Free calculator

Dissolved-Oxygen Sag — Streeter–Phelps

Enter the deoxygenation and reaeration constants, the ultimate BOD of the mixture, and the initial deficit — the calculator finds the critical time, the critical deficit, and the minimum dissolved oxygen, and checks the deficit at any travel time you choose. Optionally estimate k₂ from stream velocity and depth first.

D(t) = k₁L₀k₂ − k₁(e^−k₁t − e^−k₂t) + D₀e^−k₂t

t_c = 1k₂ − k₁ · ln [k₂k₁(1 − D₀(k₂ − k₁)k₁L₀)] DO_min = DO_sat − D(t_c)

k₁, k₂	deoxygenation / reaeration constants, day⁻¹, base e, same time base
L₀	ultimate BOD of the mixture at the outfall (mg/L)
D₀	oxygen deficit at the mixing point (mg/L)
DO_sat	saturation DO at stream temperature (≈ 9.1 mg/L at 20°C)
Requires k₂ > k₁	otherwise the sag never recovers and the equations have no physical solution

Optional: estimate k₂ from velocity and depth (O’Connor–Dobbins)

Unit system Stream velocity V (m/s) Mean depth H (m)

k₂ = 3.9 V^0.5/H^1.5 (SI, day⁻¹) or 12.9 V^0.5/H^1.5 (US, day⁻¹) — base e, at 20°C.

Sag calculation

k₁ deoxygenation (day⁻¹, base e) k₂ reaeration (day⁻¹, base e) Ultimate BOD L₀ (mg/L) Initial deficit D₀ (mg/L) Saturation DO (mg/L) Check deficit at time t (days)

For study and checking only. These calculators run entirely in your browser — nothing is sent anywhere. They are meant to verify your hand calculations while you study; on the exam you will work by hand with the NCEES reference handbook.

Dissolved-Oxygen Sag — Streeter–Phelps

Optional: estimate k2 from velocity and depth (O’Connor–Dobbins)

Sag calculation

Optional: estimate k₂ from velocity and depth (O’Connor–Dobbins)