There are many types of anaerobic models ranging from steady-state models to single-, double-, or multistep dynamic models. Steady-state models can be applied in systems where the fluctuations in the feed characteristics and organic loading rate are minimized. This basis of static design modeling has been employed in several textbooks (Tchobanoglous and Burton, 1991). In most cases, however, the model should provide information about the dynamics of the system toward changes in the input of the system. Dynamic models can be utilized 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).

The typical steps usually included in an anaerobic digestion model are:

Table 10.2 refers to various models developed before 2002, some of which were the precursor of a generic model which was developed in 2002 by a group of scientists expert on AD modeling. They constructed the AD model (ADM1) to be a frame model basis for several applications in AD (Batstone et al., 2002). The model has been used as a reference basis for many extensions made by several researchers afterward to utilize it in specific applications, such as the AD of brewery wastewater in a full-scale high-rate system (Ramsay and Pullammanappallil, 2005) or the start-up of a manure digester (Normak et al., 2015). The model is considered to be complicated (consisting of 29 processes; 19 biochemical and 10 physicochemical) and lacks steps such as sulfate reduction, acetate oxidation, homoacetogenesis, and solids precipitation. Extensions to include sulfate reduction have rarely been attempted because of the complicated biochemical reactions taking place and the antagonism between the sulfate reduction bacteria (SRB) and the methanogens. Moreover, the electron donors for sulfate reduction are not only hydrogen but also the volatile fatty acids as reported by Barrera et al. (2015), who succeeded in simulating the anaerobic digestion of a high organic strength and sulfate-rich wastewater. The model could predict the experimental results with medium (10–30%) to high (±10%) accuracy, and more importantly, the model was able to predict the system limits when tested under overloading conditions. Carbonate precipitation and acetogenesis of isovalerate have been considered in the modifications of ADM1 proposed by Batstone and Keller (2003) and Batstone et al. (2003), respectively. In the case of semidry and dry feedstocks, Esposito et al. (2008) found that the effect of the surface of the particulate matter can be taken into account by modifying the disintegration constant included in the ADM1. Another important modification involved the correlation of stoichiometry with thermodynamics to account for the shifts in the metabolic pathways during AD (Rodriguez et al., 2006).

Integration of activated sludge model 1 (ASM1) and ADM1 resulted in Benchmark Simulation Model No 2 (BSM2) to simulate an integrated sewage treatment plant including a primary clarifier, an activated sludge system consisting of five compartments, a secondary clarifier, a gravitational thickener, an anaerobic digester, a dewatering unit, and a sludge storage tank (Jeppsson et al., 2007). Since ASM1 and ADM1 are based on different substrates, the interface between these models was very crucial to be defined properly.

Moreover, several researchers prefer to use simpler models. The basis for simplifying a model is the “rate-limiting step” concept, that is, the slowest step in a sequence of reactions that determines the overall rate of a multistep process. The two slowest steps recognized in anaerobic systems are hydrolysis and acetoclastic methanogenesis (Pavlostathis and Gossett, 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 the rate-limiting step, considering the preceding steps to be at a pseudo steady state. Models derived from mass balances as the ones described above can give a better insight of how the process evolves. On the other hand, models based on the black box concept may also be accurate provided that their main parameters are tuned on a constant basis (Premier et al., 1997). In a recent work by Lopez et al. (2015), a simple model (with first-order kinetics of the methane production and a time delay transfer function) was validated over a continuous experiment. These models are particularly useful for control purposes due to their simplicity.

The ADM1 type models consist of equations describing the biochemical and the physicochemical parts. In the biochemical part of the model, the kinetic relationships expressing the reaction rates are very important. There is a wide range of kinetics that can be applied in each step of the AD (Pavlostathis and Giraldo-Gomez, 1991), but the most common relationship is the Monod kinetics:

Eq. [10.1] can be extended to include any inhibition or regulation mechanisms if required (Batstone, 2006):

The physicochemical part of the model is important to assess the gas transfer and calculate the pH (if required in the biochemical part). The gas transfer can be modeled 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). The 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. A modification of the ADM1 tested in the framework of BSM2 involves the effect of the ionic strength on the predictions of the model. The chemical activities (influenced by the ionic strength) are used instead of molar concentrations (Solon et al., 2015). The results of this work show that this correction is necessary for cases of high ionic strength (>0.2 mol/L).

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 to 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 modeling biofilm bioreactors. Several parts of the bioreactor can be considered to be homogeneous, as in UASB reactors modeled by Bolle et al. (1986), thus a nonhomogeneous 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.