A. D. Polyanin

Convective diffusion to a reacting particle in a fluid. Nonlinear surface reaction kinetics

Abstract The paper is concerned with convective diffusion to a spherical particle in a uniform Stokes flow at large Peclet numbers in the case of a chemical reaction occurring on the particle surface with the rate being finite and arbitrarily dependent on concentration. Integral equations for the local diffusion flow and the surface concentration are obtained and a numerical method of their solution is developed based on the use of the appropriate asymptotics in the neighbourhood of the forward stagnation point. The effect of the reaction rate constant and the reaction kinetics on the total diffusion flux to the particle surface is studied. A simple approximate formula is suggested allowing determination of the mean Sherwood number with adequate accuracy. A chain of reacting particles is considered and a corresponding integral equation is obtained for the local diffusion flux on their surface. A qualitative analysis of mass transfer for the chain of spheres is performed and it is shown that interaction of the diffusion wakes and boundary layers of particles in this kind of the ordered systems results in a substantial decrease in the mass transfer rate. Approximate expressions are obtained for integral fluxes to the chain particles. The proposed method is extended to an arbitrary three-dimensional flow around particles (or drops) of arbitrary shape.

Diffusion to a chain of drops (bubbles) at large péclet numbers

The problem of diffusion of a substance, dissolved in a flow, to absorbing drops (bubbles) moving one after another in a viscous incompressible fluid is investigated. An approximate analytic expression is obtained for the differential and integral flows of the substance to the surface of each drop with consideration of the changes of the concentration and velocity fields due to the presence of other drops. A chain of spherical drops of equal radius arranged on the axis of a uniform forward flow is examined. It is shown that if the distance between drops, referred to the radius of the drops, satisfies the inequality 1≪l≪P1/2 (P is the Péclet number), then the integral inflow of the substance to the surface of the second drop of the chain is 2.41 times less than the integral inflow to the first (the drops are enumerated along the flow); the total diffusion flow to the surface of an arbitrary drop with number k is determined by the expression Ik=I1[k1/2 − (k−1)1/2], where Ik is the total flow to the first drop of the chain. The case of diffusion interaction of a solid particle and drop is examined. It is shown that for particles moving one after another with the same velocity in a fluid at rest the presence of a drop before the solid particle leads to a marked decrease of the total diffusion flow of the solid particle [by O(P1/6) times], whereas the presence of a solid particle before a drop does not affect (in the main approximation with respect to the characteristic diffusion parameter) the total flow of the latter.Ik=Ii[k1/2−(k−1)1/2]

Convective diffusion to a solid particle in the case of nonlinear kinetics of heterogeneous chemical reaction

Abstract An approximate solution is obtained for a stationary problem of mass transfer between a moving solid spherical particle and a laminar gas flow at low finite Peclet and Reynolds numbers. The case of a chemical surface reaction which depends arbitrarily on reagent concentration at the particle surface is considered. The results obtained may be used, in particular, to determine the rate of mass transfer between a particle and a flow for the integral- and fractional-order reactions. The solution of the problem has been found by the method of matching outer and inner asymptotic expansions in the Peclet number. The concentration field has been determined. The dependence of the total reagent flow at the particle surface on the reaction kinetics, rate constant and the Peclet number has been obtained.

Mass transfer of particles located on the flow axis at large peclet number

Diffusional influx of a substance dissolved in a medium into absorbent particles moving relative to one another in a viscous incompressible liquid is examined. An approximate analytical expression is obtained for the differential and integral flux of material into the surface of each particle, accounting for variations in the velocity and concentration fields due to the presence of the other particles. The results obtained can be applied to a lattice of spheres washed directly and uniformly in an infinite flow and located at distance 1 ≪l ≪ P1/3 relative to each other. It is shown that the diffusional flux of material into the first sphere is almost twice as large as into the other, and for a large number of spheres k the total diffusion flux tends to zero inversely as the 1/3 power of k.

Heat and mass transfer to a reacting particle in a gas flow for arbitrary temperature dependence of the transport coefficients

A study is made of simultaneous heat and mass transfer to a reacting particle of any shape in a translational (and shear) flow of a viscous heat conducting compressible gas for which the thermal conductivity and diffusion coefficient, and also the specific heat depend on the temperature. The first two terms of the asymptotic expansion with respect to the small Reynolds number are obtained for the mean Sherwood and Nusselt numbers. The case of a power-law temperature dependence of the gas viscosity is considered in detail.

Experimental data processing by means of ‘asymptotic coordinates’

Abstract A new simple method for processing experimental data is suggested which allows one, in many cases, to reveal universal relations in different fields of science and technology. The method also makes it possible to effectively model and forecast qualitatively similar phenomena and processes that occur under identical conditions. A number of concrete examples are considered of the suggested method of application to the analysis of some experimental results obtained when studying different processes of convective heat and mass transfer, chemical technology and hydrodynamics.

Approximate integration op nonstationary equations of a diffusion or thermal boundary layer

An approximate method is proposed for integrating the nonstationary equations of a diffusion or thermal boundary layer using the known steady solution in the planar or axisymmetric case. It is shown that the proposed method is exact in problems involving mass or heat transfer of reacting drops and bubbles in a laminar flow of a viscous incompressible fluid and also particles moving in an ideal fluid. An integral equation is obtained for the local diffusion or heat flux in the case of abrupt “activation” of a reaction on the surface of a particle.

Some general invariance relations in problems of convective heat and mass transfer at large péclet numbers

Some general invariance relations are obtained for the integral diffusion fluxes of the reactant on the surface of one or several reacting particles of arbitrary shape in Stokes flow of a viscous incompressible fluid around the particles at large Péclet numbers. The case of irrotational flow is also considered.

Macrokinetics of surface reactions in a liquid or gas flow and an approximate method of calculating the mass-transfer rate of the reacting particles

A simple approximate formula is proposed for calculating the mass-transfer coefficient for a moving reactive particle, the surface of which is the site of a heterogeneous chemical reaction with arbitrary kinetics.

Dissolving of a chain of drops (bubbles) in a fluid stream

The simplest model problem of the dissolving of a chain of drops (bubbles) over which a stream of viscous incompressible fluid flows at low Reynolds numbers is considered. The dlffusional interaction of the drops due to the presence of diffusional tracks [i, 2] is allowed for. The radii of the drops and the dissolving rate are determined as a function of their position in the chain and time; the characteristic dissolving times of the drops are obtained. It is shown that the diffusional interaction in chains leads to significant slowing of the dissolving process. For a drop with the serial number k the total dissolving time t k is determined by the formula