ENV-47

Sludge age

Mean cell residence time.

Reading speed

GoverningSludge age

Governing equation

\theta_x=X V/(X_e Q_e+X_w Q_w)

$Qw \in \left[2, 400\right]$

where

X: X (mg/L)
V: V (m³)
Xe: Xe (mg/L)
Qe: Qe (m³/d)
Xw: Xw (mg/L)
Qw: Qw (m³/d)
tx: Sludge age (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

\theta_x=X V/(X_e Q_e+X_w Q_w)

$Qw \in \left[2, 400\right]$

$Qw = 2$ $tx = 32.47$

Along the curve

②

Domain of the variable — blue

Qw_{\min} = 2,\quad Qw_{\max} = 400

$Qw \in \left[2, 400\right]$

$Qw = 2$ $Qw = 2$

Along the curve

③

Response curve — green

\theta_x=X V/(X_e Q_e+X_w Q_w)

$Qw \in \left[2, 400\right]$

$Qw = 2$ $tx = 32.47$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}tx}{\mathrm{d}Qw}

$Qw \in \left[2, 400\right]$

$Qw = 4.503$ $tx^{\prime} = -2.555$

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-47 — Sludge age) is the form associated with Sludge age. Working symbols: $X$ , $V$ , $Xe$ , $Qe$ , $Xw$ , $Qw$ $\rightarrow$ $tx$ . Mean cell residence time. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

tx

from

X

V

Xe

Qe

Xw

Qw

in Environmental via

\theta_x=X V/(X_e Q_e+X_w Q_w)

Mean cell residence time. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

X = 2500.000\,\mathrm{mg/L}

V = 800.000\,\mathrm{m^{3}}

Xe = 12.000\,\mathrm{mg/L}

Qe = 3800.000\,\mathrm{m^{3}/d}

Xw = 8000.000\,\mathrm{mg/L}

Qw = 40.000\,\mathrm{m^{3}/d}

, the governing relation

\theta_x=X V/(X_e Q_e+X_w Q_w)

yields

tx = 5.470\,\mathrm{d}

. 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

X X2500 mg/LV V800 m³Xe Xe12 mg/LQe Qe3800 m³/dXw Xw8000 mg/LQw Qw40 m³/d

Outputs

Sludge age tx5.470 d

Reading speed

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Narration of this film

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

Mean cell residence time. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube