PII: S0273-1223(96)OO669-S

e> Pergamon Wal.Sci.T«11. VoL 34, No.5-6,pp.383-392,1996.Copyright C 1996fAWQ.Published by Elsevier Science Ltd

Printed in GreatBritain. AD rightsreserved.0273-1223196 S15-00 + 0-00

A MATHEMATICAL MODEL FOR THEOPTIMAL DESIGN AND OPERATION OFAN ANAEROBIC CO-DIGESTION PLANT

N. A. Bozinis", I. E. Alexiou** and E. N. Pistikopoulos*

• Centrelor Process SystemJ Engineering. Department olChemical Engineering.Imperial College. London SW72BY, UK•• Department 01Civil Engineering, University olNewcastle upon Tyne,Newcastle upon Tyne. NEJ 7RU. UK

ABSTRACT

The anaerobic co-digestion of high strength industrial wastewaters in centralised plants receiving efflu­ents from various local small scale industries is considered to be superior than individual small scaletreatment plants. An approa.ch is proposed to systematically tackle co-digestion problems, with respectto both biological and engineering models. In a steady-state mathematical model presented, the effectof combining wastewaters in mixtures, based on major component groups (i. e. lipids, hydrocarbonsand proteins), is systematically accounted for. The model is then solved via a non-linear optimisationprogram (NLP) which results in the determination of an optimal design of the necessary reactors andtheir respective layout, by balancing the natural trade-offs between reactor and post-treatment costs(due to high pollutants emissions). The preparation of appropriate mixtures is the major parametercontrolled. The applicability and potential of the model are demonstrated with a small example.Copyright@ 1996IAWQ. Published by ElsevierScienceLtd

KEYWORDS

anaerobic treatment, co-digestion, industrial wastewaters, mathematical modelling, mathematical pro­gramming, optimal design, scheduling.

INTRODUCTION

Food processing industries use as raw materials various fruits and vegetables that are seasonalin nature.Some industries have a specific product that is seasonally produced, while others vary their productionduring the year. In either case, wastewaters generated from these industries are varying significantlyduring the year, both in quantity and characteristics. Individual treatment units can be prohibitivelyexpensive for small scale food processing industries when compared to total investment. In addition, theseasonal nature of the wastewater generated makes the operation of individual treatment plants harder,due to long start-up times required. A favourable alternative is to construct centralised wastewatertreatment plants strategically situated, directing there all the wastewaters from local food processingplants. This will render the treatment of the generated wastewaters much more economical while atthe same time securing a stable year-round operation.

383

384 N.A.BOZINIS tt al,

The availability of various organic wastewaters in such a plant facilitates the potential use of co­digestion, where wastewaters are combined and treated as mixtures. The digestion of various mixed­source wastewaters is agreeable as long as it enhances the treatment efficiency (i.e. retention time,methane yield, etc.) compared to the treatment of individual wastewaters. Combining wastewatersin appropriate recipes can supply either nutrients or dilution of inhibitory compounds and/or charac­teristics (e.g. heavy metals, ammonia, high or low pH, etc.], necessary for the effective degradationof the organic load by the biomass. As it seems, "proper" wastewater mixing improves the treatmentefficiency; however, there are cases where the mixture has worse performance compared to individualwastewater treatment and should be avoided. Clearly there is a need to determine which mixtures arebeneficial and which are not, in order to exploit co-digestion in full.

Some anaerobic treatment plants using co-digestion have been built but operate mainly on an empiricalbasis; a rigorous tool that enables the selection of optimal wastewater mixing is still lacking. There areseveral diverse aspects in the problem of applied co-digestion that necessitate inter-disciplinary com­bined effort in order to be tackled. On one hand there is the need to perform experiments and derive amathematical model for the degradation of mixed wastewaters. On the other hand, given some wastew­ater sources and their expected variation in a year, a process systems (decision-making) approach isnecessary to determine the configuration of reactors in the centralised treatment plant (design stage)and the exact wastewater mixing strategy in each operational period, utilising a mathematical modelfor the mixed-sources biodegradation, with the objective to minimise capital and operational costs.This work focusses on the second part of the co-digestion task, based on available expressions for thebiodegradation, developed within the FlexiBLE, a EU industrial research project aiming to developtechniques to enable the practical utilisation of co-digestion (EU report, 1995).

MATHEMATICAL MODELLING

Biological models

A lot of research effort has focussedon the development ofmathematical models describing the biodegra­dation of single-source wastewaters (Mosey 1983, Bryers 1985, Angelidaki et al. 1993, Siegrist et al.1993, Vavilin et al. 1993). A mathematical model is effectivelyan expression or a set of expressions thatlink the rate of organic load removal, rate of biogas production and other useful effluent variables towastewater and reactor design characteristics (reactor type, pH, temperature, etc.) in a mathematicallyquantifiable manner. The various models that have been proposed vary in the details involved and arefurther distinguished into steady-state and dynamic models; most are limited to simple reactor con­figurations (i.e. CSTR). Steady-state models are relatively simpler, linking influent-effluent variablesof interest under steady-state conditions, allowing for different operational parameters. Such modelscannot describe the temporal changes in bacteria and products during transitions from one steady-stateto another but are widely used to predict residual substrate levels, biogas production and optimumretention time (RT) for the degradation of different wastewaters. More complex models are required inorder to describe the transitional or dynamic effects present in start-up and other operational upsetsof anaerobic digesters. A detailed chemical path has to be assumed for all stages from hydrolysis tomethanogenesis, formulated in differential equations, leading usually to models that cannot be math­ematically solved; iterative algorithms are necessary for the solution, conveniently performed with acomputer implementation of the model. Dynamic models are used in computer simulations of tran­sitional situations, pin-pointing potentially dangerous operational conditions, where accumulation ofintermediate degradation products (e.g. acids) may lead to significant inhibition of the process or evencomplete failure. Simulations are thus attempting to minimise the necessity of expensive experimentalidentification of such potential hazards by building pilot digesters.

In any anaerobic digester there is a substrate whose composition is partially known and then often interms which have no precise chemical meaning (i.e. COD content). The degradation of the organic load

Anaerobic co-digestion plant 38S

proceeds through a very complicated, and yet eluding precise identification, chemical path, catalyzedby a mixed bacterial population of largely unknown bacteria in undefined numbers, yet all interacting.

Furthermore, mass transfer phenomena similar to those involved in other catalytic reactors, where thesubstrate to be converted has to diffuse towards complex bacterial conglomerations of diverse sizes,add another complicating element, apart from the already complex pure chemical part. Under thisperspective it is clear that all mathematical models are simplifications, assumptions and abstractions ofthe actual phenomena and their "theoretically" valid constants have to be obtained by mathematicallyfitting available sets of experimental data. Their validity and prediction capacity is therefore limitedto the conditions that the experiments were conducted, mainly in terms of substrate composition. Thevariations in data are such that curves can be fitted using equations that have no biological basisyet still follow the general trend exhibited by the data set. Models with different conceptual make­up may fit equally adequately; it is thus questionable whether there is any need to come up withmodels of increased complexity, requiring more data in order to work and being constrained in a smallapplicability range, as far as different systems are concerned (Hobson & Wheatley, 1993). Finally,models that can accommodate mixtures of wastewaters, so as to be useful in a co-digestion context,are practically inexisting.

Economic and engineering models

An area where simpler aggregate models have been appreciated is the design and construction of anaer­obic digesters, where details of the metabolism are one small (yet significant) part of the whole problem,which has to consider reactor volume and layout, operating costs, environmental legislation and similarmanagerial aspects. These models are commonly referred to as economic and engineering models, andnecessarily have to rely on biogas production data, reduction in pollutants in effluent streams andso on, obtained from either experimental observations or from detailed biological models, describingsteady-state conditions. The key idea in these models is that most operational parameters (e.g. pH,temperature, etc.] are already fixed from the examination of the pure biological model, therefore lessinformation is required. Important objectives are the maximisation of biogas production and pollutantsremoval, while negotiating the increased reactor volumes, by appropriately choosing the retention timeor a similar control variable. An example in this area is the computer model developed by Hawkes &Rosser (1983) that takes into consideration capital and operating costs of the digester and ancillaryequipment, plus more specialised factors like governmental grants available, taxes, etc.

Although engineering models are correct in incorporating simplified expressions of the biological phe­nomena, sometimes they fail to consider an objective that will be affected by the processing systemas a whole, by using partial objectives like the maximisation of biogas production alone. This omis­sion stems from the traditionally encountered difficulty in solving engineering models, due to the largenumber of variables and equations involved when a process is regarded as a whole; however, the adventof powerful computers and rigorous numerical solution algorithms has recently enabled to tackle whatwas once an impossible task.

Mathematical programming. In general, the design problem of any chemical plant involves the selectionof the variables (degrees of freedom) that will minimise a cost function, while meeting at the sametime process mass and energy balances plus various operational constraints. The general form of amathematical model for the design of a process may be generalised as:

mino(x)III

s.t. h;(x) =0 (1)9j(X) :5 0

where x is the vector of all process variables, o(x) is the economic objective to be minimised, h;(x) theprocess equations (material and energy balances, etc.) and 9j(x) the process constraints imposed byboth the operation itself (i.e. the maximum allowabledilution rate in a CSTR digester) and the designer

386 N. A.BOZINIS etat,

(i.e. the maximum allowable effluent COD content). This approach ensures the true optimality of thedesign and avoids bottlenecks that could result in partial optimality in case some arbitrary objectiveis selected, not fully representing the whole picture of the plant design. Furthermore, if all physicallimitations are correctly inserted in the model in (1) the designer is no longer required to empiricallyfix selected variables.

The formulation and solution of mathematical programs as the one in (1) is the task of Process SystemsEngineering. In such programs, the whole biological step might be represented by as little as a singleequation linking the reactor influent and effluent streams. The solution of these programs that suppliesthe values of the control variables for achieving optimal operation (in terms of minimum operationaland annualised capital costs) is quite difficult, especially when the number of variables and constraintsinvolved rises. A whole division of mathematics (constrained optimisation) and numerical analysis isdedicated for the solution of such programs.

Modelling of co-digestion

The problem of co-digestion has two facets. First, one has to determine the effect of mixing two ormore individual wastewaters in terms of treatment efficiency, in a clear and quantifiable manner. Thefirst problem to overcomeis the apparent lack of adequate composition characterisation of wastewaters.A comprehensive characterisation prototype is required that will be substantially finer than the crudeCOD content of typical wastewaters. The hypothesis that serves as the backbone of the proposed co­digestion model is that the composition characterisation based on the lipids, protein and hydrocarbonconcentrations is adequate to convey the effect of mixing different wastewaters. This effectivelymeansthat each wastewater is regarded as a different combination of these three fundamental biopolymers,and this can be extended for actual mixtures of individual wastewaten.

The next step is to select an appropriate kinetic model. Following the guidelines for simplicity advo­cated previously, the aggregate biodegradation is assumed to follow simple uninhibited Monod kineticsat steady-state. The two required parameters, namely the maximum growth coefficient /-'mGZ and thehalf saturation constant K.. are regarded as functions of the composition in lipids, proteins and hydro­carbons alone. The exact form of these functions remains to be determined by conducting experimentswhere the composition will be varied and the organic load removal will be monitored. It must be clar­ified that no attempt is made to explain the actual phenomena; the sole objective is to obtain a usefulcorrelation that can direct the selection of the most favourable mixing policies. It is expected that thederived steady-state expressions will hold well within the limited range of wastewaters under consid­eration. This is further supported by the fact that all wastewaters must be adequately compatible, ifthey are to be considered within a co-digestion scheme.

The second facet of the co-digestion problem is the seasonal nature of the wastewaters. At each giventime, the selection of the best mixing policy will be based on the wastewater quantities that are avail­able in the centralised plant. Furthermore during the operational year the incoming wastewaters willvary, both in quality and quantity, while some of them might be available only in a certain periodof the year. This seasonal pattern is known a priori and it is straightforward to subdivide the wholeoperational year into periods, so as to ensure that within each period all influent wastewater parametersare constant. The peakperiod is defined as the one that the highest organic loads and/or maximumnumber of wastewaters is exhibited. The design of the centralised plant can be selected so as to beable to handle this peak period; then the same set of reactors is guaranteed to be adequate for theremaining reduced-load periods.

Anaerobic co-digestion plant

STATEMENT OF THE PROPOSED ALGORITHM

387

(2)

(3)

All these partial flows must sum up

An engineering model is proposed for the optimal design and operation of an anaerobic co-digestionplant. A set of N wastewaters and their seasonal variation pattern is given. The operation of such atreatment plant is realized as a series of consecutive periods, each of which is characterised by constantvalues in all wastewater-related parameters (flow, COD content etc.]. The operation in each one oftheseperiods is completely independent and can be determined by considering factors that are specific toeach period. This approach enables the decoupling of the design and operation stages of the treatmentplant. The heart of the plant is a complex of CSTR reactors operating in parallel that treat variousrecipes of wastewater mixtures according to the determined seasonal feeding pattern. The key controlvariable is the appropriate mixing of the individual wastewaters; some recipes are favourable due toincreased treatment efficiency.

Each one of the N incoming wastewaters is held each in its own storage tank, and is accompaniedby a parameter set (vector) comprising of the volumetric flow rate QUI of the wastewater, the overallanaerobic digestible organic content SUI plus the compositional breakdown of the total COD in thethree major parameters that were chosen to characterise its co-digestion features {IUl,hUl,pUl}, wherethe symbols represent the lipid, hydrocarbon and protein mass fractions, respectively. A total of Nreactors is assumed to cover the most extreme case of each wastewater treated in an individual reactor;during the solution of the model a fewer number of necessary reactors may actually be selected. Inthe most general case each of the N reactors will treat a mixture composed of all wastewaters presentin varying proportions. Elementary mass balances give the total volumetric flow Qr, COD content S"and mixture composition {zr,hr ,pr}of each reactor r as:

1Qr =E Q~, s:=QrE Q~SUI' Vr

w w

E Q:,S",I", E Q';"Swh", E Q';"SUlP",zr = VI hr _ VI r - VI VQrSr' - QrSr ' p - QrSr ' r

where Q:, is the volume of the wastewater w entering reactor r,

to the available quantity of each individual wastewater:

(4)

The operation of each reactor is assumed to be governedby simple Monod kinetics, where the maximumgrowth rate p:;'.. and half saturation constant K: are general functions of the overall composition ofthe mixture selected to be treated in the given reactor:

p:;'.. = fl(zr,hr ,pr), VrK: =h(r,hr,pr), Vr

(5)(6)

where fl and h are appropriate empirical expressions that describe the experimental data of co­digestion. The dilution rate Dr and the volume vr of each reactor are linked via the simple expression:

Qr~=&,* m

The volumeof each reactor (or equivalently its dilution rate), is a free variable that is determined duringthe optimisation procedure by observing that as the reactor volume is increased (and consequently theretention time) its capital cost increases but a greater portion of the organic load is removed at the sametime. By considering a penaltJl charge for emitting high COD contents with the treated wastewaters, itis possible to identify the optimal reactor volume that will exactly balance the competitive characterof capital versus penalty costs. Penalty costs are inflicted whenever the effluent COD content ~ fromreactor r is higher than the maximum allowable limit So set out by environmental legislation and/or

388 N. A. BOZINIS tt al.

due to post-treatment requirements. A steady-state substrate balance in the reactor r supplies theeffluent COD content as:

S' = D'K~ Yro JI:;'"z - D"

(8)

The above system of constraints (equations 2-8) comprises the full constraint set of the model, whichis a NLP mathematical program. The objective is to minimise the annualised capital cost of buildingthe reactors plus the penalty cost incurred from operation in high dilution rates:

(9).penalty

,.capital

,'---._---" .....--._--'

where a, {J and 'Y are constants.

The solution of this optimisation problem will provide the required number of reactors and their respec­tive volumes. In addition the feeding pattern of each selected reactor {Qt, Q;, ... ,Q'N} is obtained forthe peak period. The same set of reactors is available for the remaining reduced-load operational peri­ods. The operation of the plant, during each period, is formulated as a separate optimisation problemto determine the feeding pattern of each reactor, for the number of reactors and volumes determined inthe design stage. The objective at this level is to maximise the extent of COD conversion by selectingthe most suitable wastewater combinations.

The same constraints set (equations 2-8) is applicable in the operating stage optimisation model, thesole difference being that the reactor volumes V' are constant. The objective function here is:

minEQ'~, (10)

This problem is again a NLPj its solution for each one of the reduced load periods i will supply thenumber of active reactors in the period and their respective feeding patterns {Qt, Q;, . . . ,Q'N}i.

An illustrative example of application

(I) SptIII canflgaratlol (b)Co-digestion funcUon

, ,--~,.-- M - -_ 1""- _, _ ""'M ...__ -', _

,I I

Z.D -- : - -,- 1-

D.Z D.4 D.I D.IUpl'" lUll tracUon

Figure 1:

The small co-digestion example depicted in figure la involves two wastewaters A and B treated byco-digestion, with expected fiowrates QA=70 m31dand Qs=250 m3ld. Both carry equivalent organicload in terms of COD content (SA=Ss=5000 mg/lt). In order to simplify the co-digestion effects itis assumed that the half saturation constant is un~ected by composition and set to K.=240 mg/lt.

Anaerobic co-digestion plant 389

Furthermore, the maximum growth rate is assumed to follow a dependence on lipids content alone, asdepicted in figure Ib, exhibiting a maximum for a lipids mass fraction 1=0.6. The subject wastewatershave lipids contents 1,(=0.65 and lB=0.04. Given these data, what is the best way to mix the availablewastewater quantities in order to minimise costs? Four different scenarios are tested as follows:

I Each waste fed to a separate reactor.

II Available waste quantities mixed and fed to a single reactor.

III Manually devised reasonable mixing policy (sub-optimal): reactor Rl is fed with a mixture ofoptimal lipids content (A+B)j remainder quantity of waste B is treated in the second reactor.

IV Global optimal mixing policy derived by the mathematical problem solution.

Case I serves as the control case, in that it determines the reactor volumes that are required for digestionif no mixing is used. Case II is similar to current ad-hoc policies, where all available quantities are mixedas are. Case III involves a sincere yet non-systematic effort to produce a mixture that is favourablefrom a co-digestion viewpoint, and finally in case IV the proposed model is allowed to select the bestmixing policy as necessary to minimise total reactor volumes. All reactors are assumed to operate witha dilution rate that maximises the rate of COD removal per unit volume (i.e. same COD removal isachieved). The results in table 1 prove the undisputed superiority of the systematic selection of themixing policy based on co-digestion performance information, where total volume savings of 28% areattained compared to individual wastewater treatment. Moreover, arbitrary or heuristic approaches donot deliver positive results, as the total volumes necessary in cases II & III indicate.

TABLE 1: MIXING POLICY AND REACTOR VOLUMEQA Qs "':;.= Qr vr VT

Case Reactor (m3/d) (m3/d) r (l/day) (m3/d) (m3) (m3)

I Rl 70.0 0.0 0.65 1.946 70.0 45.8R2 0.0 250.0 0.04 0.246 250.0 1292.0 1337.8

II Rl 70.0 250.0 0.17 0.286 320.0 1423.6 1423.6

III Rl 70.0 6.25 0.6 2.0 76.25 48.4R2 0.0 243.75 0.04 0.246 243.75 1259.7 1308.1

IV Rl 70.0 117.66 0.27 0.661 187.66 361.2R2 0.0 132.34 0.04 0.246 132.34 683.9 1045.1

CONCLUSIONS

An initial approach to tackle the co-digestion problem has been presented, using an NLP mathematicalprogram that can deliver (via computer solution) the optimal design and operational schedule of ananaerobic co-digestion plant, thus offering substantial savings in the reactor volumes necessary for theremoval of the wastewater pollutants. The proper mixing of wastewaters, according to a quantitativemodel based on major component groups, is identified u a major control variable that is not generallypresent in typical single-waste treatment processes. The engineering model proposed for the designdoes not consider at this stage various operational costs relating to e.g.heating, biogas utilisation, etc.,yet their insertion is quite straightforward once the necessary cost factors are available.

The hypothesis that simple Monod-type coefficients,and their dependence on the lipids, hydrocarbonsand proteins contents of the wastewaters, are adequate for the quantification of the mixing effect hasto be validated once experimental data become available. It is recognised that a large number of ex­periments will be necessary in order to get dense and therefore reliable data to base the mathematicalfitting of the unknown functions It and 12, so as to cover a wide range of the possible mixture re­alisations. However, this is the only way to obtain expressions that will be reliable for consideration

390 N. A. BOZINIS etat.

within reactor design. Experiments might indicate that a greater number of composition-related pa­rameters is required to describe the perceived mixing effects, yet the general co-digestion frameworkpresented is capable to include any mathematically conceivable expression, once it is stated in a clearand quantifiable manner.

Finally, no attempt has been made to model changeover effects, that occur in between consecutiveperiods when a reactor switches to treatment of different wastewater mixtures with possibly differentretention times, too. Generally speaking, valuable time will be lost while bacteria are acclimatising tothe new nutritional conditions; the exact length of this latent period will depend on the two mixturesinvolved, the method applied for the changeover, etc. At this initial stage of the algorithm it is assumedthat the number of necessary changeoverswill be small, thus the lost time will be insignificant comparedto the operational year.

A user-friendly computer implementation of the proposed modelling algorithm is currently under de­velopment (Bozinis & Pistikopoulos, 1995).

ACKNOWLEDGEMENT

The authors gratefully acknowledge financial support from EU Brite/Euram II program, under theresearch contract BFr6090 FlexiBLE.

REFERENCES

Angelidaki 1., Ellegaard L. and Ahring B.K (1993). A Mathematical Model for Dynamic Simulationof Anaerobic Digestion of Complex Substrates: Focussing on Ammonia Inhibition. Biotechnologyand Biomgineering, 42, 159-166.

Bryers J.D. (1985). Structured Modellingof the Anaerobic Digestion of Biomass Particulates. Biotech-nology and Bioengineering, 21, 638-649.

Bozinis N.A. and Pistikopoulos E.N. (1995). Optimal Design and Operation of a Multi-Purpose Co­Digestion Plant under Seasonal Variation. Internal report, Imper ial College, London.

FlexiBLE BFr6090 2nd year report (1995). Submitted to the EU.Hawkes D.L. and Rosser B.L. (1983). In Proc. :l~ Int. Symp. An. Digestion. 3r~ Int. Symp. An. Dig.,

99 Erie St ., Cambridge, Mass., USA. p.465.Hobson P.N. and Wheatley A.D. (1993). Anaerobic Digestion: Modem Theory and Practice. Elsevier

applied science, London.McCarty P.L. and Mosey F.E. (1991). Modelling of Anaerobic Digestion Processes: A Discussion of

Concepts. Wat. Sci. Tech., 24, 17-33.Mosey F.E. (1983). Mathema.tical Modelling of the Anaerobic Digestion Process: Regula.tory Mecha­

nisms for the Formation of Short-Chain Volatile Acids from Glucose. Wat. Sci. Tecb., 15,209-232.

Anaerobic co-digestion plant 391

Siegrist H., Renggli D. and Gujer W. (1993). Mathematical Modelling of Anaerobic Mesophilic SewageSludge Treatment. Wat. Sci. Tech., 27,25-36.

Vavilin V.A., Vasiliev V.B. and Rytow S.V. (1993). Modelling o] Organic Matter Destruction byMicroorganism Community. Nauka publishers, Moscow.