ENVIRONMENTAL ENGINEERING SCIENCE Volume 23, Number 6, 2006 © Mary Ann Liebert, Inc.
Evaluation of First-Order, Second-Order, and Surface-Limiting Reactions in Anaerobic Hydrolysis of Cattle Manure M. Myint and N. Nirmalakhandan* Civil Engineering Department New Mexico State University Las Cruces, NM 88003
ABSTRACT Three kinetic models were evaluated in this study for suitability in describing anaerobic hydrolysis of particulate wastes. The three models evaluated were: first-order reaction in particulate substrate concentration, second-order reaction in acidogenic biomass and particulate substrate concentrations, and a two-parameter, surface-limiting reaction model. Process models incorporating the three hydrolysis reaction models were developed to describe the hydrolysis-acidogenesis phase in the fermentation of cattle manure. Batch reactors were run with cattle manure as the substrate under five different conditions to calibrate and validate the process models. The two-parameter, surface-limiting reaction model and the single-parameter, second-order reaction model were found to fit the experimental results better than the simple first-order reaction model with r2 values of 0.914, 0.913, and 0.881, respectively. Key words: anaerobic hydrolysis; first-order hydrolysis model; second-order hydrolysis model; surfacelimiting hydrolysis model; cattle manure INTRODUCTION
A
NAEROBIC TECHNOLOGY has been proven to be energy-efficient in stabilizing organic waste streams. Reports from several laboratory studies and full-scale projects have documented successful applications of this technology in stabilizing liquid waste streams and generating energy in the form of gaseous methane. However, large-scale application of this technology in stabilizing particulate wastes to produce energy has been hindered by the poor kinetics of the overall process (Omstead et al., 1980). Conversion of particulate organic wastes to gaseous methane involves multiple steps in series and parallel, di-
verse groups of microorganisms, and different environments. The following have been recognized as important stages in the process. In the first stage, acidogenic organisms solubilize particulate substrates extracellularly by enzymatic hydrolysis. In the second stage, acidogenic organisms catabolize the products of the first stage into volatile organic acids, carbon dioxide, and hydrogen. In the next stage, acetogenic organisms convert the products of the second stage to acetic acid. Finally, methanogenic organisms convert the acetic acid to carbon dioxide and methane (Gujer and Zehnder, 1983; Pavlostathis and Giraldo-Gomez, 1991; Ruel et al., 2002). The overall rate of the anaerobic process of gasification of particulate wastes by the above scheme is depen-
*Corresponding author: Civil Engineering Department, New Mexico State University, Las Cruces, NM 88003. Phone: 505646-5378; Fax: 505-646-6049; E-mail:
[email protected]
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EVALUATION OF FIRST-ORDER, SECOND-ORDER, AND SURFACE-LIMITING REACTIONS dent on the rate limiting step (Pavlostathis and GiraldoGomez, 1991). Several studies have identified the hydrolysis step as the rate-limiting step (Eastman and Ferguson, 1981; Gossett and Belser, 1982; Pavlostathis and Giraldo-Gomez, 1991; Veeken et al., 2000; Vavilin et al., 2002). These researchers and others have reported alternate models to describe this step (Veeken et al., 2000). The objective of this paper is to compare three of the more common hydrolysis reaction models reported in the literature to identify the appropriate one for use in modeling the hydrolysis/acidogenesis step in the digestion of cattle manure. In our preliminary studies, we have observed that the temporal COD solubilization curve consisted of two distinct segments. We presuppose that this is due to two distinct components of cattle manure—a readily hydrolyzable fraction composed primarily of hemicellulose, and a slowly hydrolyzable fraction, composed primarily of cellulose. Our premise is supported by the reports of other workers (Robbins et al., 1979; Chandler and Jewell et al., 1980; Ong et al., 2000), who have identified these two fractions as the major constituents of cattle manure, each component hydrolyzing at a different rate (Ray et al., 1989). Our process model incorporates this feature, with appropriate hydrolysis and acidogenesis process parameters for the two fractions. Our model also incorporates an optional term to account for hydrolysis-enhancing agents such as enzymes that could be added to the reactors. A family of enzymes that act to hydrolyze cellulose has been identified as cellulase. Huang (1975) reported that cellulase secreted by Trichoderma viride as the most suitable enzyme for degrading insoluble cellulose to soluble sugars. According to Gaudy and Gaudy (1988), various strains of fungi also produce cellulase that hydrolyzes cellulose to glucose, which in turn, can be metabolized by fungi and many other microorganisms as well. Poulsen and Peterson (1985) have used crude cellulase from cellulolytic bacterium to hydrolyze carboxymethylcellulose. Ruminant animals consume cellulose, hemicellulose, and pectin from different sources, which are broken down by mixed cultures of rumen bacteria. For example, cellulolytic and noncellulolytic populations found in rumen bacteria can degrade cellulose into low-molecular weight carbohydrates (Gray and Pilgrim, 1952; Howard, 1957; Gray and Weller, 1958; Dehority et al., 1962; Dehority 1965; Pavlostathis and Gossett, 1988). Barlaz et al. (1990) reported that cellulolytics can grow under both anaerobic and aerobic conditions (facultative). Sleat et al. (1989) showed that cellulose-degrading cellulolytics and hemicellulsoe-degrading hemicellulolytics were present in landfills at respective concentrations of 104 to 106 and 105 to 107 in 1 g of
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wet waste. Our modeling approach builds upon the above reports and presumes that naturally existing cellulolytic and hemicellulolytic organisms in cattle manure can hydrolyze the particulate forms of cellulose and hemicellulose contained therein.
HYDROLYSIS MODELS Three of the common models proposed in the literature to describe the anaerobic hydrolysis step, viz. the first-order, second-order, and surface-limiting model (also known as the Contois kinetic model) are briefly outlined in the following sections.
First-order model Pavlostathis and Giraldo-Gomez (1991) have reviewed the literature on hydrolysis published up to 1990, all of which had used a first-order model to describe anaerobic hydrolysis of particulate wastes. According to these studies, hydrolysis rate is modeled by the equation: dP K1P dt
(1)
where, P is the remaining concentration of the biodegradable particulate substrate (g COD/g bulk solids) and K1, is the first-order rate constant (1/day). In a recent report, Mahmoud et al. (2004) have adopted a similar model in the study of anaerobic stabilization of primary sludge.
Second-order model The second-order model for the hydrolysis rate proposed by Noykova et al. (2002) can be expressed as: dP K1PX dt
(2)
where, X is the acidogenic biomass concentration (cell g VSS/g bulk solids); and K1 is the second-order rate constant (g/g acidogenic biomass-day). Noykova et al. (2002) have used this second-order model to describe the hydrolysis process in the digestion of fresh cow dung with total solids ranging from 4.5 to 12.65%.
Surface-limiting model (also known as the Contois kinetic model) Munch et al. (1999) observed that the hydrolysis rate was reduced when the biomass concentration increased above a certain level. They attributed this observation to the limited surface area, causing mass transfer limitations. They rearranged the equation from the Contois kinetic model (Contois, 1959) and presented it as a surfacelimiting model to account for the mass transfer limitations ENVIRON ENG SCI, VOL. 23, NO. 6, 2006
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due to the limited surface area. The hydrolysis rate in this case can be expressed as:
dP P/X K1 X dt K1s P/X
(3)
where, K1 is the rate constant [1/day], and K1s is the saturation coefficient for hydrolysis (). Ruel et al. (2002) have adapted this approach in the study of fermentable fractions in wastewater.
The quantities of manure samples, water, cellulase enrichment, and biocide added to each reactor are shown in Table 1. All the reactors were placed in a water bath maintained at 37 2°C. Liquid samples from the reactors were withdrawn periodically to measure pH (using a Cole-Palmer, Chicago, IL, pH electrode probe). The samples were filtered with a 0.45-m membrane filter and the COD of the filtrate was measured following Standard Methods 522D to determine dissolved COD (APHA, AWWA, WEF, 1998; Miron et al., 2000).
EXPERIMENTAL APPROACH Laboratory batch experiments were conducted with cattle manure as the substrate to evaluate the three hydrolysis models and to validate the new hydrolysis-acidogenesis model developed in the next section. In addition, enhancement of hydrolysis by enrichment with enzymes was also investigated. In this study, we have used cellulase as an enhancer; the purpose of using the enhancer in this study is solely to validate the process model. Samples of cattle manure were obtained from a pile of filtered manure wash at a nearby dairy. The age of the test samples in the pile was 2 days. The tests were conducted in batch mode in 600 mL glass bottles. Five batch reactors (labeled Reactor 1, 2, 3, 4, and 5) were run, each with a duplicate. Reactor 1 contained raw manure sample, topped with water. Reactors 2 to 5 contained raw manure, enriched with different amounts of the enhancer— cellulase—and topped with water. Another set of experiments was conducted to verify the presumption that active hydrolytic organisms were already present in the manure samples. In these experiments, two 600-mL batch reactors (labeled Reactors 6 and 7) were run in duplicate. Both these reactors were filled with raw manure samples and water; Reactor 7, however, was dosed with a biocide (1.9 g/L HgCl2) to inhibit bacterial activity (Tellez, 1994).
Table 1.
MODEL DEVELOPMENT Our model for the hydrolysis-acidogenesis phase is based on the following simplifying assumptions: 1. the particulate hemicellulose and cellulose fractions are hydrolyzed by acidogens; 2. the acidogens grow attached to the solid matrix, utilizing the dissolved form of hemicellulose and cellulose as substrate; 3. The enhancers hydrolyze the particulate hemicellulose and cellulose in proportion to their respective initial concentrations. The solubilization efficiency of the enhancer used in this study—cellulase—was estimated to be 15% through a curve-fitting process. A sensitivity analysis showed that the change in COD production was within 1% when the efficiency ranged from 10 to 20%. 4. negligible methanogenic activity in the reactors. Thus, the only processes occurring in the reactors are assumed to be hydrolysis and acidogenesis. The modeling framework incorporating these assumptions is illustrated in Fig. 1. Based on the above assumptions, the rate of change of particulate species i (in the forms of hemicellulose (h)
Contents of test reactors. Reactors
Description Amount of wet cattle manure (g) Moisture content in wet manure (%) Amount of cellulase added (g) Amount of biocide added (g) Water added to reactor (L) Cellulase-to-manure ratio (mg/g) Manure-to-liquid ratio (g/L)
1
2
3
4
5
6
7
120 78.7 0.00 0.00 0.40 0.00 51.68
120 78.7 0.05 0.00 0.45 2.00 46.93
120 78.7 0.10 0.00 0.45 3.90 46.93
120 78.7 0.15 0.00 0.45 5.90 46.93
120 78.7 0.20 0.00 0.45 7.80 46.93
100 77.3 0.00 0.00 0.43 NA 42.83
100 77.3 0.00 1.00 0.43 NA 42.83
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Regardless of the hydrolysis reaction model, the utilization rates of dissolved hemicellulose and cellulose by the acidogenic biomass can be expressed as follows according to the two-substrate–one-biomass model (derived in Appendix II). Uptake rate of hemicellulose by acidogens: ShX dS c h kh Ksh 1 S Sh MLR (8) dt u Ksc
Uptake rate of cellulose by acidogens:
dS c dt
kc
u
ScX Sh Ksc 1 Sc Ksh
MLR
(9)
Therefore, the net rate of change of dissolved species (i hemicellulose or cellulose) in the reactor is
dS dP dS i i MLR i dt dt dt Figure 1. Conceptual model of batch system; 1- hydrolysis of hemicellulose; 2- hydrolysis of cellulose; 3- biouptake of hemicellulose; 4- biouptake of cellulose.
and cellulose (c)) due to hydrolysis can be expressed as follows for the three hydrolysis reaction models: First-order reaction model
dP Pi,0 i K1i(Pi) dt Pt,0
(4)
dP Pi,0 i K1i(Pi)(X) dt Pt,0
Thus, the rate of change of dissolved COD in the reactor is d(COD) ih dS i dt dt ic
(5)
(11)
The rate of growth of acidogenic biomass can be expressed as dX ih dt ic
Second-order reaction model
(10)
u
⎣
dS i ai dt
⎦
kdX
(12)
Surface-limiting reaction model:
Pi/X Pi,0 dP i K1i X dt K1si Pi/X Pt,0
(6)
All the variables are expressed in COD basis and are defined in Appendix I. In the above equations, the second term on the right-hand side represents the enhancement of hydrolysis by the enhancers—cellulase—where, is the solubilization rate of the enhancer. While this term is zero for Reactor 1, for Reactors 2 to 5, is expressed as Ce EMR
(7)
where, Ce is the specific COD conversion rate of the enhancer (g COD/g enhancer-day) and EMR is the enhancer-to-manure ratio (g enhancer/g manure). The specific COD conversion rate of cellulase was obtained from the supplier as 2.074 g COD/g cellulase-day and used without any verification (http://www.Wothingtonbiochem.com/CEL/cat.html, Aug 6, 2004).
Figure 2. Comparisons of dissolved COD production with/without biocide.
ENVIRON ENG SCI, VOL. 23, NO. 6, 2006
974 where, ai is the yield coefficient []; and kd is the death rate [1/day]. The model equations contain up to four hydrolysis process parameters (K1h and K1sh for hemicellulose and K1c and K1sc for cellulose) and four biological process parameters (kh and Ksh for hemicellulose and kc and Ksc for cellulose). Since these parameters were not measured through independent experiments, a curve-fitting process was used to estimate them. Measured COD data from Reactor 1 was used in the curve-fitting process to estimate the eight parameters, which were then validated using measured COD data from Reactors 2, 3, 4, and 5. The model equations were solved using dynamic simulation program (ithink® by HPS Systems, Inc., Hanover, NH) to generate the COD profile as a function of time. The literature was surveyed to determine yield coefficients for acidogenic growth on soluble forms of hemicellulose (ah) and cellulose (ac). Since the substrates, experimental conditions, and the data analysis methods varied from study to study, it was not possible to reconcile and corroborate those values. For example, typical yield coefficients ranged as follows: 0.026 g COD of VSS/g COD for cattle manure wastewater (Simeonov et al., 1996); 0.047 g COD of VSS/g COD for amino acids and sugars (Bryers, 1985); 0.057 g COD of VSS /g COD for activated sludge (Pavlostathis and Gossett, 1988); 0.051 g COD of VSS /g COD for molasses wastewater (Denac et al., 1988; Kalyuzhnyi, 1997), and 0.100 g COD of VSS/g COD for with amino acid, sugars, and fatty acid (Ruel et al., 2002). For the purpose of this study in comparing the three hydrolysis models, it was not crucial to
Figure 3. Average pH in the liquid phase. Data points represent average of duplicates.
MYINT AND NIRMALAKHANDAN
Figure 4. Enhancement of hydrolysis by cellulase. Data points represent average of duplicates.
establish absolute values for the yield coefficients; rather, it was deemed sufficient to use the same values for the three models, but of appropriate magnitude. Thus, we used the following values for the yield coefficients: 0.084 and 0.042 g COD of VSS/g COD, for ah and ac, respectively.
RESULTS AND DISCUSSION Verification of modeling assumptions Figure 2 shows the impact of the biocide dose on dissolved COD production in Reactors 6 and 7. In Reactor 6, which was not dosed with the biocide, COD production continued to increase with time while that in Reactor 7, which was dosed with the biocide, was significantly lower. The slight increase in COD production in Reactor 7 might have been due to abiotic processes or inadequate biocide dosage. This finding served to support the presumption that the production of dissolved COD was primarily due to active hydrolytic organisms naturally present in cattle manure. Figure 3 shows that the pH in all the reactors remained below 5.5. Based on literature reports, methanogenic activity could be considered negligible in this pH range. For example, Yu et al. (2003) demonstrated that methanogenic activity was negligible at pH 4.5; 90% inhibited at pH 4.75, and, 40% inhibited at pH 5.5. This finding supports the modeling assumption that the only processes occurring in the reactors are hydrolysis and acidogenesis. However, in this study, it was not corroborated by gas phase measurements.
EVALUATION OF FIRST-ORDER, SECOND-ORDER, AND SURFACE-LIMITING REACTIONS Table 2.
Best-fit values of model parameters.
Parameter For hemicellulose: K1h (day1) K1sh () Kh (day1) Ksh (g COD/L) For cellulose: K1c (day1) K1sc () Kc (day1) Ksc (g COD/L)
First-order reaction model
Second-order reaction model
0.05 0.006 — 1.0 0.12 2.2 0.26
1.2 0.104 — 0.9 0.078 2.5 0.22
1.4 28.0 1.8 15.0
0.12 2.0 0.16 1.29
0.28 0.024 — 0.5 0.044 55.0 4.8
0.09 1.5 0.8 100.0
0.008 0.13 0.07 8.6
0.019 2.28 — 0.75 0.09 75.0 9.0
Figure 4 shows the enhancement of the hydrolysis process as reflected by the increase in COD generation due to increasing doses of cellulase as the enhancer. COD generation increased 25 to 30% when the cellulase-to manure ratio increased from 2 mg/g to 3.9 mg/g. However, further increase of cellulase-to-manure ratio up to 7.8 mg/g did not result in any increase in COD release. This saturation was probably due to mass transfer limitations in the hydrolysis step as discussed by Munch et al. (1999) in the development of the surface-limiting model.
Estimation/calibration/validation of model parameters The best-fit values for the eight parameters were found by a curve-fitting process to match the COD data measured in Reactor 1 that did not receive any enhancers. The best-fit parameters found for the three hydrolysis models are tabulated in Table 2. It can be noted that the hydrolysis parameters were found to be differ-
Table 3.
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Surface limiting reaction model
ent for the three models as the underlying mechanisms are different. In the case of the biological parameters, the estimated ki values are comparable for the three models as expected; while the Ksi values are comparable for the first-order and second-order models, the corresponding values for the surface-limiting model are an order of magnitude higher. This latter anomaly is under further investigation. The parameters found through curve-fitting were further validated with COD data from Reactors 2, 3, 4, and 5 that received various doses of the enhancer. The COD profiles predicted by the three models for Reactors 2, 3, 4, and 5 generally agreed well with the measured data as summarized in Table 3. The overall goodness of fit between COD measured experimentally and the COD predicted by the process model incorporating the three hydrolysis models was good (r2 0.85 for 120 data points, with the probability of the correlation, p 0.001). This agreement supports the validity of the proposed model and the supposition that the observed two-segment COD
Quality of prediction of the three models. Reactor
Model First-order reaction (data points 12) Second-order reaction (data points 12) Surface-limiting reaction (data points 12)
r2 F p r2 F p r2 F p
1
2
3
4
5
Overall
0.932 137.28 0.001 0.97 324.74 0.001 0.992 1187.74 0.001
0.851 56.93 0.001 0.903 93.56 0.000 0.921 115.65 0.000
0.925 122.88 0.001 0.938 151.11 0.001 0.932 137.27 0.001
0.903 93.46 0.001 0.935 143.96 0.001 0.937 149.68 0.001
0.894 83.95 0.001 0.913 113.08 0.001 0.911 102.09 0.001
0.881 427.48 0.001 0.913 607.03 0.001 0.914 619.77 0.001
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profiles was due to the two components in cattle manure—hemicellulose and cellulose.
Comparison of hydrolysis reaction models The COD profiles predicted using the three hydrolysis reaction models are compared against the measured COD data for Reactors 1 to 5 in Figures 5 to 9. Overall, the surface-limiting reaction and the second-order reaction models fitted the measured data better than the firstorder reaction model as shown in Figure 10. However, the trend and the goodness of fit of both the first-order and second-order reaction models deteriorated with time, while the trend and the goodness of fit of the surfacelimiting reaction model was consistent throughout the full range of the tests. This finding supports the contention that the substrate-to-microorganism ratio, Pi/X, may be a
Figure 6. Comparison of three hydrolysis models- Reactor 2. Inset shows measured COD vs. predicted COD.
Figure 5. Comparison of three hydrolysis models- Reactor 1. Inset shows measured COD vs. predicted COD.
limiting factor in the hydrolysis of particulate substrates, rather than the remaining substrate concentration Pi as modeled by the first-order reaction model. As Pi decreases with time, soluble COD and X increase with time, and the ratio Pi/X decreases. By incorporating this limitation, the surface limiting model is able to predict the COD better than the other two models. This is further corroborated by the fact that the COD generation did not increase in proportion to the enhancer dose as illustrated in Fig. 4. Our results that the three hydrolysis models can predict COD generation reasonably well are in agreement with the suggestion of Vavilin et al. (1996), who had also concluded that different types of hydrolysis kinetics could fit experimental data well. Veeken et al. (2000) presumed that the finding of Vavilin et al. (2002) justified the broad application of first-order kinetics, since it
EVALUATION OF FIRST-ORDER, SECOND-ORDER, AND SURFACE-LIMITING REACTIONS
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Ce
specific COD conversion rate of enhancers (g COD/g enhancer-day) EMR enhancer-to-manure ratio (g enhancer/g manure) kc maximum soluble substrate utilization rate of cellulose (1/day) kh maximum soluble substrate utilization rate of hemicellulose (1/day) kd biomass death rate (1/day) K1i hydrolysis rate constant for component i (1/day or g/g biomass-day) K1s,i half saturation coefficient for hydrolysis of component i (g COD/g COD) Ksc half saturation coefficient for biomass uptake of cellulose (g COD/L) Ksh half saturation coefficient for biomass uptake of hemicellulose (g COD/L) MLR manure-to-liquid ratio (g manure/L water)
Figure 7. Comparison of three hydrolysis models- Reactor 3. Inset shows measured COD vs. predicted COD.
is the simplest way to describe the hydrolysis process. However, based on the results of this study, we propose that the single-parameter, second-order model would be more realistic than the first-order reaction model and as easy to apply. Even though the surface-limiting reaction model fitted the data with a slightly better quality of fit over the range tested, it involves two parameters that have to be established experimentally.
APPENDIX I: NOMENCLATURE ac ah
biomass yield coefficient with cellulose as substrate (g COD/g COD) biomass yield coefficient with hemicellulose as substrate (g COD/g COD)
Figure 8. Comparison of three hydrolysis models- Reactor 4. Inset shows measured COD vs. predicted COD.
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MYINT AND NIRMALAKHANDAN X
concentration of acidogenic biomass (g COD/g solids) solubilization rate of enhancer (g COD/g manureday)
APPENDIX II: DERIVATION OF EQUATIONS (7), (8), AND (11) Consider single biomass utilizing two substrates, Sc and Sh with enzymatic interaction: k1
k3
k4
k6
E Sc i ESc E P and E Sh i ESh E P k k 2
5
At steady state, d[ESc] 0 k1[E][Sc] k2[ESc] k3[ESc] dt d[ESh] 0 k4[E][Sh] k5[ESh] k6[ESh] dt
Figure 9. Comparison of three hydrolysis models- Reactor 5. Inset shows measured COD vs. predicted COD.
p Pc Ph Pi Pi,0 Pt,0 Sc Sh
probability that the regression coefficient would be as extreme as reported concentration of cellulose in particulate form (g COD/g manure) concentration of hemicellulose in particulate form (g COD/g manure) concentration of component i in particulate form (g COD/g manure) initial concentration of component i in particulate form (g COD/g manure) initial concentration of total components in particulate form (g COD/g manure) oncentration of cellulose in dissolved form (g COD/L) concentration of hemicellulose in dissolved form (g COD/L)
Figure 10. Overall comparisons of the three hydrolysis models.
EVALUATION OF FIRST-ORDER, SECOND-ORDER, AND SURFACE-LIMITING REACTIONS [E][Sc] [E][Sc] (k2 k3 ) and [ESh] Ksc k1
Hence, [ESc]
[E][Sh] [E][Sh] (k5 k6) Ksh k4
From conservation of E, [E0] [E] [ESc] [ESh] [E0] or, [E] [Sc] [Sh] 1 Ksc Ksh [E0][Sc] Hence, [ESc] and [Sh] Ksc 1 [Sc] Ksh
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hancement techniques and microbial dynamics. Crit. Rev. Environ. Control 19(6), 557–584. BRYERS, J.D. (1985). Structure D modeling of the anaerobic digestion of biomass particulate, Biotechnol. Bioeng. 27, 638–649. CHANDLER, J.A., and JEWELL, W.J. (1980). Predicting Methane Fermentation Biodegradability: Final Report. Ithaca, NY: Department of Agricultural Engineering, Cornell University. CONTOIS, D.E. (1959). Kinetics of bacterial growth: Relationship between population density and specific growth rate of continuous cultures. J. Gen. Microbiol. 21, 40–50. DEHORITY, B.A. (1965). Degradation and utilization of isolated hemicellulose lose by pure cultures of cellulolytic rumen bacteria. J. Bacteriol. 80(6), 1515–1520.
[E0][Sh] [ESh] [Sc] Ksh 1 [Sh] Ksc
DEHORITY, B.A., JOHNSON, R.R., and CONRAD, H.R. (1962). Digestibility of forage hemicellulose and pevtin by rumen bacteria in vitro and the effect of lignigication thereon. J. Dairy Sci. 45, 508–512.
Now, the rate of consumption of Sc for example, can be found from:
DENAC, M., MIGUEL, A., and DUNN, I.J. (1988). Modeling dynamic experiments on the anaerobic degradation of molasses wastewater. Biotechnol. Bioeng. 31, 1–10.
d[Sc] k1[E][Sc] k2[ESc] k3[ESc] dt [E0][Sc] k3 [Sh] Ksc 1 [Sc] Ksh
By comparison, Sc X Sh Sc ac Ksc 1 u Ksh
dS c dt
and
ShX Sc Sh ah Ksh 1 u Ksc
dS h dt
Therefore,
dt dX
dS dS ac c ah h growth dt dt
and,
dt dX
net
dSh dS ac c ah kdX dt dt
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