# Simulation of Single Vapor Bubble Condensation with Sharp Interface Mass Transfer Model

^{*}

## Abstract

**:**

## 1. Introduction

^{3}) is density. The subscripts l and g represent the liquid and vapor phases, respectively. The volumetric mass flux ${\dot{m}}^{\u2034}$ (kg/m

^{3}s) depends highly on the relaxation parameter ${r}_{c}$ (s

^{−1}). A wide range between $0.1$ and ${10}^{6}$ s

^{−1}is proposed and successfully used for ${r}_{c}$ in previous studies [23]. Li et al. [24] derived a correlation for the relaxation parameter and showed the dependence of ${r}_{c}$ on the temperature, physical properties, and phase volume fraction of the grid element.

^{2}s) using Hertz–Knudsen equation assuming a jump in the temperature and pressure across the interface ${T}_{sat}\left(\right)open="("\; close=")">{p}_{l}={T}_{g}$:

^{3}) is the cell volume, and A (m

^{2}) is the interfacial area in each cell obtained from the interface reconstruction technique (see Section 2.2). In Equations (2) and (3), the computed mass flux depends on the empirical parameter $\gamma $. The $\gamma =0.1-1$ is suggested for dynamically renewing water surfaces such as jets and moving films and $\gamma <0.1$ for stagnant surfaces [27]. Samkhaniani and Ansari [28] report that the bubble lifetime is highly sensitive to the choice of $\gamma $ in vapor condensation simulations and suggest that an appropriate value must be selected for simulations in comparison with experiments.

## 2. Methodology and Validation

- Mass conservation$$\frac{\partial \rho}{\partial t}+\nabla \xb7\left(\right)open="("\; close=")">\rho \mathbf{u}$$
- Momentum conservation$$\frac{\partial \rho \mathbf{u}}{\partial t}+\nabla \xb7\left(\right)open="("\; close=")">\rho \mathbf{uu}$$
- Energy conservation$$\frac{\partial \rho {c}_{p}T}{\partial t}+\nabla \xb7\left(\right)open="("\; close=")">\rho {c}_{p}T\mathbf{u}={\dot{m}}^{\u2034}{h}_{lg},$$
- Phase-fraction transport equation$$\frac{\partial {\alpha}_{l}}{\partial t}+\nabla \xb7\left(\right)open="("\; close=")">{\alpha}_{l}\mathbf{u}$$

#### 2.1. Stefan Problem

^{2}/s) is the vapor thermal diffusivity, and $\eta $ is obtained from:

#### 2.2. Parasitic Current

^{3}, ${\nu}_{l}={\mu}_{l}/{\rho}_{l}={10}^{-6}$ m

^{2}/s, ${\rho}_{g}=1$ kg/m

^{3}, ${\nu}_{g}={\mu}_{g}/{\rho}_{g}={10}^{-6}$ m

^{2}/s, $\sigma =0.1$ N/m. The pressure value at the boundaries is considered as uniformly constant and equal to zero and the Neumann boundary condition is assigned for velocity.

## 3. Results and Discussion

#### 3.1. Problem Definition

^{3}, $\nu =1.66\times {10}^{-5}$ m

^{2}/s, $k=0.0259$ W/m K, ${c}_{p}=2110.7$ J/kg K, and for liquid water $\rho =953.1$ kg/m

^{3}, $\nu =2.75\times {10}^{-7}$ m

^{2}/s, $k=0.68$ W/m K, ${c}_{p}=4224.4$ J/kg K. The surface tension $\sigma =0.057$ N/m, latent heat ${h}_{lg}=2237$ kJ/kg, and gravity $\mathbf{g}=9.81$ m/s

^{2}are used [43,44]. The computational domain size is $2{D}_{0}\times 4{D}_{0}$ and filled with $100\times 200$ uniform hexahedral cells, which corresponds to 50 cells per diameter. The initial diameter of the vapor bubble is ${D}_{0}=1.008$ mm, located at the middle line with distance ${D}_{0}$ from the bottom patch. There is a thin thermal region around the interface where the temperature smoothly changes from a saturated temperature inside the bubble to the subcooled temperature of the surrounding liquid. The temperature profile inside this region is initialized with [45]:

#### 3.2. Validation

#### 3.3. Thermal Boundary Layer

#### 3.4. Bubble Lifetime

^{2}K and ${f}_{3}=0.12$ s/m

^{1/3}. This correlation reveals how bubble lifetime varies with bubble diameter, subcooled temperature, and latent heat. The details of the derivation can be found in Appendix A.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Bubble Lifetime Estimation

^{2}K) and $\Delta {T}_{sub}={T}_{sat}-{T}_{inf}$ (K). Moreover, this assumes that the bubble remains in spherical shape during condensation, it is a good approximation for a small-size bubble. The surface tension force tends to keep a bubble in a spherical shape and it is dominant force in sub-millimetre bubbles. Then, the volume and interfacial area for a bubble with diameter d is $V=\pi {d}^{3}/6$ and $A=\pi {d}^{2}$, respectively. Therefore,

## References

- Mao, N.; Zhuang, J.; He, T.; Song, M. A critical review on measures to suppress flow boiling instabilities in microchannels. Heat Mass Transf.
**2021**, 57, 889–910. [Google Scholar] [CrossRef] - Liang, G.; Mudawar, I. Review of channel flow boiling enhancement by surface modification, and instability suppression schemes. Int. J. Heat Mass Transf.
**2020**, 146, 118864. [Google Scholar] [CrossRef] - Tang, J.; Sun, L.; Liu, H.; Liu, H.; Mo, Z. Review on direct contact condensation of vapor bubbles in a subcooled liquid. Exp. Comput. Multiph. Flow
**2022**, 4, 91–112. [Google Scholar] [CrossRef] - Evans, R. The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids. Adv. Phys.
**1979**, 28, 143–200. [Google Scholar] [CrossRef] - Stephan, S.; Liu, J.; Langenbach, K.; Chapman, W.G.; Hasse, H. Vapor- Liquid Interface of the Lennard-Jones Truncated and Shifted Fluid: Comparison of Molecular Simulation, Density Gradient Theory, and Density Functional Theory. J. Phys. Chem. C
**2018**, 122, 24705–24715. [Google Scholar] [CrossRef] - Najafi, M.; Maghari, A. On the calculation of liquid–vapor interfacial thickness using experimental surface tension data. J. Solut. Chem.
**2009**, 38, 685–694. [Google Scholar] [CrossRef] - Yang, C.; Li, D. A method of determining the thickness of liquid-liquid interfaces. Colloids Surfaces Physicochem. Eng. Asp.
**1996**, 113, 51–59. [Google Scholar] [CrossRef] - Srebnik, S.; Marmur, A. Negative Pressure within a Liquid–Fluid Interface Determines Its Thickness. Langmuir
**2020**, 36, 7943–7947. [Google Scholar] [CrossRef] - Baidakov, V.G.; Protsenko, S.P.; Bryukhanov, V.M. Relaxation processes at liquid-gas interfaces in one-and two-component Lennard-Jones systems: Molecular dynamics simulation. Fluid Phase Equilibria
**2019**, 481, 1–14. [Google Scholar] [CrossRef] - Baidakov, V.; Protsenko, S. Molecular-dynamics simulation of relaxation processes at liquid–gas interfaces in single-and two-component lennard-jones systems. Colloid J.
**2019**, 81, 491–500. [Google Scholar] [CrossRef] - Stephan, S.; Schaefer, D.; Langenbach, K.; Hasse, H. Mass transfer through vapour–liquid interfaces: A molecular dynamics simulation study. Mol. Phys.
**2021**, 119, e1810798. [Google Scholar] [CrossRef] - Heinen, M.; Vrabec, J. Evaporation sampled by stationary molecular dynamics simulation. J. Chem. Phys.
**2019**, 151, 044704. [Google Scholar] [CrossRef] [PubMed] - Lotfi, A.; Vrabec, J.; Fischer, J. Evaporation from a free liquid surface. Int. J. Heat Mass Transf.
**2014**, 73, 303–317. [Google Scholar] [CrossRef] - Mirjalili, S.; Jain, S.S.; Dodd, M. Interface-capturing methods for two-phase flows: An overview and recent developments. Cent. Turbul. Res. Annu. Res. Briefs
**2017**, 2017, 13. [Google Scholar] - Xu, Q.; Liang, L.; She, Y.; Xie, X.; Guo, L. Numerical investigation on thermal hydraulic characteristics of steam jet condensation in subcooled water flow in pipes. Int. J. Heat Mass Transf.
**2022**, 184, 122277. [Google Scholar] [CrossRef] - Lee, M.S.; Riaz, A.; Aute, V. Direct numerical simulation of incompressible multiphase flow with phase change. J. Comput. Phys.
**2017**, 344, 381–418. [Google Scholar] [CrossRef] [Green Version] - Tryggvason, G.; Esmaeeli, A.; Al-Rawahi, N. Direct numerical simulations of flows with phase change. Comput. Struct.
**2005**, 83, 445–453. [Google Scholar] [CrossRef] [Green Version] - Ningegowda, B.M.; Ge, Z.; Lupo, G.; Brandt, L.; Duwig, C. A mass-preserving interface-correction level set/ghost fluid method for modeling of three-dimensional boiling flows. Int. J. Heat Mass Transf.
**2020**, 162, 120382. [Google Scholar] [CrossRef] - Liu, Z.; Sunden, B.; Wu, H. Numerical modeling of multiple bubbles condensation in subcooled flow boiling. J. Therm. Sci. Eng. Appl.
**2015**, 7, 031003. [Google Scholar] [CrossRef] - Bureš, L.; Sato, Y. Direct numerical simulation of evaporation and condensation with the geometric VOF method and a sharp-interface phase-change model. Int. J. Heat Mass Transf.
**2021**, 173, 121233. [Google Scholar] [CrossRef] - Badillo, A. Quantitative phase-field modeling for boiling phenomena. Phys. Rev. E
**2012**, 86, 041603. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lee, W.H. Pressure iteration scheme for two-phase flow modeling. In Multiphase Transport: Fundamentals, Reactor Safety, Applications; Hemisphere Publishing Corporation: London, UK, 1980; pp. 407–432. [Google Scholar]
- Samkhaniani, N.; Ansari, M.R. The evaluation of the diffuse interface method for phase change simulations using OpenFOAM. Heat Transf. Res.
**2017**, 46, 1173–1203. [Google Scholar] [CrossRef] - Li, H.; Tian, M.; Tang, L. Axisymmetric numerical investigation on steam bubble condensation. Energies
**2019**, 12, 3757. [Google Scholar] [CrossRef] [Green Version] - Tanasawa, I. Advances in condensation heat transfer. In Advances in heat Transfer; Elsevier: Amsterdam, The Netherlands, 1991; Volume 21, pp. 55–139. [Google Scholar]
- Schrage, R.W. A theoretical study of interphase mass transfer. In A Theoretical Study of Interphase Mass Transfer; Columbia University Press: New York, NY, USA, 1953. [Google Scholar]
- Marek, R.; Straub, J. Analysis of the evaporation coefficient and the condensation coefficient of water. Int. J. Heat Mass Transf.
**2001**, 44, 39–53. [Google Scholar] [CrossRef] - Samkhaniani, N.; Ansari, M. Numerical simulation of bubble condensation using CF-VOF. Prog. Nuclear Energy
**2016**, 89, 120–131. [Google Scholar] [CrossRef] - Liu, H.; Tang, J.; Sun, L.; Mo, Z.; Xie, G. An assessment and analysis of phase change models for the simulation of vapor bubble condensation. Int. J. Heat Mass Transf.
**2020**, 157, 119924. [Google Scholar] [CrossRef] - Kunkelmann, C. Numerical Modeling and Investigation of Boiling Phenomena. Ph.D. Thesis, Technische Universität, Darmstadt, Germany, 2011. [Google Scholar]
- Pan, Z.; Weibel, J.A.; Garimella, S.V. Spurious current suppression in VOF-CSF simulation of slug flow through small channels. Numer. Heat Transf. Part Appl.
**2015**, 67, 1–12. [Google Scholar] [CrossRef] - Kunkelmann, C.; Stephan, P. CFD simulation of boiling flows using the volume-of-fluid method within OpenFOAM. Numer. Heat Transf. Part Appl.
**2009**, 56, 631–646. [Google Scholar] [CrossRef] - Kunkelmann, C.; Stephan, P. Numerical simulation of the transient heat transfer during nucleate boiling of refrigerant HFE-7100. Int. J. Refrig.
**2010**, 33, 1221–1228. [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] - Son, J.H.; Park, I.S. Temperature changes around interface cells in a one-dimensional Stefan condensation problem using four well-known phase-change models. Int. J. Therm. Sci.
**2021**, 161, 106718. [Google Scholar] [CrossRef] - Shang, X.; Zhang, X.; Nguyen, T.B.; Tran, T. Direct numerical simulation of evaporating droplets based on a sharp-interface algebraic VOF approach. Int. J. Heat Mass Transf.
**2022**, 184, 122282. [Google Scholar] [CrossRef] - Deshpande, S.S.; Anumolu, L.; Trujillo, M.F. Evaluating the performance of the two-phase flow solver interFoam. Comput. Sci. Discov.
**2012**, 5, 014016. [Google Scholar] [CrossRef] - Weller, H.G. A New Approach to VOF-Based Interface Capturing Methods for Incompressible and Compressible Flow; Report TR/HGW; OpenCFD Ltd.: Bracknell, UK, 2008; Volume 4, p. 35. [Google Scholar]
- Roenby, J.; Bredmose, H.; Jasak, H. A computational method for sharp interface advection. R. Soc. Open Sci.
**2016**, 3, 160405. [Google Scholar] [CrossRef] [Green Version] - Gamet, L.; Scala, M.; Roenby, J.; Scheufler, H.; Pierson, J.L. Validation of volume-of-fluid OpenFOAM® isoAdvector solvers using single bubble benchmarks. Comput. Fluids
**2020**, 213, 104722. [Google Scholar] [CrossRef] - Dai, D.; Tong, A.Y. Analytical interface reconstruction algorithms in the PLIC-VOF method for 3D polyhedral unstructured meshes. Int. J. Numer. Methods Fluids
**2019**, 91, 213–227. [Google Scholar] [CrossRef] - Greenshields, C. Interface Capturing in OpenFOAM. 2020. Available online: https://cfd.direct/openfoam/free-software/multiphase-interface-capturing/ (accessed on 22 May 2022).
- Zeng, Q.; Cai, J.; Yin, H.; Yang, X.; Watanabe, T. Numerical simulation of single bubble condensation in subcooled flow using OpenFOAM. Prog. Nucl. Energy
**2015**, 83, 336–346. [Google Scholar] [CrossRef] - Kamei, S.; Hirata, M. Condensing phenomena of a single vapor bubble into subcooled water. Exp. Heat Transf. Int. J.
**1990**, 3, 173–182. [Google Scholar] [CrossRef] - Magnini, M. CFD Modeling of Two-Phase Boiling Flows in the Slug Flow Regime with an Interface Capturing Technique. Ph.D. Thesis, Alma Mater Studiorum University of Bologna, Bologna, Italy, May 2012. [Google Scholar]
- Stephan, S.; Hasse, H. Enrichment at vapour–liquid interfaces of mixtures: Establishing a link between nanoscopic and macroscopic properties. Int. Rev. Phys. Chem.
**2020**, 39, 319–349. [Google Scholar] [CrossRef] - Stephan, S.; Langenbach, K.; Hasse, H. Enrichment of components at vapour-liquid interfaces: A study by molecular simulation and density gradient theory. Chem. Eng. Trans.
**2018**, 69. [Google Scholar] [CrossRef] - Chakraborty, S.; Ge, H.; Qiao, L. Molecular Dynamics Simulations of Vapor–Liquid Interface Properties of n-Heptane/Nitrogen at Subcritical and Transcritical Conditions. J. Phys. Chem. B
**2021**, 125, 6968–6985. [Google Scholar] [CrossRef] - Yin, Z.; Wen, J.; Wu, Y.; Wang, Q.; Zeng, M. Effect of non-condensable gas on laminar film condensation of steam in horizontal minichannels with different cross-sectional shapes. Int. Commun. Heat Mass Transf.
**2016**, 70, 127–131. [Google Scholar] [CrossRef] - Qu, X.H.; Tian, M.C.; Zhang, G.M.; Leng, X.L. Experimental and numerical investigations on the air–steam mixture bubble condensation characteristics in stagnant cool water. Nucl. Eng. Des.
**2015**, 285, 188–196. [Google Scholar] [CrossRef] - Sideman, S.; Hirsch, G. Direct contact heat transfer with change of phase: Condensation of single vapor bubbles in an immiscible liquid medium. Preliminary studies. AIChE J.
**1965**, 11, 1019–1025. [Google Scholar] [CrossRef]

**Figure 1.**Stefan problem: (

**a**) schematic, (

**b**) comparison of the present numerical simulation (Fourier model) with the exact analytical solution.

**Figure 2.**The spurious current contour around gas bubble interface in stagnant liquid in zero gravity $\left|\mathbf{g}\right|=0$ m/s

^{2}at t = 1 ms for various VoF reconstruction methods. Please note the one order of magnitude difference in the color bar.

**Figure 6.**The vapor bubble lifetime at different subcool temperatures; points correspond to the numerical simulations (Tanasawa model), solid lines are plotted based on Equation (12), where the fitting coefficients are ${f}_{1}$ = 32,440 W/m

^{2}K and ${f}_{3}$ = 0.12 s/m

^{1/3}.

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

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Samkhaniani, N.; Stroh, A.
Simulation of Single Vapor Bubble Condensation with Sharp Interface Mass Transfer Model. *Thermo* **2022**, *2*, 149-159.
https://doi.org/10.3390/thermo2030012

**AMA Style**

Samkhaniani N, Stroh A.
Simulation of Single Vapor Bubble Condensation with Sharp Interface Mass Transfer Model. *Thermo*. 2022; 2(3):149-159.
https://doi.org/10.3390/thermo2030012

**Chicago/Turabian Style**

Samkhaniani, Nima, and Alexander Stroh.
2022. "Simulation of Single Vapor Bubble Condensation with Sharp Interface Mass Transfer Model" *Thermo* 2, no. 3: 149-159.
https://doi.org/10.3390/thermo2030012