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April  2020, 19(4): 2333-2346. doi: 10.3934/cpaa.2020101

Analysis of an anaerobic digestion model in landfill with mortality term

1. 

Interdisciplinary Laboratory for Natural Resources and Environment, Ibn Tofaïl University, Kénitra, Morocco

2. 

IRIMAS, University of Haute-Alsace, Mulhouse, France, University of Strasbourg, France

3. 

LBE, University of Montpellier, INRA, Narbonne, France

4. 

MISTEA, University of Montpellier, INRA, Montpellier SupAgro, Montpellier, France

5. 

University of Haute-Alsace, IRIMAS UR 7499, F-68100 Mulhouse, France, University of Strasbourg, France

*Corresponding author

Dedicated to Professor Tomás Caraballo on the occasion of his 60-th birthday

Received  August 2019 Revised  October 2019 Published  January 2020

We study a mathematical model of anaerobic digestion with biomass recirculation, dedicated to landfill problems, and analyze its asymptotic behavior. We show that the global attractor is composed of an infinity of non-hyperbolic equilibria. For non-monotonic growth functions, this set is non connected, which impacts the performances of the bioprocess.

Citation: S. Ouchtout, Z. Mghazli, J. Harmand, A. Rapaport, Z. Belhachmi. Analysis of an anaerobic digestion model in landfill with mortality term. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2333-2346. doi: 10.3934/cpaa.2020101
References:
[1]

J. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707–723.

[2]

J. Arzate, M. Kirstein, F. Ertem, E. Kielhorn, H. Malule, P. Neubauer, M. Cruz-Bournazou and S. Junne, Anaerobic digestion model (AM2) for the description of biogas processes at dynamic feedstock loading rates, Chemie Ingenieur Technik, 89 (2017), 686–695.

[3]

I. Barbalat, Systèmes d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267–270.

[4]

G. Bastin and D. Dochain, On-line Estimation and Adaptive Control of Bioreactors, Dynamics of Microbial Competition, Elsevier Science Publishers, New-York, 1991.

[5]

B. Benyahia, T. Sari, B. Cherki and J. Harmand, Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, Journal of Process Control, 22 (2012), 1008–1019.

[6]

O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi and J. P. Steyer, Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnology and Bioengineering, 75 (2001), 424–438.

[7]

J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, 1981.

[8]

D. Chenu, Modélisation des transferts réactifs de masse et de chaleur dans les installations de stockage de déchets ménagers: application aux installations de type bioréacteur, PhD thesis, Institut National Polytechnique de Toulouse, France, 2007.

[9]

I. DidiH. Dib and B. Cherki, A Luenberger-type observer for the AM2 model, Journal of Process Control, 32 (2015), 117-126. 

[10]

D. Dochain, Automatic Control of Bioprocesses Control systems, John Wiley and Sons, 2010.

[11]

G. Dollé, O. Duran, N. Feyeux, E. Frénod, M. Giacomini and C. Prud'Homme, Mathematical modeling and numerical simulation of a bioreactor landfill using Feel++, ESAIM: Proceedings and Surveys, 55 (2016), 83–110.

[12]

R. Fekih-SalemJ. HarmandC. LobryA. Rapaport and T. Sari, Extensions of the chemostat model with flocculation, Journal of Mathematical Analysis and Applications, 397 (2013), 292-306.  doi: 10.1016/j.jmaa.2012.07.055.

[13]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganisms Cultures, ISTE Wiley, 2017.

[14]

S. Hassam, E. Ficara, A. Leva and J. Harmand, A generic and systematic procedure to derive a simplified model from the anaerobic digestion model, No. 1 (ADM1), Biochemical Engineering Journal, 99 (2015), 193–203.

[15]

M. Hmissi, J. Harmand, V. Alcaraz-Gonzalez and H. Shayeb, Evaluation of alkalinity spatial distribution in an up-flow fixed bed anaerobic digester, Water Science and Technology, 77 (2018), 948–959.

[16]

S. J. Jang and J. Baglama, Nutrient-plankton models with nutrient recycling, Computers and Mathematics with Applications, 49 (2005), 375–387. doi: 10.1016/j.camwa.2004.03.013.

[17]

M. Loreau, Material cycling and the stability of ecosystems, The American Naturalist, 143 (1994), 508–513.

[18]

J. Monod, La technique de la culture continue: Théorie et applications, Ann. Inst. Pasteur, Lille, 79 (1950), 390–410.

[19]

L. Perko, Differential Equations and Dynamical Systems, Springer, 3rd ed., 2011. doi: 10.1007/978-1-4684-0392-3.

[20]

A. Rapaport, T. Bayen, M. Sebbah, A. Donoso-Bravo and A. Torrico, Dynamical modelling and optimal control of landfills, Mathematical Models and Methods in Applied Sciences, 26 (2016), 901–929. doi: 10.1142/S0218202516500214.

[21]

A. Rapaport, T. Nidelet, S. El Aida and J. Harmand, About biomass overyielding of mixed cultures in batch processes, Prepint hal, (2019).

[22]

A. Rapaport, T. Nidelet and J. Harmand, About biomass overyielding of mixed cultures in batch processes, in, 8th IFAC Conference on Foundations of Systems Biology in Engineering (FOSBE), Valencia, Spain, 15-18 Oct., (2019).

[23]

M. Rouez, Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations, PhD thesis, Institut National des Sciences Appliquées, Lyon, France, 2008.

[24]

W. Walter, Ordinary Differential Equations, Springer, 1998. doi: 10.1007/978-1-4612-0601-9.

show all references

References:
[1]

J. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707–723.

[2]

J. Arzate, M. Kirstein, F. Ertem, E. Kielhorn, H. Malule, P. Neubauer, M. Cruz-Bournazou and S. Junne, Anaerobic digestion model (AM2) for the description of biogas processes at dynamic feedstock loading rates, Chemie Ingenieur Technik, 89 (2017), 686–695.

[3]

I. Barbalat, Systèmes d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267–270.

[4]

G. Bastin and D. Dochain, On-line Estimation and Adaptive Control of Bioreactors, Dynamics of Microbial Competition, Elsevier Science Publishers, New-York, 1991.

[5]

B. Benyahia, T. Sari, B. Cherki and J. Harmand, Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, Journal of Process Control, 22 (2012), 1008–1019.

[6]

O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi and J. P. Steyer, Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnology and Bioengineering, 75 (2001), 424–438.

[7]

J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, 1981.

[8]

D. Chenu, Modélisation des transferts réactifs de masse et de chaleur dans les installations de stockage de déchets ménagers: application aux installations de type bioréacteur, PhD thesis, Institut National Polytechnique de Toulouse, France, 2007.

[9]

I. DidiH. Dib and B. Cherki, A Luenberger-type observer for the AM2 model, Journal of Process Control, 32 (2015), 117-126. 

[10]

D. Dochain, Automatic Control of Bioprocesses Control systems, John Wiley and Sons, 2010.

[11]

G. Dollé, O. Duran, N. Feyeux, E. Frénod, M. Giacomini and C. Prud'Homme, Mathematical modeling and numerical simulation of a bioreactor landfill using Feel++, ESAIM: Proceedings and Surveys, 55 (2016), 83–110.

[12]

R. Fekih-SalemJ. HarmandC. LobryA. Rapaport and T. Sari, Extensions of the chemostat model with flocculation, Journal of Mathematical Analysis and Applications, 397 (2013), 292-306.  doi: 10.1016/j.jmaa.2012.07.055.

[13]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganisms Cultures, ISTE Wiley, 2017.

[14]

S. Hassam, E. Ficara, A. Leva and J. Harmand, A generic and systematic procedure to derive a simplified model from the anaerobic digestion model, No. 1 (ADM1), Biochemical Engineering Journal, 99 (2015), 193–203.

[15]

M. Hmissi, J. Harmand, V. Alcaraz-Gonzalez and H. Shayeb, Evaluation of alkalinity spatial distribution in an up-flow fixed bed anaerobic digester, Water Science and Technology, 77 (2018), 948–959.

[16]

S. J. Jang and J. Baglama, Nutrient-plankton models with nutrient recycling, Computers and Mathematics with Applications, 49 (2005), 375–387. doi: 10.1016/j.camwa.2004.03.013.

[17]

M. Loreau, Material cycling and the stability of ecosystems, The American Naturalist, 143 (1994), 508–513.

[18]

J. Monod, La technique de la culture continue: Théorie et applications, Ann. Inst. Pasteur, Lille, 79 (1950), 390–410.

[19]

L. Perko, Differential Equations and Dynamical Systems, Springer, 3rd ed., 2011. doi: 10.1007/978-1-4684-0392-3.

[20]

A. Rapaport, T. Bayen, M. Sebbah, A. Donoso-Bravo and A. Torrico, Dynamical modelling and optimal control of landfills, Mathematical Models and Methods in Applied Sciences, 26 (2016), 901–929. doi: 10.1142/S0218202516500214.

[21]

A. Rapaport, T. Nidelet, S. El Aida and J. Harmand, About biomass overyielding of mixed cultures in batch processes, Prepint hal, (2019).

[22]

A. Rapaport, T. Nidelet and J. Harmand, About biomass overyielding of mixed cultures in batch processes, in, 8th IFAC Conference on Foundations of Systems Biology in Engineering (FOSBE), Valencia, Spain, 15-18 Oct., (2019).

[23]

M. Rouez, Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations, PhD thesis, Institut National des Sciences Appliquées, Lyon, France, 2008.

[24]

W. Walter, Ordinary Differential Equations, Springer, 1998. doi: 10.1007/978-1-4612-0601-9.

Figure 1.  Overall scheme of the anaerobic degradation of organic matter
Figure 2.  Graphs of Monod and Haldane functions
Figure 3.  Graph of the Haldane function considered in the example
Figure 4.  Graph of $ S(\cdot) $ for initial conditions $ X_0 = 340 $ to $ X_0 = 360 $ with a step of 5
Figure 5.  Threshold on $ X_0 $ as a function of $ K_d $ for model parameters indicated in Table 1
Figure 6.  Biogas production as a function of $ X_0 $ for $ K_d = 0.02 $
Figure 7.  $ \lambda^- $, $ \lambda^+ $ (on top) $ {\rm Biogas}^- $ and $ {\rm Biogas}^+ $ (on bottom) as functions of $ K_d $
$ K_h$ $Y $ $f_1 $ $ f_2$ $ \alpha$
0.176 0.05 0.7 0.76 0.9
$ K_h$ $Y $ $f_1 $ $ f_2$ $ \alpha$
0.176 0.05 0.7 0.76 0.9
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