Aggregation source term

Thus, the source terms for each environment S(c) and Sk ((/)) will be closed. Of particular interest are the local nucleation rates /(c ). As discussed in Wang and Fox (2004), due to poor micromixing the local nucleation rates can be much larger than those predicted by the average concentrations /((c)). This results in a rapid increase in the local particle number density mo due to the creation of a very large number of nuclei. As discussed below, this will have significant consequences on the local rate of aggregation. 

A fairly general framework has been formulated for the source terms considering particle breakage, fluid particle coalescence, solid particle agglomeration/aggregation and similar processes (e.g., [109, 80, 81, 37, 114, 43, 25, 94]). Detailed discussions of the particle breakage and coalescence modeling and the mathematical properties of the constitutive equations can be found in the papers by Barrow [4], Laurencot and Mischler [64, 65]. 

As it is possible to see, the drift term has disappeared since the continuous growth of particle size does not change the total number concentration (if Gl > 0). However, N is influenced by the rate of formation of particles (e.g. nucleation), and the rates of aggregation and breakage, which cause appearance and disappearance of particles. These processes are all contained in the source term /tL.o The third-order moment mL,3 is related to the fraction of volume occupied by particles with respect to the suspending fluid and can be easily found fromEq. (2.18) ... 

In order to account for variable particle numbers, we generalize the collision term iSi to include changes in IVp due to nucleation, aggregation, and breakage. These processes will also require models in order to close Eq. (4.39). This equation can be compared with Eq. (2.16) on page 37, and it can be observed that they have the same general form. However, it is now clear that the GPBE cannot be solved until mesoscale closures are provided for the conditional phase-space velocities Afp)i, (Ap)i, (Gp)i, source term 5i. Note that we have dropped the superscript on the conditional phase-space velocities in Eq. (4.39). Formally, this implies that the definition of (for example) [Pg.113]

Note that the sign of the source term will depend on whether particles are created or destroyed in the system. Note also that the spatial transport term in Eq. (4.46) will generally not be closed unless, for example, all particles have identical velocities. The transport equation in Eq. (4.46) is mainly used for systems with particle aggregation and breakage (i.e. when N(t, x) is not constant). In such cases, it will typically be coupled to a system of moment-transport equations involving higher-order moments. 

Ip = [ 3 ( p3ji/3 nonlinear transformation results in a Jacobian equal to p[ p -. The source term for aggregation is then... 

In the case of aggregation, when the internal coordinate f is purely additive during one aggregation event, the PBE has the source term given in Eq. (7.9) and the evolution equation for the moment of order k is... 

Because the accuracy of the quadrature approximation strongly depends on the application, let us discuss some of these issues in detail for specific examples. In the case of standard nucleation, (positive) growth, aggregation, and breakage, application of the QBMM to the source term in Eq. (7.96) yields... 

The source function, which represents the rate of formation of particles of mass X by aggregation of smaller particles, is computed as follows. From conservation of mass, we have particles of mass x — x (= x in terms of the notation used in the general considerations leading to (3.3.2)) aggregating with particles of mass x to produce particles of mass x. Clearly as x varies between 0 and X, so also does x — x so that each pair in the set [x — x, x ] 0 < x < x is considered twice (i.e., 5 — 2 in (3.3.2)). Thus the source term becomes ... 

We first identify the population balance equation for the discrete range. Since the smallest particles of volume do not possess a source term by aggregation, their number density will satisfy an equation slightly different from those of volume Thus,/i will satisfy... 

Laplace transforms are particularly suitable for obtaining analytical solutions for certain forms of population balance equations. In aggregating systems, the population balance equation in particle mass (or volume) features a convolution integral in the source term which makes it amenable to solution by Laplace transforms. We shall illustrate the solution of the aggregation problem represented by Eq. (3.3.5), for suitably selected aggregation frequencies. We recall the population balance equation (3.3.5) as... 

Since the issue has to do with the birth terms for aggregation and breakage, we focus our attention on these. Denoting the total source term in (4.5.1) as v, t], we may identify it as... 

Most frequently used is the Lucassen [118] approach with monodisperse micelles (of aggregation number m) this approach is related to the studies of bulk micellization kinetics by Kresheck et al. [119], Muller [120], and Hoffinann et al. [121]. The model provides the following expressions for the source terms in Eq. (18) ... 

Sometimes, an adsorbing substance is present in die form of various species, e.g., as single molecules and in an associated form, e.g., in aggregates or micelles. If that is the case, die various transport equations are coupled through the association-dissociation equihbria by several source terms in them. These are generally... 

The recent history of the world use of coal roughly follows that of the United States for two reasons. First, the United States and the industrial nations have had, in the aggregate, similar energy behavior in terms of energy sources. Second, the United States itself accounts for about one quarter of world energy rise. Thus, world energy use patterns reflect, to a considerable degree, those of the United States. 

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