A battery of cylindrical hollow-fiber membranes is operated at steady state to concentrate a bacterial suspension from a fermenter. 350 kg min-1 broth is pumped through a stack of hollow-fiber membranes as shown. The broth contains 1% bacteria (by weight); the rest may be considered water. Buffer solution enters the annular space around the membrane tubes at a flow 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. The aim of the membrane system is produce a cell suspension containing 6% biomass.
(a) What is the flow rate of cell suspension from the membrane tubes?
(b) What is the flow rate from the annular space? Assume the cells are not active, i.e. they do not grow. Assume further that the membrane does not allow any molecules other than water to pass from annulus to inner cylinder, or vice versa.