# Comparing Internal Flow in Freezing and Evaporating Water Droplets Using PIV

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Experimental Setup

#### 2.2. Experimental Procedures

**Heating**

- The heating of the surface started.
- At ${T}_{plate}$ = 313.15, 323.15 or 333.15 K (visually determined from the computer screen):
- -
- The pipette was filled with the DI water and seeding particle suspension.
- -
- The camera and the laser were switched on.
- -
- The laser sheet was fine tuned to the center of the droplet (while the droplet was still hanging from the pipette).
- -
- Finally, the droplet was released.

- The camera and laser light were turned off after about 60 s after the droplet has hit the surface.
- The surface was cleaned and a new experiment could begin.

**Freezing**

- The cooling of the surface started.
- At ${T}_{plate}$ = 261.15 K or 265.15 (visually determined from the computer screen), the pressurized air was switched on and turned off again when RH was around 50%. This took about 60 s.
- The pipette was filled with the DI water and seeding particle suspension.
- The camera and the laser were switched on. The position of the light sheet was fined tuned to the center of the droplet (while the droplet was still hanging from the pipette).
- The droplet was released when ${T}_{plate}$ reached 265.15 K (or 261.15 K) again (visually determined from the computer screen), which occurred approximately 30 s from when the pressurized air was switched off.
- The cooling was turned off when the droplet was completely frozen.
- The surface was cleaned and dried when ${T}_{plate}$ > 273.15 K and at ${T}_{plate}$ = 277.15 K a new experiment could begin.

#### 2.3. Uncertainty Analysis

## 3. Results and Discussion

#### 3.1. Impact of Release

#### 3.2. Evaporation

#### 3.2.1. Evaporation until “Steady State”

#### 3.2.2. Evaporation after “Steady state”

#### 3.3. Freezing

#### 3.4. Comparing the Flows within Freezing and Evaporating Droplets

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

A | surface contact area (m${}^{2}$) |

d | diameter (m) |

h | height (m) |

r | radius (m) |

t | time (s) |

T | temperature (K) |

RH | relative humidity (%) |

$\mu $ | viscosity (kg/ms) |

$Ma$ | Marangoni number |

$Stk$ | Stokes number |

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**Figure 1.**Direction of Marangoni driven flow for a constant gradient: (

**Left**) ${T}_{1}$ > ${T}_{2}$, evaporation; and (

**Right**) ${T}_{1}$ < ${T}_{2}$, freezing.

**Figure 3.**The mean velocity in the five points used in the repeatability study at t = 5 and 15 s for 10 experiments when ${T}_{plate}$ = 333.15 K. The precision error with a 95% confidence interval in each point is shown with error bars. Point 1 is located at the heated surface and Point 5 is located at the top of the corrected data. Points 2–4 are found in between Points 1 and 5.

**Figure 4.**The magnitude of the mean velocity along the symmetry line for ${T}_{plate}$ = 313.15, 323.15, and 333.15 K when t = 1–50 s.

**Figure 5.**Internal flow patterns with normalized velocity vectors for the corrected data when ${T}_{plate}$ = 333.15 K (Case 1) at t = 1–15 s.

**Figure 6.**The magnitude of the velocity along the symmetry line for ${T}_{plate}$ = 333.15 K (Case 1) when t = 1–15 s and t = 7–15 s.

**Figure 7.**The spread (standard deviation) in velocity along the symmetry line during evaporation for ${T}_{plate}$ = 313.15, 323.15 and 333.15 K (all case) when t = 1–15 s shown using error bars. The solid line is the mean velocity of all times.

**Figure 8.**Internal flow patterns with normalized velocity vectors for the corrected data when ${T}_{plate}$ = 333.15 K (Case 1) at t = 20–50 s.

**Figure 9.**The magnitude of the velocity along the symmetry line for ${T}_{plate}$ = 333.15 K (Case 1) when t = 20–50 s.

**Figure 10.**The spread (standard deviation) in velocity along the symmetry line during evaporation for ${T}_{plate}$ = 313.15, 323.15 and 333.15 K (all case) when t = 20–50 s shown using error bars. The solid line is the mean velocity of all times.

**Figure 11.**Internal flow patterns with normalized velocity vectors for the corrected data when ${T}_{plate}$ = 261.15 K at t = 1–5 s.

**Figure 12.**The magnitude of the velocity along the symmetry line for ${T}_{plate}$ = 261.15 K when t = 1–5 s.

**Figure 13.**The spread (standard deviation) in velocity along the symmetry line during freezing for ${T}_{plate}$ = 265.15 and 261.15 K when t = 1–4 s and t = 1–5 s, respectively, shown using error bars. The solid line is the mean velocity of all times.

Case | Particles | Diameter, d (m) | $\mathbf{Stk}$ | Conc. of Particles |
---|---|---|---|---|

Heating | Latex (Magsphere Inc., Pasadena, CA, USA | 5.1 × 10${}^{-6}$ | <5 × 10${}^{-4}$ | 9.9 × 10${}^{-7}$ m${}^{3}$ DI water and |

PS Red Fluorescent, 1055 kg/m${}^{3}$) | 0.1 × 10${}^{-7}$ m${}^{3}$ seeding particles | |||

Freezing | Rhodamine B (microParticles | 3.16 × 10${}^{-6}$ | <5 × 10${}^{-7}$ | 9.8 × 10${}^{-7}$ m${}^{3}$ DI water and |

GmbH PS-FluoRed, 1050 kg/m${}^{3}$) | 0.2·10${}^{-7}$ m${}^{3}$ seeding particles |

Case | Plate Temperature, ${\mathit{T}}_{\mathit{p}\mathit{l}\mathit{a}\mathit{t}\mathit{e}}$ | RH in Chamber | Temperature in Chamber, ${\mathit{T}}_{\mathit{c}\mathit{h}\mathit{a}\mathit{m}\mathit{b}\mathit{e}\mathit{r}}$ | Sampling Rate | Recording Times |
---|---|---|---|---|---|

Heating | 313.15 K ± 0.22 K, 323.15 K ± 0.11 K, 333.15 K ± 0.05 K | 47.2% ± 1.2% | 298.55 K ± 0.77 K | 50–54 Hz | 60 s |

Freezing | 265.07 K ± 0.12 K, 261.11 K ± 0.06 K | 50.4% ± 4.5% | 289.85 K ± 1.7 K | 50, 54 Hz | Dependent on the freezing times of the droplets |

Case (${\mathit{T}}_{\mathit{p}\mathit{l}\mathit{a}\mathit{t}\mathit{e}}$-Temperature) | Droplet Height, h (m) | Droplet Radius, r (m) | Contact Area at Surface, A (m${}^{2}$) |
---|---|---|---|

Evaporation | |||

313.15 K: Case 1 | 0.00137 | 0.00188 | 11.1 × 10${}^{-6}$ |

313.15 K: Case 2 | 0.00143 | 0.00186 | 10.9 × 10${}^{-6}$ |

323.15 K: Case 1 | 0.00145 | 0.00186 | 10.9 × 10${}^{-6}$ |

323.15 K: Case 2 | 0.00147 | 0.00184 | 10.6 × 10${}^{-6}$ |

333.15 K: Case 1 | 0.00148 | 0.00192 | 11.6 × 10${}^{-6}$ |

333.15 K: Case 2 | 0.00145 | 0.00185 | 10.7 × 10${}^{-6}$ |

Freezing | |||

261.15 K | 0.00178 | 0.00156 | 7.65 × 10${}^{-6}$ |

265.15 K | 0.00142 | 0.00171 | 9.15 × 10${}^{-6}$ |

**Table 4.**Values of the droplets geometry in the repeatability study at t = 5 and 15 s when ${T}_{plate}$ = 333.15 K. Note that the radius of the droplets and contact areas at the heated surface are the same for both times.

Case | Droplet Radius, r (m) | Contact Area at Surface, A (m${}^{2}$) | Droplet Height, h at t = 5 s (m) | Droplet Height, h at t = 15 s (m) |
---|---|---|---|---|

1 | 0.00180 | 10.1 × 10${}^{-6}$ | 0.00143 | 0.00139 |

2 | 0.00186 | 10.9 × 10${}^{-6}$ | 0.00137 | 0.00133 |

3 | 0.00191 | 11.5 × 10${}^{-6}$ | 0.00149 | 0.00144 |

4 | 0.00178 | 9.98 × 10${}^{-6}$ | 0.00157 | 0.00152 |

5 | 0.00189 | 11.2 × 10${}^{-6}$ | 0.00143 | 0.00138 |

6 | 0.00186 | 10.8 × 10${}^{-6}$ | 0.00145 | 0.00142 |

7 | 0.00187 | 11.0 × 10${}^{-6}$ | 0.00144 | 0.00140 |

8 | 0.00177 | 9.80 × 10${}^{-6}$ | 0.00145 | 0.00142 |

9 | 0.00192 | 11.6 × 10${}^{-6}$ | 0.00147 | 0.00142 |

10 | 0.00184 | 10.6 × 10${}^{-6}$ | 0.00148 | 0.00143 |

Position | t = 5 s | t = 15 s |
---|---|---|

At heated surface | 1.00% | 0.81% |

25% | 0.25% | 1.78% |

50% | 5.37% | 0.42% |

75% | 2.30% | 0.22% |

Top | 2.68% | 0.02% |

Case | Time (s) |
---|---|

313.15 K: Case 1 | 6 |

313.15 K: Case 2 | 6 |

323.15 K: Case 1 | 4 |

323.15 K: Case 2 | 4 |

333.15 K: Case 1 | 3 |

333.15 K: Case 2 | 2 |

© 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**

Karlsson, L.; Ljung, A.-L.; Lundström, T.S.
Comparing Internal Flow in Freezing and Evaporating Water Droplets Using PIV. *Water* **2020**, *12*, 1489.
https://doi.org/10.3390/w12051489

**AMA Style**

Karlsson L, Ljung A-L, Lundström TS.
Comparing Internal Flow in Freezing and Evaporating Water Droplets Using PIV. *Water*. 2020; 12(5):1489.
https://doi.org/10.3390/w12051489

**Chicago/Turabian Style**

Karlsson, Linn, Anna-Lena Ljung, and T. Staffan Lundström.
2020. "Comparing Internal Flow in Freezing and Evaporating Water Droplets Using PIV" *Water* 12, no. 5: 1489.
https://doi.org/10.3390/w12051489