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Prediction of Contact Angle of Nanofluids by Single-Phase Approaches^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Nanofluids

_{2}and Al

_{2}O

_{3}nanoparticles were utilized. Au, Al

_{2}O

_{3}and GO NFs here employed were fully characterized in [19] where preparation and thermophysical properties evaluation including density are also reported.

_{2}NF, silica raw material with density of 2000 kg/m

^{3}at 20 °C (SIPERNAT

^{®}22S, Evonik Industries AG, Germany) was dispersed unfractionated. NF had pH- value of 10.5 with stabilization by KOH. Mean agglomerate size of particles was measured as 177 nm by using Dynamic Light Scattering analysis (Zetasizer Nano ZS, Malvern Instruments GmbH, Germany). Density of SiO

_{2}NF was calculated with the mixture rule (Equation (1)) [25]. Thermal conductivity of SiO

_{2}nanofluid was measured by using 3ω method [26]. Measurements were conducted at 24 °C with three repetitions. Main information for all nanofluids are compiled in Table 1 showing the difference in concentration, size, shape and density of nanoparticles, and density and thermal conductivity of NF.

#### 2.2. Experiments on Contact Angle Measurements and Determination of Geometrical Parameters

_{2}O

_{3}and GO NFs were collected at İzmir Kâtip Çelebi University (İKÇÜ), Universitat Jaume I Castelló (UJI) and Université Rennes 1 (UR1) and were taken from [19]. CA of SiO

_{2}nanofluid was measured at ILK-Dresden (ILK). Sessile drop method was used in all institutes. Devices used in institutes and experimental conditions such as temperature T, droplet volume V and relative humidity RH used by each institute for the different liquids are described in Table 2.

_{2}O

_{3}and GO NF) and İKÇÜ (SiO

_{2}NF; Table 3). In both institutes, the pendant drop method was employed to define the ${\sigma}_{lg}$. DSA-30 Drop Shape Analyzer (KRÜSS GmbH, Germany) and Biolin Scientific, (Sweden/Finland) were used at UR1 and İKÇÜ, respectively. In İKÇÜ, surface tension of DIW was measured before SiO

_{2}NF. For both working fluids, the volume was kept constant at 10 µL and three measurements were carried out. There is no significant deviation between replicates and between experimental values of DIW with NIST data [29] at a given temperature. The comprehensive description of the experimental procedure used for the surface tension measurement at UR1 can be found in [30].

_{d}and R

_{0}) used in the models that are described in the next section were obtained by CA image analysis. Such an analysis has been performed with an image processing program called Fiji [31] as shown in Figure 1. Pixel-to-mm ratio with respect to a reference dimension for the images from the institutes is defined to spatial calibration of each data set.

#### 2.3. Single-Phase Models for Prediction of Contact Angle

_{d}, $\delta $ and R

_{0}are the droplet baseline radius, apex height and curvature, respectively. ${\theta}_{d}$ is the contact angle of the droplet. Droplet volume equation (2) was reformulated by using non-dimensional numbers: Bond number (Bo), which is the ratio of gravitational and surface forces, two geometrical similarity simplexes—G

_{1}and G

_{2}—which describe the droplet geometry and V* as the non-dimensional droplet volume.

_{s}and its baseline radius by r

_{ds}.

_{1}, and validity was studied for nanofluids.

#### 2.4. Prediction of Droplet Shape

## 3. Results and Discussion

**○**) were used for experimental results. Due to the measurement uncertainties and in order to evaluate their influence, numerical results from the models were corrected with increasing R

_{0}by 2% and it was presented as a red cross in the figures (✕).

#### 3.1. Contact Angle Prediction with Single-Phase Models

_{2}O

_{3}NF shows also reasonable good agreement with Ω = 2.07% and Ω = 4.07%, respectively. However, the VP model seems to be not valid for GO (Ω = 8.52%) and SiO

_{2}NFs (Ω = 30.46%). Large graphene flakes in GO NF and higher concentration (3.935 wt.%) of SiO

_{2}NF may be the reason of preventing the application of the VP model.

_{0}. To illustrate this, R

_{0}was increased artificially by just 2% and additional results presented in Figure 2 for GO NF, and Figure 3 for DIW and SiO

_{2}. CA data of GO NF fit much better with the VP model with this correction. Ω decreased from 8.52% to 4.19% for GO NF. Unlike GO NF, CA of SiO

_{2}NF was not affected from the increase of R

_{0}(Ω = 27.79%). This shows the influence of nanoparticle content on the applicability of V–P model with nanofluids.

_{2}O

_{3}and GO nanofluids. As shown in Figure 4, all the data was in the range of the ±10% error band. However, the results for the SiO

_{2}NF (Figure 5) show again that a single-phase model was not appropriate for a highly concentrated nanofluid.

_{2}O

_{3}NF (Ω = 3.78%) and Au NF (Ω = 7.18%). Vafaei and Podowski [32] mentioned that as volume decreases the shape of the droplet is more spherical. Although the volume range was narrow for this study, change of the droplet shape with volume was obviously clear. For GO nanofluid at the highest volume, CA was underpredicted compared to experimental results with Ω = 7.35%. The reason could be the shape and size of the graphene flakes. Different from the other theoretical models, predicted SiO

_{2}data with the SD model was much collapsed with the experimental data with Ω = 10.71% where DIW had Ω = 3.90%.

_{2}O

_{3}NF (Ω = 9.78%) predicted by the W model is different for smaller volumes. However, increase in volume results in more accurate results with the W model. The reason could be that the concentration highly affects on CA at lower volumes for this model. Similar behavior was observed for also SiO

_{2}NF. The W model was not suitable for highly concentrated nanofluids as shown in SiO

_{2}NF (Ω = 144.46%) results in Figure 9.

#### 3.2. Droplet Shape Prediction

_{2}NF, droplets that had maximum, minimum and average contact angles were chosen (Figure 11).

## 4. Conclusions

_{2}O

_{3}NF) could be predicted by both the VP model and energy balance-based YK model. Moreover, the empirical W model was valid for dilute NFs (Au and GO NF) and higher volumes of Al

_{2}O

_{3}NF. However, significant differences were obtained for highly concentrated SiO

_{2}NF in these theoretical models and empirical model. Additional effects like disjoining pressure, convective flows inside the droplet, etc. could be the reasons of differences in results of different models. Beside them, the SD model is suitable for almost all samples due to the spherical shape of droplets as a result of smaller volumes. However, higher error band of SD model is the limitation for accurate prediction of CA. It was also shown that the droplet shape of all NFs could be well predicted from a model based on force balance.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Bo | Bond Number (–) |

c | Coefficients |

g | Gravitational Acceleration (m/s ^{2}) |

G | Geometrical Similarity Simplex (–) |

k | Thermal Conductivity (W/mK) |

r | Droplet Wetting Radius (m) |

R | Radius of Curvature (m) |

RH | Relative Humidity (%) |

V | Volume (m ^{3}) |

W | Width of the droplet (m) |

T | Temperature (°C) |

## Abbreviations

CA | Contact Angle |

DIW | Distilled Water |

İKÇÜ | Izmir Katip Çelebi University |

ILK | ILK-Dresden |

NF | Nanofluid |

UJI | Universitat Jaume I Castelló |

UR1 | Université Rennes 1 |

## Subscripts

0 | At the apex |

d | Droplet |

e | Effective |

f | Base Fluid |

lg | Liquid–Gas |

p | Particle |

s | Spherical |

v | Volumetric |

## Greek Letters

$\sigma $ | Surface Tension (mN/m) |

$\delta $ | Location of Apex (m) |

$\rho $ | Density (kg/m ^{3}) |

$\theta $ | Contact Angle (°) |

$\mathsf{\Omega}$ | Mean Absolute Percentage Error (%) |

$\varphi $ | Concentration of NF (%) |

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**Figure 2.**VP model validity for fluids at various volumes. Working fluids are (

**a**) distilled water (DIW), (

**b**) Au nanofluid (NF), (

**c**) GO NF and (

**d**) Al

_{2}O

_{3}NF. Colors indicate: orange—Universitat Jaume I Castelló (UJI) and green—Université Rennes 1 (UR1).

**Figure 3.**VP model validity for fluids at constant volume. Working fluids are (

**a**) DIW and (

**b**) SiO

_{2}NFs.

**Figure 4.**Yonemoto and Kunugi (YK) model validity for fluids at various volumes. Working fluids are (

**a**) DIW, (

**b**) Au NF, (

**c**) GO NF and (

**d**) Al

_{2}O

_{3}NF. Colors indicate: orange—UJI, green—UR1 and purple—İzmir Kâtip Çelebi University (İKÇÜ).

**Figure 6.**Spherical dome (SD) model validity for fluids at various volumes. Working fluids are (

**a**) DIW, (

**b**) Au NF, (

**c**) GO NF and (

**d**) Al

_{2}O

_{3}NF. Colors indicate: orange—UJI, green—UR1 and purple—İKÇÜ.

**Figure 7.**SD model validity for fluids at constant volume. Working fluids are (

**a**) DIW and (

**b**) SiO

_{2}NF.

**Figure 8.**Wong et al. (W) model validity for fluids at various volumes. Working fluids are (

**a**) DIW, (

**b**) Au NF, (

**c**) GO NF and (

**d**) Al

_{2}O

_{3}NF. Colors indicate: orange—UJI, green—UR1 and purple—İKÇÜ.

**Figure 9.**W model validity for fluids at constant volume. Working fluids are (

**a**) DIW and (

**b**) SiO

_{2}NF.

**Figure 10.**Droplet shape prediction at various volumes (—shows Equation (14) and—shows the ±5% errors).

**Figure 11.**Droplet shape prediction at constant volume (—shows Equation (14) and—shows the ±5% errors).

Nanoparticle | Particle Size | Particle Shape | ${\mathit{\varphi}}_{\mathit{p}}$ | Density of Nanoparticle (kg/m ^{3}) | Density of NF (kg/m ^{3}) | Thermal Conductivity Ratio (k _{NF}/k_{DIW}) |
---|---|---|---|---|---|---|

Gold (Au) | Particle diameter: 8.34 nm | Spherical | 0.001 wt. % (0.000052 vol.%) | 1,9300 | 997.25 | 0.999 |

Silica (SiO_{2}) | Particle diameter: 117 nm | Spherical | 3.935 wt. % (2 vol.%) | 2000 | 1017.06 | 1.008 |

Graphene oxide (GO) | Extension of particle: 770 to 900 nm Thickness: 2 nm to 10 nm | Flake | 0.01 wt. % (0.005679 vol.%) | 1500–1900 | 997.25 | 0.9964 |

Alumina (Al_{2}O_{3}) | Particle Diameter: 123 ± 2 nm | Spherical | 0.4 wt. % (0.1 vol. %) | 3987 | 997.25 | 0.9961 |

**Table 2.**Devices, temperature, relative humidity and droplet volume of experimental study in each institute.

Institutions & Devices | Working Fluids | ||||
---|---|---|---|---|---|

DIW | Au NF | GO NF | SiO_{2} NF | Al_{2}O_{3} NF | |

İKÇÜ Attention Theta Goniometer (Biolin Scientific, (Sweden/Finland)) | T = 24.2 °C RH = 40% V = 4.6 µL | T = 23.7 °C RH = 40% V = 9.8 µL | T = 23.1 °C RH = 36% V = 4.1 µL | ||

ILK Lab-made device | T = 22.0 °C RH = 67% V = 10 µL | T = 25.0 °C RH = 64.5 % V = 10 µL | |||

UJI Lab-made device | T = 24.0 °C RH = 54% V = 5.1–71.1 µL | T = 24.0 °C RH = 54% V = 5.3–68.6 µL | T = 24.0 °C RH = 54% V = 8.4–35.4 µL | T = 24.0 °C RH = 54% V = 5.5–28.8 µL | |

UR1 DSA-30 Drop Shape Analyzer (KRÜSS GmbH, Germany) | T = 21.0 °C RH = 24% V = 22.3 µL | T = 21.0 °C RH = 24% V = 24.1 µL | T = 21.0 °C RH = 24% V = 34.3 µL | T = 21.0 °C RH = 24% V = 20.9 µL |

Working Fluid | T (°C) | RH (%) | ${\mathit{\sigma}}_{\mathit{l}\mathit{g}}$ (mN/m) | Standard Deviation |
---|---|---|---|---|

DIW | 21 | 24 | 72.960 | 0.06 |

GO NF | 21 | 24 | 73.345 | 0.125 |

Au NF | 21 | 24 | 72.77 | 0.13 |

Al_{2}O_{3} NF | 21 | 24 | 72.005 | 0.255 |

SiO_{2} NF | 22.8 | 40 | 70.13 | 0.25 |

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**MDPI and ACS Style**

Çobanoğlu, N.; Karadeniz, Z.H.; Estellé, P.; Martínez-Cuenca, R.; Buschmann, M.H.
Prediction of Contact Angle of Nanofluids by Single-Phase Approaches. *Energies* **2019**, *12*, 4558.
https://doi.org/10.3390/en12234558

**AMA Style**

Çobanoğlu N, Karadeniz ZH, Estellé P, Martínez-Cuenca R, Buschmann MH.
Prediction of Contact Angle of Nanofluids by Single-Phase Approaches. *Energies*. 2019; 12(23):4558.
https://doi.org/10.3390/en12234558

**Chicago/Turabian Style**

Çobanoğlu, Nur, Ziya Haktan Karadeniz, Patrice Estellé, Raul Martínez-Cuenca, and Matthias H. Buschmann.
2019. "Prediction of Contact Angle of Nanofluids by Single-Phase Approaches" *Energies* 12, no. 23: 4558.
https://doi.org/10.3390/en12234558