TR(BR)-152 : Effects of dispersion on non-conservation substances

Abstract

A common practice in modelling of water quality of a river is to assume immediate cross sectional mixing and to neglect longitudinal dispersion; for example, the derivation of the widely used Streeter-Phelps equation. Researchers had explained that the propagation of pollutants in a moving water is because of the differential advection and cross sectional dispersion and, thus represented by the Advection-Dispersion equation.

For nonconservative or decaying substances, the spatial distribution of pollutants are usually estimate considering the equation which represents Advection-Dispersion-Decay of pollutants. Most of the water quality models have thus been developed assuming pollutants are completely mixed just after the point of release and neglecting the effects of dispersion.

Points arise; i) could pollutants reach to the other bank (if release is at the one bank) at the point of discharge ? ii) since the decay of nonconservative substances depends upon the incoming pollution load at any point ( expression being, Ct/Co =exp(-K.t) ; Ct = concentration at any desire time, Co = incoming concentration of pollution, K = decay coefficient, t = time}, would the estimate give the correct picture when it is assumed pollutants are completely mixed at the point of release ?

Literatures reveal that once the pollutants are completely mixed, the first order decay dominates the concentration profiles more than the longitudinal dispersion coefficient. And the effect of longitudinal dispersion could then be neglected. But in the initial period both decay and cross sectional dispersivity govern the concentration distribution. Analytical solution of pollutants transport for the initial period is difficult. A numerical analysis could be a best alternative.

A "Numerical Model" using the finite difference technique for the conceptualized stream tubes generated on the basis of the equi-velocity lines , has been developed, and solved forming the tridiagonal matrices considering Alternate Direction Implicit Explicit (IADIE) technique. Two types of convergence criteria ; i) convergence w.r.t. IADIE, ii) convergence w.r.t. time have been used for obtaining the spatial distribution of pollutant's concentration. The factors which govern the stability of the solution are; i) size of the segment, ii) selection of time step, ii) dispersivity coefficient, and iii) transverse velocity profile.

For given cross sectional and vertical velocity distribution at a specific width and depth respectively, and with a pre-determined dispersivity coefficient, the model can be used for estimating the concentration profile at any time step and at any location within the initial period. The model has been verified with published data and found satisfactory results. The results reported in this study are based on the continuous release of pollutants at one bank. Further study assuming centre line injection of pollutants and with different dispersivity coefficients are suggested for generalization of the solution. The report also addresses a comparative pictures of effects of dispersion on conservative and non conservative substances.