Coalescence random collisions

Equation (17) indicates that the entire distribution may be determined if one parameter, av, is known as a function of the physical properties of the system and the operating variables. It is constant for a particular system under constant operating conditions. This equation has been checked in a batch system of hydrosols coagulating in Brownian motion, where a changes with time due to coalescence and breakup of particles, and in a liquid-liquid dispersion, in which av is not a function of time (B4, G5). The agreement in both cases is good. The deviation in Fig. 2 probably results from the distortion of the bubbles from spherical shape and a departure from random collisions, coalescence, and breakup of bubbles. 

Kolev [46] discussed the validity of these relations for fluid particle collisions considering the obvious discrepancies resulting from the different nature of the fluid particle collisions compared with the random molecular collisions. The basic assumptions in kinetic theory that the molecules are hard spheres and that the collisions are perfectly elastic and obey the classical conservation laws do not hold for real fluid particles because these particles are deformable, elastic and may agglomerate or even coalescence after random collisions. The collision density is thus not really an independent function of the coalescence probability. For bubbly flow Colella et al [15] also found the basic kinetic theory assumption that the particles are interacting only during collision violated, as the bubbles influence each other by means of their wakes. 

More complex computational models using Monte Carlo methods have attempted to predict bubble size distributions for a combination of breakup and coalescence. These models typically treat bubble coalescence by analogy with the kinetic theory where bubbles are assumed to act as solid particles [18,19]. They use a binary collision rate (probability) and a collision efficiency factor to account for collisions that do not lead to coalescence. Since collision is assumed to be a random process in these models, turbulence of the same scale as the bubbles or smaller would increase collisions and, therefore, also increase the coalescence rate. 

The droplet velocity and the droplet frequency must be adjusted in a coupled way in order to generate a droplet train of adequate quality. At a certain droplet velocity, a too low droplet frequency leads to an excessively long tail behind the primary droplet, which leads to satellite droplets surrounding the primary droplet. On the contrary, a too high droplet frequency yields an insufficient distances between adjacent liquid pulses, which further results in random collisions and coalescence of primary droplets. Optimum combinations are droplet velocities from 2 to 3 m/s, and drop frequencies from 50 to 150 Hz. The resulting droplet diameters range from 1.4 to 2.0 mm. 

Modeling and Simulation subsection.) It is necessary to determine both the mechanism and kernels which describe growth. For fine powders within the noninertial regime of growth, all collisions result in successful coalescence provided binder is present. Coalescence occurs via a random, size-independent kernel which is only a func tion of liquid loading, or... 

The coalescence-redispersion (CRD) model was originally proposed by Curl (1963). It is based on imagining a chemical reactor as a number population of droplets that behave as individual batch reactors. These droplets coalesce (mix) in pairs at random, homogenize their concentration and redisperse. The mixing parameter in this model is the average number of collisions that a droplet undergoes. 

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

Kapur and Fuerstenau (K6) have presented a discrete size model for the growth of the agglomerates by the random coalescence mechanism, which invariably predominates in the nuclei and transition growth regions. The basic postulates of their model are that the granules are well mixed and the collision frequency and the probability of coalescence are independent of size. The concentration of the pellets is more or less fixed by the packing... 

The frequency of collision relates to the flow pattern and gas volume fraction The more random the flow pattern or the higher the gas volume fraction, the higher the frequency. The efficiency of coalescence relates to physical properties of the gas-liquid system. Some systems, such as air-water, have a high efficiency of coalescence and are often called coalescing systems. Other systems, such as gas-alcohol or gas-salt solution, have a low efficiency of coalescence and are called noncoalescing systems. ... 

It is easily observed that at high aerosol concentrations, individual particles coalesce to form larger chains or floes made up of many par-tides. The process of coagulation may be brought about solely by the random motion and subsequent collision of partides (often called thermal coagulation) or the collisions could be caused by such external forces as turbulence or electricity. In general, these external forces will act to increase the rate of coagulation. 

The Bailes and Larkai model incorporates a number of assumptions such as the use of a monodispersion and uniform interdroplet spacing. However, developing a model incorporating a typical droplet distribution with random droplet spacing would be significantly more complicated. No attempt is made, either, to incorporate file effects of flow velocity or regime, and the experimental results do not indicate whether tests were carried out in laminar or turbulent flow (though laminar flow can be deduced). These parameters would also have had an effect on droplet collision frequency, and therefore the rate of coalescence. 

Emulsions that contain more effective stabilizing additives such as one of those described above may be stable for hours, days, months, or even years. In such systems the action of random or induced motion and droplet collision will continue, but the rheological properties of the continuous phase will slow down such processes and/or interfacial layers will posses sufficient strength and rigidity so that coalescence will occur on a relatively long timescale. 

Here, (jf n) indicates (p value on the nth particle. In the coalescence/dispersion model proposed by Curl (1%3), the state of/at f + Zif is calculated from the collision frequency co. The compositions of a pair of particles, which are selected at random (denoted by wi and W2), change as ... 

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