Batch Grinding in Laboratory Ball Mills: Selection Function

Transcription

1 1560 Chem. Eng. Technol. 2009, 32, No. 10, Gordana Matiašić 1 Antun Glasnović 1 1 University of Zagreb, Faculty of Chemical Engineering and Technology, Zagreb, Croatia. Research Article Batch Grinding in Laboratory Ball Mills: Selection Function The selection functions and the breakage distribution functions are based on the experimentally-determined particle size distribution on the basis of comminution of one size fraction particles. Therefore, to obtain a clear picture of the product properties during comminution of the real polydisperse sample, a number of experiments are needed. This work introduces the tested methodology for the selection function determination based on the starting and maximal values of the selection function. The principle was tested on the planetary ball mill and the horizontal laboratory ball mill, and according to the results obtained, it can be concluded that the proposed methodology can be useful for the evaluation of the selection function during batch comminution in different mills. Keywords: Comminution, Kinetics, Particle size, Selection function Received: March 6, 2009; revised: June 4, 2009; accepted: June 9, 2009 DOI: /ceat Introduction There are numerous methods used to describe comminution process kinetics and particle breakage. One of the frequently used principles is the so-called conventional approach with the knowledge of the selection function and breakage distribution function. Defining those functions, the particle size distribution of the product formed by comminution can be predicted. The concept of the selection function and the breakage distribution function make it possible to express the population balance of batch grinding [1, 2]: dm i t dt ˆ Xn 1 ˆ1 b i; S m S t i m i t n i 1 (1) where S 1) is the selection function representing the probability of particle in the size interval to break under the lower size limit of any size interval, (x ). If the grinding is treated as a rate process, breakage of the given size fraction usually follows the first-order law [3]. Thus, the breakage rate of material from the initial size interval (i = 1) can be expressed as: dm 1 t dt ˆ m 1 t (2) Correspondence: Dr. sc. G. Matiašić (gmatias@fkit.hr), University of Zagreb, Faculty of Chemical Engineering and Technology, Marulicev trg 19, Zagreb, Croatia. 1) List of symbols at the end of the paper. Eq. (2) shows the decrease rate of the initial size interval, where mass can also be expressed as the mass fraction: dw 1 t dt or ˆ w 1 t w 1 w t 1 ˆ S 0 1 t (3) In general, the best way to obtain the selection function for certain materials is to perform experiments. Nevertheless, the experiments are not always possible and desirable, so there are several models developed for the estimation of the selection function under different conditions. The commonly used equation is the one proposed by Austin and co-workers [4 8]. That is a simple power equation assuming that comminution is conducted using a narrow particle size range. The selection function is exponentially decreasing or increasing with particle size. On the other hand, models suggested for polydisperse samples are polynomial equations [9, 10], not always applicable for a particular material. 2 Experimental 2.1 Material Dolomite samples were obtained from Samoborka Ltd, Croatia. Material was fractioned in eleven size intervals: lm (M1), lm (M2), lm (M3), lm (M4), lm (M5), lm (M6),

2 Chem. Eng. Technol. 2009, 32, No. 10, Batch Grinding lm (M7), lm (M8), lm (M9), lm (M10), and lm (M11). Each experiment was conducted using new feed material to avoid losses during the sieve analysis. After each grinding period, samplesp were taken out of the mill and sieved using ASTM sieves with 2 screen size intervals corresponding to the initial size fractions. Easier handling and greater effectiveness was enhanced with the addition of a continuous phase. Wet comminution experiments were performed at 40 vol.-% solid content, obtained as optimal in a previous work [11]. 2.2 Comminution Devices The comminution process was conducted using two different ball mills. The planetary ball mill, Pulverisette 6, Fritsch GmbH (Germany) consists of one pot vertically mounted on the rotating disc. The pot and disc are rotating in different directions resulting in intensive motion of the balls and, consequently, high impact energies. Experimental conditions are shown in Tab. 1. Ceramic laboratory ball mill was made beforehand in the Department of Mechanical and Thermal Process Engineering, Faculty of Chemical Engineering and Technology, University of Zagreb, Croatia. The mill has two rollers and a horizontally laid ceramic ar. The power density in the tumbling ball mill is significantly smaller than in the planetary ball mill, since the power consumption is limited by the critical number of revolutions. [12] Experimental conditions are shown in Tab. 2. Table 1. Experimental conditions planetary ball mill (Pulverisette 6, Fritsch GmbH). Pot and ball material 99.7 % Al 2 O 3 Pot volume 500 cm 3 Revolution speed Pot diameter Disc radius Ball diameter Number of grinding balls rpm 10 cm 13.5 cm 20 mm Pot/disc revolution ratio 0.82 Table 2. mill. Pot material Experimental conditions horizontal laboratory ball porcelain Ball material steel Pot volume 2300 cm 3 Internal pot diameter Critical revolution speed Optimal revolution speed Number of grinding balls 109 Ball diameter 14.5 cm 111 rpm 80 rpm 16 mm 3 Results and Discussion 3.1 Determination of Selection Function Results showed that the breakage of all size intervals can be treated as a first-order rate process. Fig. 1 shows the dependence of unbroken feed material over time. Linear dependence confirms first-order breakage. The selection functions were determined for eleven size intervals (M1 M11) as slopes of the straight lines shown in Fig. 1. Whereas the selection function is increasing significantly toward smaller particle sizes, it leads to the problem of the rapid disappearance of some size intervals (M5, M6, and M7). Consequently, those intervals were exposed to a shorter grinding time to establish According to the results obtained (see Tab. 3), the specific breakage rate significantly increases with particle size decrease, attains a maximum at M7 size interval, and then decreases until the M11 interval to a value that is almost the same as M1 interval. Figure 1. Mass fraction of feed remaining during comminution time; size interval M1 ( lm), M5 ( lm), and M10 ( lm); planetary ball mill.

3 1562 G. Matiašić et al. Chem. Eng. Technol. 2009, 32, No. 10, Table 3. Selection functions of all size intervals obtained experimentally in the planetary ball mill. S [min 1 ] M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M The breakage rate decreases as feed size becomes smaller. An important condition for particles to be broken is that particles need to be captured and stressed between two grinding media (e.g., balls) [13, 14]. The probability of particles to be captured and broken decrease as they become smaller and able to slip between two grinding balls. The smaller particles are more difficult to be ground, the energy utilization decreases, and the result is the decrease of the specific breakage rate with initial size. 3.2 Model E1 Hosten [10] indicates that properly chosen grinding ball top sizes and feeds with top sizes less than 1 mm can avoid the turning point for the selection function. Nevertheless, it is not always possible to use a narrow particle size distribution of feed material. Anyway, Hosten underlines that the commonly used model for the selection function given by Raamani and Herbst [9] is the satisfying solution for materials with larger particles involved: S ˆ a 1 x b 1 x 2 (4) Eq. (4) is a second-order polynomial with a 1 and b 1 as coefficients that are adustable. is the selection function of the top size interval, and and x are, respectively, the geometric mean sizes for the top size interval and the interval indexed as. The literature model proposed by Raamani and Herbst [9] was tested on the set of selection functions obtained experimentally in the planetary ball mill. Parameters a 1 and b 1 were adusted toward the best correspondence to experimental results, minimizing the sum of squared errors. Literature and proposed models were evaluated using the sum of squared errors (SSE): SSE ˆ XN ˆ1 2 S ;exp S ;mod (5) Fig. 2 shows a comparison of selection functions experimentally obtained and those calculated according to model E1. Unfortunately, SSE according to model E1 [9] was 2.189, which limits its application. Since there was no adequate literature model that could describe the selection function behavior obtained with a change in particle size under the given conditions, authors proposed a combination of models that can be used across the large particle size range. The experiments in this work were conducted on the large particle size range which provided a reliable model testing. 3.3 Model E2 Model E2 was yielded as a part of model E1 that refers to particles larger than 1 mm. The guiding fact was the tendency of the same alteration of the selection function in both areas, from its maximal value to larger particles as well as to smaller particles. That idea leads to the formulation of two equations. Eq. (6) is applicable in the range of larger particles until the maximum is reached (size intervals = 1, 2,..., 6). Upon the maximal value of the selection function, Eq. (7) can be used (size intervals = 7, 8,..., 11). S ˆ a 2 x ˆ a 2 ˆ 1:23 (6) S ˆ b 2 x c 2 ˆ b 2 ˆ 1:27 c 2 ˆ 4:68 (7) Coefficients a 2, b 2, and c 2 were obtained on the basis of an experimental set of data using the least squares method. Fig. 3 shows a procedure for the determination of the model coefficients. The ordinate in Fig. 3 represents the logarithmic values of the selection function ratio obtained by dividing the selection function of the -th size interval by the selection function of the first size interval. Values are plotted against the ratio of the lower size limits of the corresponding size interval. According to the results, two straight lines were obtained as can be seen in Fig. 3. The left one corresponds to the interval of small- Figure 2. Experimental and model E1-based selection functions.

4 Chem. Eng. Technol. 2009, 32, No. 10, Batch Grinding 1563 er particle sizes, includes intervals M7 to M11, which were used for determining the coefficients, b 2 and c 2. The slope of the right straight line in Fig. 3 was used to evaluate coefficient a 2. Reduction in the sum of squared errors was noticeable; according to model E2. Fig. 4 shows agreement between experimental values of the selection function and those calculated according to model E2, and coefficients obtained as described through Fig. 3. The estimation of the selection functions based on model E2 assumes that the selection function for the first size interval (M1 largest particle size) is known. Both Eqs. (6) and (7) are based on the value of the selection function of the fist size interval (M1). This imposes the dependence of small particles (less than 300 lm) to the breakage rate of the largest particles in the system. In the research conducted, Eqs. (6) and (7) can be used for the determination of the selection function, but on the other hand, it is clear that the dependence mentioned could be a maor disadvantage in the case when the first size interval is not in the size range, lm ( 2360 lm). Therefore, model E3 was proposed as the outcome of complete separation of the area of larger and smaller particles. 3.4 Model E3 Figure 3. Determination of coefficients in model E2. Data from M1 to M6 were used for the coefficient in Eq. (6) and data from M7 to M11 were used for the coefficients in Eq. (7). Figure 4. Experimental and model E2-based selection functions. Model E3 consists of two equations as well. For particles larger than 300 lm, Eq. (6) was used. The second equation (see Eq. (9)) of model E3 is similar to Eq. (8), but S 7 (selection function of interval M7) was used as the reference value since that is the starting value of the so-called "area of smaller particles". S ˆ a 3 x ˆ a 3 ˆ 1:23 (8) S S 7 ˆ b 3 x x 7 ˆ b 3 ˆ 1:36 (9) Fig. 5 shows the procedure for determination of model E3 coefficients. The ordinate in Fig. 3 represents the logarithmic values of the selection function ratio obtained by dividing the selection function of the -th size interval with the selection function of the starting size interval. Since model E3 was divided into the area of larger and smaller particle size, respectively, starting values correspond to those areas. Therefore, the starting selection function for the area of larger particles corresponds to the value of the first size interval (M1) and the starting selection function for the area of smaller particles corresponds to the value of the maximal selection function, in this case to the value of size interval M7. Values are plotted against the ratio of the lower size limits of the corresponding size interval. According to the results, two straight lines were obtained as can be seen in Fig. 5. Data from the M1 to M6 interval were used to evaluate coefficient a 3 through linear regression analysis and data from the M7 to M11 interval was used for determination of the coefficient b 3. The values of selection functions and lower limits of size intervals,, were divided by the starting values (S start, x start ) corresponding to the size intervals, M1 or M7, depending on the size area. According to model E3, the SSE was furthermore minimized up to Very good agreement between experimental and calculated values can be seen in Fig. 6, where selection functions are plotted against the lower size limit of the interval index as. There was

5 1564 G. Matiašić et al. Chem. Eng. Technol. 2009, 32, No. 10, no maor difference between results obtained through model E2 or E3, but it was clearly shown that the model, E3, requires less empirical coefficients (only a 3 and b 3 ) and certainly goes through at least two empirical points used as starting points ( and S 7 ). Fig. 7 shows a comparison of experimental values of the selection functions and those obtained according to all three models. The values of SSE showed that model E3 fits experimental data best, but model E2 is also applicable. The advantage of model E2 is definitely the need for less experimental work, but as mentioned earlier, that can be applied only when the first size interval is lm. On the other hand, the limitation of model E3 is the fact that the size interval of the maximal selection function has to be known. Nevertheless, large sets of experiments in previous studies show that the M7 interval ( lm) is the size interval of the maximal selection function for dolomite and, in that case, model E3 would have the advantage [15]. 3.5 Testing of Model E3 Applicability to the Data obtained in Horizontal Laboratory Ball Mill Figure 5. Determination of coefficients in model E3. Data from M1 to M6 were used for the coefficient in Eq. (8) and data from M7 to M11 were used for the coefficient in Eq. (9). To assign applicability of model E3 to the planetary ball mill only, experiments were also carried out in the horizontal laboratory ball mill. All size intervals (M1 M11) were independently ground at optimal rotational speed (approximately 70 % of the critical revolution speed; 80 rpm). Selection functions were calculated on the basis of the content of unbroken material left in the initial interval over time. Results are shown in Tab. 4. The values obtained were significantly smaller, because the power consumption is limited by the critical number of revolutions and the power density is too small to cause highly effective breakage. Nevertheless, the maximal selection function value was obtained at the same size interval (M7) and the same alteration of results was observed. Fig. 8 represents the comparison of selection functions obtained experimentally in the laboratory ball mill and values obtained from the equations of model E3 (see Eqs. (8) and (9)). Selection functions were plotted against the lower limit of the size intervals. The advantage of model E3 is once more confirmed. Calculated values will always coincide with two experimental points, and S 7, the starting and maximal values. It Figure 6. Experimental and model E3-based selection functions. would be expected that coefficients obtained in the planetary ball mill could not be used for the evaluation of selection functions in the laboratory ball mill because of the different stress Table 4. Selection functions of all size intervals obtained experimentally in the horizontal laboratory ball mill. S [min 1 ] M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M

6 Chem. Eng. Technol. 2009, 32, No. 10, Batch Grinding 1565 Figure 7. Comparison of experimental values of selection functions and those obtained according to models E1, E2, and E3; planetary ball mill. energies and different grinding behavior. Nevertheless, results showed that coefficient b 3 for the planetary ball mill (1.36) can efficiently be applied to the horizontal laboratory ball mill in the range of smaller particles (less than 300 lm). The same value of coefficient b 3 was obtained using linear regression analysis of experimental data obtained in the laboratory ball mill. The sum of squared errors for that set of data was Selection functions for particles larger than 300 lm were calculated according to Eq. (8). Although the derived coefficient, a 3 = 1.23, could be used, it is necessary to make some adustments in order to minimize error. A new coefficient, designated as a 3K, was derived using selection functions for the M1 and M7 size intervals. Fig. 9 shows the results of linear regression using the least squares method. a 3K was obtained using only two points in the regression analysis. The first point represents the logarithmic value of the ratio of the selection function of interval M7 and the selection function of interval M1, and the second point is 0. It can also be seen how the rest of the experimental values fall between two marginal values. Regression using two points, the starting and maximal values, showed that the experimental procedure could be reduced to only two experiments that will determine the starting and maximal selection function values. The corrected a 3K coefficient which referred to the horizontal ball mill was 1.43 and the resulting SSE was decreased to in the area of large particles. Fig. 10 represents a comparison of the model and experimental selection function values. It can be seen, as confirmed by SSE, that correction of the model E3 coefficient resulted in better agreement between the experimental and model data. According to the results, model E3 is applicable for both tested mills but with some limitations. Regarding other models used (E1 and E2), model E3 will enable calculation of selection functions for all size intervals used, if the starting and maximal values are known. It was also shown and confirmed that in the range of smaller particles, coefficient b 3 is independent of process parameters and equipment, and once determined can be used for other comminution conditions. As opposed to smaller particles, breakage of larger particles is influenced by material properties and process parameters. Transferring data from one comminution machine to the other demands knowledge of the starting and maximal selection function values in order to derive coefficient a 3 (see Eq. (8)). This principle was named the principle of starting and maximal selection functions. Applicability was confirmed using dolomite samples, and planetary and horizontal laboratory ball mills. Despite positive results of transferring data, such a principle has its own disadvantages. For example, if other material rather than dolomite is used, a lot of experiments are needed in order to find the size interval at which the maximal value of the selection function is achieved. But nevertheless, if selection functions for two size intervals are known values from previous research or literature, the principle of the starting and maximal values will definitely ease up determination of the selection function in various comminution conditions. 4 Conclusions Figure 8. Testing model E3 applicability to the data obtained experimentally in the horizontal laboratory ball mill. Determination of the selection function through experimental work is often difficult and a time-consuming process, even more complicated due to the

7 1566 G. Matiašić et al. Chem. Eng. Technol. 2009, 32, No. 10, starting and maximal value, was proposed, tested, and confirmed using dolomite and two laboratory ball mills. Further research is needed to acknowledge usage of this principle on other materials and comminution equipment. Symbols used S [s 1 ] selection function x [lm] particle size m [kg] mass t [s] time b [ ] breakage distribution function w [ ] mass fraction Figure 9. Correction of model E3 in the area of large particles the for horizontal laboratory ball mill. Determination of new the coefficient in Eq. (8). Greek symbols a [ ] coefficient of the corresponding model b [ ] coefficient of the corresponding model c [ ] coefficient of the corresponding model Figure 10. Comparison of experimental values of selection functions and those obtained according to model E3 and the corrected model E3; horizontal laboratory ball mill. lack of methods for preparing narrow size intervals. Therefore, a great number of models for selection function evaluation were proposed. This work showed that some of the tested models could be used to evaluate selection functions of dolomite wet comminution in the planetary ball mill and horizontal laboratory ball mill, but with some limitations when transferring data between two different comminution machines. New methodology for selection function estimation, named the principle of References [1] L. G. Austin, P. Bagga, Powder Techol. 1981, 28, 83. [2] N. Kotake, K. Suzuki, S. Asahi, Y. Kanda, Powder Technol. 2002, 122, 101. [3] D. W. Fuerstenau, A. De, P. C. Kapur, Int. J. Miner. Process. 2004, 74S, S317. [4] V. Deniz, T. Onur, Int. J. Miner. Process. 2002, 67, 71. [5] L. G. Austin, R. R. Klimpel, P. T. Luckie, Process Engineering of Size Reduction: Ball Milling AIMESME, New York [6] R. Hogg, A. J. Dynys, H. Cho, Powder Technol. 2002, 122, 122. [7] L. G. Austin, P. T. Luckie, Powder Technol. 1971/ 72, 5, 215. [8] F. Müller, R. Polke, M. Schäfer, Powder Technol. 1999, 105, 243. [9] R. K. Raamani, J. A. Herbst, Instn. Min. Metall. Sec. 1984, C93, C74. [10] Ç. Hosten, Miner. Eng. 2005, 18, 489. [11] G. Matiašić, K. Žižek, A. Glasnovic, Chem. Eng. Res. Des. 2008, 86, 384. [12] A. Kwade, Powder Technol. 1999, 105, 14. [13] A. Kwade, J. Schwedes, Powder Technol. 2002, 122, 109. [14] A. Kwade, J. Schwedes, in Handbook of Powder Technology (Eds: A. D. Salman, M. Ghadiri, M. J. Hounslow), 1st ed., Elsevier, Amsterdam [15] G. Matiašić et al., Chem. Eng. Process. 2009, 48, 846.