STUDY OF SOLUTE TRANSPORT IN A RIVER

Abstract

River water gets polluted by the effluents from municipal, industrial, and agricultural areas, and sometimes due to incidental spills of radioactive tracer from nuclear power plants. Study of solute transport in a river has relevance to monitoring of water quality, regulating pollution sources, evaluating risks from incidental releases, and forecasting of water quality. When a solute cloud is carried downstream in a watercourse, the solute disperses, the cloud domain lengthens and the peak concentration reduces as the solute gets distributed in the ever-increasing volume of water. Based on Fick's law of diffusion, Sir.G. I. Taylor had described the processes of mixing mechanisms primarily by the two basic phenomena: differential advection and diffusion. For the case of river, solute transport eventually becomes a one-dimensional process. This process of solute transport is represented by the advection-dispersion equation, which is known as Fickian dispersion model. Ogata and Banks (1961) gave the analytical solution of the Fickian dispersion model for step input. Many investigators had reported the practical difficulties in application of the Fickian model to field conditions because of the limiting assumptions underlying in its derivation in respect of river geometry, flow condition and nature of the solute. A prior estimation of mean flow velocity, u and longitudinal dispersion coefficient, DL are required for computation of the concentration in the river. In most cases, u , can be measured or estimated quite accurately from a knowledge of gauged flow or by application of flow resistance equation. The basic problem is determination of DL, whether we estimate it by analytical methods or experimental methods or using empirical formulae. The theoretical and experimental bases for estimation of DL do not provide satisfactory results. The empirical formulae estimate DL that varies in a wide range. Abstract Cells In Series (CIS) model is popular among alternate methods. Banks (1974) derived an expression for the concentration of the effluent routed through 'n' number of thoroughly mixed reservoirs, each of size equal to xo for a step function change in concentration imposed at the first reservoir. Replacing the integrand appearing in the expression of solute concentration of the effluent of the nth cell, which has the form of Poisson distribution, by an integrand having the form of a Gaussian distribution, which is valid only for a large value of 'n', Banks (1974) has shown that the expression of solute concentration derived using CIS model is identical to that derived by Ogata and Banks (1961) and Dl is equivalent to —x u . The value 'n' has not been identified for which the integrand having Poisson distribution characteristic can be replaced by an integrand having the Gaussian distribution. CIS model, thus, does not compute concentration, which is governed by dispersion, in the (n-1) cells. Stefan and Demetracopoulos(1981) had suggested that CIS model describes the dispersive properties but does not reproduce persistence skewness which is usually observed in tracer graph. The skewness coefficient decreases inversely with square root of distance. Many other investigators have applied the CIS model for studying the problem of sub-surface leaching and solute transport in streams and have identified that the CIS model has the problems in fixing the relationship between number of mixing cells, travel time of solute and the dispersive properties. Beer and Young (1984) introduced a variant on the CIS model called the aggregated dead zone (ADZ) model for analyzing longitudinal dispersion. The main difference in the ADZ model from the CIS model is that in later model a pure time delay is introduced into the input concentration, which allows advection and dispersion to be decoupled. The practical difficulty of the ADZ model is to determine and estimate the number of model coefficients. Considering the advantage of CIS model, a hybrid CIS model to study solute transport of a conservative tracer in a river for steady and uniform flow condition is developed in the present investigation assuming river to be composed of a series of equal basic units. Each basic unit is comprised of a plug flow zone and two thoroughly mixed reservoirs of unequal filling time. The three zones in a basic unit are connected in series. in Abstract To account for the persistence skewness near the boundary source as depicted by Ogata and Banks equation, a bypass component is initially introduced to the 2nd thoroughly mixed reservoir. It is found that at time, t = a i.e., the time when the tracer leaves the plug flow zone and enters into the first reservoir, there is a sudden jump, hence a discontinuity, in the graph of the unit impulse response of the hybrid unit which has a bypass component. In reality, the response of a river to unit impulse perturbation can not exhibit discontinuity. Hence, it is reasoned that a bypass component does not form a part of the hybrid cells. The proposed model has three parameters, viz., (i) a = the time required to replace the fluid in the plug flow zone, (ii) Ti ( = Vi/Q) = the filling time ofthe first thoroughly mixed reservoir, and (iii) T2 ( = Vjj/Q)"58 the filling time ofthe 2nd thoroughly mixed reservoir, in which Vi and V2 are the volumes of the 1st and the 2nd reservoir, Q = the uniform flow rate of the fluid. The mathematical formulation of the model is based on conservation of mass in each zone. The unit impulse response function depicts a rising limb, a falling limb, a peak and a time to peak and represents the fundamental characteristics of a system. Therefore, the three parameters are estimated making use of unit impulse response function. Three methods, namely, i) method of moments, ii) method using first moment, time to peak and peak concentration, and iii) least squares optimization, are suggested for estimation of the model parameters. Error in the tail of the concentration graph introduces significant error in the computation ofmoments of higher order. Therefore, partial moments were used for the estimation of parameters. Also, instead of using the complete concentration graph, part of the concentration graph is considered in method of optimization. It is found that using part ofthe observed concentration graph, i.e., up to to/tp =2, where to =time of observation, and tp = time to peak, parameters could be estimated by all the three methods. With these estimated parameters, the hybrid CIS model simulates a concentration graph which matches with the entire observed graph. The main task in the proposed hybrid CIS model is the determination of the basic unit size, Ax. For this purpose, synthetic observation data generated by Ogata and Banks at x = Ax, for given u and DL are used for estimation of the hybrid CIS model parameters. IV Abstract Using the parameters estimated, the response of the hybrid model at n Ax, n = 1,2,3,... are found and compared with the corresponding concentration graphs of Ogata and Banks. It is found that for given u, and DL, a size satisfying the condition Ax > (4 DL/ u ) is the appropriate size of the basic unit. With this basic unit, the concentration graphs predicted by the hybrid CIS model using convolution at nAx are identical to that of the Ogata and Banks, where n is an integer. In other words, when the sum of the non-dimensional time parameters, i.e, (a + Ti + T2)/(Ax/ u) -» 1, i.e., the residence time of the solute in one unit of the hybrid cell is equal to the travel time of flow, the concentration graphs produced by the proposed model at n Ax, and by the analytical solution match closely. In field situation, the velocity of flow may vary from reach to reach. A comparison of the parameters estimated for two different flow velocities for a given Ax shows that for a given Ax the parameters vary linearly with flow velocity. The hybrid CIS model has been tested using field data published by Nordin and Sabol (1974). These data include the mass of a conservative solute injected instantaneously at a specific location in a river, concentration graphs in the downstream after injection, distances of the sampling sites from the point of injection, flow velocities and discharges in the river at locations of observation. The observation made at the first sampling site was used to find the model parameters, a, Ti and T2. Applying necessary correction in the parameters for change of velocity the concentration graphs are predicted at other sampling sites. It is found that the predicted concentration graphs match satisfactorily with observed concentration graphs. The parameters of the hybrid CIS model can be estimated from concentration graph observed at one site. The model simulates solute transport as per advection and dispersion at all hybrid cells. The performance of hybrid CIS model has been studied adopting non-equilibrium adsorption isotherm in all the three zones of the basic unit.