Inverse problems breakage

The mathematical statement of the inverse problem is as follows Given measurements of F x, t), the cumulative volume (or mass) fraction of particles of volume ( x) at various times, determines, b x), the breakage frequency of particles of volume x, and G x x ), the cumulative volume fraction of fragments with volume ( x) from the breakage of a parent particle of volume x. Obviously, the experimental data on F x, t) would be discrete in nature. We assume that G x x ) is of the form (5.2.9) and rely on the development in Section 5.2.1.1 using the similarity variable z = b x)t. Self-similarity is expressed by the equation F x, t) = 0(z), which, when substituted into (6.1.1), yields the equation... 

In what follows, we let u represent the ratio of the breakage rate of the fragment to that of the parent particle. The statement of the inverse problem lies in calculating the unknown function g u) over the unit interval and the constant P given the self-similar curve in the form of 6 versus z. Since g u)... 

The expression in (6.1.13) above was obtained from the correlation developed by Narsimhan et al, (1984) (referenced in Chapter 5) from drop size distribution data on the breakage of liquid drops in a manner very similar to that of Sathyagal et al (1995). Although their methodology of Narsimhan et al (1984) was similar to that of Sathyagal et al (1995), their solution to the inverse problem was vastly improved in the latter publication. 

FIGURE 6.1.4 Comparison of inverse problem solution with the actual cumulative size distribution for breakage fragments (from Sathyagal et a/., 1995). (Reprinted with permission from Elsevier Science.)... 

With the parameter jS determined as the value of inverse problem solution for Pg(u) at M = 1, the complete inversion of the problem is at hand. Figure 6.1.6 shows the breakage frequency function b x which is indistinguishable from the function used to simulate the data. 

Narsimhan et al. (1980) 1980 Similarity hypothesis Inverse problem Experimental DSD data for dilute system used to extract breakage frequency led to validation of similarity hypothesis. 

The Inverse Breakage Problem Determination of Breakage Functions 223... 

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