During the spilling period, gas is carried into the stilling phase by the plunging jet of water, leading to a higher concentration of TDG than the solubility under ambient temperature and atmospheric pressure, commonly known as TDG supersaturation. The theoretical saturation concentration of TDG is mainly influenced by water temperature and bubble pressure. The higher the water temperature, the lower the TDG saturation solubility, and there is an exponential relationship between the water temperature and the mass transfer coefficient. The saturation solubility of TDG also varies at different water pressures: the higher the water pressure, the higher the saturation solubility. Specifically, each 1.00 m increase in head increased the saturation solubility by 10% compared to the saturation solubility at atmospheric pressure (Qu et al., 2011a). Theoretically, TDG saturation of water is affected primarily by the air gas concentration when water temperature and pressure are constant (Schierholz et al., 2006).

Due to uncertainty regarding the size, distribution, and trajectory of bubbles, predictions of TDG saturation based on the mass transfer process are generally biased. For a higher prediction accuracy, therefore, the water body and bubbles in the stilling phase were considered as a whole. The TDG generation process could then be divided into three parts: variation in TDG saturation during the process of flowing through air, the gas super-saturation process in the stilling phase, and the process of rapid release at the stilling phase exit. Meanwhile, the gas-liquid mass transfer process in the stilling phase occurs at the bubble and water interface. Thus, when the level of water downstream of the dam and the spilling flow rate are approximately regarded as constants, the TDG saturation in the spilling basin will equilibrate dynamically and reach the steady-state TDG saturation (C _ss, %). Based on the theory of mass balance, the mass transfer process of TDG generation in the spilling basin can be described by Eq. 1. d C = k L a b ( C s ∗ − C ss ) d t + k S a s ( C sat − C ss ) d t − ( C ss − C u ) Q s V d t = 0 (1) where V is the volume of the stilling phase (m³); Q _s is the spilling flow rate (m³/s); k _L a _b and k _S a _s are the mass transfer rate coefficients for the bubble and water surfaces, respectively (s⁻¹); C _s ^* is the bubble liquid-phase equilibrium TDG saturation (%); C _sat is the saturation concentration of TDG in water at local atmospheric pressure and temperature, usually set as 100%; C is the actual water TDG saturation (%); and C _u is the TDG saturation of the plunging water jet (%).

Eq. 1 can be simplified as C ss = k L a b V C s ∗ + k S a s V C sat + Q s C u k L a b V + k S a s V + Q s (2) If the numerator and denominator on the right side of Eq. 2 are divided by Q _s, the C _ss as a function of the residence time of the water in the stilling phase can be expressed by Eq. 3 C ss = k L a b T r C s ∗ + k S a s T r C sat + C u k L a b T r + k S a s T r + 1 (3) where T _r is the residence time of the water in the stilling phase (s), T _r = V/Q _s.

Previous research found that bubbles are the principal mediators of gas transfer (Demoyer et al., 2003). In addition, mass transfer at the water surface is negligible when compared to transfer at bubble surfaces due to the much slower transfer efficiency. The mass transfer coefficient k _a (s⁻¹) can be introduced to substitute the mass transfer rate coefficient for the bubble surfaces, thus simplifying Eq. 3 to C ss = k a T r C s ∗ + C u k a T r + 1 (4) The mass transfer coefficient is primarily related to liquid properties, water temperature, air content, water pressure, and mass transfer time. Air content is relatively high during the process of TDG super-saturation, and pressure and bubble retention time are key factors in this process. Thus, the mass transfer coefficient can be expressed by k a = f ( ρ , η , T r , h ) (5) f ( k a , ρ , η , T r , h ) = 0 (6) where ρ is the density (kg/m³); η is the dynamic viscosity (Pa·s); and h is the water depth of the stilling phase (m).

According to the dimensional analysis method and π-theorem, the equation for the mass transfer coefficient based on T _r can be written as k a T r = f ( ν T r h 2 ) (7) where ν is the kinematic viscosity (m²/s), ν = η/ρ.

The mass transfer coefficient can then be rearranged as follows: k a = f ( ν T r h 2 ) 1 T r (8)

The experimental setup included an air compressor, air inlet pipe, water inlet pipe, water outlet pipe, rotor flowmeters, pressure gauge, pressure reducing valve, pressure regulating valve, bubble diffuser, and pressure tank (Figure 1). At the beginning of the experiment, tap water and air were injected into a 0.30-m diameter × 1.13-m deep cylindrical pressure tank, and the flow rate was controlled using the rotor flowmeters. An air compressor was used to force air to dissolve in the water by increasing the pressure within the pressure tank, thereby generating super-saturated TDG water. In order to ensure the water body to fully contact with the bubbles, a 0.18-m diameter circular, flat, titanium alloy bubble diffuser was installed at 0.10 m from the bottom of the tank. The pressure regulating valve was installed between the air compressor and the pressure tank to ensure a steady airflow rate. The TDG saturation detector (Hydrolab DS5X Water Quality Multiprobes; Hach Company, United States) was attached to the inside wall of the tank to automatically record the TDG saturation and temperature of the water in real-time. TDG saturation was measured using the TDG pressure probe, which had a measurement range of 400–1,300 mm Hg and accuracy of ±0.1%. The temperature probe measurement range was −5 to 50°C, with an accuracy of ±0.1°C. The pressure inside the tank was measured using the water pressure probe, which had a measurement range of 0–50 m and an accuracy of ±0.1 m.

To match the residence time of the water in the stilling phase under experimental conditions with the T_r of the dams, the water body renewal time of the stilling phase was determined at Xiluodu Dam, Xiaowan Dam, and Jinping first-level Dam. They are all typical super-large hydropower stations in China. Jinping first-level Dam (height: 305 m) and Xiluodu Dam (height: 285.5 m) are in the Jinsha River. Xiaowan Dam (height: 292 m) locates in Lancang River. The 3 dams can provide comprehensive data sets of dam T _r . The results are shown in Table 1. In typical projects, the residence time of the water in the stilling phase varies greatly under different discharge rates, ranging from 2 to 13 min. To determine the airflow rate (Q _a) in the subsequent experiments, the steady-state TDG saturation was compared at Q _a values of 60, 180, 300, 600, and 900 L/h. These analyses indicated that the mass transfer process tended to remain stable at Q _a of 900 L/h. Therefore, the following experiments were carried out at a Q _a of 900 L/h. Experimental scenarios were designed with a water volume (V) of 30, 40, 50, or 60 L and a flow rate (Q _s) ranging from 180 to 750 L/h. A scenario without water renewal was examined as a control. The mass transfer coefficient was calculated for each of the scenarios using Eq. 8.

The mass transfer rate, E, was introduced to evaluate the variation in TDG saturation: E = C D − C U C s ∗ − C U (9) where C _D is the TDG saturation of the outflow (%); C _U is the TDG saturation of the inflow (%); and C _s ^* is the theoretical TDG saturation at the pressure and temperature within the tank (%), determine the value by checking the table (Colt, 2012).

TDG field observations were conducted 3.0 km downstream of the Xiluodu Dam at 0,800 h and 1,400 h from June 23 to July 6, 2017. TDG saturation and water temperature were measured using a Hydrolab DS5. During the monitoring period, the water temperature below the Xiluodu dam varied in the range of 20.4–21.2°C, with corresponding changes in the theoretical saturated dissolved concentration of each gas in the water column. The atmospheric pressure was set to 724 mm Hg, and the TDG probe monitoring frequency and duration were set to 1 and 30 min, respectively. TDG saturation of the flow in the stilling phase was calculated using Eq. 14 based on the measured results because the TDG saturation at the monitoring point (C _M; Figure 2) represented the mixing of the spill flow and tail water. C M = Q s C s + Q p C p Q s + Q p (10) where C _M is the TDG saturation at the monitoring point (%); C _s and C _p represent the TDG saturation of the flow in the stilling phase and the power flow, respectively (%); and Q _s and Q _p represent the spilling flow rate and power flow rate, respectively (m³/s).

Using the above inputs, a TDG generation prediction model using a multiple linear regression method (Eqs 4, 8) was proposed for Xiluodu Dam. During the field observation period, data regarding the flow rate and water level downstream of the Xiluodu Dam were obtained from the China Three Gorges Corporation.

The empirical model (Eq. 11) (Anderson et al., 1998) and mechanical model (Eqs 12, 13) (MaQian, 2016) were compared with the developed estimation model using field observation data. The empirical model was developed based on the spilling flow rate. However, the mechanical model was developed based on gas transfer theories.

Empirical model: C ss = a + b ⋅ exp ( c Q s ) (11) Mechanical model: C ss = 100 { 1 + Δ P P 0 [ 1 − exp ( − 0.08 t R ) ] } (12) t R = 27.73 λ 0.49 ( h k g ) 0.5 ( l 0 l ) − 0.25 (13) where ΔP is the average pressure in the stilling phase, which consists of static pressure and hydrodynamic pressure (m); P ₀ is the local atmospheric pressure (m); t _R is the retention time of aerated water in the stilling phase (s); h _k is the water depth in the stilling phase (m); g is acceleration due to gravity (m²/s); l ₀ is the horizontal distance between the downstream toe of the dam and the impact point of the jet (m); l is the length of the stilling phase (m); and a, b, and c are dimensionless constants that can be fitted from field observation results, the results were a = 154.02, b = −34.15, and c = −2.76 × 10^–4.

The standard error (SE), Normalized mean error (NME), Mean multiplicative error (MME), Nash-Sutcliffe efficiency coefficient (NSE), and coefficient of determination (R ²) were used to evaluate the performance of the predictive equations (Jha et al., 2010; Disley et al., 2015). Because the maximum TDG saturation level represents the most serious threat to the habitat, the error of maximum TDG saturation level (RE _MAX) was evaluated for each model. The errors were determined using the following equations: S E = [ ∑ i = 1 N ( C P − C M ) i 2 N ] 1 / 2 (14) N M E = 100 % N ∑ i = 1 N ( C P − C M C M ) i (15) M M E = exp [ ∑ i = 1 N | ln ( C P / C M ) i | N ] (16) N S E = 1 − ∑ i = 1 N ( C M i − C P i ) 2 ∑ i = 1 N ( C M i − C M ¯ ) 2 (17) R 2 = [ ∑ i = 1 N ( C M i − C M ¯ ) ( C P i − C P ¯ ) ∑ i = 1 N ( C M i − C M ¯ ) 2 ∑ i = 1 N ( C P i − C P ¯ ) 2 ] 2 (18) R E MAX = | C PMAX − C MMAX C MMAX | × 100 % (19) where C _P and C _M denote the predicted and measured TDG saturation levels (%); C _PMAX and C _MMAX are the maximum predicted and measured TDG saturation levels (%); N represents the number of values; and i is the time step.

The dynamics of the mass transfer rate at different T _r are summarized in Figures 3A–D. The mass transfer rate increased with increasing T _r and eventually stabilized. In the absence of water renewal, the mass transfer rate gradually approached a value of 1.0 (the theoretical saturation state). In contrast, under steady-state conditions, the mass transfer rate was <1.0 with water renewal, and the TDG saturation was lower than the theoretical saturation state. A comparison of different scenarios (Table 2) revealed that the mass transfer rate in the steady-state increased with increasing T _r . In addition, the experimentally derived mass transfer coefficients were calculated and plotted against T _r (Figure 4). These data showed that the mass transfer coefficient decreased with increasing T _r .

tBased on the established basic form of the mass transfer coefficient equation (Eq. 8), a nonlinear regression analysis of the mass transfer coefficient, pressure head and T _r (Table 2) for each working condition was performed to obtain the mass transfer coefficient equation under the experimental condition (Eq. 20), with a minimum correlation R ² = 0.87. k a = [ 2.8088 l n ( ν T r h 2 ) + 17.315 ] 1 T r (20)

The mass transfer coefficient equation (Eq. 20) was coupled with the stable-state TDG super-saturation prediction equation (Eq. 4) to establish the model for predicting TDG super-saturation under experimental conditions. Figure 5 and Table 3 show a comparison of calculated and measured TDG super-saturation values under various experimental conditions. The comparison showed that the Relative Error (RE)between the calculated and measured values was <5%. In addition, the Root Mean Squared Error (RMSE), R ² , and NSE values of 1.21%, 0.94, and 0.94, respectively, indicated that the developed water renewal model exhibits good performance in predicting TDG super-saturation generated by water under experimental conditions.

Results of monitoring TDG saturation levels at the Xiluodu Dam are listed in Table 4. The TDG saturation at the monitoring point during the observation period was generally >100%. The TDG saturation reached a maximum of 126.73% at 1,400 h on July 5 and a minimum of 98.63% at 0,800 h on June 23. According to Eq. 11, the TDG saturation at the monitoring point was transferred to the TDG saturation in the stilling phase under the dam. Data indicated that the maximum TDG saturation in the stilling phase reached 155.76% at 1,400 h on July 5, and the minimum TDG saturation in the basin reached 98.63% at 0,800 h on June 23.

To analyze the effect of the water renewal model in the engineering practice, this section generates the measured values of each parameter based on the saturation of the near zone below the Xiluodu hydropower station dam, and establishes the fitting equation for the mass transfer coefficient of the near zone below the Xiluodu dam. The water renewal model (Eq. 4) is used to calculate the mass transfer coefficient ka corresponding to each monitoring condition, and the relationship between the mass transfer coefficient ka and the T _r is shown in Figure 6. The mass transfer coefficient ka and the related parameters under each monitoring condition are shown in Table 4. Based on the basic form of the established mass transfer coefficient equation (Eq. 8), a nonlinear regression analysis of the calculated mass transfer coefficient ka, pressure head (water depth of the stilling phase) and T _r (Table 4) was conducted to obtain the expression of the mass transfer coefficient equation in the stilling phase of Xiluodu Hydropower Station (Eq. 21), with a minimum correlation R ² = 0.82. k a = [ − 0.052 l n ( ν T r h 2 ) − 0.0345 ] 1 T r (21)

By coupling the established Eq. 21 with Eq. 4, a model for predicting TDG super-saturation at Xiluodu Dam was obtained, and this model was designated the “water renewal model for Xiluodu Dam.”

TDG super-saturation values estimated based on field data using the empirical model, mechanical model, and water renewal model developed in this study are shown in Figure 7. The error of the mechanism model was larger than that of the other two models due to larger calculation results. The SE (1.81), NME (0.16), MME (1.00), and RE _MAX (2.26) values were the lowest, and the NSE (0.90) and R ² (0.93) values were the highest for the water renewal model in Table 5, indicating that the performance of the water renewal model is superior to that of the other models.