Binary coalescences

Coalescing neutron star binaries. Coalescing of neutron stars (or black holes) is foreseen to be the the most powerful source of detectable gw. The frequency of such events is estimated to be ly D/200 Mpc) and their amplitude will allow detection of sources as far as 50 Mpc. We are thus waiting for about one event every 60 years with the current sensitivity of detectors. 

For the coalescence between bubbles of class, di, and bubbles of class, dj, the binary coalescence rate sink is expressed as ... 

Fig. 9.2. A sketch of the three consecutive stages of the binary coalescence process. Two bubbles are approaching each other. The bubble surfaces deform and a thin liquid film is created between them. The liquid drains thinning the film, and hydrodynamic instabilities cause film rupture. The final result of binary bubble coalescence is a new larger bubble.

The overall coalescence rate of a dispersion/emulsion in a separator is the most important design criterion. Unfortunately, this rate is a product of several complex mechanisms like binary coalescence, interfacial coalescence, and set-tling/creaming. Each of these mechanisms is further related to other even more complex processes/factors like hydrodynamic micro- and macro-motion, droplet size distribution, and interfacial components. In order to understand the overall coalescence rate one must also understand the interactions between these mechanisms. This makes it difficult to separate the overall rate into a sum of distinct rates, and is probably the reason why there exists no generalized coalescence model for concentrated dispersions with a sound theoretical foundation. 

Binary coalescence is coalescence between droplets that are settling/creaming or packed in the dispersion band, while interfacial coalescence is the coalescence of a droplet with its own phase (a droplet with infinite dimensions). In both cases a liquid film of continuous phase separates the dispersed droplets and this film has to be drained and broken in order to complete the coalescence process. Hartland describes this draining process in detail... 

The pressure term represents the turbulent energy input across a valve. An interesting feature of this equation is that it shows a dependence of dispersed phase fraction. This equation is of a more empirical nature than the others, and the extra term could include both binary coalescence in the downstream region of the valve and the possibility that a larger dispersed mass would absorb and dissipate energy internally at larger eddy sizes. It addresses, however, the nondilute situation usually encountered in crude oil/water separation processes. 

Binary coalescence is a discrete event where two specific bubbles merge to form a single larger one. Thus the number of bubbles, N , existing after k coalescence events in the absence of breakup is N< > = N< > - k, where is the initial number of bubbles. Then the mean bubble volume, after k coalescences can be... 

This, in fact, is true in all binary coalescence, regardless of randomness or the initial distribution. If the original bubbles are of uniform size, V< >, the total gas volume is Then the mean volume ratio, p, referenced to the initial individual bubble... 

Now consider all possible binary coalescences of a population evolving from an initial group of N " uniform bubbles. Let M be the volume (measured in integer multiples of the initial volume) of a bubble selected at random from those existing after k random coalescences with probabilities given by (6) and (7). Then the... 

Functions (8) and (9) define the discrete probability density of the Random Binary Coalescence (RBC) distribution. Its mean volume after k events is found by summing (9) over all possible volumes (from 1 to - 1) ... 

The bubble size distribution from a completely random, binary coalescence process is modeled well by the geometric and exponential distributions. We now develop a simple model for non-binary, clusterwise coalescence. In the random binary model (13), that leads to the exponential distribution, the effect of coalescence is linear. But now assume that coalescence occurs not between pairs of bubbles, but simultaneously among clusters of bubbles. Then the change in the number of bubbles with volume, m, is the product of the number in the cluster (dN dN, ) and the change in the number of clusters with volume. That is. 

That the Pareto model fits the data much better than the binary exponential model is shown very dramatically in Figures 10 and 11, which compare predicted to measured probabilities for all five data sets. The Pareto clustering coalescence model shows a surprisingly good match over the entire range of bubble sizes. On the other hand, the exponential binary coalescence model hardly shows any correlation to the data. 

The bubble size distribution resulting from purely random binary coalescence is well-represented by the geometric and exponential distributions. But these distributions completely miss the behavior of measured bubble size distributions that show relatively fewer of the smallest bubbles and more of the very largest ones. But a simple cluster coalescence model follows the Pareto distrihution, which matches these characteristic trends quite well. We conclude that multiple bubble interaction... 

FIGURE 2 The intensity of some astrophysical sources. CB, compact binaries WDB, white dwarf binaries CBC, compact binary coalescence SN, supernovae a, coalescence of binary black holes with 10 /W b, black hole formations with 10 Mq c, black hole binary with 10 M d, black hole-black hole with 10 /W . 

---