Unsteady Diffusion Explorer A, 3 Tabs (embed)

Start here: Play, Pause and Reset time control the run. Change any input and watch the field, the profile and the chart respond.

Geometry

Material and surface

D (m²/s)

D = cm²/s

Half-thickness or radius x₁ (m), log

k_c (m/s)

Infinite k_c (m = 0)

c₀ (mol/m³)

c₁ (mol/m³)

K

Example set-up

The bulk is well mixed and constant. Kc₁ is the body concentration in equilibrium with the bulk fluid. k_c is the fluid-film coefficient: surface flux = k_c(c₁ − c_s/K), so Bi = k_c x₁/(KD).

Time and probe

Time t (s)

The time slider spans X = 0.001 to 5, recalculated as D and x₁ change.

Position n = x/x₁, centre to surface

Speed One-term approximation

Playback speed

Time t

X = Dt/x₁²

m = 1/Bi

Y at probe

c at probe

Y at centre

Fractional change E = 1 − Ȳ

Animated schematic

Concentration profile

Drag the red probe. Keyboard: focus view, then use ← and →.

Gurney-Lurie chart

Log Y versus X.

Curves n

Curves m

Chart X max

Live n,m curve Live point Chart families

Read the chart mode

Read XRead Y

Enable read mode, then click the chart or enter X and Y.

Time to reach a target `Y`

Target Y at probet (s)

The solver uses the full series, regardless of the one-term display toggle.

Equations, assumptions and references

Governing equations

Mass-transfer analogues of the heat-conduction solutions in Perry’s Chemical Engineers’ Handbook, Section 5. The tags give the heat-conduction equation numbers.

Perry’s Eqs. 5-24, 5-25 (analogue)

\frac{\partial c}{\partial t} = D \nabla^{2} c

X = \frac{D t}{{x_{1}}^{2}}, n = \frac{x}{x_{1}}, Bi = \frac{k_{c} x_{1}}{K D}, m = \frac{1}{Bi}

x₁ is the half-thickness (slab) or radius (cylinder, sphere); for the semi-infinite medium it is a reference depth. k_c is the fluid-film coefficient, so the surface flux is k_c(c₁ − c_s/K).

Unaccomplished change

Y = \frac{K c_{1} - c}{K c_{1} - c_{0}}, c = K c_{1} - (K c_{1} - c_{0}) Y

Kc₁ is the body concentration in equilibrium with the bulk fluid. The colour field and the dots show c.

Finite bodies: series solution (56 terms)

Y = \sum_{k = 1} C_{k} \exp (- λ_{k}^{2} X) F_{k} (n)

Infinite slab

λ_{k} \tan (λ_{k}) = Bi, F = \cos (λ_{k} n), C = \frac{4 \sin (λ_{k})}{2 λ_{k} + \sin (2 λ_{k})}

Infinite cylinder

\frac{λ_{k} J_{1} (λ_{k})}{J_{0} (λ_{k})} = Bi, F = J_{0} (λ_{k} n), C = \frac{2 J_{1} (λ_{k})}{λ_{k} [{J_{0} (λ_{k})}^{2} + {J_{1} (λ_{k})}^{2}]}

Sphere

1 - λ_{k} \cot (λ_{k}) = Bi, F = \frac{\sin (λ_{k} n)}{λ_{k} n}, C = \frac{4 (\sin (λ_{k}) - λ_{k} \cos (λ_{k}))}{2 λ_{k} - \sin (2 λ_{k})}

For Bi → ∞ the eigenvalues are (2k − 1)π/2 (slab), the zeros of J₀ (cylinder) and kπ (sphere).

Perry’s Eq. 5-28; Tables 5-1, 5-2: one-term approximation

Y \approx C_{1} \exp (- λ_{1}^{2} X) F_{1} (n), X \geq X_{c}

X_c = 0.24 (slab), 0.21 (cylinder), 0.18 (sphere).

Perry’s Eq. 5-29: first-eigenvalue correlation

λ_{1} \approx \frac{λ_{\infty}}{{[1 + {(\frac{λ_{\infty}}{\sqrt{d Bi}})}^{p}]}^{1 / p}}

d = 1, 2, 3 and p = 2.139, 2.238, 2.314 for the slab, cylinder and sphere.

Average change and run times

E = 1 - \bar{Y}, \bar{Y} = \sum_{k = 1} C_{k} \exp (- λ_{k}^{2} X) ⟨ F_{k} ⟩

E (t_{50}) = 0.5, E (t_{90}) = 0.9

Volume weights for ⟨F_k⟩: slab 1, cylinder 2n, sphere 3n². The run readouts t₅₀ and t₉₀ are found by inverting E(t) with the full series.

Perry’s Eq. 5-30: semi-infinite medium, Bi → ∞

Y = \erf (η), η = \frac{x}{2 \sqrt{D t}}

Perry’s Eq. 5-31: semi-infinite medium, finite surface transfer

Y = \erf (η) + \exp (\frac{k^{'} x}{D} + \frac{{k^{'}}^{2} t}{D}) erfc (η + k^{'} \sqrt{\frac{t}{D}}), k^{'} = \frac{k_{c}}{K}

Semi-infinite medium: penetration depth and uptake

δ = \sqrt{12 D t}, δ = x_{1} when t = \frac{{x_{1}}^{2}}{12 D}

\frac{M / A}{K c_{1} - c_{0}} = 2 \sqrt{\frac{D t}{π}} (Bi → ∞), \frac{1}{h} [e^{h^{2} D t} erfc (h \sqrt{D t}) - 1 + \frac{2 h}{\sqrt{π}} \sqrt{D t}], h = \frac{k^{'}}{D}

A finite body of half-thickness x₁ behaves as semi-infinite while δ < x₁.

Assumptions. Constant D, fixed geometry, uniform c₀, linear equilibrium K, symmetric exposure, a well-mixed bulk fluid at constant c₁, no reaction or swelling. The ideal surface step at t = 0 is a boundary discontinuity when m = 0, so the series needs many terms at very small X. When Bi < 0.2, internal gradients are small; compare with the lumped model. “Infinite” slab or cylinder means infinite lateral extent or length.

References

Perry’s Chemical Engineers’ Handbook, 9th ed., Section 5 (Eqs. 5-24 to 5-31; Tables 5-1, 5-2).
J. Crank, The Mathematics of Diffusion, 2nd ed., Oxford (1975), Chs. 2, 4, 5 and 6.
C. J. Geankoplis, Transport Processes and Separation Process Principles, 4th ed., Prentice Hall (2003), Ch. 7 (unsteady-state diffusion).

Verification (self-tests)

Show values

Use Run tests to verify analytical benchmarks.

Tests compare the full-series solution with independent textbook reference values (Perry’s and Geankoplis-style benchmarks). They are general self-checks, not homework answers.

How the animated schematic is scaled

The colour field is the exact computed concentration. At every frame the series solution (or erf/erfc for the semi-infinite medium) is evaluated at 401 evenly spaced positions n = 0 to 1 at the current X, with the same terms and one-term switch as the readouts. The colour runs linearly from pale (c₀) to blue (Kc₁), the same scale as the colour bar; between the 401 points it is interpolated linearly.

Dark dots inside the body are drawn with density proportional to c/c_max at their position; dots in the fluid are drawn at c₁ on the same scale. Dots crossing the surface appear at a rate proportional to the computed uptake rate dE/dt, so none move when the run is paused.

Time base

A run plays physical time linearly from t = 0. Each run lasts about 16 s on screen at 1× playback (8 s at 2×, 4 s at 4×, 32 s at ½×) and ends when 99% of the change is complete (E = 0.99, or X = 5 if that comes first); for the semi-infinite medium it ends at X = 0.25, when δ = 1.73x₁. Changing D, x₁, k_c, K or the geometry starts a new run from t = 0, because the time scale x₁²/D or the dimensionless problem changes. Changing c₀ or c₁ only rescales the concentration, so the run continues. Switching the one-term display during a run marks it “inputs changed” and the earlier part of the uptake trace turns grey. Play and Pause start and stop the run, Reset time goes back to t = 0 with the same inputs, and Clear plot clears the uptake trace. The page always opens paused, at t = 0 (or at the preset time for a preset link); press Play to start. Playback speed changes only how fast the run plays on screen; every time shown stays physical time.

Uptake trace

The small plot beside the body is E versus t for the current run (dark line, recorded as it plays) and the exact curve for the current inputs (dashed).

The body outline is drawn to fit the panel: sizes are not to scale.

Material and surface

Time and probe

3. Geometry and multidimensional probe

Four user times for c(x)

Concentration profile

Gurney-Lurie chart

Time to reach a target `Y`

4. One-dimensional factor charts

5. Centre response comparison

Equations, assumptions and references

Governing equations

References

Verification (self-tests)

How the animated schematic is scaled

Time base

Uptake trace

Material and surface

Time and probe

3. Geometry and multidimensional probe

Four user times for c(x)

Concentration profile

Gurney-Lurie chart

Time to reach a target Y

4. One-dimensional factor charts

5. Centre response comparison

Equations, assumptions and references

Governing equations

References

Verification (self-tests)

How the animated schematic is scaled

Time base

Uptake trace

Time to reach a target `Y`