Model for Total Dissolved Gas Supersaturation from Plunging Jets in High Dams

Lu, Jingying; Li, Ran; Ma, Qian; Feng, Jingjie; Xu, Weilin; Zhang, Faxing; Tian, Zhong

doi:10.1061/(ASCE)HY.1943-7900.0001550

Open access

Technical Papers

Oct 31, 2018

Model for Total Dissolved Gas Supersaturation from Plunging Jets in High Dams

Authors: Jingying Lu [email protected], Ran Li [email protected], Qian Ma, Ph.D. [email protected], Jingjie Feng [email protected], Weilin Xu [email protected], Faxing Zhang [email protected], and Zhong Tian [email protected]Author Affiliations

Publication: Journal of Hydraulic Engineering

Volume 145, Issue 1

https://doi.org/10.1061/(ASCE)HY.1943-7900.0001550

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Abstract

The total dissolved gas (TDG) supersaturation that results from dam spillage may cause adverse effects, including increases in the risk of gas-bubble disease and mortality in fish. The accurate prediction of TDG levels is necessary in the exploration of measures for ameliorating the effects of TDG supersaturation. Based on an analysis of the mechanisms of hydropower projects with a plunging jet that produces high TDG levels, the process of TDG generation is divided into three stages. In Stage 1, TDG levels return to normal during jet spillage in air; in Stage 2, gas is dissolved in the stilling basin under high pressure; and in Stage 3, the TDG is abruptly released at the outlet of the stilling basin. According to previous research on Stage 1, the TDG level of water entering stilling basins can reach 100%. Experiments were carried out to estimate the TDG levels in Stage 2 under different pressures and retention times, and these experiments indicated that a TDG level above equilibrium saturation (

Δ G_{0}

) displays a linear relationship with the average pressure (

Δ P

) and a negative exponential relationship with retention time (

t_{R}

). Experiments were also conducted using physical models of the Songta and Yangfanggou dam projects in China to develop a method for estimating the retention time in stilling basins. The resulting formula for estimating the retention time is a function of the water depth in the stilling basin (

h_{k}

), length of the stilling basin (

l

), distance between the toe of the dam and impact point of the jet (

l_{0}

), and the dimensionless number at the stilling basin outlet

λ

. For Stage 3, in which the abrupt release of TDG occurs, field measurements were used to determine the values of the parameters used in the abrupt release expression contained in the model. By combining the results for the three stages, a predictive model of TDG levels was obtained. TDG observations collected at six different hydropower projects in China were used for validation. Substantial agreement between predictions and measurements was found. This work may provide a scientific basis for the production of precise predictive models of TDG levels, and it has considerable application value in assessing the effects of TDG and minimizing the risks posed by elevated TDG levels to aquatic life.

Introduction

When dam spillage occurs, large amounts of air are entrained into the discharged water. Under the high pressures that occur in stilling basins, much air is dissolved, causing total dissolved gas (TDG) supersaturation relative to atmospheric pressure. Many studies have demonstrated that gas-bubble disease or mortality may occur in fish as a result of TDG supersaturation (Weitkamp and Katz 1980; Dawley and Ebel 1975; Geist et al. 2013; Weitkamp et al. 2003). The mortality rate of fish increases rapidly under long exposures to TDG supersaturation, with negative effects on ecosystems and aquaculture development. Furthermore, adequately managing TDG requires additional operational constraints on hydroelectric plants operating to meet the demands of regional power systems (Fidler 1996; Water Resources Protection Bureau of Yangtze River 1983; Hu et al. 2014).

To take full advantage of hydroenergy resources, many dams with heights exceeding 200 m, which might easily cause TDG supersaturation, have been built or put into operation. Energy dissipation of high dams with jet flow has always been adopted, owing to its high efficiency of energy dissipation and constructing conditions. Meanwhile, plunging jets could entrained large amounts of air (Diao et al. 2003). Accurate methods for evaluating the detrimental effects of TDG supersaturation are critical for studies that attempt to identify measures to mitigate the effects of TDG supersaturation.

Predictive models of TDG levels can be divided into three categories: empirical formulas, numerical models, and mechanical models. Four empirical formulations with different forms that are based on measurements in the Columbia River and its tributaries have been proposed by USACE (1997). Columbia Basin Research (2000) adopted these empirical formulations, which depend on the spilling discharge or the tailwater depth, to predict TDG saturation in version 1.6 of the Columbia River Salmon Passage (CRiSP) model. Because all of the empirical formulas are based on simple fits to data for specific projects, they have limited generality.

With respect to numerical models, Orlins and Gulliver (2000) first proposed a two-dimensional (2D) one-phase turbulent model that depends on bubble and air–water surface transfer. Urban et al. (2008) established an improved TDG generation model after analyzing gas holdup as a function of flow characteristics by area. A three-dimensional (3D) model was later developed by Politano et al. (2009, 2011, 2012); in their model, variations in bubble size are taken into account using a bubble density equation. In some cases, the extreme complexity of flow fields downstream of dams and the lack of understanding of the related bubble mass transfer has limited the development of numerical models.

As a necessary supplement to prediction methods, mechanical models play an important role in the prediction of TDG levels. Roesner et al. (1972) incorporated the mean hydrostatic pressure and dam structural characteristics into a predictive model of TDG levels. Johnson (1984) developed a predictive model of TDG levels using field data collected at 24 different structures, and they obtained the mass transfer coefficient through performing fits to data. The effective water depth and gas transfer efficiency were introduced into a predictive model of TDG levels by Hibbs and Gulliver (1997). Geldert et al. (1998) considered the transfer process between bubbles and at the water–air interface in the estimation of TDG levels. Li et al. (2009) divided the process by which elevated TDG levels are generated into two stages and presented a mechanical model of TDG. In Stage 1, large amounts of air are dissolved in the stilling basin under high pressure. In Stage 2, the TDG is abruptly released at the outlet of the stilling basin. The hydropressure and water depth at the outlet in the stilling basin are used as input variables in the mechanical model. Politano et al. (2017) proposed a TDG generation model based on bubble trajectory assumptions. In addition, a reduced-order TDG uptake equation developed by Witt et al. (2017b) mainly considered the entrainment of TDG into water in the jet region.

Few of the previous mechanical models of TDG generation are focused on the production of elevated TDG levels from plunging jet. Some mechanical models are suitable for spillage from high dams but do not consider the effect of the jet process in air (Li et al. 2009). Moreover, the influence of flow retention time in the stilling basin is neglected, and the estimation of pressure and parameter values results in large uncertainties. This study analyzes the processes by which elevated TDG levels are produced, and laboratory experiments were conducted to investigate the effects of pressure and retention time in stilling basins on the production of elevated TDG levels. A more accurate predictive model of TDG supersaturation is proposed.

Analysis of TDG Generation Processes

The mechanical model of TDG developed by Li et al. (2009) accounts for two stages in the production of elevated TDG levels. However, changes in TDG saturation in air caused by jets is neglected. Thus, the process by which TDG supersaturation occurs should be divided into three stages, as shown in Fig. 1. In Stage 1, the TDG level changes in air during dam spillage with a plunging jet by different release structures, like orifices, discharge tunnels, and spillways. In Stage 2, oversaturation occurs under high pressure in the stilling basin. In Stage 3, the TDG is rapidly released at the outlet of the stilling basin, where substantial decreases in pressure and water depth occur.

With respect to the TDG change in air that occurs in Stage 1, the TDG level in the spilling jet when the jet enters the stilling basin is always assumed to be 100% (Li et al. 2009). According to the results of field measurements, high TDG levels often occur in forebays due to the cumulative effects of cascade hydropower stations (Qu et al. 2011; Witt et al. 2017a). To determine whether the forebay TDG level has an impact on the production of elevated TDG levels, Ma (2016) conducted a series of experiments to investigate the relationship between the TDG level and conditions of jets when they enter stilling basins. In addition, a model [Eq. (1)] is presented to calculate the TDG level before entering the stilling basin. The relevant variables are identified in Fig. 2

G_{1} = G_{f} - (G_{f} - 100) [1 - \exp (- 1.12 [\sqrt{(v_{0}^{2} \sin^{2} α + 2 g H_{P})} + v_{0} \sin α] / g)]

(1)

where

G_{f}

= degree of saturation within the forebay and is expressed as a percentage equivalent to the ratio of measured TDG pressure to ambient atmospheric pressure;

H_{P}

= falling height (m), which is equal to the difference between the outlet elevation and water level downstream;

α

= jet angle at the outlet and is positive when upward;

g

= gravitational acceleration (

m^{2} / s

); and

v_{0}

= outlet velocity (m/s), which can be calculated using Eq. (2)

v_{0} = φ_{α} \sqrt{2 g H_{u}}

(2)

where

φ_{α}

= velocity coefficient, which takes values is in the range of 0.75–0.95; and

H_{u}

= water head between the water level in the reservoir and outlet elevation of the release structure (m).

Fig. 2. Enlarged view of Stage 1 in the production of elevated TDG levels.

The results of Ma (2016) demonstrate that the inflow TDG levels can approach 100% for tall dams and high flow rates. Based on the research of Ma (2016), the prediction of elevated TDG levels in Stages 2 and 3 are the focus of this study.

Development of a Model for Predicting TDG Levels

TDG Levels under Different Retention Times and High Pressures

It is assumed that the effects of mass transfer at the free surface are weak compared with the effect of mass transfer inside the stilling basin. Therefore, to quantitatively analyze the TDG levels achieved at various retention times of aerated flows and pressures inside the stilling basin, an experimental study was conducted at the State Key Laboratory of Hydraulics and Mountain River Engineering in Chengdu, China.

Experiment Description of TDG Levels under Different Retention Times and High Pressures

The experimental setup shown in Fig. 3 includes a water pump, air compressor, and set of steel pipes with eight outlets. Outlets 1–8, which are distributed at different points along the pipes, were used to collect TDG-supersaturated water. In the experiment, air and water were pumped and mixed under high pressure in the steel pipes so that TDG-supersaturated water could be generated. The different outlets represent different distances from the opening of the steel pipes and reflect different retention times of aerated flow water under high pressure.

The TDG level of the collected water is measured by a PT4 Tracker (Point Four Systems, Coquitlam, Canada).

Expression of TDG Saturation Level as a Function of Retention Time and Pressure

Fig. 4 shows the measured TDG saturation level at four different outlets under different pressures. A clear linear relationship between TDG saturation and water pressure was obtained. These results demonstrate that gas dissolution increases as pressure increases at a given temperature, consistent with the conclusion of Jiang et al. (2008).

Fig. 4. TDG saturation level varying with pressure changes.

The relationship between the measured TDG saturation and the retention time at various pressures is shown in Fig. 5. The results indicate that the retention time of aerated water at a given pressure has a strong effect on TDG saturation. At the same time, the negative exponential tendency of TDG saturation as retention time increases demonstrates that first-order kinetics are appropriate for this laboratory experiment.

Fig. 5. Changes in TDG saturation with the retention time of water in the steel pipes.

According to the experimental results, the TDG level above equilibrium saturation (

Δ G_{0} = G_{s 0} - G_{1}

) is linearly related to the average pressure (

Δ P

) in the stilling basin and displays a negative exponential relationship with the retention time (

t_{R}

), as reflected by the following equations:

\frac{Δ G_{0}}{G_{e q}} \propto \frac{Δ P}{P_{0}}

(3)

\frac{Δ G_{0}}{G_{e q}} \propto [1 - exp(- a_{1} t_{R})]

(4)

where

Δ G_{0}

= degree of oversaturation (%);

G_{s 0}

= TDG level under average pressure in the stilling basin (%);

P_{0}

= local atmospheric pressure (m

H_{2} O

);

t_{R}

= retention time of aerated water in the stilling basin (s);

a_{1}

= correction coefficient for the retention time that is determined by fits to experimental data (

s^{- 1}

);

G_{1}

= TDG level before the jet enters the stilling basin (%); and

G_{e q}

= equilibrium saturation under atmospheric pressure (%). Both

G_{1}

(%) and

G_{e q}

(%) are assumed to be 100%, as mentioned previously. Here,

Δ P

(m

H_{2} O

) is the average pressure in the stilling basin, which consists of static pressure and hydrodynamic pressure. Hydrodynamic pressure

Δ P_{m}

(m

H_{2} O

) is calculated by the formula of Liu (2010) as follows:

Δ P_{m} = (0.95 \sin^{2} β + 0.175) \frac{q^{\frac{2}{3}} H^{\frac{1}{3}}}{g^{\frac{1}{3}} h_{k}^{\frac{1}{3}}}

(5)

where

β

= possible impact of the water entry angle of the jet;

H

= difference between the water level upstream from the dam and water level downstream from the dam (m);

q

= unit-width flow rate when water enters the stilling basin (

m^{3} / s

); and

h_{k}

= water depth in the stilling basin (m).

According to the multivariate nonlinear regression method,

a_{1} = 0.08

. Moreover, the following expression is obtained:

G_{s 0} = 100 {1 + \frac{Δ P}{P_{0}} [1 - \exp (- 0.08 t_{R})]}

(6)

The coefficient of determination,

R^{2}

, is 0.96.

Investigation of the Retention Time in Stilling Basins

The retention time in Eq. (6) has been shown to be a key factor in the production of TDG supersaturation. No investigations of the retention time of aerated flow in stilling basins have been identified. Because of the complexity of the flow field in stilling basins (Fig. 6), the study of retention time is a challenging aspect of assessing the production of elevated TDG levels.

An experimental method similar to that described by Levenspiel (1962) was applied to assess the effects of the retention time in stilling basins using physical models of several high dams with a plunging jet. The effects of both the operational parameters of each dam (i.e., flow rate and water head) and their structural characteristics (i.e., type of energy dissipation used and size of the stilling basin) were investigated. Moreover, other parameters such as the water level downstream of each dam were also monitored.

Physical Experiment Description of the Retention Time in Stilling Basins

To examine the effects of the retention time of aerated flow in stilling basins, hydraulic tests were carried out using physical models of the Yangfanggou and Songta hydropower projects. The features of these projects are listed in Table 1. The scale of the model of the Yangfanggou project is 1:50, whereas the scale of the model of the Songta project is 1:80. Photographs of these physical models are shown in Fig. 7.

Table 1. Features of the Yangfanggou and Songta hydropower projects

Number	Project	River	Dam type	Maximum dam height (m)	Release structures
1	Yangfanggou	Yalongjiang	Concrete hyperbolic arch dam	155	Surface and middle orifices
2	Songta	Nujiang	Concrete hyperbolic arch dam	295	Surface and bottom orifices

Fig. 7. Laboratory experiments for (a) Yangfanggou; and (b) Songta hydropower projects.

A tracing method was used to track the flow trajectory during flood discharges. When dam spillage occurred, a pulse of a high-salinity solution was introduced at the orifice outlet. The saline solution was carried into the stilling basin by the spilling water and moved to the outlet after migrating within the stilling basin over an extended period. A conductivity meter was used to monitor changes in conductivity at the outlet of the stilling basin. The change in the electrical conductivity reflects the retention time of aerated flow in the stilling basin. The weighted average of duration and conductivity was adopted to obtain the final retention time.

Estimation Formula for Retention Time

The conductivity at the outlet as time progresses approximately follows a Poisson distribution, as shown in Fig. 8.

Fig. 8. Electrical conductivity versus time in different spillage cases: (a) middle orifices; (b) surface orifices of the Yangfanggou project; and (c) surface orifices of the Songta project.

Based on a mechanical analysis, the retention time can be expressed as follows:

g (t_{R}, l, l_{0}, v_{2}, h_{k}, g) = 0

(7)

where

t_{R}

= retention time of aerated flow (s);

l

= length of the stilling basin (m);

l_{0}

= distance between the dam site and the impact point of the jet (m);

h_{k}

= water depth in the stilling basin (m);

g

= acceleration due to gravity (

m / s^{2}

); and

v_{2}

= water velocity at the outlet of the stilling basin (m/s), which is calculated

v_{2} = \frac{Q}{h_{r} B}

(8)

where

Q

= spilling discharge (

m^{3} / s

);

h_{r}

= water height above the subsidiary dam (m); and

B

= width of outlet (m).

Using dimensional analysis and the

π

theorem, a nondimensional form of this equation is obtained

t_{R} = \sqrt{\frac{h_{k}}{g}} f (\frac{l_{0}}{l}, \frac{v_{2}}{\sqrt{g h_{k}}})

(9)

Furthermore, considering the possible impact of the water entry angle of the jet

β

,

\cot β

is added

λ = \cot β \frac{v_{2}}{\sqrt{g h_{k}}}

(10)

where

λ

= dimensionless number at the stilling basin outlet. Combining Eqs. (9) and (10), the final expression for

t_{R}

is obtained

t_{R} = \sqrt{\frac{h_{k}}{g}} f (\frac{l_{0}}{l}, λ)

(11)

The experimental results indicate that there is a certain linear correlation between the logarithmic values of the water retention time and the parameters in Eq. (11) (Fig. 9). Under the same dam spillage method, the logarithmic value of

t_{R}

increases as the logarithmic value of

l_{0} / l

decreases, whereas

λ

increases.

Fig. 9. Relationship between the retention time and relative spill parameters of different projects: (a) surface orifices; (b) middle orifices of the Songta project; (c) surface orifices; and (d) middle orifices of the Yangfanggou project.

Therefore, the relationship between

t_{R}

and the relevant parameters can be written

\ln (t_{R}) = \ln (b_{0}) + b_{1} \ln (\frac{l_{0}}{l}) + b_{2} \ln (λ) +ln (\sqrt{\frac{h_{k}}{g}})

(12)

Using the multiple linear regression method, the undetermined coefficients in Eq. (12) are determined to be

b_{0} = 27.73

,

b_{1} = - 1.82

, and

b_{2} = 0.49

. Therefore, Eq. (12) can be written

t_{R} = 27.73 {(λ)}^{0.49} {(\frac{h_{k}}{g})}^{0.5} {(\frac{l_{0}}{l})}^{- 1.82}

(13)

where

t_{R}

= retention time of aerated flow in the stilling basin (s);

l

= length of the stilling basin (m);

h_{k}

= water depth in the stilling basin (m);

g

= acceleration due to gravity (

m / s^{2}

);

λ

= dimensionless number at the stilling basin outlet, which is obtained using Eq. (10); and

l_{0}

= horizontal distance between the downstream toe of the dam and the impact point of the jet (m). According to Liu (2010), this quantity can be computed as follows:

l_{0} = \frac{v_{0}^{2} \cos α \sin α}{g} {\pm 1 + \sqrt{1 + \frac{2 g (H_{p} + 0.5 h_{k} \cos α)}{v_{0}^{2} \sin^{2} α}}}

(14)

where

v_{0}

= water velocity at the outlet section of the release structure (m/s);

H_{p}

= water head between the outlet elevation of the release structure and water elevation in the stilling basin (m);

h_{k}

= water depth in the stilling basin (m); and

α

= outflow angle of the discharge flow.

If

α

in Eq. (14) is zero, the jet exits the release structure horizontally, and

l_{0}

can be rewritten

l_{0} = \frac{v_{0}}{2 g} \sqrt{2 g (H_{p} + 0.5 h_{k})}

(15)

Verification of the Retention Time Formula

Two physical experiments based on the Yebatan and Huangdeng hydropower projects were used to verify the retention time formula. In these verification experiments, the same tracing and evaluated method referred to previously was applied. A total of 91 test cases were evaluated, and the characteristics of the Yebatan and Huangdeng projects are presented in Table 2.

Table 2. Features of the Yebatan and Huangdeng projects

Number	Project	Model scale	River	Dam type	Maximum dam height (m)	Release structures
1	Huangdeng	1:50	Jinshajiang	Concrete hyperbolic arch dam	217	Surface and bottom discharge orifices
2	Yebatan	1:80	Langcangjiang	Concrete gravity dam	202	Overflowing and bottom discharge orifices

The retention times for Yebatan are shorter than those for Huangdeng because of the larger

l_{0} / l

and smaller

λ

. In the Huangdeng experiment, the retention time under surface orifice discharge is approximately 600 s, whereas the retention time is from 180 to 380 s under bottom orifice discharge. This difference is mainly caused by the gap in trajectory distance. Furthermore, the maximum difference between the calculated and experimental results is 52.1 s under the bottom orifices of Huangdeng discharge, whereas the experimental retention time is 229.81 s. The average relative error is 8.54%. The correlation coefficient (

R^{2}

) between the simulations and measurements is 0.98 (Fig. 10); thus, the error of this predictive model of TDG levels is acceptable, and the model can be used to predict the retention time of aerated flow in stilling basins.

Fig. 10. Calculated retention time versus experimental results.

Abrupt Release of TDG at the Outlet of the Stilling Basin

Analysis of the Processes That Produce Abrupt Releases of TDG

In Stage 3 of the production of TDG supersaturation, the abrupt release of TDG occurs at the outlet due to the sharp decreases in the pressure and water depth (Fig. 1). A predictive model for this stage was developed based on field observations of TDG supersaturation in a series of hydropower projects.

During the abrupt release of TDG, the final TDG level

G_{s}

is related to the generated TDG saturation level under average pressure in the stilling basin

G_{s 0}

, water depth above the outlet

h_{r}

, and velocity at the outlet

v_{2}

. According to the research of TDG dissipation by the US Army Corps of Engineers, the dissipation of oversaturated TDG follows a first-order kinetics equation

G_{s 0} - G_{s} = (G_{s 0} - G_{e q}) \cdot \exp [- f (h_{r})]

(16)

Including the effects of

v_{2}

on Stage 3 in Eq. (16), the following equation is obtained:

G_{s 0} - G_{s} = (G_{s 0} - G_{e q}) \cdot \exp [- f (\frac{h_{r}}{v_{2}})]

(17)

where

G_{e q}

= percentage of the equilibrium TDG concentration at the local atmospheric pressure.

Parameter Calibration Using Field Observations of TDG Supersaturation

The parameters involved in assessing the abrupt release of TDG supersaturation for hydropower projects were calibrated using field observations made by Sichuan University since 2006. The detailed locations of the observed sections and the TDG measurements made at these projects with a plunging jet are given in Table 3. It is supposed that the spill and power flow is completely mixed at the observed sections.

Table 3. Measurements of TDG at high-dam hydropower projects

Number	Project	Release structure	Discharge rate, $Q_{s}$ ( $m^{3} / s$ )	Power flow, $Q_{p}$ ( $m^{3} / s$ )	Observed TDG, $G$ (%)	Distance downstream from the dam where observations were made (km)
1	Xiluodu	Bottom orifice	5,397	6,848	119.00	4.2
2	Dagangshan	Discharge tunnel	166	1,222	110.00	1.0
3	Dagangshan	Discharge tunnel	1,320	451	116.00	1.0
4	Dagangshan	Discharge tunnel	1,180	919	115.00	1.0
5	Ertan	Middle orifice	2,044	1,726	123.90	2.0
6	Manwan	Discharge tunnel	540	1,968	121.00	4.0
7	Zipingpu	Discharge tunnel	170	0	111.17	0.5
8	Zipingpu	Discharge tunnel	330	0	111.97	0.5
9	Zipingpu	Discharge tunnel	210	0	111.27	0.5
10	Xiaowan	Surface orifice	1,288	1,604.00	109.50	1.5

Based on the operational conditions and structural characteristics, Eq. (6) is used to calculate

G_{s 0}

in Eq. (17) in each spillage case. Meanwhile, to consider the effects of power flow,

G_{s}

in Eq. (17) can be modeled by Eq. (18), based on the TDG levels measured in the field (

G

)

G_{s} = \frac{Q_{s} - Q_{p}}{Q_{s}} G - \frac{Q_{p}}{Q_{s}} G_{f}

(18)

where

Q_{s}

= spilling flow rate (

m^{3} / s

);

Q_{p}

= power flow rate (

m^{3} / s

); and

G_{f}

= forebay TDG level (%).

Thus, the change in the TDG level (

Δ G = G_{s 0} - G_{s}

) in Stage 3 is obtained. In addition, the relationships between

Δ G

and

G_{s 0}

, as well as

h_{r} / v_{2}

, are shown in Fig. 11.

Fig. 11. Relationships between (a) $Δ G$ and $G_{s 0}$ ; and (b) $Δ G / (G_{s 0} - G_{e q})$ and $h_{r} / v_{2}$ .

Fig. 11(a) shows the relationship achieved for the change level accomplished by abrupt release. It can be seen that

Δ G

clearly increases linearly as

G_{s 0}

increases. Fig. 11(b) displays a clear decrease in the dimensionless factor

Δ G / (G_{s 0} - G_{e q})

with

h_{r} / v_{2}

. Therefore, Eq. (17) can be expressed

Δ G = (G_{s 0} - G_{e q}) \cdot a_{3} \cdot \exp [- b_{3} (\frac{h_{r}}{v_{2}})]

(19)

where

b_{3}

and

a_{3}

= correction coefficients.

In this abrupt release model, the correction coefficient

a_{3}

is determined to be 0.91, whereas the TDG dissipation coefficient

b_{3}

at the outlet of the stilling basin is determined to be 0.03. The coefficient of determination (

R^{2}

) obtained using the calibration data is 0.84, reflecting strong agreement between the simulation and the measurements. After introducing the coefficients, the final expression of this predictive model for the abrupt release process is obtained as follows:

Δ G = (G_{s 0} - G_{e q}) \cdot 0.91 \cdot \exp [- 0.03 (\frac{h_{r}}{v_{2}})]

(20)

Predictive Model of TDG Supersaturation for High-Dam Projects

Combining the analysis of these three stages, the final quantitative relationship used to predict TDG supersaturation is presented, and variables can be seen in Fig. 12

G = \frac{G_{s} Q_{s} + G_{f} Q_{p}}{Q_{s} + Q_{p}} G_{s} = G_{s 0} - (G_{s 0} - G_{e q}) \cdot 0.91 \cdot \exp [- 0.03 (\frac{h_{r}}{v_{2}})] G_{s 0} = 100 {1 + \frac{Δ P}{P_{0}} [1 - \exp (- 0.08 t_{R})]} t_{R} = 27.73 {(λ)}^{0.49} {(\frac{h_{k}}{g})}^{0.5} {(\frac{l_{0}}{l})}^{- 1.82}

(21)

Model Validation and Error Analysis

Model Validation

The modified predictive model of TDG levels was validated using field observations conducted by Sichuan University in recent years. The discharge flow by these projects are all plunging jet, and the features of the projects are given in Table 4. The simulation results were compared with the field measurements, as shown in Fig. 13. The maximum difference in the TDG levels between the computed results and field data (which corresponds to the spillway discharge in the Zipingpu project) is approximately 7%, and the average absolute error is 2.34%. Under these 13 sets of operating conditions, 12 of the model predictions are within

\pm 5 %

of the observations, indicating good agreement between the measurements and the predictions.

Table 4. Characteristics of the hydropower projects for model validation

Project	Release structure	Discharge rate ( $m^{3} / s$ )	Power flow ( $m^{3} / s$ )	Water head (m)	Observed forebay TDG (%)	Observed TDG downstream from the dam (%)
Xiluodu	Bottom orifice	5,362	5,928	191.50	100.00	119.00
Dagangshan(a)	Discharge tunnel	1,160	1,148	170.02	108.00	117.00
Dagangshan(b)	Discharge tunnel	416	1,224	170.67	108.00	115.00
Dagangshan(c)	Discharge tunnel	2,390	312	169.51	108.00	123.00
Ertan(a)	Middle orifice	2,054	1,815	178.61	107.00	123.30
Ertan(b)	Middle orifice	2,026	1,732	176.52	107.00	122.80
Manwan(a)	Surface orifice	1,810	1,930	88.98	106.00	115.00
Manwan(b)	Discharge tunnel	880	2,034	87.22	105.00	121.00
Zipingpu(a)	Discharge tunnel	170	0	121.27	100.00	107.27
Zipingpu(b)	Discharge tunnel	170	0	121.28	100.00	115.17
Zipingpu(c)	Discharge tunnel	193	0	119.90	100.00	111.97
Zipingpu(d)	Spillway	210	0	119.36	107.00	131.00
Xiaowan	Surface orifice	2,097	1,493	228.29	100.50	108.90

Fig. 13. Comparison between computed and observed TDG saturation levels.

Error and Uncertainty Analysis

In this model, the pressure in the stilling basin consists of the hydrodynamic pressure and the static pressure. The hydrodynamic pressure was determined using an empirical formula proposed by Liu (2010), which uses the structural characteristics of hydropower projects. However, for the large spill rates associated with high dams, this empirical formula may not display satisfactory accuracy in calculating pressures. Therefore, to avoid large errors in the prediction of TDG levels, more accurate physical models should be used in order to determine the pressure distribution as accurately as possible.

In estimating the retention time of the aerated flow in stilling basins, multiple linear regression was used to develop the relationship between the retention time and the relevant hydrodynamic variables. Due to the extremely complex nature of the flow regime in stilling basins, regression of the experimental data may lead to errors.

Field observations were used directly in the model calibration. The accuracy of this calibration is limited by the observational conditions; some observational locations are located far from the dams. All field observation data were based on the hypothesis that powerhouse and spill flows are completely mixed. In reality, it is possible that the mixing level of powerhouse and spill flows is not appropriate. Therefore, the representation of the observational data is related to the model error and affects the application of this model. Furthermore, the number of observed cases is not large enough to show validation comprehensive, which may cause uncertainty.

Conclusions and Future Work

This work summarized the development of a predictive model for TDG supersaturation downstream of high-dam hydropower projects with plunging jet flow. A three-stage generation process was proposed. In Stage 1, the TDG levels change in air when dam spillage occurs; in Stage 2, oversaturation develops under high pressure in the stilling basin; and, in Stage 3, the TDG is rapidly released at outlet of the stilling basin owing to the substantial decreases in pressure and water depth that occur there. The effects of dam spillage on the initial TDG level as the water enters the stilling basin were considered in this model. Meanwhile, experiments that assessed TDG levels under different pressures and retention times were performed, and experiments on retention time in the stilling basin were conducted to produce an estimation formula. Field measurements were adopted to determine the values of the parameters that control the abrupt release of TDG at the outlet of the stilling basin. Experimental investigations of the stages of TDG generation were used to produce a mechanical model of TDG generation. This research improves the generality of mechanical models and provides a critical tool for use in explorations of measures that can be used to mitigate the adverse environmental impacts caused by TDG supersaturation.

The effects of the abrupt release of TDG at the outlets of stilling basins should be related to the structural characteristics of the secondary dam to further improve the TDG mechanical model. Moreover, the mixing level between the spilling flow and power flow of field data collected point should also be confirmed. This model is based on the water characteristics caused by a plunging jet. Therefore, different spill types need to be studied further. To improve the predictive model of TDG levels, additional work on the three-dimensional structure of aerated flow in stilling basins, the distributions of bubbles in stilling basins, and the characteristics of mass transfer at the free surface and air–bubble interfaces are needed.

Acknowledgments

This material is based upon work supported by the National Key R&D Program of China (Grant No. 2016YFC0401710).

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Journal of Hydraulic Engineering

Volume 145 • Issue 1 • January 2019

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.

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Received: Jan 11, 2018

Accepted: Jul 5, 2018

Published online: Oct 31, 2018

Published in print: Jan 1, 2019

Discussion open until: Mar 31, 2019

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Jingying Lu [email protected]

Ph.D. Student, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan Univ., 24 South Section 1, Ring Rd. No. 1, Chengdu, Sichuan 610065, P.R. China. Email: [email protected]

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Ran Li [email protected]

Professor, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan Univ., 24 South Section 1, Ring Rd. No. 1, Chengdu, Sichuan 610065, P.R. China (corresponding author). Email: [email protected]

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Qian Ma, Ph.D. [email protected]

Assistant Professor, Southwestern Hydraulics Engineering Research Institute for Water Transport, Chongqing Jiaotong Univ., Xuefu Ave. No. 66, Nan’an, Chongqing 400016, P.R. China. Email: [email protected]

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Jingjie Feng [email protected]

Associate Professor, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan Univ., 24 South Section 1, Ring Rd. No. 1, Chengdu, Sichuan 610065, P.R. China. Email: [email protected]

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Weilin Xu [email protected]

Professor, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan Univ., 24 South Section 1, Ring Rd. No. 1, Chengdu, Sichuan 610065, P.R. China. Email: [email protected]

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Faxing Zhang [email protected]

Associate Professor, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan Univ., 24 South Section 1, Ring Rd. No. 1, Chengdu, Sichuan 610065, P.R. China. Email: [email protected]

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Zhong Tian [email protected]

Associate Professor, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan Univ., 24 South Section 1, Ring Rd. No. 1, Chengdu, Sichuan 610065, P.R. China. Email: [email protected]

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Abstract

Introduction

Analysis of TDG Generation Processes

Development of a Model for Predicting TDG Levels

TDG Levels under Different Retention Times and High Pressures

Experiment Description of TDG Levels under Different Retention Times and High Pressures

Expression of TDG Saturation Level as a Function of Retention Time and Pressure

Investigation of the Retention Time in Stilling Basins

Physical Experiment Description of the Retention Time in Stilling Basins

Estimation Formula for Retention Time

Verification of the Retention Time Formula

Abrupt Release of TDG at the Outlet of the Stilling Basin

Analysis of the Processes That Produce Abrupt Releases of TDG

Parameter Calibration Using Field Observations of TDG Supersaturation

Predictive Model of TDG Supersaturation for High-Dam Projects

Model Validation and Error Analysis

Model Validation

Error and Uncertainty Analysis

Conclusions and Future Work

Acknowledgments

References

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