When calculated by means of Rosin, Rammler and Sperling's Equation (2) the residue of crushed product, R, equals l, in case x/x=0, and 0 in case x/x=∞.
But actually, particle size of the crushed product is smaller than that of the raw material, viz., the initial particle size x0, and R equals 0 in case x/x=x0/x<∞. Therefore, the values obtained from Eq. (2) does not show good agreement with the values measured of extremely coarse grains. Whereupon, the authors have derived the following equations as an expression for the size distribution of the crushed product:
R=e-(x'/x')ⁿ (4)
where l/x'=l/x-l/x₀ (3)
From these, it is obvious that R=0 in case x=x0. Eq. (4) is especially suited for the calculation of the values of extremely coarse grains. For the finer grains, x<<x0, x' x, Eq. (4)≈Eq. (2). Again from Eqs. (4), (6) and (8), the authors have derived the following equations showing the relation among the net work input to crushing Ar, the surface increment ΔS, and the size distribution of the crushed product:
ΔS∝Ar∝W0/x' (9)
R=e^-(x'/x')n=e-k(A_r/W₀)n·x'n (10)
Most of the laws of crushing or grinding give the relation between the work required for crushing and the magnitude of the increased surface of the crushed product. But, regrettably enough, the work required for crushing is not satisfactorily defined in any of them, neither is it clearly expressed. Strictly speaking, it is often the case that we can hardly put such laws to any practical use. Even Rittinger's law will not work well, either.
According to the authors' idea, the surface increment is to be considered directly proportional to the net work input for crushing, and Rittinger's law coincided with the authors', if the work required for crushing in his law is defined as the net work input for crushing.
The authors assumed that the work proportional to the net work input for crushing by drop weight impact might be given by Eqs. (14) and (15).
The results obtained by means of Eqs. (4), (10) and (14) give good agreement with those of the experiments as shown in Fig. 2-25.

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