Dry grinding of gibbsite is conducted using tumbling ball mills with different diameters from 7.8 to 19.9 cm in order to investigate the effect of mill diameter on the size reduction rate of the sample, in correlation with the ball impact energy simulated on the basis of the Particle Element Method. The size reduction rate increases with an increase in the rotational speed of the mill, Subsequently, it rapidly falls over about 90 % of the critical rotational speed. The size reduction rate reaches a maximum value at about 90 % of the critical rotational speed, and the maximum value becomes large as the mill diameter increases. The same trend is seen in the relation between the specific impact energy of balls and the rotational speed of the mill, and the size reduction rate is proportional to the specific impact energy regardless of the mill scale. Therefore, the size reduction rate of gibbsite in tumbling ball mills scaled-up can be fairly estimated by simulating the specific impact energy of balls.