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A stoichiometric model of bacterial growth
½'t.er Research Vol. 15. pp. 1363 to 1373. 1981 Printed in Great Britain. All rights reserved
IX)43-13M 81 121363-11S02.00:0 Copyright ~) 1981 Pergamon Press Lid
A STOICHIOMETRIC MODEL OF BACTERIAL GROWTH R, W. DENNIS 1 and R. L. IRVINE 2 1Exxon Research and Engineering Co., Florham Park, NJ 07932 and 2Department of Civil Engineering, University of Notre Dame, Notre Dame, IN 46556, U.S.A. Alastract--Numerous investigators have reported that the standard Monod model of bacterial growth does not adequately describe the transient response of activated sludge systems. By incorporating intracellular storage and extracellular release of shunted soluble carbon compounds in a model of bacterial growth and substrate utilization, a more accurate picture of the activated sludge process is obtained. The basic premise of the model developed in this paper is that a heterotrophic culture of bacteria will react with a carbonaceous substrate in the presence of oxygen and nutrients to produce more bacteria, carbon dioxide, water, various shunt products and storage products. The biomass is structured into three components: active biomass, stored biomass, and inert biomass. Three soluble components are considered: primary substrate, a secondary, Shunted, biodegradable substrate and a shunted non-biodegradable substrate. Stoichiometric coefficients and rate forms have been proposed for the formation of each of these components and for the rate of oxygen utilization. The model is shown to be effective in simulating the step response to a recycle reactor, a batch test with a high initial substrate concentration and the step response of a chemostat.
ger & Grady (1977l provides a review of the shunt phenomenon in biological waste treatment systems. Backoround Perhaps the most obvious and well known example of In biological waste treatment, two reactions, one shunt product formation in activated sludge is describing growth and the other death, are usually the production of ammonia nitrogen from a waste "considered in standard modeling approaches. The containing a high concentration of organic nitrogen rate forms for both reactions are customarily assumed i relative to the concentration of organic carbon. Folto be first order with respect to biomass. In addition, lowing are further examples. A number of experiments by Gaudy indicated the in the growth reaction, the substrate dependency is either first order or of the hyperbolic, Michaelis- extent of shunted carbon compounds in activated sludge. In 1973, Obayashi & Gaudy 11973) ran experiMenten form. Numerous investigators have reported that this ments to determine if the shunted waste products of standard approach does not accurately describe the growing cells were biodegradable. Using batch experitransient response of perturbed bacterial cultures ments, they concluded that the "microbial waste (Mateles et al., 1965; Koga & Humphrey, 1967; products are readily amenable to biological waste Storer & Gaudy, 1969). Although the standard treatment." The data suggested that there was little or approach is descriptive of steady state chemostats no storage of these waste components during their with pure cultures and pure feed (Monod, 1949), and removal from solution. In 1976, Krislman & Gaudy mixed cultures operating at steady state with a pure (1976) ran chemostat perturbation studies using a feed (Chin et al., 1972), the complex physical and bio- mixed culture and glucose feed. They stepped the feed logical processes that occur in a dynamic activated concentration by factors of 3 and 5 and noted the sludge system are not adequately addressed by this response of the system measuring both the soluble classical model. Two phenomena that occur in acti- COD attributable to glucose and the total soluble vated sludge systems have recently received consider- COD. In virtually all of their experiments, they obable attention in the literature. The first is intracellu- served a higher total COD than glucose COD at all lar storage of substrate. The second is the release to times. This was attributed to the "transient leakage of solution of shunted soluble carbon compounds. By organic carbon" and "metabolic products" produced incorporating storage and shunt into a model of bac- by the cell. terial growth, a more accurate picture of the activated In 1977, Gaudy et al. 11977) ran batch experiments to measure the volatile loss of carbon from an aerated sludge process is obtained. system fed glucose. He found that the volatile loss was Shunt products not substantial, however he noted a large difference For years, the fermentation industries have con- between the glucose COD and the total soluble COD. trolled shunt product manufacture. The production of The difference was attributed to "metabolic intervinegar, citric acid, and penicillin are examples of mediates", the concentrations of which reached a products shunted from a cell during aerobic mixing. maximum at the time of glucose depletion. The Ethanol manufacture is, of course, a time honored amount of glucose converted to intermediates ranged' example of anaerobic shunt product formation. Daig- from 12 to 86%. INTRODUCTION
1363
1364
R.W. DENNisand R. L. IRVlNE
Grady et al. (1972) examined the effluent from a mixed culture chemostat fed glucose. He noted that the effluent contained non-glucose COD and attributed this to intermediates and shunt products released from the cells. He stated that it would be logical to expect the excretion of products to be influenced by both the rate of substrate utilization per cell and the number of cells in the system. Grady & Williams (1975) demonstrated that the effluent COD concentration from a mixed culture chemostat fed a non-specific substrate was proportional to the inlet COD. He found that these effects could be satisfactorily modeled by the Michaelis-Menten equation in which the proportionality constant was a function of the inlet feed concentration. He later (Daigger & Grady, 1977) modeled these chemostat systems by using a curve fit, polynomial expression for the unit rate of shunt product formation.
ton (1971} tracked tagged glucose in batch cultures and found an immediate removal of glucose from solution (t5s) without a concomitant generation of tagged carbon dioxide until the glucose concentration in solution was low. In 1973, Speece et al. (1973) analyzed batch cultures for DNA and total mass concentrations. They reasoned that DNA concentration was indicative of the active cell number and storage could therefore be defined as an increase in total mass concentration without an increase in DNA concentration. More recently, Krui (1977) examined the activity of Zoogloea ramigera growing in flocs and suspensions and found that "substrate respiration may not be equal to the glucose uptake rate." In 1967, Waiters (1966) and Waiters et al. (1968) measured both the PHB and glycogen content in a mixed batch fed culture of activated sludge. He noted an increase in both storage products when substrate was present and a decrease when substrate was not Stora#e products present. When glycogen rich bacteria were aerated A number of researchers'have explained the appar- without substrate, the glycogen content decreased ently anamolous behavior of activated sludge systems while the capacity to remove glucose from solution in terms of a storage function. Using fill and draw increased. Waiters also observed that carbohydrate reactors treating dairy waste, Porges demonstrated was the ideal substrate for storage product formation. that the rate of oxidation was not equivalent to the In 1970. Takii (1970) ran batch experiments with glurate of substrate removal (Porges et al., 1955). In a cose and glucose acclimated activated sludge He series of articles (Hoover et al., 1951; Porges, 1955), came to the following conclusions: (1) "Activated that were more or less summarized in Biological sludge rapidly accumulated carbohydrate in the Treatment of Sewage and Industrial Wastes (Porges et sludge as soon as exogenous glucose was added. The al., 1956), Porges came to the following conclusion: carbohydrate was mainly composed of glycogen like "It is apparent that in the rapid purification of a polyglucose". (2) Lack of nitrogen and phosphorus carbohydrate waste, a portion of the chemical oxygen did not affect the accumulation of polyglucose but did demand (COD) is converted to COz, another portion retard the rate of polyglucose utilization during staris synthesized into cell complex with a low immediate vation. (3) During starvation, the ability to remove oxygen demand while the remainder is converted and glucose from solution increased with an increase in stored as an insoluble glycogen-like substance. The the capacity to accumulate polyglucose. (4) In actioxygen demand of the mixed sludge continues at a vated sludge obtained from conventional sewage high rate while the storage carbohydrate is oxidized" treatment plants, the capacity to accumulate polyglu(Porges et al., 1956). Porges measured the glycogen cose was lower than the glucose acclimated sludge, content of the ceils and found that it decreased dur- but still present. ing endogenous respiration. In 1977. Takii (1977) continued his work by analyzThe two major forms of storage products are poly- ing the bacteria from industrial waste treatment B-hydroxybutyric acid (PHB) and glycogen (Macrae plants and found that storage products (i.e. polyglu& Wilkinson, t955; Law & Slepecky, 1961; Dawes & cose) were common in activated sludge plants treating Ribbons, 1964; Williamson & Wilkinson, 1958). In carbohydrate wastes. The types of industries were 1966, Crabtree et al. (1966) tried to correlate intra- confectionery, fruit juice, soft drink and brewery. He cellular PHB concentration with the settling charac- also concluded that PHB was not a major storage teristics of activated sludge. He concluded that the product, rapid accumulation of PHB by Zoogloea ramioera is Many kinetic models of activated sludge have inintimately associated with flocculation. This was later cluded the concept of substrate storage within the questioned by Forster (1976), however Crabtree did biomass. In 1970, Dohonyas et al. (1970) proposed a observe the accumulation of PHB in the floc during detailed model that included reversible substrate acperiods of high substrate tension and the utilization of cumulation in the cell, reversible substrate storage from this cellular substrate pool and irreversible subPHB during endogenuous metabolism. In the same year, Rao (1966) observed that "the strate metabolism from the stored substrate pool or uptake of carbohydrate which occurs when starved the cellular substrate pool. Other researchers (Jacceils are placed in a fresh medium requires less energy quart et al., 1973; Busby & Andrews, 1975; Andrews and therefore consumes less oxygen than the synthetic & "lien, 1977; Tsuro & Goda, 1978) have proposed processes which take place after the cells have models of activated sludge that included storage. adjusted to the medium." In 1971. Robarts & Kemp- Most of these structured models assume that sub-
A stoichiometric model of bacterial growth strate passes through a storage phase prior to metabolism. If the storage products are intracellular polyglucose/glycogen compounds, as Suggested by Waiters (1966) and Waiters et al. (1968) and Takii (1970), little biochemical justification can be found for insisting that these complex macromolecules be produced prior to active growth and reproduction of the bacteria. A more logical assumption is that polyglucose type storage products are produced via a pathway that is independent of the primary metabolic pathway. The fact that the current storage models are descriptive of bacterial growth has been demonstrated previously. In 1967, Williams (1967) proposed a structured model of bacterial growth that included the conversion of external substrate to small metabolites followed by the conversion of small metabolites to macromolecules. WiUiam's model is stoichiometrically identical to these storage models. Clifft & Andrews (1981) have recently proposed a model that includes direct soluble substrate metabolism with concurrent storage of substrate. The model has been shown to be effective in predicting the dynamics of oxygen utilization in the activated sludge process. STOICHIOMETRY AND. KINETICS
Procedures
The modeling procedure used in this paper has been discussed by Irvine et al. (1980). The procedure involves proposing a reaction system that describes the relations among the physically important Variables. Each reaction represents a biochemical function of the bacterial culture (e.g. endogenous metabolism, shunt product manufacture, bacterial growth etc.). The reactions with yield coefficients define the stoichiometry of the model. Associated with each reaction is a rate that can be determined experimentally. This approach assumes that all the reactions occurring in a biological system can be lumped into a reasonal~le number of descriptive equations. Once the stoichiometry and kinetic rates have been established, mathematical expressions for the rates of formation of each component are obtained by multiplying the component yield coefficient by the rate of the reaction. Rates of component formation can be either negative (reactant) or positive (product) depending upon t h e sign of the yield coefficient. The total component rate is obtained by summing the rates from each reaction. These rates are included as either source or sink terms in differential equations describing the hydraulic regime. The final set of differential equations has been traditionally referred to as the "model". In order to develop consistent stoichiometry, the concentrations of the components should be expressed in equivalent terms. In wastewater treatment systems, the substrate concentrations are usually expressed as oxygen equivalents while the biomass concentrations are expressed as weight mass. Because carbon is the main ingredient in substrate and the backbone of the bacterial cell, a simple and logical
1365
approch is to consider all the concentrations in terms of the carbon variable. Thus, the components of the biomass and the soluble substrate are expressed as carbon equivalents. The relationship between oxygen uptake rate and carbon utilization can be determined by the use of the respiratory quotient concept (RQ). The RQ is defined as the ratio of carbon dioxide produced to oxygen consumed on a molar basis. Varma et al. (1975) has shown that the average RQ of mixed heterOtrophic bacteria treating domestic sewage is 0.92. The RQ is known to be a function of the substrate composition, decreasing as the complexity of the substrate increases. Carbohydrates ordinarily exhibit a n RQ of 1.0, while proteins and fats have lower RQ's of 0.80 and 0.72 respectively. With knowledge of the oxygen yield coefficient and a postulated RQ for a given reaction, a carbon balance on the reaction can be completed. Stoichiometr y
The basic premise of this model is that an aerobic, heterotrophic culture of bacteria will react with a carbonaceous substrate in the presence of oxygen and nutrients to produce more bacteria, carbon dioxide, water, various shunt products and storage products. The biomass is structured into three components: active biomass, stored biomass and inert biomass. Three soluble components are considered: primary substrate, secondary shunted substrate and a non-bindegradable shunted substrate. Because these components are expressed as carbon equivalents, the total organic carbon in solution is simply the sum of the primary, secondary and non-biodegradable substrates. The mixed liquor suspended solids concentration can be calculated using an assumed formula for biomass and storage products. Biomass is approx. 53% carbon while storage products are assumed to be polyglucose compounds and thus 40'7,,, carbon. This approach was used theoretically by Andrews & Tien (1977) and analytically by Therien & Perdrieux (1978). Figure 1 shows the system of reactions. As this figure illustrates, primary substrate (SI) can be converted to storage products (XS), shunt products (52) or directly metabolized to active biomass (XA). Secondary shunted substrate ($2) and storage products (XS) can be metabolized to active biomass (XA). Active biomass (XA) degrades to inert biomass (XI) or is oxidized to provide endogenous maintenance energy. Assoicated with each metabolic reaction, some nonbiodegradabe substrate (SNB) is leaked from the cell. Table 1 is a summary of the system stoichiometry with reaction descriptions and system components. Yield coefficients are subscripted with the reaction number and the ~mq~,,~.,,~. These coefficients are negative for reactants, positive for products and zero (0) for components not involved in a given reaction. The combining ratio of any two components in one reaction can be obtained by dividing their respective
1366
R. W. DENNIS and R. L. IRVINE S2
CO z ,
r
H20
7f i
SI
st S2
. F ~ m w y ~ -
~ra~
XA XI
• i m r t biomoss
Fig. 1. Conceptual model.
Table 1. Stoichiometric reactions Rate R I Storage product manufacture ( - i)SI + (0)XA + ( + Yz,xs) XS + ( + Yt.s.~a)SNB = 0 R2 Storage product oxidation ( - I)XS + ( - Y2.o.,) 02 + ( + Y2.x~)XA 4- ( + Y2.co3 CO., = 0 R3 Shunt product formation ( - I)SI + (0)XA + ( + Y3.s2) S2 + (+ Y3.s,~n)SNB = 0 R4 Primary substrate oxidation ( - I)SI + ( - Y,,.o,I 02 + ( + Y,.x~ XA + ( + E,.s-~) SNB 4- ( + Y4.co3 CO2 = 0 R5 Secondary suhstrate oxidation ( - 1) $2 + ( - Ys.o,) Oz + ( + Ys.xO XA ÷ (+ Ys.s~n) SNB + (+ Ys.co:) CO2 = 0 R6 Endogenous maintenance ( - I) XA + ( - Y~.o,) + ( + Y~.co) = 0 R7 Natural death ( - I ) XA + ( + Y-.xL) XI = 0 Model components: S! = primary substrat¢ $2 -- secondary shunted substrate S N B = non-biod,z'wadable shunted substrate XA = active biomass XS = storage products XI = inert biomass Y~.Mc = yield coefficient for model component MC in reaction k,
rl r2 r3 r4 r5 r6 r-:
A stoichiometric model of bacterial growth
1367
Table 2. Summary of yield coefficients Reaction R1 R2 R3 R4
R5
R6 R7
Coefficients YLxs ffi L0 Yij~s = 0.0 Yz.o2= 2.67 Y2.xA= 0.0 Y2.co~= 1.0 Yals2 = 1.0 Y3.SNB= 0.0 Y4.o~= 0.92 Y4.xA = 0.64 Y4.sNs = 0,02 Y4.co~ = 0.34 Y4.o2 = 0:92 YS.xA= 0.64 Y4:sNs = 0.02 Ys.co~ = 0.34 Yt.o: = 3.7 Y6.co~= 1.0 YT.xl = 1.0
Assumptions No release of non-biodegradable carbon during storage No growth associated with storage product utilization. Storage products are oxidized at a respiratory quotient of 1.0 to supply the energy of maintenance during starvation No release of non-biodegradable carbon during shunt
4 mol of ATP are produced mol- x of oxygen. 10.5g of dry cells are produced mol- i of ATP. The RQ = I. The yield coefficient for SNB production is 0.02
The respiratory coefficient is equal to 0.72 No shunt during natural death
yield coefficients. For example, Yk.co]Y~.o~ is the tration, the rate of shunt product manufacture (r3) grams of carbon dioxide produced as carbon per was considered to be zero. Storage product manufacgram of oxygen utilized in reaction k. This ratio is ture (r~) was theorized to occur only wh©n the conparticularly important in the reactions because it centration of primary substrate (S1) was above some defines the RQ. As described by Irvine et al. (1980) the limiting value and the storage capacity of the biomass yield coefficient for the major reactant in each reac- was not exceeded. Other researchers (Andrews & tion has been arbitrarily assigned a value of (-1). Tien, 1977) have suggested that the maximum storage Thus, the yield coefficients can be thought of as the capacity of the biomass can be as much as 40% of the grams of component combining with or produced active biomass on a carbon basis. Storage products from 1 g of the major reactant. With the exception of were considered to be the preferred endogenous suboxygen, the yield coeffg'ients are expressed in terms of strate during starvation. Therefore, storage products the carbon variable. The oxygen yield coefficients could be oxidized (r2) only when the primary and (Yk,o2) have units of grams of oxygen utilized per gram secondary substrate concentrations were below some limiting values. Because the oxidation of the storage of major reactant consumed. Table 2 summarizes the yield coefficients for the products met the endogenous maintenance energy resystem of reactions along with assumptions used to obtain the yields. The numbers are general and will vary depending upon organism type and substrate. Table 3. Kinetic expressions The detailed procedures for yield coefficient estimation are included in Dennis (1978). Using the carr, = K I C x ^ C s 1 * r 2 = K2CxA~ bon variable, the stoichiometric consistency of t h e r 3 = KaCxACsl + model can be easily checked, In each reaction, only r , = (K(CxACsl)/(Ks4 + Csll one reactant contains carbon and the yield coefficient r5 = (KsCxACs2J/(Ks5 + Csz + KiCsl) for this reactant is ( - 1). The sum of the product yield r 6 = K6CxA § coefficients is therefore (+1) for each reaction. In r7 -- K~CxA reaction No. 2, storage product oxidation, the yield of * Equal to zero when the primary substrate concenactive biomass (Yz.xA) was taken as zero. In this tration is below a limiting value or the storage capacity of model, storage products are assumed to be oxidized the system is exceeded. to provide energy of maintenance during starvation. t Equal to zero when exogenous substrate (either Use of storage products for growth can be easily primary or secondary) is present or when storage products ~ccommodated in the model by adjustment of this are depleted. Equal to zero when the primary substrate concen¢ield coefficient. tration is below a limiting value. § Equal to zero when r2 is not equal to zero. Kinetics Definitions: rh -- kinetic rate form for reaction k (rag I - i h - I) Rate forms for each reaction in the network are CMc = concentration of model component MC (rag I- ~} given in Table 3. The rationale for and development K~ = kinetic rate coefficient for reaction k of these rate forms are included in Dennis (1978). A Ks~,Kss = half rate constants for reactions 4 and 5 reswitching function was employed for some of the spectively (mg I- i) K~ = inhibition constant (dimensionless}. reactions. Below a limiting primary substrate concen-
1368
R.W. DENNISand R. L. IRVINE
quirement, the rate of endogenouS maintenance from Bruner et al. (1978) investigated an industrial waste active biomass (r6) was zero whenever r~ was positive. treatment plant treating corn milling wastes and conDuring growth and after depletion of the storage cluded that shunt product formation and glucose products, the energy of maintenance was supplied by repression were intimately related: It has been shown the oxidation of active biomass (r6) according to the that Zoogleea Ramiflera secretes galactose (Obayashi & Gaudy, 1973)and under certain conditions, glucose Herbert hypothesis (Herbert, 1958). All rates are dependent upon substrate type and the can inhibit the removal of galactose in a mixed culbacterial culture. Some substrates may be more easily ture (Ghosh et al., 1972). Orhon & Tunay (1979) supstored or shunted than others, while some species of ports the use of inhibition kinetics in biological waste bacteria may shunt or store more readily than others. treatment processes. Neufeld & Valiknac (1979) used The substrate and culture can affect not only the con- inhibition to describe toxic effects in activated sludge stants used in the rate forms, but also the forms of the plants. Given the complexity of a dynamically operrate equations. For example (Clifft & Andrews, 1981) ated mixed culture system treating a multi-component have suggested that the rate of storage product manu- substrate, some form of inhibition and/or repression facture is a function of the difference between the is likely occurring. As stated previously, the rate of formation of any maximum and actual storage product concentration. Because the activated sludge process uses a mixed model component is obtained by summing the culture with a complex substrate, the precise kinetics products of the component yield coefficient and the of any reaction are unknown. The most important respective rate equation. With the exception of oxyaspect of the modeling effort, however, is that the gen uptake, all rates are in units of mass of carbon per model structure is flexible enough to include phenom- volume, time. ena such as storage and shunt, even though the associated rate forms can only be approximated. r:.~c = ~. (Yh.~c)rk With the exception of shunt product oxidation (rs), the rate forms are standard. For the utilization of r:.Mc = rate of formation of model component MC primary substrate, the Michaelis-Menten equation Yk.~c = yield coefficient for model component MC has been selected. The other rate forms are either first in reaction k order with respect to active biomass concentration or r~ = rate of reaction k: first order with respect to active biomass and primary For example, the rate of formation of primary subsubstrate concentrations. This model includes the strate is given by, potential for primary substrate inhibition of secondr:.st = ( - 1 ) r l + (-1)r3 + ( - l ) r 4 , ary substrate removal. The extent of this competitive inhibition is controlled by the magnitude of the inhi- Primary substrate is always a reactant with a yield bition constant (K;) in rs. This rate form can be de- coefficient of ( - I). The rate of formation of storage rived from the quasi-steady state hypothesis of products is given by, enzyme kinetics. The use of inhibition kinetics is not a new concept in the field of biological waste treatment. r:.xs = (+ Yl.xs)rl + ( - l)r2.
k=l
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I'
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/ A I
I 2
I 3
, I 4
I 5
Time (h) Fig. 2. Recycle reactor simulation without storage.
I 6
I 7
A stoichiometric model of bacterial growth Table 4. Kinetic rate coefficients used in dynamic simulations Recycle reactor
Coefficients K I (1 mg- i h - t ) K2 (h" t) K 3 ( 1 mg- t h - t ) Ka (h- I) Ks4 (mg I - t ) Ks (h- t) Kss (mg I- 1) Ks (h- l) K~ (h- ') K~ (dimensionless)
Batch test
0.0, 0.05 0.018 0.0 0.3
Chemostat
0.0 0.0 0.003 0.1
1.5
0.0 0.018 0.0, 0.0 I 0.66 5.0 0.15 200.0 0.013 0.005 0.0
1.5
0.0 0.0 0.013 0.005 0.0
1.0 500.0 0.013 0.005 0.0
The rate of formation of active biomass is given by
rs.x^ = ( + Yz.x^)r2 + ( + Y4,x^r4 + ( + Ys.xA)rs + (-- l)re + (--l)rT. The oxygen uptake rate is expressed in mass of oxygen consumed per volume time. The rate is,
rs,o= = (-- Y2,o,)r2 + ( - Y,.o,)r,
+ (-- Ys.o,)rs + (-- Y6.o,)r 6. The oxygen uptake rate is negative because oxygen is a reactant in each reaction. RESULTS When the component rates of formation are placed into differential equations describing the hydraulic regime, closed form, analytical solutions are difficult to obtain. The equations were therefore solved numerically by fourth order Runge-Kutta integration (Dennis 1978). Three dynamic simulations were examined: (1) step feed to completely mixed reactor with
1369
recycle; (2) batch test; (3) step feed to a chemostat. The kinetic constants used in these simulations were chosen to demonstrate the capability of the model. The kinetics are incidental however, when compared to the selection of the major reactions that are described in Table 1. These reactions were chosen only after reviewing both the literature and considerable experimental data (Dennis, 1978).
Completely mixed recycle reactor Investigators (Therien & Perdrieux, 1978; BUsch, 1971) have found that when a recycle reactor is forced by a substrate step function, often there is no discernable difference in the soluble carbon concentration in the reactor. This is in contrast to the Monod model of bacterial growth and was explained (Busch, 1971) by the "reaction potential concept". However, the observation can also be explained by storage. Figure 2 shows the response of a recycle reactor when the soluble feed concentration is stepped from 60 to 240 mg l - ~ of total organic carbon (TOC). Kinetic constants are given in Table 4. Storage and shunt product manufacture were not considered in this simulation and the response of the system is the same as that predicted by the standard Monod model. Substrate concentration increases as soon as the step is imposed (1 h)while the biomass concentration adjusts more slowly. Figure 3 is a simulation of an identical system when storage is included in the response. At the time the step is imposed, the mixed liquor suspended solids concentration increases while the TOC in solution remains constant indicating a rapid incorporation of substrate into storage products. If the step function is large, the storage capacity of the biomass may be exceeded with a subsequent breakthrough in mixed liquor soluble TOC.
2300
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-
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-
s
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2000
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2(
J
,s
1900 i
~800L
~C
o
I I
I 2
I 3
I 4
I 5
Time(h)
Fig. 3. Recycle reactor simulation with storage,
l 6
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1370
R.W. DENNISand R. L. IRVINE
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.
.
.
.
.
n
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Primary substrate Sectmdary substmte Tolol solul~ carbon
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strate. In a low loaded, continuous flow system, reactions involving secondary substrate may not be appreciable because the concentration of primary substrate may never be high enough to trigger the shunt reaction. Therefore, kinetics developed for continuous flow systems should represent the metabolism of primary substrate. Because the rates of reaction for primary and secondary substrate metabolism are likely different, the kinetics obtained from a batch test may not be directly applicable to continuous flow systems. The need to understand the important reactions in order to interpret properly data collected from an experimental system is clearly demonstrated in this example.
2
3
4
Time ( h ) Fig. 4. Batch test simulation• Batch test
Controversy exists over the use of batch tests to obtain kinetic rate constants. The procedure for evaluation of the constants is straightforward. An acclimated culture is placed in an aerated reactor with a relatively high concentration of wastewater. The rate of substrate consumption is then monitored by soluble organic carbon, COD or BODs. Busch (1971), however, has found that the overall average unit rate of substrate removal in a continuous flow stirred tank reactor is consistently higher than predicted by batch results. The discrepancy may be attributable to the production and utilization of shunt products during the batch test. Figure 4 is a plot of primary, secondary and total substrate during a simulated batch test. Kinetic constants are given in Table 4. In this example, primary substrate is consumed in R4 and R3 and is quickly removed from solution. Storage was not considered in this simulation. Secondary substrate concentration increases steadily until the rate of utilization (rs) exceeds production (r3) and then it slowly decreases. After approx. 1 h, there is little difference between total substrate in solution and secondary substrate. If an experimeter is monitoring only soluble carbon and using the data to develop kinvtic constants, the constants so developed may not represent the ~ t i c s of primary suhstrate removal. In this case, the :kinvtic constants obtained from the total soluble carbon curve approximate the removal rate of secondary sub-
In bench scale reactors, evidence of shunt products is often obtained as a difference between the residual primary substrate and the actual soluble organic carbon concentration. Such analysis requires a pure, easily identified primary substrate such as glucose. Experiments by Krishnan and Gaudy (1976) on perturbed chemostats with glucose feed illustrate the production and utilization of shunt products in continu-, ous culture. These researchers stepped the inlet feed concentration of glucose and noted a slight increase in the reactor glucose concentration, but a substantial increase in the reactor soluble carbon concentration. Figure 5 shows the predicted response of a chemostar forced by a step increase in influent waste strength from 500 to 1500mg1-1 TOC. Secondary shunt products are not considered in this simulation, however non-biodegradable shunt is included: Kinetic constants are given in Table 4. Notice that the primary substrate concentration increases to a high level before tbe biomass concentration adjusts and the system returns to steady state. Figure 6 shows the response of an identical system when shunt product manufacture and utilization are considered• Notice the production of a non-primary substrate during the transient response. Similar responses have been observed in bench scale chemostats. These data can only be explained by the production and utilization of a secondary substrate.
CONCLUSIONS
Any model is essentially an attempt to describe a physical system. In mathematical modelling, the description is in the form of equations. The equations must therefore represent phenomena that are known to be present in the system. The microbiology and sanitary engineering literature include many examples of shunt product manufacture and utilization, and storage product formation and oxidation• By including these phenomena in a model o f bacterial growth and substrate utilization, one is better prepared to understand and predict the dynamics of activated sludge.
A stoichiometric model of bacterial growth
1371 17OO
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8
_e IlO0 ~ O (/3
150 900
too
Tolol r,dulNe corbon
Primory subltrote
°lo
,'5
2'o
°°
Time (h) Fig. 5. Chemostat simulation without secondary substrate production.
In any given system, the extent to which storage and/or shunt affect system dynamics will be dependent upon the bacterial population and the substrate. A particular hydraulic regime or aeration policy may favor the development of a population with a tendency to store, while under other conditions the selected population may have less capacity to store.
Some substrates may be readily shunted while others rapidly metabolized. The nature and degree of inhibition is dependent upon the primary and secondary substrate. Thus, each of the kinetic coefficients will vary depending upon' the type of organisms and the waste components. By adjusting the model coefficients and noting the response of the system on a 1700
500 Ks=0. I
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-17oo 1 25
Time (h)
Fig. 6. Chemostat simulation with secondary substrate production.
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R.W. DENNISand R. L. IRVINE
computer, one can determine whether or not shunt and/or storage are important considerations in the real system under examination. An example of this is given in Fig. 6. The response of the perturbed chemostat can only be explained by the production and utilization of a secondary substrate, and any model of these chemostats that does not consider shunt product formation is not representative of the real situation. Critics of structured, stoichiometric models may argue that the kinetic coefficients are too difficult to evaluate for a given system and thus, the models have little practical significance. However, by attempting to quantify natural phenomena, such as storage and shunt, in biological waste treatment systems, a practitioner may be led directionally toward an operating policy that would otherwise I~e overlooked. One may not be able to determine the kinetic coefficients precisely. If, however, the stoichiometry is correct and the kinetics are descriptive, one is better able to understand system dynamics than if "precise '° kinetics are used with inadequate stoichiometry. A more fundamental understanding will naturally lead to more reliable design procedures and more effective control strategies.
REFERENCES
Andrews G. F. & Tien C. (1977) New approach to bacterial kinetics in waste water. J. enrir. Engng Dir. Am. $oc. cir. Engrs 103, 1057. Bruner R. L., Smith J. H. & Schroeder A. C. 11978) Novel biochemical pathways in the big-oxidation of corn wetmilling wastes. 33rd Ind. Waste Conf., Purdue Unit'. Busby J. B. & Andrews J. F. 11975) A dynamic model and control strategies for the activated sludge process. J. War. Pollut. Control. Fed. 47, 1055. Busch A. W. (1971) Aerobic Biological Treatment o.I"Waste Waters. Oligodynamics Press, Houston, TX. Chiu S. Y., Eriekson L. E., Fam L. T. & Kao i. C. (19721 Kinetic model identification in mixed populations using continuous culture data. Biotech. Bioengng 14, 207. Clifft R. C. & Andrews J. F. (19811 A model for predicting the dynamics of oxygen utilization in the activated sludge process. J. Wat. Pollut. Control Fed. In press. Crabtree K.. Boyle W.. McCoy E. & Rohlich G. A. 119661 A mechanism of floc formation by Zoogloea ramlgera. J. 14~tt. Pollut. Control Fed. 3g, 1968. Daigger G. T. & Grady C. P. L. 119771 A model for the big-oxidation process based on product formation concepts. Water Res. 11, 1049. Dawes E. A. & Ribbons D. W. 119641 Some aspects of the endogenous metabolism of bacteria. Bact. Rer. 28, 126. Dennis R. W. 119781 A dynamic mathematical model of bacterial growth--application to sequencing batch biological reactors. Doctoral Dissertation. University of Notre Dame. Notre Dame, IN. Dohanyos M., Grau P. & Chudoba J. (19701 Kinetic assessment of glucose removal and saccharide accumulation capacities in activated sludge. Adeances in Water Pollution Research (Edited by Jenkins S: H3, Pergamon Press. New York. Forster D. F. (19761 Bioflooculation in the activated sludge process. Water S.A. 2, ll9. Ghosh S., Pohland F. A. & Gate W. F. (19721 Phasic utiliz-
ation of substrate by aerobic cultures. J. Wat. Pollut. Control Fed. 44, 376. Grady C. P. L. & Williams D. R. 119751 Effects of influent substrate concentration on the kinetics of natural microbial populations in continuous culture. Water Res. 9, 171. Grady C. P. U, Harlow L. J. & Riesing R. R. 119721 Effects of growth rate and influent substrate concentration on effluent quality from chemostats containing bacteria in pure and mixed culture. Biotech. Bioengng 14, 391. Gaudy A. F., Goswami S. R., Manickam F. & Gaudy E. 119771 Formation of strippable metabolic products in biological waste treatment. Biotech. Bioengng 19, 1239. Herbert D. (19581 Some principles of continuous culture. Recent Progress in Microbiology (Edited by Tunewall G.). Blackwell Scientific, Oxford. Hoover S., Pepinsky J. B., Jasewicz L. & Porges N. 119511 Aeration as a partial treatment for dairy wastes. Proc. 16th Ind. Waste Conf., Purdue Unit,. lrvine R. L., Alleman J. E., Miller G. & Dennis R. W. 0980) A basic approach to stoichiometry and kinetics using nitrogen removal in SBRs as an example. J. War. Pollut. Control Fed. 52, 1997. Jacquart J. C., Lefort D. & Rovel J. M. (19731 An attempt to take account of biological storage in the mathematical analysis of activated sludge behavior. Advances in Water Pollution Research (Edited by Jenkins S. H.). Pergamon Press, New York. Koga S. & Humphrey A. E. 119671 Study of the dynamic behavior of the chemostat system, Biotechnol. Bioengng 9, 345. Krishnan P. & Gaudy A. F. Jr 119761 Response of activated sludge to quantitative shock loading. J. Wat. Pollut. Control Fed. 411, 906. Krul J. M. (19771 Activity of Zoogloea ramigera growing in flocs and in suspension. Water Res. l !, 45. Law J. H. & Slepecky R. A. (1961) Synthesis and degradation of poly-B-hydroxybutyric acid in connection with sporulation of bacillus megaterium. J. Bact. 82, 37. Macrae R. W. & Wilkinson J. F. (1955) Poly-B-hydroxybutyrate metabolism in washed suspensions of bacillus cereus and bacillus megaterlum. J. gen. Microbiol. 19. 210. Mateles R. 1.. Ryu D. Y. & Yasuda T. (19651 Measurement of unsteady state growth rates of micro-organisms. Nature 208, 263. Monod J. 119491 The growth of bacterial cultures. A. Rer. Microbiol. 3. 371. Neufeld R. D. & Valiknac T. 11979) Inhibition of phenol biodegradation by thiocyanate. J. Wat. Pollut. Control Fed. 51, 2283. Obayashi A. W. & Gaudy A. F. Jr i1973) Aerobic digestion of extracellular microbial polysaceharides. J. Wat. Pollut. Control Fed. 45, 1584. Orphan D. & Tunay O. 119791 Mathematical models of biological waste treatment processes for the design of aeration tanks---discussion. Water Res. 13, 553. Porges N. 119551 Rapid big-oxidation method of waste disposal. Proc. Am. Soc. Brewing Chem. Porges N.. Jasewicz L. & Hoover S. (1955l Biochemical oxidation of dairy wastes VIII purification, oxidation. synthesis, and storage. Proc. lOth Ind. Waste Conf.. Purdue Unir. Porges N.. Jasewicz L. & Hoover S.. R. (19561 Principles of biological oxidation. Biological Treatment of Sewage and Industrial Wastes (Edited by MeCabe J. & Eckenfelder W. W.k Vol. 1, Reinhold. New York. Robarts R. D. & Kempton A. G. (19711 Use of C~4-glucose to study substrate removal by activated sludge. J, Wat. Pollut. Control Fed. 44, 1499. Rag B. S. & Gaudy A. F. (19661 Effects of sludge concentration on various aspects of biological activity in activated sludge. J. Wat. Poltut. Control Fed. 38. 794.
A stoichiometric model of bacterial growth Spcece R. E., Englebrecht R. S. & Aukamp D. R. (1973) Cell replication and biomass in the activated sludge process. Water Res. 7, 361. Storer F. F. & Gaudy A. F. (1969) Computational analysis of transient response to quantitative shock Ioadings of heterogenous populations in continuous culture. Envir. Sci. Technol. 3, 143. Takii S. (1970) Accumulation of glycogen-like polyglucose in glucose-acclimated activated sludge. J. Ferment. Technol. 48, 350. Takii S. (1977) Saccharide in activated sludge treating carbohydrate wastes. Water Res. 11, 78. Therien N. & Perdrieux S. (1978) Characterization of the dynamics of the activated sludge wastewater treatment process in terms of the carbon variable. 85th National Meeting AIChE, Philadelphia. Tsuno H. & Goda T. (1978) Kinetic model of activated
1373
sludge metabolism and its application to the response oT qualitative shock load. Water Res. 12, 513. Varma M. M., Wcekes M. C. & Calvert A. F. (1975) Kinetics of assimilation of mixed substrates by heterogeneous bacteria. J. War. Pollut. Control Fed. 47, 1913. Waiters C. F. (1966) Biochemical storage phenomena in activated sludge. Unpublished Doctoral Dissertation, University of Illinois, Urbana, 1L. Wahers C. F., Engelbrecht R. S. & Speece R. E. (1968) Microbial substrate storage in activated sludge. J. sanit. Engno Die. Am. Soc. cir. Engrs 94, 257. Williams F. M. (1967) A model of cell growth dynamics. J. theoret. Biol. 15, 190. Williamson D. H. & Wilkinson J. F. (1958) The isolation and estimation of the poly-B hydroxybutyrate inclusion of bacillus species. J. gen. Mierobiol. 19, 198.
IX)43-13M 81 121363-11S02.00:0 Copyright ~) 1981 Pergamon Press Lid
A STOICHIOMETRIC MODEL OF BACTERIAL GROWTH R, W. DENNIS 1 and R. L. IRVINE 2 1Exxon Research and Engineering Co., Florham Park, NJ 07932 and 2Department of Civil Engineering, University of Notre Dame, Notre Dame, IN 46556, U.S.A. Alastract--Numerous investigators have reported that the standard Monod model of bacterial growth does not adequately describe the transient response of activated sludge systems. By incorporating intracellular storage and extracellular release of shunted soluble carbon compounds in a model of bacterial growth and substrate utilization, a more accurate picture of the activated sludge process is obtained. The basic premise of the model developed in this paper is that a heterotrophic culture of bacteria will react with a carbonaceous substrate in the presence of oxygen and nutrients to produce more bacteria, carbon dioxide, water, various shunt products and storage products. The biomass is structured into three components: active biomass, stored biomass, and inert biomass. Three soluble components are considered: primary substrate, a secondary, Shunted, biodegradable substrate and a shunted non-biodegradable substrate. Stoichiometric coefficients and rate forms have been proposed for the formation of each of these components and for the rate of oxygen utilization. The model is shown to be effective in simulating the step response to a recycle reactor, a batch test with a high initial substrate concentration and the step response of a chemostat.
ger & Grady (1977l provides a review of the shunt phenomenon in biological waste treatment systems. Backoround Perhaps the most obvious and well known example of In biological waste treatment, two reactions, one shunt product formation in activated sludge is describing growth and the other death, are usually the production of ammonia nitrogen from a waste "considered in standard modeling approaches. The containing a high concentration of organic nitrogen rate forms for both reactions are customarily assumed i relative to the concentration of organic carbon. Folto be first order with respect to biomass. In addition, lowing are further examples. A number of experiments by Gaudy indicated the in the growth reaction, the substrate dependency is either first order or of the hyperbolic, Michaelis- extent of shunted carbon compounds in activated sludge. In 1973, Obayashi & Gaudy 11973) ran experiMenten form. Numerous investigators have reported that this ments to determine if the shunted waste products of standard approach does not accurately describe the growing cells were biodegradable. Using batch experitransient response of perturbed bacterial cultures ments, they concluded that the "microbial waste (Mateles et al., 1965; Koga & Humphrey, 1967; products are readily amenable to biological waste Storer & Gaudy, 1969). Although the standard treatment." The data suggested that there was little or approach is descriptive of steady state chemostats no storage of these waste components during their with pure cultures and pure feed (Monod, 1949), and removal from solution. In 1976, Krislman & Gaudy mixed cultures operating at steady state with a pure (1976) ran chemostat perturbation studies using a feed (Chin et al., 1972), the complex physical and bio- mixed culture and glucose feed. They stepped the feed logical processes that occur in a dynamic activated concentration by factors of 3 and 5 and noted the sludge system are not adequately addressed by this response of the system measuring both the soluble classical model. Two phenomena that occur in acti- COD attributable to glucose and the total soluble vated sludge systems have recently received consider- COD. In virtually all of their experiments, they obable attention in the literature. The first is intracellu- served a higher total COD than glucose COD at all lar storage of substrate. The second is the release to times. This was attributed to the "transient leakage of solution of shunted soluble carbon compounds. By organic carbon" and "metabolic products" produced incorporating storage and shunt into a model of bac- by the cell. terial growth, a more accurate picture of the activated In 1977, Gaudy et al. 11977) ran batch experiments to measure the volatile loss of carbon from an aerated sludge process is obtained. system fed glucose. He found that the volatile loss was Shunt products not substantial, however he noted a large difference For years, the fermentation industries have con- between the glucose COD and the total soluble COD. trolled shunt product manufacture. The production of The difference was attributed to "metabolic intervinegar, citric acid, and penicillin are examples of mediates", the concentrations of which reached a products shunted from a cell during aerobic mixing. maximum at the time of glucose depletion. The Ethanol manufacture is, of course, a time honored amount of glucose converted to intermediates ranged' example of anaerobic shunt product formation. Daig- from 12 to 86%. INTRODUCTION
1363
1364
R.W. DENNisand R. L. IRVlNE
Grady et al. (1972) examined the effluent from a mixed culture chemostat fed glucose. He noted that the effluent contained non-glucose COD and attributed this to intermediates and shunt products released from the cells. He stated that it would be logical to expect the excretion of products to be influenced by both the rate of substrate utilization per cell and the number of cells in the system. Grady & Williams (1975) demonstrated that the effluent COD concentration from a mixed culture chemostat fed a non-specific substrate was proportional to the inlet COD. He found that these effects could be satisfactorily modeled by the Michaelis-Menten equation in which the proportionality constant was a function of the inlet feed concentration. He later (Daigger & Grady, 1977) modeled these chemostat systems by using a curve fit, polynomial expression for the unit rate of shunt product formation.
ton (1971} tracked tagged glucose in batch cultures and found an immediate removal of glucose from solution (t5s) without a concomitant generation of tagged carbon dioxide until the glucose concentration in solution was low. In 1973, Speece et al. (1973) analyzed batch cultures for DNA and total mass concentrations. They reasoned that DNA concentration was indicative of the active cell number and storage could therefore be defined as an increase in total mass concentration without an increase in DNA concentration. More recently, Krui (1977) examined the activity of Zoogloea ramigera growing in flocs and suspensions and found that "substrate respiration may not be equal to the glucose uptake rate." In 1967, Waiters (1966) and Waiters et al. (1968) measured both the PHB and glycogen content in a mixed batch fed culture of activated sludge. He noted an increase in both storage products when substrate was present and a decrease when substrate was not Stora#e products present. When glycogen rich bacteria were aerated A number of researchers'have explained the appar- without substrate, the glycogen content decreased ently anamolous behavior of activated sludge systems while the capacity to remove glucose from solution in terms of a storage function. Using fill and draw increased. Waiters also observed that carbohydrate reactors treating dairy waste, Porges demonstrated was the ideal substrate for storage product formation. that the rate of oxidation was not equivalent to the In 1970. Takii (1970) ran batch experiments with glurate of substrate removal (Porges et al., 1955). In a cose and glucose acclimated activated sludge He series of articles (Hoover et al., 1951; Porges, 1955), came to the following conclusions: (1) "Activated that were more or less summarized in Biological sludge rapidly accumulated carbohydrate in the Treatment of Sewage and Industrial Wastes (Porges et sludge as soon as exogenous glucose was added. The al., 1956), Porges came to the following conclusion: carbohydrate was mainly composed of glycogen like "It is apparent that in the rapid purification of a polyglucose". (2) Lack of nitrogen and phosphorus carbohydrate waste, a portion of the chemical oxygen did not affect the accumulation of polyglucose but did demand (COD) is converted to COz, another portion retard the rate of polyglucose utilization during staris synthesized into cell complex with a low immediate vation. (3) During starvation, the ability to remove oxygen demand while the remainder is converted and glucose from solution increased with an increase in stored as an insoluble glycogen-like substance. The the capacity to accumulate polyglucose. (4) In actioxygen demand of the mixed sludge continues at a vated sludge obtained from conventional sewage high rate while the storage carbohydrate is oxidized" treatment plants, the capacity to accumulate polyglu(Porges et al., 1956). Porges measured the glycogen cose was lower than the glucose acclimated sludge, content of the ceils and found that it decreased dur- but still present. ing endogenous respiration. In 1977. Takii (1977) continued his work by analyzThe two major forms of storage products are poly- ing the bacteria from industrial waste treatment B-hydroxybutyric acid (PHB) and glycogen (Macrae plants and found that storage products (i.e. polyglu& Wilkinson, t955; Law & Slepecky, 1961; Dawes & cose) were common in activated sludge plants treating Ribbons, 1964; Williamson & Wilkinson, 1958). In carbohydrate wastes. The types of industries were 1966, Crabtree et al. (1966) tried to correlate intra- confectionery, fruit juice, soft drink and brewery. He cellular PHB concentration with the settling charac- also concluded that PHB was not a major storage teristics of activated sludge. He concluded that the product, rapid accumulation of PHB by Zoogloea ramioera is Many kinetic models of activated sludge have inintimately associated with flocculation. This was later cluded the concept of substrate storage within the questioned by Forster (1976), however Crabtree did biomass. In 1970, Dohonyas et al. (1970) proposed a observe the accumulation of PHB in the floc during detailed model that included reversible substrate acperiods of high substrate tension and the utilization of cumulation in the cell, reversible substrate storage from this cellular substrate pool and irreversible subPHB during endogenuous metabolism. In the same year, Rao (1966) observed that "the strate metabolism from the stored substrate pool or uptake of carbohydrate which occurs when starved the cellular substrate pool. Other researchers (Jacceils are placed in a fresh medium requires less energy quart et al., 1973; Busby & Andrews, 1975; Andrews and therefore consumes less oxygen than the synthetic & "lien, 1977; Tsuro & Goda, 1978) have proposed processes which take place after the cells have models of activated sludge that included storage. adjusted to the medium." In 1971. Robarts & Kemp- Most of these structured models assume that sub-
A stoichiometric model of bacterial growth strate passes through a storage phase prior to metabolism. If the storage products are intracellular polyglucose/glycogen compounds, as Suggested by Waiters (1966) and Waiters et al. (1968) and Takii (1970), little biochemical justification can be found for insisting that these complex macromolecules be produced prior to active growth and reproduction of the bacteria. A more logical assumption is that polyglucose type storage products are produced via a pathway that is independent of the primary metabolic pathway. The fact that the current storage models are descriptive of bacterial growth has been demonstrated previously. In 1967, Williams (1967) proposed a structured model of bacterial growth that included the conversion of external substrate to small metabolites followed by the conversion of small metabolites to macromolecules. WiUiam's model is stoichiometrically identical to these storage models. Clifft & Andrews (1981) have recently proposed a model that includes direct soluble substrate metabolism with concurrent storage of substrate. The model has been shown to be effective in predicting the dynamics of oxygen utilization in the activated sludge process. STOICHIOMETRY AND. KINETICS
Procedures
The modeling procedure used in this paper has been discussed by Irvine et al. (1980). The procedure involves proposing a reaction system that describes the relations among the physically important Variables. Each reaction represents a biochemical function of the bacterial culture (e.g. endogenous metabolism, shunt product manufacture, bacterial growth etc.). The reactions with yield coefficients define the stoichiometry of the model. Associated with each reaction is a rate that can be determined experimentally. This approach assumes that all the reactions occurring in a biological system can be lumped into a reasonal~le number of descriptive equations. Once the stoichiometry and kinetic rates have been established, mathematical expressions for the rates of formation of each component are obtained by multiplying the component yield coefficient by the rate of the reaction. Rates of component formation can be either negative (reactant) or positive (product) depending upon t h e sign of the yield coefficient. The total component rate is obtained by summing the rates from each reaction. These rates are included as either source or sink terms in differential equations describing the hydraulic regime. The final set of differential equations has been traditionally referred to as the "model". In order to develop consistent stoichiometry, the concentrations of the components should be expressed in equivalent terms. In wastewater treatment systems, the substrate concentrations are usually expressed as oxygen equivalents while the biomass concentrations are expressed as weight mass. Because carbon is the main ingredient in substrate and the backbone of the bacterial cell, a simple and logical
1365
approch is to consider all the concentrations in terms of the carbon variable. Thus, the components of the biomass and the soluble substrate are expressed as carbon equivalents. The relationship between oxygen uptake rate and carbon utilization can be determined by the use of the respiratory quotient concept (RQ). The RQ is defined as the ratio of carbon dioxide produced to oxygen consumed on a molar basis. Varma et al. (1975) has shown that the average RQ of mixed heterOtrophic bacteria treating domestic sewage is 0.92. The RQ is known to be a function of the substrate composition, decreasing as the complexity of the substrate increases. Carbohydrates ordinarily exhibit a n RQ of 1.0, while proteins and fats have lower RQ's of 0.80 and 0.72 respectively. With knowledge of the oxygen yield coefficient and a postulated RQ for a given reaction, a carbon balance on the reaction can be completed. Stoichiometr y
The basic premise of this model is that an aerobic, heterotrophic culture of bacteria will react with a carbonaceous substrate in the presence of oxygen and nutrients to produce more bacteria, carbon dioxide, water, various shunt products and storage products. The biomass is structured into three components: active biomass, stored biomass and inert biomass. Three soluble components are considered: primary substrate, secondary shunted substrate and a non-bindegradable shunted substrate. Because these components are expressed as carbon equivalents, the total organic carbon in solution is simply the sum of the primary, secondary and non-biodegradable substrates. The mixed liquor suspended solids concentration can be calculated using an assumed formula for biomass and storage products. Biomass is approx. 53% carbon while storage products are assumed to be polyglucose compounds and thus 40'7,,, carbon. This approach was used theoretically by Andrews & Tien (1977) and analytically by Therien & Perdrieux (1978). Figure 1 shows the system of reactions. As this figure illustrates, primary substrate (SI) can be converted to storage products (XS), shunt products (52) or directly metabolized to active biomass (XA). Secondary shunted substrate ($2) and storage products (XS) can be metabolized to active biomass (XA). Active biomass (XA) degrades to inert biomass (XI) or is oxidized to provide endogenous maintenance energy. Assoicated with each metabolic reaction, some nonbiodegradabe substrate (SNB) is leaked from the cell. Table 1 is a summary of the system stoichiometry with reaction descriptions and system components. Yield coefficients are subscripted with the reaction number and the ~mq~,,~.,,~. These coefficients are negative for reactants, positive for products and zero (0) for components not involved in a given reaction. The combining ratio of any two components in one reaction can be obtained by dividing their respective
1366
R. W. DENNIS and R. L. IRVINE S2
CO z ,
r
H20
7f i
SI
st S2
. F ~ m w y ~ -
~ra~
XA XI
• i m r t biomoss
Fig. 1. Conceptual model.
Table 1. Stoichiometric reactions Rate R I Storage product manufacture ( - i)SI + (0)XA + ( + Yz,xs) XS + ( + Yt.s.~a)SNB = 0 R2 Storage product oxidation ( - I)XS + ( - Y2.o.,) 02 + ( + Y2.x~)XA 4- ( + Y2.co3 CO., = 0 R3 Shunt product formation ( - I)SI + (0)XA + ( + Y3.s2) S2 + (+ Y3.s,~n)SNB = 0 R4 Primary substrate oxidation ( - I)SI + ( - Y,,.o,I 02 + ( + Y,.x~ XA + ( + E,.s-~) SNB 4- ( + Y4.co3 CO2 = 0 R5 Secondary suhstrate oxidation ( - 1) $2 + ( - Ys.o,) Oz + ( + Ys.xO XA ÷ (+ Ys.s~n) SNB + (+ Ys.co:) CO2 = 0 R6 Endogenous maintenance ( - I) XA + ( - Y~.o,) + ( + Y~.co) = 0 R7 Natural death ( - I ) XA + ( + Y-.xL) XI = 0 Model components: S! = primary substrat¢ $2 -- secondary shunted substrate S N B = non-biod,z'wadable shunted substrate XA = active biomass XS = storage products XI = inert biomass Y~.Mc = yield coefficient for model component MC in reaction k,
rl r2 r3 r4 r5 r6 r-:
A stoichiometric model of bacterial growth
1367
Table 2. Summary of yield coefficients Reaction R1 R2 R3 R4
R5
R6 R7
Coefficients YLxs ffi L0 Yij~s = 0.0 Yz.o2= 2.67 Y2.xA= 0.0 Y2.co~= 1.0 Yals2 = 1.0 Y3.SNB= 0.0 Y4.o~= 0.92 Y4.xA = 0.64 Y4.sNs = 0,02 Y4.co~ = 0.34 Y4.o2 = 0:92 YS.xA= 0.64 Y4:sNs = 0.02 Ys.co~ = 0.34 Yt.o: = 3.7 Y6.co~= 1.0 YT.xl = 1.0
Assumptions No release of non-biodegradable carbon during storage No growth associated with storage product utilization. Storage products are oxidized at a respiratory quotient of 1.0 to supply the energy of maintenance during starvation No release of non-biodegradable carbon during shunt
4 mol of ATP are produced mol- x of oxygen. 10.5g of dry cells are produced mol- i of ATP. The RQ = I. The yield coefficient for SNB production is 0.02
The respiratory coefficient is equal to 0.72 No shunt during natural death
yield coefficients. For example, Yk.co]Y~.o~ is the tration, the rate of shunt product manufacture (r3) grams of carbon dioxide produced as carbon per was considered to be zero. Storage product manufacgram of oxygen utilized in reaction k. This ratio is ture (r~) was theorized to occur only wh©n the conparticularly important in the reactions because it centration of primary substrate (S1) was above some defines the RQ. As described by Irvine et al. (1980) the limiting value and the storage capacity of the biomass yield coefficient for the major reactant in each reac- was not exceeded. Other researchers (Andrews & tion has been arbitrarily assigned a value of (-1). Tien, 1977) have suggested that the maximum storage Thus, the yield coefficients can be thought of as the capacity of the biomass can be as much as 40% of the grams of component combining with or produced active biomass on a carbon basis. Storage products from 1 g of the major reactant. With the exception of were considered to be the preferred endogenous suboxygen, the yield coeffg'ients are expressed in terms of strate during starvation. Therefore, storage products the carbon variable. The oxygen yield coefficients could be oxidized (r2) only when the primary and (Yk,o2) have units of grams of oxygen utilized per gram secondary substrate concentrations were below some limiting values. Because the oxidation of the storage of major reactant consumed. Table 2 summarizes the yield coefficients for the products met the endogenous maintenance energy resystem of reactions along with assumptions used to obtain the yields. The numbers are general and will vary depending upon organism type and substrate. Table 3. Kinetic expressions The detailed procedures for yield coefficient estimation are included in Dennis (1978). Using the carr, = K I C x ^ C s 1 * r 2 = K2CxA~ bon variable, the stoichiometric consistency of t h e r 3 = KaCxACsl + model can be easily checked, In each reaction, only r , = (K(CxACsl)/(Ks4 + Csll one reactant contains carbon and the yield coefficient r5 = (KsCxACs2J/(Ks5 + Csz + KiCsl) for this reactant is ( - 1). The sum of the product yield r 6 = K6CxA § coefficients is therefore (+1) for each reaction. In r7 -- K~CxA reaction No. 2, storage product oxidation, the yield of * Equal to zero when the primary substrate concenactive biomass (Yz.xA) was taken as zero. In this tration is below a limiting value or the storage capacity of model, storage products are assumed to be oxidized the system is exceeded. to provide energy of maintenance during starvation. t Equal to zero when exogenous substrate (either Use of storage products for growth can be easily primary or secondary) is present or when storage products ~ccommodated in the model by adjustment of this are depleted. Equal to zero when the primary substrate concen¢ield coefficient. tration is below a limiting value. § Equal to zero when r2 is not equal to zero. Kinetics Definitions: rh -- kinetic rate form for reaction k (rag I - i h - I) Rate forms for each reaction in the network are CMc = concentration of model component MC (rag I- ~} given in Table 3. The rationale for and development K~ = kinetic rate coefficient for reaction k of these rate forms are included in Dennis (1978). A Ks~,Kss = half rate constants for reactions 4 and 5 reswitching function was employed for some of the spectively (mg I- i) K~ = inhibition constant (dimensionless}. reactions. Below a limiting primary substrate concen-
1368
R.W. DENNISand R. L. IRVINE
quirement, the rate of endogenouS maintenance from Bruner et al. (1978) investigated an industrial waste active biomass (r6) was zero whenever r~ was positive. treatment plant treating corn milling wastes and conDuring growth and after depletion of the storage cluded that shunt product formation and glucose products, the energy of maintenance was supplied by repression were intimately related: It has been shown the oxidation of active biomass (r6) according to the that Zoogleea Ramiflera secretes galactose (Obayashi & Gaudy, 1973)and under certain conditions, glucose Herbert hypothesis (Herbert, 1958). All rates are dependent upon substrate type and the can inhibit the removal of galactose in a mixed culbacterial culture. Some substrates may be more easily ture (Ghosh et al., 1972). Orhon & Tunay (1979) supstored or shunted than others, while some species of ports the use of inhibition kinetics in biological waste bacteria may shunt or store more readily than others. treatment processes. Neufeld & Valiknac (1979) used The substrate and culture can affect not only the con- inhibition to describe toxic effects in activated sludge stants used in the rate forms, but also the forms of the plants. Given the complexity of a dynamically operrate equations. For example (Clifft & Andrews, 1981) ated mixed culture system treating a multi-component have suggested that the rate of storage product manu- substrate, some form of inhibition and/or repression facture is a function of the difference between the is likely occurring. As stated previously, the rate of formation of any maximum and actual storage product concentration. Because the activated sludge process uses a mixed model component is obtained by summing the culture with a complex substrate, the precise kinetics products of the component yield coefficient and the of any reaction are unknown. The most important respective rate equation. With the exception of oxyaspect of the modeling effort, however, is that the gen uptake, all rates are in units of mass of carbon per model structure is flexible enough to include phenom- volume, time. ena such as storage and shunt, even though the associated rate forms can only be approximated. r:.~c = ~. (Yh.~c)rk With the exception of shunt product oxidation (rs), the rate forms are standard. For the utilization of r:.Mc = rate of formation of model component MC primary substrate, the Michaelis-Menten equation Yk.~c = yield coefficient for model component MC has been selected. The other rate forms are either first in reaction k order with respect to active biomass concentration or r~ = rate of reaction k: first order with respect to active biomass and primary For example, the rate of formation of primary subsubstrate concentrations. This model includes the strate is given by, potential for primary substrate inhibition of secondr:.st = ( - 1 ) r l + (-1)r3 + ( - l ) r 4 , ary substrate removal. The extent of this competitive inhibition is controlled by the magnitude of the inhi- Primary substrate is always a reactant with a yield bition constant (K;) in rs. This rate form can be de- coefficient of ( - I). The rate of formation of storage rived from the quasi-steady state hypothesis of products is given by, enzyme kinetics. The use of inhibition kinetics is not a new concept in the field of biological waste treatment. r:.xs = (+ Yl.xs)rl + ( - l)r2.
k=l
230C FTOC
220CL
70I
1
1
1
M ~ B
60
S
•
~ ~5o
I'
:~ 2000 ~ 3oL20t--
I'
1900
~"
~o~-
,.ooL 4
/ A I
I 2
I 3
, I 4
I 5
Time (h) Fig. 2. Recycle reactor simulation without storage.
I 6
I 7
A stoichiometric model of bacterial growth Table 4. Kinetic rate coefficients used in dynamic simulations Recycle reactor
Coefficients K I (1 mg- i h - t ) K2 (h" t) K 3 ( 1 mg- t h - t ) Ka (h- I) Ks4 (mg I - t ) Ks (h- t) Kss (mg I- 1) Ks (h- l) K~ (h- ') K~ (dimensionless)
Batch test
0.0, 0.05 0.018 0.0 0.3
Chemostat
0.0 0.0 0.003 0.1
1.5
0.0 0.018 0.0, 0.0 I 0.66 5.0 0.15 200.0 0.013 0.005 0.0
1.5
0.0 0.0 0.013 0.005 0.0
1.0 500.0 0.013 0.005 0.0
The rate of formation of active biomass is given by
rs.x^ = ( + Yz.x^)r2 + ( + Y4,x^r4 + ( + Ys.xA)rs + (-- l)re + (--l)rT. The oxygen uptake rate is expressed in mass of oxygen consumed per volume time. The rate is,
rs,o= = (-- Y2,o,)r2 + ( - Y,.o,)r,
+ (-- Ys.o,)rs + (-- Y6.o,)r 6. The oxygen uptake rate is negative because oxygen is a reactant in each reaction. RESULTS When the component rates of formation are placed into differential equations describing the hydraulic regime, closed form, analytical solutions are difficult to obtain. The equations were therefore solved numerically by fourth order Runge-Kutta integration (Dennis 1978). Three dynamic simulations were examined: (1) step feed to completely mixed reactor with
1369
recycle; (2) batch test; (3) step feed to a chemostat. The kinetic constants used in these simulations were chosen to demonstrate the capability of the model. The kinetics are incidental however, when compared to the selection of the major reactions that are described in Table 1. These reactions were chosen only after reviewing both the literature and considerable experimental data (Dennis, 1978).
Completely mixed recycle reactor Investigators (Therien & Perdrieux, 1978; BUsch, 1971) have found that when a recycle reactor is forced by a substrate step function, often there is no discernable difference in the soluble carbon concentration in the reactor. This is in contrast to the Monod model of bacterial growth and was explained (Busch, 1971) by the "reaction potential concept". However, the observation can also be explained by storage. Figure 2 shows the response of a recycle reactor when the soluble feed concentration is stepped from 60 to 240 mg l - ~ of total organic carbon (TOC). Kinetic constants are given in Table 4. Storage and shunt product manufacture were not considered in this simulation and the response of the system is the same as that predicted by the standard Monod model. Substrate concentration increases as soon as the step is imposed (1 h)while the biomass concentration adjusts more slowly. Figure 3 is a simulation of an identical system when storage is included in the response. At the time the step is imposed, the mixed liquor suspended solids concentration increases while the TOC in solution remains constant indicating a rapid incorporation of substrate into storage products. If the step function is large, the storage capacity of the biomass may be exceeded with a subsequent breakthrough in mixed liquor soluble TOC.
2300
7C -
2200
BC
-
FTOC MLSS KI " 0.05 -
-
s
5(
E
I I
210C I
if) .J
s
2000
3C ,s"
2(
J
,s
1900 i
~800L
~C
o
I I
I 2
I 3
I 4
I 5
Time(h)
Fig. 3. Recycle reactor simulation with storage,
l 6
I 7
1370
R.W. DENNISand R. L. IRVINE
35C .
.
.
.
.
.
n
- -
3OC
Primary substrate Sectmdary substmte Tolol solul~ carbon
F"
~, 2so E g o
20(3
es
_= 0 ,(t)
I 5(3 50
!(" ~'~
/~
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"
oo/"~ •
i
50
~
\\
,
\
\ o o
I
strate. In a low loaded, continuous flow system, reactions involving secondary substrate may not be appreciable because the concentration of primary substrate may never be high enough to trigger the shunt reaction. Therefore, kinetics developed for continuous flow systems should represent the metabolism of primary substrate. Because the rates of reaction for primary and secondary substrate metabolism are likely different, the kinetics obtained from a batch test may not be directly applicable to continuous flow systems. The need to understand the important reactions in order to interpret properly data collected from an experimental system is clearly demonstrated in this example.
2
3
4
Time ( h ) Fig. 4. Batch test simulation• Batch test
Controversy exists over the use of batch tests to obtain kinetic rate constants. The procedure for evaluation of the constants is straightforward. An acclimated culture is placed in an aerated reactor with a relatively high concentration of wastewater. The rate of substrate consumption is then monitored by soluble organic carbon, COD or BODs. Busch (1971), however, has found that the overall average unit rate of substrate removal in a continuous flow stirred tank reactor is consistently higher than predicted by batch results. The discrepancy may be attributable to the production and utilization of shunt products during the batch test. Figure 4 is a plot of primary, secondary and total substrate during a simulated batch test. Kinetic constants are given in Table 4. In this example, primary substrate is consumed in R4 and R3 and is quickly removed from solution. Storage was not considered in this simulation. Secondary substrate concentration increases steadily until the rate of utilization (rs) exceeds production (r3) and then it slowly decreases. After approx. 1 h, there is little difference between total substrate in solution and secondary substrate. If an experimeter is monitoring only soluble carbon and using the data to develop kinvtic constants, the constants so developed may not represent the ~ t i c s of primary suhstrate removal. In this case, the :kinvtic constants obtained from the total soluble carbon curve approximate the removal rate of secondary sub-
In bench scale reactors, evidence of shunt products is often obtained as a difference between the residual primary substrate and the actual soluble organic carbon concentration. Such analysis requires a pure, easily identified primary substrate such as glucose. Experiments by Krishnan and Gaudy (1976) on perturbed chemostats with glucose feed illustrate the production and utilization of shunt products in continu-, ous culture. These researchers stepped the inlet feed concentration of glucose and noted a slight increase in the reactor glucose concentration, but a substantial increase in the reactor soluble carbon concentration. Figure 5 shows the predicted response of a chemostar forced by a step increase in influent waste strength from 500 to 1500mg1-1 TOC. Secondary shunt products are not considered in this simulation, however non-biodegradable shunt is included: Kinetic constants are given in Table 4. Notice that the primary substrate concentration increases to a high level before tbe biomass concentration adjusts and the system returns to steady state. Figure 6 shows the response of an identical system when shunt product manufacture and utilization are considered• Notice the production of a non-primary substrate during the transient response. Similar responses have been observed in bench scale chemostats. These data can only be explained by the production and utilization of a secondary substrate.
CONCLUSIONS
Any model is essentially an attempt to describe a physical system. In mathematical modelling, the description is in the form of equations. The equations must therefore represent phenomena that are known to be present in the system. The microbiology and sanitary engineering literature include many examples of shunt product manufacture and utilization, and storage product formation and oxidation• By including these phenomena in a model o f bacterial growth and substrate utilization, one is better prepared to understand and predict the dynamics of activated sludge.
A stoichiometric model of bacterial growth
1371 17OO
MLSS 30C
1500
%-o.o
t
1300
v
e-
E
8
_e IlO0 ~ O (/3
150 900
too
Tolol r,dulNe corbon
Primory subltrote
°lo
,'5
2'o
°°
Time (h) Fig. 5. Chemostat simulation without secondary substrate production.
In any given system, the extent to which storage and/or shunt affect system dynamics will be dependent upon the bacterial population and the substrate. A particular hydraulic regime or aeration policy may favor the development of a population with a tendency to store, while under other conditions the selected population may have less capacity to store.
Some substrates may be readily shunted while others rapidly metabolized. The nature and degree of inhibition is dependent upon the primary and secondary substrate. Thus, each of the kinetic coefficients will vary depending upon' the type of organisms and the waste components. By adjusting the model coefficients and noting the response of the system on a 1700
500 Ks=0. I
c E
1300
8 E
8 II00
_=
~U~ i
O ¢,/)
900
To~ ~ 0
0
PrimorySUbS~TOm
I
I
I
5
I0
15
[ 20
-17oo 1 25
Time (h)
Fig. 6. Chemostat simulation with secondary substrate production.
1372
R.W. DENNISand R. L. IRVINE
computer, one can determine whether or not shunt and/or storage are important considerations in the real system under examination. An example of this is given in Fig. 6. The response of the perturbed chemostat can only be explained by the production and utilization of a secondary substrate, and any model of these chemostats that does not consider shunt product formation is not representative of the real situation. Critics of structured, stoichiometric models may argue that the kinetic coefficients are too difficult to evaluate for a given system and thus, the models have little practical significance. However, by attempting to quantify natural phenomena, such as storage and shunt, in biological waste treatment systems, a practitioner may be led directionally toward an operating policy that would otherwise I~e overlooked. One may not be able to determine the kinetic coefficients precisely. If, however, the stoichiometry is correct and the kinetics are descriptive, one is better able to understand system dynamics than if "precise '° kinetics are used with inadequate stoichiometry. A more fundamental understanding will naturally lead to more reliable design procedures and more effective control strategies.
REFERENCES
Andrews G. F. & Tien C. (1977) New approach to bacterial kinetics in waste water. J. enrir. Engng Dir. Am. $oc. cir. Engrs 103, 1057. Bruner R. L., Smith J. H. & Schroeder A. C. 11978) Novel biochemical pathways in the big-oxidation of corn wetmilling wastes. 33rd Ind. Waste Conf., Purdue Unit'. Busby J. B. & Andrews J. F. 11975) A dynamic model and control strategies for the activated sludge process. J. War. Pollut. Control. Fed. 47, 1055. Busch A. W. (1971) Aerobic Biological Treatment o.I"Waste Waters. Oligodynamics Press, Houston, TX. Chiu S. Y., Eriekson L. E., Fam L. T. & Kao i. C. (19721 Kinetic model identification in mixed populations using continuous culture data. Biotech. Bioengng 14, 207. Clifft R. C. & Andrews J. F. (19811 A model for predicting the dynamics of oxygen utilization in the activated sludge process. J. Wat. Pollut. Control Fed. In press. Crabtree K.. Boyle W.. McCoy E. & Rohlich G. A. 119661 A mechanism of floc formation by Zoogloea ramlgera. J. 14~tt. Pollut. Control Fed. 3g, 1968. Daigger G. T. & Grady C. P. L. 119771 A model for the big-oxidation process based on product formation concepts. Water Res. 11, 1049. Dawes E. A. & Ribbons D. W. 119641 Some aspects of the endogenous metabolism of bacteria. Bact. Rer. 28, 126. Dennis R. W. 119781 A dynamic mathematical model of bacterial growth--application to sequencing batch biological reactors. Doctoral Dissertation. University of Notre Dame. Notre Dame, IN. Dohanyos M., Grau P. & Chudoba J. (19701 Kinetic assessment of glucose removal and saccharide accumulation capacities in activated sludge. Adeances in Water Pollution Research (Edited by Jenkins S: H3, Pergamon Press. New York. Forster D. F. (19761 Bioflooculation in the activated sludge process. Water S.A. 2, ll9. Ghosh S., Pohland F. A. & Gate W. F. (19721 Phasic utiliz-
ation of substrate by aerobic cultures. J. Wat. Pollut. Control Fed. 44, 376. Grady C. P. L. & Williams D. R. 119751 Effects of influent substrate concentration on the kinetics of natural microbial populations in continuous culture. Water Res. 9, 171. Grady C. P. U, Harlow L. J. & Riesing R. R. 119721 Effects of growth rate and influent substrate concentration on effluent quality from chemostats containing bacteria in pure and mixed culture. Biotech. Bioengng 14, 391. Gaudy A. F., Goswami S. R., Manickam F. & Gaudy E. 119771 Formation of strippable metabolic products in biological waste treatment. Biotech. Bioengng 19, 1239. Herbert D. (19581 Some principles of continuous culture. Recent Progress in Microbiology (Edited by Tunewall G.). Blackwell Scientific, Oxford. Hoover S., Pepinsky J. B., Jasewicz L. & Porges N. 119511 Aeration as a partial treatment for dairy wastes. Proc. 16th Ind. Waste Conf., Purdue Unit,. lrvine R. L., Alleman J. E., Miller G. & Dennis R. W. 0980) A basic approach to stoichiometry and kinetics using nitrogen removal in SBRs as an example. J. War. Pollut. Control Fed. 52, 1997. Jacquart J. C., Lefort D. & Rovel J. M. (19731 An attempt to take account of biological storage in the mathematical analysis of activated sludge behavior. Advances in Water Pollution Research (Edited by Jenkins S. H.). Pergamon Press, New York. Koga S. & Humphrey A. E. 119671 Study of the dynamic behavior of the chemostat system, Biotechnol. Bioengng 9, 345. Krishnan P. & Gaudy A. F. Jr 119761 Response of activated sludge to quantitative shock loading. J. Wat. Pollut. Control Fed. 411, 906. Krul J. M. (19771 Activity of Zoogloea ramigera growing in flocs and in suspension. Water Res. l !, 45. Law J. H. & Slepecky R. A. (1961) Synthesis and degradation of poly-B-hydroxybutyric acid in connection with sporulation of bacillus megaterium. J. Bact. 82, 37. Macrae R. W. & Wilkinson J. F. (1955) Poly-B-hydroxybutyrate metabolism in washed suspensions of bacillus cereus and bacillus megaterlum. J. gen. Microbiol. 19. 210. Mateles R. 1.. Ryu D. Y. & Yasuda T. (19651 Measurement of unsteady state growth rates of micro-organisms. Nature 208, 263. Monod J. 119491 The growth of bacterial cultures. A. Rer. Microbiol. 3. 371. Neufeld R. D. & Valiknac T. 11979) Inhibition of phenol biodegradation by thiocyanate. J. Wat. Pollut. Control Fed. 51, 2283. Obayashi A. W. & Gaudy A. F. Jr i1973) Aerobic digestion of extracellular microbial polysaceharides. J. Wat. Pollut. Control Fed. 45, 1584. Orphan D. & Tunay O. 119791 Mathematical models of biological waste treatment processes for the design of aeration tanks---discussion. Water Res. 13, 553. Porges N. 119551 Rapid big-oxidation method of waste disposal. Proc. Am. Soc. Brewing Chem. Porges N.. Jasewicz L. & Hoover S. (1955l Biochemical oxidation of dairy wastes VIII purification, oxidation. synthesis, and storage. Proc. lOth Ind. Waste Conf.. Purdue Unir. Porges N.. Jasewicz L. & Hoover S.. R. (19561 Principles of biological oxidation. Biological Treatment of Sewage and Industrial Wastes (Edited by MeCabe J. & Eckenfelder W. W.k Vol. 1, Reinhold. New York. Robarts R. D. & Kempton A. G. (19711 Use of C~4-glucose to study substrate removal by activated sludge. J, Wat. Pollut. Control Fed. 44, 1499. Rag B. S. & Gaudy A. F. (19661 Effects of sludge concentration on various aspects of biological activity in activated sludge. J. Wat. Poltut. Control Fed. 38. 794.
A stoichiometric model of bacterial growth Spcece R. E., Englebrecht R. S. & Aukamp D. R. (1973) Cell replication and biomass in the activated sludge process. Water Res. 7, 361. Storer F. F. & Gaudy A. F. (1969) Computational analysis of transient response to quantitative shock Ioadings of heterogenous populations in continuous culture. Envir. Sci. Technol. 3, 143. Takii S. (1970) Accumulation of glycogen-like polyglucose in glucose-acclimated activated sludge. J. Ferment. Technol. 48, 350. Takii S. (1977) Saccharide in activated sludge treating carbohydrate wastes. Water Res. 11, 78. Therien N. & Perdrieux S. (1978) Characterization of the dynamics of the activated sludge wastewater treatment process in terms of the carbon variable. 85th National Meeting AIChE, Philadelphia. Tsuno H. & Goda T. (1978) Kinetic model of activated
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sludge metabolism and its application to the response oT qualitative shock load. Water Res. 12, 513. Varma M. M., Wcekes M. C. & Calvert A. F. (1975) Kinetics of assimilation of mixed substrates by heterogeneous bacteria. J. War. Pollut. Control Fed. 47, 1913. Waiters C. F. (1966) Biochemical storage phenomena in activated sludge. Unpublished Doctoral Dissertation, University of Illinois, Urbana, 1L. Wahers C. F., Engelbrecht R. S. & Speece R. E. (1968) Microbial substrate storage in activated sludge. J. sanit. Engno Die. Am. Soc. cir. Engrs 94, 257. Williams F. M. (1967) A model of cell growth dynamics. J. theoret. Biol. 15, 190. Williamson D. H. & Wilkinson J. F. (1958) The isolation and estimation of the poly-B hydroxybutyrate inclusion of bacillus species. J. gen. Mierobiol. 19, 198.