Multiplicity,stability and dynamics for isothermal autocatalytic reactions in CSTR

Multiplicity, stability and dynamics for autocatalytic reactions of the type A? → rR + ? with overall rate expression r_A = kC_A^mC_Rⁿ in a continuous stirred tank reactor (CSTR) are studied. Exact multiplicity criteria are defined in the parameter space. Stability analysis shows that no periodic oscillation is possible for the system. When multiplicity occurs, some minimum of R present initially in the reactor is required in order for the high steady state to be achieved. Loci of transient extreme for product are also investigated. Multiplicity and uniqueness criteria are further compared with reported experimental data in literature.     

Exact criteria for uniqueness and multiplicity of an nth order chemical reaction via a catastrophe theory approach

The development of a simple, generalized technique for the exact determination of regions of unique and multiple solutions to certain nonlinear equations via a catastrophe theory-implicit function theorem approach, is presented. The application of this technique to the nth order chemical reaction in the nonadiabatic and adiabatic CSTR yields exact, explicit bounds for all n ≥ 0. To our knowledge, this is the first report of exact, explicit bounds for these systems, except for n = 0, 1 for the adiabatic CSTR, and n = 1 for the nonadiabatic CSTR. For the nonadiabatic CSTR, these bounds show that the higher the reaction order, the smaller the region in parameter space for which multiplicity can occur for all γ and x_2c, (dimensionless activation energy and coolant temperature, respectively). This behavior is similar to that reported by Van den Bosch and Luss[1] for the adiabatic CSTR. The zeroth order reaction in the nonadiabatic CSTR exhibits more complex behavior and assumes characteristics of both high and low reaction orders insofar as increasing and/or decreasing the uniqueness space, in comparison to all other n > 0.An exact implicit bound between regions of uniqueness and multiplicity is also derived for the nth order reaction in a catalyst particle with an intraparticle concentration gradient and uniform temperature, and is fully demonstrated for the first order reaction. In addition, explicit criteria, sufficient for uniqueness and multiplicity of the catalyst particle steady state, stronger than those of Van den Bosch and Luss, are also developed by combining the present technique with bounds suggested by these authors.     

Conformational behaviour of (styrene-p-chlorostyrene) triblock copolymers in dilute solutions. 2. Temperature dependence of conformational behaviour in a selective solvent

Intrinsic viscosity, osmotic second virial coefficient and light scattering of the B_mA_nB_m and A_mB_nA_m copolymers (A, styrene monomeric unit; B, p-chlorostyrene monomeric unit, m and n denote the number of units) in cumene which is a good solvent for polystyrene but a θ solvent for poly(p-chlorostyrene) at 59.0°C, were studied over the temperature range 65° to 15°C. The results suggested that conformational transition from a random coil form to a segregated form occurs at a critical temperature which appears to be in the range 40° to 30°C, depending on the composition, molecular weight and structure of the block copolymers; the θ condition could not be attained by cooling the block copolymer solutions, and micelle formations due to intermolecular associations were found in some cases below the transition temperature.     

Crystal structure and thermal expansion of ammonium pentaborate NH<Subscript>4</Subscript>B<Subscript>5</Subscript>O<Subscript>8</Subscript>

Anhydrous ammonium pentaborate NH₄B₅O₈ has been synthesized by thermal dehydration of larderellite NH₄[B₅O₇(OH)₂] · H₂O at a temperature of 290°C for 7 h. The crystal structure has been determined from the X-ray powder diffraction data: a = 7.58667(5) Å, b = 12.00354(8) Å, c = 14.71199(8) Å, R _p = 6.23, R _wp = 7.98, R _B = 12.7, R _F = 8.95, and β-KB₅O₈ structure type. The double interpenetrating framework is formed by pentaborate groups, each consisting of a boron-oxygen tetrahedron and four triangles, in which all oxygen atoms are bridging. The thermal behavior of the NH₄B₅O₈ compound has been investigated using thermal X-ray diffraction. As for other pentaborates of this type, the thermal expansion of the NH₄B₅O₈ compound is anisotropic and reaches a maximum along the a axis. The thermal expansion coefficients are as follows: α _a = 39 × 10^?6, α _b = 6 × 10^?6, α _c = 20 × 10^?6, and α _V = 65 × 10^?6 °C^?1.     

Effects of macromolecular architecture on the micellization behavior of complex block copolymers

The effect of macromolecular architecture on the aggregative behavior of A–B block copolymers with different complex structures in selective solvents was studied by Dissipative Particle Dynamics. We focus on two types of diblock copolymers, (I) asymmetric linear diblocks B_yA_x and (II) miktoarm stars (B_y)_n(A_x)_m, where A block is solvophilic and B block solvophobic. Note that y and x are the block lengths of A and B blocks; n and m denote arm numbers of A and B blocks in the star. For type I linear copolymer with a given ratio of y/x, the aggregation number <p> varies with the total length (x + y) for y/x > 1 but is essentially independent of the total length for y/x ? 1. For type II star copolymer with m · x = 24 and n · y = 24, the aggregation number varies with the branch number m at a given number of solvophobic blocks n. There exists a minimum aggregation number at m ≈ 4 so that <p> declines first and then grows with increasing the branch number. Moreover, <p> increases as the polymer concentration is increased. Our simulation results indicate that at a given chemical composition, the micelle properties such as aggregation number and micellar morphology may vary with the macromolecular architecture.     

Reversible autocatalytic reactions in an isothermal CSTR: Multiplicity,stability and relaxation times

In a recent paper, Gray and Scott have considered the autocatalytic reactions: A → B; rate ∝ abⁿ, n = 0, 1 or 2 where a and b are the concentrations of A and B respectively. Interest centred mainly on irreversible systems but for which the catalytic species was not indefinitely stable, decaying instead by a rate proportional to its concentration b. In practice all chemical reactions are to some extent reversible. The present work investigates the effect of reversibility for the cases in which B does not decay. Bounds are established on the ranges of residence-time t_res for which multiple stationary-states are possible. The stability of the different solutions is assessed and the relaxation times t_* characterizing the decay of perturbations are given. For deceleratory reactions reversibility enhances the decay rates (decreases t_*): for autocatalytic systems the decay rates may be unaltered or even reduced by a finite reverse reaction-rate. Also treated is the influence of non-zero inlet concentrations b₀ of the autocatalyst. This may lead to greater changes in the patterns of behaviour possible than those observed in the absence of reversibility. The algebraic analysis remains tractable throughout so the various effects can readily be interpreted in physical terms.     

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.     

Uniqueness and multiplicity criteria for an nth order chemical reaction

New and very strong criteria are presented for a priori prediction of the conditions for which the steady-state lumped parameter model of an nth order chemical reaction (n ≥ 0) in an adiabatic CSTR has either a unique or multiple solutions. The criteria show that the higher the order of the reaction the smaller is the region in the parameters space for which multiplicity can occur.New uniqueness and multiplicity criteria are developed also for an nth order reaction in a porous catalyst using a model, which accounts for intraparticle concentration gradients, while assuming a uniform intraparticle temperature different from the ambient one. The region in the parameters space for which steady state multiplicity can occur for this model is smaller than that for a corresponding lumped model, which ignores the intraparticle concentration gradients.     

Synthesis of (ABCB)n type ternary amphiphilic multiblock copolymer via poly(ethylene oxide) macro-chain transfer agent

A novel (ABCB)_n type ternary amphiphilic multiblock copolymer was synthesized by stepwise insertion of monomers into the trithiocarbonate-embedded poly(ethylene oxide) (PEO) macro-chain transfer agent (PEO-CTA)_n. (PEO-CTA)_n was synthesized first by coupling of α,ω-dihydroxyl PEO with dicarboxylic trithiocarbonate, then styrene (St) and t-butyl acrylate (^tBA) were inserted into the (PEO-CTA)_n successively to yield (PEO-b-PS)_n and (PEO-b-PS-b-P^tBA-b-PS)_n, respectively. After hydrolysis of the (PEO-b-PS-b-P^tBA-b-PS)_n, the final product (PEO-b-PS-b-PAA-b-PS)_n was obtained.     

Crystallization behavior of oxyethylene/oxybutylene diblock and triblock copolymers

The crystallization behavior of poly(oxyethylene)-b-poly(oxybutylene) block copolymers with different compositions, morphologies and architectures (E_mB_n diblock copolymers and E_mB_nE_m, B_nE_mB_n triblock copolymers) were investigated and the effect of volume fraction and architecture on the crystallization temperature (T_c) in non-isothermal crystallization was determined. It is found that the E_mB_n diblock copolymers having long E blocks exhibit similar crystallization temperatures, irrespective of volume fraction and morphology, but for the block copolymers with shorter E blocks the crystallization temperature increases with both the volume fraction, φ_E, and the length, m, of the E block. Some block copolymers with extremely low T_c, which fall into the temperature range normally associated with homogenous nucleation, were chosen for time-resolved small-angle X-ray scattering (SAXS) and isothermal crystallization kinetics experiments. The results show that breakout crystallization occurs in all these block copolymers. Therefore, unlike E_mB_n/B_h blends, there is no obvious relationship between T_c and crystallization behavior in neat block copolymers and homogeneous nucleation does not definitely lead to confined crystallization. The values of χ_c/χ_ODT for all the block copolymers with hex and bcc morphology were also calculated. It is found that all the block copolymers have χ_c/χ_ODT<3, in agreement with the previously reported critical value and consistent with their breakout crystallization behavior.     

Adsorption equilibrium constants of methyl oleate and methyl linoleate in vapor phase on supported copper and nickel catalysts

Adsorption equilibrium constants for methyl oleate and methyl linoleate in vapor phase on supported copper and nickel catalysts have been determined using the technique of pulse gas chromatography. The results are discussed in relation to selectivity in fat hydrogenation. Notation: A, column cross-section, m₂ ; a_n,b_n, nth Fourier coefficients; c, concentration of adsorbate in bulk flow, mol/m³ ; c^* = c/ ∫ ₀ ^∞ cdt, normalized concentration of adsorbate in bulk flow; C_i, concentration of adsorbate in catalyst pores, mol/m³ ; c_a, concentration of adsorbate on catalyst surface, mol/kg; c_TOT, active area of catalyst as measured by hydrogen adsorption, mol/kg; D_e, effective diffusion coefficient of adsorbate in catalyst, m²/s; D_ea, axial dispersion coefficient based on void cross-section, m²/s; h_n, nth coefficient in Hermite polynomial expansion; H_n nth Hermite polynomial; ΔH_A, adsorption enthalpy, kJ/mol; ΔH_vap , heat of vaporization, kJ/moll; k_a, adsorption rate constant, m³/kgs;K_A, adsorption equilibrium constant, m³/kg; K₀ , preexponential factor defined in Eqn. 8, m³/kg; k_f, mass transfer coefficient, m/s; L, bed length, m; q, flow rate, m³/s; R, particle radius, m; R_g, gas constant; t, time, s; T, temperature, K; T_F, period of Fourier expansion, s; u = q/A, linear velocity, m/s; z, length coordinate in packed column, m. Greek symbols: δ(t), Dirac delta function; ∈_B, void fraction of bed; ∈_-p, particle void fraction, ρ_rp, particle density, kg/m³ ; ξ, radial coordinate in particle, m; μ₁, first absolute moment, μ₂, second central moment.     

The anodic dissolution process on active iron in alkaline solutions

Steady state anodic polarization curves were taken for Armco iron and in some experiments for high purity Puratronic iron in KOH solutions in the concentration range 5 x 10^?2-5M. After the 30 min cathodic pretreatment, well reproducible anodic Tafel plots are obtained. The experimentally obtained diagnostic criteria b_a = 67–70 mV dec^?1, n_OH? = 1.1 and nHFeO₂^?= ?0.45 are interpreted by the anodic reaction mechanism in which FeOH_adsand Fe(OH)_2,ads appear as the intermediates adsorbed under Temkin conditions, the primary stable product of the electrode reaction being HFeO₂^?, and the final product Fe (OH)₂, formed by precipitation.     

Transformation and thermal stability of Li-α-sialon ceramics

Two phase α/β and single phase α lithium sialons with different m and n values were produced by hot pressing at 1730–1750°C at 30 MPa for 30–40 min in a graphite resistance furnace. When the two-phase samples were heat-treated at lower (1200–1450°C) temperatures in different packing powders, an increase in the amount of α was observed, due to β-sialon in the as-sintered material reacting with grain boundary liquid to form more α. β→α transformation at low temperatures has not been reported previously in any sialon system and in the present case is believed to occur because the α-sialon phase field in the lithium sialon system shifts slightly towards the β-sialon line at lower temperatures. The thermal stability of lithium α-sialon is good in the centre of the single-phase α region when surrounded by a Li-containing powder bed. However, towards the edges of the single-phase region, compositional changes occur on heat-treatment. Thus, samples with high m, n values decompose into β-sialon plus other Li-containing phases. During heat-treatment of other compositions when surrounded by a BN powder bed, the composition of the α-sialon phase continually readjusts towards the α/β sialon phase region as a function of time and this is followed by decomposition of the α phase. Evaporation of the Li⁺ stabilising cation is believed to be the main reason for this behaviour. The effects of m and n value, heat treatment parameters and packing powder on the thermal stability of Li α-sialons are discussed.     

Magnetic field influence on mass transport phenomena

The oxidation reactions of hexacyanoferrate(II) and hydroquinone in KCl media were studied on disk platinum electrodes using chronoamperometry under a strong magnetic field (1.74 T). The limiting current measured under magnetic field i_B can be expressed as a function of parameters that control the mass transfer phenomenon by i_B = KC^aD^bd^cν^e?^fB^gn^h. C represents the electroactive species concentration, D the diffusion coefficient of the electroactive species, d the working electrode diameter, ν the kinematic viscosity of the electrolyte, ? the dielectric constant of the solution, B the magnetic field strength, n the number of electrons involved in the redox process and K is a proportionality constant. Contribution of B to the limiting current is well established (g = 1/3), whereas the contribution of D has to be confirmed (b = 1). The aim of this work was to specify the influence of the other parameters for which various results have been published in recent literature. We concluded that i_B = KC^4/3Dd^5/3ν^−2/3?^−7/4B^1/3n, quantifying for the first time, to our knowledge, the drastic influence of the electrolyte dielectric constant.     

Acid dissociation constants of anilinium and mono-nitro anilinium ions in formamide

The acid dissociation constant of anilinium ion and those of o-, m- and p-nitro anilinium ions have been determined accurately in formamide over the temperature range 10–50°C (283.15–323.15 K) from the measurement of the emf of the cells, Pt, H₂/B(m₁), BHCl(m₂)/AgClAg.The dissociation constants at 25°C (298.15 K) have been checked by determining the same with the help of the cells, Pt, Q-QH₂/B(m₁), BHCl(m₂/AgClAg.Thermodynamical changes ΔG⁰, ΔH⁰ and ΔS⁰ for the dissociation processes have been evaluated. The results show that the dissociation constants of these conjugate acids are in general lower (pK_a's higher) in formamide than their corresponding values in water.     

Evolution of core–rim structures and phase transformations in infra-red transmitting Y-α-SiAlON ceramics

Core–rim structures were observed as common features in Y-α-SiAlON ceramics hot-pressed between 1550?1950 °C. We found most dopants were taken into α’-rims, and a transition layer grown first on α-cores from liquid-phase over-saturated with metal solutes. Elongated β’-grain were formed as minor phase with α’- or AlN-cores thus only after the α’ matrix had consumed up all Y solutes, revealing that the α’ → β’ transformation is controlled by the transient liquid-phase and similar defects and dangling bonds could be detected in both SiAlON phases by cathodoluminescence. Quantitative assessment of Y_m/3Si_12?(m+n)Al_m+nO_nN_16?n demonstrates the multiphase evolution, initiated by over-saturation of Y solutes at low temperatures thus retaining α-phase as cores to lower the infra-red transmittance, dictated by homogenization of Al solutes at higher temperature. The elimination of those phase boundaries leads to better dopant and sintering design for achieving transparent and high-performance SiAlON ceramics.     

Thermogravimetric investigations in prediction of coking behaviour and coke properties derived from inertinite rich coals

The coal quality, temperature, pressure, heating rate, various processes and reactor type affect coking behaviour of coal and resulting coke properties. Several petrographic and chemical methods were proposed earlier for prediction of coking behaviour of coals. Inertinite rich coal samples (I^mmf>30 vol%) having different petrographic compositions were selected for thermogravimetric investigations (DTA, DTG and TGA) and their coking behaviour was studied. The petrographic build up, micro-structural properties (porosity and cell wall thickness) and mechanical strength of the resulted cokes were also investigated. ΔH and ΔH_max (the main endothermic area of heat absorption and fast absorbing main endothermic area, respectively) were distinguished in DTA curves. ΔA and ΔA_max (the main decomposition area and fast disintegrating main decomposition area) under DTG curves were identified. ΔH_max/ΔA_max shows good correlation with Roga's indices (RI, caking properties) as well as with petrographic caking ratio (PCR). The coarse mosaic content of cokes seem to depend on LΔT_max/ΔT_max (ratio of weight loss during fast decomposing main reaction to temperature difference) under DTG. L^mΔT/ΔT (ratio of weight loss during main decomposing reaction to temperature difference) under DTG exhibits correlation with p₁ (mean pore size) and t₁ (mean cell wall thickness) of cokes. ΔA_max/(L^mΔT_max) also indicates good relationship with mechanical strength of cokes. (L^mΔT_A−T_B)/(L^mΔT) (i.e. ratio of weight loss during main endothermic reaction under DTA to weight loss during main decomposing reaction) appears to have relationship with DD (compactness) of cokes. The course of main endothermic reaction/main decomposition reaction under DTA, DTG and TGA seems to govern coking behaviour and the resulting coke strength, which in turn is controlled by microlithotypes.     

Crystal structure of new polymorphic modification β-Ca<Subscript>3</Subscript>B<Subscript>2</Subscript>SiO<Subscript>8</Subscript>, β-α phase transition and thermal expansion of α- and β-modifications

Single crystals of the β-Ca₃B₂SiO₈ new monoclinic modification have been obtained by cooling the melt of a stoichiometric composition. The crystal structure has been determined from the single crystal X-ray diffraction data and refined with R = 0.059 (wR = 0.069) in the monoclinic space group P2₁/m. The thermal behavior of the synthetic borosilicate has been studied. At 472 ± 5°С, a reversible phase transition of the first order occurs, leading to the formation of the orthorhombic α-Ca₃B₂SiO₈ modification. The thermal expansion of α- and β-modifications of Ca₃B₂SiO₈ is anisotropic: (α₁₁ = 15, α₂₂ = 16, α₃₃ =–1, α _V = 30 × 10^–6°С–1) and α₁₁ = 9, α₂₂ = 28, α₃₃ = 1, α _V = 38 × 10^–6°C–1, respectively.