ENV-03

First-order BOD remaining

Lt = L0 e^{−k t}.

Reading speed

KineticsPhelps

Governing equation

L_t = L_0 e^{-kt}

$t \in \left[0.1, 20\right]$

where

L_0: Ultimate BOD (mg/L)
k: Rate constant (1/d)
t: Time (d)
L_t: BOD remaining (mg/L)
y_t: BOD exerted (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.

00:0 / 00:08

①

Governing law — red

L_t = L_0 e^{-kt}

$t \in \left[0.1, 20\right]$

$t = 0.1$ $L_t = 244.3$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0.1,\quad t_{\max} = 20

$t \in \left[0.1, 20\right]$

$t = 0.1$ $t = 0.1$

Along the curve

③

Response curve — green

L_t = L_0 e^{-kt}

$t \in \left[0.1, 20\right]$

$t = 0.1$ $L_t = 244.3$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}L_t}{\mathrm{d}t}

$t \in \left[0.1, 20\right]$

$t = 0.225$ $L_t^{\prime} = -54.61$

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-03 — First-order BOD remaining) is the form associated with Phelps. Working symbols: $L_0$ , $k$ , $t$ $\rightarrow$ $L_t$ , $y_t$ . Phelps modelled biochemical oxygen demand as a first-order decay of the remaining carbonaceous load.

Purpose

Purpose: compute

L_t

y_t

from

L_0

k

t

in Environmental via

L_t = L_0 e^{-kt}

Lt = L0 e^{−k t}. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

L_0 = 250.000\,\mathrm{mg/L}

k = 0.230\,\mathrm{1/d}

t = 5.000\,\mathrm{d}

, the governing relation

L_t = L_0 e^{-kt}

yields

L_t = 79.159\,\mathrm{mg/L}

y_t = 170.841\,\mathrm{mg/L}

. Constant k, no nitrification lag. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Ultimate BOD L_0250 mg/LRate constant k0.23 1/dTime t5 d

Outputs

BOD remaining L_t79.159 mg/L
BOD exerted y_t170.841 mg/L

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

Constant k, no nitrification lag.

Phelps modelled biochemical oxygen demand as a first-order decay of the remaining carbonaceous load.

Reading speed

Watch on YouTube