Streeter-Phelps River Dissolved Oxygen (DO) Sag Curve & Critical Deficit Calculator

Compute critical downstream time t_c (days) and minimum river Dissolved Oxygen (mg/L).

Minimum Critical Dissolved Oxygen (DO_min)
2.81 mg/L
Critical Downstream Travel Time (t_c) 2.66 days
Maximum Critical DO Deficit (D_c) 6.29 mg/L
Self-Purification Ratio (f = k_r / k_d) 1.8 ratio
Inputs
⚡ Scenarios:
mg/L
River saturation Dissolved Oxygen DO_sat at water temperature (mg/L).
mg/L
Initial DO deficit D₀ at mixing point (mg/L).
mg/L
Initial mixed ultimate BOD L₀ at mixing point (mg/L).
1/day
BOD deoxygenation rate constant k_d (1/day, base e).
1/day
Atmospheric reaeration rate constant k_r (1/day, base e).
Results & Graph
Minimum Critical Dissolved Oxygen (DO_min)
2.81 mg/L
Critical Downstream Travel Time (t_c)
2.66 days
Maximum Critical DO Deficit (D_c)
6.29 mg/L
Self-Purification Ratio (f = k_r / k_d)
1.8 ratio

Minimum Critical Dissolved Oxygen (DO_min)

Graph 1 • Streeter-Phelps DO Sag Curve • Domain Architecture & Live Telemetry Maps Saturation Do Mg L, Initial Oxygen Deficit D0 Mg L, Initial Ultimate Bod L0 Mg L through Streeter-Phelps DO Sag Curve Formula.
INPUT PARAMETERS • Saturation Do Mg L: 9.1 mg/L • Initial Oxygen Def: 1.5 mg/L • Initial Ultimate B: 22 mg/L ∑ F(x) Streeter-Phelps DO Sag Cur DESIGN OUTPUT 2.81 mg/L Critical Downstream Trav: 2.66 days Maximum Critical DO Defi: 6.29 mg/L
Graph 2 • Minimum Critical Dissolved Oxygen (DO_min) Across Saturation Do Mg L Sweep (40% → 160%) Evaluates t_c = [1 / (k_r − k_d)] × ln[ (k_r / k_d) × (1 − D_0·(k_r − k_d)/(k_d·L_ across 7 operating points (current input highlighted).
0 mg/L 4.55 mg/L 9.10 mg/L 0 3.64mg/L 0 5.46mg/L 0.990 7.28mg/L 2.81 9.10mg/L 4.63 10.92mg/L 6.45 12.74mg/L 8.27 14.56mg/L
Graph 3 • Joint Sensitivity: Saturation Do Mg L × Initial Oxygen Deficit D0 Mg L Compares Minimum Critical Dissolved Oxygen (DO_min) when Initial Oxygen Deficit D0 Mg L is shifted +25% (solid) vs. −25% (dashed).
0 mg/L 3.11 mg/L 6.22 mg/L --- Initial Oxygen Deficit D0 Mg L @ 75% 0 -30% 1.33 -15% 2.70 +0% 4.06 +15% 5.43 +30%
Summary: Critical oxygen sag occurs at t_c = 2.66 days downstream with maximum deficit D_c = 6.29 mg/L, dropping river Dissolved Oxygen to 2.81 mg/L.
Summary: Critical oxygen sag occurs at t_c = 2.66 days downstream with maximum deficit D_c = 6.29 mg/L, dropping river Dissolved Oxygen to 2.81 mg/L.
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Formula & Mathematical Methodology: Streeter-Phelps DO Sag Curve Formula

tc = [1 / (kr − kd)] × ln[ (kr / kd) × (1 − D0·kr − kdkd·L0) ]
Dc = (kd / kr)·L0·e−kd·tc
t_c = [1 / (k_r − k_d)] × ln[ (k_r / k_d) × (1 − D_0·(k_r − k_d)/(k_d·L_0)) ]; D_c = (k_d / k_r)·L_0·e^(−k_d·t_c)
  • saturationDoMgL: Saturation Do Mg L (mg/L)
  • initialOxygenDeficitD0MgL: Initial Oxygen Deficit D0 Mg L (mg/L)
  • initialUltimateBodL0MgL: Initial Ultimate Bod L0 Mg L (mg/L)
  • deoxygenationRateKdPerDay: Deoxygenation Rate Kd Per Day (1/day)
  • reaerationRateKrPerDay: Reaeration Rate Kr Per Day (1/day)

Rounding: Engineering outputs round to 2–3 decimal places; discrete material bag/unit counts round up via ceiling.

Worked Example: Engineering Scenario: Streeter-Phelps DO Sag Curve

Compute critical downstream time t_c (days) and minimum river Dissolved Oxygen (mg/L).

  1. Enter your system dimensions, electrical/RF ratings, or environmental conditions.
  2. Evaluate the governing equation: t_c = [1 / (k_r − k_d)] × ln[ (k_r / k_d) × (1 − D_0·(k_r − k_d)/(k_d·L_0)) ]; D_c = (k_d / k_r)·L_0·e^(−k_d·t_c).
  3. Inspect the primary design output, supporting allowances, and 3 SVG schematics.
Result: Computes Streeter-Phelps DO Sag Curve via t_c = [1 / (k_r − k_d)] × ln[ (k_r / k_d) × (1 − D_0·(k_r − k_d)/(k_d·L_0)) ]; D_c = (k_d / k_r)·L_0·e^(−k_d·t_c).

Frequently Asked Questions

How does the Streeter-Phelps River Dissolved Oxygen (DO) Sag Curve & Critical Deficit Calculator calculate results?

It evaluates t_c = [1 / (k_r − k_d)] × ln[ (k_r / k_d) × (1 − D_0·(k_r − k_d)/(k_d·L_0)) ]; D_c = (k_d / k_r)·L_0·e^(−k_d·t_c) directly in your browser and updates three custom SVG visualizations in real time.

Can I use Goal Seek to solve backwards for a required dimension or rating?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input parameter.

Does this engineering calculator work offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Science & Engineering · Environmental & Agricultural Science

Streeter-Phelps River Dissolved Oxygen (DO) Sag Curve & Critical Deficit Calculator: inputs, method, and how to read the result

Compute critical downstream time t_c (days) and minimum river Dissolved Oxygen (mg/L).

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Saturation Do Mg L: River saturation Dissolved Oxygen DO_sat at water temperature (mg/L).
  • Initial Oxygen Deficit D0 Mg L: Initial DO deficit D₀ at mixing point (mg/L).
  • Initial Ultimate Bod L0 Mg L: Initial mixed ultimate BOD L₀ at mixing point (mg/L).
  • Deoxygenation Rate Kd Per Day: BOD deoxygenation rate constant k_d (1/day, base e).

What the calculation includes

With the default values shown, Minimum Critical Dissolved Oxygen (DO_min) is 2.81 mg/L. The calculation follows Streeter-Phelps DO Sag Curve Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Saturation Do Mg L from 9.1 mg/L to 10.01 mg/L changes Minimum Critical Dissolved Oxygen (DO_min) from 2.81 mg/L to 3.72 mg/L. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses standard NREL, IEEE, ASHRAE, and IRC design equations.
  • Verify final structural, electrical, and mechanical specifications against local building codes.

Scope: Global Engineering & Building Code Standard Check the assumptions and limitations against your situation before relying on the estimate.

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