Molecular weight distribution of polyethylene terephthalate in homogeneous,continuous-flow-stirred tank reactors

Poly(ethylene terephthalate) (PET) formation in homogeneous, continuous-flow-stirred tank reactors (HCSTRs) operating at steady state has been simulated. The feed to the reactor is assumed to consist of the monomer bis-(hydroxyethyl) terephthalate and monofunctional compound (MF₁) cetyl alcohol. The overall polymerization is assumed to consist of the polycondensation, reaction with monofunctional compounds, redistribution, and cyclization reactions. At a given time, the reaction mass consists of polyester molecules (P_n), polyester molecules with an ending of molecules of monofunctional compound (MF_n), and cyclic polymers (C_n). A mass balance for each of these species in the reactor gives rise to a set of algebraic equations to be solved simultaneously. The MWD calculations show that the redistribution reaction plays a major role and cannot be ignored, This result is in contrast lo the observation for semi-batch reactors, for which redistribution becomes important when the cyclization reaction is included. For the same residence times of semi-batch and HCSTRs, the latter gives considerably lower-number average molecular weight, N_av, and polydispersity index, ρ. However, for the same conversions, the ρ for CSTR is higher. The concentration of the monofurctional compound, [MF₁]₀, in the feed and the reactor temperature both influence ρ, but the effect is small within the range studied.     

Preparation and analysis of a flexible curing agent for epoxy resin

A kind of novel aromatic amine bis(4‐nonyl‐2,5‐diamine‐penoxyl)alkylate (RA_n) as curing agents for epoxy resins were prepared through three steps of reactions using nonyl phenol and dibromoalkylate as materials. Dynamic mechanical analysis (DMA) indicated that the secondary relaxation for the resins cured by RA_n were generated by the nonyls in RA_n molecules when temperature was below ?50°C. Comparing with other reference resins, the enhancement for toughness of RA_n cured‐resins were at least 15%, which were contributed by such secondary relaxation. Furthermore, stiffness of the networks and thermal properties of the resins were not influent by the flexible groups (nonyl) in RA_n after curing, since the groups were located only in the branched chains of the networks. The mechanical and thermal properties of the new material have been significantly enhanced. The relevant method and procedure developed through this research have been granted Chinese patent recently (Yang and Gong, Chin. Pat. CN1978483A, 2007). © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

In silico mechanically mediated atom transfer radical polymerization: A detailed kinetic study

Mechanically mediated atom transfer radical polymerization (mechanoATRP) utilizing ultrasound to generate activators and improve the diffusivity of macromolecular chains is introduced as an innovative externally controlled ATRP. Herein, a comprehensive kinetic model with free volume theory based “series” encounter pair model accounting for diffusional limitations on termination, activation, and deactivation is developed for the mechanoATRP of methyl acrylate. Comparative study by using different diffusion models, for example, w_p model and reduced composite k_t model, as well as constant apparent k_j^app model confirms the goodness of the as-developed model. Critically, mechanochemically induced reduction rate coefficient k_r,s as a key kinetic parameter is associated with experimental conditions excluding the sonication effect by a fitting equation for the first time. In silico tracking of polymer dispersity with the help of kinetic model shows a better result compared with that by the classical dispersity equation. By defining an ultrasonic factor γ_j, a qualitative analysis for the effect of ultrasound conditions on the diffusional limitation in mechanoATRP is presented.     

A kinetic approach to multifunctional polymerization with cyclization

A kinetic model for the step growth polymerization of multifunctional monomers RA_f valid up to the gel point has been presented which accounts for the intramolecular reactions. From the Jacobson-Stockmeyer equation, the average rats constant for this step has been derived and is found to be chain length dependent. A computation scheme has been evolved by which the molecular weight distribution of the linear, branched as well as cyclized polymer can be determined up to the gel point. It is found that even though the molecular weight distribution of cyclic oligomers is little affected for the range of the values of the intramolecular rate constant (k_c) studied, the average chain length μ_n and the polydispersity index p of the polymer formed are extremely sensitive to k_c near the gel point. Numerical computations confirm that as the gel point is approached, larger polymer chains are formed which have a higher tendency to react intramolecularly. However, even at high conversions the number of the cyclic rings per polymer chain remains small, which is consistent with experimentally observed results.     

Irreversible step growth polymerization having segmental diffusion limitations in HCSTRs—simulation and steady state multiplicity

The effect of various parameters on the overall rate constant, number average chain length, and polydispersity index of isothermal, irreversible, step growth polymerization of ARB monomers, exhibiting segmental diffusional limitations and carried out in HCSTRs, are studied. A phenomenological model which has proved useful for chain polymerization is adapted for step growth polymerizations and simulations for several values of the different parameters are carried out. The number average chain length μ_n and the polydispersity index ρ are obtained as a function of the dimensionless time t^*. It is found that, even though the behavior of μ_n and ρ vs. t^* depends on the values of the parameters, the plot of μ_n vs. ρ is unique as expected from statistical considerations. An attempt has been made to investigate the multiplicity of steady states under conditions of infinitely rapid heat transfer from a jacket, using singularity and bifurcation theory. It is found from the analysis that there is no multiplicity of steady states for this system, which is confirmed by simulations using a wide range of parameter values.     

Chemical kinetic modeling of i‐butane and n‐butane catalytic cracking reactions over HZSM‐5 zeolite

A chemical kinetic model for i‐butane and n‐butane catalytic cracking over synthesized HZSM‐5 zeolite, with SiO₂/Al₂O₃ = 484, and in a plug flow reactor under various operating conditions, has been developed. To estimate the kinetic parameters of catalytic cracking reactions of i‐butane and n‐butane, a lump kinetic model consisting of six reaction steps and five lumped components is proposed. This kinetic model is based on mechanistic aspects of catalytic cracking of paraffins into olefins. Furthermore, our model takes into account the effects of both protolytic and bimolecular mechanisms. The Levenberg–Marquardt algorithm was used to estimate kinetic parameters. Results from statistical F‐tests indicate that the kinetic models and the proposed model predictions are in satisfactory agreement with the experimental data obtained for both paraffin reactants. © 2011 American Institute of Chemical Engineers AIChE J, 58: 2456–2465, 2012     

Parametric Analysis of the Continuous Stirred Tank Reactor Model

A parametric analysis is carried out for the continuous stirred tank reactor model involving a single exothermic reaction. Local bifurcations of the steady state, including Andronov–Hopf bifurcations, are analyzed for a first-order reaction. The steady-state multiplicity and neutrality lines are described in explicit form for various combinations of dimensionless parameters. Typical parametric and phase portraits of the dynamic system are presented. Parametric analysis is also carried out for the nth-order reaction nA B, the oxidation reaction A + O₂ B, and an arbitrary kinetic function.     

Polymerization of melamine and formaldehyde in homogeneous continuous-flow stirred-tank reactors using functional group approach: Part A: Conversion of functional groups

A comprehensive kinetic model using the functional group approach has been proposed for the polymerization of melamine and formaldehyde. The kinetic model is consistent with the basic chemistry of polymerization and involves five rate constants which have been estimated using the experimental data of Tomita. Homogeneous continuous-flow stirred-tank reactors (HCSTRs) have been modelled and the mole balance relations for various functional groups have been written. The performance of HCSTRs is governed by algebraic equations and, for any specified residence time, is found by the method of successive substitution using the Brown's algorithm. The computations show that as long as free formaldehyde is present, the reaction mass would consist predominantly of substituted melamine molecules. However, after formaldehyde is completely reacted, larger oligomers are formed in larger concentrations. On comparison of results with batch reactors, it is found that for the same reaction time HCSTRs yield polymer with higher branching.     

The mechanism of the ethane hydrogenolysis—a reconsideration of some kinetic data

In order to justify the deduction made previously from isokinetic effects of ethane hydrogenolysis, i.e., that the rate determining step of this reaction on Pt and Ni is the C-C bond breaking of the chemisorbed C₂H₅ unit, some kinetic data from literature are scrutinized. It turns out that the variation of the observed rate with ethane and/or hydrogen pressure is such as predicted by the simple mechanism C₂H₅* + H* +n* CH₃* + CH₃* +n*, where n means the number of free sites (*) involved. From an analysis of the kinetic data it is shown thatn=1.     

Influence of Mixing System Design and Operating Parameters on Dissolution Process

Over recent years, the solid dissolution rate in liquid has been the subject of intense and profitable scientific developments. This work aims to study the influence of hydrodynamics on the dissolution rate of solid in water. The impact of the mixing system design was also investigated. The dissolution process can be followed through several parameters. First, experimental results were validated by kinetic models based on mass transfer coefficients such as the Hixson‐Crowell equation. The Weibull model and n‐order model were also tested. Then, the dissolution time t_d,x% was defined as the time needed to reach a given dissolution degree (X). Finally, the dimensionless dissolution number (N t_d) was defined and correlated with the Reynolds number. This correlation provides a more direct and simple link between operating conditions and process efficiency.     

Polymerization of melamine and formaldehyde in homogeneous continuous-flow stirred-tank reactors using functional group approach: Part B: Molecular weight distribution

Polymerization of melamine and formaldehyde in homogeneous continuous-flow stirred-tank reactors (HCSTRs) is reversible and leads to formation of branched polymer due to the hexafunctional nature of melamine. The reversed reaction of branched molecules depends upon the chain structure, and herein a simple model is presented to account for this. The functional group analysis of part A of this series has been extended and the molecular weight distribution (MWD) relations for HCSTRs have been solved. The MWD of the polymer is extremely sharp and the molecules are more branched compared to those for batch reactors and can be explained as follows. A given reacted bond can react with water as well as free formaldehyde through the reversed reaction, and the rate constant for the latter step is much larger. Consequently, the chain growth is limited as long as there is free formaldehyde in the reaction mass. Lower conversion of melamine and formaldehyde in HCSTRs is used to explain the behavior observed.     

Optimal Stopping for I.I.D. Random Variables Based on the Sequential Information of the Location of Relative Records Only

Abstract

Let X _j , j = 1,…, n be independent and identically distributed random variables. Like in the classical secretary problem, the optimal stopper only observes Y _j  = 1, if X _j is a (relative) record, and Y _j  = 0, otherwise. The actual X _j values are not revealed. The goal is to maximize the expected X value at which one stops. We show that the optimal number of observations one should skip before considering stopping depends heavily on the underlying distribution.     

V-L-S multiphase equilibrium in bitumen-diluent systems

A multiphase flash algorithm is developed for the prediction of V-L-L-S equilibrium. The algorithm uses the Patel-Teja equation of state for the fluid phases and a separate model, based on vapour pressure, for the solid phase. A novel technique is presented for avoiding the phase disappearance problem commonly encountered in multiphase equilibrium calculations involving multiple fugacity models. The predictions are shown to be in good agreement with the data for several CO₂ and hydrocarbons V-L-S systems. This paper also presents new experimental data for the solubility of Peace River bitumen in n-pentane, n-heptane and n-octane at atmospheric conditions, and these are compared with the model predictions.     

Forced oscillations in continuous flow stirred tank reactors with nonlinear step growth polymerization

The self-step growth polymerization of RA_f monomers in homogeneous, continuous flow stirred tank reactors (HCSTRs) is simulated under conditions of periodic feed concentration (with frequency ω and amplitude α). By having periodic operation, the polydispersity index of the polymer is found to increase by about 35% over the values at steady state. Periodic operation of HCSTRs is found to lead to gelation only for certain values of the frequency and the dimensionless residence time τ^*. Gelling envelopes have been obtained to give conditions under which HCSTRs should be operated. These envelopes can be described in terms of two critical dimensionless residence times, τ and τ such that nongelling operation is always ensured when τ^* < τ. For τ^* > τ, periodic operation always leads to gelation, and HCSTRs cannot be used. For τ < τ^* < τ, the gelling behavior is found to depend on the functionality f, amplitude α, and the dimensionless residence time τ^*.     

Study of sawdust pyrolysis and its devolatilisation kinetics

Pyrolysis of sawdust was studied using a thermogravimetric analyser (TGA) to understand the devolatilisation process and to obtain its global kinetic parameters. The influences of particle size, initial weight of the sample and heating rate on the devolatilisation of sawdust particles have been studied. Results from proximate analysis show that smaller particle size has more ash content compared to larger particle size. The TG and derivative TG curve for variation in particle size and initial weight of the sample showed significant difference in the third stage of the pyrolysis. In addition, the pyrolysis of sawdust differed significantly for variation in heating rate. As the heating rates increased, the char yield also increased. The devolatilisation kinetics was studied considering different stages of pyrolysis. The kinetic parameters for thermal devolatilisation of the sawdust were determined through a nonlinear optimisation method of two independent parallel nth‐order reaction models. The kinetic parameters such as activation energy, frequency factor and order of the reaction for the two stages considered in the model were: E₂ = 79.53 (kJ/mol), E₃ = 60.71 (kJ/mol); k₀₂ = 1.90 × 10⁶ (1/min), k₀₃ = 1.01 × 10³ (1/min); n₂ = 0.91, n₃ = 1.78, respectively. The results show good agreement between the proposed model and the experimental data of the sawdust pyrolysis.     

A simple model of hyperbranched polymerisation involving AB<Subscript>2</Subscript> and B<Subscript><Emphasis Type="Italic">f</Emphasis></Subscript> core monomers and methods of narrowing the molecular size distribution

Summary A simple kinetic model of modelling hyperbranched polymerisation involving AB₂ and B_f core monomers is developed and analytical solutions of the model are presented. The number- and weight- average polymerisation degrees can be calculated using the model as functions of polymerisation time or conversion degree. Application of the model is tested for predicting the broadness of molecular size distribution of the hyperbranched polymers obtained by varying the procedures of feeding AB₂ monomer to polymerisation reactor.     

Electrochemical reduction of silver thiosulphate complexes Part II Mechanism and kinetics

In this paper the thermodynamic data for complex formation between Ag⁺ and S₂O_{3 2–} ions, determined previously, are applied to kinetic investigation of the reduction of silver thiosulphate complexes. Both electrochemical (linear sweep voltammetry on a rotating disc electrode) and surface analytical (Auger electron spectroscopy) techniques are used. The deposits resulting from the electrodeposition of silver thiosulphate complexes are shown to be composed of silver and to be polycrystalline. The reduction follows a mechanism involving mass and charge transfer and chemical reaction steps. The relevant kinetic parameters are calculated and a rate equation describing the kinetics of the reduction is given.List of symbols a activity (M) - c concentration (M) - j current density (A m^–2) - j _c current density of charge transfer (A m^–2) - j _m current density of mass transfer (A m^–2) - k rate constant (m s^–1) - y activity coefficient (molarity scale) - D diffusion coefficient against gradient of concentration (m² s^–1) - D diffusion coefficient against gradient of electrochemical potential (m² s^–1) - E electrode potential vs NHE (V) - I ionic strength (M) - T temperature (K) Greek symbols a transfer coefficient - _1n stability constant of Ag(S₂O₃) _n (^2n–1)- - kinematic viscosity (m² s^–1) - rotation speed of the electrode (rad s^–1) Indices b bulk of the solution - f free (= uncomplexed) - 1,n related to complex Ag(S₂O₃)_n ^(n–1) - t total Constants F Faraday constant (96486 A s mol^–1) - R universal gas constant (8.3145 Jmol^–1 K^–1)