Optimal Dilution Rate (D_opt) for Max Productivity (No Maintenance)

Calculation Explanation

In a chemostat (continuous stirred-tank reactor at steady state), the biomass productivity (rate of biomass production per unit volume, \(D \cdot c_x\)) depends on the dilution rate (\(D\), which equals the specific growth rate \(\mu\) at steady state). There is an optimal dilution rate (\(D_{opt}\)) that maximizes this productivity.

Assuming standard Monod kinetics for growth and, crucially, **neglecting substrate consumption for cell maintenance (i.e., \(m_s = 0\))**, the optimal dilution rate can be calculated analytically:
D_opt = μ_max * ( 1 - √( K_s / (c^f_s + K_s) ) )

Where:
- D_opt: Optimal dilution rate for maximum biomass productivity (units typically h⁻¹ or day⁻¹).
- μ_max: Maximum specific growth rate (same units as D_opt).
- K_s: Monod constant or saturation constant (substrate concentration units, e.g., g/L).
- c^f_s: Concentration of the limiting substrate in the feed (same units as K_s).

Important Note: This equation provides a useful estimate but is only valid when maintenance requirements are negligible. When maintenance (\(m_s > 0\)) is included, finding \(D_{opt}\) becomes more complex, typically requiring numerical methods or solving a higher-order polynomial equation. The true optimal dilution rate considering maintenance will generally be lower than the value calculated here.