By ultrafiltration and diafiltration, 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 100-fold. The relationship between the permeation rate and the bulk protein concentration has been determined experimentally as follows: 270 Q = 500 In () co where Q = permeation rate (liters/h) cb = bulk protein concentration (g/liter) First, determine the operation batch volume that will give the minimum time for diafiltration. Then determine the times required for (a) the first reduction of the volume to the level at which diafiltration is to be carried out, (b) diafiltration, and (c) the last reduction of the volume to the final concentration. (Hint) You might need to use a calculator or software (Mathematica, MS-Excel, Matlab, etc.) to analytically solve an equation, or to employ a numerical method