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The Collision of Water and n-Heptane Droplets on Heated Sintered Porous Stainless-Steel Surfaces
by
Nicholas Lipson
A thesis submitted in conformity with the requirements for the degree of Master of Applied Science
Department of Mechanical and Industrial Engineering
University of Toronto
© Copyright by Nicholas Lipson, 2018
ii
The Collision of Water and n-Heptane Droplets on Heated Sintered Porous Stainless-Steel Surfaces
Nicholas Lipson
Master of Applied Science
Department of Mechanical and Industrial Engineering
University of Toronto
2018
Abstract
The impact and evaporation of droplets impinging on heated porous substrates is relevant to
applications including fire suppression by sprinkler systems, spray cooling of heated surfaces, and
the deposition of fuel droplets on combustor walls. An experimental study was completed where
pure water and n-heptane droplets were deposited onto hot porous, stainless steel surfaces. Initial
surface temperatures were varied from 60°C to 300°C. The impact dynamics were photographed
and analyzed via a high-speed camera. The Leidenfrost temperature was found to increase with an
increase in the substrates porosity and was achieved with the n-heptane on the 5μm surface at
225oC and on the 100μm surface at 285oC. An empirical model was employed that accurately
predicts the evaporation time of the n-heptane droplet in the film boiling regime. Surface
temperature variations were measured at the point of impact and heat transfer coefficients were
found to decrease with an increase in substrate porosity.
iii
Acknowledgments
First, I would like to express my deepest appreciation and gratitude towards my supervisor,
Professor Sanjeev Chandra, for his constant support and the guidance provided while conducting
this research. This work would not have been possible without his direction and insight.
I would also like to thank my colleagues and friends for the support and encouragement provided
throughout this academic journey. Thank you, Jordan Bouchard, Dr. Michael Gibbons, Chen Feng,
Marina Curak, Khalil Sidawi, Dr. Larry Pershin, and everyone else within CACT, MIE machine
shop, and the University of Toronto who I have encountered over the last two years. Special thanks
to Dr. Josh Powles, for the words of encouragement which helped shape my decision to pursue a
research master’s once finishing my undergraduate degree.
Finally, I would not be where I am today without the endless support from my parents, and siblings.
I owe a special acknowledgment to my beloved partner, Angela Bugayong, for putting up with me
throughout this process, providing constant support and encouragement to overcome any hurdles
encountered.
Research was conducted at the Centre for Advanced Coating Technologies (CACT) at the
University of Toronto. Funding for the project was provided by the Natural Sciences and
Engineering Research Council of Canada.
iv
Table of Contents
Acknowledgments ........................................................................................................................ iii
Table of Contents ......................................................................................................................... iv
List of Tables ............................................................................................................................... vii
List of Figures ............................................................................................................................. viii
Nomenclature ............................................................................................................................. xvi
Chapter 1 Introduction..................................................................................................................1
1.1 Motivation ...........................................................................................................................1
1.2 Literature Review ..............................................................................................................4
1.3 Research Objectives ...........................................................................................................7
1.4 Organization of Thesis .......................................................................................................7
Chapter 2 Characterization of Porous Sintered Stainless-Steel Test Substrates .....................8
2.1 Introduction ........................................................................................................................8
2.2 Porosity .............................................................................................................................10
2.2.1 Substrate Preparation .........................................................................................10
2.2.2 Results of Measured Porosity .............................................................................11
2.3 Surface Roughness ...........................................................................................................12
2.3.1 Results of Measured Surface Roughness ...........................................................12
2.4 Permeability......................................................................................................................13
2.4.1 Experimental Setup .............................................................................................14
2.4.2 Measurement Uncertainty...................................................................................17
2.4.3 Data Reduction and Analysis ..............................................................................18
2.5 Thermal Conductivity .....................................................................................................22
2.5.1 Experimental Setup .............................................................................................22
2.5.2 Data Reduction and Analysis ..............................................................................29
v
Chapter 3 Droplet Impact and Evaporation on Heated Sintered Porous Stainless-Steel
Substrates .................................................................................................................................37
3.1 Introduction ......................................................................................................................37
3.2 Experimental Setup .........................................................................................................38
3.2.1 Impact Conditions ................................................................................................38
3.2.2 Heating of Substrate ............................................................................................41
3.2.3 Substrate \ Thermal Mass Apparatus ................................................................43
3.2.4 Scale Accuracy .....................................................................................................44
3.2.5 Substrate Preparation .........................................................................................45
3.3 Data Reduction and Analysis ..........................................................................................45
3.3.1 Experimental Observations of Impact Dynamics on Cold Substrates ............45
3.3.2 Volume Analysis Accuracy..................................................................................61
3.3.3 Droplet Lifetime Evaporation Plots ...................................................................63
3.3.4 Experimental Observations of Impact Dynamics on Substrates at
Increased Temperatures Below the Leidenfrost Point .....................................66
3.3.5 Experimental Observations of Impact Dynamics on Substrates at
Temperatures Above the Leidenfrost Point ......................................................77
Chapter 4 Predicting Heat Transfer Coefficients Between Droplets and Heated Sintered
Porous Stainless-Steel Substrates ..........................................................................................91
4.1 Introduction ......................................................................................................................91
4.2 Experimental Setup .........................................................................................................91
4.2.1 Impact Conditions ................................................................................................91
4.2.2 Thin-Film Fast Response Thermocouple ...........................................................92
4.2.3 Thermocouple Thin Film ....................................................................................95
4.2.4 Thermocouple Calibration ..................................................................................97
4.2.5 Thermocouple Uncertainty .................................................................................98
4.3 Data Reduction and Analysis ..........................................................................................99
vi
4.3.1 Determining Heat Transfer Coefficients .........................................................103
Chapter 5 Conclusions ...............................................................................................................109
5.1 Future Research .............................................................................................................110
References ...................................................................................................................................112
Appendix A Permeability Apparatus Measurement Uncertainty Sample Calculation ............118
Thermal Conductivity Apparatus Measurement Uncertainty Sample Calculation .....................122
Thin-Film Fast Response Thermocouple Measurement Uncertainty Sample Calculation .........126
Appendix B MATLAB Code ......................................................................................................129
Mettler Toledo AG245 Scale Weight Logger Program: .............................................129
Droplet Volume Calculator Program: .........................................................................131
Vapor Film Thickness Solver Program: ......................................................................133
Arduino Uno Code .......................................................................................................................136
TDS2002B Oscilloscope Trigger Program: .................................................................136
Appendix C Data ......................................................................................................................137
vii
List of Tables
Table 2.1: Summary of substrate volumes and porosities. ........................................................... 12
Table 2.2: Summary of average roughness measurements. .......................................................... 13
Table 2.3: Summary of the porous substrates intrinsic permeabilities and their associated
uncertainties. ................................................................................................................................. 22
Table 2.4: Thermal conductivities for the various materials used in the apparatus. The values are
stated at 25oC. ............................................................................................................................... 25
Table 2.5: Summary of thermal resistance values for the impermeable stainless-steel substrate,
and the thermal graphite sheet. These values are stated for an applied clamping pressure to the
stack of 265 kPa. ........................................................................................................................... 28
Table 2.6: Summary of the calculated properties and applied average heat fluxes for the
calibration cylinder and two porous substrates. ............................................................................ 33
Table 3.1: Summary of the fluid properties and impact conditions. ............................................. 40
Table A.1: Summary of the design-stage uncertainties for the flow meters depending on the flow
rate being examined. ................................................................................................................... 120
Table A.2: Summary of the design-stage uncertainties for the signal amplifier and oscilloscope
for the substrate being investigated. ............................................................................................ 127
Table A.3: Summary of the design-stage uncertainties for the fast response thermocouple. ..... 128
viii
List of Figures
Figure 1.1: Evaporation time as a function of the substrates wall temperature illustrating the
different heat transfer regimes [6]. .................................................................................................. 2
Figure 1.2: Photograph taken by Worthington of a water droplet released from 137 cm into a
milk-water mixture [7]. ................................................................................................................... 3
Figure 2.1: SEM images of the sintered porous stainless-steel samples (5 μm and 100 μm
average pore size) used during experimentation (top). Full-scale images of samples (bottom).
Substrate dimensions: 45 mm x 45 mm by 2.0 mm thick. .............................................................. 9
Figure 2.2: Experimental setup employed to determine the permeability of the porous substrates
with key components labeled (top, middle). Schematic of the experimental setup (bottom). ...... 15
Figure 2.3: 3d rendering of bracket used to mount porous samples into settling chamber with key
components labeled. ...................................................................................................................... 17
Figure 2.4: Apparent permeability as a function of the reciprocal of the mean pressure for the 5
μm porous substrate over the full flow range investigated. .......................................................... 20
Figure 2.5: Apparent permeability as a function of the reciprocal of the mean pressure for the
100 μm porous substrate over the full flow range investigated. ................................................... 20
Figure 2.6: Klinkenberg permeability plot for both the 5 μm and 100 μm porous substrates.
Visco-inertial flows have been excluded. At 1/Pm = 0 the intrinsic permeability for the substrate
is found. The intrinsic permeability is indicated for the 5 μm substrate. ...................................... 21
Figure 2.7: Experimental setup employed to determine the thermal conductivity of the porous
substrates with key components labeled (top). Schematic of the experimental setup (bottom). .. 24
Figure 2.8: Thermal resistance as a function of the applied pressure for the thermal interface
graphite sheets as provided by Panasonic [44]. ............................................................................ 27
ix
Figure 2.9: Thermocouple temperatures a function of the time the test was running, showing the
last hour for the calibration sample. A diagram illustrating thermocouple position in the
aluminum bars is shown................................................................................................................ 30
Figure 2.10: Thermocouple temperature as a function of its position in the hot and cold
aluminum bars for the calibration sample. The equations for each linear trendline are shown. The
temperatures at the interfaces between the thermal graphite sheets and the hot (TH) and cold (TC)
aluminum bars are indicated. ........................................................................................................ 31
Figure 2.11: Thermocouple temperature as a function of its position in the hot and cold
aluminum bars for the 5 and 100 µm porous substrates. The equations for each linear trendline
are shown. The temperatures at the interfaces between the thermal graphite sheets and the hot
(TH) and cold (TC) aluminum bars are indicated........................................................................... 32
Figure 2.12: Various model predictions for the effective thermal conductivity as a function of the
materials porosity. The thermal conductivities obtained through experimentation for the
calibration cylinder and two porous substrates are shown for comparison. ................................. 35
Figure 3.1: Experimental setup A and B (top, middle). Schematic of experimental setup
(bottom): (1) Syringe pump coupled with a 10 ml syringe, (2) Vertical height adjustment, (3)
Hypodermic needle, (4) High-speed camera, (5) Wireless thermocouple connector, (6) Light
source, (7) Light diffuser, (8) Thermocouple, (9) Substrate, (10) Thermal mass coupled with
100W cartridge heater, (11) PC logging scale data, and monitoring substrate temperature, (12)
Digital scale, (13) Temperature controller, (14) 120V Variac, (15) PC capturing high-speed
camera images, (16) Thermal mass coupled with two, 200W cartridge heaters. (a) Setup used at
surface temperatures ranging from 60oC to 120oC, (b) Setup used at surface temperatures above
120oC............................................................................................................................................. 40
Figure 3.2: Weight decrease as a function of time for a pure water droplet evaporating on the
impermeable substrate, at surface temperatures of 60oC, 80oC and 100oC. Images of the droplet
are shown for 100oC at 1.2s, 18s, and 35s during the evaporation process. ................................. 42
Figure 3.3: 3D printed apparatus used to hold thermal mass and substrate during evaporation
time measurements at low wall temperatures. All dimensions shown are in millimeters (mm). . 44
x
Figure 3.4: Evaporation time as a function of the initial surface temperature for a pure water
droplet evaporating on the impermeable stainless-steel substrate. Both scale and high-speed
camera times are shown. The standard deviation is shown for the scale times. ........................... 45
Figure 3.5: Water droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces.
Surface temperature Tw = 23oC, Vi = 0.9 m/s ±5 %, do = 2.5 mm ±2%, We = 29, Photograph
Angle = 30o. .................................................................................................................................. 47
Figure 3.6: Overview of different stages of droplet impact. (a) depicts a droplet with initial
diameter Do impacting on a smooth surface spreading to its maximum diameter, Dmax. (b)
Deformation that occurs at the solid-liquid interface in the initial stage of impact whereby due to
the compression of the air between the droplet and the surface a dimple is formed that encloses
around the air and forms a bubble [60]. ........................................................................................ 48
Figure 3.7: n-Heptane droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces.
Surface temperature Tw = 23oC, Vi = 0.9 m/s ±9 %, do = 2.5 mm ±2%, We = 65, Photograph
Angle = 30o. .................................................................................................................................. 49
Figure 3.8: 1 ms after impact of the water and n-heptane droplet on the (a) impermeable, (b) 5
μm and (c) 100 μm surfaces at 23oC. The single bubble is shown in the water and n-heptane
droplet on the impermeable surface, but not on the porous surfaces. ........................................... 51
Figure 3.9: n-Heptane on the impermeable surface at 10.6, 10.8 and 11 ms after impact showing
the release of the entrapped air bubble. ........................................................................................ 51
Figure 3.10: Spread factor as a function of the dimensionless time for both water and n-heptane
on the impermeable and two porous substrates at room temperature (23oC). .............................. 53
Figure 3.11: Pinned water droplet taking the form of a spherical cap after impact on the 5 μm
surface. An example inlay of the droplets mask produced by the program is shown, along with a
model (not to scale) showing how the masked area is broken down into circular volumetric
segments used to determine the total volume of the droplet sitting on the substrates surface. .... 55
Figure 3.12: V/Vo as a function of time for the water droplet on the 5 µm surface. .................... 56
Figure 3.13: V/Vo as a function of time for the water droplet on the 100 µm surface. ................ 56
xi
Figure 3.14: Water droplet impact on the 5 μm surface showing entrapped air bubbles. Surface
temperature Tw = 23oC, Vi = 0.9 m/s ±5 %, do = 2.5 mm ±2%, We = 29, Photograph Angle = 30o.
....................................................................................................................................................... 58
Figure 3.15: Water droplet impact on the 5 μm surface showing entrapped air bubbles 4.6 ms
after impact. Surface temperature Tw = 23oC, Vi = 0.9 m/s ±5 %, do = 2.5 mm ±2%, We = 29,
Photograph Angle = 0o. ................................................................................................................. 59
Figure 3.16: Water droplet impact on the 100 μm surface showing entrapped air bubbles. Surface
temperature Tw = 23oC, Vi = 0.9 m/s ±5 %, do = 2.5 mm ±2%, We = 29, Photograph Angle = 0o.
....................................................................................................................................................... 59
Figure 3.17: V/Vo as a function of time for the n-heptane droplet on the 5 µm surface. ............. 60
Figure 3.18: n-heptane droplet at its maximum spread factor 7.4 ms after impact on the 5 μm
surface. Droplet photographed at approximately 30o. The lowered center film with larger
surrounding rim can be observed. ................................................................................................. 61
Figure 3.19: V/Vo as a function of time for the water droplet on the impermeable surface. ........ 62
Figure 3.20: Weight decrease a function of time for both the water and n-heptane droplet on the
impermeable surface at room temperature (23oC ±3oC). Equations for the line are shown for both
fluids where the slope of the line yields the evaporation rate. ...................................................... 63
Figure 3.21: Evaporation time as a function of the surface temperature for the impermeable, 5
μm and 100 μm substrates using pure water as the working fluid. A graph insert is shown at
surface temperatures ranging from 150oC to 300oC using a log time scale to show evaporation
time differences between the surfaces. ......................................................................................... 64
Figure 3.22: Evaporation time as a function of the surface temperature for the impermeable, 5
μm and 100 μm surfaces using n-heptane as the working fluid. A graph insert is shown at surface
temperatures ranging from 150oC to 300oC using a log time scale to show evaporation time
differences between the surfaces. ................................................................................................. 64
xii
Figure 3.23: Water droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces.
Surface temperature Tw = 80oC (film evaporation regime), Vi = 0.9 m/s ±5%, do = 2.5 mm ±2%,
We = 29. ........................................................................................................................................ 68
Figure 3.24: n-Heptane droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm
surfaces. Surface temperature Tw = 80oC (film evaporation regime), Vi = 0.9 m/s ±9%, do = 2.5
mm ±2%, We = 65. ....................................................................................................................... 69
Figure 3.25: n-Heptane droplet on the impermeable surface at 80oC showing the bubble growth
and movement through the liquid film at 2.4, 2.8, 3.2, and 7.4 ms. ............................................. 70
Figure 3.26: n-Heptane droplet on the impermeable surface at 80oC showing thermo-capillary
convection cells at 22.8, 31, 46.6, and 110.6 ms. ......................................................................... 70
Figure 3.27: n-Heptane droplet on the 5 µm surface at 80oC showing thermo-capillary
convection cells at 11, 12.8, 15.8, and 18.6 ms. ........................................................................... 71
Figure 3.28: Water droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces.
Surface temperature Tw = 150oC (nucleate boiling regime), Vi = 0.9 m/s ±5%, do = 2.5 mm ±2%,
We = 29. ........................................................................................................................................ 72
Figure 3.29: n-Heptane droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm
surfaces. Surface temperature Tw = 150oC (nucleate boiling regime), Vi = 0.9 m/s ±9%, do = 2.5
mm ±2%, We = 65. ....................................................................................................................... 74
Figure 3.30: Water and n-heptane droplet on the (a) impermeable, (b) 5µm, and (c) 100 µm
surfaces at 150oC showing secondary droplets produced during boiling at 15 ms after impact. No
secondary droplets are observed with the water on the porous surfaces. ..................................... 76
Figure 3.31: Water droplet on the impermeable surface at 150oC showing the “pagoda-like”
formation ejecting a secondary drop at 56.8, 57, and 57.2 ms after impact. ................................ 76
Figure 3.32: n-Heptane droplet on the impermeable surface at 150oC showing “pagoda-like”
formation at 30.6 ms and 46 ms after impact. The “pagoda-like” formations are enlarged on the
right. .............................................................................................................................................. 77
xiii
Figure 3.33: Water droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces.
Surface temperature Tw = 250oC (film boiling regime), Vi = 0.9 m/s ±5%, do = 2.5 mm ±2%, We
= 29. .............................................................................................................................................. 79
Figure 3.34: n-Heptane droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm
surfaces. Surface temperature Tw = 250oC (film boiling regime), Vi = 0.9 m/s ±9%, do = 2.5 mm
±2%, We = 65. .............................................................................................................................. 81
Figure 3.35: n-Heptane droplet on the impermeable surface at 250oC showing fingers as a result
of a Kelvin-Helmholtz instability merging due to surface tension and cohesive forces. ............. 82
Figure 3.36: Model of the droplet used for determining the evaporation time in the film boiling
regime. .......................................................................................................................................... 84
Figure 3.37: Vapor thickness between the substrate and the base of the n-heptane droplet as a
function of the initial surface temperature in the film boiling regime for the impermeable, 5 μm
and 100 μm surfaces. .................................................................................................................... 88
Figure 3.38: Vapor thickness between the substrate and the base of the water droplet as a
function of the initial surface temperature in the predicted film boiling regime for the 5 μm and
100 μm surfaces. Left shows results when the vapor properties were selected at 170oC, and the
right, at 220oC. Vapor thicknesses are shown for the impermeable surface in the experimentally
observed film boiling regime for comparison. The Leidenfrost temperatures are indicated. ....... 89
Figure 3.39: n-Heptane droplet lifetime as a function of the initial surface temperature in the film
boiling regime for the impermeable, 5 μm and 100 μm surfaces. The empirical model output is
shown for comparison. .................................................................................................................. 90
Figure 4.1: Experimental setup employed to measure surface temperature variations directly
under the droplet. .......................................................................................................................... 92
Figure 4.2: Top down view (top) \ profile view (bottom) schematic of the fast response
thermocouple used to measure surface temperature variations directly under the droplet. .......... 94
xiv
Figure 4.3: SEM images of the impermeable and two porous substrates with the constantan
thermocouple wire cemented in place before and after light sanding with 1200 grit sandpaper. . 96
Figure 4.4: Surface temperature of the impermeable, and two porous substrates as a function of
their respective thermocouple output. The heat up and cool down data is shown along with the
equations for the heat up trendlines. ............................................................................................. 97
Figure 4.5: Surface temperature as a function of time for the water droplet on the impermeable, 5
μm and 100 μm surfaces at the point of impact at an initial surface temperature of 120oC.
Temperatures are shown up until the droplet reached its maximum spread factor on the
impermeable surface. .................................................................................................................. 100
Figure 4.6: Surface temperature as a function of time for the n-heptane droplet on the
impermeable, 5 μm and 100 μm surfaces at the point of impact at an initial surface temperature
of 120oC. Temperatures are shown up until the droplet reached its maximum spread factor on the
impermeable surface. .................................................................................................................. 100
Figure 4.7: Comparison between the impermeable, 5 µm and 100 µm surfaces with the drop in
surface temperature after 4 ms, the maximum time taken for the water droplet (at 23oC) to spread
on the impermeable surface, as a function of the initial surface temperature. The temperature was
measured at the point of impact. Standard deviations in ∆Tw are shown. .................................. 102
Figure 4.8: Comparison between the impermeable, 5 µm and 100 µm surfaces with the drop in
surface temperature after 13.4 ms, the maximum time taken for the n-heptane droplet (at 23oC) to
spread on the impermeable surface, as a function of the initial surface temperature. The
temperature was measured at the point of impact. Standard deviations in ∆Tw are shown. ....... 103
Figure 4.9: Surface temperature as a function of time for the water droplet on the impermeable, 5
µm and 100 µm surfaces. Times are shown until the droplet reached its maximum spread factor
on the respective surface. Predicted temperatures using equation 4.7 are shown. Initial surface
temperature was 120oC. .............................................................................................................. 105
Figure 4.10: Surface temperature as a function of time for the n-heptane droplet on the
impermeable, 5 µm and 100 µm surfaces. Times are shown until the droplet reached its
xv
maximum spread factor on the respective surface. Predicted temperatures using equation 4.7 are
shown. Initial surface temperature was 120oC. ........................................................................... 106
Figure 4.11: Heat transfer coefficient for the water during droplet spreading on the impermeable,
5 µm and 100 µm surfaces as a function of the initial surface temperature. .............................. 107
Figure 4.12: Heat transfer coefficient for the n-heptane during droplet spreading on the
impermeable, 5 µm and 100 µm surfaces as a function of the initial surface temperature. ....... 108
xvi
Nomenclature
All notation used in this study is outlined in this section. A list of acronyms and their meanings is
presented first, followed by all mathematical symbols. The acronyms and mathematical symbols
are listed in the order they first appear.
Acronym Description
SEM Scanning Electron Microscope
ISO International Organization for Standardization
PVC Polyvinyl Chloride
ASTM American Society for Testing and Materials
DAQ Data Acquisition Module
PC Personal Computer
RSS Root-Sum-Squares
DC Direct Current
A/D Analog to Digital
CPSM Cubic-Parallel-Series Model
CSPM Cubic-Series-Parallel Model
TUCM Tetrahedral Unit Cell Model
LED Light Emitting Diode
EMI Electromagnetic Interference
PLA Polylactic Acid
RFI Radio Frequency Interference
NIST National Institute of Standards and Technology
Symbol Description Units
휀 Porosity [%]
𝑉𝑉 Substrate Void Space Volume [𝑚3] 𝑊𝑆 Saturated Weight of Substrate [𝑘𝑔] 𝑊𝐷 Dry Weight of Substrate [𝑘𝑔]
𝜌𝐻 Density of n-Heptane [𝑘𝑔
𝑚3]
𝑉𝐵 Substrate Bulk Volume [𝑚3]
𝑅𝑎 Surface Roughness Average [𝜇𝑚]
𝑞 Fluid Discharge per Unit Area [𝑚
𝑠]
𝜅 Intrinsic Permeability [𝑚2] 𝜇 Dynamic Viscosity [𝑃𝑎 ∙ 𝑠]
∇𝑝 Pressure Gradient [𝑃𝑎
𝑚]
𝑢0 Zero-Order Uncertainty
𝑢𝑐 Instrument Uncertainty
𝑢𝑑 Design-Stage Uncertainty
𝑃 Pressure [𝑃𝑎] 𝑃𝑚 Mean Pressure [𝑃𝑎]
𝑄 Volumetric Flow Rate [𝑚3
𝑠]
xvii
𝑄𝑚 Mean Volumetric Flow Rate [𝑚3
𝑠]
𝑃𝑐ℎ𝑎𝑚𝑏𝑒𝑟 Chamber Pressure [𝑃𝑎] 𝑃𝑎𝑡𝑚 Atmospheric Pressure [𝑃𝑎]
𝜅′ Apparent Permeability [𝑚2] 𝐴 Substrate Cross Sectional Area [𝑚2] 𝐿 Substrate Thickness [𝑚]
𝑘 Thermal Conductivity [𝑊
𝑚 ∙ 𝐾]
𝑅 Thermal Resistance [𝑐𝑚2 ∙ 𝐾
𝑊]
∆𝑇 Temperature Difference [°𝐶]
𝑞" Heat Flux [𝑘𝑊
𝑚2]
𝑘𝐴𝐿 6061 Aluminum Thermal Conductivity [𝑊
𝑚 ∙ 𝐾]
∆𝑥 Spacing Between Thermocouples [𝑚] 𝑇 Total Torque [𝑁 ∙ 𝑚] 𝐾 Torque Coefficient
𝐹𝑖 Clamping Force [𝑁] 𝐷 Major Bolt Diameter [𝑚]
𝑅𝐺𝑆 Graphite Sheet Thermal Resistance [𝑐𝑚2 ∙ 𝐾
𝑊]
𝑘𝑅𝑒𝑢𝑠𝑠 Serial Model Effective Thermal Conductivity [𝑊
𝑚 ∙ 𝐾]
𝑘𝑓 Fluid Thermal Conductivity [𝑊
𝑚 ∙ 𝐾]
𝑘𝑠 Solid Thermal Conductivity [𝑊
𝑚 ∙ 𝐾]
𝑘𝑉𝑜𝑖𝑔𝑡 Parallel Model Effective Thermal Conductivity [𝑊
𝑚 ∙ 𝐾]
𝑘𝑇𝑈𝐶𝑀 Tetrahedral Unit Cell Model Effective Thermal
Conductivity [
𝑊
𝑚 ∙ 𝐾]
𝑅𝐴, 𝑅𝐵, 𝑅𝐶 , 𝑅𝐷 Tetrahedral Unit Cell Model Variables [𝑚 ∙ 𝐾
𝑊]
𝑑, 𝑒 Tetrahedral Unit Cell Model Variables
We Weber Number
𝜌𝑓 Droplet (fluid) Density [𝑘𝑔
𝑚3]
𝑉𝑖 Initial Droplet Impact Velocity [𝑚
𝑠]
𝑑𝑜 Initial Droplet Diameter [𝑚]
𝜎 Droplet Surface Tension [𝑁
𝑚]
𝐶𝑎 Capillary Number
𝜷 Spread Factor
xviii
𝐷𝑓𝑖𝑙𝑚 Dynamic Spread Diameter [𝑚] 𝑡∗ Dimensionless Time
𝑡 Spread Time [𝑠]
𝑣𝑟 Opposing Liquid Rim Velocity [𝑚
𝑠]
ℎ Thickness of Expanding Liquid Film [𝑚]
𝑣𝑙 Lamella Velocity [𝑚
𝑠]
𝑣 Velocity of Liquid Film [𝑚
𝑠]
𝐾𝐸1 Kinetic Energy of Droplet [𝐽]
𝑆𝐸1, 𝑆𝐸2 Surface Energy of Droplet Before and After
Impact [𝐽]
𝑊 Work Done During Droplet Spreading [𝐽] 𝐸𝑝 Energy Associated with Fluid Penetration [𝐽]
𝐷𝑚𝑎𝑥 Maximum Spread Diameter [𝑚] 𝜃𝑎 Advancing Contact Angle [°] 𝑅𝑒 Reynolds Number
𝑡𝑒 Droplet Evaporation Time [𝑠] 𝑇𝑤 Surface Temperature [℃]
𝑉𝑠𝑒𝑔 Volume of Droplet Segment [𝑚𝑚3] 𝐷𝑐ℎ𝑜𝑟𝑑 Diameter of Volume Segment [𝑚𝑚]
𝚤𝑝𝑙 Pixel to Length Conversion [𝑝𝑖𝑥𝑒𝑙
𝑚𝑚]
𝑄𝑣𝑎𝑝𝑜𝑟𝑖𝑧𝑒 Heat Required to Vaporize Droplet [𝐽] ℎ𝑓𝑔 Heat of Vaporization [𝐽] 𝑚 Mass of Droplet [𝑘𝑔]
𝑄𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 Radiation from Surface [𝐽]
𝜎𝑆𝐵 Stefan-Boltzmann Constant [𝑊
𝑚2𝐾4]
ℎ Substrate Thickness [𝑚] 𝛿 Vapor Thickness [𝑚]
𝜖 Ratio of Velocity Gradients at the Porous-
Vapor Interface
𝐴1, 𝐴2, 𝐵1, 𝐵2 Constants from Integration
𝑅 Radius of Droplet [𝑚] 𝑥 Radial Position from Droplet Center [𝑚]
𝜌𝑣 Vapor Density [𝑘𝑔
𝑚3]
�̇�𝑣, �̇�𝑢 Mass Flow Rate Through Vapor Film and
Porous Substrate [𝑘𝑔
𝑠]
𝑣0 Vapor Velocity [𝑚
𝑠]
𝑉 Droplet Volume [𝑚3] 𝛼 Ratio of the Substrates Permeability to Porosity
𝑦 Distance from Porous Surface [𝑚]
xix
𝑘𝑣 Thermal Conductivity of Vapor [𝑊
𝑚 ∙ 𝐾]
𝑔 Acceleration Due to Gravity [𝑚
𝑠2]
𝜇𝑣 Vapor Dynamic Viscosity [𝑃𝑎 ∙ 𝑠]
𝑇𝑖𝑚𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑙𝑒 , 𝑇5𝜇𝑚, 𝑇100𝜇𝑚 Surface Temperature for the Impermeable, 5
µm and 100 µm Substrates [°𝐶]
𝑇𝑠,0, 𝑇𝑠 Initial / Substrate Temperature [°𝐶] 𝑇𝑑,0 Initial Droplet Temperature [°𝐶]
𝛼𝐻𝑇𝐶 Heat Transfer Coefficient [𝑊
𝑚2 ∙ 𝐾]
𝑐 Specific Heat Capacity [𝐽
𝑘𝑔 ∙ 𝐾]
1
Chapter 1 Introduction
1.1 Motivation
The study of liquid droplet impact on surfaces has been a topic of discussion for over a hundred
years. This seemingly insignificant phenomenon, when looked upon closely, reveals a world of
beauty and complexity as the droplet collides with a surface. Even after over a century of research
on the subject, droplet impact phenomena are still not fully understood and attracts the interests of
physicists, engineers and mathematicians alike. Droplet-surface interactions occur in a wide range
of technical applications as well as in nature. To list a few, droplet impact occurs in combustion
systems including direct-injection diesel engines, intake ducts of gasoline engines, and incinerators
where the evaporation of fuel droplets on heated surfaces is of critical importance. Optimizing
droplet evaporation times is important in ensuring improved combustion efficiencies and reduced
gaseous emissions during the combustion process. An understanding of droplet impact is also
important in the spray cooling of semiconductor chips, and electronic devices; in spray painting
and coating systems and in fire extinguishment by means of sprinkler systems.
In the fire extinguishment process, water can be used to both extinguish flames, and prevent flame
spread by cooling surfaces that have not yet been ignited. However, the use of large amounts of
water can cause secondary fire damage encouraging research into this area to reduce the amount
of water used [1]. As fires can involve the burning of porous materials (e.g. wood, fabrics, papers),
understanding the thermal effects of droplet impingement on hot porous surfaces will assist in
predicting the quantity of water needed for sufficient cooling of the surface while minimizing
secondary damage. In nature, the impact of rain droplets is partly responsible for the erosion of
soils, and the aeration of lakes and oceans as a result of the formation of air bubbles in the fluid.
The prevalence of such occurrences proves the importance of fully understanding the droplet-
surface interaction [2]. The droplets behavior upon collision with a surface depends on many
factors including the droplet size, impact orientation and velocity, fluid and surface properties, and
environmental conditions which can all vary significantly.
2
In systems where heat transfer is involved, to understand at what temperatures significant changes
in the droplets behavior occurs, droplet lifetime as a function of initial surface temperature plots
can be employed (see Figure 1.1). Previous studies concerning droplet impact on porous surfaces
employed traditional imaging techniques and timers to capture the droplets evaporation time [3-
5]. Reduced estimates of the evaporation time can occur for fluids that can easily penetrate the
surface pores. This becomes more substantial at lower wall temperatures as the fluid can
completely penetrate the substrate making it invisible to photographs being taken from above. This
study removes that uncertainty and employs a method to measure the droplets evaporation time by
capturing the droplets weight decrease as a function of time as it evaporates on the substrate.
Figure 1.1: Evaporation time as a function of the substrates wall temperature illustrating the different heat transfer
regimes [6].
In addition to capturing evaporation times, a systematic approach to photograph and visualize the
impact phenomena is needed. Some of the first notable work completed to visualize droplet impact
was produced by Worthington [7] in 1908. Worthington studied splashing fluids, creating a novel
setup to photograph the moments immediately after a droplet and or solid-ball came into contact
with liquid pools (see Figure 1.2). Assumptions were made that if the falling object collides with
the surface under exactly the same conditions, the dynamics of the fluid each time would be
identical. This allowed Worthington to impact droplets repeatedly against the surface of the water
and capture successive photographs, freezing different stages of impact. A final image sequence
3
could be pieced together at the end to visualize the behavior. A spark produced by a capacitor
created as Worthington put it, a “dazzling flash” illuminating the droplet during impact. The timing
of such photographs was done through means of a falling metal sphere that passed between
terminals closing the connection and triggering the flash. A camera armed with a sensitive plate
was directed at the surface of the water capturing the impact phenomena. Single shot and high-
speed photography techniques have been utilized with great success by several others including
Savic and Boult in 1955 [8], Wachters and Westerling in 1966 [9], Toda in 1974 [10], Inada et al.
in 1985 [11] and Chandra and Avedisian in 1991 [12]. Advancements in technology has led to the
modern day high-speed imaging cameras and multi-intensity lighting systems used in this study
allowing complete control to photograph the droplets movement as it impacts and spreads over the
substrates surface at a sub-millisecond timescale resolution [13 - 15].
Figure 1.2: Photograph taken by Worthington of a water droplet released from 137 cm into a milk-water mixture [7].
4
When designing spray cooling systems, quick estimates of heat transfer during droplet impact are
often needed. Due to the complexity of analyzing spray systems, single droplet impact experiments
can be done as a starting point during analysis. Along with observing the droplets evaporation
time, to experimentally analyze the heat transfer between a surface of the substrate and a liquid
droplet requires measurements of how the substrate temperature varies directly under the droplets
base after impact.
Pasandideh-Fard et al. [16] measured substrate temperature variations at the point of impact using
a commercially available fast response eroding thermocouple and developed an analytical model
to determine the cooling effectiveness of a droplet impact on an impermeable heated surface. The
problem with sheathed thermocouples is they can be quite large, and when embedded into the
substrate can alter the surface properties changing the droplet-substrate interaction. Wang et al.
[17] investigated enhanced surfaces and their effect on water droplet dynamics and cooling
performance. K-type thermocouples were installed 2.5 mm below the top surface to measure
substrate surface temperature variations. Thermocouples embedded below the surface can have
delayed responses to immediate changes in temperatures at the liquid-solid interface.
Kim and Lee [18] studied the impact of water droplets on porous beds of glass beads while varying
substrate permeability, wall temperature, and droplet impingement velocity. Attempts were made
to observe surface temperature variation at the point of impact using an IR camera. Due to
measurements taken through a liquid film being inaccurate, temperatures were analyzed once the
droplet completely permeated into the surface pores. Assumptions were made about temperature
variations at the point of impact when the droplet still rested on the surface. Few studies have
examined surface temperature variations directly at the point of impact of a droplet on a porous
surface. To determine the heat transfer coefficients, this study employs a thin-filmed fast response
thermocouple capable of measuring temperature changes on the order of nanoseconds. The film is
small enough such that it has negligible effect on the droplets interaction with the surface.
1.2 Literature Review
Numerous experimental studies have been concerned with droplet impact and heat transfer on
impermeable surfaces [1, 8-12, 16-17, 19-25]. Various liquids, impact conditions, substrate
properties, and environmental conditions have been examined. By comparison, few studies have
been completed investigating experimentally the impact and heat transfer between droplets and
5
porous surfaces. Avedisian and Koplik [3] did an experimental and analytical study of the film
boiling of methanol droplets on ceramic porous surfaces with porosities of 10%, 25% and 40%.
Evaporation times were measured in the wetting and film boiling regimes where initial surface
temperatures were varied to 347oC. It was found that the Leidenfrost temperatures increased with
an increase in surface porosity as the vapor film can escape into the surface pores as well as
radially. Similarly, evaporation times were faster on the porous surfaces as compared to the
impermeable surface in the film boiling regime at the same initial surface temperature because the
droplet rested closer to the heated surface (i.e. smaller vapor film thickness). Analysis was
presented to model the film boiling process.
Chandra and Avedisian [4] deposited n-heptane droplets on a ceramic porous surface with a
porosity of 25%. Initial surface temperatures were varied from 22oC to 200oC. Photographs were
taken to observe the droplets impact morphology at different initial surface temperatures. Droplet
dimensions during impact and their evaporation times were recorded. It was found that the
spreading rate of a droplet on a porous surface at 22oC was slower than on an impermeable
stainless-steel surface. Similarly, the spread factor and spread rate were found to be independent
of the surface temperature immediately after impact due to negligible surface tension and viscous
effects. The maximum spread factor was found to decrease from the impermeable to porous surface
at the same initial surface temperature.
M. Abu-Zaid [5,26] examined the evaporation times of gasoline and diesel droplets impinged on
impermeable surfaces made from aluminum, stainless-steel, carbon-steel, and porous surfaces
made from ceramic (porosity of 25%), and kaolin (porosity of 28%). Initial surface temperatures
were varied from 60oC to 500oC. It was found that as the thermal diffusivity of the material
increased, the temperature at the minimum evaporation time decreased. Evaporation times were
measured using an IRWIN crystal timer with a 0.01 second resolution. A study was also completed
looking at the effects of radiant heat on the evaporation of water droplets of varying sizes deposited
on porous and non-porous ceramic surfaces. Temperatures were scaled from 75oC to 250oC. It was
found that evaporation times and surface temperature recovery times were shorter for the porous
surfaces as compared to the non-porous surfaces as the droplet could penetrate the surface pores
contacting a larger heated area. Smaller droplets were the most efficient at cooling the porous
surfaces. They reasoned this was due to evaporation of larger droplets attaining a quasi-steady
state which produces intensive local cooling. Due to the ceramic solids having a low thermal
6
conductivity, heat recovery from other portions of the solid is limited. The extra water in the larger
droplets thus increase the evaporation time. When compared to evaporation times and recovery
times for surfaces heated from the bottom, radiant heating had significant effect of reducing the
evaporation and recovery times due to the surface receiving additional heat transfer from the
radiant heaters.
Yu et al. [27] experimentally observed water droplets impacting on porous substrates in the film
boiling regime with a porosity of 34% and average pore size of 76 nm. A surface temperature of
300oC was examined. It was found that collision of the water droplet on the porous surface shows
similar behavior to the droplet on the non-porous surface and attributed this to the pore size being
small. They did find however that the residence time was longer with the droplet on the porous
surface.
Kim et al. [28] looked at custom fabricated surfaces varying the surface roughness, wettability and
nano-porosity to see how that effects water droplet impacts in the film boiling regime. Evaporation
times were obtained at initial surface temperatures of approximately 100oC to 500oC. It was found
that nano-porosity was the essential feature to offset the Leidenfrost point as it prevented a stable
vapor film from establishing caused by heterogeneous nucleation of bubbles.
Singh et al. [29] experimentally investigated the wetting and evaporation of water and ethanol
droplets on alumina porous ceramic surfaces varying the pore distribution morphologies and the
pore sizes. Average pore sizes chosen were 70 nm and 120 nm. Relative humidity was varied
inside the control volume to regulate the rate of droplet evaporation. Results were compared to
the droplets on an impermeable surface made from borosilicate glass and Teflon. Contact angles
and the droplets spreading process were recorded using high-speed imaging. They found that the
rate of evaporation of the droplet does not decrease linearly with an increase in relative humidity
and that the evaporation rate depends on the initial contact angle and radius of the droplet. It was
also found that the wetted area increases for the droplets on the porous surfaces and increases
further with more randomness in the distribution of the substrate pores along with increasing the
pore size.
7
1.3 Research Objectives
The present work reports the results of an experimental study involving the impact and evaporation
of pure water and n-heptane (C7H16) droplets deposited on hot porous substrates with different
pore sizes, and varying surface temperatures. n-Heptane was chosen since it has lower surface
tension, liquid-solid contact angle and latent heat of vaporisation than water and penetrated easier
into the surface pores. There were four principle objectives for this work:
➢ Measure droplet evaporation times using a weight-time approach at low wall temperatures
to remove the uncertainty associated when using imaging techniques.
➢ Compare the effects of fluid surface tension and latent heat on the impact and evaporation
of a droplet when impinged on a porous substrate with different surface temperatures.
➢ Photograph and analyze droplet impact dynamics on the heated porous substrates.
➢ Measure substrate surface temperature variations at the point of impact and determine heat
transfer coefficients.
1.4 Organization of Thesis
The thesis consists of five chapters. Chapter 1 outlines the motivation and objectives for the work
completed and reviews the current state of literature, summarizing what has been accomplished.
Chapter 2 covers the experimentation performed to characterize the properties of the two porous
substrates used in this study. These properties include the porosity, surface roughness,
permeability, and the substrates thermal conductivity. Chapter 3 compares the effects of pure water
and n-heptane droplets evaporating on the porous substrates when heated and looks at how that
differs from the droplets behavior on an equal sized heated impermeable substrate made from the
same material. An analytical model is presented that predicts n-heptane’s evaporation time in the
film boiling regime. Chapter 4 compares the surface temperature variations at the point of impact
of a pure water and n-heptane droplet on the three heated substrates and determines the heat
transfer coefficients immediately following droplet impact during spreading. The final Chapter 5
will summarize specific conclusions drawn from the work completed in Chapters 3 and 4 and
proposes some work that could be done in the future.
8
Chapter 2 Characterization of Porous Sintered Stainless-Steel Test Substrates
2.1 Introduction
This chapter covers the characterization of properties for the porous substrates used during
experimentation. The properties include the porosity, surface roughness, intrinsic permeability,
and the thermal conductivity. The porous substrates were commercially available (Mott
Corporation, Farmington, Connecticut, United States) ones created by sintering 316 stainless-steel
powders. Sintering involves the compaction of the steel powders at temperatures approaching the
melting point. The controlled atmosphere heating process reduces surface oxide films from the
powder particles and protects particle surfaces throughout the sintering process. The sintering time,
temperature and atmosphere are the critical parameters to help control density, permeability and
pore size of the porous substrate where the details of these processes are often proprietary and can
vary depending on the manufacturer. Two porous substrates were used, one with an average pore
size of 5 μm and the other with an average pore size of 100 μm. The pore structure was visualized
using a table top SEM (TM3000, Hitachi High-Technologies Canada Inc., Etobicoke, Ontario,
Canada) and can be seen in Figure 2.1. A third, impermeable 316 stainless-steel substrate was also
used for comparison. All surfaces were square, approximately 45 mm x 45 mm in size and 2.0 mm
thick. The samples used in this study were selected for their thermal properties (i.e. high effective
thermal conductivities) and differences in porosity, and permeabilities rather than for their use in
a particular application. The porosities and permeabilities were sufficiently different that a
comparison could be made on their effect on the evaporation of an impinging droplet.
The porosity, surface roughness, permeability, and thermal conductivity values obtained in this
chapter are used in Chapter 3 when qualitatively comparing the behavior of the droplets interaction
with the porous surfaces seen through photographs. Similarly, the porosity and permeability values
are used in an empirical model presented to determine the lifetime of a droplet in the film boiling
regime. In Chapter 4 the thermal conductivity values obtained for the substrates will be used when
determining the heat transfer coefficients.
9
5 μm 100 μm
Figure 2.1: SEM images of the sintered porous stainless-steel samples (5 μm and 100 μm average pore size) used
during experimentation (top). Full-scale images of samples (bottom). Substrate dimensions: 45 mm x 45 mm by 2.0
mm thick.
10
2.2 Porosity
The porosity for each of the porous substrates were determined using an imbibition technique
similar to that of the one described by Engblom et al. [30]. They characterized the porosity of their
rock samples by submerging them in degassed water and leaving the sample to allow for the fluid
to enter into the vacant space. Once fully saturated, the wet sample was weighed, and the volume
of fluid absorbed into the rock was determined. The porosity, 휀, is a measure of the fraction of
void space within the material and is deduced by taking the ratio of the void volume contained
within the rock to the dry bulk volume of the rock, and is defined as,
휀 =𝑣𝑜𝑖𝑑 𝑣𝑜𝑙𝑢𝑚𝑒 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑑 𝑤𝑖𝑡ℎ𝑖𝑛 𝑝𝑜𝑟𝑜𝑢𝑠 𝑠𝑎𝑚𝑝𝑙𝑒
𝑏𝑢𝑙𝑘 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑝𝑜𝑟𝑜𝑢𝑠 𝑠𝑎𝑚𝑝𝑙𝑒 (2.1)
A similar technique was applied when characterizing the stainless-steel porous substrates used in
this study. n-Heptane was chosen as the working fluid, due to its lower surface tension. As a result,
the fluid was able to better wet the surface and penetrate the surface pores to fully saturate the
substrate.
2.2.1 Substrate Preparation
Before testing, the porous samples were prepared as follows:
1. Ethanol Bath: The samples were submerged in ethanol for 5 minutes, removed and
dabbed dry.
2. Compressed Air Dry: Dry compressed air was forced through the substrate pores to
remove any fluid and dissolved agents residing in them.
3. Acetone Bath: The samples were submerged in acetone for another 5 minutes, removed
and dabbed dry. The drying process in Step 2 was repeated.
4. Distilled Water Rinse: Distilled water was rinsed through the sample pores to remove
any dissolved agents that might persist after completing Step 2.
11
5. Oven Dry: The substrates were heated to 80oC ±0.5oC in an oven (LAC 1-38, Despatch,
Minneapolis, Minnesota, United States) and left for 1 hour to allow for any residual
cleaning agents to evaporate out of the pores.
Ethanol and Acetone work as a solvent that can dissolve organic compounds that might reside on
the materials surface [31].
2.2.2 Results of Measured Porosity
To determine the measured porosity the sample was submerged in n-heptane for 5 minutes, to
ensure that the fluid entered all the vacant pores. Once removed from the n-heptane, any excess
fluid was carefully scraped off the sample. The sample was immediately moved to a tray sitting
on a laboratory digital weight scale (AG245, Mettler Toledo, Mississauga, Ontario, Canada) and
the saturated sample, WS, is weighed. Knowing the dry weight of the sample, WD, and the density
of n-heptane, 𝜌𝐻, the volume of void space within the sample, VV (i.e. volume of n-heptane within
the pores), can be determined as follows,
𝑉𝑉 =𝑊𝑆−𝑊𝐷
𝜌𝐻 (2.2)
The bulk volume of the sample can be determined by,
𝑉𝐵 = 𝐿𝑒𝑛𝑔𝑡ℎ 𝑥 𝑊𝑖𝑑𝑡ℎ 𝑥 𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 (2.3)
The porosity can then be deduced,
휀 =𝑉𝑉
𝑉𝐵 (2.4)
The saturated sample weight measurement was repeated 51 times. A summary of the exact plate
dimensions used to determine the bulk volumes, the void volumes, and the porosities, along with
their associated standard deviations are listed in the following Table 2.1.
12
Table 2.1: Summary of substrate volumes and porosities.
Average Pore Size
[μm]
Substrate Dimensions
(L x W x T) [mm]
Void Volume (VV)
[mm3]
Porosity (𝜺) [%] Standard Deviation
[%]
5 44.91 x 44.84 x 1.67 1028.9 30.6 ±1.9
100 45.45 x 45.12 x 2.46 2235.3 44.3 ±0.7
As n-heptane is a volatile liquid, to estimate how much fluid evaporated off the surface of the
saturated substrate while being transitioned to the weigh tray, the sample was left on the laboratory
digital weight scale for 30 seconds where the weight as a function of time was recorded. As the
weight decreased linearly with time, the slope of the line yielded an evaporation rate of 0.175 mg/s.
The average time taken to transition the substrate to the weigh tray was approximately 2 seconds,
resulting in a potential evaporation of 0.35 mg of n-heptane. The net effect on the calculated
porosity was determined to be negligible at 0.02%.
2.3 Surface Roughness
The smoothness or irregularities inherent in a substrates surface can be defined by a surface
roughness parameter. While there are several different roughness parameters that can be used to
characterize a surface, typically, in engineering applications the roughness average, Ra, is chosen.
Ra, as defined by ISO 4287-1998 and reviewed by Kai et al., is the arithmetic average of the
absolute values of the profile heights over the evaluation length [32].
2.3.1 Results of Measured Surface Roughness
A skid-referenced piloter Surfometer (Precision Devices Inc., Milan, Michigan, United States) was
used to measure the Ra values for the porous and impermeable substrates. The substrates were
placed on a rubber mat to prevent them from sliding as the Surfometer’s piloter travelled across
the surface. The tip radius of the piloter was 2.5 μm and the stroke length was set to 10.16 mm.
The measurements were repeated 10 times at different locations as to get an average Ra value
representative for the entire surface, for each substrate. Table 2.2 summarizes the measured Ra
values for the three substrates along with their associated standard deviations.
13
Table 2.2: Summary of average roughness measurements.
Average Pore Size
[μm]
Roughness Average
(Ra) [μm]
Standard Deviation
[μm]
Impermeable 0.100 ±0.004
5 4.7 ±0.1
100 12.2 ±1.6
2.4 Permeability
The intrinsic permeability is a measure of the ease with which a fluid can pass through a porous
medium. The Darcy flow model, or Darcy’s Law, is an accepted relationship to determine a porous
substrates permeability when the velocity of the fluid flowing through the substrate is low. First
introduced by Henry Darcy [33, 34], Darcy’s Law relates the rate of fluid filtration to the pressure
gradient in a saturated isotropic porous medium, and is described as [35]:
𝑞 = −𝜅
𝜇∇𝑝 (2.5)
Where q is the fluid discharge per unit area, 𝜅 is the permeability of the substrate, 𝜇 is the dynamic
viscosity of the fluid, and ∇𝑝 is the pressure gradient.
An experimental setup similar to that of the one employed by Venter and Jermy [36] was used to
measure the permeability of the porous substrates. In their experimental setup they measured the
flow rate of methane and nitrogen gas passing through roof shingles along with the pressure drop
across the roof shingles. Using an extended version of Darcy’s Law that accounts for the
compressibility of a gas, the intrinsic permeability was deduced; a similar method will be used in
this study. Klinkenberg [37], studied the effect of using gases instead of liquids to determine a
porous medias intrinsic permeability and discovered that when using a compressible fluid (i.e. a
gas), the permeability will be greater than the value obtained with a liquid. Klinkenberg reasoned
this effect was due to “gas slippage”, where liquids had a zero velocity at the solid surface while
gases exhibited some finite velocity resulting in a higher flow rate at a given pressure differential.
The resultant apparent permeability can be corrected to yield the intrinsic permeability
characteristic of the porous medium.
14
2.4.1 Experimental Setup
2.4.1.1 Equipment Overview
Figure 2.2 shows the experimental setup employed to measure the permeability of the porous
substrates. Dry air supplied by a compressor (Airtower 26 TU 26, Kaeser Compressors Canada
Inc., Boisbriand, Quebec, Canada) at a constant pressure of 745 kPa (108 PSI) was used as the
working fluid circulated through the setup. Two mass flow meters were used to measure the
volumetric flow rate of the dry air into the system. At low flow rates, between 0 and 30 L/min, a
flow meter (FMA5526A, Omega Engineering, Laval, Quebec, Canada) with a range of 0 to 30
L/min was employed. At high flow rates, between 30 and 220 L/min a flow meter (FMA1843,
Omega Engineering, Laval, Quebec, Canada) with a range of 0 to 220 L/min was used. The flow
rate was controlled using a precision flow adjustment valve supplied by McMaster-Carr (Elmhurst,
Illinois, United States). The FMA5526A was used for low flow rates as it offers a measurement
accuracy of ±1.5% over the full flow range while the FMA1843 offers the same accuracy within
the range of 40 to 220 L/min, and an accuracy of ±3% at flow rates below 40 L/min. The flow
meters geometry of the primary conduit and the sensor tube are designed to ensure that the flow
that reaches the sensor is laminar [38, 39]. The dry air travels into a settling chamber machined
from 6061 Aluminum where it rises and passes through the porous substrate. The chamber consists
of two parts, a lid (height: 7.7 cm, outer diameter: 15.2 cm, wall thickness: 1.4 cm) that has a 10.1
cm diameter opening on the top and threads onto the chambers body (height: 10.3 cm, outer
diameter: 12.4 cm, wall thickness: 1.3 cm, total thread height: 3.7 cm) securing the substrates
mounting bracket in place. A differential pressure gauge (DPG409-050DWU, Omega
Engineering, Laval, Quebec, Canada) was used to measure the pressure drop across the sample,
where one input was fastened in series with the fluid flow line going into the chamber, and the
other input was open to the atmosphere. Thread sealing tape was used on all tapered threaded
connections in the system to assist with preventing fluid leakage. All tubing in the system was
made from Polyvinyl Chloride (PVC) and is 6 mm in diameter. The substrate was prepared before
testing using the same procedure outlined in section 2.2.1.
15
Figure 2.2: Experimental setup employed to determine the permeability of the porous substrates with key components
labeled (top, middle). Schematic of the experimental setup (bottom).
16
2.4.1.2 Substrate Mounting Bracket
Figure 2.3 shows a detailed view of the bracket assembly used to fix the porous substrate in place.
A two-piece mounting bracket consisting of a sample holder and clamping bracket was created.
The sample holder was machined from 6061 Aluminum and has a 11.6 cm diameter, large enough
to cover the opening in the settling chamber. The holder’s thickness is 0.9 cm allowing the settling
chambers lid when tightened to push down on it and seal the settling chambers opening due to two
o-ring rubber gaskets (2.3 mm thickness, 113 mm outer diameter) that wrap around the top and
bottom faces of the holder, preventing gas leakage. It has a 3.0 cm square routed straight through
the center to allow the dry air to pass. A section (6.0 cm x 6.0 cm x 0.6 cm deep) was routed into
the sample holder where the sheet gaskets and substrate will be secured. The clamping bracket (5.9
cm x 5.9 cm x 0.46 cm thick) was machined from 316 Stainless-Steel to offer increased rigidity
when tightened down onto the substrate using 4, M2 screws. It has the same sized 3.0 cm square
opening aligning with the pass through in the sample holder. A 1.02 mm thick super-cushioning
rubber sheet gasket cut to match the size of the substrate with the same 3.0 cm square hole in the
center is fit around the opening of the holder. The substrate will sit on the gasket and another gasket
cut to the same size will sit on top. A gasket made from the same material is cut such that it wraps
around the sides of the substrate slightly folding over the top and bottom. The gaskets are used to
prevent the gas from escaping out the sides of the substrate and force it through the 3.0 cm square
opening. The clamping bracket is tightened compressing the gaskets around the sample and sealing
it in place until no air is detected leaking from the sides. A mixture of soap and water was applied
to all connections and openings where air could escape to test for leakage. All gaskets were
supplied by McMaster-Carr.
17
Figure 2.3: 3d rendering of bracket used to mount porous samples into settling chamber with key components labeled.
2.4.2 Measurement Uncertainty
To define the uncertainty with the results produced by the permeability measurement apparatus, a
design-stage uncertainty analysis similar to the approach outlined by Figliola and Beasley [40] was
performed. The system consists of three major components. The pressure gauge, used to measure
the pressure drop across the substrate, the flow meters, used to measure the volumetric flow rate
of the dry air passing through the system, and finally the caliper, used to measure the thickness of
the substrates. For each component, the errors fell into two categories: the error based on the
resolution of the instrument, estimated by the zero-order uncertainty, 𝑢0; and the error based on
the manufacturer’s statement defining the instrument uncertainty, 𝑢𝑐. The design-stage uncertainty
for each major component, 𝑢𝑑, can then be deduced by using a root-sum-squares (RSS) method to
combine the individual uncertainties and is defined as:
𝑢𝑑 = √𝑢02 + 𝑢𝑐
2 (2.6)
This method to break down the errors associated with each measurement component into a zero-
order uncertainty, and instrument uncertainty will be followed for future uncertainty analyses
presented in this study.
18
The uncertainties associated with the pressure gauge, flow meters, and caliper are as follows:
Omega Pressure Gauge:
Accuracy: ±0.08% which includes errors associated with linearity, hysteresis, and repeatability.
Resolution: 68.95 Pa (0.01 PSI).
Omega Flow Meters:
FMA5526A:
Accuracy: ±1.5% over the full-scale flow range.
Resolution: Negligible.
FMA1843:
Accuracy: ±3.0% over 0-20% of the flow range, and ±1.5% over 20-100% of the flow range.
Resolution: Negligible.
Mitutoyo Caliper:
Accuracy: ±0.02 mm.
Resolution: Negligible.
The accuracy accounts for the instruments uncertainty while the resolution accounts for the zero-
order uncertainty. A sample calculation to determine the uncertainty on the final calculated
permeability result can be seen in Appendix A.
2.4.3 Data Reduction and Analysis
To determine the intrinsic permeability of the porous substrates the volumetric flow rates of the
dry air entering the system were scaled from 0 to 220 L/min in increments of 5 L/min. At each
flow rate the pressure differential across the substrate was recorded. For a compressible fluid, the
volumetric flow rate, Q, varies with a change in pressure, P, according to the following
relationship:
𝑃𝑄 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝑃𝑚𝑄𝑚 (2.7)
19
Pm is the mean pressure and is defined as:
𝑃𝑚 =𝑃𝑐ℎ𝑎𝑚𝑏𝑒𝑟 + 𝑃𝑎𝑡𝑚
2 (2.8)
Where 𝑃𝑐ℎ𝑎𝑚𝑏𝑒𝑟 is the pressure inside the settling chamber, and 𝑃𝑎𝑡𝑚 is atmospheric pressure. The
apparent permeability can be deduced by simplifying equation (2.5), and using equations (2.7) and
(2.8), can be integrated directly to relate the volumetric flow rate to the pressure gradient in the
flow direction which yields:
𝑄 =𝜅′𝐴(𝑃𝑐ℎ𝑎𝑚𝑏𝑒𝑟
2−𝑃𝑎𝑡𝑚2)
2𝜇𝐿𝑃𝑎𝑡𝑚 (2.9)
Where A is the porous surface cross sectional area perpendicular to the flow, 𝜅′ is the apparent
permeability, and L is the thickness of the substrate. Rearranging for the apparent permeability the
following expression is obtained:
𝜅′ =𝑄2𝜇𝐿𝑃𝑎𝑡𝑚
𝐴(𝑃𝑐ℎ𝑎𝑚𝑏𝑒𝑟2−𝑃𝑎𝑡𝑚
2) (2.10)
To account for gas slippage a Klinkenberg permeability plot was employed where the apparent
permeability, 𝜅′, is plotted against the reciprocal of the mean pressure, Pm, at the respective flow
rate. The straight line plot is extrapolated to the infinite mean pressure, where the gas behaves like
a liquid (1/Pm = 0), to determine the intrinsic permeability [41]. Figures 2.4 and 2.5 show plots of
the apparent permeability as a function of the reciprocal of the mean pressure for the 5 μm and 100
μm porous substrates over the full flow range investigated. Apparent permeability values that
deviate from a straight-line need to be excluded from the analysis as they represent visco-inertial
flows and can yield a lower value than the true permeability. Figure 2.6 shows a Klinkenberg
permeability plot for both the 5 μm and 100 μm porous substrates. The visco-inertial flows are
excluded from the plot to yield only the linear data.
20
Figure 2.4: Apparent permeability as a function of the reciprocal of the mean pressure for the 5 μm porous substrate
over the full flow range investigated.
Figure 2.5: Apparent permeability as a function of the reciprocal of the mean pressure for the 100 μm porous substrate
over the full flow range investigated.
0.00E+00
5.00E-14
1.00E-13
1.50E-13
2.00E-13
2.50E-13
3.00E-13
3.50E-13
4.00E-13
0.00 2.00 4.00 6.00 8.00 10.00
Ap
par
ent
Per
mea
bili
ty k
' [m
2]
1/Pm [mm2/N]
0.00E+00
5.00E-12
1.00E-11
1.50E-11
2.00E-11
2.50E-11
3.00E-11
0.00 2.00 4.00 6.00 8.00 10.00
Ap
par
en
t P
erm
eab
ility
k' [
m2]
1/Pm [mm2/N]
21
Figure 2.6: Klinkenberg permeability plot for both the 5 μm and 100 μm porous substrates. Visco-inertial flows have
been excluded. At 1/Pm = 0 the intrinsic permeability for the substrate is found. The intrinsic permeability is indicated
for the 5 μm substrate.
The intrinsic permeability for the 5 μm substrate was determined to be 3.3x10-14 m2. A negative
extrapolated intrinsic permeability yielded from the 100 μm substrates Klinkenberg permeability
plot. As this is not possible, the outcome is likely a result of not achieving a high enough pore
pressure due to the limitations of the experimental setup. Due to this, the apparent permeabilities
produced at the high pressures investigated were still in the non-linear visco-inertial regime. The
linear trend line was fit to a non-linear portion of the dataset resulting in the erroneous intrinsic
permeability. An estimate of the intrinsic permeability for the 100 μm substrate was thus obtained
from averaging the last 5 apparent permeabilities produced and was determined to be 7.0x10-13 m2.
As expected due to the increased porosity of the 100 μm substrate the intrinsic permeability was
approximately an order of magnitude larger than the 5 μm substrates permeability. The increase in
porosity made it easier for the gas to flow through the porous structure where the pressure
differential across the substrate decreased, resulting in an increase in the permeability.
A summary of the intrinsic permeabilities and their associated uncertainties can be seen in Table
2.3.
-1.20E-12
-7.00E-13
-2.00E-13
3.00E-13
8.00E-13
1.30E-12
0.00 2.00 4.00 6.00 8.00 10.00
Ap
par
ent
Per
mea
bili
ty k
' [m
2]
1/Pm [mm2/N]
100 μm - Linear
5 μm - Linear
Intrinsic Permeability = 3.3x10-14 m2
22
Table 2.3: Summary of the porous substrates intrinsic permeabilities and their associated uncertainties.
Average Pore Size
[μm]
Intrinsic
Permeability (𝜿)
[m2]
Uncertainty
(95%) [m2]
5 3.3x10-14 ±3.3x10-15
100 7.0x10-13 (estimate) -
2.5 Thermal Conductivity
The rate at which heat transfers within the material is defined by the thermal conductivity. To
determine the thermal conductivity, 𝑘, in the porous samples used in this study, an experimental
technique similar to that of the one outlined in ASTM E1225 was employed [42]. This method
assumes the samples being investigated are homogeneous solids where the effective thermal
conductivity lies between the approximate range of 0.2 < 𝑘 < 200 𝑊
𝑚∙𝐾 over the approximate
temperature range of -183 to 1027oC. The method involves loading a sample between two
specimens of a material with known thermal properties. A constant heat source is applied to one
end of the stack, where a temperature gradient across the stack is produced. When the system
reaches steady state, the known temperature gradients from the hot and the cold side, along with
the thermal properties of the known specimens can be used to determine the thermal conductivity
of the sample.
2.5.1 Experimental Setup
2.5.1.1 Stack Arrangement
Figure 2.7 shows the experimental setup employed to determine the thermal conductivity of the
porous substrates. The two specimens with known thermal properties were chosen as 6061
Aluminum with a thermal conductivity of 167.0 𝑊
𝑚∙𝐾 and were supplied by McMaster-Carr
(Elmhurst, Illinois, United States). Each aluminum bar was 15.24 cm (6”) long and 5.08 cm (2”)
in diameter, with the type J thermocouples spaced 2.54 cm (1”) apart starting at 1.27 cm (½”) from
the base of each bar. The thermocouples were inserted 0.635 cm (¼”) into the bar. 15.24 cm lengths
were chosen for each bar to allow adequate spacing between the thermocouples to ensure a
temperature gradient would be sensed. The entire stack was held together using 4, 30.48 cm (12”)
23
long 304 stainless steel rods that were bolted to the four corners of the heat sink and the heater
block. 304 stainless steel was chosen for the rods material due to the lower thermal conductivity
as compared to that of aluminum and copper. The lower thermal conductivity of the stainless steel
will reduce the heat transfer into the columns as compared to the larger aluminum bars. Similarly,
a small air gap was left between the rod ends and the heater block and heat sink to act as an
insulating layer. The thermal conductivity for the 304 stainless steel rods is 14.4 𝑊
𝑚∙𝐾 and were
supplied by McMaster-Carr.
24
Figure 2.7: Experimental setup employed to determine the thermal conductivity of the porous substrates with key
components labeled (top). Schematic of the experimental setup (bottom).
2.5.1.2 Heating and Cooling of Stack
A 101 Copper heater block (20.32 cm (8”) x 20.32 cm (8”) by 1.59 cm (5/8”) thick) equipped with
10, 500 W cartridge heaters powered by two DC power supplies (PS-1000-55, Epsco Incorporated,
25
Bloomingdale, Illinois, United States) were used to heat the stack. Each power supply was set to
15V at 3A for a total power input of 90W. An aluminum, liquid-cooled, heat sink sized to the same
dimensions as the heater block was used to remove heat from the top of the stack. The liquid was
cooled by a chiller (N0772025, PolyScience, Niles, Illinois, United States) circulating (1.27 cm
(½”) hosing, 0.95 cm (3/8”) bores through aluminum block) SCIEX coolant set to maintain a fluid
temperature of 20oC. This fluid temperature was empirically determined to produce the best results
and induce a measurable temperature gradient across the stack. Thermally conductive silicone
paste (OT-201, Omega Engineering, Laval, Quebec, Canada) was applied at the interface between
the aluminum bars and the heater block, and heat sink and is rated for continuous use between
40oC and 200oC [43]. The heater block was placed on a sheet of Pyrogel XTE insulation (Pyrogel
XTE, Aspen Aerogels Incorporated, Northborough, Massachusetts, United States) and were
housed in a rubber insulating mold (Mold Max 60 Silicone Rubber Compound, Smooth-On,
Macungie, Pennsylvania, United States). Similarly, the stack was wrapped in a 2.54 cm (1”) thick
layer of Superwool Plus insulation (Superwool Plus blanket, Morgan Advanced Materials,
Burlington, Ontario, Canada). The insulation used served as a means to reduce the heat transfer to
the table and surroundings minimizing the time taken for the system to reach a steady state. A
summary of the thermal conductivities associated with the various materials used in the apparatus
are listed in the following Table 2.4.
Table 2.4: Thermal conductivities for the various materials used in the apparatus. The values are stated at 25oC.
Material
Thermal
Conductivity (𝒌)
[𝑾
𝒎∙𝑲]
6061 Aluminum 167.0
304 Stainless-Steel 14.4
101 Copper 390.9
Superwool Plus 0.05
Pyrogel XTE 0.02
Mold Max 60 Rubber 0.35
Air 0.03
26
2.5.1.3 Thermal Interface Material
Two thermal interface graphite sheets (EYGS0909ZLX2, Panasonic Canada Inc., Mississauga,
Ontario, Canada) were used between the substrate and the aluminum bars to reduce the contact
resistance and enhance the heat transfer. Due to a small substrate sample thickness, the thermal
resistance across the graphite sheets is not negligible as compared to the thermal resistance across
the substrates and needs to be accounted for in the analysis. A constant clamping force was applied
to the stack by tightening the 4 bolts fastening the heat sink to the 4, 304 stainless steel rods. The
clamping force is needed to further reduce any contact resistance and minimize the thermal
resistance across the graphite sheets. This would have the effect of making the final thermal
conductivity’s calculation for the substrates less sensitive to small changes in the measured thermal
resistance of the graphite sheets, improving the accuracy of the given value.
The thermal resistance is a measure of a materials ability to resist the flow of heat. In this study it
is defined as the temperature difference, ∆𝑇, at steady state between the two sides of thermal
graphite sheets in contact with the aluminum bars, divided by the heat flow through a unit area,
q”:
𝑅 = ∆𝑇
𝑞" (2.11)
With the known thermal conductivity value of the aluminum bars, 𝑘𝐴𝐿, and knowing the
temperature difference, and spacing, ∆𝑥, between the inserted thermocouples, the heat flow
through a unit area applied to the sample, or heat flux can be determined using Fourier’s law of
heat conduction integrated for a homogeneous material of 1-D geometry at constant temperature:
𝑞" = −𝑘𝐴𝐿∆𝑇
∆𝑥 (2.12)
For all experiments conducted a heat flux was determined in both the hot and cold aluminum bars
in the stack and an average was taken to determine the heat flux applied to the sample.
The clamping force was determined based on Figure 2.8, pulled from the specification datasheet
for the thermal graphite sheets [44]. Figure 2.8 shows the thermal resistance, for a single graphite
sheet as a function of the applied pressure. The thermal resistance was confirmed empirically at
the chosen applied pressure to ensure the values are within the same order of magnitude. The
27
empirically determined value would be used in place of the value obtained from Figure 2.8 when
calculating the thermal resistances of the substrates. A constant pressure of 265 kPa was chosen to
be applied to the stack. From Figure 2.8, this would yield an approximate total thermal resistance
value for both graphite sheets of 1.2 𝑐𝑚2∙𝐾
𝑊. Empirically, a thermal resistance of 1.3
𝑐𝑚2∙𝐾
𝑊 was
deduced, resulting in an 8% difference from the value obtained from Figure 2.8. A pressure greater
than 265 kPa was not applied to the stack due to the potential strain that would be put on the
apparatus and the risk of deforming either the aluminum heat sink or the copper heater block.
Applying the pressure over the diameter of the aluminum bar, 5.08 cm, a clamping force of 537 N
was determined.
Figure 2.8: Thermal resistance as a function of the applied pressure for the thermal interface graphite sheets as
provided by Panasonic [44].
To achieve this clamping force, the torque applied to the 4 screws fastening the heat sink to the
304 stainless steel rods can be obtained using the following expression as outlined by Budynas and
Nisbett [45]:
𝑇 = 𝐾 ∙ 𝐹𝑖 ∙ 𝐷 (2.13)
Where T is the total torque applied to the system and K is the torque coefficient and depends on
the coefficient of friction between the two materials being threaded into one another. In this case
steel bolts were being threaded into steel rods, where K = 0.8. Fi is the desired clamping force.
28
Finally, D is the major bolt diameter. Four 8-32 bolts were used, having a major bolt diameter of
4.17 mm (0.1640”). The resultant total torque was determined to be 1.8 Nm. A slip-release
adjustable torque-limiting screwdriver was used to apply a quarter of this calculated torque to each
bolt, as 4 bolts were used. A summary of the thermal resistance values obtained for the thermal
graphite sheets can be seen in the following Table 2.5.
Table 2.5: Summary of thermal resistance values for the impermeable stainless-steel substrate, and the thermal
graphite sheet. These values are stated for an applied clamping pressure to the stack of 265 kPa.
Material Empirical Thermal
Resistance (R)
[𝒄𝒎𝟐∙𝑲
𝑾]
Uncertainty (95%)
[𝒄𝒎𝟐∙𝑲
𝑾]
Figure 2.8 Thermal
Resistance (R)
[𝒄𝒎𝟐∙𝑲
𝑾]
Thermal Graphite
Sheet x2
1.3 ±0.1 1.2
2.5.1.4 Data Collection
Six gauge 24, type J thermocouples were inserted into each aluminum bar, capable of measuring
temperatures within the range of 0oC to 750oC [46]. Six thermocouples were chosen to confirm
the linearity of the temperature gradient once steady state was reached, and to have a contingency
in the event one of the thermocouples was reading inaccurately. This ensures a linear trendline can
still be constructed with the remaining thermocouples and the heat flux can be deduced. The heat
transfer is considered to be one dimensional from the heater block to the liquid cooled heat sink.
The thermocouple voltages were read by a data acquisition module (DAQ) (OMB-DAQ-56,
Omega Engineering, Laval, Quebec, Canada) and logged directly to a personal computer (PC)
running Personal DaqViewTM. The software read the voltage, converted the voltage to a
temperature, and logged the temperature to a file every 2 seconds.
2.5.1.5 Measurement Uncertainty
A design-stage uncertainty analysis on the thermal conductivity apparatus was performed. It will
be assumed for this analysis that the system consists of three major components. The DAQ, used
to read in the thermocouple voltages, the type J thermocouples, used to sense the temperature
gradient in the aluminum bars, and finally the caliper, used to measure the thickness of the
29
substrates. The procedure for breaking down the errors for each component was outlined in section
2.4.2.
The uncertainties associated with the DAQ, type J thermocouples, and caliper are as follows:
Omega DAQ:
Resolution and Accuracy: The DAQ has a built in 22-bit analog to digital (A/D) converter
allowing for high-resolution measurements accurate to 0.01% of the reading plus 0.002% of the
range [47].
Type J Thermocouple:
Accuracy: ±2.2oC.
Resolution: Personal DaqViewTM limits the resolution of type J thermocouples to ±0.001oC.
Mitutoyo Caliper:
Accuracy: ±0.02 mm.
Resolution: Negligible.
A sample calculation to determine the uncertainty on the final calculated thermal conductivity
result can be seen in Appendix A.
2.5.2 Data Reduction and Analysis
Four sets of experiments were conducted. The first experiment was to determine the thermal
resistance of the thermal graphite sheets. This resistance would be subtracted from the total
resistance values (2x thermal graphite sheets + sample) determined for the calibration sample and
porous substrates before deducing their thermal conductivities. The second experiment was to
measure the thermal conductivity of a calibration sample with a known thermal conductivity. Due
to the cylinder having a thermal resistance an order of magnitude larger than the thermal graphite
sheets, the apparatus could be tested to ensure it was operating correctly. Small changes in the
resistances obtained for the thermal graphite sheets would have a small impact on the thermal
conductivity obtained for the calibration sample. The third and fourth tests were to measure the
thermal conductivities of the porous substrates. Each experiment ran until the temperatures
measured by the thermocouples in both aluminum bars reached steady state where an observed
30
change in temperature over the last hour of operation was approximately 1oC or less. Figure 2.9
shows the 12 thermocouple temperatures inserted into the aluminum bars as a function of the time
the test was running for the calibration sample, where the last hour of data is shown.
Figure 2.9: Thermocouple temperatures a function of the time the test was running, showing the last hour for the
calibration sample. A diagram illustrating thermocouple position in the aluminum bars is shown.
2.5.2.1 Calibration Sample
A 316 stainless-steel calibration cylinder, 3.81 cm long (1.5”) by 5.08 cm (2”) in diameter, with a
nominal thermal conductivity of 15.9 𝑊
𝑚∙𝐾 specified by the manufacturer was used to test the
experimental setup. A constant average heat flux of 16.7 𝑘𝑊
𝑚2 was applied to the sample. The average
was taken from the heat flux obtained in the hot aluminum bar, and the cold aluminum bar. The
difference in heat fluxes were less than 5% where the change can be attributed to heat being lost
to the surroundings. Figure 2.10 shows the temperature gradient in the hot and cold aluminum
bars. The two temperature points before and after the sample at the interface between the thermal
graphite sheets and the aluminum bars were extrapolated from the linear trendlines shown on the
plot.
0
20
40
60
80
100
120
8500 9500 10500 11500 12500 13500
The
rmo
cou
ple
Tem
pe
ratu
re [
oC
]
Time [s]
(1)
(12)
TH
RO
UG
H
31
Figure 2.10: Thermocouple temperature as a function of its position in the hot and cold aluminum bars for the
calibration sample. The equations for each linear trendline are shown. The temperatures at the interfaces between the
thermal graphite sheets and the hot (TH) and cold (TC) aluminum bars are indicated.
Knowing the temperature difference, TH – TC, across the two thermal graphite sheets and sample,
and the applied heat flux, the total thermal resistance for the system, using equation 2.11, was
determined to be 30.3 𝑐𝑚2∙𝐾
𝑊. To determine the thermal conductivity of the calibration sample
equation 2.11 can be expressed in terms of the samples thickness and its thermal conductivity:
𝑅 = ∆𝑇
𝑞"=
𝐿
𝑘 (2.14)
Where 𝐿, is the thickness of the calibration sample, and 𝑘 is its thermal conductivity. Rearranging
equation 2.14 to solve for the thermal conductivity, and subtracting the thermal resistance of the
two graphite sheets from the total thermal resistance, the expression to determine the thermal
conductivity of the calibration sample is as follows:
𝑘 =𝐿
𝑅−2∙𝑅𝐺𝑆 (2.15)
y = -0.102x + 102.525
y = -0.098x + 54.975
0
20
40
60
80
100
120
0 50 100 150 200 250 300 350
Tem
per
atu
re [
oC
]
Spacing [mm]
Hot Aluminum Bar
Cold Aluminum Bar
TH
TC
Extrapolated temperatures at
the interface between the
cylinders and the thermal
interface material.
32
Where RGS is the thermal resistance across a single thermal graphite sheet. The thermal
conductivity for the calibration sample was calculated to be 13.1 𝑊
𝑚∙𝐾 which is an 18% decrease
from the calibrated value given from the manufacturer for the 316 stainless-steel cylinder.
Considering the uncertainty on the thermal conductivity calculation, being ±1.7 𝑊
𝑚∙𝐾, the percent
decrease can be reduced to less than 7%.
2.5.2.2 Porous Substrates
A constant average heat flux of 22.2 and 19.2 𝑘𝑊
𝑚2 were applied to the 5 µm and 100 µm porous
substrates respectively. Figure 2.11 shows the temperature gradient in the hot and cold aluminum
bars for both porous samples. The equations for the linear trendlines fit to the data points, and the
extrapolated temperatures before and after the sample at the interface between the thermal graphite
sheets and the aluminum bars are shown.
Figure 2.11: Thermocouple temperature as a function of its position in the hot and cold aluminum bars for the 5 and
100 µm porous substrates. The equations for each linear trendline are shown. The temperatures at the interfaces
between the thermal graphite sheets and the hot (TH) and cold (TC) aluminum bars are indicated.
y = -0.139x + 82.569
y = -0.113x + 105.521
0.00
20.00
40.00
60.00
80.00
100.00
120.00
0.000 50.000 100.000 150.000 200.000 250.000 300.000
Tem
per
atu
re [
oC
]
Spacing [mm]
5 µm - Hot Side5 µm - Cold Side100 µm - Hot Side100 µm - Cold Side
TC
TC
TH
TH
33
The higher average temperature observed on the hot side of the 100 µm substrate system is a result
of a larger thermal resistance across the substrate and graphite sheets. Following the same
procedure outlined in section 2.5.2.1 the total thermal resistances and thermal conductivities
determined for the 5 µm and 100 µm substrate systems were 7.7 and 24.4 𝑐𝑚2∙𝐾
𝑊 and 2.7 and 1.0
𝑊
𝑚∙𝐾 respectively.
A summary of the total thermal resistances, thermal conductivities, applied average heat fluxes,
and their associated uncertainties for the calibration cylinder and two porous substrates can be seen
in the following Table 2.6. All uncertainties are stated at the 95% level.
Table 2.6: Summary of the calculated properties and applied average heat fluxes for the calibration cylinder and two
porous substrates.
Stainless Steel 316
plate 5 µm Porous Substrate 100 µm Porous Substrate
Total Thermal Resistance, R
[𝒄𝒎𝟐 ∙ 𝑲
𝑾]
30.3 ±3.7 7.7 ±0.7 24.4 ±2.6
Thermal Conductivity, k
[𝑾
𝒎 ∙ 𝑲] 13.1 ±1.7 2.7 ±0.3 1.0 ±0.1
Applied Average Heat Flux, q’’
[𝒌𝑾
𝒎𝟐 ] 16.7 ±2.0 22.2 ±2.0 19.2 ±2.0
2.5.2.3 Analytical Models
Several analytical models have been developed to predict the effective thermal conductivity of
composite materials providing a range in which the results are guaranteed. Reuss [48] and Voigt
[49] were among the first to introduce models representing the upper and lower bounds of the
effective properties of composite materials. Reuss [48] deduced a model for composite materials
considering the phases as serial, while Voigt [49] considered them as parallel paths. Wiener [50]
and Egli [51] took these ideas and applied them towards heat conduction problems.
The serial model assumes that the composite phases within the material are orientated
perpendicular to the direction of the heat flux where the fluid and solid phases alternate, leaving
34
no direct path in either of the phases leading from the hot to the cold side. This represents the lower
bound for what the thermal conductivity can take on and is shown in the following equation (2.16):
𝑘𝑅𝑒𝑢𝑠𝑠 =1
𝜀
𝑘𝑓+
(1−𝜀)
𝑘𝑠
(2.16)
Where kf and ks are the thermal conductivities associated with the fluid and solid phases within the
composite material. 휀 is the porosity of the composite material. The parallel model assumes that
the composite phases are orientated laterally with the direction of the heat flux where the phase
layers alternate and are equal in size. A direct path in both phases exist between the hot and cold
sides of the material. This represents the upper bound for what the thermal conductivity can take
on and is shown in the following equation (2.17):
𝑘𝑉𝑜𝑖𝑔𝑡 = 휀𝑘𝑓 + (1 − 휀)𝑘𝑠 (2.17)
The Tetrahedral Unit Cell model (TUCM) developed by Boomsma and Poulikakos [52] was
developed for use with open celled metallic foams. The solid phase consists of tetrakaidecahedron
shaped cells filled with the fluid phase. The TUCM is as follows:
𝑘𝑇𝑈𝐶𝑀 =√2
2(𝑅𝐴+𝑅𝐵+𝑅𝐶+𝑅𝐷) (2.18)
Where RA to RD are:
𝑅𝐴 =4𝑑
(2𝑒2+𝜋𝑑(1−𝑒))𝑘𝑠+(4−2𝑒2−𝜋𝑑(1−𝑒))𝑘𝑓 (2.19)
𝑅𝐵 =(𝑒−2𝑑)2
(𝑒−2𝑑)𝑒2𝑘𝑠+(2𝑒−4𝑑−(𝑒−2𝑑)𝑒2)𝑘𝑓 (2.20)
𝑅𝐶 =(√2−2𝑒)
2
2𝜋𝑑2(1−2𝑒√2)𝑘𝑠+2(√2−2𝑒−𝜋𝑑2(1−2𝑒√2))𝑘𝑓
(2.21)
𝑅𝐷 =2𝑒
𝑒2𝑘𝑠+(4−𝑒2)𝑘𝑓 (2.22)
35
Finally, variables d and e:
𝑑 = √√2(2−(
5
8)𝑒3√2−2 )
𝜋(3−4𝑒√2−𝑒 (2.23)
𝑒 = 0.339
The following Figure 2.12 shows the various model predictions for the effective thermal
conductivities over a porosity range of 0 to 100%. The values obtained through experimentation
are shown for comparison. For all models shown, the solid phase was chosen as 316 stainless-steel
while the fluid phase was air. Their associated thermal conductivities are 13.1 𝑊
𝑚∙𝐾 and 0.03
𝑊
𝑚∙𝐾
respectively.
Figure 2.12: Various model predictions for the effective thermal conductivity as a function of the materials porosity.
The thermal conductivities obtained through experimentation for the calibration cylinder and two porous substrates
are shown for comparison.
As expected the experimentally obtained effective thermal conductivities lie between the upper
and lower bounds predicted by Reuss and Voigt. The TUC model appears to trend close to the
values being observed experimentally. While the SEM images presented in section 2.1 show a
0
5
10
15
20
0 20 40 60 80 100
Effe
ctiv
e Th
erm
al C
on
du
ctiv
ity
[W/m
K]
Porosity [%]
Measured Thermal Conductivity
Series (Reuss) Model
Parallel (Voigt) Model
Tetrahedral Unit Cell Model (TUCM)Calibration Cylinder
5 µm Substrate
100 µm Substrate
36
chaotic pore structure due to the compaction of the stainless-steel powders, the heat conduction
through the geometry must behave similarly to the assumptions made in the TUC model where
tetrakaidecahedron shaped cells are employed.
37
Chapter 3 Droplet Impact and Evaporation on Heated Sintered Porous Stainless-Steel Substrates
3.1 Introduction
The total evaporation time of a droplet is dependent on several key parameters. Bernardin et al.
[53] pointed out that the heat transfer rate to the droplet is governed by the fluid and solid thermal
properties, as well as the surface roughness and temperature. The surface temperature was
described as the most significant parameter as it is used to define what heat transfer regime exists
(e.g. evaporation, nucleate boiling, transition boiling, or film boiling). However, when dealing
with porous surfaces, the porosity and permeability parameters become important as they alter the
impact dynamics and heat transfer to the droplet. A droplet on a porous surface will not only spread
over the surface but penetrate into the surface pores, which may alter the heat transfer to the fluid.
The critical parameter influencing the impact behavior of the droplet is the Weber number, We,
which is the ratio of the inertia to surface tension forces and is defined as:
𝑊e = 𝜌𝑓𝑉𝑖
2𝑑𝑜
𝜎 (3.1)
Where 𝜌𝑓 is the density of the fluid, 𝑉𝑖 is the droplet velocity before impact, 𝑑𝑜 is the droplet’s
initial diameter, and 𝜎 is the surface tension of the fluid.
Experiments were performed looking at the impact and evaporation of deionized (pure) water, and
n-heptane droplets on heated sintered porous stainless-steel substrates. The evaporation times were
obtained using two techniques, a weight-time measurement approach, and high-speed imagining.
Similarly, high-speed imaging was employed to observe and analyze the impact dynamics of the
droplet. There were three principal parameters: the surface temperature, varied from 23oC to
300oC; the fluid properties, specifically the surface tension and latent heat difference between the
water and n-heptane; and finally the two porous substrates investigated (5 μm, and 100 μm average
pore sizes). The results were compared with those for droplet impact and evaporation on the
impermeable substrate.
38
3.2 Experimental Setup
3.2.1 Impact Conditions
Figure 3.1 shows the experimental setup employed to observe the impact and evaporation of a pure
water and n-heptane droplet on the porous and impermeable substrates. A gauge 24, type K
thermocouple, capable of measuring temperatures within the range of -200oC to 1250oC [46], was
fixed using high temperature cement (CC High Temperature Cement, Omega Engineering, Laval,
Quebec, Canada) to the surface of each of the substrates. The cement can resist temperatures up to
843oC and is thermally conductive [54] ensuring the thermocouple is the same temperature as the
surface of the substrate once steady state conditions have been reached. The surface temperatures
were monitored using a wireless thermocouple connector (MWTC-D-K-915, Omega Engineering,
Laval, Quebec, Canada) transmitting to a PC.
Using a syringe pump (NE-1000, New Era Pump Systems, Inc., Farmingdale, New York, United
States), 2.54 mm diameter droplets were formed with a 26 gauge blunt hypodermic needle for the
pure water, and a 16 gauge blunt needle for the n-heptane. The droplets formed on the needle tip,
and once large enough would detach under their own weight. Measurements of the droplet weight
showed a drop-to-drop variation of less than ±2%. General purpose PVC tubing (0.635 cm (1/4”)
outer diameter, 0.317 cm (1/8”) inner diameter) was ran from the needle to the 10 ml plastic syringe
used to dispense the water. When working with n-heptane, due to the fluids solvent characteristics,
a 10 ml glass syringe, along with stainless steel fittings and chemical resistant PVC tubing (0.635
cm (1/4”) outer diameter, 0.317 cm (1/8”) inner diameter) were used to prevent contamination of
the working fluid. The impact velocity was controlled by adjusting the vertical distance from the
tip of the needle to the top of the substrate and was fixed at 50 mm ±1 mm for the experiments.
The impact velocity was chosen such that no splash would occur on any of the three substrates
investigated when the substrate was at room temperature. Room temperature for this study was
defined as 23oC ±3oC. The impact velocities for the pure water and n-heptane droplets at the time
of impact were 0.9 m/s. The velocity measurements were determined from high speed imaging
where they varied by ±5% for the water and ±9% for the n-heptane. The Weber numbers associated
with these drop velocities are 29 and 65 respectively. A summary of the impact conditions and
fluid properties are outlined in the following Table 3.1.
39
40
Figure 3.1: Experimental setup A and B (top, middle). Schematic of experimental setup (bottom): (1) Syringe pump
coupled with a 10 ml syringe, (2) Vertical height adjustment, (3) Hypodermic needle, (4) High-speed camera, (5)
Wireless thermocouple connector, (6) Light source, (7) Light diffuser, (8) Thermocouple, (9) Substrate, (10) Thermal
mass coupled with 100W cartridge heater, (11) PC logging scale data, and monitoring substrate temperature, (12)
Digital scale, (13) Temperature controller, (14) 120V Variac, (15) PC capturing high-speed camera images, (16)
Thermal mass coupled with two, 200W cartridge heaters. (a) Setup used at surface temperatures ranging from 60oC
to 120oC, (b) Setup used at surface temperatures above 120oC.
Table 3.1: Summary of the fluid properties and impact conditions.
Fluid Fluid Properties at 23oC Impact Conditions at 50 mm ±1 mm Drop Height
𝝆 (kg/m3) 𝝈 (mN/m) 𝝁 (Pa∙s) do (mm) Vi (m/s) We Needle Gauge
Water 998 72.9 8.9x10-4
2.5 ±2%
0.9 ±5% 29 26
n-Heptane 684 20.1 3.9x10-4 0.9 ±9% 65 16
41
3.2.2 Heating of Substrate
At surface temperatures ranging from 60oC to 120oC, the setup shown in Figure 3.1a was employed
to capture the droplets evaporation. A 3D printed plastic holder was created to hold an aluminum
thermal mass (45 mm x 45 mm x 20 mm thick) in which the substrate would sit. The thermal mass
was heated using a single 100W cartridge heater. The voltage applied to the heater was adjusted
with a 120V variac and the substrate surface temperature was regulated using a temperature
controller (CN9000A, Omega Engineering, Laval, Quebec, Canada) and could be controlled to
±0.1oC. Droplet evaporation was measured by placing the heated apparatus equipped with the
surface on a digital scale (AG245, Mettler Toledo, Mississauga, Ontario, Canada) and recording
the weight decrease as a function of time as the droplet evaporated. Figure 3.2 shows a plot of the
data produced by the digital scale where a pure water droplet is evaporating on the impermeable
stainless-steel substrate at surface temperatures of 60oC, 80oC, and 100oC. The MATLAB code
used to record the scale times can be seen in Appendix B.
With surface temperatures above 120oC, due to limitations with the 3D printed plastic having a
maximum operating temperature of approximately 150oC [55], and the scale having a maximum
sample rate of 3Hz, setup Figure 3.1b was employed. This setup used a larger aluminum thermal
mass (70 mm wide x 75 mm long x 40 mm thick) equipped with two, 200W cartridge heaters wired
in parallel and powered by a 120V variac used to regulate the heater output where the surface
temperature was regulated using the omega temperature controller. The aluminum thermal mass
was wrapped in Superwool Plus insulation to prevent heat loss from the sides and bottom. Surface
temperatures of 150oC and above were in the nucleate boiling regime for both water and n-heptane,
and evaporation occurred fast enough to prevent significant spreading of the absorbed liquid. As a
result, the evaporation time of the deposited liquid coincided with the disappearance of the dry out
front enabling the use of the high-speed camera (Fastcam SA5, Photron, Tokyo, Japan) equipped
with a bellows and 105 mm lens, to capture evaporation times at frame rates of 1000 to 5000 fps.
The shutter speed was set to 1/10000 seconds. A 55W light emitting diode (LED) (AOS 55W
Offboard LED, AOS Technologies AG, Baden, Switzerland) was used as the lighting source.
All droplet photography was done with the same high-speed camera and lighting setup before
proceeding with recording the droplets evaporation time.
42
For both setups, when the system was cold, the substrates were initially heated to their first
temperature where the heater cycled for 30 minutes ensuring the system reached a steady state.
For temperature set points following the first temperature, the system was left to reach a steady
state for 10 minutes. For Setup A, the scale weighing sensor was sensitive to electromagnetic
interference (EMI), requiring the heaters to be shutoff once the evaporation time trial began. The
single cartridge heater was removed from the thermal mass and the whole apparatus was placed
inside the scale with no wires exiting the glass walls. To follow the same procedure when running
the experiments using Setup B, the heaters were switched off before the time trial at each
temperature began. For both setups the substrate was heated 3 degrees above the set point
temperature, thus when the heaters were switched off, the substrate would cool, and once it reached
the set point temperature the experiment began.
Figure 3.2: Weight decrease as a function of time for a pure water droplet evaporating on the impermeable substrate,
at surface temperatures of 60oC, 80oC and 100oC. Images of the droplet are shown for 100oC at 1.2s, 18s, and 35s
during the evaporation process.
-1.0E-03
1.0E-03
3.0E-03
5.0E-03
7.0E-03
9.0E-03
1.1E-02
0 50 100 150 200 250 300
Wei
ght
[g]
Time [s]
60oC
80oC
100oC
t=1.2s
t=18s
t=35s
43
3.2.3 Substrate \ Thermal Mass Apparatus
Figure 3.3 shows the apparatus (key dimensions labeled in figure) used to hold the thermal mass
and substrate at low wall temperatures (< 150oC). The apparatus was printed from polylactic acid
(PLA) using a 3D printer (Replicator 2, MakerBot Industries, Brooklyn, New York, United States)
that has a print resolution of 0.1 mm [56]. A carrier that housed the thermal mass and substrate
had 3 mm walls that acted as the insulation reducing heat loss from the sides and bottom of the
thermal mass. The cut-out portions of the side walls were included to assist with inserting and
removing the thermal mass when needed. Superwool Plus insulation was used to plug the cut-outs
to reduce heat loss from those regions during experimentation. A 2.0 mm high by 1.5 mm deep
groove was printed around the top of the carrier to allow a location for the thermocouple wire to
rest which reduced wire vibrations that could occur if the wire was floating. These vibrations would
be sensed by the scale introducing noise into the signal. Any excess wire not sitting in the groove
was bundled together and secured inside the legs which housed the thermocouple transmitter. PLA
was chosen for the apparatus material due to its light weight, and ability to survive temperatures
close to 150oC before major deformation begins. The maximum weight of the apparatus including
the thermal mass, substrate, thermocouple, and the thermocouple wireless transmitter was 197 g.
This weight occurred when the apparatus was loaded with the impermeable substrate. The total
weight decreased to 189 g when loaded with either of the porous substrates. As the scale has a
maximum load capacity of 210 g [57], the total weight of the apparatus had to weigh under this
capacity to ensure the weight of the droplet could still be sensed.
44
Figure 3.3: 3D printed apparatus used to hold thermal mass and substrate during evaporation time measurements at
low wall temperatures. All dimensions shown are in millimeters (mm).
3.2.4 Scale Accuracy
To ensure the scale was producing accurate times, the high-speed camera was used in conjunction
with the weight-time measurements of a pure water droplet evaporating on the impermeable
substrate at initial surface temperatures ranging from 60oC to 120oC. At each temperature 10 trials
were completed with the scale, recording the weight decrease as a function of time. The times were
compared against the evaporation times observed using the high-speed camera. Figure 3.4 shows
the evaporation time as a function of the initial surface temperature for both the scale and high-
speed camera. The standard deviation is shown with the scale times. The average difference
between the scale times as compared to the times obtained from the high-speed camera was less
than 7%.
45
Figure 3.4: Evaporation time as a function of the initial surface temperature for a pure water droplet evaporating on
the impermeable stainless-steel substrate. Both scale and high-speed camera times are shown. The standard deviation
is shown for the scale times.
3.2.5 Substrate Preparation
The substrate was prepared before testing using the same procedure outlined in section 2.2.1.
Between trials on the impermeable substrate a cotton swab soaked in ethanol was used to clear any
residue that might have formed on the surface. Due to the roughness and porosity of the porous
surfaces this was not possible, and care was taken to ensure only deionized water, and pure n-
heptane was used during tests to prevent contamination of the samples.
3.3 Data Reduction and Analysis
3.3.1 Experimental Observations of Impact Dynamics on Cold Substrates
Figure 3.5 shows an image sequence of the first 25 ms after a water droplet impacts on the
impermeable, 5 μm, and 100 μm substrates at room temperature (23oC). Immediately following
impact on all three surfaces, a sudden increase in pressure at the point of impact results in a liquid
sheet taking shape and jetting outwards radially from the point of impact [12, 58]. The liquid sheet
0
50
100
150
200
250
50 70 90 110 130
Evap
ora
tio
n T
ime
[s]
Initial Surface Temperature [oC]
Scale
High-Speed Camera
46
can be seen taking shape as early as 0.2 ms after impact but is best observed at times greater than
1 ms. The sheets diameter on the impermeable surface at this time is 5.3 mm, and on the 5 μm and
100 μm surfaces, 4.3 mm and 3.3 mm respectively. The increased surface roughness on the porous
surfaces inhibits the outwards motion of the liquid sheet resulting in the decrease in the initial sheet
diameter.
Ripples can be seen forming in the water (see Figure 3.5(a, b, c) at 1 ms) due to capillary wave
generation that travel to the top of the drop as a result of the surface tension acting to minimize the
free surface area [59].
A lone bubble can be seen trapped inside the droplet at the center on the impermeable surface (see
Figure 3.5(a) at 1 ms) right up until the droplet settles after impact at 25 ms. This phenomenon
was observed by Chandra and Avedisian [12] where they discussed the mechanisms for its
formation as being a result of air entrapment at the solid-liquid interface. The bubble is absent in
the droplet on the 5 μm, and 100 μm surfaces during the spreading process because the airs able
to escape into the porous structure following the droplets impact. C.W. Visser et al. [60] presented
a nice illustration depicting the possible liquid deformation that occurs at the center of the droplet
impact resulting in entrapped air (see Figure 3.6).
47
Surface Temperature [23oC]
a) impermeable b) 5 µm c) 100 µm
0.2
1
2.4
5
10
15
20
25
Figure 3.5: Water droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces. Surface temperature Tw
= 23oC, Vi = 0.9 m/s ±5 %, do = 2.5 mm ±2%, We = 29, Photograph Angle = 30o.
Tim
e [m
s]
48
Figure 3.6: Overview of different stages of droplet impact. (a) depicts a droplet with initial diameter Do impacting on
a smooth surface spreading to its maximum diameter, Dmax. (b) Deformation that occurs at the solid-liquid interface
in the initial stage of impact whereby due to the compression of the air between the droplet and the surface a dimple
is formed that encloses around the air and forms a bubble [60].
Water first takes the form of a flattened disc in the center with a toroidal ring (liquid rim) around
the free advancing edge on the impermeable, 5 μm and 100 μm surfaces at 3 ms. The rim forms at
the free edge where the opposing surface tension forces are not balanced, and the motion of the
rim propagates in a direction opposite to the motion of the film flow [2, 61]. If the opposing forces
are large relative to the inertial forces, this rim will begin to grow early on following impact as the
droplet spreads as seen in Figure 3.5(a, b, c) at 2.4 ms onwards. It can be seen on the 100 μm
surface that the formation of the rim is disturbed by the roughness of the substrate. This disturbance
of the free edge begins as early as 1 ms after impact.
Figure 3.7 shows an image sequence of the first 25 ms after an n-heptane droplet impacts on the
impermeable, 5 μm, and 100 μm substrates at room temperature (23oC). While a similar liquid
sheet can be seen forming on the three surfaces, at 1 ms there is a delay in the sheets formation on
the porous substrates which was not observed with the water. This again is likely a result of the
increased surface roughness slowing the outwards motion of the liquid sheet.
49
Surface Temperature [23oC]
a) impermeable b) 5 µm c) 100 µm
0.2
1
2.4
5
10
15
20
25
Figure 3.7: n-Heptane droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces. Surface temperature
Tw = 23oC, Vi = 0.9 m/s ±9 %, do = 2.5 mm ±2%, We = 65, Photograph Angle = 30o.
Tim
e [m
s]
50
Surface ripples were not observed with the n-heptane on all three surfaces. This is thought to have
occurred due to the free surface energy being relieved entirely by the outwards motion of the liquid
sheet (i.e. inertial forces dominate). Renardy et al. [62] developed criterion to determine when
these waves would be generated as 𝑊𝑒 ∙ 𝐶𝑎 < 1, where We is the weber number, and Ca is the
capillary number defined as:
𝐶𝑎 = 𝜇𝑉
𝜎 (3.2)
Where 𝜇 and 𝜎 are the dynamic viscosity and surface tension of the impacting fluid, and V is the
impact velocity. The capillary number is less than one for both fluids leaving the Weber number
to dictate whether the ripples would be observed.
The lone bubble can be seen again inside the n-heptane droplet in Figure 3.7(a) at 0.2 ms where it
formed at the point of impact and rose above the surface into the liquid film. A closeup of the 1
ms timestep is shown for clarity for both fluids on the impermeable, 5 μm, and 100 μm substrates
in Figure 3.8. The droplet proceeds to spread thin enough such that the thickness of the film
decreases to below the trapped bubbles diameter resulting in the release of the gas (see Figure 3.9).
n-Heptane takes the form of the flattened disc in the center with a toroidal ring (liquid rim) around
the free advancing edge on the impermeable and 5 μm surfaces at 4.4 ms (see Figure 3.7(a, b) at 5
ms). On the 100 μm surface the flattened disc occurs at 3.4 ms where no rim ever forms as the
fluid rapidly absorbs into the surface pores disrupting the spreading process. The rim first begins
to form on the impermeable surface at 4.4 ms. On the 5 μm surface the rim begins forming at 1.4
ms and at 2.4 ms the rim is more defined with a clear break between the expanding film and
forming rim. A decrease in the film spreading rate on the 5 μm surface could explain why n-
heptane’s liquid rim began forming earlier during the spreading process. The opposing liquid rim
velocity is a function of the fluid properties only and has been shown to be equal to 𝑣𝑟 = √(2𝜎
𝜌𝑓ℎ)
assuming the expanding film thickness is uniform with time, where h is the thickness of the
expanding liquid film. Similarly, the velocity at which the lamella (includes centralized liquid film
and rim) travels at has been shown to be equal to 𝑣𝑙 = 𝑣 − 𝑣𝑟, where v is the velocity of the
expanding liquid film [63]. As the rim velocity is a function of the fluid properties it would stay
constant between surfaces leaving the film velocity, v, to decrease due to the surface conditions
51
(i.e. surface roughness and porosity). As a result, this film velocity would decrease sooner during
the spreading process as the kinetic energy dissipates compared to the film velocity on the
impermeable surface resulting in the earlier formation of the liquid rim.
Water n-Heptane
a)
imp
erm
eab
le
b)
5 µ
m
c)
100 µ
m
Figure 3.8: 1 ms after impact of the water and n-heptane droplet on the (a) impermeable, (b) 5 μm and (c) 100 μm
surfaces at 23oC. The single bubble is shown in the water and n-heptane droplet on the impermeable surface, but not
on the porous surfaces.
10.6 ms 10.8 ms 11 ms
Figure 3.9: n-Heptane on the impermeable surface at 10.6, 10.8 and 11 ms after impact showing the release of the
entrapped air bubble.
The water and n-heptane liquid films spread on the surface until viscous dissipation, surface
tension, surface roughness, and in the case of the porous substrates porosity effects dissipate the
Bubble
Bubble Bubble
Bubble Shadow
52
kinetic energy and bring the droplet to a stop whereby the droplet will begin its recoil.
Disentangling the effects of surface roughness from porosity on the droplets spreading behavior
can be complicated. Attempts have been made to empirically characterize surface roughness
effects and how that influences the spreading behavior [64, 65] but no generalized solution exists.
The work presented by Liu et al. and Tang et al. showed that as surface roughness increases (Liu:
from Ra = 0.5 to 70 µm, Tang: from Ra = 0.025 to 6.3 µm) the maximum spread factor and the rate
of spreading decreases as the roughness can increase frictional losses during droplet spreading.
Several other models have been deduced predicting the maximum spread factors on solid surfaces
that do not factor in surface conditions directly, but rather the entire surface-fluid interaction is
carried in the fluids advancing contact angle [12, 66, 67]. These models can accurately predict
spread factors on smooth non-porous surfaces where the Ra is generally less than 5 µm [64].
Figure 3.10 shows the spread factor for the water and n-heptane on the impermeable, 5 μm, and
100 μm substrates as a function of the dimensionless time at room temperature (23oC). The spread
factor (𝜷) is the dynamic spread diameter of the droplet after impact divided by the initial droplet
diameter (𝜷 = Dfilm/do). Similarly, the dimensionless time (t*) is defined as t*= tVi/do, where t is the
time in seconds following the droplets impact. It is clear from Figure 3.10 that as the porosity
increases, and thus the surface roughness, the maximum spread diameter and rate of spreading
decreases for both the water and n-heptane. The water reached a maximum spread diameter on the
impermeable surface of 6.0 mm, on the 5 μm surface, 5.2 mm and on the 100 μm surface, 5.0 mm
at 4, 3.8, 3.6 ms respectively. Similarly, n-heptane reached a maximum spread diameter on the
impermeable surface of 12.6 mm, on the 5 μm surface, 8.0 mm, and on the 100 μm surface 4.9
mm at 12.8, 7.4, and 4 ms respectively.
53
Figure 3.10: Spread factor as a function of the dimensionless time for both water and n-heptane on the impermeable
and two porous substrates at room temperature (23oC).
The model presented by M. Pasandideh-Fard et al. [66] accurately predicts the spread factor for both
the water and n-heptane on the impermeable surface as the substrate is smooth and is presented as
follows:
Before impact the kinetic energy (KE1) and surface energy (SE1) of the droplet are given by:
𝐾𝐸1 = (1
2𝜌𝑓𝑉𝑖
2) (𝜋
6𝑑𝑜
3) (3.3)
𝑆𝐸1 = 𝜋𝑑𝑜2𝜎 (3.4)
Where 𝜌, 𝑉𝑖, 𝑑𝑜 , and 𝜎 are the droplets density, initial impact velocity, initial impact diameter and
surface tension respectively. When the droplet reaches its maximum spread diameter the kinetic
energy goes to 0 where the surface energy becomes:
𝑆𝐸2 =𝜋
4𝐷𝑚𝑎𝑥
2 𝜎(1 − cos 𝜃𝑎) (3.5)
Where 𝜃𝑎 is the advancing contact angle, and Dmax is the maximum droplet spread diameter after
impact. The work done during droplet spreading to overcome viscosity is:
0
1
2
3
4
5
6
0 1 2 3 4 5
Spre
ad F
acto
r, β
[Dfi
lm/d
o]
Dimensionless Time, t* [tVo/do]
Water - 100 μm Water - 5 μmWater - ImpermeableHeptane - 100 μmHeptane - 5 μmHeptane - Impermeable
Maximum
Spread
Increased
Spreading Rate
54
𝑊 =𝜋
3𝜌𝑓𝑉𝑖
2𝑑𝑜𝐷𝑚𝑎𝑥2 1
√𝑅𝑒 (3.6)
Where Re is the Reynolds number (Re = 𝜌𝑓𝑉𝑜𝐷𝑜/𝜇) of the impacting droplet. An energy balance
yields:
𝐾𝐸1 + 𝑆𝐸1 = 𝑆𝐸2 + 𝑊 (3.7)
Finally substituting equations 3.3 to 3.6 into 3.7 we can deduce the maximum spread factor, 𝛽𝑚𝑎𝑥,
as:
𝛽𝑚𝑎𝑥 = √𝑊𝑒+12
3(1−cos 𝜃𝑎)+4(𝑊𝑒
√𝑅𝑒) (3.8)
From literature, using a contact angle for n-heptane/stainless steel as 20o and for water/stainless-steel
as 110o, results are in close agreeance with values obtained experimentally where 𝛽𝑤𝑎𝑡𝑒𝑟 = 2.53
𝛽ℎ𝑒𝑝𝑡𝑎𝑛𝑒 = 4.23. They differ by approximately 8% and 15% respectively. From the previous studies
outlined above, it is clear that the decrease in the spread factor seen in Figure 3.10 on the impermeable
and 100 μm surfaces are partly responsible due to the change in surface roughness. To deduce
whether the spread factor is also affected as a result of the substrates porosity, consider the following.
Chandra and Avedisian [4] put forward the following energy balance to account for the fluid
penetration into the pores:
𝐾𝐸1 + 𝑆𝐸1 = 𝑆𝐸2 + 𝑊 + 𝐸𝑝 (3.9)
This equation is identical to equation 3.7 except for the additional Ep term to account for the energy
associated with fluid penetration into the surface. The net effect of fluid penetration into the pores is
a reduction in the volume of fluid available to spread over the top surface, decreasing the maximum
spread factor.
A direct solution for Ep is difficult to obtain but as a first approximation, a look at the volume decrease
once the droplet pins to the surface can be accomplished through the analysis of photographs taken
during the impact process. It will be assumed the droplet spreads axially symmetric which is in good
agreement with observations from photographs. Following the spreading and pinning of the water
droplets lamella, the droplet oscillates up and down until the inertial effects dissipate, where the
55
droplet settles on the surface taking the form of a spherical cap (see Figure 3.11 and Figure 3.5 (a, b,
c) at 5 ms to 25 ms). The volume of the cap can be deduced computationally by incrementing 1 pixel
at a time from the base to the apex of the droplet and determining the chord length at each increment.
Each chord length represents the diameter of a circular volumetric segment (𝑉𝑠𝑒𝑔 = 𝜋𝐷𝑐ℎ𝑜𝑟𝑑
2
4∙𝚤𝑝𝑙), where
𝐷𝑐ℎ𝑜𝑟𝑑 is the diameter of the segment, and 𝚤𝑝𝑙 is the pixel to length conversion. Recognizing that the
sum of the volumetric segments equates to the total volume of the spherical cap, the volume of the
droplet sitting on the surface of the substrate can be determined. Refer to Appendix B for the
MATLAB code.
Figure 3.11: Pinned water droplet taking the form of a spherical cap after impact on the 5 μm surface. An example
inlay of the droplets mask produced by the program is shown, along with a model (not to scale) showing how the
masked area is broken down into circular volumetric segments used to determine the total volume of the droplet sitting
on the substrates surface.
1 second worth of data, or 5000 separate frames were processed for both porous substrates and the
volume of the droplet sitting on the surface divided by the initial volume of the droplet (V/Vo) was
plotted as a function of time. The volume was found to decrease linearly following the completion of
droplet spreading over this time. A trendline was fit to the data once the droplet settled and pinned to
the surface and was used to extrapolate back to the time when the maximum spread factor (i.e. 4 ms)
was observed to determine whether fluid penetration into the surface occurred at impact and during
droplet spreading. Figures 3.12 and 3.13 show V/Vo as a function of time for water on the 5 μm and
100 μm surfaces respectively.
56
Figure 3.12: V/Vo as a function of time for the water droplet on the 5 µm surface.
Figure 3.13: V/Vo as a function of time for the water droplet on the 100 µm surface.
0.0
0.2
0.4
0.6
0.8
1.0
1.00 10.00 100.00 1000.00
V /
Vo
[mm
3/
mm
3]
Time [ms]
3% Volume Loss
0.0
0.2
0.4
0.6
0.8
1.0
1 10 100 1000
V /
Vo
[mm
3/
mm
3]
Time [ms]
Maximum Spread
Factor Time
57
The volume difference relative to the initial volume of the droplet released before impact was
determined to be 3% ±0.02% (95%). It should be noted that the uncertainty quoted for this volume
decrease establishes a random uncertainty about the fit of the linear regression stated at the 95%
confidence interval (t estimator). Refer to Section 3.3.5 covering the accuracy of the image analysis
code to calculate the droplets volume. The reduced volume sitting on top of the surface would result
in a new initial droplet diameter of 2.52 mm. Using the model outlined by M. Pasandideh-Fard et al.,
an estimate on the effect this would have on the maximum spread factor can be determined assuming
smooth surface conditions. Using the same parameters outlined above but now with the new
equivalent droplet diameter the maximum spread factor, 𝛽𝑤𝑎𝑡𝑒𝑟 was determined to be 2.53. To
properly compare this with the original 𝛽𝑤𝑎𝑡𝑒𝑟 produced by the model the equivalent maximum
spread diameter based on the new initial droplet diameter needs to be deduced and divided by our
original droplets diameter of 2.54 mm before impact as we know our droplet size never changes
between experiments. This yields an adjusted maximum spread factor, 𝛽𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 of 2.51 which is less
than 1% reduction in 𝛽 due to volume absorption effects. It can be reasonably asserted that the change
in the maximum spread diameter observed on the 5 μm surface relative to the impermeable surface
was a result of the substrates surface roughness.
Performing a similar analysis with the water droplet on the 100 μm surface, the decreased volume
estimate was determined to be 17% ±1% (95%) less than the initial volume of the droplet prior to
impact. Determining the new surface volumes equivalent initial droplet diameter at 2.38 mm and
again using M. Pasandideh-Fard et al. model to look at how this would affect the maximum spread
factor on a smooth surface, 𝛽𝑤𝑎𝑡𝑒𝑟 was determined to be 2.49. Adjusting 𝛽𝑤𝑎𝑡𝑒𝑟 based on our original
droplet diameter yields a 𝛽𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 of 2.33. This is an 8% decrease in 𝛽 due to volume absorption
effects. This change is too large to ignore and could account for a portion of the change seen in 𝛽
experimentally as 𝛽𝑤𝑎𝑡𝑒𝑟 only decreases by 17% from the impermeable to the 100 μm surface.
A study by J.B. Lee et al. [68, 69] looking at water droplet impingement on porous stones with
volumetric porosities of 27%, 17% and 5%, at Ra values of 10.3, 9.1, and 4.4 μm respectively,
concluded that the droplets absorption into the surface at impact, and during spreading can be
neglected. They reasoned this was due to the existence of an air layer which was entrained between
the droplet and the porous substrate during the spreading process increasing the contact resistance.
The existence of visible air pockets trapped between the base of the droplet and the substrate were
58
observed on both porous surfaces and can be seen in Figures 3.14 to 3.16 which is in agreement
with J.B. Lee et al. observations. However, due to the 100 μm surface having a larger porosity (i.e.
44%), fluid penetration during droplet impact and spreading occurred. The 5 μm surface has a
porosity of 31%, closer to the porosity of the surfaces investigated by J.B. Lee et al. Interestingly,
the rate of volume absorption into the 100 μm surface immediately following droplet spreading is
slower than on the 5 μm surface. A possible explanation for this is the capillary force driving the
fluid into the surface is smaller on the 100 μm surface due to the larger pore size.
4.8 ms 7.8 ms
11 ms 50.6 ms
Figure 3.14: Water droplet impact on the 5 μm surface showing entrapped air bubbles. Surface temperature Tw =
23oC, Vi = 0.9 m/s ±5 %, do = 2.5 mm ±2%, We = 29, Photograph Angle = 30o.
Air Bubbles
Air Bubbles
59
Figure 3.15: Water droplet impact on the 5 μm surface showing entrapped air bubbles 4.6 ms after impact. Surface
temperature Tw = 23oC, Vi = 0.9 m/s ±5 %, do = 2.5 mm ±2%, We = 29, Photograph Angle = 0o.
4.6 ms 5.2 ms
Figure 3.16: Water droplet impact on the 100 μm surface showing entrapped air bubbles. Surface temperature Tw =
23oC, Vi = 0.9 m/s ±5 %, do = 2.5 mm ±2%, We = 29, Photograph Angle = 0o.
Figure 3.17 shows V/Vo as a function of time for n-heptane on the 5 μm surface starting at the
maximum spread factor (7.4 ms) until complete absorption into the surface pores. The full droplet
impact and surface penetration, or 74 separate frames were initially recorded. As seen in Figure 3.18
and Figure 3.7(b), after the droplet reaches its maximum spread factor, instead of recoiling and
forming a cap on the surface due to surface tension, the droplet remains flat with round tapered edges
resembling a truncated spherical cap. Immediately following 𝛽𝑚𝑎𝑥𝑖𝑚𝑢𝑚, at 14.8 ms the droplet fully
absorbs into the substrate.
60
Figure 3.17: V/Vo as a function of time for the n-heptane droplet on the 5 µm surface.
The central film of the lamella sits lower than the rim around it, where the volume produced using the
MATLAB program will give an overestimate of the volume present on the surface as it sees a constant
thickness across the film. The volume determined was 13% ±8% (95%) smaller than the initial
droplets volume. This results in a 6% reduction on the adjusted 𝛽ℎ𝑒𝑝𝑡𝑎𝑛𝑒 (𝛽𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 = 3.99). As there
was a 36% decrease in the maximum spread factor observed experimentally on the 5 μm surface
relative to the impermeable surface, this suggests that 𝛽ℎ𝑒𝑝𝑡𝑎𝑛𝑒 on the 5 μm surface is influenced by
both the substrates surface roughness and porosity.
Due to porosity effecting the droplet spreading on the 5 μm surface, it can be reasonably asserted that
volume absorption into the substrate during spreading would increase on the 100 μm surface.
0.0
0.2
0.4
0.6
0.8
1.0
0.00 5.00 10.00 15.00 20.00
V /
Vo
[m
m3
/ m
m3]
Time [ms]
61
Figure 3.18: n-heptane droplet at its maximum spread factor 7.4 ms after impact on the 5 μm surface. Droplet
photographed at approximately 30o. The lowered center film with larger surrounding rim can be observed.
3.3.2 Volume Analysis Accuracy
A 3.125 mm ±0.00254 mm (1/8” ±0.0001”) precision stainless-steel ball bearing obtained through
McMaster-Carr was placed in frame for the first set of images taken and used as a means to
calibrate a known distance allowing for the conversion of pixel measurements to their real-world
values. The conversion value was obtained manually through Image J. The high-speed camera
took images with a resolution of 1024 pixels by 1024 pixels. For the n-heptane volume analysis,
this resulted in a viewing window of 15.7 mm x 15.7 mm and for the water volume analysis, 10.5
mm x 10.5 mm (sizes were obtained before image cropping). Their respective pixel sizes in both
directions are therefore 15.3 µm and 10.2 µm.
To determine how accurate the MATLAB program was at computing the volume of the droplet
sitting on the substrates surface, it was tested against a water droplet of known volume sitting on
an impermeable surface at 23oC. Figure 3.19 shows V/Vo as a function of time where the volume
was shown to be within on average 3% of the true volume with a frame to frame volume variation of
±2% over the 127 ms (351 frames) range investigated.
Liquid Rim
Liquid Film
62
Figure 3.19: V/Vo as a function of time for the water droplet on the impermeable surface.
To determine the contribution of evaporation effects on the surface volume measurements taken
during times associated with the maximum spread factor (i.e. 4 ms to 7.4 ms), both a water and n-
heptane droplet were impinged on the impermeable surface. All experiments were performed in the
same environment under the same ambient conditions. The weight decrease was recorded as a
function of time using the laboratory digital weight scale, and after approximately 20 seconds the
evaporation rates were deduced. The volatility of n-heptane resulted in a linearly decreasing weight
with time, where the slope of the line yielded an evaporation rate of 0.198 mg/s. This results in a
mass evaporated over 0.1 s of 0.02 mg or 0.3% of the total mass of the droplet. As spreading occurs
with the n-heptane at timescales approximately 10x shorter than 0.1 s (i.e. under 10 ms)
evaporation effects can be neglected. Similarly, for the water with an evaporation rate 0.004 mg/s,
evaporation effects can be ignored. Figure 3.20 shows the weight decrease as a function of time
data used to deduce the evaporation rates over the 20 second time period investigated for both the
water and n-heptane.
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80 100 120 140
V /
Vo
[m
m3
/ m
m3]
Time [ms]
63
Figure 3.20: Weight decrease a function of time for both the water and n-heptane droplet on the impermeable surface
at room temperature (23oC ±3oC). Equations for the line are shown for both fluids where the slope of the line yields
the evaporation rate.
3.3.3 Droplet Lifetime Evaporation Plots
As the substrate heats up, a droplet lifetime as a function of the initial surface temperature plots
can be used to identify at which temperatures changes in the impact dynamics and evaporation
process occurs. Figures 3.21 and 3.22 show the droplet lifetime plots for both water and n-heptane
on the impermeable, 5 μm, and 100 μm surfaces. A graph insert is shown in each of the plots,
utilizing a log time scale to clearly illustrate the difference in evaporation times between the three
surfaces at temperatures ranging from 150oC to 300oC. Refer to Appendix D for the droplet lifetime
data for both water and n-heptane.
y = -4.277E-06x + 8.591E-03
y = -1.985E-04x + 9.500E-03
0.00E+00
1.00E-03
2.00E-03
3.00E-03
4.00E-03
5.00E-03
6.00E-03
7.00E-03
8.00E-03
9.00E-03
0 10 20 30 40 50
Wei
ght
[g]
Time [s]
n-Heptane
Water
64
Figure 3.21: Evaporation time as a function of the surface temperature for the impermeable, 5 μm and 100 μm
substrates using pure water as the working fluid. A graph insert is shown at surface temperatures ranging from 150oC
to 300oC using a log time scale to show evaporation time differences between the surfaces.
Figure 3.22: Evaporation time as a function of the surface temperature for the impermeable, 5 μm and 100 μm surfaces
using n-heptane as the working fluid. A graph insert is shown at surface temperatures ranging from 150oC to 300oC
using a log time scale to show evaporation time differences between the surfaces.
-51535557595
115135155175195215235
50 70 90 110 130 150 170 190 210 230 250 270 290 310
Evap
ora
tio
n T
ime,
te
[s]
Initial Surface Temperature, Tw [oC]
100μm Impermeable Surface 5μm
Leidenfrost
Temperature
0
5
10
15
20
25
30
35
50 70 90 110 130 150 170 190 210 230 250 270 290 310
Evap
ora
tio
n T
ime,
te
[s]
Initial Surface Temperature, Tw [oC]
Impermeable Surface 5μm 100μm
0.01
0.10
1.00
10.00
100.00
145 165 185 205 225 245 265 285 305
0.01
0.10
1.00
10.00
100.00
145 165 185 205 225 245 265 285 305
65
As the surface temperature, Tw was raised from room temperature the evaporation time for both
the water and n-heptane decreased as a result of increased heat transfer to the droplet. It was
observed at temperatures below water and n-heptane’s saturation temperature (water at 100oC, n-
heptane at 98.4oC) slower evaporation times on average occurred with water on the 100 μm surface
as compared to the impermeable and 5 μm surfaces and with n-heptane on both the 5 μm and 100
μm surfaces as compared to the impermeable surface. This could be explained by the decrease in
the substrates thermal conductivity at increased porosities where the liquid cools the porous
surfaces more than the impermeable surface. n-Heptane’s lower surface tension promotes pore
penetration enhancing this effect.
As Tw increased above the saturation temperature for both fluids, boiling inside the droplet at the
solid-liquid interface occurred where the bubbles grow on imperfections that exist on the surface.
The evaporation times with the n-heptane are lower than the times observed with the water due to
n-heptane’s smaller heat of vaporization (for water 2257 kJ/kg, for n-heptane 318 kJ/kg) where
less energy is required to cause a phase change.
Evaporation times continue to decrease with an increase in Tw until the critical heat flux which was
observed to occur at an initial surface temperature of approximately 150oC for both water and n-
heptane on all three surfaces. Beyond this temperature, times increase as the transition to the film
boiling regime occurs where increased vapor formation between the liquid-solid interface slows
the heat transfer as the fluid can no longer consistently wet the surface.
The Leidenfrost point, characterized as the longest evaporation time in the film boiling regime,
was reached with the water droplet on the impermeable surface at 235oC. Similarly, the Leidenfrost
point was successfully reached with the n-heptane droplet at surface temperatures of 190oC
(impermeable surface), 225oC (5 μm surface), and 285oC (100 μm surface) respectively. As the
porosity is increased vapor flow into the surface pores reduces the vapor pressure between the
droplets base and the substrate surface delaying the onset of the Leidenfrost point. Section 3.3.5
goes into further detail regarding the behavior of water and n-heptane droplets in the film boiling
regime.
66
3.3.4 Experimental Observations of Impact Dynamics on Substrates at Increased Temperatures Below the Leidenfrost Point
Figures 3.23 shows the water droplet on the impermeable, 5 μm, and 100 μm surfaces at an
increased surface temperature of 80oC. The water behaves very similarly on the three surfaces as
it did at room temperature. The single bubble can be seen again entrapped at the point of impact
on the impermeable surface at 2.4 ms onwards (see Figure 3.23(a)). On the 5 μm surface what
appears to be air pockets are again trapped between the fluid and the surface as a potential factor
for the slower fluid permeation into the pores. The water does not appear to exhibit any visible
changes revealed through photographs on the 100 μm surface.
Figure 3.24 shows the n-heptane droplet on the impermeable, 5 μm, and 100 μm substrates at an
increased surface temperature of 80oC. A bubble ring can be seen in the n-heptane similar to the
one observed by Chandra and Avedisian [12] around an entrapped bubble in the center at the point
of impact on the impermeable surface at 1 ms (see Figure 3.24(a)). By 5 ms several other bubbles
are visible within the liquid film as it spreads. When looked upon closer the bubbles that can be
seen at 5 ms start forming as early as 2.4 ms and spread outwards with the expanding liquid film
as shown in Figure 3.25. The liquid-solid contact angle has been shown to increase with an increase
in surface temperature [12]. This increase in the contact angle could result in more air being
entrapped by the spreading liquid film and explain the appearance of the bubble ring. The lack of
bubbles on the porous surfaces could be a result of the air on impact and during spreading escaping
into the surface pores.
At 10 ms, cell structures start forming in the liquid film on the impermeable and 5 μm surfaces
that do not occur at room temperature. The lack of cell formation on the 100 μm surface could be
due to the substrates surface roughness and increased porosity whereby rapid capillary uptake is
disrupting the spreading process and the fluid fully absorbs into the surface before the cell
structures can form. As the cells form at increased temperatures suggests that they are a result of
thermo-capillary convection created by temperature gradients in the liquid film [12, 70]. On the
impermeable surface at 22.8 ms the liquid in the cell bottoms has nearly fully evaporated leaving
just the thicker ridges and rim. The recoil of the liquid rim continues its motion inwards rewetting
the dried cell bottoms and merging with pockets of n-heptane formed in the center. At 46.6 ms a
rim and a single pool of liquid in the center can be seen until finally a single puddle is formed at
67
67.4 ms after impact. Figure 3.26 shows an image sequence of the thermo-capillary convection
cells and the rewetting of the evaporated bottoms to form a single pool of liquid. Several of the
bubbles that formed in the liquid film during the spreading process can be seen in the pool of liquid
at 110.6 ms after impact.
68
Surface Temperature [80oC]
a) impermeable b) 5 µm c) 100 µm
0.2
1
2.4
5
10
15
20
25
Figure 3.23: Water droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces. Surface temperature
Tw = 80oC (film evaporation regime), Vi = 0.9 m/s ±5%, do = 2.5 mm ±2%, We = 29.
Tim
e [m
s]
69
Surface Temperature [80oC]
a) impermeable b) 5 µm c) 100 µm
0.2
1
2.4
5
10
15
20
25
Figure 3.24: n-Heptane droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces. Surface temperature
Tw = 80oC (film evaporation regime), Vi = 0.9 m/s ±9%, do = 2.5 mm ±2%, We = 65.
Tim
e [m
s]
70
2.4 ms 2.8 ms
3.2 ms 7.4 ms
Figure 3.25: n-Heptane droplet on the impermeable surface at 80oC showing the bubble growth and movement
through the liquid film at 2.4, 2.8, 3.2, and 7.4 ms.
22.8 ms 31 ms
46.6 ms 110.6 ms
Figure 3.26: n-Heptane droplet on the impermeable surface at 80oC showing thermo-capillary convection cells at
22.8, 31, 46.6, and 110.6 ms.
Bubble Ring
Bubble Formation
Dried Cell Bottoms Ridges
Rim
71
On the 5 μm surface cell bottoms are nearly fully gone by 11 ms (see Figure 3.27), happening
approximately twice as fast as the n-heptane on the impermeable surface taking 22.8 ms. This can
be explained by the substrates porosity where the fluid is evaporating and penetrating the surface
pores simultaneously. The rim can be seen recoiling and penetrating the surface pores until it stops
at 13.4 ms where the remaining liquid pins to the surface, penetrates and evaporates. This recoil
does not occur on the 5 μm surface at room temperature. A possible explanation for this is when
the fluid evaporates in the cell locations there is no liquid pressure to restrain the inward motion
of the rim [12]. Similarly, at increased surface temperatures vapor pressures would increase in the
substrate pores which could slow liquid penetration allocating time for the rim to recoil. The fluid
fully penetrates the surface pores 27.8 ms after impact, where it only took 14.8 ms at room
temperature. This is both a result of the thermo-capillary convection, where fluid moves into the
ridges forming locations where the film is thicker, along with increased vapor pressures slowing
the penetration process.
11 ms 12.8 ms
15.8 ms 18.6 ms
Figure 3.27: n-Heptane droplet on the 5 µm surface at 80oC showing thermo-capillary convection cells at 11, 12.8,
15.8, and 18.6 ms.
Figure 3.28 shows the water droplet on the impermeable, 5 μm and 100 μm surfaces above the
liquid boiling temperature at a surface temperature of 150oC. At this temperature heterogeneous
nucleation occurs on imperfections on the surface within water immediately on the impermeable
72
Surface Temperature [150oC]
a) impermeable b) 5 µm c) 100 µm
0.2
1
2.4
5
10
15
20
25
Figure 3.28: Water droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces. Surface temperature
Tw = 150oC (nucleate boiling regime), Vi = 0.9 m/s ±5%, do = 2.5 mm ±2%, We = 29.
Tim
e [m
s]
73
surface seen starting as early as 0.2 ms after impact, while on the porous surfaces it is delayed
starting at 1 ms and 2.4 ms respectively. This delay could be explained by the thermal conductivity
decrease that occurs when increasing the porosity of the substrate (impermeable: 13 W/m∙K, 5 μm:
3 W/m∙K, 100 μm: 1 W/m∙K), slowing the heat transfer to the droplet.
Figure 3.29 shows the n-heptane droplet on the impermeable, 5 μm and 100 μm surfaces above the
liquid boiling temperature at a surface temperature of 150oC. Nucleation can be seen starting with
the n-heptane at 0.2 ms on the impermeable surface, and as early as 0.4 ms and 0.6 ms on the 5
μm and 100 μm surfaces respectively due to n-heptane’s specific heat capacity and heat of
vaporization being significantly smaller than that of waters. The white reflection seen in the center
of the n-heptane droplet on the 5 μm surface at 1 ms is a large vapor bubble which lifts the liquid
film off the substrate by 2.4 ms where it finally breaks apart and settles back on the surface by 10
ms. The same lifting of the liquid film begins on the 100 μm surface at 2.4 ms but by 10 ms the
film has completely blown apart, where portions of the bulk fluid are violently flung into the air
and off the substrate. Parts of the liquid rim are left to boil on the surface until complete
evaporation occurs. It is thought this violent behavior is seen on the porous surfaces due to an
increased volume of gas trapped under the lamella because of the substrates porosity which
subsequently expands through the liquid film causing the disintegration.
At 15 ms a plethora of secondary droplets can be seen being ejected from the liquid film on all
three surfaces which is not observed with water in the early stages of impact on the porous surfaces
and less pronounced at 15 ms on the impermeable surface. Chaves et al. [71] observed similar
behavior with ethanol droplets impacted on polished aluminum ingots and suggested that the
mechanism for their formation is due to vapor bubble explosions breaking through the liquid film.
The delay seen with water is likely a result of increased cooling of the surface where heat needs to
conduct back into the contact zone slowing the heat transfer. It was also stated by Cossali et al.
[72] that an increase in liquid viscosity could decrease the local Reynolds number where a decrease
in the convective heat transfer from the wall to the liquid occurs depressing vaporization. Figure
3.30 better shows the phenomenon in question. At later timescales a very specific structure forms
in the boiling water and n-heptane that is a bubble with a liquid jet shooting out the top and can be
clearly observed on the impermeable surface.
74
Surface Temperature [150oC]
a) impermeable b) 5 µm c) 100 µm
0.2
1
2.4
5
10
15
20
25
Figure 3.29: n-Heptane droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces. Surface temperature
Tw = 150oC (nucleate boiling regime), Vi = 0.9 m/s ±9%, do = 2.5 mm ±2%, We = 65.
Tim
e [m
s]
75
This formation has been termed the “pagoda-like” bubbles, observed by Cossali et al. [72, 73].
These “pagoda-like” bubbles could be one of the mechanisms for secondary drop formation though
it was not clearly observed with either fluid on the porous surfaces and the theoretical explanation
for their existence appears to still be unknown. Furthermore, it was difficult to catch the occurrence
of them with n-heptane on the impermeable surface. This could be a result of nucleation being less
significant with the n-heptane because of the smaller contact angle resulting in smaller bubbles,
along with an insufficient camera setup (i.e. image resolution, frame rate, and shutter speed)
required to capture and clearly freeze the phenomenon in photographs, explaining the significant
blur with the frames that were obtained. Similarly, on the porous surfaces the vapor was able to
escape into the porous structure preventing large bubbles from forming and suppressing this
formation. Figures 3.31 and 3.32 shows the “pagoda-like” bubbles for the water and n-heptane on
the impermeable surface.
76
Water n-Heptane
a)
imp
erm
eab
le
b)
5 µ
m
c) 1
00 µ
m
Figure 3.30: Water and n-heptane droplet on the (a) impermeable, (b) 5µm, and (c) 100 µm surfaces at 150oC showing
secondary droplets produced during boiling at 15 ms after impact. No secondary droplets are observed with the water
on the porous surfaces.
56.8 ms 57 ms 57.2 ms
Figure 3.31: Water droplet on the impermeable surface at 150oC showing the “pagoda-like” formation ejecting a
secondary drop at 56.8, 57, and 57.2 ms after impact.
Bubble
Liquid Jet
Ejected Secondary Drop
Secondary
Droplets
77
30.6 ms
46 ms
Figure 3.32: n-Heptane droplet on the impermeable surface at 150oC showing “pagoda-like” formation at 30.6 ms
and 46 ms after impact. The “pagoda-like” formations are enlarged on the right.
3.3.5 Experimental Observations of Impact Dynamics on Substrates at Temperatures Above the Leidenfrost Point
Figure 3.33 shows the water droplet on the impermeable, 5 μm and 100 μm surfaces above the
Leidenfrost temperature on an impermeable surface at an initial surface temperature of 250oC. The
onset of nucleation was observed to happen on the impermeable surface at 0.4 ms after impact but
can clearly be seen at 1 ms within the droplet. Boiling can be seen disrupting and breaking apart
the spreading liquid film at 5 ms while simultaneously a vapor layer begins to form between the
liquid and the substrates surface. On the 5 μm surface vapor bubbles can be seen forming at 1 ms
where by 5 ms it is clear the bubbles are not disrupting the liquid film as seen with the water on
the impermeable surface. This could be a result of the water cooling the surface slowing the growth
of the vapor bubbles and subsequent rise through the liquid film caused by the substrates lower
thermal conductivity. Vapor bubbles also have a path through the porous substrate instead of
breaking off and migrating through the bulk of the droplet. On the 100 μm surface similar behavior
Liquid Jet
Bubble
78
appears to be happening. The disrupted rim edges seen at times 1 to 5 ms also occurred at room
temperature suggesting they are a result of the substrates surface roughness. By 10 ms the lamella
on the impermeable surface is recoiling into a distorted droplet while levitating on a vapor cushion
where its inwards momentum results in the droplet rebounding into the air. This is characteristic
behavior that occurs with a droplet impacting an impermeable surface in the film boiling regime
[74]. Bubbling within the droplet stops as seen in times greater than 10 ms, due to the liquid no
longer wetting the surface preventing nucleation from occurring on surface imperfections. At later
times the droplet has settled and taken the form of an oblate spheroid shown at 200 ms. Once the
droplet is levitating, little bubbles can be seen trapped inside the fluid that were created during the
nucleation process.
On the 5 μm surface the droplet can be seen in the air at 20 and 25 ms where its inwards momentum
in combination with the vapor pressures produced at the liquid-solid interface were large enough
to unpin the droplet from the substrate surface before settling back down, shown at 200 ms. The
droplet on the 100 μm surface also does not levitate, shown pinned to the surface boiling at 200
ms. Droplet rebound off the substrate does not occur on the 100 μm surface because of the lower
vapor pressure produced at the liquid-solid interface due to the substrates increased permeability
(5 μm: 3.3x10-14 m2, 100 μm: 7.0x10-13 m2). On both porous substrates the vapor pressures
produced between the liquid and the substrate were not high enough to adequately levitate the
droplet at increased times due to vapor loss into the surface pores. This allowed the droplet to
remain on the surface, reducing the evaporation times as shown in Figure 3.21 because direct heat
transfer across a solid-liquid interface is higher than across a vapor gap.
79
Surface Temperature [250oC]
a) impermeable b) 5 µm c) 100 µm
0.2
1
2.4
5
10
15
20
25
200
Figure 3.33: Water droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces. Surface temperature
Tw = 250oC (film boiling regime), Vi = 0.9 m/s ±5%, do = 2.5 mm ±2%, We = 29.
Tim
e [m
s]
80
Figure 3.34 shows the n-heptane droplet on the impermeable, 5 μm and 100 μm surfaces above the
Leidenfrost temperature on an impermeable surface at an initial surface temperature of 250oC. The
behavior of n-heptane at this surface temperature results in some interesting comparisons. Starting
as early as 1 ms after impact a liquid instability at the advancing rim can be seen forming on all
three surfaces. This behavior appears to start with water at the same time on the impermeable
surface but shortly thereafter the lamella gets distorted due to vapor bubbles forming at the liquid-
solid interface. It is interesting to note that on the impermeable and 5 μm surfaces no nucleation
occurs where the n-heptane vaporizes immediately upon impact. This is a result of n-heptane’s
low heat of vaporization which is approximately 7 times less than that of water. It should also be
noted that at a surface temperature of 250oC we are approaching the critical temperature of n-
heptane (266.8oC) where only a single phase exists at which the heat of vaporization for the fluid
approaches 0 (i.e. instant change from liquid to vapor). On the 100 μm surface nucleation appears
to have begun by 2.4 ms, though it is difficult to discern from photographs, but by 25 ms it is clear
no levitation occurs, and the droplet is pinned to the substrate surface.
The Kelvin-Helmholtz theory explains the fingering instability witnessed on the three surfaces due
to a velocity difference between the falling droplet and the escaping gas underneath (i.e. a shear
gradient). Shear instabilities occur between the n-heptane liquid and vapor that evaporates off the
base of the droplet upon approaching the hot substrate which fills the gap between the liquid and
the solid and subsequently escapes from between the two interfaces. This could also partly explain
why what appears to be an instability forming in the water film on the impermeable surface at 1
ms, is absent on the 5 μm surface. As the droplet impacts the porous substrate, the air/water vapor
can escape into the surface reducing the shear stress and suppressing instability. This behavior was
recently observed by Liu et al. [75] where they witnessed splash while impacting droplets on
smooth surfaces initially impermeable. When adding pores under where the droplet impacted,
splash was suppressed. Similarly, Xu et al [76] experimentally observed droplet impacts on dry
smooth surfaces where the surrounding air pressure was varied from 100 kPa down to 17.2 kPa
and witnessed that when droplet splash occurred at 100 kPa no visible signs of splash were present
at 17.2 kPa suggesting that the surrounding gas plays a significant role in droplet instability.
81
Surface Temperature [250oC]
a) impermeable b) 5 µm c) 100 µm
0.2
1
2.4
5
10
15
20
25
200
Figure 3.34: n-Heptane droplet impact on the (a) impermeable, (b) 5 μm and (c) 100 μm surfaces. Surface temperature
Tw = 250oC (film boiling regime), Vi = 0.9 m/s ±9%, do = 2.5 mm ±2%, We = 65.
Tim
e [m
s]
82
The finger formations on the three surfaces grow in size but decrease in the quantity around the
liquid rim as the droplet spreads, seen at 2.4 ms (impermeable: ~22 fingers, 5 μm: ~17 fingers, 100
μm: 16 fingers) and then at 10 ms (impermeable: ~12 fingers, 5 μm: ~11 fingers, 100 μm: 7
fingers). The decrease in the quantity are a result of them merging when surface tension, and
cohesive forces begin to play a role. An example of the finger merging that occurs on all three
surfaces is shown in Figure 3.35 for the impermeable surface. Due to the low surface energy of n-
heptane, the fingers are a source of satellite droplets that can be seen breaking off starting at 10 ms
on the three surfaces as the droplet recoils. The decrease in the number of fingers seen on the
porous surfaces relative to the impermeable surface could be a direct result of the vapor loss into
the substrate upon droplet impact. The decrease in the density in the thin vapor layer would have
a direct effect on the dominant wave number and thus the number of fingers formed. This is due
to a reduction in the restraining pressure of the gas at the liquid-gas interface [76].
4.4 ms 5.6 ms 7.6 ms
Figure 3.35: n-Heptane droplet on the impermeable surface at 250oC showing fingers as a result of a Kelvin-
Helmholtz instability merging due to surface tension and cohesive forces.
The low heat of vaporization of n-heptane results in the droplet levitating on the impermeable and
5 μm surfaces as seen at 200 ms but on 100 μm surface the droplet pins to the surface. On the 100
μm surface the vapor pressures between the liquid and the substrate are insufficient to levitate the
mass of the droplet.
83
3.3.5.1 Leidenfrost Evaporation Empirical Model
To get an understanding of the droplet dynamics that occur in the film boiling regime on a porous
surface and to predict the time it takes for a droplet to evaporate, an empirical model first
introduced by Avedisian and Koplik [3] will be employed. The solution presented by Avedisian
and Koplik assumes stokes flow and potential flow exist in the gas phase. Other model assumptions
are as follows:
▪ Gas is assumed to be fully developed laminar flow between the droplet and the substrate.
▪ The thickness of the vapor film is small with respect to the radius of the droplets base such
that the vertical vapor velocity in the film can be neglected with respect to the horizontal
velocity.
▪ Conduction dominates across the vapor film, and radiative effects can be neglected.
▪ Evaporation around the top and sides of the droplet can be neglected.
▪ Uniform film thickness.
▪ Substrate temperature remains constant.
▪ Substrates properties are isotropic (i.e. porosity, and permeability).
To determine whether conduction dominates and that this assumption is applicable, first the total
heat required to vaporize the droplet assuming the droplet has already been levitated and heated to
the saturation temperature can be deduced as follows:
𝑄𝑣𝑎𝑝𝑜𝑟𝑖𝑧𝑒 = ℎ𝑓𝑔 ∙ 𝑚 (3.10)
Where ℎ𝑓𝑔 is the heat of vaporization for the fluid, and m is the mass of the droplet. Solving this
for the n-heptane droplet with an average mass of 5.9 mg, yields:
𝑄𝑣𝑎𝑝𝑜𝑟𝑖𝑧𝑒 ≈ 2 𝐽
Radiation effects can be determined as follows:
𝑄𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 = 𝜎𝑆𝐵𝐴(𝑇𝑆4 − 𝑇𝐷
4)∆𝑡 (3.11)
Where 𝜎𝑆𝐵 is the Stefan-Boltzmann constant, 5.67x10-8 W/m2K4, A is the area of the base of the
droplet where the droplets assumed to be a cylinder with a diameter of 2.54 mm, Ts and TD are the
substrate and droplet temperatures respectively, and ∆𝑡 is the time it takes for the droplet to
evaporate. Obtaining the Leidenfrost temperatures and evaporation times from Figure 3.22,
𝑄𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 on the impermeable, 5 μm, and 100 μm surfaces yield:
84
𝑄𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛_𝐼𝑚𝑝 ≈ 0.09 𝐽
𝑄𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛_5μm ≈ 0.13 𝐽
𝑄𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛_100μm ≈ 0.14 𝐽
The 100 μm surface would see the largest effects due to radiation, which represents only 7% of
the total heat required to vaporize the droplet.
The droplet was modeled as a cylinder of constant radius where the height decreases due to
evaporation (see Figure 3.36). The one-dimensional momentum equations for flow in a porous
medium are as follows:
𝑑2𝑣
𝑑𝑦2 =1
𝜇
𝑑𝑃
𝑑𝑥 (3.12)
𝑑2𝑢
𝑑𝑦2 =𝜇
𝑑𝑃
𝑑𝑥+
𝑢
𝛽 (3.13)
Figure 3.36: Model of the droplet used for determining the evaporation time in the film boiling regime.
Where 𝛼 = 𝜅/휀, 𝜅 is the permeability and 휀 is the porosity. Applying the following boundary
conditions:
𝑣(𝛿) = 0 (a)
𝑢(0) = 𝑣(0) (b)
85
𝑑𝑢
𝑑𝑦= 𝜖
𝑑𝑣
𝑑𝑦 (at the vapor solid interface, y = 0) (c)
𝑢(−ℎ) = 0 (d)
No slip conditions were chosen at the base of the droplet and at the bottom wall of the substrate.
Boundary condition (c) is a fluid shear stress coupling condition at the vapor-solid interface where
𝜖 can be less than 1 if the momentum transport at the vapor-solid interface is not continuous.
According to Avedisian and Koplik this would occur if the vapor flow between the droplet and the
substrate were to transfer momentum to both the vapor in the substrate and to the substrate itself.
They found evaporation results were insensitive to changes in 𝜖 within the range of 0 to 1, so 𝜖 =
1 will be used for the current study.
Integrating equations (3.12) twice yields the velocity solution in the vapor film:
𝑣 =1
𝜇
𝑑𝑃
𝑑𝑥(
𝑦2
2+ 𝐴1𝑦 + 𝐴2) (3.14)
Equation (3.13) is a second order linear non-homogenous differential equation where the solution
yields the velocity of the vapor in the porous substrate:
𝑢 =1
𝜇
𝑑𝑃
𝑑𝑥(𝐵1𝑒𝑦/√ 𝛼 + 𝐵2𝑒−𝑦/√ 𝛼 − 휀𝛼) (3.15)
Where 𝑑𝑃
𝑑𝑥 is the change in pressure within the vapor film, A1, A2 and B1, B2 are constants from
integration.
Applying the boundary conditions, and solving for the integration constants yields:
𝐴1 =
(2𝛼휀 − 𝛿2) (1 − 𝑒−2
ℎ
√𝛼) − 4𝛼휀𝑒−
ℎ
√𝛼
2 (√𝛼 (1 − 𝑒−2
ℎ
√𝛼) + 𝛿 (1 + 𝑒−2
ℎ
√𝛼))
𝐴2 =
−√𝛼𝛿 (1 − 𝑒−2
ℎ
√𝛼) + 4𝛿𝛼𝑒−
ℎ
√𝛼 − 2휀𝛿𝛼 (1 + 𝑒−2
ℎ
√𝛼)
2 (√𝛼 (1 − 𝑒−2
ℎ
√𝛼) + 𝛿 (1 + 𝑒−2
ℎ
√𝛼))
86
𝐵1 =
√𝛼 (2휀𝛼 (1 − 𝑒−
ℎ
√𝛼) − 𝛿2) + 2휀𝛿𝛼𝑒−
ℎ
√𝛼
2 (√𝛼 (1 − 𝑒−2
ℎ
√𝛼) + 𝛿 (1 + 𝑒−2
ℎ
√𝛼))
𝐵2 =
√𝛼 (2휀𝛼𝑒−
ℎ
√𝛼 + 𝑒−2
ℎ
√𝛼(𝛿2 − 2휀𝛼)) + 2휀𝛿𝛼𝑒−
ℎ
√𝛼
2 (√𝛼 (1 − 𝑒−2
ℎ
√𝛼) + 𝛿 (1 + 𝑒−2
ℎ
√𝛼))
To ensure continuity with the mass flow in the system (i.e. within vapor gap and porous substrate),
and obtain our mass balance equation, we start by integrating our velocity equations (3.14) and
(3.15) to obtain mass flow rates through the vapor film and porous substrate. The velocity functions
will be integrated once over the perimeter of the droplet base, A, and integrated a second time with
respect to y, taking the limit from 0 to our vapor thickness, 𝛿, for flow in the vapor gap, and -h
(i.e. substrate thickness) to 0 for the flow in the porous substrate:
�̇�𝑣 = ∬ 𝜌𝑣𝑣 ∙ 𝑑𝐴𝐴
(3.16)
�̇�𝑢 = ∬ 𝜌𝑣𝑢 ∙ 𝑑𝐴𝐴
(3.17)
Solving equations (3.16) and (3.17) yields:
�̇�𝑣 = 2𝜋𝑥𝜌𝑣1
𝜇
𝑑𝑃
𝑑𝑥(
𝛿3
6+ 𝐴1
𝛿2
2+ 𝐴2𝛿) (3.18)
�̇�𝑢 = 2𝜋𝑥𝜌𝑣1
𝜇
𝑑𝑃
𝑑𝑥(𝐵1√𝛼 (1 − 𝑒
−ℎ
√𝛼) − 𝐵2√𝛼 (1 − 𝑒ℎ
√𝛼) − 휀ℎ𝛼) (3.19)
The flow in the vapor film and porous substrate is fed by evaporation at the base of the droplet.
The mass balance yields:
𝜌𝑣𝑣0𝜋𝑥2 = �̇�𝑣 + �̇�𝑢 (3.20)
Where the vapor velocity, 𝑣0, can be obtained assuming the heat supplied to vaporize the base of
the droplet is done so purely through conduction through the vapor gap. An energy balance at the
liquid-vapor interface yields:
𝑣0 =𝑘𝑣∆𝑇
𝜌𝑣ℎ𝑓𝑔𝛿 (3.21)
Where ∆𝑇 is the difference in temperature between the substrate and the droplet, ℎ𝑓𝑔 is the heat of
vaporization for the fluid, 𝜌𝑣 is the vapor density, and 𝑘𝑣 is the thermal conductivity of the vapor.
87
The droplets weight is balanced by the force supplied by a pressure distribution in the vapor film,
hence:
∫ (𝑃 − 𝑃𝑜) ∙ 2𝜋𝑥𝑑𝑥 = 𝑔(𝜌𝑓 − 𝜌𝑣)𝑉𝐴
(3.22)
Equations (3.18) to (3.21) can be combined and integrated to determine ∆𝑃 in the vapor film and
substituted into equation (3.22). Recognizing that ℎ
√𝛼 >> 1, making 𝑒
−ℎ
√𝛼 << 1, which subsequently
drops out of all expressions. This yields a relationship to determine the vapor film thickness
between the droplets base and the substrate:
(1
6) 𝛿5 + (
2
3) √𝛼𝛿4 + 𝛼(𝜅 + 1)𝛿3 + 2𝛼(ℎ휀 − √𝛼)𝛿2 − 2 (Δ𝑇𝐵𝑉
1
3 + 휀𝛼3
2(2√𝛼 − ℎ)) 𝛿 + 2𝐵√𝛼𝑉1
3Δ𝑇 = 0(3.23)
Where B is:
𝐵 =1
8(
3𝑓2
32
4𝜋𝑓1)
43
∙𝜋𝑘𝑣𝜇𝑣
ℎ𝑓𝑔𝜌𝑓𝜌𝑣𝑔
Equation (3.23) was solved numerically in MATLAB using Mullers Method, a root finding
algorithm that can be used to solve higher order polynomial expressions equal to 0. Refer to
Appendix B for the MATLAB code used to solve for the film thickness, along with a description
of Mullers Method.
Figure 3.37 shows the vapor film thickness as a function of the initial surface temperature for the
n-heptane droplet on the three surfaces in the film boiling regime. As the porosity increases the
droplet rests closer to the substrate as more vapor is lost into the surface increasing the heat transfer
and evaporation rates. Furthermore, the vapor thickness increases with initial surface temperature.
Analyzing equation (3.21) shows us that when increasing the surface temperature, the vapor
thickness needs to increase to maintain evaporation which creates the vapor flow and pressure
gradient required to levitate the droplet [3]. Equation (3.23) predicts a Leidenfrost temperature on
the 5 μm surface to be 232oC, approximately 7oC higher than what was observed experimentally.
This difference could be attributed to the selection of the vapor properties for the fluid which were
88
chosen at the average temperature of the substrate and droplet. Similarly, for the 100 μm surface
the predicted Leidenfrost temperature was 290oC, 5oC higher than what was observed
experimentally. This was determined when no roots were returned for Equation (3.23) implying at
the fluid properties and substrate temperature being examined no vapor film would form between
the droplet and the surface.
Figure 3.37: Vapor thickness between the substrate and the base of the n-heptane droplet as a function of the initial
surface temperature in the film boiling regime for the impermeable, 5 μm and 100 μm surfaces.
Equation (3.23) is unable to predict the temperature at which the water droplet will levitate on the
porous surfaces. As the experimentally observed Leidenfrost temperatures are unknown, the
average vapor properties at the appropriate temperature cannot be selected which can have
substantial effects on the predicted values. Figure 3.38 illustrates this and shows the vapor
thickness and predicted Leidenfrost temperatures for the water droplet on the 5 μm and 100 μm
surfaces when the vapor properties were selected at the average between the Leidenfrost
89
temperature and the droplet temperature on the impermeable surface (~170oC). Similarly, the
change that occurs is shown when these properties are selected at a 50oC increase (~220oC) which
is plausible for water on the porous surfaces. Qualitatively the plots reveal that water’s Leidenfrost
point, even at the assumed vapor properties on the 5 μm and 100 μm surfaces occur at a temperature
substantially higher than the temperature range investigated. This could explain why droplet
levitation was not achieved. The porosity and permeability of the substrates were too high to
support both the vapor flow into the surface and the pressure gradient under the droplet required
for levitation.
All fluid properties used in this study were obtained from the National Institute of Standards and
Technology [77].
Figure 3.38: Vapor thickness between the substrate and the base of the water droplet as a function of the initial surface
temperature in the predicted film boiling regime for the 5 μm and 100 μm surfaces. Left shows results when the vapor
properties were selected at 170oC, and the right, at 220oC. Vapor thicknesses are shown for the impermeable surface
in the experimentally observed film boiling regime for comparison. The Leidenfrost temperatures are indicated.
Knowing that heat diffuses across the vapor gap, and assuming the vapor thickness and droplet
radius is independent of time, equation (3.24) can be used to predict the evaporation time in the
film boiling regime.
𝑑𝑚
𝑑𝑡= −
𝑘𝑣
ℎ𝑓𝑔
∆𝑇
𝛿𝜋𝑅2 (3.24)
498oC 384oC 235oC 235oC 609oC 813oC
90
Where R is the initial radius of the droplet taken as 1.27 mm, and the initial mass of the droplet
taken as 5.9 mg. The evaporation times determined without any adjustable parameter
underpredicted the evaporation times. This is due to the assumption made that the initial contact
area is 𝜋𝑅2 which will overpredict the heat transferred to the droplet (see Figure 3.34(a, b) at 200
ms). A fitting parameter was employed reducing the contact area until times matched the
experimental data. Contact areas were reduced on average between the three surfaces by 33%.
Figure 3.39 shows the predicted evaporation time of the n-heptane droplet as a function of the
initial surface temperature on the impermeable, 5 μm and 100 μm surfaces in the film boiling
regime. The experimental evaporation times are shown for comparison. The results are in good
agreeance with the experimental observations.
Figure 3.39: n-Heptane droplet lifetime as a function of the initial surface temperature in the film boiling regime for
the impermeable, 5 μm and 100 μm surfaces. The empirical model output is shown for comparison.
91
Chapter 4 Predicting Heat Transfer Coefficients Between Droplets and Heated Sintered Porous Stainless-Steel Substrates
4.1 Introduction
When a cold droplet impinges on a hot solid surface, heat will transfer from the substrate to the
drop, cooling the surface. Droplets on porous substrates can not only spread over the top surface
but also penetrate into surface pores. Understanding the degree of cooling involves knowledge of
the substrate surface temperature variations at the point of impact.
Experiments were performed looking at surface temperature variations at the point of impact of a
deionized (pure) water and n-heptane droplets on heated sintered porous stainless-steel substrates.
Similar to Chapter 3, there were three principal parameters, the surface temperature, varied from
60oC to 290oC; the fluid properties, specifically the surface tension and latent heat difference
between the water and n-heptane; and finally the two porous substrates investigated (5 μm, and
100 μm average pore sizes). Heat transfer coefficients were determined for water and n-heptane
on the substrates at the varying surface temperatures. The results were compared with those for
droplet cooling on the impermeable substrate.
4.2 Experimental Setup
4.2.1 Impact Conditions
Figure 4.1 shows the experimental setup employed to impact droplets and measure the surface
temperature variations under the droplet during evaporation. The droplet impact conditions
presented in this chapter were the same as the conditions outlined in Chapter 3 with the exception
that the surface temperatures were varied from 60oC to 290oC. Furthermore, the photography and
droplet impact portion of the experimental setup is the same as the system outlined in section 3.2,
Figure 3.1, setup B with the exception in this case that the entire temperature range was examined
using this single setup. The larger aluminum thermal mass used to heat the substrate was modified
to include a 5 mm diameter hole bored straight through the center to allow for the thermocouple
wire embedded in the substrate to travel through when the substrate would sit on the thermal mass.
92
Figure 4.1: Experimental setup employed to measure surface temperature variations directly under the droplet.
4.2.2 Thin-Film Fast Response Thermocouple
Figure 4.2 shows schematics of the thermocouple design employed to measure the surface
temperature variations directly under the droplet. A 600 μm hole was drilled in the center of each
of the substrates. A separate sheathed 250 μm diameter constantan wire was positioned through
the hole of each of the substrates and omega high temperature cement was filled around the wire
the secure it in place. A sheathed wire was chosen to ensure electrical isolation from the substrate
and thermal mass. Constantan was chosen as one of the thermocouple junctions due to it having a
large voltage response when a temperature differential is induced across the wire [78]. A second
508 μm diameter wire was chosen to be 316 stainless-steel matching the material of the substrate.
Thus, the substrate itself formed the second junction. A 600 μm hole was drilled towards the edge
of each of the substrates and the stainless-steel wire was inserted where the hole was pinched
closed using a hammer and a punch securing the connection. The other end of the stainless-steel
wire was welded (Hot Spot TC Welder, DCC Corporation, Pennsauken Township, New Jersey,
United States) to a constantan wire and the junction was inserted into a vacuum flask filled with
ice and water. The cold junction temperature was fixed at 0oC ±1oC, confirmed using a type-k
thermocouple temperature reader (HH12, Omega Engineering, Laval, Quebec, Canada), for all
experiments performed.
93
The two Constantan wires, one from the substrate and the other from the cold junction were run
through a signal amplifier (Omni-amp III, Omega Engineering, Laval, Quebec, Canada) boosting
the signal by a 100x before being outputted to the oscilloscope (TDS2002B, Tektronix, Beaverton,
Oregon, United States). The connection between the substrate and embedded constantan wire was
bridged using a thin layer of silver conductive paste (735825-25G, Sigma-Aldrich Chemicals
Company, St. Louis, Missouri, United States). The droplet was impacted with its center over this
junction, positioned using the X-Y stages along with the high-speed camera setup capturing the
impact at 5000 fps. When the droplet detached from the needle tip, it would fall and pass through
a 9.53 mm slotted optical switch (OPB910W55Z, TT Electronics, Woking, United Kingdom). The
optical switch signal was monitored by an Arduino Uno microcontroller that would send a 1.6 V
DC [79] signal to trigger the oscilloscope and begin logging the voltage output from the
thermocouple. The trigger was delayed by 59 ms, the time taken for the droplet to travel from the
optical switch to just before it touched the thermocouple junction. The delay time was observed
using the high-speed camera setup capturing the fall at 5000 fps. The code used to trigger the
oscilloscope can be seen in Appendix B.
94
Figure 4.2: Top down view (top) \ profile view (bottom) schematic of the fast response thermocouple used to measure
surface temperature variations directly under the droplet.
95
4.2.3 Thermocouple Thin Film
After the Constantan thermocouple wire was cemented in place for each of the substrates, the
excess wire was cut, and 1200 grit sandpaper was used to smooth the junction (see Figure 4.3).
Compressed air was used to blow off any excess material removed during sanding. Using a gauge
26 needle tip and a magnifying glass, a small amount of silver conductive paste was dabbed onto
the center of the junction. Enough paste was added to ensure it made a connection with the
surrounding substrate. The substrate was put into an oven at 150oC for 45 minutes to cure [80].
The film diameter was measured from SEM images and varied from 700 to 900 μm. The total
resistance of the thermocouple circuit as suggested by Heichal et al. [78] was kept below 70 Ω to
reduce electrical noise. Similarly, care was taken to weld electrical junctions where possible and
minimize wire lengths as long wires can act as antennae that can pick up EMI and radio frequency
interference (RFI). Finally, the aluminum thermal mass the substrate rested on was grounded.
96
Before Light Sanding After Light Sanding
(a)
Imp
erm
eab
le
(b)
5 μ
m
(c)
100 μ
m
Figure 4.3: SEM images of the impermeable and two porous substrates with the constantan thermocouple wire
cemented in place before and after light sanding with 1200 grit sandpaper.
97
4.2.4 Thermocouple Calibration
Calibration for each of the three thermocouples was achieved by increasing the substrate
temperature from room temperature up to 300oC in approximately 50oC increments. The heating
was done in an oven where the substrate was left until steady state was reached. A gauge 24 type-
k thermocouple was fixed inside the oven directly above the junction being tested and observed
using an omega k-type thermocouple reader. Steady state was called when the type-k thermocouple
output stayed constant for 10 minutes. At each set point the temperature and voltage output from
the fast response thermocouple were recorded. The substrate was then allowed to cool, and the
voltage and temperature were recorded at approximately 50oC decrements. These steps were
repeated twice for each substrate. Figure 4.4 shows the surface temperature as a function of the
fast response thermocouples voltage output during the heat up and cool down for the impermeable,
5 μm and 100 μm substrates.
Figure 4.4: Surface temperature of the impermeable, and two porous substrates as a function of their respective
thermocouple output. The heat up and cool down data is shown along with the equations for the heat up trendlines.
The average difference in the heat up voltages at each respective set point temperature between
the two trials for the three substrates were less than 3%. The average difference between the heat
y = 223.58x + 6.8767R² = 0.9979
y = 232.04x + 1.6772R² = 0.9978
y = 225.88x + 3.5102R² = 0.9966
0
50
100
150
200
250
300
350
0.0 0.5 1.0 1.5
Surf
ace
Tem
per
atu
re [
oC
]
Thermocouple Voltage [V]
Impermeable - Heat UpImpermeable - Cool Down5 μm - Heat Up5 μm - Cool Down100 μm - Heat Up100 μm - Cool Down
98
up and cool down voltages at each set point temperature for the three substrates were less than 9%.
The increased difference between the heat up and cool down voltages can be attributed to the
thermal inertia associated with the rack and brick the substrate sat on inside the oven. When the
oven was cooling, the surrounding environment inside the oven reached the desired set point
temperature before the rack and brick, where they acted as a thermal mass conducting heat to the
substrate increasing the output voltage. Due to this, trendlines were fit to the heat up data only for
the three substrates. The trendlines fell on top of one another which is to be expected as the two
junction metals, 316 stainless-steel and constantan, are the same for the three substrates.
The equations used to convert the thermocouple voltage to a surface temperature are shown on
Figure 4.4, and are as follows:
𝑇𝑖𝑚𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑙𝑒 = 223.58(𝑉) + 6.8767 (4.1)
𝑇5𝜇𝑚 = 232.04(𝑉) + 1.6772 (4.2)
𝑇100𝜇𝑚 = 225.88(𝑉) + 3.5102 (4.3)
4.2.5 Thermocouple Uncertainty
A design-stage uncertainty analysis on the fast response thermocouple was performed. The system
consists of the signal amplifier to boost the thermocouples output, the oscilloscope used to read
the amplified voltages, and finally the type-k thermocouple and the thermocouple reader used to
calibrate the voltages. The procedure for breaking down the errors for each component was
outlined in section 2.4.2.
The uncertainties associated with the signal amplifier, oscilloscope, and type K thermocouple and
thermocouple reader are as follows:
Omega Signal Amplifier:
Accuracy: The signal amplifier’s gain was set to 100x where there was a potential drift of 0.01%
over the full-scale range per oC [81].
99
Tektronix Oscilloscope:
Accuracy: ± [3% x (reading + vertical position) + 1% of vertical position + 0.2 div + 7 mV].
Resolution: The oscilloscope has an 8-bit A/D converter.
Type K Thermocouple:
Accuracy: ±2.2oC.
Omega Type K Thermocouple Reader:
Accuracy: ± 0.3% x reading + 1oC.
Resolution: 1oC.
A sample calculation to determine the uncertainty on the thin-film fast response thermocouple
output can be seen in Appendix A.
4.3 Data Reduction and Analysis
Figures 4.5 and 4.6 illustrate the thermocouples response for a water and n-heptane droplet on the
impermeable, 5 μm and 100 μm surfaces at the point of impact at an initial surface temperature of
120oC. Temperatures are shown until the droplet reached its maximum spread factor on the
impermeable surface.
100
Figure 4.5: Surface temperature as a function of time for the water droplet on the impermeable, 5 μm and 100 μm
surfaces at the point of impact at an initial surface temperature of 120oC. Temperatures are shown up until the droplet
reached its maximum spread factor on the impermeable surface.
Figure 4.6: Surface temperature as a function of time for the n-heptane droplet on the impermeable, 5 μm and 100 μm
surfaces at the point of impact at an initial surface temperature of 120oC. Temperatures are shown up until the droplet
reached its maximum spread factor on the impermeable surface.
70
80
90
100
110
120
130
-0.5 0.5 1.5 2.5 3.5 4.5
Sub
stra
te S
urf
ace
Tem
per
atu
re [
oC
]
Time [ms]
100 μm
5 μm
Impermeable
75
80
85
90
95
100
105
110
115
120
125
-0.5 4.5 9.5 14.5
Sub
stra
te S
urf
ace
Tem
per
atu
re [
oC
]
Time [ms]
Impermeable
5 μm
100 μm
101
Figures 4.7 and 4.8 show the decrease in surface temperature, ∆Tw, on the impermeable,
impermeable, 5 μm and 100 μm surfaces at the point of impact with the water and n-heptane
droplets respectively. At each initial surface temperature (60oC to 290oC), 5 trials were performed
where the standard deviation is shown on the plots. In many applications the droplet does not
remain stationary on the surface after impact, but rather rebounds and falls off where the heat
transfer takes place during these initial impacts and spread times. Due to this, the temperature
drops are shown at the time taken for the droplet to reach its maximum spread factor on the
impermeable surface (4 ms for water, 13.4 ms for n-heptane). The same times were chosen for the
porous surfaces as to compare the cooling effects within the same time scales.
Water on the 5 μm surface resulted in the largest temperature drop at all initial surface temperatures
investigated as compared to the impermeable and 100 μm surface. This is reasoned to be due to
the decreased thermal conductivity of the substrate in combination with a minimal pore size
resulting in sufficient liquid-solid coverage to promote surface cooling. While the increased pore
size of the 100 μm surface and minimal fluid penetration resulted in the smallest temperature drop
as not enough water contacted the solid surface. An increase in cooling occurs on the impermeable
surface even above the Leidenfrost temperature (235oC) suggesting the water wets the surface
before the vapor cushion forms between the droplet and the substrate. This agrees with
observations shown in Figure 3.33(a) at 1 ms to 5 ms where surface nucleation can be seen taking
place before droplet levitation.
102
Figure 4.7: Comparison between the impermeable, 5 µm and 100 µm surfaces with the drop in surface temperature
after 4 ms, the maximum time taken for the water droplet (at 23oC) to spread on the impermeable surface, as a function
of the initial surface temperature. The temperature was measured at the point of impact. Standard deviations in ∆Tw
are shown.
n-Heptane on the 5 μm and 100 μm surfaces resulted in the largest temperature drop as compared
to the impermeable surface. Due to the improved wetting characteristics (i.e. lower surface tension)
of n-heptane, the fluid penetrates the surface pores during spreading enhancing the liquid-solid
contact area improving substrate cooling. The cooling was less on the impermeable surface with
n-heptane as compared to water evident with the smaller ∆Tw because while n-heptane had a larger
liquid-solid contact area and longer time to cool the surface, the specific heat capacity and heat of
vaporization is significantly larger for water. However, on the 5 μm and 100 μm surfaces n-
heptane’s fluid penetration into the pores results in lower surface temperatures until approximately
160oC on the 5 μm surface and 200oC on the 100 μm surface. At these increased temperatures
rapid vapor production occurs between the n-heptane and the substrate, where the liquid film often
explodes launching fluid off the surface (see Figure 3.29(b and c)) leaving less of the bulk fluid
behind to boil and remove heat. Decreased cooling continues on all three substrates above 160oC.
Interestingly, increased cooling occurs with n-heptane on the 5 μm surface above the Leidenfrost
temperature (235oC) as compared to the 100 μm surface. This could be explained by the loss of
0
20
40
60
80
100
120
0 50 100 150 200 250 300 350
∆T w
4 m
s af
ter
imp
act
[oC
]
Initial Surface Temperature [oC]
Impermeable
5 μm
100 μm
103
bulk fluid on the 100 μm surface. Similarly, the standard deviations on the 100 μm surface are
larger and overlapping with temperature measurements on the 5 μm surface. Increased temperature
drops on the 100 μm surface were measured potentially on trials when less fluid loss occurred.
Figure 4.8: Comparison between the impermeable, 5 µm and 100 µm surfaces with the drop in surface temperature
after 13.4 ms, the maximum time taken for the n-heptane droplet (at 23oC) to spread on the impermeable surface, as a
function of the initial surface temperature. The temperature was measured at the point of impact. Standard deviations
in ∆Tw are shown.
4.3.1 Determining Heat Transfer Coefficients
During initial droplet impact and spreading heat transfer to the droplet can be considered one-
dimensional. The change in the surface temperature at the point of impact can be described using
the following expression:
𝜕𝑢
𝜕𝑡= 𝑎
𝜕2𝑢
𝜕𝑥2 (4.4)
The above expression assumes there is no internal heat generation and the thermal conductivity of
the substrate is isotropic, where 𝑢 is the temperature defined as 𝑇𝑠 − 𝑇𝑠,0, and 𝑎 is the thermal
0
10
20
30
40
50
60
0 50 100 150 200 250 300 350
∆T w
13
.4 m
s af
ter
imp
act
[oC
]
Initial Surface Temperature [oC]
Impermeable
5 μm
100 μm
104
diffusivity of the substrate, 𝑘𝑠
𝑐𝜌. 𝑘𝑠 is the thermal conductivity, 𝜌 is the density, and 𝑐 is the specific
heat capacity. To determine the density of the porous substrates, the property for the impermeable
substrate was taken and multiplied by (1-휀) where 휀 is the porosity of the specific porous substrate.
During these timescales the substrate can be regarded as a semi-infinite solid, where we can apply
the following boundary conditions:
𝑢(∞, 𝑡) = 0 (a)
−𝑘𝑠𝜕𝑢
𝜕𝑥= 𝛼𝐻𝑇𝐶(𝑢𝑑,0 − 𝑢) (𝑎𝑡 𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 𝑎𝑛𝑑 𝑑𝑟𝑜𝑝𝑙𝑒𝑡 𝑏𝑎𝑠𝑒, 𝑥 = 0) (b)
Where 𝛼𝐻𝑇𝐶 is the convective heat transfer coefficient due to the droplet suddenly flowing over
the substrates surface, 𝑢𝑑,0, (𝑇𝑑,0 − 𝑇𝑠,0), is the initial droplet temperature assumed to be
isothermal during impact.
Using an initial condition of 𝑢(𝑥, 0) = 0, equation (4.4) can be solved using a Laplace transform.
Once the transform is carried out and the initial condition is applied the solution to equation (4.4)
takes the form of:
𝑈(𝑥, 𝑠) = 𝐶1𝑒√
𝑠
𝑎∙𝑥
+ 𝐶2𝑒−√
𝑠
𝑎∙𝑥
(4.5)
Accounting for boundary condition (a), C1 is found to be 0.
Performing a Laplace transform on boundary condition (b) and substituting equation (4.5) along
with its derivative with respect to x, C2 can be solved as follows:
𝐶2 = (𝑇𝑑,0 − 𝑇𝑠,0)𝛼𝐻𝑇𝐶/𝑘𝑠
𝑠(√𝑠
𝑎+
𝛼𝐻𝑇𝐶𝑘𝑠
)
(4.6)
Equation (4.6) can now be substituted into equation (4.5), and performing an inverse Laplace
transform and evaluating at x=0, the analytical solution for the transient, one-dimensional heat
conduction equation for the substrate can be determined [82]:
𝑇𝑠 = 𝑇𝑠,0 + (𝑇𝑑,0 − 𝑇𝑠,0) [1 − 𝑒𝑟𝑓𝑐 (𝛼𝐻𝑇𝐶
𝑘𝑠√𝑎 ∙ 𝑡) ∙ exp (
𝛼𝐻𝑇𝐶2∙𝑎∙𝑡
𝑘𝑠2 )] (4.7)
105
Where 𝑇𝑠,0 is the initial substrate temperature, 𝑇𝑠 is the substrate temperature at time, t.
For this analysis the heat transfer coefficient is determined just during droplet spreading because
after spreading the droplet may come to rest, recoil or splash where its behavior is dependent on
the surface wettability, roughness, and orientation which are application specific [16].
The heat transfer coefficient was determined by performing a least-squares fit to match the
predicted surface temperature to the experimental surface temperature during the time it takes for
the water and n-heptane droplet to reach its maximum spread factor. Figures 4.9 and 4.10 show
the least squares fit of the predicted temperatures as a function of time for both the water and n-
heptane on all three surfaces at an initial surface temperature of 120oC respectively.
Figure 4.9: Surface temperature as a function of time for the water droplet on the impermeable, 5 µm and 100 µm
surfaces. Times are shown until the droplet reached its maximum spread factor on the respective surface. Predicted
temperatures using equation 4.7 are shown. Initial surface temperature was 120oC.
95
100
105
110
115
120
125
0 1 2 3 4 5
Surf
ace
Tem
per
atu
re [
oC
]
Time [ms]
Impermeable - MeasuredImpermeable - Predicted5 µm - Measured5 µm - Predicted100 µm - Measured100 µm - Predicted
106
Figure 4.10: Surface temperature as a function of time for the n-heptane droplet on the impermeable, 5 µm and 100
µm surfaces. Times are shown until the droplet reached its maximum spread factor on the respective surface. Predicted
temperatures using equation 4.7 are shown. Initial surface temperature was 120oC.
Figures 4.11 and 4.12 show the heat transfer coefficients as a function of the initial surface
temperature for the water and n-heptane droplet on the three surfaces. As the porosity of the
substrate increases the heat transfer coefficients decrease with both water and n-heptane due a
decrease in the thermal conductivity of the substrate. This could also explain why increased
evaporation times were observed with n-heptane at lower wall temperatures (< 150oC) on the
porous surfaces as compared to the impermeable surface seen in Figure 3.22. Similar behavior was
also observed for the water on the 100 μm surface at lower wall temperatures (see Figure 3.21).
The significance of this decrease in the heat transfer coefficients observed with an increase in the
substrates porosity suggests that at a given surface temperature the rate at which either water or n-
heptane can extract heat from the surface decreases on impact. This would have a direct impact on
the fluids effectiveness to cool the surface. The heat transfer coefficient for water on the
impermeable surface peaks at approximately the Leidenfrost temperature (~240oC) and declines
thereafter. Interestingly, at surface temperatures above 240oC, increased surface cooling is still
measured (see Figure 4.7), though the heat transfer coefficients decline. This behavior could be
explained by a reduction in the average temperature measured over the droplets spread time where
90
95
100
105
110
115
120
125
0 5 10 15
Surf
ace
Tem
per
atu
re [
oC
]
Time [ms]
Impermeable - MeasuredImpermeable - Predicted5 µm - Measured5 µm - Predicted100 µm - Measured100 µm - Predicted
107
vapor formation is preventing the water from directly wetting the thermocouple, resulting in its
delayed response.
Figure 4.11: Heat transfer coefficient for the water during droplet spreading on the impermeable, 5 µm and 100 µm
surfaces as a function of the initial surface temperature.
0
10000
20000
30000
40000
50000
60000
0 50 100 150 200 250 300 350
Hea
t Tr
ansf
er C
oef
fici
ent
[W/m
2K
]
Initial Surface Temperature [oC]
Impermeable
5 µm
100 µm
108
Figure 4.12: Heat transfer coefficient for the n-heptane during droplet spreading on the impermeable, 5 µm and 100
µm surfaces as a function of the initial surface temperature.
0
2000
4000
6000
8000
10000
12000
14000
16000
0 50 100 150 200 250 300 350
Hea
t Tr
ansf
er C
oef
fici
ent
[W/m
2K
]
Initial Surface Temperature [oC]
Impermeable
5 µm
100 µm
109
Chapter 5 Conclusions
This study investigated the impact and evaporation of water and n-heptane droplets on two sintered
porous 316 stainless-steel substrates with average pore diameters of 5 µm and 100 µm respectively.
The results were compared to the droplets impact and evaporation on a 316 impermeable stainless-
steel substrate. The porous surfaces were characterized by their porosity, surface roughness,
permeability, and thermal conductivity. The droplet impact process was photographed and
analyzed, and droplet lifetimes were measured on all three substrates at initial surface temperatures
varying from 60oC to 300oC. A digital scale was successfully employed to measure evaporation
times at low wall temperatures ranging from 60oC to 120oC removing the uncertainty associated
when using imaging techniques due to liquid penetration into the surface pores. A simple empirical
model was presented that can predict the evaporation time of the n-heptane droplet in the film
boiling regime. Furthermore, surface temperature variations were measured at the point of droplet
impact on all three surfaces at initial surface temperatures ranging from 60oC to 290oC. Heat
transfer coefficients were determined for the water and n-heptane on the three surfaces
immediately following the droplets impact during spreading. Several specific conclusions were
drawn from this study and are as follows:
1. Preliminary work was done to show the change in the maximum spread factor observed
with water at room temperature on the 5 µm substrate was a result of surface roughness
only and on the 100 µm substrate a result of both substrate porosity and the surface
roughness. n-Heptane’s change in the maximum spread factor observed on the porous
substrates was determined to be a result of both surface roughness and substrate porosity.
2. Evaporation times were on average an order of magnitude longer with the water on all three
surfaces as compared to the n-heptane due to n-heptane’s lower heat of vaporization. At
high wall temperatures (> 150oC) the porous surfaces increased the heat transfer to both
the water and n-heptane droplets resulting in the lowest evaporation times as the wetted
contact area increased due to fluid penetration into the surface pores.
110
3. The Leidenfrost temperature with both water and n-heptane was observed to be delayed on
the porous surfaces as compared to the impermeable surface. The Leidenfrost temperature
was not reached with water on the porous surfaces while it was successfully reached with
n-heptane again due to n-heptane’s lower heat of vaporization. The Leidenfrost
temperature was shown to increase with substrate porosity (and thus permeability). This
delay was concluded to be a result of the vapor flow into the porous surface where a larger
surface temperature is needed to build the appropriate pressure acting on the droplets base
necessary to support the droplets weight for levitation. This conclusion was confirmed by
an empirical model which successfully predicted the evaporation times of the n-heptane
droplet in the film boiling regime.
4. As the substrates porosity increased the heat transfer coefficients decreased during
spreading for both the water and n-heptane on the impermeable, 5 µm, and 100 µm surfaces
due to the significant decrease in the substrates thermal conductivities. This suggests that
at a given surface temperature the rate at which either water or n-heptane can extract heat
from the surface decreases on impact. This would have a direct impact on the fluids
effectiveness to cool the surface.
5.1 Future Research
Recommendations for work that could be completed in the future are as follows:
1. Further investigate the role of porosity and surface roughness on the droplets spread factor.
Several models exist in literature that accurately predict the droplets spread factor on smooth
impermeable surfaces, but few offer any solution for accurately predicting the spread factor
when the surface roughness is increased, and porosity is introduced.
2. Several phenomena were observed through photography that could be investigated further.
Namely, the mechanism(s) causing secondary atomization as seen in Figure 3.31 and 3.32
and first described by Cossali et al. [72, 73]. This formation has been termed the “pagoda-
like” bubbles and the theoretical explanation for their existence appears to still be
unknown.
3. Examine the cooling effectiveness of water and n-heptane on the porous surfaces. This
would require an understanding of the liquid coverage inside the porous substrate. As many
applications operate at increased temperatures where phase change occurs (i.e. nucleate
111
boiling regime), investigating how the cooling effectiveness changes in these regimes
could be done.
112
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Appendix A
Permeability Apparatus Measurement Uncertainty Sample Calculation
To determine the overall measurement uncertainty for results produced by the permeability
apparatus, first, the design-stage uncertainties for the pressure, and flow measurements taken
during experimentation needs to be determined. This requires deducing the zero-order uncertainty
and instrument uncertainty for both the pressure gauge and flow meters. The key assumptions
being made are, the instrument uncertainties are reported at the 95% confidence level, and the data
follows a normal distribution.
The zero-order uncertainty for the pressure gauge is obtained from the resolution given as 68.95
Pa. Thus, the zero-order uncertainty can be obtained by doing the following:
𝑢0∆𝑝=
1
2𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (A.1)
𝑢0∆𝑝=
1
268.95 𝑃𝑎 = ±34.48 𝑃𝑎
The instruments uncertainty is obtained from the accuracy given as ±0.08% of the full-scale
range, 0 to 350 kPa (0 to 50 PSI) and is defined as:
𝑢𝑐∆𝑝= 0.0008 ∙ 350 𝑘𝑃𝑎 = ±0.28 𝑘𝑃𝑎
The design-stage uncertainty for the pressure gauge is then determined by:
𝑢𝑑∆𝑝= √(𝑢0𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒
)2 + (𝑢𝑐𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒)2 (A.2)
𝑢𝑑∆𝑝= √(34.48 𝑃𝑎)2 + (280 𝑃𝑎)2
𝑢𝑑∆𝑝= ±282.1 𝑃𝑎
119
Two flow meters were used during experimentation. The error due to the resolution of the
instrument for both flow meters was considered negligible. The instrument uncertainty for the
FMA5526A was given as ±1.5% over the full-scale flow range and can be defined as follows:
𝑢𝑐5526A= 0.015 ∙ 30
𝐿
𝑚𝑖𝑛= ±0.45
𝐿
𝑚𝑖𝑛
𝑢𝑐5526A= ±7.5𝑥10−6
𝑚3
𝑠
The design-stage uncertainty for the FMA5526A is then:
𝑢𝑑5526A= ±7.5𝑥10−6
𝑚3
𝑠
The instrument uncertainty associated with the FMA1843 is broken down into two respective flow
ranges. At flows between 0-20% of the full-scale range the accuracy was given as ±3%. At flows
between 20-100% of the full-scale range the accuracy was given as ±1.5%. The full-scale flow
range is 200 L/min. Therefore, the instrument uncertainties can be defined as:
𝑢𝑐184320% = 0.03 ∙ 200
𝐿
𝑚𝑖𝑛= ±6.0
𝐿
𝑚𝑖𝑛
𝑢𝑐184320% = ±0.0001
𝑚3
𝑠
𝑢𝑐1843100% = 0.015 ∙ 200
𝐿
𝑚𝑖𝑛= ±3.0
𝐿
𝑚𝑖𝑛
𝑢𝑐1843100% = ±5.0𝑥10−5
𝑚3
𝑠
The design-stage uncertainties for the FMA1843 are then:
𝑢𝑑184320% = ±0.0001
𝑚3
𝑠
𝑢𝑑1843100% = ±5.0𝑥10−5
𝑚3
𝑠
120
The design-stage uncertainty associated with the flow meters that will be used will depend on the
flow rate being examined. A summary of the flow rate ranges and the uncertainty that will be used
is shown in the following Table A.1.
Table A.1: Summary of the design-stage uncertainties for the flow meters depending on the flow rate being examined.
Instrument Flow Range
[L/min (m3/s)]
Uncertainty
[L/min (m3/s)]
FMA5526A 0 to 30 (0 to 5.0x10-4) ±0.45 (7.5x10-6)
FMA1843 31 to 40
(5.2x10-4 to 6.7x10-4)
±6.0 (1.0x10-4)
FMA1843 41 to 200
(6.8x10-4 to 3.3x10-3)
±3.0 (5.0x10-5)
Based on the manufacturer’s specification, the calipers used to measure the thickness of the porous
substrates have a measurement uncertainty of:
𝑢𝑑𝐿= ±0.02 𝑚𝑚
𝑢𝑑𝐿= ±2.0𝑥10−5 𝑚
From equation 2.10 the uncertainty of the calculated apparent permeability can be determined by
performing a RSS on the partial derivatives of the variables in the permeability equation that have
an attached uncertainty.
𝜅′ =𝑄2𝜇𝐿𝑃𝑎𝑡𝑚
𝐴(𝑃𝑐ℎ𝑎𝑚𝑏𝑒𝑟2−𝑃𝑎𝑡𝑚
2) (A.3)
𝑈𝜅′ = √(𝜕𝜅′
𝜕𝑄 ∙ 𝑢𝑑𝑄
)2
+ (𝜕𝜅′
𝜕𝐿 ∙ 𝑢𝑑𝐿
)2
+ (𝜕𝜅′
𝜕𝑃𝑐ℎ𝑎𝑚𝑏𝑒𝑟 ∙ 𝑢𝑑∆𝑝
)2
(A.4)
It should be noted that 𝑢𝑑𝑄 will be selected from Table A.1 depending on the flow rate being
examined.
The intrinsic permeability is deduced from a Klinkenberg permeability plot. A weighted least
squares fit is performed to add a straight trendline to the linear portion of the dataset where the
121
uncertainties in the individual data points are considered. Data points with larger assigned
uncertainties get a smaller weighting, while points with smaller uncertainties are given a larger
weighting when the least squares fit is performed. As a result, a larger emphasis on fitting a line
to the accurate datapoints occurs. R is used to compute the statistics on the trendline where the
standard error on the y-intercept (i.e. intrinsic permeability) is reported.
122
Thermal Conductivity Apparatus Measurement Uncertainty Sample Calculation
To determine the overall measurement uncertainty for the thermal conductivity apparatus, first,
the design-stage uncertainty for the temperature measurements taken during experimentation
needs to be determined. This is done by defining the design-stage uncertainties for each individual
component associated with temperature measurements. This includes the zero-order uncertainty
and instrument uncertainty for both the DAQ and the type J thermocouples. The key assumptions
being made are, the instrument uncertainties are reported at the 95% confidence level, and the data
follows a normal distribution.
The zero-order uncertainty for the DAQ can be estimated using the following relationship:
𝑢0𝐷𝐴𝑄=
1
2
𝐸𝐹𝑆𝑅
2𝑀 (A.5)
𝑢0𝐷𝐴𝑄=
1
2
20 𝑉
222 𝑏𝑖𝑡= ±2.38 𝜇𝑉
Where 𝐸𝐹𝑆𝑅 is the full-scale voltage range over which the DAQ operates, in this case 0 to 20 V. M
is the register size, 22-bit, for the digital side of the A/D converter. This allows the register to
represent 2M different binary numbers. To convert the zero-order uncertainty to its equivalent
temperature if produced by a type J thermocouple, a reference table provided by the National
Institute of Standards and Technology (NIST) can be used [83]. It is assumed that the cold junction
temperature is 0oC which yields a thermocouple temperature of ±0.047𝑜𝐶. The instruments
uncertainty was given by the manufacturer such that readings are accurate to 0.01% plus 0.002%
of the range. The mean temperature sensed by the thermocouples in the system was 70oC. Where
the full-scale input range for type J thermocouples is 0 to 800oC. The following instrument
uncertainty can then be determined:
𝑢𝑐𝐷𝐴𝑄= 𝑢𝑐𝐷𝐴𝑄𝑅𝑒𝑎𝑑𝑖𝑛𝑔
+ 𝑢𝑐𝐷𝐴𝑄𝑅𝑎𝑛𝑔𝑒 (A.6)
𝑢𝑐𝐷𝐴𝑄= 0.0001 ∙ 70𝑜𝐶 + 0.00002 ∙ 800𝑜𝐶 = ±0.023𝑜𝐶
The design-stage uncertainty for the DAQ is then determined by:
123
𝑢𝑑𝐷𝐴𝑄= √(𝑢0𝐷𝐴𝑄
)2 + (𝑢𝑐𝐷𝐴𝑄)2 (A.7)
𝑢𝑑𝐷𝐴𝑄= √(0.047𝑜𝐶)2 + (0.023𝑜𝐶)2
𝑢𝑑𝐷𝐴𝑄= ±0.052𝑜𝐶
For the type J thermocouples used in the system, Personal DaqViewTM limited the resolution of
the signal to ±0.001oC, which gives the zero-order uncertainty. The accuracy of the measurements
produced by the thermocouple is ±2.2oC. Therefore 𝑢0𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒, and 𝑢𝑐 𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒
are as
follows:
𝑢0 𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒= ±0.001𝑜𝐶
𝑢𝑐 𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒= ±2.2𝑜𝐶
The design-stage uncertainty for the type J thermocouple is then determined by:
𝑢𝑑 𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒= √(𝑢0𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒
)2 + (𝑢𝑐 𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒)2 (A.8)
𝑢𝑑 𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒= √(0.001𝑜𝐶)2 + (2.2𝑜𝐶)2
𝑢𝑑 𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒= ±2.2𝑜𝐶
Using the RSS method to combine the DAQ and type J thermocouple design-stage uncertainties,
the design-stage uncertainty associated with temperature measurements can be deduced as follows:
𝑢𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒= √(𝑢𝑑𝐷𝐴𝑄
)2 + (𝑢𝑑 𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒)2 (A.9)
𝑢𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒= √(0.052𝑜𝐶)2 + (2.2𝑜𝐶)2
𝑢𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒= ±2.2𝑜𝐶
124
The uncertainty as a result of the measurement error induced by the DAQ is negligible as compared
to the uncertainty in the measurements produced by the type J thermocouples.
To determine the final uncertainty in the thermal conductivity measurement, first the uncertainty
of the calculated heat flux passing through the substrate is determined. Using equation 2.12 the
heat flux was determined on both the hot and cold sides of the stack and averaged together.
𝑞"𝐻𝑜𝑡 = −𝑘𝐴𝐿𝑇1−𝑇6
𝑥1−𝑥6 (A.10)
𝑞"𝐶𝑜𝑙𝑑 = −𝑘𝐴𝐿𝑇7−𝑇12
𝑥7−𝑥12 (A.11)
T1 through T12 are the temperatures in oC produced by the twelve thermocouples inserted in the
aluminum bars. 𝑘𝐴𝐿 is the thermal conductivity of the 6061 Aluminum, and x1 through x12 is the
position, in meters, of the thermocouples in the aluminum bars. Position 0 m starts at the first
thermocouple on the hot side. For this analysis the thermal conductivity for the aluminum bars is
considered to be exact, as no uncertainty was given from the manufacturer. Similarly, as the
machining for the aluminum bars was done professionally, the position of the inserted
thermocouple is considered to be exact, as this error will be negligible compared to the error in the
thermocouples. As a result, these values will be excluded from the uncertainty analysis. The
uncertainty for the heat flux measurements is as follows:
𝑈𝑞"𝐻𝑜𝑡= √(
𝜕𝑞"𝐻𝑜𝑡
𝜕𝑇1 ∙ 𝑢𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
)2
+ (𝜕𝑞"𝐻𝑜𝑡
𝜕𝑇6 ∙ 𝑢𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
)2
(A.12)
𝑈𝑞"𝐶𝑜𝑙𝑑= √(
𝜕𝑞"𝐶𝑜𝑙𝑑
𝜕𝑇7 ∙ 𝑢𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
)2
+ (𝜕𝑞"𝐶𝑜𝑙𝑑
𝜕𝑇12 ∙ 𝑢𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
)2
(A.13)
As an average of the hot and cold heat fluxes was taken, the overall uncertainty on the heat flux
can be determined as follows:
𝑈𝑞′′̅̅ ̅̅ = √(𝜕𝑞"̅̅ ̅
𝜕𝑞"𝐻𝑜𝑡 ∙ 𝑈𝑞"𝐻𝑜𝑡
)2
+ (𝜕𝑞"̅̅ ̅
𝜕𝑞"𝐶𝑜𝑙𝑑 ∙ 𝑈𝑞"𝐶𝑜𝑙𝑑
)2
(A.14)
Once the temperatures of the thermal interface sheets are known, the total thermal resistance of
the sheets and substrate can be determined using equation 2.11.
125
𝑅 = 𝑇𝐻−𝑇𝐶
𝑞"̅̅ ̅ (A.15)
Where TH and TC are the temperatures at the interfaces between the hot and cold aluminum bars
and the thermal interface sheets. 𝑞"̅ is the average heat flux through the thermal interface sheets
and substrate. The uncertainty in the total thermal resistance is then:
𝑈𝑅 = √(𝜕𝑅
𝜕𝑇𝐻 ∙ 𝑢𝑇𝐻
)2
+ (𝜕𝑅
𝜕𝑇𝐶 ∙ 𝑢𝑇𝐶
)2
+ (𝜕𝑅
𝜕𝑞"̅̅ ̅ ∙ 𝑈𝑞"̅̅ ̅)2
(A.16)
TH and TC uncertainties were obtained from using Excel to compute the statistics on the linear
trendlines fit to the temperature data points. These trendlines were used to extrapolate the TH and
TC temperatures in the hot and cold aluminum bars.
The thermal conductivity of the porous substrates can be determined using equation 2.15:
𝑘 =𝐿
𝑅−2∙𝑅𝐺𝑆 (A.17)
Thus, the uncertainty in the thermal conductivity is:
𝑈𝑘 = √(𝜕𝑘
𝜕𝐿 ∙ 𝑢𝑑𝐿
)2
+ (𝜕𝑘
𝜕𝑅 ∙ 𝑈𝑅)
2
+ (𝜕𝑘
𝜕𝑅𝐺𝑆 ∙ 𝑈𝑅𝐺𝑆
)2
(A.18)
The uncertainty in the thermal resistance of the graphite sheets was deduced using the same
methodology outlined in this section. The uncertainty in the thickness measurement was due to the
caliper instrument uncertainty outlined in Permeability Apparatus Measurement Uncertainty
Sample Calculation section:
𝑢𝑑𝐿= ±2.0𝑥10−5 𝑚
126
Thin-Film Fast Response Thermocouple Measurement Uncertainty Sample Calculation
The design-stage uncertainty for the temperature measurements taken using the fast response
thermocouple needs to be determined. Similar to the analysis performed with the permeability
apparatus, and thermal conductivity apparatus, this is done by defining the design-stage
uncertainties for each individual component associated with the temperature measurements. As
before, the key assumptions being made are, the instrument uncertainties are reported at the 95%
confidence level, and the data follows a normal distribution.
The design-stage uncertainty for the signal amplifier is a result of the 0.01% (GAIN is 100x) signal
drift that occurs over the full-scale range per oC. The full-scale range is ±9Vdc, and the amplifier
is plugged in 30 to 60 minutes before operation where the amplifiers temperature change after this
time is expected to be negligible and is set to 1oC for analysis:
𝑢𝑑𝐴𝑚𝑝= 0.0001 ∙ 18𝑉𝑑𝑐 ∙ 1𝑜𝐶
𝑢𝑑𝐴𝑚𝑝= ±1.8 𝑚𝑉
The zero-order uncertainty for the oscilloscope can be estimated using equation (A.5):
𝑢0𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒=
1
2
𝐸𝐹𝑆𝑅
2𝑀 (A.19)
𝑢0𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒=
1
2
4 𝑉
28 𝑏𝑖𝑡= ±7.81 𝑚𝑉
The instruments uncertainty was given by the manufacturer as, ± [3% x (reading + vertical
position) + 1% of vertical position + 0.2 div + 7 mV]. The average voltage response of the fast
response thermocouple over the temperature range investigated was 0.77 V which corresponded
to a vertical position of 7.7 div at 100 mV/div:
𝑢𝑐𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒= ±[0.03 ∙ (0.77 V + 7.7 div) + 0.01 ∙ 7.7 div + 0.2 𝑑𝑖𝑣 + 7 𝑚𝑉]
𝑢𝑐𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒= ±7.53 𝑚𝑉
The design-stage uncertainty for the oscilloscope is:
127
𝑢𝑑𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒= √(𝑢0𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒
)2 + (𝑢𝑐𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒)2 (A.20)
𝑢𝑑𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒= √(7.81 𝑚𝑉)2 + (7.53 𝑚𝑉)2
𝑢𝑑𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒= ±10.85 𝑚𝑉
The design-stage uncertainties in the voltage output for the signal amplifier and the oscilloscope
reading can be converted to their equivalent temperatures using the calibration equations outlined
in section 4.2.4, equations (4.1 to 4.3), for the impermeable, 5 µm and 100 µm substrates
respectively:
Table A.2: Summary of the design-stage uncertainties for the signal amplifier and oscilloscope for the substrate being
investigated.
Substrate Signal Amplifier
Uncertainty [oC]
Oscilloscope
Uncertainty [oC]
impermeable ±0.40 ±2.43
5 µm ±0.42 ±2.52
100 µm ±0.41 ±2.45
The uncertainty for the signal amplifier and the oscilloscope will be selected from Table A.2
depending on the substrate being investigated.
The design-stage uncertainty for the type K thermocouple is:
𝑢𝑑 𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒= ±2.2𝑜𝐶
The zero-order uncertainty for the type K thermocouple reader can be obtained from the readers
resolution given as 1oC. Using equation (A.1):
𝑢0𝑇𝐶𝑅=
1
21𝑜𝐶
𝑢0𝑇𝐶𝑅= ±0.5𝑜𝐶
128
The type k thermocouple readers instrument uncertainty was given by the manufacturer as ± 0.3%
x reading + 1oC, where the average temperature sensed by the thermocouple during calibration
was 180oC:
𝑢𝑐 𝑇𝐶𝑅= 0.003 ∙ 180𝑜𝐶 + 1𝑜𝐶
𝑢𝑐 𝑇𝐶𝑅= ±1.54𝑜𝐶
The design-stage uncertainty for the thermocouple reader is then:
𝑢𝑑 𝑇𝐶𝑅= √(𝑢0𝑇𝐶𝑅
)2 + (𝑢𝑐 𝑇𝐶𝑅)2 (A.21)
𝑢𝑑 𝑇𝐶𝑅= √(0.5𝑜𝐶)2 + (1.54𝑜𝐶)2
𝑢𝑑 𝑇𝐶𝑅= 1.62𝑜𝐶
The total design-stage uncertainty associated with the fast response thermocouple for the
impermeable, 5 µm and 100 µm substrates can be deduced using:
𝑢𝑑𝐹𝑅𝑇= √(𝑢𝑑𝐴𝑚𝑝
)2 + (𝑢𝑑𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒)2 + (𝑢𝑑 𝑇ℎ𝑒𝑟𝑚𝑜𝑐𝑜𝑢𝑝𝑙𝑒
)2 + (𝑢𝑑 𝑇𝐶𝑅)2 (A.22)
Where 𝑢𝑑𝐴𝑚𝑝 and 𝑢𝑑𝑂𝑠𝑐𝑖𝑙𝑙𝑜𝑠𝑐𝑜𝑝𝑒
will be selected from Table A.2 depending on the substrate being
investigated. The final design-stage uncertainties are as follows:
Table A.3: Summary of the design-stage uncertainties for the fast response thermocouple.
Substrate Fast Response
Thermocouple
Uncertainty [oC]
impermeable ±3.68
5 µm ±3.74
100 µm ±3.69
129
Appendix B
MATLAB Code
Mettler Toledo AG245 Scale Weight Logger Program:
(*Will work with other models as the commands should be standard).
Program will poll the scale weight until a key press is made where a plot of the weight as a function
of time will be displayed at the end before the program terminates. The weights and times will be
logged in real-time to a text file. %%Created by: Nick Lipson
%%Date: 11-04-2016
%Close all figures before starting
close all
%Clear all variables from memory
Clearvars
%Clear all unused handles to open serial ports
delete(instrfindall);
%Open desired serial port
s = serial('COM7');
%Setup serial data line
set(s,'BaudRate',2400,'DataBits',7, 'Parity', 'even', 'StopBits', 1)
%Open the port
fopen(s);
%Setup handle to our text file so we can record our weight values
fileID = fopen('C:\Users\Ansel Jr\Desktop\WeighingResults.txt','wt');
%timeStep = 0.20; %Time step in seconds
counter = 1;
time = 0;
global KEY_IS_PRESSED
KEY_IS_PRESSED = 0;
%Returns the current figure handle.
%If a figure does not exist, then gcf creates a figure and returns its handle
gcf
set(gcf, 'KeyPressFcn', @myKeyPressFcn)
hold all
xlabel('Time (seconds)'), ylabel('Weight (g)');
title('Mettler Toledo AG-245 Weight Values');
ylim([4 12]);
tic %Start seconds counter
while ~KEY_IS_PRESSED
%SEND A COMMAND TO THE SCALE
%This will request the scale to send us the weight value with…
%currently displayed units immediately
fprintf(s,'SIU')
130
%RECEIVE DATA FROM SCALE
out = fscanf(s);
time = toc;
splitString = strread(out,'%s');
weight(counter) = str2double(splitString(3));
%Create string with time and weight along with their respective units
strfile = char(strcat(num2str(time),{' '},{'s'},{','},...
splitString(3),{' '},splitString(4),{','}));
%Write weight and time to file
fprintf(fileID,'%s\n', strfile);
%plot(time, weight(counter), '.', 'MarkerSize', 15);
%drawnow %Update Figures
%Setup time array for post loop graphing
timeArray(counter) = time;
%Increment counters
%time = time + timeStep;
counter = counter + 1;
%pause(timeStep); %Pause
end
hold off
%Replot all data using a line to better show the trend
plot(timeArray, weight, '-.','MarkerSize', 15);
legend('Mettler Toledo AG-245 Weight Values');
xlabel('Time (seconds)'), ylabel('Weight (g)');
title('Mettler Toledo AG-245 Weight Values');
%Find max weight value and store index
indexmax = find(max(weight) == weight);
%Store the max weight and associated time values
xmax = timeArray(indexmax);
ymax = weight(indexmax);
strmax = ['Max: ',num2str(ymax)];
%Add maximum weight text to the plot
text(xmax,ymax,strmax,'HorizontalAlignment','right');
%Close handle to serial port and clean up
fclose(s);
delete(s);
clear s;
fclose(fileID);
disp('FINISHED WEIGHING...')
131
Droplet Volume Calculator Program:
Program will read in a directory containing an image sequence of a droplet sitting on a surface.
The volume of the droplet contained in each frame will be computed. Only image sequences where
the viewing angle is 0o (i.e. camera lens is parallel with the plate) can be used.
%% Created by: Khalil Sidawi & Nick Lipson %% Date: 08-1-2018
clear all close all clc
% set path to folder pathFolder = ' ';
% get images Images = dir(fullfile(pathFolder, '\*.tif')); Image = {Images.name}';
% import image and prerpocess (sharpen, brighten, and convert to 0-1) X = im2double(imread(fullfile(pathFolder, char(Image(1)))));
% threshold image level_0 = multithresh(X,20);
% get binary image, elimates anything smaller than 15 pixels, close % droplet, and fill holes XX = imfill(imclose(imbinarize(X, level_0(15)),strel('disk',12)),'holes');
% overlay images for comparison figure(1); imshowpair(X, XX); figure(2); imshowpair(X, XX, 'montage');
% conversion ratio from pixels to mm conversion = 97.6; %pixels/mm (5mn) % conversion = 64.96; %pixels/mm (100mn)
V = zeros(1, 5000); Uv = zeros(1, 5000); Upixel = 1; %+/- 0.5 pixels UpixelConversion = Upixel/conversion;
% image number to analyze for t = 12:5012%numel(Image) %5mn folder %for t = 16:5016%numel(Image) %100mn folder % importing image and preprocessing (sharpen, brighten, and convert to
0-1) I = im2double(imsharpen(imadjust(imread(fullfile(pathFolder,
char(Image(t)))))));
% threshold image level_0 = multithresh(I,20);
132
% get binary image, elimates anything smaller than 15 pixels, close % droplet, and fill holes II = imfill(imclose(imbinarize(I,level_0(15)),strel('disk',12)),'holes');
% subtrate initial frame, X, which should be preprocessed and binarized. III = imsubtract(XX,II);
III(III<0) = 0; %removes area smaller than 20 III = bwareaopen(III,20);
VI = imclose(III,strel('disk',20)); %5mn %VI = imclose(III,strel('disk',100)); %100mn
%figure(3);
%imshowpair(I, VI);
%finds blob blobMeasurements = regionprops(VI,'Centroid','PixelList','Area');
j = 0; sum_dv_da = 0;
% find the index associated with the blob that has the maximum area (i.e. % our drop) f = find(cat(1,blobMeasurements.Area) ==
max(cat(1,blobMeasurements.Area)));
% loop from min y value to max y value for i =
min(blobMeasurements(f).PixelList(:,2)):max(blobMeasurements(f).PixelList(:,2
)) j = j + 1; % length of chrod between min and max x for a given y value L(j) = (numel(nonzeros(blobMeasurements(f).PixelList(:,2) == i))) /
conversion; Uda_dd(t, j) = (pi * L(j) / 2) * UpixelConversion; Udv_da(t, j) = (pi * L(j) / (2 * conversion)) * Uda_dd(t, j);
sum_dv_da = sum_dv_da + Udv_da(t, j)^2; end
Uv(t) = sqrt(sum_dv_da);
% calculate volume V(t) = pi * sum(L.^2) / 4 * (1 / conversion);
V_t = V.'; U_v = Uv.';
clearvars L; end
133
Vapor Film Thickness Solver Program:
Program will read in the solved coefficients for equation (3.23) and pass them into a root finding
algorithm (i.e. Mullers Method) to solve for the vapor film thickness at each initial surface
temperature within the film boiling regime for the n-heptane droplet on the impermeable, 5 µm,
and 100 µm substrates. The vapor film thicknesses are plotted as a function of the initial surface
temperature. These vapor film thicknesses are then passed to our conduction model to solve for
the evaporation time. The evaporation time is then plotted as a function of the initial surface
temperature within the film boiling regime for each substrate. The experimental evaporation time
data is plotted on the same graph as the empirical model data for comparison.
PlotPub [83] was utilized to format the plots shown for this data.
Mullers Method is an iterative procedure that takes three initial “guess” film thicknesses, and
computes the associated y values from the original function. A quadratic function that goes
through these three y values is constructed where the x-intercepts for the quadratic function are
determined. The x-intercepts are substituted back into the original function. The x-intercept that
results in the output from the original function coming closest to 0 becomes our new point, where
the first point guessed on the quadratic curve is discarded. This process repeats until a film
thickness (i.e. real root) is determined resulting in the output of our original function equating to
approximately 0 (within 1x10-8 m). This quadratic function essentially moves over our original
polynomial solving for all the roots. The film thickness function used in this study only ever has a
single logical positive root, which is our film thickness. It should be noted that Mullers Method
will only work on a function, f(x) that is continuous.
%% Created by: Nick Lipson %% Date: 04-18-2018
clear all; close all; clc;
%% Experimental data for Heptane err0mn_time = [0.54, 0.63, 0.42, 0.98, 0.55]; hep_exp_time = [11.48, 10.67, 10.08, 8.80, 7.85]; hep_exp_surfaceT = [190, 195, 200, 250, 300]; err5mn_time = [0.32, 0.64, 0.53, 0.73, 0.80]; hep5mn_exp_time = [10.69, 10.64, 10.04, 9.37, 7.55]; hep5mn_exp_surfaceT = [225, 230, 235, 250, 300]; err100mn_time = [0.38, 0.27, 0.30, 0.34]; hep100mn_exp_time = [6.41, 6.00, 5.91, 5.76]; hep100mn_exp_surfaceT = [285, 290, 295, 300];
num_columns = 7; temperature_points = 300-225; filename = "Model.xlsx"; plates = ["Impermeable - Avedisian Model", "5mn [\phi = 0.31] - Avedisian
Model", "100mn [\phi = 0.44] - Avedisian Model"];
hold on p1 = figure(1);
134
%Gives us info on the excel file [status,sheets,xlFormat] = xlsfinfo(filename); C = zeros(temperature_points+1, num_columns); e = zeros(temperature_points+1, numel(sheets)); %time = zeros(temperature_points+1, numel(sheets)); iTemperatures = zeros(temperature_points+1, numel(sheets));
ls = '-';
for c=1:numel(sheets) excelData = xlsread(filename, sheets{c});
%Read in model coefficients C = excelData(20:95,:); iTemperatures(:,c) = C(:,1);
for i=1:length(C(:,1)) %Call Muller() root finding algorithm, Function Parameters:
%(Function coefficients, Initial guess [x1, x2, x3]) if(c==1) e(i, c) = C(i,2)*1000000; else
Root = Muller([C(i, 2), C(i, 3), C(i, 4), C(i, 5), C(i, 6), C(i,
7)], [1 2 3]);
%Store film thickness for each plate e(i, c) = Root*1000000; end end
%Plot and store handle h(c) = plot(C(:, 1), e(:, c), ls, 'LineWidth', 2); %Build legend legnd = plates; end
%Label plots plt = Plot(); plt.Legend = legnd; plt.XLabel = 'Initial Surface Temperature [^oC]'; plt.YLabel= 'Vapour Thickness [mn]'; plt.BoxDim = [7, 4]; %[width, height] plt.LegendBox = 'on'; plt.Colors = { % three colors for three data set [0, 0, 0] % data set 1 [0, 0, 1] % data set 2 [1, 0, 0] % data set 3 };
%% Estimate evaporation time time = TimeEstimate(iTemperatures, (real(e))/1000000);
p2 = figure(2); hold on %Plot empirical model evaporation times [Columns 4:6] h2(1,:) = plot(iTemperatures(:,1:3), time(:,1:3), ls, 'LineWidth', 2);
135
%Plot experimental evaporation times with error bars a1 = errorbar(hep_exp_surfaceT, hep_exp_time,
err0mn_time,'o','MarkerSize',6,'MarkerEdgeColor','black','MarkerFaceColor','b
lack'); a2 = errorbar(hep5mn_exp_surfaceT, hep5mn_exp_time,
err5mn_time,'s','MarkerSize',6,'MarkerEdgeColor','blue','MarkerFaceColor','bl
ue'); a3 = errorbar(hep100mn_exp_surfaceT, hep100mn_exp_time,
err100mn_time,'^','MarkerSize',6,'MarkerEdgeColor','red','MarkerFaceColor','r
ed');
%Format our plots plt = Plot(); plt.LineStyle = {'--', '--', '--', 'none', 'none', 'none'}; % three line
styles plt.Legend = [legnd(1:3), 'Impermeable - Experimental','5mn [\phi = 0.31] -
Experimental','100mn [\phi = 0.44] - Experimental']; plt.XLabel = 'Initial Surface Temperature [^oC]'; plt.YLabel = 'Evaporation Time [s]'; plt.BoxDim = [7, 4]; %[width, height] plt.LegendBox = 'on'; plt.Colors = { [0, 0, 0] [0, 0, 1] [1, 0, 0] [0, 0, 0] [0, 0, 1] [1, 0, 0] };
136
Arduino Uno Code
TDS2002B Oscilloscope Trigger Program:
Program will poll the slotted optical switch signal until it reads a LOW where a trigger can be
sent to either the FASTCAM SA5 high-speed camera or the TDS2002B oscilloscope to begin
operation.
137
Appendix C
Data
Pure water droplet lifetime data.
Impermeable 100 mn 5 mn
Temperature (oC)
Average Scale Evaporation
Time (s)
Average Video Evaporation
Time (s)
Difference (s)
Standard
Deviation (s)
Average Scale Evaporation
Time (s)
Average Video Evaporation
Time (s)
Standard
Deviation (s)
Average Scale Evaporation
Time (s)
Average Video Evaporation
Time (s)
Standard
Deviation (s)
60 212.86 218.5 -5.64 16.91 188.76 16.22 133.67 3.08
70 121.42 123.85 -2.43 12.37 128.65 8.09 82.72 4.32
80 83.29 76.32 6.97 7.37 84.58 4.61 49.99 2.56
90 52.87 54.13 -1.26 4.70 62.05 2.94 34.30 1.39
100 34.59 36.44 -1.85 2.12 43.62 4.26 22.53 1.08
110 24.58 20.20 4.38 3.41 33.30 2.54 15.17 0.51
120 13.73 13.09 0.64 2.22 22.15 4.67 11.68 0.82
150 3.18 0.722 0.03 0.01 0.58 0.04
200 2.68 1.06 0.05 0.01 0.34 0.01
210 3.93 7.22
220 14.13 18.07
225 25.34 23.40
230 67.31 2.38
235 70.91 3.66
240 68.69 4.66
250 64.05 8.27 0.09 0.04 0.40 0.03
275 59.38 1.40
300 53.68 1.84 0.20 0.01 0.67 0.18
138
n-Heptane droplet lifetime data.
Impermeable 100 mn 5 mn
Temperature (oC)
Average Scale
Evaporation Time
(s)
Average Video
Evaporation Time
(s)
Standard
Deviation (s)
Average Scale
Evaporation Time
(s)
Average Video
Evaporation Time
(s)
Standard
Deviation (s)
Average Scale
Evaporation Time
(s)
Average Video
Evaporation Time
(s)
Standard
Deviation (s)
60 21.3 1.32 32.07 1.69 29.05 2.78
70 15.47 1.39 20.6 0.88 20.08 1.02
80 11.66 0.94 13.56 0.76 14.43 1.22
90 9.06 0.61 9.05 0.60 9.86 0.72
100 6.37 0.80 6.58 0.24 6.07 0.53
110 3.75 0.48 4.49 0.44 3.41 0.52
120 2.03 0.42 2.54 0.51 1.73 0.38
130 0.58 0.13 0.16 0.04 0.29 0.04
150 0.15 0.02 0.07 0.01 0.09 0.01
165 0.23 0.01 0.08 0.01 0.10 0.02
175 0.32 0.01
180 0.43 0.03
185 9.88 1.35 0.09 0.01 0.15 0.03
190 11.48 0.54
195 10.67 0.63
200 10.08 0.42 0.14 0.01 0.34 0.02
215 0.49 0.02
220 1.41 0.96
225 0.21 0.01 10.69 0.32
230 10.64 0.64
235 10.04 0.53
240
250 8.80 0.98 0.47 0.11 9.37 0.73
265 0.71 0.25
270 1.27 1.06
275 3.85 2.39
280 4.95 2.00
285 6.41 0.38
290 6.00 0.27
295 5.91 0.30
300 7.85 0.55 5.76 0.34 7.55 0.80
