# Modelling Microlayer Formation in Boiling Sodium

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Our Current Understanding of the Microlayer

_{2}substrate (transparent to both visible and infrared light). Using a He-Ne laser as a light source for an integrated optical technique of total internal reflection and thin film interferometry, they were able to simultaneously observe the liquid–vapor phase distribution beneath a growing steam bubble and the fringe pattern of the wedge-shaped microlayer existing on the liquid phase, as well as typical bubble growth images captured from the side, as shown in Figure 2. Complementary sequential measurements of the temperature distribution at the solid–fluid interface beneath the bubble indicated substantial cooling of the surface due to evaporation of the microlayer, as indicated by the temperature and wall heat flux distribution measurements reproduced, in Figure 2, from Jung et al’s work [3].

## 3. Simulation MethodologyS

#### 3.1. Interface-Capturing Model

**f**is the body force, including gravity and surface tension forces. The latter, including wall adhesion in the near-wall cells, is computed with the well-established continuum surface force method of [23].

#### 3.2. Simulation Setup

#### 3.3. Model Validation Using Water Boiling Data

## 4. Modelling Microlayer Formation in Boiling of Sodium

#### 4.1. Likely Bubble Growth Regime in Sodium; Differences with the Case of Water Boiling

#### 4.2. Parametric Trends of Simulation Results

#### 4.2.1. Effect of Bubble Growth Rate

#### 4.2.2. Effect of Liquid Viscosity

#### 4.2.3. Effect of Surface Tension Coefficient

## 5. Comparison of Simulation Results with an Extant Analytical Model of Microlayer Formation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Jung, S.; Kim, H. Hydrodynamic formation of a microlayer underneath a boiling bubble. Int. J. Heat Mass Transf.
**2018**, 120, 1229–1240. [Google Scholar] [CrossRef] - Giustini, G.; Jung, S.; Kim, H.; Ardron, K.H.; Walker, S.P. Microlayer evaporation during steam bubble growth. Int. J. Therm. Sci.
**2019**, 137, 45–54. [Google Scholar] [CrossRef] - Jung, S.; Kim, H. An experimental method to simultaneously measure the dynamics and heat transfer associated with a single bubble during nucleate boiling on a horizontal surface. Int. J. Heat Mass Transf.
**2014**, 73, 365–375. [Google Scholar] [CrossRef] - Giustini, G.; Jung, S.; Kim, H.; Walker, S.P. Evaporative thermal resistance and its influence on microscopic bubble growth. Int. J. Heat Mass Transf.
**2016**, 101, 733–741. [Google Scholar] [CrossRef] - Hänsch, S.; Walker, S. Microlayer formation and depletion beneath growing steam bubbles. Int. J. Multiph. Flow
**2019**, 111, 241–263. [Google Scholar] [CrossRef] - Urbano, A.; Tanguy, S.; Huber, G.; Colin, C. Direct numerical simulation of nucleate boiling in micro-layer regime. Int. J. Heat Mass Transf.
**2018**, 123, 1128–1137. [Google Scholar] [CrossRef] [Green Version] - Utaka, Y.; Kashiwabara, Y.; Ozaki, M. Microlayer structure in nucleate boiling of water and ethanol at atmospheric pressure. Int. J. Heat Mass Transf.
**2013**, 57, 222–230. [Google Scholar] [CrossRef] - Yabuki, T.; Nakabeppu, O. Microscale wall heat transfer and bubble growth in single bubble subcooled boiling of water. Int. J. Heat Mass Transf.
**2016**, 100, 851–860. [Google Scholar] [CrossRef] - Kim, J. Review of nucleate pool boiling bubble heat transfer mechanisms. Int. J. Multiph. Flow
**2009**, 35, 1067–1076. [Google Scholar] [CrossRef] - Gerardi, C.; Buongiorno, J.; Hu, L.-W.; McKrell, T. Study of bubble growth in water pool boiling through synchronized, infrared thermometry and high-speed video. Int. J. Heat Mass Transf.
**2010**, 53, 4185–4192. [Google Scholar] [CrossRef] - Sugrue, R.; Buongiorno, J. A modified force-balance model for prediction of bubble departure diameter in subcooled flow boiling. Nucl. Eng. Des.
**2016**, 305, 717–722. [Google Scholar] [CrossRef] - Guion, A.; Afkhami, S.; Zaleski, S.; Buongiorno, J. Simulations of microlayer formation in nucleate boiling. Int. J. Heat Mass Transf.
**2018**, 127, 1271–1284. [Google Scholar] [CrossRef] [Green Version] - Hänsch, S.; Walker, S. The hydrodynamics of microlayer formation beneath vapour bubbles. Int. J. Heat Mass Transf.
**2016**, 102, 1282–1292. [Google Scholar] [CrossRef] - Kottowski, H.M.; Savatteri, C. Fundamentals of liquid metal boiling thermohydraulics. Nucl. Eng. Des.
**1984**, 82, 281–304. [Google Scholar] [CrossRef] - Coventry, J.; Andraka, C.; Pye, J.; Blanco, M.; Fisher, J. A review of sodium receiver technologies for central receiver solar power plants. Sol. Energy
**2015**, 122, 749–762. [Google Scholar] [CrossRef] [Green Version] - Ardron, K.H.; Giustini, G.; Walker, S.P. Prediction of dynamic contact angles and bubble departure diameters in pool boiling using equilibrium thermodynamics. Int. J. Heat Mass Transf.
**2017**, 114, 1274–1294. [Google Scholar] [CrossRef] - Tryggvason, G.; Scardovelli, R.; Zaleski, S. Direct Numerical Simulations of Gas–Liquid Multiphase Flows; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar] [CrossRef]
- Issa, R.I. Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys.
**1986**, 62, 40–65. [Google Scholar] [CrossRef] - Issa, R.I.; Gosman, A.D.; Watkins, A.P. The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme. J. Comput. Phys.
**1986**, 62, 66–82. [Google Scholar] [CrossRef] - Greenshields, C. OpenFOAM User Guide; The OpenFOAM Foundation Ltd.: London, UK, 2017. [Google Scholar]
- Mikic, B.B.; Rohsenow, W.M.; Griffith, P. On bubble growth rates. Int. J. Heat Mass Transf.
**1970**, 13, 657–666. [Google Scholar] [CrossRef] - Hardt, S.; Wondra, F. Evaporation model for interfacial flows based on a continuum-field representation of the source terms. J. Comput. Phys.
**2008**, 227, 5871–5895. [Google Scholar] [CrossRef] - Brackbill, J.U.; Kothe, D.B.; Zemach, C. A continuum method for modeling surface tension. J. Comput. Phys.
**1992**, 100, 335–354. [Google Scholar] [CrossRef] - Ferrari, A.; Magnini, M.; Thome, J.R. Numerical analysis of slug flow boiling in square microchannels. Int. J. Heat Mass Transf.
**2018**, 123, 928–944. [Google Scholar] [CrossRef] - Magnini, M.; Pulvirenti, B.; Thome, J.R. Numerical investigation of hydrodynamics and heat transfer of elongated bubbles during flow boiling in a microchannel. Int. J. Heat Mass Transf.
**2013**, 59, 451–471. [Google Scholar] [CrossRef] - Denner, F.; van Wachem, B.G.M. Numerical time-step restrictions as a result of capillary waves. J. Comput. Phys.
**2015**, 285, 24–40. [Google Scholar] [CrossRef] [Green Version] - Damiàn, S.M. An Extended Mixture Model for the Simultaneous Treatment of Short and Long Scale Interfaces. Ph.D. Thesis, Universidad Nacional del Litoral, Santa Fe, Argentina, 2013. [Google Scholar]
- Nabil, M.; Rattner, A.S. interThermalPhaseChangeFoam—A framework for two-phase flow simulations with thermally driven phase change. SoftwareX
**2016**, 5, 216–226. [Google Scholar] [CrossRef] [Green Version] - Cooper, M.G.; Lloyd, A.J.P. The microlayer in nucleate pool boiling. Int. J. Heat Mass Transf.
**1969**, 12, 895–913. [Google Scholar] [CrossRef]

**Figure 1.**Observations of the microlayer in boiling at a surface. (

**a**) Typical bubble shape during the early stages of growth. (

**b**) Interferometric measurements of phase distribution on the solid surface used to detect the presence of a liquid film. (

**c**) Measured microlayer thickness as a function of radial distance from the center of the bubble base. Data [1] for pool boiling of water in atmospheric conditions at 10 K of wall superheat.

**Figure 2.**Complementary observations of the early stages of bubble growth at a solid surface [3]. From top to bottom: bubble side view, phase distribution on the heated surface, temperature distribution at the solid surface, and surface normal heat flux distribution derived from the temperature distribution.

**Figure 3.**Sequential measurements of the radial distribution of microlayer thickness, adapted from Ref [1].

**Figure 5.**Visualization of indicative distribution of the volumetric rate of mass transfer imposed at the bubble curved surface in order to mimic bubble growth due to evaporation. The black line represents the vapor–liquid interface; the shaded region indicates where mass transfer is imposed.

**Figure 6.**(

**a**) Typical modelled bubble shapes at a sequence of times, namely at t = 0 and at 0.05, 0.1, 0.15, and 0.2 ms into bubble growth. Magnification (

**b**) of the near-wall region shows the microlayer being left behind on the no-slip wall boundary.

**Figure 8.**Comparison between modelled and measured microlayer thicknesses for the water boiling case.

**Figure 11.**Effect of bubble growth rate on microlayer formation. Bubble and microlayer shapes are compared at constant bubble volume.

**Figure 12.**Effect of liquid viscosity on microlayer formation. Bubble and microlayer shapes are compared at the same instant of time into bubble growth.

**Figure 14.**Comparison of current simulation results with the model of [12]; effect of bubble growth rate. Radial distributions of microlayer thickness corresponding to the same bubble volume, presented in Figure 11, are compared with predictions of the model of [12]. Panels (

**a**–

**c**) show microlayer profiles for different values of the bubble growth rate.

**Figure 15.**Comparison of current simulation results with the model of [12]; effect of liquid viscosity. Radial distributions of microlayer thickness at a sequence of times into bubble growth are compared with predictions of the model of [12]. Panels (

**a**–

**c**) show microlayer profiles for different values of the liquid viscosity.

Properties of Saturated Water at 1 bar | ||
---|---|---|

Property | Vapour | Liquid |

Dynamic viscosity $\mu \left[\mathrm{Pa}\xb7\mathrm{s}\right]$ | $12.2\times {10}^{-6}$ | $281.6\times {10}^{-6}$ |

Density $\rho \left[\mathrm{kg}/{\mathrm{m}}^{3}\right]$ | 0.6 | 958.4 |

Specific heat capacity $c\left[\mathrm{J}/\mathrm{Kg}/\mathrm{K}\right]$ | 2077.5 | 4216.6 |

Thermal conductivity $k\left[\mathrm{W}/\mathrm{m}/\mathrm{K}\right]$ | $24.8\times {10}^{-3}$ | $677.8\times {10}^{-3}$ |

Surface tension coefficient $\sigma \left[\mathrm{N}/\mathrm{m}\right]$ | 0.059 | |

Latent heat of vaporization ${h}_{fg}\left[\mathrm{J}/\mathrm{Kg}\right]$ | $2258.0\times {10}^{3}$ |

**Table 2.**Properties of saturated sodium at 1200K in near-atmospheric conditions typical of reactor operation.

Properties of Saturated Sodium at 1200K, 1.48 bar | ||
---|---|---|

Property | Vapour | Liquid |

Dynamic viscosity $\mu \left[\mathrm{Pa}\xb7\mathrm{s}\right]$ | $1.8\times {10}^{-11}$ | $152.9\times {10}^{-6}$ |

Density $\rho \left[\mathrm{kg}/{\mathrm{m}}^{3}\right]$ | 0.39 | 732.0 |

Specific heat capacity $c\left[\mathrm{J}/\mathrm{Kg}/\mathrm{K}\right]$ | 2750.0 | 1250.0 |

Thermal conductivity $k\left[\mathrm{W}/\mathrm{m}/\mathrm{K}\right]$ | $48.0\times {10}^{-3}$ | $47.2$ |

Surface tension coefficient $\sigma \left[\mathrm{N}/\mathrm{m}\right]$ | 0.115 | |

Latent heat of vaporization ${h}_{fg}\left[\mathrm{J}/\mathrm{Kg}\right]$ | $3840.0\times {10}^{3}$ |

$\mathbf{Liquid}\mathbf{Dynamic}\mathbf{Viscosity}\mathit{\mu}\left[\mathrm{Pa}\mathit{\xb7}s\right]$ | Growth Rate A [m/s] | $\mathbf{Surface}\mathbf{Tension}\mathbf{Coefficient}\mathit{\sigma}\left[\mathbf{N}/\mathbf{m}\right]$ |
---|---|---|

$152.9\times {10}^{-6}$ | 3.39 | 0.115 |

Overview of Microlayer Formation Cases | |||
---|---|---|---|

Section | Growth Rate Value | Liquid Viscosity Value | Surface Tension Coefficient Value |

Section 4.2.1 | 0.5$\times $, base, 2$\times $ | base | base |

Section 4.2.2 | base | 0.5$\times $, base, 2$\times $ | base |

Section 4.2.3 | base | base | 0.25$\times $, 0.5$\times $, base, 2$\times $ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Giustini, G.; Kim, H.; I. Issa, R.; J. Bluck, M.
Modelling Microlayer Formation in Boiling Sodium. *Fluids* **2020**, *5*, 213.
https://doi.org/10.3390/fluids5040213

**AMA Style**

Giustini G, Kim H, I. Issa R, J. Bluck M.
Modelling Microlayer Formation in Boiling Sodium. *Fluids*. 2020; 5(4):213.
https://doi.org/10.3390/fluids5040213

**Chicago/Turabian Style**

Giustini, Giovanni, Hyungdae Kim, Raad I. Issa, and Michael J. Bluck.
2020. "Modelling Microlayer Formation in Boiling Sodium" *Fluids* 5, no. 4: 213.
https://doi.org/10.3390/fluids5040213