First‑order BOD decay assumes the remaining ultimate demand decreases exponentially with time: L(t)=L0·e−kt.
- L0 is the ultimate BOD at t=0 (mg/L).
- k is the first‑order decay constant (1/day).
- L(t) is remaining ultimate demand after time t (mg/L).
- y(t) is exerted demand: y(t)=L0−L(t)=L0(1−e−kt).
Optional temperature correction uses k(T)=k20·θ(T−20), where k20 is at 20°C and θ is a dimensionless coefficient.
- Select Forward to predict remaining and exerted demand.
- Enter L0, t, and either k or temperature fields.
- Enable temperature correction if you have k20, θ, and T.
- Optionally generate a decay table by setting duration and step.
- Press Calculate. Results appear above the form.
- Use Download CSV or Download PDF for reporting.
Sample inputs and outputs for quick verification.
| L0 (mg/L) | k20 (1/day) | θ | T (°C) | t (day) | k(T) (1/day) | L(t) (mg/L) | y(t) (mg/L) | Remaining (%) |
|---|---|---|---|---|---|---|---|---|
| 200 | 0.23 | 1.047 | 25 | 5 | 0.289375 | 47.060856 | 152.939144 | 23.5304 |
Values shown are rounded for display and may vary slightly.
BOD Decay Modeling in Water Quality
1) Understanding biochemical oxygen demand
Biochemical oxygen demand (BOD) represents the oxygen microorganisms consume while breaking down biodegradable organic matter in water. In streams, reactors, and effluent channels, BOD helps estimate how quickly oxygen can be depleted. This calculator uses a first-order approach to predict how the remaining ultimate BOD changes with time.
2) First-order decay relationship
First-order decay assumes the removal rate is proportional to the remaining ultimate BOD, L. The remaining ultimate BOD at time t is modeled as L(t)=L0e-kt. The exerted BOD is y(t)=L0−L(t), which is commonly compared to measured BOD over an incubation period.
3) Inputs used by the calculator
You can enter the initial ultimate BOD L0 (mg/L), the base rate constant at 20 °C, k20 (1/day), a temperature coefficient θ, the water temperature T (°C), and time t (days). Optional fields let you compute L0 from a measured BOD value and incubation time.
4) Temperature correction for k
Reaction rates typically increase with temperature. A common correction is k(T)=k20θ(T−20). For carbonaceous BOD, θ is often around 1.047, but values can vary with wastewater characteristics and microbial activity. The calculator reports both k20 and the temperature-adjusted k(T).
5) Interpreting the outputs
The results show remaining ultimate BOD L(t), exerted BOD y(t), and the percent remaining. A larger k means faster decay, so L(t) drops sooner and y(t) rises more quickly. The optional decay table provides stepwise values to support plotting or process checks.
6) Typical ranges and quick checks
In practice, k20 is often between about 0.05 and 0.5 day−1 for many natural and engineered systems. If your model predicts negative values, extremely large k, or more exerted BOD than L0, recheck units, time bases, and whether you are modeling carbonaceous versus total BOD.
7) Where this model is applied
First-order BOD decay is used in river dissolved oxygen sag analysis, aeration basin sizing, stabilization pond planning, and compliance evaluations where oxygen demand must be estimated over travel time. It is also useful for scenario testing, such as temperature changes, altered detention time, or influent strength shifts.
8) Assumptions and data quality
The first-order model is a simplification: it assumes a single decay rate, constant mixing, and no re-aeration or nitrification effects unless handled separately. For best results, calibrate k using site measurements and keep the same basis for time and temperature. Use multiple samples when possible, because BOD tests can exhibit notable variability.
FAQs
1) What is the difference between L(t) and y(t)?
L(t) is the remaining ultimate BOD at time t. y(t) is the exerted BOD up to time t, calculated as L0 minus L(t).
2) Which k value should I use, k20 or k(T)?
Use k20 when comparing results at the standard reference temperature. Use k(T) when modeling actual water temperature conditions.
3) What units should I enter for time?
Enter time in days. If you have hours, divide by 24. Keep k in 1/day so the exponential term remains consistent.
4) How do I estimate L0 from a measured BOD test?
If you know BOD measured at time t and a decay rate k, you can compute L0 using L0 = BODt / (1 − e^(−kt)).
5) What theta value is typical for carbonaceous BOD?
A commonly used value is around 1.047, but wastewater type and microbial conditions can shift theta. Use local calibration when available.
6) Why does my model show very slow decay?
Slow decay usually comes from a small k value or a low temperature. Confirm you did not enter k in 1/hour or time in hours by mistake.
7) Is first-order decay always accurate?
It is a practical approximation. Multi-stage behavior, nitrification, toxicity, or changing biomass can require more detailed models and calibrated parameters.