Movement of gas bubbles in a liquid layer in the presence of diffusion and chemical reactions

The problem of a bubbling reactor, in which gas and liquid are mixed by the passage of bubbles of gas through a liquid layer, is discussed. We give the results of a numerical solution of the system of equations describing the processes occurring in the reactor in the case where there are no chemical reactions, and also in the case where chemical reactions take place at constant temperature.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 131–135, January–February, 1971.The authors thank L. A. Chudov for advice and interest in the work.     

Experimental realization of unstable regimes of ethanol oxidation into acetaldehyde on a copper-oxide catalyst in a chemical flow reactor

Unstable regimes of operation of a chemical reactor have been stabilized in an experiment by means of a control system with feedback. It has been shown that the concentration limits of the region of multiplicity of reactor steady states do not in general coincide with the concentration limits that can be obtained without the use of stabilizing systems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 8–13, March–April, 1984.     

Influence of longitudinal mixing on the degree of reactant conversion in a chemical flow reactor

The method of small perturbations in the approximation of a weak chemical reaction (low dimensionless reaction rate) is used to obtain, to terms of third order, an approximate analytic solution to the problem of the concentration distribution in a one-dimensional chemical flow reactor. This solution makes it possible to analyze the dependence of the degree of conversion of the original reactant on the longitudinal diffusion and other factors. An autocatalytic reaction, in which the degree of conversion depends nonmonotonically on the Péclet number, is considered as an example. The investigation shows that for different values of the parameters of the problem longitudinal mixing can both increase and decrease the degree of conversion. The results make it possible to identify ranges of variation of the parameters which characterize the operation of the reactor in which longitudinal mixing has different influences on the degree of chemical conversion and find the degree of longitudinal mixing which ensures optimal operation of the chemical reactor.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 84–90, March–April, 1981.     

Study of the stable and unstable operating regimes of a chemical reactor

The problem of the stabilization of the unstable operating regimes of a flow reactor is considered. It is shown that for the same values of the external parameters up to five steady-state regimes may exist, and that without the use of stabilizing systems it is impossible to realize not only the unstable but also some stable regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 54–59, May–June, 1988.     

Thermomechanical instability of a heat releasing granular layer

A study is made of a homogeneous granular layer of thickness h and porosity in which heat is released uniformly with bulk power q. The heat is taken up by a uniform gas flow through the layer in the transverse direction of the y axis at velocity v ° produced by a given pressure difference across the layer. This scheme corresponds to the simplest model of a gas-cooled nuclear reactor, in which the heat release does not depend on the thermogasdynamic state. The case of a chemical reactor with heat release of Arrhenius type is considered in [1, 2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 45–53, March–April, 1979.     

Operating regimes of a chemical reactor with longitudinal and transverse mixing in the case of large Péclet numbers

A two-dimensional model of a chemical reactor with longitudinal and transverse mixing is investigated in the case of large Péclet numbers calculated from the effective thermal conductivity in the transverse direction. For this model the existence of at least one steady-state regime has been demonstrated [1], sufficient criteria of its uniqueness have been determined, an asymptotic expansion of the solution has been constructed in the case of small Péclet numbers, and the critical ignition and quenching parameters have been found. In this paper the other limiting case of the model, in which heat is propagated in the transverse direction much more slowly than it is transported by the flow along the reactor (large Péclet numbers), is analyzed in detail. An asymptotic expansion of the solution which closely coincides with the data of numerical calculations is constructed. The critical quenching and ignition conditions of the process are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 120–127, January–February, 1987.     

Investigation of the aerothermochemical process in a fixed-bed catalytic reactor

The transport processes in a fixed-bed catalytic reactor are investigated numerically with allowance for the interaction of the thermal and hydrodynamic effects. The problem is solved in the two-dimensional formulation over the entire interval of laminar viscous gas flow. The effect of voidage nonuniformities near the wall on the aerodynamic, temperature and concentration patterns is studied. The possible coexistence of different heterogeneous catalysis regimes over the cross section of the apparatus, depending on the gas dynamic and thermophysical parameters, is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 116–123, March–April, 1991.     

Use of a Monte Carlo PDF method in a study of the influence of turbulent fluctuations on selectivity in a jet-stirred reactor

A model for single-phase turbulent reacting flow is presented and a solution algorithm is described. The model combines the standardk - model for the velocity field with a transport equation for the probability density function (PDF) of the thermochemical variables. In this equation terms describing spatial transport by velocity fluctuations and mixing on the smallest scales are modelled. The essential advantage of this approach is that the effect of nonlinear kinetics appears in closed form and that the influence of turbulent fluctuations on mean reaction rates is included. A stochastic algorithm for the solution of the PDF transport equation, essentially due to Pope, is described. Cylindrical symmetry is assumed. The PDF is represented by ensembles ofN representative values of the thermochemical variables in each cell of a nonuniform finite-difference grid and operations on these elements representing convection, diffusion, mixing and reaction are derived. A simplified model and solution algorithm which neglects the influence of turbulent fluctuations on mean reaction rates is also described. Both algorithms are applied to a selectivity problem in a real reactor studied earlier by Liu and Barkelew. Spatial profiles of mean species mole fractions and of relative selectivity to the target product are obtained. The profiles are clearly different in both models but at the end of the reactor the same selectivity is predicted.Presented at the Shell Conference on Computational Fluid Dynamics for Petrochemical Process Equipment, Hoenderloo, December 10–12, 1989.     

Numerical investigation of liquid-gas system behavior in a variable acceleration vector field by a discontinuity capturing technique

A mathematical model of the behavior of a liquid in a partially filled vessel exposed to a varying acceleration vector field is presented, together with a numerical approach for solving the problem based on a discontinuity capturing technique. The results of calculations are also presented.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 129–134, September–October, 1994.     

Calculation of a Heat Flow from a Spherical Particle in a Diatomic Gas

The problem of heat flow from a uniformly heated spherical particle in a diatomic gas is considered. The paper reports results of numerical calculations for an analog of the Bhatnagar–Gross–Krook model of the collision integral under for purely diffuse reflection of the gas molecules from the surface.     

Solution to problem of strong evaporation of a monatomic gas by the Monte Carlo method

The Monte Carlo method has been used to obtain a numerical solution to the problem of strong evaporation of a monatomic gas in which the molecules are modeled by pseudo-Maxwellian and hard spheres. A comparison with the results of other authors is made. The results agree well with the solution of the problem obtained on the basis of the model Bhatnagar—Gross—Krook kinetic equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 185–188, January–February, 1984.I should like to thank M. N. Kogan for discussing the results.     

Model of the internal circulation of a liquid drop in a gas stream

An analytical model for calculating the internal flow in a drop is proposed. The model is checked by numerically solving the exact boundary-value problem for the Navier-Stokes equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 167–170, November–December, 1989.     

Spin-down of a fluid in a low cylinder at large Reynolds numbers

The problem of the axisymmetric motion of a fluid between infinite disks is solved by the method of matched asymptotic expansions without introducing model assumptions. For the strongly nonlinear stage of spin-down solutions are found that correspond to initial states different from rigid-body rotation, when the boundary layer is not a Kármán layer. The experimental results obtained are in qualitative and quantitative agreement with the theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 39–46, May–June, 1986.The authors wish to thank A. M. Obukhov and F. V. Dolzhanskii for formulating the problem and for constructive discussion.     

Calculation of the main characteristics of a turbulent stream of incompressible fluid in a pipe

The reports [1–5] are devoted to the calculation of the characteristics of the steady turbulent flow of an incompressible fluid in a straight round pipe using the model of A. N. Kolmogorov, Additional assumptions are introduced in these reports, such as not allowing for energy diffusion or molecular viscosity, dividing the region of flow into arbitrary layers, etc. In the present report the problem is solved in a more general formulation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–163, September–October, 1977.     

Experimental study of stable and unstable regimes of ethanol-to-acetaldehyde oxidation on a copper catalyst in a flow reactor

The oxidation of ethanol to acetaldehyde by atmospheric oxygen on a stationary copper catalyst in a flow reactor is investigated. It is shown that by operating in unstable rather than stable regimes the specific efficiency of conversion w of ethanol to the target product — acetaldehyde — can be increased by 13%. Operating in unstable regimes makes it possible, without reducing w (as compared with the most efficient stable regime) to increase the total productivity of the reactor by a factor of 2.89.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–17, May–June, 1989.     

Optimal vorticity regime in the flow tract of a hydroturbine

A study is made of the problem of maximizing the power taken from the shaft of the working rotor of a hydroturbine for a fixed available energy difference in the framework of a two-dimensional axisymmetric flow model. Necessary conditions of optimality of first and second order are derived and used to set up an algorithm for numerical solution of the problem. The results of calculations are given, and a comparison is made with optimal solutions obtained using two- and one-dimensional models of axisymmetric flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 68–73, May–June, 1984.     

Influence of transport processes on steady regimes of a chemical flow reactor with reaction of the type a → b,a+b→2c

An approximate analytic solution is obtained to the problem of the concentration distribution in a one-dimensional chemical flow reactor for a reaction of the type A B, A + B 2C in the approximation of a weak reaction. The method of small perturbations up to terms of third order is used. A study is made of the influence of the intensity of longitudinal mixing on the degree of transformation, the selectivity, and the yield. It is found that the optimal level of longitudinal mixing in the weak reaction approxmation is determined by a dimensionless parameter that is a combination of the dimensionless rates of the chemical reactions. The dependence of the optimal Péclet number on this parameter is found. Characteristic regions are determined on the plane of the determining parameters in which the interaction of the physical and chemical processes in the reactor leads to different results, which must be taken into account when choosing the type and the parameters of a chemical reactor.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 106–113, November–December, 1982.     

Influence of non-Newtonian properties of oil on the operation of gushers

A study is made of the problem of the conditions of stable operation of gushers that produce non-Newtonian oils. It is shown that the increasing branch of the lift characteristic always corresponds to stable regimes of operation of the gusher. A working point on the decreasing branch is stable when the indicator diagram of the stratum has at this point a slope less steep than the lift characteristic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 163–165, July–August, 1984.     

Electromagnetic field generated in air by a nonstationary gamma-ray source in an ideally conducting plane

We consider a model problem on the generation of a radio signal by a nonstationary gamma-ray source. The problem is essentially two-dimensional in space but is reduced to a number of one-dimensional nonstationary problems. The results of a numerical solution of the problem are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 18–26, May–June, 1975.In conclusion, the authors wish to thank A. A. Milyutin and I. E. Dinaburg for working out the numerical methods used for the solution of this problem, and also thank I. N. Mikhailov and G. M. Gandel'man for their participation in the development of the problem.     

Steady states of a continuous-flow adiabatic chemical reactor

Flow reactors are widely used in the chemical industry for purposes of catalytic reactions [1,2]. Calculation of reactors of this type, even in one-dimemional approximation, is complicated and possible only with the use of numerical methods [1, 3]. Such calculations make it possible to find the steady-state distribution of temperature and concentration in the chemical reactor if one exists; in general, however, there may be other steady-state regimes which may be preferable from the standpoint of obtaining a different degree of conversion of the starting product, operating stability, etc.In this connection special interest attaches to the question of the existence and number of steady-state solutions of the system of equations describing the reactor process.This problem was previously considered in [4–7]. Thus, in [4, 5] it was pointed out that in certain special cases more than one steady-state regime may exist. In [6, 7] the question of sufficient conditions of uniqueness was investigated. In [7] it was shown that the steady-state regime is unique in the ease of short reactors or a dilute mixture of reactants. In [8] the problem of the existence and uniqueness of the steady-state regime was examined for a chain reaction model with direct application of the general theorems of functional analysis.The present paper includes an analysis of a very simple mathematical model of an adiabatic chemical reactor in which an exothermic or endothermie reaction takes place. It is established that in the case of an endothermic process a unique steady-state regime always exists. In the exothermic case the problem of the steady-state regime also always has a solution which, however, may be nonunique; the possibility of the existence of several steady-state regimes, associated with the form of the temperature dependence of the heat release rate, is substantiated.The authors thank G. I. Barenblatt, A. I. Leonov, L. M. Pis'men, and Yu. I. Kharkats for discussing and commenting on the work.