# Existing Empirical Kinetic Models in Biochemical Methane Potential (BMP) Testing, Their Selection and Numerical Solution

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## Abstract

**:**

## 1. Introduction

- Start with arbitrary, but reasonable values of all coefficients of an expected equation. It is an array, called a “seed”, of all coefficients like ${V}_{\infty}$ (biomethane potential), $k$ (rate constant) and others if any existed in a model, represented as vector ($\overrightarrow{a}$):$$\overrightarrow{a}=\left({a}_{0},{a}_{1},\dots ,{a}_{m}\right)$$
- Calculate for each data point:
- The theoretical estimation of the $y$ value (either accumulated biomethane yield or production rate):$$\widehat{{y}_{i}}=f\left({x}_{i},\overrightarrow{a}\right)=f\left({x}_{i},{a}_{0},{a}_{1},\dots ,{a}_{m}\right)$$
- The value of error function based on experimental values (${y}_{i}$) and theoretically estimated values ($\widehat{{y}_{i}}$). Please note that this is not a Gauss error function:$${\epsilon}_{i}=f\left({y}_{i},\widehat{{y}_{i}}\right)$$

- Sum the error values across all measured datapoints, as an accumulated error which is also referred as an objective function:$${\rm E}={\displaystyle \sum}_{i=1}^{N}{\epsilon}_{i}$$
- Based on the accumulated error value ${\rm E}$, change the initial set of parameters to minimize ${\rm E}$.
- Repeat steps 2 to 4 until ${\rm E}$ stops decreasing or decrease is negligible.

- An indefinite number of iterations before achieving the solution;
- It results in an approximate solution, rather than an analytically precise solution;
- The precision of calculation improves with higher number of observations;
- The calculated result may be dependent on utilized error function.

- The exact algebraic form (shape) of equation;
- The range of variables, where models are mathematically possible (reasonable).

## 2. Representation of Existing Empirical Biomethane Models for Numerical Computation

- An even integer number, which will turn the ${V}_{t}$ to a negative number, which will have no physical value;
- A fractional number, which will result in taking a root from negative number and lead to complex number calculations, which also have no physical value.

## 3. Existing Empirical Models of Biomethane Evolution and Their Representation for Numerical Solution

## 4. Optimization of Calculation Process

- Minimize number of arithmetic operations in formulae;
- Narrow the search area (minimize the intervals of equation coefficient existence).

#### 4.1. Minimization the Number of Arithmetic Operations

#### 4.2. Narrowing the Search Area

- Those based on the BMP experimental set-up;
- Those based on the re-parametrization of model.

## 5. Criteria to Compare Models “inter se”

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Glossary/Nomenclature/Abbreviations

${V}_{t}$ | Accumulated biomethane yield at moment $t$ from beginning of experiment |

${V}_{\infty}$ | Maximum biomethane yield, a.k.a. (bio)methane potential |

${V}_{\infty}^{\prime}$ | Corrected accumulated biomethane yield |

$k$ | Kinetic constant |

${k}_{R}$ | Kinetic constant for rapidly decomposing compounds |

${k}_{S}$ | Kinetic constant for slowly decomposing compounds |

${k}_{H}$ | Kinetic constant for hydrolysis |

${k}_{VFA}$ | Kinetic constant for conversion of volatile fatty acids |

${k}_{CH}$ | Chen-Hashimoto dimensionless coefficient |

$t$ | Time from beginning of experiment |

${t}_{lag}$ | Duration of lag-phase |

${t}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ | Time of half-decomposition of substrate |

${\mu}_{os}$ | Specific initial growth rate on slowly decomposing compounds |

$\gamma $ | The order of time dependency |

$N$ | Number of measurements |

$M$ | Number of parameters fit by the model |

$\upsilon $ | Gas production rate (volumetric or molar on timely basis) |

${\upsilon}_{max}$ | Maximum gas production rate |

$e$ | Euler number |

COD | Chemical Oxygen Demand |

VS | Volatile Solids |

$R$ | Gas constant |

$T$ | Absolute temperature |

$p$ | Absolute pressure |

${\mu}_{or}$ | Specific initial growth rate on rapidly decomposing compounds |

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**Table 1.**Interpretation of biomethane formation models for numerical fitting for single-step digestion.

Model Name | Model Equations | Bounds | Reference |
---|---|---|---|

First-order kinetics ^{1} | ${V}_{t}=\{\begin{array}{cc}{V}_{\infty}\left[1-{e}^{-k\left(t-{t}_{lag}\right)}\right]& t>{t}_{lag}\\ 0& t\le {t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ k\ge 0\end{array}$ | [2,6,7,8,13,16,20,21,22,23,24,25] |

Specific time model ^{2} | ${V}_{t}=\{\begin{array}{cc}{V}_{\infty}{e}^{-\frac{k}{t-{t}_{lag}}}& t>{t}_{lag}\\ 0& t\le {t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ k>0\end{array}$ | [25,26] |

First-order with variable order of time dependency ^{3} | ${V}_{t}=\{\begin{array}{cc}{V}_{\infty}\left[1-{e}^{-k{\left(t-{t}_{lag}\right)}^{\gamma}}\right]& t>{t}_{lag}\\ 0& t\le {t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ k\ge 0\\ \gamma \in \forall \end{array}$ | [17,25] |

Weibull | ${V}_{t}=\{\begin{array}{cc}{V}_{\infty}\left(1-{e}^{-{\left(k\left(t-{t}_{lag}\right)\right)}^{\gamma}}\right)& t>{t}_{lag}\\ 0& t\le {t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ k\ge 0\\ \gamma \in \forall \end{array}$ | [8,27] |

Monod type | ${V}_{t}=\{\begin{array}{cc}{V}_{\infty}\frac{k\left(t-{t}_{lag}\right)}{k\left(t-{t}_{lag}\right)+1}& t>{t}_{lag}\\ 0& t\le {t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ k>0\end{array}$ | [2,28] |

Quadratic Monod | ${V}_{t}=\{\begin{array}{cc}{V}_{\infty}\frac{{\left(t-{t}_{lag}\right)}^{2}}{{\left(t-{t}_{lag}\right)}^{2}+{k}_{1}\left(t-{t}_{lag}\right)+{k}_{2}}& t\ge {t}_{lag}\\ 0& t<{t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ {k}_{1}>0\\ {k}_{2}>0\end{array}$ | [17,29] |

Michaelis–Menten | ${V}_{t}=\{\begin{array}{cc}{V}_{t}={V}_{\infty}\frac{{\left(t-{t}_{lag}\right)}^{n}}{{\left(t-{t}_{lag}\right)}^{n}+{t}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{n}}& t>{t}_{lag}\\ 0& t\le {t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ n>0\\ {t}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}>0\end{array}$ | [6] |

France | ${V}_{t}=\{\begin{array}{c}{V}_{\infty}\left(1-{e}^{{k}_{1}\left({t}_{lag}-t\right)+{k}_{2}\left(\sqrt{{t}_{lag}}-\sqrt{t}\right)}\right),t\ge {t}_{lag}\\ 0,t{t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ {k}_{1}>0\\ {k}_{2}>0\end{array}$ | [8,30] |

Cauchy ^{4} | ${V}_{t}=\{\begin{array}{cc}\frac{2{V}_{\infty}}{\pi}\mathrm{arctan}\left(k\left(t-{t}_{lag}\right)\right)& t>{t}_{lag}\\ 0& t\le {t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ k>0\end{array}$ | [8,27] |

Feller | ${V}_{t}=\frac{2{V}_{\infty}}{\pi}\mathrm{arctan}\left({e}^{k\left(t-{t}_{lag}\right)}\right)$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ k>0\end{array}$ | [8,31,32] |

Fitzhugh ^{5} | ${V}_{t}=\{\begin{array}{cc}0& t\le {t}_{lag}\\ {V}_{\infty}{\left(1-{e}^{-k\left(t-{t}_{lag}\right)}\right)}^{n}& t>{t}_{lag};n\ge 0\\ {V}_{\infty}{\left(1+{e}^{-k\left(t-{t}_{lag}\right)}\right)}^{n}& t>{t}_{lag};n<0\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ k>0\\ n\in \forall \end{array}$ | [8,20,33] |

Autocatalytic ^{6} | ${V}_{t}=\frac{{V}_{\infty}}{1-{e}^{-K\left(t-{t}_{lag}\right)}}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ k>0\end{array}$ | [34,35] |

Logistic ^{7} | ${V}_{t}=\frac{{V}_{\infty}}{1+{e}^{2-\frac{4{\upsilon}_{max}}{{V}_{\infty}}\left(t-{t}_{lag}\right)}}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ {\upsilon}_{max}>0\\ {t}_{lag}\ge 0\end{array}$ | [8,10] |

Gompertz ^{8} | ${V}_{t}={V}_{\infty}\xb7{e}^{-{e}^{-\frac{e\xb7{\upsilon}_{max}}{{V}_{\infty}}\left(t-{t}_{lag}\right)+1}}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ {t}_{lag}\ge 0\\ {\upsilon}_{max}>0\end{array}$ | [2,10,22,24] |

Corrected Gompertz | ${V}_{t}={V}_{\infty}^{\prime}\xb7\left({e}^{-{e}^{-\frac{e\xb7{\upsilon}_{max}}{{V}_{\infty}^{\prime}}\left(t-{t}_{lag}\right)+1}}-{e}^{-{e}^{\frac{e\xb7{\upsilon}_{max}\xb7{t}_{lag}}{{V}_{\infty}^{\prime}}+1}}\right)$ | $\{\begin{array}{c}{V}_{\infty}^{\prime}>0\\ {t}_{lag}\ge 0\\ {\upsilon}_{max}>0\end{array}$ | |

Stannard Richards | ${V}_{t}={V}_{\infty}\xb7{\left(1+d\xb7{e}^{1+d}\xb7{e}^{\frac{{\upsilon}_{max}}{{V}_{\infty}}\left(1+d\right)\left(1+\frac{1}{d}\right)\left({t}_{lag}-t\right)}\right)}^{-\frac{1}{d}}$ | $\{\begin{array}{c}{\upsilon}_{max}\\ {V}_{\infty}\\ {t}_{lag}\ge 0\\ 0<d<1\end{array}$ | [10,34] |

Cone | ${V}_{t}=\{\begin{array}{cc}\frac{{V}_{\infty}}{1+{\left(k\xb7\left(t-{t}_{lag}\right)\right)}^{-n}}& t>{t}_{lag}\\ 0& t\le {t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}>0\\ k>0\\ n>0\\ {t}_{lag}\ge 0\end{array}$ | [20,24,36,37] |

Chen–Hashimoto | ${V}_{t}=\{\begin{array}{c}{V}_{\infty}\left(1-\frac{{K}_{CH}}{{\mu}_{max}\xb7\left(t-{t}_{lag}\right)+{K}_{CH}-1}\right),t{t}_{lag}\\ 0,t\le {t}_{lag}\end{array}$ | $\{\begin{array}{c}{V}_{\infty}\\ {\upsilon}_{max}\\ {K}_{CH}\\ {t}_{lag}\ge 0\end{array}$ | [11,38] |

Levi-Minzi | ${V}_{t}=\{\begin{array}{c}k{\left(t-{t}_{lag}\right)}^{m},t{t}_{lag}\\ 0,t\le {t}_{lag}\end{array}$ | $\{\begin{array}{c}k>0\\ 0<m<1\\ {t}_{lag}\ge 0\end{array}$ | [25] |

^{1}The first-order kinetics model is also met under other names: Exponential, Malthus, Monomolecular, Transference function, Transfer function, Reaction-curve type, Simple Mitscherlich equation. Very often this model is used with no lag-phase, i.e., when ${t}_{lag}=0$.

^{2}The specific time model can be also found under name of first-order inverse time model,

^{3}also called a first-order kinetics with modified time dependency.

^{4}Cauchy distribution can be met under name of Lorentzian function.

^{5}Fitzhugh sometimes is used as simplified version, where $n>0$, actually such modification is Bertalanffy model.

^{6}Autocatalytic function is considered to be a predecessor of logistic function.

^{7}Logistics function is sometimes considered as a special case of Richards functions, where $d=1$.

^{8}The Gompertz equation for biomethane is also called as Zwietering modification of Gompertz equation, Zwietering-Gompertz equation or modified Gompertz equation.

Model Name | Model Equation | References |
---|---|---|

First-zero-order kinetic model ^{1} | ${V}_{t}={V}_{\infty}\xb7\left(x\left(1-{e}^{-{k}_{R}t}\right)+\left(1-x\right){k}_{S}t\right)$ | [25,53] |

First-first-order kinetic model ^{2} | ${V}_{t}={V}_{\infty}\xb7\left(1-x\xb7{e}^{-{k}_{R}t}-\left(1-x\right){e}^{-{k}_{S}t}\right)$ | [2,16,21,25,54] |

Modified Gompertz | ${V}_{t}={V}_{\infty}\xb7{e}^{-\frac{{\mu}_{or}}{{k}_{R}}{e}^{-{k}_{R}t}-\frac{{\mu}_{os}}{{k}_{S}}{e}^{-{k}_{S}t}}$ | [6,21,25,55,56] |

First-order two-step reaction | ${V}_{t}={V}_{\infty}\left(1+\frac{{k}_{H}\xb7{e}^{-{k}_{VFA}\xb7t}-{k}_{VFA}\xb7{e}^{-{k}_{H}\xb7t}}{{k}_{VFA}-{k}_{H}}\right)$ | [16,57] |

First-order two substrate two -step reaction | ${V}_{t}={V}_{\infty}\left(x\left(1+\frac{{k}_{R}\xb7{e}^{-{k}_{VFA}\xb7t}-{k}_{VFA}\xb7{e}^{-{k}_{R}\xb7t}}{{k}_{VFA}-{k}_{R}}\right)+\left(1-x\right)\left(1+\frac{{k}_{S}\xb7{e}^{-{k}_{VFA}\xb7t}-{k}_{VFA}\xb7{e}^{-{k}_{S}\xb7t}}{{k}_{VFA}-{k}_{S}}\right)\right)$ | [16] |

^{1}The first-zero-order kinetic model is also met under the names combined first- and zero-order kinetics;

^{2}The first-first-order kinetic model is also met under names: combination of two first-order kinetics, two-pool first-order and pseudo-parallel first-order.

Criterion | Calculation | Used in |
---|---|---|

Residual Sum of Squares (RSS) | $RSS={\displaystyle {\displaystyle \sum}_{i=1}^{N}}{\left({y}_{i}-\widehat{{y}_{i}}\right)}^{2}$ | [62] |

Root Mean Square Error (RMSE) ^{1} | $RMSE=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{N}{\left({y}_{i}-\widehat{{y}_{i}}\right)}^{2}}{N}}$ | [47,53,63] |

Relative Root Mean-Squared Error (rRMSE) | $rRMSE=\frac{1}{\overline{y}}\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{N}{\left({y}_{i}-\widehat{{y}_{i}}\right)}^{2}}{N}}$ | [64] |

Mean Absolute Percentage Error (MAPE) ^{2} | $MAPE=\frac{1}{N}{\displaystyle {\displaystyle \sum}_{i=1}^{N}}\frac{\left|{y}_{i}-\widehat{{y}_{i}}\right|}{{y}_{i}}$ | [17,64] |

Mean Square Percentage error (MSPE) | $MSPE=\frac{1}{N}{\displaystyle {\displaystyle \sum}_{i=1}^{N}}{\left(\frac{{y}_{i}-\widehat{{y}_{i}}}{{y}_{i}}\right)}^{2}$ | [18] |

^{1}RMSE can be also met under the names: Root-Mean-Square Deviation (rMSD), Root Mean Square Prediction Error (rMSPE), Standard Error of Estimate.

^{2}MAPE is also known as Relative Absolute Error (rAE).

Criterion Name | Formulae | Reference |
---|---|---|

Akaike Information Criterion | $AIC=\{\begin{array}{cc}N\xb7\mathrm{ln}\left(\frac{RSS}{N}\right)+2M+\frac{2M\left(M+1\right)}{N-M-1}& \frac{N}{M}<40\\ N\xb7\mathrm{ln}\left(\frac{RSS}{N}\right)+2M& \frac{N}{M}\ge 40\end{array}$ | [6,20,47,53,62] |

Bayesian Information Criterion | $BIC=\mathrm{ln}\left(\frac{RSS}{N}\right)+M\xb7\mathrm{ln}\left(N\right)$ | [47,53] |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pererva, Y.; Miller, C.D.; Sims, R.C.
Existing Empirical Kinetic Models in Biochemical Methane Potential (BMP) Testing, Their Selection and Numerical Solution. *Water* **2020**, *12*, 1831.
https://doi.org/10.3390/w12061831

**AMA Style**

Pererva Y, Miller CD, Sims RC.
Existing Empirical Kinetic Models in Biochemical Methane Potential (BMP) Testing, Their Selection and Numerical Solution. *Water*. 2020; 12(6):1831.
https://doi.org/10.3390/w12061831

**Chicago/Turabian Style**

Pererva, Yehor, Charles D. Miller, and Ronald C. Sims.
2020. "Existing Empirical Kinetic Models in Biochemical Methane Potential (BMP) Testing, Their Selection and Numerical Solution" *Water* 12, no. 6: 1831.
https://doi.org/10.3390/w12061831