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MBW:Kinetic investigations of methane co-fermentation

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Kinetic investigations of methane co-fermentation

Executive Summary

This page focuses on the mathematical model used to investigate the cofermentation of two types of waste: sewage sludge and organic fraction of municipal solid waste (OFMSW). This problem is important to study and model because of production of useful biogas during the cofermentation that can be used as a renewable energy source. It is also important because a large amount of sewage sludge and OFMSW is produced all of the world and currently is not being utilized as a high density carbon source. The studies were carried out in a large scale 40 L laboratory reactor to try and mimic industrial scale processes. Cofermentation is an ideal process to study for the utilization of sewage sludge and OFMSW because it allows for the ideal carbon to nitrogen ratio to be achieved. During anaerobic digestion, bacteria utilizes carbon 25-30 times faster than nitrogen and as such should be present in amounts relative to the rate of utilization. By cofermenting different waste sources, the anaerobic process can be carried out at the ideal carbon to nitrogen ratio and should lead to the production of more biogas and to the optimum utilization of carbon from sewage sludge and OFMSW.

The mathematical model focuses on carbon division in anaerobic fermentation. The author made certain assumptions in the model so that a simple system of ordinary differential equations could be used. A major assumption made was that the cofermentation occurred uniformly. This means that the composition of the fermenting biomass was assumed spatial constant and that the bacterial population was considered constant spatially within the biomass. While this assumption for a large scale industril process would probably not being the best due to poor mixing of highly viscous material, it is probably a safe assumption for the 40 L experimental fermentor. The model is based on the kinetics of anaerobic digestion. The equations included in the model correspond to: the carbon content in the total suspended solids, the carbon content in the intermediate substrate, the production rate of methane, and the production rate of carbon dioxide. The system of differential equation follows carbon through the process of cofermentation to determine the rate of biogas formation.

The model incorporated hydrolysis/acidogenesis and methanogenesis as the major rate limiting steps of anaerobic digestion. Rate limiting steps determine the rate of the reaction and are therefore the most important steps to include in a kinetic model. The parameters were estimated experimentally and applied to the mathematical model. The initial biomass concentration was estimated using C_{{5}}H_{{7}}O_{{2}}N as a formula for the carbon and nitrogen content. The formula is empirically based on the nitrogen content of the original inoculum.


Sosnowski, P., Klepacz-Smolka, A., Kaczorek, K., Ledakowicz, S., 2007, Kinetic investigations of methane co-fermentation of sewage sludge and organic fraction of municipal wastes, Bioresource Technology, 99 (13) (2008) 5731-5737


Biogas (mainly methane, carbon dioxide and other gases) is produced by micro organisms, which cause the biological breakdown of organic matter in absence of oxygen. The reaction called anaerobic digestion can occur in digestive systems, marshes, rubbish dumps, septic tanks and the Arctic Tundra. In an industrial scale reactors with a large volume are used. Any biodegradable material can be used for the fermentation process. Some examples are biomass, sewage sludge, municipal waste, manure, energy crops, etc.

Biogas is a renewable energy fuel, which makes it an interesting alternative to fossil fuels. It might be the best option to use biogas as a stationary fuel instead of as a mobile fuel since the gas is very hard to compress. The energy would go to waste. Biogas can be used in for cooking, heating, light or even refrigerators. Generating electricity from biogas requires a high energy input. [1]


The paper “Kinetic investigations of methane co-fermentation of sewage sludge and organic fraction of municipal wastes” compares three batch experiments using different feedstock to produce methane in large laboratory scale. The feed in the first experiment was organic fraction of municipal waste, in the second experiment sewage sludge was used and the third feed was a mixture of the previous two. The goal of the experiments was to find a simple kinetic model of anaerobic digestion by determining the carbon balances over time.

Co-digestion is the anaerobic decomposition of two or more organic substrates mixed together. Recent research results have very often shown that co-digestion has a positive effect on the process stability as well as the production rate. The experiments carried out by Sosnowski, Klepacz-Smolka, Kaczorek and Ledakowicz gave similar results. Even though the biogas production using municipal solid waste was very fast and reached the highest maximum of all, the average biogas production rate (GTR) was highest during the co-fermentation process.

During the experiment using municipal solid waste an accumulation of volatile fatty acids (VTA) was observed. Due to the resulting decrease of the pH-value the biogas production was inhibited. Furthermore the whole process was unstable. A different behaviour was noticed for the co-fermentation. The co-substrates acted as buffers, so that the production of biogas was not influenced by varying pH-values. This way the decomposition was stabilized.

To be able to estimate the carbon contents in the compounds (total suspended solids, volatile fatty acids, methane and carbon dioxide) an elemental analysis had to be applied. The carbon content of the total suspended solids was analysed and the other contents were estimated by their stoichiometric content according to the chemical formula.

Since biological processes are known to be very complex, some simplifications had to be made when developing a mathematical model. The substrate, intermediate (volatile fatty acids) and the final product (biogas) were defined by their carbon contents. The model was based on the following stages. The first stage describes hydrolytic bacteria hydrolyzing the organic compounds into simple soluble compounds as well as acid forming bacteria forming volatile fatty acids. In the second stage acetogenic bacteria and methanogenic bacteria convert the volatile fatty acids into methane and carbon dioxide. The hydrolysis was described by the first order reaction kinetics and the methanogenesis kinetics was treated as a Monod-like reaction. As possible limiting rates polymer hydrolysis, acidogenesis and methanogenesis were included.

The following differential equations describe the system for sewage sludge fermentation and co-fermentation:

{\frac  {dS}{dt}}=-k\cdot S

{\frac  {dV}{dt}}=Y_{{{\frac  {V}{S}}}}\cdot k\cdot S-\nu _{{V}}\cdot {\frac  {V}{K_{{s}}+V}}\cdot {X_{{0}}}

{\frac  {dCH_{{4}}}{dt}}=Y_{{{\frac  {CH_{{4}}}{V}}}}\cdot \nu _{{V}}\cdot {\frac  {V}{K_{{s}}+V}}\cdot X_{{0}}

{\frac  {dCO_{{2}}}{dt}}=Y_{{{\frac  {CO_{{2}}}{S}}}}\cdot k\cdot S+Y_{{{\frac  {CO_{{2}}}{V}}}}\cdot \nu _{{V}}\cdot {\frac  {V}{K_{{s}}+V}}\cdot X_{{0}}

The following parameters were defined in the paper:

k Constant of first-order reaction 0.11
Y_{{{\frac  {V}{s}}}} Yield factor VFA from substrate 0.72
K_{{S}} Saturation constant 11.24
\nu _{{V}} Maximum specific utilization of VFA rate 2.08
Y_{{{\frac  {CH_{{4}}}{V}}}} Yield factor CH_{{4}} from VFA 0.71
Y_{{{\frac  {CO_{{2}}}{S}}}} Yield factor CO_{{2}} from substrate 0.17
Y_{{{\frac  {CO_{{2}}}{V}}}} Yield factor CO_{{2}} from VFA 0.22
Table 1: Kinetic parameters used in the model for digested sludge fermentation and co-digestion

Initial conditions:





The differential equations described below characterize the system for municipal solid waste fermentation:

{\frac  {dS}{dt}}=-k\cdot S{\frac  {1}{1+{\frac  {V}{K_{{IS}}}}}}

{\frac  {dV}{dt}}=Y_{{{\frac  {V}{S}}}}\cdot k\cdot S{\frac  {1}{1+{\frac  {V}{K_{{IS}}}}}}-\nu _{{V}}\cdot {\frac  {V}{K_{{s}}+V}}\cdot {\frac  {1}{1+{\frac  {V}{K_{{IV}}}}}}\cdot X_{{0}}

{\frac  {dCH_{{4}}}{dt}}=Y_{{{\frac  {CH_{{4}}}{V}}}}\cdot \nu _{{V}}\cdot {\frac  {V}{K_{{s}}+V}}\cdot {\frac  {1}{1+{\frac  {V}{K_{{IV}}}}}}\cdot X_{{0}}

{\frac  {dCO_{{2}}}{dt}}=Y_{{{\frac  {CO_{{2}}}{S}}}}\cdot k\cdot S\cdot {\frac  {1}{1+{\frac  {V}{K_{{IS}}}}}}+Y_{{{\frac  {CO_{{2}}}{V}}}}\cdot \nu _{{V}}\cdot {\frac  {V}{K_{{s}}+V}}\cdot {\frac  {1}{1+{\frac  {V}{K_{{IV}}}}}}\cdot X_{{0}}

The parameters were defined as follows:

k Constant of first-order reaction 0.17
Y_{{{\frac  {V}{s}}}} Yield factor VFA from substrate 0.71
K_{{S}} Saturation constant 11.25
\nu _{{V}} Maximum specific utilization of VFA rate 0.66
Y_{{{\frac  {CH_{{4}}}{V}}}} Yield factor CH_{{4}} from VFA 0.66
Y_{{{\frac  {CO_{{2}}}{S}}}} Yield factor CO_{{2}} from substrate 0.29
Y_{{{\frac  {CO_{{2}}}{V}}}} Yield factor CO_{{2}} from VFA 0.19
Table 2: Kinetic parameters used in model for municipal waste fermentation

Initial conditions:






The diagrams below show the different behaviors of anaerobic digestion.

The fermentation process using sewage sludge shows a decrease of total suspended solids (TSS), while CH4 and CO2 are produced. When the reaction is started the production of volatile fatty acids (VFA) can be observed, which influence the pH-value. However the concentration of the VFA is minimal compared to the co-fermentation process in Figure 2. However this increase of the pH-value has not got a big effect on the systems stability, since the co-substrate acts as buffer. The biogas production is higher for co-fermentation after 25 days. For the fermentation model using municipal waste it was necessary to solve the equations separately in acidogenic and methanogenic phases.

Figure 1: Comparison of experimental data and simulated values for sludge fermentation.

Figure 2: Comparison of experimental data and simulated values for co-fermentation.

Figure 3: Comparison of experimental data and simulated values for OFMSW fermentation.

Simbiology Results


Figure 4: Sewage Sludge Fermentation (Carbon [g/dm3] vs. time)


Figure 5: Co-fermentation (Carbon [g/dm3] vs. time)


Figure 6: Municipal Solid Waste – Acidogenic (Carbon [g/dm3] vs. time)


Figure 7: Municipal Solid Waste – Methanogenic (Carbon [g/dm3] vs. time)

Matlab Code

In this section, the matlab code to regenerate the above pictures is provided. Methane.m

Project is based on:

This project is based on mathematical used.

Applications of this Work in Literature

The work in Sosnowski et al. 2008 is cited as an example of a model for the anaerobic production of biogas in other publications.

In Lo et al. 2010, biogas production was studied for anaerobic digestion to determine the effect of MSW incinerator fly ash and bottom ash on the cofermentation of organic fraction solid municipal waster (OFSMW). The model in Lo et al. 2010 was not heavily based on fermentation kinetics but rather sought to model biogas production using linear, exponential, and Gaussian plots. The linear model was used for the assumption that biogas production would increase linearly until a maximum and then decrease linearly to zero. The exponential model was used to simulate if biogas production increased exponentially until a maximum and then decreased exponentially. The Gaussian model simulated the situation if biogas production rate and microbial growth follow a normal distribution over the digestion period. While this paper used a different model and process of fitting different equations to experimental data, the authors cited the Sosnowski et al. 2008 paper to give an example of a model where biogas production was based on kinetic models. The authors used the results and findings in Sosnowski et al. 2008 to help predict which equations should be used to try and fit the data. They noted that the kinetic models show a linear increase and then an exponential decrease which validated their decision to use a linear and exponential equation to fit the data in Lo et al. 2010.

In Zhang et al. 2010, the effect of surfactants, such as sodium dodecylbenzene sulfonate (SDBS), on the products from fermentation of waste activated sludge was studied. The authors sought to investigate this effect using a kinetic model of the system. They modeled the hydrolysis step based on the method and model in Sosnowski et al. 2008. The hydolysis step was modeled using first order reaction kinetics according to Sosnowski et al. 2008. The Sosnowski et al. 2008 model for hydrolysis applies to the work because the authors in Zhang et al. 2010 are also studying anaerobic degradation of heterogeneous substrates.

[1] Lo, H.M. et al., (2010). Modeling biogas production from organic fraction of MSW co-digested with MSWI ashes in anaerobic bioreactors. Bioresource Technology 101: 6329-6335.

[2] Zhang, P. et al., (2010). "Effect of surfactant on hydrolysis products accumulation and short-chain fatty acids (SCFA) production during mesophilic and thermophilic fermentation of waste activated sludge: Kinetic studies". Bioresource Technology 101: 6902-6909.

Relating this article to MBW:A Mathematical Model Comparing Solute Kinetics In Low- And High-BMI Hemodialysis Patients, Sai'd Azzam

Both article analyze the metabolism system using a mathematical analogy. In both articles, they are using differential equations to solve the systems.
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