(392f) Two Separate Methodologies for Identification of Breakage Probability and Distribution Parameters From Nonlinear Breakage Events

Breakage of particles in some comminution systems such as roller mills and in particle bed breakage tests can be treated as a series of breakage events. The output particle size distribution can be algebraically obtained from the feed size distribution in these systems. Due to multi-particle interactions, breakage probability of particles of a given size is expected to be affected by the presence of all surrounding particles and thus by the whole mass density distribution function. In fact, recent work on particle bed breakage tests [1,2] clearly indicated the significance of these multi-particle interactions. On the other hand, the traditional population balance model (PBM) for particle breakage in a variety of mathematical forms is linear, and cannot account for these interactions.

Bilgili and co-workers [3,4] have recently presented a general phenomenological framework by which nonlinear effects emanating from multi-particle interactions are mathematically treated within the context of a nonlinear population balance model. The model decomposes the specific breakage rate into a size-dependent apparent specific rate function k and a population density dependent functional F. In other words, the birth and death terms (constitutive relations) in the population balance equation account for the multi-particle interactions phenomenologically. The population balance equations were derived for batch [3, 5] and continuous milling operations [6], and the numerical simulation results demonstrated the predictive capability of the novel nonlinear model in explaining the experimentally observed complex breakage behavior.

Bilgili [7] extended the model to account for nonlinear effects in event-based systems by deriving the population balance model in the small-time interval limit and defining a breakage probability. In the same study, a preliminary method for identification of breakage probability and distribution parameters was indicated. Here, we propose two separate methods for the identification of the parameters and solution of the inverse problem. If mono-sized feed breakage tests can be performed on each and every possible size-cut in a particle population, then exact solutions for the unknown linear model parameters can be obtained. Then, by performing breakage tests on binary, tertiary, or natural sized feed mixtures, one can determine the nonlinear functional parameters.

We also present a more elegant and cost-effective method for the identification problem. Instead of a significant number of breakage tests on both mono-sized and multi-sized feeds, we propose to perform only a few natural-sized feed experiments. Avoiding the separation of particles in many size classes and the use of more advanced and detailed particle size distribution characterization methodology, one can determine the PBM parameters more accurately. Moreover, the number of experiments needed to determine the model parameters is only a few. A fully numerical scheme that is based on the minimization of squared relative residuals between experimental mass fraction distribution and modeled distribution is developed in Matlab environment. A non-linear optimizer ?fmincon? minimizes the error while calling the analytical solution of the PBM for n-breakage events. Our numerical simulations using artificially generated data with random noise indicate the success of the proposed fully numerical scheme.

REFERENCES: 1. Baxter, J., Abu-Nahar, A., Tuzun, U. Understanding intra-mixture interactions in the breakage of dense particulate mixtures. Proc. 5th World Cong. Particle Technol., Paper No: 75e, Orlando, FL, Apr. 2006. 2. Baxter, J., Abu-Nahar, A., Tuzun, U. The breakage matrix approach to inadvertent particulate degradation: dealing with intra-mixture interactions, Powder Technol. 2004, 143-144: 174-178. 3. Bilgili, E., Scarlett, B. Population balance modeling of non-linear effects in milling processes. Powder Technol. 2005: 153, 59?71. 4. Bilgili, E. Yepes, J., Scarlett, B. Formulation of a non-linear framework for population balance modeling of batch grinding: beyond first-order kinetics. Chem. Eng. Sci. 2006, 61: 33?44. 5. Bilgili, E. On the consequences of non-first-order breakage kinetics in comminution processes: absence of self-similar size spectra, Part. Part. Syst. Char. 2007, 24: 12?17. 6. Bilgili, E., Scarlett, B. Numerical simulation of open-circuit continuous mills using a non-linear population balance framework: incorporation of non-first-order effects. Chem. Eng. Technol. 2005, 28: 153?159. 7. Bilgili, E. A unified approach to mathematical treatment of multi-particle interactions during size reduction: extending the nonlinear population balance model. AIChE Annu. Mtg., Paper No: 712a, Philadelphia, PA, Nov. 2008.