Introducing a Method for Calculating the Ball Size Distribution in a Grinding Mill

After the mill has been in operation for a period of time, balls must be replenished every 7 to 10 days, which often leads to a highly disordered ball-size distribution. The longer the mill runs, the greater the number of ball sizes present, making it increasingly difficult to calculate the optimal ball gradation. For small mills, when the mill is emptied for maintenance, the desired ball mix can be accurately calculated, the balls can be sorted by size and then added in accordance with these calculations, resulting in a ball gradation that closely matches the theoretical design. In contrast, for large mills, the sheer volume of grinding media makes size sorting extremely time-consuming, thereby complicating production scheduling. As a result, most operators do not follow this systematic approach; instead, they manually remove worn balls, iron slag, and undersized balls according to their own specifications, then compare the actual replenishment with the standard addition rate. If the replenishment falls short, they typically add the largest-sized balls or, based on experience, supplement with other ball sizes. Consequently, both the ball-size distribution and the average ball diameter are only estimates and are inherently imprecise.

In practice, I have employed a simple estimation method that combines probabilistic analysis with mathematical induction for sampling calculations, which has certain reference value. The method is described as follows: Construct a 500 mm–sided square frame using No. 8 steel wire; empty the mill of its charge, open the mill door to enter the mill chamber, and select two locations—one at the inlet and one at the outlet of the grinding chamber—and take three radial measurements at each location.

Count the number of steel balls of various sizes that have more than half of their diameter exposed within the square grid, record the results, and then organize and analyze the data to obtain a reasonably accurate steel-ball gradation. An example of measurement data is shown in Table 1.

Table 1: Measurement Results Inside the Mill for a Specific Instance

　　Note: All figures in the table are the sum of three points.

　　The calculated theoretical total weight (with Φ100–95 mm balls treated as Φ100 mm balls) is 295.8 kg.

　　Proportion of various steel balls:

　　　　　　26 × 4.115
　　Φ100 = ────── × 100% = 36.2%
　　　　　　　295.8

　　　　　28 × 3.111
　　Φ90=──────×100％＝29.4％
　　　　　　295.8

　　　　　33.5 × 2.107
　　Φ80=──────×100％＝23.9％
　　　　　　295.8

　　　　　21.5 × 1.498
　　Φ70 = ────── × 100% = 10.9%
　　　　　　295.8

　　The total grinding media charge is 22 t, from which the weights of the steel balls in each size class can be calculated (see Table 2).

Table 2: Weights of Steel Balls at Each Grade

The weight-average particle diameter can then be calculated as: 89.4 mm.

The calculated average particle diameter is:

26×100+28×90+33.5×80+21.5×70 9305

────────────────── ＝────≈85.4mm

26+28+33.5+21.5 109

The calculation results were broadly consistent with the actual situation. At the time, the fineness of the product leaving the mill was excessively coarse; during the mill shutdown for adjustment, calculations indicated that 5 tonnes of Φ100–95 mm grinding balls should be removed and replaced with 1 tonne of Φ90 mm balls and 4 tonnes of Φ80 mm balls, thereby bringing the fineness up to specification. Trial runs demonstrated that this approach is relatively accurate and effective for monitoring the internal conditions of the mill and resolving operational issues.