(216a) Some Fundamental Insights from Late Professor Leonard Austin’s Research on the PBM of Milling Processes

Professor Leonard Austin significantly contributed to the development of grinding–milling unit operation and its modeling in a similar manner to that of the development of other chemical engineering unit operations such as distillation, absorption, etc. Until his retirement in 1996, he was a professor in Fuel and Mineral Engineering at the Pennsylvania State University; he remained an active researcher–engineer until he passed away on Nov. 21, 2022. Interestingly, despite being a mineral engineering professor, his work in population balance modeling (PBM) as well as process development and scale-up of milling processes reflect the style of a first-class chemical engineer incorporating all fundamental approaches and principles of chemical engineering especially those that pertain to residence time distributions (RTDs) and reactor design–kinetics. In fact, his chemical engineering style of process analysis can be easily seen in his most celebrated review paper [1] and his book on ball milling [2], which became a classic in the field. Perhaps as a secondary point to the above, Prof. Austin was a visiting professor in chemical engineering departments, published in chemical engineering journals, and attended AIChE meetings. Hence, it is of no surprise that a session at the AIChE Annual Meeting honors his contributions to the development of comminution processes. It may not be an exaggeration to state that Prof. Austin was one of the most influential figures in the field of comminution processes. Any chemical engineer who modeled comminution processes used the PBM equations in the formalism of Prof. Austin, used his breakage distribution function, and/or used his direct–indirect methods for determining the PBM parameters. Obviously, this presentation cannot do justice to all of Prof. Austin’s monumental contributions; therefore, it is not intended to serve as a review. Instead, my main goal is to distill from his work the most important theoretical developments within the PBM framework and discuss the fundamental insights gained that are significant and relevant to this day.

First part of my talk is devoted to Prof. Austin’s most celebrated review paper on the treatment of grinding as a rate process [1], wherein wide variety of models developed until 1970 were shown to be inter-convertible versions of a single model, but with differences in formalism. This systematization of the modeling approaches for milling processes was and is of great importance because even to this day chemical, civil, mineral, and mechanical engineers appear to use PBMs for milling processes that mainly differ in their formalism, which makes communication and knowledge transfer between these disciplines less efficient. In other words, Austin’s review was the first attempt to unify various PBMs with differing formalisms. In essence, the PBMs for batch/plug-flow milling processes were categorized based on (i) the particle distribution (PSD): cumulative vs. density–frequency distributions, (ii) size-continuous vs. size-discrete forms, and (iii) time-continuous vs. time-discrete forms. Moreover, the size-discrete form of the PBM was presented in both the index notation and the matrix notation.

The second part of my presentation is dedicated to the estimation of the PBM parameters. By and large, all PBMs for batch/plug-flow continuous milling are based on two fundamental functions (kernels) [3,4]: a specific breakage rate parameter S_i and a breakage distribution parameter b_ij (or its cumulative counterpart B_ij). For a well-mixed batch ball mill or a plug-flow continuous mill at the steady state, the time (retention time)-continuous, size-discrete form of the PBM assuming first-order (linear) breakage is given as follows [5]:

dM_i /dt = –S_iM_i + ∑ b_ijS_jM_j with M_i(0) = M_i,ini (1)

In Eq. (1), i and j are size class indices that extend from size class 1, which contains the coarsest particles of size x₁, to size class N, which contains the finest particles of size x_N. t is milling time, and M_i is mass fraction in size class i. M_i,ini is the mass fraction of feed material in size class i. To estimate S_i and B_ij parameters, Prof. Austin developed various direct calculation methods including short duration, single-fraction milling tests and the BI, BII, and BIII methods with increasing accuracy [6,7]. In addition, he spearheaded the optimization-based back-calculation (indirect methods) of the parameters [8,9]. Pros/cons of these different approaches will be discussed in this talk.

The third part of my presentation is dedicated to the most elusive phenomenon in milling kinetics: non-linear breakage. Prof. Austin categorized this complicated phenomenon into two groups: “environment effects” and “material effects”. The latter was treated by considering first-order breakage of soft and hard components or fast-breaking and slowly-breaking components of a mixture [10,11]. On the other hand, in general, the environment effects were found to be more elusive and complex. During ball milling of cement, e.g., Austin and Bagga [12] noted the slowing down of the breakage of all particle sizes and attributed this behavior to cushioning action of the fines. A non-first-order PBM based on the concept of warped time was developed to account for the cushioning phenomenon. To the best knowledge of the author, the approach to such elusive environmental effects was through the development of time-dependent specific breakage rate functions in PBMs until a true non-linear PBM was first developed in 2005 [13,14]. In fact, the so-called Model D of this non-linear theory [14] recovers Austin’s approach as a limiting case (so-called uniform non-linear kinetics).

The last part of the talk is devoted to a brief discussion on recent advances in PBM parameter estimation [15–17] and the use of PBM–DEM (discrete element method) simulations [18,19] to gain mechanistic insights into the cushioning phenomenon. I demonstrate that DEM simulations shed light on the origin of such elusive breakage phenomenon like cushioning. Hence, the scientific journey into the realm of non-linear breakage, which was contributed significantly by Prof. Austin, continues. Overall, this talk highlights the key contributions of Prof. Austin’s research to the modeling of milling processes within the context of PBM and their relevance and importance to current research in particle breakage and milling.

References

[1] L.G. Austin, A review: introduction to the mathematical description of grinding as a rate process, Powder Technol. 5 (1971), 1-17.

[2] L.G. Austin, R.R. Klimpel, P.T. Luckie, Process Engineering of Size Reduction: Ball Mill, Society of Mining Engineers of the AIME: Littleton, USA, 1984.

[3] B. Epstein, The mathematical description of certain breakage mechanisms leading to the logarithmico-normal distribution, J. Franklin Inst. 244 (1947) 471-477.

[4] B. Epstein, Logarithmico-normal distribution in breakage of solids, Ind. Eng. Chem. 40 (1948) 2289-2291.

[5] K. Sedlatschek, L. Bass, Contribution to the theory of milling processes, Powder Metal. Bull. 6 (1953) 148-153.

[6] L.G. Austin, P.T. Luckie, Methods for determination of breakage distribution parameters, Powder Technol. 5 (1971/72) 215-222.

[7] L.G. Austin, V.K. Bhatia, Experimental methods for grinding studies in laboratory mills, Powder Technol. 5 (1971/72) 261-266.

[8] R.R. Klimpel, L.G. Austin, Determination of selection–for–breakage functions in the batch grinding equation by nonlinear optimization, Ind. Eng. Chem. Fund. 9 (1970) 230-237.

[9] R.R. Klimpel, L.G. Austin, The back-calculation of specific rates of breakage and non-normalized breakage distribution parameters from batch grinding data, Int. J. Miner. Process. 4 (1977) 7-32.

[10] L.G. Austin, T. Trimarchi, N.P. Weymont, An analysis of some cases of non-first-order breakage rates, Powder Technol. 17 (1977) 109–113.

[11] L.G. Austin, K. Shoji, D. Bell, Rate equations for non-linear breakage in mills due to material effects, Powder Technol. 31 (1982) 127–133.

[12] L.G. Austin, P. Bagga, An analysis of fine dry grinding in ball mills, Powder Technol. 28 (1981) 83–90.

[13] E. Bilgili, B. Scarlett, Population balance modeling of non-linear effects in milling processes, Powder Technol. 153 (2005) 59–71.

[14] E. Bilgili, J. Yepes, B. Scarlett, Formulation of a non-linear framework for population balance modeling of batch grinding: beyond first-order kinetics, Chem. Eng. Sci. 61 (2006) 33–44.

[15] M. Capece, E. Bilgili, R. Dave, Identification of the breakage rate and distribution parameters in a non-linear population balance model for batch milling, Powder Technol. 208 (2011) 195-204.

[16] E. Bilgili, M. Capece, A. Afolabi, Modeling of milling processes via DEM, PBM, and microhydrodynamics, in: P. Pandey, R. Bharadwaj (Eds.), Predictive Modeling of Pharmaceutical Unit Operations, Woodhead Publishing, Cambridge, 2017, pp. 159–203.

[17] N. Muanpaopong, R. Davé, E. Bilgili, A comparative analysis of steel and alumina balls in fine milling of cement clinker via PBM and DEM, Powder Technol. 421 (2023) 118454.

[18] M. Capece, R.N. Davé, E. Bilgili, A pseudo-coupled DEM–non-linear PBM approach for simulating the evolution of particle size during dry milling, Powder Technol. 323 (2018) 374-384.

[19] M. Capece, R.N. Davé, E. Bilgili, On the origin of non-linear breakage kinetics in dry milling, Powder Technol. 272 (2015) 189-203.