J. Cent. South Univ. (2014) 21: 1995−2003 DOI: 10.1007/s11771-014-2148-4

Experimental and field study on dissipation coefficient of supersaturated total dissolved gas

FENG Jing-jie(冯镜洁), LI Ran(李然), MA Qian(马倩), WANG Le-le(王乐乐)

State Key Laboratory of Hydraulics and Mountain River Engineering (Sichuan University), Chengdu 610065, China

© Central South University Press and Springer-Verlag Berlin Heidelberg 2014

Abstract: The elevated supersaturation of total dissolved gas (TDG) downstream of a high-dam spill has deleterious effects on fish in a large range. A one-dimensional (1-D) longitudinal model is optimal for the prediction of supersaturated TDG dissipation over a long distance. The key issue of the model is to determine the dissipation coefficient accurately. In agreement with field observations and experiment data, dimensional analysis and regression were performed to propose a formula for estimating the dissipation coefficient of supersaturated TDG in various rivers and reservoirs, and it involves the effects of the turbulence intensity, the hydro-pressure and the solid-liquid interface. The friction velocity, water depth, hydraulic radius and Froude number are independent variables in the formula which are easy to determine in practical applications. The 1-D longitudinal model is implemented to calculate the dissipation of TDG in a reach of the Jinsha River. Good agreement is found between the calculated results and field data for both the dissipation coefficient and the dissipation process. Key words: total dissolved gas; dissipation coefficient; longitudinal model; hydrodynamics

1 Introduction

Total dissolved gas (TDG) levels are supersaturated downstream of high-dam spills and may cause bubble disease to fish and increase fish mortality [1−3]. Mitigation measures are expected to be developed to accelerate the dissipation of supersaturated TDG. Developing effective mitigation techniques is hampered by limitations in present models of TDG dissipation processes.

ROESNER and NORDON [4] firstly proposed a physically based model of TDG transportation in a stilling basin in which the variation of TDG level was controlled by the transfer coefficient and residence time of the bubbles in the stilling basin. JOHNSON et al et al [5−6] modified the transfer coefficient and the residence time in the model and used it to model observed field data. An approach to predict the oxygen saturation concentration at a spillway plunge pool was presented by HIBBS and GULLIVER [7], in which the effective saturation concentration and effective bubble depth were incorporated to estimate the transfer efficiency for oxygen. The physics of bubble and free surface gas transfer were separately treated by GELDERT [8], where AZBEL’s theory for turbulent bubble movement was used to develop a model for the mass transfer process

across the bubble-liquid interface [9]. Combining field observation with first-order kinetics theory, Pickett et al [10] proposed a longitudinal model to predict the TDG dissipation process along the river far from the spillway. Their model considered river velocity, depth and molecular diffusion with field data from the Columbia River to estimate the TDG dissipation coefficient. FENG et al [11] developed a formula on the basis of field observation results on several rivers in China by defining a comprehensive coefficient that combines turbulent and molecular diffusion to take the place of the molecular diffusion coefficient in the formula proposed by PICKETT. Applications show that the comprehensive coefficient varies significantly for different rivers and reservoirs. In recent years, with the development of numerical simulation techniques, multidimensional models have been employed to predict the dissipation process of TDG. ORLINS and GULLIVER [12] developed a two-dimensional, laterally averaged model to predict the TDG concentration within 300 m downstream from a dam, in which both convection and turbulent diffusion through bubbles and the free surface are incorporated into the TDG mass transport [12]. POLITANO [13−14] have simulated the TDG transport process with a two- phase flow model, using a bubble density transport equation to predict the bubble size [13−16]. In their work, the TDG concentrations within

Foundation item: Projects(51179111, 51279115) supported by the National Natural Science Foundation of China; Project(20110181110073) supported by

the Doctoral Programs Foundation of Ministry of Education of China Received date: 2012−12−12; Accepted date: 2013−05−20 Corresponding author: LI Ran, Professor; Tel: +86−28−85407780; E-mail: liran@scu.edu.cn

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1000 m below a spillway were calculated with a bubble-liquid mass transfer equation involving the gas volume fraction and bubble size; however, the mass transfer across the air-water interface was neglected. FU et al performed an unsteady simulation with a two-phase model to simulate the TDG transport of 3 km downstream of Gezhouba Dam [17], where the source term in the TDG transport equation was expressed as the mass transfer through both the bubble-liquid interface and the air-water surface. The mass transfer coefficient of the surface was determined by the bubble velocity and the turbulent kinetic energy, as discussed by TAKEMURA and YABE [18].

Theoretically, a multidimensional model, especially a multiphase model, can achieve more accurate results than a 1-D model by simulating a more comprehensive hydrodynamic field and TDG distribution. However, such models are time consuming to develop, calibrate and validate, requiring extensive data from expensive field campaigns to represent spatial heterogeneity. Furthermore, the utility of such models remains unproven, as existing numerical studies have rarely focused on mass transfer across the air-water surface, which is dominant in TDG dissipation downstream of stilling basins. In practice, a 1-D model parameterized from empirical data is more suitable for the prediction of the supersaturated TDG transport process in natural rivers or reservoirs using typically available data. In a 1-D model, the key issue is to determine a bulk dissipation coefficient that represents the overall effects of processes integrated over the river cross-section. However, existing methods are generally either based on field observations for a single river or the parameters used in the models are difficult to specify. In addition, the dissipation coefficient of TDG has often been approximated by the re-aeration coefficient of dissolved oxygen (DO), an approach which leads to significant

misstatement of dissipation rates [19]. A general and accurate expression is expected to

determine the dissipation rate of supersaturated TDG. By conducting many field observations and laboratory experiments, a formula for estimating the dissipation coefficient of supersaturated TDG is developed and it is expressed with basic hydrodynamic characteristics, which are analyzed herein. 2 Field observation

In prior research, a series of field observations of supersaturated TDG dissipation were conducted during a spilling period downstream of dams on the Min River, Yangtze River, Yalong River, Jinsha River, Lancang River and Dadu River in China (Fig. 1). The TDG percent saturation levels were measured with a YSI5200 TGP/DO detector made in the USA and a PT4 TGP detector made in Canada. The data indicate that the supersaturated TDG dissipated slowly such that its influence was observed for tens to hundreds of kilometers downstream of the spillway. Along more than 600 kilometers downstream of the Three Gorges Dam, the TDG saturation was observed to be over 110% [11].

If we assume that the cross-section TDG saturation is relatively uniform and provides a reasonable basis for modeling longitudinal dynamics, a 1-D model for the supersaturated TDG dissipation process following first-order kinetics theory is

eq

T eq

d( )( )

d

G Gk G G

t

(1)

where G denotes the percent saturation of TDG at time t, Geq represents the equilibrium percent saturation of TDG and is assumed to be 100%, and kT is the dissipation coefficient.

FENG et al [11] analyzed the TDG field data by

Fig. 1 Locations of field observation projects

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employing the curve-fitting method with the longitudinal dissipation model. The dissipation coefficients were computed for different rivers, denoted as cases No. 1 to No. 11 as listed in Table 1. The same methods were used on an additional data set to compute the dissipation coefficients of Dachaoshan Reservoir, Dadu River and Tongjiezi Reservoir, which are also listed in Table 1.

The values in Table 1 show that the dissipation coefficients vary by two orders of magnitude between different rivers and reservoirs. The maximum is 0.650 h−1, for Zouma River, and the minimum is 0.003 h−1, for Dachaoshan Reservoir. In general, the dissipation coefficients of large-scale rivers or reservoirs are significantly smaller than those of small-scale rivers, indicating that the velocity and depth are key factors affecting supersaturated TDG dissipation. The data indicate the dissipation coefficient is directly related to velocity and inversely related to depth, which matches both intuition and previous work: turbulence generally helps release supersaturated gas and shallower depths improve the overall rate of release. 3 Experimental study in open channel flow 3.1 Introduction of experiment

Dissipation experiments on supersaturated TDG in open channel flow were conducted at the State Key Laboratory of Hydraulics and Mountain River Engineering (SKLH). The open channel was composed of Plexiglas, with a length of 18 m, a width of 0.15 m and a height of 1 m. The slope of the channel was designed to be 0.4%, as shown in Fig. 2. A flow of water with supersaturated TDG was generated using an autoclave. Air and water were pumped into the autoclave through different inlets and mixed under high pressure so

that supersaturated conditions were generated. The flow was directed into the experimental open channel through an outlet pipe on the autoclave. The flow rate was controlled by a valve on the autoclave, and the TDG concentration was controlled by both the pressure valve and the flow rate. Two observation sections were located upstream and downstream of the channel with a distance of 17 m. The percent saturation of TDG was measured with PT4 TGP detectors made in Canada. The water depth was measured with a steel ruler. Experimental cases were designed with different flow rates and TDG supersaturation levels, as listed in Table 2. 3.2 Experimental results and analysis

The experimental results (Table 2) show that the percent saturation of TDG declines significantly from upstream to downstream, and the maximum delta was 6.5%. A comparison of the dissipation coefficients of each case showed that the dissipation coefficient increased mainly with the increase of the flow rate and the turbulence intensity, as shown in Fig. 3. The dissipation coefficients of the experimental cases varied from 1.38 h−1 to 2.77 h−1, which were greater than those of natural rivers by two orders of magnitude (as shown in Table 1). 4 Formula for dissipation coefficient of

supersaturated TDG 4.1 Proposal of formula

Previous work, as well as the field and experimental results reported herein, indicates that velocity, depth and turbulence are factors affecting the dissipation coefficient of supersaturated TDG. Arguably, the relative extent of

Table 1 Dissipation coefficients of supersaturated TDG in natural rivers and reservoirs

Case No. Location Observation date Flow rate/(m3·s−1) Dissipation coefficient, kT/h−1

1 Yalong River 2007−07−25 5858 0.099

2 Yalong River 2007−07−28 2997 0.097

3 Jinsha River 2008−07−14 7230 0.062

4 Jinsha River 2008−07−15 7280 0.060

5 Yangtze River 2007−07−09 39600 0.014

6 Yangtze River 2007−07−09 39600 0.020

7 Baitiao River 2006−12−28 210 0.563

8 Zouma River 2006−12−28 210 0.650

9 Lancang River 2008−07−28 2352 0.106

10 Lancang River 2008−07−30 3737 0.096

11 Lancang River 2008−07−31 2970 0.101

12 Dadu River 2009−08−05 3275 0.201

13 Tongjiezi Reservoir on Dadu River 2009−08−04 4110 0.094

14 Dachaoshan Reservoir on Lancang River 2008−07−30 3737 0.003

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Fig. 2 Sketch of experimental facility for supersaturated TDG dissipation in an open channel

Table 2 Supersaturated TDG dissipation results in experimental open channel flow

Case No. Flow rate/

(L·s−1)

TDG saturation upstream/

%

TDG saturation

downstream/%

Delta of TDG

saturation/%

Dissipation

coefficient kT/h−1

1 0.33 138.8 135.5 3.2 1.38

2 0.38 133.6 130.5 3.1 1.44

3 0.50 137.1 134.0 3.1 1.57

4 0.52 152.0 147.2 4.8 1.57

5 0.56 137.1 132.6 4.5 2.04

6 0.57 146.3 140.9 5.4 2.12

7 0.64 137.4 134.0 3.4 1.65

8 0.64 139.8 134.7 5.1 2.26

9 0.71 142.3 135.8 6.5 2.77

10 0.69 142.9 138.4 4.5 1.78

Fig. 3 Relation between dissipation coefficient and flow rate in

experiments

the solid-liquid interface may also affect the dissipation rate, which may be denoted by either the wet perimeter (X) or the hydraulic radius (R). We adopt the shear velocity (u*) and Froude number (Fr) to represent turbulence, velocity, and hydraulic radius (R) is introduced to represent the effect of the solid-liquid interface. The relationship is written as follows:

* 1

T ( , , , )k f u Fr H R (2)

A dimensional analysis shows that

*

T 3600 ( )m nu Hk Fr

R R (3)

where (H/R)m represents the correction factor due to the influence of the wall, H represents the depth of water, and m, n and φ are dimensionless constants that can be fitted from field and experimental results.

By employing regression analysis on data in Table 3, the optimal values for m, n and φ are found to be 2.02, 1.73 and 0.0037, and the correlation coefficient of the regression is as high as 0.99. Substituting the values above into Eq. (3), the formula for the dissipation coefficient of supersaturated TDG is expressed as follows:

*

2.02 1.73T 3600 ( )

u Hk Fr

R R (4)

Equation (4) is used to calculate the dissipation

coefficient of the field and experimental cases (shown in Table 3), and the calculated results are listed in Table 4. A comparison of the measured dissipation coefficients with calculated ones, as shown in Fig. 4, shows reasonable agreement, with a maximum relative error of less than 35%.

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Table 3 Field and experimental data

Case No. Observation location Shear velocity/(m·s−1) Froude number Water depth/m Hydraulic radius/% Remarks

1 Yalong River I 0.462 0.344 10.0 9.6 Natural river

2 Yalong River II 0.394 0.336 7.4 7.0 Natural river

3 Jinsha River I 0.304 0.340 10.5 10.1 Natural river

4 Jinsha River II 0.304 0.340 10.5 10.2 Natural river

5 Yangtze River I 0.340 0.192 19.9 19.6 Natural river

6 Yangtze River II 0.308 0.231 16.6 16.1 Natural river

7 Baitiao River 0.354 0.530 3.2 3.1 Natural river

8 Zouma River 0.343 0.547 3.0 2.9 Natural river

9 Lancang River I 0.366 0.298 6.7 6.4 Natural river

10 Lancang River II 0.407 0.314 8.4 8.0 Natural river

11 Lancang River III 0.387 0.304 7.5 7.2 Natural river

12 Dadu River 0.260 0.358 4.2 4.0 Natural river

13 Tongjiezi Reservoir 0.695 0.054 25.2 23.2 Natural reservoir

14 Dachaoshan Reservoir 0.338 0.248 6.9 6.8 Natural reservoir

15 Open cannel I 0.032 0.110 0.040 0.026 Experimental channel

16 Open channel II 0.032 0.125 0.040 0.026 Experimental channel

17 Open channel III 0.034 0.142 0.045 0.028 Experimental channel

18 Open channel IV 0.034 0.146 0.045 0.028 Experimental channel

19 Open channel V 0.035 0.138 0.050 0.030 Experimental channel

20 Open channel VI 0.035 0.141 0.050 0.030 Experimental channel

21 Open channel VII 0.036 0.139 0.055 0.032 Experimental channel

22 Open channel VIII 0.036 0.140 0.055 0.032 Experimental channel

23 Open channel IX 0.037 0.137 0.060 0.033 Experimental channel

24 Open channel X 0.037 0.134 0.060 0.033 Experimental channel

Table 4 Verification of the supersaturated TDG dissipation coefficients

Case No. Observation location Flow rate/

(m3·s−1)

Measured dissipation

coefficients/h−1

Calculated dissipation

coefficients/h−1 Error rate/% Remarks

1 Yalong River I 5.9×103 0.099 0.108 9.2 Natural river

2 Yalong River II 3.0×103 0.097 0.127 30.8 Natural river

3 Jinsha River I 7.2×103 0.062 0.066 6.0 Natural river

4 Jinsha River II 7.3×103 0.060 0.066 9.4 Natural river

5 Yangtze River I 4.0×104 0.014 0.013 −3.7 Natural river

6 Yangtze River II 4.0×104 0.020 0.021 5.5 Natural river

7 Baitiao River 0.2×103 0.563 0.535 −4.9 Natural river

8 Zouma River 0.2×103 0.650 0.589 −9.4 Natural river

9 Lancang River I 2.3×103 0.106 0.101 −4.8 Natural river

10 Lancang River II 3.7×103 0.096 0.101 4.8 Natural river

11 Lancang River III 4.0×103 0.101 0.100 −1.3 Natural river

12 Dadu River 3.3×103 0.201 0.160 −20.6 Natural river

13 Tongjiezi Reservoir 4.1×103 0.094 0.061 −34.8 Natural reservoir

14 Dachaoshan Reservoir 3.7×103 0.003 0.003 1.0 Natural reservoir

15 Open channel I 3.3×10−4 0.876 0.856 −2.3 Experimental channel

(to be continued)

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(continued)

Case No. Observation location Flow rate/

(m3·s−1)

Measured dissipation

coefficients/h−1

Calculated dissipation

coefficients/h−1 Error rate/% Remarks

16 Open channel II 3.8×10−4 1.092 1.070 −2.0 Experimental channel

17 Open channel III 5.0×10−4 1.380 1.399 1.4 Experimental channel

18 Open channel IV 5.2×10−4 1.260 1.457 15.6 Experimental channel

19 Open channel V 5.6×10−4 1.338 1.404 4.9 Experimental channel

20 Open channel VI 5.7×10−4 1.260 1.443 14.5 Experimental channel

21 Open channel VII 6.4×10−4 1.488 1.490 0.1 Experimental channel

22 Open channel VIII 6.4×10−4 1.446 1.506 4.2 Experimental channel

23 Open channel IX 7.1×10−4 1.602 1.533 −4.3 Experimental channel

24 Open channel X 6.9×10−4 1.572 1.466 −6.8 Experimental channel

Fig. 4 Comparison between measured and calculated

supersaturated TDG dissipation coefficients

4.2 Uncertainty analysis of formula The dependent variables in the formula for kT are u*,

R, H, Fr and φ. Normally, the variables u*and Fr can be represented as functions of the acceleration of gravity (g), slope (J), hydraulic radius (R) and average velocity (v). Thus, Eq. (2) can be rewritten as kT=f(H, R, v, g, J). Uncertainty in the formula arises due to instrument accuracy when measuring the average depth (δH), hydraulic radius (δR) and average velocity (δv). The combined standard uncertainty (

Tk ) for kT is as a function of (H, R, v), which is expressed as follows:

T

2 2 2T T TR( ) ( ) ( )k H v

k k k

H R v

(5)

The combined standard uncertainty for kT is

computed using Eq. (5) in each field and experimental case. The results are listed in Table 5 and presented in Fig. 5 using error bars. The ratio of uncertainty and measured dissipation coefficient was calculated to be less than 0.25, which indicates that the instrument accuracy has little impact on the dissipation coefficients. All the analyses above demonstrated that the formula in Eq. (4) is sufficiently accurate to estimate the dissipation coefficient of supersaturated TDG universally.

4.3 Validity discussion of the formula The Froude number in Eq. (4) is a key parameter for

estimating the dissipation coefficient of TDG. It is the ratio of the inertia force on a fluid unit to the weight of the unit and it is of important effects on flows with free surfaces since gravity principally decides the type of flow. In large scale reservoirs, inertia force gradually declines with water depth raising and velocity decreasing, resulting in a quite small Froude number. Usually, in such case, the stably stratified phenomena appear in reservoirs, such as obvious distribution of temperature, hydrodynamics and constituent concentration.

Densimetric Froude number, that is the ratio of the inertia force on a fluid unit to the buoyancy force, has been proposed to discriminate the stability of stratification in a reservoir. Its definition involves overall physical and hydrodynamic characteristics of the reservoir as following:

1/ 2

0

( )

uFr

gd

(6)

where u is the mean velocity, u=Q/bd; d denotes the mean depth; b is the mean width; Q represents the flow rate; Δρ is the delta of density in range of the depth；and ρ0 indicates reference density.

Previous research shows that the stratification starts to appear in a reservoir when the densimetric Froude number is less than 1.0 and the stable stratification will be formed in a reservoir when densimetric Froude number is less than 0.1. The appearance of vertical stratification is against the assumption of the 1-D longitudinal model of TDG dissipation, which the stratification of hydrodynamic characteristics and TDG concentrations are not significant. From this aspect, it is reasonable to limit the validity of Eq. (4) for the dissipation coefficient of supersaturated TDG in rivers or reservoirs which the densimetric Froude number is greater than 1.0.

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Table 5 Uncertainty analysis of formula for dissipation coefficient

Case No. Observation location Flow rate/(m3·s−1) Measured dissipation coefficients/h−1 Standard uncertainty/h−1

1 Yalong river I 5858 0.099 0.007

2 Yalong river II 2997 0.097 0.010

3 Jinsha river I 7230 0.062 0.004

4 Jinsha river II 7280 0.060 0.004

5 Yangtze river I 39600 0.014 0.001

6 Yangtze river II 39600 0.020 0.001

7 Baitiao river 210 0.563 0.073

8 Zouma river 210 0.650 0.085

9 Lancang river I 2352 0.106 0.010

10 Lancang river II 3737 0.096 0.008

11 Lancang river III 2970 0.101 0.009

12 Dadu river 3275 0.201 0.020

13 Tongjiezi reservoir 4110 0.094 0.006

14 Dachaoshan reservoir 3737 0.003 0.001

15 Open channel I 3.3×10−4 0.876 0.205

16 Open channel II 3.8×10−4 1.092 0.258

17 Open channel III 5.0×10−4 1.380 0.300

18 Open channel IV 5.2×10−4 1.260 0.312

19 Open channel V 5.6×10−4 1.338 0.271

20 Open channel VI 5.7×10−4 1.260 0.278

21 Open channel VII 6.4×10−4 1.488 0.261

22 Open channel VIII 6.4×10−4 1.446 0.264

23 Open channel IX 7.1×10−4 1.602 0.246

24 Open channel X 6.9×10−4 1.572 0.236

Fig. 5 Standard uncertainty of each case: (a) Results of field cases; (b) Results of experimental cases

5 Application of formula for dissipation

coefficient

The formula for the dissipation coefficient is

implemented in the prediction of the TDG dissipation process over a reach of Jinsha River. The reach starts at the Yajin (Yalong River and Jinsha River) confluence and ends at Longjie ferry, with a distance of 100 km. The transportation process of supersaturated TDG is

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predicted by the 1-D longitudinal dissipation model. The hydrodynamic characteristics are simulated by Eq. (7), the essence of which is the calculation of the water surface. TDG dissipation is modeled by Eq. (1).

)11

(2

)11

(2 2

12

2

221

2

1

iiii

iiKK

sQ

AAg

QZZ (7)

iii ARn

K 3

21

(8)

where Zi and Zi+1 are the water surface elevations at the upstream and downstream sections, respectively, Q is the flow rate in the reach, A is the area of the cross section, K is the average discharge modulus of the reach, n is the roughness, and R is the hydraulic radius.

The actual topographical data are used in the calculation. The inflow rate is defined as the actual flow rate on July 15, 2008, and the TDG percent saturation of inflow is defined as the measured value at that time.

The simulated result of the water surface elevation is shown in Fig. 6. This figure shows that the hydraulic radius varies in the range of 13.3 m to 6.0 m, the average depth varies in the range of 14.1 m to 6.0 m, the average velocity varies in the range of 5.2 m/s to 6.0 m/s, and the Froude number varies in the range of 0.65 to 0.19.

Fig. 6 Water surface elevation along calculation reach

The dissipation coefficients are calculated by

substituting the hydrodynamic parameters into Eq. (4). The dissipation coefficient fluctuates as a consequence, with a maximum value of 0.25 h−1 and a minimum value of 0.03 h−1. The average is 0.07 h−1, which agrees well with field results of 0.60 h−1 and 0.62 h−1, as shown in Table 1 for Case 3 and Case 4.

The longitudinal transportation of supersaturated TDG is modeled with Eq. (1). The dissipation process is drawn in Fig. 7, which shows that the TDG supersaturation level declines gradually. Within the reach that is approximately 100 km long, the maximum delta between the upstream and downstream TDG levels is

7.3%, while the residence time of TDG from upstream to downstream is approximately 10 h. Thus, the supersaturated TDG dissipates fairly slowly in nature, which warns us that the problem of supersaturated TDG must be given much more attention.

Fig. 7 Comparison between TDG according to field

observation and calculation along reach

Figure 7 reveals that the calculation results are in

excellent agreement with the field observation data, with a maximum difference of 15%. Thus, this practical application demonstrates that the formula for the dissipation coefficient of supersaturated TDG can be employed to predict the 1-D longitudinal TDG dissipation process accurately. 6 Conclusions

1) The turbulence intensity (represented by the flow rate and velocity) and hydro-pressure (represented by the water depth) are found to affect the dissipation process of TDG significantly. Additionally, experiments show that increasing the area of the solid-liquid interface in water promotes the dissipation of TDG.

2) A formula for predicting the dissipation coefficient of supersaturated TDG, in which the factors of friction velocity, the hydraulic radius, the water depth and the Froude number are involved, is developed with the use of dimensional analysis and regression with the measured data.

3) The developed 1-D longitudinal dissipation model is employed to calculate an actual dissipation process of supersaturated TDG in a field observed reach in Jinasha River. Good agreement is found between data of calculation and field observation, both in dissipation coefficients and dissipation processes.

4) Additional research is required to perform in predicting the long-term transportation and dissipation of supersaturated TDG in large scale reservoirs by a multidimensional model.

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(Edited by HE Yun-bin)