Yu.A. Buyevich

On the thermal conductivity of granular materials

Abstract Stationary heat transfer in a granular material consisting of a continuous medium containing spherical granules of other substances is considered under the assumption that the spatial distribution of granules is random. The effective thermal conductivity characterizing macroscopic heat transfer in such a material is expressed as a certain function of the conductivities and volume fractions of the medium and dispersed substances. For reasons of mathematical analogy all the results obtained for the thermal conductivity are valid while computing the effective diffusivity of some admixture in granular materials as well as for evaluation of the effective electric conductivity or the mean dielectric and magnetic permeabilities of granular conductors and dielectrics.

Bubble formation at a submerged orifice in reduced gravity

Abstract We consider gas injection through a circular plate orifice into an ideally wetting liquid which results in the successive detachment of bubbles, each of which is regarded as a separate entity. At normal gravity and at relatively low gas flow rates, the growing bubble is modelled as a spherical segment that touches the orifice perimeter during the whole time of its evolution. If the gas flow rate exceeds a certain threshold value, a second stage of the detachment takes place that follows the first spherical segment stage. In this second stage, a nearly cylindrical stem forms at the orifice that lengthens as the bubble rises above the plate, and this stems feeds an almost spherical gas envelope stituated at the stem upper end. At high gas flow rates, bubble shape resembles that of a mushroom, and its upper envelope continues to grow until the gas supplied through the stem is completely cut off. This second stage always develops when gravity is sufficiently low, irrespective of the gas flow rate. There are two major factors that determine the moment of bubble detachment: the buoyancy force and a force due to the momentum flowing into the bubble with the injected gas. The buoyancy force dominates the process at normal gravity whereas the inflowing momentum force plays the key role under negligible gravity conditions. As gravity fluctuates, the interplay of these forces drastically influences bubble growth and detachment. At sufficiently low gravity, the bubble formation frequency is proportional to gas flow rate whereas the bubble detachment volume is independent of gas flow rate. At normal and moderately reduced gravity conditions, when the gas flow rate grows, bubble formation frequency slightly decreases and bubble detachment volume increases almost linearly. Effects of other parameters, such as the orifice radius, gas and liquid densities and surface tension coefficient are discussed.

Random fluctuations in a fluidized bed

Abstract A mathematical model is worked out to treat random small-scale fluctuations of particles and a fluid in macroscopically uniform disperse mixture. The particles are assumed large enough to ensure the interparticle exchange of momentum and energy to be effected through direct collisions, so that the particle fluctuations are nearly istropic. The model privides for closure of equations of conversation that govern the flow of both mixture phases. Statistical properties of the fluctuations are actually studied under conditions of unidirectional flow in two extreme cases when the concentration-dependent hydraulic resistance of the particles to the relative fluid flow is either linear or quadratic in the fluid slip velocity. Overall phase volume fluxes and the hydraulic resistance coefficient happen to be sensitive to the fluctuations and differ from those specific to the same mixture without fluctuations. The particle pressure and the bulk modulus of elasticity of the dispersed phase are shown to be increasing functions of the mean concentration, which makes macroscopically uniform states of the mixture thermodynamically stable. The energy dissipation due to the collisions, however insignificant in most situations of practical importance, results in a noticeable decrease in the bulk modulus of elasticity and can therefore cause the break of stability under otherwise identical conditions.

Fluid dynamics of coarse dispersions

Abstract Dispersions of identical spherical particles in a fluid are considered with allowance for random fluctuations that both, particles and the fluid, are involved in. The particles are assumed to be sufficiently large in the sense that the interparticle exchange by momentum and energy of the particle fluctuations is carried out mainly through direct collisions, and the role of interphase interaction in this process is negligible. Then the particles may be regarded as statistically independent, and the fluctuation energy is almost uniformly distributed over their translational degrees of freedom. Equations of mass and momentum conservation for flow of the dispersed phase as well as an equation of fluctuation energy transfer are stated similarly to those obtained by the classical method by Enskog in the kinetic theory of gases. The equations involved additional terms due to the interphase interaction and the energy dissipation by collisions. These equations, together with those of mass and momentum conservation of the continuous phase, constitute a complete set of equations which govern the macroscopic flow of disperse mixtures. A closure problem for this set is discussed in detail. It is reduced, in fact, to the determination of a single quantity characterizing the particle fluctuation intensity in a macroscopically homogeneous state of a disperse mixture.

Cooling of a superheated surface with a jet mist flow

The previously developed model of dynamic and thermal interaction of dilute mist flow with hot bodies is applied to study heat exchange in the circumstance when an axisymmetric two-phase jet falls normally onto a plate, the temperature of which exceeds the boiling temperature of the dispersed liquid. Depending on their approach velocity, impinging droplets either rebound, or come in direct contact with the plate and eventually evaporate, thus providing for an essential increase in the total heat removal. Such a crisis causes the occurrence of a temperature interval in which heat transfer from the plate decreases as the plate temperature grows. The very existence and properties of the mentioned anomalous region are explained in a good agreement with experimental findings. The results are also generalized to heat transfer involving polydisperse spray jets.

Dynamics of vapour bubbles in nucleate boiling

Abstract This paper considers the behaviour of a vapour bubble formed at a nucleation site on a heated horizontal wall. This bubble is modeled as a spherical segment which is separated from the wall by a microlayer of intervening liquid. The liquid is presumed to be at rest at great distances from the bubble. In order to avoid unwarranted assumptions about forces acting on the bubble which are specific to all known models of bubble growth and detachment, we derive equations that govern bubble behaviour in a rigorous way from the variational equation that describes mechanical energy conservation for the whole system, which includes both the bubble and the liquid. The variational equation leads to a set of two mutually independent strongly nonlinear equations which govern bubble expansion and the motion of its centre of mass. Because these equations contain an extra unknown variable (the bubble vapour pressure), a supplementary equation that defines bubble vapour temperature must be formulated with allowance made for heat transfer to the bubble both from the bulk of the surrounding liquid and through the microlayer. The most important conclusion of this paper consists in the fact that surface tension effects result in an effective force that tends to transform the bubble into a sphere, thereby facilitating bubble detachment. This conclusion absolutely nullifies the generally, however erroneously, held belief that this effective force presses the bubble to the wall. By way of example, we consider the evolution of bubbles whose growth is thermally controlled. Our analysis provides quite a natural explanation for a number of repeatedly observed phenomena, such as the influence of gravity and surface tension on bubble growth rate and the dependence of bubble detachment size on thermophysical parameters.

Particle distribution in suspension shear flow

Abstract The model of this paper is based on an earlier proposed constitutive equation that factors in all normal stresses originated by random particle fluctuations. This equation is used to describe the joint effect of thermal and shear-induced fluctuations on concentrational distributions in suspension flow. Averaged products of fluctuation velocity components are evaluated on the basis of a rational mechanics approach combined with a simple kinematic consideration. The momentum conservation equation for the dispersed phase of a suspension closed by this constitutive equation is applied to unidirectional shear flow in the gravity field and to rotational Couette flow. Coupling the thermal and shear-induced fluctuations results in a situation where the total volume of particles suspended in a given shear flow reaches a minimum at a finite particle size, all other things being equal. Additionally, the developed model provides a reasonable explanation for the particle distributions observed in Couette flow. For these flows, the momentum comservation equation can also be reformulated to yield a diffusion-like equation for the suspended particles. However, coefficients of mutual diffusion due to both thermal and shear-induced fluctuations are drastically different from corresponding self-diffusivities as regards both their scaling and their concentrational dependence.

INSTABILITY AND UNSTEADY PROCESSES OF THE BULK CONTINUOUS CRYSTALLIZATION.I, LINEAR STABILITY ANALYSIS

Abstract The physical nature of the instability of stationary regimes of continuous crystallization from supersaturated solutions is explained. The system of equations with distributed parameters, comprising the mass and number balance laws expressed in terms of the crystal size distribution function, is reduced to a single functional integro-differential equation describing the evolution of the saturation for arbitrary mass flux of a supersaturated solution into a volume under consideration, nucleation and crystal growth kinetics and withdrawal rate of crystals out of the volume. The mechanism of instability of a stationary crystallization process is studied. Dependencies of the neutral stability surface and of the period of initial self-oscillations occurring as a result of the instability upon physico-chemical and operating parameters are obtained. A comparison of theory and experiments is presented.

Interaction of a dilute mist flow with a hot body

Abstract The flow of an aerosol containing liquid droplets around an overheated body is considered. The liquid mass flux is assumed small enough to prevent formation of a liquid film on the body surface. Depending on the relative normal velocity, impinging droplets are either captured by the surface and ultimately evaporated or almost elastically thrown away, this change in the droplet behaviour causing the onset of a heat transfer crisis. The theoretical description of the dynamic and thermal interaction between the droplets and the surface is reduced to solving two independent problems. The first problem consists in the analysis of the dynamic Leidenfrost phenomenon and further calculation of the critical normal velocity of a single droplet as a function of physical and process parameters. The second problem involves determination of the field of droplet trajectories around the body on the basis of the conventional theory of inertial capture of suspended particles and subsequent calculation of the total liquid mass flux onto the surface, conditioned by a requirement that the droplet fall velocity exceeds the indicated critical value. Both these problems are studied. The distributions of the specific coefficient of heat removal due to evaporation over the sphere, cylinder and plate surfaces in a uniform aerosol flow are obtained under different circumstances.

Interphase interaction in fine suspension flow

Abstract A self-consistent model of moderately concentrated fine suspensions of identical spherical particles is developed. It provides for a completely closed fluid dynamic scheme of suspensions neglecting random particle fluctuations. Above all, it brings forward a set of conservation equations governing the mean flow of both phases of a suspension in the continuum approximation. The model allows also to obtain all constitutive rheological relations (“equations of state”) which determine terms of those equations as functions of unknown variables and physical parameters. The rheological relations are found in an explicit form for effective stresses and different constituents of the interphase interaction force in weakly unsteady flow of a moderately concentrated suspension. The stresses are shown to take shape as a result of a certain relaxation process. The force constituents are of the same origin and have basically the same meaning as those for a single particle in an unbounded fluid. In particular, the contents of the paper bring to an end a recent scholastic and rather fruitless debate as to how the drag and buoyancy constituents have to be singled out from the total force acting on a particle of a uniform fluidized bed at steady conditions. It is demonstrated in an unequivocal way that it is the density of the fluidized bed as a whole that must be used while expressing the buoyancy force, but not that of the fluidizing fluid alone.