UNIVERSITY OF CALIFORNIA,
IRVINE
Droplet Evaporation in an Active
Turbulence Grid Wind Tunnel
THESIS
submitted in partial satisfaction of the requirements
for the degree of
MASTER OF SCIENCE
in Mechanical and Aerospace Engineering
by
Ferran Marti Duran
Thesis Committee:
Professor Derek Dunn-Rankin, Chair
Professor John LaRue
Professor Roger Rangel
2012
c 2012 Ferran Marti Duran
TABLE OF CONTENTS
Page
LIST OF FIGURES
iv
LIST OF TABLES
viii
ACKNOWLEDGMENTS
ix
ABSTRACT
x
1 Introduction
1
2 Two Methods for Measuring the Evaporation Rate
3
3 Experimental Apparatus
3.1 Set up Components . . . . . . . . . . . . . . . . .
3.1.1 Introduction . . . . . . . . . . . . . . . . .
3.1.2 Wind Tunnel . . . . . . . . . . . . . . . .
3.1.3 Active Grid . . . . . . . . . . . . . . . . .
3.1.4 Electronic Equipment . . . . . . . . . . . .
3.1.5 Drop Generator . . . . . . . . . . . . . . .
3.2 Contributions to the Experimental Apparatus . .
3.2.1 Housing for the needle . . . . . . . . . . .
3.2.2 Heater . . . . . . . . . . . . . . . . . . . .
3.2.3 Perturbation damper for the syringe pump
3.2.4 Pitot tube and temperature transducer . .
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4 Theoretical Simulations
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14
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5 Results and Discussion
5.1 Hot Wire Characterization of the Turbulence in the Wind Tunnel . .
5.1.1 Isotropic Turbulence . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Theoretical and Experimental Results for Drops Suspended on Wires
5.2.1 Holding the drops . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Sizing methods . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
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33
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40
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41
42
5.3
5.2.3 Results and discussion . . . . . . . . . . . . . . . . . .
Theoretical and Experimental Suggestions for Traveling Drops
5.3.1 Drops and air flow moving in the gravity direction . . .
5.3.2 Drops and air flow moving against gravity direction . .
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43
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50
53
6 Conclusion
58
Bibliography
60
Appendices
A
Surface Tension . . . . . . . . . . . . . . . . . . .
B
Gas / Liquid Properties . . . . . . . . . . . . . .
B.1
Vapor Pressure . . . . . . . . . . . . . . .
B.2
Heat Capacity - Cpg . . . . . . . . . . . .
B.3
Dynamic Viscosity . . . . . . . . . . . . .
B.4
Thermal Conductivity . . . . . . . . . . .
C
Imaging Technology . . . . . . . . . . . . . . . . .
C.1
Working technology used to size the drops
C.2
Other studied technologies for sizing drops
D
Definitions . . . . . . . . . . . . . . . . . . . . . .
62
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69
69
81
90
iii
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LIST OF FIGURES
Page
3.1
3.2
3.3
3.4
3.5
3.6
3.7
4.1
4.2
4.3
4.4
This figure shows the vertical wind tunnel, the active grid to generate isotropic
turbulence, the droplet generation system, and the sizing technology. Refer
to this image while reading this chapter. . . . . . . . . . . . . . . . . . . . .
Converging nozzle and beginning of the test section of the wind tunnel. We
can see too the active grid detailed in section 3.1.3. . . . . . . . . . . . . . .
Beginning of the test section of the wind tunnel (transparent walls). Active
grid at the top on the picture. We see here how a hot wire is characterizing
the flow in the tunnel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sketch of the housing that goes around the 90 degree needle. . . . . . . . . .
Picture of a drop after leaving the housing that covers the needle. This technique of sizing the drops is explained in section 5.3 and Appendix C. We can
see a reflection of the green laser used for triggering the electronic equipment
explained in section 3.1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sketch of the damper. The spring method used here is the air compressing in
the reservoir. The on/off valves just help to fill the reservoir with an initial
level of liquid. The final destination of the damped flow rate could be a beaker,
as it shows in the image, or a needle in the experiment. . . . . . . . . . . . .
Flow rate with and without the damper for a brand new syringe pump. . . .
Heptane droplet at 300K and ambient pressure. BY and Bq versus TW . When
the Lewis number is 1, BY = Bq . B and Ts are the values from the intersection
from both curves which will be used for further calculations as the Spalding
Number and the Surface Temperature of the drop respectively. . . . . . . .
Equation 4.15 for heptane droplets with changes in the ambient temperature
at sea level pressure. Static drop with still air. . . . . . . . . . . . . . . . .
d for heptane droplets of 1 mm of initial diameter at sea level pressure and
300 K. Static drop with different velocities of mean flow. . . . . . . . . . . .
d for heptane droplets of 1 mm of initial diameter at sea level pressure and
10m/s flow. Static drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
7
8
10
15
16
18
18
24
26
27
28
4.5
4.6
4.7
d for heptane droplets of 1 mm of initial diameter at sea level pressure and
300K. The drop is initially at 0 ground velocity. Two cases are simulated,
for no air flow around the drop, and for a constant flow going downward at
15m/s. Notice how the evaporation is almost identical for both cases. This is
because, except at the very beginning, the evaporation of both drops occurs at
the same relative velocity with the air flow since both achieve terminal velocity. 30
d vs. position x (initial position (x = 0)) for heptane droplets of 1 mm
of initial diameter at sea level pressure and 300K. The drop is initially at 0
ground velocity. Two cases are simulated, for no air flow around the drop, and
for a constant flow going downward at 15m/s. This plot is interesting, since
it provides information about the length of the wind tunnel that is required
to measure a certain evaporation depending on the velocity of the air flow. . 31
Ground and relative velocities for the two cases plotted with no air, and
−15m/s air flow at 300K. Notice how for the case of no air flow, the ground
velocity and the relative velocity are the same. We can see how, after a certain
time from the beginning, both drops reach the same terminal relative velocity
to the flow, so they evaporate at the same rate. Notice too how the terminal
velocity keeps changing since the drop keeps on reducing in size. . . . . . . 32
5.1
Mean velocity in the wind tunnel with respect to the downstream position x
from the grid. M is the active grid spacing. Two tests, one of 40s and another
one of 4.2min. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Mean velocity in the wind tunnel with respect to the downstream position x
from the grid. M is the active grid spacing. Two tests, one of 40s and another
one of 4.2min. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Mean velocity in the wind tunnel with respect to the downstream position x
from the grid. M is the active grid spacing. Two tests, one of 40s and another
one of 4.2min. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Energy dissipation for the case of 40s. . . . . . . . . . . . . . . . . . . . . .
5.5 Energy dissipation for the case of 4.2min. . . . . . . . . . . . . . . . . . . .
5.6 Kolmogorov scale of turbulence for both of our tests along the wind tunnel.
5.7 Droplet suspended on two crossing wires. Heptane. Still air at 296K and sea
level pressure. Direct light imaging. 3mm drop. . . . . . . . . . . . . . . . .
5.8 Droplet suspended on two crossing wires. Heptane. Still air at 296K and sea
level pressure. Shadow imaging. 2mm drop. . . . . . . . . . . . . . . . . . .
5.9 D2 for heptane drops at room temperature. One wire configuration; still air.
Each marker shows a different experimental case. . . . . . . . . . . . . . . .
5.10 D2 for heptane drops at room temperature. Two wire configuration. Still air.
Each marker shows a different experimental case. . . . . . . . . . . . . . . .
5.11 Air flow at 2.1 − 2.3m/s. Active grid off. Room temperature 22.5C - 24C.
Heptane. One wire configuration. Each marker shows a different experimental
case. Simulation is blue solid line (initial diameter of 2.7mm). . . . . . . . .
5.12 Air flow at 2.1 − 2.3m/s. Active grid off. Room temperature 22.5C - 24C.
Heptane. Two wire configuration. Each marker shows a different experimental
case. Simulation is blue solid line (initial diameter of 1.7mm). . . . . . . . .
v
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38
39
39
40
42
43
44
45
46
47
5.13 Air flow at 2.1 − 2.3m/s. Active grid on. Room temperature 22.5C - 24C.
Heptane. One wire configuration. Each marker shows a different experimental
case. Simulation is blue solid line (initial diameter of 2.5mm). . . . . . . . .
5.14 Air flow at 2.1 − 2.3m/s. Active grid on. Room temperature 22.5C - 24C.
Heptane. Two wire configuration. Each marker shows a different experimental
case. Simulation is blue solid line (initial diameter of 1.8mm). . . . . . . . .
5.15 Traveling drops. The image in the left and the one in the center are taken with
the flash behind the drop at a speed of 500ns. They show the deformation of
the drops. The pictures are shadows of the drops and the inner light circle is
just a reflection. The picture on the right is taken using a flash from the front
at a speed of 1/4000s, it is blurred because the exposure on the camera was
not short enough. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.16 Sketch of the set up for sizing traveling drops in the order of mm. . . . . . .
5.17 Free fall of a 2mm drop with zero initial velocity and flow speed of 2m/s
going in the direction of the drop. Room temperature of 300K and sea level
pressure. Picture when droplet has traveled 1.5m (end of the wind tunnel). .
5.18 Free fall of a 2mm drop with zero initial velocity and flow speed of 2m/s going
in the direction of the drop. Room temperature of 300K and sea level pressure.
5.19 Drop diameter vs distance traveled (x is positive since now the drop moves
against gravity). 0.2mm drop shot up at 15m/s. Air flow is going up at 2m/s.
300K and sea level pressure. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.20 Drop velocity vs distance traveled. 0.2mm drop shot up at 15m/s. Air flow
is going up at 2m/s. 300K and sea level pressure. . . . . . . . . . . . . . .
A.1 Surface tension variation with temperature for Butane and Heptane. Eq.
(A.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 ∆p across the surface of a spherical Heptane droplet variation with the droplet
diameter and the temperature. Young-Laplace Eq. . . . . . . . . . . . . . .
C.3 Extension Tubes: System used between the lens and the camera. They can
be mounted individually, or in combinations of the three of them. . . . . . .
C.4 Sketch of the camera with the integrated flash and droplet position for sizing
suspended drops on wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.5 Image of the ruler set where the drops hang from the two crossed wires. . .
C.6 Sketch of the camera, back lighting coming from the flash and droplet position
for sizing traveling drops. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.7 Shadowgraphy of the metallic sphere taken with the system from Figure C.6.
C.8 Black and white image with line to plot the gray value along that line. . . .
C.9 Plot of the gray value along the yellow line of pixels from Figure C.8 . . . .
C.10 Plot of the gray value along the yellow line of pixels from Figure C.8. The
slope of the gray value across the edge of the sphere is highlighted in a red
bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.11 Threshold of the in focus metal ball. . . . . . . . . . . . . . . . . . . . . . .
C.12 Outlines that ImageJ detects as the edges of the metal ball. The result of the
sizing process will be the number of pixels inside these outlines. . . . . . . .
C.13 Image of the metal ball when is out of focus. We can see the blur in the edges.
vi
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48
51
53
54
55
56
56
63
64
71
72
73
74
76
77
77
78
79
79
80
C.14 Plot of the gray value across the diameter of the metal ball from Figure C.13.
C.15 Sketch to size drops with parallel light. The parallel beam is coming from the
left of the image, the drop could be in any position along the beam and its
shadow on the screen will always be the same. The camera, set behind the
screen and focusing on it will take a picture of the projected shadow. . . . .
C.16 The f number of a light source is defined as B/A. For a lens, large f numbers
means less light will get to them. To calculate the f number of the Xenon arc
lamp, the B and A measurements have been done at different position of the
beam (vertical red, blue and black lines) to average the results. . . . . . . . .
C.17 Generating a parallel beam of white light with a point of light. If the source
is not a point source it will be hard to obtain parallel light. The light source
has to be placed at the focal distance of the lens. . . . . . . . . . . . . . . .
C.18 Image of the shadow of the washer projected on the screen. The washer is 5
cm from the screen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.19 Image of the shadow of the washer projected on the screen. The washer is
15 cm from the screen. We cannot know the real position of the washer since
its size looks like the one in Figure C.18 eventhough the washer is now 10 cm
further from the camera. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.20 Train of two lenses to increase the width of the laser beam. . . . . . . . . .
C.21 Image taken on a screen like Figure C.15. The shadow is a washer instead of
a drop. Notice the speckle in the image. . . . . . . . . . . . . . . . . . . . .
C.22 Sketch of the system made of two splitters and two mirrors that would split
one original laser shot in two separated a few nanoseconds. . . . . . . . . .
C.23 Same image as Figure C.21 after the impact of two shots. The speckle is
reduced even if the diffraction at the edges of the washer is still there. The
image is taken with an angle with respect to Figure C.21 but that does not
change the level of speckle. . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
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82
83
84
85
86
86
87
88
89
LIST OF TABLES
Page
3.1
Conditions at x/M = 20, where M = 0.03m is the active grid spacing. . . . .
viii
11
ACKNOWLEDGMENTS
I would like to thank the Balsells Fellowships Program for supporting me during my Master
studies and the National Science Foundation for funding the project.
The help and advice of Professor Derek Dunn-Rankin and Dr. John Garman, as well as the
contribution of Kamil Samaan, Khizar Karwa and Sina Hashemi has been essential for the
good development of the research.
I would like to thank Professor John LaRue and his lab for the support in the wind tunnel
construction and hot wire anemometry tasks.
I also thank Dr. Sunny Karnani and Dr. Alireza Mirsepassi for the conversations and ideas
about the evolution of the project. Thanks to Dr. Jeff Jiang for his help in machining parts
for the experiment.
ix
ABSTRACT OF THE THESIS
Droplet Evaporation in an Active
Turbulence Grid Wind Tunnel
By
Ferran Marti Duran
Master of Science in Mechanical and Aerospace Engineering
University of California, Irvine, 2012
Professor Derek Dunn-Rankin, Chair
In this thesis, we develop and test an experimental apparatus capable of examining the effect
of turbulence on the evaporation of droplets when the Kolmogorov scale is smaller than the
size of the drop (where size refers to the nominal droplet diameter). Traveling drop and
suspended drop cases have been studied experimentally and the evaporation rate has also
been predicted based on classical computations of transport rates. Cases involving still air,
a mean flow with turbulence, and a mean flow without turbulence have been tested. The
experimental data matches the theoretical computations for the case of still air and mean
flow with no turbulence. There is no accepted computational model that can account for
turbulent flow, but the experimental conditions tested so far do not show any measurable
effect of turbulence on the evaporation rate. These findings suggest that turbulence is a much
smaller factor in evaporation than is mean flow over the droplet. Hence, one suggestion for
future work is to study traveling droplets with zero relative velocity to the air in order to
focus the analysis on the effects of turbulence with no mean flow.
x
Chapter 1
Introduction
As it can be read in [4], liquid fuels provide a large share (35%) of world energy consumption
at present and for the foreseeable future. Furthermore, liquid fuels remain the most important source of energy for all modes of transportation because few alternatives have properties
that can compete with those of liquid fuels (e.g. energy/unit mass and energy/unit volume).
Globally, the transportation sector accounts for 74% of the total projected increase in liquid
fuels use from the present to 2030, with the industrial sector accounting for the remainder [1].
All practical liquid fuel combustion devices, mobile or stationary, use atomizers to produce
sprays of fine droplets. In order to release their stored chemical energy the fuel droplets must
first vaporize before their vapor mixes with the surrounding air. This mixing is followed by
a chemical reaction between the fuel vapor and air, which converts the chemical bonding
energy into thermal energy, resulting in the volumetric expansion of the gas mixture. This
volumetric expansion produces the desired mechanical energy for rotary or reciprocating
engines. The four critical processes in the above sequence are atomization, vaporization,
mixing, and chemical reaction. In almost all practical liquid fuel combustion devices, these
processes take place inside a combustion chamber where the flow is turbulent and mixing
rates are high. Often, therefore, the vaporization rate is the main controlling mechanism of
1
the entire combustion process. Turbulence controls the dispersion of the droplets and the
rates of mass and heat transfer, and consequently the vaporization rate. On the other hand,
droplets can modify the turbulence structure ([2]; [3]). Thus, understanding these two-way
interactions and the physical details of the vaporization and mixing processes in such a turbulent flow is an essential prerequisite to understanding the chemical reaction process and
the eventual control/optimization of the energy conversion process. What is less studied and
understood, however, is the case when the turbulence scale is small relative to the droplet
diameter. In this case, the small scale effects are less dispersive but may produce modified
transport behavior.
The proposed study will ideally examine, therefore, the turbulence effect on the evaporation
rate of free fall droplets where the smallest length scale of the turbulence is smaller than the
drops. Also, it will serve as a reference to compare numerical studies that are being performed
in other laboratories of UCI using DNS. Before accomplishing this ultimate goal, however,
our specially designed active-grid wind tunnel approach must be verified by measuring the
evaporation rates of droplets in still air, and in laminar mean flows.
2
Chapter 2
Two Methods for Measuring the
Evaporation Rate
Two methods have been used to measure the evaporation rate of droplets under turbulence
effects. The first, consists in holding a droplet on a wire through surface tension. This
technique is interesting since the air velocity around the drop is constant, and the effect of
turbulence can be studied for each velocity. It is a case where sizing the drop is easy as well
since it is always in the same position so there is no problem associated with image focus.
By taking a sequence of images, the evolution of the diameter of the drop with respect to
time can be recorded. The problem this technique has is the interaction between the droplet
and the wire that holds it. For example, the wire can prevent the drop from free deformation
when eddies are around it, it blocks part of the wind impacting on the drop, and it changes
the shape of the drop so that it is not spherical. Different materials of wires have been tried.
Also, different ways to hold the drops on them have been tested. The cross technique, where
two orthogonal wires in the same horizontal plane hold the drop works well when it comes
to holding spherical drops. However, it is hard to generate an imaging code that sizes automatically the area of the drop in each picture since the wires go through the center of the
3
drop making it hard to distinguish them from the drop in an automatic way. Using just one
wire makes it easy to generate an image processing code that automatically sizes the area of
the drop in every picture since the drop hangs from the wire and it can be assumed that the
wire is on the edge of the drop. However, the drop for this case is neither axisymmetric nor
spherical. The big advantage of using drops suspended on wires is that the residence time
is infinite so we can record the full evaporation of the drop, even if measurements become
uncertain when the size of the drop is in the order of the size of the wire.
The second technique is the traveling drop approach. The drop can travel up or down the
wind tunnel and it can have different initial sizes. The traveling drop method is advantageous
since the drop can deform without any wire blocking its natural displacement. So it is a
test that can theoretically show better the effect of turbulence. Also, if the drop reaches
terminal velocity in the tunnel, and if the final drop velocity is arranged to be close to the
same velocity of the mean air flow (small terminal velocity implies a small drop), the drop
will be then in fluctuating flow turbulence only and this will be the only effect recorded since
almost no relative velocity will be affecting the evaporation rate. That is, it will allow a
direct comparison with suspended droplet evaporation in still air. However, this technique
is more complicated since the drops have some dispersion in their path and some problems
appear related with depth of focus and sizing drops. Also, the residence time is limited to
the length of the tunnel and the speed at which the drop is traveling. All the pros and cons
of these methods and the way each method is used will be discussed more deeply in the
following chapters.
4
Chapter 3
Experimental Apparatus
3.1
3.1.1
Set up Components
Introduction
In order for the drop to be able to deform we want to generate drops that will be as large as
possible so that surface tension will not keep them spherical. Also, we want the Kolmogorov
scale to be smaller than the drop. If it is larger than the droplet, then the droplet may not be
aware of the small scale turbulence since it will simply be carried by the eddies in the case of
a traveling drop. Generating small Kolmogorov scale turbulence is a mechanical problem [4]
since high velocities and special grids are needed. So for both reasons we would want a fairly
large droplet (order of 1 mm). However dealing with a large drop size has some problems
too. For the suspended droplet case, a large drop will not have enough surface tension force
to hold it to the suspending wire and it will fall. For a large traveling droplet the residence
time is going to be short, and as we will see later we would need a very long wind tunnel
in order to see any evaporation that we can measure and distinguish from errors. Appendix
5
A provides a brief explanation of some simulations regarding the surface tension of a drop.
This will help to understand the facility for suspended drops to hold to wires, as well the
resistance to deformation created by eddies.
Figure 3.1 shows a basic sketch of the elements that integrate the experimental set up and
that will be explained in detail in this chapter. The sketch is shown for the particular case
of large traveling drops (2-3 mm in diameter) moving in the gravity direction. However for
a case of suspended drops the set up will be the same, with the difference that the camera
would be focused on the wires, and we would not be using the syringe pump system as a
droplet generator. More details of each case are in the following sections.
3.1.2
Wind Tunnel
Several wind tunnel options are available at UCI, but the planned configuration is an invertible wind tunnel that can be rotated 180 degrees in order to obtain upward or downward
flow conditions. It is vertically oriented, and driven by house compressed air. The tunnel
cross section is 0.15m ∗ 0.15m and the length is 2m. A substantial flow rate is available
(750scf m = 0.354m3 /s) which can provide the desired flow velocity. The converging section
has a system similar to a shower head that distributes the compressed air coming from the
tank all over the beginning of the converging nozzle of the tunnel. For the remaining parts,
the design is similar to any existing wind tunnel. The walls of the test section are transparent to be able to size the drop optically. In section 3.2 there is more information about the
recent efforts to improve the wind tunnel’s performance.
6
Evaporation rate of traveling droplets in a
Ferran Marti Duran
Derek Dunn‐Rankin, Mechanical and Aerospace Engineerin
The Henry Samueli School of Engineering, UC Ir
Sponsored by National Science Foundation and Balsells
Period
labora
General apparatus
Vertical wind tunnel:
Pressurized air at 0‐
10 m/s. Ambient conditions.
Active grid: Generate isotropic turbulence at a 12% intensity.
Droplet generation: A syringe pump connected to a perturbation damper
provides constant flow rate to a needle inside the tunnel.
Drop release: An optical sensor measures the elapsed time between drops which is transferred to a laptop. The sensor also triggers the camera to take a picture of the traveling drop.
Sizing drop: A camera catches the drop at different positions of the tunnel. An external flash is required. Shadowgraphy method is used for the pictures. An image software is used to measure the drop sizes.
Data system: A pulse generator is used to trigger the camera and the flash with the signal from the optical sensor (drop release). Figure 3.1: This figure shows the vertical wind tunnel, the active grid to generate isotropic
turbulence, the droplet generation system, and the sizing technology. Refer to this image
while reading this chapter.
7
The mass weighted the deriva
and apply
out the no
the follow
Remedy
A flexible valve will pump to p
Damping will be a f
rate and reservoir.
Figure 3.2: Converging nozzle and beginning of the test section of the wind tunnel. We can
see too the active grid detailed in section 3.1.3.
8
3.1.3
Active Grid
In order to create test flows with Taylor’s Reynolds number, Reλ , in the range of 100 or
higher with isotropic turbulence, active rotating vane grids based on the design used by [5]
are fabricated. At the beginning of the test section, there is a grid made of 16 perpendicular rods which are equipped with triangular vanes and randomly rotated using computer
controlled stepper motors. The mesh or separation between rods is M = 0.03m. The wind
tunnel conditions were measured using a hot-wire anemometer (Section 5.1). The active grid
turbulence generator is shown at the top of Figure 3.3. Another type of grid that may be
used is a passive grid. The passive grid is composed of a biplanar mesh of rods secured by
a frame. In both cases the grid produces large turbulent eddies which decompose to smaller
eddies in the downstream direction. During this process the turbulence energy is dissipating. The eddy size continues to reduce until viscous and shear flow takes over converting
the turbulence energy into heat. The advantage of using an active grid over a passive grid
is that it will produce higher turbulence intensity. The main reason is that the root mean
square velocity developing downstream of a passive grid is lower than with an active grid [6].
Estimation of the Kolmogorov Turbulence Scale:
Following the designs of [5] to design the active grid, a level of turbulence can be expected
knowing the mean flow velocity. With x the distance downstream from the grid, for a value
of x/M = 20, it is known from experience [7] that with this grid we can expect a Reλ of
150 − 275, and a turbulence intensity of around 12%. Defining the following relations:
Turbulence intensity:
1/2
hu2 i
u =
U
0
(3.1)
9
Figure 3.3: Beginning of the test section of the wind tunnel (transparent walls). Active grid
at the top on the picture. We see here how a hot wire is characterizing the flow in the tunnel.
10
Mean velocity:
Taylor Reynolds number:
Turbulence intensity:
Taylor length scale:
Dissipation rate of turbulent kinetic energy:
Kolmogorov length scale:
U = 5 − 15m/s
Reλ = 150 − 275
u0 = 12%
λ0 = 0.033 − 0.022m
ε0 = 5.4 − 145m2 /s3
η0 = (155 − 70)10−6 m
Table 3.1: Conditions at x/M = 20, where M = 0.03m is the active grid spacing.
Taylor Reynolds number:
1/2 λ0
Reλ = hu2 i
(3.2)
ν
Dissipation rate of turbulent kinetic energy:
ε0 = 15ν
hu2 i
U2
(3.3)
Kolmogorov length scale:
η0 =
ν3
ε0
1/4
(3.4)
the Kolmogorov length scale can be estimated based on the previous equations assuming
the mean flow velocity, the expected values of Reλ and turbulence intensity. Table 3.1
summarizes the previous explanations and has been extracted from [4].
3.1.4
Electronic Equipment
An electronic system for taking pictures of the drops at the desired instant and position
is designed. It is useful for both cases, the suspended drop and the traveling drop. The
electronic equipment is comprised of a laser, an optical sensor, a TTL pulse generator, an
11
oscilloscope, a National Instruments Card and a laptop. The laser is pointing at the optical
sensor so that when a drop cuts the laser beam the sensor can trigger other devices. The
optical sensor is plugged into a filter and an amplifier. The amplified signal goes to the
oscilloscope to help the user visualize what is going on, and to the TTL generator. The TTL
generator has 4 more channels. One is used for a camera and another one for a flash. By
changing the delay, imaging the drop at different positions while it travels is possible. Also,
there is another channel from the TTL generator that goes to the Labview code in the laptop
through the NI card. That channel is a TTL pulse with the same time between pulses that
time between drops. We can use this to measure the time when drops are pinching off.
In the following sections there is a deeper explanation of the use of these electronic timing
devices to size the drops. Appendix C carries an explanation of the technology used for
sizing.
3.1.5
Drop Generator
Case of suspended drop:
A regular needle can be used by hand to hang the drops from the wire. The drops pinch
off from the needle when their weight and surface tension cannot hold them any more (since
weight scales a diameter cubed and surface tension forces decrease with the area, there is
always a point where the droplet will fall). By counting how many droplets from the needle
will generate the desired larger droplet that will be held by the wire, we make sure that our
drops will have a similar initial size. For the case of evaporation in still air, the initial size
of the drop is not important since the results can be non-dimensional after dividing them by
their initial size. However for a case with mean flow it will be explained in Chapter 4 how the
evaporation rate depends on the Reynolds number, and in consequence on the drop diameter.
12
For our experiments the drops that were generated using this method were on the order of
mm. If the initial size of the drop must be in the order of µm other methods should be used.
Case of traveling drop:
If we want drops of initial size in the order of mm for a traveling drop the set up to generate
those includes a needle inside the wind tunnel connected to an external syringe pump. Drops
fall either because the surface tension is not able to hold them, or in the case of external
flow because it forces them to pinch off. Some improvements for generating droplets in this
way will be discussed in Section 3.2.
Naturally, if the initial size of the drop must be smaller than the size of the drop that
automatically pinches off from the needle, other methods should be used. Such a case
occurs when we are looking for drops in the order of 100-300 microns. The method to
generate small droplets follows the approach of [8] for creating a monodisperse stream of
droplets by piezoelectric perturbation of a round liquid jet. The laboratory where the current
experiments were conducted also has extensive experience with this kind of droplet generator,
including the control of droplet size (e.g., [9]; [10]), the flow interactions of closely spaced
droplets [11], detailed diagnostics during combustion of such droplets (e.g., [12]; [13]), and
the charging and detection of droplets to modify the interdroplet spacing if needed [14]. By
changing the injection velocity, fluid properties, and disturbance frequency, it is possible to
adjust droplet size and spacing to ensure isolated behavior of the droplets in the turbulent
flow. The current experiments described in this thesis have used drops of initial size of 2-3
mm, so the method to generate traveling drops with a syringe pump has been used. Future
work will likely involve the smaller droplets and droplet generation methods.
13
3.2
Contributions to the Experimental Apparatus
Although the wind tunnel was an existing experimental apparatus, some improvements have
been made to the experimental set up. Those are detailed in the following sections.
3.2.1
Housing for the needle
A local flow shield housing is extremely useful for measuring traveling droplets generated
with the syringe pump. Those are drops around 2mm that travel down the tunnel. Since
the way of generating those drops is using a syringe pump connected to a needle, when
turbulence is present the drops start shaking and they pinch off at different sizes. To make
the measurements more consistent we have built a housing that goes around the needle and
prevents eddies from hitting the drop while it is being generated. Figure 3.4 shows a sketch
of the housing system. Figure 3.5 shows an image of the shadow of the housing and a drop
after its release.
3.2.2
Heater
A 400V heater has been installed in the air line feeding the tunnel. It is controlled so that
the user can select the air temperature. This way, we will increase the evaporation rate of
the drops and examine non-isothemal evaporation conditions.
3.2.3
Perturbation damper for the syringe pump
A full research effort has been made in order to understand syringe pumps. Their working
mechanisms and devices to improve their performance have been studied extensively as part
14
40mm 200mm
l point of both lenses Air flow
Housing Needle
Droplet
Figure 3.4: Sketch of the housing that goes around the 90 degree needle.
15
Figure 3.5: Picture of a drop after leaving the housing that covers the needle. This technique
of sizing the drops is explained in section 5.3
16and Appendix C. We can see a reflection of
the green laser used for triggering the electronic equipment explained in section 3.1.4.
of this thesis work. A majority of the syringe pumps that are in the market use a screw
as a pumping mechanism to push the liquid out of the syringe. Periodical oscillations in
the flow rate have been measured associated with small asymmetries in the screw. The
period of each oscillation in the flow rate is the time it takes for the screw of the pump to
make a full revolution. Several pumps were collected and tested, and this sample presented
perturbations of more than a 60% of the cases with standard deviation values up to 12%
of the average flow rate. A damping mechanism was studied and a conference paper [15]
and a patent disclosure were filed on this topic. The damper has been able to reduce the
perturbations down to standard deviations of 1%. Its basic mechanism is a simple springdamping system. We need a spring system that is able to deform when a perturbation is
coming, and a damping system that prevents that deformation from expanding. Figure 3.6
shows a sketch of the damper. Other dampers have been designed. We have used a needle
valve as the part that creates friction, and a flexible element as the spring component. In
Figure 3.6 the spring is provided by the compression of air in the reservoir. In other dampers,
the spring part was a flexible latex tube that could deform and absorb the perturbations.
Figure 3.7 shows a plot of the flow rate with and without the damper recorded for the same
syringe pump. It was a brand new syringe pump. The flow rate was tested by collecting the
liquid in a beaker which was at the same time on top of a scale. The derivative of the mass
in the beaker over time is the instantaneous flow rate.
3.2.4
Pitot tube and temperature transducer
The system used now to measure velocity is a portable anemometer which has a temperature
sensor integrated. It has a screen to read the values of temperature and velocity. With the
pitot tube and the temperature transducer that we just installed, we will be able to record
the temperature and flow velocity with a computer and study their variation in time to
17
On/Off Valves
Air/Water Reservoir
T connection Needle Valve
From Syringe Pump To Beaker Figure 3.6: Sketch of the damper. The spring method used here is the air compressing in
the reservoir. The on/off valves just help to fill the reservoir with an initial level of liquid.
The final destination of the damped flow rate could be a beaker, as it shows in the image,
or a needle in the experiment.
From External Pump 25
No Damper
Damper
Flow rate (ml/h)
Water Column
20 Overflowed water
15
To Beaker 10
5
0
500
1000
Time (s)
1500
2000
Figure 3.7: Flow rate with and without the damper for a brand new syringe pump.
18
prevent possible oscillations.
There are also other improvements in the set up that have not been installed yet, but they
have been ordered and will be installed in the near future. A flow regulator with a sonic
disc in order to control the flow rate linearly with the pressure change has been ordered and
should be installed since some perturbations have been measured in the flow rate. Those
come from the pressure tank that provides air to the wind tunnel.
After having explained the different elements that compose the experiment, the following
chapters will be dedicated to analyze the results obtained from it, as well as to simulate
droplet evaporation predictions based on classic theory of evaporating droplets.
19
Chapter 4
Theoretical Simulations
The theory of evaporation from spherical droplets with no mean flow has been studied for
many years. Also, there are some experimental and phenomenological models that relate
the evaporation rate of a drop in still air with that of a drop in a mean flow. The following
equations describe the calculation of evaporation of droplets either during free fall or in still
air under variations in the temperature or velocity of the air.
Several books [16] [17] [18] describe the following equations. From the mass transfer equations
for an evaporating droplet, the transfer number BY is:
BY =
YF ∞ − YF W
YF W − YF R
(4.1)
Where Y is the mass fraction, W is at the surface of the drop, F is Fuel or the liquid that
the drop is composed of, ∞ is far from the drop and R means reservoir, referring to the mass
fraction of the fuel inside the drop. In our case, YF ∞ = 0 and YF R = 1. So the mass transfer
20
number is then:
BY =
YF W
1 − YF W
(4.2)
Using the transfer number BY , the evaporation rate of a static droplet is expressed as:
ṁ = 4πrs ρg Dln(1 + BY )
(4.3)
Where rs is the radius of the drop, ρg is the density of the gas mixture, and D is the mass
diffusivity of the gas mixture. From the energy transfer equations for an evaporating droplet,
the thermally associated transfer number Bq is:
Bq =
Cpg (T∞ − TW )
L + Cl (TW − TR )
(4.4)
In equation 4.4 Cpg is the heat capacity at constant pressure of the mixture gas, T∞ is the
air temperature, TW is the temperature at the surface of the drop, L is the latent heat of
vaporization of the fuel, Cl is the heat capacity of the liquid fuel and TR is the temperature
of the drop or reservoir. Since L is two orders of magnitude larger than Cl (TW − TR ), we
can simplify equation 4.4 and obtain:
Bq =
Cpg (T∞ − TW )
L
(4.5)
21
Using the transfer number Bq , the evaporation rate of a static droplet is expressed as:
ṁ =
4πrs kg
ln(1 + Bq ) = 4π rs ρg α ln(1 + Bq )
Cpg
(4.6)
kg is the thermal conductivity of the gas mixture, and α is the thermal diffusivity of the gas
mixture.
In all the evaporating liquids used in the experiments the Lewis number is close to 1. By
setting Le = α/D = 1, and by equaling the evaporation rate from equations 4.3 and 4.6, we
find that BY = Bq .
By then assuming that the liquid and vapor phases are in equilibrium at the evaporating
fuel surface, for a binary gas mixture of species 1 and 2, the sum of partial densities is equal
to the total density.
ρ1 + ρ2 = ρ
(4.7)
Assuming that air and fuel vapor behave as ideal gases, ρi = Pi Mi /R T , i = 1, 2. Pi is the
partial pressure of the ith species and Mi is its molecular weight. Substituting this result
22
into equation 4.7,
PM
P 1 M1 P 2 M2
+
=
RT
RT
RT
(4.8)
where P and M respectively are the total mixture pressure and the molecular weight. The
mass fraction of species 1 is defined as the ratio of the partial density of species 1 to the
mixture density.
Y1 =
ρ1
ρ1
=
=
ρ
ρ1 + ρ2
1
(4.9)
P 2 M2
1+
P 1 M1
By Dalton’s law, P = P1 + P2 . Hence
1
Y1 =
M2
1+
M1
(4.10)
(P/P1 − 1)
Letting species 1 be the fuel and species 2 be the air, since P1 = Pvapor.f uel
1
YF W =
Mair
1+
MF
(4.11)
(P/Pvapor.f uel − 1)
23
0.25
By
Bq
Bq, By
0.2
0.15
0.1
B
0.05
Ts
0
200
220
240
260
T wall (K)
280
300
Figure 4.1: Heptane droplet at 300K and ambient pressure. BY and Bq versus TW . When
the Lewis number is 1, BY = Bq . B and Ts are the values from the intersection from both
curves which will be used for further calculations as the Spalding Number and the Surface
Temperature of the drop respectively.
Plugging equation 4.11 into 4.2 we get a new expression for BY which depends on the ambient pressure P , the fuel, and the ambient temperature since Pvapor.f uel is a function of the
ambient temperature and the surface temperature of the drop TW .
Equation 4.5 also depends on the surface temperature of the drop and the ambient temperature since L and Cpg depend on those temperatures. We have then, two equations [4.11
with 4.2] and [4.5] and two unknowns, TW and B. An iterative process is required to find
the solution. Figure 4.1 shows a plot of BY and Bq , and shows how to find the B for Lewis
number 1, and the associated Ts (surface temperature of the drop that makes BY = Bq ).
The process to find Pvapor.f uel , Cpg , L, and kg is explained in Appendix B; they are also
included in the iteration loop to find Ts and B.
24
To simulate the evaporation in time of a suspended drop we can use the evaporation rate ṁ
from Equation 4.6. Once we know B and Ts , using Ts we can find the coefficients Cpg , kg ,
and we can replace B for the term Bq in Equation 4.6. ∆t is the step in time, and mi the
mass of the drop at time i,
mi+1 = mi − ṁ ∆t
(4.12)
with ρl the density of the liquid fuel, and di the diameter of the spherical drop at time i so
that
di+1
6
= di − ṁ ∆t
πρl
3
1/3
(4.13)
For the case of still air, plotting the diameter squared with time appears as a linear relation.
Moreover, it is a classical result often referred as the d-squared law:
d2 = d0 2 − K t
(4.14)
or in a non-dimensional form:
t
d2
2 = 1−K
d0
d0 2
(4.15)
25
1
280 K
300 K
320 K
340 K
d2/d20
0.8
0.6
0.4
0.2
0
0
20
40
60
80
2
(s/mm )
time/d2
0
100
120
Figure 4.2: Equation 4.15 for heptane droplets with changes in the ambient temperature at
sea level pressure. Static drop with still air.
where d0 is the initial diameter of the drop, and K is the evaporation constant:
K=
8 kg
ln(B + 1)
ρl Cpg
(4.16)
For a case where the evaporation rate could change, instead of using ṁ in equation 4.13, we
would use ṁi . This would happen for a case where the drop is not in still air but is exposed
to an air flow.
For a case where the drop is exposed to a mean and constant flow, empirical correlations
that correct the evaporation rate have been found. The empirical correlation of Ranz and
Marshall (1952), is one well-known example (Equation 4.17). In their expression, Re is the
Reynolds number based on the radius of the drop and the relative velocity of the drop with
26
1
0 m/s
0.1 m/s
1 m/s
10 m/s
d (mm)
0.8
0.6
0.4
0.2
0
0
10
20
30
time (s)
40
50
60
Figure 4.3: d for heptane droplets of 1 mm of initial diameter at sea level pressure and 300
K. Static drop with different velocities of mean flow.
the air, and P r is the Prandlt number. ṁss stands for the evaporation rate for a spherically
symmetrical drop in still air as calculated from Equation 4.6. ṁ is the new evaporation
rate which considers mean flow. Notice that we have to consider ṁi from Equation 4.17
to use Equation 4.13 since the evaporation rate will change as the drop evaporates. Figure
4.3 shows the evaporation of a 1 mm heptane drop for different velocities, while Figure 4.4
shows the change in temperature for a 10m/s flow. Notice how for other cases than a static
drop with still air we plot d rather than d2 /d20 since the evaporation will depend on the Re
and for each diameter, the droplet diameter change will not follow a d2 − law.
ṁ = ṁss 1 + 0.3P r1/3 (2Re)1/2
(4.17)
If we want to take into account a more realistic free fall case where accelerations play a large
27
1
280 K
300 K
320 K
340 K
d (mm)
0.8
0.6
0.4
0.2
0
0
5
10
time (s)
15
20
Figure 4.4: d for heptane droplets of 1 mm of initial diameter at sea level pressure and 10m/s
flow. Static drop.
role, the following model, which includes drag and gravity, should be used. For the y axis
positive against gravity, we define the y direction with the following equations:
X
1
F~ = m ~a = −W − D = −m g − ρg vrel 2 SF CD sign(vrel )
2
(4.18)
where m is the mass of the drop, g = 9.8m/s2 , ρg is the gas density of the mixture gas, SF
is the front surface of the drop which is the area of a circle, CD is the drag coefficient and
vrel = vground − vair , where vground is the absolute velocity of the drop with respect to the
ground, and vair the flow velocity at which the wind tunnel is set.
Equation 4.18 is used to extract the acceleration ~a that will be used to find the velocity of
the drop at the next ∆t. It is an equation inside the loop that calculates the evaporation of
28
the drop with time. The important coefficients refresh at each iteration in time, changing
everything that depends on the size of the drop as its size keeps on reducing. The calculations
of ρg are in Appendix B. To calculate CD as a function of the Re the equation of Renksizbulut
and Yuen (1983) in equation 4.19 has been used for Re smaller than 30. For Re of 30 and
above, the expressions from Chiang et al. (1992) have been used (Equation 4.20). However,
the two relations of the CD with Re must be treated carefully in order to avoid non-smooth
behavior in the transition from one equation to the other (Re = 30).
CD =
24
Re(1 + B)
CD = (1 + B)−0.27
(4.19)
24.432
Re0.721
(4.20)
Figures 4.5, 4.6, and 4.7 show cases for a freely falling drop with the equations explained
here. All are for a 1mm heptane drop and 300 K at sea level pressure. Two different cases
are simulated, one for a freely falling drop with no initial velocity and no air flow around it,
and a second case with 0 initial velocity but −15m/s (velocity is negative since it moves in
the direction of gravity) flow around the drop. To understand those figures, velground is the
absolute velocity of the drop with respect to the ground, and velrel is the relative velocity,
which is velrel = velground − velair f low .
29
1
no air flow
−15 m/s air flow
d (mm)
0.8
0.6
0.4
0.2
0
0
5
10
time (s)
15
20
Figure 4.5: d for heptane droplets of 1 mm of initial diameter at sea level pressure and
300K. The drop is initially at 0 ground velocity. Two cases are simulated, for no air
flow around the drop, and for a constant flow going downward at 15m/s. Notice how the
evaporation is almost identical for both cases. This is because, except at the very beginning,
the evaporation of both drops occurs at the same relative velocity with the air flow since
both achieve terminal velocity.
30
1
no air flow
−15 m/s air flow
d (mm)
0.8
0.6
0.4
0.2
0
−300
−250
−200
−150
x (m)
−100
−50
0
Figure 4.6: d vs. position x (initial position (x = 0)) for heptane droplets of 1 mm of initial
diameter at sea level pressure and 300K. The drop is initially at 0 ground velocity. Two
cases are simulated, for no air flow around the drop, and for a constant flow going downward
at 15m/s. This plot is interesting, since it provides information about the length of the wind
tunnel that is required to measure a certain evaporation depending on the velocity of the air
flow.
31
15
velground=velrel (no air flow)
10
vel
ground
velrel (air flow −15 m/s)
5
vel (m/s)
(air flow −15m/s)
0
−5
−10
−15
−20
0
5
10
time (s)
15
20
Figure 4.7: Ground and relative velocities for the two cases plotted with no air, and −15m/s
air flow at 300K. Notice how for the case of no air flow, the ground velocity and the relative
velocity are the same. We can see how, after a certain time from the beginning, both drops
reach the same terminal relative velocity to the flow, so they evaporate at the same rate.
Notice too how the terminal velocity keeps changing since the drop keeps on reducing in size.
32
Chapter 5
Results and Discussion
5.1
Hot Wire Characterization of the Turbulence in
the Wind Tunnel
Upon the completion of the manufacturing of the vertical wind tunnel, a characterization
process was performed by the laboratory led by Professor John Larue to measure the time
dependent flow field velocities as a function of axial position. The characterization was done
using a hot wire anemometry system. The velocity and the velocity derivatives were recorded
at different locations along the centerline of the wind tunnel. The analysis of the collected
data produced several plots that were used to evaluate the degree of isotropy of turbulence.
Based on these measurements several modifications were proposed if isotropic turbulence
needs to be achieved. Independent of any changes, however, these results provide a measure
of the turbulence at different locations in the tunnel.
33
5.1.1
Isotropic Turbulence
Turbulence is introduced to the flow via an active grid consisting of an array of rotating
vanes, as has been explained in section 3.1.3. Isotropic turbulence is an ideal special case
where the interaction between eddies is uniform in all directions. This state can be identified
using three indicators: velocity skewness, velocity derivative skewness, and the behavior of
the energy dissipation. Skewness is a term used to describe the symmetry of the distribution
of the frequency plot around the mean. If a plot is skewed, it is not symmetric about the
mean or it is shifted away from the mean. For ideal isotropic turbulence, the skewness
of the velocity frequency plot must be equal to zero. The second indicator of isotropic
turbulence is the skewness of the velocity derivative. For isotropic turbulence, it has been
determined experimentally that the value of the skewness of the spatial velocity derivative
must approach -0.5 as seen in [7]. The last indicator of isotropic turbulence is the turbulent
energy dissipation. According to [7], energy dissipation for isotropic turbulence may be
calculated using the appropriate form of the turbulent kinetic energy equation for isotropic
turbulence and may also be computed using the velocity time derivative. In the first case
the equation to use is:
εu ∗ = −
1 dq 2
2 dt
(5.1)
where q 2 = u2 + v 2 + w2 where u, v, w are the three components of the perturbation velocity
vector of magnitude q. This equation can be simplified using Taylor’s hypothesis and the
assumption of isotropy and homogeneity in the decay power-law region (i.e. u2 = v 2 = w2 )
34
to:
εu ∗ = −U
3 du2
2 dx
(5.2)
where U is the mean velocity, and x is the downstream direction.
In the second case an independent estimate of the dissipation is obtained from the measured
time derivative of the downstream velocity, Taylors hypothesis, and the assumption of local
isotropy. The corresponding expression is:
15 ν
εu = − 2
U
∂u
∂t
2
(5.3)
where ν is the kinematic viscosity.
While both expressions (Equations 5.2 and 5.3) are expected to be in error close to the grid
where the flow is not isotropic nor locally isotropic, further downstream where the flow should
have reached near isotropy, homogeneity and local isotropy, the two values must match to
indicate isotropic turbulence.
5.1.2
Data Collection
Data was collected using a Labview program and a hot wire system. The Labview program
and equipment are capable of measuring three channels simultaneously. The hot wire is cal35
ibrated using a Pitot tube before and after data collection at five velocities (1, 2, 3, 4, 5m/s).
The calibration point was two inches away from the active grid. Data was collected at ten
different locations along the centerline of the wind tunnel starting at the two-inch location.
The points were six-inches apart, which is also one duct diameter. At a sampling frequency
of 12kHz, the wind tunnel was characterized twice. Data was collected for a period of 40
seconds the first time but it was suspected that the flow did not reach stationarity. A stationarity test was run and a data collection period of 4.2 minutes was deemed sufficient.
Data was collected a second time for 4.2 minutes at each location.
5.1.3
Results
The collected data included the measurement of the velocity and the velocity derivative.
This data was postprocessed resulting in several plots that define the characteristics of the
flow in the wind tunnel. The computed data was compared to published data from [7] that
used a passive grid and from [5] that used an active grid. Both seem to have achieved an
isotropic condition. The first plot produced was the mean velocity vs. axial location down
the wind tunnel centerline 5.1. The distance x is normalized by the mesh length M. The
velocity deviates 7.5% and 2.3% from the mean for the 40 second run and the 4.2 minute
run respectively. Wind tunnels typically have a 2% maximum velocity variation from the
mean for good performance. The velocity fluctuation may have been high for the 40 second
run because the period was too short to capture the behavior of the flow. High-pressure air
used to produce the airflow may have a variable pressure due to the compressor cycling, a
behavior that we have found disruptive in later tests.
The skewness of the velocity distribution was also calculated and plotted vs. the normalized
distance from the active grid and is shown in Figure 5.2. This plot shows that the skewness
of the velocity distribution does not approach zero until the last station where the turbulence
36
2.5
40 s run
4.2 min run
U mean (m/s)
2.4
2.3
2.2
2.1
2
1.9
10
20
30
40
50
x/M
60
70
80
90
Figure 5.1: Mean velocity in the wind tunnel with respect to the downstream position x from
the grid. M is the active grid spacing. Two tests, one of 40s and another one of 4.2min.
level would have dropped appreciably. There is wide scatter in the skewness values across
the test section.
Both runs do match the data from [5] fairly well. The data published by [7] approaches the
isotropic condition closer than does that in [5].
The second indicator of isotropic turbulence is the skewness of the velocity derivative. Figure
5.3 shows the computed values from both runs and the published values from [7]. The
published data follows the trend of approaching -0.5 at the isotropic condition. The skewness
of the velocity derivative for our data behaved in an opposite manner. The last indicator is
the energy dissipation of the turbulent flow. The turbulent energy dissipation is calculated
using the velocity and velocity derivative data as explained above. The two values are
compared in Figures 5.4 and 5.5 for the 40 sec and the 4.2 min data, respectively. The ratio
of these two values is also plotted and it is clear that these ratios are not approaching 1,
indicating that the isotropic condition was not met based on the energy dissipation.
37
0.6
[8]
[6]
Our test 4.2 min
Our test 40 s
Velocity Skewness
0.5
0.4
0.3
0.2
0.1
0
−0.1
10
20
30
40
50
x/M
60
70
80
90
Skewness of the Velocity Derivate
Figure 5.2: Mean velocity in the wind tunnel with respect to the downstream position x from
the grid. M is the active grid spacing. Two tests, one of 40s and another one of 4.2min.
0
[8]. Reλ 28.37
−0.1
[8]. Reλ 41.6
−0.2
Our test 4.2 min
Our test 40 s
−0.3
−0.4
−0.5
−0.6
−0.7
10
20
30
40
50
x/M
60
70
80
90
Figure 5.3: Mean velocity in the wind tunnel with respect to the downstream position x from
the grid. M is the active grid spacing. Two tests, one of 40s and another one of 4.2min.
38
Energy Dissipation (m2/s3)
10
Dissipation Isotropy
Dissipation Decay
Isotropy/Decay
8
6
4
2
0
10
20
30
40
50
x/M
60
70
80
90
Figure 5.4: Energy dissipation for the case of 40s.
Energy Dissipation (m2/s3)
12
Dissipation Isotropy
Dissipation Decay
Isotropy/Decay
10
8
6
4
2
0
10
20
30
40
50
x/M
60
70
Figure 5.5: Energy dissipation for the case of 4.2min.
39
80
90
Kolmogorov scale (µm)
300
40 s test
4.2 min test
250
200
150
100
10
20
30
40
50
x/M
60
70
80
90
Figure 5.6: Kolmogorov scale of turbulence for both of our tests along the wind tunnel.
We can obtain then the Kolmogorov scale using equation 3.4 with the turbulent energy
dissipation from equation 5.3. Figure 5.6 shows a plot of the Kolmogorov scale in µm for
the test of 40s and 4.2min along the wind tunnel.
5.1.4
Discussion
The wind tunnel characterization results show that the fully developed isotropic condition
was not achieved in the test section of the wind tunnel. After careful consideration of the
issue, it was suggested to place a passive grid after the active grid to try to improve the
level of isotropy. It is also suggested to use higher mean velocity (3 − 4 m/s) to avoid
clipping of the velocity distribution on the low side and to calibrate the hot wire from the
absolute minimum velocity up to 9m/s to capture the tails of the velocity distribution more
accurately. These changes will be made in future work. For the time being, however, the
study will assume that the turbulence condition is known (even if it is not isotropic) and we
will continue to examine the effect of turbulence on droplet evaporation.
40
5.2
Theoretical and Experimental Results for Drops
Suspended on Wires
It was interesting to test a case where the drop is not traveling, since it is easier to size it, as
there are no problems of depth of focus or exposure time due to speed of the drop generating
blur. Also, there are no problems either about residence time since the drop is always in the
same place and we can measure the entire evaporation process. It was also interesting to
set the wind tunnel at a constant speed and do test the effect of each velocity on the drop
evaporation rate. However, on the other hand there may be some interference in the way of
holding the drops that could affect the results for evaporation.
5.2.1
Holding the drops
In the literature we can find different techniques that people have used in the past to hold
drops. Basically, they consist in using a thin wire and making the drop stick to it with
surface tension. We have tried taking measurements in two different ways. The first one,
was using a horizontal wire across the wind tunnel and hanging drops from it. This method
worked well since the drop was completely out of the wire surface and automated software
could be used to process the area of the drop. The problem with this method is that the
drop is not close to being spherical nor axisymmetric. That could change the evaporation
rate, and in fact the results showed that it did.
The other method used, is a crossed wire technique. We have put two perpendicular crossing
wires in a horizontal plane inside the test section of the tunnel. The fact that they are crossing
generates a larger and more homogeneous surface for the drop to hold itself. This way, the
shape is more spherical and axisymmetric. The problem in this case is sizing the drops since
the wires are not any more at the edge of the drop (as they were in the first case) but they
41
Figure 5.7: Droplet suspended on two crossing wires. Heptane. Still air at 296K and sea
level pressure. Direct light imaging. 3mm drop.
go through the drop. The method used has been manual post processing to avoid imaging
ambiguities.
In both cases however the wire generates interference since it is blocking part of the oncoming air, preventing the drop from free movement and deformation, and changing the inner
circulation inside the liquid drop. Figure 5.7 shows a picture of the two wire configuration
method. The picture has been taken with the method explained in Appendix C.
5.2.2
Sizing methods
For the case of stationary drops, two methods have been used for sizing, both providing
similar results. There is no possible blur since the drops are stationary so the only possible
blur could be created by a long exposure time from the camera that catches different sizes
42
Figure 5.8: Droplet suspended on two crossing wires. Heptane. Still air at 296K and sea
level pressure. Shadow imaging. 2mm drop.
of the drop due to its evaporation. However, an easy calculation identifies the maximum
exposure time that we want to avoid. The two methods used are one with direct light, and
another one with back light. These methods involve using a SLR camera with its own flash
(Figure 5.7) or using the same camera but placing a flash looking to the camera to catch the
shadow of the drop (Figure 5.8). There is a deeper explanation of the methodology used for
sizing the drops in Appendix C.
5.2.3
Results and discussion
Case of still air
43
8
6
2
d (mm )
7
2
5
4
3
2
0
10
20
30
40
time (s)
50
60
70
80
Figure 5.9: D2 for heptane drops at room temperature. One wire configuration; still air.
Each marker shows a different experimental case.
The experiment has been done with heptane drops at a distance of 2 feet downstream of
the active grid. The results for the case with one wire holding the drops show consistency
in the evaporation rate for the 5 tests done. The results are plotted in Figure 5.9 . The D2
result has a form similar to linear, as would be expected from the theory. The plots are in
dimensional form to make it easier to visualize all of them.
The evaporation constant that can be extracted from the slope of the D2 curve for droplets
in still air is consistent for the cases tested. It varies from 0.0405mm2 /s up to 0.0587mm2 /s
for the one wire supported heptane droplets at room temperature. Its average is 0.049mm2 /s
and the standard deviation 0.00728mm2 /s, which is 14.86% of the average value.
Since there was a possibility that the evaporation rate was significantly different from the
one a spherical drop should have, the cross wire technique has been used. Figure 5.10 shows
the evaporation of heptane droplets in still air at room temperature using the cross wire
method. We measure temperature variations between 22.5C and 24C.
44
7
2
5
2
d (mm )
6
4
3
2
0
10
20
30
40
time (s)
50
60
70
80
Figure 5.10: D2 for heptane drops at room temperature. Two wire configuration. Still air.
Each marker shows a different experimental case.
The evaporation constant that can be calculated from the slope of the D2 law from Figure
5.10 changes from 0.0216mm2 /s to 0.0271mm2 /s. Its average is 0.0239mm2 /s and the
standard deviation 0.00223mm2 /s, which is 9.3% of the average value.
The simulations for a drop of 1 mm initial size is a linear plot for the D2 law. The evaporation constant, does not depend on the size of the drop for still air. At 300K, for heptane
drops we simulate an evaporation constant of 0.0172mm2 /s. There is a big difference in
the evaporation constant using one wire or two. However, using two wires we obtain an
evaporation constant closer to the simulated value than when using one wire. This seems
mostly due to the fact that for the two wire case the drop is closer to being a sphere, which
is the simulated case.
Case of constant air flow but no turbulence
Using the same wire configurations and positions as for the case of still air, the measurements
45
3
d (mm)
2.5
2
1.5
0
5
10
15
20
time (s)
25
30
35
40
Figure 5.11: Air flow at 2.1 − 2.3m/s. Active grid off. Room temperature 22.5C - 24C. Heptane. One wire configuration. Each marker shows a different experimental case. Simulation
is blue solid line (initial diameter of 2.7mm).
of evaporation have been taken in a steady mean flow. Figure 5.11 shows the comparison of
the evaporation for the one wire configuration and the simulation. Now, the plot is not D2
with time but D since the D2 law is just valid for still air evaporation droplets. We can see
how the curves of the simulation and the experimental values do not match for the one wire
configuration case.
Figure 5.12 shows a plot of the evaporation of droplets for the cross wire method. Again,
the markers are experimental values, and the solid blue line is the simulation for a 1.7mm
heptane drop at 300K and 2.1m/s air flow. The simulation for this case mathes better the
experimental results than for the one wire case, again highlighting the importance of the
spherical geometry.
Case of constant air flow with turbulence
46
2
d (mm)
1.5
1
0.5
0
5
10
15
20
time (s)
25
30
35
40
Figure 5.12: Air flow at 2.1 − 2.3m/s. Active grid off. Room temperature 22.5C - 24C. Heptane. Two wire configuration. Each marker shows a different experimental case. Simulation
is blue solid line (initial diameter of 1.7mm).
For this case the active grid is energized. There are no simulations since there are no models
to simulate the effect of turbulence. This is one of the reasons why this research is being done.
However, the simulation for air with no turbulence has been plotted with the experimental
data for turbulence-influenced evaporation to see if there was a noticeable difference between
the cases. Figure 5.13 shows the values of evaporation for heptane droplets at room pressure
and temperature for the one wire configuration. The markers are the experimental values,
and the solid blue line is the simulation for air without turbulence.
Figure 5.14 is the plot of the case with crossed wires. We see, as in the previous cases, that
the cross wire technique produces experimental results closer to the simulations. In this case
we see more dispersion in the experimental results than for the case of no turbulence.
The fact that the simulations with no turbulence (or the experimental data with no turbulence) does not change significantly with respect to the experimental values with turbulence
suggests the following:
47
2.8
2.6
d (mm)
2.4
2.2
2
1.8
1.6
1.4
0
5
10
15
time (s)
20
25
30
Figure 5.13: Air flow at 2.1 − 2.3m/s. Active grid on. Room temperature 22.5C - 24C. Heptane. One wire configuration. Each marker shows a different experimental case. Simulation
is blue solid line (initial diameter of 2.5mm).
2
1.8
d (mm)
1.6
1.4
1.2
1
0.8
0
5
10
time (s)
15
20
25
Figure 5.14: Air flow at 2.1 − 2.3m/s. Active grid on. Room temperature 22.5C - 24C. Heptane. Two wire configuration. Each marker shows a different experimental case. Simulation
is blue solid line (initial diameter of 1.8mm).
48
• The tunnel with the active grid off still generates some turbulence since the blades are
set parallel to the flow for a no turbulence case but may generate too much disturbance.
A hot wire characterization should be done in the tunnel when the grid is off.
• The wires generate too much interference. The drop cannot deform in a free way and
that means that the turbulence does not generate a significant effect. A traveling drop
could fix this problem.
• The speed is too high in order to see a difference from a case with turbulence or a case
with no turbulence. A free fall case is suggested where the relative speed between the
drop velocity and the air velocity is small to compare then ideally just the effect of
turbulence with the case of still air.
For all of these reasons, additional research is needed for the case of a traveling drop.
5.3
Theoretical and Experimental Suggestions for Traveling Drops
For the reasons exposed in the previous section, there is a special interest in using a traveling
drop method. Two different cases have been studied. The first one, is the case of a drop
traveling in the direction of gravity, and the vertical wind tunnel moving the air in the same
direction. Some experimental and computational results will be exposed here. The second
method, is generating drops that travel against gravity with the wind tunnel upside down
so the air travels up too. For this part, computational results will be explained to describe
the future work that should be done.
49
5.3.1
Drops and air flow moving in the gravity direction
For this case, the experimental set up has been with the tunnel pumping the air down and
the drops used here have been drops around 2mm in diameter. To generate the drops, as
explained in section 3.1 a syringe pump with its damper (section 3.2) has been used to pump
the fuel into the tip of a needle fixed in the center section of the wind tunnel. A housing
(section 3.2) has been used to reduce the time between drops and make it easier to size
with the electronic equipment. This case of 2mm drops going down the tunnel has been
useful to understand how the drops move and deform under conditions of turbulence with
scales smaller than the size of the drop. However, this configuration is not useful to measure
evaporation due to the long length that the tunnel would require before sufficient size change
occurred to provide an accurate measurement.
Figure 5.15 shows some images of traveling drops taken with different imaging methods. It is
intersting to see the non symmetric deformation in the drops. Different methods have been
studied for sizing the drops since they could work in a longer wind tunnel. The methods that
have been studied are attached in Appendix C. However, the following explains the sizing
method that could best be used for a case with a longer wind tunnel.
Sizing the drops after their release
Thanks to the housing used, the time between drops is similar. To size the drops after their
release from the needle (or the housing if we use one) a picture has been taken with a camera.
The camera is connected to the electronic equipment from section 3.1 so it can be triggered
with the signal generator. The room must be dark. Drops pinch off every 3 seconds with a
variability of 0.5 seconds. The exposure time of the camera is set to 1 second, and the delay
in the signal generator to 2.5 seconds. With this system, when the drop i pinches off and
50
Figure 5.15: Traveling drops. The image in the left and the one in the center are taken with
the flash behind the drop at a speed of 500ns. They show the deformation of the drops. The
pictures are shadows of the drops and the inner light circle is just a reflection. The picture
on the right is taken using a flash from the front at a speed of 1/4000s, it is blurred because
the exposure on the camera was not short enough.
cuts the laser beam (set up section 3.1) it triggers the camera to take a picture of drop i + 1.
This system works very well thanks to the combination of the housing and the electronic
equipment. The idea of triggering the image capture of drop i with the same drop does not
work since the camera has a fixed mechanical delay, so that by the time the drop cuts the
laser beam if we send a signal to the camera to take a picture of the drop with no electronic
delay, it has moved about 35 cm from the release position. Figure 3.5 showed a picture of a
drop after it left the housing. There we can see how the drops are axisymmetric even if they
are not spherical due to oscillations after they pinch off from the needle. Those oscillations
are a known effect and they damp out fairly quickly.
Sizing the drops at a middle position or at the end of the tunnel
Since the drop is traveling in a turbulent environment it will never follow the same path.
Ideally it should not deviate far from the centerline since the turbulence that we are looking
for is isotropic and modest. However, a way to size the drops even if they experience some
51
dispersion is needed. The idea of having a parallel beam of light and taking a picture of
the shadow of the drop would give us the same shadow no matter where the drop is since
the light beam is parallel. However, we have faced some problems trying to reach a suitably
parallel beam of white light. In the case of using a laser we have observed some speckle
and diffraction problems. Finally, the method used consists in fixing the camera focusing
on a point and taking pictures of many drops. A code has been written so that in the post
process the program looks for the drop, finds its boundary and detects if it is in focus. If it
is, then it measures the area of the drop. The method used to take the pictures and post
process them is by flashing on the camera so that we catch a shadow of the drop and the
rest of the image is bright. In Figure 5.15 the left and center pictures have been taken with
this method. Figure 5.16 is a sketch of the configuration of the camera and flash to catch
the shadow of the drops. Appendix C goes deeper in the methodology used for sizing the
drops, however we must have an idea of which is going to be the exposure time needed to
catch the drop with no blur. If a 2mm droplet travels 1m in free fall, it will reach a speed
around 4m/s. If we want to get less than 1% of blur in the image, the exposure time should
be 5µs or less. See equation 5.4
Exposure time =
Drop diameter ∗ P ercentage blur
Drop speed
(5.4)
The right image in Figure 5.15 is taken at 1/4000s = 250µs, however the other two images
in that figure are taken at 0.3µs. The image shows clearly the blur issue when the exposure
time is too long. The equipment used is detailed in Appendix C.
Figure 5.15 shows some drops caught while they were traveling where the turbulence is
significant enough to deform them. However, for our wind tunnel, by the time the drops
reach to the end they are no longer deformed. It is not because they are smaller since the
52
Needle pumping system
Camera
Spot‐S flash Drop Wind tunnel
Wind tunnel On focus plane for the camera
Figure 5.16: Sketch of the set up for sizing traveling drops in the order of mm.
evaporation in that time is around 2% of the initial size of the drop, but because their speed
is high enough so that they become wider and shorter due to drag. Figure 5.17 shows this
effect. Figure 5.18 is a plot that simulates the evaporation of a 2 mm drop that is traveling
in the direction of gravity with a flow of 2m/s going in the same direction.
Figure 5.18 shows that to measure a change in diameter of the 2mm drop of 25%, it has to
travel 50m; the test section of our tunnel is just 2m. This is the reason that we must use a
smaller drop that travels against gravity in our future work.
5.3.2
Drops and air flow moving against gravity direction
This part of the experiment has not yet been done but it has been studied theoretically.
Flipping the wind tunnel and having the flow going upward is the first step. Then, we need
to generate smaller drops. They should be around 200µm. Reference [19] shows a similar set
53
Figure 5.17: Free fall of a 2mm drop with zero initial velocity and flow speed of 2m/s going
in the direction of the drop. Room temperature of 300K and sea level pressure. Picture
when droplet has traveled 1.5m (end of the wind tunnel).
54
2
d (mm)
1.5
1
0.5
0
−300
−250
−200
−150
x (m)
−100
−50
0
Figure 5.18: Free fall of a 2mm drop with zero initial velocity and flow speed of 2m/s going
in the direction of the drop. Room temperature of 300K and sea level pressure.
up to the one we want to build. To generate drops of that size, a different system is required.
Section 3.1 explains a droplet generation system in order to make such small drops. These
drops would be shot upward in the wind tunnel at a speed of around 15m/s. Figures 5.19
and 5.20 show some plots of the simulations done for this configuration.
As we can see for this example, this configuration would work in our wind tunnel. The
tunnel is 2m long, so from Figure 5.19 we can see how in 2m the drop has a size of around
0.1µm which means an evaporation of 50% in diameter. This large evaporation is easy to
distinguish. If the evaporation, for instance, would be 2% it would not be possible to distinguish it from just error in the measurement. Looking at the plot from Figure 5.20 we
can see how the drop is shot up at 15m/s and how it decelerates, reaching terminal velocity
in 0.5m. This would be interesting since after this point the relative velocity of the drop
and the air would be almost zero, which makes the turbulence effect predominant. This way
we could just study the effect of turbulence with no mean flow at least for part of its lifetime.
55
0.2
d (mm)
0.15
0.1
0.05
0
0
0.5
1
1.5
x (m)
2
2.5
3
Figure 5.19: Drop diameter vs distance traveled (x is positive since now the drop moves
against gravity). 0.2mm drop shot up at 15m/s. Air flow is going up at 2m/s. 300K and
sea level pressure.
velground (m/s)
15
10
5
0
0
0.5
1
1.5
x (m)
2
2.5
3
Figure 5.20: Drop velocity vs distance traveled. 0.2mm drop shot up at 15m/s. Air flow is
going up at 2m/s. 300K and sea level pressure.
56
The only problem we could experience, is that we are interested in cases where the Kolmogorov scale is smaller than the size of the drop. The Kolmogorov scale keeps on increasing
as we have seen in Figure 5.6 (even if that plot is for the tunnel with the air going down,
and now we are talking about flipping the tunnel and flowing the air upwards). We must
make sure that there is a range where the drop is going to be larger than the Kolmogorov
scale, and at the same time it will have reached terminal velocity. Since the Kolmogorov
scale gets smaller by increasing the velocity in the tunnel, higher velocities should be tried
for the 0.2mm drop.
57
Chapter 6
Conclusion
The evaporation of droplets is a topic that has been studied for many years but there have
been no experimental studies that examine the effect of turbulence on evaporation when the
scale of the turbulence is small relative to the droplet size. At the extreme case of very large
droplets and very small turbulence scale, the results will approximate those for turbulent
evaporation of liquid pools. The conditions of the present study, however, are focused on
a situation more closely associated with combustion or spray-drying, where the droplets
are relatively small and the turbulence fairly intense. Mimicking this condition with a windtunnel requires a challenging balance between droplet residence time, droplet size, and active
grid turbulence generation. A numerical calculation based on classical droplet mass transport
equations has predicted the evaporation of free fall droplets under flow conditions where there
is no turbulent free stream. Experimental and analytical results of evaporation have been
compared showing a high correlation. The results show further that the turbulent effect may
be a small contribution to the evaporation rate when there is mean flow around the drop.
In retrospect this is not so surprising in that the mean flow reduces the boundary layer and
enhances mass transport. To capture the turbulence effect separately from any mean flow,
therefore, it is important that for future work the configuration of the experiment is changed.
58
Specifically, the traveling drops should be injected with the air flow but against gravity. The
drops also need to be small enough to evaporate sufficiently for a reasonable measurement of
size reduction during their residence time. No matter what the configuration, however, the
droplet will be experiencing an unsteady mean flow effect coupled with a turbulence effect
except for small time windows when the droplet velocity happens to match the flow velocity
(a difficult condition to extend for long times when the droplets are large since their settling
velocity is relatively high). That is, to measure the effect of isotropic turbulence by itself, we
need the drop to have a small Stokes number and to be carried by the flow. For droplets of
the desired size, gravitational setting will not be important, so the relative velocity with the
air will be close to zero. The challenge is that for droplets of this size, the Kolmogorov scale
will need to be exceedingly small, which is difficult in a short low-speed wind tunnel. With
some care, and based on our current findings, the appropriate conditions can be achieved
by injecting 100 micrometer droplets upward in air flow of 5 m/s with the active grid on in
our tunnel. This upward fired experiment is the next series of experiments planned for this
facility.
59
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[10] M. Rohani, D. Iobbi, F. Jabbari, and D. Dunn-Rankin. Controlling liquid jet breakup
with practical piezoelectric devices. Atomization and Sprays, 19:135–155, 2009.
[11] C. Connon and D. Dunn-Rankin. Flow behavior near an infinite droplet stream. Experiments in Fluids, 21:80–86, 1996.
[12] J. Zhu and D. Dunn-Rankin. Using cars to probe the temperature field of a combusting
droplet stream. Applied Optics, 30:2672–2674, 1991.
60
[13] C. Connon, C. Choi, R. Dimalanta, and D. Dunn-Rankin. Lif measurements of fuel
vapor in an acetone droplet stream. Combust. Sci. and Tech., 129:197–216, 1997.
[14] Q. Nguyen and D. Dunn-Rankin. Experiments examining drag in linear droplet packets.
Experiments in Fluids, 12:157–165, 1992.
[15] F. Marti-Duran, J. Garman, K. Samaan, and K. Kakavand. Observations of periodic
oscillatory behavior in flow rates of laboratory syringe pumps and proposed remedies.
Western States Section Technical Meeting, Riverside USA, 2011.
[16] Stephen R. Turns. An Introduction to Combustion. Concepts and Applications.
McGraw-Hill, 2nd edition, 2000.
[17] A. Murty Kanury. Introduction to Combustion Phenomena. Gordon and Breach, Science
Publishers, Inc., 1977.
[18] William A. Sirignano. Fluid Dynamics and Transport of Droplets and Sprays. Cambridge
University Press, 1999.
[19] S. Horender and M. Sommerfeld. Evaporation of nearly monosized droplets of hexane,
heptane, decane and their mixtures in hot air and an air/steam mixture. International
jounal of spary and combustion dynamics, 4:123–154, 2012.
[20] Bruce E. Poiling, John M. Prausnitz, and John P. O’Connell. The Properties of Gases
and Liquids. McGraw-Hill, 5th edition, 2001.
[21] G. Castane, P. Dunand, O. Caballina, and F. Lemoine. High-speed shadow imagery to
characterize the size and velocity of the secondary droplets produced by drop impacts
onto a heated surface. 16th Int Symp on Applications of Laser Techniques to Fluid
Mechanics. Lisbon, Portugal, 2012.
61
Appendices
A
Surface Tension
Surface tension and deformation of the surface:
From Curl and Pitzer, 1958; Pitzer, 1995 [20] the following expression for surface tension is
provided:
γ = Pc
2/3
Tc
1/3 1.86
+ 1.18w
19.05
3.75 + 0.91w
0.291 − 0.08w
2/3
(1 − Tr )11/9
(A.1)
The Pc and Tc are Critical Pressure and Temperature, w is the acentric factor and Tr = T /Tc .
All units are in bar and Kelvin, and the result for the surface tension is in dyn/cm. The
surface tension changes for each liquid with temperature but it does not depend on the size
of the drop. Figure (A.1) shows the evolution of Eq. (A.1) for n-Heptane and n-Butane.
However, from the YoungLaplace equation for a sphere, the change of pressure between the
62
Surface Tension(dyn/cm)
35
Butane
Heptane
30
25
20
15
10
200
220
240
260
Temperature (K)
280
300
Figure A.1: Surface tension variation with temperature for Butane and Heptane. Eq. (A.1)
internal pressure of the drop and the external pressure is expressed by:
∆p = γ
2
R
(A.2)
Where ∆p is the change in pressure across the surface of the drop, R is the radius and γ is
the surface tension from Eq. (A.1). An eddy that impacts the drop, will generate a change
in the external pressure of the drop generating a local change of ∆p, and then, a deformation
in the shape.
The smaller the drop, the larger the ∆p. So for a fixed eddy that impacts our drop and
changes the external pressure, the deformation in the shape will be more noticeable for a
bigger drop than for a small one. Since, from Eq.(A.2) ∆p = γ R2 , a perturbation in ∆p
across the surface of the drop due to an eddy would be proportional to
∆p0 = −γ
2 0
R
R2
(A.3)
63
dP (atm)
0.06
0.04
0.02
0
300
250
200 2
1.5
1
0.5
0
Diameter (mm)
Temperature (K)
Figure A.2: ∆p across the surface of a spherical Heptane droplet variation with the droplet
diameter and the temperature. Young-Laplace Eq.
So the variations in radius, R will be proportional to Ap0 R2 . Dividing Eq.(A.3) over (A.2),
R0
∆p0
=−
∆p
R
(A.4)
So in order to deform two drops of different size the same percentage, the change in ∆p has
to be also for both cases the same percentage. Since a small drop has a larger ∆p than a
big drop, it must have too a larger ∆p0 . This means, that small drops will keep their shape
more spherical than large drops when both face an eddy of the same intensity.
B
Gas / Liquid Properties
To find the liquid and gas properties of some species, some models have been applied. Here
there is a description of the model used for each property in the simulation of the droplet
evaporation. All the information has been extracted from [20].
64
B.1
Vapor Pressure
Depending on the fuel that we want to simulate, and the range of temperatures, different
expressions of the vapor pressure are used, with different coefficients. Those expressions
change with the temperature T :
log10 (Pvap ) = A −
B
T + C − 273.15
(B.5)
B
+ 0.43429xn + E x8 + F x12
T + C − 273.15
(B.6)
Another expression is:
log10 (Pvap ) = A −
Here, X = (T − t0 − 273.15)/Tc . The last expression, is:
ln(Pvap ) = ln(Pc ) +
Tc
(a y + b y 1.5 + c y 2.5 + d y 5 )
T
(B.7)
where y = 1−T /Tc . For equations B.5, B.6 and B.7 the parameters A, B, C, E, F, t0 , Tc , Pc , a, b, c, d
are extracted from the tables in the appendix of [20].
65
B.2
Heat Capacity - Cpg
The Heat Capacity at constant presure of the gas mixture is calculated in the following way
Cpg = Cpg.f uel YF W + Cpg.air (1 − YF W )
(B.8)
YF W from equation 4.11. Cpg.air has been found interpolating from values of heat capacity
and temperature. Cpg.f uel has been obtained from [20] in the shape of equation B.9 where
a0 a1 a2 a3 a4 are constants for each fuel, R is 8.314J/K mol and T temperature.
Cpg.f uel /R = a0 + a1 T + a2 T 2 + a3 T 3 + a4 T 4
B.3
(B.9)
Dynamic Viscosity
From [20], we can find some correlations for low pressure gas. The one used in this code is
Lucas’ correlation. First we calculate a dimensionless viscosity ξ in (µP )−1
ξ = 0.176
Tc
M 3 Pc 4
1/6
(B.10)
where Tc is critical temperature, in Kelvins, M is molar mass in g/mol, and Pc is critical
66
pressure in bars. To obtain the viscosity η in µP , the following expression is used
η ξ = [0.807Tr 0.618 − 0.357exp(−0.449Tr ) + 0.340exp(−4.058Tr ) + 0.018] FP FQ (B.11)
where ξ is defined from equation B.10, Tr = T /Tc and FP and FQ are correction factors to
account for polarity or quantum effects. FQ is used only for quantum gases like He, H2 and
D2 . So it will not be used in our calculations. To compute FP , a reduced dipole moment
must be defined. Lucas defines this quantity as
µr = 52.46
µ 2 Pc
Tc 2
(B.12)
µ is the dipole moment expressed in debyes. The FP values are found based on
0 ≤ µr < 0.022
FP = 1,
FP = 1 + 30.55(0.292 − Zc )1.72 ,
0.022 ≤ µr < 0.075
FP = 1 + 30.55(0.292 − Zc )1.72 abs(0.96 + 0.1(Tr − 0.7)),
(B.13)
0.075 ≤ µr
However, since for all the Alkanes tested the dipole moment was 0, the FP will be 1, as the
FQ was, so equation B.11 is shorter now and only equations B.10 and B.11 will be needed.
67
B.4
Thermal Conductivity
As for the viscosity, the same reference has been used. Some models estimate the thermal
conductivity, but here the one used has been the method of Chung. The thermal conductivity
λ in W/(m K) is
λ=
3.75 R ψ η
M0
(B.14)
where M 0 is molecular weight in kg/mol, η is the low pressure gas viscosity from the previous
section, and it is expressed here in N s/m2 , R = 8.314J/(mol K), and αβψ are as follows
α=
Cv 3
−
R
2
(B.15)
β = 0.7862 − 0.7109w + 1.3168w2
ψ =1+α
(B.16)
0.215 + 0.28288α − 1.061β + 0.26665Z
0.6366 + βZ + 1.061αβ
(B.17)
Z is defined as Z = 2 + 10.5 Tr 2 with Cv expressed in J/(mol K) and obtained from Cv =
Cp − R and w is the Pitzer acentric factor.
68
C
Imaging Technology
Different technology has been studied to size the drops. Finally, two different methods have
been used: one for droplets suspended on wires, and a second one for traveling drops. Here
you will find a detailed explanation of the techniques used for both cases, and other methods
that have been studied and that could be used for future experiments since they have not
been ideal for sizing our droplet.
C.1
Working technology used to size the drops
For droplets suspended on wires:
This is a simpler case since there are no depth of focus problems. The drops is always
suspended from the same point. For the one wire technique, the drop may move along the
wire due to vibrations and turbulence. However, we focus the camera on the vertical plane
that contains the wire so that the drop will be always in focus. For the two crossed wires
method, the drop will always be in the intersection of both wires, so we will not experience
a droplet movement. Due to the thickness of the wires used, the largest drops that we can
suspend are 2-3 mm diameter droplets. As seen in previous sections, the total evaporation
time of such droplets is in the order of 100 seconds. Our SLR camera has a minimum
exposure time of 250 µs. In this exposure time, the evaporation of the droplet is in the
order of 10−6 , 10−5 % of its total evaporation time. Because of this, using a digital SLR
camera is possible since no blur will be present. Also, these cameras have more resolution
than high speed cameras so it will give us more pixels inside our drops. (digital SLR camera
(Nikon D80) has 3872 ∗ 2592 pixels, Phantom high speed camera v4.3 has 800 ∗ 600 pixels).)
69
The lens used has been a 70-300 mm MACRO lens. This lens, with the Nikon camera, would
provide us with magnifications of 1.2X. This means, that the sensor size is 1.2 time larger
than the real size of the field of view. However, this ideal case of magnification has a focal
distance that is too long for us. If we try to get closer to the drop so that more pixels fit
in the image, we cannot reduce the focal distance and the image becomes blurred. To solve
this, we have used spacers or extension tubes. They are tube combinations that are mounted
between the lens and the camera, and they reduce the focusing distance and increase the
magnification. With those, we have reached magnifications of 1.5X at short focal distances
of 10 cm. Figure C.3 has an image of the extension tubes technology.
With this combination, we have been able to measure 200-300 pixels in the diameter of a 2
mm droplet, which reduces any possible error caused by uncertainty in finding the edges of
the drop in the image. The lighting system used has been the flash integrated in the SLR
camera. The configuration of the set up (camera plus flash and droplet) is shown in Figure
C.4.
No automatic post process has been used since the wires have an important interference with
the drops and it is hard to build an automatic process that detects and sizes the drops. The
software ImageJ has been manually used to post process the images. The results are the
ones shown in Section 5.2.3. For any kind of post process, ImageJ will give us the number
of pixels inside the drop, which we have to relate to length units. A calibration image is
required. Figure C.5 shows a picture of a ruler set in the plane where the drop hangs from
the wires which will be used to find the number of pixels per inch.
To know at which time the images were taken, so that an accurate description of the droplet
size in time could be achieved, we have used the triggering system described in Section 3.1.
Since the electronic equipment records the time between drops, we have manually activated
the system every 2-3 seconds, keeping track of the time between drops. Figure 5.8 is an
example of the images that have been obtained with this technique.
70
Figure C.3: Extension Tubes: System used between the lens and the camera. They can be
mounted individually, or in combinations of the three of them.
71
Drop Focus plane
Drop Camera with flash
Figure C.4: Sketch of the camera with the integrated flash and droplet position for sizing
suspended drops on wires.
72
Figure C.5: Image of the ruler set where the drops hang from the two crossed wires.
For traveling droplets:
This is a more complicated case since we experience depth of focus problems, and we need a
system with less exposure time than before to reduce the blur created by the droplet motion.
Other authors [21] have used similar techniques to solve this problem. Our efforts in this
part have been towards sizing traveling droplets in the order of mm in diameter. However,
as explained in previous sections, we cannot measure evaporation in traveling drops that
are so large in such a short wind tunnel. Recommendations on future techniques that could
be used for sizing traveling droplets with diameters in the order of 100 µm have been done
in previous sections. Our results however, show droplet deformation in a non-axisymmetric
way due to turbulence. The left and center images of Figure 5.15 show this deformation.
The drops travel at a maximum speed in the tunnel of 4 m/s. At that speed, if we want a 2
73
Focus plane
Parallel light Flash light Camera
Drop Figure C.6: Sketch of the camera, back lighting coming from the flash and droplet position
for sizing traveling drops.
mm drop to have a blur of 1 % we would need an exposure time of 5 µs. As explained before,
we cannot use the SLR camera as a time exposure system since we still would catch some
blur like in the right image of Figure 5.15. The method used consists in taking a picture in
a dark room, with a long exposure time in the camera, but with a really short flash. This
way, the only thing we will see will be what happened during that short flash. The camera
Focus plane
used still has been the SLR camera since it has more resolution, and we can still use the
tubes. The flash used has been a SPOT-S flash from Prism Science
advantages of extension
which has a flash duration of 0.5 µs. The configuration of the set up (camera, flash and
droplet) is shown in Figure C.6.
This technique generates a shadow of the droplet as the one seen in Figure 5.17. The camera
Drop with the drop release as explained in Section 3.1. Another
and the flash are both triggered
Camera with flash
problem of the traveling drops is the variable position of the drops along their path. This
causes many drops to be out of focus since the camera is set to focus on a single vertical
plane. This problem has been studied, and different ideas to solve will be explained in this
Appendix. However, the method that has given best results has been using the configura74
tion in Figure C.6 and since some drops will be out of focus, a program has been written to
automatically post process the images that are in focus only.
For each horizontal plane of the vertical wind tunnel where we want to size droplets, we will
take a large number of images triggered by the initial droplet release. Since we are zooming
in almost at the full capacity that the lenses used allow us, the field of view is small (5
droplets diameter in the horizontal width of the image; the droplet diameter is 2 mm). The
drops have a large dispesion due to turbulence, so a number of drops will not be caught
in our field of view, and some of the ones caught will be out of focus. Since we are taking
many images, we will go through them deleting the ones where there are no drops in them.
From the ones that have a drop, a code has been written so that it looks for the drop in the
picture, and analyzes the edge profile of colors. If the drop is on focus, the change in color
from the bright background (Back lighting) and the shadow of the drop should be in one
pixel. Otherwise, this change will occur gradually in some pixels. By calculating this slope
we can decide which images will be in focus. The following example shows how the code
works for a metallic sphere that is in focus, and for another one that is out of focus. This
case has been done with a solid sphere with still air and free fall to check the method. This
way, we have been able to intentionally generate images in focus and out of focus.
Example:
Figure C.7 shows an image of the sphere in focus.
Initially, the code transforms the image into black and white, so that the post process will
be easier. Then, a part of the code looks for the inner overexposed reflection of the flash
filaments inside the drop ( we can see that reflection in Figure 5.17, however the sphere used
to test the performance of the code is solid, so light cannot go thorough it and we cannot see
that reflection). That is used to find the center of the drop. Then, the code automatically
75
Figure C.7: Shadowgraphy of the metallic sphere taken with the system from Figure C.6.
draws a horizontal line across the drop or sphere in this case. Figure C.8 would be the result
of this process.
ImageJ, has a function, which is integrated in the code which plots the gray value of each
pixel along that line. Figure C.9 is this plot for the case of the sphere in focus. We can see
how around pixel number 100, is where we have a jump in the gray value.
Out of the edges of the drop / metal ball, the image is bright (gray values of around 150),
but inside the drop / metal ball, the gray value is low (close to 0). Evaluating the slope of
the curve highlighted in Figure C.10 we can decide if the drop will be in focus or not. If the
slope has a large magnitude, then it will be in focus.
Next, the code generates a threshold based on the color. Everything below or above a color
gray level is filtered out. The result of this for the same metallic sphere would be Figure C.11
where the red part of the image has been filtered out. If all the images are taken under the
76
Figure C.8: Black and white image with line to plot the gray value along that line.
200
Gray Value
150
100
50
0
0
500
1000
Distance (pixels)
1500
Figure C.9: Plot of the gray value along the yellow line of pixels from Figure C.8
77
200
Gray Value
150
100
50
0
0
500
1000
Distance (pixels)
1500
Figure C.10: Plot of the gray value along the yellow line of pixels from Figure C.8. The
slope of the gray value across the edge of the sphere is highlighted in a red bubble.
same illumination case, the threshold is valid for the same set of images and the automation
of the process is possible.
After it, the code tells us how many pixels are inside the drop / metal ball. We choose
the option of including holes (to capture the inner empty hole in the drop), and some other
basic options to just obtain the size of our drop, and not also of other small dots around the
picture. Figure C.12 shows the outer edges that the code finds as the metal ball edges. It
gives us as a result the number of pixels that are inside, and it continues post processing the
next image.
In case a drop is not in focus, when the magnitude of the slope of the gray value change
across the edge of the drop is too low, the image is not post processed any more, and the
code continues with the next one.
Figure C.13 is an out of focus image of the same sphere analyzed before.
78
Figure C.11: Threshold of the in focus metal ball.
Figure C.12: Outlines that ImageJ detects as the edges of the metal ball. The result of the
sizing process will be the number of pixels inside these outlines.
79
Figure C.13: Image of the metal ball when is out of focus. We can see the blur in the edges.
80
140
120
Gray value
100
80
60
40
20
0
0
200
400
600
800
Distance (pixels)
1000
1200
1400
Figure C.14: Plot of the gray value across the diameter of the metal ball from Figure C.13.
In this case, the gray scale plot across its surface is shown in Figure C.14. We can see already
how the slope across the sphere edge looks different than in the in focus case. The slope for
this case is -2.21 and for the in focus drop was -24.55
Other methods were tried but did not give such a good performance as the ones explained
above. Those methods, in order to show ideas that could work for other projects, are
explained in the following section.
C.2
Other studied technologies for sizing drops
The other methods used all have the idea of generating a shadow of the drop, projecting it
into a screen, and sizing its contour by taking a picture with the SLR camera or with a high
speed camera, depending on the velocity of the droplet. If a parallel beam of light is cut by
the drop, the shadow of this drop will be the same no matter its position since the beam is
parallel. Figure C.15 shows the configuration of the camera and lighting to size the drops
81
Screen Parallel light Camera
Drop Figure C.15: Sketch to size drops with parallel light. The parallel beam is coming from the
left of the image, the drop could be in any position along the beam and its shadow on the
screen will always be the same. The camera, set behind the screen and focusing on it will
take a picture of the projected shadow.
with parallel light.
Focus plane
Using white light
Parallel light Flash light Camera
First we have to generate a parallel beam of white light. A Xenon arc lamp produces a nice
cone of light with an f number of approximately 1.3. Figure C.16 explains what the f number
is.
By placing a lens with theDrop same f number and at the focal distance of the lens we want to
create a thick parallel beam. (of 2 cm diameter since we expect this dispersion in the drops
from images taken experimentally). Figure C.17 is an sketch of this idea.
We have tried different lenses. However, large lenses (around 10 cm in diameter) had a too
long focal distance to maintain the f number required by the lamp to make parallel light.
The amount of light per square foot out of these lenses was small so we could not create a
82
Focus plane
Light point source A
B
Figure C.16: The f number of a light source is defined as B/A. For a lens, large f numbers
means less light will get to them. To calculate the f number of the Xenon arc lamp, the B
and A measurements have been done at different position of the beam (vertical red, blue
and black lines) to average the results.
clear shadow. On the other hand, reducing the size of the lens, and so reducing the focal
distance, the light was more intense since the beam was narrower but because of the short
focal distance we were experiencing spherical and chromatic aberrations. In Appendix D, it
is explained what those aberrations mean and how to fix them. An example of the results
could be found in Figures C.18 and C.19. Using the system from Figure C.15, we have taken
pictures of washer at different positions from the screen. We have placed the washer of 1
cm in diameter where our drop would be in respect to Figure C.15. The washer is held by a
flat vertical plate so in the images we just can see the upper half of it. We move the washer
a distance of 10 cm without touching the camera, the screen or the lighting system, and it
seems to keep the same size. Since the arc lamp used was a point of light and the lens used
for this case had a long focal distance (reduces chromatic and spherical aberrations) we have
reached perfect parallel light. However, these pictures have been taken with long exposure
times to show the good performance of this method for applications where longer exposure
83
Lens Light point source Beam of parallel light
Figure C.17: Generating a parallel beam of white light with a point of light. If the source is
not a point source it will be hard to obtain parallel light. The light source has to be placed
at the focal distance of the lens.
Light point source A
B
84
Figure C.18: Image of the shadow of the washer projected on the screen. The washer is 5
cm from the screen.
times can be used. We were unable to get a parallel beam with enough light intensity to
capture enough light in the camera at our needed short exposure time.
Using monochromatic light
The same idea of catching the shadow of the drop is applied here. The difference is that the
parallel beam will be coming from an Nd:YAG laser; the laser ensures plenty of energy so
we will be able to see light now in our camera even if we have a short exposure time. We
need a train of lenses to enlarge the width of the beam, so that it will be wider than our
drop. The sketch of the train of lenses is represented in Figure C.20.
The results show some speckle and diffraction. Figure C.21 shows the speckle of the beam
85
Figure C.19: Image of the shadow of the washer projected on the screen. The washer is 15
cm from the screen. We cannot know the real position of the washer since its size looks like
the one in Figure C.18 eventhough the washer is now 10 cm further from the camera.
From laser To the object
40mm 200mm
Focal point of both lenses Figure C.20: Train of two lenses to increase the width of the laser beam.
86
Figure C.21: Image taken on a screen like Figure C.15. The shadow is a washer instead of
a drop. Notice the speckle in the image.
on the screen after generating a shadow of a washer. The speckle is not an issue, since we
can shoot more than one pulse and that will help to reduce it. We can build a simple mirror
system that sends two pulses of light separated a few nanoseconds.
Figure C.22 shows an sketch of the idea. A shot would be sent from the laser at the left side
of Figure C.22. Then, the beam gets through a first splitter. Half of the original beam will
go straight, and the other half will take a longer way to the target. This longer way has a
U shape and two mirrors and another splitter guarantee that a second shot will impact our
target. The delay between both shots, is in the order of nanoseconds. The speed of light
can be estimated as 1 foot/ nanosecond, if the longer way that the second beam takes is a
foot longer than the short path, the time between the impact of the two laser shots will be
1 nanosecond. Even for a traveling drop that is moving at 10 m/s, and has a diameter as
87
Splitters
Laser Target
Mirrors
Figure C.22: Sketch of the system made of two splitters and two mirrors that would split
one original laser shot in two separated a few nanoseconds.
small as 100 µ m, the blur from the two shots would be 0.01 % of the diameter. The drop
will not move in such a short time and we will reduce the speckle. To try its effectiveness,
we first shoot more than one pulse into a fixed object to study how the speckle improves
with more than one shot. The frequency of the laser is 10 Hz, so we can shoot two shots for
example, and fix the exposure time in the camera at 0.2 seconds. Then, we are sure that we
will catch two shots on our shadow and we will be able to compare it with the one shot case.
If the system works as expected, we then can build a mirror system so that the laser would
just shoot once, but the mirror system will generate two pulses spaced in a few nanoseconds
40mm between them. Figure C.23 shows the washer shadow after two shots. However, reducing
200mm
the diffraction that generates blur at the edge of the drops is more complicated. This is
the reason why he could not use this method, since the edges of the washer present some
Focal point of both lenses diffraction which is harder to fix.
88
Air flow
Figure C.23: Same image as Figure C.21 after the impact of two shots. The speckle is
reduced even if the diffraction at the edges of the washer is still there. The image is taken
with an angle with respect to Figure C.21 but that does not change the level of speckle.
89
D
Definitions
Critical temperature:
The critical temperature of a substance is the temperature at and above which vapor of the
substance cannot be liquefied, no matter how much pressure is applied.
Boiling point:
The boiling point of a substance is the temperature at which the vapor pressure of the liquid
equals the environmental pressure surrounding the liquid.The normal boiling point (also
called the atmospheric boiling point or the atmospheric pressure boiling point) of a liquid
is the special case in which the vapor pressure of the liquid equals the defined atmospheric
pressure at sea level, 1 atmosphere. At that temperature, the vapor pressure of the liquid
becomes sufficient to overcome atmospheric pressure and allow bubbles of vapor to form
inside the bulk of the liquid.
Evaporation:
Evaporation is a type of vaporization of a liquid that occurs only on the surface of a liquid.
The other type of vaporization is boiling, which, instead, occurs on the entire mass of the
liquid. On average, the molecules in a glass of water do not have enough heat energy to
escape from the liquid. With sufficient heat, the liquid would turn into vapor quickly. When
the molecules collide, they transfer energy to each other in varying degrees, based on how
they collide. Sometimes the transfer is so one-sided for a molecule near the surface that it
ends up with enough energy to escape (evaporate).
Vapor pressure:
Vapor pressure or equilibrium vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a
closed system. The equilibrium vapor pressure is an indication of a liquid’s evaporation rate.
It relates to the tendency of particles to escape from the liquid (or a solid). A substance
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with a high vapor pressure at normal temperatures is often referred to as volatile.
Vapor:
A vapor is a substance in the gas phase at a temperature lower than its critical point. This
means that the vapor can be condensed to a liquid or to a solid by increasing its pressure
without reducing the temperature.
Heat of vaporization:
The heat of vaporization is the amount of energy required to convert or vaporize a saturated
liquid (i.e., a liquid at its boiling point) into a vapor.
Kolmogorov microscale:
Kolmogorov microscales are the smallest scales in turbulent flow. At the Kolmogorov scale
viscosity dominates and the turbulent kinetic energy is dissipated into heat.
Lenses:
Lenses are optical devices used to converge or diverge a light beam. Their faces ideally follow
a shape that make all the light focus at a point called focal point. If a system would be made
just with air and an infinite lens where the contact between both has a parabolic shape, then
all the parallel light from the air impacting in the parabolic face of the lens would converge
into a focal point inside itself. Because lenses have two faces and are not infinite, in order to
make all light from a horizontal parallel beam of light converge into a focal point the faces
wont follow parabolic shapes. An aspherical lens is a type of lens optimized to converge all
the light in its focal point. It would be the closest to what we could call an ideal lens for
monochromatic light. The fact that it is cheaper to build spherical lenses than aspherical
generates spherical aberrations in the image of the lenses. Optical theory says that if we
design a spherical lens, it will work as an ideal lens in terms of having a narrow focal point
until we start using the exterior part of the lens. The inner part, actually, around 67% of
the diameter is the part that will not generate any spherical aberration. The outer part of
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the lens, when it gets parallel light will not focus it into a single point but into an area. This
aberration makes blur in the image and generates magnification errors. So, in order to avoid
spherical aberration there are some things to do:
• Keeping the same focal length: either using an aspherical lens, or using a larger lens
in diameter (change in the f number) and just using the center part of it (67% of the
diameter).
• Changing the focal length: for the same size of lens (Diameter), the longer the focal
length the less curvature the lens face will have and so, out of the 67% inner diameter
the difference in the spherical lens and the aspherical one will be less noticeable than
in lenses with high curvature.
There is another important aberration in lenses called chromatic aberration. Chromatic
aberration occurs because the material of the lens has a different index of refraction for each
of the colors that form white light. So when a parallel beam of white light impact on a lens,
their colors separate inside it, and the image the lens produces is blur, with a clear rainbow
and with magnification errors. The different colors of the incoming horizontal beam do not
converge into a point at the focal point but into an area. There are two ways to correct
chromatic aberration:
• Because the focal distance of a lens depends on the refractive index, and each color
that form white light have a different refractive index for a lens, they will have different
focal distances too. This is what generates the chromatic aberration. The longer the
focal distance of the lens (bigger radius of the faces of the lens) the less effect has a
change in the index of refraction. So using lenses with high focal distances will reduce
this aberration.
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• Using an achromat. An achromat is a composition of more than one lens glued together.
In the case of a doublet lens, it is two lenses with different indexes of refraction glued
together and combined to reduce this aberration.
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