# Spray Cooling Heat Transfer above Leidenfrost Temperature

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}and air pressures in the range of 214 to 480 kPa. They pointed out that an increase in the water flow causes an increase in droplet size and results in partial evaporation. Then a big portion of liquid cools the surface ineffectively. As the very fine droplets cannot penetrate the vapour layer, the optimal droplet size together with optimal droplet velocity must be found. The speed of the cooled surface movement under the nozzles plays a role. They also observed significant differences in droplet behaviour (motion and interactions with the surface) due to the impingement position relative to the nozzle axis, which causes a change in heat transfer.

^{−1}and air pressures in the range of 2.05 to 3.20 bar, increasing air pressure at a constant water flow causes finer and faster droplets, which intensifies heat transfer. The authors suggest setting water flow rate to a critical value and then adjusting air pressure to get an air-to-water volume flow rate ratio A/W of above 10.

_{i}is water flow rate in kg·m

^{−2}s

^{−1}, µ is dynamic viscosity and d

_{32}[m] is the Sauter diameter of the droplet. It must be distinguished if correlations use Reynolds for spray or for droplets.

_{w}[m

^{3}·m

^{−2}s

^{−1}] 3.5 × 10

^{−3}to 9.96 × 10

^{−3}m

^{3}·m

^{−2}s

^{−1}, v

_{w}is mean water droplet velocity at the nozzle exit in a range from 10 to 30 m·s

^{−1}, ΔT = T

_{s}−T

_{w}, T

_{s}up to 530 °C.

_{w}[m

^{3}·m

^{−2}s

^{−1}] 0.58 × 10

^{−3}to 3.5 × 10

^{−3}m

^{3}·m

^{−2}s

^{−1}, Ts up to 530 °C, d

_{32}from 0.137 to 1.35 mm.

^{−3}] is droplet number in the range 3.77 × 10

^{7}–1.48 × 10

^{8}, d

_{30}in [m] in the range from 83 to 206 µm, v

_{w}is volume weighted mean velocity of water droplets from 6.8 to 15.6 m·s

^{−1}. According to Nasr et al. [13] HTC in a stable film boiling regime can be expressed as follows:

_{w}is water density, the range of Q

_{w}[L·m

^{−2}s

^{−1}] is not specified, d

_{32}in [µm] is from 125 to 520 µm, v

_{10}arithmetic mean velocity is from 0.2 to 20.8 m·s

^{−1}.

_{w}[L·m

^{−2}s

^{−1}] from 2 to 106 L·m

^{−2}s

^{−1}; v

_{w}is volume weighted mean velocity in spray direction 9.3–45.8 m·s

^{−1}, d

_{30}[µm] 19–119 µm and T

_{s}750–1200 °C. This formula was based on experimental work where the sample was held at a steady temperature, but according to the authors it is usable for less than 5 L·m

^{−2}s

^{−1}.

_{32}can be evaluated by the use of Lefebvre’s correlation [15], which is also found in [16] for flat jet nozzles, and by Estes et al. [17] or Lefebvre et al. [18] for full cone nozzles:

_{0}[18,19]:

_{w}is in [kg·m

^{−2}s

^{−1}].

^{−2}s

^{−1}) and was motivation for the presented experimental investigation.

- Spray properties: water flow rate, air pressure (for mist nozzles), water temperature, air temperature, nozzle types and their set-up (nozzle numbers, overlap, angles and heights).
- Surface properties: surface structure and material (thermal properties), roughness, surface temperature and movement.
- Ambient conditions that can change heat transfer or fluid flow (ambient air pressure, ambient temperature or air flow).

## 2. Experiment

#### 2.1. Experimental Plan

^{−1}.

#### 2.2. Heat Transfer Coefficient Measurement

^{−1}in this study. The test plate is insulated from all sides except the sprayed surface and is made of austenitic steel to protect the surface from oxidation. Austenitic steel is advantageous for inverse heat conduction tasks because of the missing material’s phase changes and subsequent steep changes in thermos-physical properties with temperature. K-type shielded thermocouples are positioned inside the plate with the tip at a distance of 2 mm from the cooled surface. The computer with the data acquisition system monitors the heating process, controls the experiment and records the data from the thermocouples and position sensor.

#### 2.3. Water Impingement Density Measurement

^{−2}s

^{−1}.

#### 2.4. Impact Pressure Measurement

#### 2.5. Droplet Size and Velocity

_{32}—Sauter mean diameter, VP—absolute mean velocity, VPx mean velocity on x axis, VPy mean velocity on y axis). The graphs shown in Figure 10 and Figure 11 are examples of the data produced from droplet size and velocity measurement.

#### 2.6. Inputs for Correlations

- Q
_{i}[L·m^{−2}s^{−1}] water impingement density, - v [m·s
^{−1}] mean droplet velocity, - d
_{32}[m] Sauter droplet diameter, - N [m
^{−2}s^{−1}] number of drops per square meter per second, - E [J] kinetic energy of droplet (for droplet with average size and speed),
- H [kg·m·s
^{−1}] droplet momentum, - Im [Pa] impact pressure
- HTC [W·m
^{−2}·K^{−1}] average heat transfer coefficient

## 3. Correlations

^{2}” contains the average square difference between the measured and correlated HTC. $Re{s}^{2}=\frac{1}{24}\sum {\left(HT{C}_{measured}-HT{C}_{cerelated}\right)}^{2}$, where 24 is the number of HTC values used. $Re{s}^{2}$ for each tested equation is also shown in Figure 13.

## 4. Discussion and Conclusions

_{i}is definitely due to the fact that this parameter is easy to measure.

_{i}and d

_{32}(used in Equation (1)).

_{32}, Re or kinetic energy E is used. Correlations where velocity and droplet diameter are used as separate parameters provide significantly better results in comparison to the equations where these parameters are included in droplet Re or in kinetic energy.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Scheme of experiment (

**a**), test plate with thermocouples and nozzle with deflector on moving trolley for the heat transfer coefficient (HTC) measurement (

**b**).

**Figure 3.**Temperatures measured at a depth of 2 mm and computed surface temperatures, data for experiments E1 and E5 in position of nozzle axis.

**Figure 11.**Droplet diameter (

**a**) and droplet velocity (

**b**) scatter plotted for E5 nozzle in position A.

**Figure 12.**Summary of averaged measured data: HTC, water impingement density, impact pressure, Sauter mean diameter of the droplet d

_{32}, droplet mean velocity v

_{p}for all experiments.

**Figure 13.**Comparison of Equations (1)–(10) based on Res

^{2}. The parameters that were used are written under each column.

**Figure 14.**HTC above Leidenfrost measured and correlated (Equation (8): $HTC=38.448\xb7{P}^{0.454}\xb7Q{i}^{0.132}$).

Experiment | Water Flowrate [L/min] | Air Pressure [Bar] | Nozzle Type |
---|---|---|---|

E1 | 11.0 | NA | Large water |

E2 | 11.0 | 1.5 | Large mist low |

E3 | 11.0 | 3.0 | Large mist high |

E4 | 6.0 | NA | Small water |

E5 | 6.0 | 0.5 | Small mist low |

E6 | 6.0 | 1.5 | Small mist high |

ID | Formula | Res^{2} |
---|---|---|

Equation (1) | $HTC=19.6\xb7Q{i}^{0.461}\xb7{v}^{0.261}\xb7{d}_{32}^{-0.208}$ | 664 |

Equation (2) | $HTC=351\xb7{N}^{0.456}\xb7{v}^{0.263}\xb7{d}_{32}^{1.164}$ | 664 |

Equation (3) | $HTC=199\xb7R{e}^{0.040}\xb7Q{i}^{0.245}$ | 5999 |

Equation (4) | $HTC=89\xb7{E}^{-0.056}\xb7Q{i}^{0.402}$ | 5536 |

Equation (5) | $HTC=113\xb7{E}^{0.221}\xb7{N}^{0.226}$ | 1402 |

Equation (6) | $HTC=51\xb7{H}^{-0.100}\xb7Q{i}^{0.588}$ | 2957 |

Equation (7) | $HTC=1.235\xb7{H}^{0.283}\xb7{N}^{0.439}$ | 672 |

Equation (8) | $HTC=38.448\xb7I{m}^{0.454}\xb7Q{i}^{0.132}$ | 340 |

Equation (9) | $HTC=41.491\text{}\xb7I{m}^{0.468}$ | 894 |

Equation (10) | $HTC=256\xb7Q{i}^{0.277}$ | 6034 |

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**MDPI and ACS Style**

Chabicovsky, M.; Kotrbacek, P.; Bellerova, H.; Kominek, J.; Raudensky, M. Spray Cooling Heat Transfer above Leidenfrost Temperature. *Metals* **2020**, *10*, 1270.
https://doi.org/10.3390/met10091270

**AMA Style**

Chabicovsky M, Kotrbacek P, Bellerova H, Kominek J, Raudensky M. Spray Cooling Heat Transfer above Leidenfrost Temperature. *Metals*. 2020; 10(9):1270.
https://doi.org/10.3390/met10091270

**Chicago/Turabian Style**

Chabicovsky, Martin, Petr Kotrbacek, Hana Bellerova, Jan Kominek, and Miroslav Raudensky. 2020. "Spray Cooling Heat Transfer above Leidenfrost Temperature" *Metals* 10, no. 9: 1270.
https://doi.org/10.3390/met10091270