. Transactions. ning at 24 From 1 226 X 242 X tequation (2), cos a = — ~finn - = , which is the equa-tion of a circle of radius „ • The center is then on the vertical axis n units above the center of the mill. From this the value of a in Table 15 can be computed: 268 FINE CRUSHING IN BALL-MILLS Table 15.—Data for Computing Values of a r Cos a a X 4 38° 18 3 53° 56 2 66° 54 1 78° 41 0 90° From the values of r and a, the curve aO, Fig. 5, can be drawn,which is the dividing line between the parabolic path an
. Transactions. ning at 24 From 1 226 X 242 X tequation (2), cos a = — ~finn - = , which is the equa-tion of a circle of radius „ • The center is then on the vertical axis n units above the center of the mill. From this the value of a in Table 15 can be computed: 268 FINE CRUSHING IN BALL-MILLS Table 15.—Data for Computing Values of a r Cos a a X 4 38° 18 3 53° 56 2 66° 54 1 78° 41 0 90° From the values of r and a, the curve aO, Fig. 5, can be drawn,which is the dividing line between the parabolic path and the circularpath of the particles. Then by use of equations (4) and (5) it is possibleto find any number of points on the curve cd. This may be done moresimply by drawing the circle through the point e and then measuringa distance x to a vertical line which will intersect the circle at thedesired point / as in Fig. 5. For this purpose, the corresponding valuesof x are added to the preceding Fig. 5 —Paths of travel of particles in an 8-ft. mill making 24 It is then possible to draw the line Od, which is the dividing linebetween the parabolic path and the circular path for all particles of thecharge. The complete cycle of any particle p is then seen to be from eto / along the parabolic path, and then from / to e along the circular Fig. 5 it is evident that the particle p acts exactly as though itwere in a mill of radius r, the lining of which is the layer of particles ofradius next larger than r. E. W. DAVIS 269 End of the Parabolic Path Since, as has been shown, x = Ar sin a cos2 a and y = — Ar sin2 a cos a,from Fig. 5, x = Ar sin a cos2 a — r sin a,and y = Ar sin2 a cos a — r cos a. As the value of r at which a particle starts on its parabolic path andends its parabolic path is the same, then . ,, y\ Ar sin2 a cos a — r cos a .. _ . sin p = — = = — (4 cos3 a — 3 cos a). r r Then sin p = — cos 3a; but — cos 3a = cos (18
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