Showing papers on "Rate equation published in 2004"
TL;DR: Methods are presented for simulating chemical reaction networks with a spatial resolution that is accurate to nearly the size scale of individual molecules, using an intuitive picture of chemical reaction systems, where each molecule is treated as a point-like particle that diffuses freely in three-dimensional space.
Abstract: Methods are presented for simulating chemical reaction networks with a spatial resolution that is accurate to nearly the size scale of individual molecules. Using an intuitive picture of chemical reaction systems, each molecule is treated as a point-like particle that diffuses freely in three-dimensional space. When a pair of reactive molecules collide, such as an enzyme and its substrate, a reaction occurs and the simulated reactants are replaced by products. Achieving accurate bimolecular reaction kinetics is surprisingly difficult, requiring a careful consideration of reaction processes that are often overlooked. This includes whether the rate of a reaction is at steady-state and the probability that multiple reaction products collide with each other to yield a back reaction. Inputs to the simulation are experimental reaction rates, diffusion coefficients and the simulation time step. From these are calculated the simulation parameters, including the 'binding radius' and the 'unbinding radius', where the former defines the separation for a molecular collision and the latter is the initial separation between a pair of reaction products. Analytic solutions are presented for some simulation parameters while others are calculated using look-up tables. Capabilities of these methods are demonstrated with simulations of a simple bimolecular reaction and the Lotka-Volterra system.
607 citations
TL;DR: In this article, a study of CO formation for steam reforming of methanol on a commercial CuO/ZnO/Al2O3 catalyst has been performed in the temperature range 230-300°C and at atmospheric pressure.
Abstract: A study of CO formation for steam reforming of methanol on a commercial CuO/ZnO/Al2O3 catalyst has been performed in the temperature range 230–300 °C and at atmospheric pressure. The reaction schemes considered in this work are the methanol–steam reforming (SR) reaction and the reverse water gas-shift (rWGS) reaction. Power rate laws for the SR and reverse WGS reactions were used in a refinement of rate equations to the experiment data. For the temperature range studied the reaction order of methanol was determined under differential conversion (less than 10%) and was found to be 0.2. The integral method (partial pressure of the reactants and products measured as a function of contact time) was then applied to determine the reaction rate constants, activation energies, and pre-exponential factors for both reactions. The experimental results of CO partial pressure as a function of contact time at different reaction temperatures show very clearly that CO was formed as a consecutive product. The implications of the reaction scheme, in particular with respect to the production of CO as a secondary product, are discussed in the framework of on-board production of H2 for fuel cell applications in automobiles. Potential chemical engineering solutions for minimizing CO production are outlined.
274 citations
TL;DR: A model describing the free-electron generation in transparent solids under high-intensity laser irradiation that follows the nonstationary energy distribution of electrons on ultrashort time scales as well as the transition to the asymptotic avalanche regime for longer irradiations.
Abstract: We develop a model describing the free-electron generation in transparent solids under high-intensity laser irradiation. The multiple rate equation model unifies key points of detailed kinetic approaches and simple rate equations to a widely applicable description, valid on a broad range of time scales. It follows the nonstationary energy distribution of electrons on ultrashort time scales as well as the transition to the asymptotic avalanche regime for longer irradiations. The role of photoionization and impact ionization is clarified in dependence on laser pulse duration and intensity.
272 citations
TL;DR: In this paper, the authors examined the water-gas-shift reaction over noble metals and metals with ceria for catalyst contact times between 0.008 and 0.05 s for temperatures from 300 to 1000 ǫ c. All metals showed considerable promotion upon ceria addition, with Pt/ceria showing the most promise in terms of promotion, stability and lack of methanation activity.
216 citations
TL;DR: In this paper, a micro-channel Pd/ZnO catalyst with effective heat exchange has been developed to evaluate catalyst performance and measure reaction kinetics, and the rate equation has been applied to a three-dimensional pseudo-homogeneous model to simulate temperature profiles in both microchannel and conventional fixed bed reactors.
Abstract: A microchannel reactor with effective heat exchange has been developed to evaluate catalyst performance and measure reaction kinetics. The reactor provides an isothermal environment for rate measurement of the endothermic methanol steam reforming reactions over a Pd/ZnO catalyst in a wide temperature range (160–310 °C). The apparent activation energy and rate equation have been determined to fit the power law expression: −r A (mmol/kg cat / s )=2.9047×10 10 e −94 800/RT p MeOH 0.715 p H 2 O 0.088 This result provides kinetic data for the design of a miniature fuel processor for small fuel cell applications. The rate equation has been applied to a three-dimensional pseudo-homogeneous model to simulate temperature profiles in both microchannel and conventional fixed bed reactors.
102 citations
TL;DR: In this paper, the scale growth of precursor-derived ceramics was investigated at 1300° and 1500°C, and it was shown that at least within the experimental times these precursors develop extremely thin scales, thinner than pure SiC or Si3N4.39N2.28 (T2/1p).
Abstract: The oxidation behavior of three precursor-derived ceramics—Si4.46BC7.32N4.40 (AMF2p), Si2.72BC4.51N2.69 (AMF3p), and Si3.08BC4.39N2.28 (T2/1p)—was investigated at 1300° and 1500°C. Scale growth at 1500°C in air can be approximated by a parabolic rate law with rate constants of 0.0599 and 0.0593 μm2/h for AMF3p and T2/1p, respectively. The third material does not oxidize according to a parabolic rate law, but has a similar scale thickness after 100 h. The results show that at least within the experimental times these ceramics develop extremely thin scales, thinner than pure SiC or Si3N4.
91 citations
TL;DR: In this article, high power quantum cascade laser operating above liquid nitrogen temperature at λ≃87 and 130μm based on a bound-to-continuum transition was reported.
Abstract: We report high power quantum cascade lasers operating above liquid nitrogen temperature at λ≃87 and 130μm based on a bound-to-continuum transition. For λ≃87μm, 56mW peak power in pulsed operation and 50mW continuous wave operation at 10K are demonstrated. At λ≃130μm, a peak power of 50mW was achieved and devices operated in continuous wave reached a maximum temperature of 53K with an optical power of 11.5mW at T=10K. Lifetimes are extracted from the scaling of the transport and laser parameters as a function of size using a simple rate equation model.
89 citations
TL;DR: In this paper, the thermal decomposition of CH4 for co-producing H2 and carbon black is examined for a vortex flow confined to a cavity receiver and directly exposed to mean radiative power fluxes of 2500 kW / m 2.
86 citations
TL;DR: In this article, the design of GaN/AlGaN quantum-cascade laser emitting at 34 and 38 μm (ΔE∼36 and 34 meV) is presented, assuming either a- or c-plane crystal growth orientation.
Abstract: Designs of GaN/AlGaN quantum-cascade lasers emitting at 34 and 38 μm (ΔE∼36 and 34 meV) are presented, assuming either a- or c-plane crystal growth orientation. In the calculation of the quasibound state energies and wave functions, we account for the intrinsic electric field induced by piezoelectric and (in case of c-plane growth) the spontaneous polarization. The quantum-cascade structures were simulated, and their output characteristics extracted, using a fully self-consistent rate equation model with all relevant intra- and interperiod scatterings included. Both electron–LO-phonon and electron–electron scattering mechanisms are taken into account. Maximal population inversions between active laser states of up to 19% for the a-plane, and up to 40% for the c-plane design, are predicted and, based on estimated modal gain and waveguide/mirror losses in suitably designed structures, these indicate the feasibility of laser action in GaN/AlGaN cascades.
86 citations
TL;DR: In this article, a rate equation model for the carrier dynamics is fitted to the experimental data obtained for the integrated intensity of the photoluminescence at different temperatures, and two thermal escape mechanisms for carriers in the dots are identified that have the onsets at 110 K and 220 K, respectively.
Abstract: Photoluminescence from InAs quantum dots in a strained Ga0.85In0.15As quantum well is investigated over a temperature range from 10 to 300 K using low intensity optical excitation. A rate equation model for the carrier dynamics is fitted to the experimental data obtained for the integrated intensity of the photoluminescence at different temperatures. It is found necessary to assume a potential barrier, possibly arising from the strain, at the interface between the dots and the quantum well that makes the carrier capture in dots less effective at low temperatures. In addition, two thermal escape mechanisms for carriers in the dots are identified that have the onsets at 110 K and 220 K, respectively. At low temperatures, the ground state photoluminescence has a large full width at half maximum, while at temperatures above 220 K, the full width decreases as emission from larger dots dominate. The activation energies for different carrier thermal escape channels are estimated using the solutions of the steady-state rate equation system.
77 citations
TL;DR: In this paper, the fractional conversion αmax at the maximum reaction rate was applied to determine reaction order of the thermal degradation of poly(methyl methacrylate) (PMMA) sample.
TL;DR: In this article, the kinetics of NO oxidation were performed over well-defined catalysts, Pt 0 /SiO 2 and Pt x + /CeZrO 2 being chosen as model zero-valent and oxidized platinum active sites, respectively.
Abstract: The kinetics of NO oxidation was performed over well-defined catalysts, Pt 0 /SiO 2 and Pt x + /CeZrO 2 being chosen as model zero-valent and oxidized platinum active sites, respectively. On both catalysts, the global rate equation was determined, leading to the partial orders in NO and O 2 , and global rate constants. For both catalysts the orders in NO and O 2 are fractional and positive. Based on these data, two sequences of elementary steps are proposed. From the corresponding catalytic cycles, detailed kinetic rate laws were established leading to relevant kinetic constants and activation energies. Transient simulations were performed using the global power rate law and the detailed rate equations on both catalysts. A good fitting was observed up to 15% conversion on Pt 0 /SiO 2 , between 420 K and 500 K, with the global rate equation. The detailed kinetic data allowed a better fitting. For higher conversion the Arrhenius plots allowed us to conclude that diffusion limitations must be considered. A good fitting was observed over Pt x + /CeZrO 2 , between 420 K and 700 K, up to 25% conversion as expected with both global and detailed equations. Thermodynamic limitations are observed, in this case, above 30% conversion.
TL;DR: A wave packet based statistical model is suggested for complex-forming reactions and is very efficient and reasonably accurate for reactions dominated by long-lived resonances, as confirmed by its application to the C((1)D)+H(2) reaction.
Abstract: A wave packet based statistical model is suggested for complex-forming reactions. This model assumes statistical formation and decay of the long-lived reaction complex and computes reaction cross sections and their energy dependence from capture probabilities. This model is very efficient and reasonably accurate for reactions dominated by long-lived resonances, as confirmed by its application to the C(1D)+H2 reaction.
TL;DR: In this paper, the temperature dependent behavior of continuous-wave and time-resolved photoluminescence of Eu-doped GaN in the visible region is measured for both the 5D0→7F2 and 5D 0→ 7F3 transitions.
Abstract: The temperature dependent behavior of continuous-wave and time-resolved photoluminescence of Eu-doped GaN in the visible region is measured for both the 5D0→7F2 and 5D0→7F3 transitions. The radiative decay of these transitions, following pulsed laser excitation of the GaN host, is monitored by a grating spectrometer and photomultiplier tube detector system. In addition to these two radiative energy transfer pathways within Eu3+, the data reveal two nonradiative energy transfer paths between Eu3+ and the host GaN. Decay constants for the relaxation processes are extracted from the data using a numerically solved rate equation model. Although the dominant radiative relaxation processes decayed with a temperature insensitive decay constant of 166 μs, a prominent role for nonradiative transfer between Eu3+ and impurities within the GaN host was deduced above 180 K.
TL;DR: In this paper, a 3D transient heat transfer model that links conduction-convection-radiation heat transfer to the chemical kinetics is formulated using wavelength and chemical composition dependent material properties and assuming the Rosseland diffusion approximation for the internal radiative transport.
TL;DR: In this paper, a laser-diode end-pumped actively Q-switched Nd:YVO/sub 4/KTP laser with acoustic-optic-modulator is realized and the experimental results agree with the numerical solutions.
Abstract: The intracavity photon density and the initial population-inversion density are assumed to be Gaussian spatial distributions in the rate equations of a laser-diode end-pumped actively Q-switched intracavity-frequency-doubling laser. In addition, the influence of the pump rate, the thermal effect in the gain medium and the change of the photon density along the cavity axis have been taken into account. These coupled rate equations are solved numerically, and the dependences of pulse width, single-pulse energy and peak power on incident pump power are obtained for the generated-green-laser pulses. In the experiment, a laser-diode end-pumped actively Q-switched Nd:YVO/sub 4//KTP laser with acoustic-optic-modulator is realized and the experimental results agree with the numerical solutions.
TL;DR: In this paper, the chemical monomolecular triangle reaction is used to illustrate the importance of detailed balance, and simultaneous rate equations for this reaction are solved using Mathematica to show the effect that different choices of rate constants has on the changes in concentrations of the three isomers with time.
Abstract: The chemical monomolecular triangle reaction is used to illustrate the importance of the principle of detailed balance. The simultaneous rate equations for this reaction are solved using Mathematica to show the effect that different choices of rate constants has on the changes in concentrations of the three isomers with time. These solutions are discussed in terms of thermodynamic equilibrium, steady states, microscopic reversibility, and oscillating reactions.
TL;DR: In this paper, the rate equation of the solution and radical polymerization of acrylamide in the presence of potassium persulfate in D2O was investigated up to high conversion by high-field 1H-NMR spectroscopy.
Abstract: Solution and radical polymerization of acrylamide in the presence of potassium persulfate in D2O was investigated up to high conversion by high-field 1H-NMR spectroscopy. The kinetics of reaction was studied according to the data obtained from the corresponding spectra at various times during the polymerization reaction progress. Processing of the data led us to derive the rate equation of this polymerization reaction and determine the reaction order of each component in the rate equation. The order, with respect to initiator, was consistent with the classical kinetic rate equation (0.45), whereas the order with respect to monomer was greater than unity (1.49). The effect of temperature on the polymerization rate was also investigated and the activation energy of 48.4 kJ mol−1 was obtained over the temperature range of 60–75°C. Also some mechanistic studies were discussed. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 93: 2007–2013, 2004
TL;DR: In this article, a laser-diode end-pumped passively Q-switched intracavity-frequency-doubling Nd:YVO4/KTP laser with Cr 4þ :YAG saturable absorber is realized.
15 Jun 2004-Materials Science and Engineering A-structural Materials Properties Microstructure and Processing
TL;DR: In this article, the hot deformation behavior of electrolytic copper in isothermal compression has been studied in the temperature range 300-950°C and strain rate range 0.001-100−s −1 with a view to evaluate the rate controlling mechanisms relevant to different ranges.
Abstract: The hot deformation behavior of electrolytic copper in isothermal compression has been studied in the temperature range 300–950 °C and strain rate range 0.001–100 s −1 with a view to evaluate the rate controlling mechanisms relevant to different ranges. The flow curves observed are typical of the occurrence of dynamic recrystallization (DRX) and have exhibited single or multiple peak in the flow stress before reaching steady state, depending on the temperature and strain rate. Three different windows of temperature and strain rate have been identified in which the kinetic rate equation is obeyed by the temperature and strain rate dependence of flow stress. Both the power-law and hyperbolic sine relation yielded similar results. In the strain rate range 0.001–1.0 s −1 and temperature range 400–600 °C, the stress exponent is 7.8 and the apparent activation energy is 159 kJ/mole, and these indicate that dislocation core diffusion is the rate controlling mechanism. In the strain rate range 0.001–1.0 s −1 and temperature range 700–950 °C, the stress exponent is 7.1 and the apparent activation energy is 198 kJ/mole which matches well with that for lattice self diffusion in copper. The switching of the rate controlling mechanisms at temperatures higher than 700 °C has been attributed to the “clogging” of the dislocation pipes due to the rapid increase in the solid solubility of oxygen at higher temperatures. In the high strain rate range 30–100 s −1 and temperature range 700–950 °C, the stress exponent is 3.5 and the apparent activation energy is 91 kJ/mole which matches with that for grain boundary diffusion in copper. The apparent activation energy in electrolytic copper in all the above temperature and strain rate windows increases with increasing oxygen content and this is attributed to the increase in the back stress due to the presence of copper oxide particles in the matrix.
TL;DR: In this paper, the rate equations for Zeeman coherences in the broad-line approximation and steady-state balance equations directly from optical Bloch equations without the use of the perturbation theory were obtained.
Abstract: In this paper we obtain the rate equations for Zeeman coherences in the broad-line approximation and steady-state balance equations directly from optical Bloch equations without the use of the perturbation theory. The broad-line approximation allows us to use the adiabatic elimination procedure in order to eliminate the optical coherences from the optical Bloch equations, but the steady-state condition allows us to derive the balance equations in a straightforward way. We compare our approach with the perturbation-theory approach as given previously and show that our approach is more flexible for analyzing various experiments. Meanwhile we also show the validity and limitations of the application of the rate equations in experiments with coherent atomic excitation when either the broad-line approximation or steady-state conditions hold. Thus we have shown the basis for modeling the coherent atomic excitation experiments by using the relatively simple rate equations, provided that certain experimental conditions hold.
TL;DR: In this paper, a detailed theoretical study of a vertical-cavity surface-emitting laser (VCSEL) with two optically coupled, active cavities is presented.
Abstract: We present a detailed theoretical study of a vertical-cavity surface-emitting laser (VCSEL) with two optically coupled, active cavities. The study is based on a rate-equation model written for carriers and photons under steady-state conditions. The model allows one to determine all the relevant parameters-carrier densities, gains, and output powers-starting from two input parameters: the injection currents in each cavity. The system of equations is solved for different operating regimes of the device and the results provided by the model are shown to be in very good qualitative and quantitative agreement with the experimental data.
TL;DR: In this article, a rhodamine-B-doped polymer optical fiber amplifier (RBDPOFA) has been successfully fabricated and tested and the time-dependent and time-independent rate equations are solved.
Abstract: A rhodamine-B-doped polymer optical fiber amplifier (RBDPOFA) has been successfully fabricated and tested. In this paper, all parameters affecting the gain of the RBDPOFA are recognized. The time-dependent and time-independent rate equations are solved. By using Runge-Kutta and finite-element methods, different conditions are described and fluctuations of the fiber laser are analyzed. From the time-dependent rate equations, the gain and level-density variations with time can be predicted. These fiber amplifiers have unstable conditions. In experiments, the decay of dye dopants in the high pump power regime has been observed. For example, in a high pump power rhodamine-B is converted from the zwitterion isomer to the colorless lactone isomer [1]. Therefore, for complete coincidence of theoretical and experimental results, this effect must be included in the analysis, which is however beyond the scope of this paper and will be discussed in the near future.
TL;DR: A kinetic model for Q-switched, cladding-pumped, high-power Yb-doped fiber lasers that is based on the rate equation, in a difference equation form, of ion population and propagation equations for both pumping and signal light is presented.
Abstract: We present a kinetic model for Q-switched, cladding-pumped, high-power Yb-doped fiber lasers that are based on the rate equation, in a difference equation form, of ion population and propagation equations for both pumping and signal light. The effects of fiber-laser parameters, such as doping, length, pump power, and repetition rate on pulse characteristics are analyzed. This model is used to analyze the performance of Q-switched multicore fiber lasers and to show the output pulses with enhanced characteristics.
Book•
17 Mar 2004
TL;DR: In this article, the Laplace Transform was used for differential rate Equations of single and multiple Reactions, elementary reactions, molecularity, and order of Reactions Irreversible Reactions Determination of the Order of Reaction and its Rate Constant Other Irre reversible Reactions Reversible Reaction Reversible and Series Reactions (Eigenvalue Method) Use of Laplace transform for Differential Rate Equations Enzyme-Substrate Reactions Acid-Base Catalyzed Degradation Kinetics Oxidation (Free Radical Reaction) Degradation in Inclusion Complexes,
Abstract: THERMODYNAMICS Ideal Gases Real Gases Thermodynamics Free Energy IONIC EQUILIBRIUM Strong Acids and Bases Monoprotic Weak Acids and Weak Bases Polyprotic Weak Acids and Weak Bases Sparingly Soluble Salts SOLUTIONS AND DISTRIBUTION Solutions of Solids and Non-volatile Liquids in Liquids Solutions of Volatile Liquids in Liquids Solutions of Gases in Liquids (Henry's Law) Colligative Properties Distribution Law (Partition Coefficient) SURFACE CHEMISTRY AND COLLOIDS Adsorption from Solutions Liquid-Liquid Systems (Emulsions) Solid-Liquid Systems (Suspensions) KINETICS Pathways of Drug Degradation Single and Multiple Reactions, Elementary Reactions, Molecularity, and Order of Reactions Irreversible Reactions Determination of the Order of Reaction and Its Rate Constant Other Irreversible Reactions Reversible Reactions Reversible and Series Reactions (Eigenvalue Method) Use of Laplace Transform for Differential Rate Equations Enzyme-Substrate Reactions Acid-Base Catalyzed Degradation Kinetics Oxidation (Free Radical Reaction) Degradation Kinetics in Inclusion Complexes, Micelles, and Liposomes Temperature Effects on Reaction Rate Constants DIFFUSION Diffusion Diffusion Through a Membrane Diffusion in a Monolithic Matrix Diffusion in a Membrane-Matrix System Diffusion During the Swelling of Matrix Diffusion in a Matrix Erosion/Degradation Diffusion in a Matrix Swelling/Erosion Diffusion with Chemical Reaction in a Membrane Surface Area and Concentration Gradient Systems Osmotically Controlled Systems POLYMER SCIENCE Classification of Polymers Molecular Weight and Molecular Weight Distribution of Polymers Characterization of Polymers Polymer Synthesis Natural Polymers and Their Modification INDEX
TL;DR: In this article, the dielectric properties of a curing epoxy/amine system have been studied from 25 to 120 °C at 2.45 GHz, and the nature of the information yielded by dielectrometry on the dynamics of the system is discussed.
Abstract: The dielectric properties of a curing epoxy/amine system have been studied from 25 to 120 °C at 2.45 GHz. The epoxy resins at different extents of curing exhibit a γ relaxation, which can be described by the Arrhenius rate law. The relaxation is attributed to the motions of the dipolar groups associated with the crosslinking system. A simple model is proposed to represent the temperature dependence of the dielectric properties. A complete description of the evolution of the parameters during the polymerization has been obtained. The nature of the information yielded by dielectrometry on the dynamics of the system is discussed.
TL;DR: In this article, the chemical kinetics of reversible association/dissociation reactions were discussed in great length and it was shown that, as long as the characteristic time for internal energy relaxation is faster (not necessarily much faster) than that for chemical reaction, there will be a period of time, perhaps only late in the reaction but before equilibrium is reached, during which phenomenological rate laws will apply with rate coefficients that satisfy detailed balance.
Abstract: In this article we discuss the chemical kinetics of reversible association/dissociation reactions at great length. We find that, as long as the characteristic time for internal-energy relaxation is faster (not necessarily much faster) than that for chemical reaction, there will be a period of time, perhaps only late in the reaction but before equilibrium is reached, during which phenomenological rate laws will apply with rate coefficients that satisfy detailed balance. The nonequilibrium factor, fne, originally introduced by Smith, McEwan, and Gilbert (J. Chem. Phys. 1989, 90, 4265−4273) is not a measure of the degree to which detailed balance is satisfied by the association and dissociation rate coefficients. It is simply the fractional contribution to the “long-time” association rate coefficient, kadd, of the slowest-relaxing eigenmode of the system. That is, 1 − fne is the fractional contribution to the same rate coefficient of the internal-energy relaxation modes. The standard practice of taking the d...
Dissertation•
30 Jul 2004
TL;DR: In this paper, the authors developed a TURBULENT FLOTATION MODEL from first-principles and evaluated its performance under a variety of different conditions.
Abstract: .........................................................................................................................II ACKNOWLEDGEMENTS....................................................................................................... III TABLE OF CONTENTS ......................................................................................................... IV LIST OF FIGURES ...............................................................................................................VII LIST OF TABLES ................................................................................................................. IX INTRODUCTION .................................................................................................................... 1 BACKGROUND.......................................................................................................... 1 MODELING................................................................................................... 1 SURFACE FORCES (ENERGIES) ..................................................................... 5 OBJECTIVES ............................................................................................................. 8 ORGANIZATION........................................................................................................ 8 NOMENCLATURE...................................................................................................... 9 REFERENCES .......................................................................................................... 11 PAPER 1 FLUID DYNAMICS OF BUBBLES AND PARTICLES UNDER TURBULENT FLOTATION CONDITIONS: A REVIEW......................................................................................... 13 ABSTRACT ............................................................................................................. 13 INTRODUCTION ...................................................................................................... 13 COLLISION FREQUENCIES....................................................................................... 14 ATTACHMENT AND DETACHMENT ENERGIES......................................................... 19 SUMMARY.............................................................................................................. 22 NOMENCLATURE.................................................................................................... 23 REFERENCES .......................................................................................................... 24 PAPER 2 DEVELOPING A TURBULENT FLOTATION MODEL FROM FIRST PRINCIPLES ..... 26 ABSTRACT ............................................................................................................. 26 INTRODUCTION ...................................................................................................... 26 MODEL................................................................................................................... 27 RATE CONSTANT ....................................................................................... 28 PARTICLE COLLECTION.............................................................................. 28 ENERGIES................................................................................................... 30 FROTH RECOVERY ..................................................................................... 31 RESULTS ................................................................................................................ 34
TL;DR: In this paper, the kinetics of nucleation-and-branching rate-limited solid-state reactions and phase transformations are described using coupled rate equations, where the first step, nucleation, is assumed to be first order in the reagent and the second step, branching, is considered to be autocatalytic.
Abstract: The kinetics of “nucleation-and-branching” rate-limited solid-state reactions and phase transformations (which include processes that are governed by nucleation and nuclei multiplication but not nuclei growth or crystal growth) are described using coupled rate equations. We treat such processes as occurring in two steps. The first step, nucleation, is assumed to be first order in the reagent. The second step, involving the multiplication of product nuclei (or “nuclei branching”), is considered to be autocatalytic; i.e., the rate of this step depends on the relative amounts of both the reagent and the product. The general rate equation developed on the basis of this two-step model allows greater flexibility in the fitting of sigmoidal kinetic curves than does the widely used Prout-Tompkins equation. Similarities between our model and the recently described general ProutTompkins (GPT) equation are discussed.
TL;DR: In this article, rank annihilation factor analysis combined with the optimization of kinetic parameter is adopted to resolve the two-way kinetic-spectral data measured online from chemical reactions, which is applied to the analysis of practical reaction systems such as alkaline hydrolysis of phenyl benzoate.