Mathematical Modeling and Optimization for Biogas Production

Abstract : Anaerobic digestion is a biological process in which organic compounds are degraded by different microbial populations into biogas (carbon dioxyde and methane), which can be used as a renewable energy source. This thesis works towards developing control strategies and bioreactor designs that maximize biogas production.The first part focuses on the optimal control problem of maximizing biogas production in a chemostat in several directions. We consider the single reaction model and the dilution rate is the controlled variable.For the finite horizon problem, we study feedback controllers similar to those used in practice and consisting in driving the reactor towards a given substrate level and maintaining it there. Our approach relies on establishing bounds of the unknown value function by considering different rewards for which the optimal solution has an explicit optimal feedback that is time-independent. In particular, this technique provides explicit bounds on the sub-optimality of the studied controllers for a broad class of substrate and biomass dependent growth rate functions. With numerical simulations, we show that the choice of the best feedback depends on the time horizon and initial condition.Next, we consider the problem over an infinite horizon, for averaged and discounted rewards. We show that, when the discount rate goes to 0, the value function of the discounted problem converges and that the limit is equal to the value function for the averaged reward. We identify a set of optimal solutions for the limit and averaged problems as the controls that drive the system towards a state that maximizes the biogas flow rate on an special invariant set.We then return to the problem over a fixed finite horizon and with the Pontryagin Maximum Principle, we show that the optimal control has a bang singular arc structure. We construct a one parameter family of extremal controls that depend on the constant value of the Hamiltonian. Using the Hamilton-Jacobi-Bellman equation, we identify the optimal control as the extremal associated with the value of the Hamiltonian which satisfies a fixed point equation. We then propose a numerical algorithm to compute the optimal control by solving this fixed point equation. We illustrate this method with the two major types of growth functions of Monod and Haldane.In the second part, we investigate the impact of mixing the reacting medium on biogas production. For this we introduce a model of a pilot scale upflow fixed bed bioreactor that offers a representation of spatial features. This model takes advantage of reactor geometry to reduce the spatial dimension of the section containing the fixed bed and in other sections, we consider the 3D steady-state Navier-Stokes equations for the fluid dynamics. To represent the biological activity, we use a 2 step model and for the substrates, advection-diffusion-reaction equations. We only consider the biomasses that are attached in the fixed bed section and we model their growth with a density dependent function. We show that this model can reproduce the spatial gradient of experimental data and helps to better understand the internal dynamics of the reactor. In particular, numerical simulations indicate that with less mixing, the reactor is more efficient, removing more organic matter and producing more biogas.
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Antoine Haddon. Mathematical Modeling and Optimization for Biogas Production. General Mathematics [math.GM]. Université Montpellier; Universidad de Chile, 2019. English. ⟨NNT : 2019MONTS047⟩. ⟨tel-02478779⟩

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