Productivity of CSTF

It would be interesting to derive the equations for the cell concentration and residence time at this maximum cell productivity. The cell productivity for a steady-state CSTF with sterile feed is 

The productivity is maximum when drx/dCx = 0. After substituting Cs = Cs. - Cx Yx/S into the preceding equation, differentiating with respect to Cx, and setting the resultant equation to zero, we obtain the optimum cell concentration for the maximum productivity as 

Substituting Eq. (6.41) into Eq. (6.38) for Cs yields the optimum residence time  

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