Abstract
Cotton dyeing is a very complex process with many variables in which different phenomena occur simultaneously. This study aimed to describe the cotton dyeing process by natural dye, using a mathematical model that consists of three mass conservation equations that depict dyeing cotton in cones, taking a representative volume element at the micro, meso, and macroscales. The first equation describes the concentration changes of the dye in the solution, taking into account the diffusive, convective, adsorptive, and reactive effects. The second equation describes the changes in dye concentration in cotton fiber, considering the diffusive, adsorptive, and reactive effects within an intermediate scale. The last equation describes changes in the concentration of dye in the solution on the macroscale, based on the characteristics of the equipment and the difference in concentration before and after passing through the fiber. In addition, a fluid continuity equation was incorporated, taking into account Darcy’s law. In the simulation of the dyeing process with synthetic dye with initial concentrations of 0.408 and 2.06 kg/m3, RMSE of 0.00221 and 0.0289 kg/m3 were obtained, respectively. For the simulation of a dyeing process with natural dyeing, a behavior similar to the experimental data was obtained.