Longitudinal Dispersion of Conservative Pollutants in Open Channels

Abstract

Whenever rivers act as sinks for intentional or accidental spillage of effluents resulting from the activities of society, it becomes necessary to know the concentration downstream of its point of injection into the river to check whether the required environmental standards are being satisfied or not. As the injected effluent travels downstream, its concentration decreases due to advection and diffusion (molecular and turbulent) processes and variation of velocity along the depth of flow in the channel. After sufficient time of the pollutant injection, the pollutant is completely mixed laterally, so that there is only a slight variation of pollutant concentration in the lateral direction. After lateral mixing has been accomplished the primary variation of concentration is in one direction and the dispersion is known as longitudinal dispersion. The time required from instant of injection to complete lateral mixing is called 'Convective period' or the 'Mixing time', and the distance travelled during this time is called 'Mixing length'. After the mixing length the spreading of the pollutant is primarily due to variation of velocity along the depth of flow, the contribution of the molecular and turbulent diffusion being negligible (Holley, 1969; Fischer et aI., 1979). However, molecular and turbulent diffusion alongwith secondary flow are associated with lateral mixing of the pollutant. The longitudinal dispersion equation for a conservative pollutant and steady and nonuniform flow after the mixing length is assumed to be of the form (Cunge et ai. 1980): where x is the distance downstream, t is time, C and U are the cross-sectional average concentration and velocity respectively, A is the flow area and DL is the longitudinal dispersion coefficient. Taylor (1954) was probably the first to study the process of dispersion. He asserted that the phenomenon of dispersion in turbulent flow can be described by one-dimensional Fickian diffusion equation. Elder (1959) extended the theory of Taylor and stated that dispersion in open channels beyond the mixing length may also be described by the onedimensional diffusion equation. Several investigators (Sayre, 1965, 1968; Fischer, 1967; Y otsukura and Cobb, 1972; Ward, 1973; McQuivey and Keefer, 1976) studied the variation of the mixing length with flow conditions and proposed relationships for the same. These relationships were mainly derived using experimental data on the mixing process in lateral and vertical directions. In the present study it has been found that the relationships for mixing length produce results of variance from the observed values in many cases. Amilytical and numerical solutions of the longitudinal dispersion equation are available in literature. These solutions were obtained under simplifying assumptions which may generally not hold good in practice. Numerical solutions have been attempted for solving the dispersion equation for real situations; the finite difference method of solution with either Split operator approach or Combined operator approach has been used quite commonly. However, in all these studies the parameters A, U and DL were considered to be invariant with distance. In the present study a new finite difference numerical scheme for solution of longitudinal dispersion equation has been proposed. The proof-of-the-concept tests for the proposed scheme have been made with Fischer's (1968) analytical model and Jaque and Ball's (1994) numerical method. The proposed scheme is also applicable in non-uniform flow and for varying dispersion coefficient. The Combined operator scheme has been adopted in the numerical solution used, wherein the exact solution of advection process is obtained by developing a variable spatial grid so that the root of the trajectory of the concentration characteristics passes through the computational nodes. Solution of the diffusion process is achieved using Crank-Nicholson scheme by using a temporal weighting coefficient. Conventionally, non-uniform flow is treated as an equivalent uniform flow and C-t curves are predicted. But in the present study it has bee~ found that large differences occur in the predicted C-t curves, predicted by treating the flow as non-uniform and approximating it as uniform. The experiments were conducted in the Hydraulics Laboratory at Civil Engineering Department of the University of Roorkee, Roorkee in a recirculating flume of 0.20 m width and 30.0 m length. First the experiments were carried out in clear-water flows and these were followed by experiments in sediment-laden flows. Rhodamine WT was used as the tracer whose concentration was monitored in the flume using a Turner Fluorometer 10-AU-005 of continuous flow type. Two uniform sands of average diameter 0.064 mm and 0.164 mm were used as sediment. The concentration of sediment in suspension was varied from 500 ppm to 11,000 ppm. These data were analysed alongwith the data available from previous studies on dispersion. Determination of the value of Dl from the observed temporal variation of concentration (C-t curve) along the river is possible by the change of moment method (Taylor, 1954), the routing procedure and diffusive transport method (Fischer, 1966). However, all of these methods suffer from some limitations. In the present study the proposed numerical scheme has been extended by incorporating in it the one-dimensional grid search method for determination of OJ. values using the observed C-t curves at two or more stations. One needs a predictor for DL in terms of flow and channel variables to be able to make calculations of (' -t curves at different downstream locations given the C-t curve at an upstream station. Several investigators have proposed predictors for DL. However, computations have revealed that the available predictors for DL do not produce satisfactory results. In the present study a new predictor has been proposed. The proposed relationship gives better results than the existing methods. Nevertheless, the predicted and the observed DL values depart from each other by a maximum factor of ten. A sensitivity analysis for DL is made, which indicates that large errors in DL cause relatively small errors in the prediction of C-t curves at downstream stations. Nordin and Sabol (1974), Day (1975), Day and Wood (1976) and Singh et al. (1992) found DL to vary with distance in the downstream direction even for constant flow conditions, although several other studies indicated DL to remain constant with distance. Therefore, there was a need to study this aspect using a wide range of data from laboratory and field studies. Analysis of field and laboratory data revealed that DL is constant along the flow distance for uniform flow, while in non-uniform flow DL varies with distance because of its strong dependence on shear velocity. If shear velocity increases, DL increases and vice-versa. Arora (1983), Samaga et al. (1986) and Umeyama and Gerritsen (1992) have studied the velocity distribution in open channels carrying suspended sediment load. They found these velocity distributions to deviate from the velocity distributions in flows without sediment. In the present experimental program also, it was observed that variation of velocity over the depth of flow from the mean velocity is more in sediment-laden flow than in corresponding clear-water flows. Dispersion is considered to be significantly affected by the convection process (Holley, 1969). Therefore, the dispersion characteristics of flows transporting suspended sediments may be expected to be different from t~ose of clear-water flows. Singh (1987) has studied the effect of suspended sediment on longitudinal dispersion through a limited series of experiments. In the present study it was found that more dispersion oc~urs in sediment-laden flows than in the corresponding clear-water flows. Dispersion is found to increase ~ith an increase in sediment concentration in the flow, as the deviation of velocity distribution from the mean velocity is more in sediment-laden flows than in clear-water flows. No effect of sediment size is observed on longitudinal dispersion within the range of experimental data used in the present study. 1\n equation is proposed for prediction of DL values in sediment-laden flows. Predicted values of DL from the proposed relationship indicate a maximum error of :t50 percent from the observed DL values.