Estimation of Transverse Mixing Coefficient in Rivers and Experimental Flumes

Transcription

1 International Research Journal of Applied and Basic Sciences 015 Available online at ISSN X / Vol 9 (): Science Eplorer Publications Estimation of Transverse Miing Coefficient in Rivers and Eperimental Flumes Hamed Goudarzi 1* Nader Brahmand Arash Jael 3 1. Graduated student Islamic Azad Universit Larestan Branch Fars Iran. Assistant Professor Islamic Azad Universit Larestan Branch Fars Iran 3. Assistant Professor Islamic Azad Universit Yasouj Branch Iran Corresponding Author Gudarzihamed@gmail.com ABSTRACT: Rivers are the most important resource of drinking water and the chipset drainage for wastewater leaving agricultural lands industrial regions and urban runoff. In later centur with increasing of civil population man problems related to the pollution and qualit of water resources have appeared. So stud of miing and translocation of materials in the rivers are the important goals of water resource management programs. In miing processes speciall in the large and big rivers longitudinal Dispersion and the transverse diffusion of pollution are the most parameters respectivel. In this paper in addition to introducing the measuring methods of transverse coefficient in rivers using collection of 19 hdraulic datasets multiple suitable equations for estimation of this coefficient have been developed and accurac of these equations compared to Fischer s equation has been investigated. Furthermore this coefficient is surveed and evaluated. Kewords: transverse miing coefficient straight rectangular flume Fischer s equation INTRODUCTION The distribution of materials has three main parts (Fischer et al. 1979) as follow Primar miing This process starts from injection and continues to the location that concentration depth is uniform. The aim of this stud is to determine vertical miing coefficient and vertical miing length. The Primar miing has two subdivisions. In the first region (output region) pollutant is injected to the flow. In fact required force for distribution is provided b pollutant s momentum. Secondar region begins at the end of output region and continues to the location in which the vertical miing is complete. In man rivers the ratio of width to depth has a high value. So the occurrence of Primar miing is more rapid than the other processes. Complete miing This process begins after the primar miing and continues to the location that the detector (pollutant) is distributed in entire cross section. The main point of stud in this process is the determination of diffusion coefficient and miing length. The mentioned length is equal to the distance between injection source and the section whose two different point s concentration difference is less than 10 %. Therefore it is clear that in rivers passed from miing length diffusion coefficient has nearl no impact on miing / Remote region This process begins after complete miing region and ends where the detector coefficient is not traceable. In addition to translocations particles the dominant process in this region is longitudinal dispersion which causes an epansion of detector cloth on the longitudinal. Studies of detector for determination of dispersion coefficient are conducted in this step. The required length for completion of ever process depends on cross section geometr pollutant momentum detector nature discharge depth and width. According to previous processes it is calculated that the miing factor is consisted of three kinds of coefficient (vertical transverse and longitudinal). From these factors transverse miing is the main point for

2 researcher s efforts. Transverse miing intensit is defined b a cause known as transverse miing coefficient. The most researchers are interested in one-dimensional models of water qualit (dispersion coefficient) and FEW studies have been conducted on two-dimensional models. So in this essa there is an investigation for the best optimum equations for estimating diffusion coefficient using available data and information. Measurement of this coefficient in rivers is ver difficult and besides there is few data available. On the other hand achieving the accurate estimation of river transverse velocit which has an important effect on this coefficient is nearl impossible in man cases and if possible the can onl be determined b field measurements. Therefore the estimations of diffusion coefficient are handled with empirical methods. The transverse miing coefficients are defined b each one of two following terms (Fischer et al. 1979). ahu * (1) bq () D W Where transverse miing coefficient; D lateral dispersion factor; H= depth of flow; a b localized coefficients; u * shear velocit; Q= discharge; and W= width of channel. a and b don t have constant values because of Various parameters such as river width influence miing process.. Comparing Eq (1) and Eq () shows that Eq. () has the most application. Meanwhile the relation between D and can be stated b following terms. D H U.. (3) m Where U= average velocit; and m =coefficient that is almost equal to unit. Determination methods of transverse miing coefficient There are several methods for estimation of miing coefficient. MOMENTUM METHOD In this method coefficients of longitude and transverse miing are obtained from following equations. t (4) t (5) Where are the variances of detector concentration curve in the longitudinal and transverse direction. The parameters and are the coefficient of longitudinal and transversal miing respectivel. From disadvantages of momentum method is that it doesn t include transverse velocit and concentration effects on the river bank. Holl and Abraham (1973) (momentum method) attained better method according to momentum method. In this new method the effect of transverse velocit and river bank are considered. The determined the change of transverse velocit as the change of adjacent sections in the transverse distribution of cumulative discharge. Nevertheless estimating transverse velocit precisel is ver difficult especiall in bends. Therefore ever mistake in estimation causes a great error in calculation of transverse miing coefficient. Determining pollution area versus concentration In this method linear slope obtained from sketching the pollutant mass area versus the logarithmic concentration is considered to be. So with having and ou can get the longitude and transverse miing coefficients (Fischer et al. 1979). 173

3 Miing determination using the detector quantit In third method for estimating miing coefficient ou need the detector value. Considering pollutant bulk have an elliptical shape we can obtain the ratio of concentration variance in length to width direction if the shape of pollutant bulk is ellipsoid. D (6) d Where D and d are the long and small diameters of ellipse respectivel. So from the following equation ou can get the miing coefficients (Fischer et al. 1979). W D 1 (7) 1. Cma t STREAM TUBE METHOD The using of this method was first proposed b Yotsukura and Cobb (197). And later it was developed b Yotsukura and Sare (1976). In stream tube method the transverse distance is replaced b the gathering discharge; that this distance is determined b following equation. (8) q u( ). h( ). d 0 In this equation u() and h() are the average velocit and average depth in transverse distance of river bank () respectivel. Measurement or estimation of discharge distribution in all sections is the main problem in application of gathering discharge method. Nevertheless this method is the most common method for estimation of transverse miing coefficient. In stream tube method the one-dimensional miing equation is changed to following relationship. c c u D (9) Wherein c is the average concentration depth. RUTHERFORD S METHOD Rutherford used the fashionable momentum method in Eq. (8) and offered the Eq. (10). Q Q c (10) c q q dq q q q dq d q0 q0 q D Q Q d cdq q cdq 0 q0 In which Q= total of river discharge and q = center of detector distribution related to gathering discharge. Also is a function that is applied for description of transverse diffusion coefficient changes. In a wa that: b (11) q dq 1 0 The research histor Fischer et al. (1979) used the eperimental investigations in straight canals (without bend and curvature) with uniform longitudinal conditions and proposed the empirical Eq. (1) for calculation of transverse diffusion coefficient. 0.15Hu * (1) 174

4 Chang (1971) found that the relationship between transverse diffusion coefficient and curvature is the ccling changes form. lao and Krishnappan (1977) founded the constant values of transverse diffusion coefficient in meander ccle. Fisher (1969) proposed the Eq. (13) to describe effects of rivers curvature on transverse diffusion coefficient. (13) U H Hu * * u Rc Wherein R c = curvature radius; Also the smmetr constant is equal to 5. Fisher pointed that this Eq. is accurate for eperimental conditions; but it has less accurac in the field ones. Yotsukura and Sare (1976) used the apparent ratio in Fischer s Eq. and proposed the new Eq. (14) for meanders. (14) U W Hu * * u Rc It is seemed that the miing coefficient increases with growth of apparent ratio W H ; because the apparent ratio relates to transverse turbulent scale in canal. Nevertheless determination of transverse diffusion coefficient b mentioned methods is ver difficult. Because: Measurement of geometr parameters such as curvature radius (R c ) and sinuosit (S) is more difficult than those of the hdraulic elements H U W u *. Also in this case there is less information available. Curve characteristics change along the channels especiall in curvature radius. Sinuosit is a more suitable criterion for estimating transverse diffusion coefficient; because determining of this coefficient is applied in the reach form. Fischer et al. (1979) presented the following Eq. for rivers. This Eq. is the most common method for estimation of transverse diffusion coefficient in rivers regardless of the curvature radius. 0.6Hu * (15) Rutherford (1994) with usage of collection of field data comprehended that the Hu * is variable between 0.1 to 0.6. Deng et al. (001) surveed the Rutherford s dimensionless data collection. The eliminated the outling number (0.847) of them and obtained Eq. (16) for calculation of diffusion coefficient in direct channels. Hu * (16) it could be Observed from Eq. (16) that this ratio is compatible to obtained amount of Fischer s Eq. (1). Also Deng et al. (001) developed the other Eq. for estimating transverse miing coefficient Wu * (17) W / H Dimensionless analsis and effective parameters Various parameters are effective on the miing coefficient which can be clustered in three general groups. Flow characteristics This group includes the averaged velocit (U) shear velocit u * hdraulic radius (R) depth of flow (H) turbulent intensit and acceleration due to gravit (g). Canal s geometr characteristics The width of channel (W) curvature radius (R c ) sinuosit (S) angle of curvature length of meander 175

5 M width of meander M L b straight partition length between two continuous curves (T) length of the reach or length of the arc (L) hdraulic kind of river-bed (smooth or rough) height of roughness K s and rough distance from each other L are pointed from this group. s Fluid characteristics This group includes the dnamic viscosit and fluid s densit. Those mentioned parameters can be summarized as Eq. (18). f g H U u* W R S M M T K L L (18) 1 c L b s s The Eq. (19) is obtained b using the Buckingham dimensionless analsis and compounding the dimensionless groups. Hu * Hu Wu * Wu W U M f H u * M Ls L Rc W K s Re H L S b (19) Wherein M meander ratio; L roughness accumulation; K s H rough averaged depth and Re= L M b Renolds number. Also Hu Wu * Wu and Hu * L s are the dimensionless coefficients of transverse miing. It is necessar to mention that the viscosit influence is ignored in man open channels. Therefore the Renolds number can be ignored. Also there is little information related to river bed form; and more investigations have been conducted in eperimental flumes. So the eperimentall-obtained values are different to actual ones; because of the complicated three-dimensional conditions and other various causes. Therefore the roughness accumulation L L s and relative height of roughness K s H can be ignored and their influence can be placed in some factors such as U u * that it is a criteria for river roughness coefficient. Also the river morpholog is a function of various factors. These parameters cause the river plan not to have a regular geometr shape. In other words the curves characteristics show the important changes along the river. In case the river sinuosit has a constant value in an especial reach. The detector eperiments are applied in long distances; therefore it is offered that sinuosit is considered for presentation of curves influence on miing coefficients. According to mentioned eplanations the Eq. (0) can be applied. W U (0) f 3 S Hu * Hu Wu * Wu H u * Nevertheless in the most of the proposed equations dimensionless transverse miing coefficients are onl and onl functions of the u * W H. In other words: U and W U (1) f 4 Hu * Hu Wu * Wu H u * Presentation and analsis of conclusions Authors started performing transverse diffusion coefficient approimation and evaluation based on available data. The eperimental of straight rectangular flumes The 136 collected data-sets (appendi A) were used to obtain an equation for miing coefficient estimation in these kinds of flumes. According to table (1) is seen that in these flumes the ratio of W H correlates onl to Wu * Wu Hu * Hu are almost independent from this and form; and dimensionless groups of and ratio. Also U u * correlates to Wu and Hu relativel well and Hu * or Wu * is independent 176

6 of U u *. According to mentioned eplanations Hu * So according to the general Eq. (1) the ratio Hu * is almost equal to a constant value. Also in table () Hu Wu * Wu and Hu * U. is the onl group that independent of W H and u * have been analzed statisticall. It can be calculated that the dimensionless diffusion coefficient ratio in straight rectangular flume doesn t have a constant value; but since the 95% of Hu * data have a range with the little variation ( ) therefore it can be calculated that Eq. (1) with correlation coefficient equal to is a good relationship for estimation of transverse diffusion coefficient. Figure (1) is comparing the calculated diffusion coefficients of mentioned equation to measurement values. Table 1. The Correlation Matri of Effective Parameters in Estimation of Transverse Miing Coefficient in Straight Rectangular Shape Flumes Dimensionless Group W H U /(Hu*) /(Wu*) /(HU ) /(WU ) W U H * u 1 * u Table. The Statistical criterion of Transverse Miing Coefficients in Straight Rectangular Flumes The 95% of Sure Distance Standard Dimensionless Group Data No. Averaged Middle Variance The Lowest Limit The Highest Limit Deviation /(Wu*) * /(Hu*) /(HU ) /(WU ) * * * Standard Error /004 0/003 Predicted 0/00 0/ /001 0/00 0/003 0/004 Observed Figure 1. The comparison of the obtained of Eq. (1) to measurement values Transverse miing coefficient in natural streams Table (3) has been drawn according to 19 series of collected hdraulic data-sets from various references (appendi B). Dimensionless Group /(Hu*) /(HU ) /(WU ) /(Wu*) Table 3. The Statistical criterion of Transverse Miing Coefficients in Rivers The 95% of Sure Distance Standard Data No. Averaged The Lowest The Highest Middle Variance Deviation Limit Limit * * Standard Error

7 It can be concluded that in rivers the dimensionless ratio of Hu * doesn t have a constant value in comparison to the common equations. In a wa that 95% of data range has a large relativel range. Therefore Hu * 0. can t be applied for estimating river transverse miing coefficient in various conditions. It is guessed 6 that the other hdraulic factors are {can be} effective on this coefficient. Presentation of new equations for rivers For estimation of river transverse miing coefficient man equations are proposed b application of nonlinear regression methods; in etension the important ones are pointed. Number Table 4. The Proposed Equations for Estimation of Transverse Miing Coefficients in Rivers Equation The Usage Parameters W H W H U u * U u * U u * W H W H W H Hu * HU HU U u * U u * HU WU U u * Hu * W U Hu * H u * Hu W U H u * U Hu u * W U Hu H u * W U Wu H u * W U Hu * H u * Eqs. ( to 7) and also Fischer s Eq. (15) are analzed and surveed using the various statistical methods according to the table (5). Table 5. The Comparison of Eqs. ( to 7) and Also (15) b Using of the 10 Statistical Parameters Eq. Number MAE MAD R MSE STD SSE Ma (e) Min (e)

8 Data Percent Eq. Number "-0.15<DR<-0.15" "-0.3<DR<0.3" Figure. The comparison of the obtained of Eqs. ( to 7) and also (15) in rivers b using the DR As observed from table (5) it can be seen that the Eq. (7) have the best correlation coefficient (R =0.6). But the correlation coefficient isn t a high value; so this parameter or other similar statistical ones can t alone indicate that the equation or equations are pleasing here the other statistical parameter known as DR must be used. This criterion is defining b following form. DR log (10) p m (8) Fig. () is evaluating the mentioned equations b using this statistical parameter. According to this figure it can be distinguished that Eq. () is the best relationship for estimation of river transverse miing coefficient. CONCLUSION In this paper in addition to definition of transverse diffusion coefficient and its common determining methods the various equations were obtained for estimation of this coefficient. For eperimental straight rectangular flumes it is be calculated that although the dimensionless ratio of transverse diffusion isn t a constant value but the 95% of data have a limit with little variance therefore b using data analsis it can be told that the Eq. (1) is a good relationship for calculation of this coefficient. But in rivers the Eq. (15) isn t a suitable relationship. So in this case 6 equations are proposed for estimation of this coefficient. Also these equations are surveed b several statistical criteria. Finall Eq. () was chosen as the best relationship for calculation of river transverse miing coefficient. REFERENCES Beltaos S Miing Processes in.natural Streams. In: Transport Processes and River Modeling Workshop. Canada Center for Inland water. Chang HH Fluvial processes in river engineering Wile New York 0 1 and Demetracopoulous AC Stefan HG Transverse miing in Wide and Shallow River: Case Stud. J. Environ. Div. Am. Soc. Civ. Eng. 109(3) Deng ZQ Singh VP Bengtsson L.001. Longitudinal dispersion coefficient in straight rivers. J. Hdraulic. Eng. 17(11) Elder JW The Dispersion of Marked Fluid in Turbulent Shear Flow. J. Fluid Mech Engmann JEO Kellerhals R Transverse Miing in an Ice Covered River. Adv. Water Resour. 10(4). Fischer HB List EJ Koh RCY Imberger J Brooks NH Miing in Inland and Coastal Waters. Academic Press New York. 483 P. Glover RE Dispersion of Dissolved or Suspended Materials in Flowing Streams. U.S. Geological Surve Professional paper Holle ER Abraham G Field Tests and Transverse Miing in Rivers. J. Hdraulic. Div. Am. Soc. Civ. Eng. 99(HY1) Lau YL Krishnappan BG Transverse Dispersion in Rectangular Channels. J. Hdraulic. Div. Am. Soc. Civ. Eng. 103(HY1O). Proc. Paper 1394 pp Meer W "Transverse Miing in Mobile River Alabama." Journal of Research U.S. Geological Surve vol.5 No.1 Miller AC Richardson EV Diffusional Dispersion in open Channel Flow. J. Hdraulic. Div. Am. Soc. Civ. Eng. 100(HY1) Nokes RI Wood JR Vertical and lateral turbulent dispersion : some eperimental results j. Fluid Mech Okoe JK Characteristics of Transverse Miing in Open-channel Flows. Report No. KH-R-3. California Institute of Technolog. 179

9 Prch EA Effects of Densit Differences on Lateral Miing in Open-Channel Flows. Report No. KH-1-1. California Institute of Technolog. Rutherford JC River miing Wile. Chi Chester U.K Sare WW Yeh TP Transverse Miing Characteristics Missouri River Downstream forms the Cooper Nuclear Station IIHR Report 145 Iowa Institute of Hdraulics Research Universit of Iowa Iowa Cit Iowa. Webel G Schatzmann M "Transverse Miing in Open Channel Flow." J. Hdraulic. Div. Am. Soc. Civ. Eng. 110(4) Yotsukura N Cobb ED Transverse Diffusion of Solutes in Natural streams. Geological Surve Professional Paper 58-C. Yotsukura N Fischer HB Sare WW Measurement of Miing Characteristics of the Missouri River between Siou Cit Lowa and Plattsmouth Nebraska." U.S. Geological Surve Water- Suppl paper 1899-G. Yotsukura N Sare WW Transverse Miing in Natural channels. Adv. Water Resour. 1(4)