# Rectified Diffusion of Gas Bubbles in Molten Metal during Ultrasonic Degassing

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## Abstract

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## 1. Introduction

## 2. Theoretical Analysis

- The involved sizes of the bubble is far beyond the molecular level [15].
- The viscosity and the compressibility of molten metal are both considered.
- The relationship between the volume and the internal pressure of gas bubbles is described by the polytropic model [21].
- Thermal damping is currently ignored. As shown in our previous work [21], this damping mechanism often does not serve as a dominant one.

_{l}is the density of the molten metal; t is the time; c

_{l}is the sound speed propagating in the molten metal; P

_{g}is the instantaneous pressure in the bubble; σ is the surface tension coefficient; μ

_{l}is the viscosity of the molten metal; P

_{∞}is the ambient pressure; R

_{0}is the initial and equilibrium radius of the given bubble; η is the polytropic exponent; P

_{A}is the amplitude of the applied acoustic field; f is the frequency of the applied acoustic field.

**u**is the velocity of the molten metal surrounding the gas bubble; D is the diffusion constant. The initial and boundary conditions are [22]

_{i}is the initial concentration or the concentration at infinity; C

_{S}is the local gas concentration at the inner bubble wall. The solubility of gases in molten metal is controlled by Sievert’s law [23] as follows

_{0}is the saturation concentration of gas in the molten metal; k

_{S}is a constant. Equations (1) to (5) with initial and boundary conditions of Equations (6) to (8) can be solved following our previous framework [9]. Because the solubility of gas in the molten metal is controlled by the Sievert’s law rather than Henry’s law, some modifications on the previous works will be performed as follows. The time derivative of the gas amount change (n) inside the bubble is given by [22]

_{b}is the bubble oscillation period; < > denotes time averages of the given parameter. By using the Sievert’s law, F

_{0}can be expressed as,

_{g}is the universal constant; T is the temperature (unit: K). Compared with previous work (e.g., [9]) based on the Henry’s law, Equations (9), and (12) are different owing to the use of Sievert’s law (e.g., the terms with P

_{g}in the Equations (9) and (12)). Based on Equations (9) to (15), one can find that the growth or dissolution rate of gas bubbles in molten metal under acoustic excitation is dependent on many parameters, e.g., acoustic pressure, amplitude, and frequency, saturation conditions, ambient pressure, temperature, bubble radius, and time. By integration of Equation (9), the quasi-equilibrium bubble radius could be obtained. If one set dR

_{0}/dt = 0 in Equation (9), the corresponding acoustic amplitude will be the threshold for the diffusion (P

_{T}) as follows

_{A}> P

_{T}and dissolve if P

_{A}< P

_{T}.

## 3. Results and Discussions

_{m}= 933.4 K; σ = 868 − 0.152(T − T

_{m}) dyn/m; ρ

_{l}= 2375 kg/m

^{3}; c

_{l}= 6187 m/s; μ

_{l}= 0.1492exp(1984.5/T) Pa·s.

^{2}/s); D

_{0}is the maximum diffusion coefficient at the infinite temperature (m

^{2}/s); Q is the activation energy for diffusion. Here, a group of values suggested by [26] are used: D

_{0}= 3.8 × 10

^{−6}m

^{2}/s; Q = 19320 J/mol. The constants are highlighted in Table 1 with the following acoustic parameters given: P

_{∞}= 1.01 × 10

^{5}Pa; f = 18 kHz; P

_{A}= 1.1 × 10

^{5}Pa. For some cases, the polytropic exponent could not be regarded as a pure constant. Instead, it will depend on the external frequency and the bubble radius. For more details for the predictions of the polytropic exponent, readers could consider our previous model [21].

_{T}) based on Equation (16) for hydrogen bubbles in molten aluminum. The threshold is minimal near resonance and increases gradually for non-resonance conditions. For oversaturation conditions (e.g., C

_{i}/C

_{0}= 1.05), the threshold is lower while for subsaturation conditions (e.g., C

_{i}/C

_{0}= 0.95), the threshold is higher. In Figure 1, Eskin’s predictions [6] were not compared with ours because Eskin’s model cannot be used to predict the threshold phenomenon of the rectified diffusion owing to the assumption of gas always flowing into bubbles embedded in his model ([6], Equation (3)). It should be emphasized that for effective ultrasonic degassing, the amplitude of applied ultrasonic fields should be strong enough (i.e., above the threshold of rectified diffusion) to facilitate the growth of a gas bubble.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Predicted threshold of acoustic pressure amplitude of rectified diffusion of hydrogen bubbles in molten aluminum. C

_{i}/C

_{0}= 0.95, 1.00 and 1.05. f = 18 kHz.

**Figure 2.**Predicted dynamic change of equilibrium bubble radius against time during rectified diffusion processes of hydrogen bubbles in molten aluminum. Initial bubble radius (R

_{0}) employed for the predictions are 14, 16 and 18 μm respectively. C

_{i}/C

_{0}= 1. f = 18 kHz. P

_{A}= 1.1 × 10

^{6}Pa.

Parameters | Values |
---|---|

${\rho}_{l}$ | 2375 kg/m^{3} |

${T}_{m}$ | 933.4 K |

${R}_{g}$ | 8.314 J/mol/K |

$T$ | 1073.15 K |

$\eta $ | 1.2 |

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**MDPI and ACS Style**

Zhang, Y.; Zhang, Y.
Rectified Diffusion of Gas Bubbles in Molten Metal during Ultrasonic Degassing. *Symmetry* **2019**, *11*, 536.
https://doi.org/10.3390/sym11040536

**AMA Style**

Zhang Y, Zhang Y.
Rectified Diffusion of Gas Bubbles in Molten Metal during Ultrasonic Degassing. *Symmetry*. 2019; 11(4):536.
https://doi.org/10.3390/sym11040536

**Chicago/Turabian Style**

Zhang, Yuning, and Yuning Zhang.
2019. "Rectified Diffusion of Gas Bubbles in Molten Metal during Ultrasonic Degassing" *Symmetry* 11, no. 4: 536.
https://doi.org/10.3390/sym11040536