Download criteria for the compasion of equations

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; ylocal 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.
REFERENCES
[1] Deng, Z.Q and Singh, V.P and Fellow and Bengtsson, L., “Longitudinal Dispersion
in Straight Rivers”. Journal of Hydraulic Engineering, (2001), Vol. 127, No. 11, 919927.
[2] Fisher, B. H., “The mechanics of dispersion in natural streams. J Hydraul. Division
ASCE, 1967;93:187–216.
[3] Fischer, B.H., Discussion on simple method for predicting dispersion in streams’ by
R.S. Mc Quiveyand T.N Keefer. J Environ Eng Div ASCE, 1975;101:453–5.
[4] J Environ Eng Div ASCE 1975;101:453–5. “The Mechanics of Dispersion in Natural
Streams”, J. Hydr. Div., ASCE, (1967), 93(6), 187-216.
[5] Guymer, I and O’Brien, R., “Longitudinal Dispersion Due to Surcharged Manhole”.
Journal of Hydraulic Engineering, (2000), Vol. 126, No. 2, 137-149.
[6] Guymer, I., “Longitudinal Dispersion in Sinuous Channel With Changes in Shape”.
Journal of Hydraulic Engineering, (1998),Vol. 124, No. 1, 33-40.
[7] Kashefipour, S.M and Falconer, R.A., “Longitudinal Dispersion in Natural
Channels”. Water Research, )2002a(, Vol. 36, 1596-1608.
[8] Kashefipour, S.M and Falconer, R.A., “Modelling Longitudinal Dispersion
Coefficient in Natural Channel Flows Using ANNs”. (2002b), International
Conference on Fluvial Hydraulics, Belgium, River Flow, 111-116.
[9] Pujol, L and Sanchez-Cabeza, J.A., “Determination of Longitudinal Dispersion
Coefficient and Velocity of Ebro River Waters (Northeast Spain) Using Tritium as a
Radiotracer”. Journal of Environmental Radioactivity, (1999), Vol. 45, 39-57.
[10] Rutherford, J.C, O’Sillivan M.J., “Simulation of Water Quality in Tarawera River. J
Environ Eng Div, ASCE, (1974), Vol. 100, 369-390.
[11] Seo, W and Cheong, T.S., “Prediction Longitudinal Dispersion Coefficient in
Natural Streams”. Journal of Hydraulic Engineering, (1998), Vol. 124, No. 1, 25-32.
[12] Taylor, G.I., “The Dispersion of Matter in Turbulent Flow Through a Pipe”, Proc.,
Royal Soc., London, (1954), Ser. A, 223, 446-468.