ENV-09

Streeter–Phelps oxygen sag

dD/dt = k_d L − k_a D. Deficit D between saturation and DO.

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Water qualityStreeter–Phelps

Governing equation

\dfrac{dD}{dt}=k_d L-k_a D

$D \in \left[0, 12\right]$

where

k_d: Deoxygenation (1/d)
k_a: Reaeration (1/d)
L: BOD remaining (mg/L)
D: Deficit now (mg/L)
D': Deficit slope (mg/L/d)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\dfrac{dD}{dt}=k_d L-k_a D

$D \in \left[0, 12\right]$

$D = 0$ $D' = 4$

Along the curve

②

Domain of the variable — blue

D_{\min} = 0,\quad D_{\max} = 12

$D \in \left[0, 12\right]$

$D = 0$ $D = 0$

Along the curve

③

Response curve — green

\dfrac{dD}{dt}=k_d L-k_a D

$D \in \left[0, 12\right]$

$D = 0$ $D' = 4$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}D'}{\mathrm{d}D}

$D \in \left[0, 12\right]$

$D = 0.075$ $D'^{\prime} = -0.5$

Along the curve

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-09 — Streeter–Phelps oxygen sag) is the form associated with Streeter–Phelps. Working symbols: $k_d$ , $k_a$ , $L$ , $D$ $\rightarrow$ $D'$ . First-order BOD decay plus atmospheric reaeration. The sag has a critical time.

Purpose

Purpose: compute

D'

from

k_d

k_a

L

D

in Environmental via

\dfrac{dD}{dt}=k_d L-k_a D

dD/dt = k_d L − k_a D. Deficit D between saturation and DO. 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_d = 0.200\,\mathrm{1/d}

k_a = 0.500\,\mathrm{1/d}

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

D = 2.000\,\mathrm{mg/L}

, the governing relation

\dfrac{dD}{dt}=k_d L-k_a D

yields

D' = 3.000\,\mathrm{mg/L/d}

. A river, a BOD slug, a DO sag curve. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Deoxygenation k_d0.2 1/dReaeration k_a0.5 1/dBOD remaining L20 mg/LDeficit now D2 mg/L

Outputs

Deficit slope D'3.000 mg/L/d

Reading speed

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Narration of this film

A river, a BOD slug, a DO sag curve.

First-order BOD decay plus atmospheric reaeration. The sag has a critical time.

Reading speed

Watch on YouTube