Circulating load estimation in closed circuits André Carlos Silva* Elenice Maria Schons Silva** Ricardo Antonio de Rezende***
* Goiás Federal University, Catalão, GO, 75704-020 Brazil (Tel: 55(64)3441-5327; e-mail:
[email protected]). ** Goiás Federal University, Catalão, GO, 75704-020 Brazil (e-mail:
[email protected]) *** Goiás Federal University, Catalão, GO, 75704-020 Brazil (e-mail:
[email protected]) Abstract: A problem for solving mass balances in mineral processing plants is the calculation of circulating load in closed circuits. A family of possible methods to the resolution of this calculation is the iterative methods, consisting of a finite loop where each iteration the initial solution is refined in order to move closer to the exact solution. The present work shows a low-complexity iterative algorithm for circulating load calculation in mineral processing closed circuits, thus enabling the construction of mass, metallurgist and water reliable balances. The proposed equations on the algorithm were obtained through the analysis of many industrial systems, taking into account the process operational parameters. A validation was performed with real industrial data, in order to ensure a greater reliability of the obtained results. Four different types of closed circuits are presented, each one with different levels of complexity, to clarify the proposed algorithm. With the obtained results it is possible to affirm that the proposed iterative algorithm can be successfully applied to any kind of closed circuit in mineral processing. The results obtained were satisfactory with respect to processing speed, convergence of the solution and the number of iterations required for the circulating load calculation. Keywords: Process Keywords: Process modeling, control and optimization; Advanced process control; Mineral processing.
1. INTRODUCTION The mass balances in mineral processing circuits are based on the equation of mass conservation, given by:
(1)
Where F Where F is is the flow (mass or volume) entering the system; C is the flow (mass or volume) of concentrate and T is is the flow (mass or volume) both coming out of the s ystem. The equation 1 assumes theoretical conditions streaming operation, whose fundamental assumption is that the mass t hat enters the system is equal to the mass that comes out of the system (Lavoisier´s law of mass conservation). Data obtained in technological characterization tests or through sampling campaigns in mineral processing plants must be consistent and reliable, being common the reconciliation of experimental data using simulation software. To this end, the mass balance of the circuit is used to confirm the quality of the data and the use of computational simulation aims to extend the knowledge of the industrial processes function and as a tool for studies of process optimization. The circulating load can be defined as a process flow (mass or volumetric) of a given material that returns to a unit operation after failing in some selection criteria. This disapproval may be due to an imperfection in the selection equipment or due the material being out of the specification accepted by any subsequent step in the process. So, while the material does not
suit the specifications of subsequent steps it will be forwarded to this unit operation. Taking the most common circulating load as example, a closed-circuit grinding (see figure 1), after going through the mill the pulp feeds a hydrocyclone for a particle size classification. The hydrocyclone overflo w heads for the subsequent step of the process and the underflow, which size is above the acceptable to follow the process, return to the mill to be comminuted again. In mineral processing plants is common practice operating mills with circulating load above 400%, that is, the material flow returning to the mill is about four times the flow of new material fed into the mill.
Fig. 1. Grinding circuit closed by a hydrocyclone. The circulating load correct calculation in a mass balance is essential to the understanding of the studied circuit, to build reliable simulations of the circuit and for the operational control. Chen et alli (2008) working with predictive control
models applied to ball mills stated that the stable control of grinding process has great importance to achieve improvements in equipment operation efficiency, for the valuable minerals recovery and significant costs reductions in production in concentration. Lestage et alli (2002) showed a supervisory system for real-time milling circuits optimization, where the circulating load was one of the most important configuration parameters of the system and its value was dependent of operational parameters such as: feed rate, pulp density and particle size distribution of the feed and product. White et alli (1977) studied algorithms used for data reconciliation from field measurements with the mass balance results. According to the authors a simple algorithm based in the least squares method could be applied to the minimize errors and should be enough to adjust the real to the theoretical data. Grinding circuit data had to be reconciled due to inability to calculate the circulating load of these circuits with good accuracy. Complex circuit analysis with data redundancy in mineral processing historically requires a wide sampling campaign and the use of matrix calculation for each mineral component present in each considered flow. Wills (1986) demonstrated how complex circuits can be solved by transforming the productive flowchart in a graph, where each node represents a unit operation. The present work shows a low complexity algorithm with high convergence speed for circulating load calculation in mineral processing closed circuits. The obtained results indicate that the proposed algorithm can be satisfactorily used for circulating load calculation in any closed circuit, regardless the complexity of the circuit, with low computing demand and high convergence speed of the result.
2. METHODOLOGY 2.1. General formulation of the algorithm Analyses of many existing mineral processing circuits were carried out, observing the behavior of minerals flows through the variations of operating parameters, such as grades, partitions and metallurgical recovery. From these observations, a correlation between all kinds of circuits was established, regardless the complexity level of the circuit. Yingling (1990) used a Markov chain to model the mineral flow in mining operations. From this work, the conception of an iterative method for the calculation of the circulating load was thought. At each iteration the circulating load (CL) is calculated using the following equation:
The circulating load calculation shown in equation 2 differs from the one proposed by Tsakalakis (2000) by combining the unit operations efficiencies in a single parameter (pi). The exact result obtained by the author with the use of monographic can also be obtained by using the equation 2, eliminating the need of this chart. The linearity hypothesis proposed by Meloy (1983) was considered to developed the iterative algorithm proposed, which assumes that in a separation process there is not particle-particle type interactions that may affect the probability of a given particle be selected for a n output stream of the process. In other words, this hypothesis indicates if the feed flow of a given unit operation is doubled or tripled the fraction of particles which possess a given selection feature will continue the same in each unit operation output stream. However , this hypothesis is not real, since an increase in the feed flow will affect the behavior of the operation itself, whether it be separation or classification. The adoption of the linearity hypothesis proposed by Meloy (1983) simplifies the proposed algorithm, since the unit operations partition varies linearly with the operation feed. However, any other model that allows the calculation of the unit operations feed partition can be adopted without any change in the proposed algorithm. Jankovic and Valery (2013) correlated the grinding and classification efficiency with the circulating load in closed circuit grinding operations. The same authors also showed that the hydrocyclones classification efficiency does not vary linearly with the circulating load in this circuit type. The error calculation is given by the difference between the circulating load calculated in k -th iteration and the circulating load calculated in iteration (k -1)-th. At the first iteration the error will be given only by the circulating load calculated on this iteration. At the beginning of each iteration the feed of each unit operation must be calculated, since they depend on the circulating load and the flow partition. Figure 2 presents the proposed iterative algorithm for the circulating load calculation in closed circuits. It can be noted that the iterations are repeated while a preset tolerance limit is not reached. This is because of the solution obtained by this method is not exact, but an approximation of the real solution. The convergence of the method is provided by calculating the error, which can be both positive and negative. The Meloy (1983) linearity hypothesis applies here, since at each iteration the feed circuit varies, keeping constant the partition of each unit operations involved. It stands out that the water flows in a given circuit can be calculated using the proposed algorithm without any additional modification. In this way, the feed flow ( f i) to be used in the algorithm can be a solid, water or pulp flow.
(2) 2.2 Application of proposed method for grinding circuits
Where f i is the flow (volumetric or mass) which feeds the i-th unit operation that contributes directly to the circulating load and pi is the flow partition in the same unit operation, calculated according to the unit operation and its operational parameters. The product pi. f i is nothing more than the output stream of the i-th unit operation that contributes directly to the circulating load.
For analysis and validation of the proposed iterative algorithm four circuits with different levels of complexity were tested. All analyzed circuits were built based on actual industrial processes present in mining companies installed in Catalão/GO/Brazil in order to be possible the comparison of results obtained by the proposed algorithm with the data provided by the companies. The first considered circuit this is
a simple closed-circuit grinding, as shown in figure 3, which is used as a secondary ball mill by VALE Fertilizers Company, Catalão/GO, Brazil. According to Furuya et alli (1971) closed-circuit comminution systems typically involve mills, classifiers and material transport equipment in various combinations, being the combination presented in figure 1 the most common. The coarse particles are separated by a classifier (in this case a hydrocyclone) and feed back into the mill.
circulating load calculation, because another output flow and two other partitions were generated. The circulating load calculation in this case will take into account the hydrocyclone partition, the pipeline partition and the magnetic separator partition. This circuit is used industrially for separation of magnetite present in the phosphate ore.
Fig. 4. Closed circuit grinding with a low-field magnetic separation.
2.3 Application of proposed method to flotation circuits In a froth flotation circuit the flow partition ( p) can be estimated by the metallurgical recovery and by the minerals grade present in flows. So, for this type of circuit the circulating load can be calculated by the j-th mineral species by:
Fig. 2. Iterative algorithm for circulating load calculation in closed circuits.
(3)
(4)
Where f ij is the flow (volumetric or mass) of the j-th mineral specie in the i-th unit operation that directly contributes to the circulating load; r ij is the metallurgical recovery the j-th mineral specie in the i-th unit operation; t inputij is the feed grade of the j-th mineral specie in the i-th unit operation and t outputij is the output grade in the considered flow of the j-th mineral specie in the i-th unit operation.
Fig. 3. A simple grinding closed circuit. The circuit shown in figure 4 is used as secondary ball mill in AngloAmerican Phosphate, Catalan/GO, Brazil and it is a grinding circuit where a low-field magnetic separator receives part of the hydrocyclone underflow. The flow partition is physically made by valves installed in the pipeline. The addition of a magnetic separator in the circuit changes the
The circuit shown in figure 5 describes the apatite froth flotation used in AngloAmerican Phosphate, Catalão/GO, Brazil, and consists of four froth flotation steps: rougher, cleaner, scavenger and recleaner. Notice that in this circuit the circulating load is influenced by four metallurgical recoveries and five grades for each mineral specie, since the output grade of an unit operation is the subsequent operation feed (for example t output1j = t imput4j and t output2j = t imput3j).
Circulating load
2,482.33 t/h 515% 482 t/h
Overflow do hidrociclone Iterations Time spent Error
Fig. 5. Circuit with four froth flotation cells with the variables required for the circulation load calculation for the j-th mineral specie.
177 3.01 s 0,00E+00
To the grinding circuit with low-field magnetic separator described in figure 4 the hydrocyclone partition parameter ( p1) adopted was equal to 76.29% and the low-field magnetic separator partition parameter ( p2) was equal to 15%. Considering the new feed equals to 303.74 t/h the calculated circulating load was 947.14 t/h, requiring 118 iterations (2.72 s). The results can be seen in the table 2. Table 2. Results of the iterative algorithm applied to the to the figure 4 milling circuit.
The last proposed circuit is used by VALE Fertilizers Company, Catalão/GO, Brazil, in the apatite froth floatation (figure 6) and it is a battery consisting of two rougher cells, two scavenger cells and a froth flotation column operating as cleaner. For the calculation of the circulating load in this circuit it is necessary to know seven metallurgical recoveries and six grades.
New feed Hydrocyclone partition ( p 1 ) Low-field magnetic separator partition ( p 2 ) Circulating load Hydrocyclone overflow Low-field magnetic separator concentrate Iterations Time spent Error
303.74 t/h 76.29 % 15.00 % 947.14 t/h 311.82% 296.58 t/h 7.16 t/h 118 2.72 s 0.00E+00
Table 3 presents data (recovery and grade) required for the calculation of the circulating load of the froth flotation battery shown in figure 5. The results are shown in table 4. For a new feed equals to 267.02 t/h the calculated circulating load was 138.87 t/h, needed 33 iterations (2.17 s) for the exact calculation of the circulating load. Table 3. Input data for circulating load calculation of the forth flotation circuit (figure 5).
Fig. 6. Circuit with four froth flotation cells and a forth flotation column.
3. RESULTS AND DISCUSSION Table 1 summarizes the results of the proposed iterative method applied to a simple grinding (figure 3). The hydrocyclone partition parameter adopted was 83.74%, indicating that 83.74% of the hydrocyclone feed was forwarded for the mill. Considering the new feed equals to 482 t/h the calculated circulating load was equal to 2,482.33 t/h, being necessary 177 iterations for the calculation. The time spent in the calculation was 3.01 seconds in a Samsung notebook RV411 Intel i3 processor 2.53 GHz, 3.0 GB RAM and Windows 7. Table 1. Results of the iterative algorithm applied to the figure 3 milling circuit. New feed Hydrocyclone partition ( p 1 )
482 t/h 83.74 %
Metallurgical Rougher Scavenger Cleaner Recleaner
recoveries 66.41% 65.41% 91.42% 90.31%
Grades Rougher feed Scavenger feed Scavenger concentrate Recleaner feed Recleaner concentrate
24.15% 15.30% 22.70% 35.45% 36.64%
Table 4. Results of the iterative algorithm applied to the froth flotation circuit (figure 5). New feed Circulating load Rougher feed Scavenger feed Cleaner feed Recleaner feed Iterations Time spent Error
267.02 t/h 138.87 t/h 52% 405.89 t/h 215.21 t/h 190.68 t/h 167.88 t/h 33 2.17 s 0.00E+00
Table 5 presents the data (recovery and grade) required for the circulating load calculation of the froth flotation battery shown in figure 6. The results are shown in table 6, as well as a comparison with the results for the mass balance calculated using the Caspeo BILCOTM software, version 3.0, which uses the Lagrange methodology to calculate mass balances and it was admitted an error of ± 5% for the mass balance closure. For a new feed equals to 287.17 t/h the calculated circulating load was equal to 97.838 t/h by the iterative method and 97.833 t/h by BILCO. Table 5. Input data for circulating load calculation of the forth flotation circuit (figure 6). Metallurgical recoveries 75.79% Rougher 1 72.05% Rougher 2 Scavenger 1 27.91% Scavenger 2 29.77% 77.07% Cleaner
Grades Roughers feed Rougher 1 concentrate Rougher 2 concentrate Scavenger 1 feed Scavenger 2 feed
8.257% 18.000% 18.260% 24.213% 27.946%
Table 6. Results of the iterative algorithm applied to the froth flotation circuit (figure 6).
New feed Circulating load Iterations Time spent
within a preset tolerance limit (here adopted as 5%). So, if the solution found is in this tolerance limit , this solution is adopted as a solution of the Lagrange method which is basically an error minimization method, similar to the method of the least squares.
Note that the number of iterations needed to calculate the circulating load changed in each of the tested circuits. This is due to the fact that the method iterates the feed value while the error converges to zero. Therefore, the smaller the circulating load, closer to the solution the adopted initially value will be and less iteration should be made. In a numerical sequence of this type , the first approximation of the solution is called seed. The proposed method is analogous to the bisector method for algebraic equations root determination. In this method the number of iterations depends on the amplitude between the two extreme points of the considered range. In the case of the proposed algorithm the number of iterations depends on the amplitude between the seed and the solution.
4. CONCLUSION
Iterative algorithm
BILCO
Difference
287.175720 t/h
287.175723
-3.00E-06
97.837758 t/h
97.833147
4.611E-03
34.068%
34.06%
0.008
24
–
–
0.45 s
–
–
The proposed iterative algorithm was valid and acceptable in relation to the iteration number and convergence speed to different circuit types and complexities. It is noticed that the iterative method converges faster to the exact solution in froth flotation circuits than in other circuits. This fact can be proven by examining the iteration number needed spent on each circuit, being the cause of such behavior the fact that the initial solutions (the algorithm seed) in froth flotation circuits were closer to the exact solutions. For all circuits studied the results were compared with the results obtained by Caspeo BILCOTM software, version 3.0, and similar results between the software and the proposed algorithm were found.
Rougher 1 feed
192.506739 t/h
192.504435 t/h
2.304E-03
Rougher 1 concentrate
62.722467 t/h
62.719372 t/h
3.095E-03
Rougher 1 tailings
129.792810 t/h
129.785063 t/h
7.747E-03
Rougher 2 feed
192.506739 t/h
192.504435 t/h
2.304E-03
Rougher 2 concentrate
66.925493 t/h
66.921965 t/h
3.528E-03
Rougher 2 tailings
125.569093 t/h
125.582470 t/h
-1.33E-02
14.679437 t/h
14.679263 t/h
1.74E-04
5. ACKNOWLEDGEMENTS
115.112238 t/h
115.105800 t/h
6.438E-03
The authors thank financial support from the Brazilian agencies CNPq, CAPES, FAPEG and FUNAPE.
13.262072 t/h
13.260988 t/h
1.084E-03
112.326512 t/h
112.321483 t/h
5.029E-03
Scavenger 1 concentrate Scavenger 1 tailings Scavenger 2 concentrate Scavenger 2 tailings
The present work shows that the calculation of the circulating load can be understood as a mathematical function where it is wanted to minimize the error and, therefore, the any errors minimization techniques or mathematical optimization can be successfully used to calculate the circulating load.
6. NOMENCLATURE C is the flow (mass or volume) of concentrate;
Cleaner feed
129.647861 t/h
129.641337
6.524E-03
Cleaner concentrate
59.751602 t/h
59.748441
3.161E-03
Cleaner tailings
69.896259 t/h
69.892896
3.363E-03
The difference between the circulating load calculation by the proposed iterative algorithm and the Lagrangean method used by BILCO is due to the fact that BILCO searches a solution
F is the flow (mass or volume) entering the system; f i is the flow (volumetric or mass) which feeds the i-th unit operation unit that contributes directly to the circulating load; f ij is the flow (volumetric or mass) of the j-th mineral specie in the i-th unit operation that directly contributes to the circulating load; pi is the flow partition in the same unit operation, calculated according to the unit operation and its operational parameters.
r ij is the metallurgical recovery the j-th mineral specie in the ith unit operation; T is the flow (mass or volume) both coming out of the system; t inputij is the feed grade of the j-th mineral specie in the i-th unit operation; t outputij is the output grade in the considered flow of the j-th mineral specie in the i-th unit operation.
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