CBOD Oxidation and Hydrolysis: Decay Rates in River Water Quality Models

Carbonaceous biochemical oxygen demand (CBOD) is the primary oxygen-consuming constituent in rivers and a key term in the dissolved oxygen balance. QUAL2K separates CBOD into two fractions — fast CBOD and slow CBOD — to represent the biphasic decay observed in natural waters.

The Streeter-Phelps Foundation

The classic Streeter-Phelps model (1925), as documented in the EPA QUAL2K framework, describes BOD decay as a first-order process. QUAL2K extends this to two fractions with separate decay rates:

Figure 1: Two-fraction CBOD decay pathway

Fast CBOD Kinetics

Fast CBOD represents the easily biodegradable organic matter (sugars, amino acids, volatile fatty acids). It decays via first-order oxidation:

\frac{dCBOD_f}{dt} = -k_{dc} \cdot CBOD_f - k_h \cdot CBOD_f + k_h \cdot CBOD_s

Where:

$k_{dc}$ — fast CBOD oxidation rate at 20°C (day⁻¹), typically 0.1–1.0
$k_h$ — hydrolysis rate transferring slow → fast CBOD (day⁻¹), typically 0.01–0.1

Slow CBOD Kinetics

Slow CBOD represents the refractory organic fraction (cellulose, lignocellulose). It decays much more slowly:

\frac{dCBOD_s}{dt} = -k_{dcs} \cdot CBOD_s - k_h \cdot CBOD_s

Where $k_{dcs}$ is the slow CBOD oxidation rate (day⁻¹), typically 0.001–0.01.

Arrhenius Temperature Correction

All kinetic rate constants are corrected for temperature using the Arrhenius equation:

k(T) = k(20) \cdot \theta^{(T - 20)}

where $\theta$ is the temperature correction coefficient. Typical values:

Temperature correction coefficients (θ)

Process	θ	Effect at 10°C	Effect at 30°C
CBOD fast oxidation	1.047	0.63× (37% slower)	1.59× (59% faster)
CBOD slow oxidation	1.047	0.63×	1.59×
Hydrolysis	1.047	0.63×	1.59×
Nitrification	1.083	0.45× (55% slower)	2.22× (122% faster)
Reaeration	1.024	0.79× (21% slower)	1.27× (27% faster)
Denitrification	1.047	0.63×	1.59×

Typical CBOD Decay Rates

Literature values for CBOD decay rate constants at 20°C

Water Body Type	k_dc (day⁻¹)	Source
Clean mountain streams	0.05 – 0.10	Thomann & Mueller (1987)
Moderately polluted rivers	0.10 – 0.30	Chapra (1997)
Untreated municipal wastewater	0.30 – 0.70	Metcalf & Eddy (2014)
Treated wastewater effluent	0.10 – 0.25	Thomann & Mueller (1987)
Industrial wastewater	0.05 – 1.00	varies widely

Discrete Implementation

QUAL2K uses the analytical solution for first-order decay over the travel time $\Delta t$ :

CBOD_f(t + \Delta t) = CBOD_f(t) \cdot e^{-k_{dc} \cdot \Delta t}

This prevents numerical instability for large time steps.

pythonkinetics.py — CBOD decay implementation

# Temperature-corrected rates
kdc = temp_correction(rates['kdc'], rates.get('kdc_theta', 1.047), temp)
kdcs = temp_correction(rates['kdcs'], rates.get('kdcs_theta', 1.047), temp)
kh = temp_correction(rates.get('khc', 0), 1.047, temp)

# Fast CBOD: oxidation + settling → loss; hydrolysis from slow → gain
loss_f = kdc + kh
conc[4] *= math.exp(-loss_f * dt)    # decay
conc[4] += kh * conc[3] * dt         # hydrolysis source

# Slow CBOD: slow oxidation + hydrolysis → loss
loss_s = kdcs + kh
conc[3] *= math.exp(-loss_s * dt)