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Register a new userOptimal optical conditions for Microalgal production in photobioreactors
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Added by
Liu-Di Lu
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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The potential of industrial applications for microalgae has motivated their recent fast development. Their growth dynamics depends on different factors that must be optimized. Since they get their energy from photosynthesis, light is a key factor that strongly influences their productivity. Light is absorbed and scattered in the liquid medium, and irradiance exponentially decreases towards the darkest part of the photobioreactor at a rate nonlinearly depending on the biomass concentration. Maximizing productivity is then a tricky problem, especially when the growth rate is inhibited by an excess of light. Productivity optimization turns out to be highly dependent on how light is distributed along the reactor, and is therefore related to the extinction rate and the background turbidity. The concept of optical depth productivity is introduced for systems where background turbidity must be accounted for and a global optimum maximizing productivity is proposed, extending the concept of the compensation condition. This optimal condition consists in compensating the algal growth rate at the bottom of the reactor by the respiration. This condition can drive the optimization of the surface biomass productivity depending on the minimum reachable depth. We develop a nonlinear controller and prove the global asymptotic stability of the biomass concentration towards the desired optimal value.
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This paper focuses on mixing strategies to enhance the growth rate in an algal raceway system. A mixing device, such as a paddle wheel, is considered to control the rearrangement of the depth of the algae cultures hence the light perceived at each lap. The dynamics of the photosystems after a rearrangement is accounted for by the Han model. Our approach consists in considering permanent regimes where the strategy is parametrized by a permutation matrix which modifies the order of the layers at the beginning of each lap. It is proven that the dynamics of the photosystems is then periodic, with a period corresponding to one lap of the raceway whatever the order of the considered permutation matrix is. An objective function related to the average growth rate over one lap is then introduced. Since N ! permutations (N being the number of considered layers) need to be tested in the general case, it can be numerically solved only for a limited number of layers. Consequently, we propose a second optimization problem associated with a suboptimal solution of the initial problem, which can be determined explicitly. A sufficient condition to characterize cases where the two problems have the same solution is given. Some numerical experiments are performed to assess the benefit of optimal strategies in various settings.
We consider a coupled physical-biological model describing growth of microalgae in a raceway pond cultivation process, accounting for hydrodynamics. Our approach combines a biological model (based on the Han model) and shallow water dynamics equations that model the fluid into the raceway pond. We present an optimization procedure dealing with the topography to maximize the biomass production over one lap or multiple laps with a paddle wheel. On the contrary to a widespread belief in the microalgae field, the results show that a flat topography is optimal in a periodic regime. In other frameworks, non-trivial topographies can be obtained. We present some of them, e.g., when a mixing device is included in the model.
This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method.
The SOR-like iteration method for solving the absolute value equations~(AVE) of finding a vector $x$ such that $Ax - |x| - b = 0$ with $ u = |A^{-1}|_2 < 1$ is investigated. The convergence conditions of the SOR-like iteration method proposed by Ke and Ma ([{em Appl. Math. Comput.}, 311:195--202, 2017]) are revisited and a new proof is given, which exhibits some insights in determining the convergent region and the optimal iteration parameter. Along this line, the optimal parameter which minimizes $|T_ u(omega)|_2$ with $$T_ u(omega) = left(begin{array}{cc} |1-omega| & omega^2 u |1-omega| & |1-omega| +omega^2 u end{array}right)$$ and the approximate optimal parameter which minimizes $eta_{ u}(omega) =max{|1-omega|, uomega^2}$ are explored. The optimal and approximate optimal parameters are iteration-independent and the bigger value of $ u$ is, the smaller convergent region of the iteration parameter $omega$ is. Numerical results are presented to demonstrate that the SOR-like iteration method with the optimal parameter is superior to that with the approximate optimal parameter proposed by Guo, Wu and Li ([{em Appl. Math. Lett.}, 97:107--113, 2019]). In some situation, the SOR-like itration method with the optimal parameter performs better, in terms of CPU time, than the generalized Newton method (Mangasarian, [{em Optim. Lett.}, 3:101--108, 2009]) for solving the AVE.
In this article we study the numerical solution of the $L^1$-Optimal Transport Problem on 2D surfaces embedded in $R^3$, via the DMK formulation introduced in [FaccaCardinPutti:2018]. We extend from the Euclidean into the Riemannian setting the DMK model and conjecture the equivalence with the solution Monge-Kantorovich equations, a PDE-based formulation of the $L^1$-Optimal Transport Problem. We generalize the numerical method proposed in [FaccaCardinPutti:2018,FaccaDaneriCardinPutti:2020] to 2D surfaces embedded in $REAL^3$ using the Surface Finite Element Model approach to approximate the Laplace-Beltrami equation arising from the model. We test the accuracy and efficiency of the proposed numerical scheme, comparing our approximate solution with respect to an exact solution on a 2D sphere. The results show that the numerical scheme is efficient, robust, and more accurate with respect to other numerical schemes presented in the literature for the solution of ls$L^1$-Optimal Transport Problem on 2D surfaces.
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