A battery of cylindrical hollow$-$fibre membranes is operated at steady state to concentrate a bacterial suspension harvested from a fermenter. Fermentation broth is pumped at a rate of $450$ kg min$-1$ through a stack of follow fibre membranes. The broth contains $0\mathrm{.}5\%$ bacteria; the rest may be considered water. Buffer solution enters the annular space around the membrane tubes at a rate of $80$ kg min$-1$; because broth in the membrane tubes is under pressure, water is forced across the membrane into the buffer. Cells in the broth are too large to pass through the membrane and pass out of the tubes as a concentrate. If the aim of the membrane system is to produce a cell suspension containing $8\%$ biomass:

$($i$)$ how much water has to be removed, per minute, from the fermentation broth?

$($ii$)$ what is the flowrate of the buffer solution that leaves the membrane system?