## 3 Modelling Cumulative Methane Production  
[PERSON] et al. [118] studied the behavior of several substrates using BMP tests and reported differences in biogas evolution behaviors based on the ease of degradation of the substrate by the microflora. [PERSON] et al. [119] also studied the biogas production from several substrates and the effect of their composition. It is reasonable to assume that the main substrate constituents will keep a close relationship with the methane yield obtained and the evolution of the gas production curve. However, factors such as accessibility to the degradable components and inhibitory conditions created during their transformation may alter the final outcome. Thus, carbohydrate-rich substrates are usually well-fitted to first-order decay models (Equation (1)) whereas, more complex substrates or those experiencing inhibitory conditions are fitted to different models capable of predicting these effects. \[B_{(t)}=B_{0}\Big{(}1-e^{k\cdot t}\Big{)} \tag{1}\]  
where \(B_{(t)}\) represents the cumulative methane yield at any time, \(t\) is the time of the batch assay, \(B_{0}\) is the maximum gas produced by the substrate and \(k\) is the first-order constant. The value of the first-order constant gives an indication of the easiness of degradation. Thus, substrates rich in carbohydrates are characterized by high \(k\) values between 0.39-0.66 1/d [119]. This model gives curves with a fast evolution of biogas during the first days of the batch assay, usually ending the fermentation in a short time and reaching a plateau very soon, coinciding with the end of the degradation. However, this fast degradation may cause operational problems in digesesters and also when evaluating BMP tests. The inoculum-to-substrate ratio used for starting up the tests greatly affects the ultimate methane production. When acidification is expected to become a problem, adding a greater amount of inoculum, and if necessary alkaline solutions, would aid in obtaining the desired results. The addition of rich-carbohydrate substrates as feed to a reactor operating under a continuous mode may cause localized acidification resulting in pH excursions. Many reactors work using a feeding recipe where the substrate is introduced at specific hours of the day. These substrates may cause a temporal variation in methane content, increasing CO\({}_{2}\) levels in biogas and leading to a greater gas production rate right after feeding. Special care should be taken regarding the type of reactor configuration based on operating conditions and substrate composition. Continuous stirred tank reactors (CSTRs) run under equivalent values of HRT and cell retention time (CRT). Therefore, imbalances may appear due to the lack of enough methanogenic microorganisms. The lower growth rate of these organisms compared with that of acidogenic bacteria may cause a washout of the first ones. Other configurations capable of retaining biomass in the reactor may be more suitable for dealing with substrates that are easily degraded, thus needing a shorter retention time such as UASB, anaerobic filters or anaerobic sequencing batch reactors (ASBR). Recent strategies for attaining biomass retention and improving degradation rates are the introduction of active filling to promote biogas production and attain higher-quality effluents thanks to the presence of magnetically active filling layers inside the reactor which are capable of reducing the nitrogen and phosphorus concentration [120]. The use of active filling containing metals (copper and iron) has also proven effective with similar results [121]. Table 2 presents the \(k\) values obtained by different authors under mesophilic conditions. There is a wide range of values reported in the literature for this parameter which is highly dependent on the characteristics of the substrate and the experimental conditions [122], such as substrate concentration, inoculum-to-substrate ratio, temperature and particle size.