A mathematical model involving the syntrophic relationship of two major populations of bacteria (acetogens and methanogens), each responsible for a stage of the methane fermentation process is proposed. A detailed qualitative analysis is carried out. The local and global stability analyses of the equilibria are performed. We demonstrate, under general assumptions of monotonicity, relevant from an applied point of view, the global asymptotic stability of a positive equilibrium point which corresponds to the coexistence of acetogenic and methanogenic bacteria.

This study critically analyses filtration process modeling in membrane bioreactor (MBR) technology. More specifically, the variety of approaches and assumptions considered within a curated selection of resistance-in-series (RIS) filtration models found in the literature is critically assessed. Aimed to move towards good filtration process modeling practices, the basis for establishing a unified framework rooted in the fundamentals of membrane fouling is defined in this work, considering fouling classifications, process dynamics, and underlying processes used by different authors for elucidating membrane fouling phenomena. Systematically analyzing these factors should be considered as a basic step for efficiently comparing the performance of different models. This involves a detailed examination of the processes applied within each model and their interplay with the involved resistances and fouling types. A lack of homogeneity in RIS-based filtration modeling has been observed. To address this, basic guidelines towards good modeling practices are proposed aimed at balancing model accuracy and complexity. Specifically, seven model processes, six resistances, and three subgroups for types of fouling, further divided into four or five categories are proposed to guide the selection of processes and state variables in the model structure. Hence, this study facilitates the understanding of different approaches to be used during the modeling exercise of membrane filtration processes within the MBR field, not only to enhance the comprehensibility of available filtration models, but also to help the comparison, implementation, and adaptation of available models and the comprehensive development of new ones.

In this work, we are interested in a reduced and simplified model of the anaerobic digestion process. We focus on the acetogenesis and hydrogenetrophic methanogenesis phases. The model describes a syntrophic relashionship between two microbial species (the acetogenic bacteria and the hydrogenetrophic methanogenic bacteria) with two input substrates (the fatty acids and the hydrogen) including both decay terms and inhibition of the acetogenic bacteria growth by an excess of hydrogen in the system. The existence and stability analysis of the steady states of the model points out the existence of a new equilibrium point which can be stable according to the operating parameters of the system. By means of operating diagrams, we show that, whatever the region of space considered, there exists only one locally exponentially stable steady state.

A two-step model of the anaerobic digestion process is mathematically and numerically studied. The focus of the paper is put on the hydrolysis and methanogenesis phases when applied to the digestion of waste with a high content of solid matter: existence and stability properties of the equilibrium points are investigated. The hydrolysis step is considered a limiting step in this process using the Contois growth function for the bacteria responsible for the first degradation step. The methanogenesis step being inhibited by the product of the first reaction (which is also the substrate for the second one), and the Haldane growth rate is used for the second reaction step. The operating diagrams with respect to the dilution rate and the input substrate concentrations are established and discussed.

As world demand for clean water increases, reverse osmosis (RO) desalination has emerged as an attractive solution. Continuous RO is the most used desalination technology today. However, a new generation of configurations, working in unsteady-state feed concentration and pressure, have gained more attention recently, including the batch RO process. Our work presents a mathematical modeling for batch RO that offers the possibility of monitoring all variables of the process, including specific energy consumption, as a function of time and the recovery ratio. Validation is achieved by comparison with data from the experimental set-up and an existing model in the literature. Energetic comparison with continuous RO processes confirms that batch RO can be more energy efficient than can continuous RO, especially at a higher recovery ratio. It used, at recovery, 31% less energy for seawater and 19% less energy for brackish water. Modeling also proves that the batch RO process does not have to function under constant flux to deliver good energetic performance. In fact, under a linear pressure profile, batch RO can still deliver better energetic performance than can a continuous configuration. The parameters analysis shows that salinity, pump and energy recovery devices efficiencies are directly linked to the energy demand. While increasing feed volume has a limited effect after a certain volume due to dilution, it also shows, interestingly, a recovery ratio interval in which feed volume does not affect specific energy consumption.

One of Tunisia’s main challenges is to conserve and protect water resources for current and future generations. Using non-conventional water in agriculture, such as treated wastewater, can be a sustainable water-saving solution. Therefore, the objectives of this study are (i) to analyze the value chain of treated wastewater for olive growing farms production and (ii) to apply mathematical modeling to maximize the olive production in optimizing irrigation distribution and nitrogen amendment in olive growing farms. The work is carried out in a peri-urban irrigated perimeter of Msaken, Sousse, which is mainly occupied by olive trees and irrigated by treated wastewater. A SWOT analysis is also applied to identify the strengths and weaknesses of reuse in this study area. Moreover, mathematical models are used to determine the optimal schedule for fertigation with treated wastewater. In this process, data on rainfall, soil, water quality and olive production were collected from local farmers, local decision makers, field and laboratory experiments. SWOT results determine farmers’ perceptions of reusing treated wastewater for irrigation. The viability analysis, in terms of soil humidity and nitrogen, shows that the nitrogen stress is not a limiting factor for olive biomass production, but water stress is. This analysis provides numerical values for the maximum irrigation rate and total amount of irrigation water to ensure maximum olive production. It was found that the maximum irrigation could be 5.77 m³/day/ha and the total annual water requirement is 1240 m³/ha. Modeling appears to be an important tool to help local decision makers and to support and encourage local farmers to reuse treated wastewater under safe conditions and without environmental risks.