This paper is concerned with the rate of transfer of heat or mass from a force-free couple-free particle immersed in fluid whose velocity far from the particle is steady and varies linearly with position. Asymptotic results for both small and large Peclet numbers are considered. There is at least a four-parameter family of different linear ambient velocity distributions, but nevertheless a comprehensive set of results for the transfer rate may be compiled by combining previously published work with some new developments. Some of these are exact results for particular linear ambient flow fields and some are approximate results for classes of linear flow fields.For small Peclet number (P), the non-dimensional additional transfer rate due to convection is equal to αN20P½, where N0 is the Nusselt number for P = 0 and the proportionality constant α is a parameter of the concentration distribution due to a steady point source in the given linear ambient flow field. A general method of determining α is developed, and numerical values are found for some particular linear ambient flow fields. It is estimated that the value of α for any linear ambient flow field in which the vorticity does not dominate the straining motion lies within 10% of 0·34 when P is defined in terms of a particular invariant of the ambient rate-of-strain tensor E.At large Peclet number the transfer rate N depends on the velocity distribution near the particle, and attention is restricted to the case of a sphere in low-Reynolds-number flow. For a rigid sphere for any ambient pure straining motion, and the Levich concentration-boundary-layer method may be used to show that β = 0·90 for both axisymmetric and two-dimensional ambient pure straining, and probably for any other pure straining motion, when P is suitably defined. When the ambient vorticity ω is non-zero, the sphere rotates, and the Levich method cannot be used. However, it is shown that the part of the velocity distribution that varies sinusoidally with the azimuthal angle around the rotation axis does not affect the transfer rate and that N is asymptotically the same as for an ambient axisymmetric pure straining motion with rate of extension in the direction of the axis of symmetry equal to Eω(= ω. E. ω/ω2). In the exceptional case Eω = 0, N approaches a constant as P → ∞.It is possible to interpolate between the asymptotic relations for small and large Peclet number with comparatively little uncertainty for any ambient pure straining motion and for any linear ambient flow field in which ω and Eω are non-zero.

Mass transfer from a particle suspended in fluid with a steady linear ambient velocity distribution

Small rigid spherical particles are suspended in fluid, and material is being transferred from the surface of each sphere by convection and diffusion. The fluid is in statistically steady turbulent motion maintained by some stirring device. It is assumed that the Peclet number of the flow around a particle is large compared with unity, so that a concentration boundary layer exists at the particle surface, and that the Reynolds number of the flow around the particle is sufficiently small for the velocity distribution near the particle surface to be given by the Stokes equations.The flow around a particle is a superposition of (a) a streaming flow due to a translational motion of the particle relative to the fluid with a velocity proportional to the density difference, and (b) a flow due to the velocity gradient in the ambient fluid. An expression for the mean transfer rate which is asymptotically exact for large Peclet numbers is obtained in terms of statistical parameters of these two superposed flow fields. As a consequence of the partial suppression of convective transfer by particle rotation, the only relevant parameters are the mean translational velocity of the particle in the direction of the ambient vorticity vector and the mean ambient rate of extension in the direction of the ambient vorticity. The former is shown to be zero in common turbulent flow fields, and an expression for the latter in terms of the mean dissipation rate e is obtained from the equilibrium theory of the small-scale components of the turbulence. The final non-dimensional expression for the transfer rate is 0·55(a2e½/κν½)1/3, where a is the particle radius. This is found to agree well with some previously published sets of data for values of less than 102.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112080000304

Mass transfer from small particles suspended in turbulent fluid

Publisher Summary The study of the diffusion equation subject to a phase change at one boundary is in a relatively satisfactory state, provided simple boundary conditions of the first, second, or third kind are specified. Mathematically interpreted, the interesting features of the problem arise from the nonlinearity, exhibited for all but a few particular boundary motions. This chapter illustrates solution of the equation for diffusion of heat, mass, or some other scalar quantity, subject to the existence of a free boundary. The chapter aims to find exact solutions regarding relative motion between the phases, ablating slabs, and growth of a vapor film. Analytic approximation methods and analog and digital computer solutions are discussed in details. Integral equation methods, variation methods, and boundary layer methods along with active and passive analog solutions and application of numerical schemes are appropriate for bringing clarity to heat conduction or diffusion with phase change. In the search of growth of small particles of a condensed phase, the quasi-steady-state theory is applied particularly where a condensed phase exists whose volume changes slowly with time. This is true, for example, in the sublimation of ice or the condensation of water vapor from air on liquid droplets. While focusing on active and passive analog solutions, freezing problems remain symmetric in one, two, and three dimensions, where the liquid is initially at the fusion temperature. The heat capacity of the solidified phase is taken into account. Thermal properties are considered to be temperature independent.

Heat Conduction or Diffusion With Change of Phase

Abstract–We review recent analytical results in the theory of transition from laminar to turbulent premixed Rame propagation. We exploit the fact that the overall activation energy is large to formally derive dynamical flame sheet models, which are then used to predict instability thresholds as functions of the various parameters in the problem, at which steps in the transition occur. Employing perturbation techniques, we then describe bifurcations from a steady, planar flame to both pulsating and cellular modes of propagation. These nonsteady, nonplanar propagation modes represent intermediate stages in the evolution from laminar to turbulent combustion.

Nonlinear Stability and Bifurcation in the Transition from Laminar to Turbulent Flame Propagation

The ignition of a gaseous reactive mixture subject to a localized energy source is analysed using large activation energy asymptotics The energy released by the source results in a thermal non-uniformity in a small region of the gas We distinguish two different regimes, non-diffusive and diffusive, depending on the dominant cooling mechanism during the ignition stage: expansion effects or heat conduction We focus on the non-diffusive ignition, considering the energy source as instantaneous We show the existence of a critical value of a Damkohler number, defined as the ratio of the characteristic times of the expansion waves and chemical reaction, such that ignition only occurs for supercritical values at a well-defined ignition time, which is calculated numerically The ignition process for a non-instantaneous energy source is also described in terms of an initial inert heating stage and a shorter reactive stage ending in thermal runaway for supercritical values of the Damkohler number

/pdf/fast-non-diffusive-ignition-of-a-gaseous-reacting-mixture-ek89uizzy1.pdf

Iu.S. Riazantsev

Papers

Diffusion on a particle in the shear flow of a viscous fluid. Approximation of the diffusion boundary layer: PMM vol. 36, n≗3, 1972, pp. 475–479

Diffusion to a particle at large péclet numbers in the case of arbitrary axisymmetric flow over a viscous fluid PMM vol. 40, n≗ 5, 1976, pp. 941-945

Asymptotic analysis of stationary propagation of the front of parallel exothermic reaction: PMM vol. 39, n≗2, 1975, pp. 306–315

Analysis of the problem of thermal flame propagation by the method of matched asymptotic expansions PMM vol. 36, n≗4, 1972, pp. 659-666

The influence of heat generation in a droplet on thermocapillary force