Breakage functions

The breakage function AB gives the size distribution of product breakage of size u into all smaller sizes k. Since some fragments from size u are large enough to remain in the range of size u, the term AB is not zero, and... 

In the investigations mentioned earlier the breakage function was assumed to be normahzable i.e., the shape was independent of Xq. Austin and Luclde [Powder Technol., 5(5X 267 (1972)] allowed the coefficient A to vary with the size of particle breaking when grinding soft feeds. 

FIG. 20-18 Experimental breakage functions. (Reid and Stewaii, Chemica meeting, 1.970.)... 

In order to describe these different mechanisms, various breakage functions have been proposed (Hill and Ng, 1995, 1996). For precipitation processes, a breakage function of the form given in equation (6.32) with h(v, Xk) being the discretized number fraction of particles broken from size v into size interval x, seems particularly suitable as both attrition - with a high probability - and particle splitting - with a low probability - are accounted for. 

The use of selection and breakage functions is much more cumbersome than working with a single index number. This concept has, therefore, been used only forthe description of comminution processes but not for the description of attrition. 

Within Eq. (7), the selection function S(y) was approximated by Sty) = Kyc with adjustable parameters K and c and the breakage function b(x, y) was simply modeled by a triangular shaped function between a minimal breaking particle size of xq = 25 nm and x with a modal value at... 

Determine the breakage functions that pertain to the size fraction of the feed material Cumulative breakage functions can be calculated [1] from the relation Bi J = Fj/Sj. Therefore, in the case of the feed-size selection function 5), the equation is B ,i = Fi/S, and the breakage functions are as follows ... 

Determine the selection and breakage functions for other size fractions of the same material Selection functions for other size intervals may be calculated via the relation... 

Similar calculations can be made to find selection functions for other size intervals. Then, cumulative breakage functions can be calculated by the relationship noted in step 3, namely, B,j = F -t/Sj. 

Given the feed size distribution, the breakage functions, and the selection functions (probabilities of breakage) for a feedstock to a grinding operation, as shown in Table 13.4, predict the size distribution for the product from the operation. 

Predict the size distribution of the product of the breakage of broken particles. This product does not exist by itself as a separate entity, of course, because the particles that become broken remain mixed with those which stay unbroken. Even so, the distribution can be calculated by postmultiplying the n x n lower triangular matrix of breakage functions B in Table 13.4 by the percentage of particles broken, the nxl matrix (S)(F), as calculated in step 1. This postmultiplication works out as follows ... 

At this time, it is instructive to note that Eqs. (34) and (42) predict that the large drops will continuously break down to a size flniax> where the breakage function g(fl ) approaches some arbitrary low value close to zero. At this point, the exponential term dominates the behavior of the breakage rate expression and a relation for Oniax is obtained similar to the one given by Eq. (18), which was derived on the basis of maximum stable drop theory. 

The population balance equation is employed to describe the temporal and steady-state behavior of the droplet size distribution for physically equilibrated liquid-liquid dispersions undergoing breakage and/or coalescence. These analyses also permit evaluation of the various proposed coalescence and breakage functions described in Sections III,B and C. When the dispersion is spatially homogeneous it becomes convenient to describe particle interaction on a total number basis as opposed to number concentration. To be consistent with the notation employed by previous investigators, the number concentration is replaced as n i,t)d i = NA( i t)dXi, where N is the total number of particles per unit volume of the dispersion, and A(xj t) dXi is the fraction of drops in increment X, to X( + dxi- For spatially homogeneous dispersions such as in a well-mixed vessel, continuous flow of dispersions, no density changes, and isothermal conditions Eq. (102) becomes... 

It is encouraging that substantial progress has been made in analyzing the hydrodynamics of droplet interactions in dispersions from fundamental considerations. Effects of flow field, viscosity, holdup fraction, and interfacial surface tension are somewhat delineated. With appropriate models of coalescence and breakage functions coupled with the drop population balance equations, a priori prediction of dynamics and steady behavior of liquid-liquid dispersions should be possible. Presently, one universal model is not available. The droplet interaction processes (and... 

Several breakage functions were early suggested [Gardner and Austin,... 

FIG. 21-67 E3q>erimental breakage functions. Reid and Stewart, Chemical meeting, 1970.)... 

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