*2.5. Mathematical Adjustment and Determinations of Results*

Once the results are obtained these are treated mathematically. Methane generation curves are treated as first-order kinetics, according to the described process by Veeken and Hamlers [32]. The disintegration constant (kdis) can be obtained directly by considering the complete anaerobic digestion process, giving information about the depth and speed of the hydrolysis process. Also, maximum amount of actual methane obtained in the process (CH4max) can be determined, adjusting the generation data by means of least squares, to a first order kinetics reaction:

$$\text{CH}\_4(\text{t}) = \text{CH}\_{4\text{ max}} \cdot \left[1 - \text{e}^{-\text{k}\_{\text{dis}} \cdot \text{t}}\right] \,, \tag{1}$$

where CH4(t) represents the production of methane on the t-day; CH4max the maximum generation of methane recorded (which can be assumed by the latest generation); and kdis the average disintegration constant measured in days−<sup>1</sup> .

The theoretical generation of methane that was to be expected after anaerobic degradation of the substrate can be obtained from the reduction of COD experienced before and after the BMP test.

$$\text{V}\_{\text{theoretical CH4}\_{\text{accumulated}}}[\text{Nml}] = \frac{(\text{COD}\_0 - \text{COD}\_f) \cdot \text{V}\_{\text{best}} \cdot \text{340}}{1000} \cdot \frac{1}{0.9869'} \tag{2}$$

where COD<sup>0</sup> and COD<sup>f</sup> represents the COD levels measured at baseline at the end of the BMP test expressed in mg/l; Vtest the test volume occupied by the mixture substrate + sludge expressed in litters; 340 the conversion factor of COD in methane; and 1/0.9869 the conversion factor from standard conditions (0 ◦C and 1 bar) to normal conditions (0 ◦C and 1 atm).

The level of degradation of the substrate or residue can be calculated analytically through the COD reduction degradation levels and provides information about the level of the degradation of the substrate, regardless of degradation inoculum.

$$\text{BD}\_{\text{residence}}[\text{\textquotedblleft}] = \frac{\left(\text{COD}\_{\text{mixture}\_0} - \text{COD}\_{\text{sludge}\_0}\right) - \left(\text{COD}\_{\text{mixture}\_f} - \text{COD}\_{\text{sludge}\_f}\right)}{\left(\text{COD}\_{\text{mixture}\_0} - \text{COD}\_{\text{sludge}\_0}\right)} \cdot 100 \,\text{.}\tag{3}$$
