MATHEMATICAL MODELS OF THE KINETICS OF

ANAEROBIC DIGESTIONÐA SELECTED REVIEW

A. HUSAIN

Ontario Hydro Technologies, 800 Kipling Avenue- KR230, Toronto, ON, M8Z 5S4, Canada

(Received 23 September 1997; revised 28 October 1997; accepted 29 October 1997)

AbstractÐA review of steady state and dynamic models of the kinetics of anaerobic digestion was car-ried out. A simpli®ed dynamic model developed by Hill (Auburn University, AL) was determined tohave the desired balance between accuracy and complexity. A steady state solution to Hill's simpli®eddynamic model was developed. It predicts the substrate and bacterial concentrations and hence the gasgeneration rate in a continuously stirred tank reactor. The solution is recommended for general use inoptimizing the gas yield from anaerobic digestion processes. # 1998 Elsevier Science Ltd. All rightsreserved

KeywordsÐAnaerobic digestion; review kinetics; mathematical models.

1. INTRODUCTION

Anaerobic digestion of farm wastes, sewage

sludge, municipal solid wastes (MSW) etc.

results in the production of digester gas con-

taining over 50% methane, the balance being

principally carbon dioxide. Utilization of the

gas in power generation more than o�sets the

energy requirements of the digester facility

and mitigates the environmental damage

resulting from methane emissions to the at-

mosphere.

The volumetric production rate of methane

must be optimized to maximize power gener-

ation revenues and hence the overall econ-

omics of digestion facilities. Optimization

can be facilitated by utilizing mathematical

models of the anaerobic digestion process.

These models predict the substrate utilization

rate, the rate of methane generation and the

composition of the gas, and help avoid pro-

cess conditions leading to unstable digester

operation and subsequent failure.

Accordingly, a review of published kinetic

models was carried out. Because of the

exhaustive literature on the subject, the

review was necessarily limited to models

which are not unduly complex and yet pre-

dict the principal features of digester per-

formance. Both steady state and dynamic

models were assessed. Although kinetic

equations have been formulated for di�erent

types of reactor con®gurations, the model

equations presented here pertain only to thecontinuously stirred tank reactor (CSTR).

This review was commissioned because ofOntario Hydro's recent involvement in thearea of anaerobic digestion and speci®callywith the commissioning of an animal wastedigestion facility at a farm in Ontario. Assuch, much of the literature examined con-cerned the modeling of animal waste digestion.Because Hill (Auburn University, AL) and co-workers have published extensively on thistopic, their work formed the main thrust ofour review.

To provide a uni®ed treatment of the sub-ject, a description of the anaerobic digestionprocess, a discussion of the characteristics andmethane productivity of major animal wastesare given followed by an overview of kineticmodels. It is further shown how the steadystate substrate and bacterial concentrations,and hence the gas generation rate in a CSTR,can be readily solved from a simpli®eddynamic model developed by Hill and co-workers. Because the description of the an-aerobic digestion process embodied in thisdynamic model is more sophisticated than thatconsidered in earlier steady state models, themodel solution developed here providesgreater insight into the digestion process. Thissolution is applicable, in general, to all typesof wastes and is recommended for use in opti-mizing gas yields from anaerobic digestionprocesses.

Biomass and Bioenergy Vol. 14, Nos. 5/6, pp. 561±571, 1998# 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0961-9534/98 $19.00+0.00PII: S0961-9534(97)10047-2

561

Fig. 1. Reaction scheme for the anaerobic digestion of animal manure.

2. DESCRIPTION OF THE ANAEROBIC DIGESTIONPROCESS

The basic microbiology of animal waste

digestion1 as illustrated in Fig. 1 consists of

four stages, namely, hydrolysis, acetogenesis,

hydrogenogenesis and methanogenesis. Extra

cellular enzymes excreted by acid forming bac-

teria (acetogens) convert the complex organic

material (represented by C6H13NO5) into sol-

uble organics (represented by glucose

C6H12O6) according to:

C6H13NO5�H2O�H�4C6H12O6�NH�4 �1�Once solubized as C6H12O6, acetogens convert

the waste into three major volatile fatty acids

(VFAs), namely, acetate, propionate and buty-

rate according to the acetogenesis reaction

C6H12O64 0:1115C5H7NO2

� 0:744CH3COOH� 0:5CH3CH2CH2OOH

� 0:5CH3CH2COOH � 0:454CO2 �2�

Note that a small fraction of the solubleorganics is consumed to maintain the bacterialpopulation represented by C5H7NO2; thenitrogen required for bacterial synthesis comesfrom the release of NH+

4 in reaction (1).Because of the presence of excess ammonia inanimal waste, the rate of reaction (2) is notlimited by the availability of ammonia.

A second bacterial group called hydrogeno-gens utilizes the propionate and butyrate toform acetate, formate, H2 and CO2 as the major

A. HUSAIN562

end products. The stoichiometry for propionateand butyrate hydrogenogenesis are as follows:

CH3CH2COOH � 1:786H2O4

0:0458C5H7NO2 � 0:924CH3COOH

� 2:778H2 � 0:924CO2 �3�

CH3CH2CH2COOH� 1:84H2O4

0:0545C5H7NO2 � 1:86CH3COOH

� 1:92H2 �4�Note as before the partial utilization of thesubstrate for bacterial upkeep. The formate, H2

and CO2 are eventually converted into methaneby methanogens (methanogenesis) or to acetateby homoacetogens (homoacetogenesis).

CO2 � 3:813H240:022C5H7NO2

� 0:89CH4 � 1:956H2O �5�

CO2 � 2:073H240:0487C5H7NO2

� 0:378CH3COOH � 1:146H2O �6�As shown in Fig. 1, 60% of the acetate isformed via reactions (3) and (4), 30% via reac-tion (2) and 10% via reaction (6). Whereas H2

methanogenesis (reaction (5)) accounts forabout 30% of the CH4 produced, the directconversion of acetate (acetate methanogenesis)to CH4 according to

CH3COOH4 0:022C5H7NO2 � 0:945CH4

� 0:06H2O� 0:945CO2 �7�accounts for the balance 70%. Yield coe�-cients for various bacterial cultures based onreactions (1)±(7) are summarized in Table 1.

While the volatile fatty acids (VFAs) areessential substrates for the reactions producingCH4, they are toxic to the bacteria when pre-sent at elevated levels. Toxicity results inreduced methane productivity and eventuallyin digester failure. An increase in acetate levelindicates a problem with the methanogenicpopulation as conversion to methane mustoccur to keep acetate level low. An increase in

propionate/butyrate level relative to the acet-ate level is signi®cant since propionate andbutyrate are the immediate precursors toabout 60% of the acetate. An increase in thepropionate to acetate or butyrate to acetateratio indicates a problem with the hydrogeno-genic bacterial population. Inhibition of thismicrobial population would signi®cantlyreduce methane formation.

Overall, the conversion of organic materialto CH4 involves a close relationship amongfour types of bacterial populations with thedynamic balance between production and util-ization of the intermediate products beingcritical to the overall success of the fermenta-tion. Disturbance of the dynamic balancewould cause accumulation of VFAs and even-tually lead to digester failure.

3. CHARACTERISTICS OF RAW ANIMAL WASTE

Mathematical representation of raw animalwaste must account for di�erences in theirchemical and biological characteristics.Because the bacterial cultures involved indigestion of di�erent animal wastes are func-tionally similar, kinetic parameters and bac-terial yield coe�cients should be independentof the type of waste. This is achieved by di�er-entiating the various types of waste accordingto their biodegradability, soluble organics con-tent, acidity and nitrogen levels. These par-ameters a�ect methane yield and in¯uenceinhibitory e�ects in the digestion process.

The biodegradability constant, B0 (g volatilesolids destroyed per g volatile solids added),represents the ultimate biodegradability of awaste as residence time approaches in®nity.Application of B0 to raw waste gives the por-tion of the waste which can serve as substrate.

Biodegradable volatile solids (BVS) aremade up of soluble organics and insolubleorganics. Soluble organics can be directly uti-lized whereas insoluble organics must ®rst behydrolysed. Thus, a solubility constant (g sol-uble organics per g BVS) applied to the biode-

Table 1. Yield coe�cients for various bacterial cultures

Yield coe�cient Reaction Value

g acetogens/g substrate C6H12O6 (2) 0.07g hydrogenogens/g substrate CH3CH2COOH (3) 0.07g hydrogenogens/g substrate CH3CH2CH2COOH (4) 0.07g homoacetogens/g substrate CO2 (6) 0.125g H2 methanogens/g substrate CO2 (5) 0.0565g acetate methanogens/g substrate CH3COOH (7) 0.042

Models of kinetics of anaerobic digestion 563

gradable portion of raw waste represents itssoluble organic content.

Inhibitory substances such as VFAs andammonia vary among the waste types. Theacidity constant, Af (g VFA per g BVS)expresses that portion of the biodegradablematerial which is initially in the acid form.The ammonia content is similarly expressed ing NH3 per g BVS.

Values of these constants for a number ofanimal wastes are given in Table 22. Note thatB0 varies from 0.36 for dairy waste to 0.9 forswine and poultry wastes. Also, because of el-evated acid and ammonia levels, the potentialfor digester failure is greater for poultry wastethan for other types of waste.

4. METHANE PRODUCTIVITY OF MAJOR ANIMALWASTES

Combination of reactions (2)±(7) results inthe overall reaction:

C6H12O64 0:24C5H7NO2 � 2:516CH4

� 2:279CO2 �8�Based on the assumption that the mass of vol-atile solids (VS) destroyed corresponds to themass of CH4 and CO2 formed, the CH4 pro-ductivity can be calculated to be 0.4 m3/kg VSdestroyed. This value di�ers from the value of0.55 m3/kg VS destroyed which Hill obtainedfrom the same set of reactions. Reasons forthe discrepancy is not clear. Based on reaction(8), the CH4 content of the digester gas is esti-mated to be 52.5%.* In a later study,3 Hillrecalculated the methane productivity value tobe 0.5 m3/kg VS destroyed; this value was aconstant independent of the type of animalwaste. In addition to expressing methane pro-ductivity on the basis of VS destroyed, Hillalso estimated methane productivity based onVS loaded (involves biodegradability), total

solids TS (involves fraction of the TS which isVS) and animal live weight LW (involves bio-degradability, VS/TS ratio and the animalwaste production factor). These data are sum-marized in Table 3 and are independent of thedigestion temperature, provided the retentiontime is su�ciently long (>14 d at 358C and>6 d at 608C).

Although methane productivity per unit VSdestroyed is independent of the type of animalwaste, the productivity for dairy and beefwastes, when expressed on the basis of VSloaded, is signi®cantly lower than that for theother two types of waste because of di�erencesin biodegradability. Compared to the methaneproductivity per kg VS loaded, the pro-ductivity per kg TS loaded is lower becausethe wastes contain refractory non-biodegrad-able material. On a LW basis, poultry wastesshow about three times the productivity ofother waste types because of the higher wasteproduction rate for poultry and its higher bio-degradability.

Note that the B0 values implicit in Table 3,i.e. 0.64 (0.32/0.5) for swine waste, 0.46 forbeef waste, 0.26 for dairy waste and 0.72 forpoultry, generally di�er (in the ®rst three casessigni®cantly) from the values in Table 2,although their relative ranking for the di�erenttypes of wastes remains the same.

The data in Table 3 su�ces for techno-econ-omic assessments of animal waste fermenta-tion processes. However, when designing oroperating digesters, the interaction of operat-ing variables such as residence time, tempera-ture, loading rate etc. on the methaneproductivity is of signi®cant interest. This isdiscussed next.

5. KINETIC MODELING OF THE ANAEROBICDIGESTION PROCESS

The complexity of the anaerobic digestionprocess is evident from the stoichiometrydescribed earlier. A complete mathematicaltreatment of the process would require the

Table 2. Characteristics of raw animal wastes

Waste type Biodegradability, (B0) Solubility Acidity, (Af) Ammonia

Swine 0.90 0.30 0.10 0.0154Beef (con®nement) 0.65 0.50 0.05 0.0154Beef (dirt) 0.56 0.45 0.05 0.0115Dairy 0.36 0.50 0.05 0.0115Poultry (broiler) 0.70 0.30 0.20 0.0308Poultry (layer) 0.87 0.30 0.20 0.0308

*As pointed out by one of the reviewers, the actualmethane content will depend on the extent of CO2 dis-solution in the digesting slurry.

A. HUSAIN564

simultaneous solution of mass balanceequations for each individual substrate andbacterial population along with interphasemass transfer between the slurry and gasphases. Clearly, such a treatment4 is extremelycomplex, yielding equations with numerousparameters. Simpler treatments have, there-fore, been developed to predict the steadystate and dynamic behaviour of digesters.Details of relevant models are presented inAppendix 1Appendix 2.

Steady state models involve a mass balancefor the total substrate5,6 or as in Chen andHashimoto's model7,8 also consider a massbalance for the total bacterial population.These models are relatively simple and havelimited applicability. Chen and Hashimoto'smodel predicts the total substrate utilizationrate as a function of residence time for wasteswith known refractory content. Although themodel predicts failure as a result of bacterialwashout, it is not capable of predicting processfailure due to the inhibitory e�ects of VFAs

and ammonia content because the model doesnot include these parameters explicitly. Thesee�ects are important when dealing with animalwastes.

Hill9 developed an empirical steady statemodel based on simulation results fromdynamic models. The model predicts thesteady state methane production rate as afunction of the substrate loading rate forspeci®c types of animal wastes. As illustratedin Fig. 2, the linear relationship betweenmethane production rate and substrate loadingrate breaks down at a certain loading ratedepending on the type of wasteÐthis is thepoint where biological stress begins to occur.Beyond this point, with further increase inloading rates, the methane production ratedecreases sharply to zero. Di�erences in thelinear responses between di�erent wastes aresolely attributable to di�erences in unstressedVS reduction or biodegradability. Althoughthe model is useful for estimating methaneproduction rates generally, its application is

Table 3. Data on methane productivity

CH4 Productivity

Waste type m3/kg VS loaded m3/kg VS destroyed m3/kg TS loaded m3/mg LW-d

Swine 0.32 0.50 0.250 1.54Beef 0.23 0.50 0.200 1.38Dairy 0.13 0.50 0.105 1.12Poultry 0.36 0.50 0.250 3.37

Fig. 2. Predicted mesophyllic methane productivity for animal wastes.

Models of kinetics of anaerobic digestion 565

limited to mesophilic only conditions because

model parameters for other conditions were

unfortunately not determined.

Compared with steady state models,

dynamic models, such as those described by

Hill and co-workers, are signi®cantly more

complex, take into account mass balances for

several distinct substrates and bacterial cul-

tures, and consider inhibitory e�ects of VFAs

and ammonia. In addition to predicting the

substrate utilization and methane generation

rate, these dynamic models are also able to de-

lineate unstable digester operating conditions.

By representing raw waste in terms of its bio-

degradability B0 and acid factor Af, depen-

dence of model parameters on the type of

waste was eliminated with the parameters

being a function only of the bacterial popu-

lations present. Therefore, the models can be

expected to apply to other types of wastes pro-

vided their B0 and Af values are determined.

A simpli®ed treatment10,11 is outlined below.

This model distinguishes between substrate

utilization by acid forming bacteria vs that by

methane forming bacteria. The acid forming

bacteria use the biodegradable portion of the

waste as a substrate whereas the methane

forming bacteria utilize VFAs as a substrate.

The following mass balances were solved for:

BVS: Rate of change in concentration, S=Netrate of change in concentration because of¯owÿ rate of consumption by acid formingbacteria

VFAs: Rate of change in concentration, VFA=Net rate of change in concentration becauseof ¯owÿRate of consumption by methaneforming bacteria + Rate of formation byacid formers

Acid formers: Rate of change in concentration, X=Netrate of change in concentration because of¯owÿDeath rate + Rate of growth byconsumption of BVS

CH4 formers: Rate of change in concentration, Xc = Netrate of change in concentration because of¯owÿDeath rate + Rate of growth byconsumption of VFAs

Expressions for the growth and death ratesof the bacterial populations are required tosolve the above mass balances. The growthrate functions depend on the substrate concen-tration (S or VFA) and incorporate termswhich account for the inhibitory e�ects ofVFAs (and ammonia if required). The deathrate functions depend on VFA concentrationswith the rates increasing with increase in VFAconcentration. In total, the four mass balancesinvolve 12 waste independent parameters.

Although the time dependent solution of theabove set of mass balances is straightforward,it still represents a formidable computationalundertaking. In many cases of practical inter-est, transient behaviour of the digester maynot be of signi®cant interest and, therefore,steady state solutions of the model equationsare su�cient. Rather than rely on simplesteady state models5±8 or on Hill's empiricalrelationship9 which can be applied only undermesophilic conditions, it was considered desir-able to develop a more general solution forthe steady state concentrations in a digester.Such a generalized steady state solution wasobtained by setting the time derivatives in theabove mass balances equal to zero and solvingthe resulting algebraic equations iteratively forthe steady state VFA concentration followedby the corresponding concentrations S, X andXc. The substrate reduction rates and hencethe gas generation rates can then be estimatedfrom these steady state concentrations. Detailsare given in Appendix 3.

For selected cases listed in the literature(Table 4 of Ref. 10), Table 4 compares theresults obtained using the iterative procedure

Table 4. Comparison between simulated results obtained by Hill and those obtained from the present solution to Hill'sdynamic model

CH4 Generation rate (l/l/d)*

Waste Type T (8C)Residence time, y

(d)Inlet volatile solids

conc. (g/l) Hill's result10 Present result

Beef (dirt) 35 20 47.5 0.69 0.56Beef (con®nement) 55 12 62.5 1.40 1.46Beef (con®nement) 55 7 82.6 3.05 3.09Beef (con®nement) 55 5 85.0 3.75 3.76Dairy manure 60 6.2 63.5 1.14 1.19Poultry 35 52.5 72.5 0.53 0.51Swine (con®nement) 35 15 60.0 1.45 1.55Swine (con®nement) 35 30 31.4 0.40 0.42

*l refers to reactor volume

A. HUSAIN566

outlined above with those computed by Hill.Clearly, the agreement between the two sets ofresults is very good. For the case with resi-dence time y of 5 days, the iterative proceduredid not converge but achieved a local mini-mum; the gas generation rate of 3.76 l/l/d (lrefers to reactor volume in liters) correspondsto this situation and evidently agrees well with

literature prediction. Similarly, for the casewith y equal to 6.2 days, convergence occurredbut outside the physically acceptable range of0 to VFA0, the in¯uent VFA concentration;once again, the calculated gas generation rateagrees well with literature prediction. Clearly,for both these cases, steady state was notachieved because of insu�cient residence timeand the gas generation rates reported in theliterature represent computational artifacts.

Figure 3 illustrates an application of thesteady state solution of Hill's model; the caseconsidered involved the digestion of swinewaste at 358C when the in¯uent volatile solidsconcentration is 50 g/l. As shown, both sub-strate concentrations (S and VFA) decreasedwith increasing residence time, whereas thebacterial concentrations (X and Xc) increasedwith increasing residence time. Although thegas generation rate decreased with residencetime, the volume of gas produced increases,albeit with diminishing returns, as residencetime increases. These results provide far moreinsight in to the digestion process than is poss-ible using the simpler steady state models.

Finally, in Hill's treatment, the use of B0

and Af implies that the substrate is instantlyavailable for use by the bacterial population.This assumption is valid only when the timerequired for hydrolysis is short compared withthe residence time. Otherwise, as proposed byThomas and Nordstedt,12 two additional sub-strate concentrations are required to represent,respectively, the breakdown of readily degrad-able and slowly degradable volatile solids; thelatter represents material which is protectedfrom rapid hydrolytic attack by the lignocellu-losic structures frequently found in crop resi-dues and other sources of biomass.

6. CONCLUSIONS

Hill's simpli®ed dynamic model provides areasonable description of the digestion processwith kinetic parameters which are intrinsicallyindependent of the type of waste. Hence, themodel is applicable in general to a variety ofwastes containing readily degradable solidsand require only two waste speci®c input par-ameters, namely, their biodegradability andacid factor.

In many cases of practical interest, transientbehaviour of the digester may not be of sig-ni®cant interest and steady state solutions suf-®ce. Although Hill's empirical expression for

Fig. 3. Steady state digester behaviour based on Hill'smodel; (a) concentrations of substrate and bacterial cul-

tures and (b) gas generation.

Models of kinetics of anaerobic digestion 567

the steady state digestion of animal wastes ful-®lls this need, the estimated parameter valuespertain only to digestion of speci®c types ofwaste at mesophyllic conditions. A more gen-eralized steady state solution procedure devel-oped here allows the steady state substrateand bacterial concentrations and hence the gasgeneration rate to be determined for any typeof waste at any desired temperature.

AcknowledgementsÐComments received from my col-leagues, Y. Nguyen and D. Brown, during the preparationof this paper are gratefully acknowledged.

REFERENCES

1. Hill, D. T., A comprehensive dynamic model for ani-mal waste methanogenesis, Transactions of the ASAE,1982, 25(5), 1374.

2. Hill, D. T., Practical and theoretical aspects of engin-eering modeling of anaerobic digestion for livestockwaste utilization systems, Transactions of the ASAE,1985, 28(3), 850.

3. Hill, D. T., Methane productivity of the major animalwaste types, Transactions of the ASAE, 1984, 27(2),530.

4. Hill, D. T. and Barth, C. L., A dynamic model forsimulation of animal waste digestion, J. WaterPollution Control Federation, 1977, 49, 2129.

5. Chowdhury, R. B. S. and Fulford, D. J., Batch andsemi-continuous anaerobic digestion systems,Renewable Energy, 1992, 2(4/5), 391.

6. Srivastava V. J., Biogasi®cation kinetics of biomass/waste blends. In Methane from Community Wastes, ed.R. Isaacson. Elsevier Science, New York, 1991.

7. Chen, Y. R. and Hashimoto, A. G., Substrate utiliz-ation kinetic model for biological treatment processes,Biotechnology and Bioengineering, 1980, XXII, 2081.

8. Chen, Y. R. and Hashimoto, A. G., Kinetics ofmethane fermentation, Biotechnology andBioengineering Symposium, 1978, 8, 269.

9. Hill D. T., Steady state mesophilic design equationsfor methane production from livestock wastes,Presented at the 1990 International Meeting Sponsoredby ASAE, Columbus, Ohio, 24±27 June 1990.

10. Hill, D. T., Simpli®ed Monod kinetics of methane fer-mentation of animal wastes, Agricultural Wastes,1983, 5, 1.

11. Hill, D. T., Tollner, E. W. and Holmberg, R. D., Thekinetics of inhibition in methane fermentation ofswine manure, Agricultural Wastes, 1983, 5, 105.

12. Thomas, M. V. and Nordstedt, R. A., Generic anaero-bic digestion model for the simulation of various reac-tor types and substrates, Transactions of the ASAE,1993, 36(2), 537.

13. Monod, J., The technique of continuous culture the-ory and application, Annales de l'Institut Pasteur,1950, 79, 390.

14. Contois, D. E., J. Gen. Microbiol., 1959, 21, 40.15. Hashimoto, A. G., Chen, Y. R. and Varel, V. H.,

Theoretical aspects of methane production: state-of-the-art, Livestock Waste: A Renewable Resource, 1980,4(86), .

APPENDIX 1

Steady state models.It is desirable to operate digesters at steady state

conditions. Unstable conditions arising from stress of the

biological population lead to a reduction in methane pro-duction. Stress of the biological population may occur asa result of short residence time leading to bacterial wash-out, inhibition by high VFA levels or toxicity due to highammonia concentrations. From design considerations, oneis interested in steady state conditions and the regionwhere this no longer exists. A number of steady statemodels are described below.

First order model.The ®rst order model assumes that the rate of substrate

removal is proportional to its concentration. The substratemass balance for a CSTR can be written as:5,6

dS=dt � �S0 ÿ S�=yÿ k:S �A1�where S is the volatile solids concentration (kg/m3); S0 isthe in¯uent volatile solids concentration; y is the hydraulicretention time (the ratio of reactor volume V to the volu-metric feed rate v) (d); t is the time (d); k is the ®rst orderrate constant (1/d).The gas production rate (m3/d) is proportional to the re-

duction in substrate concentration and is, therefore, givenby

g � g:B0:�S0 ÿ S�:v �A2�where y is the speci®c methane productivity, m3/kg VSdestroyed. Substituting the steady state concentrations,[obtained by setting the time derivative in equation (A1)equal to zero] in equation (A2) and re-arranging yields

y � g:B0:V:S0=gÿ 1=k �A3�It follows that a plot of y vs B0�V�S0/g should yield astraight line with slope g and intercept 1/k.

Simple Monod kinetics.In contrast to the ®rst order model, consideration of

Monod kinetics13 for the growth rate of bacteria requiresthe solution of at least two mass balance equations, onefor the substrate and the other for the bacteria feeding onthat substrate. The mass balances for the biodegradablesubstrate S and bacteria X are, respectively, given by

dS=dt � �S0 ÿ S�=yÿ mX=Y �A4�

dX=dt � mX ÿ X=y �A5�where Y is a yield coe�cient (g organisms/g substrate) andm, the bacterial growth rate, is according to Monod kin-etics given by

m � mm S=�Ks � S� �A6�where mm is the maximum speci®c growth rate (1/d) andKs is the half velocity constant (i.e. the value of S corre-sponding to m= mm/2).Note that equation (A5) assumes the in¯uent bacterial

concentration to be zero. At steady state, equations (A4)and (A5) yield,

m � 1=y �A7�

X � Y�S0 ÿ S� �A8�

S � Ks=�ymm ÿ 1� �A9�Substitution of m in equation (A6) gives

S � Ks=�ymm ÿ 1� �A9�As can be seen from equation (A9), simple Monod kineticsleads to the anomalous conclusion7 that the e�uent sub-strate concentration is independent of the in¯uent sub-strate concentration.

A. HUSAIN568

Chen and Hashimoto's model.

Because of the di�culty with simple Monod kinetics dis-cussed above, Chen and Hashimoto7,8 developed a kineticmodel based instead on the Contois description14 of bac-terial growth rate according to which m is given by

m � mm S=�bX � S� �A10�where b is a kinetic parameter. Unlike Monod's descrip-tion, the Contois expression incorporates a dependence onX.

The basic mass balances for the substrate S and bacterialculture X are, as before, given by equations (A4) and(A5); substitution of X according to equation (A8) in theContois expression yields

S=S0 � K=�mmyÿ 1� K� �A11�where K equals bY. Unlike equation (A9), the Contoisdescription leads to a more satisfactory outcome for thesubstrate concentration, namely, its direct proportionalityto the in¯uent substrate concentration S0 when K is a con-stant. The value of K increases above this constant valuewhen S0 increases above a certain threshold concentrationand causes system overload.

Considering the refractory component of the substrate Sr,the total substrate concentration in the reactor St andin¯uent, St,0 can be expressed as

St � S � Sr �A12�

St;0 � S0 � Sr �A13�Further, de®ning R as Sr/St,0 leads to the following ex-pression for the substrate utilization rate:

�St;0 ÿ St�=y �f�1ÿ R�St;0=ygf1ÿ K=�mmyÿ 1� K�g �A14�

Fig. A1 illustrates the variation of the substrate utilizationrate with y. At y< ymax, the substrate reduction ratedecreases and ultimately reaches zero (i.e. digester failureoccurs) as washout of the microorganisms occurs. Thiscan also be understood from the alternate representationof equation (A11) which after combining with equation(A7) has the form:

m=mm � S=S0=fK � �1ÿ K�S=S0g �A15�According to equation (A15), when S4 S0 at washout, mwill approach mm regardless of the value of K. Therefore,according to equation (A10), X will tend to 0.

Simple empirical relationship for predicting methane pro-duction Rates.

Hill9 described the overall nonlinear behaviour depicted inFig. 2 by the relationship

a � gB0sl �A16�where a is the CH4 production rate (l CH4/l�d where lrefers to reactor volume); Y is the CH4 productivity whichhas a value of 0.5 l CH4/g VS destroyed (see Table 3); s isthe loading rate (g VS loaded/l�d), and I represents a pro-ductivity index with a value of 1.0 in the unstressed regionbut which decreases to zero as stress develops and leads todigester failure. I is given by

l � 0:5� A tanf�tÿ s�=0:211g=2:95 �A17�where the constant t represents a stress index (value of swhere I= 0.5). The Atan term is essentially equal to 0.5when s is below the loading rate value where stress beginsto occur. Note that the argument of the Atan term is inradians.

Table A1 summarizes the values for the two constants inequation (A16). Note that B0 values di�er signi®cantlyfrom those in Table 2, although their relative ranking forthe di�erent animal wastes remains the same. Althoughswine and poultry waste are almost identical in CH4 pro-ductivity in the unstressed region, poultry waste, as aresult of its higher NH3 content experiences inhibition orstress before swine wasteÐthis is re¯ected in the lower tvalue for poultry waste.

APPENDIX 2

Dynamic models.

Dynamic models predict the transient behaviour of adigester based on a set of di�erential equations represent-ing the mass balances for the various substrates and bac-terial cultures. Details of a simpli®ed model developed byHill and co-workers10,11 and its extension to substratescontaining slowly degradable solids are described below.

Simpli®ed model of methane fermentation of animal wastesproposed by Hill.

In 1983, Hill and co-workers published a Monod basedkinetic model11,12 wherein the waste was represented bytwo lumped parameters, namely, B0 and Af. The modelwas capable of predicting dynamic process failure underconditions of microbial washout or inhibition. It distin-guished between substrate utilization by acid forming bac-teria vs substrate utilization by methane forming bacteria.The acid forming bacteria use the biodegradable portionof the waste as a substrate whereas the methane formingbacteria utilise VFAs as a substrate. Accordingly, the fol-lowing mass balances were solved for:

BVS:

dS=dt � �S0 ÿ S�=yÿ mX=Y �A18�Acid forming bacteria:

dX=dt � �mÿ Kd ÿ 1=y�X �A19�Methane forming bacteria:

Fig. A1. Volumetric substrate utilization rate as a functionof retention time according to Chen and Hashimoto's

model.

Table A1. Values of the biodegradability factor B0 andstress index t

Type of waste B0 t

Swine 0.6312 6.69Poultry 0.6259 4.89Beef 0.4830 9.21Dairy 0.2292 10.12

Models of kinetics of anaerobic digestion 569

dXc=dt � �mc ÿ Kdc ÿ 1=y�:Xc �A20�VFAs:

dVFA=dt � �VFA0 ÿ VFA�=y� mX�1ÿ Y�=Y ÿ mcXc=Yc �A21�

The form of equations (A19) and (A20) di�ers from thatof equation (A5) because of the inclusion of the deathterm. Thus, at steady state, the growth rate of the bacteriais balanced by its death rate and the rate at which the bac-terial population ¯ows out of the reactor.

Based on Monod kinetics, the growth rate of the acid andmethane forming bacteria are, respectively, given by

m � mm=�Ks=S � 1� VFA=Ki� �A22�

mc � mmc=�Ksc=VFA� 1� VFA=Kic� �A23�where Ki and Kic are the VFA inhibition coe�cients foracid and methane forming bacteria, respectively.

Similarly, the following expressions for death rates wereproposed based on Monod's function:

Kd � Kdm=�1� Kid=VFA� �A24�

Kdc � Kdmc=�1� Kidc=VFA� �A25�where Kdm and Kdmc were assumed to be, respectively,equal to mm and mmc and Kid and Kidc represent ``half vel-ocity'' death rate constants. Thus, as VFA concentrationincreases, the death rates increase. mm and mmc wereassumed equal and estimated from the temperature depen-dent function of Hashimoto et al15

mm�0:013:T��C�ÿ0:129 �20<T<60� �A26�Table A2 summarizes the values of the various constantsused in the model.

The initial conditions (t = 0) required to solve the abovedi�erential equations are:

S � S0 � VSin:B0 �A27�

VFA � VFA0 � S0:Af �A28�

X � Xc � Xs �A29�where VSin is the in¯uent volatile solids concentration andXs represents some seed concentration introduced into thereactor to initiate the fermentation.

Because the only means available for removing organiccarbon in anaerobic digestion is the conversion of the or-ganic material to CH4 and CO2, a direct correlation existsbetween solids destruction and CH4 formation. The rateof volatile solids destruction is represented by mcXc/Yc (gVFA/l/d) and, therefore, the volumetric CH4 productivity(l CH4/l/d) equals 0.5 mcXc (1ÿYc)/Yc where the factor

(1ÿYc) takes into account the portion of VFAs which isconverted into bacterial mass and not into methane.Solution of the dynamic mass balance equations

(A18)±(A21) is not necessary if the objective is to ascertainthe operating envelop within which stable digester beha-viour can be expected. Stable behaviour is ensured whenthe derivative dX/dt or the real growth rate (RGR) of thebacterial population given by

RGR � mÿ Kd ÿ 1=y �A30�is greater than zero.2 In equation (A30), the expression form may include the inhibitory e�ects of both VFAs andammonia, Kd depends on the concentration of the deathagent (VFA for methanogenic culture) and 1/y is the wash-out rate. Thus, the bacterial mass increases if RGR > 0,is at steady state when RGR= 0 and decreases whenRGR< 0. The speci®c combinations of VFA and NH3

levels which de®ne the RGR< 0 envelop represent con-ditions where stable digester operation cannot be sus-tained.Plots of RGR as a function of VFA and NH3 levels can

be constructed using a range of y values for both metha-nogenic culture and acid formers; note that in the lattercase, the substrate is not an inhibitory substance. Theseplots can provide valuable insight in to the operation ofthe digester. For wastes which are high in VFA or NH3

(swine and poultry) content, y can be increased to o�setthe slower growth encountered in inhibiting environments.

Thomas and Nordstedt's model.In Hill's treatment, the use of B0 and Af implies that the

substrate is instantly available for use by the bacterialpopulation. This assumption is valid only when the timerequired for hydrolysis is short compared with y.Otherwise, as proposed by Thomas and Nordstedt,12 twoadditional substrate concentrations R and L are requiredto represent, respectively, the breakdown of readilydegradable (assumed half-life of 2 d) and slowly degrad-able volatile solids (assumed half-life of 20 d).Accordingly, the decomposition of R and L can be rep-resented by

R � R0 exp �ÿ0:693 t=2� �A31�

L � L0 exp �ÿ0:693 t=20� �A32�Thomas and Nordstedt developed a generic anerobicdigestion model to represent a wide variety of anaerobicreactor types and substrates. Model equations for a CSTRare summarized below where terms with the multiplier0.693 represent time derivatives of R or L.

Balance on acetogens:

dX=dt � �mÿ Kd ÿ 1=y�: �A33�Balance on methanogens:

dXc=dt � �mc ÿ Kdc ÿ 1=y�:X �A34�Balance on L:

dL=dt��L0ÿL�=yÿ0:693=20 L �A35�Balance on R:

dR=dt � �R0 ÿ R�=y� 0:693=20 Lÿ 0:693=2 R �A36�Soluble organics:

dS=dt ��S0 ÿ S�=y� 0:693=2 R

� kd:X � kdc:Xc ÿ m:X=y �A37�VFAs:

dt ��VFA0 ÿ VFA�=y� m X yhac

ÿ mc Xc=yc �A38�

Table A2. Parameter values in Hill's model

Parameter Units Value

Kid g VFA/L 16Kidc g VFA/L 16Kdm, Kdmc 1/d Equal to mm, mmc

mm, mmc 1/d equation (A26)Y g organism/g BVS 0.1Yc g organism / g VFA 0.0315Ks g BVS/L 9.0Ksc g VFA/L 3.0Ki g VFA/L 9.0Kic g VFA/L 11.0

A. HUSAIN570

yhac in equation (A38) is the acid yield coe�cient withunits of g VFA/g acetogens. Note the distinction betweenthe terms mXyhac and mX/yÐthe former represents VFAsproduced by acetogens whereas the latter represents theconsumption of the soluble organics (C6H12O6) substrate.Growth and death rates were estimated according toequations (A22)±(A25).

Besides the initial conditions stated in equations (A27)±(A29), two additional initial conditions are required tosolve the mass balances:

L � L0 � VSin B0 Lf �A39�

R � R0 � VSin B0 Rf �A40�where Lf and Rf, respectively, represent the slowly degrad-able and the rapidly degradable fractions of the biodegrad-able solids.

Because, animal wastes are characterized by Lf valueswhich are essentially zero, calculations based on theThomas and Nordstedt model should be essentially identi-cal to those based on the simpli®ed Hill's model. Not sur-prisingly, prediction of methane generation rates agreedwell with the results of several animal waste studies. Thismodel, however, can be expected to better represent thedigestion behavior of other biomass such as water hya-cinths for which Lf value is signi®cantly greater than zero.

APPENDIX 3

Steady state solution to Hill's dynamic model.

A steady state solution to the dynamic model developedby Hill and co-workers can be obtained by setting thetime derivatives in equations (A18)±(A21) equal to zeroand combining the resulting equations with equations(A22)±(A25). These equations are summarized below:

�S0 ÿ S�=yÿ mX=Y � 0 �A41�

mÿ Kd ÿ 1=y � 0 �A42�

mc ÿ Kdc ÿ 1=y � 0 �A43�

�VFA0ÿVFA�=y�mX�1ÿY�=YÿmcXc=Yc�0 �A44�

m � mm=�Ks=S � 1� VFA=Ki� �A22�

mc � mmc=�Ksc=VFA� 1� VFA=Kic� �A23�

Kd � Kdm=�1� Kid=VFA� �A24�

Kdc � Kdmc=�1� Kidc=VFA� �A25�Combining equations (A43), (A23) and (A25) yields

Kdmc=�1� Kidc=VFA� � 1=y �mmc=�Ksc=VFA� 1� VFA=Kic� �A45�

which can be iteratively solved for the steady state VFAconcentration. Substituting VFA into equation (A46)

Kdm=�1� Kid=VFA� � 1=y �mm=�Ks=S � 1� VFA=Ki� �A46�

which is obtained by combining equations (A42), (A22)and (A24) allows the steady state substrateconcentration S to be solved for. The steady state bac-terial concentrations X and Xc can then be solved forsuccessively using equations (A41) and (A44). Themethane generation rate can then be calculated as pre-viously described.

Note that equation (A45) is a cubic equation and hencewill yield three solutions for VFA. Based on physicalreasoning, values of VFA outside the range of 0 to VFA0

should be discarded because the steady state VFA concen-tration must be lower than VFA0. Based on computationstodate, only one solution was found in this range, exceptwhen y was low in which case no solution was found. Thelatter behaviour is consistent with the observation ofdigester failure and hence the absence of a steady statewhen the residence time is too low.

Models of kinetics of anaerobic digestion 571