$4\mathrm{.}9$ Concentration and Diafiltration with an Ultrafiltration System A $2000-$liter volume of a $30$ g $/$ liter protein solution needs to be concentrated to $200$ g $/$ liter. Also, the initial salt concentration must be reduced one hundredfold. The relationship between the permeation rate and the bulk protein concentration has been determined experimentally as follows: Q$=500$ ln$(270/$c$\_$b$)182//$ BIOSEPARATIONS SCIENCE AND ENGINEERING where Q$=$ permeation rate $($liters$/$h$)$ c$\_$b$=$ bulk protein concentration $($g$/$liter$)$ Determine the operation batch volume that will give the minimum time for diafiltration. Then determine the times required for $($a$)$ concentration to the volume at which diafiltration is to be carried out, $($b$)$ diafiltration, and $($c$)$ concentration to the final concentration desired.