Bubble Systems

The general concept of the two-phase systems and their classification is presented. Basic methods of two-phase flows description are discussed: method of detailed description, unit cell models, single continuum models, models of multispeed medium. The main problems and limitations of existing approaches to the development of mathematical models are shown. It turned out that the correct choice of the coordinate system in many cases can simplify the description of the two-phase systems. The examples describing the typical two-phase systems behaviour in ‘laboratory coordinate system’ and in ‘native coordinate system’ are presented. The detailed study of the problem of the proper assignment of the ‘coupling conditions’, which are 'inner' boundary conditions written on the boundaries of coexisting phases is carried out. The general equations governing the coupling conditions were derived, and then reduced for a broad set of individual cases of practical and scientific interest. The basic approach to the determination of the intensity of nonequilibrium phase transitions in two-phase flows is presented. Finally, the structure of the book is described and a brief content of the individual chapters is presented.

We solve the fundamental problems of dynamic growth, collapse and pulsations of bubbles under the influence of the inertia, surface tension and viscosity forces. For all these problems it proved possible to obtain analytic solutions, study the degenerate (asymptotic) branches, and use them to construct practical interpolation formulas. In this way we build the solution of the gas bubble collapse problem, give a detailed analysis of the laws of variation of its radius and of the pressure distribution in the liquid. The surface tension forces are shown as having predominant influence on the dynamics of a bubble in terms of the variation of the initial pressure drop. A general solution of the dynamic bubble growth problem is obtained with allowance for the viscosity and surface tension forces. It is shown that, in the absence of the viscosity and surface tension forces (the dynamic-inertia-controlled growth law), the bubble growth rate increases in time, in the initial period, according to the linear law, and only then asymptotically turns to the concluding stage, in which the growth rate is constant. An analysis is given of the initial growth period duration of vapour nucleus in equilibrium with the ambient superheated liquid as a function of the initial perturbation. A conclusion is made that there exists a peculiar “incubation period” of bubble growth, within which the bubble growth is not very fast. The effect of the viscosity forces on the bubble growth process is studied. The effect of viscosity is shown to degenerate both in the initial stages of bubble growth and for large growth times. Nevertheless, in the intermediate growth stages the effects of viscosity may have a substantial effect on the bubble growth even for relatively low-viscous liquids like water. The effect of viscosity exhibits the most powerful manifestation when a vapour nucleus grows in the regions of liquid superheats close to spinodal.

An analytic investigation of the problem of gas bubble pulsations under oscillations of external pressure is obtained; the similarity numbers governing this process are presented. It is shown that the variation of the gas characterisics inside a bubble is determined by two parameters; the adiabatic index for the gas and the Fourier number, whereas the complete solution of the problem is a function of three similarity numbers (the adiabatic index, the Fourier number, and the reduced bubble radius). A conclusion is made that in the presence of heat exchange it is impossible to describe the real process of bubble oscillations in the framework of polytropic approximation. An analysis of the laws of pulsations of a bubble in dimensionless coordinates is given. It is shown that in the presence of heat exchange a real process of bubble oscillations cannot be described “in detail” in the framework of the polytropic approximation; also it is impossible to describe the process of forced oscillations of bubble “in the mean”. If the dependence for polytropic exponent is chosen so as to describe the amplitude of the pulsations of the pressure, it cannot adequately describe the amplitude of the pulsations of temperature, and vice versa. An explicit dependence for the resonance oscillation frequency of a bubble is obtained for the first time. It is shown that under resonance conditions the pressure homogeneity assumption inside a bubble holds practically always, except in the direct vicinity of the thermodynamic critical point. Adiabatic pulsations of bubbles during stepwise variation of pressure are studied extensively. Analytical solutions were obtained by using a dynamic analysis approach as well as by energy balance methods. The shape of pulsation is presented.

The mathematical formulation of the heat input governed vapor bubble growth in a bulk of uniformly heated liquid is presented. Using the theory of dimensions, the structure of the solution was analyzed qualitatively. A historical survey of theoretical works devoted to the considered problem is presented. The set of degenerate solutions of the problem is obtained and studied systematically. The complete analytical solution of the problem is obtained for the first time. Formulas for the calculation of the bubble growth rate in the whole domain of possible variations in regime parameters are presented. The conclusion is made that the influence of permeability of the interface has a significant effect on the bubble growth rate. It is shown that the Plesset–Zwick formula, which is commonly accepted in computational practice, is not applicable at both small and large Jakob numbers and its good agreement with the experiment is determined to a large extent by a combination of the imperfectness of the theoretical analysis and the experimental error. The conclusion is made that, for many liquids, the ultimately achievable value of the dimensionless superheating parameter (Stefan number) can exceed unity. In this case, the regularities in the bubble growth acquire some features unexplored to date.

The survey the experimental studies on the rates of growth, condensation and dissolution of vapour (gaseous) bubbles in turbulent flows is presented. The experimental data on the condensation of single bubbles (water vapour, high-speed filming, pressure 0.1–10.0 MPa, velocities 0.25–1.5 m/s, liquid subcooling up to 13 K), the growth (water vapour, the double exposure method, 0.3–4.0 MPa, 8–30 m/s, liquid superheat 0.7–2.5 K), and the dissolution of gaseous bubbles (carbon dioxide, 0.1–0.3 MPa, 0.22–2.82 m/s) show that the interfacial exchange processes on the surface of bubbles are substantially intensified in a turbulent flow. The dynamics of bubbles carried by the turbulent flow is determined to a large extent by the velocities of the liquid forced motion. We show that the previously proposed semi-empirical design formulas are built basing on single groups of experimental points and hence may not pretend to describe the entire set of data. A detailed analysis of the inner structure of turbulent flows is carried out. Formulas for the magnitude of turbulent dissipation of energy with forced motion of a bubble two-phase mixture in pipes are offered. With the help of the surface renewal model formulas for bubble dynamics are derived describing the full set of the available experimental data and encompassing a wide range of similarity parameters (Re = 1.81 × 10³ − 1.9 × 10⁶, Pr = 0.8 − 568, Nu = 60 − 5300). A detailed analysis of the adopted assumptions enabled us to determine the boundaries of the region of applicability of the model developed, covering the entire region of existence of the turbulent flow. The relations obtained below may be considered as providing the basis for theoretical description of the vapour generation processes as well as the processes of vapour condensation in nonequilibrium bubble flows that are realized in discharge of flashing liquid, in channels of forced heat-exchange systems, in cavitating flows, and so on.

The existing methods for critical discharge of flashing liquid and the results of experimental measurements are surveyed. The principal problems on the consistency of experimental data are formulated and the set of the most reliable experiments is presented. The existing approaches to the determination of the intensity of vapour generation in high-speed flows of superheated liquid are considered. A model for critical discharge of saturated and subcooled liquid from relatively long cylindrical channels is put forward. A comparison of numerical calculations with the set of experimental data is seen to give a good agreement not only in terms of the flow rate, but also in the distribution of pressure and reactive forces. By analyzing the effect of principal regime parameters a variation of the channel length was shown as having a much stronger effect on the critical discharge, than a variation of its diameter—hence the relative channel length cannot serve as a criterion which uniquely determines its discharge characteristics. A conclusion is made that the conjecture made by several researchers that there exist two critical pressure ratios does not at all agree with reality and can be explained by a relatively slight dependence of the specific flow rate versus the back pressure in the near-critical region. Considering the variation of thermophysical properties of phases and saturation parameters over the channel length, it is shown that a substantiated similarity equation for the critical flow rate can be built only on the basis of a generalized method of describing dependences of properties on the reduced parameters of state. A universal dependence for the critical flow rate of various liquids is obtained for the first time, extending both the numerical results and the set of existing experimental data. For water, this dependence applies for channels of diameters from 1 to 1000 mm, with sharp or smooth inlet edges, in the region of reduced pressures exceeding 0.6 (for water up to 14 MPa) and for a wide range of subcoolings. A comparison of theoretical results with the data obtained for five various liquids (water, Freon-11, Freon-12, propane, n-hexane) provided a further justification for this approach. Integral mass and momentum conservation equations are employed to show that for the known hydraulic characteristics of the channel inlet and its geometry, the reactive force is uniquely related with the flow rate of flashing liquid. An analytic dependence is derived expressing the reaction displacement principle, from which one may determine the reaction of a discharging jet without recourse to calculating the flow parameters in the channel exit section. This dependence describes nearly all the presently available experimental data on the discharge of several saturated or subcooled liquids or a two-phase mixture through channels of various geometries: from capillaries of diameter about tenth of a millimeter up to industrial pipes whose diameter approaches tens of centimeters.

The theory of boiling shock is developed. A boiling shock is shown to be a rarefaction shock wave. An analysis is carried out of the boiling shock adiabat. The entropy is shown to increase across the shock front. The stability of the boiling shock is investigated. Two possible situations are singled out: a shock formed under transient conditions following a seal failure of high pressure vessels (the U-shock) and a shock formed when the limiting (spinodal) superheats are reached in the flow (the S-shock). The mechanisms of flow choking with formation of an S-shock are examined. The model proposed is shown to adequately describe the experimental data on the discharge from diaphragms and short nozzles. The structure of the S-shock front is studied. The appearance of the S-shock is shown to be always accompanied by the formation of oscillations specific to this class of problems. A conclusion is made that under certain conditions the process of boiling of liquid acquires a self-accelerating character, when the development of the process of phase transitions triggers a growth of liquid superheat. Besides, discharge regimes with radial jet expansion also appear, when the jet of boiling liquid acquires a specific daisy-shaped form. Moreover, in these regimes the reactive force abruptly drops down to zero or to small negative values. The gas dynamics of these discharge regimes is given a detailed treatment. Patterns of shock waves accompanying discharge process are obtained. Mechanisms of propagation of the U-shock in a bulk of superheated liquid are considered. The velocity of its propagation is shown to be well described by the theory developed. The requirement of the stability of the U-shock leads to a well-defined quantity of superheat ahead of its front (the pressure undershot), which is an unambiguous function of the initial liquid temperature and its properties. All the conclusions of the above analysis are well supported by the experimental evidence.

Methods of similarity theory are employed to analyze the general features of emersion of vapour bubbles in a bulk of still liquid. A set of similarity criteria is obtained governing the process under study. An analysis of experimental data on emersion is carried out. Three typical cases are singled out: rise of spherical bubbles, rise of ellipsoidal bubbles, and rise of bubbles in the form of spherical caps. For each of these cases, we perform an analysis by the methods of similarity theory; give the results of analytical solutions and the available semiempirical formulas. It is shown that in many instances, a sequential application of the methods of similarity theory is capable of delivering a solution of the problem under study up to a universal dimensionless constant. The cases of rise of solid spherical particles and gas spheres in the field of gravity force are considered. The effect of surfactant impurities on emersion of bubbles is analyzed. General design formulas are obtained capable of describing the motion of both solid and gas spheres over the entire possible range of Reynolds numbers, both in the presence and in the absence of surfactant impurities. An explanation is given of the absence of the effect of surfactants on the rise velocity of large bubbles. A detailed clarification is given of the mechanism of the formation of bubbles in the form of spherical caps, as well as of the mechanisms governing their ascent motion. From the above analysis, a general formula is derived describing the rise velocity of gaseous (vapour) bubbles. This formula takes into account the effect of all parameters governing the gravitational ascent of bubbles, encompasses the entire possible range of variation of similarity numbers, and justifies the required passages to the limit. The formula can be used both for pure liquids and in the presence of surfactant impurities. An influence of congregate effects on the emersion of bubbles is analyzed. It is shown that during intensive bubbling the ascent rate of vapour (gas) phase can differ by many times from rise velocity of single bubbles. A detailed analysis is given of the physical mechanisms of this phenomenon and principal approaches to the problem of bubbling hydrodynamics.

The breakup of bubbles rising in gravity field is governed by one of the three mechanisms: breakup of bubbles due to instability of the interface surface, breakup due to centrifugal forces, and disintegration of bubbles due to the direct impact of turbulent pulsations during intensive bubbling. Mechanisms of wave formation and evolution on the liquid surface are considered in detail, both in the presence or absence of relative motion of phases. Formulas for the boundaries of the stable motion region are derived, as well as for the growth rate of perturbations, both for the Rayleigh–Taylor instability and for the Kelvin-Helmholtz instability. A new physical model of bubble breakup as a result of evolution of instability of the interface surface is put forward. Dependence is obtained for calculating the development time of the instability until the bubble breakup. This quantity is shown to be uniquely depending on the properties a two-phase system and the rate of initial perturbations. Contrary to a widespread opinion, the effect of the Kelvin-Helmholtz instability is shown to be practically always negligible in the process of breakup of bubbles rising in a large volume of still liquid. Formulas for determination the maximal size of a stable bubble in a mass bubbling condition are obtained. The agreement between theory and experiment is quite good. For moderate vapour holdups, the intensity of turbulent pulsations is shown to be governed by the effects in the wake of the rising bubble, and for large vapour holdups, by the universal equilibrium region of the energy spectrum of turbulent pulsations. A model of bubbles breakup due to centrifugal forces is developed. It is shown that the curvature radius of the bubble surface at the time of bubble breakup cannot be determined from the balance of the liquid inertia forces and the surface tension, as was done in previous studies. The agreement between the formula for the maximal size of a stable bubble thus obtained and experiment for elevated pressures is fairly good.

We give a coherent treatment of the hydrodynamic theory of heat exchange based on the perceptions about the unity of the mechanisms of turbulent transfer of heat and momentum (the Reynolds analogy). A conclusion is made that for nonequilibrium two-phase flows the Reynolds analogy is the natural method of describing heat transfer processes, which is capable of applying the results of theoretical analysis to obtain relatively simple analytic solutions. Two extreme cases of motion of two-phase bubble mixture in pipes are singled out: the sliding bubble flow and the coring bubble flow. For each of these limiting regimes of bubble flow, it proved possible to construct, a closed hydrodynamic model of the flow, which is capable of determining both the velocity profiles and the magnitudes of the hydrodynamic drag. Approximate formulas describing the results of accurate analytic studies are obtained. The calculations results and the set of available experimental data were found to be in a good agreement. The relations obtained are used to derive equations for the Reynolds analogy for nonequilibrium flows of a two-phase mixture. A new similarity criterion is obtained taking into account the relative role of the convective and conductive heat transfer mechanisms for flows with “double disequilibrium” (superheated near-wall layer of liquid– subcooled liquid in the core). The potency of this approach is illustrated by solving a number of problems on physics of surface boiling. Besides, it proved possible to obtain analytic solutions of such problems, no introduction of empirical constants matched from experimental data begin made. A solution to the problem of maximal (over the growth—condensation cycle) diameter of parietal bubbles is put forward. A comparison with the set of available experimental data with liquid subcoolings 3–80 K, flow velocities 0.2–9.2 m/s, and densities of heat flux 0.38–8.53 MW/m² showed a good agreement. The solution of the problem of pressure losses for flow boiling of subcooled liquid showed a good matching both for parameters that are characteristic of power units (velocities 0.5–2.0 m/s, heat fluxes up to 1.5 MW/m², pressures up to 14 MPa) and for high-performance heat exchange systems (velocities up to 20 m/s, heat fluxes up to 40 MW/m², pressures up to 14.7 MPa). A consistent use of the Reynolds analogy has enabled us to solve the problem of heat transfer and hydrodynamics of film boiling under forced motion conditions. The relation thus obtained shows a good match with the available experimental data for water (pressures 0.1–21.6 MPa, flow velocities up to 14.2 m/s, densities of heat flux up to 81.2 MW/m²), liquid helium and nitrogen. It is shown that for elevated velocities and subcoolings of liquid, in the regime of film boiling one may, without destruction of the heat-transfer surface, remove heat fluxes exceeding 100 MW/m². A conclusion is made that the application of the Reynolds analogy holds the key to solving a number of other problems in physics of nucleate boiling.