The gas entrainment by a plunging liquid jet may occur in many problems of practical interest. A good understanding of the air carryunder and bubble dispersion process associated with a plunging liquid jet is vital if one is to be able to quantify such phenomena as sea surface chemistry, the meteorological and ecological significance of (breaking) ocean waves, the performance of certain type of chemical reactors, and the "Greenhouse Effect" (i.e., the absorption of CO_{2} by the oceans). Indeed, the absorption of greenhouse gases into the ocean has been hypothesized to be highly dependent upon the air carryunder that occurs due to breaking waves. This process can be approximated with a plunging liquid jet (Monahan, 1991; Kerman, 1984).
A number of prior experimental studies have been performed in which axisymmetric plunging liquid jets have been used to investigate the air carryunder process. These include the work of Lin and Donnelly (1966), Van De Sande and Smith (1973), and McKeogh and Ervine (1981). Recent experiments include those of Chanson and Cummings (1994) and Bonetto and Lahey (1993).
As shown in Figure 1, a converging nozzle oriented vertically produces an axisymmetric liquid jet. This jet impacts orthogonally a pool of water and, when a threshold velocity was exceeded, the plunging liquid jet causes significant air entrainment (Bonetto and Lahey, 1993). In agreement with the observations of McKeogh and Ervine (1981), different two-phase jet characteristics were noted by Bonetto and Lahey (1993), depending on the turbulence intensity of the plunging liquid jet. For a laminar axisymmetric liquid jet (i.e., one having a turbulence intensity less than about 0.8%), the diameter of the entrained bubbles was in the range of 15-300 μm. On the other hand, for liquid jet turbulence intensities of about 3%, the entrained air bubbles had diameters in the range of 1-3 mm. (See Turbulence and Turbulent Flow.)
Two different methods were used for the measurement of void fraction in the two-phase jet: a KfK impedance probe and a DAN-TEC Fiber-optic Phase Doppler Anemometer (FPDA). The turbulence intensity of the liquid jet at the nozzle exit was found to be one of the most important parameters affecting jet roughness and the size of the bubbles entrained by the axisymmetric plunging liquid jet. As can be seen in Figure 1, an arrangement of honeycombs and screens was used to control the turbulence intensity of the flow entering the nozzle.
Figure 2 depicts a contour plot of the two-dimensional probability density function of the bubble diameters (D_{b}) and axial velocities (v_{z}) for an axisymmetric nozzle. Quantitatively the most probable value of peak no. 1 (liquid velocities) was at D_{b} = 5 mm, v_{z} = 4.05 m/s, and the most probable value of peak no. 2 (gas velocities) was at D_{b} = 125 mm, v_{z} = 3.5 m/s.
Figure 2. Contour plot of the two-dimensional probability density function-smooth jet (z = 35.1 mm, w_{1} = 0.143 kg/s; h = 9.0 mm; y = 0).
Figure 3 presents the mean liquid and gas velocities and the turbulent intensity of the liquid and gas velocities as a function of lateral position (y) for the liquid jet and for h = 8.5 mm, w_{1} = 1.8 kg/s and z = 31.0 mm. We see that at the edge of the spreading two-phase jet the gas (bubble) velocity is negative, indicating buoyancy-driven countercurrent flow. It was also found that the two-phase jet was more dispersed than the corresponding single-phase flow case and that the turbulence intensity was higher. This turbulence enhancement is presumably due to bubble-induced turbulence. In this case the bubble-induced turbulence accounts for about 30% of the total turbulence level.
As noted previously, when the liquid jet impacts the pool surface, air entrainment occurs around the jet's perimeter. In Figure 4athe measured local void fraction is presented as a function of y for z = 1.0 mm (i.e., with the probe 1 mm under the undisturbed liquid level) for a planar liquid jet (Bonetto and Lahey, 1994). We see that the void fraction has a maximum at y approximately equal to 3 mm, and this peak disperses with z. Obviously, the air entrainment process is responsible for this effect. In the high-speed video visualization of these experiments it was rare to observe bubbles at the liquid jet's centerline for z < 10 mm. However, once the air was entrained, lateral dispersion of the gas phase occurred as z was increased. Figure 4b shows how the void peak was dispersed at z = 26.0 mm. We see that the maximum now occurs at y equal to 5.5 mm. Moreover, we see that there is significant void fraction at y = 0 (i.e., the jet's centerplane) because of the void dispersion process. Figure 4c shows the void fraction profile at z = 58.8 mm. Significantly, the curve now tends towards a maximum at the centerplane of the jet (y = 0). Again, this is a direct result of the void dispersion process in the two-phase jet.
For a turbulent (i.e., rough) liquid jet the entrained bubble sizes were of the order of D_{b } 3 mm, and the slip ratio was close to calculated values based on the terminal rise velocity of a single bubble. Moreover, the turbulence intensity of the liquid jet had two components, one due to shear-induced turbulence and the other due to bubble-induced pseudo-turbulence. Both components of turbulence were of the same order to magnitude.
The mechanism that produces the gas entrainment currently is not well understood. Lezzi and Prosperetti (1991) have proposed that the instability responsible for the air entrainment was caused by the gas viscosity. In particular, they studied the linear stability of a vertical film of a viscous gas bounded by an inviscid liquid in uniform motion on one side, and by inviscid liquid at rest on the other side. They also obtained the marginal stability boundary with the gas gap width, d, as the control parameter (i.e., they numerically compute for a given d the range of wavelengths that makes the system unstable).
Bonetto et al. (1994) proposed that a Helmholtz-Taylor instability is responsible for the air entrainment. They assumed the liquid jet, the liquid in the pool and the gas in the gap to be inviscid fluids. A linear stability analysis was performed on the system, the perturbation being a sinusoidal wave on the liquid jet/gas gap interface. The celerity of the perturbation was calculated using the appropriate conservation equations. Also the sinusoidal perturbation imposed on the liquid jet/gas gap interface produces a sinusoidal perturbation on the gas gap/pool liquid interface with the same celerity and wave number and is out-of-phase (i.e., phase = 180°). Figure 5 presents a schematic of the air entrainment process. Two interfacial waves of small amplitude and wavelength λ_{d}) grow as they move with celerity C. At a certain position they may touch each other entrapping a volume of air proportional to the shaded area. The shaded area, Λ, is given by Λ = λ_{d}δ.
Figure 6 shows the volumetric flow rate of air, , measured by McKeogh and Ervine (1981) as a function of the liquid jet velocity, v_{z} for a liquid jet diameter of, D_{jet} = 0.0051 m. The jet turbulence level in these experiments was 3% and the distance of the nozzle above the pool's surface was h = 0.03 m. The solid line corresponds to the theoretical predictions by Bonetto et al. (1994) for δ = 0.291 mm. Significantly, no value of δ used with the results of Lezzi and Prosperetti (1991) agrees with McKeogh and Ervine experiments, thus it appears that Helmholtz-Taylor instability is the dominant air entrainment mechanism.
The spreading of a bubbly two-phase jet involves the interaction between the liquid turbulence and the bubbles. For the computation of jet dispersion it is important to appropriately model the turbulent intensity of the continuous phase. Rodi (1984) presented results using the classical k-ε model of Gibson and Launder (1976), and showed that k-ε models do not accurately predict jet spreading. Sini and Dekeyser (1987) solved the single-phase turbulent jet using Rodi's k-ε model. Significantly, it has been found that single-phase turbulent jet data can be used for the assessment of Turbulence Models because one does not have to constitute complicated turbulent closure laws, such as those required near solid (no slip) boundaries. Indeed, due to the absence of walls and the associated shear boundary conditions the turbulent jet is probably the simplest non-trivial case to analyze. Interestingly, the same conclusions can be reached for a two-phase turbulent jet.
Most researchers have analyzed liquid jets using a parabolic scheme instead of an elliptic one. Unfortunately, using a parabolic scheme one cannot compute the pressure distribution field accurately, more importantly one cannot compute recirculating flows (such as occur in vertical two-phase downflowing jets) and the downstream pressure has to be (arbitrarily) specified. Solving the partial differential equations as an elliptic system increases the complexity of the numerical problem but provides more detailed and accurate information on the flow field (pressure) than a parabolic scheme.
We note that the turbulence present in the liquid has two components in a two-phase jet. One component, the shear-induced turbulence, is due to viscosity and it is present in both single- and two-phase flows. The other component is the bubble-induced turbulence due to slip between the bubbles and the surrounding liquid, and it only occurs in two-phase flows.
A state-of-the-art multidimensional two-fluid model obtained using ensemble averaging has been derived and was closed using cell average model (Arnold et al., 1989). This approach provides equations for multiphase flows that are mechanistically based (as opposite to being empirical). The two-fluid model's conservation equations and the associated k-ε turbulence model were numerically integrated using a CFD code, PHOENICS. Figure 7a shows the computed v_{z} at z = 0 and Figure 7b at z = 31 mm. The open circles are experimental points. The spreading of the jet is well predicted. Significantly, the underprediction at the centerline is similar to that observed in single-phase flows (Rodi, 1984). Figure 7c shows the computed vz at z = 59 mm and the trends are similar to Figure 7b. The agreement with the experimental data is quite good, indicated that suitable formulated two-fluid models are able to predict two-phase jet flows.
Figure 7a. Axial velocity as a function of the lateral coordinate (z = 0.0). |
Figure 7b. Axial velocity as a function of the lateral coordinate (z = 31 mm). |
Figure 7c. Axial velocity as a function of the coordinate (z = 59 mm). |
Figure 7.
REFERENCES
Arnold, G., Drew, D. A., and Lahey, R. T., Jr. (1989) Derivation of constitutive equations for interfacial force and reynolds stress for a suspension of spheres using ensemble averaging, J. Chem. Eng. Comm., 86, 43-54.
Bonetto, F. and Lahey, R. T., Jr. (1993) An experimental study on air carryunder due to a plunging liquid jet, Int. J. Multiphase Flow, 19, 2, 281-294. DOI: 10.1016/0301-9322(93)90003-D
Bonetto, F., Drew, D. A., and Lahey, R. T., Jr. (1994) The analysis of a plunging liquid jet—the air entrainment process, J. Chem. Eng. Comm., 130, 11-29.
Bonetto, F. and Lahey, R. T., Jr. (1994) Experimental Results of a Two-Phase Planar Jet, in preparation.
Chanson, H. and Cummings, P. (1994) Modeling Air Bubble Entrainment by Plunging Breakers, Proceedings of the Int. Symp. Waves, 783-793, held in Univ. of Columbia, Vancouver, Canada, Aug. 21-24.
Delhaye, J. M. (1968) Equations Fondamentales des Ecoulement Dipha-siques, Parts I and II, CEA-R-3429, Centre d'Etudes Nucleaires de Grenoble, France.
Drew, D. A. and Lahey, R. T., Jr. (1979) Application of general constitutive principles to the derivation of multidimensional two-phase flow equations, Int. J. Multiphase Flow, 5, 423-264. DOI: 10.1016/0301-9322(79)90024-7
Ishii, M. (1975) Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles.
Lezzi, A. M. and Prosperetti, A. (1991) The stability of an air film in a liquid flow, J. Fluid Mech., 226, 319-347.
Lin, T. J. and Donnelly, H. O. (1966) Gas bubble entrainment by plunging laminar liquid jets, AJChE J., 12-3, 563.
McKeogh, E. J. and Ervine, D. A. (1981) Air entrainment rate and diffusion pattern of plunging liquid jets, Chem. Eng. Sci., 36, 1161. DOI: 0009-2509(81)85064-6
Park, J., W. (1992) Void wave propagation in two-phase flow, Ph.D. Thesis, Rensselaer Polytechnic Institute.
Rodi, W. (1984) Turbulence Models and Their Application in Hydraulics, IAHR/AIRH Monograph.
Sini, J. F., and DeKeyser, I. (1987) Numerical prediction of turbulent plane jets and forced plumes by use of the k-e model of turbulence, Int. J. Heat Mass Transfer, 30, 9, 1787-1801. DOI: 0017-9310(85)90187-5
Les références
- Arnold, G., Drew, D. A., and Lahey, R. T., Jr. (1989) Derivation of constitutive equations for interfacial force and reynolds stress for a suspension of spheres using ensemble averaging, J. Chem. Eng. Comm., 86, 43-54. DOI: 10.1080/00986448908940362
- Bonetto, F. and Lahey, R. T., Jr. (1993) An experimental study on air carryunder due to a plunging liquid jet, Int. J. Multiphase Flow, 19, 2, 281-294. DOI: 10.1016/0301-9322(93)90003-D
- Bonetto, F., Drew, D. A., and Lahey, R. T., Jr. (1994) The analysis of a plunging liquid jetâ€”the air entrainment process, J. Chem. Eng. Comm., 130, 11-29.
- Bonetto, F. and Lahey, R. T., Jr. (1994) Experimental Results of a Two-Phase Planar Jet, in preparation.
- Chanson, H. and Cummings, P. (1994) Modeling Air Bubble Entrainment by Plunging Breakers, Proceedings of the Int. Symp. Waves, 783-793, held in Univ. of Columbia, Vancouver, Canada, Aug. 21-24.
- Delhaye, J. M. (1968) Equations Fondamentales des Ecoulement Dipha-siques, Parts I and II, CEA-R-3429, Centre d'Etudes Nucleaires de Grenoble, France.
- Drew, D. A. and Lahey, R. T., Jr. (1979) Application of general constitutive principles to the derivation of multidimensional two-phase flow equations, Int. J. Multiphase Flow, 5, 423-264. DOI: 10.1016/0301-9322(79)90024-7
- Ishii, M. (1975) Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles.
- Lezzi, A. M. and Prosperetti, A. (1991) The stability of an air film in a liquid flow, J. Fluid Mech., 226, 319-347. DOI: 10.1017/S0022112091002409
- Lin, T. J. and Donnelly, H. O. (1966) Gas bubble entrainment by plunging laminar liquid jets, AJChE J., 12-3, 563. DOI: 10.1002/aic.690120331
- McKeogh, E. J. and Ervine, D. A. (1981) Air entrainment rate and diffusion pattern of plunging liquid jets, Chem. Eng. Sci., 36, 1161. DOI: 0009-2509(81)85064-6
- Park, J., W. (1992) Void wave propagation in two-phase flow, Ph.D. Thesis, Rensselaer Polytechnic Institute.
- Rodi, W. (1984) Turbulence Models and Their Application in Hydraulics, IAHR/AIRH Monograph.
- Sini, J. F., and DeKeyser, I. (1987) Numerical prediction of turbulent plane jets and forced plumes by use of the k-e model of turbulence, Int. J. Heat Mass Transfer, 30, 9, 1787-1801. DOI: 0017-9310(85)90187-5