On a mechanism for autocatalysis

The mechanism proposed by Walles and Platt for an autocatalytic reaction A → B + C is analysed by the method of singular perturbations. This has some features of interest in that the stable root of the degenerate problem changes during the course of the reaction and an intermediate solution with a different dependence on the small perturbation parameter must be introduced. Comparison between the calculated and asymptotic solution shows good agreement with only the first two terms of the perturbation expansion.     

Oxygen transport at porous flow-by air electrodes: theoretical approach

The theory of oxygen transport at porous flow-by air electrodes is treated mathematically. The three transport mechanisms of diffusion and convection in the gas channel, pore diffusion, and diffusion in the electrolyte are considered, and combined in the derivation of an asymptotic solution to electrode performance when U_avef²/L → ∞ ¹. Together with a further asymptote for the condition of oxygen exhaustion, ieU_avef/L → 0, this effectively defines the limit of real operation of an air electrode.     

The combined flux technique for diffusion—reaction problems in partial equilibrium: Application to the facilitated transport of carbon dioxide in aqueous bicarbonate solutions

A systematic method is derived for the simplification of a system of diffusion—reaction equations in which the kinetics fall into two distinct classes, E fast reactions and R — E slow reactions. The method has its origins in the “combined flux” technique introduced for the modelling of laminar flames by Dixon-Lewis et al. (1975, Proc. R. Soc. A346, 261–278). In using this method the E fast reactions are assumed to be in local equilibrium throughout some region of space, and this allows a reduction by E in the number of differential equations which must be simultaneously solved to compute multicomponent concentration profiles and fluxes. By eliminating the fast reactions this approach alleviates the stiffness problem that typically accompanies a reaction network in which the characteristic timescales vary by several orders of magnitude and provides what is essentially the lowest order term of the asymptotic expansion in the outer region,.Once derived, this technique is applied to the problem of the facilitated transport of carbon dioxide in aqueous bicarbonate solution to determine, in particular, the effect of induced electric fields on membrane permeability. The theoretical results obtained show that diffusion potentials in the uncatalyzed membrane in partial reaction equilibrium, i.e. R ⪢ E, are several orders of magnitude lower than those predicted for the carbonic anhydrase catalyzed membrane which attains full reaction equilibrium, i.e. R = E. Furthermore, in the example studied the induced electric field is shown to have a negligibly small influence on the net flux of CO₂ regardless of the assumed kinetic rates.     

Fast and very slow reactions: phase transfer catalysis

The effect of phase transfer catalysts (PTC), such as tricaprylmethyl ammonium chloride (Aliquat-336), cetyl trimethyl ammonium bromide, etc. on the rate of alkaline hydrolysis of different formate and acetate esters was studied. In the case of formate esters extraction is accompanied by fast pseudo-first order reaction in diffusion film; for acetate esters the reaction is insufficiently fast to occur in the diffusion film.The values of the volumetric rate of extraction, R_Aa, and the specific rate of extraction, R_A, with or without PTC, were measured in a fully baffled mechanically agitated contactor and a constant interfacial area stirred cell, respectively. The alkaline hydrolysis reaction was carried out with aqueous solutions of sodium hydroxide or aqueous lime slurry and the effect of speed of agitation and PTC concentration on the rate of hydrolysis was studied. A remarkable increase in the value of R_A was realized in the presence of PTC and the enhancement factor for the formate ester hydrolysis reaction ranged from 20 to over 200.     

Multiple steady states in two-phase reactors under boiling conditions

In this paper, we analyze the nonlinear behavior of two-phase reactors under boiling conditions. First we focus on a simple nth-order reaction of the form A→B, which allows a rigorous analytical treatment. Three necessary conditions for the existence of multiple steady states have been identified: the reactant A has to be the light-boiling component, the difference in boiling point temperatures between the reactant A and the product B has to be sufficiently large, and the order of the reaction has to be less than some physical parameter α. This parameter α can be interpreted as a measure for the phase-equilibrium-driven self-inhibition of the reaction mechanism. Thus, we have found an elegant explanation for the occurrence of multiplicities. Analytical and therefore general quantitative criteria identifying the regions of multiplicity for the model system are presented. Practical relevance of our results is demonstrated by means of two examples, the Monsanto process for the production of acetic acid and the ethylene glycol reactive distillation system.     

Theory for double potential step chronoamperometry for any potential values at spherical electrodes: Simultaneous determination of the diffusion coefficients of the electroactive species

We present a general and explicit analytical solution for double potential step chronoamperometry with any applied potential values (E₁, E₂) corresponding to a reversible charge transfer process at spherical electrode. This solution is essential to analyze double pulse electrochemical techniques such as RPV and DPV. We consider unequal diffusion coefficients, initial presence of both electroactive species and that the reaction product can dissolve in the electrolytic solution or in the electrode.From the analytical equation obtained it is possible to deduce interesting simplified expressions for some particular cases: both species soluble in the electrolytic solution with equal diffusion coefficients, planar electrodes, ultramicroelectrodes when both species are soluble in the electrolytic solution, and double potential step chronoamperometry with limit current potentials (E₁−E^°′→−∞, E₂−E^°′→+∞). In this last case, when reaction product is not initially present it is pointed that planar electrodes and ultramicroelectrodes cannot be used for determining both diffusion coefficients. This interesting practical consequence can be demonstrated by means of the analytical expression deduced here, which represents a notable advantage in front of numerical results.     

Analysis of multiple gas solid reactions

Multiple gas solid reactions involving one solid and N gaseous reactants are investigated in this study by using a matched asymptotic expansion technique. Two cases are particularly studied. In the first case all N chemical reaction rates are faster than the diffusion rate. While in the second case only M (M <N) chemical reaction rates are faster than the diffusion rate and the rates of the remaining (N-M) chemical reactions are comparable to that of diffusion. For these two cases the solid concentration profile behaves like a travelling wave. In the first case the wave front velocity is contributed linearly by all gaseous reactants (additive law) while in the second case this law does not hold.     

Approximate design of loop reactors

A loop reactor (LR) is an N-unit system composed of a loop with gradually shifted inlet/outlet ports. This system was shown in our previous study [Sheintuch M., Nekhamkina O., 2005. The asymptotes of loop reactors. A.I.Ch.E. Journal 51, 224-234] to admit an asymptotic model for a loop of a fixed length with N→∞. Both the finite-unit and the asymptotic model exhibit a quasi-frozen or a frozen rotating pulse (FP) solution, respectively, within a certain domain of parameters that becomes narrower as feed concentration declines.In the present paper we derive approximate solutions of the ignited pulse properties in an LR as a function of the external forcing (switching) rate. Analysis of these solutions enable us to determine the maximal temperature in the system, as well as the boundaries of the FP domain. For the optimal solution we determine the maximal temperature and conversion dependencies on the reactor length and on N. The approximate solutions are verified by comparison with direct simulations of the asymptotic model and a good agreement was found. The obtained results can be successfully used for prediction of the finite unit LR.     

Effects of temperature and pressure gradients on catalyst pellet effectiveness factors—I

The dusty gas model is used to establish the effects of temperature and pressure gradients on catalyst pellet effectiveness factors for reaction systems in which species molecular weights and transport coefficients are indistinguishable and Σν_i = 0. For this class of reactions, the total molar flux of species i is shown to be expressible simply as N_i = ?cD ??_i in terms of the molar concentration, the Bosanguet diffusivity, and the mole fraction gradient. The effects of temperature and pressure gradients are reflected only in variations in molar concentration and diffusivity. Furthermore, the temperature-pressure relationship is shown to be given by the thermal transpiration equation for a pure gas.Typical numerical results are reported for first order reactions in spherical pellets under diffusive conditions ranging from the Knudsen through the bulk diffusion regimes.The variation in diffusion regime is shown to be controlled by an additional parameter α, the Knudsen to bulk diffusion ratio. Comparisons are made with the classical Weisz-Hicks nonisothermal pellet solutions based on N_i = ?D_eff?c_i. For highly exothermic reactions, effectiveness factors are 18% lower in the Knudsen regime and 30% higher in the bulk diffusion regime than are the Weisz—Hicks values. For highly endothermic reactions with a significant diffusion limitation, the effectiveness factors are 30% lower than the Weisz—Hicks values.The classical Damköehler relationship for pellet temperature rise is shown to apply in the Knudsen regime, with the maximum dimensionless center temperature given by (1 + β), where β is the heat of reaction parameter. This temperature is accompanied by a maximum dimensionless center pressure of (1 + β)^{$\text{1}\text{2}$}.In the bulk diffusion regime, the maximum center temperature is shown to be increased by the additional term β²/4. This additional temperature rise accounts for the 42% increased bulk diffusion effectiveness for highly exothermic reactions.     

Residue curve maps of reactive membrane separation

A batch reactive membrane separation process is analysed and compared with a batch reactive distillation process by means of residue curve maps. In both processes, the chemical reaction takes place (quasi-) homogeneously in the liquid bulk phase and vapour-liquid equilibrium is assumed to be established. Additionally, in the reactive membrane separation process, selective vapour phase permeation through a membrane is incorporated.A model is formulated which describes the autonomous dynamic behaviour of reactive membrane separation at non-reactive and reactive conditions when vacuum is applied on the permeate side. The kinetic effect of the chemical reaction is characterized by the Damköhler number Da, while the kinetic effect of multicomponent mass transfer through the membrane is characterized by the matrix of effective mass transfer coefficients. The process model is used to elucidate the effect of selective mass transfer on the singular points of reactive membrane separation for non-reactive conditions (Da=0), for kinetically controlled reaction (0<Da<∞), and for equilibrium controlled reaction (Da→∞). Scalar, diagonal and non-diagonal mass transfer matrices are considered. As examples, the simple reaction A⇔B+C in ideal liquid phase, and the cyclization of 1,4-butanediol to tetrahydrofurane in non-ideal liquid phase are investigated.     

Multiplicity and uniqueness for binary,exothermic reaction in a non-adiabatic continuous stirred tank reactor

Exact multiplicity and uniqueness criteria for steady state in a non-adiabatic continuous stirred tank reactor are studied through simple tangent analysis for binary, exothermic reaction of the type A + bB → Products with rate expression r_A = kC_A^m C_Bⁿ, where A is the limiting reactant. Important parameters for multiplicity criteria are reaction orders m and n, stoichiometric coefficient b, the ratio p of feed concentration of A to that of B, dimensionless activation energy α, dimensionless heat of reaction β, dimensionless heat transfer coefficient γ and dimensionless coolant temperature. Necessary conditions for the system to have multiple exit conversions (temperatures) are defined in the (m, n, b, p, α, β, γ) space. Multiplicity is guaranteed by limiting the dimensionless space time ? in a proper range in addition to the necessary conditions. Effects of various parameters on multiplicity and uniqueness are numerically calculated and graphically represented. Theoretical prediction for multiplicity are further compared with multiplicity data reported in literatures.     

Effect of surface diffusion on the selectivity of catalytic reactions

Analysis is made on the effect of surface diffusion on the selectivity of a catalyst for a consecutive reaction A→B→C in a well-mixed stirred tank reactor. The catalyst is composed of an α region despersed on the β region. A→B is assumed to occur on the α region and B→C on the β region. Migration of B from α to B proceeds both by surface diffusion and gas phase transport. Influence of the flow rate through the reactor, the crystallite size of α, and the loading of the catalyst on the selectivity for C in the presence of surface diffusion are discussed. Under otherwise identical conditions, selectivity is increased by surface diffusion. The optimum condition for the production of C is also discussed.     

Eigenvalue-eigenfunction analysis of infinitely fast reactions and micromixing regimes in regular and chaotic bounded flows

We analyze the dynamics of a single irreversible reaction A+B→ Products, occurring in a bounded incompressible flow. Within the limits of infinitely fast kinetics, the system is reduced to an advection-diffusion equation for the scalar φ, representing the difference between the reactant concentrations. By the linearity of the governing PDE, the system evolution is determined by the properties of the eigenvalue-eigenfunction spectrum associated with the advection-diffusion operator. In particular, the dependence of the dominant eigenvalue Λ—yielding the time-scale controlling the asymptotic reactant decay—as a function of the molecular diffusivity, , for different stirring protocols is analyzed. We find , where the exponent α∈[0,1] depends upon the kinematic features of the stirring flow. When the kinematics is regular within most of the flow domain (e.g. two-dimensional autonomous flows or time-periodic protocols possessing large quasiperiodic islands) a purely diffusive scaling, α=1 settles as . The singular scaling α=0 is found in the case of globally chaotic kinematics, whereas mixed regimes, 0<α<1, occur in flows that are characterized by the coexistence of quasiperiodic and chaotic behavior. The analysis of spectral properties of the advection-diffusion operator provides a new classification of micromixing regimes, and new mixing indices for quantifying homogenization performances in the presence of diffusion.     

Buoyancy-driven pattern formation in reactive immiscible two-layer systems

Buoyancy-driven convection can be induced by concentration and temperature gradients near the interface between two immiscible fluids filling a vertical Hele–Shaw cell, each of them containing a reactant of an exothermic A+B→C reaction taking place in the bulk of the lower layer. A chemo-hydrodynamical pattern appears then due to the combined action of Rayleigh–Taylor, diffusive layer convection and Rayleigh–Bénard instabilities occurring either below or above the interface. The mathematical model we develop to describe such dynamics consists in a set of reaction–diffusion–advection equations ruling the evolution of concentrations and temperature coupled to Navier–Stokes equation, written in a Hele–Shaw approximation. We perform numerical simulations of the non-linear system and study the influence on pattern formation of changes in the Damköhler number of the problem and in the ratio of initial reactant concentrations.     

Blending of irregularly shaped particles

Existing blending theories have mostly been verified in the past by mixing of monosized, smooth, spherical particles. In this study, binary mixing of non-spherical particles with rough surfaces is shown to adhere to diffusional mixing equations as long as the mean diameters of the two fractions (A and B) are identical, i.e.d_A = d_B. When this is not the case, blending rates (diffusional coefficients of blending, D) reduce drastically when the small particles are not of such a size that they will fit into the interstices between the larger particles. It is shown also that the final standard deviation in these cases is many orders of magnitude higher than the predicted random standard deviation. When d_A = d_B, the diffusion coefficient will depend on the magnitude of d_A. The diffusion coefficient is not a function of the ratio of A to B. In blending in a horizontal mixer, D decreases with addition of lubricant. This is not the case in a V-blender, where bulk mass transfer appears to be the controlling step. In all other respects, V-blending is identical, functionally, to the profiles obtained in cylindrical blending.     

Emission analysis of CdO–Bi2O3–B2O3 glasses doped with Eu and Tb

This article reports on the emission properties of cadmium bismuth borate (CdBiB) glasses as a function of doping concentrations of Eu³⁺ and Tb³⁺ ions. The functional groups present in the glasses have been identified by analyzing FT-IR spectra. The emission spectra of Eu³⁺ and Tb³⁺:CdBiB glasses have shown reddish green emissions at 616 nm (⁵D₀→⁷F₂) under the excitation at 465 nm and at 547 nm (⁵D₄→⁷F₅) under the excitation at 485 nm, respectively. The Judd–Ofelt (J–O) theory was applied to evaluate the J–O intensity parameters from the absorption and the emission spectra; by using the J–O intensity (Ω_λ) parameters, spontaneous emission transition probability (A), total radiative transition rate (A_T), radiative lifetime (τ_R) and branching ratios (β) of the various emission transitions have been computed for both Eu³⁺ and Tb³⁺:CdBiB glasses. The quenching behavior in the emission intensity with increased concentration of Eu³⁺ and Tb³⁺ was observed, which could be useful for optimizing the compositions toward practical applications.     

Estimation of activation energies during hydrodesulfurization of middle distillates

Hydrodesulfurization (HDS) of different petroleum distillates was carried out in a batch reactor using commercial CoMo catalyst and reaction conditions similar to industrial practice. Various experiments (agitation, particle size and amount of catalyst tests) were conducted with different hydrocarbon feeds to assure the operation under kinetic regime. Reaction orders and activation energies for each feed were determined by two approaches (linear and non-linear regressions). Both kinetic parameters (n and E_A) were found to follow a direct relationship with sulfur content in the feed. Reaction orders ranged between 1.96 and 3.36 and activation energies from 21.49 to 41.96 kcal/mol, which were within the range of those reported in the literature. High values of reaction order were attributed to contributions of HDS reaction of each individual compound exhibiting very different reaction rate constants.     

The effect of stoichiometric ratio λ on the performance of a polymer electrolyte fuel cell

The effect of stoichiometric ratio λ on performance of a polymer electrolyte fuel cell is rationalized. λ→∞ corresponds to uniform oxygen concentration along the channel; the limiting current density then is determined by transport of oxygen across the cell. Finite λ corresponds to decreasing oxygen concentration along the channel. In that case j-axis of a voltage-current plot is “compressed” by a factor and apparent value of limiting current density appears to be f_λ times lower. The function f_λ→∞ as λ→1; physically, in this limit all available oxygen is consumed close to the channel inlet and apparent limiting current tends to zero. Recently derived expression for the voltage-current curve of the cell is extended to take into account the effect of λ. Fitting of experimental voltage-current curves with the new relation gives an estimate of basic parameters of the cell.     

Thermally safe operation of liquid-liquid semibatch reactors Part II: Single diffusion controlled reactions with arbitrary reaction order

The thermally safe operation of an indirectly cooled semibatch reactor in which an exothermic liquid-liquid reaction occurs corresponds to conditions of potentially very high macrokinetic conversion rates compared with the supply rate of the coreactant, which accumulation in the system remains consequently low. This leads to the definition of a target temperature that can be compared with the real temperature-time profile, in order to develop boundary diagrams which summarize all the possible thermal behaviors of the reactor and can be used for safe scale-up purposes. The variable parameters which appear in such diagrams are an exothermicity and a reactivity number derived from the expressions of the conversion rates in the kinetically or diffusion controlled regime, respectively.In this work the influence of the microkinetic rate of reaction on the shape and location of the boundary diagrams for single liquid-liquid diffusion controlled reaction systems is discussed, extending to this regime the results previously obtained for kinetically controlled reactions.Also in the case of diffusion controlled reactions, it is shown that for many practical systems, using boundary diagrams based on (1,1) reaction orders can lead to both unsafe or not necessary low production operating conditions. Consequently, a number of new boundary diagrams for arbitrary reaction orders is presented and some rules-of-thumb useful to their application are discussed.