Isothermal Chemical System Research Articles

Robust multi‐objective model predictive control for constrained nonlinear systems with disturbances

AbstractThis work develops a new robust multi‐objective model predictive control (MoMPC) strategy for constrained non‐linear systems with bounded disturbances. The multiple objectives are always contradictory, and the presence of disturbances may result in the violation of state constraints and instability of the controlled system. Firstly, the conflict between multiple objectives is reconciled by minimizing the distance of the cost function vector to the vector of independently minimised objectives obtained by solving a set of steady‐state optimisation problems. Then the contractive state set, which takes into account the effect of disturbances on system states, is constructed to guarantee the robust satisfaction of state constraints. Finally, a stability constraint updated online by solving an auxiliary optimization problem is established to ensure the robust stability of the system under MoMPC. An isothermal chemical reactor system is employed to verify the effectiveness of the controller proposed here.

A comparative study on solving fractional cubic isothermal auto-catalytic chemical system via new efficient technique

Abstract In this paper, we examine a cubic isothermal auto-catalytic chemical system (CIACS) with the help of the newly developed technique. Classical model of this system is transformed into a new fractional forms by using three different and special fractional operators. The new model is therefore called as fractional cubic isothermal auto-catalytic chemical system (FCIACS). Then, the new systems are solved by optimal perturbation iteration method. Obtained results are compared to get an idea about the new derivative operators and optimal perturbation iteration method.

Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense

In this paper, we have generalized the fractional cubic isothermal auto-catalytic chemical system (FCIACS) with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional time derivatives, respectively. We apply the Homotopy Analysis Transform Method (HATM) to compute the approximate solutions of FCIACS using these fractional derivatives. We study the convergence analysis of HATM by computing the residual error function. Also, we find the optimal values of h so we assure the convergence of the approximate solutions. Finally we show the behavior of the approximate solutions graphically. The results obtained are very effectiveness and accuracy.

A reliable analytical algorithm for space–time fractional cubic isothermal autocatalytic chemical system

In this paper, we present an algorithm by using the homotopy analysis method (HAM), Adomian decomposition method (ADM) and variational iteration method (VIM) to find the approximate solutions of the space–time fractional cubic isothermal autocatalytic chemical system (STFCIACS). The HAM, ADM and VIM approximate solutions are evaluated and compared by using the computation program Mathematica and excellent results are obtained.

Analysis of reaction–diffusion system via a new fractional derivative with non-singular kernel

In this paper, we obtain analytical solutions for the fractional cubic isothermal auto-catalytic chemical system with Caputo–Fabrizio and Atangana–Baleanu fractional time derivatives in Liouville–Caputo sense. We utilize the q-homotopy analysis transform method to compute the approximate solutions. We find the optimal values of h so we assure the convergence of the approximate solutions. Finally, we compare our results numerically with the finite difference method and excellent agreement is found.

Comparing the Caputo, Caputo-Fabrizio and Atangana-Baleanu derivative with fractional order: Fractional cubic isothermal auto-catalytic chemical system

In this work we extend the standard model for a cubic isothermal auto-catalytic chemical system (CIACS) to a new model of a fractional cubic isothermal auto-catalytic chemical system (FCIACS) based on Caputo (C), Caputo-Fabrizio (CF) and Atangana-Baleanu in the Liouville-Caputo sense (ABC) fractional time derivatives, respectively. We present approximate solutions for these extended models using the q -homotopy analysis transform method (q -HATM). We solve the FCIACS with the C derivative and compare our results with those obtained using the CF and ABC derivatives. The ranges of convergence of the solutions are found and the optimal values of h , the auxiliary parameter, are derived. Finally, these solutions are compared with numerical solutions of the various models obtained using finite differences and excellent agreement is found.

An effective homotopy analysis method to solve the cubic isothermal auto-catalytic chemical system

We established an effective algorithm for the homotopy analysis method (HAM) to solve a cubic isothermal auto-catalytic chemical system (CIACS). Our solution comes in a rapidly convergent series where the intervals of convergence given by h-curves and to find the optimal values of h, we used the averaged residual errors. The HAM solutions are compared with the solutions obtained by Mathematica in-built numerical solver. We also show the behavior of the HAM solution.

Nonequilibrium thermodynamic formalism of nonlinear chemical reaction systems with Waage–Guldberg’s law of mass action

Macroscopic entropy production rate σ(tot) in the general nonlinear isothermal chemical reaction system with mass action kinetics is decomposed into a free energy dissipation rate and a house-keeping heat dissipation rate: σ(tot)=σ(fd)+σ(hk); σ(fd)=-dA/dt, where A is a generalized free energy function. This yields a novel nonequilibrium free energy balance equation dA/dt=-σ(tot)+σ(hk), which is on a par with celebrated entropy balance equation dS/dt=σ(tot)+η(ex) where η(ex) is the rate of entropy exchange with the environment. For kinetic systems with complex balance, σ(fd) and σ(hk) are the macroscopic limits of stochastic free energy dissipation rate and house-keeping heat dissipation rate, which are both nonnegative, in the Delbrück–Gillespie description of the stochastic chemical kinetics. A full kinetic and thermodynamic theory of chemical reaction systems that transcends mesoscopic and macroscopic levels emerges.

Theoretical analyses of chemical dissolution‐front instability in fluid‐saturated porous media under non‐isothermal conditions

SummaryThis paper mainly deals with the theoretical aspects of chemical dissolution‐front instability problems in two‐dimensional fluid‐saturated porous media under non‐isothermal conditions. In the case of the mineral dissolution ratio (that is defined as the ratio of the dissolved‐mineral equilibrium concentration in the pore fluid to the molar concentration of the dissolvable mineral in the solid matrix of the fluid‐saturated porous medium) approaching zero, the corresponding critical condition has been mathematically derived when temperature variation effects are considered. As a complementary tool, the computational simulation method is used to simulate the morphological evolution of chemical dissolution fronts in two‐dimensional fluid‐saturated porous media under non‐isothermal conditions. The related theoretical and numerical results have demonstrated that: (i) a temperature increase in a non‐isothermal chemical dissolution system can have some influence on the propagation speed of the planar chemical dissolution front in the system. Generally, the chemical dissolution front in the non‐isothermal chemical dissolution system propagates slower than that in the counterpart isothermal chemical dissolution system when the temperature of the non‐isothermal chemical dissolution system is higher than that of the counterpart isothermal chemical dissolution system; (ii) a temperature increase in the non‐isothermal chemical dissolution system can stabilize the chemical dissolution front propagating in the system, because it can cause a decrease in the Zhao number of the system but does not affect the critical Zhao number of the system; and (iii) the temperature gradient in the upstream direction of a chemical dissolution front is smaller than that in the downstream direction of the chemical dissolution front when the non‐isothermal chemical dissolution system is supercritical. Copyright © 2014 John Wiley & Sons, Ltd.

Traveling waves solutions of isothermal chemical systems with decay

This article studies propagating traveling waves in a class of reaction–diffusion systems which include a model of microbial growth and competition in a flow reactor proposed by Smith and Zhao [17], and isothermal autocatalytic systems in chemical reaction of order m with a decay order n, where m and n are positive integers and m≠n. A typical system in autocatalysis is A+2B→3B (with rate k1ab2) and B→C (with rate k2b), where m=2 and n=1, involving two chemical species, a reactant A and an auto-catalyst B whose diffusion coefficients, DA and DB, are unequal due to different molecular weights and/or sizes. Here a is the concentration density of A, b that of B and C an inert chemical species. The two constants k1 and k2 are material constants measuring the relative strength of respective reactions.It is shown that there exist traveling waves when m>1 and n=1 with suitable relation between the ratio DB/DA, traveling speed c and rate constants k1, k2. On the other hand, it is proved that there exists no traveling wave when one of the chemical species is immobile, DB=0 or n>m for all choices of other parameters.

The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay

In this paper, we study a reaction–diffusion system for an isothermal chemical reaction scheme governed by a quadratic autocatalytic step A+B→2B and a decay step B→C, where A, B, and C are the reactant, the autocatalyst, and the inner product, respectively. Previous numerical studies and experimental evidences demonstrate that if the autocatalyst is introduced locally into this autocatalytic reaction system where the reactant A initially distributes uniformly in the whole space, then a pair of waves will be generated and will propagate outwards from the initial reaction zone. One crucial feature of this phenomenon is that for the strong decay case, the formation of waves is independent of the amount of the autocatalyst B introduced into the system. It is this phenomenon of KPP-type which we would like to address in this paper. To study the propagation of reactant and autocatalyst analytically, we first use the tail behavior of waves to construct a pair of generalized super-/sub-solutions for the approximate system of the autocatalytic reaction system. Note that the autocatalytic reaction system does not enjoy comparison principle. Together with a family of truncated problems, we can establish the existence of a family of traveling waves with the minimal speed. Second, we use this pair of generalized super-/sub-solutions to show that the propagation of waves is fully determined by the rate of decay of the initial data at infinity in the sense of Aronson–Weinberger formulation, which in turn confirms the aforementioned numerical and experimental results.

Stationary patterns in a two-cell coupled isothermal chemical system with arbitrary powers of autocatalysis

Stationary patterns of a two-cell coupled isothermal chemical system with arbitrary powers of autocatalysis are considered. Firstly, we obtain the stability of the unique positive constant equilibrium solution for the system. Then, based on a priori estimates, non-existence and existence of nontrivial steady state solutions are shown by using implicit function theorem and topological degree theory, respectively. The effects of autocatalysis order and diffusion coefficients to the pattern formation are discussed.

Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

The reaction-diffusion systems which are based on an isothermal autocatalytic chemical reaction involving both an autocatalytic step of the ( m + 1 ) (m+1) th order ( A + m B → ( m + 1 ) B A+mB\rightarrow (m+1)B ) and a decay step of the same order ( B → C B\rightarrow C ) have very rich and interesting dynamics. Previous studies in the literature indicate that traveling waves play a key role in understanding these interesting dynamical phenomena. However, there is a lack of rigorous proof of the existence of traveling waves to this system. Here we generalize this isothermal autocatalytic chemical reaction model and provide a rigorous proof of the existence of traveling waves for the resulting reaction-diffusion system which also includes the systems arising from epidemiology and the microbial growth in a flow reactor.

Oscillations and multiscale dynamics in a closed chemical reaction system: Second law of thermodynamics and temporal complexity

We investigate the oscillatory reaction dynamics in a closed isothermal chemical system: the reversible Lotka-Volterra model. The second law of thermodynamics dictates that the system ultimately reaches an equilibrium. Quasistationary oscillations are analyzed while the free energy of the system serves as a global Lyapunov function of the dissipative dynamics. A natural distinction between regions near and far from equilibrium in terms of the free energy can be established. The dynamics is analogous to a nonlinear mechanical system with time-dependent increasing damping. Near equilibrium, no oscillation is possible as dictated by Onsager's reciprocal symmetry relation. We observe that while the free energy decreases in the closed system's dynamics, it does not follow the steepest descending path.

Fluctuation-induced and Nonequilibrium-induced Bifurcations in a Thermochemical System

We study the Semenov thermochemical model at different levels of description. This simple model of combustion includes the description of slow oxidation with stabilization of the system around a low temperature and thermal explosion leading the system to a high temperature. The results of the mesoscopic description based on the master equation and microscopic simulations of the particle dynamics reveal deviations from the macroscopic deterministic dynamics. Fluctuation effects observed in small systems change the domains of mono- and bistability of the system. These are noise-induced bifurcations, already known in isothermal chemical systems. Another effect is related to the departure of particle velocity distribution from its equilibrium shape. The departure from partial equilibrium originating at the level of the velocity distribution also leads to changes of the bifurcation diagram. By analogy, we call this phenomenon a nonequilibrium-induced bifurcation.

Enhanced sensitivity of a thermochemical system to microscopic perturbations

We study a thermochemical system at different description levels. The results of the mesosocopic description based on the master equation and microscopic simulations of the particle dynamics show deviations from the deterministic predictions. Fluctuation effects observed in the vicinity of bifurcations change the domains of mono- and bistability of the system. These are noise-induced transitions, already known in isothermal chemical systems. Another effect is related to the departure of particle velocity distribution from its equilibrium shape which modifies the rates of the processes. The resulting deviation from the deterministic dynamics is much stronger in the thermochemical system than in the isothermal case. The departure from partial equilibrium originating at the level of the velocity distribution also leads to changes of the bifurcation diagram. By analogy, we call this phenomenon a nonequilibrium-induced transition.

The global dynamics of isothermal chemical systems with critical nonlinearity

In this paper, we study the Cauchy problem of a cubic autocatalytic chemical reaction system u1,t = u1,xx − uα1 uβ2, u2,t = du2,xx + uα1 uβ2 with non-negative initial data, where the exponents α,β satisfy 1<α,β<2, α+β = 3 and the constant d>0 is the Lewis number. Our purpose is to study the global dynamics of solutions under mild decay of initial data as |x|→∞. We show the exact large time behaviour of solutions which is universal.

The propagation of travelling waves in an open cubic autocatalytic chemical system

The reaction-diffusion travelling waves that can be initiated in an open isothermal chemical system governed by cubic autocatalytic kinetics are discussed. The system is shown to be capable of sustaining up to three spatially uniform steady states, the (trivial) unreacted state, which is always stable (a node), and two nontrivial states, one of which is always unstable (a saddle point). The third state can change its stability through Hopf bifurcation (both subcritical and supercritical). This allows the possibility of two sorts of travelling wave being established; there are wave profiles which connect the unreacted state ahead to the nontrivial state at the rear, and wave profiles (pulse waves) which have the unreacted state at both the front and rear. The conditions under which a particular wave is initiated are considered by both a discussion of the (ordinary) differential equations governing the travelling waves and by numerical integrations of an initial-value problem. This treatment also reveals the possibility of a stable travelling wave propagating through the system, leaving behind a temporally unstable stationary state. Under these conditions, spatiotemporal chaotic behaviour is seen to develop after the passage of the wave.

Steady state thermodynamics for homogeneous chemical systems

We contribute to the development of steady state thermodynamics for isothermal and homogeneous chemical systems, through a generalized Einstein fluctuation relation, by utilizing a local steady state interpretation of static concentration fluctuations of reaction intermediates. For systems of constrained total chemical affinity the ‘‘next-particle ratio’’ of the probability density, q=P(N+1)/P(N), derived from a usual chemical master equation is employed for the construction of chemical potential steady state laws. This analysis is based on the identification of the exponent of the distribution, as a generalized availability of local fluctuations. Further, pressure steady state laws are derived through a generalized Gibbs–Duhem equation, restricted to constant ‘‘disequilibrium variables.’’ Such variables are introduced as state variables, in addition to the classic ones, for the characterization of steady states. They relate to externally controlled generalized forces or affinities, which induce the flows of mass through the system. Within the local steady state approach, the state laws enable the construction of the generalized availability, as state function for quasisteady state processes beginning from a reference state. This quantity is found to provide a Liapounov function for the deterministic evolution of the system towards stationary states in analogy to a previously developed local equilibrium theory. The analysis is applied to two-variable chemical systems of high stoichiometry change, but should be capable of extension to general hydrodynamic systems.

Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis

The propagation of planar reaction-diffusion waves in the isothermal autocatalytic system A + m B → ( m + 1)B, rate kab m , is considered. Attention is paid to the case when the diffusion coefficient of reactant A is much less than that of the autocatalyst; the case when A is completely immobilized is discussed in detail. Permanent-form travelling waves are initiated and their structure is treated. It is found that for sufficiently large values of m this steady wave structure becomes unstable to longitudinal disturbances and an oscillatory wave structure develops. The structure of this propagating oscillatory reaction-diffusion front is also discussed.