Optimal Estimation of Mass Partition in a Centrifugal Separator by Metallurgical Balance

Mineral processing plants make use of several types of equipments for size separation (size classification) to segregate ore particles by size. A particular type of such equipments is the centrifugal separator, which is intended to receive an input stream of ore slurry to be partitioned into two output streams: a coarse and a fines one. The coarse stream contains most of the coarse solids particles of the slurry, whereas the fines stream contains most of the fine particles. Although a centrifugal separator intends to perform a physical segregation of the solids particles by their size, a chemical segregation also results, in such a way that the chemical content of the coarse and the fines streams are normally different from the content of the input stream. When evaluating the performance of the separation process, three fundamental aspects should be analyzed: 1) the size distribution of the solids particles in each stream; 2) the amount of solids mass from the input stream that goes to the coarse stream and to the fines stream, that is, the mass partition; and 3) the chemical content of each stream. This work presents the application of the Least Squares method of optimization to calculate the mass partition, based on the measured chemical content of the streams, and on the metallurgical balance equations of the separation process. Index Terms — least squares, mass balance, mass partition, metallurgical balance, size separation, optimization.


I. INTRODUCTION
AUXITE ore is the basic raw material for the aluminium production chain [1], which has four major steps, normally implemented as separate industrial plants: 1) processing of the raw ore from the bauxite mines, to produce concentrated ore; 2) refining the concentrated bauxite ore through the Bayer process [2], to produce high-purity smelter-grade alumina; 3) electrolytic reduction of the smelter-grade alumina in electrolytic furnaces, to produce primary aluminium in the form of ingots, billets, or slabs; and 4) smelting and conformation of the primary aluminium, to generate aluminium products.
The main chemical compound of interest in the bauxite ore is the alumina (Al2O3), but some other chemicals are also considered to characterize the chemical profile of the ore: silica (SiO2), iron dioxide (FeO2), and titanium dioxide (TiO2).The concentrations of chemicals in an ore can be measured by specific methods like X-ray mass spectrometry, atomic absorption spectroscopy, inductively coupled plasma spectrometry, or chemical reaction methods.Moreover, a certain amount of the content is characterized as "loss on ignition" (LoI), which represents the amount of moisture or impurities lost when the ore is ignited under specific conditions.For purposes of chemical content characterization, the LoI can be regarded as an additional chemical compound.
The Bayer process is the most widely used in industrial scale to produce high-purity alumina from bauxite ore.Only part of the total alumina in the ore, called valuable alumina, can be extracted by reagents in the Bayer process, to produce smeltergrade alumina.And conversely, a certain portion of the total silica in the ore, known as reactive silica, is a harmful compound because it competes with the valuable alumina by the reagents, thus impairing the extraction of this latter.
Due to this fact, bauxite ore plants are required to produce, as input for the Bayer process plants, concentrated ore with higher content of valuable alumina and lower content of reactive silica.This requirement should be extended to all mineral processing equipments in an ore concentration plant.Another important requirement is that the equipments should be able to convert most of the feeding (input) ore into product (output), thus reducing the amount of reject (waste) material.The ratio between the product and the feeding masses is referred as massic recovery, and is a measure of equipment performance.
In this context, this article describes the determination of the mass partition performed by a centrifugal separator [3].The main contribution of this work is to demonstrate how to determine the the massic recovery of a mineral processing equipment from chemical compound measurements of the streams, when it is not feasible or impractical to directly measure the mass of ore in the streams.The article is organized as follows: Section II presents the material processing performed by a centrifugal separator with three streams (feeding, product, and reject).Section III the metallurgical balance problem relating the mass partition and the chemical compound measurements of the streams.Section IV describes the solution of the metallurgical balance problem using an optimization method, in order to compute the mass partition performed by the centrifugal separator.Finally, Section V discuss the results obtained.

II. PROCESSING OF BAUXITE ORE IN A CENTRIFUGAL SEPARATOR
Mineral processing plants make use of several types of size classification (size separation) equipments to segregate ore particles, either in the form of bulk solids or slurries, by size [4].Size separation is needed due to several reasons like: 1) to feed coarse particles back to breakage processes; 2) to meet product size requirements; or 3) to segregate different mineral species in the ore to meet chemical content requirements.For the processing of ore slurries, a particular type of classification equipment is the centrifugal separator [3], which is intended to divide an input stream of ore slurry into two output streams: a coarse stream and a fines stream.The coarse stream contains most of the coarse solids particles of the slurry, whereas the fines stream contains most of the fine particles.Fig. 1 shows an example of centrifugal separator for industrial applications, and a block diagram of its separation process.
The size separation process performed by a centrifugal separator (indeed, by any type of separation equipment) is not perfect, so that it will exist a certain amount of fine particles in the coarse stream, as well as a certain (although very much lower) amount of coarse particles in the fines stream, as illustrated in Fig. 2. The main goal of a centrifugal separator is to produce a coarse stream with considerably less fine particles than the input stream.The size distribution of the particles in the coarse and the fines stream will depend not only on the separation performance of the equipment, but also on the size distribution of the input stream.
Although a centrifugal separator intends to perform a physical segregation of the solids particles by their size, a chemical segregation also results, in such a way that the chemical contents of the coarse and the fines streams are normally different from the content of the input stream.This is because the coarse particles typically result from the hardest mineral species in the ore, whereas the fine particles result from the softest mineral species.The main mineral species that form our bauxite ore are the gibbsite and the kaolinite.The first is hard and contains most of the valuable alumina, while the latter is soft and contains most of the reactive silica.
Table I shows the chemical contents of each stream in a centrifugal separator, measured by inductively coupled plasma spectrometry, for nine tests.Note that, for each stream, the sum of the concentrations of all its chemicals, including the LoI, does not result equal to the ideal value 100%.This is due to: 1) the existence of other disregarded chemicals with very small concentration values, which are not representative of the ore, and so were not taken into account in the content measurements; and 2) measurement errors from the plasma spectroscopy (even if those errors are very small, they will actually exist).The values that make the sum of the concentrations of all the chemicals exactly equal to 100% are the residue values.

III. THE METALLURGICAL BALANCE PROBLEM
Let  denotes any chemical compound of interest in the bauxite ore, that is,   {Al2O3 , SiO2 , FeO2 , TiO2 , "LoI"}.Moreover, let mI, mC, and mF denote the input, coarse, and fine solids mass, respectively; and cI, cC, and cF denote the concentration of any chemical compound  in the input, coarse, and fines stream.
The performance of a mineral processing equipment is often characterized by its metallurgical recovery, which is a measure of how much a chemical compound pass from the input stream to an output stream.As an example, the metallurgical recovery of alumina to the coarse stream is given by: In a generic form, the metallurgical recoveries of any chemical compound , for the coarse and the fines stream are, respectively: As described in the previous section, although a centrifugal separator performs a physical segregation of the solids particles by their size, a chemical segregation also results.If each stream in a separator is characterized by its solids mass and its chemical content, a problem that arises is: by knowing (measuring) the chemical content of the three streams, is it possible to determine the solids mass in each stream, allowing the calculation of the metallurgical recovery?To answer this question, one needs to understand the material balance equations of the separation process.As the input mass mI is partitioned by the centrifugal separator to the coarse solids mass mC and the fine solids mass mF, with no accumulation of mass inside of the equipment, the following mass balance relationship must be satisfied: Moreover, the metallurgical balance equation for any chemical compound  is: Therefore, for all the chemicals of interest, the following set of equations is derived from (4) and ( 5): The goal would be to solve (6) for {mC, mF, mI}, by knowing the content values {cC(), cF(), cI()} for all chemicals, and then calculate the metallurgical recovery by (1).Nevertheless, it's not possible to find an unique solution {mC ≠ 0, mF ≠ 0, mI ≠ 0} for (6), because it is a homogeneous set of equations and the chemical content values have some measurement errors, so that the only possible solution would be the trivial solution {mC = 0, , where k is a positive real number, is also a solution.In other words, there will be infinite solutions, and any particular solution will be found only if one of the three masses , allowing to solve the set of equations for the two others.Fortunately, the terms mC/mI and mF/mI in ( 2) and (3) make unnecessary to know the actual values of these masses, allowing to work in percent basis, by setting mI = 100% and then solving the set of equations for {mC, mF}.By this way, equation ( 6) can be rewritten as:  (7) or in matrix form: where C is the matrix of output concentrations, M is the vector of output masses (to be determined), K is the vector of input concentrations, N = 6 is the number of equations in (7), and mC and mF are in percent basis.Now, the goal is to solve (8) to find M = [mC mF] T .

IV. SOLVING THE METALLURGICAL BALANCE PROBLEM BY OPTIMIZATION
Due to the fact that the chemical content values {cC(), cF(), cI()} in ( 8) have some measurement errors, and that C is not a square matrix, the best way to solve (8) for M is by using an optimization method, like the Least Squares method [4;5;6;7].This is explained in the following.Let M ˆ denotes an estimate of M, and M C K   the corresponding estimate of K, according to (8).The estimation error is given by: Each component ei in vector E is the error between the estimated value i k ˆ in K ˆ and its corresponding measurement ki in K.As an indicator of the quality of the estimate M ˆ, one considers the mean squared error, denoted by J: The principle of Least Squares states that M ˆ will be the best or optimal estimate if it minimizes the error indicator J, that is: Therefore, the optimal solution for ( 10) is given by: The solution M ˆ in (12) will only exist if C T C is non-singular (invertible).Normally, the measurement errors in the chemical content values will cause C T C to be non-singular.Equation ( 11) means that matrix C T C must be non-negative definite.If one wants that the solution M ˆ makes the first equation in (7) to be exact up to some degree of accuracy, one can consider an extension of the basic Least Squares, called Weighted Least Squares [4,5], which includes a diagonal matrix W whose diagonal elements are weights for each individual equation in (7).In this case, the optimal solution is given by: where  should be a high value empirically chosen to make the first equation in (7) as exact as desired.Typical values for  are 10 5  10 9 .Equation ( 12) is a particular case of (13), when W is an identity matrix, that is, when all individual equations have the same weight, equal to the unity.
Although the Least Squares method is simple, it is very sensible to the presence of outliers in the measured data (the chemical content values).Outliers are data values containing high measurement errors, which will impair the accuracy of the solution M ˆ given by (13).Therefore, it's important to assure that the measured chemical content values have a good accuracy.A simpleand limitedway to check the accuracy of the chemical content values is by the magnitude of their residues (Table I).Although a low magnitude of a residue does not mean the absence of high errors, if a residue has a significant magnitude, then at least one chemical content value has a high error.In this case, a more accurate measurement of that chemical content values should be performed.

V. RESULTS
The computational implementation of the Least Squares method to evaluate (13) was done in VBA™ (Visual Basic for Applications), for use in MS-Excel™.
Firstly, it was investigated the influence of  on the solution M ˆ, according to (13).Several values for w11 =  were tried, in order to satisfy the first equation in (7).This was achieved for w11 = 10 8 , or higher.Table II shows some results, for the data from test T-01 in Table I.Note that when the total mass reaches the ideal value 100%, the coarse and the fine solids masses remain unchanged even if  is further increased.
Upon verified the influence of  on the results, it was fixed to  = 10 8 .Then, the estimate of M ˆ and its respective error indicator J were calculated for all the measurement tests, as shown in Table III.The calculated masses of ore in the coarse and fines streams are expressed in percent of the feeding stream, assumed as 100% according to (7).The estimates of M ˆ were then used to calculate the metallurgical recoveries of alumina and silica, according to (2) and (3).
To better understand why the calculated masses in Table III are the "best" or "optimal" estimates for M ˆ, Fig. 3 shows the error indicator J as a function of the masses mC and mF, for Test T-01.The shape of J depends on matrix C and vector K, according to equations ( 9) and (10), and it clearly indicates that J has a global minimum.This minimum is indicated by a dot in Fig. 4, which is just the optimal estimate for M ˆ, as stated by the principle of Least Squares.Moreover, the straight diagonal gray line in Fig. 4 represents the constraint mG + mF = 100, which is the first equation in (7).The optimal estimate will stay on that line whenever it meets this constraint, by a proper choice of w11 = .
As the quality of the results (estimates of the coarse and the fine solids mass) are influenced by the quality of the data (measured chemical content values), and considering that the  sum of the concentrations of all the chemicals is not equal to the ideal value 100% (Table I), the following question arose: as the residue values cause the sum of the measured chemical content values to be exactly 100%, will the results be improved if the residue values are also considered in the computations?To answer this question, one needs to include the following equation into (7): This is like to consider the residues as an additional chemical compound that allows the sum of the measured chemical content values to be exactly 100%.
The results obtained for M ˆ after including (14) in (7) are shown in Table IV, together with the results from Table III

VI. CONCLUSION
This article presented the application of the Least Squares method to estimate optimal values of the mass partition performed by a centrifugal separator, according to its material balance equations, in order to evaluate its metallurgical   recovery of alumina and silica.The methodology presented should be useful when it's not feasible or impractical to measure the actual mass values processed by the equipment.In this case, one can simply consider the masses as a percent value of the input stream, according to equation (7).The methodology can also be used for other types of equipments with similar processing characteristics.
Although the main result of interest is the mass partition estimate M ˆ, the error indicator J is also an important parameter, as it measures the quality of the estimate.Besides the usual formulation of the Least Squares method presented in this article, an alternative formulation based on Lagrange Multipliers can also be used to solve the optimization problem presented.The user can choose either method for use depending on his familiarity with the mathematical formulations, the complexity of the application problem, and the computational resources to implement the method.
, for better comparison.The value of J increased significantly for almost all the tests, except for T-01 and T-08.The mean and the standard deviation of J for all the nine tests, indicate that the estimates of M ˆ are normally impaired when the residue values are taken into account in the computations.Therefore, the final optimal estimates of M = [mC mF] T are those calculated without the residues.

Fig. 4 .
Fig. 4. Level curves of J with the point at which J is a minimum.

Fig. 3 .
Fig. 3. Shape of the error indicator J, for the data from test T-01.

TABLE I MEASURED
CHEMICAL CONTENTS OF EACH STREAM IN A CENTRIFUGAL SEPARATOR, FOR NINE MEASUREMENT TESTS.

TABLE III ESTIMATES
OF THE COARSE AND FINE SOLIDS MASSES, ERROR INDICATOR J, AND METALLURGICAL RECOVERIES OF ALUMINA AND SILICA.

TABLE II INFLUENCE
OF  ON THE TOTAL MASS, FOR THE DATA FROM TEST T-01.

TABLE IV INFLUENCE
OF THE RESIDUE VALUES ON THE ESTIMATES OF M ˆ.