Arbitrary Kinetics Research Articles

Taylor series solutions to steady‐state non‐isothermal diffusion–reaction problems for porous catalyst pellets with arbitrary kinetics

In this study, we present Taylor series solutions for steady‐state non‐isothermal diffusion–reaction problems pertaining to porous catalyst pellets exhibiting arbitrary kinetics. Using the Damkohler relation, the system of two nonlinear differential equations is reduced to a single differential equation subject to the algebraic constraint. We derive the novel semi‐analytical and closed‐form explicit approximate solutions for the reactant concentration and temperature in catalyst pellets of planar, cylindrical, and spherical geometries. The derived semi‐analytical and explicit approximations give insight into the effect of process parameters on concentration and temperature profiles. The proposed methods are verified numerically for isothermal and non‐isothermal steady‐state problems with power‐law kinetics. They can serve as a practicable alternative to numerical schemes.

Rate Equations of Structure-Sensitive Catalytic Reactions with Arbitrary Kinetics

A general mathematical framework for the quantitative description of the cluster size dependence in heterogeneous catalytic reactions has been developed based on an analysis of the Gibbs energy of elementary reactions. The methodology was illustrated for a generic linear sequence of elementary reactions with three steps, a multi-step mechanism of ethanol oxidation comprising linear, nonlinear and quasi-equilibria steps and a network of parallel reactions in transformations of furfural.

Conservation Laws for Chemical Reactions with Arbitrary Kinetics in a Partially Stirred Reactor

Conservation Laws for Chemical Reactions with Arbitrary Kinetics in a Partially Stirred Reactor

Relaxation Times of Chemical Reactions with Arbitrary Kinetics

Relaxation Times of Chemical Reactions with Arbitrary Kinetics

A structural property for reduction of biochemical networks

Large-scale biochemical models are of increasing sizes due to the consideration of interacting organisms and tissues. Model reduction approaches that preserve the flux phenotypes can simplify the analysis and predictions of steady-state metabolic phenotypes. However, existing approaches either restrict functionality of reduced models or do not lead to significant decreases in the number of modelled metabolites. Here, we introduce an approach for model reduction based on the structural property of balancing of complexes that preserves the steady-state fluxes supported by the network and can be efficiently determined at genome scale. Using two large-scale mass-action kinetic models of Escherichia coli, we show that our approach results in a substantial reduction of 99% of metabolites. Applications to genome-scale metabolic models across kingdoms of life result in up to 55% and 85% reduction in the number of metabolites when arbitrary and mass-action kinetics is assumed, respectively. We also show that predictions of the specific growth rate from the reduced models match those based on the original models. Since steady-state flux phenotypes from the original model are preserved in the reduced, the approach paves the way for analysing other metabolic phenotypes in large-scale biochemical networks.

Detailed Balance = Complex Balance + Cycle Balance: A Graph-Theoretic Proof for Reaction Networks and Markov Chains

We further clarify the relation between detailed-balanced and complex-balanced equilibria of reversible chemical reaction networks. Our results hold for arbitrary kinetics and also for boundary equilibria. Detailed balance, complex balance, “formal balance,” and the new notion of “cycle balance” are all defined in terms of the underlying graph. This fact allows elementary graph-theoretic (non-algebraic) proofs of a previous result (detailed balance = complex balance + formal balance), our main result (detailed balance = complex balance + cycle balance), and a corresponding result in the setting of continuous-time Markov chains.

SOLVING INVERSE PROBLEM OF CHEMICAL KINETICS WITH USE OF CUBIC SPLINES

A simple effective method for solving the inverse problem of chemical kinetics based on non-stationary experiments for multistage reactions occurring in an isothermal reactor of ideal mixing is described. The idea of the method is based on taking into account the distinctive features (informativeness) of different fragments of relaxation curves for chemical reactions with arbitrary (non-monotonic) kinetics and their as accurate approximation as possible. For this purpose, non-linear (cubic) splines are used to describe different informative fragments of relaxation curves, which allow to approximate and interpolate experimental data as accurately as possible. An additional advantage of cubic splines, from the point of view of the implementation of the described method, is their continuity at all given points up to and including second-order derivatives (smoothness). This allows us to calculate with good accuracy not only the concentration of reagents, but also the instantaneous rate of change at any time. The consequence of this is the possibility of a sufficiently accurate solution of the inverse problem based on the data of non-stationary experiments. The correctness of the mathematical model used and the stability of the method were tested using variations of the original data. An example of using the method for determining the intervals of physical values of the rate constants of the stages of a two-stage reaction is given. The influence of the method of selecting the reference points (structure) of the spline and measurement errors (noise) of experimental data on the error of determining the speed constants of the stages is estimated. The efficiency of application and good accuracy of the method for solving the inverse problem of chemical kinetics of multistage reactions occurring in non-gradient systems with taking into account of noise is shown.

Modeling mafic carbonation efficiency using mafic rock chemistries from Nevada, USA

Mineral carbonation is one of the many ways that are being actively investigated to sequester point-source carbon dioxide emissions. However, relations between reaction conditions, variations in reactant mineral composition, and carbon sequestration potential are poorly understood. In this study we used reaction path geochemical modeling to evaluate carbon sequestration potential during ex-situ mineral carbonation of ten mafic rock samples from Nevada, USA. Models were run using arbitrary dissolution kinetics at temperatures between 0 and 200 °C. A subset of models were run using true dissolution kinetics. In the models, carbon is sequestered in 5 mineral phases: magnesite, siderite, dolomite, calcite, and dawsonite, with magnesite and dolomite the most abundant. Dawsonite sequesters carbon at T > 150 °C in most of the arbitrary kinetics models but is not a significant carbon sink in the models using true dissolution kinetics. The arbitrary kinetics models resulted in 4.5–13 mol of carbon sequestered per kg of reacted mafic rock. True kinetics models only resulted in 1–2 mol of carbon sequestered, but the models only reacted 12.5 to 15 wt percent of the mafic rock inputs. Product minerals using the arbitrary kinetics model have volumes 150%–470% larger than the reactant volumes, whereas using true kinetics the models have a modest increase. Modeling presented herein confirms several areas of Nevada as having potential for mafic rock carbonation, some of which are located near existing coal- and natural gas-fired power plants. As shown in this study, reaction path modeling is a vital and inexpensive tool to help optimize costs and reaction conditions for ex-situ mafic rock carbonation projects.

Mathematical modelling of nucleation and growth of crystals with buoyancy effects

A complete analytical solution of the integro-differential model describing the nucleation of crystals and their subsequent growth in a binary system with allowance for buoyancy forces is constructed. An exact analytical solution of the Fokker-Planck-type equation for the three-parameter density distribution function is found for arbitrary nucleation kinetics. Two important cases of the Weber–Volmer–Frenkel–Zel’dovich and Meirs kinetics are considered in some detail. It is shown that the solute concentration decreases and the distribution function increases with increasing the melt supercooling (with increasing the depth of a metastable system). It is demonstrated that the distribution function attains its minimum at a certain size of crystals owing to buoyancy forces.

Impedance of Mediated Electrochemical Processes. Novel Impedance Element for Unequal Diffusivities

The faradaic impedance of the catalytic EC′ processes is given in an analytical form together with the potential dependence of all impedance parameters. The exact solution derived in the rigorous treatment can be used in the analysis of catalytic systems with arbitrary kinetics. Unusual impedance characteristics have been found for systems having unequal diffusivities of the oxidized and reduced forms of the redox mediator. Depending on the ratio of diffusion coefficients, either the spectra are markedly deformed at low frequency range or an inductive loop appears at the complex plane. This behavior is related to a novel impedance element, which reduces to Gerischer impedance when both diffusion coefficients are equal. The resistance of the charge transfer, Rct, has been found to be potential-independent at far anodic or far cathodic overpotentials for electrocatalytic oxidation and reduction, respectively. It is shown that the key parameters of catalytic systems, such as both heterogeneous and homogeneou...

Analytical theory of the catalytic mechanism in square wave voltammetry at disc electrodes

A simple analytical expression is presented for the study of the first-order catalytic mechanism using Square Wave Voltammetry (SWV) at disc electrodes. These electrodes are extensively used in electrochemical studies but modelling the electrochemical response at this geometry is complex and usually requires the use of sophisticated numerical methods. By contrast, the analytical solution presented in this work is easy to compute and it is applicable to any size of the disc and for arbitrary kinetics of the catalytic reaction. The effects of the electrode size, the homogeneous rate constants, the frequency and the square wave amplitude on the SWV response are analyzed. Criteria are given for the detection of the steady-state response as well as procedures for the extraction of the catalytic rate constant from the value of the peak current. The theory is applied to obtain the kinetics of the reduction of the anion nitrite by an electrogenerated heteropolyanion [P(W(3)O(10))(4)](4-) at gold microdiscs.

Parametric study of shock-wave formation in a spatially inhomogeneous self-igniting medium for arrhenius combustion kinetics

An approach to describing the formation and propagation of small gas-dynamic perturbations in a spatially inhomogeneous self-igniting medium for arbitrary chemical reaction kinetics was developed in a previous paper. In the present paper, the proposed approach is illustrated by studying the problem for a simple reaction with an Arrhenius reaction rate. A comparison of calculation results with the solution of the complete system of gas dynamics equations shows that the approach provides good accuracy of quantitative estimates.

Transport equation for gas-dynamic perturbations in a spatially nonuniform self-igniting medium

An approach is developed to describe the formation and propagation of small gas-dynamic perturbations in a spatially nonuniform self-igniting medium for arbitrary chemical-reaction kinetics. An asymptotic equation describing the variation in the amplitude of the gas-dynamic perturbation along the characteristic is derived, and the conditions of its applicability are defined. Zel’dovich’s spontaneous flame wave is shown to be a natural zeroth approximation to the solution of the equations of gas dynamics. A method is proposed to determine the point in space at which a shock wave is formed.

New classes of exact solutions to nonlinear sets of equations in the theory of filtration and convective mass transfer

New classes of exact solutions to nonlinear sets of equations encountered in the theory of filtration and convective mass transfer of reacting media are described. Focus is placed on general first-order sets in which the chemical reaction rates depend on arbitrary functions. General solutions to some first-order systems with power-law nonlinearities are found. A set of new exact solutions with a functional separation of variables involving arbitrary functions is constructed. The results obtained are used for solving the problems of the theory of filtration of one-component and multicomponent suspensions with an arbitrary kinetics of particle accumulation.

An analysis of the morphological transitions during nonequilibrium growth of a cylindrical crystal from solution

Using linear stability analysis and the principle of maximum entropy production, the complete morphological phase diagrams (including stable, metastable, and labile phase regions) were calculated for the morphology selection during the continuous growth of an infinite cylindrical crystal from solution. The calculation was performed with an allowance for an arbitrary kinetics at the growth surface and linear dependence of the growth rate on the degree of supersaturation. It was found that the three and more morphological phases may coexist in the system under kinetically controlled and intermediate growth conditions.

Solving design equations for a hollow fiber bioreactor with arbitrary kinetics

An approach for solving the hollow fiber bioreactor design equations is presented. The original set of differential mass balance equations is cast into an equivalent system of integral equations by generating the appropriate Green’s functions. Mathematical features common to all hollow fiber bioreactors (HFBRs) operating with laminar flow are imbedded in the corresponding Green’s functions on the lumen side, and thus separated from specific aspects arising from mass transport through the permeable wall. On the spongy matrix side, the appropriate Green’s functions are expressed in terms of the mass transfer properties without involving any chemical kinetic parameters; this avoids repetitive computational effort when treating different reaction kinetics. The derived integral equations are numerically solved on an appropriately transformed coordinate system. The numerical method is well suited for problems where steep gradients of concentration cause an inaccurate numerical integration and low rates of convergence if the equations are solved with a uniform rectangular grid on the original coordinate system. The effectiveness of the proposed approach for the simulation of HFBRs with power-law, Michaelis–Menten and zero-order kinetics is demonstrated. The method is readily extendible to treat problems with chemical kinetics described by any arbitrary functional form.

Parametric Analysis of the Simplest Model of the Theory of Thermal Explosion—the Zel'dovich—Semenov Model

Parametric analysis is performed for the Zel'dovich–Semenov model, a basic model of the theory of thermal explosion that describes the dynamics of an exothermic reaction with arbitrary kinetics in a continuous stirred tank reactor. Particular attention is given to the cases of first‐ and nth‐order reactions, oxidation reaction, and reaction with arbitrary kinetics. Parametric dependences of steady states on dimensionless parameters, curves of multiplicity and neutrality of steady states, and parametric and phase portraits of the system are constructed. Regions of multiple steady states and self‐oscillations and a region of technological safety are distinguished.

DYNAMICS OF ADSORPTIVE REACTOR WITH A BIMOLECULAR REACTION

A theoretical analysis is used to gain insight in the dynamics of an adsorptive fixed-bed reactor where an irreversible bimolecular reaction of the type A + B→P takes place. One of the reactants is assumed to be preferentially adsorbed with a favorable isotherm, while the other is adsorbed with a linear isotherm. The reaction follows an arbitrary kinetics where the rate is dependent on the concentrations of A and B. An analytic solution based on the method of characteristics is obtained for the case of local adsorption equilibrium. The dynamic behavior is affected by an intimate coupling of reaction kinetics and adsorption and breakthrough of the favorably adsorbed species is delayed by the occurrence of the reaction. The theoretical analysis is compared to experimental results obtained for an immobilized-enzyme reactor system for the esterification of propionic acid and isoamyl alcohol in hexane. The local adsorption equilibrium solution obtained for this system is in qualitative agreement with the experimental results. Quantitative agreement is obtained taking into account numerically the finite rate of adsorption and axial dispersion.

Effective potential for the reaction-diffusion-decay system

In previous work [cond-mat/9904207,cond-mat/9904215] we have developed a general method for casting stochastic partial differential equations (SPDEs) into a functional integral formalism, and have derived the one-loop effective potential for these systems. In this paper we apply the same formalism to a specific field theory of considerable interest, the reaction-diffusion-decay system. When this field theory is subject to white noise we can calculate the one-loop effective potential (for arbitrary polynomial reaction kinetics) and show that it is one-loop ultraviolet renormalizable in 1, 2, and 3 space dimensions. For specific choices of interaction terms the one-loop renormalizability can be extended to higher dimensions. We also show how to include the effects of fluctuations in the study of pattern formation away from equilibrium, and conclude that noise affects the stability of the system in a way which is calculable.

Modal analysis and optimization of isothermal autocatalytic reactions

A comprehensive and systematic analysis of optimization of autocatalytic reactions is presented in this article considering a very general objective function. The objective functions of common interest, such as maximum yield, maximum selectivity, maximum productivity and minimal reaction time, are some of the special cases of this general objective function. The following reaction scheme is considered in this work: A+ nB ⇌ ( n+1)B, A ⇌ B, B→C. The problem of maximization of the general objective function is posed as that of the optimal control of a variable-volume reactor with time-variant feed and effluent flow rates as the control variables. The systematic ‘modal’ analysis enables identification of feasible modes of reactor operation and conditions under which each feasible mode is admissible. Of all the possible modes, only a few are proved to be feasible and only certain transitions are permissible from each feasible mode. The policy for programmed manipulation of the feed and effluent flow rates is obtained employing the optimal control theory. Candidate optimal policies are identified as individual feasible modes or certain sequences of these. A singular fed-batch operation in which the feed rate is decided exclusively by the state of the reactor is part of some of the candidate optimal policies. Conditions for admissibility and feasibility of singular control and the feeding policy during singular control interval are obtained analytically for arbitrary kinetics of these reactions. The concentration trajectories in a fed-batch operation converge to or diverge from one or more limit points. Operation at each limit point is characterized by invariance of concentrations of all species, the feed rate being an exponential function of time. A linearized stability analysis, supported by numerical illustrations, reveals that some or all of the limit points may be accessible during fed-batch operation. The optimal operation strategy is linked to the disposition of batch reactor trajectories in the space of concentration variables. The rich behavioral patterns of autocatalytic reactions are revealed by specific numerical illustrations.