BOD Removal — First-Order Kinetics
First-order BOD removal in a continuously stirred reactor (CSTR) or plug-flow reactor (PFR). Applies to activated sludge basins, oxidation ponds, and trickling-filter effluent prediction. SI and US customary units.
Defaults: 200 mg/L influent BOD (medium-strength domestic), k = 0.25/day (typical activated sludge at 20°C), 6-hour aeration tank. Concentrations are unit-agnostic; θ in hours is consistent with k in 1/day via internal conversion.
CSTR vs PFR — when to use which
Plug-flow reactors are mathematically equivalent to a long thin pipe where each fluid parcel ages in sequence. They give the highest treatment efficiency for first-order kinetics at a given detention time. Real long-rectangular aeration basins approach plug-flow behavior, especially with serial baffles.
CSTR (completely mixed) reactors assume instantaneous mixing — every parcel sees the effluent concentration. They tolerate shock loads better but require more volume than PFR for the same effluent quality. Most circular activated-sludge basins, oxidation ditches, and SBRs in fill-and-react mode behave as CSTR.
The ratio between CSTR and PFR volume needed for the same effluent S grows with kτ. At 90% removal, CSTR needs ~3.9× the PFR volume. Engineers exploit this by series-staging CSTRs (3-4 in series approaches PFR behavior) to gain efficiency.
Typical first-order rate constants
- Activated sludge, BOD: k = 0.10 to 0.40 /day at 20°C (depends on F:M ratio, MLSS, aeration intensity).
- Trickling filter, BOD: k = 1.0 to 2.5 /day (modified Velz formula uses different exponents but k is comparable).
- Oxidation pond, BOD: k = 0.05 to 0.15 /day (slow, but cheap).
- Aerated lagoon, BOD: k = 0.10 to 0.30 /day.
- Anaerobic digester, COD: k = 0.05 to 0.15 /day.
Temperature dependence
Biological reaction rates roughly double for every 10°C rise. The Arrhenius factor θ = 1.04 for BOD removal is the textbook value (Metcalf & Eddy). Some references use 1.05 for trickling filters and 1.07 for nitrification. Always check the design temperature — winter low-flow conditions often govern POTW design even though peak summer biomass is more dramatic.
Calibration vs design k
For an existing plant, back-calculate k from observed effluent quality at known temperature. For a new design, take k from a textbook range and apply a conservative factor of 0.7 to 0.8. The rate is sensitive to MLSS, mixing, aeration efficiency, and influent biodegradability — none of which are captured in a single k.
Reference: Metcalf & Eddy / AECOM (2014). Wastewater Engineering: Treatment and Resource Recovery, 5th ed., McGraw-Hill, ch. 8. Tchobanoglous, G., et al. (2003). Wastewater Engineering, 4th ed., ch. 5.