The Activated Sludge Process (ASP) is a suspended-growth biological treatment system widely utilized to remove dissolved organic pollutants from municipal wastewater. Aerobic microorganisms within an aerated reactor consume biodegradable organic matter (measured as BOD) to synthesize cellular biomass, carbon dioxide, and water. The operational control of a continuous-flow completely mixed activated sludge reactor relies on key kinetic metrics, primarily the Food-to-Microorganism (F/M) Ratio : $$\text{F/M} = \frac{Q \cdot S_0}{V \cdot X}$$ Where $Q$ is the influent flow rate, $S_0$ is the influent substrate (BOD) concentration, $V$ is the aeration tank volume, and $X$ is the Mixed Liquor Volatile Suspended Solids (MLVSS) concentration representing active microbial biomass. The overall biological retention within the system is governed by the Mean Cell Residence Time ($\theta_c$ or Sludge Age) , which defines the average time micro-organisms are retained inside the system: $$...