ENV-31

WASP first-order reach

C = C0 exp(−k x / U). Advection plus first-order decay, WASP snapshot.

Reading speed

Water qualityEPA WASP

Governing equation

C=C_0\exp(-k x/U)

$x \in \left[0.2, 80\right]$

where

C_0: Upstream conc. (mg/L)
k: Decay rate (1/d)
x: Distance (km)
U: Velocity (m/s)
C: Downstream conc. (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

C=C_0\exp(-k x/U)

$x \in \left[0.2, 80\right]$

$x = 0.2$ $C = 11.99$

Along the curve

②

Domain of the variable — blue

x_{\min} = 0.2,\quad x_{\max} = 80

$x \in \left[0.2, 80\right]$

$x = 0.2$ $x = 0.2$

Along the curve

③

Response curve — green

C=C_0\exp(-k x/U)

$x \in \left[0.2, 80\right]$

$x = 0.2$ $C = 11.99$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}C}{\mathrm{d}x}

$x \in \left[0.2, 80\right]$

$x = 0.702$ $C^{\prime} = -0.069$

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-31 — WASP first-order reach) is the form associated with EPA WASP. Working symbols: $C_0$ , $k$ , $x$ , $U$ $\rightarrow$ $C$ . WASP solves 1-D advection–dispersion with kinetic sources. This sheet drops dispersion and keeps a decaying plug.

Purpose

Purpose: compute

C

from

C_0

k

x

U

in Environmental via

C=C_0\exp(-k x/U)

C = C0 exp(−k x / U). Advection plus first-order decay, WASP snapshot. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

C_0 = 12.000\,\mathrm{mg/L}

k = 0.200\,\mathrm{1/d}

x = 8.000\,\mathrm{km}

U = 0.400\,\mathrm{m/s}

, the governing relation

C=C_0\exp(-k x/U)

yields

C = 11.457\,\mathrm{mg/L}

. A reach, a decaying C profile, a travel arrow. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Upstream conc. C_012 mg/LDecay rate k0.2 1/dDistance x8 kmVelocity U0.4 m/s

Outputs

Downstream conc. C11.457 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

A reach, a decaying C profile, a travel arrow.

WASP solves 1-D advection–dispersion with kinetic sources. This sheet drops dispersion and keeps a decaying plug.

Reading speed

Watch on YouTube