(75e) Understanding Intra-Mixture Interactions in the Breakage of Dense Particulate Mixtures

Understanding intra-mixture interactions in the breakage of dense particulate mixtures.

There are two categories of degradation, which are intentional and unintentional degradation. Intentional degradation is refereed as a desired breakage where the process is intentional to bring about the desired particle size as for instance in milling and pulverisation. Unintentional degradation or synonymously attrition is refereed, as an accidental breakage in which the resulting size reduction is often undesirable.  Unlike intentional degradation where the area has been intensively researched over the past century aiming to address energy deficiency in comminution industry, study on the inadvertent degradation is a relatively recent development.

Breakage Matrix Approach

Breakage matrix is a general technique useful to model any degradation process. It has long successful history over more than a century since Broadbent & Calcott; first introduced it in 1956 in predicting intentional degradation process of coal. It illustrates the input size distribution as a vector of an event and the resulting breakage process as a matrix multiplying with that vector to give output size distribution (as a further vector)

It is ?time invariant-time average? type of approach where the mass balance is accounted at the beginning and at the end of the process (or within set boundaries within a larger process) as a single breakage event. A ?single breakage event? may consist of a single or series of consecutive process depending on the boundaries define by the user. The breakage boundary is established based on the available data, which is sufficient to represent the input and output of a single breakage event. It is practically convenient in such process where sampling is only possible at certain points (as it is known to be impossible to obtain data from each inlet and outlet for most of the systems) within the process.  PSDs of such an event can be expressed mathematically via matrix equation as:

M*f=O                                                   (1)

Where f and O are the feed and product vectors of PSD entering and leaving a unit process, and M is the breakage matrix, relating the two by using matrix multiplication. Here, vector is defined as a series of numbers, which follow a certain order. Conveniently, elements in the vector can often be obtained directly from size analysis from example from mechanical/ hand sieving.

Quantifying inter-particle interaction

Via this approach, successful prediction as for instance on the milling performance can be achieved by comparing the output vector value obtained from a degradation experiment of a binary mixture with the output vector calculated from the breakage matrix prediction. If the value of the both vectors coincide with each other, then the individual particles within the mixture is supposed to break independently from its surrounding particles. Therefore, the breakage matrix is constructed from a combination of the successive column vectors of the corresponding mono-size breakage test at a time that will be appropriate to a given mixture under that condition. Assuming independent breakage however is sensible only for certain degradation scenarios as for instance once through grinding milling processes, where each and every particle passing the roller mill independently or lean phase pneumatic conveyor where thin streams of particles hitting on pipe bends one at a time. Equation (1) is therefore can be re-written as:

M*f=O_M                                                                                                                                                  (2-a)

M=M_null if there is no interaction within the mixture.            (2-b)

and O_M=O_E                                                                                                                                        (2-c)

In which O_M=O_E signify output vector obtained from breakage matrix calculation and experimental observation respectively and M_nul is appropriate breakage matrix for non-interacting mixture.

Any deviation that exists between the two pairs of output vectors is an indication of inter-particle interaction that may decelerate or accelerate the breakage of the top size interval in the mixture (coarsest size fraction).;see [1]

In this paper, we propose a more simple analytical technique in which intra-mixture interaction within a mixture could be quantified by measuring the coefficient of distance of two output vectors obtained by breakage matrix calculation and experimental observation. Coefficient of distance that is chosen is based on the concept of Canberra Distance approach (http://people.revoledu.com/kardi/tutorial/Similarity/CanberraDistance.html) to indicate the degree of dissimilarity between two compared vectors. The approach is total summation of ratio differences between two compared vectors.

The relationship between the coefficient of dissimilarity and the two output vectors can be expressed as:

                                                  (3)

Where C_D denotes the Canberra distance coefficient, O_E&O_M are the i-th element vectors obtained from experiment and the breakage matrix calculations respectively.

We aim to demonstrate the usefulness of Canberra distance approach as a classifying tool in which the unknown interacting mixture (experimental observation) could be distinguished into two groups of insignificant and significant interacting mixtures. The robustness of Canberra distance approach is also tested by varying the basic mixture properties such as size ratio and fines composition under different applied stress condition.

Experimental Verification: The experimental  work has involved two types of testers: i)  a model impact degradation tester to simulate the lean phase condition of insignificant interacting mixture and ii) compression tester to represent the condition of high number density of inter-particle contacts within a dense-phase assembly. The tests are performed under applied stresses of 1.6 & 3.2 MPa for both testers. Throughout the tests, granulated sugar with mean sizes of 850, 600 and 300 microns are used as individual size classes. For the mono size and binary mixture tests, an appropriate sample mass is subjected to the different breakage mechanisms of interest. For binary mixture tests, 2 different size classes; 850 and 600 micron are mixed thoroughly prior to the tests with a size class of 300 micron to give size ratios of 3.0 and 2.0 correspondingly. For each binary mixture test, the composition of fines is varied from 20, 50 to 80% to signify fines continuous, non-continuous and coarse continuous mixtures respectively. [2].

Figure 1: Coefficient of distance at 2 different mechanism under size ratio of 3.0 and applied stress of 1.6Mpa.

Figure 1 shows significant separation/gap between compression and impact mechanisms according to the coefficient of distance value calculated via Canberra distance approach under size ratio of 3.0 and applied stress of 1.6MPa. The value of coefficient of distance under the compression mechanism always exhibits a far greater value (over 40%) as opposed to the mixtures subjected to the impact mechanism.  This suggests that for dense phase mixture assemblies, the interaction within the particles in the mixture contribute larger differentiation/ dissimilarity between the two compared output vectors of unknown (observed experiment) and known with null interaction (breakage matrix calculation).

In contrast, with lean phase mixture assemblies where the interaction is insignificant the dissimilarity is small and thus resulting in a smaller value of the coefficient of dissimilarity. This concrete evidence supports the adequacy of the Canberra distance approach as an analytical classifying tool to distinguish two groups of clusters of significant interaction and insignificant interacting mixtures. In addition, increasing the fines composition from 0.2 to 0.5 and 0.8 is shown to have a direct effect on the degree of interaction for both impact and compression mechanisms.

Figure 2(a): Coefficient of distance at 2 different               Figure2(b): Coefficient of distance at 2 different

applied stresses under size ratio of 3.0.                           applied stresses under size ratio of 2.0.

We then investigate the effect of varying the size ratio on the value coefficient of distance. The influence of varying size ratio on the value of coefficient of distance is presented in Figure 2(a)&(b). Again, similar finding is also observed for both figures in which a clear gap between impact and compression tests is exhibited. Nevertheless, the effect of separation (total area of separation) is much clearer when the size ratio is increased from 2.0 to 3.0.  The results are found to be consistent as portrayed in both graphs 2(a)&(b).  This is due to the fact that, as the size ratio is increased, the total number of fines particles shielding the coarse particles is increased which results in significant dissimilarity between the two compared output vectors particularly under the compression mode of breakage.

Figure 3(a): Coefficient of distance at 2 different               Figure3(b): Coefficient of distance at 2 different size ratio under applied stress of 1.6MPa.                                              size ratio under applied stresses of 3.2MPa

Figure 3(a)&(b) indicate the effect of varying applied stress on the calculated values  of the coefficient of distance under two different size ratios of 2.0 and 3.0 accordingly. As we can see, the value of the coefficient of distance is increased with increasing applied stress on the mixture assemblies with a size ratio of 3.0. However, for the smaller size ratio (2.0) the difference is hardly noticeable whilst increasing the applied stress does not yield significant influence on the value of the coefficient of distance. This is due to the fact that at the higher size ratio, more breakage is obtained when breaking the individual size class of the coarsest particles present in the mixture.

References

1-) Baxter J., Abu-Nahar A., Tuzun U., (2004), Powder Technology, 143-144, pp:174-178

2-) Tuzun U., Artega P., (1992), Chem. Eng. Sci., 47, pp:1619-1630