🧱 Transient Diffusion in Solids Calculator
Compute one-dimensional unsteady diffusion in slabs, long cylinders, and spheres using Heisler-style Fourier-series solutions. Outputs C(x,t), average concentration, and total absorbed or desorbed mass.
🧬 Transient Diffusion Schematic
📝 Configuration
Key Equations:
Fourier number: Fo = D t / Lc²
Mass Biot number: Bim = hmLc/D
Normalized concentration:
θ = (C - Cs)/(C0 - Cs)
Series form:
θ(x,t) = Σ An Φn(x*) exp(-λn² Fo)
Fourier number: Fo = D t / Lc²
Mass Biot number: Bim = hmLc/D
Normalized concentration:
θ = (C - Cs)/(C0 - Cs)
Series form:
θ(x,t) = Σ An Φn(x*) exp(-λn² Fo)
📊 Results & Visualization
Configure inputs and click Calculate to view results.
ℹ️ About Transient Diffusion in Solids
This calculator solves unsteady one-dimensional diffusion in common solid geometries. It is useful for moisture uptake, drying, solvent absorption, carburizing-style concentration penetration, and polymer/additive diffusion estimates.
This calculator solves unsteady one-dimensional diffusion in common solid geometries. It is useful for moisture uptake, drying, solvent absorption, carburizing-style concentration penetration, and polymer/additive diffusion estimates.
📘 Calculation Methodology
Mathematical Model & Theory
The solution assumes one-dimensional Fickian diffusion with constant diffusivity. Geometry is treated using Fourier eigenfunction series analogous to Heisler charts for transient conduction and mass diffusion.
Worked Engineering Example
Polymer slab: a 4 mm thick sheet exposed on both sides has L = 2 mm. With D = 1×10−11 m²/s and t = 3600 s, Fo = Dt/L² = 0.009. The calculator evaluates the series, profiles C(x,t), and integrates the average concentration to obtain absorbed mass.
Assumptions & Notes
- Constant D and constant equilibrium surface concentration Cs.
- Symmetry at centerline for slab, cylinder axis, or sphere center.
- Finite surface resistance handled through Bim; very large Bi approximates prescribed surface concentration.
- Reported mass sign is positive for absorption when Cs > C0, negative for desorption/drying.