$\textit{Saccharomyces cerevisiae}$ is cultured in a chemostat (continuous fermentation) at a dilution rate of $0.5 \: h^{-1}$. The feed substrate concentration is $10 \: g \cdot L^{-1}$. The biomass concentration in the chemostat at steady state will be _________ $g \cdot L^{-1}$.
Assumptions: Feed is sterile, maintenance is negligible and maximum biomass yield with respect to substrate is $0.4$ ($g$ biomass per $g$ ethanol).
Microbial growth kinetics is given by $\mu = \dfrac{\mu _m s}{K_s+s}$ where $\mu$ is specific growth rate $(h^{-1})$, $\mu_m = 0.7 \: h^{-1}$, $K_s = 0.3 \: g \cdot L^{-1}$ and $s$ is substrate concentration $(g \cdot L^{-1})$.