The modelling of a textile dyeing process utilizing the method of volume averaging (original) (raw)

Modelling of the dyeing process of packed cotton threads using reactive dyes

Transport in Porous Media, 2007

The Method of Volume Averaging is used to model the process of dyeing textile threads on bobbins. This analysis allows one to upscale the relevant information at the micro-scale, composed of the textile fibres of the thread in contact with the dyeing bath fluid, to the macro-scale, consisting of the bobbins of threads inside the equipment. The final mathematical model consists of two equations, one for the fluid phase external to the thread and the other for the fluid phase internal to the thread. In order to solve the partial differential equations obtained in the mathematical model, the authors developed a computation code using the Method of Finite Volumes. This code utilized a system of generalized coordinates to facilitate application of the boundary conditions to different bobbin geometries. The numerical results for the kinetics of dyeing packed cotton threads with reactive dyes are compared to the experimental results obtained in Brazilian textile industries, leading to good agreement between theory and experiment. This demonstrates that the model developed in this paper is able to predict the operational conditions to be used in the textile industries, minimizing the consumption of dyes and other auxiliaries necessary for the dyeing process.

Numerical Simulation of Dyeing Process of Cotton with Natural Dye

Processes, 2021

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 ...

Modeling of the Diffusion During Polyester Dyeing-A Survey

A common goal of all dyers is to achieve the correct shade as early as the first dyeing. However, the disperse dyeing process is very complex, being characterized by the diffusion-controlled sorption of dyes that depends on several physicochemical parameters such as dye concentration and temperature of dyebath solution. The literature reveals no equation that expresses the influence of experimental and measurable parameters on the diffusion coefficient value. Thus, a mathematical model describing the evolution of this diffusion coefficient for disperses dyes as a function of time, initial dye concentration, temperature and the nature of the dyebath (finite, infinite) is proposed. This model applies to the dyeing of conventional polyester fibers and microfibers with disperse dyes. The critical discussion is developed dealing with the possibility of the model industrial application and its effectiveness.

Modeling of Mass Transfer Process’ Kinetics During Dyeing and Printing of Textile Fibers

Nauka ì tehnìka sʹogodnì, 2023

We can single out two approaches currently used in the theory of adsorption in general. First of all, it is microscopic , based on the application of methods of statistical physics. Based on the idea of an averaged molecular field characterizing the interaction of a sorbing molecule with a sorbent surface, and then using the grand canonical ensemble method, it is possible in principle to determine the number of molecules adsorbed on the surface. The main difficulty here is the calculation of the interaction energy, since the method of successive approximations is inapplicable. In the framework of the so-called lattice model, attempts to calculate the interaction energy in the case of sorption of nonpolar adsorbent molecules were made, and it was shown that the dipole-dipole interaction was not taken into account , since the contribution of the dipole-quadrupole interaction is not small. An attempt to simulate the real structure of an active carbon grain, presenting it as a set of slitlike and cylindrical pores, and on this basis to calculate adsorption using the Lennard-Jones potential, was made in the work of Everett. A method was developed for calculating the interaction energy of polar sorbate molecules with an ionic cubic lattice. The comparability of the calculated and experimental data, therefore, largely depends on the adequacy of the chosen model to the real structure. Accounting for № 2(16) 2023 348 other types of interactions, which, generally speaking, take place during adsorption, is associated with significant mathematical difficulties. Within the framework of this approach, the theory of sorption can also be constructed based on the kinetic consideration of the process, based on the use of the so-called chain of BBGKY equations, which considers the behavior of partial distribution functions in time that describe the correlation between molecules. The solution of such a system of equations is currently known only for simple cases. Significant difficulties that arise when describing the adsorption process in a statistical way force us to use a phenomenological approach that does not require detailed knowledge of the sorption mechanism. Based on general thermodynamic relationships, he establishes a relationship between the amount of sorbed substance and thermodynamic quantities: chemical affinity, heat of adsorption, free energy, etc. We will dwell on its application in describing adsorption from aqueous solutions in more detail , especially since the kinetics of the process in porous structures is also considered within the framework of this phenomenological approach.

Numerical simulation of dyebath and the influence of dispersion factor on dye transport

Proceedings of the 37th …, 2005

In order to model the mass transfer in a fluid, a "dispersion" model is frequently used. When the flow behavior does not drastically deviate from plug flow, the model performs well. The dispersive flow properties of liquids within fibrous textile assemblies however, have not been fully explored. In the mass transfer model, the dispersive flow is assumed to reduce the concentration gradient of dye liquor flowing through the package. This paper illustrates the influence of dispersion term on dye transport based on numerical simulation of dyebath. The transfer of dye through the package is described by a set of time-dependent partial differential equations, which govern convection, dispersion, and adsorption of dyes in the dyebath and across the yarn package. The simulation results prove, theoretically, that the inclusion of the dispersion term in the dyeing model improves the results of the dyeing process in terms of dye uptake and levelness.

Theoretical study of the influence of dispersion factor on dye transport during the dyeing process

Coloration Technology, 2006

During dyeing, dispersive flow is assumed to reduce the concentration gradient of dye liquor flowing through a yarn package, resulting in a more even distribution of dye in the liquor within the package. However, little is known about the degree of significance of dispersive transport, as a proportion of the total transport of dye. This study investigates the influence of dispersion on dye transport during dyeing, based on a finite element method approach utilised in a mathematical model. The model has been applied to establish the influence of several parameters during the course of the dyeing process. This study theoretically discusses the role of dispersion on the rate of dyeing and the uniformity of dye distribution.

Numerical study of the adsorption of dyes from textile effluents

Applied Mathematical Modelling, 2008

The textile industry is one of the greatest generators of liquid effluents, due to the high quantities of water used in the dyeing process. Fixed bed adsorption columns using activated carbon have been widely used in industrial processes for the removal of contaminants from textile effluents. In this study we present the modelling of an adsorption process applied to textile dyes, using fixed bed columns. This model permits the prediction of the dye concentration at the adsorption column outlet, considering the influence of various operational parameters. The adsorption isotherms of the dye Basic Green 4, over granular activated carbon at 25°C in an aqueous solution, were determined experimentally through a comprehensive series of tests. The Langmuir and Radke-Prausnitz models gave the best results for the adsorption isotherms. Tests were carried out in fixed bed columns to determine the breakthrough curves, with variations in feed rate, feed concentration, diameter of the column and mass transfer coefficient. The experimental conditions were simulated using a transient mathematical model. The data obtained numerically and experimentally were compared to validate the mathematical model proposed.

Modelling of Colour Yield for Selected Reactive Dyes in Dyeing Cotton Cloth by Two Phase Pad-steam Method

Iranian Polymer Journal, 2006

M athematical Modelling can lead to a better understanding of the role of variables in real dyeing process. The aim of this study was to model colour yield in dyeing cotton cloth with selected reactive dyes via two phase pad-steam fixation method, as a function of steam temperature, steaming time and alkali concentration. Factorial design was employed and at first, polynomial regression models were produced for the selected monochlorotriazine, dichlorotriazine, trichloropyrimidine and vinylsulphone reactive dyes. As these models did not meet the necessary requirements, orthogonal polynomial regression was employed. This led to the acceptable models except in one case. The models for the selected dyes in each group of monochlorotriazine, dichlorotriazine, trichloropyrimidine showed similarity but not for vinylsulphone dyes. Response surfaces of the models can be used to find the optimum conditions that maximize the colour yield of each dye.