# Free-Flowing Shear-Thinning Liquid Film in Inclined μ-Channels

^{*}

## Abstract

**:**

_{O}= 1200 μm) made of brass which can be set to various inclination angles. The liquid film characteristics were measured by a non-intrusive technique that is based on the features of a micro Particle Image Velocimetry (μ-PIV) system. Relevant computational fluid dynamics (CFD) simulations revealed that the volume average dynamic viscosity over the flow domain is practically the same as the corresponding asymptotic viscosity value, which can thus be used in the proposed design equations. Finally, a generalized algorithm for the design of FFMRs, containing non-Newtonian shear thinning liquids, is suggested.

## 1. Introduction

_{2}absorption into a wavy falling film of NaOH aqueous solution on a micro-baffled plate using two methods for visualizing the liquid film thickness. The first method uses the light reflection from a small angle (5°), that illuminates the liquid surface of a liquid film at another angle (45°). The second one is a particle injection method. More specifically the particles are PMMA slurry that can be detected and form the liquid film layer. Lokhat et al. [9], also studied, via CO

_{2}absorption experiments, the influence of a reaction plate orientation and gas flow rate on liquid phase mass transfer coefficient. They proposed correlations that are based on Nusselt’s condensation theory. Tourvieille et al. [6], have visualized the liquid film thickness using fluorescence confocal microscopy, and they proposed correlations for determining the mass transfer coefficient, using Nusselt and Kapitza numbers. Yang et al. [10], who have investigated the liquid film thickness and the shape of the interface on an open channel FFMR using stereo digital microscopy, also proposed an empirical correlation, that predicts their experimentally measured liquid film thickness with 7% deviation. Patel et al. [11], have proposed another technique for the characterization of the interface between the liquid and the gas phase in a microchannel, using tracing particles and a microscope, and proved that this technique can reach the accuracy of 1.06 μm, without proposing any correlations for this specific method. Yu et al. [7], have experimentally measured the meniscus shape and the characteristics of the flow in inclined open rectangular microgrooves heat sinks using micro Particle Image Velocimetry (μ-PIV), and found that the meniscus shape is a parabolic arc.

## 2. Experimental Procedure

#### 2.1. Experimental Setup

^{®}, Sarasota, FL, USA) to feed the liquid and the μ-PIV system. The μ-PIV system used consists of a high sensitivity charge-coupled device (CCD) camera (Hisense MkII, Dantec Dynamics

^{®}, Skovlunde, Denmark), which is connected to a microscope (Nikon Eclipse LV150, Nikon Corporation

^{®}, Tokyo, Japan), while the acquired images were processed by the Flow Manager Software (Dantec Dynamics, v4.00). Prior to measurements the fluids were traced by adding Nile red fluorescent carboxylate microspheres (Invitrogen, Carlsbad, MA, USA) with mean diameter of 1 μm. The measurements were conducted 30 diameters downstream from the inlet of the microchannel, where fully developed flow is established. The test section was placed on the microscope stage, which can be moved along its vertical axis with ten-micron accuracy. To obtain sufficiently magnified images a 20× air immersion objective with numerical aperture (NA) equal to 0.45 was used, which results in 3 μm depth of field. The maximum recording frame rate of the camera is 12.2 fps and that restricts the maximum frequency of the measurements to 12 Hz. The field of view was covering the whole channel width.

_{O}= 1200 μm), made of brass. The test section was constructed by employing ultrahigh precision micromachining techniques, which have the unique advantage of being able to manufacture geometrically complex miniature components to high accuracy, with fine surface finish, in a wide range of engineering materials. The test-section used in the present was constructed using a five-axis ultra-precision micro milling machine that ensures both minimal surface roughness and straightness of the channels. Typical overall values of repeatability, surface finish and straightness achievable by this procedure are in the range of 1.0 μm, 0.040 μm Ra, and 0.2 μm over 100 mm of travel respectively. The channel was milled using a 500 μm cutter at 200 μm steps, while to ensure minimal surface roughness, a final 20 μm cut was performed.

^{®}, Helsinki, Finland), and dynamic viscosity, μ, measured by a magnetic rheometer (AR-G2, TA Instruments, Sussex, UK). All properties were evaluated at room temperature (20–22 °C), and the experiments were conducted under the same conditions. The viscosity of the shear-thinning non-Newtonian fluids used can be accurately described by the Herschel–Bulkley model [14]:

#### 2.2. Measuring Procedure

## 3. Results and Discussion

#### 3.1. Liquid Film Thickness Calculation

_{O}) as the characteristic length and the superficial velocity (U

_{S}) defined as the volumetric flow rate divided by cross section of the meniscus. By inserting the aforementioned quantities, the dimensionless numbers are defined as follows:

_{φ}being the component of the acceleration of gravity that acts in the direction of flow, i.e., g

_{φ}= g sin(φ) and φ is the inclination angle from the horizontal.

#### 3.2. Effective Viscosity Prediction

^{®}(v. 19, ANSYS Inc., Canonsburg, PA, USA). ANSYS CFX

^{®}provides a finite volume method, a fully coupled solver for pressure and velocity coupling. The Direct Numerical Simulation (DNS) method for laminar flow was employed for the solution, as the flow in the μ-channel does not present any turbulent phenomena. Figure 8, shows the dynamic viscosity distribution at cross section, which was placed 100 widths downstream from the inlet to minimize inlet disturbances.

#### 3.3. Meniscus Shape

_{f}− H

_{O}

_{f}is the height where the three-phase contact is pinned, which for shallow μ-channels equals the channel’s depth and H is the height of the meniscus (Figure 6a) that can be calculated by Equation (2).

_{∞}) can be used for calculating the liquid layer characteristics in place of volume averaged viscosity (μ

_{ave}), estimated by CFD. More precisely, the shape of the meniscus calculated by Equations (2)–(7) using either μ

_{∞}or μ

_{ave}, is compared with the one defined experimentally. Although the curve corresponding to μ

_{ave}fits the experimental data better, the curve calculated using μ

_{∞}, which is a priori known, can also predict the meniscus shape with reasonable accuracy, i.e., ±8%, compared with the one based on the experimental data.

_{O}and W

_{O}are parameters defined in Figure 6a. The liquid phase cross section (A) was calculated by:

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | liquid phase cross section | μm^{2} |

Ca | capillary number | Dimensionless |

Fr | Froude number | Dimensionless |

g | acceleration of gravity | m/s^{2} |

H_{f} | height of the microchannel | μm |

H_{o} | height of the meniscus | μm |

H | liquid film thickness | μm |

L | length of the interface (meniscus) | μm |

M | objective lenses magnitude | Dimensionless |

m | mass flow rate | kg/s |

NA | numerical aperture | Dimensionless |

Q | volumetric flow rate | mL/h |

Re | Reynolds number | Dimensionless |

S | gas–liquid interfacial area | μm^{2} |

V | total fluid volume | μm^{3} |

X | distance from vertical wall | μm |

Y | distance from the channel bottom | μm |

W_{O} | width of microchannel | μm |

Greek symbols | ||

α | specific surface | m^{2}/m^{3} |

μ | liquid viscosity | Pa·s |

σ | surface tension | N/m |

τ | residence time/channel length | s/m |

ρ | liquid density | kg/m^{3} |

φ | inclination angle | ° |

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**Figure 5.**Typical measurements of the liquid film (nW, 40 mL/h, inclination angle 25°) where X is the distance from vertical wall and Y the distance from the channel bottom.

**Figure 6.**(

**a**) Geometrical characteristics of the liquid film; (

**b**) Typical photo of the film at the channel exit.

**Figure 8.**Dynamic viscosity distribution on a cross section located 30 W

_{O}downstream from the channel entrance (

**left**to

**right**): nW_40; nG20_40.

**Figure 9.**Comparison between experimental and calculated by Equation (2) liquid film thickness, H, normalized with respect to the channel width for the various non-Newtonian liquids, liquid flow rates and inclination angles tested.

**Figure 10.**Comparison of the shape of the interface calculated by Equation (6) with relevant experimental data.

**Figure 12.**Comparison of experimental data with curves calculated using asymptotic and effective viscosity (nW, 40 mL/h, 25°).

**Figure 15.**Comparison of specific surface between the flat interface and the meniscus shape calculated by the proposed correlation Equation (6) for nW, 40 mL/h, 25°.

Index | Liquid | Refractive Index | Contact Angle | Density | Surface Tension | Viscosity |
---|---|---|---|---|---|---|

- | (°) | (kg/m^{3}) | (mN/m) | (Pa·s) | ||

nW | 100% water + 0.03% xanthan gum | 1.340 | 74 | 998 | 72.1 | μ = 0.003698γ^{−1} + 0.004339γ^{−}^{0.1819} |

nG20 | 75% water + 25% glycerol w/w + 0.03% xanthan gum | 1.360 | 74 | 1059 | 66.7 | μ = 0.002952γ^{−1} + 0.006295γ^{−0.1535} |

Fluid | μ_{∞} (Pa·s) | μ_{ave} (Pa·s) | % Difference |
---|---|---|---|

nG20_80 | 0.0025 | 0.00260 | 4.0 |

nG20_40 | 0.0025 | 0.00270 | 8.0 |

nW_80 | 0.0014 | 0.00145 | 4.0 |

nW_40 | 0.0014 | 0.00150 | 7.1 |

Constants for Equation (1) | Present Work | Previous Work [13] |
---|---|---|

a | 0.50 | 0.50 |

b | 0.01 | 0.01 |

c | 3.90 | 2.04 |

d | −0.56 | −0.56 |

f | −0.86 | −0.86 |

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**MDPI and ACS Style**

Koupa, A.T.; Stergiou, Y.G.; Mouza, A.A.
Free-Flowing Shear-Thinning Liquid Film in Inclined *μ*-Channels. *Fluids* **2019**, *4*, 8.
https://doi.org/10.3390/fluids4010008

**AMA Style**

Koupa AT, Stergiou YG, Mouza AA.
Free-Flowing Shear-Thinning Liquid Film in Inclined *μ*-Channels. *Fluids*. 2019; 4(1):8.
https://doi.org/10.3390/fluids4010008

**Chicago/Turabian Style**

Koupa, Angeliki T., Yorgos G. Stergiou, and Aikaterini A. Mouza.
2019. "Free-Flowing Shear-Thinning Liquid Film in Inclined *μ*-Channels" *Fluids* 4, no. 1: 8.
https://doi.org/10.3390/fluids4010008