ENV-38

Fick diffusive flux

J = −D (C2−C1)/Δx. Steady one-dimensional diffusion.

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Water qualityFick 1855

Governing equation

J=-D\dfrac{C_2-C_1}{\Delta x}

$\Delta x \in \left[0.001, 2\right]$

where

D: Diffusivity (m²/s)
C_1: Conc. 1 (mg/L)
C_2: Conc. 2 (mg/L)
\Delta x: Distance (m)
J: Flux toward 2 (mg/(m²·s))

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

J=-D\dfrac{C_2-C_1}{\Delta x}

$\Delta x \in \left[0.001, 2\right]$

$\Delta x = 0.001$ $J = 0.008$

Along the curve

②

Domain of the variable — blue

\Delta x_{\min} = 0.001,\quad \Delta x_{\max} = 2

$\Delta x \in \left[0.001, 2\right]$

$\Delta x = 0.001$ $\Delta x = 0.001$

Along the curve

③

Response curve — green

J=-D\dfrac{C_2-C_1}{\Delta x}

$\Delta x \in \left[0.001, 2\right]$

$\Delta x = 0.001$ $J = 0.008$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}J}{\mathrm{d}\Delta x}

$\Delta x \in \left[0.001, 2\right]$

$\Delta x = 0.014$ $J^{\prime} = -0.306$

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-38 — Fick diffusive flux) is the form associated with Fick 1855. Working symbols: $D$ , $C_1$ , $C_2$ , $\Delta x$ $\rightarrow$ $J$ . Fick's first law: flux is minus diffusivity times the concentration gradient. The start of mixing models.

Purpose

Purpose: compute

J

from

D

C_1

C_2

\Delta x

in Environmental via

J=-D\dfrac{C_2-C_1}{\Delta x}

J = −D (C2−C1)/Δx. Steady one-dimensional diffusion. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

D = 1.000e-9\,\mathrm{m^{2}/s}

C_1 = 10.000\,\mathrm{mg/L}

C_2 = 2.000\,\mathrm{mg/L}

\Delta x = 0.050\,\mathrm{m}

, the governing relation

J=-D\dfrac{C_2-C_1}{\Delta x}

yields

J = 1.600e-4\,\mathrm{mg/(m^{2}·s)}

. Two wells, a gradient, a flux arrow. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Diffusivity D1e-9 m²/sConc. 1 C_110 mg/LConc. 2 C_22 mg/LDistance \Delta x0.05 m

Outputs

Flux toward 2 J0.000160 mg/(m²·s)

Reading speed

Watch on YouTube

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Narration of this film

Two wells, a gradient, a flux arrow.

Fick's first law: flux is minus diffusivity times the concentration gradient. The start of mixing models.

Reading speed

Watch on YouTube