Efficiency of Monofunctional Reactive Dyeing : A Mathematical Analysis

Abstract

We present a mathematical model for the diffusion-reaction process, describing the dyeing of textile fibers with monofunctional reactive dyes. In particular, we have studied the behavior of the fixation efficiency for various dyeing conditions. The behavior of a reactive dye moving from the dyebath into the interior of the textile substrate depends primarily on four processes: the convectional diffusion in the dyebath to the fiber surface, the molecular diffusion through the hydrodynamic boundary layer and within the fiber, the adsorption, and the chemical reaction. The latter two processes cause immobilization of the diffusing species on the polymer chains of the fibers. The overall effect of these four steps determines the levelness of the dyeing, the spatial distribution of the fixed dye throughout the fiber cross section, and also the efficiency of the dyeing. The efficiency is given by the ratio of the total amount of dye that has reacted with the substrate versus the total amount of the dye that has hydrolysed in the external (= dyebath) and internal (= pore liquid) solutions.Up to now, various expressions for efficiency have been suggested and discussed [ 8,9,13 ] . Our contribution derives the efficiency by rigorous mathematical analysis of the interplay between diffusive transport phenomena within a textile substrate and the competitive chemical rate processes encountered in reactive dyeing. The question as to whether or not hydrodynamic effects in the dyebath can also influence the efficiency will be the subject of a subsequent study.The overall rate of reactive dyeing with textile fibers can be expressed by F'ick's second equation in its original meaning of a material balance:In this equation, Dreai is the real, concentration-independent diffusion coefficient' of the dye in solution, b is a factor that takes into account the random configuration of the pores (tortuosity factor = lr3 for an anisotropic medium), and 72 denotes the Laplacian. The right-hand side of Equation t represents the difference between flux into and out of a volume element and is therefore concerned only with the concentration of the mobile reactive dye [D,*] in the pore liquid of the textile fibers ( ~ ), whereas the left-hand side represents the total amount of molecules accumulated in that volume element: Here, [Do.] and [D,°j are the concentrations of adsorbed and covalent-bonded dye, respectively; [Dh,,.*] is the concentration of dye hydrolyzed in the pore liquid ; and Pdenotes the porosity. Combining Equations I and 2 gives Equation 3:Because the hydrolyzed dye must be washed off very easily, reactive dyes have to possess low affinity. Thus, a linear adsorption isotherm can be assumed:where K is the inner adsorption equilibrium constant. Because the adsorption of the reactive dye on the pore surface within the fiber is much faster than the diffusion and the chemical reaction with the functional groups of the textile substrate, the approximation (Equation 4) reduces Equation 3 to Equation 5:As a first approximation, we can ass...

Molecular Diffusion in Microporous Materials: Formalisms and Mechanisms

“…An additional source of dye transport cannot be discarded for more recent synthetic polymers of negligible pore volume (water uptake) dyed near or above the glass transition temperature: Segment mobility, i.e., the contribution of motion of polymer segments. However, the pore mechanism remains a basic element in the treatment of dyeing phenomena and procedures (Rys, 1973; Rohner and Zollinger, 1986;Rys and Sperb, 1989).…”

Application of Taguchi and Full Factorial Experimental Design to Model the Color Yield of Cotton Fabric Dyed with Six Selected Direct Dyes

“…A summary of these works follows next. A mathematical model was proposed by Rys and Sperb and described the behavior of the fixation efficiency of monofunctional reactive dyes for various dyeing conditions [7]. Cegarra and Puente produced isoreactivity equations to determine the conditions of the temperature to achieve a constant sorption at different sorption times [8].…”

Application of central composite design to model the color yield of six diazo direct dyes on cotton fabric

“…A summary of these works follows. A mathematical model was proposed by Rys and Sperb (1989) describing the behavior of the fixation efficiency of mono-functional reactive dyes for various dyeing conditions. Cegarra and Puente (1976) produced isoreactivity equations to determine the conditions of temperature -time to achieve constant sorption at different sorption times.…”