(216g) Microhydrodynamic Scaling Analysis of a Vertical Wet Stirred Media Mill: Justifying PBM Parameters on Physical Grounds

Ore grades are decreasing due to the more complex form/mineralogy of the ore bodies over the years, which necessitates finer grind sizes to produce liberated minerals. According to various plant surveys [1,2], energy demand increases with product fineness exponentially. Therefore, selection of an appropriate grinding technology is important. Owing to their simple operation and design, conventional ball mills have been widely used, although they are known to be energy inefficient. The inefficiency of ball mills becomes especially obvious in <75 µm range [3]. Energy is imparted to the media by a rotating shell whose speed is in the range of 72%–80% of the critical speed, which limits the power input [4]. Wet stirred media milling (WSMM) has been used in the mining industry since 1980s; its use is on the rise due to higher demand for finer grinds [5]. In WSMM, a suspension of the ore particles is fed through a horizontal or vertical milling chamber which houses a rotor and beads upon filling. The high-speed rotation of the rotor induces turbulent flow and frequent collisions of the beads, which capture the particles in between and cause their compression–shear resulting in particle breakage. WSMM offers 30–40% energy saving compared to ball milling operated at the same size reduction ratio [6].

Modeling can help develop WSMM processes and provide quantitative insights into the impacts of process–design parameters [7]. The models include CFD–DEM [8], the stress intensity–stress number model [9], and the microhydrodynamic model [10,11]. Although they simulate the bead–bead collisions, particle capture efficiency, and their roles in WSMM with different levels of fidelity and complexity, they cannot simulate the evolution of the particle size distribution (PSD) in a transient process or predict the product PSD in a steady-state operation. To this end, population balance modeling (PBM) is used in which the particle size domain is discretized into N classes with sizes represented by x_i with i = 1, 2,...N [12]. The PBM for milling describes how fast particles of size x_i break and where their broken pieces end up in the size domain or what the sizes of the progeny particles are. These quantitative descriptions are respectively provided by two fundamental breakage functions: the specific breakage rate function S_i and the cumulative breakage distribution function B_ij.

Many challenges exist when engineers are confronted with the inverse problem, i.e., estimation of S_i and B_ij from experimental data. The “direct measurement methods” include the one-size fraction method [13] and BI, BII, and BIII methods [14], respectively, and treat experimental batch ball milling data. They entail separate milling of multiple narrowly-sized feed samples covering the size range of interest; hence, they are experimentally laborious and time-consuming [12]. To mitigate some of these issues, the “back-calculation method” [15] has been used for the WSMM process [16], wherein a local optimizer along with the analytical or numerical solution of the PBM is used to estimate S_i and B_ij parameters simultaneously. The optimizer minimizes the sum of squared residuals (SSR) between the model predicted PSD and the experimental PSD. However, a major challenge since the early applications of the back-calculation method is to find a unique set of the PBM parameters [12,15]. A local optimizer could get stuck at one of the local minima, leading to an erroneous set of parameters despite yielding a deceptively reasonable fit to the PSD data. The non-uniqueness of the solution is revealed when the set of initial guesses for the parameters to be estimated is varied, which leads to different parameter estimates [17]. Therefore, the use of a global optimization scheme in the back-calculation with PBMs is warranted.

Not only should an industrially relevant PBM describe and fit existing plant data, but also it should be able to predict the product PSD under different processing conditions for different feed PSDs. Moreover, its S_i and B_ij parameters must be sufficiently accurate and physically plausible to inform engineers about the breakage mechanisms and dependence of S_i on the process–design parameters. On the other hand, such a PBM could have up to a dozen S_i and B_ij parameters to be estimated, and even the use of a global optimizer on a sufficiently dense data set cannot guarantee the accuracy and the uniqueness of the estimated parameters. There are two possible ways to resolve this challenge: (i) reduce the number of parameters using a hybrid approach and/or (ii) use modeling approaches that are more mechanistic along with PBMs as well as physical plausibility arguments to justify the parameters estimated. In the hybrid approach (i), a direct measurement method is used first to estimate the B_ij parameters and then the so-obtained B_ij was input to the back-calculation method to estimate the S_i parameters solely [18]. This presentation focuses on the second approach (ii).

Using a microhydrodynamic scaling analysis, we aimed to justify the S_i parameters of a PBM, which was used to model an industrial-scale vertical wet stirred media mill. Feed and product particle size distributions (PSDs) of copper ore were measured at various steady-state conditions. Starting with the microhydrodynamic theory [11] and the definition of S_i for impact breakage [19], a simple power-law relation between S_i and the rotation rate of the rotor (ω), % ore content (w_p), and particle size (x_i) was derived. The impact of the suspension volumetric flow rate Q_s was accounted for via the mean residence of the PBM. The PBM incorporated 1-large–2-small-tanks model and the power-law specific breakage rate function S_i. Parameters of S_i and a non-normalized cumulative breakage function B_ij were estimated by a back-calculation method with global optimization. Fig. 1 shows the product PSDs in 18 different processing conditions fitted reasonably well by the PBM. Results suggest that higher rotor speed, lower solids loading, and use of steel rotor vs. rubber rotor and coarser beads led to faster breakage. The exponents revealed by the microhydrodynamic scaling analysis were within 17% of the exponents estimated by the back-calculation. Overall, we have demonstrated the use of the microhydrodynamic theory to justify the physical plausibility of the PBM parameters.

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Acknowledgement: This work was equally contributed by all authors.