Semi-analytical solutions for a Gray-Scott reaction-diffusion cell with an applied electric field

An ionic version of the Gray-Scott chemical reaction scheme is considered in a reaction-diffusion cell, with an applied electric field, which causes migration of the reactant and autocatalyst in a preferred direction. The Galerkin method is used to reduce the governing partial differential equations to an approximate model consisting of ordinary differential equations. This is accomplished by approximating the spatial structure of the reactant and autocatalyst concentrations. Bifurcation analysis of the semi-analytical model is performed by using singularity theory to analyse the static multiplicity and a stability analysis to determine the dynamic multiplicity. The application of the electric field causes variation in the parameter regions, in which multiple steady-state and oscillatory solutions occur. Moreover, as the reactor is not symmetric, reversal of the direction of the electric field can cause bifurcation in the reactor between high and low conversion states. Comparisons with numerical solutions of governing partial differential equations confirms the accuracy and usefulness of the semi-analytical model.     

Multiplicity,stability, and oscillatory dynamics of the tubular reactor

Numerical bifurcation techniques were developed for studying the multiplicity, stability, and oscillatory dynamics of the nonadiabatic tubular reactor with a single A → B reaction. The techniques illustrate the existence of one, three, five, or seven steady states and bifurcating periodic solutions. We present numerical procedures for computing the Hopf bifurcation formulas which can determine the stability and location of the oscillation without integrating the parabolic partial differential equations. The combination of our Hopf techniques with steady state bifurcation methods enables us to determine all possible steady and stable oscillatory solutions exhibited by distributed parameter models such as the tubular reactor.     

An approximate solution to the model of a sparged packed bed electrode reactor

An approximate analytical solution to the mathematical model of a sparged packed bed electrochemical reactor (SPBER) in which a gaseous reactant undergoes direct anodic oxidation is presented. The model is based on two strongly nonlinear partial differential equations and is solved by the inverse operator method (IOM). The solution is used to study the direct anodic oxidation of propylene. The approximate IOM solution is useful in that it enables the direct investigation of the effect of model parameters on the operation of the reactor. Current density against electrode potential, polarization data, for the anodic oxidation of propylene in the SPBER is presented. Nonlinear parameter estimate methods are used to fit the model to experimental data to obtain physically meaningful values of kinetic parameters.     

The bifurcation behavior of tubular reactors

Methods for studying the bifurcation behavior of tubular reactors have been developed. This involves the application of static and Hopf bifurcation theory for PDE's and the very precise determination of steady state profiles. Practical computational methods for carrying out this analysis are discussed in some detail. For the special case of a first order, irreversible reaction in a tubular reactor with axial dispersion, the bifurcation behavior is classified and summarized in parameter space plots. In particular the influence of the Lewis and Peclet numbers is investigated. It is shown that oscillations due to interaction of dispersion and reaction effects should not exist in fixed bed reactors and moreover, should only occur in very short “empty” tubular reactors. The parameter study not only brings together previously published examples of multiple and periodic solutions but also reveals a hitherto undiscovered wealth of bifurcation structures. Sixteen of these structures, which come about by combinations of as many as four bifurcations to multiple steady states and four bifurcations to periodic solutions, are illustrated with numerical examples. Although the analysis is based on the pseudohomogeneous axial dispersion model, it can readily be applied to other reaction diffusion equations such as the general two phase models for fixed bed reactors.     

Dynamic modelling of a heterogeneously catalysed system with stiff Hopf bifurcations

We report the results of a detailed dynamic modelling of CO oxidation on a platinum catalytic wire, carried out via an iterative parameter estimation scheme which employs both experimental bifurcation-to-oscillation data as well as the actual oscillatory data. By exploiting the stiffness of the system's model equations, some algebraic equations that describe the oscillatory boundaries in the feed temperature/partial pressure parameter space are extracted. The resulting model is shown to fit quantitatively these boundaries and the dynamic oscillographs of the wire temperature and simulate, with satisfactory accuracy, the unfitted dynamic data. Following an extension of the independent parameters space of the model beyond the reported experimental one, interesting bifurcation sequences (including multiple steady states) obtained via a computational study are predicted. The utility of both the resulting model in catalytic reactor design and the novel iteration scheme in general parameter estimation work on catalytic models is discussed.     

Fast binary reactions in a heterogeneous catalytic batch reactor

When heterogeneous chemical reaction is sufficiently fast, transport of reactants becomes limiting. In a well stirred, batch reactor, macroscopic concentration gradients can be eliminated as a factor limiting the rate of reaction, leaving only the mesoscopic mass transfer of reactants to the surface of the catalyst as limiting, if the reaction does not occur inside a porous support. Here, a transformation of the governing equations for the time-dependence of bulk and surface concentrations results in second order ODE in time and a single nonlinear constraint with boundary values at the initial and infinite times for two auxiliary variables termed modified Thiele moduli. This system of two equations—one differential, one algebraic—and two unknowns is an exact consequence of the governing equations (three ODEs and three algebraic constraints). The power of this formulation is demonstrated by analytic solutions for irreversible and nearly irreversible theories. These solutions are corroborated by full nonlinear numerical computations of the boundary value problem, for the case when asymmetric mass transfer coefficients admit the possibility that the mode of operation switches from relative surface depletion of one reactant to depletion of the other in a binary reaction. The modified Thiele modulus formulation reveals the time scale for the switch over, as well as giving a reliable prediction for the time scale for 99% conversion based on the switch time identified from the irreversible theory.     

Behaviour of an elementary oxidation reaction in a semi-batch reactor

The dynamics of a non-isothermal bi-molecular gas-phase reaction in a semi-batch reactor is investigated. It is assumed that one of the reactants flows into a reactor containing the second. A reduced model is obtained by making a ‘pool-chemical’ approximation on the concentration of the reactant initially in the reactor. The region in parameter space in which oscillations are observable in the full transient model is estimated by determining the Hopf bifurcation locus in the reduced system. The contribution of the current work is its comparative study of the behaviour of the full system to that of the pool-chemical model. Although the reaction scheme is symmetric with respect to the reactants the regions of oscillatory behaviour are not identical because the reactants have different heat capacities.     

Enhancement of catalytic reaction by pressure swing adsorption

A theoretical study of an adsorptive reactor which combines multibed pressure swing adsorption and chemical reaction is presented; such a reactor is referred to as a pressure swing reactor, or PSR. Studies have concentrated on an asymptotic case in which there is the ideal propagation of concentration waves within the reactor beds; the method of characteristics was employed in the solution of the governing PSR equations. The studies assessed the effects of operating conditions, and cycle configurations, on the PSR performance. Calculations indicate enhanced reactant conversion when compared to conventional steady state plug flow operation. In particular, for some reversible reactions, substantial improvements over equilibrium yields have been calculated. For example, for the dissociation reaction 2A B + C, and where B is the only adsorbing component, approximately two-fold improvements over the equilibrium yield of product B have been predicted. Such reaction enhancement can be attributed to the limitation of the backward reaction, which results from the separation of the product species B and C.

In addition to the method of characteristics, a cells-in-series method for the asymptotic case has been developed, and found to yield calculations consistent with the method of characteristics solutions. In a third numerical approach, the spatial discretisation technique of orthogonal collocation on finite elements was applied to the governing PSR equations, and the resulting system of ordinary differential equations solved by a standard integration algorithm. In this case, many of the simplifying model assumptions were relaxed, allowing, for example, the simulation of a non-isothermal PSR with finite mass transfer rates.

One practical significance of reaction enhancement by pressure swing adsorption is a lower temperature of operation than in a conventional reactor. This would lead to savings in the energy requirements of the reactor, and limit the rate and degree of catalyst deactivation due to coke deposition or sintering.     

Dynamics of a CSTR for styrene polymerization initiated by a binary initiator system

The steady state and dynamic behavior of a continuous stirred tank reactor has been analyzed for free radical solution polymerization of styrene initiated by a mixture of two initiators having different thermal stabilities. From the steady state analysis of the reactor model with a mean residence time as a bifurcation parameter, four unique regions of steady state solutions are identified in an operating parameter space for a given initiator feed composition. A variety of complex bifurcation behavior such as multiple steady states, Hopf bifurcation and limit cycles have been observed and their stability characteristics have been analyzed. The effects of feed initiator composition and the concentration of the initiator in the feed stream on the reactor dynamics are also presented.     

Mathematical modelling of the transient behaviour of cstrs with reactive particulates: Part 1 — The population balance framework

The population balance model appears to be the best approach to model particulate systems where multiple heterogeneous reactions occur. This work demonstrates a mathematical formulation that is based on the population balance model, and aims at simulating the non steady-state behaviour of a single-stage CSTR under isothermal operation. The chemical reaction system is a typical example from the field of hydrometallurgy with two parallel reactions, one being leaching, the other precipitation with simultaneous reactant regeneration. The solution of the resulting system of the partial and ordinary differential equations is achieved by combining the moment transformation of the population balance equations with the numerical method of lines, using the Mathematica® software. Finally, examples are given for a reactor startup in two cases: a single leaching reaction, and simultaneous leaching and precipitation reactions. In the first case, the difference between simultaneous and sequential feeding in achieving steady-state is also discussed.     

Modeling of Shallow Fluidized Bed Reactors: I. analytic solution of a simplified model

A fairly general dynamic model for shallow fluidized bed reactors is developed, and analytic solutions for the governing equations of the model are obtained after some simplification. The bubble size, the lateral dispersion coefficient of solids, the level of the excess fluid reactant and the structure of the bed are examined to determine their effects on the conversion of solids. It has been shown that the conversion of solids is influenced profoundly by the bubble size and that it is advantageous to employ a shallow fluidized bed reactor if a high conversion of solids is required.     

非线性分布参数系统状态估计的最佳测量位置

研究了分布参数系统状态估计中特有的最佳测量位置问题．建立了基于后集中方法的分布参数系统的非线性状态估计器,包括状态估计偏微分方程和微分灵敏度矩阵偏微分方程,并用适当的数值计算方法实现状态估计器的求解;以一个最小化的空间域上积分函数表达最佳测量位置的目标函数,并相应地用非线性约束优化方法求解系统具有一个或多个测量时的最佳测量位置．还以壁冷式单管固定床反应器为例,讨论了各种因素对最佳测量位置的影响及其灵敏度,并得出了一些有普遍意义的结论．     

A GENERALIZED DIFFERENTIAL TRANSFORM METHOD FOR COMBINED FREE AND FORCED CONVECTION FLOW ABOUT INCLINED SURFACES IN POROUS MEDIA

In this article, we apply the differential transform method (DTM) to obtain approximate analytical solutions of combined free and forced (mixed) convection about inclined surfaces (or wedges) in a saturated porous medium. Both aiding and opposing flows are considered. It is found that the parameter mixed convection from inclined surfaces in porous media is Gr/Re, where Gr is the local Grashof number and Re is the local Reynolds number. DTM solutions are obtained for mixed convection from an isothermal vertical flat plate as well as an inclined plate with constant heat flux having an inclination of 45°. Temperature and velocity profiles for these two cases at different values of Gr/Re are presented. The similarity transformations are applied to reduce the governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. DTM is used to solve the nonlinear differential equations governing the problem in the form of series with easily computable terms. Thereafter a Padé approximant is applied to the solutions to increase the convergence of the given series. Excellent correlation between DTM-Padé and numerical quadrature (shooting) solutions is achieved. The DTM-Padé simulation is shown to be a robust benchmarking tool providing an excellent means of validation of numerical methods. The study has applications in geothermal energy systems, chemical engineering filtration systems, and packed beds.     

Dynamic simulation of chemical processes described by distributed and lumped parameter models

Currently available dynamic simulation routines are limited to processes in which the units can be described by lumped parameter models. In this paper, a general procedure is presented which permits the units to be described by both distributed and lumped parameter models. The partial differential equations resulting from a distributed parameter model are transformed into a set of ordinary differential equations by discretizing the spatial variable. The coupling of this set of discretized equations to the sets from other units is accomplished through the boundary conditions which represent the inlet and outlet of the unit. The resulting large set of time-dependent ordinary differential equations is solved simultaneously using a Gear-type integrator. The method is demonstrated by considering the start-up of a process composed of a tubular reactor, a gas absorption column and a completely mixed tank.     

NETWORK NUMERICAL SIMULATION OF HYDROMAGNETIC MARANGONI MIXED CONVECTION BOUNDARY LAYERS

The study of a steady coupled dissipative layer, known as the Mangaroni mixed convection boundary layer, in the presence of a magnetic field is presented. The mixed convection boundary layer is generated when in addition to Marangoni (thermocapillary) effects there are also buoyancy effects due to gravity and external pressure gradient effects. In the model considered the Marangoni coupling condition has been included in the boundary conditions at the interface. Similarity transformations are utilized to transform the governing partial differential conservation equations into nondimensional ordinary differential equations in a single independent space variable (η) and solved using the network simulation method (NSM) using an electronic circuit simulator, Pspice. NSM is founded on the classical thermoelectric analogy between thermal and electrical variables. A set of finite-differential equations, one for each control volume, was obtained by spatial discretization of the transformed equations. The solutions obtained are compared with earlier computations using other numerical techniques, showing excellent agreement. The influence of the Marangoni mixed parameter and Hartmann number on the velocity and temperature functions are studied in detail. The effectiveness of utilizing magnetic fields to control heat transfer in Marangoni convection boundary layers is identified. An increase in Hartmann hydromagnetic number (M) is found to strongly decelerate the flow but increase temperatures. An increase in Marangoni mixed convection parameter (λ) for the scenario opposing Marangoni flow (Γ > 0) considerably accelerates the flow but decreases temperatures in the boundary layer. Conversely, an increase in Marangoni mixed convection parameter (λ) for the case favorable to the Marangoni flow (Γ < 0) decelerates the flow but enhances temperatures in the boundary layer. Applications of the model include semiconductor crystal hydromagnetic heat transfer control.     

Complex nonlinear behaviour of a fixed bed reactor with reactant recycle

The fixed bed reactor with reactant recycle investigated in this paper can exhibit periodic solutions. These solutions bifurcate from the steady state in a Hopf bifurcation. The Hopf bifurcation encountered at the lowest value of the inlet concentration turns the steady state unstable and marks the emergence of a stable periodic solution. This periodic solution in turn undergoes a period doubling leaving it unstable and giving rise to a stable period 2 solution. It is know that if the system possesses one period doubling it often also has the possibility of posessing a chaotic attractor. It is shown, that the dynamic behaviour of a fixed bed reactor with reactant recycle is much more complex than previously reported.     

SIMILARITY SOLUTION FOR UNSTEADY HEAT AND MASS TRANSFER FROM A STRETCHING SURFACE EMBEDDED IN A POROUS MEDIUM WITH SUCTION/INJECTION AND CHEMICAL REACTION EFFECTS

An analysis is presented to investigate the effects of chemical reaction on unsteady free convective heat and mass transfer on a stretching surface in a porous medium. The governing partial differential equations have been transformed by a similarity transformation into a system of ordinary differential equations, which are solved numerically using an efficient tri-diagonal implicit finite-difference method. The results obtained show that the flow field is influenced appreciably by the presence of unsteadiness parameter, chemical reaction parameter, permeability parameter, and suction/injection parameter.     

Transient behaviour of an adiabatic fixed-bed methanator—I: Experiments with binary feeds of CO or CO2 in hydrogen

The transient behaviour of an adiabatic fixed-bed catalytic reactor has been studied experimentally as well as theoretically using the hydrogenation of small amounts of CO and CO₂ to methane as the test reactions. The axial temperature profile in a 0·5-litre reactor containing a Ni/Al₂O₃ catalyst was measured as a function of time after applying changes of the carbon oxide concentration in the feed and of the feed temperature. Besides, a quasi-homogeneous model was developed to simulate the transient behaviour of the reactor; its partial differential equations were solved numerically using the Crank-Nicholson algorithm.The agreement between measured and calculated reactor responses was quite good at carbon oxide concentrations up to 2·2 vol %, an inlet temperatures below 250°C and in the space velocity range of 5000–25000 h^?1, under which conditions the reaction is so slow that mass and heat transfer limitations do not occur.     

HYDROMAGNETIC STAGNATION POINT FLOW TOWARDS A POROUS STRETCHING SHEET WITH VARIABLE SURFACE HEAT FLUX IN THE PRESENCE OF HEAT GENERATION

The influence of heat generation or absorption on the steady, two-dimensional flow of an electrically conducting fluid near a stagnation point on a stretching permeable surface with variable surface heat flux in the presence of a magnetic field is investigated. The governing system of partial differential equations describing the problem are converted into highly non-linear ordinary differential equations using similarity transformation. Numerical solutions of these equations are obtained using the fourth-order Runge-Kutta integration scheme with the shooting method. The effects of the heat generation or absorption parameter and the velocity ratio parameter on the velocity and the temperature are displayed graphically and discussed. The numerical values of the local skin-friction coefficient and the local Nusselt number for various values of physical parameters are presented through tables and discussed.     

Stagnation point flow towards a stretching/shrinking sheet in a micropolar fluid with a convective surface boundary condition

The problem of a steady laminar two‐dimensional stagnation point flow towards a stretching/shrinking sheet in a micropolar fluid with a convective surface boundary condition is studied. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically using the Runge–Kutta–Fehlberg method with shooting technique. The effects of the material parameter and the convective parameter on the fluid flow and heat transfer characteristics are disscussed. It is found that the skin friction coefficient and the heat transfer rate at the surface decrease with increasing values of the material parameter. Moreover, dual solutions are found to exist for the shrinking case, while for the stretching case, the solution is unique. © 2011 Canadian Society for Chemical Engineering