Understanding the metabolic reactions and the kinetics that govern them provides the quantitative framework for predicting system performance, calculating oxygen demand, and managing sludge production. These concepts transform biological treatment from a qualitative art into a quantitative science, enabling engineers and operators to design and manage processes with precision.

The metabolic activity within a biological reactor can be broadly categorized into three primary phases:

1. Oxidation: Microorganisms respire, breaking down organic or inorganic matter to release the energy required for their life processes.
2. Synthesis: Using the energy from oxidation, microorganisms convert a portion of the substrate into new cell material (protoplasm). It is estimated that approximately one-third of the organic matter removed is oxidized to provide energy for the synthesis of the remaining two-thirds into new cell material.
3. Endogenous Respiration: In the absence of an external food source, microorganisms undergo a process of self-destruction, breaking down their own cellular material to obtain maintenance energy.

These phases are captured by two fundamental equations that describe the overall biological metabolism:

Organic matter metabolized = Protoplasm synthesized + Energy for synthesis

Net protoplasm accumulation = Protoplasm synthesized – Endogenous respiration

The speed at which these reactions occur is described by microbial growth kinetics. The most widely accepted model relating the specific growth rate of microorganisms (μ) to the concentration of the substrate (S) is the Monod equation:

μ = (μ_max * S) / (K + S)          (Eq. 5)

Where:

• μmax is the maximum specific growth rate, achievable when the substrate is not a limiting factor.
• K is the half-saturation constant (also known as the half-velocity coefficient), representing the substrate concentration at which the specific growth rate is half of μ_max.

Two other critical kinetic coefficients are the growth yield coefficient (Y), which describes the mass of new cells synthesized per mass of substrate removed, and the microbial decay rate (b), which quantifies the rate of biomass loss due to endogenous respiration.

These coefficients are used to calculate the net specific growth rate (μ), which accounts for both cell synthesis and decay:

Net Specific Growth Rate (μ) = [(q_max * Y * S) / (K + S)] – b          (Eq. 15)

Finally, the impact of temperature on these biological reaction rates is a critical operational consideration. The relationship is typically described by an Arrhenius-type equation, which shows that reaction rates increase with temperature:

k_T = k₂₀ * θ^{(T – 20)}          (Eq. 16)

Where θ is the temperature coefficient, a value that quantifies the multiplier effect on the reaction rate for a given change in temperature from a 20°C baseline.

These theoretical kinetic models form the basis for the practical design and operational parameters used to control real-world biological treatment systems.