Dye Meets Fiber
Textile dyeing is fundamentally a diffusion process. Dye molecules in the bath contact the fiber surface, adsorb, and then slowly diffuse inward through the fiber's amorphous regions. The concentration gradient between the dye-rich surface and the dye-poor core drives the process. This simulation models the radial diffusion into a cylindrical fiber using Crank's classical solution to Fick's second law.
The Mathematics of Diffusion
For a cylinder of radius a, the fractional uptake over time follows an infinite series solution involving Bessel functions. The key parameter is the dimensionless group Dt/a². When this ratio is small, only the fiber surface is dyed; as it increases, dye penetrates deeper. The half-dyeing time (50% uptake) occurs at Dt/a² = 0.0492 — a universal constant independent of fiber type or dye chemistry.
Temperature: The Master Variable
The diffusion coefficient follows the Arrhenius equation, approximately doubling for every 10°C increase. At room temperature, dyeing polyester would take days. At 130°C under pressure, the same process completes in 30-60 minutes. Temperature control is the single most important process variable in industrial dyeing, and precise control prevents uneven dyeing (barré defects).
From Lab to Mill
Industrial dyeing scales these principles to tons of fabric per day. Jet dyeing machines circulate fabric through a dye liquor at controlled temperature profiles. The heating rate, hold time, and cooling rate are all designed around diffusion kinetics. Too fast a heating rate can cause uneven initial strike; too short a hold time leaves under-dyed fiber cores that bleed in washing. This simulator helps visualize the kinetic constraints that drive process design.