Introduction
Two-phase flows are widely used in industry. In the overwhelming majority of cases, various technological processes involving such flows are organized in such a way that the carrier fluid flow is directed bottom-up. Separation regimes of such flows with one part of solid phase moving with the flow and another part moving counter the flow are rather widespread. For example, processes of minerals concentration, bulk materials fractionating by particles sizes, processes in boiling bed, powder materials drying, most mass-exchange processes are organized in this way.
Visual observations and rapid filming of two-phase flows with a poly-fractional solid phase in a transparent vertical channel show an absolutely chaotic motion of solid particles. They move upwards and downwards, to the left and to the right, take part in vortical motions of the carrier medium, collide with each other and with channel walls, form aggregates that permanently appear and decompose. All this makes it impossible to predict the direction and velocity of even one particle, not to speak about the behaviour of solid phase on the whole.
This can explain why scientists have not attained theoretical understanding of such flows until now. It is not surprising, because until today there is no acceptable general theory even for purely one-phase liquid and gas flows. It is known that many prominent specialists have been working for many years in this field in many countries around the world. Numerous theoretical schemes for pure turbulent flows have not led to generalizing results, but only to a grim joke saying that ‘turbulence is a cemetery of theories’.
However, at the same time, dams, hydroelectric power station, pneumo- and hydro-transport systems, and more are being built and operated. They all successfully function and practically always reach their design characteristics.
As known, it is a result of fragmentary, and not general regularities of flows, which have not been combined into a unified theory yet. Such regularities fix only some ‘patch-like’ particulars of flows. For one-phase flows, such fragmentary regularities include so-called similarity criteria derived from rather simple mathematical models. At the same time, these criteria possess an explicit physical meaning. In hydrodynamics, they are, as a rule, dimensionless, and each of them consists of parameters reflecting some characteristic features of flows. These parameters can acquire an infinitely large number of numerical values, while the criterion can remain constant.
Such relations are usually called invariants. It is a rather broad notion, which can be defined as follows:
Invariant is a system of values, parameters, or regularities describing a group of phenomena remaining unchanged at one or another transformation of parameters inside this system.
It is noteworthy that invariants are not obligatorily dimensionless. Even graphical relations can serve as invariants at a respective choice of coordinates, so that all experimental or estimated values of parameters fitted the same curve.
It is very important to study the possibility of finding such invariants for two-phase flows, which are, for good reason, much more complicated than one-phase flows. This will make it possible to start the development of a general approach to the construction of their theory.
The initial study of the process started with the study of the behaviour of solitary particles. First publications on the topic appeared in the second half of the nineteenth century. First, regularities of settling of solitary spherical particles in an unlimited motionless liquid were studied. In numerous subsequent papers devoted to the study of this phenomenon, the effect of various factors (fluid and material densities, final velocities of particles settling, their drag coefficients, etc.) on these regularities was revealed. A transition to materials with irregular particles shape has been so complicated that publications dealing with these problems are still appearing nowadays. An attempt of applying main regularities obtained when studying the behaviour of solitary particles in moving medium to real processes has not given positive results.
Today all the problems of two-phase flows are solved at an empirical level. Numerous researches are continuously performed on models of actual industrial units. The transfer of the obtained results onto pilot plants is not always successful. A huge volume of experimental studies has allowed the development of numerous empirical computation methods for specific apparatuses. However, the best and the most grounded empirical formula can be applied with satisfactory results only within a limited range determined by the conditions of its derivation. Frequently used extrapolation of such dependencies beyond the experimental range limits leads to bad mistakes. Besides, empirical relations do not reflect the regularities of the phenomena under study, but only provide their quantitative characteristics. Any carefully conducted experiment does not allow taking into account the diversity of constant and random factors—both quantitative and qualitative ones. The latter can be taken into account least of all.
Since it is unrealistic at present to discuss the possibility of developing a general theory of two-phase flows, the efforts should be concentrated on the search for physically grounded invariants for them. The chaotic disorder in the motion of solid particles in two-phase flows by no means points to the absence of general regularities. On the contrary, rigid internal regularities can manifest themselves in such systems only through a general chaotization, as it was established, for example, during the development of kinetic theory of gases. This theory was developed within the framework of thermodynamics. To understand the logics and the essence of the approach to the problem that led to the development of the kinetic theory of gases, we will try to understand how the development of thermodynamics led to this theory.
Initially it was known that fire releases heat that can lead to an increase in the volume of any bodies—solid, liquid, and gases. As a result, work is performed. Fire led people to a steam engine, which became the basis of the industrial society development in the beginning of the nineteenth century. To create steam engines, it was necessary to establish the connection between heat and work. In search of a solution to this problem, energy conservation law (the first law of thermodynamics) was formulated. It implies that heat is of the same nature as energy. In a heat-engine, heat is converted into work, but the amount of energy is conserved. The law of total potential and kinetic energy conservation was established much earlier, in classical mechanics developed by Newton, Lagrange, and Hamilton. It was outstandingly successful and is still developing at present. One of its distinctive features is a strict determinism.
Classical mechanics can predict the behaviour of a system with either a small number of elements or with many symmetrically arranged elements. If the initial conditions for a system are specified, and forces acting on it are determined, one can follow the changes in this system both in the future and in the past using classical mechanics.
An especially important conclusion of this mechanics was that due to Newton's works it has become clear that nature obeys simple universal laws that are cognizable and can be expressed using exact mathematical language. Since then, experiments, quantitative study of various physical values in their interconnection, mathematical relations between them have formed the basis for the insight into the secrets of nature.
Using classical mechanics, it has become possible to carry the problem of celestial bodies motion to its conclusion, but it proved to be absolutely helpless facing the problem of three bodies motion in a general case, not to speak about mass motion of many bodies.
In the seventeenth and eighteenth centuries, Newton's works had such a high prestige that insistent attempts were made to reduce all laws of nature, discovered by that time, to classical mechanics. In the long run, they were unsuccessful. However, at least in one field of science these attempts led to excellent results, namely, in the theory of heat phenomena, but it happened only in the middle of the nineteenth century. By that time, confidence in the molecular structure of substances started strengthening. Therefore, efforts were made to find out how macroscopic properties of a substance depend on the behaviour of imaginary molecules it is composed of. The first successes in this direction concerned gases in which molecules interact with a negligible force. The obtained results allowed expressing pressure, temperature, and other macroscopic parameters of gas through such an averaged characteristic of molecules as their kinetic energy. Therefore, this theory was called a kinetic theory or statistical mechanics. It is based on fundamental Boltzmann's works, where speculative analysis allowed establishing statistical relations between the molecular structure of substance and its microscopic properties at a microscopic level.
Statistical mechanics was based on classical mechanics, but it established new relations and introduced new concepts. A starting point for Boltzmann was atomistic theory stating that matter consists of immense number of small moving balls.
In principle, it is possible to describe the motion of a system of independent particles from the standpoint of classical mechanics. Writing a differential equation for each particle, one can obtain comprehensive information on...
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