A protein chromatography column that processes $15kg$ of protein in $60h,$ is cycled every $6h,$ can be loaded with $10$ grams of protein per liter of column volume, has a maximum flow rate of $50$ liters$/$ min at a pressure of $4$ bar, and is packed with an incompressible resin having a particle diameter of $90m$ and void fraction of $0\mathrm{.}35.$ The viscosity can be assumed to be that of water.

i$.$ What are the dimensions required for the column?

ii$.$ If the flow rate is $1$ column volume per minute and the pressure rating is still $4$ bar, what column dimensions are required? Is the resolution equivalent to that for Part $($i$)?$

iii. If the pressure rating for the column were raised to $12$ bar, should the column dimensions change? Is so$,$ what should the dimensions be$?$ Do they make sense?

iv$.$ What is the best way to increase the productivity of the column packing?