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2014, 11(5): 1139-1166. doi: 10.3934/mbe.2014.11.1139

On optimization of substrate removal in a bioreactor with wall attached and suspended bacteria

1. 

EAWAG, Swiss Federal Institute of Aquatic Science and Technology, Überlandstrasse 133, P.O. Box 611, CH-8600 Dübendorf,, Switzerland

2. 

Biophysics Interdepartmental Program and Department, of Mathematics and Statistics, University of Guelph, Guelph ON, N1G 2W1, Canada

Received  February 2013 Revised  March 2014 Published  June 2014

We investigate the question of optimal substrate removal in a biofilm reactor with concurrent suspended growth, both with respect to the amount of substrate removed and with respect to treatment process duration. The water to be treated is fed externally from a buffer vessel to the treatment reactor. In the two-objective optimal control problem, the flow rate between the vessels is selected as the control variable. The treatment reactor is modelled by a system of three ordinary differential equations in which a two-point boundary value problem is embedded. The solution of the associated singular optimal control problem in the class of measurable functions is impractical to determine and infeasible to implement in real reactors. Instead, we solve the simpler problem to optimize reactor performance in the class of off-on functions, a choice that is motivated by the underlying biological process. These control functions start with an initial no-flow period and then switch to a constant flow rate until the buffer vessel is empty. We approximate the Pareto Front numerically and study the behaviour of the system and its dependence on reactor and initial data. Overall, the modest potential of control strategies to improve reactor performance is found to be primarily due to an initial transient period in which the bacteria have to adapt to the environmental conditions in the reactor, i.e. depends heavily on the initial state of the dynamic system. In applications, the initial state, however, is often unknown and therefore the efficiency of reactor optimization, compared to the uncontrolled system with constant flow rate, is limited.
Citation: Alma Mašić, Hermann J. Eberl. On optimization of substrate removal in a bioreactor with wall attached and suspended bacteria. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1139-1166. doi: 10.3934/mbe.2014.11.1139
References:
[1]

F. Abbas, R. Sudarsan and H. J. Eberl, Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates,, Math. Biosc. Eng., 9 (2012), 215.  doi: 10.3934/mbe.2012.9.215.  Google Scholar

[2]

F. Abbas and H. J. Eberl, Analytical substrate flux approximation for the Monod boundary value problem,, Appl. Math. Comp., 218 (2011), 1484.  doi: 10.1016/j.amc.2011.05.102.  Google Scholar

[3]

M. M. Ballyk, D. A. Jones and H. L. Smith, The biofilm model of Freter: A review,, in Structured Population Models in Biology and Epidemiology, (1936), 265.  doi: 10.1007/978-3-540-78273-5_6.  Google Scholar

[4]

M. M. Ballyk, D. A. Jones and H. L. Smith, Microbial competition in reactors with wall attachment,, Microb. Ecol., 41 (2001), 210.   Google Scholar

[5]

J. T. Betts, Practical Methods for Optimal Control and Estimation using Nonlinear Programming,, SIAM series Adv. Design and Control, (2010).  doi: 10.1137/1.9780898718577.  Google Scholar

[6]

C. Carathéodory, Vorlesungen über reelle Funktionen,, 3rd edition, (1968).   Google Scholar

[7]

N. G. Cogan, B. Szomolay and M. Dindo, Effect of periodic disinfection on persisters in a one-dimensional biofilm model,, Bull. Math. Biol., 75 (2013), 94.  doi: 10.1007/s11538-012-9796-z.  Google Scholar

[8]

T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms,, 2nd edition, (2001).   Google Scholar

[9]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Springer-Verlag New York Inc, (1975).   Google Scholar

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R. Freter, H. Brickner, J. Fekete, M. Vickerman and K. Carey, Survival and implantation of Escherichia coli in the intestinal tract,, Infect. Immun., 39 (1983), 686.   Google Scholar

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P. Gajardo, J. Harmand, H. Ramírez C. and A. Rapaport, Minimal time bioremediation of natural water resources,, Automatica, 47 (2011), 1764.  doi: 10.1016/j.automatica.2011.03.001.  Google Scholar

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A. Göpfert and R. Nehse, Vektoroptimierung,, BSB Teubner Verlagsgesellschaft, (1990).   Google Scholar

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E. V. Grigorieva and E. N. Khailov, Minimization of pollution concentration on a given time interval for the waste water cleaning plant,, J. Control Sci. Eng. Article, 2010 (2010), 1.  doi: 10.1155/2010/712794.  Google Scholar

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B. Houska, H. J. Ferreau and M. Diehl, ACADO toolkit - An open-source framework for automatic control and dynamic optimization,, Optim. Contr. Appl. Met., 32 (2010), 298.  doi: 10.1002/oca.939.  Google Scholar

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I. Ivanovic and T. O. Leiknes, Particle separation in Moving Bed Biofilm Reactor: Applications and opportunities,, Separ. Sci. Technol., 47 (2012), 647.  doi: 10.1080/01496395.2011.639590.  Google Scholar

[16]

J. Jahn, Vector Optimization: Theory, Applications and Extensions,, 2nd edition, (2011).  doi: 10.1007/978-3-540-24828-6.  Google Scholar

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T. L. Johnson, J. P. McQuarrie and A. R. Shaw, Integrated Fixed-film Activated Sludge (IFAS): The new choice for nitrogen removal upgrades in the United States,, Proceedings WEFTEC Session, (2004), 296.  doi: 10.2175/193864704784147214.  Google Scholar

[18]

D. Jones, H. V. Kojouharov, D. Le and H. Smith, The Freter model: A simple model of biofilm formation,, J. Math. Biol., 47 (2003), 137.  doi: 10.1007/s00285-003-0202-1.  Google Scholar

[19]

J. B. Kaplan, Biofilm dispersal: Mechanisms, clinical implications, and potential therapeutic uses,, J. Dent. Res., 89 (2010), 205.  doi: 10.1177/0022034509359403.  Google Scholar

[20]

I. Klapper, Productivity and equilibrium in simple biofilm models,, Bull. Math. Biol., 74 (2012), 2917.  doi: 10.1007/s11538-012-9791-4.  Google Scholar

[21]

Z. Lewandowski and H. Beyenal, Fundamentals of Biofilm Research,, CRC Press, (2007).   Google Scholar

[22]

A. Mašić and H. Eberl, Persistence in a single species CSTR model with suspended flocs and wall attached biofilms,, Bull. Math. Biol., 74 (2012), 1001.  doi: 10.1007/s11538-011-9707-8.  Google Scholar

[23]

A. Mašić and H. Eberl, A modeling and simulation study of the role of suspended microbial populations in nitrification in a biofilm reactor,, Bull. Math. Biol., 76 (2014), 27.  doi: 10.1007/s11538-013-9898-2.  Google Scholar

[24]

The Mathworks, MATLAB online documentation, http://www.mathworks.com/help/matlab/,, accessed on January 18, (2013).   Google Scholar

[25]

J. Moreno, Optimal time control of bioreactors for the wastewater treatment,, Optim. Control Appl. Meth., 20 (1999), 145.   Google Scholar

[26]

E. Morgenroth, M. C. M. van Loosdrecht and O. Wanner, Biofilm models for the practitioner,, Water Sci. Technol., 41 (2000), 509.   Google Scholar

[27]

L. S. Pontryagin, N. G. Boltyanski, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Wiley, (1962).   Google Scholar

[28]

B. E. Rittmann and P. L. McCarty, Environmental Biotechnology,, McGraw-Hill, (2001).   Google Scholar

[29]

M. E. Roberts and X.-M. Zhan, Moving-medium Biofilm Reactors,, Rev. Environ. Sci. Biotechnol., 2 (2003), 213.   Google Scholar

[30]

I. M. Ross, A beginner's guide to DIDO: A MATLAB application package for solving optimal control problems,, Elissar Global, (2007).   Google Scholar

[31]

I. Y. Smets and J. F. Van Impe, Optimal control of (bio-)chemical reactors: Generic properties of time and space dependent optimization,, Math. Comput. Simulat., 60 (2002), 475.  doi: 10.1016/S0378-4754(02)00034-4.  Google Scholar

[32]

E. D. Stemmons and H. L. Smith, Competition in a chemostat with wall attachment,, SIAM J. Appl. Math., 61 (2000), 567.  doi: 10.1137/S0036139999358131.  Google Scholar

[33]

B. Szomolay, I. Klapper and M. Dindos, Analysis of adaptive response to dosing protocols for biofilm control,, SIAM J. Appl. Math., 70 (2010), 3175.  doi: 10.1137/080739070.  Google Scholar

[34]

M. von Sperling, Activated Sludge and Aerobic Biofilm Reactors,, IWA Publishing, (2007).   Google Scholar

[35]

W. Walter, Gewöhnliche Differentialgleichungen,, 7th ed., (2000).  doi: 10.1007/978-3-642-57240-1.  Google Scholar

[36]

O. Wanner, H. Eberl, E. Morgenroth, D. R. Noguera, C. Picioreanu, B. Rittmann and M. van Loosdrecht, Mathematical Modeling of Biofilms,, Scientific and Technical Report No.18, (2006).   Google Scholar

[37]

M. I. Zelikin and V. F. Borisov, Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering,, Systems & Control: Foundations & Applications, (1994).  doi: 10.1007/978-1-4612-2702-1.  Google Scholar

show all references

References:
[1]

F. Abbas, R. Sudarsan and H. J. Eberl, Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates,, Math. Biosc. Eng., 9 (2012), 215.  doi: 10.3934/mbe.2012.9.215.  Google Scholar

[2]

F. Abbas and H. J. Eberl, Analytical substrate flux approximation for the Monod boundary value problem,, Appl. Math. Comp., 218 (2011), 1484.  doi: 10.1016/j.amc.2011.05.102.  Google Scholar

[3]

M. M. Ballyk, D. A. Jones and H. L. Smith, The biofilm model of Freter: A review,, in Structured Population Models in Biology and Epidemiology, (1936), 265.  doi: 10.1007/978-3-540-78273-5_6.  Google Scholar

[4]

M. M. Ballyk, D. A. Jones and H. L. Smith, Microbial competition in reactors with wall attachment,, Microb. Ecol., 41 (2001), 210.   Google Scholar

[5]

J. T. Betts, Practical Methods for Optimal Control and Estimation using Nonlinear Programming,, SIAM series Adv. Design and Control, (2010).  doi: 10.1137/1.9780898718577.  Google Scholar

[6]

C. Carathéodory, Vorlesungen über reelle Funktionen,, 3rd edition, (1968).   Google Scholar

[7]

N. G. Cogan, B. Szomolay and M. Dindo, Effect of periodic disinfection on persisters in a one-dimensional biofilm model,, Bull. Math. Biol., 75 (2013), 94.  doi: 10.1007/s11538-012-9796-z.  Google Scholar

[8]

T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms,, 2nd edition, (2001).   Google Scholar

[9]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Springer-Verlag New York Inc, (1975).   Google Scholar

[10]

R. Freter, H. Brickner, J. Fekete, M. Vickerman and K. Carey, Survival and implantation of Escherichia coli in the intestinal tract,, Infect. Immun., 39 (1983), 686.   Google Scholar

[11]

P. Gajardo, J. Harmand, H. Ramírez C. and A. Rapaport, Minimal time bioremediation of natural water resources,, Automatica, 47 (2011), 1764.  doi: 10.1016/j.automatica.2011.03.001.  Google Scholar

[12]

A. Göpfert and R. Nehse, Vektoroptimierung,, BSB Teubner Verlagsgesellschaft, (1990).   Google Scholar

[13]

E. V. Grigorieva and E. N. Khailov, Minimization of pollution concentration on a given time interval for the waste water cleaning plant,, J. Control Sci. Eng. Article, 2010 (2010), 1.  doi: 10.1155/2010/712794.  Google Scholar

[14]

B. Houska, H. J. Ferreau and M. Diehl, ACADO toolkit - An open-source framework for automatic control and dynamic optimization,, Optim. Contr. Appl. Met., 32 (2010), 298.  doi: 10.1002/oca.939.  Google Scholar

[15]

I. Ivanovic and T. O. Leiknes, Particle separation in Moving Bed Biofilm Reactor: Applications and opportunities,, Separ. Sci. Technol., 47 (2012), 647.  doi: 10.1080/01496395.2011.639590.  Google Scholar

[16]

J. Jahn, Vector Optimization: Theory, Applications and Extensions,, 2nd edition, (2011).  doi: 10.1007/978-3-540-24828-6.  Google Scholar

[17]

T. L. Johnson, J. P. McQuarrie and A. R. Shaw, Integrated Fixed-film Activated Sludge (IFAS): The new choice for nitrogen removal upgrades in the United States,, Proceedings WEFTEC Session, (2004), 296.  doi: 10.2175/193864704784147214.  Google Scholar

[18]

D. Jones, H. V. Kojouharov, D. Le and H. Smith, The Freter model: A simple model of biofilm formation,, J. Math. Biol., 47 (2003), 137.  doi: 10.1007/s00285-003-0202-1.  Google Scholar

[19]

J. B. Kaplan, Biofilm dispersal: Mechanisms, clinical implications, and potential therapeutic uses,, J. Dent. Res., 89 (2010), 205.  doi: 10.1177/0022034509359403.  Google Scholar

[20]

I. Klapper, Productivity and equilibrium in simple biofilm models,, Bull. Math. Biol., 74 (2012), 2917.  doi: 10.1007/s11538-012-9791-4.  Google Scholar

[21]

Z. Lewandowski and H. Beyenal, Fundamentals of Biofilm Research,, CRC Press, (2007).   Google Scholar

[22]

A. Mašić and H. Eberl, Persistence in a single species CSTR model with suspended flocs and wall attached biofilms,, Bull. Math. Biol., 74 (2012), 1001.  doi: 10.1007/s11538-011-9707-8.  Google Scholar

[23]

A. Mašić and H. Eberl, A modeling and simulation study of the role of suspended microbial populations in nitrification in a biofilm reactor,, Bull. Math. Biol., 76 (2014), 27.  doi: 10.1007/s11538-013-9898-2.  Google Scholar

[24]

The Mathworks, MATLAB online documentation, http://www.mathworks.com/help/matlab/,, accessed on January 18, (2013).   Google Scholar

[25]

J. Moreno, Optimal time control of bioreactors for the wastewater treatment,, Optim. Control Appl. Meth., 20 (1999), 145.   Google Scholar

[26]

E. Morgenroth, M. C. M. van Loosdrecht and O. Wanner, Biofilm models for the practitioner,, Water Sci. Technol., 41 (2000), 509.   Google Scholar

[27]

L. S. Pontryagin, N. G. Boltyanski, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Wiley, (1962).   Google Scholar

[28]

B. E. Rittmann and P. L. McCarty, Environmental Biotechnology,, McGraw-Hill, (2001).   Google Scholar

[29]

M. E. Roberts and X.-M. Zhan, Moving-medium Biofilm Reactors,, Rev. Environ. Sci. Biotechnol., 2 (2003), 213.   Google Scholar

[30]

I. M. Ross, A beginner's guide to DIDO: A MATLAB application package for solving optimal control problems,, Elissar Global, (2007).   Google Scholar

[31]

I. Y. Smets and J. F. Van Impe, Optimal control of (bio-)chemical reactors: Generic properties of time and space dependent optimization,, Math. Comput. Simulat., 60 (2002), 475.  doi: 10.1016/S0378-4754(02)00034-4.  Google Scholar

[32]

E. D. Stemmons and H. L. Smith, Competition in a chemostat with wall attachment,, SIAM J. Appl. Math., 61 (2000), 567.  doi: 10.1137/S0036139999358131.  Google Scholar

[33]

B. Szomolay, I. Klapper and M. Dindos, Analysis of adaptive response to dosing protocols for biofilm control,, SIAM J. Appl. Math., 70 (2010), 3175.  doi: 10.1137/080739070.  Google Scholar

[34]

M. von Sperling, Activated Sludge and Aerobic Biofilm Reactors,, IWA Publishing, (2007).   Google Scholar

[35]

W. Walter, Gewöhnliche Differentialgleichungen,, 7th ed., (2000).  doi: 10.1007/978-3-642-57240-1.  Google Scholar

[36]

O. Wanner, H. Eberl, E. Morgenroth, D. R. Noguera, C. Picioreanu, B. Rittmann and M. van Loosdrecht, Mathematical Modeling of Biofilms,, Scientific and Technical Report No.18, (2006).   Google Scholar

[37]

M. I. Zelikin and V. F. Borisov, Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering,, Systems & Control: Foundations & Applications, (1994).  doi: 10.1007/978-1-4612-2702-1.  Google Scholar

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