Planning of Extremal Experiments in the Study of Ash Granule Technology
Abstract Generated abstract
This paper applies experimental design methods to optimize the pelletizing of finely dispersed ash from Tomsk GRES-2 on a semi-industrial plate granulator. Using simplex planning, the study first evaluates granule quality as a weighted function of strength, bulk density, and porosity to identify near-optimal operating conditions, then investigates maximum granulator productivity under constraints on granulate elasticity. In an almost stationary region, an orthogonal second-order plan is used to derive an adequate regression model for productivity as a function of clay, peat, and ash charge composition. The resulting calculations indicate a maximum productivity of about 5.306 kg/min, with clay and peat contents of approximately 12.6% and 4%, respectively.
Full Text
PLANNING EXTREME EXPERIMENTS IN THE STUDY OF ASH GRANULE TECHNOLOGY
V. M. Vitortsin, G. G. Vertun, A. I. Ruban
(Presented by the scientific seminar of the Department of GCT)
The application of planning methods makes it possible to obtain mathematical models and to estimate the optimal conditions of processes and the extreme operating regimes of apparatus with considerably lower labor expenditures than with ordinary experimental methods. In the present work, the study of the pelletizing of finely dispersed ash from the Tomsk GRES-2 was carried out on a plate granulator of semi-industrial type (plate diameter 1000 mm, rim height 100 mm). To obtain a strong, lightweight granulate that swells well during firing, additives of clay, peat, magnetite slag, sulfite-spirit stillage (SSB), etc., were introduced into the ash. The task was to determine the optimal operating regime of the plate granulator by the method of experimental design. In solving this problem, the following were considered as independent variables: the angle of inclination of the granulator—\((Z_1)\); pelletizing time—\((Z_2)\); the percentage content of components of the ash batch: clay—\((Z_3)\) and magnetite slag—\((Z_4)\).
The quality of the granules obtained was judged by the following indices: crushing strength—\(y_1\); dropping strength—\(y_2\); bulk density—\(y_3\); porosity—\(y_4\).
The quality function, determined on the basis of the above indices, was represented in the form
\[ J = 0.5y_1 + y_2 - 0.7y_3 + 0.7y_4. \tag{1} \]
The weighting coefficients were introduced into the equation on technological grounds. The basic levels of the input independent variables and the intervals of variation are given in Table 1.
Table 1
| Factors | Basic level | Variation interval |
|---|---|---|
| $Z_1$ | 45 | 1 |
| $Z_2$ | 15 | 2 |
| $Z_3$ | 5 | 0.5 |
| $Z_4$ | 5 | 0.5 |
Determination of the values of the factors $Z_1, Z_2, Z_3, Z_4$ corresponding to the maximum value of the quality function ($J$) was carried out by the method of simplex planning of the experiment. The initial simplex of four independent variables included 5 experiments. Table 2 gives the values of the quality function obtained experimentally at the points of the simplex. As can be seen from Table 2, the worst in the first series of 5 experiments is experiment I. In accordance with the methodology of simplex planning [1], point I can be replaced by a new point corresponding to experiment 6. However, the quality function for this experiment is worse than for the rejected one; therefore we return to the initial simplex and “reflect” the second-worst point (experiment 4).
Table 2
| No. | $Z_1$ | $Z_2$ | $Z_3$ | $Z_4$ | $J$ |
|---|---|---|---|---|---|
| I | 45°30 | 15 min 34.7 sec | 5.1 | 5.08 | 186.50 |
| 2 | 44°30 | 15 min 34.7 sec | 5.1 | 5.08 | 272.14 |
| 3 | 45° | 13 min 50.6 sec | 5.1 | 5.08 | 245.80 |
| 4 | 45° | 15 min | 4.69 | 5.08 | 223.13 |
| 5 | 45° | 15 min | 5 | 4.68 | 291.68 |
| 6 | 44°06 | 14 min 07.8 sec | 4.84 | 4.84 | 165.00 |
| 7 | 45° | 15 min | 5.47 | 4.88 | 252.30 |
| 8 | 44°18 | 14 min 13 sec | 5.23 | 4.78 | 146.54 |
| 9 | 45° | 16 min 50 sec | 5.23 | 4.78 | 150.40 |
The quality function in experiment 7 proves to be higher than for the reflected point. This indicates a displacement of the center of the simplex toward an increase in the quality function. Experiments 8 and 9, which are reflections of experiments I and 3, give a quality function lower than at the reflected points. Consequently, on the basis of the experiments carried out it may be considered that
the operating parameters of the granulator that are close to optimal correspond to experiment 5 (the quality function for this experiment has the greatest value):
\[ L = 45^\circ; \qquad \tau = 15\ \text{min}; \]
\[ \text{clay content} - 5\%; \]
\[ \text{sludge content} - 4.68\% \text{ in the ash charge.} \]
The second stage of the research was aimed at determining the maximum productivity of the granulator at a specified granulate quality. The following were adopted as independent parameters:
\[ Z_3 \text{ — clay content in the charge }(\%), \]
\[ Z_5 \text{ — peat content in the charge }(\%), \]
\[ Z_6 \text{ — CFB content in the charge }(\%). \]
In evaluating the optimum value of the plant productivity, the quality of the granulate was also taken into account in terms of the magnitude of the modulus of elasticity of the granulate, \(P\) [2], on which, for technological reasons, restrictions of the form
\[ 0.08 \le P \le 0.15. \]
were imposed. The basic levels and intervals of variation of the independent variables are presented in Table 3.
Table 3
| Factors | Basic level | Interval of variation |
|---|---|---|
| \(Z_3\) | 10 | 2 |
| \(Z_5\) | 5 | 1 |
| \(Z_6\) | 85 | 5 |
The experiments were carried out by the method of simplex planning. The results of the experiments are presented in Table 4.
Table 4
| I | \(Z_3\) | \(Z_5\) | \(Z_6\) | \(C\), kg/min |
|---|---|---|---|---|
| 1 | 11,0 | 5,27 | 83,02 | 4,2 |
| 2 | 9,0 | 5,27 | 86,02 | 2,5 |
| 3 | 10,0 | 4,42 | 86,07 | 4,3 |
| 4 | 10,0 | 5,00 | 81,94 | 4,0 |
| 5 | 11,6 | 4,52 | 83,38 | 4,8 |
| 6 | 10,0 | 4,48 | 81,00 | 3,2 |
| 7 | 11,8 | 5,46 | 81,20 | 3,9 |
| 8 | 11.8 | 5.46 | 81.20 | 3.9 |
|---|---|---|---|---|
It should be noted that, for each charge composition, a series of experiments was carried out with different rates of raw-material feed to the granulator. As the quality function, the charge consumption corresponding to the limiting value of the modulus of elasticity of the granulate \((P = 0.008)\) was used.
As can be seen from Table 4, in the experiments of the initial simplex (the first 4 experiments), point 2 is the worst. We “reflect” this point when setting up experiment 5. Then, in experiments 6 and 7, we “reflect” points 4 and 1 of the simplex. Since the quality function obtained both in experiment 6 and in experiment 7 is lower than at the initial points, it may be assumed that the experiments are being conducted in an “almost stationary region.” This is also confirmed by reflecting point 3 in the initial simplex \((1,3,4,5)\). The quality function again turns out to be lower than at the reflected point. Consequently, it may be considered that the granulator productivity, close to the maximum, is \(4.8\ \text{kg/min}\) (experiment 5).
For the purpose of a mathematical description of the granulation process in the “almost stationary region,” an orthogonal second-order plan was drawn up. The data of experiment 5 were taken as the basic levels.
Table 5
| Factors | Basic level | Variation intervals |
|---|---|---|
| \(X_1\) | 11.6 | 1.0 |
| \(X_2\) | 4.5 | 0.5 |
| \(X_3\) | 83.4 | 2.5 |
The values of the levels and variation intervals are presented in Table 5. The planning matrix after transformation of the quadratic variables takes the form presented in Table 6.
Table 6
| No. | \(X_1\) | \(X_2\) | \(X_3\) | \(X_4\) | \(X_5\) | \(X_6\) | \(X'_1\) | \(X'_2\) | \(X'_3\) | \(G\) |
|---|---|---|---|---|---|---|---|---|---|---|
| \(n/n\) | \(X_1\) | \(X_2\) | \(X_3\) | \(X_1X_2\) | \(X_1X_3\) | \(X_2X_3\) | \(X_1^2\) | \(X_2^2\) | \(X_3^2\) | |
| 1 | + | + | + | + | + | + | 0.27 | 0.27 | 0.27 | 4.95 |
| 2 | + | + | − | + | − | − | 0.27 | 0.27 | 0.27 | 4.60 |
| 3 | + | − | + | − | + | − | 0.27 | 0.27 | 0.27 | 5.15 |
| No. | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 4 | − | + | + | − | − | + | 0.27 | 0.27 | 0.27 | 3.70 |
| 5 | − | − | − | + | + | + | 0.27 | 0.27 | 0.27 | 4.00 |
| 6 | − | − | + | + | − | − | 0.27 | 0.27 | 0.27 | 4.50 |
| 7 | + | − | − | − | − | + | 0.27 | 0.27 | 0.27 | 5.00 |
| 8 | − | + | − | − | + | − | 0.27 | 0.27 | 0.27 | 3.90 |
| 9 | \(+\ell\) | 0 | 0 | 0 | 0 | 0 | 0.748 | −0.73 | −0.73 | 5.10 |
| 10 | \(-\ell\) | 0 | 0 | 0 | 0 | 0 | 0.748 | −0.73 | −0.73 | 4.15 |
| 11 | 0 | \(+\ell\) | 0 | 0 | 0 | 0 | −0.73 | 0.748 | −0.73 | 4.92 |
| 12 | 0 | \(-\ell\) | 0 | 0 | 0 | 0 | −0.73 | 0.748 | −0.73 | 4.70 |
| 13 | 0 | 0 | \(+\ell\) | 0 | 0 | 0 | −0.73 | −0.73 | 0.748 | 4.95 |
| 14 | 0 | 0 | \(-\ell\) | 0 | 0 | 0 | −0.73 | −0.73 | 0.748 | 4.20 |
| 15 | 0 | 0 | 0 | 0 | 0 | 0 | −0.73 | −0.73 | −0.73 | 4.80 |
The regression equation for the granulation process, taking into account double interactions of the parameters, can be written in the form
\[ G=b_0X_0+b_1X_1+b_2X_2+b_3X_3+b_{12}X_1X_2+b_{13}X_1X_3+ \]
\[ +\,b_{23}X_2X_3+b_{11}X_1^2+b_{22}X_2^2+b_{33}X_3^2 \tag{2} \]
The coefficients of the equation were calculated on the basis of the planning matrix by formula [3].
\[ b_i=\frac{(X_i,G)}{(X_i,X_i)}, \qquad i=1,2,3,\ldots,N \tag{3} \]
where \(X_i\) are the components of the columns corresponding to the linear terms and interaction effects. By an analogous formula, the coefficients for the quadratic terms are also calculated.
After calculating the coefficients of the equation and assessing their significance, the regression equation takes the form
\[ =4.481+0.411X_1+0.136x_2+0.133x_3-0.109x_1^2- \]
\[ -\,0.162x_3^2 \tag{4} \]
Checking by the \(F\)-criterion established the adequacy of the mathematical description of the process by this equation.
On the basis of the equation obtained, the problem of finding the extremum of the function was also solved, subject to constraints imposed on the variables:
\[ G=f(x_1,x_2,x_3)=\max_{x_1;\,x_2;\,x_3,} \tag{5} \]
under the constraints \(1 \ge X_i \ge -1\).
This problem was solved both analytically and on a BESM-4 computer. The results of the analytical calculation and of the computations performed on the machine agree satisfactorily. According to the calculations, the maximum productivity of the granulator is 5.306 kg/min. The contents of clay and peat in the ash charge in this case are, respectively, 12.6 and 4%.
Literature
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V. G. Gorskii, V. Z. Brodskii. The simplex method for planning extreme experiments. “Zavodskaya laboratoriya,” No. 7, 1965.
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V. M. Vityugin, M. P. Talkhaev, V. P. Gusev. Study of the operating efficiency of a pan granulator. Proceedings of Higher Educational Institutions, “Ferrous Metallurgy,” No. 6, 1972.
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V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extreme Experiments. “Nauka,” Moscow, 1965.