2011, 8(4): 953-971. doi: 10.3934/mbe.2011.8.953

Effects of spatial structure and diffusion on the performances of the chemostat

1. 

UMR INRA/SupAgro 'MISTEA' and EPI INRA/INRIA 'MODEMIC', 2, pl. Viala 34060, Montpellier, France

2. 

UMR Analyse des Systèmes et Biométrie, INRA, EPI INRA/INRIA 'MODEMIC', 2 pl. Viala 34060 Montpellier

3. 

UMR INRA/SupAgro/CIRAD/IRD 'Eco&Sols', 2, pl. Viala 34060, Montpellier, France

Received  November 2010 Revised  April 2011 Published  August 2011

Given hydric capacity and nutrient flow of a chemostat-like system, we analyse the influence of a spatial structure on the output concentrations at steady-state. Three configurations are compared: perfectly-mixed, serial and parallel with diffusion rate. We show the existence of a threshold on the input concentration of nutrient for which the benefits of the serial and parallel configurations over the perfectly-mixed one are reversed. In addition, we show that the dependency of the output concentrations on the diffusion rate can be non-monotonic, and give precise conditions for the diffusion effect to be advantageous. The study encompasses dead-zone models.
Citation: Ihab Haidar, Alain Rapaport, Frédéric Gérard. Effects of spatial structure and diffusion on the performances of the chemostat. Mathematical Biosciences & Engineering, 2011, 8 (4) : 953-971. doi: 10.3934/mbe.2011.8.953
References:
[1]

C. de Gooijer, W. Bakker, H. Beeftink and J. Tramper, Bioreactors in series: An overview of design procedures and practical applications,, Enzyme and Microbial Technology, 18 (1996), 202.   Google Scholar

[2]

C. de Gooijer, H. Beeftink and J. Tramper, Optimal design of a series of continuous stirred tank reactors containing immobilised growing cells,, Biotechnology Letters, 18 (1996), 397.   Google Scholar

[3]

P. Doran, Design of mixing systems for plant cell suspensions in stirred reactors,, Biotechnology Progress, 15 (1999), 319.   Google Scholar

[4]

A. Dramé, J. Harmand, A. Rapaport and C. Lobry, Multiple steady state profiles in interconnected biological systems,, Mathematical and Computer Modelling of Dynamical Systems, 12 (2006), 379.   Google Scholar

[5]

A. Dramé, C. Lobry, J. Harmand, A. Rapaport and F. Mazenc, Multiple stable equilibrium profiles in tubular bioreactors,, Mathematical and Computer Modelling, 48 (2008), 1840.   Google Scholar

[6]

S. Foger, "Elements of Chemical Reaction Engineering,", 4th edition, (2008).   Google Scholar

[7]

A. Grobicki and D. Stuckey, Hydrodynamic characteristics of the anaerobic baffled reactor,, Water Research, 26 (1992), 371.   Google Scholar

[8]

L. Grady, G. Daigger and H. Lim, "Biological Wastewater Treatment,'' 3nd edition,, Environmental Science and Pollution Control Series, (1999).   Google Scholar

[9]

D. Gravel, F. Guichard, M. Loreau and N. Mouquet, Source and sink dynamics in metaecosystems,, Ecology, 91 (2010), 2172.   Google Scholar

[10]

I. Hanski, "Metapopulation Ecology,'', Oxford University Press, (1999).   Google Scholar

[11]

J. Harmand, A. Rapaport and A. Trofino, Optimal design of two interconnected bioreactors-some new results,, American Institute of Chemical Engineering Journal, 49 (1999), 1433.   Google Scholar

[12]

J. Harmand, A. Rapaport and A. Dramé, Optimal design of two interconnected enzymatic reactors,, Journal of Process Control, 14 (2004), 785.   Google Scholar

[13]

J. Harmand and D. Dochain, Towards a unified approach for the design of interconnected catalytic and auto-catalytic reactors,, Computers and Chemical Engineering, 30 (2005), 70.   Google Scholar

[14]

G. Hill and C. Robinson, Minimum tank volumes for CFST bioreactors in series,, The Canadian Journal of Chemical Engineering, 67 (1989), 818.   Google Scholar

[15]

W. Hu, K. Wlashchin, M. Betenbaugh, F. Wurm, G. Seth and W. Zhou, "Cellular Bioprocess Technology, Fundamentals and Frontier,'', Lectures Notes, (2007).   Google Scholar

[16]

O. Levenspiel, "Chemical Reaction Engineering,'', 3nd edition, (1999).   Google Scholar

[17]

R. Lovitt and J. Wimpenny, The gradostat: A bidirectional compound chemostat and its applications in microbial research,, Journal of General Microbiology, 127 (1981), 261.   Google Scholar

[18]

K. Luyben and J. Tramper, Optimal design for continuously stirred tank reactors in series using Michaelis-Menten kinetics,, Biotechnology and Bioengineering, 24 (1982), 1217.   Google Scholar

[19]

R. MacArthur and E. Wilson, "The Theory of Island Biogeography,'', Princeton University Press, (1967).   Google Scholar

[20]

K. Mischaikow, H. Smith and H. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions,, Transactions of the American Mathematical Society, 347 (1995), 1669.  doi: 10.2307/2154964.  Google Scholar

[21]

J. Monod, La technique de la culture continue: Théorie et applications,, Annales de l'Institut Pasteur, 79 (1950), 390.   Google Scholar

[22]

S. Nakaoka and Y. Takeuchi, Competition in chemostat-type equations with two habitats,, Mathematical Bioscience, 201 (2006), 157.   Google Scholar

[23]

M. Nelson and H. Sidhu, Evaluating the performance of a cascade of two bioreactors,, Chemical Engineering Science, 61 (2006), 3159.   Google Scholar

[24]

A. Novick and L. Szilard, Description of the chemostat,, Science, 112 (1950), 715.   Google Scholar

[25]

A. Rapaport, J. Harmand and F. Mazenc, Coexistence in the design of a series of two chemostats,, Nonlinear Analysis, 9 (2008), 1052.   Google Scholar

[26]

E. Roca, C. Ghommidh, J.-M. Navarro and J.-M. Lema, Hydraulic model of a gas-lift bioreactor with flocculating yeast,, Bioprocess and Biosystems Engineering, 12 (1995), 269.   Google Scholar

[27]

G. Roux, B. Dahhou and I. Queinnec, Adaptive non-linear control of a continuous stirred tank bioreactor,, Journal of Process Control, 4 (1994), 121.   Google Scholar

[28]

A. Saddoud, T. Sari, A. Rapaport, R. Lortie, J. Harmand and E. Dubreucq, A mathematical study of an enzymatic hydrolysis of a cellulosic substrate in non homogeneous reactors,, Proceedings of the IFAC Computer Applications in Biotechnology Conference (CAB 2010), (2010), 7.   Google Scholar

[29]

A. Scheel and E. Van Vleck, Lattice differential equations embedded into reaction-diffusion systems,, Proceedings of the Royal Society Edinburgh Section A, 139 (2009), 193.   Google Scholar

[30]

H. Smith and P. Waltman, "The Theory of Chemostat. Dynamics of Microbial Competition,'', Cambridge Studies in Mathematical Biology, 13 (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[31]

G. Stephanopoulos and A. Fredrickson, Effect of inhomogeneities on the coexistence of competing microbial populations,, Biotechnology and Bioengineering, 21 (1979), 1491.   Google Scholar

[32]

R. Schwartz, A. Juo and K. McInnes, Estimating parameters for a dual-porosity model to describe non-equilibrium, reactive transport in a fine-textured soil,, Journal of Hydrology, 229 (2000), 149.   Google Scholar

[33]

C. Tsakiroglou and M. Ioannidis, Dual-porosity modelling of the pore structure and transport properties of a contaminated soil,, European Journal of Soil Science, 59 (2008), 744.   Google Scholar

[34]

F. Valdes-Parada, J. Alvarez-Ramirez and A. Ochoa-Tapia, An approximate solution for a transient two-phase stirred tank bioreactor with nonlinear kinetics,, Biotechnology Progress, 21 (2005), 1420.   Google Scholar

[35]

K. Van't Riet and J. Tramper, "Basic Bioreactor Design,'', Marcel Dekker, (1991).   Google Scholar

show all references

References:
[1]

C. de Gooijer, W. Bakker, H. Beeftink and J. Tramper, Bioreactors in series: An overview of design procedures and practical applications,, Enzyme and Microbial Technology, 18 (1996), 202.   Google Scholar

[2]

C. de Gooijer, H. Beeftink and J. Tramper, Optimal design of a series of continuous stirred tank reactors containing immobilised growing cells,, Biotechnology Letters, 18 (1996), 397.   Google Scholar

[3]

P. Doran, Design of mixing systems for plant cell suspensions in stirred reactors,, Biotechnology Progress, 15 (1999), 319.   Google Scholar

[4]

A. Dramé, J. Harmand, A. Rapaport and C. Lobry, Multiple steady state profiles in interconnected biological systems,, Mathematical and Computer Modelling of Dynamical Systems, 12 (2006), 379.   Google Scholar

[5]

A. Dramé, C. Lobry, J. Harmand, A. Rapaport and F. Mazenc, Multiple stable equilibrium profiles in tubular bioreactors,, Mathematical and Computer Modelling, 48 (2008), 1840.   Google Scholar

[6]

S. Foger, "Elements of Chemical Reaction Engineering,", 4th edition, (2008).   Google Scholar

[7]

A. Grobicki and D. Stuckey, Hydrodynamic characteristics of the anaerobic baffled reactor,, Water Research, 26 (1992), 371.   Google Scholar

[8]

L. Grady, G. Daigger and H. Lim, "Biological Wastewater Treatment,'' 3nd edition,, Environmental Science and Pollution Control Series, (1999).   Google Scholar

[9]

D. Gravel, F. Guichard, M. Loreau and N. Mouquet, Source and sink dynamics in metaecosystems,, Ecology, 91 (2010), 2172.   Google Scholar

[10]

I. Hanski, "Metapopulation Ecology,'', Oxford University Press, (1999).   Google Scholar

[11]

J. Harmand, A. Rapaport and A. Trofino, Optimal design of two interconnected bioreactors-some new results,, American Institute of Chemical Engineering Journal, 49 (1999), 1433.   Google Scholar

[12]

J. Harmand, A. Rapaport and A. Dramé, Optimal design of two interconnected enzymatic reactors,, Journal of Process Control, 14 (2004), 785.   Google Scholar

[13]

J. Harmand and D. Dochain, Towards a unified approach for the design of interconnected catalytic and auto-catalytic reactors,, Computers and Chemical Engineering, 30 (2005), 70.   Google Scholar

[14]

G. Hill and C. Robinson, Minimum tank volumes for CFST bioreactors in series,, The Canadian Journal of Chemical Engineering, 67 (1989), 818.   Google Scholar

[15]

W. Hu, K. Wlashchin, M. Betenbaugh, F. Wurm, G. Seth and W. Zhou, "Cellular Bioprocess Technology, Fundamentals and Frontier,'', Lectures Notes, (2007).   Google Scholar

[16]

O. Levenspiel, "Chemical Reaction Engineering,'', 3nd edition, (1999).   Google Scholar

[17]

R. Lovitt and J. Wimpenny, The gradostat: A bidirectional compound chemostat and its applications in microbial research,, Journal of General Microbiology, 127 (1981), 261.   Google Scholar

[18]

K. Luyben and J. Tramper, Optimal design for continuously stirred tank reactors in series using Michaelis-Menten kinetics,, Biotechnology and Bioengineering, 24 (1982), 1217.   Google Scholar

[19]

R. MacArthur and E. Wilson, "The Theory of Island Biogeography,'', Princeton University Press, (1967).   Google Scholar

[20]

K. Mischaikow, H. Smith and H. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions,, Transactions of the American Mathematical Society, 347 (1995), 1669.  doi: 10.2307/2154964.  Google Scholar

[21]

J. Monod, La technique de la culture continue: Théorie et applications,, Annales de l'Institut Pasteur, 79 (1950), 390.   Google Scholar

[22]

S. Nakaoka and Y. Takeuchi, Competition in chemostat-type equations with two habitats,, Mathematical Bioscience, 201 (2006), 157.   Google Scholar

[23]

M. Nelson and H. Sidhu, Evaluating the performance of a cascade of two bioreactors,, Chemical Engineering Science, 61 (2006), 3159.   Google Scholar

[24]

A. Novick and L. Szilard, Description of the chemostat,, Science, 112 (1950), 715.   Google Scholar

[25]

A. Rapaport, J. Harmand and F. Mazenc, Coexistence in the design of a series of two chemostats,, Nonlinear Analysis, 9 (2008), 1052.   Google Scholar

[26]

E. Roca, C. Ghommidh, J.-M. Navarro and J.-M. Lema, Hydraulic model of a gas-lift bioreactor with flocculating yeast,, Bioprocess and Biosystems Engineering, 12 (1995), 269.   Google Scholar

[27]

G. Roux, B. Dahhou and I. Queinnec, Adaptive non-linear control of a continuous stirred tank bioreactor,, Journal of Process Control, 4 (1994), 121.   Google Scholar

[28]

A. Saddoud, T. Sari, A. Rapaport, R. Lortie, J. Harmand and E. Dubreucq, A mathematical study of an enzymatic hydrolysis of a cellulosic substrate in non homogeneous reactors,, Proceedings of the IFAC Computer Applications in Biotechnology Conference (CAB 2010), (2010), 7.   Google Scholar

[29]

A. Scheel and E. Van Vleck, Lattice differential equations embedded into reaction-diffusion systems,, Proceedings of the Royal Society Edinburgh Section A, 139 (2009), 193.   Google Scholar

[30]

H. Smith and P. Waltman, "The Theory of Chemostat. Dynamics of Microbial Competition,'', Cambridge Studies in Mathematical Biology, 13 (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[31]

G. Stephanopoulos and A. Fredrickson, Effect of inhomogeneities on the coexistence of competing microbial populations,, Biotechnology and Bioengineering, 21 (1979), 1491.   Google Scholar

[32]

R. Schwartz, A. Juo and K. McInnes, Estimating parameters for a dual-porosity model to describe non-equilibrium, reactive transport in a fine-textured soil,, Journal of Hydrology, 229 (2000), 149.   Google Scholar

[33]

C. Tsakiroglou and M. Ioannidis, Dual-porosity modelling of the pore structure and transport properties of a contaminated soil,, European Journal of Soil Science, 59 (2008), 744.   Google Scholar

[34]

F. Valdes-Parada, J. Alvarez-Ramirez and A. Ochoa-Tapia, An approximate solution for a transient two-phase stirred tank bioreactor with nonlinear kinetics,, Biotechnology Progress, 21 (2005), 1420.   Google Scholar

[35]

K. Van't Riet and J. Tramper, "Basic Bioreactor Design,'', Marcel Dekker, (1991).   Google Scholar

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