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ESTIMATING LONGITUDINAL DISPERSION COEFFICIENT IN RIVERS AYYOUBZADEH S, A Water Structure Department, Tarbiat Modarres University, Jalale Ale Ahmad Ave Tehran, 14115-11, Iran FARAMARZ, M Water Structure Department, Tarbiat Modarres University, Jalale Ale Ahmad Ave Tehran, 14115-11, Iran MOHAMMADI, K Irrigation Department, Tarbiat Modarres University, Jalale Ale Ahmad Ave Tehran, 14115-11, Iran In this research the eight available empirical methods to predict longitudinal dispersion coefficient are analyzed and effect of river shape and hydraulic parameters in predicting dispersion characteristics are studied. The amount of longitudinal dispersion coefficient was computed using hydraulic and geometric parameters of 95 river data in USA gathered from different published sources. Statistical measures are employed to compare the results obtained from empirical and theoretical methods and then methods with superior results are selected. Furthermore sensitivity analysis indicates that among hydraulic and geometric parameters appeared in the available equations, the river flow velocity has the highest effect on longitudinal dispersion coefficient of the rivers. INTRODUCTION For steady uniform flow, the 1D advection dispersion equation (ADE) is widely used to predict distribution of water quality concentration in rivers and channels, with the form of equation as [9]: c c 2c u D 2 Sc t x x (1) where c= concentration; u= velocity; D= dispersion coefficient, x= direction of flow and Sc is source/sink term. Clearly, Equation (1) demonstrates the direct dependency of pollutants transport to longitudinal dispersion coefficient. Owing to the importance of longitudinal dispersion coefficient in water quality related issues, many studies have been developed to predict longitudinal dispersion coefficient in rivers after the first study of Taylor in 1953. When the 1D dispersion model is applied to predict concentration variation in rivers, the selection of a proper dispersion coefficient is the most important 1 2 task. When mixing and dispersion characteristics are unknown, the dispersion coefficient can only be estimated using empirical or theoretical equations. RESEARCH REVIEW Taylor (1954) first introduced a concept of longitudinal dispersion coefficient in the straight circular tube in turbulent flow and derived an equation to predict the dispersion coefficient as follows [11], [4], [5], [8]: D 10.1rU * (2) Where r = pipe radius and U*= shear velocity =(. Elder (1959) extended Taylor’s method for uniform flow in open channels. He derived the equation to predict the dispersion coefficient as [1]: D 5.93HU * (3) where H= flow depth. Fischer (1967) showed that Elder’s equation significantly underestimates the dispersion coefficient, because it does not take the transverse variation of the velocity profile across the river into account. He obtained an integral relation to predict the dispersion coefficient in natural streams using the lateral velocity profile instead of vertical velocity profile as follows [2], [3]: D 1 B y 1 y h( y ).u ( y ) 0 h( y ).u ( y ).dydydy A 0 yh 0 (4) where B=channel width; h(y)=local water depth; A=cross sectional area; y= coordinate in lateral direction; ylocal transverse mixing coefficient and u´(y)=deviation of local depth mean velocity from the cross sectional mean velocity. The fundamental difficulty in determining D from equation (4) is the lack of knowledge of transverse profiles of both velocity and depth. Hence, Fisher (1975) developed a simpler equation by introducing a reasonable approximation of the triple integration, velocity deviation and transverse turbulent diffusion coefficient as follows: D 0.011 U 2B2 HU * (5) McQuivey and Keefer (1974) developed a simple equation to predict the dispersion coefficient using the similarity between 1D solute transport equation and 1D flow equation for the Froude number less than 0.5 as follows [6]: 3 D 0.058 HU s (6) Liu (1977) obtained a dispersion coefficient equation using Fischer’s equation with take the role of lateral velocity gradient in dispersion in rivers into account as [1]: D U 2B2 (7) HU * in which a parameter represent a function of the channel cross section shape and the velocity distribution across the stream and can be computed as: U* U 1.5 0.18 (8) Iwasa and Aya (1991) derived an equation to predict the dispersion coefficient in natural streams using laboratory data and previous field data as [1]: B 2.0 HU H D 1.5 (9) * Seo and Cheong (1998) developed an equation to predict the dispersion coefficient using dimensional analysis and a regression analysis for the one step Huber method as[10]: B 5.915 * HU H D 0.620 U * U 1.428 (10) Deng et al (2001) developed their equation to predict longitudinal dispersion coefficient using the transverse mixing coefficient. [1]. They showed that the derived equation containing the improved transverse mixing coefficient predict the longitudinal dispersion coefficient of natural rivers more accurately. They equation is: D HU * 0.15 B 8 r 0 H 5 3 U * U 2 (11) in which r0= transverse mixing coefficient and can be calculated as: 1 U B * 3520 U H r 0 0.145 1.38 (12) 4 Kashefipour and Falconer (2002) developed an equation to predicting dispersion coefficient in rivers, based on collected data and obtained data from rivers in USA. [6], [7]. Their equation can be written as: U D 10.612 HU * U (13) also they combined the equation (13) with that’s proposed by Seo and Cheong (1998) and derived an equation as: 0.620 U* B D 7.428 1.775 U H 0.572 HU U U * (14) CRITERIA FOR THE COMPASION OF EQUATIONS The analyses here included an evaluation of the methods described in the previous section for determining dispersion coefficient in a river. These are Elder (1959), Fisher (1957), Liu (1977), Iwasa and Aya (1991), Seo and Cheong (1998), Deng et al. (2001), Kashefipour and Falconer (2002) and Seo and Cheong (1998). Longitudinal dispersion coefficient is mainly affected by channel geometry and flow characteristics. The amount of longitudinal dispersion coefficient differs from a specific river to another one and its range is considerable. Unreliable estimation of longitudinal dispersion coefficient lead to a great error; therefore use of statistical measures based on observed data is necessary in definition a certain longitudinal dispersion coefficient. In this research comparison and assessment of different equations using measured data is performed. The river data include river width, flow velocity, flow depth and dispersion coefficient for each river. Statistical measures include discrepancy ratio, , Mean Absolute Error (MAE) and Root Mean Square (RMS) are used to evaluate the empirical and theoretical methods in predicting river longitudinal dispersion coefficients. These measures are defined as follows: log Dp Dm RMS 1 ME 1 log( D p Dm ) N ( D Dm ) i N i 1 p N ( D p Dm ) i (15) 2 (16) (17) in which Dp and Dm are predicted and measured dispersion coefficient respectively, and N is number of data. The discrepancy ratio equals to zero indicates equality of predicted value with measured value. For values of greater than zero, the model overestimates the value of the dispersion coefficient, and conversely for values less than zero the model 5 underestimates the parameter. The accuracy of each model is categorized by the number of values between –0.3 and 0.3, relative to the total number of data values. This range was selected due to maximum acceptable error in the predicting dispersion coefficient from the corresponding measured values (±100%). COMPARISON OF RESULTS Obtained results from the available equations indicate that Elder's equation, which relates longitudinal dispersion coefficient just to depth, takes the discrepancy ration below zero and hence it falls out of accuracy range. The results of statistical measures for all considered methods are shown in Table (1). As seen in the Table, the Deng et al (2001) equation produces smallest MAE accompanied by maximum regression coefficient indicating better results compared to other equations. Table 1. Results of used statistical methods to comparison of various models (%) Method and associated equation MAE RMS <-0.3 -0.3-0 0-0.3 >0.3 R2 Elder (1959), Eq. (3) 99.03 0.97 0.0 0.0 2.01 0.20 0.35 Fisher (1975), Eq. (5) 27.18 16.5 16.5 39.81 0.48 0.06 0.52 Liu (1977), Eq. (7) 17.84 21.36 36.10 31.07 0.39 0.05 0.47 Iwasa and Aya (1991)Eq. (9) 12.62 27.18 21.36 27.18 0.35 0.04 0.44 Seo and Cheong (1998), Eq. (10) 12.62 20.39 37.86 29.13 0.32 0.04 0.61 Deng et al. (2001), Eq. (11) 13.59 25.34 33.98 27.18 0.30 0.04 0.62 Kashefipour and Falconer (2002) وEq. (13) 29.13 38.83 18.45 13.59 0.38 0.05 0.54 Seo and Cheong (1998), Eq. (14) 20.39 33.98 26.21 19.42 0.32 0.04 0.59 The distribution of discrepancy ratio is shown in Figures (1) and (2). As shown in the Figure 1.Distribution of percentage of discrepancy ratios for methods evaluated 6 Figure 2.Comparison of discrepancy ratio in different models Figures, the results obtained from equations (13) and (14) were skewed towards negative values while those obtained from equations (5), (10) and (13) were skewed towards positive values. Also the results show that Equation (11), Deng et al (2002), produces the best results with the percentage of values between 0.5and 2 equals to 60%. The variation of discrepancy ratio for different shape factors (ratio of river width to flow depth) are shown in figure (3). The equations (3), (13) and (14) underestimate the dispersion coefficient with variation of shape factor and the equations (5), (10), and (11) overestimates the corresponding values with variation of shape factor. 7 Figure 3. Variation of discrepancy ratios with shape factor in different methods SENSITIVITY ANALYSIS The sensitivity analysis was done. The results of sensitivity analysis indicate that the velocity is the most sensitive variable among the four variables. The same change of 20% in velocity causes the greatest variation in the dispersion coefficient and causes the increase 20% in the dispersion coefficient value. The channel width is next in importance. CONCLUSION Eight methods for determination of longitudinal dispersion coefficient have been examined with reference to river data. The comparison was based on 95 natural water course data of USA rivers. The most reliable method for predicting longitudinal dispersion coefficient is the equation of Deng et al. (2001). The predicted values by this equation show highest regression coefficient, 62%, and lowest mean absolute error of 0.31. Furthermore, discrepancy ratios of the results obtained from the equation of Deng et al. (2001) show that about 60% of the data is in the error band of 0.5-2. Equations presented by Elder(1959) and Kashefipour and Falconer (2002) underestimate the 8 dispersion coefficients while equations of Fisher (1957) and Seo and Cheong (1998) overestimates the actual values depending on the values of width to depth ratio. The sensitivity analysis performed here indicates that the dispersion coefficient is highly dependent on flow velocity in such a way that a 20% increment in flow velocity affects 40% increasing in dispersion coefficient. 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