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A 2D model for hydrodynamics and biology coupling applied toalgae growth simulations

Published online by Cambridge University Press:  30 July 2013

Olivier Bernard ,
Anne-Céline Boulanger ,
Marie-Odile Bristeau  and
Jacques Sainte-Marie
Olivier Bernard
Affiliation:
Inria, team BioCore, BP93, 06902 Sophia-Antipolis Cedex, France.. [email protected]
Anne-Céline Boulanger
Affiliation:
Inria, team ANGE, B.P. 105, 78153 Le Chesnay Cedex, France.; [email protected] UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France.; [email protected]
Marie-Odile Bristeau
Affiliation:
Inria, team ANGE, B.P. 105, 78153 Le Chesnay Cedex, France.; [email protected] UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France.; [email protected]
Jacques Sainte-Marie
Affiliation:
Inria, team ANGE, B.P. 105, 78153 Le Chesnay Cedex, France.; [email protected] UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France.; [email protected] CETMEF, 2 boulevard Gambetta, 60200 Compiègne, France.; [email protected]
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Abstract

Cultivating oleaginous microalgae in specific culturing devices such as raceways is seenas a future way to produce biofuel. The complexity of this process coupling non linearbiological activity to hydrodynamics makes the optimization problem very delicate. Thelarge amount of parameters to be taken into account paves the way for a usefulmathematical modeling. Due to the heterogeneity of raceways along the depth dimensionregarding temperature, light intensity or nutrients availability, we adopt a multilayerapproach for hydrodynamics and biology. For free surface hydrodynamics, we use amultilayer Saint–Venant model that allows mass exchanges, forced by a simplifiedrepresentation of the paddlewheel. Then, starting from an improved Droop model thatincludes light effect on algae growth, we derive a similar multilayer system for thebiological part. A kinetic interpretation of the whole system results in an efficientnumerical scheme. We show through numerical simulations in two dimensions that ourapproach is capable of discriminating between situations of mixed water or calm andheterogeneous pond. Moreover, we exhibit that a posteriori treatment ofour velocity fields can provide lagrangian trajectories which are of great interest toassess the actual light pattern perceived by the algal cells and therefore understand itsimpact on the photosynthesis process.

Keywords

Hydrostatic Navier–Stokes equationsSaint–Venant equationsfree surface stratified flowsmultilayer systemkinetic schemedroop modelracewayhydrodynamics and biology couplingalgae growth
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1387 - 1412
Copyright
© EDP Sciences, SMAI, 2013

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References

E Audusse, Modelisation hyperbolique et analyse numerique pour les ecoulements en eaux peu profondes. Ph.D. thesis. Université Pierre et Marie Curie - Paris VI (2004).
Audusse, E. and Bristeau, M.-O., Transport of pollutant in shallow water flows: A two time steps kinetic method. ESAIM: M2AN 37 (2003) 389–416. Google Scholar
Audusse, E. and Bristeau, M.-O., A well-balanced positivity preserving second-order scheme for shallow water flows on unstructured meshes. J. Comput. Phys. 206 (2005) 311–333. Google Scholar
Audusse, E., Bristeau, M.-O., Pelanti, M. and Sainte-Marie, J., Approximation of the hydrostatic Navier-Stokes system for density stratified flows by a multilayer model. Kinetic interpretation and numerical validation. J. Comput. Phys. 230 (2011) 3453–3478. Google Scholar
Audusse, E., Bristeau, M.-O., Perthame, B. and Sainte-Marie, J., A multilayer saint–venant system with mass exchanges for shallow water flows. Derivation and numerical validation. ESAIM: M2AN 45 (2011) 169–200. Google Scholar
Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. and Perthame, Be., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050–2065. Google Scholar
S.-D. Ayata, M. Lévy, O. Aumont, A. Sciandra, J. Sainte-Marie, A. Tagliabue and O. Bernard, Phytoplankton growth formulation in marine ecosystem models: should we take into account photo-acclimation and variable stochiometry in oligotrophic areas? To appear in J. Marine Syst.
Baklouti, M., Diaz, F., Pinazo, C., Faure, V. and Queguiner, B., Investigation of mechanistic formulations depicting phytoplankton dynamics for models of marine pelagic ecosystems and description of a new model. Progr. Oceanogr. 71 (2006) 1–33. Google Scholar
Barré de Saint-Venant, A.-J.-C., Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et làintroduction des marées dans leur lit. Comptes Rendus des Séances de l’Académie des Sciences, Paris 73 (1871) 147–154. Google Scholar
Bernard, O., Hurdles and challenges for modelling and control of microalgae for co2 mitigation and biofuel production. J. Process Control 21 (2011) 1378–1389. Google Scholar
Bernard, O. and Gouzé, J.-L., Transient behavior of biological loop models, with application to the Droop model. Math. Biosci. 127 (1995) 19–43. Google ScholarPubMed
Bernard, O. and Gouzé, J.-L., Global qualitative behavior of a class of nonlinear biological systems: application to the qualitative validation of phytoplankton growth models. Artif. Intel. 136 (2002) 29–59. Google Scholar
A.-C. Boulanger and J. Sainte-Marie, Analytical solutions for the free surface hydrostatic euler equations. Submitted to Nonlinearity (2011).
J.-F. Bourgat, P. Le Tallec, F. Mallinger, B. Perthame, Y. Qiu, C. boltzmann and navier-stokes, Research Report RR-2281, Projet MENUSIN. INRIA (1994).
Bristeau, M.-O. and Sainte-Marie, J., Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems. DCDS(B) 10 (2008) 733–759. Google Scholar
Bristeau, M.-O., Goutal, N. and Sainte-Marie, J., Numerical simulations of a non-hydrostatic shallow water model. Comput. Fluids 47 (2011) 51–64. Google Scholar
Casulli, V., A semi–implicit finite difference method for non-hydrostatic, free–surface flows. Int. J. Numer. Methods Fluids 30 (1999) 425–440. Google Scholar
Chisti, Y., Biodiesel from microalgae. Biotech. Adv. 25 (2007) 294–306. Google ScholarPubMed
Droop, M.R., Vitamin B12 and marine ecology. IV. the kinetics of uptake growth and inhibition in Monochrysis lutheri. J. Mar. Biol. Assoc. 48 (1968) 689–733. Google Scholar
Droop, M.R., 25 years of algal growth kinetics, a personal view. Botanica Marina 16 (1983) 99–112. Google Scholar
Dugdale, R.C., Nutrient limitation in the sea: dynamics, identification and significance. Limnol. Oceanogr. 12 (1967) 685–695. Google Scholar
Esposito, S., Botte, V., Iudicone, D. and Ribera d’Alcala, M., Numerical analysis of cumulative impact of phytoplankton photoresponses to light variation on carbon assimilation. J. Theor. Biol 261 (2009) 361–371. Google Scholar
Geider, R.J., MacIntyre, H.L. and Kana, T.M., A dynamic regulatory model of phytoplanktonic acclimation to light, nutrients, and temperature. Limnol Oceanogr 43 (1998) 679–694. Google Scholar
Gerbeau, J.-F., and Perthame, B., Derivation of viscous saint–venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89–102. Google Scholar
Grobbelaar, J.U., Soeder, C.J. and Stengel, E., Modeling algal productivity in large outdoor cultures and waste treatment systems. Biomass 21 (1990) 297–314. Google Scholar
Guterman, H., Vonshak, A. and Ben-Yaakov, S., A macromodel for outdoor algal mass production. Biotechnol. Bioengineer. 35 (1990) 809–819. Google ScholarPubMed
Han, B.P., Photosynthesis-irradiance response at physiological level: a mechanistic model. J. Theoret. Biol. 213 (2001) 121–127. Google ScholarPubMed
Han, B.P., A mechanistic model of algal photoinhibition induced by photodamage to photosystem-ii. J. Theoret. Biology 214 (2002) 519–527. Google ScholarPubMed
J.-M. Hervouet, Hydrodynamics of Free Surface Flows: Modelling With the Finite Element Method. John Wiley and Sons (2007).
Huggins, D.L., Piedrahita, R.H. and Rumsey, T., Analysis of sediment transport modeling using computational fluid dynamics (cfd) for aquaculture raceways. Aquacult. Engrg. 31 (2004) 277–293. Google Scholar
Huggins, D.L., Piedrahita, R.H. and Rumsey, T., Use of computational fluid dynamics (cfd) for aquaculture raceway design to increase settling effectiveness. Aquacult. Engrg. 33 (2005) 167–180. Google Scholar
James, S.C. and Boriah, V., Modeling algae growth in an open–channel raceway. J Comput. Biol. 17 (2010) 895–906. Google Scholar
B. Khobalatte and B. Perthame, Maximum principle on the entropy and minimal limitations for kinetic schemes. Research Report RR-1628, Projet MENUSIN. INRIA (1992).
Lange, K. and Oyarzun, F.J., The attractiveness of the Droop equations. Math. Biosci. 111 (1992) 261–278. Google ScholarPubMed
Luo, H.-P. and Al-Dahhan, M.H., Analyzing and modeling of photobioreactors by combining first principles of physiology and hydrodynamics. Biotechnol. Bioengineer. 85 (2004) 382–393. Google Scholar
Metting, F.B., Biodiversity and application of microalgae. J. Indust. Microbiol. Biotechnol. 17 (1996) 477–489. Google Scholar
Peeters, J. C. H. and Eilers, P., The relationship between light intensity and photosynthesis: a simple mathematical model. Hydrobiol. Bull. 12 (1978) 134–136. Google Scholar
Perner, I., Posten, C. and Broneske, J., Cfd-aided optimization of a plate photobioreactor for cultivation of microalgae. Chemie Ingenieur Technik 74 (2002) 865–869. Google Scholar
Perner-Nochta, I. and Posten, C., Simulations of light intensity variation in photobioreactors. J. Biotechnol. 131 (2007) 276–285. Google ScholarPubMed
B. Perthame, Kinetic formulation of conservation laws. Oxford lecture series in mathematics and its applications. Oxford University Press (2002).
Pruvost, J., Pottier, L. and Legrand, J., Numerical investigation of hydrodynamic and mixing conditions in a torus photobioreactor. Chemical Engineer. Sci. 61 (2006) 4476–4489. Google Scholar
Rodolfi, L., Zittelli, G.C., Bassi, N., Padovani, G., Biondi, N., Bonini, G. and Tredici, M.R., Microalgae for Oil: Strain Selection, Induction of Lipid Synthesis and Outdoor Mass Cultivation in a Low-Cost Photobioreactor. Biotechnol. Bioeng. 102 (2009) 100–112. Google Scholar
Rosello Sastre, R., Coesgoer, Z., Perner-Nochta, I., Fleck-Schneider, P. and Posten, C., Scale–down of microalgae cultivations in tubular photo-bioreactors – a conceptual approach. J. Biotechnol. 132 (2007) 127–133. Google ScholarPubMed
Sainte-Marie, J., Vertically averaged models for the free surface euler system. derivation and kinetic interpretation. Math. Models Methods Appl. Sci. 21 (2011) 459–490. Google Scholar
Sciandra, A. and Ramani, P., The limitations of continuous cultures with low rates of medium renewal per cell. J. Exp. Mar. Biol. Ecol. 178 (1994) 1–15. Google Scholar
Sukenik, A., Falkowski, P.G. and Bennett, J.. Potential enhancement of photosynthetic energy conversion in algal mass culture. Biotechnol. Bioengineer. 30 (1987) 970–977. Google ScholarPubMed
Sukenik, A., Levy, R.S., Levy, Y., Falkowski, P.G. and Dubinsky, Z., Optimizing algal biomass production in an outdoor pond: a simulation model. J. Appl. Phycol. 3 (1991) 191–201. Google Scholar
Vejrazka, C., Janssen, M., Streefland, M. and Wijffels, R.H., Photosynthetic efficiency of chlamydomonas reinhardtii in flashing light. Biotechnol. Bioengineer. 108 (2011) 2905–2913. Google ScholarPubMed
Wijffels, R.H. and Barbosa, M.J., An outlook on microalgal biofuels. Science 329 (2010) 796–799. Google ScholarPubMed
Williams, P.J.B. and Laurens, L.M.L., Microalgae as biodiesel and biomass feedstocks: Review and analysis of the biochemistry, energetics and economics. Energy Environ. Sci. 3 (2010) 554–590. Google Scholar