The biochemical reactions within a biological reactor can be categorized into three main phases: oxidation (respiration), synthesis (cell growth), and endogenous respiration.

2.1. Fundamental Metabolic Reactions

• Oxidation (Respiration): Microorganisms oxidize approximately one-third of the organic matter into carbon dioxide and water to generate energy.
• Synthesis (Protoplasm): The energy from oxidation is used to synthesize the remaining two-thirds of the organic matter into new cell material.
• Endogenous Respiration: In the absence of an external food source, microorganisms metabolize their own cellular material for maintenance energy, leading to a net decrease in biomass.

2.2. Key Metabolic Equations

The overall biological metabolism can be summarized by two basic equations:

1. Organic matter metabolized = Protoplasm synthesized + Energy for synthesis
2. Net protoplasm accumulation = Protoplasm synthesized – Endogenous respiration

2.3. Kinetic Models of Microbial Growth

Mathematical models are used to describe the rate of microbial growth and substrate utilization.

• Monod Equation: This widely accepted model describes the relationship between the specific growth rate of microorganisms (μ) and the substrate concentration (S) in a substrate-limited system: μ = (μ_max * S) / (K + S) where μ_max is the maximum specific growth rate and K is the half-velocity coefficient.
• Haldane Equation: This model is used for systems where the substrate itself becomes inhibitory at high concentrations. It modifies the Monod equation to account for this inhibition: μ = (μ_max * S) / (S + K_s + (S² / K_i)) where K_i is the inhibition constant. A smaller K_i value indicates greater inhibition.

2.4. Substrate Utilization and Yield

The relationship between microbial growth and substrate removal is defined by the growth yield coefficient (Y), which represents the mass of new cells synthesized per unit mass of substrate removed (ΔX = Y * ΔS). The rate of substrate utilization per unit of biomass is denoted as q.

2.5. Effect of Temperature

Temperature is a significant parameter influencing biological reaction rates. The effect is typically modeled using the following equation: k_T = k_20 * θ^(T-20) where k_T is the rate constant at temperature T, k_20 is the rate at 20°C, and θ is the temperature coefficient (often around 1.035).