Breakage binary

Furthermore, the assumption is made that the motion of the centers of mass of daughter droplets to be formed (binary breakage) is similar to the relative motion of two lumps of fluid in a turbulent flow field as described by Batchelor (B6). Thus, for the inertial subrange eddies... 

Batch, semibatch, or continuous-flow operation can be simulated. The continuous phase is assumed well mixed. Particle movement was either random or followed the flow direction of the sum of the local average fluid velocity and the particle gross terminal velocity. The probability of droplet breakup is assigned based on droplet size. Binary breakage was assumed to form two randomly sized particles whose masses equal the parent drop. The probability of coalescence exists when two drops enter the same grid location. Particles are added and removed to simulate flow. 

The main contribution from the work of Luo [95, 96] was a closure model for binary breakage of fluid particles in fully developed turbulence flows based on isotropic turbulence - and probability theories. The author(s) also claimed that this model contains no adjustable parameters, a better phrase may be no additional adjustable parameters as both the isotropic turbulence - and the probability theories involved contain adjustable parameters and distribution functions. Hagesaether et al [49, 50, 51, 52] continued the population balance model development of Luo within the framework of an idealized plug flow model, whereas Bertola et al [13] combined the extended population balance module with a 2D algebraic slip mixture model for the flow pattern. Bertola et al [13] studied the effect of the bubble size distribution on the flow fields in bubble columns. An extended k-e model was used describing turbulence of the mixture flow. Two sets of simulations were performed, i.e., both with and without the population balance involved. Four different superficial gas velocities, i.e., 2,4,6 and 8 (cm/s) were used, and the superficial liquid velocity was set to 1 (cm/s) in all the cases. The population balance contained six prescribed bubble classes with diameters set to = 0.0038 (m), d = 0.0048 (m), di = 0.0060 (m), di = 0.0076 (m), di = 0.0095 (m) and di = 0.0120 (m). 

In this section the macroscopic population balance formulation of Prince and Blanch [92], Luo [73] and Luo and Svendsen [74] is outlined. In the work of Luo [73] no growth terms were considered, the balance equation thus contains a transient term, a convection term and four source terms due to binary bubble coalescence and breakage. 

Usually binary breakage is assumed for which r (x, r, Y,t) = v = 2. However, experimental determination of this variable is recommended. The function Pb, r x, r, Y, t) should also be determined from experimental observations. 

Considering binary breakage only, Tsouris and Tavlarides [114] postulated that the probability of formation of a daughter particle of size d is inversely proportional to the energy required to split a parent particle of size d into a particle of size d and its complementary particle of size d = (P — d... 

Considering binary interactions only, as were assumed in the previous section, the particle breakage terms can be written as ... 

Most daughter distribution functions can be easily extended to bivariate problems. Let us consider two examples. In the first example particles with two components A and B are described. The particulate system is defined in terms of the size of these particles dp and the composition of the particles 0, expressed for example as the mass fraction of component A in the particle. When a particle breaks we can assume for example that the amount of component A is partitioned among the daughters proportionally to the mass of the fragments. Under these hypotheses, and the additional assumption of binary breakage following the beta distribution, the resulting bivariate distribution is... 

This procedure is very easy in the case of simple daughter distribution functions (e.g. symmetric binary breakage), whereas additional random numbers might have to be selected for more complex daughter distribution functions (e.g. beta, uniform, etc.)... 

As an example, consider a monodisperse population of particles characterized by mass as internal coordinate and moments m = mjt(O) = 1 with k = 0,..., 2N 1. This population of particles is continuously fed to a system wherein particles undergo aggregation and symmetric binary breakage with constant kernels. The equations describing the evolution of the moments are... 

The foregoing example is interesting because it shows population balance models can account for the occurrence of physicochemical processes in dispersed phase systems simultaneously with the dispersion process itself. Shah and Ramkrishna (1973) also show how the predicted mass transfer rates vary significantly from those obtained by neglecting the dynamics of drop breakage. The model s deficiencies (such as equal binary breakage) are deliberate simplifications because its purpose had been to demonstrate the importance of the dynamics of dispersion processes in the calculation of mass transfer rates rather than to be precise about the details of drop breakup. 

As an example, consider a pure binary breakage process in which an average breakage time is selected as the discretization step. Of course if breakage slows down, as particles become smaller, the discretization step... 

Vigil and ZilF (1989) dispense with the assumption of binary breakage in their analysis of self-similarity but appear to assume a constant mean number of fragments independently of the size of the fragmenting particle. 

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