Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 2 / i / Solution

Write the distributed fragment power-law size distribution as $n(D_f)=K_f{D}_f^{-{\alpha }_f}$. A spherical fragment has mass $\pi \rho D_f^3/6$. Since this population contains one half of the target mass,and henceThe distributed fragments have total geometric cross-sectionThe single fragment containing the other half of the mass has diameter $2^{-1/3}{D}_t$ and cross-section $\pi D_t^2/(42^{2/3})$. Therefore the exact result within the model isFor $D_{\max ,f}\gg D_{\min ,f}$ and $3<{\alpha }_f<4$, area is dominated by the smallest distributed fragments while mass is dominated by the largest, so normally the first term is negligible and