ENV-39

Oxygen deficit

Reaeration-only deficit decay.

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GoverningOxygen deficit

Governing equation

D=D_0 e^{-k_a t}

$t \in \left[0.1, 10\right]$

where

D0: D0 (mg/L)
ka: ka (1/d)
t: t (d)
D: Oxygen deficit (mg/L)

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

D=D_0 e^{-k_a t}

$t \in \left[0.1, 10\right]$

$t = 0.1$ $D = 5.765$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0.1,\quad t_{\max} = 10

$t \in \left[0.1, 10\right]$

$t = 0.1$ $t = 0.1$

Along the curve

③

Response curve — green

D=D_0 e^{-k_a t}

$t \in \left[0.1, 10\right]$

$t = 0.1$ $D = 5.765$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}D}{\mathrm{d}t}

$t \in \left[0.1, 10\right]$

$t = 0.162$ $D^{\prime} = -2.249$

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-39 — Oxygen deficit) is the form associated with Oxygen deficit. Working symbols: $D0$ , $ka$ , $t$ $\rightarrow$ $D$ . Reaeration-only deficit decay. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

D

from

D0

ka

t

in Environmental via

D=D_0 e^{-k_a t}

Reaeration-only deficit decay. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

D0 = 6.000\,\mathrm{mg/L}

ka = 0.400\,\mathrm{1/d}

t = 2.000\,\mathrm{d}

, the governing relation

D=D_0 e^{-k_a t}

yields

D = 2.696\,\mathrm{mg/L}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

D0 D06 mg/Lka ka0.4 1/dt t2 d

Outputs

Oxygen deficit D2.696 mg/L

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Reaeration-only deficit decay. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube