Breakage discrete

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. 

Hill, P.J. and Ng, K.M., 1995. New discretization procedure for the breakage equation. American Institution of Chemical Engineers Journal, 41(5), 1204-1217. 

In other cases, several discrete relaxation times or distributions of relaxation times can be found [39]. This is typically the case if the stress relaxation is dominated by reptation processes [42 ]. The stress relaxation model can explain why surfactant solutions with wormlike micelles never show a yield stress Even the smallest applied stress can relax either by reptation or by breakage of micelles. For higher shear rates those solutions typically show shear thinning behaviour and this can be understood by the disentanglement and the orientation of the rod-like micelles in the shear field. 

The death rate in grinding is also referred to as the breakage rate and, for the population balance on a discrete mass basis, is given by [11]... 

Lee et al [66] and Prince and Blanch [92] adopted the basic ideas of Coulaloglou and Tavlarides [16] formulating the population balance source terms directly on the averaging scales performing analysis of bubble breakage and coalescence in turbulent gas-liquid dispersions. The source term closures were completely integrated parts of the discrete numerical scheme adopted. The number densities of the bubbles were thus defined as the number of bubbles per unit mixture volume and not as a probability density in accordance with the kinetic theory of gases. 

Hagesffither et al [28, 29, 30] extended the model by Luo and Svendsen [74], but the resulting breakage model was still not completely conservative. To ensure number and mass conservation they thus adopted a numerical procedure redistributing the bubbles on pivot points in accordance with the discrete solution method [94]. 

It is finally remarked that working with the macroscopic and average microscopic model formulations, care should be taken as the discrete macroscopic breakage rate models are sometimes erroneously assumed equal to... 

Luo and Svendsen [74] derived a discrete expression for the breakage density of a particle of diameter d into two daughter particles of size dj and (d — d3) /3 respectively, using energy arguments similar to those employed by Tsouris and Tavlarides [114]. 

Vanni, M. 1999 Discretized procedure for the breakage equation. AIChE Journal 45, 916-919. 

The size distribution of products from various types of size-reduction equipment can be predicted by a computer simulation of the comminution process. This makes use of two basic concepts, that of a grinding-rate function S and a breakage function A5 a. The material in a mill or crusher at any time is made up of partides of many different sizes, and they all interact with one another during the size-reduction process, but for purposes of computer simulation the material is imagined to be divided into a number of discrete fractions (such as the ones retained on the various standard screens) and that particle breakage occurs in each fraction more or less independently of the other fractions. 

Paper [84] discusses the solution of this equation in the form of the sum of independent solutions with discrete spectra. Special cases of monodisperse and uniform over some interval of initial distributions are examined. On the basis of the obtained solutions, the first four moments of the volume distribution of drops are found, and a Pearson diagram (see Fig. 11.1) is used to show that the solution converges to a lognormal distribution. It is consistent with the result of [86], where it was supposed that breakage frequency /(V) is constant and does not depend on the size of drops. 

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