Hydrocyclone performance and grinding-circuit mass balance: 24-MMP-B5 Mineral Processing Design and Operations, May 2016, Q4

Question 4: Hydrocyclone performance and grinding-circuit mass balance (4/6)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Cyclone overflow (COF) 40% solids by weight; cyclone feed (CF) 60% solids by weight; cyclone underflow (CUF) 75% solids by weight, all in a closed grinding circuit where the underflow returns to the mill.

Find. (a) The three characteristics read from a corrected partition (performance) curve, its axes, the meaning of points w/x/y/z, and the three underflow discharge patterns. (b) The mechanism of ball-mill size control via the cyclone, plus the solids recovery to underflow and the circulating load ratio.

Fig. 2 — Idealized corrected partition (performance) curve for a hydrocyclone: partition to underflow vs. particle size. w = fine-particle bypass (water-split floor); x = corrected cut point d50c (50% partition size); y = the steep mid-slope region (sharpness of separation); z = the coarse-particle asymptote (essentially 100% reports to underflow).

Approach. (a) is descriptive, read directly off the standard corrected partition curve; (b) closes a two-product solids balance around the cyclone using the three given % solids values.

(a) A hydrocyclone performance (corrected partition) curve plots the partition/recovery of each size fraction to the underflow (vertical axis, 0-1 or 0-100%) against particle size (horizontal axis, on a log or linear scale). Three characteristics are read from it directly: (1) the cut point d50c — the size at which a particle has a 50:50 chance of reporting to either product, marking the effective separation size (labelled x in Fig. 2); (2) the sharpness of separation — the steepness of the curve through its mid-section (labelled y), a shallow slope meaning a poorly-selective cyclone that misplaces both fine and coarse material; and (3) the bypass fraction — the non-zero partition value at very fine sizes (labelled w), representing fines that are carried to underflow purely by the water split rather than by true classification, with the curve's coarse-size asymptote (labelled z) confirming that essentially all coarse particles do report to underflow. Operators observe three underflow discharge patterns at the apex: a rope discharge (dense, twisted, high-underflow-density condition — the normal, efficient operating state), a spray discharge (a wide-angle fan/umbrella spray indicating the apex is oversized or underflow density is too low, causing fines to short-circuit to underflow and lose classification efficiency), and an intermediate transitional discharge between the two, generally avoided as unstable.

Ball-mill particle-size control mechanism. The hydrocyclone classifies mill discharge by size: oversize (coarse, under-ground) material reports to the underflow and is returned to the mill for further grinding, while properly-sized material reports to the overflow and exits the circuit as final product. Because the cyclone's cut point x depends on feed density, pressure and apex/vortex-finder geometry, an operator controls the product size (P80) by adjusting cyclone feed density (via sump water addition) or feed pressure — a denser or higher-pressure feed pushes the cut point coarser, sending more material to overflow and coarsening the product; more sump dilution or lower pressure does the opposite. This is why the closed mill-cyclone loop is the primary size-control lever in most grinding circuits, rather than the mill itself.
Two-product solids balance (basis: cyclone feed F = 1). With $f_s=0.60$ (CF), $o_s=0.40$ (COF), $u_s=0.75$ (CUF), and $F=U+O$, $Ff_s=Uu_s+Oo_s$: $$U=\frac{f_s-o_s}{u_s-o_s}=\frac{0.60-0.40}{0.75-0.40}=0.5714 \qquad O=1-0.5714=0.4286$$
Solids recovery to underflow. $$\text{Recovery}=\frac{Uu_s}{Ff_s}\times100\%=\frac{0.5714\times0.75}{1\times0.60}\times100\%=\boxed{71.4\%}$$
Circulating load ratio. The underflow U recirculates to the mill; the overflow O is the circuit product: $$\text{C.L.}=\frac{U}{O}\times100\%=\frac{0.5714}{0.4286}\times100\%=\boxed{133.3\%}$$

Final Results — Question 4
Quantity	Value
Solids recovery to underflow	71.4%
Circulating load ratio	133.3%

Topic: Hydrocyclone classification — corrected partition curve and closed-circuit grinding mass balance.

Key relations: Two-product balance $F=U+O$, $Ff_s=Uu_s+Oo_s$; recovery $=Uu_s/(Ff_s)$; C.L. $=U/O\times100\%$.

Why this works: Percent-solids data alone is enough to close a mass balance around a splitter because water and solids are each independently conserved; solving the solids balance with the overall balance pins down the two unknown split fractions without needing the actual tonnage.

Common pitfall: Reporting a spray-discharge apex condition as "more efficient" because it looks like it removes more fines — in reality a spray discharge signals loss of classification control and excessive fines misplacement to underflow.

Source: Wills' Mineral Processing Technology, 8th ed., §9.3-9.5 (hydrocyclones, corrected efficiency curves, closed-circuit grinding).

Question 4 of 8: Hydrocyclone performance and grinding-circuit mass balance

Question 4: Hydrocyclone performance and grinding-circuit mass balance (4/6)