Two-phase fluctuations

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

On compression, a gaseous phase may condense to a liquid-expanded, L phase via a first-order transition. This transition is difficult to study experimentally because of the small film pressures involved and the need to avoid any impurities [76,193]. There is ample evidence that the transition is clearly first-order there are discontinuities in v-a plots, a latent heat of vaporization associated with the transition and two coexisting phases can be seen. Also, fluctuations in the surface potential [194] in the two phase region indicate two-phase coexistence. The general situation is reminiscent of three-dimensional vapor-liquid condensation and can be treated by the two-dimensional van der Waals equation (Eq. Ill-104) [195] or statistical mechanical models [191]. 

If we consider the scattering from a general two-phase system (figure B 1.9.10) distinguished by indices 1 and 2) containing constant electron density in each phase, we can define an average electron density and a mean square density fluctuation as ... 

It is generally preferable to meter each of the individual components of a two-phase mixture separately prior to mixing, since it is difficult to meter such mixtures accurately. Problems arise because of fluctuations in composition with time and variations in composition over the cross section of the channel. Information on metering of such mixtures can be obtained from the following sources. 

Poor flow distributions may result in localized dry hotspots which, absent control of the temperature fluctuations, may cause rapid overheating. Temperature and pressure fluctuations, and poor flow distribution, are the main problems that accompany the use of two-phase micro-channels. 

Unlike at adiabatic conditions, the height of the liquid level in a heated capillary tube depends not only on cr, r, pl and 6, but also on the viscosities and thermal conductivities of the two phases, the wall heat flux and the heat loss at the inlet. The latter affects the rate of liquid evaporation and hydraulic resistance of the capillary tube. The process becomes much more complicated when the flow undergoes small perturbations triggering unsteady flow of both phases. The rising velocity, pressure and temperature fluctuations are the cause for oscillations of the position of the meniscus, its shape and, accordingly, the fluctuations of the capillary pressure. Under constant wall temperature, the velocity and temperature fluctuations promote oscillations of the wall heat flux. 

In Chapter 3 the steady-state hydrodynamic aspects of two-phase flow were discussed and reference was made to their potential for instabilities. The instability of a system may be either static or dynamic. A flow is subject to a static instability if, when the flow conditions change by a small step from the original steady-state ones, another steady state is not possible in the vicinity of the original state. The cause of the phenomenon lies in the steady-state laws hence, the threshold of the instability can be predicted only by using steady-state laws. A static instability can lead either to a different steady-state condition or to a periodic behavior (Boure et al., 1973). A flow is subject to a dynamic instability when the inertia and other feedback effects have an essential part in the process. The system behaves like a servomechanism, and knowledge of the steady-state laws is not sufficient even for the threshold prediction. The steady-state may be a solution of the equations of the system, but is not the only solution. The above-mentioned fluctuations in a steady flow may be sufficient to start the instability. Three conditions are required for a system to possess a potential for oscillating instabilities ... 

Evaluation of p and Km2 requires determination of the void fraction and the two-phase pressure drop. Crossflow is determined from the appropriate lateral momentum balance equation. The interchange due to mixing, represented by w is determined by the turbulent transverse fluctuating flow rate per foot of axial length (lb/hr ft), where... 

Jones, O. C., Jr., N. Abuof, G. A. Zimmer, and T. Feierabuend, 1981, Void Fluctuation Dynamics and Measurement Techniques, in Two-Phase Flow Dynamics, A. E. Bergles and S. Ishgai, Eds., Hemisphere, New York. (3)... 

Moody, F. J. 1975, Maximum Discharge Rate of Liquid-Vapor Mixtures from Vessels, in Non-equilibrium Two-Phase Flows, R. T. Lahcy, Jr., and G. B. Wallis, Eds., ASME, New York. (3) Moore, F. D., and R. B. Mesler, 1961, The Measurement of Rapid Surface Temperature Fluctuations during Nucleate Boiling of Water, AIChE J. 7 620-624. (2)... 

Figure 8.10. The real two-phase system (a) and the transition into an ideal system (c) by removal of the density fluctuation background, Ipi, and of a transition layer of thickness dz between the hard and the soft domains. The elongated white region indicates a void...

Here w is a weighting factor. Asr (s) is the absorption factor (in this case for symmetrical absorption), Hz (5) and In consider the non-ideal character of the two-phase topology (cf. p. 124, Fig. 8.10) by consideration of a smooth phase transition zone and density fluctuations inside the phases. 

The same pseudo-ensemble concept has been used by Camp and Allen [44] to obtain a pseudo-Gibbs method in which particle transfers are substituted by volume fluctuations of the two phases. The volume fluctuations are unrelated to the ones required for pressure equality (10.7) but are instead designed to correct imbalances in the chemical potentials of some of the components detected, for example, by test particle insertions. 

The decomposition of a solution with composition outside the spinodal region but within the metastable region can be analyzed in a similar way. Let us assume that a sample with composition in this region is cooled to low temperatures. Small fluctuations in composition now initially lead to an increase in the Gibbs energy and the separation of the original homogeneous solution must occur by nucleation of a new phase. The formation of this phase is thermally activated. Two solutions with different composition appear, but in this case the composition of the nucleated phase is well defined at all times and only the relative amount of the two phases varies with time. 

The energy increase related to a compositional fluctuation resulting in a two-phase splitting may be considered as an energy threshold of activation of the de-mixing process. The spinodal curve thus defines a kinetic limit, not a phase boundary line. 

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