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METHODOLOGIES FOR THE EVALUATION OF GRINDING MEDIA CONSUMPTION RATES AT FULL PLANT SCALE Dr. Jaime Sepúlveda Moly-Cop Grinding Systems Santiago, Chile
[email protected] ABSTRACT
Due recognition of the significant impact of the cost of grinding media on the overall economics of comminution facilities worldwide has created the need for reliable, practical methodologies to compare – over extended control periods - different operating conditions, arising from eventual changes in ore type, ball supplier or simply, a trial of new products from the same supplier. The current publication describes the main aspects to be considered in the planning and execution of a full scale evaluation campaign, the actual data to be recorded and the required calculation routines, including the theoretical framework justifying their applicability. The Linear Wear Model, herein described, provides a theoretical framework for the best estimations of comparative grinding media wear performance (in the absence of ball breakage) in any /(kWh/ton)), derived given application, on the basis of the Specific Wear Rate Constant, kdE (( µm /(kWh/ton)) from the Specific Consumption Rate (g/kWh), corrected by actual make-up ball size (mm). When significant ball breakage is to be expected – as in semiautogenous grinding (SAG) applications – an expanded, conceptual model, based on pilot Drop Ball Testing Testing (DBT) results is gaining acceptance as a way to incorporate breakage as a possible media consumption mechanism. Ideally, evaluations should be conducted in parallel grinding sections, in order to have the option of establishing multiple sequential, concurrent or cross-reference comparisons. The evaluation period should cover at least 6 months after the complete “purge” of the string of balls being substituted, which may well take from 3 to 6 months. As a result, the evaluation period required for reliable conclusions should typically exceed 9 to 12 months of fairly undisrupted, normal operation. The methodologies here proposed make intensive use of Moly-Cop Tools, a software software package developed by the international Moly-Cop Grinding Systems organization (formerly, ARMCO Worldwide Grinding Systems) with the specific purpose of helping process engineers characterize and evaluate the operating efficiency of any given grinding circuit, following standardized methodologies and widely accepted evaluation criteria.
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Methodologies for the Evaluation of Grinding…
INTRODUCTION
Primary and secondary grinding circuit operators worldwide are fully aware of the significant impact of grinding media consumption on the overall cost structure of any particular, full scale grinding facility. The cost associated with this essential consumable depends mainly on two factors : delivered price and durability (quality) of the grinding media. When comparing different operating conditions, arising from eventual changes in ore type or ball supplier (or simply, a trial of new products from the same supplier), there is a well accepted evaluation criterion that accounts for both media price and quality, referred to as the Effective Grinding Cost or the Cost-Effectiveness of the application. Any given operating condition is considered to be cost-effective when its unit grinding media cost – normally expressed in $/ton ground - is to some extent reduced, with respect to a nominal reference condition : Grinding Media Cost = Ball Price x Ball Consumption
($ /ton ground)
($/ton balls)
(1)
(ton balls/ton ground)
Under this criterion, an alternative, higher-price grinding media product could be cost-effective if its associated consumption rate is sufficiently lower than the reference media, to yield an also lower grinding cost, as dictated by Equation 1. For the proper application of the above criterion, it is then a basic requirement to maintain continuously updated and representative indicators of the performance of any particular grinding media type being utilized. In each particular case, media price should always be a known, well defined variable; however, it is not so evident how media performance (quality) differences amongst alternative product types or operating conditions could be assessed with reasonable accuracy and precision. In such context, the current publication proposes a functional theoretical framework – based on the Linear Wear Model, expanded to incorporate a fairly conceptual, simple Breakage Model – to derive the most representative indicators of grinding media quality in any given grinding application.
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REFERENTIAL THEORETICAL FRAMEWORK GRINDING MEDIA WEAR KINETICS.
The most widely accepted approach to characterize the slow, sustained consumption (wear) kinetics of grinding bodies in rotary tumbling mills is known as the Linear Wear Theory (1,2); according to which - at every instant ‘t’ after the grinding body was thrown into the mill charge - its rate of weight loss will be directly proportional to its surface area exposed to gradual abrasion and/or corrosion wear mechanisms :
Ωt = d (m)/d (t) = - k m Ab
(2)
where :
Ωt m Ab km
= = = =
media consumption rate, kg/h ball weight, kg; after t hours in the mill 2 surface area of the ball exposed to wear, m 2 mass wear rate constant, kg/h/m .
Equivalently, taking into account the geometry of the grinding body (sphere or cylinder), Equation 2 converts to : d (d)/d (t) = - 2 k m /
ρb = - kd
(3)
where : d
ρb kd
= size (diameter) of the grinding body, after t hours in the mill charge, mm 3 3 = density of the grinding body, g/cm or ton/m = linear wear rate constant, mm/h.
The above first-order differential equation may be easily integrated for the particular and most frequent practical case in which kd remains constant with time - that is, k d is not a function of the instantaneous ball diameter (Linear Kinetics) - and the mill is continuously recharged with media of a single size dR (monorecharge). In such case : R
d = d - kd t
(4)
indicating that the rate at which the grinding body loses diameter is constant with time. In other words, if a ball loses 1.0 mm in diameter during the first 100 hours in the mill charge, it will also lose 1.0 mm in the second 100 hours in the charge, and in the tenth 100 hours, and so on, until the ball is totally consumed or rejected from the mill. In those much less frequent cases – when kd is a function of the instantaneous ball diameter (Non-Linear Kinetics) – more complex, but also available (3), models should be applied. Equation 4 is the basis for the simultaneous, experimental determination of k d constants of many different types of balls, all present in any given mill charge at the same time; a methodology referred to as Marked Ball Wear Testing (MBWT) (4).
APPLICATION OF THE LINEAR WEAR MODEL TO FULL SCALE MILLS.
In order to maintain a constant inventory (hold-up) of grinding media in the mill - normally measured by the ratio Jb of the apparent volume of balls (i. e., including interstitial spaces in between the balls) to the total effective internal mill volume – operators must continuously compensate for the PROCEMIN 2004, Santiago, Chile. www.procemin.cl
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Methodologies for the Evaluation of Grinding…
steel being consumed by recharging new media, preferentially of a single size dR. In this regard, the typical practice of recharging on a shift-per-shift or day-per-day basis may well be considered continuous, for all modeling purposes. Regardless of the recharge frequency - as a reflection of the linear nature of Equation 4 above - the size distribution of the media in the mill charge should approximate the so-called uniform size distribution. For example, consider the case of an operator charging 1000 new balls of 50 mm diameter, every day. By the following day, those balls will have a diameter of say (50 – 0.1) = 49.9 mm and he will be adding 1000 new balls of 50 mm. By the next following day, the first charged balls will have a diameter of (50 – 2*0.1) = 49.8 mm and the balls charged just the day before will have a size of (50 – 0.1) = 49.9 mm. So, after 3 days, not accounting for all the older media, the mill will contain 1000 new 50 mm balls, 1000 almost new 49.9 mm balls and 1000 not so new 49.8 mm balls, assuming kd remained constant at 0.1 mm/day. If this practice is maintained for a sufficiently large number of days, the operator will find that, in the same period that he recharges another 1000 new balls, he will be losing from the mill charge 1000 old ball cores or nuclei that reached a small enough critical rejection (scrap) size. From then on, the mill will contain a ‘string’ of equal number (1000 balls) of every possible size; that is, a uniform distribution of sizes that can be mathematically described by the simple expression : R
S
f 0(d) = 1 / (d – d )
S
R
; for d < d < d
(5)
where dS (mm) represents the scrap or rejection size, characteristic of the design and operating conditions of each particular application. In Equation 5, the density function f 0(d) is such that f 0(d)d (d) = d (d)/(dR – d S) represents the number fraction of balls in the mill whose size falls in the infinitesimal range [d, d + d (d)]. The mass size distribution F3(d), corresponding to the fraction of the total weight W b (not number) of balls in the ‘string’ with size smaller than ‘d’, may be determined from the population balance relationship : d
W b F3(d) =
∫ ρb (πd3/6) N f 0(d) d (d)
(6)
S
d
S
R
The (W b/N) ratio is derived from the integration of Equation 6 over the whole range of sizes [d , d ] imposing that, by definition, F3(dR) = 1.0 to obtain : R 4
S 4
R
S
(W b/N) = ( ρb π/24) [(d ) – (d ) ]/[(d ) – (d )]
(7)
Then, upon proper substitution and integration Equation 6 reduces to : F3(d) = [d4 – (dS)4]/[(dR)4 – (dS)4]
; for dS < d < dR
(8)
In the special case, when dS → 0, the above equation reduces to : F3(d) = (d/dR)4
; for 0 < d < dR
(9)
Regarding this simpler case, it is interesting to mention that the well known F. C. Bond (5) empirically determined and proposed a value of 3.8 for the exponent in the above equation; which is fairly close to 4.0, the theoretically derived value. Further, the relatively high value of the exponent in Equation 9 indicates that most of the weight of balls in the charge is distributed in sizes not much smaller than the original make-up size, dR. Given that the wear rate of each grinding body is proportional to its own exposed surface S R area, the integration of Equation 2 – over the whole range of sizes [d , d ] – demonstrates that the PROCEMIN 2004, Santiago, Chile. www.procemin.cl
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overall grinding media consumption rate ? t (kg steel/operating hours), corresponding to the ensemble of balls in the mill charge, is consequently proportional to the total area exposed A (m 2) by the ‘string’ :
Ωt = - km A = - ρb kd A / 2
(10)
For a monorecharge policy, with balls of size dR, the total area of the charge may be obtained from the expression : A =
dR ∫ (πd2) N f 0(d) d (d) dS
(11)
which upon substitution of Equations 5 and 7 plus integration yields a simple expression for the so called Specific Surface Area of the ‘string’ : a = (A/Vap) = 8000 (1 - f v) [(dR)3 – (dS)3]/[(dR)4 – (dS)4]
(12)
where : a Vap Wb f v
2
3
= specific surface area of the charge, m /m (apparent) = apparent mill volume occupied by the charge (including 3 interstitial spaces), m , calculated as W b/ρb/(1-f v) = total weight of balls in the charge, ton = volumetric fraction of interstitial voids; typically 35-40%.
According to this equation, the total surface area exposed by the mill charge is inversely proportional to the make-up ball size. On this basis, smaller balls of same “intrinsic quality” (i. e., same value of kd) will wear off faster just because of the correspondingly larger total surface area exposed. Therefore, when comparing two alternative operating conditions, any observed difference in Ωt does not imply the same corresponding difference in media quality if the balls are not of exactly the same size. In those special cases when two different ball sizes (d 1R and d2R) are continuously charged to the mill, in a proportion r 1 : r 2 (by weight), the combined area exposed so generated may be calculated with the expression : a = (A/Vap) = v 1 a1 + (1 - v 1) a2
(13)
where : v 1 = r 1 a2 / [(1 - r 1) a1 + r 1 a2] R
R
R
R
and the specific areas a1 and a2 are obtained from Equation 12 above, for d = d1 and d = d2 , respectively. Equation 13 arises from recognizing that each make-up size generates its own independent ‘string’ and that, in order for them to be consumed in the r 1 : r 2 (by weight) proportion, their total exposed areas (m2) must be in the same proportion inside the mill. Finally, substitution in Equation 10 above yields :
Ωt = - 4000 kd [ρb (1 - f v) Vap] [(dR)3 – (dS)3]/[(dR)4 – (dS)4]
(14)
or equivalently, R 3
S 3
R 4
S 4
? Ωt = - 4000 kd W b [(d ) – (d ) ]/[(d ) – (d ) ] PROCEMIN 2004, Santiago, Chile. www.procemin.cl
(15)
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On this basis, kd may be easily back-calculated from actual operating records or estimates of Ωt, W b, dR and dS. S Again, as d → 0, Equation 15 reduces to the much simpler form : R
? Ωt = - 4000 kd W b /d
(16)
which re-emphasizes the fact that smaller balls will produce higher consumption rates (kg/h) for the same ball quality, when characterized by k d. THE SPECIFIC ENERGY AS A CONTROLLING WEAR KINETICS PARAMETER.
By direct analogy to mineral particle breakage kinetics, it appears reasonable to postulate that an even more representative and scaleable quality indicator than kd is the Energy Specific Wear E Rate Constant [kd , µm/(kWh/ton)], defined through the expression : kd = kdE (Pb/W b) / 1000
(17)
where the power intensity ratio (Pb/W b) corresponds to the contribution to mill net power draw P b (kW) of every ton of balls in the charge (W b) to the total net power draw P net (kW) of the mill. The underlying theoretical claim is that grinding balls will wear faster in a more power intensive E environment. In other words, kd is equivalent to kd, but proportionally corrected by how much power E is being absorbed by each ton of balls in the charge. Therefore, it is to be expected that k d should be more insensitive than kd to variations in mill operating conditions (that may affect P b and/or W b) that may, in turn, produce higher or lower media consumption rates (kg/h), not caused by variations in grinding media quality. As a practical evaluation criterion, it should then be accepted that the top quality grinding media, in any given application, will be the one that exhibits the lowest E value of the Energy Specific Wear Rate Constant k d , regardless of the mill operating conditions.
Due application of Equation 17 creates the need for a mathematical representation of the total Net Power Draw of the mill in terms of its main dimensions and basic operating conditions. And also, how each component of the mill charge (balls, rocks (if any) and slurry) contributes to this total net power demand. An expanded version of the simple Hogg and Fuerstenau (6,7,8) model serves such
purpose well : Pnet = η Pgross = 0.238 D3.5 (L/D) Nc ρap (J - 1.065 J2) sinα where : Pgross
η
D L Nc J
α
= gross power draw of the mill (kW) = P net / η = overall mechanical and electrical transmission efficiency, °/1 = effective internal diameter of the mill, ft = effective internal length of the mill, ft = rotational mill speed; expressed as a fraction (°/1) of its critical centrifugation speed : Ncrit = 76.6/D0.5 = apparent mill filling, °/1 (including balls, rocks (if any), slurry and the interstitial spaces in between the balls and the rocks, with respect to the total effective mill volume) = charge lifting angle (defines the dynamic positioning of the center of gravity of the mill load (the ‘kidney’ ) with respect to the vertical direction, typically with values in the range of 35° to 45°. PROCEMIN 2004, Santiago, Chile. www.procemin.cl
(18)
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3
and where ρap denotes the apparent density of the charge (ton/m ), which may be evaluated on the basis of the indicated charge components (balls, rocks a nd interstitial slurry) (8) : ? ρap = { (1-f v) ρb Jb + (1-f v) ρm (J – Jb) + ρp Jp f v J } / J
(19)
with : f v Jb Jp
ρm ρp
= volume fraction (°/1) of interstitial voids in between the balls (typically assumed to be 35-40% of the volume apparently occupied by the balls). = apparent ball filling (°/1) (including balls and the interstitial voids in between such balls). = interstitial slurry filling (°/1), corresponding to the fraction of the available interstitial voids (in between the balls and rocks in the charge) actually occupied by the slurry of finer particles. = mineral particle density, ton/m 3. 3 = slurry density (ton/m ), directly related solids of the slurry (f s) by : 1/[(f s/ρm) + (1 - f s)].
to
the
weight
%
Substitution of Equation 19 into Equation 18 allows for the decomposition of the total net power draw of the mill, in terms of the charge components (8). In particular, the contribution by the balls in the charge becomes : Pb = [(1-f v) ρb Jb / ρap J] · Pnet
(20)
Similarly, the contribution to the net mill power by the rocks in the charge becomes : Pr = [ (1-f v) ρm (J - Jb) / ρap J] · Pnet
(21)
and finally, just for completeness, the contribution of the slurry in the charge becomes : Ps = [ ρp Jp f v J / ρap J] · Pnet
(22)
Referring back to Equations 15 and 17, an additional formula for the Energy Specific Media Consumption Rate, Ω E (g of steel / kWh drawn) , may now be derived :
ΩE = 1000 Ωt / Pb
(23)
equivalent to :
ΩE = 4000 kdE [(dR)3 – (dS)3]/[(dR)4 – (dS)4]
(24)
On this basis, kdE may be easily back-calculated from actual operating records or estimates of ΩE, d R and dS; recalling that the top quality grinding media - in any given application - will be the one that E exhibits the lowest value of the Energy Specific Wear Rate Constant kd , regardless of the mill(s) operating conditions. S
Again, as d → 0, Equation 24 reduces to : E
R
? ΩE = 4000 kd /d
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(25)
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Methodologies for the Evaluation of Grinding…
which re-emphasizes the fact that smaller balls will produce higher consumption rates (g/kWh) for the same ball quality, when properly characterized by kdE.
PURGE PERIOD.
When comparing two different types of grinding media, an accurate evaluation of their relative performance must necessarily discard all data collected during the so-called ‘purge’ period of the mill; that is, the lapse of time required for the complete consumption of the last ball charged prior to the beginning of the test with the alternative media or the time required for the complete consumption of the first new ball charged at the beginning of the test with the alternative media, whichever is longer. Such periods may be readily estimated from Equation 4, setting d = 0 : tmax
R
= d / kd
(26)
If the test balls are of similar quality, the cumulative grinding media consumption during the whole S purge period may be estimated from Equation 14 above (for d → 0) :
Ωt tmax /1000 = 4 ρb (1 - f v) Vap = 4 W b
!!!
(27)
concluding that the purge period is roughly the time required to consume an amount of steel equivalent to 4 times the tons of balls in the mill load. As a result, a quick calculation of the purge period (months), in any specific application, is simply : [ 4W b / (average monthly consumption)]. The resulting value could be as long as 6 to 8 months. In practice, it is considered acceptable to purge the mill for the equivalent of only 2W b as, by that time, there should be no more than 10% of the old ‘string’ remaining.
MATHEMATICAL CHARACTERIZATION OF IMPACT BREAKAGE AS A MECHANISM FOR GRINDING MEDIA CONSUMPTION.
In operations where noticeable ball breakage is to be expected – like in high-impact, semiautogenous grinding (SAG) applications – an expanded, conceptual model, based on pilot Drop Ball Testing (DBT) results has been proposed to incorporate breakage as a potentially significant grinding media consumption mechanism. Assessment of Impact Breakage Resistance at Pilot Scale.
The Drop Ball Tester (DBT) is a standard, pilot scale testing procedure, originally designed by the U. S. Bureau of Mines (9,10) and later adapted by the international Moly-Cop Grinding Systems organization to assess the resistance of any given sample or lot of balls to repeated severe ball-to-ball impacts. Briefly, the DBT facility (Figure 1) consists of a 10 m-high, J-shaped tube of slightly larger internal diameter than the size of the balls being tested. The curved, bottom part of the tube is filled with a constant number of balls (for instance, 24 when testing 5" balls). When another ball is dropped through the tube from a height of 10 m above, the top ball retained below in the tube suffers the direct impact of the falling ball, which is replicated through the whole line of balls retained in the curve at the bottom of the J-tube, originating the removal (through the lower tip of the J-tube) of the first ball in the line, which is so replaced by the last ball dropped. The balls removed from the tube are continuously lifted - via a bucket elevator - back to the top of the tube to be dropped down once again. The DBT is run until a certain maximum number of balls are broken (say, 5 balls) or a reasonable number of total cycles have been completed (say, 20,000 drops).
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J. Sepulveda
Figure 1.
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Schematic representation of a typical Drop Ball Testing (DBT) facility.(From Ref. 10)
Maximum Impact Energy in Full Scale Mills.
Figure 2 illustrates the most critical, outer trajectory of a ball being lifted to a position defined by the angle ? 1 in the upper-right quadrant of the section of a mill of diameter D (ft) and then allowed to free-fall down to impact the toe of the mill charge ‘kidney’ (or worse, to impact any exposed mill lining part) at a position ? 2, in the left-lower quadrant. The mill is rotating at a speed N, related to its critical centrifugation speed N c by the expression (see Equation 18) : N = Nc (76.6/D0.5)
(28)
Tangential Velocity :
φ1 φ2
N, rpm
υ = π D N Rotational Speed : N = N c (76.6/D0 ) .5
D
Figure 2. Illustration of the outer, most critical trajectory of a ball in a mill.
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Methodologies for the Evaluation of Grinding…
Therefore, at the time of release, the ball is moving at a linear speed υ (m/sec) determined by : ???? υ? = π D N = π D Nc (76.6/D0 ) (0.305/60) .5
= 0.3894 π N c D 0 .5
(29)
The free-fall height (m) of the ball is simply determined by : ?? ∆ h = 0.305 [ (D/2) cos φ1 + (D/2) cos φ2 ]
(30)
A high-impact condition would then be represented by angle values around φ1 ≅ φ2 ≅ 30° and conversely, low-impact conditions by values φ1 ≅ φ2 ≅ 60°. In such context, the maximum energy of a ball of mass ‘m’ at the time of release from the upper shoulder of the ‘kidney’ may be estimated as the sum of its kinetic energy plus its potential energy with respect to the point of impact : Emax
= Kinetic Energy + Potential Energy = m υ2/2 + mg ∆ h
(31)
= 0.07581 mπ2Nc 2D + 0.305 mg[(D/2)cosφ1 + (D/2)cosφ2] This total energy is assumed to be of a similar magnitude as the maximum energy delivered at impact, not taking into account any rotational kinetic energy or friction losses. The equivalent impact energy imposed in the DBT facility is simply given by : EDBT,std = mg h DBT, std
(32)
where hDBT, std (m) represents the free-height of the standard J-tube. Finally, the equivalent DBT height to attain equal impact energy at both scales (pilot and industrial) is obtained from matching Equations 31 and 32 above and recalling that g = 9.806 m/sec2 : 2
hDBT, eq /D = 0.0763 Nc + 0.153 (cosφ1 + cosφ2)
(33)
independent of ball size (!). For typical values of N c = 75%, the above ratio takes on values in the range of 0.6 to 1.0 for low and high impact conditions, respectively.
Projection of DBT Results to Full Scale.
The first necessary step in correlating pilot DBT results to industrial scale performance is to 3 derive estimates of the ‘lifting capacity’ of the cavities in between lifters; that is, how many m /hr of charge are being lifted up by each cavity; in order to then proceed to assess the number rate of balls being lifted from the ‘kidney’ and subjected to the most critical impact conditions. With reference to Figure 3 below, such volume could be approximated as : 2
VL = 0.0000983 hL [tanδ2 – tanδ1] L
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(34)
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where VL (m 3/lifter) represents the volume of charge being lifted by each lifter bar of height h L (in), in a mill of effective length L (ft).
δ2
δ1
Figure 3. Schematic view of a lifter bar pulling rocks and balls away from the ‘kidney’.
Therefore, the ‘lifting capacity’ CL (m 3 (app)/h) of the lining system would be : CL = 0.0059 nL N hL2 [tanδ2 – tanδ1] L
(35)
where nL represents the number of lifter bars in a cross section of the mill. A proportion Jb/J of CL will correspond to lifted balls : 3
CL, balls (m (app)/h) = (Jb/J) CL
(36)
In terms of mass of balls : CL, balls (ton/h) = ρb (1-f v) (Jb/J) CL
(37)
and in terms of number of balls (as per Equation 8, mostly rather new balls of size close to d R) : R
3
CL, balls (#/h) = ρb (1-f v) (Jb/J) CL /[ ρb π? d /1000) /6]
(38)
This expression allows for the estimation of the total number of balls being lifted (therefore, impacted) per unit of time. But certainly not all of these impacts will be critical enough to cause breakage of the ball. In this regard, they are considered to be c r i t i c a l impacts (i. e., those causing potential breakage) when balls hit balls mostly surrounded by other balls ; that is, without the ‘cushioning effect’ of surrounding rocks. In other words, a ball would have a sizeable probability of breakage when such ball is impacted directly against another ball, surrounded by other balls. And the probability of such breakage is scaled up from the DBT result as : DBTstd = [broken balls/(drop * # balls in J-tube)] DBTeq = DBT std * ( hDBT, eq/ h DBT, std )
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(39)
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Methodologies for the Evaluation of Grinding…
Furthermore, the probability of a ball directly hitting other ball (and not a rock) is proportional to the (Jb/J) ratio and the probability of a ball being surrounded by other balls is assumed to be also proportional to the same ratio, in such a way that the rate of ball breakage would be estimated by : (see Equation 38) [Broken Balls/hr] = DBT eq * [CL, balls (#/hr)] * (J b/J)2
(40)
which may then be converted into a mass rate (ton/hr) consumption of balls. Again, for this latter R calculation, it is assumed that all broken balls were of size close to the make-up size, d . Since [CL, balls (# / h)] is proportional to the (J b/J) ratio, Equation 40 shows that this relative filling ratio has a very significant influence to the power 3.0. Recent trends in operating practice of many SAG mills worldwide are signaling towards increased ball fillings, J b, at lower total fillings, J; that is, increased (Jb/J) ratios. Consequently, higher media consumption rates are to be expected in such cases, for even the same quality of balls. In closing, the underlying claims of the just described theoretical framework are that, in terms of gradual wear resistance, the top quality grinding media - in any given application - will be the one that exhibits the lowest value of the Energy Specific Wear Rate Constant k dE, regardless of the mill(s) operating conditions and that, in terms of impact breakage resistance, the top quality media will be the one with the lowest DBTstd parameter . The series of cumbersome calculations involved in the assessment of the combined effect of both mechanisms of grinding media consumption, may be conveniently performed with the aid of the Media Charge_Impact&Wear spreadsheet, included in the Moly-Cop Tools (3) software package.
ANALYSIS OF FULL SCALE RESULTS In the analysis of plant scale data, it is not yet normal practice to base comparisons on estimates of wear rate or breakage rate constants like those described in the preceding section. Instead, one or more of the following consumption indicators are used : Consumption by unit of energy consumed, Ω E (g / kWh) (see Equation 24), Consumption by unit of operating time, Ω t (kg / h) (see Equation 15), Consumption by unit of ore ground, Ω M (g / ton) : As indicated above, starting from available plant data and corresponding DBT results, the Media Charge_Impact&Wear spreadsheet of Moly-Cop Tools (3) facilitates the simultaneous evaluation of each of these indicators plus the estimate of the more intrinsic wear rate parameters, indicative of the actual grinding media performance in the specific application under consideration. No doubt, the most commonly used indicator – and unfortunately, the least representative of all – is the specific consumption rate ΩM normally expressed in (g of steel/ton ground), that may be also calculated from the expression : ? ΩM
=
( g/ton )
ΩE ( g/kWh )
* E ( kWh/ton )
(41)
where E - the specific energy consumed (kWh) per each ton of ore ground - depends exclusively on the operating conditions and intrinsic characteristics of the ore, with no relation whatsoever to the quality of the media being used. It is so concluded that any variation in E could be mistakenly interpreted as a variation in media quality, if the ΩM indicator were to be adopted for comparative evaluations. It is likewise concluded that a much better indicator of media quality is ΩE, properly PROCEMIN 2004, Santiago, Chile. www.procemin.cl
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corrected by media size dR - as suggested by Equation 24 - to arrive at the Energy Specific Wear E Rate Constant, kd (µm/(kWh/ton)).
EXAMPLE OF APPLICATION Consider the hypothetical case of Fair Mining Co. operating two parallel 36’ φx17’ SAG lines. Attachment A contains operational records for both lines in the period July ’02 thru June ’03. The standard 5” balls are being supplied by Forge+. However, since November ’02, management decided to start an evaluation with the alternative supplier, NKOB. In the interest of evaluating which supplier provides better quality balls, it must be first recognized that there are multiple ways of establishing such comparison. In fact, discarding 4 months (November ’02 thru February ’03) as ‘purge’ period, at least the following options could be selected : Sequential Evaluation , comparison of historical consumption rates of the same mill (SAG 2), before and after the purge period. Concurrent Evaluation , comparison of consumption rates of a test mill (SAG 2) against a standard mill (SAG 1), both operating in parallel, for exactly the same time period, once the purge period has been completed. In the first case, the following Sequential Evaluation could be established :
SAG SAG SAG 22 SAG 22 Pr P ree PPur urge g e Post Post Purge Purge ORE ORE THROUGHPUT THROUGHPUT ton/hr t on/ hr ENERGY ENERGY CONSUMPTION CONSUMPTION kkW W ((nnet e t )) kWh/ton kWh/ton BALLS BALLS CONSUMPTION CONSUMPTION ggrr//ttoon n
VVaarriiaat t iioonn % %
11,,225544
11, , 441100
1122..44
1122,,005588 99..6622
1111,,669911 88..2299
((33..00)) ((1133..88))
555522
550011
((99..22))
If the analysis was to be based only on ΩΜ (g/ton), the first conclusion to be drawn would be that the alternative NKOB balls are 9.2% better than the standard Forge + balls (?). Even more, the process analyst (or the NKOB vendor) could claim that, thanks to the new balls, throughput was increased by 12.4% ! But the analyst should not ignore that there has been a very significant change in E (kWh/ton), before and after the purge period. However, assuming similar DBTstd performance of 5 broken balls every 20,000 drops, for both type of media, the following more detailed comparison may be established, for the same Sequential Evaluation above, using the spreadsheet Media Charge_Impact&Wear of Moly-Cop Tools (3) :
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T16 - 14
Methodologies for the Evaluation of Grinding…
SAG SAG SAG 22 SAG 22 PPrree Pu e Post P urg r ge P ost Purge Purge ORE ORE THROUGHPUT THROUGHPUT ton/hr t on/ hr ENERGY ENERGY CONSUMPTION CONSUMPTION kkW W ((nnet e t )) kWh/ton kWh/t on BALLS BALLS CONSUMPTION CONSUMPTION ggrr/ / ttoonn kg/hr kg/hr gr/kWh gr/kWh SSp. p. W e a r CConstant, o n s t a n t , kkddEE Wear
VVaarriiaattiioonn % %
11,,225544
11, , 441100
1122..44
1122,,005588 99..6622
1111,,669911 88..2299
((33..00)) ((1133..88))
555522 669922 5577..44
550011 770077 6600..44
((99..22)) 2. 2 .22 5. 5 .22
22..80 80
22..995 5
55.3 .3
The additional indicators reveal that, in reality, the alternative media would be, not 9.2% better, but 5.3% worse in performance, as compared to the standard media; proving that ΩM (g/ton) is not a reliable media quality indicator, because is distorted by changes in E (kWh/ton) strictly related to process variables and totally independent of media quality. A Concurrent Evaluation , based on the same data base of Attachment A, shows :
SAG SAG SAG 11 SAG 22 Post Purge Post Post Purge P ost Purge Purge ORE ORE THROUGHPUT THROUGHPUT ton/hr t on/ hr ENERGY ENERGY CONSUMPTION CONSUMPTION kkW W ((nnet e t )) kWh/ton kWh/t on BALLS BALLS CONSUMPTION CONSUMPTION ggrr/ / ttoonn kg/hr kg/hr gr/kWh gr/kWh E
SSp. p. W e a r CConstant, o n s t a n t , kkdd E Wear
VVaarriiaattiioonn % %
11,,229999
11,,441100
88..55
1111,,779911 99..0088
1111,,669911 88..2299
447799 662211 5522..77
550011 770077 6600..44
44..66 1133..88 1144..66
22..55 55
22. . 9955
1155.4 .4
((00..88)) ((88..77))
that is, much more significant differences against the NKOB balls than those indicated by the previous comparison. Which one to believe : the sequential or the concurrent evaluations ? There is not a clear cut answer to such question. The merits of the concurrent evaluation is that the comparison is based on the performance of parallel lines, operating for the same periods in time, most likely being fed with the same type of ore. However, no two lines in any Concentrator are really identical and in that regard, the sequential evaluation assures that the comparison is not being affected by intrinsic differences in the process equipment. A third option for comparisons, referred to as Cross Reference Evaluation, can be stated in two totally equivalent ways : Cross Reference A : difference in the consumption rate of the test mill (SAG 2) - before and after the purge period - minus the same difference for the standard mill (SAG 1), normalized with respect to the standard mill wear constant, before the purge period : [(kdE)SAG2,after
-
(kdE)SAG2,before]
-
[(k dE)SAG1,after
-
(kdE)SAG1,before]/(kdE) A,before (42)
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J. Sepulveda
T16 - 15
Cross Reference B : difference in the consumption rate of both mills (SAG 1 and SAG 2) before and after the purge period - normalized with respect to the standard mill wear constant, before the purge period : E
[(kd )SAG2,after
-
E
(kd )SAG1,after ]
-
E
[(k d )SAG2,before
E
-
E
(kd )SAG1,before]/(kd ) A,before (43)
For the particular example under consideration, such Cross Reference (A and B are mathematically equivalent) would be :
Specific Wear Rate Constant, k dE SAG 1 P r e P u r g e Post Purge
2.68 2.55
SAG 2 2.80 2.95 1 0 . 4 %
indicating that while - because of a change in ore properties or process conditions - SAG1 was reducing the wear rate, SAG 2 could not do the same because of the different grinding media being charged. From a different but equivalent perspective, while SAG 2 was performing slightly worse than SAG 1, prior to the purge period, that difference was incremented after the purge period because of the lower media quality. In both cases, the most reliable estimate of media quality difference is 10.4% + worse than the standard Forge balls in SAG 1 before the purge period. And if the NKOB balls didn’t break at all ? In such case, all of the observed difference in E
performance would have to be explained by an even larger value of k d associated to the NKOB balls :
Specific Wear Rate Constant, k dE SAG 1 P r e P u r g e Post Purge
2.68 2.55
SAG 2 2.80 3.11 1 6 . 4 %
Or if the NKOB balls were slightly larger, say 130 mm ? In such case, going back to the
assumption of similar DBT std performance, the observed difference in performance would also have to E be explained by a larger value of k d for the NKOB balls : S p e c i f iic c W e a r R a t e C o n s t a n t , kk d E S A G 1 P r e P u r g e Post Purge
2 .6 8 2 .5 5
S A G 2 2 .8 0 3 .0 2
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%
T16 - 16
Methodologies for the Evaluation of Grinding…
The practice of making balls slightly larger than their nominal size ( ‘overweight’ ) is quite common amongst grinding media manufacturers in their continuous efforts to maximize equipment availability and production capacity. It should have no significant effect on the grinding efficiency of the media but, in terms of wear consumption, it could help hide 3-5% differences in relative performance, if the consumption indicators are not properly corrected for ‘overweight’ .
CONCLUDING REMARKS The Linear Wear Model provides a theoretical framework for the most reliable estimations of comparative grinding media wear performance (in the absence of breakage) for any given application, E on the basis of the Specific Wear Rate Constant, k d . Because of natural process fluctuations and measurement errors, it is not possible to develop a single comparative performance indicator. In such context, evaluations should be ideally conducted in parallel lines, in order to have the option to establish the preferred Cross Reference comparisons. A fairly simple Ball Breakage Model - of particular interest in SAG applications – is also available in Moly-Cop Tools (Version 2), in order to account for this consumption mechanism, whenever suspected significant.
REFERENCES 1.
Prentice, T. K., “Ball Wear in Cylindrical Mills” , Journal of the Chem., Met. and Mining Society of South Africa, Jan – Feb 1943, pp. 99-131.
2.
Norquist, D. E. and Moeller, J. E., “Relative Wear Rates of Various Diameter Grinding Balls in Production Mills” , Trans. AIME, Vol. 187, 1950, pp. 712-714.
3.
Moly-Cop Tools, Version 1.0, “Software for the Assessment and Optimization of Grinding Circuit Performance” , available upon request at
[email protected]
4.
Sepúlveda, J. E., “Experimental Protocol for Marked Ball Wear Tests” , Internal Report, MolyCop Grinding Systems, April 2003, available upon request at
[email protected]
5.
Bond, F. C., “Crushing and Grinding Calculations” , Part I, Rev. Jan 2, 1961, Allis Chalmers Co. Publication.
6.
Hogg and Fuerstenau, “Power Relations for Tumbling Mills” , Trans. SME-AIME, Vol. 252, 1972, pp. 418-432.
7.
Gutiérrez, L. and Sepúlveda, J. E., “Dimensionamiento y Optimización de Plantas Concentradoras Mediante Técnicas de Modelación Matemática” , CIMM – Chile Publication, 1986.
8.
Sepúlveda, J. E., “A Phenomenological Model of Semiautogenous Grinding Proceses in a Moly-Cop Tools Environment” , Proceedings SAG 2001 Conference, Vol. 4, pp. 301-315, Vancouver, B. C., Canada.
9.
R. Blickenderfer and J. H. Tylczak, "A Large-Scale Impact Spalling Test" , WEAR, vol. 84, 1983, pp 361-373.
10.
R. Blickensderfer and J. H. Tylczak, "Laboratory Tests of Spalling, Breaking, and Abrasion of Wear-Resistant Alloys Used in Mining and Mineral Processing" , USBM RI 8968 [Report], 1985, pp 1-17. PROCEMIN 2004, Santiago, Chile. www.procemin.cl
J. Sepulveda
T16 - 17
Attachment A EXAMPLE OF APPLICATION Fair Mining Co. Operational Records
Unit Unit: : Mill MillDiam. Diam. Mill Lenght Mill Lenght %%Critical Critical Ore OreDensity Density
SAG SAG11
Ore Ore Throughput Throughput ton/month ton/month Jul '02 Jul '02 Aug Aug Sep Sep Oct Oct Nov Nov Dec Dec Jan '03 Jan '03 Feb Feb Mar Mar Apr Apr May May Jun Jun Average Average
Unit Unit: :
Mill MillDiam. Diam. Mill MillLenght Lenght %%Critical Critical Ore OreDensity Density
Make-up Make-upBalls Balls: :5.0" 5.0"φφ %%Balls (Nominal) 14 Balls (Nominal) 14 %%Charge (Nominal) 26 Charge (Nominal) 26 %%Solids 74 Solids(Nominal) (Nominal) 74
36 36ftft 17 17ftft 76 76%% 2.8 2.8ton/m3 ton/m3 Operating Operating hours hours hr/month hr/month
Grinding Grinding Capacity Capacity ton/hr ton/hr
Energy Energy Consumption Consumption MWh/month MWh/month
Mill Mill Power Power kW kW
Balls Consumption Balls Consumption ton/month ton/month
gr/ton gr/ton
kg/hr kg/hr
gr/kWh gr/kWh
1,017,541 1,017,541 915,593 915,593 908,071 908,071 718,227 718,227 703,180 703,180 852,259 852,259 995,836 995,836 1,014,800 1,014,800 864,302 864,302 935,336 935,336 867,843 867,843 747,636 747,636
721.0 721.0 644.0 644.0 715.0 715.0 643.0 643.0 627.0 627.0 695.0 695.0 718.0 718.0 691.0 691.0 639.0 639.0 699.0 699.0 661.0 661.0 631.0 631.0
1,411 1,411 1,422 1,422 1,270 1,270 1,117 1,117 1,121 1,121 1,226 1,226 1,387 1,387 1,469 1,469 1,353 1,353 1,338 1,338 1,313 1,313 1,185 1,185
8,533 8,533 7,639 7,639 8,576 8,576 7,506 7,506 6,960 6,960 7,712 7,712 7,872 7,872 7,814 7,814 7,606 7,606 8,231 8,231 8,071 8,071 7,103 7,103
11,836 11,836 11,862 11,862 11,994 11,994 11,674 11,674 11,100 11,100 11,096 11,096 10,964 10,964 11,308 11,308 11,903 11,903 11,775 11,775 12,210 12,210 11,256 11,256
499.21 499.21 375.07 375.07 480.04 480.04 425.99 425.99 358.08 358.08 444.01 444.01 513.25 513.25 464.15 464.15 400.83 400.83 400.84 400.84 436.64 436.64 396.00 396.00
491 491 410 410 529 529 593 593 509 509 521 521 515 515 457 457 464 464 429 429 503 503 530 530
692 692 582 582 671 671 663 663 571 571 639 639 715 715 672 672 627 627 573 573 661 661 628 628
58.5 58.5 49.1 49.1 56.0 56.0 56.8 56.8 51.5 51.5 57.6 57.6 65.2 65.2 59.4 59.4 52.7 52.7 48.7 48.7 54.1 54.1 55.8 55.8
878,385 878,385
673.7 673.7
1,304 1,304
7,802 7,802
11,581 11,581
432.84 432.84
493 493
643 643
55.5 55.5
SAG SAG22
Operating Operating hours hours hr/month hr/month
Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge +
φ Make-up Make-upBalls Balls: :5.0" 5.0"φ %%Balls (Nominal) 1 Balls (Nominal) 144 %%Charge (Nominal) 2 Charg e ( No minal) 266 %%Solids (Nominal) 7 Solids (Nominal) 744
3366ftft 1177ftft 7766%% 2.8 2.8t ot onn/ m3 / m3
Ore Ore Throughput Throughput ton/month ton/month
Supplier Supplier
Grinding Grinding Capacity Capacity ton/hr ton/hr
Energy Energy Consump tion Consu mption M Wh/m onth M Wh/ m on th
Mill Mill Powe r Pow er kW kW
Balls Consumption Balls Consumption ton/month ton/month
gr/ton gr/ton
kg/hr kg/hr
g r/ k Wh g r/ k Wh
Jul '02 Jul '02 A u g A u g S ep Se p O ct O ct N ov N ov D ec D ec Jan '03 Jan '03 F eb Fe b M ar M ar A pr A pr M ay M a y J un J un
755,228 755,228 866,067 866,067 845,614 845,614 951,123 951,123 985,943 985,943 701,282 701,282 877,346 877,346 916,566 916,566 915,974 915,974 976,000 976,000 856,863 856,863 1,073,551 1,073,551
632.0 632.0 715.0 715.0 691.0 691.0 688.0 688.0 710.9 710.9 549.0 549.0 723.2 723.2 661.1 661.1 678.0 678.0 692.2 692.2 640.9 640.9 700.4 700.4
1,195 1,195 1,211 1,211 1,224 1,224 1,382 1,382 1,387 1,387 1,277 1,277 1,213 1,213 1,386 1,386 1,351 1,351 1,410 1,410 1,337 1,337 1,533 1,533
7,401 7,401 8,314 8,314 8,879 8,879 8,275 8,275 8,629 8,629 6,208 6,208 8,459 8,459 7,719 7,719 7,904 7,904 8,329 8,329 7,296 7,296 8,170 8,170
11,711 11,711 11,628 11,628 12,849 12,849 12,027 12,027 12,140 12,140 11,308 11,308 11,697 11,697 11,675 11,675 11,657 11,657 12,033 12,033 11,385 11,385 11,666 11,666
401.29 401.29 483.39 483.39 494.29 494.29 507.00 507.00 370.84 370.84 470.29 470.29 431.54 431.54 443.47 443.47 412.35 412.35 457.80 457.80 511.82 511.82 534.28 534.28
531 531 558 558 585 585 533 533 376 376 671 671 492 492 484 484 450 450 469 469 597 597 498 498
635 635 676 676 715 715 737 737 522 522 857 857 597 597 671 671 608 608 661 661 799 799 763 763
54.2 54.2 58.1 58.1 55.7 55.7 61.3 61.3 43.0 43.0 75.8 75.8 51.0 51.0 57.5 57.5 52.2 52.2 55.0 55.0 70.1 70.1 65.4 65.4
Average Average
893,463 893,463
673.5 673.5
1,327 1,327
7,965 7,965
11,828 11,828
459.86 459.86
515 515
683 683
57.7 57.7
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Supplier Supplier Forge + Forge + Forge + Forge + Forge + Forge + Forge + Forge + NKOB N KO B NKOB N KO B NKOB N KO B NKOB N KO B NKOB N KO B NKOB N KO B NKOB N KO B NKOB N KO B
