ENV-27

Weir oxygen transfer

r = 1 + 0.38 a b H (1−0.11 H)(1+0.046 T). Deficit ratio across a weir.

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Water qualityGameson

Governing equation

r=1+0.38abH(1-0.11H)(1+0.046T)

$H \in \left[0.1, 4\right]$

where

a: Weir type a (—)
b: Water quality b (—)
H: Drop (m)
T: Temperature (°C)
C_s: Saturation DO (mg/L)
C_{up}: Upstream DO (mg/L)
r: Deficit ratio (—)
C_{down}: Downstream DO (mg/L)

The figure, part by part

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①

Governing law — red

r=1+0.38abH(1-0.11H)(1+0.046T)

$H \in \left[0.1, 4\right]$

$H = 0.1$ $r = 1.066$

Along the curve

②

Domain of the variable — blue

H_{\min} = 0.1,\quad H_{\max} = 4

$H \in \left[0.1, 4\right]$

$H = 0.1$ $H = 0.1$

Along the curve

③

Response curve — green

r=1+0.38abH(1-0.11H)(1+0.046T)

$H \in \left[0.1, 4\right]$

$H = 0.1$ $r = 1.066$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}r}{\mathrm{d}H}

$H \in \left[0.1, 4\right]$

$H = 0.125$ $r^{\prime} = 0.65$

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-27 — Weir oxygen transfer) is the form associated with Gameson. Working symbols: $a$ , $b$ , $H$ , $T$ , $C_s$ , $C_{up}$ $\rightarrow$ $r$ , $C_{down}$ . Gameson correlated the aeration ratio r = (Cs−Cup)/(Cs−Cdown) with drop H, weir type a and water quality b.

Purpose

Purpose: compute

r

C_{down}

from

a

b

H

T

C_s

C_{up}

in Environmental via

r=1+0.38abH(1-0.11H)(1+0.046T)

r = 1 + 0.38 a b H (1−0.11 H)(1+0.046 T). Deficit ratio across a weir. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

a = 1.300\,\mathrm{—}

b = 0.800\,\mathrm{—}

H = 1.200\,\mathrm{m}

T = 15.000\,\mathrm{^{\circ}C}

C_s = 10.000\,\mathrm{mg/L}

C_{up} = 6.000\,\mathrm{mg/L}

, the governing relation

r=1+0.38abH(1-0.11H)(1+0.046T)

yields

r = 1.696\,\mathrm{—}

C_{down} = 7.641\,\mathrm{mg/L}

. A weir nappe, upstream and downstream DO ticks. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Weir type a a1.3 —Water quality b b0.8 —Drop H1.2 mTemperature T15 °CSaturation DO C_s10 mg/LUpstream DO C_{up}6 mg/L

Outputs

Deficit ratio r1.696 —
Downstream DO C_{down}7.641 mg/L

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Narration of this film

A weir nappe, upstream and downstream DO ticks.

Gameson correlated the aeration ratio r = (Cs−Cup)/(Cs−Cdown) with drop H, weir type a and water quality b.

Reading speed

Watch on YouTube