ENV-36

Oxygen transfer KLa

C(t) = Cs − (Cs−C0) e^{−KLa t}. Two-film aeration.

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Water qualityASCE oxygen transfer

Governing equation

C=C_s-(C_s-C_0)e^{-K_La t}

$t \in \left[0.02, 6\right]$

where

K_La: KLa (1/h)
C_s: Saturation DO (mg/L)
C_0: Initial DO (mg/L)
t: Time (h)
C: Dissolved oxygen (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

C=C_s-(C_s-C_0)e^{-K_La t}

$t \in \left[0.02, 6\right]$

$t = 0.02$ $C = 2.546$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0.02,\quad t_{\max} = 6

$t \in \left[0.02, 6\right]$

$t = 0.02$ $t = 0.02$

Along the curve

③

Response curve — green

C=C_s-(C_s-C_0)e^{-K_La t}

$t \in \left[0.02, 6\right]$

$t = 0.02$ $C = 2.546$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}C}{\mathrm{d}t}

$t \in \left[0.02, 6\right]$

$t = 0.058$ $C^{\prime} = 22.64$

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-36 — Oxygen transfer KLa) is the form associated with ASCE oxygen transfer. Working symbols: $K_La$ , $C_s$ , $C_0$ , $t$ $\rightarrow$ $C$ . Lewis–Whitman two-film: flux = KL (Cs−C) a. The lumped KLa is the aeration rate of a basin.

Purpose

Purpose: compute

C

from

K_La

C_s

C_0

t

in Environmental via

C=C_s-(C_s-C_0)e^{-K_La t}

C(t) = Cs − (Cs−C0) e^{−KLa t}. Two-film aeration. 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_La = 4.000\,\mathrm{1/h}

C_s = 9.100\,\mathrm{mg/L}

C_0 = 2.000\,\mathrm{mg/L}

t = 0.500\,\mathrm{h}

, the governing relation

C=C_s-(C_s-C_0)e^{-K_La t}

yields

C = 8.139\,\mathrm{mg/L}

. A basin, bubbles, a rising DO curve. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

KLa K_La4 1/hSaturation DO C_s9.1 mg/LInitial DO C_02 mg/LTime t0.5 h

Outputs

Dissolved oxygen C8.139 mg/L

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Narration of this film

A basin, bubbles, a rising DO curve.

Lewis–Whitman two-film: flux = KL (Cs−C) a. The lumped KLa is the aeration rate of a basin.

Reading speed

Watch on YouTube