# Toward the Design of a Representative Heater for Boiling Flow Characterization under PWR’s Prototypical Thermalhydraulic Conditions

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## Abstract

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## Featured Application

**This work is related to boiling studies in PWR conditions, it will lead to the design of an experimental setup to characterise boiling flow and develop safety models over boiling crisis.**

## Abstract

## 1. Introduction

- The experimental setup and associated measurement techniques are introduced in the Section 2.
- The thermal behavior of the heater is then studied through a 1D approach considering realistic time-dependent boundary conditions that properly simulate nucleate boiling in Section 3.
- In the Section 4, a 2D extension analysis of the thermal behavior of the heater is introduced; some discussions concerning space meshing of the method and enhancement will be presented.
- Finally, the Section 5 concludes this study by highlighting the main findings and suggestions for further research.

## 2. Experimental Setup

## 3. Thermal Study of the Heater

#### 3.1. Methodology

- (1)
- In the first step, the direct heat conduction problem is solved within the plate for a given couple of boundary conditions expressed on both sides of the calculation domain:
- (a)
- On the dry side: $x=0$, ${q}^{\u2033}=0$.
- (b)
- On the wet side: $x=e$, ${T}_{d,wet}=f\left(t\right)$.

The objective is to calculate ${T}_{d,dry}=T(x=0,t)$ The subscript d indicates that the direct problem is being solved. - (2)
- In the second step, the associated inverse heat conduction problem is studied by solving the diffusion equation for a given couple of boundary conditions expressed on the dry side of the plate:
- (a)
- On the dry side: $x=0$, ${q}^{\u2033}=0$.
- (b)
- On the dry side: $x=0$, ${T}_{i,dry}(0,t)={T}_{d,dry}(0,t)$.

The objective is to calculate the wet temperature ${T}_{i,wet}$ and to check whether ${T}_{i,wet}={T}_{d,wet}$. The subscript i means that the inverse problem is being solved. One should notice that this configuration is very close to the one that will be encountered during the tests since ${T}_{d,dry}$ will effectively be measured by IR thermography. - (3)
- In the third step, a sensitivity analysis to the uncertainties of ${T}_{d,dry}$ will be performed.

#### 3.2. Direct Problem

#### 3.2.1. Modeling

#### 3.2.2. Preliminary Magnitude through Analytical Analysis

#### 3.2.3. 1D Solving

#### 3.2.4. Thickness Calculation

#### 3.3. Inverse Problem

#### 3.3.1. Measurement Techniques

#### 3.3.2. Methodology

#### 3.3.3. Results

## 4. Spatial Resolution of the Measurements

#### 4.1. 2D Diffusion Influence

#### 4.2. Thermal Influence Area of a Bubble Growth

- The first step consists of identifying the thickness corresponding to the camera-readable amplitude of the dry-side temperature variation (example Figure 20; criteria 1 gives a 170 µm of thickness, criteria 2 gives 110 µm of thickness.),
- The second step determines the dry-side influence area considering the two temperature criteria (see Figure 21; criteria 1’s thickness is valid (${d}_{1}$ = 60 µm), but criteria 2 correlates with 100 µm of thickness (${d}_{2}$ = 60 µm)).

#### 4.3. Discretization between Two Bubbles Growing on the Plate

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PWR | pressurized water reactor |

CEA | Commissariat à l’Energie Atomique et aux Energies Alternatives |

CFD | computational fluid dynamics |

ONB | onset of nucleate boiling |

DNB | departure from nucleate boiling |

ITO | indium tin oxide |

## Appendix A. Unal Frequency Correlation

**Table A1.**Range of validity of Unal correlation [3].

Pressure | [1–10] bar |

Heat flux | [0.47–4.5] MW/m${}^{2}$ |

Subcooling | [20–72] K |

Velocity | [0.08–3.05] m/s |

Time | [0.175–5] ms |

Diameter | [0.19–0.9] mm |

## References

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**Figure 1.**Diagram of the heat flux wall partitioning (Kurul and Podowski [1]), ${q}_{e}^{"}$ is the evaporation heat flux, ${q}_{q}^{"}$ is the quenching heat flux, and ${q}_{c}^{"}$ is the single-phase convective heat flux.

**Figure 2.**Sapphire substrate coated with ITO. Reprinted with permission from Richenderfer et al. [6].

**Figure 3.**The 3D schematic of the future test section. The heated element is drawn in red: The heated length L ≈ 5 × 10${}^{-1}$ m, heated width l ≈ 1 × 10${}^{-2}$ m, and thickness of the metallic plate are to be determined.

**Figure 6.**The prototypical time evolution of the temperature beneath a nucleation site, ${t}_{g}$ is the growth time and ${t}_{w}$ is the waiting time between a bubble take-off and the appearance of a new nucleation, based on Wang and Podowski [9].

**Figure 9.**Correspondence between the harmonic variation and the temperature variation imposed in the initial problem.

**Figure 10.**Variation of $\theta $ within the plate, the blue curve shows the signal attenuation as a function of the thickness, the red curve shows the time delay on the initial information as a function of the thickness.

**Figure 11.**The 1D steady-state problem–solution: the temperature within the plate (thickness 10 µm).

**Figure 13.**Direct problem–solution: temperature on the external face (dry temperature) as a function of the thickness of the plate. This problem–solution was solved with the following values: $N=1000$, $n=50$, ${t}_{g}=0.005\phantom{\rule{0.166667em}{0ex}}$s, ${t}_{w}=0.01\phantom{\rule{0.166667em}{0ex}}$s, $\mathsf{\Delta}t=0.00031\phantom{\rule{0.166667em}{0ex}}$s, $P=1\phantom{\rule{0.166667em}{0ex}}$bar, $I=400\phantom{\rule{0.166667em}{0ex}}$A, $\mathsf{\Delta}T$ = 5 ${\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C, ${C}_{P}=502\phantom{\rule{0.166667em}{0ex}}$J/K·kg, $\lambda =15\phantom{\rule{0.166667em}{0ex}}$W/m·K, $\rho =7960\phantom{\rule{0.166667em}{0ex}}$kg/m${}^{3}$.

**Figure 14.**Steps to solve the inverse problem, (

**a**) solving the direct problem by imposing the wet temperature (blue curve, 1) to get the dry temperature (2), (

**b**) solving the inverse problem by imposing the dry temperature (red curve, 3), calculated before (2) to get the wet temperature (4).

**Figure 15.**Illustration of the noise amplification solving the inverse problem for various thicknesses. This problem–solution was solved with the following values: $N=1000$, $n=50$, ${t}_{g}=0.005\phantom{\rule{0.166667em}{0ex}}$s, ${t}_{w}=0.01\phantom{\rule{0.166667em}{0ex}}$s, $\mathsf{\Delta}t=0.00031\phantom{\rule{0.166667em}{0ex}}$s, $P=1\phantom{\rule{0.166667em}{0ex}}$bar, $I=400\phantom{\rule{0.166667em}{0ex}}$A, $\mathsf{\Delta}T=5{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C, ${C}_{P}=502$ J/K·kg, $\lambda =15$ W/m·K, $\rho =7960$ kg/m${}^{3}$.

**Figure 18.**Illustration of the 2D diffusion effect over the dry temperature for various thicknesses. This problem–solution was solved with the following values: ${N}_{x}=50$, ${N}_{y}=200$, $n=50$, ${t}_{g}=0.005\phantom{\rule{0.166667em}{0ex}}$s, ${t}_{w}=0.01\phantom{\rule{0.166667em}{0ex}}$s, $\mathsf{\Delta}t=0.00031\phantom{\rule{0.166667em}{0ex}}$s, $P=1\phantom{\rule{0.166667em}{0ex}}\mathrm{bar}$, $I=400\phantom{\rule{0.166667em}{0ex}}$A, $\mathsf{\Delta}$T = 5 ${}^{\circ}$C, ${C}_{P}$ = 502 J/K·kg, $\lambda $ = 15 W/m·K, $\rho $ = 7960 kg/m${}^{3}$.

**Figure 19.**Influence area during a bubble growth on a nucleation site. This area can be characterized by the size d and the amplitude of the thermal disturbance A for the dry side of the plate.

**Figure 20.**Example of the determination of the limit thickness at $y={y}_{0}$ for $20\phantom{\rule{0.166667em}{0ex}}\mathrm{bar}$ pressure, a 120 µm bubble diameter was imposed. This problem–solution was solved with the following values: ${N}_{x}=50$, ${N}_{y}=200$, $n=50$, ${t}_{g}=0.005\phantom{\rule{0.166667em}{0ex}}$s, ${t}_{w}=0.01\phantom{\rule{0.166667em}{0ex}}$s, $\mathsf{\Delta}t=0.00031\phantom{\rule{0.166667em}{0ex}}$s, $P=1\phantom{\rule{0.166667em}{0ex}}\mathrm{bar}$, $I=400\phantom{\rule{0.166667em}{0ex}}$A, $\mathsf{\Delta}$T = 5 ${}^{\circ}$C, ${C}_{P}$ = 502 J/K·kg, $\lambda $ = 15 W/m·K, $\rho $ = 7960 kg/m${}^{3}$.

**Figure 21.**Example of the verification of the zone of influence of the temperature variation according to criteria 1 and 2. Here, for a pressure of 20 bar, for a thickness of 170 µm (

**left**), criterion 1 is respected, the zone of influence is 60 µm; for the second criterion (

**right**), it is necessary to reach 100 µm of thickness to detect the influence area on the dry side. This problem–solution was solved with the following values: ${N}_{x}=50$, ${N}_{y}=200$, $n=50$, ${t}_{g}=0.005\phantom{\rule{0.166667em}{0ex}}$s, ${t}_{w}=0.01\phantom{\rule{0.166667em}{0ex}}$s, $\mathsf{\Delta}t=0.00031\phantom{\rule{0.166667em}{0ex}}$s, $P=1\phantom{\rule{0.166667em}{0ex}}\mathrm{bar}$, $I=400\phantom{\rule{0.166667em}{0ex}}$A, $\mathsf{\Delta}$T = 5 ${}^{\circ}$C, ${C}_{P}$ = 502 J/K·kg, $\lambda $ = 15 W/m·K, $\rho $ = 7960 kg/m${}^{3}$.

**Figure 22.**Example of the determination of the minimum gap for two bubbles to be discernible. For the first example (

**left**), the bubbles are too close, and the temperature on the dry side will not allow discretizing two objects except for one big bubble growing on the plate. For the second example (

**right**), after expanding the distance between the bubbles, the temperature field allows detection of two different spots.

Bubble Size µm | Pressure bar | First Criterion: Detection Thickness µm | Second Criterion: Measurement Thickness µm |
---|---|---|---|

800 | 1 | 250 | 200 |

120 | 20 | 170 | 110 |

70 | 50 | 100 | 60 |

50 | 80 | 70 | 40 |

40 | 110 | 65 | 35 |

Thickness | Bubble Size | Minimum Gap for Detection |
---|---|---|

µm | µm | µm |

70 | 200 | 60 |

70 | 50 | 120 |

50 | 200 | 40 |

50 | 50 | 100 |

30 | 200 | 20 |

30 | 50 | 60 |

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**MDPI and ACS Style**

Bernadou, L.; François, F.; Bottin, M.; Djeridi, H.; Barre, S. Toward the Design of a Representative Heater for Boiling Flow Characterization under PWR’s Prototypical Thermalhydraulic Conditions. *Appl. Sci.* **2023**, *13*, 1534.
https://doi.org/10.3390/app13031534

**AMA Style**

Bernadou L, François F, Bottin M, Djeridi H, Barre S. Toward the Design of a Representative Heater for Boiling Flow Characterization under PWR’s Prototypical Thermalhydraulic Conditions. *Applied Sciences*. 2023; 13(3):1534.
https://doi.org/10.3390/app13031534

**Chicago/Turabian Style**

Bernadou, Louise, Fabrice François, Manon Bottin, Henda Djeridi, and Stephane Barre. 2023. "Toward the Design of a Representative Heater for Boiling Flow Characterization under PWR’s Prototypical Thermalhydraulic Conditions" *Applied Sciences* 13, no. 3: 1534.
https://doi.org/10.3390/app13031534