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This work proposes a mathematical model on partial nitritation/anammox (PN/A) granular bioreactors, with a particular interest in the start-up phase. The formation and growth of granular biofilms is modelled by a spherical free boundary problem with radial symmetry and vanishing initial value. Hyperbolic PDEs describe the advective transport and growth of sessile species inhabiting the granules. Parabolic PDEs describe the diffusive transport and conversion of soluble substrates, and the invasion process mediated by planktonic species. Attachment and detachment phenomena are modelled as continuous and deterministic fluxes at the biofilm-bulk liquid interface. The dynamics of planktonic species and substrates within the bulk liquid are modelled through ODEs. A simulation study is performed to describe the start-up process of PN/A granular systems and the development of anammox granules. The aim is to investigate the role that the invasion process of anaerobic ammonia-oxidizing (anammox) bacteria plays in the formation of anammox granules and explore how it affects the microbial species distribution of anaerobic ammonia-oxidizing, aerobic ammonia-oxidizing, nitrite-oxidizing and heterotrophic bacteria. Moreover, the model is used to study the role of two key parameters in the start-up process: the anammox inoculum size and the inoculum addition time. Numerical results confirm that the model can be used to simulate the start-up process of PN/A granular systems and to predict the evolution of anammox granular biofilms, including the ecology and the microbial composition. In conclusion, after being calibrated, the proposed model could provide quantitatively reliable results and support the start-up procedures of full-scale PN/A granular reactors.

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A local sensitivity analysis was performed for a S0-driven two-step denitrification model, accounting for NO2 - accumulation, biomass growth and S0 solubilization. The model sensitivity was aimed at verifying the model stability, understanding the identifiability of the model structure and evaluating the model parameters to be further optimized. The sensitivity analysis identified the mass specific area of the sulfur particles (a*) and hydrolysis kinetic constant (k1) as the dominant parameters. Additionally, the maximum growth rate of the denitrifying biomass on NO3 - (μmax 2,3) and NO2 - (μmax 2,4) were detected as the most sensitive kinetic parameters. Further calibration would be performed for the sensitive model parameters to optimize the quality of the model.

This dissertation relates to the applications of a one-dimensional mathematical model for multispecies biofilm formation and growth. The model consists of a system of nonlinear hyperbolic partial differential equations, describing the growth of microbial species in biofilms, and a system of semilinear parabolic partial differential equations, which governs substrate diffusion from the surrounding aqueous phase into the biofilm. Overall, this leads to a free boundary value problem, essentially hyperbolic. In a first study, the analysis and simulations of the initial phase of biofilm growth have been addressed. The resulting mathematical problem has been discussed by using the method of characteristics and the fixed-point theorem has been used to obtain existence, uniqueness and properties of solutions. A second aspect of the thesis deals with the analysis and prediction of population dynamics in multispecies biofilms for wastewater treatment. The model has been applied to simulate the bacterial competition and to evaluate the influence of substrate diffusion on microbial stratification for a nitrifying multispecies biofilm including Anammox bacteria and a sulfate-reducing biofilm. In both cases, the method of characteristics has been used for numerical purposes and the mass conservation equation plays a crucial role in checking the accuracy of simulations. The simulation results reveal that the model is able to evaluate properly the effects that boundary conditions exert on bacterial competition. Finally, the biofilm model has been extended to include the colonization phenomenon. The new model is able to take into account the invasion of new species diffusing from bulk liquid to biofilm, still based on a set of nonlinear hyperbolic partial differential equations for what concerns growth process. Indeed, the biological invasion process of new species into the biofilm has been modeled by a system of nonlinear parabolic partial differential equations. The invasion model has been successfully applied to simulate the invasion of heterotrophic bacteria in a constituted autotrophic biofilm and viceversa; Cette thèse s'intéresse à l'application d'un modèle mathématique unidimensionnel de formation et de croissance de biofilms multi-espèces. Le modèle se compose d'un système d'équations non linéaires aux dérivées partielles hyperboliques, décrivant la croissance d'espèces microbiennes dans le biofilm, et un système d'équations semi-linéaires aux dérivées partielles paraboliques, qui régit la diffusion de substrat de la phase aqueuse vers la matrice du biofilm. L'ensemble conduit à un problème de valeur limite libre, essentiellement hyperbolique. Dans une première étude, l'analyse et la simulation de la phase initiale de croissance du biofilm ont été examinées. Le problème mathématique résultant a été discuté en utilisant la méthode des caractéristiques et le théorème du point fixe a été utilisé pour déterminer l'existence et l'unicité des solutions mathématiques. Un deuxième aspect de la thèse porte sur l'analyse et la prévision de la dynamique des populations microbienne dans plusieurs types biofilms pour le traitement des eaux usées. Le modèle a été appliqué pour simuler la compétition bactérienne et évaluer l'influence de la diffusion du substrat sur la stratification microbienne des biofilms multi-espèces, en incluant les bactéries nitrifiantes, Anammox et bactéries sulfato-réductrices. Dans les deux cas, la méthode des caractéristiques a été utilisée à des fins numériques et l'équation de conservation de masse joue un rôle crucial pour vérifier l'exactitude des simulations. Les résultats des simulations montrent que le modèle est en mesure d'évaluer correctement les effets des conditions limites qui s'exercent sur la concurrence bactérienne. Enfin, ce modèle a été étendu pour inclure le phénomène de colonisation microbienne. Le nouveau modèle est capable de prendre en compte l'invasion de nouvelles espèces en se basant sur un ensemble d'équations non linéaires aux dérivées partielles hyperboliques pour ce qui concerne le processus de croissance. De plus, le processus d'invasion biologique d'espèces nouvelles dans le biofilm a été modélisé par un système d'équations non linéaires aux dérivées partielles paraboliques. Ce modèle d'invasion a été appliqué avec succès pour simuler l'invasion des bactéries hétérotrophes dans les biofilms autotrophes

The aim of this study was to develop a mathematical model to assess the effect of soluble microbial products production and external carbon source addition on the performance of a sulfur-driven autotrophic denitrification (SdAD) process. During SdAD, the growth of autotrophic biomass (AUT) was accompanied by the proliferation of heterotrophic biomass mainly consisting of heterotrophic denitrifiers (HD) and sulfate-reducing bacteria (SRB), which are able to grow on both the SMP derived from the microbial activities and on an external carbon source. The process was supposed to occur in a sequencing batch reactor to investigate the effects of the COD injection on both heterotrophic species and to enhance the production and consumption of SMP. The mathematical model was built on mass balance considerations and consists of a system of nonlinear impulsive differential equations, which have been solved numerically. Different simulation scenarios have been investigated by varying the main operational parameters: cycle duration, day of COD injection and quantity of COD injected. For cycle durations of more than 15 days and a COD injection after the half-cycle duration, SdAD represents the prevailing process and the SRB represent the main heterotrophic family. For shorter cycle duration and COD injections earlier than the middle of the cycle, the same performance can be achieved increasing the quantity of COD added, which results in an increased activity of HD. In all the performed simulation even in the case of COD addition, AUT remain the prevailing microbial family in the reactor.

Nitrate represents one of the most common pollutants for groundwater and industrial wastewaters. In recent years, many studies demonstrated that sulfur-based autotrophic denitrification with elemental sulfur represents a valid solution for the removal of nitrate in case of lack of organic substances. However, in presence of organic carbon the autotrophic species compete with heterotrophic denitrifying bacteria for nitrate and nitrite. Moreover, the formation of sulfate induced by the autotrophic denitrification may lead to the growth of sulfate reducing bacteria, which compete with denitrifying bacteria for organic carbon. These microbial interactions result in both beneficial and controversial effects. For instance, heterotrophic denitrification could induce a further reduction of nitrate in the effluent; while sulfate reducing bacteria could contribute to the reduction of sulfate concentration in the effluent and the consequent recovery of elemental sulfur. The aim of the work is to introduce a mathematical model to assess the competition between the autotrophic denitrifying bacteria, the heterotrophic denitrifying bacteria and the sulfate reducing bacteria. The heterotrophic ones are supposed to grow on soluble microbial products, released during the mentioned biological processes. The mathematical model is defined based on mass balance considerations and leads to consider a system of impulsive differential equations, which are solved numerically by implementing an original software on the Matlab platform. Numerical simulations are performed to assess process efficiency by varying different operational parameters.

The work presents a contribution to the mathematical modelling of formation and growth of multispecies biofilms in the framework of continuum approach, without claiming to be complete. Mathematical models for biofilms often lead to consider free boundary value problems for nonlinear PDEs. The emphasis is on the qualitative analysis, uniqueness and existence of solutions and their main properties. Biofilm life is a complex biological process formed by several phases from the formation, development of colonies, attachment and detachment of microbial mass from (to) biofilm to (from) bulk liquid. Most of these processes are modelled and discussed. Moreover, some problems of interest for engineering and biological applications are considered. Indeed, we discuss the free boundary value problem related to biofilm reactors extensively used in wastewater treatment, and the invasion of new species into an already constituted biofilm with the successive colonizations. The main mathematical methodology used is the method of characteristics. The original differential problem is converted to integral equations. Then, the fixed point theorem is applied.

International audience; In this study, a new model based on anaerobic digestion model no. 1 (ADM1) approach has been proposed to simulate trace elements (TEs) complexation, precipitation and their effect on the anaerobic batch methane production. TEs complexation reactions with VFAs and EDTA have been incorporated in an extended ADM1 model which considers TE precipitation/dissolution reactions as well as biodegradation processes. The kinetic model tracks the dynamics of 90 state variables which constitute the components of the proposed anaerobic digestion (AD) model. The incorporation of the complexation reactions required the definition of new inorganic components (EDTA species) and new complexation process rates in the ADM1 framework. The charge balance was modified accordingly to consider the effects of the additional components. The new model is able to predict: a) the effect of TE-EDTA/VFA complexation on methane production, and b) the effect of the initial calcium and magnesium concentrations on process performance