Li-ping CHEN*, Jing-tao CHENG, Guang-fa DENG

1. College of Urban Construction and Safety Engineering, Nanjing University of Technology, Nanjing 210009, P. R. China

2. Jiangsu Frontier Electric Technology Co., Ltd., Nanjing 211102, P. R. China

Anisotropic diffusion of volatile pollutants at air-water interface

Li-ping CHEN*1, Jing-tao CHENG1, Guang-fa DENG2

1. College of Urban Construction and Safety Engineering, Nanjing University of Technology, Nanjing 210009, P. R. China

2. Jiangsu Frontier Electric Technology Co., Ltd., Nanjing 211102, P. R. China

The volatile pollutants that spill into natural waters cause water pollution. Air pollution arises from the water pollution because of volatilization. Mass exchange caused by turbulent fluctuation is stronger in the direction normal to the air-water interface than in other directions due to the large density difference between water and air. In order to explore the characteristics of anisotropic diffusion of the volatile pollutants at the air-water interface, the relationship between velocity gradient and mass transfer rate was established to calculate the turbulent mass diffusivity. A second-order accurate smooth transition differencing scheme (STDS) was proposed to guarantee the boundedness for the flow and mass transfer at the air-water interface. Simulations and experiments were performed to study the trichloroethylene (C2HCl3) release. By comparing the anisotropic coupling diffusion model, isotropic coupling diffusion model, and non-coupling diffusion model, the features of the transport of volatile pollutants at the air-water interface were determined. The results show that the anisotropic coupling diffusion model is more accurate than the isotropic coupling diffusion model and non-coupling diffusion model. Mass transfer significantly increases with the increase of the air-water relative velocity at a low relative velocity. However, at a higher relative velocity, an increase in the relative velocity has no effect on mass transfer.

volatile pollutant; interfacial mass transfer; anisotropic diffusion; STDS; anisotropic coupling diffusion model

1 Introduction

It is significant to accurately predict the temporal and spatial distribution of volatile pollutants after leakage. It can provide an important theory for health diagnosis and management of the hydrosphere ecosystem. Mass transfer of volatile pollutants at the air-water interface is a complex process, in which the liquid phase transforms into the gas phase. Whitman (1962) and Liss and Slater (1974) proposed film models to study the mass transfer mechanism of volatile pollutants at the air-water interface. The film models are suitable for the calculation of the exchange capacity of volatile pollutants at the air-water interface of lakes.But they are not suitable for highly turbulent conditions, in which new surfaces are continuously formed by breaking waves, air bubbles entrapped in the water, and water droplets ejected into the air. Danckwerts (1955) proposed a surface renewal model, in which it is assumed that turbulent flow of water transports pollutants to the air-water interface and the mass transfer processes continue until another turbulent flow rolls the pollutants away. The surface renewal model suggests an exponential relationship between the ratio of the exchange velocity in air to that in water and the ratio of the molecular diffusivity in air to that in water. The exponent is 0.5 in Danckwerts’ model, and 0.57 in Schwarzenbach et al. (2002). In order to calculate the exposure time, Lamourelle and Sandall (1972) proposed a large eddy simulation (LES) model, Lamont and Scott (1970) developed a small eddy simulation model, and Komori et al. (1989) and Rashidi et al. (1991) described a turbulent impulse model. However, these models have not been widely used, because the turbulent characteristic parameters required cannot be obtained, especially for those far from the air-water interface. A boundary layer model developed by Deacon (1977) suggests that the exponent is 2/3 for the relationship between the exchange coefficient and Schmidt number. The exponent changes to–0.5 at a wind speed of 5 m/s (J?hne and Haubecker 1998).

Mass transfer of volatile pollutants at the air-water interface is affected by the concentration distribution of pollutants in water and air or vice versa. Therefore, the concentrations of pollutants in water and air are coupled with each other. However, none of the models previously mentioned take into account the coupling characteristic of mass transfer at the air-water interface. These uncoupled models suggest that the mass transfer processes at the air-water interface are mainly controlled by the liquid phase, and the effect of the gas phase can be negligible (Bade 2009; Hardt and Wondra 2008). Chen et al. (2009) found that low-volatility pollutants have larger gas phase resistance at low wind velocities and the impact of the gas phase should not be ignored. Hasegawa and Kasagi (2009) considered the coupling characteristic of momentum, but did not take the coupling characteristic of concentration into account in simulating mass transfer at the air-water interface.

The smoothed particle hydrodynamics (SPH), level-set function, and volume of fluid (VOF) methods can be used to capture the free surface (Marrone et al. 2010). But the SPH method requires a higher computational cost. The level-set function method smears out the sharp interface so that it can distort the large gradient. The VOF method is widely used to track the air-water interface because simultaneous calculations for water and gas can be completed (Afkhami et al. 2009). An air-water coupling diffusion model based on the VOF method for volatile pollutants has been developed. Mass exchange caused by turbulent fluctuation is stronger in the direction normal to the air-water interface than that in other directions due to the large density difference between air and water. Thus, turbulent diffusion of the volatile pollutants at the air-water interface shows anisotropic diffusion characteristics. However, the diffusion of volatile pollutants in the air is isotropic. The conventionaldiscretization scheme cannot be applied to anisotropic diffusion. In this study, a second-order accurate smooth transition differencing scheme (STDS) was proposed to guarantee the boundedness for the flow and mass transfer at the air-water interface. Simulations and experiments were performed to study the trichloroethylene (C2HCl3) release. By comparing the anisotropic coupling diffusion model, isotropic coupling diffusion model, and non-coupling diffusion model, the features of the transport of volatile pollutants varying with the air-water relative velocity were determined.

2 Model development

2.1 VOF model

The continuity equation, the momentum equation, and the equation of fraction of volume occupied by water are written as follows:

where the subscriptsiandjare equal to 1, 2, or 3, representing the three directions in the Cartesian coordinate system;uis the velocity of the computational cell;ρis the density of the computational cell, andρ=αρL+(1?α)ρG, whereρLandρGare the densities of water and air, respectively, andαis the fraction of volume occupied by water in the computational cell;uLanduGare the velocities of water and air, respectively;pis the pressure of the computational cell;ηtis the turbulent dynamic viscosity;fis the body force (the gravity is the only body force in this study); andtis time.

2.2 Realizablek-εmodel

For a large strain rate, the realizablek-εmodel is more effective than the standardk-εmodel. The governing equations for the turbulent kinetic energykand the dissipation rateεin the realizablek-εmodel are

2.3 Air-water coupling diffusion model for volatile pollutants

whereCis the total pollutant concentration, andC=αCL+(1?α)CG;EiandDare the turbulent mass diffusivities of pollutants in water and air, respectively; and the third item on the left side of Eq. (7) indicates the effect of the air-water relative velocity onC.

where Khis the mass transfer rate, m1= –0.22, and G denotes the velocity gradient at the air-water interface. Lee and Saylor (2010) proposed a method for the calculation of the mass transfer rate Kh.

3 Solutions to equations

The finite volume method (FVM) was used to discretize equations. The pressure implicit with splitting of operators (PISO) algorithm was used for pressure-velocity coupling. The implicit scheme was used for the temporal discretization.

The conventional discretization scheme can not be used to deal with anisotropic diffusion. Based on the normalized variable diagram (NVD), Jasak et al. (1999) developed the Gamma discretization scheme for unstructured grids to guarantee the boundedness of physical variables. In Fig. 1(a), point P is the center of the computational cell, point E is the center of the downstream computational cell, and e denotes the interface between the two computational cells . At point P, the normalized variable is defined as follows:

Fig. 1Computational cell and differencing schemes in NVD

4 Model verification

4.1 Conditions for simulations and experiments

Several groups of numerical simulations and experiments were performed on C2HCl3leakage. The flume was 14.0 m long, 0.4 m wide, and 0.4 m deep, and had a bed slope of 3/700. The water depth was 0.1 m. The average velocities of water and air were 0.2 m/s and 1.2 m/s, respectively. The flow was in thexdirection, and the flume width was in theydirection. Thezdirection was upward and the bottom of the flume was atz= 0. The coordinate system is shown in Fig. 2(a). The experimental facility is shown in Fig. 2.

Fig. 2Experimental facility

50 mL of C2HCl3at an initial concentration ofC0= 2 g/L was instantaneously released into the flume at the pointx= 1.25 m,y= 0.2 m, andz= 0.1 m. The sampling cross-sections were atx= 2.25, 2.65, 2.85, 3.45, and 3.75 m. The multi-point synchronous sampling shown in Fig. 2(b) was adopted. The C2HCl3concentration in water was measured by the gas chromatography method and the C2HCl3concentration in the air was measured by the pyridine-alkali colorimetric method.

4.2 Results and analysis of simulations and experiments

4.2.1 Grid independence

Grid independence is associated with the accuracy or even rationality of numerical results. The computational zone was 14 m × 0.4 m × 0.4 m in the flume. Fig. 3 shows theC/C0variations at the pointx= 2.25 m,y= 0.225 m,z= 0.065 m for four different computational zones in the air in sizes of 14 m × 3.6 m × 1.5 m, 14 m × 4.8 m × 1.95 m, 14 m × 6.4 m × 2.6 m, and 14 m × 8.5 m × 3.5 m, whereC/C0denotes the ratio of the C2HCl3concentration to the initial concentration. Hexahedron cells were used in the simulations. The number of grid cells for the four computational zones was 59 738, 106 200, 188 800, and 335 644, respectively. The simulation results for computational zones of 14 m × 3.6 m × 1.5 m and 14 m × 4.8 m × 1.95 m are higher than the experimental results. The simulation results for computational zones of 14 m × 6.4 m × 2.6 m and 14 m × 8.5 m × 3.5 m are close to the experimental results. The size of 14 m × 6.4 m × 2.6 m in the air was used in the following study.

Fig. 4 shows theC/C0variations at the pointx= 2.25 m,y= 0.225 m,z= 0.065 m for four sets of grids. The simulation results of 188 800 and 251 730 grid cells tend towards identical: they are in good agreement with experimental results. The grid system of 188 800 cells is considered grid-independent.

Fig. 3Variations ofC/C0at (2.25 m, 0.225 m, 0.065 m) for four computational zones in air

Fig. 4Variations ofC/C0at (2.25 m, 0.225 m, 0.065 m) for four sets of grids

4.2.2 Comparison of anisotropic coupling diffusion model, isotropic coupling diffusion model, and non-coupling diffusion model

Fig. 5Variations ofin water simulated with three models

Fig. 6Variations ofat pointsy= 0.225 m,z= 0.248 m at differentxvalues

4.2.3 Effect of air-water relative velocity on anisotropic coupling diffusion

Fig. 7Variations ofat different relative velocities

Fig. 8Concentration distributions of C2HCl3aty= 0.2 m at different relative velocities

Fig. 8 shows the calculated concentration distribution of C2HCl3at the longitudinal cross-sectiony= 0.2 m at different relative velocitiesuG?uL=?0.2 m/s (case 1), 0.2 m/s(case 2), 0.6 m/s (case 3), and 1.0 m/s (case 4), at 8 seconds. It can be seen that the C2HCl3concentrations in the gas and liquid phases are solved simultaneously. The concentration distribution at the air-water interface shows no abrupt change, indicating that the transition function in STDS is valid. The concentration distribution in case 1 is similar to that in case 2. The space occupied by gas phase C2HCl3is much larger in case 3 and case 4 than in other cases. The concentration of gas-phase C2HCl3is much higher in case 4 than in other cases, which explains why the relative velocity has a significant effect on the concentration of gas-phase C2HCl3in the air.

5 Conclusions

(1) Momentum and mass exchange caused by turbulent fluctuation is stronger in the direction normal to the air-water interface than in other directions due to the large density difference between air and water. Therefore, mass transfer of pollutants at the air-water interface shows anisotropic diffusion properties. The relationship between the velocity gradient and the mass transfer rate was used to calculate the turbulent mass diffusivity in the direction normal to the air-water interface in this study.

(2) The air-water coupling diffusion model for volatile pollutants was used to simulate the temporal and spatial distribution of C2HCl3concentration after C2HCl3was released. STDS was proposed to connect the first-order upwind differencing and central differencing smoothly, which not only ensures the boundedness of the air-water interface but also maintains the sharpness of the interface. The simulation results of the anisotropic coupling diffusion model are closer to the experimental results, compared with those of the isotropic coupling diffusion model and non-coupling diffusion model.

(3) Mass transfer of volatile pollutants at the air-water interface is affected by the relative velocity of water and air. At a low relative velocity, the gas phase resistance is close to the liquid phase resistance. Therefore, the gas phase resistance decreases and mass transfer increases obviously with the increase of the relative velocity. However, at a higher relative velocity, the increase of the relative velocity has no effect on mass transfer because the gas phase resistance is much smaller than the liquid phase resistance.

Afkhami, S., Zaleski, S., and Bussmann, M. 2009. A mesh-dependent model for applying dynamic contact angles to VOF simulations. Journal of Computational Physics, 228(15), 5370-5389. [doi:10.1016/ j.jcp.2009.04.027]

Bade, D. L. 2009. Gas exchange at the air-water interface. Encyclopedia of Inland Waters, 3, 70-78. [doi:10.1016/B978-012370626-3.00213-1]

Chen, L. P., Jiang, J. C., and Han, D. M. 2009. Model for coupling diffusion of volatile poisons in water and air. Advances in Water Science, 20(4), 549-553. (in Chinese)

Danckwerts, P. V. 1955. Gas absorption accompanied by chemical reaction. AIChE Journal, 1(4), 456-463. [doi:10.1002/aic.690010412]

Deacon, E. L. 1977. Gas transfer to and across an air-water interface. Tellus, 29(4), 363-374. [doi:10.1111/j.2153-3490.1977.tb00746.x]

Hardt, S., and Wondra, F. 2008. Evaporation model for interfacial flows based on a continuum-field representation of the source terms. Journal of Computational Physics, 227(11), 5871-5895. [doi: 10.1016/j.jcp.2008.02.020]

Hasegawa, Y., and Kasagi, N. 2009. Hybrid DNS/LES of high Schmidt number mass transfer across turbulent air-water interface. International Journal of Heat and Mass Transfer, 52(3-4), 1012-1022. [doi:10.1016/ j.ijheatmasstransfer.2008.07.015]

J?hne, B., and Haubecker, H. 1998. Air-water gas exchange. Annual Review of Fluid Mechanic, 30, 443-468. [doi:10.1146/annurev.fluid.30.1.443]

Jasak, H., Weller, H. G., and Gosman, A. D. 1999. High resolution NVD differencing scheme for arbitrarily unstructured meshes. International Journal of Numerical Methods in Fluids, 31(2), 431-449. [doi:10.1002/(sici)1097-0363(19990930)31:2<431::aid-fld884>3.3.co;2-k]

Komori, S., Murakami, Y., and Ueda, H. 1989. The relationship between surface-renewal and bursting motions in an open-channel flow. Journal of Fluid Mechanics, 203, 103-123. [doi:10.1017/ S0022112089001394]

Lamont, J. C., and Scott, D. S. 1970. An eddy cell model of mass transfer into the surface of a turbulent liquid. AIChE Journal, 16(4), 513-519. [doi:10.1002/aic.690160403]

Lamourelle, A. P., and Sandall, O. C. 1972. Gas absorption into a turbulent liquid. Chemical Engineering Science, 27(5), 1035-1043. [doi:10.1016/0009-2509(72)80018-6]

Law, C. N. S., and Khoo, B. C. 2002. Transport across a turbulent air-water interface. AIChE Journal, 48(9), 1856-1868. [doi:10.1002/aic.690480904 ]

Lee, R. J., and Saylor, J. R. 2010. The effect of a surfactant monolayer on oxygen transfer across an air/water interface during mixed convection. International Journal of Heat and Mass Transfer, 53(3), 3405-3413. [doi:10.1016/j.ijheatmasstransfer.2010.03.037]

Liss, P. S., and Slater, P. G. 1974. Flux of gases across the air-sea interface. Nature, 247(5438), 181-184. [doi:10.1038/247181a0]

Magnaudet, J., and Calmet, I. 2006. Turbulent mass transfer through a flat shear-free surface. Journal of Fluid Mechanics, 553,155-185. [doi:10.1017/S0022112006008913]

Marrone, S., Colagrossi, A., Touzé, D. L, and Graziani, G. 2010. Fast free-surface detection and level-set function definition in SPH solvers. Journal of Computational Physics, 229(10), 3652-3663. [doi:10.1016/ j.jcp.2010.01.019]

Rashidi, M., Hetsroni, G., and Banerjee, S. 1991. Mechanism of heat and mass transport at gas-liquid interfaces. International Journal of Heat and Mass Transfer, 34(7), 1799-1810. [doi:10.1016/0017-9310(91)90155-8]

Schwarzenbach, R. P., Gschwend, P. M., and Imboden, D. M. 2002. Environmental Organic Chemistry. 2nd ed. Wiley-Interscience.

Whitman, W. G. 1962. The two-film theory of gas absorption. International Journal of Heat and Mass Transfer, 5(5), 429-433. [doi:10.1016/0017-9310(62)90032-7]

(Edited by Yan LEI)

This work was supported by the National Natural Science Foundation of China (Grant No. 51109106) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 11KJB570001).

*Corresponding author (e-mail: clpjoy@njut.edu.cn)

Received Feb. 14, 2012; accepted Dec. 21, 2012