# A Review of Bubble Dynamics in Liquid Metals

^{*}

## Abstract

**:**

## 1. Introduction

_{C}and L

_{C}are the characteristic velocity and length, respectively. For this, different values are employed, though for bubbly flows, the bubble size, expressed as the diameter of a sphere with the same volume and the relative velocity between bubble and liquid, is commonly utilized. If other quantities are used, this is explicitly mentioned in the text.

## 2. Measurement Methods

#### 2.1. Theoretical Investigations

#### 2.2. Experimental Measurements

#### 2.3. Direct Numerical Simulation

^{9/4}. This implies that computations are limited to relatively small computational areas with low turbulence (low Re) and only a few bubbles.

## 3. Bubble Formation Mechanisms

#### 3.1. Bubble Generation at Single Nozzles

_{or}, where the nozzle diameter and the exit velocity are the characteristic scales, and depend on the orifice inner diameter d

_{ni}. In contrast to that, at turbulent outflow conditions, the bubble size is considerably smaller and almost independent of the Reynolds number and the orifice diameter.

_{g}flow rates up to 250 cm

^{3}/s:

_{17}Pb

_{83}[16], GaInSn [22], or aluminum (Al) [48] were investigated. Thus, a broad range of physical properties is covered.

_{ni}in Equations (4)–(6) has to be replaced by the outer diameter d

_{no}in non-wetting systems. Irons and Guthrie [18] made similar observations, but found less satisfying agreement with the correlation of Davidson and Amick (Equation (5)) or Mersmann (Equation (6)). This was attributed to the fact that the experimental setup, in particular the chamber volume of the injector, differed between the experiments, which had a large influence on the resulting bubble size. Interestingly, both measurements confirm the observations from the water experiments that the bubble size becomes independent of the fluid’s material properties for sufficiently large flow rates. Thus, this observation seems to be valid for systems with significantly higher surface tensions and density differences like molten metals. Based on the correlation of Mersmann (Equation (6)), Mori et al. [46] derived a first correlation of the bubble size in liquid metals (note that this correlation was first published in 1977 in Japanese by Sano and Mori and is therefore often called Sano and Mori equation):

_{2}O. It should be noted, however, that this correlation was derived for very small flow rates Q

_{g}≤ 70 cm

^{3}/s. This is primarily due to the indirect measurement technique. This has the additional disadvantage that no conclusions can be drawn whether the bubbles disintegrate during injection or on their rising path. Therefore, subsequent studies used resistivity measurements.

_{0}is the distance between the nozzle and the mathematical origin of the plume in m (correlation given in [47]).

**Table 1.**Studies on bubble generation in liquid metals (Pres Pul = indirect pressure pulse method, Ac = indirect acoustic method, Res = Electric resistivity, d

_{no}= Outer diameter of the nozzle, d

_{ni}= Inner diameter of the nozzle, BSD = Bubble size distribution).

Ref. | Year | Method | Nozzle [cm] | Flowrate [cm^{3}/s] | Fluid | Result |
---|---|---|---|---|---|---|

Sano & Mori [15] | 1976 | Pres Pul | d_{no}: 0.22–0.82d _{ni}: 0.1–0.3 | 0.0167–70 | Hg/Ag | Correlations from water can be used in case the O.D. instead of the I.D is used (non-wettability) |

Andreini et al. [17] | 1977 | Ac | d_{ni}: 0.015–0.1 | Cu/Pb/Sn | Bubble size depend on We and Fr, not on Re. Water model correlations cannot be used | |

Irons & Guthrie [18] | 1978 | Ac | d_{no}: 0.64–5.1, d _{ni}: 0.16–0.64 | 0.5–1000 | Fe | Use of O.D because of non-wettability, BSD becomes independent of fluid properties at higher flow rates |

Mori et al. [46] | 1979 | Pres. Pul | d_{no}: 0.32–0.68d _{ni}: 0.12–0.33 | 0.1–36 | Fe | Empirical Equation (7) for various liquids. At high flow rates, BSD becomes independent of the fluid |

Sano & Mori [42] | 1980 | Res | d_{no}: 0.4–1.0d _{ni}: 0.2–0.6 | 50–1330 | Hg | Good agreement with water model correlation, BSD is independent of fluid properties, empirical correlation Equation (9) |

Mori et al. [20] | 1982 | Res | d_{ni}: 0.1–0.4 | 0.05–4500 | Hg | Good agreement with Equation (6) at higher flow rates |

Xie et al. [47] | 1992 | Res | d_{ni}: 0.2–0.5 | 100–1200 | Woods | BSD follows log-normal distribution, BSD becomes independent of nozzle, BSD is result of breakup |

Tsuchiya et al. [16] | 1993 | Pres Pul | d_{no}: 0.2–0.64d _{ni}: 0.1–0.4 | 0.5–10 | H_{2}0/HgCH _{3}OHLi _{17}Pb_{83} | Fitting parameters of Equation (7) depend on nozzle orientation and fluid properties |

Iguchi et al. [21] | 1995 | Res | d_{no}: 0.6d _{ni}: 0.1 | 50–100 | Fe | Larger bubbles than predicted by Equation (7) |

Iguchi et al. [26] | 1995 | X-ray | d_{no}: 0.13–0.45d _{ni}: 0.09–0.19 | 20–413 | Fe | At high flow rates, BSD depend on I.D, not O.D. BSD depends on fluid and gas properties |

Iguchi et al. [50] | 1997 | Res | d_{ni} = 0.005 | 60 | Woods | Confirmed Equation (10) |

Gnyloskurenko & Nakamura [48] | 2003 | X-ray | d_{no}: 0.2–1d _{ni}: 0.1–0.4 | 0.43–12 | Al | Non-wettability of the nozzle increases bubble size |

#### 3.2. Bubble Generation from Porous Plugs

_{max}/r

_{or}, where y

_{max}is the maximum radius of the contact line, which depends on the wetting angle θ. The contact line is the place at which the plug, the bubble, and the liquid are in three-phase contact. The relation between y

_{max}and θ is summarized in [53].

_{b,32}and the superficial gas velocity u

_{sg}. In the heterogeneous regime, there are inconsistent results, but most studies found that the bubble size increases with increasing flow rate. In this regime, the bubble size is mainly determined by coalescence and breakup, which is influenced by many different parameters [3]. For the heterogenous regime, Mohagheghian et al. [62] argued that the Ohnesorg number, Oh, is a function of the Morton and the Capillary number, Ca:

_{pl}is the plug Weber number (u

^{2}

_{sg}d

_{po}ρ

_{l}/ε

^{2}σ) and Fr

_{pl}is the plug Froude number (u

^{2}

_{sg}/ε

^{2}gd

_{po}). Another correlation for intermediately contaminated systems was proposed as well. However, since the influence of contaminations in liquid metals have not been studied yet, it is not given here.

#### 3.3. Bubble Generation from Slot Plugs

_{sl}is the slot Weber number (u

^{2}

_{sg}w

_{sl}ρ

_{l}/σ).

#### 3.4. Bubble Size in a Steel Casting Ladle

_{d}can be assumed to be 8/3. However, this reasoning is based on extrapolation of correlations for single nozzles, so the bubble size is probably overestimated by the authors. For the most probable range, the bubbles are mostly ellipsoids, so the drag coefficient cannot be assumed to be constant as discussed in the subsequent sections.

## 4. Bubble Deformation

_{eq}, is defined as the diameter of a sphere with the same volume as the distorted bubble. A simplified analytical solution for inviscid flows was derived by Moore [78] by balancing the normal stresses caused by dynamic pressure and surface tension at the stagnation points and on the equatorial plane. This analysis yields:

_{D}, the bubble Reynolds number, the Morton number, the Reynolds number inside the bubble, and the density ratio as dimensionless groups. If the density and viscosity of the gas is considered to be negligible, the number of dimensionless groups can be further reduced to three. Nowadays the bubble Reynolds number, Eötvös number, and Morton number are commonly used, although some correlations have been derived for the Weber number.

_{10}(Mo)~−13), Mishima et al. [28] found a good qualitative agreement between the predicted and observed bubble shapes.

**Figure 6.**Grace diagram to approximate the bubble shape as a function of the Eötvös, Reynolds, and Morton number (original in French, reproduced from [88], with permission from Elsevier, 2021).

_{10}(Mo) < 2.3, which was collected by different researchers. Using Equation (27) as f-function, they correlated:

_{b}≤ 7 mm) or ellipsoidal cap shaped (d

_{b}> 7 mm), depending on their size and their Eötvös and Reynolds numbers. However, when evaluating these results, it should be noted that no turbulence model was used in both studies, but the mesh resolution used is too coarse for a real DNS. Therefore, the models can be classified as implicit large eddy turbulence model (LES) without a subgrid model, which underestimates the draining effect of the small-scale eddies on the energy cascade. Therefore, it is likely that the resolved turbulence is overestimated in both studies, which certainly has some influence on the bubble shape.

## 5. Interfacial Force Closure

#### 5.1. Drag Force

_{D}, which depends on bubble properties and the flow conditions. The effect of bubble deformation is usually lumped into c

_{D}as well. By dimensionless analysis, it can be shown that the drag coefficient depend on the bubble Reynolds number, the Eötvös number, and the Morton number [82].

_{D}. However, they are not applicable for most industrial processes, since the bubbles are usually deformed and the fluid system is contaminated to some extent, which increase the real drag coefficient.

_{D}are experimental measurements. Usually, the rise of single bubbles in stagnant (u

_{l}= 0), mostly aqueous liquids, is studied. After the bubble reaches its terminal rising velocity, the drag force is in equilibrium with the buoyancy force. The drag coefficient is then given by:

_{D}, based on a DNS, was proposed by Dijkhuizen et al. [89]:

_{D}(Re) is the analytical solution by Mei and Klausner (Equation (39)) and c

_{D}(Eo) is given by:

#### 5.1.1. Influence of Contaminants

_{D}and thus contaminants also no longer have an influence on the rising velocity. However, contaminants not only affect the flow boundary condition at the bubble surface, but also the deformation and vortex shedding. Therefore, Tasoglu et al. [101] found that contaminants are relevant for a broad range of Eötvös numbers, but the effect is much more pronounced for smaller bubbles. A problem here is that DNS studies usually investigate very small Reynolds number flows and large Morton numbers, since the computational costs increases with Re

^{9/4}. Experimental studies, on the other hand, face the problem that it is practically impossible to generate completely uncontaminated systems. Therefore, sound studies on the effect of contaminates for large bubbles at relevant Reynolds numbers are largely lacking. A further critique of the Tomiyama et al. [95] approach is that for most engineering applications, there is no a priori definition of the degree of contamination of the system.

#### 5.1.2. Influence of Surrounding Bubbles

_{b}≤ 7 mm). For larger bubbles, the drag increases up to a critical void fraction of approximately 15% and decreases afterwards. Thus, Equation (53) is only valid for an air–water system, where bubble diameters are between 5 and 10 mm, with the restriction that for void fractions above 15%, the bubble diameter has to be larger than 7 mm. Simonnet et al. [106] assumed that the increase is caused by a hindrance effect by surrounding bubbles while the decrease might be explained by the aspiration in the bubble wakes of preceding bubbles.

_{b}was 8.5 mm. Thus, this assumption seems to be oversimplified. Lau et al. [114] used a mapping technique to determine the local void fraction. However, within this approach, the result were dependent on the size of the mapping window, n. For sizes n > d

_{b}, however, this effect became negligible.

#### 5.2. Lift Force

_{L}is the lift coefficient. The first analytical solution for the lift coefficient was derived by Saffman [116] for rotating, solid spheres in creeping flows (Re→0) and infinite shear rates, which could later be extended numerically by McLaughlin [117] to arbitrary shear rates:

_{L}so that Equation (55), which was originally derived for shear-induced lift forced only, can be employed to describe both effects.

_{L}was carried out in Tomiyama’s much-acclaimed work [85]. A rotating belt was used to induce a laminar shear flow in distilled water–glycerol mixtures of different viscosities. In this shear flow, the rise of bubbles with different diameters was analyzed in the ranges of −5.5 ≤ log

_{10}(Mo) ≤ −2.8, 1.39 ≤ Eo ≤ 5.75, and 0 ≤ ω ≤ 8.3 s

^{−1}. Measurement in systems with lower viscosity were not feasible, because the rotating belt requires a sufficient viscosity to induce a linear shear flow. The measurements yield the following empirical correlation:

_{d}is the modified Eötvös number, using the major axis of deformed bubbles as characteristic length:

_{L}is a monotonic function of the modified Eötvös number. However, it is questionable whether it is physically justified to extrapolate the correlation for very large bubbles, because the bubble size has a significant influence on the wake structure.

_{10}(Mo) for water at room temperature is about -10.6. Nonetheless, though the study was conducted for higher viscous systems (−5.5 ≤ log

_{10}(Mo) ≤ −2.8), a very good agreement with the experimental measurements in air–water systems [115,122,123] and Equation (61) was found. Because of that, the authors concluded that their correlation can be extrapolated to air–water systems.

_{L}values and a limit of 0.5 for small Eötvös numbers. For a range of −10.8 < log

_{10}(Mo) < −1.8, 0 < Eo

_{d}< 20, 0 < Eo < 12, 0 < Re < 1500, Dijkhuizen et al. [132] found, via numerical simulation, that Equation (61) is only applicable in the range that the measurements actually took place. They showed that for systems of higher viscosity or for very small Reynolds numbers (Re < 10 and E < 0.95), the predicted values can vary quite significantly from the simulated ones. For the latter case of small Reynolds numbers, Dijkhuizen et al. [132] found good agreement with the correlation of Legende and Magnaudet [121] (Equation (60)) rather than the plateau proposed by Tomiyama et al. [85]. On the other hand, they found that Equation (60) overestimates c

_{L}significantly in case the bubble shape deviates from a sphere. Based on their results, Dijkhuizen et al. [132] proposed:

^{−2}< Re < 1.2 × 10

^{2}, −6.6 ≤ log

_{10}(Mo) ≤ −3.2, 2.2 × 10

^{−2}< Eo < 5.0, 3.4 × 10

^{−2}< Sr < 3.5) by Aoyama et al. [129] using a similar rotating belt system as Tomiyama et al. [85]. Similar to Dijkhuizen et al. [132], it was found that Equation (61) is only applicable in a limited range. Moreover, it was found that none of the tested dimensionless numbers (Re, Eo, We, Eo

_{d}, Ca) alone can be used to correlate an accurate equation for the lift coefficient. The most promising attempts in this direction were made with the Reynolds and modified Eötvös numbers, though for both approaches, the Morton number has to be taken into account, too.

_{L}for a broader range of Morton numbers. In addition, it was found that the instantaneous lift force can vary quite significantly in water systems. Combining their results with those of Aoyama et al. [129] suggests that there is always a sign change and a linear behavior of c

_{L}around the sign change, as shown in Figure 15. However, the modified Eötvös number at which the sign change occurs as well as the slope of the linearity around the sign change depend on the Morton number. The sign change occurs at smaller modified Eötvös numbers with a decreasing Morton number. However, this trend reverses at a Morton number not exactly known yet. In addition, it was found that c

_{L}asymptotically approaches an upper and a lower limit for high and low modified Eötvös numbers, respectively. These limits again seem to depend on the Morton number, too.

_{L}is increasing in case of very small Re numbers. However, these small Reynolds numbers could not be produced in an air–water system. For the investigated air–water system, Ziegenhein et al. [133] correlated, via second-order polynomial regression, their results:

_{10}(Mo) < −6.6 and −10.5 > log

_{10}(Mo), which should be complemented to make correlations more applicable to a wider range. For Morton numbers corresponding to the range of liquid metals, studies are missing.

_{L}accurately for a wider range of physical properties.

#### Influence of Contamination

_{L}) on small bubbles is mainly driven by pressure components (c

_{LP}), while increasing surfactant levels decrease its impact until it disappears entirely, so that the small negative lift force is due to viscous stress (c

_{LV}). Overall, the effect of surfactants was attributed to the Marangoni effect.

_{10}(Mo) < −4, 0 < Sr < 1, 1.38 ≤ La ≤ 13.8, 0 < Ha < 41). Similarly, they found that the lift coefficient decreases with increasing Langmuir number. The main effect, however, depends on the Hatta number. In case of large Hatta numbers, which means that the absorption of contaminants is much faster than the bubble rising, the decrease of the lift coefficient is attributed to a decrease of the effective surface tension by:

_{L}increases due to this additional contribution of the Magnus effect, which itself depends on the shear rate. In contrast, Hessenkemper et al. [138] found no influence of tracer particles in a linear shear flow. Until further research has been conducted, one can only speculate about this discrepancy. Reasons can either be that the dimensionless shear rates investigated by Rastello were quite high (Sr > 0.2), the flow was rotating, the number of tracers attached to the bubble in the experiment by Hessenkemper et al. [138] were too small, or the tracer sizes were different and therefore the surface tension interaction of the particles with the bubbles was significantly different. Hessenkemper et al. [138] investigates the influence of inorganic surfactants. In contrast to organic surfactance, it was found that they increase the lift coefficient.

_{L}would be predicted, which is likely to lead to unphysical behavior and stability problems in numerical models. Therefore, in particular, the negative asymptotical limit of c

_{L}for large bubble in liquid metals is of great interest. The discussion is further complicated by the large uncertainty of the prediction of the bubble’s deformation analyzed above, which is necessary for the calculation of the modified Eötvös number.

#### 5.3. Virtual Mass Force

_{VM}, by which all influences on the force are represented. If c

_{VM}is too small, the bubble acceleration may become too large, which can cause numerical instability depending on the solution process and settings. If c

_{VM}is too large, the acceleration phase becomes unphysically long and smaller oscillations in the bubble’s rising path may be suppressed. For a spherical bubble in stagnant liquids, an analytical value of c

_{VM}= 0.5 was derived [140]. This value is also used in almost all numerical studies. There are only a small number of quantitative studies on the exact value of c

_{VM}in real flow conditions. These suggest that the value of c

_{VM}also depends on the void fraction [141,142] and the bubble deformation ([143] found in [144]). However, if acceleration effects are not significant in the flow, then the virtual mass force is not important for the flow either. This applies, for example, to bubble reactors [142]. Therefore, the exact knowledge of c

_{VM}plays a minor role in this type of flow.

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Symbol | Description |

c_{D} | Drag coefficient |

c_{1–4} | Fitting parameter |

c_{L} | Lift coefficient |

c_{VM} | Virtual mass coefficient |

c_{∞} | Far-field concentration of contaminants, mol/m^{3} |

d_{b} | Arithmetic mean equivalent bubble diameter, m |

d_{b,32} | Bubble Sauter mean diameter, m |

d_{eq} | Equivalent bubble diameter, m |

d_{max} | Length of bubble major axis, m |

d_{min} | Length of bubble minor axis, m |

d_{ni} | Orifice inner diameter, m |

d_{no} | Orifice outer diameter, m |

d_{po} | Pore diameter |

E | Bubble eccentricity |

Eo | Eötvös number |

Eo_{d} | Modified Eötvös number |

Fr | Froude number |

F_{i} | Force i on bubble, N |

F_{D} | Drag force, N |

F_{L} | Lift force, N |

F_{VM} | Virtual mass force, N |

f | Friction factor in Bozzano drag model |

g | Gravity, 9.81 m/s^{2} |

H_{0} | Distance between nozzle and the mathematical origin of the nozzle, m |

H_{Fill} | Filling height, m |

K | Empirical constant in Mersmann correlation |

k | Adsorption rate, m^{3}/mol∙s |

Lc | Characteristic length, m |

Mo | Morton number |

m | Fitting parameter |

m_{B} | Bubble mass, kg |

n | Number of activated orifices |

Q_{g} | Gas flow rate in m^{3}/s |

Q_{min} | Minimum gas flow rate at which an orifice gets activated, m^{3}/s |

R_{g} | Universal gas constant, 8.3145 J/mol∙K |

Re | Reynolds number |

r_{or} | Orifice radius, m |

Sr | Dimensionless shear rate |

S_{br} | Breakup rate |

S_{c} | Coalescence rate |

T | Temperature, K |

Ta | Tadaki number |

u_{c} | Characteristic velocity, m/s |

u_{sg} | Superficial gas velocity |

u_{b} | Bubble velocity, m/s |

u_{l} | Liquid velocity, m/s |

V_{b} | Bubble volume, m^{3} |

We | Weber number |

w_{sl} | Slot width, m |

ŷ | Fitting parameter |

z | Height, m |

α | Global void fraction |

α_{i} | Fitting parameter |

α_{loc} | Local void fraction |

β | Desorption rate, mol/m^{3} |

Γ | Contaminant concentration, mol/m^{2} |

ε | Porosity of a porous plate |

θ | Fitting parameter |

κ | Permeability of a porous plug |

λ_{sl} | Distance between active bubble formation sites on slot nozzles, m |

μ_{g} | Gas viscosity, Ns/m^{2} |

μ_{l} | Liquid viscosity, Ns/m^{2} |

ρ_{b} | Bubble density, kg/m^{3} |

ρ_{l} | Liquid density, kg/m^{3} |

σ | Surface tension, N/m |

φ | Shape factor |

ω | Shear rate, 1/s |

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**Figure 2.**Bubble generation from a wetted (

**a**) and non-wetted (

**b**) sieve tray (Reproduced from [53], with permission from Elsevier, 2021).

**Figure 3.**Modes of gas dispersion in porous plug injection: (

**a**) Quiescent column of discrete bubbles, (

**b**) onset of coalescence, and (

**c**) blanketing (Reproduced from [58], with permission from Springer, 2021).

**Figure 4.**Regimes of bubble formation at a slot nozzle (Adapted from [70], with permission from Elsevier, 2021).

**Figure 11.**Schematic overview of the effect of contaminants at (

**a**) pure, (

**b**) slightly contaminated, and (

**c**) fully contaminated systems (Γ* = dimensionless contaminant concentration).

**Figure 12.**Bubble rising velocity of air in water in dependency of the degree of contamination of the fluid.

**Figure 13.**Drag coefficient as a function of the Langmuir number (adapted from [105] with permission from AIP publishing, 2021).

**Figure 14.**Lift coefficient of a spherical bubble in dependence of the dimensionless shear rate (Sr) and the Reynolds number (Re) according to the model of Legendre and Magnaudet [121].

**Figure 15.**Experimental lift coefficients as a function of the modified Eötvös number (adapted from [133], with permission from Elsevier, 2021).

**Figure 17.**Effect of the Langmuir number on the lift coefficient (c

_{L}) and its pressure (c

_{LP}) and viscosity (c

_{VL}) components (adapted from [105] with permission from AIP publishing, 2021).

**Table 2.**Bubble size correlations and their corresponding value in soft bubbling in a 185 t ladle (Q

_{g}= 200Nl/min, T = 1600 °C, H

_{Fill}= 3.23 m, D

_{pl}= 0.1 m).

System | Ref. | Equation | d_{b,ladle} |
---|---|---|---|

Analytical | [39] | ${\mathrm{d}}_{\mathrm{b},\mathrm{R}}{=\mathrm{d}}_{\mathrm{b},\mathrm{M}}\sqrt{\frac{{\mathsf{\sigma}}_{\mathrm{R}}{\left({\mathsf{\rho}}_{\mathrm{l}}-{\mathsf{\rho}}_{\mathrm{b}}\right)}_{\mathrm{M}}}{{\mathsf{\sigma}}_{\mathrm{M}}{\left({\mathsf{\rho}}_{\mathrm{l}}-{\mathsf{\rho}}_{\mathrm{b}}\right)}_{\mathrm{R}}}}$ | 8.5 mm |

Single Nozzles | [47] | ${\mathrm{d}}_{\mathrm{b}}=0.000146{\left[{10}^{4}\frac{{\mathrm{Q}}_{\mathrm{g}}}{{\mathrm{z}+\mathrm{H}}_{0}}\right]}^{0.1}$ | 15.8 mm |

[15] | ${\mathrm{d}}_{\mathrm{b}}=0.01\xb7{\left[{\left({10}^{6}\xb7\frac{6\mathsf{\sigma}\xb7{\mathrm{d}}_{\mathrm{no}}}{{\mathsf{\rho}}_{\mathrm{l}}\mathrm{g}}\right)}^{2}+0.0242{\left({10}^{7}{\mathrm{Q}}_{\mathrm{g}}{\mathrm{d}}_{\mathrm{no}}^{0.5}\right)}^{1.734}\right]}^{1/6}$ | 31.2 mm | |

[42] | ${\mathrm{d}}_{\mathrm{b},32}=0.00091{\left({10}^{6}\frac{\mathsf{\sigma}}{{\mathsf{\rho}}_{\mathrm{l}}}\right)}^{0.5}{\left(\frac{{200\mathrm{Q}}_{\mathrm{g}}}{{\mathsf{\pi}\mathrm{d}}_{\mathrm{ni}}^{2}}\right)}^{0.44}$ | 96.8 mm | |

Porous Plugs (heterogenous regime) | [64] | ${\mathrm{d}}_{\mathrm{b}}=0.0703\xb7{\left({10}^{6}\xb7\frac{{\mathrm{d}}_{\mathrm{po}}\mathsf{\sigma}}{{\mathsf{\rho}}_{\mathrm{l}}\mathrm{g}}\right)}^{1/3}{\left(\frac{{\mathsf{\rho}}_{\mathrm{l}}{\mathrm{gd}}_{\mathrm{po}}^{2}}{\mathsf{\sigma}}\right)}^{0.364}{\left(\frac{{\mathrm{Fr}}_{\mathrm{pl}}}{\sqrt{{\mathrm{We}}_{\mathrm{pl}}}}\right)}^{0.133{\left(\frac{{\mathsf{\rho}}_{\mathrm{l}}{\mathrm{gd}}_{\mathrm{po}}^{2}}{\mathsf{\sigma}}\right)}^{-0.14}}$ | 13.5 mm |

[62] | ${\mathrm{d}}_{\mathrm{b},32}=0.2477\frac{{\mathsf{\mu}}_{\mathrm{l}}^{2}}{{\mathsf{\rho}}_{\mathrm{l}}\mathsf{\sigma}}{\left(\frac{{\mathrm{gu}}_{\mathrm{sg}}{\mathsf{\mu}}_{\mathrm{l}}^{5}}{{\mathsf{\rho}}_{\mathrm{l}}{\mathsf{\sigma}}^{4}}\right)}^{-0.4}$ | 0.65 mm | |

Slot plug | [71] | ${\mathsf{\lambda}}_{\mathrm{sl}}=6.5{\mathrm{w}}_{\mathrm{sl}}{\mathrm{We}}_{\mathrm{sl}}^{0.466}$ ${\mathrm{d}}_{\mathrm{b}}=0.01\xb7{\left[{\left({10}^{6}\xb7\frac{6\mathsf{\sigma}\xb7{\mathrm{d}}_{\mathrm{no}}}{{\mathsf{\rho}}_{\mathrm{l}}\mathrm{g}}\right)}^{2}+0.0242{\left({10}^{7}{\mathrm{Q}}_{\mathrm{g}}{\mathrm{d}}_{\mathrm{no}}^{0.5}\right)}^{1.734}\right]}^{1/6}$ | 13.2 mm |

System | m | (h/b)_{cap} | Ta_{1} | Ta_{2} | c_{1} | c_{2} | c_{3} | c_{4} |
---|---|---|---|---|---|---|---|---|

3Dcontaminated | 3 | 0.24 | 1 | 40 | 0.81 | 0.2 | 2.0 | 0.8 |

3Dpure | 3 | 0.24 | 0.3 | 20 | 0.81 | 0.2 | 1.8 | 0.4 |

Fluid | Density [kg/m^{3}] | Viscosity [Ns/m^{2}] | Surface Tension [N/m] | log10 (Mo) |
---|---|---|---|---|

GaInSn | 6330 | 0.00234 | 0.585 | −12.6 |

Steel | 6900 | 0.00506 | 1.5 | −12.6 |

Mercury | 13,550 | 0.00155 | 0.487 | −13.4 |

Silver | 9510 | 0.00389 | 0.92 | −12.5 |

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**MDPI and ACS Style**

Haas, T.; Schubert, C.; Eickhoff, M.; Pfeifer, H.
A Review of Bubble Dynamics in Liquid Metals. *Metals* **2021**, *11*, 664.
https://doi.org/10.3390/met11040664

**AMA Style**

Haas T, Schubert C, Eickhoff M, Pfeifer H.
A Review of Bubble Dynamics in Liquid Metals. *Metals*. 2021; 11(4):664.
https://doi.org/10.3390/met11040664

**Chicago/Turabian Style**

Haas, Tim, Christian Schubert, Moritz Eickhoff, and Herbert Pfeifer.
2021. "A Review of Bubble Dynamics in Liquid Metals" *Metals* 11, no. 4: 664.
https://doi.org/10.3390/met11040664