Variability of the Ball Mill Bond’s Standard Test in a Ta Ore Due to the Lack of Standardization

Abstract

There is no doubt about the practical interest of Fred Bond’s methodology in the field of comminution, not only in tumbling mills design and operation but also in mineral raw materials grindability characterization. Increasing energy efficiency in comminution operations globally is considered a significant challenge involving several Sustainable Development Goals (SDGs). In particular, the Bond work index (w_i) is considered a critical parameter at an industrial scale, provided that power consumption in comminution operations accounts for up to 40% of operational costs. Despite this, the variability of w_i when performing the ball mill Bond’s standard test is not always understood enough. This study shows the results of a variability analysis (a 3³ factorial design) performed to elucidate the influence on w_i of several parameters obtained from the particle size distribution (PSD) in feed and product. Results showed a clear variability in the work and grindability indexes with some of the variables considered.

1. Introduction

There is no doubt about the importance of Fred Bond’s methodology [1,2,3,4,5] and its practical value in the field of comminution, not only in tumbling mills design and operation but also in the characterization of mineral raw materials grindability. The Third Law of Comminution, also known as the Bond’s Law, is summarized in Equation (1) [5].

$$W=10·w_i·\left(\frac{1}{\sqrt{P_{80}}}-\frac{1}{\sqrt{F_{80}}}\right)$$

(1)

wherein:

W is the specific power consumption [kWh/t];

w_i is the Bond work index [kWh/t];

P₈₀ is 80% passing size in the grinding product particle size distribution (PSD);

F₈₀ is 80% passing size in the feed PSD.

Increasing energy efficiency in comminution operations globally is considered a significant challenge involving several SDGs, especially goals 7 (affordable and clean energy), 9 (industry innovation and infrastructure), 12 (responsible consumption and production) and 13 (climate action), since the increasing energy efficiency reduces waste and emissions production and increases energy availability. In particular, the Bond work index (w_i) is considered a critical parameter at an industrial scale, for power consumption in comminution operations accounts for up to 40% of operational costs [6,7,8]. Moreover, w_i should be one of the key parameters to consider in a potential process plant digitalization action, using adequate measurable parameters correlation. Despite this, the variability of w_i when performing the ball mill Bond’s standard test is not always considered or understood at an industrial scale [9,10,11,12,13]. In the study presented by Mosher and Tague [9], they addressed the variability of Bond test results independent of sampling or procedural variation. They discussed test sensitivity and detailed test procedures to maximize the accuracy and precision of the test, concluding that the Bond tests within one laboratory showed repeatability of less than ±4% at two standard deviations. They also recommended not to report Bond work indices beyond 0.1 kWh/t, based on the precision of the test and suggested that determination of the reproducibility of w_i can be improved significantly by accurate determination of the fresh feed and product PSD. Rodríguez et al. [11] studied this extent, showing that the methodology used for F₈₀ and P₈₀ determination by interpolation significantly affects w_i calculation.

In the case of the research presented in [10], the results of this research, carried out on a porphyry copper ore, concluded that the Bond work index values differ with different Bond ball mills and with different grinding ball charge distributions, but variations were higher when comparing different Bond ball mills than when comparing different ball charges in the same mill. Maximum variations of 8.6% with different mills and 6.2% with different grinding ball charges were measured.

The authors could not find a precedent comprising a variability study on the Bond standard test itself; mineral processing engineers sometimes attribute the w_i variations to ore grindability changes, while the reason can yield in feed PSD variations. Recently, it has been evidenced that, for a given ore, the grindability function (variation of the Maxon index, gbp, with P₁₀₀) can present a regular shape while the w_i function with P₁₀₀ can be pretty erratic [14]. Some lack of standardization in the so-called standard test can be the most probable cause of w_i variability. This work presents the result of a careful experimental design defined to elucidate the influence of several parameters obtained from the particle size distribution (PSD) in feed and product on w_i determination.

2. Materials and Methods

2.1. Materials

In order to carry out the series of tests, a 400 kg Ta-Nb-Sn ore sample from the tailings deposit of former mining activities in the Penouta mine (Orense, Spain) was received. A detailed characterization of this ore sample can be found in previous research works [15,16,17]. The sample was fully sieved in the following size intervals (µm): 3150/2500; 2500/2000; 2000/1600; 1600/1250; 1250/800; 800/500; 500/400; 400/200; 200/160; 160/100. With adequate blending, using the aforementioned size intervals, nine composite feed samples were prepared to fulfil the requirements posed by the multivariate design. In each case, the composite sample was homogenized and divided, checking by PSD analysis that aliquots verified the requirements in each case (Figures S1–S27 at the Supplementary Materials).

2.2. Methods

2.2.1. Bond Ball Mill Standard Test

The procedure to carry out the Bond grindability test [1,18] is described below. The test is performed in the so-called Bond’s standard ball mill, a laboratory mill 12′′ × 12′′, running at 70 rpm (BICO, San Francisco, CA, USA) with rounded inner edges and without lifters. The grinding charge is comprised of a steel balls distribution;

Table 1. Evolution of the ball grinding charge distributions proposed by Bond.

Ball Charge Distribution 1961				Ball Charge Distribution 1999
Ball Size		Balls		Ball Size		Balls
inch	cm	Number	Weight (g)	inch	cm	Number	Weight (g)
1.45	3.683	43	8803	1.500	3.810	25	5690
1.17	2.972	67	7206	1.25	3.175	39	5137
1.00	2.540	10	672	1.000	2.540	60	4046
0.75	1.905	71	2011	0.875	2.223	68	3072
0.61	1.549	94	1433	0.750	1.905	93	2646
	Total	285	20,125		Total	285	20,592

shows the distribution proposed by Bond in 1961 [5] and that proposed in 1999 [19]; the latter was selected for this test.

The mill feed must be prepared by controlled crushing to 100% passing 6 Tyler mesh (3.35 mm). The first grinding cycle feed must be 700 cm³, and this volume’s weight is fixed as the mill charge in all subsequent cycles. Fresh feed PSD is obtained to calculate the 80% passing size (F₈₀) and undersize weight already present in the feed. The test procedure consists of performing several dry grinding cycles to simulate a continuous closed-circuit operation with a 250% circulating load. The circuit is closed by a sieve (P₁₀₀) selected according to the industrial grinding size target, always between 28 and 325 Tyler mesh (600–45 microns). The detailed grinding cycles procedure can be found in [5,18].

Once finished the grinding cycles, a minimum of five, the ball mill Bond’s work index w_i [kWh/sht] can be calculated using Equation (2). In order to express it in metric tons, the corresponding conversion factor must be used.

$$w_i=\frac{44.5}{P_{100}^{0.23}·gb{p}^{0.82}·\left(\frac{10}{\sqrt{P_{80}}}-\frac{10}{\sqrt{F_{80}}}\right)}$$

(2)

where:

w_i is the ball mill Bond’s work index [kWh/sht];

P₁₀₀ is the mesh size used to close the grinding circuit [µm];

gbp is the grindability index [g/rev].

It has been recently proposed gbp be renamed as the Maxson index [14]. Walter Maxson led the first research in which gbp was named as the grindability index [1], and was also Fred Bond’s mentor at the beginning of his successful career.

2.2.2. Multivariate Experimental Design

The standard test states tight conditions to some test parameters, while others can rest in broad validity ranges. For instance, F₈₀ and P₁₀₀ only limitations are being less than 3.35 mm and 600 microns, respectively. Moreover, the undersize content in the ore feed sample is considered by some authors as a variability source. Accordingly, with the same ore, minor differences under correct sampling procedures or even internal procedures in different laboratories could lead to different w_i values. Following the considerations above, the selected variables to perform a variability analysis on the Bond’s ball mill standard test were the following:

• Feed particle size, F₈₀
• Closing circuit sieve (should coincide with maximum size in the closed-circuit product, P₁₀₀
• Undersize percentage in the feed for each P₁₀₀, % < P₁₀₀

It is important to notice that F₈₀ and the undersize percentage in the feed (% < P₁₀₀) variations could occur easily due to changes in material preparation; changes in P₁₀₀ should be justified due to changes in the ore liberation size, which is not a strange event in mine operations over time.

Table 2. Three levels multivariate experimental design.

Variables		Levels
		1	2	3
F80 (µm)	D	2500	2000	1250
P100 (µm)	C	500	400	200
% < P100 (%)	F	0	10	20

shows the variables coding (D, C, F) and their values (level 1, 2 or 3) in each case. A total of 27 combinations of variables and levels defined the conditions of the 27 Bond standard tests. Enough ore feed was carefully prepared to fulfil D and F requirements (nine different feed samples prepared), and the Bond standard test was carried out at C value of P₁₀₀ (three levels). It must be understood that, with the same ore and with no further specifications, each of the 27 possibilities fulfils the standard test requirements and the corresponding w_i should be considered with the same validity. The basis and practical use of the ANOVA (SPSS, IBM, Amonk NY, USA) test can be found in Navidi [20].

3. Results and Discussion

Table 3. Experimental results of w_i and gbp.

	C1
	D1-F1	D1-F2	D1-F3	D2-F1	D2-F2	D2-F3	D3-F1	D3-F2	D3-F3
wi [kWh/t]	7.82	8.54	8.96	8.69	9.09	9.50	11.25	11.95	12.13
gbp [g/rev]	6.552	6.008	5.668	6.432	6.265	5.809	6.110	6.046	5.773
	C2
	D1-F1	D1-F2	D1-F3	D2-F1	D2-F2	D2-F3	D3-F1	D3-F2	D3-F3
wi [kWh/t]	8.07	8.39	8.45	8.49	8.84	8.80	10.16	10.79	10.69
gbp [g/rev]	5.427	5.220	4.995	5.504	5.383	5.332	5.506	5.377	5.241
	C3
	D1-F1	D1-F2	D1-F3	D2-F1	D2-F2	D2-F3	D3-F1	D3-F2	D3-F3
wi [kWh/t]	8.85	9.15	9.29	9.24	9.33	9.46	10.93	11.01	10.50
gbp [g/rev]	3.300	3.157	3.044	3.264	3.235	3.121	3.087	3.082	3.121

collects the results of Bond work index, w_i determination after performing the resulting 27 Bond standard tests; the Mosher and Tague repeatability estimation was considered adequate [9], lower than ±4% at two standard deviations, after checking it with preliminary tests. In Table 3 the gbp value obtained in each test is also included. Full details of the performed tests can be found in the spreadsheet file provided as Supplementary Materials.

The first glance at Table 3 evidences a variability in both w_i and gbp values; this variability should be explained due to the sole effect of variables combination in each test. It must be highlighted again that feed preparation was performed carefully, and feed variations among synthetic feeds and a naturally taken feed could be similar to those produced in the field sampling process. In all cases, test conditions fulfilled the Bond standard test requirements (which, in passing, are very open; the only limitation is that feed top size must be under 3.35 mm). Therefore, in summary, the different nine synthetic feeds could be the result of different sampling procedures performed on the same deposit without enough representativity, provided that a tailings pond could show differences in the spatial distribution of particle sizes. Results are also depicted in the Supplementary Materials Figures S28–S30 in the case of w_i, and Figures S31–S33 in the case of gbp.

A formal analysis of results was carried out employing the ANOVA test [20], both on w_i and gbp.

Table 4. Analysis of variance (ANOVA) test results on w_i.

Source of Variance	Sum of Squares	Degrees of Freedom	Mean Square	F-Ratio	p-Value
Main effects
C	1.9777	2	0.9888	44.99	0
D	30.2763	2	15.1381	688.76	0
F	1.1734	2	0.5867	26.69	0.0003
Interactions
C&D	2.2241	4	0.5560	25.30	0.0001
C&F	0.5885	4	0.1471	6.69	0.0114
D&F	0.1495	4	0.0374	1.70	0.2422
Residual	0.1758	8	0.0220
Total (corrected)	36.5654	26

garners the ANOVA test results in the case of w_i.

Table 4 breaks down the variability of w_i into contributions due to individual variables effects and the binary interactions among them. Considering the sum of squares values and p-values in the case of individual variables and binary interactions, variable D (F₈₀) is identified as the primary source of variability among the studied ones. The second source of variability stems from C and D interaction, that is, F₈₀ and P₁₀₀ combined effect, which surprisingly has more significant influence than C alone effect. From a w_i variability point of view, F (undersize feed content) was identified as the third variable in importance. In the case of D and F interaction, the p-value is not less than 0.05, so this combination does not have a statistically significant effect on w_i, at the 95.0% confidence level.

Similarly, another ANOVA test was carried out on Maxson grindability index values, and the results are provided in

Table 5. ANOVA test results on gbp.

Source of Variance	Sum of Squares	Degrees of Freedom	Mean Square	F-Ratio	p-Value
Main effects
C	41.3668	2	20.6834	3653.30	0
D	0.0724	2	0.0362	6.39	0.0220
F	0.5276	2	0.2638	46.59	0
Interactions
C&D	0.0656	4	0.0164	2.90	0.0937
C&F	0.1921	4	0.0480	8.48	0.0056
D&F	0.0941	4	0.0235	4.15	0.0413
Residual	0.0453	8	0.0057
Total (corrected)	42.3638	26

. In this case, variable C (P₁₀₀) is identified as the most relevant source of variability; despite D, F and C and F having a p-value more than 0.05 (in consequence, they have a statistically significant effect on gbp, at the 95.0% confidence level), the difference in the sum of squares values lets us affirm that C can be considered as almost the only source of variability in this case.

Results suggest that, under the conditions considered in the multivariate design described, the Maxson grindability index, gbp, represents more robustly the intrinsic grindability properties of the ore, being its source of variation the Bond standard test condition, P₁₀₀. This result reinforces the concept, first proposed by Maxson et al. [1] and subsequently adopted and disseminated by Bond [3,4,5], that gbp was the best index in characterizing the ore comminution amenability. This fact also justifies the proposal of renaming gbp as the Maxson grindability index.

On the other side, Bond work index variability has a more profound influence from feed PSD conditions (mainly F₈₀ value), even to a far greater extent than P₁₀₀ values. As the standard test established relatively frugal recommendations about feed PSD conditions (maximum feed size, F₁₀₀, less than 3.35 mm), it can be qualified as a worrying source of w_i variation, and the following additional recommendations should be taken into account:

• To establish desirable Bond test conditions, always consider performing feed preparation according to the planned/expected industrial conditions (for instance, by product size estimation on the previous comminution stage—fine crushing or coarse grinding);
• When reporting w_i results, P₁₀₀ and F₈₀ values in the test should always be indicated, especially F₈₀, which seems more responsible for w_i variability than P₁₀₀ itself.

4. Conclusions

The following conclusions were derived from this research work and considering the tested ore:

• It was evidenced that the considered parameters induced variability in both Bond work index, w_i, and Maxson grindability index, gbp.
• The ANOVA test results suggested that, in the case of w_i, the primary source of variability is F₈₀, followed by the binary interaction F₈₀ and undersize (<P₁₀₀) feed content.
• In the case of gbp, the ANOVA test showed that almost the only source of variability is P₁₀₀, with almost no influence of feed PSD.
• The following additional recommendations should be taken into account:
• To establish desirable Bond test conditions, always consider performing feed preparation according to the planned/expected industrial conditions
• When reporting w_i results, P₁₀₀ and F₈₀ values should always be indicated in the test.