Liquid droplet theory

Bohr called his idea the liquid droplet theory. He suggested that a mass of atoms behave like a drop of liquid—clustering around one another to form the drop. When bombarded by a neutron, the drop forms the shape of a dumbbell and then splits into two, forming two new drops. During the split into two drops, energy is released. Bohr s theory provided scientists with a visualization of how the actual process of fission would be achieved. 

The quasi-steady-state theory has been 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 (M3, M4). In the condensation of water vapor onto a spherical drop of radius R(t), the concentration of water vapor in the surrounding atmosphere may be approximated by the well-known spherically symmetric solution of the Laplace equation ... 

The pressure due to surface tension is given in standard physics texts, and the derivation of the general equation need not be given hej-e in detail. Some phases of the general theory of capillary pressure, however, are worth mentioning. Assume a liquid droplet in contact with a plane solid surface and assume further that the surface tensions of three surfaces which separate the solid and liquid, liquid and gas, and solid and gas, are known. 

The objective of the analysis given in this section is to illustrate the use of equation (2) by considering the problem of determining the combustion efficiency of a variable-area, quasi-one-dimensional rocket chamber such as that illustrated in Figure 11.1, in which M different kinds of liquid droplets are present. In order to avoid considering the behavior of the gas, we must assume that the material burns to completion as soon as it evaporates. The amount of heat released will then be proportional to the mass evaporated, thus making it possible to relate the combustion efficiency to the mass of the spray present. Even in this case, the equations contain parameters, such as Rj, which depend on the local gas properties. However, estimates of these parameters are often obtainable without solving for the gas flow, so that, while the theory is essentially incomplete, it is not entirely useless. 

In the approach adopted in my first edition, the derivation and use of the general dynamic equation for the particle size distribution played a central role. This special form of a population balance equation incorporated the Smoluchowski theory of coagulation and gas-to-panicle conversion through a Liouville term with a set of special growth laws coagulation and gas-to-particle conversion are processes that take place within an elemental gas volume. Brownian diffusion and external force fields transport particles across the boundaries of the elemental volume. A major limitation on the formulation was the assumption that the panicles were liquid droplets that coalesced instantaneously after collision. 

The theory developed in Section 12.6 assumed that the plate acts as a perfect equilibrium stage for separation. However, in practice it is found that this is only an approximation. While the temperature of the vapour leaving the plate is likely to be the same as that of the liquid leaving the plate, the difference in compositions of the vapour and the liquid is often not as great as implied by equation (12.63), and the plate has therefore not been such an efficient separation stage. The deviation from equilibrium is accounted for partly by gross physical phenomena such as carryover of liquid droplets in the vapour leaving the plate, but a more fundamental reason is the restricted rate of mass transfer between the vapour and liquid phases. A column efficiency, r/r, may be defined as the ratio of the number of plates needed in theory. Nr, to the number needed in practice, Np, to achieve the desired separation ... 

Nucleation plays a fundamental role whenever condensation, precipitation, crystallization, sublimation, boiling, or freezing occur. A transformation of a phase a, say, a vapor, to a phase p, say, a liquid, does not occur the instant the free energy of p is lower than that of a. Rather, small nuclei of p must form initially in the a phase. This first step in the phase transformation, the nucleation of clusters of the new phase, can actually be very slow. For example, at a relative humidity of 200% at 20°C (293 K), far above any relative humidity achieved in the ambient atmosphere, the rate at which water droplets nucleate homogeneously is about 10 54 droplets per cm3 per second. Stated differently, it would take about 1054 s (1 year is 3 x 107 s) for one droplet to appear in 1 cm3 of air. Yet, we know that droplets are readily formed in air at relative humidities only slightly over 100%. This is a result of the fact that water nucleates on foreign particles much more readily than it does on its own. Once the initial nucleation step has occurred, the nuclei of the new phase tend to grow rather rapidly. Nucleation theory attempts to describe the rate at which the first step in the phase transformation process occurs—the rate at which the initial very small nuclei appear. Whereas nucleation can occur from a liquid phase to a solid phase (crystallization) or from a liquid phase to a vapor phase (bubble formation), our interest will be in nucleation of trace substances and water from the vapor phase (air) to the liquid (droplet) or solid phase. 

Abstract A bquid droplet may go through shape oscillation if it is forced out of its equilibrium spherical shape, while gas bubbles undergo both shape and volume oscillations because they are compressible. This can happen when droplets and bubbles are exposed to an external flow or an external force. Liquid droplet oscillation is observed during the atomization process when a liquid ligament is first separated from a larger mass or when two droplets are collided. Droplet oscillations may change the rate of heat and mass transport. Bubble oscillations are important in cavitation problems, effervescent atomizers and flash atomization where large number of bubbles oscillate and interact with each other. This chapter provides the basic theory for the oscillation of liquid droplet and gas bubbles. 

When the contact angle is 90° and the liquid droplet forms a little hemisphere on the surface, then no work is done by the wetting, and the work of adhesion in liquid is the same as in vapor. But if the liquid wets the surface then adhesion must be reduced. This is equivalent to Young s original theory of 1805, which he did not express in symbols but only in words. All the results above were consistent with this equation. Thus the second law of adhesion is true for wetting liquids. 

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