Autocatalytic reactions in the isothermal,continuous stirred tank reactor: Isolas and other forms of multistability

Autocatalytic reactions are often complicated, and analyses of their behaviour in open systems can seem too particular to permit useful generalisation. We study here the simplest of circumstances (uniform temperatures and concentrations in the isothermal CSTR) and the simplest of reaction schemes: (i) quadratic autocatalysis (A + B→2B); and (ii) cubic autocatalysis (A + 2B→3B). The catalyst B may be stable or have a finite lifetime (B→ inert products). Allowing for this finite lifetime adds another dimension to the interest.The phenomena encountered include multistability, hysteresis, critical extinctions, critical ignitions, and anomalous relaxation times (though infinite values do not arise). Patterns of stationary states as function of residence time can show isolas and mushrooms. All these aspects yield to simple algebraic analysis. The presence of the catalyst B in the inflow can make qualitative differences of a kind parallelled by an additional, non-catalytic reaction of the same stoichiometry (e.g. A→B). Invoking the reversibility of the reactions neither increases nor diminishes their variety, and thermodynamic considerations have little to do with the many different patterns of reactivity displayed.The local stability of the various stationary states has also been characterized. Quadratic autocatalysis shows limited variety (stable node, stable focus); cubic autocatalysis generates all the kinds of stationary state possible in a two-variable system. Again all the algebra is straightforward if not always simple. Sustained oscillatory behavior (limit cycles) also occur.All these remarks relate to isothermal systems, but there are the most striking parallels between isothermal autocatalysis and the exothermic, first-order reaction in the CSTR. Behaviour with an autocatalyst of complete stability corresponds to perfect heat insulation (adiabatic operation) in the non-isothermal, exothermic system.     

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.     

Periodic operation of isothermal plug-flow reactors with input multiplicities

The performance of isothermal plug-flow tubular reactors under periodic inlet concentrations is theoretically analyzed for improvement in yield for homogeneous consecutive reactions A→ B → C which, under conventional steady-state operation, show input multiplicities in the flow rate on the yield of B. Two values of flow rate give an identical yield of B under steady-state operation. For periodic operation, a rectangular pulse is assumed for the inlet concentration. It is shown in three numerical examples that under concentration forcing the yield is significantly different for these two inlet flow rates. Under periodic operation the larger value of flow rate gives a higher yield of B.     

The effect on selectivity of the macroscopic distribution of the components in a dual function catalyst

This paper is concerned with the effect on selectivity of the macroscopic distribution of the components X and Y of a dual function catalyst. Three catalyst formulations are considered: (1) catalyst I where catalyst X 1: distributed at the pore mouth, (2) catalyst II where catalyst Y is distributed at the pore mouth, and (3) nonporous catalyst III where catalysts X and Y are mixed together. The selectivities of each catalyst are compared for the consecutive reaction scheme A_X→^k₁B_Y→^k₂C_X→^k₁D and factors influencing the optimum catalyst composition are discussed. The results show that under certain circumstances the macroscopic distribution of catalyst functions greatly affects the yields of intermediate products and suggest that the macroscopic distribution of functions can be an important variable in catalyst design.     

Model-based,thermo-physical optimisation for high olefin yield in steam cracking reactors

The steam cracking practice seems to have reached a stage of maturity which makes it increasingly difficult to improve ethylene yield. In order to determine if there is still scope for yield improvements it is helpful to know what the optimal reaction conditions for the steam cracking process are. This work presents a model-based synthesis approach that enables to determine the optimal thermal and physical reaction conditions for a particular feed, maximising olefin yield. A distributive reaction-mixing synthesis model has been combined with an industrially proven large kinetic scheme, SPYRO^®, which contains over 7000 reactions between 218 molecular and 27 radical species. The model combination allows optimising the following degrees of freedom with respect to olefin yield: feed distribution, product removal, macro-mixing, along a reaction volume coordinate. The reaction temperature upper limit is put at 1300 K, exceeding the current (metallurgical) bound by 100 K. For the cracking of ethane a linear-concave unconstrained temperature profile with a maximum temperature of ∼1260 K proves optimal which is lower than allowed while all ethane should be supplied at the entrance of the reaction volume. For propane and heavier feedstocks an isothermal profile at the upper temperature bound, with dips at the beginning and the middle of the reaction coordinate is optimal, while distribution of the hydrocarbon feed along the reactor coordinate results in higher yields. The theoretical maximum achievable ethylene yield for ethane cracking is found to be 66.8 wt% while in conventional cracking typically 55 wt% is considered to be the maximum value. This optimum is constrained by the pressure which is at its lower bound. The resulting residence time is in the same order as with current technology for ethane cracking. For the more heavy feedstocks these times are one order of magnitude smaller which will be a challenge for designing.     

The selectivity of porous catalysts: parallel reactions

The influence of intraparticle mass diffusion on the overall and relative rates of the parallel reactions A → B and A → C is examined for the case of an isothermal catalyst particle with no external concentration or temperature gradients. Calculations for three pairs of reaction orders: (0,1), (0,2) and (1,2), show that internal concentration gradients can cause the yield of the product formed in the higher-order reaction to decrease by as much as 80 percent. Extension of the analysis to other reaction orders is discussed. Criteria for close approach to various asymptotic limits are presented, together with an asymptotic expression which allows the maximum effect of pore diffusion on selectivity to be estimated for any pair of reaction orders.     

Effect of dilution on the activity of diluted porous catalysts

For a composite catalyst, consisting of active catalyst beads distributed uniformly within a less active but more porous matrix of a diluent, the effectiveness factor exhibits a maximum as a function of the amount of active catalyst. Conditions have been established for the existence of an optimum dilution, and expressions have been developed for its estimation for an isothermal first order reaction. A nonisothermal analysis of the same problem, for an exothermic reaction, indicates a shift in the optimum catalyst dilution towards higher dilutions. The analysis has been extended to the catalytic cracking process, where such catalysts are used. In this process large hydrocarbon molecules decompose to an intermediate product (A→B) in the diluent (silica-alumina), while the subsequent reactions

are carried out in the active microparticles (zeolite). Using for the sake of simplicity first order kinetics, the analysis demonstrates the occurrence of a maximum in the rate of production of component C as a function of the amount of the active catalyst. Finally, it is shown in the paper that dilution can also affect the selectivity of a catalyti     

Optimal oxygen concentration strategy through an isothermal oxidative coupling of methane plug flow reactor to obtain a high yield of C2 hydrocarbons

An optimal oxygen concentration trajectory in an isothermal OCM plug flow reactor for maximizing C₂ production was determined by the algorithm of piecewise linear continuous optimal control by iterative dynamic programming (PLCOCIDP). The best performance of the reactor was obtained at 1,085 K with a yield of 53.9%; while, at its maximum value, it only reached 12.7% in case of having no control on the oxygen concentration along the reactor. Also, the effects of different parameters such as reactor temperature, contact time, and dilution ratio (N₂/CH₄) on the yield of C₂ hydrocarbons and corresponding optimal profile of oxygen concentration were studied. The results showed an improvement of C₂ production at higher contact times or lower dilution ratios. Furthermore, in the process of oxidative coupling of methane, controlling oxygen concentration along the reactor was more important than controlling the reactor temperature. In addition, oxygen feeding strategy had almost no effect on the optimum temperature of the reactor. Finally, using the optimal oxygen strategy along the reactor has more effect on ethylene selectivity compared to ethane.     

The influence of axial pressure on relaxor properties of BaBi2Nb2O9 ceramics

On the basis of our studies it results that dielectric properties of BaBi₂Nb₂O₉ ceramics are sensitive to axial pressure applied. The pressure causes an increase of dispersion in the real part of dielectric permittivity ?′(T,f) and a rise in the temperature T_m at which the maximum in ?′(T,f) dependence occurs. The applied pressure induces in the ?′(T) dependence an additional step-like anomaly, which appears at the temperature T_A < T_m. The applied pressure shifts both T_m and T_A at the same rate, i.e. dT_A/dX = dT_m/dX = +14 °C/kbar at high axial pressure range, above the threshold pressure X_thresh. The Vogel–Fulcher relationship is employed to determine the axial pressure influence on relaxor properties of BBN ceramics. The simulated order parameter q takes non-zero values below Burn‘s temperature T_B, where the polar clusters appear on cooling. For pressures higher than 0.8 kbar, the T_B changes at the rate dT_B/dX = −200 °C/kbar. The decrease in the difference between Burn's T_B and the freezing T_f temperatures induced by the applied axial pressure is observed. This could be ascribed to the narrowing of temperature range of relaxor behavior.     

Theoretical study of the application of porous membrane reactor to oxidative dehydrogenation of n-butane

This paper is the theoretical study of the oxidative dehydrogenation of n-butane in porous membrane reactors. Performance of the membrane reactors was compared with that of conventional fixed-bed reactors. The porous membrane was employed to add oxygen to the reaction side in a controlled manner so that the reaction could take place evenly.Mathematical models for the fixed-bed reactor and the membrane reactor were developed considering non-isothermal condition and both radial heat and mass dispersion. From this study, it was found that the hot spot problem was pronounced particularly near the entrance of the conventional fixed-bed reactor. In addition, the assumption of plug flow condition did not adequately represent the reaction system. The effect of radial dispersion must be taken into account in the modelling.The use of the porous membrane to control the distribution of oxygen feed to the reaction side could significantly reduce the hot spot temperature. The results also showed that there were optimum feed ratios of air/n-butane for both the fixed-bed reactors and the membrane reactors. The membrane reactor outperformed the fixed-bed reactor at high values of the ratio. In addition, there was an optimum membrane reactor size. When the reactor size was smaller than the optimum value, the increased reactor size increased the reaction and heat generation and, consequently, the conversion and the selectivity to C₄ increased. However, when the reactor size was larger than the optimum value, oxygen could not reach the reactant near the stainless steel wall. It was consumed to react with the product, C₄. As a result, the yield dropped. Finally, it was found that the increase of wall temperature increased the yield and that the feed air temperature could help control the temperature profile of the reaction bed along the reactor length.     

The comparative kinetic study of non-isothermal and isothermal dehydration of swollen poly(acrylic acid) hydrogel using the Weibull probability function

The isothermal and non-isothermal dehydration of swollen poly(acrylic acid) hydrogel has been investigated. The possibility of calculating the isothermal conversion curves from the non-isothermal conversion curves was examined. It was found that the experimental conversion curves under non-isothermal and isothermal conditions can be described by the Weibull probability distribution function. Changes in the values of distibution parameters with heating rate (non-isothermal dehydration) and operating temperature (isothermal dehydration) were identified. The mathematical procudere for determining the non-isothermal kinetic conversion curves, based on the functional relationship between distribution parameters and the heating rate was developed. In addition, the new mathematical procedure for calculating the isothermal kinetic curves from the non-isothermal ones at a certain temperature was established. It was found that the large deviation exists between the experimental and calculated isothermal conversion curves. In order to explain the difference in the mechanism of hydrogel dehydration under non-isothermal and isothermal conditions, the dependence of the apparent activation energy on the degree of conversion (E_a = E_a(α)) was performed using the isoconversional (model-free) methods. It was found that in the case of non-isothermal dehydration, the apparent activation energy exhibits a concave decrease with the degree of conversion (α), while for the isothermal dehydration, the apparent activation energy exhibits a convex increase with α. Certain changes in a variety of E_a with α are explained on the basis of different relaxation processes, that take place during non-isothermal and isothermal hydrogel dehydration. The characteristic features of density distribution functions of the apparent activation energies are derived in the case of non-isothermal and isothermal dehydration. It was found that the differences in characteristic properties of derived distribution functions are a consequence of different relaxation processes, which taking place under various experimental conditions.     

Yellow-emitting (Tb1−xCex)3Al5O12 phosphor powder and ceramic (0≤x≤0.05): Phase evolution,photoluminescence, and the process of energy transfer

(Tb_1-xCe_x)₃Al₅O₁₂ yellow phosphors (0≤x≤0.05) were calcined from their coprecipitated carbonate precursors, and the effects of the temperature and atmosphere (air and H₂) of calcination on the sequence of phase evolution and the characteristics of the powders were investigated in detail. The activation energy for the crystallite growth during calcination was estimated to be ~39 kJ/mol. The powder calcined at 1000 °C showed good reactivity and was sintered into a ceramic plate of ~97% dense (average grain size: ~1.3 µm) in a H₂/Ar gas mixture at the relatively low temperature of 1500 °C. The phosphors simultaneously exhibit the 4f⁸→4f⁷5d¹, ⁷F₆→⁵D₃ and ⁷F₆→⁵D₄ excitations of Tb³⁺ and the 4f¹→5d¹ excitation of Ce³⁺ when monitoring the yellow emission of Ce³⁺ at 560 nm, suggesting the presence of efficient Tb³⁺→Ce³⁺ energy migration. The optimal Ce³⁺ content for luminescence was found to be x=0.015 and 0.01 under the direct excitation of Ce³⁺ and through Tb³⁺→Ce³⁺ energy transfer, respectively, and concentration quenching of luminescence was analyzed to be resulted from exchange interaction. Luminescence features of the phosphors, including excitation, emission, quantum yield, fluorescence lifetime, color coordinates and color temperature, were thoroughly investigated against the processing temperature and Ce³⁺ content, with an in-depth discussion on the process of energy transfer among the optically active Tb³⁺ and Ce³⁺ ions. The materials may find application in blue-light excited white LEDs.     

Autocatalytic reactions in the isothermal,continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C

The prototype, cubic autocatalytic reaction (A + 2B → 3B) forms the basis for the simplest homogeneous system to display “exotic” behaviour. Even under well-stirred, isothermal, open conditions (CSTR) we may find multistability, hysteresis, extinction, ignition and anomalous relaxation times. Allowing for the finite lifetime of the catalyst (B→inert products) adds another dimension. The dependence of the stationary-states on residence-time now yields isolas or mushrooms. Sustained oscillations (stable limit cycles) are also possible. The onset of oscillation corresponds to a change in the character of the stationary-state (from stable focus to unstable focus) and the conditions for this change can be evaluated analytically. The period of the oscillations and their amplitudes increase as the residence time is lengthened. A total of nine different phase-portraits in the a–b plane has been found.The isothermal system is less complex than the exothermic, first-order reaction in a CSTR, but there are strong analogies between the two.     

Modelling of non-premixed swirl burner flows using a Reynolds-stress turbulence closure

In this computational study, the performance of a differential Reynolds-stress turbulence model has been assessed in predicting a turbulent, non-premixed combusting swirling flow of the type frequently found in practical combustion systems. Calculations are also performed using the widely employed eddy-viscosity based k-ε turbulence model in order to examine the relative performances of these two closure models. The predictions are compared against the experimental data of mean axial and tangential velocities, turbulence quantities, gas temperatures and oxygen concentration collected in a 400 kW semi-industrial scale combustor fired with coke-oven gas using an industry-type swirl burner at the International Flame Research Foundation [17]. Computations of a corresponding non-combusting flow are also carried out and the predictions are compared with limited data available. The overall agreement between the measurements and the predictions obtained with both the k-ε and Reynolds-stress turbulence models are reasonably good, in particular, the flame properties. However, some features of the isothermal and combusting flow fields, and the flame are better predicted by the Reynolds-stress model. The subcritical nature of the isothermal flow and the effects of combustion on the size and shape of the swirl-induced internal recirculation zone in the corresponding combusting flow are well simulated by this model. The k-ε model fails to reproduce the subcritical nature of the isothermal flow. The predictions of this model erroneously show a general trend of the mean tangential velocity distribution to assume a forced-vortex profile. The levels of gas temperature and oxygen concentration in the internal recirculation zone and the enveloping shear region are on the whole better predicted by the Reynolds-stress model.     

An empirical model for kinetics of boron removal from boron-containing wastewaters by ion exchange in a batch reactor

In this study, it was investigated boron removal from boron containing wastewaters prepared synthetically. The experiments in which Amberlite IRA 743, boron specific resin was used was carried out in a batch reactor. The ratio of resin/boron solution, boron concentration, stirring speed and temperature were selected as experimental parameters. The obtained experimental results showed that percent of boron removal increased with increasing ratio of resin/boron solution and with decreasing boron concentration in the solution. Stirring speed and temperature had not significant effects on the percent of total boron removal, but they increased the starting boron removal rate. As a result, it was seen that about 99 % of boron in the wastewater could be removed at optimum conditions. On the other hand, the process kinetics were predicted by using heterogeneous fluid-solid reaction models. It was seen statistically that the kinetics of this process agreed the pseudo- second order model, as follows: X_Bl(1−X_B) = 11,241.5[OH][C]^−1.76[S/L]^2.17exp(−19,57l.2/RT)t^1.24     

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.     

Minimum end time policies for batchwise radical chain polymerization—III: The initiator addition policies

For batchwise radical chain solution polymerization, the minimum end time problem is studied by considering the initiator concentration (or feed rate) and temperature as two control variables. The kinetic model used is based on a general mechanism and involving the gel effect. It is found that the optimal initiator addition policy is to make the rate of initiation constant and rate of conversion nearly constant throughout the entire process. This results in a small variation of $\text{X}$_n and $\text{X}$_w with time and therefore in a nearly constant polydispersity (about 1.6 for styrene/toulene system) in the absence of chain transfer agent. For isothermal case, the initiator concentration should remain constant; analytical expression for temperature and initiator concentration are obtained. For nonisothermal case with high conversion, the temperature increases to a maximum and then drops, while the initiator concentration progresses oppositely. Further calculation shows that the optimal initiator addition processes are more efficient than the optimal processes with one-charge of initiator. Experimental verifications on the isothermal initiator addition policies for the polymerizations of styrene/toluene system show promising. ca*|Author to whom correspondence should be addressed.     

Kinetics of lignite pyrolysis in an entrained-flow,isothermal furnace

An entrained-flow, isothermal furnace was used to study rapid pyrolysis in nitrogen of a Texas lignite. Weight losses for fixed residence times up to 0.4 s were independent of particle size over the range of mean diameter from 41 to 201 μm and increased with temperature elevation over the range 700 to 1000 °C. The maximum yield of volatiles, 66.7 wt% of the dry-ash-free coal, was particle size independent and relatively temperature independent over the range of operating conditions. The maximum yield represented a fractional increase of 1.30 over the ASTM volatile matter. This yield enhancement was associated with a reduction in the preponderence of secondary char-forming reactions of the volatiles and not directly with the increase in heating rate from 20 °C s^?1 in the ASTM test to >10⁴ °C s^?1 in the isothermal furnace. Pyrolysis was multi-stage; initial rapid devolatilization to 50% weight loss was followed by slower latter-stage devolatilization. First stage pyrolysis could be fitted by a single, first-order kinetic expression whether total weight loss was taken as just that for the first stage or for complete devolatilization. Rate constants for the two cases were 0.6 × 10³exp(? 45 kJ/mol/RT) and 0.9 × 10²exp(? 32 kJ/mol/RT) s^?1, respectively. One or more additional equations would be necessary to describe the completion of pyrolysis. It is argued that the relatively low activation energy is not necessarily indicative of physical rate control and therefore not necessarily in contradiction with the absence of particle size effects.     

Modification of the pyrolysis/carbonization of PPTA polymer by intermediate isothermal treatments

Pyrolysis/carbonization of poly (p-phenylene terephtalamide) (PPTA) was investigated, studying the possibility of modifying the pyrolysis/carbonization behavior and hence the carbon yield by introducing intermediate isothermal treatments. Thermogravimetric analysis (TG) was used to establish the main degradation steps of the material. It showed that the yield of solid residue at 950 °C increases by more than 15 wt.% by introducing an isothermal step at 500 °C for at least 50 min. Intermediate decomposition products at different temperatures/times of PPTA decomposition were characterized by X-ray diffraction (XRD), elemental microanalysis and diffuse reflectance infrared Fourier transform spectroscopy (DRIFTS). XRD results showed that carbonization progresses during the isothermal step, so that the material is degraded loosing its crystallinity in a continuous way. DRIFTS spectra showed that PPTA undergoes a rupture of polymeric chains during the isothermal stage enhancing aryl nitrile formation. This favors crosslinking reactions that take place with increasing temperature, yielding a solid residue with a higher nitrogen content and higher char yield.     

Heterogeneous catalytic reactor design with optimum temperature profile II: application of non-uniform catalyst

A new methodology has been developed to design non-isothermal, non-adiabatic heterogeneous catalytic fixed bed and tubular reactors with optimal temperature profiles inside a reactor. Catalyst characteristics such as pellet diameter, shape and activity distributions inside a pellet are considered simultaneously for reactor design. Various types of non-uniform activity distributions inside a pellet are modelled and optimised for the maximisation of an objective such as yield or selectivity. Dirac-δ, layered and general non-uniform distribution profiles such as egg-shell, egg-yolk and middle peak distributions are applied for the reactor design. The research demonstrates that different catalyst distribution profiles can approach the optimum performance. Whilst it is known that the Dirac-δ profile (and its step-function equivalent) always gives the best performance for clean catalyst, other profiles can approach this performance and might offer advantages in catalyst manufacture and under degraded conditions. A profile-based synthesis approach is applied to generate various shapes of activity profiles for multiple sections along the reactor during the optimisation of non-uniform catalyst pellets. A case study with the ethylene oxidation process illustrates that the catalyst characteristics, such as activity distribution profiles inside a pellet, sizes and shapes can be manipulated to control the temperature through the reactor very effectively, leading to significant improvements in selectivity or yield. The non-uniform catalyst pellet is further applied to various reactor configurations such as inert mixing and side stream distributions. This work is the first to consider all of these effects simultaneously.