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4 min read**Fig. 11.10** Simulation of TAN (XsubNHx-N,1/sub) in [mg/l] over 2 days = 2880 min with Q = 300 l/min (blue) and Q = 200 l/min (orange)

**Fig. 11.11** Simulation of nitrate-N (XsubNO3-N,1/sub) in [mg/l] over 50 days = 72,000 min with QsubExc/sub = 300 l/day (yellow), QsubExc/sub = 480 l/day (orange) and QsubExc/sub = 600 l/day (blue)

Anaerobic digestion (AD) of organic material is a process that involves the sequential steps of hydrolysis, acidogenesis, acetogenesis and methanogenesis (Batstone et al. 2002). The anaerobic digestion of a mixture of proteins, carbohydrates and lipids is visualized in Figure 11.11. Most often, hydrolysis is considered as the rate-limiting step in the anaerobic digestion of complex organic matter (Pavlostathis and GiraldoGomez 1991). Thus, increasing the hydrolysis reaction rate will most likely lead to a higher anaerobic digestion reaction rate. However, increasing the reaction rates needs further understanding of the related process. Further understanding can be obtained via experimentation and/or mathematical modelling. As there are many factors influencing, for instance, the hydrolysis process, such as ammonia concentration; temperature; substrate composition; particle size; pH; intermediates; degree of hydrolysis; i.e. the potential of hydrolysable content; and residence time, it is almost impossible to evaluate the total effect of the factors on the hydrolysis reaction rate through experimentation. Mathematical modelling could therefore be an alternative, but as a result of all the uncertainties in model formulation, rate coefficients and initial conditions, no unique answers can be expected. But, a mathematical modelling framework would allow sensitivity and uncertainty analyses to facilitate the modelling process. As mentioned before, hydrolysis is just one of the steps in anaerobic digestion. Consequently, understanding and optimization of the full anaerobic digestion process needs connections from hydrolysis to the other processes taking place during anaerobic digestion and interactions between all these steps.

The well-known and widely used ADM1 (anaerobic digestion model #1) is a structured model including disintegration and hydrolysis, acidogenesis, acetogenesis and methanogenesis steps. Disintegration and hydrolysis are two extracellular steps. In the disintegration step, composite particulate substrates are converted into inert material, particulate carbohydrates, protein and lipids. Subsequently, the enzymatic hydrolysis step decomposes particulate carbohydrates, protein and lipids to monosaccharides, amino acids and long-chain fatty acids (LCFA), respectively (Batstone et al. 2002) (see Fig. 11.12).

ADM1 is a mathematical model that describes the biological processes and physicochemical processes of anaerobic digestion as a set of differential and algebraic equations (DAE). The model contains 26 dynamic state variables in terms of concentrations, 19 biochemical kinetic processes, 3 gas-liquid transfer kinetic processes and 8 implicit algebraic variables for each process unit. As an alternative, Galí et al. (2009) described the anaerobic process as a set of differential equations with 32 dynamic state variables in terms of concentrations and an additional 6 acid-base kinetic processes per process unit. For an overview of the modelling of anaerobic digestion processes, we refer to Ficara et al. (2012). However, in what follows and for some first insights into the AD process, we will present a simple nutrient-balance model of AD in a sequencing batch reactor (SBR).

The nutrient mineralization can be calculated using the following equation (Delaide et al. 2018):

$NR=100\% \times (\frac{DN__{out}-DN{in}}{TN__{in}-DN_{in}})$ (11.15a)

**Fig. 11.12** A simplified scheme for the anaerobic digestion of complex particulate organic matter (based on El-Mashad 2003)

where NR is the nutrient recovery at the end of the experiment in percent, DNsubout/sub is the total mass of dissolved nutrient in the outflow, DNsubin/sub is the total mass of dissolved nutrient in the inflow and TNsubin/sub is the total mass of dissolved plus undissolved nutrients in the inflow (see also Fig. 11.13).

The organic reduction performance of the reactor can be calculated using the following equation:

$η__{OM}=1-\frac{\Delta OM+T__{OM\ out}}{T_{OM\ in}}$ (11.15b)

where ΔOM is the organic matter (i.e. COD, TS, TSS, etc.) inside the reactor at the end of the experiment minus the one at the beginning of the experiment, TsubOM out/sub is the total OM outflow and TsubOM in/sub is the total OM inflow (see also Fig. 11.14).

**Fig. 11.13** Overall reactor scheme for determining the mineralization potential, where DN are the dissolved nutrients in the water, UN the undissolved nutrients in the sludge (i.e. TN-DN) and TN the total nutrients

**Fig. 11.14** Overall reactor scheme for determining the organic material reduction potential, where TsubOM/sub is the total organic matter and ΔOM the change of organic matter inside the reactor