Electron. J. Differential Equations,
Vol. 2019 (2019), No. 84, pp. 1-26.
Shape optimization of spatial chemostat models
Maria Crespo, Benjamin Ivorra, Angel Manuel Ramos, Alain Rapaport
Abstract:
In this work, we study the shape optimization of a continuous bioreactor in which a
substrate is degraded by a microbial ecosystem in a nonhomogeneous environment.
The bioreactor considered here is a three-dimensional vertically oriented
cylindrical tank. The behavior of reactants is described with a spatial
chemostat model based on an Advection-Diffusion-Reaction system while the
fluid flow is modeled using incompressible Navier-Stokes equations.
We consider that the reaction rate between biomass and nutrient
shows either monotonic or non-monotonic behavior. We tackle an optimization problem
which aims to minimize the considered total reactor volume, with an output concentration
(at stationary state) maintained below a desired threshold, by choosing a suitable
bioreactor shape. We propose a methodology to create three different discrete
parametrizations of the bioreactor geometry and obtain the optimized shapes with
the help of a Hybrid Genetic Algorithm. We show that the optimized reactors
exhibit height much larger than width and their exterior wall is concavely curved
(the concavity at the upper part of the exterior wall being more pronounced for
non-monotonic functions).
Submitted February 1, 2019. Published July 11, 2019.
Math Subject Classifications: 35Q93, 35K40, 65K10, 92E20.
Key Words: Shape optimization; optimal design; continuous bioreactor;
spatial chemostat model; advection-diffusion-reaction;
hybrid genetic algorithm.
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María Crespo
Departamento de Matematica Aplicada a la Ingeniería Industrial
ETSII - Universidad Politécnica de Madrid
C José Gutiérrez Abascal 2, 28006 Madrid, Spain
email: maria.crespo@upm.es, Phone +34 913377370
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Benjamin Ivorra
Instituto de Matemática Interdisciplinar &
Departamento de Análisis Matemático y Matemática Aplicada
Universidad Complutense de Madrid
Plaza de Ciencias, 3, 28040 Madrid, Spain
email: ivorra@ucm.es
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Angel Manuel Ramos
Instituto de Matemática Interdisciplinar &
Departamento de Análisis Matemático y Matemática Aplicada
Universidad Complutense de Madrid
Plaza de Ciencias, 3, 28040 Madrid, Spain
email: angel@mat.ucm.es
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Alain Rapaport
MISTEA, Université Montpellier
INRA, Montpellier SupAgro. 2 Place P.Viala
34060 Montpellier, France
email: rapaport@supagro.inra.fr
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