Analytical solution of solute dispersion in the pipe flow under the wall absorption condition

During the prolonged operation of water supply pipelines, the deposition of sediment and the proliferation of microorganisms frequently lead to the formation of a biofilm on the pipe walls. Since the biofilm has a significant adsorption effect on solute transport, this study focuses on understanding the migration and diffusion characteristics of solutes in the laminar flow of water pipelines with wall adsorption. Based on the multi-scale perturbation method, an exact analytical solution for the two-dimensional solute concentration distribution is derived from the nondimensionalized advection–diffusion equation. The analytical results of the mean concentration distributions show excellent agreement with the corresponding numerical results obtained by the finite difference method, demonstrating that the proposed analytical solution can accurately estimate the solute concentration profile in the given flows. Then, the validated analytical solution is used to explore the spatiotemporal evolution of the solute within a circular pipe under the combined effects of advection and molecular diffusion. Results indicate that the solute concentration displays a notable nonuniform distribution along the radial direction, and the timescale required to reach uniform distribution is primarily dependent on the cross-sectional location and the strength of wall adsorption. The timescale τ = 4 is considered to be the moment at which the radial concentration achieves a uniform distribution across the typical cross sections (ζ/Pe = −0.1, 0, and 0.1), when the wall absorption strength α′ is equal to 0.1. This study can provide some theoretical guidance for understanding the solute transport and diffusion laws in pipe flows.