Mathematical Model of the Diffusion and Concentration of a Dye

Abstract

This paper examines the diffusion and concentration of a dye through a laboratory experiment involving a long thin glass tube filled with water and sealed at both ends. The tube is divided into two equal sections by a thin membrane placed down the center. Before the tube was sealed, different colors of dye were injected into each section. A mathematical model was developed to describe the diffusion of the dye within the tube, and a function  representing the concentration of the dye per unit volume was evaluated. Subsequently, this concentration value was used to determine a critical point for the fastest rate of change of the diffusion and as well as its limiting value of dye concentration as time approaches infinity. The model was simulated using maple software to see the dye behavior under various conditions. Results shows that, a high rate of diffusion gradually decrease as the concentration gradient reduces, oscillation behaviors were also observed when . However, when , the dye concentration remain at steady rate, irrespective of the changes in the value of time, which shows the influence of temperature on medium properties. The findings from this paper have several real-life applications in various industries, including textiles, food processing, and pharmaceuticals.