Nth Order Chemical Reaction Research Articles

High-temperature infrared kinetics of transition-metal-catalyzed chemical reactions in solid state complexes of polybutadienes with palladium chloride

High-temperature infrared spectroscopy to 180°C identifies irreversible chemical changes that occur in solid state complexes of 1,2-polybutadiene and palladium chloride, prior to the onset of oxidation. The decrease in infrared absorption intensities at 910, 994, 1418 and 1640 cm −1 signifies the consumption of carbon–carbon double bonds in the polymer whose microstructure is 80% 1,2-vinyl. The loss of CC functionality in the sidegroup at elevated temperatures can be explained by palladium-catalyzed dimerization reactions. Transient experiments between 100°C and 140°C reveal that the reduction in the CC infrared signal at 1640 cm −1 follows an approximate 1st-order rate law which is consistent with a previous calorimetric study of exothermic kinetics over the same temperature range. Characteristic nth-order chemical reaction time constants for solid films of 1,2-polybutadiene with 4 mol% palladium chloride have been calculated at 100°C, 125°C and 140°C via infrared spectroscopy. These reaction time constants decrease at higher temperature with an apparent Arrhenius activation energy of ≈ 43 kJ/mol. Palladium complexes with cis-polybutadiene reveal that π back-donation of electron density from the metal center into the antibonding orbitals of the alkene group in the main chain shifts the CC absorption from 1653 to 1543 cm −1. The uncomplexed signal at 1653 cm −1 is insensitive to high-temperature annealing whereas the palladium- π-complexed signal at 1543 cm −1 is severely attenuated after annealing for a few minutes at 150°C. This suggests that the formation of a π-complex is required before high-temperature irreversible chemical reactions. Infrared-determined characteristic nth-order chemical reaction time constants for solid films of cis-polybutadiene with 4 mol% palladium chloride at 100°C, 125°C and 140°C are five-fold longer than the corresponding reaction time constants for 1,2-polybutadiene with 4 mol% PdCl 2. This kinetic mismatch is consistent with the trend in reaction rates for mono-substituted vs. di-substituted small-molecule alkenes, and suggests that a mixing strategy is required to compatibilize 1,2-polybutadiene and cis-polybutadiene via PdCl 2 at high temperatures.

Long-Term Durability of Fiber/Matrix Interfacial Bonding in Hygrothermal Environments

This study investigated the long-term durability of the interfacial bond in carbon fiber /polyetheretherketone (C/PEEK) and carbon fiber/polysulfone (C/PSF) com posites after exposure to hygrothermal environments. A single fiber pull-out test was uti lized to quantitatively determine the ultimate bond strength (UBS) of the samples following exposure. Samples were tested at three temperatures for six time periods and in two en vironments (dry and salt water). A mathematical model based on nth order chemical reac tion kinetics was applied to describe the long-term durability of the interface. The results of this study indicate that interfacial bond strength in C/PSF and C/PEEK (380 grade) composites is significantly influenced by long-term exposure to salt water, especially at high temperatures. The results indicate that salt water saturation reduces interfacial bond strength in C/PSF and C/PEEK (380 grade) composites as a function of both time and temperature, with bond strength apparently reaching a stable temperature-dependent equi librium level.

Numerical length estimation for tubular flow reactors

This work was motivated by the engineering problem of determining the optimal length for tubular flow reactors. We considered an isothermal nth-order chemical reaction occuring in a tubular flow reactor with axial missing. The mathematical model describing the problem was a free boundary value one. The free boundary represented the reactor length. In this context we proved a theorem characterizing an upper bound for the free boundary. In order to solve the problem in point we introduced two different numerical methods. First a noniterative method was considered in order to obtain an upper bound of the free boundary. Then we used this upper bound in order to start an iterative method that allowed us to find an approximation of the free boundary. In closing we obtained numerical results either for an even or an odd order reaction.

A COMPARISON OF APPROXIMATE AND EXACT SOLUTIONS FOR HOMOGENEOUS IRREVERSIBLE CHEMICAL REACTION IN THE LAMINAR BOUNDARY LAYER

A comparison of approximate and exact solutions for homogeneous irreversible chemical reaction in the laminar boundary layer flow has been made. By using the Method of Weighted Residuals, approximate analytical expressions for the velocity and concentration profiles were developed for the case of a laminar boundary layer flow over a flat plate at zero incidence angle, where isothermal, homogeneous, nth order chemical reaction takes place. Both the Subdomain and Galerkin methods were employed to examine the influence of the choice of the weighting function on the predictions, and to provide a means for improving the solutions systematically. The problem was also solved numerically for the case of first order reaction by using a similarity solution for the hydrodynamic flow and a power series expansion method for the mass transfer. The analytical results were compared with the exact solutions in order to evaluate the accuracy of the approximate analytical solutions. The Method of Weighted Residuals provided...

Determination of initial rates for a general class of chemical reactions: A methodology

AbstractThe initial rate of any general nth order (n not necessarily integer) chemical reaction can be accurately and easily computed from the slope of a chord joining two points on the progress curve. Expressions for calculating the intermediate concentration corresponding to this initial rate are provided. Examples of the technique applied to second‐ and third‐order reactions as well as an example of a reaction with fractional order (n = 0.6) are given.

Mass transfer with Nth order chemical reaction around spheres in the presence of surfactants

A numerical technique has been used to solve the nonlinear equation describing the mass transfer taking place for single and multiple particles for Nth order chemical reactions in the presence of a surfactant. A generalized form is given for the Sherwood number as a function of the Peclet number for reaction orders of 1/2, 1 and 2 in the presence of surfactants, (Sh = aPe b ). The parameters (a) and (b) are dependent on the reaction orders and the amount of surfactant present. The coefficient (b) had an average value of 0.46 with the parameter (a) increasing with larger reaction rate constants. The numerical technique involved DISPL, a computer software package for the solution of partial differential equations by the method of lines.

Exact universal uniqueness criteria for the adiabatic tubular packed bed reactor

Exact, universal a priori bounds and regions of multiplicity for the entire tubular packed bed reactor are developed by application of a technique reported in Chang and Calo, Chem. Engng Sci. 1979 34 285 to a cascade two-phase cell model for an nth order chemical reaction with interphase resistance to mass and heat transport, Le ≠ 1 (or Le = 1) and either lumped parameter catalyst particles or with intraparticle concentration gradients with uniform temperature. Both the more common case of interphase heat transfer greater than interphase mass transfer rate and the inverse case of particle over-temperature have been considered. In all cases it has been shown that the reactor conservation equations can be decoupled at certain points along the bed determined by the Lewis number, and that questions of multiplicity and uniqueness reduce to consideration of a single algebraic equation which is actually a form of the two-phase adiabatic CSTR. Also as for the CSTR, the topology of the adiabatic packed bed reactor is shown to be the simple cusp catastrophe. The application of the resultant criteria is quite simple and represents a practical step in the design procedure for highly exothermic reactions in packed beds. A flow chart of a suggested procedure is included.

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 2 c , (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.

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.

The critical conditions in thermal explosion theory with reactant consumption

The energy and chemical kinetic equations describing the behaviour of a symmetrically heated reactive material with finite heat of reaction have been analysed. For first order reactions the relationship between the parameters at criticality, and the critical induction period as a function of the parameters are determined. It is shown how the general nth order chemical reaction may be dealt with. The analysis results in an exact ignition temperature and a lower bound to the adiabatic temperature rise at criticality which increase with the order of the reaction.