Cooling of a composite slab

A mixed boundary value problem associated with the diffusion equation, that involves the physical problem of cooling of an infinite parallel-sided composite slab, is solved completely by using the Wiener-Hopf technique. An analytical expression is derived for the sputtering temperature at the quench front being created by a cold fluid moving on the upper surface of the slab at a constant speed v. The dependence of the various configurational parameters of the problem under consideration, on the sputtering temperature, is rather complicated and representative tables of numerical values of this important physical quantity are prepared for certain typical values of these parameters. Asymptotic results in their most simplified forms are also obtained when (i) the ratio of the thicknesses of the two materials comprising the slab is very much smaller than unity, and (ii) the quench-front speed v is very large, keeping the other parameters fixed, in both the cases.     

Cooling of a composite slab

A mixed boundary value problem associated with the diffusion equation, that involves the physical problem of cooling of an infinite parallel-sided composite slab, is solved completely by using the Wiener-Hopf technique. An analytical expression is derived for the sputtering temperature at the quench front being created by a cold fluid moving on the upper surface of the slab at a constant speed v. The dependence of the various configurational parameters of the problem under consideration, on the sputtering temperature, is rather complicated and representative tables of numerical values of this important physical quantity are prepared for certain typical values of these parameters. Asymptotic results in their most simplified forms are also obtained when (i) the ratio of the thicknesses of the two materials comprising the slab is very much smaller than unity, and (ii) the quench-front speed v is very large, keeping the other parameters fixed, in both the cases.     

Variational-Lagrangian irreversible thermodynamics of initially-stressed solids with thermomolecular diffusion and chemical reactions

A general principle of virtual dissipation in irreversible thermodynamics is applied to a solid under initial stress with small non-isothermal incremental deformations and coupled thermomolecular diffusion and chemical reactions. Dynamical field and Lagrangian equations are obtained directly by variational procedures. In addition, the treatment embodies new fundamental concepts and methods in the thermodynamics of open systems and thermochemistry. The new concept of ‘thermobaric potential’ is briefly outlined. The theory is also applicable to porous solids with ‘diffusionlike’ behaviour of pore-fluid mixtures. General validity of viscoelastic correspondence for chemical or other relaxation processes with internal coordinates is indicated in acoustic propagation and seismic problems.     

Self-similar problems of convective diffusion in the presence of heterogeneous chemical reactions with allowance for thermal diffusion effects

A number of studies have been made of the problem of the effect of a temperature gradient on mass transfer in a forced viscous fluid flow. The question of allowing for thermal diffusion effects has been examined in connection with flow around bodies [1–4], duct flow [5], and jet flows [6,7]. Recently, in addition to the problem of thermal diffusion separation, the attention of investigators has been claimed by the problem of taking into account the effect of thermal diffusion on mass transfer in a convective flow in the presence of chemical reactions on the flow surfaces [4].     

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.     

Consecutive chemical reactions in a turbular reactor with turbulent flow

Homogeneous consecutive chemical reactions in turbulent flow system is analyzed in this paper. Arbitrary chemical reaction order is considered. The governing nonlinear material balance equations are solved numerically on a digital computer. It is seen from the preliminary numerical results that high Reynolds and Schmidt numbers significantly increase both reaction velocities.Nomenclature c _i concentration of i species, g. mole/cm³ - C _i dimensionless concentration of i species, c _i /c _A0 - c _A0 inlet concentration of A, g. mole/cm³ - D _i molecular diffusivity of i species, cm²/sec. - f Fanning friction factor - k ₁ first reaction rate coefficient, (g. mole/cm³)^1–m /sec. - k ₂ second reaction rate coefficient, (g. mole/cm³)^1–n /sec. - K ₁ dimensionless first reaction rate coefficient, k ₁ r ₀ ² c _A0 ^–1 /D _A - K ₂ dimensionless second reaction rate coefficient, k ₂ r ₀ ² c _A0 ^–1 /D _A - m order of the first chemical reaction - n order of the second chemical reaction - r radial distance, cm - r ₀ reactor radius, cm - R dimensionless radial distance, r/r ₀ - Re Reynolds number, r ₀ / - Sc Schmidt number, D/ - u local velocity, cm - bulk velocity, cm - U dimensionless local velocity, u/ - X, Y dimensionless concentrations of A and B, c _i /c _A 0 - xxx0058;, xxx0059; dimensionless bulk concentrations of A and B - z axial distance, cm - Z dimensionless axial distance, zD _A /r ₀ ² - ratio of eddy mass diffusivity to eddy viscosity, / - eddy mass diffusivity, cm²/sec. - kinematic viscosity, cm²/sec.     

A diffusion problem with chemical reaction

Summary A substance is subjected to cylindrical Poiseuille flow, diffusion and a chemical reaction. In this paper the stationary state will be considered. It is assumed that the axial component of the diffusion can be neglected. Then the problem can be reduced to an eigenvalue problem determining an orthogonal set of eigenfunctions. Also a simplified model will be considered where the cylindrical wall is replaced by a flat wall. This problem can be solved by means of Laplace transformation. An explicit solution for the concentration at the wall has been obtained.     

A further note on a diffusion problem with chemical reaction

Summary The problem of chemical reaction in an isothermal liminar-flow tubular reactor has been studied by Lauwerier for a first order irreversible reaction. The analysis of Lauwerier is extented in this work to include the case of consecutive irreversible first order reactions. Also the important numerical parameters have been calculated and are tabulated in this paper.     

Viscous Rayleigh-Taylor instability with and without diffusion effect

The approximate but analytical solution of the viscous Rayleigh-Taylor instability(RTI) has been widely used recently in theoretical and numerical investigations due to its clarity. In this paper, a modified analytical solution of the growth rate for the viscous RTI of incompressible fluids is obtained based on an approximate method. Its accuracy is verified numerically to be significantly improved in comparison with the previous one in the whole wave number range for different viscosity ratios and Atwood numbers. Furthermore, this solution is expanded for viscous RTI including the concentration-diffusion effect.     

Flow of a viscous gas in a shock layer with equilibrium chemical reactions

Steady flow of supersonic air over a sphere is examined, allowing for viscosity, heat conduction, and actual physical and chemical processes. Flow in the shock layer at flight speeds in the range 3 km/sec V10 km/sec (10⁴R10⁶) is investigated, under the assumption of local thermodynamic equilibrium. The flow is described by simplified Navier-Stokes equations, which are solved by a finite difference method. The case of a cooled surface is examined. The distribution of gasdynamic parameters is obtained in different flow regimes. The distribution of heat flux and friction coefficient is investigated as a function of the oncoming-stream parameters and the sphere radius. The shape and position of the shock wave are determined, and the stream lines and sonic lines are constructed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 150–153, July–August, 1970.The authors thank Yu. P. Lun'kin and F. D. Popov for their help in formulating the problem and their constant interest.     

Formation of gas flows with account for the effects of heat conduction,viscosity, diffusion,and chemical reactions

In this paper we examine the formation of one-dimensional-nonsteady gas flows with account for the influence of thermal conduction, viscosity, diffusion, and chemical reactions. An analogous problem but without account for the effect of diffusion and chemical reactions was examined in [1].It is shown that the effect of chemical reactions is not significant for short times-the gas behaves as an incompressible fluid. The results obtained may be used to calculate gas flows with discontinuous initial conditions with subsequent numerical integration of the Navier-Stokes equations.As examples we consider the flows of binary gas mixtures which arise upon application of thermal flux to the flat end of a moving piston (1) and during the decay of an arbitrary discontinuity (2).It is assumed that the internal degrees of freedom of the gas molecules are in equilibrium with the translational degrees of freedom.The author wishes to thank Yu. A. Dem'yanov for his guidance in this study.     

Boundary conditions for flows with chemical reactions occurring on a surface

With the use of a solution of a model Boltzmann equation for a binary mixture in the Knudsen layer, we obtain the boundary conditions for the equations of gas dynamics when the reactionl _iA_il _jA_j (l _i molecules of A_i change intol _j molecules of A_j, and vice versa) is occurring on a surface. The boundary condition that we obtain differs from those that are usually applicable by the presence of terms of the same order. This confirms the conclusion arrived at by the authors in [1], where it was shown that if the Knudsen layer is left out of account, which is precisely what is usually done, it is impossible to obtain correct boundary conditions.Moscow. Translated from Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 129–138, January–February, 1972.     

Consecutive chemical reactions in an annular reactor with non-newtonian flow

The purpose of this paper is to analyze the homogeneous consecutive chemical reactions carried out in an annular reactor with non-Newtonian laminar flow. The fluids are assumed to be characterized by a Ostwald-de Waele (powerlaw) model and the reaction kinetics is considered of general order. Effects of flow pseudoplasticity, dimensionless reaction rate constants, order of reaction kinetics and ratio of inner to outer radii of reactor on the reactor performances are examined in detail.Nomenclature c _A concentration of reactant A, g.mole/cm³ - c _B concentration of reactant B, g.mole/cm³ - c _A0 inlet concentration of reactant A, g.mole/cm³ - C ₁ dimensionless concentration of A, c _A/c _A0 - C ₂ dimensionless concentration of B, c _B/c _A0 - C ₁ dimensionless bulk concentration of A - C ₂ dimensionless bulk concentration of B - D _A molecular diffusivity of A, cm²/sec - D _B molecular diffusivity of B, cm²/sec - k _A first reaction rate constant, (g.mole/cm³)^1–m /sec - k _B second reaction rate constant, (g.mole/cm³)^1–n /sec - K ₁ dimensionless first reaction rate constant, k _A r ₀ ² c _A0 ^m–1 /D _A - K ₂ dimensionless second reaction rate constant, k _B r ₀ ² c _A0 ^n–1 /D _B - K apparent viscosity, dyne(sec) ^m /cm² - m order of reaction kinetics - n order of reaction kinetics - P pressure, dyne/cm² - r radial coordinate, cm - r _i radius of inner tube, cm - r _max radius at maximum velocity, cm - r _o radius of outer tube, cm - R dimensionless radial coordinate, r/r _o - s reciprocal of rheological parameter for power-law model - u local velocity, cm/sec - u _max maximum velocity, cm/sec - u bulk velocity, cm/sec - U dimensionless velocity, u/u - z axial coordinate, cm - Z dimensionless axial coordinate, zD _A/r ₀ ² /u - ratio of molecular diffusivity, D _B/D _A - ratio of inner to outer radius of reactor, r _i/r _o - ratio of radius at maximum velocity to outer radius, r _max/r _o