Aggregation frequency

The chief phenomenological instrument of the population balance model of an aggregation process is the aggregation frequency. It represents the probability per unit time of a pair of particles of specified states aggregating. Alternatively, it represents the fraction of particle pairs of specified states aggregating per unit time. This interpretation must, however, be modified for the aggregation frequency commonly used in population balance models in which the population is regarded as well-mixed and external coordinates do not appear explicitly in the population density. We will subsequently derive this modified frequency from the quantity that we have just defined. 

Alternatively, we recognize that a x, r x, r Y, t) is the fraction of particle pairs of states (x, r) and (x, r ) aggregating per unit time. The aggregation frequency is defined for an ordered pair of particles, although from a physical viewpoint the ordering of particle pairs should not alter the value of the frequency. In other words, a x, r x, r Y, t) satisfies the symmetry property... 

It is essential to consider only one of the above order for a given pair of particles. The explicit time dependence in (3.3.1) in the aggregation frequency is generally not a desirable feature in models and is eliminated in the remaining treatment. 

The aggregation frequencies in the discrete and the continuous ranges are generally related by a,j = a(Xj, Xj). The distinctive notations for particles in the discrete and continuous ranges serve as a reminder as to whence the aggregating pair arises. 

Such estimates can be made by using a uniform upper bound for the aggregation frequency and solving the population balance equation analytically for the density at the exit of the bed. 

Thus, the determination of the aggregation frequency depends on that of the function P r", t + i). It would appear from (3.3.42) that the aggregation frequency is dependent on the time elapsed since the instant t. We shall return to this issue presently. 

In obtaining the Brownian coalescence frequency, we had assumed that particles move independently of one another even when they are in the immediate proximity of each other. Thus, the foregoing analysis does not account for any correlation between the movement of particles as a result of interparticle forces and/or viscous forces in the intervening fluid. We next outline the manner in which such effects may be included in the derivation of the aggregation frequency. 

The Fokker-Planck equation for the stochastic differential equation (3.3.47) will differ from Equation (3.3.36) because of the drift term. However, since the calculation of the aggregation frequency depends on the function P(r", t + t) as defined in Section 3.3.5.2, we will directly proceed to the differential equation in P(r", t + t). Recognizing spherical symmetry, we have... 

The boundary conditions are the same as in Section 3.3.5.2. This problem has been addressed in detail by Simons et al. (1986). The resulting aggregation frequency shows sizable deviations from that obtained by splicing the Brownian and gravitational coagulation frequencies. 

Aggregation frequency = Collision frequency x Aggregation efficiency. 

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... 

Similarly, solutions for the product frequency can also be obtained by the method of Laplace transforms (Scott, 1968). For an integrated treatment of the constant, sum and product aggregation frequencies, the reader is referred to Hidy and Brock. 

Some latitude exists here in the choice of the average value withdrawn from the integral so that slightly different forms of (4.5.9) can also be envisaged. In this approach, one is then required to calculate the integral of the aggregation frequency at each step. 

To Hounslow and his co-workers (1988) must go the credit, however, for attending to particle numbers as well as particle mass which was accomplished through the instrument of a correction factor. The factor did not depend on the aggregation frequency, however. 

Establish directly by solving Eq. (5.2.16) via the method of Laplace transforms for the case of constant aggregation frequency, given by a x, x ) = the self-similar solution il/ rj) = e. (Hint Recognize the convolution on the right-hand side of (5.2.16). Letting = ij/ where ij/ is the derivative of the Laplace transform ij o ij/ respect to the transform variable s, obtain and solve a (separable) differential equation for the derivative of 0 with respect to ij/). 

THE INVERSE AGGREGATION PROBLEM DETERMINATION OF THE AGGREGATION FREQUENCY... 

Notice in particular the acquired time dependence of the newly defined aggregation frequency. Equation (6.2.2) converts (6.2.1) into the population balance equation... 

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