Yang Fluidization, Solids Handling, and Processing

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The above findings are quite significant since they indicate a clear way to design solvent evaporation in the bed when the binder is a solution or the cooling time when the binder is a melt. One has to stress, however, that the above estimates are probably not very general and may require corrections for different pairs of powders and binders. Since the experimental procedure and the instrument are quite simple and straightforward to use, such measurements should be easy to perform for each specific case.

3.5Granule Consolidation, Attrition and Breakage

Characterizing Fracture Properties of Particulate Systems. A granule may be viewed as a nonuniform physical composite of primary particulate material, or grains. It possesses certain macroscopic mechanical properties, such as anisotropic yield stress and inherent flaw distribution. It has long been realized that materials, especially brittle ones, can fail by the propagation of cracks which act as points of stress concentration. Therefore, the failure stress of a granule may differ significantly from its true yield stress, which is more a function of inherent bond strength. Therefore, bulk attrition tests of granule strength measure both the inherent bond strength of the granule as well as its flaw distribution. In addition, the mechanism of granule breakage or attrition often is a strong function of materials properties of the granule itself, as well as the type of loading used in the mechanical testing. Thus, there is often appreciable variation in the results of different tests, which may also be different from the observed breakage in actual process handling.

The approach taken here is to employ standard materials characterization tests to measure the materials properties of the granulated product. With this information, the mechanism of attrition, i.e., breakage versus erosion, is determined. The rate of attrition can then be related, semi-empirically, to material properties of the formulation and the operating variables of the process, such as bed depth and fluidizing velocity.

It is inherently difficult to measure the “strength” of a material since this is strongly influenced by the microstructure of the material, i.e., the distribution of flaws which strongly influence the propagation of cracks. This concept is illustrated in Fig. 31, where the elastic stress distribution in an ideally elastic, brittle material is seen to become infinite as the crack tip is approached. The key properties which characterize the strength of a material are:

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specimens each with identical geometry and material will have a wide scatter whereas the fracture toughness will not. This scatter is due to the wide variation of flaws and not the inherent strength of the material itself.

Using the picture shown in Fig. 31, the fracture toughness defines the stress distribution in the body just before fracture and is given by:

where σf is the applied fracture stress, c is the length of the crack in the body and Sf is a shape factor to account for the different geometries of the specimens. Since the elastic stress cannot exceed the true yield stress of the material, there is a region of local yielding at the crack tip known as the process zone. The characteristic size of this zone, rp, is given by:

The above results are derived from linear elastic fracture mechanics and are strictly valid for ideally brittle materials with the limit of the process zone size going to zero. In order to apply this simple framework of results, Irwin (1957) proposed that the process zone, rp, be treated as an effective increase in crack length, δc. With this modification, the fracture toughness becomes

As discussed above, because materials often fail by the propagation of cracks, particularly brittle ones, it is difficult to measure the true yield stress, σy, of a material. Instead one measures a fracture stress, σf, which is related to the size of the process zone and actual crack length by:

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Equation (32) clearly illustrates the point that as rp becomes very small relative to c, i.e., for large cracks, the fracture stress may be orders of magnitude smaller than the true yield stress of the material.

The final fracture property of interest here is the critical strain energy release rate, Gc. This is the energy equivalent to fracture toughness. The energy analysis of fracture was the original approach to fracture mechanics first proposed by Griffith (1920). Fracture toughness and critical strain energy release rate are related by:

As stated previously, in order to determine fracture properties in a reproducible manner, very specific test geometry must be used. Two traditional methods for measuring fracture properties are the three point bend test and the indentation fracture test. In the case of the three point bend test, toughness is determined from the variance of fracture stress on induced crack length given by Eq. (31). For the indentation test, one can determine the hardness, H, from the area of the residual plastic impression, A, and the indentation load, P. The fracture toughness can be determined from the hardness and the length of cracks propagating from the indentation as a function of P. The following relationships are used for determining H and Kc from the indentation tests:

The Relationship Between Fracture Mechanics and Granule Attrition. Fracture properties play a very significant role in a variety of phenomena involving particles and granules. These properties control breakage and wear of particles in fluidized beds. Table 5 compares some fracture properties of some typical agglomerated materials. When comparing the values in Table 5 to those found for typical polymers and ceramics, we can make the following generalizations:

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(i)Fracture toughness is less than for typical polymers and ceramics

(ii)Critical strain energy release rates are typical for ceramics but much less than for polymers

(iii)The process zone sizes are large and typical for polymeric materials

(iv)Critical displacements of 10-7–10-8 are typical for polymers while 10-9 is more typical of ceramics.

Table 5. Comparison of Fracture Properties of Agglomerated Materials

To summarize, we can say that granulated materials should behave in a brittle manner, similar to a ceramic, in the sense of having small yield strains and small critical displacements but their behavior should also be ductile, similar to polymers, in the sense of having large process zones. The consequence of having large process zones is important since it implies that

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two colliding granules are capable of absorbing elastic energy through plastic deformation thereby dissipating gross fracture. On the other hand, the energy dissipation within the process zone is expected to occur via diffuse microcracking, as would be the case for ceramic materials. The size of the process zone plays an important role in determining the mechanism of attrition. There exists a critical specimen size, xc, for a granule which has been shown to be related to the process zone size (Ennis and Sunshine, 1993), by the following approximate relationship

If a granule is smaller thanxc, then there is insufficient granule volume to concentrate enough elastic energy to propagate a gross fracture during a collision. When this is the case the most likely mechanism of attrition is one of wear or erosion. On the other hand, when the granule size is larger than xc, gross fracture or breakage of the granule is much more likely to occur during collision. From the data presented in Table 5, it appears that the dominant mechanism for attrition should be wear or erosion for all the granules listed with the exception of Bladex 90 DF, which would appear to be susceptible to gross fracture during collision. This concept is investigated further below.

By considering wear to occur by the intersection of subsurface lateral cracks which depend on the load and material properties, Evans and Wilshaw (1976) determined the volumetric wear rate, Vwear, to be

where d is the indentor diameter, P is the applied load, l is the wear displacement and Ais the apparent area of contact of the indentor. This result implies a very specific dependence of wear rate on fracture toughness and hardness. The dependence of these two material properties, as predicted by Eq. (36), is compared in Figures 32 and 33 for bar wear rate and erosion in a fluidized bed. It is interesting to note that the wear rates of materials with radically different structures (polymer-glass vs. herbicide formulated granules) collapse onto the single line, predicted by Eq. (36).