Environ Geol (2009) 57:899–908 DOI 10.1007/s00254-008-1369-y
ORIGINAL ARTICLE
Estimation of stream water quality parameter using regime channel theory K. Muthukrishnavellaisamy Æ G. C. Mishra Æ M. L. Kansal Æ N. C. Ghosh
Received: 24 November 2007 / Accepted: 5 May 2008 / Published online: 3 June 2008 Ó Springer-Verlag 2008
Abstract For predicting and forecasting fate of nonconservative pollutants downstream from source using advection–dispersion–decay equation (ADDE), estimation of three parameters; mean flow velocity (U), longitudinal dispersion co-efficient (DL) and decay rate co-efficient (k), is required a priori. In this three parameters model, estimation of DL holds difficulties and draws interest towards it. The empirical formulae use the field and experimental data of channel and flow characteristics to estimate DL. In this paper, an innovative approach has been proposed towards the estimation of DL using regime channel concept. Having known discharge of flow and silt factor of the riverbed material, hydraulic parameters of the channel can be determined theoretically, which in turn can be used to estimate DL appropriately making use of a suitable empirical formula. Keywords Regime channel concept Longitudinal dispersion co-efficient Pollutants transport River Brahmani India
Introduction In recent years, the growing industrialization and urbanization are the main reasons of increasing pollution threats and public concerns toward health related issues. With the
K. Muthukrishnavellaisamy (&) G. C. Mishra M. L. Kansal Water Resources Development and Management, Indian Institute of Technology, Roorkee 247667, India e-mail:
[email protected] N. C. Ghosh National Institute of Hydrology, Roorkee 247667, India
advent of sophisticated computational tools and investigation technologies, much attention is being paid to derive pollutant’s transport phenomena more close to reality. Rivers are being used as a sink of pollutants. This in turn deteriorates the water quality and spoils the associated ecosystem of the rivers because rivers have limited assimilation capacity to the pollutants. Disposal of pollutants in excess to the assimilative capacity of the river would not only affect the health of water but would also damage the aquatic life marinating by the river. Thus, study of pollutant’s transport in a stream/river is essential to correctly evaluate the fate of pollution threat at downstream locations from the source and regulating disposal of pollutants in rivers such that the eco-system remains safe from any incoming threat. Although, the pollutants’ transport processes are three-dimensional, particularly near to the source; however away from the source it becomes onedimensional process of coupled action of advection–dispersion along the longitudinal direction (Taylor 1953, 1954; Elder 1959; Fischer 1968; Banks 1974; Beer and Young 1983). Further, most of the researches on pollutant transport to date pertain to the low flow period in which flows in rivers remain nearly steady (Runkel 1998). For non-conservative pollutants the one-dimensional transport equation is given by (Ogata and Banks 1961; Fischer et al. 1979): oC oC o2 C ¼U þ DL 2 k C ot ox ox
ð1Þ
where, U is the mean flow velocity, DL is the longitudinal dispersion co-efficient, C is the pollutant concentration, k is the decay rate co-efficient, x is distance measured downstream of source and t is time. To compute the concentration of constituent of interest along the river reach, whether by applying an analytical
123
900
Environ Geol (2009) 57:899–908
method or by any numerical method, on Eq. (1), parameters; U, DL and k are required a priori. Estimation of parameters, U and k are straightforward, such as U can be measured in the field or can be obtained using flow resistance equation and k can be determined from the laboratory experimentation of field samples. However, estimation of DL is not straightforward. A number of approaches have been suggested by many investigators (Taylor 1954; Fischer 1966; 1967; Fischer et al. 1979; Gray and Pinder 1976; Jain 1976; Beltaos 1978; Liu 1978; Asai et al. 1991; Seo and Cheong 1998) to estimate DL. These can be grouped as: (1) analytical approaches, (2) experimental approaches, and (3) empirical formulae. Each approach has its own merits and demerits. Interestingly, the value of DL of a river if obtained using any of the method does not match with the value of the other methods with hydraulic properties remaining same (EPA 1985). Out of these three groups, the third group, i.e., empirical formulae, which mainly suggest estimation of DL from river’s bulk flow properties and geometry, are less complicated to use. The other two approaches require intensive database. With the advancement of science and technology, it is preferred that a method being developed should be simple in practice and involves less cost. The paper is thus focused to investigate a method of determining stream bulk flow properties, which would ease out estimation of DL with reasonable accuracy.
Theoretical background With the following initial and boundary conditions, Ogata and Banks (1961) have given the analytical solution of Eq. (1). Cðx; 0Þ ¼ 0;
x [ 0;
Cð0; tÞ ¼ CR ; t 0; Cð1; tÞ ¼ 0; t 0: CR xU x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Cðx; tÞ ¼ exp U þ 4DL k exp 2DL 2DL 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! x U 2 þ 4DL k t pffiffiffiffiffiffiffiffi erfc 2 DL t x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 2 þ 4DL k þ exp 2DL pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !) xþ U 2 þ 4DL k t pffiffiffiffiffiffiffiffi erfc ð2Þ 2 DL t where C(x, t) is concentration of constituent at distance x and over time t; DL is longitudinal dispersion coefficient, U is mean flow velocity, erfc(z) is complimentary error
123
function
of z = 1 - erf(z); the error function Rz 2 erf(z) = p2ffiffip eu du; and t is time measured since injec0
tion of input. For a unit step input applied continuously at the boundary, i.e., CR = 1, Eq. (2) becomes unit step response function, K(x, t). The unit impulse response function, k(x, t), is the derivative of the unit step response function, and is given by: " # x ðx UtÞ2 kðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp kt ð3Þ 4DL t 2t pDL t
Theoretical and experimental methods for determination of DL Theoretical method requires detailed measurement of velocity distribution along three Cartesian coordinate systems. Expression to estimate the dispersion co-efficient for uniform flow in an infinitely broad channel suggested by Elder (1959) has restriction on its applicability for estimation of DL in case of natural streams (McQuivey and Keefer 1974; Fischer et al. 1979; Seo and Cheong 1998). The analytical method suggested by Fischer et al. (1979) also has difficulties in estimation of DL because of the data requirements and accuracy needed in the estimation of transverse mixing co-efficient. Experimental bases require measurement of concentration–time distributions at least at two locations. The theoretical and experimental bases for estimation of DL thus are not encouraging in terms of data requirement.
Empirical methods for determination of DL There are numerous empirical formulae available mostly representing relationship of DL ¼ a0 HU where, H is depth pffiffiffiffiffiffiffiffiffiffi of flow and the shear flow velocity U* = gRh S; Rh is hydraulic radius, S is the bed slope, g is acceleration due to gravity and a0 is a constant, which is found to have a value ranging from 8.6 to 7,500 (Bansal 1971). The available empirical formulae suggested by various investigators are tabulated in Table 7 in Annexure. From these empirical formulae, it can be noticed that a number of formulae are of the type DL=U H is constant. However, several researchers (refer to Annexure) have shown that the ratio DL=U H is not constant. This makes difficulty in estimation of DL. Empirical formulae eventually dictate that DL is a function of stream flow characteristic and stream geometry. By using the non-dimensional analysis (Seo and Cheong 1998) or by reasonable approximation of the integral relating the dispersion co-efficient in natural streams (Fischer 1967),
Environ Geol (2009) 57:899–908
901
the functional relationship pertinent to the dispersion coefficient has been obtained as b c DL W U ¼a ð4Þ H U U H where, W is the channel width, H is the depth of the flow, U is the average velocity, U* is the shear velocity, and a, b, c are constants. The values of these constants are given in the Table 1. The dimensionless logarithmic plots of DL=U H versus W= having U= ¼ 25 and DL= U U H versus =U having H U W= ¼ 30 for the values of a, b and c as suggested by H different investigators are shown in Figs. 1 and 2, respectively. The figures also show the average of distributions computed by different empirical formulae. From the figures it can be noted that the average, of distributions computed using different formulae, is found approximately matching with the values estimated using the formula suggested by Seo and Cheong (1998). Having the data sets given in Table 2, the values of DL have been estimated using different empirical formulae and presented in Fig. 3. From Fig. 3, it can be noted that the formulae, suggested by Seo and Cheong (1998), gives DL close to observed values. Thus, Seo and Cheong (1998) equation of the following form has been chosen for estimation of DL: 0:62 1:428 W U DL ¼ 5:915H ð5Þ H U0:428 Regime channel concept Lindley’s (1919) defined that when an artificial channel is used to carry silty water, both the bed and banks scour or fill, depth, gradient and width changes until a state of balance is attained at which the channel is said to be in regime. Many investigators (Pierre 2002; Lindley 1919; Lacey 1930) recognized that sediment size plays an important role in determining the channel geometry. The
Table 1 Values of the constants a, b and c by various investigators Investigators
a
b
c
Fischer (1966)
0.011
2
2
Thackston and Krenkel (1967)
7.25
0.25
0
Beltaos (1978)
0.066
2
0
Liu (1978)
0.18
2
0.5
Marivoet and Craenenbroec (1986)
0.0021
2
2
Asai et al. (1991)
2
1.5
0
Koussis and Mirasol (1998)
0.6
2
0
Seo and Cheong (1998) Kezhong and Yu (2000)
5.915 3.5
0.62 1.125
1.428 0.25
approximate relationship, suggested by Lindley (1919) to find silt factor, is fL = 8Dm, where Dm is the grain size in inches. Lacey (1930) has suggested couple of empirical relationships to estimate the channel geometry having known the stream flow rate and the silt factor. For a regime channel having average stream flow rate (Q) and silt factor (fL), the channel geometry can be approximated as follows; 1 1
U ¼ 0:438Q6 fL3
ð6Þ
1 1 3 3
Rh ¼ 0:47Q fL 5 6
ð7Þ
13
A ¼ 2:28Q fL P ¼ 4:818Q
ð8Þ
1 2
ð9Þ 1 5
S ¼ 0:0003Q6 fL3
ð10Þ
where, U is the mean flow velocity (m/s), Rh is the hydraulic mean radius (m), A is the cross-sectional area of flow (m2), P is the wetted perimeter (m) and S is the dimensionless slope (m/m). The mean grain size is given by: Dm ¼ Dg exp 0:5 ln ln rg ð11Þ where, Dm is the mean grain size in mm, the geometric
D50 þ standard deviation rg = 12 DD84:1 D15:9 ; and the geometric 50 1
mean diameter Dg = ðD84:1 D15:9 Þ2 : Henderson (1966) has also suggested the concept of regime channel, which considers the values of constants K1, K2 and K3 according to channel type. Although both the concepts are applicable for wider channel, however, the Lacey’s regime channel concept holds some advantages over the Henderson’s (1966) theory. These are: (1) field measurement of stream’s cross-section can be ignored; (2) sampling and analyses of streambed material are easier than stream cross-section measurement, and (3) simple in handling. It is clearly evident that if mean grain size of the streambed sediments is known then silt factor, fL, can be determined. Knowing silt factor and Q, and by using Eqs. (6)–(10), the channel’s characteristics and geometry can be ascertained. Field application Study area and flow data To investigate the potentiality of the proposed concept, a stretch of the Brahmani River down below the Rengali dam in Orissa, India, which is under the threat of pollution, has been considered. The study stretch between two locations Rengali Dam and Talcher having lat-long, 85°020 E– 21°170 N and 85°130 E–20°570 N, respectively, is presented
123
902
Environ Geol (2009) 57:899–908
Fig. 1 Variations of dimensionless dispersion (DL/HU*) with W/H for U/U* = 25
Fig. 2 Variations of dimensionless dispersion (DL/HU*) with U/U* for W/H = 30
in Fig. 4. Bed samples from pre-decided locations along the length have been collected and analyzed in the laboratory to obtain the grain size distribution. Sieve analysis (dry test) and hydrometer analysis (wet test) were carried out to obtain grain size distribution. The plot of particle size versus percent of finer is shown in Fig. 5. The mean grain size estimated using the Eq. (11) is found to be 0.149 mm. The silt factor corresponding to this grain size estimated using the relation fL = 8 Dm is 0.6141. In order to define the low flow period, daily discharge data of two locations (Talcher and Jenapur) have been analyzed and the plots of daily discharge data are presented in Figs. 6 and 7. The river stretch between Rengali dam and Talcher, which is shown in Fig. 4, is the reach of interest in this paper. In Figs. 6 and 7, the average flow period has been grouped into three categories. They are
123
pre-monsoon (January–June), monsoon (July–September) and post-monsoon (October–December). For each season, the average flow rate (Q) has been calculated. To check the consistency of the flow data standard deviation from the mean has been calculated for each period, and found, it fits to the normal distribution. Main focus of this paper is on the low flow period (October–December and January–June). The channel geometry and flow characteristics have been estimated using Eqs. (6)–(10) and presented in Tables 3 and 4 by having the flow rate (Q) and silt factor (fL). Verification of regime characteristics Regime channel width corresponding to flow rate, Q = 2,500 m3/s which has been observed during a
Environ Geol (2009) 57:899–908
903
Fig. 3 Comparison of observed and computed dispersion coefficients (lines indicate the observed values of DL and bullet marks indicate the empirically computed DL)
Table 2 Observed dispersion, river geometry and flow characteristics data (adapted from Nordin and Sabol 1974) Streams
Width in m
Depth in m
Slope
Velocity in m/s
Shear velocity in m
Observed DL in m2/s
Antietam Creek 1
12.8
0.305
0.00095
0.42
0.057
17.5
Antietam Creek 2
11.89
0.66
0.00095
0.43
0.085
20.9
Tickfau River
14.94
0.59
0.00117
0.27
0.08
10.3
Sabina River
12.19
0.51
0.00018
0.23
0.03
14.7
Fig. 4 Location map showing River Brahmani and its tributary Tikira (digitized from Google Earth)
monsoon period is computed as 242.37 m. From satellite image (Google Earth) the channel width during that particular monsoon period near Talcher is measured and found to be 250.59 m which has been presented in Fig. 8. The regime channel depth could not be compared with the actual depth of flow due to absence of observations. For the purpose of analysis of pollutant transport, the regime channel width and depth computed for different flow are assumed to be appropriate.
Simulation of pollutant concentration Adopting the empirical equation proposed by Seo and Cheong (1998), the longitudinal dispersion co-efficient has been estimated to have computed channel geometry and flow characteristics and tabulated in Tables 5 and 6. DL changes between source and Jenapur (153.75 km) due to the change in channel geometry for a particular season due to the inflow from external sources. To obtain the concentration–space
123
904
Environ Geol (2009) 57:899–908
Fig. 5 Plot of grain size distribution obtained from sieve and hydrometer analysis of the bed sediment samples collected from the River Brahmani, Orissa, India (source: Muthukrishnavellaisamy 2007)
Fig. 6 Ten years average discharge data of river flow for the location Talcher of Brahmani River, Orissa
Fig. 7 Twelve years average discharge data of river flow for the location Jenapur of Brahmani River, Orissa
profiles along the river reach, the values are considered from the Tables 5 and 6 as follows: for 0 \ x B 26 km (up to Talcher); U = 1.0276 m/s and DL = 784.042 m2/s for x [ 26 km (up to Jenapur); U = 1.0412 m/s and DL = 845.245 m2/s. The Tikira River discharging
123
pollutants to the River Brahmani at 25 km down-stream of Rengali dam. Having observed water quality and flow data of both rivers, the boundary input concentration (CR) has been found as 24.03 mg/L. The decay rate (k) of non-conservative pollutant (BOD) is determined as 0.001901 per min. To
Environ Geol (2009) 57:899–908
905
Table 3 Channel geometry estimate having discharge and silt factor (Location: Talcher)
Table 5 Longitudinal dispersion co-efficient estimated (Location: Talcher)
Q in m3/s
U in m/s
A in m2
H in m
S
U* in m/s
U in m/s
208.7
0.9067
229.875
3.279
70.084
5.465E-05
0.04193
0.9067
1097.66
1.1957
916.738
5.704
160.720
4.144E-05
0.04816
1.1957
333.5
0.9804
339.705
3.835
88.590
5.054E-05
0.04360
0.9804
Average
1.0276
W in m
Table 4 Channel geometry estimate having discharge and silt factor (Location: Jenapur) Q in m3/s
U in m/s
203.73
0.9031
1352.05
1.2379
337.94
0.9825
A in m2 225.286 1090.64 343.469
H in m
W in m
S
3.254
69.241
5.48E-05
6.114
178.375
4.00E-05
3.851
89.178
5.04E-05
illustrate the procedure, using the Eq. (2), for average values U = 1.0344 m/s, DL = 814.644 m2/s and with the above data, the concentration profile has been simulated for predicting the threat of pollutant, which is discharged in to the stream at the source to the down-stream locations. Due to the change of the reach length from source to Talcher (26 km) and 24 km stretch down-stream of Talcher, the concentration-time profile has to be convoluted over a time limit 0 to t as follows.
C¼
Zt
I ðsÞK ðt sÞds
0
C ðx; tÞ ¼
Zt
W in m 70.1166 160.911 88.69494
dK ðx; tÞ K ðx; t sÞds ds
H in m
DL in m2/s
3.2805
448.4691
5.705 3.8353
1298.229 605.4255 784.0412
ð12Þ
0
where, s is the dummy variable. For each stream reach, the values of U and DL vary as given in Tables 5 and 6. Having these values, using the Eq. (12), and the concentration–time profile for step input have been simulated and plotted in Fig. 9 for specific locations to predict the fate of pollutant’s threat to the down-stream. Similarly, the impulse responses have been plotted in the Fig. 10 over time using Eq. (3) for the estimated values of U and DL for the specific locations.
Results and discussion Figures 1 and 2 show a close match between DL estimated by Eq. (5) and average distribution of different empirical formulae. Eq. (5) shows good agreement with observed value of DL, which can be noticed from Fig. 3. From the Tables 3 and 4, for the maximum and minimum seasonal
Fig. 8 Satellite image (Google Earth) showing the average width of river at Talcher
123
906
Environ Geol (2009) 57:899–908
Table 6 Longitudinal dispersion co-efficient estimated (Location: Jenapur) U* in m/s
U in m/s
W in m
H in m
0.04185
0.9031
69.241
0.04899
1.2379
178.375
6.114
0.04365
0.9825
89.178
3.851
Average
1.0412
3.254
DL in m2/s 441.5591 1483.600 610.5745 845.245
Fig. 10 Concentration-time profile at selected locations, for CR = 24.03 mg/L (impulse input), U = 1.0344 m/s, DL = 814.644 m2/s, k = 0.001901 per day
Conclusions
Fig. 9 Concentration-time profile at selected locations, for CR = 24.03 mg/L (step input), U = 1.0344 m/s, DL = 814.644 m2/ s, k = 0.001901 per day
average flow rate at Talcher and Jenapur, the channel geometry and flow characteristics has been estimated theoretically using Regime channel equations (6)–(10).Using these DL has been estimated reasonably. From the Tables 5 and 6, it can be noted that the DL value varies with W/H ratio and U/U* which mainly depends on the flow rate (Q). Using the Eqs. (2) and (3), pollutants concentration has been predicted for the downstream locations for the estimated DL. Figure 9 shows the pollution status of river at specific locations (15, 27 and 50 km down steam of pollutant disposal site). From Fig. 9, it can be noted that the pollutant concentration reaches its maximum value of 15.34 mg/L after 500 min for the location Samal, which is approximately equal to the observed value of 15 mg/L (Mishra et al. 2007). Figure 10 shows the concentration– time profile for the locations 15, 27 and 50 km down steam of pollutant disposal site for impulse input.
123
The approach, which has been proposed here to ascertain stream’s geometry and hydraulic properties for estimation of DL, is employed mainly on regime channel concept. Having known stream geometry and hydraulic properties, and making use of a suitable empirical formula, one can estimate DL appropriately. The silt factor and discharge of flow may vary in the stream along direction of flow. Change of these factors indicates change in stream geometry also. For changing velocity, normally caused due to non-uniform flow, DL will also change. Having the value of U and DL estimated for non-uniform flow, one can predict/ forecast the probable threat at the downstream for a given/ known pollution load at the source. The concept of regime channel has successfully been extended for ascertaining stream’s geometry including estimation of DL of the river Brahmani down below the Rengali dam, Orissa at some selected locations Talcher and Jenapur. The upshot of pollutants threat to the downstream has been evaluated by plotting the concentration–space and concentration–time profiles for the unit step input of pollutant injected at the source continuously from the Eq. 2 by having estimated D L.
Annexure Table 7
Environ Geol (2009) 57:899–908 Table 7 The empirical equations to estimate DL from river geometry (Source: Prasad 2002)
907
S.No Investigator
Empirical formulae
1
Taylor (1954)
DL = 10.1 U*r; where r is the radius of the pipe
2
Elder (1959)
DL = 6.3 U*H; where H is the depth of flow
3
Yotsukura and Fiering (1964)
DL = 9.0 to 13.0 U*H
4
Fischer (1966)
DL = 0.011 U2 W2/U*H
5
Thackston and Krenkel (1967)
DL = 7.25 U*H{U/U*}
6
Sooky (1969)
DL = K10 + K0 + K0 K10 ¼ 0:2222fU =j02 gfHm =ag K0 = aK00 K 00 ¼ aj0 U Hm =9
7
Sumer (1969)
DL = 6.23U*H
8
Fukuoka and Sayre (1973)
DL/RhU* = 0.8{rc2/LBH}1.4
9
McQuivey and Keefer (1974)
DL = 0.058Q/SW
10
Jain (1976)
DL = U2 W2/kAU*
11
Hou and Christensen (1976)
DL/RhU* = 1.5239 – 0.1395 log{Rh/Ks}+10.0081{log(Rh/ K)}2
12
Beltaos (1978)
DL/RhU* = a{W/Rh}2
13
Liu (1978)
DL = Q2/2U*Rh3{U*/U}2
14
Magazine (1983)
DL/RbU* = DL/RwU* = 75.86(Pr)1.632 pffiffiffi where Pr = Cw/ g{x/h}0.3{x10 /b}0.3
15
Marivoet and Craenenbroec (1986) DL = 0.0021U2 W2/U*H
16
Asai et al (1991)
DL/U*H = 2.0{W/Rh}1.5
17
Ranga Raju et al (1997)
DL/qS = 0.4Pt
18
Koussis and Mirasol (1998)
where Pt = {W/Rh}2.16{U/U*}-0.82{S}-0.2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi DL = U ðgRh SÞ/H{W}2
19
Seo and Cheong (1998)
DL/U*H = 5.915{W/H}0.628{U/U*}1.428
20
Kezhong and Yu (2000)
DL/U*H = 3.5{W/H}1.125{U/U*}0.25
{1.5 + e/h}
References Asai K, Fujisaki K, Awaya Y (1991) Effect of aspect ratio on longitudinal dispersion coefficient. In: Lee, Cheung (eds) Environmental Hydraulics, vol 2. Balkema, Rotterdam Banks RB (1974) A mixing cell model for longitudinal dispersion in open channel. Water Resour Res 10(2):357–358 Bansal MK (1971) Dispersion in natural streams. J Hydraul Div ASCE 11:1867–1886 Beer T, Young PC (1983) Longitudinal dispersion in natural streams. J Environ Eng Div ASCE 109(5):1049–1067 Beltaos S (1978) An interpretation of longitudinal dispersion data in rivers. Report No. SER 78-3, Transportation and surface water div, Alberta Research Council, Edmonton, Canada Elder JW (1959) The dispersion of marked fluid in turbulent shear flow. J Fluid Mech 5(4):544–560 Environmental Protection Agency (1985) Rates constants and kinetics formulations in surface water quality modeling 2e, Environmental Research Laboratory, US EPA Fischer HB (1966) Longitudinal dispersion in laboratory and natural streams. Report No KH-R–12, W. M. Keck Laboratory of Hydraul and Water Resour, California Institute of Technology, California Fischer HB (1967) The mechanics of dispersion in natural streams. J Hydraul Div ASCE 93:187–216 Fischer HB (1968) Dispersion predictions in natural streams. J Sanit Eng Div 94:927–943
Fischer HB, Imberger J, List EJ, Koh RCY, Brooks NH (1979) Mixing in inland and coastal waters. Academic Press, New York, p 483 Fukuoka S, Sayre WW (1973) Longitudinal dispersion in sinuous channels. J Hydraul Div ASCE 99:195–218 Gray WG, Pinder GF (1976) An analysis of the numerical solution of the transport equation. Water Resour Res 12:547–555 Henderson FM (1966) Open channel flow. Macmillan, New York Hou H, Christensen BA (1976) Influence of equivalent sand roughness on the dispersion coefficient in laboratory and natural stream. 3rd Annual Symp. Waterways, Harbors and Coastal Engineering Div, ASCE, Colorado State University, Fort Collins, Rivers 2:1179–1198 Jain SC (1976) Longitudinal dispersion coefficient for streams. J Environ Engg ASCE 102(2):465–474 Kezhong H, Yu H (2000) A new empirical equation of longitudinal dispersion co-efficient. In: Wang, Hu (eds) Stochastic Hydraulics 2000. Balkema, Rotterdam Koussis AD, Mirasol JR (1998) Hydraulic estimation of dispersion coefficient for streams. J Hydraul Engg ASCE 124(3):317–320 Lacey G (1930) Stable channels in alluvium, vol 229. Min Proc Inst Civ Eng London, Paper No 4736 Lindley ES (1919) Regime channels. In: Proceedings of Punjab Engineering Congress, vol 7 Liu H. (1978) Closure of discussion on predicting dispersion coefficient of streams. J Environ Eng Div ASCE 104(4):825–828
123
908 Magazine MK (1983) Effect of bed and side roughness on dispersion and diffusion in open channels. PhD thesis, University of Roorkee Marivoet JL, Craenenbroeck WV (1986) Longitudinal dispersion in ship canals. J Hydraul Res 24(2):123–133 McQuivey RS, Keefer TN (1974) Simple method for predicting dispersion in streams. J Environ Eng Div ASCE 100(4):997– 1011 Mishra GC, Kansal ML, Muthukrishnavellaisamy K (2007) Minimum flow requirement of river system for ecology. Project Report No. WRC–1004/2002-2003, Dept. of Water Resources Development and Management, Indian Institute of Technology, Roorkee Muthukrishnavellaisamy K (2007) A study on pollutant transport in a stream. PhD thesis, Dept. of WRD & M, Indian Institute of Technology, Roorkee Nordin CF Jr, Sabol GV (1974) Empirical data on longitudinal dispersion in rivers, USGS, Water Resources Investigations Ogata A, Banks RB (1961) A solution of the deferential equation of longitudinal dispersion in porous media. USGS Professional paper no.411-A Pierre YJ (2002) River mechanics. Cambridge University Press, London 434 p
123
Environ Geol (2009) 57:899–908 Ranga Raju KG, Kothyari UC, Ahmad Z (1997) Dispersion of conservative pollutant. Dept of Civil Engg, University of Roorkee Runkel RL (1998) One dimensional transport with inflow and storage (OTIS): a solute transport model for streams and rivers. USGS Water Resour Invest Report No 98-4018, Denver, Colorado Seepana Bala Prasad (2002) Simulation of solute transport in unsteady stream flows. PhD Thesis, WRDTC, IIT, Roorkee Seo IW, Cheong TS (1998) Predicting longitudinal dispersion coefficient in natural streams. J Hydraul Eng ASCE 124(1):25–32 Sooky AA (1969) Longitudinal dispersion in open channels. J Hydraul Div ASCE 95(4):1327–1345 Sumer M (1969) On the longitudinal dispersion co-efficient for a broad open channel. J Hydraul Res 7(1):129–135 Taylor GI (1953) Dispersion of soluble matter in solvent flowing slowly through a tube. Proc R Soc Lond A 219:186–203 Taylor GI (1954) The dispersion of matter in turbulent flow through a pipe. Proc R Soc Lond A 219:446–468 Thackston EL, Krenkel PA (1967) Longitudinal mixing in natural streams. J Sanit Eng Div ASCE 93(5):67–90 Yotsukura N, Fiering HB (1964) Numerical solution to a dispersion equation. J Hydraul Engg Div ASCE 90(5):83–104
