Start here: pick a geometry tab, then press Play (or drag time t). Red dots fill the body as c rises.
1. Material and boundary
2. Time and position
The time slider spans X = 0.001 to 5, recalculated as D and x₁ change.
3. Geometry and multidimensional probe
Reference x₁ above is radius R. Full cylinder height = 2L. The map shows r ≥ 0, z ≥ 0.
Reference x₁ above is half-side a. Other half-sides b and c have independent diffusion scales. Each sealed choice covers a pair of opposite faces.
A sealed direction means BOTH opposite faces are insulated. One-sided sealing is not modelled by replacing a symmetric factor with 1.
Four user times for c(x)
Edit any time, in seconds. The red marker is the live probe.
Chart → time solver
Dr. Ray’s Learning Center | Transport Phenomena
3. Live cross-section
4. Gurney-Lurie chart
Log Y versus X. Thin curves are selected families; teal is the current n and m; red is the live state.
Live n,m curve Live point Chart families
4. One-dimensional factor charts
Each Gurney-Lurie curve uses its own Xᵢ = Dt/lengthᵢ², nᵢ, and mᵢ = KD/(k_c lengthᵢ). The red point and teal curve use the full 1D series at the selected probe; the thin blue curve is the m = 0 reference. If one-term is selected, the main body readout uses the approximation and is flagged when outside its validity range.
5. Centre response comparison
Centre Y versus physical time (logarithmic t axis); curves show the whole body and each unsealed 1D factor. The red point is the current body-centre value.
Equations, assumptions and references
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.
Product rule requires uniform c₀, constant D, constant uniform c₁ and K, and linear separable boundary conditions. Different symmetric face-pair k_c values could be accommodated. An arbitrary nonuniform initial profile does not generally permit this simple product.
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
| Case | Computed | Expected | Result |
|---|
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
Concentration field
The red dot density inside the body is the computed concentration c/cmax at the current X for the selected geometry (exact series solution, product solution for short cylinders and bricks, sampled at 101 points per direction). Blue dots in the surrounding fluid are drawn at the bulk concentration c1 on the same scale.
Surface exchange
With a finite Biot number the surface concentration stays below the equilibrium value Kc1; the readout shows both. Gold dots cross the surface at a rate proportional to dE/dX, the computed uptake rate per unit Fourier number.
Schematic, not to scale.