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281606 Impact of Multi-Particle Interactions On the Attainment of A Self-Similar Particle Size Distribution During Dense-Phase Dry Milling

In dense-phase comminution systems, particle concentration is high and particles with a multitude of sizes undergo mechanical

*multi-particle interactions*through collisions or enduring contacts. Austin and Bagga (1981) determined that presence of fines has a cushioning effect during a dry ball milling process causing the specific rate of breakage, particularly of the coarsest particles, to decrease. Gutsche and Fuerstenau (1999) also observed the retardation in the breakage of coarse particles caused by the addition of fines in particle bed compression experiments. They explained that fine particles distribute the force-flux over the surface of coarse particles and stabilize the particle preventing breakage. They also showed that the stabilization of a coarse particle is a function of the size and the mass fraction of the fines. In other words, multi-particle mechanical interactions among particles of different sizes at the particle ensemble scale lead to a population-dependent breakage probability. Other studies (Austin et al., 1981; Meloy and Williams, 1992; Rajamani and Guo, 1992) led to similar findings for both dry and wet milling processes. Traditional time-continuous linear population balance models (TCL-PBMs) cannot explain the impact of the aforementioned multi-particle interactions.

Time-continuous non-linear population balance models (TCNL-PBMs), which have been recently introduced due to the limitations of the TCL-PBMs, describe the impact of the multi-particle interactions and thus are capable of predicting many types of complex non-first-order breakage kinetics during size reduction operations (Bilgili and Scarlett, 2005). Bilgili et al. (2006) decomposed the specific breakage rate function *S _{i}* into a first-order specific breakage rate function

*k*and a population dependent functional

_{i}*F*[ ].

_{i}**The functional**

*F*[ ] describes different types of non-linear (non-first-order) breakage kinetics and explicitly accounts for multi-particle interactions. TCNL-PBM recovers the TCL-PBM in the limit of weak multi-particle interactions, i.e.,

_{i}*F*[ ]→1, and the linear time-variant PBMs in the limit of uniform kinetics, i.e.,

_{i}*F*[ ]→

_{i}*f*(

*t*), where

*f*(

*t*) is a time-dependent function (Bilgili et al., 2006). Numerical simulations using the TCNL-PBM predicted the experimentally observed complex and non-intuitive breakage behavior such as cushioning effect, crossing-effect, and acceleration effect in batch and continuous milling operations.

In this study, we investigate the evolution of the particle size distribution toward a self-similar distribution during dry milling via computer simulations. Three models were considered in the simulations: a TCL-PBM, Model B (TCNL-PBM with non-uniform kinetics), and Model D (TCNL-PBM with uniform kinetics). A power-law function for the first-order specific breakage rate and difference-similar (normalized) form of the breakage distribution function were assumed for all three models. Evolution of the particle size distribution during batch dry milling of quartz in a tumbling ball mill (Austin et al., 1990) was fitted to determine the parameters adopting the approach in Capece et al. (2011). Goodness-of-fit and statistical significance of the parameters estimated were evaluated to discriminate the models. The experimental data exhibited strong slowing-down effect due to the cushioning action of the finer particles on the coarser ones. Both Model B and Model D, being non-linear, fitted the data much better than the TCL-PBM. Having statistically significant parameters, the non-linear models exhibited a similar fit. It was not possible to differentiate Model D (uniform kinetics) from Model B (non-uniform kinetics) based on goodness-of-fit and statistical significance.

In the second part of this study, we performed simulations with different initial particle size distributions (Gaussian distributions with different mean and standard deviation and bimodal distributions) a) to investigate their effects on the attainment of a self-similar size distribution and b) to assess if Model B and Model D can be discriminated. The model parameters were kept the same as those obtained from the data fitting, as mentioned above. The TCL-PBM solution attained a self-similar particle size distribution very fast, whereas Models B and D solutions tended towards such size distribution at prolonged milling times. In other words, the non-linear effects (slowing-down phenomenon due to the multi-particle interactions) led to a delay of the attainment of a self-similar size distribution. In fact, the delay was more pronounced in the case of Model B than in Model D. Surprisingly, both Model B and Model D simulations show similar predictions for some initial size distributions, thus making the discrimination of these two non-linear models extremely challenging. Our simulations suggest that these non-linear models can only be discriminated by fitting the evolution of the particle size distribution for multiple initial (feed) particle size distributions simultaneously in the parameter estimation stage.

**References:**

Austin, L.G., Bagga, P., 1981. An analysis of fine dry grinding in ball mills. Powder Technology 28, 83–90.

Austin, L.G., Shah, J., Wang, J., Gallagher, E., Luckie, P.T., 1981. An analysis of ball-and-race milling. Part I. The Hardgrove mill. Powder Technology 29, 263–275.

Austin, L.G., Yekeler, M., Dumm, T., Hogg, R., 1990. Kinetics and shape factors of ultrafine dry grinding in a laboratory tumbling ball mill. Particle & Particle Systems Characterization 7, 242–247.

Bilgili, E., Scarlett, B., 2005. Population balance modeling of non-linear effects in milling processes. Powder Technology 153, 59–71.

Bilgili, E., Yepes, J., Scarlett, B., 2006. Formulation of a non-linear framework for population balance modeling of batch grinding: beyond first-order kinetics. Chemical Engineering Science 61, 33–44.

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

Gutsche, O., Fuerstenau, D.W., 1999. Fracture kinetics of particle bed comminution—ramifications for fines production and mill optimization. Powder Technology 105, 113–118.

Meloy, T.P., Williams, M.C., 1992. Problems in population balance modeling of wet grinding. Powder Technology 71, 273–279.

Rajamani, R.K., Guo, D., 1992. Acceleration and deceleration of breakage rates in wet ball mills. International Journal of Mineral Processing 34, 103–118.

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