d m a n y have been p r o p o s e d a n d tested ( G o o d m a n a n d Englande, 1974; G a u d y a n d K i n c a n n o n , 1977; Lawrence a n d M c C a r t y , 1980; C h e n a n d H a s h i m o t o , 1980). Even the simplest models, based o n M o n o d kinetics for cell growth ( G r a d y a n d Lim, 1980) offer a useful guide to sizing aeration tanks a n d illustrate the i m p o r t a n c e o f m e a n cell residence time as a control parameter. M o r e a d v a n c e d versions like t h a t produced by the I A W P R C task g r o u p (Henze et al., 1987) m a y ultimately allow a rational choice to be m a d e between process variants (contact stabilization, extended aeration etc.) for a particular wastewater.
1535
1536
NANDAN PADUKONEand GRAHAM F. ANDREWS
A truly comprehensive model would be extremely complex. Some of the difficulties that have been incorporated into one or more models are as follows: (1) As with all wastewater treatment processes the substrate is a complex and variable mixture of chemicals. Some of these may not be biodegradable and those that are may be degraded simultaneously or sequentially (Vavilin and Vasiliev, 1985). This is normally dealt with by the "pseudo-single solute" assumption by which the substrate concentration is defined in terms of a single measurement such as BOD or TOC. Of particular importance for activated sludge is the distinction between soluble organic material, which is biodegraded normally, and colloidal material, which adsorbs rapidly on the biomass and is subsequently slowly degraded (Fujie et al., 1988). (2) Both the concentration and composition of wastewaters change with time. Thus, although quasi-steady state models can be quite successful, only a transient model can give a truly accurate description of the process. {3) The biological flocs are also complex and variable mixtures of different species of heterotrophic bacteria (including filamentous organisms implicated in sludge bulking), autotrophic nitrifying bacteria, protozoa, extra-cellular polysaccharide material produced by the bacteria to bind them together, adsorbed colloidal substrate and inert material produced by decay of the cells or by adsorption of non-biodegradable wastewater constituents (both particulate and soluble; Tsezos and Bell, 1988). An unstructured kinetic model that considers these flocs just as "biomass" can never give an accurate description of their behaviour, so many types of structured models have been proposed. Some attempt to describe the interaction between the various species of bacteria with the many types of substrate in the wastewater (Vavilin and Vasiliev, 1985; Benefield and Mole, 1984). Mikesell (1981, 1984) focussed his attention on the production of biopolymers in the floc. Work on the contact stabilization process has stressed the importance of substrate stored in the flocs in the contact tank and then oxidized in the stabilization tank, although the complexities inherent in the use of these structured models have not always been appreciated [see the discussion by Orhon (1977) of the model proposed by Benefield and Randall (1976)]. The IAWPRC model (Henze et al., 1987) divides the flocs into active heterotrophic biomass, active autotrophic biomass, slowly degradable adsorbed colloidal matter and inert matter.
(4) The secondary settler is an integral part of the activated sludge process and must be included in a comprehensive process model (Busby and Andrews, 1975; Sheintuch, 1987). This requires knowledge of how the settling velocity of the floes depends on their size and composition and how these depend on process variables such as loading and sludge residence time. This is an area in which only generalizations are presently available; the presence of filamentous bacteria is associated with bulking sludge, contact stabilization tends to produce good settling behaviour while extended aeration may generate small floes that are hard to settle. Until new experimental techniques (Da Hong and Gaczarcyzk, 1988) measure settling velocity as a function of floc composition and structured kinetic models can predict floc size and composition as a function of proce3s variables (including shear in the aeration tank) most models will continue to treat the settler as a "separation point" with no biomass in the overflow. They will provide little information about the common problem of settler overloading. The complexities listed above mean that a workable comprehensive model will need to be assembled from the work of individual researchers, each working on one facet of the problem. The original objective of the work reported here was to develop the simplest possible kinetic model for the contact stabilization process that could incorporate both the storage/metabolism hypothesis for why this process works (Ulrich and Smith, 1951: Jones, 1970) and the growth/decay model of Orhon and Jenkins (1973). The subsequent development of the model was found to illustrate some more general problems and possibilities in the use of structured models, and to suggest a way in which a floc's settling is related to its composition. MODEL DEVELOPMENT
A complete model consists of four things; a description of its "structure", that is the components into which the substrate and the biomass are to be divided; a set of rate equations describing how rapidly one component is coverted into another by the various physical and biological processes; a description of the stoichiometry showing how much of a component is consumed or generated by a process: and finally a set of conservation equations describing the mass balance for each component in the particular type of reactor chosen. Structure and stoichiometry
The simplest structured kinetic model for activated sludge describes the substrate by a single parameter (BOD) and divides the biomass into an active cell (protoplasm) fraction and a stored substrate fraction
Model for activated sludge processes (Andrews and Tien, 1977; Busby and Andrews, 1975). The "reactions" can be described by:
Substrate S
upt~k~ Stored substrateq
XS
1537
Applying intuition to the process of adsorption of colloids onto the floc suggests that the adsorption
Growth
Active cell mass (protoplasm)
t~/ Y
XA
Maintenance
~ Respiration
+ Respiration products
(1) products.
k~
Two categories of "stored substrate" are usually identified, substrate which is adsorbed on the floc and intracellular storage products such as PHB and glycogen. However some part of the extracellular biopolymers that hold the floc together should also probably be included. The polymers are mainly polysaccharides and it seems reasonable to suppose that, if faced with starvation, the bacteria would re-hydrolyse them to make substrate. The distinction between the three categories of stored substrate is then mainly one of time scale. If flocs are aerated in the absence of fresh substrate for a few hours, as in a stabilization tank, the adsorbed material and intracellular storage products are consumed but there is little change in floc volume. If aeration is continued for tens of hours, as in aerobic digestion, the polymers are degraded and there is a marked reduction in volume. Rate equations
The rates in equation (1) (q,#/Y, km) are the specific rates (i.e. per unit active biomass) with which substrate is being taken up, used for growth and used for maintenance, respectively. Rate equations describe how these processes are affected by the substrate concentration, the condition of the floc etc. In choosing a set of rate equations there are three sources of information. The first is our understanding and intuition about how the process behaves. Second, and much less used, is our accumulated knowledge about how bacteria behave during "balanced growth". Balanced growth can be defined as the situation where the conditions to which the floc is exposed change so slowly that the composition of the biomass always remains perfectly acclimated to them. In this model the composition of a floc is described by the variable y = Ms/MA (i.e. the ratio of stored substrate to active biomass in a floc) and, mathematically, balanced growth corresponds to dy/dt ~ O. If we require that during balanced growth # is described by the Monod equation, then this imposes severe constraints on the form of rate equations that can be adopted. The third source of information is experimental data. Unfortunately this ultimate test may not be as conclusive as we would like; models based on two different sets of rate equations can sometimes describe a single set of data.
rate per unit of active biomass should be proportional to the surface area not already occupied by adsorbed material and to the colloid concentration in the liquid. The resulting rate equation has the form: q = k~S(1 - Y/Ym).
(2)
This form also seems reasonable for soluble substrate and is therefore adopted for the total substrate uptake. Ym is the maximum amount of stored substrate, as adsorbed material, intracellular storage products and bio-polymers, that can be associated with a unit of active biomass. Since the model [equation (1)] sees the bacteria growing on the stored substrate, it is reasonable to make the specific growth rate proportional to the amount of stored substrate available to a unit of protoplasm: # =k2y.
(3)
Balanced growth
Balanced growth occurs when the external conditions to which a cell is exposed (S in the model) change so slowly that its composition (y in the model) remains perfectly acclimated to them. For wastewater treatment we should also specify that a floc has been exposed to a set of conditions for long enough so that its microbial composition (the ratios of the different types of micro-organisms it contains) has reached a steady-state that is optimal for the particular wastewater. There are several ways to find what a particular set of rate equations predicts about balanced growth. The most convenient is to follow what happens when a fioc particle containing a mass MA0 of active biomass and Ms0 of stored substrate enters a completely mixed tank where it is exposed to a constant substrate concentration S. Note that these values are not concentrations but the mass per floc, and that if the floc breaks up in the tank then they include the total active biomass and stored substrate in all the resulting fragments. These masses will change with the time, t, that the floc spends in the reactor.
1538
NANDANPADUKONEand GRAHAMF. ANDREWS
The mass conservation equations for this floc are:
shows that they are identical, with equivalent parameters
active biomass: kd =km Y dMA
dt
(4a)
= taMA
tam = k2.)'m+km Y K = k2Ym/k ~Y.
stored substrate: (4b)
dt --MA q - - Y boundary condition: MA=MA0
Ms=Ms0
at t =0.
These can be reduced to a single equation for the floc composition y = M s / M A. Substituting from rate equations (2) and (3) and converting to dimensionless variables gives this equation as: dv'
"-dt'
=
Eye,
-
Ey'
S'-km
y(,---
-
y,2
E=
S ' "q- 1
S'+I
(5)
YYm
Physically, we know that there will be lag period during which the floc will become acclimated to the (constant) conditions. After this lag period y ' will remain constant at a "balanced growth" value y ~, which can be found by setting dy'/dt = 0 in equation (5) and solving the resulting quadratic equation [the same result can be obtained by solving equation (5) and letting t ' ~ ~ , but this is unnecessarily complicated] (6) Since YYm is of order 0.2 the inequality ( S ' q'- 1) 2 -
YYm<< -
( s ' - k;.)
(7)
will certainly be true both for large and small (S '-~ kin) substrate concentrations. It follows that y~<
tab=k2yb=k2Ym -
(8)
Comparing this with the usual Monod model with a cell-decay term: tab
tamS K+S
kd
(9)
(10)
This equivalence is very useful both as a demonstration that the assumed form of the rate equations is feasible (it does not prove they are correct; other forms may also reduce to the Monod equation) and as a way of finding values for the model parameters from known values of the Monod parameters. Typical values for the Monod parameters for the floc in a system treating domestic sewage have been taken as K = 35 mg carbon/l, tam = 0.1 h -1. kd = 0.0025 h -j, Y = 0.30. Assuming a value of Ym= 0.4 (Andrews and Tien, 1977) allows us to calculate from equations (9) that k~ =0.0t l/(mgh), k 2 = 0 . 2 6 h -~, k m= 0.008 h ~. These values have been used in subsequent analyses. The fact that the maintenance term in the structured model becomes a biomass decay term in the unstructured model is not unreasonable. In the structured model "biomass" consists of both protoplasm and stored substrate, so the consumption of stored substrate for maintenance does represent a net decay of biomass. Mass consert~ation equations in a stirred tank
The reactors in an activated sludge system are usually modelled as completely mixed tanks or a series of such tanks. However, a rigorous analysis of the problem of combining structured kinetic models with the wide residence time distribution of stirred tanks shows that the apparent simplicity of such models can be misleading. The difficulty lies in the floc composition, given in this model by the parameter y. Taking a sample of biomass from the reactor or the effluent (they would be the same in a completely mixed reactor) would show some flocs had been in the reactor for a very short time, so that their composition had hardly changed from that of the incoming flocs (Y0)- Other flocs will have been in the reactor for a very long time and their composition will approach the quasi-steady state value (Yb) given by equation (8). Most of the flocs will have compositions distributed between these two extremes. With such a wide variation of floc composition [unless Y0 =Yb as it is in a complete-mix activated sludge (CMAS) system if metabolic activity in the settler is ignored] how can we calculate the substrate uptake rate? Consider the simplest case, a completely mixed aeration tank whose influent contains a concentration n flocs/l all of which are identical in size and composition. It will be necessary to assume that if a floc breaks up due to shear or the wash-off of new cells from the surface, then all fragments from a given floc leave the reactor at the same time. Since the residence
1539
Model for activated sludge processes time distribution function for complete mixing is
mathematical fact that the average value of a function
e-'/°/O (Levenspiel, 1974), the number of floes that
f ( x ) in some range of x equals the function evaluated at the average value of x, only if f ( x ) is a linear
have spent a time between t and (t + dt) in the reactor is Vn e -"° dt/O (all fragments resulting from a single floc being counted as one). The total concentrations of active biomass and stored substrate in the reactor are related to the mass in each floc particle by: XA= ~
MA e-'/°dt
X s = ~n f ; Mse-'/°dt.
(11)
The specific growth rate and substrate uptake rate (/~ and q) for a particular floe particle are functions of the floe composition (y) and therefore vary from floc to floc depending on the time that the floe particle has spent in the reactor (t). The total growth rate and substrate uptake rate must be found by integrating over all possible values of t. The mass balance equations are therefore: substrate: So - S = n
f;
M Aq e-'/° dt
(12a)
X k - XA0 = n ; ; MA# e -t/° dt
(12b)
active biomass:
function. For all other types of rate equation, setting the average values of q and # equal to the values at the average biomass composition can only be approximately correct. This caution applies to, among others, the IAWPRC model in which the hydrolysis rate of adsorbed organics is a non-linear, Monod-type function of y (called Xs/Xau by Henze et al., 1987). Using the "average composition" approximation with these more complex rate equations may be the only way to make the problem mathematically tractable [solving equations (4) for the IAWPRC model would be difficult], but it is not exact and would introduce an unknown amount of error. Note that in an ideal plug flow tank all the floes have identical residence times so no distribution of floe compositions will develop and modelling becomes, at least from this viewpoint, simpler. Unfortunately a real tank not only has a mixing condition somewhere between the extremes of plug flow and complete mixing, but also may have a distribution of floc sizes and compositions in the influent. A rigorous model for this situation would be impossibly complex. Equations (12) and (13) give the solution in dimensionless variables as" XA -
stored substrate:
Xs -- Xso = n
f0 (
M A q -- ~
f0
MA#e-'/°dt =k2n
k2n
f;
f;
MAye-t/°dt
M s e -'/° dt = Ok2Xs = Ok2ymXA
(13a)
n f f M Aq e-'/°dt =klSn fo~ M a ( 1 - y ) e-t/°dt = klSn f f M a e - ~ / ° - Ms - - e -I, o dt Ym =k'SnO[XA---~m]=OXAkIS(1--YA~V']ym/" (13b) This proves that it is valid to evaluate q and/~ using the rate equations for a floc [equations (2) and (3)] and the average value of floc composition YAV XS/XA" However, it must be emphasized that this is only true because the rate equations adopted here are linear in both XA and X s. It results from the =
1 =
Y~v =
Equations (12) are true for any form of the rate equations. The rigorous procedure for solving them would be to solve equations (4) for y and X A as functions of time and to use the resulting expressions to evaluate the integrals in (12). Fortunately the specific form of the rate equations used here [equations (2) and (3)] allows a considerable simplification of this procedure.
n
-
XA0
So
--
S
-
-
(14a)
-
1 -- O'y'AV
y~ + EO'y~, 1 + O'(E
1 = O'X'Ao
(14b)
+y~)
1 -Y~v ,
l -- OyAv
.
(14C)
As expected, equation (14b) gives Y~v = Y~ for a short residence time (0 -~ 0), and it reduces to Y~v = Y(, for very long residence times (0 ~ ~ ) if E >>y~. (Recall that the balanced-growth situation discussed previously reduced to y ~ = y~, if E>>y~,; the condition is slightly different for Y~v because it is affected by the residence time distribution of the floes.) In the aeration tank of a conventional CMAS system y~ = Y~v (assuming no metabolic activity in the settler) and (14b) reduces to the exact balanced-growth equation for y ' [a quadratic equation with the solution given by equation (6)]. Equations (14) contain five dimensionless parameters, the hydraulic residence time, 0, the amount and condition of the incoming cells, X~0 and y~, and E and y~, which are related to the ultimate "storage capacity" of the biomass and its "storage capacity" under balanced growth conditions. Obtaining numerical solutions is complicated by the fact that E and y~, both depend on S'. The best procedure depends on what values are specified. In a design situation the required S ' value is known so E and y~, can be calculated knowing Ym and the Monod parameter values. Then if the condition of the incoming biomass
1540
NANDAN PADUKONE and GRAHAM F. ANDREWS
is specified, )'~,v can be calculated from 14(b) for a given residence time, and 14(c) gives the relationship between the inlet substrate concentration $6 and the amount of biomass X~0 required to achieve the given S' value. If the objective is to calculate the S' value for given values of 0', S~, X~,0, y~, Ymand the Monod parameters, then y l v can be eliminated between 14(b) and (c) leaving a complex quadratic equation for S'. FLOC COMPOSITION AND SLUDGE SETTLING
The only type of activated sludge system in which all growth is balanced is the conventional, CMAS system operating at steady state. A floc may be recycled around the system but it encounters the same S value throughout the aeration tank and, if metabolic activity in the settler is ignored, all around the recycle loop. It therefore remains perfectly acclimated in a way that is not possible in a contact stabilization unit, or in a plug-flow system in which conditions vary along the aeration tank and flocs acclimated to the aeration tank effluent conditions are recycled to the inlet. If there are no cells in the settler overflow, the growth rate of active biomass in the C M A S system (VpXA) must be balanced by the wastage rate. Using equation (3) for p gives the mean cell residence time as:
1
0~= ~ .
(15)
When combined with the previous discussion about including extra-cellular biopolymer in the stored substrate this equation may clarify the poor settling behaviour and appearance of pin-point floc observed at very long mean cell residence times (Bisogni and Lawrence, 1971). As 0x --, ~ , y ~ 0, there are no polysaccharides to hold the flocs together and therefore no settling. This could never actually happen, even with no deliberate wasting of biomass, because it would be a self-limiting process. If flocculation stopped, more cells would appear in the settler overflow thus reducing the effective value of 0X, increasing y [equation (11)] and allowing flocculation to resume. Real extended aeration systems would operate at this limit. This highly simplified picture of the effect of sludge age on floc composition and thus on sludge settling is complicated by the presence of the three different categories of stored substrate discussed previously. These can be distinguished qualitatively in terms of time scales. Adsorbed colloids are short-term storage, quickly adsorbed and quickly degraded. Intracellular compounds such as PHB represent medium term storage. They require a greater investment of time and energy for the cell to manufacture, and would not be used until no more liquid-phase or adsorbed substrate was available. Extracellular polymers are long-term storage produced late in the growth cycle perhaps only after the cell contains significant
amounts of intracellular storage products. This would explain the apparent correlation between bioflocculation and PHB content reported by Crabtree et al. (1966) and the results of Parsons and Dugan (1971). The extracellular polymers are relatively resistant to biodegradation by non-acclimated cells but "old" cultures become acclimated to them resulting in floc dispersion (Pavoni et al., 1972). This suggests a systematic variation of stored substrate composition with 0~. At small 0x it is virtually all adsorbed and intracellular material because the cells have no time to produce extracellular polymers. This accounts for the poor settling observed with very small 0x in conventional systems and for why it can be improved by adding a sludge stabilization tank to provide the necessary time for converting one form of stored substrate to another. At larger 0x there is less total stored substrate [equation (15)] but a higher proportion of it is biopolymer so the flocs can be more cohesive and settle better. At the limit of extended aeration virtually all the stored substrate may be biopolymer, but enzymes are being induced to depolymerize it, and there is not enough of it to ensure good settling (Bisogni and Lawrence, 1971). This very qualitative discussion shows a general agreement between what the model predicts about the composition of the flocs, and what is known about their settling behaviour. The agreement could obviously be improved if interchanges between different categories of stored substrate were incorporated into the model. However, quantitative predictions about settling would require modelling the floc size, which depends not only on its composition and cohesiveness but also on the intensity of shear in the reactor. CONTACT STABILIZATION
Unlike the conventional CMAS and extended aeration systems, contact stabilization (and to a lesser extent plug flow and step-feed systems) is specifically designed to take advantage of unbalanced growth. Substrate stored in the contact tank is degraded in the stabilization tank resulting in large changes in floc composition. Unstructured kinetic models can give little useful information but the ability to predict, at least semi-quantitatively, the performance of a contact stabilization system provides a useful first test for a structured model. A contact stabilization system consists of two tanks, usually modelled as completely mixed reactors. Mass balances can be written either over one tank and the entire system, or over each tank separately. The latter approach is preferred because the equations [14(a), (b) and (c)] are the same for each tank. Only the parameter values need to be changed. The contact tank
Equations (14) are plotted in Figs 1 and 2 as S/So and y vs dimensionless space time, 0', with the condition of the incoming cells (stored substrate
Model for activated sludge processes concentration, y~), as a parameter. The inlet concentrations S~ = 33.7 and X~0 = 160 are typical of those for a system treating domestic sewage (Metcalf & Eddy, 1977; Committee on Water Pollution Management, 1980). It should be noted that these are not concentrations based on the raw wastewater or the recycle stream but on the concentrations of substrate and cells when these two streams are mixed. Consider first a conventional system. This has a space time, t = 5 h (this is the actual space time = tank volume/total inflow rate including recycle flow) which, with the parameter values stated previously, corresponds to 0' = 0.5. Figure 1 shows that the condition of the incoming biomass (value of y0) has virtually no effect on the substrate removal. It follows that aerating the recycle stream to decrease Y0 under these conditions would be unproductive. However, if the space time, t, is reduced to 0.5 h or less, as it is in the contact stabilization process, the above conclusion will no longer be valid. Suppose 0' is set at 0.05. If the recycle sludge is not aerated then we require that the sludge entering the tank has the same composition as that which leaves it (y~ = y ' ) again assuming no activity in the settler. Figure 2 shows that at 0' = 0.5 this happens at y ' = 0.4. Now, going back to Fig. 1, it can be seen that the corresponding substrate removal is approx. 82%. By aerating the recycle stream in a stabilization tank most of the stored substrate will be either converted to biopolymer (see previous section) or removed from the sludge, altogether making Y0 smaller although not zero since this would imply deflocculation. As Y0 decreases the substrate removal in the contact tank increases towards 90% (Fig. 1) so there is now a definite advantage to aerating the recycled flocs. It is clear from Fig. 1 that this advantage increases rapidly as the space time of the contact tank decreases.
1541
1.0
S~=33.7 x~o= 16o
0.9 .~_ 0.8 P
0.7 0.6 0.5 0.4
"8
o 0.5
'~ 0.2 u_ 0.1
~
~o.4 i
,
,
,
,
,
,
,,
0.2
Oo
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0._0 0.45 0.50 Space time (8')
Fig. 2. Biomass composition leaving the contact tank.
The stabilization tank The major difference between the stabilization tank and the contact tank is that SG is lower and -¥~,0 is higher in the stabilization tank. Figures 3 and 4 show the predictions of the model for X~ = 375 which is typical for the stabilization tank. With this high cell/substrate ratio it is not surprising that Fig. 3 shows rapid removal of substrate from the liquid phase. (Note that, as in Fig. 1, the model predicts that S drops not to zero but to kin.) Figure 4, which is more important for design, shows how large a stabilization tank needs to be to remove specific amounts of stored substrate from the sludge (the same graph, with an extended abscissa, could in principle be used for the design of aerobic digestion systems). The system
"-
lo [
s0= 33T
0.9 ~
X~O=160
0.7'
1.0
o,m -i: 2 0.5 ~ 0 . 4 0.4 ~ 0 . 2 °.°
I \\l~-~--
0
~ono~ . . . . .
0.05 0.10 0.15 0.20 0.z5 0.30 0.35 0.40 0.45 0.50
Space time (8') Fig. 1. Substrate c o n c e n t r a t i o n leaving the contact tank.
To illustrate how Figs 1--4 describe the performance of the whole system, assume a contact tank size of 0 ' = 0.05 (corresponding to 0 = 0.05 h) and assume that y~ = 0.05 is the smallest desirable value consistent with proper flocculation of the incoming flocs. Figure 1 predicts 89% removal which fixes the inlet substrate concentration to the stabilization tank as S~ = 3.6 which is the value for which Figs 3 and 4 were drawn. The biomass composition leaving the contact tank, y ' = 0.15 (Fig. 2), becomes the inlet value for the stabilization tank. Interpolating on Fig. 4 shows that Y0 = 0.15 and Y~v = 0.05 requires a space time of approx. 0' = 0.2 or 0 = 2 h. The value of 5 h typical for the stabilization tank seems too conservative in this case. An interesting feature of this analysis is that it does not contain the usual parameters of recycle ratio, R, and mean cell residence time, 0x. These quantities can be calculated from mass balances at the wastage
1542
NANDANPADUKONEand GRAHAMF. ANDREWS 1.0
point, and the point where the recycle stream mixes with the incoming wastewater (concentration S~,) g0A1 + R ) = S~, + RSs XAoAI + R) = RXAs
XAc (I +YAvc)
0x =
(16c)
XA0cX~0s XA~ XA~
1
0.6
~o51 04
However 0~ does not appear as a central feature of the model in the way that it does for analyses of conventional CMAS systems. In fact, application of the 0x concept to the contact stabilization process represents a convenient extrapolation from experience with the CMAS system (Alexander et al., 1980) which has little theoretical justification since the underlying mechanisms (balanced vs unbalanced growth) are different. Greater basic insight into the operation of the contact stabilization system may show that its design and control should in fact be based on some other parameter. CONCLUSIONS Structured kinetic models can give some quantitative insight into the effect of the operating conditions on the composition of the biomass and thus on its settling behaviour. The mathematics can be simplified by lumping adsorbed colloidal material, intracellular storage products and extracellular polysaccharides into a single "stored substrate" term but this obscures some important, time-dependent differences between these categories. When choosing the rate equations for a structured kinetic model, the form to which they reduce
10
os ~'~'~'-
0.2
I.L
10
01
0.4 0.0
I 005
I 0.10
I I 015 020 Space
I 025 time
I I I I 0.30 0.35 0.40 0 4 5
102 0.50
(8')
Fig. 4. Biomass composition leaving the stabilization tank.
under balanced growth conditions, where floc composition is a function only of its local environment, must be considered. This is particularly important for modelling the conventional CMAS process in which the floes are inherently in balanced growth. The contact stabilization process involves unbalanced growth, and the combination of structured kinetic models and floc residence time distributions found in stirred tanks must be treated cautiously. The usual approach of basing the overall substrate uptake and growth rates on the average floc compositions is inexact for non-linear rate equations. A kinetic model has been presented that reduces to the Monod equation under balanced growth conditions, and for which the "average composition" approach is exact. It is believed to be the simplest model capable of giving a realistic description of the contact stabilization process. REFERENCES
s~=3.6
0.9
X;,o=37~
~1~ o.e
0.7 ~ 0.6 0u 0.5
~y~ = 1.O /0.8 0.4 ~ ! !O.6 .ID 0.3
0.4 02
0.2
0.0
0"1~ I ~ 1 0
XA0=575
0.7 c o0
XA0c ( 1 + YAVs)
S0=3.6
0.8
(16b)
0c+0~
b.
0,9
(16a)
0,05 010
=
t
i
=
~
,
~
I
0.15 0 . 2 0 0 . 2 5 0 . 3 0 0.35 0 , 4 0 0.45 0 . 5 0 Spoce
time
(8')
Fig. 3. Substrate composition leaving the stabilization tank.
Alexander W. V., Ekama G. A. and Marais G. R. (1980) The activated sludge process, Part 2. Application of the general kinetic model to the contact stabilization process. Wat. Res. 14, 1737 1747. Andrews G. F. and Tien C. (1977) New approach to bacterial kinetics in wastewater. J. envir. Engng Div. Am. Soc. cir. Engrs 103, 1057-1074. Benefield L. D. and Randall C. W. (1976) Design procedure for a contact stabilization activated sludge process. J. Wat. Pollut. Control Fed. 48, 147-152. Benefield L. and Mole F. (1984) A model for the activated sludge process. Biotechnol. Bioengng 26, 352-361. Bisogni J. J. and Lawrence A. W. (1971) Relationships between biological solids retention time and settling characteristics of activated sludge. War. Res. 5, 753-763. Busby J. B. and Andrews J. F. (1975) Dynamic modelling and control strategies for the activated sludge process. J. Wat. Pollut. Control Fed. 47, 1055-1080. Chen Y. R. and Hashimoto A. G. (1980) Substrate utilization kinetic model for biological treatment processes. Biotechnol. Bioengng 22, 2081-2095.
Model for activated sludge processes Committee on Water Pollution Management of the Environmental Engineering Division (1980) Engineering design variables for activated sludge processes. J. envir. Engng Div., Am. Soc. cir. Engrs 106, 473-503. Crabtree K., Boyle W., McCoy E. and Rohlich G. A. (1966) A mechanism of floc formation by Zooglea ramigera. J. Wat. Pollut. Control Fed. 38, 1968-1980. Da Hong J. and Ganczarczyk J. L. (1988) Stroboscopic determination of settling velocity, size and porosity of activated sludge flocs. War. Res. 21, 257-262. Fujie K., Tsubone T., Moriya H. and Kubota H. (1988) A simplified kinetic model to simulate soluble organic substances removal in an activated sludge aeration tank. War. Res. 22, 29-36. Gaudy A. F. Jr and Kincannon D. F. (1977) Comparing design models for activated sludge. Wat. Swge Wks 123, 66-70. Goodman B. L. and Englande A. J. (1974) A unified model of the activated sludge process. J. War. Pollut. Control Fed. 46, 312-332. Grady C. and Lira H. (1980) Biological Wastewater Treatment: Theory and Applications. Dekker, New York. Henze M., Grady C. P. L., Gujer W., Marais G. V. R. and Matsuo T. (1987) A general model for single sludge wastewater treatment systems. War. Res. 21, 505-515. Jones P. H. (1970) A mathematical model for the contact stabilization modification of the activated sludge process. Adv. Wat. Pollut. Res. 5, Paper II-5. Lawrence A. W. and McCarty P. L. (1980) Unified basis for biological treatment and design. J. sanit. Engng Div., Am. Soc. civ. Engrs 96, 757-778.
WR 23/12--F
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Levenspiel O. (1974) Chemical Reaction Engineering, p. 260. Wiley, New York. Metcalfe L. & Eddy H. P. (1977) Wastewater Engineering. Metcalf & Eddy. Mikesell R. (1981) Aerobic bacterial metabolic processes. J. envir. Engng Div., Am. Soc. cir. Engrs 107, 1261-1275. Mikesell R. (1984) New activated sludge theory: steady state. J. envir. Engng 110) 141-149. Orhon D. M. (1977) Discussion on design procedure for contact stabilization activated sludge process. J. Wat. Pollut. Control Fed. 49, 865-869. Orhon D. M. and Jenkins D. (1973) The mechanism and design of the contact stabilization activated sludge system. Adv. War. Pollut. Res. 7, 353-366. Parsons A. B. and Dugan P. R. (1971) Production of extracellular polysaccharide matrix by Zooglea ramigera. Appl. Microbiol. 21, 657-661. Pavoni J. L., Tenney M. W. and Echelberger W. F. (1972) Bacterial exocellular polymers and biological floculation. J. War. Pollut. Control Fed. 44, 414-431. Sheintuch M. (1987) Steady-state modelling of reactorsettler interaction. War Res. 21, 1463-1472. Tsezos M. and Bell J. P. (1988) Significance of biosorption for the hazardous organics removal efficiency of a biological reactor. Wat. Res. 22, 391-394. Ulrich A. H. and Smith M. W. (1951) The biosorption process of sewage and waste treatment. Swge lndust. Wastes 23, 1248-1253. Vavilin A. and Vasiliev L. (1985) A multispecies model of biological treatment with sequential pollutant oxidation. Biotechnol. Bioengng 27, 491-497.