A packed bed composed of crushed rock having a density of $175 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}$ is to be used as a filter. The size and shape of the rock particles is such that the average surface area to volume ratio is $50 \mathrm{in.}^{2} / \mathrm{in} .^{3}$ and the bed porosity is 0.3 . A lab test using the slurry to be filtered is run on a small bed of the same particles, which is $6 \mathrm{in}$. deep and $6 \mathrm{in}$. in diameter. The slurry is pumped through this bed at a constant filtrate rate of $10 \mathrm{gpm}$, and it is found that after $5 \mathrm{~min}$ the pressure drop is $5 \mathrm{psi}$, while after $10 \mathrm{~min}$ it is 8 psi. Calculate how long it would take to filter $100,000 \mathrm{gal}$ of filtrate from the slurry in a full-scale bed that is $10 \mathrm{ft}$ in diameter and $2 \mathrm{ft}$ deep, if the slurry is maintained at a depth of $2 \mathrm{ft}$ above the bed and drains through it by gravity. Assume the slurry densities to be the same as water.