Abstract:

Objective Dyeing is a very complex process with many variables in which different phenomena occur simultaneously. In order to further understand the fluid flow and chemical reaction process of dye solution infiltrating into fabrics during the initial stage of dyeing, and to study the effects of physical properties of fabric and dye solution on fluid flow and dye concentration during the dyeing process, a relevant mathematical model has been developed and experimentally validated to ensure its accuracy.

Method The fiber orientation probability density function was established as a mathematical model for the inter-fiber capillary radius. The Stokes-Einstein equation defined the effective diffusion rate of dye molecules in the fluid. By deriving a kinetic model for fluid flow and material exchange in the fabric by combining the Hagen-Poiseuille equation, the mathematical and kinetic models were validated by dyeing cotton fabric samples in a Reactive Red 195 dye solution at a concentration of 0.03 g/L, a temperature of 40 ℃, and a dyeing time of 15 minutes.

Results The actual flow rate of the dyeing solution flow in the fabric was measured by percolation dyeing experiments to verify the accuracy of the capillary flow model constructed. Comparison between the flow rate predicted by equation with the actual flow rate revealed that the predicted flow rate and the actual flow rate were basically matched. Datacolor 800 spectrophotometer was used to measure the K/S values of the sample fabrics at the end of the dyeing experiment. The K/S values were normalized the predicted concentrations for comparison. The results showed that the predicted concentrations were in general agreement with the actual concentrations. These results validated the model used in this work, breaking down the critical phenomena and stages of the dyeing process, such as diffusion and adsorption. The numerical simulation results showed that as the fiber volume fraction increases, the capillary flow rate within the fabric decreases and the rise height of the dyeing solution decreases when equilibrium was reached(Fig. 3). Moreover, under these conditions, the dye concentration within the fabric reached a steady state much more quickly (Fig. 4). Contact angle analysis revealed that the size of the contact angle had minimal impact on the capillary flow rate but primarily affected the material exchange rate within the fabric (Fig. 5). However, if the contact angle exceeds π/2, there is no capillary effect in the fabric. A reduction in contact angle resulted in a slower material exchange rate, thus delaying the dyeing process. Conversely, an increase in surface tension would increase the flow rate within the fabric, but it would decrease the penetration height and dye concentration at the same location upon reaching equilibrium (Fig. 7 and Fig. 8). It was discovered that viscosity of the dyeing solution plays a critical role in determining equilibrium between permeation process and dyeing process within fabric. When viscosity is low, the permeation process and dyeing process could easily achieve equilibrium (Fig. 9 and Fig. 10).

Conclusion A scientific and effective method for describing the fluid flow and chemical reaction process of dye solution infiltrating into fabrics during the initial stage of dyeing is explored. The simulation results generated by this flow model provided valuable information regarding the velocity and concentration distribution of capillary flow within the fabric. These results were validated through experimental validation. The kinetic model enables the rapid assessment of how variables such as porosity, contact angle, surface tension, and viscosity influence the dyeing process and its outcomes. The numerical simulation results showed that the fiber volume fraction has the greatest influence on the whole dyeing process. This method can also be applied to the description and analysis of the dyeing process and results for different fabrics, fluids, and dyes, researchers can effectively regulate and optimize the entire dyeing process and its results for specific applications.

Key words: dyeing, dye concentration, probability density function, capillary flow, fabric structure, dyeing kinetic