Population balance with aggregation

When the particles are formed at high temperatures the particles are often liquid droplets. These droplets stick together when they collide, altering the particle size distribution produced. Accounting for aggregation in the population balance in gas phase reactors is performed in the following way  

The first term in this partial differential equation describes the temporal change of the population tj the second term describes the atomistic growth of the particles (which assumes that G is independent of particle size r), and finally the last two terms account for the birth and death of particles of size r by an aggregation mechanism. The birth fimction describes the rate at which particles enter a particle size range r to r + Ar, and the death function describes the rate at which the particles leave this size range. In the case of continuous nucleation, an additional birth rate term is used for the production of atoms (or molecules) of product by chemical reaction. In this case, the size of the nuclei are the size of a single atom (or molecule) and the rate of their production is identical to the rate of chemical reaction, kfi, where C is the reactant concentration, giving 

Chapter 7 Powder Synthesis with Gas Phase Reactants 

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Population Balances. Three different models based on two approximations regarding the mode of breakage and two approximations regarding the size dependence of growth rate have been examined. The differential equations for modeling the size distribution are based on a population balance on aggregates of size L which, for a CSTR at steady state, mean residence time x, and with no particles in the feed, reduces to... 

The dispersed model after (16) with a modified solution technique was used to calculate the kinetic parameters from the experimental data in the dispersed zone. At steady state the population balance with no particle aggregation or breakage is given by the following differential equation... 

Let us now consider aggregation processes in srrspertsiorts with arbitrary particle size distributions. At first the particle derrsity of the volume-based poprrlation balance will be converted into the particle density of a population balance with the particle size as the chosen parameter. In Fig. 8.4-11 the combination of a particle... 

F. Gmy, Population balance for aggregation coupled with morphology changes. Colloids Surf. 

Gruy F (2011) Population balance for aggregation coupled with morphology changes. Colloids Surf A 374(13) 69-76. doi 10.1016/j.colsurfa.2010.11.010... 

At the simplest level, the rate of flow-induced aggregation of compact spherical particles is described by Smoluchowski s theory [Eq. (32)]. Such expressions may then be incorporated into population balance equations to determine the evolution of the agglomerate size distribution with time. However with increase in agglomerate size, complex (fractal) structures may be generated that preclude analysis by simple methods as above. 

Again the assumption on the aggregation rate are that the most frequent collisions are between the larger particle and the small particles. This partial differential equation can be approximated by an ordinary one by creating a new characteristic time variable, t (= t -RIG = R G], which is constant. With this variable change the population balance becomes... 

Silva, L. E. L. R., Rodrigues, R. C., Mitre, J. E. Lage, R L. C. 2010 Comparison of the accuracy and performance of quadrature-based methods for population balance problems with simultaneous breakage and aggregation. Computer and Chemical Engineering 34, 286-297. 

This is the population balance equation (PBE). Its principal form was first given by Smoluchowski (1916). In Eq. (4.1), denotes the aggregate number concentration, while Kjj is a kinetic constant related with the collision between clusters of mass i and j, and Etj is the sticking probability of the sub-clusters. Commonly, it is assumed that any successful collision (i.e. surface contact) leads to a stable particle bond and that Ey can be, thus, set to 1. 

We have in this chapter developed the various features of formulation of population balance. Section 2.11 discussed several examples in which the different features were demonstrated. However, in most of the examples, the net birth term could be dealt with through the boundary conditions. In the next chapter it will be our concern to investigate closely the nature of the birth and death terms in population balance due to breakage and aggregation processes... 

Apply the method of successive approximations to solve the aggregation problem for the constant frequency. Assume that the population balance equation is given by Eq. (4.3.4) in terms of the dimensionless variables of Section 4.3 with a(x, x ) = 1 and that the initial distribution is given by (5(x — 1). 

One encounters similar constraints with aggregation processes to generate closed integral moment equations, i.e., a constant aggregation rate and at most a linear growth rate. In order to demonstrate the moment equations for this case, we recall the population balance equation (3.3.5) for the constant aggregation rate, a x, x ) = a, incorporate a linear growth term X(x, t) = kx, and take moments. The result is... 

In either of the preceding categories, since the integrand contains the unknown number density, the mathematically rigorous choice for the pivot, which is consistent with the mean value theorem is of course not accessible. The finer the interval, the less crucial would be the location of the pivot in /f. The fineness required would depend on the extent to which the phenomenological functions of the population balance model such as the aggregation and breakage functions vary in the interval. 

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