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...