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<p>In textile dyeing, understanding adsorption kinetics is essential for optimizing process efficiency and product quality. However, although traditional empirical models have been widely used, often fail to provide clear physical interpretations of parameters. This study introduces a Discrete-Time Markov Chain (DTMC) model to overcome this limitation. By analogizing fibers as cities, dyebaths as suburbs, and dye molecules as migrating populations, the DTMC model offers a probabilistic framework that captures the temporal evolution of dyeing processes at discrete intervals. The DTMC model was validated across six typical dyeing processes, involving five types of dyes (reactive, basic, acid, direct, and disperse) and five types of fibers (cotton, polyamide, wool, viscose, and polyester). This validation demonstrated its superior performance relative to five traditional models, as evaluated by the coefficient of determination, root mean square error, residual violin plots and time series residual plots. Notably, the DTMC model requires only one linear fitting step to obtain gauge-invariant transition probabilities, providing clear physical meanings for adsorption and desorption effects. For instance, during the dyeing of cotton fabric with C.I. Basic Blue 9, the model yielded an adsorption probability of 0.00180 and a desorption probability of 0.00219. This indicates that 0.180% of the dye molecules in the dyebath are expected to transfer to the fiber per second, while 0.219% of the those on the fiber may return to the dyebath. This innovative approach advances the mechanistic understanding of dyeing kinetics and facilitates model unification, offering significant potential for optimizing textile dyeing processes.</p>

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Discrete-Time Markov Chain-Based Adsorption Kinetic Model for Textile Exhaust Dyeing

  • 期刊论文

  • 发表日期：

    2025年5月27日

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  • Dapeng Lei,
  • Jianhua Huang

摘要

In textile dyeing, understanding adsorption kinetics is essential for optimizing process efficiency and product quality. However, although traditional empirical models have been widely used, often fail to provide clear physical interpretations of parameters. This study introduces a Discrete-Time Markov Chain (DTMC) model to overcome this limitation. By analogizing fibers as cities, dyebaths as suburbs, and dye molecules as migrating populations, the DTMC model offers a probabilistic framework that captures the temporal evolution of dyeing processes at discrete intervals. The DTMC model was validated across six typical dyeing processes, involving five types of dyes (reactive, basic, acid, direct, and disperse) and five types of fibers (cotton, polyamide, wool, viscose, and polyester). This validation demonstrated its superior performance relative to five traditional models, as evaluated by the coefficient of determination, root mean square error, residual violin plots and time series residual plots. Notably, the DTMC model requires only one linear fitting step to obtain gauge-invariant transition probabilities, providing clear physical meanings for adsorption and desorption effects. For instance, during the dyeing of cotton fabric with C.I. Basic Blue 9, the model yielded an adsorption probability of 0.00180 and a desorption probability of 0.00219. This indicates that 0.180% of the dye molecules in the dyebath are expected to transfer to the fiber per second, while 0.219% of the those on the fiber may return to the dyebath. This innovative approach advances the mechanistic understanding of dyeing kinetics and facilitates model unification, offering significant potential for optimizing textile dyeing processes.