ENV-01

Streeter–Phelps oxygen sag

DO deficit from first-order BOD and reaeration.

Reading speed

River qualityStreeter–Phelps 1925

Governing equation

D=\dfrac{k_1 L_a}{k_2-k_1}(e^{-k_1 t}-e^{-k_2 t})+D_0 e^{-k_2 t}

where

k_1: Deoxygenation (1/d)
k_2: Reaeration (1/d)
L_a: Initial BOD (mg/L)
D_0: Initial deficit (mg/L)
t: Travel time (d)
D: DO deficit (mg/L)

Lecture brief

Historical brief

Streeter–Phelps (1925) oxygen sag, settling theory and Guldberg–Waage kinetics made water and air quality a rate problem. The lab computes sag, overflow and a snapshot of reactor mass balance. This sheet (ENV-01 — Streeter–Phelps oxygen sag) is the form associated with Streeter–Phelps 1925. Working symbols: $k_1$ , $k_2$ , $L_a$ , $D_0$ , $t$ $\rightarrow$ $D$ . Streeter and Phelps coupled dL/dt = −k1 L with dD/dt = k1 L − k2 D, giving the classic sag curve.

Purpose

Purpose: compute

D

from

k_1

k_2

L_a

D_0

t

in Environmental via

D=\dfrac{k_1 L_a}{k_2-k_1}(e^{-k_1 t}-e^{-k_2 t})+D_0 e^{-k_2 t}

DO deficit from first-order BOD and reaeration. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

k_1 = 0.300\,\mathrm{1/d}

k_2 = 0.600\,\mathrm{1/d}

L_a = 20.000\,\mathrm{mg/L}

D_0 = 1.000\,\mathrm{mg/L}

t = 2.000\,\mathrm{d}

, the governing relation

D=\dfrac{k_1 L_a}{k_2-k_1}(e^{-k_1 t}-e^{-k_2 t})+D_0 e^{-k_2 t}

yields

D = 5.254\,\mathrm{mg/L}

. Point source, steady, well-mixed 1-D reach. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Deoxygenation k_10.3 1/dReaeration k_20.6 1/dInitial BOD L_a20 mg/LInitial deficit D_01 mg/LTravel time t2 d

Outputs

DO deficit D5.254 mg/L

Reading speed

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Narration of this film

Point source, steady, well-mixed 1-D reach.

Streeter and Phelps coupled dL/dt = −k1 L with dD/dt = k1 L − k2 D, giving the classic sag curve.

Reading speed

Watch on YouTube