Oxygen transfer characteristics of bubbly jet in regular waves ＊

Ze-gao Yin , De-chun Liu, Yuan Li, Yan-xu Wang

College of Engineering, Ocean University of China, Qingdao 266100, China

Abstract: The dissolved oxygen level is an important index of the water environment, and in this paper, the oxygen transfer of the bubbly jet in regular waves is investigated numerically and experimentally. The Reynolds-averaged Navier-Stokes equations, the re-normalisation group k-ε equations, and the volume of fluid (VOF) technique are used along with a 2-D CFD model to simulate the wave and bubble motions as well as the turbulence, and a dissolved oxygen transport equation is used to model the oxygen transfer behavior both through the bubbly interface and the wave surface. A series of experiments are conducted to validate the mathematical model, with good agreement. In addition, a group of dimensionless parameters are defined from the wave parameter and the aeration parameter, and their relationships with the total oxygen transfer coefficient are explored. Furthermore, the dimensional analysis and the least squares methods are used to derive simple prediction formulas for the total oxygen transfer coefficient, and they are validated with the related experimental data.

Key words: Bubbly jet, regular wave, mathematical model, total oxygen transfer coefficient

Introduction

The dissolved oxygen (DO) plays an important role in the survival and the reproduction of fishes and other species in water, and it is closely related to the environment and the ecology of the oceans. In recent years, the sea hypoxia aggravates due to the water eutrophication and the climate warming tendency[1-2].Artificial aeration is widely used to increase the oxygen in water of lakes, reservoirs and wastewater

treatment systems[3-4]. Tojo and Miyanami[5]and Mueller et al.[6]investigated the oxygen transfer characteristics in the gas-liquid jet tank experimentally. Baylar and Emiroglu[7]studied the entrainment rate and the oxygen transfer efficiency of the air bubbles through circular nozzles. Surinder[8]experimentally examined the oxygen transfer efficiency of multiple inclined plunging jets in a water pool. Bagatur and Onen[9]presented new formulations to explore the flow characteristics in the plunging water jet system. Jahromi and Khiadani[10]proposed two equations to estimate the oxygen transfer efficiency for a single plunging jet in a turbulent cross-flow. Politano et al.[11]presented a two-phase transport equation for the total dissolved gas downstream a spillway, with the bubble/liquid mass transfer expressed as a function of the air volume fraction and the bubble size distribution. Witt and Gulliver[12]proposed an improved model for predicting the oxygen transfer efficiency at gated sills,with the oxygen transfer coefficient estimated with consideration of the effects of the mean bubble diameter, the air entrainment rate and the turbulent energy dissipation rate. In addition, the Runge-Kutta-Fehlberg method[13-14]was extensively used to obtain the numerical solutions of nonlinear governing equations for flow, heat and mass transfer problems.

In the aforementioned investigations, the oxygen transfer characteristics of the bubbly jet were obtained,but mostly based on experimental data and field engineering, little work is currently available in the literature on the mathematical simulation of the oxygen transfer both through the bubbly interface and the wave surface. In this paper, the oxygen transfers through the bubbly interface and the wave surface are considered, and with the introduction of the total oxygen transfer coefficient (TOTC) to investigate its relationship with the wave and aeration parameters numerically and experimentally, to provide some insight into the oxygen transfer performance of the bubbly jet in waves.

1. Mathematical model

1.1 Governing equations

The 2-D CFD mathematical model is built in the ANSYS FLUENT 16.0 software. The Reynoldsaveraged Navier-Stokes (RANS) equations are used to study the hydrodynamic behavior of regular waves.As a simple turbulent model without major adjustment work of constants or functions, the re-normalization group (RNG)k-εmodel is used to predict the wave turbulence. The volume of fluid (VOF) technique is adopted for the wave surface tracking. For the DO transport in the water, the following governing equation is adopted

whereρis the density of the water and air bubble mixture,Cis the DO concentration,tis the time,uiis the velocity component inidirection,xiis the coordinate inidirection,cΓis the turbulent diffusion coefficient,Sis the source term representing the oxygen mass transfer between the water and the air, and it is equal to the sum of oxygen mass transfer terms through the bubbly interfaceSBand through the wave surfaceSW.SBcan be modeled as follows[15]

whereKL,BaBis the oxygen transfer coefficient for the air bubbles.Cs,Bis the saturated DO concentration at the bubble location, related to the temperature and the pressure[16],KL,Bis the liquid film mass transfer coefficient for the air bubbles and it is modeled as the maximum value for the bubbles rising in a stagnant liquid[17], and the bubbles in the turbulent flow[18-19]

whereDmis the oxygen molecular diffusion coefficient,νis the kinematic viscosity coefficient of the water,Peis the bubble Peclet number,Reis the bubbly Reynolds number andRis the bubbly radius, and by assuming the spherical bubble shape, it can be expressed as follows[20]

wheregis the gravitational acceleration,σis the surface tension coefficient,wμis the dynamic viscosity of the water,Uais the surface air velocity.aBis the specific interfacial area of the bubbles,expressed as

For the oxygen transfer on the wave surface,similar toSBas illustrated in Eq. (2),SWis expressed as follows

whereKL,WaWis the oxygen transfer coefficient on the wave surface,KL,Wis the liquid film mass transfer coefficient on the free wave surface,aWis the specific surface area for the free wave andCs,Wis the saturation DO concentration at the wave surface for the water body.

KL,Wunder the wave condition was not well studied due to its complexity. However, by using experimental results and data in literature, Daniil and Gulliver[21]obtained a formula as follows

whereHis the wave height,Tis the wave period andScis the Schmidt number. Eq. (7) is relatively simple, however, it works well in a number of cases[22].

In the Fluent 16.0 software, the DEFINE_SOURCE and DEFINE_DIFFU -SIVITY macros are used to implement Eqs. (1) through (7) for computing the DO concentration distribution combined with the RANS equations, the RNGk-εequations and the VOF technique.

1.2 Wave generation, absorption methods and boundary conditions

The desired wave surfaceη(x,t) for regular linear waves is determined as follows

whereX0is the stroke distance of the paddle,ωis the angular frequency of the wave,xis the horizontal directional coordinate, with the origin at the location of the wave generator,Kis the wave number andhis the still water depth. To generate the desired wave surface as illustrated in Eq. (8) in the mathematical model, the pushing paddle method is used, and the paddle pushing speed as in a simple harmonic motion,U(t), was expressed as

To avoid the unwanted wave reflection at the end of the flume, an artificial damping momentum source term is added to the momentum equations to absorb the waves and it can be expressed as follows[23]

wherex1andx2are the start and end locations of the damping zone,Kxis a coefficient.

The boundary conditions are set up as shown in Fig. 1. A′B′ is set as a dynamic wall to generate the desired wave, and its speed is defined as in Eq. (9).The bottom boundary, B′C′, is defined as a straight wall boundary with zero velocities for the water and the air. The right boundary, C′D′, is set up as the wave absorbing boundary. The top boundary, A′D′, is set as the pressure inlet boundary, and its pressure is set as 0.At the bubbly jet boundary, E′F′, the air velocity is determined by the air flow rate and the wave flume width, and the gradient of the DO concentration is set as 0 according to the Neumann boundary condition.The still water depthhis 0.4 m for all following mathematical and experimental scenarios.

1.3 Discretization error examination

The grid generator GAMBIT 2.4.6 is used to generate the 2-D structured grid system. The grid independence is tested by using the grid index technique,which is a recommended and widely-used method in various CFD simulations[24-25]. Due to the relatively high-accuracy requirement of the bubbly behavior, the refined grids are used within the range of 0.5 m to the bubble. The numbers of refined girds are 91 260,46 260 and 22 860 for grid 1, grid 2 and grid 3. In the relative far field, the coarser grids are selected as 0.01 m, 0.02 m and 0.04 m for grid 1, grid 2 and grid 3, respectively, and the total grid numbers are 150 060,75 060 and 37 560, respectively. The Richardson extrapolation method is used to compute the extrapolated relative error (ERE) and the grid convergence index (GCI). Table 1 shows the discretization error of the wave height at the locationx= 17 m for the incident wave heightH= 0.10 m ,the wave periodT=2.0s , the air flow rate per unit widthq=0.0025m2/s and the length of the air injection inletd=0.004 m. It is found that the error is as small as 1.40% between grid 2 and grid 3. Thus,the grid size of grid 2 can be said to be relatively independent of the computational results. Therefore,grid 2 is chosen for the following computations after considering the balance between the computational accuracy and the speed. The iterative convergence is assumed to be reached when the normalized residuals are less than three orders of magnitude (10-3). To obtain accurate solutions, the time step is set as small as 0.005 s, and 2×104iterations are sufficed for all scenarios involved.

1.4 Experiments and mathematical model validation

1.4.1 Physical experiments

Fig. 1 Boundary conditions for mathematical model and experimental setup

Table 1 Analysis of discretization error

To validate our mathematical model, a series of experiments are conducted in a wave flume at the hydraulic lab of Ocean University of China. The length, the width and the height of the wave flume are 25.0 m, 1.0 m and 1.2 m, respectively. A piston type wave generator is used to generate the desired waves on the left side of the flume. And a large amount of porous material is used to absorb the incident waves and avoid unwanted wave reflections on the right side of the flume. The DO probes #1, #2 and #3 are installed at the locations 11 m, 12 m and 14 m to measure the DO concentrations after a proper calibration using the Winkler technique[26]. The wave gauges I, II and III are fixed at the locations 7 m, 8 m and 17 m to measure the wave heights. The aeration system consists of an air compressor of OTS-1500*4-160L type to supply the air, a valve to control the air flow, an air flow meter of SIARGO MF5700 type to measure the flow rate, and a transverse polyvinyl chloride (PVC) circular pipe with aeration holes to inject bubbles into the water at the location 13 m. The PVC pipe length is 1.0 m, as consistent with the flume width, its inner diameter is 0.02 m, 40 circular orifices are drilled over the pipe top for the air injection, and the distance between adjacent orifices is about 0.025 m. The major components are assembled as shown in Fig. 1. In addition, Table 2 shows the experimental summary.

Table 2 Experimental and numerical summaries

Before the start of an experiment, the flume is filled with the fresh tap water until the still water depth reaches 0.4 m. The water temperature is 16.1°C.With the water volume, the present and desired DO concentrations being taken into consideration,the suitable quantities of Na2SO3and CoCl2are determined, and they are uniformly put into the water flume to consume the DO. Subsequently, the wave generator starts to mix the water, Na2SO3and CoCl2completely, the DO concentrations are measured by the DO probes, and if they are decreased, it is indicated that Na2SO3is not reacted to consume the DO in the water adequately. Only when the measured DO concentrations increase stably showing that Na2SO3is reacted adequately, the air compressor will start to operate, the air bubbles are injected into the water flume through the holes in the horizontal pipe after going through the controlling valve and the air flow meter, subsequently, a continuous curtain of air bubbles forms across the flume section as a result.Then, the wave gauges measure the wave height histories. Note that all wave parameters aforementioned are consistent with the rule of the model limits suggested by Frostick et al.[26].

1.4.2 Mathematical model validation

Figure 2 shows the numerical wave surface elevation validation with experimental data forq=0.0025m2/s . It is seen that the numerical values agree well with the experimental data, with discrepancies less than 10%, showing that the aforementioned mathematical model can reliably be used to investigate the wave parameters in the wave flume.

Figure 3 shows the numerical DO concentrations in comparison to the experimental data measured by the DO probes #1 and #2, whenH=0.06 m, 0.10 m,T=1.0 s, 2.0 s,d=0.004 m andq=0.0025m2/s .It is seen that their relative deviations are less than 10%, indicating a good agreement, and that the mathematical model can be used to predict the DO transfer with an acceptable accuracy. Therefore, the 3-D bubbly jet and the DO transfer can be reasonably investigated by the 2-D mathematical model without large deviations. In addition, it is expected that the DO concentration at the DO probe #2 is higher than that at the DO probe #1, due mainly to the closer distance from the DO probe #2 to the bubbles. It is also observed the increase of the DO concentration whenH= 0.10 m , T = 2.0 s is larger than that whenH=0.06 m andT=1.0s to some extent,confirming the hypothesis of better oxygen transfer performance of the cases with higher wave height and period. Subsequently, a series of numerical simulations are conducted by using the aforementioned mathematical model, and the summary is shown in Table 2.

Fig. 2 Numerical wave surface elevation history compared with experimental data

2. Results and discussions

The oxygen transfer coefficients for the bubbly jetKL,BaBand the wave surfaceKL,WaWare important parameters to predict the DO concentration variation in the water. However, it is not easy to determine them due mainly to the complex characteristics of the bubbles and the waves.Consequently, a relative saturation coefficientEis usually used instead to consider the oxygen transfer effect through the bubbly interface and the wave surface as an alternative parameter, and it is expressed as follows

whereC0is the average DO concentration at initial time for the flume water, C1is the average DO concentration at the present time for the flume water andCsis the average value of the saturated DO concentration for the flume water.

In order to simplify the expression, the TOTCKais defined asKa=KL,BaB+KL,WaW, and Eq.(11) is rewritten as

To further investigate the effects of the wave parameter and the aeration parameter on the TOTC, a series of dimensionless parameters are defined asH/d,gT2/handq2/ gh3, respectively, and their intrinsic relationship with the TOTC is explored as in the following sections.

2.1 The effects of the wave and aeration parameters on the TOTC

Figure 4 shows the TOTC relationship withH/dfor variousTscenarios whenq=0.00333 m2/s. It is seen that the TOTC increases with the increase ofH/dto some extent. A possible explanation is that with the increase ofHfor a givendscenario, the water particle velocity and the oscillation strength near the wave surface increase, the surrounding and far-field water is mixed significantly, and the high DO is more prone to transport to the far-field hypoxia water, and the TOTC increases as a result. With the increase ofdfor a givenHscenario, the bubbly size increases in general, the specific surface area decreases along withKL,BaB, and the TOTC decreases consequently.

Fig. 4 TOTC relationship with H/d

Figure 5 illustrates theTOTCrelationship withgT2/hfor variousHanddscenarios. It is found that the TOTC increases significantly with the increase ofgT2/h. A possible reason is that with the increased periodfor a givenhscenario, the wave velocity increases according to the disperse relationship for the linear wave, and the bubbly oscillation range and the bubbly retention time in the water increase, respectively, the oxygen transfer amount from the bubbles to the water increases, and the TOTC increases as a result. Additionally, Fig. 5 shows that the TOTC increases with the decrease ofd, as is well consistent with the trend as shown in Fig.4.

Figure 6 shows the TOTC relationship withq2/gh3for variousHandTscenarios. It is found that theTOTCincreases with the increase ofq2/gh3.A possible explanation is that with the increase ofq,the air volume fraction increases along with the specific interfacial area of the bubbles, and the bubbles are prone to transfer the oxygen for a givenhscenario. In addition, the TOTC increases with the increase ofHas expected, agreeing well with the tendency as illustrated in Fig. 4.

2.2 The spacing effect between two pipes on TOTC

Fig. 5 TOTC relationship with gT2/h

Note that the aforementioned results are based on a single pipe with aeration holes, however, the oxygen transfer behavior for the aeration of multiple pipes is also very interesting due to its high performance, and it deserves a consideration. In this section, a simple numerical investigation is made for the TOTC in cases of bubbly jets passing through the aeration holes on the tops of two pipes, and their spacinglis from 20dto 60d, 100d, 140dand 180dwithd=0.004 m , i.e., from 0.08 m to 0.24 m, 0.40 m,0.56 m and 0.72 m, respectively.

Fig. 6 TOTC relationship with q 2 / g h3

Figure 7 shows the relationship between the TOTC andl/hfor variousHandTscenarios,and the air flow rate per unit width for every pipe is assumed asq/2 withq=0.0033m2/s to simplify the problem. In addition, the TOTC is also presented for the single pipe scenario (i.e.,l/h=0) with the sameq. It is found that the TOTC increases with the increase ofl/hto some extent, showing that the bubbly jet of two pipes performs better in the oxygen transfer than that of a single pipe. In the scenariosl/h＜0.2, the TOTC increases significantly. For smallTscenarios as shown in Figs. 7(a), 7(b), the TOTC increases slightly in the ranges 0.2＜l/h＜0.6 andl/h＞1.0, while it increases significantly in the range 0.6＜l/h＜ 1.0. A possible reason is that in the scenarios 0.2＜l/h＜0.6, the bubbles from two pipes are prone to convergence near the pipes, the relative large bubbles occur, and their specific interfacial area decreases along with the TOTC. In the scenariosl/h＞ 1.0, the bubbles from two pipes are arduous to converge during their ascent process accompanied with oscillation, the bubbly size and motion path vary little with the increase ofl/h, and the TOTC increases little as a result. In the scenarios 0.6＜l/h＜1.0, on the other hand, the convergence location of the bubbles from two pipes ascends significantly with the increase ofl/h, and the TOTC increases consequently. In highTscenarios as shown in Figs. 7(c), 7(d), the TOTC increases steadily in the scenariosl/h＞0.2 due mainly to the steady ascent of the bubbly convergence location, and the increased amplitude is declined to zero as a result of the gradual separation of two bubbly jets. Consequently, the bubbly convergence location plays a significant role in the TOTC for the bubbly jet of two pipes in regular waves.

Fig. 7 TOTC relationship with l/ h for two pipes’ aeration

2.3 Prediction formula for the TOTC

Due to the complex effect of the wave and aeration parameters on the oxygen transfer behavior, it is desirable to develop a simple theoretical formula for the TOTC. The wave heightH, the wave periodT,the still water depthhand the gravitational accelerationgare considered to account for the wave effect. The air flow rate per unit widthq, the length of the air injection inletdand the pipe’s spacingl(only for two pipes) are considered to account for the aeration effect. The TOTC formula can thus be expressed as follows

The dimensional analysis, the least squares methods and the numerical results are used to determine the following formulas:

where Eqs. (14) and (15) are for a single pipe aeration and the aeration of two pipes, respectively, and their correlation coefficients are 0.91, 0.89, respectively.Figure 8 shows a comparison between Eqs. (14), (15)and Eq. (12) along with the experimental data. It is found that all relative deviations between them are smaller than 10%, showing that the results of Eqs. (14)and (15) agree well with the experimental data.Consequently, they are acceptable in the oxygen transfer performance assessment for the bubbly jet in regular waves without large deviations.

Fig. 8 Equations (14), (15) comparison with Eq. (12) by using experimental data

3. Conclusions

Based on the RANS equations, the RNGk-εequations and the VOF technique, a 2-D DO transport mathematical model is built to investigate the oxygen-supply characteristics of the bubbly jet in regular waves, and the oxygen transfers both through the bubbly interface and the wave surface are taken into account. A series of experiments are conducted to validate the mathematical model with good agreement.The TOTC relationships with the wave parameters and the aeration parameters as well as with the two pipes’ spacing are examined. It is found that the TOTC increases with the increase ofH/d,gT2/handq2/gh3to some extent. The TOTC increases generally with the increase ofl/hfor the aeration of two pipes. The dimensional analysis and the least squares methods are used to derive simple prediction formulas for the TOTC, and they are validated with the related experimental data. The results of this paper may help understanding the oxygen transfer characteristics of the bubbly jet in regular waves,particularly, help the development and the calibration of CFD and oxygen transfer models.

The present research has some limitations: the still water depth is assumed to be a constant to simplify the problem, and its effect on the bubbly behavior and the oxygen transfer should be further investigated in future studies. In addition, the bubbly characteristic parameters (e.g., the shape, the size and the diameter) should be measured in the experiments,and they should be used in the future model to improve the computational accuracy.

Acknowledgement

This work was supported by the State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, China (Grant No. 1602).