MASTER'S THESIS
Experimental Investigation of the Internal Flow in Evaporating and Freezing Water
Droplets
Tea Eriksson 2016
Master of Science in Engineering Technology Engineering Physics and Electrical Engineering
Luleå University of Technology
Department of Engineering Sciences and Mathematics
ABSTRACT
This report presents a fundamental study on the flow inside water droplets as they evaporate or freeze. The internal flow is visualized with the particle image velocimetry (PIV) method, and the results are analyzed from the viewpoint that thermal gradients in one way or another induces the flow. Light refraction at the droplet surface causes a distorted flow field, which is corrected by using an algorithm based on the ray tracing method.
Room temperatured 10 µL droplets (base diameter of about 3-4 mm) are evaporated on two different surfaces; sapphire and polycarbonate. The surface temperature is varied between 40, 50 and 60 °C. The results show that the internal velocities are greater when the droplet lies on the sapphire plate, which has higher thermal conductivity than polycarbonate. The velocities also increase when the plate temperature is increased. The flow is mainly directed upwards through the droplet symmetry axis and downwards along the surface. This implies that natural convection is the dominating mechanism of the internal flow in evaporating droplets.
The same droplet volume is examined in the freezing experiments. The surface material is sapphire and the plate temperature is varied between -10.0 and -12.8°C. Two different types of freezing are observed. Either the droplet starts to freeze the moment it touches the cold surface, and the ice climbs from the droplet bottom to the top, or the droplet stays in the liquid phase for some time even after it is placed on the surface. Once the droplet starts to freeze, the droplet surface freezes first, in the wink of an eye, followed by ice climbing from the bottom to the top inside the droplet. Since the droplet surface freezes first, no internal flow can be seen. The internal flow in the first mentioned type of freezing is directed upwards along the centerline and downwards along the surface, which indicates thermo-capillary convection.
The main differences between the internal flow in freezing and evaporating water droplets are the velocity magnitudes and the locations of the velocity maxima. Up to five times higher velocities are found inside freezing water droplets. The highest velocities are found close to the surface in freezing droplets, while in evaporating droplets, the velocity maximum is located on the symmetry axis.
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PREFACE
This master thesis has been carried out at the Division of Fluid and Experimental Me- chanics at Lule˚a University of Technology. It is the final work of the Master of Science programme in Engineering Physics and Electrical Engineering.
I would like to thank my supervisor and examiner Anna-Lena Ljung for support during the project, and Henrik Lycksam for help with the experimental setup. In addition, I would like to thank students Emilie Gousse, Martin H¨aggstr¨om and Bernhard Wullt for sharing the results of their thermal camera experiments with me.
Finally, I would like to thank all the cheerful people at the division, who made these weeks such a pleasure.
Tea Eriksson Lule˚a, June 2016
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CONTENTS
Chapter 1 – Introduction 1
1.1 Background . . . . 1
1.2 Literature review . . . . 1
1.3 Thesis objectives . . . . 3
1.3.1 Limitations . . . . 3
Chapter 2 – Theory 5 2.1 Droplet shape . . . . 5
2.1.1 Capillary length . . . . 6
2.2 Driving mechanisms of the internal flow . . . . 6
2.2.1 Evaporation-induced flow . . . . 6
2.2.2 Natural convection . . . . 6
2.2.3 Thermo-capillary convection . . . . 7
2.3 Particle image velocimetry . . . . 9
2.4 Distortion correction . . . . 9
Chapter 3 – Method 11 3.1 Experimental setup . . . . 11
3.1.1 Light source . . . . 11
3.1.2 Experimental box . . . . 12
3.1.3 Recording system . . . . 13
3.2 Experiments . . . . 14
3.3 Evaluation . . . . 14
Chapter 4 – Results and discussion 15 4.1 Isothermal experiment . . . . 15
4.2 Velocity correction . . . . 16
4.3 Evaporation . . . . 18
4.3.1 Sapphire plate . . . . 18
4.3.2 Polycarbonate plate . . . . 21
4.3.3 Comparison with simulated droplets . . . . 23
4.3.4 Thermal camera results . . . . 24
4.4 Freezing . . . . 25
4.5 Comparison between freezing and evaporation . . . . 28
Chapter 5 – Conclusions 29 5.1 Conclusions . . . . 29 5.1.1 Sources of error . . . . 30
Appendix A – Matlab code 31
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CHAPTER 1 Introduction
The background to the project is here described, followed by a literature review and the thesis objectives.
1.1 Background
Freezing and evaporation of droplets occur both in nature and in engineering applications.
For example, subcooled rain freezes as it hits the road, and droplet evaporation is used in spray cooling, where the endothermic process of evaporation is used to cool down heated surfaces. One example of when the internal flow in evaporating droplets is of interest is when particle deposition patterns are studied. Stains are formed when droplets with dispersed particles dries out. Examples are the coffee-ring stain [1] and stains left on drying dishes.
What motivates the study of evaporating droplets in this thesis is a project between the university and the dishwasher manufacturing company Electrolux, who are interested in the drying of droplets. The evaporation experiments done in this project will be used to validate CFD (Computational Fluid Dynamics) simulations. By studying the fundamental process of a drying droplet, the long-term goal is to improve drying in dishwashers.
There are many examples of when ice formation is an unwanted effect, for example ice on wind turbine blades. A fundamental understanding of the mechanisms driving droplet freezing is therefore desirable.
1.2 Literature review
Particle image velocimetry is a method with which velocity fields in fluids can be de- termined through tracing of seeding particles. One problem that occurs when the PIV method is used to measure velocity fields inside droplets is however that light refracts
1
2 Introduction
at the droplet surface and the images received are distorted. In 2004, Kang et al. [2]
presented an algorithm to correct this distortion. The algorithm is based on the ray tracing method and two ways to correct the distortion were presented. The first one is called the image mapping method and is used to correct the original images before they are analyzed. The other method is called the velocity mapping method and it corrects the velocity vectors retrieved from analysis of the raw images. In 2006, Minor et al. [3]
proved that the algorithm formulated by Kang et al. only works well for hemispherical shaped droplets, why they improved the original work.
He and Qiu [4] investigated the internal flow in evaporating droplets with PIV. They compared the result from a pure water droplet with that from droplets with different ethanol/water ratios and found that pure water only exhibits one flow direction while droplets containing alcohol has three stages of flow, namely downward, transitional and upward. Their conclusion was that in the multicomponent droplets, solutal Marangoni convection caused the downward vortices flow, and once most ethanol had evaporated, the flow changed direction and was then caused by natural convection. Hu and Larson [5] also noted that, even though theoretically expected, thermo-capillary flow in water droplets is weak. In octane droplets, however, thermo-capillary flow can be observed and due to this flow, dispersed particles will mainly deposit at the droplet center, rather than at the edge.
Droplet freezing has been visualized by Enr´ıques et al. [6] who deposited a water drop on a -20 °C cold surface and recorded how the ice front moved from the bottom of the droplet to the top. The initial shape of the droplet was that of a spherical cap, and over the course of freezing the droplet expanded vertically. In the end, a pointy tip was formed at the top of the droplet.
Kawanami et al. [7] studied the internal flow in freezing water droplets qualitatively.
A numerical model where surface tension, density and the moving freezing front were considered was validated with experiments and the agreement was good. The conclusion was that both natural and thermo-capillary convection influence the internal flow.
The internal flow in freezing water droplets has also been studied by Karlsson et al. [8].
Droplets of three different sizes (8, 10 and 12 µL) and two different plate temperatures (-7.0 °C and -11.4 °C) were investigated and results showed that the magnitude of the internal velocities lies in the range 0.05-1 mm/s. The flow direction furthermore indicated that thermo-capillary convection was the main mechanism causing the flow. Another phenomenon that was observed was that the droplet did not always start to freeze directly when it hit the cold surface. Subcooling of the droplet might be one reason for this.
Water turns into ice starting with a small crystallite, as described by Sastry [9]. This first step in phase change is called ice nucleation and it is a random process, but lowering the temperature and/or adding solid particles to the liquid hurries the process. Therefore, the roughness of the solid surface on which the droplets freeze is important. Jung et al. [10] compared the freezing delay of supercooled droplets on surfaces with different roughness and wettability. The result was that the droplets stayed in liquid phase for the
1.3. Thesis objectives 3
longest time when the surface was hydrophilic and smooth (nanometer-scale roughness).
1.3 Thesis objectives
The goal with this project is to determine and evaluate the flow inside both evaporating and freezing water droplets. An experimental setup has been developed to study freezing water droplets [8], and one of the tasks in this project is to reconstruct the equipment and apply it to heated droplets. For both evaporating and freezing water droplets, the aim is to be able to determine what induces the internal flow. Focus lies on evaporating droplets, where also the effect of the surface material will be investigated.
To get accurate flow fields, a method to correct the distortion generated by light re- fraction at the droplet surface will be implemented in Matlab.
1.3.1 Limitations
The whole freezing process will be studied, but only the first 30 seconds of evaporation.
This is because the velocities are highest in the beginning and the flow pattern is more distinct. As the droplet evaporates, its height decreases and less velocity information can be retrieved from the inside.
The humidity and temperature around the droplet probably affect both evaporation rate and freezing time, but it would require an environmental chamber to control them.
Therefore, these values will be noted, but not controlled.
CHAPTER 2 Theory
This chapter contains an overview of the relevant theory. First, the definition of the contact angle is given and the shape of a droplet on a substrate is discussed. Then the main causes of the internal flow in evaporating or freezing droplets are presented. The chapter ends with a short description of the PIV method and the image distortion that eventuates.
2.1 Droplet shape
A liquid drop on a solid substrate is called a sessile droplet. The shape of the droplet depends on its volume, properties of the substrate and the surface tension of the liquid, which in its turn depends on temperature and the vapor surrounding it. The unit of surface tension is energy per unit area or force per unit length. The droplet shape is characterized by the contact angle θc, which is defined as the angle between the solid surface and the tangent of the droplet surface at the point of contact, see Figure 2.1.
Figure 2.1: Definition of contact angle.
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6 Theory
If θc < 90°, the substrate is called hydrophilic and otherwise it is called hydrophobic.
If the contact angle is larger than 150°, the surface is called superhydrophobic.
2.1.1 Capillary length
There is a characteristic length, κ−1, defined as κ−1 =
r γ
ρg, (2.1)
where γ, ρ and g are the surface tension, density and gravitational constant, respec- tively. It is called the capillary length, and if a sessile droplet’s radius is less than this, the droplet has the shape of a spherical cap [11]. The capillary length for water at room temperature is approximately 2.7 mm. If the droplet radius is larger than this, gravity affects the droplet’s shape and the result is a flattened droplet.
2.2 Driving mechanisms of the internal flow
The primary causes of the internal flow in evaporating droplets are [12]
• Evaporation-induced flow
• Natural convection
• Thermo-capillary convection
The two last-mentioned mechanisms are relevant for freezing water droplets as well.
The flow inside a droplet is the result of competition between the above mentioned phenomena.
2.2.1 Evaporation-induced flow
The rate of evaporation in a droplet on a hot surface is higher at the bottom than it is at the top when the contact angle is less than 90° [12]. Water leaves the surface as vapor, and to compensate for this a radially outward flow is produced, see Figure 2.2. This is, unlike the other two flows listed above, not a circulating flow, and the result of this flow is that particles mixed in the water will deposit at the three phase contact line between air, liquid and solid, and leave a ring-like stain on the surface.
2.2.2 Natural convection
Free convection is fluid motion due to buoyancy forces. The thermal gradient in an evaporating droplet causes density differences. Warmer water is lighter and will therefore rise, while the colder, heavier fluid will descend. Figure 2.3 shows the densities of water for different temperatures. Note that water is heaviest at around +4 °C.
2.2. Driving mechanisms of the internal flow 7
Figure 2.2: Evaporation-induced flow is radially outward.
Figure 2.3: Water density as a function of temperature at atmospheric pressure.
2.2.3 Thermo-capillary convection
The surface temperature of an evaporating or freezing water droplet is non-uniform, which causes surface tension gradients. The result is flow from warmer regions, where the surface tension is less, to colder regions. The surface tension of water is shown in Figure 2.4.
When the apex is the coldest part of the droplet surface, the flow goes in the direction shown in Figure 2.5, otherwise according to Figure 2.6.
8 Theory
Figure 2.4: Surface tension of water as a function of temperature.
Figure 2.5: Downward vortices.
Figure 2.6: Upward vortices.
2.3. Particle image velocimetry 9
2.3 Particle image velocimetry
Particle image velocimetry (PIV) is an optical method for measuring velocity fields in fluids [13]. The fluid that is to be investigated is seeded with tiny particles that follow the flow well. In order to do this, they should be small and have a density close to the fluid’s.
A plane in the flow is illuminated by a laser light sheet, either pulsed or continuous, and the light scattered (or emitted, in the case of fluorescent tracer particles) is recorded by a camera. If a continuous laser is used, a movie is recorded and two neighboring frames are compared to determine the particle displacements in order to get the velocity field.
Often, some image pre-processing is done to enhance contrast in the images. A com- puter program is used to divide every image in smaller subareas, called interrogation areas. The number of tracer particles in each interrogation area should be 6-8. If the time step between two images is small enough, the assumption that the particles have moved homogeneously is valid, and a function calculates the most probable local displace- ment vector. This is done for all interrogation areas and the result is a two dimensional velocity field. Figure 2.7 shows the basic principle. The field can be calibrated if the magnification and frame rate is known.
Figure 2.7: The basic principle of PIV analysis.
2.4 Distortion correction
The light emitted from the fluorescent particles in the droplet refract at the droplet surface. The image is therefore distorted and needs to be corrected in order to get an accurate flow field inside the droplet. A correction algorithm based on the ray tracing method was first formulated by Kang et al. [2] and later on improved by Minor et al. [3]
The fundamental assumption that the whole correction algorithm builds upon is that
10 Theory
the droplet is axisymmetric and that its shape can be described by the function given in Eq. 2.2.
F (r, θ) = r− (R + b cos θ) = 0 (2.2) Figure 2.8 shows droplet shapes when the base radius R = 10 and b = 0.2R and b = −0.2R. If b = 0, the droplet has the shape a hemisphere and the contact angle is 90°.
(a) R = 10 and b = 2. Contact angle: 102° (b) R = 10 and b = -2. Contact angle: 78°
Figure 2.8: Droplet shapes.
Of the two correction methods proposed by Kang et al. [2], the velocity mapping method is chosen because it recovers more velocity vectors than the image mapping method. The Matlab code that performs the correction is given in Appendix A. The equations used are those given in [3], except for the expression for the factor f (equation number 18 in [3]), which has a printing error in it. The correct expression is
f = 1
p(r + b cos θ)2sin2θ sin2φ + (r cos θ− b sin2θ)2. (2.3)
CHAPTER 3 Method
The basic principle of the experimental method is that fluorescent particles are added to the water. A laser sheet lights up a plane in the middle of the droplet from underneath, and the light emitted by the particles is captured by a camera. In this chapter, the experimental setup is presented along with a description of how the experiments were post-processed.
3.1 Experimental setup
In Figure 3.1, a picture of the experimental setup is shown. The arrangement can be divided into three parts, namely
• Light source
• Experimental box
• Recording system
3.1.1 Light source
The laser is a continuous DPSS green laser from Altechna with wavelength λ = 532 nm and maximum power 150 mW. The laser is controlled by a power supply unit where the output power is set to 50 mW, motivated by previously done experiments, e.g. [4] and [8]. To prevent exposure from the powerful laser, the beam is passed through a beam shutter to block the laser beam in-between experiments. A polarizing beam splitter cube connected to a beam dump is another safety precaution used to control the amount of light that is passed through. A cylinder lens transforms the laser beam to a laser sheet with a thickness much smaller than the diameter of the droplets. Finally, an optical window is used to adjust the sheet in the up-down direction. The sheet is then guided underneath the droplet by means of a mirror.
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12 Method
Figure 3.1: The main parts of the experimental setup are (1) water tank, (2) camera, (3) zoom lens, (4) filter, (5) laser controller, (6) laser, (7) cylinder lens, (8) optical window, (9) pipette and (10) experimental box.
3.1.2 Experimental box
The surface on which the droplets are placed is a 3 mm thick transparent plate with a diameter of 5 cm, see Figure 3.2. The plate is in its turn placed on aluminum in contact with a Peltier element. The current through the Peltier element is controlled by DC power supply and the temperature of the plate is measured with a K-type thermocouple and recorded in LABview SignalExpress 2011. For the freezing experiments, the cold side of the Peltier element is upwards and ice cold water is circulated though the cooling fins connected to the hot side. For the evaporation experiments, when the hot side is upwards, room temperatured water is circulated through the cold side.
To avoid air flow around the droplet, a transparent box made of plexiglass is placed over the experimental setup. The droplets are released from a pipette though a hole in the top of the plexiglass cover. The temperature and relative humidity inside the box are measured with a 6000 Series Therma-Hygrometer from ETI.
Seeding particles
Prior to the experiments, the water is seeded with fluorescent Rhodamine B with diameter 3.5 µm. The concentration of seeding particles is adjusted to meet the requirement given
3.1. Experimental setup 13
Figure 3.2: Close-up view of the circular plate. The plate rests on a block of aluminum. Under the hole in the middle is the mirror that guides the incoming light sheet up through the droplet center. The picture also shows the attached thermocouple.
in section 2.3. The bottle containing the water and particles is put in an ultrasonic bath for 20 minutes before each set of experiments since the particles tend to lump together.
Before each experiment the bottle is also shaken to facilitate uniform distribution of fluorescent particles.
3.1.3 Recording system
The yellow light emitted from the fluorescent particles is recorded with an iDS camera with resolution 1280 x 1024 pixels. In front of the camera is a zoom lens from Navitar and a filter that filters out the green laser light. In the µEye software, the exposure time is set to 5 ms. The frame rate is varied between 30 and 60 frames per second for different experiments.
14 Method
3.2 Experiments
Two different plate materials are used in the evaporation experiments; sapphire (thermal conductivity ks = 40.81 mW·K) and polycarbonate (kp = 0.2 mW·K). For both plates, the temperature is varied between 40, 50 and 60 °C. The procedure for each experiment is:
the DC power supply is set to a specific current to reach the desired plate temperature.
When the plate temperature and the temperature inside the plexiglass cover are stabi- lized, a room temperatured 10 µL droplet is released from a pipette placed approximately 0.5 cm above the plate. The evaporation process is recorded for at least 30 s. Afterwards the plate is cleaned with paper. Two evaporation experiments are conducted without the plexiglass cover, but instead with a thermographic camera to show the surface tem- perature of the droplet. The plate temperature is 60 °C in those two experiments and the plate is sapphire.
The freezing process is investigated on the sapphire plate, where the plate temperature is varied between -10.0 and -12.8 °C.
Several experiments were performed with the same apparent conditions, and the results presented are representative of each case.
3.3 Evaluation
Image pairs are extracted from the recorded movies and analyzed in the GUI based PIV tool PIVlab in Matlab [14]. The procedure for each image pair is as follows:
• A mask is applied to exclude parts of the image, i.e. the parts that are not droplet, from the analysis
• A highpass filter is added to remove low frequency background information
• A multipass correlation scheme with decreasing window size (64 x 64 pixels in the first pass and 32 x 32 in the second) and 50 % overlap is selected
• The image pairs are analyzed
• The velocity field is calibrated using an external picture of a ruler placed in the same plane as the droplet center, and the time step is determined by the frame rate of the video
• The velocity field is post-processed by applying a standard deviation filter with threshold 7, and missing data in the vector field are interpolated
• Apparent outliers are removed
• .mat-files are exported for additional post-processing in Matlab
CHAPTER 4 Results and discussion
In this chapter, the experimental results are presented and discussed. First, the result of an isothermal experiment is given and then the distortion is visualized. After that, the evaporation results are presented, followed by the results from the freezing experiments.
The chapter ends with a comparison between the results from evaporation and freezing.
4.1 Isothermal experiment
In order to clarify that the particle motion inside the droplets is not caused by buoyancy between seeding particles and water or impact-induced flow, the result of an isothermal experiment is first presented. The temperatures of the plate, water and air are initially 20.5, 20.8 and 21.0 °C, respectively. In Figure 4.1, the droplet and part of its reflection in the plate is seen. The picture also shows the droplet shape function, Eq. 2.2, in blue.
Figure 4.1: Droplet with contact angle 77°. The base diameter is 3.5 mm and the droplet height is 1.4 mm.
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16 Results and discussion
There is an impact-induced flow during the first few seconds after the droplet is placed on the surface. Once the flow has settled, the fluorescent particles moves with a velocity magnitude in the order of 10−6 m/s. This means that it would take more than 15 minutes for a particle at the top of the droplet to fall down to the bottom. The corresponding velocity field and hence the slow descent of tracer particles is shown in Figure 4.2.
Figure 4.2: Velocity distribution in a droplet with approximately the same temperature as the plate and the surrounding air.
4.2 Velocity correction
The main difference between the uncorrected velocity fields (generated from PIV analysis of the raw images) and the corrected velocity vectors (obtained by using transformation equations developed by Kang et al. and Minor et al.) is that the velocity vectors are moved closer together in the corrected version. Figure 4.3 shows an example of an uncorrected velocity field and Figure 4.4 shows the corresponding corrected field. The solid black line represents the droplet shape. It becomes evident that velocity information is missing close to the surface. The velocities along the symmetry line, however, says a lot about the internal flow in the droplet. If they are negative, the flow is going downwards through the droplet center and if they are positive the flow is upwards, compare Figures 2.5 and 2.6.
4.2. Velocity correction 17
Figure 4.3: Velocity field from raw image.
Figure 4.4: Corrected velocity field. The solid line represents the droplet shape.
The curved droplet surface has a magnifying effect, which means that the corrected velocities are slightly lower than the original. This phenomenon is displayed in Figure 4.5, where corrected and uncorrected velocities along the centerline in the y-direction are compared at fifteen corresponding points. A maximum difference of about 12 % is observed between the corrected and uncorrected values.
A freezing water droplet expands because ice has lower density than water. In the beginning, the droplet shape is well described by Eq. 2.2, but in the end of freezing the droplet gets a singular shape. No velocity correction is therefore performed for the freezing experiments, so that the flow field evolution during the whole freezing process can be studied consistently.
18 Results and discussion
Figure 4.5: Comparison between fifteen uncorrected and corrected velocities in the y-direction along the symmetry line.
4.3 Evaporation
All initial contact angles are less than 90°, and the droplets evaporate with constant contact radius and decreasing height during the investigated time interval. The flow is seemingly random for the first one or two seconds after the droplets hit the surface, and this time interval is therefore left out of the analysis. The air temperature and relative humidity inside the plexiglass cover is presented in Table 4.1.
Table 4.1: Conditions inside the experimental box.
Plate temperature (°C) Air temperature (°C) Initial relative humidity (%)
40 28 16
50 32 15
60 37 14
4.3.1 Sapphire plate
Figure 4.6 shows the corrected mean velocities along the centerline for different temper- atures of the sapphire plate. The graphs show that the velocities are higher when the plate temperature is warmer and that for most of the time, the flow is directed upwards along the symmetry line (i.e. the velocities are greater than zero). Another observation
4.3. Evaporation 19
from the graphs is that the flow sometimes is directed downwards along the centerline in the beginning, but only for a few seconds.
Figure 4.6: Corrected mean velocities along the symmetry line for different plate temperatures when plate material is sapphire.
In Figure 4.7, the flow direction and velocity magnitudes are shown for different sap- phire plate temperatures. The velocities are taken 10 s into the evaporation process.
When the plate temperature is 50 or 60 °C, the internal flow forms two convection cells, but when the plate temperature is 40 degrees, the pattern is not as distinct. Increasing the plate temperature from 50 to 60 furthermore increases the maximum velocity with more than 50 percent.
20 Results and discussion
(a) Velocity distribution when Tplate = 60 °C.
(b) Velocity distribution when Tplate= 50 °C.
(c) Velocity distribution when Tplate= 40 °C.
Figure 4.7: Internal velocities at time t = 10 s in droplets placed on the sapphire plate.
4.3. Evaporation 21
4.3.2 Polycarbonate plate
There is practically no internal flow when polycarbonate is subjected to a plate temper- ature of 40 °C. Figure 4.8 shows the corrected mean velocities inside the whole droplet when the plate temperature is 50 and 60 °C. Compared to sapphire, the internal ve- locities are both lower (for a given plate temperature) and the flow pattern is a little different. The downward flow that can be observed in Figure 4.6 lasts for a longer time in droplets placed on polycarbonate. In Figure 4.9, the velocity distribution is shown at three different times for a plate temperature of 50 °C.
Figure 4.8: Corrected mean velocities inside droplet placed on polycarbonate plate.
22 Results and discussion
(a) Velocity distribution after 5 s.
(b) Velocity distribution after 10 s.
(c) Velocity distribution after 20 s.
Figure 4.9: Velocities inside droplet placed on a 50°C polycarbonate plate.
4.3. Evaporation 23
4.3.3 Comparison with simulated droplets
Preliminary simulations of the internal flow in an evaporating droplet, performed in an- other project, confirm the experimental results. The simulated droplet has rigid bound- aries with a contact angle of 90°. A temperature boundary condition is applied at the bottom and at the surface heat loss due to evaporation is accounted for. A transient simulation including the isolated effect of buoyancy on the internal flow show that the flow in the beginning is going downwards through the middle of the droplet (see Figure 4.10) and after a few seconds it turns around, see Figure 4.10 (b). The temperature distributions seen in the right halves of the images can possibly explain this. In (a), the upwards buoyancy force must be larger closer to the surface than it is in the middle, since a larger part of the fluid is warmer there. Therefore, the fluid will rise along the surface and go down in the middle. In (b), the temperature distribution is reversed and the flow goes in the other direction. Evaporative cooling cools the surface near the triple line and the result is that the temperatures along the bottom are warmer in the middle than closer to the surface. General conclusions regarding transient effects and magnitudes of velocities are subject of future work, but the results indicate that the internal velocities retrieved from experiments show large similarities with natural convection induced flow.
(a) Downward flow along centerline (b) Upward flow along centerline
Figure 4.10: Velocity and temperature distribution in simulated water droplets.
24 Results and discussion
4.3.4 Thermal camera results
To be able to image the small droplet, extender rings were used with the thermal camera.
These extender rings produce a radially distorted temperature field, which is not corrected here. However, the droplet lies in the center of the image where there is hardly no distortion. Figure 4.11 (a)-(d) shows the surface temperature at 0.5, 1, 1.5 and 2 s after the droplet hit the surface. The view is obliquely from above and results show that the surface temperature is very unsteady during the first two seconds.
(a) Surface temperature after 0.5 s (b) Surface temperature after 1 s
(c) Surface temperature after 1.5 s (d) Surface temperature after 2 s
Figure 4.11: Surface temperature of the droplet during the first two seconds. The droplet occu- pies approximately 1/9 of the picture.
Fifteen seconds after the droplet is placed on the 60 °C sapphire plate, the surface temperatures are those according to Figure 4.12. The temperature difference between the apex and the triple line is around 5-10 degrees.
4.4. Freezing 25
Figure 4.12: Surface temperature after 15 s.
4.4 Freezing
Two types of freezing are observed when a droplet is placed on a cold sapphire plate. Ei- ther ice starts to form instantly, and the droplet freezes from bottom to top, as described by [6]. The other case occurs when the plate surface is smooth and no rough layer of frost has formed. The droplet then stays in liquid phase for anything in-between a few seconds up to a couple of minutes. The freezing process then starts when, in one instant, the droplet surface freezes to ice. This is then followed by ice climbing from the bottom to the top inside the droplet, as in the first case. It is hard to beforehand determine if the surface is frosty enough to result in the first type of freezing, which is desired for the velocity observation.
Figure 4.13 shows how the freezing time of a 10 µL droplet depend on both surface temperature and contact angle. The different contact angles are obtained by releasing the droplets from different heights, but also the climate inside the experimental box affects the contact angle. The results show that the freezing time increases with the contact angle in a rather linear manner.
When ice is formed, the PIV program still detects small velocities within the ice. These are probable errors. Therefore, in the freezing experiments, the maximum velocity along the centerline better represents the deceleration of the flow than the mean velocities. In Figure 4.14, a representative experiment of a freezing droplet is presented. The plate temperature and the surrounding air are -11.2 °C and 12 °C, respectively. The contact angle is 93° and the freezing time is 18 s. The graph shows that the internal velocities are high during the first part of the freezing process. The flow gradually slows down and in the end, the internal velocities are close to zero.
26 Results and discussion
Figure 4.13: Measured freezing times for 10 µL droplets.
Figure 4.14: Maximum velocities along centerline for a 10 µL droplet freezing on -11.2 °C sapphire plate.
In the beginning of the freezing process, the internal velocities are higher than for evaporating droplets, but the flow slows down quickly. The surface temperature is lowest closest to the ice front, which means that the surface tension is higher there than at the top of the droplet. The flow direction where warmer water at the top is pulled downwards
4.4. Freezing 27
along the droplet surface therefore implies thermo-capillary convection. Figure 4.15 shows the uncorrected internal velocities at different times during the freezing process. One interesting observation is that at t = 6 s, the flow is going in the other direction, which can also be noticed in Figure 4.14.
t = 2 s t = 4 s
t = 6 s t = 18 s
Figure 4.15: Uncorrected internal velocities at different times during the freezing process.
28 Results and discussion
4.5 Comparison between freezing and evaporation
The main differences between the internal flow in freezing and evaporating water droplets is visualized in Figure 4.16. The plots show the uncorrected velocity fields, where (a) is an evaporating droplet placed on a 60 °C sapphire plate, and (b) shows the velocity distribution inside a freezing droplet placed on a -11.2 °C sapphire plate. Note that the colorbars have different scales.
(a) Evaporating droplet (b) Freezing water droplet
Figure 4.16: Velocity distribution in evaporating and freezing water droplets.
In an evaporating droplet, the highest velocities are found close to the symmetry axis, while for a freezing water droplet, the largest velocities are located closer to the surface.
During the first few seconds of freezing, the internal velocities are up to a couple of mm per second, but in a droplet evaporating on a 60°C sapphire plate, the maximum internal velocity is one fifth of that.
In both evaporating and freezing water droplets, the flow is mainly directed upwards along the symmetry axis. In evaporating droplets, this indicates flow due to buoyancy forces, while the flow direction in freezing water droplets indicates influence of surface tension gradients.
CHAPTER 5 Conclusions
5.1 Conclusions
The internal flow in evaporating sessile droplets has been investigated for different plate materials and temperatures. The results show that the internal velocities increase with increasing plate temperature and that the thermal properties of the plate affect both flow direction and velocity magnitudes. For droplets placed on a sapphire plate, which has higher thermal conductivity than polycarbonate, the flow direction is mainly upwards through the droplet center. In some experiments, a downward flow along the symmetry axis is seen during the first few seconds. In droplets placed on the polycarbonate plate, the flow field is less distinct and the downward flow along the centerline lasts for a longer time in these droplets.
The upward flow at the centerline and the velocity distribution suggests that natural convection is the main mechanism driving the flow in evaporating water droplets. Warmer water at the bottom is lighter than the colder water at the top. The flow then goes up through the middle of the droplet and down on the sides. However, since the exact temperature distribution inside the droplet is unknown, natural convection induced flow might in some cases produce downward vortices. More delicate experiments are required to be able to estimate the temperatures in the droplet and to draw accurate conclusions about what mechanism is really driving the flow.
There are two ways that a droplet can freeze on a cold surface. Either, the droplet starts to freeze immediately as it hits the cold surface. The ice front moves from the bottom to the top and the first part of freezing results in a distinct flow on the inside, possibly originating from thermo-capillary convection, with velocities up to a few mm/s in a 10 µL droplet placed on a -11.2 °C surface. The internal velocities decrease as the ice climbs upwards and in the end, no internal flow is visible. The other type of freezing occurs when the droplet stays in liquid form for some time even after it is placed on the surface. The droplet surface then freezes in an instant, followed by ice accretion inside the droplet. No internal flow could be seen in these cases.
29
30 Conclusions
Higher velocities are found inside freezing water droplets than in evaporating droplets and the velocity distributions are different. In evaporating droplets, the velocity maxi- mum is located on the symmetry axis. In a freezing water droplet, the highest velocities are found closer to the droplet surface.
5.1.1 Sources of error
There are some inexact parameters in the experiments. First of all, the droplet volume might not be exactly the same in all experiments since liquid residues are sometimes left in the pipette tip. Errors also originate from the temperature measurements. The plate temperature was measured approximately 1 cm from the edge. This is especially important to note when looking at the experimental results from the polycarbonate plate.
As described in section 3.1, the plate edges are warmed from underneath. Polycarbonate has low thermal conductivity, and the measured temperature at the sensor might differ from the temperature in the middle of the plate where the droplet is placed.
A few problems are related to the PIV measurements. The first problem is that the depth of focus of the camera might be larger than the light sheet thickness and particle motion from other planes than the central plane might be captured. Another source of error concerns the velocity magnitudes. They are calculated by using an external calibration picture of a ruler located in the plate center. But the droplets weren’t posed on the exact same spot in all experiments.
APPENDIX A Matlab code
1 function [xo, yo, uo, vo, Rm, bm, Oxm, Oym] = ...
2 VelocityMapping(xi, yi, ui, vi, R, b, Ox, Oy)
3 % Velocity mapping function based on articles written by Kang et al. and
4 % Minor et al.
5
6 % Correction equation Number in Kang et al.
7 % xo = xi + zs*Bx 4a
8 % yo = yi + zs*By 4b
9 % uo = ui + ui*d(zs*Bx)/dxi + vi*d(zs*Bx)/dyi 5a
10 % vo = vi + ui*d(zs*By)/dxi + vi*d(zs*By)/dyi 5b
11
12 % ---
13 % INPUT:
14 % xi - x coordinate matrix from exported .mat file from PIVlab
15 % yi - y coordinate matrix -"-
16 % ui - u velocity matrix -"-
17 % vi - v velocity matrix -"-
18 % R - droplet base radius in pixels (original image)
19 % b - droplet up/down displacement in pixels (original image)
20 % Ox - x origin in pixels (original image)
21 % Oy - y origin in pixels (original image)
22
23 % OUTPUT:
24 % xo - x coordinates in object plane
25 % yo - y coordinates in object plane
26 % uo - corrected u velocities
27 % vo - corrected v velocities
28 % Rm - base radius (real distance in m)
29 % bm - b value (real distance in m)
30 % Oxm - (real distance in m)
31 % Oym - (real distance in m)
32 % ---
31
32 Appendix A
33 % The input matrices are not 1024 x 1280, so scaling is required. xi (and
34 % ui) are in m (and m/s), so R, b, Ox and Oy need to be converted.
35 [M, N] = size(xi);
36 k = M / 1024; % scaling factor
37 dx = xi(1,2) - xi(1,1);
38
39 Rm = k*R*dx;
40 bm = k*b*dx;
41 Oxm = k*Ox*dx;
42 Oym = k*Oy*dx;
43
44 % Origin is at the center of the droplet base circle
45 yi = Oym - yi;
46 xi = xi - Oxm;
47
48 % Refractive index of air and water
49 na = 1;
50 nd = 1.33;
51
52 % Distance from origin to point Ps on surface
53 rs = (Rm + sqrt(Rmˆ2 + 4*bm*yi))/2;
54
55 % Initial values
56 zs = NaN(M,N);
57 Bx = zeros(M,N);
58 By = zeros(M,N);
59 xo = xi; yo = yi;
60
61 % LOOP TO CALCULATE CORRECTION PARAMETERS
62 for i = 1:M
63 for j = 1:N
64
65 % If inside droplet
66 if ((rs(i,j) >= sqrt(xi(i,j)ˆ2 + yi(i,j)ˆ2)) && (yi(i,j) >= 0))
67
68 % z coordinate of Ps
69 zs(i,j) = sqrt(rs(i,j)ˆ2 - xi(i,j)ˆ2 - yi(i,j)ˆ2);
70
71 % Spherical coordinates angles
72 theta s = acos(yi(i,j) / rs(i,j));
73 phi s = asin(xi(i,j) / sqrt(xi(i,j)ˆ2 + zs(i,j)ˆ2));
74
75 % Angles of incidence and refraction
76 psi a = acos((rs(i,j) + bm * cos(theta s)) * ...
77 sin(theta s) * cos(phi s) / ...
78 sqrt(rs(i,j)ˆ2 + bmˆ2 * sin(theta s)ˆ2));
79 psi d = asin((na/nd) * sin(psi a));
80 81 82
33
83 % f, Bx and By
84 f = 1 / sqrt((rs(i,j) + bm * cos(theta s))ˆ2 * ...
85 sin(theta s)ˆ2 * sin(phi s)ˆ2 + ...
86 (rs(i,j) * cos(theta s) - bm * sin(theta s)ˆ2)ˆ2);
87
88 Bx(i,j) = -f * tan(psi a - psi d) * ...
89 (rs(i,j) + bm * cos(theta s)) * sin(theta s) * sin(phi s);
90 By(i,j) = -f * tan(psi a - psi d) * ...
91 (rs(i,j) * cos(theta s) - bm * sin(theta s)ˆ2);
92
93 % ---
94 % Coordinate transformation
95 % ---
96 xo(i,j) = xi(i,j) + zs(i,j) * Bx(i,j);
97 yo(i,j) = yi(i,j) + zs(i,j) * By(i,j);
98 end
99 end
100 end
101
102 % ---
103 % Velocity transformation
104 % ---
105 A = zs.*Bx;
106 B = zs.*By;
107
108 % EXPAND MATRICES
109 % x-direction
110 Ax = [zeros(M,1), A, zeros(M,1)];
111 Bx = [zeros(M,1), B, zeros(M,1)];
112
113 % y-direction
114 Ay = [zeros(1,N); A; zeros(1,N)];
115 By = [zeros(1,N); B; zeros(1,N)];
116
117 % First derivative (central difference approximation)
118 for i = 1:M
119 for j = 1:N
120
121 dAdx(:,j) = (Ax(:,j+2) - Ax(:,j)) ./ (2*dx);
122 dBdx(:,j) = (Bx(:,j+2) - Bx(:,j)) ./ (2*dx);
123
124 dAdy(i,:) = -(Ay(i+2,:) - Ay(i,:)) ./ (2*dx);
125 dBdy(i,:) = -(By(i+1,:) - By(i,:)) ./ (2*dx);
126
127 end
128 end
129
130 vi = -vi;
131 132
34 Appendix A
133 % ---
134 % Velocity transformation
135 % ---
136 uo = ui + ui.*dAdx + vi.*dAdy;
137 vo = vi + ui.*dBdx + vi.*dBdy;
138
139 % ---
140 % The outermost vectors are incorrect and the code below finds these "end
141 % points" and sets the velocities there to NaN.
142 % ---
143
144 % FIND LEFT END POINTS
145 Lrow = [];
146 Lcolumn = [];
147
148 for rows = 1:M
149 for columns = 2:N
150 if (isnan(uo(rows,columns-1)) && ~isnan(uo(rows,columns)))
151 Lrow = [Lrow rows];
152 Lcolumn = [Lcolumn columns];
153 end
154 end
155 end
156
157 % FIND RIGHT END POINTS
158 Rrow = [];
159 Rcolumn = [];
160
161 for rows = 1:M
162 for columns = 1:N-1
163 if (isnan(uo(rows,columns+1)) && ~isnan(uo(rows,columns)))
164 Rrow = [Rrow rows];
165 Rcolumn = [Rcolumn columns];
166 end
167 end
168 end
169
170 % FIND TOP END POINTS
171 Trow = [];
172 Tcolumn = [];
173
174 for rows = 2:M
175 for columns = 1:N
176 if (isnan(uo(rows-1,columns)) && ~isnan(uo(rows,columns)))
177 Trow = [Trow rows];
178 Tcolumn = [Tcolumn columns];
179 end
180 end
181 end
182
35
183 % FIND BOTTOM END POINTS
184 Brow = [];
185 Bcolumn = [];
186
187 for rows = 1:M-1
188 for columns = 1:N
189 if (isnan(uo(rows+1,columns)) && ~isnan(uo(rows,columns)))
190 Brow = [Brow rows];
191 Bcolumn = [Bcolumn columns];
192 end
193 end
194 end
195
196 % SET END POINTS TO NaN
197 for aa = 1:length(Lrow)
198 uo(Lrow(aa),Lcolumn(aa)) = NaN;
199 vo(Lrow(aa),Lcolumn(aa)) = NaN;
200 end
201
202 for bb = 1:length(Rrow)
203 uo(Rrow(bb),Rcolumn(bb)) = NaN;
204 vo(Rrow(bb),Rcolumn(bb)) = NaN;
205 end
206
207 for cc = 1:length(Trow)
208 uo(Trow(cc),Tcolumn(cc)) = NaN;
209 vo(Trow(cc),Tcolumn(cc)) = NaN;
210 end
211
212 for dd = 1:length(Brow)
213 uo(Brow(dd),Bcolumn(dd)) = NaN;
214 vo(Brow(dd),Bcolumn(dd)) = NaN;
215 end
216 end
REFERENCES
[1] R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, and T. A. Witten,
“Capillary flow as the cause of ring stains from dried liquid drops,” Nature, vol. 389, no. 6653, pp. 827–829, 1997.
[2] K. H. Kang, S. J. Lee, C. M. Lee, and I. S. Kang, “Quantitative visualization of flow inside an evaporating droplet using the ray tracing method,” Measurement Science and Technology, vol. 15, no. 6, pp. 1104–1112, 2004.
[3] G. Minor, P. Oshkai, and N. Djilali, “Optical distortion correction for liquid droplet visualization using the ray tracing method: further considerations,” Measurement Science and Technology, vol. 18, no. 11, pp. L23–L28, 2007.
[4] M. He and H. Qiu, “Internal flow patterns of an evaporating multicomponent droplet on a flat surface,” International Journal of Thermal Sciences, vol. 100, pp. 10–19, 2016.
[5] H. Hu and R. G. Larson, “Marangoni effect reverses coffee-ring depositions,” The Journal of Physical Chemistry B Letters, vol. 110, pp. 7090–7094, 2006.
[6] O. R. Enr´ıques, ´Alvaro G. Mar´ın, K. G. Winkels, and J. H. Snoeijer, “Freezing singularities in water drops,” Physics of Fluid, vol. 24, no. 9, 2012.
[7] T. Kawanami, M. Yamada, S. Fukusako, and H. Kawai, “Solidification characteris- tics of a droplet on a horizontal cooled wall,” Heat Transfer – Japanese Research, vol. 26, no. 7, pp. 469–483, 1997.
[8] L. Karlsson, H. Lycksam, A.-L. Ljung, P. Gren, and T. S. Lundstr¨om, Experimental study of the internal flow in freezing water droplets on a cold surface. Licentiate thesis, Lule˚a University of Technology (LTU), Lule˚a, Sweden, 2015.
[9] S. Sastry, “Ins and outs of ice nucleation,” Nature, vol. 438, pp. 746–747, 2005.
37
38
[10] S. Jung, M. Dorrestijn, D. Raps, A. Das, C. M. Megaridis, and D. Poulikakos,
“Are superhydrophobic surfaces best for icephobicity?,” Langmuir, vol. 27, no. 6, pp. 3059–3066, 2011.
[11] P.-G. de Gennes, F. Brochard-Wyart, and D. Qu´er´e, Capillarity and wetting phe- nomena: drops, bubbles, pearls, waves. Springer, 2004.
[12] D. Brutin, ed., Droplet wetting and evaporation: from pure to complex fluids. Aca- demic Press, 2015.
[13] M. Raffel, C. E. Willert, S. T. Wereley, and J. Kompenhans, Particle Image Ve- locimetry A practial guide. Springer, 2007.
[14] W. Thielicke and E. J. Stamhuis, “PIVlab – Towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB,” Journal of Open Research Software, vol. 2, 2014.
