Power law agglomerates

Figure 8.2 Autocoirelaiion function for a power-law agglomerate as a function of distance r. The fractal dimension i.s calculated from the straight-line portion for small values of r. The tail-off for large values of r corresponds to the edge of the agglomerate.

For-inatiy practical applications, the power law relationship (8.1 requires an appropriate proportionality constant or coefficient. Consider the case of nionodisperse primaty particles of radius ft,o which fonn power law agglomerates according to a process whose statistical features are independent of r ,o. The statistical properties of the agglomerates produced by this process do not depend on the magnitude of fl o. That is, if were multiplied by a factor of 10, the value of Df would not be affected nor would all of the other higher-order particle correlation functions that are not considered in this analysis. This means that the system should scale as R/apo so that (8,3) becomes... 

SMOLUCHOWSKI EQUATION COLLISION KERNELS FOR POWER LAW AGGLOMERATES... 

Hence for power law agglomerates the collision kernel becomes... 

The collision kernels for power law agglomerates, (8.11) and (8.13), are homogeneous functions of the volumes of the colliding particles ... 

Smoiuchowski Equation Collision Kernels for Power Law Agglomerates 230... 

Furthermore, the agglomeration and disruption kernels are also assumed to depend on the supersaturation in power law form (Zauner and Jones, 2000a)... 

The mass-fractal dimension of the ramified agglomerates is determined from the slope of the weak power law decay in between the power law regimes that follow Porod s law (Equation 10.9) ... 

Figure 2.3 Cjkulaied dilTusiitn cueflictcms for power-law fractai-l ike) agglomerates normalized by the Sioke.s-Ein.<>tcin diffusion coefiidem for a primary particle. The total number of primary parttdes in the aggregate is and Df is the fractal dimetision. The results hold for the continuum regime, Op, ip. (After Tandon and Rasner, 1995.)...

It is found experimentally that in many cases of practical interest, the total number of primary particles Np in an agglomerate is related to R through a power law expression... 

The collision cross section for agglomerates is ba.sed on the relationship fora power law (fractal-like) agglomerate (8.4) which can also be expressed as follows ... 

The expressions (8.23) and (8.24) for ii and R cannot hold for short times when the agglomerates are composed of few primary particles, before the power law structure is established. However, these expressions can be used to examine qualitatively the transition from small to large agglomerates as shown in Fig. 8.7b. The results indicate that there is a... 

In these analyses, it has been assumed that the power law exponent (fractal dimension) is constant during the agglomeration process. This is not necessarily the case as the experimental observations di.scussed in the next section show. 

The action of the enzyme rennet on milk, known to destabllze a K-caseln and trigger agglomeration of a multl-caseln micellar suspension (ot,3,Y E <) to produce coagulated milk for cheese-making, has been shown to produce a reaction suspension with power law behavior also described by eqns (2.4-2.5)(Tucznlckl and Scott Blair). ... 

Fractal aggregate, fractal agglomerate aggregates or agglomerates with a non-uniform distribution of the constiment particles, which typically coincides with a very porous, branch-like morphology fractal aggregates are characterised by a power-law decrease of the pair-correlation density function g(v) (Eq. (4.8)) and a power-law relationship between mass and size (Eq. (4.9)), in which the exponent is less than the Euclidean dimension fractal aggregates are not ideal fractal objects, but rather obey the fractal relationships only in a statistical sense (cf. Sect. 4.2.1). 

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