Determination of the Effective Dispersion Coefficient in A One-Dimensional Advection-Dispersion model using A Random Walk Particle Tracking Approach Feng Shang and Jonathan Burkhardt
To accurately model the transport of water quality constituents within a pipe network, the one-dimensional advection-dispersion (AD) equation is often solved. However, determining the effective dispersion coefficient is challenging because the effective dispersion is the result of both radial variations in velocity and molecular/turbulent diffusion in the radial direction. To better understand this challenge, a random walk particle tracking approach was developed to analyze the advection and dispersion processes of solute transport in a circular pipe and check the validity of applying a one-dimensional AD model to simulate solute transport within a pipe. Stochastic difference equations were derived to model a solute particle’s two-dimensional random movement in the cross-section of a pipe. Effective dispersion coefficients are calculated based on the spatial distribution of the particles. This two-dimensional modeling approach can be applied to any mixing time and can accurately model the longitudinal distribution of the solute concentration as a function of the mixing time. The simulation results approach the asymptotic solution analytically derived when the mixing time is long enough. Simulations confirm that, under laminar conditions, the impact of the molecular diffusion on the solute transport is negligible for short mixing times. For turbulent flow, simulations showed that the longitudinal dispersion of the solute is very sensitive to the cross-sectional velocity profiles assumed. It is found that when the mixing time is short, effective dispersion coefficients and the one-dimensional AD model cannot model solute transport accurately. The presented random walk particle tracking approach is easy to implement programmatically and unconditionally stable. It can be used to evaluate the validity of one-dimensional AD modeling and support detailed water quality modeling within water distribution systems and premise plumbing systems.
