Mathematical models have been developed to improve understanding of the complex dynamics of the anaerobic digestion process and to predict the response of the anaerobic systems to changes in operating conditions (hydraulic retention time, organic load, temperature, etc.). Models are tools for process design, control strategies, diagnosis or prediction of system performance under conditions of increasing or decreasing load and variation of feeding characteristics.

There are many types of anaerobic models ranging from steady-state models to single — or double — or multi-step dynamic models. Steady-state models can be applied in systems where the fluctuations in the feed characteristics and organic loading are minimised. This basis of static design modelling has been employed in several text books (Tchobanoglous and Burton, 1991). In most cases, however, the model should provide information about the dynamics of the system towards changes in the input of the system. Dynamic models can be utilised successfully in control schemes or for simulation purposes. Depending on the purpose, the model should be simple enough including only the basic steps for describing the dynamics of the core process (control) or more complex including as many steps as possible making it widely applicable (simulation). Table 12.1 refers to various models developed in the last three decades.

The basis for simplifying a model is the ‘rate limiting step’ concept, that is, the last slow step in a sequence of reactions that determines the overall rate of a multistep process. The two slowest steps recognised in anaerobic systems are hydrolysis and acetoclastic methanogenesis (Gossett and Belser, 1982; Pavlostathis and Gossett, 1986, 1988). When the feedstock contains particulate organic matter (sludge, organic fraction of municipal solid wastes, solid residues, etc.), the rate of hydrolysis usually determines the overall rate. In this case, the steps that follow are usually considered to be at pseudo steady state and can be described by algebraic equations reducing the degree of complexity of the model. In the absence of particulate matter in the feedstock, acetoclastic methanogenesis is

Table 12.1 Steps involved in various models of anaerobic digestion developed in the last three decades Hydrolysis Acidogenesis Acetogenesis Methanogenesis Source Volatile fatty acids Graef and Andrews (acetate) -»■ CH4, C02 (1974) Particulate organics -»■ soluble organics Soluble organics -»■ volatile fatty acids Volatile fatty acids Hill and Barth (1977) (glucose) Glucose -»■ butyrate, propionate, Butyrate, propionate (acetate) -»■ CH4, C02 Acetate -> CH4 Hill (1982) acetate -»■ acetate H2, co2 CH4 H2, C02 -»■ acetate Particulate organics -»■ aminoacids, sugars, Aminoacids, sugars, fatty Propionate -»■ acetate Acetate -> CH4 Bryers (1985) fatty acids acids -»■ propionate, acetate H2, co2 CH4 Glucose -»■ butyrate, propionate, Butyrate, propionate Acetate -> CH4 Mosey (1983) acetate -»■ acetate H2, C02 CH4 Pullammanappallil etal. (1991) Particulate organics (fats, carbohydrates, Soluble organics -»■ acetate Acetate -» CH4 Kleinstreuer and proteins) -»■ soluble organics Soluble organics (glucose) -»■ volatile fatty acids (acetate) Acetate -> CH4 Powegha (1982) Moletta et al. (1986) Easily biodegradable biomass -»■ soluble Soluble organics -»■ volatile fatty acids Volatile fatty Smith et al. (1988) organics Glucose -»■ lactate, butyrate, Butyrate, propionate acids -> CH4 Acetate CH4 Costello et al. (1991) propionate, acetate Lactate -»■ propionate, acetate -»■ acetate H2, co2 CH4 Particulate carbohydrates -»■ soluble Soluble carbohydrates -»■ butyrate, Butyrate, propionate Acetate -> CH4 Angelidaki etal. carbohydrates propionate, acetate -»■ acetate (1993) Particulate carbohydrates, proteins, fats -»■ Aminoacids and sugars -»■ Propionate -»■ acetate Acetate CH4 Siegrist et al. (2002) aminoacids, sugars, fatty acids propionate, acetate fatty acids -»■ acetate H2, co2 CH4 Particulate carbohydrates, proteins -»■ Soluble carbohydrates and proteins, Propionate -»■ acetate Acetate CH4 Gavala et al. (1996) soluble carbohydrates and proteins other organics -»■ propionate, acetate Particulate organics^carbohydrates, Aminoacids and sugars -»■ butyrate, Butyrate, propionate Acetate CH4 Batstone etal. (2002) proteins, fats -»■ aminoacids, sugars, fatty propionate, acetate -»■ acetate H2, C02 CH4 acids Fatty acids -»■ acetate

the rate limiting step, considering the preceding steps to be at a pseudo steady state.

On the other hand, in the case of multistep models, the steps usually included are:

• Hydrolysis of particulate matter: Although the mechanisms of the individual hydrolysis steps are known, the hydrolysis step is usually lumped as a single first order process (Eastman and Ferguson, 1981; Pavlostathis and Giraldo — Gomez, 1991).

• Acidogenesis of soluble organic matter: Modelling of sugar fermentation is challenging due to the variety of the possible fermentation products and the determination of the stoichiometry (subjected to the regulation mechanisms prevailed in the heterogeneous group of acidogens). The main pathways acknowledged to take place are towards formation of butyrate, acetate, ethanol and acetate, as well as propionate and acetate as end products (Ren et al, 1997; Batstone et al., 2002). Lactate has been also considered important to be in­cluded among the sugar fermentation products (Costello et al., 1991). In mixed fermentation processes, the mechanisms that regulate the composition of the fermentation product mixture have not been elucidated completely and as a result, modelling of this step has not yet been effective (Mosey, 1983; Costello et al, 1991; Ruzicka, 1996). This limitation has become critical due to the increasing interest concerning the production of biohydrogen produced along with the other sugar fermentation metabolic products. As far as the modelling of amino acid fermentation is concerned, the pathways based on Stickland reactions have been proposed (Ramsay and Pullammanappallil, 2001).

• Acetogenesis and methanogensis: Both steps have been extensively and successfully simulated. However, the incorporation of hydrogen, free ammonia and pH effects on the kinetics of both steps can be further improved.

In the biochemical part of the model, the kinetic relationships expressing the bioreaction rates are very important. There is a wide range of kinetics that can be applied in each step of the anaerobic digestion (Pavlostathis and Giraldo-Gomez, 1991), but the most common relationship is the Monod kinetics:

where p is the consumption rate of the substrate, km is the maximum specific consumption rate constant, KS is the saturation constant, S is the concentration of the substrate and X is the concentration of the microorganisms that consume the substrate.

Equation 12.1 can be extended to include any inhibition or regulation mechanisms if required (Batstone, 2006):

where Ij, I2, . . In are functions expressing inhibition mechanisms can include classic non-competitive or competitive inhibition, or empirical formulas. Modification of Monod kinetics to account for all kinds of product, cell and substrate inhibition has been extensively applied in biochemical engineering (Levenspiel, 1980; Han and Levenspiel, 1988).

Moreover, apart from the biochemical part of the model, it is important to include a physicochemical part to assess the gas transfer and calculate the pH (if required in the biochemical part). The gas transfer can be modelled by applying the gas-liquid transfer theory for each gas. Equilibrium can also be assumed for those gases that are practically insoluble in water, such as hydrogen and methane. The total gas production rate can be calculated as the sum of individual gas production rates. Gas flow can also be derived by setting a pressure difference between the headspace and the atmosphere (Batstone, 2006). pH calculation requires solving algebraic equations derived from the equilibrium of weak acids and bases as well as charge balance. Dissociation of acids and bases can also be considered as dynamic processes evolving at a high rate.

In 2002, a group of scientists expert on anaerobic digestion modelling constructed the anaerobic digestion model (ADM1) to be a frame model basis for several applications in anaerobic digestion (Batstone et al., 2002). The model has been used as a reference basis for many extensions made by several researchers afterwards to utilise it in specific applications, such as, the anaerobic digestion of brewery wastewater in a full scale high rate system (Ramsay and Pullammanappallili, 2005).

Depending on the bioreactor design (homogeneous or heterogeneous system), simple hydraulic or more complex models taking into account mass transfer phenomena can be developed. Mass transfer is important in the case of ‘biofilm’ bioreactors where microorganisms are attached on the surface of an inert material (anaerobic filters) or attached on each other (UASB). There are different degrees of complexity that can be entailed in modelling biofilm bioreactors. Several parts of the bioreactor can be considered to be homogeneous, as in UASB reactors modelled by Bolle et al. (1986), thus a non homogeneous system can be depicted by a combination of the homogeneous systems connected. In a more complex model design, the layers composing the biofilm in a filter or the granule in a UASB are taken into account, with each layer being formed by a specific group of microorganisms. Many UASB models assume that the granules are spherical and the relative concentration of the acidogens and methanogens remain constant in the granule. The density of the granules is also assumed to remain constant. Saravanan and Sreekrishnan (2006) review the various model approaches available for biofilm reactors extensively.