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), 202-219.

[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), 397-402.

[3]

P. Doran, Design of mixing systems for plant cell suspensions in stirred reactors, Biotechnology Progress, 15 (1999), 319-335.

[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), 379-393.

[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), 1840-1853.

[6]

S. Foger, "Elements of Chemical Reaction Engineering," 4th edition, Prentice Hall, New York, 2008.

[7]

A. Grobicki and D. Stuckey, Hydrodynamic characteristics of the anaerobic baffled reactor, Water Research, 26 (1992), 371-378.

[8]

L. Grady, G. Daigger and H. Lim, "Biological Wastewater Treatment,'' 3nd edition, Environmental Science and Pollution Control Series, Marcel Dekker, New York, 1999.

[9]

D. Gravel, F. Guichard, M. Loreau and N. Mouquet, Source and sink dynamics in metaecosystems, Ecology, 91 (2010), 2172-2184.

[10]

I. Hanski, "Metapopulation Ecology,'' Oxford University Press, 1999.

[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), 1433-1450.

[12]

J. Harmand, A. Rapaport and A. Dramé, Optimal design of two interconnected enzymatic reactors, Journal of Process Control, 14 (2004), 785-794.

[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), 70-82.

[14]

G. Hill and C. Robinson, Minimum tank volumes for CFST bioreactors in series, The Canadian Journal of Chemical Engineering, 67 (1989), 818-824.

[15]

W. Hu, K. Wlashchin, M. Betenbaugh, F. Wurm, G. Seth and W. Zhou, "Cellular Bioprocess Technology, Fundamentals and Frontier,'' Lectures Notes, University of Minesota, 2007.

[16]

O. Levenspiel, "Chemical Reaction Engineering,'' 3nd edition, Wiley, New York, 1999.

[17]

R. Lovitt and J. Wimpenny, The gradostat: A bidirectional compound chemostat and its applications in microbial research, Journal of General Microbiology, 127 (1981), 261-268.

[18]

K. Luyben and J. Tramper, Optimal design for continuously stirred tank reactors in series using Michaelis-Menten kinetics, Biotechnology and Bioengineering, 24 (1982), 1217-1220.

[19]

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

[20]

K. Mischaikow, H. Smith and H. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Transactions of the American Mathematical Society, 347 (1995), 1669-1685. doi: 10.2307/2154964.

[21]

J. Monod, La technique de la culture continue: Théorie et applications, Annales de l'Institut Pasteur, 79 (1950), 390-410.

[22]

S. Nakaoka and Y. Takeuchi, Competition in chemostat-type equations with two habitats, Mathematical Bioscience, 201 (2006), 157-171.

[23]

M. Nelson and H. Sidhu, Evaluating the performance of a cascade of two bioreactors, Chemical Engineering Science, 61 (2006), 3159-3166.

[24]

A. Novick and L. Szilard, Description of the chemostat, Science, 112 (1950), 715-716.

[25]

A. Rapaport, J. Harmand and F. Mazenc, Coexistence in the design of a series of two chemostats, Nonlinear Analysis, Real World Applications, 9 (2008), 1052-1067.

[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), 269-272.

[27]

G. Roux, B. Dahhou and I. Queinnec, Adaptive non-linear control of a continuous stirred tank bioreactor, Journal of Process Control, 4 (1994), 121-126.

[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), Leuven, Belgium, July 7-9, 2010.

[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), 193-207.

[30]

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

[31]

G. Stephanopoulos and A. Fredrickson, Effect of inhomogeneities on the coexistence of competing microbial populations, Biotechnology and Bioengineering, 21 (1979), 1491-1498.

[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), 149-167.

[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), 744-761.

[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), 1420-1428.

[35]

K. Van't Riet and J. Tramper, "Basic Bioreactor Design,'' Marcel Dekker, New York, 1991.

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), 202-219.

[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), 397-402.

[3]

P. Doran, Design of mixing systems for plant cell suspensions in stirred reactors, Biotechnology Progress, 15 (1999), 319-335.

[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), 379-393.

[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), 1840-1853.

[6]

S. Foger, "Elements of Chemical Reaction Engineering," 4th edition, Prentice Hall, New York, 2008.

[7]

A. Grobicki and D. Stuckey, Hydrodynamic characteristics of the anaerobic baffled reactor, Water Research, 26 (1992), 371-378.

[8]

L. Grady, G. Daigger and H. Lim, "Biological Wastewater Treatment,'' 3nd edition, Environmental Science and Pollution Control Series, Marcel Dekker, New York, 1999.

[9]

D. Gravel, F. Guichard, M. Loreau and N. Mouquet, Source and sink dynamics in metaecosystems, Ecology, 91 (2010), 2172-2184.

[10]

I. Hanski, "Metapopulation Ecology,'' Oxford University Press, 1999.

[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), 1433-1450.

[12]

J. Harmand, A. Rapaport and A. Dramé, Optimal design of two interconnected enzymatic reactors, Journal of Process Control, 14 (2004), 785-794.

[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), 70-82.

[14]

G. Hill and C. Robinson, Minimum tank volumes for CFST bioreactors in series, The Canadian Journal of Chemical Engineering, 67 (1989), 818-824.

[15]

W. Hu, K. Wlashchin, M. Betenbaugh, F. Wurm, G. Seth and W. Zhou, "Cellular Bioprocess Technology, Fundamentals and Frontier,'' Lectures Notes, University of Minesota, 2007.

[16]

O. Levenspiel, "Chemical Reaction Engineering,'' 3nd edition, Wiley, New York, 1999.

[17]

R. Lovitt and J. Wimpenny, The gradostat: A bidirectional compound chemostat and its applications in microbial research, Journal of General Microbiology, 127 (1981), 261-268.

[18]

K. Luyben and J. Tramper, Optimal design for continuously stirred tank reactors in series using Michaelis-Menten kinetics, Biotechnology and Bioengineering, 24 (1982), 1217-1220.

[19]

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

[20]

K. Mischaikow, H. Smith and H. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Transactions of the American Mathematical Society, 347 (1995), 1669-1685. doi: 10.2307/2154964.

[21]

J. Monod, La technique de la culture continue: Théorie et applications, Annales de l'Institut Pasteur, 79 (1950), 390-410.

[22]

S. Nakaoka and Y. Takeuchi, Competition in chemostat-type equations with two habitats, Mathematical Bioscience, 201 (2006), 157-171.

[23]

M. Nelson and H. Sidhu, Evaluating the performance of a cascade of two bioreactors, Chemical Engineering Science, 61 (2006), 3159-3166.

[24]

A. Novick and L. Szilard, Description of the chemostat, Science, 112 (1950), 715-716.

[25]

A. Rapaport, J. Harmand and F. Mazenc, Coexistence in the design of a series of two chemostats, Nonlinear Analysis, Real World Applications, 9 (2008), 1052-1067.

[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), 269-272.

[27]

G. Roux, B. Dahhou and I. Queinnec, Adaptive non-linear control of a continuous stirred tank bioreactor, Journal of Process Control, 4 (1994), 121-126.

[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), Leuven, Belgium, July 7-9, 2010.

[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), 193-207.

[30]

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

[31]

G. Stephanopoulos and A. Fredrickson, Effect of inhomogeneities on the coexistence of competing microbial populations, Biotechnology and Bioengineering, 21 (1979), 1491-1498.

[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), 149-167.

[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), 744-761.

[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), 1420-1428.

[35]

K. Van't Riet and J. Tramper, "Basic Bioreactor Design,'' Marcel Dekker, New York, 1991.

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