# Sensible Heat Transfer during Droplet Cooling: Experimental and Numerical Analysis

^{1}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Objectives

#### 1.2. Motives

#### 1.3. Research Importance

#### 1.4. Research Questions

- Is it possible to reproduce high spatial-temporal resolution experiments numerically? If yes, which are the limitations?
- Can the numerical results describe properly the sensible heat transfer in droplet cooling?
- How is the flow field within the droplet connected with the heat removed?
- Gap in knowledge 1: Validation of a numerical solver has never been conducted comparing high resolution, transient, experimental and numerical results for the entire temperature field that is developed in the solid region during the droplet impact.
- Gap in knowledge 2: The relation between the flow field within the droplet and the heat removed during droplet spreading has not been accurately described yet.

#### 1.5. Literature Review

^{2}·K [2] and up to 660,000 W/m

^{2}·K [3] have been reported in literature, thus confirming that even without boiling, the sensible heat transfer during droplet impact is still suitable for cooling applications.

#### 1.6. State of the Art

## 2. Materials and Methods

#### 2.1. IR Thermography—High Speed Visualization Set-Up

#### 2.2. Experimental Methods

_{amb}= 293 ± 2 K. The droplet diameter before the impact was ${D}_{0}$ = 2.6 ± 0.1 mm. The temperature of the droplet before impact was assumed to be equal to the ambient temperature. The test consisted in electrically heating the stainless-steel foil up to the desired temperature and then let the droplet impact on it, by action of gravity. During droplet impact, simultaneous but not synchronized high-speed and high-speed thermography images were recorded. The acquisition frequency and resolution were 2000 fps and 512 × 512 px

^{2}for the high-speed images and 1000 fps and 150 × 150 px

^{2}for the IR images. For each experimental condition considered here, five tests were performed to assure reproducibility of the experiments. Care was taken to assure that the initial surface temperature and wetting conditions were reproducible before each new droplet impact. The radial temperature profiles were obtained after post processing the IR images using a custom made, in-house developed MatLab code which allowed conversion of the raw IR images to temperature data. The high speed images were used to evaluate the spreading diameter i.e., the contact diameter and impact velocity using a code previously developed MatLab [40]. An example of the IR post processed images, temperature profiles as well as the raw and post-processed high speed images, is shown in Figure 3.

#### 2.3. Measurement Uncertainties

#### 2.4. Numerical Methodology

_{r}is the artificial compression velocity which is calculated from the following relationship:

_{f}is the cell surface normal vector, φ is the mass flux, S

_{f}is the surface area of the cell, and C

_{γ}is a coefficient the value of which can be set between 1 and 4. U

_{r}is the relative velocity between the two fluid phases due to the density and viscosity change across the interface. In Equation (6) the divergence of the compression velocity U

_{r}, ensures the conservation of the volume fraction α, while the term α (1 − α) limits this artificial compression approach only near the interface, where 0 < α < 1 [42]. The level of compression depends on the value of C

_{γ}[42,43]. For the simulations of the present investigation, initial, trial simulations indicated that a value of C

_{γ}= 1 should be used, in order to maintain a quite sharp interface without at the same time having unphysical results.

#### 2.5. Numerical Set Up

^{2}fluid domain in the X-Y plane was chosen to avoid the influence of the boundaries in the fluid flow. The solid domain dimensions were 8 × 0.020 mm

^{2}(20 µm thickness of the heated foil). The mesh consisted of 640,000 hexahedral cells in the fluid domain and 4000 in the solid domain. In the fluid domain, the mesh progressively coarsens away from the initial droplet position by a grading factor of 5 in both X and Y directions (last to first cell dimension in each direction is equal to 5). This leads to a minimum cell size of 4 μm and a maximum cell size of 20 μm. These cell dimensions were selected, for the solution to be mesh independent. Before every simulation an arbitrary thermal boundary layer was patched in the domain to facilitate the initial convergence of the coupling between the solid and liquid temperatures. A droplet with the same diameter and velocity as the experimental conditions was patched as well, at the time instant just before it touches the heated surface. The solid was considered as a volumetric heat source. Constant contact angle was assumed between the fluid and the solid with a value of θ = 81.7°, following the experimentally measured equilibrium contact angle value, for the sample. As aforementioned, a static contact angle was chosen instead of an advancing contact angle, due to the reasons explained in Section 2.1. A sensitivity analysis was performed on adiabatic droplet impact, which revealed that up to the maximum spreading, using a static contact angle instead of an advancing one would lead to errors within the uncertainty of the experimental tests, so this decision is shown not to affect significantly the results of the simulations, for the conditions tested here. The boundary conditions and the initial configuration of the simulation are summarized in Figure 4.

## 3. Results

#### 3.1. Comparison Between Experimental and Numerical Results

#### 3.2. Relation Between Heat Tranfer and Droplet Dynamics

^{2}·K was evaluated at t = 1 ms after the impact and at a radial distance r = 1840 µm. The corresponding minimum value was around 3900 W/m

^{2}·K at t = 4 ms after impact and at a radial distance r = 2240 µm.

#### 3.3. Theoretical Analysis of Heat and Mass Tranfer in the Rim Region

- A characteristic dimension for the vorticity field is the height of the droplet measured at the position of the minima or maxima of the vorticity, ${H}_{droplet\left(r+\Delta {s}_{numerical},t\right)}$. The height of the droplet corresponds to the maximum value of the Y coordinate, measured for a volume fraction of $\alpha \ge 0.5$. Therefore, the height of the droplet coincides with the vertical position of the liquid-air interface. ${H}_{droplet\left(r+\Delta {s}_{numerical},t\right)}$ substitutes the distance $\Delta y$ in Equation (17).
- The shifts relate to phenomena occurring in the bulk of the rim. Hence, the variation of the average temperature of the fluid measured along the shift dimension, should be accounted as ${\overline{T}}_{f\left(r+\Delta {s}_{numerical},t\right)}-{\overline{T}}_{f\left(r,\text{}t\right)}$. Here ${\overline{T}}_{f\left(r,\text{}t\right)}$ is the average temperature where the local minima or maxima of the heat transfer coefficient are, while ${\overline{T}}_{f\left(r+\Delta {s}_{numerical,t}\right)}$ is the average temperature at the corresponding local maxima or minima of the vorticity. The average temperature of the fluid is obtained as the average of the calculated temperatures in the vertical direction of the droplet for $\alpha \ge 0.5$. ${\overline{T}}_{f\left(r+\Delta {s}_{numerical},t\right)}-{\overline{T}}_{f\left(r,t\right)}$ replaces $\Delta {T}_{f}$ in the initially proposed formulation (Equation (17)).
- A characteristic velocity of the droplet can be defined as ${\frac{dr}{dt}|}_{t}$ measured in time by the forward finite difference method, from the previously obtained spreading factors. This has been chosen since the minima or maxima for the vorticity and heat transfer coefficient in the region of the contact line, move together with the contact line and consequently with a velocity similar to ${\frac{dr}{dt}|}_{t}$. The proposed velocity also accounts for the bulk conditions of the droplet. This characteristic velocity substitutes ${\overline{V}}_{r}$ in Equation (17).

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic view and (

**b**) photograph of the experimental set up used in the present work.

**Figure 2.**(

**a**) Detail of the stainless-steel foil and relative position of the IR camera; (

**b**) Foil support and heating assembly.

**Figure 3.**Illustrative representation of a raw and post-processed high-speed image (

**a**). Corresponding infrared thermography and relative temperature profile along the radial direction (

**b**). The blue line in the infrared thermography indicates the radius along which the temperature is measured. The green curve in the post-processed high-speed image, indicates the definition of the interface of the droplet.

**Figure 4.**Schematic view of the computational domain, boundary conditions and initial condition for the simulations.

**Figure 5.**Qualitative comparison between high-speed recorded and numerically predicted droplet shape and between experimentally measured and numerically calculated temperature field (Tw) at the bottom of the solid sample. Comparison is presented for different time instants after impact at test conditions We = 24, Tw(in) = 353.15 K.

**Figure 6.**Evolution of the spreading factor versus time for all the experimental and the corresponding numerical tests performed.

**Figure 7.**Distribution of dimensionless temperature within the solid domain, along the dimensionless radial distance for 1, 2, 3 and 4 ms after impact, for all the calculated and measured conditions.

**Figure 8.**Example of lamella fingering as captured in the high speed recorded images and in the correspondent IR thermography images, taken from the bottom of the heated stainless steel foil. The blue circle highlights the region where fingering is observed. The reported images refer to We = 151, Tw(in) = 333.15 K and were taken at a time after impact t = 4 ms.

**Figure 9.**Measured and calculated temperature distribution along the dimensioless radial distance (We = 24, Tw(in) = 353.15 K).

**Figure 10.**Details of the temperature distribution in the rim region as predicted by the numerical simulation and measured from the IR image (top). Spreading stage of the droplet from corresponding high speed image.

**Figure 11.**Numerically obtained temperature, radial velocity and z-vorticity fields within the fluid domain in the rim region. For We = 24 and Tw(in) = 353.15 K at 4 ms after impact. The z-axis is perpendicular to the plane of view. The white line identifies interface between the liquid and gas phases (alpha = 0.5).

**Figure 12.**Vorticity in the Z direction and heat transfer coefficient numerically evaluated along the droplet radial distance, at different times after impact, for We = 24 and Tw(in) = 353.15 K. In the plots, the squares in the curves highlight the local maxima and minima of the numerically evaluated vorticity and heat transfer coefficients. The green line identifies the rim entrance and the purple line identifies the contact line.

**Figure 13.**(

**a**) Schematic view of the rim region and representation of the parameters chosen for the final formulation of the theoretical spatial shift between local maxima and minima of heat transfer coefficient and vorticity. The bright grey area represents the fluid domain and the dark grey area the solid domain; (

**b**) Corresponding local minimum in heat transfer coefficient and local maximum in vorticity for which the theoretical shift is calculated.

Parameter | Uncertainties (rel. or abs) | Evaluation Method |
---|---|---|

Temperature T (K) | ${U}_{T}=\pm \text{}1\text{}\mathrm{K}$ | ${U}_{T}$ for ${T}_{amb}$ is the one given by manufacturer. ${U}_{T}$ for ${T}_{w}$ is calculated after black body recalibration of the Infrared Camera. No standard deviation was accounted. |

Temperature difference ΔT (K) | ${U}_{\Delta T}=\pm \text{}1.4\text{}\mathrm{K}$; ${u}_{\Delta T}max=\pm \text{}14\%\text{}at\text{}\left(\Delta T=10\text{}\mathrm{K}\right)$; ${u}_{\Delta T}min=\pm \text{}1.7\%\text{}at\text{}\left(\Delta T=78\text{}\mathrm{K}\right)$ | ${U}_{\Delta T}=\sqrt[2]{{\left({U}_{T}\right)}^{2}+{\left({U}_{T}\right)}^{2}}$; ${u}_{\Delta T}={U}_{\Delta T}/\Delta T$ |

Non-dimensional temperature T* (-) | ${U}_{{T}^{\ast}}max=\pm \text{}0.04\text{}at\text{}\left({T}^{\ast}=1\right)$; ${U}_{{T}^{\ast}}min\text{}=\pm \text{}0.05\text{}at\text{}\left({T}^{\ast}=0.35\right)$; ${u}_{{T}^{\ast}}max=\pm \text{}14\%\text{}at\text{}\left({T}^{\ast}=0.35\right);$ ${u}_{{T}^{\ast}}max=\pm \text{}4\%\text{}at\text{}\left({T}^{\ast}=1\right)$ | ${U}_{{T}^{\ast}}=\sqrt[2]{{\left({U}_{\Delta T}\right)}^{2}+{\left({U}_{\Delta T}\right)}^{2}}$; ${u}_{{T}^{\ast}}=\sqrt[2]{{\left({u}_{\Delta T}\right)}^{2}+{\left({u}_{\Delta T}\right)}^{2}}$ |

Imposed volumetric heat rate ${{q}_{V}}^{\u2034}\text{}\left(\mathrm{W}/{\mathrm{m}}^{3}\right)$ | $u{{q}_{V}}^{\u2034}\text{}max=\pm \text{}12\%\text{}at\text{}$ $({q}^{\u2034}=6.5\times {10}^{6}\text{}\left(\mathrm{W}/{\mathrm{m}}^{3}\right))$ | $u{{q}_{V}}^{\u2034}=\sqrt[2]{{\left(\frac{{U}_{I}}{I}\right)}^{2}+{\left(\frac{{U}_{V}}{V}\right)}^{2}+{\left(\frac{{U}_{t}}{t}\right)}^{2}+{\left(\frac{{U}_{L}}{L}\right)}^{2}+{\left(\frac{{U}_{W}}{W}\right)}^{2}}$ |

Radial distance r (mm) (infrared) | ${U}_{r}=\pm \text{}200\text{}\mathsf{\mu}\mathrm{m}$ | ${U}_{r}$ is evaluated according to the resolution in the arrangement used in the study |

Parameter | Uncertainties (rel. or abs) | Evaluation Method |
---|---|---|

Contact Diameter $D\text{}\left(\mathrm{mm}\right)$ Droplet diameter before impact ${D}_{0}\text{}\left(\mathrm{mm}\right)$ | ${U}_{D}=\pm \text{}160\text{}\mathsf{\mu}\mathrm{m}$ ${U}_{{D}_{0}}=\pm \text{}160\text{}\mathsf{\mu}\mathrm{m}$ | The uncertainty of post processed $D$ and ${D}_{0}$ was given as the maximum uncertainty of the image post processing in the definition of the interface of the droplet. |

Spreading ratio $D/{D}_{0}$ (-) | ${u}_{D/{D}_{0}}max=\pm \text{}37\%\text{}at\text{}(D/{D}_{0}=0.17)$ $uD/{D}_{0}min=\pm \text{}7\%\text{}at\text{}(D/{D}_{0}=3.86)$ | ${u}_{D/{D}_{0}}=\sqrt[2]{{\left(\frac{{U}_{D}}{D}\right)}^{2}+{\left(\frac{{U}_{{D}_{0}}}{{D}_{0}}\right)}^{2}}$ |

Impact velocity ${V}_{0}\text{}\left(\mathrm{m}/\mathrm{s}\right)$ | ${U}_{{V}_{0}}=0.08\text{}\left(\mathrm{m}/\mathrm{s}\right)$ | ${U}_{{V}_{0}}={{V}_{0}}_{\Delta y}\pm {{V}_{0}}_{\Delta y\pm Uy}$ In which ${{V}_{0}}_{\Delta y}$ is the measured value of velocity considering a vertical displacement of the droplet $\Delta y$ in the time interval between three consecutive frames i.e., 1.5 ms. ${{V}_{0}}_{\Delta y\pm Uy}$ is the velocity that would result from a displacement equal to $\Delta y\pm Uy$ in the same interval between three consecutive frames. $Uy$ is considered to be ±1 pixel (40 µm) in the optical arrangement used. |

Weber number We (-) | ${u}_{We}\text{}max=\pm \text{}15\%\text{}at\text{}\left(We=24\right)$ ${u}_{We}\text{}min=\pm \text{}8\%\text{}at\text{}\left(We=151\right)$ | ${u}_{We}=\sqrt[2]{{\left(\frac{{U}_{D}}{D}\right)}^{2}+2{\left(\frac{{U}_{{V}_{0}}}{{V}_{0}}\right)}^{2}}$ |

Material Properties | $\mathit{\rho}$ (kg/m^{3}) | ${\mathit{c}}_{\mathit{p}}$ (J/kg·K) | $\mathit{k}$ (W/m·K) | ν (m^{2}/s) | σ (N/m) |
---|---|---|---|---|---|

Air | 1 | 1006.4 | 0.025874 | 0.0000148 | - |

Water | 1000 | 4184 | 0.59844 | 0.000001 | 0.007 |

Stainless steel | 7880 | 477 | 18 | - | - |

${\mathit{q}}^{\u2034}$ (W/m^{3}) | Tw(in) (K) | We (-) |
---|---|---|

98.6 × 10^{6} | 353.15 | 24 |

65.4 × 10^{6} | 333.15 | 151 |

122.8 × 10^{6} | 373.15 | 151 |

**Table 5.**Variables of interest and numerical versus theoretical shifts for We = 24 and Tw(in) = 353.15 K taken at 1, 2, 3 and 4 ms after droplet impact.

Time after Impact = 1 ms | ||||

$\Delta {s}_{numerical}\text{}\left(\mu m\right)$ | $\Delta {s}_{theory}\text{}\left(\mu m\right)$ | ${H}_{droplet\left(r+\Delta {s}_{numerical}\right)}\text{}\left(\mu m\right)$ | ${\overline{T}}_{f\left(r+\Delta {s}_{numerical}\right)}-{\overline{T}}_{f\left(r\right)}\text{}\left(K\right)$ | ${\frac{dr}{dt}|}_{t}\text{}\left(\frac{m}{s}\right)$ |

160 | 516 | 238 | 1.3 | 0.67 |

160 | 616 | 179 | 2 | 0.67 |

80 | 72 | 89 | 1.5 | 0.67 |

Time after Impact = 2 ms | ||||

160 | 121 | 240 | 0.2 | 0.45 |

320 | 202 | 298 | 0.3 | 0.45 |

0 | 0 | 29 | 0 | 0.45 |

Time after Impact = 3 ms | ||||

80 | 104 | 209 | 0.3 | 0.26 |

240 | 626 | 327 | 0.5 | 0.26 |

0 | 0 | ≈0 | 0 | 0.26 |

Time after Impact = 4 ms | ||||

80 | 20 | 149 | 0.2 | 0.08 |

320 | 2811 | 327 | 3 | 0.08 |

400 | 806 | 358 | 3.2 | 0.08 |

320 | 101 | 120 | 1.2 | 0.08 |

0 | 0 | 0 | 0 | 0.08 |

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**MDPI and ACS Style**

Teodori, E.; Pontes, P.; Moita, A.; Georgoulas, A.; Marengo, M.; Moreira, A.
Sensible Heat Transfer during Droplet Cooling: Experimental and Numerical Analysis. *Energies* **2017**, *10*, 790.
https://doi.org/10.3390/en10060790

**AMA Style**

Teodori E, Pontes P, Moita A, Georgoulas A, Marengo M, Moreira A.
Sensible Heat Transfer during Droplet Cooling: Experimental and Numerical Analysis. *Energies*. 2017; 10(6):790.
https://doi.org/10.3390/en10060790

**Chicago/Turabian Style**

Teodori, Emanuele, Pedro Pontes, Ana Moita, Anastasios Georgoulas, Marco Marengo, and Antonio Moreira.
2017. "Sensible Heat Transfer during Droplet Cooling: Experimental and Numerical Analysis" *Energies* 10, no. 6: 790.
https://doi.org/10.3390/en10060790