Problem $1(50$ points$)$ A process stream containing $2m\frac{g}{m}l$ of a target protein is to be concentrated using three ultrafiltration units connected in series. Further process considerations suggest that $20\%$ of the stream is removed as the filtrate in unit $1\mathrm{.}30\%$ in units $2$ and $3.$ Washing buffer may be added in between units $1$ and $2$ and in between units $2$ and $3$ to eliminate $($or alleviate $)$ concentration polarization problems. The washing cannot exceed $40\frac{L}{h}m($total$).$ Two membranes are available a retention of the target protein of either $0\mathrm{.}90$ or $0\mathrm{.}95.$ Both membranes work properly up to a bulk protein concentration of $4m\frac{g}{m}l.$ The minimum concentration fold is $2$:$00.$ Choose the appropriate configuration. Consider the following steps,

a$)$ Write a mass balance for the protein

b$)$ Assume $R$ at various combinations of washing

c$)$ Calculate concentration fold and yield

d$)$ Check for concentration polarization problems.