Wave-scattering processes: path-integrals designed for the numerical handling of complex geometries.

Relying on Feynman-Kac path-integral methodology, we present a new statistical perspective on wave single-scattering by complex three-dimensional objects. The approach is implemented on three models-Schiff approximation, Born approximation, and rigorous Born series-and familiar interpretative difficulties such as the analysis of moments over scatterer distributions (size, orientation, shape, etc.) are addressed. In terms of the computational contribution, we show that commonly recognized features of the Monte Carlo method with respect to geometric complexity can now be available when solving electromagnetic scattering.

Mode solver based on Gegenbauer polynomial expansion for cylindrical structures with arbitrary cross sections.

We present a modal method for the computation of eigenmodes of cylindrical structures with arbitrary cross sections. These modes are found as eigenvectors of a matrix eigenvalue equation that is obtained by introducing a new coordinate system that takes into account the profile of the cross section. We show that the use of Hertz potentials is suitable for the derivation of this eigenvalue equation and that the modal method based on Gegenbauer expansion (MMGE) is an efficient tool for the numerical solution of this equation. Results are successfully compared for both perfectly conducting and dielectric structures. A complex coordinate version of the MMGE is introduced to solve the dielectric case.

Reformulation of the modal method based on Gegenbauer polynomial expansion for cylindrical structures with arbitrary cross sections.

The work presented here focuses on the numerical modeling of cylindrical structure eigenmodes with an arbitrary cross section using Gegenbauer polynomials. The new eigenvalue equation leads to considerable reduction in computation time compared to the previous formulation. The main idea of this new formulation involves considering that the numerical scheme can be partially separated into two independent parts and the size of the eigenvalue matrix equation may be reduced by a factor of 2. We show that the ratio of the computation times between the first and current versions follows a linear relation with respect to the number of polynomials.

Coupling radiative, conductive and convective heat-transfers in a single Monte Carlo algorithm: A general theoretical framework for linear situations.

It was recently shown that radiation, conduction and convection can be combined within a single Monte Carlo algorithm and that such an algorithm immediately benefits from state-of-the-art computer-graphics advances when dealing with complex geometries. The theoretical foundations that make this coupling possible are fully exposed for the first time, supporting the intuitive pictures of continuous thermal paths that run through the different physics at work. First, the theoretical frameworks of propagators and Green's functions are used to demonstrate that a coupled model involving different physical phenomena can be probabilized. Second, they are extended and made operational using the Feynman-Kac theory and stochastic processes. Finally, the theoretical framework is supported by a new proposal for an approximation of coupled Brownian trajectories compatible with the algorithmic design required by ray-tracing acceleration techniques in highly refined geometry.

Dynamics of long-term continuous culture of Limnospira indica in an air-lift photobioreactor.

MELiSSA (Microecological Life Support System Alternative) is a developing technology for regenerative life support to enable long-term human missions in Space and has developed a demonstration Pilot Plant. One of the components of the MELiSSA Pilot Plant system is an 83L external loop air-lift photobioreactor (PBR) where Limnospira indica (previously named Arthrospira sp. PC8005) is axenically cultivated in a continuous operation mode for long-periods. Its mission is to provide O2 and consume CO2 while producing edible material. Biological and process characterization of this PBR is performed by analysing the effect of two main variables, dilution rate (D) and PFD (Photon Flux Density) illumination. A maximum oxygen productivity ( r O 2 ) of 1.35 mmol l-1 h-1 is obtained at a D of 0.025 h-1 and PFD of 930 µmol m-2 s-1 . Photoinhibition can occur when a 1 g l-1 cell density culture is exposed to PFD higher than 1700 µmol m-2 s-1 . This process is reversible if the illumination is returned to dim light (150 µmol m-2 s-1 ), proving the cell adaptability and capacity to respond at different illumination conditions. Influence of light intensity in cell composition is also described. Specific photon flux density (qPFD) has a direct effect on phycobiliproteins and chlorophyll content causing a decrease of 62.5% and 47.8%, respectively, when qPFD increases from 6.1 to 19.2 µmol g-1 s-1 . The same trend is observed for proteins and the opposite for carbohydrate content. Morphological and spiral structural features of L. indica are studied by confocal microscopy, and size distribution parameters are quantified. A direct effect between trichome width and CDW/OD ratio is observed. Changes in size distribution are not correlated with environmental factors, further confirms the adaptation capacity of the cells. The systematic analysis performed provides valuable insights to understand the key performance criteria of continuous culture in air-lift PBRs.

Radiative analysis of luminescence in photoreactive systems: Application to photosensitizers for solar fuel production.

Most chemical reactions promoted by light and using a photosensitizer (a dye) are subject to the phenomenon of luminescence. Redistribution of light in all directions (isotropic luminescence emission) and in a new spectral range (luminescence emission spectrum) makes experimental and theoretical studies much more complex compared to a situation with a purely absorbing reaction volume. This has a significant impact on the engineering of photoreactors for industrial applications. Future developments associated with photoreactive system optimization are therefore extremely challenging, and require an in-depth description and quantitative analysis of luminescence. In this study, a radiative model describing the effect of luminescence radiation on the calculation of absorptance is presented and analyzed with the multiple inelastic-scattering approach, using Monte Carlo simulations. The formalism of successive orders of scattering expansion is used as a sophisticated analysis tool which provides, when combined with relevant physical approximations, convenient analytical approximate solutions. Its application to four photosensitizers that are representative of renewable hydrogen production via artificial photosynthesis indicates that luminescence has a significant impact on absorptance and on overall quantum yield estimation, with the contribution of multiple scattering and important spectral effects due to inelastic scattering. We show that luminescence cannot be totally neglected in that case, since photon absorption lies at the root of the chemical reaction. We propose two coupled simple and appropriate analytical approximations enabling the estimation of absorptance with a relative error below 6% in every tested situation: the zero-order scattering approximation and the gray single-scattering approximation. Finally, this theoretical approach is used to determine and discuss the overall quantum yield of a bio-inspired photoreactive system with Eosin Y as a photosensitizer, implemented in an experimental setup comprising a photoreactor dedicated to hydrogen production.

Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors.

An in-depth investigation of how various illumination conditions influence microalgal growth in photobioreactors (PBR) has been presented. Effects of both the light emission spectrum (white and red) and the light incident angle (0° and 60°) on the PBR surface were investigated. The experiments were conducted in two fully controlled lab-scale PBRs, a torus PBR and a thin flat-panel PBR for high cell density culture. The results obtained in the torus PBR were used to build the kinetic growth model of Chlorella vulgaris taken as a model species. The PBR model was then applied to the thin flat-panel PBR, which was run with various illumination conditions. Its detailed representation of local rate of photon absorption under various conditions (spectral calculation of light attenuation, incident angle influence) enabled the model to take into account all the tested conditions with no further adjustment. This allowed a detailed investigation of the coupling between radiation field and photosynthetic growth. Effects of all the radiation conditions together with pigment acclimation, which was found to be relevant, were investigated in depth. © 2016 American Institute of Chemical Engineers Biotechnol. Prog., 32:247-261, 2016.

Design, operation, and modeling of a membrane photobioreactor to study the growth of the Cyanobacterium Arthrospira platensis in space conditions.

A membrane photobioreactor was designed, implemented and used to grow the cyanobacterium Arthrospira platensis PCC 8005 in batch mode. Growth was followed directly by monitoring optical density and indirectly by measuring pressure increase due to the oxygen produced and separated from the liquid phase by diffusion through a hydrophobic membrane, and pH increase due to carbon consumption. When the pressure attained an upper limit, valves opened automatically, and the oxygen in the gas chamber was flushed out with nitrogen. As expected, two growth phases were observed, a short exponential phase followed by a linear phase, indicating limitation by light transfer. Growth rate during the second phase was measured easily and accurately, and consistency of optical density, pressure and pH data values was checked using a model of the system. Pressure measurement was found best suited to monitoring and measuring growth rate in space in terms of accuracy, precision and reliability.

Modeling stability of photoheterotrophic continuous cultures in photobioreactors.

Continuous cultures of the purple non-sulfur bacterium Rhodospirillum rubrum were grown in a cylindrical photobioreactor in photoheterotrophic conditions, using acetate as carbon source. A new kinetic and stoichiometric knowledge model was developed, and its ability to simulate experimental results obtained under varying incident light fluxes and residence times is discussed. The model accurately predicts the stable, unstable, or oscillating behavior observed for the reactor productivity. In particular, the values of residence time corresponding to a subcritical bifurcation with a typical hysteresis effect are calculated and analyzed. The robustness of the proposed model allows the engineering operating domain of the photobioreactor function to be set and offers a promising tool for the design and control of such photoheterotrophic processes.

Kinetic modeling of the photosynthetic growth of Chlamydomonas reinhardtii in a photobioreactor.

The aim of this study was to establish and validate a model for the photosynthetic growth of Chlamydomonas reinhardtii in photobioreactors (PBRs). The proposed model is based on an energetic analysis of the excitation energy transfer in the photosynthesis apparatus (the Z-scheme for photosynthesis). This approach has already been validated in cyanobacteria (Arthorspira platensis) and is extended here to predict the volumetric biomass productivity for the microalga C. reinhardtii in autotrophic conditions, taking into consideration the two metabolic processes taking place in this eukaryotic microorganism, namely photosynthesis and respiration. The kinetic growth model obtained was then coupled to a radiative transfer model (the two-flux model) to determine the local kinetics, and thereby the volumetric biomass productivity, in a torus PBR. The model was found to predict PBR performances accurately for a broad set of operating conditions, including both light-limited and kinetic growth regimes, with a variance of less than 10% between experimental results and simulations.

A model-based method for investigating bioenergetic processes in autotrophically growing eukaryotic microalgae: application to the green algae Chlamydomonas reinhardtii.

A constraint-based modeling approach was developed to investigate the metabolic response of the eukaryotic microalgae Chlamydomonas reinhardtii under photoautotrophic conditions. The model explicitly includes thermodynamic and energetic constraints on the functioning metabolism. A mixed integer linear programming method was used to determine the optimal flux distributions with regard to this set of constraints. It enabled us, in particular, to highlight the existence of a light-driven respiration depending on the incident photon flux density in photobioreactors functioning in physical light limitation.

Experimental and theoretical assessment of maximum productivities for the microalgae Chlamydomonas reinhardtii in two different geometries of photobioreactors.

The validity of a simple, reliable, and useful recently published formula enabling to calculate the maximum volumetric biomass productivities in photobioreactors (PBRs) was investigated through the cultivation of the microalga Chlamydomonas reinhardtii. Experimental maximum kinetic performances accurately obtained in two different, artificially lightened torus-plane and cylindrical reactors having the same specific illuminated area confirmed the availability, power, and robustness of such formula. The predictive kinetic parameters previously proposed and validated with cyanobacteria were then proved general and robust in case of eukaryotic microalgae, as postulated in the founding article. In this case, an additional criterion requiring rigorous control of the working illuminated fraction gamma = 1 +/- (15%) inside the reactor is demonstrated. For this, the usefulness and reliability of a generalized two-flux model accurately describing the radiation field inside turbid culture media of C. reinhardtii were also established in this article. These important results contribute to identify the main engineering factors governing light-limited PBRs functioning and then to clarify some misinterpretations widely reported in the literature. Together with the referenced previous work, this article gives a framework toward optimal conception of PBRs on a strong physical basis.

A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors.

This article establishes and discusses the consistency and the range of applicability of a simple but general and predictive analytical formula, enabling to easily assess the maximum volumetric biomass growth rates (the productivities) in several kinds of photobioreactors with more or less 15% of deviation. Experimental validations are performed on photobioreactors of very different conceptions and designs, cultivating the cyanobacterium Arthrospira platensis, on a wide range of volumes and hemispherical incident light fluxes. The practical usefulness of the proposed formula is demonstrated by the fact that it appears completely independent of the characteristics of the material phase (as the type of reactor, the kind of mixing, the biomass concentration...), according to the first principle of thermodynamics and to the Gauss-Ostrogradsky theorem. Its ability to give the maximum (only) kinetic performance of photobioreactors cultivating many different photoautotrophic strains (cyanobacteria, green algae, eukaryotic microalgae) is theoretically discussed but experimental results are reported to a future work of the authors or to any other contribution arising from the scientific community working in the field of photobioreactor engineering and potentially interested by this approach.

Spectral kinetic modeling and long-term behavior assessment of Arthrospira platensis growth in photobioreactor under red (620 nm) light illumination.

The ability to cultivate the cyanobacterium Arhtrospira platensis in artificially lightened photobioreactors using high energetic efficiency (quasi-monochromatic) red LED was investigated. To reach the same maximal productivities as with the polychromatic lightening control conditions (red + blue, P/2e(-) = 1.275), the need to work with an optimal range of wavelength around 620 nm was first established on batch and continuous cultures. The long-term physiological and kinetic behavior was then verified in a continuous photobioreactor illuminated only with red (620 nm) LED, showing that the maximum productivities can be maintained over 30 residence times with only minor changes in the pigment content of the cells corresponding to a well-known adaptation mechanism of the photosystems, but without any effect on growth and stoichiometry. For both poly and monochromatic incident light inputs, a predictive spectral knowledge model was proposed and validated for the first time, allowing the calculation of the kinetics and stoichiometry observed in any photobioreactor cultivating A. platensis, or other cyanobacteria if the parameters were updated. It is shown that the photon flux (with a specified wavelength) must be used instead of light energy flux as a relevant control variable for the growth. The experimental and theoretical results obtained in this study demonstrate that it is possible to save the energy consumed by the lightening device of photobioreactors using red LED, the spectral range of which is defined according to the action spectrum of photosynthesis. This appears to be crucial information for applications in which the energy must be rationalized, as it is the case for life support systems in closed environments like a permanent spatial base or a submarine.

Dynamic aspects and controllability of the MELiSSA project: a bioregenerative system to provide life support in space.

Manmade ecosystems differ from their prototype biosphere by the principle of control. The Earth Biosphere is sustainable by stochastic control and very large time constants. By contrast, in a closed ecosystem such as the micro-ecological life support system alternative (MELiSSA system) developed by the European Space Agency for space exploration, a deterministic control is a prerequisite of sustainable existence. MELiSSA is an integrated sum of interconnected biological subsystems. On one hand, all unit operations in charge of the elementary functions constitutive of the entire life support system are studied until a thorough understanding and mathematical modelling. On the other hand, the systemic approach of complex, highly branched systems with feedback loops is performed. This leads to study in the same perspective, with the same degree of accuracy and with the same language, waste degradation, water recycling, atmosphere revitalisation and food production systems prior to the integration of knowledge-based control models. This paper presents the mathematical modelling of the MELiSSA system and the interface between the control strategy of the entire system and the control of the bioreactors.

Modelling continuous culture of Rhodospirillum rubrum in photobioreactor under light limited conditions.

Rhodospirillum rubrum was grown continuously and photoheterotrophically under light limitation using a cylindrical photobioreactor in which the steady state biomass concentration was varied between 0.4 to 4 kg m(-3) at a constant radiant incident flux of 100 W m(-2). Kinetic and stoichiometric models for the growth are proposed. The biomass productivities, acetate consumption rate and the CO2 production rate can be quantitatively predicted to a high level of accuracy by the proposed model calculations.