Hydrocyclone performance and grinding-circuit mass balance: 24-MMP-B5 Mineral Processing Design and Operations, Undated paper, 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, in a closed grinding circuit where the underflow returns to the mill.

Find. (a) The prime 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. 3 — Closed ball-mill/hydrocyclone grinding circuit: cyclone feed pump box feeds the cyclone; overflow (COF) reports to mineral separation, underflow (CUF) recycles through the mill and back to the pump box.

Fig. 4 — Idealized corrected partition (performance) curve for a hydrocyclone: recovery 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 recovery/partition of each size fraction to the underflow (vertical axis, 0-1 or 0-100%) against particle size (horizontal axis). The prime characteristics surveyed from it are: (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. 4); (2) the sharpness of separation — the steepness of the curve through its mid-section (labelled y), where a shallow slope means 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 carried to underflow purely by the water split rather than true classification, with the curve's coarse-size asymptote (labelled z) confirming essentially all coarse particles 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 too low, short-circuiting fines to underflow and losing classification efficiency), and a choked/plugged discharge (an overloaded apex rejecting solids in intermittent clumps, signalling the cyclone is hydraulically overloaded beyond its capacity).

Ball-mill particle-size control mechanism. The hydrocyclone classifies mill discharge by size: oversize (coarse, under-ground) material reports to the underflow and returns to the mill for further grinding, while properly-sized material reports to the overflow and exits the circuit as final product. Because the cut point x depends on feed density, pressure and apex/vortex-finder geometry, an operator controls 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 (coarsening the product), while more sump dilution or lower pressure does the opposite. This closed mill-cyclone loop is therefore 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 $F=U+O$, $Ff_s=Uu_s+Oo_s$: $$U=\frac{f_s-o_s}{u_s-o_s}=\frac{0.60-0.40}{0.75-0.40}=\boxed{0.5714}\qquad O=1-0.5714=0.4286$$
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 together 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)