Problems in Population Balances

1 The problem thus stated is akin to the familiar question in fluid mechanics of how Eulerian observations, that are more conveniently made, can be converted to Lagrangian information that is often more relevant and the desired quantity. This is because Eulerian observations are made at a fixed point in space whereas Lagrangian measurements require tracking a specific particle in motion. 

The modeling of a breakage process has been discussed in Section 3.2 of Chapter 3. We assume that no particle growth occurs and that aggregation 

There are examples in the literature of fitting parameters to single particle models in both aggregation and breakage processes until an experimentally measured equilibrium particle size distribution is closely matched by the solution to the population balance equation. The rationality of such a procedure is much in question, as it is clearly not sensitive to the time scales of breakage and aggregation. 

Furthermore, numerical regularization procedures, to be referred to subsequently, are required to restore well-posedness to the inversion problem. 

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 

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Wright, H. and Ramkrishna, D., 1992. Solutions of inverse problems in population balance aggregation kinetics. Computers and Chemical Engineering, 16(2), 1019-1030. 

Wright, H., and Ramkrishna, D., Solutions of Inverse Problems in Population Balances, I Aggregation Kinetics, Computers Chem. Eng. 16 (12) 1019 (1992). 

We conclude this section with the observation that many problems in population balance may feature particles distributed only according to their size or some other scalar variable. We shall exploit the simplicity of such problems to demonstrate concepts applicable to the more general problems of population balance. 

Sathyagal, A. N., D. Ramkrishna and G. Narsimhan, Solution of Inverse Problems in Population Balances — II. Particle Break-up, Comp. Chem. Eng. 19, 437-451 (1995). 

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