A simple isothermal model for homogeneous-heterogeneous reactions in boundary-layer flow. I Equal diffusivities

A simple model for homogeneous-heterogeneous reactions in stagnation-point boundary-layer flow is constructed in which the homogeneous (bulk) reaction is assumed to be given by isothermal cubic autocatalator kinetics and the heterogeneous (surface) reaction by first order kinetics. The possible steady states of this system are analysed in detail in the case when the diffusion coefficients of both reactant and autocatalyst are equal. Hysteresis bifurcations leading to multiple solutions are found. The temporal stability of these steady states is then discussed.     

Melting heat transfer in Cu-water and Ag-water nanofluids flow with homogeneous-heterogeneous reactions

This article addresses melting heat transfer in magnetohydrodynamics(MHD)nanofluid flows by a rotating disk. The analysis is performed in Cu-water and Ag-water nanofluids. Thermal radiation, viscous dissipation, and chemical reactions impacts are added in the nanofluid model. Appropriate transformations lead to the nondimensionalized boundary layer equations. Series solutions for the resulting equations are computed.The role of pertinent parameters on the velocity, temperature, and concentration is analyzed in the outputs. It is revealed that the larger melting parameter enhances the velocity profile while the temperature profile decreases. The surface drag force and heat transfer rate are computed under the influence of pertinent parameters. Furthermore, the homogeneous reaction parameter serves to decrease the surface concentration.     

Chemically reactive and radiative von Kármán swirling flow due to a rotating disk

A new mathematical model is presented to study the heat and mass transfer characteristics of magnetohydrodynamic (MHD) Maxwell fluid flow over a convectively heated stretchable rotating disk. To regulate the fluid temperature at the surface, a simple isothermal model of homogeneous-heterogeneous reactions is employed. The impact of nonlinear thermal radiative heat flux on thermal transport features is studied. The transformed nonlinear system of ordinary differential equations is solved numerically with an efficient method, namely, the Runge-Kutta-Felberg fourth-order and fifth-order (RKF45) integration scheme using the MAPLE software. Achieved results are validated with previous studies in an excellent way. Major outcomes reveal that the magnetic flux reduces the velocity components in the radial, angular, and axial directions, and enhances the fluid temperature. Also, the presence of radiative heat flux is to raise the temperature of fluid. Further, the strength of homogeneous–heterogeneous reactions is useful to diminish the concentration of reaction.     

Bifurcation analysis of a diffusive ratio-dependent predator–prey model

In this paper, a ratio-dependent predator–prey model with diffusion is considered. The stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied. Necessary and sufficient conditions for the stability of the positive constant equilibrium are explicitly obtained. Spatially heterogeneous steady states with different spatial patterns are determined. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. For the steady state bifurcation, the normal form shows the possibility of pitchfork bifurcation and can be used to determine the stability of spatially inhomogeneous steady states. Some numerical simulations are carried out to illustrate and expand our theoretical results, in which, both spatially homogeneous and heterogeneous periodic solutions are observed. The numerical simulations also show the coexistence of two spatially inhomogeneous steady states, confirming the theoretical prediction.     

Diffusion separation of chemical elements on a catalytic surface

An asymptotic expansion of the solution, for large Schmidt numbers, of the system of equations of a chemically nonequilibrium multicomponent boundary layer on the catalytic surface of a blunt body [1] is used to obtain expressions for the diffusion fluxes of the reaction products and chemical elements and the heat flux as functions of the gradients of the reaction product concentrations, chemical element concentrations and enthalpy across the boundary layer. It is shown that when the body is exposed to a supersonic air flow, the diffusion separation of the chemical element oxygen depends importantly on the atom concentration at the outer edge of the boundary layer and the nature of the homogeneous and heterogeneous catalytic reactions. If the surface promotes the rapid recombination of oxygen atoms and is chemically neutral with respect to nitrogen atoms, then an excess of the chemical element oxygen is formed on the body. Otherwise we get an enhanced concentration of the element nitrogen. As distinct from the case of an ideally catalytic wall [2–4], on a surface possessing the property of catalytic selectivity the diffusion separation of chemical elements takes place even when only atoms are present at the outer edge of the boundary layer. On a chemically neutral surface diffusion separation may be caused by homogeneous recombination reactions between oxygen and nitrogen atoms if their rate constants are essentially different.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–121, July–August, 1988.     

Strong explosion in a combustible gas mixture

Assume that a planar, cylindrical, or spherical point explosion takes place in a combustible mixture of gases. As a result of the explosion a strong shock wave develops and triggers chemical reactions with the release of heat. The solution of the problem for the case in which the thickness of the heat release zone is neglected (the infinitely thin detonation wave model) was obtained in [1–3].It was emphasized in [4] that these solutions can be considered only as asymptotic solutions for time and distance scales which are large in comparison with the scales which are characteristic for the chemical reactions, and under the assumption that as the overdriven detonation wave which is formed in the explosion is weakened by the rarefaction waves it does not degenerate into an ordinary compression shock. Here the question remains open of the possibility of obtaining such asymptotic solutions with account for finite chemical-reaction rates.In conclusion the authors wish to thank E. Bishimov for carrying out most of the computations for this study.     

New Directions in Advanced Modeling and in Situ Measurements Near Reacting Surfaces

Multidimensional numerical modeling and in situ spatially-resolved measurements of gas-phase thermoscalars over the catalyst boundary layer have fostered fundamental investigation of the heterogeneous and homogeneous chemical reaction pathways and their coupling at realistic operating conditions. The methodology for validating catalytic and gas-phase reaction mechanisms is firstly outlined for industrially-relevant fuels. Combination of advanced modeling and in situ near-wall species and velocity measurements is then used to address the intricate interplay between interphase fluid transport (laminar or turbulent) and hetero-/homogeneous kinetics. Controlling parameters of this interplay are the homogeneous ignition chemistry, flame propagation characteristics, competition between the catalytic and gaseous pathways for fuel consumption, diffusional imbalance of the limiting reactant, flow laminarization due to heat transfer from the hot catalytic walls, and fuel leakage through the gaseous reaction zone. Dynamic reactor operation and intrinsic flame dynamics driven by interactions between homogeneous kinetics and catalytic walls are outlined using detailed transient simulation. It is shown that the presence of catalytic reactions moderates flame instabilities. Future directions for transient modeling and for temporally-resolved in situ near-wall measurements are finally summarized.     

Analysis of Differential Equations Modelling the Reactive Flow through a Deformable System of Cells

A system of model equations coupling fluid flow, deformation of solid structure and chemical reactions is formulated starting from processes in biological tissue. The main aim of this paper is to analyse this non-standard system, where the elasticity modules are functionals of a concentration and the diffusion coefficients of the chemical substances are functions of their concentrations. A new approach and new methods are required and adapted to these nonlinearities and the transmission conditions on the interface solid–fluid. Strong solutions for the initial and boundary value problem are constructed under suitable regularity assumptions on the data, and stability estimates of the solutions with respect to the initial and boundary values are proved. These estimates imply uniqueness directly. The approach of the paper can be used in more general problems modeling reactive flow and transport and its interaction with elastic cell structures. In a forthcoming paper the approach of this paper is used for getting the upscaled system modeling reactive flow through biological tissue on the macroscopic scale, starting from a system on the cell level.     

Wall effects on close-to-equilibrium flow in a tubular reactor

The deviation from local homogeneous chemical equilibrium flow due to non-catalytic heterogeneous reactions in a tubular reactor is analysed by an asymptotic expansion for large Damköhler numbers. Close to the reacting surface a boundary layer is shown to exist where the effects of the homogeneous and the heterogeneous reaction have an influence of the same order of magnitude. The extension of this boundary layer into the equilibrium flow is proportional to the inverse of the square root of the Damköhler number. The boundary layer equation is derived and solved for a particular example.     

Conjugated heat and mass transfer between a reactive solid and a gas in the presence of nonequilibrium chemical reactions

An investigation of a multicomponent boundary layer taking account of nonequilibrium chemical reactions has been made in a number of publications [1–3]; here, the temperature of the solid was assumed to be known or was determined from the condition of the conservation of energy at the interface between the gas and the solid, taking account of the solution of the equation of thermal conductivity in the solid phase. At the same time, heating of the material of a coating is an unavoidable step in any mechanism of thermokinetic decomposition and, in view of this, it is necessary to take account of the lag of the heat-transfer process inside the solid. Therefore, it is necessary to solve the equation of the energy balance in the solid phase simultaneously with the system of the equations of the boundary layer, i.e., the conjugate problem. The present article discusses the problem of flow around a solid in the vicinity of a frontal critical point, taking account of the dependence of the processes taking place in the solid body on the time, in the presence of two heterogeneous and one homogeneous reactions. The distributions of the velocity, the temperature, and the concentrations in the boundary layer are obtained, as well as the mass rate of entrainment of the material at different moments of time. The time of the change between kinetic and diffusion conditions of the course of the heterogeneous chemical reactions (the ignition time) is determined. It is established that, in the presence of a homogeneous chemical reaction, the mass rate of entrainment is less than with a frozen flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 121–128, March–April, 1974.     

Three-dimensional chemically nonequilibrium viscous shock layer on a catalytic surface with allowance for associated heat transfer

A three-dimensional flow of dissociating air past blunt bodies is investigated in the framework of the thin viscous shock layer theory. Multicomponent diffusion and homogeneous chemical reactions, including dissociation, recombination, and exchange reactions, are taken into account. The generalized Rankine-Kugoniot conditions are specified on the shock wave and the conditions which take into account the heterogeneous catalytic reactions, on the surface of the body. The viscous shock layer equations are solved together with the heat equations inside the coating, which is carbon with a deposited thin film of SiO₂, or quartz. The case of a thermally insulated surface is also considered. The problem for the case of the motion of a body along the re-entry trajectory into Earth's atmosphere is investigated numerically. The temperature of the surface and the heat flux toward it are given as a dependence on the height (tine) of the flight for different cases of the specification of the catalytic reactions. It is shown that the difference between the heat fluxes towards the thermally insulated surface and the fluxes toward the heat-conducting surface in the neighborhood of the stagnation point is of the order of 6–12% for all the cases considered. This makes it possible to decouple the solution of the problem of heat conduction in the body.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 140–146, November–December, 1985.deceased     

Theoretical investigation of ignition and detonation of coal-particle gas mixtures

A mathematical model of ignition of a coal-particle gas suspension is developed within the framework of mechanics of reacting heterogeneous media. Some qualitative features are studied, which reveal different typical variants of heat dynamics of the mixture: heterogeneous ignition caused by the coke residue ignition reaction, homogeneous ignition induced by volatile oxidation in the gas phase, and hybrid ignition due to the simultaneous effect of surface and volume reactions. Verification of the model is performed using available experimental data on ignition delay times for coal-particle suspensions in air and oxygen behind reflected shock waves. The model is extended to detonation processes in the suspension, verified by experimental data on the relation between the propagation velocity and initial particle concentration. Stationary detonation regimes are studied, and specific features of detonation structures are discussed.This paper was presented at the 18th International Colloquium on the Dynamics of Explosions and Reactive System, Seattle, Washington, USA, August, 2001     

Numerical Simulation of Pulverized Coal MILD Combustion Using a New Heterogeneous Combustion Submodel

The scope of this investigation is the application and analysis of a recently developed submodel (Schulze et al., Oil Gas Science Technol, 2013, doi:10.2516/ogst/2012069) for char particle combustion and gasification. The distinguishing feature of this model is a detailed representation of the diffusion and convection processes as well as the homogeneous reactions in the boundary layer around the char particle. These processes are fully coupled to the heterogeneous particle kinetics. The model was implemented into the CFD code ANSYS-Fluent. The coupled solver is used for simulating the IFRF full scale pulverized coal combustion MILD furnace, for which detailed experimental data are available for model evaluation (Orsino et al., IFRF Doc. No F46/y/3, 2000) The new model yields improved agreement with measured data as compared to the standard modeling approach. This can be directly related to the prediction of the char burnout rate. For further analysis, the mixing field in the IFRF furnace is investigated in detail by introducing four mixture fractions for pyrolysis products, char burn-off gases, primary and secondary air, respectively. The solutions of the respective transport equations are used to define the local stoichiometry both in the gas phase and on the particle surface in such a multi-stream system. The conditions in the particle surrounding gas phase as well as on the particle surface are used to define the regime of particle-gas interaction based on the simulations with the new submodel. It can be shown that for certain conditions the homogeneous reactions in the particle boundary must be accounted for.     

Two-dimensional stability of the combustion of condensed systems

The question of the combustion stability of condensed systems relative to curvature of the front is investigated in a linear approximation. Two of the simplest combustion models are examined, a gasless system and a model of flameless combustion of a solid fuel. In the first case, the combustion products are condensed, just as are the initial materials, and in the second the solid fuel is converted into a gas in which no chemical reactions occur. Boundaries of the stability of the stationary combustion mode are found. It is shown that gasless systems are less stable with respect to two-dimensional perturbations than to one-dimensional perturbations. For the flameless combustion model the result depends on the relationship between the thermophysical constants of the initial material and the products. The question of the influence of heat emission on the one-dimensional stability of the gasless composites is considered. An increase in the heat emission diminishes the stable combustion region, where a one-dimensional instability originates earlier than collapse of combustion occurs because of strong heat emission to the wall.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 51–59, September–October, 1971.     

A hypersonic chemically nonequilibrium viscous shock layer oh wings with a catalytic surface

Within the framework of the theory of a hypersonic viscous shock layer a study is made of flow round wings of infinite span with blunt leading edges at various angles of attack and slip. Account is taken of multicomponent diffusion, and homogeneous chemical reactions, including dissociation-recombination reactions and exchange reactions. On the shock wave the generalized Rankine-Hugoniot conditions are given, and on the surface of the body conditions which allow for heterogeneous catalytic reactions of the first order with reaction rate constants depending [1] or not depending [2] on the temperature. The cases of an ideally catalytic and a noncatalytic surface are also considered. The surface of the body is assumed to be heatinsulated. A numerical study was made of the problem in a broad range of variation in the angles of attack and slip for different cases of prescribed constants representing the rates of the heterogeneous reactions. The conditions of the flow corresponded to the motion of a body which possess a lifting force along the trajectory of entry into the Earth's atmosphere [3]. The dependences are given of the equilibrium temperature of the surface along the stagnation line of the wing on the height of the flight and the distribution of this temperature along the surface of wings with parabolic and hyperbolic contours. It is shown that for flow regimes with a relatively high degree of dissociation in cases when the proportion of atoms recombined on the surface of the body is small, the dependences of the heat flow and the temperature of the surface on the angle of slip are of a nonmonotonic nature.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza., No. 6, pp. 127–135, November–December, 1984.     

Particle-scale and sub-grid drag models coupled CFD for simulating the CO methanation in a CFB riser

The diffusion and chemical reactions inside the catalyst particles and the heterogeneous flow structure in the computational cells are key factors to affect the accuracy of the coarse-grid simulation in circulating fluidized bed (CFB) methanation reactors. In this work, a particle-scale model is developed to calculate the effective reaction rate considering the transient diffusion and chemical reactions in the particle scale, i.e., the scale of the single catalyst particle. A modified sub-grid drag model is proposed to consider the effects of the meso-scale and chemical reactions on the heterogeneous gas-solid interaction, where the meso-scale is between the single particle and the whole reactor and featured with the particle cluster. Subsequently, a coupled model is developed by integrating the particle-scale and modified sub-grid drag models into CFD. Moreover, the coupled model is validated to achieve accurate predictions on the CO methanation process in a CFB riser. Notably, the coupled model can be performed with a coarse grid (∼58 times particle diameter) and a large time step (0.005 s) to accelerate the simulation. By simply changing the reaction kinetics, different gas-solid catalytic reaction systems can be simulated by using the coupled model.     

Investigation of the Influence of Different Heterogeneous Recombination Mechanisms on the Heat Fluxes to a Catalytic Surface in Dissociated Carbon Dioxide

A kinetic model of heterogeneous recombination in dissociated carbon dioxide on high-temperature heat-shield coatings is developed; the model takes into account the nonequilibrium adsorption-desorption reactions of oxygen atoms and their recombination in the Eley-Rideal and Langmuir-Hinshelwood reactions. On the basis of a comparison of the calculated heat fluxes in dissociated carbon dioxide with those measured in the VGU-3 plasma generator of the Institute for Problems in Mechanics of the Russian Academy of Sciences (IPM RAS) and the available literature data, the parameters of the catalysis model are chosen for the glassy coating of the Buran orbiter tile heat shield based on the SiO₂–B₂O₃–SiB₄ system. The effects of heterogeneous recombination proceeding in accordance with the Langmuir-Hinshelwood mechanism, as well as the processes involving carbon atoms and those involving physically adsorbed oxygen atoms, on the heat fluxes to the glassy coating are analyzed on the surface temperature range from 300 to 2000 K.     

Steady growth of a crack with a rate and temperature sensitive cohesive zone

The steady-state dynamic propagation of a crack in a heat conducting elastic body is numerically simulated. Specifically, a mode III semi-infinite crack with a nonlinear temperature dependent cohesive zone is assumed to be moving in an unbounded homogeneous linear thermoelastic continuum. The numerical results are obtained via a semi-analytical technique based on complex variables and integral transforms. The relation between the thermo-mechanical properties of the failure zone and the resulting crack growth regime are investigated. The results show that temperature dependent solutions are substantially different from purely mechanical ones in that their existence and stability strongly depends on the cohesive zone thermal properties.