Oxygen Sag Curve Calculator
A river with mixed ultimate BOD 20 mg/L, initial deficit 1 mg/L and reaeration twice the deoxygenation rate reaches a critical deficit of 5.26316 mg/L after about 2.8 days.
9.1 mg/L at 20 °C, 8.3 at 25 °C, 7.6 at 30 °C (fresh water).
5.26315789473684 mg/L
Full answer, then cut to fewer decimals (not rounded):
- 5.26315789473684 mg/L
- 5.2631578947368 mg/L
- 5.263157894736 mg/L
- 5.26315789473 mg/L
- 5.2631578947 mg/L
- 5.263157894 mg/L
- 5.26315789 mg/L
- 5.2631578 mg/L
- 5.263157 mg/L
- 5.26315 mg/L
- 5.2631 mg/L
- 5.263 mg/L
- 5.26 mg/L
- 5.2 mg/L
- Minimum dissolved oxygen DO_sat − D_c
- 3.836842105 mg/L
- Critical time t_c
- 2.79066907 days
- Critical distance downstream
- 72.3341423 km
- Self-purification factor f = k_r/k_d
- 2
- Note
- DO falls below about 4 mg/L, the level most fish need.
| Day | Deficit (mg/L) | DO (mg/L) |
|---|---|---|
| 0 | 1 | 8.1 |
| 0.5 | 2.731 | 6.369 |
| 1 | 3.896 | 5.204 |
| 1.5 | 4.634 | 4.466 |
| 2 | 5.054 | 4.046 |
| 3 | 5.252 | 3.848 |
| 4 | 4.953 | 4.147 |
| 5 | 4.428 | 4.672 |
| 7 | 3.239 | 5.861 |
| 10 | 1.814 | 7.286 |
Step-by-step solution
- t_c = [1/(k_r − k_d)] ln{(k_r/k_d)[1 − D₀(k_r − k_d)/(k_d L₀)]} = 2.79066907 days
- D_c = (k_d/k_r) L₀ e^(−k_d t_c) = 5.263157895 mg/L
- x_c = u × t_c = 72.3341423 km
Oxygen Sag Curve Formula
D_t = (k_d L₀/(k_r − k_d))(e^(−k_d t) − e^(−k_r t)) + D₀e^(−k_r t)
t_c = ln[(k_r/k_d)(1 − D₀(k_r − k_d)/(k_d L₀))]/(k_r − k_d); D_c = (k_d/k_r)L₀e^(−k_d t_c)
Worked Example
A river with mixed L₀ = 20 mg/L, D₀ = 1 mg/L, k_d = 0.23 and k_r = 0.46 per day:
- t_c = [1/(k_r − k_d)] ln{(k_r/k_d)[1 − D₀(k_r − k_d)/(k_d L₀)]} = 2.79066907 days
- D_c = (k_d/k_r) L₀ e^(−k_d t_c) = 5.263157895 mg/L
- x_c = u × t_c = 72.3341423 km
Critical time t_c: 2.79066907 days · Critical distance downstream: 72.3341423 km · Self-purification factor f = k_r/k_d: 2 · Note: DO falls below about 4 mg/L, the level most fish need.
| Day | Deficit (mg/L) | DO (mg/L) |
|---|---|---|
| 0 | 1 | 8.1 |
| 0.5 | 2.731 | 6.369 |
| 1 | 3.896 | 5.204 |
| 1.5 | 4.634 | 4.466 |
| 2 | 5.054 | 4.046 |
| 3 | 5.252 | 3.848 |
| 4 | 4.953 | 4.147 |
| 5 | 4.428 | 4.672 |
| 7 | 3.239 | 5.861 |
| 10 | 1.814 | 7.286 |
GATE tip
Correct both rate constants for temperature first: k_d with θ = 1.047 and k_r with θ = 1.024. Saturation DO also falls with temperature, which lowers the minimum DO further.
More Water Quality Calculators
-
BOD Calculator
BOD_t = L₀(1 − e^(−kt)), BOD₅ = (D₁ − D₂)/P
k_T = k₂₀ × 1.047^(T − 20); P = sample volume ÷ bottle volume
Example: Wastewater with ultimate BOD 250 mg/L and k = 0.23 per day at 20 °C, after 5 days: BOD at day 5 = 170.841 mg/L.
-
Mixing Mass Balance Calculator
C = (Q₁C₁ + Q₂C₂) / (Q₁ + Q₂)
flows in any consistent unit; concentrations in any consistent unit
Example: A river of 10 m³/s with 2 mg/L BOD receiving 0.5 m³/s of effluent at 150 mg/L: Mixed concentration C = 9.04762.
-
Sodium Adsorption Ratio Calculator
SAR = Na / √((Ca + Mg)/2)
concentrations in meq/L; mg/L ÷ equivalent weight (Na 22.99, Ca 20.04, Mg 12.15)
Example: Water with 230 mg/L sodium, 60 mg/L calcium and 24 mg/L magnesium: Sodium adsorption ratio SAR = 6.34704.
-
Water Hardness Calculator
TH = 2.497 Ca + 4.118 Mg (mg/L as CaCO₃)
carbonate hardness = min(TH, alkalinity); non-carbonate = TH − carbonate
Example: Water with 60 mg/L calcium, 24 mg/L magnesium and 150 mg/L alkalinity: Total hardness = 248.669 mg/L as CaCO₃.
Frequently asked questions
What is the self-purification factor?
f = k_r/k_d. Fast, shallow, turbulent streams have high reaeration and recover quickly; slow deep rivers have low f and sag more.
How do I find the critical distance?
Multiply the critical time by the stream velocity: 2.8 days at 0.3 m/s is about 72 km.
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Oxygen Sag Curve Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/oxygen-sag-calculator/
IEEE
Scientific Converters, "Oxygen Sag Curve Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/oxygen-sag-calculator/