Machine

The machine is responsible for production of bubble swarms and effecting bubble–particle collision. Recent work has related the machine “factor” to the BSAF. BSAF (S_b) is the rate at which bubble surface area moves through the cell per unit of cell cross-sectional area with units m² m⁻² s⁻¹ or simply s⁻¹. Introduced in Section 12.5, to remind, it is estimated from the measurements of superficial gas velocity (the volumetric gas flowrate per unit area of cell, J_g), and the bubble size, usually taken as the Sauter mean diameter (D₃₂):

$S_{\mathrm{b}}=\frac{6J_{\mathrm{g}}}{D_{32}}$ (12.7)

(12.7)

Since J_g is typically quoted in cm s⁻¹ D₃₂ must be in cm. To give a sense of magnitude, at a typical value of J_g=1 cm s⁻¹ and D₃₂=1 mm (0.1 cm) then S_b=60 s⁻¹.

Both J_g and D₃₂ are measurable using suitable probes (see Section 12.14.2). The S_b can also be predicted using a correlation developed by Gorain et al. (1999).

Gorain et al. (1997) and Alexander et al. (2000) showed for shallow froth depths that the first order rate constant was linearly related to the BSAF. In addition, this relationship was shown to be independent of cell size and operating parameters. This independence is illustrated in Figure 12.33, which shows that the relationship measured in a pilot scale 60 L cell was essentially identical to that measured in a parallel 100 m³ cell. Letting the proportionality constant be P, the pulp zone rate constant (k_p) can be expressed as follows:

$k_{\mathrm{p}}=P{S}_{\mathrm{b}}$ (12.29)

(12.29)

There is support for Eq. (12.29) taking a first principles approach to bubble–particle interaction (Finch, 1998), but the relationship is largely empirical.

Particles

The relationship expressed in Figure 12.33 introduces a new engineering measure of floatability, the proportionality constant P, a dimensionless quantity if k_p and S_b are in expressed in the same reciprocal time units (e.g., s⁻¹ as in Figure 12.33). The value of P is a reflection of particle properties: size, composition (liberation), and hydrophobicity. While treated as a constant, P is likely an inverse function of bubble size (Yoon, 1993; Hernandez-Aguilar, 2011); nevertheless the model (Eq. (12.29)) represents an advance over using the rate constant alone in analyzing flotation systems. Based on these findings, flotation performance can be considered to arise from the interaction of a stream property—the particle floatability (P)—with parameters that characterize the operating conditions of the pulp (S_b).

Making the analogy of the bubble/particle reaction to a chemical reaction means the derived kinetic model applies to the pulp (or slurry) zone, not to the overall cell, which includes froth.

Froth

Equation (2.29) can be modified by introducing a froth zone recovery factor. Figure 12.34 shows the concept of two interacting zones, pulp and froth, with dropback (1 − R_f) considered to be combined with the fresh feed to the cell (Finch and Dobby, 1990). Letting the feed (dry) solids mass flowrate to the pulp zone be X and introducing R_p as pulp zone recovery and R_f as froth zone recovery, the resulting solids flowrates are as indicated on the figure. A mass balance across the dashed line block gives the incoming feed F and thus the overall flotation cell recovery R_fc as:

$F=X{R}_{\mathrm{p}}{R}_{\mathrm{f}}+X(1-R_{\mathrm{f}})$ (12.30)

(12.30)

$R_{\mathrm{fc}}=\frac{R_{\mathrm{p}}{R}_{\mathrm{f}}}{R_{\mathrm{p}}{R}_{\mathrm{f}}+(1-R_{\mathrm{f}})}$ (12.31)

(12.31)

Substituting Eq. (12.28) into Eq. (12.31), the overall flotation rate constant (k_fc) is given by:

$k_{\mathrm{fc}}=k_{\mathrm{p}}{R}_{\mathrm{f}}$ (12.32)

(12.32)

and substituting Eq. (12.29) into Eq. (12.32) gives the expression relating the overall flotation cell rate constant to the three contributing factors: particles (ore), machine, and froth.

$k_{\mathrm{fc}}=P{S}_{\mathrm{b}}{R}_{\mathrm{f}}$ (12.33)

(12.33)

The fully mixed first-order kinetic model then becomes:

$R_{\mathrm{fc}}=\frac{P{S}_{\mathrm{b}}{R}_{\mathrm{f}}\tau }{1+P{S}_{\mathrm{b}}{R}_{\mathrm{f}}\tau }$ (12.34)

(12.34)

Techniques to quantify the froth recovery factor continue to be developed (e.g., Seaman et al., 2004; Yianatos et al., 2008; Rahman et al., 2012). However, most methods are either intrusive in the froth or subject to assumptions (e.g., no entrainment). A method initially developed for batch flotation cells by Feteris et al. (1987) was later modified by Vera et al. (1999) to determine R_f directly on industrial flotation cells. In this approach, froth zone recovery (R_f) is estimated by determining the cell recovery at a measured froth depth (hence giving the overall cell first order rate constant, k_fc) relative to the cell recovery at no froth depth (and hence giving the pulp (collection) zone rate constant, k_p). The cell recovery at no-froth depth cannot be measured directly, but can be estimated by extrapolation of results obtained at four or more froth depths (Amelunxen and Runge, 2014).

Froth zone recovery is related to froth stability. As was the case in characterizing frothers’ froth stabilizing function (Section 12.5.6), the problem is one of definition and measurement. The notion is that there is an optimum froth stability that will give the best performance from a flotation cell, for example, maximum recovery at given grade, or maximum grade at given recovery (Farrokhpay, 2011). A promising measure of froth stability is air recovery, the fraction of the air delivered to the cell that overflows as unburst bubbles. Air recovery α can be calculated using the following expression:

$\alpha =\frac{\xi \cdot v_{\mathrm{f}}\cdot h\cdot w}{Q_{\mathrm{g}}}$ (12.35)

(12.35)

where Q_g is the total air flowrate into the cell, ζ the GH in the froth zone (usually assumed to be 1), v_f the overflowing froth velocity, h the froth height over the lip, and w the length of the lip where froth is overflowing. Froth velocity is measured by a camera monitoring the surface of the froth as it overflows using image processing software. A concern could be raised regarding propagation of error involved in calculation from several independent measurements (what is a typical error, e.g., 95% confidence interval, on α?) plus the fact that some air may leave with the cell tailings. Nevertheless, air recovery has been successfully used. By manipulating air rate to cells to operate at maximum (or peak) air recovery (PAR) improved bank performance was obtained, marking a new approach in flotation circuit control and optimization (Hadler and Cilliers, 2009; Hadler et al., 2010).

Fine Particles

Particle recovery by true flotation involves collision, attachment and formation of stable attachment to bubbles, often expressed as probabilities (or efficiencies). Collision probability is governed by the physics of the process (or “hydrodynamics”), while attachment probability and stability of attachment include a chemical component (particle hydrophobicity). The decline in recovery at the fine end is usually ascribed to low collision probability, fine particles having insufficient inertia to cross the water streamlines around the coursing bubble. Note this does not say there is a lower particle size limit to flotation, only that the rate of flotation will decrease as particle size decreases. Figure 12.36 reinforces this conclusion, showing the fines have a low but finite rate constant. (Pease et al. (2006) make the point quite forcefully that there is not a lower size limit to flotation, just slowing kinetics.)

The collision-limited explanation of low recovery of fines implies that fine particles will be strongly influenced by flotation time, which is supported in Figure 12.35, and perhaps less affected by reagent additions that alter hydrophobicity, which can be the case (Trahar, 1981; Senior et al., 1994). Chemistry is not completely absent of course: fines are subject to surface oxidation and other contamination effects (e.g., adsorption of metal ions), and fines can cause aggregation (slime coating), issues that do respond to chemistry (Grano et al., 2014).

Increasing the rate of flotation of fine particles has obvious practical benefits. The requirement is to increase exposure to bubbles in order to provide more collision events and bubble surface area to transport collected particles. In an individual cell, increasing air rate and decreasing bubble size appear to offer means to achieve this increased exposure, the combination increasing S_b (Eq. (12.7)), which predicts an increase in rate constant (and the effect may be even greater than Eq. (12.7) predicts if P is inversely related to bubble size (Yoon, 1993)). Improved fine Pt-mineral recovery by frother addition to decrease bubble size is an example of successful implementation of this approach (Hernandez-Aguilar et al., 2006).

Increasing air or decreasing bubble size, however, also increases GH and there is this noted limit, about 15%, before a cell “boils.” The options to increase kinetics using air rate and bubble size in a cell are, therefore, limited. The practical option to increasing exposure to bubbles is to float for longer. In the laboratory the effect of time on increasing fine particle recovery is easy to demonstrate (Figure 12.35); in the plant, surveying cells cumulatively down a flotation bank will show the impact of increasing number of cells (i.e., time) on increasing recovery of fines (Trahar, 1981; Grano et al., 2014). The installed flotation capacity is often dictated by the slow floating fine particles, including the use of a scavenger stage to provide increased flotation time.

Rather than relying on increased exposure, the energy of interaction with the bubble could be increased to “force” the particles across the water streamlines to intercept the bubble. The positive effect of increasing energy input on fines flotation is known, which recent work using a specially designed cell to give uniform distribution of energy throughout the pulp has reinforced (Safari et al., 2013). Increased energy input can be achieved by increasing impeller speed in mechanical cells; an option sometimes exercised to increase fines recovery is to increase impeller speed down a bank of cells. The reactor–separator flotation cells aim to inject this energy more directly to the collection of fine particles, as noted in Eq. (12.36).

Coarse Particles

At the coarse end of the particle size spectrum, the decline in recovery probably includes a contribution from decreased liberation, but in terms of bubble–particle interaction events is usually attributed to decreased stability of attachment, that is, increased probability of detachment (Jameson, 2012). Dissipation of energy as fluid eddies causes the bubble–particle aggregate, such as depicted in Figure 12.1, to spin and the centrifugal force can dislodge the particle. Another factor is higher froth dropback of coarse particles (Finch and Dobby, 1990; Rahman et al., 2012). Detachment of the particle and froth dropback introduce a chemical aspect: addition of collector to increase hydrophobicity (increase contact angle in Figure 12.2) should increase stability of the bubble–particle aggregate in the pulp zone and particle retention in the froth zone and thus increase recovery. The positive impact of chemistry on the recovery of coarse particles is seen in the recovery–size data of Trahar (1981) and Senior et al. (1994). Loss of free coarse particles to tailings may indicate that altered chemistry is required.

In contrast to fine particles, high energy input is evidently detrimental to coarse particle recovery, as Safari et al. (2013) illustrate. To accommodate the different needs of fine and coarse particles some strategies have been implemented. Distributing collector along the bank is one (Bazin and Proulx, 2001). Low collector concentration at the head of the bank is sufficient for the fast floating intermediate-size particles and upon their removal further collector dosage down the bank can target recovering the slower floating coarse particles. Classifying feed into fine and coarse fractions is often discussed but there are few examples. At the Mt Keith operation the feed was first split into two size fractions then into three (Senior and Thomas, 2005), and the improved Ni recovery paid for the project within a month. The authors emphasized the benefits of understanding particle size effects. New cells combining teeter bed (hindered settling) and flotation principles are aimed expressly at enhancing coarse particle recovery (see Section 12.13.4).

The trend in the recovery vs. size shown in Figure 12.35 is qualitatively explained by the product of collision probability (that increases as particle size increases) and detachment probability (that decreases as particle size increases).

Liberation

Recovery-by-size-by-liberation data are not common, a situation set to change with the increasing spread of automated mineralogy technology (Chapter 17). A set of such data is given by Welsby et al. (2010), where liberation is considered to be surface liberation. Derived from their data, Figure 12.38(a) shows rate constant as a function of size and liberation. Figure 12.38(b) replots the data as rate constant relative to the maximum rate constant, k/k_max showing there is consistency to the rate constant–size relationship independent of liberation. Jameson (2012) noted this consistency in the data, showing the rate constant of liberation class k_x relative to the rate constant of fully liberated mineral, k_lib (L=k_x/k_lib), was a unique function of liberation (surface exposure, x) independent of particle size, that is, L=fn(x), introduced as a liberation function.

In predicting the rate constant for locked (composite) particles, a first approximation is that the rate constant of the liberation class k_l is a weighted average (Nesset, 2011); for example, for a particle with three surface exposed minerals, A, B, and C the k_l is given by:

$k_{\mathrm{l}}=x_{\mathrm{A}}{k}_{\mathrm{A}}+x_{\mathrm{B}}{k}_{\mathrm{B}}+x_{\mathrm{C}}{k}_{\mathrm{C}}$ (12.39)

(12.39)

where x is fractional surface exposure and k refers to rate constant of liberated mineral. From Eq. (12.39), a refinement of the liberation function might be to consider locked particle type, binary locked A, B particles as distinct from binary locked A, C particles, etc.

Recovery-size-liberation data offer the promise to interface comminution and flotation models (and other separation process models). The comminution step produces the size-liberation spectrum and the associated recovery (or rate constant) data for a separation unit can be used to predict the outcome (grade and recovery) from that unit. The common simplification is to assume that recovery–size data are independent of the size distribution of the feed to the separation unit and how the size distribution was produced. This simplified approach to interfacing comminution with flotation has been used over the years (Ramirez-Castro and Finch, 1980; Bazin et al., 1994; Bradshaw and Vos, 2013). A related assumption is that the size-liberation data are also independent of how the size distribution is generated. With the consistency evident in the rate constant-size-liberation relationship (Figure 12.38(b)), a way to include liberation in interfacing comminution and flotation may be forthcoming.

Role of Cleaner

The rougher stage contributes the initial (and often large) entrainment and the cleaners play an important role in reducing it. Figure 12.40 illustrates an example of the cleaner’s effect on recovery-by-size for a floatable mineral (sphalerite) and entrained mineral (NSG). The rougher inputs significant entrained fine (<10 μm) NSG, which the combination of cleaner stages (or cleaner block) rejects (and at the same time the scavenger increases fine sphalerite recovery due to increased retention time, as discussed above). Cleaners do not fully eliminate entrainment; even after several stages of flotation there is evidence that operational changes can still be effective in reducing its impact (Cooper et al., 2004).

This role of cleaners in rejecting entrainment is one argument against by-passing products direct to concentrate. Candidate products include concentrates from the first cell in the rougher bank, a unit cell ahead of the roughers or a flash flotation stage inside the grinding circuit. These products might be “high grade” (e.g., exceeding final concentrate grade) but can still carry substantial entrained (−50 μm) material that would be easily rejected in a cleaner stage with little loss of value recovery. Not cleaning these products means the remaining feed to the cleaners has to be upgraded sufficiently to compensate, which is not always achievable.

Pulp Dilution

Increasing pulp dilution reduces water recovery (Johnson and Munro, 2002). It may seem counter-intuitive, but recall that water transport is via the bubbles. Diluting the feed to a flotation stage means increasing the feed rate of water, but since the bubbling rate is not changed (S_b is unchanged) then the rate of water carried to the concentrate does not change, at least to a first approximation. Since water recovery is the ratio of the concentrate water rate to feed water rate, the recovery goes down. Not necessarily practical in the roughing stage because of the volumes involved, pulp dilution to cleaner stages is widely practiced.

An illustration of the merit of cleaner feed dilution is described by Shannon et al. (1993). In that example dilution enabled the operation to take full advantage of increased liberation through regrinding by controlling entrainment of the additional fines produced. A reassignment of cleaner cell capacity retained the necessary retention time. Retrofitting pulp dilution while retaining the required retention time (e.g., by reassignment of cells) may not always be feasible and it is best to plan pulp dilution at the plant design stage.

Particles tend to stabilize froths and increase water recovery. This is one reason why flotation usually aims to float the mineral in least amount in the feed, to reduce entrainment (and entrapment) of unwanted minerals. Commonly the mineral in least amount is the valuable mineral and is floated, but in some cases such as processing iron ores, the gangue (silica) is in lesser amount and reverse flotation is practiced. In cleaner stages the floatable mineral becomes the majority and sometimes reverse flotation, for example, pyrite from base metal concentrates, is worth considering.

Air Rate

Knowing water transport is via bubbles introduces manipulating air rate to control entrainment. Cooper et al. (2004) found by lowering air rate in the first cells of a Zn-cleaner bank (and increasing air rate down the bank to maintain Zn recovery) improved the grade of bank concentrate. The improvement was attributed to decreased entrainment of NSG in the first cells due to the lower air rate.

Froth Depth and Wash Water

Increasing froth depth is a direct way to reduce entrainment and manipulation of froth depth is a common grade control strategy. Deep froths with wash water addition to control entrainment was probably the prime advantage of flotation columns when these units reemerged in the 1980s (Finch and Dobby, 1990). Wash water is added into the froth to replace the water coming from the pulp and thus eliminates the entrained particles coming with the pulp water. By replacing the water draining from the froth under gravity, the addition of wash water enables deep froths to be built. Entrainment control using wash water is successful on mechanical cells as well (Kaya et al., 1990).

Since reducing water consumption is a concern in most locations, wash water use is under some scrutiny. How to introduce wash water to minimize use while achieving the benefits remains a challenge (Neethling et al., 2006). Most operations appear to favor introducing wash water from over-head drip pans rather than submerged devices in order to facilitate visual checks. An example wash water addition system is shown in Section 12.13.2.

Recovery

Introduced in Section 12.9, recovery in a single cell has two mathematical extremes depending on material transport, namely plug flow (Eq. (12.27)) and fully mixed flow (Eq. (12.28)).

For mechanical cells and flotation columns the fully mixed approximation is usually adopted. Figure 12.41 compares recovery as a function of kτ for the plug flow and fully mixed cases: it is evident, since k is the same for both, that more time is needed in a fully mixed cell to achieve the same recovery as under plug flow. For example, for a target recovery of 90% (0.9) the fully mixed kτ is 9 (solving Eq. (12.28), see Figure 12.41) and for plug flow kt is 2.3 (Eq. (12.27)). This means retention time is 9/2.3 (i.e., 3.9) times longer in the fully mixed case compared to plug flow: for equivalent recovery this translates to 3.9 times the cell volume being required in the fully mixed case compared to plug flow.

This difference in recovery between fully mixed and plug flow transport is well understood, one consequence is the need to scale up lab determined flotation time (i.e., equivalent to plug flow) to the continuous (industrial) case. A common scale up factor is 2.5 to estimate plant flotation time in sizing cells (Wood, 2002). (Increasing flotation time by 2.5 is equivalent to scaling down the rate constant by 2.5, noting that it is always the product kτ in the formulae.) Less realized perhaps is the impact of mixing degree on separation (selectivity) of floatable minerals.

Selectivity

To illustrate the impact of mixing on separation, consider two minerals A and B with rate constant k_A=1 min⁻¹ and k_B=0.1 min⁻¹ and take as the measure of efficiency the difference in recovery, that is, SE=R_A – R_B (Eq. (1.2)). Solving Eqs. (12.27) and (12.28) for these k values, we find that SE in the fully mixed case is 43% (0.43) and in the plug flow case SE is 69% (0.69) (Table 12.6). The lower selectivity in the fully mixed case can be seen in Figure 12.41; the slope is the rate of change in R with kτ (i.e., δR/δkτ) and it is evident that slope is higher for the plug flow case, meaning greater selectivity compared to the fully mixed case. (A convenient reference point is to compare δR/δkτ at R=0.5.) In summary, therefore, not only is more time (i.e., more cell volume) required in the fully mixed case to reach the same recovery as in plug flow but the SE is also less.

The analysis of selectivity can be addressed by writing Eq. (12.28) for both A and B and combining to eliminate τ (which is clearly the same for both minerals) to express R_B as a function of R_A. For the fully mixed case the result is:

$R_{\mathrm{B}}=\frac{1}{1+\left(\frac{k_{\mathrm{A}}}{k_{\mathrm{B}}}\right)\left(\frac{1-R_{\mathrm{A}}}{R_{\mathrm{A}}}\right)}$ (12.40)

(12.40)

Introducing S as the relative rate constant (k_A/k_B) Eq. (12.40) becomes:

$R_{\mathrm{B}}=\frac{1}{1+S\left(\frac{1-R_{\mathrm{A}}}{R_{\mathrm{A}}}\right)}$ (12.41)

(12.41)

Figure 12.42 shows the trend in R_B with R_A for S varying from “difficult” separation (S=2) to “easy” separation (S=10, equivalent to the situation above where k_A=1 and k_B=0.1). An example of a difficult separation is sphalerite from pyrite, where Cooper et al. (2004) reported S ~ 2–3. The figure shows the increasing rate of recovery of B with respect to A as recovery of A is increased.

Instead of expressing S as a relative rate constant, we could express in terms of relative floatability by expanding S in terms of the k–S_b relationship (Eq. (12.33)):

$S=\frac{k_{\mathrm{A}}}{k_{\mathrm{B}}}=\frac{P_{\mathrm{A}}\cdot S_{\mathrm{b}}\cdot R_{\mathrm{fA}}}{P_{\mathrm{B}}\cdot S_{\mathrm{b}}\cdot R_{\mathrm{fB}}}$ (12.42)

(12.42)

Since S_b is the same for both minerals and there is evidence that R_fA=R_fB for floatable minerals (Vera et al., 1999) then S ~ P_A/P_B.

Using SE and relative floatability, S, the following optimization problem can be addressed: to maximize SE (maximize selectivity) at target recovery. This optimization problem can be stated as follows:

Maximize

$\mathrm{SE}=R_{\mathrm{A}}-R_{\mathrm{B}}$ (12.43)

(12.43)

by searching on R_A subject to

$R_{\mathrm{A}}=R_{\mathrm{Atarget}}\mathrm{performance\; constraint}$ (12.44)

(12.44)

and

$R_{\mathrm{B}}=\frac{1}{1+S\left(\frac{1-R_{\mathrm{A}}}{R_{\mathrm{A}}}\right)}\mathrm{process\; constraint}$ (12.41)

(12.41)

Maximizing SE is equivalent to minimizing R_B at the target R_A or maximizing grade (specifically grade of A relative to B).

Applying to a single cell it is evident that the optimization problem cannot be solved: in a single cell it is not possible to satisfy two constraints, maximum SE and target R_A (Maldonado et al., 2011).

Three limitations of the single fully mixed cell are therefore identified: the need for increased flotation residence time (i.e., cell volume) to attain the same recovery as in plug flow; the lower selectivity between floatable minerals compared to plug flow; and the inability to maximize SE at a target recovery. These limitations are addressed by placing cells in series; that is, to form a bank (or line or row).

Recovery

The calculation of bank recovery from individual cell recoveries is illustrated in Figure 12.43. Taking mass feed rate to the bank of unity (F=1) as a basis, the mass flows around each cell are calculated based on the cell recovery (relative to the feed to that cell). In Figure 12.43(a) the mass flows are shown for three cells; and in Figure 12.43(b) the general solution for N cells is given for the case where all cell recoveries are equal, R_i. The bank (total) recovery R_bk is given by the summation of the individual cell mass flows (since F=1, the mass flows are equivalent to fractional recoveries). In case (b) where all cells have equal recovery (R_i) the bank recovery can be derived directly from a mass balance around the dashed block:

$R_{\mathrm{bk}}=1–{(1–R_i)}^N$ (12.45)

(12.45)

Assuming equal cell recovery might seem a mathematical contrivance to provide a simple solution, but a justification for equal recoveries will be given when selectivity is considered. By substituting Eq. (12.28) into Eq. (12.45) and, noting that τ_i=T/N where T is the total bank retention time, the following is derived:

$R_{\mathrm{bk}}=1-{\left(\frac{N}{N+kT}\right)}^N$ (12.46)

(12.46)

Figure 12.44 illustrates the impact of N: increasing N from 1 to 4 gives a significant increase in recovery for a given kT; in the limit as N → ∞, Eq. (12.46) becomes the plug flow solution, R_bk=1 – exp(−kT). Recovery approaches the plug flow result as the number of cells is increased because transport approaches plug flow (Figure 12.45).

Banks clearly give an advantage in recovery over one large cell of equivalent total volume. A question that arises is: how many cells should the bank comprise?

The analysis above implicitly assumed the individual cells are isolated and there is no short circuiting or back mixing between cells. The modern circuit of tank cells approaches the assumption: in that case Figure 12.44 suggests a bank of four or five cells is sufficient. Some care in this interpretation is required, however. Eq. (12.46) derives from Eq. (12.45), where all cells have equal recovery, making this the basis of the calculations. Significant deviation from equal recovery invalidates the solution, potentially making the four to five cell bank too short. As a resolution, six to eight cells to a bank may offer a practical compromise (Wood, 2002). A caveat is that in practice cells are often paired in a bank and act as one unit and this may represent an effective reduction in number of cells in the bank. (Note that in some plants “bank” refers to this pairing of cells, the whole being a row or line.) With older trough style banks of cells the impact of short circuiting between cells was countered by having long banks, up to 20 or more cells (Wood, 2002).

Figure 12.44 is a general solution independent of the magnitude of k: for example, if k is low this is compensated by selecting larger cells to increase T and maintain the kT value. In practice, longer banks (more cells) are usually recommended if k is considered low (Wood, 2002).

Short banks are encouraged by the increase in the size of mechanical cells, which over the past 50 years has gone from about 6 to 600 m³ (see Section 12.13.1). The advantage of large cells in reducing the number required and thus reducing capital and operating costs is an obvious incentive. To retain the recovery advantage of banks with large cells it is still necessary to keep a minimum number of cells in the bank, at least four, and preferably six to eight. The cost advantage of large cells is retained by reducing the number of parallel banks needed to reach plant throughput.

In constructing Figure 12.44 it was assumed that the retention time in each cell was the same, that is, each cell was the same volume. Maldonado et al. (2012) showed that for a given total bank capacity, cells of the same volume give the highest recovery; that is, equal size cells make best use of the installed volume. At the design stage there is little incentive to mix cell sizes, but there is less constraint in a plant expansion when a larger cell (usually at the head of the bank) can give the added volume perhaps more cheaply.

Selectivity

Gaining recovery for a given installed cell volume is one impact of the bank over a single cell. There is also a gain in separation between true floating minerals. Applying the optimization problem to a bank of cells, maximizing SE is solved by searching on R_Ai subject to the performance constraint (Eq. (12.44)) and the process constraint (Eq. (12.41)). Maldonado et al. (2011) illustrated the solution for two- and three-cell banks, which are amenable to trial-and-error solution, and went on to solve the general problem of an N-cell bank. The result is that the maximum in bank SE (SE_{bk, max}) occurs when all cells have equal recovery, referred to as a balanced or flat bank recovery profile.

The physical interpretation derives from Eq. (12.41) and Figure 12.42. The relationship between R_B and R_A means that a cell recovering more A than its neighbor incurs a penalty of incremental B recovery that is not offset by another cell under-recovering A: if no cell should recover more than its neighbors then all cells must have the same recovery. There is evidence that this flat recovery profile does offer a metallurgical advantage over other profiles (Maldonado et al., 2012).

Figure 12.46 shows the impact of number of cells in the bank on SE: SE_bk increases and levels off at about N=7 or 8. The physical interpretation is again based on Figure 12.42: increasing the number of cells means the recovery per cell is lowered which means the slope of the R_B–R_A relationship (δR_B/δR_A) is lowered; or in other words, less R_B is recovered per R_A recovery and thus selectivity is increased. There is evidently a potential loss in SE by having a bank shorter than about four cells, and, remember, the calculation in Figure 12.46 assumes the bank is operated with a flat recovery profile: a short bank with a nonflat profile risks even larger loss in SE than Figure 12.46 suggests. Again, we may have a practical compromise of six to eight cells in a bank. For older circuits with long banks, some up to 20 cells, balancing recovery is not really a concern; differences in recovery between so many cells will tend to be small and the profile will approach balanced.

The increase in SE_bk as N is increased is reminiscent of the effect of N on recovery (Figure 12.44) and for the same reason: as N increases the bank approaches plug flow transport, which represents the theoretical maximum SE that can be achieved by the bank (Figure 12.46). Assuming plug flow for the bank and setting up the problem in the same fashion as in deriving Eq. (12.41), it can be shown that:

$R_{\mathrm{Bbk}}=1-{(1-R_{\mathrm{Abk}})}^{1/S}$ (12.47)

(12.47)

In the limit (plug flow) setting a target bank recovery of A determines the bank recovery of B for a known S and thus the maximum bank SE can be calculated.

Note that the analysis does not consider cell volume, only recovery. Having cells of different size complicates balancing cell recoveries, as a larger cell in the bank will naturally tend to recover more than its smaller neighbors. If plant expansion (or some other reason) suggests adding cells to a bank, the effect on selectivity, along with the noted effective use of cell volume in terms of recovery, is an argument in favor of the added cell having the same volume.

The optimization problem was to find a maximum in SE for a target bank recovery. We could extract the benefit in a different way: by setting a target SE_bk the flat recovery profile will give the maximum bank recovery. The difficulty is knowing beforehand if the target SE_bk is achievable.

Singh and Finch (2014), using JKSimFloat, explored two aspects not addressed in the analytical solution: variable S and impact of entrainment. The relative rate constant S will tend to decrease down the bank as high rate constant A particles are recovered in the first cells; in the example described they showed that this variation in S did not alter the flat recovery solution. Their examination of entrainment suggested the optimum operating condition was equal mass flowrate (mass pull) from each cell (flat mass pull profile), but that the flat recovery profile was close in performance. Entrainment is a significant factor in the roughing stage (e.g., Figure 12.40), perhaps less so in the cleaning stage when selectivity between floatable minerals is the focus. It is possible that different bank profiling strategies should be adopted for roughers and cleaners.

A review of standard texts does not reveal guidelines on bank operation. The flat recovery profile is offered as an operating strategy: the question then is how to achieve it. Some options discussed by Maldonado et al. (2012) are briefly reviewed.

Controlling a Flat Recovery Profile

Recovery profiling requires local cell control. One way, and the most rapid in response, is manipulation of air rate. Cooper et al. (2004) found using air to redistribute recovery down a Zn-cleaner bank gave improved performance (higher bank grade at target recovery). High recovery in the first cell of a bank is common because the high proportion of fast floating mineral renders recovery “easy.” Regardless of whether the flat recovery profile can be achieved, making sure the first cells do not “over-recover” is perhaps the principal practical outcome of the analysis. That observation holds whether maximizing selectivity or minimizing entrainment is the target; and is an argument that if a larger cell is to be added to a bank for some purpose that it not be made the first cell.

The Cooper et al. experience was on a cleaner bank where the first cells are probably carrying capacity limited rather than kinetically controlled. Nevertheless, there was an advantage to redistributing recovery down the bank suggesting that balancing recovery is a robust guideline.

Manipulating air rate down a bank became known as air rate (or J_g) profiling, part of a general strategy of air distribution management. An extension is PAR profiling (Hadler et al., 2012). In this strategy air rate to each cell in the bank is adjusted to achieve the maximum air recovery (Eq. (12.35)), which has been shown to maximize bank performance. Air distribution management to banks is gaining traction in flotation control strategies (Shean and Cilliers, 2011).

Online control of the recovery profile might be effected by monitoring froth velocity using froth imaging technology. If a relationship between froth velocity and recovery can be identified it would allow air to be manipulated to achieve the target froth velocity profile. As a first approach it may be easier to link froth velocity to mass pull rather than recovery, and consider the flat mass pull profile strategy identified by Singh and Finch (2014).

Variables other than air that might be considered for local cell control include froth depth (level), reagents, and impeller speed. Froth depth has been incorporated into air profiling (Gorain, 2005) and in the PAR strategy (Hadler et al., 2012). (In some control strategies froth depth is the principal manipulated local variable with air rate for fine tuning.) Distributing reagents along the bank and increasing impeller speed down the bank, both practiced to recover slow floating fractions (Section 12.10.1), have the effect of balancing recoveries. In general, reagents do not offer local control, as additions in one location will affect units downstream (and upstream through recycle). At one operation, reduced frother dosage was used to “slow down” the first cell in a bank when air rate could not be lowered sufficiently (Blonde et al., 2013).

Selectivity

A basic circuit is shown in Figure 12.47, compromising two stages, the second being a scavenger to the first (i.e., treating the tailings from the first stage). This arrangement is familiar in the rougher–scavenger arrangement, and in the cleaner–cleaner/scavenger arrangement (the intended circuit depicted in the Figure). Circuit recovery can be solved by iteration, but also in an analogous fashion to that applied to Figure 12.34, this time introducing X as the mineral mass flowrate into the cleaner stage and deriving the (mineral) mass flows around the circuit (Figure 12.47). By mass balancing at the feed node (indicated by the circle), F can be deduced and the circuit recovery R_circ solved:

$F=X[1–R_{\mathrm{cs}}(1–R_{\mathrm{c}})]$ (12.48)

(12.48)

$R_{\mathrm{circ}}=\frac{R_{\mathrm{c}}}{1-R_{\mathrm{cs}}(1-R_{\mathrm{c}})}$ (12.49)

(12.49)

The solution will be referred to as an “analytical” solution (an alternative name is “transfer function” (Williams and Meloy, 2007)). The impact of the circuit on SE can be judged by setting R_c and varying R_cs for the target mineral. Assuming the banks are long enough for plug flow to apply, the result for S=5 is shown in Figure 12.48(a). The calculation proceeds as follows: for A: select R_c,A (e.g., 0.2, 0.5, etc.), vary R_cs,A (e.g., from 0 to 1), calculate R_circ,A using Eq. (12.49); and for B: from R_c,A and R_cs,A calculate R_c,B and R_cs,B using Eq. (12.47) with S=5, and solve for R_circ,B using Eq. (12.49); finally, solve for SE_circ=R_circ,A − R_circ,B. The Figure shows circuit SE increases as R_cs increases. This increase in SE_circ is related to the increase in circulating load of A, given by:

${\mathrm{CL}}_{\mathrm{A}}=\frac{R_{\text{cs},A}(1-R_{c,A})}{1-R_{\text{cs},A}(1-R_{c,A})}$ (12.50)

(12.50)

The CL_A is further increased by lowering R_c,A, as Eq. (12.50) indicates, and this further increases SE_circ as Figure 12.48(a) indicates. Putting all this together, Figure 12.48(b) shows the advance in SE going from single cell to bank (plug flow) to the two-stage circuit.

According to Figure 12.48(b), the circuit makes a significant increase in SE. This is indeed a circuit “property”, to improve selectivity. The magnitude of the impact shown, however, is exaggerated as the analysis implicitly assumed the recycled materials have the same recoverability as the fresh feed to the circuit. Figure 12.49, an example of rate constant versus size and liberation based on Figure 12.38(a), illustrates why this is not the case. The dashed horizontal lines indicate possible divisions for the cleaner and cleaner/scavenger stages; it is evident that the circulated material (CS con) comprises low rate constant material, namely fines, coarse and low grade (i.e., locked) particles. To take this variation in material properties into account requires simulators.

The analysis, nevertheless, does confirm why circuits are used in flotation: they increase selectivity between floatable particles (in addition to cleaners damping entrainment, Section 12.10.2). The benefit of the circuit, the circulating load, is also the “cost”: circulating loads mean increased stage (i.e., cell) volume is required. The target circulating load needs to be established at the design stage.

To optimize a circuit, Agar et al. (1980) suggested using the maximum SE at each stage. Based on Jowett (1975), they showed that the maximum in bank SE coincides with the cell in the bank that produces concentrate of grade equal to the feed grade; thus this cell defines the length of the bank. This is equivalent to selecting the length of bank such that no cell is yielding concentrate of lower grade than the feed to the bank, making it a tempting logical choice. However, using SE alone does not allow for the performance constraint of reaching a target recovery, nor does the strategy say how the bank is to be operated. It was later noted that maximizing stage SE does not result in maximizing circuit SE (Jowett and Sutherland, 1985; Lauder, 1992). The solution proposed here is to set the target circuit recovery of mineral A, then select the stage recoveries that meet that target and operate the stages with a balanced recovery profile. This will result in the maximum circuit SE at the target circuit recovery of A.

Varying target A recovery will generate the predicted relationship between SE_circ and R_circ,A (such as Figure 12.48) to aid reaching a decision on where to operate (which could include an economic assessment) that will then give the corresponding stage recoveries. Repeating the calculations for the other mineral components (B, etc.) and summing gives total solids flows around the circuit and coupled with estimates of the water content (% solids) slurry volume flows can be calculated and stage cell volume requirements assessed. Agar and Kipkie (1978) used this approach, employing bench tests to determine stage recovery data for the mineral components (referred to as split factors) which were used to estimate the flow of mineral around a circuit by an iterative procedure. An example calculation using the analytical solution for the circuit in Figure 12.47 is included below (Example 12.1). The analytical approach helps understand the interactions between stages, which is further illustrated later. Once the circuit is decided, control of the circulating load is key to performance.

Example 12.1

For the circuit in Figure 12.47 calculate:

1. the mass flow of A and B,

2. the grade and recovery of A to cleaner concentrate.

for the following conditions:

Mass flow (F)=100 t h⁻¹; mineral feed grades f_A=5% (0.05) and f_B=95% (0.95).

Stage recoveries: R_cA=0.75, R_cB=0.10, R_csA=0.90, R_csB=0.15.

Solution

1. Mass flows.
From the mass balance:

$F=X[1-R_{\mathrm{cs}}(1-R_{\mathrm{c}})]$

this becomes the scaling factor to determine mass flows for any stream; note, for the individual minerals F=F f where f=f_A of f_B. The table below summarizes the calculations.

Stream	Formula	Mass Flow (t h−1)
		A	B
Cleaner concentrate	$\frac{Ff{R}_{\mathrm{c}}}{[1-R_{\mathrm{cs}}(1-R_{\mathrm{c}})]}$	4.84	10.98
Cleaner tails	$\frac{Ff(1-R_{\mathrm{c}})}{[1-R_{\mathrm{cs}}(1-R_{\mathrm{c}})]}$	1.61	98.84
Cleaner/scavenger concentrate (circulating load)	$\frac{Ff{R}_{\mathrm{cs}}(1-R_{\mathrm{c}})}{[1-R_{\mathrm{cs}}(1-R_{\mathrm{c}})]}$	1.45	14.83
Cleaner feed (as a check)	$\frac{Ff}{[1-R_{\mathrm{cs}}(1-R_{\mathrm{c}})}$	6.45	109.83

The cleaner feed calculation offers a check, where cleaner feed=fresh feed + circulating load:
For A: 6.45=5 + 1.45 (i.e., checks); and for B: 109.83=95 + 14.83 (i.e., checks).

2. Grades of and recoveries to cleaner concentrate.
From the table these are readily calculated:

Grade A=4.84/(4.84+10.98)=0.3058 or 30.58%

Recovery A=4.84/5=0.9677 or 96.77%

While simplified, the calculation method offers a quick assessment, and for teaching purposes provides a calculation route that does not require a simulator to complete.

The arrangement in Figure 12.47 represents a component of a circuit. Some circuit options are considered.

Circuit Options

Table 12.7 shows some basic circuits. Simplifying by assuming equal recovery in each stage, the analytical solution for the circuit recovery can be found and is included in the Table. (To derive, a suitable starting point is required—e.g., in the R-S-C circuit letting X be feed to rougher, and in the R-C-CS circuit letting X be feed to the cleaner—and then proceeding as for Figure 12.47). Noble and Luttrell (2014) have considered over 30 circuits using this same basic approach.

Table 12.7

Some Common Circuit Arrangements, with Analytical Solution and Measure of Selectivity, the Slope (dR_circ/dR)

	Circuit	Circuit Recovery (all R Equal)	(dRcirc/dR)
R-S-C		$\frac{R^2}{R^2+{(1-R)}^2}$	$\frac{2R(1-R)}{{[R^2+{(1-R)}^2]}^2}$
R-C-CS		$\frac{R^2}{R+{(1-R)}^2}$	$\frac{2R-R^2}{{[R+{(1-R)}^2]}^2}$
R-C3-CS		$\frac{R^4}{R^3+{(1-R)}^2}$	$\frac{R^6+2R^5-6R^4+4R^3}{{[R^3+{(1-R)}^2]}^2}$
R-S-C3		$\frac{R^4}{(R^2-R+1)(3R^2-3R+1)}$	$\frac{-2R^3(3R^3-7R^2+6R-2)}{{(3R^4-6R^3+7R^2-4R+1)}^2}$
(R-C-CS)2		$\frac{R^2(1-R)}{R+{(1-R)}^2}$	$\frac{R^2(3R^5-8R^4+10R^3-5R^2-R+1)}{{[R+{(1-R)}^2]}^2}$

The first two circuits in Table 12.7 represent the rougher–scavenger–cleaner circuit (R-S-C) and the rougher–cleaner–cleaner/scavenger circuit (R-C-CS). (In the first circuit it is traditional to refer to “scavenger” not rougher/scavenger.) The R-S-C circuit can be treated as the conventional arrangement providing as it does some buffering against the inevitable feed changes, enabling a level of manual supervision in preautomatic control days. The interaction between the stages, however, creates difficulties in control, as any action in one stage impacts the other stages. This made for a debate on advantages of open versus closed circuits (Lauder, 1992).

The R-C-CS circuit does partially open the circuit by isolating the rougher stage from the cleaner block (C-CS) with the CS tail being sent to final tail. The rougher function is now recovery (still at maximized SE) and the cleaner block function is grade. Independent control action appropriate to the two functions can then be implemented. In that regard the R-C-CS circuit is more “controllable” or “operable” than the R-S-C circuit. The stages may use different flotation machines, for example, flotation columns as the cleaner stage with mechanical cells in the cleaner/scavenger stage (Dobby, 2002).

It is common to regrind recycled material which will contain middling (composite) particles. Regrind is shown on the R-C-CS circuit (RG, which today would likely be a stirred mill) and could be included in the R-S-C circuit by combining the scavenger concentrate and cleaner tails. Regrind should always be considered. It may also permit a coarse primary grind if the mineral is readily floated, the lower mass of material making further grinding (regrinding) more energy efficient. Treatment of high tonnage porphyry copper ores often takes advantage of this approach.

Figure 12.50(a) compares R_circ for the R-S-C and R-C-CS circuits as a function of R, the stage recovery. Two features are: (1) the R_circ of the R-C-CS always lags that of the R-S-C and is less than the stage recovery (specifically the rougher stage) as there is no recycle to the rougher; and (2) the R-S-C circuit reduces recovery when R < 0.5 and enhances recovery when R > 0.5. This pivoting at R=0.5 points to how the R-S-C circuit increases SE, increasing the recovery of target mineral A with R_A > 0.5 while decreasing the recovery of B with R_B < 0.5. Selectivity is related to the slope of the R_circ–R relationship, that is, (δR_circ/δR), and is included in Table 12.7 and plotted in Figure 12.50(b). Note that at high stage recovery (R > 0.7) the slope for R-C-CS circuit is greater than for R-S-C. This means that the selectivity between two minerals that have high and close recovery (i.e., low S) is greater in the R-C-CS circuit than in the R-S-C circuit. The R-C-CS circuit may not only be more “operable” but also offer an advantage in difficult selectivity cases, that is, ones with low relative rate constant, S. This general result is not altered by relaxing the assumption of equal stage recoveries, simply the magnitude of the SE changes (as illustrated in Figure 12.48).

This advantage of the R-C-CS circuit in difficult separation situations is more directly evident in Figure 12.51, which compares the SE of a single stage to that for the R-S-C and R-C-CS circuits under conditions of (a) “difficult” separation, S=2, and (b) “easy” separation, S=10. It is evident for the difficult case that at R_A > 0.8 the R-C-CS circuit gives higher SE_circ than the R-S-C circuit. Since high recovery for the target mineral, i.e., high R_A, is the target the R-C-CS circuit seems favored, the R-S-C circuit even failing to achieve the SE for a single stage at R_A > 0.85. For easy separations (b), the R-S-C circuit seems the better choice; this would include when separation is primarily against entrained mineral, as in the case of recovering talc, coal, or bitumen.

In the late 1980s Brunswick Mine implemented the R-C-CS type circuit, with significant impact on Zn metallurgy, increasing the selectivity of sphalerite over pyrite (Shannon et al., 1993). This separation would be classed as difficult with S ~ 2–3 (Cooper et al., 2004). Sending CS tail to final tailings removed material that actually decreased performance if recycled to the rougher. Including regrind (ball mill/hydrocyclone) on the feed to the cleaner block (as in Table 12.7) to increase liberation, and diluting the pulp to control entrainment of the additional fines produced (Section 12.10.2) added to the advantage of the R-C-CS circuit. The circuit modification, which was a major simplification over the prior flowsheet, included removing a thickener. Thickeners do offer surge capacity and densification of feed to subsequent stages (regrind or flotation) but in some sulfide flotation circuits they may not be desirable by giving opportunity for oxidation and introducing a time lag that makes control difficult (Bulatovic et al., 1998). The modifications at Brunswick Mine were the start of an impressive series of improvements documented by Orford et al. (2005). During the 1990s the Noranda group moved to adopt the R-C-CS circuit across operations. The R-C-CS circuit is also common in Cu–Mo plants using columns as cleaners and mechanical cells as cleaner/scavengers (Bulatovic et al., 1998).

Opening the cleaner block by sending CS tails to final tails is not a decision taken lightly. Recirculating streams can appear to contain substantial mineral values but upon opening the circuit the flow diminishes and the amount of mineral lost to final tails represented by the CS tails is minor (and, as noted, trying to recover the contained values may be counter-productive). If there is concern over potential loss represented by the CS tails the stream could be considered for alternative treatments, upgrading by pyro- or hydrometallurgical processes, such as the Galvanox™ process (Dixon et al., 2007).

The next two circuits in Table 12.7 are developments on the first two. The R-C₃-CS circuit introduces two additional cleaner stages to the R-C-CS network with cleaner 2 and cleaner 3 tails combined and recycled to the first cleaner via regrind. (Brunswick Mine had up to four cleaner stages in this arrangement.) The R-S-C₃ circuit shows the conventional countercurrent flow arrangement. In the R-S-C₃ circuit it is not easy to regrind all the recycled streams, which is a disadvantage. The analytical solution is also quite complex (to solve, incidentally, you have to start with feed to the last cleaner and work backwards through the circuit) compared to the solution for R-C₃-CS, suggesting a strong interaction between stages, which makes for more difficult control. The number of stages could get quite large: prior to replacing with flotation columns the Cu–Mo separation circuit at Mines Gaspe had up to 14 countercurrent cleaning stages of mechanical cells (Finch and Dobby, 1990). Incidentally, rather than being referred to as cleaner 1, 2, etc. the cleaner stages may have local names, such as recleaner.

The last circuit supplements the basic R-C-CS circuit by repeating it, let’s call it (R-C-CS)₂. This option allows for reagent additions into the second circuit aimed at the slow floating fractions. In that case the two concentrates will be combined. The same arrangement also serves when two minerals are to be recovered, for minerals A and B making the circuits (R-C-CS)_A and (R-C-CS)_B with separate concentrates.

Plants which concentrate two or more minerals may repeat circuits for each mineral or produce a bulk concentrate (e.g., a concentrate of at least two valuable minerals) then perform selective flotation on the bulk concentrate. Options are illustrated in Figure 12.52 for a Cu–Pb–Zn ore ranging from (a) “easy” milling ore where separate concentrates can be produced, to (b) more difficult ore where a bulk Cu–Pb concentrate is produced and then separated into Cu and Pb concentrates, to (c) where a bulk Zn–Pb concentrate is an additional final concentrate.

Circuit design is a combination of experience and increasingly the use of simulators, at least for the non-chemistry aspects (Herbst and Harris, 2007; Mendez et al., 2009; Ghobadi et al., 2011). While search algorithms may find optimal circuit configurations based on metallurgical performance criteria, in the end the choice has to be a circuit that is readily operated.

Circuit Flexibility

The decision having been reached to design a flotation circuit according to a certain scheme, it is necessary to provide for fluctuations in the flowrate and grade of ore to the plant.

The simplest way of smoothing out grade fluctuations and of providing a smooth flow to the flotation plant is by interposing a large agitated storage tank (agitator) between the grinding section and the flotation plant:

$\mathrm{Grind}\to \mathrm{Storage\; Agitator}\to \mathrm{Flotation\; Plant}$

Any minor variations in grade and tonnage are smoothed out by the agitator, from which material is pumped at a controlled rate to the flotation plant. The agitator can also be used as a conditioning tank, reagents being fed directly into it. It is essential to precondition the pulp sufficiently with the reagents (including sometimes air, Section 12.8) before feeding to the flotation banks, otherwise the first few cells in the bank act as an extension of the conditioning system, and poor recoveries result.

Provision must be made to accommodate any major changes in flowrate that may occur; for example, grinding mills may have to be shut down for maintenance. This is achieved by splitting the feed into parallel banks of cells (Figure 12.53). Major reductions in flowrate below the design target can then be accommodated by shutting off the feed to the required number of banks. The optimum number of banks required will depend on the ease of control of the particular circuit. More flexibility is built into the circuit by increasing the number of banks, but the problems of controlling large numbers of banks must be taken into account. The move to very large unit processes, such as grinding mills, flotation machines, etc., in order to reduce costs and facilitate automatic control, has reduced the need for many parallel banks.

In designing each flotation bank, the number of cells required must be assessed: should a few large cells be incorporated or many small cells giving the same total capacity?

Some theoretical considerations have been introduced (Section 12.11.2), but there is a practical aspect as well: if a small cell in a bank containing many such cells has to be shut down, then its effect on production and efficiency is not as large as that of shutting down a large cell in a bank consisting of only a few such cells.

Flexibility can include having “extra” cells in a bank. It is often suggested that the last cell in the bank normally should not be producing much overflow, thus representing reserve capacity for any increase in flowrate or grade of bank feed. This reserve capacity would have to be factored in when selecting the length of the bank (number of cells) and how to operate it, for example, trying to take advantage of recovery or mass pull profiling. If the ore grade decreases, it may be necessary to reduce the number of cells producing rougher concentrate, in order to feed the cleaners with the required grade of material. A method of adjusting the “cell split” on a bank is shown in Figure 12.54. If the bank shown has, say, 20 cells (an old-style plant), each successive four cells feeding a common launder, then by plugging outlet B, 12 cells produce rougher concentrate, the remainder producing scavenger concentrate (assuming a R-S-C type circuit). Similarly, by plugging outlet A, only eight cells produce rougher concentrate, and by leaving both outlets free, a 10–10 cell split is produced. This approach is less attractive on the shorter modern banks. Older plants may also employ double launders, and by use of froth diverter trays cells can send concentrate to either launder, and hence direct concentrate to different parts of the flowsheet. An example is at the North Broken Hill concentrator (Watters and Sandy, 1983).

Rather than changing the number of cells, it may be possible to adjust air (or level) to compensate for changes in mass flowrate of floatable mineral to the bank. To maintain the bank profile at Brunswick Mine, total air to the bank was tied to incoming mass flowrate of floatable mineral so that changes would trigger changes in total air to the bank, while maintaining the air distribution profile (Cooper et al., 2004).

Locked Cycle Tests

These are multiple-step flotation tests designed to measure the effect of circulating materials to provide a better insight into the expected performance from multistage processing of an ore. To incorporate the effect of recycle streams, the tests are performed multiple times with the recycled stream from a previous interaction added in the subsequent test. Normally at least six cycles are required before the circuit reaches equilibrium and a complete material balance should be made on each cycle. Since the reagents are in solution, it is essential that liquids as well as solids recirculate, so any liquid used to adjust pulp density must be circuit liquid obtained from decantation or filtration steps. Cycle tests are very laborious to carry out, and often the test fails to reach steady state. Procedures have been developed to simulate the locked cycle test steady state result from individual batch flotation test data (Agar and Kipkie, 1978; Runge, 2010).

Two other test procedures which use multiple flotation steps are release analysis (Dell, 1964) and tree flotation (Pratten et al., 1989). Devised to determine coal washability (i.e., the liberation-limited separation of coal from ash), they serve to show how close the actual operation is to the best possible performance. In establishing the effectiveness of new flotation machines, one way is to show the separation approximates the release or tree flotation result (e.g., Adel et al., 1991).

High-Confidence Flotation Testing

This method for has become a standard in some flotation laboratories (Lotter, 1995a,b; Lotter and Fragomeni, 2010). The approach is based on the following criteria:

The layout of the method is shown in Figure 12.57. Apart from its focus on representative sampling to produce reliable test material, which is shown as Section 1 in Figure 12.57, it has a structure of replicate flotation tests with associated quality controls that identify outlier tests and reject these at the 95% confidence level, as shown in Section 2 of the figure. The number of replicate tests required is dependent on the sample standard deviation that is typical of the ore type in question, and of the format of the flotation test. If the ore is unknown, it is best to perform a set of baseline rougher float tests and empirically obtain an estimate of the sample standard deviation.

The acceptable error in the table of 3.3% is based on experience with S. African Pt ores; the acceptable error needs to be established for the ore in question, using techniques introduced in Chapter 3.

Additionally, investigators should have a clear idea of the differences in grade and recovery that they may want to demonstrate (Napier-Munn, 2012). For base metal ores, triplicates are often suitable. However for precious metal bearing ores, or in cases where the ore is polymetallic bearing both base metal and precious metals, the number of replicates may be as high as five. In either case, the sample standard deviation and relative standard deviation (RSD) are calculated from these replicate tests for the first flotation concentrate mass. If this exceeds an RSD of 5%, the quality control triggers a check for outliers using the small data set model after Grubbs (1969). Outliers are then rejected, leaving the original set of replicate tests short.

Repeat flotation tests are then performed to replace the rejected outliers. Thereafter, acceptable replicates are taken into a set of composite concentrates and tailings for assaying and mass balancing. This amounts to the equivalent use of averages instead of singleton observations, a tenet of the Central Limit Theorem in statistics, which reduces overall random error and leaves any residual error distributed almost in the pattern of a normal distribution (Box et al., 2005). Use of this method puts the investigator in a position to compare smaller differences in grade and recovery between tests using different conditions, for example, reagent suite A versus B.

In Figure 12.57, the trial metal balance produces an independent estimate of the ore grade (the built-up head) from the weights and metal grades of the flotation test products. This built-up head value is compared to the sample mean assay head produced earlier in Section 1 of the model. The percent difference, using the assay head as the 100% basis, may trigger some repeat investigation of assays because of excessive error in the trial metal balance (i.e., in excess of ±3.3% in the case of Figure 12.57), calling for new subsamples of the assay heads and flotation test products (legend “1”). If this does not resolve the metal balance error, final resort activities indicated by legend “2” are triggered, calling for a new set of flotation tests.

The largest risk in any greenfield project is the scale-up from laboratory to operations at commissioning. The High Confidence Flotation Testing methodology has demonstrated accurate scale-up properties in the development and prediction of a flowsheet for a new orebody (Lotter and Fragomeni, 2010). For example, in the flowsheet development testwork for the Ivanhoe Kamoa copper project, which may be used as a relevant case study of this practice, the error in the copper metal balances averaged −0.3% with 80% of the individual errors falling between ±2% (Lotter et al., 2013).

Presenting Data

In addition to the common methods of presenting separation data, which include grade–recovery, recovery–mass recovery, and recovery–recovery plots (Runge, 2010), in flotation testing for scale-up time is also used a variable. Appendix VI gives a set of data on flotation of a Pb-ore collected as a function of time. The data can serve to illustrate all the data presentation methods, but here just time-based plots are presented, first to illustrate locating the maximum SE, following Agar et al. (1980) as introduced previously (Section 12.11.3), and second to estimate the flotation rate constant.

1. Maximum SE
This is located in two ways: at the time when the difference in recovery, in this case of galena versus the rest (lumped as one gangue G), is maximum; and at the time when the increment of grade is equal to the feed grade.
The computations (Appendix VI) involve determining cumulative recovery of Pb (galena) and gangue G. To determine the latter, the Pb assay has to be converted to galena mineral assay (see Example 1.1) with G determined by difference, 100−%galena. The resulting plot is shown in Figure 12.58. It is evident that the time when maximum SE occurs does correspond to the time when the increment in grade equals the feed grade. (Note that the increment of grade is plotted against the average of the time interval.) Translating to the case of a bank, it led to the notion that the optimum length of the bank would be that cell producing the same grade of concentrate as the feed (Section 12.11.3).

2. Determining flotation rate constant
In batch flotation the assumption of first order kinetics gives the following result (model) from Section 12.9.1:

$R=1–\mathrm{e}\mathrm{x}\mathrm{p}(-kt)$ (12.26)

(12.26)

It is important that the conditions do not change while performing the test, and this includes physical conditions such as air rate as well as chemical conditions. Figure 12.59(a) illustrates the “disappearance” plot (ln (1−R) vs. t), with slope k. Taking the data up to about 80% recovery gives k=0.88 min⁻¹, which Figure 12.59(b) shows gives a reasonable fit to the recovery–time data.

Extending the disappearance plot to all the data, it is evident that a single k does not cover the full range (the reader should test this). To improve the fit has resulted in various modifications to the simple form of Eq. (12.26). Two modifications are: (1) introducing a maximum recovery less than 100% by rewriting as:

$R=R_{\infty }[1–\mathrm{e}\mathrm{x}\mathrm{p}(-kt)]$ (12.51)

(12.51)

where R_∞ refers to recovery after infinite flotation time; and (2) introducing the notion of fast and slow floating components:

$R={Ø}_{\mathrm{f}}(1–\exp (-k_{\mathrm{f}}t)+(1-{Ø}_{\mathrm{f}})(1–\mathrm{e}\mathrm{x}\mathrm{p}(-k_{\mathrm{s}}t)$ (12.52)

(12.52)

where Ø_f refers to the fraction of fast floating mineral (the fraction of slow floating mineral is then (1 − Ø_f)), and subscripts f and s refer to fast and slow. This could be expanded to consider a nonfloating component as well. These refinements are referred to as a floatability component model (FCM).

There are any number of curve-fitting routines to estimate the parameters in these models, including Excel Solver. For example, the best-fit parameters for model Eq. (12.51) are R_∞=97.62% and k=0.95 min⁻¹. Plotting, however, remains a useful visual check.

What is being recognized in these modifications is that the rate constant has a distribution of values, which has led to attempts to describe this distribution with mathematical functions (Weiss, 1985). Klimpel (1980) assumed a rectangular distribution, which leads to the following solution:

$R=R_{\infty }[1-\frac{1}{Kt}(1-\exp (-Kt))]$ (12.53)

(12.53)

where K is the maximum rate constant in the distribution. Klimpel made use of this model in plant evaluations by determining the effect of variables, for example, collector, on R_∞ and K noting sometimes K would increase but R decrease and vice versa, the so-called “R–K trade-off.”

The distribution of k values derives from the effect of particle size and liberation illustrated in Figure 12.38; that is, it is “property based.” It would seem best to determine the rate constant on a size-liberation basis. This represents a large experimental effort for one condition and may become prohibitive if dependence on other conditions, for example, process chemistry, is to be considered. Welsby et al. (2010) show these property-based parameters can be related to floatability components—fast, slow, and nonfloating—and thus the simpler FCM can be used. Using an FCM, it should be possible to reassign floatability components following a change in feed stream properties (such as particle size), instead of rederiving them, a large reduction in experimental effort. It must be appreciated that data fitting exercises are accompanied by a degree of uncertainty in the derived parameters that can be high (Sandoval-Zambrano and Montes-Atenas, 2012). Simulations provide trends in metallurgy rather than precise predictions.

Cell Sizing (Example 12.2)

With the target retention time established, calculation of cell volume is relatively straightforward. Based on Wood (2002), the computation is as follows:

$N{V}_{\mathrm{eff}}=\frac{F}{60}TEP$ (12.54)

(12.54)

where,