Comparison between K and alpha sampling constants with the Pierre Gy’s factors for bauxite

Abstract Sampling is an important tool for gathering information about mineral deposits, helping to estimate the ore grade, and has great relevance at any stage of a mining project, to ensure reliable results and consequently, to project a better mining planning. There are several errors to which the sampling is exposed and among them we can mention the fundamental sampling error (FSE). This error cannot become zero as it is related to the constitutional heterogeneity of the ore, so it represents the minimum sampling error. This study applies Pierre Gy’s theory of sampling (TOS) to estimate the values of K and α sampling constants at different deposits of bauxite in Brazil and compare them with the calculated factors, then calculate the relative standard deviation of the fundamental sampling error s(FSE) based on the results of the heterogeneity test (HT) in each deposit. The deposits are located in: Barro Alto - State of Goiás, Juruti - State of Pará, and Poços de Caldas - State of Minas Gerais. The weathering process is the main reason of the alteration of the physical and chemical factors diversifying bauxites in Brazil and worldwide. However, when sampling is carried out correctly, the Gy’s formula becomes a powerful tool for improving mining processes as long as the factors are estimated correctly for each ore.


Mining Mineração
Aluminum ore, or simply bauxite, was discovered in 1821 by the Frenchman Pierre Berthier, who discovered it near the village of Lês Baux, in the south of France.It is a reddish land that containes high-grade aluminum ore, with about 40% alumina (Al 2 O 3 ) (Sampaio et al., 2008).Currently, bauxite is the main raw material in the production of aluminum metal, being composed basically of gibbsite, diaspora and boehmite, which are considered aluminum oxyhydroxides (Arenare, 2008).
According to Monteiro and Silva (2018), the total of the world's bauxite reserves is around 27.9 billion tons, with Guinea and Australia being the largest holders.Brazil is ranked third, with approximately 2.7 billion tons.Among the states of the federation, Pará represents 89.5% of the national bauxite production, and has great potential for growth.
In the case of the evaluation of mineral deposits, or process control and marketing of the product, the importance of sampling is irrefutable (Luz et al., 2004).
Sampling aims at the estimation of a quality parameter of a population, such as the grade.However, the mineral constituents of a deposit vary from point to point, making it unlikely that a single sample accurately represents the overall composition of the sampled unit (Arioli and Andriotti, 2007).It should be noted that bad sampling can result in considerable losses or distortions of results, with unpredictable technical and financial consequences.Sampling is one of the most complex operations and can easily introduce errors in the metallurgical and mining industries (Luz et al., 2004).
The current emphasis on understanding the natural variability of the deposit and source of sampling errors has arisen out of the theory of sampling (TOS) by Pierre Gy, who, in 1951, wrote an article entitled "Minimum mass of a sample needed to represent a mineral lot" (Minnitt et al., 2007).
According to Gy (1998), the representativeness must be maintained at all stages of the sampling process.By ensuring the precision and accuracy of the samples taken at all stages of the mineral chain, the results will be auditable and reproducible.
Among the sampling errors that contribute to the non-representativity of the samples, Pitard (2005) listed ten: in situ nugget effect (NE); fundamental sampling error (FSE); grouping and seg-
According to Gy (1982) the total sampling error (TSE) can be divided into separate components: random errorswhich can be reduced, but never eliminated = NE + FSE + GSE + QFE1 + QFE2, and systematic errors -which can be eliminated when correct sampling procedures are performed = IDE+IEE+IWE+IPE+AE.
The fundamental sampling error is one of the errors that cannot be eliminated because it is related to the constitutional heterogeneity of the ore, which has been pointed out as the source of all sampling errors (Gy, 1998).It is worth noting that the constitutional heterogeneity constant factor IH L can be calculated by obtaining several other factors, according to Gy's formula in Equation (1) (Minnitt et al., 2007):

Bibliographic survey 2.2 Heterogeneity test (HT)
For the calibration of the K and α sampling constants, the methodology proposed by Pitard (2004) was applied for indirectly estimating the constitutional heterogeneity constant factor (IH L ).
Initially, the samples from deposits 1 and 2 were collected by the companies in different quantities so that they could be submitted to HT.Moreover, they were crushed and arranged in an elongated pile (Fig. 1) for homogenization and primary quartering.The samples from deposit 3 were the same as those used in the article from Alves et al. (2020).
The ideal mass to perform the test was collected from the pile, with this being variable from company to company.In sequence, the fraction collected was screened into four size fractions for the samples from Barro Alto and Juruti, and in three fractions for the sample from Poços de Caldas, resulting in the nominal top-size of the fragment (d N ).It is worth mention-ing, that the native bauxite from Poços de Caldas has a very fine granulometry when compared to the typical bauxite from other Brazilian states and, therefore, only three size fractions were used (Alves et al., 2020).
According to Pitard (1993), the method consists in the extraction of a Q number of fragments, randomly collected from all size fractions of the investigated material.In the case of ores with metal of interest in low concentration, Q ≥ 100 is used.For ores such as bauxite, with high concentration, Q ≥ 50 is used.It is suggested that the number (N) of fragments present in each particle size fraction be at least 10 times larger than the number of fragments (Q) multiplied by the number of groups collected (p), Equation (2): After confirming the total number of fragments per particle size fraction, an individual fragment was weighed, and the minimum sample mass was calculated to compose each fraction.Each sample had to be homogenized and placed in a matrix with a square mesh (totalling 50 squares) (Fig. 2).Care should be taken to avoid over-lapping fragments or even empty spaces; in other words, the fragments should be scattered so that every fragment has the same chance of selection.where c is the mineralogical factor, f is the shape factor, g is the granulometric factor, and l is the liberation factor obtained using the liberation diameter d l , d N is the fragment nominal diameter or top size, M S is the sample mass, and M L is the lot mass.As mentioned, these factors can be replaced by the constant K.However, due to the great difficulty in estimating these factors, HT is used, which allows the calculation through the calibration of the heterogeneity parameters K and α, where K=cfgl and α = exponent of d.

N > 10 Qp
In summary, the objective of this study was to estimate the values of K and α sampling constants in different deposits of bauxite from Brazil, and to compare them with the calculated Gy's factors for each deposit.
As mentioned in the previous section, a bibliographical survey was first carried out for the characterization of the bauxite deposits of the three study sites: Barro Alto, Juruti and Poços de Caldas known as deposits 1, 2 and 3, respectively.
The bauxite deposit located in the region of Barro Alto, state of Goiás, was discovered at the end of the 90s (Oliveira, 2011).Its genesis occurred on neoproterozoic anorthosites (Santos, 2011), located in the northern portion of the Mafic Ultramafic Complex of Barro Alto.
According to the climate classification of Köppen-Geiger (Kottek et al., 2006), for being in a tropical climate, hot and rainy, the weather had a great impact on the anorthosite, generating deposits of bauxite with grades above 50% of usable Al 2 O 3 and a reserve estimated at 100 million tons.This deposit is unique, complex, and peculiar, with a profile of alteration represented by a bauxite saprolite located close to the important nickel laterite deposit of the same municipality (Santos, 2011).Santos (2011) states that the profiles are highly evolved, constituting a body of ore that has five lithologies from top to bottom: A. Colluvium bauxite: fragments of bauxite disaggregated from the preexisting bauxitic facies.
B. Massive bauxite: formed from the remobilization of aluminum in a second phase of bauxitization, in which the microcrystalline texture of gibbsite fills the pores and fractures of primary bauxite.
C. Clay with bauxite blocks: constituting the portion in which a higher volume of kaolinite is associated with gibbsite.
D. Porous Bauxite: considered as isalterite with preserved structures of the anorthosite.
This bauxite deposit is located in the sub-basin of the lower Amazon, Alter do Chão formation, in the town of Juruti, state of Pará (Carvalho, 1989).According to Patterson (1967) and Carvalho (1989), the Amazonian bauxite deposits are classified as "Blankets", formed "in situ" by processes of weathering on clastic sedimentary rocks in humid tropical climate conditions, Köppen-Geiger classification in Kottek et al. (2006).These deposits have thicknesses that vary up to 10 meters, presenting notable lateral variations in the contents of aluminum, silica, and iron.
The typical lithological profile of the deposit 2, from top to bottom, is described as in Bortoleto et al. (2014): A. Yellow clay (known as Belterra): consisting of a very uniform and permeable cover of kaolinitic yellow clay.
B. Nodular bauxite: composed of fine gibbsite nodules of varying sizes and distributed in a kaolinitic matrix.
C. Laterite: corresponding to a low silica horizon, presenting considerable variations in hardness, texture, color, iron-aluminum ratio, and silica grade, which can be classified as ferruginous bauxite in some places.
D. Massive bauxite (bauxite layer): horizon composed of gibbsite, hematite, and kaolinite with economic value.The hardness, texture and color of the material varies according to the iron and clay content.
E. Variegated clay: consisting of a horizon composed of variegated kaolinitic clay, usually reddish to pink, with whitish spots and purple hues, possibly containing gibbsite.

Deposit 2, State of Pará
In the next step, one fragment of each square was collected, composing a group, or sub-sample, comprising 50 fragments.The previous procedure was repeated 50 times, forming 50 groups of at least 50 fragments per size fraction.The groups of fragments or sub-samples were then identified, weighed, separated, and placed in plastic bags.Each group was weighed to obtain the mass (M q ) and sent for chemical analysis for determination of the grade (a q ) of usable alumina.All the data were tabulated and used for the construction of the graphs that will be discussed later.
The alkaline plateau of Poços de Caldas has an area of approximately 800 km², being located on the border between the states of São Paulo and Minas Gerais.The region is composed basically of phonolites and nepheline syenite, which makes this plateau one of the largest alkaline plateaus in the world (Ellert, 1959).
The bauxite deposits of this region are associated with alkaline intrusive rocks and are classified into two types: massive edge bauxite and plateau baux-ite.The bauxite formation in the area is reported as a direct and continuous process whose most important factors of influence are the topography, the nature of the bedrock, and the climatic conditions (Carvalho, 1989).
It is worth mentioning that the main mineral found in the region is gibbsite which is located mainly in the north of the plateau.In addition, the bauxite profiles can be classified as (Leonardi et al., 2010): A. Reworked bauxite profiles: presence of bauxite gravel, similar to those found in mountain and plain profiles.
B. Bauxite profiles of mountain: located in the high altitudes of the plateau, they have a high concentration of aluminum, and are thick and well developed.
C. Bauxite plain profiles: located in the low altitudes of the plateau, they have small thickness, a low grade of aluminum, and are very clayey materials.
In order to obtain the fundamental sampling error and the values of the K and α constants, the grades (a q ) of available alumina and the masses (M q ) resulting from the 50 samples obtained from each of the established size fractions were first determined.The size fractions used were (Silva, 2019): • -38.1 mm + 25.4 mm; • -25.4 mm + 12.7 mm; • -12.7 mm + 6.3 mm; • -6.3 mm + 1.2 mm.The four size fractions were used for the bauxite samples of Barro Alto and Juruti (Bortoleto et al., 2014) and the last three for the aluminum ore from Poços de Caldas (Alves et al., 2020).
After this, the value of the estimated IH L was calculated for each of the size fractions according to Equation (3) (Pitard, 1993).It is worth remembering that the value to be used for the granulometric factor (g) is 0.55 for calibrated materials (Chieregati and Pitard, 2018).
where, EST IH L = estimated constitutional heterogeneity constant factor, g = granulometry factor, a q = individual sample grade, a Q = average grade, M q = mass, and M Q = total mass.
To obtain the values of M Q and a Q , Equations (4 and 5) were used (Koyama et al., 2010).
Then, it was necessary to obtain the nominal top size (d N ) of the fragments in the different size fractions to establish a correlation with the constant factor of heterogeneity, according to Equation ( 6) (Minnitt and Assibey-Bonsu, 2010).The results are shown in Table 1.
Subsequently, the K and α constants were estimated by plotting the bi-log graph and power regression lines of the results in Table 1.
The trend line of the graph provides an equation that allows calibrat-in which d1 and d2 represent the openings of the upper and lower sieves of each size fraction.
After obtaining the sampling constants, the relative standard deviation of the fundamental sampling error s(FSE) can be determined.Pitard (1993) suggests a maximum standard deviation of: • s(FSE) ≤ ± 0.5% for commercial sampling.
• s(FSE) ≤ ± 16.0% for environmental and exploratory sampling.ing the K and α sampling parameters for the different bauxite deposits, replacing the Gy's factors according to Equation ( 7): 3.3 Comparison between sampling protocols using Pierre Gy's factors and HT

Conclusion
Table 3 shows the values of Pierre Gy's factors (Pitard, 1993) for gibbsite bauxite described in Equation (1).The mineralogical factor (c) consists of two components, the densities of the mineral of interest and gangue, and the lot grade.In this case, the factor c was admitted as being a standard value as shown in Table 3, since all alumina refineries in Brazil use gibbsite bauxite (Bortoleto et al., 2019).This article presents two different methods to estimate the relative standard deviation of the s(FSE) by applying the Pierre Gy's formula and using the results of the HT.HT allows obtaining the sampling parameters K and α, which were compared with Gy's estimated parameters of the gibbsite bauxite.
Through the theory of sampling (TOS), it was possible to determine the estimated constitutional heterogeneity constant factor (EST IH L ) present in each step of a sampling protocol and find the associated errors.An appropriate sampling protocol for any type of ore ensures the reliability of the process.
The evaluation of sampling bias should be done by incorporating all the principles discussed here to guarantee that the sampling process is equi-probabilistic.When dealing with sampling variability, the s(FSE) in each step of the sampling protocol should be observed to diagnose the critical sampling stages and act proactively to minimise the total error, and moreover, it is important to determine what is the minimum M S to represent all size fractions.
In Table 2, the comparison between K and α parameters allow to conclude that there is a greater heterogeneity associated with the deposit 1.In the bauxite from deposits 2 and 3, the K and α parameters obtained experimentally were close to the estimated parameters using Gy's factors: K=0.0085 and α=2.5, resulting in identical total values of Gy's s(FSE) and HT s(FSE) for the bauxite from Poços de Caldas.
The primary quartering step of the sampling protocol from deposit 1 showed greater contrast because of intrinsic heterogeneity.Therefore, it is possibly incorrect to assume the mineralogical factor (c) as a standard value for gibbsite bauxite in Gy's formula, as shown in Table 3.
Thus, it can be concluded that the biggest influence on the physical and chemical characteristics of the bauxite is the impact of the weathering process.
As part of the last step of this study, the relative standard deviation of the fundamental sampling error s(FSE) was estimated using the Pierre Gy's factors as well as the parameters of K and α obtained experimentally by the HT.The standard deviation was calculated for each step of the sampling and sample preparation protocol for each of the three deposits as shown in Table 4.
Considering Equation (7), a maximum relative standard deviation of the fundamental sampling error s(FSE) of 5.0% and a high value for the initial lot mass (M L ), the minimum sample mass (M S ) to represent the pri-mary sample at the mine, where d N is 5 cm, was calculated and is presented in Table 5.Table 4 -The sampling protocol and the relative standard deviation of the s(FSE) of each stage using Pierre Gy's factors as well as the results of HT.

Parameter
Table 5 -Minimum sample masses (M S ) for the primary sample at the mine.
Fundamentally, the climatic factor and the exposure time in which the parent rock was subjected to adverse conditions of temperature and pressure can alter the shape, granulometric, mineralogical, and liberation factors, diversifying bauxites in Brazil and worldwide.
Optimizing sampling protocols is not a simple task.The theory of sampling proposed by Pierre Gy suggests that additional experiments are necessary to complement the ones pro-posed in international standards and to increase the reliability of the sample results, although the Gy's formula proved to be very effective in optimizing bauxite sampling protocols, mainly for deposits 2 and 3.

Figure 1 -
Figure 1 -Example of the elongated homogenization pile, where x represents width, z length and y height.Source: Drawn by the author using SketchUp Software.
First, a bibliographical survey was carried out to survey the mineral-ogical characteristics of the ore from three bauxite deposits of different geological formations, belonging to different companies.

Figure 2 -
Figure2-Illustration of the square mesh for arrangement and collection of the fragments that make up the final sample whose dimension x is variable.Source: Drawn by the author using SketchUp Software.
Estimated parameters K and α for each bauxite mine in this study.where s²(FSE) represents the variance of the fundamental sampling error, K is the sampling constant for each ore type, d N the nominal diameter, α is the exponent of the original cubic formula of Pierre Gy, and M S and M L are, respectively, the masses of the sample and of the lot.The bi-log correlations are shown in Fig. 3, Fig. 4, and Fig. 5.

Figure 3 -
Figure 3 -Correlation between EST IH L and d N of the deposit 1.

Figure 4 -
Figure 4 -Correlation between EST IH L and d N of the deposit 2.

Figure 5 -
Figure 5 -Correlation between EST IH L and d N of the deposit 3.

Table 1 -
Heterogeneity factors correlated to each nominal topsize.