Interaction between a Liquid Surface and an Impinging Gas Jet

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1. Introduction

Impinging of gas jets in liquid surfaces is employed for different purposes in several industries, e.g. metallurgical, gas cleaning, mixing, and so on. Interaction between a liquid surface and an impinging gas jet has been studied for a long time. Different approaches have been used to study this problem: analytical [1], numerical [2], and experimental [3].

During the interaction between a liquid surface and an impinging gas jet, three main stages are present. These are dimpling, splashing, and penetration [2] [4] [5]. Dimpling is the formation of a shallow depression in the liquid surface at low velocity of the gas jet or at high lance injection height. During splashing, due to momentum transfer, shear stresses exceed surface tension and gravitational forces, and as a consequence drops are ejected from the liquid surface. The splashing phenomenon arises when the jet velocity is increased beyond a critical value, which depends on the geometric configuration of the system and the physical properties of the involved phases. The above phenomenon has been previously numerically analyzed by some of the authors of the present work [6] [7]. In the penetration stage, the depth of the cavity is significantly increased. The cavity depth is directly proportional to the jet velocity and inversely proportional to the liquid density and the height of the injection lance.

In the vast majority of published works, it is considered that the ratio between the height of the injection lance above the liquid surface (H) and the diameter of such lance (D), as is depicted in Figure 1, is greater than 10, *i.e.* H/D > 10. There are few studies reported for H/D < 10. Among the latter is a recent work [8], in which the water-air and the Wood’s metal-air systems interactions are experimentally and numerically simulated for H/D values of 0.8, 1.7 and 3.

Here, experimental work reported in [8] is simulated by means of Computational Fluid Dynamics (CFD) to investigate the effect of the velocity of the air jet, the height of the injection lance, and the density of the liquid on the depth of the formed cavity. The results on the cavity depth are compared with results of other authors. In addition, the emergence of the splashing phenomenon is predicted in terms of the critical velocity of each system. The oscillation of the liquid surface is not considered here since the turbulence model used generates only average

Figure 1. The liquid-air systems considered [8].

values of the variables involved. The system that is simulated in this work, and that is the same one used in [8], is shown in Figure 1.

2. Mathematical Model and Computer Simulations

Computational Fluid Dynamics (CFD) is employed to study the interaction between a liquid surface and an impinging air jet. The air jet is vertically impinged onto a liquid (water or Wood’s metal) surface through a nozzle located at the top of the liquid surface. The process is considered isothermal. The equations of continuity and momentum, the K-*ε* model for turbulence and the Volume of Fluid model for multiphase flow are employed in 2D CFD transient numerical simulations. Meshing of the 2D geometrical model yielded 69,824 trilateral cells.

A blowing number (*N _{B}*) is defined as follows in [9] [10] [11]:

${N}_{B}=\frac{{\rho}_{g}{\left(\eta {u}_{j}\right)}^{2}}{2\sqrt{{\sigma}_{l}g{\rho}_{l}}}$ (1)

where *u _{j}* is the jet velocity,

${u}_{cr}={\left\{\frac{48g{\sigma}_{l}\left({\rho}_{l}-{\rho}_{g}\right)}{{\rho}_{g}^{2}}\right\}}^{0.25}$ (2)

3. Results for Water

3.1. Splashing

In the CFD simulations, the physical properties of the fluids involved are shown in Table 1.

For the air-water system, and using the above values of the physical properties, Equation (2) yields a critical velocity for splashing *u _{cr}* = 12.53 m/s. As in [8], three jet velocities (

Table 1. Physical properties of the considered fluids [8].

Table 2. Blowing number, drop generation rate, and splashing conditions for the considered jet velocities.

splashing are satisfied, *i.e.* *N _{B}* > 1 and

${G}_{d}=\frac{{Q}_{g}{N}_{B}^{3.2}}{{\left[2.6\times {10}^{6}+2.0\times {10}^{-4}{N}_{B}^{12}\right]}^{0.2}}$ (3)

where *Q _{g}* is the air volumetric flow rate, m

${Q}_{g}=\left(\frac{\pi {D}^{2}}{4}\right){u}_{j}$ (4)

Table 2 shows that for *u _{j}* = 9.46 m/s the drop generation rate is not significant, and therefore neither the splashing is significant.

3.2. Cavity Depth

The cavity depth *h*, as is depicted in Figure 1, was determined from the results of the CFD simulations, and compared with the experimental results of [8]. Besides, *h* was estimated from the expressions proposed in [3] and [12]. In [3] *h* is estimated from a modified Froude number *Fr _{m}*:

$h=4.1D{\left(F{r}_{m}\right)}^{1/3}$ (5)

In the above equation, *Fr _{m}* is defined as

$F{r}_{m}=\frac{{\left(4/\pi \right)}^{2}{\rho}_{g}{u}_{i}^{2}}{{\rho}_{l}gD}$ (6)

where *u _{i}* is the impact velocity of the air jet. The impact velocities of the jets were obtained by scanning the vertical components of the jet velocity in the vicinity of the liquid-gas interface, as is depicted in Figure 2. Due to jet decaying, the impact velocity is less than the nominal jet velocity.

In [12] an implicit expression for describing the cavity depth is proposed:

$\frac{M\mathrm{cos}\theta}{{\rho}_{l}gn{H}^{3}}=\frac{\pi h}{2{K}^{2}\eta H}\left[{\left(1+\frac{h}{H}\right)}^{2}+\frac{4{\beta}_{*}{\sigma}_{l}}{{\rho}_{l}g{H}^{2}}\right]$ (7)

where *M *is the momentum flow rate of the air jet, *θ* is the lance inclination angle, *n* is the number of nozzles,*K* = 11.5 is a constant, *η* is a constant which depends on the lance height *H* (see Figure 1), and *β*_{*} = 125 is a constant too. *M* is determined from the expression:

Figure 2. Impact velocity of the air jets for *H* = 0.008 m. Jet velocity: Black = 9.46 m/s, red = 18.5 m/s, blue = 27.4 m/s.

$M=\left(\frac{\pi}{4}\right){\rho}_{g}{D}^{2}{u}_{j}^{2}$ (8)

For its part *η* depends on *H* in an empirical way as follows [12]:

$\eta =a{H}^{2}+bH+c$ (9)

where *a* = 15.84127, *b* = −2.57058, and *c* = 0.48989. Besides, in Equation (7) *θ* = 0 and *n* = 1 for the liquid-air system of Figure 1 considered here.

A comparison is made for the cavity depth among the CFD results of this work and the experimental results of [8] and the analytical results of [3] [12]. This comparison is observed in Table 3, and is graphically depicted in Figure 3. It is observed that as the jet velocity is increased, a small underestimation (less than 0.005 m) of the results of [3] and this work arises related to the experimental results of [8]. Inversely, an overestimation of around 0.005 m is present between the depth cavity results of [12] and the experimental results of [8].

The dependence of the cavity depth on the lance height and the jet velocity was numerically explored. Results are depicted in Figure 4. Presumably, cavity depth is promoted by high momentum transfer, consequently cavity depth is increased as the jet velocity is increased or the height of the injection lance is decreased. This is corroborated in Figure 4.

3.3. Water Surface-Air Jet Interaction

During the interaction between a liquid surface and an impinging gas jet three main successive stages arise. These stages are dimpling, splashing, and penetration [2] [4] [5]. Dimpling is the formation of a shallow depression in the liquid surface at a low velocity of the gas jet or at a high height of the injection lance (see Figure 1). As the jet velocity is increased and it exceeds a certain critical value given by Equation (2), the splashing phenomenon arises. During

Table 3. Comparison of the cavity depth among the CFD results of this work and those reported by other researchers, for *H* = 0.008.

Figure 3. Comparison of the cavity depth among the CFD results of this work and those reported by other researchers, for *H* = 0.008. Experimental [8] = black, based on Froude number [3] = red, based on jet momentum [12] = green, CFD (this work) = blue.

Figure 4. Cavity depth as function of lance height. Jet velocity: Black = 9.46 m/s, red = 18.5 m/s, blue = 27.4 m/s.

splashing small liquid drops are ejected from the liquid surface due to that shear stresses in the liquid surpass gravitational and surface tension forces. In the penetration stage the cavity depth is directly proportional to the jet velocity and inversely proportional to the lance height.

In order to study the interaction between the water surface and the impinging air jet, several CFD transient 2D numerical simulations were carried out. As in [8], three values of the jet velocity were considered, namely 9.46, 18.5 and 27.4 m/s, respectively. Besides, three values of the lance height were assumed: 0.008, 0.017 and 0.03 m. In accordance to Table 2, no splashing was expected for a jet velocity of 9.46 m/s. This is appreciated in Figure 5, in which just a small depression is formed and no splashing is detected.

A sequential dynamic behavior consisting in dimpling, splashing and penetration stages during water surface-air jet interaction is predicted if jet velocity becomes beyond the critical value of 12.53 m/s, as was explained in Section 3.1. Figure 6

Figure 5. Phases distribution and streamtraces for a jet velocity of 9.46 m/s and a lance height of 0.008 m. Red phase is water, blue phase is air.

Figure 6. Dimpling, splashing and penetration stages. Jet velocity = 27.4 m/s, lance height = 0.008 m. Time from the start of blowing: Left = 0.02 s, center = 0.1 s, right = 0.4 s. Red phase is water, blue phase is air.

depicts the above behavior for a jet velocity of 27.4 m/s and a lance height of 0.008 m.

Figure 7 summarizes the water surface-air jet interaction for the values of the jet velocity *u _{j}*(9.46, 18.5, 27.4 m/s), and the values of the lance height

Figure 7. Water surface-air jet interaction for several values of jet velocity and lance height. First column: *u _{j}* = 9.46 m/s, second column:

the key to explain that interaction. The first column corresponds to *u _{j}* = 9.46 m/s, for which

4. Results for Wood’s Metal

CFD numerical simulations were also carried using Wood’s metal instead of water. The physical properties of the Wood’s metal are shown in Table 1. Using these properties in Equation (2) yields *u _{cr}* = 34.50 m/s for this liquid.

Figure 8. Phases distribution for *u _{j}* = 27.4 m/s and

Figure 9. Cavity depth as function of the jet velocity for Wood’s Metal and water considering a lance height of 0.008 m. Black = water, red = Wood’s metal.

Figure 8 shows a comparison of the interactions between the water-air system (left) and the Wood’s metal-air system (right) using the same jet velocity, 27.4 m/s, and the same lance height, 0.008 m, in the computer runs. For water, intense splashing is expected, as is observed in the left side of Figure 8. For the Wood’s metal no splashing is expected whatever the value of the jet velocity given that at no time it is verified that *u _{j}* >

Figure 9 shows the cavity depth as function of the jet velocity for Wood’s Metal and water considering a lance height of 0.008 m. Since the density of the Wood’s metal is much higher than that of the water, the cavity height is expected to be much lower for the first one than for the second one regardless of the height of the lance used. This is because the gravitational forces exerted on Wood’s metal are greater than those exerted on water. The surface tension, in accordance to [8], is not relevant to determine the cavity depth.

5. Conclusions

The system proposed by Sato *et al.* [8] was analyzed by means of CFD simulations to study the interaction between a liquid surface and an impinging air jet under the near field blowing conditions. The effect of the air jet velocity, the height of the injection lance, and the density of the liquid in the cavity depth was analyzed using water and Wood’s metal as liquids. In addition, the emergence of the splashing phenomenon was predicted in terms of critical velocity.

From the results of the numerical simulations the following conclusions arise:

1) The three stages (dimpling, splashing, penetration) experimentally and numerically reported in the literature during the interaction of a liquid surface with an impinging gas jet are adequately reproduced by the CFD simulations.

2) The critical velocity of the water-air system provides a useful criterion to predict the splashing phenomenon. For jet velocity below 12.53 m/s no splashing arises.

3) The blowing number provides correct information, at least qualitatively, on the drop generation rate.

4) The variables studied here (jet velocity, injection lance height, liquid density) significantly affect the depth of the cavity.

5) The depth of the cavity is directly proportional to the speed of the jet and inversely proportional to the height of the lance and the density of the liquid.

6) Under the operating conditions simulated here, a very high density of the liquid can inhibit the splashing phenomenon.

References

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