Equations Of Chemical Kinetics Research Articles

Quantifying uncertainty in the chemical master equation.

We describe a novel approach to quantifying the uncertainty inherent in the chemical kinetic master equation with stochastic coefficients. A stochastic collocation method is coupled to an analytical expansion of the master equation to analyze the effects of both extrinsic and intrinsic noise. The method consists of an analytical moment-closure method resulting in a large set of differential equations with stochastic coefficients that are in turn solved via a Smolyak sparse grid collocation method. We discuss the error of the method relative to the dimension of the model and clarify which methods are most suitable for the problem. We apply the method to two typical problems arising in chemical kinetics with time-independent extrinsic noise. Additionally, we show agreement with classical Monte Carlo simulations and calculate the variance over time as the sum of two expectations. The method presented here has better convergence properties for low to moderate dimensions than standard Monte Carlo methods and is therefore a superior alternative in this regime.

Microwave fast-tracking biodiesel production

Three kinds of biodiesel production processes are compared and it is shown that the microwave heated acid process has higher productivity than the conventional heating alkaline or acid processes. The reaction time to reach the maximum yield in microwave-irradiated process is reduced to 50% of the time required by conventional heating processes. An optimization of process variables was done to identify the optimum microwave process irradiation. Searching an explanation for this non thermal microwave effect, a statistical design to stablish the complete chemical kinetic equation for soya oil transesterification with ethanol catalyzed by methane sulfonic acid was realized. The kinetical results show that this process has a complex mechanism that can be seen as a pseudo -1.6 global order reaction with low activation energy. Precisely, the order is 0.004 with respect to ethanol and -1.6 to soya oil. An analysis of this negative order was done and it brings a little more information to understand how to get best returns from biodiesel operations.

Experimental Polymer Mechanochemistry and its Interpretational Frameworks.

Polymer mechanochemistry is an emerging field at the interface of chemistry, materials science, physics and engineering. It aims at understanding and exploiting unique reactivities of polymer chains confined to highly non-equilibrium stretched geometries by interactions with their surroundings. Macromolecular chains or their segments become stretched in bulk polymers under mechanical loads or when polymer solutions are sonicated or flow rapidly through abrupt contractions. An increasing amount of empirical data suggests that mechanochemical phenomena are widespread wherever polymers are used. In the past decade, empirical mechanochemistry has progressed enormously, from studying fragmentations of commodity polymers by simple backbone homolysis to demonstrations of self-strengthening and stress-reporting materials and mechanochemical cascades using purposefully designed monomers. This progress has not yet been matched by the development of conceptual frameworks within which to rationalize, systematize and generalize empirical mechanochemical observations. As a result, mechanistic and/or quantitative understanding of mechanochemical phenomena remains, with few exceptions, tentative. In this review we aim at systematizing reported macroscopic manifestations of polymer mechanochemistry, and critically assessing the interpretational framework that underlies their molecular rationalizations from a physical chemist's perspective. We propose a hierarchy of mechanochemical phenomena which may guide the development of multiscale models of mechanochemical reactivity to match the breadth and utility of the Eyring equation of chemical kinetics. We discuss the limitations of the approaches to quantifying and validating mechanochemical reactivity, with particular focus on sonicated polymer solutions, in order to identify outstanding questions that need to be solved for polymer mechanochemistry to become a rigorous, quantitative field. We conclude by proposing 7 problems whose solution may have a disproportionate impact on the development of polymer mechanochemistry.

Modelling of the temporal evolution of the gas temperature in N2 discharges

The time-dependent evolution of the energy transfer to gas heating in a pure N2 discharge produced in a cylindrical tube at low pressures (1–10 Torr) is studied for different fixed values of the reduced electric field and electron density. We consider a model based on the self-consistent solutions to the time-dependent gas thermal balance equation coupled to the electron, vibrational, and chemical kinetic equations for the most important heavy species produced in N2 plasma discharges. The results of this model provide the temporal variation of the radially averaged value of the gas temperature, as well as the corresponding gas heating mechanisms. It is shown that the pooling reactions N2(A) + N2(A) → N2(B) + N2 and N2(A) + N2(A) → N2(C) + N2 are responsible for a smooth increase in the gas temperature before the first millisecond. For longer times, gas heating is found to be mainly caused by vibrational energy exchanges from non-resonant vibration–vibration (V–V) processes between N2 molecules and by vibration–translation (V–T) N2–N collisions. The heating rates of these different gas heating mechanisms and the gas temperature are calculated for a reduced electric field of 50 and 100 Td (1 Td = 10−17 Vcm2), an electron density of 1010 and 1011 cm−3, and a pressure of 1 and 10 Torr. The fractional power converted to gas heating from electronic and vibrational excitation is also calculated for these parameters, being respectively ∼2% and in the range 10%–35%. The effect of having a contribution of non-resonant V–V processes to gas cooling within the time interval 0.1–1 ms is analysed. The role of the gas temperature on the temporal evolution of the vibrational distribution of N2(X, v) molecules is also discussed.

Surface Assisted Combustion of Hydrogen-Oxygen Mixture in Nanobubbles Produced by Electrolysis

The spontaneous combustion of hydrogen–oxygen mixture observed in nanobubbles at room temperature is a puzzling phenomenon that has no explanation in the standard combustion theory. We suggest that the hydrogen atoms needed to ignite the reaction could be generated on charged sites at the gas–liquid interface. Equations of chemical kinetics augmented by the surface dissociation of hydrogen molecules are solved, keeping the dissociation probability as a parameter. It is predicted that in contrast with the standard combustion, the surface-assisted process can proceed at room temperature, resulting not only in water, but also in a perceptible amount of hydrogen peroxide in the final state. The combustion time for the nanobubbles with a size of about 100 nm is in the range of 1–100 ns, depending on the dissociation probability.

The H-theorem for the chemical kinetic equations with discrete time and for their generalizations

In this paper the generalizations of equations of chemical kinetics, including classical and quantum chemical kinetics, is considered. We make time discrete in these equations and prove the H-theorem.

Modeling the flow of activated H2+ CH4mixture by deposition of diamond nanostructures

Algorithm of the direct simulation Monte Carlo method for the flow of hydrogen and methane mixture in a cylindrical channel is developed. Heterogeneous reactions on tungsten channel surfaces are included into the model. Their effects on flows are analyzed. A one-dimensional approach based on the solution of equilibrium chemical kinetics equations is used to analyze gas-phase methane decomposition. The obtained results may be useful for optimization of gas-dynamic sources of activated gas diamond synthesis.

Chemical Reactions Using a Non-Equilibrium Wigner Function Approach

A three-dimensional model of binary chemical reactions is studied. We consider an ab initio quantum two-particle system subjected to an attractive interaction potential and to a heat bath at thermal equilibrium at absolute temperature T > 0 . Under the sole action of the attraction potential, the two particles can either be bound or unbound to each other. While at T = 0 , there is no transition between both states, such a transition is possible when T > 0 (due to the heat bath) and plays a key role as k B T approaches the magnitude of the attractive potential. We focus on a quantum regime, typical of chemical reactions, such that: (a) the thermal wavelength is shorter than the range of the attractive potential (lower limit on T) and (b) ( 3 / 2 ) k B T does not exceed the magnitude of the attractive potential (upper limit on T). In this regime, we extend several methods previously applied to analyze the time duration of DNA thermal denaturation. The two-particle system is then described by a non-equilibrium Wigner function. Under Assumptions (a) and (b), and for sufficiently long times, defined by a characteristic time scale D that is subsequently estimated, the general dissipationless non-equilibrium equation for the Wigner function is approximated by a Smoluchowski-like equation displaying dissipation and quantum effects. A comparison with the standard chemical kinetic equations is made. The time τ required for the two particles to transition from the bound state to unbound configurations is studied by means of the mean first passage time formalism. An approximate formula for τ, in terms of D and exhibiting the Arrhenius exponential factor, is obtained. Recombination processes are also briefly studied within our framework and compared with previous well-known methods.

Respiration and ethylene generation modeling of “Hass” avocado and feijoa fruits and application in modified atmosphere packaging

ABSTRACTA proper description of the respiration and ethylene generation is important in the development of package systems for the preservation of fresh fruits and vegetables. In this work, a model based on both Michaelis-Menten and chemical kinetics equations was developed and assessed in order to describe the respiration and ethylene generation of avocado (Persea americana cv. Hass) and feijoa fruits (Acca sellowiana Berg) from experimental data obtained at different temperatures by a closed system method. The temperature effect in both processes was described using Arrhenius-type relationships. After this, avocadoes and feijoas were packed in perforated low-density polyethylene (LDPE) and polypropylene (PP) bags for 8 days at 12°C and 80% RH to validate the usefulness of the model to describe the gas evolution in modified atmosphere packaging (MAP). For avocado fruits, respiration rates of 2680-3030 cm3 kg-1 d-1 were obtained at 24 °C and normal atmosphere, and 3650-4230 cm3 kg-1 d-1 for feijoa fruits. As for ethylene production rates, at 24 °C were obtained values of 3.11 cm3 kg-1 d-1 for avocado and 0.50 cm3 kg-1 d-1 for feijoa. At 6 °C, respiration and ethylene production rates were up to 6 times lower. It was possible to describe properly the ethylene generation using a Michaelis-Menten simple equation and the respiration rates using a Michaelis-Menten uncompetitive kinetics for both fruits with coefficients of determination above 0.9 in each case. The overall model was validated in the MAP test being possible to predict successfully the O2, CO2 and C2H4 levels inside the packages.

Dynamic model of the ladle treatment of steel

The basic factors that affect the ladle treatment of steel are analyzed. Balance equations, chemicalkinetic equations, thermodynamic equations, and hydrodynamic and gas-dynamic equations are written for the treatment of steel in a ladle–furnace unit with a vacuum-degassing system. A cylindrically symmetric calculation grid is developed. The equations are adapted on the basis of the finite-element method.

Modeling of the chemical composition of DC atmospheric pressure air discharge plasma in contact with aqueous solutions of sodium dodecylbenzenesulfonate

The results of calculation of the chemical composition of plasma of the DC atmospheric pressure discharge in air in contact with aqueous solutions of sulfonol (C12H25C6H4SO3Na) in the concentration range of 0–10 g/L at a discharge current of 40 mA are presented. At modeling, the combined solution of Boltzmann equation for the electron gas, equations of vibrational kinetics for ground states of N2, O2, H2O, and NO molecules, equations of chemical kinetics and plasma conductivity equation were used. It was shown that sodium atoms appearing as a result of transfer processes from the liquid cathode affect the chemical composition of the plasma and the characteristics of the electron gas.

ERENA: A fast and robust Jacobian-free integration method for ordinary differential equations of chemical kinetics

We herein propose a fast and robust Jacobian-free time integration method named as the extended robustness-enhanced numerical algorithm (ERENA) to treat the stiff ordinary differential equations (ODEs) of chemical kinetics. The formulation of ERENA is based on an exact solution of a quasi-steady-state approximation that is optimized to preserve the mass conservation law through use of a Lagrange multiplier method. ERENA exhibits higher accuracy and faster performance in homogeneous ignition simulations compared to existing popular explicit and implicit methods for stiff ODEs such as VODE, MTS, and CHEMEQ2. We investigate the effects of user-specified threshold values in ERENA, to provide trade-off information between the accuracy and the computational cost.

Laws of the oxidation of carbon isotopes in plasma processes under magnetic field

From law of quantum mechanics it follows that spin precession phase of unpaired electron in external magnetic field cannot be determined. It uncertainty necessary take into account in different physical and chemical processes. The expression of the rate constant of a chemical reaction based on the number of discrete spin states was obtained. The equations of chemical kinetics of plasma oxidation of carbon isotopes in the magnetic field were given.

Two-dimensional numerical modeling of combustion of Jordanian oil shale

ABSTRACTA two-dimensional (2-D) modeling of the burning process of Jordanian oil shale in a circulating fluidized bed (CFB) burner was done in this study. The governing equations of continuity, momentum, energy, mass diffusion, and chemical combustion reactions kinetics were solved numerically using the finite volume method. The numerical solution was carried out using a high-resolution 2-D mesh to account for the solid and gaseous phases, k-ε turbulence, non-premixed combustion, and reacting CFD model with the same dimensions and materials of the experimental combustion burner used in this work. The temperature distribution and evolution of species were also computed.Proximate and ultimate analyses were also performed to evaluate the air–fuel ratio and ash content. The required thermophysical properties, such as heating value, density, and porosity were obtained experimentally, while the activation energy was obtained from published literature.It was found that the temperature contours of the combustion process showed that the adiabatic flame temperature was 1080 K in a vertical burner, while the obtained experimental results of maximum temperature at various locations of the burner in actual, non-adiabatic, non-stoichiometric combustion reached 950 K, showing good agreement with the model.

Pyrolysis model of single biomass pellet in downdraft gasifier

By coupling the heat transfer equation with semi-global chemical reaction kinetic equations, a onedimensional, unsteady mathematical model is developed to describe the pyrolysis of single biomass pellet in the pyrolysis zone of downdraft gasifier. The simulation results in inert atmosphere and pyrolysis zone agree well with the published experimental results. The pyrolysis of biomass pellets in pyrolysis zone is investigated, and the results show that the estimated convective heat transfer coefficient and emissivity coefficient are suitable. The mean pyrolysis time is 15.22%, shorter than that in inert atmosphere, and the pellet pyrolysis process in pyrolysis zone belongs to fast pyrolysis. Among the pyrolysis products, tar yield is the most, gas the second, and char the least. During pyrolysis, the temperature change near the center is contrary to that near the surface. Pyrolysis gradually moves inwards layer by layer. With the increase of pyrolysis temperature and pellet diameter, the total pyrolysis time, tar yield, char yield and gas yield change in different ways. The height of pyrolysis zone is calculated to be 1.51–3.51 times of the characteristic pellet diameter.

Estimating the parameters of chemical kinetics equations from the partial information about their solution

The inverse problem of identifying the parameters of sets of ordinary differential equations using experimental measurements of three functions that correspond to some components in the vector solution of a set is considered. A private case that is important for applications of chemical and biochemical kinetics when reduced equations linearly depend on the combinations of initial unknown parameters has been studied. An analysis and the numerical results are presented for two types of sets of chemical kinetics equations, such as the Lotka–Volterra model that describes the coexistence of a predator and a prey and the chemical kinetics equations that model enzyme catalysts reactions, including the Michaelis–Menten equations. The search for unknown parameters is confined to the problem of minimizing a quadratic function. In this case, the reduced differential equations of systems are used instead of their vector solutions, which are unknown in most cases. The cases of both stable and unstable search for unknown parameters are analyzed.

Multiphase Modeling of Bottom-Stirred Ladle for Prediction of Slag–Steel Interface and Estimation of Desulfurization Behavior

Ladle furnace is a key unit in which various phenomena such as deoxidation, desulfurization, inclusion removal, and homogenization of alloy composition and temperature take place. Therefore, the processes present in the ladle play an important role in determining the quality of steel. Prediction of flow behavior of the phases present in the ladle furnace is needed to understand the phenomena that take place there and accordingly control the process parameters. In this study, first a mathematical model is developed to analyze the transient three-phase flow present. Argon gas bottom-stirred ladle with off-centered plugs has been used in this study. Volume of fluid method is used in a computational fluid dynamics (CFD) model to capture the behavior of slag, steel, and argon interfaces. The results are validated with data from literature. Eye opening and slag–steel interfacial area are calculated for different operating conditions and are compared with experimental and simulated results cited in literature. Desulfurization rate is then predicted using chemical kinetic equations, interfacial area, calculated from CFD model, and thermodynamic data, obtained from the Thermo-Calc software. Using the model, it is demonstrated that the double plug purging is more suitable than the single plug purging for the same level of total flow. The advantage is more distinct at higher flow rates as it leads higher interfacial area, needed for desulfurization and smaller eye openings (lower oxygen/nitrogen pickup).

Influence of Diffusion on the Kinetics of Multisite Phosphorylation

When an enzyme modifies multiple sites on a substrate, the influence of the relative diffusive motion of the reactants cannot be described by simply altering the rate constants in the rate equations of chemical kinetics. We have recently shown that, even as a first approximation, new transitions between the appropriate species must also be introduced. The physical reason for this is that a kinase, after phosphorylating one site, can rebind and modify another site instead of diffusing away. The corresponding new rate constants depend on the capture or rebinding probabilities that an enzyme-substrate pair, which is formed after dissociation from one site, reacts at the other site rather than diffusing apart. Here we generalize our previous work to describe both random and sequential phosphorylation by considering inequivalent modification sites. In addition, anisotropic reactive sites (instead of uniformly reactive spheres) are explicitly treated by using localized sink and source terms in the reaction-diffusion equations for the enzyme-substrate pair distribution function. Finally, we show that our results can be rederived using a phenomenological approach based on introducing transient encounter complexes into the standard kinetic scheme and then eliminating them using the steady-state approximation.

A chemical reaction network solver for the astrophysics code NIRVANA

Context. Chemistry often plays an important role in astrophysical gases. It regulates thermal properties by changing species abundances and via ionization processes. This way, time-dependent cooling mechanisms and other chemistry–related energy sources can have a profound influence on the dynamical evolution of an astrophysical system. Modeling those effects with the underlying chemical kinetics in realistic magneto-gasdynamical simulations provide the basis for a better link to observations.Aims. The present work describes the implementation of a chemical reaction network solver into the magneto-gasdynamical code NIRVANA. For this purpose a multispecies structure is installed, and a new module for evolving the rate equations of chemical kinetics is developed and coupled to the dynamical part of the code. A small chemical network for a hydrogen–helium plasma was constructed including associated thermal processes which is used in test problems.Methods. Evolving a chemical network within time-dependent simulations requires the additional solution of a set of coupled advection-reaction equations for species and gas temperature. Second-order Strang-splitting is used to separate the advection part from the reaction part. The ordinary differential equation (ODE) system representing the reaction part is solved with a fourth-order generalized Runge-Kutta method applicable for stiff systems inherent to astrochemistry.Results. A series of tests was performed in order to check the correctness of numerical and technical implementation. Tests include well-known stiff ODE problems from the mathematical literature in order to confirm accuracy properties of the solver used as well as problems combining gasdynamics and chemistry. Overall, very satisfactory results are achieved.Conclusions. The NIRVANA code is now ready to handle astrochemical processes in time-dependent simulations. An easy-to-use interface allows implementation of complex networks including thermal processes. In combination with NIRVANA’s self-gravity solver, its efficient solver for dissipation terms and its adaptive mesh refinement capability challenging astrophysical problems come into reach with the code.

Mathematics analysis of polymerase chain reaction kinetic curves.

The paper reviews different approaches to the mathematical analysis of polymerase chain reaction (PCR) kinetic curves. The basic principles of PCR mathematical analysis are presented. Approximation of PCR kinetic curves and PCR efficiency curves by various functions is described. Several PCR models based on chemical kinetics equations are suggested. Decision criteria for an optimal function to describe PCR efficiency are proposed.