Aggregates solid fraction

Fig. 24 Mean aggregate solid fraction vs filler volume fraction of N339 in E-SBR- and S-SBR-composites, respectively, as obtained from TEM-analysis (in-rubber state)...

J>P(CDBP) Primary aggregate solid fraction from CDBP-absorption with data from [26]... 

Fig. 12. Effect of pressure drop on the solid fraction of soot cake deposits as a function of soot aggregate size.

The nonideal nature of G/S fluidization, shown as the bulging curve for zero solids flow in the voidage-velocity plot of Fig. 60, is considered to be caused by gas aggregation at low velocities and solids aggregation at high velocities. Alternately, the G/S system may be visualized to be equivalent to a binary mixture of the aggregated gas bubbles with a volume fraction of (e — 0) on the one hand, and of the aggregated solid particles in the form of clusters with a volume fraction of (1 — e) on the other. At low gas velocities,... 

Different concentration limits of the filler arise from the CCA concept [22]. With increasing filler concentration first an aggregation limit O is reached. For >+, the distance of neighboring filler particles becomes sufficiently small for the onset of flocculation and clusters with solid fraction A are formed. Dependent on the concentration of filler particles, this flocculation process leads to spatially separated clusters or, for 0>0, a through going filler network that can be considered as a space-filling configuration of fractal CCA-clusters. The different cases for spherical filler particles are shown schematically in Fig. 1. 

The structure or amount of specific voids of the three grades is almost the same and differs between 1 cm3/g and 1.2 cm3/g, only. Since the specific weight of carbon black is almost twice that for DBP, this corresponds to a factor of 2 for the void volume as compared to the solid volume of the aggregates. It means that about 2/3 of the aggregate volume is empty space, i.e., the solid fraction p of the primary aggregates is relatively small (Op 0.33). 

It is shown below that p fulfills a scaling relation which involves the size and mass fractal dimension of the primary aggregates. Due to significant deviations of the solid fraction p from 1, the filler volume fraction of carbon black in rubber composites has to be treated as an effective one in most applications, i.e., 0eff=0/0p (compare [22]). 

Here, Va is the solid volume and NA is the number of particles or primary aggregates of size d in the clusters of size . p is the solid fraction of primary aggregates considered in Sect. 3.2.2. For spherical filler particles it equals... 

Accordingly, we expect a power law behavior G,0 (O/Op)3 5 of the small strain elastic modulus for 0>0. Thereby, the exponent (3+df [j)/(3—df)w3.5 reflects the characteristic structure of the fractal heterogeneity of the filler network, i.e., the CCA-clusters. The strong dependency of G 0 on the solid fraction Op of primary aggregates reflects the effect of structure on the storage modulus. 

In Chapter 7, we observed that U and H are state properties of a species that is. their values depend only on the state of the species—primarily on its temperature and state of aggregation (solid, liquid, or gas) and, to a lesser extent, on its pressure (and for mixtures of some species, on its mole fraction in the mixture). A state property does not depend on how the species reached its state. Consequently, when a species passes from one state to another, both Af/ and for the process are independent of the path taken from the first state to the second one. 

Fractal dimension In gel networks or aggregates of nanoparticles in which the mass M inside any sphere of radius R, about a centre chosen at random in the gel network or aggregate, increases statistically with R as, M k Ef, the number/is usually a noninteger and is termed the fractal dimension. Aerogels are examples of fractal solids Fractional function of the first kind /o-a( ) Fraction of the total black-body radiation intensity having wavelengths between 0 and A... 

Whatever is the packing state of aggregates into agglomerates, and the compaction degree of agglomerates into pellets, there is a certain degree of "voids" such that the solid fraction of carbon black pellets is given by ... 

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