Lactic acid is produced by a Lactobacillus species cultured in a CSTR.To increase the cell concentration and production rate , most of the cells in the reactor outlet are recycled to the CSTR, such that the cell concentration in the product stream is 10% of cell concentratior in the reactor. Find the optimum dilution rate that will maximize the rate of lactic acid production in the reac- tor. How does this optimum dilution rate change if the exit cell concentration fraction is changed? [r_(p)=(alpha mu +beta )C_(c)] mu _(max)=0.5h^-1,K_(s)=2.0g/dm^3,alpha =0.2g/g, C_(j0)=50g/dm^3 beta =0.1g/gcdot h, Y_(x/s)=0.2g/g,Y_(p/s)=0.3g/g

To find the optimum dilution rate that will maximize the rate of lactic acid production in the reactor, we need to maximize the production rate equation $r_{p}=(\alpha \mu +\beta )C_{c}$.<br /><br />Given:<br />$\mu _{max}=0.5h^{-1},K_{s}=2.0g/dm^{3},\alpha =0.2g/g,\quad C_{j0}=50g/dm^{3}$<br />$\beta =0.1g/g\cdot h,\quad Y_{x/s}=0.2g/g,Y_{p/s}=0.3g/g$<br /><br />The production rate equation can be rewritten as:<br />$r_{p}=(\alpha \mu +\beta )C_{c}=(\alpha \mu +\beta )(\frac{C_{j0}}{1+K_{s}C_{j0}})$<br /><br />To maximize $r_{p}$, we need to maximize $\mu$. Since $\mu$ is limited by the maximum specific growth rate $\mu_{max}$, the optimum dilution rate is when $\mu = \mu_{max}$.<br /><br />Therefore, the optimum dilution rate is $\mu_{max} = 0.5 h^{-1}$.<br /><br />If the exit cell concentration fraction is changed, the optimum dilution rate will also change. The new optimum dilution rate can be found by substituting the new exit cell concentration fraction into the production rate equation and solving for $\mu$.<br /><br />In summary, the optimum dilution rate that will maximize the rate of lactic acid production in the reactor is $\mu_{max} = 0.5 h^{-1}$. If the exit cell concentration fraction is changed, the optimum dilution rate will also change and can be found by substituting the new exit cell concentration fraction into the production rate equation and solving for $\mu$.