Dr. Ray's Learning Center

Unsteady Diffusion Explorer: 1D Geometries

Unsteady Diffusion Explorer, 1D geometries and Gurney-Lurie charts.

TRANSPORT PHENOMENA | INTERACTIVE SIMULATOR

Unsteady Diffusion Explorer: 1D Geometries and Gurney-Lurie Charts

Semi-infinite medium, infinite slab, infinite cylinder and sphere: drag time and position, watch the cross-section and the chart marker move together.

1. Material and boundary

D = cm²/s
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).

2. Time and position

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

Time t
Fourier number X
Bi = k_c x₁/(KD)
m = 1/Bi
Y at probe
c at probe, mol/m³
Y at centre
Fractional change E = 1 − Ȳ

3. Live cross-section

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

4. Gurney-Lurie chart

Log Y versus X. Thin curves are selected families; gold is the current n and m; red is the live state.

Position curves n
Surface resistance curves m = 1/Bi

Live n,m curve Live point Chart families

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

Chart → time solver

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

5. Governing equations

Mass-transfer analogues of Perry’s heat-transfer equations. Equation tags refer to the heat-conduction forms in Perry’s Chemical Engineers’ Handbook, Section 5; the mass-transfer forms follow by analogy.

Perry’s Eqs. 5-24 / 5-25 (analogue)∂c/∂t = D ∇²c; X = Dt/x₁²; n = x/x₁; Bi = k_c x₁/(KD); m = 1/Bi = KD/(k_c x₁).
Normalized concentrationY = (Kc₁ − c)/(Kc₁ − c₀); c = Kc₁ − (Kc₁ − c₀)Y.
Finite-body series and Perry’s Tables 5-1 / 5-2Y = Σₖ Cₖ exp(−λₖ²X) Fₖ(n), k = 1, 2, …
Infinite slabλ tan λ = Bi; F = cos(λn); C = 4 sin λ/(2λ + sin 2λ).
Infinite cylinderλ J₁(λ)/J₀(λ) = Bi; F = J₀(λn); C = 2J₁(λ)/[λ(J₀² + J₁²)].
Sphere1 − λ cot λ = Bi; F = sin(λn)/(λn); C = 4(sin λ − λ cos λ)/(2λ − sin 2λ).
Bi → ∞λ: slab (2k−1)π/2; cylinder J₀ zeros; sphere kπ.
Perry’s Eq. 5-28; Tables 5-1 / 5-2One term: Y ≈ C₁ exp(−λ₁²X)F₁(n). Validity guide: X ≥ Xc.
Perry’s Eq. 5-29 (correlation)λ₁ ≈ λ∞/[1 + (λ∞/√(d Bi))ᵖ]^(1/p); d = 1, 2, 3; p = 2.139, 2.238, 2.314 (slab, cylinder, sphere).
Perry’s Eq. 5-30, p. 5-7Semi-infinite, perfect surface transfer: Y = erf[x/(2√(Dt))].
Perry’s Eq. 5-31, p. 5-7Semi-infinite, finite transfer: Y = erf(η) + exp(k′x/D + k′²t/D) erfc(η + k′√(t/D)), η = x/(2√(Dt)), k′ = k_c/K.
Average changeE = 1 − Ȳ. Ȳ = Σ Cₖ exp(−λₖ²X) ⟨Fₖ⟩. Volume weights: slab 1, cylinder 2n, sphere 3n².

Assumptions: constant D, fixed geometry, uniform c₀, linear equilibrium K, symmetric exposure, well-mixed external reservoir, no reaction or swelling. Semi-infinite E is not a finite-volume average.

The label “infinite slab/cylinder” means infinite lateral extent or length, not infinite diffusion thickness or radius.

6. Verification

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.

Dr. Ray’s Learning Center | Transport Phenomena

Lessons: The Semi-Infinite Solid; Slabs, Cylinders and Spheres: Gurney-Lurie Charts.