Identifiable Transport Groups and Bayesian Experimental Design in Finite Bath Dyeing

Abstract

Finite bath dyeing models can fit trajectories precisely while obscuring geometric ambiguity and binding error. We derive a conservative diffusion–binding model with uncertain thickness, pore volume, film transfer and optical scale. An exact scaling symmetry identifies Dref/L² and kf/L as observable transport groups. A validated polynomial emulator supports nine-parameter kinetic Bayesian inference. Under an equal 108-vessel budget, prior-averaged information design reduces aggregate group-estimation error by 18.9% and 23.5% in 200 paired campaigns for two illustrative fibre systems. A separate goal-oriented design study minimizes uncertainty in three log groups. In 200 new same-family campaigns per system, it reduces group RMSE by 29.4% and 22.7%, and unseen-condition loading error by 31.7% and 35.1%. Richer-binding campaigns can reverse the predictive benefit and expose severe interval undercoverage. Two published wool studies supply 30 kinetic or concentration observations; independent natural-dye degradation controls and nine deposited cotton optical observations challenge interpreting bath loss or colour as uptake. Analytical verification, mesh refinement, solver audits and attributed source records distinguish numerical reliability from material validation. The contribution connects observable groups, decision-aligned measurement allocation and mechanism-dependent reliability in a finite inventory. Independent absolute-unit multi-temperature natural-dye calibration remains necessary for material transfer