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Whence Dynamical Systems
Published in LM Pismen, Working with Dynamical Systems, 2020
Another kind of an industrial chemical reactor is a plug flow reactor (PFR). An ideal PFR is a flow-through reactor, where each volume element of the inlet flow spends in the reactor identical time. The flow elements entering the PFR at different moments of time do not mix, and therefore each one can be considered as an infinitesimal batch reactor. Therefore in a PFR operating in a stationary regime, the clock time is replaced by the residence time that has passed from the moment of entry of a particular flow element. The residence time t can be related to the current spatial position x by the differential relationship dx = v dt, where v is the local flow velocity. Although a batch reactor and a PFR are not at all similar physically, a stationary PFR is described by exactly the same equations as a batch reactor, and we shall consider both together. The similarity is lost when nonstationary operation of a PFR is considered, or when deviations of a real flow pattern in the reactor from the ideal model are taken into account.
Introduction and Overview
Published in John F. Rekus, Complete Confined Spaces Handbook, 2018
Some spaces, like underground environmentally controlled telecommunications equipment chambers, present few, if any, hazards. Conversely, chemical reactors may be plagued with a litany of hazards, including mechanical agitators, steam jackets, nitrogen inerting systems and a network of pipes which could flood unsuspecting workers with solvents, acids, caustics or other hazardous materials. Between these two extremes are innumerable tanks, pits, containers and vessels, each of which presents its own unique set of hazards.
Interactions of mixing and reaction kinetics of depolymerization of cellulose to renewable fuels
Published in Chemical Engineering Communications, 2018
In processes involving synthesis of biofuels, mixing in a chemical reactor is of utmost importance since it increases conversion, yield, and selectivity of a complex chemical or biochemical system. In general, it is a physical phenomena performed to achieve uniform spatial distribution of components in a reaction vessel. In most cases, mixing significantly influences the product formation and distribution rates. It is essential to know the rate and extent of the reaction and diffusion during the spatial distribution of reacting species in the vessel that alter with the mixing effects in the vessel. Therefore, in-depth knowledge of the mixing process is essential to improve and control the process conditions inside the reactor (Gyenis, 1999). Lacey (1954) has described different mixing mechanisms to explain the transfer of components from one location to the other. It was proposed that convective mixing happens because of the transfer of bigger size particles in the mixture from one position to other in the reactor (Lacey, 1954). Mixing due to diffusion of particles was described as the spatial distribution of reacting particles because of the newly formed surfaces (Lacey, 1954), which is similar to molecular diffusion of the components in the reactor. However, random motion of the particles, their dispersion, and interactions with other particles were seen to be analogous to eddy diffusion.
Combined microfluidics and drying processes for the continuous production of micro-/nanoparticles for drug delivery: a review
Published in Drying Technology, 2023
Ankit Patil, Pritam Patil, Sagar Pardeshi, Preena Shrimal, Norma Rebello, Popat B. Mohite, Aniruddha Chatterjee, Arun Mujumdar, Jitendra Naik
Microreactor is a chemical reactor consisting of confinement known as microchannels in which the chemical reaction occurs. Microchannels must be fabricated using PDMS (Poly-Di-Methyl-Siloxane), Glass, Silicon, Quartz, etc. material depending on its chemical compatibility. Among them, glass is widely accepted due to its inertness. The diameter of these confinements is significantly smaller. Thus, microreactors have a large surface-to-volume ratio, which efficiently controls highly exothermic reactions and reaction conditions, i.e., temperature with reasonable safety. It also provides high mass transfer and heat transfer area, a good flow regime, etc.
Optimal control of parabolic differential inclusions in one space dimension
Published in International Journal of Control, 2022
Over the past decades, great progress has been made in various areas of optimal control problems described by ordinary (Aubin, 2009; Ferrara et al., 2019; Frigon, 1998; Jackson & Horn, 1965; Jiang et al., 2019; Kidambi et al., 2020; Mahmudov, 2011, 2015, 2020a, 2020b; Mahmudov & Mardanov, 2020c; Mordukhovich, 2006; Vijayakumar, 2018a, 2018b; Zimenko et al., 2020) and partial differential equations/inclusions (Ahmed-Ali et al., 2016; Aubin, 2009; Augusta et al., 2015; Baker & Christofides, 2000; Cernea, 2001; Cheng et al., 2011; De Blasi & Pianigiani, 2007; Ferrara et al., 2019; Frigon, 1998; Iannizzotto, 2011; Jackson & Horn, 1965; Karafyllis, 2021; Karafyllis et al., 2019a; Karafyllis & Krstic, 2014, 2016a, 2016b; Karafyllis & Krstic, 2019b; Kidambi et al., 2020; Kisielewicz, 2008; Mahmudov, 2011, 2006, 2007, 1990, 1987, 2013a, 2013b; Mazenc & Prieur, 2011; Mophou et al., 2020; Pazy, 1983; Sakthivel et al., 2008; Stafford et al., 2011). A great many problems in economic dynamics, as well as classical problems on optimal control, differential games, and so on, can be reduced to such investigations (Mahmudov, 2011; Mordukhovich, 2006). For one-dimensional parabolic differential equations (PDEs) (Karafyllis & Krstic, 2016b), with disturbances on both boundaries and distributed disturbances, estimates of input data state stability (ISS) in different norms are given. Due to the absence of the Lyapunov ISS functional for boundary perturbations, the proof methodology uses (i) expansion of the solution in terms of eigenfunctions and (ii) a finite difference scheme. The ISS estimate for the sup-norm leads to a refinement of the well-known maximum principle for the heat equation. The paper (Karafyllis, 2021) presents a methodology for the construction of simple Control Lyapunov Functionals for boundary controlled parabolic PDEs. The proposed methodology provides functionals that contain only simple (and not double or triple) integrals of the state. Existence and uniqueness results one-dimensional parabolic PDEs (Karafyllis & Krstic, 2019b), which are in feedback interconnection with a system of ODEs are provided in the chapter. The obtained results allow the presence of nonlinear and non-local terms and guarantee the existence of classical solutions. The existence/uniqueness results are utilised in two applications with non-local terms: a chemical reactor in which an exothermic chemical reaction is taking place and a water tank. Finally, the case of the presence of boundary inputs is also investigated.