# Modelling of Boiling Flows for Nuclear Thermal Hydraulics Applications—A Brief Review

## Abstract

**:**

## 1. Introduction

## 2. The ‘Eulerian–Eulerian’ Two-Fluid Simulation Approach

#### 2.1. Our Current Understanding of Vertical Upward Subcooled Flow Boiling

#### 2.2. Overview of a Practical Simulation Method for Component-Scale Analysis of Boiling

## 3. An Assessment of the Basic Wall-Boiling Model

#### 3.1. Current Understanding of the Cycle of Processes Associated with Boiling at a Surface

#### 3.2. Basic Wall-Boiling Model for Eulerian–Eulerian Simulation

#### 3.3. Manual Assessment of the RPI Model

## 4. Development of Boiling Models and Their Implementation in CFD Simulation

#### 4.1. Importance of the Wall-Boiling Model

#### 4.2. Development of Wall-Boiling Models

_{DF}, lift force F

_{L}and liquid reaction to bubble expansion F

_{H}), surface tension, gravity and wall-adhesion forces.

## 5. Development of Physics-Based Microscopic Models of Boiling

#### 5.1. The Interface-Capturing Simulation Approach

- -
- With the volume of fluid (VOF) method [37], the volume fraction of the ‘primary’ fluid is used to distinguish the two phases;
- -
- The level set (LS) method [38] identifies the interface as the zero level of a function representing the shortest distance from the interface;
- -
- The front tracking (FT) [39] method describes the interface as a set of massless particles moved around by the fluid velocity field.

#### 5.2. Further Developments for Extension to Boiling Conditions

#### 5.2.1. Modelling Mass Transfer

#### 5.2.2. Modelling Bubble–Wall Interaction

#### 5.3. Application of Interface-Capturing Simulation

#### 5.3.1. Computation of Bubble Departure Diameters and Frequencies

#### 5.3.2. Surface Phenomena

#### 5.3.3. Towards CHF Prediction: Modelling the Collective Behaviour of a Small Population of Bubbles

## 6. Outlook—Future Issues

- (1)
- The two-fluid approach to modelling flow boiling presented in this review relies on a number of approximations and empirical parameters that limit the applicability of the approach, which should be considered obsolete, as more capable and physically consistent methods have now been developed.
- (2)
- Application of modern interface capturing methods to problems typical of boiling in laboratory conditions, typically in low-pressure, low-subcooling pool boiling mode, enabled gaining unprecedented insight on the process of bubble formation and release at a surface. However, none of the interface capturing methods discussed in this review are yet applicable to highly turbulent subcooled bubbly flows typical of reactor operations.
- (3)
- Extensions to modelling the behaviour of the solid–liquid–vapour contact line at the base of a steam bubble are in their early stages of development, and a general model that is applicable to any fluid or surface material does not yet exist.
- (4)
- Modelling of evaporation at the phase boundary (e.g., the curved surface of a bubble) has to date been possible only for the case of thermally driven evaporation in near-equilibrium conditions. Thus, efforts should be pursued to extend current modelling capabilities to non-equilibrium conditions and cases where both dynamic and thermal effects determine bubble behaviour.

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Our current understanding of nucleate flow boiling in nuclear reactor conditions [2]. The bottom panel shows a diagram of the likely distribution of steam bubbles generated at a heated wall in vertical subcooled boiling. In the top panel, the likely distribution of cross-section averaged void fraction is shown. Flow regimes are identified depending on void fraction values: ‘single-phase convection’—zero void fraction; ‘highly subcooled’—bubbles are present only near the wall and condense as soon as released into the flow; ‘low subcooled’—permanence of steam bubbles in a body of partly subcooled liquid; ‘saturated boiling’—the fluid is entirely at or above the saturation temperature and bubbles fill up the pipe cross-section.

**Figure 2.**(

**a**) Current understanding of the cycle of processes associated with boiling; (

**b**) basic representation of the main mechanisms of wall heat transfer. From Thakrar et al. [15].

**Figure 4.**Comparison between measured wall temperatures, their values predicted using manual evaluation of the RPI model and boiling heat transfer correlations, for a typical boiling validation case. From [15].

**Figure 5.**Volume fraction distribution predicted with computational fluid dynamics (CFD) simulation using RPI modelling of wall-boiling. Note the mismatch between the size of the simulation domains, more than one metre in length, and the likely much smaller characteristic size of the steam bubbles generated at the heated wall, of perhaps only 10–100 micron in diameter. The computational mesh used to resolve the volume fraction distribution indicated in the figure consists of cells much larger than the single bubbles, which are replaced by a corresponding continuous distribution of the vapour phase. The image shows results for two different benchmark cases [22] (‘bar2’ and ‘bar4’ in the figure), using in in the CFD model the bubble departure diameter correlations of Tolubinskiy et al. [23] (panels (

**a**,

**c**) and Kocamustafaogullari et al. [21] (panels (

**b**,

**d**). For the same test case, different correlations for computing the bubble departure diameter return radically discrepant volume fraction distributions. From Colombo et al. [4].

**Figure 7.**An example of the accuracy of current bubble departure models used to generate input data for RPI-based CFD simulation of boiling flows. The blue squares indicate application [32] of the energy based model of Ardron et al. [30]. The black lines indicate results of application of the Klausner [25] and Yun [34] models, and of the empirical fit to the Klausner model by Sugrue et al. [28]. From [32].

**Figure 8.**Predicted bubble departure diameter (

**a**) and wall temperature (

**b**) in typical vertical subcooled flow boiling conditions, from Colombo et al. [27]. Lines correspond to different methods to compute bubble departure diameters: Tolubinskiy et al. correlation [23] (yellow lines), Kocamustafaogullari et al. correlation [21] (green lines), Klausner et al. model [25] (red lines), Sugrue et al. model [28] (black lines), Yun et al. model [34] (blue lines). In panel (

**b**), squares denote experimental wall temperature values [22].

**Figure 9.**Example of interface-capturing simulation of boiling at the scale of the single steam bubbles, from Tryggvason et al. [43]. Computational meshes used for this kind of simulation are required to resolve the details of the vapour–liquid interface (grey surface in the figure), typically resulting in cell sizes of a few micrometres in realistic boiling conditions.

**Figure 11.**Panels (

**a**,

**b**) indicate, respectively, the bubble shape and heat flux distribution at the solid surface reconstructed from measurement. Panel (

**c**) shows comparison of heat flux distributions at the solid surface beneath a bubble from interface-capturing simulation (solid lines) and from the experiment (squares). The dashed lines represent predictions of the heat flux from the solid surface to the liquid obtained with the physical model used for RPI simulation of boiling at the component scale. From Giustini et al. [69,74].

**Figure 12.**Simulation of the collective behaviour of steam bubbles in pool boiling [56] at a heat flux of 300 KW/m

^{2}, tracking the evolution of a few bubbles for two seconds until the bubbles merge and form a vapour blanket on the solid surface.

Authors | Interface Capturing Method | Mass Transfer Model | Application |
---|---|---|---|

Welch et al. [50] | VOF | Heat flux balance | Film boiling |

Son et al. [51] | Level Set | Conduction in interface cells | Film boiling, single bubble growth |

Gibou et al. [52] | Level Set | Heat flux balance | Film boiling |

Tryggvason et al. [43,53,54] | Front Tracking | Heat flux balance | Film boiling, nucleate boiling |

Sato et al. [55,56] | ‘Constrained Interpolation Profile’ VOF | Heat flux balance | Nucleate boiling |

Hardt et al. [45,46,47,48] | VOF | Kinetic model | Film boiling, single bubble growth, boiling in microchannels |

Badillo [49,57] | Phase Field, VOF | Asymptotic relaxation model | Single bubble growth |

Nichita [58] | VOF + Level Set | Conduction in interface cells | Single bubble growth |

Ganapathy et al. [59] | VOF | Conduction in interface cells | Boiling in microchannels |

Sun et al. [60] | VOF | Conduction in liquid cells | Film boiling |

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**MDPI and ACS Style**

Giustini, G.
Modelling of Boiling Flows for Nuclear Thermal Hydraulics Applications—A Brief Review. *Inventions* **2020**, *5*, 47.
https://doi.org/10.3390/inventions5030047

**AMA Style**

Giustini G.
Modelling of Boiling Flows for Nuclear Thermal Hydraulics Applications—A Brief Review. *Inventions*. 2020; 5(3):47.
https://doi.org/10.3390/inventions5030047

**Chicago/Turabian Style**

Giustini, Giovanni.
2020. "Modelling of Boiling Flows for Nuclear Thermal Hydraulics Applications—A Brief Review" *Inventions* 5, no. 3: 47.
https://doi.org/10.3390/inventions5030047